Reciprocity among elemental relaxation and driven-flow problems for - PowerPoint PPT Presentation

Reciprocity among elemental relaxation and driven-flow problems for a rarefied gas Shigeru TAKATA ( 髙田 滋 ) In collaboration with Masashi Oishi Department of Mechanical Engineering and Science, (also Advanced Research Institute of Fluid Science and Engineering) Kyoto University takata.shigeru.4a@kyoto-u.ac.jp http://www.mfd.me.kyoto-u.ac.jp/eng-menu.htm 1

Motivation: Fluid-dynamical and thermal phenomena are mutually inductive in rarefied gases. Example 1: Steady flow mass flux heat flux Poiseuille flow Thermal transpiration heat flux mass flux identity e.g. Loyalka (1971) Onsager reciprocity 2

e.g. Example 2: steady flow Roldughin (1994) (slow) uniform flow identity Non-uniform A force acting on temperature the particle field is induced. Thermal polarization Thermophoresis What kind of relations (or identities) does hold, in general , between two problems described by the linearized Boltzmann equation? 3

Our recent theory • Steady case – T. (2009) J. Stat. Phys (symmetry argument) – T. (2009) J. Stat. Phys (relation to Onsager’s reciprocity) (cf) for entropy production, Onsager’s reciprocity Waldmann, Kuscer, Roldughin , Loyalka, … • Unsteady case – T. (2010) J. Stat. Phys. TODAY : Numerical demonstration of a simple example 4

Problem: Rarefied gas flows in a channel time dependent (DP) (DT) D D gravity (0,- g ,0) (RP) (RT) Initial state Initial state D D Uniform flow (0, U ,0) Uniform heat flow (0, q ,0) 5

Problem: Rarefied gas flows in a channel Basic assumption  Boltzmann equation  kinetic boundary condition (with the detailed balance)  linearization (weakly perturbed system ) around a reference resting equilibrium state Numerical computation  Bhatnagar-Gross-Krook model (Boltzmann-Krook-Welander model)  diffuse reflection boundary condition 6

Dimensionless formulation 1 7 (Kn: Knudsen number)

(DP) (DT) Temperature gradient gravity Temperature gradient (RP) (RT) Initial state Initial state Uniform flow Uniform heat flow 8

Reciprocity of fluxes

Symmetric relation (time-dependent) Time-dependent linearized Boltzmann equation b.c. i.c. T (2010) J. Stat. Phys Symmetric relation : convolution 10

Apply the symmetric relation for unsteady problems T (2010) J. Stat. Phys [NOTE]  For any t and any k  3! = 6 identities 11

Dimensionless mass flow Dimensionless heat flow (DP) (DT) Temperature gradient gravity Temperature gradient (RP) (RT) Initial state Initial state Uniform flow Uniform heat flow 12

Numerical results

Numerical results 1. Flux reciprocity

Dimensionless mass flow Dimensionless heat flow (DP) (DT) Temperature gradient gravity Temperature gradient k = 1 Initial state Initial state (RP) (RT) Uniform flow Uniform heat flow 15

Dimensionless mass flow k = 1 Dimensionless heat flow (DP) (DT) Temperature gradient gravity Temperature gradient Initial state Initial state (RP) (RT) Uniform flow Uniform heat flow 16

Dimensionless mass flow k = 1 Dimensionless heat flow (DP) gravity Initial state (RP) Uniform flow 17

Dimensionless mass flow k = 1 Dimensionless heat flow (DT) Temperature gradient Temperature gradient Initial state (RT) Uniform heat flow 18

Numerical results 2. Velocity distribution function

(DP) (DT) Temperature gradient gravity Temperature gradient (RP) (RT) Initial state Initial state Uniform flow Uniform heat flow 20

Discontinuities propagate into a gas along the characteristic lines Sugimoto & Sone (1990) (RP) (RT) Initial state Initial state Uniform flow Uniform heat flow 21

(DP) (DT) Temperature gradient gravity Temperature gradient No discontinuities 22

Marginal VDF k = 1, x 1 =0.3965 Propagation of discontinuities (RP) Initial state Uniform flow

Marginal VDF k = 1, x 1 =0.3965 (DP) gravity

Marginal VDF k = 1, x 1 =0.3965 (DP) gravity No discontinuity (BUT) Propagation of derivative discontinuities

Relaxation problems vs driven-flow problems Relaxation Problems 1. discontinuity g : 2. Identity should hold at the level of profile Driven-flow Problems derivative discontinuity 26

(DP) (DT) Temperature gradient gravity Temperature gradient Time integral (RP) (RT) Initial state Initial state Uniform flow Uniform heat flow 27

Dimensionless mass flow Dimensionless heat flow k = 1 (DT) Temperature gradient Temperature gradient (RT) Initial state Uniform heat flow 28

Dimensionless mass flow Dimensionless heat flow (DP) gravity (RP) Initial state Uniform flow k = 1 29

Summary • Linearized Boltzmann Eq. (rarefied gases) • Reciprocity among – Relaxations from uniform mass (RP) and heat flows (RT) – Gravity-driven (DP) and temperature-driven (DT) flows • Numerical demonstration (BGK+diffuse reflection) – Identities among the fluxes – Discontinuity of VDF in (RP) and (RT) – Derivative discontinuity of VDF in (DP) and (DT) – (DP) and (DT) = Time integral of (RP) and (RT) 30