In this work, improved simplified and highly stable lattice Boltzmann methods (SHSLBMs) are developed for incompressible flows. The SHSLBM is a newly developed scheme within the lattice Boltzmann method (LBM) framework, which utilizes the fractional step technology to resolve the governing equations recovered from lattice Boltzmann equation (LBE) and reconstructs the equations in the Lattice Boltzmann frame. By this treatment, the SHSLBM directly tracks the macroscopic variables in the evolution process rather than the distribution functions of each grid node, which greatly saves virtual memories and simplifies the implementation of physical boundary conditions. However, the Chapman–Enskog expansion analysis reveals that the SHSLBM recover the weakly compressible Navier–Stokes equations with the low Mach number assumption. Therefore, the original SHSLBM can be regarded as an artificial compressible method and may cause some undesired errors. By modifying the evolution equation for the density distribution function, the improved SHSLBMs can eliminate the compressible effects. The incompressible SHSLBMs are compared with the original SHSLBM in terms of accuracy and stability by simulating several two-dimensional steady and unsteady incompressible flow problems, and the results demonstrate that the present SHSLBMs ensure the second order of accuracy and can reduce the compressible effects efficiently, especially for the incompressible flows with large pressure gradients. We then extended the present SHSLBMs to study the more complicated two-dimensional lid-driven flow and found that the present results are in good agreement with available benchmark results.

中文翻译：

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用于不可压缩流动的改进的简化和高度稳定的格子 Boltzmann 方法

在这项工作中，为不可压缩流动开发了改进的简化和高度稳定的格子 Boltzmann 方法 (SHSLBMs)。SHSLBM 是格子玻尔兹曼方法 (LBM) 框架内新开发的方案，它利用分数步技术求解从格子玻尔兹曼方程 (LBE) 恢复的控制方程，并在格子玻尔兹曼框架中重构方程。通过这种处理，SHSLBM直接跟踪演化过程中的宏观变量，而不是每个网格节点的分布函数，大大节省了虚拟内存，简化了物理边界条件的实现。然而，查普曼-恩斯科格展开分析表明，SHSLBM 恢复了具有低马赫数假设的弱可压缩 Navier-Stokes 方程。所以，原始的 SHSLBM 可以看作是一种人工可压缩的方法，可能会导致一些不希望的错误。通过修改密度分布函数的演化方程，改进的SHSLBMs可以消除可压缩效应。通过模拟若干个二维定常和非定常不可压缩流动问题，将不可压缩SHSLBMs与原SHSLBMs的精度和稳定性进行了比较，结果表明，现有的SHSLBMs保证了二阶精度，并且可以有效地降低可压缩效应，特别是对于压力梯度较大的不可压缩流动。然后，我们扩展了目前的 SHSLBM 以研究更复杂的二维盖子驱动流动，并发现目前的结果与可用的基准结果非常一致。