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Research Papers: Fundamental Issues and Canonical Flows

Is the Lattice Boltzmann Method Applicable to Rarefied Gas Flows? Comprehensive Evaluation of the Higher-Order Models

[+] Author and Article Information
Minoru Watari

LBM Fluid Dynamics Laboratory,
3-2-1 Mitahora-higashi,
Gifu 502-0003, Japan
e-mail: watari-minoru@kvd.biglobe.ne.jp

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 10, 2014; final manuscript received June 28, 2015; published online August 21, 2015. Assoc. Editor: John Abraham.

J. Fluids Eng 138(1), 011202 (Aug 21, 2015) (18 pages) Paper No: FE-14-1654; doi: 10.1115/1.4031000 History: Received November 10, 2014

Lattice Boltzmann method (LBM) whose equilibrium distribution function contains higher-order terms is called higher-order LBM. It is expected that nonequilibrium physics beyond the Navier–Stokes can be accurately captured using the higher-order LBM. Relationship between the level of higher-order and the simulation accuracy of rarefied gas flows is studied. Theoretical basis for constructing higher-order LBM is presented. On this basis, specific higher-order models are constructed. To confirm that the models have been correctly constructed, verification simulations are performed focusing on the continuum regime: sound wave and supersonic flow in Laval nozzle. With applications to microelectromechanical systems (MEMS) in mind, low Mach number flows are studied. Shear flow and heat conduction between parallel walls in the slip flow regime are investigated to confirm the relaxation process in the Knudsen layer. Problems between concentric cylinders are investigated from the slip flow regime to the free molecule regime to confirm the effect of boundary curvature. The accuracy is discussed comparing the simulation results with pioneers' studies. Models of the fourth-order give sufficient accuracy even for highly rarefied gas flows. Increase of the particle directions is necessary as the Knudsen number increases.

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Figures

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Fig. 1

Tuning result, projected average particle speed

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Fig. 2

Schematic view of speed of sound simulation

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Fig. 3

Wave propagation in the speed of sound simulation: Δx = 0.01 and Δt = 0.0002

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Fig. 4

Simulation results of the speed of sound: Δx = 0.1 and Δt = 0.002. Maximum error (L1 norm) relative to cs = 1.29 at T = 1 is 0.8%.

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Fig. 5

Generalized coordinate system fitted with nozzle shape. The grids are 101 × 21. Δx = 0.1 and Δt = 0.001.

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Fig. 6

Mach contours in the supersonic nozzle simulation for Model 4–10

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Fig. 7

Ratios of the averaged flow parameters: A/A*,ρ/ρ*,T/T*, and p/p* versus Mach. Maximum error in pressure relative to the value at the throat is 1%.

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Fig. 8

Schematic view of velocity slip and temperature jump simulations

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Fig. 9

Nodes for the velocity slip and temperature jump simulations: Δx = 0.005 and Δt = 0.0002

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Fig. 10

Typical profile of velocity slip and temperature jump

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Fig. 11

Grid convergence for the velocity slip simulation. The dotted line indicates the gradient 2 in log–log scale.

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Fig. 12

Simulation results of the velocity slip simulation. Maximum error of Model 10–24 relative to uyw = 0.01 is 0.3% for the velocity slip and 0.5% for the Knudsen profile.

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Fig. 13

Simulation results of the temperature jump simulation. Maximum error of Model 10–24 relative to ΔTw=0.01 is 0.6% for the temperature jump and 0.3% for the Knudsen profile.

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Fig. 14

Discontinuity in the velocity distribution near a convex boundary

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Fig. 15

Schematic view of shear flow and heat conduction between concentric cylinders

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Fig. 16

Grid of the physical space used in the simulations: Δr = 0.005,Δθ = π/100, and Δt = 0.00025

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Fig. 17

Simulation results. Azimuthal velocity uθ contours for Model 4–24.

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Fig. 18

Simulation results. Azimuthal velocity uθ contours for Models 4–60.

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Fig. 19

Azimuthal velocity uθ profiles for Models 4–24 and 4–60 at θ = 0 deg and 7.2 deg. For Models 4–60, since the flow is uniform, only θ = 0 deg is shown.

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Fig. 20

Comparison of azimuthal velocity profiles uθ at θ=0 deg between Models 4 and 6

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Fig. 21

Azimuthal velocity profiles are compared with previous studies. Models 4–10 and 4–24 for Kn≥0.5 are averaged profiles. The analytical solution of free molecule limit is not discernible overlapped with the Kn = 100. Maximum error of Models 4–60 relative to uθw=0.01 is 1–3%.

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Fig. 22

Distribution function gki (k = 1) of Models 4–60 for Kn = 0.5 along θ = 0 at radial positions r=1.0…2.0. Cross symbols are the positions of discontinuity. Right figure shows the ranges of velocity distribution affected by emissions from the inner cylinder.

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Fig. 23

Standard deviations in azimuthal velocity divided by inner cylinder speed σuθ/uθw

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Fig. 24

Shear stress prθ×r2 versus radial position for Models 4–60

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Fig. 25

Shear stress prθ×r2 versus Knudsen number. Theoretical curve is constructed by combining studies by Ai, Cercignani and Sernagiotto, and analytical solution of the free molecule limit.

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Fig. 26

Simulation results. Temperature T contours for Models 4–24.

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Fig. 27

Simulation results. Temperature T contours for Models 4–60.

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Fig. 28

Temperature T profiles for Models 4–24 and 4–60 at θ = 0 deg and 7.2 deg. For Models 4–60, since the flow is uniform, only θ = 0 deg is shown.

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Fig. 29

Temperature and density profiles for all models are compared with previous studies. The analytical solution of free molecule limit is not discernible overlapped with the Kn = 160. Maximum error in T of Models 4–60 relative to Tw1-Tw2=0.1 is 1–2%.

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Fig. 30

Standard deviations in temperature divided by temperature difference σT/ΔTw

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Fig. 31

Heat flux qr×r versus Knudsen number

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