zbMATH — the first resource for mathematics

An exact Riemann solver for compressible two-phase flow models containing non-conservative products. (English) Zbl 1216.76044
Summary: In this article we present a new numerical procedure for solving exactly the Riemann problem of compressible two-phase flow models containing non-conservative products. These products appear in the expressions for the interactions between the two phases. Thus, in the compressible limit, the governing equations are hyperbolic but can not be written as conservation laws, i.e. in divergence form. In general, the solution to the Riemann problem of these models contains six distinct centered waves. According to the relative position of these waves in the \(x-t\) plane, the possible solutions can be classified into four principal configurations. The Riemann solver we propose herein investigates sequentially each of these configurations until an admissible solution is calculated. Special configurations, corresponding to coalescence of waves, are also analyzed and included in the solver. Further, we examine the accuracy and robustness of three known methods for the integration of the non-conservative products, via a series of numerical tests. Finally, the issue of existence and uniqueness of solutions to the Riemann problem is discussed.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76T99 Multiphase and multicomponent flows
Full Text: DOI
[1] Drew, D.A.; Passman, S.L., Theory of multicomponent fluids, (1998), Springer New York · Zbl 0919.76003
[2] Enwald, H.; Peirano, E.; Almstedt, A.E., Eulerian two-phase flow theory applied to fluidization, Int. J. multiphase flow, 222, 21-66, (1986) · Zbl 1135.76409
[3] Baer, M.R.; Nunziato, J.W., A two-phase mixture theory for the deflagration-to-detonation transition in reactive granular materials, Int. J. multiphase flow, 12, 861-889, (1986) · Zbl 0609.76114
[4] Powers, J.M.; Stewart, D.S.; Krier, H., Theory of two-phase detonation – part II: structure, Combust. flame, 80, 280-303, (1990)
[5] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. comp. phys., 150, 425-467, (1999) · Zbl 0937.76053
[6] Bdzil, J.B.; Menikoff, R.; Son, S.F.; Kapila, A.K.; Stewart, D.S., Two-phase modelling of deflagration-to-detonation transition in granular materials: a critical examination of modelling issues, Phys. fluids, 11, 378-402, (1999) · Zbl 1147.76317
[7] Papalexandris, M.V., A two-phase model for compressible granular flows based on the theory of irreversible processes, J. fluid mech., 517, 103-112, (2004) · Zbl 1063.76102
[8] Gonthier, K.A.; Powers, J.M., A high-resolution numerical method for a two-phase model of deflagration-to-detonation transition, J. comput. phys., 163, 376-433, (2000) · Zbl 0995.76062
[9] Papalexandris, M.V., Numerical simulations of detonations in mixtures of gases and solid particles, J. fluid mech., 507, 95-142, (2004) · Zbl 1065.76199
[10] Dal Maso, G.; Lefloch, P.G.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068
[11] Lefloch, P.G.; Tzavaras, A.E., Representation of weak limits and definition of nonconservative products, SIAM J. math. anal., 30, 1309-1342, (1999) · Zbl 0939.35115
[12] Crasta, G.; Lefloch, P.G., Existence result for a class of nonconservative and nonstrictly hyperbolic systems, Commun. pure appl. anal., 1, 1-18, (2002) · Zbl 1031.35096
[13] Andrianov, N.; Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model, J. comput. phys., 195, 434-464, (2004) · Zbl 1115.76414
[14] Embid, P.; Baer, M., Mathematical analysis of a two-phase continuum mixture theory, Continuum mech. thermodyn., 4, 279-312, (1992) · Zbl 0760.76096
[15] Saurel, R.; Lemetayer, O., A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation, J. fluid mech., 431, 239-271, (2001) · Zbl 1039.76069
[16] Gavrilyuk, S.; Saurel, R., A compressible multiphase model with microinertia, J. comput. phys., 175, 326-360, (2002)
[17] 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
[18] Schwendeman, D.W.; Wahle, C.W.; Kapila, A.K., The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J. comput. phys., 212, 490-526, (2006) · Zbl 1161.76531
[19] Lax, P.D., Hyperbolic systems of conservation laws II, Commun. pure appl. math., 10, 537-567, (1957) · Zbl 0081.08803
[20] Keyfitz, B.L.; Kranzer, H.C., A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. rat. mech. anal., 72, 219-241, (1980) · Zbl 0434.73019
[21] Keyfitz, B.L.; Kranzer, H.C., The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. differ. equations, 72, 35-65, (1983) · Zbl 0521.35035
[22] Schaeffer, D.G.; Shearer, M., The classification of 2×2 systems of non-strictly hyperbolic conservation laws with applications to oil recovery, Commun. pure appl. math., 40, 141-178, (1987) · Zbl 0673.35073
[23] Isaacson, E.; Temple, B., Nonlinear resonance in systems of conservation laws, SIAM J. appl. math., 52, 1260-1278, (1992) · Zbl 0794.35100
[24] Colella, P.; Glaz, H.M., Efficient solution algorithms for the Riemann problem for real gases, J. comput. phys., 59, 264-289, (1985) · Zbl 0581.76079
[25] Fryxell, B.; Olson, K.; Ricker, P.; Timmes, F.X.; Zingale, M.; Lamb, D.Q.; MacNeice, P.; Rosner, R.; Truran, J.W.; Tufo, H., FLASH: an adaptive-mesh hydrodynamics code for modelling astrophysical thermonuclear flashes, Astroph. J. supp. series, 131, 273-334, (2000)
[26] B. Braconnier, Université Bordeaux, Private communication, 2005.
[27] van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223
[28] Lefloch, P.G.; Thanh, M.D., The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun. math. sci., 1, 763-796, (2003) · Zbl 1091.35044
[29] Andrianov, N., Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations, Int. J. numer. meth. fluids, 47, 825-831, (2005) · Zbl 1134.76402
[30] Dafermos, C., The entropy rate admissibility criterion for solutions of hyperbolic conservations laws, J. differ. equations, 14, 202-212, (1973) · Zbl 0262.35038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.