Revised Chapman-Enskog analysis for a class of forcing schemes in the lattice Boltzmann method

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 $\mathbf{u}={\sum }_{\alpha }{\mathbf{e}}_{\alpha }{f}_{\alpha }/\rho$, while the actual fluid velocity is defined as $\widehat{\mathbf{u}}=\mathbf{u}+{\delta }_t\mathbf{F}/\left(2\rho \right)$. 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)] 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.

Figure 1

Numerical (symbols) and analytical (solid line) results of Womersley flow at different times $t=nT/16$ with (a) $n=4,6,8,10$ and (b) $n=0,2,12,14$. The numerical results are obtained by Wagner's forcing scheme.

Figure 2

Simulation of circular interfaces: steady-state density contours $\left(\tau =0.7\right)$ obtained by the forcing schemes proposed by (a) Wagner [12], (b) Kupershtokh et al. [13], and (c) Ladd [10].