Hyperbolic relaxation models for granular flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) no. 2, pp. 371-400.

In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow 16 (1986) 861-889]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marching algorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fraction is updated in order to take into account the equilibrium source term. The whole procedure is entropy dissipative. For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volume fraction stays within its natural bounds.

DOI : https://doi.org/10.1051/m2an/2010006
Classification : 76M12,  65M12
Mots clés : two-phase flow, hyperbolicity, relaxation, finite volume, entropy
@article{M2AN_2010__44_2_371_0,
author = {Gallou\"et, Thierry and Helluy, Philippe and H\'erard, Jean-Marc and Nussbaum, Julien},
title = {Hyperbolic relaxation models for granular flows},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {371--400},
publisher = {EDP-Sciences},
volume = {44},
number = {2},
year = {2010},
doi = {10.1051/m2an/2010006},
mrnumber = {2655954},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010006/}
}
Gallouët, Thierry; Helluy, Philippe; Hérard, Jean-Marc; Nussbaum, Julien. Hyperbolic relaxation models for granular flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) no. 2, pp. 371-400. doi : 10.1051/m2an/2010006. http://www.numdam.org/articles/10.1051/m2an/2010006/

[1] 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. Multiph. Flow 16 (1986) 861-889. | Zbl 0609.76114

[2] F. Coquel, K. El Amine, E. Godlewski, B. Perthame and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136 (1997) 272-288. | Zbl 0893.76052

[3] F. Coquel, T. Gallouët, J.-M. Hérard and N. Seguin, Closure laws for a two-fluid two-pressure model. C. R. Math. Acad. Sci. Paris 334 (2002) 927-932. | Zbl 0999.35057

[4] 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. | Zbl 1177.76428

[5] D. Gidaspow, Multiphase flow and fluidization - Continuum and kinetic theory descriptions. Academic Press Inc., Boston, USA (1994). | Zbl 0789.76001

[6] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences 118. Springer-Verlag, New York, USA (1996). | Zbl 0860.65075

[7] A. Goldshtein, M. Shapiro and C. Gutfinger, Mechanics of colisional motion of granular materials. Part 3: Self similar shock wave propagation. J. Fluid Mech. 316 (1996) 29-51. | Zbl 0876.76069

[8] P.S. Gough, Modeling of two-phase flows in guns. AIAA 66 (1979) 176-196.

[9] V. Guillemaud, Modélisation et simulation numérique des écoulements diphasiques par une approche bifluide à deux pressions. Ph.D. Thesis, Université Aix-Marseille I, France (2007).

[10] 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. | Zbl 0565.65051

[11] J.-M. Hérard and O. Hurisse, A simple method to compute standard two-fluid models. Int. J. Comput. Fluid Dyn. 19 (2005) 475-482. | Zbl 1184.76744

[12] A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son and D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13 (2001) 3002-3024. | Zbl 1184.76268

[13] K.K. Kuo, V. Yang and B.B. Moore, Intragranular stress, particle-wall friction and speed of sound in granular propellant beds. J. Ballistics 4 (1980) 697-730.

[14] J. Nussbaum, Modélisation et simulation numérique d'un écoulement diphasique de la balistique intérieure. Ph.D. Thesis, Université de Strasbourg, France (2007).

[15] J. Nussbaum, P. Helluy, J.-M. Hérard and A. Carriére, Numerical simulations of gas-particle flows with combustion. Flow Turbulence Combust. 76 (2006) 403-417. | Zbl 1145.76050

[16] V.V. Rusanov, The calculation of the interaction of non-stationary shock waves with barriers. Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961) 267-279.

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

[18] E.F. Toro, Riemann-problem based techniques for computing reactive two-phase flows, in Proc. Third Intl. Conf. on Numerical Combustion, A. Dervieux and B. Larrouturou Eds., Lecture Notes in Physics 351, Springer, Berlin, Germany (1989) 472-481.