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Relaxation and numerical approximation of a two-fluid two-pressure diphasic model

Published online by Cambridge University Press:  09 October 2009

Annalisa Ambroso ,
Christophe Chalons ,
Frédéric Coquel  and
Thomas Galié
Annalisa Ambroso
Affiliation:
DEN/DANS/DM2S/SFME/LETR CEA-Saclay, 91191 Gif-sur-Yvette, France. [email protected]
Christophe Chalons
Affiliation:
DEN/DANS/DM2S/SFME/LETR CEA-Saclay, 91191 Gif-sur-Yvette, France. [email protected] Université Paris 7-Denis Diderot and UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. [email protected]
Frédéric Coquel
Affiliation:
Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. [email protected] CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Thomas Galié
Affiliation:
DEN/DANS/DM2S/SFME/LETR CEA-Saclay, 91191 Gif-sur-Yvette, France. [email protected]
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Abstract

This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows.We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative termsthat are inherently present in the set of convective equations and that couple the two phases.In particular, the proposed approximate Riemann solver is given by explicit formulas, preservesthe natural phase space, and exactly captures the coupling waves between the two phases.Numerical evidences are given to corroborate the validity of our approach.

Keywords

Two-phase flowstwo-fluid two-pressure modelhyperbolic systemsfinite volume methodsrelaxation schemesRiemann solvers.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1063 - 1097
Copyright
© EDP Sciences, SMAI, 2009

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References

Abgrall, R. and Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361–396. CrossRef
Ambroso, A., Chalons, C., Coquel, F., Galié, T., Godlewski, E., Raviart, P.-A and Seguin, N., The drift-flux asymptotic limit of barotropic two-phase two-pressure models. Comm. Math. Sci. 6 (2008) 521–529. CrossRef
N. Andrianov, Analytical and numerical investigation of two-phase flows. Ph.D. Thesis, Univ. Magdeburg, Germany (2003).
Andrianov, N. and Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195 (2004) 434–464. CrossRef
Andrianov, N., Saurel, R. and Warnecke, G., A simple method for compressible multiphase mixtures and interfaces. Int. J. Numer. Methods Fluids 41 (2003) 109–131. CrossRef
Baer, M.R. and Nunziato, J.W., A two phase mixture theory for the deflagration to detonation (DDT) transition in reactive granular materials. Int. J. Multiphase Flows 12 (1986) 861–889. CrossRef
Berthon, C., Braconnier, B., Nkonga, B., Numerical approximation of a degenerate non-conservative multifluid model: relaxation scheme. Int. J. Numer. Methods Fluids 48 (2005) 85–90. CrossRef
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhauser (2004).
B. Braconnier, Modélisation numérique d'écoulements multiphasiques pour des fluides compressibles, non miscibles et soumis aux effets capillaires. Ph.D. Thesis, Université Bordeaux I, France (2007).
Buffard, T., Gallouët, T. and Hérard, J.M., A sequel to a rough Godunov scheme. Application to real gas flows. Comput. Fluids 29 (2000) 813–847. CrossRef
Castro, C.E. and Toro, E.F., Riemann, A solver and upwind methods for a two-phase flow model in nonconservative form. Int. J. Numer. Methods Fluids 50 (2006) 275–307. CrossRef
Chalons, C. and Coquel, F., Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes. Numer. Math. 101 (2005) 451–478. CrossRef
Chalons, C. and Coulombel, J.F., Relaxation approximation of the Euler equations. J. Math. Anal. Appl. 348 (2008) 872–893. CrossRef
Coquel, F., El Amine, K., Godlewski, E., Perthame, B. and Rascle, P., Numerical methods using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136 (1997) 272–288. CrossRef
F. Coquel, E. Godlewski, A. In, B. Perthame and P. Rascle, Some new Godunov and relaxation methods for two phase flows, in Proceedings of the International Conference on Godunov methods: theory and applications, Kluwer Academic, Plenum Publisher (2001).
Coquel, F., Gallouët, T., Hérard, J.M. and Seguin, N., Closure laws for a two-phase two-pressure model. C. R. Math. 334 (2002) 927–932. CrossRef
Dubois, F. and LeFloch, P.G., Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71 (1988) 93–122. CrossRef
Embid, P. and Baer, M., Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4 (1992) 279–312. CrossRef
T. Galié, Couplage interfacial de modèles en dynamique des fluides. Application aux écoulements diphasiques. Ph.D. Thesis, Université Pierre et Marie Curie, France (2008).
Gallouët, T., Hérard, J.M. and Seguin, N., Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Mod. Meth. Appl. Sci. 14 (2004) 663–700. CrossRef
Gavrilyuk, S. and Saurel, R., Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175 (2002) 326–360. CrossRef
Glimm, J., Saltz, D. and Sharp, D.H., Two phase flow modelling of a fluid mixing layer. J. Fluid Mech. 378 (1999) 119–143. CrossRef
P. Goatin and P.G. LeFloch, The Riemann problem for a class of resonant nonlinear systems of balance laws. Ann. Inst. H. Poincaré, Anal. Non linéaire 21 (2004) 881–902. CrossRef
E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag (1996).
V. Guillemaud, Modélisation et simulation numérique des écoulements diphasiques par une approche bifluide à deux pressions. Ph.D. Thesis, Université de Provence, Aix-Marseille 1, France (2007).
Jin, S. and Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995) 235–276. CrossRef
Kapila, A.K., Son, S.F., Bdzil, J.B., Menikoff, R. and Stewart, D.S., Two phase modeling of DDT: structure of the velocity-relaxation zone. Phys. Fluids 9 (1997) 3885–3897. CrossRef
Karni, S., Kirr, E., Kurganov, A. and Petrova, G., Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN 38 (2004) 477–493. CrossRef
LeFloch, P.G., Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form. Commun. Partial Differ. Equ. 13 (1988) 669–727.
P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint # 593, Institute for Math. and its Appl., Minneapolis, USA (1989).
LeFloch, P.G. and Thanh, M.D., The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Comm. Math. Sci. 1 (2003) 763–796. CrossRef
Munkejord, S.T., Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation. Comput. Fluids 36 (2007) 1061–1080. CrossRef
V.H. Ransom, Numerical benchmark tests, in Multiphase science and technology, Vol. 3, G.F. Hewitt, J.M. Delhaye and N. Zuber Eds., Washington, USA, Hemisphere/Springer (1987) 465–467.
Rusanov, V.V, Calculation of interaction of non-steady shock waves with obstacles. J. Comp. Math. Phys. USSR 1 (1961) 267–279.
Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. CrossRef
Saurel, R. and Lemetayer, O., A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation. J. Fluid Mech. 431 (2001) 239–271. CrossRef
Schwendeman, D.W., Wahle, C.W. and Kapila, A.K, The Riemann problem and high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212 (2006) 490–526. CrossRef