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

Abstract

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.