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.
Mots-clés : Godunov scheme, Euler system, contact discontinuities, thermodynamics, conservative schemes
@article{M2AN_2002__36_6_1133_0,
author = {Gallou\"et, Thierry and H\'erard, Jean-Marc and Seguin, Nicolas},
title = {A hybrid scheme to compute contact discontinuities in one-dimensional {Euler} systems},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1133--1159},
publisher = {EDP-Sciences},
volume = {36},
number = {6},
year = {2002},
doi = {10.1051/m2an:2003009},
mrnumber = {1958662},
zbl = {1137.65419},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003009/}
}
TY - JOUR AU - Gallouët, Thierry AU - Hérard, Jean-Marc AU - Seguin, Nicolas TI - A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2002 SP - 1133 EP - 1159 VL - 36 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003009/ DO - 10.1051/m2an:2003009 LA - en ID - M2AN_2002__36_6_1133_0 ER -
%0 Journal Article %A Gallouët, Thierry %A Hérard, Jean-Marc %A Seguin, Nicolas %T A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 1133-1159 %V 36 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003009/ %R 10.1051/m2an:2003009 %G en %F M2AN_2002__36_6_1133_0
Gallouët, Thierry; Hérard, Jean-Marc; Seguin, Nicolas. A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 6, pp. 1133-1159. doi : 10.1051/m2an:2003009. http://www.numdam.org/articles/10.1051/m2an:2003009/
[1] , How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125 (1995) 150-160. | Zbl
[2] and, Computations of compressible multifluids. J. Comput. Phys. 169 (2001) 594-623. | Zbl
[3] , and, A five equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577-616.
[4] , and, 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. | Zbl
[5] , and, 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.
[6] , and, Some schemes to compute unsteady flashing flows. AIAA J. 40 (2002) 905-913.
[7] and, Approximation of thermodynamic properties for subcooled water and superheated steam. Polish Academy of Sciences (1991).
[8] and, Physical aspects of the relaxation model in two phase flows. Proc. Roy. Soc. London A 428 (1990) 379-397. | Zbl
[9] , and, A reinterpretation of the results of the moby dick experiments in terms of the non equilibrium model. J. Fluid Eng. 112 (1990) 212-217.
[10] , and, Schéma VFRoe en variables caractéristiques. Principe de base et applications aux gaz réels. EDF-DER Report HE-41/96/041/A (1996) in French.
[11] , and, A sequel to a rough Godunov scheme. Application to real gas flows. Comput. & Fluids 29 (2000) 813-847. | Zbl
[12] and, Extension of Roe's solver for multi species real gases. LMFA report, École Centrale de Lyon, Lyon, France (1995).
[13] , Accurate computation of contact discontinuities in flows with general equations of state. Comput. Methods Appl. Mech. Engrg. 178 (1999) 245-255. | Zbl
[14] , Numerical simulation of the homogeneous equilibrium model for two phase flows. J. Comput. Phys. 161 (2000) 354-375. | Zbl
[15] and, 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). | Zbl
[16] ,, and, Computation of flashing flows in variable cross-section ducts. Int. J. Comput. Fluid Dyn. 13 (2000) 365-391. | Zbl
[17] ,, and, A non oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid approach). J. Comput. Phys. 152 (1999) 457. | MR | Zbl
[18] , and, Some recent Finite Volume schemes to compute Euler equations using real gas EOS. Internat. J. Numer. Methods Fluids 39-12 (2002) 1073-1138. | Zbl
[19] , and, An hybrid scheme to compute contact discontinuities in Euler systems. LATP Report 01-027, Université de Provence, France (2001).
[20] , and, On the use of some symmetrizing variables to deal with vacuum (submitted).
[21] and, Mathematical and numerical modelling of two phase compressible flows with inertia. J. Comput. Phys. 175 (2002) 326-360. | Zbl
[22] and, Numerical approximation for hyperbolic systems of conservation laws. Springer Verlag (1996). | MR | Zbl
[23] , A difference method for numerical calculation of discontinous equations of hydrodynamics. Sbornik (1959) 271-300 (in Russian).
[24] and FLoch, Why non conservative schemes converge to wrong solutions. Math. Comp. 62 (1994) 497-530. | Zbl
[25] , Numerical evaluation of an energy relaxation method for inviscid real fluids. SIAM J. Sci. Comput. 21 (1999) 340-365. | Zbl
[26] , Méthodes numériques pour les équations de la dynamique des gaz complexes et écoulements diphasiques. Ph.D. thesis, Université Paris VI, France (1999).
[27] , Thermo-fluid dynamic theory of two-phase flows. Collection de la Direction des Etudes et Recherches d'Electicité de France (1975). | Zbl
[28] ,,, and, Two-phase modelling of DDT: structure of the velocity relaxation zone. Phys. Fluids 9 (1997) 3885-3897.
[29] , Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112 (1994) 31-43. | Zbl
[30] , Hybrid multifluid algorithms. SIAM J. Sci. Comput. 17 (1996) 1019-1039. | Zbl
[31] and, Ghost fluid for the poor: a single fluid algorithm for multifluid. Oberwolfach (2001). | MR
[32] , and, Chemkin: a general purpose, problem independant transportable fortran chemical kinetics code package. SAND Report 80-8003, Sandia National Laboratories.
[33] , 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).
[34] , Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. thesis, Université Paris VI, France (2000).
[35] and, Le problème de Riemann en fluide quelconque. CEA-DMT Report 93/451 (1993) in French.
[36] , Numerical methods for conservation laws. Birkhauser (1992). | MR | Zbl
[37] , 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).
[38] and, Interface utilisateur de Thetis - THErmodynamique en Tables d'InterpolationS. EDF-DER Report HT-13/95021B, Clamart, France (1995) in French.
[39] , Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357-372. | Zbl
[40] , Modélisation mathématique et numérique d'écoulements diphasiques compressibles. Ph.D. thesis, Université de Toulon et du Var, France (2000).
[41] and, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. | Zbl
[42] and, A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21 (1999) 1115-1145. | Zbl
[43] , Level set methods. Cambridge University Press (1996). | MR | Zbl
[44] , A fluid mixture type algorithm for compressible multicomponent flow with Van der Waals equation of state. J. Comput. Phys. 156 (1999) 43-88. | Zbl
[45] , Shock waves and reaction diffusion equations. Springer Verlag (1983). | MR | Zbl
[46] , Riemann solvers and numerical methods for fluid dynamics. Springer Verlag (1997). | MR | Zbl
[47] , Contribution à la modélisation numérique des écoulements diphasiques eau-vapeur. Thèse d'habilitation, Université Paris Sud, France (2000).
Cité par Sources :
