1. Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University
Intensive Lecture Series
(Postech, June 20-21, 2011)
Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3)
Kazuo Aoki
Dept. of Mech. Eng. and Sci.
Kyoto University

2. Transition regime and Numerical methods

3. Stochastic (particle) method
Transition regime
arbitrary
Numerical Methods for 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.

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 representation of linearized collision term
(Hard-sphere molecules)

8. Linearized boundary condition (diffuse reflection)

9. Poiseuille flow and thermal transpiration
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),
CPAM 60, 147

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

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

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

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

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

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

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

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

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

22. Iteration scheme for large

24. Linearized Boltzmann eq. Diffuse reflection
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)

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

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

27. Velocity distribution function

28. Drag Force
Stokes drag
Small Kn
viscosity

29. Stochastic (particle) method
Transition regime
arbitrary
Numerical Methods for 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.

30. Model Boltzmann equation

31. Finite difference (BGK model)
Outline (2D steady flows) [dimensionless]
Marginal distributions
Independent variables

32. Eqs. for
Discretization
Grid points

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

34. Flow caused by discontinuous wall temperature
Example
Flow caused by discontinuous
wall temperature
A, Takata, Aikawa, & Golse
(2001), Phys. Fluids 13, 2645
BC Diffuse reflection
Discontinuity in

35. Discontinuity in velocity distribution function
Boltzmann eq. (steady flows)
Sone & Takata (1992),
TTSP 21, 501
Cercignani (2000),
TTSP 29, 607

36. Discontinuous boundary data
Finite difference +
Characteristic
Sone & Sugimoto
(1992, 1993, 1995)
Takata, Sone, & A (1993),
Sone, Takata, & Wakabayashi (1994)
A, Kanba, & Takata (1997), …
Mathematical theory
Boudin & Desvillettes (2000), Monatsh. Math. 131, 91
IVP of Boltzmann eq.
A, Bardos, Dogbe, & Golse (2001), M3AS 11, 1581
BVP of a simple transport eq.
C. Kim (2010) BVP of Boltzmann eq.

37. Method

38. F-D eq. along characteristics
(line of discontinuity)

39. Induced gas flow
Arrows:

40. Arrows:

41. Arrows:

43. Isothermal lines

44. Isothermal lines

45. Marginal velocity distribution

46. Marginal velocity distribution

47. (Model of radiometric force)
Example
(Model of radiometric force)
Taguchi & A (2011)

48. Radiometer

49. Induced gas flow
Arrows:

50. Induced gas flow
Arrows:

51. Force acting on plate