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 terms that 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, preserves the natural phase space, and exactly captures the coupling waves between the two phases. Numerical evidences are given to corroborate the validity of our approach.
Mots-clés : two-phase flows, two-fluid two-pressure model, hyperbolic systems, finite volume methods, relaxation schemes, Riemann solvers
@article{M2AN_2009__43_6_1063_0,
author = {Ambroso, Annalisa and Chalons, Christophe and Coquel, Fr\'ed\'eric and Gali\'e, Thomas},
title = {Relaxation and numerical approximation of a two-fluid two-pressure diphasic model},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1063--1097},
publisher = {EDP-Sciences},
volume = {43},
number = {6},
year = {2009},
doi = {10.1051/m2an/2009038},
mrnumber = {2588433},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2009038/}
}
TY - JOUR AU - Ambroso, Annalisa AU - Chalons, Christophe AU - Coquel, Frédéric AU - Galié, Thomas TI - Relaxation and numerical approximation of a two-fluid two-pressure diphasic model JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 1063 EP - 1097 VL - 43 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009038/ DO - 10.1051/m2an/2009038 LA - en ID - M2AN_2009__43_6_1063_0 ER -
%0 Journal Article %A Ambroso, Annalisa %A Chalons, Christophe %A Coquel, Frédéric %A Galié, Thomas %T Relaxation and numerical approximation of a two-fluid two-pressure diphasic model %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 1063-1097 %V 43 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009038/ %R 10.1051/m2an/2009038 %G en %F M2AN_2009__43_6_1063_0
Ambroso, Annalisa; Chalons, Christophe; Coquel, Frédéric; Galié, Thomas. Relaxation and numerical approximation of a two-fluid two-pressure diphasic model. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1063-1097. doi : 10.1051/m2an/2009038. http://www.numdam.org/articles/10.1051/m2an/2009038/
[1] and , Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361-396. | MR | Zbl
[2] , , , , , and , The drift-flux asymptotic limit of barotropic two-phase two-pressure models. Comm. Math. Sci. 6 (2008) 521-529. | MR | Zbl
[3] , Analytical and numerical investigation of two-phase flows. Ph.D. Thesis, Univ. Magdeburg, Germany (2003).
[4] and , The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195 (2004) 434-464. | MR | Zbl
[5] , and , A simple method for compressible multiphase mixtures and interfaces. Int. J. Numer. Methods Fluids 41 (2003) 109-131. | MR | Zbl
[6] and , A two phase mixture theory for the deflagration to detonation (DDT) transition in reactive granular materials. Int. J. Multiphase Flows 12 (1986) 861-889. | Zbl
[7] , , , Numerical approximation of a degenerate non-conservative multifluid model: relaxation scheme. Int. J. Numer. Methods Fluids 48 (2005) 85-90. | Zbl
[8] , Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhauser (2004). | MR | Zbl
[9] , Modélisation numérique d'écoulements multiphasiques pour des fluides compressibles, non miscibles et soumis aux effets capillaires. Ph.D. Thesis, Université Bordeaux I, France (2007).
[10] , and , A sequel to a rough Godunov scheme. Application to real gas flows. Comput. Fluids 29 (2000) 813-847. | MR | Zbl
[11] and , A Riemann solver and upwind methods for a two-phase flow model in nonconservative form. Int. J. Numer. Methods Fluids 50 (2006) 275-307. | MR | Zbl
[12] and , Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes. Numer. Math. 101 (2005) 451-478. | MR | Zbl
[13] and , Relaxation approximation of the Euler equations. J. Math. Anal. Appl. 348 (2008) 872-893. | MR | Zbl
[14] , , , and , Numerical methods using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136 (1997) 272-288. | MR | Zbl
[15] , , , and , 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). | MR | Zbl
[16] , , and , Closure laws for a two-phase two-pressure model. C. R. Math. 334 (2002) 927-932. | MR | Zbl
[17] and , Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71 (1988) 93-122. | MR | Zbl
[18] and , Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4 (1992) 279-312. | MR | Zbl
[19] , Couplage interfacial de modèles en dynamique des fluides. Application aux écoulements diphasiques. Ph.D. Thesis, Université Pierre et Marie Curie, France (2008).
[20] , and , Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Mod. Meth. Appl. Sci. 14 (2004) 663-700. | MR | Zbl
[21] and , Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175 (2002) 326-360. | MR | Zbl
[22] , and , Two phase flow modelling of a fluid mixing layer. J. Fluid Mech. 378 (1999) 119-143. | MR
[23] and , The Riemann problem for a class of resonant nonlinear systems of balance laws. Ann. Inst. H. Poincaré, Anal. Non linéaire 21 (2004) 881-902. | EuDML | Numdam | MR | Zbl
[24] and , Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag (1996). | MR | Zbl
[25] , 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).
[26] and , The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995) 235-276. | MR | Zbl
[27] , , , and , Two phase modeling of DDT: structure of the velocity-relaxation zone. Phys. Fluids 9 (1997) 3885-3897.
[28] , , and , Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN 38 (2004) 477-493. | EuDML | Numdam | MR | Zbl
[29] , Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form. Commun. Partial Differ. Equ. 13 (1988) 669-727. | MR | Zbl
[30] , Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint # 593, Institute for Math. and its Appl., Minneapolis, USA (1989).
[31] and , The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Comm. Math. Sci. 1 (2003) 763-796. | MR | Zbl
[32] , Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation. Comput. Fluids 36 (2007) 1061-1080. | Zbl
[33] , 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.
[34] , Calculation of interaction of non-steady shock waves with obstacles. J. Comp. Math. Phys. USSR 1 (1961) 267-279. | MR
[35] and , A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. | MR | Zbl
[36] and , A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation. J. Fluid Mech. 431 (2001) 239-271. | Zbl
[37] , and , The Riemann problem and high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212 (2006) 490-526. | MR | Zbl
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