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Hyperbolic relaxation models for granular flows

Published online by Cambridge University Press:  27 January 2010

Thierry Gallouët ,
Philippe Helluy ,
Jean-Marc Hérard  and
Julien Nussbaum
Thierry Gallouët
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
Philippe Helluy
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
Jean-Marc Hérard
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
Julien Nussbaum
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
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Abstract

In this work we describe an efficient model for the simulation of atwo-phase flow made of a gas and a granular solid. The starting point is the two-velocitytwo-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented bya relaxation source term in orderto take into account the pressure equilibrium between the two phases andthe granular stress in the solid phase. We show that the relaxationprocess 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 marchingalgorithm 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 fractionis 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 volumefraction stays within its natural bounds.

Keywords

Two-phase flowhyperbolicityrelaxationfinite volumeentropy

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 2 , March 2010 , pp. 371 - 400
Copyright
© EDP Sciences, SMAI, 2010

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References

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. Multiph. Flow 16 (1986) 861–889. CrossRef
Coquel, F., El Amine, K., Godlewski, E., Perthame, B. and Rascle, P., A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136 (1997) 272–288. CrossRef
Coquel, F., Gallouët, T., Hérard, J.-M. and Seguin, N., Closure laws for a two-fluid two-pressure model. C. R. Math. Acad. Sci. Paris 334 (2002) 927–932. CrossRef
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. CrossRef
D. Gidaspow, Multiphase flow and fluidization – Continuum and kinetic theory descriptions. Academic Press Inc., Boston, USA (1994).
E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences 118. Springer-Verlag, New York, USA (1996).
Goldshtein, A., Shapiro, M. and Gutfinger, C., Mechanics of colisional motion of granular materials. Part 3: Self similar shock wave propagation. J. Fluid Mech. 316 (1996) 29–51. CrossRef
Gough, P.S., Modeling of two-phase flows in guns. AIAA 66 (1979) 176–196.
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).
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. CrossRef
Hérard, J.-M. and Hurisse, O., A simple method to compute standard two-fluid models. Int. J. Comput. Fluid Dyn. 19 (2005) 475–482. CrossRef
Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F. and Stewart, D.S., Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13 (2001) 3002–3024. CrossRef
Kuo, K.K., Yang, V. and Moore, B.B., Intragranular stress, particle-wall friction and speed of sound in granular propellant beds. J. Ballistics 4 (1980) 697–730.
J. Nussbaum, Modélisation et simulation numérique d'un écoulement diphasique de la balistique intérieure. Ph.D. Thesis, Université de Strasbourg, France (2007).
Nussbaum, J., Helluy, P., Hérard, J.-M. and Carriére, A., Numerical simulations of gas-particle flows with combustion. Flow Turbulence Combust. 76 (2006) 403–417. CrossRef
Rusanov, V.V., The calculation of the interaction of non-stationary shock waves with barriers. Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961) 267–279.
Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. CrossRef
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.