**On Turbulence and its Effects on Aerodynamics of Flow through Turbine Stages**

Galina Ilieva Ilieva

[8] Fernández Oro, J.M., Argüelles Díaz, K.M., Blanco Marigorta, E., 2009, "Non-Deterministic Kinetic Energy within the Rotor Wakes and Boundary Layers of Low-Speed Axial Fans: Frequency-Based Decomposition of Unforced Unsteadiness and

[9] Davidson, L., Dahlström, S., (2005), "Hybrid LES–RANS: An Approach to Make LES Applicable at High Reynolds Numbers", International Journal of Computational Fluid

[10] Galdo Vega, M., Rodríguez Lastra, M., Argüelles Díaz, K.M., Fernández Oro, J.M., (2012), "Application of Deterministic Correlations in the Analysis of Rotor–Stator Interactions in Axial Flow Fans". Proceedings of the ASME 2012 Fluids Engineering Summer Meeting,

[11] Fernández Oro, J.M., Argüelles Díaz, K.M., Rodríguez Lastra, M., Galdo Vega, M., Pereiras García, B., (2014), "Converged Statistics for Time-Resolved Measurements in Low-Speed Axial Fans using High-Frequency Response Probes", Experimental Thermal

[12] Persico, G., Rebay, S., (2012), "A Penalty Formulation for the Throughflow Modeling of

[13] Tropea, C., Yarin, A.L., Foss, J.F., (eds.), Handbook of Experimental Fluid Mechanics. Ch. 10. "Measurement of Turbulent Flows", pp. 745-855, Springer, New York, (2008).

[14] Fernández Oro, J.M., Argüelles Díaz, K.M., Santolaria Morros, C., Blanco Marigorta, E., (2007), "On the Structure of Turbulence in a Low–Speed Axial Fan with Inlet Guide

[15] Adamczyk, J.J., (1996), "Modelling the Effect of Unsteady Flows on the Time Average Flow Field of a Bladerow embedded in an Axial Flow Multistage Turbomachine", VKI Lecture Series, on Unsteady Flows in Turbomachines, 1996-05, Von Karmán Institute for

[16] Meneveau, C., Katz, J., (2002), "A Deterministic Stress Model for Rotor-Stator Interactions in Simulations of Passage-Averaged Flow", ASME Journal of Fluids Engineering, 124,

[17] Stollenwerk, S., Kügeler, E., (2013), "Deterministic Stress Modeling for Multistage Compressor Flowfields", Proceedings of the ASME Turbo Expo 2013, GT2013-84860,

[18] Porreca, L., Hollestein, M., Kalfas, A.I., Abhari, R.S., (2005) "Turbulence Measurements and Analysis in a Multistage Axial Turbine", Proceedings of the 17th International Symposium on Air Breathing Engines, ISABE2005-1032, September 4-9, Munich

[21] Tennekes, H., Lumley, J.L., A First Course in Turbulence, MIT Press, Massachusetts, (1972).

[19] Pope, S.B., Turbulent Flows, Cambridge University Press, New York, (2000).

[20] Bailly, C., Comte-Bellot, G., Turbulence, Springer, Switzerland, (2015).

Turbulence", Journal of Turbulence, 10, N28.

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FEDSM2012-72450, July 8-12, Puerto Rico (USA).

Turbomachinery". Computers and Fluids, 60, pp. 86-98.

Vanes", Experimental Thermal and Fluid Science, 32, pp. 316-331.

Dynamics, 19 (6), pp. 415-427.

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Fluid Mechanics, Brussels (Belgium).

June 3-7, San Antonio (USA).

pp. 550-554.

(Germany).

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68205

## **Abstract**

In reality, the flows encountered in turbines are highly three‐dimensional, viscous, tur‐ bulent, and often transonic. These complex flows will not yield to understanding or pre‐ diction of their behavior without the application of contemporary and strong modeling techniques, together with an adequate turbulence model, to reveal effects of turbulence phenomenon and its impact on flow past turbine blades. The discussion primarily targets the turbulence features and their impact on fluid dynamics; streaming of blades, and effi‐ ciency performance. Turbulence as a phenomenon, turbulence effects and the transition onset in turbine stages are discussed. Flow parameters distribution past turbine stages, approaches to turbulence modeling, and how turbulent effects change efficiency and require an innovative design, among others are presented. Furthermore, a comparison study regarding the application and availability of various turbulence models is fulfilled, showing that every aerodynamic effect, encountered of flow pass turbine blades can be predicted via different model. This work could be very helpful for researchers and engi‐ neers working on prediction of transition onset, turbulence effects, and their impact on the overall turbine performance.

**Keywords:** interaction effects, separation, transition, turbine blade, turbulence, vortices

## **1. Introduction**

The phenomenon known as "turbulence" was already recognized as a distinct fluid behavior more than 500 years ago [1]. **Figure 1** shows a sketch of L. Da Vinci, related to observations of free‐stream turbulence.

Turbulence is defined as flow regime, characterized by changes in pressure and veloc‐ ity, boundary layer separation, creation of vortex structures, and flow disturbances, etc.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Figure 1.** A sketch of turbulence by Leonardo Da Vinci, picture taken from [2].

Turbulence can be discovered in our everyday life and surrounding phenomena such as ocean waves, wind storms and smoke coming up from a chimney, among many others. Most flows observed in nature and existing in various machines and aggregates, such as pumps, fans, turbines, dryers, cyclones, stirred vessels, propulsion systems and their elements (propellers as an example), combustion chambers, and many others, are turbulent too.

In particular, a turbulent flow can be expected to exhibit all of the following specifics: dis‐ organized, chaotic, seemingly random behavior; nonrepeatability in its structures; various and large range of length and timescales; enhanced mixing and dissipation, depending on the viscosity; three dimensionality, time dependence and rotationality; intermittency in both space and time [1].

Here, it is better to clarify that the so‐called von Karman vortex street (behind a cylin‐ der) is a vortex flow regular and coherent and cannot be referred to as a turbulent flow, **Figure 2**.

**Figure 2.** Von Karman streets behind a cylinder in a nonrotating 2D flow for *Re* = 140, fluorescein visualization, picture taken from [3].

The onset of turbulence can be predicted by the Reynolds number, a dimensionless parame‐ ter; it characterizes the ratio between inertial and viscous forces. Reynolds number is equal to

$$\text{Re} = \left(\rho \cdot \mathcal{U} \cdot \mathcal{L}\right) / \mu,\tag{1}$$

In this expression *ρ* and *μ* are, respectively, fluid density (kg/m<sup>3</sup> ) and dynamic viscosity (Pa.s), *U* is flow velocity (m/s), and *L* is length scale (m).

When a subvolume of fluid is characterized by excessive kinetic energy, which is higher than the dampening effect of the fluid's viscosity, in other words, when very high *Re* number is realized in that subvolume, turbulence will appear. That is why turbulence is easier for observation in low viscosity fluids, but more difficult for observation in highly viscous fluids. If Reynolds is less than a definite critical value, damping friction forces prevent turbulent movement and the flow is laminar.

In a turbulent flow, vortex structures of various sizes and frequencies could be found. Large vortex structures, influenced by the domain boundaries and the global flow field, break up into smaller structures, characterized of higher frequencies. Small vortex structures are character‐ ized by less frequency. However, due to the flow aerodynamic character and domain boundar‐ ies, small vortices can form bigger vortex structures and vice versa. At the same time, vortices in a volume, being at continuous interaction, can exchange energy among them, change their energy levels and travel in the flow volume, changing their so‐called "mixing length."

Turbulence can be discovered in our everyday life and surrounding phenomena such as ocean waves, wind storms and smoke coming up from a chimney, among many others. Most flows observed in nature and existing in various machines and aggregates, such as pumps, fans, turbines, dryers, cyclones, stirred vessels, propulsion systems and their elements (propellers

In particular, a turbulent flow can be expected to exhibit all of the following specifics: dis‐ organized, chaotic, seemingly random behavior; nonrepeatability in its structures; various and large range of length and timescales; enhanced mixing and dissipation, depending on the viscosity; three dimensionality, time dependence and rotationality; intermittency in both

Here, it is better to clarify that the so‐called von Karman vortex street (behind a cylin‐ der) is a vortex flow regular and coherent and cannot be referred to as a turbulent flow,

**Figure 2.** Von Karman streets behind a cylinder in a nonrotating 2D flow for *Re* = 140, fluorescein visualization, picture

as an example), combustion chambers, and many others, are turbulent too.

**Figure 1.** A sketch of turbulence by Leonardo Da Vinci, picture taken from [2].

144 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

space and time [1].

**Figure 2**.

taken from [3].

Vortices, which contain the highest amount of kinetic energy, are described by the Taylor scale. In the inertial range, the vortex breakup can be described by inertial effects, thus viscous effects are negligible. Small vortices contain a low amount of energy but contribute mostly to the dissipation. The smallest turbulent vortices are defined by the Kolmogorov microscale [1].

In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, exchanging energy, as a result drag increases due to friction effects.

The level of turbulence has significant impact on the stability of boundary and shear lay‐ ers. High stream turbulence scales can contribute to earlier laminar‐turbulent transition. Small turbulence scales, observed in the boundary layer, are very important for the skin fric‐ tion levels and can exert the separation due to adverse pressure gradients. Thus, separated boundary layer and vortices lead to turbulent fluctuations and increase the level of unsteadi‐ ness. Shear layers, between the separated and main flow, act as cores of further developed turbulence levels and turbulent scales.

Turbulence of flow through turbine channels is a prerequisite to problems in blades' streaming; they lead to less levels of aerodynamic efficiency; changes in flow regimes; significant pressure fluctuations, causing vibrations; and variable forces acting on blade surfaces, among many oth‐ ers. Aforementioned, lead to worse efficiency, possible blades destruction, and problems related to other processes and elements, which make part of the turbine aggregate and installation.

The aim of this work is to discuss the complexity of flows through turbine passages with a particular emphasis on turbulence and its mechanisms and to explain their effects on turbine aerodynamics and efficiency. The chapter discusses some current state of the art in regard to modeling and prediction of turbulence features, adequacy of turbulence models to achieve physically correct picture for flow parameters distribution in turbine stages. The aim of this chapter is to discuss the follow characteristics and to provide the reader to a better under‐ standing of turbulence mechanisms, their impact on other phenomena in turbines, on the design of turbine components and on the working regimes. Also, this work could be very helpful for researchers and engineers working on prediction of transition onset, turbulence effects, and their impact on the overall turbine performance.

## **2. Turbulence and its relation to various processes and effects**

## **2.1. Turbulence and interaction effects and losses**

Turbulence in turbine stages depends on many aerodynamic features and flow conditions. In this chapter, the author is trying to shed light, based on previously conducted researched works, on the unsteady effects and loss mechanisms in turbines, how they depend on the tur‐ bulence effects and how the aerodynamic performance could be compromised.

In [4], an extensive review of loss generating mechanisms in turbomachinery is presented. The three principal sources of losses in turbine stage, are described as: viscous shear in boundary layers, shear layers and mixing processes; nonequilibrium processes such as shock waves and heat transfer processes.

Boundary layers are known as highly viscous regions, also could be referred to as regions of steep velocity gradients and shear stresses. In the boundary layer, a higher amount of energy losses is produced in the areas where the steepest gradients are found [5].

In [6], a parameter called "dissipation coefficient "Cd is described. Its variation for laminar and turbulent boundary layers, with Reynolds numbers in the range of 300 < *Reθ* < 1000, shows that the phenomenon of transition prediction is very important in the assessment of losses in turbomachinery boundary layers.

Stator and rotor blades interact, thus unsteady flow perturbations will appear both in station‐ ary and rotating frames of reference. Various aerodynamic effects, such as wake shedding at the trailing edge, secondary flows in radial direction, blade vibrations, flow leakage in axial gaps, shock waves and effects at trailing edge in transonic stages, angles of attack at the lead‐ ing edges, etc., can seriously affect the rotor blades and their efficiency performance. Blades loading and forces, acting on blades, are significantly lower under aforementioned condi‐ tions, leading to less efficiency [7].

Vortex shedding, as a phenomenon, will appear if flow detaches periodically from the back of a body, forming a Von Kármán vortex street. Such flow picture can occur at the trailing edge of the blade profile. The resulting frequency can be estimated based on the Strouhal number, Eq. (2).

$$St \quad = \begin{cases} \{\mathfrak{e} \cdot L\} / v \end{cases} \tag{2}$$

In Eq. (2) parameters are as follows: *L* is the characteristic length in (m), the excitation fre‐ quency is *fe*, and *v* is the flow velocity (m/s).

modeling and prediction of turbulence features, adequacy of turbulence models to achieve physically correct picture for flow parameters distribution in turbine stages. The aim of this chapter is to discuss the follow characteristics and to provide the reader to a better under‐ standing of turbulence mechanisms, their impact on other phenomena in turbines, on the design of turbine components and on the working regimes. Also, this work could be very helpful for researchers and engineers working on prediction of transition onset, turbulence

Turbulence in turbine stages depends on many aerodynamic features and flow conditions. In this chapter, the author is trying to shed light, based on previously conducted researched works, on the unsteady effects and loss mechanisms in turbines, how they depend on the tur‐

In [4], an extensive review of loss generating mechanisms in turbomachinery is presented. The three principal sources of losses in turbine stage, are described as: viscous shear in boundary layers, shear layers and mixing processes; nonequilibrium processes such as shock waves and

Boundary layers are known as highly viscous regions, also could be referred to as regions of steep velocity gradients and shear stresses. In the boundary layer, a higher amount of energy

In [6], a parameter called "dissipation coefficient "Cd is described. Its variation for laminar and turbulent boundary layers, with Reynolds numbers in the range of 300 < *Reθ* < 1000, shows that the phenomenon of transition prediction is very important in the assessment of

Stator and rotor blades interact, thus unsteady flow perturbations will appear both in station‐ ary and rotating frames of reference. Various aerodynamic effects, such as wake shedding at the trailing edge, secondary flows in radial direction, blade vibrations, flow leakage in axial gaps, shock waves and effects at trailing edge in transonic stages, angles of attack at the lead‐ ing edges, etc., can seriously affect the rotor blades and their efficiency performance. Blades loading and forces, acting on blades, are significantly lower under aforementioned condi‐

Vortex shedding, as a phenomenon, will appear if flow detaches periodically from the back of a body, forming a Von Kármán vortex street. Such flow picture can occur at the trailing edge of the blade profile. The resulting frequency can be estimated based on the Strouhal number,

*St* = (*fe* . *L*)/*v* (2)

effects, and their impact on the overall turbine performance.

146 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

**2.1. Turbulence and interaction effects and losses**

losses in turbomachinery boundary layers.

tions, leading to less efficiency [7].

Eq. (2).

heat transfer processes.

**2. Turbulence and its relation to various processes and effects**

bulence effects and how the aerodynamic performance could be compromised.

losses is produced in the areas where the steepest gradients are found [5].

Turbulence intensity decays much slower than the velocity deficit in the wake [8]. The velocity deficit in an upstream blade wake can be perceived as an incidence variation by the down‐ stream blade row. Turbulence levels in the wake can change the boundary layer from laminar to turbulent, leading to additional losses. The influence of wake will maintain through the blade rows far in a downstream direction.

Flow, characterized by low velocities, has normal velocity component toward the suction side of the downstream blade, thus an upstream wake with low velocities and higher turbulence levels can move toward the suction side of the blade. In a similar way, a fluid characterized by higher velocity can move toward the pressure side of the downstream blades. This movement of fluid particles has the following major effects on the downstream blade row, as described in [9]: change in boundary layer characteristics of a profile through its effect on the transition processes; affects the secondary flow generation through the blades in downstream direc‐ tion; and influences the wake mixing losses due to the phenomenon of wake stretching or compression.

One of the first studies related to the interaction between streamwise vortices and down‐ stream blades is described in [10]. A significant increase in random unsteadiness at the front part of rotor blades, in regions associated with stator secondary flow, was observed. This is a result of vortex breakdown, and is proposed that it occurred due the vortex filling and cutting, and the strong deformation of the vortex cross‐sectional area, at the moment when the vortex enters into the rotor interblade channels [10]. The vortex energy is converted into energy of random fluctuations during the process of vortex breakdown. Later, more detailed mecha‐ nism for the vortex‐rotor interaction, was developed, see Ref. [11]. According to that model, a moment after the vortex is separated from the rotor blade, thus disturbance will create and will start propagate along the vortex axis, at the local speed of sound, whilst simultaneously being swept downstream, at the local convection velocity. The place where vortex arises is located close to the pressure‐side stagnation region [11]. In [12], for one and a half stage, it is demonstrated that flow passing through the first row of rotor blades is highly unsteady and is much influenced by the flow generated between two adjacent stator blades. Upstream, flow disturbances can convect through the rotor blades, without interaction with last mentioned, for a short time and vice versa [12].

In a passage of low aspect ratio blades, secondary flows generated in the form of streamwise vortices, are significant across the blade and can take more than 1/3 part of the blade span [13]. These vortices are convected downstream toward the next blade row where they interact with the main flow.

In [14], for first time, results of studies related to the origin and mechanisms of secondary flows generation, on the basis of the analytical modeling, **Figure 3**, are described. The incom‐ ing boundary layer is modeled as a vortex filament ab and is shown that when this filament convects in downstream blade rows to def, it produces three forms of vorticity: secondary, trailing filament, and trailing shed vorticity. The distributed secondary vorticity is result of

**Figure 3.** Model of Hawthorne for secondary flows, realized in 1955, picture taken from [14].

the turning of the inlet vortex filament; the trailing filament vorticity forms due to the differ‐ ences in velocity between the concave and convex surfaces. The last one, the so‐called trail‐ ing shed vorticity, is formed as a result of the spanwise variation of the blade circulation. Furthermore, it was stated that these mechanisms contribute to the overall secondary flow generation [14]. The Hawthorne's approach, applied and described in [13], is result of many assumptions; however, it can shed some light on the origin of the secondary flows. Similar models were developed and presented by other scientists [15, 16].

Every wake is initially represented as a perturbation in the uniform flow, as stated in [17, 18]. Wakes are transported with the main flow and are cut up into separate segments by the down‐ stream blades. Inside the blade passage, the wake continues to behave as a jet, taking energy from the main flow. The velocity induced by that jet leads to generation of wake flow over the convex blade surface; the induced wake structure over the concave blade surface disappears. As a result of the blade circulation, in axial turbines, formed wake structures can stretch and shear over the blade surfaces or along the channel, they are travelling through [19].

Wake, before entering the blade passage, is subjected to "bowing" due to the higher velocities in the middle of the passage, in comparison to the near blade surfaces [20]. Moreover, wake experiences "shearing" near the suction surface and "stretching" near the pressure blade sur‐ face. It is due to the fact that the part adjacent to the convex blade surface convects more rapidly in comparison with the part adjacent to the concave surface. As a result of these pro‐ cesses of bowing, stretching, shearing, and distortion, the wake is moved to the convex blade surfaces and its tail is stretched in direction back to the leading edge of rotor blades. This wake transport is a prerequisite to losses generated from mixing of wakes in the downstream direction [15, 16, 19, 21, 22].

Secondary flows lead to increase in losses, formed at the end walls, and to nonuniform distri‐ bution of exit flow angles. In turbomachines, the upstream end walls are rotating relatively to a blade row, thus the inlet boundary layer is expected to be skewed. Experiments, described in [23–25], show that already mentioned skew significantly can affect the existing passage vortices and losses. The streamwise vorticity, introduced to the flow, by the skewed bound‐ ary layer at the inlet, will strengthen the streamwise vorticity, observed at the exit; all that is visible in the direction of the turbine rotation.

Existing radial gaps lead to increase in the leakage mass flow rate and causes efficiency losses. Leakage flows are formed as a result of pressure differences between concave and convex blade surfaces, dominated by formed trailing vortices, shed in the downstream direction. These vortices can reduce the local turning, performed by the blade, and can gen‐ erate decrease in the extracted mechanical energy. As a consequence of the viscous effects in the tip clearance, entropy increases. The second major feature is the subsequent mixing of flow, which passes through the tip clearance gap with that coming from the main flow. Flow structure in the tip region is studied and explained by a number of researchers; see Ref. [26–29].

Low pressure, generated immediately behind the trailing edge, leads to very high losses at the trailing edges. Flow expands around the trailing edge to that low pressure values and is then recompressed after a strong shock wave, where meet flows coming from suction and pres‐ sure sides [30, 31]. The interaction between shocks and boundary layers can lead to unsteady boundary layer separations and increase in loss for transonic velocities [32, 33].

## **2.2. Turbulence and condensation effects in two‐phase flows in turbine stages**

the turning of the inlet vortex filament; the trailing filament vorticity forms due to the differ‐ ences in velocity between the concave and convex surfaces. The last one, the so‐called trail‐ ing shed vorticity, is formed as a result of the spanwise variation of the blade circulation. Furthermore, it was stated that these mechanisms contribute to the overall secondary flow generation [14]. The Hawthorne's approach, applied and described in [13], is result of many assumptions; however, it can shed some light on the origin of the secondary flows. Similar

Every wake is initially represented as a perturbation in the uniform flow, as stated in [17, 18]. Wakes are transported with the main flow and are cut up into separate segments by the down‐ stream blades. Inside the blade passage, the wake continues to behave as a jet, taking energy from the main flow. The velocity induced by that jet leads to generation of wake flow over the convex blade surface; the induced wake structure over the concave blade surface disappears. As a result of the blade circulation, in axial turbines, formed wake structures can stretch and

Wake, before entering the blade passage, is subjected to "bowing" due to the higher velocities in the middle of the passage, in comparison to the near blade surfaces [20]. Moreover, wake experiences "shearing" near the suction surface and "stretching" near the pressure blade sur‐ face. It is due to the fact that the part adjacent to the convex blade surface convects more rapidly in comparison with the part adjacent to the concave surface. As a result of these pro‐ cesses of bowing, stretching, shearing, and distortion, the wake is moved to the convex blade surfaces and its tail is stretched in direction back to the leading edge of rotor blades. This wake transport is a prerequisite to losses generated from mixing of wakes in the downstream

Secondary flows lead to increase in losses, formed at the end walls, and to nonuniform distri‐ bution of exit flow angles. In turbomachines, the upstream end walls are rotating relatively to a blade row, thus the inlet boundary layer is expected to be skewed. Experiments, described in [23–25], show that already mentioned skew significantly can affect the existing passage

shear over the blade surfaces or along the channel, they are travelling through [19].

models were developed and presented by other scientists [15, 16].

**Figure 3.** Model of Hawthorne for secondary flows, realized in 1955, picture taken from [14].

148 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

direction [15, 16, 19, 21, 22].

In two‐phase flows in turbine stages, the liquid phase could be presented both as water film, along the streamed blade walls, and as droplets, travelling with the main flow. The inter‐ face between primary and second phases is characterized by many aerodynamic and thermal effects.

Developed models for annular two‐phase flows take into account differences and specif‐ ics for both the liquid and vapor phases and also introduce continuous and dispersed fields. Knowledge of the turbulence characteristics related to the continuous vapor phase, modified due to the presence of the droplet field, is required to introduce closure for those models. Vapor core turbulence is known to influence interfacial shear and affects transport and specifics in structure of the dispersed liquid droplet field, and ultimately interfacial shear effects and droplets dynamics and deposition are affected by vapor core turbulence.


$$\mathbf{u}\_{i} = \mathbf{u}\_{i\leftharpoons} \mathbf{+u}\_{i}^{\prime} \tag{3}$$

$$u'\_{\parallel} = \Pi \, (\mathfrak{D}k/\mathfrak{J})^{\mathbb{I}\mathbb{Z}} \tag{4}$$

where *П* provides normally distributed random number, which allows the fluctuating com‐ ponent *ui′* to take different values around a mean velocity *ui*.

The mean component of the fluid velocity will affect the average trajectory of flow particles. Two identical particles can have different trajectories due to the specifics in flow aerodynam‐ ics, such as secondary flows, turbulence, condensation, etc. In the case of turbulent two‐phase flows, the fluctuating velocity component leads to particles' trajectory deviations.

Particles with *Re* < 400 tend to suppress turbulence and its effects; particles with *Re* > 400, enhance turbulence as a result of vortex shedding [34].

In [35] turbulence measurements in flows with liquid film interface are reported; it was shown that turbulence intensities were higher than those in a pipe with wall roughness equivalent to that of the film interface.

Aerodynamic and thermal losses in turbine stages, due to the presence of second phase and turbulence effects, are as follows:


The influence of direct losses in nonequilibrium flows grows with the number of stages and reaches its maximum at the end of the steam turbine. The indirect losses cause changes in aerodynamic loss by departures from equilibrium.


Other specifics of the two‐phase flow pass turbine surfaces are as follows:


From structural point of view, coarse droplets together with the turbulence impact blade sur‐ faces, leading to erosion and efficiency decrease [36].

## **2.3. Turbulence and cooling in gas turbine passages**

*ui* = *ui*

ponent *ui′* to take different values around a mean velocity *ui*.

150 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

enhance turbulence as a result of vortex shedding [34].

*ui*

that of the film interface.

lowing blade row;

and separation;

layer;

turbulence effects, are as follows:

\_+ *ui*

where *П* provides normally distributed random number, which allows the fluctuating com‐

The mean component of the fluid velocity will affect the average trajectory of flow particles. Two identical particles can have different trajectories due to the specifics in flow aerodynam‐ ics, such as secondary flows, turbulence, condensation, etc. In the case of turbulent two‐phase

Particles with *Re* < 400 tend to suppress turbulence and its effects; particles with *Re* > 400,

In [35] turbulence measurements in flows with liquid film interface are reported; it was shown that turbulence intensities were higher than those in a pipe with wall roughness equivalent to

Aerodynamic and thermal losses in turbine stages, due to the presence of second phase and

• Water film losses, originating from the movement of the water film on the blade surfaces; • Drag loss due to the presence of coarse water droplets: small fraction of the overall losses

• Coarse water braking losses: kinetic energy of droplets decreases by impinging on the fol‐

• Fog droplet loss due to the interphase velocity slip, entropy increases due to the viscous drag;

• Fog droplet deposition or collection loss, turbine efficiency performance is affected by de‐

The influence of direct losses in nonequilibrium flows grows with the number of stages and reaches its maximum at the end of the steam turbine. The indirect losses cause changes in

• Changes in flow‐incidence angle, leading to increased losses, especially at the tip regions

• Changes in profile losses due to the variations in surface shear stress, leading to transition

• Interactions between both the condensation‐induced shock waves and formed boundary

• Nonuniformity effects—pressure changes along streamed profiles change the working

originates from acceleration of the large coarse droplets at the trailing edges;

creased kinetic energy of fog droplets, merged on the blade surfaces.

aerodynamic loss by departures from equilibrium.

with thin leading edge blade profiles;

conditions for the downstream blade row.

flows, the fluctuating velocity component leads to particles' trajectory deviations.

' (3)

′ = *П* (2*k*/3 )1/2 (4)

Turbine blades, as part of gas turbine aggregates, must be cooled as they are highly exerted by temperature loads, deformations and stresses. Furthermore, unsteady effects, rotationally induced forces (Corriolis and centrifugal forces), and secondary flows and change in flow parameters, among others complicate the flow aerodynamics within the interblade channels.

The flow field is characterized by higher levels of turbulence, transition effects, secondary flows, unsteadiness, particularly due to the cooling flows and effects. Turbulence results due to wakes from upstream stages; rotationally induced forces; cooling fluid (coolant) that mixes with the main stream in turbine channels; interaction effects between film cooling jets and the mainstream flow; complex flow conditions from the combustor; separation of cooling film, separation effects associated with cooling jets and cavities, etc.

Heat transfer coefficients increase by the enhancement of flow turbulence levels and bound‐ ary layer separation effects. Last mentioned are accompanied by increase in the examined pressure drop [37].

The effect of turbulence levels on heat transfer, aerodynamic performance, drag forces, skin friction and flow parameters distribution has been studied by many researchers in last decades [38–42].

High turbulence levels are provoked due to the presence of upstream wakes and rotation effects [43–45]. High levels of turbulence lead to increased temperatures over blade surfaces, particularly at leading edges where temperatures reach maximum values [46].

As an effect of the interaction between cross‐flow and film cooling, higher turbulence levels present in turbine flows. These effects depend on the diameter of holes that supply the cool‐ ant jets and their exact place on turbine blade surfaces. Experiments on cooling via inclined holes, characterized by ratio (*δ*1/*d<<*1), show that a higher percent of turbulence levels can be achieved immediately in the downstream direction of a row of film cooling holes [47]. Turbulence effects and their increase are found even after film cooling jet dissipation. By application of hot‐wire measurement techniques is also concluded that in the case of (*δ*1/*d<<*11) jet‐turbulent fluid could dominate the boundary layer in the downstream direc‐ tion [48].

In [49], authors have studied how circular wall jets can increase the free‐stream turbulence intensity. It is found that the increased turbulence levels in the free‐stream can cause high levels of mixing and quick dissipation in the film layer.

An increase in heat transfer levels is related to moderate turbulence intensity levels and rela‐ tively small length scales recognized in the fluid.

Mainstream turbulence intensity and length scales have significant impact on the film cooling jets and their intensity in the flow, also lead to additional interactions between cooling jets and core flow and affect the effectiveness of film cooling in the downstream direction [50].

Free‐stream turbulence can decrease the efficiency of film cooling, in the downstream direc‐ tion and can increase it between injection holes, due to the enhanced mixing [51].

An increase in mixing of momentum, that effectively thins the boundary layer, will result in an increase in the film cooling efficiency in the downstream direction. At the same time, an increase in the rate of cooling film disintegration is a reason for less efficiency performance of the film cooling. In general, which case will be dominant depends on the "blowing ratio" [37, 52]. In [53–55], the effects of increased turbulence intensity on film cooling performance, at leading edge were examined and the effects of increased turbulence intensity are in a strong relation to the blowing ratio.

The efficiency of performed film cooling varies rapidly with the blowing ratio and is in func‐ tion of the spanwise angle [56].

At higher blowing ratios and cylindrical film cooling holes, the effectiveness in downstream direction will increase if the jet angle is deflected in a close proximity to the wall.

An increase in turbulent mixing levels, due to free‐stream turbulence, leads to improved spanwise mixing between holes and better film cooling performance [57].

Film layers characterized with low blowing ratios could be rapidly dispersed by increased turbulence intensity, while increasing the local heat transfer coefficient. High blowing ratio film layers were found to be relatively insensitive to increase in the turbulence intensity levels, consistent with previous studies with flat plates.

Many numerical and experimental studies have been conducted to determine the relation between turbulence and the Nusselt number. A numerical study of both heat transfer char‐ acteristics and flow field of a slot turbulent jet impinging, on a concave surface with constant heat flux, has been carried out and discussed in [58]. In [59] attention is paid on flat plate with varying curvature, results are discussed in [59]. Computational results show that maximum value of *Nu* is attained at the stagnation point and gains higher values as the Reynolds num‐ ber increases.

## **3. Transition modeling in turbine stages**

By application of hot‐wire measurement techniques is also concluded that in the case of (*δ*1/*d<<*11) jet‐turbulent fluid could dominate the boundary layer in the downstream direc‐

In [49], authors have studied how circular wall jets can increase the free‐stream turbulence intensity. It is found that the increased turbulence levels in the free‐stream can cause high

An increase in heat transfer levels is related to moderate turbulence intensity levels and rela‐

Mainstream turbulence intensity and length scales have significant impact on the film cooling jets and their intensity in the flow, also lead to additional interactions between cooling jets and core flow and affect the effectiveness of film cooling in the downstream

Free‐stream turbulence can decrease the efficiency of film cooling, in the downstream direc‐

An increase in mixing of momentum, that effectively thins the boundary layer, will result in an increase in the film cooling efficiency in the downstream direction. At the same time, an increase in the rate of cooling film disintegration is a reason for less efficiency performance of the film cooling. In general, which case will be dominant depends on the "blowing ratio" [37, 52]. In [53–55], the effects of increased turbulence intensity on film cooling performance, at leading edge were examined and the effects of increased turbulence intensity are in a strong relation to

The efficiency of performed film cooling varies rapidly with the blowing ratio and is in func‐

At higher blowing ratios and cylindrical film cooling holes, the effectiveness in downstream

An increase in turbulent mixing levels, due to free‐stream turbulence, leads to improved

Film layers characterized with low blowing ratios could be rapidly dispersed by increased turbulence intensity, while increasing the local heat transfer coefficient. High blowing ratio film layers were found to be relatively insensitive to increase in the turbulence intensity levels,

Many numerical and experimental studies have been conducted to determine the relation between turbulence and the Nusselt number. A numerical study of both heat transfer char‐ acteristics and flow field of a slot turbulent jet impinging, on a concave surface with constant heat flux, has been carried out and discussed in [58]. In [59] attention is paid on flat plate with varying curvature, results are discussed in [59]. Computational results show that maximum value of *Nu* is attained at the stagnation point and gains higher values as the Reynolds num‐

direction will increase if the jet angle is deflected in a close proximity to the wall.

spanwise mixing between holes and better film cooling performance [57].

tion and can increase it between injection holes, due to the enhanced mixing [51].

levels of mixing and quick dissipation in the film layer.

152 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

tively small length scales recognized in the fluid.

tion [48].

direction [50].

the blowing ratio.

ber increases.

tion of the spanwise angle [56].

consistent with previous studies with flat plates.

## **3.1. Transition as a phenomenon: transition in turbine stages**

Turbulence modeling has been a subject of intensive research for many years. Turbulent models are able to predict turbulence occurrence and its effects in turbine passages. However, in the case of modeling, with all specific features, such as stator‐rotor interac‐ tion, phase changes (if any), compressible and viscosity effects, pressure gradients, among others, turbulent model lacks of applicability is not able to provide accurate predictions of turbulence effects on the mean flow characteristics. Models should be more powerful and enriched by specific terms related to existing specific aerodynamic observations due to turbulent effects.

One of the most important and most difficult turbulent phenomenon, to model and resolve, is the so‐called transition.

Reynolds is the first scientist who worked on the transition phenomenon; he investigated the transition from laminar to turbulent flow by injecting a dye streak into a flow through a pipe having smooth transparent walls [1].

Prediction of the onset of boundary layer transition is one of the most important concerns in the area of fluid mechanics. There is a great interest to transition as it plays a major role in many engineering applications and raises important questions to the flow physics, also could serve as an ingesting example for determinism and chaos.

The so‐called viscous instability of a laminar boundary layer was for the first time taken into account and studied by Tollmien. Under low free‐stream turbulence conditions, instability is observed in the case of two‐dimensional unstable Tollmien‐Schlichting waves are formed and propagate in the streamwise direction. These waves lead to additional 3D aerodynamic effects to appear in the flow structure, such as peaks, stronger secondary flow effects, hairpin vorti‐ ces and transition effects. Turbulent spots are formed in the regions of vorticity peaks and can develop to continuously spreading turbulence. A turbulent spot model to describe the specif‐ ics of a transitional flow is proposed in [60]. Later, turbulent spots generated over a flat plate surfaces, without imposed pressure gradients, were also visualized [61]. Recently, scientists have been working on more accurate transition length predictions, based on measurement of transition length in a field of adverse pressure gradients and of triggered turbulent spots, see Ref. [62]. It was found that spot characteristics, in the case of adverse pressure gradients, are different from those formed in the case of zero or favorable pressure gradients. Also, it became clear that in the presence of adverse pressure gradient, a spot can be formed at the center of a highly amplified transverse waves and is convected at lower velocity than under a zero pressure gradient, as discussed in [63].

Laminar to turbulent transition is proved as a phenomenon, which seriously affects the effi‐ ciency performance of various machines. The transition effects contribute to additional drag and lift forces, also heat fluxes that are crucial for overall working principles of different types of machines and installations.

A particular field, in which there is a specific interest to the transition phenomenon and its physics, is the area of turbomachinery. Transitional flows can be seen in flows past turbine blades, mainly in low pressure turbines. Transition is observed when various geometry blades are streamed at variable flow parameters and boundary conditions. Turbulence and transition effects significantly decrease the aerodynamic performance of turbine stages and must be studied and understood in detail [64].

Speaking about transition, one must stress that there are different types of transition [65]. The first one is called "natural" transition. It begins with a weak instability in the laminar bound‐ ary layer, as was described years ago by Tollmien and Schlichting [6], next proceeds through various stages of amplified instability to fully turbulent flow. The second type of transition is the "bypass"" transition, defined by Morkovin [84]. Bypass transition is caused by high lev‐ els of disturbances in the external flow (such as free‐stream turbulence) and can completely bypass "natural transition." The third type of transition is the so called "separated‐flow" transition. It exists in a separated laminar boundary layer and may or may not reveals some instabilities of the Tollmien‐Schlichting type.

As it is related to the "bypass" transition, some researchers presumed that it could appear as an instantaneous turbulent breakdown with zero length of transitional flow. However, the bypass transition does not always exclude instability processes. Only the long region of two‐ dimensional wave amplification preceding the appearance of three‐dimensional disturbances (spanwise periodicity) in low turbulence flow is bypassed, as clarified in detail in [63, 66]. In more detail, during the phase of the so‐called "bypass transition," not all specific laminar breakdown processes would be recognized. Separated‐flow transition occurs when a laminar boundary layer separates and transitions in the free shear layer, above the already formed bubble. Transition due to separated flow could develop at the leading edge and close to the place where minimum pressure on the suction surface sides is formed. This type of transition is extremely harmful for low pressure turbines and leads to early separation of the bound‐ ary layer. There is another type of transition, reverse transition, which is known as "relami‐ narization." Relaminarization is possible to appear at places where a previously turbulent or transitional region is affected by strong favorable pressure gradients, and as a result of that, it transfers to laminar again. Depending on the profile section geometry and the flow regime, near the leading edge, laminar flow followed by a wake‐induced or shock‐induced transition could be visualized. Last described phenomenon could be replaced by a relaminarization with subsequent transition to turbulence, occurring at multiple locations simultaneously [63].

The phenomenon of "separated‐flow transition" could arise after the so‐called boundary layer trip wires as a result of laminar separation under strong adverse pressure gradients. Thus, the flow can reattach as turbulent, forming laminar separation/turbulent‐reattachment "bubble", on the surface under consideration. In gas turbine stages, the transition of separated flow could be seen in the so‐called overspeed region close to the leading edge of the profile, over the convex or concave side, or both, and near the place where minimum pressure on the convex side is observed. What will be the bubble size depends on the transition process within the free shear layer and may involve all of the stages for a natural transition type. For bubbles with bigger size and characterized by low free‐stream turbulence levels, flow in the bubble is dominantly laminar and instabilities could be observed [67]. Big bubbles, along the blade surfaces, produce losses and act as a prerequisite to exit flow angles deviation. Small bubble configurations are an effective way to increase the turbulence levels and can possibly control the blade aerodynamics [63].

A particular field, in which there is a specific interest to the transition phenomenon and its physics, is the area of turbomachinery. Transitional flows can be seen in flows past turbine blades, mainly in low pressure turbines. Transition is observed when various geometry blades are streamed at variable flow parameters and boundary conditions. Turbulence and transition effects significantly decrease the aerodynamic performance of turbine stages and must be

154 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

Speaking about transition, one must stress that there are different types of transition [65]. The first one is called "natural" transition. It begins with a weak instability in the laminar bound‐ ary layer, as was described years ago by Tollmien and Schlichting [6], next proceeds through various stages of amplified instability to fully turbulent flow. The second type of transition is the "bypass"" transition, defined by Morkovin [84]. Bypass transition is caused by high lev‐ els of disturbances in the external flow (such as free‐stream turbulence) and can completely bypass "natural transition." The third type of transition is the so called "separated‐flow" transition. It exists in a separated laminar boundary layer and may or may not reveals some

As it is related to the "bypass" transition, some researchers presumed that it could appear as an instantaneous turbulent breakdown with zero length of transitional flow. However, the bypass transition does not always exclude instability processes. Only the long region of two‐ dimensional wave amplification preceding the appearance of three‐dimensional disturbances (spanwise periodicity) in low turbulence flow is bypassed, as clarified in detail in [63, 66]. In more detail, during the phase of the so‐called "bypass transition," not all specific laminar breakdown processes would be recognized. Separated‐flow transition occurs when a laminar boundary layer separates and transitions in the free shear layer, above the already formed bubble. Transition due to separated flow could develop at the leading edge and close to the place where minimum pressure on the suction surface sides is formed. This type of transition is extremely harmful for low pressure turbines and leads to early separation of the bound‐ ary layer. There is another type of transition, reverse transition, which is known as "relami‐ narization." Relaminarization is possible to appear at places where a previously turbulent or transitional region is affected by strong favorable pressure gradients, and as a result of that, it transfers to laminar again. Depending on the profile section geometry and the flow regime, near the leading edge, laminar flow followed by a wake‐induced or shock‐induced transition could be visualized. Last described phenomenon could be replaced by a relaminarization with subsequent transition to turbulence, occurring at multiple locations simultaneously [63].

The phenomenon of "separated‐flow transition" could arise after the so‐called boundary layer trip wires as a result of laminar separation under strong adverse pressure gradients. Thus, the flow can reattach as turbulent, forming laminar separation/turbulent‐reattachment "bubble", on the surface under consideration. In gas turbine stages, the transition of separated flow could be seen in the so‐called overspeed region close to the leading edge of the profile, over the convex or concave side, or both, and near the place where minimum pressure on the convex side is observed. What will be the bubble size depends on the transition process within the free shear layer and may involve all of the stages for a natural transition type. For bubbles with bigger size and characterized by low free‐stream turbulence levels, flow in the

studied and understood in detail [64].

instabilities of the Tollmien‐Schlichting type.

Speaking about transition and its effects on the entire flow field, it is necessary to mention that flow passing through a turbine stage is essentially turbulent and unsteady. The nonstationary pulsations are obtained as a result from the stator‐rotor interaction, mainly. Periodic phenome‐ non, caused by stator‐rotor interaction effects, excites both the flow, passing over blade surfaces, and boundary layer characteristics [63]. This results in an increased production of the so‐called turbulent spots and shifts the location of laminar‐turbulent transition in the upstream direction. This laminar‐turbulent transition phenomenon is known as "wake‐induced transition" [65].

**Figure 4**, a picture of possible boundary layer development over surfaces of high pressure blade is shown, see [65]. On the suction side, it is usually expected that in the downstream direction of the initial laminar part, a boundary layer will transfer to turbulent (2) in **Figure 4**. The size of the transition zone is related to the place where transition phenomenon could be observed ‐ in upstream direction or downstream direction of the place of minimum pressure. In the upstream direction, the zone of transition is expected to comprehend bigger area. If a laminar separation bubble occurs in the front part of the suction side (1), then the presence of high pressure gradients will force the boundary layer to develop as laminar again in down‐ stream direction; forward transition will take place [63]. The reverse transition may appear on the suction surface [67–69]. In the case of research on film‐cooled gas turbine blades, the tran‐ sition is expected to appear at the places where cooling jets are injected in the main flow [63]. In downstream, a reverse transition process also could be recognized. This fact could affect the heat transfer distribution over surfaces of film‐cooled blades. On the profile pressure side ‐ if a separation bubble occurs, the reattached turbulent boundary layer may become again laminar like, (2) in **Figure 4**. In the case of lack of separation bubble, a forward transition zone, followed by a reverse one, in the rear part of the profile, could be observed, (1) in **Figure 4**.

**Figure 4.** Boundary layer development on high pressure turbine blades [65].

In high pressure turbines, the effect of transition on losses is usually small, because the aero‐ dynamic losses are mainly related to the turbulent flow development after the moment of transition. In low pressure turbines, the flow in interblade channels is characterized by low *Re*. Especially for gas turbines, as part of aircraft engines, the operating Reynolds numbers are low at high altitudes to begin with and a further decrease can cause separation before transition.

In regions where expansion occurs, the fluid is highly accelerated and the boundary layer has small thickness due to the favorable pressure gradients.

At high Reynolds numbers, transition occurs far in the upstream direction, flow is mainly turbulent over the profile. Near the trailing edge, in function of the blade profile geometry, the boundary layer will separate forced by turbulent levels. When Re number decreases, turbu‐ lent separation disappears and transition (the "bypass transition") moves in the downstream direction; at that moment losses are minimal. If *Re* number decreases more, laminar separa‐ tion ahead of the transition region could appear. In the case of no separation, the bubble is small enough so that the flow could reattach to the blade surface. In this case, aerodynamical losses are slightly higher than the previously described case. For lower Reynolds numbers the increase of laminar shear layer and transition length, until reattachment, before the trailing edge, is no longer possible and thus a complete separation occurs, **Figure 5**.

#### **3.2. What transition depends on?**

In general, the turbulence Reynolds number (*Re*θt) increases with increase in acceleration or with decrease in the free‐stream turbulence levels. The effect of acceleration is significant for low turbulence levels. However, for turbulence levels found in gas turbine stages, it has a negligible value. In the case of high turbulence levels, the transition onset is controlled by the free‐stream turbulence [65].

**Figure 5.** Transition on a low‐pressure turbine airfoil at various Reynolds number.

For accelerating flows, length of transition is variable for thermal and momentum boundary layers [70, 71].

In high pressure turbines, the effect of transition on losses is usually small, because the aero‐ dynamic losses are mainly related to the turbulent flow development after the moment of transition. In low pressure turbines, the flow in interblade channels is characterized by low *Re*. Especially for gas turbines, as part of aircraft engines, the operating Reynolds numbers are low at high altitudes to begin with and a further decrease can cause separation before

In regions where expansion occurs, the fluid is highly accelerated and the boundary layer has

At high Reynolds numbers, transition occurs far in the upstream direction, flow is mainly turbulent over the profile. Near the trailing edge, in function of the blade profile geometry, the boundary layer will separate forced by turbulent levels. When Re number decreases, turbu‐ lent separation disappears and transition (the "bypass transition") moves in the downstream direction; at that moment losses are minimal. If *Re* number decreases more, laminar separa‐ tion ahead of the transition region could appear. In the case of no separation, the bubble is small enough so that the flow could reattach to the blade surface. In this case, aerodynamical losses are slightly higher than the previously described case. For lower Reynolds numbers the increase of laminar shear layer and transition length, until reattachment, before the trailing

In general, the turbulence Reynolds number (*Re*θt) increases with increase in acceleration or with decrease in the free‐stream turbulence levels. The effect of acceleration is significant for low turbulence levels. However, for turbulence levels found in gas turbine stages, it has a negligible value. In the case of high turbulence levels, the transition onset is controlled by the

edge, is no longer possible and thus a complete separation occurs, **Figure 5**.

**Figure 5.** Transition on a low‐pressure turbine airfoil at various Reynolds number.

small thickness due to the favorable pressure gradients.

156 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

**3.2. What transition depends on?**

free‐stream turbulence [65].

transition.

The local point where boundary layer can separate is not in a close relation to the levels of free‐stream turbulence. In the case of observed increase in levels of free‐stream turbulence, the separation bubble size decreases and turbulent transition point moves in upstream in regard to the flow direction. The separation bubble decreases with increase of the Reynolds number [72]. At high free‐stream turbulence intensity, the streak structures can be observed in the upstream direction and are related to the place of boundary layer separation, showing that bypass transition of an attached boundary layer can be realized at the high Reynolds number. In the case of low free‐stream turbulence intensity, velocity fluctuations are seen within the shear layer of the separation bubble.

The roughness over streamed surfaces is nonuniform and characterized by significant varia‐ tions in streamwise and spanwise directions [73, 74]. Modifications in the behavior of the boundary layer due to the presence of definite values of surface roughness can decrease the aerodynamic efficiency [75–77] and can also increase the heat transfer [78–80]). Levels of heat transfer may also be affected by changes in material properties or in the case of eroded or broken protective coatings.

The effect of geometry curvature and/or streamlines curvature on the transition was studied and described in [81, 82]. In [81], it was found that a laminar boundary layer formed over a concave surface becomes unstable as a result of acting centrifugal forces, three‐dimensional disturbances and streamwise vortices presented in the boundary layer. Liepmann in [82] showed that transition on a convex surface is only slightly delayed, but can occur earlier over the concave surface.

The increase in transition Reynolds number is caused by the Görtler vortices, which can increase velocity gradients near the wall and thus can delay the transition. For highly curved surfaces, this effect dominates that one caused by turbulence. A concave curvature can decrease or can increase the transition The Reynolds number depends on the turbulence intensity and the curvature.

Heating or cooling can seriously affect boundary layer transition at low free‐stream turbu‐ lence. Heat transfer through a laminar boundary layer formed over the concave blade sur‐ face is influenced both by Taylor‐Görtler vortices and the main flow turbulence levels [83]. Transition occurs when these factors surpass the tendency of boundary layer to remain laminar in the presence of higher pressure gradients. If spot production is not affected by the heat transfer, at high free‐stream turbulence intensities, transition would not be observed.

Film cooling affects the boundary layer formed on the streamed surfaces of gas turbine blades. At places where cooling fluid is injected, holes are usually much larger than the boundary layer thickness, thus the injection of coolant through these holes disrupts the flow close to the surfaces and provides higher turbulence levels within the downstream developing boundary layer [63]. Therefore, it may be said that film cooling effect is to "trip" a laminar boundary layer and initiates transition to turbulence.

In the case of acceleration, sufficient to cause reverse transition in the downstream direction of the injection, the heat transfer intensity approaches that for laminar flow [69]. This implies that even though injection can initiate transition, a subsequent strong acceleration can cause the flow to become laminar again. Such a situation is common for film‐cooled blades of first gas turbine stages. Heat transfer measurements on a stator vane, presented in [85, 86], indi‐ cated that the behavior of the boundary layer transition along the suction side of the vane showed dependency to the film‐cooling injection place.

## **4. Numerical modeling of turbulence in turbine stages**

Many numerical and experimental research works have been performed, and various codes and approaches have been developed and applied, for the purposes of modeling and research of origin and mechanisms of laminar‐turbulent transition and how it exerts the fluid dynamics in turbine stages. However, the phenomenon in turbomachinery flows is not well understood. Transition modeling still limits the performance of nowadays CFD codes, and problems in estimation of the transition onset and extension of the transition affect the efficiency by sev‐ eral percent and the component life by more than an order of magnitude [87, 88]. Transition as a phenomenon and its understanding is of huge importance to arrive to appropriate design of blade geometry and increased aerodynamic performance.

A brief history of development of turbulence models dates back more than 140 years ago and is shown in [89].

Many of existing turbulence models are applied for modeling and research of turbulence effects in turbine stages. The performances of standard *k* – ε model, RNG *k* – ε model, realiz‐ able *k* – ε model, SST *k* – ω model, and LRR Reynolds stress transport models for the pur‐ poses of research on the convective heat transfer, during slot jet impingement over flat and concave cylindrical surfaces, were evaluated against available experimental data [90, 91]). Near‐wall models such as equilibrium wall function and two‐layer enhanced wall treatment were applied to the boundary layer to obtain physically correct results. When the impinge‐ ment surface is within the potential core of the jet, applied models overpredict the Nusselt number in the impingement region, and at the same time, *Nu* is not very correct in the wall jet region. The RNG *k* – ε model, applied together with the enhanced wall treatment, also the SST *k* – ω model, gives better Nusselt number distribution in comparison with other models for flat plate and concave surface impingement cases. However, mean velocity profiles are not well predicted by the SST *k* – ω model for the concave surface impingement case. Results for velocity profiles, obtained with RNG *k* – ε model, agree very well with the experiment. The Reynolds stress model could not give better prediction, compared to other eddy viscosity models [91].

In [58, 90], performance of various turbulence models to predict the convective heat transfer for slot jets impinged on flat and concave surfaces was under consideration. The outcomes as a result of application of more specific model are found very accurate in the case of an impingement surface placed outside of the jet core. In the case of surface placed inside the jet core, results obtained for *Nu* demonstrated larger discrepancies and variation in the impinge‐ ment region, as their models overpredicted the *Nu* in that region. However, prediction of values for *Nu* is fairly accurate in the wall jet region.

In the case of acceleration, sufficient to cause reverse transition in the downstream direction of the injection, the heat transfer intensity approaches that for laminar flow [69]. This implies that even though injection can initiate transition, a subsequent strong acceleration can cause the flow to become laminar again. Such a situation is common for film‐cooled blades of first gas turbine stages. Heat transfer measurements on a stator vane, presented in [85, 86], indi‐ cated that the behavior of the boundary layer transition along the suction side of the vane

Many numerical and experimental research works have been performed, and various codes and approaches have been developed and applied, for the purposes of modeling and research of origin and mechanisms of laminar‐turbulent transition and how it exerts the fluid dynamics in turbine stages. However, the phenomenon in turbomachinery flows is not well understood. Transition modeling still limits the performance of nowadays CFD codes, and problems in estimation of the transition onset and extension of the transition affect the efficiency by sev‐ eral percent and the component life by more than an order of magnitude [87, 88]. Transition as a phenomenon and its understanding is of huge importance to arrive to appropriate design

A brief history of development of turbulence models dates back more than 140 years ago and

Many of existing turbulence models are applied for modeling and research of turbulence effects in turbine stages. The performances of standard *k* – ε model, RNG *k* – ε model, realiz‐ able *k* – ε model, SST *k* – ω model, and LRR Reynolds stress transport models for the pur‐ poses of research on the convective heat transfer, during slot jet impingement over flat and concave cylindrical surfaces, were evaluated against available experimental data [90, 91]). Near‐wall models such as equilibrium wall function and two‐layer enhanced wall treatment were applied to the boundary layer to obtain physically correct results. When the impinge‐ ment surface is within the potential core of the jet, applied models overpredict the Nusselt number in the impingement region, and at the same time, *Nu* is not very correct in the wall jet region. The RNG *k* – ε model, applied together with the enhanced wall treatment, also the SST *k* – ω model, gives better Nusselt number distribution in comparison with other models for flat plate and concave surface impingement cases. However, mean velocity profiles are not well predicted by the SST *k* – ω model for the concave surface impingement case. Results for velocity profiles, obtained with RNG *k* – ε model, agree very well with the experiment. The Reynolds stress model could not give better prediction, compared to other eddy viscosity

In [58, 90], performance of various turbulence models to predict the convective heat transfer for slot jets impinged on flat and concave surfaces was under consideration. The outcomes as a result of application of more specific model are found very accurate in the case of an impingement surface placed outside of the jet core. In the case of surface placed inside the jet

showed dependency to the film‐cooling injection place.

**4. Numerical modeling of turbulence in turbine stages**

158 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

of blade geometry and increased aerodynamic performance.

is shown in [89].

models [91].

Other interesting numerical studies on rib‐roughened channels are related to measurements and simulations with standard *k* – Ɛ and nonlinear *k* – Ɛ turbulence models [92].

The predicted Nusselt number depends on the selected turbulence model. An improvement in the predicted Nusselt number was found when comparing the LES with a standard *k* – Ɛ turbulence model [93].

The standard *k* – Ɛ and RNG *k* – Ɛ models can accurately predict streaming over an impinge‐ ment surface [94].

As a result of the numerical modeling and analysis on turbulent flow field and heat transfer in 3D ribbed ducts, it is found that heat transfer predictions obtained using the *v*<sup>2</sup> – *f* cannot cover well the experimental data for the 3D ribbed duct. On the wall of the duct where ribs exist, predicted heat transfer agrees well with the experimental data for all configurations, heat transfer predictions on the smooth‐side wall do not cover well the experimental data. It is mainly due to the presence of strong secondary flow structures which might not be properly simulated with turbulence models based on eddy viscosity.

The standard *k* – Ɛ turbulence model with wall functions cannot predict very well heat transfer, due to the presence of large separation regions in the flow field [95]. In [96], it was found that the *k* – Ɛ model can give only reasonable qualitative agreement with the experi‐ mental data. The application of standard *k* – Ɛ model to complex 3D problems is computa‐ tionally expensive and leads to wrong results [97, 98]. Application of two‐layer *k* – Ɛ with the effective viscosity model gives bad predictions for heat transfer in rotating ribbed pas‐ sages [99]. Computations with low‐*Re* models could give good heat transfer predictions by introducing a differential version of the Yap length scale correction term [100].

Since low‐*Re* models cannot correctly capture the separation and reattachment that takes place between ribs, *v*<sup>2</sup> – *f* and Spalart‐Allmaras (S‐A) were taken under consideration in many research works. The *v*<sup>2</sup> – *f* turbulence model was successfully applied to separated flow in [100], to 3D boundary layers [101], impinging jets, [102, 103] and prediction of flow character‐ istics and heat transfer in 3D duct with ribs and in model configuration, resembling the tip of an axial turbine blade [104].

The model of Spalart and Allmaras [105] was proven to be robust and accurate in aerody‐ namic applications [106].

As discussed before, the phenomenon of transition is very complicated and depends on many parameters, such as free‐stream turbulence, roughness, curvature, heat transfer and film cool‐ ing, among others, and needs specific mathematical models and turbulence closures for the purposes of its research and analysis.

Due to the strong accelerations and decelerations of flow in turbine cascades, the local value of free‐stream turbulence, at the location of boundary layer transition onset, may significantly vary from first to the last turbine stage. Currently applied transition onset correlations involve data from many scientists, who have adopted different approaches to define the free‐stream turbulence values.

Laminar separation bubbles can result from laminar separation followed by sufficiently early transition in the separated shear layer and subsequent turbulent reattachment. Errors in pre‐ dicting the length of these bubbles will lead to failures in the blade design and wrong solu‐ tions. Early attempts at describing bubble development and separation, see [107, 108], were based on semi‐empirical models. In those models, constant pressure for the region of the separated laminar shear layer, instantaneous transition, and linear variations in free‐stream velocities during the phase of turbulent reattachment was assumed. An integral boundary layer computation procedure was applied, the location of the transition onset was found with correlations for separated laminar layer, in function of the momentum thickness and *Re*.

Flows with transition and separation phenomena could be modeled with application of con‐ temporary models for accurate description of all aerodynamic effects, which are expected to be observed, together with innovative and very correct transition model. Modeling of bubble dynam‐ ics is important for the purposes of research and prediction of separated flows with transition [63].

In [109–113] various predictive techniques were described in detail in the area of turbulence in turbines and stressed on the need for application of improved and correct transition modeling.

In the literature, many papers discuss experiments and their outcomes related to turbine blades. Mostly, they present research in more global aspect, a small number of experiments provide detailed results useful for turbulence modeling. This is related to the fact that it is very difficult to obtain sufficiently thick boundary layers to perform detailed measurements on the suction and pressure sides. Many results are obtained after modeling and measure‐ ments related to research on a flat plate with a pressure gradient, imposed by the external wall [70, 114, 115] for negative pressure gradients; also, after application of Görtler vortex on the concave plate [116]. Results for the streaming effects of blade convex side, are shown in [12]. Studies in [117] discuss results of measurements on the suction surface of blade under conditions of very low Reynolds number.

There are mainly two approaches used to model bypass transition in industry [65]. The first is to apply low‐Reynolds number turbulence models in which wall‐damping functions imple‐ mented into the turbulent transport equations were applied to obtain the moment when boundary layer transition will occur [129, 130]. Research activities have proved that this approach cannot predict very well the influence of various factors, such as pressure gradients, free‐stream turbulence, and wall roughness to predict the transition onset [131]. Damping functions, optimized to damp the turbulence in the viscous sublayer, cannot give reliable pre‐ diction of the transition when subjected different and complicated processes [132].

The second approach is application of experimental correlations related to the free‐stream turbulence intensity and to the transition Reynolds number, with included boundary layer momentum thickness [65, 118]. The last approach is proved as accurate, but very challeng‐ ing, ‐ actual momentum‐thickness, Reynolds numbers must be compared with their critical values, obtained from correlations, included into the mathematical model, applied for the purposes to arrive to physically correct numerical solution. There are additional difficulties related to application of unstructured mesh, not well‐defined boundary layer, and various approaches to attain numerical solution [64].

## **5. An example of modeling of turbine stage with twisted rotor blade with three different turbulence models**

The main target of this research in which geometry modeling, numerical set‐up, and conver‐ gence problem solution are described in detail in [119, 120] is to define the flow parameters distribution in a 3D turbine stage with twisted rotor blade. For the purposes of the turbulence modeling, the standard *k* – *ε* turbulence model, RNG *k* – *ε*, standard *k* – *ε*, for the case of research on radial gap, and RSM (Reynolds stress model) models are applied, in regard to the flow conditions.

The Reynolds stress model (RSM) is applicable for modeling effects of additional vortices, found in flow and shear stress effects over fluid particles [122].

The standard *k* – *ε* model gives quite good values, especially for the turbulent kinetic energy, in the core flow see [119, 121, 122]. In [123], results show that the advantage of using the RSM in regions of flow separation; however, the main flow features were still good enough, cap‐ tured by the *k* – *ε* model.

The RNG model gives the highest prediction of lift and maximal lift angle [124, 127].

The *k* – *ε* turbulence models are appropriate for flows characterized by high adverse pressure and intensive separation. This model allows for a more accurate near wall treatment with an automatic switch from wall function to low‐Reynolds number formulation, based on grid spacing see [125–128].

In the current study, it is found that depending on the specific flow feature, under consider‐ ation, different turbulence model have to be applied.

Qualitative results are shown in **Figure 6**–**Figure 9**.

vary from first to the last turbine stage. Currently applied transition onset correlations involve data from many scientists, who have adopted different approaches to define the free‐stream

160 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

Laminar separation bubbles can result from laminar separation followed by sufficiently early transition in the separated shear layer and subsequent turbulent reattachment. Errors in pre‐ dicting the length of these bubbles will lead to failures in the blade design and wrong solu‐ tions. Early attempts at describing bubble development and separation, see [107, 108], were based on semi‐empirical models. In those models, constant pressure for the region of the separated laminar shear layer, instantaneous transition, and linear variations in free‐stream velocities during the phase of turbulent reattachment was assumed. An integral boundary layer computation procedure was applied, the location of the transition onset was found with correlations for separated laminar layer, in function of the momentum thickness and *Re*.

Flows with transition and separation phenomena could be modeled with application of con‐ temporary models for accurate description of all aerodynamic effects, which are expected to be observed, together with innovative and very correct transition model. Modeling of bubble dynam‐ ics is important for the purposes of research and prediction of separated flows with transition [63]. In [109–113] various predictive techniques were described in detail in the area of turbulence in turbines and stressed on the need for application of improved and correct transition modeling. In the literature, many papers discuss experiments and their outcomes related to turbine blades. Mostly, they present research in more global aspect, a small number of experiments provide detailed results useful for turbulence modeling. This is related to the fact that it is very difficult to obtain sufficiently thick boundary layers to perform detailed measurements on the suction and pressure sides. Many results are obtained after modeling and measure‐ ments related to research on a flat plate with a pressure gradient, imposed by the external wall [70, 114, 115] for negative pressure gradients; also, after application of Görtler vortex on the concave plate [116]. Results for the streaming effects of blade convex side, are shown in [12]. Studies in [117] discuss results of measurements on the suction surface of blade under

There are mainly two approaches used to model bypass transition in industry [65]. The first is to apply low‐Reynolds number turbulence models in which wall‐damping functions imple‐ mented into the turbulent transport equations were applied to obtain the moment when boundary layer transition will occur [129, 130]. Research activities have proved that this approach cannot predict very well the influence of various factors, such as pressure gradients, free‐stream turbulence, and wall roughness to predict the transition onset [131]. Damping functions, optimized to damp the turbulence in the viscous sublayer, cannot give reliable pre‐

The second approach is application of experimental correlations related to the free‐stream turbulence intensity and to the transition Reynolds number, with included boundary layer momentum thickness [65, 118]. The last approach is proved as accurate, but very challeng‐ ing, ‐ actual momentum‐thickness, Reynolds numbers must be compared with their critical values, obtained from correlations, included into the mathematical model, applied for the

diction of the transition when subjected different and complicated processes [132].

turbulence values.

conditions of very low Reynolds number.

**Figure 6** shows velocity field distribution in the case of rotating rotor blade and activated the standard *k* – *ε* turbulence model.

**Figure 7** presents vorticity magnitude values, in radial direction, for the turbine stage under consideration.

Numerical results for pressure distribution, in the case of applied standard *k* – *ε* model, are shown in **Figure 8**.

**Figure 9** visualized vortices, in radial direction, due to difference between the pressure field values for hub and shroud sections in the turbine stage, in the case of applied Reynolds stress model (RSM). The area occupied by this vortex is bigger than the one formed in the case of standard *k* – *ε* turbulence model, **Figure 7**.

**Figure 6.** Flow velocity field distribution in control sections, in radial direction.

**Figure 7.** Vorticity magnitude values in control sections, in radial direction.

**Figure 7.** Vorticity magnitude values in control sections, in radial direction.

**Figure 6.** Flow velocity field distribution in control sections, in radial direction.

162 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

**Figure 8.** Static pressure distribution ‐ by control sections (a), for the turbine blade (b) in the case of rotating rotor blade.

**Figure 9.** Vorticity magnitude values by control sections, in radial direction.

The outcomes of the performed research are as follows:


## **6. Conclusion**

On the basis of previously performed numerical and experimental works by many research‐ ers and by the author, transition onset, and turbulence origin, their effects and impact on efficiency, flow parameters distribution, and possible blades design. Various modeling tech‐ niques, turbulence models and their application are discussed to obtain physically correct prediction of turbulence effects in flow past turbine blades. Various turbulence features, and fluid dynamics specifics, streaming of blades and their efficiency performance are discussed.

The chapter presents contemporary approaches to turbulence modeling and the adequacy of turbulence models to obtain flow characteristics, also.

This work could be very helpful for engineers working on prediction of transition onset, tur‐ bulence effects and their impact on the overall turbine performance. Moreover, it could be a basis for future research works related to innovative models and advanced numerical tech‐ niques to turbulence modeling and analysis.

## **Acknowledgements**

This research is supported by the National Science Fund of Bulgaria and the Technical University ‐ Varna.

## **Author details**

Galina Ilieva Ilieva

Address all correspondence to: galinaili@yahoo.com

Technical University, Varna, Bulgaria

## **References**

The outcomes of the performed research are as follows:

**Figure 9.** Vorticity magnitude values by control sections, in radial direction.

164 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

namic features at the leading edge.

**6. Conclusion**

• The RNG model is acceptable to study both the shear stress and streamlines curvature effects. It presents vortices formed at the trailing edge and also provides results for aerody‐

• In the case of applied RSM model, a relative decrease of 1.308% for turbine stage efficiency is observed. This is a result of taking into account of all pulsations and vortex structures near the wall regions, boundary layer separation, viscosity, and compressibility effects. • The RNG *k* – *ε* turbulence model leads to increased values for turbulent intensity and less turbulent viscosity. This is a prerequisite for decrease of the left‐hand side term values in the momentum equations, furthermore causes relative increase of stage efficiency with

On the basis of previously performed numerical and experimental works by many research‐ ers and by the author, transition onset, and turbulence origin, their effects and impact on efficiency, flow parameters distribution, and possible blades design. Various modeling tech‐ niques, turbulence models and their application are discussed to obtain physically correct prediction of turbulence effects in flow past turbine blades. Various turbulence features, and fluid dynamics specifics, streaming of blades and their efficiency performance are discussed.

0.147%, in a comparison with the case of implemented RSM turbulence model.


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## **RANS Modelling of Turbulence in Combustors**

## Lei‐Yong Jiang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68361

#### Abstract

Turbulence modelling is a major issue, affecting the precision of current numerical simulations, particularly for reacting flows. The RANS (Reynolds-averaged Navier-Stokes) modelling of turbulence is necessary in the development of advanced combustion systems in the foreseeable future. Therefore, it is important to understand advantages and limitations of these models. In this chapter, six widely used RANS turbulence models are discussed and validated against a comprehensive experimental database from a model combustor. The results indicate that all six models can catch the flow features; however, various degrees of agreement with the experimental data are found. The Reynolds stress model (RSM) gives the best performance, and the Rk-ε model can provide similar predictions as those from the RSM. The Reynolds analogy used in almost all turbulent reacting flow simulations is also assessed in this chapter and validated against the experimental data. It is found that the turbulent Prandtl/Schmidt number has a significant effect on the temperature field in the combustor. In contrast, its effect on the velocity field is insignificant in the range considered (0.2–0.85). For the present configuration and operating conditions, the optimal turbulent Prandtl/Schmidt number is 0.5, lower than the traditionally used value of 0.6–0.85.

Keywords: turbulence modelling, RANS, momentum and scalar modelling, combustor

## 1. Introduction

Turbulence is one of the principal unsolved problems in physics today [1] and its modelling is one of the major issues that affect the precision of current numerical simulations in engineering applications, particularly for reacting flows. Turbulence is characterized by irregularity or randomness, diffusion, vortices and viscous dissipation, and involves a wide range of time and length scales. Despite the rapid development of computing power, large eddy simulations are limited to benchmark cases with relatively simple geometries, while direct numerical simulations of turbulent flows remain practical only at low Reynolds numbers [2–4]. It is particularly

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

true for turbulent reacting flows. Even without turbulence, combustion is a complicated process and can consist of hundreds of species and thousands of element reactions, where numerical difficulties occur [5]. Consequently, it is necessary to utilize turbulence models for the development of advanced combustion systems in the foreseeable future.

Much effort has been made to the development of turbulence modelling in the last six decades. Advances focused on constant density flows have been reviewed or described by a number of researchers [6–8] and brought up to date for the second momentum closure in reacting flows [9]. Various algebraic, one- and two-equation turbulence models were systematically evaluated [2, 3] against a number of well-documented non-reacting flows, including freeshear, boundary-layer and separated flows. Some guidelines regarding applications of these models were provided. Recently, six eddy-viscosity and two variants of Reynolds stress turbulence models were used to study the flow field around a ship hull [10]. It was found that the two Reynolds stress models were able to reproduce all the salient features and the predicted Reynolds stresses and turbulence kinetic energy were in good agreement with the experimental results. Despite the considerable progresse in turbulence modelling, no universal turbulence model is available for all flows at the current time. Therefore, it is important to understand advantages and shortcomings of these models and select the best one for defined engineering problems.

In contrast to turbulence momentum transfer modelling, studies on turbulence scalar transfer modelling are limited, but are of great engineering interest. From the 1970s (CFD pioneer work) up to present, in almost all turbulent reacting flow simulations, the Reynolds analogy concept has been used to model turbulence scalar transfers (mixture fraction, species and energy or temperature). In this approach, the turbulent Prandtl (Prt) and Schmidt (Sct) numbers are used to link the turbulence scalar transfers to the momentum transfer that is calculated by a selected turbulence model. The main advantage of this approach is that the turbulence scalar transfers can be effectively computed from the modelled momentum transfer without solving a full-second moment closure for both momentum and scalar transportations.

The Reynolds analogy concept was first postulated over a century ago on the similarity between wall shear and heat flux in boundary layers [11]. This original hypothesis has been considerably amended and applied to general 3D (three-dimensional) turbulent heat and species transfers [12, 13]. Recently, its applications to high-Mach-number boundary layers [14], turbine flows [15] and film cooling [16] have been studied.

In most turbulent reacting or mixing flow simulations, it has become a common practice to set Prt ¼ Sct [17]. Traditionally, a constant value of Prt ¼ Sct ¼ 0.6–0.85 has been used in jet and gas turbine flows [18–21], and these values are consistent with numerous measurements performed in the 1930s–1980s [12, 13]. However, low Prt and Sct numbers from 0.20 to 0.5 have been used by a large number of authors for simulating gas turbine combustors. Effort was made to validate a two-dimensional finite difference code against a number of isothermal and reacting flow measurements [22], and a value of Prt ¼ Sct ¼ 0.5 was recommended for recirculation zone simulations. The numerical results of five RQL (rich burn, quick quench and lean burn) low-emission combustor designs were calibrated against CARS (coherent anti-Stokes Raman spectroscopy) temperature measurements, and good agreement was found by using Prt ¼ Sct ¼ 0.2 [23]. An entire combustor from the compressor diffuser exit to the turbine inlet was successfully studied, and a low value of 0.25 was used for Prt and Sct since it consistently demonstrated better agreement with the fuel/air mixing results [24]. Moreover, the turbulence scalar mixing of a gaseous jet issued into a cross airflow was investigated, and in comparison with the available experimental data, Prt ¼ Sct ¼ 0.2 was recommended [25]. The above examples suggest that for reacting flow modelling, the scalar transfer modelling or Reynolds analogy has to be investigated.

Although there are a large number of publications on numerical simulations of practical combustion systems [for example, 20–24, 26–29], systematic assessment and validation of turbulence models in combustor flow fields against well-defined comprehensive experimental results are rare.

This chapter focuses on most widely used turbulence models in practical engineering, that is RANS (Reynolds-averaged Navier-Stokes) models, including the Reynolds stress model (RSM), a second moment closure and five popular two-equation eddy-viscosity models, the standard k-ε, renormalization group (RNG) k-ε, realizable k-ε (Rk-ε), standard k-ω and shearstress transport (SST) k-ω model. The contents are based on the author's experience and publications accumulated over many years on turbulent reacting flow studies, related to gas turbine combustion systems [30–36].

A benchmark case, a model combustor, is used as technology demonstration. Although the model combustor geometry is simple, the complex phenomena, such as jet flows, wall boundary layers, shear layers, flow separations and reattachments, as well as recirculation zones, are involved, which are fundamental features in practical combustion systems. In addition, because the model combustor geometry is much simpler than practical combustors, its boundary conditions can be well defined. More importantly, a comprehensive experimental database is available, and then the assessment of the above issues is appropriate.

In the following sections, firstly, the governing equations, turbulence models and Reynolds analogy are discussed and then the other physical models and experimental measurements are briefly described, followed by the benchmark results. Finally, a few conclusions are highlighted.

## 2. Governing equations, turbulence models and Reynolds analogy

#### 2.1. Governing equations

true for turbulent reacting flows. Even without turbulence, combustion is a complicated process and can consist of hundreds of species and thousands of element reactions, where numerical difficulties occur [5]. Consequently, it is necessary to utilize turbulence models for

Much effort has been made to the development of turbulence modelling in the last six decades. Advances focused on constant density flows have been reviewed or described by a number of researchers [6–8] and brought up to date for the second momentum closure in reacting flows [9]. Various algebraic, one- and two-equation turbulence models were systematically evaluated [2, 3] against a number of well-documented non-reacting flows, including freeshear, boundary-layer and separated flows. Some guidelines regarding applications of these models were provided. Recently, six eddy-viscosity and two variants of Reynolds stress turbulence models were used to study the flow field around a ship hull [10]. It was found that the two Reynolds stress models were able to reproduce all the salient features and the predicted Reynolds stresses and turbulence kinetic energy were in good agreement with the experimental results. Despite the considerable progresse in turbulence modelling, no universal turbulence model is available for all flows at the current time. Therefore, it is important to understand advantages and shortcomings of these models and select the best one for defined engineering

In contrast to turbulence momentum transfer modelling, studies on turbulence scalar transfer modelling are limited, but are of great engineering interest. From the 1970s (CFD pioneer work) up to present, in almost all turbulent reacting flow simulations, the Reynolds analogy concept has been used to model turbulence scalar transfers (mixture fraction, species and energy or temperature). In this approach, the turbulent Prandtl (Prt) and Schmidt (Sct) numbers are used to link the turbulence scalar transfers to the momentum transfer that is calculated by a selected turbulence model. The main advantage of this approach is that the turbulence scalar transfers can be effectively computed from the modelled momentum transfer without

solving a full-second moment closure for both momentum and scalar transportations.

turbine flows [15] and film cooling [16] have been studied.

The Reynolds analogy concept was first postulated over a century ago on the similarity between wall shear and heat flux in boundary layers [11]. This original hypothesis has been considerably amended and applied to general 3D (three-dimensional) turbulent heat and species transfers [12, 13]. Recently, its applications to high-Mach-number boundary layers [14],

In most turbulent reacting or mixing flow simulations, it has become a common practice to set Prt ¼ Sct [17]. Traditionally, a constant value of Prt ¼ Sct ¼ 0.6–0.85 has been used in jet and gas turbine flows [18–21], and these values are consistent with numerous measurements performed in the 1930s–1980s [12, 13]. However, low Prt and Sct numbers from 0.20 to 0.5 have been used by a large number of authors for simulating gas turbine combustors. Effort was made to validate a two-dimensional finite difference code against a number of isothermal and reacting flow measurements [22], and a value of Prt ¼ Sct ¼ 0.5 was recommended for recirculation zone simulations. The numerical results of five RQL (rich burn, quick quench and lean burn) low-emission combustor designs were calibrated against CARS (coherent anti-Stokes

the development of advanced combustion systems in the foreseeable future.

176 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

problems.

The first-moment Favre-averaged conservation equations for mass, momentum, species, mixture fraction and total enthalpy may be expressed in a coordinate-free form as [37–39]

$$\nabla \cdot (\overline{\rho} \,\, \tilde{\mathbf{V}}) = \mathbf{0} \tag{1}$$

$$\nabla \cdot (\overline{\rho} \,\, \tilde{\mathbf{V}} \tilde{\mathbf{V}}) = -\nabla \, \overline{\rho} + \nabla \cdot \mathbf{T} - \nabla \cdot (\overline{\rho \,\, \mathbf{v''} \mathbf{v''}}) \tag{2}$$

$$\nabla \cdot (\overline{\rho} \,\, \tilde{\mathbf{V}} \tilde{Y}) = \nabla \cdot (\rho D\_i \nabla \tilde{Y}\_i) - \nabla \cdot (\overline{\rho \,\, \mathbf{v''} Y''\_i}) + \omega\_i \tag{3}$$

$$\nabla \cdot (\overline{\rho} \,\, \tilde{\mathbf{V}} \tilde{f}) = \nabla \cdot (\rho D \nabla \tilde{f}) - \nabla \cdot (\overline{\rho \,\, \mathbf{v}'' f''}) \tag{4}$$

$$\nabla \cdot (\overline{\rho} \,\, \tilde{\mathbf{V}} \tilde{h}) = \nabla \cdot \left(\frac{\mu}{P} \,\, \nabla \tilde{h}\right) - \nabla \cdot (\overline{\rho \,\, \mathbf{v}'' \boldsymbol{h}''}) + \mathbf{S}\_h \tag{5}$$

where V~ stands for the mean velocity vector, ρ represents the mean density, v<sup>00</sup> is the fluctuation velocity vector, <sup>T</sup> <sup>¼</sup> <sup>μ</sup>½∇V<sup>~</sup> þ ð∇V<sup>~</sup> <sup>Þ</sup> <sup>T</sup>� � <sup>2</sup> <sup>3</sup> <sup>μ</sup><sup>∇</sup> � V I <sup>~</sup> represents the viscous stress tensor with I a unit tensor and µ molecular viscosity, Yi stands for the mass fraction of the ith species, f denotes the mixture fraction, p is the pressure, h stands for the total enthalpy, ρ v00v<sup>00</sup> is the Reynolds stresses and D and Pr are molecular diffusivity and Prandtl number, respectively. Note that in all equations, the symbols with straight overbars are time-averaged variables, and the symbols with curly overbars stand for Favre-averaged variables.

The source terms in Eqs. (2)–(5) should be modelled or determined in order to close these equations. A combustion model is chosen to compute the species source term, ω<sup>i</sup> in Eq. (3). In Eq. (5), the energy source term, SH contains radiation heat transfer and viscous heating. Turbulence momentum transfer or Reynolds stresses, ρ v00v<sup>00</sup> , in Eq. (2) are calculated by a selected turbulence model, whilst turbulence scalar transfers in Eqs. (3)–(5), ρ v00Y<sup>00</sup> , ρ v00f <sup>00</sup> and ρ v00h<sup>00</sup> are computed by means of Reynolds analogy. That is

$$-\nabla \cdot \overline{\rho \text{ } \nabla'' \phi''} \cong \nabla \cdot \left(\frac{\mu\_t}{\Gamma t} \nabla \tilde{\phi}\right) \tag{6}$$

where φ stands for species, mixture fraction or enthalpy, µt is the turbulence viscosity that is computed from the chosen turbulence model and Γt represents the turbulence Prandtl (Prt) or Schmidt (Sct) number. Note that in Eq. (6), the turbulence scalar transfer coefficients, µt/Гt, are simply the products of the turbulence momentum transfer coefficient (µt) and 1/Гt.

#### 2.2. Turbulence models

The main features of these turbulence models are outlined here, and detailed description and formation of each model can be found in the references mentioned below. The Boussinesq hypothesis is utilized to model Reynolds stresses for the five two-equation eddy-viscosity turbulence models,

$$-\overline{\rho}\,\overline{u\_i\,^\circ u\_j}^\* = \mu\_t \left(\frac{\partial \mathcal{U}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \mathcal{U}\_j}{\partial \mathbf{x}\_i}\right) - \frac{2}{3}\delta\_{\vec{\eta}} \left(\mu\_t \frac{\partial \mathcal{U}\_k}{\partial \mathbf{x}\_k} + \rho \,\vec{k}\right) \tag{7}$$

Using this approach, the turbulence viscosity, µt, for high Reynolds number flows is given by the expression for the standard k-ε, RNG, Rk-ε models,

$$
\mu\_t = \frac{\overline{\rho} \, \mathrm{C}\_{\mu} k^2}{\varepsilon} \tag{8}
$$

where Cµ is a constant and the values of turbulence kinetic energy, k, and dissipation rate, ε, are calculated from a pair of transport differential equations. The detailed description of the standard k-ε turbulence model is given in Ref. [40], which represents a pioneer work in turbulence modelling.

<sup>∇</sup> � ð<sup>ρ</sup> <sup>V</sup><sup>~</sup> <sup>Y</sup><sup>~</sup> Þ ¼ <sup>∇</sup> � ðρDi∇Y~<sup>i</sup>

178 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

<sup>∇</sup> � ð<sup>ρ</sup> <sup>V</sup><sup>~</sup> <sup>~</sup>hÞ ¼ <sup>∇</sup> � <sup>μ</sup>

fluctuation velocity vector, <sup>T</sup> <sup>¼</sup> <sup>μ</sup>½∇V<sup>~</sup> þ ð∇V<sup>~</sup> <sup>Þ</sup>

ρ v00h<sup>00</sup> are computed by means of Reynolds analogy. That is

�ρ ui}uj} ¼ μ<sup>t</sup>

the expression for the standard k-ε, RNG, Rk-ε models,

Favre-averaged variables.

2.2. Turbulence models

turbulence models,

<sup>∇</sup> � ð<sup>ρ</sup> <sup>V</sup><sup>~</sup> <sup>~</sup><sup>f</sup> Þ ¼ <sup>∇</sup> � ðρD∇~<sup>f</sup> Þ � <sup>∇</sup> � ð<sup>ρ</sup> <sup>v</sup>00<sup>f</sup>

P r <sup>∇</sup>~<sup>h</sup> 

where V~ stands for the mean velocity vector, ρ represents the mean density, v<sup>00</sup> is the

tensor with I a unit tensor and µ molecular viscosity, Yi stands for the mass fraction of the ith species, f denotes the mixture fraction, p is the pressure, h stands for the total enthalpy, ρ v00v<sup>00</sup> is the Reynolds stresses and D and Pr are molecular diffusivity and Prandtl number, respectively. Note that in all equations, the symbols with straight overbars are time-averaged variables, and the symbols with curly overbars stand for

The source terms in Eqs. (2)–(5) should be modelled or determined in order to close these equations. A combustion model is chosen to compute the species source term, ω<sup>i</sup> in Eq. (3). In Eq. (5), the energy source term, SH contains radiation heat transfer and viscous heating. Turbulence momentum transfer or Reynolds stresses, ρ v00v<sup>00</sup> , in Eq. (2) are calculated by a

selected turbulence model, whilst turbulence scalar transfers in Eqs. (3)–(5), ρ v00Y<sup>00</sup> , ρ v00f

�<sup>∇</sup> � <sup>ρ</sup> <sup>v</sup>00φ<sup>00</sup> ffi <sup>∇</sup> � <sup>μ</sup><sup>t</sup>

simply the products of the turbulence momentum transfer coefficient (µt) and 1/Гt.

∂Ui ∂xj þ ∂Uj ∂xi 

where φ stands for species, mixture fraction or enthalpy, µt is the turbulence viscosity that is computed from the chosen turbulence model and Γt represents the turbulence Prandtl (Prt) or Schmidt (Sct) number. Note that in Eq. (6), the turbulence scalar transfer coefficients, µt/Гt, are

The main features of these turbulence models are outlined here, and detailed description and formation of each model can be found in the references mentioned below. The Boussinesq hypothesis is utilized to model Reynolds stresses for the five two-equation eddy-viscosity

Using this approach, the turbulence viscosity, µt, for high Reynolds number flows is given by

� 2 3 δij μ<sup>t</sup> ∂Uk ∂xk

þ ρ k 

Γt <sup>∇</sup>φ<sup>~</sup>

<sup>T</sup>� � <sup>2</sup>

Þ � ∇ � ðρ v00Y<sup>00</sup>

<sup>3</sup> μ∇ � V I

<sup>i</sup>Þ þ ω<sup>i</sup> ð3Þ

00Þ ð4Þ

~ represents the viscous stress

<sup>00</sup> and

ð6Þ

ð7Þ

� ∇ � ðρ v00h00Þ þ Sh ð5Þ

The RNG turbulence model is originated from a re-normalization group theory [41]. The major difference between the standard k-ε and RNG models is that the coefficient of the destruction term in the turbulence dissipation rate equation is not a constant, but a function of flow mean strain rate and turbulence parameters of k and ε. Moreover, an analytical formula to account for variations of turbulent Prandtl and Schmidt numbers for the energy and species equations is provided. These modifications have made this model more responsive to the effects of strain rate and streamline curvature than the standard k-ε model. Using this model, good agreement between the numerical and experimental results is observed for the isothermal flow over a backward facing step [41].

The main improvement of the Rk-ε turbulence model is that Reynolds stresses comply with physics. That is, turbulence normal stresses always remain positive and shear stresses obey Schwarz inequality [42]. In addition, instead of a constant, Cµ in Eq. (8) is a function of the turbulence parameters and mean strain and rotation rates. The turbulence dissipation rate equation is obtained from the dynamic equation of the mean-square vorticity fluctuation at high Reynolds numbers. Some advantages have been observed with this model for flows with separations and recirculation zones, as well as jet spread rates over the standard k-ε model [42, 43].

The details of the standard k-ω and SST models are given in Refs. [2, 3, 44]. The corresponding turbulence viscosity for high Reynolds number flows is obtained by the following two expressions, respectively,

$$
\mu\_t = \frac{\overline{\rho} \, k}{\omega} \tag{9}
$$

$$
\mu\_t = \frac{\overline{\rho}k}{\omega} \frac{1}{\max\left[1, \frac{SF}{a\omega}\right]}\tag{10}
$$

In Eqs. (9) and (10), a pair of transportation differential equations is used to obtain turbulence kinetic energy, k, and specific dissipation rate, ω; F equals zero in the free-shear layer and one in the near-wall region; 'a' is a constant and S stands for the strain rate magnitude.

For the above two-equation models, the linear relationship of Reynolds stresses with mean strain rate and isotropic eddy viscosity is presumed, as implied in Eqs. (7)–(10). For turbulent flow simulations with the RSM, a transportation differential equation is solved for each Reynolds stress component in the flow field. Therefore, it is expected that this second-moment closure model is more 'applicable' than the two-equation, eddy-viscosity models. To convert Reynolds stress equations into a closed set of equations, unknown terms must be modelled by mean flow variables and/or Reynolds stresses [45].

#### 2.3. Reynolds analogy

By reducing the three-dimensional conservation equations (2)–(5) to two-dimensional steady boundary flows and neglecting the streamwise pressure gradient, molecular viscous terms and source terms, Eqs. (11) and (12) are obtained and the rationale and limitation of Reynolds analogy can be revealed.

$$
\overline{\rho}\check{u}\frac{\partial\check{u}}{\partial x} + \overline{\rho}\check{v}\frac{\partial\check{u}}{\partial y} \approx \frac{\partial}{\partial y}\left(\mu\_t\frac{\partial\check{u}}{\partial y}\right) \tag{11}
$$

$$
\overline{\rho}\check{u}\frac{\partial\check{\phi}}{\partial x} + \overline{\rho}\check{v}\frac{\partial\check{\phi}}{\partial y} \approx \frac{\partial}{\partial y}\left(\frac{\mu\_t}{\Gamma t}\frac{\partial\check{\phi}}{\partial y}\right) \tag{12}
$$

In Eqs. (11) and (12), the turbulence viscosity concept is applied to both streamwise momentum and scalar transfers. With Гt ¼ 1, the two equations become identical. That is, under appropriate boundary conditions, the solution of all these flow parameters can be obtained from a single-partial differential equation or the momentum and scalar fields are similar. For wall boundary flows, the original form of Reynolds analogy can be deduced [15],

$$\frac{\text{2St}}{\text{c}\_{\text{f}}} = \frac{(h/\rho \text{ C}\_{p} \text{ U}\_{\text{es}})}{(\text{\tau}\_{w}/\rho \text{ U}\_{\text{es}}^{2})} \approx 1\tag{13}$$

where cf <sup>¼</sup> <sup>τ</sup>w=ð0:5ρU<sup>2</sup> <sup>∞</sup>Þ represents the wall friction coefficient, and St ¼ h=ðρCpU∞Þ denotes the Stanton number. From Eq. (13), the turbulence heat transfer coefficient can be calculated from the measured pressure loss owing to friction in the flow.

The above analysis suggests that the Reynolds analogy method can be used to adequately calculate turbulence scalar transfers in a boundary type of flows, such as free jets, wall boundary layers and shear layers, where the effects of the streamwise pressure gradient, viscous and source terms are minor. However, it should be cautious to apply it to general complex three-dimensional flows. Its failure to disturbed turbulent boundary flows has been reported by a number of authors [46–48].

#### 2.4. Other physical models and numerical methods

For combustion modelling, the laminar flamelet, probability density function (PDF) and eddydissipation (EDS) models were considered. The previous benchmark study on combustion models indicated that for mixing-control dominated diffusion flames the temperature and velocity fields could be fairly well captured by these three combustion models [33].

A major advantage of the flamelet model over the probability density function and eddydissipation models is that detailed more realistic chemical kinetics can be incorporated into turbulent reacting flows [49]. For the present case, the propane-air chemical mechanism from Ref. [50] was used. This mechanism was consisted of 228 element reactions, and 31 chemical species, i.e. O2, N2, CO2, H2O, C3H8, CH4, H2, CHO, CH2O, CH2CO, CH3CO, CH3CHO, C2H, C2H2, C2H3, C2H4, C2H5, C2H6, C3H6, O, OH, H, H2O2, HO2, CO, CH, CH2, CH3, C2HO, N\*C3H7 and I\*C3H7.

stress equations into a closed set of equations, unknown terms must be modelled by mean flow

By reducing the three-dimensional conservation equations (2)–(5) to two-dimensional steady boundary flows and neglecting the streamwise pressure gradient, molecular viscous terms and source terms, Eqs. (11) and (12) are obtained and the rationale and limitation of Reynolds

In Eqs. (11) and (12), the turbulence viscosity concept is applied to both streamwise momentum and scalar transfers. With Гt ¼ 1, the two equations become identical. That is, under appropriate boundary conditions, the solution of all these flow parameters can be obtained from a single-partial differential equation or the momentum and scalar fields are similar. For

> <sup>¼</sup> <sup>ð</sup>h=<sup>ρ</sup> Cp <sup>U</sup>∞<sup>Þ</sup> <sup>ð</sup>τw=<sup>ρ</sup> <sup>U</sup><sup>2</sup>

the Stanton number. From Eq. (13), the turbulence heat transfer coefficient can be calculated

The above analysis suggests that the Reynolds analogy method can be used to adequately calculate turbulence scalar transfers in a boundary type of flows, such as free jets, wall boundary layers and shear layers, where the effects of the streamwise pressure gradient, viscous and source terms are minor. However, it should be cautious to apply it to general complex three-dimensional flows. Its failure to disturbed turbulent boundary flows has been

For combustion modelling, the laminar flamelet, probability density function (PDF) and eddydissipation (EDS) models were considered. The previous benchmark study on combustion models indicated that for mixing-control dominated diffusion flames the temperature and

A major advantage of the flamelet model over the probability density function and eddydissipation models is that detailed more realistic chemical kinetics can be incorporated into

velocity fields could be fairly well captured by these three combustion models [33].

wall boundary flows, the original form of Reynolds analogy can be deduced [15],

2St cf

from the measured pressure loss owing to friction in the flow.

<sup>∂</sup><sup>y</sup> <sup>μ</sup><sup>t</sup>

μt Γt

∂u~ ∂y � �

∂φ~ ∂y !

<sup>∞</sup>Þ represents the wall friction coefficient, and St ¼ h=ðρCpU∞Þ denotes

<sup>∞</sup><sup>Þ</sup> <sup>≈</sup> <sup>1</sup> <sup>ð</sup>13<sup>Þ</sup>

ð11Þ

ð12Þ

<sup>ρ</sup>u<sup>~</sup> <sup>∂</sup>u<sup>~</sup> ∂x þ ρ~v ∂u~ <sup>∂</sup><sup>y</sup> <sup>≈</sup> <sup>∂</sup>

180 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

<sup>ρ</sup>u<sup>~</sup> <sup>∂</sup>φ<sup>~</sup> ∂x þ ρ~v ∂φ~ <sup>∂</sup><sup>y</sup> <sup>≈</sup> <sup>∂</sup> ∂y

variables and/or Reynolds stresses [45].

2.3. Reynolds analogy

analogy can be revealed.

where cf <sup>¼</sup> <sup>τ</sup>w=ð0:5ρU<sup>2</sup>

reported by a number of authors [46–48].

2.4. Other physical models and numerical methods

The PDF combustion model is based on the mixture fraction approach with an assumption of fast chemistry [51]. It offers some advantages over the EDS or EDS-finite-rate models and allows intermediate (radical) species prediction, more thorough turbulence-chemistry coupling and dissociation effect. Eighteen species were considered for the PDF model, including C3H8, CO2, H2O, O2, N2, CO, HO, H, O, H2, C3H6, C2H6, C2H4, CH4, CH3, CH2, CH and C(s). The selection of these species was based on the basic chemical kinetics and requirements for pollutant predictions [52]. As full chemical equilibrium gave considerable errors in temperature on the rich side of hydrocarbon flames [53, 54], to avoid this, a partial equilibrium approach was applied in the rich flame region. When the instantaneous equivalence ratio exceeded 1.75, the combustion reaction was considered extinguished and unburned fuel coexisted with reacted products.

The EDS model is widely accepted in diffusion flame modelling [53]. For this model, the reaction rate is governed by turbulent mixing, or the large-eddy mixing timescale, k/ε [55]. In the present case, the heat radiation from the hot gas mixture to the surroundings was computed by the discrete ordinates model [56], where the local species mass fractions were used to calculate the absorption coefficient of gaseous mixture in the flow. At all wall boundaries, an enhanced wall boundary approach was utilized, where the traditional two-layer zonal model was improved by smoothly combining the viscous sub-layer with the fully turbulent region.

The specific heat of species was calculated by polynomials as a function of temperature. For the case of the flamelet and PDF models, the polynomials were determined from the JANAF tables [57]; while in the case of the EDS model, the polynomials from Rose and Cooper [58] were used, where the chemical dissociation was considered. For other thermal properties such as molecular viscosity, thermal conductivity and diffusivity, the values of air at 900 K were used.

A segregated solver with a second-order-accuracy scheme from a commercial software, Fluent, was used to resolve the flow fields. The results were well-converged, and the normalized residuals of the flow variables were about or less than 10–<sup>5</sup> for all test cases. The axial velocities monitored in shear layers of the flow fields were unchanged for the first four digits. A LINUX cluster with eight nodes and 64-GB RAM/node was employed to perform all numerical simulations.

## 3. Benchmark experimental measurements

A series of experimental measurements on a diffusion flame model combustor were carried out at the National Research Council of Canada (NRCC). The results provided a comprehensive database for the evaluation and development of various physical models, including mean and fluctuation velocity components, mean temperature, wall temperature, radiation heat flux through walls, as well as species concentrations [59].

The test apparatus and model combustor are shown in Figure 1 where all dimensions are in mm. The model combustor consists of the air and fuel inlet section, combustion chamber and contracted exhaust section. Fuel entered the combustion chamber through the centre of the bluff body, while air flowed into the chamber around a disc flame-holder. The combustor was mounted on a three-axis traversing unit with a resolution of 100 µm. Fuel was commercial grade propane, and air was from a dry air supply. Both air and fuel flow rates were regulated by Sierra Side-Trak mass-flow controllers with 2% accuracy of the full scale (100 l/min for fuel and 2550 l/min for air).

A 25.4-mm thick fibre blanket of Al2O3 was wrapped around the combustion chamber in order to reduce the heat losses through the combustor walls. The optical and physical access to the combustion chamber was through four windows. The viewing area of the windows measured 17 mm in width, 342 mm in length and 44–388 mm from the disk flame-holder in the axial direction. Interchangeable sets of stainless steel and fused silica windows were used, the former for physical probing with gas sampling probes, radiometers and thermocouples, and the latter for optical probing with a laser Doppler anemometer (LDA).

Both two- and three-component LDA systems operating in a back-scattering mode were used to measure flow velocities. The restricted optical access in the lower section of the combustion chamber forced the use of a single fibre head to measure axial and tangential velocities, and a

Figure 1. The test apparatus and model combustor.

complete three-component LDA system was applied in the upper section of the chamber. An uncoated 250-µm diameter, type 'S' thermocouple held by a twin-bore ceramic tube was used to measure gas temperatures in the flow field. The wall temperatures were measured by the thermocouples embedded in and flush with the wall. Gas sample was obtained by a sampling probe and the species were measured using a Varian model 3400 gas chromatograph. The measured major species were CO, CO2, H2O and C3H8. In addition, minor species fractions, such as CH4 and C2H2, were also obtained. NO<sup>x</sup> and NO were collected through the same probe but analysed using a Scintrix NO<sup>x</sup> analyser.

## 4. Application of RANS turbulence models

The test apparatus and model combustor are shown in Figure 1 where all dimensions are in mm. The model combustor consists of the air and fuel inlet section, combustion chamber and contracted exhaust section. Fuel entered the combustion chamber through the centre of the bluff body, while air flowed into the chamber around a disc flame-holder. The combustor was mounted on a three-axis traversing unit with a resolution of 100 µm. Fuel was commercial grade propane, and air was from a dry air supply. Both air and fuel flow rates were regulated by Sierra Side-Trak mass-flow controllers with 2% accuracy of the full scale (100 l/min for fuel

A 25.4-mm thick fibre blanket of Al2O3 was wrapped around the combustion chamber in order to reduce the heat losses through the combustor walls. The optical and physical access to the combustion chamber was through four windows. The viewing area of the windows measured 17 mm in width, 342 mm in length and 44–388 mm from the disk flame-holder in the axial direction. Interchangeable sets of stainless steel and fused silica windows were used, the former for physical probing with gas sampling probes, radiometers and thermocouples, and

Both two- and three-component LDA systems operating in a back-scattering mode were used to measure flow velocities. The restricted optical access in the lower section of the combustion chamber forced the use of a single fibre head to measure axial and tangential velocities, and a

the latter for optical probing with a laser Doppler anemometer (LDA).

182 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

and 2550 l/min for air).

Figure 1. The test apparatus and model combustor.

## 4.1. Computational domain and boundary conditions

The computational domain covers the entire combustor flow field from the fuel and air inlets to the exhaust exit, as shown in Figure 2. The two-dimensional quadrilateral meshes were generated because of the axisymmetric geometry. To resolve the recirculation region, fine grids were created behind the flame-holder in the combustion chamber. Fine grids were also laid in the shear layers between the fuel and air jets and recirculation region, and the gap between the flame-holder edge and air inlet section wall as well. In the solid stainless steel wall and ceramic blanket regions, coarse grids were generated. A number of meshes were created and tested to check mesh independence issue. Finally, a mesh with 74,100 cells was used for most of the simulations. The skewness in the flow-field domain was less than 0.2 and the aspect ratio was less than 12 for 99.5% cells. Effort was made to keep the wall parameter, y<sup>þ</sup> ( ffiffiffiffiffiffiffiffiffiffi τw=ρ p y = υ), in the desired range of 30–60.

The air and fuel flow rates were 550 and 16.2 g/s, respectively, and the corresponding overall equivalence ratio was 0.46. The inlet temperature for both flows was 293 K. The Reynolds number based on the flame-holder diameter and air entry velocity was 1.9 � <sup>10</sup><sup>5</sup> . To estimate Reynolds stress components and turbulence dissipation or specific-dissipation rates at the fuel and air inlets, the turbulence intensity of 10% and hydraulic diameters were specified. The effect of the inlet turbulence intensity assignment on the flow field was examined by comparing the simulation results from three inlet turbulence intensity settings, 2, 5 and 10%. The effect is only observable for turbulence variables, for example, the change in turbulent kinetic energy was seen in the fuel inlet path and a small portion at x ≈ 80 mm along the combustor axis, with a maximum difference of 2.3%. For the mean flow variables along the combustor axis, such as temperature and axial velocity, the deviation from the experimental data was negligibly small.

At the upstream edges of the ceramic insulation and combustion chamber and at the inlet section walls, the room temperature of 293 K was defined. Along the outer boundary of the ceramic insulation, a linear temperature profile from 294 to 405 K was assigned. The temperature of the external boundary of the contract section was set to 960 K. The same temperature was given to the downstream edge of the combustion chamber since the metal heat resistance was much smaller than the ceramic insulation. For the downstream edge of the insulation, a

Figure 2. Computational domain.

linear temperature profile from 960 to 405 K was defined. Finally, the atmospheric pressure was set at the combustor exit.

## 4.2. Velocity distributions

The predicted distributions of velocity, temperature and species inside the combustor chamber are presented in the following sections. These results are obtained with the flamelet combustion model and an optimized turbulent Prandtl/Schmidt number of 0.5 (please see Section 5). The advantages and limitations of the six turbulence models can be found by comparing the numerical results with the experimental data.

The numerical results of axial velocity contours and flow path lines for six turbulence models are shown in the upper halves of six plots in Figure 3, respectively, while the experimental data with the zero axial velocity lines specified are displayed in the lower halves. Owing to the limited number of measured data points, no flow path lines are drawn for the experimental

Figure 3. Axial velocity contours and flow path lines.

linear temperature profile from 960 to 405 K was defined. Finally, the atmospheric pressure

184 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

The predicted distributions of velocity, temperature and species inside the combustor chamber are presented in the following sections. These results are obtained with the flamelet combustion model and an optimized turbulent Prandtl/Schmidt number of 0.5 (please see Section 5). The advantages and limitations of the six turbulence models can be found by comparing the

The numerical results of axial velocity contours and flow path lines for six turbulence models are shown in the upper halves of six plots in Figure 3, respectively, while the experimental data with the zero axial velocity lines specified are displayed in the lower halves. Owing to the limited number of measured data points, no flow path lines are drawn for the experimental

was set at the combustor exit.

numerical results with the experimental data.

4.2. Velocity distributions

Figure 2. Computational domain.

plots. It is noted that all models can catch the main flow features or patterns in the combustion chamber. There are two recirculation zones behind the flame-holder, i.e. a central recirculation zone (CRZ) created by the central fuel jet flow and an annular recirculation zone (ARZ) induced due to the annular air jet flow. The CRZ is completely buried inside the ARZ region, which indicates that the laminar and turbulent diffusions across the ARZ are only mechanisms for fuel transportation into the main flow field. Each recirculation zone is divided to two regions by the zero axial velocity line, and the gas mixture moves downstream in one region and moves upstream in the other. In addition, at the upper left corner of the combustion chamber, another small recirculation zone is formed due to the same reason, flow passage increases suddenly.

Various degrees of agreement with the experimental data are illustrated among the six models for predicting the reattachment points or lengths of the ARZ and CRZ. As shown in the first plot for the standard k-ε model, both ARZ and CRZ lengths are considerably under-predicted. This type of shortcoming is also pointed out by other researchers for non-reacting flow studies [60, 61]. The Rk-ε model illustrates superior performance over the standard k-ε model. It can properly predict the ARZ length and give a moderate result for the CRZ length. The RNG model underestimates the ARZ length slightly, but the CRZ length considerably. The results from the two k-ω models are poorer than those from the Rk-ε and RNG models. The k-ω model underestimates the ARZ length, while overestimates the CRZ length. In terms of the SST model, it considerably over-predicts the ARZ length though it gives a good result for the CRZ length. The RSM model, as shown in the last plot of Figure 3, illustrates the best performance, where both ARZ and CRZ lengths are satisfactorily provided.

A few parameters that may be valuable to combustion emission and stability studies can be obtained from the above RSM results. It is found that the gas mixture flow rate re-circulated in the ARZ is equal to 5.5% of the total inlet airflow, and the ARZ length is 1.7 times longer than the diameter of the flame-holder.

The axial velocity profiles along the combustor axis are illustrated in Figure 4 for the six turbulence models and compared with the experimental data. As shown in the figure, the peak value of measured negative axial velocity is �10 m/s, located at x ≈ 80 mm. The error bars representing 4% measurement accuracy are included in the figure. Considerable differences are found in the upper stream region from 10 to 80 mm for the k-ε and k-ω models, and in the downstream region from 80 to 360 mm for the k-ε and SST models. The RSM, Rk-ε and RNG models reasonably well predict the axial velocities along the central axis. Generally, the RSM gives the best results among the six turbulence models, which is consistent with the fact that only the RSM can adequately calculate both recirculation zones. For the five two-equation models, the Rk-ε illustrates better performance than the other four models.

The axial velocity profiles from x ¼ 20 to 240 mm are displayed in Figure 5 at selected seven cross-sections. Among these sections, three pass through the recirculation zones, one nearly cuts through the annular stagnation point and the remaining three are located further downstream. The features and magnitudes predicted from these models are generally close to the experimental measurements, but considerable differences are found at x ¼ 40 – 200 mm sections for the k-ε model, and x ≥ 160 mm downstream sections for the SST model. The best performance at the three upstream sections is given by the SST model because it adequately

Figure 4. Axial velocities along the combustor central axis.

Figure 5. Axial velocity profiles at cross-sections, x ¼ 20–240 mm.

from the two k-ω models are poorer than those from the Rk-ε and RNG models. The k-ω model underestimates the ARZ length, while overestimates the CRZ length. In terms of the SST model, it considerably over-predicts the ARZ length though it gives a good result for the CRZ length. The RSM model, as shown in the last plot of Figure 3, illustrates the best per-

A few parameters that may be valuable to combustion emission and stability studies can be obtained from the above RSM results. It is found that the gas mixture flow rate re-circulated in the ARZ is equal to 5.5% of the total inlet airflow, and the ARZ length is 1.7 times longer than

The axial velocity profiles along the combustor axis are illustrated in Figure 4 for the six turbulence models and compared with the experimental data. As shown in the figure, the peak value of measured negative axial velocity is �10 m/s, located at x ≈ 80 mm. The error bars representing 4% measurement accuracy are included in the figure. Considerable differences are found in the upper stream region from 10 to 80 mm for the k-ε and k-ω models, and in the downstream region from 80 to 360 mm for the k-ε and SST models. The RSM, Rk-ε and RNG models reasonably well predict the axial velocities along the central axis. Generally, the RSM gives the best results among the six turbulence models, which is consistent with the fact that only the RSM can adequately calculate both recirculation zones. For the five two-equation

The axial velocity profiles from x ¼ 20 to 240 mm are displayed in Figure 5 at selected seven cross-sections. Among these sections, three pass through the recirculation zones, one nearly cuts through the annular stagnation point and the remaining three are located further downstream. The features and magnitudes predicted from these models are generally close to the experimental measurements, but considerable differences are found at x ¼ 40 – 200 mm sections for the k-ε model, and x ≥ 160 mm downstream sections for the SST model. The best performance at the three upstream sections is given by the SST model because it adequately

formance, where both ARZ and CRZ lengths are satisfactorily provided.

186 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

models, the Rk-ε illustrates better performance than the other four models.

the diameter of the flame-holder.

Figure 4. Axial velocities along the combustor central axis.

Figure 6. Turbulence kinetic energy profiles at cross-sections, x ¼ 60–240 mm.

estimates the central recirculation zone, as shown in Figure 3. The results of the Rk-ε and RNG models are close to those obtained from the RSM. The improvement of the RSM over the Rk-ε model is observed in the regions of x ¼ 120–200 mm and R ¼ 30–40 mm.

The comparisons between the numerical and experimental results for the turbulence kinetic energy at four cross-sections from x ¼ 60 to 240 mm are presented in Figure 6, and the measurement accuracy of 8% is shown by error bars. Again, only the RSM provides encouraging predictions at all sections. In addition, the promising results of three normal turbulence stresses and one shear stress from the RSM model are also obtained. The numerical results agree reasonably well with the experimental data, particularly for the shear-stress profiles. Please see Ref. [31] for detail.

In Figure 6, the turbulence kinetic energy is considerably overestimated by the RNG model at all sections and the k-ε model at the upstream sections. For the Rk-ε and k-ω models, except for the k-ω model at section x ¼ 100 mm, reasonable agreement with the experimental data is observed. The SST model fairly well predicts the turbulence kinetic energy in the central area, and however overestimates its value away from the combustor axis. These comply again with the above observation shown in Figure 3 that the SST model can properly calculate the CRZ, but fails to assess the ARZ. Note that since the x ¼ 60 mm section crosses both recirculation zones where the two peak regions of turbulence kinetic energy are located, it represents a challenging task for numerical prediction. Unfortunately, none of these models can properly capture the central peak value.

In short, in terms of velocity flow-field prediction, the RSM is superior over the five twoequation models and in general, the Rk-ε model illustrates better performance than the other four two-equation models.

## 4.3. Temperature distributions

estimates the central recirculation zone, as shown in Figure 3. The results of the Rk-ε and RNG models are close to those obtained from the RSM. The improvement of the RSM over the Rk-ε

The comparisons between the numerical and experimental results for the turbulence kinetic energy at four cross-sections from x ¼ 60 to 240 mm are presented in Figure 6, and the measurement accuracy of 8% is shown by error bars. Again, only the RSM provides encouraging predictions at all sections. In addition, the promising results of three normal turbulence stresses and one shear stress from the RSM model are also obtained. The numerical results agree reasonably well with the experimental data, particularly for the shear-stress profiles.

In Figure 6, the turbulence kinetic energy is considerably overestimated by the RNG model at all sections and the k-ε model at the upstream sections. For the Rk-ε and k-ω models, except for the k-ω model at section x ¼ 100 mm, reasonable agreement with the experimental data is observed. The SST model fairly well predicts the turbulence kinetic energy in the central area, and however overestimates its value away from the combustor axis. These comply again with the above observation shown in Figure 3 that the SST model can properly calculate the CRZ, but fails to assess the ARZ. Note that since the x ¼ 60 mm section crosses both recirculation

model is observed in the regions of x ¼ 120–200 mm and R ¼ 30–40 mm.

Figure 6. Turbulence kinetic energy profiles at cross-sections, x ¼ 60–240 mm.

188 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

Please see Ref. [31] for detail.

The upper halves of Figure 7 present the temperature contour results of the six turbulence models. The stoichiometric line of the mean mixture fraction is superimposed in each plot, and it starts from the edge of the disk flame-holder, passes through the high-temperature region and ends at the combustor axis. As illustrated in the figure, the flame is ignited at the downstream end of the flame-holder edge and propagates downstream over the annular recirculation zone envelope. Along the envelope, the gaseous mixture of fuel and hot gas re-circulated from downstream mixes with the fresh air from the inlet section and the combustion takes place. During the experimental testing, a carbon deposit was observed at the disk edge of the flame-holder, which is consistent with the numerical prediction.

The comparisons of the numerical results with the experimental data in the lower halves of Figure 7 have shown that the size and location of the high-temperature region are in good agreement with the experimental data for the RSM and Rk-ε models, while the RSM performance is a little better than the Rk-ε model. As shown in the figure, the standard k-ε and RNG models underestimate high-temperature region and the high-temperature regions are shifted upstream. On the contrary, the k-ω and SST models considerably overestimate high-temperature region, and the high-temperature regions are shifted downstream.

Figure 8 quantitatively compares the calculated temperature profiles along the combustor central axis with the experimental data, where the measurement error is about 5%. The calculated trends are close to the experimental values along the combustor central axis from 50 to 350 mm. However, in the central portion, the experimental profile is almost flat, while the predicted profiles display peak values. Overall, the better performance is given by the RSM and Rk-ε models among the six models. The k-ε and RNG models predict higher temperature than the experimental data in the upstream area and lower temperature in the downstream area. On the contrary, the k-ω and SST models underestimate the temperature in the upstream and significantly overestimate it in the downstream.

As indicated in Figures 7 and 8, the calculated temperatures are higher than the measured values in the centre region from x ≈ 140 to 250 mm, and the maximum deviation is about 150 K. Three reasons are anticipated. Firstly, as mentioned earlier, the temperature was measured by thermocouples. Due to the radiation and conduction heat losses from the thermocouple, the measurement error could be higher than 100 K at the locations where the flow velocity was low and the gas temperature was high [62]. Secondly, the temperature probe could modify local flow structure, thus increases local turbulent mixing and causes an increase in local temperature [62]. Thirdly, the turbulence kinetic energy (Figure 6) may not be properly calculated in

Figure 7. Temperature contours and flow path lines.

Figure 8. Temperature profiles along the combustor central axis.

the local region, and consequently the combustion and temperature calculation could be affected.

The temperature profiles at seven cross-sections, x ¼ 52–353 mm are presented in Figure 9 for the six models. For the RSM and Rk-ε models, the predicted results agree reasonably well with the experimental data, besides the most upstream section and the small local area near the combustor wall. The RNG model illustrates the similar performance at sections from x ¼ 82 to 293 mm.

Figure 9. Temperature profiles at cross-sections, x ¼ 52–353 mm.

For the other three models, poor performance is found at upstream sections from x ¼ 52 to 112 mm for the k-ε model, x ¼ 82, 232–353 mm sections for the k-ω model, and most sections for the SST model.

Similar to the velocity predictions observed above, in general, the predicted temperature results from the RSM and Rk-ε models fairly agree with the experimental data.

#### 4.4. Species distributions

the local region, and consequently the combustion and temperature calculation could be

The temperature profiles at seven cross-sections, x ¼ 52–353 mm are presented in Figure 9 for the six models. For the RSM and Rk-ε models, the predicted results agree reasonably well with the experimental data, besides the most upstream section and the small local area near the combustor wall. The RNG model illustrates the similar performance at sections from x ¼ 82 to 293 mm.

affected.

Figure 7. Temperature contours and flow path lines.

190 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

Figure 8. Temperature profiles along the combustor central axis.

Figure 10 presents the CO2 mole fraction profiles at five cross-sections, from x ¼ 21 to 171 mm for six turbulence models, and compares them with the experimental data. These results are obtained with the flamelet combustion model (31 species) and the estimated error for species measurements is about 5%, as shown by error bars in the figure. The five cross-sections are chosen to show the main trends of chemical reactions, that is two passing through the central recirculation zone, two across the annular recirculation zone and the last one after the recirculation region (see Figure 3).

Carbon dioxide is one final major species of propane-air combustion. The Rk-ε and RSM predictions agree fairly well with the experimental data, except for the central region at x ¼ 111 mm and the middle region at x ¼ 81 mm, as illustrated in Figure 10. Poor prediction

Figure 10. CO2 profiles at cross-sections, x ¼ 21–291 mm.

is found at x ¼ 51 mm section for the k-ε model, x ¼ 81 and 171 mm sections for the k-ω model and x ¼ 171 mm section for the SST model.

Carbon monoxide, CO, is one major immediate species in hydrocarbon fuel combustion. The CO radial profiles are represented in Figure 11, where the numerical results are compared with the experimental data. Similar to the CO2 case, all models properly estimate the CO profile at the most upstream section, except for the SST model showing a small bump at R ≈ 38 mm. The RSM and Rk-ε predictions agree fairly with the experimental data at x ¼ 51 and 171 mm. It is found that the two models over-predict the CO mole fraction in the central region, while under-predict the CO2 at the two other sections, x ¼ 81 and 111 mm. However, the sum of CO2 and CO of the predicted results is close to the sum of the experimental data for CO2 and CO. This indicates that the calculated oxidization of CO at these two sections is lag behind. Poor prediction is found again for the k-ε model at section x ¼ 51 mm, k-ω model at section x ¼ 171 mm and SST model at sections x ¼ 111 and 171 mm.

Figure 11. CO profiles at cross-sections, x ¼ 21–291 mm.

is found at x ¼ 51 mm section for the k-ε model, x ¼ 81 and 171 mm sections for the k-ω model

Carbon monoxide, CO, is one major immediate species in hydrocarbon fuel combustion. The CO radial profiles are represented in Figure 11, where the numerical results are compared with the experimental data. Similar to the CO2 case, all models properly estimate the CO profile at the most upstream section, except for the SST model showing a small bump at R ≈ 38 mm. The RSM and Rk-ε predictions agree fairly with the experimental data at x ¼ 51 and 171 mm. It is found that the two models over-predict the CO mole fraction in the central region, while under-predict the CO2 at the two other sections, x ¼ 81 and 111 mm. However, the sum of CO2 and CO of the predicted results is close to the sum of the experimental data for CO2 and CO. This indicates that the calculated oxidization of CO at these two sections is lag behind. Poor prediction is found again for the k-ε model at section x ¼ 51 mm, k-ω model at section

and x ¼ 171 mm section for the SST model.

Figure 10. CO2 profiles at cross-sections, x ¼ 21–291 mm.

192 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

x ¼ 171 mm and SST model at sections x ¼ 111 and 171 mm.

In brief, the Rk-ε and RSM results are consistent with the experimental data except for some local regions, and better than the other models. The species predictions are encouraging in general.

The above qualitative and quantitative comparisons of velocity, temperature and species distributions inside the combustor between the numerical results and experimental database clearly indicate that the RSM model, a second-moment closure, is better than the eddy-viscosity models. This agrees with the findings from other authors, such as Ref. [10] for a nonreacting flow and Ref. [27] for a reacting flow. Furthermore, the Rk-ε model illustrates better performance than other four two-equation models. Instead of the RSM, utilization of the Rk-ε model for practical gas turbine combustor simulations can avoid some numerical problems, such as stability and time-consuming issues.

For the SST model, it can provide good solutions in many non-reacting flows, such as the NACA 4412 air foil, backward-facing step and adverse pressure gradient flows [44]. However, the model considerably overestimates the high-temperature region and annular recirculation zone in the combustion chamber and this type of result is also found in the simulations of a real-world gas turbine combustor [36]. Two reasons are anticipated. Firstly, the testing cases used for model validation are isothermal or almost isothermal flows, and the considerable thermal expansion and chemical reaction may not be adequately accounted for in the model [3, 44]. This may justify that the features in the central recirculation zone can be appropriately estimated by the SST model, as seen from Figures 3, 5 and 6, because the temperature is low in the central recirculation zone, as seen from Figure 7. Secondly, multiple large vortices or recirculation regions play an important role in fuel-air mixing and combustion management in the combustion chamber, and this type of flow is more complex than single vortex flows, such as the backward-facing step flow. For example, in the present case, the whole central recirculation zone is buried inside the annular recirculation zone.

## 5. Application of Reynolds analogy

Most of these results are obtained with the RSM turbulence model and PDF combustion model. By comparing the numerical results with the experimental database, the Reynolds analogy approach can be assessed and the optimized turbulence Prandtl/Schmidt number for the combustor flow-fields can be identified.

#### 5.1. Velocity distributions

The predicted axial velocity contours are illustrated in Figure 12 for Lt ¼ Prt ¼ Sct ¼ 0.85, 0.50 and 0.25, respectively, and compared with the experimental data. The flow patterns in the combustion chamber are well captured in all three plots and two reattachment points or lengths of the two recirculation zones are properly predicted.

As mentioned earlier, turbulence scalar transfers are calculated based on the modelled turbulent momentum transfer, and however the former may also affect the latter since they are coupled. As shown in Figure 12, the effect of Γt on velocity field for Γt ¼ 0.85–0.25 is minor. For Γt ¼ 0.25, the length and volume of the annular recirculation zone are slightly reduced in comparison with those from Γt ¼ 0.50 and 0.85.

The similar trends for the axial velocity profiles along the combustor central axis, and the axial velocity, turbulence kinetic energy and shear-stress profiles at a number of cross-sections are also observed [32]. These results indicate that the effect of Гt variation on the predicted velocity

Figure 12. Axial velocity contours and flow path lines, Γt ¼ 0.85, 0.50 and 0.25.

field is minor, and the predicted velocity fields agree fairly well with the experimental data for Гt > 0.35.

#### 5.2. Temperature distributions

clearly indicate that the RSM model, a second-moment closure, is better than the eddy-viscosity models. This agrees with the findings from other authors, such as Ref. [10] for a nonreacting flow and Ref. [27] for a reacting flow. Furthermore, the Rk-ε model illustrates better performance than other four two-equation models. Instead of the RSM, utilization of the Rk-ε model for practical gas turbine combustor simulations can avoid some numerical problems,

For the SST model, it can provide good solutions in many non-reacting flows, such as the NACA 4412 air foil, backward-facing step and adverse pressure gradient flows [44]. However, the model considerably overestimates the high-temperature region and annular recirculation zone in the combustion chamber and this type of result is also found in the simulations of a real-world gas turbine combustor [36]. Two reasons are anticipated. Firstly, the testing cases used for model validation are isothermal or almost isothermal flows, and the considerable thermal expansion and chemical reaction may not be adequately accounted for in the model [3, 44]. This may justify that the features in the central recirculation zone can be appropriately estimated by the SST model, as seen from Figures 3, 5 and 6, because the temperature is low in the central recirculation zone, as seen from Figure 7. Secondly, multiple large vortices or recirculation regions play an important role in fuel-air mixing and combustion management in the combustion chamber, and this type of flow is more complex than single vortex flows, such as the backward-facing step flow. For example, in the present case, the whole central

Most of these results are obtained with the RSM turbulence model and PDF combustion model. By comparing the numerical results with the experimental database, the Reynolds analogy approach can be assessed and the optimized turbulence Prandtl/Schmidt number for

The predicted axial velocity contours are illustrated in Figure 12 for Lt ¼ Prt ¼ Sct ¼ 0.85, 0.50 and 0.25, respectively, and compared with the experimental data. The flow patterns in the combustion chamber are well captured in all three plots and two reattachment points or

As mentioned earlier, turbulence scalar transfers are calculated based on the modelled turbulent momentum transfer, and however the former may also affect the latter since they are coupled. As shown in Figure 12, the effect of Γt on velocity field for Γt ¼ 0.85–0.25 is minor. For Γt ¼ 0.25, the length and volume of the annular recirculation zone are slightly reduced in

The similar trends for the axial velocity profiles along the combustor central axis, and the axial velocity, turbulence kinetic energy and shear-stress profiles at a number of cross-sections are also observed [32]. These results indicate that the effect of Гt variation on the predicted velocity

such as stability and time-consuming issues.

5. Application of Reynolds analogy

the combustor flow-fields can be identified.

comparison with those from Γt ¼ 0.50 and 0.85.

5.1. Velocity distributions

recirculation zone is buried inside the annular recirculation zone.

194 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

lengths of the two recirculation zones are properly predicted.

The temperature contours for Гt ¼ 0.85, 0.50 and 0.25 are presented and compared with the experimental database in Figure 13. Strong turbulent mixing makes the temperature in the recirculation region relatively uniform. Along the envelope of the annular recirculation zone, combustion occurs intensively. In comparison with the experimental data, for the large Гt number of 0.85, the high-temperature region is enlarged and shifted downstream, while for the small Гt number of 0.25, the high-temperature region is considerably reduced and shifted upstream. The best results are observed for Гt ¼ 0.50. Generally, as Гt decreases, the turbulence transfers of fuel into the airflow and then the chemical reaction are enhanced. As a result, the high temperature region is smaller and shifts upstream. Figure 13 shows that the predicted maximum temperature in the high temperature region is higher than the measured values, which has been explained earlier.

Figure 14 illustrates the effect of Гt on the predicted flame length. In the upper plot, the stoichiometric line of the mean mixture fraction (~<sup>f</sup> <sup>¼</sup> <sup>0</sup>:0603) is used to signify the flame length, and in the lower plot, the OH mole fraction contour lines are employed. The effect of Гt on the flame volume or length is clearly observed. Both the flame volume and length are substantially decreased as Гt decreases from 0.85 to 0.25, and the flame length reduces more than three times from 365 to 110 mm.

The predicted temperature profiles along the combustor central axis are compared with the experimental data in Figure 15. The limited effect of Гt is shown in the upstream region (x < �80 mm); the predicted values for Гt ¼ 0.50 and 0.85 agree well with the measured data.

Figure 13. Temperature contours, Гt ¼ 0.85, 0.50, 0.25.

Figure 14. Variation of predicted flame length with Гt.

Conversely, the strong effect of Гt is observed in the downstream region. This is because the fuel distribution or chemical reaction in the upstream region is mainly controlled by the size and location of the annular recirculation zone, which is created by complex flow interactions among the central fuel jet, annular airflow and two recirculation zones. In other words, the flow field is dominated by convection. However, the turbulent diffusion or transfer in the downstream region plays a major role in the fuel spreading away from the axis of symmetry, as shown in Figure 12 where the flow path lines are almost parallel to each other. As shown in Figure 15, Гt ¼ 0.50 gives the best results although the predicted profile shows a peak in the middle portion, while the measurements tend to be flat.

Figure 16 presents the temperature profiles for Гt ¼ 0.85, 0.50 and 0.25 at five cross-sections from x ¼ 52 to 233 mm. The temperature profiles become flatter or the fuel spreading becomes faster as Гt decreases at all sections. Generally, the predicted results for Гt ¼ 0.50 are close to the measured data, except for the local regions around the combustor wall. In these near-wall regions, the temperature is underestimated and it may indicate the under-prediction of fuel spreading in these regions. Poor prediction is found for Гt ¼ 0.25 at sections x ¼ 82 and 233 mm, and Гt ¼ 0.85 at section x ¼ 52 mm. The effect of Гt is strong at all five sections.

Figure 15. Temperature profiles along the combustor central axis.

### 5.3. Wall temperature distribution

Conversely, the strong effect of Гt is observed in the downstream region. This is because the fuel distribution or chemical reaction in the upstream region is mainly controlled by the size and location of the annular recirculation zone, which is created by complex flow interactions among the central fuel jet, annular airflow and two recirculation zones. In other words, the flow field is dominated by convection. However, the turbulent diffusion or transfer in the downstream region plays a major role in the fuel spreading away from the axis of symmetry, as shown in Figure 12 where the flow path lines are almost parallel to each other. As shown in Figure 15, Гt ¼ 0.50 gives the best results although the predicted profile shows a peak in the

Figure 16 presents the temperature profiles for Гt ¼ 0.85, 0.50 and 0.25 at five cross-sections from x ¼ 52 to 233 mm. The temperature profiles become flatter or the fuel spreading becomes faster as Гt decreases at all sections. Generally, the predicted results for Гt ¼ 0.50 are close to the measured data, except for the local regions around the combustor wall. In these near-wall regions, the temperature is underestimated and it may indicate the under-prediction of fuel spreading in these regions. Poor prediction is found for Гt ¼ 0.25 at sections x ¼ 82 and 233 mm,

and Гt ¼ 0.85 at section x ¼ 52 mm. The effect of Гt is strong at all five sections.

middle portion, while the measurements tend to be flat.

Figure 13. Temperature contours, Гt ¼ 0.85, 0.50, 0.25.

196 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

Figure 14. Variation of predicted flame length with Гt.

Variation of combustor wall temperature with Гt is shown in Figure 17, and the numerical results are evaluated against the experimental data. Unsurprisingly, the predicted wall temperature increases as Гt decreases. The best agreement with the experimental data is obtained with Гt ¼ 0.35 although the wall temperature is a little overestimated in the upstream region and underestimated in the downstream region. Note that this Гt number is different from the preferred value of 0.50 for the temperature prediction inside the combustor as discussed earlier. It may indicate that instead of a constant number, varying Гt should be considered in turbulent reacting flow simulations.

In order to thoroughly assess the Reynolds analogy issue, numerical simulations were also carried out with the eddy-dissipation (EDS) and laminar flamelet combustion models. A large amount of numerical results and figures were generated, with a Гt increment of 0.05 or even 0.01. The trends and magnitudes of velocity, turbulence kinetic energy, Reynolds stresses and temperature distributions are similar to those obtained from the PDF combustion model. Although the results are not presented in this chapter, the optimized Гt numbers are given in Table 1 for the purpose of comparison.

As shown in the above results, the optimal Гt number for the temperature prediction inside the combustor is 0.50 for all three combustion models. This number is different from 0.20 [23] and 0.25 [24] for gas turbine combustor studies, and 0.20 for a cross-jet flow simulation [25]. However, it is the same as recommended by Syed and Sturgess [22] for recirculation zone simulations.

Two important facts are revealed from the above all examples. Firstly, the Гt number optimized is smaller than the conventionally accepted value of 0.6–0.85. Secondly, most likely the

Figure 16. Temperature profiles at sections x ¼ 52–233 mm.

combustor configuration and possibly the operating conditions affect the optimal value of Гt. Therefore, a priori optimization of Гt is preferred in order to confidently predict temperature and species distributions inside combustors.

These observations may be attributed to the following reasons. Firstly, theoretically, Eqs. (11)–(13) are only valid for boundary layer flows, where the streamwise pressure gradient, viscous and source terms can be neglected. Certainly, its application to complex turbulent reacting flows is questionable. Secondly, the experimental numbers of Гt (�0.7) are measured from fully developed boundary or pipe flows [12, 13, 63], which are quite different from practical three-dimensional

Figure 17. Temperature profiles along the combustor wall.


Table 1. Optimal Prandtl/Schmidt number.

combustor configuration and possibly the operating conditions affect the optimal value of Гt. Therefore, a priori optimization of Гt is preferred in order to confidently predict temperature

These observations may be attributed to the following reasons. Firstly, theoretically, Eqs. (11)–(13) are only valid for boundary layer flows, where the streamwise pressure gradient, viscous and source terms can be neglected. Certainly, its application to complex turbulent reacting flows is questionable. Secondly, the experimental numbers of Гt (�0.7) are measured from fully developed boundary or pipe flows [12, 13, 63], which are quite different from practical three-dimensional

and species distributions inside combustors.

Figure 16. Temperature profiles at sections x ¼ 52–233 mm.

198 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

turbulent reacting flows. In terms of the relative strength between the turbulence momentum and scalar transfers for the numerical methods and models used today, a low value of Гt is true, and it may change with flow configurations.

Thirdly, the gradient-type diffusion assumption used in Eqs. (11)–(13) has been questioned by a large number of researchers, in particular for turbulent energy and heat transfer. As pointed out in Ref. [12], to adequately model turbulence scalar transfers, not only the gradient-based diffusion from small-scale turbulence, but also the convection effect from large-scale turbulent motion should be taken into consideration. This may imply that the gradient-based diffusion method is appropriate for turbulent boundary flows, but it may not be proper for complex flows, such as practical combustion systems.

As a summary, the Reynolds analogy approach has been applied to general three-dimensional flow-field simulations from the 1970s. In order to accurately predict the turbulence scalar transfers without a prior optimization, the improvement of the current approach is necessary, and new ideas should be considered.

## 6. Conclusions

Turbulence modelling is one of major issues, which affects the precision of current numerical simulations in engineering applications, particularly for reacting flows. To systematically study and validate various physical models, a series of experimental measurements have been carried out at the National Research Council of Canada on a model combustor, and a comprehensive database has been obtained. The combustor simulations with the interior and exterior conjugate heat transfers have been carried out with six turbulence models, i.e. the standard k-ε, re-normalization group k-ε, realizable k-ε, standard k-ω, shear-stress transport (SST) and Reynolds stress models. The laminar flamelet, PDF and EDS combustion models and the discrete ordinate radiation model as well are also used.

All six turbulence models can capture the flow features or patterns; however, for the quantitative predictions of velocity, temperature and species fields, different levels of performance are revealed. The RSM model gives the best performance, and it is the only one that can accurately predict the lengths of both recirculation zones and offer reasonable prediction on the turbulence kinetic energy distribution in the combustor. In addition, the performance of the Rk-ε model is better than other four two-equation models, and it can give similar results as those from the RSM under the present configuration and operating conditions.

The effect of the turbulent Prandtl/Schmidt number on the flow field of the model combustor has also been numerically studied. In this chapter, some of the results obtained with turbulent Prandtl/Schmidt number varying from 0.85 to 0.25 have been presented and discussed. It has a strong effect on the temperature fields, particularly downstream in the combustor. This is also true for the temperature profile along the combustor wall. On the contrary, its effect on the velocity field is limited.

For all three combustion models, the optimal Гt is 0.5 for temperature prediction in the combustor, while for predicting temperature at the combustor wall the optimal value alters from 0.35 to 0.50. With Гt ¼ 0.50, except for some local regions, the velocity, temperature and major species fields in the combustor are fairly well simulated.

As a final point, considering the foundation and shortcoming of the Reynolds analogy, to accurately predict temperature and species distributions in turbulent reacting flow fields without an optimization of turbulent Prt and Sct numbers, the Reynolds analogy approach should be enhanced and new ideas should be considered.

## Author details

Lei-Yong Jiang

Address all correspondence to: lei-yong.jiang@nrc-cnrc.gc.ca

Gas Turbine Laboratory, Aerospace, National Research Council of Canada, Ottawa, Ontario, Canada

## References

6. Conclusions

velocity field is limited.

Author details

Lei-Yong Jiang

Canada

Turbulence modelling is one of major issues, which affects the precision of current numerical simulations in engineering applications, particularly for reacting flows. To systematically study and validate various physical models, a series of experimental measurements have been carried out at the National Research Council of Canada on a model combustor, and a comprehensive database has been obtained. The combustor simulations with the interior and exterior conjugate heat transfers have been carried out with six turbulence models, i.e. the standard k-ε, re-normalization group k-ε, realizable k-ε, standard k-ω, shear-stress transport (SST) and Reynolds stress models. The laminar flamelet, PDF and EDS combustion models and the

All six turbulence models can capture the flow features or patterns; however, for the quantitative predictions of velocity, temperature and species fields, different levels of performance are revealed. The RSM model gives the best performance, and it is the only one that can accurately predict the lengths of both recirculation zones and offer reasonable prediction on the turbulence kinetic energy distribution in the combustor. In addition, the performance of the Rk-ε model is better than other four two-equation models, and it can give similar results as those

The effect of the turbulent Prandtl/Schmidt number on the flow field of the model combustor has also been numerically studied. In this chapter, some of the results obtained with turbulent Prandtl/Schmidt number varying from 0.85 to 0.25 have been presented and discussed. It has a strong effect on the temperature fields, particularly downstream in the combustor. This is also true for the temperature profile along the combustor wall. On the contrary, its effect on the

For all three combustion models, the optimal Гt is 0.5 for temperature prediction in the combustor, while for predicting temperature at the combustor wall the optimal value alters from 0.35 to 0.50. With Гt ¼ 0.50, except for some local regions, the velocity, temperature and

As a final point, considering the foundation and shortcoming of the Reynolds analogy, to accurately predict temperature and species distributions in turbulent reacting flow fields without an optimization of turbulent Prt and Sct numbers, the Reynolds analogy approach

Gas Turbine Laboratory, Aerospace, National Research Council of Canada, Ottawa, Ontario,

discrete ordinate radiation model as well are also used.

200 Turbulence Modelling Approaches - Current State, Development Prospects, Applications

from the RSM under the present configuration and operating conditions.

major species fields in the combustor are fairly well simulated.

should be enhanced and new ideas should be considered.

Address all correspondence to: lei-yong.jiang@nrc-cnrc.gc.ca


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