8.6 Symmetry Plane 对称面[cfd-8-6]

如果流动相对于某条直线或某个平面是对称的，那么首先必须满足的条件是边界上没有通量通过。这等价于要求垂直于对称边界的速度为零。此外，下列梯度必须为零：

标量在垂直于边界方向上的梯度；
切向速度在垂直于边界方向上的梯度；
法向速度沿边界方向的梯度（因为 $\vec{v}\cdot\vec{n}=0$ ）。

If the flow is to be symmetrical with respect to a line or a plane, the first condition which must be met is that there is no flux across the boundary. This is equivalent to the requirement that the velocity normal to the symmetry boundary is zero. Furthermore, the following gradients have to vanish:

gradient of a scalar quantity normal to the boundary,
gradient of the tangential velocity normal to the boundary,
gradient of the normal velocity along the boundary (since $\vec{v}\cdot\vec{n}=0$ ).

这些条件可以写为en

We can write these conditions as

\begin{aligned} \vec{n}\cdot\vec{\nabla}U&=0\\ \vec{n}\cdot\vec{\nabla}\left(\vec{v}\cdot\vec{t}\right)&=0\\ \vec{t}\cdot\vec{\nabla}\left(\vec{v}\cdot\vec{n}\right)&=0\,, \end{aligned} \tag{8.40}

其中 $$U$$ 表示标量变量， $\vec{t}$ 表示与对称边界相切的向量。en

where $$U$$ stands for a scalar variable and $\vec{t}$ denotes a vector tangential to the symmetry boundary.

Cell-Centred Scheme 单元中心格式

采用虚单元可以大大简化对称边界条件的实现。虚单元中的流动变量利用镜像单元（reflected cells）的概念得到。也就是说，虚单元中密度、压力等标量取为对面内部单元中的值，即en

The implementation of the symmetry boundary condition can be largely simplified by employing dummy cells. The flow variables in the dummy cells are obtained using the concept of reflected cells. This means that scalar quantities like density or pressure in the dummy cells are set equal to the values in the opposite interior cells, i.e.,

U_1=U_2\quad\text{and}\quad U_0=U_3\,. \tag{8.41}

记号与图8.2一致。速度分量按式(8.10)所示相对于边界作反射。虚单元中法向速度的法向梯度与对面内部单元中的相等，但符号相反。en

The notation corresponds to that in Fig. 8.2. The velocity components are reflected with respect to the boundary as indicated in Eq. (8.10). The normal gradient of the normal velocity in the dummy cell equals to that in the opposite interior cell, but it has a reversed sign.

Cell-Vertex Scheme (Dual Control Volume) 单元顶点格式（对偶控制体）

可以遵循两种不同的做法。一种可能是通过对边界上的网格作镜像来补出控制体缺失的一半，然后像内部区域一样，用反射后的流动变量计算通量和梯度（见上文）。第二种方法对减半的控制体计算通量（但不跨越边界），然后把残差中垂直于对称面的分量置零。此外，还必须修正控制体中触及边界的那些面的法向量（如图8.4中点 $2^{*}$ 处），修正办法是去掉面向量中所有垂直于对称面的分量。梯度还须按式(8.40)进行修正。en

Two different approaches can be followed. One possibility is to construct the missing half of the control volume by mirroring the grid on the boundary. The fluxes and the gradients are then evaluated like in the interior using reflected flow variables (see above). The second methodology computes the fluxes for the halved control volume (but not across the boundary). The components of the residual normal to the symmetry plane are then zeroed out. It is also necessary to correct normal vectors of those faces of the control volume, which touch the boundary (like at point $2^{*}$ in Fig. 8.4). The modification consists of removing all components of the face vector, which are normal to the symmetry plane. The gradients have also to be corrected according to Eq. (8.40).