Musavi S Hossein, Ashrafizaadeh Mahmud
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Feb;91(2):023310. doi: 10.1103/PhysRevE.91.023310. Epub 2015 Feb 17.
A meshless lattice Boltzmann numerical method is proposed. The collision and streaming operators of the lattice Boltzmann equation are separated, as in the usual lattice Boltzmann models. While the purely local collision equation remains the same, we rewrite the streaming equation as a pure advection equation and discretize the resulting partial differential equation using the Lax-Wendroff scheme in time and the meshless local Petrov-Galerkin scheme based on augmented radial basis functions in space. The meshless feature of the proposed method makes it a more powerful lattice Boltzmann solver, especially for cases in which using meshes introduces significant numerical errors into the solution, or when improving the mesh quality is a complex and time-consuming process. Three well-known benchmark fluid flow problems, namely the plane Couette flow, the circular Couette flow, and the impulsively started cylinder flow, are simulated for the validation of the proposed method. Excellent agreement with analytical solutions or with previous experimental and numerical results in the literature is observed in all the simulations. Although the computational resources required for the meshless method per node are higher compared to that of the standard lattice Boltzmann method, it is shown that for cases in which the total number of nodes is significantly reduced, the present method actually outperforms the standard lattice Boltzmann method.
提出了一种无网格格子玻尔兹曼数值方法。与通常的格子玻尔兹曼模型一样，格子玻尔兹曼方程的碰撞算子和流输运算子是分离的。虽然纯局部碰撞方程保持不变，但我们将流输运方程重写为纯对流方程，并使用时间上的拉克斯 - 温德罗夫格式和空间上基于增强径向基函数的无网格局部彼得罗夫 - 伽辽金格式对所得的偏微分方程进行离散化。所提出方法的无网格特性使其成为一种更强大的格子玻尔兹曼求解器，特别是对于那些使用网格会在解中引入显著数值误差的情况，或者当提高网格质量是一个复杂且耗时的过程时。为了验证所提出的方法，对三个著名的基准流体流动问题进行了模拟，即平面库埃特流、圆形库埃特流和突然启动的圆柱流。在所有模拟中，均观察到与解析解或与文献中先前的实验和数值结果具有极好的一致性。尽管与标准格子玻尔兹曼方法相比，无网格方法每个节点所需的计算资源更高，但结果表明，对于节点总数显著减少的情况，本方法实际上优于标准格子玻尔兹曼方法。
