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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.

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Presentation on theme: "Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University."— Presentation transcript:

1 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)

2 Transition regime and Numerical methods

3 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

4 Linearized Boltzmann equation

5 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 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

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

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

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

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

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

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

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

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

21 Iteration method with convergence proof Takata & Funagane (2011), J. Fluid Mech. 669, 242 EQ for : BC for :

22 Iteration scheme for large

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24 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. ]

25 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

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

27 Velocity distribution function

28 Drag Force Stokes drag viscosity Small Kn

29 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

30 Model Boltzmann equation

31 Finite difference (BGK model) Outline (2D steady flows) [dimensionless] Marginal distributions Independent variables

32 Eqs. for Discretization Grid points

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

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

35 Discontinuity in velocity distribution function Boltzmann eq. (steady flows) Sone & Takata (1992), TTSP 21, 501 Cercignani (2000), TTSP 29, 607

36 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.

37 Method

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

39 Induced gas flow Arrows:

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41

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43 Isothermal lines

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45 Marginal velocity distribution a b c d

46 a b c d

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

48 Radiometer

49 Induced gas flow Arrows:

50 Induced gas flow Arrows:

51 Force acting on plate


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