**3.1.1 Numerical results**

The cavitating flow in the venturi type duct was characterized using the pressure coefficient, *Cp*, the cavitation number, *σ*, and the Strouhal number, *St*, defined as:

$$\mathcal{C}\_{p} = \frac{p - p\_{ref}}{1/2 \cdot \rho\_l \cdot v\_1^2}, \quad \sigma = \frac{p\_1 - p\_v}{1/2 \cdot \rho\_l \cdot v\_1^2} \quad \text{and} \quad \text{St} = \frac{f\_{cav} \cdot \overline{l\_{cav}}}{v\_1} \tag{17}$$

where ݈௩ തതതതത is the time average of *lcav*.

Preliminary 2D computations suggest a strong interaction of the turbulence and the unsteady cavitation at the venturi throat.

### **3.1.1.1 RNG** κ−ε **turbulence model**

The first unsteady simulations were carried out using the classical RNG κ–ε model. These numerical results did not reproduce the well-known instabilities of this configuration. The results show, after an unsteady numerical behaviour, a steady cavitation with *lcav* constant for all the times. This cavity remains attached on the lower wall downstream of the venturi throat. This vapour region grows when σis reduced, but it remains stable.

The cavitation detachment could not be modelled appropriately. The reason is that RNG κ– ε model overestimates the turbulent viscosity in the cavitation region. Hence, the re-entrant jet is stopped at cavitation sheet closure and then, it could not incite the cavitation break off. The RNG κ–ε model was originally conceived to fully incompressible fluids, and no

particular correction was applied in the case of the highly compressible two-phase mixture. Therefore, the fluid compressibility is only taken into account in the turbulence equations through the mixture density changes (Wilcox, 1998).

## **3.1.1.2 RNG** κ−ε **modified turbulence model**

On the other hand, other calculations were performed using the RNG κ–ε modified model. The results show an unstable flow due to the vapour detachment, as it was observed experimentally. The modification of turbulent viscosity allows the detachment of the cavitation caused by a re-entrant jet, which appears on the lower divergent wall. The reentrant jet goes from cavitation sheet closure toward the venturi throat in the opposite direction to the main flow, which lets the detachment of the cavitation from the wall, see Fig. 3. Thus, the cavitation can be convected in the main flow, where it will be collapsed downstream.

Fig. 3. Contours of vapour fraction (α ≥ 10%) and velocity vectors near to venturi throat at *t=0.273·tref* and σ*=2.45*. Calculations using RNG κ–ε modified model

The cavitation detachment cycles can be observed, for σ values of 2.45, 2.34, and 2.30, in Fig. 4. This figure shows that the cavitation length, *lcav*, increases while the cavitation number, σ, and the detachment frequency, *fcav*, decrease. Generally, the cavitation detachment cycle is composed of the following steps, see the cycle of σ=2.45 in Fig. 4:

The cavitating flow in the venturi type duct was characterized using the pressure coefficient,

Preliminary 2D computations suggest a strong interaction of the turbulence and the

The first unsteady simulations were carried out using the classical RNG κ–ε model. These numerical results did not reproduce the well-known instabilities of this configuration. The results show, after an unsteady numerical behaviour, a steady cavitation with *lcav* constant for all the times. This cavity remains attached on the lower wall downstream of the venturi

The cavitation detachment could not be modelled appropriately. The reason is that RNG κ– ε model overestimates the turbulent viscosity in the cavitation region. Hence, the re-entrant jet is stopped at cavitation sheet closure and then, it could not incite the cavitation break off. The RNG κ–ε model was originally conceived to fully incompressible fluids, and no particular correction was applied in the case of the highly compressible two-phase mixture. Therefore, the fluid compressibility is only taken into account in the turbulence equations

On the other hand, other calculations were performed using the RNG κ–ε modified model. The results show an unstable flow due to the vapour detachment, as it was observed experimentally. The modification of turbulent viscosity allows the detachment of the cavitation caused by a re-entrant jet, which appears on the lower divergent wall. The reentrant jet goes from cavitation sheet closure toward the venturi throat in the opposite direction to the main flow, which lets the detachment of the cavitation from the wall, see Fig. 3. Thus, the cavitation can be convected in the main flow, where it will be collapsed

α

*=2.45*. Calculations using RNG κ–ε modified model

4. This figure shows that the cavitation length, *lcav*, increases while the cavitation number,

and the detachment frequency, *fcav*, decrease. Generally, the cavitation detachment cycle is

σ

σ

<sup>ଶ</sup> ܽ݊݀ ܵݐ ൌ ݂௩⋅݈௩

is reduced, but it remains stable.

≥ 10%) and velocity vectors near to venturi throat at

values of 2.45, 2.34, and 2.30, in Fig.

σ,

σ

=2.45 in Fig. 4:

തതതതത ଵݒ

(17)

ଵ െ ௩ ଵݒ ߩ ʹͳȀ

*Cp*, the cavitation number, *σ*, and the Strouhal number, *St*, defined as:

<sup>ଶ</sup> ǡ ߪൌ

െ

ଵݒ ߩ ʹͳȀ

തതതതത is the time average of *lcav*.

unsteady cavitation at the venturi throat.

throat. This vapour region grows when

through the mixture density changes (Wilcox, 1998).

**3.1.1.2 RNG** κ−ε **modified turbulence model** 

Fig. 3. Contours of vapour fraction (

The cavitation detachment cycles can be observed, for

composed of the following steps, see the cycle of

σ

**3.1.1.1 RNG** κ−ε **turbulence model** 

**3.1.1 Numerical results** 

where ݈௩

downstream.

*t=0.273·tref* and

ൌ ܥ

Fig. 4. Contours of vapour volume fraction (*α*≥10%). Vapour detachment cycle calculated for three cavitation numbers. Calculations using RNG κ–ε modified model, *tref=65.8E-3 s*

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 187

(a)

(b)

 *Tcav (s) fcav (Hz) lcav St 2.61 9.8e-3 102.0 20.90 0.27 2.45 16.5e-3 60.6 34.77 0.27 2.34 27.3e-3 36.6 57.20 0.27 2.30 38.6e-3 25.9 81.02 0.27* 

σ

*=*2.45). (b) Comparison of

Fig. 5. (a) Spectral analysis of the upstream static pressure (

Table 2. Characteristics of the cycles of vapour detachment

σ

numerical results and experimental data (Stutz & Reboud, 2000)


Fig. 5(a) shows the spectral analysis of the upstream static pressure behaviour *(*σ*=2.45*) which is disturbed by the cavitation detachment. The cavitating flow has a cyclic behaviour with a frequency of cavitation detachment of about *fcav=60 Hz*. The numerical results were compared with the experimental data obtained for the same geometry. Although the vapour volume fraction obtained numerically is higher than the reference data, there is a good agreement if the vapour detachment frequencies are compared, see Fig. 4(b).

Table 2 summarizes the main characteristics of the vapour detachment cycles. According to the experimental data, the detachment frequencies are inversely proportional to the cavitation lengths, and so the Strouhal number remains constant for all the cavitation numbers, *St*=0.27. The cavitation length was measured at twelve instants of the cavitating period and so, the average value was obtained.

#### **3.2 Numerical analysis of the unsteady cavitating flow in two blades cascades**

This part presents the numerical study carried out on two blades cascades: first, on a twoblade aircraft inducer with a blade tip angle of 4°, and then, on a three-blade industrial inducer with a blade tip angle of 8°, see Fig. 1.

#### **3.2.1 Geometrical model and grid generation**

The numerical domain, for both cases, has been divided into three sub-domains in order to impose moving mesh conditions. Fig. 6(a) shows the three computational sub-domains of whole numerical model, defined as: upstream region (A), inter-blades region (B), and downstream region (C). Tangential velocity was imposed in the moving region (B) using a sliding mesh technique, whereas (A) and (C) regions were defined as static.

Boundary conditions at the domain inlet and outlet were imposed far enough (15*·l)*, from the leading and trailing edges, respectively, in order to avoid influencing the final results. The used boundary conditions are the following:


By way of example, the Fig. 6(b) shows the computational grid of two-blade inducer. The mesh was generated with a rectangle–like structured grid. An independence grid study was

1. A very small vapour region appears at the venturi throat (from *t=*0.023*·tref* to *t=*0.046*·* 

2. The attached vapour region grows at the downstream venturi throat (from *t=*0.068*·tref* to

3. A jet flow is generated along the lower wall in the vapour region and goes from cavitation closure towards the venturi throat. The interaction between this re-entrant jet and the interface liquid-vapour causes the detachment of the cavitation (from *t*=0.160*·tref*

4. The generated vapour region is convected in the main flow (from *t*=0.251*·tref* to

which is disturbed by the cavitation detachment. The cavitating flow has a cyclic behaviour with a frequency of cavitation detachment of about *fcav=60 Hz*. The numerical results were compared with the experimental data obtained for the same geometry. Although the vapour volume fraction obtained numerically is higher than the reference data, there is a good

Table 2 summarizes the main characteristics of the vapour detachment cycles. According to the experimental data, the detachment frequencies are inversely proportional to the cavitation lengths, and so the Strouhal number remains constant for all the cavitation numbers, *St*=0.27. The cavitation length was measured at twelve instants of the cavitating

This part presents the numerical study carried out on two blades cascades: first, on a twoblade aircraft inducer with a blade tip angle of 4°, and then, on a three-blade industrial

The numerical domain, for both cases, has been divided into three sub-domains in order to impose moving mesh conditions. Fig. 6(a) shows the three computational sub-domains of whole numerical model, defined as: upstream region (A), inter-blades region (B), and downstream region (C). Tangential velocity was imposed in the moving region (B) using a

Boundary conditions at the domain inlet and outlet were imposed far enough (15*·l)*, from the leading and trailing edges, respectively, in order to avoid influencing the final results.

a. Constant velocity at the inlet. The nominal flow corresponds to a null incidence angle. b. Constant static pressure at the outlet. This value has been modified for to get diverse

e. Cyclic conditions were applied at two or three successive blades, depending of the

By way of example, the Fig. 6(b) shows the computational grid of two-blade inducer. The mesh was generated with a rectangle–like structured grid. An independence grid study was

σ*=2.45*)

Fig. 5(a) shows the spectral analysis of the upstream static pressure behaviour *(*

**3.2 Numerical analysis of the unsteady cavitating flow in two blades cascades** 

sliding mesh technique, whereas (A) and (C) regions were defined as static.

*tref*).

*t=*0.137*·tref*).

to *t*=0.228*·tref*).

*t*=0.273*·tref*) and (from *t*=0.023*·tref* to *t*=0.068*·tref*).

period and so, the average value was obtained.

inducer with a blade tip angle of 8°, see Fig. 1.

**3.2.1 Geometrical model and grid generation** 

The used boundary conditions are the following:

c. Non-slip conditions at the blades boundaries.

d. Sliding conditions at the interfaces (A)–(B) and (B)–(C).

cavitation conditions.

analyzed inducer.

5. The cavitation collapses downstream (from *t*=0.091*·tref* to *t*=0.114*·tref*).

agreement if the vapour detachment frequencies are compared, see Fig. 4(b).

Fig. 5. (a) Spectral analysis of the upstream static pressure (σ*=*2.45). (b) Comparison of numerical results and experimental data (Stutz & Reboud, 2000)


Table 2. Characteristics of the cycles of vapour detachment

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 189

carried out on non-cavitating flow. Three meshes were tested: a coarse mesh (300X50), a fine mesh (500X50), and a refined mesh (650X50). The first coarse mesh presented the backflow at outlet domain because of very important aspect ratio upstream and downstream regions. The fine and refined meshes presented similar results, but the fine mesh reaches the solution faster than the refined mesh; for this reason, the fine mesh was selected for realize all calculations. Boundary layer meshing was used to ensure sufficient refinement near the walls and thus a small dimensionless factor *y+*. First cell distance was imposed to 1 *mm*, with

<sup>ଶ</sup> ܽ݊݀ ߔ ൌ ሺܥሻଵ

*,* on the leading edge of suction side. The regions of vapour are symmetric, for all

*=0.056*, the regions of vapour corresponding to diverse values of

σ

σ

*=0.042* presents the same patron of cavitation than for

. The vapour region increases as

Calculations of unsteady cavitating flow were carried out for four flow rates over the blades cascade of a two–blade inducer and for three flow rates over the blades cascade of a three– blade inducer, see Fig. 7. En general, the results present two small regions of vapour, at high

regions of vapour become large enough to block the flow channel, resulting in the

figure shows that the cavitation begins with very small regions of vapour which appear on the leading edge of the blade, on the suction side. The regions of vapour have a steady

characterized by a symmetrical cavitation attached to each blade. The cavitation length

α≥*10%* of vapour takes, approximately, *50%* of the blade spacing, *h*. Consequently, the

symmetrical on both blades and large enough to produce the performances drop of inducer. Alternate blade cavitation is a phenomenon in which the cavitation length on the blades changes alternately from blade to blade. According to (Tsujimoto, 2005) the alternate blade cavitation starts to develop when the cavitation length exceeds about *65%* of the blade spacing: *(lcav/h)≥65%*. So, the incidence angle to the neighbouring blade decreases and, hence the cavitation length on the neighbouring blade decreases also. Then the incidence angle of

*=0.*056; after a symmetrical blade cavitation, the alternate blade cavitation starts as soon as

*=0.723* and

σ

*=0.140*. After that, for

σ

σ

௧ܥ

σ

, and the flow coefficient,

decreases, until these

σ*.* This

*=0.174* and this

Ψ

σ

decreases even more, the regions of vapour

*=0.219*, the cavitation length containing

*=0.219*. This stable behaviour is

σ

*=0.114*, the cavitation becomes

, and the

(18)

a growth rate of 1.2 which allowed values of *y+* between 6 and 51.

Strouhal number, *St*, defined in (17); and by the head coefficient,

<sup>Ψ</sup> ൌ ܲଶ െ ܲଵ ଵݒ ߩ ʹͳȀ

σ

between

alternate blade cavitation appears when the cavitation number decreases to

σ

the original blade increases and the cavitation length on it also increases.

σ*=0.156*.

Φ

σ

The unsteady cavitating flow was characterized by the cavitation number,

**3.2.2 Numerical results** 

**3.2.2.1 Stable blade cavitation** 

the time and all the values of

behaviour, for values of

performance drop of inducer, see Fig. 7.

Φ

grows gradually as σ decreases. When

asymmetrical cavitation continue to

The cavitation behaviour for

the *lcav/h* ratio is higher than 65%, i.e. to

Φ

increase and it obstruct the flow channel. So, at

Φ

, given by:

values of

σ

Fig. 8 presents, for

(c) Boundary condition and near-wall mesh resolution

Fig. 6. Blades cascades corresponding to the inducers studied, see Fig. 1

carried out on non-cavitating flow. Three meshes were tested: a coarse mesh (300X50), a fine mesh (500X50), and a refined mesh (650X50). The first coarse mesh presented the backflow at outlet domain because of very important aspect ratio upstream and downstream regions. The fine and refined meshes presented similar results, but the fine mesh reaches the solution faster than the refined mesh; for this reason, the fine mesh was selected for realize all calculations. Boundary layer meshing was used to ensure sufficient refinement near the walls and thus a small dimensionless factor *y+*. First cell distance was imposed to 1 *mm*, with a growth rate of 1.2 which allowed values of *y+* between 6 and 51.

### **3.2.2 Numerical results**

188 Mass Transfer - Advanced Aspects

(c) Boundary condition and near-wall mesh resolution

Fig. 6. Blades cascades corresponding to the inducers studied, see Fig. 1

(a) Two-blade cascade (b) Three-blade cascade

The unsteady cavitating flow was characterized by the cavitation number, σ, and the Strouhal number, *St*, defined in (17); and by the head coefficient, Ψ, and the flow coefficient, Φ, given by:

$$\Psi = \frac{P\_2 - P\_1}{1/2 \cdot \rho\_l \cdot v\_1^2} \quad \text{and} \quad \Phi = \frac{(\mathcal{C}\_a)\_1}{\mathcal{C}\_t} \tag{18}$$

#### **3.2.2.1 Stable blade cavitation**

Calculations of unsteady cavitating flow were carried out for four flow rates over the blades cascade of a two–blade inducer and for three flow rates over the blades cascade of a three– blade inducer, see Fig. 7. En general, the results present two small regions of vapour, at high values of σ*,* on the leading edge of suction side. The regions of vapour are symmetric, for all the time and all the values of σ. The vapour region increases as σ decreases, until these regions of vapour become large enough to block the flow channel, resulting in the performance drop of inducer, see Fig. 7.

Fig. 8 presents, for Φ*=0.056*, the regions of vapour corresponding to diverse values of σ*.* This figure shows that the cavitation begins with very small regions of vapour which appear on the leading edge of the blade, on the suction side. The regions of vapour have a steady behaviour, for values of σ between σ*=0.723* and σ*=0.219*. This stable behaviour is characterized by a symmetrical cavitation attached to each blade. The cavitation length grows gradually as σ decreases. When σ decreases even more, the regions of vapour increase and it obstruct the flow channel. So, at σ*=0.219*, the cavitation length containing α≥*10%* of vapour takes, approximately, *50%* of the blade spacing, *h*. Consequently, the alternate blade cavitation appears when the cavitation number decreases to σ*=0.174* and this asymmetrical cavitation continue to σ*=0.140*. After that, for σ*=0.114*, the cavitation becomes symmetrical on both blades and large enough to produce the performances drop of inducer.

Alternate blade cavitation is a phenomenon in which the cavitation length on the blades changes alternately from blade to blade. According to (Tsujimoto, 2005) the alternate blade cavitation starts to develop when the cavitation length exceeds about *65%* of the blade spacing: *(lcav/h)≥65%*. So, the incidence angle to the neighbouring blade decreases and, hence the cavitation length on the neighbouring blade decreases also. Then the incidence angle of the original blade increases and the cavitation length on it also increases.

The cavitation behaviour for Φ*=0.042* presents the same patron of cavitation than for Φ*=0.*056; after a symmetrical blade cavitation, the alternate blade cavitation starts as soon as the *lcav/h* ratio is higher than 65%, i.e. to σ*=0.156*.

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 191

However, the alternate blade cavitation was not detected in the three-blade inducer, this agrees with experimental observations which have noted this instability only on inducers

with pair blades number.

Fig. 8. Curve of the performance drop (

high values of the cavitation number (

σ

**3.2.2.2 Unstable blade cavitation** 

diverse forms which vary as

was decreased to

rate (Φ

(*α≥10%*) corresponding to diverse values of

σ

Fig. 9 shows the contours of vapour fraction (*α=10%* and

; e.g., for

Φ

Φ

cavitation became symmetrical on both blades, for cavitation numbers lower than

σ

σ

The numerical results show the appearance of the rotating blade cavitation to a low flow

the purpose of observing a cycle of the rotating blade cavitation, *Tcav*. In monitoring the evolution of the cavitation on blade **1**, it is observed that the cavitation length is the same on both blades at *t=0.5·Tcav* and *t=1.0·Tcav*. In the beginning of the cycle, the cavitation length on the blade **1**, *lcav-b1*, decreases with the time. So, at *t=0.267·Tcav*, the size of *lcav-b1* is the smallest on the blade **1**, while the size of *lcav-b2* becomes the largest on the blade **2**. The cavitation

*=0.039*). As it was observed for the flow rates analysed previously, the cavitation has

*=0.258*, the rotating blade cavitation was occurred. After that, the

σ

*=0.056*) and pictures of the regions of vapour

. Calculations using RNG κ–ε model

*=0.039*, symmetrical cavitations were observed for

σ*<0.185*.

*=0.258*) at different times, with

*≥0.294)*. However, as soon as the cavitation number

Fig. 7. Curves of the performance drop inducer

However, the alternate blade cavitation was not detected in the three-blade inducer, this agrees with experimental observations which have noted this instability only on inducers with pair blades number.

Fig. 8. Curve of the performance drop (Φ*=0.056*) and pictures of the regions of vapour (*α≥10%*) corresponding to diverse values of σ. Calculations using RNG κ–ε model
