1. Theory and numerical approach in kinetic theory of gases (Part 3)
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. 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
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 Proof:
Takata (2009), …
Proof:
Similarity sol.

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

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

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

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

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

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

25. Computation of
Basis functions
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. 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)

30. 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

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. 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.

35. Model Boltzmann equation I:
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. Hot topic in micro fluid dynamics
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. A thin plate with one side heated in a rarefied gas
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)
gas
Discontinuous wall temperature
Sharp edges
Flow and force
???
Assumptions:
BGK model (nonlinear)
Arbitrary Knudsen number
Gas-surface interaction Diffuse reflection
Numerical analysis by finite-difference method

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

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

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

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), M3AS 11, 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. Velocity distribution function
Result of computation
marginal
Velocity distribution function

47. Velocity distribution function
marginal

48. Induced gas flow
Arrows:

49. Induced gas flow
Arrows:

50. Induced gas flow
Arrows:

51. Induced gas flow
Arrows:

52. Induced gas flow
Arrows:

53. Temperature field
Isothermal lines

54. Temperature field
Isothermal lines

55. Pressure field
Isobaric lines

56. Pressure field
Isobaric lines

57. Force acting on the plate

58. Normal stress on the plate
right surface
left surface
Normal stress Thermal stress

59. Usual explanation
Correct when collisions
between molecules are
not frequent
Force
cold
hot
What about when
collisions are frequent?
Number of incident
molecules is reduced.
No force?
cold
hot
True in the middle

60. What about when
collisions are frequent?
Number of incident
molecules is reduced.
No force?
cold
hot
True in the middle
Near the edge
Incident molecules from side
Reduction of incident molecules
Compensated by molecules
from side
Force
Edge effect is important!
cold
hot

61. Model Boltzmann equation II:
Decay of pendulum

62. Spatially 1D time-dependent problem VDF:
Molecular velocity
Damping rate of
linear pendulum
in full space
External force
(Hooke’s law)
Gas-body coupling
Collision-less gas
Caprino, Cavallaro,
& Marchioro, M3AS 17 (2007)
Tsuji & K.A, J. Stat. Phys.
146 (2012)
Collisional gas
Unsteady behavior of gas with interest in decay rate
BGK model + Diffuse reflection Numerical

63. BC: Diffuse reflection on plate
Gas:
EQ:
IC:
External force
(Hooke’s law)
BC: Diffuse reflection on plate

64. Singularity in VDF’s
Solution: determined along characteristic line

65. cf. Discontinuity around a convex body in steady flows
Singularity in VDF’s
Type 1
(discontinuity)
cf. Discontinuity around
a convex body in steady
flows
Sone & Takata, TTSP (1992)

66. Numerical method I: Method of characteristics
T. Tsuji & K.A., J. Comp. Phys. 250, 574 (2013)
- Accurate description of singularities in VDF
- Computationally expensive
Not suitable for long-time computation
Numerical method II: Semi-Lagrangian method
G. Russo & F. Filbet, KRM 2, 231 (2009)
[T. Tsuji & K.A., Phys. Rev. 89, (2014) ]
- Unable to describe singularities
- Accurate description for macroscopic quantities
- Computationally cheap
Suitable for long-time computation

67. Numerical method II: Semi-Lagrangian method
G. Russo & F. Filbet, KRM 2, 231 (2009)
[T. Tsuji & K.A., Phys. Rev. E 89, (2014) ]
Space coordinate relative to
Molecular velocity relative to
Plate at rest
External force term Curved characteristics

69. Numerical result
Semi-Lagrangian method Long-time computation
Parameters
Knudsen number
(dimensionless) plate density
(dimensionless) initial displacement
(dim-less) initial plate velocity

70. Results
Decay of displacement
Gradient

71. Results
Decay of displacement
Gradient

72. Slower than collision-less case
Numerical evidence (?)
Slower than collision-less case
Linear pendulum with a spherical body in a Stokes fluid
Cavallaro & Marchioro, M3AS (2010)