Abstract
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 u=∑_{α}e_{α}f_{α}/ρ, while the actual fluid velocity is defined as u[over ̂]=u+δ_{t}F/(2ρ). 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. E 74, 056703 (2006)10.1103/PhysRevE.74.056703] 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.
Highlights
The lattice Boltzmann (LB) method, which is a special discrete solver for mimicking the kinetic Boltzmann equation, has been developed into an efficient mesoscopic numerical approach for simulating fluid flow and heat transfer [1,2]
It can be found that these two forcing schemes share the following feature: The velocity in the equilibrium density distribution function is given by u = α eαfα/ρ, while the actual fluid velocity is defined as u = u + δt F/(2ρ), where F is the force and δt is
In the above-mentioned class of forcing schemes, the actual fluid velocity is defined as u = u + δt F/2ρ
Summary
The lattice Boltzmann (LB) method, which is a special discrete solver for mimicking the kinetic Boltzmann equation, has been developed into an efficient mesoscopic numerical approach for simulating fluid flow and heat transfer [1,2]. Wagner showed [12] that his forcing scheme was obtained based on the Taylor expansion analysis for recovering the correct macroscopic equations. To clarify such an issue, in the present study we demonstrate that the usual Chapman-Enskog analysis in the literature for the aforementioned class of forcing schemes should be modified in order to derive the actual macroscopic equations recovered from these forcing schemes. The revised Chapman-Enskog analysis shows that Wagner’s forcing scheme can correctly recover the macroscopic equations at the Navier-Stokes level, confirming the consistency between the Chapman-Enskog analysis and the Taylor expansion analysis.
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