**Meet the editor**

Dr Giovanna Cavazzini received her degree in Mechanical Engineering in 2003 from the University of Padova, and her doctorate with European Label in Energetics in 2007 from the University of Padova. She started working with PIV in 2005 at the Ecole Nationale Superieure Arts et Métiers ParisTech de Lille with emphasis to the application of the PIV in the turbo-machinery field with

focus to the study of unsteady turbulent phenomena. In 2009 she recevied the qualification of Maitre de Conference from the Ministère de l'Enseignement supérieur et de la Recherche in France and from 2011 she is aggregate professor of Turbomachinery and Energetic Systems at the University of Padova.

Contents

**Preface IX** 

Chapter 1 **Stereoscopic PIV and Its Applications on** 

Chapter 2 **Limits in Planar PIV Due to Individual** 

**Characteristics, Limits and Post-Processing Methods 1** 

**Flow Fields for Unsteady Flows in Turbomachines 97**  G. Cavazzini, A. Dazin, G. Pavesi, P. Dupont and G. Bois

**Reconstruction Three-Dimensional Flow Field 3** 

**Variations of Particle Image Intensities 29** 

Chapter 3 **PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 51**  Gabriel Dan Ciocan and Monica Sanda Iliescu

Chapter 4 **Post-Processing Methods of PIV Instantaneous** 

Chapter 5 **PIV Measurements on Oxy-Fuel Burners 123**  Boushaki Toufik and Sautet Jean-Charles

Brian A. Maicke and Joseph Majdalani

**Vortex Using Particle Image Velocimetry 149** 

**Velocimetry to the Couette-Taylor Flow 177**  Innocent Mutabazi, Nizar Abcha, Olivier Crumeyrolle

Chapter 6 **Characterization of the Bidirectional** 

Chapter 7 **Application of the Particle Image** 

and Alexander Ezersky

**Section 1 The PIV Technique:** 

L. Gan

Holger Nobach

**Section 2 PIV Applications 121** 

## Contents

### **Preface XI**


Innocent Mutabazi, Nizar Abcha, Olivier Crumeyrolle and Alexander Ezersky

X Contents


Chapter 9 **PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow 229**  Chong Y. Wong, Graham J. Nathan, Richard M Kelso

#### **Section 3 Micro-PIV Applications 259**

	- **Section 4 PIV and PTV 319**

V. Pesarino, F. Lalli and G.P. Romano

## Preface

The Particle Image Velocimetry is undoubtedly one of the most important technique in Fluid-dynamics since it has a spatial character that allows to obtain a direct and instantaneous visualization of the flow field in a non-intrusive way. It represents the point of arrival of an intense research on the laser techniques and its origin dates back at the beginning of 80's when Meynart, a doctoral student of the Von Karman Institute, while explorying the Laser Speckle Velocimetry technique, guessed the importance of capturing the movements of individual particles instead of speckles. The rising PIV technique immediately captured the scientific world interest but its spread was initially limited by technical problems related to the limits of the available technologies.

A first significant boost to the spreading of the PIV technique was provided by the migration from photographic to video-graphic recording. This allowed not only to increase the maximum number of PIV images that could be acquired in a single experimental test but also to improve the interrogation process. However, the applications were still limited to analyses of flow fields having two-dimensional characteristics since the technique allowed to determine only the two velocity components belonging to the laser sheet plane.

In the midi-90's the introduction of the stereographic PIV gave a further great impulse to the use and popularity of the PIV technique since it enabled the possibility of determining by stereographic imaging of the particles the third velocity component, normal to the planar laser sheet. This further improvement, combined with the rising and continuous development of the available computational capacity, greatly favored the spread of the technique in a wide number of research fields, from aerodynamics to medicine, from biology to turbulence researches, from aerodynamics to combustion processes.

The book is aimed at presenting the PIV technique and its wide range of possible applications so as to provide a reference for researchers who intended to exploit this innovative technique in their research fields. Several aspects and possible problems in the analysis of large- and micro-scale turbulent phenomena, two-phase flows and polymer melts, combustion processes and turbo-machinery flow fields, internal waves and river/ocean flows were considered.

#### X Preface

The book is managed as follows. The first part is focused on the characteristics, advantages and limits of the PIV technique, with reference both to 2D and 3D analyses. Moreover, the latest improvements in the post-processing analysis methods are presented with particular reference to the turbo-machinery field.

The other sections present an excursus of the possible applications of the PIV technique. In particular, the second section focuses on the studies of large-scale flows considering combustion processes, fluid-dynamic phenomena and flows in polymer melts. In the third section two examples of micro-piv applications are presented, whereas in the forth section the combination of the PIV technique with the Particle Tracking Velocimetry for the definition of velocities and trajectories in internal waves, stratified ocean and river flows is considered.

**Eng. Cavazzini Giovanna, Ph.D.**

Department of Industrial Engineering University of Padova Italy

X Preface

The book is managed as follows. The first part is focused on the characteristics, advantages and limits of the PIV technique, with reference both to 2D and 3D analyses. Moreover, the latest improvements in the post-processing analysis methods

The other sections present an excursus of the possible applications of the PIV technique. In particular, the second section focuses on the studies of large-scale flows considering combustion processes, fluid-dynamic phenomena and flows in polymer melts. In the third section two examples of micro-piv applications are presented, whereas in the forth section the combination of the PIV technique with the Particle Tracking Velocimetry for the definition of velocities and trajectories in internal waves,

> **Eng. Cavazzini Giovanna, Ph.D.** Department of Industrial Engineering

> > University of Padova

Italy

are presented with particular reference to the turbo-machinery field.

stratified ocean and river flows is considered.

**Section 1** 

**The PIV Technique: Characteristics,** 

**Limits and Post-Processing Methods** 

## **Section 1**

## **The PIV Technique: Characteristics, Limits and Post-Processing Methods**

L. Gan

**1**

*United Kingdom*

*Department of Engineering, University of Cambridge*

Various concepts involving in the stereoscopic PIV are very briefly summarised in this section. For more details, readers are recommended to read Lavision (2007); Prasad (2000); Raffel et al. (2007). Stereoscopic PIV adopts two digital cameras viewing at the same laser illuminated plane1 from two different angles to resolve the three velocity components on the plane; see figure 1. Sometimes it is also called 2D3C (two-dimension three-component) PIV. The basic principle of stereoscopic PIV is similar to a pair of human eyes simultaneously observing an object to capture its movement in a plane as well as in the third direction. One major difference to the two-dimensional PIV is that the illuminated plane cannot be too thin, because the third component needs to be resolved. It should allow most of the particles to remain in the illuminated volume after the PIV Δ*t*, to give valid cross-correlation signals for calculating

**Stereoscopic PIV and Its Applications on** 

**Reconstruction Three-Dimensional Flow Field** 

In this arrangement, the two cameras simultaneously accept a pair of laser exposures to do normal two-dimensional PIV independently. Because the common field of view (FOV) of the

Fig. 1. A schematic diagrame of a typical stereoscopic PIV setup. Picture taken from Prasad

<sup>1</sup> Strictly speaking, it is not a plane, but a very thin volume with a typical thickness of 2 <sup>−</sup> <sup>5</sup>*mm*.

Ca libr at ion plate inyspecti −6422046on−6:: r 0ed =pla−4ne 1, gre−2en =plaxne 2, both 20mapped on40to same 60plane for view ing ease

**1. Introduction of stereoscopic PIV**

the third component.

**1.1 Principle**

(2000).

## **Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field**

L. Gan

*Department of Engineering, University of Cambridge United Kingdom*

### **1. Introduction of stereoscopic PIV**

Various concepts involving in the stereoscopic PIV are very briefly summarised in this section. For more details, readers are recommended to read Lavision (2007); Prasad (2000); Raffel et al. (2007). Stereoscopic PIV adopts two digital cameras viewing at the same laser illuminated plane1 from two different angles to resolve the three velocity components on the plane; see figure 1. Sometimes it is also called 2D3C (two-dimension three-component) PIV. The basic principle of stereoscopic PIV is similar to a pair of human eyes simultaneously observing an object to capture its movement in a plane as well as in the third direction. One major difference to the two-dimensional PIV is that the illuminated plane cannot be too thin, because the third component needs to be resolved. It should allow most of the particles to remain in the illuminated volume after the PIV Δ*t*, to give valid cross-correlation signals for calculating the third component.

#### **1.1 Principle**

In this arrangement, the two cameras simultaneously accept a pair of laser exposures to do normal two-dimensional PIV independently. Because the common field of view (FOV) of the

Fig. 1. A schematic diagrame of a typical stereoscopic PIV setup. Picture taken from Prasad (2000).

<sup>1</sup> Strictly speaking, it is not a plane, but a very thin volume with a typical thickness of 2 <sup>−</sup> <sup>5</sup>*mm*.

Fig. 2. Reconstructions of displacements seen from the two cameras.

two cameras are maximized2, the resultant velocity vector fields from the two cameras are combined and from which the 2D3C field is reconstructed, provided a successful stereoscopic calibration having been applied.

The relationships between the real particle displacements (Δ*x* and Δ*z*) and the pixel displacements seen in each of the cameras (*d*<sup>1</sup> and *d*2) can be worked out simply by considering the geometries, as shown in figure 2. For camera 1,

$$d\_1 = \Delta z \cos \mathfrak{a} + \Delta \mathfrak{x} \sin \mathfrak{a};\tag{1}$$

Ca libr at ion platey−642 in2046specti −60on:: red−4 =plane−2 1, green =pxlane 2,0 both ma pped on40to s6ame plne for view ing ease

Fig. 3. Calibration plate images viewed from the two cameras. The viewing angle of the two

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 5

To calibrate the cameras for stereoscopic PIV, one typically needs to take an image of the calibration plate (a plate with a two-dimensional array of circular or cross-shaped markers, which is necessary for correcting distortions), then shift the plate in the plane-normal direction for a small amount and take a second image of it. The amount of shift should be comparable to the laser sheet thickness, as mentioned above, typically around 2 − 5*mm*, depending on and limiting by various situations. Because the amount of shift needs to be very accurate, it is recommended to use a two-level calibration plate as it shown in figure 3, without any need to

To take into account the possible distortion, a third-ordered polynomial fitting function is applied for mapping the global coordinates of the markers on the plate to the associated pixel locations of them. One can also adopts higher ordered polynomial functions, but usually third-order is sufficient. If a third-ordered polynomial function is used, the mapping function

where *X* and *Y* are physical coordination array of the markers on the calibration plate, with

*<sup>s</sup>* <sup>=</sup> <sup>2</sup>(*<sup>x</sup>* <sup>−</sup> *xo*) *nx <sup>t</sup>* <sup>=</sup> <sup>2</sup>(*<sup>y</sup>* <sup>−</sup> *yo*)

To solve this matrix for the unknown coefficients, one needs sufficient number of markers; if there are more markers than necessary, a least-squares fit will be applied. This fitting process is carried out for each image in the two different normal (z) locations for both cameras (two sets of coefficients for each camera). After the calibration is done, the angles of the line-of-sight

the image size *nx* × *ny*, the origin location (*xo*, *yo*) and the normalized pixel location:

<sup>2</sup> + *a*6*t*

<sup>2</sup> + *b*6*t*

<sup>3</sup> + *a*7*st* + *a*8*s*2*t* + *a*9*st*<sup>2</sup>

, (4)

<sup>3</sup> + *b*7*st* + *b*8*s*2*t* + *b*9*st*<sup>2</sup>

*ny* . (5)

cameras in this case is about 30*<sup>o</sup>* each.

typically looks like equation 4, as used in Lavision (2007).

*ao* + *a*1*s* + *a*2*s*<sup>2</sup> + *a*3*s*<sup>3</sup> + *a*4*t* + *a*5*t*

*bo* + *b*1*s* + *b*2*s*<sup>2</sup> + *b*3*s*<sup>3</sup> + *b*4*t* + *b*5*t*

**1.2 Calibration**

shift.

 *X Y* = 

for camera 2,

$$d\_2 = \Delta z \cos \alpha - \Delta x \sin \alpha;\tag{2}$$

therefore the true displacements can be re-written in terms of pixel displacements, once the viewing angle is known, as

$$
\Delta \mathbf{x} = \frac{d\_1 - d\_2}{2 \cos \alpha}
$$

$$
\Delta z = \frac{d\_1 + d\_2}{2 \sin \alpha}.\tag{3}
$$

The displacements in the physical space can be obtained by a simple calibration of *d*<sup>1</sup> and *d*<sup>2</sup> to their corresponding displacements in physical space.

However, in most situations, the viewing angles are difficult to measure and moreover, if the experiments are carried out in water, as the viewing angle deviates from 90*o*, the image distortion becomes severer. This requires a robuster calibration process.

<sup>2</sup> The two cameras can be located either on the same side (FB [forward-backward scattering] setup, like it shown in figure 1) or on different sides (FF [forward-forward scattering] or BB [backward-backward scattering] setup) of the plate. In the first case, the two FOVs are not possible to be the same; while the second type of setup it is possible to make the FOV almost the same.

Fig. 3. Calibration plate images viewed from the two cameras. The viewing angle of the two cameras in this case is about 30*<sup>o</sup>* each.

#### **1.2 Calibration**

2 Will-be-set-by-IN-TECH

two cameras are maximized2, the resultant velocity vector fields from the two cameras are combined and from which the 2D3C field is reconstructed, provided a successful stereoscopic

The relationships between the real particle displacements (Δ*x* and Δ*z*) and the pixel displacements seen in each of the cameras (*d*<sup>1</sup> and *d*2) can be worked out simply by

therefore the true displacements can be re-written in terms of pixel displacements, once the

<sup>Δ</sup>*<sup>x</sup>* <sup>=</sup> *<sup>d</sup>*<sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>2</sup> 2 cos *α* <sup>Δ</sup>*<sup>z</sup>* <sup>=</sup> *<sup>d</sup>*<sup>1</sup> <sup>+</sup> *<sup>d</sup>*<sup>2</sup>

The displacements in the physical space can be obtained by a simple calibration of *d*<sup>1</sup> and *d*<sup>2</sup>

However, in most situations, the viewing angles are difficult to measure and moreover, if the experiments are carried out in water, as the viewing angle deviates from 90*o*, the image

<sup>2</sup> The two cameras can be located either on the same side (FB [forward-backward scattering] setup, like it shown in figure 1) or on different sides (FF [forward-forward scattering] or BB [backward-backward scattering] setup) of the plate. In the first case, the two FOVs are not possible to be the same; while the

Fig. 2. Reconstructions of displacements seen from the two cameras.

considering the geometries, as shown in figure 2. For camera 1,

to their corresponding displacements in physical space.

distortion becomes severer. This requires a robuster calibration process.

second type of setup it is possible to make the FOV almost the same.

calibration having been applied.

viewing angle is known, as

for camera 2,

Laser plane

Ca libr at ion plateyin−6422046specti −6on:: r0ed−4 =plane1, g−2reen =plaxne 2, both 20ma pped on40to same pl60ane for viewing ease

*d*<sup>1</sup> = Δ*z* cos *α* + Δ*x* sin *α*; (1)

*d*<sup>2</sup> = Δ*z* cos *α* − Δ*x* sin *α*; (2)

2 sin *<sup>α</sup>* . (3)

To calibrate the cameras for stereoscopic PIV, one typically needs to take an image of the calibration plate (a plate with a two-dimensional array of circular or cross-shaped markers, which is necessary for correcting distortions), then shift the plate in the plane-normal direction for a small amount and take a second image of it. The amount of shift should be comparable to the laser sheet thickness, as mentioned above, typically around 2 − 5*mm*, depending on and limiting by various situations. Because the amount of shift needs to be very accurate, it is recommended to use a two-level calibration plate as it shown in figure 3, without any need to shift.

To take into account the possible distortion, a third-ordered polynomial fitting function is applied for mapping the global coordinates of the markers on the plate to the associated pixel locations of them. One can also adopts higher ordered polynomial functions, but usually third-order is sufficient. If a third-ordered polynomial function is used, the mapping function typically looks like equation 4, as used in Lavision (2007).

$$
\begin{bmatrix} X \\ Y \end{bmatrix} = \begin{bmatrix} a\_0 + a\_1s + a\_2s^2 + a\_3s^3 + a\_4t + a\_5t^2 + a\_6t^3 + a\_7st + a\_8s^2t + a\_9st^2 \\ b\_0 + b\_1s + b\_2s^2 + b\_3s^3 + b\_4t + b\_5t^2 + b\_6t^3 + b\_7st + b\_8s^2t + b\_9st^2 \end{bmatrix}, \tag{4}
$$

where *X* and *Y* are physical coordination array of the markers on the calibration plate, with the image size *nx* × *ny*, the origin location (*xo*, *yo*) and the normalized pixel location:

$$s = \frac{2(x - x\_o)}{nx}$$

$$t = \frac{2(y - y\_o)}{ny}.\tag{5}$$

To solve this matrix for the unknown coefficients, one needs sufficient number of markers; if there are more markers than necessary, a least-squares fit will be applied. This fitting process is carried out for each image in the two different normal (z) locations for both cameras (two sets of coefficients for each camera). After the calibration is done, the angles of the line-of-sight

Ca libr at ion ple iny−6422046specti −6on:: r0ed =p−4lane 1, g−2reen =plaxne 2, bot0h ma pped 40onto sam60e ane for view ing ease

Fig. 5. The concept of disparity vectors, due to the mis-alignment of the laser sheet and the

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 7

Stereoscopic self-calibration is useful when there is an imperfection in the calibration process discussed in section 1.2: the laser sheet may not align perfectly with the calibration plate surface. In this case, disparity vectors will appear when the images of camera 1 and camera 2 are dewarped<sup>3</sup> and combined by the original calibration; see figure 5. The original calibration coefficients (in the third-ordered polynomial function) will be modified and updated after self-calibration iterations such that the length of the disparity vectors will be minimized

The prerequisite for self-calibration is that the calibration process should be done fairly well and one has about 20 or more good quality4 particle field images from the stereoscopic recordings for the purpose of statistical convergence. A single particle image is typically divided into a grid of 7 × 7 (or more) sub-windows. In each of these windows, a two-dimensional disparity vector is calculated by seeking the highest disparity signal in the cluster (with respect to the centre of the window). A typical disparity map of an image is shown in figure 6. Based on the disparity map generated by such figures, the calibration coefficients are updated. Then this updated calibration coefficients are used to compute an updated disparity map, and after which calibration will be updated again. Usually a good self-calibration process requires two to three such iterations. To search the possible matching particles, a method similar to particle tracking technique is used (Wieneke, 2005; 2008).

Because the two cameras view the FOV at an angle, usually Scheimpflug adapters are needed. When the object, lens and image planes intersect, the FOV in the object plane will be in focus. Depending on the limits of different types of the Scheimpflug adapters, compensation between the camera aperture size and the laser light intensity needs to be considered. The

<sup>3</sup> The concept of dewarping is not introduced in this chapter, readers are referred to Lavision (2007);

<sup>4</sup> Good quality particle images should be well focused, with proper particle size and density and low background noise level, etc. Image preprocessing is also recommended before applying self-calibration.

calibration plate surface. Figure taken from Lavision (2007).

(ideally zero disparity vectors everywhere).

**1.3 Self-calibration**

**1.4 Focusing problem**

Prasad (2000); Raffel et al. (2007).

Fig. 4. The flowchart of the computation of 2D3C vector fields adopted in Lavision (2007).

(*α* in equation 3) of each camera should be known, or one can use this mapping function to reconstruct the 2D3C velocity field directly based on the two-dimensional PIV vectors calculated in each camera, e.g. figure 4 illustrates how Lavision (2007) does this computation.

Fig. 5. The concept of disparity vectors, due to the mis-alignment of the laser sheet and the calibration plate surface. Figure taken from Lavision (2007).

#### **1.3 Self-calibration**

4 Will-be-set-by-IN-TECH

Fig. 4. The flowchart of the computation of 2D3C vector fields adopted in Lavision (2007).

(*α* in equation 3) of each camera should be known, or one can use this mapping function to reconstruct the 2D3C velocity field directly based on the two-dimensional PIV vectors calculated in each camera, e.g. figure 4 illustrates how Lavision (2007) does this computation. Stereoscopic self-calibration is useful when there is an imperfection in the calibration process discussed in section 1.2: the laser sheet may not align perfectly with the calibration plate surface. In this case, disparity vectors will appear when the images of camera 1 and camera 2 are dewarped<sup>3</sup> and combined by the original calibration; see figure 5. The original calibration coefficients (in the third-ordered polynomial function) will be modified and updated after self-calibration iterations such that the length of the disparity vectors will be minimized (ideally zero disparity vectors everywhere).

The prerequisite for self-calibration is that the calibration process should be done fairly well and one has about 20 or more good quality4 particle field images from the stereoscopic recordings for the purpose of statistical convergence. A single particle image is typically divided into a grid of 7 × 7 (or more) sub-windows. In each of these windows, a two-dimensional disparity vector is calculated by seeking the highest disparity signal in the cluster (with respect to the centre of the window). A typical disparity map of an image is shown in figure 6. Based on the disparity map generated by such figures, the calibration coefficients are updated. Then this updated calibration coefficients are used to compute an updated disparity map, and after which calibration will be updated again. Usually a good self-calibration process requires two to three such iterations. To search the possible matching particles, a method similar to particle tracking technique is used (Wieneke, 2005; 2008).

#### **1.4 Focusing problem**

Because the two cameras view the FOV at an angle, usually Scheimpflug adapters are needed. When the object, lens and image planes intersect, the FOV in the object plane will be in focus. Depending on the limits of different types of the Scheimpflug adapters, compensation between the camera aperture size and the laser light intensity needs to be considered. The

<sup>3</sup> The concept of dewarping is not introduced in this chapter, readers are referred to Lavision (2007); Prasad (2000); Raffel et al. (2007).

<sup>4</sup> Good quality particle images should be well focused, with proper particle size and density and low background noise level, etc. Image preprocessing is also recommended before applying self-calibration.

where *Up* is the piston velocity and *L* is the length of the discharged slug, Γ*slug* is the slug

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 9

Vortex rings which are well-formed at the nozzle or orifice exit, under certain conditions, undergo an instability along the core circumference in the form of azimuthal waves. This type of instability is an important feature of the transition from laminar to turbulent rings. Although previous investigations of this type of waviness have been made in laminar rings or rings during the transition stage from laminar to turbulent Maxworthy (1977); Saffman (1978); Shariff et al. (1994); Widnall & Tsai (1977), the results show that azimuthal waves are also observed in fully turbulent vortex rings and this technique can successfully capture this

Taylor's hypothesis (Taylor, 1938; Townsend, 1976) states that "if the velocity of the airstream which carries the eddies is very much greater than the turbulent velocity, one may assume that the sequence of changes in U at the fixed point are simply due to the passage of an unchanging pattern of turbulent motion over the point". In other words, if the relative turbulence intensity

*u*�

the time-history of the flow signal from a stationary probe can be regarded as that due to advection of a frozen spatial pattern of turbulence past the probe with the mean advection

where Δ*t* is the time delay and should not be a too large value. Taylor's hypothesis is effectively a method to transfer the time dependent measurement results to a spatial domain. The selection of the correct velocity scales relevant to turbulent vortex rings are discussed in

All the figures presented in this chapter are plotted in the similarity coordinates derived from the similarity model (Glezer & Coles, 1990). The reason is because the original model was derived from Taylor's hypothesis. Any structure plotted in the similarity coordinates contains not only spatial but also temporal information: it can be transferred back to the physical coordinates at any point in a series of time history. It has been proved in Gan & Nickels

If a proper pair of spatial and temporal origins *zo*, *to* can be found, the length and velocity scales in a turbulent vortex ring in physical space can be shown to be self-similar, and they

*η* = *r*

 *ρ I* (*t* − *to*)  1 4

; (9)

 1 4

(2010) that rings produced in this study are well predicted by this model.

 *ρ I* (*t* − *to*)

can be written in terms of the similarity variables, i.e.

*ξ* = (*z* − *zo*)

*<sup>U</sup>* � 1, (7)

*u* (*x*, *t*) = *u* (*x* − *U*Δ*t*, *t* + Δ*t*), (8)

*u*� is assumed to be small enough compared to the mean advection speed *U*:

circulation and *ν* is the kinematic viscosity of the fluid.

feature.

speed *U*, i.e.

section 2.3.

**2.1.2 Similarity model**

**2.1 Theoretical background 2.1.1 Taylor's hypothesis**

Fig. 6. A typical disparity map. The image window is divided into 7 × 7 sub-windows. From the peak intensity of each sub-window, a two-dimensional disparity vector can be defined, with respect to the centre of the sub-window. Each sub-window shown in this figure is typically 11 <sup>×</sup> <sup>11</sup> *pixel*2.

viewing angle between a single camera and the laser sheet normal direction usually should not be larger than 60*o*, above which the distortion can be too strong and the focusing is difficult even a Scheimpflug adapter is used. In some circumstances, prisms (Prasad, 2000) can also be adopted to help reduce the distortion.

#### **2. An application of stereoscopic PIV: three-dimensional velocity field reconstruction**

This section presents an application of stereoscopic PIV to reconstruct a three-dimensional turbulent vortex ring. It is a critical assessment of such an application because a turbulent vortex ring is fully three-dimensional with no preferential direction in the azimuthal plane. Moreover, there is an important property- the translational velocity of a ring is unambiguous.

A vortex ring in laboratory studies is usually generated by an impulsive ejection of fluid through a nozzle or an orifice into a quiescent environment. Circulation based Reynolds number is defined as:

$$Re = \frac{\Gamma\_{\text{slug}}}{\nu} = \frac{\mathcal{U}\_p \mathcal{L}}{\mathcal{2}\nu}.\tag{6}$$

where *Up* is the piston velocity and *L* is the length of the discharged slug, Γ*slug* is the slug circulation and *ν* is the kinematic viscosity of the fluid.

Vortex rings which are well-formed at the nozzle or orifice exit, under certain conditions, undergo an instability along the core circumference in the form of azimuthal waves. This type of instability is an important feature of the transition from laminar to turbulent rings. Although previous investigations of this type of waviness have been made in laminar rings or rings during the transition stage from laminar to turbulent Maxworthy (1977); Saffman (1978); Shariff et al. (1994); Widnall & Tsai (1977), the results show that azimuthal waves are also observed in fully turbulent vortex rings and this technique can successfully capture this feature.

#### **2.1 Theoretical background**

#### **2.1.1 Taylor's hypothesis**

6 Will-be-set-by-IN-TECH

Fig. 6. A typical disparity map. The image window is divided into 7 × 7 sub-windows. From the peak intensity of each sub-window, a two-dimensional disparity vector can be defined, with respect to the centre of the sub-window. Each sub-window shown in this figure is

viewing angle between a single camera and the laser sheet normal direction usually should not be larger than 60*o*, above which the distortion can be too strong and the focusing is difficult even a Scheimpflug adapter is used. In some circumstances, prisms (Prasad, 2000) can also be

This section presents an application of stereoscopic PIV to reconstruct a three-dimensional turbulent vortex ring. It is a critical assessment of such an application because a turbulent vortex ring is fully three-dimensional with no preferential direction in the azimuthal plane. Moreover, there is an important property- the translational velocity of a ring is unambiguous. A vortex ring in laboratory studies is usually generated by an impulsive ejection of fluid through a nozzle or an orifice into a quiescent environment. Circulation based Reynolds

*<sup>ν</sup>* <sup>=</sup> *UpL*

<sup>2</sup>*<sup>ν</sup>* . (6)

**2. An application of stereoscopic PIV: three-dimensional velocity field**

*Re* <sup>=</sup> <sup>Γ</sup>*slug*

typically 11 <sup>×</sup> <sup>11</sup> *pixel*2.

**reconstruction**

number is defined as:

adopted to help reduce the distortion.

Taylor's hypothesis (Taylor, 1938; Townsend, 1976) states that "if the velocity of the airstream which carries the eddies is very much greater than the turbulent velocity, one may assume that the sequence of changes in U at the fixed point are simply due to the passage of an unchanging pattern of turbulent motion over the point". In other words, if the relative turbulence intensity *u*� is assumed to be small enough compared to the mean advection speed *U*:

$$\frac{u'}{U} \ll 1,\tag{7}$$

the time-history of the flow signal from a stationary probe can be regarded as that due to advection of a frozen spatial pattern of turbulence past the probe with the mean advection speed *U*, i.e.

$$
\mu\left(\mathbf{x}, t\right) = \mu\left(\mathbf{x} - \mathcal{U}\Delta t, t + \Delta t\right), \tag{8}
$$

where Δ*t* is the time delay and should not be a too large value. Taylor's hypothesis is effectively a method to transfer the time dependent measurement results to a spatial domain. The selection of the correct velocity scales relevant to turbulent vortex rings are discussed in section 2.3.

#### **2.1.2 Similarity model**

All the figures presented in this chapter are plotted in the similarity coordinates derived from the similarity model (Glezer & Coles, 1990). The reason is because the original model was derived from Taylor's hypothesis. Any structure plotted in the similarity coordinates contains not only spatial but also temporal information: it can be transferred back to the physical coordinates at any point in a series of time history. It has been proved in Gan & Nickels (2010) that rings produced in this study are well predicted by this model.

If a proper pair of spatial and temporal origins *zo*, *to* can be found, the length and velocity scales in a turbulent vortex ring in physical space can be shown to be self-similar, and they can be written in terms of the similarity variables, i.e.

$$\xi = (z - z\_o) \left[ \frac{\rho}{I \left( t - t\_o \right)} \right]^{\frac{1}{4}} \qquad \qquad \eta = r \left[ \frac{\rho}{I \left( t - t\_o \right)} \right]^{\frac{1}{4}};\tag{9}$$

The motion of the piston is driven by a stepper motor. The motor is able to drive the piston to move at a constant speed of up to 1000*mms*−<sup>1</sup> with an acceleration and deceleration of about

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 11

The effective *Re* is set to 41280 and *L*/*D* = 3.43 in order to match the conditions in Glezer & Coles (1990) and Gan & Nickels (2010), because parts of the reconstruction validation will rely on the data in them. The piston velocity programme is approximately a top-hat shape: the piston acceleration/deceleration time is about 10% of the piston total movement duration. Rings produced at this combination of *Re* and *L*/*D* are well in the turbulent region on the

In this arrangement, the PIV (laser sheet) plane is located 6D (six orifice diameters) downstream of the orifice exit. The PIV system is provided by LaVision Ltd: a pair of high-speed Photron APX cameras are used as the image recording devices and the particle illumination is realised by a Pegasus PIV Laser which consists of a dual-cavity diode pumped Nd:YLF laser head and is capable of emitting a beam of 527*nm* wavelength and 10*m J* energy. The laser beam is converted to a sheet by passing through a cylindrical diverging lens. The thickness of the sheet can be adjusted by changing the separation of a pair of telescope lenses housed before the cylindrical lens and is set to about 4 − 5*mm*. The flow is seeded by 50*μm*

After the changes in refractive index are considered, the effective angle between the two cameras is approximately 120*o*. The two cameras run in single-frame single-exposure mode, and the operation frequency is set at *f* = 600*Hz* giving a Δ*t* = 1.67 *ms*. The interrogation window size for time series cross-correlation process is set to 16 <sup>×</sup> <sup>16</sup> *pixel*<sup>2</sup> with a 25% overlap to give a spatial resolution of Δ*x* = Δ*y* ≈ 1.70*mm* (based on vector spacing, ≈ 3.4% D) in the

An important prerequisite for the reconstruction of a turbulent vortex ring using Taylor's hypothesis is to identify appropriate scales of the fluctuating and convective velocities in

The relevant convective velocity in this work is the ring advection speed, or celerity, and is obtained from the independent two-dimensional PIV results in Gan & Nickels (2010). The centroid-determined vortex ring radii *r* and celerities *ut* for 50 realisations are reproduced in figure 8. From this figure the spatially averaged celerity ( data) at various streamwise locations is obtained between *z* = 5.5*D* to *z* = 6.5*D* and the ensemble averaged ring celerity

The ring celerity also affects the spatial resolution in the *z* direction, Δ*z*. The reconstruction of the planar velocity fields in the *z* direction is selected to give a similar spatial resolution in the

The most straight-forward reconstruction process requires the entire cross-sectional area of the ring bubble to pass normally through the PIV plane with constant celerity. Figure 8 shows that

is obtained by a least-squares fit (blue line) giving: *ut* <sup>=</sup> 270.34 *mms*−<sup>1</sup> <sup>≈</sup> 0.54 *Up*.

1500*mms*−2.

transition map (see Glezer, 1988).

**2.3 Accuracy justification**

equation 7.

*x* − *y* plane.

diameter, silver-coated hollow glass spheres.

PIV plane. The FOV of the PIV plane is about 230 <sup>×</sup> <sup>164</sup> *mm*2.

Fig. 7. Schematic diagram of the vortex ring generator. The diagram is not to scale. The coordinate system adopted in this experiment is also shown, where *z* = 0 is set at the orifice exit.

and from the streamfunction,

$$\mathcal{U} = \mathfrak{u}\left(\frac{\rho}{I}\right)^{\frac{1}{4}}(t - t\_o)^{\frac{3}{4}} \qquad \qquad V = \mathcal{v}\left(\frac{\rho}{I}\right)^{\frac{1}{4}}(t - t\_o)^{\frac{3}{4}}\,. \tag{10}$$

where *I* is the hydrodynamic impulse (assumed invariant); *ρ* is the density (a constant when incompressibility is assumed); *u*, *v* are the radial and axial velocity components respectively; *U*, *V* are the corresponding non-dimensional velocities. *ξ* and *η* are the non-dimensional similarity quantities for the length scales.

#### **2.2 Experimental setup**

A simple sketch of the ring generator is presented in figure 7.

The rectangular tank is made of 15*mm* thick perspex with a bottom cross-sectional area of 750*mm* × 750*mm* and a height of 1500*mm*. The top of the tank is uncovered. The tank is therefore transparent from all directions of views. Other important geometrical parameters of the vortex generator are labelled in figure 7 and is described in Gan et al. (2011).

The effective *Re* is set to 41280 and *L*/*D* = 3.43 in order to match the conditions in Glezer & Coles (1990) and Gan & Nickels (2010), because parts of the reconstruction validation will rely on the data in them. The piston velocity programme is approximately a top-hat shape: the piston acceleration/deceleration time is about 10% of the piston total movement duration. Rings produced at this combination of *Re* and *L*/*D* are well in the turbulent region on the transition map (see Glezer, 1988).

In this arrangement, the PIV (laser sheet) plane is located 6D (six orifice diameters) downstream of the orifice exit. The PIV system is provided by LaVision Ltd: a pair of high-speed Photron APX cameras are used as the image recording devices and the particle illumination is realised by a Pegasus PIV Laser which consists of a dual-cavity diode pumped Nd:YLF laser head and is capable of emitting a beam of 527*nm* wavelength and 10*m J* energy. The laser beam is converted to a sheet by passing through a cylindrical diverging lens. The thickness of the sheet can be adjusted by changing the separation of a pair of telescope lenses housed before the cylindrical lens and is set to about 4 − 5*mm*. The flow is seeded by 50*μm* diameter, silver-coated hollow glass spheres.

After the changes in refractive index are considered, the effective angle between the two cameras is approximately 120*o*. The two cameras run in single-frame single-exposure mode, and the operation frequency is set at *f* = 600*Hz* giving a Δ*t* = 1.67 *ms*. The interrogation window size for time series cross-correlation process is set to 16 <sup>×</sup> <sup>16</sup> *pixel*<sup>2</sup> with a 25% overlap to give a spatial resolution of Δ*x* = Δ*y* ≈ 1.70*mm* (based on vector spacing, ≈ 3.4% D) in the PIV plane. The FOV of the PIV plane is about 230 <sup>×</sup> <sup>164</sup> *mm*2.

### **2.3 Accuracy justification**

8 Will-be-set-by-IN-TECH

Fig. 7. Schematic diagram of the vortex ring generator. The diagram is not to scale. The coordinate system adopted in this experiment is also shown, where *z* = 0 is set at the orifice

<sup>4</sup> *V* = *v*

where *I* is the hydrodynamic impulse (assumed invariant); *ρ* is the density (a constant when incompressibility is assumed); *u*, *v* are the radial and axial velocity components respectively; *U*, *V* are the corresponding non-dimensional velocities. *ξ* and *η* are the non-dimensional

The rectangular tank is made of 15*mm* thick perspex with a bottom cross-sectional area of 750*mm* × 750*mm* and a height of 1500*mm*. The top of the tank is uncovered. The tank is therefore transparent from all directions of views. Other important geometrical parameters of

the vortex generator are labelled in figure 7 and is described in Gan et al. (2011).

 *ρ I* 1 4

(*t* − *to*) 3

<sup>4</sup> , (10)

exit.

and from the streamfunction,

**2.2 Experimental setup**

*U* = *u*

similarity quantities for the length scales.

 *ρ I* 1 4

A simple sketch of the ring generator is presented in figure 7.

(*t* − *to*) 3

Ca libr at ion pley−642 in2046specti −60on:: r ed−4 =plane−2 1, greenx =plane 20, both ma 40pped onto60 same plne for view ing ease

> An important prerequisite for the reconstruction of a turbulent vortex ring using Taylor's hypothesis is to identify appropriate scales of the fluctuating and convective velocities in equation 7.

> The relevant convective velocity in this work is the ring advection speed, or celerity, and is obtained from the independent two-dimensional PIV results in Gan & Nickels (2010). The centroid-determined vortex ring radii *r* and celerities *ut* for 50 realisations are reproduced in figure 8. From this figure the spatially averaged celerity ( data) at various streamwise locations is obtained between *z* = 5.5*D* to *z* = 6.5*D* and the ensemble averaged ring celerity is obtained by a least-squares fit (blue line) giving: *ut* <sup>=</sup> 270.34 *mms*−<sup>1</sup> <sup>≈</sup> 0.54 *Up*.

> The ring celerity also affects the spatial resolution in the *z* direction, Δ*z*. The reconstruction of the planar velocity fields in the *z* direction is selected to give a similar spatial resolution in the *x* − *y* plane.

> The most straight-forward reconstruction process requires the entire cross-sectional area of the ring bubble to pass normally through the PIV plane with constant celerity. Figure 8 shows that

ξRe1

ξRe1

(d): <sup>−</sup>*∂<sup>v</sup> ∂t* 

terms

−*∂<sup>u</sup> ∂t* and

−25.8

−24.6

−25.8

−25.4

−25

−25.4

−25

−24.6

<sup>η</sup>Re1 −0.5 0 0.5

<sup>η</sup>Re1 −0.5 0 0.5

reconstruction is perfect, contours of these two terms will be identical.

*ut ∂u ∂z* and *ut ∂v ∂z* 

Fig. 9. Test of Taylor's hypothesis by comparing (a):

this purpose. The convection terms

−*∂v ∂t* 

> *ut ∂***u** *∂z* = *ut ∂***u** *∂z*

> −*∂***<sup>u</sup>** *∂t* = −*∂***<sup>u</sup>** *∂t*

the rings' time history where the similarity transformation holds.

<sup>η</sup>Re1 −0.5 0 0.5

(c) (d)

(a) (b)

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 13

−2000

−600 −400 −200 0 200 400 600

−1000

0

1000

2000

<sup>η</sup>Re1 −0.5 0 0.5

> <sup>−</sup>*∂<sup>u</sup> ∂t* , (c): *ut ∂v ∂z* and

are compared with the acceleration

<sup>4</sup> . (12)

*ut ∂u ∂z* , (b):

in their non-dimensional forms in figure 9, where

(*t* − *to*) 7 4

(*t* − *to*) 7

 *ρ I* 1 4

 *ρ I* 1 4

Figure 9 shows excellent agreement between the ensemble averaged acceleration and convection terms which lends confidence to the validity of the reconstruction. The good agreement obtained in the similarity coordinates means that Taylor's hypothesis is valid over

. Data obtained from two-dimensional PIV measurements. If the Taylor's

Fig. 8. Ring radius *r* and celerities *ut* as functions of streamwise distances *z* for the ensemble of 50 realisations from two-dimensional PIV measurements. The grey lines are the traces of each of the 50 realisations; : the ensemble averaged quantities; −: the first order least squares fit of . Data are taken from Gan & Nickels (2010).

this condition can be approximated by the ensemble averaged values.5 The celerities of single realisations are denoted by the grey traces and are always scattered as the ring centroid is changed by the turbulence. This results in a variation of ±10% of the value at each location.

If the averaged Reynolds stresses −*V*�*V*� of the ring bubble is used (from Gan & Nickels, 2010), taking the square root of these and the similarity value of the ring celerity, *Ut* = 6.4 (which can be scaled from *ut* in physical space by equation 10), for rings with a *Re* = 41280 (see Table 1 in Gan & Nickels, 2010), equation 7 gives:

$$\frac{u'}{\underline{U}} = \frac{V'}{\underline{U}\_t} < \frac{0.78}{6.4} \approx 0.12.\tag{11}$$

The maximum error can be estimated using the maximum fluctuation within the ring bubble *V*� which gives *V*� /*Ut* ≈ 0.33. However, only small contour regions within the ring bubble have this value and these regions will have the largest uncertainty in the reconstruction. A rough estimation of the contribution from *ut* fluctuation can also be given by figure 8, which is *u*� *<sup>t</sup>*/*ut <* ±10% , which according to equation 11 gives a value of 0.2.

A more rigorous method to assess the validity of Taylor's reconstruction is to check whether the material derivative of the velocity vector *D***u**/*Dt* ≈ 0 which is satisfied if *ut* � *u*� , *v*� neglecting pressure and viscous terms. The 2D PIV results in Gan & Nickels (2010) are used for

<sup>5</sup> Ideally the instantaneous convection velocity of the ring should be used. This requires simultaneous measurements normal to the measurement plane which were not available.

10 Will-be-set-by-IN-TECH

5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5

z/D

5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5

z/D

Fig. 8. Ring radius *r* and celerities *ut* as functions of streamwise distances *z* for the ensemble of 50 realisations from two-dimensional PIV measurements. The grey lines are the traces of each of the 50 realisations; : the ensemble averaged quantities; −: the first order least

this condition can be approximated by the ensemble averaged values.5 The celerities of single realisations are denoted by the grey traces and are always scattered as the ring centroid is changed by the turbulence. This results in a variation of ±10% of the value at each location. If the averaged Reynolds stresses −*V*�*V*� of the ring bubble is used (from Gan & Nickels, 2010), taking the square root of these and the similarity value of the ring celerity, *Ut* = 6.4 (which can be scaled from *ut* in physical space by equation 10), for rings with a *Re* = 41280 (see Table

The maximum error can be estimated using the maximum fluctuation within the ring bubble

have this value and these regions will have the largest uncertainty in the reconstruction. A rough estimation of the contribution from *ut* fluctuation can also be given by figure 8, which

A more rigorous method to assess the validity of Taylor's reconstruction is to check whether the material derivative of the velocity vector *D***u**/*Dt* ≈ 0 which is satisfied if *ut* � *u*�

neglecting pressure and viscous terms. The 2D PIV results in Gan & Nickels (2010) are used for

<sup>5</sup> Ideally the instantaneous convection velocity of the ring should be used. This requires simultaneous

/*Ut* ≈ 0.33. However, only small contour regions within the ring bubble

6.4 <sup>≈</sup> 0.12. (11)

, *v*�

squares fit of . Data are taken from Gan & Nickels (2010).

*u*� *<sup>U</sup>* <sup>=</sup> *<sup>V</sup>*� *Ut <* 0.78

*<sup>t</sup>*/*ut <* ±10% , which according to equation 11 gives a value of 0.2.

measurements normal to the measurement plane which were not available.

1 in Gan & Nickels, 2010), equation 7 gives:

0.5 0.6 0.7 0.8 0.9

0

0.5

ut / Up

*V*� which gives *V*�

is *u*�

1

r / D

Fig. 9. Test of Taylor's hypothesis by comparing (a): *ut ∂u ∂z* , (b): <sup>−</sup>*∂<sup>u</sup> ∂t* , (c): *ut ∂v ∂z* and (d): <sup>−</sup>*∂<sup>v</sup> ∂t* . Data obtained from two-dimensional PIV measurements. If the Taylor's reconstruction is perfect, contours of these two terms will be identical.

this purpose. The convection terms *ut ∂u ∂z* and *ut ∂v ∂z* are compared with the acceleration terms −*∂<sup>u</sup> ∂t* and −*∂v ∂t* in their non-dimensional forms in figure 9, where

$$
\begin{split}
\left< u\_{l} \frac{\partial \mathbf{u}}{\partial z} \right> &= \left( u\_{l} \frac{\partial \mathbf{u}}{\partial z} \right) \left( \frac{\rho}{I} \right)^{\frac{1}{4}} (t - t\_{o})^{\frac{7}{4}} \\
\left< -\frac{\partial \mathbf{u}}{\partial t} \right> &= \left( -\frac{\partial \mathbf{u}}{\partial t} \right) \left( \frac{\rho}{I} \right)^{\frac{1}{4}} (t - t\_{o})^{\frac{7}{4}} .
\end{split}
\tag{12}
$$

Figure 9 shows excellent agreement between the ensemble averaged acceleration and convection terms which lends confidence to the validity of the reconstruction. The good agreement obtained in the similarity coordinates means that Taylor's hypothesis is valid over the rings' time history where the similarity transformation holds.

−25 −20 −15 −10 −5 0 5 10 15 20 25

Fig. 11. The core radius variation as the ring passes through the testing station for all the 50 realisations from two-dimensional PIV measurements. The grey lines are the traces of each of the 50 realisations. *r*(*δt*) denotes the radius of a single realisation as a function of the time *t*; �*r*� denotes the average radius of all the 50 rings. The threshold to determine the radius is set as 100*s*−1, which is about 50% of the peak vorticity intensity, see figure 18. Note that the core

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 15

δ t/ms

−600 −400 −200 0 200 400 600

Fig. 12. The circulation variation as the ring passes through the testing station for all the 50 realisations from two-dimensional PIV measurements. The grey lines are the *β*(*t*) traces for each of the 50 realisations, which is calculated by equation 13. The ±10% lines are also

As the largest velocity fluctuations occur around the vortex core, the spatially averaged velocity of 50 realisations can be compared with the reconstructed value. To estimate the

> *C*

where Γ is the circulation obtained along a closed loop *C* enclosing an area of 2*D* and 1.5*D* in streamwise and spanwise directions respectively around each vortex core similar to that shown in figure 10. (*t*) is the time when the ring passes through the 6*D* station, similar to that shown in (a), (b) and (c) of figure 10. The subscript 're' indicates the reconstructed ring at the 6*D* station as shown in (d) of figure 10. The quantity *β*(*t*) is determined for 50 realisations and

**u** (*t*) · d**l** −

 *C* **u***re* · d**l**

 C **u**re · d**l**

, (13)

=

spatially averaged velocity error, a quantity *β*(*t*) is introduced:

*<sup>β</sup>*(*t*) = <sup>Γ</sup> (*t*) <sup>−</sup> <sup>Γ</sup>*re* Γ*re*

δ t/ms

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.5

shown by the −− traces.

shown in figure 12.

0

β(t)

0.5

passes through the testing station within ±19*ms*.

[r(δ t)−<r>]/<r>

Fig. 10. An individual vortex ring recorded by two-dimensional PIV is used to test the Taylor's reconstruction process. (a), (b), (c): the three continuous snapshots showing the left core of a ring passing through the 6*D* station. The layers of the two-component velocity information at the 6*D* station are then stacked to reconstruct the core, in (d), in which the ensemble averaged ring celerity 270.34 *mms*−1, instead of the one for this particular ring, is used. The vorticity contour levels: -240.0 (10.0) -20.0; → denotes the in-plane velocity vectors.

Figure 10 (a) to (c) plots an image sequence showing a single vortex ring core passing through the 6D station which are then used to reconstruct the vortex core in (d). The velocity vectors are overlaid with vorticity contours. Comparing (a) to (c) with (d) shows that the peak vorticity of the core is basically retained. The vorticity distribution of the inner core shows a slight change in figure 10 (c) but these are small changes and close to the maximum spatial resolution. The instantaneous vorticity is naturally very sensitive to the velocity fluctuation however the core radius remains similar, within ±10%, see figure 11. The core is compact being about 0.2*D* in size, passes through the 6D station within about 37 *ms*. The vorticity surrounding the core is weaker and undergoing shedding-reattachment processes at a fairly fast pace (Gan, 2010). Any high speed fluctuations in the weaker regions are less frozen than the core but their features are well captured in the reconstructed image in figure 10 (d).

12 Will-be-set-by-IN-TECH

−6.5

−5.5

−6.5

−6

−6

−5.5

−1 −0.5

y / D −1 −0.5

−1 −0.5

(a) (b)

y / D

(c) (d)

Fig. 10. An individual vortex ring recorded by two-dimensional PIV is used to test the Taylor's reconstruction process. (a), (b), (c): the three continuous snapshots showing the left core of a ring passing through the 6*D* station. The layers of the two-component velocity information at the 6*D* station are then stacked to reconstruct the core, in (d), in which the ensemble averaged ring celerity 270.34 *mms*−1, instead of the one for this particular ring, is used. The vorticity contour levels: -240.0 (10.0) -20.0; → denotes the in-plane velocity vectors. Figure 10 (a) to (c) plots an image sequence showing a single vortex ring core passing through the 6D station which are then used to reconstruct the vortex core in (d). The velocity vectors are overlaid with vorticity contours. Comparing (a) to (c) with (d) shows that the peak vorticity of the core is basically retained. The vorticity distribution of the inner core shows a slight change in figure 10 (c) but these are small changes and close to the maximum spatial resolution. The instantaneous vorticity is naturally very sensitive to the velocity fluctuation however the core radius remains similar, within ±10%, see figure 11. The core is compact being about 0.2*D* in size, passes through the 6D station within about 37 *ms*. The vorticity surrounding the core is weaker and undergoing shedding-reattachment processes at a fairly fast pace (Gan, 2010). Any high speed fluctuations in the weaker regions are less frozen than the core but their features are well captured in the reconstructed image in figure 10 (d).

−1 −0.5

z / D

z / D

−6.5

−6

−6.5

−5.5

−6

−5.5

Fig. 11. The core radius variation as the ring passes through the testing station for all the 50 realisations from two-dimensional PIV measurements. The grey lines are the traces of each of the 50 realisations. *r*(*δt*) denotes the radius of a single realisation as a function of the time *t*; �*r*� denotes the average radius of all the 50 rings. The threshold to determine the radius is set as 100*s*−1, which is about 50% of the peak vorticity intensity, see figure 18. Note that the core passes through the testing station within ±19*ms*.

Fig. 12. The circulation variation as the ring passes through the testing station for all the 50 realisations from two-dimensional PIV measurements. The grey lines are the *β*(*t*) traces for each of the 50 realisations, which is calculated by equation 13. The ±10% lines are also shown by the −− traces.

As the largest velocity fluctuations occur around the vortex core, the spatially averaged velocity of 50 realisations can be compared with the reconstructed value. To estimate the spatially averaged velocity error, a quantity *β*(*t*) is introduced:

$$\beta(t) = \frac{\Gamma(t) - \Gamma\_{\rm re}}{\Gamma\_{\rm re}} = \frac{\oint\_{\mathbb{C}} \mathbf{u}\left(t\right) \cdot \mathbf{dl} - \oint\_{\mathbb{C}} \mathbf{u}\_{\rm re} \cdot \mathbf{dl}}{\oint\_{\mathbb{C}} \mathbf{u}\_{\rm re} \cdot \mathbf{dl}},\tag{13}$$

where Γ is the circulation obtained along a closed loop *C* enclosing an area of 2*D* and 1.5*D* in streamwise and spanwise directions respectively around each vortex core similar to that shown in figure 10. (*t*) is the time when the ring passes through the 6*D* station, similar to that shown in (a), (b) and (c) of figure 10. The subscript 're' indicates the reconstructed ring at the 6*D* station as shown in (d) of figure 10. The quantity *β*(*t*) is determined for 50 realisations and shown in figure 12.

As the loops are of the same size, and the circulation is conserved, *β*(*t*) compares the spatially-averaged variation between **u**(*t*) of each realisation with the reconstructed one. The resulting error is less than ±10%, a similar value to the core dispersion in figure 8 and radius variation in figure 11. The results from these tests show that the expected error of the reconstruction is within ±10%.

A more direct assessment is to investigate the divergence field of the reconstructed three-dimensional vortex ring. Figure 13 shows the p.d.f. (probability density function) of the divergence of the flow field, where in (b) and (d) the divergence is normalised by the norm of the local vector gradient tensor (∇**u** : ∇**u**) 1/2; in (c) *κ* is defined by:

$$\kappa = \frac{\left(\partial u/\partial \mathbf{x} + \partial v/\partial y + \partial w/\partial \mathbf{z}\right)^2}{\left(\partial u/\partial \mathbf{x}\right)^2 + \left(\partial v/\partial y\right)^2 + \left(\partial w/\partial \mathbf{z}\right)^2}.\tag{14}$$

−2 0 2

∇⋅u/(∇u:∇u)1/2

1/2; contour levels

−2 0 2

∇⋅u/(∇u:∇u)1/2

0

0.02

0.04

PDF

(a) (b)

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 17

(∇u:

0

1/2 and (∇**<sup>u</sup>** : <sup>∇</sup>**u**)

1

2

3

4

5

6

(d)

∇u)1/2 (s−1)

−20 −10 0 10 20

divergence free

0 0.2 0.4 0.6 0.8

p.d.f. of *κ*. (d) the joint p.d.f. of ∇ · **u**/ (∇**u** : ∇**u**)

κ

∂u/∂x (s−1)

(c)

Fig. 13. The divergence of the reconstructed three-dimensional velocity field. (a) the joint p.d.f. between *∂u*/*∂x* and − (*∂v*/*∂y* + *∂w*/*∂z*), contour levels [0.0002 0.0007 0.002 0.007 0.02 0.1]; the straight line indicates divergence free. (b) the p.d.f. of ∇ · **<sup>u</sup>**/(∇**<sup>u</sup>** : <sup>∇</sup>**u**)−1/2. (c) the

−20

0

[0.0003 : 0.0003 : 0.003].

0.02

0.04

0.06

PDF

0.08

0.1

−15

−10

−5

0

−(∂v/∂y+∂w/∂z) (s−1)

5

10

15

20

0.06

It can be seen that most of the data points locate in regions very closed to divergence free, bearing in mind that no experiment is divergence free, due to the finite spatial resolution.

#### **2.4 The reconstructed velocity field**

A typical instantaneous stereoscopic PIV velocity - vorticity field is shown in figure 14. Only the in-plane velocity vectors are plotted from which the overlaid vorticity contours *ω<sup>k</sup>* were determined. The level of turbulence is clearly captured by the asymmetry in the vector field and the fluctuations of vorticity.

In order to visualise the vortex ring structure, an appropriate scalar which might be used after Taylor's reconstruction of velocity field is the vorticity (∇ × *u* ) magnitude. The structure of the vortex ring bubble and the wake can then be observed in figure 15 with the strong level of turbulence illustrated by the degree of the wrinkling along the isosurfaces and streamlines. It must be pointed out that in order to visualise the ring bubble by streamlines, the ring needs to be put in a stationary frame of reference. This is achieved by simply subtracting the mean advection velocity of the ring at the PIV testing location<sup>6</sup> from the instantaneous velocity at every data point which has no effect on the vorticity field. In figure 15 the extent of irregularity in the streamline patterns can be distinguished upstream and downstream of the ring bubble with localised areas of high vorticity found in the wake. A wavy core is also observed, confirming the existence of azimuthal waves.

In order to visualise the relationship between the high and low vorticity fluctuations around the vortex ring core and the outer bubble respectively, two sections of the selected bubble vorticity isosurface are shown in figure 16. As expected the low intensity wraps around the core. Tube shaped vorticity isosurfaces of low intensity are shed into the wake showing some agreements with the numerical simulations of Bergdorf et al. (2007) and Archer et al. (2008) which show hairpin vortices being shed from the vortex bubble into the wake. Although, hairpin structures were not clearly observed in the current study. This could be due to the lower spatial resolution of the experiment or the different Reynolds numbers and initial conditions compared with the numerical simulations.

<sup>6</sup> It is assumed here that the ring advection velocity in the vicinity of the PIV testing location is constant. Figure 8 shows that it is a fairly reasonable assumption.

14 Will-be-set-by-IN-TECH

As the loops are of the same size, and the circulation is conserved, *β*(*t*) compares the spatially-averaged variation between **u**(*t*) of each realisation with the reconstructed one. The resulting error is less than ±10%, a similar value to the core dispersion in figure 8 and radius variation in figure 11. The results from these tests show that the expected error of the

A more direct assessment is to investigate the divergence field of the reconstructed three-dimensional vortex ring. Figure 13 shows the p.d.f. (probability density function) of the divergence of the flow field, where in (b) and (d) the divergence is normalised by the

*<sup>κ</sup>* <sup>=</sup> (*∂u*/*∂<sup>x</sup>* <sup>+</sup> *<sup>∂</sup>v*/*∂<sup>y</sup>* <sup>+</sup> *<sup>∂</sup>w*/*∂z*)

<sup>2</sup> + (*∂v*/*∂y*)

It can be seen that most of the data points locate in regions very closed to divergence free, bearing in mind that no experiment is divergence free, due to the finite spatial resolution.

A typical instantaneous stereoscopic PIV velocity - vorticity field is shown in figure 14. Only the in-plane velocity vectors are plotted from which the overlaid vorticity contours *ω<sup>k</sup>* were determined. The level of turbulence is clearly captured by the asymmetry in the vector field

In order to visualise the vortex ring structure, an appropriate scalar which might be used after Taylor's reconstruction of velocity field is the vorticity (∇ × *u* ) magnitude. The structure of the vortex ring bubble and the wake can then be observed in figure 15 with the strong level of turbulence illustrated by the degree of the wrinkling along the isosurfaces and streamlines. It must be pointed out that in order to visualise the ring bubble by streamlines, the ring needs to be put in a stationary frame of reference. This is achieved by simply subtracting the mean advection velocity of the ring at the PIV testing location<sup>6</sup> from the instantaneous velocity at every data point which has no effect on the vorticity field. In figure 15 the extent of irregularity in the streamline patterns can be distinguished upstream and downstream of the ring bubble with localised areas of high vorticity found in the wake. A wavy core is also

In order to visualise the relationship between the high and low vorticity fluctuations around the vortex ring core and the outer bubble respectively, two sections of the selected bubble vorticity isosurface are shown in figure 16. As expected the low intensity wraps around the core. Tube shaped vorticity isosurfaces of low intensity are shed into the wake showing some agreements with the numerical simulations of Bergdorf et al. (2007) and Archer et al. (2008) which show hairpin vortices being shed from the vortex bubble into the wake. Although, hairpin structures were not clearly observed in the current study. This could be due to the lower spatial resolution of the experiment or the different Reynolds numbers and initial

<sup>6</sup> It is assumed here that the ring advection velocity in the vicinity of the PIV testing location is constant.

(*∂u*/*∂x*)

1/2; in (c) *κ* is defined by:

<sup>2</sup> . (14)

2

<sup>2</sup> + (*∂w*/*∂z*)

reconstruction is within ±10%.

**2.4 The reconstructed velocity field**

and the fluctuations of vorticity.

norm of the local vector gradient tensor (∇**u** : ∇**u**)

observed, confirming the existence of azimuthal waves.

conditions compared with the numerical simulations.

Figure 8 shows that it is a fairly reasonable assumption.

Fig. 13. The divergence of the reconstructed three-dimensional velocity field. (a) the joint p.d.f. between *∂u*/*∂x* and − (*∂v*/*∂y* + *∂w*/*∂z*), contour levels [0.0002 0.0007 0.002 0.007 0.02 0.1]; the straight line indicates divergence free. (b) the p.d.f. of ∇ · **<sup>u</sup>**/(∇**<sup>u</sup>** : <sup>∇</sup>**u**)−1/2. (c) the p.d.f. of *κ*. (d) the joint p.d.f. of ∇ · **u**/ (∇**u** : ∇**u**) 1/2 and (∇**<sup>u</sup>** : <sup>∇</sup>**u**) 1/2; contour levels [0.0003 : 0.0003 : 0.003].

Ca libr at ion plye−6422046 inspecti −60on:: r−4ed =pla−2ne 1, grexen =plane 20, both ma 40pped on60to same plne for view ing ease

Fig. 16. Isosurfaces of the three-dimensional vorticity magnitude, showing the high intensity core and two portions of the low intensity vorticity blobs wrapping around the core and shed to the wake. Isosurface intensities: 150*s*−<sup>1</sup> and 90*s*−1. Colour code on the core surface

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 19

The three components of the vorticity, *ω<sup>i</sup> ω<sup>j</sup> ω<sup>k</sup>* in the two central cross-sectional planes *y* − *z* and *x* − *z* are plotted in figure 17. The reconstructed azimuthal vorticity contours in (a) and (b) can be compared with the instantaneous vorticity contours obtained from the two-dimensional PIV results, shown in figure 18. Note that Figure 17 and 18 belong to different realisations. Because the rings are very turbulent, contour shapes are noisier especially for low level vorticity, and are expected to be different for different realisations due to the stochastic nature of the flow. However, the main features and the maximum intensities show good agreements with the main difference being that the reconstructed fields show less low level azimuthal vorticity. A long streamwise vorticity *ω<sup>k</sup>* structure is observed in the three-dimensional view in figure 19, which agrees with the findings in Bergdorf et al. (2007). In the ring bubble region, the streamwise vorticity are found to wrap around the vortex core, in a manner such that the positive and the negative valued vortices are separated by each other. This can be seen by the colour code on the core isosurface in figure 16 and shows reasonable agreement with the results in Archer et al. (2008). However, the structure of the vorticity is difficult to identify in the three-dimensional view due to the effect of turbulence which breaks the isosurfaces thresholded at *ω<sup>k</sup>* = 30*s*−<sup>1</sup> into pieces. The colour code on the *ω<sup>k</sup>* isosurface indicates the strength of the streamwise stretching term *Sk* = (*ω* · ∇)*uk*. The positive value of *ω<sup>k</sup>* reflects the stretching of the streamwise vortex tubes. The negative valued streamwise vortex tubes are not shown but the structure and the stretching strength of these tubes was

indicates the streamwise vorticity *ωk*.

found to be similar.

Fig. 14. A presentation of instantaneous velocity and vorticity in azimuthal plane in physical coordinates, when the ring centre is about to reach the PIV plane. Only the velocity vectors in the azimuthal plane are shown. The vorticity is in the streamwise direction, *ωk*, zero level bypassed.

Fig. 15. A three-dimensional vorticity magnitude contour. Isosurface levels are 150*s*−1, 100*s*−1, 50*s*−1. Streamlines are shown in the second figure. The colour bar shows the streamwise velocity level on the core surface.

16 Will-be-set-by-IN-TECH

x/mm

Fig. 14. A presentation of instantaneous velocity and vorticity in azimuthal plane in physical coordinates, when the ring centre is about to reach the PIV plane. Only the velocity vectors in the azimuthal plane are shown. The vorticity is in the streamwise direction, *ωk*, zero level

> Ca libr at ion pleyin−6422046specti −6on:: r0ed−4 =plane−2 1, green =xplane 2,0 both ma 40pped onto60 same plne for view ing ease

−100

−50

0

50

100

−60 −40 −20 0 20 40 60

Fig. 15. A three-dimensional vorticity magnitude contour. Isosurface levels are 150*s*−1, 100*s*−1, 50*s*−1. Streamlines are shown in the second figure. The colour bar shows the

y/mm

−60

streamwise velocity level on the core surface.

bypassed.

−40

−20

0

20

40

60

Fig. 16. Isosurfaces of the three-dimensional vorticity magnitude, showing the high intensity core and two portions of the low intensity vorticity blobs wrapping around the core and shed to the wake. Isosurface intensities: 150*s*−<sup>1</sup> and 90*s*−1. Colour code on the core surface indicates the streamwise vorticity *ωk*.

The three components of the vorticity, *ω<sup>i</sup> ω<sup>j</sup> ω<sup>k</sup>* in the two central cross-sectional planes *y* − *z* and *x* − *z* are plotted in figure 17. The reconstructed azimuthal vorticity contours in (a) and (b) can be compared with the instantaneous vorticity contours obtained from the two-dimensional PIV results, shown in figure 18. Note that Figure 17 and 18 belong to different realisations. Because the rings are very turbulent, contour shapes are noisier especially for low level vorticity, and are expected to be different for different realisations due to the stochastic nature of the flow. However, the main features and the maximum intensities show good agreements with the main difference being that the reconstructed fields show less low level azimuthal vorticity. A long streamwise vorticity *ω<sup>k</sup>* structure is observed in the three-dimensional view in figure 19, which agrees with the findings in Bergdorf et al. (2007). In the ring bubble region, the streamwise vorticity are found to wrap around the vortex core, in a manner such that the positive and the negative valued vortices are separated by each other. This can be seen by the colour code on the core isosurface in figure 16 and shows reasonable agreement with the results in Archer et al. (2008). However, the structure of the vorticity is difficult to identify in the three-dimensional view due to the effect of turbulence which breaks the isosurfaces thresholded at *ω<sup>k</sup>* = 30*s*−<sup>1</sup> into pieces. The colour code on the *ω<sup>k</sup>* isosurface indicates the strength of the streamwise stretching term *Sk* = (*ω* · ∇)*uk*. The positive value of *ω<sup>k</sup>* reflects the stretching of the streamwise vortex tubes. The negative valued streamwise vortex tubes are not shown but the structure and the stretching strength of these tubes was found to be similar.

y/D

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 21

Fig. 18. The vorticity contour of an instantaneous realisation from the two-dimensional PIV

Fig. 19. The streamwise vorticity wraps around the core. The core position is shown by the yellow coloured isosurface of vorticity magnitude *ω<sup>m</sup>* = 140*s*−1. The vorticity isosurfaces wrapping around the core are *ω<sup>k</sup>* = 30*s*−1. Colour code on *ω<sup>k</sup>* surface indicates the

positive-valued streamwise (axial) vortex stretching term *Sk*.

−200 −150 −100 −50 0 50 100 150 200

−1 <sup>0</sup> <sup>1</sup> −7

z/D

experiments.

−6

−5

−4

−3

Fig. 17. The three components of vorticity in the two central cross-sectional planes *y* − *z* and *x* − *z*. (a) and (b): azimuthal vorticity contours, *ωi*, *ωj*; (c) and (d): streamwise vorticity contours, *ωk*; (e) and (f): spanwise vorticity contours, *ωj*, *ωi*.

18 Will-be-set-by-IN-TECH

(a) (b)

(c) (d)

(e) (f)

z/D

−7

−3

z/D

−7

−3

z/D

−7

−6

−5

−4

y/D

x/D −1.5 −1 −0.5 0 0.5 1 1.5

Fig. 17. The three components of vorticity in the two central cross-sectional planes *y* − *z* and *x* − *z*. (a) and (b): azimuthal vorticity contours, *ωi*, *ωj*; (c) and (d): streamwise vorticity

−200

−80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80

−100

0

100

200

−1.5 −1 −0.5 0 0.5 1 1.5

contours, *ωk*; (e) and (f): spanwise vorticity contours, *ωj*, *ωi*.

−6

−5

−4

−6

−5

−4

−3

Fig. 18. The vorticity contour of an instantaneous realisation from the two-dimensional PIV experiments.

Fig. 19. The streamwise vorticity wraps around the core. The core position is shown by the yellow coloured isosurface of vorticity magnitude *ω<sup>m</sup>* = 140*s*−1. The vorticity isosurfaces wrapping around the core are *ω<sup>k</sup>* = 30*s*−1. Colour code on *ω<sup>k</sup>* surface indicates the positive-valued streamwise (axial) vortex stretching term *Sk*.

#### **2.5 The azimuthally averaged quantities**

Statistically speaking, an ensemble-averaged vortex ring bubble tends to be axisymmetric7. It is interesting to compare the azimuthally averaged velocity components to those by two-dimensional ensemble averaged ones, which may also serve as a validation of the technique.

The azimuthally averaging process is applied in a cylindrical coordinate system in which the process begins at *θ* = 0 and ends at *θ* = 2*π*. The first step is to locate the 'best' centre point, or a proper axis of symmetry as the results from the azimuthally averaging process will depend heavily on the location of this axis. The axis is found by a best r.m.s. fit of the core area to a circle. After this, data format in Cartesian coordinates is converted to cylindrical coordinates. More details can be found in Gan et al. (2011).

The velocities are non-dimensionalised by equation 10 and presented in the similarity coordinates calculated from equation 9. Figure 20 shows the dimensionless velocities *U<sup>θ</sup>* (radial), *<sup>V</sup><sup>θ</sup>* (axial), *<sup>W</sup><sup>θ</sup>* (azimuthal) and the dimensionless vorticity *<sup>ω</sup><sup>θ</sup>* , where:

$$
\hat{\omega}\_{\theta} = \frac{\partial V\_{\theta}}{\partial \eta} - \frac{\partial \mathcal{U}\_{\theta}}{\partial \xi} \, \, \, \tag{15}
$$

ξRe

ξRe

azimuthally averaging process.

directions, *Si*, *Sj*, *Sk*:

on *ωi*, *ωj*, *ω<sup>k</sup>* respectively.

−25.9

−25.7

−25.5

−25.3

−25.1 −25

−25.9

−25.7

−25.5

−25.3

−25.1 −25

0 0.2 0.4

<sup>η</sup>Re 0 0.2 0.4

turbulence quantities, one can refer to Gan et al. (2011) for more details.

⎛ ⎝ *Si Sj Sk* ⎞

0 0.2 0.4

−14 −12 −10 −8 −6 −4 −2 0 2

(b)

100

⎠ , (17)

200

300

400

500

600

<sup>η</sup>Re 0 0.2 0.4

−25.9

−6 −4 −2 0 2 4 6

−25.9

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

averaging process: averaging along the *θ* direction in similarity coordinates. Contours shown

If the azimuthally averaged velocity components make sense it is also possible to compare the

The turbulence production is believed to be closely related to the vortex stretching, this mechanism is expected to take place in the ring core and bubble windward regions (Gan & Nickels, 2010). The stretching effect can be assessed via the vorticity equation. In Cartesian coordinates, the stretching term (*ω* · ∇)*u* represents the vortex stretching in three principal

⎠ = (*ω* · ∇)

Because the flow field of a vortex ring is close to axisymmetric, the stretching of the vortex tubes in the windward bubble surface is expected to be orientated in the radial direction (due to the mean velocity direction). To better illustrate the stretching effect, the vorticity and the stretching vector in equation 17 are transferred from Cartesian coordinates to cylindrical coordinates, as for velocities: *ω* (*i*, *j*) �→ *ω* (*θ*,*r*), *S* (*i*, *j*) �→ *S* (*θ*,*r*), *ω<sup>k</sup>* and *Sk* are unaffected.

⎛ ⎝ *ui uj uk* ⎞

Fig. 20. The mean structures of velocity and vorticity calculated by the azimuthally

are *<sup>U</sup><sup>θ</sup>* (radial), *<sup>V</sup><sup>θ</sup>* (axial), *<sup>W</sup><sup>θ</sup>* (azimuthal) and *<sup>ω</sup>*�*<sup>θ</sup>* (azimuthal) in (a), (b), (c) and (d), respectively; the subscript *θ* denotes that the corresponding quantity is calculated by the

−25.7

−25.5

−25.3

−25.1 −25

(c) (d)

−25.7

−25.5

−25.3

−25.1 −25

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 23

(a)

and *θ* here denotes an azimuthally averaging result, *ξ*, *η* denote the axial and radial direction in cylindrical coordinates respectively.

In figure 20, noise increases towards the centre due to the reduced number of data points for averaging8. However, the main area of interest is obviously the core region. The central region exhibits low azimuthal velocities in figure 20 c): a long region of positive mean velocity is observed in the wake while a weak negative mean velocity can be observed in the core centre region. The presence of a mean velocity in the core region is not surprising, a number of researchers have also observed such behaviour, nevertheless the magnitude is considerably smaller than both the convection velocity *V<sup>θ</sup>* and the radial velocity *U<sup>θ</sup>* components shown in figure 20 a) and b). The opposite sensed mean velocity in the inner region can be partly due to the noise and partly due to a possible mechanism of conservation of angular momentum of the vortex ring bubble. The streamline pattern in figure 15 also suggests the presence of a mean azimuthal velocity.

The dimensionless circulation Γ*<sup>θ</sup>* can be computed by equation 16 from the contour plot in figure 20 d).

$$
\Gamma\_{\theta} = \int\_{S} \widehat{\omega}\_{\theta} \, d\xi d\eta\_{\prime} \tag{16}
$$

where *S* denotes the entire area of the FOV in the similarity coordinates. A value of 7.08 is found, which is similar to the circulation from the two-dimensional PIV results, 6.87 (see Gan & Nickels, 2010).

<sup>7</sup> If one produces a large number of turbulent vortex rings from a circular orifice, and does an ensemble average, the resultant velocity field is approximately axisymmetric.

<sup>8</sup> Recall that the raw data is stored in the Cartesian coordinates; towards smaller radii, there are less data points.

Fig. 20. The mean structures of velocity and vorticity calculated by the azimuthally averaging process: averaging along the *θ* direction in similarity coordinates. Contours shown are *<sup>U</sup><sup>θ</sup>* (radial), *<sup>V</sup><sup>θ</sup>* (axial), *<sup>W</sup><sup>θ</sup>* (azimuthal) and *<sup>ω</sup>*�*<sup>θ</sup>* (azimuthal) in (a), (b), (c) and (d), respectively; the subscript *θ* denotes that the corresponding quantity is calculated by the azimuthally averaging process.

If the azimuthally averaged velocity components make sense it is also possible to compare the turbulence quantities, one can refer to Gan et al. (2011) for more details.

The turbulence production is believed to be closely related to the vortex stretching, this mechanism is expected to take place in the ring core and bubble windward regions (Gan & Nickels, 2010). The stretching effect can be assessed via the vorticity equation. In Cartesian coordinates, the stretching term (*ω* · ∇)*u* represents the vortex stretching in three principal directions, *Si*, *Sj*, *Sk*:

$$
\begin{pmatrix} S\_i \\ S\_j \\ S\_k \end{pmatrix} = (\vec{\omega} \cdot \nabla) \begin{pmatrix} u\_i \\ u\_j \\ u\_k \end{pmatrix}, \tag{17}
$$

on *ωi*, *ωj*, *ω<sup>k</sup>* respectively.

20 Will-be-set-by-IN-TECH

Statistically speaking, an ensemble-averaged vortex ring bubble tends to be axisymmetric7. It is interesting to compare the azimuthally averaged velocity components to those by two-dimensional ensemble averaged ones, which may also serve as a validation of the

The azimuthally averaging process is applied in a cylindrical coordinate system in which the process begins at *θ* = 0 and ends at *θ* = 2*π*. The first step is to locate the 'best' centre point, or a proper axis of symmetry as the results from the azimuthally averaging process will depend heavily on the location of this axis. The axis is found by a best r.m.s. fit of the core area to a circle. After this, data format in Cartesian coordinates is converted to cylindrical coordinates.

The velocities are non-dimensionalised by equation 10 and presented in the similarity coordinates calculated from equation 9. Figure 20 shows the dimensionless velocities *U<sup>θ</sup>*

and *θ* here denotes an azimuthally averaging result, *ξ*, *η* denote the axial and radial direction

In figure 20, noise increases towards the centre due to the reduced number of data points for averaging8. However, the main area of interest is obviously the core region. The central region exhibits low azimuthal velocities in figure 20 c): a long region of positive mean velocity is observed in the wake while a weak negative mean velocity can be observed in the core centre region. The presence of a mean velocity in the core region is not surprising, a number of researchers have also observed such behaviour, nevertheless the magnitude is considerably smaller than both the convection velocity *V<sup>θ</sup>* and the radial velocity *U<sup>θ</sup>* components shown in figure 20 a) and b). The opposite sensed mean velocity in the inner region can be partly due to the noise and partly due to a possible mechanism of conservation of angular momentum of the vortex ring bubble. The streamline pattern in figure 15 also suggests the presence of a

The dimensionless circulation Γ*<sup>θ</sup>* can be computed by equation 16 from the contour plot in

where *S* denotes the entire area of the FOV in the similarity coordinates. A value of 7.08 is found, which is similar to the circulation from the two-dimensional PIV results, 6.87 (see Gan

<sup>7</sup> If one produces a large number of turbulent vortex rings from a circular orifice, and does an ensemble

<sup>8</sup> Recall that the raw data is stored in the Cartesian coordinates; towards smaller radii, there are less data

Γ*<sup>θ</sup>* = *S*

average, the resultant velocity field is approximately axisymmetric.

*∂η* <sup>−</sup> *<sup>∂</sup>U<sup>θ</sup>*

*∂ξ* , (15)

*<sup>ω</sup><sup>θ</sup> <sup>d</sup>ξdη*, (16)

(radial), *<sup>V</sup><sup>θ</sup>* (axial), *<sup>W</sup><sup>θ</sup>* (azimuthal) and the dimensionless vorticity *<sup>ω</sup><sup>θ</sup>* , where:

*<sup>ω</sup><sup>θ</sup>* <sup>=</sup> *<sup>∂</sup>V<sup>θ</sup>*

**2.5 The azimuthally averaged quantities**

More details can be found in Gan et al. (2011).

in cylindrical coordinates respectively.

mean azimuthal velocity.

figure 20 d).

& Nickels, 2010).

points.

technique.

Because the flow field of a vortex ring is close to axisymmetric, the stretching of the vortex tubes in the windward bubble surface is expected to be orientated in the radial direction (due to the mean velocity direction). To better illustrate the stretching effect, the vorticity and the stretching vector in equation 17 are transferred from Cartesian coordinates to cylindrical coordinates, as for velocities: *ω* (*i*, *j*) �→ *ω* (*θ*,*r*), *S* (*i*, *j*) �→ *S* (*θ*,*r*), *ω<sup>k</sup>* and *Sk* are unaffected.

Γ and the advection speed *ut*. Simply increasing Reynolds number will not improve the situation. Certain treatments independent of the ring will be necessary. A possible way is to move the PIV measurement (cameras and the laser sheet together) against the ring advection direction - an 'active scanning' process. Equation 11 shows that by simply moving the PIV measurement plane at the same speed as the ring, it can bring down the ratio *u*

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 25

significantly (doubling the denominator). Moreover, to resolve the wake correctly, an active

The next question is that how fast the scan speed is optimal. Definitely scan with an infinite speed gives no error. However, first it is not allowed, even it is capable to; second, it may not be necessary in applications where Reynolds number is not too high. A proper quantity can be searched to judge what scan speed gives satisfactory accuracy level. For instance, a possible candidate can be the spatial velocity correlation tensor, or the structure function, which is expected to converge as the scan speed increases. Thus, a scan speed can be considered adequate and economic (in terms of hardware requirement and computational expense), above which the increase of the accuracy is less significant while the cost-benefit ratio is high. In different flow problems, the optimal scan speed is expected to vary. Similar to the accuracy assessment given in section 2.3, a crude estimation of the scan speed could be made from the eddy turn over time: if the mean flow is of the only interest, a moderate scan speed should be

It has been pointed out that there exists an upper limit of the active scanning speed, it is not allowed to go to infinity. In other words, it is never possible to capture a truly frozen flow field. It is because if the flow is really frozen, there will be no particle displacement, thus the PIV fails to work. In order to allow a particle displacement to compute the velocity, a minimum time duration of PIV Δ*t* has to be given at one measurement station along the scanning path. In this aspect, this PIV Δ*t*, despite its shortness - in order of millisecond, limits the highest

Nevertheless there is a potential solution for this limitation: a second stereoscopic PIV system can be introduced, aligning with the first system in the scanning path, and with a spacing *l* to the first one; see figure 22.9 Thus the value of *l*/*Uscan* effectively gives the PIV Δ*t*. Theoretically, this dual-system arrangement allows the choice of the *Uscan* value to a much higher value: at a fixed *Uscan*, PIV Δ*t* can be adjusted by setting the spacing *l* between the two systems (two laser sheets) very carefully. The highest scan speed in this case is then limited by the required spatial resolution in the scan direction, which is much more relaxed than the limit from the particle displacements. The high speed cameras available nowadays can work at several *kHz* frame rates, which allows very high scan speed while maintaining good spatial resolution in the scanning direction. By choosing a very high value of *Uscan*, one approaches

The state-of-the-art tomographic PIV is also a candidate to provide truly instantaneous three-dimensional velocity information of a flow field. Its working principle is developed

<sup>9</sup> Vortex ring study is definitely not the only customer of this method. It can be applied to many general

enough; if small eddies are to be resolved, higher speed is required.

a truly frozen and instantaneous flow structure.

flow problems. Figure 22 shows an application on a turbulent jet flow.

**3.2 Comparison to tomographic PIV**

scanning seems to be compulsory.

scan speed.

/*ut*

Fig. 21. Vortex ring core *ωθ* <sup>=</sup> <sup>−</sup>140*s*−<sup>1</sup> shown by the yellow coloured isosurface; and the radial vorticity *<sup>ω</sup><sup>r</sup>* <sup>=</sup> <sup>±</sup>30*s*−1. The colour code on the *<sup>ω</sup><sup>r</sup>* isosurface shows the magnitudes of the radial stretching term |*Sr*|, with the positive valued *Sr* dyed on the positive *ω<sup>r</sup>* and negative valued *Sr* dyed on the negative *ωr*.

The radial vorticity *ω<sup>r</sup>* is plotted in figure 21, with the radial direction stretching term *Sr* coloured on the isosurface. Figure 21 supports the expectation that vortices are being stretched in the radial direction and this stretching mechanism prevails on the windward side of the bubble. The compression of vorticity, i.e. the correlation of the opposite sensed vorticity and the stretching value, is much weaker (less than 20%) compared with stretching and is not shown here. The vorticity in the windward bubble region can also be seen in figure 17, in Cartesian coordinates. The streamwise stretching, which has been mentioned in section 2.4, is believed to be responsible for the turbulence production in the wake region however this is very small overall.

#### **3. Conclusions**

The first part of this chapter introduces the 2D3C working principle of stereoscopic PIV and its calibration procedure and its self-calibration to increase the accuracy level. The second part of the chapter presents an application of stereoscopic PIV for reconstructing a fully three-dimensional vortex ring. It has shown that by fixing the laser sheet position and hence the two cameras' FOV, and reconstructing the fully three-dimensional velocity field by Taylor's hypothesis, the accuracy of the results is limited. In other words, due to the intrinsic nature of Taylor's hypothesis, the resultant velocity field is never really instantaneous. However, there are various aspects which can increase the potential accuracy of such reconstruction and thus deserve further discussion.

#### **3.1 Possible accuracy improvements**

In order to freeze the ring structure better, the results of equation 11 needs to be closer to zero. Nevertheless, it is expected that for fully turbulent vortex ring, the level of turbulence intensity *u*� scales with Reynolds number (because *u*� ∼ *Up*), hence scales with vortex ring circulation 22 Will-be-set-by-IN-TECH

Fig. 21. Vortex ring core *ωθ* <sup>=</sup> <sup>−</sup>140*s*−<sup>1</sup> shown by the yellow coloured isosurface; and the radial vorticity *<sup>ω</sup><sup>r</sup>* <sup>=</sup> <sup>±</sup>30*s*−1. The colour code on the *<sup>ω</sup><sup>r</sup>* isosurface shows the magnitudes of the radial stretching term |*Sr*|, with the positive valued *Sr* dyed on the positive *ω<sup>r</sup>* and

The radial vorticity *ω<sup>r</sup>* is plotted in figure 21, with the radial direction stretching term *Sr* coloured on the isosurface. Figure 21 supports the expectation that vortices are being stretched in the radial direction and this stretching mechanism prevails on the windward side of the bubble. The compression of vorticity, i.e. the correlation of the opposite sensed vorticity and the stretching value, is much weaker (less than 20%) compared with stretching and is not shown here. The vorticity in the windward bubble region can also be seen in figure 17, in Cartesian coordinates. The streamwise stretching, which has been mentioned in section 2.4, is believed to be responsible for the turbulence production in the wake region however this is

The first part of this chapter introduces the 2D3C working principle of stereoscopic PIV and its calibration procedure and its self-calibration to increase the accuracy level. The second part of the chapter presents an application of stereoscopic PIV for reconstructing a fully three-dimensional vortex ring. It has shown that by fixing the laser sheet position and hence the two cameras' FOV, and reconstructing the fully three-dimensional velocity field by Taylor's hypothesis, the accuracy of the results is limited. In other words, due to the intrinsic nature of Taylor's hypothesis, the resultant velocity field is never really instantaneous. However, there are various aspects which can increase the potential accuracy

In order to freeze the ring structure better, the results of equation 11 needs to be closer to zero. Nevertheless, it is expected that for fully turbulent vortex ring, the level of turbulence intensity *u*� scales with Reynolds number (because *u*� ∼ *Up*), hence scales with vortex ring circulation

negative valued *Sr* dyed on the negative *ωr*.

of such reconstruction and thus deserve further discussion.

**3.1 Possible accuracy improvements**

very small overall.

**3. Conclusions**

Ca libr at ion plye−6422046 inspecti −60on:: r −4ed =pla−2ne 1, greenx =plane 20, both ma 40pped onto60 same plne for view ing ease Γ and the advection speed *ut*. Simply increasing Reynolds number will not improve the situation. Certain treatments independent of the ring will be necessary. A possible way is to move the PIV measurement (cameras and the laser sheet together) against the ring advection direction - an 'active scanning' process. Equation 11 shows that by simply moving the PIV measurement plane at the same speed as the ring, it can bring down the ratio *u* /*ut* significantly (doubling the denominator). Moreover, to resolve the wake correctly, an active scanning seems to be compulsory.

The next question is that how fast the scan speed is optimal. Definitely scan with an infinite speed gives no error. However, first it is not allowed, even it is capable to; second, it may not be necessary in applications where Reynolds number is not too high. A proper quantity can be searched to judge what scan speed gives satisfactory accuracy level. For instance, a possible candidate can be the spatial velocity correlation tensor, or the structure function, which is expected to converge as the scan speed increases. Thus, a scan speed can be considered adequate and economic (in terms of hardware requirement and computational expense), above which the increase of the accuracy is less significant while the cost-benefit ratio is high. In different flow problems, the optimal scan speed is expected to vary. Similar to the accuracy assessment given in section 2.3, a crude estimation of the scan speed could be made from the eddy turn over time: if the mean flow is of the only interest, a moderate scan speed should be enough; if small eddies are to be resolved, higher speed is required.

It has been pointed out that there exists an upper limit of the active scanning speed, it is not allowed to go to infinity. In other words, it is never possible to capture a truly frozen flow field. It is because if the flow is really frozen, there will be no particle displacement, thus the PIV fails to work. In order to allow a particle displacement to compute the velocity, a minimum time duration of PIV Δ*t* has to be given at one measurement station along the scanning path. In this aspect, this PIV Δ*t*, despite its shortness - in order of millisecond, limits the highest scan speed.

Nevertheless there is a potential solution for this limitation: a second stereoscopic PIV system can be introduced, aligning with the first system in the scanning path, and with a spacing *l* to the first one; see figure 22.9 Thus the value of *l*/*Uscan* effectively gives the PIV Δ*t*. Theoretically, this dual-system arrangement allows the choice of the *Uscan* value to a much higher value: at a fixed *Uscan*, PIV Δ*t* can be adjusted by setting the spacing *l* between the two systems (two laser sheets) very carefully. The highest scan speed in this case is then limited by the required spatial resolution in the scan direction, which is much more relaxed than the limit from the particle displacements. The high speed cameras available nowadays can work at several *kHz* frame rates, which allows very high scan speed while maintaining good spatial resolution in the scanning direction. By choosing a very high value of *Uscan*, one approaches a truly frozen and instantaneous flow structure.

#### **3.2 Comparison to tomographic PIV**

The state-of-the-art tomographic PIV is also a candidate to provide truly instantaneous three-dimensional velocity information of a flow field. Its working principle is developed

<sup>9</sup> Vortex ring study is definitely not the only customer of this method. It can be applied to many general flow problems. Figure 22 shows an application on a turbulent jet flow.

limit of tomographic PIV is normally much lower than that of stereoscopic PIV with typical

Stereoscopic PIV and Its Applications on Reconstruction Three-Dimensional Flow Field 27

Moreover, although tomographic PIV can also provide temporal-resolved information, its working principle requires iteration and can only give *the most likely* (not true) particle distribution in the FOV (due to the intersection of lines of sight giving ghost particles, although the cross-correlation signals of which are weaker). This is a considerable error source of tomographic PIV, among others. In addition, compared with the active scanning method, with the same amount of information10, to resolve temporal information, it must either be

Archer, P. J., Thomas, T. G. & Coleman, G. N. (2008). Direct numerical simulation of vortex ring

Bergdorf, M., Koumoutsakos, P. & Leonard, A. (2007). Direct numerical simulations of vortex

Elsinga, G. E., Scarano, F., Wieneke, B. & van Oudheusden, B. W. (2006). Tomographic particle

Gan, L. (2010). *PhD Dissertation: An experimental study of turbulent vortex rings using particle*

Gan, L. & Nickels, T. B. (2010). An experimental study of turbulent vortex rings during their

Gan, L., Nickels, T. B. & Dawson, J. R. (2011). An experimental study of turbulent vortex rings during: a three-dimensional representation, *Exp. Fluids* 51: 1493–1507.

Glezer, A. & Coles, D. (1990). An experimental study of turbulent vortex ring., *J. Fluid Mech.*

Maxworthy, T. (1977). Some experimental studies of vortex rings., *J. Fluid Mech.* 81: 465–495. Prasad, A. K. (2000). Stereoscopic particle image velocimetry., *Exp. Fluids* 29: 103–116.

Raffel, M., Willert, C., Wereley, S. & Kompenhans, J. (2007). *Particle Image Velocimetry-A*

Saffman, P. G. (1978). The number of waves on unstable vortex rings., *J. Fluid Mech.*

Shariff, K., Verzicco, R. & Orlandi, P. (1994). A numerical study of three-dimensional

Widnall, S. E. & Tsai, C. Y. (1977). The instability of the thin vortex ring of constant vorticity.,

<sup>10</sup> If two scan systems are used, four cameras are needed, which is the same as tomo-PIV; while only two cameras are needed for one system scanning, which is only half of the information amount.

Taylor, G. I. (1938). The spectrum of turbulence., *Proc. R. Soc. London, Ser. A* 164: 476–490. Townsend, A. A. (1976). *The structure of turbulent shear flow. Second Edition.*, Cambridge

vortex ring instabilities: viscous corrections and early nonlinear stage., *J. Fluid Mech.*

Glezer, A. (1988). On the formation of vortex rings., *Phys. Fluids* 31: 3532–3542.

Lavision (2007). *Product Manual-Davis 7.2 Software.*, Lavision GmbH, Gottingen.

*Practical Guide, Second Edition*, Springer-Verlag, Berlin.

evolution from the laminar to the early turbulent regime., *J. Fluid Mech.* 598: 201–226.

*voxel* − *pixel* ratio of unit.

**4. References**

211: 243–283.

84: 625–639.

279: 351–375.

University Press, Cambridge.

*Philo Trans R Soc Lond A* 287: 273–305.

more computational expensive or less accurate.

rings at Re<sup>Γ</sup> =7500., *J. Fluid Mech.* 581: 495–505.

image velocimetry., *Exp. Fluids* 41: 933–947.

*image velocimetry*, University of Cambridge.

early development, *J. Fluid Mech.* 649: 467–496.

Fig. 22. A simple sketch of the proposed scan system including dual stereo recording PIV systems, denoted in the figure by S1 (M1) and S2 (M2). The two systems can be mounted on a rail-carriage system, so that they can move together, at a designed speed *Uscan*. M1 can be a beam splitter. *l* denotes the spacing of the two systems, *U* denotes the scan speed *Uscan*. This scan system can be applied to any flow problem. A sample flow field - turbulent jet flow - is shown.

from the medical tomographic applications. The illuminated particle volume is viewed by several cameras (the more cameras used, the more accurate the results will be) at the same time. Then the *most-likely* three-dimensional particle distribution in the volume is reconstructed by the tomographic algorithm, based on the intersections of lines of sight giving the estimated particle location; see Elsinga et al. (2006) for more details. Albeit its robustness of measuring the truly instantaneous three-dimensional velocity field, its major weakness lies in its relatively low signal-noise ratio compared to the two-dimensional and stereoscopic PIV. It is because tomographic reconstruction is an inverse problem, i.e. the three-dimensional information is reconstructed from its two-dimensional projections, regardless of how many cameras are used, the error or the noise (in particular ghost particles, which will be discussed later below) is inevitable.

In addition to this intrinsic problem, tomographic PIV would also encounter some difficulties when a large volume of flow is to be examined. To reach the same FOV as the current experiment, a minimum required FOV will be 100*mm*3. This means that first of all, this volume needs to be illuminated; the emission intensity of practical lasers would be very weak when it is diffused to such a large volume.

Second, it would be extremely difficult for the cameras to be focused on such a deep FOV (which means the aperture needs to be very small) while accepting enough light during very short laser emission time. Therefore, if tomographic PIV is to be used, one can only produce small scaled flows, but small scaled flow reduces the spatial resolution, probably to an undesired level. The spatial resolution is also limited by the interrogation volume (similar to interrogation window for planner PIV configuration) size. For the practical sparsity level of particle field, the commonly acceptable interrogation volume size is typically 48 *voxel*3, while the planner PIV can easily reach 16 *pixel*2. Thus the absolute spatial resolution limit of tomographic PIV is normally much lower than that of stereoscopic PIV with typical *voxel* − *pixel* ratio of unit.

Moreover, although tomographic PIV can also provide temporal-resolved information, its working principle requires iteration and can only give *the most likely* (not true) particle distribution in the FOV (due to the intersection of lines of sight giving ghost particles, although the cross-correlation signals of which are weaker). This is a considerable error source of tomographic PIV, among others. In addition, compared with the active scanning method, with the same amount of information10, to resolve temporal information, it must either be more computational expensive or less accurate.

#### **4. References**

24 Will-be-set-by-IN-TECH

Fig. 22. A simple sketch of the proposed scan system including dual stereo recording PIV systems, denoted in the figure by S1 (M1) and S2 (M2). The two systems can be mounted on a rail-carriage system, so that they can move together, at a designed speed *Uscan*. M1 can be a beam splitter. *l* denotes the spacing of the two systems, *U* denotes the scan speed *Uscan*. This scan system can be applied to any flow problem. A sample flow field - turbulent jet flow - is

from the medical tomographic applications. The illuminated particle volume is viewed by several cameras (the more cameras used, the more accurate the results will be) at the same time. Then the *most-likely* three-dimensional particle distribution in the volume is reconstructed by the tomographic algorithm, based on the intersections of lines of sight giving the estimated particle location; see Elsinga et al. (2006) for more details. Albeit its robustness of measuring the truly instantaneous three-dimensional velocity field, its major weakness lies in its relatively low signal-noise ratio compared to the two-dimensional and stereoscopic PIV. It is because tomographic reconstruction is an inverse problem, i.e. the three-dimensional information is reconstructed from its two-dimensional projections, regardless of how many cameras are used, the error or the noise (in particular ghost particles, which will be discussed

In addition to this intrinsic problem, tomographic PIV would also encounter some difficulties when a large volume of flow is to be examined. To reach the same FOV as the current experiment, a minimum required FOV will be 100*mm*3. This means that first of all, this volume needs to be illuminated; the emission intensity of practical lasers would be very weak when it

Second, it would be extremely difficult for the cameras to be focused on such a deep FOV (which means the aperture needs to be very small) while accepting enough light during very short laser emission time. Therefore, if tomographic PIV is to be used, one can only produce small scaled flows, but small scaled flow reduces the spatial resolution, probably to an undesired level. The spatial resolution is also limited by the interrogation volume (similar to interrogation window for planner PIV configuration) size. For the practical sparsity level of particle field, the commonly acceptable interrogation volume size is typically 48 *voxel*3, while the planner PIV can easily reach 16 *pixel*2. Thus the absolute spatial resolution

shown.

later below) is inevitable.

is diffused to such a large volume.


<sup>10</sup> If two scan systems are used, four cameras are needed, which is the same as tomo-PIV; while only two cameras are needed for one system scanning, which is only half of the information amount.

**0**

**2**

Holger Nobach

*Germany*

**Limits in Planar PIV Due to Individual Variations**

The basic algorithm of digital particle image velocimetry (PIV) processing (Keane & Adrian, 1992; Utami et al., 1991; Westerweel, 1993; Willert & Gharib, 1991) utilizes the cross-correlation of image sub-spaces (interrogation windows) for local displacement estimation from two consecutively acquired images of a tracer-particle-laden flow. A variety of image processing techniques using sub-pixel interpolations has been applied in the past to significantly improve both, the accuracy of the particle displacement measurement beyond the nominal resolution of the optical sensor and the spatial resolution beyond the nominal averaging size of image

• sub-pixel interpolation of the correlation planes, e. g. the peak centroid (center-of-mass) method (Alexander & Ng, 1991; Morgan et al., 1989), the Gaussian interpolation (Willert & Gharib, 1991), a sinc interpolation (Lourenco & Krothapalli, 1995; Roesgen, 2003) or a polynomial interpolation (Chen & Katz, 2005), which reduce the "pixel locking" or "peak locking" effect (Christensen, 2004; Fincham & Spedding, 1997; Lourenco & Krothapalli,

• windowing functions, vanishing at the interrogation window boundaries (Gui et al., 2000; Liao & Cowen, 2005), reducing the effect of particle image truncation at the edges of the

• iterative shift and deformation of the interrogation windows (Fincham & Delerce, 2000; Huang et al., 1993b; Lecordier, 1997; Scarano, 2002; Scarano & Riethmuller, 2000) with different image interpolation schemes as e. g. the widely used, bi-linear interpolation, or more advanced higher-order methods (Astarita, 2006; Astarita & Cardone, 2005; Chen & Katz, 2005; Fincham & Delerce, 2000; Lourenco & Krothapalli, 1995; Roesgen, 2003)

interrogation windows to be correlated (Nogueira et al., 2001; Westerweel, 1997) • direct correlation with a normalization, which so far has been realized in three ways: asymmetrically, with a small interrogation window from the first image correlated with a larger window in the second image (Fincham & Spedding, 1997; Huang et al., 1997; 1993a; Rohály et al., 2002), symmetrically, with two interrogation windows of the same size (Nobach et al., 2004; Nogueira et al., 1999) or bi-directional, combining an asymmetric direct correlation as above and a second direct correlation with a small interrogation window from the second image correlated with a larger window in the first image (Nogueira et al., 2001), originally introduced as a "symmetric" method, but nonetheless

**1. Introduction**

sub-spaces to be correlated. These include:

1995; Prasad et al., 1992; Westerweel, 1998)

using image sub-spaces of different sizes

**of Particle Image Intensities**

*Max Planck Institute for Dynamics and Self-Organization*


## **Limits in Planar PIV Due to Individual Variations of Particle Image Intensities**

Holger Nobach *Max Planck Institute for Dynamics and Self-Organization Germany*

#### **1. Introduction**

26 Will-be-set-by-IN-TECH

28 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Wieneke, B. (2005). Stereo-piv using self-calibration on particle images., *Exp. Fluids*

Wieneke, B. (2008). Volume self-calibration for 3d particle image velocimetry., *Exp. Fluids*

39: 267–280.

45: 549–556.

The basic algorithm of digital particle image velocimetry (PIV) processing (Keane & Adrian, 1992; Utami et al., 1991; Westerweel, 1993; Willert & Gharib, 1991) utilizes the cross-correlation of image sub-spaces (interrogation windows) for local displacement estimation from two consecutively acquired images of a tracer-particle-laden flow. A variety of image processing techniques using sub-pixel interpolations has been applied in the past to significantly improve both, the accuracy of the particle displacement measurement beyond the nominal resolution of the optical sensor and the spatial resolution beyond the nominal averaging size of image sub-spaces to be correlated. These include:


(a) First exposure (b) Second exposure

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 31

Note that the effects of intensity variations are different from extern large scale illumination variations (Huang et al., 1997), the intensity variations only due to the different particle locations within the light sheet without relative changes between the exposures (Westerweel, 2000), or the loss-of-pairs and the degradation of the correlation peak due to out-of-plane motion (Keane & Adrian, 1990; 1992; Keane et al., 1995; Westerweel, 2000). While the loss-of-pairs and the degradation of the correlation peak increase the susceptibility to noise and the probability of outliers, the effect discussed here occurs additionally and directly affects the position of the correlation maximum and is a dominant limitation of the achievable accuracy in correlation-based image processing of planar PIV (Nobach, 2011; Nobach &

This study generally applies to Standard-PIV (two-dimensional, two-component, planar), independent of its application. This error principally applies also for Micro-PIV, where the particle images are large and may strongly overlap. However, the particle density and the intensity variations between consecutive images are small yielding a small effect of intensity variations. For Stereo-PIV, the errors are expeted to increase further compared to Standard-PIV due to the necessary coordinate transforms. Furthermore, the errors from the two perspectives are dependent due to the observation of identical particles within the same illumination sheet. In Tomo-PIV a reconstruction of three-dimensional particle locations preceds a three-dimentional correlation analysis. Since the overlap of particle images occurs in the projections only, the three-dimentional correlation should not be affected by this error. However, detailed studies about this error in Micro-PIV, Stereo-PIV and Tomo-PIV are still

In PIV, the displacement of particle patterns between consecutive images is obtained from the peak position in the two-dimensional cross-correlation plane of the two images or image sub-spaces (interrogation windows). Assuming (i) a certain number of imaged particles in

Fig. 2. Examples demonstrating individual particle intensity variations (marked regions, detail of public PIV images from the PIV challenge 2003, case A, axisymmetric turbulent jet

in stagnant surrounding, images A001a and A001b)

Bodenschatz, 2009).

pending.

**2. Effect of varying intensities**

including the Whittaker interpolation (Scarano & Riethmuller, 2000; Whittaker, 1929), also known as sinc or cardinal interpolation and the bi-cubic splines, which have found wide acceptance

• image deformation techniques (Astarita, 2007; 2008; Jambunathan et al., 1995; Lecuona et al., 2002; Nogueira et al., 1999; Nogueira, Lecuona & Rodriguez, 2005; Nogueira, Lecuona, Rodriguez, Alfaro & Acosta, 2005; Scarano, 2004; Schrijer & Scarano, 2008; Tokumaru & Dimotakis, 1995), where the entire images are deformed accordingly to the assumed velocity field before the sub-division into interrogation windows to be correlated, also using different image interpolation techniques.

With iterative window shift and deformation or image deformation techniques, an accuracy of the order of 0.01 pixel or better has been reported (Astarita & Cardone, 2005; Lecordier, 1997; Nobach et al., 2005) based on synthetic test images. In contrast, the application to real images from experiments shows less optimistic results, where the limit usually observed is about 0.1 pixel. Only under special conditions, like in two-dimensional flows with carefully aligned light sheets, can better accuracy be achieved (Lecordier & Trinité, 2006).

(a) Particles having an out-of-plane velocity component

(b) Two-dimensional flow aligned with the light sheet plane (only in-plane velocity components)

#### Fig. 1. Particles moving through a light sheet with an intensity profile

One reason for the different achievable accuracies in simulations and experiments may be the fact that in experiments, particles usually change their position within the light sheet (Fig. 1a). Therefore, the particles are illuminated differently in the two consecutive exposures. Additionally, the different illumination is individually different for each particle due to their different starting positions perpendicular to the light sheet plane. The result is an individual variation of particle intensities (further denoted as "intensity variations"), even in a homogeneous flow without any velocity gradient. Intensity variations can easily be seen in images from a variety of PIV applications, where some particles become brighter between the two exposures, whereas other particles, even if close by, become darker (Fig. 2). Simulations often assume that different particles can have different intensities, but not that the intensities can vary between subsequent exposures. This scenario can be realized in experiments only in two-dimensional flows with light sheets exactly aligned parallel to the flow field (Fig. 1b). Other sources of intensity variations could be an offset between the light sheets of the two illumination pulses or fluctuating scattering properties of the particles, e. g. non-spherical particles rotating in the flow.

2 Will-be-set-by-IN-TECH

• image deformation techniques (Astarita, 2007; 2008; Jambunathan et al., 1995; Lecuona et al., 2002; Nogueira et al., 1999; Nogueira, Lecuona & Rodriguez, 2005; Nogueira, Lecuona, Rodriguez, Alfaro & Acosta, 2005; Scarano, 2004; Schrijer & Scarano, 2008; Tokumaru & Dimotakis, 1995), where the entire images are deformed accordingly to the assumed velocity field before the sub-division into interrogation windows to be correlated,

With iterative window shift and deformation or image deformation techniques, an accuracy of the order of 0.01 pixel or better has been reported (Astarita & Cardone, 2005; Lecordier, 1997; Nobach et al., 2005) based on synthetic test images. In contrast, the application to real images from experiments shows less optimistic results, where the limit usually observed is about 0.1 pixel. Only under special conditions, like in two-dimensional flows with carefully

(a) Particles having an out-of-plane velocity component

(b) Two-dimensional flow aligned with the light sheet plane (only in-plane

One reason for the different achievable accuracies in simulations and experiments may be the fact that in experiments, particles usually change their position within the light sheet (Fig. 1a). Therefore, the particles are illuminated differently in the two consecutive exposures. Additionally, the different illumination is individually different for each particle due to their different starting positions perpendicular to the light sheet plane. The result is an individual variation of particle intensities (further denoted as "intensity variations"), even in a homogeneous flow without any velocity gradient. Intensity variations can easily be seen in images from a variety of PIV applications, where some particles become brighter between the two exposures, whereas other particles, even if close by, become darker (Fig. 2). Simulations often assume that different particles can have different intensities, but not that the intensities can vary between subsequent exposures. This scenario can be realized in experiments only in two-dimensional flows with light sheets exactly aligned parallel to the flow field (Fig. 1b). Other sources of intensity variations could be an offset between the light sheets of the two illumination pulses or fluctuating scattering properties of the particles, e. g. non-spherical

aligned light sheets, can better accuracy be achieved (Lecordier & Trinité, 2006).

Fig. 1. Particles moving through a light sheet with an intensity profile

also using different image interpolation techniques.

velocity components)

particles rotating in the flow.

acceptance

including the Whittaker interpolation (Scarano & Riethmuller, 2000; Whittaker, 1929), also known as sinc or cardinal interpolation and the bi-cubic splines, which have found wide

Fig. 2. Examples demonstrating individual particle intensity variations (marked regions, detail of public PIV images from the PIV challenge 2003, case A, axisymmetric turbulent jet in stagnant surrounding, images A001a and A001b)

Note that the effects of intensity variations are different from extern large scale illumination variations (Huang et al., 1997), the intensity variations only due to the different particle locations within the light sheet without relative changes between the exposures (Westerweel, 2000), or the loss-of-pairs and the degradation of the correlation peak due to out-of-plane motion (Keane & Adrian, 1990; 1992; Keane et al., 1995; Westerweel, 2000). While the loss-of-pairs and the degradation of the correlation peak increase the susceptibility to noise and the probability of outliers, the effect discussed here occurs additionally and directly affects the position of the correlation maximum and is a dominant limitation of the achievable accuracy in correlation-based image processing of planar PIV (Nobach, 2011; Nobach & Bodenschatz, 2009).

This study generally applies to Standard-PIV (two-dimensional, two-component, planar), independent of its application. This error principally applies also for Micro-PIV, where the particle images are large and may strongly overlap. However, the particle density and the intensity variations between consecutive images are small yielding a small effect of intensity variations. For Stereo-PIV, the errors are expeted to increase further compared to Standard-PIV due to the necessary coordinate transforms. Furthermore, the errors from the two perspectives are dependent due to the observation of identical particles within the same illumination sheet. In Tomo-PIV a reconstruction of three-dimensional particle locations preceds a three-dimentional correlation analysis. Since the overlap of particle images occurs in the projections only, the three-dimentional correlation should not be affected by this error. However, detailed studies about this error in Micro-PIV, Stereo-PIV and Tomo-PIV are still pending.

#### **2. Effect of varying intensities**

In PIV, the displacement of particle patterns between consecutive images is obtained from the peak position in the two-dimensional cross-correlation plane of the two images or image sub-spaces (interrogation windows). Assuming (i) a certain number of imaged particles in

(a) Varying relative intensity of well separated particle images

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 33

(b) Varying relative intensity of overlapping particle images yielding a correlation peak with a

Fig. 4. Intensity and cross-correlation function (CC with lines of zero displacement in *x* and in *y* direction respectively and with the correlation maximum marked with a black dot) of

out-of-plane motion of the particles (where the illumination of individual particles changes between the two light pulses). This is almost independent of the particle number density as

While different intensities of particle images obviously occur if the particles move out-of-plane in e. g. a Gaussian illumination profile, this effect also occurs for a top-hat profile, if one of the two particle images is present in only one of the images (drop-off), as it occurs if one particle moves out of or enters the illumination plane. With a top-hat illumination profile, the amplitude of one of the particle images stays constant between the two exposures while the other particle image is absent in one of the two images. For well separated particle images (Fig. 5a) the correlation has its maximum at the correct position. As soon as the two particle images (in one of the two images) overlap, the correlation maximum is shifted (Fig. 5b).

To derive the dependence of the achievable accuracy on the intensity variations, computer simulated images have been used with varying parameters. The simulated particles are uniformly distributed within the light sheet and over the observation area. To consider the diffraction-limited imaging of small particles, the simulated particle images are represented by Airy functions (diameter given by the first zero value), integrated over the sensitive sensor

shifted maximum location

shown below.

**3. Accuracy**

two images (I and II), each consisting of two particle images

the interrogation window, each with different intensity, but with the same relative intensity in the two consecutive images and (ii) no truncation at the edges of the interrogation windows, the correlation peak is at the correct position, even if the particle images overlap and if the intensity of one entire image is scaled by a constant factor. Note the different meaning of "images", which are the entire images to be correlated, and "particle images", which are the spots at the particle positions. For demonstration, in Fig. 3a two images, each consisting of two well separated particle images (Airy discs), are correlated. The particles are at identical positions in the two images (no displacement between the images). The correct position of the correlation maximum at zero displacement can be seen clearly even for overlapping particle images and also with a constant scaling of one image (Fig. 3b).

(a) Same intensity of the particle images in the two images with well separated particle images

(b) One image intensity scaled and with overlapping particle images

Fig. 3. Intensity and cross-correlation function (CC with lines of zero displacement in *x* and in *y* direction respectively and with the correlation maximum marked with a black dot) of two images (I and II), each consisting of two particle images

This holds true also for the correlation of images with different relative amplitudes of the particle images, as long as the particle images do not overlap (Fig. 4a). With overlapping particle images and varying relative amplitudes (Fig. 4b), the maximum position of the correlation peak is shifted, yielding a biased displacement estimate, depending on the amplitudes of the particle images, widths, and overlap.

The consequence for PIV image processing is an additional error in displacement estimates, if the intensities of particle images vary between the consecutive PIV images, while the particle images overlap. This error is especially large for de-focussed particle images (where the particle images tend to overlap) and in the case of misaligned light sheets or flows with

(a) Varying relative intensity of well separated particle images

(b) Varying relative intensity of overlapping particle images yielding a correlation peak with a shifted maximum location

Fig. 4. Intensity and cross-correlation function (CC with lines of zero displacement in *x* and in *y* direction respectively and with the correlation maximum marked with a black dot) of two images (I and II), each consisting of two particle images

out-of-plane motion of the particles (where the illumination of individual particles changes between the two light pulses). This is almost independent of the particle number density as shown below.

While different intensities of particle images obviously occur if the particles move out-of-plane in e. g. a Gaussian illumination profile, this effect also occurs for a top-hat profile, if one of the two particle images is present in only one of the images (drop-off), as it occurs if one particle moves out of or enters the illumination plane. With a top-hat illumination profile, the amplitude of one of the particle images stays constant between the two exposures while the other particle image is absent in one of the two images. For well separated particle images (Fig. 5a) the correlation has its maximum at the correct position. As soon as the two particle images (in one of the two images) overlap, the correlation maximum is shifted (Fig. 5b).

### **3. Accuracy**

4 Will-be-set-by-IN-TECH

the interrogation window, each with different intensity, but with the same relative intensity in the two consecutive images and (ii) no truncation at the edges of the interrogation windows, the correlation peak is at the correct position, even if the particle images overlap and if the intensity of one entire image is scaled by a constant factor. Note the different meaning of "images", which are the entire images to be correlated, and "particle images", which are the spots at the particle positions. For demonstration, in Fig. 3a two images, each consisting of two well separated particle images (Airy discs), are correlated. The particles are at identical positions in the two images (no displacement between the images). The correct position of the correlation maximum at zero displacement can be seen clearly even for overlapping particle

(a) Same intensity of the particle images in the two images with well separated particle images

(b) One image intensity scaled and with overlapping particle images

Fig. 3. Intensity and cross-correlation function (CC with lines of zero displacement in *x* and in *y* direction respectively and with the correlation maximum marked with a black dot) of

This holds true also for the correlation of images with different relative amplitudes of the particle images, as long as the particle images do not overlap (Fig. 4a). With overlapping particle images and varying relative amplitudes (Fig. 4b), the maximum position of the correlation peak is shifted, yielding a biased displacement estimate, depending on the

The consequence for PIV image processing is an additional error in displacement estimates, if the intensities of particle images vary between the consecutive PIV images, while the particle images overlap. This error is especially large for de-focussed particle images (where the particle images tend to overlap) and in the case of misaligned light sheets or flows with

images and also with a constant scaling of one image (Fig. 3b).

two images (I and II), each consisting of two particle images

amplitudes of the particle images, widths, and overlap.

To derive the dependence of the achievable accuracy on the intensity variations, computer simulated images have been used with varying parameters. The simulated particles are uniformly distributed within the light sheet and over the observation area. To consider the diffraction-limited imaging of small particles, the simulated particle images are represented by Airy functions (diameter given by the first zero value), integrated over the sensitive sensor

0.01

0.001

interpolation

0.01

total RMS error (pixel)

0.1

1

1 1.5 2 2.5 3 3.5 4 4.5 5

no noise, only in−plane displacement noise, only in−plane displacement no noise, 25% out−of−plane displacement

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 35

particle image diameter (pixel)

line key as in subfigure a

1 1.5 2 2.5 3 3.5 4 4.5 5

particle image diameter (pixel)

(b) Iterative sub-pixel interrogation window shift with bi-cubic splines image

To demonstrate the dominating influence of the intensity variations on the accuracy of correlation-based PIV algorithms, in Fig. 6 the total RMS error over the particle image diameter is shown for three test cases: (i) for only in-plane motion (without noise), (ii) for only in-plane motion, but with strong photon noise (1000 photo electrons for the brightest particles

Fig. 6. Total RMS error of the displacement estimate as a function of the particle image diameter (particle number density: 0.05 pixel−2, interrogation window size: 16×16 pixels)

(a) A simple FFT estimation with full-pixel shift

0.1

total RMS error (pixel)

1

(a) Drop-off with well separated particle images

(b) Drop-off with overlapping particle images yielding a correlation peak with a shifted maximum location

Fig. 5. Intensity and cross-correlation function (CC with lines of zero displacement in *x* and in *y* direction respectively and with the correlation maximum marked with a black dot) of two images (I and II), one consisting of two particle images and one with only one particle image (particle image drop-off)

areas (pixels). The pixels are assumed to have a square shape with uniform sensitivity with a fill-factor of 1 (no gaps between the sensitive areas). All particle images get a random maximum intensity, equally distributed between zero and 1000 photo electrons (see comments about the noise below), corresponding to e. g. different sizes or reflectivity. The maximum intensity does not change between the exposures for only in-plane motion. With an out-of-plane motion, the particles change their position relative to the light sheet plane yielding different illumination of each individual particle in the two exposures. In this simulation, a top-hat profile of the light sheet illumination intensity is simulated, where the illumination changes only, if a particle enters or leaves the light sheet. The Airy functions of overlapping particle images are linearly superimposed. To investigate the error of the displacement estimation, a series of 1000 individual image pairs is generated for each of the following test cases. The displacement of the particles between the two exposures is randomly chosen between −1 and +1 pixel simulating a variety of sub-pixel displacements. Larger in-plane displacements can easily be eliminated by full-pixel shift of the interrogation windows (Scarano & Riethmuller, 1999; Westerweel, 1997; Westerweel et al., 1997). To isolate the effect of intensity variations from additional effects by e. g. velocity gradients, the simulated displacement is constant for all particles, imitating a homogeneous velocity field.

6 Will-be-set-by-IN-TECH

(a) Drop-off with well separated particle images

(b) Drop-off with overlapping particle images yielding a correlation peak with a shifted maximum

Fig. 5. Intensity and cross-correlation function (CC with lines of zero displacement in *x* and in *y* direction respectively and with the correlation maximum marked with a black dot) of two images (I and II), one consisting of two particle images and one with only one particle

areas (pixels). The pixels are assumed to have a square shape with uniform sensitivity with a fill-factor of 1 (no gaps between the sensitive areas). All particle images get a random maximum intensity, equally distributed between zero and 1000 photo electrons (see comments about the noise below), corresponding to e. g. different sizes or reflectivity. The maximum intensity does not change between the exposures for only in-plane motion. With an out-of-plane motion, the particles change their position relative to the light sheet plane yielding different illumination of each individual particle in the two exposures. In this simulation, a top-hat profile of the light sheet illumination intensity is simulated, where the illumination changes only, if a particle enters or leaves the light sheet. The Airy functions of overlapping particle images are linearly superimposed. To investigate the error of the displacement estimation, a series of 1000 individual image pairs is generated for each of the following test cases. The displacement of the particles between the two exposures is randomly chosen between −1 and +1 pixel simulating a variety of sub-pixel displacements. Larger in-plane displacements can easily be eliminated by full-pixel shift of the interrogation windows (Scarano & Riethmuller, 1999; Westerweel, 1997; Westerweel et al., 1997). To isolate the effect of intensity variations from additional effects by e. g. velocity gradients, the simulated displacement is constant for all particles, imitating a homogeneous velocity field.

location

image (particle image drop-off)

(b) Iterative sub-pixel interrogation window shift with bi-cubic splines image interpolation

Fig. 6. Total RMS error of the displacement estimate as a function of the particle image diameter (particle number density: 0.05 pixel−2, interrogation window size: 16×16 pixels)

To demonstrate the dominating influence of the intensity variations on the accuracy of correlation-based PIV algorithms, in Fig. 6 the total RMS error over the particle image diameter is shown for three test cases: (i) for only in-plane motion (without noise), (ii) for only in-plane motion, but with strong photon noise (1000 photo electrons for the brightest particles

0.01

0.01

window shift with bi-cubic splines image interpolation

0.1

total RMS error (pixel)

1

16×16 32×32 64×64 128×128

no noise, only in−plane displacement noise, only in−plane displacement no noise, 25% out−of−plane displacement

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 37

)

line key as in subfigure a

interrogation area size (pixel<sup>2</sup>

0.001 0.01 0.1 1

(b) Total RMS error as a function of the particle number density

Fig. 7. Total RMS error of the displacement estimates for an iterative sub-pixel interrogation

noise, intensity interpolation over the pixel areas or by errors during image interpolation. The individual errors average over all particles in the interrogation window, yielding an RMS error decreasing with the square root of the particle number density. This complies with Westerweel (2000) (there for low particle densities). This error has a lower bound caused by interpolation errors in the correlation plane, which are independent of the particle number density. The image noise has been used to provoke large RMS errors in Fig. 7b, to make the range of RMS

particle number density (pixel−2)

(a) Total RMS error as a function of the size of the interrogation windows

0.1

total RMS error (pixel)

1

giving about 32 electrons noise), read-out noise (RMS of 20 electrons) and quantization noise (10 electrons per count, yielding a mean gray value of 102 for the above mentioned 1000 photo electrons incl. read-out noise) and (iii) for an out-of-plane component of 25 % of the light sheet thickness. Fig. 6a shows the results for a simple displacement estimation utilizing the peak position of the cross-correlation of two interrogation windows with 16×16 pixels obtained by means of the fast Fourier transform (FFT). The sub-pixel location of the maximum is obtained by fitting a Gaussian function to the maximum of the correlation and its two direct neighbors in *x* and *y* direction separately. In Fig. 6b an iterative window shift method has been used alternatively. Starting with the displacement estimate obtained from the simple FFT-based method above, in the next and all following iteration steps, the two consecutive PIV images are re-sampled at positions shifted symmetrically by plus/minus half the pre-estimated displacement. For re-sampling the images at sub-pixel positions, bi-cubic splines are used for interpolation, widely accepted as one of the best methods so far (Raffel et al., 2007; Stanislas et al., 2008). The interpolation has been realized here with an 8 × 8 pixels kernel, requiring also the environment of the 16×16 pixels large interrogation window to be simulated. To keep the investigations simple and to isolate the influence of intensity variations, window deformation has not been implemented here to avoid other well known effects, such as limited spatial resolution or dynamic range issues, which may additionally influence the results. However, the conclusions are equally applicable to the case of velocity fields with gradients. In that case the other error sources sum.

The difference between the simulated displacement and that estimated by the above procedures gives individual estimation errors. >From the series of individual errors, an averaged RMS error is derived. In the interesting range of particle image diameters of 2 pixels and larger, for both algorithms, the influence of the out-of-plane displacements is significantly larger that the error due to the noise, making the intensity variations a dominating limitation of the achievable accuracy of planar PIV displacement estimation. The uncertainty of the estimated RMS values is about 21 % of the actual value. This value has been derived assuming independent estimates, yielding an estimation variance of the variance estimate of 2*σ*<sup>4</sup> /*N* with *N* the number of estimates (1000 image pairs) and *σ* the true RMS value. The uncertainty of the shown graphs is then <sup>√</sup><sup>4</sup> 2/*<sup>N</sup> <sup>σ</sup>*.

The estimation accuracy can be improved in all three test cases by increasing the size of the interrogation windows, because the displacement errors average (Fig. 7a). The particle image diameter is set fix to 3 pixels. All other simulation and estimation parameters remain unchanged from the simulation above. The results are shown representatiovely for an iterative sub-pixel interrogation window shift with bi-cubic splines image interpolation only. For a constant particle number density, the RMS value decreases as the inverse of the linear dimension of the interrogation window. For large interrogation window sizes a transition towards a lower bound of the total RMS error is indicated. This lower bound is due to remaining interpolation errors in the correlation plane, which are independent of the size of the interrogation windows, and agrees with the findings in Fig. 6.

In contrast, varying the particle number density (Fig. 7b) has almost no effect on the RMS error in the case with an out-of-plane displacement. With only in-plane motion, the number of successfully correlated particle images increases linearly with the particle number density. For each particle, the correlation of the images has a small stochastic error, caused e. g. by image 8 Will-be-set-by-IN-TECH

giving about 32 electrons noise), read-out noise (RMS of 20 electrons) and quantization noise (10 electrons per count, yielding a mean gray value of 102 for the above mentioned 1000 photo electrons incl. read-out noise) and (iii) for an out-of-plane component of 25 % of the light sheet thickness. Fig. 6a shows the results for a simple displacement estimation utilizing the peak position of the cross-correlation of two interrogation windows with 16×16 pixels obtained by means of the fast Fourier transform (FFT). The sub-pixel location of the maximum is obtained by fitting a Gaussian function to the maximum of the correlation and its two direct neighbors in *x* and *y* direction separately. In Fig. 6b an iterative window shift method has been used alternatively. Starting with the displacement estimate obtained from the simple FFT-based method above, in the next and all following iteration steps, the two consecutive PIV images are re-sampled at positions shifted symmetrically by plus/minus half the pre-estimated displacement. For re-sampling the images at sub-pixel positions, bi-cubic splines are used for interpolation, widely accepted as one of the best methods so far (Raffel et al., 2007; Stanislas et al., 2008). The interpolation has been realized here with an 8 × 8 pixels kernel, requiring also the environment of the 16×16 pixels large interrogation window to be simulated. To keep the investigations simple and to isolate the influence of intensity variations, window deformation has not been implemented here to avoid other well known effects, such as limited spatial resolution or dynamic range issues, which may additionally influence the results. However, the conclusions are equally applicable to the case of velocity

The difference between the simulated displacement and that estimated by the above procedures gives individual estimation errors. >From the series of individual errors, an averaged RMS error is derived. In the interesting range of particle image diameters of 2 pixels and larger, for both algorithms, the influence of the out-of-plane displacements is significantly larger that the error due to the noise, making the intensity variations a dominating limitation of the achievable accuracy of planar PIV displacement estimation. The uncertainty of the estimated RMS values is about 21 % of the actual value. This value has been derived assuming

independent estimates, yielding an estimation variance of the variance estimate of 2*σ*<sup>4</sup>

*N* the number of estimates (1000 image pairs) and *σ* the true RMS value. The uncertainty of

The estimation accuracy can be improved in all three test cases by increasing the size of the interrogation windows, because the displacement errors average (Fig. 7a). The particle image diameter is set fix to 3 pixels. All other simulation and estimation parameters remain unchanged from the simulation above. The results are shown representatiovely for an iterative sub-pixel interrogation window shift with bi-cubic splines image interpolation only. For a constant particle number density, the RMS value decreases as the inverse of the linear dimension of the interrogation window. For large interrogation window sizes a transition towards a lower bound of the total RMS error is indicated. This lower bound is due to remaining interpolation errors in the correlation plane, which are independent of the size

In contrast, varying the particle number density (Fig. 7b) has almost no effect on the RMS error in the case with an out-of-plane displacement. With only in-plane motion, the number of successfully correlated particle images increases linearly with the particle number density. For each particle, the correlation of the images has a small stochastic error, caused e. g. by image

/*N* with

fields with gradients. In that case the other error sources sum.

of the interrogation windows, and agrees with the findings in Fig. 6.

the shown graphs is then <sup>√</sup><sup>4</sup> 2/*<sup>N</sup> <sup>σ</sup>*.

(a) Total RMS error as a function of the size of the interrogation windows

(b) Total RMS error as a function of the particle number density

Fig. 7. Total RMS error of the displacement estimates for an iterative sub-pixel interrogation window shift with bi-cubic splines image interpolation

noise, intensity interpolation over the pixel areas or by errors during image interpolation. The individual errors average over all particles in the interrogation window, yielding an RMS error decreasing with the square root of the particle number density. This complies with Westerweel (2000) (there for low particle densities). This error has a lower bound caused by interpolation errors in the correlation plane, which are independent of the particle number density. The image noise has been used to provoke large RMS errors in Fig. 7b, to make the range of RMS

0.001

outlier probability (%)

0 20 40 60 80 100

iterative window shift, bi−linear interpolation iterative window shift, Whittaker interpolation iterative window shift, bi−cubic splines interpolation

FFT, full−pixel shift FFT, full−pixel shift, window direct correlation, normalization

out−of−plane displacement (%)

0 20 40 60 80 100

out−of−plane displacement (%)

(b) Probability of outliers

Experimental verification of the results given above requires a PIV setup with an adjustable beam shape (and width) and an adjustable out-of-plane component of the real velocity field. The first requirement can be realized with a video projector imaging different intensity profiles

displacement (in percent of the light sheet thickness) for various PIV procedures (particle

Fig. 8. Properties of the displacement estimates as a function of the out-of-plane

number density: 0.05 pixel−2, interrogation window size: 16×16 pixels)

(a) Total RMS error

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 39

line key as in subfigure a

0.01

total RMS error (pixel)

0.1

1

errors decreasing with the square root of the particle number density on top of the lower bound visible.

If there is a certain out-of-plane displacement, the previous errors are superimposed by the strong influence of intensity variations of overlapping particle images. In contrast to the number of successfully correlated particle images, the probability of overlapping particle images increases with the square of the particle number density. Each of these pairs of overlapping particle images contributes a stochastic error to the correlation. After averaging the individual errors of overlapping pairs of particle images over the number of particles in the interrogation window, these two contributions exactly compensate, and the observed error becomes independent of the particle number density.

A better view onto the influence of the out-of-plane displacement can be achieved by investigating the total RMS error as a function of the out-of-plane displacement (Fig. 8a). Here, a variety of commonly used algorithms has been simulated for comparison: a simple FFT-based estimation with full-pixel shift as above, the same algorithm but with a triangular window function applied to the interrogation windows, a symmetric direct correlation with normalization and iterative sub-pixel shift of the interrogation windows with either bi-linear, Whittaker or bi-cubic splines image interpolation. The different algorithms show the smallest errors for only in-plane motion, however, they have a large variation of achievable accuracy in this case. With increasing out-of-plane displacement, the total error increases approximately exponentially with decreasing difference (on the log scale) between the various algorithms. The large error of the method with the window function applied to the interrogation windows is originated in the smaller "effective" window size, which is for the triangular weighting function about half the nominal size of the window, amplifying the susceptibility to intensity variations. Also the iterative window shift with bi-linear interpolation shows large errors due to the pure quality of the bi-linear interpolation scheme.

For large displacements, also outliers occur. To separate the RMS error due to the limited accuracy and the dominating influence of outliers a simple outlier detection algorithm has been implemented. All displacement estimates outside a range of ±1 pixel around the expected value are assumed to be outliers and are not taken into account for the calculation of the RMS error. From the number of outliers the probability of outliers is estimated. More reliable outlier detection algorithms based on statistical properties of the surrounding vector field as e. g. in Westerweel & Scarano (2005) could not be used in this simulation because only single displacement vectors are simulated. Starting at about 50 % out-of-plane displacement, the probability of outliers increases rapidly (Fig. 8b), limiting the useful range to a maximum out-of-plane displacement of about half the light sheet thickness for the given particle number density and interrogation window size. For the algorithm making use of the window function the onset of outliers is at smaller out-of-plane displacements due to the smaller effective size of the interrogation window. For the symmetric direct correlation with normalization the onset is shifted to larger displacements due to the better robustness of this procedure. The uncertainty of the estimated RMS values again is about 21 % of the actual value for small out-of-plane displacements (√<sup>4</sup> 2/*N*<sup>1</sup> *<sup>σ</sup>*, where *<sup>N</sup>*<sup>1</sup> is the number of validated estimates) and increases with larger out-of-plane displacements as *N*<sup>1</sup> the number of validated estimates decreases. The uncertainty of the outlier probability is <sup>√</sup>*P*(1−*P*)/*<sup>N</sup>* with the true value *<sup>P</sup>* of the outlier probability and *N* the total number of estimates (1000 image pairs).

10 Will-be-set-by-IN-TECH

errors decreasing with the square root of the particle number density on top of the lower

If there is a certain out-of-plane displacement, the previous errors are superimposed by the strong influence of intensity variations of overlapping particle images. In contrast to the number of successfully correlated particle images, the probability of overlapping particle images increases with the square of the particle number density. Each of these pairs of overlapping particle images contributes a stochastic error to the correlation. After averaging the individual errors of overlapping pairs of particle images over the number of particles in the interrogation window, these two contributions exactly compensate, and the observed error

A better view onto the influence of the out-of-plane displacement can be achieved by investigating the total RMS error as a function of the out-of-plane displacement (Fig. 8a). Here, a variety of commonly used algorithms has been simulated for comparison: a simple FFT-based estimation with full-pixel shift as above, the same algorithm but with a triangular window function applied to the interrogation windows, a symmetric direct correlation with normalization and iterative sub-pixel shift of the interrogation windows with either bi-linear, Whittaker or bi-cubic splines image interpolation. The different algorithms show the smallest errors for only in-plane motion, however, they have a large variation of achievable accuracy in this case. With increasing out-of-plane displacement, the total error increases approximately exponentially with decreasing difference (on the log scale) between the various algorithms. The large error of the method with the window function applied to the interrogation windows is originated in the smaller "effective" window size, which is for the triangular weighting function about half the nominal size of the window, amplifying the susceptibility to intensity variations. Also the iterative window shift with bi-linear interpolation shows large errors due

For large displacements, also outliers occur. To separate the RMS error due to the limited accuracy and the dominating influence of outliers a simple outlier detection algorithm has been implemented. All displacement estimates outside a range of ±1 pixel around the expected value are assumed to be outliers and are not taken into account for the calculation of the RMS error. From the number of outliers the probability of outliers is estimated. More reliable outlier detection algorithms based on statistical properties of the surrounding vector field as e. g. in Westerweel & Scarano (2005) could not be used in this simulation because only single displacement vectors are simulated. Starting at about 50 % out-of-plane displacement, the probability of outliers increases rapidly (Fig. 8b), limiting the useful range to a maximum out-of-plane displacement of about half the light sheet thickness for the given particle number density and interrogation window size. For the algorithm making use of the window function the onset of outliers is at smaller out-of-plane displacements due to the smaller effective size of the interrogation window. For the symmetric direct correlation with normalization the onset is shifted to larger displacements due to the better robustness of this procedure. The uncertainty of the estimated RMS values again is about 21 % of the actual value for small out-of-plane displacements (√<sup>4</sup> 2/*N*<sup>1</sup> *<sup>σ</sup>*, where *<sup>N</sup>*<sup>1</sup> is the number of validated estimates) and increases with larger out-of-plane displacements as *N*<sup>1</sup> the number of validated estimates decreases. The uncertainty of the outlier probability is <sup>√</sup>*P*(1−*P*)/*<sup>N</sup>* with the true value *<sup>P</sup>* of

the outlier probability and *N* the total number of estimates (1000 image pairs).

becomes independent of the particle number density.

to the pure quality of the bi-linear interpolation scheme.

bound visible.

Fig. 8. Properties of the displacement estimates as a function of the out-of-plane displacement (in percent of the light sheet thickness) for various PIV procedures (particle number density: 0.05 pixel−2, interrogation window size: 16×16 pixels)

Experimental verification of the results given above requires a PIV setup with an adjustable beam shape (and width) and an adjustable out-of-plane component of the real velocity field. The first requirement can be realized with a video projector imaging different intensity profiles

0.01

0

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40

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80

100

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out−of−plane displacement (%)

(b) Probability of outliers

to derive the mean displacement and the velocity gradients in *y* direction. The second PIV analysis is done with standard interrogation windows (32×32 pixels) in a 352×288 pixels large window, centered within the original observation area of 480×480 pixels. This area coincides with the area that is taken for the reference estimation. Based on the difference

Fig. 10. Comparison of the experiment and the simulation for an iterative sub-pixel interrogation window shift with bi-cubic splines image interpolation as a function of the

(a) Total RMS error

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 41

line/symbol key as in subfigure a

0.1

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1

into the measurement volume using an additional collimation lens (Fig. 9). To achieve stable illumination, LCD technology is preferred. The projector with DLP technology used here realizes individual gray values by pulse width modulation, which causes illumination problems with PIV cameras at short exposure (integration) times. In the present study the exposure time has been set to 0.25 s, which corresponds to 30 illumination cycles of the DLP chip, since it works at a frame rate of 120 Hz. This long exposure time requires small velocities, which have been realized by moving a solid glass block on a 3D translation stage (Newport CMA12PP stepping motors and ESP300 controller). The glass block has a size of 5 cm × 5 cm × 8 cm and includes 54 000 randomly distributed dots in the inner 3 cm <sup>×</sup> 3 cm <sup>×</sup> 6 cm volume, corresponding to a particle density of 1 mm−3.

Fig. 9. Sketch of the experimental setup: A video projector is imaging different illumination profiles into the measurement volume, which is observed by a digital camera. A glass block with internal markers is translated vertically through the measurement volume.

Furthermore, an accurate synchronization of the in-plane and the out-of-plane translation through the light sheet is required. To avoid synchronization problems, the system has been inverted. The glass block moves along one axis of the translation stage, while the plane of illumination is tilted with respect to the axis of motion. During the translation of the glass block with a constant velocity of 0.1 *mm*/s through the observation area of the camera (Phantom V10), a series of 80 images of 480×480 pixels size has been taken at a frame rate of 0.8 Hz. By choosing the number of frames between the two images to be correlated, different out-of-plane components can be imitated. For details of the experiment see Nobach & Bodenschatz (2009). The images are available from the author. For a comparison to the previous simulation, the images taken with a 4mm wide top-hat illumination profile with a slope of 0.75 have been re-processed in this study.

Unfortunately, the precision of the translation stage and the motion of the glass block are not satisfactory. An *a priory* analysis discovered a frame-to-frame variation of the displacement. Additionally, a small perspective error has been found generating a velocity gradient in *y* direction. To compensate the displacement variations and the velocity gradient within the observation field, for each image pair, an *a priori* analysis with two large interrogation windows (352×192 pixels) with 50 % overlap in *y* direction has been taken as a reference 12 Will-be-set-by-IN-TECH

into the measurement volume using an additional collimation lens (Fig. 9). To achieve stable illumination, LCD technology is preferred. The projector with DLP technology used here realizes individual gray values by pulse width modulation, which causes illumination problems with PIV cameras at short exposure (integration) times. In the present study the exposure time has been set to 0.25 s, which corresponds to 30 illumination cycles of the DLP chip, since it works at a frame rate of 120 Hz. This long exposure time requires small velocities, which have been realized by moving a solid glass block on a 3D translation stage (Newport CMA12PP stepping motors and ESP300 controller). The glass block has a size of 5 cm × 5 cm × 8 cm and includes 54 000 randomly distributed dots in the inner

Fig. 9. Sketch of the experimental setup: A video projector is imaging different illumination profiles into the measurement volume, which is observed by a digital camera. A glass block

Furthermore, an accurate synchronization of the in-plane and the out-of-plane translation through the light sheet is required. To avoid synchronization problems, the system has been inverted. The glass block moves along one axis of the translation stage, while the plane of illumination is tilted with respect to the axis of motion. During the translation of the glass block with a constant velocity of 0.1 *mm*/s through the observation area of the camera (Phantom V10), a series of 80 images of 480×480 pixels size has been taken at a frame rate of 0.8 Hz. By choosing the number of frames between the two images to be correlated, different out-of-plane components can be imitated. For details of the experiment see Nobach & Bodenschatz (2009). The images are available from the author. For a comparison to the previous simulation, the images taken with a 4mm wide top-hat illumination profile with a slope of 0.75 have been

Unfortunately, the precision of the translation stage and the motion of the glass block are not satisfactory. An *a priory* analysis discovered a frame-to-frame variation of the displacement. Additionally, a small perspective error has been found generating a velocity gradient in *y* direction. To compensate the displacement variations and the velocity gradient within the observation field, for each image pair, an *a priori* analysis with two large interrogation windows (352×192 pixels) with 50 % overlap in *y* direction has been taken as a reference

with internal markers is translated vertically through the measurement volume.

re-processed in this study.

3 cm <sup>×</sup> 3 cm <sup>×</sup> 6 cm volume, corresponding to a particle density of 1 mm−3.

Fig. 10. Comparison of the experiment and the simulation for an iterative sub-pixel interrogation window shift with bi-cubic splines image interpolation as a function of the out-of-plane displacement

to derive the mean displacement and the velocity gradients in *y* direction. The second PIV analysis is done with standard interrogation windows (32×32 pixels) in a 352×288 pixels large window, centered within the original observation area of 480×480 pixels. This area coincides with the area that is taken for the reference estimation. Based on the difference

To proof the gain of resolution by image deformation a series of 100 pairs of PIV images with 512×512 pixels each has been generated with a random in-plane displacement on a pixel-resolution (Gaussian distribution for each component and for each pixel with an RMS value of 0.5 pixel) and no out-of-plane motion. The particle images have random maximum intensities, equally distributed between zero and 1000 photo electrons, and Airy disk intensity

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 43

The images have been analyzed with an iterative window shift and first-order deformation technique (Scarano, 2002) with 32×32 and 16×16 pixels window size and an iterative image deformation with a triangular weighting applied to each PIV window of 32×32 pixels size. Except for the interrogation window size, the window function is identical to that in Nogueira et al. (1999), who apply the square of the triangular window to the product of the two PIV windows. To isolate the effect of decreasing the effective window size by weighting, the triangular weighting function has also been applied to the iterative window shift and deformation with a 32×32 pixels window. All methods use 10 iteration and a velocity estimation grid of 8×8 pixels corresponding to 75 % overlap for 32×32 pixels windows and

>From the individual displacement estimates, which are interpolated with bi-cubic splines and re-sampled at all pixel positions, and the simulated displacement, which originally is

est,*ijU*sim,*ij* + *V*<sup>∗</sup>

sim,*ijU*sim,*ij* + *V*<sup>∗</sup>

is calculated, where *U*sim,*ij* and *V*sim,*ij* are the two-dimensional Fourier transforms of the simulated *u* and the *v* displacement fields, *U*est,*ij* and *V*est,*ij* are the estimated counterparts, the <sup>∗</sup> denotes the conjugate complex and �� denotes the ensemble average. The products and the coherent frequency function are calculated element-wise for the two-dimensional functions. From the two-dimensional coherent frequency response function a common (one-dimensional) one is derived by iteratively optimizing a one-dimensional function *ci* so that the component-wise products *cicj* fit best the two-dimensional function *Cij* with

Fig. 11a shows the frequency response function for only in-plane motion for the four investigated estimation procedures. With a rectangular weighting window, the frequency response clearly drops below zero at 1/32 pixel or 1/16 pixel corresponding to the interrogation window size of 32×32 or 16×16 pixels respectively. The triangular weighting window applied to a 32×32 pixels interrogation window leads to a frequency response function with only non-negative values, while the resolution increases beyond the nominal resolution of the interrogation window size, reaching almost an effective window size of half the nominal window size. The image deformation technique can further improve the spatial resolution, which then is limited by the velocity grid of 8×8 pixels. Fig. 12a shows the obtained bandwidth (−3 dB limit) as a function of the overlap of interrogation windows. Clearly, the image deformation technique gains most by increasing the density of the velocity estimation grid. Note, that the overlap of interrogation windows for a given grid of velocity estimates

est,*ijV*sim,*ij*

sim,*ijV*sim,*ij* (1)

given for all pixel positions, a two-dimensional coherent frequency response

 *U*∗

 *U*∗

*Cij* =

profiles with 3 pixels diameter (defined by the first zero value of the Airy disk).

50 % overlap for the 16×16 pixels window.

minimum L2 norm.

between the PIV analysis with standard interrogation windows and the reference estimation with large interrogation windows the RMS error is calculated. To suppress effects from the edges, the RMS analysis uses only valid vectors from a further reduced window (160×96 pixels), yielding 5×3 displacement estimates with interrogation windows of 32×32 pixels size. From these estimates the universal outlier detection (Westerweel & Scarano, 2005) can be used as a validation criterion with a threshold of 0.5 pixel plus 2 times the found median RMS derived from the neighboring vectors. For better statistics, all validated displacement vectors from all image pairs with the same number of frames between them, selected from the original series of 80 images, have been averaged.

For direct comparison of the experimental and simulated results in Fig. 10, the numerical simulation has been repeated with simulation parameters and processing and validation methods as for the experimental images (particle number density: 0.013 pixel−2, interrogation window size: 32×32 pixels, iterative window shift and deformation, universal outlier detection). The uncertainty of the results of the simulation again are <sup>√</sup><sup>4</sup> 2/*N*<sup>1</sup> *<sup>σ</sup>* for the RMS values and <sup>√</sup>*P*(1−*P*)/*<sup>N</sup>* for the outlier probability. The pendents for the measurements change with the distance between frames since the number of image pairs decreases with increasing distance between frames. Here error bars are given, showing the interval of plus/minus the RMS of expected uncertainty. Note that the expected uncertainty represents only random errors. Systematic errors or non-detected outliers are not included.

Except for a small shift of large probabilities of outliers towards smaller out-of-plane displacements, the results of simulated and experimentally obtained data agree, verifying both the effect of the intensity variations and the simulation procedure. Remaining deviations are possibly originated in cross-illumination of markers, interference and camera noise.

#### **4. Resolution**

To increase the spatial resolution of PIV processing beyond the size of the interrogation windows, overlapping the interrogation windows is an appropriate mean. Of course, this has limitations, since the image data of overlapping windows is not independent, however for moderate overlaps of about 50 % this works fine for all PIV algorithms, for PIV algorithms with windowing functions or iterative image deformation techniques the overlap can be even larger to obtain further increased spatial resolution. In the latter case, the deformation's degree of freedom is related to the grid of velocity estimates, independent of the interrogation window size. With a high overlap of neighboring interrogation windows the spatial resolution of iterative image deformation is governed by the grid spacing without loosing the robustness of the large interrogation windows. Therefore, this method is gained to improve the achievable spatial resolution of the PIV processing. Instabilities of this technique, occurring for high overlaps of interrogation windows due to negative responses in certain frequency ranges (Nogueira et al., 1999; Scarano, 2004) can be avoided either by applying appropriate spatial filters to the estimated velocity field or the application of appropriate windowing functions to the interrogation windows, which then have frequency responses with only non-negative values. Investigations of stability and spatial resolution of iterative image deformation applying either spatial filters or window functions can be found in Astarita (2007); Lecuona et al. (2002); Nogueira, Lecuona & Rodriguez (2005); Nogueira, Lecuona, Rodriguez, Alfaro & Acosta (2005); Scarano (2004); Schrijer & Scarano (2008).

14 Will-be-set-by-IN-TECH

between the PIV analysis with standard interrogation windows and the reference estimation with large interrogation windows the RMS error is calculated. To suppress effects from the edges, the RMS analysis uses only valid vectors from a further reduced window (160×96 pixels), yielding 5×3 displacement estimates with interrogation windows of 32×32 pixels size. From these estimates the universal outlier detection (Westerweel & Scarano, 2005) can be used as a validation criterion with a threshold of 0.5 pixel plus 2 times the found median RMS derived from the neighboring vectors. For better statistics, all validated displacement vectors from all image pairs with the same number of frames between them, selected from the

For direct comparison of the experimental and simulated results in Fig. 10, the numerical simulation has been repeated with simulation parameters and processing and validation methods as for the experimental images (particle number density: 0.013 pixel−2, interrogation window size: 32×32 pixels, iterative window shift and deformation, universal outlier detection). The uncertainty of the results of the simulation again are <sup>√</sup><sup>4</sup> 2/*N*<sup>1</sup> *<sup>σ</sup>* for the RMS values and <sup>√</sup>*P*(1−*P*)/*<sup>N</sup>* for the outlier probability. The pendents for the measurements change with the distance between frames since the number of image pairs decreases with increasing distance between frames. Here error bars are given, showing the interval of plus/minus the RMS of expected uncertainty. Note that the expected uncertainty represents only random

Except for a small shift of large probabilities of outliers towards smaller out-of-plane displacements, the results of simulated and experimentally obtained data agree, verifying both the effect of the intensity variations and the simulation procedure. Remaining deviations are possibly originated in cross-illumination of markers, interference and camera noise.

To increase the spatial resolution of PIV processing beyond the size of the interrogation windows, overlapping the interrogation windows is an appropriate mean. Of course, this has limitations, since the image data of overlapping windows is not independent, however for moderate overlaps of about 50 % this works fine for all PIV algorithms, for PIV algorithms with windowing functions or iterative image deformation techniques the overlap can be even larger to obtain further increased spatial resolution. In the latter case, the deformation's degree of freedom is related to the grid of velocity estimates, independent of the interrogation window size. With a high overlap of neighboring interrogation windows the spatial resolution of iterative image deformation is governed by the grid spacing without loosing the robustness of the large interrogation windows. Therefore, this method is gained to improve the achievable spatial resolution of the PIV processing. Instabilities of this technique, occurring for high overlaps of interrogation windows due to negative responses in certain frequency ranges (Nogueira et al., 1999; Scarano, 2004) can be avoided either by applying appropriate spatial filters to the estimated velocity field or the application of appropriate windowing functions to the interrogation windows, which then have frequency responses with only non-negative values. Investigations of stability and spatial resolution of iterative image deformation applying either spatial filters or window functions can be found in Astarita (2007); Lecuona et al. (2002); Nogueira, Lecuona & Rodriguez (2005); Nogueira, Lecuona,

Rodriguez, Alfaro & Acosta (2005); Scarano (2004); Schrijer & Scarano (2008).

original series of 80 images, have been averaged.

**4. Resolution**

errors. Systematic errors or non-detected outliers are not included.

To proof the gain of resolution by image deformation a series of 100 pairs of PIV images with 512×512 pixels each has been generated with a random in-plane displacement on a pixel-resolution (Gaussian distribution for each component and for each pixel with an RMS value of 0.5 pixel) and no out-of-plane motion. The particle images have random maximum intensities, equally distributed between zero and 1000 photo electrons, and Airy disk intensity profiles with 3 pixels diameter (defined by the first zero value of the Airy disk).

The images have been analyzed with an iterative window shift and first-order deformation technique (Scarano, 2002) with 32×32 and 16×16 pixels window size and an iterative image deformation with a triangular weighting applied to each PIV window of 32×32 pixels size. Except for the interrogation window size, the window function is identical to that in Nogueira et al. (1999), who apply the square of the triangular window to the product of the two PIV windows. To isolate the effect of decreasing the effective window size by weighting, the triangular weighting function has also been applied to the iterative window shift and deformation with a 32×32 pixels window. All methods use 10 iteration and a velocity estimation grid of 8×8 pixels corresponding to 75 % overlap for 32×32 pixels windows and 50 % overlap for the 16×16 pixels window.

>From the individual displacement estimates, which are interpolated with bi-cubic splines and re-sampled at all pixel positions, and the simulated displacement, which originally is given for all pixel positions, a two-dimensional coherent frequency response

$$\mathbf{C}\_{lj} = \frac{\left\langle U^\*\_{\text{est,}j} \mathcal{U}\_{\text{sim},ij} + V^\*\_{\text{est,}ij} V\_{\text{sim},ij} \right\rangle}{\left\langle U^\*\_{\text{sim},ij} \mathcal{U}\_{\text{sim},ij} + V^\*\_{\text{sim},ij} V\_{\text{sim},ij} \right\rangle} \tag{1}$$

is calculated, where *U*sim,*ij* and *V*sim,*ij* are the two-dimensional Fourier transforms of the simulated *u* and the *v* displacement fields, *U*est,*ij* and *V*est,*ij* are the estimated counterparts, the <sup>∗</sup> denotes the conjugate complex and �� denotes the ensemble average. The products and the coherent frequency function are calculated element-wise for the two-dimensional functions. From the two-dimensional coherent frequency response function a common (one-dimensional) one is derived by iteratively optimizing a one-dimensional function *ci* so that the component-wise products *cicj* fit best the two-dimensional function *Cij* with minimum L2 norm.

Fig. 11a shows the frequency response function for only in-plane motion for the four investigated estimation procedures. With a rectangular weighting window, the frequency response clearly drops below zero at 1/32 pixel or 1/16 pixel corresponding to the interrogation window size of 32×32 or 16×16 pixels respectively. The triangular weighting window applied to a 32×32 pixels interrogation window leads to a frequency response function with only non-negative values, while the resolution increases beyond the nominal resolution of the interrogation window size, reaching almost an effective window size of half the nominal window size. The image deformation technique can further improve the spatial resolution, which then is limited by the velocity grid of 8×8 pixels. Fig. 12a shows the obtained bandwidth (−3 dB limit) as a function of the overlap of interrogation windows. Clearly, the image deformation technique gains most by increasing the density of the velocity estimation grid. Note, that the overlap of interrogation windows for a given grid of velocity estimates

0

0

1/32

1/16

 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

**5. RMS error versus resolution**

bandwidth (pixel−1)

0.05 pixel−2)

1/32

1/16

0 20 40 60 80 100

32x32 window deformation (rectangular weight) 16x16 window deformation (rectangular weight) 32x32 window deformation (triangular weight) 32x32 image deformation (triangular weight)

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 45

overlap (%)

line/symbol key as in subfigure a

0 20 40 60 80 100

overlap (%)

(b) Out-of-plane displacement of 25 % of the light sheet thickness

Fig. 12. Bandwidth as a function of the window overlap obtained from a series of simulated

However, taking into account the RMS errors, a significant influence of the intensity variations can be seen. Fig. 13a shows the obtained total RMS errors against the bandwidth with overlaps of interrogation windows varied between 0 and 87.5 %. The various methods cover different

PIV images with a random in-plane displacement field (particle number density:

(a) Only in-plane motion

 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

bandwidth (pixel−1)

(b) Out-of-plane displacement of 25 % of the light sheet thickness

Fig. 11. Coherent frequency response for the different estimation procedures for a velocity estimation grid of 8×8 pixels corresponding to 75 % overlap for 32×32 pixels windows and 50 % overlap for the 16×16 pixels window (particle number density: 0.05 pixel−2)

changes with the size of the interrogation windows yielding a shifted overlap for the method with the 16×16 pixels interrogation window compared to the other methods.

Figs. 11b and 12b show the corresponding results for an out-of-plane displacement of 25 % of the light sheet thickness. There is no significant difference compared to Figs. 11a and 12a. Therefore, one can conclude that the intensity variations have no significant influence on the achievable resolution.

16 Will-be-set-by-IN-TECH

32x32 window deformation (rectangular weight) 16x16 window deformation (rectangular weight) 32x32 window deformation (triangular weight) 32x32 image deformation (triangular weight)

0.001 0.01 0.1 1

0.001 0.01 0.1 1

(b) Out-of-plane displacement of 25 % of the light sheet thickness

changes with the size of the interrogation windows yielding a shifted overlap for the method

Figs. 11b and 12b show the corresponding results for an out-of-plane displacement of 25 % of the light sheet thickness. There is no significant difference compared to Figs. 11a and 12a. Therefore, one can conclude that the intensity variations have no significant influence on the

Fig. 11. Coherent frequency response for the different estimation procedures for a velocity estimation grid of 8×8 pixels corresponding to 75 % overlap for 32×32 pixels windows and

50 % overlap for the 16×16 pixels window (particle number density: 0.05 pixel−2)

with the 16×16 pixels interrogation window compared to the other methods.

spatial frequency (pixel−1)

1/32 1/16 1/8

(a) Only in-plane motion

spatial frequency (pixel−1)

1/32 1/16 1/8

line key as in subfigure a

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

frequency response (−)

achievable resolution.

frequency response (−)

(b) Out-of-plane displacement of 25 % of the light sheet thickness

Fig. 12. Bandwidth as a function of the window overlap obtained from a series of simulated PIV images with a random in-plane displacement field (particle number density: 0.05 pixel−2)

#### **5. RMS error versus resolution**

However, taking into account the RMS errors, a significant influence of the intensity variations can be seen. Fig. 13a shows the obtained total RMS errors against the bandwidth with overlaps of interrogation windows varied between 0 and 87.5 %. The various methods cover different

deformation technique with a triangular window function the bandwidth further increases up to about 75 % overlap, reaching almost the bandwidth of the window shift and deformation with a rectangular window function of half the size corresponding to an effective window size, which is half as large as the nominal size. Again, the RMS error increases rapidly for further increased overlaps. With the image deformation technique and strong overlap of interrogation windows the bandwidth can be increased further. The RMS error increases much less then

Limits in Planar PIV Due to Individual Variations of Particle Image Intensities 47

The picture changes completely in the presence of an out-of-plane component. Fig. 13b shows the results for an out-of-plane component of 25 % of the light sheet thickness. Again a rapid increase of the RMS error can be seen beyond 50 % overlap for iterative window shift and deformation techniques with rectangular window functions, respectively 75 % for a triangular window function. The image deformation reaches the highest bandwidth at strong overlaps. However, with the out-of-plane component the errors are much larger than with only in-plane motion and, additionally, the results for the various methods do not overlay any more. For the window shift and deformation techniques the achievable accuracy depends on the window function and the interrogation window size, as one has seen in Fig. 7. For the image deformation, the bandwidth continuously increases with the overlap, and the RMS error nearly linearly increases with the obtained bandwidth, but here for the prize of a larger RMS error in the entire range of overlaps and bandwidths compared to the other methods.

The effect of particle image intensities varying individually between the two consecutive images on the obtainable accuracy of a PIV system has been reviewed. Such intensity variations occur in experiments due to the motion of the particles in the intensity profile of the light sheet, misalignments of the two light pulses or changes of the particle's scattering properties between the two exposures. The error has been quantified for several commonly used PIV processing methods. This effect limits the obtainable accuracy of PIV measurements, even under otherwise ideal conditions and is much stronger than noise or in-plane loss of particle images. The commonly used best practice parameters for PIV experiments (particle image diameter around 3 pixels and out-of-plane components of not more than 25 % of the light sheet thickness) and the usually observed limit of about 0.1 pixel could be re-produced. This error is almost independent of the particle number density, but it strongly increases with increasing out-of-plane displacements, and decreases with increasing interrogation window size. In summary, besides under-sampling, the variations of the particle image intensities are an additional error, dominating the range of particle image diameters of larger than 2 pixels. This error leads to a basic limitation of the planar PIV technique and explains the accuracy limit of PIV of about 0.1 pixel usually seen in experiments. High-resolution image deformation techniques as in Nogueira et al. (1999) or Schrijer & Scarano (2008), with their small effective interrogation windows are especially affected in terms of the achievable

accuracy, even if the achievable resolution does not change with intensity variations.

centroid estimation, *Opt. Eng.* 30: 1320–1331.

Alexander, B. F. & Ng, K. C. (1991). Elimination of systematic error in sub-pixel accuracy

with the other methods.

**6. Conclusion**

**7. References**

(b) Out-of-plane displacement of 25 % of the light sheet thickness

Fig. 13. Total RMS error against the bandwidth (particle number density: 0.05 pixel−2)

ranges of obtainable bandwidths and RMS errors yielding a lower bound of about 0.02 pixel, slightly increasing with the obtainable bandwidth.

For the window shift and deformation techniques with rectangular window function, the achievable bandwidth basically depends on the size of the interrogation windows. The bandwidth increases slightly with the overlap up to about 50 % overlap. For higher overlaps, the bandwidth stays constant and the RMS error rapidly increases. With the window shift and deformation technique with a triangular window function the bandwidth further increases up to about 75 % overlap, reaching almost the bandwidth of the window shift and deformation with a rectangular window function of half the size corresponding to an effective window size, which is half as large as the nominal size. Again, the RMS error increases rapidly for further increased overlaps. With the image deformation technique and strong overlap of interrogation windows the bandwidth can be increased further. The RMS error increases much less then with the other methods.

The picture changes completely in the presence of an out-of-plane component. Fig. 13b shows the results for an out-of-plane component of 25 % of the light sheet thickness. Again a rapid increase of the RMS error can be seen beyond 50 % overlap for iterative window shift and deformation techniques with rectangular window functions, respectively 75 % for a triangular window function. The image deformation reaches the highest bandwidth at strong overlaps. However, with the out-of-plane component the errors are much larger than with only in-plane motion and, additionally, the results for the various methods do not overlay any more. For the window shift and deformation techniques the achievable accuracy depends on the window function and the interrogation window size, as one has seen in Fig. 7. For the image deformation, the bandwidth continuously increases with the overlap, and the RMS error nearly linearly increases with the obtained bandwidth, but here for the prize of a larger RMS error in the entire range of overlaps and bandwidths compared to the other methods.

#### **6. Conclusion**

18 Will-be-set-by-IN-TECH

32x32 window deformation (rectangular weight) 16x16 window deformation (rectangular weight) 32x32 window deformation (triangular weight) 32x32 image deformation (triangular weight)

0 0.01 0.02 0.03 0.04 0.05 0.06

line/symbol key as in subfigure a

bandwidth (pixel−1)

0 0.01 0.02 0.03 0.04 0.05 0.06

bandwidth (pixel−1)

(b) Out-of-plane displacement of 25 % of the light sheet thickness

ranges of obtainable bandwidths and RMS errors yielding a lower bound of about 0.02 pixel,

For the window shift and deformation techniques with rectangular window function, the achievable bandwidth basically depends on the size of the interrogation windows. The bandwidth increases slightly with the overlap up to about 50 % overlap. For higher overlaps, the bandwidth stays constant and the RMS error rapidly increases. With the window shift and

Fig. 13. Total RMS error against the bandwidth (particle number density: 0.05 pixel−2)

(a) Only in-plane motion

0

 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

slightly increasing with the obtainable bandwidth.

total RMS error (pixel)

0.05

0.1

0.15

total RMS error (pixel)

0.2

0.25

The effect of particle image intensities varying individually between the two consecutive images on the obtainable accuracy of a PIV system has been reviewed. Such intensity variations occur in experiments due to the motion of the particles in the intensity profile of the light sheet, misalignments of the two light pulses or changes of the particle's scattering properties between the two exposures. The error has been quantified for several commonly used PIV processing methods. This effect limits the obtainable accuracy of PIV measurements, even under otherwise ideal conditions and is much stronger than noise or in-plane loss of particle images. The commonly used best practice parameters for PIV experiments (particle image diameter around 3 pixels and out-of-plane components of not more than 25 % of the light sheet thickness) and the usually observed limit of about 0.1 pixel could be re-produced. This error is almost independent of the particle number density, but it strongly increases with increasing out-of-plane displacements, and decreases with increasing interrogation window size. In summary, besides under-sampling, the variations of the particle image intensities are an additional error, dominating the range of particle image diameters of larger than 2 pixels. This error leads to a basic limitation of the planar PIV technique and explains the accuracy limit of PIV of about 0.1 pixel usually seen in experiments. High-resolution image deformation techniques as in Nogueira et al. (1999) or Schrijer & Scarano (2008), with their small effective interrogation windows are especially affected in terms of the achievable accuracy, even if the achievable resolution does not change with intensity variations.

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**3** 

*Canada* 

**PIV Measurements Applied to** 

Gabriel Dan Ciocan and Monica Sanda Iliescu *Université Laval, Laboratoire de Machines Hydrauliques* 

**and Cavitation-Free Flows** 

**Hydraulic Machinery: Cavitating** 

Hydraulic machinery is an ideal field of application for Particle Image Velocimetry in terms of scientific interest, due to the complexity of the flow behaviour and to the need of detailed unsteady experimental data simultaneously recorded over large sections of the flow field. Within the same machine, a whole range of phenomena are encountered in the different components: wake patterns, separation, rotating vortex structures, vortex breakdown, etc. The unsteady interactions between the stationary and rotating frames, both upstream and downstream the runner, contribute to the efficiency loss. The generated quasi-periodic fluctuations overlay onto the average flow field, which may be symmetrical or not, issuing an unpredictable dynamic behaviour with respect to the operating regime. Whilst the intrinsic parameters of the local phenomena (e.g. sheared flow mixing length) vary, the flow topology may be modified drastically for conditions situated relatively close to one another in terms of head, flow rate and efficiency. The mapping of the unsteady velocity fields and corresponding turbulence levels is thus an essential tool in the analysis of these complex phenomena. The PIV technique opens large perspectives in the analysis of internal flows in hydraulic machinery, providing valuable insight towards an extensive understanding of the underlying physical mechanisms. Nevertheless, the use of PIV systems in this context is one of the most challenging applications, due to the structural constrains related to the optical access to the measurement areas, to the spatial and temporal scales of the phenomena that are to be investigated, two phase flow structure in cavitating regime and also due to the

In this book chapter it is proposed for presentation a development for hydraulic machines, based on our extensive experience and on the current bibliography in the field. The focus will be on two directions: rotor-stator interactions and two-phase flows in hydraulic

The research on hydraulic machinery is mainly performed on reduced scale models. The internal geometry needs to be respected because it is essential for the step-up of flow phenomena taking place on the model to actual prototype conditions. Furthermore, the model operation conditions must comply with the IEC 60193 Standard, which rules the

turbines taking into account flow periodicity, turbulence and cavitation.

**1. Introduction** 

industrial aspects of the application.


## **PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows**

Gabriel Dan Ciocan and Monica Sanda Iliescu *Université Laval, Laboratoire de Machines Hydrauliques Canada* 

### **1. Introduction**

22 Will-be-set-by-IN-TECH

50 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Westerweel, J. (1993). *Digital Particle Image Velocimetry: Theory and Application*, Delft University

Westerweel, J. (1997). Fundamentals of digital particle image velocimetry, *Meas. Sci. Technol.*

Westerweel, J. (1998). Effect of sensor geometry on the performance of PIV interrogation, *Proc.*

Westerweel, J. (2000). Theoretical analysis of the measurement precision in particle image

Westerweel, J. & Scarano, F. (2005). Universal outlier detection for PIV data, *Exp. in Fluids*

Westerweel, J., Dabiri, D. & Gharib, M. (1997). The effect of discrete window offset on

Whittaker, J. M. (1929). The Fourier theory of the cardinal functions, *Proc. - R. Soc. Edinburgh*

Willert, C. E. & Gharib, M. (1991). Digital particle image velocimetry, *Exp. in Fluids*

*9th Int. Symp. on Appl. of Laser Techn. to Fluid Mechanics*, Lisbon, Portugal. paper 1.2.

the accuracy of cross-correlation analysis of digital PIV recordings, *Exp. in Fluids*

Press, Delft, The Netherlands.

velocimetry, *Exp. in Fluids* 29: S3–S12.

8: 1379–1392.

39: 1096–1100.

*Sect. A Math.* 1: 169–176.

23: 20–28.

10: 181–193.

Hydraulic machinery is an ideal field of application for Particle Image Velocimetry in terms of scientific interest, due to the complexity of the flow behaviour and to the need of detailed unsteady experimental data simultaneously recorded over large sections of the flow field. Within the same machine, a whole range of phenomena are encountered in the different components: wake patterns, separation, rotating vortex structures, vortex breakdown, etc. The unsteady interactions between the stationary and rotating frames, both upstream and downstream the runner, contribute to the efficiency loss. The generated quasi-periodic fluctuations overlay onto the average flow field, which may be symmetrical or not, issuing an unpredictable dynamic behaviour with respect to the operating regime. Whilst the intrinsic parameters of the local phenomena (e.g. sheared flow mixing length) vary, the flow topology may be modified drastically for conditions situated relatively close to one another in terms of head, flow rate and efficiency. The mapping of the unsteady velocity fields and corresponding turbulence levels is thus an essential tool in the analysis of these complex phenomena. The PIV technique opens large perspectives in the analysis of internal flows in hydraulic machinery, providing valuable insight towards an extensive understanding of the underlying physical mechanisms. Nevertheless, the use of PIV systems in this context is one of the most challenging applications, due to the structural constrains related to the optical access to the measurement areas, to the spatial and temporal scales of the phenomena that are to be investigated, two phase flow structure in cavitating regime and also due to the industrial aspects of the application.

In this book chapter it is proposed for presentation a development for hydraulic machines, based on our extensive experience and on the current bibliography in the field. The focus will be on two directions: rotor-stator interactions and two-phase flows in hydraulic turbines taking into account flow periodicity, turbulence and cavitation.

The research on hydraulic machinery is mainly performed on reduced scale models. The internal geometry needs to be respected because it is essential for the step-up of flow phenomena taking place on the model to actual prototype conditions. Furthermore, the model operation conditions must comply with the IEC 60193 Standard, which rules the

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 53

Interface design criteria and optimisation of the optical access taking into account local

 Experiment setup for measurements in the static and rotating frames: synchronisation of the acquisition with the predominant physical phenomena; adjustment of the acquisition parameters with respect to the local flow conditions and to the global

 Requirements for image quality optimisation in 2D/3D-PIV experiments in single and two-phase flows: uniformization of the laser illumination in the measuring section, solutions to prevent localised reflections in the active viewing area, specificities related

Image processing in two-phase flows: filtering methodology and morphological

Data processing tools to determine the 3D velocity fields, with emphasis on particle

Accuracy study and validation of measurement results, in terms of velocity fields and

Physical analysis of the flow (wake propagation, vortex core detection techniques,

To conclude the chapter, best practice guidelines are provided for the setup of PIV experiments in hydraulic turbo-machinery, comparatively in single-phase flows and

All this works are done in the frame of PhD works of the authors: (Ciocan 1998) performed at Institute National Politechnique de Grenoble, France and (Iliescu 2007) performed at Ecole Polytechnique Federale de Lausanne, Switzerland. Other ulterior developments were performed (Tridon et al. 2008), (Tridon et al. 2010), (Gagnon et al. 2008), (Beaulieu et al. 2009) and (Houde et al. 2011) in collaboration or under the coordination of the authors.

A Dantec MT 3D-PIV system is used for measuring the three-dimensional unsteady velocity fields. The equipment presented herein corresponds to a classical medium-frequency PIV system. The system components belong to four categories of devices: illumination, particles

Two laser units with individual power supplies and cooling loops deliver a high-energy laser beam, which is transformed into a plane for locally illuminating the flow seeded with

The laser has an Yttrium Aluminum Garnet crystal, doped with triply ionized Neodymium, as lasing medium (Nd:YAG). While stimulated with a flash lamp, it emits photons of 1064 nm wavelength. The infrared output light is converted to the visible spectrum by frequency doubling with a birefringent crystal, up to the green radiation wavelength, 532 nm. The laser delivers pulses of 250 μs at a frequency of 8Hz. A Q-switch mechanism releases only 8 ns

introduced in the flow, image recording and data processing unit – see Fig. 3.

Calibration devices and practical procedures for accuracy assessment;

to measurements in two phase flows such as background illumination;

detection on textured backgrounds and related masking techniques;

geometrical features extracted in two-phase flow conditions;

**2. General setup for hydraulic machines applications** 

optical distortions and perspective effects;

operations to extract the relevant flow features;

reconstruction of vapours volume, etc).

cavitation conditions.

**2.1 General experimental set-up** 

**2.2 Illumination system** 

tracer particles.

operating range of the machine;

methodology for model acceptance testing of hydraulic turbines, storage pumps and pumpturbines, and provides guidelines for the model-to-prototype transposition.

Fig. 1. Cross-section of a Francis turbine scale model

Fig. 2. Development of a cavitating helical vortex downstream the runner of a Francis turbine at partial load

An example of hydraulic geometry of a medium-head radial turbine is presented in Fig. 1. For this kind of turbine, the runner has constant-pitch fixed blades and the guide vanes are adjustable. The environment around the machine is also complex because it concerns the operating elements of the machine, supports of the test bench, model instrumentation to operate the machine. The working fluid is water; also thus optical interfaces are required for measurements based on imaging techniques. Their design must take into account the internal geometry of the model, while minimizing distortion.

Another major topic to be considered is the flow topology and diversity, with various unsteady phenomena taking place: wake propagation, vortex breakdown and machine-circuit resonance, rotor-stator interaction, runner flow behaviour. The 2D PIV and 3D PIV, in cavitation-free or two phase flow are the ideal tool to characterize these complex flows – see Fig. 2.

The proposed chapter covers all aspects of the development of a stereoscopic PIV experiment for the investigation of unsteady flows in hydraulic machinery, illustrated by the main applications of the authors' experience (see bibliography). The following topics are discussed in details:


To conclude the chapter, best practice guidelines are provided for the setup of PIV experiments in hydraulic turbo-machinery, comparatively in single-phase flows and cavitation conditions.

All this works are done in the frame of PhD works of the authors: (Ciocan 1998) performed at Institute National Politechnique de Grenoble, France and (Iliescu 2007) performed at Ecole Polytechnique Federale de Lausanne, Switzerland. Other ulterior developments were performed (Tridon et al. 2008), (Tridon et al. 2010), (Gagnon et al. 2008), (Beaulieu et al. 2009) and (Houde et al. 2011) in collaboration or under the coordination of the authors.

### **2. General setup for hydraulic machines applications**

### **2.1 General experimental set-up**

A Dantec MT 3D-PIV system is used for measuring the three-dimensional unsteady velocity fields. The equipment presented herein corresponds to a classical medium-frequency PIV system. The system components belong to four categories of devices: illumination, particles introduced in the flow, image recording and data processing unit – see Fig. 3.

### **2.2 Illumination system**

52 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

methodology for model acceptance testing of hydraulic turbines, storage pumps and pump-

Fig. 2. Development of a cavitating helical vortex downstream the runner of a Francis

internal geometry of the model, while minimizing distortion.

An example of hydraulic geometry of a medium-head radial turbine is presented in Fig. 1. For this kind of turbine, the runner has constant-pitch fixed blades and the guide vanes are adjustable. The environment around the machine is also complex because it concerns the operating elements of the machine, supports of the test bench, model instrumentation to operate the machine. The working fluid is water; also thus optical interfaces are required for measurements based on imaging techniques. Their design must take into account the

Another major topic to be considered is the flow topology and diversity, with various unsteady phenomena taking place: wake propagation, vortex breakdown and machine-circuit resonance, rotor-stator interaction, runner flow behaviour. The 2D PIV and 3D PIV, in cavitation-free or

The proposed chapter covers all aspects of the development of a stereoscopic PIV experiment for the investigation of unsteady flows in hydraulic machinery, illustrated by the main applications of the authors' experience (see bibliography). The following topics are

two phase flow are the ideal tool to characterize these complex flows – see Fig. 2.

turbines, and provides guidelines for the model-to-prototype transposition.

Fig. 1. Cross-section of a Francis turbine scale model

turbine at partial load

discussed in details:

Two laser units with individual power supplies and cooling loops deliver a high-energy laser beam, which is transformed into a plane for locally illuminating the flow seeded with tracer particles.

The laser has an Yttrium Aluminum Garnet crystal, doped with triply ionized Neodymium, as lasing medium (Nd:YAG). While stimulated with a flash lamp, it emits photons of 1064 nm wavelength. The infrared output light is converted to the visible spectrum by frequency doubling with a birefringent crystal, up to the green radiation wavelength, 532 nm. The laser delivers pulses of 250 μs at a frequency of 8Hz. A Q-switch mechanism releases only 8 ns

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 55

Focus lens focus adjuster module

The enlighten field is visualized by two double-frame progressive scan interline CCD cameras, with an active matrix of 1280x1024 with 8 bit depth, see Table 4. The active area of light-sensitive cells is doubled by a second array of storage cells, for increasing the data transfer rate. The acquisition frequency in double-frame mode is 4.5 Hz. The CCD chip is cooled, which gives a higher sensitivity and signal-to-noise ratio enhancement in low lighting conditions. Two Nikon objectives with focal length of 24mm and 60mm are used depending on the geometrical characteristics of the camera setup. measurement area

The control and synchronization of the laser, cameras and external trigger input, as well as the raw vector field processing, are realized with a specific processor, Dantec MT's FlowMap2200. The main advantage is the integrated correlator unit, which performs realtime raw vector maps processing by a cross-correlation technique applied on the doubleframe images. In this way a qualitative vector field validation can be rapidly performed

Dantec MT's FlowManager software version 4.5, along with 3D-PIV software module, have been used for data acquisition, validation and processing. Data post-processing and

Spherical particles in borosilicate glass, are used as flow tracers; a silver coating improves their scattering characteristic. The relative density of 1.4 against the water one and the average size of 10m allow these particles to accurately follow the flow. Their refractive index is 1.52. The melting point is high, 740°C, which makes them suitable for a broad range of applications.

For the correct evaluation of the 3D displacement of the particles, a mapping of the measurement volume onto the two cameras' images and the definition of the overlapping

Active area 1280 x 1024 Resolution 8-12 bit

3.4μm x 6.7μm Frame rate 4.5 Hz in double-

frame mode

Narrow angle lens opening 20° Wide angle lens opening 40° Thickness adjuster compression factor for 2D PIV 0.67 Thickness adjuster expansion factor for 3D PIV 1.5

Pixel width and

pitch

**2.4 Acquisition control and data processing** 

visualization tools are realized in Matlab.

zone of the two fields of view are necessary.

Table 3. Laser sheet optics assembly specifications

**2.3 Image recording** 

dimensions and optical path.

cooling system, anti-blooming protection, pixels binning

during acquisition.

**2.5 Seeding** 

**2.6 Calibration** 

Table 4. CCD camera specifications

CCD chip

Fig. 3. Typical layout of a 3D-PIV system

out of the total pulse duration, thus producing a stroboscopic effect on the particles in the test area. For obtaining the desired delay between two pulses, two laser cavities with the same characteristics are used. Thus, the time interval between two successive impulses can be easily adjusted within 1 μs ÷ 100 ms range, depending on the local flow characteristics or the phenomenon, which is to be captured. The characteristics of the illumination system used for the current experiments are summarized in Table 1.


Table 1. Laser unit specifications

For facilitating the optical access to the test section, the laser beam is transmitted through an articulated light guide with high transfer ratio and resistant to the high laser energy employed in water experiments. The characteristics of the beam guide are given in Table 2.


Table 2. Beam-guiding arm specifications

A beam expander mounted at the end of the arm transforms the input laser beam into a light sheet. A series of lenses allow adjusting the thickness and divergence angle of the laser sheet. The characteristics of the optics assembly are given in Table 3.


Table 3. Laser sheet optics assembly specifications

### **2.3 Image recording**

54 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

out of the total pulse duration, thus producing a stroboscopic effect on the particles in the test area. For obtaining the desired delay between two pulses, two laser cavities with the same characteristics are used. Thus, the time interval between two successive impulses can be easily adjusted within 1 μs ÷ 100 ms range, depending on the local flow characteristics or the phenomenon, which is to be captured. The characteristics of the illumination system

Peak energy 60 mJ Pulse duration 8 ns

6mm Pulse separation rate 300 ns to 300 ms

Nd:YAG Power 25 MW Pulse repetition rate 8 to 20 Hz

For facilitating the optical access to the test section, the laser beam is transmitted through an articulated light guide with high transfer ratio and resistant to the high laser energy employed in water experiments. The characteristics of the beam guide are given in Table 2.

Maximum input pulse energy 500 mJ for 10 ns pulse and 6-12 mm

A beam expander mounted at the end of the arm transforms the input laser beam into a light sheet. A series of lenses allow adjusting the thickness and divergence angle of the laser

beam

Type 5 flexible joints mirrors

Fig. 3. Typical layout of a 3D-PIV system

Type NewWave

Lasing medium

Output wavelength Gemini

Table 1. Laser unit specifications

532nm Beam

Table 2. Beam-guiding arm specifications

used for the current experiments are summarized in Table 1.

diameter

Optical transmission >90%

Maximum input beam diameter 12 mm

sheet. The characteristics of the optics assembly are given in Table 3.

The enlighten field is visualized by two double-frame progressive scan interline CCD cameras, with an active matrix of 1280x1024 with 8 bit depth, see Table 4. The active area of light-sensitive cells is doubled by a second array of storage cells, for increasing the data transfer rate. The acquisition frequency in double-frame mode is 4.5 Hz. The CCD chip is cooled, which gives a higher sensitivity and signal-to-noise ratio enhancement in low lighting conditions. Two Nikon objectives with focal length of 24mm and 60mm are used depending on the geometrical characteristics of the camera setup. measurement area dimensions and optical path.


Table 4. CCD camera specifications

### **2.4 Acquisition control and data processing**

The control and synchronization of the laser, cameras and external trigger input, as well as the raw vector field processing, are realized with a specific processor, Dantec MT's FlowMap2200. The main advantage is the integrated correlator unit, which performs realtime raw vector maps processing by a cross-correlation technique applied on the doubleframe images. In this way a qualitative vector field validation can be rapidly performed during acquisition.

Dantec MT's FlowManager software version 4.5, along with 3D-PIV software module, have been used for data acquisition, validation and processing. Data post-processing and visualization tools are realized in Matlab.

### **2.5 Seeding**

Spherical particles in borosilicate glass, are used as flow tracers; a silver coating improves their scattering characteristic. The relative density of 1.4 against the water one and the average size of 10m allow these particles to accurately follow the flow. Their refractive index is 1.52. The melting point is high, 740°C, which makes them suitable for a broad range of applications.

#### **2.6 Calibration**

For the correct evaluation of the 3D displacement of the particles, a mapping of the measurement volume onto the two cameras' images and the definition of the overlapping zone of the two fields of view are necessary.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 57

000 100 010 001 110 101 011

*A A X A Y A Z A XY A XZ A YZ*

222 3 2 2

2 22 *Y Z A XZ A YZ A XYZ* <sup>102</sup> <sup>012</sup> <sup>111</sup>

The accuracy of the model in transposing the space coordinates in the image plane and the accuracy in detecting the 3rd velocity component have been analysed, taking into account the effect of the geometry and refractive index of the optical interface. In addition, a comparison between the two types of calibration has been made. The 3rd order polynomial model in XY, parabolic in Z has been chosen. The relative errors between the three methods

The target is placed in the test section and its spatial position is checked to fit the

 The test section is filled with water, the cameras are positioned in stereoscopic arrangement The position and angle of the cameras, relatively to the target surface are adjusted and the lenses are chosen such that the field of view recovers the most part of the target surface and their overlapping area is maximized. The perspective correction is realized by rotating the CCD plane relatively to the lens plane until all the camera's

 Images of the target are acquired for different transversal positions of the target. The target surface is evenly illuminated with high-power projectors. The image accuracy is checked by applying an image-processing algorithm for markers detection. The coefficients of the transfer function, for mapping the real-world coordinates onto the

 The laser sheet is aligned with the target surface and its opening angle and thickness are adjusted taking into account the uniform energy distribution over the entire

The calibration target is removed from the test section, the hydraulic circuit is refilled,



An error analysis is made for a same measurement position at the outlet of the draft tube, using the two calibration targets successively. For the same operating conditions, the comparison between mean 3D velocity fields obtained with the two calibrations is shown in

camera image, are calculated with the position information.

and seeding particles are introduced in the flow.

For the calibration, two targets were tested – see Table 5:

in five vertical positions at 1 mm +/-0.01 distance;

Fig. 4 for the linear transform and for the polynomial transfer functions.

*A X A Y A Z A X A XY A XZ*

200 020 002 300 210 201

are in the range 2 to 10 %, with higher errors near the measurement zone boundary.

3 2 030 120 021

The calibration process includes the following steps:

*A Y A XY A*

*x*

*y*

accuracy limits.

field of view is in focus.

measurement zone.

**2.6.3 Calibration target definition** 

levels at an offset of 5 mm.

(2)

The camera calibration consists in defining the coefficients of a mathematical model that relates the real spatial locations in the measurement plane to the corresponding positions in the recording plane. This model includes the geometrical and optical characteristics of the cameras set-up, taking into account the optical distortions due to perspective imaging, lens aberrations, interposed media with different refractive indices. The images of a plane target with equally spaced markers, moved in five transversal positions (corresponding to the laser sheet thickness) are stored to have volume information. The corresponding positions of the points in all the image plane allows to determine the optical transfer function by a least squares fitting algorithm.

#### **2.6.1 Optical distortion correction**

For the optical access of the cameras and of the laser, the test model is equipped with polymethyl methacrylate (PMMA) windows, with a refractive index of 1.44. The inner face of the windows follows the hydraulic profile of the test model, while their external face is flat, for minimizing the optical distortions. The optical interfaces used for the PIV measurements in the cone will be detailed in the next paragraphs.

The Scheimpflung correction is applied for perspective imaging rectification. It consists in rotating the CCD plane relatively to the lens plane for reducing the perspective effect by balancing the optical path difference between points near and far away from the camera axis. The focus plane, lens plane and CCD sensor plane are made coincident using a mechanism which allows individual rotation of these components.

The angle at which the CCD chip plane needs to be tilted about the lens plane can be calculated with the camera's focal length and its geometrical position, i.e. the camera tilt angle and the distance from the lens to the measurement plane. In our case, this value can only be used as a rough approximation, because the optical path is distorted while passing through air, PMMA and water. The final adjustment is then realized by compensating the blur on the lateral edges of the image and bringing the entire view into focus.

#### **2.6.2 Optical transfer functions**

The optical path from the measurement zone to the camera crosses 3 media with different refractive indices: water 1.33. PMMA 1.44 and air 1. In our particular case, the optical windows are not parallel to the cameras' plane. To correct the optical distortion between the measurement zone and the corresponding image, two analytical functions have been tested:

direct linear transform :

$$
\begin{bmatrix} k\_x \\ k\_y \\ k\_0 \end{bmatrix} = \begin{bmatrix} A\_{11} & A\_{12} & A\_{13} & A\_{14} \\ A\_{21} & A\_{22} & A\_{23} & A\_{24} \\ A\_{31} & A\_{32} & A\_{33} & A\_{34} \end{bmatrix} \cdot \begin{bmatrix} X \\ Y \\ Z \\ 1 \end{bmatrix} \tag{1}
$$

 3rd order polynomial for the XY directions in the plane of the target and parabolic in the out-of-plane direction Z:

$$
\begin{split}
\widehat{\begin{bmatrix}X\\Y\end{bmatrix}} &= \overrightarrow{A\_{000}} + \left(\overrightarrow{A\_{100}} \cdot X + \overrightarrow{A\_{010}} \cdot Y + \overrightarrow{A\_{001}} \cdot Z\right) + \left(\overrightarrow{A\_{110}} \cdot XY + \overrightarrow{A\_{101}} \cdot XZ + \overrightarrow{A\_{011}} \cdot YZ\right) + \\
&+ \left(\overrightarrow{A\_{200}} \cdot X^2 + \overrightarrow{A\_{020}} \cdot Y^2 + \overrightarrow{A\_{002}} \cdot Z^2\right) + \left(\overrightarrow{A\_{300}} \cdot X^3 + \overrightarrow{A\_{210}} \cdot X^2Y + \overrightarrow{A\_{201}} \cdot X^2Z\right) + \\
&+ \left(\overrightarrow{A\_{000}} \cdot Y^3 + \overrightarrow{A\_{120}} \cdot XY^2 + \overrightarrow{A\_{021}} \cdot Y^2Z\right) + \left(\overrightarrow{A\_{102}} \cdot XZ^2 + \overrightarrow{A\_{012}} \cdot YZ^2 + \overrightarrow{A\_{111}} \cdot XZZ\right)
\end{split}
\tag{2}
$$

The accuracy of the model in transposing the space coordinates in the image plane and the accuracy in detecting the 3rd velocity component have been analysed, taking into account the effect of the geometry and refractive index of the optical interface. In addition, a comparison between the two types of calibration has been made. The 3rd order polynomial model in XY, parabolic in Z has been chosen. The relative errors between the three methods are in the range 2 to 10 %, with higher errors near the measurement zone boundary.

The calibration process includes the following steps:

56 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

The camera calibration consists in defining the coefficients of a mathematical model that relates the real spatial locations in the measurement plane to the corresponding positions in the recording plane. This model includes the geometrical and optical characteristics of the cameras set-up, taking into account the optical distortions due to perspective imaging, lens aberrations, interposed media with different refractive indices. The images of a plane target with equally spaced markers, moved in five transversal positions (corresponding to the laser sheet thickness) are stored to have volume information. The corresponding positions of the points in all the image plane allows to determine the optical transfer function by a least

For the optical access of the cameras and of the laser, the test model is equipped with polymethyl methacrylate (PMMA) windows, with a refractive index of 1.44. The inner face of the windows follows the hydraulic profile of the test model, while their external face is flat, for minimizing the optical distortions. The optical interfaces used for the PIV

The Scheimpflung correction is applied for perspective imaging rectification. It consists in rotating the CCD plane relatively to the lens plane for reducing the perspective effect by balancing the optical path difference between points near and far away from the camera axis. The focus plane, lens plane and CCD sensor plane are made coincident using a

The angle at which the CCD chip plane needs to be tilted about the lens plane can be calculated with the camera's focal length and its geometrical position, i.e. the camera tilt angle and the distance from the lens to the measurement plane. In our case, this value can only be used as a rough approximation, because the optical path is distorted while passing through air, PMMA and water. The final adjustment is then realized by compensating the

The optical path from the measurement zone to the camera crosses 3 media with different refractive indices: water 1.33. PMMA 1.44 and air 1. In our particular case, the optical windows are not parallel to the cameras' plane. To correct the optical distortion between the measurement zone and the corresponding image, two analytical functions have been tested:

> 11 12 13 14 21 22 23 24 0 31 32 33 34 1

3rd order polynomial for the XY directions in the plane of the target and parabolic in the

*k AAAA*

*k AAAA*

*k AAAA*

*X*

*Y*

*Z*

(1)

measurements in the cone will be detailed in the next paragraphs.

mechanism which allows individual rotation of these components.

*x y*

blur on the lateral edges of the image and bringing the entire view into focus.

squares fitting algorithm.

**2.6.1 Optical distortion correction** 

**2.6.2 Optical transfer functions** 

direct linear transform :

out-of-plane direction Z:


#### **2.6.3 Calibration target definition**

For the calibration, two targets were tested – see Table 5:


An error analysis is made for a same measurement position at the outlet of the draft tube, using the two calibration targets successively. For the same operating conditions, the comparison between mean 3D velocity fields obtained with the two calibrations is shown in Fig. 4 for the linear transform and for the polynomial transfer functions.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 59

 the tangential velocity component (perpendicular to the main flow direction) is small compared with the axial flow velocity. only 10%, and thus the error is amplified. Taking into account these considerations, the final choice was to use the 2D target, with a spatial resolution of 5mm for the in-plane positions and 1mm in the out-of-plane direction.

A slight displacement of the target during calibration has been noticed. In this context, a theoretical and an experimental study have been performed on the stability of the target in the fluid medium, both in rotation and in translation, and on its influence on the

Denoting C the velocity in the reference frame OXYZ and C' the velocity in a frame rotated

 1 100 [%] *<sup>C</sup> C*

 1 1 cos 1 sin

 1 1 cos 1 sin

 1 1 cos 1 sin

*C C*

*C C*

*C C*

*y z y z y z*

*x z x z x z*

*x y x y x y*

*C C* (7)

*C C* (6)

*C C* (5)

(4)

by an angle δ around one of the axes, the relative error for each component is:

cos sin

cos sin

cos sin

angles ranging between 0 and 2°. The black line denotes 3% error limit.

Cy is sensitive if Cz<Cy, but the admissible angle drops under 1° if Cz>Cy;

Cz is the most sensitive component, which gives a tolerance of 0.1-0.2°.

;

; 

Thus the error depends only on the rotation angle and velocity components ratio. These ratios vary in the range [0 1] for Cz/Cy, [1 20] for Cz/Cx, and [1 30] for the in-plane components Cy/Cx. The relative errors for rotation around each axis are given in Fig. 5, for

Cz is very sensitive to rotation if Cz<<Cy, but for Cz~=Cy the tolerance approaches 1.7°;

;

 100 [%] *C C <sup>C</sup>*

*y y z z y <sup>z</sup> C C C C C <sup>C</sup>*

*x xz z x <sup>z</sup> C CC C C C*

*x xy y x <sup>y</sup> C CC C C C*

the optical access window is not parallel to the image plane;

**2.6.4 Target positioning system** 

measurement accuracy.




For rotation around the OX axis:

For rotation around the OY axis:

**Target rotation** 


Table 5. 2D and volumetric calibration targets dimensions

Fig. 4. Relative error for calibration with the 2D and the 3D targets, in the case of linear (top row) and polynomial (bottom row) transfer functions

$$\mathcal{E} = \frac{\mathbf{C}\_{\text{2Dtaget}} - \mathbf{C}\_{\text{3Dtager}}}{\mathbf{C}\_{\text{2Dtager}}} \cdot \mathbf{100} \left[\% \right] \tag{3}$$

The high error values obtained for the comparison between the measurements obtained with the two targets shows that the 3D target cannot be used in our configuration. The reasons, which explain this difference of accuracy, are the following:


Taking into account these considerations, the final choice was to use the 2D target, with a spatial resolution of 5mm for the in-plane positions and 1mm in the out-of-plane direction.

#### **2.6.4 Target positioning system**

A slight displacement of the target during calibration has been noticed. In this context, a theoretical and an experimental study have been performed on the stability of the target in the fluid medium, both in rotation and in translation, and on its influence on the measurement accuracy.

#### **Target rotation**

58 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Plane target Volumetric target

Fig. 4. Relative error for calibration with the 2D and the 3D targets, in the case of linear (top

 2Dtarget 3Dtarget 2Dtarget

The high error values obtained for the comparison between the measurements obtained with the two targets shows that the 3D target cannot be used in our configuration. The

 for the volumetric target, the dots positions cover only 2 traverse planes on the vertical direction (Z), thus only a linear correction can be performed along Z, and the larger

the position of the cameras is not symmetrically placed relatively to the target plane, by

spacing of the dots results in a lack of information for the in-plane positions;

*C C*

100 %

*<sup>C</sup>* (3)

active area 200x200 mm 270x190 mm dot spacing 5 mm 20 mm dot diameter 2 mm 5 mm reference marker diameter 2.7 mm 7 mm axis marker diameter 1.3 mm 5 mm Level spacing - 4 mm

Table 5. 2D and volumetric calibration targets dimensions

row) and polynomial (bottom row) transfer functions

accessibility reasons;

reasons, which explain this difference of accuracy, are the following:

Denoting C the velocity in the reference frame OXYZ and C' the velocity in a frame rotated by an angle δ around one of the axes, the relative error for each component is:

$$\mathcal{L} = \left| \frac{\mathbf{C} - \mathbf{C}'}{\mathbf{C}} \right| \cdot 100 \text{ [\%]} \quad \text{s} = \left| 1 - \frac{\mathbf{C}'}{\mathbf{C}} \right| \cdot 100 \text{ [\%]} \tag{4}$$


$$
\begin{bmatrix} \mathbf{C}\_{y} \\ \mathbf{C}\_{x} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{y} & -\mathbf{C}\_{x} \\ \mathbf{C}\_{x} & \mathbf{C}\_{y} \end{bmatrix} \cdot \begin{bmatrix} \cos \delta \\ \sin \delta \end{bmatrix}\_{\circ} \begin{bmatrix} \mathcal{E}\_{y} \\ \mathcal{E}\_{x} \end{bmatrix} = \begin{bmatrix} 1 & -\mathbf{C}\_{x}/\mathbf{C}\_{y} \\ 1 & \mathbf{C}\_{y}/\mathbf{C}\_{x} \end{bmatrix} \cdot \begin{bmatrix} 1 - \cos \delta \\ \sin \delta \end{bmatrix} \tag{5}
$$


$$
\begin{bmatrix} \mathbf{C}\_{x}^{'} \\ \mathbf{C}\_{x}^{'} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{x} & -\mathbf{C}\_{x} \\ \mathbf{C}\_{x} & \mathbf{C}\_{x} \end{bmatrix} \cdot \begin{bmatrix} \cos\delta \\ \sin\delta \end{bmatrix}\_{\circ} \begin{bmatrix} \mathcal{E}\_{x} \\ \mathcal{E}\_{x} \end{bmatrix} = \begin{bmatrix} 1 & -\mathbf{C}\_{x}/\mathbf{C}\_{x} \\ 1 & \mathbf{C}\_{x}/\mathbf{C}\_{x} \end{bmatrix} \cdot \begin{bmatrix} 1 - \cos\delta \\ \sin\delta \end{bmatrix} \tag{6}
$$


$$
\begin{bmatrix} \mathbf{C}\_x \\ \mathbf{C}\_y \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_x & -\mathbf{C}\_y \\ \mathbf{C}\_y & \mathbf{C}\_x \end{bmatrix} \cdot \begin{bmatrix} \cos \delta \\ \sin \delta \end{bmatrix}\_\circ \begin{bmatrix} \varepsilon\_x \\ \varepsilon\_y \end{bmatrix} = \begin{bmatrix} 1 & -\mathbf{C}\_y / \mathbf{C}\_x \\ 1 & \mathbf{C}\_x / \mathbf{C}\_y \end{bmatrix} \cdot \begin{bmatrix} 1 - \cos \delta \\ \sin \delta \end{bmatrix} \tag{7}
$$

Thus the error depends only on the rotation angle and velocity components ratio. These ratios vary in the range [0 1] for Cz/Cy, [1 20] for Cz/Cx, and [1 30] for the in-plane components Cy/Cx. The relative errors for rotation around each axis are given in Fig. 5, for angles ranging between 0 and 2°. The black line denotes 3% error limit.

For rotation around the OX axis:


For rotation around the OY axis:

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 61

The transversal and out-of-plane velocity components (Cx and Cz) are the most sensitive to accidental rotation of the target during calibration, imposing a limit of 0.1° in rotation about

The necessary resolution of the target displacement in the out-of-plane direction has been evaluated by accuracy analysis for calibrations with 1 mm and 0.5 mm spacing between

1mm 0.5mm

(8)

1mm 100 % *C C*

The error distributions show that the uncertainty does not exceed 3% for the velocity components in the cross-section plane, Cx and Cz, and is smaller than 0.5 % for the component in the main flow direction, Cy. On the other hand, the polynomial transfer function linear in Z and parabolic in XY is very sensitive to the target positions spacing in the out-of-plane direction, reaching 5% for the small velocity components Cx and Cz. The calibration with 1 mm target displacement in the out-of-plane direction has been chosen for the PIV measurements in most of the turbine sections. The offset was reduced to 0.5mm in configurations where drastic tolerances were required for the 3rd velocity component.

A specific system for placing the target inside the draft tube has been developed. For an uncertainity of 3% on the velocity field measurement, the tolerances are 0.1° in rotation and

A typical PIV experiment procedure consists in setting several acquisition parameters, choosing and testing the trigger signal, acquiring the raw image and/or velocity data, validating the vector field, calculating the 3rd velocity component and finally data output.

For the CCD camera used in this experiment, the first frame's exposure takes up to 132 μs, while the second frame is exposed during the entire read-out sequence of the first frame, 111 ms. It means that, for short time delays between laser pulses, the background gray level on the second frame will increase sensibly. In order to broaden the dynamic range, the ambient light level is set to minimum during the data acquisition. As the time delay varies according to the local flow field characteristics, the laser's energy level is balanced for each experiment, in order to reach uniform brightness on both images. Moreover, the cameras are equipped

*C*

0.01 mm for the relative target displacement in the out-of-plane direction.

The parameters that need to be adjusted for each measurement setup are:

with high-pass filters for the emission wavelength of the laser: 532 +/- 5 nm.




the three axes.

**Target transversal displacement** 

**Target traversing system** 

**2.7 Acquisition parameters** 

**2.7.1 Image quality** 

successive positions of the target along the Z axis.


Fig. 5. Relative error for an accidental target rotation during calibration

The transversal and out-of-plane velocity components (Cx and Cz) are the most sensitive to accidental rotation of the target during calibration, imposing a limit of 0.1° in rotation about the three axes.

### **Target transversal displacement**

60 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

 Cx is the less sensitive if Cz<Cx (tolerance >2°), but the limit decreases under 0.2° if Cz>>Cx, slowly until Cz=3.5Cx (1.7° to 0.5°) and steeply as the ratio Cz/Cx increases; Cz is very sensitive to rotation if Cz<Cx (limit 0.1°), but for Cz>>Cx the tolerance

Cx is the most sensitive component, being the smallest, which gives a tolerance of 0.2°.

exceeds 1.7°;

For rotation around the OZ axis:

Cy is almost insensitive if Cy>Cx;

Cx is more sensitive as the ratio Cy/Cx increases;

Cx is the most sensitive component, tolerance 0.1°.

Fig. 5. Relative error for an accidental target rotation during calibration

The necessary resolution of the target displacement in the out-of-plane direction has been evaluated by accuracy analysis for calibrations with 1 mm and 0.5 mm spacing between successive positions of the target along the Z axis.

$$\mathcal{L} = \frac{\mathbf{C}\_{1\text{mm}} - \mathbf{C}\_{0.5\text{mm}}}{\mathbf{C}\_{1\text{mm}}} \cdot 100 \,\mathrm{[\,\%\text{]}}\tag{8}$$

The error distributions show that the uncertainty does not exceed 3% for the velocity components in the cross-section plane, Cx and Cz, and is smaller than 0.5 % for the component in the main flow direction, Cy. On the other hand, the polynomial transfer function linear in Z and parabolic in XY is very sensitive to the target positions spacing in the out-of-plane direction, reaching 5% for the small velocity components Cx and Cz. The calibration with 1 mm target displacement in the out-of-plane direction has been chosen for the PIV measurements in most of the turbine sections. The offset was reduced to 0.5mm in configurations where drastic tolerances were required for the 3rd velocity component.

#### **Target traversing system**

A specific system for placing the target inside the draft tube has been developed. For an uncertainity of 3% on the velocity field measurement, the tolerances are 0.1° in rotation and 0.01 mm for the relative target displacement in the out-of-plane direction.

#### **2.7 Acquisition parameters**

A typical PIV experiment procedure consists in setting several acquisition parameters, choosing and testing the trigger signal, acquiring the raw image and/or velocity data, validating the vector field, calculating the 3rd velocity component and finally data output.

The parameters that need to be adjusted for each measurement setup are:


### **2.7.1 Image quality**

For the CCD camera used in this experiment, the first frame's exposure takes up to 132 μs, while the second frame is exposed during the entire read-out sequence of the first frame, 111 ms. It means that, for short time delays between laser pulses, the background gray level on the second frame will increase sensibly. In order to broaden the dynamic range, the ambient light level is set to minimum during the data acquisition. As the time delay varies according to the local flow field characteristics, the laser's energy level is balanced for each experiment, in order to reach uniform brightness on both images. Moreover, the cameras are equipped with high-pass filters for the emission wavelength of the laser: 532 +/- 5 nm.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 63

The time delay between two laser pulses must be adapted to the local characteristics of the flow. The prior knowledge of the local velocity range and flow field structure is useful for

The time interval must be chosen as small as possible to insure a good sensitivity for small particle displacements, to fit into the limits of the sub-pixel interpolation resolution, and as large as possible that the particle remains in the laser sheet width and within the interrogation area limits during the two camera exposures. The time delay optimization is particularly challenging in zones with strong velocity gradients, such as backflow zones or

In a turbine, the runner rotation forces the periodic behavior of the flow. One way to reconstruct the shape of a signal acquired by an unsteady measurement technique is the phase-locking, i.e. triggering the acquisition with a reference signal at different time delays within the event's period. In our specific case, the reference signal is the runner frequency, delivered by an optical encoder mounted on the shaft. The optical encoder delivering a signal per rotation has been used for acquisition synchronization for the operating points

The PIV laser can be operated in window triggering mode, which means that the laser burst can be advanced if the trigger signal comes within a fixed time interval before the expected bursting moment. Although, the acquisition frequency is limited by the camera frequency in double-frame mode: 4.5 Hz. This slow-down only influences the experiment duration for

The synchronization of the laser and cameras with the external event is insured by the

Fig. 6. Timing diagram for the PIV data acquisition and synchronization with an external

The raw 2D vector fields resulting from cross-correlation can contain velocity values that have not been correctly detected, which bias the subsequent statistical analysis. A series of

vortices. Typical values in our experimental conditions are between 50 – 300 μs.

achieving convergence, since the acquired signal is sampled at the same phase.

processor unit. The timing diagram is presented in Fig. 6.

**2.8 Data validation and post-processing** 

validation methods have been used for removing the outliers:

**2.8.1 Data validation criteria** 

**2.7.4 Time interval** 

**2.7.5 Synchronization** 

frequency

near the best efficiency conditions.

setting an initial guess for the time delay setting.

 Another important topic is the uneven energy distribution along the laser sheet, which can have several causes:


All these parameters are taken into account and adjusted accordingly for each measurement setup, prior to data acquisition.

### **2.7.2 Spatial resolution**

The measurement zone is divided in small analysis areas, for which a local velocity is calculated with the mean displacement of seeding particles between two successive frames, and the vector's application point is chosen at the center of the area. This grid defines the spatial resolution of the PIV measurement in the 2D case.

In a 2D configuration, the spatial resolution would be given by the distance between two successive vector anchor points in pixels, divided by the scale factor pixels/mm. If a linear dependence exists between the CCD chip dimensions and the measurement zone dimensions, then the scale factor depends on the field of view, on which the camera is focused, i.e. the aperture setup and the optical characteristics of the encountered media.

In a stereoscopic setup, the correspondence image – field of view is not linear anymore, because of the geometrical and perspective distortions due to camera setup. The velocities are calculated on a grid defined on the common zone of the two fields of view, by applying the calibration relationship on the neighboring vectors in both 2D fields, thus solving a linear or nonlinear system of four equations with three unknowns.

The spatial resolution is chosen such as the interpolation bias has a minimum influence on the vector value uncertainty – 2.4x2.5x3mm.

### **2.7.3 Seeding density**

For a measurement to be successful, a minimum of two matching particles should be present in the analysis areas on both frames. According to the Nyquist criterion, the average number of correlated particles in the analysis window should not exceed five. This can be verified statistically checking the overall distribution of the correlation peaks width, which corresponds to the number of matching particle pairs in the correspondent analysis areas. It is checked that the average value fits within 3 – 5 for each measurement setup.

For the instantaneous velocity fields it is very difficult to insure a homogeneous seeding distribution of particles in all interrogation windows, particularly in measurements zones with strong velocity gradients or vortices. To improve the particles traceability, an overlapping of 25% of the analysis areas has been considered. The lack of particles or bad correlations is accounted for during vector field post-processing by correlation peak height and vector size criteria.

### **2.7.4 Time interval**

62 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Another important topic is the uneven energy distribution along the laser sheet, which can


All these parameters are taken into account and adjusted accordingly for each measurement

The measurement zone is divided in small analysis areas, for which a local velocity is calculated with the mean displacement of seeding particles between two successive frames, and the vector's application point is chosen at the center of the area. This grid defines the

In a 2D configuration, the spatial resolution would be given by the distance between two successive vector anchor points in pixels, divided by the scale factor pixels/mm. If a linear dependence exists between the CCD chip dimensions and the measurement zone dimensions, then the scale factor depends on the field of view, on which the camera is focused, i.e. the aperture setup and the optical characteristics of the encountered media.

In a stereoscopic setup, the correspondence image – field of view is not linear anymore, because of the geometrical and perspective distortions due to camera setup. The velocities are calculated on a grid defined on the common zone of the two fields of view, by applying the calibration relationship on the neighboring vectors in both 2D fields, thus solving a

The spatial resolution is chosen such as the interpolation bias has a minimum influence on

For a measurement to be successful, a minimum of two matching particles should be present in the analysis areas on both frames. According to the Nyquist criterion, the average number of correlated particles in the analysis window should not exceed five. This can be verified statistically checking the overall distribution of the correlation peaks width, which corresponds to the number of matching particle pairs in the correspondent analysis areas. It

For the instantaneous velocity fields it is very difficult to insure a homogeneous seeding distribution of particles in all interrogation windows, particularly in measurements zones with strong velocity gradients or vortices. To improve the particles traceability, an overlapping of 25% of the analysis areas has been considered. The lack of particles or bad correlations is accounted for during vector field post-processing by correlation peak height

is checked that the average value fits within 3 – 5 for each measurement setup.




spatial resolution of the PIV measurement in the 2D case.

linear or nonlinear system of four equations with three unknowns.

the vector value uncertainty – 2.4x2.5x3mm.

**2.7.3 Seeding density** 

and vector size criteria.

have several causes:


refractive index of the material.

setup, prior to data acquisition.

**2.7.2 Spatial resolution** 

The time delay between two laser pulses must be adapted to the local characteristics of the flow. The prior knowledge of the local velocity range and flow field structure is useful for setting an initial guess for the time delay setting.

The time interval must be chosen as small as possible to insure a good sensitivity for small particle displacements, to fit into the limits of the sub-pixel interpolation resolution, and as large as possible that the particle remains in the laser sheet width and within the interrogation area limits during the two camera exposures. The time delay optimization is particularly challenging in zones with strong velocity gradients, such as backflow zones or vortices. Typical values in our experimental conditions are between 50 – 300 μs.

### **2.7.5 Synchronization**

In a turbine, the runner rotation forces the periodic behavior of the flow. One way to reconstruct the shape of a signal acquired by an unsteady measurement technique is the phase-locking, i.e. triggering the acquisition with a reference signal at different time delays within the event's period. In our specific case, the reference signal is the runner frequency, delivered by an optical encoder mounted on the shaft. The optical encoder delivering a signal per rotation has been used for acquisition synchronization for the operating points near the best efficiency conditions.

The PIV laser can be operated in window triggering mode, which means that the laser burst can be advanced if the trigger signal comes within a fixed time interval before the expected bursting moment. Although, the acquisition frequency is limited by the camera frequency in double-frame mode: 4.5 Hz. This slow-down only influences the experiment duration for achieving convergence, since the acquired signal is sampled at the same phase.

The synchronization of the laser and cameras with the external event is insured by the processor unit. The timing diagram is presented in Fig. 6.

Fig. 6. Timing diagram for the PIV data acquisition and synchronization with an external frequency

#### **2.8 Data validation and post-processing**

#### **2.8.1 Data validation criteria**

The raw 2D vector fields resulting from cross-correlation can contain velocity values that have not been correctly detected, which bias the subsequent statistical analysis. A series of validation methods have been used for removing the outliers:

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 65

However, even if all of the previous considerations are fulfilled, the uncertainty of 3% holds only if the optical deformations of the image, due to the optical interfaces, are sufficiently small to be corrected by the polynomial function. To check this condition, a comparison was made between the PIV measurements and LDV measurements see Fig. 7. In this way, the

The similar spatial resolution for the two measurement systems does not induce a supplementary error source: the measurement volume for the PIV measurement is a parallelepiped of 2.4 x 2.4 x 3 mm3 and for the LDV measurement is an ellipsoid of 2.4 x 0.2 x 0.2 mm3. The result shows a good agreement, within 3%, for the entire PIV measurement

The rotor-stator interaction is a complex set of phenomena stemming from the nature of the machine itself – such as periodic rotation instabilities and blade interactions – (Ciocan et al. 1996). The unsteady phenomena related to the interactions between the static and mobile parts often affect machine efficiency. Under the influence of periodic constraints, pressure and velocity fluctuations generate stress fluctuations and induce vibrations that contribute to material fatigue of the components and may also issue hydraulic noise. Blade interactions differentiate from other phenomena encountered in turbomachinery by their independence of the operation conditions. Whichever the turbomachine type and its operating regime, these interactions occur under the form of an unsteady secondary flow superposed onto the

In hydraulic machinery (pumps, turbines and pump-turbines), the following phenomena, briefly described hereafter, are to be considered: potential interactions, wake interactions, von Kármán vortex interactions, three-dimensional viscous interactions and instabilities of

 Potential interactions are characterized by a non-uniform distribution of the unsteady pressure field in the gap between the static and rotating parts (e.g. guide-vanes and runner). This type of interaction has a non-convective character and its influence extends towards upstream as well as downstream the gap – see (Mesquita et al. 1999). Wake interactions are determined by the non-uniformity of the velocity profiles downstream blade casades (e.g. at the runner outlet in a centrifugal pump) – see (Iliescu et al 2004). The main source of this non-uniformity is the shear layer created at the trailing edge by the combination of the boundary layers developed on the blade's pressure and suction sides. The velocity defect decays due to viscous effects, and its mixing length depends on the blade shape, local flow velocity and turbulence level. This phenomenon is of purely convective nature and it has no influence upstream its

 Von Kármán vortex street interactions take place downstream blade cascades with blunt trailing edge, in low turbulence conditions. Their shedding frequency is generally expressed in terms of Strouhal number, which depends on the characteristic blade dimensions and on the local flow velocity. If the vortex shedding frequency reaches resonance with the natural frequencies of the system, it may lead to structural vibrations.

global accuracy of the PIV measurement in this configuration is assessed.

**3. PIV measurements for rotor-stator interaction investigations** 

**3.1 Phenomenology in pumps and turbines** 

average stationary flow.

the efficiency curve:

location.

field.


$$\overline{\mathbf{C}}(\mathbf{x}, \mathbf{y}) = \frac{1}{mn} \sum\_{i=x-\frac{m-1}{2}}^{x+\frac{m-1}{2}} \sum\_{j=y-\frac{n-1}{2}}^{y+\frac{n-1}{2}} \mathbf{C}(i, j) \tag{9}$$

 if the difference between them exceeds a percentage of the maximum difference in the vector field,

$$\left\|\mathbf{C}(\mathbf{x},\boldsymbol{y}) \cdot \overline{\mathbf{C}}(\mathbf{x},\boldsymbol{y})\right\| > a \max\_{\mathbf{x},\boldsymbol{y}} \left\|\mathbf{C}(\mathbf{x},\boldsymbol{y}) \cdot \overline{\mathbf{C}}(\mathbf{x},\boldsymbol{y})\right\|\tag{10}$$

This filter has a smoothing effect, thus it is used with caution when strong velocity gradients are present.

#### **2.8.2 Validation with LDV data**

Systematic errors, coming from calibration accuracy, image quality, cross-correlation, vector validation, interpolation for 3rd component calculation, repetitiveness and reference frame transform, have been addressed in the previous chapters and are evaluated to 3% of the mean velocity value.

Fig. 7. PIV-LDV data comparison in the inlet and outlet sections of the cone of the draft tube

However, even if all of the previous considerations are fulfilled, the uncertainty of 3% holds only if the optical deformations of the image, due to the optical interfaces, are sufficiently small to be corrected by the polynomial function. To check this condition, a comparison was made between the PIV measurements and LDV measurements see Fig. 7. In this way, the global accuracy of the PIV measurement in this configuration is assessed.

The similar spatial resolution for the two measurement systems does not induce a supplementary error source: the measurement volume for the PIV measurement is a parallelepiped of 2.4 x 2.4 x 3 mm3 and for the LDV measurement is an ellipsoid of 2.4 x 0.2 x 0.2 mm3. The result shows a good agreement, within 3%, for the entire PIV measurement field.

### **3. PIV measurements for rotor-stator interaction investigations**

### **3.1 Phenomenology in pumps and turbines**

64 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications




<sup>1</sup> (,) (, ) *m n x y*

if the difference between them exceeds a percentage of the maximum difference in the

, ( , )- ( , ) max ( , )- ( , ) *x y*

This filter has a smoothing effect, thus it is used with caution when strong velocity gradients

Systematic errors, coming from calibration accuracy, image quality, cross-correlation, vector validation, interpolation for 3rd component calculation, repetitiveness and reference frame transform, have been addressed in the previous chapters and are evaluated to 3% of the

Fig. 7. PIV-LDV data comparison in the inlet and outlet sections of the cone of the draft tube

*m n ix jy Cxy Ci j mn*

(9)

*C x y C x y C x y C x y* (10)

illumination and good spot-background contrast;

length or its components may be limited;

its m×n neighbors,

**2.8.2 Validation with LDV data** 

mean velocity value.

vector field,

are present.

The rotor-stator interaction is a complex set of phenomena stemming from the nature of the machine itself – such as periodic rotation instabilities and blade interactions – (Ciocan et al. 1996). The unsteady phenomena related to the interactions between the static and mobile parts often affect machine efficiency. Under the influence of periodic constraints, pressure and velocity fluctuations generate stress fluctuations and induce vibrations that contribute to material fatigue of the components and may also issue hydraulic noise. Blade interactions differentiate from other phenomena encountered in turbomachinery by their independence of the operation conditions. Whichever the turbomachine type and its operating regime, these interactions occur under the form of an unsteady secondary flow superposed onto the average stationary flow.

In hydraulic machinery (pumps, turbines and pump-turbines), the following phenomena, briefly described hereafter, are to be considered: potential interactions, wake interactions, von Kármán vortex interactions, three-dimensional viscous interactions and instabilities of the efficiency curve:


PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 67

Fig. 9. Overview of the PIV setup. Detail on the right side shows the image reflected by the

To obtain a sharp image with a CCD camera, the focusing plane must be in the plane of the laser sheet. Another requirement for a 2D-PIV experiment is to set the camera imaging axis normal to the measurement plane. To ensure minimum optical distortion, the different interfaces traversed by the scattered light must be smooth and parallel to the laser sheet as well. In the present case, the camera is installed on a mobile support with two degrees of freedom – translation and rotation. Considering the camera's focal length and visual access difficulties, a plane mirror was placed in front of the window to redirect the optical path

The mirror must meet high quality standards (flatness of the surface and homogeneity of the reflective coating), to avoid additional errors in the measurement process. The solution adopted for this experiment was a price-quality compromise: Pyrex wafer with ALMGF2 coating, λ/2 flatness. The mirror was mounted on the same support as the camera, aligned at 45° with respect to the lens. The CCD-mirror assembly was oriented parallel to the laser sheet and the support axis was secured horizontally. The final adjustment was made by optimizing the sharpness of the acquired images, with an estimated maximum uncertainty

Regarding the laser sheet positioning, in a first approach the optical path of the laser sheet was calculated and it was planned to adjust its position in-situ using a device attached to the light-guiding arm. The precision was not satisfactory, thus it has been decided to mark directly the position of the plane directly onto the wicket gates, for each vertical position. The laser sheet contour was traced on the opposite faces of two guide vanes that form a channel. First tests revealed scattering problems on the shroud, runner blades and guide vanes. The issue was corrected by painting these bright metallic surfaces with a matte black dye. Several tests were made to determine the appropriate dye for the intended objectives – reflection attenuation and durability under repeated laser pulse operation. A black

The measurements have been acquired synchronously with the runner rotation, for multiple phase angles corresponding to portions of a single runner inter-blade channel. Ten angular

inclined mirror, as seen from the CCD camera standpoint

horizontally towards the camera, as illustrated in Fig. 9.

of 0.6°.

permanent marker was chosen.

**3.3 Main results** 

 Viscous three-dimensional interactions are secondary flows in the inter-blade channels, such as passage vortex, corner vortex, horseshoe vortex and tip gap vortices. They are closely related to the shape of the different hydraulic passages, which force variations in the flow direction and local modifications of the boundary layer topology. It is likely that these phenomena are the main contribution to local efficiency loss in specific areas of the operating range. Certain machines (pumps and pump-turbines) often exhibit an unstable zone of the efficiency characteristic, with discontinuities of up to 3%, accompanied or not by hysteresis – see (Ciocan et al. 2001).

The accurate prediction of these phenomena is essential at the design stage, and PIV measurements are the ideal tool to investigate them. The first experiment set up to analyze these interactions is described in (Ciocan et al. 2006).

### **3.2 Experimental set-up**

2D-PIV measurements have been performed in the guide-vanes channel of a pump-turbine model of specific speed nq=66. Several operating conditions were investigated in both pump and turbine regimes, covering a large portion of the operating range.

To obtain the velocity field evolution in the guide-vane channels, the measurement section has been chosen such as to cover the space between two wicket gates. This area is determined by the 'visibility' zone, i.e. the area which is accessible by a laser sheet through windows embedded in the spiral casing walls – see Fig. 8. Two stay vanes that obstructed the visual access to the measurement section have been removed. Views through two separate windows were necessary in order to cover the full extent of the measurement domain.

Fig. 8. Top and side views of the measurement sections and laser sheet positions for the PIV experiment in the guide-vanes channel of a pump-turbine

Five sections have been analyzed along the height of the channel. Considering the important optical constraints imposed by previous LDV investigations in the same section, the imaging window was tilted by 9° with respect to the machine's axis. To enable the most adequate conditions for the PIV experiment, the laser planes were also tilted of 9° in the radial direction to match the central window level – see Fig. 9.

Fig. 9. Overview of the PIV setup. Detail on the right side shows the image reflected by the inclined mirror, as seen from the CCD camera standpoint

To obtain a sharp image with a CCD camera, the focusing plane must be in the plane of the laser sheet. Another requirement for a 2D-PIV experiment is to set the camera imaging axis normal to the measurement plane. To ensure minimum optical distortion, the different interfaces traversed by the scattered light must be smooth and parallel to the laser sheet as well. In the present case, the camera is installed on a mobile support with two degrees of freedom – translation and rotation. Considering the camera's focal length and visual access difficulties, a plane mirror was placed in front of the window to redirect the optical path horizontally towards the camera, as illustrated in Fig. 9.

The mirror must meet high quality standards (flatness of the surface and homogeneity of the reflective coating), to avoid additional errors in the measurement process. The solution adopted for this experiment was a price-quality compromise: Pyrex wafer with ALMGF2 coating, λ/2 flatness. The mirror was mounted on the same support as the camera, aligned at 45° with respect to the lens. The CCD-mirror assembly was oriented parallel to the laser sheet and the support axis was secured horizontally. The final adjustment was made by optimizing the sharpness of the acquired images, with an estimated maximum uncertainty of 0.6°.

Regarding the laser sheet positioning, in a first approach the optical path of the laser sheet was calculated and it was planned to adjust its position in-situ using a device attached to the light-guiding arm. The precision was not satisfactory, thus it has been decided to mark directly the position of the plane directly onto the wicket gates, for each vertical position. The laser sheet contour was traced on the opposite faces of two guide vanes that form a channel. First tests revealed scattering problems on the shroud, runner blades and guide vanes. The issue was corrected by painting these bright metallic surfaces with a matte black dye. Several tests were made to determine the appropriate dye for the intended objectives – reflection attenuation and durability under repeated laser pulse operation. A black permanent marker was chosen.

#### **3.3 Main results**

66 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

 Viscous three-dimensional interactions are secondary flows in the inter-blade channels, such as passage vortex, corner vortex, horseshoe vortex and tip gap vortices. They are closely related to the shape of the different hydraulic passages, which force variations in the flow direction and local modifications of the boundary layer topology. It is likely that these phenomena are the main contribution to local efficiency loss in specific areas of the operating range. Certain machines (pumps and pump-turbines) often exhibit an unstable zone of the efficiency characteristic, with discontinuities of up to 3%,

The accurate prediction of these phenomena is essential at the design stage, and PIV measurements are the ideal tool to investigate them. The first experiment set up to analyze

2D-PIV measurements have been performed in the guide-vanes channel of a pump-turbine model of specific speed nq=66. Several operating conditions were investigated in both pump

To obtain the velocity field evolution in the guide-vane channels, the measurement section has been chosen such as to cover the space between two wicket gates. This area is determined by the 'visibility' zone, i.e. the area which is accessible by a laser sheet through windows embedded in the spiral casing walls – see Fig. 8. Two stay vanes that obstructed the visual access to the measurement section have been removed. Views through two separate windows were necessary in order to cover the full extent of the measurement

> Laser beam

90 % 10 % 30 % 50 % 70 %

Laser plane (9°)

Camera

Mirror

Fig. 8. Top and side views of the measurement sections and laser sheet positions for the PIV

Five sections have been analyzed along the height of the channel. Considering the important optical constraints imposed by previous LDV investigations in the same section, the imaging window was tilted by 9° with respect to the machine's axis. To enable the most adequate conditions for the PIV experiment, the laser planes were also tilted of 9° in the radial

experiment in the guide-vanes channel of a pump-turbine

direction to match the central window level – see Fig. 9.

accompanied or not by hysteresis – see (Ciocan et al. 2001).

and turbine regimes, covering a large portion of the operating range.

these interactions is described in (Ciocan et al. 2006).

**3.2 Experimental set-up** 

domain.

The measurements have been acquired synchronously with the runner rotation, for multiple phase angles corresponding to portions of a single runner inter-blade channel. Ten angular positions were investigated, evenly distributed at 7.2°. For each spatial position, downstream the runner and between the guide vanes, the two dominant velocity components have been measured, corresponding to the radial and tangential directions. The average velocities are calculated with 250 PIV frame pairs by phase.

According to the ergodicity assumption for a stationary flow, the mean velocity was estimated by the temporal mean of the ensemble (eq. 11), and the standard deviation (eq. 12) for each component of the mean velocity is calculated as follows:

$$
\mathfrak{c} = \Sigma \mathfrak{c}(\mathfrak{i}) / N \tag{11}
$$

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 69

Radial velocity Tangential velocity

Velocity angle Velocity field

Fig. 10. Mean velocity field in the guide vanes channel for optimum flow rate Qn in turbine

Radial velocity Tangential velocity

Velocity angle Mean velocity field

Fig. 11. Mean velocity field in the guide vanes channel for optimum flow rate Qn in pump

guide vane

guide vane

guide vane

runner inlet

runner inlet

mode; mid-channel height; flow direction towards the runner

guide vane

mode; mid-channel height; flow direction exiting the runner

$$
\sigma^2 = \left[ \sum (c(i) - c)^2 \right] \left[ (N - 1) \right] \tag{12}
$$


Using the rms values, the turbulent kinetic energy can be estimated. Having only two components, it is assumed that isotropy conditions are met, i.e. the rms of the missing component is of the same order of magnitude as the two measured components. Furthermore, the periodic velocity fluctuations (deterministic) are negligible with respect to the turbulent fluctuations (random), thus the specific energy of the turbulent structures in the flow may be calculated using the individual standard deviation of the velocity components. In these conditions, the turbulent kinetic energy can be calculated as follows:

$$k = \bigvee\_{4} \left(\sigma\_{r}^{2} + \sigma\_{t}^{2}\right) \tag{13}$$


A typical result for the steady velocity field in turbine mode is presented in Fig. 10. For the measured range of operating points, 20% around the nominal flow rate, the guide vanes direct correctly the mean flow towards the runner inlet and back flows or detachment zones have not been observed.

The mean velocity profiles in pump mode, in a guide vanes channel ad mid height, are presented in Fig. 11 for a stable operating condition, while Fig. 12 shows the velocity profiles for an unstable portion of the operating range.

A phase averaging technique is used to analyze the periodic fluctuations synchronuous with the runner. An optical encoder provides the time basis, as a reference runner position. According to the Reynolds decomposition scheme (eq. 14), an unsteady velocity field can split in three parts: the temporal mean of the ensemble *c* , a periodic fluctuation with respect to the mean ( ) *c c* and a random turbulent fluctuation ' *c* . The process is illustrated in Fig. 13. A comparison between PIV and LDV phase average technique is presented in Fig. 14

$$\mathcal{L}\{\mathbf{i}\} = \overline{\mathcal{c}} + (\tilde{\mathcal{c}} - \overline{\mathcal{c}}) + \mathcal{c}' \tag{14}$$

positions were investigated, evenly distributed at 7.2°. For each spatial position, downstream the runner and between the guide vanes, the two dominant velocity components have been measured, corresponding to the radial and tangential directions. The

According to the ergodicity assumption for a stationary flow, the mean velocity was estimated by the temporal mean of the ensemble (eq. 11), and the standard deviation (eq. 12)

> *c ci N*

Using the rms values, the turbulent kinetic energy can be estimated. Having only two components, it is assumed that isotropy conditions are met, i.e. the rms of the missing component is of the same order of magnitude as the two measured components. Furthermore, the periodic velocity fluctuations (deterministic) are negligible with respect to the turbulent fluctuations (random), thus the specific energy of the turbulent structures in the flow may be calculated using the individual standard deviation of the velocity components. In these conditions, the turbulent kinetic energy can be calculated as follows:

2 2 <sup>3</sup>

A typical result for the steady velocity field in turbine mode is presented in Fig. 10. For the measured range of operating points, 20% around the nominal flow rate, the guide vanes direct correctly the mean flow towards the runner inlet and back flows or detachment zones

The mean velocity profiles in pump mode, in a guide vanes channel ad mid height, are presented in Fig. 11 for a stable operating condition, while Fig. 12 shows the velocity profiles

A phase averaging technique is used to analyze the periodic fluctuations synchronuous with the runner. An optical encoder provides the time basis, as a reference runner position. According to the Reynolds decomposition scheme (eq. 14), an unsteady velocity field can split in three parts: the temporal mean of the ensemble *c* , a periodic fluctuation with respect to the mean ( ) *c c* and a random turbulent fluctuation ' *c* . The process is illustrated in Fig. 13. A

comparison between PIV and LDV phase average technique is presented in Fig. 14

*k* - turbulent kinetic energy for the synchronous velocity component

( )/ (11)

² ( ) ² ( 1) *ci c N* (12)

<sup>4</sup> *r t k* (13)

*ci c c c c* () ( ) ' (14)

average velocities are calculated with 250 PIV frame pairs by phase.

for each component of the mean velocity is calculated as follows:

*c(i)* – measured instantaneous velocity *c* – statistical average velocity

*r* - rms of the mean radial velocity *t* - rms of the mean tangential velocity

for an unstable portion of the operating range.

have not been observed.

*N* – number of samples in each spatial position - standard deviation (rms) of the mean value

Fig. 10. Mean velocity field in the guide vanes channel for optimum flow rate Qn in turbine mode; mid-channel height; flow direction towards the runner

Fig. 11. Mean velocity field in the guide vanes channel for optimum flow rate Qn in pump mode; mid-channel height; flow direction exiting the runner

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 71

All the measurements can be superposed in a representative channel due to the good periodicity of the blade-to-blade channels; this periodicity is checked for each measurement

The specific turbulent kinetic energy is the summation of the normal stress components. Direct measurement provides two orthogonal components of the velocity field. By assuming turbulence isotropy, the 3 normal stress components have the same order of magnitude, the

\* '2 '2 <sup>3</sup> ( )

\* '2 '2 <sup>3</sup> ( )

Distinction is made between the total kinetic energy (eq. 15) and the turbulent kinetic energy content of the periodic fluctuations (eq. 16). This calculation is a better approach of the specific turbulent kinetic energy because the velocity fluctuation synchronous with the runner position is a deterministic phenomenon and this fluctuation does not concern the

important for the design operating point and more important for the off-design operating

Fig. 15 and Fig. 16 show the unsteady flow propagation in the guide-vane channel for 5 runner positions. This unsteady velocity field distribution shows that the fluctuation of the

The blade passage perturbs the flow and therefore leads to a pressure fluctuation at the runner inlet. This pressure fluctuation induces a velocity fluctuation synchronous with the runner position. The direction of propagation of this perturbation is not the same for the two components: in the radial direction for the radial velocity and in tangential direction for the tangential velocity in the same direction with the runner rotation. The suction side of the blade produces a suction effect and an increase of the radial velocity. The pressure side of the blade induces a decrease of the radial velocity and a deviation of the flow direction. The fluctuating component *c c* is the same on the two velocity components and corresponds to

For the operating conditions corresponding to the unstable region of the efficiency curve, the same wake phenomenon is observed in the bottom half of the channel, but presents

In the top half of the channel, the flow is uneven. Analyzing a sequence of synchronous velocity fields, von Karman vortex structures are detected, propagating in the guide-vanes channel, as well as flow tendency to re-enter the runner. The von Karman vortex frequency does not seem to relate to the runner rotation frequency, and the runner wake isn't present either. A comparison of the flow topology in different operation conditions is presented in

higher fluctuations than the optimum operation condition – see Fig.17.

velocity components at the runner inlet is the same in all the guide vanes channel.

<sup>4</sup> *x z k cc* (15)

<sup>4</sup> *x z k cc* (16)

calculations is close to 15%, less

point.

specific turbulent kinetic energy becomes:

At the runner inlet the difference between the \* *k* and \* *k*

specific turbulent energy.

30% of the mean velocity.

points.

Fig. 18.

Fig. 12. Mean velocity field in the guide vanes channel for partial load 80%Qn, in pump mode corresponding to an unstable zone on the efficiency curve; planes at 90% and 10% of the guide-vanes channel's height respectively

Fig. 13. Reynolds decomposition of a velocity signal

Fig. 14. Comparison of phase averaging technique for a PIV and a LDV experiment; data collected in a same spatial location and same operating conditions

section Mean velocity field

section Mean velocity field

~*c c* 2°

t

'

Radial velocity Top

Radial velocity Bottom

the guide-vanes channel's height respectively

*c* ~

<sup>~</sup> <sup>T</sup>

Fig. 13. Reynolds decomposition of a velocity signal

 *c c*

C

C


Fig. 12. Mean velocity field in the guide vanes channel for partial load 80%Qn, in pump mode corresponding to an unstable zone on the efficiency curve; planes at 90% and 10% of

**SYNCHRONISATION IMPULSE**

Fig. 14. Comparison of phase averaging technique for a PIV and a LDV experiment; data

collected in a same spatial location and same operating conditions

All the measurements can be superposed in a representative channel due to the good periodicity of the blade-to-blade channels; this periodicity is checked for each measurement point.

The specific turbulent kinetic energy is the summation of the normal stress components. Direct measurement provides two orthogonal components of the velocity field. By assuming turbulence isotropy, the 3 normal stress components have the same order of magnitude, the specific turbulent kinetic energy becomes:

$$
\overline{k}^\* = \frac{3}{4} (\overline{c}\_x^{'2} + \overline{c}\_z^{'2}) \tag{15}
$$

$$
\tilde{k}^\* = \frac{3}{4} (\tilde{c}\_x'^2 + \tilde{c}\_z'^2) \tag{16}
$$

Distinction is made between the total kinetic energy (eq. 15) and the turbulent kinetic energy content of the periodic fluctuations (eq. 16). This calculation is a better approach of the specific turbulent kinetic energy because the velocity fluctuation synchronous with the runner position is a deterministic phenomenon and this fluctuation does not concern the specific turbulent energy.

At the runner inlet the difference between the \* *k* and \* *k* calculations is close to 15%, less important for the design operating point and more important for the off-design operating points.

Fig. 15 and Fig. 16 show the unsteady flow propagation in the guide-vane channel for 5 runner positions. This unsteady velocity field distribution shows that the fluctuation of the velocity components at the runner inlet is the same in all the guide vanes channel.

The blade passage perturbs the flow and therefore leads to a pressure fluctuation at the runner inlet. This pressure fluctuation induces a velocity fluctuation synchronous with the runner position. The direction of propagation of this perturbation is not the same for the two components: in the radial direction for the radial velocity and in tangential direction for the tangential velocity in the same direction with the runner rotation. The suction side of the blade produces a suction effect and an increase of the radial velocity. The pressure side of the blade induces a decrease of the radial velocity and a deviation of the flow direction. The fluctuating component *c c* is the same on the two velocity components and corresponds to 30% of the mean velocity.

For the operating conditions corresponding to the unstable region of the efficiency curve, the same wake phenomenon is observed in the bottom half of the channel, but presents higher fluctuations than the optimum operation condition – see Fig.17.

In the top half of the channel, the flow is uneven. Analyzing a sequence of synchronous velocity fields, von Karman vortex structures are detected, propagating in the guide-vanes channel, as well as flow tendency to re-enter the runner. The von Karman vortex frequency does not seem to relate to the runner rotation frequency, and the runner wake isn't present either. A comparison of the flow topology in different operation conditions is presented in Fig. 18.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 73

Fig. 16. Velocity field synchronous with the runner rotation *c* for nominal flow rate Qn

Fig. 17. Instantaneous velocity fields at 80% of the channel height: optimum flow rate Qn

0 10 20 30 40 50 60 70 80 90 100

<sup>0</sup> *c*( )

<sup>0</sup> *c*( 43.2 )

<sup>0</sup> *c*( 21.6 )

0 10 20 30 40 50 60 70 80 90 100

(left) and partial load 0.8Qn (right)

Fig. 15. Velocity field synchronous with the runner rotation *c* for different runner positions – turbine operation Qn ; tangential velocity on left column and radial velocity on right column

0

runner inlet runner inlet

0+14.4°

0+28.8°

0+43.2°

0+57.6°

Fig. 15. Velocity field synchronous with the runner rotation *c* for different runner positions – turbine operation Qn ; tangential velocity on left column and radial velocity on

right column

Fig. 16. Velocity field synchronous with the runner rotation *c* for nominal flow rate Qn

Fig. 17. Instantaneous velocity fields at 80% of the channel height: optimum flow rate Qn (left) and partial load 0.8Qn (right)

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 75

For these measurements, rhodamine (Tetraethylrhodamine hydrochloride - RhB) particles of 1-10m diameter, receiving 532nm and emitting at 594nm wavelength are introduced in the flow. Using fluorescent particles, which are excited by the laser wavelength, but emit at a higher wavelength, along with a longpass filter, prevents reflections in the laser wavelength to reach the cameras. In this way, the strong reflections from the rope interface, from residual bubbles in the flow or due to the optical interfaces are strongly attenuated and the vector field can be determined by cross-correlation from the second camera. The first camera has an antireflection-coated filter, focused on the laser wavelength 532 ±5 nm, to record the reflections in the laser light from the rope-water interface, and the second camera has a cutoff filter, 580nm, on the emission wavelength of fluorescent particles. The vortex core boundary can then be extracted by image processing of the first camera image and the

The laser, cameras and processor used for the 2D-PIV experiment are adapted to the present measurement set-up. Optical filters have been mounted on the cameras. The two cameras'

fields of view have been made coincident through a mirrors system, see Fig. 20.

velocity field is extracted of the second camera image.

Laser

CCD

LDV

Fig. 20. Experimental setup for 2D-PIV measurements in two-phase flow

Since the PIV system is set up in 2D configuration, the calibration would only be necessary for establishing the scale factor. Nevertheless, in this case, the distortions due to the cone's wall shape give an uneven distribution of the scale factor over the image. The distortions are diminished by a flat external surface, but the internal geometry had to be preserved for the hydraulic path, see Fig. 21. This conical surface induces distortion near the image edges.

For this reason, a polynomial model is found to be suitable for these measurements.

**4.1.2 Measurement set-up** 

**4.1.3 Calibration** 

Fig. 18. Instantaneous velocity fields at 80% of the channel height: optimum flow rate Qn (left) and partial load 0.8Qn (right)

### **4. PIV measurements in two phase flow**

### **4.1 Two-dimensional PIV in two-phase flow**

### **4.1.1 Principle**

PIV has been applied successfully to two-phase flows in the case of bubble flows or mixing experiments. Starting from these examples, a new two-phase PIV application was developed for simultaneous measurement of the flow velocity field and the volume of a compact unsteady vapors cavity - the rope that forms downstream the runner of a hydraulic turbine in low-pressure conditions – see Fig. 19. Specific image acquisition and filtering procedures are implemented for the investigation of cavity volume and its evolution related to the under-pressure level in the draft tube to quantify the development of the rope in the diffuser cone of a Francis turbine scale model.

The new technique is an adaptation of the PIV method for two-phase flows. Using fluorescent particles and corresponding cut-off filters on the two cameras, the two wavelengths can be separated. Fluorescence is an optical phenomenon that occurs when a molecule or atom goes to a higher energy level under the influence of incident radiation and emits radiation as the system relaxes, at a higher wavelength. The emission can be in the visible or ultraviolet spectrum if an electronic transition is involved or in the infrared range if it is a vibrational transition.

Fig. 19. Vapors-core vortex development in the cone at part load – example of measured unsteady velocity field overlaid on the corresponding rope image

For these measurements, rhodamine (Tetraethylrhodamine hydrochloride - RhB) particles of 1-10m diameter, receiving 532nm and emitting at 594nm wavelength are introduced in the flow. Using fluorescent particles, which are excited by the laser wavelength, but emit at a higher wavelength, along with a longpass filter, prevents reflections in the laser wavelength to reach the cameras. In this way, the strong reflections from the rope interface, from residual bubbles in the flow or due to the optical interfaces are strongly attenuated and the vector field can be determined by cross-correlation from the second camera. The first camera has an antireflection-coated filter, focused on the laser wavelength 532 ±5 nm, to record the reflections in the laser light from the rope-water interface, and the second camera has a cutoff filter, 580nm, on the emission wavelength of fluorescent particles. The vortex core boundary can then be extracted by image processing of the first camera image and the velocity field is extracted of the second camera image.

### **4.1.2 Measurement set-up**

74 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Fig. 18. Instantaneous velocity fields at 80% of the channel height: optimum flow rate Qn

PIV has been applied successfully to two-phase flows in the case of bubble flows or mixing experiments. Starting from these examples, a new two-phase PIV application was developed for simultaneous measurement of the flow velocity field and the volume of a compact unsteady vapors cavity - the rope that forms downstream the runner of a hydraulic turbine in low-pressure conditions – see Fig. 19. Specific image acquisition and filtering procedures are implemented for the investigation of cavity volume and its evolution related to the under-pressure level in the draft tube to quantify the development of the rope in the diffuser

The new technique is an adaptation of the PIV method for two-phase flows. Using fluorescent particles and corresponding cut-off filters on the two cameras, the two wavelengths can be separated. Fluorescence is an optical phenomenon that occurs when a molecule or atom goes to a higher energy level under the influence of incident radiation and emits radiation as the system relaxes, at a higher wavelength. The emission can be in the visible or ultraviolet spectrum if an electronic transition is involved or in the infrared range

Fig. 19. Vapors-core vortex development in the cone at part load – example of measured

unsteady velocity field overlaid on the corresponding rope image

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

**4. PIV measurements in two phase flow 4.1 Two-dimensional PIV in two-phase flow** 

(left) and partial load 0.8Qn (right)

cone of a Francis turbine scale model.

if it is a vibrational transition.

**4.1.1 Principle** 

The laser, cameras and processor used for the 2D-PIV experiment are adapted to the present measurement set-up. Optical filters have been mounted on the cameras. The two cameras' fields of view have been made coincident through a mirrors system, see Fig. 20.

Fig. 20. Experimental setup for 2D-PIV measurements in two-phase flow

### **4.1.3 Calibration**

Since the PIV system is set up in 2D configuration, the calibration would only be necessary for establishing the scale factor. Nevertheless, in this case, the distortions due to the cone's wall shape give an uneven distribution of the scale factor over the image. The distortions are diminished by a flat external surface, but the internal geometry had to be preserved for the hydraulic path, see Fig. 21. This conical surface induces distortion near the image edges. For this reason, a polynomial model is found to be suitable for these measurements.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 77

follow the rope interface for a brief period, for less than 10% of the rope period but the minimum interval for two successive PIV measurements is limited by the maximum CCD cameras frequency, i.e. 4.5 Hz, such as to avoid new acquisitions triggered by the bubble passage just after the rope passage in front of the pressure sensor. The PIV acquisition is performed at constant phase delay value with respect to the vortex trigger signal – see Fig. 23. The influence of the vortex period variation for this kind of phase average calculation is

checked, and fits within the same uncertainty range as the measurement method: 3%.

Fig. 22. Waterfall diagram of the power spectra of the wall pressure fluctuations

Rope Position in front of the sensor

Phase

Fig. 23. Trigger signals diagram

LDV Signal

or

Pressure Signal

Fixed Time Interval

PIV frequency

Runner Position

B A

Rope Position in the measurement area

C

a) calibration setup b) calibration target position Fig. 21. Image calibration setup for 2D-PIV measurements in two-phase flow

### **4.1.4 Acquisition parameters**

The laser energy distribution, camera opening and exposure, seeding density, time interval between pulses and number of acquisitions are adjusted as described in the paragraph 2.

#### **Synchronization**

For determining the helical shape of the rope, it is necessary to synchronize the image acquisition with the rope position. The frequency of rope precession is influenced by the σ cavitation level and vapours content in its core. In certain operating conditions it can change over one revolution. Therefore, the triggering system cannot be based on the runner rotation but should detect the rotation frequency of the rope.

The technique to detect the rope precession is based on the measurement of the pressure pulsation generated at the cone wall by the precession of the rope. It has the advantage to work even for cavitation-free conditions when the rope is no longer visible, and, for this reason, it has been selected as trigger for the PIV data acquisition. To characterise the cavitation level, the - cavitation number is defined:

$$\sigma = \frac{P\_a + P\_d - P\_v}{\frac{1}{2}\rho C^2} \tag{17}$$

Pa = atmospheric pressure, Pd = water pressure, Pv = water vapour pressure, C = flow velocity and ρ = water density

The pressure signals power spectra corresponding to the σ values are represented in the waterfall diagram in Fig. 22. For this operating point, the frequency of the rope precession is decreasing with the σ value.

The correspondence between the wall pressure signal breakdown and the rope spatial position has been validated through the optical detection of the rope passage using a LDV probe. By reducing the gain, the photomultiplier of the LDV system delivers a signal each time the rope boundary or bubbles intersect the LDV measuring volume. Small bubbles

a) calibration setup b) calibration target position

The laser energy distribution, camera opening and exposure, seeding density, time interval between pulses and number of acquisitions are adjusted as described in the paragraph 2.

For determining the helical shape of the rope, it is necessary to synchronize the image acquisition with the rope position. The frequency of rope precession is influenced by the σ cavitation level and vapours content in its core. In certain operating conditions it can change over one revolution. Therefore, the triggering system cannot be based on the runner rotation

The technique to detect the rope precession is based on the measurement of the pressure pulsation generated at the cone wall by the precession of the rope. It has the advantage to work even for cavitation-free conditions when the rope is no longer visible, and, for this reason, it has been selected as trigger for the PIV data acquisition. To characterise the

> 1 <sup>2</sup> 2

(17)

Pa = atmospheric pressure, Pd = water pressure, Pv = water vapour pressure, C = flow

The pressure signals power spectra corresponding to the σ values are represented in the waterfall diagram in Fig. 22. For this operating point, the frequency of the rope precession is

The correspondence between the wall pressure signal breakdown and the rope spatial position has been validated through the optical detection of the rope passage using a LDV probe. By reducing the gain, the photomultiplier of the LDV system delivers a signal each time the rope boundary or bubbles intersect the LDV measuring volume. Small bubbles

*PPP adv C* 

Fig. 21. Image calibration setup for 2D-PIV measurements in two-phase flow

**4.1.4 Acquisition parameters** 

velocity and ρ = water density

decreasing with the σ value.

but should detect the rotation frequency of the rope.

cavitation level, the - cavitation number is defined:

**Synchronization** 

follow the rope interface for a brief period, for less than 10% of the rope period but the minimum interval for two successive PIV measurements is limited by the maximum CCD cameras frequency, i.e. 4.5 Hz, such as to avoid new acquisitions triggered by the bubble passage just after the rope passage in front of the pressure sensor. The PIV acquisition is performed at constant phase delay value with respect to the vortex trigger signal – see Fig. 23. The influence of the vortex period variation for this kind of phase average calculation is checked, and fits within the same uncertainty range as the measurement method: 3%.

Fig. 22. Waterfall diagram of the power spectra of the wall pressure fluctuations

Rope Position in front of the sensor

Rope Position in the measurement area

Fig. 23. Trigger signals diagram

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 79

By spatial filtering, the zone of the image where the rope appears is extracted from the input image, see Fig. 24. The image-processing algorithm will only be applied on this zone, with the benefit of reducing the computing time, due to smaller amount of data. Since almost 50% of the input image is rejected, the contribution to the image histogram of dark background pixels and of the noise coming from seeding particles in the discarded area is strongly

Fig. 26. First step of image processing: spatial filtering. Graylevel histogram for the input

reduced, thus improving the grey-level distribution, see Fig. 26.

Fig. 25. Image processing flowchart

image and after the spatial filter is applied

**Spatial filtering** 

### **4.1.5 Image processing**

Image processing is applied to the first camera image, to obtain the rope shape. The distortion correction is performed with the calibration transform prior to the image processing step. Using a polynomial optical transfer function, the deformed raw image is mapped onto the real measurement cross-section, cf. Fig. 24. Each pixel on the image has now a linear dependency to its real-worls coordinates through a constant scale factor.

Fig. 25 presents the steps that are considered for filtering the image noise and for calibrating the contour relatively to the image brightness and rope boundary reflection:


Fig. 24. Raw image of the rope, after distorsion compensation with a polynomial transfer function

Fig. 25. Image processing flowchart

#### **Spatial filtering**

78 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Image processing is applied to the first camera image, to obtain the rope shape. The distortion correction is performed with the calibration transform prior to the image processing step. Using a polynomial optical transfer function, the deformed raw image is mapped onto the real measurement cross-section, cf. Fig. 24. Each pixel on the image has now a linear dependency to its real-worls coordinates through a constant scale factor.

Fig. 25 presents the steps that are considered for filtering the image noise and for calibrating







Fig. 24. Raw image of the rope, after distorsion compensation with a polynomial transfer


the contour relatively to the image brightness and rope boundary reflection:


towards higher values, weighted logarithmically;

position and the rope diameter are available;

rope area and rope diameter.

on sliding neighborhoods of 8x8 pixels;

**4.1.5 Image processing** 

spatial filtering;

outside the rope;

function

By spatial filtering, the zone of the image where the rope appears is extracted from the input image, see Fig. 24. The image-processing algorithm will only be applied on this zone, with the benefit of reducing the computing time, due to smaller amount of data. Since almost 50% of the input image is rejected, the contribution to the image histogram of dark background pixels and of the noise coming from seeding particles in the discarded area is strongly reduced, thus improving the grey-level distribution, see Fig. 26.

Fig. 26. First step of image processing: spatial filtering. Graylevel histogram for the input image and after the spatial filter is applied

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 81

optical filter on the camera prevents image blurring, by reflecting part of the incident light scattered by the vapors-core boundary, as well as by residual bubbles and tracer particles in the flow. Nevertheless, the zone where the laser sheet reaches the rope boundary can be identified as the area with concentrated high intensity values. Thus, the best approximation of the real rope diameter, at the intersection with the inclined laser plane, would be the

The influence of the reflection zone where the laser sheet reaches the rope boundary is overcame by cutting the gray intensity profile along the diameter line found in the first approximation at 94% of the maximum value, and keeping the value on the right as the new starting point for the diameter line, see Fig. 29. The centre of the rope in the measurement

At last, a statistical analysis on the position and diameter of the rope in the measurement plane is performed for each phase – see (Iliescu et al. 2003). The coherence of the result is

distance between edges along a line inclined at ~30° about the horizontal, see Fig. 28.

plane is considered to be the centre of mass of the final diameter line, see Fig. 29.

based on dimensions criteria check – Fig. 30. The rate of validated images is 95%.

Fig. 28. Rope diameter detection and validation

### **Noise reduction**

Since the camera records the reflection in the laser wavelength, the reflections from the seeding particles will be apparent on the image as well. In order to reduce the noise, a median filter is applied on sliding neighbourhoods adapted to the particle reflections size on the image. This filter replaces the central pixel with the median value of the surrounding ones. It has the advantage of preserving the shape and location of edges, but removes the "salt and pepper" like noise induced by seeding particles by reducing and spreading the grey intensity values. The dynamic range of the input image intensities will not be extended, as the median filter does not generate new grey values, Fig. 27 left.

#### **Brightness adjustment**

The nonlinear filtering is followed by an intensity adjustment through a linear transfer function for the grey values of the input and output images, Fig. 27 centre. The result is presented in Fig. 27 right.

Fig. 27. Image processing steps: noise removal (left), intensity adjustment (centre) and nonlinear filtering (right). State of the graylevel histogram before&after each step

#### **Rope diameter detection**

Due to pressure and velocity fields' unsteady variations in the cone in low-charge operating conditions, as well as vapors compressibility, the rope surface exhibits irregularities. The laser sheet, encountering this uneven surface of the vapor-water interface, is reflected diffusely and gives a strongly illuminated area on the image. An antireflection-coated

Since the camera records the reflection in the laser wavelength, the reflections from the seeding particles will be apparent on the image as well. In order to reduce the noise, a median filter is applied on sliding neighbourhoods adapted to the particle reflections size on the image. This filter replaces the central pixel with the median value of the surrounding ones. It has the advantage of preserving the shape and location of edges, but removes the "salt and pepper" like noise induced by seeding particles by reducing and spreading the grey intensity values. The dynamic range of the input image intensities will not be extended,

The nonlinear filtering is followed by an intensity adjustment through a linear transfer function for the grey values of the input and output images, Fig. 27 centre. The result is

Fig. 27. Image processing steps: noise removal (left), intensity adjustment (centre) and nonlinear filtering (right). State of the graylevel histogram before&after each step

Due to pressure and velocity fields' unsteady variations in the cone in low-charge operating conditions, as well as vapors compressibility, the rope surface exhibits irregularities. The laser sheet, encountering this uneven surface of the vapor-water interface, is reflected diffusely and gives a strongly illuminated area on the image. An antireflection-coated

as the median filter does not generate new grey values, Fig. 27 left.

**Noise reduction** 

**Brightness adjustment** 

presented in Fig. 27 right.

**Rope diameter detection** 

optical filter on the camera prevents image blurring, by reflecting part of the incident light scattered by the vapors-core boundary, as well as by residual bubbles and tracer particles in the flow. Nevertheless, the zone where the laser sheet reaches the rope boundary can be identified as the area with concentrated high intensity values. Thus, the best approximation of the real rope diameter, at the intersection with the inclined laser plane, would be the distance between edges along a line inclined at ~30° about the horizontal, see Fig. 28.

The influence of the reflection zone where the laser sheet reaches the rope boundary is overcame by cutting the gray intensity profile along the diameter line found in the first approximation at 94% of the maximum value, and keeping the value on the right as the new starting point for the diameter line, see Fig. 29. The centre of the rope in the measurement plane is considered to be the centre of mass of the final diameter line, see Fig. 29.

At last, a statistical analysis on the position and diameter of the rope in the measurement plane is performed for each phase – see (Iliescu et al. 2003). The coherence of the result is based on dimensions criteria check – Fig. 30. The rate of validated images is 95%.

Fig. 28. Rope diameter detection and validation

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 83

Fig. 31. Image processing steps: edge detection (a), edge dilation (b), edge linking (c), filling connected edges (d), bubble removal from the background (e) and extracted rope contour

For the velocity field calculation the same processing and validation methods like in paragraph 4.1.6. are applied, after the extraction of the rope mask from the image – see Fig.

overlaid on the input graylevel image (f)

**4.1.6 Vector field processing** 

34 and Fig. 35.

Fig. 29. Intensity profile along the approximate diameter

Fig. 30. Lower and upper thresholds for discarding aberrant diameter values

For each image, the intersection of the laser plane with the rope is detected and the values of the rope centre and diameter are transformed into real coordinates, for each validated image. Considering the mean values for each phase, the rope volume is reconstructed by the phase averaging technique.

#### **Binary mask extraction**

The binary image also provides a mask for the velocity fields. The pixel values on areas corresponding to each interrogation area for the velocity vector evaluation are averaged and then converted back to binary values. This digital mask thus obtained will be multiplied with the vector values and the zone covered by the rope is eliminated.

Due to the uneven intensity distribution on the zone occupied by the rope, particular image processing steps should be considered for separating the bright rope area from the dark background pixels.

First, the Canny edge-detection method – see (Canny 1986) is applied on the gray level image resulting from the previous steps. This method is based on the computation of the intensity gradient in both horizontal and vertical directions. Applying two threshold values on it gives a binary map of the contours between contrasting portions of the image, Fig. 31a. The boundaries are than thickened by extrapolation, Fig. 31b. Vertical and horizontal nonlinear filters are applied for linking the edges Fig. 31c. After holes filling, Fig. 31d is obtained. Spurious bubbles in the binary image are eliminated and only the largest foreground area is retained, Fig. 31e. Fig. 32f presents the rope contour overlaid on the image.

Fig. 29. Intensity profile along the approximate diameter

phase averaging technique.

**Binary mask extraction** 

background pixels.

Fig. 30. Lower and upper thresholds for discarding aberrant diameter values

with the vector values and the zone covered by the rope is eliminated.

retained, Fig. 31e. Fig. 32f presents the rope contour overlaid on the image.

For each image, the intersection of the laser plane with the rope is detected and the values of the rope centre and diameter are transformed into real coordinates, for each validated image. Considering the mean values for each phase, the rope volume is reconstructed by the

The binary image also provides a mask for the velocity fields. The pixel values on areas corresponding to each interrogation area for the velocity vector evaluation are averaged and then converted back to binary values. This digital mask thus obtained will be multiplied

Due to the uneven intensity distribution on the zone occupied by the rope, particular image processing steps should be considered for separating the bright rope area from the dark

First, the Canny edge-detection method – see (Canny 1986) is applied on the gray level image resulting from the previous steps. This method is based on the computation of the intensity gradient in both horizontal and vertical directions. Applying two threshold values on it gives a binary map of the contours between contrasting portions of the image, Fig. 31a. The boundaries are than thickened by extrapolation, Fig. 31b. Vertical and horizontal nonlinear filters are applied for linking the edges Fig. 31c. After holes filling, Fig. 31d is obtained. Spurious bubbles in the binary image are eliminated and only the largest foreground area is

Fig. 31. Image processing steps: edge detection (a), edge dilation (b), edge linking (c), filling connected edges (d), bubble removal from the background (e) and extracted rope contour overlaid on the input graylevel image (f)

#### **4.1.6 Vector field processing**

For the velocity field calculation the same processing and validation methods like in paragraph 4.1.6. are applied, after the extraction of the rope mask from the image – see Fig. 34 and Fig. 35.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 85

process is the same as for 2D-PIV in two-phase flow. The image acquisition is triggered by the rope passage in front of a pressure sensor mounted on the cone's wall – see paragraph

The gray level distribution on the raw image histogram, Fig. 33, shows three separate peaks

A series of image processing steps, summarized in Fig. 34, are applied on both cameras' raw

The first objective is to extract the rope shape from the recorded images, and transform it into a binary mask used for outliers' removal from the vector field. The second one is to detect the rope diameter at its intersection with the laser plane within the measurement area limits.

The distortion correction is performed with the calibration transform prior to the image processing sequence. Using a polynomial optical transfer function, the deformed raw image is mapped onto the real measurement position – Fig. 35a. Each pixel on the image depends

Fig. 32. Experimental setup for 3D-PIV measurements in two-phase flow

corresponding to the rope shadow, background light and particles.

Fig. 33. Raw image of the rope and corresponding histogram

now linearly on the real coordinates through a constant scale factor.

images, to obtain the rope contour.

4.1.4.

**4.2.4 Image processing** 

### **4.2 Tree-dimensional PIV in two-phase flow**

### **4.2.1 Principle**

The new method is an adaptation for two-phase flows of the stereoscopic PIV technique. It allows obtaining simultaneously the unsteady 3D velocity field and the rope shape. Using fluorescent particles, which are excited by the laser wavelength, but emit at a higher wavelength, along with high-pass filters mounted on the cameras, prevents reflections in the laser wavelength to reach the CCD chip and allows recording the light scattered by particles. In this way, the strong reflections from the rope boundary, from residual bubbles in the flow or due to the optical interfaces are eliminated.

Furthermore, using backward illumination in a third wavelength renders a darker rope shape on a brighter background with a good contrast. It makes the cavity profile sharp enough for an accurate detection of the rope edges using fewer processing steps for image enhancement. The constraint consists in adjusting the luminosity level of the background, such that the bright reflections from seeding particles to be displayed with a good contrast as well.

### **4.2.2 Measurement equipment**

The laser, cameras and processor used for the 3D-PIV experiment are adapted to the present measurement set-up. Optical filters have been mounted on the cameras' lenses. For these measurements, rhodamine (RhB) particles of 1-10m diameter, receiving 532nm and emitting at 594nm wavelength are used as flow-field tracers.


Table 6. LED-array characteristics

Two panels of LEDs, Osram OS-LM01A-Y –Table 6–, placed in front of each camera behind the cone, insure the backside illumination. Arrays of 22x24 LEDs wired in parallel are mounted on two plates, and connected to the same power supply system and synchronization board. A diffusing screen is placed in front of the LED panels.

#### **4.2.3 Calibration & acquisition parameters**

The distortions due to the cone wall shape, giving an uneven distribution of the scale factor over the image, are integrated in a calibration model. The distortions are diminished by a flat external surface, but the internal geometry of the hydraulic path had to be preserved, see Fig. 32. This conical surface induces distortion near the image edges. For this reason, a polynomial model, parabolic in Z, is found to be suitable for these measurements.

The laser energy distribution, camera opening and exposure, seeding density, time interval between pulses and number of acquisition are carefully adjusted. The synchronization process is the same as for 2D-PIV in two-phase flow. The image acquisition is triggered by the rope passage in front of a pressure sensor mounted on the cone's wall – see paragraph 4.1.4.

Fig. 32. Experimental setup for 3D-PIV measurements in two-phase flow

### **4.2.4 Image processing**

84 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

The new method is an adaptation for two-phase flows of the stereoscopic PIV technique. It allows obtaining simultaneously the unsteady 3D velocity field and the rope shape. Using fluorescent particles, which are excited by the laser wavelength, but emit at a higher wavelength, along with high-pass filters mounted on the cameras, prevents reflections in the laser wavelength to reach the CCD chip and allows recording the light scattered by particles. In this way, the strong reflections from the rope boundary, from residual bubbles

Furthermore, using backward illumination in a third wavelength renders a darker rope shape on a brighter background with a good contrast. It makes the cavity profile sharp enough for an accurate detection of the rope edges using fewer processing steps for image enhancement. The constraint consists in adjusting the luminosity level of the background, such that the bright reflections from seeding particles to be displayed with a good contrast

The laser, cameras and processor used for the 3D-PIV experiment are adapted to the present measurement set-up. Optical filters have been mounted on the cameras' lenses. For these measurements, rhodamine (RhB) particles of 1-10m diameter, receiving 532nm and

> Viewing angle

Two panels of LEDs, Osram OS-LM01A-Y –Table 6–, placed in front of each camera behind the cone, insure the backside illumination. Arrays of 22x24 LEDs wired in parallel are mounted on two plates, and connected to the same power supply system and

The distortions due to the cone wall shape, giving an uneven distribution of the scale factor over the image, are integrated in a calibration model. The distortions are diminished by a flat external surface, but the internal geometry of the hydraulic path had to be preserved, see Fig. 32. This conical surface induces distortion near the image edges. For this reason, a

The laser energy distribution, camera opening and exposure, seeding density, time interval between pulses and number of acquisition are carefully adjusted. The synchronization

587 nm 15 nm 280 mcd 120° 10.5 VDC 300 mA 3.2 W

synchronization board. A diffusing screen is placed in front of the LED panels.

polynomial model, parabolic in Z, is found to be suitable for these measurements.

Operating voltage

Operating current

Power consumption

**4.2 Tree-dimensional PIV in two-phase flow** 

in the flow or due to the optical interfaces are eliminated.

emitting at 594nm wavelength are used as flow-field tracers.

Luminous intensity

**4.2.1 Principle** 

as well.

Nominal wavelength

**4.2.2 Measurement equipment** 

Spectral bandwidth

Table 6. LED-array characteristics

**4.2.3 Calibration & acquisition parameters** 

The gray level distribution on the raw image histogram, Fig. 33, shows three separate peaks corresponding to the rope shadow, background light and particles.

Fig. 33. Raw image of the rope and corresponding histogram

A series of image processing steps, summarized in Fig. 34, are applied on both cameras' raw images, to obtain the rope contour.

The first objective is to extract the rope shape from the recorded images, and transform it into a binary mask used for outliers' removal from the vector field. The second one is to detect the rope diameter at its intersection with the laser plane within the measurement area limits.

The distortion correction is performed with the calibration transform prior to the image processing sequence. Using a polynomial optical transfer function, the deformed raw image is mapped onto the real measurement position – Fig. 35a. Each pixel on the image depends now linearly on the real coordinates through a constant scale factor.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 87

a) b)

c) d)

e) f)

g) h)

i) j)

of particle density on horizontal slices (j)

Fig. 35. Raw image of the rope, after distorsion correction (a); only a section of the active area is used for subsequent image proccessing (b); Image processing stages: noise removal (c) and respective histogram of the filtered image (d); binary image obtained by adaptive thresholding (e); binary mask extraction (f); masking of the rope area (g); detection of particle locations (h); particles selection based on maximum diameter criteria (i); calculation

#### Fig. 34. Image processing flowchart

#### **Image enhancement**

The zone where the rope appears is extracted from the input image by spatial filtering – Fig. 35b. Almost 40% of the input image is rejected and the image processing algorithm will only be applied on this zone. A median filter is then applied on sliding neighborhoods of 8x8 pixels of the image, in order to remove the particles from the background, Fig. 35c, and smooth the corresponding peak at high intensity levels on the histogram - Fig. 35d.

#### **Binary mask extraction**

An adaptive threshold – denoted with a vertical line in Fig. 35d – is applied on the graylevel distribution for separating the rope area from the background pixels. The resulting binary image is presented in Fig. 35e. Lower gray values corresponding to the contour of the rope are converted to black, while brighter intensities are converted to white values.

In the next step, spurious spots are removed from the white background and bright zones corresponding to reflections from the rope boundary facing the camera are removed from the rope foreground, giving a smooth rope area – Fig. 35f. This binary image will be used as mask on the corresponding velocity field obtained by cross-correlation from the same raw image and its pair.

#### **Rope diameter detection**

In the current case, the passage of the rope through the laser plane does not provide reflections on the image like in the case of 2D two-phase PIV, thus another solution for detecting this zone had to be used. Particles are present on the image around the rope, except at the rear of the zone where the compact vapors volume obstructs the laser light passage. Therefore, calculating the particles density on the image leads to the expected solution.

The zone occupied by the rope is subtracted from the image by applying the mask in Fig. 35f to the raw image in Fig. 35c. An adaptive threshold applied on the resulting image, Fig. 35g, gives the positions of seeding particles surrounding the rope profile - Fig. 35h. Furthermore, larger objects are eliminated from the image based on area selection criteria –

The zone where the rope appears is extracted from the input image by spatial filtering – Fig. 35b. Almost 40% of the input image is rejected and the image processing algorithm will only be applied on this zone. A median filter is then applied on sliding neighborhoods of 8x8 pixels of the image, in order to remove the particles from the background, Fig. 35c, and

An adaptive threshold – denoted with a vertical line in Fig. 35d – is applied on the graylevel distribution for separating the rope area from the background pixels. The resulting binary image is presented in Fig. 35e. Lower gray values corresponding to the contour of the

In the next step, spurious spots are removed from the white background and bright zones corresponding to reflections from the rope boundary facing the camera are removed from the rope foreground, giving a smooth rope area – Fig. 35f. This binary image will be used as mask on the corresponding velocity field obtained by cross-correlation from the same raw

In the current case, the passage of the rope through the laser plane does not provide reflections on the image like in the case of 2D two-phase PIV, thus another solution for detecting this zone had to be used. Particles are present on the image around the rope, except at the rear of the zone where the compact vapors volume obstructs the laser light passage. Therefore, calculating the particles density on the image leads to the expected

The zone occupied by the rope is subtracted from the image by applying the mask in Fig. 35f to the raw image in Fig. 35c. An adaptive threshold applied on the resulting image, Fig. 35g, gives the positions of seeding particles surrounding the rope profile - Fig. 35h. Furthermore, larger objects are eliminated from the image based on area selection criteria –

smooth the corresponding peak at high intensity levels on the histogram - Fig. 35d.

rope are converted to black, while brighter intensities are converted to white values.

Fig. 34. Image processing flowchart

**Image enhancement** 

**Binary mask extraction** 

image and its pair.

solution.

**Rope diameter detection** 

Fig. 35. Raw image of the rope, after distorsion correction (a); only a section of the active area is used for subsequent image proccessing (b); Image processing stages: noise removal (c) and respective histogram of the filtered image (d); binary image obtained by adaptive thresholding (e); binary mask extraction (f); masking of the rope area (g); detection of particle locations (h); particles selection based on maximum diameter criteria (i); calculation of particle density on horizontal slices (j)

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 89

For the velocity field calculation, the raw vectors maps are processed by cross-correlation of the two frames from each camera. The raw values are filtered based on range and peak validation criteria. The distortions of position coordinates and particles displacements are corrected with the calibration transform. In order to eliminate the aberrant vectors in the region of the rope or due to the shadow produced by bubbles, the binary mask obtained previously by image processing is applied on the vector maps by multiplication. The

The 2D- and 3D-PIV measurements in two-phase flow described herein were the first experiments of this kind in hydraulic turbomachinery operating under cavitation conditions – see Fig. 37, see (Iliescu et al 2003). They provide data on the rope volume and the surrounding velocity field, acquired simultaneously and in unsteady regime. The spatial position of the rope was also obtained, both in cavitation and cavitation-free conditions. This set of measured geometrical characteristics served to derive an analytical expression (eq. 18-

Fig. 38. Reconstructed vortex centreline for a Francis turbine operating at partial load: single

phase flow (blue) and in cavitation conditions (red)

statistical convergence is achieved at 1200 velocity fields for 3% uncertainty.

eq. 20) for the pseudo-temporal vortex rope pattern – Fig. 38.

**4.2.5 Vectors field processing** 

**4.3 Main results** 

Fig. 35i. The particle density is then calculated on horizontal slices; their distribution is shown in Fig. 35j. An adaptive threshold based on local minima of particle density gives the intersection limit of the rope with the laser sheet – illustrated by a horizontal red line in Fig. 35j.

The boundary of the rope is recovered from black-white transitions in the binary mask in Fig. 35f. The contour of the rope projection onto the camera imaging plane is outlined in red on the raw image in Fig. 36. A linear fit is applied on the median line (blue) in the proximity of the horizontal limit determined previously. The rope diameter at the intersection with the laser sheet is calculated within the rope boundary limits in the direction perpendicular on the linear fit – yellow line in Fig. 36 – and the center is considered at their intersection. The geometrical parameters of the intersection of the rope with the laser plane (diameter and centre position) are detected for each image, and the values are transformed into real coordinates through the scale factor. Considering the mean values for each phase, the rope shape is reconstructed spatially by the phase averaging technique, see (Iliescu et al 2008).

Fig. 36. Rope diameter detection (yellow), normal to the centerline of the vapours core (blue/black), at the intersection with the laser sheet (red)

Fig. 37. Phase averaged vectors field and measured vapors core rope for extreme cavitation factor at partial load

#### **4.2.5 Vectors field processing**

For the velocity field calculation, the raw vectors maps are processed by cross-correlation of the two frames from each camera. The raw values are filtered based on range and peak validation criteria. The distortions of position coordinates and particles displacements are corrected with the calibration transform. In order to eliminate the aberrant vectors in the region of the rope or due to the shadow produced by bubbles, the binary mask obtained previously by image processing is applied on the vector maps by multiplication. The statistical convergence is achieved at 1200 velocity fields for 3% uncertainty.

### **4.3 Main results**

88 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Fig. 35i. The particle density is then calculated on horizontal slices; their distribution is shown in Fig. 35j. An adaptive threshold based on local minima of particle density gives the intersection

The boundary of the rope is recovered from black-white transitions in the binary mask in Fig. 35f. The contour of the rope projection onto the camera imaging plane is outlined in red on the raw image in Fig. 36. A linear fit is applied on the median line (blue) in the proximity of the horizontal limit determined previously. The rope diameter at the intersection with the laser sheet is calculated within the rope boundary limits in the direction perpendicular on the linear fit – yellow line in Fig. 36 – and the center is considered at their intersection. The geometrical parameters of the intersection of the rope with the laser plane (diameter and centre position) are detected for each image, and the values are transformed into real coordinates through the scale factor. Considering the mean values for each phase, the rope shape is reconstructed spatially by the phase averaging technique, see (Iliescu et al 2008).

Fig. 36. Rope diameter detection (yellow), normal to the centerline of the vapours core

Fig. 37. Phase averaged vectors field and measured vapors core rope for extreme cavitation

(blue/black), at the intersection with the laser sheet (red)

factor at partial load

limit of the rope with the laser sheet – illustrated by a horizontal red line in Fig. 35j.

The 2D- and 3D-PIV measurements in two-phase flow described herein were the first experiments of this kind in hydraulic turbomachinery operating under cavitation conditions – see Fig. 37, see (Iliescu et al 2003). They provide data on the rope volume and the surrounding velocity field, acquired simultaneously and in unsteady regime. The spatial position of the rope was also obtained, both in cavitation and cavitation-free conditions. This set of measured geometrical characteristics served to derive an analytical expression (eq. 18 eq. 20) for the pseudo-temporal vortex rope pattern – Fig. 38.

Fig. 38. Reconstructed vortex centreline for a Francis turbine operating at partial load: single phase flow (blue) and in cavitation conditions (red)

$$
\vec{r} = \vec{r} \left( \Theta \right) = x(\Theta)\vec{i} + y(\Theta)\vec{j} + z(\Theta)\vec{k} \tag{17}
$$

$$r = r\_o b^{(0-\theta\_o)/2\pi} \tag{18}$$

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 91

Fig. 40. Vortex position variation for two values versus the vortex phase

completely parameterized – see (Ciocan et al 2009).

the radius.

The unsteady analysis of the results the standard deviation of the vortex centre position representative for the flow stability. The decrease of the standard deviation of the rope centre with the σ value, Fig. 39, shows that the flow becomes unstable when the volume of the rope cavity increases. For σ = 0.380, the location of the rope center has an unsteady spatial variation of ~8% of the local radius of the cone – see Fig. 40. The resonance at this σ value induces a lost of stability of the rope shape and spatial position are illustrated by the increase of standard deviation of the vortex center position – twice, from 2% to 4.1% of

Fitting a linear equation to the vapors-core vortex radius in the measured positions, added to the vortex filament model, allows reconstructing the rope volume in the draft tube – see Fig. 41. In this way, starting with the experimental results, the rope shape and position is

$$\begin{cases} x = r\_o b^{9/2\pi} \cos \theta \\ y = r\_o b^{9/2\pi} \sin \theta \\ z = z\_O - r\_o (b^{9/2\pi} - 1) / \tan \beta \end{cases} \tag{19}$$

The main parameters are given in Table 7, for the investigated operation conditions: cavitation-free flow and maximum vapours core volume.


Table 7. Conical helical vortex model parameters

Fig. 39. Rope centre variations versus the cavitation number

( )2 *<sup>o</sup>*

2

The main parameters are given in Table 7, for the investigated operation conditions:


cos sin

2 2

 

*o o O o*

*x rb y rb z z rb*

Initial radius ro 0.09 0.15 Initial depth zo -0.615 -1.2 Rate of radial growth b 3.2 4

Cone angle β 17° 25.5°

cavitation-free flow and maximum vapours core volume.

Table 7. Conical helical vortex model parameters

Fig. 39. Rope centre variations versus the cavitation number

*rr x iy jz k* (17)

Cavitation-free vortex Vapors-core vortex

*<sup>o</sup> r rb* (18)

(19)

Fig. 40. Vortex position variation for two values versus the vortex phase

The unsteady analysis of the results the standard deviation of the vortex centre position representative for the flow stability. The decrease of the standard deviation of the rope centre with the σ value, Fig. 39, shows that the flow becomes unstable when the volume of the rope cavity increases. For σ = 0.380, the location of the rope center has an unsteady spatial variation of ~8% of the local radius of the cone – see Fig. 40. The resonance at this σ value induces a lost of stability of the rope shape and spatial position are illustrated by the increase of standard deviation of the vortex center position – twice, from 2% to 4.1% of the radius.

Fitting a linear equation to the vapors-core vortex radius in the measured positions, added to the vortex filament model, allows reconstructing the rope volume in the draft tube – see Fig. 41. In this way, starting with the experimental results, the rope shape and position is completely parameterized – see (Ciocan et al 2009).

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 93

Fig. 42. Rope diameter variations versus the vortex phase

Fig. 43. Rope diameter variations versus the value

The PIV measurement system is a valuable asset for flow dynamics investigations in turbomachines. The PIV measurements, either 2D or 3D, give access to steady values of velocity and turbulent kinetic energy, as well as to their periodic fluctuations. With the help of specific data analysis tools, a wide range of phenomena can be analysed, such as vortex detection and pattern tracking, wake propagation and dissipation, rotor-stator interactions.

**5. Conclusions** 

Fig. 41. Analytical representation of the vortex rope

In the cone cross-section, the rope center radius is increasing following the phase evolution: the rope goes closer to the wall with the cone depth – see Fig. 42. This evolution is confirmed by the increasing of the wall pressure fluctuation synchronous with the rope rotation, with the cone depth. The standard deviation corresponding to the rope position is quasi-constant for all phases (depth) in the measurement zone.

The procedure to establish the phase average of the rope diameter, already described, is applied for 7 σ values. The rope diameter is represented versus the σ value, as well as the rope center position in the measurement zone. Associated with these values, the standard deviation of the rope diameter and vortex center position is calculated.

The rope diameter is decreasing with the σ value – Fig. 43. The standard deviation of the rope diameter is related to the rope diameter fluctuations and represents the rope volume variations. The physical significance of this calculation is related to the axial pressure waves that produce a volume variation of the rope, due to the local changing of the pressure distribution. The standard deviations are quasi-constant for all σ values at 2.5% of R, except for the value 0.380, where it increases at 4.3% of R. For σ = 0.380 the rope area reported to the local cone section area has a variation between 0.5% and 1.2%.

In fact, for the 0.380 σ value it was pointed out, by hydro-acoustic simulation, that a pressure source located in the inner part of the draft tube elbow induces a forced excitation. This excitation represents the synchronous part of the vortex rope excitation. An eigen frequency of the hydraulic system is also excited at 2.5 of the runner rotation frequency. The plane waves generated by the pressure source, propagating in all the hydraulic circuit, induce consecutively a decreasing and an expansion of the vapors volume of the rope, which explains the increasing of the rope diameter standard deviation for this σ value. Reported to the phase evolution, the rope volume and its standard deviation remain quasi-constant.

Fig. 42. Rope diameter variations versus the vortex phase

Fig. 43. Rope diameter variations versus the value

#### **5. Conclusions**

92 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

In the cone cross-section, the rope center radius is increasing following the phase evolution: the rope goes closer to the wall with the cone depth – see Fig. 42. This evolution is confirmed by the increasing of the wall pressure fluctuation synchronous with the rope rotation, with the cone depth. The standard deviation corresponding to the rope position is quasi-constant

The procedure to establish the phase average of the rope diameter, already described, is applied for 7 σ values. The rope diameter is represented versus the σ value, as well as the rope center position in the measurement zone. Associated with these values, the standard

The rope diameter is decreasing with the σ value – Fig. 43. The standard deviation of the rope diameter is related to the rope diameter fluctuations and represents the rope volume variations. The physical significance of this calculation is related to the axial pressure waves that produce a volume variation of the rope, due to the local changing of the pressure distribution. The standard deviations are quasi-constant for all σ values at 2.5% of R, except for the value 0.380, where it increases at 4.3% of R. For σ = 0.380 the rope area reported to

In fact, for the 0.380 σ value it was pointed out, by hydro-acoustic simulation, that a pressure source located in the inner part of the draft tube elbow induces a forced excitation. This excitation represents the synchronous part of the vortex rope excitation. An eigen frequency of the hydraulic system is also excited at 2.5 of the runner rotation frequency. The plane waves generated by the pressure source, propagating in all the hydraulic circuit, induce consecutively a decreasing and an expansion of the vapors volume of the rope, which explains the increasing of the rope diameter standard deviation for this σ value. Reported to the phase evolution, the rope volume and its standard deviation remain quasi-constant.

deviation of the rope diameter and vortex center position is calculated.

the local cone section area has a variation between 0.5% and 1.2%.

Fig. 41. Analytical representation of the vortex rope

for all phases (depth) in the measurement zone.

The PIV measurement system is a valuable asset for flow dynamics investigations in turbomachines. The PIV measurements, either 2D or 3D, give access to steady values of velocity and turbulent kinetic energy, as well as to their periodic fluctuations. With the help of specific data analysis tools, a wide range of phenomena can be analysed, such as vortex detection and pattern tracking, wake propagation and dissipation, rotor-stator interactions.

PIV Measurements Applied to Hydraulic Machinery: Cavitating and Cavitation-Free Flows 95

Ciocan G.D., Kueny J-L, Mesquita A.A., (1996) "Steady and unsteady flow pattern between

Ciocan G.D., Mauri S., Arpe J., Kueny J-L., (2001) "Etude du champ instationnaire de vitesse

Gagnon J.M., Ciocan G.D., Deschênes C., Iliescu M.S., (2008) "Experimental investigation of

Guedes A., Kueny J-L., Ciocan G.D., Avellan F., (2002) "Unsteady rotor-stator analysis of a

Houde S., Iliescu M.S., Fraser R., Deschênes C., Lemay S., Ciocan G.D., (2011) "Experimental

Iliescu M.S., Ciocan G.D., Avellan F., (2008) "Two Phase PIV Measurements of a Partial Flow

Iliescu M.S. (2007) "Large scale hydrodynamic phenomena analysis in turbine draft tubes",

Iliescu M.S., Ciocan G.D., Avellan F. (2004): "Experimental Study of the Runner Blade-to-

Iliescu M., Ciocan G.D., Avellan F., (2003) "2 Phase PIV Measurements at the Runner Outlet

Mesquita A.A., Ciocan G.D., Kueny J-L., (1999) "Experimental Analysis of the Flow between

Susan-Resiga R., Ciocan G.D., Anton I., Avellan F., (2006) "Analysis of the Swirling Flow

Tridon S., Barre S., Ciocan G.D., Leroy P., Ségoufin C. (2010), "Experimental Investigation of

Tridon S., Barre S., Ciocan G.D., Tomas L., (2010) "Experimental Analysis of the Swirling

Valencia, Spain, 16-19 September

p.767-780, Lausanne, Suisse, 9-12 September

2202, Volume 130, Issue 2, pp. 146-157

Blanche no. 2, p. 46-59

October 27-31

24-29

EPFL Thesis n° 3775

Sweden, June 29 – July 2

Paper Contest

no. 4, p. 580-588

Environmental Science 12

177-189

stay and guide vanes in a pump-turbine" - Proceedings of the XVIII International Symposium on Hydraulic Machinery and Cavitation, IAHR, vol. 1, p. 381-390,

en sortie de roue de turbine - Etude expérimentale et numérique" - La Houille

runner outlet flows in an axial turbine using LDV and stereoscopic PIV", 24th IAHR Symposium on Hydraulic Machinery and Systems, Foz do Iguassu, Brazil,

hydraulic pump-turbine – CFD and experimental approach" – Proceedings of the XXI International Symposium on Hydraulic Machinery and Cavitation, IAHR,

and Numerical Analysis of the Cavitating Part Load Vortex Dynamics of Low-Head Hydraulic Turbines" Proceedings of ASME-JSME-KSME Joint Fluids Engineering Conference, paper AJK2011-FED, Hamamatsu, Shizuoka, Japan, July

Rate Vortex Rope in a Francis Turbine" – Journal of Fluids Engineering, ISSN: 0098-

Blade Shear Flow Turbulent Mixing in the Cone of Francis Turbine Scale Model", 22nd IAHR Symposium on Hydraulic Machinery and Systems, Stockholm,

in a Francis Turbine" The 2003 Joint US ASME-European Fluids Engineering Summer Conference, Honolulu, Hawaii, USA, July 6-10, – Award in the Student

Stay and Guide Vanes of a Pump-Turbine in Pumping Mode" – Journal of the Brazilian Society of Mechanical Sciences and Engineering, ISSN: 1678-5878, vol. 21,

Downstream a Francis Turbine Runner" – Journal of Fluids Engineering, vol. 128, p.

Draft Tube Flow Instability", Institute of Physics (IoP) Conf. Series: Earth and

Flow in a Francis Turbine Draft Tube: Focus on Radial Velocity Component

However, results with the required accuracy are only possible by rigorous developments. The main parameters influencing the measurement accuracy have been identified and their impact on the final result analyzed. A careful choice of the most favorable optical configuration, flawless calibration setup and adequate adjustment of the acquisition parameters are the ingredients of a successful PIV experiment. Furthermore, for investigations in hydraulic turbomachinery the original geometry of the hydraulic profile should be preserved. This requirement adds a degree of complexity due to strong optical deformations and possible local discontinuities in the optical interface, which are difficult to compensate.

Through a specific implementation of the standard PIV technique in two-phase flows, it was possible to evaluate the unsteady flowfield generated by the partial load vortex rope which develops downstream a turbine runner in cavitation conditions, together with the vortex core topology. Specific developments were necessary for the PIV application in cavitation conditions, both in the 2D and the 3D configurations. New image processing tools have been created, allowing simultaneous measurements of the velocity field and related vapours core characteristics. Based on the experimental results, an analytical description of the rope has been determined. The impact of resonance on the rope behaviour was analysed as well.

These measurements represent a valuable database used for theoretical developments as well – see (Susan-Resiga et al. 2006), (Ciocan et al. 2008) the numerical simulations as boundary conditions or validation data – see (Guedes et al. 2002) and (Ciocan et al. 2007).

#### **6. References**


However, results with the required accuracy are only possible by rigorous developments. The main parameters influencing the measurement accuracy have been identified and their impact on the final result analyzed. A careful choice of the most favorable optical configuration, flawless calibration setup and adequate adjustment of the acquisition parameters are the ingredients of a successful PIV experiment. Furthermore, for investigations in hydraulic turbomachinery the original geometry of the hydraulic profile should be preserved. This requirement adds a degree of complexity due to strong optical deformations and possible local

Through a specific implementation of the standard PIV technique in two-phase flows, it was possible to evaluate the unsteady flowfield generated by the partial load vortex rope which develops downstream a turbine runner in cavitation conditions, together with the vortex core topology. Specific developments were necessary for the PIV application in cavitation conditions, both in the 2D and the 3D configurations. New image processing tools have been created, allowing simultaneous measurements of the velocity field and related vapours core characteristics. Based on the experimental results, an analytical description of the rope has been determined. The impact of resonance on the rope behaviour was analysed as well.

These measurements represent a valuable database used for theoretical developments as well – see (Susan-Resiga et al. 2006), (Ciocan et al. 2008) the numerical simulations as boundary conditions or validation data – see (Guedes et al. 2002) and (Ciocan et al. 2007).

Beaulieu S., Deschenes C., Iliescu M., Ciocan G.D., (2009) "Study of the Flow Field Through

Canny J., (1986) "A Computational Approach to Edge Detection," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-8, No. 6, pp. 679-698. Ciocan G.D., (1998) "Contribution à l'analyse des écoulements 3D complexes en turbomachines", PhD Thesis, Institut National Polytechnique de Grenoble Ciocan G.D., Avellan F., (2004) "Flow Investigations in a Francis Draft Tube: Advanced

Ciocan G.D., Iliescu M.S., (2009) "3D PIV Measurements in two phase flow", 4th

Ciocan G.D., Iliescu M.S. (2008) "3D PIV Measurements in Two Phase Flow and Rope

Ciocan G.D., Iliescu M.S., Vu T., Nnennemann B., Avellan F., (2007) "Experimental Study

Ciocan G.D., Kueny J-L., (2006) "Experimental Analysis of the Rotor-Stator Interaction in a

Machinery and Systems, Brno, Czech Republic, October 14-16

Hydropower Engineers, Bucarest, Roumanie, Mai 28-29

Fluid Engineering, ISSN: 0098-2202, vol. 129, p. 146-158

Systems, Foz do Iguassu, Brazil, October 27-31

Yokohama, Japan, October 17 – 21

Symposium on Green Energy, May, 14-16 – invitated paper

the Runner of a Propeller Turbine using Stereoscopic PIV" 3rd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic

Experimental Methods" Proceedings of the 3rd Conference of Romanian

International Conference on Energy and Environment, The 1st International

Parametrical Modeling", 24th IAHR Symposium on Hydraulic Machinery and

and Numerical Simulation of the Flindt Draft Tube Rotating Vortex" - Journal of

Pump-Turbine", 23rd IAHR Symposium on Hydraulic Machinery and Systems,

discontinuities in the optical interface, which are difficult to compensate.

**6. References** 


**4** 

*1Italy 2France* 

**Post-Processing Methods of** 

**PIV Instantaneous Flow Fields for** 

*2Laboratoire de Mécanique de LILLE (UMR CNRS 8107),* 

*Arts et Métiers ParisTech, École Centrale de Lille,* 

**Unsteady Flows in Turbomachines** 

G. Cavazzini1, A. Dazin2, G. Pavesi1, P. Dupont2 and G. Bois2 *1Department of Mechanical Engineering, University of Padova, Padova,* 

Among the experimental techniques, the particle image velocimetry (PIV) is undoubtedly one of the most attractive modern methods to investigate the fluid flow in a non-intrusive way and allows to obtain instantaneous fluid flow fields by correlating at least two sequential exposures. This technique was successfully applied in several fields in order to study high complex three-dimensional flow velocity fields and to provide a significant

However, on one side, experimental limits and possible perturbing phenomena could negatively affect the PIV experimental accuracy, altering the real physics of the studied fluid flow field. On the other side, the huge amount of data obtainable by means of the PIV technique requires properly post-processing tools to be exploited in an in-depth study of the

To avoid misinterpretation of the phenomena, complex cleaning techniques were developed and applied at the different steps of the PIV processing, starting from the acquired images (background subtraction, mask application, etc.) so as to increase the signal to noise ratio, and finishing to the instantaneous flow fields by means of statistical methods applied in order to identify residual spurious vectors [Raffel et al., 2002]. Even though all these methods allows to obtain a good filtering of the instantaneous flow fields, however they are not able to completely eliminate all the outliers in the results since the removal criteria are always dependent on the choice of a threshold value [Heinz et al., 2004; Westerweeel, 1994;

To overcome this problem, the most common approach is to average the instantaneous PIV flow fields so as to improve the quality of the resulting flow field reconstruction and to more

Several averaging methods were proposed and applied in literature. However their effectiveness in reducing the spurious vector number is strictly connected with the flow field

easily identify the flow field characteristics in the investigated area.

experimental data base for the validation of combined numerical analysis models.

**1. Introduction** 

fluid-dynamical phenomena.

Westerweel and Scarano, 2005].

Determination" - European Journal of Mechanics – B/Fluids, ISSN: 0997-7546, vol. 25, Issue 4, pp. 321-335

Tridon S., Ciocan G.D., Barre S., Tomas L., (2008) "3D Time-resolved PIV Measurements in a Francis Draft Tube Cone", 24th IAHR Symposium on Hydraulic Machinery and Systems, Foz do Iguassu, Brazil, October 27-31

## **Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines**

G. Cavazzini1, A. Dazin2, G. Pavesi1, P. Dupont2 and G. Bois2 *1Department of Mechanical Engineering, University of Padova, Padova, 2Laboratoire de Mécanique de LILLE (UMR CNRS 8107), Arts et Métiers ParisTech, École Centrale de Lille, 1Italy 2France* 

### **1. Introduction**

96 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Tridon S., Ciocan G.D., Barre S., Tomas L., (2008) "3D Time-resolved PIV Measurements in a

25, Issue 4, pp. 321-335

Systems, Foz do Iguassu, Brazil, October 27-31

Determination" - European Journal of Mechanics – B/Fluids, ISSN: 0997-7546, vol.

Francis Draft Tube Cone", 24th IAHR Symposium on Hydraulic Machinery and

Among the experimental techniques, the particle image velocimetry (PIV) is undoubtedly one of the most attractive modern methods to investigate the fluid flow in a non-intrusive way and allows to obtain instantaneous fluid flow fields by correlating at least two sequential exposures. This technique was successfully applied in several fields in order to study high complex three-dimensional flow velocity fields and to provide a significant experimental data base for the validation of combined numerical analysis models.

However, on one side, experimental limits and possible perturbing phenomena could negatively affect the PIV experimental accuracy, altering the real physics of the studied fluid flow field. On the other side, the huge amount of data obtainable by means of the PIV technique requires properly post-processing tools to be exploited in an in-depth study of the fluid-dynamical phenomena.

To avoid misinterpretation of the phenomena, complex cleaning techniques were developed and applied at the different steps of the PIV processing, starting from the acquired images (background subtraction, mask application, etc.) so as to increase the signal to noise ratio, and finishing to the instantaneous flow fields by means of statistical methods applied in order to identify residual spurious vectors [Raffel et al., 2002]. Even though all these methods allows to obtain a good filtering of the instantaneous flow fields, however they are not able to completely eliminate all the outliers in the results since the removal criteria are always dependent on the choice of a threshold value [Heinz et al., 2004; Westerweeel, 1994; Westerweel and Scarano, 2005].

To overcome this problem, the most common approach is to average the instantaneous PIV flow fields so as to improve the quality of the resulting flow field reconstruction and to more easily identify the flow field characteristics in the investigated area.

Several averaging methods were proposed and applied in literature. However their effectiveness in reducing the spurious vector number is strictly connected with the flow field

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 99

The experimental results, on which the validation procedure was tested, were obtained in a 2D/2C PIV measurement campaign carried out on one diffuser blade passage of a

Fig. 1. Schematic representation of the centrifugal pump. The dotted line indicates the

All the details about the test rig and the measurement devices, being outside the interest of this work, are not here reported, but can be found in previous studies [Wuibaut et al., 2001-

As regards the images acquisition and processing, two single exposure frames were taken each two complete revolutions of the impeller and 400 instantaneous flow fields were determined for various operating conditions at different heights (Fig. 2). A home-made software was used to treat and process the images so as to increase the signal to noise ratio (background subtraction, mask application, etc.) and a detailed cleaning procedure was

Since the turbulent phenomena under investigation were expected to be periodically associated with the impeller passage frequency, a phase-averaging technique based on this

The first parameter to be considered to verify the meaningfulness of an averaged flow field is undoubtedly the number of acquired images, whose choice is generally affected by two

applied to the instantaneous flow field to remove possible spurious vectors.

frequency was applied the instantaneous flow fields.

**2. Validation method of PIV results** 

investigated diffuser blade passage.

**2.1 Convergence history** 

2002].

centrifugal pump (fig. 1).

characteristics, the experimental set-up and the acquisition characteristics of the PIV instrumentation.

The most simple but less accurate averaging procedure is undoubtedly the classical time average of a suitable number of instantaneous velocity fields, whose effectiveness is greatly affected by the quality of the starting velocity fields. To overcome the limits of this classical method still maintaining a similar approach, Meinhart et al. (2000) proposed to determine the time average of the instantaneous correlation functions so as to determine with greater precision the correlation peak and hence the average velocity. Even though this method allows to increase the quality of the resulting averaged flow field, however it is not able to overcome the essential limit of the time-averaging methods, that is their inapplicability to unsteady flow fields and in particular to fluid-dynamical structures having a formation rate different from the framing rate of the camera. In addition to this, the method loses all the information about the evolution in time of the flow field, allowing to obtain only the averaged one.

To overcome these limits of the time-averaging methods and in particular their dependence from the framing rate of the camera, several phase-averaging methods were developed [Geveci et al. 2003; Perrin et al. 2007; Raffel et al. 1995, 1996; Schram and Riethmuller, 2001- 2002; Ullum et al. 1997; Vogt et al. 1996; Yao and Pashal 1994]. These methods reorder and average the instantaneous flow fields on the basis of a proper phase, characterizing the development of the investigated phenomena so as to obtain a phase-averaged time series. These approaches, even though partially overcome the limits of the time-averaging methods, do not represent an universal solution to the problem of the data validation, since they require the characteristic frequencies of the phenomena to be known beforehand or to be determinable by combination with further experimental measurements (for example, pressure signals post-processed by spectral analysis). Moreover, they fail in case of notperiodical or frequency-combined structures, developing in the flow field.

In the first part of the chapter, a validation method of PIV results was proposed to critically analyse the quality and the meaningfulness of the experimental results in a PIV analysis on unsteady turbulent flow fields, commonly developing in turbomachines. The procedure was tested on the results of a classical phase-averaging method and was subdivided into three main steps: a convergence analysis to verify the fairness of the number of acquired images; an analysis of the probability density distribution to verify the repeatability of the velocity data; an evaluation of the maxima errors associated with the velocity averages to quantitatively analyse their trustworthiness. The procedure allowed to statistically verify the meaningfulness of the average flow field in unsteady flow conditions and to identify possible zones characterized by a low accuracy of the averaging method results.

In the second part of the chapter, a particular averaging method of PIV velocity fields was proposed to experimentally capture and visualize the unsteady flow field associated with an instability developing in a turbomachine with a known movement velocity. According to this method, the PIV flow fields was properly spatially moved according to its development velocity and was averaged on the basis of their new location. This procedure allowed to combine and average the flow fields in a frame moving with the instability so as to obtain a global visualization of the instability characteristics.

### **2. Validation method of PIV results**

98 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

characteristics, the experimental set-up and the acquisition characteristics of the PIV

The most simple but less accurate averaging procedure is undoubtedly the classical time average of a suitable number of instantaneous velocity fields, whose effectiveness is greatly affected by the quality of the starting velocity fields. To overcome the limits of this classical method still maintaining a similar approach, Meinhart et al. (2000) proposed to determine the time average of the instantaneous correlation functions so as to determine with greater precision the correlation peak and hence the average velocity. Even though this method allows to increase the quality of the resulting averaged flow field, however it is not able to overcome the essential limit of the time-averaging methods, that is their inapplicability to unsteady flow fields and in particular to fluid-dynamical structures having a formation rate different from the framing rate of the camera. In addition to this, the method loses all the information about the evolution in time of the flow field, allowing to obtain only the

To overcome these limits of the time-averaging methods and in particular their dependence from the framing rate of the camera, several phase-averaging methods were developed [Geveci et al. 2003; Perrin et al. 2007; Raffel et al. 1995, 1996; Schram and Riethmuller, 2001- 2002; Ullum et al. 1997; Vogt et al. 1996; Yao and Pashal 1994]. These methods reorder and average the instantaneous flow fields on the basis of a proper phase, characterizing the development of the investigated phenomena so as to obtain a phase-averaged time series. These approaches, even though partially overcome the limits of the time-averaging methods, do not represent an universal solution to the problem of the data validation, since they require the characteristic frequencies of the phenomena to be known beforehand or to be determinable by combination with further experimental measurements (for example, pressure signals post-processed by spectral analysis). Moreover, they fail in case of not-

In the first part of the chapter, a validation method of PIV results was proposed to critically analyse the quality and the meaningfulness of the experimental results in a PIV analysis on unsteady turbulent flow fields, commonly developing in turbomachines. The procedure was tested on the results of a classical phase-averaging method and was subdivided into three main steps: a convergence analysis to verify the fairness of the number of acquired images; an analysis of the probability density distribution to verify the repeatability of the velocity data; an evaluation of the maxima errors associated with the velocity averages to quantitatively analyse their trustworthiness. The procedure allowed to statistically verify the meaningfulness of the average flow field in unsteady flow conditions and to identify

In the second part of the chapter, a particular averaging method of PIV velocity fields was proposed to experimentally capture and visualize the unsteady flow field associated with an instability developing in a turbomachine with a known movement velocity. According to this method, the PIV flow fields was properly spatially moved according to its development velocity and was averaged on the basis of their new location. This procedure allowed to combine and average the flow fields in a frame moving with the instability so as to obtain a

periodical or frequency-combined structures, developing in the flow field.

possible zones characterized by a low accuracy of the averaging method results.

global visualization of the instability characteristics.

instrumentation.

averaged one.

The experimental results, on which the validation procedure was tested, were obtained in a 2D/2C PIV measurement campaign carried out on one diffuser blade passage of a centrifugal pump (fig. 1).

Fig. 1. Schematic representation of the centrifugal pump. The dotted line indicates the investigated diffuser blade passage.

All the details about the test rig and the measurement devices, being outside the interest of this work, are not here reported, but can be found in previous studies [Wuibaut et al., 2001- 2002].

As regards the images acquisition and processing, two single exposure frames were taken each two complete revolutions of the impeller and 400 instantaneous flow fields were determined for various operating conditions at different heights (Fig. 2). A home-made software was used to treat and process the images so as to increase the signal to noise ratio (background subtraction, mask application, etc.) and a detailed cleaning procedure was applied to the instantaneous flow field to remove possible spurious vectors.

Since the turbulent phenomena under investigation were expected to be periodically associated with the impeller passage frequency, a phase-averaging technique based on this frequency was applied the instantaneous flow fields.

#### **2.1 Convergence history**

The first parameter to be considered to verify the meaningfulness of an averaged flow field is undoubtedly the number of acquired images, whose choice is generally affected by two

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 101

Fig. 3. Convergence history in a point located at the entrance of the diffuser passage at mid-

Fig. 4. Convergence history in a point located in the middle of the diffuser passage at mid-

span.

span near the blade pressure side.

conflicting aims. On one side, the meaningfulness of the averaged flow field that is favoured by a great number of acquired images; on the other side, the reduction of the acquisition time and of the required data storage capacity, increasing with the images number.

Fig. 2. Seeding of the blade passage as seen by PIV cameras with an overlapping (black parts are the walls of the diffuser passage).

So, to determine a suitable number of images to be acquired, a convergence analysis, similar to that suggested by Wernert and Favier (1999), has to be applied.

This analysis studies the evolution in time of the average *C x <sup>N</sup>* (,) *y* and of standard deviation N(x,y) of the absolute velocity C(x,y) over an increasing number of flow fields:

$$\overline{\mathbb{C}\_{N}}(\mathbf{x}, y) = \frac{1}{N} \sum\_{i=0}^{N} \mathbb{C}(\mathbf{x}, y, t\_0 + i\Delta t) \tag{1}$$

$$\varphi\_N(\mathbf{x}, y) = \sqrt{\frac{1}{N - 1} \sum\_{i=1}^{N} \left( \overline{\mathbf{C}\_i}(\mathbf{x}, y) - \overline{\mathbf{C}\_{N \max}}(\mathbf{x}, y) \right)^2} \tag{2}$$

where N is the progressive number of flow fields (N=1,…Nmax), Nmax is the total number of determined flow fields, t is the sampling period, t0 is the initial instant, C(x,y,t0+it) is the absolute velocity at the coordinates (x, y) of the flow field (i+1), (,) *C xy <sup>i</sup>* is the average of the absolute velocity determined over 'i' flow fields at the coordinates (x, y) and max *C xy <sup>N</sup>* (,) is the average of the absolute velocity over the total number of acquired flow fields at the coordinates (x, y).

The analysis of the evolution in time of the average velocity and of its standard deviation allows to verify the existence of a minimum number of flow fields to be averaged so as to obtain a meaningful averaged flow field. For example, the convergence history of fig. 3 is characterized by an asymptotic behaviour of the average and standard deviation with a asymptotic value reached after about 300 flow fields. This number represents the minimum number of flow fields to be determined in order to obtain a meaningful result. A greater number would not change the resulting average velocity and would not increase its meaningfulness.

A different behaviour characterized the convergence history of fig. 4, where both the average velocity and its standard deviation are not clearly stabilized after 400 flow fields. The velocity tends to zero and the standard deviation is of the order of the average velocity,

conflicting aims. On one side, the meaningfulness of the averaged flow field that is favoured by a great number of acquired images; on the other side, the reduction of the acquisition

Fig. 2. Seeding of the blade passage as seen by PIV cameras with an overlapping (black parts

So, to determine a suitable number of images to be acquired, a convergence analysis, similar

This analysis studies the evolution in time of the average *C x <sup>N</sup>* (,) *y* and of standard deviation

0 <sup>1</sup> (,) (,, ) *N*

where N is the progressive number of flow fields (N=1,…Nmax), Nmax is the total number of determined flow fields, t is the sampling period, t0 is the initial instant, C(x,y,t0+it) is the absolute velocity at the coordinates (x, y) of the flow field (i+1), (,) *C xy <sup>i</sup>* is the average of the absolute velocity determined over 'i' flow fields at the coordinates (x, y) and max *C xy <sup>N</sup>* (,) is the average of the absolute velocity over the total number of acquired flow fields at the

The analysis of the evolution in time of the average velocity and of its standard deviation allows to verify the existence of a minimum number of flow fields to be averaged so as to obtain a meaningful averaged flow field. For example, the convergence history of fig. 3 is characterized by an asymptotic behaviour of the average and standard deviation with a asymptotic value reached after about 300 flow fields. This number represents the minimum number of flow fields to be determined in order to obtain a meaningful result. A greater number would not change the resulting average velocity and would not increase its

A different behaviour characterized the convergence history of fig. 4, where both the average velocity and its standard deviation are not clearly stabilized after 400 flow fields. The velocity tends to zero and the standard deviation is of the order of the average velocity,

*i C xy Cxyt i t <sup>N</sup>*

1 <sup>1</sup> (,) (,) (,) <sup>1</sup> *N N i N i x y C xy C xy <sup>N</sup>*

0

max

(1)

2

(2)

N(x,y) of the absolute velocity C(x,y) over an increasing number of flow fields:

to that suggested by Wernert and Favier (1999), has to be applied.

*N*

coordinates (x, y).

meaningfulness.

are the walls of the diffuser passage).

time and of the required data storage capacity, increasing with the images number.

Fig. 3. Convergence history in a point located at the entrance of the diffuser passage at midspan.

Fig. 4. Convergence history in a point located in the middle of the diffuser passage at midspan near the blade pressure side.

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 103

Fig. 5. Examples of probability density distributions of velocity values

highlighting a great perturbation of the instantaneous velocity values around the average one.

Before indistinctly increasing the number of images to be acquired, a critical analysis of the convergence history is necessary so as to consider the reasons of the non-convergent trend. The development of turbulence flows or unsteady structures in the zone of the reference point and/or experimental problems such as laser reflections or seeding problems should be considered. In fig. 4, as the reference point is located near the pressure side of the pump diffuser blade, laser reflection problems as well as spurious vectors owing to the boundarylayer development could not be excluded. The trend to zero of the progressive averaged velocity and the great values of the standard deviation support this hypothesis. In this case, the acquisition of a higher number of images would not probably guarantee the achievement of a meaningful averaged velocity value.

Hence, the convergence analysis appears to provide useful information about the proper choice of the number of images to be acquired, but allows only preliminary hypotheses on the quality of the results.

### **2.2 Probability density distribution**

A phase-averaging method is based on the hypothesis that the experimental values to be averaged are repeated measures of the same experimental quantity. However, the repeatability of the experimental measurements in an investigated area could be invalidated by the possible development of non-periodical fluid-dynamical phenomena and by possible experimental problems. The lack of this repeatability, negatively affecting the accuracy of the phase-averaging method, is highlighted by a non-Gaussian probability density distribution of the experimental values. Therefore, the second step of the validation procedure is the analysis of the probability density distribution of the determined velocity values.

Since the aim of the analysis was to verify the Gaussianity of this distribution, no hypothesis on its form can be done. Hence, the probability density function has to be estimated using non-parametric kernel smoothing methods, with no hypothesis on the original distribution of the data [Bowman and Azzalini, 1997].

Figure 5 reports three examples of possible probability density distributions of velocity values, translated to have zero mean value. In fig. 5a, the classical symmetric bell-shape of the Gaussian distribution testifies the repeatability of the corresponding experimental measures. Moreover, the great values of the probability density function demonstrates the meaningfulness of the determined average velocity. In contrast, fig. 5b shows an asymmetric distribution of the data with multiple maxima and a wide dispersion of the values. In this case, the velocity average, corresponding to the abscissa μ = 0, cannot be considered as meaningful, even if a higher number of images will be acquired, as the repeatability of the measures is not guaranteed.

The analysis of the probability density distribution could also allow to identify possible experimental problems. Indented bell-shaped distribution, as that reported in fig. 5c, clearly indicates the presence of peak-locking problems in the images acquisition. The peak-locking does not affect the mean velocity flow field but only its fluctuating part [Christensen, 2004]

highlighting a great perturbation of the instantaneous velocity values around the average

Before indistinctly increasing the number of images to be acquired, a critical analysis of the convergence history is necessary so as to consider the reasons of the non-convergent trend. The development of turbulence flows or unsteady structures in the zone of the reference point and/or experimental problems such as laser reflections or seeding problems should be considered. In fig. 4, as the reference point is located near the pressure side of the pump diffuser blade, laser reflection problems as well as spurious vectors owing to the boundarylayer development could not be excluded. The trend to zero of the progressive averaged velocity and the great values of the standard deviation support this hypothesis. In this case, the acquisition of a higher number of images would not probably guarantee the

Hence, the convergence analysis appears to provide useful information about the proper choice of the number of images to be acquired, but allows only preliminary hypotheses on

A phase-averaging method is based on the hypothesis that the experimental values to be averaged are repeated measures of the same experimental quantity. However, the repeatability of the experimental measurements in an investigated area could be invalidated by the possible development of non-periodical fluid-dynamical phenomena and by possible experimental problems. The lack of this repeatability, negatively affecting the accuracy of the phase-averaging method, is highlighted by a non-Gaussian probability density distribution of the experimental values. Therefore, the second step of the validation procedure is the analysis of the probability density distribution of the determined velocity

Since the aim of the analysis was to verify the Gaussianity of this distribution, no hypothesis on its form can be done. Hence, the probability density function has to be estimated using non-parametric kernel smoothing methods, with no hypothesis on the original distribution

Figure 5 reports three examples of possible probability density distributions of velocity values, translated to have zero mean value. In fig. 5a, the classical symmetric bell-shape of the Gaussian distribution testifies the repeatability of the corresponding experimental measures. Moreover, the great values of the probability density function demonstrates the meaningfulness of the determined average velocity. In contrast, fig. 5b shows an asymmetric distribution of the data with multiple maxima and a wide dispersion of the values. In this case, the velocity average, corresponding to the abscissa μ = 0, cannot be considered as meaningful, even if a higher number of images will be acquired, as the repeatability of the

The analysis of the probability density distribution could also allow to identify possible experimental problems. Indented bell-shaped distribution, as that reported in fig. 5c, clearly indicates the presence of peak-locking problems in the images acquisition. The peak-locking does not affect the mean velocity flow field but only its fluctuating part [Christensen, 2004]

achievement of a meaningful averaged velocity value.

the quality of the results.

values.

**2.2 Probability density distribution** 

of the data [Bowman and Azzalini, 1997].

measures is not guaranteed.

one.

Fig. 5. Examples of probability density distributions of velocity values

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 105

Once verified its trustworthiness, the goodness-of-fit test has to be applied to each point of the investigated area so as to obtain a global visualization of the non-Gaussian critical zones of the flow field. Figure 6 reports an example of the Anderson-Darling application to the velocity flow fields determined in the diffuser passage. Cores of non-Gaussian distribution can be clearly identified at the entrance of the diffuser passage, near the blade walls and also in the mean flow (light grey circles in fig. 6). Even though it is always quite difficult to discriminate between fluid-dynamical and experimental problems, these results allow some preliminary hypotheses about the origin of these critical areas. At the diffuser entrance, the position of the impeller blade close to diffuser blade leading edge lets suppose the development of non-periodical turbulent phenomena coming from the impeller discharge,

such as impeller blade wakes and/or rotor-stator interaction phenomena.

Fig. 6. Example of the results of the Anderson-Darling test applied to the pump diffuser blade passage (the black line represents the blade profile; the dark grey line represents the limit of the mask applied to the grid and the light grey circles are marks to identify the non-

The non-Gaussian cores near the blade sides could be due to possible laser reflections problems or seeding problems, whereas laser reflections can be excluded for the cores in the mean flow because of their distance from the walls. These cores could be probably due to non-periodical phenomena proceeding in the passage, but this hypothesis has to be verified

The validation procedure of the experimental results is completed by a measure of the reliability of the averaged flow field, that is obtained by estimation of the confidence interval of the determined averaged velocities, that depends on the effective distribution of

Gaussian zones in the figure)

the experimental data.

by numerical analysis of the flow field.

**2.3 Confidence interval of the measured values** 

and is attributable to both the choice of the sub-pixel estimator and the under-resolved optical sampling of the particle images. So, even though this problem does not affect the meaningfulness of the average velocity value, its identification during a preliminary study could allow to correct the test rig set-up and to increase the quality of the instantaneous flow fields.

Even though the subjective visual analysis of the probability density distributions allowed a preliminary analysis of the measurement repeatability and of the presence of possible experimental problems, to effectively verify the normality of the velocity data distributions in the investigated area, it is necessary to apply a goodness-of-fit test. This test allows to verify the acceptance of the so-called 'null' hypothesis (i.e. if the data follow a specific theoretical distribution) or of the alternative hypothesis (i.e. if the data do not follow the specified distribution). The null hypothesis is rejected with a confidence level , if the test statistic is greater than a critical value fixed at that confidence level. The greater the difference between the statistic and the critical value is, the greater the probability that the data do not follow the specified distribution.

In literature, several tests with different characteristics and powers are available (Pearson χ2, Anderson–Darling, Kolmogorov, Cramer–von Mises–Smirnov, and so on). The hypotheses of these tests can be divided into simple ones, when the parameters of the theoretical distribution to be checked are known, and complex ones, when the parameters of the theoretical distribution are determined using the same data sample to be tested. Complex hypotheses are typical of a PIV experimental analysis since the average and the standard deviation of the theoretical distribution are generally unknown and the average *C xy <sup>N</sup>* (,) and the standard deviation N(x,y) of the sample are used as distribution parameters.

Among the available goodness-of-fit tests, the choice of the most proper one depends on the sampling conditions of the data. Lemeshko et al. (2007) compared the power of several goodness-of-fit tests by means of statistical modelling methods and based the choice on the desired confidence level and on the range of number of samples. For example, for 400 velocity values, as those of the experimental analysis considered as test case, and for a confidence level of 0.05 (5 per cent), the suggested goodness-of-fit test is the Anderson– Darling test.

However, the sampling conditions of an experimental analysis cannot be generalized in ranges and hence the trustworthiness of a goodness-of-fit test must be properly verified with a dedicated simulation. To do this, the test must be applied several times on random Gaussian samples having the same number of values of the experimental sample. The number of test applications has to be great enough to avoid a dependence of the simulation results on it. As this test is not time-expensive, the choice of a very high number of applications, such as 100 000, guarantees its negligible influence on the simulation results.

The trustworthiness of the goodness-of-fit test is verified once the error percentage of its application on Gaussian samples is lower or at most equal to the desired confidence level. In the example considered earlier, the Anderson–Darling test was applied 100 000 times on random Gaussian samples of 400 data, giving an error value of 5 per cent, which is not greater than the desired confidence level. This result demonstrates the applicability of the Anderson–Darling test to the experimental analysis sampling conditions.

and is attributable to both the choice of the sub-pixel estimator and the under-resolved optical sampling of the particle images. So, even though this problem does not affect the meaningfulness of the average velocity value, its identification during a preliminary study could allow to correct the test rig set-up and to increase the quality of the instantaneous flow

Even though the subjective visual analysis of the probability density distributions allowed a preliminary analysis of the measurement repeatability and of the presence of possible experimental problems, to effectively verify the normality of the velocity data distributions in the investigated area, it is necessary to apply a goodness-of-fit test. This test allows to verify the acceptance of the so-called 'null' hypothesis (i.e. if the data follow a specific theoretical distribution) or of the alternative hypothesis (i.e. if the data do not follow the specified distribution). The null hypothesis is rejected with a confidence level , if the test statistic is greater than a critical value fixed at that confidence level. The greater the difference between the statistic and the critical value is, the greater the probability that the

In literature, several tests with different characteristics and powers are available (Pearson χ2, Anderson–Darling, Kolmogorov, Cramer–von Mises–Smirnov, and so on). The hypotheses of these tests can be divided into simple ones, when the parameters of the theoretical distribution to be checked are known, and complex ones, when the parameters of the theoretical distribution are determined using the same data sample to be tested. Complex hypotheses are typical of a PIV experimental analysis since the average and the standard deviation of the theoretical distribution are generally unknown and the average *C xy <sup>N</sup>* (,) and the standard deviation N(x,y) of the sample are used as distribution parameters.

Among the available goodness-of-fit tests, the choice of the most proper one depends on the sampling conditions of the data. Lemeshko et al. (2007) compared the power of several goodness-of-fit tests by means of statistical modelling methods and based the choice on the desired confidence level and on the range of number of samples. For example, for 400 velocity values, as those of the experimental analysis considered as test case, and for a confidence level of 0.05 (5 per cent), the suggested goodness-of-fit test is the Anderson–

However, the sampling conditions of an experimental analysis cannot be generalized in ranges and hence the trustworthiness of a goodness-of-fit test must be properly verified with a dedicated simulation. To do this, the test must be applied several times on random Gaussian samples having the same number of values of the experimental sample. The number of test applications has to be great enough to avoid a dependence of the simulation results on it. As this test is not time-expensive, the choice of a very high number of applications, such as 100 000, guarantees its negligible influence on the simulation results. The trustworthiness of the goodness-of-fit test is verified once the error percentage of its application on Gaussian samples is lower or at most equal to the desired confidence level. In the example considered earlier, the Anderson–Darling test was applied 100 000 times on random Gaussian samples of 400 data, giving an error value of 5 per cent, which is not greater than the desired confidence level. This result demonstrates the applicability of the

Anderson–Darling test to the experimental analysis sampling conditions.

fields.

Darling test.

data do not follow the specified distribution.

Once verified its trustworthiness, the goodness-of-fit test has to be applied to each point of the investigated area so as to obtain a global visualization of the non-Gaussian critical zones of the flow field. Figure 6 reports an example of the Anderson-Darling application to the velocity flow fields determined in the diffuser passage. Cores of non-Gaussian distribution can be clearly identified at the entrance of the diffuser passage, near the blade walls and also in the mean flow (light grey circles in fig. 6). Even though it is always quite difficult to discriminate between fluid-dynamical and experimental problems, these results allow some preliminary hypotheses about the origin of these critical areas. At the diffuser entrance, the position of the impeller blade close to diffuser blade leading edge lets suppose the development of non-periodical turbulent phenomena coming from the impeller discharge, such as impeller blade wakes and/or rotor-stator interaction phenomena.

Fig. 6. Example of the results of the Anderson-Darling test applied to the pump diffuser blade passage (the black line represents the blade profile; the dark grey line represents the limit of the mask applied to the grid and the light grey circles are marks to identify the non-Gaussian zones in the figure)

The non-Gaussian cores near the blade sides could be due to possible laser reflections problems or seeding problems, whereas laser reflections can be excluded for the cores in the mean flow because of their distance from the walls. These cores could be probably due to non-periodical phenomena proceeding in the passage, but this hypothesis has to be verified by numerical analysis of the flow field.

### **2.3 Confidence interval of the measured values**

The validation procedure of the experimental results is completed by a measure of the reliability of the averaged flow field, that is obtained by estimation of the confidence interval of the determined averaged velocities, that depends on the effective distribution of the experimental data.

In the hypothesis of normal distribution, the confidence interval of the average velocity *C xy <sup>N</sup>* (,) for a confidence level (1-) is:

$$\left| \left( \overline{\mathbb{C}\_{N}} - \frac{\varepsilon\_{N}}{\sqrt{N}} \boldsymbol{\Phi}^{-1} \left( 1 - \frac{a}{2} \right) \overline{\mathbb{C}\_{N}} + \frac{\varepsilon\_{N}}{\sqrt{N}} \boldsymbol{\Phi}^{-1} \left( 1 - \frac{a}{2} \right) \right| \tag{3}$$

where is the normal cumulative distribution function and N the standard deviation of the average velocity (Montgomery and Runger, 2003). So, in this hypothesis, the maximum error in the estimation of the average velocity is:

$$\frac{\varepsilon\_N}{\sqrt{N}} \Phi^{-1} \left( 1 - \frac{a}{2} \right) \tag{4}$$

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 107

passage and could be attributed to the combination of the boundary-layer development with experimental problems such as reflection or seeding problems and to vortical cores

The further proof of the possible development of unsteady phenomena could be obtained by the spectral analysis of the velocity signals. Figure 8 reports the fast Fourier transform (FFT)

Fig. 8. FFT of the velocity components signals determined in three points of the diffuser passage: at the entrance near the suction side (blue line), near the blade pressure side (red line) and in the second half of the diffuser passage far from the blade profiles (black line). a)

Cx b) Cy

coming from the impeller discharge on the suction side.

When the goodness-of-fit test highlights not-normal distributions of the velocity values, to correctly determine the corresponding confidence interval, the effective distribution of the experimental data should be investigated. However, according to the central limit theorem, the procedure for estimating the confidence interval of normal samples can be also applied, with approximation, to not-normal samples if their dimension is sufficiently great (Montgomery and Runger, 2003). This estimation, even approximate, is useful to critically analyse the meaningfulness of the experimental results and to identify the problematic zones of the investigated area.

Figure 7 reports the distribution of the maxima errors (eq. (4)) that can be made in the evaluation of the averaged velocity in the diffuser blade passage. As it can be seen, the maxima errors are localized near the diffuser blade profiles in the inlet throat of the diffuser

Fig. 7. Example of distribution of the maxima errors of the averaged velocity in the diffuser passage (the black line represents the blade profile; the red lines represents the limit of the mask applied to the grid)

In the hypothesis of normal distribution, the confidence interval of the average velocity

where is the normal cumulative distribution function and N the standard deviation of the average velocity (Montgomery and Runger, 2003). So, in this hypothesis, the maximum

<sup>1</sup> 1

When the goodness-of-fit test highlights not-normal distributions of the velocity values, to correctly determine the corresponding confidence interval, the effective distribution of the experimental data should be investigated. However, according to the central limit theorem, the procedure for estimating the confidence interval of normal samples can be also applied, with approximation, to not-normal samples if their dimension is sufficiently great (Montgomery and Runger, 2003). This estimation, even approximate, is useful to critically analyse the meaningfulness of the experimental results and to identify the problematic

Figure 7 reports the distribution of the maxima errors (eq. (4)) that can be made in the evaluation of the averaged velocity in the diffuser blade passage. As it can be seen, the maxima errors are localized near the diffuser blade profiles in the inlet throat of the diffuser

Fig. 7. Example of distribution of the maxima errors of the averaged velocity in the diffuser passage (the black line represents the blade profile; the red lines represents the limit of the

*N N C C N N N N*

*N N* 

1 1 1, 1 2 2

2

 

(4)

(3)

*C xy <sup>N</sup>* (,) for a confidence level (1-) is:

zones of the investigated area.

mask applied to the grid)

error in the estimation of the average velocity is:

passage and could be attributed to the combination of the boundary-layer development with experimental problems such as reflection or seeding problems and to vortical cores coming from the impeller discharge on the suction side.

The further proof of the possible development of unsteady phenomena could be obtained by the spectral analysis of the velocity signals. Figure 8 reports the fast Fourier transform (FFT)

Fig. 8. FFT of the velocity components signals determined in three points of the diffuser passage: at the entrance near the suction side (blue line), near the blade pressure side (red line) and in the second half of the diffuser passage far from the blade profiles (black line). a) Cx b) Cy

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 109

Fig. 9. Comparison between experimental and numerical results: average velocity profiles in

Fig. 10. Effects of the modification of the laser sheet direction on the experimental results

PIV is now a method widely used in the field of Turbomachinery. It has proved its ability to provide useful experimental data for various research topics: rotor stator interaction in radial pumps or fans [Cavazzini et al., 2009; Meakhal & Park 2005; Wuibaut et al., 2002], tipleakage vortex in axial flow compressors [Voges et al., 2011, Yu & Liu, 2007], swirling flow in hydraulic turbines [Tridon et al., 2010]. Nevertheless, in most cases, PIV was efficiently applied to catch phenomena which were correlated with the impeller rotation: PIV images were taken phase locked with the rotor. Consequently, with this kind of acquisition technique, the measurements were not able to treat phenomena, such as rotating stall or surge, whose frequencies are not constant or simply not linked with the impeller speed.

**3. A new averaging procedure for instabilities visualization** 

some sections of the diffuser passage

quality

of the velocity components signals determined in three significant points of the flow field of fig. 7: near the suction side at the entrance of the diffuser passage (blue line), near the pressure side in the zone of maximum error (red line) and in the second half of the diffuser passage far from the blade profiles (black line). The FFT results are reported as a function of the ratio between the frequency f and the sampling frequency fs of the data.

Concerning the velocity component in the mean flow direction Cx (fig. 8a), the points near the blade profiles (points 1 and 2) present peaks having amplitudes much greater than those of the point placed in the mean flow, whereas the velocity component in the direction normal to the mean flow Cy presents low FFT peaks for all the three points (fig. 8b). This strengthens the hypothesis of intense unsteady velocity fluctuations near blade profiles, proceeding in the mean flow direction, and hence confirms the results of the previous statistical analysis.

### **2.4 Comparison between experimental and numerical results: A critical validation**

In fluid-dynamical investigations, the experimental results generally represent a significant reference database for the validation of combined numerical analysis models. However, since the experimental analysis could be negatively affected by experimental problems or post-processing limits, the possible discrepancies between experimental and numerical results have to be critically analysed in order to correctly identify the real error sources.

Numerical error sources, such as the grid resolution and the choice of the turbulent model, have to be considered in the comparison, but may not be the only causes. Experimental problems due to the test-rig or to unsteady phenomena could be also possible reasons for discrepancies. In this context, the validation procedure is extremely useful since it allows to identify the problematic zones of the investigated flow field and to appreciate the meaningfulness of the velocity averages for a critical comparison between the numerical and experimental results.

Figure 9 shows a comparison between the experimental results of fig. 7 and the results of a numerical analysis carried out on the same machine at the same operating conditions. Averaged velocity profiles determined in some sections of the diffuser blade passage are compared. The agreement is quite good, but there are some discrepancies near the diffuser blades (y/l=0 and y/l=1). Numerical error sources, such as a low stream-wise grid resolution or an improper choice of the turbulent model have to be considered as possible causes. However, the validation procedure previously applied to the PIV flow fields highlighted a low trustworthiness of the experimental averaged flow field near the blade profiles (fig. 7), indicating that the discrepancies between numerical and experimental results can be also due to experimental limits, such as reflection problems or difficult seeding near the blades.

In a preliminary study, this combined analysis could be also exploited to modify the set-up of the test rig so as to increase the experimental results quality in the problematic zones. For example, the possible reflection problems near the blade profiles of fig. 9 could be reduced modifying the laser configuration (fig. 10).

of the velocity components signals determined in three significant points of the flow field of fig. 7: near the suction side at the entrance of the diffuser passage (blue line), near the pressure side in the zone of maximum error (red line) and in the second half of the diffuser passage far from the blade profiles (black line). The FFT results are reported as a function of

Concerning the velocity component in the mean flow direction Cx (fig. 8a), the points near the blade profiles (points 1 and 2) present peaks having amplitudes much greater than those of the point placed in the mean flow, whereas the velocity component in the direction normal to the mean flow Cy presents low FFT peaks for all the three points (fig. 8b). This strengthens the hypothesis of intense unsteady velocity fluctuations near blade profiles, proceeding in the mean flow direction, and hence confirms the results of the previous

**2.4 Comparison between experimental and numerical results: A critical validation** 

In fluid-dynamical investigations, the experimental results generally represent a significant reference database for the validation of combined numerical analysis models. However, since the experimental analysis could be negatively affected by experimental problems or post-processing limits, the possible discrepancies between experimental and numerical results have to be critically analysed in order to correctly identify the real error

Numerical error sources, such as the grid resolution and the choice of the turbulent model, have to be considered in the comparison, but may not be the only causes. Experimental problems due to the test-rig or to unsteady phenomena could be also possible reasons for discrepancies. In this context, the validation procedure is extremely useful since it allows to identify the problematic zones of the investigated flow field and to appreciate the meaningfulness of the velocity averages for a critical comparison between the numerical and

Figure 9 shows a comparison between the experimental results of fig. 7 and the results of a numerical analysis carried out on the same machine at the same operating conditions. Averaged velocity profiles determined in some sections of the diffuser blade passage are compared. The agreement is quite good, but there are some discrepancies near the diffuser blades (y/l=0 and y/l=1). Numerical error sources, such as a low stream-wise grid resolution or an improper choice of the turbulent model have to be considered as possible causes. However, the validation procedure previously applied to the PIV flow fields highlighted a low trustworthiness of the experimental averaged flow field near the blade profiles (fig. 7), indicating that the discrepancies between numerical and experimental results can be also due to experimental limits, such as reflection problems or difficult

In a preliminary study, this combined analysis could be also exploited to modify the set-up of the test rig so as to increase the experimental results quality in the problematic zones. For example, the possible reflection problems near the blade profiles of fig. 9 could be reduced

the ratio between the frequency f and the sampling frequency fs of the data.

statistical analysis.

experimental results.

seeding near the blades.

modifying the laser configuration (fig. 10).

sources.

Fig. 9. Comparison between experimental and numerical results: average velocity profiles in some sections of the diffuser passage

Fig. 10. Effects of the modification of the laser sheet direction on the experimental results quality

### **3. A new averaging procedure for instabilities visualization**

PIV is now a method widely used in the field of Turbomachinery. It has proved its ability to provide useful experimental data for various research topics: rotor stator interaction in radial pumps or fans [Cavazzini et al., 2009; Meakhal & Park 2005; Wuibaut et al., 2002], tipleakage vortex in axial flow compressors [Voges et al., 2011, Yu & Liu, 2007], swirling flow in hydraulic turbines [Tridon et al., 2010]. Nevertheless, in most cases, PIV was efficiently applied to catch phenomena which were correlated with the impeller rotation: PIV images were taken phase locked with the rotor. Consequently, with this kind of acquisition technique, the measurements were not able to treat phenomena, such as rotating stall or surge, whose frequencies are not constant or simply not linked with the impeller speed.

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 111

A 2D/3C High Speed PIV combined with pressure transducers was used to study the flow field inside the vaneless diffuser at several flow rates and at three different heights in the hub to shroud direction (0.25, 0.5 and 0.75 of the diffuser width) with an impeller rotation

The laser illumination system consists of two independent Nd:YLF laser cavities, each of

Two CMOS cameras (1680 x 930 pixel2), equipped with 50 mm lenses, were properly synchronized with the laser pulses. They were located at a distance of 480 mm from the measurement regions with an angle between the object plane and the image plane of about

All the details about the experimental set-up, being outside the interest of this work, are not

The image treatment was performed by a software developed by the Laboratoire de Mecanique de Lille. The cross-correlation technique was applied to the image pairs with a correlation window size of 32 x 32 pixels2 and an overlapping of 50%, obtaining flow fields of 80 x 120mm2 and 81 x 125 velocity vectors. The correlation peaks were fitted with a three points Gaussian model. Concerning the stereoscopic reconstruction, the method first proposed by Soloff et al. (1997) was used. A velocity map spanned nearly all the diffuser extension in the radial direction, whereas in the tangential one was covering an angular

Each PIV measurement campaign was carried out for a time period of 1.6 seconds, corresponding to 32 impeller revolutions at a rotation speed of 1200 rpm. Since the temporal resolution of the acquisition was of 980 velocity maps per second, the time period of 1.6 second allowed obtaining 1568 consecutives velocity maps, corresponding to about 49

As regards the pressure measurements, two Brüel & Kjaer condenser microphones (Type 4135) were placed flush with the diffuser shroud wall at the same radial position (1.05 of diffuser inlet radius r3) but at different angular position (=75°). The unsteady pressure measurements, acquired with a sampling frequency of 2048 Hz, were properly synchronized

At partial load and in particular at 0.26 Qdes, previous analyses showed that a rotating

The existence of this instability was determined through the analysis of the cross-power spectra of the pressure signals acquired by the two microphones at partial load and at design flow rate (fig. 13) [Dazin et al 2008, 2011]. The spectrum at design flow rate was clearly dominated by the blade passage frequency fb (7·fimp). At partial load (Q=0.26Qdes) the frequency spectrum was overcome by several peaks in the frequency band between 0.5fimp and 2.0fimp, particularly by the frequency fri = f/fimp=0.84, that was demonstrated to be the fundamental frequency of a rotating instability composed by three cells rotating around the impeller discharge with an angular velocity equal to 28% of the impeller rotation velocity

them producing about 20 mJ per pulse at a pulse frequency of 980 Hz.

here reported, but can be found in a previous paper [Dazin et al., 2011].

speed of 1200 rpm.

portion of about 14°.

[Dazin et al 2008].

velocity maps per impeller revolution.

with the PIV image acquisition system.

**3.2 Constant angular velocity phenomena** 

instability developed in the vaneless diffuser [Dazin et al., 2008].

45°.

The recent development of high speed PIV offers new perspective for the application of the PIV technique in Turbomachinery. Van den Braembussche et al. (2010) have recently proposed a original experiment in which the PIV acquisition system was rotating with a simplified rotating machinery. However, this technique does not overcome the problem of studying rotating phenomena whose frequency was not determined before the experiment.

To catch such type of phenomena, an original averaging procedure of the data based on a frequency or time-frequency analysis of a signal characteristic of the phenomenon was developed. The procedure was applied on two different test cases presented below: a constant rotating phenomenon and an intermittent one.

### **3.1 Experimental set-up of the test cases**

The experimental results presented above were obtained in a PIV experimental analysis carried out on the so-called SHF impeller (fig. 11) coupled with a vaneless diffuser. The tests were made in air with a test rig developed for studying the rotor-stator interaction phenomena (fig. 12).

Fig. 11. SHF impeller

Fig. 12. Experimental set-up

A 2D/3C High Speed PIV combined with pressure transducers was used to study the flow field inside the vaneless diffuser at several flow rates and at three different heights in the hub to shroud direction (0.25, 0.5 and 0.75 of the diffuser width) with an impeller rotation speed of 1200 rpm.

The laser illumination system consists of two independent Nd:YLF laser cavities, each of them producing about 20 mJ per pulse at a pulse frequency of 980 Hz.

Two CMOS cameras (1680 x 930 pixel2), equipped with 50 mm lenses, were properly synchronized with the laser pulses. They were located at a distance of 480 mm from the measurement regions with an angle between the object plane and the image plane of about 45°.

All the details about the experimental set-up, being outside the interest of this work, are not here reported, but can be found in a previous paper [Dazin et al., 2011].

The image treatment was performed by a software developed by the Laboratoire de Mecanique de Lille. The cross-correlation technique was applied to the image pairs with a correlation window size of 32 x 32 pixels2 and an overlapping of 50%, obtaining flow fields of 80 x 120mm2 and 81 x 125 velocity vectors. The correlation peaks were fitted with a three points Gaussian model. Concerning the stereoscopic reconstruction, the method first proposed by Soloff et al. (1997) was used. A velocity map spanned nearly all the diffuser extension in the radial direction, whereas in the tangential one was covering an angular portion of about 14°.

Each PIV measurement campaign was carried out for a time period of 1.6 seconds, corresponding to 32 impeller revolutions at a rotation speed of 1200 rpm. Since the temporal resolution of the acquisition was of 980 velocity maps per second, the time period of 1.6 second allowed obtaining 1568 consecutives velocity maps, corresponding to about 49 velocity maps per impeller revolution.

As regards the pressure measurements, two Brüel & Kjaer condenser microphones (Type 4135) were placed flush with the diffuser shroud wall at the same radial position (1.05 of diffuser inlet radius r3) but at different angular position (=75°). The unsteady pressure measurements, acquired with a sampling frequency of 2048 Hz, were properly synchronized with the PIV image acquisition system.

### **3.2 Constant angular velocity phenomena**

110 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

The recent development of high speed PIV offers new perspective for the application of the PIV technique in Turbomachinery. Van den Braembussche et al. (2010) have recently proposed a original experiment in which the PIV acquisition system was rotating with a simplified rotating machinery. However, this technique does not overcome the problem of studying rotating phenomena whose frequency was not determined before the experiment. To catch such type of phenomena, an original averaging procedure of the data based on a frequency or time-frequency analysis of a signal characteristic of the phenomenon was developed. The procedure was applied on two different test cases presented below: a

The experimental results presented above were obtained in a PIV experimental analysis carried out on the so-called SHF impeller (fig. 11) coupled with a vaneless diffuser. The tests were made in air with a test rig developed for studying the rotor-stator interaction

constant rotating phenomenon and an intermittent one.

**3.1 Experimental set-up of the test cases** 

phenomena (fig. 12).

Fig. 11. SHF impeller

Fig. 12. Experimental set-up

At partial load and in particular at 0.26 Qdes, previous analyses showed that a rotating instability developed in the vaneless diffuser [Dazin et al., 2008].

The existence of this instability was determined through the analysis of the cross-power spectra of the pressure signals acquired by the two microphones at partial load and at design flow rate (fig. 13) [Dazin et al 2008, 2011]. The spectrum at design flow rate was clearly dominated by the blade passage frequency fb (7·fimp). At partial load (Q=0.26Qdes) the frequency spectrum was overcome by several peaks in the frequency band between 0.5fimp and 2.0fimp, particularly by the frequency fri = f/fimp=0.84, that was demonstrated to be the fundamental frequency of a rotating instability composed by three cells rotating around the impeller discharge with an angular velocity equal to 28% of the impeller rotation velocity [Dazin et al 2008].

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 113

Fig. 15. Averaging computation results after 1, 10, 80 and 175 velocity maps for the

tangential velocity component at mid span (in m/s)

Fig. 14. Mesh used for the averaging procedure.

The identification and visualization of the topology of these instability cells was not immediate since the angular span of one PIV map (about 14° of the whole diffuser) was much smaller than the size of an instability cell (about 75°).

To overcome this limit, a new averaging method was developed so as to combine the PIV velocity maps on the basis of the determined instability precession velocity and to obtain an averaged flow field in a reference frame rotating with the instability.

The knowledge of the instability angular speed was needed to be able to apply the PIV averaging procedure.

Fig. 13. Cross-power spectra of pressure signals acquired by the microphones at the design flow rate Qdes (red line) and at 0.26Qdes (black line)

First, since the measurements were not synchronized with the instability rotation, the velocity maps could not be exactly superimposed at each impeller revolution. So it was necessary to create a mesh (Fig. 14), having the same dimensions of the diffuser (0<<360°, 0.257<r<0.390 m), to be used as reference grid for the combination of the PIV maps. To have an almost direct correspondence between this mesh and the PIV grid, the size of one cell of the mesh was fixed roughly equal to the size of one cell of the PIV grid.

Then, the first velocity map was bi-linearly interpolated on the new grid, as shown for the tangential velocities in fig. 15a. The velocity values of the mesh were fixed equal to zero (green in the figure) except in the zone corresponding to the first PIV map properly interpolated on the reference grid. Since the reference frame was fixed to rotate with the instability, the second velocity map was added in the new mesh after a rotation of an angle

Fig. 14. Mesh used for the averaging procedure.

The identification and visualization of the topology of these instability cells was not immediate since the angular span of one PIV map (about 14° of the whole diffuser) was

To overcome this limit, a new averaging method was developed so as to combine the PIV velocity maps on the basis of the determined instability precession velocity and to obtain an

The knowledge of the instability angular speed was needed to be able to apply the PIV

0.26 Qd

Qd

fb

0 2 4 6 810 St = f / fimp

Fig. 13. Cross-power spectra of pressure signals acquired by the microphones at the design

First, since the measurements were not synchronized with the instability rotation, the velocity maps could not be exactly superimposed at each impeller revolution. So it was necessary to create a mesh (Fig. 14), having the same dimensions of the diffuser (0<<360°, 0.257<r<0.390 m), to be used as reference grid for the combination of the PIV maps. To have an almost direct correspondence between this mesh and the PIV grid, the size of one cell of

Then, the first velocity map was bi-linearly interpolated on the new grid, as shown for the tangential velocities in fig. 15a. The velocity values of the mesh were fixed equal to zero (green in the figure) except in the zone corresponding to the first PIV map properly interpolated on the reference grid. Since the reference frame was fixed to rotate with the instability, the second velocity map was added in the new mesh after a rotation of an angle

the mesh was fixed roughly equal to the size of one cell of the PIV grid.

much smaller than the size of an instability cell (about 75°).

fri

flow rate Qdes (red line) and at 0.26Qdes (black line)

averaging procedure.

10-9

10-8

10-7

10-6

**2**

10-5

10-4

10-3

averaged flow field in a reference frame rotating with the instability.

Fig. 15. Averaging computation results after 1, 10, 80 and 175 velocity maps for the tangential velocity component at mid span (in m/s)

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 115

Fig. 16. Results of the averaging procedure: a) radial velocity; b) tangential velocity; c) axial

velocity [m/s]

equal to the instability velocity multiplied by the sampling period of the PIV measurements. As this second velocity map overlapped the first one, in the overlapping zone the velocity values were properly averaged. This operation was repeated for the following velocity maps till a complete revolution of the instability, corresponding to 175 maps, was made. Afterwards, the maps were averaged with the ones of the previous revolution(s). Examples of the averaging computation results respectively after 10, 80 and 175 velocity maps are reported in fig. 15(b-d). At the end of the procedure, 120 velocity vectors were averaged in each point of the reference grid, obtaining a mean velocity vector. The standard deviation was of the order of 2 m/s and the corresponding 95% confidence interval for each averaged velocity component *<sup>i</sup> c* was:

$$\left[\overleftarrow{c\_i} \pm 0.4m/s\right]$$

The procedure described above allowed to obtain averaged flow fields in a reference frame rotating with the instability for the three velocity components (fig. 16). Because of laser sheet reflections on the impeller blades, several instantaneous flow fields were negatively affected at the diffuser inlet by the proximity of the impeller blades. For this reason, the averaged flow fields are presented only for r> 0.3 m. The average flow field of the radial velocity component shows three similar patterns composed of two cores, respectively of inward and outward radial velocities, located near the diffuser outlet (fig 16a). In correspondence to these two cores, a zone of negative tangential velocity is identifiable near the diffuser inlet (fig. 16b) and a zone of slightly positive axial velocity is outlined within the diffuser (fig 16c).

So, the averaging procedure allowed to clearly visualize the topology of the instability rotating in the diffuser and to obtain several information about its fluid-dynamical characteristics.

#### **3.3 Intermittent phenomena**

In the same pump configuration at a greater flow rate (0.45 Qdes), rotating instabilities were still identified in the diffuser, but resulted to be characterized by two competitive lowfrequency modes.

The first mode, which dominated the spectrum, corresponded to an instability composed by two cells rotating at imp = 0.28, whereas the second mode corresponded to an instability composed by three cells rotating at imp = 0.26 [Pavesi et al., 2011]. Moreover, the timefrequency analysis, carried out on the pressure signals, highlighted that these two competitive modes did not exist at the same time but were present intermittently in the diffuser (fig. 17).

Consequently, the averaging procedure defined in §3.2 could not be immediately applied to the PIV results but it was adapted to the new intermittent characteristics of the fluiddynamical instability. In particular, the results of the time frequency analysis were used to determine the time periods of the acquisition process during which only one mode was dominant. Then, the PIV averaging procedure, described in §3.2, was applied only to the flow fields determined in those time periods characterized by the presence of one mode. In this way, two averaged flow fields corresponding to the two competitive modes were obtained (fig. 18).

equal to the instability velocity multiplied by the sampling period of the PIV measurements. As this second velocity map overlapped the first one, in the overlapping zone the velocity values were properly averaged. This operation was repeated for the following velocity maps till a complete revolution of the instability, corresponding to 175 maps, was made. Afterwards, the maps were averaged with the ones of the previous revolution(s). Examples of the averaging computation results respectively after 10, 80 and 175 velocity maps are reported in fig. 15(b-d). At the end of the procedure, 120 velocity vectors were averaged in each point of the reference grid, obtaining a mean velocity vector. The standard deviation was of the order of 2 m/s and the corresponding 95% confidence interval for each averaged

[ 0.4 / ] *<sup>i</sup> c ms*

The procedure described above allowed to obtain averaged flow fields in a reference frame rotating with the instability for the three velocity components (fig. 16). Because of laser sheet reflections on the impeller blades, several instantaneous flow fields were negatively affected at the diffuser inlet by the proximity of the impeller blades. For this reason, the averaged flow fields are presented only for r> 0.3 m. The average flow field of the radial velocity component shows three similar patterns composed of two cores, respectively of inward and outward radial velocities, located near the diffuser outlet (fig 16a). In correspondence to these two cores, a zone of negative tangential velocity is identifiable near the diffuser inlet (fig. 16b) and a zone

So, the averaging procedure allowed to clearly visualize the topology of the instability rotating in the diffuser and to obtain several information about its fluid-dynamical

In the same pump configuration at a greater flow rate (0.45 Qdes), rotating instabilities were still identified in the diffuser, but resulted to be characterized by two competitive low-

The first mode, which dominated the spectrum, corresponded to an instability composed by two cells rotating at imp = 0.28, whereas the second mode corresponded to an instability composed by three cells rotating at imp = 0.26 [Pavesi et al., 2011]. Moreover, the timefrequency analysis, carried out on the pressure signals, highlighted that these two competitive modes did not exist at the same time but were present intermittently in the

Consequently, the averaging procedure defined in §3.2 could not be immediately applied to the PIV results but it was adapted to the new intermittent characteristics of the fluiddynamical instability. In particular, the results of the time frequency analysis were used to determine the time periods of the acquisition process during which only one mode was dominant. Then, the PIV averaging procedure, described in §3.2, was applied only to the flow fields determined in those time periods characterized by the presence of one mode. In this way, two averaged flow fields corresponding to the two competitive modes were

of slightly positive axial velocity is outlined within the diffuser (fig 16c).

velocity component *<sup>i</sup> c* was:

characteristics.

frequency modes.

diffuser (fig. 17).

obtained (fig. 18).

**3.3 Intermittent phenomena** 

Fig. 16. Results of the averaging procedure: a) radial velocity; b) tangential velocity; c) axial velocity [m/s]

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 117

For example, the first mode resulted to be dominant in a time period of about 0.4 s at the beginning of the simultaneous pressure and PIV acquisitions. Consistently with the Fourier spectra analysis, the PIV averaged velocity map (fig. 18a) obtained on this time period presents two instability cells diametrically located, similar to those obtained at the lowest flow

For the second mode, the longer time period identified was of only about 0.1 s. The averaged velocity procedure applied on this time period gives the velocity map plotted on fig 18b. For this mode, the expected number of cells was three, whereas the averaged velocity map presents only two clear cells (surrounded by a solid line). The third cell of this mode is hardly visible (inside the dashed lines), most probably because of a too-short period

This work presents two different post-processing procedures suitable to be applied to PIV

The first procedure was focused on the PIV experimental accuracy and was aimed at the validation of the averaged flow fields in a PIV analysis. This procedure combines several

a convergence analysis to verify that the number of acquired images allowed to obtain a

the analysis of the probability density distribution to verify the repeatability of the

 the estimation of the confidence interval to evaluate the maxima errors associated with the determined velocity averages and hence to quantitatively analyze their

This validation procedure can be considered not only as a necessary critical analysis of the meaningfulness of experimental PIV results, but also as a possible preliminary study for improving the test rig before starting time- and work-intensive measurement campaigns.

The second part of the chapter is focused on the averaging techniques and presents an original averaging procedure of PIV flow fields for the study of unforced unsteadiness. Since the spectral characteristics of the instability and in particular its precession velocity has to be known, the procedure is necessarily combined with a spectral analysis of

On the basis of the spectrally determined instability velocity, the PIV flow fields were properly combined and averaged, obtaining an average flow field in the reference frame of the instability to be studied. This result allows to capture and visualize the topology of the phenomenon and to obtain more in-depth information about its fluid-dynamical

The procedure was also developed and adapted for intermittent instability configurations,

characterized by competitive modes alternatively present in the flow field.

measurements and to identify the critical area of the investigated flow field.

instantaneous flow fields characterized by the development of unsteady flows.

for the application of the PIV averaging procedure.

statistical tools and can be summarized in three main steps:

meaningful averaged flow field

simultaneously acquired pressure signals.

development and characteristics.

trustworthiness.

rate.

**4. Conclusions** 

Fig. 18. PIV averaging procedure results for the two modes identified at Q/Qdes = 0.45

For example, the first mode resulted to be dominant in a time period of about 0.4 s at the beginning of the simultaneous pressure and PIV acquisitions. Consistently with the Fourier spectra analysis, the PIV averaged velocity map (fig. 18a) obtained on this time period presents two instability cells diametrically located, similar to those obtained at the lowest flow rate.

For the second mode, the longer time period identified was of only about 0.1 s. The averaged velocity procedure applied on this time period gives the velocity map plotted on fig 18b. For this mode, the expected number of cells was three, whereas the averaged velocity map presents only two clear cells (surrounded by a solid line). The third cell of this mode is hardly visible (inside the dashed lines), most probably because of a too-short period for the application of the PIV averaging procedure.

### **4. Conclusions**

116 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Fig. 17. Detail of the wavelet analysis of the pressure signals acquired at Q/Qdes = 0.45

Fig. 18. PIV averaging procedure results for the two modes identified at Q/Qdes = 0.45

This work presents two different post-processing procedures suitable to be applied to PIV instantaneous flow fields characterized by the development of unsteady flows.

The first procedure was focused on the PIV experimental accuracy and was aimed at the validation of the averaged flow fields in a PIV analysis. This procedure combines several statistical tools and can be summarized in three main steps:


This validation procedure can be considered not only as a necessary critical analysis of the meaningfulness of experimental PIV results, but also as a possible preliminary study for improving the test rig before starting time- and work-intensive measurement campaigns.

The second part of the chapter is focused on the averaging techniques and presents an original averaging procedure of PIV flow fields for the study of unforced unsteadiness. Since the spectral characteristics of the instability and in particular its precession velocity has to be known, the procedure is necessarily combined with a spectral analysis of simultaneously acquired pressure signals.

On the basis of the spectrally determined instability velocity, the PIV flow fields were properly combined and averaged, obtaining an average flow field in the reference frame of the instability to be studied. This result allows to capture and visualize the topology of the phenomenon and to obtain more in-depth information about its fluid-dynamical development and characteristics.

The procedure was also developed and adapted for intermittent instability configurations, characterized by competitive modes alternatively present in the flow field.

Post-Processing Methods of PIV Instantaneous Flow Fields for Unsteady Flows in Turbomachines 119

Raffel, M.; Willert, C.; Werely, S. & Kompenhans, J. (2002). Particle image velocimetry – a

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Vogt, A.; Baumann, P.; Gharib, M. & Kompenhans, J. (1996). Investigations of a wing tip

Wernert, P. & Favier, D. (1999). Considerations about the phase-averaging method with

Westerweel, J. (1994). Efficient detection of spurious vectors in particle image velocimetry

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**PIV Applications** 

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## compressor stage*. Experimental Thermal and Fluid Science*, vol. 31, pp. 1049–1060 **Section 2**

**PIV Applications** 

120 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Yu, X.J. & Liu, B.-J. (2007). Stereoscopic PIV measurement of unsteady flows in an axial

**5** 

*France* 

**PIV Measurements on Oxy-Fuel Burners** 

*2CORIA, CNRS-Université et INSA de Rouen, Saint Etienne du Rouvray* 

*1ICARE-CNRS, Avenue de la Recherche Scientifique, Orléans, University of Orléans,* 

This chapter concerns the application of the PIV measurements in a semi-industrial combustion system. The emphasis is on oxy-fuel burners with multi-jets. The mixing and the

Nowadays, combustion occupies a prominent place to meet the increasing needs of our economic world, in fields as diverse as: the production of electrical energy; the space heating, the development of building materials, the metallurgy; the land and air transport; the synthesis of many chemicals in flames; the production of hydrocarbons from crude oil in refineries, etc. Mastering combustion obtained under as varied conditions requires: a knowledge of more advanced fundamental phenomena governing the reaction processes; a " know-how" to an optimal implementation in terms of energy (efficiency) and in terms of

The evolution of pollution standards and the optimization of combustion chamber performances require a development of new burner types and the improvement of combustion techniques. The industrialists are turning to a new burner generation with separate injection of fuel and oxidant. Design of these burners requires the knowledge of mechanisms controlling the stabilization of flame and the production of pollutants. The present study concerns the control of turbulent natural gas-oxygen flames resulting from burners with aligned separated jets. One possibility to understand the structure and mechanism of flame stabilization more accurately is to analyze the structure of the flow by means of particle image velocimetry (PIV) (Raffel and al., 1999). This method allows direct velocity measurements without and with combustion. The application of PIV in industrial scale flame has been already demonstrated on a 1MW experimental boiler (Honoré et al.,

Oxy-fuel combustion, air is substituted by pure oxygen, is characterized by a higher adiabatic flame temperature, a higher flame velocity, a lower ignition temperature, and a wider flammability range that is the case of combustion with air (Baukal and Gebhart, 1997; GEFGN, 1983; Perthuis, 1983; Ivernal and Marque, 1975). This oxy-fuel combustion allows to have a better thermal efficiency and a better stabilization of flame. Oxy-fuel burners have been adopted in a wide range of industrial furnaces to improve productivity and fuel efficiency, to reduce emissions of pollutants, and, in some applications, to improve product quality and yield or to eliminate the capital and maintenance costs of air preheaters

dynamic field for both reacting and non-reacting flows are investigated.

**1. Introduction** 

pollution (air pollution, noise).

2001).

Boushaki Toufik1 and Sautet Jean-Charles2

## **PIV Measurements on Oxy-Fuel Burners**

Boushaki Toufik1 and Sautet Jean-Charles2

*1ICARE-CNRS, Avenue de la Recherche Scientifique, Orléans, University of Orléans, 2CORIA, CNRS-Université et INSA de Rouen, Saint Etienne du Rouvray France* 

### **1. Introduction**

This chapter concerns the application of the PIV measurements in a semi-industrial combustion system. The emphasis is on oxy-fuel burners with multi-jets. The mixing and the dynamic field for both reacting and non-reacting flows are investigated.

Nowadays, combustion occupies a prominent place to meet the increasing needs of our economic world, in fields as diverse as: the production of electrical energy; the space heating, the development of building materials, the metallurgy; the land and air transport; the synthesis of many chemicals in flames; the production of hydrocarbons from crude oil in refineries, etc. Mastering combustion obtained under as varied conditions requires: a knowledge of more advanced fundamental phenomena governing the reaction processes; a " know-how" to an optimal implementation in terms of energy (efficiency) and in terms of pollution (air pollution, noise).

The evolution of pollution standards and the optimization of combustion chamber performances require a development of new burner types and the improvement of combustion techniques. The industrialists are turning to a new burner generation with separate injection of fuel and oxidant. Design of these burners requires the knowledge of mechanisms controlling the stabilization of flame and the production of pollutants. The present study concerns the control of turbulent natural gas-oxygen flames resulting from burners with aligned separated jets. One possibility to understand the structure and mechanism of flame stabilization more accurately is to analyze the structure of the flow by means of particle image velocimetry (PIV) (Raffel and al., 1999). This method allows direct velocity measurements without and with combustion. The application of PIV in industrial scale flame has been already demonstrated on a 1MW experimental boiler (Honoré et al., 2001).

Oxy-fuel combustion, air is substituted by pure oxygen, is characterized by a higher adiabatic flame temperature, a higher flame velocity, a lower ignition temperature, and a wider flammability range that is the case of combustion with air (Baukal and Gebhart, 1997; GEFGN, 1983; Perthuis, 1983; Ivernal and Marque, 1975). This oxy-fuel combustion allows to have a better thermal efficiency and a better stabilization of flame. Oxy-fuel burners have been adopted in a wide range of industrial furnaces to improve productivity and fuel efficiency, to reduce emissions of pollutants, and, in some applications, to improve product quality and yield or to eliminate the capital and maintenance costs of air preheaters

PIV Measurements on Oxy-Fuel Burners 125

1999; Tamburello and Amitay, 2008) and flaps (Susuki et al., 1999). It has been proven in the literature that jet actuators have drastic effects on mixing and flow dynamics. In fact, jets actuators are capable to change the flow structure, to act on mixing between the reactants, and thus on the flame characteristics such as stability and flame size, as well as pollutant production. The perpendicular arrangement of tube actuators at the periphery of a main jet can confer to the flow a helical movement (swirl). This kind of swirl with actuators has very significant effects on the flow. Ibrahim et al. (2002) and Faivre and Poinsot (2004) indicated that a radial fluid injection into the main jet enhances mixing with the surrounding air. Béer and Chigier (1972), and Feikema et al. (1990) showed that the addition of swirl significantly changes the aerodynamic pattern and can be used to stabilize the flame. The helical fluid flow creates a recirculation zone, allowing dilution with combustion products, and the decrease of the flame temperature limiting the NOx production (Syred and Béer, 1974;

The present chapter reports the results of an experimental investigation of the dynamic field on a burner with 25 kW power composed of 3 jets, one central jet of natural gas and two side jets of pure oxygen. The velocity measurements were carried out using Particle Image Velocimetry (PIV) in both cases of non-reacting flow and reacting flow inside the combustion chamber. Two control systems, one passive and one active, are developed and added to the basic burner to improve the combustion process and to ensure the stabilization of flame and pollutant reductions. The passive control is based on the slope of side oxygen jets towards the central natural gas jet in a triple jet configuration. The active control concerns the use of four small jet actuators, placed tangentially to the exit of the main jets to generate a swirling flow. These actuators are able to strongly modify the flow structure and

to act on mixing between the reactants and consequently on the flame behavior.

M initial velocity ratio ( 0 0 ( /) *MU U ng ox* ) oxygen jet angle, (°)

u' longitudinal velocity fluctuation ms-1 ng natural gas jet V radial mean velocity, m.s-1 ox oxygen jet v' radial velocity fluctuation, m.s-1 tot total

*m* mass flow rate, kg.s-1µ dynamic viscosity, g.m.s-1

gas density, kg.m-3

d tube internal diameter, mm *Greek symbols*

*U d*

S separation distance between the jets, mm Subscripts SW2J Burner configuration with 2 jets ac actuator SW3J Burner configuration with 3 jets jet main jet U0 nozzle exit velocity, m.s-1 cl centerline U longitudinal mean velocity, m.s-1 lo lift-off

*Nomenclature*

r ratio of volumetric flow rate (*rm m act tot* / )

Re Reynolds number 0 ( /)

x, z radial and longitudinal coordinate, mm

Schmittel et al., 2000; Coghe et al., 2004).

(Baukhal, 2003). Furthermore, the use of oxy-fuel combustion in separated-jet burners open interesting possibilities in the NOx reduction, and the modularity of flame properties such as stabilization, topology and flame length (Sautet et al., 2006, Boushaki et al., 2007).

Flames from burner with multiple jets have many practical situations. Several studies have been published on the structure and development of non-reacting multiple jets (Krothapalli et al., 1980; Raghunatan et al., 1980; Pani and Dash, 1983; Simonich 1986; Yimer et al., 1996; Moawad et al., 2001). However, the studies of multiple jet flames are mostly limited, for example, flame developing in still air without confinement (Leite et al., 1996) or in a wind tunnel with cross flow (Menon and Gollahali, 1998). Lee et al. (2004) studied the blowout limit considering the interaction of multiple non premixed jet flames and giving a number of variables such as distance between the jets, the number of jets and their arrangements. Lenze et al. (1975) have studied the mutual influence of three and five jet diffusion flame, with town gas and natural gas burners. Their measurements concern concentrations, flame length, and flame width in free and confined multiple flames. The burner configuration used in this work is composed of three round jets, one central jet of natural gas and two lateral jets of pure oxygen.

For the separated jet burner, the principle is based on the geometrical separation of its nozzles. The separation of jets allows high dilution of reactants by combustion products in the combustion chamber. This dilution leads to a lower flame temperature and homogenization of temperature throughout the volume of the flame, and consequently a decline in NOx production. However, the configuration of separated jets, which contributes to the dilution of reactants, may become unfavorable to the stabilization of flame. Indeed, this dilution may be controlled by various burner parameters, such as exit velocities of reactants and the distance between the jets. In previous papers of the authors (Boushaki et al., 2007), the characteristics of flames in burners with separated jets have been studied versus the burner parameters such as exit velocities, separation distance between jets and angle of injection. It is interesting to note that the inclination of jets allows to improve the flame stability and above all reduces significantly the NOx (Boushaki et al. 2008). The present chapter focuses on the dynamic behavior of three jet interactions in more detail by varying angle of the side oxygen jets.

Controlling the flows has been the focus of numerous investigations. The objectives of flow control (passive and active) differ according to the considered industrial application. Among these aims are the improvement of mixing with ambient air (Davis, 1982 ; Denis et al., 1999), the limitation of combustion instabilities (Lang et al., 1987; Candel, 1992), the enhancement of heat transfer of flame (Candel, 1992), the decrease of pollutant emissions (Delabroy, 1998; Demayo, 2002) and the reduction of noise engendered in some combustion chambers (Barrère and Williams, 1968; Strahle, 1978). The two dominant methods of passive flow control include noncircular nozzles (e.g., Ho and Gutmark, 1987; Gutmark and Grinstein, 1999, Gollahalli et al., 1992) or the use of tabs at the nozzle exit (e.g., Bradbury and Khadem, 1975; Ahuja, 1993; Hileman et al., 2003). The active control consists in injecting external energy through actuators .The quality of the control depends directly on the design of the actuators. Some of them are specific to combustion applications but most actuation techniques are encountered in both reactive and non-reactive applications, such as loudspeakers (Bloxsidge et al., 1987, McManus et al., 1993), small jet actuators (Lardeau et al., 2002 ; Faivre and Poinsot, 2004, Boushaki et al., 2009), synthetic jets (Davis, A. Glezer

(Baukhal, 2003). Furthermore, the use of oxy-fuel combustion in separated-jet burners open interesting possibilities in the NOx reduction, and the modularity of flame properties such

Flames from burner with multiple jets have many practical situations. Several studies have been published on the structure and development of non-reacting multiple jets (Krothapalli et al., 1980; Raghunatan et al., 1980; Pani and Dash, 1983; Simonich 1986; Yimer et al., 1996; Moawad et al., 2001). However, the studies of multiple jet flames are mostly limited, for example, flame developing in still air without confinement (Leite et al., 1996) or in a wind tunnel with cross flow (Menon and Gollahali, 1998). Lee et al. (2004) studied the blowout limit considering the interaction of multiple non premixed jet flames and giving a number of variables such as distance between the jets, the number of jets and their arrangements. Lenze et al. (1975) have studied the mutual influence of three and five jet diffusion flame, with town gas and natural gas burners. Their measurements concern concentrations, flame length, and flame width in free and confined multiple flames. The burner configuration used in this work is composed of three round jets, one central jet of natural gas and two

For the separated jet burner, the principle is based on the geometrical separation of its nozzles. The separation of jets allows high dilution of reactants by combustion products in the combustion chamber. This dilution leads to a lower flame temperature and homogenization of temperature throughout the volume of the flame, and consequently a decline in NOx production. However, the configuration of separated jets, which contributes to the dilution of reactants, may become unfavorable to the stabilization of flame. Indeed, this dilution may be controlled by various burner parameters, such as exit velocities of reactants and the distance between the jets. In previous papers of the authors (Boushaki et al., 2007), the characteristics of flames in burners with separated jets have been studied versus the burner parameters such as exit velocities, separation distance between jets and angle of injection. It is interesting to note that the inclination of jets allows to improve the flame stability and above all reduces significantly the NOx (Boushaki et al. 2008). The present chapter focuses on the dynamic behavior of three jet interactions in more detail by

Controlling the flows has been the focus of numerous investigations. The objectives of flow control (passive and active) differ according to the considered industrial application. Among these aims are the improvement of mixing with ambient air (Davis, 1982 ; Denis et al., 1999), the limitation of combustion instabilities (Lang et al., 1987; Candel, 1992), the enhancement of heat transfer of flame (Candel, 1992), the decrease of pollutant emissions (Delabroy, 1998; Demayo, 2002) and the reduction of noise engendered in some combustion chambers (Barrère and Williams, 1968; Strahle, 1978). The two dominant methods of passive flow control include noncircular nozzles (e.g., Ho and Gutmark, 1987; Gutmark and Grinstein, 1999, Gollahalli et al., 1992) or the use of tabs at the nozzle exit (e.g., Bradbury and Khadem, 1975; Ahuja, 1993; Hileman et al., 2003). The active control consists in injecting external energy through actuators .The quality of the control depends directly on the design of the actuators. Some of them are specific to combustion applications but most actuation techniques are encountered in both reactive and non-reactive applications, such as loudspeakers (Bloxsidge et al., 1987, McManus et al., 1993), small jet actuators (Lardeau et al., 2002 ; Faivre and Poinsot, 2004, Boushaki et al., 2009), synthetic jets (Davis, A. Glezer

as stabilization, topology and flame length (Sautet et al., 2006, Boushaki et al., 2007).

lateral jets of pure oxygen.

varying angle of the side oxygen jets.

1999; Tamburello and Amitay, 2008) and flaps (Susuki et al., 1999). It has been proven in the literature that jet actuators have drastic effects on mixing and flow dynamics. In fact, jets actuators are capable to change the flow structure, to act on mixing between the reactants, and thus on the flame characteristics such as stability and flame size, as well as pollutant production. The perpendicular arrangement of tube actuators at the periphery of a main jet can confer to the flow a helical movement (swirl). This kind of swirl with actuators has very significant effects on the flow. Ibrahim et al. (2002) and Faivre and Poinsot (2004) indicated that a radial fluid injection into the main jet enhances mixing with the surrounding air. Béer and Chigier (1972), and Feikema et al. (1990) showed that the addition of swirl significantly changes the aerodynamic pattern and can be used to stabilize the flame. The helical fluid flow creates a recirculation zone, allowing dilution with combustion products, and the decrease of the flame temperature limiting the NOx production (Syred and Béer, 1974; Schmittel et al., 2000; Coghe et al., 2004).

The present chapter reports the results of an experimental investigation of the dynamic field on a burner with 25 kW power composed of 3 jets, one central jet of natural gas and two side jets of pure oxygen. The velocity measurements were carried out using Particle Image Velocimetry (PIV) in both cases of non-reacting flow and reacting flow inside the combustion chamber. Two control systems, one passive and one active, are developed and added to the basic burner to improve the combustion process and to ensure the stabilization of flame and pollutant reductions. The passive control is based on the slope of side oxygen jets towards the central natural gas jet in a triple jet configuration. The active control concerns the use of four small jet actuators, placed tangentially to the exit of the main jets to generate a swirling flow. These actuators are able to strongly modify the flow structure and to act on mixing between the reactants and consequently on the flame behavior.


PIV Measurements on Oxy-Fuel Burners 127

=10° =30°

=0°

The active control is applied using four small jet actuators arranged slightly upstream the exit of the main jets. Fig. 3 shows the oxy-fuel burner fitted with actuators. Each nozzle is equipped with four small control jets placed tangentially around the main jets. This kind of tangential control generates a swirling motion in the flow, its intensity being controlled by the flow rate in the actuators. The exit diameter of the jet actuators (*dact*) is 1 mm and their position relative to the main jet exit is z = -1mm. In the present work, the rotating flow intensity is quantified by the ratio of volumetric flow rates of actuators ( *mact* ) and total

**O2 NG O2**

*act tot m r m*

where *mmm tot jet act* and *mjet* is the flow rate of the main jet ( *m m tot NG* for the fuel flow rate and *m m tot Ox* for the oxygen flow rate) . The subscripts "tot" and "act" are valuable for both reactants (oxygen and natural gas). For the present configurations, the swirl number

(*Sn*) characterizing rotating flows is an increasing function of the flow rate ratio (*r)*.

Fig. 3. Schematic view of main nozzles with tangential tube actuators

(1)

Fig. 2. Photo of burners with angle injection 0°, 10° and 30°

( *mtot* ), which varies from 0 to 30%; it is given by:

**2.3 Active control system** 

### **2. Burner and operating conditions**

#### **2.1 Basic configuration of the burner**

The basic configuration of the burner consists of three round tubes, one central of natural gas and two laterals of pure oxygen. A schematic of the oxy-fuel burner apparatus is shown in Fig. 1. This burner can operate with only two jets if the entire oxygen flow rate is injected into a single tube. Fuel and oxidizer flow rates are constant for all experiments to ensure constant power flames of 25 kW ( 3 1 556 10 *m k ng g.s* , 3 1 *mOx* 1964 10 kg.s ). The natural gas ( <sup>3</sup> <sup>ρ</sup>ng 0.83 kg.m ) flows from the central tube (diameter dNG=6 mm, length 250 mm) and pure oxygen ( <sup>3</sup> <sup>ρ</sup>ox 1.35 kg. m ) flows from the two side jets (diameter dOx=6 mm or 8 mm, length 250 mm). The natural gas composition is: 85% CH4; 9% C2H6; 3% C3H8; 2%N2; 1% CO2; plus traces of higher hydrocarbon species. The flow rate of natural gas is controlled by a regulator of mass flow rate TYLAN RDM 280; the oxygen is regulated by sonic throats calibrated by a classical flowmeter in function of pressure.

Fig. 1. Schematic diagram of oxy-fuel burner

In the case of reacting flow, the oxy-fuel flames takes place inside a combustion chamber of 1 m-high with square cross section of 6060 cm. The lateral walls are refractory lined inside and water cooled outside of the combustion chamber. Optical access is provided by quartz windows at many vertical positions of the combustion chamber. The burner is located at the centre of the bottom wall of the combustion chamber (more details, see Boushaki et al. 2009).

#### **2.2 Passive control system**

The control technique consists in inclining the side oxygen jets towards the natural gas jet as shown in Fig. 2. The angle of oxygen jets () compared to the vertical direction varies from 0° to 30° (0, 10, 20, 30°). The internal diameters of natural gas and oxygen tubes ( d and ng dox ) are both 6 mm. Exit velocities for the natural gas and the oxygen jets without control (=0°) inferred from the flow rates are respectively are <sup>0</sup> *UNG* 23.7 m/s and <sup>0</sup> *UOx* 25.7 m/s. The separation distance between the jets (S) is fixed at 12 mm. The exit Reynolds numbers of the natural gas jet and the oxygen jet are Reng= 10772 and Reox= 10936 respectively.

The basic configuration of the burner consists of three round tubes, one central of natural gas and two laterals of pure oxygen. A schematic of the oxy-fuel burner apparatus is shown in Fig. 1. This burner can operate with only two jets if the entire oxygen flow rate is injected into a single tube. Fuel and oxidizer flow rates are constant for all experiments to ensure constant power flames of 25 kW ( 3 1 556 10 *m k ng g.s* , 3 1 *mOx* 1964 10 kg.s ). The natural gas ( <sup>3</sup> <sup>ρ</sup>ng 0.83 kg.m ) flows from the central tube (diameter dNG=6 mm, length 250 mm) and pure oxygen ( <sup>3</sup> <sup>ρ</sup>ox 1.35 kg. m ) flows from the two side jets (diameter dOx=6 mm or 8 mm, length 250 mm). The natural gas composition is: 85% CH4; 9% C2H6; 3% C3H8; 2%N2; 1% CO2; plus traces of higher hydrocarbon species. The flow rate of natural gas is controlled by a regulator of mass flow rate TYLAN RDM 280; the oxygen is regulated by

In the case of reacting flow, the oxy-fuel flames takes place inside a combustion chamber of 1 m-high with square cross section of 6060 cm. The lateral walls are refractory lined inside and water cooled outside of the combustion chamber. Optical access is provided by quartz windows at many vertical positions of the combustion chamber. The burner is located at the centre of the bottom wall of the combustion chamber (more details, see Boushaki et al. 2009).

The control technique consists in inclining the side oxygen jets towards the natural gas jet as shown in Fig. 2. The angle of oxygen jets () compared to the vertical direction varies from 0° to 30° (0, 10, 20, 30°). The internal diameters of natural gas and oxygen tubes ( d and ng dox ) are both 6 mm. Exit velocities for the natural gas and the oxygen jets without control (=0°) inferred from the flow rates are respectively are <sup>0</sup> *UNG* 23.7 m/s and <sup>0</sup> *UOx* 25.7 m/s. The separation distance between the jets (S) is fixed at 12 mm. The exit Reynolds numbers of

the natural gas jet and the oxygen jet are Reng= 10772 and Reox= 10936 respectively.

sonic throats calibrated by a classical flowmeter in function of pressure.

**2. Burner and operating conditions 2.1 Basic configuration of the burner** 

Fig. 1. Schematic diagram of oxy-fuel burner

**2.2 Passive control system** 

Fig. 2. Photo of burners with angle injection 0°, 10° and 30°

#### **2.3 Active control system**

The active control is applied using four small jet actuators arranged slightly upstream the exit of the main jets. Fig. 3 shows the oxy-fuel burner fitted with actuators. Each nozzle is equipped with four small control jets placed tangentially around the main jets. This kind of tangential control generates a swirling motion in the flow, its intensity being controlled by the flow rate in the actuators. The exit diameter of the jet actuators (*dact*) is 1 mm and their position relative to the main jet exit is z = -1mm. In the present work, the rotating flow intensity is quantified by the ratio of volumetric flow rates of actuators ( *mact* ) and total ( *mtot* ), which varies from 0 to 30%; it is given by:

$$r = \frac{\dot{m}\_{\text{act}}}{\dot{m}\_{\text{tot}}} \tag{1}$$

where *mmm tot jet act* and *mjet* is the flow rate of the main jet ( *m m tot NG* for the fuel flow rate and *m m tot Ox* for the oxygen flow rate) . The subscripts "tot" and "act" are valuable for both reactants (oxygen and natural gas). For the present configurations, the swirl number (*Sn*) characterizing rotating flows is an increasing function of the flow rate ratio (*r)*.

Fig. 3. Schematic view of main nozzles with tangential tube actuators

The swirl number is a dimensionless quantity defined as (Beèr and Chigier, 1972; Sheen et al. 1996):

$$\mathbf{S}\_n = \frac{\mathbf{G}\_\varphi}{\mathbf{R}\mathbf{G}\_\chi'} \tag{2}$$

PIV Measurements on Oxy-Fuel Burners 129

Particle Image Velocimetry (PIV) is a laser diagnostic method which has been developed in parallel with Laser Sheet Visualization (LSV) ). As with LSV, The PIV is based on the collection of images of Mie scattering of fine particles seeded in the flow. Processing of these particle images provides instantaneous data on two velocity components in a plane crossing the flow. Then, the mean and root-mean-square (rms) velocity fields can be easily deduced

Fig. 4 shows a schematic diagram of the PIV system. It includes a laser sheet that illuminates the zone of flow studied, a CCD camera, a PC for data acquisition and a control unit for synchronization. The laser used is double-pulsed Nd-YAG (Big Sky CFR200 Quantel) with a wavelength of 532 nm and a frequency of 10 Hz. Laser energy is adjustable and can be increased up to 150 mJ per pulse with pulse duration of 8 ns. The laser sheet is formed by a first divergent cylinder lens, which spreads out the beam then by second convergent spherical lens, which focuses the sheet. The signal of Mie scattering emitted by particles is collected perpendicularly by a CCD camera FlowMaster of Lavision (12-bit dynamic and 12801024 pixels resolution) with a 50 mm lens F/1.2 Nikkon. In reacting flow measurements, to reject the bright luminosity from the oxy-flame, an interference filter (532 nm centre, 3 nm bandwidth) was placed in front of the imaging lens. The time delay between the laser pulses varies from 8 to 20 µs according to the case. For each operation

**3. PIV system: Experimental set-up and procedure** 

from statistical studies of the instantaneous measurements sequences.

condition, up to 400 pairs of instantaneous images were collected.

Fig. 4. Schematic view of a PIV system. The CCD camera and the laser sheet are

For the non-reacting flow, the jets are seeded with olive oil particles ( 3 to 4 µm in diameter) whereas for reacting flow they are seeded with zirconium oxide (ZrO2) particles ( 5 µm diameter). The reasons for the choice of these ZrO2 particles are: a high resistance of high temperatures generated by the oxy-combustion (melting point 2715°C), a good refractive index (2.2) which allows a good light scattering, and a little size for a better mixing

perpendicular

where G is the axial flux of the tangential momentum, <sup>φ</sup> Gx is the axial flux of axial momentum, and *R* is the exit radius of the burner nozzle.

$$G\_{\wp} = \bigcap\_{0}^{R} (\mathsf{V}\mathsf{V}r) \mathsf{p} \mathsf{U} \mathsf{I} \mathsf{2}.\pi \, r \, dr \tag{3}$$

$$\mathbf{G}\_{\mathbf{x}} = \bigcap\_{0}^{R} \mathbf{U} \boldsymbol{\varrho} \boldsymbol{U} \boldsymbol{\mathcal{Q}} \boldsymbol{\omega} \, \boldsymbol{r} \, \mathrm{d}\mathbf{r} \tag{4}$$

*U* and *W* are the longitudinal and tangential components of the velocity respectively.

The corresponding geometrical swirl number, defined following the previous work (Boushaki et al. 2009) as:

$$\mathbf{S}\_{n} = \frac{\frac{\dot{\dot{m}}\_{\text{act}}}{\dot{\dot{m}}\_{\text{jet}}}}{1 + \frac{\dot{\dot{m}}\_{\text{act}}}{\dot{\dot{m}}\_{\text{jet}}}} \left[ \frac{2\left(\mathbf{d}\_{\text{act}}^{2} + 3\mathbf{R}\left(\mathbf{R} - \mathbf{d}\_{\text{act}}\right)\right)}{3\mathbf{R}\left(2\mathbf{R} - \mathbf{d}\_{\text{act}}\right)}\right] \tag{5}$$

is calculated from the flow rate of main and actuator jets ( *mjet* and *mact* ), the radius of the main jet and the jet actuators (*R* and *dact*). From *r*=0 to 30%, the swirl number varies from 0 to 0.25 for R=3 mm and to 0.26 for R=4 mm.

The exit parameters of the two studied configurations are listed in Table 1. *U°*, *Re* and *M* are the jet exit velocity, Reynolds number and initial velocity ratio ( 0 0 *U U NG Ox* / ) respectively. In the presence of jet actuators, the exit velocity of the main jet decreases when the flow rate ratio (*r*) increases since a portion of the total flow rate is injected into the tube actuators. Conversely, the exit velocity of jet actuators increases with *r*.


Table 1. Burner configurations and parameters. The notation SW3J means Swirl with three main jets (1NG and 2O2)

The swirl number is a dimensionless quantity defined as (Beèr and Chigier, 1972; Sheen et

n

0

0

*U* and *W* are the longitudinal and tangential components of the velocity respectively.

*R G U U r dr <sup>x</sup>*

 *R*

where G is the axial flux of the tangential momentum, <sup>φ</sup> Gx

momentum, and *R* is the exit radius of the burner nozzle.

n

Conversely, the exit velocity of jet actuators increases with *r*.

20

20

Jet S (mm) Gas <sup>d</sup>

 

*m m m m*

*act jet act jet*

S

0.25 for R=3 mm and to 0.26 for R=4 mm.

Number of

3 1 NG, 2 O2

2 1 NG, 1 O2 φ

G S =

( )ρ 2

ρ 2

The corresponding geometrical swirl number, defined following the previous work

3R 2R d <sup>1</sup>

is calculated from the flow rate of main and actuator jets ( *mjet* and *mact* ), the radius of the main jet and the jet actuators (*R* and *dact*). From *r*=0 to 30%, the swirl number varies from 0 to

The exit parameters of the two studied configurations are listed in Table 1. *U°*, *Re* and *M* are the jet exit velocity, Reynolds number and initial velocity ratio ( 0 0 *U U NG Ox* / ) respectively. In the presence of jet actuators, the exit velocity of the main jet decreases when the flow rate ratio (*r*) increases since a portion of the total flow rate is injected into the tube actuators.

Table 1. Burner configurations and parameters. The notation SW3J means Swirl with three

 <sup>2</sup> act act

*( )* 

2 d 3R R d

act

(mm)

Natural gas dng=6 23.7 10772

Oxygen dox=8 14.45 8198

Natural gas dng=6 23.7 10772

Oxygen dox=8 28.91 16400

U0

(m/s) Re

*( )*

*G Wr U r dr*

x

RG (2)

(3)

(4)

is the axial flux of axial

(5)

M= 0 0 *U U NG Ox* /

1.64

0.82

al. 1996):

(Boushaki et al. 2009) as:

Burner configuration

SW3J

SW2J

main jets (1NG and 2O2)

### **3. PIV system: Experimental set-up and procedure**

Particle Image Velocimetry (PIV) is a laser diagnostic method which has been developed in parallel with Laser Sheet Visualization (LSV) ). As with LSV, The PIV is based on the collection of images of Mie scattering of fine particles seeded in the flow. Processing of these particle images provides instantaneous data on two velocity components in a plane crossing the flow. Then, the mean and root-mean-square (rms) velocity fields can be easily deduced from statistical studies of the instantaneous measurements sequences.

Fig. 4 shows a schematic diagram of the PIV system. It includes a laser sheet that illuminates the zone of flow studied, a CCD camera, a PC for data acquisition and a control unit for synchronization. The laser used is double-pulsed Nd-YAG (Big Sky CFR200 Quantel) with a wavelength of 532 nm and a frequency of 10 Hz. Laser energy is adjustable and can be increased up to 150 mJ per pulse with pulse duration of 8 ns. The laser sheet is formed by a first divergent cylinder lens, which spreads out the beam then by second convergent spherical lens, which focuses the sheet. The signal of Mie scattering emitted by particles is collected perpendicularly by a CCD camera FlowMaster of Lavision (12-bit dynamic and 12801024 pixels resolution) with a 50 mm lens F/1.2 Nikkon. In reacting flow measurements, to reject the bright luminosity from the oxy-flame, an interference filter (532 nm centre, 3 nm bandwidth) was placed in front of the imaging lens. The time delay between the laser pulses varies from 8 to 20 µs according to the case. For each operation condition, up to 400 pairs of instantaneous images were collected.

Fig. 4. Schematic view of a PIV system. The CCD camera and the laser sheet are perpendicular

For the non-reacting flow, the jets are seeded with olive oil particles ( 3 to 4 µm in diameter) whereas for reacting flow they are seeded with zirconium oxide (ZrO2) particles ( 5 µm diameter). The reasons for the choice of these ZrO2 particles are: a high resistance of high temperatures generated by the oxy-combustion (melting point 2715°C), a good refractive index (2.2) which allows a good light scattering, and a little size for a better mixing

PIV Measurements on Oxy-Fuel Burners 131

units of pixel was performed. A range may be specified for each component of velocity U and V (range of ± value). Any vectors outside this range are removed. In some cases, in particular for instantaneous images, filters to refine the results are used as local median filter

For the inclined injectors (see Fig. 2), the shape of exit nozzles is elliptical rather than round. Therefore, the profile of velocity for inclined jet at the exit nozzle is quite different to the one of a straight jet. That is why results of the exit velocities are provided; they are very useful for numerical studies. Fig. 6 shows the profiles of mean velocities near the burner exit (z=3 mm) in the non-reacting flow. For the side jets, the longitudinal mean velocity, U, decreases with the angle of the oxygen jets, however, the radial velocity, V, increases as a result of the deflection of injectors. It is noted that from the velocity profiles of PIV measurements, the flow rates at the exit nozzles are calculated and are in very good agreement with the flow

> -20 -15 -10 -5 0 5 10 15 20

a b Fig. 6. Radial profiles of longitudinal (a) and radial (b) velocities at exit nozzles (z=3mm) in

Fig. 7 shows an example of instantaneous velocity fields, taken among 400 instantaneous fields, for oxygen jet angles 0° and 30°, with the longitudinal velocity U (along the vertical direction) in color scale. The burner configuration is highly three-dimensional, since some discontinuous aspects of streamlines are observed on the instantaneous velocity fields. However, this aspect is not visible in Fig. 8 because the images averaging remove the discontinuities and high gradients on the velocity distribution. For the central jet, the longitudinal velocity varies weakly and the radial velocity is nearly zero in the near nozzle field. More downstream, the side jets affect the central jet and its radial velocity is no longer zero. In the far field, vortices appeared in the region of the mixing layer between the jet and the ambient air (for θ=30). At the merging zone of three the jets, gradients in the velocity

V (m/s)


 0° 10° 20° 30°

or regional median filter based on the neighbouring vectors.

**4. PIV measurements on burners with inclined jets** 

0° 10° 20° 30°

**4.1 Instantaneous and mean velocity fields** 


non-reacting flow for the oxygen jet angle, = 0°, 10°, 20° and 30°

rate injected.

U (m/s)

in the flow. Fig.5 illustrates the two systems of seeding particles. For olive oil particles, the seeding system is an atomizer based on Venturi principle to produce fine particles. For ZrO2 particles, the system consists of tubes equipped with porous plates, placed at the level of exit gases. The gas passes through the porous medium and drags a certain quantity of particles providing a uniform seeding. Concentration of particles is controlled by valves through the gas flow rate in the line of seeding. A particular attention has been carried in the drying of particles before their injection in the seeding system in order to limit agglomerates. The measurements were conducted with seeding rates relatively low to avoid perturbing the flow. Particles in fact can modify the characteristics of the flow if they are introduced in high quantity, and can even blow out the flame. The criterion assuring a good track of flow by particles is respected here, since the Stokes number, defined as the ratio between the response time of particles and a time characteristic of the flow, obtained in the present experiments is much lower than unity (St 1). For example, *pt* 0.015ms in the case of oxygen and oxide zirconium particles ( <sup>2</sup> <sup>4</sup> 2 /9 *t r p pp CH* : <sup>3</sup> 5600 . *<sup>p</sup> kg m* , 0.5 *pr µm* , 6 <sup>4</sup> 20,18.10 . *CH Pa s* ), with the frequency of 1000Hz, the Stokes number is about 0.015.

Fig. 5. Seeding systems, for olive oil particles in non-reacting flow (left), for ZrO2 particles in reacting flow (right)

The Davis Lavision software package uses a cross-correlation technique to find the average particle displacement in each subregion (3232 pixels) of the image. The velocity is found by dividing the particle displacement by the time between laser pulses. The sub-pixel displacement was estimated by means of Gaussian peak of fitting (Lecordier and Trinité, 2003). With a maximum displacement of 8 pixels, this would correspond to less than 2% uncertainty in final velocity measurement. It was necessary to carry out a post processing to detect and correct the aberrant vectors which appear after cross-correlation calculations. Detection of false vectors can be done by the size of the vector. In this case, it is necessary to locate all vectors above a certain threshold of velocity according to the expected results. The direction of vectors can also help to identify false vectors knowing a priori the direction of flow. For that the *allowable vector range*, restricting the filtered vectors to a user specified in

in the flow. Fig.5 illustrates the two systems of seeding particles. For olive oil particles, the seeding system is an atomizer based on Venturi principle to produce fine particles. For ZrO2 particles, the system consists of tubes equipped with porous plates, placed at the level of exit gases. The gas passes through the porous medium and drags a certain quantity of particles providing a uniform seeding. Concentration of particles is controlled by valves through the gas flow rate in the line of seeding. A particular attention has been carried in the drying of particles before their injection in the seeding system in order to limit agglomerates. The measurements were conducted with seeding rates relatively low to avoid perturbing the flow. Particles in fact can modify the characteristics of the flow if they are introduced in high quantity, and can even blow out the flame. The criterion assuring a good track of flow by particles is respected here, since the Stokes number, defined as the ratio between the response time of particles and a time characteristic of the flow, obtained in the present experiments is much lower than unity (St 1). For example, *pt* 0.015ms in the case of

> <sup>4</sup> 2 /9 *t r p pp*

*CH Pa s* ), with the frequency of 1000Hz, the Stokes number is about 0.015.

Fig. 5. Seeding systems, for olive oil particles in non-reacting flow (left), for ZrO2 particles in

The Davis Lavision software package uses a cross-correlation technique to find the average particle displacement in each subregion (3232 pixels) of the image. The velocity is found by dividing the particle displacement by the time between laser pulses. The sub-pixel displacement was estimated by means of Gaussian peak of fitting (Lecordier and Trinité, 2003). With a maximum displacement of 8 pixels, this would correspond to less than 2% uncertainty in final velocity measurement. It was necessary to carry out a post processing to detect and correct the aberrant vectors which appear after cross-correlation calculations. Detection of false vectors can be done by the size of the vector. In this case, it is necessary to locate all vectors above a certain threshold of velocity according to the expected results. The direction of vectors can also help to identify false vectors knowing a priori the direction of flow. For that the *allowable vector range*, restricting the filtered vectors to a user specified in

 

*kg m* , 0.5 *pr µm* , 6

*CH* : <sup>3</sup> 5600 . *<sup>p</sup>* 

oxygen and oxide zirconium particles ( <sup>2</sup>

<sup>4</sup> 20,18.10 .

reacting flow (right)

units of pixel was performed. A range may be specified for each component of velocity U and V (range of ± value). Any vectors outside this range are removed. In some cases, in particular for instantaneous images, filters to refine the results are used as local median filter or regional median filter based on the neighbouring vectors.

#### **4. PIV measurements on burners with inclined jets**

#### **4.1 Instantaneous and mean velocity fields**

For the inclined injectors (see Fig. 2), the shape of exit nozzles is elliptical rather than round. Therefore, the profile of velocity for inclined jet at the exit nozzle is quite different to the one of a straight jet. That is why results of the exit velocities are provided; they are very useful for numerical studies. Fig. 6 shows the profiles of mean velocities near the burner exit (z=3 mm) in the non-reacting flow. For the side jets, the longitudinal mean velocity, U, decreases with the angle of the oxygen jets, however, the radial velocity, V, increases as a result of the deflection of injectors. It is noted that from the velocity profiles of PIV measurements, the flow rates at the exit nozzles are calculated and are in very good agreement with the flow rate injected.

Fig. 6. Radial profiles of longitudinal (a) and radial (b) velocities at exit nozzles (z=3mm) in non-reacting flow for the oxygen jet angle, = 0°, 10°, 20° and 30°

Fig. 7 shows an example of instantaneous velocity fields, taken among 400 instantaneous fields, for oxygen jet angles 0° and 30°, with the longitudinal velocity U (along the vertical direction) in color scale. The burner configuration is highly three-dimensional, since some discontinuous aspects of streamlines are observed on the instantaneous velocity fields. However, this aspect is not visible in Fig. 8 because the images averaging remove the discontinuities and high gradients on the velocity distribution. For the central jet, the longitudinal velocity varies weakly and the radial velocity is nearly zero in the near nozzle field. More downstream, the side jets affect the central jet and its radial velocity is no longer zero. In the far field, vortices appeared in the region of the mixing layer between the jet and the ambient air (for θ=30). At the merging zone of three the jets, gradients in the velocity

PIV Measurements on Oxy-Fuel Burners 133

From the initial state where =0° to inclined states, the structure of the dynamic field changes. As observed in the figure, the mixing of jets is more upstream for the inclined jet configurations. The merging region starts at 15 mm high for the straight jets (=0°), and more upstream for inclined jets, 7 mm for =10 ° and nearby to the burner for 20° and 30° (z=3 mm). The combined region, in which the velocity profiles combine to form a single jet profile, also starts much more upstream when the jets are angled. An increase of velocities between the jets with increasing jet angle is noted. At the height z=10 mm, the longitudinal

Radial profiles of the mean longitudinal velocity (U) at different heights from the burner for the configurations = 0°, 10°, 20° and 30° are shown in Fig. 9. For the straight jets, a classical behavior of multiple jets is found for the distribution of longitudinal velocity, maxima in the centre of jets and minima between the jets. In the near burner region (e.g. z=15 mm), the distribution of velocity shows maxima and minima corresponding to the three jets and that

> > U (m/s)

Fig. 9. Radial profiles of mean longitudinal velocity for jet angles 0°, 10°, 20° and 30° in non-

U (m/s)



z=15 mm z=35 mm z=55 z=75 mm z=95 mm z=115 mm

**= 10°**

**= 30°**

velocity is zero between the jets for =0°, whereas it is around 12 m/s for =20°.

**= 0°**

**= 20°**

**4.2 Radial profiles of mean velocities and fluctuations** 

z=15 mm z=35 mm z=55 mm z=75 mm z=95 mm z=115 mm



reacting flow at different positions from the burner

U (m/s)

U (m/s)

values are noted characterizing the three-dimensionality, particularly when the side jets are inclined. This is due to the transverse flow of jets and the elliptical shape of inclined nozzles, as it was shown in the paper of Gutmark and Grinstein (1999) where the entrainment rate and the mixing for elliptic jets are more significant compared to the round jets. Mean velocity fields, obtained by averaging 400 instantaneous images, in non-reacting flow for the jet angles 0° and 30° are illustrated in Fig.8. These results show that increasing jet angle leads to a decrease of longitudinal velocity and an increase of radial velocity for the side jets.

Fig. 7. Example of instantaneous velocity fields for = 0° (left) and = 30° (right) in nonreacting flow

Fig. 8. Mean velocity fields for = 0° (left) and = 30° (right) in non-reacting flow (with longitudinal velocity in color scale)

values are noted characterizing the three-dimensionality, particularly when the side jets are inclined. This is due to the transverse flow of jets and the elliptical shape of inclined nozzles, as it was shown in the paper of Gutmark and Grinstein (1999) where the entrainment rate and the mixing for elliptic jets are more significant compared to the round jets. Mean velocity fields, obtained by averaging 400 instantaneous images, in non-reacting flow for the jet angles 0° and 30° are illustrated in Fig.8. These results show that increasing jet angle leads to a decrease of longitudinal velocity and an increase of radial velocity for the side jets.

z (mm)

Fig. 7. Example of instantaneous velocity fields for = 0° (left) and = 30° (right) in non-

z (mm)

Fig. 8. Mean velocity fields for = 0° (left) and = 30° (right) in non-reacting flow (with

x (mm)

U (m/s)

x (mm)



U (m/s)

x (mm)

U (m/s)

x (mm)

longitudinal velocity in color scale)



U (m/s)

z (mm)

z (mm)

reacting flow

From the initial state where =0° to inclined states, the structure of the dynamic field changes. As observed in the figure, the mixing of jets is more upstream for the inclined jet configurations. The merging region starts at 15 mm high for the straight jets (=0°), and more upstream for inclined jets, 7 mm for =10 ° and nearby to the burner for 20° and 30° (z=3 mm). The combined region, in which the velocity profiles combine to form a single jet profile, also starts much more upstream when the jets are angled. An increase of velocities between the jets with increasing jet angle is noted. At the height z=10 mm, the longitudinal velocity is zero between the jets for =0°, whereas it is around 12 m/s for =20°.

#### **4.2 Radial profiles of mean velocities and fluctuations**

Radial profiles of the mean longitudinal velocity (U) at different heights from the burner for the configurations = 0°, 10°, 20° and 30° are shown in Fig. 9. For the straight jets, a classical behavior of multiple jets is found for the distribution of longitudinal velocity, maxima in the centre of jets and minima between the jets. In the near burner region (e.g. z=15 mm), the distribution of velocity shows maxima and minima corresponding to the three jets and that

Fig. 9. Radial profiles of mean longitudinal velocity for jet angles 0°, 10°, 20° and 30° in nonreacting flow at different positions from the burner

PIV Measurements on Oxy-Fuel Burners 135

to 11 m/s when varies from 0° to 30°. For the inclined jets the velocity profile is composed of two parts, one positive and one negative with a passage by zero corresponding

Fig. 11 shows the radial distribution of the turbulence intensity (u'/U) with the jet angle at vertical positions z=15 and 75 mm. Results of u'/U highlight the interaction zones between the jets and surrounding air as well as between the jets themselves. In the initial region of flow (z=15mm), four zones of high turbulence are noticed. Two of them take place on the outer region of the side jets, involving a dilution of the oxygen jets by ambient air. The other two zones are located between the jets representing the jets mixing. Near the nozzle exits, the outer zones of turbulence do not seem to be affected by the increase of jet angle, while the inner zones decrease in intensity since the merging region is reached faster by the deflection of jets. Further downstream, when the jets merge, it is found that only the outer zones of turbulence behave as a single jet. At z=95 mm, except for straight jets, it is noted that whatever is the jet angle, the turbulence intensity profile u'/U is similar owing to the


Fig. 11. Turbulence intensity (u'/U) at the vertical positions z=15mm and z=95mm for the jet

Fig. 12 shows the mean velocity fields in the reacting flow (with combustion) for jet angles 0° and 30°. These vector fields show a significant difference between non-reacting and reacting flow. The first remark concerns the more significant velocities above the stabilization point in the reacting flow. Indeed, for the straight jets (=0°), the longitudinal velocity (U) decreases along the flow, however, this decrease is less significant compared to non-reacting flow. For the inclined jets, the longitudinal velocity in combustion keeps higher values even at more significant heights from the burner. The hot environment and the presence of a reaction zone lead to a fast expansion of gases due to the presence of flame, and therefore to an acceleration of the flow. The second remark concerns a greater radial

expansion in combustion in particularly above the stabilization zone of the flame.


0° 10° 20° 30°

0

Z=95 mm

0,5

u'/U

1

1,5

2

respectively to the side jets (left and right) and the central jet of natural gas.

complete merging of jets at this position.


angles 0°, 10°, 20° and 30° in non reacting flows

0° 10° 20° 30°


0

Z=15 mm

**4.3 Velocities in oxy-fuel flames** 

0,5

u'/U

1

1,5

2

the maximum velocity decreases when the jet angle increases. More downstream, when inclining the jets, the extreme velocities begin to disappear into a single maximum located along the axis of the center jet. This combined zone of jets, characterizing a single jet, is reached earlier when the jet angle increases.

Without control (=0°), the combined zone is not reached even at z=115 mm, while for =30° it is already occurred at z=35 mm. With control, in the combined zone it is noted an acceleration of the jet along the axis of the flow. In fact, at z=75 mm, from =0° to 10° the centerline longitudinal velocity (U) increases from around 15 to 19 m/s. However, once the combined region is reached, the velocity decreases with the angle of jets. The expansion of the flow decreases with the angle in the region near to the burner, and then increases downstream the flow.

The radial velocity profiles are shown in Fig.10 at different positions from the burner in nonreacting flow. For the straight jets (=0°), the radial velocity is low and ranges from -1 to 1 m/s. The deflection of jets leads to an increase of the radial velocity of the oxygen jets, particularly near the burner since at z=15 mm, the maximum value of V varies from 0.6 m/s

Fig. 10. Radial profiles of mean radial velocity for jet angles 0°, 10°, 20° and 30° in nonreacting flow at different positions from the burner

the maximum velocity decreases when the jet angle increases. More downstream, when inclining the jets, the extreme velocities begin to disappear into a single maximum located along the axis of the center jet. This combined zone of jets, characterizing a single jet, is

Without control (=0°), the combined zone is not reached even at z=115 mm, while for =30° it is already occurred at z=35 mm. With control, in the combined zone it is noted an acceleration of the jet along the axis of the flow. In fact, at z=75 mm, from =0° to 10° the centerline longitudinal velocity (U) increases from around 15 to 19 m/s. However, once the combined region is reached, the velocity decreases with the angle of jets. The expansion of the flow decreases with the angle in the region near to the burner, and then increases

The radial velocity profiles are shown in Fig.10 at different positions from the burner in nonreacting flow. For the straight jets (=0°), the radial velocity is low and ranges from -1 to 1 m/s. The deflection of jets leads to an increase of the radial velocity of the oxygen jets, particularly near the burner since at z=15 mm, the maximum value of V varies from 0.6 m/s

> -6 -4 -2 0 2 4 6


Fig. 10. Radial profiles of mean radial velocity for jet angles 0°, 10°, 20° and 30° in non-

V (m/s)

V (m/s)



z=15 mm z=25 mm z=35 mm z=55 z=75 mm z=95 mm

**= 10°**

**= 30°**

**= 0°**

**= 20°**

reached earlier when the jet angle increases.



reacting flow at different positions from the burner

z=15 mm z=25 mm z=35 mm z=55 mm z=75 mm z=95 mm

downstream the flow.


V (m/s)


0

1

V (m/s)

to 11 m/s when varies from 0° to 30°. For the inclined jets the velocity profile is composed of two parts, one positive and one negative with a passage by zero corresponding respectively to the side jets (left and right) and the central jet of natural gas.

Fig. 11 shows the radial distribution of the turbulence intensity (u'/U) with the jet angle at vertical positions z=15 and 75 mm. Results of u'/U highlight the interaction zones between the jets and surrounding air as well as between the jets themselves. In the initial region of flow (z=15mm), four zones of high turbulence are noticed. Two of them take place on the outer region of the side jets, involving a dilution of the oxygen jets by ambient air. The other two zones are located between the jets representing the jets mixing. Near the nozzle exits, the outer zones of turbulence do not seem to be affected by the increase of jet angle, while the inner zones decrease in intensity since the merging region is reached faster by the deflection of jets. Further downstream, when the jets merge, it is found that only the outer zones of turbulence behave as a single jet. At z=95 mm, except for straight jets, it is noted that whatever is the jet angle, the turbulence intensity profile u'/U is similar owing to the complete merging of jets at this position.

Fig. 11. Turbulence intensity (u'/U) at the vertical positions z=15mm and z=95mm for the jet angles 0°, 10°, 20° and 30° in non reacting flows

#### **4.3 Velocities in oxy-fuel flames**

Fig. 12 shows the mean velocity fields in the reacting flow (with combustion) for jet angles 0° and 30°. These vector fields show a significant difference between non-reacting and reacting flow. The first remark concerns the more significant velocities above the stabilization point in the reacting flow. Indeed, for the straight jets (=0°), the longitudinal velocity (U) decreases along the flow, however, this decrease is less significant compared to non-reacting flow. For the inclined jets, the longitudinal velocity in combustion keeps higher values even at more significant heights from the burner. The hot environment and the presence of a reaction zone lead to a fast expansion of gases due to the presence of flame, and therefore to an acceleration of the flow. The second remark concerns a greater radial expansion in combustion in particularly above the stabilization zone of the flame.

PIV Measurements on Oxy-Fuel Burners 137

=20° at z=28 mm is observed. This behavior is attributed to the merging of three jets

In the case with combustion, the velocity Ucl increases with z distance in the first zone up to a maximum, then it conserves a high value, and finally, it slightly decreases along the centerline. The oxy-fuel combustion considerably influences the flow throughout the studied domain. In fact, in comparison with the non-reacting flow in the initial region, the flow is slower for both fluids, and then accelerates under the influence of high flame temperatures. For =20°, starting from z=40 mm, the centerline velocity is higher in the case of the reacting flow. After the stabilization point, the flow velocity is higher in the reactive case than that in the non reactive case. This is due both to the rapid expansion of burnt gases which accelerates the flow and to a retardation of mixing due to the presence of a flame.

In order to examine the effects of the jet actuators on the flow, the case without control (*r*=0) and a case with a control parameter of *r*=0.15 are discussed (*r* represents the ratio of volumetric flow rates of actuators *mact* and total *mtot* ). Instantaneous velocity fields for the both configurations SW3J and SW2J (see Fig.3 and table 1) in the non reacting flow are shown in Fig. 14 and 16. The mean velocity fields are illustrated in Fig. 15 and 17. The instantaneous or mean fields clearly show the significant effect of the jet actuators on the flow structure. With the activation of actuation, the length of the potential core of the jets decreases and the longitudinal velocity decays more rapidly. This happens in favour of a high jet spreading and an enhancement of mixing, which furthermore leads to the flame stabilization more upstream, as previously indicated. This confirms the results previously reported in the literature on the efficiency of swirl on the mixing between the jet and the surrounding fluid (Syred and Béer, 1974; Faivre and Poinsot, 2004). Without control (r=0),

r=0 r=0.15

z (mm)

10

O2 jet Ng jet O2 jet O2 jet Ng jet O2 jet Fig. 14. Samples of instantaneous velocity fields for the flow rate ratio *r*=0 (without control)

and *r*=0.15 (with control) in non-reacting flow for the SW3J configuration

20

30

40

50

60

x (mm)


leading to an acceleration of flow at the beginning of the combined region.

**5. PIV measurements on burners with swirling jets** 

**5.1 Velocity fields in nonreacting flow** 

x (mm)


z (mm)

10

20

30

40

50

60

Fig. 12. Mean velocity fields for the oxygen jet angle =0° (left) and =30° (right) in reacting flow

The longitudinal velocities along centerlines (Ucl) in reacting and nonreacting flow are shown in Fig.13. The results concern the central jet of natural gas and one of the two side jets of oxygen for =0° and =20°. In the case of nonreacting flow, for the straight jets (=0°), the centerline velocity (Ucl) follows a classical decrease for the central and the side jets: first, a very slight decrease down to 26 mm corresponding to the potential core length of the jet, then a high decay of Ucl which corresponds to the merging of mixing layers, and finally a slow decay down to 65 mm for the natural gas jet and 75 mm for the oxygen jet. When the side jets are inclined (=20°, nonreacting), a decrease of the length of potential core is noted, and therefore a fast decrease of the longitudinal velocity appears in the first zone of the flow in particular for the central jet. After this first part of the flow, an increase of Ucl for the cases

Fig. 13. Mean longitudinal velocity (Ucl) along the centerline of the central jet (left) and the side jet (right). A comparison between the reacting and the non reacting flow for the angles =0° and =20°

z (mm)

10

Fig. 12. Mean velocity fields for the oxygen jet angle =0° (left) and =30° (right) in reacting

The longitudinal velocities along centerlines (Ucl) in reacting and nonreacting flow are shown in Fig.13. The results concern the central jet of natural gas and one of the two side jets of oxygen for =0° and =20°. In the case of nonreacting flow, for the straight jets (=0°), the centerline velocity (Ucl) follows a classical decrease for the central and the side jets: first, a very slight decrease down to 26 mm corresponding to the potential core length of the jet, then a high decay of Ucl which corresponds to the merging of mixing layers, and finally a slow decay down to 65 mm for the natural gas jet and 75 mm for the oxygen jet. When the side jets are inclined (=20°, nonreacting), a decrease of the length of potential core is noted, and therefore a fast decrease of the longitudinal velocity appears in the first zone of the flow in particular for the central jet. After this first part of the flow, an increase of Ucl for the cases

5

Fig. 13. Mean longitudinal velocity (Ucl) along the centerline of the central jet (left) and the side jet (right). A comparison between the reacting and the non reacting flow for the angles

10

15

20

Ucl (m/s)

25

30

35

20

30

40

50

60

70

x (mm)

0 20 40 60 80 100 120 z (mm)

0°Ox-non-reacting 20°Ox-non-reacting

0°Ox 20°Ox


30 m/s

x (mm)

0 20 40 60 80 100 120 z (mm)

0°NG-non-reacting 20°NG-non-reacting

0° NG 20°NG


z (mm)

10

flow

5

=0° and =20°

10

15

20

Ucl (m/s)

25

30

35

20

30

40

50

60

70

=20° at z=28 mm is observed. This behavior is attributed to the merging of three jets leading to an acceleration of flow at the beginning of the combined region.

In the case with combustion, the velocity Ucl increases with z distance in the first zone up to a maximum, then it conserves a high value, and finally, it slightly decreases along the centerline. The oxy-fuel combustion considerably influences the flow throughout the studied domain. In fact, in comparison with the non-reacting flow in the initial region, the flow is slower for both fluids, and then accelerates under the influence of high flame temperatures. For =20°, starting from z=40 mm, the centerline velocity is higher in the case of the reacting flow. After the stabilization point, the flow velocity is higher in the reactive case than that in the non reactive case. This is due both to the rapid expansion of burnt gases which accelerates the flow and to a retardation of mixing due to the presence of a flame.

#### **5. PIV measurements on burners with swirling jets**

#### **5.1 Velocity fields in nonreacting flow**

In order to examine the effects of the jet actuators on the flow, the case without control (*r*=0) and a case with a control parameter of *r*=0.15 are discussed (*r* represents the ratio of volumetric flow rates of actuators *mact* and total *mtot* ). Instantaneous velocity fields for the both configurations SW3J and SW2J (see Fig.3 and table 1) in the non reacting flow are shown in Fig. 14 and 16. The mean velocity fields are illustrated in Fig. 15 and 17. The instantaneous or mean fields clearly show the significant effect of the jet actuators on the flow structure. With the activation of actuation, the length of the potential core of the jets decreases and the longitudinal velocity decays more rapidly. This happens in favour of a high jet spreading and an enhancement of mixing, which furthermore leads to the flame stabilization more upstream, as previously indicated. This confirms the results previously reported in the literature on the efficiency of swirl on the mixing between the jet and the surrounding fluid (Syred and Béer, 1974; Faivre and Poinsot, 2004). Without control (r=0),

Fig. 14. Samples of instantaneous velocity fields for the flow rate ratio *r*=0 (without control) and *r*=0.15 (with control) in non-reacting flow for the SW3J configuration

PIV Measurements on Oxy-Fuel Burners 139

actuators, since many discontinuities in the velocity contours are observed. The mixing point (where the jets begin to interact) moves upstream in the flow with increasing *r*. In addition, when the distance (S) between the jets decreases, the mixing point gets closer to

r=0 r=0.15

z (mm)

x (mm)


U (m/s)

O2 jet Ng jet O2 jet Ng jet

Fig. 17. Mean velocity fields for the flow rate ratio *r*=0 (without control) and *r*=0.15 (with

Fig.18 shows radial profiles of the longitudinal and radial velocities at different heights from the burner, in the non reacting flow for the SW2J configuration. These profiles are presented for the both cases without control (*r*=0) and with control (*r*=0.15). Without jet actuators, the classical distribution of the longitudinal velocity in a multiple jet configuration is found, with maxima in the centre of the jets and minima between the jets. In the initial zone, each jet follows its own evolution, and then the jets start to interact. The activation of control jets changes the flow behavior and the distribution of jet velocities. The control leads to a faster velocity decay and alters the shape of radial profiles near the burner. Indeed, at z = 35 mm for example, the maximum longitudinal velocity is 34 m/s without control, whereas in the controlled configuration it is 17 m/s. The control also accelerates the longitudinal velocity decay, as expected in as swirling flow. Near the injector exits (z=5 mm), it is observed that the maximum velocity remains high and even slightly higher when *r* increases. It can be explained by the involvement of the tangential velocity component, which increases with the flow rate control and compensates the decrease induced in main jets when *r* increases. This situation was also observed in the work of Faivre and Poinsot (2004) on a single jet. Furthermore, the asymmetry of the velocity profiles in the initial zone of flow induced by the tangential activation of control jets is observed. Far from the burner, this asymmetry

The influence of the jet actuators on the flow behaviours also concerns the other velocity components, in particular in the initial zone of the flow. Fig.18.c-d shows that the radial velocity increases in the case of controlled jets. Without control, V is low (-1 V +1 m/s), whereas with control the maximum radial velocity can reach 6 m/s for *r*=0.15 and 12 m/s

the burner.

z (mm)

x (mm)

control) in non-reacting flow for the SW2J configuration

however disappears and the profiles become axisymmetric.

for *r*=0.2 near the burner.


U (m/s)

Fig. 15. Mean velocity fields for the flow rate ratio *r*=0 (without control) and *r*=0.15 (with control) in non-reacting flow for the SW3J configuration

Fig. 16. Samples of instantaneous velocity fields for the flow rate ratio *r*=0 (without control) and *r*=0.15 (with control) in non-reacting flow for the SW2J configuration

the jets are well organized with a proper development for each jet in the initial zone of the flow and have a slight velocity variation at the centre. With control (r=0.15), instantaneous flow becomes highly disorganized and velocity gradients become important. In addition, the acceleration and the deflection of velocity vectors clearly appear in the whole field of view. The potential core of the controlled jets is definitely disturbed and its length decreases as a result of the jet actuators. For high swirl intensity (*r* 0.2), the jets are more affected by the control, highly disorganized and develop also in the transverse plane (x,y); consequently the generated flame would be very oscillating and unstable (see Boushaki et al.2009). It is also noted the strong three-dimensional aspect of the flow, especially when using the jet

r=0 r=0.15

z (mm)

O2 jet Ng jet O2 jet O2 jet Ng jet O2 jet Fig. 15. Mean velocity fields for the flow rate ratio *r*=0 (without control) and *r*=0.15 (with

r=0 r=0.15

z (mm)

O2 jet Ng jet O2 jet Ng jet

and *r*=0.15 (with control) in non-reacting flow for the SW2J configuration

Fig. 16. Samples of instantaneous velocity fields for the flow rate ratio *r*=0 (without control)

the jets are well organized with a proper development for each jet in the initial zone of the flow and have a slight velocity variation at the centre. With control (r=0.15), instantaneous flow becomes highly disorganized and velocity gradients become important. In addition, the acceleration and the deflection of velocity vectors clearly appear in the whole field of view. The potential core of the controlled jets is definitely disturbed and its length decreases as a result of the jet actuators. For high swirl intensity (*r* 0.2), the jets are more affected by the control, highly disorganized and develop also in the transverse plane (x,y); consequently the generated flame would be very oscillating and unstable (see Boushaki et al.2009). It is also noted the strong three-dimensional aspect of the flow, especially when using the jet

x (mm)

x (mm)



U (m/s)

30 m/s

x (mm)

x (mm)


control) in non-reacting flow for the SW3J configuration


U (m/s)

z (mm)

z (mm)

actuators, since many discontinuities in the velocity contours are observed. The mixing point (where the jets begin to interact) moves upstream in the flow with increasing *r*. In addition, when the distance (S) between the jets decreases, the mixing point gets closer to the burner.

Fig. 17. Mean velocity fields for the flow rate ratio *r*=0 (without control) and *r*=0.15 (with control) in non-reacting flow for the SW2J configuration

Fig.18 shows radial profiles of the longitudinal and radial velocities at different heights from the burner, in the non reacting flow for the SW2J configuration. These profiles are presented for the both cases without control (*r*=0) and with control (*r*=0.15). Without jet actuators, the classical distribution of the longitudinal velocity in a multiple jet configuration is found, with maxima in the centre of the jets and minima between the jets. In the initial zone, each jet follows its own evolution, and then the jets start to interact. The activation of control jets changes the flow behavior and the distribution of jet velocities. The control leads to a faster velocity decay and alters the shape of radial profiles near the burner. Indeed, at z = 35 mm for example, the maximum longitudinal velocity is 34 m/s without control, whereas in the controlled configuration it is 17 m/s. The control also accelerates the longitudinal velocity decay, as expected in as swirling flow. Near the injector exits (z=5 mm), it is observed that the maximum velocity remains high and even slightly higher when *r* increases. It can be explained by the involvement of the tangential velocity component, which increases with the flow rate control and compensates the decrease induced in main jets when *r* increases. This situation was also observed in the work of Faivre and Poinsot (2004) on a single jet. Furthermore, the asymmetry of the velocity profiles in the initial zone of flow induced by the tangential activation of control jets is observed. Far from the burner, this asymmetry however disappears and the profiles become axisymmetric.

The influence of the jet actuators on the flow behaviours also concerns the other velocity components, in particular in the initial zone of the flow. Fig.18.c-d shows that the radial velocity increases in the case of controlled jets. Without control, V is low (-1 V +1 m/s), whereas with control the maximum radial velocity can reach 6 m/s for *r*=0.15 and 12 m/s for *r*=0.2 near the burner.

PIV Measurements on Oxy-Fuel Burners 141

U cl (m/s)

Fig. 19. Mean longitudinal velocity along the centerline of the natural gas jet (left) and the oxygen jet (right) for flow rate ratios (*r*), 0, 0.1, 0.15 and 0.2. Case: configuration SW2J, non-

The activation of control jets changes the flow behaviour and the distribution of jet velocities. The decrease of centreline velocity is faster near the burner, and then it further slows downstream. It appears that the greater the flow rate ratio is, the higher the centreline velocity decays. Another important element concerns the potential core of jets, which is strongly affected by the control. Indeed, for *r*=0.1, the length of the potential core is very small compared to the cases without actuators. Between the two cases (*r*=0 and 0.1), the length of the potential core for the oxygen jet decreases from 4dox to 2.25dox. Over for *r*=0.1, the plateau of the potential core is completely disappeared and the centres of jets are reached closer to the exit nozzles, which is caused by jet actuators impacting the main jets with high velocities.

The radial profiles of rms velocities, u' (longitudinal velocity fluctuations) and v' (radial velocity fluctuations) for the SW2J configuration at various heights of the flow with and without control are shown in Fig.20. The results indicate the significant effect of the jet actuators, since the intensity of velocity fluctuations increases in the case *r*=0.15. In the case without actuators, the radial distribution of u' and v' with peaks of turbulence is found. The maximum values of u' are obtained close to the burner, and are of about 6 and 7 m/s for natural gas jet and oxygen jet, respectively (a and c). Further downstream, the intensity of the fluctuations is attenuated in turbulence zones. Indeed, at z=75 mm, u' is about 3 and 4 m/s in the natural gas and oxygen jets, respectively. In the case with actuators, the values of u' and v' are more significant (b and d). In fact, for *r*=0.15, the maximal values of u' are 10 and 6 m/s for the fuel and oxidizer. For z=15 mm, a high increase of fluctuations appears along the jets axis, affected by the tangential jets. This shows the increase of turbulence zones near the nozzle

The control effect on the fluctuations of radial velocity (v') is more pronounced as shown in Fig.20.d. The fact of introducing tangentially a portion of jet flow rate tends to influence the initial zone of the flow and induces high radial velocity fluctuations. It is shown that maximum values of v' are of the order of 4 to 5.5 m/s with actuators (d), instead of 1.5 to 2.5

exits, favouring mixing with the ambient air and between the jets themselves.

m/s for the case without jets actuators (c) in the zone close to the burner.

0 20 40 60 80 100 z (mm)

Ox-0 Ox-0.1 Ox-0.15 Ox-0.2

NG-0 NG-0.1 NG-0.15

reacting flow

U cl (m/s)

> 0 20 40 60 80 100 z (mm)

**5.3 Radials profiles of velocity fluctuations** 

a) U (*r* = 0) b) U (*r* = 0.15)

Fig. 18. Radial profiles of mean velocities for the SW2J configuration in the non reacting flow, without control (*r*=0) and with control (*r*=0.15); a-b) longitudinal component (U), c-d) radial component (V).

#### **5.2 Axial profiles of mean velocities**

c) V (*r* = 0) d) V (*r* = 0.15)

Fig.19 shows longitudinal velocities along centrelines of natural gas and oxygen jets for the flow rate ratios r=0, 0.1, 0.15 and 0.2 in the non-reacting flow for the SW2J configuration. Without control, the velocity evolution along the centreline (Ucl) is almost similar to the one of a simple jet. First, it can exhibit a plateau corresponding to the potential core of the jet, followed by a pronounced decay of Ucl (mixing layers merging), and finally a slow decay. It is noted that the initial plateau is longer for the oxygen jet on account of greater jet diameter (dox= 8 mm, dng= 6 mm) with a slight difference in injection velocities of the two fluids.


O2 jet O2 jet NG jet NG jet

Fig. 18. Radial profiles of mean velocities for the SW2J configuration in the non reacting flow, without control (*r*=0) and with control (*r*=0.15); a-b) longitudinal component (U), c-d)

Fig.19 shows longitudinal velocities along centrelines of natural gas and oxygen jets for the flow rate ratios r=0, 0.1, 0.15 and 0.2 in the non-reacting flow for the SW2J configuration. Without control, the velocity evolution along the centreline (Ucl) is almost similar to the one of a simple jet. First, it can exhibit a plateau corresponding to the potential core of the jet, followed by a pronounced decay of Ucl (mixing layers merging), and finally a slow decay. It is noted that the initial plateau is longer for the oxygen jet on account of greater jet diameter (dox= 8 mm, dng= 6 mm) with a slight difference in injection

V (m/s)

U (m/s)


z=55 mm z=75 mm

z=5 mm z=15 mm z=35 mm


z=5 mm z=15 mm z=35 mm z=55 mm z=75 mm z=95 mm

*r = 0.15*

NG jet

*r = 0.15*

*r = 0*

O2 jet O2 jet NG jet

*r = 0*



V (m/s)

radial component (V).

velocities of the two fluids.

0

1

2

U (m/s)


z=55 mm z=75 mm


**5.2 Axial profiles of mean velocities** 

a) U (*r* = 0) b) U (*r* = 0.15)

c) V (*r* = 0) d) V (*r* = 0.15)

z=5 mm z=15 mm z=35 mm

z=5 mm z=15 mm z=35 mm z=55 mm z=75 mm z=95 mm

Fig. 19. Mean longitudinal velocity along the centerline of the natural gas jet (left) and the oxygen jet (right) for flow rate ratios (*r*), 0, 0.1, 0.15 and 0.2. Case: configuration SW2J, nonreacting flow

The activation of control jets changes the flow behaviour and the distribution of jet velocities. The decrease of centreline velocity is faster near the burner, and then it further slows downstream. It appears that the greater the flow rate ratio is, the higher the centreline velocity decays. Another important element concerns the potential core of jets, which is strongly affected by the control. Indeed, for *r*=0.1, the length of the potential core is very small compared to the cases without actuators. Between the two cases (*r*=0 and 0.1), the length of the potential core for the oxygen jet decreases from 4dox to 2.25dox. Over for *r*=0.1, the plateau of the potential core is completely disappeared and the centres of jets are reached closer to the exit nozzles, which is caused by jet actuators impacting the main jets with high velocities.

#### **5.3 Radials profiles of velocity fluctuations**

The radial profiles of rms velocities, u' (longitudinal velocity fluctuations) and v' (radial velocity fluctuations) for the SW2J configuration at various heights of the flow with and without control are shown in Fig.20. The results indicate the significant effect of the jet actuators, since the intensity of velocity fluctuations increases in the case *r*=0.15. In the case without actuators, the radial distribution of u' and v' with peaks of turbulence is found. The maximum values of u' are obtained close to the burner, and are of about 6 and 7 m/s for natural gas jet and oxygen jet, respectively (a and c). Further downstream, the intensity of the fluctuations is attenuated in turbulence zones. Indeed, at z=75 mm, u' is about 3 and 4 m/s in the natural gas and oxygen jets, respectively. In the case with actuators, the values of u' and v' are more significant (b and d). In fact, for *r*=0.15, the maximal values of u' are 10 and 6 m/s for the fuel and oxidizer. For z=15 mm, a high increase of fluctuations appears along the jets axis, affected by the tangential jets. This shows the increase of turbulence zones near the nozzle exits, favouring mixing with the ambient air and between the jets themselves.

The control effect on the fluctuations of radial velocity (v') is more pronounced as shown in Fig.20.d. The fact of introducing tangentially a portion of jet flow rate tends to influence the initial zone of the flow and induces high radial velocity fluctuations. It is shown that maximum values of v' are of the order of 4 to 5.5 m/s with actuators (d), instead of 1.5 to 2.5 m/s for the case without jets actuators (c) in the zone close to the burner.

PIV Measurements on Oxy-Fuel Burners 143

moves upstream with the control and locates in the zone inter-jets where the velocity is low. Note that the stabilization point has been deduced from OH\* emission measurements (see Bouhaki et al. 2009) and represents the region where the combustion starts. Without actuators (*r*=0), the flow velocity decreases along the axis but this decrease is slower than in the case of non-reacting flow, in particular for oxygen jet. Between z=5 mm and 95 mm, the maximum velocity of the oxygen jet decreases from 33 to 27 m/s, whereas in non-reacting flow, the maximum velocity passes from 36 to 18 m/s. In non-reactive flow, mixing and turbulence develop faster which generates a faster decrease of velocity. It is shown that in the initial zone of the flow, the maximal velocity is higher in non-reacting flow than in the reacting flow. The velocity profiles in combustion are slightly more flattened and more open, due to the high heat release from oxy-fuel flame. With control, the longitudinal velocity decay and radial spreading are more significant when the flow rate ratio *r* increases. As in the case without control, the velocity decay is slower with control in the reacting flow compared to the non-reacting flow. Above the stabilization point, the flow keeps a higher velocity owing to the fast expansion of hot gas by the reaction. As an example, for z=55 mm, the maximum velocity of reacting flow is about 21 against 12 m/s in the non-reacting flow. The presence of flame may decrease the entrainment of ambient fluid, which accelerates the flow. This result was observed by Takaji et al. (1981) on a turbulent H2-N2/Air flame by

The enhancement of mixing by the jet actuators leads to a decrease in lift-off heights and a better stability of the flame as shown in Boushaki et al. (2009). However, the flame length decreases with the flow rate of jet control since the mixing is improved by using swirling flow. In practical systems the flame length is an important factor since it defines the distance on which the heat transfer is transmitted. On the other hand, for this control system in the range 0 *r* 0.2, the flame length remains relatively higher. Moreover, in this range of low

r=0 r=0.15

z (mm)

Fig. 21. Mean velocity fields for the SW2J configuration in reacting flow, without control (left) and with control. The point in pink color represents the position of the flame base

x (mm)


comparison between non-reacting and reacting flow.

x (mm)


100 U (m/s)

z (mm)

flow rate ratio, globally NOx production decreases when *r* increases.

Fig. 20. Radial profiles of rms velocities for the SW2J configuration in non reacting flows without control (*r*=0) and with control (*r*=0.15); a-b) longitudinal velocity fluctuations (u'), cd) radial velocity fluctuations (v')

#### **5.4 Velocity fields in oxy-fuel flames**

In this section, the measurements of velocities in oxy-fuel combustion by PIV technique in the furnace are presented. Fig.21 shows the mean velocity fields (with the longitudinal velocity in color scale) for the SW2J configuration without control (*r*=0) and with control (*r*=0.15). The radial profiles are shown in Fig.22. These results indicate that the velocity field is also affected by the use of jet actuators. In the presence of jet actuators, a higher decrease and wider spreading of the flow are observed when the flow rate control *r* increases. The merging and combined zones of the jets occur more and more upstream with increasing *r*. Therefore, the stabilization point (small squares in pink color in the figure) of the flame

v' (m/s)

Fig. 20. Radial profiles of rms velocities for the SW2J configuration in non reacting flows without control (*r*=0) and with control (*r*=0.15); a-b) longitudinal velocity fluctuations (u'), c-

In this section, the measurements of velocities in oxy-fuel combustion by PIV technique in the furnace are presented. Fig.21 shows the mean velocity fields (with the longitudinal velocity in color scale) for the SW2J configuration without control (*r*=0) and with control (*r*=0.15). The radial profiles are shown in Fig.22. These results indicate that the velocity field is also affected by the use of jet actuators. In the presence of jet actuators, a higher decrease and wider spreading of the flow are observed when the flow rate control *r* increases. The merging and combined zones of the jets occur more and more upstream with increasing *r*. Therefore, the stabilization point (small squares in pink color in the figure) of the flame

u' (m/s)

NG jet NG jet


O2 jet

O2 jet

z=5 mm z=15 mm z=35 mm z=75 mm


z=5 mm z=15 mm z=35 mm z=75 mm

*r = 0.15*

*r = 0.15*

NG jet

*r = 0*

*r = 0*

NG jet

v' (m/s)

u' (m/s)


O2 jet

z=5 mm z=15 mm z=35 mm z=75 mm


O2 jet

d) radial velocity fluctuations (v')

**5.4 Velocity fields in oxy-fuel flames** 

c) v' (*r* = 0) d) v' (*r* = 0.15)

a) u' (*r* = 0) b) u' (*r* = 0.15)

z=5 mm z=15 mm z=35 mm z=75 mm moves upstream with the control and locates in the zone inter-jets where the velocity is low. Note that the stabilization point has been deduced from OH\* emission measurements (see Bouhaki et al. 2009) and represents the region where the combustion starts. Without actuators (*r*=0), the flow velocity decreases along the axis but this decrease is slower than in the case of non-reacting flow, in particular for oxygen jet. Between z=5 mm and 95 mm, the maximum velocity of the oxygen jet decreases from 33 to 27 m/s, whereas in non-reacting flow, the maximum velocity passes from 36 to 18 m/s. In non-reactive flow, mixing and turbulence develop faster which generates a faster decrease of velocity. It is shown that in the initial zone of the flow, the maximal velocity is higher in non-reacting flow than in the reacting flow. The velocity profiles in combustion are slightly more flattened and more open, due to the high heat release from oxy-fuel flame. With control, the longitudinal velocity decay and radial spreading are more significant when the flow rate ratio *r* increases. As in the case without control, the velocity decay is slower with control in the reacting flow compared to the non-reacting flow. Above the stabilization point, the flow keeps a higher velocity owing to the fast expansion of hot gas by the reaction. As an example, for z=55 mm, the maximum velocity of reacting flow is about 21 against 12 m/s in the non-reacting flow. The presence of flame may decrease the entrainment of ambient fluid, which accelerates the flow. This result was observed by Takaji et al. (1981) on a turbulent H2-N2/Air flame by comparison between non-reacting and reacting flow.

The enhancement of mixing by the jet actuators leads to a decrease in lift-off heights and a better stability of the flame as shown in Boushaki et al. (2009). However, the flame length decreases with the flow rate of jet control since the mixing is improved by using swirling flow. In practical systems the flame length is an important factor since it defines the distance on which the heat transfer is transmitted. On the other hand, for this control system in the range 0 *r* 0.2, the flame length remains relatively higher. Moreover, in this range of low flow rate ratio, globally NOx production decreases when *r* increases.

Fig. 21. Mean velocity fields for the SW2J configuration in reacting flow, without control (left) and with control. The point in pink color represents the position of the flame base

PIV Measurements on Oxy-Fuel Burners 145

The results about the active control show that the presence of tangential jet actuators appreciably acts on the structure of the flow and consequently on the flame behaviour. The control by jet actuators induces a decrease in the length of potential core of jets, more important spreading of flow and higher longitudinal velocity decay. It is observed that for *r* (flow rate ratio) > 0.1 the potential core disappears completely due to the jet actuation. The center of the jet is reached closer to the exit of nozzles as a result of jet actuators that impact with high velocity the main jets. Moreover, the increase of control flow rate accelerates the merging of the jets and the flow rapidly reaches the characteristics of a single jet. Near the burner, the transverse velocity increases with the control and can reach 10 m/s due to the tangential injection by actuators. Velocity fluctuations (longitudinal and radial) increase with the control, and therefore the layers of turbulence widen significantly and promote mixing with the ambient medium and between the jets themselves. From non controlled jets (*r*=0) to controlled jets (*r*>0), the turbulence intensity along the jet axis highly increases, in particular near the exit of jets (from 5 to 25% in the configuration SW2J). Measurements in combustion revealed some changes on the flow velocity fields. The dynamic development of oxy-fuel flame is highly slowed by the presence of the high temperature environment, since the decrease in velocities is slower compared to the nonreactive case. The evolution of the longitudinal velocity shows that even in the presence of control, the values of Ucl remain

high near the burner, and then decrease slowly along the flow development.

This work was supported by the CRCD (Centre de Recherche Claude-Delorme) of Air Liquide, Jouy-en-Josas, France. The authors are grateful to Bernard Labegorre for useful

Ahuja, K.K. (1993). Mixing enhancement and jet noise reduction through tabs plus ejectors.

Barrère, M. & Williams, F. (1968). Comparison of Combustion Instabilities found in Various

Baukal, CE. & Gebhart, B. (1997). Oxygen-enhanced/natural gas flame radiation.

Beér, J. M. & Chigier N. A. (1972). Combustion Aerodynamics, ed. Krieger, Malabar, Florida. Bloxsidge, G J; Dowling, A P.; Hooper, N. & and Langhorne, P.J. (1987) Active control of an acoustically driven combustion instability *J. Theor. Appl. Mech.* 6, pp. 161–75. Boushaki, T.; Sautet, J.C; Salentey, L. & Labegorre, B. (2007)., The behaviour of lifted oxy-

Boushaki, T; Mergheni, M-A. ; Sautet, JC. & Labegorre, B. (2008). Effects of inclined jets on

*International Journal of Heat and Mass Transfer* 40(11): 2539-2547.

Baukal, CE. (2003). Industrial burners handbook, CRC Press

Types of Combustion Chambers, *Proceedings of the Combustion Institute*, 12, pp. 169-

fuel flames in burners with separated jets, *International communication in Heat and* 

turbulent oxy-flame characteristics in a triple jet burner, *Exp. Therm. Fluid Sci*.

**7. Acknowledgment** 

*AIAA Paper*, 93-4347.

*Mass Transfer*, 34, pp 8-18.

32:1363-1370.

discussions.

**8. References** 

181.

Fig. 22. Radial profiles of mean longitudinal velocity for the SW2 configuration in reacting flow, a) without control (*r*=0), b) with control (*r*=0.15).

#### **6. Conclusion**

The dynamic study by PIV measurements on oxy-fuel burners with separated jets is investigated in this chapter. Measurements by PIV technique enable the characterization of the behavior and structure of jets and the distribution of velocity fields. Particular attention has been paid to the particle seeding, oil particles for non-reacting configuration and solid ZrO2 particles in combustion. The measurements of the velocity fields are fulfilled on a burner composed of three jets, one central jet of natural gas and two side jets of pure oxygen. Measurements concern the nonreacting flow and the reacting flow in turbulent diffusion flames inside a combustion chamber. Two types of control of flows are developed and applied on the basic configuration of burner. A passive control based on the deflection of jets, and an active control that consists of four small jet actuators, placed tangentially to the exit of the main jets to generate a swirling flow.

Results of passive control show that the inclination of side jets towards the central jet improves the mixing and thus accelerates the merging, and then the combining of jets where velocity profiles become uniform to form a single jet profile. The deflection of injectors induces a radial velocity in the first zone of flow increasing with the jet angle. Along the flow, this velocity decreases since the side jets impact the central jet and the flow forms a single jet. With deflection of jets, the potential core length decreases, and therefore the longitudinal velocity decreases rapidly in the first zone of the flow with the jet angle. The oxy-fuel combustion generates significant changes on the distribution of the dynamic fields. The velocities are higher compared to non-reacting flow, in particular above the stabilization region. This is due both to the rapid expansion of burnt gases which accelerates the flow and a retardation of mixing due to the presence of the flame. Also, a greater radial expansion in reacting flow, particularly above the lift-off position of flame, was observed.

Fig. 22. Radial profiles of mean longitudinal velocity for the SW2 configuration in reacting

The dynamic study by PIV measurements on oxy-fuel burners with separated jets is investigated in this chapter. Measurements by PIV technique enable the characterization of the behavior and structure of jets and the distribution of velocity fields. Particular attention has been paid to the particle seeding, oil particles for non-reacting configuration and solid ZrO2 particles in combustion. The measurements of the velocity fields are fulfilled on a burner composed of three jets, one central jet of natural gas and two side jets of pure oxygen. Measurements concern the nonreacting flow and the reacting flow in turbulent diffusion flames inside a combustion chamber. Two types of control of flows are developed and applied on the basic configuration of burner. A passive control based on the deflection of jets, and an active control that consists of four small jet actuators, placed tangentially to

Results of passive control show that the inclination of side jets towards the central jet improves the mixing and thus accelerates the merging, and then the combining of jets where velocity profiles become uniform to form a single jet profile. The deflection of injectors induces a radial velocity in the first zone of flow increasing with the jet angle. Along the flow, this velocity decreases since the side jets impact the central jet and the flow forms a single jet. With deflection of jets, the potential core length decreases, and therefore the longitudinal velocity decreases rapidly in the first zone of the flow with the jet angle. The oxy-fuel combustion generates significant changes on the distribution of the dynamic fields. The velocities are higher compared to non-reacting flow, in particular above the stabilization region. This is due both to the rapid expansion of burnt gases which accelerates the flow and a retardation of mixing due to the presence of the flame. Also, a greater radial expansion in reacting flow, particularly above the lift-off position of

U (m/s)


z=5 mm z=15 mm z=35 mm z=55 mm z=75 mm z=95 mm

*r = 0.15*

*r = 0*

**6. Conclusion** 

flame, was observed.

U (m/s)


a) U, (*r*=0) b) U, (*r*=0.15)

flow, a) without control (*r*=0), b) with control (*r*=0.15).

the exit of the main jets to generate a swirling flow.

z=5 mm z=15 mm z=35 mm z=55 mm z=75 mm z=95 mm The results about the active control show that the presence of tangential jet actuators appreciably acts on the structure of the flow and consequently on the flame behaviour. The control by jet actuators induces a decrease in the length of potential core of jets, more important spreading of flow and higher longitudinal velocity decay. It is observed that for *r* (flow rate ratio) > 0.1 the potential core disappears completely due to the jet actuation. The center of the jet is reached closer to the exit of nozzles as a result of jet actuators that impact with high velocity the main jets. Moreover, the increase of control flow rate accelerates the merging of the jets and the flow rapidly reaches the characteristics of a single jet. Near the burner, the transverse velocity increases with the control and can reach 10 m/s due to the tangential injection by actuators. Velocity fluctuations (longitudinal and radial) increase with the control, and therefore the layers of turbulence widen significantly and promote mixing with the ambient medium and between the jets themselves. From non controlled jets (*r*=0) to controlled jets (*r*>0), the turbulence intensity along the jet axis highly increases, in particular near the exit of jets (from 5 to 25% in the configuration SW2J). Measurements in combustion revealed some changes on the flow velocity fields. The dynamic development of oxy-fuel flame is highly slowed by the presence of the high temperature environment, since the decrease in velocities is slower compared to the nonreactive case. The evolution of the longitudinal velocity shows that even in the presence of control, the values of Ucl remain high near the burner, and then decrease slowly along the flow development.

#### **7. Acknowledgment**

This work was supported by the CRCD (Centre de Recherche Claude-Delorme) of Air Liquide, Jouy-en-Josas, France. The authors are grateful to Bernard Labegorre for useful discussions.

#### **8. References**


PIV Measurements on Oxy-Fuel Burners 147

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Menon, R. & Gollahali, SR. (1998). Combustion characteristics of interacting multiple jets in

Moawad Ahmed K.; Rajaratnam, N. & Stanley, SJ. (2001). Mixing with multiple circular

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Sautet, JC; Boushaki, T.; Salentey, L. & Labegorre, B. (2006). Oxy-combustion properties of

Schmittel, P.; Günther, B.; Lenze, B.; Leuckel, W. & Bockhorn, H. (2000). Turbulent swirling

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**0**

**6**

**Characterization of the Bidirectional Vortex**

The practical applications and theoretical difficulties associated with swirl-dominated flow fields have turned them into a source of constant scientific inquiry. Although some modern research efforts can be traced back to the piecewise formulation developed by Rankine (1858), an understanding of fundamental aspects of vortex dynamics has been demonstrated as far back as in ancient Greece (Vatistas, 2009). In fact, a multitude of naturally occurring phenomena such as hurricanes and tornadoes (Penner, 1972), as well as celestial events such as galactic jets are often modeled as vortex-dominated spiraling motions (Kirshner, 2004; Königl,

Confined vortex modeling involves small-scale helical motions that can be affected by the presence of viscous resistance. In naturally occurring vortices, an inviscid model is often sufficient as the natural length scales of the vortex substantially exceed the dimensions of the fluid regions influenced by viscous effects. However, in a confined vortex environment, the length scales are greatly reduced and therefore accounting for viscous effects becomes vital,

A wealth of literature exists on confined vortex studies. For example, the efficiency of cyclone separators has been the focus of an investigation by ter Linden (1949). Bloor & Ingham (1987) have also introduced an incompressible formulation for a conical separator in spherical coordinates. In addition to these practical applications, the confined vortex possesses important academic value. As far as stability of unidirectional vortices is concerned, Rusak et al. (1998) describe the evolution of a perturbed vortex in a pipe in an attempt to characterize axisymmetric vortex breakdown. This work is further extended by Rusak & Lee (2004) to include compressible vortices. The intention of these studies is to not only characterize confined vortex breakdown, but to also extend the mechanisms entailed in

While only few analytical models have been proposed for describing the various swirl-dominated solutions of a confined vortex, there exists a significant body of literature that is devoted to experimental investigations. These studies can be roughly separated depending on the methods employed in their data collection: probes, Laser Doppler Velocimetry (LDV),

**1. Introduction**

especially near the axis of rotation.

**1.1 Experimental studies**

and Particle Image Velocimetry (PIV).

generalized vortex breakdown to unconfined vortices.

1986).

**Using Particle Image Velocimetry**

Brian A. Maicke and Joseph Majdalani *University of Tennessee Space Institute*

*United States of America*


## **Characterization of the Bidirectional Vortex Using Particle Image Velocimetry**

Brian A. Maicke and Joseph Majdalani *University of Tennessee Space Institute United States of America*

#### **1. Introduction**

148 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Tamburello, DA. & Amitay, M. (2008). Manipulation of an Axisymmetric Jet by a Single Synthetic Jet Actuator. *International Journal of Heat and Fluid Flow* 29, pp. 967-984. Yimer, I.; Becker, HA. & Grandmaison, EW. (1996). Development of flow from multiple jet

burner, *A.I.C.H.E Journal* 74, pp. 840-851.

The practical applications and theoretical difficulties associated with swirl-dominated flow fields have turned them into a source of constant scientific inquiry. Although some modern research efforts can be traced back to the piecewise formulation developed by Rankine (1858), an understanding of fundamental aspects of vortex dynamics has been demonstrated as far back as in ancient Greece (Vatistas, 2009). In fact, a multitude of naturally occurring phenomena such as hurricanes and tornadoes (Penner, 1972), as well as celestial events such as galactic jets are often modeled as vortex-dominated spiraling motions (Kirshner, 2004; Königl, 1986).

Confined vortex modeling involves small-scale helical motions that can be affected by the presence of viscous resistance. In naturally occurring vortices, an inviscid model is often sufficient as the natural length scales of the vortex substantially exceed the dimensions of the fluid regions influenced by viscous effects. However, in a confined vortex environment, the length scales are greatly reduced and therefore accounting for viscous effects becomes vital, especially near the axis of rotation.

A wealth of literature exists on confined vortex studies. For example, the efficiency of cyclone separators has been the focus of an investigation by ter Linden (1949). Bloor & Ingham (1987) have also introduced an incompressible formulation for a conical separator in spherical coordinates. In addition to these practical applications, the confined vortex possesses important academic value. As far as stability of unidirectional vortices is concerned, Rusak et al. (1998) describe the evolution of a perturbed vortex in a pipe in an attempt to characterize axisymmetric vortex breakdown. This work is further extended by Rusak & Lee (2004) to include compressible vortices. The intention of these studies is to not only characterize confined vortex breakdown, but to also extend the mechanisms entailed in generalized vortex breakdown to unconfined vortices.

#### **1.1 Experimental studies**

While only few analytical models have been proposed for describing the various swirl-dominated solutions of a confined vortex, there exists a significant body of literature that is devoted to experimental investigations. These studies can be roughly separated depending on the methods employed in their data collection: probes, Laser Doppler Velocimetry (LDV), and Particle Image Velocimetry (PIV).

Using Particle Image Velocimetry 3

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 151

consider the full geometry of the separator including the inlet, hoppers, and other supporting hardware. Moreover, their experimental investigation is accompanied by a turbulent computational solution. In addition to verifying turbulent vortex models, their study aims

Along similar lines, an investigation into the turbulent kinetic energy of a confined vortex is reported in a forward scatter LDV study by Yan et al. (2000). In this work, data collected at a wide range of Reynolds numbers is used to validate their empirically derived solutions. These particular models rely on scaling laws to reduce the problem's dependence in each case to one or two key parameters and these tend to involve some combination of the inlet flow rate and

PIV is another minimally intrusive technique that will be discussed in the remainder of this chapter. Much like LDV, PIV employs particle seeding to collect velocity measurements. The primary difference between the two techniques stands in how the data is gathered. Whereas LDV relies on two focused beams to generate interference patterns, PIV uses optics to create a laser sheet that illuminates a plane in the chamber. High-speed cameras are then utilized to capture images of the illuminated particles at two closely spaced intervals such that a net

By way of comparison, both PIV and LDV methods are used by Sousa (2008) to determine the velocity field that accompanies vortex breakdown in a closed container. Sousa finds that accurate measurements may be acquired using either method; he also reports several challenges that may be associated with PIV techniques. The fully three-dimensional nature of the flow field can lead to a decrease in correlation accuracy as seed particles move normal to the light sheet. Sousa accounts for this factor by shortening the duration between laser pulses and by slightly thickening the laser sheet to increase the chances that the particles of interest

In the spirit of improvement, Zhang & Hugo (2006) use a stereoscopic PIV setup to investigate the vortex motion in a pipe. Stereoscopic PIV can be used to capture the fully three-dimensional flow field; however, it requires an additional high speed camera and more elaborate calibration skills to ensure that both cameras will target the same area. This obviously leads to an increase in post-experimental processing as the images from two cameras have to be analyzed for each exposure, effectively doubling the amount of data acquired. Finally, Zhang & Hugo (2006) develop and implement an improved calibration technique to reduce the optical distortion caused by refraction through the fluid and the

While a large body of literature may be reported on vortical motions, the segment devoted to analytical modeling remains much smaller in comparison. For this reason, a brief introduction

In examining the classic models for describing swirl-dominated flowfields, it is helpful to further discriminate between solutions by specifying whether the attendant equations allow for unidirectional or multidirectional swirl. Unidirectional models are characterized

velocity profile may be deduced from the cross-correlation of these images.

at improving the prediction of cyclone efficiency.

the contraction ratio.

**1.1.3 Particle Image Velocimetry**

will remain in the area of investigation.

to some of these classical models will follow.

curved chamber wall.

**1.2 Classical solutions**

#### **1.1.1 Probes**

Within the context of cyclone separators, the experimental study by Smith (1962b) employs a glass tube filled with smoke particles to capture the general structure of a confined swirling flow. Smith also utilizes a special slender probe stretched across the diameter of the chamber to determine the magnitude and direction of the velocity in the cyclone. This setup allows the measurement of the axial and tangential components of the velocity. In this case, the radial velocity is assumed to be so small that it can be inferred from continuity. In a companion paper, Smith (1962a) combines analytical methods with experimental measurements to characterize the dynamics and possible instabilities that occur in a confined vortex.

In a later investigation into the behavior of cyclone combustors, Vatistas et al. (1986) conduct a similar experiment in which a prismatic pitot tube is used to capture the velocity and pressure maps within a cyclonic chamber. These researchers compare their findings to an experimentally correlated inviscid model and find what may be essentially viewed as a favorable agreement. Their study highlights a key realization in confined vortex modeling, namely, that swirl velocity variations in the axial direction are so small that they may be ignored (see Ogawa, 1984; Reydon & Gauvin, 1981, among others). This simplification is common in the analytical modeling of vortices.

Furthermore, these studies provide early insights into the conditions arising in a confined vortex; however, some deficiencies must be noted. Even with proper calibration, the minimally intrusive probes can introduce disturbances into the flow, and these, in turn, can lead to potentially misleading results. This is especially important when investigating dynamic effects such as vortex instability.

#### **1.1.2 Laser Doppler Velocimetry**

Improvements in technology give rise to increasingly sophisticated experimental techniques that help to provide valuable information regarding confined swirl velocities without the intrusion invariably present with even the smallest probes. For instance, LDV minimizes flow disruptions by seeding the fluid domain with particles and then using a focused laser to scatter light off those particles. The interference patterns are then correlated to velocity measurements obtained in localized regions. Subsequently, the corresponding subvolumes are summed together to reconstruct the overall velocity profile of a given flow pattern.

Hoekstra et al. (1999) take an increasingly common approach of pairing a CFD solution with LDV measurements to validate their proposed turbulence models. Their experimental setup uses a back-scatter LDV to collect the axial and tangential velocity profiles in small volumes and these are then combined and correlated to provide an overall velocity profile. In this effort, however, the turbulent cross-correlation of the LDV measurements is found to be problematic because of the finite wall thickness of the cyclonic chamber which, in itself, can cause refraction and dissimilar levels of distortion based on the spatial location within the chamber. Without proper accounting for these optical disparities, a perfect correlation between the acquired signal and the flow profile will be difficult to realize. The quality of the seeding in the core region also proves to be an issue, as the natural motion in the cyclone tends to separate particles from the flow.

Hu et al. (2005) conduct a similar study for industrial-size cyclone separators. Whereas Hoekstra et al. (1999) focus on the separation section of the cyclone, Hu et al. (2005) consider the full geometry of the separator including the inlet, hoppers, and other supporting hardware. Moreover, their experimental investigation is accompanied by a turbulent computational solution. In addition to verifying turbulent vortex models, their study aims at improving the prediction of cyclone efficiency.

Along similar lines, an investigation into the turbulent kinetic energy of a confined vortex is reported in a forward scatter LDV study by Yan et al. (2000). In this work, data collected at a wide range of Reynolds numbers is used to validate their empirically derived solutions. These particular models rely on scaling laws to reduce the problem's dependence in each case to one or two key parameters and these tend to involve some combination of the inlet flow rate and the contraction ratio.

### **1.1.3 Particle Image Velocimetry**

2 Will-be-set-by-IN-TECH

Within the context of cyclone separators, the experimental study by Smith (1962b) employs a glass tube filled with smoke particles to capture the general structure of a confined swirling flow. Smith also utilizes a special slender probe stretched across the diameter of the chamber to determine the magnitude and direction of the velocity in the cyclone. This setup allows the measurement of the axial and tangential components of the velocity. In this case, the radial velocity is assumed to be so small that it can be inferred from continuity. In a companion paper, Smith (1962a) combines analytical methods with experimental measurements to

In a later investigation into the behavior of cyclone combustors, Vatistas et al. (1986) conduct a similar experiment in which a prismatic pitot tube is used to capture the velocity and pressure maps within a cyclonic chamber. These researchers compare their findings to an experimentally correlated inviscid model and find what may be essentially viewed as a favorable agreement. Their study highlights a key realization in confined vortex modeling, namely, that swirl velocity variations in the axial direction are so small that they may be ignored (see Ogawa, 1984; Reydon & Gauvin, 1981, among others). This simplification is

Furthermore, these studies provide early insights into the conditions arising in a confined vortex; however, some deficiencies must be noted. Even with proper calibration, the minimally intrusive probes can introduce disturbances into the flow, and these, in turn, can lead to potentially misleading results. This is especially important when investigating

Improvements in technology give rise to increasingly sophisticated experimental techniques that help to provide valuable information regarding confined swirl velocities without the intrusion invariably present with even the smallest probes. For instance, LDV minimizes flow disruptions by seeding the fluid domain with particles and then using a focused laser to scatter light off those particles. The interference patterns are then correlated to velocity measurements obtained in localized regions. Subsequently, the corresponding subvolumes are summed together to reconstruct the overall velocity profile of a given flow pattern.

Hoekstra et al. (1999) take an increasingly common approach of pairing a CFD solution with LDV measurements to validate their proposed turbulence models. Their experimental setup uses a back-scatter LDV to collect the axial and tangential velocity profiles in small volumes and these are then combined and correlated to provide an overall velocity profile. In this effort, however, the turbulent cross-correlation of the LDV measurements is found to be problematic because of the finite wall thickness of the cyclonic chamber which, in itself, can cause refraction and dissimilar levels of distortion based on the spatial location within the chamber. Without proper accounting for these optical disparities, a perfect correlation between the acquired signal and the flow profile will be difficult to realize. The quality of the seeding in the core region also proves to be an issue, as the natural motion in the cyclone tends

Hu et al. (2005) conduct a similar study for industrial-size cyclone separators. Whereas Hoekstra et al. (1999) focus on the separation section of the cyclone, Hu et al. (2005)

characterize the dynamics and possible instabilities that occur in a confined vortex.

common in the analytical modeling of vortices.

dynamic effects such as vortex instability.

**1.1.2 Laser Doppler Velocimetry**

to separate particles from the flow.

**1.1.1 Probes**

PIV is another minimally intrusive technique that will be discussed in the remainder of this chapter. Much like LDV, PIV employs particle seeding to collect velocity measurements. The primary difference between the two techniques stands in how the data is gathered. Whereas LDV relies on two focused beams to generate interference patterns, PIV uses optics to create a laser sheet that illuminates a plane in the chamber. High-speed cameras are then utilized to capture images of the illuminated particles at two closely spaced intervals such that a net velocity profile may be deduced from the cross-correlation of these images.

By way of comparison, both PIV and LDV methods are used by Sousa (2008) to determine the velocity field that accompanies vortex breakdown in a closed container. Sousa finds that accurate measurements may be acquired using either method; he also reports several challenges that may be associated with PIV techniques. The fully three-dimensional nature of the flow field can lead to a decrease in correlation accuracy as seed particles move normal to the light sheet. Sousa accounts for this factor by shortening the duration between laser pulses and by slightly thickening the laser sheet to increase the chances that the particles of interest will remain in the area of investigation.

In the spirit of improvement, Zhang & Hugo (2006) use a stereoscopic PIV setup to investigate the vortex motion in a pipe. Stereoscopic PIV can be used to capture the fully three-dimensional flow field; however, it requires an additional high speed camera and more elaborate calibration skills to ensure that both cameras will target the same area. This obviously leads to an increase in post-experimental processing as the images from two cameras have to be analyzed for each exposure, effectively doubling the amount of data acquired. Finally, Zhang & Hugo (2006) develop and implement an improved calibration technique to reduce the optical distortion caused by refraction through the fluid and the curved chamber wall.

#### **1.2 Classical solutions**

While a large body of literature may be reported on vortical motions, the segment devoted to analytical modeling remains much smaller in comparison. For this reason, a brief introduction to some of these classical models will follow.

In examining the classic models for describing swirl-dominated flowfields, it is helpful to further discriminate between solutions by specifying whether the attendant equations allow for unidirectional or multidirectional swirl. Unidirectional models are characterized

Using Particle Image Velocimetry 5

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 153

In what concerns bidirectional behavior, an unbounded bipolar solution to the Navier–Stokes equations is provided by Sullivan (1959). For what is essentially a two-celled vortex, Sullivan's inner region exhibits a descending axial velocity coupled with an outward radial motion. Conversely, the outer cell flows inwardly and up. The model itself can be written in an integral

d*t*; *f*(*t*) = −*t* + 3

As before, *A* denotes the suction strength and *ν*, the kinematic viscosity. The corresponding

It is the combination of the axial and radial velocities that makes the Sullivan vortex a suitable

A comparison of the above-mentioned swirl velocities is presented in Fig. 1. In all cases, the equations are normalized such that their peak velocities occur at a dimensionless radius of one. This is accomplished by dividing the radius by *δc*, which is the distance from the axis of rotation to the point where the maximum swirl velocity occurs. Traditionally, a diameter of 2*δ<sup>c</sup>* may be used to define the thickness of the forced viscous core. While all of the models capture similar trends, a significant amount of variability in the profiles may be seen. The Rankine solution displays an abrupt change in behavior at the peak velocity. The remaining models exhibit smooth contours, with the Sullivan profile concentrating the swirl velocity to a narrower region than that of Burgers-Rott. Although not depicted, the Lamb-Oseen velocity becomes identical to that of Burgers-Rott especially that time dependence is not featured in

�� ; *<sup>u</sup>*¯*r*(*r*¯) = <sup>−</sup>*Ar*¯ <sup>+</sup>

012345

Fig. 1. A comparison of selected swirl velocity models, rescaled so that the peak velocity

*r*/ *<sup>c</sup>*

; *<sup>δ</sup>* <sup>=</sup> <sup>√</sup>2*ν*/*<sup>A</sup>*

� *t* 0 � 1 − *e*

> 6*ν r*¯ �

 Rankine (1858) Burgers-Rott (1948) Sullivan (1959)

<sup>−</sup>*y*� <sup>d</sup>*<sup>y</sup> y*

<sup>1</sup> <sup>−</sup> exp �

− *r*¯2 *δ*2 (4)

�� (5)

**1.2.2 Bidirectional solutions**

⎧ ⎪⎪⎪⎨

*<sup>u</sup>*¯*<sup>θ</sup>* (*r*¯) = <sup>Γ</sup>

*<sup>H</sup>*(*x*) = � *<sup>x</sup>*

axial and radial components may be expressed as

�

candidate for modeling tornadoes (Wu, 1986).

0

occurs at *r* = 1.

0.5

max ( ) *u u*

1

2*πr*¯

0 *ef*(*t*)

<sup>1</sup> <sup>−</sup> 3 exp �

1 *H*(∞) *H* � *r*¯<sup>2</sup> *δ*2 �

− *r*¯ 2 *δ*2

⎪⎪⎪⎩

*u*¯*z*(*r*¯, *z*¯) = 2*Az*¯

this figure.

representation using

by a one-directional axial velocity, whereas multidirectional motions exhibit a reverse flow character in which the axial velocity switches polarity. In practical applications, multidirectional flows are reduced to a bidirectional profile in which the axial velocity reverses only once.

#### **1.2.1 Unidirectional models**

One of the earliest unidirectional models is provided by Rankine (1858). Accordingly, the swirl velocity is formulated as a simple piecewise solution that is radially dependent. A normalized representation of this model may be written as

$$\frac{\vec{u}\_{\theta}}{(\vec{u}\_{\theta})\_{\text{max}}} = \begin{cases} \frac{\vec{r}}{\delta\_{\mathcal{C}}} & \vec{r} \le \delta\_{\mathcal{C}}\\ \frac{\delta\_{\mathcal{C}}}{\vec{r}} & r > \delta\_{\mathcal{C}} \end{cases} \quad \text{or} \quad u\_{\theta} = \begin{cases} r & r \le 1\\ r^{-1} & r > 1 \end{cases} \tag{1}$$

where overbars denote dimensional quantities and the radius is referenced to *δc*, the distance from the origin to the point at which *u*¯*<sup>θ</sup>* reaches its peak value, (*u*¯*<sup>θ</sup>* )max. The profile consists of a forced core that exhibits solid body rotation. The outer vortex varies with the inverse of the radius, so the swirl velocity diminishes away from the axis of rotation. Rankine's model is often used as a baseline solution for comparison or initialization owing to its simplicity. Nonetheless, Eq. (1) is somewhat limited in that the optimal matching location, *δc*, cannot be specified a priori, but requires data for anchoring. Moreover, the model is not differentiable at the matching location, which may be undesirable in subsequent analysis.

The Lamb-Oseen vortex consists of another solution that incorporates a time-dependent decay of the vortex motion (Wendt, 2001). This makes the model particularly suitable for capturing the behavior of wing-tip vortices. The dimensional representation of its swirl velocity may be expressed as

$$\vec{u}\_{\theta}(\vec{r},\vec{t}) = \frac{\Gamma}{2\pi\bar{r}} \left[ 1 - \exp\left( -\frac{\bar{r}^2}{\delta^2} \right) \right] \tag{2}$$

Here Γ refers to the circulation and *δ* = 2 √ *ν*¯*t*, to the characteristic radius which is dependent on time, ¯*t*, and the kinematic viscosity, *ν*. Equation (2) starts as a potential vortex, behaving as 1/*r*¯ away from the centerline before smoothly switching to a linear dependence on *r*¯ in the forced vortex core evolving around *r*¯ = 0. As time elapses, the vortex decays exponentially.

The Burgers-Rott vortex (Burgers, 1948) is similar in form to the Lamb-Oseen profile with two notable exceptions. First, rather than a time-dependent decay, the exponential function here is governed by the suction parameter, *A*. Secondly, the Burgers-Rott vortex possesses well-defined relations for the axial and radial velocities. It can be written as

$$
\begin{split}
\bar{u}\_{\theta}(\bar{r}) &= \frac{\Gamma}{2\pi\bar{r}} \left[1 - \exp\left(-\frac{\bar{r}^2}{\delta^2}\right)\right] \\
\bar{u}\_{\bar{r}}(\bar{r}) &= -A\bar{r}; \qquad \bar{u}\_{\bar{z}}(\bar{z}) = 2A\bar{z}
\end{split}
\tag{3}
$$

where *<sup>δ</sup>* <sup>=</sup> <sup>√</sup>2*ν*/*A*. The presence of an axial velocity and a suction parameter has proven useful in applications related to the modeling of thunderstorms.

#### **1.2.2 Bidirectional solutions**

4 Will-be-set-by-IN-TECH

by a one-directional axial velocity, whereas multidirectional motions exhibit a reverse flow character in which the axial velocity switches polarity. In practical applications, multidirectional flows are reduced to a bidirectional profile in which the axial velocity reverses

One of the earliest unidirectional models is provided by Rankine (1858). Accordingly, the swirl velocity is formulated as a simple piecewise solution that is radially dependent. A normalized

where overbars denote dimensional quantities and the radius is referenced to *δc*, the distance from the origin to the point at which *u*¯*<sup>θ</sup>* reaches its peak value, (*u*¯*<sup>θ</sup>* )max. The profile consists of a forced core that exhibits solid body rotation. The outer vortex varies with the inverse of the radius, so the swirl velocity diminishes away from the axis of rotation. Rankine's model is often used as a baseline solution for comparison or initialization owing to its simplicity. Nonetheless, Eq. (1) is somewhat limited in that the optimal matching location, *δc*, cannot be specified a priori, but requires data for anchoring. Moreover, the model is not differentiable

The Lamb-Oseen vortex consists of another solution that incorporates a time-dependent decay of the vortex motion (Wendt, 2001). This makes the model particularly suitable for capturing the behavior of wing-tip vortices. The dimensional representation of its swirl velocity may be

�

on time, ¯*t*, and the kinematic viscosity, *ν*. Equation (2) starts as a potential vortex, behaving as 1/*r*¯ away from the centerline before smoothly switching to a linear dependence on *r*¯ in the forced vortex core evolving around *r*¯ = 0. As time elapses, the vortex decays exponentially. The Burgers-Rott vortex (Burgers, 1948) is similar in form to the Lamb-Oseen profile with two notable exceptions. First, rather than a time-dependent decay, the exponential function here is governed by the suction parameter, *A*. Secondly, the Burgers-Rott vortex possesses

1 − exp

� − *r*¯2 *δ*2 ��

or *u<sup>θ</sup>* =

�

*r r* ≤ 1

*ν*¯*t*, to the characteristic radius which is dependent

*<sup>r</sup>*−<sup>1</sup> *<sup>r</sup> <sup>&</sup>gt;* <sup>1</sup> (1)

(2)

*r*¯ ≤ *δ<sup>c</sup>*

*<sup>r</sup>*¯ *<sup>r</sup> <sup>&</sup>gt; <sup>δ</sup><sup>c</sup>*

only once.

expressed as

**1.2.1 Unidirectional models**

representation of this model may be written as

Here Γ refers to the circulation and *δ* = 2

*u*¯*θ* (*u*¯*<sup>θ</sup>* )max

=

⎧ ⎪⎪⎨

*r*¯ *δc*

*δc*

at the matching location, which may be undesirable in subsequent analysis.

*<sup>u</sup>*¯*<sup>θ</sup>* (*r*¯, ¯*t*) = <sup>Γ</sup>

well-defined relations for the axial and radial velocities. It can be written as

2*πr*¯

�

1 − exp

where *<sup>δ</sup>* <sup>=</sup> <sup>√</sup>2*ν*/*A*. The presence of an axial velocity and a suction parameter has proven

� − *r*¯2 *δ*2 ��

*u*¯*r*(*r*¯) = −*Ar*¯; *u*¯*z*(*z*¯) = 2*Az*¯ (3)

*<sup>u</sup>*¯*<sup>θ</sup>* (*r*¯) = <sup>Γ</sup>

useful in applications related to the modeling of thunderstorms.

2*πr*¯

√

⎪⎪⎩

In what concerns bidirectional behavior, an unbounded bipolar solution to the Navier–Stokes equations is provided by Sullivan (1959). For what is essentially a two-celled vortex, Sullivan's inner region exhibits a descending axial velocity coupled with an outward radial motion. Conversely, the outer cell flows inwardly and up. The model itself can be written in an integral representation using

$$\begin{cases} \begin{aligned} \vec{u}\_{\theta}(\vec{r}) &= \frac{\Gamma}{2\pi \vec{r}} \frac{1}{H(\infty)} H\left(\frac{\vec{r}^{2}}{\delta^{2}}\right); \quad \delta = \sqrt{2v/A} \\\\ H(\mathbf{x}) &= \int\_{0}^{\mathbf{x}} e^{f(t)} \mathbf{d}t; \quad f(t) = -t + 3 \int\_{0}^{t} \left(1 - e^{-y}\right) \frac{\mathbf{d}y}{y} \end{aligned} \end{cases} \tag{4}$$

As before, *A* denotes the suction strength and *ν*, the kinematic viscosity. The corresponding axial and radial components may be expressed as

$$\vec{u}\_{2}(\vec{r},\vec{z}) = 2A\vec{z}\left[1 - 3\exp\left(-\frac{\vec{r}^{2}}{\delta^{2}}\right)\right]; \quad \vec{u}\_{l}(\vec{r}) = -A\vec{r} + \frac{6\nu}{\vec{r}}\left[1 - \exp\left(-\frac{\vec{r}^{2}}{\delta^{2}}\right)\right] \tag{5}$$

It is the combination of the axial and radial velocities that makes the Sullivan vortex a suitable candidate for modeling tornadoes (Wu, 1986).

A comparison of the above-mentioned swirl velocities is presented in Fig. 1. In all cases, the equations are normalized such that their peak velocities occur at a dimensionless radius of one. This is accomplished by dividing the radius by *δc*, which is the distance from the axis of rotation to the point where the maximum swirl velocity occurs. Traditionally, a diameter of 2*δ<sup>c</sup>* may be used to define the thickness of the forced viscous core. While all of the models capture similar trends, a significant amount of variability in the profiles may be seen. The Rankine solution displays an abrupt change in behavior at the peak velocity. The remaining models exhibit smooth contours, with the Sullivan profile concentrating the swirl velocity to a narrower region than that of Burgers-Rott. Although not depicted, the Lamb-Oseen velocity becomes identical to that of Burgers-Rott especially that time dependence is not featured in this figure.

Fig. 1. A comparison of selected swirl velocity models, rescaled so that the peak velocity occurs at *r* = 1.

Using Particle Image Velocimetry 7

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 155

Moving beyond the Bloor–Ingham approximation, Vyas & Majdalani (2006) introduce a bidirectional model with a reversing axial character. Their complex-lamellar solution, which constitutes the basis for the upcoming analysis, seeks to describe the bulk gaseous motion in the Vortex Combustion Cold-Wall Chamber (VCCWC) developed by Chiaverini et al. (2002). In this vortex-driven engine, the swirling motion of the oxidizer insulates the sidewalls against thermal loading, thus leading to a substantial reduction in engine weight. The swirling motion also has a mitigating impact on pressure oscillations in the chamber as shown by Batterson & Majdalani (2011a;b). The mathematical character of this application is described in the

The remainder of this chapter is focused on the synergistic combination of analytical and experimental methods in the context of confined vortex flows. First, two distinct analytical frameworks are devised with the intent of modeling the swirl velocity of a confined vortex. These models introduce control parameters that guide the construction of an experimental apparatus. A PIV setup is then used to capture the swirl velocity in the vortex chamber. Finally, the data is compared to the analytical frameworks in an effort to reconcile between

The vortex chamber, shown in Fig. 3, is modeled as a right circular cylinder of radius *a* and length *L*. The origin of the coordinate system is fixed at the center of the inert headwall, while the aft section is partially left open with a radius of *b*. The radial and axial coordinates are denoted by *r*¯ and *z*¯, respectively, while the characteristic geometric parameters are the

A single phase, non-reactive fluid is injected tangentially at *z*¯ = *L*. The fluid spirals up the outer annular region towards the headwall, thus forming an outer vortex (see Fig. 3). Once the fluid reaches the top of the chamber, it reverses axial direction and funnels out of the chamber as an inner vortex in the region 0 *< r*¯ *< b*. These two vortices are separated by a spinning, non-translating layer called the 'mantle.' This fluid interface materializes at *r*¯ = *b*.

*r*¯ = *a*, *z*¯ = *L*, *u*¯*<sup>θ</sup>* = *U* tangential velocity at entry (10)

*z*¯ = 0, ∀*r*¯, *u*¯*<sup>z</sup>* = 0 impervious headwall (11) *r*¯ = 0, ∀*z*¯, *u*¯*<sup>r</sup>* = 0 no flow across centerline (12) *r*¯ = *a*, ∀*z*¯, *u*¯*<sup>r</sup>* = 0 impervious sidewall (13)

*r*¯ = 0, ∀*z*¯, *u*¯*<sup>θ</sup>* = 0 forced vortex center (15) *r*¯ = *a*, *z*¯ *< L*, *u*¯*<sup>θ</sup>* = 0 no slip condition at the sidewall (16)

The last two conditions are only used when accounting for viscous effects in the chamber. The *Q*¯*<sup>i</sup>* term represents the inlet volumetric flow rate and *Q*¯ <sup>0</sup> is the outlet volumetric flow rate,

*u*¯*z*(*r*¯, *L*)*r*¯d*r*¯ = *Q*¯*<sup>i</sup>* axial outflow matching tangential source (14)

Mathematically, these conditions translate into the following boundary conditions:

following section.

theory and experiment.

2*π a b*

calculated from the integral in (14).

**2. Analytical formulations**

outflow fraction *β* = *b*/*a* and the aspect ratio *l* = *L*/*a*.

In work related to cyclone separators, a study by Bloor & Ingham (1987) leads to one of the most frequently cited models. The resulting inviscid solution arises in the context of a conical cyclone (see Fig. 2). Bloor and Ingham solve the Bragg-Hawthorne equation in spherical coordinates to the extent of producing a stream function of the form

$$\psi = \sigma R^2 \left\{ \left[ \csc^2 a + \ln \left( \tan \frac{a}{2} \right) - \csc a \cot a \right] \sin^2 \phi - \sin^2 \phi \ln \left( \tan \frac{\phi}{2} \right) + \cos \phi - 1 \right\} \tag{6}$$

Equation (6) translates into the following velocity components

$$\mu\_R = 2\sigma R \left\{ \left[ \csc^2 a + \ln \left( \tan \frac{a}{2} \right) - \csc a \cot a \right] \cos \phi - \cos \phi \ln \left( \tan \frac{\phi}{2} \right) - 1 \right\} \tag{7}$$

$$u\_{\phi} = \frac{2\psi}{R^2 \sin \phi}; \quad u\_{\theta} = \frac{1}{R \sin \phi} \sqrt{1 - \frac{\vec{Q}\_i^2 \sigma \psi}{\pi^2 a^4 L^2}} \tag{8}$$

Here *Q*¯*<sup>i</sup>* denotes the volumetric flow rate through the cyclone, *U* and *W* stand for the average swirl and axial velocities at the entrance, *α* represents the taper angle of the cyclone, and *σ* refers to the dimensionless swirl parameter described by

$$
\sigma = \frac{\pi a\_0^2 \mathcal{U}^2}{\bar{Q}\_i W} \tag{9}
$$

Here *a*<sup>0</sup> is the outer radius of the virtual inlet (see Fig. 2). Equations (6)– (9) constitute an improvement on previous work (Bloor & Ingham, 1973), where use of the Polhausen technique leads to a solution that is insensitive to injection conditions. It should be noted that Eqs. (6)– (9) represent a corrected form of the Bloor–Ingham solution according to Barber & Majdalani (2009).

Fig. 2. Bloor-Ingham solution domain and geometry.

Moving beyond the Bloor–Ingham approximation, Vyas & Majdalani (2006) introduce a bidirectional model with a reversing axial character. Their complex-lamellar solution, which constitutes the basis for the upcoming analysis, seeks to describe the bulk gaseous motion in the Vortex Combustion Cold-Wall Chamber (VCCWC) developed by Chiaverini et al. (2002). In this vortex-driven engine, the swirling motion of the oxidizer insulates the sidewalls against thermal loading, thus leading to a substantial reduction in engine weight. The swirling motion also has a mitigating impact on pressure oscillations in the chamber as shown by Batterson & Majdalani (2011a;b). The mathematical character of this application is described in the following section.

The remainder of this chapter is focused on the synergistic combination of analytical and experimental methods in the context of confined vortex flows. First, two distinct analytical frameworks are devised with the intent of modeling the swirl velocity of a confined vortex. These models introduce control parameters that guide the construction of an experimental apparatus. A PIV setup is then used to capture the swirl velocity in the vortex chamber. Finally, the data is compared to the analytical frameworks in an effort to reconcile between theory and experiment.

### **2. Analytical formulations**

6 Will-be-set-by-IN-TECH

In work related to cyclone separators, a study by Bloor & Ingham (1987) leads to one of the most frequently cited models. The resulting inviscid solution arises in the context of a conical cyclone (see Fig. 2). Bloor and Ingham solve the Bragg-Hawthorne equation in spherical

− csc *α* cot *α*

Here *Q*¯*<sup>i</sup>* denotes the volumetric flow rate through the cyclone, *U* and *W* stand for the average swirl and axial velocities at the entrance, *α* represents the taper angle of the cyclone, and *σ*

*<sup>σ</sup>* <sup>=</sup> *<sup>π</sup>a*<sup>2</sup>

Here *a*<sup>0</sup> is the outer radius of the virtual inlet (see Fig. 2). Equations (6)– (9) constitute an improvement on previous work (Bloor & Ingham, 1973), where use of the Polhausen technique leads to a solution that is insensitive to injection conditions. It should be noted that Eqs. (6)– (9) represent a corrected form of the Bloor–Ingham solution according to Barber

sin<sup>2</sup> *<sup>φ</sup>* <sup>−</sup> sin<sup>2</sup> *<sup>φ</sup>* ln

*R* sin *φ*

0*U*<sup>2</sup>

 tan *<sup>φ</sup>* 2 

*<sup>i</sup> σψ*

 tan *<sup>φ</sup>* 2 − 1 

*<sup>Q</sup>*¯*iW* (9)

cos *φ* − cos *φ* ln

<sup>1</sup> <sup>−</sup> *<sup>Q</sup>*¯ <sup>2</sup>

+ cos *φ* − 1

*<sup>π</sup>*2*a*4*U*<sup>2</sup> (8)

(6)

(7)

coordinates to the extent of producing a stream function of the form

Equation (6) translates into the following velocity components

*<sup>u</sup><sup>φ</sup>* <sup>=</sup> <sup>2</sup>*<sup>ψ</sup>*

refers to the dimensionless swirl parameter described by

Fig. 2. Bloor-Ingham solution domain and geometry.

 tan *α* 2 

− csc *α* cot *α*

*<sup>R</sup>*<sup>2</sup> sin *<sup>φ</sup>*; *<sup>u</sup><sup>θ</sup>* <sup>=</sup> <sup>1</sup>

*ψ* = *σR*<sup>2</sup>

*uR* = 2*σR*

& Majdalani (2009).

csc<sup>2</sup> *α* + ln

 tan *α* 2 

csc<sup>2</sup> *α* + ln

The vortex chamber, shown in Fig. 3, is modeled as a right circular cylinder of radius *a* and length *L*. The origin of the coordinate system is fixed at the center of the inert headwall, while the aft section is partially left open with a radius of *b*. The radial and axial coordinates are denoted by *r*¯ and *z*¯, respectively, while the characteristic geometric parameters are the outflow fraction *β* = *b*/*a* and the aspect ratio *l* = *L*/*a*.

A single phase, non-reactive fluid is injected tangentially at *z*¯ = *L*. The fluid spirals up the outer annular region towards the headwall, thus forming an outer vortex (see Fig. 3). Once the fluid reaches the top of the chamber, it reverses axial direction and funnels out of the chamber as an inner vortex in the region 0 *< r*¯ *< b*. These two vortices are separated by a spinning, non-translating layer called the 'mantle.' This fluid interface materializes at *r*¯ = *b*. Mathematically, these conditions translate into the following boundary conditions:


The last two conditions are only used when accounting for viscous effects in the chamber. The *Q*¯*<sup>i</sup>* term represents the inlet volumetric flow rate and *Q*¯ <sup>0</sup> is the outlet volumetric flow rate, calculated from the integral in (14).

Using Particle Image Velocimetry 9

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 157

This free vortex form is common to many swirl-dominated inviscid flows. However, to realistically model the bidirectional vortex, viscous corrections will be required especially at

With the swirl velocity successfully decoupled from the remaining two velocities, the Stokes

Applying (24) and the vorticity transport equation (for additional detail see Vyas & Majdalani,

Equation (25) can be solved via separation of variables. After applying the boundary

*ψ* = *κz* sin(*πr*

The stream function provides the axial and radial velocity profiles which are key quantities in

With the choice of normalization being based on the average tangential speed at entry, the small size of *κ* ensures that these two quantities remain of secondary importance relative to

The irrotational form in (23) accurately describes an inviscid vortex. However, for a confined vortex, some deficiencies arise, notably the singularity at the centerline where the velocity approaches infinity, and at the sidewall where slippage occurs. Bloor & Ingham (1987) note that the unbounded behavior at the centerline appears to be a common characteristic of inviscid models to the extent of becoming archetypical (cf. Leibovich, 1984). To treat this core singularity, the second-order viscous terms in the *θ*-momentum equation must be considered.

where *Re* represents the mean flow Reynolds number, *μ*, the dynamic viscosity, and *ρ*, the density. In this particular case, *Re* is of the order of 105. Thus, a perturbation parameter may be formed by setting *ε* ≡ 1/*Re*. For simplicity, the dimensionless angular momentum is

> d d*r*

1 *r* d*ξ* d*r* +

where *κ* is a constant determined from the mass conservation boundary condition in (14)

; *uz* <sup>=</sup> <sup>1</sup> *r ∂ψ*

+ *C*2*r*

<sup>2</sup>)/*r*; *uz* = 2*πκz* cos(*πr*

*∂* (*ru<sup>θ</sup>* ) *∂r*

; *Re* <sup>≡</sup> *<sup>ρ</sup>Ua*

*κ* sin(*πr*2) *r*2

d*ξ*

*<sup>∂</sup><sup>r</sup>* (24)

<sup>2</sup>*ψ* = 0 (25)

<sup>2</sup>) (26)

<sup>2</sup>) (28)

*<sup>μ</sup>* (29)

<sup>d</sup>*<sup>r</sup>* <sup>=</sup> 0 (30)

*κ* = *Qi*/(2*πl*) (27)

*ur* <sup>=</sup> <sup>−</sup><sup>1</sup> *r ∂ψ ∂z*

> *∂*2*ψ <sup>∂</sup>r*<sup>2</sup> <sup>−</sup> <sup>1</sup> *r ∂ψ ∂r*

the centerline.

*u<sup>θ</sup>* .

stream function may be introduced using

conditions, the solution becomes

Retention of these terms leads to

*ε* d d*r* 1 *r* d*ξ* d*r* <sup>−</sup> *ur r* d*ξ*

*ur ∂u<sup>θ</sup> ∂r* + *uru<sup>θ</sup> <sup>r</sup>* <sup>=</sup> <sup>1</sup> *Re ∂ ∂r* 1 *r*

2006) leads to the following relation in *ψ*, namely,

*∂*2*ψ <sup>∂</sup>z*<sup>2</sup> <sup>+</sup>

the calculation of the pressure. They may be expressed as

*ur* = −*κ* sin(*πr*

consolidated into a single variable, *ξ* ≡ *ru<sup>θ</sup>* ; this turns Eq. (29) into

<sup>d</sup>*<sup>r</sup>* <sup>=</sup> 0 or *<sup>ε</sup>*

Fig. 3. Sketch of the bidirectional vortex chamber and corresponding coordinate system.

To facilitate comparisons to existing models, it is beneficial to use a non-dimensional form of the governing equations. This is achieved by setting

$$r = \frac{\vec{r}}{a}; \; z = \frac{\Xi}{a}; \; \delta = \frac{\vec{\delta}}{a}; \; \nabla = a\bar{\nabla};\tag{17}$$

$$\mu\_r = \frac{\vec{u}\_r}{\mathcal{U}};\ u\_\theta = \frac{\vec{u}\_\theta}{\mathcal{U}};\ u\_z = \frac{\vec{u}\_z}{\mathcal{U}};\ p = \frac{\vec{p}}{\rho \mathcal{U}^2};\ Q\_i = \frac{\vec{Q}\_i}{\mathcal{U}a^2} = \frac{A\_i}{a^2} \tag{18}$$

Here all spatial variables are normalized by the chamber radius and the velocities by the injection velocity. The variable *δ* represents the characteristic length scale used in the viscous analysis.

#### **2.1 Laminar core model**

The laminar core model is a solution to the Navier–Stokes equations using a stream function approximation. An inviscid base flow is obtained initially, followed by a boundary layer type correction at the centerline. In modeling confined vortices, it is common to assume that the flow is both axisymmetric and that the swirl velocity is axially invariant (see Leibovich, 1984). The equations for the inviscid flow, after applying these assumptions, reduce to:

$$\frac{1}{r}\frac{\partial}{\partial r}\frac{(ru\_r)}{\partial r} + \frac{\partial u\_z}{\partial z} = 0 \tag{19}$$

$$
\mu\_r \frac{\partial u\_r}{\partial r} + \mu\_z \frac{\partial u\_r}{\partial z} - \frac{u\_\theta^2}{r} = -\frac{1}{\rho} \frac{\partial p}{\partial r} \tag{20}
$$

$$
u\_r \frac{\partial u\_\theta}{\partial r} + \frac{u\_r u\_\theta}{r} = 0\tag{21}$$

$$
\mu\_r \frac{\partial u\_z}{\partial r} + u\_z \frac{\partial u\_z}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial z} \tag{22}
$$

Equation (21) clearly shows the tangential velocity as fully decoupled from the radial and axial components such that it can be directly retrieved. Further application of the boundary condition from (10) gives

$$
\mu\_{\theta} = \frac{1}{r} \tag{23}
$$

8 Will-be-set-by-IN-TECH

*Qi*

*Qo*

*<sup>a</sup>*<sup>2</sup> (18)

; <sup>∇</sup> <sup>=</sup> *<sup>a</sup>*∇¯ ; (17)

*Ua*<sup>2</sup> <sup>=</sup> *Ai*

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>0</sup> (19)

*<sup>r</sup>* <sup>=</sup> <sup>0</sup> (21)

*<sup>r</sup>* (23)

*<sup>∂</sup><sup>r</sup>* (20)

*<sup>∂</sup><sup>z</sup>* (22)

*L*

; *<sup>z</sup>* <sup>=</sup> *<sup>z</sup>*¯ *a*

*<sup>U</sup>* ; *uz* <sup>=</sup> *<sup>u</sup>*¯*<sup>z</sup>*

The equations for the inviscid flow, after applying these assumptions, reduce to:

+ *uz ∂ur <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>u</sup>*<sup>2</sup> *θ <sup>r</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> *ρ ∂p*

> + *uz ∂uz <sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> *ρ ∂p*

Equation (21) clearly shows the tangential velocity as fully decoupled from the radial and axial components such that it can be directly retrieved. Further application of the boundary

*<sup>u</sup><sup>θ</sup>* <sup>=</sup> <sup>1</sup>

*ur ∂u<sup>θ</sup> ∂r* + *uru<sup>θ</sup>*

*ur ∂uz ∂r*

*∂* (*rur*) *∂r*

1 *r*

*ur ∂ur ∂r*

Fig. 3. Sketch of the bidirectional vortex chamber and corresponding coordinate system.

To facilitate comparisons to existing models, it is beneficial to use a non-dimensional form of

; *<sup>δ</sup>* <sup>=</sup> ¯ *δ a*

Here all spatial variables are normalized by the chamber radius and the velocities by the injection velocity. The variable *δ* represents the characteristic length scale used in the viscous

The laminar core model is a solution to the Navier–Stokes equations using a stream function approximation. An inviscid base flow is obtained initially, followed by a boundary layer type correction at the centerline. In modeling confined vortices, it is common to assume that the flow is both axisymmetric and that the swirl velocity is axially invariant (see Leibovich, 1984).

> + *∂uz*

*<sup>U</sup>* ; *<sup>p</sup>* <sup>=</sup> *<sup>p</sup>*¯

*<sup>ρ</sup>U*<sup>2</sup> ; *Qi* <sup>=</sup> *<sup>Q</sup>*¯*<sup>i</sup>*

*r*

*a b*

analysis.

**2.1 Laminar core model**

condition from (10) gives

*z*

the governing equations. This is achieved by setting

*ur* <sup>=</sup> *<sup>u</sup>*¯*<sup>r</sup>*

*<sup>r</sup>* <sup>=</sup> *<sup>r</sup>*¯ *a*

*<sup>U</sup>* ; *<sup>u</sup><sup>θ</sup>* <sup>=</sup> *<sup>u</sup>*¯*<sup>θ</sup>*

This free vortex form is common to many swirl-dominated inviscid flows. However, to realistically model the bidirectional vortex, viscous corrections will be required especially at the centerline.

With the swirl velocity successfully decoupled from the remaining two velocities, the Stokes stream function may be introduced using

$$u\_I = -\frac{1}{r}\frac{\partial \psi}{\partial z};\ u\_z = \frac{1}{r}\frac{\partial \psi}{\partial r} \tag{24}$$

Applying (24) and the vorticity transport equation (for additional detail see Vyas & Majdalani, 2006) leads to the following relation in *ψ*, namely,

$$\frac{\partial^2 \psi}{\partial z^2} + \frac{\partial^2 \psi}{\partial r^2} - \frac{1}{r} \frac{\partial \psi}{\partial r} + \mathcal{C}^2 r^2 \psi = 0 \tag{25}$$

Equation (25) can be solved via separation of variables. After applying the boundary conditions, the solution becomes

$$
\psi = \kappa z \sin(\pi r^2) \tag{26}
$$

where *κ* is a constant determined from the mass conservation boundary condition in (14)

$$
\kappa = \mathbb{Q}\_{\text{i}} / (2\pi l) \tag{27}
$$

The stream function provides the axial and radial velocity profiles which are key quantities in the calculation of the pressure. They may be expressed as

$$u\_r = -\kappa \sin(\pi r^2)/r; \qquad u\_z = 2\pi\kappa z \cos(\pi r^2) \tag{28}$$

With the choice of normalization being based on the average tangential speed at entry, the small size of *κ* ensures that these two quantities remain of secondary importance relative to *u<sup>θ</sup>* .

The irrotational form in (23) accurately describes an inviscid vortex. However, for a confined vortex, some deficiencies arise, notably the singularity at the centerline where the velocity approaches infinity, and at the sidewall where slippage occurs. Bloor & Ingham (1987) note that the unbounded behavior at the centerline appears to be a common characteristic of inviscid models to the extent of becoming archetypical (cf. Leibovich, 1984). To treat this core singularity, the second-order viscous terms in the *θ*-momentum equation must be considered. Retention of these terms leads to

$$\ln \mu\_r \frac{\partial u\_\theta}{\partial r} + \frac{u\_r u\_\theta}{r} = \frac{1}{Re} \frac{\partial}{\partial r} \left[ \frac{1}{r} \frac{\partial \left( r u\_\theta \right)}{\partial r} \right]; \text{ Re} \equiv \frac{\rho l Ia}{\mu} \tag{29}$$

where *Re* represents the mean flow Reynolds number, *μ*, the dynamic viscosity, and *ρ*, the density. In this particular case, *Re* is of the order of 105. Thus, a perturbation parameter may be formed by setting *ε* ≡ 1/*Re*. For simplicity, the dimensionless angular momentum is consolidated into a single variable, *ξ* ≡ *ru<sup>θ</sup>* ; this turns Eq. (29) into

$$
\varepsilon \frac{\mathbf{d}}{\mathrm{d}r} \left(\frac{1}{r} \frac{\mathrm{d}\xi}{\mathrm{d}r}\right) - \frac{\mu\_r}{r} \frac{\mathrm{d}\xi}{\mathrm{d}r} = 0 \quad \text{or} \quad \varepsilon \frac{\mathrm{d}}{\mathrm{d}r} \left(\frac{1}{r} \frac{\mathrm{d}\xi}{\mathrm{d}r}\right) + \frac{\kappa \sin(\pi r^2)}{r^2} \frac{\mathrm{d}\xi}{\mathrm{d}r} = 0 \tag{30}
$$

By converting the independent coordinate using *<sup>η</sup>* <sup>≡</sup> *<sup>π</sup>r*2, Eq. (30) simplifies into

$$\frac{\varepsilon}{\kappa} \frac{\mathbf{d}^2 \tilde{\xi}}{\mathbf{d} \eta^2} + \frac{\sin \eta}{2 \eta} \frac{\mathbf{d} \tilde{\xi}}{\mathbf{d} \eta} = 0 \tag{31}$$

Using Particle Image Velocimetry 11

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 159

For a complete physical description of *V*, the reader may consult with Majdalani & Chiaverini (2009). The injection mass flow rate appears as *m*˙ *<sup>i</sup>* while the density and viscosity are given by *ρ* and *μ*. The outer constant *C*¯ may be procured from the tangential injection condition. This

<sup>1</sup> <sup>−</sup> exp

Equation (38) provides a laminar representation of the swirl velocity in the bidirectional vortex chamber in which slip is permitted at the sidewall. To reconcile between this model and

Before leaving this subject, it should be noted that a similar asymptotic analysis can be used to obtain the sidewall boundary layer correction. However, given the scope of this chapter, the attendant analysis is omitted. This is partly due to the fidelity of the PIV procedure and the data near the sidewall being too coarse to resolve the wall boundary layer. The additional

The constant shear stress model provides a piecewise swirl velocity solution that may be used to model confined vortex motions. The basis for the model is that a free vortex of the 1/*r* type develops away from the core of the vortex; in contrast, equilibrium is maintained near the core between the shear and pressure terms. This balance leads to a model that is equally valid for both laminar and turbulent conditions. It is important to note that the corresponding flow is not turbulent per se as there are no unsteady effects included, but rather a mean velocity

The justification for this approach can be seen mathematically from the conservation of

At the centerline, the tangential velocity vanishes and this leaves the pressure and shear stress terms available to balance each other (Townsend, 1976). The flow under consideration for this work has a zero tangential pressure gradient; therefore, the shear stress in the tangential direction may be assumed to have a constant value. The equation for the dominant shear

where *�* is the same viscous parameter 1/*Re*. Because the flow is axisymmetric, the *θ*

*∂ ∂r v r* 

(U · ∇) U = −∇*p* + ∇ · τ (39)

= *C*<sup>1</sup> (41)

(40)

−1 4*Vr*<sup>2</sup>

*<sup>L</sup>* <sup>=</sup> *<sup>ρ</sup>UAi*

*<sup>μ</sup><sup>L</sup>* <sup>=</sup> *<sup>m</sup>*˙ *<sup>i</sup>*

*<sup>μ</sup><sup>L</sup>* (36)

(38)

(1) = 1 or *<sup>C</sup>*¯ � <sup>1</sup> (37)

where *V* stands for the vortex Reynolds number,

and so, when suitably combined, one has

**2.2 Constant shear stress model**

momentum, namely,

stress becomes

derivative is eliminated such that

implies

*<sup>V</sup>* <sup>≡</sup> <sup>1</sup>

*εσ<sup>l</sup>* <sup>=</sup> *Re σ a*

*ξ*(*ci*)

*<sup>u</sup><sup>θ</sup>* � <sup>1</sup> *r* 

experimental data, additional steps are required as it will be shown below.

complexity engendered by the sidewall correction is hence disregarded.

profile that can be used to represent the bulk flow field in a turbulent regime.

*τr<sup>θ</sup>* = *�*

*� ∂v <sup>∂</sup><sup>r</sup>* <sup>−</sup> *<sup>v</sup> r* 

 1 *r ∂u ∂θ* <sup>+</sup> *<sup>r</sup>*

In order to bring the swirl velocity to zero along the chamber axis, one must account for the rapid changes caused by the local emergence of viscous stresses (see Fig. 4). To do so, one may introduce the slowly varying scale, *s* ≡ *η*/*δ*(*ε*). The stretching transformation maps the region of non-uniformity about the centerline to an interval of order unity. Applying the transformation and linearizing the equation near the core where *s* ≈ 0 yields

$$\frac{\varepsilon}{\kappa \delta} \frac{\mathbf{d}^2 \tilde{\xi}}{\mathbf{d}s^2} + \frac{1}{2} \left[ 1 - \frac{\delta^2 s^2}{3!} + O(\delta^2 s^2) \right] \frac{\mathbf{d} \tilde{\xi}}{\mathbf{d}s} = 0 \tag{32}$$

The diffusive and convective terms strike a balance near the core. This occurs in (32) when *δ* ∼ *ε*/*κ*. Having identified the distinguished limit, the core boundary layer equation becomes

$$\frac{\mathbf{d}^2 \xi^{(i)}}{\mathbf{d}s^2} + \frac{1}{2} \frac{\mathbf{d} \xi^{(i)}}{\mathbf{d}s} = 0 \tag{33}$$

where the superscript (*i*) stands for the inner, near-core approximation. Using a standard perturbation series, successive viscous corrections can be determined to any desired order. For this chapter, a one-term inner solution is sought. Hence, by integrating (33) and insisting on a forced vortex near the core, one retrieves

$$\mathfrak{F}^{(i)} = \mathbb{C}\_0 \left[ \exp \left( -\frac{1}{2}s - 1 \right) \right] \tag{34}$$

The remaining constant may be determined through matching with the outer expansion. Using Prandtl's matching principle, the outer limit of the inner solution may be equated to the inner limit of the free vortex. This process results in a composite solution that is valid everywhere except in the close vicinity of the sidewall. After transforming back to the original coordinate, the composite inner solution collapses into

Fig. 4. Schematic of the dual end point boundary layers present in the confined vortex.

where *V* stands for the vortex Reynolds number,

10 Will-be-set-by-IN-TECH

sin *η* 2*η*

In order to bring the swirl velocity to zero along the chamber axis, one must account for the rapid changes caused by the local emergence of viscous stresses (see Fig. 4). To do so, one may introduce the slowly varying scale, *s* ≡ *η*/*δ*(*ε*). The stretching transformation maps the region of non-uniformity about the centerline to an interval of order unity. Applying the

d*ξ*

3! <sup>+</sup> *<sup>O</sup>*(*δ*2*<sup>s</sup>*

d*ξ*(*i*)

The diffusive and convective terms strike a balance near the core. This occurs in (32) when *δ* ∼ *ε*/*κ*. Having identified the distinguished limit, the core boundary layer equation becomes

> 1 2

where the superscript (*i*) stands for the inner, near-core approximation. Using a standard perturbation series, successive viscous corrections can be determined to any desired order. For this chapter, a one-term inner solution is sought. Hence, by integrating (33) and insisting

The remaining constant may be determined through matching with the outer expansion. Using Prandtl's matching principle, the outer limit of the inner solution may be equated to the inner limit of the free vortex. This process results in a composite solution that is valid everywhere except in the close vicinity of the sidewall. After transforming back to the original

> −1 4*Vr*<sup>2</sup>

2) d*ξ*

<sup>d</sup>*<sup>η</sup>* <sup>=</sup> <sup>0</sup> (31)

<sup>d</sup>*<sup>s</sup>* <sup>=</sup> <sup>0</sup> (32)

(34)

(35)

<sup>d</sup>*<sup>s</sup>* <sup>=</sup> <sup>0</sup> (33)

By converting the independent coordinate using *<sup>η</sup>* <sup>≡</sup> *<sup>π</sup>r*2, Eq. (30) simplifies into

*ε κ* d2*ξ* <sup>d</sup>*η*<sup>2</sup> <sup>+</sup>

transformation and linearizing the equation near the core where *s* ≈ 0 yields

d2*ξ*(*i*) <sup>d</sup>*s*<sup>2</sup> <sup>+</sup>

*ξ*(*i*) = *C*<sup>0</sup>

*ξ*(*ci*) = *C*¯

 1 − exp

Fig. 4. Schematic of the dual end point boundary layers present in the confined vortex.

 exp −1 <sup>2</sup> *s* − 1 

<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*2*s*<sup>2</sup>

1 2 

*ε κδ*

on a forced vortex near the core, one retrieves

coordinate, the composite inner solution collapses into

d2*ξ* <sup>d</sup>*s*<sup>2</sup> <sup>+</sup>

$$V \equiv \frac{1}{\epsilon \sigma l} = \frac{\text{Re}}{\sigma} \frac{a}{L} = \frac{\rho lIA\_i}{\mu L} = \frac{\dot{m}\_i}{\mu L} \tag{36}$$

For a complete physical description of *V*, the reader may consult with Majdalani & Chiaverini (2009). The injection mass flow rate appears as *m*˙ *<sup>i</sup>* while the density and viscosity are given by *ρ* and *μ*. The outer constant *C*¯ may be procured from the tangential injection condition. This implies

$$\mathfrak{L}^{(\text{cj})}(1) = 1 \text{ or } \vec{\mathbb{C}} \simeq 1 \tag{37}$$

and so, when suitably combined, one has

$$
\mu\_{\theta} \simeq \frac{1}{r} \left[ 1 - \exp\left( -\frac{1}{4} V r^2 \right) \right] \tag{38}
$$

Equation (38) provides a laminar representation of the swirl velocity in the bidirectional vortex chamber in which slip is permitted at the sidewall. To reconcile between this model and experimental data, additional steps are required as it will be shown below.

Before leaving this subject, it should be noted that a similar asymptotic analysis can be used to obtain the sidewall boundary layer correction. However, given the scope of this chapter, the attendant analysis is omitted. This is partly due to the fidelity of the PIV procedure and the data near the sidewall being too coarse to resolve the wall boundary layer. The additional complexity engendered by the sidewall correction is hence disregarded.

#### **2.2 Constant shear stress model**

The constant shear stress model provides a piecewise swirl velocity solution that may be used to model confined vortex motions. The basis for the model is that a free vortex of the 1/*r* type develops away from the core of the vortex; in contrast, equilibrium is maintained near the core between the shear and pressure terms. This balance leads to a model that is equally valid for both laminar and turbulent conditions. It is important to note that the corresponding flow is not turbulent per se as there are no unsteady effects included, but rather a mean velocity profile that can be used to represent the bulk flow field in a turbulent regime.

The justification for this approach can be seen mathematically from the conservation of momentum, namely,

$$(\mathbf{U} \cdot \nabla)\mathbf{U} = -\nabla p + \nabla \cdot \boldsymbol{\tau} \tag{39}$$

At the centerline, the tangential velocity vanishes and this leaves the pressure and shear stress terms available to balance each other (Townsend, 1976). The flow under consideration for this work has a zero tangential pressure gradient; therefore, the shear stress in the tangential direction may be assumed to have a constant value. The equation for the dominant shear stress becomes

$$\pi\_{r\theta} = \varepsilon \left[ \frac{1}{r} \frac{\partial u}{\partial \theta} + r \frac{\partial}{\partial r} \left( \frac{v}{r} \right) \right] \tag{40}$$

where *�* is the same viscous parameter 1/*Re*. Because the flow is axisymmetric, the *θ* derivative is eliminated such that

$$
\varepsilon \left( \frac{\partial v}{\partial r} - \frac{v}{r} \right) = \mathbb{C}\_1 \tag{41}
$$

The traditional forced vortex model can be recovered by setting the constant equal to zero; nonetheless, the model examined here will retain the general constant. After integration, the inner swirl velocity becomes

$$\mu\_{\theta}^{(i)} = r \left[ \frac{\mathbb{C}\_1}{\mathfrak{e}} \ln(r) + \mathbb{C}\_2 \right] \tag{42}$$

Using Particle Image Velocimetry 13

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 161

the base of the chamber is retrofitted with a simple convergent nozzle with a throat diameter of 1.27 cm and an inlet radius of curvature of 0.635 cm. Figure 5 showcases the modular

The working fluid for the PIV experiment is gaseous nitrogen. A number of configurable injectors are used to provide injection pressure drops that range from 10% to 30% of the chamber pressure. This, in turn, generates an increasing set of injection velocities. The nitrogen flow is seeded by a Corona Integrated Technologies Colt4 smoke generator. The

The imaging system consists of a 250 mJ/pulse Nd:Yag laser to illuminate the seed particles. For this study, a typical pulse separation of 1 *μ*s yields adequate resolution for the anticipated swirl velocities. The laser is focused through a series of adjustable optical devices to produce a sheet in the *r*–*θ* plane that can be positioned at three different axial locations. The particle images are captured by a LaVision 1280 x 1024 Flowmaster 3 camera at 1 *μ*s increments which are spaced out over a thirty second run time. The images are then cross-correlated to deduce the swirl velocity at three axial locations in the chamber. A schematic of the setup is provided

The cross-correlation is carried out with LaVision DaVis 6.2 software. The two images, separated by 1 *μ*s, are analyzed through deformed interrogation windows, initially 64 x 64 pixels in size. The windows decrease to 16 x 16 pixels during successive passes over the image. For each experimental configuration, ten sets of images are acquired and correlated to

In a related study by Rom et al. (2004), the PIV apparatus is supplemented with a modified end cap that is fitted with pressure taps. The taps are spaced at intervals of 15% of the radius with two additional taps at *r* = 0.9 and 0.967 to capture the near wall behavior. The pressure

The modularity of the experimental apparatus will permit for a widely varying range of trials. For all trials, the radius of the chamber is kept fixed at 1.27 cm. According to this arrangement, the smallest aspect ratio allows sampling at the midpoint of the chamber only. Table 1(a)

In addition to the geometric variability, the injectors are configurable for three separate injection pressure drops. This is achieved by varying the available port area of the incoming fluid. All trials are conducted using eight equally spaced tangential injection ports. Details of

After a successful trial, the ten pairs of image files are cross-correlated with DaVis 6.2 to produce a vector field for the swirl velocity, represented by a 128 x 160 matrix. Further data analysis is furnished via Matlab scripts which act upon the matrix exported by the DaVis software. The scripts average the ten raw velocity magnitudes at each axial location such that a radial profile of the swirl velocity may be reconstructed. The swirl velocity profile is an ideal

candidate for comparison to the analytical models developed in Sec. 2.

measurements provide an additional avenue to verify the analytical approximations.

process creates liquid particles of 0.2 *μ*m diameter.

provides the geometric variability of the test chamber.

the port construction are available in Table 1(b).

chamber.

in Fig. 6.

ten velocity fields.

**3.1 Trial overview**

**4. Results**

It may be interesting to note that each of the two undetermined constants, *C*<sup>1</sup> and *C*<sup>2</sup> , has a clear physical meaning: while the first relates to the swirl strength of the velocity component generating the stress, the second corresponds to the swirl strength of a flow undergoing solid body rotation. The two undetermined constants can be manipulated to match the inner solution with the outer, free vortex expression at their intersection point. This is achieved by equating the velocity and its derivative to the outer vortex at a specific matching radius. However, since the matching radius is not known a priori, it must be carefully specified. For the moment, the matching point *X* is left arbitrary. The equation to match the velocities at *X* translates into

$$X\frac{\mathbf{C}\_1}{\epsilon}\ln(X) + X\mathbf{C}\_2 = \frac{1}{X} \tag{43}$$

Equation (43) represents an effort to match the inner solution from (42) to the outer, free vortex solution. The same procedure can be used on the derivatives to provide

$$\frac{\mathbb{C}\_1}{\mathfrak{c}} \left[ 1 + \ln(X) \right] + \mathbb{C}\_2 = \frac{1}{X^2} \tag{44}$$

After solving (43) and (44) for (*C*1, *C*2) and substituting back into (42), the result may be expressed as

$$u\_{\theta} = \begin{cases} \frac{r}{X^2} \left[ 1 - \ln \left( \frac{r^2}{X^2} \right) \right]; & r \le X \\ 1, & r > X \end{cases} \tag{45}$$

At this point in the analysis, the value of *X* is still unknown although it produces a smooth function due to the matching conditions imposed through (43) and (44). The actual value of *X* can be determined in a number of different ways, the foremost being a comparison to experimental data.

#### **3. Experimental setup**

With the analytical models for the confined vortex determined, it is possible to design a meaningful experiment through which collections of data can be obtained and compared directly to their theoretical predictions. Having identified the vortex Reynolds number, *V*, as the key parameter that governs the shape of the swirl velocity, any experimental setup must be able to incorporate a number of different values for *V*. When limited to a single working fluid, two ways exist for inducing variations in *V*: changing the aspect ratio or the injection velocity. Rom (2006) uses this information to design a PIV experiment through which the character of the bidirectional vortex is carefully investigated.

To provide the necessary geometric flexibility, a modular chamber is used that allows for a variety of vortex Reynolds numbers. The chamber itself is made from a quartz cylinder and top plate. Four different chamber lengths provide a range of aspect ratios from 2.8 to 8.8. The bottom plate is acrylic and can be modified for a number of injection conditions. Additionally, the base of the chamber is retrofitted with a simple convergent nozzle with a throat diameter of 1.27 cm and an inlet radius of curvature of 0.635 cm. Figure 5 showcases the modular chamber.

The working fluid for the PIV experiment is gaseous nitrogen. A number of configurable injectors are used to provide injection pressure drops that range from 10% to 30% of the chamber pressure. This, in turn, generates an increasing set of injection velocities. The nitrogen flow is seeded by a Corona Integrated Technologies Colt4 smoke generator. The process creates liquid particles of 0.2 *μ*m diameter.

The imaging system consists of a 250 mJ/pulse Nd:Yag laser to illuminate the seed particles. For this study, a typical pulse separation of 1 *μ*s yields adequate resolution for the anticipated swirl velocities. The laser is focused through a series of adjustable optical devices to produce a sheet in the *r*–*θ* plane that can be positioned at three different axial locations. The particle images are captured by a LaVision 1280 x 1024 Flowmaster 3 camera at 1 *μ*s increments which are spaced out over a thirty second run time. The images are then cross-correlated to deduce the swirl velocity at three axial locations in the chamber. A schematic of the setup is provided in Fig. 6.

The cross-correlation is carried out with LaVision DaVis 6.2 software. The two images, separated by 1 *μ*s, are analyzed through deformed interrogation windows, initially 64 x 64 pixels in size. The windows decrease to 16 x 16 pixels during successive passes over the image. For each experimental configuration, ten sets of images are acquired and correlated to ten velocity fields.

In a related study by Rom et al. (2004), the PIV apparatus is supplemented with a modified end cap that is fitted with pressure taps. The taps are spaced at intervals of 15% of the radius with two additional taps at *r* = 0.9 and 0.967 to capture the near wall behavior. The pressure measurements provide an additional avenue to verify the analytical approximations.

### **3.1 Trial overview**

12 Will-be-set-by-IN-TECH

The traditional forced vortex model can be recovered by setting the constant equal to zero; nonetheless, the model examined here will retain the general constant. After integration, the

It may be interesting to note that each of the two undetermined constants, *C*<sup>1</sup> and *C*<sup>2</sup> , has a clear physical meaning: while the first relates to the swirl strength of the velocity component generating the stress, the second corresponds to the swirl strength of a flow undergoing solid body rotation. The two undetermined constants can be manipulated to match the inner solution with the outer, free vortex expression at their intersection point. This is achieved by equating the velocity and its derivative to the outer vortex at a specific matching radius. However, since the matching radius is not known a priori, it must be carefully specified. For the moment, the matching point *X* is left arbitrary. The equation to match the velocities at *X*

ln(*X*) + *XC*<sup>2</sup> <sup>=</sup> <sup>1</sup>

Equation (43) represents an effort to match the inner solution from (42) to the outer, free vortex

*�* [<sup>1</sup> <sup>+</sup> ln(*X*)] <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>

After solving (43) and (44) for (*C*1, *C*2) and substituting back into (42), the result may be

At this point in the analysis, the value of *X* is still unknown although it produces a smooth function due to the matching conditions imposed through (43) and (44). The actual value of *X* can be determined in a number of different ways, the foremost being a comparison to

With the analytical models for the confined vortex determined, it is possible to design a meaningful experiment through which collections of data can be obtained and compared directly to their theoretical predictions. Having identified the vortex Reynolds number, *V*, as the key parameter that governs the shape of the swirl velocity, any experimental setup must be able to incorporate a number of different values for *V*. When limited to a single working fluid, two ways exist for inducing variations in *V*: changing the aspect ratio or the injection velocity. Rom (2006) uses this information to design a PIV experiment through which the

To provide the necessary geometric flexibility, a modular chamber is used that allows for a variety of vortex Reynolds numbers. The chamber itself is made from a quartz cylinder and top plate. Four different chamber lengths provide a range of aspect ratios from 2.8 to 8.8. The bottom plate is acrylic and can be modified for a number of injection conditions. Additionally,

� *r*<sup>2</sup> *X*<sup>2</sup> ��

; *r > X*

; *r* ≤ *X*

ln(*r*) + *C*<sup>2</sup>

�

(42)

(45)

*<sup>X</sup>* (43)

*<sup>X</sup>*<sup>2</sup> (44)

� *C*1 *�*

*u*(*i*) *<sup>θ</sup>* = *r*

*<sup>X</sup> <sup>C</sup>*<sup>1</sup> *�*

solution. The same procedure can be used on the derivatives to provide

*C*1

⎧ ⎪⎪⎨

*r X*<sup>2</sup> � 1 − ln

1 *r*

⎪⎪⎩

*u<sup>θ</sup>* =

character of the bidirectional vortex is carefully investigated.

inner swirl velocity becomes

translates into

expressed as

experimental data.

**3. Experimental setup**

The modularity of the experimental apparatus will permit for a widely varying range of trials. For all trials, the radius of the chamber is kept fixed at 1.27 cm. According to this arrangement, the smallest aspect ratio allows sampling at the midpoint of the chamber only. Table 1(a) provides the geometric variability of the test chamber.

In addition to the geometric variability, the injectors are configurable for three separate injection pressure drops. This is achieved by varying the available port area of the incoming fluid. All trials are conducted using eight equally spaced tangential injection ports. Details of the port construction are available in Table 1(b).

### **4. Results**

After a successful trial, the ten pairs of image files are cross-correlated with DaVis 6.2 to produce a vector field for the swirl velocity, represented by a 128 x 160 matrix. Further data analysis is furnished via Matlab scripts which act upon the matrix exported by the DaVis software. The scripts average the ten raw velocity magnitudes at each axial location such that a radial profile of the swirl velocity may be reconstructed. The swirl velocity profile is an ideal candidate for comparison to the analytical models developed in Sec. 2.

Using Particle Image Velocimetry 15

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 163

Parameter Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Injector Pressure Drop, 27.6 55.2 82.8 55.2 27.6 55.2

Aspect Ratio, *L* 2.4 2.4 4.4 3.4 4.4 4.4 Average Injection Speed, 68.73 77.72 89.61 88.78 74.81 88.31

Modified Swirl Number, 2.81 4.10 5.26 4.10 2.81 4.10

Inflow Parameter, 0.0239 0.0164 0.0069 0.0114 0.0129 0.0088

Vortex Reynolds Number, 47 150 36 540 30 160 29 150 27 650 22 370

Six cases are chosen from the available trials for comparison with the laminar core and constant shear stress models. Data fidelity is the primary consideration in selection, as many of the other trials exhibit an increasing amount of scatter to the data, especially in the core region. Of the six cases considered, the first three are chosen to develop the empirically based correlations needed to match the data to the analytical approximation. The remaining three cases are held in reserve, to provide verification of the corrected models. The experimental

In order to compare the analytical models to the experimental data, some additional effort is required. For the laminar core case, a simple least-squares regression will permit the vortex Reynolds number to be tuned to the experimental conditions. This correlation is necessary as the flow conditions in the experiment are beyond the range of validity of the laminar model.

(b) Injector configurations

Δ*p* Port Dia. Aggregate Injection (% of *pc*) (cm) Area (cm2) 10 0.605 2.299 20 0.500 1.571 30 0.442 1.228

(a) Geometric configurations Length Aspect Ratio Axial Sampling (cm) Locations (*z*/*L*)

3.56 2.80 0.5

6.10 4.80 0.5

8.64 6.80 0.5

11.2 8.82 0.5

Δ*p*¯ (kPa)

*U* (m/s)

*σ* = *a*2/*Ai*

*κ* = 1/(2*πσL*)

*V* = *m*˙ *<sup>i</sup>*/(*L*0*μ*)

**4.1 Laminar core correlation**

Table 1. Available configurations for experimental trials.

Table 2. Operational parameters for the PIV experiments.

parameters for these test cases are furnished in Table 2.

0.2

0.7 0.2

0.7 0.2

0.7

Fig. 5. Modular vortex chamber for the PIV experiment.

Fig. 6. Schematic of the PIV setup.



Fig. 5. Modular vortex chamber for the PIV experiment.

sonic choke

> smoke machine

low pressure manifold

> laser system

data acquisition and control

camera

vortex chamber laser sheet optics

high pressure cylinders

Fig. 6. Schematic of the PIV setup.

high pressure manifold

pneumatic control


20 0.500 1.571 30 0.442 1.228

Table 1. Available configurations for experimental trials.


Table 2. Operational parameters for the PIV experiments.

Six cases are chosen from the available trials for comparison with the laminar core and constant shear stress models. Data fidelity is the primary consideration in selection, as many of the other trials exhibit an increasing amount of scatter to the data, especially in the core region. Of the six cases considered, the first three are chosen to develop the empirically based correlations needed to match the data to the analytical approximation. The remaining three cases are held in reserve, to provide verification of the corrected models. The experimental parameters for these test cases are furnished in Table 2.

#### **4.1 Laminar core correlation**

In order to compare the analytical models to the experimental data, some additional effort is required. For the laminar core case, a simple least-squares regression will permit the vortex Reynolds number to be tuned to the experimental conditions. This correlation is necessary as the flow conditions in the experiment are beyond the range of validity of the laminar model.

Using Particle Image Velocimetry 17

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 165

space changes every time that a new radius is calculated. The iterative approach continues to calculate new values of *X* until the difference between successive radii falls below a user-specified tolerance. Since the data comprises a limited set of discrete points, the solution converges rapidly. The final radius is checked against neighboring values to ensure that the

Another fundamental issue associated with the shear stress framework is that the model is not written in terms of the vortex Reynolds number, *V*. The laminar core analysis highlights the importance of the vortex Reynolds number as it incorporates the geometry of the chamber, *l*, as well as the swirl dominated nature of the flow through the modified swirl number, *σ*. This parameter clearly determines the behavior of the swirl velocity near the core. To determine the vortex Reynolds number dependence, the constant shear stress model is matched to the laminar core solution such that both representations will return the same maximum velocity.

= 0 (47)

<sup>√</sup>*<sup>V</sup>* (48)

<sup>2</sup> *<sup>e</sup>*−1/2 (49)

<sup>√</sup>*<sup>V</sup>* (50)

yes

print radius

<sup>√</sup>*e*. The same procedure applied to the laminar model yields

*<sup>e</sup>*−1/2 � 2.242

<sup>−</sup>1, <sup>−</sup><sup>1</sup>

� 3.802

within tolerance?

no

d*u<sup>θ</sup>* d*r r*=*r*max

<sup>−</sup><sup>1</sup> <sup>−</sup> 2pln

<sup>−</sup>1, <sup>−</sup><sup>1</sup> 2

> 2

<sup>−</sup>1, <sup>−</sup><sup>1</sup>

<sup>−</sup>1, <sup>−</sup><sup>1</sup>

<sup>−</sup>*<sup>V</sup>*

<sup>1</sup> <sup>+</sup> 2pln

<sup>2</sup> *<sup>e</sup>*<sup>−</sup> <sup>1</sup> 2 

<sup>2</sup> *<sup>e</sup>*<sup>−</sup> <sup>1</sup> 2

With the maximum velocity locations known, these values may be substituted back into their

<sup>2</sup> *<sup>e</sup>*−1/2)

<sup>1</sup> <sup>+</sup> 2pln

1 <sup>2</sup> <sup>+</sup> pln

compare radius with previous value

solution is in fact fully optimized.

Mathematically, this requires taking

Solving Eq. (47) gives *r*max = *X*/

2

initialize data

*Xe*1/2 <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>*

*<sup>X</sup>* <sup>=</sup> <sup>2</sup> <sup>√</sup>*eV*

evaluate regression

trim data

Fig. 7. Flowchart of the piecewise least-squares algorithm.

*r*max =

 2 *V* 

respective expressions for velocity and then equated to produce

 −2 

<sup>2</sup> <sup>+</sup>pln(−1,<sup>−</sup> <sup>1</sup>

<sup>1</sup> <sup>−</sup> exp

1

Equation (49) may be solved for *X* in terms of *V* to obtain

**4.2.2 Viscous correlation**


Table 3. Least-squares parameters for laminar core and constant shear stress frameworks.

However, since the experimental vortex Reynolds number is also known, the test results can be linked to the analytical vortex Reynolds number through an empirically based viscosity ratio of the form *<sup>t</sup>* = *μ*/*μt*, with *μ<sup>t</sup>* denoting a turbulent viscosity. In this manner, the vortex Reynolds number definition may be modified such that

$$V = \frac{\rho LIA\_i}{\mu L} = \frac{\rho LIA\_i}{\mu\_t \ell\_t L} = \frac{V\_t}{\ell\_t} \tag{46}$$

An average eddy viscosity may be calculated with this correlation that will properly match the laminar, analytical model to the high-speed experimental counterpart.

The standard least-squares regression scheme is used to determine the laminar vortex Reynolds number that best fits the model to the data. This value is then used to calculate the corresponding *<sup>t</sup>* that will rectify the laminar and experimental vortex Reynolds numbers. The three trials provide 879 data points for the regression, producing an average *<sup>t</sup>* = 151.8. Results of the least-squares regression runs for the individual trials are given in Table 3. Note that the three sets of experiments yield rather consistent values of *<sup>t</sup>* = 150, 154, and 151, respectively.

#### **4.2 Constant shear stress correlation**

The constant shear stress model requires a more elaborate analysis to achieve the same sort of comparison. First, the model itself consists of a piecewise function and the optimization parameter appears in both the solution and the boundary. To properly match the model to the experiment, a modified least-squares routine is developed using an iterative process that is capable of accounting for the moving boundary in the optimization.

#### **4.2.1 Piecewise least-squares regression**

The piecewise least-squares code contains several distinct components. The first element is a rewrite of the standard least-squares technique in a manner to incorporate the piecewise nature of the function into the derivative calculations. The function returns the optimized parameter, in this case the matching radius, *X*. The second function is simply a truncation function that adjusts the data set to reflect the new value of the optimization parameter. Finally, a control function loops over the data set, calling the least-squares function and comparing the new radius to the previous trial, *X*, until a satisfactory tolerance is reached, such as 0.0001 in this case. For the reader's convenience, a flow chart of the numerical procedure is provided in Fig. 7.

This iterative procedure is necessary because of the nature of this particular piecewise solution. For most piecewise solutions, a standard least-squares algorithm is sufficient. However, in this case, the optimization parameter coincides with the matching radius that determines the boundary between the inner and outer solutions. As a result, the optimization space changes every time that a new radius is calculated. The iterative approach continues to calculate new values of *X* until the difference between successive radii falls below a user-specified tolerance. Since the data comprises a limited set of discrete points, the solution converges rapidly. The final radius is checked against neighboring values to ensure that the solution is in fact fully optimized.

#### **4.2.2 Viscous correlation**

16 Will-be-set-by-IN-TECH

*Vt <sup>X</sup>*<sup>0</sup> *<sup>X</sup>* 50/√*Vt <sup>t</sup>* 47 150 49.04 0.243 0.230 150.3 36 540 49.63 0.267 0.262 154.1 30 160 50.67 0.314 0.288 151.0

Table 3. Least-squares parameters for laminar core and constant shear stress frameworks.

*<sup>V</sup>* <sup>=</sup> *<sup>ρ</sup>UAi*

the laminar, analytical model to the high-speed experimental counterpart.

is capable of accounting for the moving boundary in the optimization.

Reynolds number definition may be modified such that

respectively.

**4.2 Constant shear stress correlation**

**4.2.1 Piecewise least-squares regression**

procedure is provided in Fig. 7.

However, since the experimental vortex Reynolds number is also known, the test results can be linked to the analytical vortex Reynolds number through an empirically based viscosity ratio of the form *<sup>t</sup>* = *μ*/*μt*, with *μ<sup>t</sup>* denoting a turbulent viscosity. In this manner, the vortex

*<sup>μ</sup><sup>L</sup>* <sup>=</sup> *<sup>ρ</sup>UAi*

An average eddy viscosity may be calculated with this correlation that will properly match

The standard least-squares regression scheme is used to determine the laminar vortex Reynolds number that best fits the model to the data. This value is then used to calculate the corresponding *<sup>t</sup>* that will rectify the laminar and experimental vortex Reynolds numbers. The three trials provide 879 data points for the regression, producing an average *<sup>t</sup>* = 151.8. Results of the least-squares regression runs for the individual trials are given in Table 3. Note that the three sets of experiments yield rather consistent values of *<sup>t</sup>* = 150, 154, and 151,

The constant shear stress model requires a more elaborate analysis to achieve the same sort of comparison. First, the model itself consists of a piecewise function and the optimization parameter appears in both the solution and the boundary. To properly match the model to the experiment, a modified least-squares routine is developed using an iterative process that

The piecewise least-squares code contains several distinct components. The first element is a rewrite of the standard least-squares technique in a manner to incorporate the piecewise nature of the function into the derivative calculations. The function returns the optimized parameter, in this case the matching radius, *X*. The second function is simply a truncation function that adjusts the data set to reflect the new value of the optimization parameter. Finally, a control function loops over the data set, calling the least-squares function and comparing the new radius to the previous trial, *X*, until a satisfactory tolerance is reached, such as 0.0001 in this case. For the reader's convenience, a flow chart of the numerical

This iterative procedure is necessary because of the nature of this particular piecewise solution. For most piecewise solutions, a standard least-squares algorithm is sufficient. However, in this case, the optimization parameter coincides with the matching radius that determines the boundary between the inner and outer solutions. As a result, the optimization

*<sup>μ</sup><sup>t</sup>tL* <sup>=</sup> *Vt*

*t*

(46)

Another fundamental issue associated with the shear stress framework is that the model is not written in terms of the vortex Reynolds number, *V*. The laminar core analysis highlights the importance of the vortex Reynolds number as it incorporates the geometry of the chamber, *l*, as well as the swirl dominated nature of the flow through the modified swirl number, *σ*. This parameter clearly determines the behavior of the swirl velocity near the core. To determine the vortex Reynolds number dependence, the constant shear stress model is matched to the laminar core solution such that both representations will return the same maximum velocity. Mathematically, this requires taking

$$\left.\frac{\mathbf{d}u\_{\theta}}{\mathbf{d}r}\right|\_{r=r\_{\text{max}}}=\mathbf{0}\tag{47}$$

Solving Eq. (47) gives *r*max = *X*/ <sup>√</sup>*e*. The same procedure applied to the laminar model yields

$$r\_{\text{max}} = \sqrt{\frac{2}{V} \left[ -1 - 2\text{pln}\left( -1, -\frac{1}{2}e^{-1/2} \right) \right]} \simeq \frac{2.242}{\sqrt{V}} \tag{48}$$

With the maximum velocity locations known, these values may be substituted back into their respective expressions for velocity and then equated to produce

$$\frac{2}{Xe^{1/2}} = 1 - e^{\frac{1}{2} + \text{pln}\left(-1, -\frac{1}{2}e^{-1/2}\right)} \sqrt{\frac{-V}{2\left[1 + 2\text{pln}\left(-1, -\frac{1}{2}e^{-1/2}\right)\right]}}\tag{49}$$

Equation (49) may be solved for *X* in terms of *V* to obtain

Fig. 7. Flowchart of the piecewise least-squares algorithm.

This solution serves a twofold purpose. First, it demonstrates how the constant shear stress model can be matched to existing data as illustrated in Fig. 8. For additional matching paradigms the reader is directed to Maicke & Majdalani (2009). Secondly, it provides an expected form of the relationship between *Vt* and the matching radius. The iterative least-squares method uses the relationship

$$X = \frac{X\_0}{\sqrt{V\_t}}\tag{51}$$

Using Particle Image Velocimetry 19

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 167

(a) *V* = 47150

(b) *V* = 36540

0 0.2 0.4 0.6 0.8 1

*r*

(c) *V* = 30160

Fig. 9. Laminar and constant shear stress models compared to experimental data from Rom

 *z*/*L* 0.2 0.5 0.7

> laminar core constant shear

0

0

0

(2006) that is used in the least-squares regression.

100

200

300

*u*

400

100

200

300

*u*

400

100

200

300

*u*

400

The algorithm optimizes the constant *X*<sup>0</sup> to produce a matching radius for each trial that incorporates the experimental vortex Reynolds number. The calculated values are shown in Table 3. Note that an average value of approximately 50 provides very good agreement with the experimental data.

#### **4.3 Graphical comparison**

Using *X*<sup>0</sup> = 50 and *<sup>t</sup>* = 151.8, a comparison is drawn in Figs. 9 and 10 between theory and experiment. The measurements collected in each trial correspond to the data acquired at three axial locations in the chamber, specifically *z* = 0.2, 0.5, and 0.7. The velocity plots in Fig. 9 display the collection of data used in the least-squares analysis. Consequently, good agreement is expected and achieved in all three figures. Figure 10 showcases the reserve data sets that were not employed in the regression algorithm, but rather saved to test the accuracy of the models at various vortex Reynolds numbers. As with the first three trials, the data in the outer, free vortex region is seen to be tightly grouped to the extent of faithfully following the shape predicted by both analytical models. Evidently, the agreement in the core region may be seen to be less appreciable. The additional scatter observed in the data is due to PIV limitations that will be discussed in Sec. 5. Nonetheless, the models do capture the general trends in the core. To accommodate the scatter, the fits for both theoretical models are weighted such that they provide an envelope in the core region, rather than a strict regression fit through the data. This weighting is chosen specifically because of the increased scatter in the core region that artificially lowers the predicted velocities.

It may be interesting to note the ability of the laminar core model with a turbulent eddy viscosity to duplicate the essential features of the flow. This behavior is consistent with

Fig. 8. Peak velocity matched models shown at *V* = 1000.

18 Will-be-set-by-IN-TECH

This solution serves a twofold purpose. First, it demonstrates how the constant shear stress model can be matched to existing data as illustrated in Fig. 8. For additional matching paradigms the reader is directed to Maicke & Majdalani (2009). Secondly, it provides an expected form of the relationship between *Vt* and the matching radius. The iterative

> *<sup>X</sup>* <sup>=</sup> *<sup>X</sup>*<sup>0</sup> <sup>√</sup>*Vt*

The algorithm optimizes the constant *X*<sup>0</sup> to produce a matching radius for each trial that incorporates the experimental vortex Reynolds number. The calculated values are shown in Table 3. Note that an average value of approximately 50 provides very good agreement with

Using *X*<sup>0</sup> = 50 and *<sup>t</sup>* = 151.8, a comparison is drawn in Figs. 9 and 10 between theory and experiment. The measurements collected in each trial correspond to the data acquired at three axial locations in the chamber, specifically *z* = 0.2, 0.5, and 0.7. The velocity plots in Fig. 9 display the collection of data used in the least-squares analysis. Consequently, good agreement is expected and achieved in all three figures. Figure 10 showcases the reserve data sets that were not employed in the regression algorithm, but rather saved to test the accuracy of the models at various vortex Reynolds numbers. As with the first three trials, the data in the outer, free vortex region is seen to be tightly grouped to the extent of faithfully following the shape predicted by both analytical models. Evidently, the agreement in the core region may be seen to be less appreciable. The additional scatter observed in the data is due to PIV limitations that will be discussed in Sec. 5. Nonetheless, the models do capture the general trends in the core. To accommodate the scatter, the fits for both theoretical models are weighted such that they provide an envelope in the core region, rather than a strict regression fit through the data. This weighting is chosen specifically because of the increased scatter in the core region that

It may be interesting to note the ability of the laminar core model with a turbulent eddy viscosity to duplicate the essential features of the flow. This behavior is consistent with

> laminar core constant shear free outer vortex

0 0.2 0.4 0.6 0.8 1

*r*

(51)

least-squares method uses the relationship

artificially lowers the predicted velocities.

0

Fig. 8. Peak velocity matched models shown at *V* = 1000.

2

4

6

8

*u*

10

the experimental data.

**4.3 Graphical comparison**

Fig. 9. Laminar and constant shear stress models compared to experimental data from Rom (2006) that is used in the least-squares regression.

Using Particle Image Velocimetry 21

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 169

0 0.2 0.4 0.6 0.8 1

Fig. 11. Comparison of laminar core and constant shear stress models to experimental and

the results of several experimental studies in which vortices are shown to exhibit an approximately constant angular velocity core with low turbulence levels to the extent of appearing as nearly laminar. With turbulent diffusion being restricted to an annular region about the laminar core, it is not surprising to see that the free vortex approximation in the

In addition to the swirl velocity distribution, the pressure measurements from Rom et al. (2004) may be used to verify the pressure predictions associated with the existing theoretical

> � *πr* 2 �� + <sup>1</sup> 4*V* � Ei � −1 4*Vr*<sup>2</sup> � <sup>−</sup> Ei � −1 2*Vr*<sup>2</sup>

where Ei(*x*) is the second exponential integral function. For the constant shear stress model,

sin2(*πr*2) 2*r*<sup>2</sup>

For more detail on the pressure relations, the reader is referred to Majdalani & Chiaverini

Figure 11 depicts a normalized form of Δ*p* in the vortex chamber. Note that along with the experimental data, a numerical solution from Murray et al. (2004) is presented for the same conditions. All data sets are normalized to a reference value at the sidewall. The agreement displayed is encouraging especially near the wall region. Both analytical models exhibit

�

4*π*2*z* +

models. For the laminar core approximation, the equation of interest is

+ *κ*<sup>2</sup> sin2

*r*4 *X*<sup>4</sup>

4*π*2*z* +

� � ln � *<sup>r</sup> X* � − 2 � �

2) � <sup>−</sup> *<sup>κ</sup>*<sup>2</sup> � *r*

 laminar core constant shear numerical experimental

<sup>−</sup> *<sup>κ</sup>*2*X*<sup>4</sup> sin2(*π<sup>r</sup>*

sin2(*πr*2) 2*r*<sup>2</sup>

; *r* ≤ *X*

�

2) �

; *r > X*

�� (52)

(53)

0

numerical pressure data for *Vt* = 47150.

outer domain continues to hold.

��

1 2*κ*4*r*<sup>2</sup>

> + 1 <sup>2</sup> <sup>−</sup> <sup>3</sup>

(2009) and Maicke & Majdalani (2009).

1 − *e* − 1 <sup>4</sup> *Vr*<sup>2</sup> �<sup>2</sup>

> � *r* 4 �

the piecewise relation for the pressure drop becomes

*<sup>X</sup>*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*<sup>2</sup>

<sup>5</sup> <sup>+</sup> ln �

�

1 + *κ*<sup>2</sup> sin2(*πr*

<sup>Δ</sup>*<sup>p</sup>* <sup>=</sup> <sup>−</sup>2*π*2*κ*2*z*<sup>2</sup>

− 1 <sup>2</sup> *<sup>r</sup>*−<sup>2</sup>

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 <sup>2</sup> <sup>−</sup> <sup>1</sup> 2*r*<sup>2</sup> �

Δ*p* =

0.2

0.4

0.6

0.8

*ref p p* Δ Δ

1

Fig. 10. Laminar and constant shear stress models compared to experimental data from Rom (2006) that is reserved for verification.

20 Will-be-set-by-IN-TECH

(a) *V* = 29150

(b) *V* = 27650

0 0.2 0.4 0.6 0.8 1

*r*

(c) *V* = 22370

Fig. 10. Laminar and constant shear stress models compared to experimental data from Rom

 0.2 0.5 0.7

 laminar core constant shear

400 *z*/*L*

0

0

0

(2006) that is reserved for verification.

100

200

300

*u*

100

200

300

*u*

100

200

300

*u*

Fig. 11. Comparison of laminar core and constant shear stress models to experimental and numerical pressure data for *Vt* = 47150.

the results of several experimental studies in which vortices are shown to exhibit an approximately constant angular velocity core with low turbulence levels to the extent of appearing as nearly laminar. With turbulent diffusion being restricted to an annular region about the laminar core, it is not surprising to see that the free vortex approximation in the outer domain continues to hold.

In addition to the swirl velocity distribution, the pressure measurements from Rom et al. (2004) may be used to verify the pressure predictions associated with the existing theoretical models. For the laminar core approximation, the equation of interest is

$$\begin{aligned} \Delta p &= -2\pi^2 \kappa^2 z^2 \\ &- \frac{1}{2} r^{-2} \left[ \left( 1 - e^{-\frac{1}{4}Vr^2} \right)^2 + \kappa^2 \sin^2 \left( \pi r^2 \right) \right] + \frac{1}{4} V \left[ \text{Ei} \left( -\frac{1}{4}Vr^2 \right) - \text{Ei} \left( -\frac{1}{2}Vr^2 \right) \right] \end{aligned} \tag{52}$$

where Ei(*x*) is the second exponential integral function. For the constant shear stress model, the piecewise relation for the pressure drop becomes

$$\Delta p = \begin{cases} \frac{1}{2\kappa^4 r^2} \left( r^4 \left\{ 5 + \ln \left( \frac{r^4}{X^4} \right) \left[ \ln \left( \frac{r}{X} \right) - 2 \right] \right\} - \kappa^2 X^4 \sin^2(\pi r^2) \right) & \text{ $\kappa$ } \\\ \quad + \frac{1}{2} - \frac{3}{X^2} - \kappa^2 \left[ 4\pi^2 z + \frac{\sin^2(\pi r^2)}{2r^2} \right]; & r \le X \\\ \frac{1}{2} - \frac{1}{2r^2} \left[ 1 + \kappa^2 \sin^2(\pi r^2) \right] - \kappa^2 \left[ 4\pi^2 z + \frac{\sin^2(\pi r^2)}{2r^2} \right]; & r > X \end{cases} \tag{53}$$

For more detail on the pressure relations, the reader is referred to Majdalani & Chiaverini (2009) and Maicke & Majdalani (2009).

Figure 11 depicts a normalized form of Δ*p* in the vortex chamber. Note that along with the experimental data, a numerical solution from Murray et al. (2004) is presented for the same conditions. All data sets are normalized to a reference value at the sidewall. The agreement displayed is encouraging especially near the wall region. Both analytical models exhibit

Using Particle Image Velocimetry 23

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 171

be difficult to capture in the region of maximum swirl, even with properly sized particles. In the core region, the particles exhibit increased drag as the swirl velocity reaches its peak. The particles also display a tendency to separate via centrifugal entrainment. Visually, this manifests itself as a reduction in the seed particles near the flow centerline. Furthermore, the core of the chamber happens to be the region of highest axial velocity; thus, the particles that are tagged in the first exposure may have traveled below the laser sheet by the time the second image is taken. This scenario can also lead to decreased confidence in the correlated data in

The verification process in this chapter is concentrated on a single aspect of the theoretical models, namely, the tangential velocity. To fully characterize the analytical approximations, the axial and radial velocities should be measured alongside the pressure. Practically, this cannot be accomplished in one trial with the current experimental configuration as the camera cannot track the fluid motion perpendicularly to the field of view. A stereoscopic PIV setup can overcome this deficiency albeit with increased cost in both equipment and procedural time as the pair of cameras require careful calibration in order to properly account for the velocities along the third axis. For this particular application, Laser Doppler Velocimetry may prove to be a better choice. In addition to providing velocity measurements in all three directions, a properly configured LDV experiment can produce data at more axial locations in a single setup. While not specific to PIV, acquiring the necessary pressure measurements may require yet another setup in which the clear headwall module can be replaced with a separate

Finally, the PIV approach cannot provide an accurate assessment of flow motion over the entire chamber. In the vicinity of the nozzle, accurate measurements are difficult to obtain as the fluid is restricted to a narrower cross-section; moreover, the character of the flow changes abruptly as it transitions from a swirl-dominated to a predominantly axial motion. In this region PIV is not sufficient to accurately resolve the velocity field; alternative techniques such as X-ray radiography may be required to more suitably investigate this region experimentally.

This chapter focuses on the use of PIV measurements in conjunction with two analytical models to describe the flow field in a bidirectional vortex chamber. Although the primary motivation for this work is to better understand the gas dynamics within a vortex-fired liquid rocket engine known as the Vortex Combustion Cold-Wall Chamber, the principal attributes associated with the resulting cyclonic motion have industrial, geophysical, meteorological,

Using a cold-flow environment as a basis for this work, the findings indicate that both PIV and theoretical frameworks converge in predicting the presence of a forced vortex core region followed by a free, irrotational vortex tail that are characteristic of not only bidirectional vortices, but of unidirectional swirl-dominated flows as well. Moreover, despite the underlying restriction to non-reactive conditions, the essential features captured here seem to be somewhat representative of those reported in combustion chambers in which chemical reactions are prominent (Fang et al., 2004). It appears that the addition of high intensity swirl leads to a robust cyclonic motion that tends to retain its fundamental structure even in the presence of chemical kinetics. Nonetheless, more effort in modeling reactive and multiphase flows in cyclonic chambers will be critically important to further illuminate the particular

and astrophysical applications that fall beyond the propulsive dimension.

the core region (see also Hu et al., 2005).

apparatus containing pressure taps.

**6. Conclusion**


Table 4. Statistical parameters for the regression of the laminar and constant shear stress models.

a slight overprediction of the pressure, but follow the experimental trend admirably. The correlation of the theory with the numerical simulation appears to be excellent, with only slight deviations near the core region.

#### **4.4 Statistical verification**

To objectively compare the accuracy of the two models, several statistical parameters may be evaluated. By comparing correlation coefficients, *rcc*, standard errors, *σe*, and total relative errors, Δ*Et*, the constant shear-based model seems to provide a slightly better fit to the data than the modified laminar distribution. The standard and total relative errors are calculated from

$$
\sigma\_{\varepsilon} = \frac{1}{\sqrt{n-1}} \sqrt{\sum\_{i=1}^{n} \left[ \mathfrak{d}\_{\theta}(r\_i) - \mathfrak{u}\_{\theta}(r\_i) \right]^2} \tag{54}
$$

and

$$
\Delta E\_t = \sum\_{i=1}^n \left[ \mathfrak{d}\_\theta(r\_i) - u\_\theta(r\_i) \right]^2 / \sum\_{i=1}^n \mathfrak{d}\_\theta^2(r\_i) \tag{55}
$$

where *n* and *u*ˆ*<sup>θ</sup>* denote the number of data points and the measured velocity at *ri*, the radius of the *i*th data point. The standard error of the estimate quantifies the spread of data about the regression line, much like the standard deviation that measures the spread about a mean value. As shown in Table 4, the total relative error falls under 3.2, 1.3, and 4.4% for the three cases associated with the constant shear stress approach. The corresponding experimental correlation coefficients are calculated at 0.88, 0.97 and 0.90, respectively. When the modified laminar core technique is used, the relative errors slightly increase to 3.5, 1.5, and 5.0%, with an equally minute reduction in *rcc*.

The graphical and statistical verifications highlight the effectiveness of the theoretical approach that is pursued. The agreement between all cases, even those held in reserve, is particularly satisfying. However, additional experiments are necessary to both confirm the correlations developed here and to extend their usefulness to a wider variety of engine configurations and injection conditions.

#### **5. Challenges and limitations**

While the data provided by the PIV experiments proves invaluable in the verification of theoretical models, the technique is not without its challenges. In this section, the challenges and limitations of PIV applications to confined vortex flows are discussed.

In most cases, the PIV technique constitutes a powerful non-intrusive method for measuring velocities. However, as demonstrated in the previous sections, the vortical flow field can be difficult to capture in the region of maximum swirl, even with properly sized particles. In the core region, the particles exhibit increased drag as the swirl velocity reaches its peak. The particles also display a tendency to separate via centrifugal entrainment. Visually, this manifests itself as a reduction in the seed particles near the flow centerline. Furthermore, the core of the chamber happens to be the region of highest axial velocity; thus, the particles that are tagged in the first exposure may have traveled below the laser sheet by the time the second image is taken. This scenario can also lead to decreased confidence in the correlated data in the core region (see also Hu et al., 2005).

The verification process in this chapter is concentrated on a single aspect of the theoretical models, namely, the tangential velocity. To fully characterize the analytical approximations, the axial and radial velocities should be measured alongside the pressure. Practically, this cannot be accomplished in one trial with the current experimental configuration as the camera cannot track the fluid motion perpendicularly to the field of view. A stereoscopic PIV setup can overcome this deficiency albeit with increased cost in both equipment and procedural time as the pair of cameras require careful calibration in order to properly account for the velocities along the third axis. For this particular application, Laser Doppler Velocimetry may prove to be a better choice. In addition to providing velocity measurements in all three directions, a properly configured LDV experiment can produce data at more axial locations in a single setup. While not specific to PIV, acquiring the necessary pressure measurements may require yet another setup in which the clear headwall module can be replaced with a separate apparatus containing pressure taps.

Finally, the PIV approach cannot provide an accurate assessment of flow motion over the entire chamber. In the vicinity of the nozzle, accurate measurements are difficult to obtain as the fluid is restricted to a narrower cross-section; moreover, the character of the flow changes abruptly as it transitions from a swirl-dominated to a predominantly axial motion. In this region PIV is not sufficient to accurately resolve the velocity field; alternative techniques such as X-ray radiography may be required to more suitably investigate this region experimentally.

### **6. Conclusion**

22 Will-be-set-by-IN-TECH

laminar core 0.887 0.592 4.91 0.962 0.276 1.47 0.870 0.391 3.45 constant shear 0.900 0.558 4.36 0.968 0.253 1.23 0.880 0.376 3.19

a slight overprediction of the pressure, but follow the experimental trend admirably. The correlation of the theory with the numerical simulation appears to be excellent, with only

To objectively compare the accuracy of the two models, several statistical parameters may be evaluated. By comparing correlation coefficients, *rcc*, standard errors, *σe*, and total relative errors, Δ*Et*, the constant shear-based model seems to provide a slightly better fit to the data than the modified laminar distribution. The standard and total relative errors are calculated

> *n* ∑ *i*=1

[*u*ˆ*<sup>θ</sup>* (*ri*) − *u<sup>θ</sup>* (*ri*)]

where *n* and *u*ˆ*<sup>θ</sup>* denote the number of data points and the measured velocity at *ri*, the radius of the *i*th data point. The standard error of the estimate quantifies the spread of data about the regression line, much like the standard deviation that measures the spread about a mean value. As shown in Table 4, the total relative error falls under 3.2, 1.3, and 4.4% for the three cases associated with the constant shear stress approach. The corresponding experimental correlation coefficients are calculated at 0.88, 0.97 and 0.90, respectively. When the modified laminar core technique is used, the relative errors slightly increase to 3.5, 1.5, and 5.0%, with

The graphical and statistical verifications highlight the effectiveness of the theoretical approach that is pursued. The agreement between all cases, even those held in reserve, is particularly satisfying. However, additional experiments are necessary to both confirm the correlations developed here and to extend their usefulness to a wider variety of engine

While the data provided by the PIV experiments proves invaluable in the verification of theoretical models, the technique is not without its challenges. In this section, the challenges

In most cases, the PIV technique constitutes a powerful non-intrusive method for measuring velocities. However, as demonstrated in the previous sections, the vortical flow field can

and limitations of PIV applications to confined vortex flows are discussed.

*<sup>σ</sup><sup>e</sup>* <sup>=</sup> <sup>1</sup>

Δ*Et* =

<sup>√</sup>*<sup>n</sup>* <sup>−</sup> <sup>1</sup>

*n* ∑ *i*=1

Table 4. Statistical parameters for the regression of the laminar and constant shear stress

models.

from

and

slight deviations near the core region.

an equally minute reduction in *rcc*.

configurations and injection conditions.

**5. Challenges and limitations**

**4.4 Statistical verification**

*Vt* 47 150 36 540 30 160 *rcc σ<sup>e</sup>* Δ*Et*% *rcc σ<sup>e</sup>* Δ*Et*% *rcc σ<sup>e</sup>* Δ*Et*%

[*u*ˆ*<sup>θ</sup>* (*ri*) − *u<sup>θ</sup>* (*ri*)]

2 / *n* ∑ *i*=1 *u*ˆ 2 <sup>2</sup> (54)

*<sup>θ</sup>* (*ri*) (55)

This chapter focuses on the use of PIV measurements in conjunction with two analytical models to describe the flow field in a bidirectional vortex chamber. Although the primary motivation for this work is to better understand the gas dynamics within a vortex-fired liquid rocket engine known as the Vortex Combustion Cold-Wall Chamber, the principal attributes associated with the resulting cyclonic motion have industrial, geophysical, meteorological, and astrophysical applications that fall beyond the propulsive dimension.

Using a cold-flow environment as a basis for this work, the findings indicate that both PIV and theoretical frameworks converge in predicting the presence of a forced vortex core region followed by a free, irrotational vortex tail that are characteristic of not only bidirectional vortices, but of unidirectional swirl-dominated flows as well. Moreover, despite the underlying restriction to non-reactive conditions, the essential features captured here seem to be somewhat representative of those reported in combustion chambers in which chemical reactions are prominent (Fang et al., 2004). It appears that the addition of high intensity swirl leads to a robust cyclonic motion that tends to retain its fundamental structure even in the presence of chemical kinetics. Nonetheless, more effort in modeling reactive and multiphase flows in cyclonic chambers will be critically important to further illuminate the particular

Using Particle Image Velocimetry 25

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry 173

to predict the effective vortex Reynolds number, *V* = *Vt*/*<sup>t</sup>*, which will reduce the overshoot in the swirl velocity of the laminar core flow model. It is gratifying that the data generated from the PIV technique leads to a consistent agreement with the theoretical predictions despite

Clearly, Particle Image Velocimetry seems to offer a vital resource in the quantification of confined vortex flows. Though not perfect, the technique captures the structure of the swirl velocity in the bidirectional vortex engine quite satisfactorily. By knowing the limitations of the method, specifically the likelihood of increased drag on particles near the peak flow velocity, it may be possible to refine the analytical models to the extent of better predicting their peak swirl velocities measured experimentally. Another aspect that must be brought into perspective is that, unlike the case of an unbounded vortex, the axial and radial velocities associated with confined vortices can have appreciable contributions in some regions of the flow domain. Then given that most studies to date emphasize the swirl velocity to the exclusion of the remaining components, more research is warranted to fully characterize the three-dimensional nature of the bidirectional vortex. This in turn will require a concerted effort between theoretical techniques and more advanced stereoscopic PIV or LDV

Because stereoscopic PIV relies on a rigorous procedure to capture the multidimensionality of complex flows, the addition of a second camera will be required to permit investigators to deduce particle velocities normal to the laser sheet. At the outset, all three components of velocity can be acquired at a given experimental section. With proper equipment, the light sheet can traverse the entire chamber to the extent of reproducing a comprehensive map of the vortex structure. In cases where the added expense of a stereoscopic apparatus proves impractical, a traditional PIV setup can still be employed to compile the necessary results,

An LDV technique may also be used in reconstructing the three-dimensional structure of the velocity field. In fact, Sousa (2008) relies on LDV measurements in his confined vortex study. However, a concern is raised in his investigation regarding the gains in spatial resolution that are offset by a substantially prolonged measurement time. In practice, each of these laser-based methods serves a different overall purpose. Whereas the PIV technique offers superb capabilities in capturing large scale velocity structures, LDV yields localized point measurements of fluid velocities. It can thus be seen that PIV may be the preferred method

In closing, it may be helpful to re-emphasize the tight balance between theory and experiment that stands behind this work. All too often, an investigation is steered towards one aspect of research to the detriment or neglect of the other. Without the theory to guide the proper design of experiments, more trials would have been necessitated to fully characterize the swirl velocity. Conversely, without the experimental values, the theoretical model could not have been refined and validated, nor could the turbulent eddy viscosity correlation been obtained. Pushing the borders of scientific inquiry certainly requires both theory and experiment to be

This material is based on work supported partly by the National Science Foundation, and partly by the University of Tennessee Space Institute and the H. H. Arnold Chair of Excellence

for visualizing large scale flow patterns such as those arising in cyclonic chambers.

the decreased fidelity in the core region.

procedures.

albeit with multiple trials.

pursued diligently and in unison.

**7. Acknowledgments**

mechanisms and parameters that control the performance of the VCCWC engine over a wide range of physical properties and geometric scales.

At this stage, the need to advance both computational and experimental capabilities in the modeling of reactive mixtures in swirl-dominated flows in general and cyclonic chambers in particular cannot be overrated. In what concerns the latter, it would be advantageous if a standard set of universal group parameters can be defined against which data acquired experimentally or numerically can be properly correlated. Thus guided by rigorous mathematical formulations that are derived from first principles, a basic set of dimensionless groupings such as the vortex Reynolds number, swirl parameter, inflow parameter, and chamber aspect ratio can be used consistently by various researchers while undertaking wide-range parametric studies. Presently, several different expressions of the swirl and vortex Reynolds numbers exist in the literature and this may be attributed to the dissimilar approaches used in defining them, be it rationalization, order-of-magnitude scaling, or conjecture. It would be strongly recommended that these are replaced by their corresponding forms that emerge naturally in analytical solutions that are obtained directly from first principles. The use of standard forms of these parameters will be essential before investing in new equipment and undertaking full-range laboratory or numerical experiments.

Novel investigations will be required to open up new lines of research inquiry and address several fundamental questions that remain unanswered. For example, it is still unclear what conditions will lead to the formation of a coherent cyclone, to vortex breakdown in a cyclonic chamber, or to the precession of the core vortex. It is also uncertain what techniques will be best suited to capture such behavior. The same may be said of acoustic and/or combustion instability of vortex-fired engines. Preliminary studies using the biglobal stability approach by Batterson & Majdalani (2011a;b) show that stability is promoted with successive increases in swirl. Nonetheless, their study is limited to a narrow range of chamber aspect ratios and the hydrodynamic instability effects only represent one of the factors that must be accounted for in studying engine stability. The sidewall and headwall boundary layers, which were omitted in this chapter, pose another challenge that future examinations are hoped to overcome. Resolving the viscous stresses along the chamber wall will be essential not only to improve the velocity field description, but also to make any sort of formulation of a Nusselt number possible. This effort will have to be carried out using a three-pronged approach that leverages analytical, computational, and experimental techniques.

In hindsight, the availability of a laminar core flow solution to describe the bidirectional vortex has proven helpful in the design of a successful PIV experiment. The converse is also true given that the production of experimental data has been quintessential in refining the analytical approximations. The need to investigate the existence of more accurate, multi-dimensional, or higher-order helical solutions is evident. So far the analytical framework has provided several dimensionless parameters, including the vortex Reynolds number which is critical in controlling the flow behavior and providing guidance to minimize the number of experimental trials. The availability of a core shear stress model has also led to an iterative least-squares method that is specifically developed to enable comparisons between piecewise functions and experiments. To mitigate the bias in the unavoidable scatter in PIV data, the measurements have been carefully averaged and distilled in the process of calculating the turbulent eddy viscosity, *μt*, and the corresponding viscosity ratio, *<sup>t</sup>* = *μt*/*μ* 152, for the simulated vortex chamber. This effort epitomizes theory and experiment working hand-in-hand. In the absence of the PIV data, it would have been virtually impossible 24 Will-be-set-by-IN-TECH

mechanisms and parameters that control the performance of the VCCWC engine over a wide

At this stage, the need to advance both computational and experimental capabilities in the modeling of reactive mixtures in swirl-dominated flows in general and cyclonic chambers in particular cannot be overrated. In what concerns the latter, it would be advantageous if a standard set of universal group parameters can be defined against which data acquired experimentally or numerically can be properly correlated. Thus guided by rigorous mathematical formulations that are derived from first principles, a basic set of dimensionless groupings such as the vortex Reynolds number, swirl parameter, inflow parameter, and chamber aspect ratio can be used consistently by various researchers while undertaking wide-range parametric studies. Presently, several different expressions of the swirl and vortex Reynolds numbers exist in the literature and this may be attributed to the dissimilar approaches used in defining them, be it rationalization, order-of-magnitude scaling, or conjecture. It would be strongly recommended that these are replaced by their corresponding forms that emerge naturally in analytical solutions that are obtained directly from first principles. The use of standard forms of these parameters will be essential before investing in

new equipment and undertaking full-range laboratory or numerical experiments.

Novel investigations will be required to open up new lines of research inquiry and address several fundamental questions that remain unanswered. For example, it is still unclear what conditions will lead to the formation of a coherent cyclone, to vortex breakdown in a cyclonic chamber, or to the precession of the core vortex. It is also uncertain what techniques will be best suited to capture such behavior. The same may be said of acoustic and/or combustion instability of vortex-fired engines. Preliminary studies using the biglobal stability approach by Batterson & Majdalani (2011a;b) show that stability is promoted with successive increases in swirl. Nonetheless, their study is limited to a narrow range of chamber aspect ratios and the hydrodynamic instability effects only represent one of the factors that must be accounted for in studying engine stability. The sidewall and headwall boundary layers, which were omitted in this chapter, pose another challenge that future examinations are hoped to overcome. Resolving the viscous stresses along the chamber wall will be essential not only to improve the velocity field description, but also to make any sort of formulation of a Nusselt number possible. This effort will have to be carried out using a three-pronged approach that leverages

In hindsight, the availability of a laminar core flow solution to describe the bidirectional vortex has proven helpful in the design of a successful PIV experiment. The converse is also true given that the production of experimental data has been quintessential in refining the analytical approximations. The need to investigate the existence of more accurate, multi-dimensional, or higher-order helical solutions is evident. So far the analytical framework has provided several dimensionless parameters, including the vortex Reynolds number which is critical in controlling the flow behavior and providing guidance to minimize the number of experimental trials. The availability of a core shear stress model has also led to an iterative least-squares method that is specifically developed to enable comparisons between piecewise functions and experiments. To mitigate the bias in the unavoidable scatter in PIV data, the measurements have been carefully averaged and distilled in the process of calculating the turbulent eddy viscosity, *μt*, and the corresponding viscosity ratio, *<sup>t</sup>* = *μt*/*μ* 152, for the simulated vortex chamber. This effort epitomizes theory and experiment working hand-in-hand. In the absence of the PIV data, it would have been virtually impossible

range of physical properties and geometric scales.

analytical, computational, and experimental techniques.

to predict the effective vortex Reynolds number, *V* = *Vt*/*<sup>t</sup>*, which will reduce the overshoot in the swirl velocity of the laminar core flow model. It is gratifying that the data generated from the PIV technique leads to a consistent agreement with the theoretical predictions despite the decreased fidelity in the core region.

Clearly, Particle Image Velocimetry seems to offer a vital resource in the quantification of confined vortex flows. Though not perfect, the technique captures the structure of the swirl velocity in the bidirectional vortex engine quite satisfactorily. By knowing the limitations of the method, specifically the likelihood of increased drag on particles near the peak flow velocity, it may be possible to refine the analytical models to the extent of better predicting their peak swirl velocities measured experimentally. Another aspect that must be brought into perspective is that, unlike the case of an unbounded vortex, the axial and radial velocities associated with confined vortices can have appreciable contributions in some regions of the flow domain. Then given that most studies to date emphasize the swirl velocity to the exclusion of the remaining components, more research is warranted to fully characterize the three-dimensional nature of the bidirectional vortex. This in turn will require a concerted effort between theoretical techniques and more advanced stereoscopic PIV or LDV procedures.

Because stereoscopic PIV relies on a rigorous procedure to capture the multidimensionality of complex flows, the addition of a second camera will be required to permit investigators to deduce particle velocities normal to the laser sheet. At the outset, all three components of velocity can be acquired at a given experimental section. With proper equipment, the light sheet can traverse the entire chamber to the extent of reproducing a comprehensive map of the vortex structure. In cases where the added expense of a stereoscopic apparatus proves impractical, a traditional PIV setup can still be employed to compile the necessary results, albeit with multiple trials.

An LDV technique may also be used in reconstructing the three-dimensional structure of the velocity field. In fact, Sousa (2008) relies on LDV measurements in his confined vortex study. However, a concern is raised in his investigation regarding the gains in spatial resolution that are offset by a substantially prolonged measurement time. In practice, each of these laser-based methods serves a different overall purpose. Whereas the PIV technique offers superb capabilities in capturing large scale velocity structures, LDV yields localized point measurements of fluid velocities. It can thus be seen that PIV may be the preferred method for visualizing large scale flow patterns such as those arising in cyclonic chambers.

In closing, it may be helpful to re-emphasize the tight balance between theory and experiment that stands behind this work. All too often, an investigation is steered towards one aspect of research to the detriment or neglect of the other. Without the theory to guide the proper design of experiments, more trials would have been necessitated to fully characterize the swirl velocity. Conversely, without the experimental values, the theoretical model could not have been refined and validated, nor could the turbulent eddy viscosity correlation been obtained. Pushing the borders of scientific inquiry certainly requires both theory and experiment to be pursued diligently and in unison.

### **7. Acknowledgments**

This material is based on work supported partly by the National Science Foundation, and partly by the University of Tennessee Space Institute and the H. H. Arnold Chair of Excellence

Using Particle Image Velocimetry 27

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**7** 

*France* 

**Application of the Particle Image Velocimetry** 

*1LOMC, UMR 6294, CNRS-Université du Havre 53, rue Prony, Le Havre Cedex,* 

For longtime, the investigation of flow regimes has been achieved using fluorescent particles or anisotropic reflective particles. Fluorescent particles are suitable for open flows (Van Dyke, 1982) such as flows in channels (Peerhossaini *et al.*, 1988) or flows behind a cylinder (to visualize Benard-von Karman street) (Provansal *et al.*, 1986; Mutabazi *et al.*, 2006). For closed flows such as flows in a rectangular cavity or in an annular cylindrical rotating cavity, fluorescent particles rapidly color the entire flow and no flow structure can be caught. Anisotropic reflective particles (aluminium, iriodin or Kalliroscope flakes) are more convenient for detection of the flow structures (Taylor, 1923; Andereck *et al*., 1986; Coles, 1965; Matisse *et al*., 1984; Dominguez-Lerma *et al*. , 1985; Thoroddsen *et al*. 1999). A laser light is used to illuminate the flow cross-sections and to detect the flow structure in the axial, radial and azimuthal directions. The motion of the seeded particles in a fluid gives a qualitative picture of flows which can be used to develop appropriate theoretical models. The development of chaotic models of fluid flows (Rayleigh-Bénard convection, Couette-Taylor flow or plane Couette flow) has benefited from observations using visualizations techniques (Bergé *et al*., 1994). Using appropriate signal processing techniques such as spacetime diagrams and complex demodulation, it is possible to obtain spatio-temporal evolution of the flows (Bot *et al*., 2000). In order to obtain quantitative data on velocity fields, different velocimetry techniques have been developed such as Laser Doppler Velocimetry (LDV, Durst *et al.*, 1976, Jensen 2004), Ultrasound Doppler Velocimetry (UDV, Takeda *et al.*, 1994) and Particle Image Velocimetry (PIV, Jensen, 2004). Nowadays, there is a lot of literature on velocimetry techniques the development of which is beyond the scope of this chapter, some of them and their applications are described in this volume. Each velocimetry technique has its advantages and own limitations depending on the flow system under consideration. For example, in the case of the Couette-Taylor flow, the LDV (Ahlers *et al.*, 1986) gives time averaged velocity in a point, the UDV measures a velocity profile along a chosen line in the flow and the PIV gives a velocity field in a limited flow cross section. The Couette-Taylor system is composed of a flow in the gap between two coaxial differential rotating cylinders. This system represents a good hydrodynamic prototype for the study of the transition to turbulence in closed systems. The experimental results obtained from this system have led

**1. Introduction** 

**to the Couette-Taylor Flow** 

Olivier Crumeyrolle1 and Alexander Ezersky2

*2M2C, UMR 6143, CNRS-University of Caen-Basse Normandie,* 

Innocent Mutabazi1, Nizar Abcha2,

Vyas, A. & Majdalani, J. (2006). Exact solution of the bidirectional vortex, *AIAA Journal* 44(10): 2208–2216.

URL: *http://dx.doi.org/10.2514/1.14872*


## **Application of the Particle Image Velocimetry to the Couette-Taylor Flow**

Innocent Mutabazi1, Nizar Abcha2, Olivier Crumeyrolle1 and Alexander Ezersky2 *1LOMC, UMR 6294, CNRS-Université du Havre 53, rue Prony, Le Havre Cedex, 2M2C, UMR 6143, CNRS-University of Caen-Basse Normandie, France* 

### **1. Introduction**

28 Will-be-set-by-IN-TECH

176 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Vyas, A. & Majdalani, J. (2006). Exact solution of the bidirectional vortex, *AIAA Journal*

Wendt, B. J. (2001). Initial circulation and peak vorticity behavior of vortices shed from airfoil

Wu, J. Z. (1986). Conical turbulent swirling vortex with variable eddy viscosity, *Proceedings of*

Yan, L., Vatistas, G. H. & Lin, S. (2000). Experimental studies on turbulence kinetic energy in

Zhang, Z. & Hugo, R. J. (2006). Stereo particle image velocimetry applied to a vortex pipe

vortex generators, *Technical Report NASA/CR–2001-211144*, NASA.

*the Royal Society of London, Series A* 403(1825): 235–268.

confined vortex flows, *Journal of Thermal Science* 9(1): 10–22.

URL: *http://dx.doi.org/10.1098/rspa.1986.0011*

flow, *Experiments in Fluids* 40: 333–346.

URL: *http://dx.doi.org/10.1007/s11630-000-0040-z*

URL: *http://dx.doi.org/10.1007/s00348-005-0071-z*

44(10): 2208–2216.

URL: *http://dx.doi.org/10.2514/1.14872*

For longtime, the investigation of flow regimes has been achieved using fluorescent particles or anisotropic reflective particles. Fluorescent particles are suitable for open flows (Van Dyke, 1982) such as flows in channels (Peerhossaini *et al.*, 1988) or flows behind a cylinder (to visualize Benard-von Karman street) (Provansal *et al.*, 1986; Mutabazi *et al.*, 2006). For closed flows such as flows in a rectangular cavity or in an annular cylindrical rotating cavity, fluorescent particles rapidly color the entire flow and no flow structure can be caught. Anisotropic reflective particles (aluminium, iriodin or Kalliroscope flakes) are more convenient for detection of the flow structures (Taylor, 1923; Andereck *et al*., 1986; Coles, 1965; Matisse *et al*., 1984; Dominguez-Lerma *et al*. , 1985; Thoroddsen *et al*. 1999). A laser light is used to illuminate the flow cross-sections and to detect the flow structure in the axial, radial and azimuthal directions. The motion of the seeded particles in a fluid gives a qualitative picture of flows which can be used to develop appropriate theoretical models. The development of chaotic models of fluid flows (Rayleigh-Bénard convection, Couette-Taylor flow or plane Couette flow) has benefited from observations using visualizations techniques (Bergé *et al*., 1994). Using appropriate signal processing techniques such as spacetime diagrams and complex demodulation, it is possible to obtain spatio-temporal evolution of the flows (Bot *et al*., 2000). In order to obtain quantitative data on velocity fields, different velocimetry techniques have been developed such as Laser Doppler Velocimetry (LDV, Durst *et al.*, 1976, Jensen 2004), Ultrasound Doppler Velocimetry (UDV, Takeda *et al.*, 1994) and Particle Image Velocimetry (PIV, Jensen, 2004). Nowadays, there is a lot of literature on velocimetry techniques the development of which is beyond the scope of this chapter, some of them and their applications are described in this volume. Each velocimetry technique has its advantages and own limitations depending on the flow system under consideration. For example, in the case of the Couette-Taylor flow, the LDV (Ahlers *et al.*, 1986) gives time averaged velocity in a point, the UDV measures a velocity profile along a chosen line in the flow and the PIV gives a velocity field in a limited flow cross section. The Couette-Taylor system is composed of a flow in the gap between two coaxial differential rotating cylinders. This system represents a good hydrodynamic prototype for the study of the transition to turbulence in closed systems. The experimental results obtained from this system have led

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 179

The experimental system consists of two vertical coaxial cylinders, immersed in a large square Plexiglas box filled with water in order to maintain a controlled temperature (Fig. 1). The square box allows to minimize distortion effects of refraction due to curvature of the outer cylinder during optical measurements. The inner cylinder made of aluminium has a radius *a* = 4 cm, the outer cylinder made of glass has a radius *b* = 5 cm, the gap between the cylinders is *d = b-a* = 1 cm and the working height is *L* = 45.9 cm. Therefore the radius ratio

avoid the end effects; the flow system is considered as an extended system. The gap is filled

Fig. 1. Experimental apparatus: scheme of visualization and data acquisition system

The cylinders are rotated by two servomotors at controlled angular rotation frequencies

Taylor number is often preferred: *Ta* = *Rei* (*d/a*)1/2. Three cameras have been implemented on

**<sup>o</sup>**. The control parameters are the Reynolds numbers defined for each cylinder : *Rei* =

*obd*/ for the inner and outer cylinder respectively. When *Re0* = 0, the

= *L*/*d* = 45.9. Such an aspect ratio is large enough to

= 9.8.10-3cm2/s at the temperature *T* = 21.2°C. Its size has

been chosen in order to obtain a good resolution in the radial direction.

**i**

OC Outer cylinder IC Inner cylinder

Aspect ratio

Radius ratio

Angular velocity, rad/s

Kinematic viscosity, m²/s

Fluid density, g/cm3

Pattern wavelength

**2. Experimental apparatus** 

= *a*/*b* = 0.8 and the aspect ratio is

with a deionized water for which

Dimensionless radial coordinate

Dimensionless axial coordinate

and 

*iad*/ and *Reo* =

to the development of powerful theoretical models for the transition to chaos (Chossat *et al* . (1994)). Beside the theoretical interpretation of patterns observed in the Couette-Taylor system, many theoretical attempts have been made to connect flow quantitative properties and visualized structures by anisotropic particles (Matisse *et al*. , 1984; Savas (1985), Gauthier *et al*., (1998)). The application of PIV in the Couette-Taylor system with a fixed outer cylinder was first performed by Wereley & Lueptow (Wereley *et al*., 1994, 1998) followed later by few authors. The question of correlation between velocimetry data and qualitative structure given by anisotropic reflective particles in the Couette-Taylor flow was addressed only recently (Gauthier *et al.*, 1998; Abcha *et al.*, 2008). However, many questions connected with the interpretation of results obtained by different techniques have not been answered. A special attention is paid to some of these unresolved problems.

This chapter illustrates how the PIV technique can be applied to the Couette -Taylor system. Two special cases are described: 1) flow patterns obtained when the outer cylinder is fixed while the inner is rotating; 2) flow patterns achieved when both cylinders are in contrarotation. A detailed comparison between PIV and visualisation by anisotropic reflective particles will be provided for illustration of the complementarity between these two techniques. The chapter is organized as follows. The experimental setup and procedure are presented in the next section. Section 3 is devoted to the flow visualization by Kalliroscope particles and the space-time diagram technique. In section 4, the description of PIV and its adaptation to the Couette-Taylor flow are described. Section 5 contains results for flow regimes when the outer cylinder is fixed (Taylor Vortex Flow (TVF) and Wavy Vortex Flow (WVF)). Section 6 gives results for spiral vortex flow when both cylinders are sufficiently counter-rotating. Section 7 summarizes the content of the chapter.

#### **List of symbols**


to the development of powerful theoretical models for the transition to chaos (Chossat *et al* . (1994)). Beside the theoretical interpretation of patterns observed in the Couette-Taylor system, many theoretical attempts have been made to connect flow quantitative properties and visualized structures by anisotropic particles (Matisse *et al*. , 1984; Savas (1985), Gauthier *et al*., (1998)). The application of PIV in the Couette-Taylor system with a fixed outer cylinder was first performed by Wereley & Lueptow (Wereley *et al*., 1994, 1998) followed later by few authors. The question of correlation between velocimetry data and qualitative structure given by anisotropic reflective particles in the Couette-Taylor flow was addressed only recently (Gauthier *et al.*, 1998; Abcha *et al.*, 2008). However, many questions connected with the interpretation of results obtained by different techniques have not been

This chapter illustrates how the PIV technique can be applied to the Couette -Taylor system. Two special cases are described: 1) flow patterns obtained when the outer cylinder is fixed while the inner is rotating; 2) flow patterns achieved when both cylinders are in contrarotation. A detailed comparison between PIV and visualisation by anisotropic reflective particles will be provided for illustration of the complementarity between these two techniques. The chapter is organized as follows. The experimental setup and procedure are presented in the next section. Section 3 is devoted to the flow visualization by Kalliroscope particles and the space-time diagram technique. In section 4, the description of PIV and its adaptation to the Couette-Taylor flow are described. Section 5 contains results for flow regimes when the outer cylinder is fixed (Taylor Vortex Flow (TVF) and Wavy Vortex Flow (WVF)). Section 6 gives results for spiral vortex flow when both cylinders are sufficiently

answered. A special attention is paid to some of these unresolved problems.

counter-rotating. Section 7 summarizes the content of the chapter.

**List of symbols** 

*d* Size gap, cm

*a* Inner cylinder radius, cm *b* Outer cylinder radius, cm

*g* Gravity acceleration *L* Cylinder length, cm *Re* Reynolds number *Ta* Taylor number

*n* Optical refraction index *vs* Sedimentation velocity

*t*res Residence time, s *Abs* Absolute value CCF Circular Couette Flow TVF Taylor Vortex Flow WVF Wavy Vortex Flow

SVF Spiral Vortex Flow

*Vr* Radial velocity component, m/s *u* Dimensionless radial velocity *Vz* Axial velocity component, m/s *w* Dimensionless axial velocity

MWVF Modulated Wavy Vortex Flow TTVF Turbulent Taylor Vortex Flow


### **2. Experimental apparatus**

The experimental system consists of two vertical coaxial cylinders, immersed in a large square Plexiglas box filled with water in order to maintain a controlled temperature (Fig. 1). The square box allows to minimize distortion effects of refraction due to curvature of the outer cylinder during optical measurements. The inner cylinder made of aluminium has a radius *a* = 4 cm, the outer cylinder made of glass has a radius *b* = 5 cm, the gap between the cylinders is *d = b-a* = 1 cm and the working height is *L* = 45.9 cm. Therefore the radius ratio = *a*/*b* = 0.8 and the aspect ratio is = *L*/*d* = 45.9. Such an aspect ratio is large enough to avoid the end effects; the flow system is considered as an extended system. The gap is filled with a deionized water for which = 9.8.10-3cm2/s at the temperature *T* = 21.2°C. Its size has been chosen in order to obtain a good resolution in the radial direction.

Fig. 1. Experimental apparatus: scheme of visualization and data acquisition system

The cylinders are rotated by two servomotors at controlled angular rotation frequencies **i** and **<sup>o</sup>**. The control parameters are the Reynolds numbers defined for each cylinder : *Rei* = *iad*/ and *Reo* = *obd*/ for the inner and outer cylinder respectively. When *Re0* = 0, the Taylor number is often preferred: *Ta* = *Rei* (*d/a*)1/2. Three cameras have been implemented on

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 181

Fig. 2. Pictures of flow regimes in the Couette-Taylor system : a) TVF, b) WVF, c) SVF with a

Fig. 3. Cross-section of : a) CCF, b) TVF, c) WVF, d) MWVF, e) TTVF, f) SVF. OC and IC

illustrated (Fig. 4) for wavy vortex flow and in Figure 5 for spiral vortex flow.

cross-section (r = *a* + *d*/2), parallel to the cylindrical axis over a length of 27.8 cm in the central part of the flow system. The intensity was sampled over a linear range of 256 values, displayed in gray levels at regular time intervals in order to produce space-time diagrams *I*(*z*,*t*) of the pattern which exhibits the temporal and spatial evolution of vortices (Fig. 4a). The radial variation of intensity *I*(*r*) was recorded using a 2-d IEEE1394 camera, and then sampled at regular time intervals to obtain the space-time diagram *I*(*r*,*t*). Examples are

A cross-section of spiral vortex flow is shown in Fig. 5-a; its space-time diagram in the axial and radial directions are shown respectively in Fig. 5-b and Fig. 6. The right and left travelling spirals merge into a single point called "sink" at *z* = *z*0 (Fig. 5-a). Using a 2-D Fast Fourier Transform (FFT), it is possible to obtain the axial wavenumbers and the frequencies

stand for outer and inner cylinder respectively.

sink.

the experimental table: a linear CCD camera of 1024 pixels that records the light reflected by anisotropic reflective particles along a line parallel to cylinder axis. The second camera is a 2-d IEEE1394 camera (A641f, Basler) that is used to record the flow motion in the (*r*,*z*) plane; this record allows a better investigation of the flow in the radial direction. The third camera is a CCD camera (Kodak) with 1034x779 pixels for PIV data recording. All cameras were connected to a computer for data recording and processing.

### **3. Visualization by Kalliroscope particles and space-time diagrams**

Particles added to the flow must have controlled characteristics such as size, distribution, and concentration. These particles must be small enough to be good flow tracers and large enough to scatter sufficient light for imaging. In the Couette-Taylor flow, the commonly used particles are Kalliroscope flakes of typical size of 30 µm x 6 µm x 0.07 µm (Matisse *et al*. 1984) with a relatively large reflective optical index *n* = 1.85 and a density of *'* = 1.62 g/cm3. A concentration of 1% to 2% reflective particles is added to water to realize a Kalliroscope AQ1000 suspension, 2% per volume of which was added to the working solution. The sedimentation of these particles remains negligible in horizontal or vertical configurations if the experiment lasts less than 10 hours (Matisse *et al*., 1984) because their sedimentation velocity is *vs* = 2 *a*2*g*('-)/(9 = 2.8 10-5cm/s. The time scales related to the particle motions (transient, rotation and diffusion) were discussed in detail by Gauthier *et al.* (1998). These particles do not modify significantly the flow viscosity and no non-Newtonian effect was detected as far as small concentrations (*c* < 5%) are used (Dominguez-Lerma *et al*., 1985). The choice of the concentration of 2% was done to ensure the best contrast in the flow. The values of the control parameters (*Reo*, *Rei*) were determined within a precision of 2%.

Increasing values of the control parameters leads to the occurrence of different patterns in the Couette-Taylor flow depending of whether both cylinders rotate or only the inner cylinder is rotating (Fig. 2, 3). A whole state diagram of flow regimes in the Couette-Taylor system has been established by Andereck (Andereck *et al*., 1986) for a configuration with radius ratio = 0.883 and aspect ratios ranging from 20 to 48. When the outer cylinder is fixed and the inner Reynolds number *Rei* is increased, the transition sequence is the following : Circular Couette Flow (CCF) bifurcates to Taylor Vortex Flow (TVF) which is formed of axisymmetric stationary vortices, then to Wavy Vortex Flow (WVF) oscillating in the azimuthal and axial directions with a frequency *f* and an azimuthal wavenumber *m*; the later bifurcates to Modulated Wavy Vortex Flow (MWVF) characterized by two incommensurate frequencies. The ultimate state is the Turbulent Taylor Vortex Flow (TTVF) iin which large scale vortices of the size of the gap and small vortices of different scales coexist. In case of counter-rotating cylinders, the bifurcation of the circular Couette flow leads to spiral vortex flow (SVF) composed of helical vortices travelling in axial and azimuthal directions, followed by interpenetrating spirals then by wavy spirals and modulated waves before transition to turbulence. Interpenetrating spirals, wavy spirals and modulated waves are characterized by incommensurate frequencies. Using a He-Ne Laser sheet (whose wavelength is 632 nm, one millimetre wide beam, spread by a cylindrical lens), it was possible to visualize the cross section of the flow in the r-z plane. Fig. 3 gives the cross section of regimes observed in the Couette-Taylor system for different values of the control parameters. Linear CCD camera of 1024 pixels was used to record a reflected light intensity *I(z)*. Records were performed at regular time intervals along a line in the centre of the flow

the experimental table: a linear CCD camera of 1024 pixels that records the light reflected by anisotropic reflective particles along a line parallel to cylinder axis. The second camera is a 2-d IEEE1394 camera (A641f, Basler) that is used to record the flow motion in the (*r*,*z*) plane; this record allows a better investigation of the flow in the radial direction. The third camera is a CCD camera (Kodak) with 1034x779 pixels for PIV data recording. All cameras were

Particles added to the flow must have controlled characteristics such as size, distribution, and concentration. These particles must be small enough to be good flow tracers and large enough to scatter sufficient light for imaging. In the Couette-Taylor flow, the commonly used particles are Kalliroscope flakes of typical size of 30 µm x 6 µm x 0.07 µm (Matisse *et al*.

A concentration of 1% to 2% reflective particles is added to water to realize a Kalliroscope AQ1000 suspension, 2% per volume of which was added to the working solution. The sedimentation of these particles remains negligible in horizontal or vertical configurations if the experiment lasts less than 10 hours (Matisse *et al*., 1984) because their sedimentation

motions (transient, rotation and diffusion) were discussed in detail by Gauthier *et al.* (1998). These particles do not modify significantly the flow viscosity and no non-Newtonian effect was detected as far as small concentrations (*c* < 5%) are used (Dominguez-Lerma *et al*., 1985). The choice of the concentration of 2% was done to ensure the best contrast in the flow. The values of the control parameters (*Reo*, *Rei*) were determined within a precision of 2%.

Increasing values of the control parameters leads to the occurrence of different patterns in the Couette-Taylor flow depending of whether both cylinders rotate or only the inner cylinder is rotating (Fig. 2, 3). A whole state diagram of flow regimes in the Couette-Taylor system has been established by Andereck (Andereck *et al*., 1986) for a configuration with

fixed and the inner Reynolds number *Rei* is increased, the transition sequence is the following : Circular Couette Flow (CCF) bifurcates to Taylor Vortex Flow (TVF) which is formed of axisymmetric stationary vortices, then to Wavy Vortex Flow (WVF) oscillating in the azimuthal and axial directions with a frequency *f* and an azimuthal wavenumber *m*; the later bifurcates to Modulated Wavy Vortex Flow (MWVF) characterized by two incommensurate frequencies. The ultimate state is the Turbulent Taylor Vortex Flow (TTVF) iin which large scale vortices of the size of the gap and small vortices of different scales coexist. In case of counter-rotating cylinders, the bifurcation of the circular Couette flow leads to spiral vortex flow (SVF) composed of helical vortices travelling in axial and azimuthal directions, followed by interpenetrating spirals then by wavy spirals and modulated waves before transition to turbulence. Interpenetrating spirals, wavy spirals and modulated waves are characterized by incommensurate frequencies. Using a He-Ne Laser sheet (whose wavelength is 632 nm, one millimetre wide beam, spread by a cylindrical lens), it was possible to visualize the cross section of the flow in the r-z plane. Fig. 3 gives the cross section of regimes observed in the Couette-Taylor system for different values of the control parameters. Linear CCD camera of 1024 pixels was used to record a reflected light intensity *I(z)*. Records were performed at regular time intervals along a line in the centre of the flow

= 2.8 10-5cm/s. The time scales related to the particle

ranging from 20 to 48. When the outer cylinder is

*'* = 1.62 g/cm3.

**3. Visualization by Kalliroscope particles and space-time diagrams** 

1984) with a relatively large reflective optical index *n* = 1.85 and a density of

connected to a computer for data recording and processing.

velocity is *vs* = 2 *a*2*g*(

radius ratio

'-)/(9

= 0.883 and aspect ratios

Fig. 2. Pictures of flow regimes in the Couette-Taylor system : a) TVF, b) WVF, c) SVF with a sink.

Fig. 3. Cross-section of : a) CCF, b) TVF, c) WVF, d) MWVF, e) TTVF, f) SVF. OC and IC stand for outer and inner cylinder respectively.

cross-section (r = *a* + *d*/2), parallel to the cylindrical axis over a length of 27.8 cm in the central part of the flow system. The intensity was sampled over a linear range of 256 values, displayed in gray levels at regular time intervals in order to produce space-time diagrams *I*(*z*,*t*) of the pattern which exhibits the temporal and spatial evolution of vortices (Fig. 4a). The radial variation of intensity *I*(*r*) was recorded using a 2-d IEEE1394 camera, and then sampled at regular time intervals to obtain the space-time diagram *I*(*r*,*t*). Examples are illustrated (Fig. 4) for wavy vortex flow and in Figure 5 for spiral vortex flow.

A cross-section of spiral vortex flow is shown in Fig. 5-a; its space-time diagram in the axial and radial directions are shown respectively in Fig. 5-b and Fig. 6. The right and left travelling spirals merge into a single point called "sink" at *z* = *z*0 (Fig. 5-a). Using a 2-D Fast Fourier Transform (FFT), it is possible to obtain the axial wavenumbers and the frequencies

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 183

Figure 5-b. Analysis of temporal and spatial spectra show that the left and right spirals have different frequencies and wavenumbers: *fR* = 0.166 Hz ≠ *fL* = 0.163 Hz and *kR* = 5.21 cm-1 *≠ kL* = 5.06 cm-1. The standard deviations on measured frequencies and wavenumbers are

The azimuthal wavenumber of the spiral flow can be determined by measuring the

*<sup>L</sup>* = *2 fL* are the corresponding frequencies, *kR* and *kL* the corresponding axial

*t k z i t kz i i RR LL <sup>m</sup> Irzt Fr Azt e Bzt e <sup>e</sup>* (1)

of the spirals and extracting the value of *m* from the formula

is the wavelength and *R* = (*a*+*b*)/2 is the

= 4.85°. Therefore, the space-time diagrams of the

 

2 2 2

2 2 2

0.002 Hz and

*2 fR* ,

their group velocity,

where *0*, 

inclination angle

  arctan( /(2 )) arctan( /( )) *m R m kR*

for *Reo*= -251 and *Rei* = 202: *m* = 2 and

*t z z*

*k* = ± 0.04 cm-1 respectively.

where

mean radius. For *Reo*= -230 and *Rei* = 174 the obtained values are *m* = 2 and

*B*(*z*,*t*) satisfy the complex Ginzburg-Landau equations (Cross *et al*., 1993):

spiral pattern in Figure 4 can be represented by the following signal [Cross *et al*., 1993]:

( , , ) ( ) Re[ ( , ) (,) ]

where *A* and *B* are the amplitudes of right-handed and left-handed spirals respectively,

 

 

0 0 <sup>0</sup> 12 3 <sup>2</sup> 11 1 1 *A A <sup>A</sup> s icA i c gi c AA ic BA*

0 0 <sup>0</sup> 12 3 <sup>2</sup> 11 1 1 *<sup>B</sup> B B s icB i c gi c BB ic AB*

the coupling constant of left and right travelling spirals, *ci* are the dispersion coefficients. These coefficients can be determined either numerically (Demay *et al*. , 1984; Tagg *et al.* 1990) or experimentally (Goharzadeh *et al*., 2010); their values depend on the control parameters

(a) (b) (c)

Fig. 7. Structure function *F(r)* of the time-averaged amplitude of the pattern *I(r,t)* for

*Reo* = -230 and *Rei* = 174: (a) before the sink; (b) in the sink; (c) after the sink.

2

2

*t z <sup>z</sup>* (2a)

*<sup>0</sup>* represent the characteristic time and characteristic length of perturbations, *s*

is the criticality, *g* the Landau constant of nonlinear saturation,

wavenumbers, *m* their azimuthal wavenumber and c.c. stands for complex conjugate. The "structure function" *F(r)* characterizes the radial dependence of the spiral pattern and vanishes at the cylindrical surfaces: *F*(*r* = *a*) = *F*(*r* = *b*) = 0 (Fig. 7). The amplitudes *A*(*z*,*t*) and

*f* = ±

= 4.88° while

*<sup>R</sup>*=

(2b)

is

Fig. 4. Space-time diagrams for Wavy Vortex Flow (*Reo* = 0; *Rei* = 880) cross section of which is shown in Fig. 2 : (a) axial distribution *I(z,t)*; (b) radial distribution *I(r,t)*.

Fig. 5. Spiral pattern for *Reo* = - 230 and *Rei* = 174 just above the critical value (*Rei*c = 160, = 0.0875): (a) cross-section of flow; (b) space- time diagrams *I(z,t)* taken in the mid-gap position (x = 0.5) over the axial length *z* = 13.8 cm

Fig. 6. Space- time diagrams *I(r,t)* of the spiral pattern : (a) before the sink; (b) in the sink; (c) after the sink.

of oscillations of the patterns. The FFT can be complemented by the complex demodulation technique (Bot *et al.* 2000) in order to determine the physical properties like phase, amplitude, frequency, and wavenumber of more complex patterns (in the presence of localized defects, sinks or sources) . These techniques were applied to the spiral pattern of

Fig. 4. Space-time diagrams for Wavy Vortex Flow (*Reo* = 0; *Rei* = 880) cross section of which

Fig. 5. Spiral pattern for *Reo* = - 230 and *Rei* = 174 just above the critical value (*Rei*c = 160, = 0.0875): (a) cross-section of flow; (b) space- time diagrams *I(z,t)* taken in the mid-gap

*z* = 13.8 cm

(a) (b) (c)

Fig. 6. Space- time diagrams *I(r,t)* of the spiral pattern : (a) before the sink; (b) in the sink;

of oscillations of the patterns. The FFT can be complemented by the complex demodulation technique (Bot *et al.* 2000) in order to determine the physical properties like phase, amplitude, frequency, and wavenumber of more complex patterns (in the presence of localized defects, sinks or sources) . These techniques were applied to the spiral pattern of

is shown in Fig. 2 : (a) axial distribution *I(z,t)*; (b) radial distribution *I(r,t)*.

position (x = 0.5) over the axial length

(c) after the sink.

Figure 5-b. Analysis of temporal and spatial spectra show that the left and right spirals have different frequencies and wavenumbers: *fR* = 0.166 Hz ≠ *fL* = 0.163 Hz and *kR* = 5.21 cm-1 *≠ kL* = 5.06 cm-1. The standard deviations on measured frequencies and wavenumbers are *f* = ± 0.002 Hz and *k* = ± 0.04 cm-1 respectively.

The azimuthal wavenumber of the spiral flow can be determined by measuring the inclination angle of the spirals and extracting the value of *m* from the formula arctan( /(2 )) arctan( /( )) *m R m kR* where is the wavelength and *R* = (*a*+*b*)/2 is the mean radius. For *Reo*= -230 and *Rei* = 174 the obtained values are *m* = 2 and = 4.88° while for *Reo*= -251 and *Rei* = 202: *m* = 2 and = 4.85°. Therefore, the space-time diagrams of the spiral pattern in Figure 4 can be represented by the following signal [Cross *et al*., 1993]:

$$I(r,z,t) = F(r)\left\{ \text{Re}\left[A(z,t)e^{i\left(\alpha\_R t - k\_R z\right)} + B(z,t)e^{i\left(\alpha\_L t + k\_L z\right)}\right]e^{-im\theta} \right\} \tag{1}$$

where *A* and *B* are the amplitudes of right-handed and left-handed spirals respectively, *<sup>R</sup>*= *2 fR* , *<sup>L</sup>* = *2 fL* are the corresponding frequencies, *kR* and *kL* the corresponding axial wavenumbers, *m* their azimuthal wavenumber and c.c. stands for complex conjugate. The "structure function" *F(r)* characterizes the radial dependence of the spiral pattern and vanishes at the cylindrical surfaces: *F*(*r* = *a*) = *F*(*r* = *b*) = 0 (Fig. 7). The amplitudes *A*(*z*,*t*) and *B*(*z*,*t*) satisfy the complex Ginzburg-Landau equations (Cross *et al*., 1993):

$$\tau\_0 \left( \frac{\partial A}{\partial t} + s \frac{\partial A}{\partial z} \right) = \varepsilon \left( 1 + ic\_0 \right) A + \xi\_0^2 \left( 1 + ic\_1 \right) \frac{\partial^2 A}{\partial z^2} - g \left( 1 + ic\_2 \right) \left| A \right|^2 A - \delta \left( 1 + ic\_3 \right) \left| B \right|^2 A \tag{2a} \tag{2a}$$

$$\tau\_0 \left( \frac{\partial B}{\partial t} - s \frac{\partial B}{\partial z} \right) = \varepsilon \left( 1 + ic\_0 \right) B + \xi\_0^2 \left( 1 + ic\_1 \right) \frac{\partial^2 B}{\partial z^2} - g \left( 1 + ic\_2 \right) \left| B \right|^2 B - \delta \left( 1 + ic\_3 \right) \left| A \right|^2 B \tag{2b}$$

where *0*, *<sup>0</sup>* represent the characteristic time and characteristic length of perturbations, *s* their group velocity, is the criticality, *g* the Landau constant of nonlinear saturation, is the coupling constant of left and right travelling spirals, *ci* are the dispersion coefficients. These coefficients can be determined either numerically (Demay *et al*. , 1984; Tagg *et al.* 1990) or experimentally (Goharzadeh *et al*., 2010); their values depend on the control parameters

Fig. 7. Structure function *F(r)* of the time-averaged amplitude of the pattern *I(r,t)* for *Reo* = -230 and *Rei* = 174: (a) before the sink; (b) in the sink; (c) after the sink.

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 185

YAG Laser sources, a MasterPiv processor (from Tecflow) and a CCD camera (Kodak) with 1034x779 pixels. The time delay between two Laser pulses varies from 0.5 to 25 ms, depending on the values of Reynolds numbers *Rei* and *Reo*. The flow in the test area of the plane (*r*,*z*) is visualized with a thin light sheet that illuminates the glass particles, the positions of which can be recorded at short time intervals. To obtain velocity field, 195 pairs of images of size 1034 x 779 pixels were recorded. Each image of a pair was sampled into windows of 32x32 pixels2 with a recovering of 50%. The velocity fields were computed using the intercorrelation function, which is implemented in the software "Corelia-V2IP" (Tecflow). The PIV measurements were performed in the CCF, in the TVF and WVF regimes in order to calibrate our data acquisition system and to fit data available in the literature for these regimes (Wereley *et al*. 1994, Wereley *et al*. 1998,

In the circular Couette flow, the spherical glass particles are uniformly distributed (Fig. 9a) while in the TVF and WVF, after 10 hours, the particles have migrated towards the vortex cores where the radial velocity vanishes (Fig. 9b). The PIV allows to visualize velocity and vorticity fields in the cross section (*r*,*z*). The results of the complete process is illustrated by 2D velocity fields of Fig. 10a, b. The inflow (arrow (2)) and outflow (arrow (3)) are clearly evidenced in the case of the TVF and WVF. The measured radial and axial velocity components *Vr*(*r*), *Vz*(*r*) at a given axial position *z* or at a given radial position *r* are plotted

The radial and axial velocity components have been fitted by a polynomial function satisfying the non-slip condition at the cylindrical walls *r* = *a* and *r = b*. The velocity data

were computed in the axial and radial directions (Fig.10). The instantaneous velocity components can be superposed chronologically at regular time intervals in order to obtain space-time diagrams in both direction (*z*,*t*) and (*r*,*t*) (Fig. 11). The resulting diagrams are

(a)

(b) Fig. 9. Cross section of flow visualized with glass particles for: a) the CCF (*Ta* = 37.5), b) the

*<sup>i</sup>* as follows : *u* =*Vr/a*

= *z*/*d*. In order to plot their profiles, time-averaged velocity components

*<sup>i</sup> w= Vz*/*a*

= (*r* – *a*)/*d* and the axial

*<sup>i</sup>*. The lengths

Abcha *et al*. 2008).

in Fig. 10c-h in scaled units.

colour-coded as in Abcha *et al*. 2008.

coordinate is

WVF (*Ta* = 565).

are scaled by the inner cylinder velocity *a*

are scaled by the gap size, the radial position becomes

*Rei* and *Reo*. The patterns shown in Fig. 5 and time-averaged amplitude profiles of which are shown in Fig. 8 correspond to the case when > 1, i.e. the wave coupling is destructive. The sink corresponds to the intersection of two amplitudes solutions given by *A*(*z*) = *B*(*z*).

Fig. 8. Spatial distribution of the time-averaged amplitude of the right and left spiral, for *Reo* = -230 and *Rei* = 174 in the neighbourhood of the sink localized in *z0* = 64 mm.

The fit of experimental data with the theoretical curves (Fig.8) 0 0 *Az A z z* ( ) tanh[( ) / ] and 0 0 *Bz A z z* ( ) tanh[ ( ) / ] gives *A0* = 0.3, = 7.14 mm = 0.714 d. The value of the coefficient *0* in the equations (2) is given by *<sup>0</sup>* = mm. This value is in a good agreement with theoretical values *<sup>0</sup>*/*d* ≈ 0.2 (Tagg *et al.*, 1990]. The structure functions in Fig. 7 show that the source weakly affect the radial flow.

#### **4. Description of the PIV and its adaptation to the Couette-Taylor flow**

The technique of space-time diagrams does not provide quantitative data on the velocity or vorticity fields that are important for the estimate of energy or momentum transfer in the different regimes. Thus it is necessary to perform particle image velocimetry in order to get more quantitative data. For PIV measurements, the working fluid was seeded with spherical glass particles of diameter 8-11 m and density *'* = 1.6 g/cm3, with a concentration of about 1ppm. The diffusion time of such particles is <sup>3</sup> *D pB* 3 / 4 500 *d kT s* where *dp* is the particle diameter. The particle Reynolds number *Rep* = *Udp*/ < 0.1 or equivalently their Stokes number *St* = *(pdp*/*d*)*Rep* < 10-3 so that they are assumed to follow the flow streamlines, i.e. they are good tracers of the flow. Here *U* is the characteristic velocity of the particle. The PIV system consists of two Nd-

*Rei* and *Reo*. The patterns shown in Fig. 5 and time-averaged amplitude profiles of which are

sink corresponds to the intersection of two amplitudes solutions given by *A*(*z*) = *B*(*z*).

Fig. 8. Spatial distribution of the time-averaged amplitude of the right and left spiral, for *Reo* = -230 and *Rei* = 174 in the neighbourhood of the sink localized in *z0* = 64 mm.

0 0 *Bz A z z* ( ) tanh[ ( ) / ] gives *A0* = 0.3, = 7.14 mm = 0.714 d. The value of the

*<sup>0</sup>* = 

The technique of space-time diagrams does not provide quantitative data on the velocity or vorticity fields that are important for the estimate of energy or momentum transfer in the different regimes. Thus it is necessary to perform particle image velocimetry in order to get more quantitative data. For PIV measurements, the working fluid was seeded with

concentration of about 1ppm. The diffusion time of such particles is

that they are assumed to follow the flow streamlines, i.e. they are good tracers of the flow. Here *U* is the characteristic velocity of the particle. The PIV system consists of two Nd-

*D pB* 3 / 4 500 *d kT s* where *dp* is the particle diameter. The particle Reynolds

The fit of experimental data with the theoretical curves (Fig.8)

**4. Description of the PIV and its adaptation to the Couette-Taylor flow** 

spherical glass particles of diameter 8-11 m and density

number *Rep* = *Udp*/ < 0.1 or equivalently their Stokes number *St* = *(*

*0* in the equations (2) is given by

Fig. 7 show that the source weakly affect the radial flow.

> 1, i.e. the wave coupling is destructive. The

0 0 *Az A z z* ( ) tanh[( ) / ]

*'* = 1.6 g/cm3, with a

*d*)*Rep* < 10-3 so

mm. This value is in a good

*<sup>0</sup>*/*d* ≈ 0.2 (Tagg *et al.*, 1990]. The structure functions in

*pdp*/

shown in Fig. 8 correspond to the case when

and

agreement with theoretical values

coefficient

 <sup>3</sup> YAG Laser sources, a MasterPiv processor (from Tecflow) and a CCD camera (Kodak) with 1034x779 pixels. The time delay between two Laser pulses varies from 0.5 to 25 ms, depending on the values of Reynolds numbers *Rei* and *Reo*. The flow in the test area of the plane (*r*,*z*) is visualized with a thin light sheet that illuminates the glass particles, the positions of which can be recorded at short time intervals. To obtain velocity field, 195 pairs of images of size 1034 x 779 pixels were recorded. Each image of a pair was sampled into windows of 32x32 pixels2 with a recovering of 50%. The velocity fields were computed using the intercorrelation function, which is implemented in the software "Corelia-V2IP" (Tecflow). The PIV measurements were performed in the CCF, in the TVF and WVF regimes in order to calibrate our data acquisition system and to fit data available in the literature for these regimes (Wereley *et al*. 1994, Wereley *et al*. 1998, Abcha *et al*. 2008).

In the circular Couette flow, the spherical glass particles are uniformly distributed (Fig. 9a) while in the TVF and WVF, after 10 hours, the particles have migrated towards the vortex cores where the radial velocity vanishes (Fig. 9b). The PIV allows to visualize velocity and vorticity fields in the cross section (*r*,*z*). The results of the complete process is illustrated by 2D velocity fields of Fig. 10a, b. The inflow (arrow (2)) and outflow (arrow (3)) are clearly evidenced in the case of the TVF and WVF. The measured radial and axial velocity components *Vr*(*r*), *Vz*(*r*) at a given axial position *z* or at a given radial position *r* are plotted in Fig. 10c-h in scaled units.

The radial and axial velocity components have been fitted by a polynomial function satisfying the non-slip condition at the cylindrical walls *r* = *a* and *r = b*. The velocity data are scaled by the inner cylinder velocity *a<sup>i</sup>* as follows : *u* =*Vr/a<sup>i</sup> w= Vz*/*a<sup>i</sup>*. The lengths are scaled by the gap size, the radial position becomes = (*r* – *a*)/*d* and the axial coordinate is = *z*/*d*. In order to plot their profiles, time-averaged velocity components were computed in the axial and radial directions (Fig.10). The instantaneous velocity components can be superposed chronologically at regular time intervals in order to obtain space-time diagrams in both direction (*z*,*t*) and (*r*,*t*) (Fig. 11). The resulting diagrams are colour-coded as in Abcha *et al*. 2008.

Fig. 9. Cross section of flow visualized with glass particles for: a) the CCF (*Ta* = 37.5), b) the WVF (*Ta* = 565).

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 187

Fig. 11. Space-time diagrams for *Ta* = 440: a) *Vr*(*z*,*t*) taken at the midgap *x* = 0.5, b) *Vr*(*r*,*t*) at outflow position. The red colour corresponds to positive values and blue to negative values

The instantaneous velocity fields of the Taylor vortex flow in the radial-axial plane, just above the transition to supercritical flow at *Rei* = 125 for 4 records (ti+1= ti+0.5s) are shown in Fig. 12. These velocity fields illustrate the dynamics of the TVF: a regime composed of stationary counter-rotating vortices characterized by spatial periodicity equal to twice the size of the gap. The traditional flow visualization of wavy vortex flow by observing the motion of small particles at the outer cylinder suggests that the vortices passing a point on

**5. Spatio-temporal structure of Taylor vortex flow and wavy vortex flow** 

Fig. 12. Instantaneous velocity fields of TVF at *Rei* =125 for 4 records (ti+1=ti+0.5s).

of the velocity.

**5.1 Velocity fields** 

Fig. 10. a); b) Velocity field from PIV measurement; c-d) axial variation of velocity component *u*() at the midgap ( = 0.5). Radial variation of velocity components: e-f) axial component *w*() in the vortex core (arrow (1)): and g-h) radial component *u*() at outflow (arrow (3)) for TVF (Ta = 62.5), WVF (Ta = 440) .

Fig. 11. Space-time diagrams for *Ta* = 440: a) *Vr*(*z*,*t*) taken at the midgap *x* = 0.5, b) *Vr*(*r*,*t*) at outflow position. The red colour corresponds to positive values and blue to negative values of the velocity.

### **5. Spatio-temporal structure of Taylor vortex flow and wavy vortex flow**

#### **5.1 Velocity fields**

186 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Fig. 10. a); b) Velocity field from PIV measurement; c-d) axial variation of velocity

) in the vortex core (arrow (1)): and g-h) radial component *u*(

= 0.5). Radial variation of velocity components: e-f) axial

) at outflow

component *u*(

component *w*(

) at the midgap (

(arrow (3)) for TVF (Ta = 62.5), WVF (Ta = 440) .

The instantaneous velocity fields of the Taylor vortex flow in the radial-axial plane, just above the transition to supercritical flow at *Rei* = 125 for 4 records (ti+1= ti+0.5s) are shown in Fig. 12. These velocity fields illustrate the dynamics of the TVF: a regime composed of stationary counter-rotating vortices characterized by spatial periodicity equal to twice the size of the gap. The traditional flow visualization of wavy vortex flow by observing the motion of small particles at the outer cylinder suggests that the vortices passing a point on

Fig. 12. Instantaneous velocity fields of TVF at *Rei* =125 for 4 records (ti+1=ti+0.5s).

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 189

to the left and right, and finally to the right only. These observations led to the conclusion that the middle vortex is losing the same amount of fluid during the second half of the cycle as it gained in the first half. The second half of the cycle (frames (v)-(vii)) appears identical to the first half of the cycle (frames (i)-(iv)) by a rule of reflective symmetry. For example, the vortex 2 in the field (vi) is a reflection of vortex 1 in the field (iii) with reversal of flow direction, that is to say that the flow of the left vortex from to the middle of the field (iii) became the flow of the middle vortex to the left the vortex of field (vi). This is called ''shift-

From velocity fields (Fig. 12, 13), different quantities of the flow perturbations in the cross

Fig. 14. Cross-section (*r*,*z*) of hydrodynamics fields for WVF on the background of vector velocity field: a) Axial velocity; b) radial velocity; c) vorticity; d) kinetic energy; e) axial

elongation; f) radial elongation; g) shear rate.

and-reflect'' symmetry (Marcus*,* 1984).

section (*r*,*z*) can be computed for each regime (Fig. 14):

**5.2 Other hydrodynamic fields** 

the outer cylinder oscillate axially. The PIV permits to visualize the significant transfer of fluid between adjacent vortices in the time and to check the time-dependent theory of shiftand-reflective symmetry in the vortex using the models of wavy vortex flow (Marcus *et al.* 1984). Although the axial motion of the vortices is evidently based on the location of the vortex centres, marked by diamonds (Fig.13), the significant transfer of fluid between adjacent vortices indicates that vortex cells are not independent. The transfer of fluid occurs in a cyclic fashion with a particular vortex gaining fluid from adjacent vortices and then losing fluid to adjacent vortices.

The cycle can be described most easily with reference to the center vortex of « vortex 0 » in Fig. 13. The cycle begins by the frame (i) and ends by the frame (vii). During the cycle the fluid moves from the inner part of the left-hand vortex flowing into the middle vortex and toward the outer cylinder. Simultaneously, fluid from the center vortex moves into the right-hand vortex and toward the inner cylinder. The flow out of the right-hand side of the middle vortex shifts as it is shown in frame (iii), so that now the middle vortex is gaining fluid from the left-hand vortex without losing any fluid. An inward flow from the righthand vortex also feeds fluid into the middle vortex (see frame (iv)). Frames from (v) to (vii) demonstrate the reversed process beginning with flow around the inner side of the middle vortex from right to left, followed by flow out of the middle vortex to the left, then flow out

Fig. 13. The instantaneous velocity fields of WVF at *Rei* = 880 for 8 records (ti+1=ti+0.5s), the time progresses from top to bottom through one complete cycle of an azimuthal wave passing the measurement plane

to the left and right, and finally to the right only. These observations led to the conclusion that the middle vortex is losing the same amount of fluid during the second half of the cycle as it gained in the first half. The second half of the cycle (frames (v)-(vii)) appears identical to the first half of the cycle (frames (i)-(iv)) by a rule of reflective symmetry. For example, the vortex 2 in the field (vi) is a reflection of vortex 1 in the field (iii) with reversal of flow direction, that is to say that the flow of the left vortex from to the middle of the field (iii) became the flow of the middle vortex to the left the vortex of field (vi). This is called ''shiftand-reflect'' symmetry (Marcus*,* 1984).

### **5.2 Other hydrodynamic fields**

188 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

the outer cylinder oscillate axially. The PIV permits to visualize the significant transfer of fluid between adjacent vortices in the time and to check the time-dependent theory of shiftand-reflective symmetry in the vortex using the models of wavy vortex flow (Marcus *et al.* 1984). Although the axial motion of the vortices is evidently based on the location of the vortex centres, marked by diamonds (Fig.13), the significant transfer of fluid between adjacent vortices indicates that vortex cells are not independent. The transfer of fluid occurs in a cyclic fashion with a particular vortex gaining fluid from adjacent vortices and then

The cycle can be described most easily with reference to the center vortex of « vortex 0 » in Fig. 13. The cycle begins by the frame (i) and ends by the frame (vii). During the cycle the fluid moves from the inner part of the left-hand vortex flowing into the middle vortex and toward the outer cylinder. Simultaneously, fluid from the center vortex moves into the right-hand vortex and toward the inner cylinder. The flow out of the right-hand side of the middle vortex shifts as it is shown in frame (iii), so that now the middle vortex is gaining fluid from the left-hand vortex without losing any fluid. An inward flow from the righthand vortex also feeds fluid into the middle vortex (see frame (iv)). Frames from (v) to (vii) demonstrate the reversed process beginning with flow around the inner side of the middle vortex from right to left, followed by flow out of the middle vortex to the left, then flow out

Fig. 13. The instantaneous velocity fields of WVF at *Rei* = 880 for 8 records (ti+1=ti+0.5s), the time progresses from top to bottom through one complete cycle of an azimuthal wave

losing fluid to adjacent vortices.

passing the measurement plane

From velocity fields (Fig. 12, 13), different quantities of the flow perturbations in the cross section (*r*,*z*) can be computed for each regime (Fig. 14):

Fig. 14. Cross-section (*r*,*z*) of hydrodynamics fields for WVF on the background of vector velocity field: a) Axial velocity; b) radial velocity; c) vorticity; d) kinetic energy; e) axial elongation; f) radial elongation; g) shear rate.

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 191

Fig. 16. Space-time diagrams for WVF (*Rei* = 880): (a) *Vr*(r,t) at outflow position, (b) *Vz*(*r*,*t*) in

Fig. 17a and Fig. 18a compare the space-time diagrams obtained from PIV measurements (Fig. 11) with those obtained from flow visualization by Kalliroscope flakes in both directions for a wavy vortex regime at *Rei* = 880. Unlike space time diagrams of velocities components, space-time diagrams obtained from the reflected light intensity do not give any information about the flow direction (upward or downward for *I(z,t)*, inward vs. outward for *I(r,t)*). That is why for comparison, the absolute values of the velocity components

At first glance, it was realized that the space-time diagrams obtained by Kalliroscope flakes are very similar with those of the radial velocity component *Vr*(*z*,*t*) and *Vr*(*r*,*t*). Fig. 17b and Fig. 18b illustrate the time-average profiles in the axial and radial directions. These plots highlight the fact that Kalliroscope particles give a signature of the radial velocity component measured

positions. A similar correspondence is obtained with the envelopes of the space-time diagrams in the axial and radial directions and leads to the same conclusion (Fig. 17b and Fig. 18b): a perfectly identical evolution in the annular space, a maximum reached in the middle of the gap and minima at the walls of two cylinders. Moreover, the absolute value of the radial velocity vanishes in the vortex core while it reaches the maximum in the outflow and in the inflow. The reflected light intensity vanishes in the vortex core because of the weak motion of Kalliroscope flakes. In the inflow and outflow where the Kalliroscope flakes are faster in the

Recent numerical simulations (Gauthier *et al*. 1998) have shown that the Kalliroscope or iriodin particles may be related to the radial velocity component but no measurements

is about 0.01 *Tp*, where the precession time *Tp* ~ *d*/*Vr* ~ *d*/*Vz* 2s for the TVF and *Tp* 1s for WVF. The time scale of the Brownian orientation in a water flow is about 100 s at room temperature (Savas, 1985); it is large enough compared to other time scales of our experiment

radial direction, the intensity is much larger than in the other parts of the flow.

were provided to sustain these arguments. The relaxation time

= 0.5). The minima and maxima are reached for identical axial

of the Kalliroscope flakes

**5.4 Intensity of light reflected by Kalliroscope** *vs***. velocity component** 

the core of anticlockwise vortex.

obtained by PIV are used.

in the centre of the gap (


$$
\rho \alpha\_{\theta} = \left( \left\| V\_r \right\| \left\| \mathbf{\hat{z}} - \left\| V\_z \right\| \left\| \mathbf{\hat{r}} \right\| \right) \; ; \; \mathbf{E} = \left( V\_r^2 + V\_z^2 \right) / \, / \, 2 \tag{3}
$$


$$\dot{\varepsilon}\_{rr} = \left\| V\_r \right\| \left\| \text{\textdegree r} \; ; \; \dot{\varepsilon}\_{zz} = \left\| V\_z \right\| \left\| \text{\textdegree z} \; ; \; \dot{\varepsilon}\_{rz} = \left( \left\| V\_r \right\| \left\| \text{\textdegree z} + \left\| V\_z \right\| \left\| r \right\| \right) / 2 \right) \tag{4}$$

The vorticity fields and velocity components show that inflow and outflow are almost symmetric in the Taylor vortex flow (Fig. 10a 10c) while they are dissymmetric in the wavy vortex flow (Fig. 10b 10d) because of the oscillations of the separatrix.

#### **5.3 Space-time dependence of velocity profiles**

In order to have the most complete information on dynamics of vector velocity field, records of instantaneous profiles of both axial and radial velocity components were superimposed chronologically at regular time intervals (Fig. 15, 16) with color code as in Figure 11. For example, Fig. 15 illustrates the space-time diagram of radial *Vr*(*z*,*t*) and axial velocity *Vz*(*z*,*t*) of TVF (*Rei* = 125). The red colour corresponds to the outflow and the blue colour to the inflow. In Fig. 16, the space-time diagram of radial *Vr*(*r*,*t*) and axial velocity *Vz*(*r*,*t*) of WVF (*Rei* = 880), were the red colour corresponds to anti-clockwise vortex core and the blue colour to clockwise vortex core.

Fig. 15. Space-time diagrams of velocity components for TVF (*Rei* =125): a) *Vr*(z,t) maeasured at the midgap = 0.5 and b)*Vz*(z,t) near the outer cylinder at = 0.75.

 

The vorticity fields and velocity components show that inflow and outflow are almost symmetric in the Taylor vortex flow (Fig. 10a 10c) while they are dissymmetric in the wavy

In order to have the most complete information on dynamics of vector velocity field, records of instantaneous profiles of both axial and radial velocity components were superimposed chronologically at regular time intervals (Fig. 15, 16) with color code as in Figure 11. For example, Fig. 15 illustrates the space-time diagram of radial *Vr*(*z*,*t*) and axial velocity *Vz*(*z*,*t*) of TVF (*Rei* = 125). The red colour corresponds to the outflow and the blue colour to the inflow. In Fig. 16, the space-time diagram of radial *Vr*(*r*,*t*) and axial velocity *Vz*(*r*,*t*) of WVF (*Rei* = 880), were the red colour corresponds to anti-clockwise vortex core and the blue

Fig. 15. Space-time diagrams of velocity components for TVF (*Rei* =125): a) *Vr*(z,t) maeasured

= 0.75.

= 0.5 and b)*Vz*(z,t) near the outer cylinder at

and the kinetic energy *E*:

*rr r V r V z V zV r* / ; / ; / / /2 *zz z rz r <sup>z</sup>* (4)

*V zV r r z* / / ; 2 2 ( )/2 *E VV r z* (3)


**5.3 Space-time dependence of velocity profiles** 

colour to clockwise vortex core.

at the midgap

vortex flow (Fig. 10b 10d) because of the oscillations of the separatrix.


Fig. 16. Space-time diagrams for WVF (*Rei* = 880): (a) *Vr*(r,t) at outflow position, (b) *Vz*(*r*,*t*) in the core of anticlockwise vortex.

#### **5.4 Intensity of light reflected by Kalliroscope** *vs***. velocity component**

Fig. 17a and Fig. 18a compare the space-time diagrams obtained from PIV measurements (Fig. 11) with those obtained from flow visualization by Kalliroscope flakes in both directions for a wavy vortex regime at *Rei* = 880. Unlike space time diagrams of velocities components, space-time diagrams obtained from the reflected light intensity do not give any information about the flow direction (upward or downward for *I(z,t)*, inward vs. outward for *I(r,t)*). That is why for comparison, the absolute values of the velocity components obtained by PIV are used.

At first glance, it was realized that the space-time diagrams obtained by Kalliroscope flakes are very similar with those of the radial velocity component *Vr*(*z*,*t*) and *Vr*(*r*,*t*). Fig. 17b and Fig. 18b illustrate the time-average profiles in the axial and radial directions. These plots highlight the fact that Kalliroscope particles give a signature of the radial velocity component measured in the centre of the gap ( = 0.5). The minima and maxima are reached for identical axial positions. A similar correspondence is obtained with the envelopes of the space-time diagrams in the axial and radial directions and leads to the same conclusion (Fig. 17b and Fig. 18b): a perfectly identical evolution in the annular space, a maximum reached in the middle of the gap and minima at the walls of two cylinders. Moreover, the absolute value of the radial velocity vanishes in the vortex core while it reaches the maximum in the outflow and in the inflow. The reflected light intensity vanishes in the vortex core because of the weak motion of Kalliroscope flakes. In the inflow and outflow where the Kalliroscope flakes are faster in the radial direction, the intensity is much larger than in the other parts of the flow.

Recent numerical simulations (Gauthier *et al*. 1998) have shown that the Kalliroscope or iriodin particles may be related to the radial velocity component but no measurements were provided to sustain these arguments. The relaxation time of the Kalliroscope flakes is about 0.01 *Tp*, where the precession time *Tp* ~ *d*/*Vr* ~ *d*/*Vz* 2s for the TVF and *Tp* 1s for WVF. The time scale of the Brownian orientation in a water flow is about 100 s at room temperature (Savas, 1985); it is large enough compared to other time scales of our experiment

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 193

so that the Brownian motion can be neglected. The comparison of the space-time diagrams obtained from flow visualization and PIV measurements performed for different flow regimes confirms that in the case of the fixed outer cylinder the reflective particles in the flow give information on the radial velocity component. Therefore, the commonly admitted conjecture that the reflective particles give information on the shear rate (Savas 1985) is in contradiction with the quantitative results. In fact, Fig. 19 shows profiles of different flow properties in the axial and radial direction. None of them has a similar behaviour as the reflected light intensity profile (Fig. 17b, 18b). These results give a more precise content on the fact the small anisotropic particles align with the flow streamlines (Savas 1985, Gauthier *et al*. 1998, Matisse *et al*. 1984) by giving the precision on the velocity component which bears these alignment.

Fig. 19. Axial profiles of the absolute values of flow characteristics measured at = 0.5: a)

 *rr* , f) *zz* , g) 

*rz* and d) kinetic energy *E*; Radial profiles of the absolute values of flow

*rz* and h) kinetic energy *E*.

 *rr* , b) *zz* , c) 

characteristics measured in the outflow: e)

Fig. 17. a) Space-time diagrams of the intensity distribution in the axial *I(z,t)* and radial *I(r,t)* direction for *Rei* = 880. b) Radial profile and axial profile of light reflected intensity / max *I II* taken at = 0.5. (1) : vortex core, (2): inflow, (3): outflow.

Fig. 18. a) Space-time diagrams of the absolute value of radial velocity component for *Rei* = 880, b) Radial and axial profiles of the absolute value *uV V r r* / max of the radial velocity component measured at = 0.5.

Fig. 17. a) Space-time diagrams of the intensity distribution in the axial *I(z,t)* and radial *I(r,t)*

= 0.5. (1) : vortex core, (2): inflow, (3): outflow.

Fig. 18. a) Space-time diagrams of the absolute value of radial velocity component for *Rei* = 880, b) Radial and axial profiles of the absolute value *uV V r r* / max of the radial velocity

direction for *Rei* = 880. b) Radial profile and axial profile of light reflected intensity

/ max *I II* taken at

component measured at

= 0.5.

so that the Brownian motion can be neglected. The comparison of the space-time diagrams obtained from flow visualization and PIV measurements performed for different flow regimes confirms that in the case of the fixed outer cylinder the reflective particles in the flow give information on the radial velocity component. Therefore, the commonly admitted conjecture that the reflective particles give information on the shear rate (Savas 1985) is in contradiction with the quantitative results. In fact, Fig. 19 shows profiles of different flow properties in the axial and radial direction. None of them has a similar behaviour as the reflected light intensity profile (Fig. 17b, 18b). These results give a more precise content on the fact the small anisotropic particles align with the flow streamlines (Savas 1985, Gauthier *et al*. 1998, Matisse *et al*. 1984) by giving the precision on the velocity component which bears these alignment.

Fig. 19. Axial profiles of the absolute values of flow characteristics measured at = 0.5: a) *rr* , b) *zz* , c) *rz* and d) kinetic energy *E*; Radial profiles of the absolute values of flow characteristics measured in the outflow: e) *rr* , f) *zz* , g) *rz* and h) kinetic energy *E*.

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 195

The velocity and vorticity fields of the spiral vortex flow (SVF) for *Reo* = -299 and *Rei* = 212 in the radial-axial plane (*r*,*z*) of the flow are shown in Fig. 21. The measurement zone is located

Fig. 21. The instantaneous velocity (arrows) and vorticity fields of the SVF for 4 records (ti+1 = ti+0.5s). The color varies from blue (minimal negative vorticity) to red (maximal

The space-time diagrams of velocity component *Vr*(*z*,*t*) and *Vz*(*r*,*t*) (Fig. 22c-d) confirmed the result from visualization using Kalliroscope flakes that the Taylor spiral vortex pattern is composed of a pair of vortices which propagate along and around the inner cylinder Fig. 22 a-b. Moreover it was revealed that the separatrix between two vortices in a spiral are inclined as in numerical simulations (Ezersky *et al.*, 2010). The radial velocity vanishes in the vortex core while its amplitude is maximal in the outflow and in the inflow. There exists an asymmetry between the inflow and outflow which is well pronounced for the spiral flow

the spiral core is located in the region near the inner cylinder (Fig. 23b). The application of the Rayleigh circulation criterion for counter-rotating cylinders shows that the potentially

0.99. This indicates that the centrifugal instability in case of counter-rotating cylinders is penetrative instability (i.e. it invades the potentially stable zone near the outer cylinder). Using the formula (3-5), the meridional kinetic energy, the radial and axial elongations and

= 0 and the nodal surface

*0* = 0.38 for

) is characterized by an asymmetry in

*<sup>0</sup>* 0.4, meaning that

 

*<sup>0</sup>* = 0.42 for

= -

 <sup>2</sup> <sup>0</sup> / 1 , where

 

= - 1.13 and

0.39 (Fig. 23d). Similarly the profile of the radial

14,17 ) from the bottom. Th e instantaneous velocity

**6. Spatio-temporal structure of spiral vortex flow** 

(Fig. 23a). In the radial direction, the axial velocity *w*(

the shear rate for the spiral vortex pattern have been computed (Fig. 24).

) presents an asymmetry, and reaches a maximum around

 = 0.8, 

the radial direction: it vanishes at

unstable zone is located between

Re /Re *o i* . In our experiment with

fields are regular and show very well the axial motion of the vortices.

**6.1 PIV velocity measurements** 

in the lower part of the system (

positive vorticity).

velocity *u(*

  One should mention that our results were verified for TVF, WVF and MWVF in which the radial velocity component has a magnitude larger than that of the axial component. For turbulent Taylor vortex flow (TTVF), no conclusive observation has been made (Fig. 20). Following the results from (Gauthier et *al.* 1998) one would expect the applicability of the present results to pre-turbulent patterns observed in flow between differentially rotating discs (Cros *et al.* 2002).

Fig. 20. Axial profiles of the absolute values of flow characteristics measured at = 0.5 for different regimes TVF, WVF, MWVF and TTVF

### **6. Spatio-temporal structure of spiral vortex flow**

### **6.1 PIV velocity measurements**

194 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

One should mention that our results were verified for TVF, WVF and MWVF in which the radial velocity component has a magnitude larger than that of the axial component. For turbulent Taylor vortex flow (TTVF), no conclusive observation has been made (Fig. 20). Following the results from (Gauthier et *al.* 1998) one would expect the applicability of the present results to pre-turbulent patterns observed in flow between differentially rotating

Fig. 20. Axial profiles of the absolute values of flow characteristics measured at

different regimes TVF, WVF, MWVF and TTVF

= 0.5 for

discs (Cros *et al.* 2002).

The velocity and vorticity fields of the spiral vortex flow (SVF) for *Reo* = -299 and *Rei* = 212 in the radial-axial plane (*r*,*z*) of the flow are shown in Fig. 21. The measurement zone is located in the lower part of the system ( 14,17 ) from the bottom. Th e instantaneous velocity fields are regular and show very well the axial motion of the vortices.

Fig. 21. The instantaneous velocity (arrows) and vorticity fields of the SVF for 4 records (ti+1 = ti+0.5s). The color varies from blue (minimal negative vorticity) to red (maximal positive vorticity).

The space-time diagrams of velocity component *Vr*(*z*,*t*) and *Vz*(*r*,*t*) (Fig. 22c-d) confirmed the result from visualization using Kalliroscope flakes that the Taylor spiral vortex pattern is composed of a pair of vortices which propagate along and around the inner cylinder Fig. 22 a-b. Moreover it was revealed that the separatrix between two vortices in a spiral are inclined as in numerical simulations (Ezersky *et al.*, 2010). The radial velocity vanishes in the vortex core while its amplitude is maximal in the outflow and in the inflow. There exists an asymmetry between the inflow and outflow which is well pronounced for the spiral flow (Fig. 23a). In the radial direction, the axial velocity *w*() is characterized by an asymmetry in the radial direction: it vanishes at 0.39 (Fig. 23d). Similarly the profile of the radial velocity *u(*) presents an asymmetry, and reaches a maximum around *<sup>0</sup>* 0.4, meaning that the spiral core is located in the region near the inner cylinder (Fig. 23b). The application of the Rayleigh circulation criterion for counter-rotating cylinders shows that the potentially unstable zone is located between = 0 and the nodal surface <sup>2</sup> <sup>0</sup> / 1 , where Re /Re *o i* . In our experiment with = 0.8, *0* = 0.38 for = - 1.13 and *<sup>0</sup>* = 0.42 for = - 0.99. This indicates that the centrifugal instability in case of counter-rotating cylinders is penetrative instability (i.e. it invades the potentially stable zone near the outer cylinder). Using the formula (3-5), the meridional kinetic energy, the radial and axial elongations and the shear rate for the spiral vortex pattern have been computed (Fig. 24).

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 197

Fig. 24. Cross-section (*r*,*z*) of hydrodynamics fields for SVF : a) kinetic energy *E*; b) axial

*rr* d) shear rate

Comparison of the space-time diagrams, reveals a strong similarity between diagrams obtained by Kalliroscope flakes and those of the axial velocity component *Vz*(*z*,*t*) and *Vz*(*r*,*t*). Fig. 25a and Fig. 25b illustrate the time-average profiles in the axial and radial directions respectively of light reflected intensity and axial velocity component. Fig. 25c demonstrates discrepancies between radial velocity and intensity *I* for this case. The minima and maxima of intensity *I* are observed at approximately the same coordinates as minima and maxima of axial velocity *Vz*. It should be noted that the absolute value of the axial velocity vanishes in the vortex core while it reaches the maximum in the outflow and in the inflow. The reflected light intensity vanishes in the vortex core because of the weak

These plots highlight the fact that Kalliroscope particles give a signature of the axial velocity

*rz* .

**6.2 Intensity of light reflected by Kalliroscope** *vs***. velocity component** 

elongation

motion of Kalliroscope flakes.

*zz* c) radial elongation

component measured in axial and radial direction.

Fig. 22. Space-time diagrams of velocity components for *Reo* = -299 and *Rei* = 212 : a) *Vr*(*z*,*t*), b) *Vz*(*z*,*t*), c) *Vr*(z,t), d) *Vz*(*x*,*t*).

Fig. 23. Instantaneous velocity profiles *u* and *w* for *Rei*=212, *Reo*=-299. Axial variation of velocity component at the midgap ( = 0.5) : a) *u*() and c) *w*(). Radial variation of velocity components: b) Radial component *u*() at outflow, d) axial component *w*() in the vortex core.

Fig. 22. Space-time diagrams of velocity components for *Reo* = -299 and *Rei* = 212 : a) *Vr*(*z*,*t*),

Fig. 23. Instantaneous velocity profiles *u* and *w* for *Rei*=212, *Reo*=-299. Axial variation of

) and c) *w*(

) at outflow, d) axial component *w*(

). Radial variation of velocity

) in the vortex core.

= 0.5) : a) *u*(

b) *Vz*(*z*,*t*), c) *Vr*(z,t), d) *Vz*(*x*,*t*).

velocity component at the midgap (

components: b) Radial component *u*(

Fig. 24. Cross-section (*r*,*z*) of hydrodynamics fields for SVF : a) kinetic energy *E*; b) axial elongation *zz* c) radial elongation *rr* d) shear rate *rz* .

#### **6.2 Intensity of light reflected by Kalliroscope** *vs***. velocity component**

Comparison of the space-time diagrams, reveals a strong similarity between diagrams obtained by Kalliroscope flakes and those of the axial velocity component *Vz*(*z*,*t*) and *Vz*(*r*,*t*). Fig. 25a and Fig. 25b illustrate the time-average profiles in the axial and radial directions respectively of light reflected intensity and axial velocity component. Fig. 25c demonstrates discrepancies between radial velocity and intensity *I* for this case. The minima and maxima of intensity *I* are observed at approximately the same coordinates as minima and maxima of axial velocity *Vz*. It should be noted that the absolute value of the axial velocity vanishes in the vortex core while it reaches the maximum in the outflow and in the inflow. The reflected light intensity vanishes in the vortex core because of the weak motion of Kalliroscope flakes.

These plots highlight the fact that Kalliroscope particles give a signature of the axial velocity component measured in axial and radial direction.

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 199

Fig. 26. Space-time diagrams of velocity components in the neighbourhood of the defect for

= 0.5), for *Rei* = 227 and

*Rei*=227 , *Reo*=-299 : a) *Vr*(z,t) , b) *Vz*(z,t), c) *Vr*(r,t) and d) *Vz*(r,t).

Fig. 27. Velocity profiles through the defect at the mid-gap (

evolution of radial and axial velocity components.

*Reo* = -299: a) The temporal evolution of radial and axial velocity components, b) Spatial

Fig. 25. a) Radial profile and axial profile of light reflected intensity *I* taken at = 0.5 b) Radial and axial profiles of the absolute value of the axial velocity component *wV V z z* / max measured at = 0.5; c) Radial and axial profiles of the absolute value *u* of the radial velocity component measured at = 0.5

#### **6.3 The velocity field in the vicinity of defects**

When the Reynolds number *Rei* is increased , the pattern of the spiral vortex flow becomes unstable and spatio-temporal defects appear as a result of vortex merging (annihilation event) or of splitting of a vortex (creation event). The creation and annihilation events appear randomly in the pattern; they are due to long wavelength modulations. The velocity field was determined in the neighborhood of the spatio-temporal defects for *Rei* = 227 and *Reo* = -299. The defect was localized as point where amplitude of velocity field is closed to zero 2 2 0 *V V z r* and phase of the field has nonzero circulation around this point in the plane (*z*,*t*) (Fig. 26a-b black ellipses near *td* = 42s and *zd* = 12 mm ). A special attention was focused on the spatiotemporal behavior of radial and axial velocity components across the defect. Fig. 27a shows the temporal evolution of the axial and radial velocity components at the position of the defect and Fig. 27b shows the velocity profile taken at the collision time.

Fig. 25. a) Radial profile and axial profile of light reflected intensity

= 0.5

**6.3 The velocity field in the vicinity of defects** 

measured at

component measured at

Radial and axial profiles of the absolute value of the axial velocity component *wV V z z* / max

When the Reynolds number *Rei* is increased , the pattern of the spiral vortex flow becomes unstable and spatio-temporal defects appear as a result of vortex merging (annihilation event) or of splitting of a vortex (creation event). The creation and annihilation events appear randomly in the pattern; they are due to long wavelength modulations. The velocity field was determined in the neighborhood of the spatio-temporal defects for *Rei* = 227 and *Reo* = -299. The defect was localized as point where amplitude of velocity field is closed to zero 2 2 0 *V V z r* and phase of the field has nonzero circulation around this point in the plane (*z*,*t*) (Fig. 26a-b black ellipses near *td* = 42s and *zd* = 12 mm ). A special attention was focused on the spatiotemporal behavior of radial and axial velocity components across the defect. Fig. 27a shows the temporal evolution of the axial and radial velocity components at the position of the defect and Fig. 27b shows the velocity profile taken at the collision time.

= 0.5; c) Radial and axial profiles of the absolute value *u* of the radial velocity

*I* taken at

= 0.5 b)

Fig. 26. Space-time diagrams of velocity components in the neighbourhood of the defect for *Rei*=227 , *Reo*=-299 : a) *Vr*(z,t) , b) *Vz*(z,t), c) *Vr*(r,t) and d) *Vz*(r,t).

Fig. 27. Velocity profiles through the defect at the mid-gap ( = 0.5), for *Rei* = 227 and *Reo* = -299: a) The temporal evolution of radial and axial velocity components, b) Spatial evolution of radial and axial velocity components.

In the neighborhood of the defect, the temporal variation of both velocity components follows a parabolic law (Fig. 27a):

$$V\_r(t) \approx \alpha (t - t\_d)^2; \quad V\_z(t) \approx \beta (t - t\_d)^2 \tag{5}$$

Application of the Particle Image Velocimetry to the Couette-Taylor Flow 201

3. Intensity of image brightness is a two dimensional field. How to project three dimensional field of particle concentration or orientation on two dimensional plane? The answers to these questions will enable researchers in Hydrodynamics to understand spatio–temporal structures of closed flows by comparing results obtained from different

This work has been benefited from a financial support from the CPER-Haute-Normandie

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Demay Y. & Iooss G. (1984). Calcul des solutions bifurquées pour le problème de Couette-

Dominguez-Lerma M.A.; Ahlers G. & Cannell D.S. (1985). Effects of Kalliroscope flow

Durst F. & Whitelaw J.H. (1976). *Principles and Practice of Laser-Doppler Anemometry*,

Egbers Ch. & Pfister G. (eds) (2000). *Physics of Rotating Fluids*, Lectures Notes in Physics 549,

Ezersky A.B.; Abcha N. & Mutabazi I. (2010). The structure of spatio-temporal defects in a spiral pattern in the Couette-Taylor flow. *Physics Letters A* Vol.374, p. 3297. Gauthier G., Gondret P. & Rabaud M. (1998). Motions of anisotropic particles : Application to visualization of three-dimensional flows, *Physics of Fluids* Vol. 10, p.2147.

Taylor avec les deux cylindres en rotation, *Journal de Mécanique Théorique et* 

visualization particles on rotating Couette-Taylor flow, *Physics of Fluids* Vol. 28,

reflected light intensity by Kalliroscope flakes and velocity field in the Couette-

1. How image brightness depends on particle concentration or particle orientation? 2. How concentration or orientation depends on velocity field characteristics?

techniques (velocimetry, visualization) and numerical simulations.

Taylor flow system, *Experiments in Fluids* Vol. 45, p.85.

*European Physics Journal* B Vol. 13, p.141.

Springer-Verlag, New-York.

*Modern Physiscs* Vol. 65, p.851.

*Appliquée*, Numéro spécial p.193.

Academic Press, New York.

p.1204.

Springer Berlin.

Bergé P., Pomeau Y. & Vidal Ch. (1984). L'ordre dans le chaos, Hermann, Paris.

**8. Acknowledgements** 

under the program THETE.

**9. References** 

while the spatial evolution is linear in the neighborhood of the defect (Fig. 27b):

$$V\_r(z) \approx a(z - z\_d) \quad ; \; V\_z(z) \approx b(z - z\_d) \tag{6}$$

The coefficients of the best fit are given in the Table. These results are in agreement with the solutions of the Ginzburg-Landau equation near a defect as was shown in Ezersky et *al.*, 2010.


Table 1. Best fit coefficients of the temporal and spatial evolution of the velocity field in the neighbourhood of a defect.

#### **7. Summary**

This chapter has made a focus on the correspondence between the intensity of reflected light by particles and the velocity components in the meridional plane (*r*,*z*). When the outer cylinder is fixed, there is a correspondence between radial velocity component and the intensity of light reflected by anisotropic particles. This result has confirmed recent numerical simulations [Gauthier 1998]. When cylinders are counter-rotating, the intensity of light reflected by anisotropic particles is related to the axial velocity component. To investigate all the aspects of the transition to turbulence in closed or open flows, visualization by particle seeding and velocimetry techniques (LDV, UDV, PIV)are very complementary as they permit to access to different flow characteristics. In fact, the reflective flakes allow to access to flow properties on a large spatial extent and for a long time, and it is possible to evidence the spatio-temporal evolution to turbulence in each directions. For example the study of defects (sources, sinks, dislocations, ...) has facilitated the application of the Ginzburg-Landau model to the study of stationary and timedependent patterns, the transition to chaos or weak turbulence has been characterized using results from visualization. The velocimetry techniques allow therefore to access to physical quantities needed in the model of turbulence (kinetic energy, rate of strain, vorticity, momentum,...) which are useful for validation of theoretical models for example in computing structure coefficients of the statistical distributions. The choice of appropriate technique depends on flow under consideration. In some cases, visualization by anisotropic particles is more preferable than LDV or PIV technique. Besides its simplicity and low cost, it is possible to visualise larger spatial extent, and to record long time data and therefore obtain a better power spectra. The problem of correlation of data obtained from anisotropic particles and velocimetry data is far from being solved, it represents a big challenge for experimental research in Hydrodynamics. Although some similarities between these two methods were found in the case of Couette-Taylor flow patterns, there are many fundamental questions that are far from being resolved:


The answers to these questions will enable researchers in Hydrodynamics to understand spatio–temporal structures of closed flows by comparing results obtained from different techniques (velocimetry, visualization) and numerical simulations.

### **8. Acknowledgements**

This work has been benefited from a financial support from the CPER-Haute-Normandie under the program THETE.

### **9. References**

200 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

In the neighborhood of the defect, the temporal variation of both velocity components

*Vt t t Vt t t r dz d* ( ) ( )² ; ( ) ( )²

*V z az z V z bz z r dz d* () ( ) ; () ( ) (6)

The coefficients of the best fit are given in the Table. These results are in agreement with the solutions of the Ginzburg-Landau equation near a defect as was shown in Ezersky et

Coefficient a b Best fit value 0.01 0.07 -0.62 0.21 Table 1. Best fit coefficients of the temporal and spatial evolution of the velocity field in the

This chapter has made a focus on the correspondence between the intensity of reflected light by particles and the velocity components in the meridional plane (*r*,*z*). When the outer cylinder is fixed, there is a correspondence between radial velocity component and the intensity of light reflected by anisotropic particles. This result has confirmed recent numerical simulations [Gauthier 1998]. When cylinders are counter-rotating, the intensity of light reflected by anisotropic particles is related to the axial velocity component. To investigate all the aspects of the transition to turbulence in closed or open flows, visualization by particle seeding and velocimetry techniques (LDV, UDV, PIV)are very complementary as they permit to access to different flow characteristics. In fact, the reflective flakes allow to access to flow properties on a large spatial extent and for a long time, and it is possible to evidence the spatio-temporal evolution to turbulence in each directions. For example the study of defects (sources, sinks, dislocations, ...) has facilitated the application of the Ginzburg-Landau model to the study of stationary and timedependent patterns, the transition to chaos or weak turbulence has been characterized using results from visualization. The velocimetry techniques allow therefore to access to physical quantities needed in the model of turbulence (kinetic energy, rate of strain, vorticity, momentum,...) which are useful for validation of theoretical models for example in computing structure coefficients of the statistical distributions. The choice of appropriate technique depends on flow under consideration. In some cases, visualization by anisotropic particles is more preferable than LDV or PIV technique. Besides its simplicity and low cost, it is possible to visualise larger spatial extent, and to record long time data and therefore obtain a better power spectra. The problem of correlation of data obtained from anisotropic particles and velocimetry data is far from being solved, it represents a big challenge for experimental research in Hydrodynamics. Although some similarities between these two methods were found in the case of Couette-Taylor flow patterns, there are many

  (5)

while the spatial evolution is linear in the neighborhood of the defect (Fig. 27b):

follows a parabolic law (Fig. 27a):

neighbourhood of a defect.

fundamental questions that are far from being resolved:

*al.*, 2010.

**7. Summary** 


**8** 

**Rheo-Particle Image Velocimetry for** 

Rodríguez- González3 and José G. González-Santos4 *1Laboratorio de Reología, Escuela Superior de Física y Matemáticas,* 

*e Industrias Extractivas, Instituto Politécnico Nacional,* 

*Instituto Politécnico Nacional,* 

*Instituto Politécnico Nacional,* 

*Instituto Politécnico Nacional,* 

*México* 

**the Analysis of the Flow of Polymer Melts** 

José Pérez-González1, Benjamín M. Marín-Santibáñez2, Francisco

*3Departamento de Biotecnología, Centro de Desarrollo de Productos Bióticos,* 

*4Departamento de Matemáticas, Escuela Superior de Física y Matemáticas,* 

*2Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Química* 

The knowledge of the flow kinematics of polymer melts is relevant for basic rheology as well as for polymer processing, particularly for the design of molds and extrusion dies. Nevertheless, the analysis of the flow behavior of polymer melts has been typically performed by using rheometrical (mechanical) measurements and numerical simulation. In spite of the large amount of publications in the field, few works have been dedicated to the analysis of the underlying flow kinematics by using velocimetry techniques. This may in part be due to the difficulties to implement velocimetry techniques during the processing of

With the advent of modern technologies that permit the efficient measurement of the velocity of particles seeded in a fluid, it has been possible to obtain velocity maps in fluids, which provide precise details of their flow kinematics. The analysis of flow fields has been mainly done by using optical techniques like laser Doppler velocimetry (LDV), particle tracking velocimetry (PTV) and particle image velocimetry (PIV), which are powerful noninvasive techniques to describe the flow kinematics in transparent fluids. However, while LDV is a single-point measurement technique and PTV requires the tracking of individual particles, PIV is a whole-field method that allows for the determination of instantaneous velocity maps of a flow region. This last approach, of common use in fluid mechanics, has been gradually implemented for the analysis of the flow behavior of polymer melts. Major limitations for the use of PIV in polymer melts rely on the design and adaptation of transparent dies and molds capable to withstand the high temperatures and pressures

polymer melts at high temperatures and high pressures.

characteristic of polymer processing operations.

**1. Introduction** 


## **Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts**

José Pérez-González1, Benjamín M. Marín-Santibáñez2, Francisco Rodríguez- González3 and José G. González-Santos4 *1Laboratorio de Reología, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, 2Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Química e Industrias Extractivas, Instituto Politécnico Nacional, 3Departamento de Biotecnología, Centro de Desarrollo de Productos Bióticos, Instituto Politécnico Nacional, 4Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, México* 

### **1. Introduction**

202 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Goharzadeh A. & Mutabazi I. (2010). Measurement of coefficients of the Ginzburg-Landau equation for patterns of Taylor spirals, *Physical Review E* Vol. 82, p.016306. Hoffmann C.; Lücke M. & Pinter A. (2005). Spiral vortices traveling between two rotating defects in the Taylor-Couette system, *Physical Review E* Vol.72, p. 056311. Jensen K. D. ( 2004). *Journal of the Brazilian Society of Mechanical Science & Engineering*, Vol.

Langford W. F.; Tagg R.; Kostelich E. J.; Swinney H. L. & Golubitsky M. (1988). Primary

Marcus P. S. (1984). Simulation of Taylor-Couette flow, Part 2: Numerical results for wavyvortex flow with one travelling wave. *Journal of Fluid Mechanics* Vol. 146, p.65. Matisse P. & Gorman M. (1984). Neutrally buoyant anisotropic particles for flow

Mutabazi I.; Wesfreid J.E. & Guyon E. (2006). *Dynamics of Spatio-Temporal Cellular Structures*,

Peerhossaini H. & Wesfreid J.E. (1988). On the inner structure of streamwise Görtler

Provansal M.; Mathis C. & Boyer L. (1987). Bénard-von Karman instability: transient and

Savas Ö (1985). On flow visualization using reflective flakes, *Journal of Fluid Mechanics* Vol.

Takeda Y.; Fischer W.E. & Sakakibara J. (1994). Decomposition of the modulated waves in

Taylor G.I. (1923). Stability of a viscous liquid contained between two rotating cylinders,

Thoroddsen ST; Bauer JM (1999). Qualitative flow visualization using colored lights and

Wereley S.T. & Lueptow R.M. (1994). Azimuthal velocity in supercritical circular Couette

Wereley S.T. & Lueptow R.M. (1998). Spatio-temporal character of non wavy and wavy

vortices, *International Journal of Heat and Fluid Flow* Vol. 9, p. 12.

*Philosophical Transactions of the Royal Society A* Vol. 223, p.289.

Taylor-Couette flow, *Journal of Fluid Mechanics* Vol. 364, p.59.

forced regimes, *Journal of Fluid Mechanics* Vol. 182, p.1.

rotating Couette system, *Science* Vol. 263, p.502.

reflective flakes, *Phys. Fluids* Vol. 11, p.1702. Van Dyke M. (1982). *An Album of Fluid Motion*, Parabolic, Stanford.

flow. *Experiments in Fluids* Vol.18, pp.1.

instabilities and bicriticality in flow between counter-rotating cylinders, *Physics of* 

XXVI(4), p. 401.

*Fluids* Vol.31, p. 776.

Springer, New York.

152, pp.235.

visualization, *Physics of Fluids* Vol.27, p.759.

The knowledge of the flow kinematics of polymer melts is relevant for basic rheology as well as for polymer processing, particularly for the design of molds and extrusion dies. Nevertheless, the analysis of the flow behavior of polymer melts has been typically performed by using rheometrical (mechanical) measurements and numerical simulation. In spite of the large amount of publications in the field, few works have been dedicated to the analysis of the underlying flow kinematics by using velocimetry techniques. This may in part be due to the difficulties to implement velocimetry techniques during the processing of polymer melts at high temperatures and high pressures.

With the advent of modern technologies that permit the efficient measurement of the velocity of particles seeded in a fluid, it has been possible to obtain velocity maps in fluids, which provide precise details of their flow kinematics. The analysis of flow fields has been mainly done by using optical techniques like laser Doppler velocimetry (LDV), particle tracking velocimetry (PTV) and particle image velocimetry (PIV), which are powerful noninvasive techniques to describe the flow kinematics in transparent fluids. However, while LDV is a single-point measurement technique and PTV requires the tracking of individual particles, PIV is a whole-field method that allows for the determination of instantaneous velocity maps of a flow region. This last approach, of common use in fluid mechanics, has been gradually implemented for the analysis of the flow behavior of polymer melts. Major limitations for the use of PIV in polymer melts rely on the design and adaptation of transparent dies and molds capable to withstand the high temperatures and pressures characteristic of polymer processing operations.

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 205

The capillary rheometer may be operated at constant pressure or constant flow rate. It consists of a reservoir that holds the fluid to be characterized and a capillary through which the fluid is forced to flow by an imposed pressure (see Fig. 1). The data obtained from this rheometer are the pressure drop (*Δp*) between capillary ends and the volumetric flow rate

momentum and mass conservation equations, respectively, along with a constitutive equation for the fluid (Bird et al., 1977). These equations are solved for isothermal conditions

The assumption of zero velocity at a solid boundary, also known as the no-slip condition, has serious implications from the basic and practical point of view. This condition is generally satisfied for the flow of Newtonian fluids, but is not necessarily valid for some non-Newtonian ones, for example, entangled polymer melts. An investigation of the validity of such condition for the extrusion flow of polymer melts is precisely one of the objectives of

( ) *rz w*

2 *<sup>w</sup> R p L*

where *τw* is the wall shear stress, *τrz* is the radial dependent shear stress, *r* and *z* are the cylindrical coordinates, *R* and *L* are the capillary radius and length, respectively. In practice, the pressure drop (*Δp*) is not linear between capillary ends due to rearrangements of the velocity profiles at the inlet and outlet region. Then, a correction for end effects is usually applied to the pressure drop (Bagley, 1957). The analysis of such a correction is beyond the

conservation of momentum, which results in a parabolic velocity profile, Eq. 3 (see Fig. 2a),

 

*r*

*r*

*R*

*r*

 

2 2

*pR r*

*L R* 

> 3 4

*Q R*

this work. Considering the previous assumptions, the shear stress is given by:

scope of this work, but its details may be found elsewhere (de Vargas et al., 1995).

() 1 <sup>4</sup> *<sup>z</sup>*

*w*

For a Newtonian fluid the constitutive equation *<sup>z</sup> <sup>v</sup>*

where *Q* is the volumetric flow rate given by:

and an expression for the shear rate at the capillary wall given by Eq. 4:

*v r*

(1)

(2)

is included in the equation of

(3)

(4)

), are obtained by solving the

(*Q*). The flow variables, wall shear stress (*τw*) and shear rate (

considering the following boundary conditions:

c. Steady and well developed flow d. No-slip at the capillary wall

a. Laminar flow b. Incompressible fluid

In this contribution, the extrusion of polymer melts is analyzed by using simultaneous rheometrical and two-dimensional particle image velocimetry measurements (2D PIV), or what has been called Rheo-PIV. First, the fundamentals of rheology and rheometry, shear flow and pressure driven flows are introduced. Then, a brief description of flow instabilities and slip in polymer melts is presented, followed by a short account of the work carried out in the study of the kinematics of Poiseuille flows of polymer melts by using optical velocimetry techniques. The geometry and materials of construction of the dies used for velocimetry measurements are highlighted and problems of actual interest in the field, as flow instabilities and slip at solid boundaries, are particularly addressed. Another section includes the basics of the PIV technique, as well as a discussion of some algorithms relevant for calculation of velocity vectors near solid boundaries and high shear gradients regions, which are very common in the flow of polymer melts. Finally, we present some applications of the PIV technique to the study of the stable and unstable Poiseuille flow under slip and no-slip boundary conditions of molten polyolefins with great practical importance, namely, low-density polyethylene (LDPE), polypropylene (PP), high-density polyethylene (HDPE) and linear low-density polyethylene (LLDPE).

### **2. Rheology**

Rheology is the science of deformation and flow of matter. Following this general definition, one might think that rheology is used to analyze any type of material. In practice, however, rheology has been mainly devoted to the study of fluids containing large molecules (macromolecules) or suspended particles, and fluids having a structure, also known as complex fluids. The flow behavior of these fluids cannot be described by the Newton's law of viscosity and are then called non-Newtonian.

Rheometers are based on shear and shear-free flows. Shear-free flows occur when there is not contact of the fluid with solid walls, for instance, during the film blowing or during the elongation of a polymer melt filament. Contrarily, shear flows may be generated by a moving surface into contact with the fluid or by an applied pressure gradient (Fig. 1). For example, that generated in a capillary rheometer.

### **2.1 The capillary rheometer**

Most polymer processing operations and fluid transport take place at high shear rates. In such cases, the rheological characterization of the fluids is performed by using rheometers based on Poiseuille flows, since rotational rheometers are limited to low shear rates. The capillary rheometer is the most used for this purpose; it has a great practical importance since it is found in different polymer processing operations, particularly in extrusion dies and mold runners.

Fig. 1. Schematic representation of a capillary rheometer

The capillary rheometer may be operated at constant pressure or constant flow rate. It consists of a reservoir that holds the fluid to be characterized and a capillary through which the fluid is forced to flow by an imposed pressure (see Fig. 1). The data obtained from this rheometer are the pressure drop (*Δp*) between capillary ends and the volumetric flow rate (*Q*). The flow variables, wall shear stress (*τw*) and shear rate ( ), are obtained by solving the momentum and mass conservation equations, respectively, along with a constitutive equation for the fluid (Bird et al., 1977). These equations are solved for isothermal conditions considering the following boundary conditions:

a. Laminar flow

204 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

In this contribution, the extrusion of polymer melts is analyzed by using simultaneous rheometrical and two-dimensional particle image velocimetry measurements (2D PIV), or what has been called Rheo-PIV. First, the fundamentals of rheology and rheometry, shear flow and pressure driven flows are introduced. Then, a brief description of flow instabilities and slip in polymer melts is presented, followed by a short account of the work carried out in the study of the kinematics of Poiseuille flows of polymer melts by using optical velocimetry techniques. The geometry and materials of construction of the dies used for velocimetry measurements are highlighted and problems of actual interest in the field, as flow instabilities and slip at solid boundaries, are particularly addressed. Another section includes the basics of the PIV technique, as well as a discussion of some algorithms relevant for calculation of velocity vectors near solid boundaries and high shear gradients regions, which are very common in the flow of polymer melts. Finally, we present some applications of the PIV technique to the study of the stable and unstable Poiseuille flow under slip and no-slip boundary conditions of molten polyolefins with great practical importance, namely, low-density polyethylene (LDPE), polypropylene (PP), high-density polyethylene (HDPE)

Rheology is the science of deformation and flow of matter. Following this general definition, one might think that rheology is used to analyze any type of material. In practice, however, rheology has been mainly devoted to the study of fluids containing large molecules (macromolecules) or suspended particles, and fluids having a structure, also known as complex fluids. The flow behavior of these fluids cannot be described by the Newton's law

Rheometers are based on shear and shear-free flows. Shear-free flows occur when there is not contact of the fluid with solid walls, for instance, during the film blowing or during the elongation of a polymer melt filament. Contrarily, shear flows may be generated by a moving surface into contact with the fluid or by an applied pressure gradient (Fig. 1). For

Most polymer processing operations and fluid transport take place at high shear rates. In such cases, the rheological characterization of the fluids is performed by using rheometers based on Poiseuille flows, since rotational rheometers are limited to low shear rates. The capillary rheometer is the most used for this purpose; it has a great practical importance since it is found in different polymer processing operations, particularly in extrusion dies and mold runners.

and linear low-density polyethylene (LLDPE).

of viscosity and are then called non-Newtonian.

example, that generated in a capillary rheometer.

Fig. 1. Schematic representation of a capillary rheometer

**2.1 The capillary rheometer** 

**2. Rheology** 


The assumption of zero velocity at a solid boundary, also known as the no-slip condition, has serious implications from the basic and practical point of view. This condition is generally satisfied for the flow of Newtonian fluids, but is not necessarily valid for some non-Newtonian ones, for example, entangled polymer melts. An investigation of the validity of such condition for the extrusion flow of polymer melts is precisely one of the objectives of this work. Considering the previous assumptions, the shear stress is given by:

$$
\tau\_{rz}(r) = \tau\_w \frac{r}{R} \tag{1}
$$

$$
\tau\_w = \frac{R \Delta p}{2L} \tag{2}
$$

where *τw* is the wall shear stress, *τrz* is the radial dependent shear stress, *r* and *z* are the cylindrical coordinates, *R* and *L* are the capillary radius and length, respectively. In practice, the pressure drop (*Δp*) is not linear between capillary ends due to rearrangements of the velocity profiles at the inlet and outlet region. Then, a correction for end effects is usually applied to the pressure drop (Bagley, 1957). The analysis of such a correction is beyond the scope of this work, but its details may be found elsewhere (de Vargas et al., 1995).

For a Newtonian fluid the constitutive equation *<sup>z</sup> <sup>v</sup> r* is included in the equation of conservation of momentum, which results in a parabolic velocity profile, Eq. 3 (see Fig. 2a), and an expression for the shear rate at the capillary wall given by Eq. 4:

$$v\_z(r) = \frac{\Delta pR^2}{4\,\mu L} \left[1 - \left(\frac{r}{R}\right)^2\right] \tag{3}$$

$$
\dot{\gamma}\_w = \frac{4Q}{\pi R^3} \tag{4}
$$

where *Q* is the volumetric flow rate given by:

Fig. 2. Velocity profiles: a) Newtonian fluid; b) Non-Newtonian fluid; c) Non-Newtonian fluid with slip at the wall (*vs*)

$$Q = 2\pi \int\_0^R v\_z(r)r dr\tag{5}$$

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 207

Amongst the different flow instabilities, the stick-slip phenomenon occurring in the extrusion of entangled linear polymers melts has received particular attention. This phenomenon starts at a critical shear stress (*τc*) and manifests itself as periodic oscillations of the pressure drop and volumetric flow rate under controlled flow rate experiments. Such oscillations are related to dynamic transitions, from stick to slip, of the boundary condition

The flow curve for a polymer that exhibits the stick-slip instability is discontinuous and is commonly divided into three regions (Fig. 3b). The flow is stable in the first region for *τ < τ<sup>c</sup>* (low shear rate branch), unstable for *τ = τc* (stick-slip region), and assumed to be stable again for *τ > τc* (high shear rate branch). Slip may occur or not in the low shear rate branch depending on the molecular characteristics of the polymer, whereas it is characteristic of the

Fig. 3. a) Extrudate distortions: i) Sharkskin, ii) Bamboo, and iii) Gross melt fracture. b) Non-

The assumption of zero relative velocity between a fluid and a solid wall, or "no-slip" condition, has been a matter of debate for a long time. Nowadays, it is widely accepted that such condition may not be satisfied during the flow of entangled polymer melts (see Fig. 2c). The calculation of the slip velocity (*vs*), for a given shear stress, has been done by using the Mooney's method (1931), which is a phenomenological correction representing the contribution of slip to the experimentally measured flow rate. According to this, *vs* is given

( )

 

dependence of *vs* on the shear stress has been modeled by a linear Navier's condition

, where *b* is a parameter with dimension of length that depends on the fluid-solid

*s D*

*v*

*<sup>w</sup>*. Following this scheme, *<sup>s</sup> v* and *<sup>f</sup> <sup>s</sup>*

the origin, respectively, of a linear plot of *app*

8 *<sup>w</sup> s fs*

represent the shear rates with slip and free of slip, respectively, for a

(9)

are calculated from the slope and ordinate to

versus *1/D* for any given shear stress. The

at a solid wall and produce bamboo-like distortions on the extrudates (Fig. 3a-ii).

high shear rate branch.

monotonic flow curve.

by:

Where *<sup>s</sup>* 

*s w v b* 

given   and *<sup>f</sup> <sup>s</sup>* 

The fluid viscosity (*μ*), which is defined as the ratio *w w* , is given by the well known Hagen-Poiseuille equation:

$$
\mu = \frac{\pi \Delta p R^4}{8LQ} \tag{6}
$$

For non-Newtonian fluids, the viscosity (*η*) is dependent on the shear rate, i. e., ( ) and a commonly used constitutive equation to describe the viscous behavior is the power-law model, *<sup>n</sup> m* , where *m* and *n* are the consistency and shear thinning index, respectively. By using this model in solving the motion equations, the fluid velocity (see Fig. 2b) and the corresponding shear rate at the capillary wall are given by:

$$w\_z(r) = \frac{nR}{1+n} \left(\frac{R\Delta p}{2mL}\right)^{\frac{1}{n}} \left[1 - \left(\frac{r}{R}\right)^{\frac{n+1}{n}}\right] \tag{7}$$

$$
\dot{\gamma}\_w = \dot{\gamma}\_{app} \frac{3n+1}{4n} \tag{8}
$$

Where *app* is given by Eq. 4 and it is referred to as the apparent shear rate when used for non-Newtonian fluids. Eq. 8 is used to calculate the true shear rate and it is obtained after using the Rabinowitsch's correction (Bird et al., 1977).

#### **2.2 Flow instabilities and slip in polymer melts**

Flow instabilities restrain rheometrical measurements and limit productivity in polymer processing operations. Unstable flow is obviously unsteady and may produce distortions of extruded materials (see Fig. 3a). Due to these facts, a great deal of work has been devoted to understand and explain the origin of flow instabilities. Most of the research done in the field in the last decades has been thoroughly reviewed by several authors, being some of the most influential reviews those by Petrie and Denn (1976), Denn (1990) and Denn (2001).

Fig. 2. Velocity profiles: a) Newtonian fluid; b) Non-Newtonian fluid; c) Non-Newtonian

0 2 () *R Q v r rdr* 

> 8 *pR LQ*

a commonly used constitutive equation to describe the viscous behavior is the power-law

By using this model in solving the motion equations, the fluid velocity (see Fig. 2b) and the

*nR R p r*

4 *w app*

non-Newtonian fluids. Eq. 8 is used to calculate the true shear rate and it is obtained after

Flow instabilities restrain rheometrical measurements and limit productivity in polymer processing operations. Unstable flow is obviously unsteady and may produce distortions of extruded materials (see Fig. 3a). Due to these facts, a great deal of work has been devoted to understand and explain the origin of flow instabilities. Most of the research done in the field in the last decades has been thoroughly reviewed by several authors, being some of the most

influential reviews those by Petrie and Denn (1976), Denn (1990) and Denn (2001).

*n mL R* 

3 1

is given by Eq. 4 and it is referred to as the apparent shear rate when used for

*n n*

( ) 1 1 2

> 

For non-Newtonian fluids, the viscosity (*η*) is dependent on the shear rate, i. e.,

 *w w* 

, where *m* and *n* are the consistency and shear thinning index, respectively.

1 1

*n n*

*n*

(8)

4

*<sup>z</sup>* (5)

(6)

, is given by the well known

 ( ) and

(7)

fluid with slip at the wall (*vs*)

Hagen-Poiseuille equation:

model, *<sup>n</sup> m*

Where *app* 

The fluid viscosity (*μ*), which is defined as the ratio

corresponding shear rate at the capillary wall are given by:

*z*

using the Rabinowitsch's correction (Bird et al., 1977).

**2.2 Flow instabilities and slip in polymer melts** 

*v r*

Amongst the different flow instabilities, the stick-slip phenomenon occurring in the extrusion of entangled linear polymers melts has received particular attention. This phenomenon starts at a critical shear stress (*τc*) and manifests itself as periodic oscillations of the pressure drop and volumetric flow rate under controlled flow rate experiments. Such oscillations are related to dynamic transitions, from stick to slip, of the boundary condition at a solid wall and produce bamboo-like distortions on the extrudates (Fig. 3a-ii).

The flow curve for a polymer that exhibits the stick-slip instability is discontinuous and is commonly divided into three regions (Fig. 3b). The flow is stable in the first region for *τ < τ<sup>c</sup>* (low shear rate branch), unstable for *τ = τc* (stick-slip region), and assumed to be stable again for *τ > τc* (high shear rate branch). Slip may occur or not in the low shear rate branch depending on the molecular characteristics of the polymer, whereas it is characteristic of the high shear rate branch.

Fig. 3. a) Extrudate distortions: i) Sharkskin, ii) Bamboo, and iii) Gross melt fracture. b) Nonmonotonic flow curve.

The assumption of zero relative velocity between a fluid and a solid wall, or "no-slip" condition, has been a matter of debate for a long time. Nowadays, it is widely accepted that such condition may not be satisfied during the flow of entangled polymer melts (see Fig. 2c). The calculation of the slip velocity (*vs*), for a given shear stress, has been done by using the Mooney's method (1931), which is a phenomenological correction representing the contribution of slip to the experimentally measured flow rate. According to this, *vs* is given by:

$$w\_s = \frac{D(\dot{\nu}\_s - \dot{\nu}\_{f-s})}{8}\Big|\_{\tau\_w} \tag{9}$$

Where *<sup>s</sup>* and *<sup>f</sup> <sup>s</sup>* represent the shear rates with slip and free of slip, respectively, for a given *<sup>w</sup>*. Following this scheme, *<sup>s</sup> v* and *<sup>f</sup> <sup>s</sup>* are calculated from the slope and ordinate to the origin, respectively, of a linear plot of *app* versus *1/D* for any given shear stress. The dependence of *vs* on the shear stress has been modeled by a linear Navier's condition *s w v b* , where *b* is a parameter with dimension of length that depends on the fluid-solid

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 209

consuming and impose limitations for the analysis of fast changing or unsteady flow. On the other hand, slit channels with flat walls have been preferred for measurements, since they do not introduce optical distortions. However, the shear rate in a slit is dependent onto two directions across the channel, so a full description of the flow field requires measurements into two directions, being the slit central plane the proper measurement place to obtain meaningful rheometrical results. In addition, slit channels are typically long and require the

Migler et al. (2001) developed a rheo-optics technique to visualize how polymer processing additives (PPA) eliminate sharkskin in a metallocene LLDPE (mLLDPE). These authors measured tracking velocity profiles near the wall and the coating process of PPA in a sapphire capillary, and showed evidence of slip in the coated die and adhesion when this was uncoated. These authors used polydimethysiloxane between the die and an external cube to minimize refractive index variations and to avoid optical distortions. Nigen et al. (2003) studied the entry flow of polybutadiene and its relation with flow instabilities in channel flow by using PIV. Polybutadiene is a low melting point polymer that may be extruded at room temperature, so their analysis does not require special fittings for temperature control. Mitsoulis et al. (2005) analyzed the flow of branched polypropylene in a quartz capillary by visualization combined with laser speckle velocimetry and simulation. These authors showed the formation of vortices in the contraction region and suggested their variation in size with increasing shear rate to be related to the presence of slip in the die. The reported velocity profiles, however, do not exhibit slip. Fournier et al. (2009) characterized the extrusion flow of polycarbonate and polystyrene in a rectangular die with quartz windows by using PIV. The measured velocity profiles in this case, showed good agreement with numerical simulations that considered the influence of slip at the die wall. The use of capillaries has been generally avoided in studies of the kinematics of polymer melts mainly because of optical distortions introduced by their curvature. However, capillary flow is very often found in rheometry and polymer processing operations. Then, the description of the flow kinematics in such geometry was required. Recently, Rodríguez-González et al. (2009) described for the first time the flow kinematics of HDPE in a glass capillary die by using PIV. These authors corroborated the main results by Münstedt et al. (2000) and Robert et al. (2004) and showed that the velocity profiles cannot become plug-like in the presence of shear thinning in the melt. In a subsequent work (2010) these authors provided clear evidence of alternating behavior between full adhesion and slip in the

Particle image velocimetry (PIV) has been developed from the early 1980's and can be applied to virtually any kind of flow, as long as the fluid is transparent to enable the imaging the suspended particles. This technique has been developed rapidly over the last three decades and the main findings have been reviewed by Adrian (1991) and Raffel et al. (2007). PIV is a non-intrusive technique commonly used to obtain instantaneous measurements of the velocity vectors in a flow (velocity maps). For two-dimensional PIV, the observation plane is illuminated by a laser light sheet, where two consecutive images of particles seeded in the fluid are obtained. Each image is divided into subsections called interrogation areas, and the statistical displacement of the seeding particles between

use of high pressures to induce flow, which limits the range of shear rates to explore.

unstable stick-slip regime of mLLDPE.

**4. Fundamentals of PIV** 

pair into contact (Denn, 2008). Another model for *vs*, more often used in numerical simulations with polymer melts, is a power-law, *<sup>p</sup> s w v k* , where *k* and *p* also depend on the fluid-solid pair.

### **3. Flow kinematics in polymer melts**

Despite the large amount of publications about the flow of polymer melts, the experimental description of their kinematics has received scarce attention. Limitations rely on the adaptation of transparent flow geometries capable to withstand high pressures and high temperatures. Following, a short account of the work done on the analysis of the kinematics of polymer melts by using optical techniques is presented.

Early works on the flow kinematics of polymer melts were mainly focused on visualization by using streak photography (Han, 2007), which mainly provides qualitative information on the characteristics of the flow field. In contrast, velocimetry techniques provide not only qualitative, but also quantitative information about the flow field. Accordingly to Mackley and Moore (1986), the pioneer work in the description of the flow kinematics in polymer melts by using velocimetry techniques goes back to the work by Kramer and Meissner, who reported the measurements of velocity profiles for LDPE flowing through and abrupt contraction in a rectangular duct by using LDV. Mackley and Moore (1986) measured the velocity profiles of HDPE during the steady state flow in a slit die with glass windows by using LDV and reported normalized velocity profiles almost insensitive to temperature and flow rate. Piau et al. (1995) carried out LDV measurements in a polybutadiene flowing through a metal slit channel with glass windows and observed slip, characterized by a nearly plug flow, when the die surface was fluorinated. Münstedt and coworkers have made an intensive use of the LDV technique to analyze the flow of polymer melts. Schmidt et al. (1999) analyzed the flow of LDPE through a metal slit channel with a glass wall. These authors obtained the velocity along the channel and measured the length for fully developed flow. Also, they obtained the velocity profiles in the slit and calculated the viscosity from a velocity profile. Later, Wassner et al. (1999) analyzed the secondary entry flow of LDPE in the same channel by using LDV. In a subsequent paper Münstedt et al. (2000) reported what is probably the first detailed description of the velocity profiles during the slit flow of HDPE under stable and unstable conditions by using LDV. Even though their work was limited to a narrow shear rate range, these authors described the typical characteristics of the velocity profiles in the three regimes of the unstable flow curve (see Fig. 3b). A more detailed description of the stick-slip kinematics of HDPE in a slit die, also by using LDV, was provided by Robert et al. (2004). These researchers reported that slip was not homogeneous across the die, then, from a numerical computation, they suggested that the measured slip velocities were of the same order of magnitude as those measured in a capillary rheometer. Combeaud et al. (2007) analyzed the flow of polystyrene under stable and unstable flow conditions at the entry of a slit channel by using LDV. These authors reported periodic oscillation of the velocity corresponding to instabilities and volume distortions in the extruded materials.

In all this set of reports, which are based on single-point measurements, the measurement region (of micrometric size) was generally located by micrometrical displacements in a direction normal to the flow to cover the whole region of interest. This may be very time

pair into contact (Denn, 2008). Another model for *vs*, more often used in numerical

Despite the large amount of publications about the flow of polymer melts, the experimental description of their kinematics has received scarce attention. Limitations rely on the adaptation of transparent flow geometries capable to withstand high pressures and high temperatures. Following, a short account of the work done on the analysis of the kinematics

Early works on the flow kinematics of polymer melts were mainly focused on visualization by using streak photography (Han, 2007), which mainly provides qualitative information on the characteristics of the flow field. In contrast, velocimetry techniques provide not only qualitative, but also quantitative information about the flow field. Accordingly to Mackley and Moore (1986), the pioneer work in the description of the flow kinematics in polymer melts by using velocimetry techniques goes back to the work by Kramer and Meissner, who reported the measurements of velocity profiles for LDPE flowing through and abrupt contraction in a rectangular duct by using LDV. Mackley and Moore (1986) measured the velocity profiles of HDPE during the steady state flow in a slit die with glass windows by using LDV and reported normalized velocity profiles almost insensitive to temperature and flow rate. Piau et al. (1995) carried out LDV measurements in a polybutadiene flowing through a metal slit channel with glass windows and observed slip, characterized by a nearly plug flow, when the die surface was fluorinated. Münstedt and coworkers have made an intensive use of the LDV technique to analyze the flow of polymer melts. Schmidt et al. (1999) analyzed the flow of LDPE through a metal slit channel with a glass wall. These authors obtained the velocity along the channel and measured the length for fully developed flow. Also, they obtained the velocity profiles in the slit and calculated the viscosity from a velocity profile. Later, Wassner et al. (1999) analyzed the secondary entry flow of LDPE in the same channel by using LDV. In a subsequent paper Münstedt et al. (2000) reported what is probably the first detailed description of the velocity profiles during the slit flow of HDPE under stable and unstable conditions by using LDV. Even though their work was limited to a narrow shear rate range, these authors described the typical characteristics of the velocity profiles in the three regimes of the unstable flow curve (see Fig. 3b). A more detailed description of the stick-slip kinematics of HDPE in a slit die, also by using LDV, was provided by Robert et al. (2004). These researchers reported that slip was not homogeneous across the die, then, from a numerical computation, they suggested that the measured slip velocities were of the same order of magnitude as those measured in a capillary rheometer. Combeaud et al. (2007) analyzed the flow of polystyrene under stable and unstable flow conditions at the entry of a slit channel by using LDV. These authors reported periodic oscillation of the velocity corresponding to instabilities and volume

In all this set of reports, which are based on single-point measurements, the measurement region (of micrometric size) was generally located by micrometrical displacements in a direction normal to the flow to cover the whole region of interest. This may be very time

*s w v k* 

, where *k* and *p* also depend on

simulations with polymer melts, is a power-law, *<sup>p</sup>*

of polymer melts by using optical techniques is presented.

**3. Flow kinematics in polymer melts** 

distortions in the extruded materials.

the fluid-solid pair.

consuming and impose limitations for the analysis of fast changing or unsteady flow. On the other hand, slit channels with flat walls have been preferred for measurements, since they do not introduce optical distortions. However, the shear rate in a slit is dependent onto two directions across the channel, so a full description of the flow field requires measurements into two directions, being the slit central plane the proper measurement place to obtain meaningful rheometrical results. In addition, slit channels are typically long and require the use of high pressures to induce flow, which limits the range of shear rates to explore.

Migler et al. (2001) developed a rheo-optics technique to visualize how polymer processing additives (PPA) eliminate sharkskin in a metallocene LLDPE (mLLDPE). These authors measured tracking velocity profiles near the wall and the coating process of PPA in a sapphire capillary, and showed evidence of slip in the coated die and adhesion when this was uncoated. These authors used polydimethysiloxane between the die and an external cube to minimize refractive index variations and to avoid optical distortions. Nigen et al. (2003) studied the entry flow of polybutadiene and its relation with flow instabilities in channel flow by using PIV. Polybutadiene is a low melting point polymer that may be extruded at room temperature, so their analysis does not require special fittings for temperature control. Mitsoulis et al. (2005) analyzed the flow of branched polypropylene in a quartz capillary by visualization combined with laser speckle velocimetry and simulation. These authors showed the formation of vortices in the contraction region and suggested their variation in size with increasing shear rate to be related to the presence of slip in the die. The reported velocity profiles, however, do not exhibit slip. Fournier et al. (2009) characterized the extrusion flow of polycarbonate and polystyrene in a rectangular die with quartz windows by using PIV. The measured velocity profiles in this case, showed good agreement with numerical simulations that considered the influence of slip at the die wall.

The use of capillaries has been generally avoided in studies of the kinematics of polymer melts mainly because of optical distortions introduced by their curvature. However, capillary flow is very often found in rheometry and polymer processing operations. Then, the description of the flow kinematics in such geometry was required. Recently, Rodríguez-González et al. (2009) described for the first time the flow kinematics of HDPE in a glass capillary die by using PIV. These authors corroborated the main results by Münstedt et al. (2000) and Robert et al. (2004) and showed that the velocity profiles cannot become plug-like in the presence of shear thinning in the melt. In a subsequent work (2010) these authors provided clear evidence of alternating behavior between full adhesion and slip in the unstable stick-slip regime of mLLDPE.

### **4. Fundamentals of PIV**

Particle image velocimetry (PIV) has been developed from the early 1980's and can be applied to virtually any kind of flow, as long as the fluid is transparent to enable the imaging the suspended particles. This technique has been developed rapidly over the last three decades and the main findings have been reviewed by Adrian (1991) and Raffel et al. (2007). PIV is a non-intrusive technique commonly used to obtain instantaneous measurements of the velocity vectors in a flow (velocity maps). For two-dimensional PIV, the observation plane is illuminated by a laser light sheet, where two consecutive images of particles seeded in the fluid are obtained. Each image is divided into subsections called interrogation areas, and the statistical displacement of the seeding particles between

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 211

centroid of the cross-correlation function in the vicinity of the data peak. ii) the digital crosscorrelation function, in the vicinity of the peak, is approximated by a known continuous function. The coefficients of the fitting function are found by least squares. The new peak position is taken where the fitting function is a maximum. A parabolic or Gaussian fitting function is used. iii) Finally, the digital auto-correlation around the peak location is

On the other hand, the cross-correlation method only produces an approximation to the velocity field and may be necessary to apply a post-processing to eliminate the wrong vectors (so-called outliers). Detection of either a valid or a spurious displacement depends on the number and spatial distribution of the particle-image pairs inside the interrogation area. In practice, a minimum of four particle-image pairs is required to obtain an

Most PIV algorithms use the simple forward difference interrogation (FDI) scheme to calculate the velocity. The velocity at time *t* is calculated from the particle images recorded at times *t* and *t+Δt*, by using the forward finite difference (Raffel et al., 2007). This approximation is accurate to order *Δt* and it can be improved by using a central difference

Wereley and Meinhart (2001) suggested a PIV technique that performs better than the conventional PIV. The main part of this technique is a central difference approximation of the flow velocity. An adaptive interrogation region-shifting algorithm is used to implement the central difference approximation. Adaptive shifting algorithms also have the advantage of helping to eliminate the velocity bias error. The adaptive central difference interrogation technique has the following advantages over the forward difference approximation with or without adaptation. This technique performs better near flow boundaries or in the presence of velocity gradients because the spatial shift between interrogation areas is based on the local and not on the global velocity. The adaptive method automatically reduces the spatial shift between interrogation areas in regions where the particle displacement is small. In addition, the adaptive CDI technique is more accurate, especially at large time delays

between camera exposures and it provides a temporally symmetric view of the flow.

The post-processing of the PIV data is an extensive area and the procedures applied to the vector field depend on the application. At least two stages are necessary for any application; validation and replacement of the incorrect data. When the number of outliers is small their validation can be done by visual inspection and the wrong velocities may be deleted in an interactive way. The spurious vectors are visually recognized by their deviation with respect to the neighbor vectors. Such analysis becomes prohibitive when a great number of recordings have to be evaluated. Then, several automatic procedures have been proposed to validate the raw data. An algorithm that can reject most of the outliers due to noise in the cross-correlation is based on the assumption that for real flow fields the vector difference between the neighboring velocity vectors is small. Typically, a *3x3* neighborhood is used and eight neighbors are used. The vector velocity is rejected if the magnitude of the difference between

computed in a refined grid using an interpolation scheme.

interrogation (CDI) scheme, which is accurate to order (*Δt)2.*

unambiguous measurement of the displacement.

**4.1 Adaptive second-order accuracy method** 

**4.2 Data validation** 

corresponding areas of the two images gives the displacement vector. The resulting displacement vector is divided by the time elapsed between the two consecutive images, *Δt*, to obtain the velocity vector. The seeding particles need to be small enough to follow the flow with minimal drag, but sufficiently large to scatter light to obtain a good particle image. The success of a good analysis is greater when the interrogation areas contain about *8-10* particle images.

Almost all algorithms for estimation of the displacement of a group of tracer particles use cross-correlation techniques. One of the most popular methods is the cross-correlation of two frames of singly exposed recordings. This is a robust method that uses the fast Fourier transform (FFT) to evaluate the cross-correlation coefficient in the interrogation windows. The cross-correlation between the two consecutive images is given by:

$$R(\mathbf{x}, y) = \sum\_{i=-M}^{M} \sum\_{j=-N}^{N} f(i, j) \mathbf{g}(i + \mathbf{x}, j + y) \tag{10}$$

for *x=0,±1,±2,…±(N-1)* and *y=0,±1,±2,…±(M-1)*. *f(i,j)* and *g(i,j)* are the gray level functions at the position (i,j) of the images taken at time *t* and *t+ Δt*, respectively. *NxM* pixels is the size of the interrogation area. By applying this operation for a range of *(x,y)* shifts a correlation plane of size *(2N-1)X(2M-1)* is obtained. The location of the first order intensity peaks in this plane is directly proportional to the mean displacement.

There are two ways to compute Eq. 10; the first is by the evaluation of the cross-correlation function directly by using sum of products. In this case the interrogation windows can have different sizes. Although this approach also reduces the sources of errors, it is rarely used due to the computational cost. The second approach to compute Eq. 10 is via the correlation theorem, which states that the correlation of two functions is equal to the inverse Fourier transform of the complex conjugate product of their Fourier transform. The two dimensional FFT transform is efficiently implemented for an image (2D array of values of the gray scale) using the fast Fourier transform, which reduces the computation from *O(N2)* to *O(N2log2N),* for a *NxN* image size. When the FFT is used, it is generally assumed that the data are periodic; this means that the image sample repeats itself in *x* and *y* directions. Besides the most common implementation of the FFT needs the interrogation areas are squares of the same size; *NxN,* with *N* a power of *2*. There may be two systematic errors in the crosscorrelation calculation. One of them is the aliasing; since the input data are considered to be periodic a correlation peak will be folded back in the correlation plane when the data contains a signal exceeding half of the sample size (*N/2*) and appears on the opposite place. Aliasing may be reduced by either increasing the size of the interrogation areas or by decreasing the laser pulse delay. Another consequence of the periodicity of the correlation data is that the correlation estimation is biased; when the shift magnitude increases less data are actually correlated with each other and their contribution to the actual correlation is limited. Computation speed requirements favor in most cases FFT-based correlation, although the aliasing effect and bias error can be present.

The cross-correlation algorithm may produce several peaks and it is necessary to obtain the position and magnitude of the strongest one. Several algorithms to determine the displacement, from the correlation data at subpixel level, have been proposed by Raffel et al. (2007). Three alternatives are the most common; i) the peak position is approximated by the centroid of the cross-correlation function in the vicinity of the data peak. ii) the digital crosscorrelation function, in the vicinity of the peak, is approximated by a known continuous function. The coefficients of the fitting function are found by least squares. The new peak position is taken where the fitting function is a maximum. A parabolic or Gaussian fitting function is used. iii) Finally, the digital auto-correlation around the peak location is computed in a refined grid using an interpolation scheme.

On the other hand, the cross-correlation method only produces an approximation to the velocity field and may be necessary to apply a post-processing to eliminate the wrong vectors (so-called outliers). Detection of either a valid or a spurious displacement depends on the number and spatial distribution of the particle-image pairs inside the interrogation area. In practice, a minimum of four particle-image pairs is required to obtain an unambiguous measurement of the displacement.

#### **4.1 Adaptive second-order accuracy method**

Most PIV algorithms use the simple forward difference interrogation (FDI) scheme to calculate the velocity. The velocity at time *t* is calculated from the particle images recorded at times *t* and *t+Δt*, by using the forward finite difference (Raffel et al., 2007). This approximation is accurate to order *Δt* and it can be improved by using a central difference interrogation (CDI) scheme, which is accurate to order (*Δt)2.*

Wereley and Meinhart (2001) suggested a PIV technique that performs better than the conventional PIV. The main part of this technique is a central difference approximation of the flow velocity. An adaptive interrogation region-shifting algorithm is used to implement the central difference approximation. Adaptive shifting algorithms also have the advantage of helping to eliminate the velocity bias error. The adaptive central difference interrogation technique has the following advantages over the forward difference approximation with or without adaptation. This technique performs better near flow boundaries or in the presence of velocity gradients because the spatial shift between interrogation areas is based on the local and not on the global velocity. The adaptive method automatically reduces the spatial shift between interrogation areas in regions where the particle displacement is small. In addition, the adaptive CDI technique is more accurate, especially at large time delays between camera exposures and it provides a temporally symmetric view of the flow.

#### **4.2 Data validation**

210 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

corresponding areas of the two images gives the displacement vector. The resulting displacement vector is divided by the time elapsed between the two consecutive images, *Δt*, to obtain the velocity vector. The seeding particles need to be small enough to follow the flow with minimal drag, but sufficiently large to scatter light to obtain a good particle image. The success of a good analysis is greater when the interrogation areas contain about

Almost all algorithms for estimation of the displacement of a group of tracer particles use cross-correlation techniques. One of the most popular methods is the cross-correlation of two frames of singly exposed recordings. This is a robust method that uses the fast Fourier transform (FFT) to evaluate the cross-correlation coefficient in the interrogation windows.

(,) (, ) ( , ) *M N*

for *x=0,±1,±2,…±(N-1)* and *y=0,±1,±2,…±(M-1)*. *f(i,j)* and *g(i,j)* are the gray level functions at the position (i,j) of the images taken at time *t* and *t+ Δt*, respectively. *NxM* pixels is the size of the interrogation area. By applying this operation for a range of *(x,y)* shifts a correlation plane of size *(2N-1)X(2M-1)* is obtained. The location of the first order intensity peaks in this

There are two ways to compute Eq. 10; the first is by the evaluation of the cross-correlation function directly by using sum of products. In this case the interrogation windows can have different sizes. Although this approach also reduces the sources of errors, it is rarely used due to the computational cost. The second approach to compute Eq. 10 is via the correlation theorem, which states that the correlation of two functions is equal to the inverse Fourier transform of the complex conjugate product of their Fourier transform. The two dimensional FFT transform is efficiently implemented for an image (2D array of values of the gray scale) using the fast Fourier transform, which reduces the computation from *O(N2)* to *O(N2log2N),* for a *NxN* image size. When the FFT is used, it is generally assumed that the data are periodic; this means that the image sample repeats itself in *x* and *y* directions. Besides the most common implementation of the FFT needs the interrogation areas are squares of the same size; *NxN,* with *N* a power of *2*. There may be two systematic errors in the crosscorrelation calculation. One of them is the aliasing; since the input data are considered to be periodic a correlation peak will be folded back in the correlation plane when the data contains a signal exceeding half of the sample size (*N/2*) and appears on the opposite place. Aliasing may be reduced by either increasing the size of the interrogation areas or by decreasing the laser pulse delay. Another consequence of the periodicity of the correlation data is that the correlation estimation is biased; when the shift magnitude increases less data are actually correlated with each other and their contribution to the actual correlation is limited. Computation speed requirements favor in most cases FFT-based correlation,

The cross-correlation algorithm may produce several peaks and it is necessary to obtain the position and magnitude of the strongest one. Several algorithms to determine the displacement, from the correlation data at subpixel level, have been proposed by Raffel et al. (2007). Three alternatives are the most common; i) the peak position is approximated by the

*iM jN R x y f <sup>i</sup> j g i x j y* (10)

The cross-correlation between the two consecutive images is given by:

plane is directly proportional to the mean displacement.

although the aliasing effect and bias error can be present.

*8-10* particle images.

The post-processing of the PIV data is an extensive area and the procedures applied to the vector field depend on the application. At least two stages are necessary for any application; validation and replacement of the incorrect data. When the number of outliers is small their validation can be done by visual inspection and the wrong velocities may be deleted in an interactive way. The spurious vectors are visually recognized by their deviation with respect to the neighbor vectors. Such analysis becomes prohibitive when a great number of recordings have to be evaluated. Then, several automatic procedures have been proposed to validate the raw data. An algorithm that can reject most of the outliers due to noise in the cross-correlation is based on the assumption that for real flow fields the vector difference between the neighboring velocity vectors is small. Typically, a *3x3* neighborhood is used and eight neighbors are used. The vector velocity is rejected if the magnitude of the difference between

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 213

A fluoropolymer polymer processing additive (FPPA, Dynamar™ FX-9613) at a concentration of *0.1 wt.%* was used to produce slip at the interface between the LLDPE melt and the capillary wall. Experiments were carried out at the temperatures given in Table 1 under continuous extrusion with a Brabender single screw extruder of *0.019* m in diameter and length to diameter ratio of *25/1*. The pressure drop (*∆p*) between capillary ends was measured with a Dynisco™ pressure transducer, whose voltage signal was sent to an independent computer, via an USB data acquisition board, in order to follow the pressure evolution while evaluating the flow kinematics. Pressure data were acquired at a rate of *100* points/s. The volumetric flow rate (*Q*) was determined by collecting and measuring the

Measurements were performed with a capillary die (*D = 0.0017 m*) made up of borosilicate glass, with an entry angle of *180º* and *L/D=20*; no corrections for end effects were performed. This type of glass capillaries has more than *90%* of light transmissibility, high resistance to

A fixture made up of stainless steel was adapted to the extruder die head in order to support a capillary die as shown in Fig. 4a. The fixture has two pairs of perpendicular windows, one pair was used to pass the laser light sheet through the flow region and the other for visualization. Image distortions produced by the curved geometry of the capillary were eliminated by using an aberration corrector, which was made up of a small rectangular prism with glass walls containing a fluid as shown in Fig. 4b. For this purpose, the refractive index (*n*) of the borosilicate glass capillary (*nglass=1.43*) was closely matched by filling the prism with glycerol (*ngly=1.47*). In order to perform the PIV measurements, one part of the capillary corresponding to a length of *15D* was kept inside the extruder die head at controlled temperature, and the other part of the capillary was inside the aberration corrector, in which the temperature was continuously monitored and supplied with pre-

Fig. 4. a) Stainless steel fixture to support glass capillary in the extruder die head: 1) borosilicate glass die, 2) cooper ring, 3) stainless steel ring, 4) high-temperature o-ring, 5) stainless steel fixture. b) Aberration corrector made up of a small rectangular prism with

wear and high dimensional stability under several processing conditions.

ejected mass as a function of time.

heated glycerol.

glass wall filled with glycerol.

the vector and the average over its neighbors is greater than certain threshold. Raffel et al. (2007) used this validation procedure to detect automatically the invalid vectors from the velocity map above an airfoil. Other data validation techniques are found in the literature, however, no general method can be offered for the problem of data validation in PIV. After removing the outliers, it is necessary to replace the missing data. This may be done, for instance, by using bilinear (or bicubic) interpolation from the valid neighbor vectors.

### **4.3 Velocity bias**

Several authors have proposed methods to reduce or to eliminate the velocity bias, which is produced by using equally-sized interrogation areas in the first and the second images. Westerwell (1994) showed that the error bias can be completely eliminated by dividing the image correlation function by the areas correlation function. Keane and Adrian (1992) suggested reduction of the bias by doing the second interrogation area larger than the first, so all particle images in the first interrogation area will likely be contained in the second. They also proposed that spatially-shifting the second interrogation area by an integer part of the displacement will substantially reduce the bias error. Haung et al. (1997) proposed an efficient algorithm to eliminate velocity bias by renormalizing the values of the correlation function in the neighborhood of the peak location prior to calculation of the subpixel peak location. These techniques improve, in general, the PIV results. However, the improvement is limited near to flow boundaries.

To reduce the errors of the displacement estimation, iterative methods have been developed which use a shift of the interrogation areas where the offset can be an integer number of pixels or a fraction (sub-pixel accuracy). If the displacement of the particles of the first interrogation area, between *t* and *t+Δt*, can be estimated, then the interrogation area from the second image can be matched via a relative offset. Scarano and Riethmuller (1999) proposed a method to obtain the displacement and optimize it within an iterative process. Since the components of the predicted displacement, *Δs=(Δx,Δy)* are an integer number of pixels, their method does not require to use interpolation.

### **5. Experimental methods**

### **5.1 Materials and rheometry**

Molten polyolefins were analyzed in this work, namely, LDPE, PP, HDPE, and mLLDPE, respectively. All the polyethylenes are reported as free of additives that might screen interactions at the die wall and PP is an industrial grade polymer. Their main characteristics are given in Table 1.


Table 1. Polymer characteristics and experimental conditions

the vector and the average over its neighbors is greater than certain threshold. Raffel et al. (2007) used this validation procedure to detect automatically the invalid vectors from the velocity map above an airfoil. Other data validation techniques are found in the literature, however, no general method can be offered for the problem of data validation in PIV. After removing the outliers, it is necessary to replace the missing data. This may be done, for

Several authors have proposed methods to reduce or to eliminate the velocity bias, which is produced by using equally-sized interrogation areas in the first and the second images. Westerwell (1994) showed that the error bias can be completely eliminated by dividing the image correlation function by the areas correlation function. Keane and Adrian (1992) suggested reduction of the bias by doing the second interrogation area larger than the first, so all particle images in the first interrogation area will likely be contained in the second. They also proposed that spatially-shifting the second interrogation area by an integer part of the displacement will substantially reduce the bias error. Haung et al. (1997) proposed an efficient algorithm to eliminate velocity bias by renormalizing the values of the correlation function in the neighborhood of the peak location prior to calculation of the subpixel peak location. These techniques improve, in general, the PIV results. However, the improvement

To reduce the errors of the displacement estimation, iterative methods have been developed which use a shift of the interrogation areas where the offset can be an integer number of pixels or a fraction (sub-pixel accuracy). If the displacement of the particles of the first interrogation area, between *t* and *t+Δt*, can be estimated, then the interrogation area from the second image can be matched via a relative offset. Scarano and Riethmuller (1999) proposed a method to obtain the displacement and optimize it within an iterative process. Since the components of the predicted displacement, *Δs=(Δx,Δy)* are an integer number of

Molten polyolefins were analyzed in this work, namely, LDPE, PP, HDPE, and mLLDPE, respectively. All the polyethylenes are reported as free of additives that might screen interactions at the die wall and PP is an industrial grade polymer. Their main characteristics

(*g/10min*) Mw (*g/mol)* <sup>ρ</sup> (*g/cm3*) Tm (°C) TExp

LDPE 1.5 --- 0.922 115 190 Aldrich PP 1.8 --- 0.9 151 200 Basell HDPE 0.25 125000 0.950 130 180 Aldrich mLLDPE 0.8 93200 0.880 60 190 Aldrich

(°C) Supplier

instance, by using bilinear (or bicubic) interpolation from the valid neighbor vectors.

**4.3 Velocity bias** 

is limited near to flow boundaries.

**5. Experimental methods 5.1 Materials and rheometry** 

Polymer Melt Index

are given in Table 1.

pixels, their method does not require to use interpolation.

Table 1. Polymer characteristics and experimental conditions

A fluoropolymer polymer processing additive (FPPA, Dynamar™ FX-9613) at a concentration of *0.1 wt.%* was used to produce slip at the interface between the LLDPE melt and the capillary wall. Experiments were carried out at the temperatures given in Table 1 under continuous extrusion with a Brabender single screw extruder of *0.019* m in diameter and length to diameter ratio of *25/1*. The pressure drop (*∆p*) between capillary ends was measured with a Dynisco™ pressure transducer, whose voltage signal was sent to an independent computer, via an USB data acquisition board, in order to follow the pressure evolution while evaluating the flow kinematics. Pressure data were acquired at a rate of *100* points/s. The volumetric flow rate (*Q*) was determined by collecting and measuring the ejected mass as a function of time.

Measurements were performed with a capillary die (*D = 0.0017 m*) made up of borosilicate glass, with an entry angle of *180º* and *L/D=20*; no corrections for end effects were performed. This type of glass capillaries has more than *90%* of light transmissibility, high resistance to wear and high dimensional stability under several processing conditions.

A fixture made up of stainless steel was adapted to the extruder die head in order to support a capillary die as shown in Fig. 4a. The fixture has two pairs of perpendicular windows, one pair was used to pass the laser light sheet through the flow region and the other for visualization. Image distortions produced by the curved geometry of the capillary were eliminated by using an aberration corrector, which was made up of a small rectangular prism with glass walls containing a fluid as shown in Fig. 4b. For this purpose, the refractive index (*n*) of the borosilicate glass capillary (*nglass=1.43*) was closely matched by filling the prism with glycerol (*ngly=1.47*). In order to perform the PIV measurements, one part of the capillary corresponding to a length of *15D* was kept inside the extruder die head at controlled temperature, and the other part of the capillary was inside the aberration corrector, in which the temperature was continuously monitored and supplied with preheated glycerol.

Fig. 4. a) Stainless steel fixture to support glass capillary in the extruder die head: 1) borosilicate glass die, 2) cooper ring, 3) stainless steel ring, 4) high-temperature o-ring, 5) stainless steel fixture. b) Aberration corrector made up of a small rectangular prism with glass wall filled with glycerol.

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 215

map. This was not made for unstable regimes where the velocity oscillates with time. In such a case, two frequencies were used, *2.0* and *6.1 Hz*, for slow and fast changes, respectively. The axial velocity component as a function of the radial position (velocity

According to discussion in subsection 4.1 an adaptive correlation algorithm with a central difference approximation was used to calculate the velocity vectors. This technique has been proved to be more accurate than conventional PIV algorithms when measuring near flow boundaries or in the presence of velocity gradients (Wereley & Meinhart, 2001), which is the case during the evaluation of velocity vectors in the neighborhood of the capillary wall. Since fully developed flow is unidirectional, each interrogation area was chosen as a long rectangle in the flow direction of *256 pixels* long and *16 pixels* wide (*642 μm x 40 μm*, radial and axial direction, respectively), with an overlap of *50%* in both axis. With this width of the interrogation area and the particle size used for seeding, the closest distance to the capillary wall at which measures were made was *40 μm*. Further approach to the wall would require the seeding with smaller particles. Finally, data validation was performed by using a

The rheometrical flow curve for the LDPE is shown in Fig. 6a, along with the one obtained from the integration of the velocity profiles according to Eq. 7. Note that both curves are well fitted by the power-law relationship in the apparent shear rate range studied (see the equation inserted in Fig. 6a). Validation of the PIV measurements is performed via its comparison with rheometrical data. In this case, the data obtained from the velocity profiles agree well with the rheometrical ones; the maximum difference in the volumetric flow rates obtained by using the two methods was *6.5%* at most (the average difference was around *3.4%*), which shows the reliability of the PIV technique to describe the behavior of the polymer melt in capillary flow. The origins of the differences between the two methods are likely found in the location of the laser light sheet with respect to the real central plane of the capillary and its thickness, as well as in the uncertainty of the flow rate measurements. Figure 6b shows the velocity maps in the capillary for different flow conditions. It is clear that almost all vectors in each map are parallel to the flow direction, which shows that the flow in the observation region was unidirectional for the different apparent shear rates studied, with a velocity field simply given by *vz=vz(r)*, as it is expected for a fully developed

The PIV velocity profiles for different apparent shear rates are shown in Figs. 7a-b along with the profiles calculated with Eq. 7. Observe that the velocity profiles in the capillary are symmetric with respect to the flow direction and that they are very well matched by Eq. 7. Also, the standard deviation of the time average of fifty profiles, which is represented by the error bars, shows variations below *5%*, which indicates that the flow was steady. It is interesting to note that all the velocity profiles in Figs. 7a-b extrapolate to zero value at the capillary wall, indicating the absence of slip. This result agrees with the well known fact that

profile) was obtained by averaging the profiles in a map.

moving average filter.

shear flow.

**6. Results and discussion** 

**6.1.1 Low-density polyethylene** 

**6.1 Analysis of stable flow conditions** 

branched polyethylene melts do not exhibit slip.

### **5.2 PIV measurements**

The study of the flow kinematics in the capillary was performed with a two dimensional (2D) PIV Dantec Dynamics system as shown in Fig. 5. The PIV system consists of a high speed and high sensitivity HiSense MKII CCD camera of *1.35 Mega-pixels*, two coupled Nd:YAG lasers of *50 mJ* with *λ = 532 nm* and the Dantec Dynamic Studio 2.1 software. The light sheet was reduced in thickness up to less than *200 µm* by using a biconvex lens with *0.05 m* of focal distance, and then sent through the center plane of the capillary by using a prism oriented at *45°* relative to the original direction of the laser beam. The prism was mounted on a rail carrier and the center plane of the capillary was found by horizontal displacements of the prism up to see the longest chord on the image plane. The particles used were solid copper spheres *< 10 μm* in diameter (Aldrich 32,6453) at a concentration of *0.5 wt.%*. This amount of particles is not expected to affect the rheological behavior of the polymer. Using the Einstein relation for spherical particles in a fluid, the increase in the viscosity due to the presence of copper particles was calculated to be less than *0.1%*, which is negligible.

Fig. 5. PIV set up: 1) Single screw extruder, 2) Extruder die head with the stainless steel fixture and the aberration corrector as shown in Fig. 6, 3) Nd:YAG lasers, 4) high-speed and high -sensitivity CCD camera with a continuously-focusable video microscope, 5) biconvex lens, 6) prism, 7) pressure transducer.

An InfiniVar™ continuously-focusable video microscope CFM-2/S was attached to the CCD camera in order to increase the spatial resolution. The depth of field of the microscope at the position of observation was measured as *40 µm*. A small depth of field reduces the true observation volume and the error introduced by out of plane particles. This fact is particularly important in experiments in which the shear stress is not homogeneous, since the wider the light sheet, the bigger the variation of the shear stress in the observation volume. Thus, the variation of the shear stress in the region associated with the depth of field of the microscope was only *5%* of *w*.

The images taken by the PIV system covered an area of *0.00171 m x 0.0032 m* and were centered at an axial position *z=17D* downstream from the contraction. Series of fifty image pairs were obtained for each flow condition. For stable conditions, the frequency was *6.1 Hz* and all the image pairs were interrogated and ensemble averaged to obtain a single velocity map. This was not made for unstable regimes where the velocity oscillates with time. In such a case, two frequencies were used, *2.0* and *6.1 Hz*, for slow and fast changes, respectively. The axial velocity component as a function of the radial position (velocity profile) was obtained by averaging the profiles in a map.

According to discussion in subsection 4.1 an adaptive correlation algorithm with a central difference approximation was used to calculate the velocity vectors. This technique has been proved to be more accurate than conventional PIV algorithms when measuring near flow boundaries or in the presence of velocity gradients (Wereley & Meinhart, 2001), which is the case during the evaluation of velocity vectors in the neighborhood of the capillary wall. Since fully developed flow is unidirectional, each interrogation area was chosen as a long rectangle in the flow direction of *256 pixels* long and *16 pixels* wide (*642 μm x 40 μm*, radial and axial direction, respectively), with an overlap of *50%* in both axis. With this width of the interrogation area and the particle size used for seeding, the closest distance to the capillary wall at which measures were made was *40 μm*. Further approach to the wall would require the seeding with smaller particles. Finally, data validation was performed by using a moving average filter.

### **6. Results and discussion**

214 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

The study of the flow kinematics in the capillary was performed with a two dimensional (2D) PIV Dantec Dynamics system as shown in Fig. 5. The PIV system consists of a high speed and high sensitivity HiSense MKII CCD camera of *1.35 Mega-pixels*, two coupled Nd:YAG lasers of *50 mJ* with *λ = 532 nm* and the Dantec Dynamic Studio 2.1 software. The light sheet was reduced in thickness up to less than *200 µm* by using a biconvex lens with *0.05 m* of focal distance, and then sent through the center plane of the capillary by using a prism oriented at *45°* relative to the original direction of the laser beam. The prism was mounted on a rail carrier and the center plane of the capillary was found by horizontal displacements of the prism up to see the longest chord on the image plane. The particles used were solid copper spheres *< 10 μm* in diameter (Aldrich 32,6453) at a concentration of *0.5 wt.%*. This amount of particles is not expected to affect the rheological behavior of the polymer. Using the Einstein relation for spherical particles in a fluid, the increase in the viscosity due to the presence of copper particles was calculated to be less than *0.1%*, which

Fig. 5. PIV set up: 1) Single screw extruder, 2) Extruder die head with the stainless steel fixture and the aberration corrector as shown in Fig. 6, 3) Nd:YAG lasers, 4) high-speed and high -sensitivity CCD camera with a continuously-focusable video microscope, 5) biconvex

> *w*.

An InfiniVar™ continuously-focusable video microscope CFM-2/S was attached to the CCD camera in order to increase the spatial resolution. The depth of field of the microscope at the position of observation was measured as *40 µm*. A small depth of field reduces the true observation volume and the error introduced by out of plane particles. This fact is particularly important in experiments in which the shear stress is not homogeneous, since the wider the light sheet, the bigger the variation of the shear stress in the observation volume. Thus, the variation of the shear stress in the region associated with the depth of

The images taken by the PIV system covered an area of *0.00171 m x 0.0032 m* and were centered at an axial position *z=17D* downstream from the contraction. Series of fifty image pairs were obtained for each flow condition. For stable conditions, the frequency was *6.1 Hz* and all the image pairs were interrogated and ensemble averaged to obtain a single velocity

**5.2 PIV measurements** 

is negligible.

lens, 6) prism, 7) pressure transducer.

field of the microscope was only *5%* of

### **6.1 Analysis of stable flow conditions**

#### **6.1.1 Low-density polyethylene**

The rheometrical flow curve for the LDPE is shown in Fig. 6a, along with the one obtained from the integration of the velocity profiles according to Eq. 7. Note that both curves are well fitted by the power-law relationship in the apparent shear rate range studied (see the equation inserted in Fig. 6a). Validation of the PIV measurements is performed via its comparison with rheometrical data. In this case, the data obtained from the velocity profiles agree well with the rheometrical ones; the maximum difference in the volumetric flow rates obtained by using the two methods was *6.5%* at most (the average difference was around *3.4%*), which shows the reliability of the PIV technique to describe the behavior of the polymer melt in capillary flow. The origins of the differences between the two methods are likely found in the location of the laser light sheet with respect to the real central plane of the capillary and its thickness, as well as in the uncertainty of the flow rate measurements.

Figure 6b shows the velocity maps in the capillary for different flow conditions. It is clear that almost all vectors in each map are parallel to the flow direction, which shows that the flow in the observation region was unidirectional for the different apparent shear rates studied, with a velocity field simply given by *vz=vz(r)*, as it is expected for a fully developed shear flow.

The PIV velocity profiles for different apparent shear rates are shown in Figs. 7a-b along with the profiles calculated with Eq. 7. Observe that the velocity profiles in the capillary are symmetric with respect to the flow direction and that they are very well matched by Eq. 7. Also, the standard deviation of the time average of fifty profiles, which is represented by the error bars, shows variations below *5%*, which indicates that the flow was steady. It is interesting to note that all the velocity profiles in Figs. 7a-b extrapolate to zero value at the capillary wall, indicating the absence of slip. This result agrees with the well known fact that branched polyethylene melts do not exhibit slip.

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 217

a)

b) Fig. 7. Velocity profiles for different apparent shear rates for LDPE at a) low and b) high

clarity. It is clear that the data obtained from the velocity profiles follow the trend of the rheometrical ones and extend well into the transition between the Newtonian and powerlaw regions, which is clearly observed in the shear viscosity curve shown in Fig. 8b. The PIV

since allows the analysis of the low shear rate region that is not accessible by using the macroscopic physical quantities provided by a single capillary. Instead, a rotational rheometer or a capillary of bigger diameter must be used to study the low shear rate

. This is a valuable fact,

shear rate regimes. Continuous lines represent the power-law model (Eq. 7).

data extend to lower shear rate values than the rheometrical *w*

behavior.

Fig. 6. a) Flow curve for LDPE obtained from rheometrical measurements and by the integration of the velocity profiles. b) Velocity maps in the capillary for different flow conditions.

#### **6.1.1.1 Determination of the flow and viscosity curves from velocity profiles**

Due to the radial distribution of the shear stress in capillaries (Eq. 1), i.e., *τ=τ(r)*, the true flow curve for a polymer may be obtained from the velocity profiles if a determination of the true shear rate as a function of the radial position is performed. There is a range of shear rates and stresses in a capillary for a given flow rate, namely, *0≤ <sup>r</sup> ≤ <sup>R</sup>* and *0≤τr≤τ w*, which enables one to obtain the flow curve from the velocity profiles and the measured wall shear stress. The local shear rate may be calculated from the numerical derivative of the velocity profiles with respect to the radial position. A central difference approximation was used in this work. Meanwhile, the corresponding shear stress was calculated from the measured *τ<sup>w</sup>* by using Eq. 1.

The flow curve calculated by using the velocity profiles corresponding to *54.78, 112.31, 167.1, 359.79* and *834.31 s-1*, respectively, is displayed in Fig. 8a along with that obtained by applying the Rabinowitsch's correction to the rheometrical data. The segments of the flow curve reconstructed from the velocity profiles are represented with different symbols for

a)

b)

Due to the radial distribution of the shear stress in capillaries (Eq. 1), i.e., *τ=τ(r)*, the true flow curve for a polymer may be obtained from the velocity profiles if a determination of the true shear rate as a function of the radial position is performed. There is a range of shear

enables one to obtain the flow curve from the velocity profiles and the measured wall shear stress. The local shear rate may be calculated from the numerical derivative of the velocity profiles with respect to the radial position. A central difference approximation was used in this work. Meanwhile, the corresponding shear stress was calculated from the measured *τ<sup>w</sup>*

The flow curve calculated by using the velocity profiles corresponding to *54.78, 112.31, 167.1, 359.79* and *834.31 s-1*, respectively, is displayed in Fig. 8a along with that obtained by applying the Rabinowitsch's correction to the rheometrical data. The segments of the flow curve reconstructed from the velocity profiles are represented with different symbols for

 *≤ <sup>R</sup>*

and *0≤τr≤τ w*, which

Fig. 6. a) Flow curve for LDPE obtained from rheometrical measurements and by the integration of the velocity profiles. b) Velocity maps in the capillary for different flow

**6.1.1.1 Determination of the flow and viscosity curves from velocity profiles** 

rates and stresses in a capillary for a given flow rate, namely, *0≤ <sup>r</sup>*

conditions.

by using Eq. 1.

b)

Fig. 7. Velocity profiles for different apparent shear rates for LDPE at a) low and b) high shear rate regimes. Continuous lines represent the power-law model (Eq. 7).

clarity. It is clear that the data obtained from the velocity profiles follow the trend of the rheometrical ones and extend well into the transition between the Newtonian and powerlaw regions, which is clearly observed in the shear viscosity curve shown in Fig. 8b. The PIV data extend to lower shear rate values than the rheometrical *w* . This is a valuable fact, since allows the analysis of the low shear rate region that is not accessible by using the macroscopic physical quantities provided by a single capillary. Instead, a rotational rheometer or a capillary of bigger diameter must be used to study the low shear rate behavior.

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 219

a)

b)

The feature of the PIV technique to produce instantaneous velocity maps, allows for the detection of rapid variations in a region of the flow field. Thus, the PIV technique permits, for example, an accurate description of the flow kinematics under unstable flow conditions occurring in some polymer processing operations. This is particularly the case for HDPE and LLDPE, which are known to exhibit extrusion instabilities like those described in

The flow curve for HDPE shown in Fig. 10a displays the typical three regions (I-III) of a nonmonotonic flow curve (see Fig. 3b). Region I corresponds to a stable flow regime

Fig. 9. a) Flow curve for PP obtained from rheometrical measurements and by the integration of the velocity profiles. b) Velocity profiles in the capillary for different flow

conditions.

**6.2 Analysis of unstable flow conditions** 

**6.2.1 High-density polyethylene** 

section 2.2 (see Fig. 3a).

The wider shear rate range covered by PIV data in Figs. 8a-b permits the fitting of the viscous behavior of the LDPE, ( ) , by a more realistic constitutive equation including the response at low shear rates, as for example the Carreau´s model (Bird et al., 1977).

Fig. 8. a) True flow curve and b) viscosity curve for LDPE obtained from local shear rate data. Dashed and continuous lines indicate the power-law and Carreau's model fit, respectively.

#### **6.1.2 Polypropylene**

PP shows a similar behavior to the observed for LDPE. The flow curve in Fig. 9a may also be well fitted by a power-law at shear stresses prior to the onset of gross melt fracture, as well as the velocity profiles in Fig 9b. Akin to the LDPE, all the velocity maps (not shown here) agree with a fully developed flow, meanwhile the profiles appear symmetric and extrapolate to zero value at the capillary wall, indicating the absence of slip.

The wider shear rate range covered by PIV data in Figs. 8a-b permits the fitting of the

a)

b) Fig. 8. a) True flow curve and b) viscosity curve for LDPE obtained from local shear rate data. Dashed and continuous lines indicate the power-law and Carreau's model fit,

PP shows a similar behavior to the observed for LDPE. The flow curve in Fig. 9a may also be well fitted by a power-law at shear stresses prior to the onset of gross melt fracture, as well as the velocity profiles in Fig 9b. Akin to the LDPE, all the velocity maps (not shown here) agree with a fully developed flow, meanwhile the profiles appear symmetric and

extrapolate to zero value at the capillary wall, indicating the absence of slip.

the response at low shear rates, as for example the Carreau´s model (Bird et al., 1977).

( ) , by a more realistic constitutive equation including

 

viscous behavior of the LDPE,

respectively.

**6.1.2 Polypropylene** 

Fig. 9. a) Flow curve for PP obtained from rheometrical measurements and by the integration of the velocity profiles. b) Velocity profiles in the capillary for different flow conditions.

#### **6.2 Analysis of unstable flow conditions**

#### **6.2.1 High-density polyethylene**

The feature of the PIV technique to produce instantaneous velocity maps, allows for the detection of rapid variations in a region of the flow field. Thus, the PIV technique permits, for example, an accurate description of the flow kinematics under unstable flow conditions occurring in some polymer processing operations. This is particularly the case for HDPE and LLDPE, which are known to exhibit extrusion instabilities like those described in section 2.2 (see Fig. 3a).

The flow curve for HDPE shown in Fig. 10a displays the typical three regions (I-III) of a nonmonotonic flow curve (see Fig. 3b). Region I corresponds to a stable flow regime

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 221

10b. Again, the velocity profiles are very well described by the power-law relationship and there is a good agreement between the rheometrical and PIV data in region I. The velocity profiles in this flow regime do not show slip, even though it is known that HDPE may slip at the die wall. This may be explained by a relatively low molecular weight of the polymer. The magnitude of slip depends on the polymer molecular weight. The higher the molecular

a)

b) Fig. 11. a) Minimum and maximum velocity profiles during one pressure oscillation at an apparent shear rate of 297 s-1. b) Evolution of pressure, maximum velocity and slip velocity for the same apparent shear rate. Reprinted with permission from Rodríguez-González et al., *Chemical Engineering Science*, Vol. 64, No. 22, (November 2009), pp. 4675-4683, ISSN 0003-

The pressure signal and the flow maps were recorded simultaneously in the stick-slip regime. The minimum and maximum velocity profiles during one pressure oscillation at an apparent shear rate of *297 s-1* are shown in Fig. 11a. There is a large difference between the maxima of the two profiles, *vmin* and *vmax*, and their values during one oscillation change

weight, the larger the slip velocity.

2509. Copyright Elsevier.

characterized by a power-law behavior. The stick-slip instability appears in region II; vertical bars represent the amplitude of the pressure oscillations. Region III begins at an apparent shear rate of *652 s-*1 and is characterized by the presence of lower amplitude and much faster pressure oscillations than those in region II, which might be related to the onset of the gross melt fracture regime.

Fig. 10. a) Flow curve for HDPE obtained from rheometrical measurements and by the integration of the velocity profiles. b) Velocity profiles in the capillary for different flow conditions. Reprinted with permission from Rodríguez-González et al., *Chemical Engineering Science*, Vol. 64, No. 22, (November 2009), pp. 4675-4683, ISSN 0003-2509. Copyright Elsevier.

The velocity profiles for the different apparent shear rates in region I are shown in Fig. 10b along with the profiles calculated by using Eq. 7. Also in this case, the flow rate data obtained from the integration of the velocity profiles are included in the flow curve in Fig.

characterized by a power-law behavior. The stick-slip instability appears in region II; vertical bars represent the amplitude of the pressure oscillations. Region III begins at an apparent shear rate of *652 s-*1 and is characterized by the presence of lower amplitude and much faster pressure oscillations than those in region II, which might be related to the onset

a)

b)

The velocity profiles for the different apparent shear rates in region I are shown in Fig. 10b along with the profiles calculated by using Eq. 7. Also in this case, the flow rate data obtained from the integration of the velocity profiles are included in the flow curve in Fig.

Fig. 10. a) Flow curve for HDPE obtained from rheometrical measurements and by the integration of the velocity profiles. b) Velocity profiles in the capillary for different flow conditions. Reprinted with permission from Rodríguez-González et al., *Chemical Engineering* 

*Science*, Vol. 64, No. 22, (November 2009), pp. 4675-4683, ISSN 0003-2509. Copyright

of the gross melt fracture regime.

Elsevier.

10b. Again, the velocity profiles are very well described by the power-law relationship and there is a good agreement between the rheometrical and PIV data in region I. The velocity profiles in this flow regime do not show slip, even though it is known that HDPE may slip at the die wall. This may be explained by a relatively low molecular weight of the polymer. The magnitude of slip depends on the polymer molecular weight. The higher the molecular weight, the larger the slip velocity.

Fig. 11. a) Minimum and maximum velocity profiles during one pressure oscillation at an apparent shear rate of 297 s-1. b) Evolution of pressure, maximum velocity and slip velocity for the same apparent shear rate. Reprinted with permission from Rodríguez-González et al., *Chemical Engineering Science*, Vol. 64, No. 22, (November 2009), pp. 4675-4683, ISSN 0003- 2509. Copyright Elsevier.

The pressure signal and the flow maps were recorded simultaneously in the stick-slip regime. The minimum and maximum velocity profiles during one pressure oscillation at an apparent shear rate of *297 s-1* are shown in Fig. 11a. There is a large difference between the maxima of the two profiles, *vmin* and *vmax*, and their values during one oscillation change

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 223

a)

b)

Fig. 12. a) Flow curves for the mLLDPE with and without FPPA. b) Slip velocity as a function of wall shear stress calculated by using Eq. 9 and from velocity profiles.

154, ISSN 0035-4511. Copyright Springer-Verlag.

Continuous lines represent kink functions (Shaw, 2007). Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2, (February 2010), pp. 145-

regions with and without slip at the die wall. This new characteristic of the stick-slip flow was recently discovered by Rodríguez-González et al. (2009) in HDPE and it is nicely visualized in the photograph of the extrudate in Fig. 14b. It is noteworthy that these peculiar details of the stick-slip flow may only be detected by instantaneous measurements of the flow field. The detection of this feature of the stick-slip flow is not possible by one point measurements, like in LDV in previous works, but it is allowed by the instantaneous recording of the PIV maps. In fact, the existence of slip during the oscillations can not be assured by LVD, unless a full velocity profile across the plane of interest is constructed.

from *vmin=0.055 m/s* up to *vmax=0.151 m/s*. Similar variations in velocity are typical of other apparent shear rates in the stick-slip regime. In addition, the velocity profiles in Fig. 11a show that the boundary conditions change from stick to slip at the capillary wall, in agreement with the term, stick-slip, used to describe this oscillating phenomenon.

Figure 11b shows the evolution of pressure together with the maximum velocity and slip velocity an apparent shear rate of *297 s-1*. Note that both, the maximum velocity and the slip velocity, increase continuously during the stick part of the oscillations. In contrast, at the beginning of the slip part of the cycle both velocities rise steeply, reach a maximum and then decrease abruptly up to a minimum at the end of the cycle.

In the high shear rate branch, pressure oscillations decreased in amplitude and became much faster than those in region II. The high frequency of the oscillations in this case did not permit the recording of consecutive velocity maps to describe one full cycle with good resolution (see Rodríguez-González et al, 2009).

### **6.2.2 Linear low-density polyethylene**

The flow curves for the mLLDPE with and without additive are shown in Fig. 12. The flow curve corresponding to the pure polymer resembles that of Fig. 10a; only regions and I-II were explored. Region I corresponds to a stable flow regime that cannot be well fitted by a simple power-law. Instead, a more complex constitutive equation seems to describe the melt behavior. The stick-slip instability appears in region II, the amplitude of pressure oscillations are represented by double points.

Figure 12a also includes the flow curve obtained for the mLLDPE containing additive. In this case, an increase in the flow rate with respect to the pure polymer is evident, as well as the absence of the stick-slip regime. This increase results from interfacial slip between the polymer melt and FPPA. It is noteworthy here that the flow curve for the mLLDPE containing additive also deviates from a power-law and should be fitted by a more complex constitutive equation (Rodríguez-González et al., 2010).

*vs* was calculated for the mLLDPE+FPPA system by comparing the data from both flow curves in Fig. 12a using Eq. 9, To use this equation, only stable data from the pure polymer before the stick-slip were considered, assuming them as free of slip. The slip velocity as a function of the shear stress is shown in Fig. 12b. The slip velocity increases with the shear stress and follows a power-law behavior at low shear stresses, but the trend changes as the shear stress is further increased. This change of the rate of increase of the slip velocity along with the shear stress has been attributed to shear thinning of the melt (Pérez-González & de Vargas, 2002).

Figure 13 shows the velocity profiles for the different apparent shear rates in region I. According to the discussion in the previous section, a single power-law relationship was not appropriate to describe all of these profiles. Note that the velocity profiles in this flow regime extrapolate to a zero value at the die wall (no slip), which supports the assumption made in using Eq. 9.

The main results for this mLLDPE agree with those described for HDPE in the previous section. However, an issue to highlight in the analysis of the stick-slip flow of this polymer is the appearance of non-homogeneous slip (Fig. 14a), i.e., the simultaneous appearance of

from *vmin=0.055 m/s* up to *vmax=0.151 m/s*. Similar variations in velocity are typical of other apparent shear rates in the stick-slip regime. In addition, the velocity profiles in Fig. 11a show that the boundary conditions change from stick to slip at the capillary wall, in

Figure 11b shows the evolution of pressure together with the maximum velocity and slip velocity an apparent shear rate of *297 s-1*. Note that both, the maximum velocity and the slip velocity, increase continuously during the stick part of the oscillations. In contrast, at the beginning of the slip part of the cycle both velocities rise steeply, reach a maximum and then

In the high shear rate branch, pressure oscillations decreased in amplitude and became much faster than those in region II. The high frequency of the oscillations in this case did not permit the recording of consecutive velocity maps to describe one full cycle with good

The flow curves for the mLLDPE with and without additive are shown in Fig. 12. The flow curve corresponding to the pure polymer resembles that of Fig. 10a; only regions and I-II were explored. Region I corresponds to a stable flow regime that cannot be well fitted by a simple power-law. Instead, a more complex constitutive equation seems to describe the melt behavior. The stick-slip instability appears in region II, the amplitude of pressure

Figure 12a also includes the flow curve obtained for the mLLDPE containing additive. In this case, an increase in the flow rate with respect to the pure polymer is evident, as well as the absence of the stick-slip regime. This increase results from interfacial slip between the polymer melt and FPPA. It is noteworthy here that the flow curve for the mLLDPE containing additive also deviates from a power-law and should be fitted by a more complex

*vs* was calculated for the mLLDPE+FPPA system by comparing the data from both flow curves in Fig. 12a using Eq. 9, To use this equation, only stable data from the pure polymer before the stick-slip were considered, assuming them as free of slip. The slip velocity as a function of the shear stress is shown in Fig. 12b. The slip velocity increases with the shear stress and follows a power-law behavior at low shear stresses, but the trend changes as the shear stress is further increased. This change of the rate of increase of the slip velocity along with the shear stress has been attributed to shear thinning of the melt (Pérez-González & de

Figure 13 shows the velocity profiles for the different apparent shear rates in region I. According to the discussion in the previous section, a single power-law relationship was not appropriate to describe all of these profiles. Note that the velocity profiles in this flow regime extrapolate to a zero value at the die wall (no slip), which supports the assumption

The main results for this mLLDPE agree with those described for HDPE in the previous section. However, an issue to highlight in the analysis of the stick-slip flow of this polymer is the appearance of non-homogeneous slip (Fig. 14a), i.e., the simultaneous appearance of

agreement with the term, stick-slip, used to describe this oscillating phenomenon.

decrease abruptly up to a minimum at the end of the cycle.

resolution (see Rodríguez-González et al, 2009).

oscillations are represented by double points.

Vargas, 2002).

made in using Eq. 9.

constitutive equation (Rodríguez-González et al., 2010).

**6.2.2 Linear low-density polyethylene** 

Fig. 12. a) Flow curves for the mLLDPE with and without FPPA. b) Slip velocity as a function of wall shear stress calculated by using Eq. 9 and from velocity profiles. Continuous lines represent kink functions (Shaw, 2007). Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2, (February 2010), pp. 145- 154, ISSN 0035-4511. Copyright Springer-Verlag.

regions with and without slip at the die wall. This new characteristic of the stick-slip flow was recently discovered by Rodríguez-González et al. (2009) in HDPE and it is nicely visualized in the photograph of the extrudate in Fig. 14b. It is noteworthy that these peculiar details of the stick-slip flow may only be detected by instantaneous measurements of the flow field. The detection of this feature of the stick-slip flow is not possible by one point measurements, like in LDV in previous works, but it is allowed by the instantaneous recording of the PIV maps. In fact, the existence of slip during the oscillations can not be assured by LVD, unless a full velocity profile across the plane of interest is constructed.

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 225

a)

b) Fig. 15. Velocity profiles obtained for the mLLDPE under strong slip conditions at a) low and b) high shear rates. Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2, (February 2010), pp. 145-154, ISSN 0035-4511. Copyright

A comparison of the slip velocity calculated by using the kink function (Eq. 11) and a power-law (see Fig. 12b) at *τw= 0.443 MPa* leads to an overestimation of *23%* in *vs* when using the power-law model. Considering the trend in Fig. 12b for the slip velocity, the error introduced by using a power-law model for this polymer will become even more significant

The extrusion of polyolefins of significant practical importance, namely, LDPE, HDPE, PP and LLDPE, was analyzed in this work by using simultaneous rheometrical and twodimensional particle image velocimetry measurements (2D PIV), or what has been called

Springer-Verlag.

**7. Conclusion** 

at higher shear stresses.

Fig. 13. Velocity profiles for the different apparent shear rates in region I of the mLLDPE flow curve. Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2, (February 2010), pp. 145-154, ISSN 0035-4511. Copyright Springer-Verlag.

Fig. 14. Non-homogeneous slip characterized by a) velocity map and b) extrudate. Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2, (February 2010), pp. 145-154, ISSN 0035-4511. Copyright Springer-Verlag.

Figures 15a-b show the velocity profiles obtained for the mLLDPE+FPPA. In contrast to the observed for the pure polymer, the velocity profiles in Figs. 15a-b exhibit a non-zero velocity at the die wall, whose magnitude increases along with the apparent shear rate (shear stress). The slip velocity values obtained from the extrapolation of the velocity profiles at the die wall are plotted in Fig. 12b along with those calculated from the rheometrical data by using Eq. 9. The agreement between both sets of data is remarkable, which provides a direct proof for the validity of Eq. 9.

The relationship between the slip velocity and wall shear stress in Fig. 12b clearly deviates from the power-law behavior at a shear stress above *0.30 MPa*. Then, a more realistic equation, that could be used in numerical calculations, may be obtained by fitting the data to a continuous "kink" function (Shaw, 2007):

$$\log v\_s = -1.4276 + 1.7716(\log \tau\_w - 5.5202) + 0.0078(1.11 - 1.7716)\ln \left[ 1 + \exp\left[\frac{\log \tau\_w - 5.5202}{0.0078}\right] \right] (11)$$

Fig. 15. Velocity profiles obtained for the mLLDPE under strong slip conditions at a) low and b) high shear rates. Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2, (February 2010), pp. 145-154, ISSN 0035-4511. Copyright Springer-Verlag.

A comparison of the slip velocity calculated by using the kink function (Eq. 11) and a power-law (see Fig. 12b) at *τw= 0.443 MPa* leads to an overestimation of *23%* in *vs* when using the power-law model. Considering the trend in Fig. 12b for the slip velocity, the error introduced by using a power-law model for this polymer will become even more significant at higher shear stresses.

#### **7. Conclusion**

224 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Fig. 13. Velocity profiles for the different apparent shear rates in region I of the mLLDPE flow curve. Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol.

Fig. 14. Non-homogeneous slip characterized by a) velocity map and b) extrudate. Reprinted with permission from Rodríguez-González et al. (2010). *Rheologica Acta*, Vol. 49, No. 2,

Figures 15a-b show the velocity profiles obtained for the mLLDPE+FPPA. In contrast to the observed for the pure polymer, the velocity profiles in Figs. 15a-b exhibit a non-zero velocity at the die wall, whose magnitude increases along with the apparent shear rate (shear stress). The slip velocity values obtained from the extrapolation of the velocity profiles at the die wall are plotted in Fig. 12b along with those calculated from the rheometrical data by using Eq. 9. The agreement between both sets of data is remarkable, which provides a direct proof

The relationship between the slip velocity and wall shear stress in Fig. 12b clearly deviates from the power-law behavior at a shear stress above *0.30 MPa*. Then, a more realistic equation, that could be used in numerical calculations, may be obtained by fitting the data

(11)

 log 5.5202 log 1.4276 1.7716(log 5.5202) 0.0078(1.11 1.7716)ln 1 exp 0.0078 *<sup>w</sup> s w v*

(February 2010), pp. 145-154, ISSN 0035-4511. Copyright Springer-Verlag.

for the validity of Eq. 9.

to a continuous "kink" function (Shaw, 2007):

49, No. 2, (February 2010), pp. 145-154, ISSN 0035-4511. Copyright Springer-Verlag.

The extrusion of polyolefins of significant practical importance, namely, LDPE, HDPE, PP and LLDPE, was analyzed in this work by using simultaneous rheometrical and twodimensional particle image velocimetry measurements (2D PIV), or what has been called

Rheo-Particle Image Velocimetry for the Analysis of the Flow of Polymer Melts 227

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### **8. Acknowledgment**

This work was supported by SIP-IPN (20111119). J. P.-G. and J. G. G.-S. are COFFA-EDI fellows. B. M. M.-S. had a fellowship under the CONACyT's program for the *Apoyos Complementarios para la Consolidación Institucional de Grupos de Investigación* (Sol. 147970).

### **9. References**


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This work was supported by SIP-IPN (20111119). J. P.-G. and J. G. G.-S. are COFFA-EDI fellows. B. M. M.-S. had a fellowship under the CONACyT's program for the *Apoyos Complementarios para la Consolidación Institucional de Grupos de Investigación* (Sol. 147970).

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**8. Acknowledgment** 

**9. References** 


**9** 

*Australia* 

**PIV as a Complement to LDA in the Study of an** 

In the last decade, particle image velocimetry (PIV) has become a standard laser diagnostic tool in numerous fluid mechanics laboratories worldwide. At first glance, it appears straight-forward to set up an off-the-shelf system for a quick investigation of the flow of interest. However, unless additional effort is taken to understand the flow, results from 'quick investigations' may lead to limited and sometimes spurious interpretation of the physical flow phenomena. This is especially so for highly three-dimensional and turbulent flows. This chapter examines the efficacy of phase-averaged particle image velocimetry results in assessing the physical phenomena occurring in highly periodic flows and how they complement results from phase-averaged laser Doppler anemometry (LDA) and surface flow visualisation techniques. A specific case study will be presented to demonstrate

The selected flow, a fluidic precessing jet, is a turbulent and highly three-dimensional jet that is used as a fluid mixing device in a combustion lance, or "burner", in rotary kilns. It has been found to achieve low-NOx (Oxides of Nitrogen; a type of Greenhouse gas) emissions as a gas-fired burner, developed by researchers at the University of Adelaide (Luxton & Nathan, 1988). The flow field lowers flame temperatures by reducing flame strain, which enhances soot formation (Nathan et al., 2006). This reduces NOx emissions by 30-40% in typical cement kilns (Manias & Nathan, 1994) compared with conventional kiln burners (free jet burners), and its enhanced radiation heat transfer also improves the product quality (Manias & Nathan, 1994) and output (Videgar, 1997) of the clinker in rotary cement kilns. Specific fuel savings of approximately 3-6% were typically reported (Videgar, 1997). The patented burner (hereinafter called the 'fluidic precessing jet' or FPJ nozzle) is commercially known as the GyrothermTM and is based on a geometrically simple nozzle

Although the FPJ nozzle is simple in design, the flow within and emerging from the nozzle is unsteady and highly three-dimensional (Fig. 1). Several experimental FPJ nozzles were developed to study the fundamental characteristics of the precessing jet. Some of the experiments employed classical flow visualisation techniques, such as particle-tracing (using glass beads or gas bubbles), shadowgraph, smoke, cotton tufts, coloured dyes and Chinaclay surface flow visualisations (Nathan, 1988), as well as quantitative methods including

**1. Introduction** 

configuration.

the complementary nature of these techniques.

 *University of Adelaide,* 

**Unsteady Oscillating Turbulent Flow** 

Chong Y. Wong\*, Graham J. Nathan, Richard M Kelso

*\*Now at CSIRO Process Science and Engineering,* 

*Time-Dependent Materials,* Vol. 3, No. 4, (December 1999), pp. 371-393, ISSN 1385- 2000.


## **PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow**

Chong Y. Wong\*, Graham J. Nathan, Richard M Kelso  *University of Adelaide, \*Now at CSIRO Process Science and Engineering, Australia* 

### **1. Introduction**

228 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

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data, *Experiments in Fluids*, Vol. 16, No. 3-4, (February 1994), pp. 236-247, ISSN 0723-

In the last decade, particle image velocimetry (PIV) has become a standard laser diagnostic tool in numerous fluid mechanics laboratories worldwide. At first glance, it appears straight-forward to set up an off-the-shelf system for a quick investigation of the flow of interest. However, unless additional effort is taken to understand the flow, results from 'quick investigations' may lead to limited and sometimes spurious interpretation of the physical flow phenomena. This is especially so for highly three-dimensional and turbulent flows. This chapter examines the efficacy of phase-averaged particle image velocimetry results in assessing the physical phenomena occurring in highly periodic flows and how they complement results from phase-averaged laser Doppler anemometry (LDA) and surface flow visualisation techniques. A specific case study will be presented to demonstrate the complementary nature of these techniques.

The selected flow, a fluidic precessing jet, is a turbulent and highly three-dimensional jet that is used as a fluid mixing device in a combustion lance, or "burner", in rotary kilns. It has been found to achieve low-NOx (Oxides of Nitrogen; a type of Greenhouse gas) emissions as a gas-fired burner, developed by researchers at the University of Adelaide (Luxton & Nathan, 1988). The flow field lowers flame temperatures by reducing flame strain, which enhances soot formation (Nathan et al., 2006). This reduces NOx emissions by 30-40% in typical cement kilns (Manias & Nathan, 1994) compared with conventional kiln burners (free jet burners), and its enhanced radiation heat transfer also improves the product quality (Manias & Nathan, 1994) and output (Videgar, 1997) of the clinker in rotary cement kilns. Specific fuel savings of approximately 3-6% were typically reported (Videgar, 1997). The patented burner (hereinafter called the 'fluidic precessing jet' or FPJ nozzle) is commercially known as the GyrothermTM and is based on a geometrically simple nozzle configuration.

Although the FPJ nozzle is simple in design, the flow within and emerging from the nozzle is unsteady and highly three-dimensional (Fig. 1). Several experimental FPJ nozzles were developed to study the fundamental characteristics of the precessing jet. Some of the experiments employed classical flow visualisation techniques, such as particle-tracing (using glass beads or gas bubbles), shadowgraph, smoke, cotton tufts, coloured dyes and Chinaclay surface flow visualisations (Nathan, 1988), as well as quantitative methods including

PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow 231

(Nathan, Hill & Luxton, 1998). Hill (1992) reported that the axial jet mode is greatly suppressed by use of a centrebody positioned near to the exit of the chamber. The presence of a centrebody blocks the path of the axial jet. Consequently, some of the flow becomes redirected back into the nozzle while the majority of the flow by-passes the centrebody and emerges in an asymmetric fashion from the exit plane resulting in the precessing-jet mode. Therefore, in order to increase the probability of the precessing jet mode for all the phaseaveraging experiments, a centrebody arrangement is used in the investigation reported here.

Fig. 1. Perspex-wire model of the conjectured flow topology from flow visualisation studies

Asymmetric Reattachment

Induced Ambient Air

Precessing Jet

Nathan and Luxton (1992) reported that a precessional instability occurs in an axisymmetric duct when the expansion ratio (*E*=*D1*/*d*) of a sudden expansion at its inlet is sufficiently large (ie. *E*>5). In addition, they noted that the chamber's length-to-diameter ratio must be about *L/D1*=2.7 and that the Reynolds number based on the inlet conditions be sufficiently high, with the precessional instability becoming dominant above a critical value (*Red*>20,000) for precession with a deflected jet to occur. Hill, Nathan and Luxton (1995) investigated, in water, the flow through an axisymmetric large sudden expansion into a long downstream duct. They found that jet precession can be generally described by the axial momentum (*M*), duct diameter (*D1*) and the fluid properties (i.e. density and kinematic viscosity) of the inlet fluid. In quantifying their results for duct diameters of *D1*=60mm and *D1*=140mm for varying expansion ratios, using a video camera with a framing rate of 30Hz, they visually counted (frame by frame) the frequency of precession by noting the oscillations in the seeded jet in water. Their inlet flow was seeded with 0.6mm diameter neutral-density polystyrene beads. The minimum precession cycle count was *Np*=3 for *E*=3.75 at *Red*=4400 while the maximum count recorded was *Np*=107 for *E*=14.2 at *Red*=56500. The small sample size, used for the low *Red* number experiments, meant that the result was not statistically conclusive. Despite the measurement difficulties, especially for the low *Red* flows, a useful relationship between expansion ratio *E* and Reynolds number *Red* was established for

(reprinted with permission and adapted from Nathan, 1988).

Three-dimensional Separation

Dependence on inlet flow conditions

Cavity Swirl

suddenly-expanded flows into long ducts.

yaw-probe meters, hot-wire anemometers, pressure probes, an entrainment shroud (Nathan, 1988), planar laser induced fluorescence (PLIF), OH-PLIF and particle image velocimetry (PIV) (Newbold, 1997; Nobes, 1997). Nevertheless, many details of the flow structure eluded researchers for some two decades and it is only recently that its phase-averaged flow structure has been revealed through the use of laser diagnostic techniques such as phaseaveraged PIV and LDA, complemented by a classical surface flow visualisation technique (Wong, Nathan & Kelso, 2008).

The chapter presents a summary both of the flow field itself and of the approaches used to investigate it. Section 2 provides details of what is known about the fluidic precessing jet flow and explains how a qualitative understandaing of the flow was used to develop a methodology for a systematic investigation of it. Finally section 3 discusses the key results and constructs a qualitative image of the flow based on these results.

### **2. A case study of an unsteady oscillating turbulent flow**

This section presents a case study in the use of PIV to investigate a fluidic precessing jet (FPJ). As discussed earlier, the FPJ produces an unsteady, oscillating turbulent flow with unique fluid mechanical features. In order to apply any laser diagnostic technique, it is necessary to understand the characteristic features of the flow so that an appropriate experimental strategy can be planned.

### **2.1 Characteristic of a fluidic precessing jet**

The FPJ nozzle, introduced above, comprises a cylindrical chamber with a diameter *D1* and a length *L* (~3*D1*) with a small axisymmetric inlet (*d*=15.79mm) at one end and an exit lip (*D2*) at the other (Fig. 3). Referring to Fig. 1, the inlet flow forms a central jet which in the jet precession mode, reattaches non-preferentially onto the curved wall of the chamber (Nathan, 1988). As a result of flow instabilities setup within the chamber, the flow precesses around the chamber wall (Nathan & Luxton, 1992). A region of swirl is formed at the upstream end of the chamber, approximately *x/L*  1/6. A larger recirculation region is also observed to feed the swirling region, originating from near to the lip. The fluid in the swirling flow is found to comprise contributions both from the recirculated fluid from the main jet and a typically smaller contribution by induced ambient fluid (Nathan, 1988; Nathan, Hill & Luxton, 1998; Parham, 2000). As the jet exits, it does not completely occupy the exit plane (Nathan & Luxton, 1992). The exit flow is then directed through a large angle (typically =45 from the nozzle axis) towards the axis and across the face of the nozzle outlet. The emerging flow is also highly three-dimensional and the precession extends for several chamber diameters downstream from the exit plane (Nathan, 1988).

#### **2.1.1 Mode switching nature**

Nathan, Hill and Luxton (1998) reported on the mode-switching behaviour of the precessing jet. They identified two major flow modes: an axial-jet mode and a precessing-jet mode. The corresponding conjectured flow patterns for the flow in either mode are also given in Nathan, Hill and Luxton (1998). When in the axial jet mode, the inlet jet emerges from the exit plane without significantly attaching to the inner walls of the chamber. A symmetric region of recirculating flow on the upper and lower sides of the jet is clearly observed (Nathan, Hill & Luxton, 1998). Hill (1992) reported that the axial jet mode is greatly suppressed by use of a centrebody positioned near to the exit of the chamber. The presence of a centrebody blocks the path of the axial jet. Consequently, some of the flow becomes redirected back into the nozzle while the majority of the flow by-passes the centrebody and emerges in an asymmetric fashion from the exit plane resulting in the precessing-jet mode. Therefore, in order to increase the probability of the precessing jet mode for all the phaseaveraging experiments, a centrebody arrangement is used in the investigation reported here.

Fig. 1. Perspex-wire model of the conjectured flow topology from flow visualisation studies (reprinted with permission and adapted from Nathan, 1988).

#### Dependence on inlet flow conditions

230 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

yaw-probe meters, hot-wire anemometers, pressure probes, an entrainment shroud (Nathan, 1988), planar laser induced fluorescence (PLIF), OH-PLIF and particle image velocimetry (PIV) (Newbold, 1997; Nobes, 1997). Nevertheless, many details of the flow structure eluded researchers for some two decades and it is only recently that its phase-averaged flow structure has been revealed through the use of laser diagnostic techniques such as phaseaveraged PIV and LDA, complemented by a classical surface flow visualisation technique

The chapter presents a summary both of the flow field itself and of the approaches used to investigate it. Section 2 provides details of what is known about the fluidic precessing jet flow and explains how a qualitative understandaing of the flow was used to develop a methodology for a systematic investigation of it. Finally section 3 discusses the key results

This section presents a case study in the use of PIV to investigate a fluidic precessing jet (FPJ). As discussed earlier, the FPJ produces an unsteady, oscillating turbulent flow with unique fluid mechanical features. In order to apply any laser diagnostic technique, it is necessary to understand the characteristic features of the flow so that an appropriate

The FPJ nozzle, introduced above, comprises a cylindrical chamber with a diameter *D1* and a length *L* (~3*D1*) with a small axisymmetric inlet (*d*=15.79mm) at one end and an exit lip (*D2*) at the other (Fig. 3). Referring to Fig. 1, the inlet flow forms a central jet which in the jet precession mode, reattaches non-preferentially onto the curved wall of the chamber (Nathan, 1988). As a result of flow instabilities setup within the chamber, the flow precesses around the chamber wall (Nathan & Luxton, 1992). A region of swirl is formed at the upstream end of the chamber, approximately *x/L*  1/6. A larger recirculation region is also observed to feed the swirling region, originating from near to the lip. The fluid in the swirling flow is found to comprise contributions both from the recirculated fluid from the main jet and a typically smaller contribution by induced ambient fluid (Nathan, 1988; Nathan, Hill & Luxton, 1998; Parham, 2000). As the jet exits, it does not completely occupy the exit plane (Nathan & Luxton, 1992). The exit flow is then directed through a large angle

=45 from the nozzle axis) towards the axis and across the face of the nozzle

outlet. The emerging flow is also highly three-dimensional and the precession extends for

Nathan, Hill and Luxton (1998) reported on the mode-switching behaviour of the precessing jet. They identified two major flow modes: an axial-jet mode and a precessing-jet mode. The corresponding conjectured flow patterns for the flow in either mode are also given in Nathan, Hill and Luxton (1998). When in the axial jet mode, the inlet jet emerges from the exit plane without significantly attaching to the inner walls of the chamber. A symmetric region of recirculating flow on the upper and lower sides of the jet is clearly observed

several chamber diameters downstream from the exit plane (Nathan, 1988).

and constructs a qualitative image of the flow based on these results.

**2. A case study of an unsteady oscillating turbulent flow** 

(Wong, Nathan & Kelso, 2008).

experimental strategy can be planned.

(typically

**2.1.1 Mode switching nature** 

**2.1 Characteristic of a fluidic precessing jet** 

Nathan and Luxton (1992) reported that a precessional instability occurs in an axisymmetric duct when the expansion ratio (*E*=*D1*/*d*) of a sudden expansion at its inlet is sufficiently large (ie. *E*>5). In addition, they noted that the chamber's length-to-diameter ratio must be about *L/D1*=2.7 and that the Reynolds number based on the inlet conditions be sufficiently high, with the precessional instability becoming dominant above a critical value (*Red*>20,000) for precession with a deflected jet to occur. Hill, Nathan and Luxton (1995) investigated, in water, the flow through an axisymmetric large sudden expansion into a long downstream duct. They found that jet precession can be generally described by the axial momentum (*M*), duct diameter (*D1*) and the fluid properties (i.e. density and kinematic viscosity) of the inlet fluid. In quantifying their results for duct diameters of *D1*=60mm and *D1*=140mm for varying expansion ratios, using a video camera with a framing rate of 30Hz, they visually counted (frame by frame) the frequency of precession by noting the oscillations in the seeded jet in water. Their inlet flow was seeded with 0.6mm diameter neutral-density polystyrene beads. The minimum precession cycle count was *Np*=3 for *E*=3.75 at *Red*=4400 while the maximum count recorded was *Np*=107 for *E*=14.2 at *Red*=56500. The small sample size, used for the low *Red* number experiments, meant that the result was not statistically conclusive. Despite the measurement difficulties, especially for the low *Red* flows, a useful relationship between expansion ratio *E* and Reynolds number *Red* was established for suddenly-expanded flows into long ducts.

PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow 233

They added that following the reappearance of the flow patterns the precession was reported to sometimes reverse direction in the azimuthal direction. That observation was similarly reported in the numerical simulations of Guo (2000) who explained that the change in precession direction was the result of the phase interaction between the pressure gradient

Summarising the previous section, the emerging precessing flow is highly threedimensional and unsteady, with both the exit angle and precession frequency exhibiting considerable cycle-to-cycle variation. To complicate matters, the precession direction and flow mode also change intermittently with time. These flow variations make the flow challenging to study. Laser-based techniques such as LDA and PIV, which were chosen to study the flow field due to their minimally-intrusive nature and capacity to resolve the flow direction, nevertheless have a limited dynamic range. Hence, the simple application of these techniques to such a flow will lead to large uncertainties and also would not provide much information of the 'instantaneous' or phase-averaged structure of the flow. To resolve the flow in a way that accounts for the variations in flow mode, exit angle, precession direction and phase, requires the use of additional flow sensors to condition, or select, the measurements obtained with these laser-based techniques. This type of conditioning allows "phase averaged" data to be recorded, as used successfully by Fick, Griffiths and O'Doherty (1997) to study the precessing vortex core in swirl burners and by Fernandes and Heitor (1998) to measure oscillating flames. Triggering of a PIV data collection system by pressure probes has also been used by Fick, Griffiths and O'Doherty (1997) to study the precessing vortex core near the exit of a swirl burner. A key element to the success of that measurement was the predictability of the precession direction of the PVC. The naturally-excited fluidic precessing jet studied here not only precesses, but is known to change direction intermittently for the Chamber-Lip (Ch-L) configuration (Nathan, 1988). Frequent directional changes in the emerging flow for a Chamber-Lip-Centrebody (Ch-L-CB) configuration were also observed in the LDA tangential velocity measurements (Wong, Nathan & O'Doherty, 2004). It is also numerically predicted by Guo, Langrish and Fletcher (2001) in a long pipe downstream from a sudden expansion inlet. However, those techniques did not require the resolution of precession direction since the directions of these oscillations are predetermined by the physical geometries of the burners. Hence their triggering devices required only one phase sensor. In investigating the unsteady FPJ flow, both precession direction and phase information should be accounted for to provide a

driving the precession and the jet momentum.

suitable trigger to resolve the phase-averaged structure of the jet.

**2.2.1 Geometric configuration of the fluidic precessing jet nozzles** 

The configuration of the FPJ nozzles investigated here is based on the specifications proposed by Hill, Nathan and Luxton (1992) and Hill (1992) for reliable jet precession. A total of nine configurations, each of which give rise to a precessing jet flow, were studied by Wong, Nathan & O'Doherty (2004). Three different inlet conditions were combined with three alternative ways of arranging the FPJ chamber (Wong, Nathan & O'Doherty, 2004). Of these nine, the Ch-L-CB configuration was chosen for further studies because its inlet flow is

**2.2 Experimental methodology** 

#### **2.1.2 Jet precession and its dependence on nozzle configuration**

Mi and Nathan (2000) conducted a parametric study to determine the influence on precession frequency of the systematic variation of key dimensional parameters. Their single stationary hot-wire study concluded that jet precession frequency was mainly a function of the chamber's length-to-diameter, lip-to-diameter, and centrebody-position-to-diameter ratios, further supporting the study conducted earlier by Nathan and Luxton (1992). Other factors included the size of the centrebody (CB), the size of the exit lip, the presence or absence of a CB, and the presence or absence of a lip. They noted that different inlet conditions, such as an orifice or a contraction inlet, also influenced precession frequency.

### **2.1.3 Variation of exit angles in the emerging precessing jet**

Nathan (1988), using flow visualization, documented that the exit angle of the emerging jet (Fig. 1) can vary from =30 to =60 relative to the geometric centreline. This adds even more variability to the flow and renders measurements in the flow more difficult. The influence of the jet exit angle on the downstream flow has been examined in further detail by Nobes (1997) who used a mechanically-rotated precessing jet (MPJ). He found that increasing the jet exit angle increases the spread of the jet helix and this results in an increase in the initial mixing rate in the jet. For every given angle, there is a critical Strouhal number, above which the jet converges to the axis of rotation and, below which, it does not. For an exit angle of 45, *Stp,cr* 0.008 (Mi and Nathan, 2005), where *St* = *fp de* / *ue*, where *fp* is the frequency of precession, *de* is the nozzle exit diameter and *ue* is the bulk mean exit velocity.

#### **2.1.4 Three-dimensional nature of flow**

The flow reversals within the FPJ chamber are caused by the adverse pressure gradient downstream from the expansion face and the large expansion (Nathan, Hill & Luxton, 1998). However, this reverse flow is not only turbulent, it also exhibits the large-scale coherent precession motion. This distinguishes it from the relatively steady asymmetry that has been observed in channel flows with similar expansion ratios (Ouwa, Watanabe & Asawo, 1981). The investigation of this three-dimensional oscillation in absolute flow direction requires a measurement technique that is able to differentiate direction in the flow velocity, as well as other standard requirements, including adequate spatial resolution and the need to minimise disturbing the flow field of interest. This rules out the use of stationary hot-wire anemometers and pressure probes, which are not able to resolve such turbulent flows reliably. Flying hot-wires, although able to resolve such unsteady flows, are very complex and are difficult to apply to flows within confined cavities. Laser diagnostic tools such as LDA and PIV are considered to be ideal because they are non-intrusive, do not suffer from directional ambiguity when measuring flow velocities (Durst, Melling & Whitelaw, 1981) and can be applied to flows within cavities when adequate optical access is provided.

#### **2.1.5 Bi-directional azimuthal direction**

In addition to the oscillation in axial direction of the flow with each cycle, the azimuthal direction of the entire flow can change. Nathan and Luxton (1992), while conducting surface flow visualisation experiments in an FPJ chamber, reported that "for no apparent reason, the surface flow patterns were occasionally destroyed for a short time and then reappeared". They added that following the reappearance of the flow patterns the precession was reported to sometimes reverse direction in the azimuthal direction. That observation was similarly reported in the numerical simulations of Guo (2000) who explained that the change in precession direction was the result of the phase interaction between the pressure gradient driving the precession and the jet momentum.

#### **2.2 Experimental methodology**

232 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

Mi and Nathan (2000) conducted a parametric study to determine the influence on precession frequency of the systematic variation of key dimensional parameters. Their single stationary hot-wire study concluded that jet precession frequency was mainly a function of the chamber's length-to-diameter, lip-to-diameter, and centrebody-position-to-diameter ratios, further supporting the study conducted earlier by Nathan and Luxton (1992). Other factors included the size of the centrebody (CB), the size of the exit lip, the presence or absence of a CB, and the presence or absence of a lip. They noted that different inlet conditions, such as an orifice or a contraction inlet, also influenced precession frequency.

Nathan (1988), using flow visualization, documented that the exit angle of the emerging jet

more variability to the flow and renders measurements in the flow more difficult. The influence of the jet exit angle on the downstream flow has been examined in further detail by Nobes (1997) who used a mechanically-rotated precessing jet (MPJ). He found that increasing the jet exit angle increases the spread of the jet helix and this results in an increase in the initial mixing rate in the jet. For every given angle, there is a critical Strouhal number, above which the jet converges to the axis of rotation and, below which, it does not. For an exit angle of 45, *Stp,cr* 0.008 (Mi and Nathan, 2005), where *St* = *fp de* / *ue*, where *fp* is the frequency of precession, *de* is the nozzle exit diameter and *ue* is the bulk mean exit velocity.

The flow reversals within the FPJ chamber are caused by the adverse pressure gradient downstream from the expansion face and the large expansion (Nathan, Hill & Luxton, 1998). However, this reverse flow is not only turbulent, it also exhibits the large-scale coherent precession motion. This distinguishes it from the relatively steady asymmetry that has been observed in channel flows with similar expansion ratios (Ouwa, Watanabe & Asawo, 1981). The investigation of this three-dimensional oscillation in absolute flow direction requires a measurement technique that is able to differentiate direction in the flow velocity, as well as other standard requirements, including adequate spatial resolution and the need to minimise disturbing the flow field of interest. This rules out the use of stationary hot-wire anemometers and pressure probes, which are not able to resolve such turbulent flows reliably. Flying hot-wires, although able to resolve such unsteady flows, are very complex and are difficult to apply to flows within confined cavities. Laser diagnostic tools such as LDA and PIV are considered to be ideal because they are non-intrusive, do not suffer from directional ambiguity when measuring flow velocities (Durst, Melling & Whitelaw, 1981) and can be applied to flows within cavities when adequate optical access is provided.

In addition to the oscillation in axial direction of the flow with each cycle, the azimuthal direction of the entire flow can change. Nathan and Luxton (1992), while conducting surface flow visualisation experiments in an FPJ chamber, reported that "for no apparent reason, the surface flow patterns were occasionally destroyed for a short time and then reappeared".

=60 relative to the geometric centreline. This adds even

**2.1.2 Jet precession and its dependence on nozzle configuration** 

**2.1.3 Variation of exit angles in the emerging precessing jet** 

=30 to

**2.1.4 Three-dimensional nature of flow** 

**2.1.5 Bi-directional azimuthal direction** 

(Fig. 1) can vary from

Summarising the previous section, the emerging precessing flow is highly threedimensional and unsteady, with both the exit angle and precession frequency exhibiting considerable cycle-to-cycle variation. To complicate matters, the precession direction and flow mode also change intermittently with time. These flow variations make the flow challenging to study. Laser-based techniques such as LDA and PIV, which were chosen to study the flow field due to their minimally-intrusive nature and capacity to resolve the flow direction, nevertheless have a limited dynamic range. Hence, the simple application of these techniques to such a flow will lead to large uncertainties and also would not provide much information of the 'instantaneous' or phase-averaged structure of the flow. To resolve the flow in a way that accounts for the variations in flow mode, exit angle, precession direction and phase, requires the use of additional flow sensors to condition, or select, the measurements obtained with these laser-based techniques. This type of conditioning allows "phase averaged" data to be recorded, as used successfully by Fick, Griffiths and O'Doherty (1997) to study the precessing vortex core in swirl burners and by Fernandes and Heitor (1998) to measure oscillating flames. Triggering of a PIV data collection system by pressure probes has also been used by Fick, Griffiths and O'Doherty (1997) to study the precessing vortex core near the exit of a swirl burner. A key element to the success of that measurement was the predictability of the precession direction of the PVC. The naturally-excited fluidic precessing jet studied here not only precesses, but is known to change direction intermittently for the Chamber-Lip (Ch-L) configuration (Nathan, 1988). Frequent directional changes in the emerging flow for a Chamber-Lip-Centrebody (Ch-L-CB) configuration were also observed in the LDA tangential velocity measurements (Wong, Nathan & O'Doherty, 2004). It is also numerically predicted by Guo, Langrish and Fletcher (2001) in a long pipe downstream from a sudden expansion inlet. However, those techniques did not require the resolution of precession direction since the directions of these oscillations are predetermined by the physical geometries of the burners. Hence their triggering devices required only one phase sensor. In investigating the unsteady FPJ flow, both precession direction and phase information should be accounted for to provide a suitable trigger to resolve the phase-averaged structure of the jet.

#### **2.2.1 Geometric configuration of the fluidic precessing jet nozzles**

The configuration of the FPJ nozzles investigated here is based on the specifications proposed by Hill, Nathan and Luxton (1992) and Hill (1992) for reliable jet precession. A total of nine configurations, each of which give rise to a precessing jet flow, were studied by Wong, Nathan & O'Doherty (2004). Three different inlet conditions were combined with three alternative ways of arranging the FPJ chamber (Wong, Nathan & O'Doherty, 2004). Of these nine, the Ch-L-CB configuration was chosen for further studies because its inlet flow is

PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow 235

Fig. 3. Coordinate system used in the PIV experiments. *d*=15.79mm, *D1*=80mm, *D2*=64mm, *Dcb*=60mm and *L*=216, tcb=16.8mm, tlip=4mm, Lcb-lip=21.6mm and rod diameter of centrebody

The TSI seeder produces a particle distribution with a modal-mean diameter of 0.6m, while

±0.5m (Kahler, Sammler and Kompenhans, 2002). Liquid droplets from the nozzle particle generator were used to seed the co-flow around the FPJ chamber and were distributed by means of a ring-type distributor located at the base of the flow conditioner. A cylindrical shroud of about 4.4 times the diameter of the FPJ nozzle was positioned such that the top edge was aligned with the exit plane of the FPJ nozzle. This is used to confine the co-flow seeding within the region of interest. The whole rig is positioned under an exhaust hood which produces a co-flow velocity of about 0.1 m/s based on PIV measurements. To reduce room draughts, all the doors and windows of the experimental laboratory (with a length, breadth

The LDA system is a Dantec two-component LDA system in the burst and back-scatter mode using a Coherent Innova 70 5-W continuous wave Argon-ion laser. In this chapter, only the axial component of velocity is reported here. The 514.5nm (Green) beam line was used. To remove directional ambiguity, one of the split beam line was frequency-shifted by 40MHz. The LDA optical head had a beam separation of 64mm and a focal length of 310mm, resulting in a probe volume with a waist diameter of 0.17mm and a length of 1.65mm. The system was mounted on a Dantec 57G15 three-axis traverse with a position accuracy of 0.05mm for all axes. Transit-time weighting method of correction was applied to all the

For phase-averaging experiments, a 3-mm diameter open-ended metal tube bevelled at 45 with its bevelled end facing upstream and protruding 10mm into the chamber was mounted half-way between the centrebody and the exit lip. The other end of the tube was connected to a pressure transducer with a 2mV/Pa sensitivity via a 300mm length of PVC tube. The

=0.026kg/(m.s)) with a nominal particle diameter of approximately 1m

90o

Exit Lip

z

Exit Plane

*x'*

a. Side View b. End View

*r*

*Dcb D2*

Lcb-lip

1800

y

=0o

=874kg/m3

270o

the nozzle particle generator uses olive oil to generate particle seeding (density,

support is 5mm.

dynamic viscosity,

**2.2.3 LDA system** 

*D1 d*

Inlet Plane Chamber Centrebody

and height of 6m, 6m and 5m respectively) were closed.

*L*

velocity data here because a burst mode data sampling was used.

uniform and well-defined, and the configuration provides reliable jet precession. Details of these are provided below.

#### **2.2.2 Experimental arrangement**

Fig. 2 shows the apparatus used in the experiment, while Fig. 3 shows the geometry of the devices and the coordinate system used for the PIV experiments. A compressor with an operating pressure of up to 650 kPa was used to deliver conditioned and compressed air to the experimental nozzle. The compressed air was regulated to maintain a constant flow rate and was divided into three sub-streams. This was done to provide a flow through the nozzle, while also providing separate seeding both to the nozzle flow and to the entrained air. This avoids the bias that would occur were only the nozzle fluid to be seeded. The first sub-stream was fed into a 6-jet particle generator (TSI Model 9306), while the second substream was diverted to a bypass valve. The two streams were re-combined at the exit of the 6-jet particle generator and transported via a flexible hose into the brass section used to condition the flow. The by-pass arrangement allows the particle generator to function optimally, whilst providing a large air flow rate to the FPJ chamber. The third sub-stream was fed into an in-house-built nozzle particle generator system.

Fig. 2. Experimental arrangement used for PIV experiments. Note that for *x'-r* plane experiments, light sheet plane is normal to camera plane as shown. For y-z plane experiments, the camera is directed at an inclined mirror positioned downstream from the FPJ exit plane (not shown here).

Fig. 3. Coordinate system used in the PIV experiments. *d*=15.79mm, *D1*=80mm, *D2*=64mm, *Dcb*=60mm and *L*=216, tcb=16.8mm, tlip=4mm, Lcb-lip=21.6mm and rod diameter of centrebody support is 5mm.

The TSI seeder produces a particle distribution with a modal-mean diameter of 0.6m, while the nozzle particle generator uses olive oil to generate particle seeding (density, =874kg/m3 dynamic viscosity, =0.026kg/(m.s)) with a nominal particle diameter of approximately 1m ±0.5m (Kahler, Sammler and Kompenhans, 2002). Liquid droplets from the nozzle particle generator were used to seed the co-flow around the FPJ chamber and were distributed by means of a ring-type distributor located at the base of the flow conditioner. A cylindrical shroud of about 4.4 times the diameter of the FPJ nozzle was positioned such that the top edge was aligned with the exit plane of the FPJ nozzle. This is used to confine the co-flow seeding within the region of interest. The whole rig is positioned under an exhaust hood which produces a co-flow velocity of about 0.1 m/s based on PIV measurements. To reduce room draughts, all the doors and windows of the experimental laboratory (with a length, breadth and height of 6m, 6m and 5m respectively) were closed.

#### **2.2.3 LDA system**

234 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

uniform and well-defined, and the configuration provides reliable jet precession. Details of

Fig. 2 shows the apparatus used in the experiment, while Fig. 3 shows the geometry of the devices and the coordinate system used for the PIV experiments. A compressor with an operating pressure of up to 650 kPa was used to deliver conditioned and compressed air to the experimental nozzle. The compressed air was regulated to maintain a constant flow rate and was divided into three sub-streams. This was done to provide a flow through the nozzle, while also providing separate seeding both to the nozzle flow and to the entrained air. This avoids the bias that would occur were only the nozzle fluid to be seeded. The first sub-stream was fed into a 6-jet particle generator (TSI Model 9306), while the second substream was diverted to a bypass valve. The two streams were re-combined at the exit of the 6-jet particle generator and transported via a flexible hose into the brass section used to condition the flow. The by-pass arrangement allows the particle generator to function optimally, whilst providing a large air flow rate to the FPJ chamber. The third sub-stream

> Exhaust Hood

> > Light shield

Laskin nozzle particle generator

Flowmeter

was fed into an in-house-built nozzle particle generator system.

Fig. 2. Experimental arrangement used for PIV experiments. Note that for *x'-r* plane experiments, light sheet plane is normal to camera plane as shown. For y-z plane

PIV camera

Computer with camera controller

Seed distributor

experiments, the camera is directed at an inclined mirror positioned downstream from the

Regulated mains air supply

TSI 6-jet particle generator

 Flow conditioner

*r*

*x'*

FPJ nozzle

these are provided below.

**2.2.2 Experimental arrangement** 

FPJ exit plane (not shown here).

SRS DG-535 Pulse Delay Generator

Nd:YAG Laser

> The LDA system is a Dantec two-component LDA system in the burst and back-scatter mode using a Coherent Innova 70 5-W continuous wave Argon-ion laser. In this chapter, only the axial component of velocity is reported here. The 514.5nm (Green) beam line was used. To remove directional ambiguity, one of the split beam line was frequency-shifted by 40MHz. The LDA optical head had a beam separation of 64mm and a focal length of 310mm, resulting in a probe volume with a waist diameter of 0.17mm and a length of 1.65mm. The system was mounted on a Dantec 57G15 three-axis traverse with a position accuracy of 0.05mm for all axes. Transit-time weighting method of correction was applied to all the velocity data here because a burst mode data sampling was used.

> For phase-averaging experiments, a 3-mm diameter open-ended metal tube bevelled at 45 with its bevelled end facing upstream and protruding 10mm into the chamber was mounted half-way between the centrebody and the exit lip. The other end of the tube was connected to a pressure transducer with a 2mV/Pa sensitivity via a 300mm length of PVC tube. The

PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow 237

DG-535 digital pulse delay generator, with a pulse-to-pulse jitter of less than 5ns. The pulse-to-pulse jitter (or *r.m.s.* fluctuation) between laser pulses was measured to be about 0.7s, slightly more than the manufacturer rated jitter of 0.5s. This value was used to estimate the precision of the PIV measurements. The pulse delay generator produced a signal whenever an internal timing trigger initiated a timing cycle. The flashlamp-to-Qswitch delay for both lasers was typically set to 300s to provide a laser energy output of about 180mJ per oscillator. The output signal from the internal timer was sent to the camera which, following an external trigger, had an internal delay of 20s before exposing the first frame. The exposure time was determined by the transfer pulse width, which was set to 304s. The first laser pulse was synchronised to illuminate the region of interest during this period. Following this, a transfer pulse delay of 1s occured before the second exposure was

activated for a fixed time period of up to 30ms. The second laser pulse fired at a time

negligible levels and background light was later subtracted from each image pair.

**2.2.4.4 PIV camera and lenses** 

local hard disk drives.

the first pulse to be recorded by frame two. Since a 532nm10nm narrow bandpass filter was not available, minimum room lighting was used to reduce the effect of background light to

The PIV camera was a Kodak Megaplus ES 1.0 (maximum frame-rate of 30Hz), which employed a charged-coupled device (CCD) with an array of 1008 by 1018 pixels (width and height) respectively. The quantum efficiency of the camera is approximately 36% for light at a wavelength of 532nm. Each pixel is approximately 9m square and has a fill factor of 60%. Each pixel contains three main regions: light sensitive region 1, light sensitive region 2 and a dark region. It was purpose designed to allow it to collect two images in rapid succession. In the "triggered double-exposure mode" or the "frame-straddling mode", light sensitive region 1 is activated to collect photons for an extremely short period. This period, known as the transfer pulse delay, can range from 1 to 999s. After the transfer pulse delay, a transfer pulse width event occurs before light sensitive region 2 is activated. This time can range from 1 to 5s. During the transfer pulse width event, integrated light signals falling on region 1 are sequentially shifted into the dark region, which is then further transferred to the random access memory (RAM) of the camera capture card. Light sensitive region 2 is activated for a fixed time of up to 30ms before the data are transferred into the dark region and ultimately into the RAM. Data in RAM are later transferred to permanent storage in

The camera was connected to an AF Zoom-Nikkor 70-300mm f/4-5.6D ED lens set at an *f*number (*f#*) of 5.6. The *f#* is the ratio of the focal length, *f*, to the aperture diameter, *Da*. This *f#* value ensured that the image distortion around the edges of the image was minimised, while allowing adequate light to the image array via the lens. Raffel, Willert and Kompenhans (1998) noted that this kind of systematic perspective distortion due to the lens arrangement is generally neglected in most experiments and the only way to quantify this error for highly three-dimensional flows is to measure all three components of velocity. This

The depth-of-field of the lens system was checked to ensure that particles within the lightsheet were adequately imaged and focused. The depth-of-field of a lens is the distance along the optical axis over which an image can be clearly focused. For PIV, it is calculated based

was not possible using two-component PIV employed in the present investigation.

on the following set of equations from Raffel, Willert and Kompenhans (1998):

*t* after

pressure signal was low-pass filtered at 10 Hz by a 6-pole Butterworth filter and the subsequent signal passed to an oscilloscope which generated a TTL trigger pulse from each falling edge of the filtered pressure pulse. The TTL signal was stored by a Dantec 57N20 enhanced Burst Spectrum Analyser as a false velocity reading that was used a a reference marker for the phase-averaging algorithm. The phase-averaging algorithm divided each 360-precession cycle into 36 segments based on the reference marker. For each radial measurement location, the axial velocity data in each segment was averaged over each 10 segment. This technique assumes that the precession mode, precession frequency and phase speed do not change. Variations in these parameters will reduce the measured phaseaveraged velocities and will increase the measured *r.m.s.* velocity fluctuations.

### **2.2.4 PIV experimental arrangement**

The PIV system consisted of a light generation system, light delivery system, shaping optics, a pulse delay generator, a PIV camera and suitable lenses.

#### **2.2.4.1 Light generation system**

Light was generated by a Quantel Brilliant Twins pulsed Nd:YAG laser system rated at 380mJ per pulse at a frequency-doubled wavelength of 532nm. The Twins has two separate laser systems which produce a fundamental harmonic wavelength of 1064nm. A reference beam and a secondary beam were initially parallel with the secondary beam offset from the reference. The secondary beam was combined down-beam by a series of mirrors and lenses to share a common geometric axis with the primary beam. The combined beams were passed through a second harmonic generator which frequency-doubled the light to generate a (green) wavelength of 532nm. These beams were used in the PIV experiments. Each beam pulse had a manufacturer-specified pulse duration of about 6ns at FWMH (full width at maximum half-height). The short pulse width provided excellent temporal resolution for the current flow. The pulse width duration was also verified by observing on an oscilloscope the electronic signals produced by a high-speed light detector positioned in front of the scattered particles. After optimising the lasers for equal beam strength, the actual power measured was typically 180mJ per pulse per laser cavity for a flashlamp-to-Q-switch delay of 300s.

#### **2.2.4.2 Light delivery and shaping optics**

The laser beam was delivered, via a series of mirrors, to the shaping optics. These comprised a diverging cylindrical lens (focal length 200mm) followed by a cylindrical focusing lens with a focal length of 260mm. Optimising this arrangement of lenses produced a suitable light sheet with a non-uniform thickness that varied between 1mm and 2mm in the region of interest. This thickness was chosen because of the significant out-of-plane motion in the emerging precessing jet. The variation in light sheet thickness was found to have little impact on the results since the calculations for particle out-of-plane motion are based on the minimum light sheet thickness (1mm) and the probe resolutions of the interrogation windows are all larger than the thickest part of the lightsheet (2mm).

#### **2.2.4.3 Pulse delay generator**

The laser system comprises two independently-controlled laser oscillators, each of which fire at a nominal rate of 10Hz. The lasers were controlled by a Stanford Research Systems DG-535 digital pulse delay generator, with a pulse-to-pulse jitter of less than 5ns. The pulse-to-pulse jitter (or *r.m.s.* fluctuation) between laser pulses was measured to be about 0.7s, slightly more than the manufacturer rated jitter of 0.5s. This value was used to estimate the precision of the PIV measurements. The pulse delay generator produced a signal whenever an internal timing trigger initiated a timing cycle. The flashlamp-to-Qswitch delay for both lasers was typically set to 300s to provide a laser energy output of about 180mJ per oscillator. The output signal from the internal timer was sent to the camera which, following an external trigger, had an internal delay of 20s before exposing the first frame. The exposure time was determined by the transfer pulse width, which was set to 304s. The first laser pulse was synchronised to illuminate the region of interest during this period. Following this, a transfer pulse delay of 1s occured before the second exposure was activated for a fixed time period of up to 30ms. The second laser pulse fired at a time *t* after the first pulse to be recorded by frame two. Since a 532nm10nm narrow bandpass filter was not available, minimum room lighting was used to reduce the effect of background light to negligible levels and background light was later subtracted from each image pair.

#### **2.2.4.4 PIV camera and lenses**

236 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

pressure signal was low-pass filtered at 10 Hz by a 6-pole Butterworth filter and the subsequent signal passed to an oscilloscope which generated a TTL trigger pulse from each falling edge of the filtered pressure pulse. The TTL signal was stored by a Dantec 57N20 enhanced Burst Spectrum Analyser as a false velocity reading that was used a a reference marker for the phase-averaging algorithm. The phase-averaging algorithm divided each 360-precession cycle into 36 segments based on the reference marker. For each radial measurement location, the axial velocity data in each segment was averaged over each 10 segment. This technique assumes that the precession mode, precession frequency and phase speed do not change. Variations in these parameters will reduce the measured phase-

The PIV system consisted of a light generation system, light delivery system, shaping optics,

Light was generated by a Quantel Brilliant Twins pulsed Nd:YAG laser system rated at 380mJ per pulse at a frequency-doubled wavelength of 532nm. The Twins has two separate laser systems which produce a fundamental harmonic wavelength of 1064nm. A reference beam and a secondary beam were initially parallel with the secondary beam offset from the reference. The secondary beam was combined down-beam by a series of mirrors and lenses to share a common geometric axis with the primary beam. The combined beams were passed through a second harmonic generator which frequency-doubled the light to generate a (green) wavelength of 532nm. These beams were used in the PIV experiments. Each beam pulse had a manufacturer-specified pulse duration of about 6ns at FWMH (full width at maximum half-height). The short pulse width provided excellent temporal resolution for the current flow. The pulse width duration was also verified by observing on an oscilloscope the electronic signals produced by a high-speed light detector positioned in front of the scattered particles. After optimising the lasers for equal beam strength, the actual power measured was typically 180mJ per pulse per laser cavity for a flashlamp-to-Q-switch delay

The laser beam was delivered, via a series of mirrors, to the shaping optics. These comprised a diverging cylindrical lens (focal length 200mm) followed by a cylindrical focusing lens with a focal length of 260mm. Optimising this arrangement of lenses produced a suitable light sheet with a non-uniform thickness that varied between 1mm and 2mm in the region of interest. This thickness was chosen because of the significant out-of-plane motion in the emerging precessing jet. The variation in light sheet thickness was found to have little impact on the results since the calculations for particle out-of-plane motion are based on the minimum light sheet thickness (1mm) and the probe resolutions of the interrogation

The laser system comprises two independently-controlled laser oscillators, each of which fire at a nominal rate of 10Hz. The lasers were controlled by a Stanford Research Systems

windows are all larger than the thickest part of the lightsheet (2mm).

averaged velocities and will increase the measured *r.m.s.* velocity fluctuations.

**2.2.4 PIV experimental arrangement** 

**2.2.4.2 Light delivery and shaping optics** 

**2.2.4.3 Pulse delay generator** 

**2.2.4.1 Light generation system** 

of 300s.

a pulse delay generator, a PIV camera and suitable lenses.

The PIV camera was a Kodak Megaplus ES 1.0 (maximum frame-rate of 30Hz), which employed a charged-coupled device (CCD) with an array of 1008 by 1018 pixels (width and height) respectively. The quantum efficiency of the camera is approximately 36% for light at a wavelength of 532nm. Each pixel is approximately 9m square and has a fill factor of 60%. Each pixel contains three main regions: light sensitive region 1, light sensitive region 2 and a dark region. It was purpose designed to allow it to collect two images in rapid succession. In the "triggered double-exposure mode" or the "frame-straddling mode", light sensitive region 1 is activated to collect photons for an extremely short period. This period, known as the transfer pulse delay, can range from 1 to 999s. After the transfer pulse delay, a transfer pulse width event occurs before light sensitive region 2 is activated. This time can range from 1 to 5s. During the transfer pulse width event, integrated light signals falling on region 1 are sequentially shifted into the dark region, which is then further transferred to the random access memory (RAM) of the camera capture card. Light sensitive region 2 is activated for a fixed time of up to 30ms before the data are transferred into the dark region and ultimately into the RAM. Data in RAM are later transferred to permanent storage in local hard disk drives.

The camera was connected to an AF Zoom-Nikkor 70-300mm f/4-5.6D ED lens set at an *f*number (*f#*) of 5.6. The *f#* is the ratio of the focal length, *f*, to the aperture diameter, *Da*. This *f#* value ensured that the image distortion around the edges of the image was minimised, while allowing adequate light to the image array via the lens. Raffel, Willert and Kompenhans (1998) noted that this kind of systematic perspective distortion due to the lens arrangement is generally neglected in most experiments and the only way to quantify this error for highly three-dimensional flows is to measure all three components of velocity. This was not possible using two-component PIV employed in the present investigation.

The depth-of-field of the lens system was checked to ensure that particles within the lightsheet were adequately imaged and focused. The depth-of-field of a lens is the distance along the optical axis over which an image can be clearly focused. For PIV, it is calculated based on the following set of equations from Raffel, Willert and Kompenhans (1998):

$$M = \frac{Z\_o}{Z\_o} \tag{2-1}$$

PIV as a Complement to LDA in the Study of an Unsteady Oscillating Turbulent Flow 239

pulsed Nd:YAG laser is that the 10-Hz flashlamp frequency must not vary by more than ±0.5Hz. The laser manufacturer advises that power output decreases dramatically when the

Fig. 4.a. Block diagram of the relationship between the triggers, laser and camera systems.

Thermal signals from fluid with different speeds

Lead low-pass filter

Lead Schmitt trigger Lag Schmitt trigger

Atmel AT90S2313-10PC microprocessor

> SRS DG-535 digital delay/pulse delay generator

> > Light signal Scattered light

Lag low-pass filter

Flashlamp #1

Frame grabber & RAM

> Flashlamp #2 Q-Switch #1

Q-Switch #2

*fl\_min*=95ms.

*fl\_nom* = 1/ *ffl\_nom*.

*fl\_nom*=Nominal flashlamp time interval

*fl\_max*=Maximum flashlamp time

The following establishes some criteria for the timing system of the microprocessor.

*fl\_nom* > 

where, *ffl\_nom*=Nominal flashlamp frequency [Hz]; *ffl\_min*=Minimum flashlamp frequency

The next step is to choose an appropriate time window that selects a particular band of precession frequency. In the present case, the precession frequency is nominally 5Hz and a frequency range between 3 and 6Hz was chosen. The 3 and 6Hz cut-off frequency range was

If the frequency range were to be shifted to a higher precession frequency envelope, i.e., from 3 and 6Hz to 4 and 7Hz, fluid structures having a higher precession frequency will be recorded. Although slightly different sizes of structures may be observed, the overall flow topology of the higher precession frequency flow is not expected to be markedly different from the lower frequency flow topology. Data collection time may be increased for the 4 to

*ffl\_min* < *ffl\_nom* < *ffl\_max* ; 9.5Hz < *ffl\_nom* < 10.5Hz, and

Laser #1

Camera with Lens

Hard-drive

Laser #2

chosen to match the –3dB frequency interval for a 5Hz precession frequency.

*fl\_max*=105ms >

*fl\_min*=Minimum flashlamp time interval [sec], and

Electrical signal

[Hz]; *ffl\_max*=Maximum flashlamp frequency [Hz];

therefore,

Legend:

Flow signal Lead Hot-wire

Lag Hot-wire

interval [sec].

[sec]; 

flashlamp frequency is outside of the nominal flashlamp frequency (Quantel, 1994).

Lead hot-wire bridge amplifier

Lag hot-wire bridge amplifier

$$f\_{\#} = \frac{f}{D\_a} \tag{2-2}$$

$$d\_{\rm diff} = 2.44 f\_{\#} (M+1) \lambda \tag{2-3}$$

$$\delta\_z = \frac{2f\_\# d\_{diff}(M+1)}{M^2} \tag{2-4}$$

where,

*M* = magnification factor, *zo* = distance between image plane and lens [m]; *Zo* = distance between object plane and lens [m]; *f#* = f-stop; *f* = focal length of lens [m]; *Da* = aperture diameter [m]; *ddiff* = diffraction limited minimum object diameter [m]; = wavelength of light used [m], (532nm for frequency doubled Nd:YAG); *<sup>z</sup>*= depth-of-field [m].

To estimate the particle image diameter, the following equation is used neglecting effects of lens aberrations:

$$d\_r = \sqrt{\left(M \, d\_p\right)^2 + d\_{diff}^2} \tag{2-5}$$

where, *d*= particle image diameter [m] and *dp* = particle diameter [m].

The depth-of-field was calculated for each experimental setup, assuming a mean particle diameter of about 1m, *f#* = 5.6, laser wavelength of 532nm, and a nominal image pixel size of 9m. In general, the depth-of-field exceeds 5.75mm for a magnification (px/mm) of 15. Since all the experiments have light sheet thicknesses of less than 3mm and a magnification of not more than 15px/mm, the depth-of-field used is appropriate and all the particles moving into or out of the light sheet were thus in focus. For a typical magnification of 10.6px/mm, the field-of-view is about 95mm wide by 95mm high.

#### **2.2.5 Phase-and-precession-direction-resolved PIV**

Various methods of triggering a data collection system based on external reference conditions have been used by researchers to study time-dependent flows. As noted above, it is necessary to develop a data-collection system that resolves both precession direction and phase. This is achieved with a triggering system using a pair of hot-wire probes to detect the phase and direction of precession. The components to be synchronised are the PIV lasers, the camera system and the triggering system. A block diagram showing the interaction between each system is presented in Fig. 4a while the timing diagram for the trigger system is presented in Fig. 4b.

#### **2.2.5.1 Overall system timing**

An ATMEL microprocessor interfaced the laser and camera system with the external hotwire sensor-trigger system. An important criterion for stable and reliable operation of the pulsed Nd:YAG laser is that the 10-Hz flashlamp frequency must not vary by more than ±0.5Hz. The laser manufacturer advises that power output decreases dramatically when the flashlamp frequency is outside of the nominal flashlamp frequency (Quantel, 1994).

$$f\_{\text{fl\\_min}} < f\_{\text{fl\\_nom}} < f\_{\text{fl\\_max}}; \text{ 9.5Hz} < f\_{\text{fl\\_nom}} < 10.5 \text{Hz}, \text{ and } \delta\_{\text{fl\\_nom}} = 1/f\_{\text{fl\\_nom}}.$$

$$\delta\_{\text{l\\_max}} = 105 \text{ms} > \delta\_{\text{l\\_nom}} > \delta\_{\text{l\\_min}} = 95 \text{ms}.$$

therefore,

238 The Particle Image Velocimetry – Characteristics, Limits and Possible Applications

*<sup>z</sup> <sup>M</sup>*

#
