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

Transition regime and Numerical methods

Transition regime arbitrary

Numerical Methods for the Boltzmann eq. or its models

Stochastic (particle) method

DSMC ( D irect S imulation M onte C arlo) 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

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 Ohwada, Sone, & A (1989),

Phys. Fluids A

Gas between two parallel plates Small pressure gradient Linearized Boltzmann eq.

Small temperature gradient Mathematical study Chen, Chen, Liu, & Sone (2007), CPAM 60 , 147

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 Similarity solution Ohwada, Sone & A (1989) EQ for : BC for :

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

Finite-difference scheme

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

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

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 Similarity solution Takata, Sone, & A (1993),

Phys. Fluids A

Linearized Boltzmann eq.

Diffuse reflection [ Sone & A (1983),

J Mec. Theor. Appl.

] Numerical solution (finite-difference)

Difficulty 1: Discontinuity Sone & Takata (1992), Cercignani (2000) of velocity distribution function (VDF) BC • VDF is discontinuous on convex body.

• Discontinuity propagates in gas along characteristics EQ Finite difference + Characteristic

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

Velocity distribution function

Drag Force Stokes drag Small Kn viscosity

Transition regime arbitrary

Numerical Methods for the Boltzmann or its models

Stochastic (particle) method

DSMC ( D irect S imulation M onte C arlo) 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

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

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

M 3

AS 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 d c b a

Marginal velocity distribution d c b a

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

Radiometer

Induced gas flow Arrows:

Induced gas flow Arrows:

Force acting on plate