Achieving thermodynamic consistency in a class of free-energy multiphase lattice Boltzmann models

The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Phys. Rev. E 83, 036707 (2010)]. The numerical error term mainly includes two parts: one comes from the discrete gradient operator and the other can be identified in a high-order Chapman-Enskog analysis. In this paper, we propose an improved scheme to eliminate the thermodynamic inconsistency of the aforementioned class of free-energy LB models. The improved scheme is constructed by modifying the equation of state of the standard LB equation, through which the discretization of $\mathbf{\nabla }\left(\rho c_s^2\right)$ is no longer involved in the force calculation and then the numerical errors can be significantly reduced. Numerical simulations are subsequently performed to validate the proposed scheme. The numerical results show that the improved scheme is capable of eliminating the thermodynamic inconsistency and can significantly reduce the spurious currents in comparison with the standard forcing-based free-energy LB model.

Figure 1

Comparison of the numerical coexistence curves with the analytical solution given by the Maxwell construction. The kinematic viscosity is given by (a) $\nu =0.15$ and (b) $0.03$.

Figure 2

Comparison of the chemical potential profiles predicted by the standard forcing-based free-energy LB model and the free-energy LB model using the improved scheme at $T=0.8T_{\mathrm{c}}$.

Figure 3

Comparison of the density profiles obtained by the standard forcing-based free-energy LB model and the free-energy LB model using the improved scheme at $T=0.8T_{\mathrm{c}}$.

Figure 4

Simulation of circular droplets. Comparison of the numerical liquid and gas densities obtained by different forcing-based free-energy LB models with the theoretical ones at $T=0.8T_{\mathrm{c}}$.

Figure 5

Simulation of circular droplets. Comparison of the numerical liquid and gas densities obtained by the free-energy LB model using the improved scheme with the theoretical ones at $T=0.7T_{\mathrm{c}}$.

Figure 6

Density contours of a moving circular droplet simulated by the free-energy LB model with the improved scheme. (a) $t=20000{\delta }_t$, (b) $40000{\delta }_t$, (c) $70000{\delta }_t$, and (d) $90000{\delta }_t$.

Figure 7

Comparison of the chemical potential profiles predicted by the Lax-Wendroff scheme [32], the forcing scheme proposed in Ref. [49], and the improved scheme devised in the present work at $\nu =0.04$. (a) $T=0.9T_{\mathrm{c}}$, (b) $T=0.85T_{\mathrm{c}}$, (c) $T=0.8T_{\mathrm{c}}$, and (d) $T=0.7T_{\mathrm{c}}$.