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A multiphase Godunov method for compressible multifluid and multiphase flows. (English) Zbl 0937.76053
Summary: We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: 1) one of the equations is written in non-conservative form; 2) non-conservative terms exist in the momentum and energy equations. We propose a robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. Several test cases provide reliable results. The method is also able to compute strong shock waves and deal with complex equations of state. $$\copyright$$ Academic Press.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 76T99 Multiphase and multicomponent flows 80A22 Stefan problems, phase changes, etc.
HLLE
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##### References:
 [1] Abgrall, R, Generalization of the roe scheme for the computation of a mixture of perfect gases, Rech. Aérospat., 6, (Dec. 1988) [2] Abgrall, R, How to prevent pressure oscillations in multicomponent flow calculations: A quasi conservative approach, J. comput. phys., 125, 150-160, (1996) · Zbl 0847.76060 [3] Baer, M.R; Nunziato, J.W, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. multiphase flow, 12, 861-889, (1986) · Zbl 0609.76114 [4] Baer, M.R, A numerical study of shock wave reflections on low density foam, Shock waves, 2, 121-124, (1992) [5] G. Baudin, Quercy, un programme de calcul thermochimique rapide pour évaluer l’état de Chapman-Jouguet et d’isentrope des explosifs condensés, 1992 [6] P. Bauer, O. Heuzé, A simple method for the calculation of detonation properties of CHNO explosives, Proceedings, International Symposium on High Dynamic Pressures, La Grande Motte, France, 1989, 225, 232 [7] Butler, P.B; Lambeck, M.F; Krier, H, Modeling of shock development and transition to detonation initiated by burning in porous propellant beds, Combust. flame, 46, 75-93, (1982) [8] Cocchi, J.P; Saurel, R, A Riemann problem based method for compressible multifluid flows, J. comput. phys., 137, 265-298, (1997) · Zbl 0934.76055 [9] G. Cochran, J. Chan, Shock initiation and detonation models in one and two dimensions, Lawrence Livermore National Laboratory Report, 1979 [10] Coquel, F; El Amine, K; Godlewski, E; Perthame, B; Rascle, P, A numerical method using upwind schemes for the resolution of two-phase flows, J. comput. phys., 136, 272-288, (1997) · Zbl 0893.76052 [11] Davis, S.F, Simplified second order Godunov type methods, SIAM J. sci. statist. comput., 9, 445-473, (1988) · Zbl 0645.65050 [12] Delhaye, J.M; Bouré, J.A, General equations and two-phase flow modeling, Handbook of multiphase systems, 1, 36-95, (1982) [13] Drew, D.A, Mathematical modeling of two-phase flows, Annu. rev. fluid mech., 15, 261-291, (1983) · Zbl 0542.76134 [14] R. P. Fedkiw, A. Marquina, B. Merriman, An isobaric fix for the overheating problem in multimaterial compressible flows, 1998 · Zbl 0933.76075 [15] R. P. Fedkiw, B. Merriman, S. Osher, Simplified upwind discretisation of systems of hyperbolic conservation laws containing advection equations, with applications to compressible flows of multiphase, chemically reacting and explosive materials, 1998 [16] E. Fried, CHEETAH 1.39 User’s Manual, Energetic Material Center, Lawrence Livermore National Laboratory, 1996 [17] Gallouet, T; Masella, J.M, Un schéma de Godunov approché, C.R. acad. sci. Paris Sér. I, 323, 77-84, (1996) · Zbl 0856.76045 [18] Godunov, S.K, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 357-393, (1959) [19] Godunov, S.K; Zabrodine, A; Ivanov, M; Kraiko, A; Prokopov, G, Résolution numérique des problèmes multidimensionnels de la dynamique des gaz, (1979) [20] H. Guillard, C. Viozat, On the behavior of upwind schemes in the low Mach number limit, Comput. & Fluids, 28, 63, 86 · Zbl 0963.76062 [21] Haas, J.F, Interaction of a weak shock wave and discrete gas inhomogeneities, (1984) [22] Haas, J.F; Sturtevant, B, Interaction of a weak shock wave with cylindrical and spherical gas inhomogeneities, J. fluid mech., 181, 41-76, (1987) [23] Harten, A; Lax, P.D; van Leer, B, On upstream differencing and Godunov type schemes for hyperbolic conservation laws, SIAM rev., 25, 33-61, (1983) · Zbl 0565.65051 [24] F. Harlow, A. Amsden, Fluid dynamics, LANL Monograph LA-4700, 1971 [25] O. Heuzé, Mémoire d’habilitation à diriger des recherches, Université d’Orléans, 1997 [26] Karni, S, Multicomponent flow calculations by a consistent primitive algorithm, J. comput. phys., 112, 31-43, (1994) · Zbl 0811.76044 [27] E. L. Lee, H. C. Horning, J. W. Kury, Adiabatic expansion of high explosives detonation products, Lawrence Radiation Lab. University of California, Livermore, 1968 [28] J. Massoni, R. Saurel, G. Demol, G. Baudin, A Mechanistic model for shock to detonation transition in solid energetic materials, Phys. Fluids · Zbl 1147.76456 [29] Marsh, S.P, LASL shock hugoniot data, (1980) [30] Powers, J.M; Stewart, D.S; Krier, H, Theory of two-phase detonation. part I. modeling, Combust. & flame, 80, 264-279, (1990) [31] V. H. Ransom, Numerical benchmark tests, Multiphase Science and Technology, HewittDelhayeZuber, Hemisphere, Washington, DC, 1987, 3 [32] Rogue, X, Expériences et simulations d’écoulements diphasiques en tube á choc, (1997) [33] Rogue, X; Rodriguez, G; Haas, J.F; Saurel, R, Experimental and numerical investigation of the shock-induced fluidization of a particle bed, Shock waves, 8, 29-45, (1998) · Zbl 0899.76024 [34] Rusanov, V.V, Calculation of interaction of non-steady shock waves with obstacles, J. comput. math. & phys. USSR, 1, 267-279, (1961) [35] Sainsaulieu, L, Finite volume approximations of two phase fluid flows based on an approximate roe-type Riemann solver, J. comput. phys., 121, 1-28, (1995) · Zbl 0834.76070 [36] Saurel, R; Larini, M; Loraud, J.C, Ignition and growth of a detonation by a high energy plasma, Shock waves, 2, 91-102, (1992) · Zbl 0775.76222 [37] Saurel, R; Daniel, E; Loraud, J.C, Two-phase flows: second order scheme and boundary conditions, Aiaa j., 32, 1214-1221, (1994) · Zbl 0811.76051 [38] Saurel, R; Larini, M; Loraud, J.C, Exact and approximate Riemann solver for real gases, J. comput. phys., 112, 126-137, (1994) · Zbl 0799.76058 [39] Saurel, R, Numerical analysis of ram accelerator employing two-phase combustion, AIAA J. propulsion & power, 12, 708-717, (1996) [40] R. Saurel, R. Abgrall, A simple method for compressible multifluid flows, SIAM J. Sci. Comput. · Zbl 0957.76057 [41] Saurel, R; Massoni, J, On Riemann problem based methods for detonation in solid energetic materials, Int. J. on numer. methods in fluids, 26, 101-121, (1998) · Zbl 0906.76057 [42] R. Saurel, T. Gallouet, Modèles et méthodes numériques pour les écoulements fluides, Cours de DEA, Centre de Mathématiques et d’Informatique, Université de Provence, 1998 [43] R. Saurel, J. P. Cocchi, P. B. Butler, A numerical study of the cavitation effects in the wake of a hypervelocity underwater projectile, AIAA J. Propulsion & Power [44] K. M. Shyue, An efficient shock-capturing algorithm for compressible multicomponent problems, J. Comput. Phys. 142, 208, 1998, 1997 · Zbl 0934.76062 [45] Strang, On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 506-517, (1968) · Zbl 0184.38503 [46] E. F. Toro, Riemann-problem based techniques for computing reactive two-phase flows, Proceedings, Third International Conference on Numerical Combustion, Antibes, France, May 1989, DervieuxLarrouturrou, Springer-Verlag, New York/Berlin, 1989, 351, 472, 481 [47] E. F. Toro, Some IVPs for which conservative methods fail miserably, Proceedings, 6th International Symposium on Computational Fluid Dynamics, Lake Tahoe, California, September 4-8, 1995. [48] Toro, E.F, Riemann solvers and numerical methods for fluid dynamics, (1997) · Zbl 0888.76001 [49] Toumi, I; Kumbaro, A, An approximate linearized Riemann solver for a two-fluid model, J. comput. phys., 124, 286-300, (1996) · Zbl 0847.76056 [50] van Leer, B, Toward the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, J. comput. phys., 32, 101, (1979) · Zbl 1364.65223
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