O Coutier-Delgosha

R Fortes-Patella

J L Reboud
email: reboud@enise.fr

3D NUMERICAL SIMULATION OF PUMP CAVITATING BEHAVIOR

The quasi-steady cavitating behavior of three pumps was investigated by 3D unsteady viscous computations. The numerical model is based on the commercial code FINE/TURBO TM , which was adapted to take into account the cavitation phenomenon. The resolution resorts to a timemarching algorithm initially devoted to compressible flows. A low-speed preconditioner is applied to treat low Mach number flows. The vaporization and condensation processes are controlled by a barotropic state law that links the void ratio evolution to the pressure variations.

A radial pump, a centrifugal pump, and a turbopump inducer were calculated and the cavitating behaviors obtained by the computations were compared to experimental measurements and visualizations. A reliable agreement is obtained for the two pumps concerning both the head drop charts and the extension of the vapor structures. A qualitative good agreement with experiments is also observed in the case of the turbopump inducer. The accuracy of the numerical model is discussed for the three geometries.

These simulations are a first attempt to simulate the complete 3D cavitating flows in turbomachinery. Results are promising, since the quasi-steady behaviors of the pumps in cavitating condition are found quantitatively close to the experimental ones. A continuing effort is pursued to improve the prediction accuracy, and to simulate unsteady effects observed in experiments, as, for example, rotating cavitation.

INTRODUCTION

To increase the performance of industrial hydraulic components, pumps should operate with high rotation speed and low inlet pressure. It results in cavitating conditions that can lead to severe disturbances, such as substantial performance losses, strong unsteady forces acting on the pump components, noise and erosion.

In rocket engine turbopumps, for example, the inducer stage is designed to allow a stable cavitation behavior at the lowest available inlet pressure. In centrifugal pumps, strong pressure gradients in the cavity wakes and associated unsteady cavitation behavior are involved in erosion phenomena, whose intensity directly acts on the machine life duration.

The general understanding of the unsteady behavior of cavitation is therefore of primary importance for the design and operation purposes. More particularly, the investigation of mechanisms that are responsible for a pump elevation drop in cavitating conditions is a necessary step to improve inducers design. These considerations have motivated the present study, which focuses on the quasi-steady effects of cavitation on the behavior of two centrifugal pumps and a turbopump inducer.

A 3D numerical model of unsteady cavitation is proposed with the final objective to predict cavitating flows in rocket engine turbopump inducers. Cavitation phenomenon was simulated by strong variations of the density of a single fluid. That model was implemented in the commercial code FINE/TURBO TM developed by NUMECA International. The numerical code solves the RANS equations, associated with a barotropic state law managing the fluid variable density. Models based on that simple law gave many promising results in 2D simulations of steady and unsteady cavitating flows ( [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF] to [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation: part 1 visualisation / part 2 dynamic pressures[END_REF]). Computations were applied in previous works to Venturi type sections designed to simulate an inducer blade suction side, and to inducer blade sections placed in a cavitation tunnel: the proposed model seemed to well describe the cavitation behavior in 2D geometries [START_REF] Coutier-Delgosha | Numerical simulation of cavitating flow in an inducer geometry[END_REF], [START_REF] Coutier-Delgosha | numerical study of the effect of the leading edge shape on cavitation around inducer blade sections[END_REF]. A first application to a centrifugal pump geometry also gave promising results [START_REF] Hofmann | Experimental and numerical studies on a centrifugal pump with 2D curved blades in cavitating conditions[END_REF].

The model is applied in the present paper to calculate 3D unsteady cavitating flows in a radial pump, in a centrifugal pump and in a rocket engine turbopump inducer. The influence of quasi-steady cavitation behavior on the pump characteristics and on the final head drop is simulated by transient calculations. Experimental and numerical results concerning pumps characteristics and performance breakdown are compared at different flow rates. Appearing types of cavitation and the spatial distribution of vapor structures inside the impellers are also analyzed for different operating points at nominal flow rate.

The investigation of these three different pump behaviors enhances the description of the performance of the numerical model, and points out its limitations at the present time. : τ/τ 0 τ 0 : reference cavitation parameter -

NOMENCLATURE

PHYSICAL AND NUMERICAL MODEL

The mean features of the applied physical and numerical models, as well as the calculation procedure are summarized in the present paper. More details are given by Coutier-Delgosha et al. [START_REF] Coutier-Delgosha | Numerical simulation of cavitating flow in an inducer geometry[END_REF].

1) Physical model

Cavitating flows are described by a single fluid model, based on previous numerical and physical work developed in LEGI by Delannoy and Kueny [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF], Reboud et al. [4]. This fluid is characterized by a density ρ that varies in the computational domain: when the density in a cell equals the liquid one (ρ l ), the whole cell is occupied by liquid, and if it equals the vapor one (ρ 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 α = (ρ -ρ l ) / (ρ v -ρ l ) 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. The barotropic law links the density to the local static pressure ρ(P). When the pressure is higher or lower than vapor pressure, the fluid is supposed to be purely liquid or purely vapor, according to the incompressible state law ρ = ρ l or ρ = ρ v . The two fluid states are joined smoothly in the vaporpressure neighborhood. It results in the evolution law presented in Figure 1, characterized mainly by its maximum slope 1/c min 2 , where c 2 = ∂P/∂ρ. c min can thus be interpreted as the minimum speed of sound in the mixture.

2) Numerical model

The numerical model of cavitating flows based on this physical description is an extension of the commercial code FINE/TURBO TM developed by NUMECA International. FINE/TURBO TM solves the time dependant Reynolds-Averaged Navier-Stokes equations on three-dimensional structured meshes. 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 low-compressible or incompressible flows by introducing a preconditioning matrix [START_REF] Hakimi | Preconditioning methods for time dependent Navier-Stokes equations[END_REF].

The discretization is based on a finite volume approach. Convection terms are treated by a second order central scheme associated with artificial 2 nd order and 4 th order dissipation terms. 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. 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 by Coutier-Delgosha et al. [START_REF] Coutier-Delgosha | Numerical simulation of cavitating flow in an inducer geometry[END_REF]. The key point of this adaptation is the modification of the fluid state law. 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 5) and is almost incompressible in the pure vapor 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, cavitation consists in a very sharp and very rapid process. The density variations in time and space are smoothed to avoid numerical instabilities. A theoretical study of the influence of the preconditionner on the stability in the two-phase regions was performed. An adaptation of the preconditionning coefficients was found to be effective in 2D configurations, and work is in progress to apply it to calculations of rotating machineries.

We use for the simulations presented in this paper a Baldwin-Lomax turbulence model. More calculations considering κ-ε turbulence models are in progress to improve physical analyses. An investigation of the respective efficiency of different turbulence models associated with the single fluid modelization of cavitation is also proposed in [START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF].

3) Initial transient treatment

First of all, a steady step is carried out, with a pseudo vapor pressure low enough to ensure non-cavitating conditions in the whole computational domain. Then, the unsteady calculation is started: the NPSH is slowly reduced by increasing smoothly the pseudo vapor pressure at each new time step. Vapor structures spontaneously appear and grow during this process, in the regions of low static pressure. In the presented computations, the NPSH is decreased until the pump head drop, due to highly cavitating conditions, is reached. The physical time-step is high enough to only simulate the quasicstatic effects of the NPSH decrease. So the unsteady effects are not modeled in the computations presented hereafter. For a given pseudo vapor pressure value, the inlet pressure does not fluctuate so far the convergence is reached, and it is directly used to compute the NPSH values reported on the charts.

CALCULATION OF A 2D RADIAL PUMP

1) Geometry

This impeller geometry has been chosen by the Laboratory of Turbomachinery and Fluid Power of the Darmstadt University of Technology (TUD) [START_REF] Hofmann | Experimental and numerical studies on a centrifugal pump with 2D curved blades in cavitating conditions[END_REF] [START_REF] Hofmann | Ein Beitrag zur Verminderung des erosiven Potentials kavitierender Strömungen[END_REF]. Measuring techniques previously developed for a single hydrofoil in a test section were adapted to the pump [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation: part 1 visualisation / part 2 dynamic pressures[END_REF]. The impeller has 5 single-curved blades with two different radii at inlet and outlet. Hub and shroud are parallel to get an almost 2-dimensional blade-to-blade channel with constant width.

A radial-symmetric housing (Figure 2) is used to obtain almost constant conditions on the impeller outlet, so that comparable cavitation conditions are simulated in each channel, without any influence of a volute casing.

Nominal conditions are a rotational speed of 36 Hz and a flow-rate Q = 210 m 3 /h. Specific speed of the runner is n s = N Q 1/2 / H 3/4 = 20 (europ. value: N in rpm, Q in m 3 /s and H in m), the outer runner 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.

Figure 2: housing of the TUD pump

2) Mesh and boundary conditions

The model was applied to the presented radial pump geometry. A 300.000 cells multi-blocks mesh of one single blade-to-blade channel was used (Figure 3). Boundary conditions applied for the simulations are the following:

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 runner 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 runner outer radius, where a uniform static pressure is imposed. Periodic conditions are applied at the lateral boundaries of the blade-to-blade channels.

The c min value in the state law was calibrated by previous 2D studies reported in [START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modelling of unsteady effects[END_REF], and it is set to the reference value 2.25 m/s in this computation.

3) Calculation

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 visualizations at nominal point of operation. 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 caused by the local curvature of the streamlines is fully consistent with the observations reported by Hofmann et al. [START_REF] Hofmann | Experimental and numerical studies on a centrifugal pump with 2D curved blades in cavitating conditions[END_REF]. The experimental visualization is compared to the result of the computation to enhance this reliable agreement (Figure 4). The cavitation number was adjusted to give the same global size of the cavitation structures than in the experiment: the numerical result then corresponds to a NPSH about 10% lower than the experimental one (7 m, instead of 8 m). The three cavitation areas are correctly simulated by the code: attached cavity on the suction side (A), extension of cavitation on the shroud along the blade (B), and cavitating flow on the inlet radius of the shroud (C). In the computation, attached sheet cavity (A) and its extension on the shroud (B) belong to the same vapor structure, while they look like two separated regions in the experiment. A more global comparison between experiments and computations is presented in Figure 5, by drawing the head drop curves for three different flow rates: namely 0.8Qn, Qn and 1.08Qn. In the presented calculations, the performance drop is represented first by a smooth decrease of the pump head, as a function of the NPSH. The final head-drop is only partially simulated because the computation rapidly becomes unstable and stops (this is more particularly the case at 1.08Qn). Our upstream boundary condition consists in imposing in a strict manner the mass flow rate passing in 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 headdrop is less progressive than in the experiments.

Results obtained from this first simulation of the pump cavitation behavior are promising: the head drop is predicted with a good homogeneity with respect to the three flow rates. The NPSH values obtained for the 3% and 10% head drop are globally overestimated with respect to experimental values (of about 1m for the 10% head drop, and 1.5 to 3 m for the 3% head drop).

CALCULATION OF THE SHF PUMP

1) Geometry and mesh

The SHF pump is a shrouded seven blades centrifugal runner (figure 6). The scale and the calculation configuration are as close as possible to the experimental conditions reported by Combes and Archer [START_REF] Combes | Etude de la cavitation dans la pompe SHF[END_REF], in order to simplify comparisons in cavitating conditions. The outlet radius is 330 mm, the inlet pipe length is 200 mm, the rotation speed equals 3000 rpm, and the nominal flow rate is 0.157 m 3 /s. Specific speed of the runner is n s ≈ 35. The experimental outlet has been slightly simplified for the numerical simulations: a rotating vaneless diffuser is used at the outlet, and the fluid leakage on the back of the shroud is not taken into account. Both hub and shroud are rotating, axial velocities are imposed at the inlet, and a static pressure is fixed at the diffuser outlet (figure 7).

Figure 7: Calculation conditions (meridional view)

A two-block mesh is applied on a single blade-to-blade channel. The influence of the mesh size was evaluated by noncavitating steady computations performed with meshes made respectively of 75000 cells, 250000 cells, 500000 cells and 10 6 cells. The effect on the total pressure head and on velocity profiles through a blade-to-blade channel was found negligible in the three last cases, and a 250000 cells mesh is applied for the non-cavitating and cavitating computations presented in this paper. Figure 8 presents a 3D view showing principally the hub treatment. For this geometry, the head drops are more completely simulated than for the previous one. Numerical instabilities that occur in highly cavitating conditions were mastered, so that 10% to 15% of head decrease could be obtained before the computation divergence. This improvement was reached mainly by eliminating the initial transient effects associated with the start of the unsteady computation, which probably interfere with the cavitation process and generate during the head drop some spurious numerical instability. The increase of the second order artificial dissipation level in the two-phase regions also removed some difficulties.

Results obtained from cavitating calculation of the pump are promising: the elevation decrease due to cavitation development, as well as the final head drop, are correctly predicted for both flow rates. This reliable agreement concerning the head drop prediction confirms the ability of the numerical model, already observed in the case of the purely radial pump presented in the previous section, to predict the quasi-steady behavior of centrifugal pumps in cavitating conditions.

3) Flow field visualizations at nominal flow rate

The head drop chart H (NPSH) obtained by the calculation at nominal flow rate is drawn on Figure 10. Figure 11 illustrates the general development of cavitation in the pump. The hub is colored in grey, the blades are represented in dark red, and the two-phase areas appear in yellow (the surface drawn corresponds to a 5% void ratio). It can be observed that cavitation mainly consists in a cavity that grows on the blades suction side, while no vapor is present on the pressure side until the three last configurations.

Figure 11: Cavity evolution on the blade suction side Vaporized areas in yellow, hub in grey, blades in dark red

The suction side cavity grows progressively: the cavitation sheet first appears near the hub, where its extension remains always larger than around the shroud (Conf. 1 and 2); then, the cavity length increases slowly until it reaches the throat (conf. 2 to 4) and after that it almost stabilizes (conf. 5 and 6). In the same time, the detachment point of the cavitation sheet moves downstream on the blade.

Another cavity suddenly appears on the pressure side in configuration 4 (not visible here) in the vicinity of the shroud, and grows rapidly (conf. 5 and 6) inside the blade-to-blade channels. This phenomenon is strongly correlated with the final head drop reported in Figure 10.

This behavior is consistent with experimental observations, which indicate at nominal flow rate a two-step cavitation evolution: as a matter of fact, vapor first appears near the hub on the blades suction side. It progressively grows towards the throat, but it does not enter the blade-to-blade channels. This first step only slightly affects the pump elevation. The second step consists in a sudden development of cavitation on the pressure side, which fills the blade-to-blade channels and induces the final head drop.

The good qualitative agreement obtained with the numerical simulation at nominal flow rate has to be associated with the accurate prediction of the head drop chart.

A comparison between the results of the computation and experimental visualizations performed at EDF R&D Division is proposed in Figure 12. The numerical results are presented with the same viewpoint as in experiments. The order of the figures corresponds to a decrease of the NPSH, from the vapor onset to the pump head drop. The experimental views are extracted from video, which leads to an uncertainty on the experimental NPSH of ± 0.5 m.

A good agreement with experiments can be observed concerning the shape of the suction side cavitation sheet, during the whole NPSH decrease. The cavity length and its position in the radial direction are systematically well predicted by the numerical model.

The main discrepancy appears at the leading edge, where the calculation shows a progressive shift of the cavity detachment point, when the NPSH decreases. This behavior is not observed in the experimental visualizations.

1) Geometry, mesh, and boundary conditions

The third geometry is a turbopump inducer, i.e. a fourblade axial runner especially designed to operate in cavitating conditions (Figure 13).

Figure 13: Inducer geometry

A 500.000 cells multi-blocks mesh of one single blade-toblade channel was used (Figure 14a). Only one channel is treated, and periodicity conditions are applied to the frontiers with the adjacent channels. Non-matching cells are used along these periodic boundaries, to avoid an excessive skewness of the grid (Figure 14b). A special attention is also paid to the mesh in the expected cavitating areas: grid is refined in the streamwise direction just after the blade leading edge, on the suction side, and also in the tip area, in the three directions. Boundary conditions applied for the simulations are the following:

Velocity is imposed at the inlet of the suction pipe, about 2 diameters upstream from the runner, and the static pressure is imposed at the outlet. Laws of the wall are imposed along solid boundaries. The relative motion between the inlet pipe walls and the inducer is taken into account. The tip gap is also modelized. Periodic conditions are applied at the lateral boundaries of the blade-to-blade channels.

The calculation was performed in hydrogen. The minimum speed of sound c min in the two-phase mixture, which is necessary to determinate the maximum slope of the barotropic law, is calibrated to ensure the similarity between the calculations (in H 2 ) and the experiments (in water). The parameter c min /V ref is thus the same in both cases, which leads for the calculations to a value c min within the celerity range (30 m/s -100m/s) calculated with the Jakobsen's law [START_REF] Jakobsen | On the mechanism of head breakdown in cavitating inducers[END_REF]. Figure 16 shows the development of cavitation corresponding to the eight operating points indicated on Figure 15. It illustrates the apparition and the growing of vapor/liquid areas on both faces of the blades, and finally the progressive filling of blade-to-blade channels by the vapor. Blades are colored in grey, the hub in blue, and the external shape of the two-phase areas (corresponding to a 5% void ratio) is represented in yellow. It can be observed that vapor first appears in the vicinity of the shroud, near the leading edge. This phenomenon slightly affects the inducer performance, which starts to decrease, even if only a little volume of vapor is present in the flow field. Vapor remains localized near the shroud and in the tip gap (config. 1 to 4) until τ' reaches 0.3. This behavior is consistent with the smooth decrease of the inducer head observed at the same time.

The development of cavitation on the blade suction side (config. 5) directly induces a severe performance breakdown. This behavior can be explained by the important blockage generated immediately by the vapor in the upstream part of the blade-to-blade channels. It is a very sudden phenomenon (config. 6 and 7), which leads to the presence of vapor structures characterized by a low void ratio, from hub to shroud and from pressure side to suction side at the channel inlet. Experimental visualisations are not yet available concerning this geometry, but the cavitating areas upstream from the inducer are consistent with behaviors observed previously in this type of machine. Calculations are in progress to investigate the influence of the inlet and outlet boundary conditions on the result.

3) Comparisons with experimental performance charts Numerical head drop charts are compared to the experimental ones at three flow rates, namely 0.8 Qn, Qn and 1.2 Qn (Figure 17). Details on the experiments can be found in [START_REF] Coutier-Delgosha | Numerical simulation of turbopump inducer cavitating behavior[END_REF].

The cavitation parameter τ' corresponding to the performance breakdown is in all cases over-predicted by the model: the gap between experimental and numerical results equals ∆τ' = 0.1. This discrepancy must be considered circumspectly. In the case of the previous centrifugal pumps, the order of magnitude of the cavitation parameter at head drop is notably higher, so that the same absolute error results in a much lower relative error (about 20% for the first pump, and less than 5% for the second one). Since an inducer has much better suction capacities (τ' at the performance breakdown is about 0.1), the order of magnitude of τ' is, in the present case, the same than the prediction error ∆τ'. As a matter of fact, in the case of an inducer, the simulation does not allow at the present time a sufficient accuracy for the useful prediction of the cavitation performance. The prediction of the head drop could be notably improved in the future by reducing the aspect ratio of the mesh in the two-phase areas, and by analyzing the influence of the artificial dissipation on the simulation of the performance breakdown.

CONCLUSION

The cavitating behaviors of a radial pump, a centrifugal pump, and an axial turbopump inducer were investigated with 3D unsteady calculations performed at various flow rates. The study of the quasi-steady development of cavitation was performed by drawing the head drop charts and visualizing the vapor areas for several operating points. The numerical simulations were compared for the three pumps to experimental results.

A good agreement was obtained in the case of the pure radial and the centrifugal pumps: head drops predictions are consistent with experimental measurements in most of the cases, and the two-phase structures observed during experiments are similar to the ones predicted by the numerical model. A good qualitative agreement was also obtained in the case of the turbopump inducer, but the very good suction capacity of this machine leads to a poor relative accuracy concerning the head drop location. More details and numerical tests concerning calculations performed with the radial 2D pump, the SHF pump, and the inducer can be found respectively in [START_REF] Hofmann | Experimental and numerical studies on a centrifugal pump with 2D curved blades in cavitating conditions[END_REF], [START_REF] Coutier-Delgosha | Etude des Ecoulements Cavitants : modélisation des comportements Instationnaires et Application Tridimensionnelle aux Turbomachines[END_REF], and [START_REF] Coutier-Delgosha | Numerical simulation of turbopump inducer cavitating behavior[END_REF]. Work is in progress to improve the prediction for this type of machine, and to modelize the unsteady effects associated with cavitation, as already performed in 2D blade cascades [START_REF] Coutier-Delgosha | Numerical Simulation of the Unsteady Cavitating Behavior of an Inducer Blade Cascade[END_REF], [START_REF] Joussellin | Cavitating inducer instabilities: experimental analysis and 2d numerical simulation of unsteady flow in blade cascade[END_REF].

Figure 1 :

 1 Figure 1: The barotropic state law ρ ρ ρ ρ(P) for water.

Figure 3 :

 3 Figure 3: View of the mesh on shroud side of the pump. (The entire pump geometry is reconstructed by rotation of the single blade-to-blade channel)

Figure 4 :

 4 Figure 4: Vapor structures on suction side (experiment NPSH=8m, computation NPSH = 7m). Calculation: iso-density contour (ρ ρ ρ ρ ≈ ≈ ≈ ≈ 0.95ρ ρ ρ ρ l : void ratio >5%) drawn in yellow, shroud in blue, blade in grey

Figure 5 :

 5 Figure 5: Head-drop curves of the TUD pump Comparison at 0.8Qn, Qn and 1.08Qn

Figure 6

 6 Figure 6: SHF pump geometry

Figure 8 :

 8 Figure 8: Reference mesh on hub side of the pump 2) Head drop charts Figure 9 presents the head drop curves H (NPSH) for two different flow rates: namely Qn and 1.3Qn.

Figure 9 :

 9 Figure 9: Head drop charts at two flow rates

Figure 10 :6

 10 Figure 10: Head drop chart at nominal flow rate Green line = onset of vapor in the flow field Squares = operating points represented on Fig. 11, 12.

Figure 12 :

 12 Figure 12: Comparison experiment/calculation of the suction side cavity extension for several NPSH values SPATIAL TURBOPUMP INDUCER

Figure 14a :Figure 15 :

 14a15 Figure 14a: View of the mesh on hub side of the inducer (The entire inducer geometry is reconstructed by rotation of a single blade-to-blade channel)

Figure 16 :

 16 Figure 16: Development of the two-phase areas as the cavitation number decreases (corresponding to red squares reported on Figure 15)

Figure 17 :

 17 Figure 17: Comparisons between numerical and experimental head drop charts

ACKNOWLEDGMENTS

This research was supported by a doctoral grant from the Education French Ministry MERT and SNECMA Moteurs DMF. The study of the radial pump is a part of a European exchange program PROCOPE, with the research teams of Grenoble and Darmstadt as members. The authors wish also to express their gratitude to the French space agency CNES (Centre National d'Etudes Spatiales) for its continuous support. EDF R&D Division and the CREMHyG Laboratory are also thanked for the experimental results, respectively in the centrifugal pump and the inducer geometry cases.


R Fortes-Patella

J L Reboud
email: reboud@enise.fr

Evaluation of the Turbulence Model Influence on the Numerical Simulations of Unsteady Cavitation

Unsteady cavitation in a Venturi-type section was simulated by two-dimensional computations of viscous, compressible, and turbulent cavitating flows. The numerical model used an implicit finite volume scheme (based on the SIMPLE algorithm) to solve Reynolds-averaged Navier-Stokes equations, associated with a barotropic vapor/liquid state law that strongly links the density variations to the pressure evolution. To simulate turbulence effects on cavitating flows, four different models were implemented (standard k-RNG; modified k-RNG; kwith and without compressibility effects), and numerical results obtained were compared to experimental ones. The standard models k-RNG and k-without compressibility effects lead to a poor description of the self-oscillation behavior of the cavitating flow. To improve numerical simulations by taking into account the influence of the compressibility of the two-phase medium on turbulence, two other models were implemented in the numerical code: a modified k-model and the kmodel including compressibility effects. Results obtained concerning void ratio, velocity fields, and cavitation unsteady behavior were found in good agreement with experimental ones. The role of the compressibility effects on turbulent two-phase flow modeling was analyzed, and it seemed to be of primary importance in numerical simulations.

Introduction

Cavitating flows in turbomachinery lead to performance losses and modifications of the blades load. These consequences can be related to the quasi-steady cavitation effects, mainly depending on the time-averaged shapes of the vaporized structures. Moreover, turbomachinery under cavitating conditions is also submitted to transient phenomena, such as compressibility effects, flow rate fluctuations, noise, vibration, erosion, in which the unsteady behavior and the two-phase structure of the cavitating flow are influent. The study and modeling of the unsteady behavior of cavitation are thus essential to estimate the hydraulic unsteadiness in turbomachinery and the associated effects.

In this context, some numerical models taking into account the two-phase structure and the unsteady behavior of cavitation have been developed. Most of them are based on a single fluid approach: the relative motion between liquid and vapor phases is neglected and the liquid vapor mixture is treated as a homogeneous medium with variable density. The mixture density is related directly to the local void fraction and is managed either by a state law ͑Delannoy and Kueny ͓1͔, Song and He ͓2͔, and Merkle et al. ͓3͔͒, using a supplementary equation relating the void fraction to the dynamic evolution of bubble cluster ͑Kubota et al. ͓4͔, and Chen and Heister ͓5͔͒, or by a multiple species approach with a mass transfer law between liquid and vapor ͑Kunz et al. ͓6͔͒. The last model can be used in a full two-fluid approach, including relative motion between the phases ͑Alajbegovic et al. ͓7͔͒.

Main numerical difficulties are related to the strong coupling between the pressure field and the void ratio and to the coexistence of the strong compressibility of the two-phase medium with the quasi-incompressible behavior of the pure liquid flow. Moreover, the influence of the turbulence on unsteady two-phase compressible flows is not yet well known.

This paper presents unsteady cavitating flow simulations performed with a two-dimensional code. It is based on the model developed by Delannoy and Kueny ͓1͔ for inviscid fluids. Several physical modifications have been investigated by Reboud and Delannoy ͓8͔,Reboud et al. ͓9͔, to increase the range of applications and improve the physical modeling.

In the numerical code, the two-phase aspects of cavitation are treated by introducing a barotropic law that strongly links the fluid density to the pressure variations. From the numerical point of view, this physical approach is associated with a pressurecorrection scheme derived from the SIMPLE algorithm, slightly modified to take into account the cavitation process. The model has been validated on numerous cases, such as Venturi, ͓9,10͔, foils, ͓8,11͔, or blade cascades, ͓12,13͔. The numerical results showed a good agreement with experiments. Mainly, the code leads to a reliable simulation of the cyclic self-oscillation behavior of unsteady cavitating flows.

The complex unsteady mechanism that governs this cyclic cavitation behavior is strongly affected by the applied turbulence model, ͓9,10͔. The aim of this paper is to study the influence of different models ͑standard k-RNG; modified k-RNG; kincluding or not compressibility effects͒ on the numerical simulation of cavitating flows. Indeed, because of the barotropic state law adopted in the model, the vaporization and condensation processes correspond mainly to highly compressible flow areas. Therefore, it is of primary importance to take into account the compressibility of the vapor/liquid mixture in the turbulence model.

The paper describes the turbulence models applied in the numerical simulations and presents comparisons between different numerical results obtained in a two-dimensional Venturi-type section, whose experimental behavior has been already studied by Stutz and Rebouch ͓14,15͔. In this geometry, the flow is characterized by an unstable cavitation behavior, with almost periodical vapor cloud shedding.

In addition to the standard k-model, three other turbulence models were applied, to investigate their influence on the twophase turbulent flow. Mainly, the turbulence model proposed by Wilcox ͓16͔, which includes compressibility considerations, was applied, and promising results are obtained. Comparisons with experimental measurements of void ratio and velocities inside the cavity, ͓9͔, are presented.

The ability of these turbulence models to predict complex twophase flow is discussed, and the role of the compressibility effects is studied.

Physical Model

The present work applies a single fluid model based on previous numerical and physical studies, ͓1,8 -10͔. The fluid density varies in the computational domain according to a barotropic state law ( P) that links the density to the local static pressure ͑Fig. 1͒. When the pressure in a cell is higher than the neighborhood of the vapor pressure ( PϾ P v ϩ(⌬ P v /2)), the fluid is supposed to be purely liquid. The entire cell is occupied by liquid, and the density l is calculated by the Tait equation, ͓17͔. If the pressure is lower than the neighborhood of the vapor pressure ( PϽ P v Ϫ(⌬ P v /2)), the cell is full of vapor and the density v is given by the perfect gas law ͑isotherm approach͒. Between purely vapor and purely liquid states, the cell is occupied by a liquid/vapor mixture, which is considered as one single fluid, with the variable density . The density is directly related to the void fraction ␣ ϭ(Ϫ l )/( v Ϫ l ) corresponding to the local ratio of vapor contained in this mixture.

To model the mixture state, the barotropic law presents a smooth link in the vapor pressure neighborhood, in the interval Ϯ(⌬ P v /2). In direct relation with the range ⌬ P v , the law is 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. Its calibration was done in previous studies, ͓10͔. The optimal value was found to be independent of the hydrodynamic conditions, and is about 2 m/s for cold water, with P v ϭ0.023 bar, and corresponding to ⌬ P v Ϸ0.06 bar ͑green chart on Fig. 1͒. That value is applied for the computations presented hereafter.

Mass fluxes resulting from vaporization 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 vapor: In the mixture regions vapor structures are supposed to be perfectly carried along 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, ͓3͔. The momentum transfers between the phases are thus strongly linked to the mass transfers.

Numerical Model

To solve the time-dependent Reynolds-averaged Navier-Stokes equations associated with the barotropic state law presented here above, the numerical code applies, on two-dimensional structured curvilinear-orthogonal meshes, the SIMPLE algorithm, modified to take into account the cavitation process. It uses an implicit method for the time-discretization, and the HLPA nonoscillatory second-order convection scheme proposed by Zhu ͓18͔. The numerical model is detailed in Coutier-Delgosha et al. ͓10͔: A complete validation of the method was performed, and the influence of the numerical parameters was widely investigated. The present paper describes mainly the different turbulence models applied.

Boundary Conditions. 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.

Initial Transients Conditions.

To start unsteady calculations, the following numerical procedure is applied: First of all, a stationary step is carried out, with an outlet pressure high enough to avoid any vapor 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 . Vapor appears during the pressure decrease. The cavitation number is then kept constant throughout the computation.

The Geometry

Numerical simulations have been performed on a Venturi-type section whose convergent and divergent angles are, respectively, about 18 deg and 8 deg ͑Fig. 2͑a͒͒. The shape of the Venturi bottom downstream from the throat simulates an inducer blade suction side with a beveled leading edge geometry ͑Kueny et al. ͓19͔͒ and a chord length L ref ϭ224 mm.

According to experimental observations, in this geometry the flow is characterized by unsteady cavitation behavior, ͓14͔, with quasi-periodic fluctuations. Each cycle is composed of the following successive steps: The attached sheet cavity grows from the Venturi throat. A re-entrant 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 vapor cloud is then convected by the main stream, until it collapses.

For a cavitation number of about 2.4 ͑based on the timeaveraged upstream pressure͒ and an inlet velocity V ref ϭ7.2 m/s, vapor shedding frequency observed experimentally is about 50 Hz for a cavity length of 45Ϯ5 mm, ͓14͔.

The standard computational grid is composed of 160ϫ50 orthogonal cells ͑Fig. 2͑a͒͒. A special stretching 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 ͑Fig. 2͑b͒͒. In the other direction, a stretching is also applied close to the walls, to obtain at the first grid point the non dimensional parameter y ϩ 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.

Turbulence Models

"a… Standard k-RNG Model. The first model applied in the numerical code is a standard k-RNG model, associated with laws of the wall, ͓9͔.

In this model, the effective viscosity applied in the Reynolds equations is defined as ϭ t ϩ l where t ϭC k 2 / is the turbulent viscosity and C ϭ0.085 ͑Yakhot et al. ͓20͔͒.

This model is originally devoted to fully incompressible fluids, and no particular correction is applied here in the case of the highly compressible two-phase mixture. Thus, the fluid compressibility is only taken into account in the turbulence equations through the mean density changes.

With this model, the unstable cavitating behavior observed experimentally is not correctly simulated: After an initial transient fluctuation of the cavity length, the numerical calculation leads to a quasi-steady behavior of the cavitation sheet, which globally stabilizes ͑Fig. 3͒.

The resulting cavity length is much too small, compared with the experimental observations ͑in this case, the error is larger than 50 percent͒. Moreover, comparisons with experimental data ob-tained by double optical probes by Stutz and Reboud ͓14,15͔ show that the numerical mean void ratio is overestimated in the main part of the cavity. Calculations give a high time-averaged void ratio in the upstream part of the cavitation sheet ͑Ͼ90 per-cent͒, abruptly falling to 0 percent in the wake, while the measured void ratio never exceeds 25 percent and decreases slowly from the cavity upstream end to its wake.

This poor agreement with the real configuration seems to be related to an overprediction of the turbulent viscosity in the rear part of the cavity. The cyclic behavior of the cloud cavitation process is strongly related to the re-entrant jet development from the cavity closure, ͓17͔. 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 in ͓10͔ 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 (264ϫ90) and first or second-order accurate time discretization schemes still lead to the same complete stabilization of the cavitation sheet. and the vapor cloud shedding, we modified the standard k-RNG model simply by reducing the mixture turbulent viscosity, mainly in the low void ratio areas:

t ϭ f ͑ ͒C k 2 / where f ͑ ͒ϭ v ϩ ͩ v Ϫ v Ϫ l ͪ n ͑ l Ϫ v ͒ nӷ1.
Indeed, according to the experimental results ͓15͔, the reentrant jet seems to be mainly composed of liquid (␣ϳ0), and thus the reduction of the mixture turbulence viscosity leads to substantial changes in the simulation. The prediction of the unsteady re-entrant jet is now obtained, and the vapor cloud shedding is well simulated ͑See Fig. 4.͒

The transient evolution observed during this unsteady calculation is shown in Fig. 5. Fig. 5͑a͒ 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 Fig. 3, it gives information concerning the vapor 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. Curve 5b presents the transient upstream pressure evolution.

The experimental self-oscillatory behavior of the cavitation sheet is correctly simulated ͑Fig. 5͒. We obtain in this case a fluctuating cavity whose mean length is about 45 mm ͑i.e., L cav /L ref ϭ0.2). The mean fluctuation period is about 0.59 T 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, respectively͒. The reduced frequency based on the reference velocity and the mean cavity length equals 0.33, which only slightly overestimates the classical Strouhal number Stϭ0.3 ͑0.28 from the experimental measurements͒. The transient evolution is almost periodic, with some random disturbances affecting the oscillation regularity. This behavior is consistent with the experimental observations, which pointed out a quite regular cavitation cycle, whose frequency fluctuated around a central value, ͓14͔.

Several tests were performed to investigate the influence of the numerical parameters on the unsteady cavitating result. The whole study, including the influence of the grid size, the time-step, the order of the time discretization scheme, the ratio v / l , and the maximum slope of the barotropic law A min , is detailed in ͓10͔. Results concerning the grid resolution and the time step influence are reported in the present paper in Table 1. The effect of the mesh size appears to be small, so far it is fine enough: the cavity oscillation frequency is almost constant with the two finest grids. In contrast, the influence of the time-step cannot be completely removed: using ⌬tϭ0.005 T ref leads to a systematic slight overestimation of the cycles frequency. Nevertheless, it must be considered that the uncertainty on experimental measurements is of the same order of magnitude.

"c… k-Model: Compressibility Effects. As presented before, the proposed numerical simulations take into account a single-flow physical model to describe cavitation phenomenon. According to the adopted barotropic state law, in the vapor/liquid mixture zones the sound celerity Aϭͱץ P/ץ is very low, the fluid is locally highly compressible. Thus the flow often reaches Mach numbers larger than 5 in the vaporization or condensation areas. As a matter of fact, Birch and Eggers ͓21͔ showed that an increase of the Mach number could modify substantially the turbulence structure in some configurations, such as mixing layers. To ana-lyze and to evaluate the compressibility effects on the turbulence modeling of cavitating flows, we implemented in the numerical code the k-turbulence model proposed by Wilcox ͓16͔. Its major advantage, compared with the previous models, consists in the corrections that are proposed by Wilcox in the turbulence equations to model the effects of density fluctuations. The main features of the models are presented hereafter. All details concerning the turbulence model can be found in ͓16͔.

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

t ϭ k ץk ץt ϩu j ץk ץx j ϭ i j ץu i ץx j Ϫ␤ w *kϩ ץ ץx j ͫ ͑ ϩ w * t ͒ ץk ץx j ͬ ץ ץt ϩu j ץ ץx j ϭ␣ w k i j ץu i ץx j Ϫ␤ w 2 ϩ ץ ץx j ͫ ͑ ϩ w t ͒ ץ ץx j ͬ
where ␣ w ϭ5/9, ␤ w ϭ3/40, ␤ w *ϭ9/100, w ϭ0.5, and w *ϭ0.5 are the coefficients of the model.

i j ϭ2 t S i j with S i j ϭ 1 2 ͑ u i, j ϩu j,i ͒
Compressibility Effects. By analyzing the reduction of the mixing layer growth rate as a function of the Mach number, Wilcox ͓16͔ observed that the phenomenon 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 concludes 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 are based on the previous studies of Sarkar et al. ͓22͔ and Zeman ͓23͔, who aimed to take into account the compressibility effects in a k-turbulence model.

He adapted this development to the kmodel to propose the following equations giving the parameters ␤ w and ␤ w * as functions of the turbulence Mach number defined as M t 2 ϭ2k/A 2 ͑where A is the local sound celerity͒.

␤ w *ϭ␤ wi * (1ϩ*F(M t )) and ␤ w ϭ␤ wi Ϫ␤ wi * *F(M t ), where the index i indicates the values of the incompressible model given before with

F͑ M t ͒ϭ ͭ ͑ M t 2 ϪM t 0 2 ͒ if M t ϾM t 0 0 if M t рM t 0
M t 0 ϭ0.25 and *ϭ1.5.

Results. Figure 6 presents calculations performed with the k-turbulence model.

In the first case ͑Fig. 6͑a͒͒, the influence of the compressibility is not taken into account in the model. The numerical results are not satisfactory: As in the case of standard k-RNG model, the unsteady cavitation behavior is not correctly simulated and the cavity length obtained by simulations is more than 50 percent smaller than the experimental one. On the other hand, numerical results obtained by taking into account the compressibility effects are in good agreement with experimental ones. They are very similar to those presented previously in the case of the modified k-RNG model: As can be seen in Fig. 6͑b͒, the vapor cloud shedding process is correctly simulated. The mean cavity length is still correct, and the shedding frequency remains almost the same.

Comparisons are investigated with experimental data obtained by using double optical probes, ͓15͔. This intrusive technique allows measurements of the local void ratio and the velocities of the two-phase structures inside the cavitation sheet. Their timeaveraged and standard deviation values are presented for four profiles in Fig. 7.

These comparisons point out a general good agreement between experimental and numerical results. The time-averaged void ratio repartition is well predicted, and the re-entrant jet structure under the cavity appears qualitatively correct. The fluctuation levels are also in reliable close agreement with measurements. The negative velocity region at station xϭ0.065 m corresponds to the rear part of the cavity, alternatively affected by the re-entrant jet progression and by the vapor cloud shedding. Both models fail to simulate these negative velocities, because of a slight underestimation of the mean cavity length in this configuration. A second discrepancy can be observed at station 0.03 m, concerning the void ratio distribution close to the wall; nevertheless, the experimental estimation of velocities and void ratio in the re-entrant jet is associated to a high uncertainty in this area, due to important fluctuations level, ͓14͔. Thus, the comparison, along the wall, between experiments and the numerical result cannot be relevant.

Conclusion

Four turbulence models have been applied to simulate unsteady cavitating flows in a Venturi-type section.

Results obtained with the standard k-RNG model and with the k-model without taking into account compressibility effects are in poor agreement with experimental observations: the models fail to reproduce the vapor cloud shedding behavior observed experimentally. Moreover, numerical tests reported in ͓10͔ and the results presented here above indicate that the global level of dissipation induced by the numerical model seems not to be responsible for its inability to simulate the unsteady cavity behavior.

On the other hand, the modified k-RNG model and the kmodel including compressibility effects lead to a very reliable simulation of the Venturi unsteady cavitation behavior. The satisfactory results obtained with the modified k-RNG model have been confirmed by simulations performed in other geometries, such as a hydrofoil, ͓11͔, a foil cascade, ͓13͔, and another Venturitype section leading to a more stable sheet of cavitation, ͓10͔. In all cases, the general experimental behavior is correctly obtained, and oscillation frequencies are accurately predicted in unsteady configurations.

Results obtained in the present paper with kmodels pointed out the relevant role of the compressibility effects on the turbulence model. Indeed, the corrections proposed to treat compressibility effects take into account the 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 the vapor/liquid mixture zones.

The modifications proposed to improve the k-RNG model had been also based on the reduction of the turbulent viscosity in the mixture zones, which are characterized by a very low sound celerity and large Mach number. Indeed, results obtained with the modified k-RNG model are similar to the ones calculated by the compressible kmodel.

According to the numerical calculations, the fluid compressibility has a strong effect on the turbulence structure, and must be taken into account to simulate unsteady cavitating flows.

Fig. 1

 1 Fig. 1 Barotropic state law "P…. Water 20°C.

"

  Fig. 4 Modification of the mixture viscosity "nÄ10…

Fig. 5

 5 Fig. 5 Transient evolution of unsteady cavitating flow in the Venturi-type duct. "a… Temporal evolution "in abscissa… of the cavity length "graduated in ordinate…. Instantaneous density distribution of attached and cloud cavities are drawn on the left at TÄ11T ref "velocity vectors are drawn only 1 cell over 2 in the two directions…. "b… Time evolution of the upstream pressure.

Fig. 6

 6 Fig. 6 Unsteady cavitation behavior calculated with the kmodel-"a… without and "b… with compressibility effects

Fig. 7

 7 Fig. 7 Time-averaged values and standard deviation of velocity "1,2… and void ratio "3,4…. Numerical results: "a… modified k-, "b… compressible k-"lines… and optical probe measurements "points…-V ref Ä7.2 mÕs. Mean cavity length 45 mm: dotted line Äexperimental external shape of the cavity, obtained from image processing "sigmaÄ2.4….

NomenclatureA

  min ϭ minimum speed of sound in the medium ͑m/s͒ Cp ϭ ( PϪ P ref )/(V ref 2 /2) pressure coefficient k ϭ turbulent kinetic energy per unit mass ͑m 2 s Ϫ2 ͒ L ref ϭ geometry reference length ͑m͒ P ϭ local static pressure ͑Pa͒ P ref ϭ reference pressure ͑Pa͒ P v ϭ vapor pressure ͑Pa͒ V ref ϭ reference velocity (ϭV upstream ) ͑ms Ϫ1 ͒ ␣ ϭ void ratio ϭ turbulence dissipation per unit mass ͑m 2 s Ϫ3 ͒ l , t ϭ laminar and turbulent dynamic viscosity ͑Pa•s͒ , t ϭ laminar and turbulent kinetic viscosity ͑m 2 /s͒ ⌬ P v ϭ range of mixture state law ͑Pa͒ ϭ mixture density ͑kg/m 3 ͒ l , v ϭ liquid, vapor density ͑kg/m 3 ͒ ref ϭ reference density (ϭ l ) ͑kg/m 3 ͒ ϭ ( P upstream Ϫ P v )/(V ref 2 /2) cavitation number ϭ specific dissipation rate ͑s Ϫ1 ͒

Table 1 Numerical tests performed on the Venturi-type section with the k-modified model

 1 Ä2.4, A

min Ä2 mÕs, v Õ l Ä0.01

				RESULTS
	⌬T/T ref	Mesh	Mean Vapor Volume (ϫ10 Ϫ4 m 3 )	Oscillation Frequency ͑Hz͒	Standard Deviation (ϫ10 Ϫ4 m 3 )
			Reference case
	0.005	160ϫ50	11.0	55	2.4
			Influence of the time-step
	0.002	160ϫ50	12.3	51	4.3
	0.005	160ϫ50	11.0	55	2.4
	0.01	160ϫ50	9.3	55	2.7
			Influence of the mesh
	0.002	264ϫ90	15.6	52	2.9
	0.005	264ϫ90	14.5	54	4.3
	0.005	160ϫ50	11.0	55	2.4
	0.005	110ϫ30	6.9	64	1.9

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

Acknowledgments

The authors wish to express their gratitude to the French Space Agency: Centre National d'Etudes Spatiales ͑CNES͒ for its continuous support.


R Al Khatib
email: reem_el_khatib@yahoo.com

B Lartiges

C Mustin

S Skali-Lami

CONTROLE HYDRODYNAMIQUE DE L'ORGANISATION DES BIOFILMS EN MILIEU EAUX USEES

Keywords: Biofilm, force de cisaillement, exopolymères, écoulement couette-poiseuille

La formation des biofilms sur les surfaces internes des réseaux d'assainissement à l'interface liquide-solide joue un rôle prépondérant dans la concentration des polluants, leur transport, leur dégradation, leur relargage et leur devenir dans le réseau. Le développement d'un biofilm est un phénomène complexe et influencé par plusieurs facteurs parmi lesquels les conditions hydrodynamiques sont les plus importantes. En effet, l'épaisseur, la quantité de biomasse fixée, la nature et la quantité d'EPS produits ainsi que la répartition des colonies bactériennes varient largement suivant le mode d'écoulement et de cisaillement appliqués. Ainsi le développement des biofilm des eaux usées sous écoulement Couette-Poiseuille sous deux vitesses de cisaillement n'est pas similaire dans les deux cas. Les observations par microscopie confocale, après marquage des biofilm par l'acridine orange, montrent que les biofilms obtenus à cisaillement 1s -1 sont épais, peu denses, poreux et présentent des élongations (streamers) qui oscillent dans le flux alors que des biofilms générés en cisaillement 6s -1 sont denses et peu épais. En appliquant une configuration d'écoulement avec un gradient pariétal nul le développement du biofilm se trouve freiné pour plus de deux jours. Cette configuration pourrait servir pour prévenir la formation de biofilm dans les différents systèmes.

Introduction

Les biofilms sont des associations de microcolonies bactériennes d'espèces différentes qui se développent sur toute surface à l'interface liquide-solide dans des conditions variées de température, apport en nutriments, conditions hydrodynamiques,… [START_REF] Eberl | A three-dimensional study on the correlation of spatial structure, hydrodynamic conditions, and mass transfer and conversion in biofilms[END_REF]. Dans les réseaux d'assainissement, la formation des biofilms joue un rôle primordial dans la transformation, le stockage et l'élimination des polluants. Les changements de conditions hydrodynamiques, le vieillissement et la quantité de substrat dans le milieu affectent les caractéristiques physicochimiques des biofilms et peuvent provoquer le détachement de ces derniers libérant ainsi les micropolluants dans le réseau [START_REF] Horn | Simulation of growth and detachement in biofilm systems under defined hydrodynamic conditions[END_REF]. Bien que la développement de biofilm soit un phénomène fréquent voire systématique à l'interface solide/liquide en présence de bactéries, sa formation passe par une séries d'étapes influencées par différents facteurs (mode d'écoulement, la nature de la surface, le type des nutriments et leurs concentrations, conditions d'aération). La formation du biofilm met donc en oeuvre plusieurs processus physicochimiques (diffusion, adhésion, …) ainsi que des processus biologiques (formation d'agrégats, croissance de microorganismes, production d'exopolymères, …) [START_REF] Celmer | Impact of shear force on the biofilm structure and performance of a membrane biofilm reactor for tertiary hydrogen-driven denitrification of municipal wastewater[END_REF][START_REF] Andersson | Characterization of bacterial biofilms for wastewater treatment[END_REF]. Jusqu'à présent les différentes études réalisées sur le mécanisme de formation de biofilm ne peuvent pas présumer d'une séquence d'événements bien définie.

Influence des conditions hydrodynamiques L'activité du biofilm est fortement liée à son volume, la quantité de biomasse fixée, sa densité, sa porosité et le transfert de nutriments à l'intérieur du biofilm [START_REF] Abrahamson | Development of an artificial biofilm to study the effects of a single microcolony on mass transport[END_REF]. La vitesse d'écoulement du liquide et les forces de cisaillement qui en résultent semble être le facteur le plus important influençant la structure du biofilm vu leur impact sur la répartition des microcolonies et la production des exopolymères [START_REF] Costerton | Discussion : Introduction to biofilm[END_REF][START_REF] Stoodley | Influence of hydrodynamics and nutrients on biofilm structure[END_REF]. En effet, les différentes études menées sur la formation des biofilms à partir des systèmes naturels ou des souches pures sous différentes conditions hydrodynamiques montrent que l'épaisseur et la quantité de masse fixée, la nature et la quantité d'EPS produits ainsi que la répartition des colonies bactériennes varient largement suivant le mode de cisaillement appliqué [START_REF] Celmer | Impact of shear force on the biofilm structure and performance of a membrane biofilm reactor for tertiary hydrogen-driven denitrification of municipal wastewater[END_REF][START_REF] Stoodley | Influence of hydrodynamics and nutrients on biofilm structure[END_REF][START_REF] Pereira | Effect of flow regime on the architecture of a Pseudomonas fluorescens biofilm[END_REF]. Les écoulements à grande vitesse voire écoulement turbulent engendrent des biofilms denses et favorisent le transfert des nutriments depuis le liquide extérieur vers le biofilm ce qui favorise la production de exopolymères notamment les polysaccharides, ainsi le biofilm détermine une structure plus rigide et la cohésion entre les cellules sera plus importante [START_REF] Decho | Microbial biofilms in itertidal systems: An overview[END_REF]. Mais ceci aura un effet défavorable sur le transfert de nutriments au sein même du biofilm ce qui va diminuer l'activité de celui-ci et les biofilms seront plus fins [START_REF] Celmer | Impact of shear force on the biofilm structure and performance of a membrane biofilm reactor for tertiary hydrogen-driven denitrification of municipal wastewater[END_REF][START_REF] Liu | The essential role of hydrodynamic force in the formation of biofilm and granular sludge[END_REF][START_REF] Wagner | Combined use of confocal laser scanning microscopy (CLSM) and Raman microscopy (RM): Investigation on EPS-Matrix[END_REF]. A faible force de cisaillement, les biofilms sont beaucoup moins denses, plus poreux et plus épais. Ces biofilms sont formés par l'agrégation de colonies bactériennes séparées par des vides interstitiels qui créent une sorte de canalisations qui assurent le transfert de nutriments à l'intérieur du biofilm ce qui augmente son activité et aboutit à des biofilms plus épais mais moins rigides car la production des EPS est moins importante [START_REF] Celmer | Impact of shear force on the biofilm structure and performance of a membrane biofilm reactor for tertiary hydrogen-driven denitrification of municipal wastewater[END_REF][START_REF] Wagner | Combined use of confocal laser scanning microscopy (CLSM) and Raman microscopy (RM): Investigation on EPS-Matrix[END_REF]. Nous nous intéressons dans ce travail, à la croissance de biofilms en milieu eaux usées urbaines dans des conditions contrôlées d'écoulement et de cisaillement. Ainsi un réacteur Couette poiseuille a été choisi pour la réalisation des expériences et la croissance des biofilms à partir des eaux usées.

Matériels et Méthodes

Réacteur de croissance

Le réacteur Couette poiseuille (figure 1) est formé de deux cylindres concentriques en polychlorure de vinyle (PVC) espacé de 3,5 mm. Le cylindre extérieur de rayon 5cm et de hauteur 10cm est statique et porte 16 coupons de prélèvement des biofilms de 16mm de diamètre. La surface concave de ceux-ci est ajustée pour s'adapter à la forme du cylindre extérieur. Le cylindre interne est plein et entraîné par un moteur à vitesse réglable de façon à contrôler sa vitesse de rotation autour de son axe. La vitesse de rotation du cylindre interne détermine en partie le profil de cisaillement du fluide qui circule entre les deux cylindres. A l'entrée du réacteur, un divergent est installé pour assurer une distribution homogène du liquide entrant sur toute la hauteur du réacteur. L'avantage de ce réacteur est de pouvoir assurer un écoulement Couette-Poiseuille qui n'est que le couplage d'un écoulement Couette avec un écoulement Poiseuille. Cet écoulement présente l'avantage de pouvoir travailler dans différentes configurations d'écoulement en variant le rapport vitesse de rotation du cylindre interne sur vitesse débitante du fluide [START_REF] Paris | Formation et organisation de biofilms en milieu eau potable. Influence du gradient de vitesse pariétale[END_REF]. Le profil de vitesse d'écoulement de Couette suivant le modèle de notre réacteur (cylindre interne mobile et cylindre externe fixe) est linéaire (figure 2). Ceci implique qu'une particule sera soumise au même gradient de vitesse (γ p ) quelle que soit sa position dans le fluide , l'expression de la vitesse sera γ p = R int Ω/e avec Ω (radians) la vitesse de rotation du cylindre interne. Cette particularité de gradient de vitesse permet de calculer la force hydrodynamique à laquelle sera soumise une particule dans l'entrefer du réacteur. En ajoutant une entrée et une sortie à notre système (figure 1B), un écoulement de Poiseuille est obtenu. Dans le cas de nos conditions expérimentales, cylindre interne mobile, l'écoulement devient un écoulement Couette-Poiseuille. L'expression du gradient de cisaillement pour des petits espaces annulaires [START_REF] Paris | Formation et organisation de biofilms en milieu eau potable. Influence du gradient de vitesse pariétale[END_REF] est donnée par :

γ= ( ) e R 1 ) 2 1 ( 6 1 int - - Ω β β= Ω int R
Vq avec V q la vitesse moyenne du fluide Le facteur β (rapport entre la vitesse débitante et la vitesse de rotation du cylindre interne) détermine le profil de vitesse et par la suite le gradient de cisaillement appliqué. L'eau usée dans la cuve est maintenue sous agitation à une vitesse de 150 tours/minutes à l'aide d'une palette repliable d'agitation en polypropylène (VWR 442-2121) L'eau est pompée dans un tuyau (Legris 1025406K02) en polyuréthane de 4mm de diamètre de la cuve à l'intérieur du réacteur à l'aide d'une pompe immergée à débit réglable (Fisher W75601) (1s -1 et 6s -1 ), (figure3). L'eau sortant du réacteur retourne ensuite à la cuve. La position du tuyau de retour joue un grand rôle dans le contrôle du débit de l'eau pompée, en effet le réglage du débit correspondant à la vitesse de cisaillementappliquée est réalisé par l'ajustement de la hauteur de ce tuyau. Le coupon est ensuite placé sur un dispositif comprenant une lame en quartz, contenant de l'eau ultrapure pour assurer un atmosphère humide pour l'échantillon, et observé en microscopie confocale sous objectif x4 et x20 (figure 4). Les observations sont stockées sous forme de piles d'images (3D). Les piles d'images obtenues seront destinées à différents logiciels de traitements d'image en commençant par l'amélioration de la résolution et l'atténuation du bruit de fond, passant par la déconvolution 3D pour arriver à la fin à la reconstruction 3D et le calcul des différents paramètres (épaisseur du biofilm, dimension fractales, volume et co-localisation, etc…).

Résultats et discussion

Les expériences dans le réacteur de croissance sont effectuées sous deux forces de cisaillements (1s -1 et 6s -1 ). Ceci est réalisé dans le but de voir l'impact de la variation des forces hydrodynamiques sur la structure et la croissance du biofilm. Les algorithmes mathématiques appliqués par l'intermédiaire du logiciel Gocad sur les piles d'image 3D obtenues par microscopie confocale ont permis de calculer les épaisseurs des biofilms, la quantité de biomasse fixée ainsi que l'estimation du volume de polymères, la surface et le volume de biofilm qui occupe la surface de coupons. Les résultats (figure 4) montrent que la formation du biofilm n'est pas similaire pour les deux gradients. Concernant l'épaisseur, les résultats montrent que les biofilms obtenus sous cisaillement 1s -1 sont plus épais que ceux obtenus à 6s -1 . L'épaisseur parait augmenter avec le temps jusqu'à atteindre une valeur au-dessous de laquelle le biofilm se détache et son épaisseur décroit. Pour le cisaillement 1s -1 , une chute de l'épaisseur est observée après 72h de mise en marche de réacteur puis elle re-augmente dans les jours qui suivent. Cette chute observée peut être due à un détachement de biofilm lors de sa croissance : le biofilm atteint une épaisseur très importante de sorte que l'espace entre les deux cylindres devient très étroit et les particules en suspension dans le fluide entrainent une partie du biofilm d'où une baisse de son épaisseur. A noter que les valeurs d'épaisseur obtenues sont probablement sous-estimées car une petite partie du biofilm se détache lors de la préparation des coupons pour l'observation en microscopie confocale. Concernant les quantités de biomasse et de polymères produits, elles sont différentes dans les deux cas. A 6 s -1 , la quantité de polymères et de bactéries dans la phase d'initiation à la formation de biofilm est beaucoup plus importante montrant que la croissance de biofilm est initialisé par la production de polymères qui vont assurer l'adhésion des bactéries. Une fois que le biofilm commence à croître, cette quantité de polymères devient proportionnelle à la quantité de bactéries et l'épaisseur du biofilm.

Pour le cisaillement de 1s -1 , le comportement est différent, la quantité de polymères et de bactéries augmentent avec la croissance du biofilm. Au dernier prélèvement, l'épaisseur du biofilm est maximale (environ 350 microns) et la quantité de biomasse est la plus élevée tandis que la quantité de polymères est très faible. Ceci peut être confirmé par les études qui trouvent une diminution de la production de polymères quand l'épaisseur du biofilm augmente [START_REF] Celmer | Impact of shear force on the biofilm structure and performance of a membrane biofilm reactor for tertiary hydrogen-driven denitrification of municipal wastewater[END_REF][START_REF] Wagner | Combined use of confocal laser scanning microscopy (CLSM) and Raman microscopy (RM): Investigation on EPS-Matrix[END_REF].

Les résultats montrent aussi que la majorité des polymères élaborés à l'intérieur du biofilm semble former une matrice indépendante tandis que les bactéries semblent formées de deux catégories équivalentes une liées aux exopolymères et l'autre sans polymères. Après 48h, un dépôt de biofilm commence à se former dont l'origine est à rechercher sur les modifications des conditions hydrodynamiques sur la surface du cylindre interne du réacteur: en effet le nouveau profil de cisaillement, s'il défavorise la formation de biofilm sur les coupons, facilite la croissance de biofilm sur le cylindre interne.

Conclusion

Les expériences réalisées dans le réacteur Couette-Poiseuillle ont permis l'étude du développement des biofilms à partir des eaux usées dans un écoulement laminaire sous différents gradients de vitesses pariétales. La reconstruction 3D et l'application des algorithmes de calcul sur les données issus des images obtenues à partir de la microscopie confocale a permis d'étudier la structure et les caractéristiques des biofilms formés. Les biofilms présentaient deux modes différents de développement ainsi que des structures différentes selon le taux de cisaillement appliqué (1s -1 , 6s -1 ). La formation de biofilm se trouve très ralentie (au moins 48h) en appliquant les conditions de « non dépôt ». Ceci s'avère très important pour toutes les applications nécessitant d'éviter le développement de biofilm.

Figure1:

  Figure1 : Coupe longitudinale (A) et transversale (B) du réacteur Couette-Poiseuille.

Figure2:V

  Figure2 : Profil de vitesse dans un écoulement Couette.

Figure 3 :

 3 Figure 3 : Schéma du montage expérimental pour la croissance des biofilms.Le suivie de l'évolution de l'eau usée est réalisé chaque heure par mesure du pH, conductivité, potentiel redox, oxygène dissous (WTW, Mutltiline F/SET) et turbidité (Hach 2100P). Un comptage bactérien sur la cellule de Thomas (VWR 720-0150) est effectué pour une évaluation du nombre de bactéries au début de la journée et au moment de renouvellement après 24h. Un coupon, sur lequel se développe le biofilm, est prélevé chaque 24h, l'eau est éliminée en égouttant le coupon. Le coupon est ensuite pesé et placé dans une petite boite à bouchon en polyéthylène (partie contenant le biofilm vers le haut) contenant au fond quelque mL d'eau ultrapure. La boite est ensuite placée à 4°C avant l'observation en microscopie confocale à balayage laser. L'observation par microscopie confocale est réalisée le plus tôt possible (1 jour après le prélèvement des coupons) afin de minimiser une évolution possible du biofilm.Observation par microscopie confocaleAvant d'observer les biofilms au CLSM, ces derniers sont colorés par l'acridine orange. Ce fluorochrome possède une longueur d'onde d'excitation et d'émission spécifique au marquage des acides nucléiques (ADN et ARN) ainsi que les exopolymères extracellulaires[START_REF] Wirtanen | Microbiological methods for testing disinfectant efficiency on Pseudomonas biofilm[END_REF][START_REF] Jana-Sur | Studies on the interaction of bacterial capsular polysaccharide-Klebsiella K16 with cationic dyes[END_REF]. Pour réaliser cette coloration, à l'aide d'une micropipette 100µL d'acridine orange sont déposés sur l'échantillon de biofilm sur le coupon, le coupon est remué doucement pour disperser le colorant sur la totalité de la surface; le coupon est ensuite placé à l'obscurité pendant 20 à 30 min pour la

Figure 4 :

 4 Figure 4 : Résultats obtenues à partir de Gocad : a) quantité de bactéries (exprimé par voxels), b) quantité de polymères (exprimé par voxels), c) épaisseur de biofilm sur le coupon

Figure 5 :

 5 Figure 5 : Images obtenues par microscopie confocale (objectif x20) : a) coupon 24h condition non dépôt, b) coupon 48h condition non dépôt, c et d) coupons respectivement 24 et 48h (condition normale, cisaillement 6s -1 ), e et f) coupons respectivement 24 et 48h (condition normale, cisaillement 1s -1 )

  -1 nous remarquons que la propagation des colonies bactériennes est surfacique, les colonies sont de petite taille mais occupent toute la surface du coupon. Pour le cisaillement 1s -1 , la prolifération des colonies se fait verticalement, le nombre de colonies occupant la surface des coupons est limité et la croissance se fait en épaisseur. Un phénomène commun observé sur tous les coupons est l'élongation des colonies dans le sens d'écoulement à un angle d'environ 45 degré. Ceci peut être corrélé aux forces de cisaillement qui entrainent les cellules filles lors de leur division dans le sens d'écoulement[START_REF] Stoodley | Influence of hydrodynamics and nutrients on biofilm structure[END_REF]. Deux expériences additionnelles, avec les mêmes gradients de cisaillement, en condition dit de « non dépôt » ont été réalisées dans le but d'avoir les conditions les plus défavorables pour la formation de biofilm. Dans ce cas, la vitesse d'écoulement est nulle à la surface du cylindre interne et maximale à la surface du cylindre externe. Les premiers résultats des observations en microscopie confocale montrent une différence nette entre les deux conditions. Que ce soit au gradient 6s -1 ou 1s -1 aucun développement de biofilm n'a été observé durant les deux premiers jours (figure5).


On Quantum Attacks

Compare to best generic attack, generic attack is accelerated, so broken classical primitive might be unbroken in a quantum setting: e.g. a primitive might not have 256-bits security against a classical adversary but might have 128bit security against a quantum one.

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Scenarios and Models

Considered Scenarios Model Q 0 classical attacks with classical computers. Model Q 1 Q 0 + access to a quantum computer. Model Q 2 Q 1 + superposition queries to a quantum cryptographic oracle (QCO). Model Q 3 Q 1 + superposition queries with the differences of a secret key in a QCO.

7/57

Model Q 0 Nothing new here.

8/57

Model Q 1 So far, the best we have obtained is a quadratic speed-up, but it can be smaller:

• If a primitive is safe in Q 0 , it will also be in Q 1 .

Does this mean that (so far) the Q 1 scenario/results are not interesting?

No! safe = no attack better than generic attack 9/57

Model Q 1

In a post-quantum future: Classical or quantum surnames will disappear: Expected security given by their best generic attack (e.g. Grover).

And security margin? → determined by the highest number of rounds cryptanalyzed with any attack more performant than generic.

Q 1 results: important information needed for determining the unique and future security margin. Amplitude Amplification (Grover's generalization)

Exhaustive search:

Given f : {0, 1} n → {0, 1}, find one element x ∈ {0, 1} n such that f (x) = 1. Classical complexity: Ω( 2 n |supp(f )| ). Quantum complexity [Brassard-Hoyer 97]: Ω( 2 n |supp(f )| ) .
In detail, we will see later: O

2 n |supp(f )| (s T + f T ) .

19/57

Quantum Counting Algorithm Distinguish a biased distribution: Given a Bernouilli distribution, determine with high probability whether it has a parameter 1/2 or 1/2 + ε.

Classical complexity: O 1 ε 2 . Quantum complexity: [Brassard-Hoyer-Tapp 98] O 1 ε .
20/57

Quantum Collision Algorithms

Collision problem: Given a random function

H :{0, 1} n → {0, 1} n , find x, y ∈ {0, 1} n with x = y such that H(x) = H(y).
Classical complexity: Ω(2 n/2 ).

Quantum complexity:

[Brassard-Hoyer-Tapp 97] O 2 n/3 in queries, in time and in quantum memory → scenario A. (Scenario B later)

Kuperberg's algorithm

Hidden Shift Problem with modular addition: Let f , g be two injective functions, (G, +) a group. Given the promise that there exists s ∈ G such that, for all x, f (x) = g(x + s), retrieve s. Many applications: e.g. generic attacks on hash functions.

(Multi-target preimage search can be seen as a particular case).

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Best known algorithms

Time Queries Qubits Classical Memory Pollard 2 n/2 2 n/2 0 O(n) Grover 2 n/2 2 n/2 O(n) 0 BHT 2 2n/3 * 2 n/3 O(n)* 2 n/3 Ambainis 2 n/3 2 n/3 2 n/3 0 26/57
Considered Model

The same one as in the previous collision quantum algorithms BUT we limit the amout of quantum memory available to a small amount O (n): scenario B instead of A.

Available small quantum computers seem like the most plausible scenario.

We are interested in the theoretical algorithm and we did not take into account yet implementation aspects. Exhaustive search for finding one element that collides: With AA, the number of iterations is:

( 2 n 2 n/3 ) 1/2 = 2 n/3 . Testing the membership with L for the superposition of states costs 2 n/3 with n qbits: Time:

2 n/3 + 2 n/3 (1 + 2 n/3 ) ≈ 2 2n/3

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Can we improve this? Let's build the list L with distinguished points e.g.

H(x i ) = 0 u ||z, for z ∈ {0, 1} n-u .
The cost of building the list is bigger: 2 n/3+u/2 . The setup of AA is bigger:

2 u/2
The membership test stays the same: |L| = 2 n/3 BUT The number of iterations is smaller: 2 n/3-u/2 Time:

2 n/3+u/2 + 2 n/3-u/2 (2 u/2 + 2 n/3 ) ≈ 2 2n/3-u/2 + 2 n/3+u/2

3-xor: Scenario A Algorithm

Same technique as before, but no need for a common prefix of zeroes. This gives QM= 2 n/5 and Time= 2 3n/10 .

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The k-xor algorithms When optimal? Not intuitive at all! ⇒ Recursive MILP program for optimality in both scenarios.

Can we reach better complexities than O 2 n/( log 2 (k) +2) when k is not a power of 2 in scenario A? Can we obtain time complexities better than classical for k ≥ 8 in scenario B?

P k 2 E k 1 ,k 2 (x) f (x) = E K (x) ⊕ P (x) → f (x) = f (x ⊕ k 1 ) Simon's algo on f ⇒ k 1 in O(n) 47/57
[Kaplan-Leurent-Leverrier-NP 16]

Simon on most authentication modes + slide attacks For example encrypt-last-block CBC-MAC:

0 m 1 E k m 2 E k m 3 E k E k τ f : {0, 1} × {0, 1} n → {0, 1} n b, x → CBCMAC(α b x) = E k E k x ⊕ E k (α b ) . CBCMAC(α 1 x ⊕ E k (α 0 ) ⊕ E k (α 1 )) = E k E k (x ⊕ E k (α 0 ) ⊕ E k (α 1 )) ⊕ E k (α 1 ) = CBCMAC(α 0 x) s = 1 E k (α 0 ) ⊕ E k (α 1 )
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[Kaplan-Leurent-Leverrier-NP 16]

Simon on most authentication modes + slide attacks For example encrypt-last-block CBC-MAC:

0 α 0 /α 1 E k x E k E k τ f : {0, 1} × {0, 1} n → {0, 1} n b, x → CBCMAC(α b x) = E k E k x ⊕ E k (α b ) . CBCMAC(α 1 x ⊕ E k (α 0 ) ⊕ E k (α 1 )) = E k E k (x ⊕ E k (α 0 ) ⊕ E k (α 1 )) ⊕ E k (α 1 ) = CBCMAC(α 0 x) s = 1 E k (α 0 ) ⊕ E k (α 1 )

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Simon and Grover on FX construction

The FX construction is a natural construction for extending the key-length n ⇒ 2n.

x k 1 E k 0 k 2 E k 0 ,k 1 ,k 2 (x)

  Some New Useful Quantum Tools New Quantum Collision Algorithm Quantum K-xor Algorithms Multicollisions Grover-meets-Simon Simon-meets-Kuperberg Framework for quantizing classical attacks Quantumly efficient DDT equivalent Miss-in-the-middle search 24/57 Collision Search with A. Chailloux, A. Schrottenloher Collision Search Problem Given a random function H :{0, 1} n → {0, 1} n , find x, y ∈ {0, 1} n with x = y such that H(x) = H(y).

  Build a list L of 2 n/3 elements (classical memory),

  time are O (2 α k n ) 41/57 k-xor algorithms: Very Recent results Related to dissection: partial solutions to subproblems n < n, k < k and combining them.

  Improved Kuperberg's algorithm and new algorithm for parallel modular additions.State size needed for 128-bit security. at least 5200 bits (but for FX) ⇒ not very realistic.Might be better to just avoid vulnerable constructions, or try different patches.

	Improvement and Simulation Results -Conclusion General Conclusion (for now) 1/2
	No reason to panic, symmetric crypto seems
	to be holding on well Our improvement: all the bits with one iteration. O(n 2 2 √ 2 log 2 (3)n ) ⇒ O(n2 √ 2 log 2 (3)n ) Bigger internal states?
	Our simulations: 0.7 × 2 1.8 √ Ideas from quantum analysis might improve n for recovering full s. Final Conclusion classical analysis
	Code available: ask Xavier Bonnetain if interested. Many things yet to do to precisely evaluate
	[Leander May 17] Combined Simon with Grover: xavier.bonnetain@inria.fr security, to find best attacks, to adjust Recently: concrete security of some Isogeny-parameters... based primitives [Bonnetain-Schrottenloher] → broken in O(2n2 n/2 ) 52/57 53/57 54/57

Simon's algorithm

Simon's problem: Given f : {0, 1} n → {0, 1} n such that ∃s | f (x) = f (y) ⇐⇒ [x = y or x ⊕ y = s], find s.

Classical complexity: Ω(2 n/2 ).

Quantum complexity [Simon 94]: O (n) .

With optimal parameters

The cost will be optimized for a certain size of L:

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Example of Applications

Hash functions: Collision and Multipreimages time from 2 n/2 to 2 2n/5 and 2 3n/7 (Q1).

Ex When both parallelized: up to 2 n/3 processors this algorithm is more time-efficient than parallel rho.

Conclusion -Collision

New efficient collision search algorithm with small quantum memory (nothing scary, new trade-offs):

First algorithm with less than 2 n/2 computations in scenario B.

Many applications in symmetric cryptography.

Open question: is it possible to meet the optimal 2 n/3 in time with small quantum memory? Given query access to a random function

For us, equivalent to the case with k different random functions.

Many applications (with k-sum, similar algorithms apply), ex.: attacks on FSB, XLS, SWIFFT; correlation attacks.

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The 3-xor problem Find 3 elements that xor to 0: not much better than collision in classical setting.

Classically, no exponential time acceleration, only logarithmic:

3-xor: Scenario B Algorithm 1st approach, distinguished point: Intuition: With a memory of 2 v + 2 v we obtain 2 2v potential collisions.

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3-xor: Scenario B Algorithm 1st approach:

2nd approach, techniques linked to "list merging": Improved time= 2 5n/14 , with