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

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

Oct 06, 2023 452 likes •770 views

Reciprocity among elemental relaxation and driven-flow problems for a rarefied gas Shigeru TAKATA ( ) In collaboration with Masashi Oishi Department of Mechanical Engineering and Science, (also Advanced Research Institute of Fluid

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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

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Motivation:

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Example 1: Steady flow identity mass flux mass flux heat flux heat flux

Poiseuille flow Thermal transpiration

Fluid-dynamical and thermal phenomena are mutually inductive in rarefied gases.

e.g. Loyalka (1971) Onsager reciprocity

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Example 2: steady flow

Thermophoresis A force acting on the particle Thermal polarization Non-uniform temperature field is induced. (slow) uniform flow

identity

What kind of relations (or identities) does hold, in general, between two problems described by the linearized Boltzmann equation?

e.g. Roldughin (1994)

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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

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(DP) (DT) (RP) (RT)

gravity (0,-g,0) D

Initial state

Uniform flow (0,U,0) D D

Initial state

Uniform heat flow (0,q,0) D

Problem: Rarefied gas flows in a channel

time dependent

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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

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Dimensionless formulation

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(Kn: Knudsen number)

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(DP) (DT) (RP) (RT)

gravity

Initial state

Uniform flow

Initial state

Uniform heat flow

Temperature gradient Temperature gradient

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Reciprocity of fluxes

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T (2010) J. Stat. Phys

Symmetric relation (time-dependent)

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: convolution

Symmetric relation

b.c. Time-dependent linearized Boltzmann equation i.c.

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Apply the symmetric relation for unsteady problems

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T (2010) J. Stat. Phys [NOTE] For any t and any k 3! = 6 identities

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Temperature gradient Temperature gradient 12

(DP) (DT) (RP) (RT)

gravity

Initial state Uniform flow Initial state Uniform heat flow

Dimensionless mass flow Dimensionless heat flow

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Numerical results

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Numerical results

  • 1. Flux reciprocity
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Temperature gradient Temperature gradient 15

(DP) (DT) (RP) (RT)

gravity

Initial state Uniform flow Initial state Uniform heat flow Dimensionless mass flow Dimensionless heat flow

k= 1

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Temperature gradient Temperature gradient 16

(DP) (DT) (RP) (RT)

gravity

Initial state Uniform flow Initial state Uniform heat flow

k= 1

Dimensionless mass flow Dimensionless heat flow

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(DP) (RP)

gravity

Initial state Uniform flow

k= 1

Dimensionless mass flow Dimensionless heat flow

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Temperature gradient Temperature gradient 18

(DT) (RT)

Initial state Uniform heat flow

k= 1

Dimensionless mass flow Dimensionless heat flow

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Numerical results

  • 2. Velocity distribution function
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(DP) (DT) (RP) (RT)

gravity

Initial state

Uniform flow

Initial state

Uniform heat flow

Temperature gradient Temperature gradient

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(RP) (RT)

Initial state

Uniform flow

Initial state

Uniform heat flow Discontinuities propagate into a gas along the characteristic lines

Sugimoto & Sone (1990)

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(DP) (DT)

gravity

Temperature gradient Temperature gradient

No discontinuities

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Marginal VDF

k= 1, x1=0.3965

Initial state

Uniform flow

(RP)

Propagation of discontinuities

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Marginal VDF

k= 1, x1=0.3965

gravity

(DP)

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Marginal VDF

k= 1, x1=0.3965

gravity

(DP)

No discontinuity

(BUT)

Propagation of derivative discontinuities

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Relaxation Problems

g:

Driven-flow Problems

Relaxation problems vs driven-flow problems

  • 1. discontinuity

derivative discontinuity

  • 2. Identity should hold

at the level of profile

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(DP) (DT) (RP) (RT)

gravity

Initial state

Uniform flow

Initial state

Uniform heat flow

Temperature gradient Temperature gradient

Time integral

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Temperature gradient Temperature gradient 28

(DT) (RT)

Initial state Uniform heat flow

k= 1

Dimensionless mass flow Dimensionless heat flow

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(DP) (RP)

gravity

Initial state Uniform flow

k= 1

Dimensionless mass flow Dimensionless heat flow

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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)

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