Boundary-value problems of the Boltzmann equation:

Asymptotic and numerical analyses(Part 3)

Kazuo Aoki

Dept. of Mech. Eng. and Sci.

Kyoto University

Intensive Lecture Series(Postech, June 20-21, 2011)

Transition regime and Numerical methods

Stochastic (particle) method

Deterministic methods

DSMC (Direct Simulation Monte Carlo) methodG. A. Bird (1963, …, 1976, …, 1994, …)

Finite-difference (or discrete-ordinate) method

Linearized Boltzmann eq. Brief outline & some examples

Model Boltzmann eq. & Nonlinear Boltzmann eq. Brief outline & some examples

Transition regime

Numerical Methods for the Boltzmann eq. or its models

arbitrary

Linearized Boltzmann equation

Linearized Boltzmann equation

Steady (or time-independent) problems

Linearized B eq.:

Linearized Boltzmann equation

Steady (or time-independent) problems

Linearized B eq.:

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

Linearized boundary condition (diffuse reflection)

Poiseuille flow and thermal transpiration

Gas between two parallel plates

Small pressure gradient

Small temperature gradient

Linearized Boltzmann eq.

Ohwada, Sone, & A(1989), Phys. Fluids A

Chen, Chen, Liu, & Sone (2007),CPAM 60, 147

Mathematical study

Similarity solution

Numerical solution (finite-difference)

Flow velocity

Heat Flow

Flow velocity

Heat Flow

Global mass-flow rate

Global heat-flow rate

Flow velocity

Heat Flow

Global mass-flow rate

Global heat-flow rate

Flow velocity

Heat Flow

Global mass-flow rate

Global heat-flow rate

Numerical method Ohwada, Sone & A (1989)

Similarity solution

EQ for :

BC for :

Grid points

Time-derivative term Long-time limit Steady sol.

Finite-difference scheme

(Suffix omitted)

known

Finite-difference scheme Finite difference in second-order, upwind

Computation of Basis functions

Piecewise quadraticfunction in

Independent of and Computable beforehand

Numerical kernels

Iteration method with convergence proof

Takata & Funagane (2011), J. Fluid Mech. 669, 242

EQ for :

BC for :

Iteration scheme for large

Slow flow past a sphere

Linearized Boltzmann eq.Diffuse reflection

Takata, Sone, & A (1993), Phys. Fluids A

Numerical solution (finite-difference)

Similarity solution [ Sone & A (1983), J Mec. Theor. Appl. ]

Difficulty 1: Discontinuity of velocity distribution function (VDF)

Sone & Takata (1992), Cercignani (2000)

• VDF is discontinuous on convex body.• Discontinuity propagates in gas along characteristics

BC

EQ

Finite difference +Characteristic

Difficulty 2: Slow approach to state at infinity

Numerical matching with asymptotic solution

Velocity distribution function

Drag Force

Stokes drag

viscosity

Small Kn

Stochastic (particle) method

Deterministic methods

DSMC (Direct Simulation Monte Carlo) methodG. A. Bird (1963, …, 1976, …, 1994, …)

Finite-difference (or discrete-ordinate) method

Linearized Boltzmann eq. Brief outline & some examples

Model Boltzmann eq. & Nonlinear Boltzmann eq. Brief outline & some examples

Transition regime

Numerical Methods for the Boltzmann or its models

arbitrary

Model Boltzmann equation

Finite difference (BGK model)

Outline (2D steady flows) [dimensionless]

Marginal distributions

Independent variables

Eqs. for

Discretization

Grid points

(Iterative) finite-difference scheme

Standard finite difference (2nd-order upwind scheme)

known

Example

BC Diffuse reflection

Discontinuity in

Flow caused by discontinuous wall temperature

A, Takata, Aikawa, & Golse(2001), Phys. Fluids 13, 2645

Discontinuity in velocity distribution function

Boltzmann eq. (steady flows)

Sone & Takata (1992),TTSP 21, 501Cercignani (2000),TTSP 29, 607

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.

Method

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

Induced gas flow

Arrows:

Arrows:

Arrows:

Isothermal lines

Isothermal lines

Marginal velocity distribution

ab

c

d

Marginal velocity distribution

ab

c

d

Example (Model of radiometric force)

Taguchi & A (2011)

Radiometer

Induced gas flow

Arrows:

Induced gas flow

Arrows:

Force acting on plate