圆柱坐标系扩散问题有限容积紧致格式研究

Compact Finite-Volume Scheme for Diffusion Problems in a Cylindrical Coordinate System

圆柱坐标系下扩散型方程由于扩散面积随半径的变化而不同,使得该坐标系下高阶紧致格式的推导十分复杂。本文通过坐标转化的思想,将圆柱坐标系下扩散型方程转化为类直角坐标的形式(Lnr型扩散型方程),并推导出Lnr型扩散型方程的四阶精度有限容积紧致格式。本文使用了显式、隐式两种方法求解方程,并与二阶精度有限容积法进行对比。计算结果表明本文所推导的紧致格式较二阶精度有限容积法在相同的网格机架点下可以得到较高的精度,同时也表明本文所提出的方程转化方法是有效可行的。

In cylindrical coordinate systems, it is not convenient to construct high-order compact schemes directly, because diffusion area varies along the radius. In this article, we propose a new governing equation （we call it Lnr-type equation） for the heat conduction problems in a cylindrical coordinate system, which shares the similar form with that in a Cartesian coordinate system. After- wards, the fourth-order compact schemes are constructed and formulated for the proposed equation by borrowing the constructing method in a Cartesian coordinate system. Subsequently, the results calculated by the proposed compact finite volume schemes are compared with those calculated by the second-order central difference （CD） schemes. The comparison shows that higher-order accuracy could be obtained by the proposed scheme on the same mesh. The study in this article confirmed that the Lnr-type governing equation is valid and feasible for constructing the high-order compact finite-volume schemes for the heat conduction problems in a cylindrical coordinate system.