Gas-kinetic schemes for flow computations

1. Gas-kinetic schemes for flow computations Kun Xu Mathematics Department Hong Kong University of Science and Technology

2. Collaborators: Changqiu Jin, Meiliang Mao, Huazhong Tang, Chun-lin Tian Acknowledgements: RGC6108/02E, 6116/03E, 6102/04E,6210/05E

3. Contents • Gas-kinetic BGK-NS flow solver • Navier-Stokes equations under gravitational field • Two component flow • MHD • Beyond Navier-Stokes equations

4. FLUID MODELING Continuum Models Molecular Models Euler Deterministic Navier-Stokes Statistical Burnett Liouville MD Chapman-Enskog DSMC Boltzmann 0.1 10 Kn 0.001 Free moleculae Continuum Transition Slip flow

5. Gas-kinetic BGK scheme for the Navier-Stokes equations fluxes

6. Gas-kinetic Finite Volume Scheme • Based on the gas-kinetic BGK model, a time dependent gas distribution function is obtained under the following IC, • Update of conservative flow variables,

7. BGK model: Equilibrium state: Collision time: A single temperature is assumed: To the Navier-Stokes order: in the smooth flow region !!!

8. Relation between and macroscopic variables • Conservation constraint

9. BGK flow solverIntegral solution of the BGK model

10. Initial gas distribution function on both sides of a cell interface. The corresponding is where the non-equilibrium states have no contributions to conservative macroscopic variables,

11. Equilibrium state

12. Equilibrium state is determined by

13. Where is determined by

14. Numerical fluxes: • Update of flow variables:

15. Double Cones Detached shock Attached shock

16. Double-cone M=9.50 (RUN 28 in experiment) Mesh: 500x100

19. Unified moving mesh method physical domain computational domain Unified coordinate system ( W.H.Hui, 1999) geometric conservation law

20. The 2D BGK model under the transformation Particle velocity macroscopic velocity Grid velocity

21. The computed paths - fluttering - - tumbling -

22. computed experiment

23. fluid force as functions of phase

24. fluid force as functions of phase

25. 3D cavity flow

26. BGK model under gravitational field: Integral solution: where the trajectory is

27. Integral solution: Gravitational potential

28. X=0 for x<0 where for x>0

29. Initial non-equilibrium state: Equilibrium state

30. The gas distribution function at a cell interface: Flux with gravitational effect: Flux without gravitational effect (multi-dimensional):

31. Steady state under gravitational potential N=500000 steps Diamond: with gravitational force term in flux Solid line: without G in flux

33. Gas-kinetic scheme for multi-component flow . and have different

34. Gas distribution function at a cell interface:

35. Shock tube test:

36. Sod test + =

37. A Ms=1.22shock wave in air hits a helium cylindrical bubble

38. Shock helium bubble interaction (Y.S. Lian and K. Xu, JCP 2000)

39. Ideal Magnetohydrodynamics Equations in 1D

40. Moments of a gas distribution function: Equilibrium state: The macroscopic flow variables are the moments of g. For example, Then, according to particle velocities, we can split flow variables as:

41. With the definition of moments: We have Recursive relation:

42. Therefore,

43. Kinetic Flux vector splitting scheme (Croisille, Khanfir, and Ghanteur, 1995) free transport j+1/2

44. Flux splitting for MHD equations:

45. free transport Construction of equilibrium state: j collision , where j+1 j+1/2

46. Equilibrium flux function: The BGK flux is a combination of non-equilibrium and equilibrium ones: (K. Xu, JCP159)

48. 1D Brio-Wu test case: Left state: Right state: x-component velocity density solid lines: current BGK scheme dash-line: Roe-MHD solver

49. y-component velocity By distribution shock Contact discontinuity +: BGK, o: Roe-MHD, *: KFVS

50. Orszag-Tang MHD Turbulence: t=0.5 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5th WENO