Abstract We present a lattice Boltzmann ( lb ) method using a rectangular, non-isotropic lattice based on d2q9 and d3q27 velocity sets in two and three dimensions. A second order multi-scale expansion ensures that the scheme correctly reproduces hydrodynamic behaviour. A novel set of basis vectors is introduced in order to allow independent adjustment of eigenvalues corresponding to second order moments as required in order to ensure correct hydrodynamic behaviour using the non-isotropic lattice. Errors are reduced compared to other rectangular grid implementations. Linear perturbation analysis indicates that our scheme has similar stability properties to the isotropic lb method. We investigate the error behaviour of our scheme by performing Taylor-Green vortex flow simulations and comparing our results to simulations using a square grid and also to analytical results. We demonstrate that our scheme is well suited to direct numerical simulation of wall bounded turbulent flows and compare to well known benchmark results.

中文翻译：

---

二维和三维使用多重弛豫时间碰撞算子的矩形格子玻尔兹曼方法

摘要 我们提出了一种基于二维和三维速度集 d2q9 和 d3q27 的矩形非各向同性晶格的晶格玻尔兹曼 (lb) 方法。二阶多尺度扩展确保该方案正确再现流体动力学行为。引入了一组新的基向量，以便根据需要独立调整对应于二阶矩的特征值，以确保使用非各向同性晶格的正确流体动力学行为。与其他矩形网格实现相比，减少了错误。线性扰动分析表明我们的方案具有与各向同性 lb 方法相似的稳定性特性。我们通过执行 Taylor-Green 涡流模拟并将我们的结果与使用方形网格的模拟以及分析结果进行比较来研究我们方案的错误行为。我们证明我们的方案非常适合壁面有界湍流的直接数值模拟，并与众所周知的基准结果进行比较。