AN APPROXIMATE NONLINEAR PROJECTION SCHEME FOR A COMBUSTION MODEL

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 tur- bulent 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.

[1]  Claude Marmignon,et al.  A Roe-type linearization for the Euler equations for weakly ionized multi-component and multi-temperature gas , 1995 .

[2]  Christian Olivier,et al.  On the numerical approximation of the K-epsilon turbulence model for two dimensional compressible flows , 1990 .

[3]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[4]  Rémi Abgrall,et al.  An extension of Roe's upwind scheme to algebraic equilibrium real gas models , 1991 .

[5]  Eitan Tadmor,et al.  A minimum entropy principle in the gas dynamics equations , 1986 .

[6]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[7]  Sharath S. Girimaji,et al.  Assumed PDF turbulence-chemistry closure with temperature-composition correlations , 2003 .

[8]  R. LeVeque.  Approximate Riemann Solvers , 1992 .

[9]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[10]  A. Noussair,et al.  Microscopic profiles of shock waves and ambiguities in multiplications of distributions , 1989 .

[11]  Travelling Wave Solutions of a Convective Diffusive System with First and Second Order Terms in Nonconservation Form , 1999 .

[12]  B. Larrouturou.  How to preserve the mass fractions positivity when computing compressible multi-component flows , 1991 .

[13]  B. Perthame,et al.  Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics , 1998 .

[14]  T. Hou,et al.  Why nonconservative schemes converge to wrong solutions: error analysis , 1994 .

[15]  David Gilbarg,et al.  The Existence and Limit Behavior of the One-Dimensional Shock Layer , 1951 .

[16]  O. Pironneau,et al.  Analysis of the K-epsilon turbulence model , 1994 .

[17]  Pierre-Arnaud Raviart,et al.  A NONCONSERVATIVE HYPERBOLIC SYSTEM MODELING SPRAY DYNAMICS. PART I: SOLUTION OF THE RIEMANN PROBLEM , 1995 .

[18]  Lionel Sainsaulieu.  Traveling Waves Solution of Convection-Diffusion Systems Whose Convection Terms are Weakly Nonconservative: Application to the Modeling of Two-Phase Fluid Flows , 1995, SIAM J. Appl. Math..

[19]  Jean-Marc Hérard,et al.  Solveur de type Godunov pour simuler les écoulements turbulents compressibles , 1997 .

[20]  P. Floch.  Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form , 1988 .

[21]  P. Raviart,et al.  The Case of Multidimensional Systems , 2020, Applied Mathematical Sciences.

[22]  Frederic COQUELtt.  CONVERGENCE OF FINITE DIFFERENCE SCHEMES FOR CONSERVATION LAWS IN SEVERAL SPACE DIMENSIONS: A GENERAL THEORY* , 1993 .

[23]  Smadar Karni,et al.  Viscous shock profiles and primitive formulations , 1992 .