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The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. (English) Zbl 1161.76531
Summary: This paper considers the Riemann problem and an associated Godunov method for a model of compressible two-phase flow. The model is a reduced form of the well-known Baer-Nunziato model that describes the behavior of granular explosives. In the analysis presented here, we omit source terms representing the exchange of mass, momentum and energy between the phases due to compaction, drag, heat transfer and chemical reaction, but retain the non-conservative nozzling terms that appear naturally in the model. For the Riemann problem the effect of the nozzling terms is confined to the contact discontinuity of the solid phase. Treating the solid contact as a layer of vanishingly small thickness within which the solution is smooth yields jump conditions that connect the states across the contact, as well as a prescription that allows the contribution of the nozzling terms to be computed unambiguously. An iterative method of solution is described for the Riemann problem, that determines the wave structure and the intermediate states of the flow, for given left and right states. A Godunov method based on the solution of the Riemann problem is constructed. It includes non-conservative flux contributions derived from an integral of the nozzling terms over a grid cell. The Godunov method is extended to second-order accuracy using a method of slope limiting, and an adaptive Riemann solver is described and used for computational efficiency. Numerical results are presented, demonstrating the accuracy of the numerical method and in particular, the accurate numerical description of the flow in the vicinity of a solid contact where phases couple and nozzling terms are important. The numerical method is compared with other methods available in the literature and found to give more accurate results for the problems considered.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76T25 Granular flows
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics
76N99 Compressible fluids and gas dynamics
Software:
HE-E1GODF
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References:
[1] 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
[2] Butler, P.B.; Krier, H., Analysis of deflagration-to-detonation transition in high-energy solid propellants, Combust. flame, 63, 31, (1986)
[3] Gokhale, S.S.; Krier, H., Modeling of unsteady two-phase reactive flow in porous beds of propellants, Prog. energy combust. sci., 8, 1, (1982)
[4] Powers, J.M.; Stewart, D.S.; Krier, H., Theory of two-phase detonation, part I: modeling, Combust. flame, 80, 264, (1990)
[5] Powers, J.M.; Stewart, D.S.; Krier, H., Theory of two-phase detonation, part II: structure, Combust. flame, 80, 280, (1990)
[6] Asay, B.W.; Son, S.F.; Bdzil, J.B., The role of gas permeation in convective burning, Int. J. multiphase flow, 23, 5, 923-952, (1996) · Zbl 1135.76350
[7] Godunov, S.K., Difference methods for the numerical calculation of the equations of fluid dynamics, Mater. sb., 47, 271-306, (1959) · Zbl 0171.46204
[8] 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
[9] Bell, J.; Colella, P.; Trangenstein, J., Higher order Godunov methods for general systems of hyperbolic conservation laws, J. comput. phys., 82, 362-397, (1989) · Zbl 0675.65090
[10] Embid, P.; Baer, M., Mathematical analysis of a two-phase continuum mixture theory, Continuum mech. thermodyn., 4, 279-312, (1992) · Zbl 0760.76096
[11] Kapila, A.K.; Son, S.F.; Bdzil, J.B.; Menikoff, R.; Stewart, D.S., Two-phase modeling of DDT: structure of the velocity-relaxation zone, Phys. fluids, 9, 12, 3885-3897, (1997)
[12] Bdzil, J.B.; Menikoff, R.; Son, S.F.; Kapila, A.K.; Stewart, D.S., Two-phase modeling of deflagration-to-detonation transition in granular materials: a critical examination of modeling issues, Phys. fluids, 11, 2, 378-402, (1999) · Zbl 1147.76317
[13] Kapila, A.K.; Menikoff, R.; Bdzil, J.B.; Son, S.F.; Stewart, D.S., Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Phys. fluids, 13, 10, 3002-3024, (2001) · Zbl 1184.76268
[14] J.B. Bdzil, S.F. Son, Engineering models of deflagration-to-detonation transition, LANL Report LA-12794-MS. · Zbl 1184.76268
[15] 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
[16] Gonthier, K.A.; Powers, J.M., A numerical investigation of transient detonation in granulated material, Shock waves, 6, 183-195, (1996) · Zbl 0867.76057
[17] 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
[18] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. comput. phys., 150, 425-467, (1999) · Zbl 0937.76053
[19] N. Andrianov, R. Saurel, G. Warnecke, A simple method for compressible multiphase mixtures and interfaces, Technical Report 4247, INRIA, 2001. · Zbl 1025.76025
[20] 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
[21] Gavrilyuk, S.; Saurel, R., Mathematical and numerical modeling of two-phase compressible flows with micro-inertia, J. comput. phys., 175, 326-360, (2002) · Zbl 1039.76067
[22] Abgrall, R.; Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures, J. comput. phys., 186, 361-396, (2003) · Zbl 1072.76594
[23] Abgrall, R., How to prevent pressure oscillations in mulicomponent flow calculations: a quasi conservative approach, J. comput. phys., 125, 150-160, (1996) · Zbl 0847.76060
[24] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004
[25] LeVeque, R.J., Numerical methods for conservation laws, (1992), Birkhäuser Basel · Zbl 0847.65053
[26] Powers, J.M., Two-phase viscous modeling of compaction in granular explosives, Phys. fluids, 16, 8, 2975-2990, (2004) · Zbl 1186.76430
[27] 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
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