The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite volume methods induce large errors when approximated the convection-diffusion extracted system. To solve this difficulty, recent works propose a nonlinear projection scheme based on cancellation phenomenon of relevant dissipation rates of entropy. Unfortunately, such a property never holds in the present framework. The nonlinear projection procedures are thus extended.
Keywords: hyperbolic systems in nonconservation form, finite volume methods, nonlinear projection method
@article{M2AN_2003__37_3_451_0,
author = {Berthon, Christophe and Reignier, Didier},
title = {An approximate nonlinear projection scheme for a combustion model},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {451--478},
year = {2003},
publisher = {EDP Sciences},
volume = {37},
number = {3},
doi = {10.1051/m2an:2003037},
mrnumber = {1994312},
zbl = {1062.65102},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an:2003037/}
}
TY - JOUR AU - Berthon, Christophe AU - Reignier, Didier TI - An approximate nonlinear projection scheme for a combustion model JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 451 EP - 478 VL - 37 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an:2003037/ DO - 10.1051/m2an:2003037 LA - en ID - M2AN_2003__37_3_451_0 ER -
%0 Journal Article %A Berthon, Christophe %A Reignier, Didier %T An approximate nonlinear projection scheme for a combustion model %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 451-478 %V 37 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an:2003037/ %R 10.1051/m2an:2003037 %G en %F M2AN_2003__37_3_451_0
Berthon, Christophe; Reignier, Didier. An approximate nonlinear projection scheme for a combustion model. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 3, pp. 451-478. doi: 10.1051/m2an:2003037
[1] , An extension of Roe's upwind scheme to algebraic equilibrium real gas models. Comput. and Fluids 19 (1991) 171-182. | Zbl
[2] and, An assumed PDF Turbulence-Chemistery closure with temperature-composition correlations. 37th Aerospace Sciences Meeting (1999).
[3] and, Travelling wave solutions of a convective diffusive system with first and second order terms in nonconservation form, Hyperbolic problems: theory, numerics, applications, vol. I, Zürich (1998) 47-54, Intern. Ser. Numer. Math. 129 Birkhäuser (1999). | Zbl
[4] and, About shock layers for compressible turbulent flow models, work in preparation, preprint MAB 01-29 2001 (http://www.math.u-bordeaux.fr/berthon). | MR
[5] and, Nonlinear projection methods for multi-entropies Navier-Stokes systems, Innovative methods for numerical solutions of partial differential equations, Arcachon (1998), World Sci. Publishing, River Edge (2002) 278-304. | Zbl
[6] , and, Entropy dissipation measure and kinetic relation associated with nonconservative hyperbolic systems (in preparation).
[7] ,, and, Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Numer. Anal. 26 (1989) 871-883. | Zbl
[8] and, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory. SIAM J. Numer. Anal. 30 (1993) 675-700. | Zbl
[9] and, A Roe-type linearization for the Euler equations for weakly ionized multi-component and multi-temperature gas. Proceedings of the AIAA 12th CFD Conference, San Diego, USA (1995).
[10] and, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35 (1998) 2223-2249. | Zbl
[11] , and, Definition and weak stability of a non conservative product. J. Math. Pures Appl. 74 (1995) 483-548. | Zbl
[12] , and, A Godunov type solver to compute turbulent compressible flows. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 919-926. | Zbl
[13] and, Hyperbolic systems of conservations laws. Springer, Appl. Math. Sci. 118 (1996). | Zbl | MR
[14] , and, On upstream differencing and Godunov type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl
[15] and, Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp. 62 (1994) 497-530. | Zbl
[16] , Modélisation et étude numérique de flamme de diffusion supersonique et subsonique en régime turbulent. Ph.D. thesis, Université Bordeaux I, France (1999).
[17] , How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys. 95 (1991) 59-84. | Zbl
[18] and, On the numerical appproximation of the K-eps turbulence model for two dimensional compressible flows. INRIA report, No. 1526 (1991).
[19] , Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. Comm. Partial Differential Equations 13 (1988) 669-727. | Zbl
[20] and, Analysis of the K-Epsilon Turbulence Model. Masson Eds., Rech. Math. Appl. (1994). | MR
[21] and, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem. Math. Models Methods Appl. Sci. 5 (1995) 297-333. | Zbl
[22] , Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357-372. | Zbl
[23] , Travelling waves solutions of convection-diffusion systems whose convection terms are weakly nonconservative. SIAM J. Appl. Math. 55 (1995) 1552-1576. | Zbl
[24] , A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2 (1986) 211-219. | Zbl
Cité par Sources :
