Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations

In this paper, a comparative study of the lattice Boltzmann (LB) models for the Allen-Cahn (A-C) and Cahn-Hilliard (C-H) equations is conducted. To this end, a new LB model for the A-C equation is first proposed, where the equilibrium distribution function and the source term distribution function are delicately designed to recover the A-C equation correctly. The gradient term in this model can be computed by the nonequilibrium part of the distribution function such that the collision process can be implemented locally. Then a detailed numerical study on several classical problems is performed to give a comparison between the present model for the A-C equation and the recently developed LB model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)] for the C-H equation in terms of tracking the interface of two-phase flow. The results show that the present LB model for the A-C equation is more accurate and more stable, and also has a second-order convergence rate in space, while the convergence rate of the previous LB model for the C-H equation is only about 1.5.

Figure 1

The phase-field contours ($\varphi =0$) of diagonal motion of a circular interface at $t=0$ (solid line) and $t=T$ (dashed line). (a) The present model; (b) the model in Ref. [7].

Figure 2

The convergence rates of the present model for the A-C equation and the LB model for the C-H equation [7].

Figure 3

The phase-field contours ($\varphi =0$) of Zalesak's disk $t=0$ (solid line) and $t=2T_f$ (dashed line). (a), (c) Results of the present model for the A-C equation at $\text{Pe}=80$ and $\text{Pe}=400$; (b), (d) results of the LB model for the C-H equation [7] at $\text{Pe}=80$ and $\text{Pe}=400$.

Figure 4

The phase-field contours ($\varphi =0$) at (a) $t=T/8$, (b) $t=T/4$, (c) $t=3T/8$, (d) $t=T/2$, (e) $t=5T/8$, (f) $t=3T/4$, (g) $t=7T/8$, and (h) $t=0$ (solid line) and $t=T$ (dashed line).

Figure 5

The phase-field contours ($\varphi =0$) of two models in the $X\text{−}Z$ plane ($Y=L0/2$) at different times ($M=0.001$ and $U0=0.02$). (a) Results of the present model for the A-C equation; (b) results of the LB model for the C-H equation [7].