Abstract

In this work, a fourth-order compact finite difference (FD) algorithm on the nine-point 2-D stencil is proposed to simulate numerically natural convection in an inclined square enclosure using the vorticity-stream function form of the Naiver-Stokes equations. The key feature of the proposed algorithm is that for all Rayleigh numbers Ra of physical interest, the point-successive overrelaxation or point-successive underrelaxation (SOR) iteration can be used. The numerical capability of the presented algorithm is demonstrated by the application to natural convection in an inclined square enclosure. The angle of inclination of the cavity is varied from $-{90}^o$ (heated from above) to ${90}^o$ (heated from below) in steps of ${15}^o.$ Computations are performed for Rayleigh numbers equal to 10³, 10⁴, 10⁵, 10⁶ and 10⁷ while the Prandtl number is kept constant (Pr = 0.71). Test results, which are presented in terms of streamlines, isotherms, isovorticities and local and average Nusselt number, indicate that the present algorithm could predict the benchmark results for temperature and flow fields on relatively coarser grids.

摘要

在这项工作中，提出了一种基于九点二维模板的四阶紧致有限差分 (FD) 算法，以使用 Naiver-Stokes 方程的涡流函数形式对倾斜方形外壳中的自然对流进行数值模拟。所提出算法的关键特征是对于所有物理感兴趣的瑞利数Ra，可以使用点连续过松弛或点连续欠松弛 (SOR) 迭代。所提出算法的数值能力通过应用于倾斜方形外壳中的自然对流来证明。腔体的倾斜角度从$-{90}^{○}$ （从上面加热）到 ${90}^{○}$ （从下面加热）的步骤 ${15}^{○}.$对等于 10 ³、10 ⁴、10 ⁵、10 ⁶和 10 ^{7 的}瑞利数进行计算，而普朗特数保持不变（Pr  = 0.71）。以流线、等温线、等涡度以及局部和平均努塞尔数表示的测试结果表明，本算法可以预测相对较粗网格上温度和流场的基准结果。