A robust entropy-satisfying finite volume scheme for the isentropic Baer-Nunziato model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, p. 165-206
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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.

DOI : https://doi.org/10.1051/m2an/2013101
Classification:  76T05,  35L60,  35F55
Keywords: two-phase flows, entropy-satisfying methods, relaxation techniques, Riemann problem
@article{M2AN_2014__48_1_165_0,
author = {Coquel, Fr\'ed\'eric and H\'erard, Jean-Marc and Saleh, Khaled and Seguin, Nicolas},
title = {A robust entropy-satisfying finite volume scheme for the isentropic Baer-Nunziato model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {1},
year = {2014},
pages = {165-206},
doi = {10.1051/m2an/2013101},
zbl = {1286.76098},
mrnumber = {3177841},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_1_165_0}
}

Coquel, Frédéric; Hérard, Jean-Marc; Saleh, Khaled; Seguin, Nicolas. A robust entropy-satisfying finite volume scheme for the isentropic Baer-Nunziato model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 165-206. doi : 10.1051/m2an/2013101. http://www.numdam.org/item/M2AN_2014__48_1_165_0/

[1] A. Ambroso, C. Chalons, F. Coquel and T. Galié, Relaxation and numerical approximation of a two-fluid two-pressure diphasic model. ESAIM: M2AN 43 (2009) 1063-1097. | Numdam | MR 2588433 | Zbl pre05636847

[2] A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P-A. Raviart and N. Seguin, The coupling of homogeneous models for two-phase flows. Int. J. Finite 4 (2007) 39. | MR 2465468

[3] A. Ambroso, C. Chalons and P.-A. Raviart, A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput. Fluids 54 (2012) 67-91. | MR 2972036 | Zbl 1291.76212

[4] N. Andrianov and G. Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195 (2004) 434-464. | MR 2046106 | Zbl 1115.76414

[5] M.R. Baer and J.W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12 (1986) 861-889. | Zbl 0609.76114

[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. | MR 2887641 | Zbl 1234.35190

[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). | MR 2128209 | Zbl 1086.65091

[8] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness. Commun. Partial Differ. Eqs. 24 (1999) 2173-2189. | MR 1720754 | Zbl 0937.35098

[9] B. Boutin, F. Coquel and P.G. Lefloch, Coupling nonlinear hyperbolic equations (iii). A regularization method based on thick interfaces. SIAM J. Numer. Anal. 51 (2013) 1108-1133. | MR 3038113 | Zbl pre06189183

[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. | MR 2815642 | Zbl pre06115936

[11] G-Q. Chen, C.D. Levermore and T-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787-830. | MR 1280989 | Zbl 0806.35112

[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. | MR 1909942 | Zbl 0999.35057

[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. | MR 1963591 | Zbl 1064.76545

[14] F. Coquel, E. Godlewski and N. Seguin, Relaxation of fluid systems. Math. Models Methods Appl. Sci. 22 (2012). | MR 2928102 | Zbl 1248.35008

[15] F. Coquel, J.-M. Hérard and K. Saleh, A splitting method for the isentropic Baer-Nunziato two-phase flow model. ESAIM: Proc., 38 (2012) 241-256. | MR 3006545

[16] F. Coquel, J.-M. Hérard, K. Saleh and N. Seguin, Two properties of two-velocity two-pressure models for two-phase flows. Commun. Math. Sci. (2013) 11. | Zbl pre06351202

[17] F. Coquel, K. Saleh and N. Seguin, A Robust and Entropy-Satisfying Numerical Scheme for Fluid Flows in Discontinuous Nozzles. http://hal.archives-ouvertes.fr/hal-00795446 (2013). | MR 3211117 | Zbl pre06322949

[18] V. Deledicque and M.V. Papalexandris, A conservative approximation to compressible two-phase flow models in the stiff mechanical relaxation limit. J. Comput. Phys. 227 (2008) 9241-9270. | MR 2463206 | Zbl 1202.76146

[19] M. Dumbser, A. Hidalgo, M. Castro, C. Parés and E.F. Toro, FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Engrg. 199 (2010) 625-647. | MR 2796172 | Zbl 1227.76043

[20] P. Embid and M. Baer, Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4 (1992) 279-312. | MR 1191989 | Zbl 0760.76096

[21] T. Gallouët, J.-M. Hérard and N. Seguin, Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Models Methods Appl. Sci. 14 (2004) 663-700. | MR 2057513 | Zbl 1177.76428

[22] S. Gavrilyuk and R. Saurel, Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175 (2002) 326-360. | MR 1877822 | Zbl 1039.76067

[23] P. Goatin and P.G. Lefloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Institut. Henri Poincaré Anal. Non Linéaire 21 (2004) 881-902. | Numdam | MR 2097035 | Zbl 1086.35069

[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). | MR 1410987 | Zbl 0860.65075

[25] B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rational Mech. Anal. 169 (2003) 89-117. | MR 2005637 | Zbl 1037.35041

[26] A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | MR 693713 | Zbl 0565.65051

[27] J.-M. Hérard and O. Hurisse, A fractional step method to compute a class of compressible gas-luiquid flows. Comput. Fluids. Int. J. 55 (2012) 57-69. | MR 2979696 | Zbl 1291.76217

[28] E. Isaacson and B. Temple, Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625-640. | MR 1331577 | Zbl 0838.35075

[29] A.K. Kapila, S.F. Son, J.B. Bdzil, R. Menikoff and D.S. Stewart, Two-phase modeling of DDT: Structure of the velocity-relaxation zone. Phys. Fluids 9 (1997) 3885-3897.

[30] S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws. Archive for Rational Mech. Anal. 174 (2004) 345-364. | MR 2107774 | Zbl 1065.35187

[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] R. Saurel and R. Abgrall, A multiphase godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. | MR 1684902 | Zbl 0937.76053

[36] D.W. Schwendeman, C.W. Wahle and A.K. Kapila, The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212 (2006) 490-526. | MR 2187902 | Zbl 1161.76531

[37] M.D. Thanh, D. Kröner and C. Chalons, A robust numerical method for approximating solutions of a model of two-phase flows and its properties. Appl. Math. Comput. 219 (2012) 320-344. | MR 2949595 | Zbl 1291.76325

[38] M.D. Thanh, D. Kröner and N.T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows. Appl. Numer. Math. 61 (2011) 702-721. | MR 2772280 | Zbl pre05865703

[39] S.A. Tokareva and E.F. Toro, HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229 (2010) 3573-3604. | MR 2609742 | Zbl pre05705265

[40] U.S. Nrc: Glossary, Departure from Nucleate Boiling (DNB). http://www.nrc.gov/reading-rm/basic-ref/glossary/departure-from-nucleate-boiling-dnb.html.

[41] U.S. Nrc: Glossary, Loss of Coolant Accident (LOCA). http://www.nrc.gov/reading-rm/basic-ref/glossary/loss-of-coolant-accident-loca.html.

[42] W-A. Yong, Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal. 172 (2004) 247-266. | MR 2058165 | Zbl 1058.35162