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Low Diffusion Kinetic Scheme For Compressible Flows

Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and Related Problems. Beijing Normal University , May 21-27 , 2014. Yibing Chen Key Laboratory of Computational Physics (Group of numerical simulations for 3D multi-material flows)

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Low Diffusion Kinetic Scheme For Compressible Flows

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  1. Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and Related Problems Beijing Normal University , May 21-27 , 2014 Yibing Chen Key Laboratory of Computational Physics (Group of numerical simulations for 3D multi-material flows) Institute of Applied Physics and Computational Mathematics(IAPCM), Beijing , China Chen_yibing@iapcm.ac.cn Low Diffusion Kinetic Scheme For Compressible Flows Collaborators: Song Jiang (IAPCM) Jiang@iapcm.ac.cn Na Liu (IAPCM) liu_na@iapcm.ac.cn

  2. Introduction Numerical scheme Numerical tests Conclusion Table of contents

  3. The Euler equations Flux vector splitting (FVS) method Gas Kinetic Scheme(GKS) Goals 1 Introduction

  4. 1.1 The euler equations Upwind method : popular

  5. FVS (Comparing with Godunov method) • simpler , more efficient • very attractive for large community of CFD codes • large hyperbolic systems with non-conservative terms(e.g compressible multiphase flows) • over diffusion: • smooth region ,(particularly) contact discontinuities and shear waves 1.2 Flux vector splitting(FVS) method <= according to eigenvalues

  6. Low diffusion fvs captures contact discontinuity as well as AUSM more robust and more accurate than AUSM Resolution of smooth region is still not satisfied (comparing to BGK)

  7. GKS : Boltzmann approach(FVS) • KFVS : Collisionless Boltzmann equation • Pullin (EFM , 1980) • BGK : BGK model • K.Xu(1993) 1.3 Gas Kinetic Scheme(GKS) Ref: Kun Xu, Gas-kinetic schemes for unsteady compressible flow simulations, VKI1998-03

  8. KFVS

  9. KFVS vs Van leer(FVS) M M M

  10. KFVS :Very similar to the Van Leer (FVS)scheme • Advantages • Simple • Robust: can deal with very strong shock waves • Disadvantages • Over diffusion , especially near the region of contact discontinuities KFVS

  11. BGK More accurate than KFVS: smooth region , contact discontinuities

  12. Traditional KFVS and BGK produce oscillation near the region of contact discontinuities • Yibing Chen and Song Jiang, Modified kinetic flux vector splitting schemes for compressible flows, Journal of Computational Physics, (2009) • Yibing Chen and Song Jiang, A non-oscillatory kinetic scheme for compressible multi-component flows with the state for a stiffened gas, Journal of Computational Mathematics,(2011) Remark 1

  13. Current GKS can not capture the stationary contact discontinuities exactly ? Remark 2 BGK KFVS

  14. Extend the traditional GKS to the framework of Toro-Vazquez to get a low diffusion kinetic scheme. • Simple: easy to implement • Robust : can treat very strong shock waves • Accurate • low diffusion in the smooth region and contact discontinuity • contact discontinuities • Oscillation free • Capture the stationary waves exactly goals

  15. Main ideas The advection part The pressure part Summary of the new scheme 2 Numerical scheme

  16. Follow the idea of Toro-Vazquez scheme , split the governor equations into two parts: advection systems and pressure systems • Utilize the upwind-type scheme to solve advection systems to reduce over diffusion • Utilize the KFVS –type scheme to solve pressure systems 2.1 main ideas

  17. 2.2 advection systems

  18. Modification : avoid oscilation near contact discontinuities Will not vanish near contact discontinuities (original GKS)

  19. Modification: reduce diffusion Introduce collision mechanism as BGK scheme

  20. 2.3 pressure systems ,

  21. Define Calcuate Compute the advection flux Compute the pressure flux Compute the intercell flux 2.4 Summary of the new scheme

  22. 1D: Compare the numerical results of • KFVS scheme • BGK scheme • Toro-Vazquez scheme • LDK(Low Diffusion Kinetic) scheme • 2D: BGK & LDK 3 Numerical tests

  23. TEST 1: ISOLATED STATIONARY CONTACT DISCONTINUITY

  24. TEST2 NON-ISOLATED STATIONARY CONTATCT DISCONTINUITY

  25. 26

  26. TEST3:LOW DENSITY

  27. 28

  28. TEST 4:COLLISION OF TWO SHOCK WAVES

  29. 30

  30. TEST 5WOODWARD COLELLA PROBLEM

  31. Second order

  32. 2D Riemann problems P.D.Lax and X.D.Liu. Solution of two-dimensional Riemann problem of gas dynamics by positive schemes. SIAM J.Sci.Comput./165:126-166,2000

  33. 800*800 cells Second order BGK LDK BGK LDK

  34. 800*800 cells Second order BGK LDK

  35. 800*800 cells Second order BGK BGK LDK LDK

  36. Follow the idea of Toro-Vazquez, we developed a low diffusion kinetic flux splitting scheme • Simple (KFVS) • Robust : can treat very strong shock waves • Accurate: • low diffusion (similar to BGK) , better than the original Toro-Vazquez scheme and KFVS • contact discontinuities : (as well as Toro-Vazquez scheme ) • Oscillation free • Capture the stationary waves exactly conclusions Further work: Large hyperbolic systems with non-conservative terms (compressible multi-phase flows)

  37. Thank you!

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