Application of a Spectral Finite Difference Scheme to a Complex Configuration

在当前的谱有限差分格式中，假设因变量以完整的谱展开（在一个空间分量中）表示，例如傅立叶展开和迪尼展开，这导致在空间中垂直于展开方向和时间的联立偏微分方程系统。结果表明，在空间分量上分解原偏微分方程时没有引入误差，因此，至少对于用简单解析函数表示的单连通区域或双连通区域上的自然对流/强迫对流/非牛顿流体流动，该格式具有更好的空间分辨率和更高的计算速度。根据目前的项目，建议是引入一个功能，它映射到一个圆的情况下，一个简单的连接区域的边界。对于几个复杂的配置，例如，一个半无限矩形腔，卡西尼腔，和一个非圆形变形腔，一个具体的映射解析函数的确定和发现得到的稳态解层流自然对流下的各种组合的参数，如Grashof数，普朗特数，和弹性数（粘弹性流体）。

In the current spectral finite difference scheme, dependent variables are assumed to be expressed in a complete spectral expansion ( in one spatial component ) such as Fourier expansion and Dini expansion, which leads to a system of simultaneous partial differential equations in space normal to the direction(s) of expansion and in time. As a result, no error is introduced in decomposing the original partial differential equations in spatial components, so that the scheme possesses better resolution in space and high computation speed in nature at least for natural convection / forced convection / non-Newtonian fluid flow in a simply-connected region or over a doubly-connected region expressed in terms of a simple analytic function. Under the current project, proposed is introduction of a functional which maps the boundary to a circle in case of a simply-connected region. For several complex configurations, e.g. a semi-infinite rectangular cavity, a Cassini cavity, and a non-circular deformed cavity, a concrete mapping analytic function is determined and found to get to a steady-state solution under laminar natural convection for various combinations of parameters such as a Grashof number, a Prandtl number, and an elatic number (for a viscoelastic fluid).