An approximate nonlinear projection scheme for a combustion model

Christophe Berthon; Didier Reignier

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 3, page 451-478
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Berthon, Christophe, and Reignier, Didier. "An approximate nonlinear projection scheme for a combustion model." ESAIM: Mathematical Modelling and Numerical Analysis 37.3 (2010): 451-478. <http://eudml.org/doc/194173>.

@article{Berthon2010,
abstract = { 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. },
author = {Berthon, Christophe, Reignier, Didier},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Hyperbolic systems in nonconservation form; finite volume methods; nonlinear projection method.; hyperbolic systems in nonconservation form; nonlinear projection method; turbulent flow; numerical results; convection-diffusion systems},
language = {eng},
month = {3},
number = {3},
pages = {451-478},
publisher = {EDP Sciences},
title = {An approximate nonlinear projection scheme for a combustion model},
url = {http://eudml.org/doc/194173},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Berthon, Christophe
AU - Reignier, Didier
TI - An approximate nonlinear projection scheme for a combustion model
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 3
SP - 451
EP - 478
AB - 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.
LA - eng
KW - Hyperbolic systems in nonconservation form; finite volume methods; nonlinear projection method.; hyperbolic systems in nonconservation form; nonlinear projection method; turbulent flow; numerical results; convection-diffusion systems
UR - http://eudml.org/doc/194173
ER -

References

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  1. R. Abgrall, An extension of Roe's upwind scheme to algebraic equilibrium real gas models. Comput. and Fluids19 (1991) 171–182.  Zbl0721.76061
  2. R.A. Baurle and S.S. Girimaji, An assumed PDF Turbulence-Chemistery closure with temperature-composition correlations. 37th Aerospace Sciences Meeting (1999).  
  3. C. Berthon and F. Coquel, 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).  Zbl0934.35030
  4. C. Berthon and F. Coquel, About shock layers for compressible turbulent flow models, work in preparation, preprint MAB 01-29 2001 ().  Zbl1109.35073URIhttp://www.math.sciences.univ-nantes.fr/
  5. C. Berthon and F. Coquel, 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.  Zbl1078.76573
  6. C. Berthon, F. Coquel and P. LeFloch, Entropy dissipation measure and kinetic relation associated with nonconservative hyperbolic systems (in preparation).  Zbl1234.35190
  7. J.F. Colombeau, A.Y. Leroux, A. Noussair and B. Perrot, Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Numer. Anal.26 (1989) 871–883.  Zbl0674.76049
  8. F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory. SIAM J. Numer. Anal.30 (1993) 675–700.  Zbl0781.65078
  9. 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 12th CFD Conference, San Diego, USA (1995).  Zbl1052.65520
  10. F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal.35 (1998) 2223–2249.  Zbl0960.76051
  11. G. Dal Maso, P. LeFloch and F. Murat, Definition and weak stability of a non conservative product. J. Math. Pures Appl.74 (1995) 483–548.  Zbl0853.35068
  12. A. Forestier, J.M. Herard and X. Louis, A Godunov type solver to compute turbulent compressible flows. C. R. Acad. Sci. Paris Sér. I Math.324 (1997) 919–926.  Zbl0881.76063
  13. E. Godlewski and P.A. Raviart, Hyperbolic systems of conservations laws. Springer, Appl. Math. Sci. 118 (1996).  Zbl0860.65075
  14. A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov type schemes for hyperbolic conservation laws. SIAM Rev.25 (1983) 35–61.  Zbl0565.65051
  15. T.Y. Hou and P.G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp.62 (1994) 497–530.  Zbl0809.65102
  16. L. Laborde, Modélisation et étude numérique de flamme de diffusion supersonique et subsonique en régime turbulent. Ph.D. thesis, Université Bordeaux I, France (1999).  
  17. B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys.95 (1991) 59–84.  Zbl0725.76090
  18. B. Larrouturou and C. Olivier, On the numerical appproximation of the K-eps turbulence model for two dimensional compressible flows. INRIA report, No. 1526 (1991).  
  19. P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. Comm. Partial Differential Equations13 (1988) 669–727.  
  20. B. Mohammadi and O. Pironneau, Analysis of the K-Epsilon Turbulence Model. Masson Eds., Rech. Math. Appl. (1994).  
  21. P.A. Raviart and L. Sainsaulieu, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem. Math. Models Methods Appl. Sci.5 (1995) 297–333.  Zbl0837.76089
  22. P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys.43 (1981) 357–372.  Zbl0474.65066
  23. L. Sainsaulieu, Travelling waves solutions of convection-diffusion systems whose convection terms are weakly nonconservative. SIAM J. Appl. Math.55 (1995) 1552–1576.  Zbl0841.35047
  24. E. Tadmor, A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math.2 (1986) 211–219.  Zbl0625.76084

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