Chapter 5: Grids with Appropriate Transformation¶
5.1 Introduction¶
标准的有限差分法需要均匀网格。而很多时候需要非均匀网格！
机翼
机翼面是非常重要的边界条件，必须在网格中得到很好的体现。但是使用均匀网格的话，机翼表面很少落到网格点上，误差很大。
- A boundary-fitted coordinate system（贴体坐标系）
- 以边界的形状做等值线，构建正交曲线坐标系
- 将物理空间的正交曲线网格转换为计算空间的均匀网格，一一对应（\(x, y \leftrightarrow \xi, \eta\)）
- 在计算空间中使用有限差分方法求解方程，再对应回物理空间
5.2 General Transformations of the Equations¶
为简单起见，考虑二维非定常流动。物理空间到计算空间的变换 \((x, y, t) \to (\xi, \eta, \tau)\)
\[ \begin{align} \xi &= \xi(x, y, t) \tag{5.1a}\\ \eta &= \eta(x, y, t) \tag{5.1b}\\ \tau &= \tau(t) \tag{5.1c} \end{align} \]
通常 \(\tau\) 不依赖于 \(x, y\)，并且通常是 \(\tau = t\). 下面研究导数的变换关系。使用链式法则
\[ \left( \frac{\partial}{\partial x} \right)_{y, t} = \left( \frac{\partial}{\partial \xi} \right)_{\eta, \tau} \left( \frac{\partial \xi}{\partial x} \right)_{y, t} + \left( \frac{\partial}{\partial \eta} \right)_{\xi, \tau} \left( \frac{\partial \eta}{\partial x} \right)_{y, t} + \left( \frac{\partial}{\partial \tau} \right)_{\xi, \eta} \left( \frac{\partial \tau}{\partial x} \right)_{y, t} \]
下标表示求偏导时保持不变的变量。由此得到变换关系
\[ \begin{align} \frac{\partial}{\partial x} &= \frac{\partial \xi}{\partial x} \frac{\partial}{\partial \xi} + \frac{\partial \eta}{\partial x} \frac{\partial}{\partial \eta} \tag{5.2} \label{eq:transform_x} \\ \frac{\partial}{\partial y} &= \frac{\partial \xi}{\partial y} \frac{\partial}{\partial \xi} + \frac{\partial \eta}{\partial y} \frac{\partial}{\partial \eta} \tag{5.3} \label{eq:transform_y} \\ \frac{\partial}{\partial t} &= \frac{\partial \xi}{\partial t} \frac{\partial}{\partial \xi} + \frac{\partial \eta}{\partial t} \frac{\partial}{\partial \eta} + \frac{\partial \tau}{\partial t} \frac{\partial}{\partial \tau} \tag{5.4} \label{eq:transform_t} \end{align} \]
导数前面的系数称为度量（metric）。下面求二阶导数。以 \(\frac{\partial^2}{\partial x^2}\) 为例，利用 \eqref{eq:transform_x}，得到
\[ \begin{aligned} \frac{\partial^2}{\partial x^2} &= \frac{\partial}{\partial x} \left( \frac{\partial \xi}{\partial x} \frac{\partial}{\partial \xi} + \frac{\partial \eta}{\partial x} \frac{\partial}{\partial \eta} \right) \\ &= \frac{\partial^2 \xi}{\partial x^2} \frac{\partial}{\partial \xi} + \frac{\partial \xi}{\partial x} \frac{\partial^2}{\partial x \partial \xi} + \frac{\partial^2 \eta}{\partial x^2} \frac{\partial}{\partial \eta} + \frac{\partial \eta}{\partial x} \frac{\partial^2}{\partial x \partial \eta} \end{aligned} \]
非结构化网格的优缺点
-
Pros
- Improved mesh quality
- Enhanced
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Cons
- Increased computational cost
- Complex data structures