Li Q, Yu Y, Huang R Z
School of Energy Science and Engineering, Central South University, Changsha 410083, China.
Phys Rev E. 2021 Jan;103(1-1):013304. doi: 10.1103/PhysRevE.103.013304.
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)10.1103/PhysRevE.83.036707]. 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 ∇(ρc_{s}^{2}) 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.
自由能格子玻尔兹曼（LB）模型是LB领域主要的多相模型之一。本研究聚焦于一类自由能LB模型，其中热力学压力张量的散度或其用化学势表示的等效形式通过一个外力项被纳入到LB方程中。尽管这类自由能LB模型在连续介质层面可能是热力学一致的，但由于数值误差，它在离散格子层面存在热力学不一致性[郭等人，《物理评论E》83，036707（2010）10.1103/PhysRevE.83.036707]。数值误差项主要包括两部分：一部分来自离散梯度算子，另一部分可在高阶查普曼-恩斯科格分析中识别出来。在本文中，我们提出一种改进方案以消除上述这类自由能LB模型的热力学不一致性。该改进方案通过修改标准LB方程的状态方程构建而成，通过这种方式，∇(ρc_{s}^{2})的离散化不再参与力的计算，进而数值误差可显著降低。随后进行了数值模拟以验证所提出的方案。数值结果表明，与基于标准外力的自由能LB模型相比，该改进方案能够消除热力学不一致性并能显著减少虚假电流。
