O Coutier-Delgosha

J L Reboud

R Fortes-Patella

NUMERICAL STUDY OF THE EFFECT OF THE LEADING EDGE SHAPE ON CAVITATION AROUND INDUCER BLADE SECTIONS

. Pressure distributions along the foil suction side and the tunnel walls were measured for different cavity lengths. Total pressure measurements along the foil suction side allow characterizing the effects of cavitation on the liquid flow. Influence of the leading edge shape on the cavitation behaviour and comparison between experiments and numerical predictions are discussed.

Nomenclature

Introduction

The prediction of the cavitation behaviour is of first importance for the design of rocket engine turbo-pumps. Indeed, this phenomenon results in strong unsteady forces acting on the pump components, and it may lead to substantial performance losses. The analysis of the vapour occurrence in the pump, as well as the study of the unsteady effects related to cavitation, could directly influence the pump inducer design.

Several studies based on multi-phase flow approaches were developed during the last 10 years [Delannoy andKueny, 1990, Kubota et al. 1992]. They lead now to the development of three-dimensional Navier-Stokes codes that take into account the cavitation process [START_REF] Takasugi | Study on cavitating flow around a finite span hydrofoil[END_REF], Alajbegovic et al., 1999[START_REF] Kunz | Multi-phase CFD analysis of natural and ventilated cavitation about submerged bodies[END_REF], Bunnel and Heister, 2000]. In this context, a numerical model, aiming to take into account the cavitation phenomenon in inducers geometry is developed in collaboration with SNECMA Moteurs and the French space agency CNES. The final objective is to provide assistance to the design and prevision of operating range of rocket engine turbo-pumps, taking into account the steadystate and unsteady effects of cavitation. The model is based on the commercial code FINE/TURBO TM , developed by NUMECA International [START_REF] Hakimi | Preconditioning methods for time dependent Navier-Stokes equations[END_REF]. To simulate the cavitation process, a single-phase flow model is implemented [Coutier-Delgosha et al., 2001a] based on previous studies in 2D [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF][START_REF] Reboud | Two-phase flow modelling of unsteady cavitation, 2nd Int. Symp. on Cavitation[END_REF][START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modelling of unsteady effects[END_REF]].

The aim of the present paper is to evaluate the ability of the numerical model to take into account the influence of the leading edge shape on cavitation behaviour in inducer geometries. The study concerns two hydrofoils simulating a section of an inducer blade, held in a cavitation tunnel with two different angles of attack. They are characterised by two different leading edge geometries, chosen to approach rocket engine inducer design [START_REF] Reboud | Two-phase flow modelling of unsteady cavitation, 2nd Int. Symp. on Cavitation[END_REF]. Experimental results are compared with 2D computations in the different cases. A three-dimensional simulation of the complete hydrofoil geometry is also performed in one case, to illustrate the 3D effects on the cavitation sheet.

Experiment

The experimental study of the cavitation behaviour of two-dimensional hydrofoils simulating a mid-span section of an inducer blade has been performed in the large cavitation tunnel of the CREMHyG laboratory (Figure 1) [START_REF] Reboud | Two-phase flow modelling of unsteady cavitation, 2nd Int. Symp. on Cavitation[END_REF].

chord 0.9 m Vref < 30 m/s

Figure 1: large cavitation tunnel (width = 0.12 m) and inducer type blade section.

The upper and lower walls of the cavitation tunnel have been designed to generate along the suction side of the foil a pressure distribution as close as possible to the one existing along the inducer blade in non-cavitating condition. The hydrofoil chords are 0.9 meter long and their relative thickness is smaller than 1.5%. A 5 mm thin streamlined leg, through which pressure taps have been drilled, holds the foils. The reference flow velocity Vref and static pressure Pref are measured in the upstream rectangular section Sref (137x120 mm 2 ), located 200 mm upstream the foil leading edge.

Those reference values are used to define the cavitation number σ. The maximum inlet flow velocity Vref may be larger than 30 m/s. No marked effect of the amount of air dissolved in water has been observed on developed cavitation. During the present experiments, the air content of the water has been kept near saturation.

The behaviours of two different leading edges have been compared with respect to the development of sheet cavitation.

Their shapes are reported on Figure 2. The first leading edge presents a parabolic shape, about 3 mm long, creating with the suction side a sharp ridge. It simulates a blade whose upstream end has been bevelled. The second one is sharper and its rounded upstream end, where the cavity detachment occurs, presents a small curvature radius close to 0.2 mm. The two profiles will respectively be named hereafter "bevelled leading edge" (BLE) foil and "sharp leading edge" (SLE) foil.

The second hydrofoil is 3 mm shorter than the first one and its relative thickness is a little smaller, but their two suction sides are identical. Two different angles of attack have been tested: 4.1 and 4.5 degrees. They have been chosen to provide flow conditions close to those predicted in a blade-to-blade channel of the inducer at its nominal operating point. Because of the small ratio between the tunnel height and the chord length, the 0.4 degrees difference has a great influence on the cavitation behaviour.

Pressure measurements have been performed along the foils suction side (10 taps) and the tunnel walls (3 taps in the reference section, 7 along the lateral and lower walls and 4 along the upper wall). Pressure taps are connected with 1 mm diameter Rilsan pipes to piezo-resistive transducers with accuracy better than +/-0.1% of the 5 bar range. The flow rate is measured by an electromagnetic flow meter, with a relative accuracy of 0.2 %. The combination of inaccuracies on pressure and velocity measurements ensures on the pressure coefficient Cp and on the cavitation number σ a global uncertainty of +/-0.02 at Vref = 10 m/s.

Total pressure measurements have been performed using a 4 holes probe along 3 axes crossing the upper channel at 37%, 70% and 95% of the chord from the leading edge (Figure 1). Velocities have been deduced from the total and static pressure measurements. Comparisons with LDA data acquired along the same axes were used to validate the method under non-cavitating and cavitating conditions.

Physical model

Two-phase flow models are based on the assumption that the fluid is present in the computational domain both as liquid and vapour. The vapour is characterized by the density ρ v and the liquid by the density ρ l . On each cell of the mesh, the unknowns are calculated for each phase, by averaging them on the volume occupied respectively by liquid and gas [for example [START_REF] Ishii | Thermo-Fluid Dynamic Theory of Two-Phase Flow[END_REF]].

In the present work, we apply a single fluid model based on previous numerical and physical work developed in LEGI [Delannoy andKueny, 1990, Reboud et al. 1998]. It considers only one fluid, characterized by a density ρ that varies in the computational domain according to a state law. When the density in a cell equals the liquid one ρ l , the whole cell is occupied by liquid, and if it equals the vapour one ρ v , the cell is full of vapour. Between these two extreme values, the cell is occupied by a liquid/vapour mixture that we still consider as one homogeneous single fluid. The void fraction α = (ρ -ρ l ) / (ρ v -ρ l ) corresponds to the local ratio of vapour contained in the mixture. In this simple model, the void ratio α is related to the state law, the fluxes between the phases are treated implicitly, and no supplementary assumptions are required.

Moreover, concerning the momentum fluxes, our model assumes that locally (in each cell), velocities are the same for liquid and for vapour: in the mixture regions gas structures are supposed to be perfectly carried along by the main flow. This hypothesis is often assessed for this problem of 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[END_REF][START_REF] Song | Numerical simulation of cavitating flows by single-phase flow approach[END_REF][START_REF] Kunz | Multi-phase CFD analysis of natural and ventilated cavitation about submerged bodies[END_REF]. The momentum transfer between the phases is thus strongly linked to the mass transfer. In the present work the empirical barotropic law ρ(P) used is presented in Figure 3. When the pressure is respectively higher or lower than the vapour pressure, the fluid is supposed to be purely liquid or purely vapour, according respectively to the Tait equation and to the perfect gas law. The two fluid states are joined smoothly in the vapourpressure neighbourhood. It results in the evolution law characterized mainly by its maximum slope 1/A min 2 , where A min 2 = ∂P/∂ρ. A min can thus be interpreted as the minimum speed of sound in the mixture.

Numerical model

To calculate cavitating flows we apply the commercial code FINE/TURBO TM developed by NUMECA International. It is a three-dimensional structured mesh code that solves the time dependant Reynolds-averaged Navier-Stokes equations. A detailed description of the code is given in [START_REF] Hakimi | Preconditioning methods for time dependent Navier-Stokes equations[END_REF].

Unsteady Calculations Time accurate resolutions use the dual time stepping approach (see [START_REF] Jameson | Time dependant calculations using multigrid, with application to unsteady flows past airfoils and wings[END_REF]). Pseudo-time derivative terms are added to the equations. They march the solution towards convergence at each physical time step. The code resorts to a multigrid strategy to accelerate the convergence, associated with a local time stepping and an implicit residual smoothing. This kind of resolution is devoted to highly compressible flows. In the case of low-compressible or incompressible flows, its efficiency decreases dramatically. This well-known problem has been addressed by many authors and solved by introducing a preconditioner [START_REF] Hakimi | Preconditioning methods for time dependent Navier-Stokes equations[END_REF]. This one is based on the studies of [Turkel, 1987, Choi and[START_REF] Choi | The application of preconditioning in viscous flows[END_REF]. It consists in multiplying the pseudo-time derivatives by a preconditioning matrix Γ -1 . Such modifications have no influence on the converged result, since these terms are of no physical meaning, and converge to zero.

The discretization is based on a finite volume approach. We use a second order central scheme, which must be associated with two artificial dissipation terms, respectively of second and fourth order. The first one is activated in the strong pressure gradient areas, such as shocks. The other one is used in the whole domain, and it results in a second order space accuracy. The pseudo-time integration is made by a four-step Runge-Kutta procedure.

The physical time-derivative terms are discretized with a second order backward difference scheme that ensures a second order accuracy in time.

Boundary Conditions

The boundary condition setting is based on a system of dummy cells. Classical incompressible types of boundary conditions are applied: imposed velocities at the inlet, and an imposed static pressure at the outlet. Numerical studies have been started in LEGI to improve these conditions, mainly by taking into account the test rig influence [START_REF] Longatte | Contribution à l'Analyse Phénoménologique des Ecoulements Instationnaires dans les Turbomachines: Etude du Couplage Pompe-Circuit et Rotor-Stator[END_REF]], but have not been applied in the cavitating case yet. Laws of the wall are applied along solid boundaries.

Initial Transient Treatment

First of all, a steady step is carried out, with a pseudo vapour pressure low enough to ensure no vapour presence in the whole computational domain. Then, this vapour pressure is increased progressively during the early time steps, until the required cavitation number σ is reached. Vapour appears progressively in the low static pressure regions during this transient. The cavitation number is then kept constant throughout the computation.

Two-dimensional results

2D computations were performed on both geometries, for the two experimental angles of attack of 4.1 and 4.5 degrees. A 10.000 cells mesh was applied in the four cases (Figure 4), and a large range of cavitation numbers was considered, according to the experimental results. In the case of the SLE foil at 4.5 degrees incidence, a second mesh has been tested, more refined near the leading edge. In all computations a cavitation sheet appears on the suction side of the foil and completely stabilises as soon as the operating cavitation number is kept constant. Then, decreasing the cavitation number σ in successive steps allows drawing the law: cavitation number σ versus cavity length. Figures 5a andb present the results for the two angles of attack, respectively in the case of the BLE and SLE foils. Experimental similar results are also provided on these figures, to enhance comparison. We obtain a reliable agreement between numerical and experimental results, the case of the SLE foil at 4.5 degrees angle of attack needing nevertheless at particular treatment, as discussed hereafter.

4.1 degree angle of attack: experiments and computations confirm that the leading edge shape has almost no influence on the cavities extend at low incidence. Visualisations under stroboscopic light show smooth and quasi-steady cavitation sheets at all the cavitation numbers tested. Only a short fluctuating wake downstream of the cavity is composed of small travelling vapour structures. 4.5 degree angle of attack: experimental observations report the unsteady behaviour of the cavities on the two foils, and a strong influence of the leading edge shape. At given cavitation number σ, the mean cavity length is about two times larger on the SLE foil than on the BLE foil, and correspondingly for the same cavity length the cavitation number is about 25% higher. This effect was first not obtained in the numerical simulations while making the computation by decreasing progressively the cavitation number: contrarily, it can be observed on Figure 5b that very short cavities are then predicted in the case of the SLE foil. That is why the computation was continued, re-increasing the cavitation number from the lowest value. A strong hysteresis phenomenon is then observed, the cavity development remaining much larger during that second phase. It can be seen on Figure 6a that the pressure distribution obtained by the calculation along the suction side is the same on BLE and SLE foils, after the first 2% of the chord. The minimum of pressure, very close to the leading edge, is a little lower on the SLE foil. The velocity fields calculated in non-cavitating condition are illustrated by Figure 6b. An area of separated flow can be seen in the case of the SLE foil. According to numerical simulations, the separated flow is progressively filled with vapour when the cavitation number is decreased. For low value of σ, the cavity length becomes much larger than the initial separated flow structure and breaks it. While increasing again the cavitation number the cavity length remains larger than in the previous phase, and the agreement with experiment becomes much better.

In fact, the hysteresis phenomenon is not observed experimentally. The hysteresis predicted by the numerical calculations can be related to the physical and numerical models that do not well describe the shear effects of separated flows and do not capture the unsteadiness of the cavitation region. Indeed, from the experimental point of view the cavities adopt on the two foils at 4.5 degree angle of attack an unsteady behaviour associated with large vapour cloud shedding. The detachment of the cavitation sheets obtained in the four cases is shown on Figure 7, for the cavitation number corresponding to a cavity length close to 15% of the chord. It can be noticed that the high angle of attack leads to a faster growing of the cavity thickness downstream of the detachment point, and to a larger void ratio along the foil surface. Quantitative comparisons are investigated between experiments and the numerical results in the case of the bevelled leading edge at high angle of attack. The static pressure evolution along the profile is represented in Figure 8 for a cavity length increasing from 5% of the chord to 20% of the chord. We observe a general good agreement between the numerical results and the experimental data: the charts confirm that the cavity length is correctly predicted. The pressure distribution downstream from the closure area is a little over estimated by the calculation, which is mainly an effect of the 2D simulation that do not take into account the obstruction effects linked to the boundary layers developing along the sides walls of the tunnel. Nevertheless, the most important discrepancy concerns the pressure increase in the cavity closure area, which appears significantly more pronounced in the calculations than in the experiments. This sharp elevation predicted by the model in the cavity wake can be related to the complete stabilization of the cavitation sheet. The real behaviour involves self oscillations of the cavitation sheet and the measured time-averaged pressure distributions downstream to the cavities appear smoothed.

Velocity profiles across the upper channel are more closely studied. Three axes almost perpendicular to the foil suction side, respectively at 37%, 70%, and 95% of the chord length were investigated by means of a 4 holes probe. The component of the velocity vectors normal to these sections is represented in Figure 9 in non cavitating and cavitating conditions. The numerical results and the experimental data are compared. Because the obstruction effect due to the boundary layers along the side walls and along the leg holding the foil is not taken into account in the 2D simulation, the experimental mean velocity in the 3 sections are about 10% higher than the numerical ones. That is why to compare the profiles, all values have been divided by the mean velocity in non cavitating condition.

In non cavitating conditions, the boundary layer appears almost symmetric on both sides of the channel. In cavitating condition, the important increase of the displacement thickness due to the presence of the cavity wake is clearly visible all along the foil suction side. The interaction between that wake and the flow at the foil trailing edge leads to the slight decrease of the flow rate passing in the upper channel in cavitating conditions, what modifies a little the flow circulation around the foil and the resulting lift [START_REF] Reboud | Two-phase flow modelling of unsteady cavitation, 2nd Int. Symp. on Cavitation[END_REF]]. The agreement between measurement and simulation is satisfactory, in non cavitating and cavitating conditions. The quantitative influence of the cavitation sheet on the suction side boundary layer is notably well predicted.

Three-dimensional computation

A three dimensional calculation taking into account the complete experimental geometry was performed, to illustrate the 3D ability of the numerical code and to estimate the effect of 2D simplifications applied in the previous cases. A 1000000 cells mesh of the SLE foil at 4.1 degrees angle of attack is used. The cavitation number is fixed equal to 0.2, which leads to a cavity length close to 5% of the chord. The cavity shape, shown in Figure 10, appears clearly 3D, with an important influence of the side walls boundary layers, also visible on the streamlines drawn from the foil leading edge.

Conclusion

A numerical study of the cavitation behaviour of two-dimensional hydrofoils simulating a section of an inducer blade was presented. Two dimensional computations were performed in different cases: two different foil leading edge shapes, two different angle of attack. A general good agreement with experiments is obtained for the cavity length at different cavitation numbers. In one case, namely the sharp leading edge foil at the higher angle of attack, the presence in non cavitation conditions of a separated flow near the sharp leading edge creates from the numerical point of view the condition of a strong hysteresis phenomenon. Only the simulation performed by increasing the cavitation number from its minimum value provides a behaviour that conforms to experimental observations. With this angle of attack, the cavity extend is larger than with the bevelled leading edge foil, in both experiments and simulations. The self oscillating behaviour of cavitation is not simulated at the present stage of the model. There is no effect of the leading edge shape at the lower angle of attack. Static pressure distributions along the foil suction side and velocity profiles in the cross sections of the foil suction side channel agree well with measurements. The presence of the cavitation sheet leads to the modification of the boundary layer all along the foil, with a probable effect on the trailing edge flow. A first 3D computation shows the influence of side wall boundary layers on the aspect of the attached cavity. Another application of the model, to 3D cavitating flows in a centrifugal pump, is presented in [START_REF] Hofmann | Experimental and numerical studies on a centrifugal pump with 2D-curved blades in cavitating condition[END_REF]].

Figure 2 .

 2 Figure 2. BLE and SLE leading edge shapes.

Figure 3 :

 3 Figure 3: The barotropic state law ρ(P)

  This numerical model was adapted to treat the cavitation process. The key point of this adaptation is the modification of the state law of the fluid. Our barotropic law implies the simultaneous treatment of two different cases: the fluid is highly compressible in the two-phase flow areas (the Mach number can be as high as 4 or 5) and it is almost incompressible in the pure vapour or pure liquid areas. So the main difficulty consisted in managing these two different states of the fluid, without creating any spurious discontinuity in the flow field. Besides, the cavitation consists in a very sharp and very rapid process. It means that the density time fluctuations and the density repartition must be smoothed, to avoid numerical instabilities. Thus, we use an under-relaxation of the variations of the density to prevent too fast changing during one pseudo-time step. We also modified the switch between the two types of artificial dissipations. The second difference term is now also activated in the strong density gradient areas, such as cavitation sheet interface. Turbulence Model We use for the calculations presented in this paper a classical Baldwin-Lomax turbulence model. Other calculations considering k-ε turbulence models are in progress to improve the physical analyses. A more detailed study of the influence of the turbulence model on unsteady cavitation simulation is proposed in[START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF]. It shows a major effect of the compressibility effect modelling on the unsteady behaviour of cloud cavitation.

Figure 4 :

 4 Figure 4: Detail of the mesh near the leading edges: BLE and SLE foil

Figure 5 .

 5 Figure 5. Cavitation number σ vs. (Lcav/Lref) laws: a) BLE foil; b) SLE foil* (*at 4.5 degree, the arrow shows the succession of calculations)

  The numerical model fails to simulate this behaviour, and on the contrary completely stabilises the cavity. This current limitation of the model could be linked to the use of standard turbulence model [Coutier-Delgosha et al. 2001b]. Work is in progress to qualify the different effects of the model used and of the numerical scheme, to improve the simulations of unsteady cloud cavitation behaviours.

Figure 6 .

 6 Figure 6. Comparison of numerical pressure distributions (a) and leading edge flow fields (b)for the two hydrofoils in non-cavitating conditions.

Figure 7 :

 7 Figure 7: shape of the cavities with Lcav = 15% Lref a) Bevelled leading edge -i=4.5° b) Bevelled leading edge -i=4.1° c) Sharp leading edge -i=4.5° d) Sharp leading edge -i=4.1°

Figure 8 :

 8 Figure 8: Static pressure distribution on the profile: numerical and experimental results (BLE foil, Vref = 16m/s).

Figure 9 .

 9 Figure 9. Velocity profiles in three sections located respectively at 37%, 70%, and 95% of the chord. All values are divided by the mean velocity in the corresponding section in non cavitating condition. (BLE foil at 4.5 degree, Vref = 16m/s). Non cavitating condition experimental points ♦ ; numerical result (full line) ; Cavitating condition (Lcav/Lref = 15%) experimental points ; numerical result (full line) ; Difference between non-cavitating and cavitating conditions experimental l; numerical result (full line).

Figure 10 .

 10 Figure 10. Cavity shape: SLE foil, σ = 0.02. Grey surface corresponds to a void ratio of 5%. Streamlines of the flow in the vicinity of the cavity surface are drawn in red.

beveled leading edge -i=4.5° -Lcav/Lref=5%

											beveled leading edge i=4.5° Lcav/Lref=10%
			0.8							0.8
			0.6							0.6
			0.4							0.4
	Cp	-0.2 0 0.2						Cp	0 0.2
		-0.4							-0.2
		-0.6							-0.4
		-0.8							-0.6
			0	0.2	0.4	0.6 X/Lref	0.8	1	1.2	0	0.2	0.4	0.6 X/Lref	0.8	1	1.2
					numerical result		experimental result				numerical result	experimental result
				beveled leading edge i=4.5° Lcav/Lref=15%			beveled leading edge i=4.5° Lcav/Lref=20%
			0.8							0.8
			0.6							0.6
			0.4							0.4
	Cp	0 0.2						Cp	0.2
		-0.2							0
		-0.4						-0.2
		-0.6						-0.4
			0	0.2	0.4	0.6 X/Lref	0.8	1	1.2	0	0.2	0.4	0.6 X/Lref	0.8	1	1.2
					numerical result	experimental result				numerical result	experimental result

Acknowledgement

This work is supported by SNECMA Moteurs (Rocket Engine Division), the French space agency CNES, and by a doctoral grant from the Education French Ministry MERT. The laboratory CREMHyG (Grenoble, France) provided experimental results.


Olivier Courtot

Jean-Luc Coutier-Delghosa

Reboud

F Joussellin

Y Courtot

O Coutier-Delgosha

J L Reboud

CAVITATING INDUCER INSTABILITIES: EXPERIMENTAL ANALYSIS AND 2D NUMERICAL SIMULATION OF UNSTEADY FLOW IN BLADE CASCADE

Introduction

To achieve operating at high rotational speed and low inlet pressure, rocket engines turbo pumps are generally equipped with an axial inducer stage. Under such operating conditions, cavitation develops on suction side of the blades and at inducer periphery near the tip. Peripheral cavitation concerns all cavitating structures that appear near the shroud casing at the inducer inlet, as backflow of the pump [START_REF] Yokata | A Study on Swirling Backflow and Vortex Structures at the inlet of Inducer[END_REF] and cavitating tip vortices [START_REF] Laborde R | Tip Clearance Cavitation in an Axial Flow Pump[END_REF]. When inlet pressure is decreased from cavitation inception to breakdown of the pump, unsteady phenomena may appears, associated with different blade cavitation patterns. In Figure 1, successive flow patterns are located on the performance curve of a liquid hydrogen inducer [START_REF] Joussellin | Experimental Analysis of Instabilities Related to Cavitation in Turbopump Inducer[END_REF]. At cavitation inception, a steady and balanced flow pattern with one short attached cavity on each blade is observed from flow visualisations. When cavitation parameter is slightly decreased, a steady and alternate cavitating configuration appears (only on four blades inducers) with alternatively one short and one long cavity. Then flow visualisations achieve to identify an unsteady flow pattern. Rotating cavitation appears at low cavitation parameter [START_REF] Kamijo | An Experimental Investigation of Cavitating Inducer Instability[END_REF] just above breakdown. Unbalanced attached cavities are observed in the different channels, their distribution rotating faster than the inducer [START_REF] Joussellin | Experimental Analysis of Instabilities Related to Cavitation in Turbopump Inducer[END_REF][START_REF] Pagnier P | Conception and experimental study of an inducer[END_REF] and leading to large radials loads on the shaft [START_REF] Goirand | Experimental Investigation of Radial Loads Induced by Partial Cavitation with Liquid Hydrogen Induce[END_REF][START_REF] Goirand | Experimental Investigation of Radial Loads Induced by Partial Cavitation with Liquid Hydrogen Induce[END_REF]. Finally near the breakdown of the inducer, a steady and balanced flow pattern with fully developed cavitation is observed. Over the last few years, numerical models have been developed to predict the cavitation instability in inducers. These models are based on stability analyses and linear approach, taking into account the total flow rate variations through a cavitating blade-to-blade channel [START_REF] Tsujimoto | A Theoretical Analysis of Rotating Cavitation Inducers[END_REF][START_REF] Cambridge | Analytical Modelling of Instabilities in a Cavitating Turbopump Inducer[END_REF], or calculating the flow around attached cavities [START_REF] Pilipenko | Theoretical and Experimental Computational Methods for Determining Volume of Cavities and Coefficients of Dynamic Equation of Cavitation Cavities[END_REF][START_REF] Watanabe | Linear Analyses of Cavitation Instabilities[END_REF]. To improve the understanding and the prediction capability of cavitation instabilities, numerical and experimental analyses are developed through collaborations between the LEGI, the Rocket Engine Division of SNECMA Moteurs and the French space agency CNES. In this paper, experimental analysis focuses on the size of peripheral cavitating structures and their influences on the inlet flow. In parallel, the cavitation behaviour of inducers is simulated using a 2D model of unsteady cavitating flow developed at LEGI [START_REF] Reboud | Two-phase flow modelling of unsteady cavitation[END_REF][START_REF] Coutier-Delgosha O | Numerical Simulation of the Unsteady Cavitating Behaviour of an Inducer Blade Cascade[END_REF].

Figure 1. Sketches of cavitation patterns for various cavitation parameters and their correspondence to the performance curve [START_REF] Joussellin | Experimental Analysis of Instabilities Related to Cavitation in Turbopump Inducer[END_REF].

Cavitation in inducer: description of different types of cavitation

There are many types of cavitation in inducers. The cavitation depends on inducer geometry (leading edge) and on operating conditions. Cavitation in such inducer could appear in 2 regions well separated (Figure 2 The case of swept back leading edge leads to small pressure gradient in radial direction. Then cavitation could appear on the entire blade, from hub to tip radius. In that case, the cavitation sheet volume is maximum at about 5% of the blade to blade volume, which is very small compare to peripheral zone where it could be really large at low flow rate [START_REF] Joussellin | 3D Cavity shape in an inducer :Experimental investigations and numerical predictions[END_REF]. The behaviour of cavitation on this blade is similar to the one appearing in 2D facility with hairpins vortices in the unsteady cavity wake (see Figure 3 a radial view obtaining with a light sheet perpendicular to the blade, following AB). In the case of a more radial leading edge design, the pressure gradient versus radius is high, and low pressure zone i.e. cavitation is limited to the periphery of inducer. At peripheral zone, cavitation is mainly related to the tip leakage flow:

The pressure difference between the 2 sides of a blade causes a tip leakage flow to pass through the tip clearance which leads to a reverse flow (against active flow) which generates 3 types of cavitation (Figure 4): On the other hand, as a shear layer is formed between the swirling reverse flow and the non-rotational inlet flow, backflow vortices are formed and generates cavitation, called "backflow vortex cavitation" (Figure 4.), due to the fact that inducers are used at low flow rate, with incidence non equal to zero:

• Bf -backflow vortex cavitation: it generates an adverse flow; this type of cavitation could be the major mode of cavitation which could grow and take all peripheral place. Interaction exists between these different types of cavitation; the main volume occupied by cavitation could be attributed to backflow phenomenon.

Experimental facility and image processing results

As we want to quantify all type of peripheral cavitation, we use only axial view as it is shown above: The size of peripheral cavitation is obtained by image processing of the front view of the inducer (Figure 5a.), using the fact that cavitation appears clearly on these pictures (Figure 5b.). For a given flow rate (Figure 7.), cavitating region, Scav, appears constant at ± 10% with respect to the cavitation parameter. On the other hand, for a given cavitation parameter (Figure 8.), Scav (image processing results, points x x), decreases linearly when the flow rate, Q/Qo, increases (where Qo is the design flow rate with a zero incidence at inducer tip). These results agree with those presented by [START_REF] Yokata | A Study on Swirling Backflow and Vortex Structures at the inlet of Inducer[END_REF]. [START_REF] Petrov | Cavitation Characteristics of High-Rotational Centrifugal Pumps with Inducers[END_REF]. This result leads to conclude that cavitation appears in backflow zone without strongly modifying its expansion. LDA measurements performed in CREMHyG laboratory [START_REF] Von | Three dimensional partial cavitating flow in a rocket turbopump inducer: numerical predictions compared with laser velocimetry measurements[END_REF], cf. point • in Figure 8, confirm that peripheral cavitation area, Scav, deduced from visualization coincides with backflow area in non-cavitating conditions.

In conclusion, the obstruction of the flow section by the cavitating backflow area modifies the inlet axial velocity Cm with respect to the uniform value Q/Sflow. That phenomenon must be kept in mind to analyse the results of 2D numerical simulations.

Unsteady cavitating flow model in 2D blade cascade

Numerical simulation of the cavitating flow in the inducer was performed with the objective to take into account the cavitation sheets attached to the blades and their unsteady behaviour. The 2D numerical model of unsteady cavitating flow, developed in previous studies [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF][START_REF] Reboud | Two-phase flow modelling of unsteady cavitation[END_REF], Reboud et al. 1998), was adapted to 2D blade cascades [START_REF] Coutier-Delgosha O | Numerical Simulation of the Unsteady Cavitating Behaviour of an Inducer Blade Cascade[END_REF]. The main features of the models are:

• The liquid-vapour mixture is described as a single fluid, whose density ρ rapidly varies between the pure liquid density ρ l and the pure vapour one ρ v when the static pressure in the flow field reaches the vapour pressure. The fluid density is managed by a barotropic state law ρ(p), as in [START_REF] Coutier-Delgosha O | Numerical study of the effect of the leading edge shape on cavitation around inducer blade sections[END_REF].

at constant cavitation parameter at constant flow rate

• 2D unsteady Reynolds averaged Navier-Stokes equations are applied to that single fluid with variable density. The turbulence model used is the classical k-ε model, associated with laws of the wall along solid boundaries.

• Other boundary conditions are: imposed velocity at the mesh inlet (the flow velocity is deduced from the classical value Q/Sflow), imposed static pressure at the outlet, and periodicity or connection conditions between the different channel of the blade cascade. These conditions are applied along non-matching boundaries, through a special treatment developed in [START_REF] Coutier-Delgosha O | Numerical Simulation of the Unsteady Cavitating Behaviour of an Inducer Blade Cascade[END_REF] to assure the best transfer of both mass and momentum fluxes. Computations are performed in a four blades cascade representing a complete inducer. The 2D blade-to-blade channels were drawn by cutting the 3D inducer geometry at constant radius equal to 70% of the tip radius R (Figure 9.). Taking into account the real radius and thickness evolutions of the stream-sheet crossing the inducer would be possible by using a S1/S2 approach, but the numerical model has not yet been adapted to that formulation. We use a 190x30 structured mesh per channel, giving a total of 22800 internal nodes when calculating the 4 blades cascade.

Single-channel computation

In that first case, only one channel is computed, associated with a periodicity condition. A slow decrease of the cavitation number, from the non-cavitating condition provides at given flow coefficient the cascade cavitation performance. Large time steps are imposed during that slow transient, to minimise the time dependant terms of the discretized equation and to obtain quasi-steady flow fields. Some numerical instability is observed when the cavitating flow is fundamentally unstable. That is more particularly the case when reaching the final head drop: the flow rate is imposed strictly at the inlet and the cavitation blockage is thus especially abrupt. When σ values decrease, cavitation performance curve shows (Figure 10.) a first drop and the increase of the head just before the final blockage. Observation of the velocity fields shows that the phenomenon is due the interaction between the attached sheet cavitation and the boundary layer in the blade-to-blade channels.

Multiple-channels computations

Unsteady computations are performed on the 4 blades mesh. The periodicity condition is then applied between the 4 th and 1 st channels, as is the real runner. The time step, mesh and turbulence model are chosen to put the attention on the low frequency fluctuations of the attached cavity, more than to the local unsteadiness in the cavitation sheet wake (cloud shedding): the time step if fixed equal to 1% of the blade passage time t ref .

Successive time accurate computations are performed at fixed cavitation number and flow rate coefficient. First cases with quite high cavitation number (σ ≈ 0.1) lead to stable and symmetrical small cavities attached on each blade (Figure 11a.), their shape is then exactly the same than in the single-channel computation. Figure 11b shows the flow rates passing the different channels, there are constant at a value around 25% of the total flow rate (the initial flow rate transient correspond to the growing of the 4 attached cavities from the non cavitating initial condition). When the cavity length is increased, an unsteady configuration spontaneously appears. Figure 12 shows that the cavities are then different on the successive blades. The unbalanced cavitating structure propagates from blade to blade in time. Figure 13 shows the repartition of the flow rate in the cascade. After the initial transient, periodic fluctuations take place with different phases in the four channels. Their amplitude increases with the decreasing cavitation number. Visual observation (Figure 12) and a FFT analysis of cavity length fluctuations signal show that in stator frame the phenomenon is about 50% faster than the inducer rotation speed. As observed in the experiments, this value slightly decreases with cavitation parameter. However, the numerical frequencies of that super synchronous phenomenon remain about 25% larger than experimental values, as already observed in analytical models [START_REF] Tsujimoto | A Theoretical Analysis of Rotating Cavitation Inducers[END_REF][START_REF] Cambridge | Analytical Modelling of Instabilities in a Cavitating Turbopump Inducer[END_REF].

For a lower cavitation parameter a stable configuration appears (Figure 14), with alternate long and short cavities on the blades and different flow rates in the successive channels. Such a configuration is observed experimentally but at a σ higher than the super synchronous range.

Figure 15 shows the cavitating performance computed by averaging the head coefficient obtained with the 4 channels simulations at fixed cavitation number and flow rate coefficient (points ∆). In the non-symmetric range, the head coefficient is then larger than the one obtained from the single-channel computation. On the other hand, the final head drop is reached at higher σ, because of the lack of symmetry.

Conclusion: comparison between experimental and numerical results

Computations were performed at different operating conditions. Numerical results allow to draw the cavitating performance of the blade cascade at different flow rates and to identify the limit of the rotating cavitation behaviour.

In the different cases, the final head drop is predicted by the model at a σ about 0.02 too high with respect to the experimental value (0.04 instead of 0.02). The cavitation parameter range of rotating cavitation ∆σ = (σ + -σ-);, increases when the flow rate coefficient decreases (Figure 16 and Table 1). That result agrees with experimental observations [START_REF] Pagnier P | Conception and experimental study of an inducer[END_REF][START_REF] Yokata | A Study on Swirling Backflow and Vortex Structures at the inlet of Inducer[END_REF]. At reference flow rate, φ ref the experimental instability range is about ∆σ exp = 0.025 (Figure 1). Numerical results of ∆σ are plotted with respect to the flow rate coefficient in

  .): on the blade (b), or at peripheral zone (p).

Figure 2 .

 2 Figure 2. Axial view of an inducer Figure 3. Radial view obtained with light sheet illumination of an inducer radius (AB).

•

  Tc-tip clearance cavitation: it is defined as the cavitation in the tip clearance (between the tip surface of the blade and the casing); it is in fact a form of blade cavitation on the tip surface and it depends on the geometry of the tip[START_REF] Laborde R | Tip Clearance Cavitation in an Axial Flow Pump[END_REF].• Tv -tip vortex cavitation: the tip leakage flow induces a shear layer that rolls up to form the tip vortex, which is similar to such of a wing. The cavitation in an inducer first appears in the tip vortex (at cavitation inception) • T sl -tip shear layer cavitation: Between blade and tip vortex, there is a shear layer formed by the tip leakage flow which is turbulent, and could cavitate.

Figure 4 .

 4 Figure 4. Scheme of cavitation in inducer: Bf-backflow vortex cavitation, Tv -tip vortex cavitation, T sl -tip shear layer cavitation, Tc-tip clearance cavitation, in inducer tip.

Figure 5b .

 5b Photoelectric cell for synchronisation

Figure 7 .

 7 Figure 7. Evolution of flow sections, Scav, versus cavitation parameter.

Figure 8 .

 8 Figure 8. Evolution of flow sections, Scav and Sback, versus flow rate (Q/Q 0 ),This behaviour is similar to the evolution of the backflow section (Sback mentioned in Figure5b.) with the flow rate (Figure8.), which has been highlighted in Russian inducers studies[START_REF] Petrov | Cavitation Characteristics of High-Rotational Centrifugal Pumps with Inducers[END_REF]. This result leads to conclude that cavitation appears in backflow zone without strongly modifying its expansion. LDA measurements performed in CREMHyG laboratory[START_REF] Von | Three dimensional partial cavitating flow in a rocket turbopump inducer: numerical predictions compared with laser velocimetry measurements[END_REF], cf. point • in Figure8, confirm that peripheral cavitation area, Scav, deduced from visualization coincides with backflow area in non-cavitating conditions.In conclusion, the obstruction of the flow section by the cavitating backflow area modifies the inlet axial velocity Cm with respect to the uniform value Q/Sflow. That phenomenon must be kept in mind to analyse the results of 2D numerical simulations.

Figure 9 .

 9 Figure 9. Description of the blade cascade obtained from the inducer geometry, and the 22800 nodes mesh.

Figure 10 .Figure 11 .

 1011 Figure 10. Single channel computation: cavitation characteristic of the cascade at φ = φ ref , and associated extension of the attached cavity (ratio 5:1 between horizontal and vertical scales).

Figure 13 .Figure 14 .

 1314 Figure 13. Flow rate repartition in the 4 channels, φ = φ ref ; a) σ ≈ 0.097 ; b) σ ≈ 0.086.

Table 1 :

 1 the comparison shows that experimental range correspond to the numerical result at φ/φ ref between 1.05 and 1.1. That result agrees qualitatively with the previous analysis that the obstruction due to the cavitating backflow modifies the cascade inlet condition. Further work is now needed to assess the prediction capability of the model. Therefore, improvements of the analysis of 3D inlet flows and study of the influence of the cascade design by numerical simulations are in progress, in parallel to the development of a full 3D model[START_REF] Coutier-Delgosha O | Numerical study of the effect of the leading edge shape on cavitation around inducer blade sections[END_REF].Figure16. Effet of the flow rate on the cavitation characteristic and on the instability range (comparison with the experimental instability range in grey)

	φ/φ ref	∆σ
	0.90	0.099
	0.95	0.072
	1.0	0.046
	1.05	0.033
	1.1	0.018
	1.2	0

Table 1 :

 1 Instability range versus flow coefficient.

Acknowledgments

The authors wish to thank SNECMA Moteurs (Rocket Engine Division) and the French space agency CNES for their support to the present work. The CREMHyG laboratory -Grenoble, France -provided the experimental results.


Regiane Fortes Patella

Guillaume Challier

Jean-Luc L Reboud

Antoine Archer

R Fortes-Patella

Cavitation erosion mechanism: numerical simulations of the interaction between pressure waves and solid boundaries

NOMENCLATURE

V pit : pit volume (m 3 ) β material : mechanical characteristic coefficient (J/mm 3

Introduction

Many works have been developed last years in order to gain better knowledge concerning cavitation erosion mechanism [START_REF] Franc | From Pressure Pulses Measurements to Mass Loss Prediction: The Analysis of a Method[END_REF][START_REF] Lecoffre | Cavitation Erosion, Hydrodynamics Scaling Laws, Practical Method of Long Term Damage Prediction[END_REF][START_REF] Kato | Possibility of Quantitative Prediction of Cavitation Erosion Without Model Test[END_REF][START_REF] Fortes-Patella | A New Approach to Evaluate the Cavitation Erosion Power[END_REF][START_REF] Pereira | Prediction of Cavitation Erosion: an Energy Approach[END_REF][START_REF] Boehm | Investigations on Possibilities to Control de Erosive Cavitation Aggressiveness by Hydrodynamic Effects[END_REF]. In the industrial point of view concerning both machine design and maintenance, the evaluation of the erosion power of cavitating flows and the prediction of the material damage remains a major concern for machinery manufacturers and users. According to previous works [START_REF] Fortes-Patella | A New Approach to Evaluate the Cavitation Erosion Power[END_REF][START_REF] Fortes-Patella | Energetical Approach and Impact Efficiency in Cavitation Erosion[END_REF], pressure waves emitted during the collapses of vapor structures seem to be the main factor contributing to cavitation damage. The emission of the pressure waves can be generated either by spherical bubble or vortex collapses (as observed by [START_REF] Vogel | Optical and Acoustic Investigations of the Dynamics of Laser-Produced Cavitation Bubbles Near a Solid Boundary[END_REF] and [START_REF] Ward | The Energies and Pressures of Acoustics Transients associated with Optical Cavitation in Water[END_REF]) as well as by micro jet formation [START_REF] Philipp | Single Bubble Erosion on a Solid Surface[END_REF]. The emitted pressure waves interact with neighboring solid surfaces, leading to material damage. Previous work [START_REF] Fortes-Patella | Energetical Approach and Impact Efficiency in Cavitation Erosion[END_REF] indicated that the damage of material due to cavitation phenomena could be related to the characteristics (mainly the energy) of the pressure wave emitted by vapor structures collapses.

In this context, the present work aims to simulate and to analyze the emission of pressure waves during bubble collapses and their interaction with neighboring solid surfaces.

The first part of the paper (section 2) presents a theoretical and numerical analysis of pressure waves emitted during spherical bubble collapses. The Keller's model [START_REF] Prosperetti | Bubble Dynamics in a Compressible Liquid. Part I: First Order Theory[END_REF] is applied to simulate numerically the dynamic behavior of cavitation bubbles in water. The characteristics of the pressure wave emitted during the bubble implosion (such as signal shape, wave passage time, wave amplitude and emitted energy) were calculated as a function of hydrodynamic conditions (i.e., surrounding pressure, air contents and bubble radius) [START_REF] Fortes-Patella | Study of Pressure Wave Emitted During Spherical Bubble Collapse[END_REF].

The second part of the paper (section 3) concerns the material damage due to the pressure wave impact. By associating the proposed pressure wave model with an elastoplastic solid code developed by [START_REF] Reboud | Impact Response of an Elasto-plastic Medium[END_REF], it was possible to build a 2D axisymmetric coupled fluid-structure model to study the interaction between the pressure wave and the material surface. The section 3 presents numerical results obtained concerning mainly the permanent surface deformation of the solid surface and the energy balance between pressure wave phenomenon and material damage. The proposed energetical approach is applied in the case of different hydrodynamic conditions (i.e., pressure wave characteristics) and solid properties (aluminum, copper, stainless steel).

The main application of our study, presented in section 4, is related to the evaluation of cavitation aggressiveness.

Based on the Knapp's approach [START_REF] Knapp | Recent Investigations of the Mechanics of Cavitation and Cavitation Damage[END_REF], which proposed to use the material as a sensor to evaluate the cavitation erosion power, we analyzed pitted samples exposed to cavitating flows. A methodology to evaluate and to correct the volume damage rates obtained by pitting analysis is also introduced to rectify the influence of test duration and of the cut-off parameters [START_REF] Fortes Patella | Cavitation Mark Measurements by 3D Laser Profilometry[END_REF]. Making use of results obtained by numerical simulation and by analyzing pitted samples, the pressure wave power density, which represents the flow cavitation aggressiveness, is defined and calculated for different flow velocities.

Pressure wave simulation a) Bubble collapse model

To simulate the spherical bubble collapses, we have applied the model proposed by [START_REF] Prosperetti | Bubble Dynamics in a Compressible Liquid. Part I: First Order Theory[END_REF], based on the Kellers's approach [START_REF] Keller | Damping of Underwater Explosion Bubble Oscillation[END_REF]. In this model, the following hypothesis are taken into account: the vapor/gas bubble is and remains spherical, the fluid is infinite, compressible and viscous; the pressure within the bubble is uniform; the non-condensable gas inside the bubble is inviscid and obeys the perfect-gas adiabatic law; the physical properties of liquid and gas are constant; heat conduction and thermal effects are not taken into account; the interaction between compressibility and viscosity, and the effect of gravity and gas diffusion are negligible. The equation of the motion, describing the bubble dynamic, is:

ρ         + + =         - +         - ∞ ∞ ∞ ∞ P R B 2 dt d R R 1 R 3 1 1 2 3 R R R 1 c c c c with ( ) R R R t P P p P g v B & µ σ 4 2 - - - + = ∞
where R is the bubble radius; C ∞ is the sound speed in the liquid (assumed constant according to the acoustical approximation); p v and P g are the vapor pressure and the gas pressure inside the bubble; µ is the liquid viscosity; σ is the surface tension; ρ is the liquid density and P ∞ is the pressure at infinity. The gas pressure inside the bubble is calculated considering adiabatic conditions:

g g P P R R =       0 3 0 γ
where P g0 and R 0 are respectively the initial gas pressure and bubble radius, and γ is the ratio of the specific heats (γ =1.4 for air).

b) Pressure signal

To simulate the time and spatial pressure distribution throughout the liquid during bubble collapse, the study considers:

• the Tait state equation given by [START_REF]Underwater Explosions[END_REF]:

        ∞ = + + ∞ ρ ρ n B P B P
where P is the liquid pressure, P ∞ the pressure at infinity, ρ the liquid density and ρ ∞ the density at infinity. B is 3010 atm and the index n is 7.15 for water.

• the Fujikawa and Akamatsu,'s model [START_REF] Fujikawa | Effects of Non-Equilibrium Condensation of Vapor on the Pressure Wave Produced by Collapse of a Bubble in Liquid[END_REF], which uses a first-order approximation of the velocity potential φ(r,t) for the liquid. The time and spatial pressure distribution in the liquid can be then calculated by the equation:

φ = - - +               ∞ 1 2 2 2 2 r R RR R R c R & && &
In the present work, the differential equations were numerically integrated by the 4 th order Runge-Kutta method. In order to validate the numerical code, we compared results obtained with ones available in the literature [START_REF] Fujikawa | Effects of Non-Equilibrium Condensation of Vapor on the Pressure Wave Produced by Collapse of a Bubble in Liquid[END_REF][START_REF] Isselin | Investigations of Material Damages Induced by an Isolated Vapor Bubble Created by Pulsed Laser[END_REF]. Results were presented in [START_REF] Fortes-Patella | Study of Pressure Wave Emitted During Spherical Bubble Collapse[END_REF].

Several simulations have been performed to study the pressure waves emitted during the bubble collapse. According to the numerical study, the emitted pressure signals can be represented by a non-dimensional distribution, as indicated by the Figure 1. Pressure waves are characterized by the maximum amplitude Pmax at a given radial distance "r" from the wave emission center, and by the wave passage time δt (given by the pressure signal width at p=Pmax/2). Dimensionless time is defined as t/δt.

Figure 1 illustrates also the comparison between the numerical and experimental results concerning the nondimensional shape of the pressure wave. Experimental work has been developed by [START_REF] Isselin | Investigations of Material Damages Induced by an Isolated Vapor Bubble Created by Pulsed Laser[END_REF] in the Laboratory LP3 of Marseille. The collapses of single bubbles generated by a laser pulse have been studied and the pressure signal emitted during bubble collapse was measured by a PVDF transducer, placed at 8.6 mm away from the center of the bubble. We have observed a good agreement between experiments and numerical results concerning the pressure wave nondimensional shape and pressure amplitudes. This study was detailed in [START_REF] Challier | Interaction Between Pressure Waves and Spherical Cavitation Bubbles : Discussions about cavitation erosion mechanism[END_REF].

c) Pressure wave energy

Pressure waves created by bubble collapses are also characterized by the emitted energy E wave , which can be evaluated by the acoustic energy approach [7-9]:

t C r 2 P max 2 4 dt p 2 C r 2 4 E wave δ ∞ ρ ςπ ≈ ∞ ρ π = ∫
where p is the transient pressure signal at a given radius "r", Pmax is the peak of the pressure signal at radius "r", C ∞ is the pressure wave celerity, ρ is the fluid density, and δt is the wave passage time. The factor ζ is related to the waveform of the pressure signal (for signal illustrated by Figure 1, ζ is equal to 1). This method was also used by [START_REF] Lovik | Basic and Applied Aspects of Scaling Cavitation Noise[END_REF] to determine the acoustic energy spectrum and the noise level due to bubble collapses.

Pressure wave/material coupling

To study the local aspects of cavitation erosion we performed a pressure wave-solid coupling code, which simulates the plastic deformation of materials subjected to the impact of spherical pressure wave emitted to a distance "L" to the solid surface during the bubble collapses [START_REF] Fortes-Patella | A New Approach to Evaluate the Cavitation Erosion Power[END_REF].

a) Solid code

The material's response to the pressure wave impact is calculated by a finite-element numerical code (SOLID Model), which involves a Lagrangian formulation in cylindrical co-ordinates [START_REF] Reboud | Impact Response of an Elasto-plastic Medium[END_REF]. It takes into account the laws of continuum mechanics (mass conservation and the fundamental law of the dynamics) and an elastoplastic constitutive equation. The material's behavior is described by four independent factors: Young's modulus E, Poisson's ratio ν, celerity of the t/δt=1 longitudinal wave C L (involving E, ν and the material density), and a yield limit for the simple shear stress S 0 . The code provides the complete transient evolution of strain, stress and energy fields within the material and mainly the permanent surface deformation resulting from pressure wave impact. One notes that the code does not take into account the mass loss, and it can describe only the incubation stage of cavitation erosion. The code considers symmetry conditions along the axis of revolution. The pressure spatial and temporal signal presented here above in the case of pressure waves emitted during bubble collapses are used by the SOLID code as the boundary conditions at the wetted surface of the material, as illustrated by Figure 2. Moreover, the other solid boundaries are considered to be non-reflecting with respect to the longitudinal and transversal stress waves.

c) Energetical approach

To improve our previous works concerning the energy balance between pressure wave energy and material damage [START_REF] Fortes-Patella | Energetical Approach and Impact Efficiency in Cavitation Erosion[END_REF], many numerical simulations have been done by considering different material properties and wave amplitudes. Three materials have been analyzed: aluminum, copper and stainless steel. Table 1 gives the mechanical characteristics used by the simulations.

According to the simulations, the solid damage represented by the pit volume V pit seems to be directly proportional to the pressure wave energy as indicated by Figure 4: E wave =β V pit We observe that the parameter β depends strongly on the solid properties but it is almost independent on the impact amplitude. It is worth noting that, for smaller impact energy, the elastic behavior of materials becomes more relevant, leading to some numerical instabilities concerning mainly solid deformation calculations. Figure 4 illustrates numerical results concerning the evaluation of the β parameter for the three materials:

β aluminum ~4 (±0.4) J/mm 3 , β copper ~20 (±3) J/mm 3 and β stainless steel ~30 (±2) J/mm 3 (1) . Indeed, the value of the coefficient β is strongly related to the characteristic of the materials, mainly to the value of the yield limit for simple shear stress S 0 . In order to evaluate the influence of this parameter on the evaluation of β, many simulations have been performed considering stainless steel properties (E, ν and C L ) for a range of 100MPa≤S 0 ≤450MPa. Results are illustrated by Figure 5. We can note that the β value increases when S 0 rises, according to a power law. The same behavior was observed in the case of aluminum and copper simulations.

S
It is worth noting that the numerical code uses material static properties. To improve quantitative analysis, more information is needed concerning the mechanical behavior of materials at very high strain rates and their dynamic properties.

The material used as a sensor

Based on the results presented here above concerning the energy balance between pressure wave and material damage, the aim of this section is to evaluate cavitation aggressiveness by analyzing pitted sample exposed to cavitating flows.

a) Experimental study

This study involved analyzing experimental results obtained by Electricité de France [START_REF] Simoneau | Transposition of Cavitation Marks on Different Hardness Metals[END_REF] in MODULAB Test Rig. Samples in several materials were damaged by water cavitating flow downstream a diaphragm, which generates jet cavitation (Figure 6). Tests were performed for a constant cavitation number σ c =0.7 and different water flow velocities (Table 2a). Water is demineralized and deaerated. The air content is controlled with an oxygenmeter and the oxygen rate is maintained less than 3 ppm.

b) Pitted sample analysis

The erosion of the sample surfaces was measured by a UBM16 laser profilometer device. According to the manufacturer, the accuracy for depth measurements is 0.06 µm. Pit radius measurements are made with 1 µm lateral resolution. Special software was developed to scan the surface of the pitted samples [START_REF] Fortes Patella | Cavitation Mark Measurements by 3D Laser Profilometry[END_REF]. This code gives the location and size (depth « H », radius « R 10% » and volume « V ») of each individual pit. It supplies also the volume damage rate « Vd » (i.e., the ratio between deformed volume, analyzed sample surface and test duration) for each analyzed sample. Generally, one assumes a linear behavior between total volume damage ΣV and test duration ∆T (i.e., Vd would be constant for different test durations). Indeed, experimental study [START_REF] Fortes Patella | Cavitation Mark Measurements by 3D Laser Profilometry[END_REF] seems to indicate dependence between Vd and ∆T for a sample surface ∆S constant. This phenomenon could be explained by the existence of superposed impacts, whose number increases with test duration ∆T. In order to perform an adequate evaluation of the volume damage rate, the measured value of Vd should be corrected to eliminate the influence of the test duration and of the analysis cut-off parameter (related to the measurement resolution threshold). Table 2b presents the values of the volume damage rate before (Vd) and after (Vd*) correction. Vd* represents the volume damage rate corresponding to a small reference time t*=1/f characteristic of the considered cavitation condition (flow velocity v, cloud shedding frequency f, …). The methods applied to evaluate Vd* are detailed in [START_REF] Fortes Patella | Cavitation Mark Measurements by 3D Laser Profilometry[END_REF] and [START_REF] Challier | Prévision de l'Erosion de Cavitation : Couplage Ecoulement et Matériau[END_REF].

c) Pressure wave power density

To analyze samples exposed to different flow hydrodynamic conditions, the pressure wave power density P wave /∆S obtained from experimental results is introduced:

( ) ( ) V * d S T V pit S T E wave S P wave β = ∆ ∆ ∑ β = ∆ ∆ ∑ = ∆
where ∆S is the analyzed sample surface (10 x 10 mm 2 ), and ∆T is the test duration (s). (ΣEwave) represents the total relevant wave energy detected by the sample surface during cavitating tests. Results concerning the evaluation of the pressure wave power density are illustrated by Figure 7. It appears that for a same cavitating flow, the wave power density is constant, independently on the sample materials: Thus, the pressure wave power density seems to well represent the cavitation aggressiveness of the flow and can be related to the flow hydrodynamic characteristics: it increases approximately as P wave /∆S~v 5 For tested stainless steel samples, S 0 ≈ 400 MPa and β ≈ 90 J/mm 3 .

Conclusion

The interaction between pressure wave emitted during vapor structures collapses and neighboring solid boundaries was analyzed by means of numerical calculations concerning spherical bubble implosions and material deformation. Pressure wave signals obtained by simulations appeared to be well represented by a universal non-dimensional distribution relating maximum pressure amplitude, wave energy and passage time.

The permanent surface deformation calculated for pressure wave impact on material surface seemed to be also well characterized by a non dimensional pit profile. According to the simulations, material damage (represented by pit volume) is directly related to the pressure wave energy via the mechanical coefficient β, characteristic of solid behavior. The analysis of pitted sampled exposed to cavitating flows associated with numerical calculations leaded to the evaluation of the pressure wave power density, characteristic of flow aggressiveness. Results indicated that the evaluation of the pressure wave power density and the mechanical coefficient β are a useful tool in the prediction of cavitation erosion concerning:

• the influence of solid properties on material damage • and the effect of flow velocity on cavitating flow aggressiveness.

Further works, requiring more experimental data, are in progress to:

-validate the proposed methodology to correct volume damage rate obtained by pitting analysis by rectifying the influence of test duration -evaluate mechanical behavior of materials at high strain rates and their dynamic properties -improve analysis concerning energy transfer between vapor structure, pressure wave emission and material damage -predict pressure wave power density from unsteady, two-phase, viscous numerical simulations of cavitating flows [START_REF] Reboud | Two-phase flow structure of cavitation : experiment and modelling of unsteady effects[END_REF].

  Liquid / Pressure wave emission: C ∞ : celerity of pressure wave (m/s) δt : wave passage time (s) E wave : pressure wave energy (J) L : distance emission center -solid boundary (m) P : peak pressure applied on solid surface (N/m 2 ) Pmax : peak of the transient pressure signal (N/m 2 ) P ∞ : initial fluid pressure (N/m 2 ) p : static pressure (N/m 2 ) r : radial distance from wave emission center (m) v : mean flow velocity (m/s) ρ : fluid density (kg/m 3 ) ρ ∞ : initial fluid density (kg/m 3 ) σ c : cavitation number σ c = (P downstream -P vapor )/P upstream • dots denote differentiation with respect to time

Figure 1 :

 1 Figure1: Non dimensional pressure signal obtained by numerical calculations and experiments[START_REF] Isselin | Investigations of Material Damages Induced by an Isolated Vapor Bubble Created by Pulsed Laser[END_REF].Dimensionless time is defined as t/δt.

Figure 2 : 2 Figure 3 :

 223 Figure 2: a) spatial pressure signal applied to the solid surface at seven different times. b) stress field (second invariant of the deviatoric part) in the material due to the pressure wave impact at two different times. Results are given in N/m 2 . The co-ordinates are cylindrical axisymmetric and only one meridian half-plane is presentedb) Material damageAccording to numerical simulations concerning different materials submitted to pressure wave impacts[START_REF] Fortes-Patella | A New Approach to Evaluate the Cavitation Erosion Power[END_REF], the material damage can be represented by a non-dimensional axisymmetric indentation, as illustrated in Figure3. The figure shows a good agreement between numerical non-dimensional pit profile and experimental ones observed on samples of different materials exposed to several cavitation conditions.

Figure 4 :

 4 Figure 4: The figure illustrates the relation between the pressure wave energy and pit volume obtained by numerical simulations concerning three materials. Range of the pressure wave characteristics adopted by simulations: 0.8 GPa ≤ P ≤ 4 GPa ; 10 ns ≤ δt ≤ 150 ns ; 10 µm ≤ L ≤ 100 µm.

Figure 5 :Figure 6 :

 56 Figure 5: Evaluation of β parameter as a function of the stress limit S 0 .

Figure 7 :

 7 Figure 7: Pressure wave power density (measured by three different materials) as a function of the flow velocity.For tested stainless steel samples, S 0 ≈ 400 MPa and β ≈ 90 J/mm 3 .

Table 1 :

 1 Material mechanical properties applied by numerical code. Calculations concerning stainless steel take into account the material strain hardening.

	Aluminum	0 (MPa) 100	E (GPa) 50	C L (m/s) 5000	ν ν 0.30
	Copper		200	120		4700	0.33
	S. Steel		200	200		5800	0.30
			5.E-03
			4.E-03			copper
		Ewave (J)	2.E-03 3.E-03			Alu S. Steel Copper Alu
			1.E-03			S. Steel
			0.E+00
			0.E+00	1.E-13	2.E-13	3.E-13
				Vpit (m3)

Table 2 :

 2 as illustrated by Figure7. Experimental tests realized in MODULAB for aluminum, copper and stainless steel samples (∆S=100mm 2 ).

	v (m/s)	∆ ∆T (s)	Vd (µ µm 3 /mm 2 /s)	Vd* (µ µm 3 /mm 2 /s)
	20	Aluminum: 120	871	1023
		Copper: 300	95.5	140
		S. Steel: 7200	3.5	69
	25	Aluminum: 60	2640	3087
		Copper: 120	414	560
		S. Steel: 3600	8.5	161
	32	Aluminum: 30	9030	10603
		Copper: 60	1110	1513
		S. Steel: 1800	28	553
	38.5	Aluminum: 21	19700	23912
		Copper: 30	3840	5042
		S. Steel: 900	59	1006
	a) Tests conditions		b) Volume damage rate

& & & & & &

Standard deviations are presented in parenthesis.

Acknowledgements

This study is supported by EDF-Electricité de France (R&D Division) and by a doctoral grant from the Education French Ministry MERT.


H Lohrberg

B Stoffel

R Fortes-Patella

J L Reboud

NUMERICAL AND EXPERIMENTAL INVESTIGATIONS ON THE CAVITATING FLOW IN A CASCADE OF HYDROFOILS

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 self-oscillation of cloud cavitation. A detailed comparison of the results allows proposing an interpretation of the interaction mechanisms.

Nomenclature

Introduction

In the scope of the European Research Program PROCOPE, researchers of the LEGI (Laboratoire des Ecoulements Géophysiques et Industriels de Grenoble), and of the TUD (Laboratory for Turbomachinery and Fluid Power of Darmstadt) 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 nose were performed. Previous results concerning the self-oscillating behaviour of cloud cavitation were presented in [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation: part 1 visualisation / part 2 dynamic pressures[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.

Experiments on the 2D-hydrofoils cascade were carried out at Darmstadt University of Technology 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 last 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 by using barotropic state law, mass transfer equation or bubble dynamic model. Besides applications of different models to quasi-steady cavitating flow, some studies focused on unsteady cloud cavitation phenomenon in Venturi type ducts [Delannoy andKueny, 1990, Reboud et al, 1998] or 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]. Numerical simulation of the stabilising effect of small obstacle placed in the cavitation sheet area [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation: part 1 visualisation / part 2 dynamic pressures[END_REF]], shows the ability of such 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, proc[END_REF]], allows to predict instabilities due to presence of sheet cavitation on the different blades, 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 the hydrofoils cascade geometry. Comparison of experimental and numerical results are performed to analyse the interaction between the cavitating flows in the different channels.

Experimental Set-up

Object of the investigations is a cascade of three hydrofoils, located in a test section with a rectangular cross section of 30 by 100 millimetres (Figure 1). The hydrofoils can be adjusted to different angles of attack in a range of ±10° according to different flow rates in a centrifugal pump. The flow can be illuminated and observed through a lateral window. Illumination war done with a stroboscopic light source, images were taken with a CCD-camera (PCO sensicam). A piezoresistive pressure transducer (Kulite XTM-190M) with a frequency range up to 125 kHz was mounted in the window too. The membrane is connected to the flow by a small, short opening. Geometry of the hydrofoils and sensor position are shown in figure 1b. The sensor is mounted above profile 1, the positions above profile 2 and 3 are used for post-processing of the numerical results. The test section is part of a closed hydraulic loop. Flow rate and static pressure can be varied in order to achieve different cavitation conditions. In the case presented hereafter, the hydrofoils are held at 5 degree angle of attack, and the inlet velocity Vref is 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 a personal computer. Every image is then related to a part of the pressure signal.

Signal treatment

First the power density spectra of the signals were calculated. They 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 signals. First the signals were split at 100 Hz into high and low frequent parts by high and low pass filtering respectively. The auto-correlation function was then calculated form both parts. The function of a fully periodic signal would show several peaks 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 variation of the frequency 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 center is the cosine of this phase difference. The frequency variation can therefore be calculated from the peak value at cycle duration. The both frequencies and their variation are shown as a function of the cavitation number σ in Figure 2. This fluctuation was extracted by filtering the original signal. The filter design based on the auto-correlation function of the signal. The mean cycle frequency plus / minus the frequency variation are taken as the edge frequencies of a band pass filter. Filtering was carried out by Fourier transformation, deleting of the parts with frequencies outside the band and inverse Fourier transformation. By this, no further delay was added to the signal. After filtering the signals were undertaken a Hilbert transformation. This transformation adds an imaginary part to the signal by shifting the Fourier coefficients by an angle of π/2 and performing the inverse Fourier transformation. The phase information can be extracted from the complex signal (Figure 3). The nonconsecutively acquired images can then be sorted into phase angle consecutive order.

With the phase information available for every image, there are two further treatments possible both used to explain the cavitation behaviour and compare the experimental and the numerical results respectively. First, the images were put together to an exemplary sequence of one cavitation cycle (Figure 4). The cycle was divided into 25 steps. On the basis of the phase angle, every image was assigned to one step. With a total image number of one thousand, every 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 by absolute value and sign thus keeping the image signals energy in every wavelet form. The inverse transformation was applied on the averaged coefficients. The second treatment extracted the mean brightness of three regions of interest, 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 mean grey values are normalised by mean value and standard deviation. The first harmonic amplitude of the brightness is calculated. The harmonic functions of each channel are shown in Figure 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 vapour variation is related to the low fluctuation cycle. The larger part is related to cloud shedding and noise.

Numerical model

To model and to analyse the unsteady behaviour of the cavitating flow in the tested hydrofoils cascade, the unsteady cavitating flow model developed at LEGI (Grenoble) [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling, Cavitation and Multiphase Flow Forum[END_REF][START_REF] Reboud | Two-phase flow modelling of unsteady cavitation[END_REF][START_REF] Reboud | Two-phase flow structure of cavitation : experiment and modelling of unsteady effects[END_REF]] was applied. The theoretical and numerical model was developed to describe the unsteady behaviour of cavitation, including vapour clouds shedding. In that model, the liquid-vapour two-phase flow is described by homogeneous flow model with a no-slip condition between liquid and vapour phases. The liquid-vapour mixture, with a void ratio α, is considered as a homogeneous medium and described as a single fluid, which varying specific mass is ρ = α ρ v + (1α) ρ l . A simple description of the vaporisation and condensation phenomena is proposed by introducing an empirical barotropic state law ρ(p) for the mixture. A smooth law was chosen, ρ rapidly varying between liquid density ρ l and vapour density ρ v when the local static pressure P is around the vapour pressure Pv. The law is characterised by its maximum slope at P = Pv, 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: part 1 visualisation / part 2 dynamic pressures[END_REF]] at 10% of inlet velocity Vref. 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 co-ordinates on a staggered mesh. An iterative resolution based on the SIMPLE algorithm was developed to deal with incompressible flow (α=0 and α = 1) and highly compressible flow (0<α<1) [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling, Cavitation and Multiphase Flow Forum[END_REF]. The liquid-vapour interfaces are described by high gradients of density, which was made possible by the use of 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-dependant elliptic problem a first order fully implicit method is used. Turbulent flows are calculated by solving the Reynolds equations using a k-ε RNG turbulence model with law of the wall. The mass flow rate is imposed constant as 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, to the value required to assure the expected cavitation number σ. Phase change then occurs spontaneously in the regions where the pressure decreases under the vapour pressure Pv. Because the experimental value of σ is based on an upstream pressure measurement as reference pressure Pref, 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 [Reboud andDelannoy, 1994, Hofmann et al., 1999]. Empirical reduction of turbulence dissipative terms in the two-phase regions was proposed, to allow the simulation the unsteady behaviour of cavitation observed experimentally: dissipation in the pure liquid phase remains provided by the classical k-ε RNG model, while stronger shear flows are allowed in the two phase regions by arbitrary reduction of the turbulence model constant C µ (multiplication by a function f(ρ), equal to 1 for ρ = ρ l and rapidly decreasing towards µ v /µ l when ρ < ρ l ) [START_REF] Reboud | Two-phase flow structure of cavitation : experiment and modelling of unsteady effects[END_REF]]. Effects 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 hereafter, the static pressures was decreased to the experimental value σ = 1.9. Vapour structures, characterised 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 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 blade, but with large amplitude and low frequency fluctuations of length. The observed low frequency behaviour shows a period of 33ms, which is a frequency of 30 Hz with an uncertainty lower than 5%. Frequency of cloud shedding process in the upper channel (named hereafter channel 1) can be estimated for the largest structures to 180 Hz +/-10%. Based on the corresponding maximum attached cavity length, the Strouhal number is found to be about 0.27 +/-10%. The two values present a quite good agreement with the experimental values obtained from Figure 2 at σ = 1.95 (respectively about .

The difference between the cavitation behaviour in channel 1 and the others is associated to the larger 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 cavities with low relative thickness develop on the two other foils (channels 2 and 3). It can be seen in Figure 6, that the fluctuations of the cavity extend (as cloud cavitation in channel 1 or attached sheet cavitation in channels 2 and 3) appear completely out of phase. Disappearance of the cavitation structures in the upper channel correspond to a maximum development of the cavitation sheets on the two other foils, and vice versa. Transient pressure coefficient distributions corresponding to three locations (Figure 1) are given in Figure 7, while two pressure fields are given at two different times of the low frequency cycle (Figure 8). Larger development of cloud cavitation in channel 1 is associated with high-pressure fluctuations. The amplitude is higher in the first channel, because of the travelling clouds passing the point corresponding to the transducer location, but pressure fluctuations can also be clearly noticed in the two other channels. Contrarily, the phases during which cavitation is more developed as stable sheets on the two other foils show a complete stabilisation of the pressure signal. Figure 9 shows the time variation of the vapour volume computed in the three channels. The out of phase behaviour between channel 1 and the two others is clearly visible. A small phase shift between the channels 2 and 3 may be noticed too. Those behaviours are in total agreement with the analysis of Figure 5

  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.

Figure 2 .Figure 3 .

 23 Figure 2. Main frequency of the low and high pass filtered pressure signal as a function of the cavitation number σ, Vref = 14 m/s.The higher frequencies correspond to the cavitation self-oscillation behaviour, characterised by 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 to the phase angle of the low frequency fluctuation.

Figure 4 .Figure 5

 45 Figure 4. Exemplary frames from a sequence of phase angle averaged images, α = 5°, σ = 1.9, Vref = 14 m/s.

  , showing the phase shifts between the brightness fluctuations in the different channels. Despite the observed different Density field at t = 160 and 172 ms (=45 and 48 tref) minimum value of density (ρ/ρ l ) in cross sections of the channels. Variation with time (Tref = Lref/Vref = 3.6 ms) of the extend of the cavitation structures in the channels.

Figure 6 .Figure 7 .

 67 Figure 6. Time evolution of the cavitation structures on the three hydrofoils (Vref = 14m/s, σ = 1.95)

Figure 8 .Figure 9 .

 89 Figure 8. Pressure distribution : a) t = 160 and b) 172 ms.

Figure 10b .

 10b Figure 10a. Angle of velocity vector, 5% upstream of the upper foil leading edge.

Figure 11 .

 11 Figure 11. Visualisation of the flow vorticity at t = 143, 153, 164 and 174 ms.

Acknowledgements

This work is part of a European exchange program 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