Skip to main content Accessibility help

A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

  • Frédéric Coquel (a1), Jean-Marc Hérard (a2), Khaled Saleh (a2) (a3) (a4) and Nicolas Seguin (a3) (a4) (a5)


We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.



Hide All
[1] Ambroso, A., Chalons, C., Coquel, F. and Galié, T., Relaxation and numerical approximation of a two-fluid two-pressure diphasic model. ESAIM: M2AN 43 (2009) 10631097.
[2] Ambroso, A., Chalons, C., Coquel, F., Godlewski, E., Lagoutière, F., P-A. Raviart and N. Seguin, The coupling of homogeneous models for two-phase flows. Int. J. Finite 4 (2007) 39.
[3] Ambroso, A., Chalons, C. and Raviart, P.-A., A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput. Fluids 54 (2012) 6791.
[4] Andrianov, N. and Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195 (2004) 434464.
[5] Baer, M.R. and Nunziato, J.W., A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12 (1986) 861889.
[6] C. Berthon, F. Coquel and P.G. LeFloch, Why many theories of shock waves are necessary: kinetic relations for non-conservative systems, in vol. 142 of Proc. R. Soc. Edinburgh, Section: A Mathematics (2012) 1–37.
[7] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004).
[8] Bouchut, F. and James, F., Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness. Commun. Partial Differ. Eqs. 24 (1999) 21732189.
[9] Boutin, B., Coquel, F. and LeFloch, P.G., Coupling nonlinear hyperbolic equations (iii). A regularization method based on thick interfaces. SIAM J. Numer. Anal. 51 (2013) 11081133.
[10] C. Chalons, F. Coquel, S. Kokh and N. Spillane, Large time-step numerical scheme for the seven-equation model of compressible two-phase flows, in vol. 4 of Springer Proceedings in Mathematics, FVCA 6 (2011) 225–233.
[11] Chen, G-Q., Levermore, C.D. and Liu, T-P., Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787830.
[12] F. Coquel, T. Gallouët, J.-M. Hérard and N. Seguin, Closure laws for a two-fluid two pressure model. C. R. Acad. Sci. I-334 (2002) 927–932.
[13] F. Coquel, E. Godlewski, B. Perthame, A. In and P. Rascle, Some new Godunov and relaxation methods for two-phase flow problems, in Godunov methods (Oxford, 1999). Kluwer/Plenum, New York (2001) 179–188.
[14] Coquel, F., Godlewski, E. and Seguin, N., Relaxation of fluid systems. Math. Models Methods Appl. Sci. 22 (2012).
[15] Coquel, F., Hérard, J.-M. and Saleh, K., A splitting method for the isentropic Baer-Nunziato two-phase flow model. ESAIM: Proc., 38 (2012) 241256.
[16] Coquel, F., Hérard, J.-M., Saleh, K. and Seguin, N., Two properties of two-velocity two-pressure models for two-phase flows. Commun. Math. Sci. (2013) 11.
[17] F. Coquel, K. Saleh and N. Seguin, A Robust and Entropy-Satisfying Numerical Scheme for Fluid Flows in Discontinuous Nozzles. (2013).
[18] Deledicque, V. and Papalexandris, M.V., A conservative approximation to compressible two-phase flow models in the stiff mechanical relaxation limit. J. Comput. Phys. 227 (2008) 92419270.
[19] Dumbser, M., Hidalgo, A., Castro, M., Parés, C. and Toro, E.F., FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Engrg. 199 (2010) 625647.
[20] Embid, P. and Baer, M., Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4 (1992) 279312.
[21] Gallouët, T., Hérard, J.-M. and Seguin, N., Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Models Methods Appl. Sci. 14 (2004) 663700.
[22] Gavrilyuk, S. and Saurel, R., Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175 (2002) 326360.
[23] Goatin, P. and LeFloch, P.G., The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Institut. Henri Poincaré Anal. Non Linéaire 21 (2004) 881902.
[24] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, in vol. 118 of Appl. Math. Sci. Springer-Verlag, New York (1996).
[25] Hanouzet, B. and Natalini, R., Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rational Mech. Anal. 169 (2003) 89117.
[26] Harten, A., Lax, P.D. and van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 3561.
[27] Hérard, J.-M. and Hurisse, O., A fractional step method to compute a class of compressible gas-luiquid flows. Comput. Fluids. Int. J. 55 (2012) 5769.
[28] Isaacson, E. and Temple, B., Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625640.
[29] 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) 38853897.
[30] Kawashima, S. and Yong, W.-A., Dissipative structure and entropy for hyperbolic systems of balance laws. Archive for Rational Mech. Anal. 174 (2004) 345364.
[31] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint 593, IMA, Minneapolis (1991).
[32] Y. Liu, Ph.D. thesis. Université Aix-Marseille, to appear in (2013).
[33] L. Sainsaulieu, Contribution à la modélisation mathématique et numérique des écoulements diphasiques constitués d’un nuage de particules dans un écoulement de gaz. Thèse d’habilitation à diriger des recherches. Université Paris VI (1995).
[34] K. Saleh, Analyse et Simulation Numérique par Relaxation d’Ecoulements Diphasiques Compressibles. Contribution au Traitement des Phases Evanescentes. Ph.D. thesis. Université Pierre et Marie Curie, Paris VI (2012).
[35] Saurel, R. and Abgrall, R., A multiphase godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425467.
[36] Schwendeman, D.W., Wahle, C.W. and Kapila, A.K., The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212 (2006) 490526.
[37] Thanh, M.D., Kröner, D. and Chalons, C., A robust numerical method for approximating solutions of a model of two-phase flows and its properties. Appl. Math. Comput. 219 (2012) 320344.
[38] Thanh, M.D., Kröner, D. and Nam, N.T., Numerical approximation for a Baer–Nunziato model of two-phase flows. Appl. Numer. Math. 61 (2011) 702721.
[39] Tokareva, S.A. and Toro, E.F., HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229 (2010) 35733604.
[40] U.S. NRC: Glossary, Departure from Nucleate Boiling (DNB).
[41] U.S. NRC: Glossary, Loss of Coolant Accident (LOCA).
[42] Yong, W-A., Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal. 172 (2004) 247266.


Related content

Powered by UNSILO

A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

  • Frédéric Coquel (a1), Jean-Marc Hérard (a2), Khaled Saleh (a2) (a3) (a4) and Nicolas Seguin (a3) (a4) (a5)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.