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

Published online by Cambridge University Press:  18 December 2013

Frédéric Coquel ,
Jean-Marc Hérard ,
Khaled Saleh  and
Nicolas Seguin
Frédéric Coquel
Affiliation:
CMAP, École Polytechnique CNRS, UMR 7641, Route de Saclay, 91128 Palaiseau Cedex, France.
Jean-Marc Hérard
Affiliation:
EDF-R&D, Département MFEE, 6 Quai Watier, 78401 Chatou Cedex, France.
Khaled Saleh
Affiliation:
EDF-R&D, Département MFEE, 6 Quai Watier, 78401 Chatou Cedex, France. UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.. [email protected] CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.
Nicolas Seguin
Affiliation:
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.. [email protected] CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France. Inria Paris-Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France.
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Abstract

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

Keywords

Two-phase flowsentropy-satisfying methodsrelaxation techniquesRiemann problem
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 165 - 206
Copyright
© EDP Sciences, SMAI 2013

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References

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) 1063–1097. Google Scholar
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. Google Scholar
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) 67–91. Google Scholar
Andrianov, N. and Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195 (2004) 434–464. Google Scholar
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) 861–889. Google Scholar
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.
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).
Bouchut, F. and James, F., Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness. Commun. Partial Differ. Eqs. 24 (1999) 2173–2189. Google Scholar
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) 1108–1133. Google Scholar
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.
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) 787–830. Google Scholar
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.
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.
Coquel, F., Godlewski, E. and Seguin, N., Relaxation of fluid systems. Math. Models Methods Appl. Sci. 22 (2012). Google Scholar
Coquel, F., Hérard, J.-M. and Saleh, K., A splitting method for the isentropic Baer-Nunziato two-phase flow model. ESAIM: Proc., 38 (2012) 241–256. Google Scholar
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. Google Scholar
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).
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) 9241–9270. Google Scholar
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) 625–647. Google Scholar
Embid, P. and Baer, M., Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4 (1992) 279–312. Google Scholar
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) 663–700. Google Scholar
Gavrilyuk, S. and Saurel, R., Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175 (2002) 326–360. Google Scholar
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) 881–902. Google Scholar
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).
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) 89–117. Google Scholar
Harten, A., Lax, P.D. and van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. Google Scholar
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) 57–69. Google Scholar
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) 625–640. Google Scholar
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. Google Scholar
Kawashima, S. and Yong, W.-A., Dissipative structure and entropy for hyperbolic systems of balance laws. Archive for Rational Mech. Anal. 174 (2004) 345–364. Google Scholar
P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint 593, IMA, Minneapolis (1991).
Y. Liu, Ph.D. thesis. Université Aix-Marseille, to appear in (2013).
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).
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).
Saurel, R. and Abgrall, R., A multiphase godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. Google Scholar
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) 490–526. Google Scholar
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) 320–344. Google Scholar
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) 702–721. Google Scholar
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) 3573–3604. Google Scholar
U.S. NRC: Glossary, Departure from Nucleate Boiling (DNB). http://www.nrc.gov/reading-rm/basic-ref/glossary/departure-from-nucleate-boiling-dnb.html.
U.S. NRC: Glossary, Loss of Coolant Accident (LOCA). http://www.nrc.gov/reading-rm/basic-ref/glossary/loss-of-coolant-accident-loca.html.
Yong, W-A., Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal. 172 (2004) 247–266. Google Scholar