A two-fluid hyperbolic model in a porous medium
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 6, pp. 1319-1348.

The paper is devoted to the computation of two-phase flows in a porous medium when 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 corresponding to 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 L1 norm of the error.

DOI : 10.1051/m2an/2010033
Classification : 76S05, 76M12, 65M12, 76T10
Mots-clés : porous medium, well-balanced scheme, analytic solution, convergence rate, two-phase flow 
@article{M2AN_2010__44_6_1319_0,
     author = {Girault, La\"etitia and H\'erard, Jean-Marc},
     title = {A two-fluid hyperbolic model in a porous medium},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1319--1348},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {6},
     year = {2010},
     doi = {10.1051/m2an/2010033},
     mrnumber = {2769060},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2010033/}
}
TY  - JOUR
AU  - Girault, Laëtitia
AU  - Hérard, Jean-Marc
TI  - A two-fluid hyperbolic model in a porous medium
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2010
SP  - 1319
EP  - 1348
VL  - 44
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2010033/
DO  - 10.1051/m2an/2010033
LA  - en
ID  - M2AN_2010__44_6_1319_0
ER  - 
%0 Journal Article
%A Girault, Laëtitia
%A Hérard, Jean-Marc
%T A two-fluid hyperbolic model in a porous medium
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2010
%P 1319-1348
%V 44
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2010033/
%R 10.1051/m2an/2010033
%G en
%F M2AN_2010__44_6_1319_0
Girault, Laëtitia; Hérard, Jean-Marc. A two-fluid hyperbolic model in a porous medium. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 6, pp. 1319-1348. doi : 10.1051/m2an/2010033. http://archive.numdam.org/articles/10.1051/m2an/2010033/

[1] 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).

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

[3] E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrodynamic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065. | Zbl

[4] 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

[5] F. Bouchut, Nonlinear stability of Finite Volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhauser (2004). | Zbl

[6] 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. | Zbl

[7] T. Buffard, T. Gallouët and J.-M. Hérard, A sequel to a rough Godunov scheme. Application to real gases. Comput. Fluids 29 (2000) 813-847. | Zbl

[8] 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/.

[9] 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. | Zbl

[10] 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. | Zbl

[11] T. Gallouët, J.-M. Hérard and N. Seguin, A hybrid scheme to compute contact discontinuities in one dimensional Euler systems. ESAIM: M2AN 36 (2002) 1133-1159. | Numdam | Zbl

[12] T. Gallouët, J.-M. Hérard and N. Seguin, Some recent Finite Volume schemes to compute Euler equations using real gas EOS. Int. J. Num. Meth. Fluids 39 (2002) 1073-1138. | Zbl

[13] T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow water equations with topography. Comput. Fluids 32 (2003) 479-513. | Zbl

[14] T. Gallouët, J.-M. Hérard and N. Seguin, Numerical modelling of two phase flows using the two-fluid two-pressure approach. Math. Mod. Meth. Appl. Sci. 14 (2004) 663-700. | Zbl

[15] 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.

[16] 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. | Numdam | Zbl

[17] 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).

[18] E. Godlewski and P.A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: 1. The scalar case. Numer. Math. 97 (2004) 81-130. | Zbl

[19] E. Godlewski, K.C. Le Thanh and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems. ESAIM: M2AN 39 (2005) 649-692. | Numdam | Zbl

[20] S.K. Godunov, Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47 (1959) 271-300.

[21] J.M. Greenberg and A.Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl

[22] 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).

[23] 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.

[24] 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/.

[25] J.-M. Hérard, A three-phase flow model. Math. Comp. Model. 45 (2007) 432-455. | Zbl

[26] J.-M. Hérard, Un modèle hyperbolique diphasique bi-fluide en milieu poreux. C. r., Méc. 336 (2008) 650-655. | Zbl

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

[28] D. Kröner and M.D. Thanh, Numerical solution to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43 (2006) 796-824. | Zbl

[29] D. Kröner, P. Le Floch and M.D. Thanh, The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section. ESAIM: M2AN 42 (2008) 425-442. | Numdam | Zbl

[30] C.A. Lowe, Two-phase shock-tube problems and numerical methods of solution. J. Comput. Phys. 204 (2005) 598-632. | Zbl

[31] 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. | Zbl

Cité par Sources :