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A hybrid scheme to compute contact discontinuitiesinone-dimensional Euler systems

Published online by Cambridge University Press:  15 January 2003

Thierry Gallouët ,
Jean-Marc Hérard  and
Nicolas Seguin
Thierry Gallouët
Affiliation:
LATP-UMR CNRS 6632, C.M.I., Université de Provence, 13453 Marseille Cedex 13, France. gallouet@cmi.univ-mrs.fr., herard@cmi.univ-mrs.fr.
Jean-Marc Hérard
Affiliation:
LATP-UMR CNRS 6632, C.M.I., Université de Provence, 13453 Marseille Cedex 13, France. gallouet@cmi.univ-mrs.fr., herard@cmi.univ-mrs.fr. Département MFTT, Électricité de France - R&D, 78401 Chatou Cedex, France. herard@chi80bk.der.edf.fr., seguin@chi80bk.der.edf.fr.
Nicolas Seguin
Affiliation:
LATP-UMR CNRS 6632, C.M.I., Université de Provence, 13453 Marseille Cedex 13, France. gallouet@cmi.univ-mrs.fr., herard@cmi.univ-mrs.fr. Département MFTT, Électricité de France - R&D, 78401 Chatou Cedex, France. herard@chi80bk.der.edf.fr., seguin@chi80bk.der.edf.fr.
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Abstract

The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict “perfectly” unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties (“perfect” representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed.

Keywords

Godunov schemeEuler systemcontact discontinuitiesthermodynamicsconservative schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1133 - 1159
Copyright
© EDP Sciences, SMAI, 2002

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References

Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125 (1995) 150-160. CrossRef
Abgrall, R. and Karni, S., Computations of compressible multifluids. J. Comput. Phys. 169 (2001) 594-623. CrossRef
Allaire, G., Clerc, S. and Kokh, S., A five equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577-616. CrossRef
Allaire, G., Clerc, S. and Kokh, S., A five equation model for the numerical solution of interfaces in two phase flows. C.R. Acad. Sci. Paris Sér. I Math. 331 (2000) 1017-1022. CrossRef
T. Barberon, P. Helluy and S. Rouy, Finite Volume simulations of cavitating flows, in Proc. of Third Symposium on Finite Volumes for Complex Applications, R. Herbin and D. Kroner Eds., Hermes Penton Science (2002) 455-462.
Barret, M., Faucher, E. and Hérard, J.M., Some schemes to compute unsteady flashing flows. AIAA J. 40 (2002) 905-913. CrossRef
S. Bilicki and D. Kardas, Approximation of thermodynamic properties for subcooled water and superheated steam. Polish Academy of Sciences (1991).
Bilicki, S. and Kestin, J., Physical aspects of the relaxation model in two phase flows. Proc. Roy. Soc. London A 428 (1990) 379-397. CrossRef
Bilicki, S., Kestin, J. and Pratt, M.M., A reinterpretation of the results of the moby dick experiments in terms of the non equilibrium model. J. Fluid Eng. 112 (1990) 212-217. CrossRef
T. Buffard, T. Gallouët and J.M. Hérard, Schéma VFRoe en variables caractéristiques. Principe de base et applications aux gaz réels. EDF-DER Report HE-41/96/041/A (1996) in French.
T. Buffard, T. Gallouët and J.M. Hérard, A sequel to a rough Godunov scheme. Application to real gas flows. Comput. & Fluids 29 (2000) 813-847.
M. Buffat and A. Page, Extension of Roe's solver for multi species real gases. LMFA report, École Centrale de Lyon, Lyon, France (1995).
Clerc, S., Accurate computation of contact discontinuities in flows with general equations of state. Comput. Methods Appl. Mech. Engrg. 178 (1999) 245-255. CrossRef
Clerc, S., Numerical simulation of the homogeneous equilibrium model for two phase flows. J. Comput. Phys. 161 (2000) 354-375. CrossRef
Coquel, F. and Perthame, B., Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics equations. SIAM J. Numer. Anal. 35 (1998) 2223-2249 (in Memory of Ami Harten). CrossRef
Faucher, E., Hérard, J.M., Barret, M. and Toulemonde, C., Computation of flashing flows in variable cross-section ducts. Int. J. Comput. Fluid Dyn. 13 (2000) 365-391. CrossRef
Fedkiw, R.P., Aslam, T., Merriman, B. and Osher, S., A non oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid approach). J. Comput. Phys. 152 (1999) 457. CrossRef
Gallouët, T., Hérard, J.M. and Seguin, N., Some recent Finite Volume schemes to compute Euler equations using real gas EOS. Internat. J. Numer. Methods Fluids 39-12 (2002) 1073-1138. CrossRef
T. Gallouët, J.M. Hérard and N. Seguin, An hybrid scheme to compute contact discontinuities in Euler systems. LATP Report 01-027, Université de Provence, France (2001).
T. Gallouët, J.M. Hérard and N. Seguin, On the use of some symmetrizing variables to deal with vacuum (submitted).
Gavrilyuk, S. and Saurel, R., Mathematical and numerical modelling of two phase compressible flows with inertia. J. Comput. Phys. 175 (2002) 326-360. CrossRef
E. Godlewski and P.A. Raviart, Numerical approximation for hyperbolic systems of conservation laws. Springer Verlag (1996).
S.K. Godunov, A difference method for numerical calculation of discontinous equations of hydrodynamics. Sbornik (1959) 271-300 (in Russian).
Hou, X. and Le FLoch, P. G., Why non conservative schemes converge to wrong solutions. Math. Comp. 62 (1994) 497-530. CrossRef
Numerical, A. In evaluation of an energy relaxation method for inviscid real fluids. SIAM J. Sci. Comput. 21 (1999) 340-365.
A. In, Méthodes numériques pour les équations de la dynamique des gaz complexes et écoulements diphasiques. Ph.D. thesis, Université Paris VI, France (1999).
M. Ishii, Thermo-fluid dynamic theory of two-phase flows. Collection de la Direction des Etudes et Recherches d'Electicité de France (1975).
Kapila, A.K., Son, S.F., Bdzil, J.B., Menikoff, R. and Stewart, D.S., Two-phase modelling of DDT: structure of the velocity relaxation zone. Phys. Fluids 9 (1997) 3885-3897. CrossRef
Karni, S., Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112 (1994) 31-43. CrossRef
Karni, S., Hybrid multifluid algorithms. SIAM J. Sci. Comput. 17 (1996) 1019-1039. CrossRef
S. Karni and R. Abgrall, Ghost fluid for the poor: a single fluid algorithm for multifluid. Oberwolfach (2001).
R. Kee, J. Miller and T. Jefferson, Chemkin: a general purpose, problem independant transportable fortran chemical kinetics code package. SAND Report 80-8003, Sandia National Laboratories.
S. Kokh, Aspects numériques et théoriques de la modélisation des écoulements diphasiques compressibles par des méthodes de capture d'interface. Ph.D. thesis, Université Paris VI, France (2001).
F. Lagoutière, Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. thesis, Université Paris VI, France (2000).
A. Letellier and A. Forestier, Le problème de Riemann en fluide quelconque. CEA-DMT Report 93/451 (1993) in French.
R. LeVeque, Numerical methods for conservation laws. Birkhauser (1992).
R. Pollak, Die thermodynamischen eigenschaften von wasser dargestellt durch eine kanonische zustands gleichung fur die fluiden homogenen und heterogenen zustande bis 1200 Kelvin und 3000 bars. Ph.D. thesis, Ruhr Universitat, Germany (1974).
P. Rascle and O. Morvant, Interface utilisateur de Thetis - THErmodynamique en Tables d'InterpolationS. EDF-DER Report HT-13/95021B, Clamart, France (1995) in French.
Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357-372. CrossRef
S. Rouy, Modélisation mathématique et numérique d'écoulements diphasiques compressibles. Ph.D. thesis, Université de Toulon et du Var, France (2000).
Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. CrossRef
Saurel, R. and Abgrall, R., A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21 (1999) 1115-1145. CrossRef
J. Sethian, Level set methods. Cambridge University Press (1996).
Shyue, K.M., A fluid mixture type algorithm for compressible multicomponent flow with Van der Waals equation of state. J. Comput. Phys. 156 (1999) 43-88. CrossRef
J. Smoller, Shock waves and reaction diffusion equations. Springer Verlag (1983).
E.F. Toro, Riemann solvers and numerical methods for fluid dynamics. Springer Verlag (1997).
I. Toumi, Contribution à la modélisation numérique des écoulements diphasiques eau-vapeur. Thèse d'habilitation, Université Paris Sud, France (2000).