Siebert D N, Hegele L A, Philippi P C
LMPT Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianopolis, SC, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Feb;77(2 Pt 2):026707. doi: 10.1103/PhysRevE.77.026707. Epub 2008 Feb 26.
Although several thermal lattice Boltzmann models have been proposed, this method has not yet been shown to be able to describe nonisothermal fully compressible flows in a satisfactory manner, mostly due to the presence of important deviations from the advection-diffusion macroscopic equations and also due to numerical instabilities. In this context, this paper presents a linear stability analysis for some lattice Boltzmann models that were recently derived as discrete forms of the continuous Boltzmann equation [P. C. Philippi, L. A. Hegele, Jr., L. O. E. dos Santos, and R. Surmas, Phys. Rev. E 63, 056702 (2006)], in order to investigate the sources of instability and to find, for each model, the upper and lower limits for the macroscopic variables, between which it is possible to ensure a stable behavior. The results for two-dimensional (2D) lattices with 9, 17, 25, and 37 velocities indicate that increasing the order of approximation of the lattice Boltzmann equation enhances stability. Results are also presented for an athermal 2D nine-velocity model, the accuracy of which has been improved with respect to the standard D2Q9 model, by adding third-order terms in the lattice Boltzmann equation.
尽管已经提出了几种热格子玻尔兹曼模型，但该方法尚未被证明能够以令人满意的方式描述非等温完全可压缩流动，这主要是由于与平流扩散宏观方程存在重大偏差，以及数值不稳定性。在此背景下，本文对一些最近作为连续玻尔兹曼方程的离散形式推导出来的格子玻尔兹曼模型进行了线性稳定性分析[P.C.菲利皮、L.A.赫格勒、L.O.E.多斯桑托斯和R.苏尔马斯，《物理评论E》63，056702(2006)]，以研究不稳定性的来源，并为每个模型找到宏观变量的上限和下限，在这两者之间可以确保稳定行为。具有9、17、25和37个速度的二维(2D)晶格的结果表明，提高格子玻尔兹曼方程的近似阶数可增强稳定性。还给出了一个无热二维九速度模型的结果，通过在格子玻尔兹曼方程中添加三阶项，其精度相对于标准D2Q9模型有所提高。
