**3.2.2.2 Unstable blade cavitation**

190 Mass Transfer - Advanced Aspects

(a) Two-blade inducer

(b) Three-blade inducer

Fig. 7. Curves of the performance drop inducer

The numerical results show the appearance of the rotating blade cavitation to a low flow rate (Φ*=0.039*). As it was observed for the flow rates analysed previously, the cavitation has diverse forms which vary as σ; e.g., for Φ*=0.039*, symmetrical cavitations were observed for high values of the cavitation number (σ*≥0.294)*. However, as soon as the cavitation number was decreased to σ*=0.258*, the rotating blade cavitation was occurred. After that, the cavitation became symmetrical on both blades, for cavitation numbers lower than σ*<0.185*.

Fig. 9 shows the contours of vapour fraction (*α=10%* and σ*=0.258*) at different times, with 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

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 193

cavitation was observed for two flow rate coefficients in the three-blade inducer. In these cases it was observed, at a certain time, two cavitations with the same size and another cavitation with a smaller size. This smaller cavitation moved in the same translational direction than the blade cascade, i.e. in the same rotational direction of the inducer; see Fig. 10(b). These cavitating fluctuations are cyclic with specific frequency. Table 3 resumes the frequencies of the rotating blade cavitation captured numerically for different flow rates and

(a) Two-blade cascade

(b) Three blade cascade

Fig. 10. Pressure fluctuation on suction side of the blades caused by the variation of the

on different configurations.

length cavitation

length on the blade **2** is inversed to the one on the blade **1**. So, *lcav-b2* decreases from *t=0.267·Tcav* to *t=0.777·Tcav,* where *lcav-b2* is the smallest on the blade **2** and *lcav-b1* is the largest on blade **1**.


Table 3. Comparison of frequencies of vapour detachment in the studied blades cascades

Fig. 9. Rotating blade cavitation on a two-blade inducer (σ=0.258 and Φ=0.039). Calculations using RNG κ-ε turbulence model

The vapour fluctuation has an unsteady cyclic behaviour, with a low frequency equal to *fcav=0.07·f*ω on one blade, see Fig. 10(a). The unsteady behaviour of the cavitation modifies the flow patterns which cause a pressure fluctuation upstream. The frequency analysis on the absolute frame gives a cavitating frequency of *fcav=0.14·f*ω, because of the cavitation fluctuation on the two blades.

Fig. 10 shows the pressure coefficients measured on the two blades for the aircraft inducer and on the three blades for the industrial inducer. These ones were measured by virtual probes on the suction side of the blades inducers. These pressure fluctuations have been caused by the variation of the cavitation size attached to each blade. Rotating blade

length on the blade **2** is inversed to the one on the blade **1**. So, *lcav-b2* decreases from *t=0.267·Tcav* to *t=0.777·Tcav,* where *lcav-b2* is the smallest on the blade **2** and *lcav-b1* is the largest

σ

**Two-blade** 0.039 0.258 9.1 *Hz* 0.068 0.110 *s* **Three-blade** 0.070 0.189 1 *Hz* 0.041 1.000*<sup>s</sup>*

Table 3. Comparison of frequencies of vapour detachment in the studied blades cascades

Fig. 9. Rotating blade cavitation on a two-blade inducer (σ=0.258 and Φ=0.039). Calculations

The vapour fluctuation has an unsteady cyclic behaviour, with a low frequency equal to

the flow patterns which cause a pressure fluctuation upstream. The frequency analysis on

Fig. 10 shows the pressure coefficients measured on the two blades for the aircraft inducer and on the three blades for the industrial inducer. These ones were measured by virtual probes on the suction side of the blades inducers. These pressure fluctuations have been caused by the variation of the cavitation size attached to each blade. Rotating blade

the absolute frame gives a cavitating frequency of *fcav=0.14·f*

on one blade, see Fig. 10(a). The unsteady behaviour of the cavitation modifies

ω

, because of the cavitation

*fcav fcav/f*

0.084 0.184 1.59 *Hz* 0.066 0.630 *s*

ω

*Tcav*

Φ

on blade **1**.

**Blade cascade** 

using RNG κ-ε turbulence model

fluctuation on the two blades.

*fcav=0.07·f*

ω

cavitation was observed for two flow rate coefficients in the three-blade inducer. In these cases it was observed, at a certain time, two cavitations with the same size and another cavitation with a smaller size. This smaller cavitation moved in the same translational direction than the blade cascade, i.e. in the same rotational direction of the inducer; see Fig. 10(b). These cavitating fluctuations are cyclic with specific frequency. Table 3 resumes the frequencies of the rotating blade cavitation captured numerically for different flow rates and on different configurations.

(a) Two-blade cascade

(b) Three blade cascade

Fig. 10. Pressure fluctuation on suction side of the blades caused by the variation of the length cavitation

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 195

σ

**3.3 Experimental study and numerical analysis of cavitating flows in three inducers** 

Fig. 13. Bank of experimental tests from DynFluid laboratory (Arts et Métiers ParisTech),

The hydrodynamic bank used for the experimental tests consists of three independent closed loops. The first loop is adapted for test on the industrial inducers, see Fig. 13. The second loop is used for experimental tests of centrifugal pumps alone or coupled to an

*=0.258* and

Φ

*=0.039*).

Fig. 12. Rotating blade cavitation on a two-blade inducer (

inducer. The third loop is adapted for aircraft-type inducers.

Calculations using RNG κ-ε modified turbulence model

**3.3.1 Experimental tests** 

loop of the industrial inducers

#### **3.2.2.3 Coupling of the intrinsic instabilities and the system instabilities**

Numerical simulations were complemented using the RNG κ–ε modified turbulence model for σ*=0.258* and Φ*=0.039*. This modification allows the interaction between the system unsteadiness and the self–oscillation of the vapour region.

Fig. 11. Temporal evolution of the cavitation length. Calculations using RNG κ-ε and RNG κ-ε modified models

Fig. 12 presents the contours of vapour fraction at different times on a blades cascade. The cavitation shows a quasi–cyclical unsteady behaviour with a cavitation detachment frequency equal to *fcav=0.06·f*ω on relative frame and *fcav=0.12·f*ωon absolute frame.

The cavity has a similar cyclical unsteady behaviour than in the analysis using RNG κ–ε model, but now, the results show the vapour detachment (*t=0.833·T* to *t=0.867·Tcav*, see Fig. 12, blade **1**), followed by its convection downstream (*t=0.900·Tcav* to *t=0.933·Tcav*, see Fig. 12, blade **1**) and then, the cavitation passing from blade **1** to blade **2** at blades cascade throat (*t=0.967·Tcav* to *t=1.0·Tcav*, see Fig. 12, blade **1** and blade **2**).

The curves of Fig. 11 show the temporal evolution of the cavitation length (*α≥10%*) measured on each blade (e.g., *t=0.033·Tcav* on Fig. 9), for both results calculations using RNG κ-ε and RNG κ-ε modified models. A negative cavitation length, for calculations using RNG κ-ε modified model, means that vapour region is attached to the pressure side of the neighbour blade. This phenomenon appears when the cavitation length of neighbour blade is large enough which produces the blockage of the channel flow. Thus, the fluid flow passes only through the other channel. All curves have a similar behaviour but the cavitation length is larger with the RNG κ–ε modified model than the RNG κ–ε model. The local length fluctuations observed on modified turbulence model are caused by the self– oscillation of the vapour region.

Numerical simulations were complemented using the RNG κ–ε modified turbulence model

Fig. 11. Temporal evolution of the cavitation length. Calculations using RNG κ-ε and RNG

Fig. 12 presents the contours of vapour fraction at different times on a blades cascade. The cavitation shows a quasi–cyclical unsteady behaviour with a cavitation detachment

ω

on absolute frame.

on relative frame and *fcav=0.12·f*

The cavity has a similar cyclical unsteady behaviour than in the analysis using RNG κ–ε model, but now, the results show the vapour detachment (*t=0.833·T* to *t=0.867·Tcav*, see Fig. 12, blade **1**), followed by its convection downstream (*t=0.900·Tcav* to *t=0.933·Tcav*, see Fig. 12, blade **1**) and then, the cavitation passing from blade **1** to blade **2** at blades cascade throat

The curves of Fig. 11 show the temporal evolution of the cavitation length (*α≥10%*) measured on each blade (e.g., *t=0.033·Tcav* on Fig. 9), for both results calculations using RNG κ-ε and RNG κ-ε modified models. A negative cavitation length, for calculations using RNG κ-ε modified model, means that vapour region is attached to the pressure side of the neighbour blade. This phenomenon appears when the cavitation length of neighbour blade is large enough which produces the blockage of the channel flow. Thus, the fluid flow passes only through the other channel. All curves have a similar behaviour but the cavitation length is larger with the RNG κ–ε modified model than the RNG κ–ε model. The local length fluctuations observed on modified turbulence model are caused by the self–

*=0.039*. This modification allows the interaction between the system

**3.2.2.3 Coupling of the intrinsic instabilities and the system instabilities** 

unsteadiness and the self–oscillation of the vapour region.

for σ

*=0.258* and

κ-ε modified models

frequency equal to *fcav=0.06·f*

oscillation of the vapour region.

ω

(*t=0.967·Tcav* to *t=1.0·Tcav*, see Fig. 12, blade **1** and blade **2**).

Φ

Fig. 12. Rotating blade cavitation on a two-blade inducer (σ*=0.258* and Φ*=0.039*). Calculations using RNG κ-ε modified turbulence model
