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![Page 1: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/1.jpg)
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)
![Page 2: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/2.jpg)
Transition regime and Numerical methods
![Page 3: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/3.jpg)
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
![Page 4: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/4.jpg)
Linearized Boltzmann equation
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Linearized Boltzmann equation
Steady (or time-independent) problems
Linearized B eq.:
![Page 6: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/6.jpg)
Linearized Boltzmann equation
Steady (or time-independent) problems
Linearized B eq.:
![Page 7: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/7.jpg)
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
![Page 11: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/11.jpg)
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
![Page 14: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/14.jpg)
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Flow velocity
Heat Flow
Global mass-flow rate
Global heat-flow rate
![Page 16: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/16.jpg)
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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 quadraticfunction 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) 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
![Page 30: Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 3) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University.](https://reader038.fdocuments.net/reader038/viewer/2022110205/56649cc25503460f94989a4f/html5/thumbnails/30.jpg)
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, 501Cercignani (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
ab
c
d
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Marginal velocity distribution
ab
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