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

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

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 Molecular Models Continuum Models Euler Navier-Stokes Burnett Deterministic Statistical MD Liouville DSMC Boltzmann Chapman-Enskog 0.001 0.110 Kn Continuum Slip flow Transition Free moleculae

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

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

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

8. Relation between and macroscopic variables Conservation constraint

9. BGK flow solver Integral 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 Attached shock Detached shock

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

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

20. The 2D BGK model under the transformation Particle velocitymacroscopic velocityGrid velocity

21. The computed paths fluttering tumbling - - - -

22. computed experiment

23. 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. where for x<0 for x>0 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. N=500000 steps Steady state under gravitational potential Diamond: with gravitational force term in flux Solid line: without G in flux

33. andhave different. Gas-kinetic scheme for multi-component flow

34. Gas distribution function at a cell interface:

35. Shock tube test:

36. = + Sod test

37. A M s =1.22 shock 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) j+1/2 free transport

44. Flux splitting for MHD equations:

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

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: density x-component velocity solid lines: current BGK scheme dash-line: Roe-MHD solver

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

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

51. t=2.0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5th WENO

52. t=3.0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy 5th WENO

53. t=8.0 (a): density (b): gas pressure (c): magnetic pressure (d): kinetic energy

54. 3D examples:

59. BGK (100^3)

60. FLUID MODELING Molecular Models Continuum Models Euler Navier-Stokes Burnett Deterministic Statistical MD Liouville DSMC Boltzmann Chapman-Enskog 0.001 0.110 Kn Continuum Slip flow Transition Free moleculae new continuum models

61. Generalization of Constitutive Relationship Gas-kinetic BGK model: Compatibility condition: Constitutive relationship:

62. is obtained by substituting the above solution into BGK eqn. The solution becomes With the assumption of closed solution of the BGK model:

63. A time-dependent gas distribution function at a cell interface where Extended Navier-Stokes-type Equations Viscosity and heat conduction coefficient

64. Argon shock structure Observation: Experiment: Alsmeyer (‘76), Schmidt (‘69),... Shock thickness: Mean free path (upstream):

65. Density distribution in Mach=9 Argon shock front Circles: experimental data (Alsmeyer, ‘76); dash-dot line: BGK-NS; solid line: BGK-Xu

66. Diatomic gas: N2 (two temperature model: bulk viscosity is replaced by temperature relaxation),

67. BGK Compatibility condition

68. M=12.9 nitrogen shock structure

69. M=11 nitrogen shock structure Efficiency: DSMC: hours Extended BGK: minutes