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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)

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Transition regime and Numerical methods

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Stochastic (particle) method Deterministic methods DSMC (Direct Simulation Monte Carlo) method G. 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

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Linearized Boltzmann equation

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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 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

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

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

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

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

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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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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. ]

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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

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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 viscosity Small Kn

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Stochastic (particle) method Deterministic methods DSMC (Direct Simulation Monte Carlo) method G. 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

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

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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), M 3 AS 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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Isothermal lines

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Marginal velocity distribution a b c d

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a b c d

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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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