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**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

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**Transition regime and Numerical methods**

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**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.

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**Linearized Boltzmann equation**

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**Linearized Boltzmann equation**

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

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**Linearized Boltzmann equation**

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

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**Kernel representation of linearized collision term**

(Hard-sphere molecules)

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**Linearized boundary condition (diffuse reflection)**

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**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

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

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

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

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

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**Numerical method Similarity solution EQ for : BC for :**

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

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

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**Finite-difference scheme**

Finite difference in second-order, upwind known

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

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**Iteration method with convergence proof**

Takata & Funagane (2011), J. Fluid Mech. 669, 242 EQ for : BC for :

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**Iteration scheme for large**

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**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)

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**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

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

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**Velocity distribution function**

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Drag Force Stokes drag Small Kn viscosity

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**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.

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**Model Boltzmann equation**

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**Finite difference (BGK model)**

Outline (2D steady flows) [dimensionless] Marginal distributions Independent variables

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Eqs. for Discretization Grid points

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**(Iterative) finite-difference scheme**

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

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**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

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**Discontinuity in velocity distribution function**

Boltzmann eq. (steady flows) Sone & Takata (1992), TTSP 21, 501 Cercignani (2000), TTSP 29, 607

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**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.

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Method

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**F-D eq. along characteristics**

(line of discontinuity)

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Induced gas flow Arrows:

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Arrows:

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Arrows:

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Isothermal lines

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Isothermal lines

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**Marginal velocity distribution**

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**Marginal velocity distribution**

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**(Model of radiometric force)**

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

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Radiometer

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Induced gas flow Arrows:

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Induced gas flow Arrows:

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Force acting on plate

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