Theory and numerical approach in kinetic theory of gases (Part 3) 2018 International Graduate Summer School on “Frontiers of Applied and Computational Mathematics” (Shanghai Jiao Tong University, July 9-21, 2018) Theory and numerical approach in kinetic theory of gases (Part 3) Kazuo Aoki Dept. of Math., National Cheng Kung University, Tainan and NCTS, National Taiwan University, Taipei

Transition regime and Numerical methods

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.

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 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

Similarity solution EQ for : EQ for : BC for : Numerical solution (finite-difference)

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

Global mass-flow rate Global heat-flow rate Symmetry relation Proof: Takata (2009), … Proof: Similarity sol.

EQ for : (a) EQ for : (b) BC for :

Properties (i) Commutation with parity operator (ii) Self-adjointness (iii)

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

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

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

Kernel representation of linearized collision term (Hard-sphere molecules)

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

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)

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

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

Velocity distribution function

Drag Force Stokes drag Small Kn viscosity

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.

Model Boltzmann equation I: Radiometric flow

Radiometer and radiometric force Crookes Radiometer (light mill) William Crookes (1874) Atmospheric pressure Effect of rarefied gas Effect of microscale mean free path

Hot topic in micro fluid dynamics Classical topic Maxwell, Reynolds, Einstein, Kennard, Loeb, … Hot topic in micro fluid dynamics Wadsworth & Muntz (1996), Ohta, et al. (2001), Selden, Muntz, Ketsdever, Gimelshein, et al. (2009, 2011) light flow force cold hot Flow induced by temperature difference Resulting force acting on vane

A thin plate with one side heated in a rarefied gas Model problem Taguchi & A, J. Fluid Mech. 694, 191 (2012) A thin plate with one side heated in a rarefied gas in a square box (2D problem) gas Discontinuous wall temperature Sharp edges Flow and force ??? Assumptions: BGK model (nonlinear) Arbitrary Knudsen number Gas-surface interaction Diffuse reflection Numerical analysis by finite-difference method

BGK model BC 2D steady flows [dimensionless] Diffuse reflection No net mass flux across boundary

BGK model BC 2D steady flows [dimensionless] Diffuse reflection Specular No net mass flux across boundary

Eqs. for BC for Marginal distributions Independent variables Discretization Grid points

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

Computational difficulty Discontinuity in velocity distribution function Finite difference + Characteristic A, Sone, Nishino, Sugimoto (1991) Sone & Sugimoto (1992, 1993, 1995) Takata, Sone, & A (1993), Sone, Takata, & Wakabayashi (1994) A, Kanba, & Takata (1997), A, Takata, Aikawa, Golse (2001), … 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 (2011), Commun. Math. Phys. IBVP of Boltzmann eq.

Method (Upper half)

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

Velocity distribution function Result of computation marginal Velocity distribution function

Velocity distribution function marginal

Induced gas flow Arrows:

Induced gas flow Arrows:

Induced gas flow Arrows:

Induced gas flow Arrows:

Induced gas flow Arrows:

Temperature field Isothermal lines

Temperature field Isothermal lines

Pressure field Isobaric lines

Pressure field Isobaric lines

Force acting on the plate

Normal stress on the plate right surface left surface Normal stress Thermal stress

Usual explanation Correct when collisions between molecules are not frequent Force cold hot What about when collisions are frequent? Number of incident molecules is reduced. No force? cold hot True in the middle

What about when collisions are frequent? Number of incident molecules is reduced. No force? cold hot True in the middle Near the edge Incident molecules from side Reduction of incident molecules Compensated by molecules from side Force Edge effect is important! cold hot

Model Boltzmann equation II: Decay of pendulum

Spatially 1D time-dependent problem VDF: Molecular velocity Damping rate of linear pendulum in full space External force (Hooke’s law) Gas-body coupling Collision-less gas Caprino, Cavallaro, & Marchioro, M3AS 17 (2007) Tsuji & K.A, J. Stat. Phys. 146 (2012) Collisional gas Unsteady behavior of gas with interest in decay rate BGK model + Diffuse reflection Numerical

BC: Diffuse reflection on plate Gas: EQ: IC: External force (Hooke’s law) BC: Diffuse reflection on plate

Singularity in VDF’s Solution: determined along characteristic line

cf. Discontinuity around a convex body in steady flows Singularity in VDF’s Type 1 (discontinuity) cf. Discontinuity around a convex body in steady flows Sone & Takata, TTSP (1992)

Numerical method I: Method of characteristics T. Tsuji & K.A., J. Comp. Phys. 250, 574 (2013) - Accurate description of singularities in VDF - Computationally expensive Not suitable for long-time computation Numerical method II: Semi-Lagrangian method G. Russo & F. Filbet, KRM 2, 231 (2009) [T. Tsuji & K.A., Phys. Rev. 89, 052129 (2014) ] - Unable to describe singularities - Accurate description for macroscopic quantities - Computationally cheap Suitable for long-time computation

Numerical method II: Semi-Lagrangian method G. Russo & F. Filbet, KRM 2, 231 (2009) [T. Tsuji & K.A., Phys. Rev. E 89, 052129 (2014) ] Space coordinate relative to Molecular velocity relative to Plate at rest External force term Curved characteristics

Numerical result Semi-Lagrangian method Long-time computation Parameters Knudsen number (dimensionless) plate density (dimensionless) initial displacement (dim-less) initial plate velocity

Results Decay of displacement Gradient

Results Decay of displacement Gradient

Slower than collision-less case Numerical evidence (?) Slower than collision-less case Linear pendulum with a spherical body in a Stokes fluid Cavallaro & Marchioro, M3AS (2010)