Theory and numerical approach in kinetic theory of gases (Part 3)

1. 2018 International Graduate Summer School on “Frontiers of Applied and Computational Mathematics” (Shanghai Jiao Tong University, July 9-21, 2018) Theory and numerical approach in kinetic theory of gases(Part 3) Kazuo Aoki Dept. of Math., National Cheng Kung University, Tainan and NCTS, National Taiwan University, Taipei

2. Transition regime and Numerical methods

3. Transition regime arbitrary Numerical Methodsfor the Boltzmann eq. or its models Stochastic (particle) method DSMC(Direct Simulation Monte Carlo) method G. A. Bird (1963, …, 1976, …, 1994, …) Deterministic methods Finite-difference(or discrete-ordinate) method Linearized Boltzmann eq. Brief outline & some examples Model Boltzmann eq. & Nonlinear Boltzmann eq. Brief outline & some examples

4. Linearized Boltzmann equation

5. Linearized Boltzmann equation Steady (or time-independent) problems Linearized B eq.:

6. Linearized Boltzmann equation Steady (or time-independent) problems Linearized B eq.:

7. Kernel representationof linearized collision term (Hard-sphere molecules)

8. Linearized boundary condition (diffuse reflection)

9. Ohwada, Sone, & A (1989), Phys. Fluids A Poiseuille flow and thermal transpiration Gas between two parallel plates Small pressure gradient Linearized Boltzmann eq. Small temperature gradient Mathematical study Chen, Chen, Liu, & Sone (2007), CPAM60, 147

10. Similarity solution EQ for : EQ for : BC for : Numerical solution(finite-difference)

11. Similarity solution Numerical solution(finite-difference) Flow velocity Heat Flow

12. Flow velocity Heat Flow Global mass-flow rate Global heat-flow rate

14. Flow velocity Heat Flow Global mass-flow rate Global heat-flow rate

16. Flow velocity Heat Flow Global mass-flow rate Global heat-flow rate

18. Global mass-flow rate Global heat-flow rate Symmetry relation Takata (2009), … Proof: Similarity sol.

19. EQ for : (a) (b) EQ for : BC for :

20. Properties (i) Commutation with parity operator (ii) Self-adjointness (iii)

21. Numerical method Ohwada, Sone & A (1989) Similarity solution EQ for : BC for :

22. (Subscript omitted) Long-time limit Steady sol. Time-derivative term Grid points Finite-difference scheme

23. Finite-difference scheme Finite difference in second-order, upwind known

24. Kernel representationof linearized collision term (Hard-sphere molecules)

25. Basis functions Computation of Piecewise quadratic function in Numerical kernels Independent of and Computable beforehand

26. Iteration method with convergence proof Takata & Funagane (2011), J. Fluid Mech. 669, 242 EQ for : BC for :

27. Iteration scheme for large

29. Slow flow past a sphere Takata, Sone, & A (1993), Phys. Fluids A Linearized Boltzmann eq. Diffuse reflection Similarity solution [ Sone & A (1983), J Mec. Theor. Appl. ] Numerical solution(finite-difference)

30. Sone & Takata (1992), Cercignani (2000) Difficulty 1: Discontinuity of velocity distribution function (VDF) BC • VDF is discontinuous on convex body. • Discontinuity propagates in gas along characteristics EQ Finite difference + Characteristic

31. Difficulty 2: Slow approach to state at infinity Numerical matching with asymptotic solution

32. Velocity distribution function

33. Drag Force Stokes drag Small Kn viscosity

34. Transition regime arbitrary Numerical Methodsfor the Boltzmann or its models Stochastic (particle) method DSMC(Direct Simulation Monte Carlo) method G. A. Bird (1963, …, 1976, …, 1994, …) Deterministic methods Finite-difference(or discrete-ordinate) method Linearized Boltzmann eq. Brief outline & some examples Model Boltzmann eq. & Nonlinear Boltzmann eq. Brief outline & some examples

35. Model Boltzmann equationI: Radiometric flow

36. Radiometer and radiometric force Crookes Radiometer (light mill) William Crookes (1874) Atmospheric pressure Effect of rarefied gas Effect of microscale mean free path

37. Classical topic Maxwell, Reynolds, Einstein, Kennard, Loeb, … Hot topic in micro fluid dynamics Wadsworth & Muntz (1996), Ohta, et al. (2001), Selden, Muntz, Ketsdever, Gimelshein, et al. (2009, 2011) light flow force cold hot Flow induced by temperature difference Resulting force acting on vane

38. Model problem Taguchi & A, J. Fluid Mech. 694, 191 (2012) A thin plate with one side heated in a rarefied gas in a square box (2D problem) Discontinuous wall temperature Sharp edges gas ??? Flow and force Assumptions: • BGK model (nonlinear) • Arbitrary Knudsen number • Gas-surface interaction Diffuse reflection Numerical analysis by finite-difference method

39. BGK model 2D steady flows [dimensionless] BC Diffuse reflection No net mass flux across boundary

40. BGK model 2D steady flows [dimensionless] BC Specular Diffuse reflection No net mass flux across boundary

41. Marginal distributions Independent variables Eqs. for BC for Grid points Discretization

42. (Iterative) finite-difference scheme Standard finite difference (2nd-order upwind scheme) known

43. Computational difficulty Discontinuity in velocity distribution function Finite difference + Characteristic A, Sone, Nishino, Sugimoto (1991) Sone & Sugimoto (1992, 1993, 1995) Takata, Sone, & A (1993), Sone, Takata, & Wakabayashi (1994) A, Kanba, & Takata (1997), A, Takata, Aikawa, Golse (2001), … Mathematical theory Boudin & Desvillettes (2000), Monatsh. Math. 131, 91 IVP of Boltzmann eq. A, Bardos, Dogbe, & Golse (2001), M3AS11, 1581 BVP of a simple transport eq. C. Kim (2011), Commun. Math. Phys. IBVP of Boltzmann eq.

44. Method (Upper half)

45. (Upper half) F-D eq.along characteristics (line of discontinuity)

46. marginal Result of computation Velocity distribution function

47. marginal Velocity distribution function

48. Induced gas flow Arrows:

49. Induced gas flow Arrows:

50. Induced gas flow Arrows: