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

Frédéric Coquel; Jean-Marc Hérard; Khaled Saleh; Nicolas Seguin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 1, page 165-206
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Coquel, Frédéric, et al. "A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.1 (2014): 165-206. <http://eudml.org/doc/273257>.

@article{Coquel2014,
abstract = {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.},
author = {Coquel, Frédéric, Hérard, Jean-Marc, Saleh, Khaled, Seguin, Nicolas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {two-phase flows; entropy-satisfying methods; relaxation techniques; Riemann problem},
language = {eng},
number = {1},
pages = {165-206},
publisher = {EDP-Sciences},
title = {A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model},
url = {http://eudml.org/doc/273257},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Coquel, Frédéric
AU - Hérard, Jean-Marc
AU - Saleh, Khaled
AU - Seguin, Nicolas
TI - A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 1
SP - 165
EP - 206
AB - 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.
LA - eng
KW - two-phase flows; entropy-satisfying methods; relaxation techniques; Riemann problem
UR - http://eudml.org/doc/273257
ER -

References

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  1. [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. Zbl05636847MR2588433
  2. [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. Zbl1140.35515MR2465468
  3. [3] A. Ambroso, C. Chalons and P.-A. Raviart, A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput. Fluids54 (2012) 67–91. Zbl1291.76212MR2972036
  4. [4] N. Andrianov and G. Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys.195 (2004) 434–464. Zbl1115.76414MR2046106
  5. [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 Flow12 (1986) 861–889. Zbl0609.76114
  6. [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. Zbl1234.35190MR2887641
  7. [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). Zbl1086.65091MR2128209
  8. [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. Zbl0937.35098MR1720754
  9. [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. Zbl06189183MR3038113
  10. [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. Zbl1246.76071MR2815642
  11. [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. Zbl0806.35112MR1280989
  12. [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. Zbl0999.35057MR1909942
  13. [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. Zbl1064.76545MR1963591
  14. [14] F. Coquel, E. Godlewski and N. Seguin, Relaxation of fluid systems. Math. Models Methods Appl. Sci. 22 (2012). Zbl1248.35008MR2928102
  15. [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. Zbl1329.76253MR3006545
  16. [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. Zbl1303.35069
  17. [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). Zbl06322949MR3211117
  18. [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. Zbl1202.76146MR2463206
  19. [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. Zbl1227.76043MR2796172
  20. [20] P. Embid and M. Baer, Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn.4 (1992) 279–312. Zbl0760.76096MR1191989
  21. [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. Zbl1177.76428MR2057513
  22. [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. Zbl1039.76067MR1877822
  23. [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 Linéaire21 (2004) 881–902. Zbl1086.35069MR2097035
  24. [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). Zbl0860.65075MR1410987
  25. [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. Zbl1037.35041MR2005637
  26. [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. Zbl0565.65051MR693713
  27. [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. Zbl1291.76217MR2979696
  28. [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. Zbl0838.35075MR1331577
  29. [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. Fluids9 (1997) 3885–3897. 
  30. [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. Zbl1065.35187MR2107774
  31. [31] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint 593, IMA, Minneapolis (1991). 
  32. [32] Y. Liu, Ph.D. thesis. Université Aix-Marseille, to appear in (2013). 
  33. [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. [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. [35] R. Saurel and R. Abgrall, A multiphase godunov method for compressible multifluid and multiphase flows. J. Comput. Phys.150 (1999) 425–467. Zbl0937.76053MR1684902
  36. [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. Zbl1161.76531MR2187902
  37. [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. Zbl1291.76325MR2949595
  38. [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. Zbl05865703MR2772280
  39. [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. Zbl05705265MR2609742
  40. [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. [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. [42] W-A. Yong, Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal.172 (2004) 247–266. Zbl1058.35162MR2058165

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