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A numerical method for 3D barotropic flows in turbomachinery Edoardo Sinibaldi ([email protected]) Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa (Italy)

Fran¸cois Beux ([email protected]) Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa (Italy)

Maria Vittoria Salvetti ([email protected]) Dip. di Ingegneria Aerospaziale, Universit` a di Pisa, via Caruso, 56122 Pisa (Italy) Abstract. A numerical method for the simulation of 3D inviscid barotropic flows in rotating frames is presented. A barotropic state law incorporating a homogeneousflow cavitation model is considered. The discretisation is based on a finite-volume formulation applicable to unstructured grids. A shock-capturing Roe-type upwind scheme is proposed for barotropic flows. The accuracy of the proposed method at low Mach numbers is ensured by ad-hoc preconditioning, preserving time consistency. An implicit time advancing only relying on the algebraic properties of the Roe flux function, and thus applicable to a variety of problems, is presented. The proposed numerical ingredients, already validated in a 1D context and applied to 3D nonrotating computations, are then applied to the 3D water flow around a typical turbopump inducer. Keywords: barotropic, cavitation, turbomachinery, unstructured grids, low Mach, preconditioning, implicit

1. Introduction The set-up of a numerical method for generic barotropic flows in rotating frames is presented, together with its application to a 3D geometry typical of turbopump inducers exploited in the feed system of liquid rocket engines. The working fluid is allowed to cavitate; a homogeneousflow cavitation model explicitly accounting for thermal cavitation effects and for the concentration of the active cavitation nuclei in the pure liquid is adopted (d’Agostino et al., 2001). Despite the model simplifications leading to a unified barotropic state law, formidable numerical problems arise from the extremely abrupt transition between wetted (nearly-incompressible) and cavitating regimes (highly-compressible) to be simulated at the same time. In order to simultaneously cope with nearly-incompressible and highlycompressible flow regions, two opposite ways can be followed: adaptation to the compressible case of numerical methods suitable for incompressible flows (see e.g. (Coutier-Delgosha et al., 2003), (Senocak and Shyy, 2002)) or, conversely, adaptation to the low Mach number c 2005 Kluwer Academic Publishers. Printed in the Netherlands.

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limit of compressible solvers. The latter approach is considered here and a finite-volume space discretisation is adopted, based on the definition of a Roe matrix (Roe, 1981) suitable for a generic barotropic fluid. Standard compressible solvers, while being more efficient than modified incompressible schemes at high Mach numbers, exhibit efficiency problems as well as accuracy problems when dealing with nearlyincompressible flows. Following (Guillard and Viozat, 1999), the latter problem is counteracted here by preconditioning. Implicit timeadvancing only based on the algebraic properties of the Roe flux function is introduced, to at least partially overcome efficiency problems. Finally, the extension to rotating frames of the resulting preconditioned implicit scheme is presented. The proposed scheme, already validated in a 1D context and then applied to 3D non-rotating frames, is applied to the 3D water flow around a typical turbopump inducer for which experimental data are available.

2. Constitutive relations and governing equations in non-rotating frames A weakly-compressible liquid at constant temperature T L is considered as working fluid. The liquid density ρ is allowed to locally fall below the saturation limit ρLsat = ρLsat (TL ) thus originating cavitation phenomena. A regime-dependent (wetted/cavitating) constitutive relation is therefore adopted. As for the wetted regime (ρ ≥ ρLsat ), a barotropic model of the form:   ρ 1 ln (1) p = psat + βsL ρLsat is adopted, psat = psat (TL ) and βsL = βsL (TL ) being the saturation pressure and the liquid isentropic compressibility, respectively. As for the cavitating regime (ρ < ρLsat ), a homogeneous-flow model explicitly accounting for thermal cavitation effects and for the concentration of the active cavitation nuclei in the pure liquid has been adopted (d’Agostino et al., 2001). According to this model, pressure and velocity differences between the liquid and the vapour phases may be neglected and the mixture density ρ is directly related to the void fraction α: ρ = ρLsat (1 − α). By assuming that the evaporation/condensation phenomena taking place at the liquid-vapour interface do not alter the global entropy of the mixture, the model exploits the energy balance to derive a (differential) barotropic relation for the cavitating mixture

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as well:     η  p dρ p 1 ? pc = (1 − α) (1 − εL ) +α + εL g 2 ρ dp p γV ρLsat aLsat

(2)

where g ? , η, γV and pc are liquid parameters, aLsat is the liquid sound speed at saturation and εL = εL (α, δT /R) is a given function representing the volume fraction (0 ≤ εL ≤ 1) of the liquid that is in thermal equilibrium with the bubbles, δT /R being a free parameter accounting for thermal effects and the concentration of the active cavitation nuclei (d’Agostino et al., 2001). By assigning the value of dp/dρ(ρ = ρ Lsat ) as provided by (1) to a2Lsat and starting the (a priori) integration of (2) from p(ρ = ρLsat ) = psat , a C 1 -continuous junction is obtained between the two branches of the constitutive relation. Thanks to the physical foundations of the pure liquid and the cavitation models, dp/dρ is always strictly positive and may be regarded to as the square of the fluid sound speed a. Despite the model simplifications leading to a unified barotropic state law (only depending on T L and δT /R), the transition between wetted and cavitating regimes is extremely abrupt. Indeed, the fluid sound speed falls from O(10 3 ) m/s in the pure liquid down to O(10−1 ÷ 100 ) m/s in the mixture, while the void fraction increment just below the saturation point is of the order of 10 −6 ÷10−4 . The corresponding Mach number variation renders this state law very stiff from a numerical viewpoint. Consistently with the fact that viscous effects are usually negligible, at least at a preliminary stage, with respect to the huge dynamic actions typical of modern hydraulic turbomachinery, the 3D Euler equations for an inviscid fluid are considered as governing equations. By virtue of the barotropic state law, the energy balance is decoupled from the mass and momentum balances; hence, it is possible to consider the following reduced set of governing equations: ~ · F(W ~ ∂t W + ∇ )=0

(3)

where W = (ρ, ρu, ρv, ρw) T is the conservative variables vector (u,v and ~ = (∂x , ∂y , ∂z )T , w denoting the components of the velocity vector ~u), ∇ T ~ F(W ) = (Fx , Fy , Fz ) and:       ρw ρv ρu     ρu2 + p    Fz (W ) =  ρuw   Fy (W ) =  ρuv Fx (W ) =  2  ρuv   ρv + p   ρvw  ρw2 + p ρuw ρvw The constitutive law p = p(ρ) provides the closing relation.

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3. Numerical discretisation in non-rotating frames The framework of our numerical formulation is based on a numerical solver (Farhat et al., 1999) dealing with compressible, viscous, perfectgas flows on tetrahedral (unstructured) grids, which has been adapted to a generic barotropic state law. 3.1. Space discretisation As far as the space discretisation is concerned, the finite-volume approach already present in the original solver has been maintained. The semi-discrete balance applied to cell C i centred at node Ni and having volume Vi reads (not accounting for boundary contributions): Vi

X dWi + Φij = 0 dt

(4)

j∈K(i)

where Wi = (ρi , ρi ui , ρi vi , ρi wi )T is the semi-discrete unknown associated with Ci , K(i) represents the set of nodes joined to N i through an edge and Φij denotes the numerical flux crossing the boundary ∂C ij shared by Ci and Cj (positive towards Cj ). Once defined ~νij as the integral over ∂Cij of the outer unit normal to the cell boundary, it is possible to approximate Φij by exploiting a 1D flux function between Wi and Wj , along the direction ~νij . By adopting the Roe numerical flux function (Roe, 1981), as for the original solver, Φij reads: Φij (Wi , Wj , ~νij ) =

 1 ~ ~ (Wj ) · ~νij − 1 |A˜ij |(Wj − Wi ) (5) F (Wi ) + F 2 2

where the Roe matrix A˜ij (Wi , Wj , ~νij ) is diagonalisable with real eigenvalues λij : A˜ij = Tij Diag(λij )Tij−1 ; |A˜ij | is consequently well-defined as |A˜ij | = Tij Diag(|λij |)Tij−1 ; Moreover, the Roe matrix should satisfy the following properties: ~ ˜ij = ∂ F (W ∗ ) · ~νij A Wi ,Wj →W ∗ ∂W lim

  ~ (Wj ) − F(W ~ i ) · ν~ij A˜ij (Wj − Wi ) = F

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A suitable Roe matrix for a generic barotropic fluid has been defined (Sinibaldi et al., 2003) (Sinibaldi et al., 2004), namely:   y x z νij 0 νij νij    x 2  y xu zu  νij a  ˜ − u ˜ X ν ˜ + X ν u ˜ ν ˜ ij ij ij ij ij ij ij ij ij ij   ˜   Aij =   y z 2 x  νy a  νij v˜ij νij v˜ij + Xij νij v˜ij  ij ˜ij − v˜ij Xij    y z 2 x z νij a ˜ij − w ˜ij Xij νij w ˜ij νij w ˜ij νij w ˜ij + Xij where u ˜ij , v˜ij and w ˜ij are the well-known “Roe averages” (Roe, 1981), y x +v z and a Xij = u ˜ij νij ˜ij νij +w ˜ij νij ˜2ij is given by (Sinibaldi et al., 2003):  p −p j i  if | ρj − ρi |> 0  ρj − ρ i 2 a ˜ij =   2 a (ρi ) = a2 (ρj ) otherwise

(6)

As far as the numerical implementation is concerned, the condition in (6) must be replaced with: | ρj − ρi |> , where  is a suitable numerical threshold. If | ρj − ρi |< , then a ˜ 2ij must be computed as a2 (¯ ρij ), where ρ¯ij is an average value tending to ρ(W ∗ ) when both Wi and Wj tend √ to the same value W ∗ (for instance ρ¯ij = ρi ρj ). 3.2. Time discretisation In order to avoid the severe time-step limitations related to common explicit time-advancing techniques, a linearised implicit time-advancing was implemented within the original solver, explicitly relying on the 1st order homogeneity of the flux function (Fezoui and Stoufflet, 1989) holding true for the perfect-gas state law. Since this property is not satisfied in the barotropic case, a new linearisation of the flux function (5) has been proposed (Sinibaldi et al., 2003), namely:  +  − ∆n Φij ∼ ∆n Wi + A˜nij ∆n Wj (7) = A˜nij n n+1 − (·)n , superscript n ˜ ˜ where 2A˜± ij = Aij ± |Aij | and ∆ (·) = (·) denoting the time-level of the fully-discrete solution. The above linearisation is based only on the algebraic properties of the Roe flux function and it is therefore applicable to a variety of problems. It is possible to show (Sinibaldi et al., 2003) that, under some regularity assumptions, the error introduced by (7) is of the order of O(δt2 , δxδt), where δt and δx are characteristic sizes of the time and

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space discretisation, respectively. By substituting (7) into the fullyimplicit scheme: X ∆n Wi Φn+1 =0 (8) Vi n + ij ∆ t j∈K(i)

a linear system is obtained, of the form:         X X X − + ˜nij  ∆n Wj = − ˜nij  ∆n Wi +  Vi I + A Φnij A ∆n t j∈K(i)

j∈K(i)

j∈K(i)

(9)

where I denotes the identity matrix. 3.3. Preconditioning at low Mach numbers It is well known that compressible solvers exhibit accuracy problems when dealing with nearly-incompressible flows. An explanation of this behaviour, for the case of the Roe flux function associated with a perfect-gas state law, has been proposed by (Guillard and Viozat, 1999) by applying an asymptotic analysis in power of a Mach number, M? , characteristic of the nearly-incompressible limit of the flow. By performing the same kind of analysis for a generic barotropic fluid in a 1D context, it has been shown (Sinibaldi et al., 2003) that the nearly-incompressible limit of the semi-discrete solution admits pressure fluctuations in space of the order of M ? , which are larger than those allowed by the analytical one, of the order of M ?2 . In order to counteract this discrepancy, a Turkel-like preconditioning has been adopted, of the same kind of that proposed in (Guillard and Viozat, 1999). A preconditioned flux function is obtained by replacing |A˜ij | in (5) with Pij−1 |Pij A˜ij |: Φprec = ij

 1 ~ ~ (Wj ) · ~νij − 1 P −1 |Pij A˜ij |(Wj − Wi ) F(Wi ) + F 2 2 ij

(10)

where Pij is a Turkel-type preconditioner (Turkel, 1987) which, as a function of the primitive variables W p = (p, u, v, w) T , reads: P (Wp ) = Diag(β 2 , 1, 1, 1) and for a generic barotropic fluid can be recast, as a function of the conservative variables W , as follows:   1 ~0T 2 P (W ) = I + (β − 1) (11) ~u O where ~0 and O denote the null vector and matrix, respectively. The matrix Pij in (10) is obtained by replacing each velocity component in (11) with its corresponding Roe average. The matrix P ij A˜ij is still

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diagonalisable with real eigenvalues; hence, |P ij A˜ij | is well-defined. It has been shown (Sinibaldi et al., 2003) that the preconditioned semidiscrete solution (i.e. that satisfying (4) with Φ prec in place of Φij ) ij recovers the same asymptotic behaviour of the analytical one, provided the free parameter β 2 is of the order of M? . In addition, it has been shown (Sinibaldi et al., 2003) that the linearisation (7) and thus the linearised implicit time-advancing (9) may be extended so as to incor ± porate the preconditioned flux at the only cost of replacing A˜nij  −1  ± with Pijn Pijn A˜nij . It is worth nothing that the proposed preconditioning technique does not destroy the time-consistency of the discrete scheme, since it only affects the “upwind” part of the flux function (10) (i.e. that associated with Wj − Wi ). As a consequence, the proposed preconditioned scheme can be exploited for unsteady computations as well. 4. Governing equations and numerical discretisation in rotating frames The method presented in sec. 3 has been extended to frames rotating with a constant angular velocity ω ~ . To the purpose, the governing equations are recast in the non-inertial frame and the domain boundary is supposed to be axisymmetrical, in order not to deal with moving meshes. The source term −ρ~ω ∧(~ω ∧~x)−2~ω ∧(ρ~u), respectively accounting for centrifugal and Coriolis effects, is added to the right-hand-side of the momentum balance in eq. (3). Once introduced the following position:   Z 1 ~xdV ~σi = −~ω ∧ ω ~∧ V i Ci the corresponding semi-discrete source term S i now appearing on the right-hand-side of eq. (4) reads:   0 Si = V i (12) ρi~σi − 2~ω ∧ ρi ~ui Once introduced the Jacobian Ji = ∂Si /∂Wi and having noticed that it does not depend on the time-level, it is possible to incorporate noninertial effects into the implicit time-advancing (9) by adding S in to right-hand side and −Ji to the coefficient multiplying ∆n Wi . Finally, as for the non-rotating case, the preconditioning technique proposed for nearly-incompressible flows can be straightforwardly incorporated,  −1  ±  ± once A˜n have been replaced with P n P n A˜n . ij

ij

ij

ij

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Figure 1. Schematic representation of the inducer geometry.

5. Applications

A preliminary validation of the proposed numerical ingredients was performed in a 1D context (Sinibaldi et al., 2003) by simulating the quasi-1D water flow in a convergent-divergent nozzle. This validated the proposed adaptation of the Roe scheme as well as the preconditioning technique, which was found to effectively counteract the accuracy problems systematically encountered when dealing with liquid flows at typical conditions (M? ∼ = 10−3 ). As for time-advancing, the proposed implicit technique was found to be very efficient (CFL coefficients up to O(102 )) for fully-wetted flows but, when cavitation occurs, severe stability time-step restrictions appear. The proposed non-rotating scheme was applied to the simulation of the 3D water flow around a NACA0015 hydrofoil (Beux et al., 2005) while the rotating scheme is applied here to the water flow around a turbopump inducer, rotating at 2000 rpm. The considered geometry is sketched in Fig. 1, where the inducer block is denoted by “I”. The nose “N” is part of an axisymmetrical ellipsoid smoothly joining “I” while the after-body “A” is a circular cylinder having a diameter equal to the base diameter of the inducer block. The flow domain is bounded by a cylindrical case, whose diameter is equal to the maximum blade tip diameter D (no tip clearance). The length of the after-body L out as well as the length of the inflow section L in are equal to 1.5D. The domain has been discretised by a tetrahedral unstructured grid having 549139 nodes (2588501 elements). As for the water state law, the free model parameters have been given the following value: TL = 23◦ C, δT /R = 0.1. The dependent parameters in (1) are psat = 2806.82 Pa, ρLsat = 997.29 kg/m3 and

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Figure 2. Pressure contours on the inducer surface: max (red) 177700 Pa, min (blue) 79700 Pa, spacing 5000 Pa.

βsL = 5 10−10 Pa−1 , while those appearing in (2) are g ∗ = 1.67, η = 0.73, γV = 1.28, pc = 2.21 107 Pa and aLsat = 1415 m/s ca. Once chosen a free-stream axial speed k~u ∞ k = 0.48 m/s (M? = 3.40 10−4 ), a non-cavitating as well as a cavitating simulation have been performed by assuming a free-stream pressure p ∞ = 1.15 105 Pa and p∞ = 3.55 104 , respectively. Dirichlet boundary conditions have been imposed at the inlet by upwinding between the inlet cells and a “ghost” state-vector obtained by representing in the rotating frame the free-stream state-vector W∞ = (ρ∞ , ρ∞ u∞ , ρ∞ v∞ , ρ∞ w∞ )T with ρ∞ = ρ(p∞ ). Homogeneous-Neumann conditions have been imposed at the outlet by upwinding between the outlet cells and “ghost” specular state-vectors (i.e. by computing the numerical flux (5) between states Wi and Wj = Wi ). Slip conditions have been weakly imposed at the solid walls (either body or external case). Despite the fact that the parameter β 2 in (11) is assumed to be a constant, a heuristic “local” preconditioner has been exploited by 2 ˜ ij k/˜ aij . replacing β 2 with β 2 (Wi , Wj ) = 1 − e−Mij where Mij = k~u Indeed, when treating rotating flows, even for fully-wetted cases, the local Mach number computed with respect to the rotating frame, may undergo substantial variations along the radial direction, due to dragging velocity and thus it is difficult to identify a unique Mach number M? which is characteristic of the whole flow domain. The non-cavitating simulation has been advanced in time up to a steady-state. The CFL coefficient has been increased during the simulation up to a value of 350, the required total CPU time (IBM Power4) amounting to 1500 hours. The qualitative agreement between

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the resulting flow field and the expected physical trend is good. Indeed, the working fluid gradually undergoes a pressure rise while flowing within the rotating blade vanes, as shown by the pressure contours reported in Fig. 2. According to the simulation, the flow region which is most prone to cavitation is located near that portion of the blades where the volutes, detaching from the hub, firstly reach the external tip diameter D. This is in agreement with experimental results which, for similar flow conditions, observe the cavitation inception exactly in the flow region under consideration. Furthermore, the notable axial backflow occurring near the blade tip where the tip diameter is less than D (i.e. the volutes are not yet completely shrouded) is correctly described by the numerical results, as shown in Fig. 3. This phenomenon is well documented in several experimental works, e.g. (Yamada et al., 2002). From a quantitative point of view, the numerical simulation overestimates the inducer pressure rise, as found in experiments, (d’Agostino, private comm.) by a factor 3 ca. This discrepancy appears to be reasonable in view of the fact that the absence of tip clearance prevents the formation of secondary flows which, in actual flow conditions, considerably reduce the pumping effect (with respect to the considered perfectly shrouded, ideal, case). Furthermore, the inviscid approximation certainly contributes to the overestimate of the inducer pressure rise. The cavitating simulation has been advanced in time only up to the inception stage. As expected, cavitation inception takes place within the minimum pressure region indicated by the non-cavitating simulation. Stability requirements, already highlighted for the nozzle as well as for the hydrofoil test-cases, have imposed to dramatically reduce the time-step (three to four orders of magnitude) thus making it hardly affordable to completely carry on the computation, unless exploiting supercomputing resources not available for the present research. Investigations involving the state law as well as the numerical scheme are in progress, in order to address and cure the efficiency problems the scheme exhibit when applied to cavitating flows.

6. Conclusions and perspectives A numerical method for computing generic barotropic flows in rotating frames has been presented, based on an existing numerical frame designed for perfect-gas flows. The adaptation to a generic barotropic case of the Roe flux function has been discussed. Furthermore, a new linearised implicit timeadvancing technique has been introduced, only relying on the algebraic

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p 175000 170000 165000 160000 155000 150000 145000 140000 135000 130000 125000 120000 115000 110000 105000 100000 95000 90000 85000

40

y

20

0

-20

-40 -20

0

20

40

60

x

Figure 3. Velocity field in a longitudinal cut plane of the flow domain (x: axial component, y: longitudinal component). Pressure contours are drawn in the background.

properties of the Roe flux function and therefore applicable to a variety of problems. The preconditioning technique required by the compressible solver when dealing with nearly-incompressible flows and already present in the original solver, has been adapted to the barotropic implicit scheme and can be exploited for unsteady computations as well. All these numerical ingredients were validated in a 1D context and then applied to the 3D water flow around a hydrofoil. The extension of the proposed scheme to rotating frames has been discussed here, together with its application to the water flow around a 3D geometry typical of turbopump inducers exploited in the feed system of liquid rocket engines. A homogeneous-flow cavitation model explicitly accounting for thermal cavitation effects and for the concentration of the active cavitation nuclei in the pure liquid has been adopted. Both non-cavitating and cavitating simulations have been performed. In non-cavitating conditions the proposed scheme exhibits good efficiency properties and it is reasonably accurate. As far as the cavitating simulations are concerned, stability problems already highlighted in a 1D context as well as in 3D non-rotating computations, impose very severe restrictions on the time-step, thus deteriorating the efficiency of the scheme. Investigations involving the state law as well

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as the numerical scheme are in progress, in order to address and to counteract these problems.

Acknowledgements Support by the Italian Space Agency (ASI) and CIRA, under the FAST2 project, is acknowledged. The authors wish to thank CINECA for the computational support and are grateful to H. Guillard and A. Dervieux for many precious discussions.

References Beux F., Salvetti M.V., Ignatyev A., Li D., Merkle C. and Sinibaldi E., A numerical study of non-cavitating and cavitating liquid flow around a hydrofoil accepted for publication in Mathematical Modelling and Numerical Analysis, 2005. Coutier-Delgosha O., Reboud J.L. and Delannoy Y., Numerical simulation of the unsteady behaviour of cavitating flows International Journal for Numerical Methods in Fluids, 42/5, 527-548, 2003. d’Agostino L., Rapposelli E., Pascarella C. and Ciucci A., A Modified Bubbly Isenthalpic Model for Numerical Simulation of Cavitating Flows. 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Salt Lake City, UT, USA, July 8-11, 2001. Farhat C., Koobus B. and Tran H., Simulation of Vortex Shedding Dominated Flows Past Rigid and Flexible Structures Computational Methods for Fluid-Structure Interaction, Tapir, 1-30, 1999. Fezoui L. and Stoufflet B., A class of implicit upwind schemes for Euler simulations with unstructured meshes Journal of Computational Physics, 84, 174-206, 1989. Guillard H. and Viozat C., On the behaviour of upwind schemes in the low Mach number limit Computers and Fluids, 28, 63-86, 1999. Roe P.L., Approximate Riemann Solvers, Parameters Vectors, and Difference Schemes Journal of Computational Physics, 43, 357-372, 1981. Senocak I. and Shyy W., A pressure-based method for turbulent cavitating flow computations Journal of Computational Physics, 176, 363-383, 2002. Sinibaldi E., Beux F. and Salvetti M.V., A preconditioned implicit Roe’s scheme for barotropic flows: towards simulation of cavitation phenomena INRIA Research Report, N o 4891, 2003. Sinibaldi E., Beux F. and Salvetti M.V., A Preconditioned Compressible Flow Solver for Numerical Simulation of 3D Cavitation Phenomena ECCOMAS 2004, Jyv¨ askyl¨ a, Finland, July 24-28, 2004. Turkel E., Preconditioned methods for solving the incompressible and low speed compressible equations Journal of Computational Physics, 72, 277-289, 1987. Yamada H., Hasegawa S., Watanabe M., Hashimoto T., Kimura T., Takita J. and Kubota I., Observation of the inner flow in the inducer The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii, February 10-14, 2002.

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