Journal of Statistical Physics

, Volume 171, Issue 3, pp 493–520 | Cite as

Chapman–Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media

  • Chen Chen
  • Like LiEmail author
  • Renwei Mei
  • James F. Klausner


The gray lattice Boltzmann equation (GLBE) method has recently been used to simulate fluid flow in porous media. It employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) in the evolution equation to account for the linear drag of the medium. Several particular GLBE schemes have been proposed in the literature and these schemes are very easy to implement; but there exists uncertainty about the need for redefining the macroscopic velocity as there has been no systematic analysis to recover the Brinkman equation from the various GLBE schemes. Rigorous Chapman–Enskog analyses are carried out to show that the momentum equation recovered from these schemes can satisfy Brinkman equation to second order in \( \epsilon \) only if \( \theta = {\rm O}\left( \epsilon \right) \) in which \( \epsilon \) is the ratio of the lattice spacing to the characteristic length of physical dimension. The need for redefining macroscopic velocity is shown to be scheme-dependent. When a body force is encountered such as the gravitational force or that caused by a pressure gradient, different forms of forcing redefinitions are required for each GLBE scheme.


Lattice Boltzmann method Brinkman equation Chapman–Enskog expansion 



This work was partially supported by the U.S. Department of Energy, ARPA-E, under Award No. DEAR0000184, and the U.S. Department of Energy, SunShot Initiative, under Award No. DE-EE0006534.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Chen Chen
    • 1
  • Like Li
    • 2
    Email author
  • Renwei Mei
    • 1
  • James F. Klausner
    • 3
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Department of Mechanical EngineeringMississippi State UniversityMississippi StateUSA
  3. 3.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA

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