[Paper Review] Appendix: Chapman-Enskog Expansion in the Lattice Boltzmann Method

Jun Li|arXiv (Cornell University)|Dec 6, 2015

Lattice Boltzmann Simulation Studies2 references3 citations

TL;DR

This paper derives the Navier-Stokes equation from the lattice Boltzmann method (LBM) using the Chapman-Enskog expansion, establishing a direct link between LBM model parameters and the kinematic viscosity. The key result is the analytical formula ν = (τ − 0.5)Δtc²/3, which enables accurate viscosity control in LBM simulations when the Mach and Knudsen numbers are small.

ABSTRACT

The Chapman-Enskog expansion was used in the lattice Boltzmann method (LBM) to derive a Navier-Stokes-like equation and a formula was obtained to correlate the LBM model parameters to the kinematic viscosity implicitly implemented in LBM simulations. The obtained correlation formula usually works as long as the model parameters are carefully selected to make the Mach number and Knudsen number small although the validity of Chapman-Enskog expansion that has a formal definition of time derivative without tangible mathematical sense is not recognized by many mathematicians.

Motivation & Objective

• To formally derive the Navier-Stokes-like equation from the lattice Boltzmann method using the Chapman-Enskog expansion.
• To establish a closed-form relationship between LBM model parameters (τ, Δt, c) and the kinematic viscosity ν.
• To validate the validity of the Chapman-Enskog expansion in LBM despite its formal mathematical definition lacking tangible meaning.
• To enable accurate simulation of incompressible fluid flows by linking model parameters to physical viscosity.
• To support the use of LBM for large eddy simulations (LES) by deriving a strain rate tensor formula from the expansion.

Proposed method

• Applies the Chapman-Enskog expansion to the lattice Boltzmann equation by expanding the distribution function fα as fα = feqα + f(1)α + f(2)α + …
• Introduces a multiple-scale time expansion ∂t = ∂t₀ + ∂t₁ + … to separate fast (t₀) and slow (t₁) dynamics.
• Uses Taylor expansion of the LBM streaming step to express the evolution equation in terms of time derivatives and spatial gradients.
• Derives the zeroth-order equation from the equilibrium distribution function, recovering mass and momentum conservation.
• Solves the first-order equation to express f(1)α in terms of the time and spatial derivatives of feqα.
• Computes the first-order moments of f(1)α and substitutes into the momentum equation to derive the viscous stress term.

Experimental results

Research questions

• RQ1 How can the Chapman-Enskog expansion be systematically applied to derive a Navier-Stokes-like equation from the lattice Boltzmann method?
• RQ2 What is the explicit analytical relationship between the LBM relaxation time τ and the kinematic viscosity ν?
• RQ3 Under what conditions is the Chapman-Enskog expansion valid in LBM, particularly regarding Mach and Knudsen numbers?
• RQ4 How does the derived viscosity formula ν = (τ − 0.5)Δtc²/3 relate to the physical viscosity in incompressible flows?
• RQ5 Can the derived strain rate tensor expression from the expansion be used effectively in large eddy simulations (LES) within LBM?

Key findings

• The Chapman-Enskog expansion successfully recovers a Navier-Stokes-like equation from the lattice Boltzmann equation under multiple-scale asymptotic analysis.
• The derived kinematic viscosity is given by ν = (τ − 0.5)Δtc²/3, which directly links model parameters to physical viscosity.
• The expansion is valid when the Mach number and Knudsen number are small, ensuring the asymptotic expansion remains consistent.
• The strain rate tensor is derived as ∑α eα,i eα,j f(1)α = −τΔt[c²ρ/3(∂ui/∂xj + ∂uj/∂xi) − ∂(ρuiujuk)/∂xk], enabling LES applications.
• The momentum equation derived from the expansion matches the standard incompressible Navier-Stokes form when density variations and nonlinear terms are negligible.
• The method ensures mass and momentum conservation through the equilibrium distribution function properties ∑α fα = ρ and ∑α eα,i fα = ρui.