Abstract

This article is devoted to the development and application of the filling cells method for the solution of hydrodynamics problems with a complicated geometry of the computational domain, in particular, a liquid domain, to increase the smoothness and accuracy of the finite-difference solution. The spatial-two-dimensional flow problem of a viscous fluid between two coaxial semicylinders and the spatial-three-dimensional problem of wave propagation in the coastal zone demonstrate the possibilities of the proposed method. The rectangular grids are used to solve these problems, taking into account the filling of cells. The approximation of problems are used to split schemes in time for physical processes and the approximation in spatial variables is made using the balance method, taking into account the filling of cells and without it. An analytical solution describing the Taylor-Couette flow is used as the reference to assess the accuracy of the numerical solution of the first problem. The simulation is performed on a a sequence of condensing computational grids with the following dimensions: 11 × 21, 21 × 41, 41 × 81, and 81 × 161 nodes in the case of using the method and without using it. In the case of the direct use of rectangular grids (stepwise approximation of boundaries), the relative error of the calculations reaches 70%; under the same conditions, the use of the proposed method allows us to reduce the error to 6%. It is shown that splitting up the rectangular grid by factors of 2 to 8 in each of the spatial directions does not lead to the same increase in the accuracy of the numerical solutions obtained taking into account the filling of the cells.

中文翻译：

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计算域复杂的流体力学问题的填充单元计算方法

摘要

本文致力于发展具有复杂几何形状的计算域（尤其是液体域）的流体力学问题的填充单元法的开发和应用，以提高有限差分法的平滑度和准确性。两个同轴半圆柱体之间的粘性流体的空间二维流动问题以及沿海岸带传播的空间三维问题证明了该方法的可行性。考虑到单元的填充，矩形网格用于解决这些问题。问题的近似用于对物理过程的时间方案进行拆分，而空间变量的近似则是使用平衡方法进行的，其中要考虑到单元的填充和没有单元的填充。描述泰勒-库特流的解析解被用作评估第一个问题数值解的准确性的参考。在使用以下方法的情况下，对具有以下尺寸的压缩计算网格序列进行仿真：11×21、21×41、41×81和81×161节点。在直接使用矩形网格的情况下（边界的逐步逼近），计算的相对误差达到70％。在相同条件下，使用建议的方法可使我们将误差降低到6％。结果表明，在每个空间方向上以2到8的倍数分割矩形网格不会导致在考虑了单元填充的情况下获得的数值解的精度得到同样的提高。