O Coutier-Delgosha

R Fortes-Patella

J L Reboud

M Hofmann

B Stoffel

Experimental and Numerical Studies in a Centrifugal Pump With Two-Dimensional Curved Blades in Cavitating Condition

In the presented study a special test pump with two-dimensional curvature blade geometry was investigated in cavitating and noncavitating conditions using different experimental techniques and a three-dimensional numerical model implemented to study cavitating flows. Experimental and numerical results concerning pump characteristics and performance breakdown were compared at different flow conditions. Appearing types of cavitation and the spatial distribution of vapor structures within the impeller were also analyzed. These results show the ability of the model to simulate the complex threedimensional development of cavitation in a rotating machinery, and the associated effects on the performance.

Introduction

The work presented in this paper was carried out in the scope of the European Research Program PROCOPE, between researchers of the TUD ͑Darmstadt University of Technology, Laboratory for Turbomachinery and Fluid Power͒ and of the LEGI ͑Laboratoire des Ecoulements Ge ´ophysiques et Industriels de Grenoble͒. The aim of this European program was to improve the understanding of the unsteady behavior of cavitating flows and the related erosive aggressiveness by experimental analyses and development of numerical models.

In this context, some research works were developed by both laboratories to analyze and to characterized erosion phenomenon, ͓1,2͔, as well as to study and to model unsteady cavitating flows around a two-dimensional hydrofoil, ͓3͔, and in a cascade of three hydrofoils, ͓4͔.

In order to extend those previous analyses, the present study consists of investigations by experimental means and numerical simulation of a special centrifugal test pump in cavitating and noncavitating conditions. Experiments were carried out at the TUD using different experimental techniques. The measurement of classical pump characteristics and performance breakdown at different flow conditions were associated with flow visualisations. Appearing types of cavitation and the spatial distribution of vapor structures within the impeller were analyzed.

Three-dimensional Navier-Stokes codes taking into account the cavitation process have been developed during the last years, ͓5-8͔, based on different multiphase flow approaches, ͓9,10͔. Industrial CFD codes are now starting to take into account cavitation models, allowing first applications to pump geometries, ͓11,12͔. In this context, a numerical model for three-dimensional cavitating flows is developed at LEGI, based on the threedimensional code FINE/TURBO™, developed by NUMECA International. This work is performed in cooperation with the Rocket Engine Division of SNECMA Moteurs and the French Space Agency CNES, with the final objective to provide accurate simulations of unsteady cavitating flows in the inducers of rocket engine turbopumps, ͓13,14͔. The application to the centrifugal pump represents a first step of validation of the model on steadystate cavitating flow in turbomachinery.

Geometry

A special impeller geometry has been chosen to easily adapt existing measuring techniques for a single hydrofoil in a test section, ͓3,15͔, to a pump test rig. Optical access in two planes was made possible to enable a view perpendicular to the blade surface on both suction and pressure side by Plexiglas windows in the housing and a Plexiglas shroud. The impeller has five singlecurved blades with two different radii at inlet and outlet. The part of the blade with the larger radius ͑the second part in flow direc-tion͒ was also made of Plexiglas to obtain optical access to the pressure side of the following blade and the entire channel, respectively.

Figure 1͑a͒ shows the intersection of the two blade radii. Another characteristic is the parallel hub and shroud to get an almost two-dimensional blade-to-blade channel with constant width. An axial-symmetric housing ͑Fig. 1͑b͒͒ is used to obtain almost constant conditions at the impeller outlet ͑if we neglect gravitational forces that are small compared to the performance of the impel-ler͒. With these preconditions we obtain comparable cavitation conditions in each channel without the influence of a volute casing.

Nominal conditions are at a rotational speed of 36 Hz and a flow rate Qϭ210 m 3 /h. Specific speed of the impeller is n s ϭnQ1/2 /H 3/4 ϭ20 ͑European units: n s and n in rpm, Q in m 3 /s and H in m͒, the outer impeller diameter equals 278 mm. Cavitation conditions are defined by NPSH value based on the upstream total pressure, water vapor pressure and density at ambient temperature.

Visualizations

Besides the measurements of the characteristics of the pump at different flow-rates and cavitation conditions, various visualisation techniques were used. All images shown in this paper were taken at nominal flow rate with various values of NPSH, where both cavitation on suction and pressure side of the blades occurs.

Stroboscopic light was used on one hand for standard imaging and high-speed video with the light-sheet illumination ͑Fig. 2͒ was applied to observe self-oscillating states of the cavitation on the blade pressure side or other unsteady effects, ͓16͔. With the aid of laser light-sheet illumination of the vapor-fluid interface, an analysis of the unsteadiness of the attached part of the cavitation in the impeller could be done at half-width of the channel. The images were taken at a rate of 2 Hz but triggered by an angular encoder and a special signal conditioning that allowed to take the image at every phase angle compared to a reference angle with a step size of 1 deg. Hence, the illuminated blade had 18 revolutions between every image.

Based on 500 images, each of them illuminated during 50 s, a mean gray value distribution was calculated to identify the statistical mean cavity on the leading edge of the blade.

The standard deviation of the gray values or its variance can serve as a quantity to determine the unsteadiness of the cavitation. In regions of these images with higher values of the variance ͑to be identified as dark regions͒, the fluctuations of the gray values and, therefore, the fluctuations of the reflecting vapor structures are larger than in regions with a smaller variance. Those regions are constantly filled with either water or vapor. The result of such a treatment is shown in Figs. 5 and6.

Mean distribution ͑Figs. 5͑a͒ and 6͑a͒͒: The attached part of the cavitation can only be identified by its two-phase interface, because the light is mainly reflected. But the mean region with a cloud shedding is also indicated by a higher mean gray value just downstream of the closure region of the cavitation sheet. For the comparison with numerical results, we just identify the attached part. As expected, the extent of both regions enlarges with decreasing NPSH value. As already mentioned, the flow has the tendency to separate at the leading edge of the blade, which is enhanced ͑Fig. 6͑a͒͒ at lower pressure conditions.

Standard deviation ͑Figs. 5͑b͒ and 6͑b͒͒: The fluctuation of the shedding becomes larger with decreasing NPSH and its extension fills almost half of the height of the channel. This is a result of the higher production of transient vapor structures from the leading edge cavitation. In contast, the interface of the attached cavitation seems to be rather stable. Probably the flow is clearly separated

Physical and Numerical Model

The main features of the physical and numerical models applied to simulate the flow field in the pump are summarized in the present paper. More details are given in ͓13,14,17͔.

Cavitating flows are described by a single-fluid model, based on previous numerical and physical work developed at LEGI, ͓9,18͔. This fluid is characterized by a density that varies in the computational domain: when the density in a cell equals that one of the liquid ( 1 ), the whole cell is occupied by liquid, and if it equals that one of the vapor ( v ), the cell is full of vapor. Between these two extreme values, a liquid/vapor mixture, still considered as one single fluid, occupies the cell. The void fraction ␣ϭ(Ϫ 1 )/( v Ϫ 1 ) can thus be defined as the local ratio of vapor contained in this homogeneous mixture.

Velocities are assumed to be locally the same for liquid and for vapor. An empirical state law is used to manage the mass fluxes resulting from vaporization and condensation processes. That barotropic law links the density to the local static pressure ( P). When the pressure is remarkably higher or lower than vapor pressure, the fluid is supposed to be purely liquid or purely vapor, according to the Tait equation or to the perfect gas law, respectively. The two fluid states are joined smoothly in the vaporpressure neighborhood. This results in the evolution law presented in Fig. 7, characterized mainly by its maximum slope 1/A min 2 ,

where A min 2 ϭץP/.ץ A min can thus be interpreted as the minimum speed of sound in the mixture.

The numerical model of cavitating flows based on that physical description is generated on the basis of the commercial code FINE/TURBO™ developed by NUMECA International. FINE/ TURBO™ is a three-dimensional structured mesh code that solves the time dependent Reynolds-averaged Navier-Stokes equations. Time-accurate resolutions of the equations use the dual-time stepping approach. Pseudo-time derivative terms are added to march the solution towards convergence at each physical time-step. The range of application is extended to weakly compressible or incompressible flows by introducing a preconditioning matrix, ͓19͔. The discretization is based on a finite volume approach. Convection terms are treated by a second-order central scheme associated with artificial dissipation terms. The pseudotime integration is made by a four-step Runge-Kutta procedure. The physical time-derivative terms are discretized with a secondorder backward difference scheme. The code resorts to a multigrid strategy to accelerate the convergence, associated with a local time-stepping and an implicit residual smoothing.

The numerical model was adapted to treat the cavitation process, ͓13,17͔. The key point of this adaptation is the modification of the state law of the fluid. Applied barotropic law implies the simultaneous treatment of two different cases: the fluid is highly compressible in the liquid/vapor mixture ͑the Mach number can be as high as 4 or 5͒ and is almost incompressible in the pure More details concerning the basic numerical method can be found in ͓20͔, and a more precise investigation of the influence of numerical parameters in cavitating conditions ͑artificial dissipation, turbulence model, physical time step, mesh size . . . ͒ is available in ͓17͔.

The model was applied to the presented centrifugal pump geometry. Several meshes were tested, as well H-I typed as H-O typed, including between 300,000 and 700,000 cells. They are composed of two blocks ͑the impeller and the inlet domain͒ in the H-I case, and three blocks ͑the inlet domain, the impeller, and the outlet domain͒ in the H-O case. Only one blade-to-blade channel is treated, and periodicity conditions are applied to the frontiers with the adjacent channels. This automatically implies that the five channels will behave the same way during the numerical simulation.

H-O meshes are particularly well adapted to the present geometry, because of the sharp leading and trailing edges ͑Fig. 8͒. H-I meshes induce more important stretchings of the cells in these areas, but they also lead to more reduced mesh sizes.

Noncavitating steady calculations were performed with the different meshes, and the results showed the weak influence of the mesh type on the pump performance and on the pressure field around the leading edge. Consequently, all the simulations presented hereafter were obtained with the H-I 300,000 cells mesh, whose structure is presented in Fig. 9. Special attention was paid to the cavitating areas ͑suction side of the blade, leading edge . . . ͒: the stability study reported in ͓17͔ shows indeed that cells characterized by a high distortion or high aspect ratio strongly penalize the numerical stability in cavitating conditions. Thus, a fine grid is applied in these areas. Figure 9 hereafter shows the radial mesh structure, a three-dimensional view of a complete blade-to-blade channel, and some details of the mesh on the hub surface.

Conditions applied for the simulations are the following:

• Turbulence model: we use for the simulations presented in this paper a Baldwin-Lomax turbulence model. A more detailed study of the influence of the turbulence model on twodimensional unsteady cavitation simulation is proposed in ͓21͔ and ͓22͔. Those works pointed out a major influence of the compressibility effect modeling on the unsteady behavior of cloud cavitation. Other three-dimensional calculations considering different k-turbulence models are in progress to improve the physical analyses. • Boundary conditions ͑Fig. 10͒: velocity is imposed at the inlet of the suction pipe. Laws of the wall are imposed along solid boundaries. The relative motion between the inlet pipe walls and the impeller is taken into account. On the other hand, the outlet housing shape is not described and the parallel walls are treated as hub and shroud extensions up to the outlet, at 1.5 times the impeller outer radius, where a uniform static pressure is imposed. • Initial transient treatment: First of all, a steady step is carried out, with a pseudo vapor pressure low enough to ensure noncavitating conditions in the whole computational domain.

Then, the NPSH is slowly lowered by increasing smoothly the pseudo vapor pressure at each new time-step up to the physical value. Vapor structures spontaneously appear and grow during that process, in the regions of low static pressure. The final NPSH value, depending on the outlet static pressure imposed, is then kept constant throughout the computation.

Noncavitating Characteristics

Experimental tests and numerical calculations were performed considering a large range of flow-rates, in non-cavitating conditions. Figure 11͑a͒ illustrates the computational result at nominal flow rate: the total pressure distribution is represented at halfwidth of the pump channels. Figure 11͑b͒ presents a comparison between the numerical and experimental performance charts. Directly based on the measurements performed, the head of the pump is defined as the difference between downstream static pressure and upstream total pressure.

Hϭ

P outlet ϪP tot inlet g where P tot inlet ϭP inlet ϩ V inlet 2 2 P inlet and P outlet are the static pressures measured, respectively, at the pump inlet and outlet, and V inlet is the mean inlet velocity. The outlet pressure was determined by taking the static wall pressure in the vaneless diffuser at 5 static pressure taps located at a diameter of 500 mm and averaged by connecting all pressure taps to a single pressure transducer.

The total pressure at pump inlet is determined from the wall pressure and the velocity head two diameters upstream of pump inlet where the velocity is calculated from measured flow rate and the area of the measuring cross section. Precise locations of the sensors in the experiments are consistent with the inlet and outlet of the computational domain in the calculation.

We observe a reliable agreement between the pump characteristics given by measurement and computation in the whole range of flow rates investigated experimentally. The model gives a better prediction when the flow rate is over 50 percent of the nominal value (Qnϭ210 m 3 /h), and the numerical simulation becomes unstable at very low partial flow rate. As a matter of fact, the numerical simulation slightly overestimates the pump head. This is an expected result, since the flow through the side chamber is not taken into account in numerical simulations. This gap flow of the pump impeller has to be added to the flow rate actually passing through the blade-to-blade channels and is therefore slightly higher in the experiments than in the model. At nominal flow rate, the head is overestimated by about 5 percent ͑40.5 m instead of 38.5 m͒. The uncertainty of the measured value was estimated to be no more than 0.8 m and corresponds to 2 percent of the value, which also has to be taken into account.

Cavitation Behavior

Numerical simulations of the pump are performed in cavitating conditions at several flow rates. First, we present some qualitative results, consisting in a visualisation of the vapor/liquid structures inside the pump at nominal flow rate and for several NPSH values. Then, a quantitative analysis is performed, using the experimental results: head drop charts are studied at three flow rates, and the shape of the cavitating areas at nominal flow rate is compared to visualisations inside a blade-to-blade channel. where the inlet total pressure was defined previously. The head drop is only partially obtained by the computation: this point will be discussed hereafter.

Figures 13 and14 show the development of cavitation corresponding to the six operating conditions indicated on the chart. Figure 13 illustrates the apparition and the growing of vapor/ We observe that cavitation appears first on the suction side ͑Fig. 13-1͒, and also quite rapidly on the pressure side ͑Fig. 13-2͒. The vapor volume increases mainly on the suction side ͑Figs. 13-3 and 13-4͒. A vapor structure also grows on the shroud, in the vicinity of the inlet, caused by the local curvature of the streamlines. It progressively joins the attached cavity on the blade.

The pressure side cavity is almost uniform from hub to shroud, and it grows slowly as the NPSH decreases. On the contrary, the suction side cavitation remains very close to the shroud, where the obstruction is mainly visible ͑Figs. 13-5 and 13-6͒.

For a better understanding of the liquid/vapor mixture behavior in the pump, we have presented on Fig. 14 the void ratio distribution on a blade-to-blade surface, close to the shroud. The velocity field is also drawn in each case. It indicates the void ratio variations in the suction side cavity, and in the two-phase areas observed on the shroud.

We observe that the void ratio obtained on the suction side of the blade increases rapidly: it reaches almost 90 percent in the second configuration, in the middle of the attached cavity. On the contrary, the vapor region on the shroud is characterized by a low void ratio, which reaches only 60 percent, and the cavity on the pressure side of the blades is mainly composed of liquid, even when its volume increases. That low void ratio is probably related to fluctuations of the vapor/liquid structures, which are not obtained by the simulation. It is worth noting that we use here a high time-step (⌬tϭ10 Ϫ4 s, i.e. 1/300 of one rotation period͒, so that the unsteady phenomena are not modelized, and a quasi-static head-drop chart can be obtained. As a consequence, each operating point represented here corresponds to a mean flow field that does not take into account the eventual instabilities generated by cavitation.

Comparisons With

Experiments. Numerical simulation of the cavitation characteristics of the pump was performed at different flow rates. The shape of the cavitating structures is first compared to the experimental visualisations presented above at nominal flow rate. When the NPSH decreases in the calculation, attached cavitation sheets grow both on the suction side and on the pressure side, as observed experimentally. Moreover, vapor structures appear at the inlet radius of the shroud. This cavitation behavior is fully consistent with the observations reported previously. The visualisation obtained in Fig. 3 is compared to the computational results to enhance the reliable agreement. The cavitation number was adjusted to give the same global extent of the cavitation structures as in the experiment: the numerical result then corresponds to a NPSH about 10 percent lower than the experimental one ͑7 m, instead of 8 m͒. The three cavitation areas are correctly simulated by the code ͑Fig. 15͒: attached cavity on the suction side ͑A͒, extent of cavitation on the shroud along the blade ͑B͒, and cavitating flow on the inlet radius of the shroud ͑C͒. In the computation, attached sheet cavity ͑A͒ and extent on the shroud ͑B͒ belong to the same vapor structure, while they look like two separated regions in the experiment.

Figure 16 shows the attached cavity on the pressure side of the blade. Its size is compared to the mean distribution obtained from gray level averaging ͑Fig. 5͒. Both experimental and numerical NPSH values are equal to 7 m. Here, the calculated cavity appears smaller than the experimental one. Moreover, only the steady attached part of the cavity on the pressure side is obtained by the computation. The transient vapor structures in the unsteady cavity closure region are not found from the simulation. Actually this is still a limitation of the physical and numerical model. The mesh size and the standard turbulence model used in the calculation do not allow catching the cloud shedding process downstream of the attached cavity. An ongoing work is pursued to improve this aspect.

In Fig. 17, we summarize the whole comparison, by drawing the head drop curves for three different flow rates: namely 0.8Qn, Qn and 1.08Qn. In non-cavitating conditions, we observe again a slight overestimation of the pump head ͑about 5 percent at nominal flow rate͒, mainly due to the fact that the gap flow through the side chamber is not considered in the simulation. From the numerical point of view, while decreasing the NPSH, the performance drop appears first as a smooth decrease of the pump head. The final head-drop is only partially simulated because the computation rapidly becomes unstable and stops ͑this is more particularly the case at 1.08Qn͒. Our upstream boundary condition consists in imposing in a strict manner the mass flow rate passing the pump. Because the coupling between the pump and the hydraulic loop is not taken into account, the effects of the cavitation blockage on the flow rate are neglected and the head-drop is less pro-gressive than in the experiments. However, all the numerical results represented in Fig. 17 refer to a converged solution at the corresponding physical time-step.

Results obtained from first simulations of the pump cavitation behavior are promising: the head drop is predicted with a good homogeneity with respect to the 3 flow rates. The NPSH values obtained for the 3 percent and 10 percent head drop are globally overestimated in respect to experimental values ͑of about 1 m for the 10 percent head drop, and 1.5 to 3 m for the 3 percent head drop, see Fig. 18͒. These results correspond to our first try of predicting the cavitation characteristic of a pump, and a study of the effect of the model parameters ͑mainly the turbulence model͒ might probably improve the quantitative agreement.

It can be observed that the head measured in experiments increases a little bit in low conditions before the head drop, for the three flow rates. This effect is not obtained in the calculations. However, it does not exceed 1 to 2 percent of the initial head, whereas the discrepancy between calculations and experiments concerning the pump performance is about 2 to 5 percent. More- over this effect, which has been observed also in the case of turbopump inducers ͓17͔, is not yet well explained, may be due to a coupling between the pump channels; such a phenomenon can not be predicted here because only one channel is considered. It probably involves slight interactions between the cavities and the boundary layer along the blade sides, which are not simulated with a sufficient accuracy by our simple turbulence model.

Conclusion

Numerical and experimental results were presented in this study, concerning a test pump having an impeller with 2D curvature blade geometry.

From the experimental point of view, besides the measurements of the cavitation characteristics of the pump in various conditions, a special visualisation set-up was developed to investigate the unsteady behavior of leading edge cavitation. Image processing and statistical treatment of the photographs taken at given impeller position allowed to quantify the attached and cloud cavitation extent.

A numerical model of three-dimensional cavitating flows, based on the three-dimensional code FINE/TURBO™, has been developed to predict the cavitation behavior in turbomachinery, ͓13,14͔. This model was applied to the centrifugal pump geometry. Noncavitating and cavitating conditions were investigated. Calculations were found to be in good agreement with experimental measurements and visualisations. Experimental and numerical results concerning the pump characteristics and performance breakdown were drawn at different flow conditions and the mean spatial distributions of vapor structures within the impeller were compared: It was found that the main features of the cavitating flow field are correctly simulated by the model. These results show the ability of the model to simulate the complex three-dimensional development of cavitation in rotating machinery, and the associated effects on the performance. This is of first importance for the purpose of pump future design. However, the fluctuating two-phase areas are not simulated yet, and work is in progress to improve the numerical model in that way. This is a necessary further step to predict more efficiently the local void ratio, since its distribution usually slightly evolves when unsteady effects are correctly predicted, ͓18͔. The final objective is to simulate the unsteady effects due to cavitation ͑cavity self-oscillation, rotating cavitation͒ in three dimensions, as we already performed in more simple twodimensional configurations, ͓4,23͔.

  The investigated flow conditions during the experiments show always unsteady behavior of the leading-edge cavitation, not only concerning the unsteadiness of the closure region of the attached cavitation but also the shedding of vapor structures in the channel. Besides the well-known attached cavitation on pressure and suction side, another type of cavitation occurs at the inlet of the impeller caused by the strong curvature of the streamlines along the front shroud. It is visible either as attached cavitation caused by the depression at the radius ͑regions B and C in Fig. 3͒, or as shear cavitation because the flow was separating at the inlet radius of the front shroud at lower values of NPSH ͑region B in Fig. 4͒. Dependent on the length of the attached part of the cavitation ͑region A in all figures͒ the extent of this shear layer cavitation spreads into the channel as it is shown in Fig. 4 at two different stages of a cavitation cycle. In Fig. 4͑a͒ the larger extent of the shear cavitation ͑region B͒ on top of the attached part ͑see region A, the illuminated interface of the cavitation sheet on the pressure side͒ is visible. In Fig. 4͑b͒ ͑with a smaller cavitation sheet͒ no shear cavitation can be seen. The contour of the interface of the attached part of the cavitation ͑region A͒ already indicates a different state of the typical self-oscillating cloud cavitation condition.

Fig. 1

 1 Fig. 1 "a… Impeller geometry, "b… housing

Fig. 4

 4 Fig. 4 Unsteady-state of blade cavitation on pressure side at two different time and NPSHÄ7 m "laser light sheet…

Fig. 7

 7 Fig. 7 The barotropic state law "P… for water

6. 1

 1 Qualitative Results. The head drop chart H͑NPSH͒ obtained by the calculation at nominal flow rate is drawn in Fig.12. The green line corresponds to the apparition of vapor in the flow field. The six blue points indicated on the chart are related to the six visualisations of the cavitating flow field presented in Fig.13and 14 hereafter. The quantity NPSH has the following definition:

Fig. 10

 10 Fig. 10 Boundary conditions

Fig. 12

 12 Fig. 12 Head drop chart at nominal flow rate. The points indicate the cavitating conditions visualized on Figs. 13 and 14. The line corresponds to the apparition of vapor.

Fig. 14

 14 Fig. 14 Void ratio evolution on a blade-to-blade surface close to the shroud, and velocity fields "corresponding to the NPSH decrease represented on Fig. 12…

Fig. 16 "

 16 Fig. 16 "a… Numerical void ratio distribution, "b… pressure side cavity, comparison with experiment "NPSHÄ7 m…

Currently at ENSTA UME/DFA, chemin de la Huniere, 91761 Palaiseau cedex, France.

Currently at LTDS/ENISE, Saint-Etienne, France.

Acknowledgments

This work is part of a European exchange program PROCOPE with the research teams of Grenoble and Darmstadt as members. The numerical cavitation model implemented in FINE/TURBO™ has been developed with the support of SNECMA Moteurs DMF ͑Rocket Engine Division͒, and the French space agency CNES. The experimental work carried out at Darmstadt was part of the research project STO 190/9-2 which was financially supported by the DFG ͑German Research Foundation͒.


O Coutier-Delgosha

R Fortes-Patella

J L Reboud
email: reboud@enise.fr

Unsteady effects associated with cavitation were investigated by numerical simulations in three configurations. The simplest one was a Venturitype section in which the cavitation sheet oscillates periodically with vapour cloud shedding. The second one was a hydrofoil whose unsteady cavitating behaviour depends on the angle of attack, and the most complex one was a cascade of three hydrofoils. In this last configuration, in addition to the unsteadiness associated with each cavity, a coupling between the three channels was also observed. These cavitating flows were simulated by 2D computations. Resolution of Reynoldsaveraged Navier-Stokes equations was based on a finite-volume discretization associated with a pressure correction algorithm. Cavitation was simulated by using a barotropic vapour/liquid state law that links the fluid density evolution to the pressure variations. As standard k-ε RNG or k-ω turbulence models were found to be weakly efficient to simulate unsteady cavitation, influence of the compressibility of the two-phase medium on turbulence was considered. Both k-ε RNG and k-ω turbulence models, including corrections of these compressibility effects, were applied and results obtained were consistent with experiments: in the three configurations, the oscillation frequencies, the cavity length, the void ratio and the velocity fields obtained by numerical simulation were in reliable agreement with the available experimental data.

Simulation of unsteady cavitation with a two-equation turbulence model including compressibility effects † 1. Introduction

In the case of cavitation, the two-phase mixture is unsteady at the microscopic scale, since vapour and liquid structures are in continuous interaction through mass and momentum exchanges. At a larger scale, typically the size of the global cavitation area, unsteady behaviours are also observed in experiments. For example, the cavitation sheets observed on the suction side of turbomachinery blades often adopt a pulsating behaviour [START_REF] Joussellin | experimental investigations on unsteady attached cavities[END_REF]. Rotating cavitation configurations are also reported, mainly in space turbopump inducers (see references [START_REF] Kamijo | An experimental investigation of cavitating inducer instability[END_REF][START_REF] De Bernardi | Experimental analysis of instabilities related to cavitation in turbopump inducer[END_REF][START_REF] Tsujimoto | A theoretical analysis of rotating cavitation inducers[END_REF][START_REF] Pagnier | Conception and experimental study of an inducer Proc[END_REF]). These consist of unbalanced attached cavities in the different channels, their distribution rotating faster than the inducer.

In such axial pumps, the tip vortex cavitation also appears highly unsteady, with no discernible arrangement. All these effects, directly due to cavitation, are responsible for spurious transient phenomena, such as flow rate fluctuations, noise, vibration, erosion, and radial or axial loads on the shaft or the casing [START_REF] Goirand | Experimental investigation of radial loads induced by partial cavitation with liquid hydrogen inducer[END_REF].

The experimental study and the numerical modelling of these unsteady effects are thus essential to understand their mechanisms and to reduce their consequences in turbomachinery.

The pulsating behaviour of cavities attached to the blades is usually studied in simplified 2D non-rotating configurations, to focus on the physical effects associated with cavitation. To simulate a 3D pump geometry, we have considered three 2D simplified geometries as illustrated in figure 1. They are a cascade of hydrofoils, a single hydrofoil, and a Venturi-type section (whose lower wall represents the suction side of a inducer blade).

In these three configurations, at low-pressure conditions a cavitation sheet appears, respectively at the leading edge of the hydrofoils, and at the throat of the Venturi-type section. When hydrofoil angle of attack and Venturi divergence angle are high enough, this cavity becomes unstable. Three behaviours, indicated in figure 2, can be observed experimentally: they are usually called respectively sheet cavitation (only the rear part of the cavity is unsteady, with a rather chaotic pulsating behaviour of low magnitude), cloud cavitation (the main part of the cavity is oscillating periodically with cloud shedding), and shear cavitation (in the case of high angle of attack, no attached cavitation sheet remains permanently at the leading edge, and only clouds of vapour appear, which are immediately carried out by the stream). An example of cloud cavitation observed experimentally on a hydrofoil suction side is given in figure 15.

The numerical simulation of cavitating flow is usually based on a single-fluid approach: the liquid/vapour mixture is considered as a homogeneous medium whose density varies from the pure vapour one to the pure liquid one. So the local void fraction α is directly linked to this density evolution, and physical cavitation models mainly differ in the way these two related variables are calculated. They can be managed either by a state law (see references [START_REF] Delannoy | Two-phase flow approach in unsteady cavitation modeling Cavitation and Multiphase Flow Forum[END_REF], [START_REF] Song | Numerical simulation of cavitating flows by single-phase flow approach[END_REF], [START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation 3rd[END_REF]), using a supplementary equation relating the void fraction to the dynamic evolution of bubble cluster (see references [START_REF] Kubota | A new modelling of cavitation flows: a numerical study of unsteady cavitation on a hydrofoil section[END_REF] and [START_REF] Chen | Modeling hydrodynamic non-equilibrium in cavitating flows[END_REF]), or by a multiple species approach with a mass transfer law between liquid and vapour [START_REF] Kunz | Multi-phase CFD analysis of natural and ventilated cavitation about submerged bodies 3rd[END_REF]. The last model can be used in a full two-fluid approach, including relative motion between the phases [START_REF] Alajbegovic | Calculation of transient cavitation in nozzle using the two[END_REF].

This paper presents two-dimensional simulations of unsteady cavitating flows. The cavitation treatment is based on the physical model developed by Delannoy and Kueny [START_REF] Delannoy | Two-phase flow approach in unsteady cavitation modeling Cavitation and Multiphase Flow Forum[END_REF] for inviscid fluids. Reynolds-averaged Navier-Stokes equations are resolved for the liquid/vapour mixture considered as a single fluid. The two-phase structure is defined by the local void ratio, which is directly governed by the fluid density, varying between the pure liquid one and the pure vapour one. Density evolution is related to local pressure variations through the fluid barotropic state law. This physical model for cavitation was implemented numerically into a pressurecorrection scheme derived from the simple algorithm. Several physical modifications have been investigated successively by Reboud and Delannoy [START_REF] Reboud | Two-phase flow modeling of unsteady cavitation[END_REF], Reboud et al [START_REF] Reboud | Two phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF] and Coutier-Delgosha et al [START_REF] Coutier-Delgosha | Numerical simulations in unsteady cavitating flows[END_REF] to increase the range of applications and improve the physical modelling.

In this physical approach, the high compressibility of cavitation areas, as well as the very low compressibility of pure liquid and pure vapour, are both taken into account. These two very different fluid properties are responsible for the main numerical difficulties that are encountered. Another crucial point is the modelization of turbulence: the effect of cavitation on the turbulence structure is not well known. Although it may have an important impact on the complex mechanisms that govern unsteady cavitation [START_REF] Reboud | Two phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF], no fully efficient model has been reported yet. However, a first attempt to take into account the influence of only one of a cavitating fluid properties, namely its high compressibility, has been proposed in a previous paper [START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF].

In the present cavitation model, the vapourization and condensation processes correspond mainly to highly compressible flow areas. Therefore, it is of primary importance to take into account the compressibility of the vapour/liquid mixture in the turbulence model. Several twoequation turbulence models were tested in the case of cavitation in a Venturi-type section, and important differences were noticed, whether empirical corrections of the compressibility effects were considered or not. It was finally found that correction terms proposed by Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF]i n the case of compressible flows, as well as arbitrary modifications investigated by Reboud et al [START_REF] Reboud | Two phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF] were both adequate to obtain the cyclic cavitation behaviour reported in experiments.

This paper presents an interpretation of this conclusion, and results obtained by applying the compressibility corrections in three different configurations, from quite simple (the Venturitype section) to more complicated (cavitation on a hydrofoil, and then in a cascade of three hydrofoils). In all cases, the unsteady behaviour reported experimentally is correctly predicted by the numerical model. Comparisons with experimental data focus on the length of the mean cavitation sheet for different flow conditions and oscillation frequencies. Several video animations are proposed to illustrate and to analyse the unsteady behaviour of cavitation.

Physical and numerical models 2.1. Physical approach

This work applies a single-fluid model based on previous numerical and physical studies (see references [START_REF] Delannoy | Two-phase flow approach in unsteady cavitation modeling Cavitation and Multiphase Flow Forum[END_REF][START_REF] Reboud | Two-phase flow modeling of unsteady cavitation[END_REF][START_REF] Reboud | Two phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF][START_REF] Coutier-Delgosha | Numerical simulations in unsteady cavitating flows[END_REF][START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF]). The fluid density ρ varies in the computational domain according to a barotropic state law ρ(P ) that links the density to the local static pressure (figure 3). When the pressure in a cell is higher than the neighbourhood of the vapour pressure P v (P>P v +(∆P v /2)), the fluid is supposed to be purely liquid. The entire cell is occupied by liquid, and its density ρ l is calculated by the Tait equation [START_REF] Knapp | Cavitation[END_REF]. If the pressure is lower than the neighbourhood of the vapour pressure (P<P v -(∆P v /2)), the cell is full of vapour and its density ρ v is given by the perfect gas law (isotherm approach). Between purely vapour and liquid states, the cell is occupied by a liquid/vapour mixture, which is considered as one single fluid with a variable density ρ. This is directly related to the void fraction α =( ρρ l )/(ρ v -ρ l ) corresponding to the local ratio of vapour contained in the mixture.

To model the mixture state, the barotropic law presents a smooth link in the vapour pressure neighbourhood, in the interval ±(∆P v /2). In direct relation with the range ∆P v , the law is characterized mainly by its maximum slope 1/A 2 min , where A 2 min = ∂P/∂ρ. A min can thus be interpreted as the minimum speed of sound in the mixture. Its calibration has been done previously [START_REF] Coutier-Delgosha | Numerical simulations in unsteady cavitating flows[END_REF]. The optimal value was found to be independent of the hydrodynamic conditions, and is about 2 m s -1 for cold water, with P v =0 .023 bar, and corresponding to ∆P v ≈ 0.06 bar (chart in dotted line on figure 3). That value is applied for the computations presented hereafter.

Mass fluxes resulting from vapourization and condensation processes are treated implicitly by the barotropic state law, and no supplementary assumptions are required. Concerning the momentum fluxes, the model assumes that locally velocities are the same for liquid and for vapour: in the mixture regions vapour structures are supposed to be perfectly carried by the main flow. This hypothesis is often assessed to simulate sheet-cavity flows, in which the interface is considered to be in dynamic equilibrium [START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation 3rd[END_REF]. The momentum transfers between the phases are thus strongly linked to the mass transfers.

Numerical resolution

To solve the time-dependent Reynolds-averaged Navier-Stokes equations associated with the barotropic state law presented above, the numerical code applies, on 2D structured curvilinearorthogonal meshes, the SIMPLE algorithm, modified to take into account the cavitation process. This uses an implicit method for the time discretization, and the HLPA non-oscillatory secondorder convection scheme proposed by Zhu [START_REF] Zhu | A low diffusive and oscillation-free convection scheme[END_REF]. The numerical model is detailed in Coutier-Delgosha et al [START_REF] Coutier-Delgosha | Numerical simulations in unsteady cavitating flows[END_REF]: a complete validation of the method was performed, and the influence of the numerical parameters was widely investigated.

In the code, the velocity field is imposed at the computational domain inlet, and the static pressure is imposed at the outlet. Along the solid boundaries, the turbulence models are associated with laws of the wall (figure 4(a)).

To start unsteady calculations, the following numerical procedure is applied: first, a stationary step is carried out, with an outlet pressure high enough to avoid any vapour in the whole computational domain. Then, this pressure is lowered slowly at each new time step, down to the value corresponding to the desired cavitation number σ defined as (P upstream -P v )/(ρV 2 upstream /2). Vapour appears during the pressure decrease. The cavitation number is then kept constant throughout the computation.

Turbulence model

In our previous studies [START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF], either the k-ω model proposed by Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF]orthek-ε RNG model presented by Yakhot et al [START_REF] Yakhot | Development of turbulence models for shear flows by a double expansion technique[END_REF] were applied to model cavitating flows. Results obtained have demonstrated that for both models, corrections of the influence of the vapour/liquid mixture compressibility on the turbulence should be taken into account to obtain the unsteady effects due to cavitation. The corrections proposed by Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF]i nt h ek-ω model were implemented in the numerical code and the first results obtained were very promising. This section presents the main features of this turbulence model.

Equations are similar to the governing equations solved in the case of the k-ε model: all the modifications result from the use of the dissipation specific rate ω instead of the turbulence dissipation ε. The turbulent viscosity expression and the k equation are modified, and the ε equation is replaced by a ω equation:

ν t = k ω ∂k ∂t + u j ∂k ∂x j = τ ij ∂u i ∂x j -β * w kω + ∂ ∂x j (ν + σ * w ν t ) ∂k ∂x j ∂ω ∂t + u j ∂ω ∂x j = α w ω k τ ij ∂u i ∂x j -β w ω 2 + ∂ ∂x j (ν + σ w ν t ) ∂ω ∂x j
where α w =5 /9, β w =3 /40, β * w =9 /100, σ w =0 .5, and σ * w =0 .5 are the coefficients of the model;

τ ij =2ν t S ij with S ij = 1 2 (u i,j + u j,i ).
Influence of the compressibility is derived from an analysis in supersonic conditions: Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF] observed that the reduction of the mixing layer growth rate as a function of the Mach number could not be well simulated only by taking into account the mean density evolution through the mixing length, and that an explicit supplementary correction must be applied. He concluded that the Reynolds-stress transport equation is directly concerned with the compressibility effects, and introduced the influence of the Mach number in the turbulence equations.

The modifications proposed by Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF] are based on the previous studies of Sarkar et al [START_REF] Sarkar | The analysis and modeling of dilatational terms in compressible turbulence NASA[END_REF] and Zeman [START_REF] Zeman | Dilatational dissipation: the concept and application in modelling compressible mixing layers[END_REF], who aimed to take into account the compressibility effects in a k-ε turbulence model. He adapted this development to the k-ω model to propose the following equations giving the parameters β w and β * w as functions of the turbulence Mach number defined as M 2 t =2 k/A 2 (where A is the local speed of sound):

β * w = β * wi (1 + ξ * F (M t )) and β w = β wi -β * wi ξ * F (M t )
, where the index i indicates the values of the incompressible model given before, with [START_REF] Stutz | Experiments on unsteady cavitation[END_REF] and ξ * =1.5.

F (M t )= (M 2 t -M 2 t 0 )i f M t >M t 0 0i f M t ≤ M t 0 M t 0 =0.
We have applied this turbulence model to perform the calculations described above concerning the geometries of a Venturi, a hydrofoil and a hydrofoil cascade.

Cavitation behaviour of the Venturi-type section 3.1. Experimental behaviour

The first geometry considered is the Venturi illustrated in figure 4(a). Convergent and divergent angles of the lower wall of the Venturi-type section are respectively about 18 • and 8 • (figure 4(a)). The shape of the Venturi bottom downstream from the throat simulates an inducer blade suction side with a bevelled leading edge geometry [START_REF] Kueny | Analysis of partial cavitation: image processing and numerical prediction Cavitation[END_REF] and a chord length L ref = 224 mm.

According to experimental observations [START_REF] Stutz | Experiments on unsteady cavitation[END_REF], in this geometry the flow is characterized by unsteady cavitation behaviour, with quasi-periodic fluctuations. Each cycle is composed of the following successive steps: the attached sheet cavity grows from the Venturi throat. A reentrant jet is generated at the cavity closure and flows along the Venturi bottom toward the cavity upstream end. Its interaction with the cavity surface results in the cavity break off. The generated vapour cloud is then convected by the main stream, until it collapses.

For a cavitation number σ of about 2.4 (based on the time-averaged upstream pressure) and an inlet velocity V ref =7.2ms -1 , vapour shedding frequency f observed experimentally is about 50 Hz for a cavity length of 45 ± 5mm [START_REF] Stutz | Experiments on unsteady cavitation[END_REF]. A more detailed study concerning these calculations is presented in Coutier-Delgosha et al [START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF]. This section summarizes the main results obtained.

Calculation with no effect of compressibility on the turbulence

The computational grid is composed of 160 × 50 orthogonal cells (figure 4(a)). A special contraction of the mesh is applied in the main flow direction just after the throat, so that the two-phase flow area is efficiently simulated: about 50 grid points are used in this direction to model the 45 mm long mean cavity obtained hereafter (figure 4(b)). In the other direction, a contraction is also applied close to the walls, to obtain at the first grid point the non-dimensional parameter y + of the boundary layer varying between 30 and 100 and to use standard laws of the walls. The grid is finer in the bottom part of the Venturi section than in its upper part, to enhance the accuracy in the cavitation domain: cavities obtained in the following sections contain about 30 cells across their thickness.

The k-ω turbulence model presented in section 2.3 is applied with wall functions. No effect of two-phase flow compressibility on the turbulence structure is considered in this section. Thus, the fluid compressibility is only taken into account in the turbulence equations through the mean density ρ change.

With this model, the unstable cavitating behaviour observed experimentally is not correctly simulated: after an initial transient fluctuation of the cavity length, the numerical calculation leads to a quasi-steady behaviour of the cavitation sheet, which globally stabilizes (figure 5).

The resulting cavity length is much too small, compared with the experimental observations (in this case, the error is larger than 50%). Moreover, comparisons with experimental data obtained by double optical probes by Stutz and Reboud [START_REF] Stutz | Experiments on unsteady cavitation[END_REF][START_REF] Stutz | Measurements within unsteady cavitation[END_REF] show that the numerical mean void ratio is over-estimated in the main part of the cavity. Calculations give a high time-averaged void ratio in the upstream part of the cavitation sheet (>90%), abruptly falling to 0% in the wake, while the measured void ratio never exceeds 25% and decreases slowly from the cavity upstream end to its wake.

This poor agreement with the real configuration seems to be related to an over-prediction of the turbulent viscosity in the rear part of the cavity. The cyclic behaviour of the cloud cavitation process is strongly related to the re-entrant jet development from the cavity closure [START_REF] Knapp | Cavitation[END_REF]. As a matter of fact, the main problem in the turbulent flow simulations consisted in the premature removal of the reverse flow along the solid wall: the re-entrant jet was stopped too early and it did not result in any cavity break off.

It is worth noting that numerical tests reported by Coutier-Delgosha et al [START_REF] Coutier-Delgosha | Numerical simulations in unsteady cavitating flows[END_REF] confirm that the mesh size, spatial scheme, and the time discretization applied in the model do not modify the results. Computations performed with a finer mesh (264×90) and first-or second-order accurate time discretization schemes still lead to the same complete stabilization of the cavitation sheet.

Calculation including the compressibility effects

As mentioned above, the proposed numerical simulations take into account a single-flow physical model to describe the cavitation phenomenon. According to the adopted barotropic state law, in the vapour/liquid mixture zones the speed of sound A = ∂P /∂ρ is very low, and the fluid is locally highly compressible. Thus the flow often reaches Mach numbers larger than 5 in the vapourization or condensation areas. In fact, Birch and Eggers [START_REF] Birch | Free turbulent shear flow[END_REF] showed that an increase of the Mach number could substantially modify the turbulence structure in some configurations, such as mixing layers. So the corrections proposed by Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF] and presented in section 2.3 are applied in the case of the Venturi-type section.

This leads to substantial changes in the simulation. The prediction of the unsteady re- entrant jet is now obtained, and the vapour cloud shedding is well simulated. The transient evolution observed during this unsteady calculation is shown in figure 16 for the following flow conditions:

V ref =7.2ms -1 , σ =2.4.
For more quantitative comparisons with experiments, figure 6 illustrates at a given time and for each cross section of the Venturi-type duct the value of the minimal density in the section. By comparison with figure 5, it gives information concerning the vapour cloud shedding process: the part of the cavity that breaks off clearly appears, and the fluctuation frequency can be evaluated. Moreover, it also supplies the maximum void ratio in each section. We obtain in this case a fluctuating cavity whose mean length is about L cav = 45 mm (i.e. L cav /L ref =0 .2, with L ref the geometry reference length). The mean fluctuation period is about 0.59T ref , and the corresponding shedding frequency equals 55 Hz. Both length and frequency are in good agreement with measurements (i.e. 45 mm and 50 Hz respectively). The transient evolution is almost periodic, with some random disturbances affecting the oscillation regularity. This behaviour is consistent with the experimental observations, which pointed to a quite regular cavitation cycle, whose frequency fluctuated around a central value [START_REF] Stutz | Experiments on unsteady cavitation[END_REF].

The reliable agreement between experiments and calculation is confirmed by figure 7: the frequency of the self-oscillation behaviour is drawn with respect to the ratio V ref /L cav . The maximum length of the attached cavity given by the numerical simulation is chosen as an estimate of the visual length reported from the experiments. It can be seen that the cavitation oscillation frequency obtained by the calculation is very close to the measured one for a large range of cavity lengths. Dividing the shedding frequency by the ratio V ref /L cav gives an almost constant Strouhal number Str = fL cav /V ref =0 .27 ± 0.02, which is a classical value for vapour cloud shedding (see references [START_REF] Joussellin | experimental investigations on unsteady attached cavities[END_REF][START_REF] Stutz | Experiments on unsteady cavitation[END_REF][START_REF] Lush | Visualisation of the cavitating flow in a Venturi-type duct using high speed cine photography[END_REF]).

Figure 17 presents two phase-averaged cavitation cycles. The upper one results from experimental visualizations: video frames acquired during a 100 ns exposure time under laser sheet light are identified and digitized into 256 grey levels. A sampling technique is applied to classify them into nine sets corresponding to the different states of the recorded quasi-periodic pressure signal. Then, averaging the grey levels pixel per pixel for each set, allows one to draw a sequence of phase-averaged images, from an initial data set of 300 frames. Maximum standard deviation computed for each set is about 2/3 of the maximum mean grey level, and is mainly observed along the attached cavity and cloud boundaries.

The bottom part of figure 17 corresponds to the same sequence, obtained by numerical simulation. This one lasts 20T ref , i.e. about 30 cycles. The same sampling technique is used: the computational result is decomposed into 30 × 9 short sequences corresponding to the nine steps of the cavitation cycle and the phase-averaging process is applied. We observe a good agreement between the two results concerning the external shape and global structure of both attached cavity and vapour cloud shedding.

Discussion

Results obtained with the k-ω turbulence model point to the relevant role of the compressibility effects on the turbulence: unsteadiness due to cavitation is obtained in the computation only if the corrections proposed by Wilcox to consider these effects are applied.

In fact, these corrections are derived from considering possible density fluctuations: a supplementary term appears from the averaged equations, increasing the turbulent dissipation. The final effect is a reduction of the turbulent viscosity in the compressible areas, i.e. in our particular case in the vapour/liquid mixture zones. The same result is obtained by reducing directly the turbulent viscosity in the two-phase areas, as proposed by Reboud et al [START_REF] Reboud | Two phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF] with a k-ε RNG turbulence model. This model is originally devoted to fully incompressible fluids, and no particular correction is applied in the case of the highly compressible two-phase mixture. To improve the turbulence modelling and to try to better simulate the re-entrant jet behaviour and the vapour cloud shedding, we modified arbitrarily the standard model simply by reducing the mixture turbulent viscosity, mainly in the low void ratio areas, as represented in figure 8:

µ t = f (ρ)C µ k 2 /ε C µ =0.0085
where

f (ρ)=ρ v + ρ v -ρ ρ v -ρ l n (ρ l -ρ v ) n ≫ 1.
The results obtained were almost identical to those presented previously for the k-ω model including compressibility effects [START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF].

The reduction of turbulent viscosity in the two-phase area, which appears to be the direct cause of the simulation improvement, is not confirmed yet by any measurements. However, this trend is consistent with the one reported by Wang et al [START_REF] Wang | 3D turbulence structure and phase distribution measurements in bubbly two-phase flows[END_REF], who mentions for bubbly flows an increase of turbulent dissipation with void ratio, and thus a possible diminution of velocity fluctuations.

Cavitation behaviour of a hydrofoil 4.1. Experimental behaviour

The second geometry is a two-dimensional foil of 150 mm chord and 80 mm span. Its cross section is composed of a flat upper surface and a convex lower surface of 195 mm radius, as illustrated by figure 9. Experiments were performed in the ENSTA cavitation tunnel whose test section is 150 mm height, 80 mm width, and 640 mm length. The foil was located at mid-height, with 5 • angle of attack, so a cavitation sheet appears on the upper face, when the pressure is decreased in the tunnel. Compared with the previous configuration, the cavitation behaviour is modified by the possible interaction between foil pressure and suction sides.

For an inlet velocity equal to 6.8 m s -1 and a cavitation parameter σ =1 .65, a cavitation sheet is obtained on the upper surface. Its mean length is about 60 ± 5 mm, but its behaviour is unsteady with large periodical vapour cloud shedding. Figure 10 shows the mean cavitation sheet.

It is worth noting that the length of this mean cavity (about 90 mm) includes the vapour Vref/Lcav Strouhal number Figure 11.

Comparison between numerical and experimental oscillation frequencies concerning the hydrofoil geometry. cloud before it begins to collapse. So it is notably higher than the mean length of the attached cavity reported during the experiment.

The cavity oscillation frequency, measured from the cavitation noise with a hydrophone, equals 33 ± 3 Hz. It was reported in previous studies [START_REF] Pham | Investigation of unsteady sheet cavitation and cloud cavitation mechanisms[END_REF] that in this configuration the vapour shedding is directly driven by the re-entrant jet that periodically flows under the cavity towards the leading edge.

Calculation

The numerical model is applied with the k-ω model including compressibility effects. The computational domain is composed of 140 × 70 cells, with a special contraction in the expected cavitating area, as indicated in the case of the Venturi-type section (figure 9). The experimental flow conditions are imposed (V ref =6.8ms -1 , σ =1.65).

The predicted flow behaviour is then in good general agreement with experiments: the cavitation sheet adopts a periodical unsteady behaviour, with a mean frequency of 30/35 Hz and a mean cavity length equal to 60 mm. These values are in reliable agreement with experimental data.

Figure 18 presents a comparison between phase-averaged cavity shapes, obtained respectively in the upper part from the calculation, and in the bottom part from pictures of the cavitation sheet. It can be noticed that the evolution of the cavity shape in both cases is very similar. In the calculation, a small-pressure side cavitation sheet is also obtained, whereas it is observed in experiments only for slightly lower pressure or higher angle of attack.

Several flow conditions were investigated by varying the cavitation number. The oscillation frequencies obtained for different mean cavity lengths are reported in figure 11, as well as the corresponding Strouhal numbers. These are all close to the standard value of 0.3 reported by many authors in such configurations (see [START_REF] Joussellin | experimental investigations on unsteady attached cavities[END_REF][START_REF] Stutz | Experiments on unsteady cavitation[END_REF]). The experimental measurement for V ref /L cav = 194 Hz must be considered circumspectly, since the visual estimate of the mean cavity length is not precise for such small sheets of cavitation. The high value of the Strouhal number in this case (Str =0.36), as well as the discrepancy with the numerical result, could be related to this uncertainty. For other flow conditions, calculation and experimental results are in good general agreement, which indicates that the physical cavitation model, associated with the k-ω turbulence model (including compressibility effects) is efficient to predict the behaviour of the cavitation sheet in this more complex configuration.

Case of a cascade of three hydrofoils

Experimental behaviour

The last test case-the most complex one-is a cascade composed of three hydrofoils, located in a test section whose size is 30 × 100 mm (figure 12). The experiments were performed at the Laboratory for Turbomachinery and Fluid Power of the Darmstadt University of Technology. The angle of attack of all foils is 5 • in the results presented hereafter, and the inlet velocity equals 14 m s -1 . A piezoresistive pressure transducer (Kulite XTM-190M) with a frequency range up to 125 kHz is used to investigate the unsteadiness of cavitation. The sensor membrane is connected to the flow by a small opening. All details concerning the experimental setup and the pressure signal treatment are given by Lohrberg et al [START_REF] Lorberg | Numerical and experimental investigations on the cavitating flow in a cascade of hydrofoils Exp[END_REF].

According to the experimental study, the cavitation phenomenon, for different values of σ, is characterized by two frequencies: the higher one is found to be related to the self-oscillation behaviour of the cavitation sheet on the upper foil, while the lower one corresponds to a coupling between the different channels.

Calculation

This section presents results obtained in this geometry by calculations for σ =1.95. More details concerning numerical simulations in this configuration are presented in reference [START_REF] Lorberg | Numerical and experimental investigations on the cavitating flow in a cascade of hydrofoils Exp[END_REF].

The turbulence model applied in this configuration is the modified k-ε RNG model, whose modifications were found in section 3 to be almost equivalent to the treatment of the compressibility effects in the k-ω model. Vapour structures, characterized in the model by small values of the density ρ, are generated along the suction side of the foils from the leading edge. After the initial transient, a special unsteady behaviour is observed (figure 13):

• On the upper foil suction side, typical cloud cavitation behaviour takes place. However, the size and shedding period of the cavitation structures show strong variations between the successive cycles. • On the two other foils the cavities appear as almost stable sheets attached to the blades, but with large amplitude and low frequency length fluctuations.

The difference between the cavitation behaviour in channel 1 and the others is associated with the larger flow angle of attack observed in front of the upper hydrofoil. This effect is due to the presence of the upper wall. The divergence angle between the wall and the foil suction side, associated with the trailing edge condition, increases the flow rate passing over the foil. An opposite effect occurs between the third foil and the lower wall. With the larger angle of attack, typical unsteady cloud cavitation takes place, while stable sheet cavities with low relative thickness develop on the two other foils (channels 2 and 3). It can be seen in figure 13 that the fluctuations of the cavitation structures (i.e. cloud cavitation in channel 1 and attached cavitation sheets in channels 2 and 3) appear with completely opposite phases. Disappearance of the cavitation structures in the upper channel corresponds to a maximum development of the cavitation sheets on the two other foils, and vice versa. To investigate the reason for the coupling at low frequency between the cavitating flows in the different channels, the time evolution of the local flow angle in front of the upper foil was first drawn (figure 14(a)). As explained before, the occurrence of large cloud cavitation in channel 1 corresponds to the increase of the incident flow angle. In contrast, periods with stable cavitation sheets on the second and third foils correspond to a small flow angle. Longitudinal velocity near the trailing edge, in the boundary layer of the suction side of the upper foil, is shown in figure 14(b) with respect to time. This shows that periods of developed cloud cavitation are followed by the passage at the foil trailing edge of travelling vortices, with negative longitudinal velocity. The interactions between these travelling flow structures and the foil trailing edge seem at to be at the origin of the decrease of the flow angle in front of the foil.

Another illustration of this phenomenon is given in figure 20, where the flow vorticity evolution is observed during several low-frequency cycles. The period of the low-frequency behaviour seems to be linked to the convection time of the large vortices generated by unsteady cavitation, to the foil trailing edge.

Conclusion

Unsteady cloud cavitation has been investigated in three configurations, of increasing complexity. Experimental measurements and visualization have been compared to the results of twodimensional computations, performed with a single-fluid cavitation model based on a barotropic state law.

The first case was a simple Venturi-type section, for which several turbulence models were applied. It was found that it is necessary to take into account corrections of the effects of the vapour/liquid mixture compressibility on the turbulence structure to obtain the unsteady cavitating behaviour observed experimentally. This influence of compressibility was tested as well with a k-ω model proposed by Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF] for compressible flows as with a k-ε RNG model modified arbitrarily. Results obtained are in reliable agreement with experimental data. In both cases, the main reason for the improvement of the prediction is a reduction of the turbulent viscosity in the two-phase flow area.

The ability of the physical cavitation model, associated with any of these two turbulence models, to predict unsteady behaviour of cavitation sheets was confirmed by two complementary test cases, namely a single hydrofoil and a cascade of three hydrofoils. In these two configurations, the numerical model correctly predicts both cavity lengths and oscillation frequencies, for given flow conditions.

Concerning the hydrofoil cascade, the numerical approach allowed us to obtain a detailed analysis of the cavitating flow, and a reliable prediction of the characteristic frequencies of the phenomena of cloud shedding and of the coupling between the three channels. This indicates that not only single oscillation behaviours, but also other types of unsteady effects due to cavitation, may be efficiently simulated by the proposed model.

Figure 1 .Figure 2 .

 12 Figure 1. Successive 2D simplifications of an inducer geometry.

Figure 3 .

 3 Figure 3. Barotropic state law ρ(P ). Water 20 • C.

Figure 4 .

 4 Figure 4. (a) Curvilinear-orthogonal mesh of the Venturi-type section (160 × 50 cells). (b) Zoom of the mesh in the throat region of the Venturi duct.

Figure 5 .

 5 Figure 5. Time evolution of the cavity length. Abscissa, time; ordinate, the X position in the tunnel of cavitation. The colours represent density values: white for pure liquid and from red to dark blue for vapour. At a given point in time and position, the colour indicates the minimum density in the corresponding cross section of the cavitation tunnel. Calculation conditions: σ ≈ 2.4; V ref = 7.2ms -1 ; mesh = 160 × 50; time step ∆t =0.005T ref (T ref = L ref /V ref )[ 17].

Figure 6 .

 6 Figure 6. Transient evolution of unsteady cavitating flow in the Venturitype duct. Temporal evolution (abscissa) of the cavity length (ordinate). Instantaneous density distribution of attached and cloud cavities is illustrated on the left at T =11T ref (velocity vectors are drawn only 1 cell over 2 in the two directions) [17].

Figure 7 .

 7 Figure 7. Frequencies of the cloud shedding obtained by experimental and numerical studies.

Figure 8 .

 8 Figure 8. Modification of the mixture turbulent viscosity (n = 10).

Figure 9 .

 9 Figure 9. Geometry of the hydrofoil and computational grid (L ref = 15 cm).

Figure 10 .

 10 Figure 10. Mean shape of the cavitation sheet observed on the hydrofoil tested in the ENSTA test rig (Angle of attack 5 • , V ref =6 .8ms -1 , σ =1 .65, length scale indicated on the foil each centimetre.).

VFigure 12 .

 12 Figure 12. Geometry of the cascade of hydrofoils tested in the Laboratory for Turbomachinery and Fluid Power, Darmstadt.

Figure 13 .

 13 Figure 13. Time evolution of the cavitation structures on the three hydrofoils (V ref =14ms -1 , σ =1.95) [31].

Figure 14 .

 14 Figure 14. (a) Angle of velocity vector, 5% upstream of the upper foil leading edge. (b) Longitudinal velocity near the upper foil trailing edge (suction side) [31].

Figure 15 .

 15 Figure 15. Example of an unsteady cloud cavitation on a foil suction side (see animation).

Figure 16 .

 16 Figure 16. Numerical simulation of unsteady oscillation behaviour in the Venturi type section (see animation).

Figure 17 .

 17 Figure 17. Comparison between numerical and experimental phase-averaged cavity shapes for the Venturi type section (see animation).

Figure 18 .

 18 Figure 18. Comparison between numerical and experimental phase-averaged cavity shapes for the hydrofoil (see animation).

Figure 19 .

 19 Figure 19. Numerical simulation of the unsteady cavitating behaviour of the cascade of hydrofoils (see animation).

Figure 20 .

 20 Figure 20. Evolution of the vorticity field in the hydrofoils cascade (see animation).

Acknowledgments

The authors wish to express their gratitude to the French Space Agency, Centre National d'Etudes Spatiales (CNES), for its continuous support. The study of the hydrofoil cascade was performed in collaboration with Dr H Lohrberg and Professor B Stoffel, from the Laboratory for Turbomachinery and Fluid Power of Darmstadt. The experimental results concerning the Venturi-type section were investigated by Dr B Stutz, at the LEGI laboratory.


R Lohrberg
email: lohrberg@tfa.tu-darmstadt.de

Darmstadt Stoffel

Regiane Fortes Patella

Olivier Coutier-Delghosa

Jean-Luc Reboud

H Lohrberg

B Stoffel

R Fortes-Patella

O Coutier-Delgosha

Numerical and experimental investigation on the cavitating flow in a cascade of hydrofoils

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Numerical and experimental investigations on the cavitating flow in a cascade of hydrofoils H. Lohrberg, B. Stoffel, R. Fortes-Patella, O. Coutier-Delgosha, J.L. Reboud Abstract The cavitating flow in a cascade of three hydrofoils was investigated by experimental means and numerical simulation. Experiments on the 2D-hydrofoils cascade were carried out at Darmstadt University of Technology in a rectangular test section of a cavitation tunnel. A numerical model developed at LEGI (Grenoble) to describe the unsteady behaviour of cavitation, including the shedding of vapour structures, was applied to the hydrofoils cascade geometry. Results of both experimental and numerical studies show a strong interaction between the cavities of each flow channel besides the typical selfoscillation of cloud cavitation. A detailed comparison of the results allows an interpretation of the interaction mechanisms to be proposed. In the scope of the European Research Program PRO-COPE, researchers of the LEGI (Laboratoire des Ecoulements Ge ´ophysiques et Industriels de Grenoble) and of the Chair of Turbomachinery and Fluid Power at Darmstadt University of Technology (TUD) work together in order to improve the understanding of the unsteady behaviour of cavitating flows and the related erosive aggressiveness. In this context, experimental investigation and numerical modelling of cavitating flows around a 2D hydrofoil with a semicircular leading edge were performed. Previous results concerning the self-oscillating behaviour of cloud cavitation were presented by [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation[END_REF], and the stabilising effect of a small obstacle modifying the foil suction side has been more particularly studied. In the present paper, the cavitating flow in a cascade of three hydrofoils was investigated by experimental means and numerical simulation.

List of symbols

Experiments on the 2D-hydrofoils cascade were carried out at TUD in a rectangular test section of a cavitation tunnel. Transient wall pressure signals were measured, treated to evaluate the dominant frequencies and used to perform conditional sampling of images of the cavitating flow. Images of the cavitating flow were taken synchronously with the pressure transients to provide further information about the cavitation dynamics.

Numerical models of unsteady cavitating flows using single fluid approaches have been developed in recent years. In such models, the same velocity is assumed for liquid and vapour phases in the two-phase zones. Vaporisation and condensation of the fluid are managed using a barotropic state law, a mass transfer equation or a bubble dynamic model. Besides applications of different models to quasi-steady cavitating flow, some studies focused on unsteady cloud cavitation phenomena in Venturi-type ducts [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF][START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modelling of unsteady effects[END_REF], around hydrofoils in cavitation tunnels [START_REF] Kubota | A new modelling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil section[END_REF][START_REF] Reboud | Two-phase flow modelling of unsteady cavitation[END_REF][START_REF] Song | Numerical simulation of cavitating flows by single-phase flow approach[END_REF][START_REF] Sauer | Unsteady cavitating flow -A new cavitation model based on modified front capturing method and bubble dynamics[END_REF] or in centrifugal pumps [START_REF] Hofmann | Experimental and numerical studies on a centrifugal pump with 2d-curved blades in cavitating condition[END_REF]. Numerical simulation of the stabilising effect of a small obstacle placed in the cavitation sheet area [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation[END_REF] shows the ability of such a model to catch localised hydrodynamic effects. Recent extension to blade cascades representative of inducer geometries [START_REF] Coutier-Delgosha | Numerical simulation of the unsteady cavitation behavior of an inducer blade cascade[END_REF] allows instabilities due to the presence of sheet cavitation on the different blades to be predicted, showing a reliable agreement with experimental observations [START_REF] Joussellin | Cavitating inducer instabilities: experimental analysis and 2D numerical simulation of unsteady flow in blade cascade[END_REF]).

In the present paper, the numerical model, developed at LEGI (Grenoble) to describe the unsteady behaviour of cavitation including the shedding of vapour structures, was applied to hydrofoil cascade geometry. Comparisons of experimental and numerical results are performed to analyse the interaction between the cavitating flows in the different channels.

Experimental set-up

The object of the investigations was a cascade of three hydrofoils, located in a test section with a rectangular cross section of 30•100 mm (Fig. 1). The hydrofoils can be adjusted to different angles of attack in a range of ±10°a ccording to different flow rates in a centrifugal pump.

The flow can be illuminated and observed through a lateral window. Illumination was done with a stroboscopic light source, and images were taken with a CCD camera (PCO sensicam). A piezoresistive pressure transducer (Kulite XTM-190 M) with a frequency range up to 125 kHz was also mounted in the window. The membrane was connected to the flow by a small, short opening. The geometry of the hydrofoils and sensor position are shown in Fig. 1b. The sensor was mounted above profile 1, the positions above profiles 2 and 3 were additionally used for postprocessing of the numerical results. The test section was part of a closed hydraulic loop. Flow rate and static pressure could be varied in order to achieve different cavitation conditions. In the case presented hereafter, the hydrofoils were held at a 5°angle of attack, and the inlet velocity V ref was equal to 14 m/s. Image illumination and acquisition and pressure signal recording were both triggered by a personal computer. Signals were sampled with 100 kHz and stored in a transient recorder. Storing started with 50% pre-trigger, thus the image was taken in the middle of the measurement. Images and measurements were read out and stored on the personal computer. Every image was then related to a part of the pressure signal. Unfortunately, parts of the black coating of the channel wall detached during image acquisition, leaving some bright stains around the hydrofoils.

Signal treatment

Power density spectra of the pressure signals showed two dominant frequencies for each signal. To extract those frequencies from the very noisy pressure signal, a more careful method was applied to the records.

First the signals were split offline at 100 Hz into highand low-frequency parts by high-and low-pass filtering respectively. The auto-correlation function was then calculated from both parts. The function of a fully periodic signal would show several peaks, each with a maximum of one. The first is located at zero shifting, the others at positive and negative multiples of cycle duration. Due to the frequency variation of a real signal, these peaks do not have a maximum of one. If the variation is seen as a phase difference between successive periods, the maximum value of the first peak next to the centre is the cosine of this phase difference. The more remote peaks decline exponentially with their distance from the centre. The frequency variation can therefore be calculated from the peak value at cycle duration. Both the frequencies and their variation are shown as functions of the cavitation number r in Fig. 2.

In contrast to frequency estimation by Fourier transformation, this process delivers a unique fluctuation frequency even of very noisy signals. Furthermore, it is possible to directly derive a frequency instability. The use of this information is described below. In power density spectra, such an unstableness appears as a broadening of the amplitude peak, which is hardly quantifiable.

The higher frequencies correspond to the cavitation self-oscillation behaviour characterised by a Strouhal number between 0.2 and 0.3. The lower frequency was presumed to be related to a coupling of the cavitation on the three hydrofoils. To verify this presumption, images were associated with the phase angle of the low-frequency fluctuation. After filtering, the signals were undertaken to a Hilbert transformation. This transformation adds an imaginary part to the signal by shifting the Fourier coefficients by an angle of p/2 and performing the inverse Fourier transformation. The phase information can be extracted from the complex signal (Fig. 3). The non-consecutively acquired images can then be sorted into phase angle consecutive order.

With the phase information available for every image, two further treatments could be used to explain the cavitation behaviour and to compare the experimental and the numerical results respectively.

First, the images were put together to an exemplary sequence of one cavitation cycle (Fig. 4). The cycle was divided into 25 steps. On the basis of the phase angle, each image was assigned to one step. With a total image number of 1,000, each step includes about forty images. These images were averaged by a special algorithm in order to let the resulting image appear like a real one-shot cavitation image. This approach makes the subsequent interpretation of the sequence easier. A wavelet transformation was applied using sigmoid wavelets. The wavelet coefficients were then averaged separately by absolute value and sign, thus keeping the image signals energy in every wavelet form. The inverse transformation was applied to the averaged coefficients.

The second treatment extracted the mean brightness of three regions of interest (ROI), each related to one channel of the cascade. As the vapour appears bright in the image, the mean brightness is a measure of the vapour volume even if the relation between both is not quantitatively known The theoretical and numerical model was developed to describe the unsteady behaviour of cavitation, including vapour cloud shedding. In that model, the liquid-vapour two-phase flow is described by a homogeneous flow model with a no-slip condition between liquid and vapour phases. The liquid-vapour mixture, with a void ratio a, is considered as a homogeneous medium and described as a single fluid whose varying specific mass is q=aq v +(1-a)q l . A simple description of the vaporisation and condensation phenomena is proposed by introducing an empirical barotropic state law q(p) for the mixture. A smooth law was chosen, q rapidly varying between liquid density q l and vapour density q v when the local static pressure P is around the vapour pressure P v . The law is characterised by its maximum slope at P=P v , which is related to the minimum speed of sound A min in the two-phase homogeneous medium. A min is the only adjustable parameter of the model, fixed in the present study as in the previous one [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation[END_REF]) at 10% of inlet velocity V ref .

Reynolds-averaged Navier-Stokes equations are solved in the case of this single fluid with variable density. A finite volume spatial discretisation is applied in curvilinear orthogonal coordinates on a staggered mesh.

An iterative resolution based on the SIMPLE algorithm was developed to deal with quasi-incompressible flow (a=0 and a=1) and highly compressible flow (0<a<1) [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF]. The liquid-vapour interfaces are described by high gradients of the mixture density q, which was made possible by using a conservative approach and the HLPA non-oscillatory second-order MUSCL scheme [START_REF] Zhu | A low diffusive and oscillation-free convection scheme[END_REF]. To solve the time-dependent elliptic problem, a first-order fully implicit method is used. Turbulent flows are calculated by solving the Reynolds equations using a k-RNG turbulence model with standard laws of the wall.

The mass flow rate is imposed constant as an upstream boundary condition. The transient static pressure, imposed as a downstream boundary condition, is slowly decreased from a high value, leading to steady non-cavitating conditions, down to the value required to assure the expected cavitation number r. Phase change then occurs spontaneously in the regions where the pressure decreases close to the vapour pressure P v . Because the experimental value of r is based on an upstream pressure measurement as reference pressure, the losses generated in the cavitation tunnel by viscous and unsteady effects have to be taken into account in the calculation to compare a posteriori the experimental and numerical cavitation numbers.

The model was previously tested in Venturi-type ducts [START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modelling of unsteady effects[END_REF]) and on single hydrofoil geometries [START_REF] Reboud | Two-phase flow modelling of unsteady cavitation[END_REF][START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation[END_REF]. Empirical modification of turbulence dissipative terms in the two-phase regions was proposed, to enhance the simulation of the unsteady behaviour of cavitation observed experimentally: turbulent viscosity in the pure liquid phase remains provided by the classical k-RNG model, while stronger shear flows are allowed in the two-phase regions by arbitrary reduction of the turbulence model constant C l (multiplication by a function f(q), equal to 1 for q=q l and rapidly decreasing towards q v /q l when q<q l ; [START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modelling of unsteady effects[END_REF]. Eects on unsteady cavitation modelling of other modifications of standard turbulence models have been investigated recently and are reported in [START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF].

Simulation of the unsteady cavitating flow in the cascade

In the case presented below, the static pressure is decreased down to the experimental value r=1.95. Vapour structures, characterised in the model by small values of the density q, are generated along the suction side of the foils from the leading edge. After the initial transient, a special unsteady behaviour is observed (Fig. 6):

• On the upper foil suction side, typical cloud cavitation behaviour takes place. However, the size and shedding period of the cavitation structures show strong variations between the successive cycles.

• On the two other foils, the cavities appear as almost stable sheets attached to the blades, but with large amplitude and low-frequency fluctuations of length.

The difference between the cavitation behaviour in channel 1 and the others is associated with the larger flow angle of attack observed in front of the upper hydrofoil. This effect is due to the presence of the upper wall. The divergence angle between the wall and the foil suction side, associated with the trailing edge condition, increases the flow rate passing over the foil. An opposite effect occurs between the third foil and the lower wall. With the larger angle of attack, typical unsteady cloud cavitation takes place, while stable sheet cavities with low relative thickness develop on the two other foils (channels 2 and 3). It can be seen in Fig. 6, that the fluctuations of the cavitation structures (i.e. cloud cavitation in channel 1 and attached cavitation sheets in channels 2 and 3) appear with completely opposite phases. Disappearance of the cavitation structures in the upper channel corresponds to a To evaluate the characteristic frequencies of the cavitating flow, the pressure signals obtained by numerical calculations were analysed with the same treatment used in the experimental study (Sect. 3). Figure 7a illustrates the time evolution of numerical pressure signal at the location of the three transducers. Figure 7b presents the autocorrelation function calculated for high-pass and low-pass filtered pressure signals.

From power density spectra, the lowest characteristic frequency obtained equals 31.42±0.98 Hz.

The frequency of the cloud shedding process in the upper channel can also be estimated by the treatment: a frequency of 262±35 Hz is clearly obtained from the signals corresponding to channels 2 and 3. The frequency content of the signal of channel 1 is much more complex, as can be seen from the auto-correlation function (Fig. 7b) and many peaks are observed in the spectra between 160 and 500 Hz.

Based on the corresponding mean attached cavity length (~0.3L ref ~1.5 cm), the inlet velocity (V ref =14 m/s) and a frequency of 262 Hz, the Strouhal number is found to be about 0.28. These values present an order of magnitude in good agreement with the experimental measurements obtained from Fig. 2 A zoom of transient pressure coefficient distributions corresponding to the locations of the three transducers (Fig. 1) is given in Fig. 8a, while two pressure fields are given at two different times of the low-frequency cycle (Fig. 8b). Larger development of cloud cavitation in channel 1 is associated with high-pressure fluctuations. The amplitude is higher in the first channel, because the travelling clouds pass the point corresponding to the transducer location, but pressure fluctuations can also be clearly noticed in the two other channels. On the contrary, the periods during which cavitation is more developed as stable sheets on the two other foils show a complete stabilisation of the pressure signal.

Figure 9a shows the time variation of the vapour volume computed in the three channels. The treatment developed in the experimental study was applied on the numerical result to evaluate the first harmonic amplitude of the vapour volume calculated for each channel (Fig. 9b).

The result obtained is similar to the experimental one drawn in Fig. 5. The opposite phase behaviour between channel 1 and the two others is clearly visible. A small phase shift between channels 2 and 3 may be noticed too. Nevertheless, the volume of vapour obtained by numerical calculation is characterised by a phase delay to the pressure signal in channel 1 that is different from the one observed in the experiments. The difference between calculations and experiments could be related to the position of the transducer. Indeed, the sensor geometric position is the same in numerical and experimental cases, but the position with regard to the cavitation structures is slightly different for each case (since the cavity length is underestimated by the numerical simulations).

The amplitude of the numerical signal energies estimated by fit of the first harmonic is higher than the ones obtained from the experimental results owing to of the better precision of the numerical calculations.

Despite the observed different cavitation behaviour in each channel, the order of magnitude of the vapour volume predicted by the simulation remains the same in the three channels.

To investigate the reason for the coupling at low frequency between the cavitating flows in the different channels, the time evolution of the local flow angle in front of the upper foil was first drawn (Fig. 10a). As explained before, the occurrence of large cloud cavitation in channel 1 corresponds to the increase in the incident flow angle. In contrast, periods with stable cavitation sheets on the second and third foils correspond to a small flow angle. Longitudinal velocity near the trailing edge, in the boundary layer of the suction side of the upper foil, is drawn Fig. 10b with respect to time. It shows that periods of developed cloud cavitation are followed by the passage of travelling vortices at the foil trailing edge, with negative longitudinal velocity. The interactions between these travelling flow structures and the foil trailing edge seem to be at the origin of the decrease in the flow angle in front of the foil.

Another illustration of this phenomenon is given in Fig. 11, where the flow vorticity is drawn at different stages of the low-frequency cycle. The period of the lowfrequency behaviour is then certainly linked to the convection time of the large vortices generated by unsteady cavitation, to the foil trailing edge.

Numerical calculations considering two other values of the cavitation number (r=2.48 and r=2.95) were also performed. Results concerning the auto-correlation functions obtained for high and low frequencies are illustrated in Fig. 12 for these two complementary cases. Characteristic frequencies can then be deduced and compared with the experimental ones (Fig. 2).

Numerical results present a good agreement with experimental values, and the code leads to a reliable simulation of the cyclic self-oscillation behaviour of unsteady cavitating flows.

Conclusion

Cavitation was studied in a cascade of three hydrofoils mounted in a cavitation tunnel. Estimation of the shedding and the coupling frequency was possible only in the autocorrelation domain due to the unsteadiness of the cycle duration and the low signal to noise ratio. Filter design on the basis of the frequency determination as described above is well suited for the extraction of interesting signal parts. Hilbert transformation and image phase averaging appeared to be a useful tool for getting detailed information about high-speed cyclic processes without the necessity of a high-speed camera.

Both the experimental and the numerical investigations show two dominant frequencies. The lower one is about 40 Hz in the experiments and does not depend on the cavitation number. It is found to be a little lower (30 Hz) by the numerical simulation. The higher fre- According to the phase-averaged videos and the computation results, the lowest frequency seems to be strongly linked to the coupling of the three channels. Analyses of numerical results indicate that vortices generated by the cloud cavitation phenomenon in the upper channel are by the mean flow to the trailing edge. The flow repartition in the different channels seems to be strongly affected by the interaction between the travelling vortices and the trailing edge flow of the upper foil, which may be the origin of the low-frequency phenomenon.

The good agreement obtained between experimental and numerical results indicates that the proposed model well describes the unsteady cavitation behaviour and complex associated hydrodynamic effects.

  L cav cavity length, m L ref reference length=chord of the foil=5 cm, m V ref reference velocity=inlet flow velocity, m/s P static pressure, Pa P ref reference pressure=inlet static pressure, Pa P v vapour pressure, Pa St Strouhal number=fL cav /V ref T ref reference time=L ref /V ref , s a angle of velocity vector r cavitation number=(P ref -P v )/(1/2q l V ref 1 Introduction

Fig

  Fig. 1a, b. Hydrofoils cascade geometry: a cut through the test section (pressure transducers are not shown); b side view of the window with one sensor (upper channel) and three post-processing positions

  . The ROI mean grey values are normalised by timeaveraged value and standard deviation. The first harmonic amplitude of the brightness is calculated. The harmonic functions of each channel are shown in Fig. 5. The amount of the signals energy of the brightness that is stored in the first harmonic (value e in the legend) is only about 15%. That means that only a small part of the vapour variation is related to the low fluctuation cycle. The larger part is related to cloud shedding and noise. 4 Numerical model To model and to analyse the unsteady behaviour of the cavitating flow in the tested hydrofoil cascade, the unsteady cavitating flow model developed at LEGI (Grenoble)

Fig. 2 .

 2 Fig. 2. Main frequency of the low-and high-pass filtered pressure signal as a function of the cavitation number r, V ref =14 m/s

Fig. 4 .

 4 Fig. 4. Exemplary frames from a sequence of phase angle averaged images, a=5°, r=1.9, V ref =14 m/s

Fig. 5 .

 5 Fig. 5. First harmonic function of the brightness of each channel

  at r=1.95 (respectively about 40-50 Hz and 110-180 Hz).

Fig. 6

 6 Fig. 6. a Time evolution of the cavitation structures on the three hydrofoils (V ref =14 m/s, r=1.95): density field at t=160 and 172 ms (45 and 48T ref , T ref =L ref /V ref =3.6 ms). b Time evolution of the cavitation structures: size of the cavitation structures in the channels

Fig. 7

 7 Fig. 7. a Time evolution of numerical pressure signal corresponding to the three post-processing positions (r=1.95). b Auto-correlation function calculated for low-pass (top) and high-pass filtered pressure signals (bottom) for the three transducers (r=1.95)

Fig. 9

 9 Fig. 9. a Time evolution of the vapour volume in the three channels. b First harmonic amplitude of the vapour volume calculated for each channel

Fig. 10

 10 Fig. 10. a Angle of velocity vector, 5% upstream of the upper foil leading edge. b Longitudinal velocity near the upper foil trailing edge (suction side)

This work is part of a European exchange programme PROCOPE with the research teams of Grenoble and Darmstadt as members. The numerical model has been developed with the support of the French space agency CNES.