Nonlinear projection methods for multi-entropies Navier–Stokes systems

Berthon, Christophe; Coquel, Frédéric

doi:10.1090/S0025-5718-07-01948-5

Nonlinear projection methods for multi-entropies Navier–Stokes systems
HTML articles powered by AMS MathViewer

by Christophe Berthon and Frédéric Coquel PDF

Math. Comp. 76 (2007), 1163-1194 Request permission

Abstract:

This paper is devoted to the numerical approximation of the compressible Navier–Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier–Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.

References

C. Berthon, Contributions to the numerical analysis of the compressible Navier-Stokes equations with two specific entropies. Applications to turbulent compressible flows. Ph.D. dissertation (in French) University Paris VI, 1999.
C. Berthon and F. Coquel, Travelling wave solutions existence for multi-entropies Navier-Stokes equations, work in preparation, see also Proceedings of the 7th International Conference on Hyperbolic Problems, Zurich, 1998.
C. Berthon and F. Coquel, About shock layers for compressible turbulent flow models, work in preparation.
J.-F. Colombeau, A. Y. LeRoux, A. Noussaïr, and B. Perrot, Microscopic profiles of shock waves and ambiguities in multiplications of distributions, SIAM J. Numer. Anal. 26 (1989), no. 4, 871–883. MR 1005514, DOI 10.1137/0726048
Frédéric Coquel and Philippe LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), no. 195, 169–210. MR 1079010, DOI 10.1090/S0025-5718-1991-1079010-2
F. Coquel and C. Marmignon, A Roe-type linearization for the Euler equations for weakly ionized multi-component and multi-temperature gas, Proceedings of the AIAA $\mbox {12}^{th}$ CFD Conference, San Diego (USA) 1995.
Frédéric Coquel and Benoît Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), no. 6, 2223–2249. MR 1655844, DOI 10.1137/S0036142997318528
Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
Alain Forestier, Jean-Marc Hérard, and Xavier Louis, Solveur de type Godunov pour simuler les écoulements turbulents compressibles, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 8, 919–926 (French, with English and French summaries). MR 1450449, DOI 10.1016/S0764-4442(97)86969-8
David Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 (1951), 256–274. MR 44315, DOI 10.2307/2372177
Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR 1410987, DOI 10.1007/978-1-4612-0713-9
Thomas Y. Hou and Philippe G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), no. 206, 497–530. MR 1201068, DOI 10.1090/S0025-5718-1994-1201068-0
S. Karni, Viscous shock profiles and primitive formulations, SIAM J. Numer. Anal. 29 (1992), no. 6, 1592–1609. MR 1191138, DOI 10.1137/0729092
B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys. 95 (1991), no. 1, 59–84. MR 1112315, DOI 10.1016/0021-9991(91)90253-H
B. Larrouturou and C. Olivier, On the numerical approximation of the K-eps turbulence model for two dimensional compressible flows, INRIA report, No 1526 1991.
Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
Philippe LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Comm. Partial Differential Equations 13 (1988), no. 6, 669–727. MR 934378, DOI 10.1080/03605308808820557
F. R. Menter, Zonal two equation $(k-\omega )$ turbulence model for aerodynamic flows, 24th AIAA fluid dynamics conference, Orlando, (1993).
F. R. Menter, Improved two equation $(k-\omega )$ turbulence model for aerodynamic flows, NASA Technical Report 103975, (1992).
B. Mohammadi and O. Pironneau, Analysis of the $k$-epsilon turbulence model, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR 1296252
Pierre-Arnaud Raviart and Lionel Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem, Math. Models Methods Appl. Sci. 5 (1995), no. 3, 297–333. MR 1330136, DOI 10.1142/S021820259500019X
P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), no. 2, 357–372. MR 640362, DOI 10.1016/0021-9991(81)90128-5
Lionel Sainsaulieu, Traveling waves solution of convection-diffusion systems whose convection terms are weakly nonconservative: application to the modeling of two-phase fluid flows, SIAM J. Appl. Math. 55 (1995), no. 6, 1552–1576. MR 1358789, DOI 10.1137/S0036139994268292
B. R. Smith, A near wall model for the $(k-l)$ two equation turbulence model, 25th AIAA fluid dynamics conference, Colorado Springs, (1994).
Eitan Tadmor, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math. 2 (1986), no. 3-5, 211–219. MR 863987, DOI 10.1016/0168-9274(86)90029-2
Eitan Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comp. 49 (1987), no. 179, 91–103. MR 890255, DOI 10.1090/S0025-5718-1987-0890255-3