# A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems

Thierry Gallouët; Jean-Marc Hérard; Nicolas Seguin

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 36, Issue: 6, page 1133-1159
- ISSN: 0764-583X

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topGallouët, Thierry, Hérard, Jean-Marc, and Seguin, Nicolas. "A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems." ESAIM: Mathematical Modelling and Numerical Analysis 36.6 (2010): 1133-1159. <http://eudml.org/doc/194143>.

@article{Gallouët2010,

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.
},

author = {Gallouët, Thierry, Hérard, Jean-Marc, Seguin, Nicolas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Godunov scheme; Euler system; contact discontinuities;
thermodynamics; conservative schemes.; thermodynamics; conservative schemes},

language = {eng},

month = {3},

number = {6},

pages = {1133-1159},

publisher = {EDP Sciences},

title = {A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems},

url = {http://eudml.org/doc/194143},

volume = {36},

year = {2010},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 6

SP - 1133

EP - 1159

AB -
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.

LA - eng

KW - Godunov scheme; Euler system; contact discontinuities;
thermodynamics; conservative schemes.; thermodynamics; conservative schemes

UR - http://eudml.org/doc/194143

ER -

## References

top- R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys.125 (1995) 150-160.
- R. Abgrall and S. Karni, Computations of compressible multifluids. J. Comput. Phys.169 (2001) 594-623.
- G. Allaire, S. Clerc and S. Kokh, A five equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys.181 (2002) 577-616.
- G. Allaire, S. Clerc and S. Kokh, 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.
- 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.
- M. Barret, E. Faucher and J.M. Hérard, Some schemes to compute unsteady flashing flows. AIAA J.40 (2002) 905-913.
- S. Bilicki and D. Kardas, Approximation of thermodynamic properties for subcooled water and superheated steam. Polish Academy of Sciences (1991).
- S. Bilicki and J. Kestin, Physical aspects of the relaxation model in two phase flows. Proc. Roy. Soc. London A428 (1990) 379-397.
- S. Bilicki, J. Kestin and M.M. Pratt, A reinterpretation of the results of the moby dick experiments in terms of the non equilibrium model. J. Fluid Eng.112 (1990) 212-217.
- 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).
- S. Clerc, Accurate computation of contact discontinuities in flows with general equations of state. Comput. Methods Appl. Mech. Engrg.178 (1999) 245-255.
- S. Clerc, Numerical simulation of the homogeneous equilibrium model for two phase flows. J. Comput. Phys.161 (2000) 354-375.
- F. Coquel and B. Perthame, 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).
- E. Faucher, J.M. Hérard, M. Barret and C. Toulemonde, Computation of flashing flows in variable cross-section ducts. Int. J. Comput. Fluid Dyn.13 (2000) 365-391.
- R.P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid approach). J. Comput. Phys.152 (1999) 457.
- T. Gallouët, J.M. Hérard and N. Seguin, Some recent Finite Volume schemes to compute Euler equations using real gas EOS. Internat. J. Numer. Methods Fluids39-12 (2002) 1073-1138.
- 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).
- S. Gavrilyuk and R. Saurel, Mathematical and numerical modelling of two phase compressible flows with inertia. J. Comput. Phys.175 (2002) 326-360.
- 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).
- X. Hou and P. G. Le FLoch, Why non conservative schemes converge to wrong solutions. Math. Comp.62 (1994) 497-530.
- A. In, Numerical 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).
- A.K. Kapila, S.F. Son, J.B. Bdzil, R. Menikoff and D.S. Stewart, Two-phase modelling of DDT: structure of the velocity relaxation zone. Phys. Fluids9 (1997) 3885-3897.
- S. Karni, Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys.112 (1994) 31-43.
- S. Karni, Hybrid multifluid algorithms. SIAM J. Sci. Comput.17 (1996) 1019-1039.
- 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.
- P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys.43 (1981) 357-372.
- S. Rouy, Modélisation mathématique et numérique d'écoulements diphasiques compressibles. Ph.D. thesis, Université de Toulon et du Var, France (2000).
- R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys.150 (1999) 425-467.
- R. Saurel and R. Abgrall, A simple method for compressible multifluid flows. SIAM J. Sci. Comput.21 (1999) 1115-1145.
- J. Sethian, Level set methods. Cambridge University Press (1996).
- K.M. Shyue, A fluid mixture type algorithm for compressible multicomponent flow with Van der Waals equation of state. J. Comput. Phys.156 (1999) 43-88.
- 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).

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