Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T06:18:13.815Z Has data issue: false hasContentIssue false
  • English
  • Français

A two-fluid hyperbolic model in a porous medium

Published online by Cambridge University Press:  10 May 2010

Laëtitia Girault  and
Jean-Marc Hérard
Laëtitia Girault
Affiliation:
EDF, R&D, Fluid Dynamics, Power Generation and Environment, 6 quai Watier, 78400 Chatou, France. [email protected] Centre de Mathématiques et Informatique, LATP, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France.
Jean-Marc Hérard
Affiliation:
EDF, R&D, Fluid Dynamics, Power Generation and Environment, 6 quai Watier, 78400 Chatou, France. [email protected]
Get access

Abstract

The paper is devoted to the computation of two-phase flows in a porous mediumwhen applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them correspondingto solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme,are shown to give wrong approximations in some cases involving sharp porous profiles.The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal. 43 (2006) 796–824]for the computation of single phase flows in varying cross section ducts,provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L 1 norm of the error.

Keywords

Porous mediumwell-balanced schemeanalytic solutionconvergence ratetwo-phase flow.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 6 , November 2010 , pp. 1319 - 1348
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.A. Raviart and N. Seguin, Working group on the interfacial coupling of models. http://www.ann.jussieu.fr/groupes/cea (2003).
Andrianov, N. and Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195 (2004) 434–464. CrossRef
Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrodynamic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. CrossRef
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. CrossRef
F. Bouchut, Nonlinear stability of Finite Volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhauser (2004).
B. Boutin, F. Coquel and E. Godlewski, Dafermos Regularization for Interface Coupling of Conservation Laws, in Hyperbolic problems: Theory, Numerics, Applications, Springer (2008) 567–575.
Buffard, T., Gallouët, T. and Hérard, J.-M., A sequel to a rough Godunov scheme. Application to real gases. Comput. Fluids 29 (2000) 813–847. CrossRef
A. Chinnayya, A.Y. Le Roux and N. Seguin, A well-balanced numerical scheme for shallow-water equations with topography: the resonance phenomena. Int. J. Finite Volumes 1 (2004) available at http://www.latp.univ-mrs.fr/IJFV/.
F. Coquel, T. Gallouët, J.M. Hérard and N. Seguin, Closure laws for a two-fluid two-pressure model. C. R. Acad. Sci. Paris. I-332 (2002) 927–932.
R. Eymard, T. Gallouët and R. Herbin, Finite Volume methods, in Handbook of Numerical Analysis VII, P.G. Ciarlet and J.L. Lions Eds., North Holland (2000) 715–1022.
Gallouët, T., Hérard, J.-M. and Seguin, N., A hybrid scheme to compute contact discontinuities in one dimensional Euler systems. ESAIM: M2AN 36 (2002) 1133–1159. CrossRef
Gallouët, T., Hérard, J.-M. and Seguin, N., Some recent Finite Volume schemes to compute Euler equations using real gas EOS. Int. J. Num. Meth. Fluids 39 (2002) 1073–1138. CrossRef
Gallouët, T., Hérard, J.-M. and Seguin, N., Some approximate Godunov schemes to compute shallow water equations with topography. Comput. Fluids 32 (2003) 479–513. CrossRef
Gallouët, T., Hérard, J.-M. and Seguin, N., Numerical modelling of two phase flows using the two-fluid two-pressure approach. Math. Mod. Meth. Appl. Sci. 14 (2004) 663–700. CrossRef
L. Girault and J.-M. Hérard, Multidimensional computations of a two-fluid hyperbolic model in a porous medium. AIAA paper 2009–3540 (2009) available at http://www.aiaa.org.
P. Goatin and P. Le Floch, The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. Henri Poincaré, Anal. non linéaire 21 (2004) 881–902.
E. Godlewski, Coupling fluid models. Exploring some features of interfacial coupling, in Proceedings of Finite Volumes for Complex Applications V, Aussois, France, June 8–13 (2008).
Godlewski, E. and Raviart, P.A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: 1. The scalar case. Numer. Math. 97 (2004) 81–130. CrossRef
Godlewski, E., Le Thanh, K.C. and Raviart, P.-A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems. ESAIM: M2AN 39 (2005) 649–692. CrossRef
Godunov, S.K., Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47 (1959) 271–300.
Greenberg, J.M. and Leroux, A.Y., A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. CrossRef
V. Guillemaud, Modélisation et simulation numérique des écoulements diphasiques par une approche bifluide à deux pressions. Ph.D. Thesis, Université Aix Marseille I, Marseille, France (2007).
P. Helluy, J.-M. Hérard and H. Mathis, A well-balanced approximate Riemann solver for variable cross-section compressible flows. AIAA paper 2009-3888 (2009) available at http://www.aiaa.org.
J.M. Hérard, A rough scheme to couple free and porous media. Int. J. Finite Volumes 3 (2006) available at http://www.latp.univ-mrs.fr/IJFV/.
Hérard, J.-M., A three-phase flow model. Math. Comp. Model. 45 (2007) 432–455. CrossRef
J.-M. Hérard, Un modèle hyperbolique diphasique bi-fluide en milieu poreux. C. r., Méc. 336 (2008) 650–655.
Kapila, A.K., Son, S.F., Bdzil, J.B., Menikoff, R. and Stewart, D.S., Two-phase modeling of a DDT: structure of the velocity relaxation zone. Phys. Fluids 9 (1997) 3885–3897. CrossRef
Kröner, D. and Thanh, M.D., Numerical solution to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43 (2006) 796–824. CrossRef
Kröner, D., Le Floch, P. and Thanh, M.D., The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section. ESAIM: M2AN 42 (2008) 425–442. CrossRef
Lowe, C.A., Two-phase shock-tube problems and numerical methods of solution. J. Comput. Phys. 204 (2005) 598–632. CrossRef
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. CrossRef