4.1.1 Two-Dimensional Case 二维情形[cfd-4-1-1]

一般地ï¼我们把平面内的流动看作三维问题的一个特例ï¼其中解关于某一坐标方向(例如 $$z$$ 方向)对称。由于对称性ï¼也为了使体积、压力等物理量具有正确的单位ï¼我们把所有网格单元和控制体的深度设为常数 $$b$$ 。于是ï¼在二维中ï¼控制体的体积等于其面积与深度 $$b$$ 的乘积。四边形的面积可以用Gauss公式精确计算。因此ï¼对于如图4.1a所示的控制体ï¼经过一些代数运算可得en

In general, we consider flow in a plane as a special case of a 3-D problem, where the solution is symmetric with respect to one coordinate direction (e.g., to the $$z$$ -direction). Because of the symmetry and in order to obtain correct physical units for volume, pressure, etc., we set the depth of all grid cells and control volumes equal to a constant value $$b$$ . The volume of a control volume results then in two dimensions from the product of its area with the depth $$b$$ . The area of a quadrilateral can be exactly calculated by the formula of Gauss. Hence, for a control volume like that displayed in Fig. 4.1a, we get after some algebra

\Omega_{I,J} = \frac{b}{2}\left[(x_1 - x_3)(y_2 - y_4) + (x_4 - x_2)(y_1 - y_3)\right]. \tag{4.4}

上式中ï¼我们假定控制体位于 $$x$$ – $$y$$ 平面内ï¼且 $$z$$ 坐标是对称轴。由于深度 $$b$$ 是任意的ï¼为方便起见可取 $$b = 1$$ 。在二维中ï¼控制体的面由直线段构成ï¼因此单位法向量沿面为常数。当我们按照方程(4.2)的近似对通量积分时ï¼需要计算面的面积 $\Delta S$ 与相应单位法向量 $\vec{n}$ 的乘积——即面向量(face vector) $\vec{S}$ en

In the above, we have assumed that the control volume is located in the $$x$$ – $$y$$ -plane and that the $$z$$ -coordinate is the symmetry axis. Since the depth $$b$$ is arbitrary, we may set $$b = 1$$ for convenience. In two dimensions, the faces of a control volume are given by straight lines and therefore the unit normal vector is constant along them. When we integrate the fluxes according to the approximation of Eq. (4.2), we have to evaluate the product of the area of a face $\Delta S$ and the corresponding unit normal vector $\vec{n}$ - the face vector $\vec{S}$

\vec{S}_m = \begin{bmatrix} S_{x,m} \\ S_{y,m} \end{bmatrix} = \vec{n}_m\,\Delta S_m\,. \tag{4.5}

由于对称性ï¼面向量(以及单位法向量)的 $$z$$ 分量为零ï¼因此从表达式中略去。图4.1a中控制体的面向量由以下关系式给出en

Because of the symmetry, the $$z$$ -component of the face vectors (and of the unit normal vector) is zero. It is therefore dropped from the expressions. The face vectors of the control volume from Fig. 4.1a are given by the relations

\begin{aligned} \vec{S}_1 &= b\begin{bmatrix} y_2 - y_1 \\ x_1 - x_2 \end{bmatrix}, &\quad \vec{S}_2 &= b\begin{bmatrix} y_3 - y_2 \\ x_2 - x_3 \end{bmatrix}, \\ \vec{S}_3 &= b\begin{bmatrix} y_4 - y_3 \\ x_3 - x_4 \end{bmatrix}, &\quad \vec{S}_4 &= b\begin{bmatrix} y_1 - y_4 \\ x_4 - x_1 \end{bmatrix}. \end{aligned} \tag{4.6}

于是ï¼第 $$m$$ 面上的单位法向量由方程(4.5)得到en

The unit normal vector at face $$m$$ is then obtained from Eq. (4.5) as

\vec{n}_m = \frac{\vec{S}_m}{\Delta S_m} \tag{4.7}

其中en

with

\Delta S_m = |\vec{S}_m| = \sqrt{S_{x,m}^2 + S_{y,m}^2}\,. \tag{5}

实际计算中ï¼对每个控制体 $\Omega_{I,J}$ 只计算并存储面向量 $\vec{S}_1$ 与 $\vec{S}_4$ 。面向量 $\vec{S}_2$ 与 $\vec{S}_3$ 则(经反号使其指向外侧)取自相应的相邻控制体ï¼以节省内存并减少点操作次数。en

In practice, only the face vectors $\vec{S}_1$ and $\vec{S}_4$ are computed and stored for each control volume $\Omega_{I,J}$ . The face vectors $\vec{S}_2$ as well as $\vec{S}_3$ are taken (with reversed signs to become outward facing) from the appropriate neighbouring control volumes in order to save memory and to reduce the number of point operations.