The more actual macroscopic equations recovered from lattice Boltzmann equation and their applications

摘要

Chapman-Enskog (C-E) expansion analysis shows that the lattice Boltzmann equation (LBE) can recover the second-order continuity and N-S equations of the weakly compressible model. However, directly solving the governing equations of weakly compressible model suffers from serious instability. The present paper aims to find the mechanism of good performance of LBE for simulation of incompressible flows by using the weakly compressible model. After detailed analysis, it is found that the macroscopic equations recovered from LBE by using Taylor series expansion, which retain some additional small terms, are slightly different from those by using C-E expansion analysis. The numerical tests indicate that those additional small terms inherently included in the LBE computation play an important role in stabilizing numerical computation, which can explain the mechanism of good performance of LBE to a large degree. On the other hand, it is found that those small terms do not have an obvious effect on the accuracy of numerical solutions. These results indicate that Taylor series expansion can recover the more actual macroscopic equations (MAMEs). Based on MAMEs, this paper presents a new solver based on the conventional finite difference method (FDM) to simulate incompressible flows. Compared with LBE, it has higher computational efficiency, competitive accuracy and acceptable stability. Besides, the drawbacks of LBE, which include the limitation of uniform mesh, coupled time step and mesh spacing, and the extra memory size, can be easily overcome in the discretized MAMEs. (C) 2020 Elsevier Inc. All rights reserved.