Chapman Enskog Analysis Essay

In the lattice Boltzmann (LB) method, the forcing scheme, which is used to incorporate an external or internal force into the LB equation, plays an important role. It determines whether the force of the system is correctly implemented in an LB model and affects the numerical accuracy. In this paper we aim to clarify a critical issue about the Chapman-Enskog analysis for a class of forcing schemes in the LB method in which the velocity in the equilibrium density distribution function is given by , while the actual fluid velocity is defined as . It is shown that the usual Chapman-Enskog analysis for this class of forcing schemes should be revised so as to derive the actual macroscopic equations recovered from these forcing schemes. Three forcing schemes belonging to the above class are analyzed, among which Wagner's forcing scheme [A. J. Wagner, Phys. Rev. E74, 056703 (2006)] is shown to be capable of reproducing the correct macroscopic equations. The theoretical analyses are examined and demonstrated with two numerical tests, including the simulation of Womersley flow and the modeling of flat and circular interfaces by the pseudopotential multiphase LB model.

  • Received 8 May 2016
  • Revised 2 September 2016


Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Fluid Dynamics

Title: Chapman-Enskog Analysis of Finite Volume Lattice Boltzmann Schemes

Authors:Nima H. Siboni, Dierk Raabe, Fathollah Varnik

(Submitted on 20 Jul 2014)

Abstract: In this paper, we provide a systematic analysis of some finite volume lattice Boltzmann schemes in two dimensions. A complete iteration cycle in time evolution of discretized distribution functions is formally divided into collision and propagation (streaming) steps. Considering mass and momentum conserving properties of the collision step, it becomes obvious that changes in the momentum of finite volume cells is just due to the propagation step. Details of the propagation step are discussed for different approximate schemes for the evaluation of fluxes at the boundaries of the finite volume cells. Moreover, a full Chapman-Enskog analysis is conducted allowing to recover the Navier-Stokes equation. As an important result of this analysis, the relation between the lattice Boltzmann relaxation time and the kinematic viscosity of the fluid is derived for each approximate flux evaluation scheme. In particular, it is found that the constant upwind scheme leads to a positive numerical viscosity while the central scheme as well as the linear upwind scheme are free of this artifact.

Submission history

From: Nima Hamidi Siboni [view email]
[v1] Sun, 20 Jul 2014 10:43:41 GMT (59kb,D)

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