**3.3.2 Numerical analysis**

196 Mass Transfer - Advanced Aspects

a) σ=0.280 b) σ=0.230 c) σ=0.140 d) σ=0.090

e) σ=0.060 f) σ=0.055 g) σ=0.050 h) σ=0.045

Fig. 14. Diverse cavitation forms obtained experimentally on a three-blade inducer with

• Two tanks with a capacity of *4 m3*, connected by pipes of diameter of *350 mm*.

• A *22 kW* motor controls the rotational velocity. It is measured using a tachometer.

• A temperature sensor. The average temperature during the tests was about *20 °C*. To capture the experimental structures of vapour through the transparent cover, it was used

• The inducer has a transparent casing allowing the observation of vapour structures

Fig. 14 shows the vapour formation at leading edge of a three blade inducer. The regions of vapour were mainly manifested in the form of three identical regions attached to each blade

*=0.280*). As the inlet pressure decreases, the cavitating structures suffer a growth phase,

σ

blade is blocked by the vapour, the performance drop of the inducer occurs suddenly

The pictures obtained during the cavitation tests are typical cavitation forms and, in general, they are consistent with those reported by (Offtinger et al., 1996), see Fig. 14. Representative pictures were obtained from different experimental tests on the three studied inducers at partial flow rate, nominal flow rate and over-flow rate. As an example, some pictures obtained on a three-blade industrial inducer with blade tip angle of 16° are commented: - The inception of the cavitation at leading edge of the tip blade, see Fig. 14(a-b). - The formation of a tip tourbillon which captures the vapour, see Fig. 14(c).


this figure, each image corresponds to a value of σ for a flow rate constant.

σ

*=0.060* and

*=0.045*). The gradual vapour apparition generates noise and vibrations. In

*=0.230* to

σ

σ

*=0.090*), which move to down

*=0.055*). When the passage inter-

The industrial inducers loop consists mainly of the follow elements:

• A vacuum pump adjusts the pressure at the free surface tank.

• A motorized valve is used to vary the fluid flow of the inducer.

β*=16°* (Φ*=0.164*)

(σ

(σ

*=0.050* and

(transparent cover).

σ

blade, see Fig. 14(d).

a digital camera under strobe lighting.

principally at tip leading edge of the blade (from

to the hub until they block the flow channel (

#### **3.3.2.1 Domain de control and grid generation**

The numerical simulations, in steady and unsteady regime, were carried out on a two-blade aircraft inducer with a blade tip angle of 4°. In order of accelerate the calculation time, for the steady numerical simulations, only one third of the inducer was modelled. By contrast, this simplification could not be considered for the unsteady calculations because the instabilities of the cavitating flow are influencing by the neighbour blades (system instabilities).

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 199

variations on the numerical results because of a poor grid used. The last two grids give approximately the same than the experimental data but the grid (d) makes a lot of computational time to find the solution. For this reason, the grid (c) was selected for carried

Finally, a grid dependence study to model the radial tip clearance was made. Fist, different grid types were tested, see Fig. 19. The first two grids were generated automatically on only one region which merges the blade to blade region and the tip clearance region. In this type of grids is very difficult to control the mesh near to wall, particularly the space formed between the tip blade and the carter, so the grid (d) on Fig. 19 was selected because it uses two blocks to defines the blade to blade and tip clearance regions and a conformal grid interfaces between both regions. Second, the grid density was varied from 5, 15, 25 and 30

 *lupstream lrotor Ldownstream*  Fig. 17. Different blocks which form whole of the computational domain

(c) Meridional view

(a) Front view (b) Lateral view

Fig. 18. Computational grid used for the numerical simulations

out the numerical results.

equidistant cells in radial direction.

Fig. 16. Vapour behaviour for different cavitation conditions and its corresponding performance drop curve obtained experimentally on a two-blade inducer (Φ*=0.005*)

Many three-dimensional hybrid grids were generated using the pre-processor *Gambit*. The computational domain was divided into four sub–blocks in order to facilitate the grid generation. The sub–blocks are located as follow, see Fig. 17:


The grid process starts by meshing, with 2D triangular type cells, the blades surfaces. Then, the blade to blade region was meshed with 3D tetrahedral type cells. The tip clearance and blade to blade regions were meshed using smaller size cells than inlet and outlet regions, which were meshed with prism type cells, see Fig. 18. One conformal grid interfaces were used at the boundary of the regions "blade to blade – tip clearance". Two non–conformal grid interfaces were used at the boundary of the regions "upstream – blade to blade" and "blade to blade – downstream", see Fig. 17. Fig. 18 shows the surface grid on the rotor and the meridional view which is noticed the tip clearance grid.

For the grid independence study, four computational grids were tested. All these were generated with the same meshing strategy, but they are different in the cells number. The grid sizes are: (a) 480,185; (b) 1, 050,154; (c) 1,528,668; and (d) 1,996,418. The numerical results in steady state were compared to the experimental data. The first two grids show

Fig. 16. Vapour behaviour for different cavitation conditions and its corresponding

• **Blade to blade region**, which includes the flow channels formed by the blades;

Many three-dimensional hybrid grids were generated using the pre-processor *Gambit*. The computational domain was divided into four sub–blocks in order to facilitate the grid

The grid process starts by meshing, with 2D triangular type cells, the blades surfaces. Then, the blade to blade region was meshed with 3D tetrahedral type cells. The tip clearance and blade to blade regions were meshed using smaller size cells than inlet and outlet regions, which were meshed with prism type cells, see Fig. 18. One conformal grid interfaces were used at the boundary of the regions "blade to blade – tip clearance". Two non–conformal grid interfaces were used at the boundary of the regions "upstream – blade to blade" and "blade to blade – downstream", see Fig. 17. Fig. 18 shows the surface grid on the rotor and

For the grid independence study, four computational grids were tested. All these were generated with the same meshing strategy, but they are different in the cells number. The grid sizes are: (a) 480,185; (b) 1, 050,154; (c) 1,528,668; and (d) 1,996,418. The numerical results in steady state were compared to the experimental data. The first two grids show

δ

Φ*=0.005*)

*<sup>t</sup>* and an axial length *lrotor*.

performance drop curve obtained experimentally on a two-blade inducer (

• **Inlet region**, lengthened *2.6 Dt* upstream of the blade leading edge;

• **Outlet region**, extended *2.4 Dt* downstream of trailing edge; and

generation. The sub–blocks are located as follow, see Fig. 17:

• **Tip clearance region**, modelled by a ring of thickness

the meridional view which is noticed the tip clearance grid.

variations on the numerical results because of a poor grid used. The last two grids give approximately the same than the experimental data but the grid (d) makes a lot of computational time to find the solution. For this reason, the grid (c) was selected for carried out the numerical results.

Finally, a grid dependence study to model the radial tip clearance was made. Fist, different grid types were tested, see Fig. 19. The first two grids were generated automatically on only one region which merges the blade to blade region and the tip clearance region. In this type of grids is very difficult to control the mesh near to wall, particularly the space formed between the tip blade and the carter, so the grid (d) on Fig. 19 was selected because it uses two blocks to defines the blade to blade and tip clearance regions and a conformal grid interfaces between both regions. Second, the grid density was varied from 5, 15, 25 and 30 equidistant cells in radial direction.

Fig. 17. Different blocks which form whole of the computational domain

Fig. 18. Computational grid used for the numerical simulations

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 201

possible to observe that, as numerical as experimentally, at the inception of cavitation, the vapour appears in form of triangle at leading edge, and it is extended at tip blade as far as to

> σ*=0.080*

> σ*=0.100*

Fig. 21. Qualitative comparison of cavitation between experimental tests and numerical

Fig. 22. Quantitative comparison of cavitation between experimental tests and numerical

The numerical results obtained in steady state were compared as much qualitatively as qualitatively to the experimental tests. Fig. 21 shows a qualitative comparison of cavitation

The cavitating analyse find a fluid flow more instable, during the numerical simulations and experimental tests, in the partial flow rates than in the over-flows. In the Fig. 20 is possible to note that, when the performance drop arrives, the flow channel is filled by the vapour for

Φ

*=0.079*) in the Fig. 21. On the other hand,

β*=8°*

behaviour on a three-blade inducer with blade tip angle of 8°.

*=0.*112) unlike a partial flow rate (

*=8°* (*α≥15%* and

β

σ*=0.047* 

σ*=0.059* 

Φ*=0.079*) σ*=0.025* 

σ*=0.046*  σ*=0.023*

σ*=0.038*

cause the flow channel blockage.

σ*=0.255* 

σ*=0.232* 

results on a three-blade inducer with

results on a three-blade inducer with

Φ

an over-flow (

σ*=0.144* 

σ*=0.175* 

σ*=0.407* 

σ*=0.419* 

Fig. 19. Tip clearance grids tested

#### **3.3.2.2 Numerical results in steady state**

The numerical simulations in steady state were carried out for the three inducers presented on the Fig. 1. The cavitating calculations were realized on one flow channel of the inducers using periodical conditions for different flow rates (partial flow rates, nominal flow rates and over-flows), and for various cavitation conditions (σ). By way of example, the Fig. 20 shows the diverse cavitation forms which rise as the cavitation number decreases. It is

Fig. 20. Vapour behaviour for different cavitation conditions and its corresponding performance drop curve obtained numerically on a three-blade inducer with β*=8°* (*α≥15* and Φ*=0.112*)

The numerical simulations in steady state were carried out for the three inducers presented on the Fig. 1. The cavitating calculations were realized on one flow channel of the inducers using periodical conditions for different flow rates (partial flow rates, nominal flow rates

shows the diverse cavitation forms which rise as the cavitation number decreases. It is

Fig. 20. Vapour behaviour for different cavitation conditions and its corresponding performance drop curve obtained numerically on a three-blade inducer with

σ

). By way of example, the Fig. 20

β

*=8°* (*α≥15* and

(a) (b)

(c) (d)

and over-flows), and for various cavitation conditions (

Fig. 19. Tip clearance grids tested

Φ*=0.112*)

**3.3.2.2 Numerical results in steady state** 

possible to observe that, as numerical as experimentally, at the inception of cavitation, the vapour appears in form of triangle at leading edge, and it is extended at tip blade as far as to cause the flow channel blockage.

Fig. 21. Qualitative comparison of cavitation between experimental tests and numerical results on a three-blade inducer with β*=8°* (*α≥15%* and Φ*=0.079*)

Fig. 22. Quantitative comparison of cavitation between experimental tests and numerical results on a three-blade inducer with β*=8°*

The numerical results obtained in steady state were compared as much qualitatively as qualitatively to the experimental tests. Fig. 21 shows a qualitative comparison of cavitation behaviour on a three-blade inducer with blade tip angle of 8°.

The cavitating analyse find a fluid flow more instable, during the numerical simulations and experimental tests, in the partial flow rates than in the over-flows. In the Fig. 20 is possible to note that, when the performance drop arrives, the flow channel is filled by the vapour for an over-flow (Φ*=0.*112) unlike a partial flow rate (Φ*=0.079*) in the Fig. 21. On the other hand,

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 203

Knowingly the hard work needed to detect the instabilities by 3D numerical simulations; the results obtained to Φ=0.0050 and σ=0.064 are presented, in different views, on the Fig. 23, Fig. 24 and Fig. 25. The pictures were obtained when all flow parameters were stabilized (after *t=130·tref*). This figure shows the temporal evolution of the vapour (*α≥20%*) in the two-blade aircraft inducer with blade tip angle of *4°*. Eighteen instants can be observed between *t=130·tref* and *t=147·tref*, each picture corresponds to one inducer turn

These three figures show the cavitation has a crown form located at the periphery of the inducer. Another vapour region, with shape of the torch is located upstream, see Fig. 24 and Fig. 25. In certain instants, both cavitations, this one located at the periphery and the other one in form of torch, are connected by a narrow vapour region formed along the leading

The cavitation develops gradually on the leading edge of the blade **2** (green blade), and it becomes bigger than on the blade **1** (blue blade) at *t=135·tref*. Later, the cavitation decreases gradually on the blade **2** until *t=140·tref*. In contrast, at this instant, the cavitation on the leading edge of the blade **1** is the biggest. Finally, the size cavitation increases again on the

α

α

*=20%*) in the two-blade inducer (

*=20%*) in the two-blade inducer (

Φ

Φ

*=0.0050* and

*=0.0050* and

(*tref*).

σ

σ

edge, from the tip blade to the inducer hub.

Fig. 24. Temporal evolution of the vapour (

Fig. 25. Temporal evolution of the vapour (

*=0.064*), isometric view

*=0.064*), lateral view

blade **2** and it decreases on the blade **1** at *t=144·tref.* 

the main difference found on the curves of the Fig. 22 was to the partial flow rate where the performance drop was predicted faster by the calculation than the experimental tests.

Fig. 22 shows a comparison of the performance drop curves obtained numerical and experimentally where is possible to observer a good coherence of the results. The differences found between numerical results and experimental data are, amongst others, attributing to that:


#### **3.3.2.3 Numerical results in unsteady state**

The numerical simulations in unsteady state were carried out from the results obtained in steady state for a two-blade inducer with tip blade angle of 4° described in the Fig. 1. The calculations were realized to partial flow rates, where more instability problems were detected in the previous experimental and numerical cavitating analyse.

Fig. 23. Temporal evolution of the vapour (α*=20%*) in the two-blade inducer (Φ*=0.0050* and σ*=0.064*), front view

The reference time was defined as the inducer rotating period, *tref=T*ω*=7.5E-3 s.* The time step, used for all numerical simulations, was calculated, after a temporal independence study, for an angular moving of 3.6°; then, the time step was defined as Δ*t=tref/100=7.5E-5 s*.

It is important to underline that the unsteady calculations needed about 80 iterations by time step, resulting in 8,000 iterations by one inducer turn. The numerical stability was noted after 50 inducer turns, i.e. it was necessary at least 400,000 iterations to start to observe the instabilities caused by the cavitating flow in the inducer… if these ones exist at these flow conditions (Φ and σ). In considering that, the calculations were developed on an 8 processors cluster, with a calculation time of about 10 *s/iteration*, the unsteady calculations can take more than two months (time of machine) for a constant flow rate and a specific cavitation number. The very long calculating time, combined with the high capacity of storage, necessary for the post-processing stage, make very difficult to realize successfully the numerical calculations in unsteady state. Moreover, the analysis of the results requires a hard work to find the instabilities.

the main difference found on the curves of the Fig. 22 was to the partial flow rate where the performance drop was predicted faster by the calculation than the experimental tests. Fig. 22 shows a comparison of the performance drop curves obtained numerical and experimentally where is possible to observer a good coherence of the results. The differences found between numerical results and experimental data are, amongst others, attributing to

• The numerical simulations were carried out on only one flow channel using cyclical conditions; consequently, the fluid flow is perfectly identical on the three flow channels.

This configuration does not consider the interaction between the flow channels. • The numerical simulations suppose that the inducer has parfait geometry, without

• The experimental back elements located upstream and downstream were negligee.

α

step, used for all numerical simulations, was calculated, after a temporal independence

It is important to underline that the unsteady calculations needed about 80 iterations by time step, resulting in 8,000 iterations by one inducer turn. The numerical stability was noted after 50 inducer turns, i.e. it was necessary at least 400,000 iterations to start to observe the instabilities caused by the cavitating flow in the inducer… if these ones exist at these

processors cluster, with a calculation time of about 10 *s/iteration*, the unsteady calculations can take more than two months (time of machine) for a constant flow rate and a specific cavitation number. The very long calculating time, combined with the high capacity of storage, necessary for the post-processing stage, make very difficult to realize successfully the numerical calculations in unsteady state. Moreover, the analysis of the results requires a

*=20%*) in the two-blade inducer (

). In considering that, the calculations were developed on an 8

Φ

ω

Δ

*=0.0050* and

*=7.5E-3 s.* The time

*t=tref/100=7.5E-5 s*.

The numerical simulations in unsteady state were carried out from the results obtained in steady state for a two-blade inducer with tip blade angle of 4° described in the Fig. 1. The calculations were realized to partial flow rates, where more instability problems were

manufacturing defects and composed of identical blades.

• The numerical model does not consider the radial tip clearance.

detected in the previous experimental and numerical cavitating analyse.

The reference time was defined as the inducer rotating period, *tref=T*

study, for an angular moving of 3.6°; then, the time step was defined as

**3.3.2.3 Numerical results in unsteady state** 

Fig. 23. Temporal evolution of the vapour (

Φ and σ

hard work to find the instabilities.

that:

σ

*=0.064*), front view

flow conditions (

Knowingly the hard work needed to detect the instabilities by 3D numerical simulations; the results obtained to Φ=0.0050 and σ=0.064 are presented, in different views, on the Fig. 23, Fig. 24 and Fig. 25. The pictures were obtained when all flow parameters were stabilized (after *t=130·tref*). This figure shows the temporal evolution of the vapour (*α≥20%*) in the two-blade aircraft inducer with blade tip angle of *4°*. Eighteen instants can be observed between *t=130·tref* and *t=147·tref*, each picture corresponds to one inducer turn (*tref*).

These three figures show the cavitation has a crown form located at the periphery of the inducer. Another vapour region, with shape of the torch is located upstream, see Fig. 24 and Fig. 25. In certain instants, both cavitations, this one located at the periphery and the other one in form of torch, are connected by a narrow vapour region formed along the leading edge, from the tip blade to the inducer hub.

The cavitation develops gradually on the leading edge of the blade **2** (green blade), and it becomes bigger than on the blade **1** (blue blade) at *t=135·tref*. Later, the cavitation decreases gradually on the blade **2** until *t=140·tref*. In contrast, at this instant, the cavitation on the leading edge of the blade **1** is the biggest. Finally, the size cavitation increases again on the blade **2** and it decreases on the blade **1** at *t=144·tref.* 

Fig. 24. Temporal evolution of the vapour (α*=20%*) in the two-blade inducer (Φ*=0.0050* and σ*=0.064*), lateral view

Fig. 25. Temporal evolution of the vapour (α*=20%*) in the two-blade inducer (Φ*=0.0050* and σ*=0.064*), isometric view

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 205



Finally, the unsteady cavitating calculations realized in three-dimensions for a two-blade inducer demonstrate the complexity to obtain and analyze the flow instabilities caused by the cavitation in these machines. The obtained results show that rotating cavitation appears for a partial flow rate, but it is less obvious in the inducer than in the blades cascade. It was noted that the shape and behaviour of cavitation is greatly disturbed by the tip radial

Bakir, F., Kouidri, S., Noguera, R. & Rey, R. (2003). *Experimental analysis of an axial inducer* 

Campos-Amezcua, R., Khelladi, S., Bakir, F., Mazur-Czerwiec, Z., Sarraf, C., & Rey, R. (2010)

Campos-Amezcua, R. (2009). *Analyse des écoulements cavitants stationnaires et instationnaires* 

Mejri, I., Bakir, F., Rey, R. & Belamri, T. (2006). *Comparison of computational results obtained* 

Kim, S.E., Mathur, S.R., Murthy, J.Y. & Chouhury, D. (1998). *A Reynolds–Average Navier–* 

Offtinger, C. Henry, C. & Morel, R. (1996). *Instabilité de fonctionnement en débit partiel d'un* 

Reboud, J., Stutz, B. & Coutier-Delgosha, O. (1998). *Two-phase flow structure of cavitation:* 

Singhal, A.K., Athavale, M.M., Li, H.Y. & Jiang, Y. (2002). *Mathematical basis and validation of* 

Stutz, B. & Reboud, J. L. (2000). *Measurements within unsteady cavitation,* Exp. in Fluids. Vol.

Tsujimoto, Y., Horiguchi, H. & Qiao, X. (2005). *Backflow from inducer and its dynamics*, 5th

Tsujimoto, Y., Kamijo, K., & Brennen, C. (2001). *Unified treatment of flow instabilities of* 

*the full cavitation model*, J. Fluids Eng. Vol. 124(3), pp. 617–624.

Pumping Machinery Symp., June 19–23, Houston, Texas.

*turbomachines,* J. propulsion and power. Vol. 17(3), pp. 636-643.

d'Energétique et Mécanique des Fluides Interne, Paris, France.

Sciences Meeting and Exhibit, Reno, NV.

pp. 31–38. Grenoble, France.

Grenoble, France.

29, pp. 245–552.

*influence of the shape of the blade leading edge on the performances in cavitating regime*, Journal of Fluids Engineering, Transactions of the ASME, Vol. 125(2), pp. 293-

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*dans les turbomachines*, PhD thesis. Arts et Métiers ParisTech - Laboratoire

*from a homogeneous cavitation model with experimental investigations of three inducers*, Journal of Fluids Engineering, Transactions of the ASME, Vol. 128(6), pp. 1308-

*Stokes solver using unstructured mesh based finite–volume scheme,* 36th Aerospace

*inducteur fretté and comparaison avec le cas non fretté,* 3ème journée on cavitation, SHF.

*Experiment and modelling of unsteady effects,* 3rd Int. Symp. Cavitation. April 7-10,

leading edge of the neighbouring blade (system instability).

clearance, which also modifies the torch which is formed upstream.

(combination of intrinsic and system instabilities).

**5. References** 

301.

1323.

224(2), pp. 223-238.

In conclusion, this fluctuation of cavitation size is almost cyclic, as observed previously in the analysis of the 2D unsteady cavitating flow. As can be seen in the figure, the cavitation length is maximal at the instants *t=135·tref* and *t=144·tref* on the blade **2** and at the instants *t=131·tref* and *t=140·tref*, and *t=149·tref* on the blade **1**.

The period of the cavitation fluctuation is *Tcav=0.0675 s* and its frequency *fcav=14.8 Hz*. The fluctuations can be driven by the cavitation torch formed upstream of the inducer. The torch turns in the same direction, but with a lower rotational velocity of the inducer. So, the torch turns 1 time while the inducer turns 9 times, as can be observed on the Fig. 23. The unsteady cavitating calculations were performed, in the first place, for σ*=0.064*. Then, from this unsteady results, the calculations were realized for σ*=0.051*, and then, for σ*=0.043*. This last calculation corresponds about *10%* of the inducer performance drop curve. In this case, one cavitation fluctuation occurs each 22 inducer turns. Thus, the cavitation fluctuation period is *Tcav=0.0600 s* and its frequency *fcav=16.7 Hz*, for σ*=0.043*.
