A high resolutive scheme for reactive flows

1. A high resolutive scheme for reactive flows Guoxi Ni IAPCM,Beijing,2014-5-22 With Xihua Xu,Song Jiang

2. Content 1. Model for reactive fluids 2. ALE type scheme on moving mesh 3. GRP solvers for Fluxes 4. Numerical examples

3. 1Model for reactive fluids Compressible model for reactive fluids where are stress,temprature and heat 。

4. For Newton flow， where and for fluids with reactive ，

5. For reactive fluids，a reactive rate should be given,usually, Such as Arrhenius function where is mass fraction，E is active energy。

6. In practice,some other complex reactive rates are to be used ,for examples, Lee-Tarver reactive rate function Or modified reactive rate function

7. This reactive rate cause many difficulties, in numerical simulation,which includes the following: 1) More complex structure in the reactive region. 2) Different fluids in the region, especially fluids with comples equation of states

8. Typical landscape for ZND.

9. Standard EOS’ are likes the following 1.ideal gas 2.stiffen 3.Mie-Gruneisen 4.JWL

10. Some numerical methods W.C.Davis, foundation work,1979 D.L.Youngs, VOF W.Bao,S.Jin,projection method M.BenArtzi,GRP scheme, A.J.Majda, R.J.Leveque,fraction steps R.Fedkiw, J.J. Quirk,Ghost fluid method R. Abgrall,Saurel,7 equation model,1999 B.N. Azarnok,T.Tang,moving meshes,2005

11. 2 ALE type scheme on moving mesh The target of the current method is to present a discrete scheme for the reactive fluids on moving mesh method. Different from previous approach, the method is constructed directly from the integral equations of the conservation laws.

12. For example,a triangle cell move with this velocity

13. Integral form governing equations for reactive fluids

14. If then we get the syetem for Euler method

15. If then we get the syetem for (modified) Lagrange method

16. Denote , then for mass equation where

17. Momentum equation where

18. Energy equation where

19. In summary

20. 3 DRP solvers for fluxes In the following , we are going to determine fluxes for the above discretize system. The strategy is to use GRP scheme for the mass,momentum and energy equations,and to update the reactive rate seperately.

21. The GRP scheme is a numerical method based on the analytic resolution of the associated generalized Riemann problem. 1. It is a high order analytic extension of the Godunov scheme 2. Main gradients are the Lax-Wendroff approach plus singularity tracking

22. Ben-Artzi, Matania; Falcovitz, Joseph’ J.Comput. Phys. 55 (1984), no. 1, 1–32. Ben-Artzi, Matania; Falcovitz, Joseph,Cambridge monographs on Applied and Computational Mathematics, 11, 2003. LeFloch, Ph.; Raviart, P.-A.: , Non LinWaire 5 (1988),no. 2, 179–207. ; Non LinWaire 6,(1989), no. 6, 437–480. LeFloch, Philippe; Li, Tatsien, Asymptotic Anal. 3 (1991), no. 4, 321–340.. Li and Z. Sun, J. Comput. Phys. 222 (2007), no. 2,796–808. M. Ben-Artzi, J. Li and G. Warnecke, J. Comput. Phys. 218(2006), no. 1, 19–43.

23. .

24. The target is to get the following values then we can get the numerical flux for mass,momentum and energy which based on the Taylor expansion of time.

25. For the reactive rate equation,consider the contact velocity, then we can get the numerical flux for reactive rate.

26. .

27. what remained is to determine the mesh velocity in the above discrete equation, a direct implementation of numerical techniques on the discretized level to prevent mesh distortion is preferred.

28. Moving meshmethod (H.Z.Tang,T.Tang 2003) Where

29. 4 Numerical tests Case 1 steady state problem pressure

30. density

31. Case 2 strong detonation pressure

32. density

33. Case 3 unsteady problem Fixed meshes

34. Results by moving meshes

35. Case 4 two dimension unsteady problem

36. pressure

37. .

38. meshes

39. Thanks 谢谢！