Techniques and Procedures

What is the Small Parameter ε in the Chapman-Enskog Expansion of the Lattice Boltzmann Method?

[+] Author and Article Information
Minoru Watari

 LBM Fluid Dynamics Laboratory, 3-2-1 Mitahora-higashi, Gifu 502-0003, Japanwatari-minoru@kvd.biglobe.ne.jp

J. Fluids Eng 134(1), 011401 (Feb 24, 2012) (7 pages) doi:10.1115/1.4005682 History: Received March 06, 2011; Revised December 07, 2011; Published February 23, 2012; Online February 24, 2012

The lattice Boltzmann method (LBM) is shown to be equivalent to the Navier-Stokes equations by applying the Chapman-Enskog (C-E) expansion, which has been established by pioneer researchers. However, it is still difficult for elementary researchers. There is no clear explanation of the small parameter ε used in the C-E expansion. There are several expressions for the viscosity coefficient; some are unclear on the relationship with ε. There are two expressions on the LBM evolution equation. Elementary researchers are perplexed as to which is correct. The LBM achieves second order accuracy by including the numerical viscosity within the physical viscosity. This is not only difficult for elementary researchers to understand but also sometimes leads senior researchers into making errors. The C-E expansion of the LBM was thoroughly reviewed and is presented as a self-contained form in this paper. It is natural to use the time step Δt as ε. The viscosity coefficient is expressed as μ∝Δxc(τ − 1/2). The viscosity relationship and the second order accuracy were confirmed by numerical simulations. The difference in the two expressions on the LBM evolution is simply one of perspective. They are identical. The difference between the relaxation parameter τD for the discrete Boltzmann equation and τ for the LBM was discussed. While τD is a quantity of time, τ is genuinely nondimensional, which is sometimes overlooked.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Velocity particles of the D2Q9 model

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Figure 2

Grid system in the LBM simulation

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Figure 3

The ux distribution in the numerical simulation

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Figure 4

Simulation result. The ux distribution versus time t.

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Figure 5

Simulation result. Kinetic viscosity ν versus lattice increment Δx.

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Figure 6

Simulation result. Kinetic viscosity ν versus collision parameter τ.

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Figure 7

Simulation result. Kinetic viscosity ν versus basic particle speed c.

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Figure 8

Simulation result. Kinetic viscosity ν versus lattice increment Δx. Combinations of (Δx, τ) are changed to give the same kinetic viscosity ν = 0.001666.

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Figure 9

Simulation result. Convergence rate of error in the kinetic viscosity ν versus lattice increment Δx.




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