8.10 Flow Gradients at Boundaries of Unstructured Grids 非结构网格边界上的流动梯度[cfd-8-10]

我们在第5.3.4小节中已经指出，对于中位对偶（median-dual）格式，流动梯度的计算需要格外小心。如果在三角形或四面体网格上用式(5.50)的Green-Gauss方法计算梯度，那么来自求解域边界的贡献（对称边界和周期边界除外）必须像式(8.12)或式(8.13)那样计算，而不是取算术平均，否则梯度将不准确。采用图8.4中的记号，边界节点1处的贡献为en

We already stated in Subsection 5.3.4 that the evaluation of the flow gradients requires some care in the case of the median-dual scheme. If the gradients are calculated on triangular or tetrahedral grids using the Green-Gauss approach in Eq. (5.50), the contributions from the boundaries of the domain (except at symmetry or periodic boundaries) must be evaluated similar to Eq. (8.12) or Eq. (8.13), instead of the arithmetic average. Otherwise, the gradient will not be accurate. Considering the notation in Fig. 8.4, the contribution to the boundary node 1 reads

\frac{1}{6}\left(5U_1+U_2\right)\vec{n}_{12}\,\frac{\Delta S_{12}}{2}\,, \tag{1}

其中 $\Delta S_{12}$ 是节点1与2之间边界面的长度（因此取其一半）。对应于式(8.13)，图8.5中三角形面1-2-3对节点1的贡献变为en

where $\Delta S_{12}$ is the length of the boundary face between node 1 and 2 (therefore halved). Corresponding to Eq. (8.13), the contribution of the triangular face 1-2-3 to node 1 in Fig. 8.5 becomes

\frac{1}{8}\left(6U_1+U_2+U_3\right)\vec{n}_{123}\,\frac{\Delta S_{123}}{3} \tag{2}

其中 $\Delta S_{123}/3$ 是三角形1-2-3中的灰色面积。在混合网格上，更合适的做法是采用带虚拟边的最小二乘法[8]（见图5.15）。en

with $\Delta S_{123}/3$ being the grey area in the triangle 1-2-3. On mixed grids, it is more appropriate to employ the least squares approach with virtual edges [8] (see Fig. 5.15).

单元中心格式在对称边界或周期边界处不需要任何特殊处理，其实现与第8.6节或第8.8节中对通量的讨论完全相同。对于中位对偶格式，如果梯度用最小二乘法计算，情况也是如此。唯一需要额外做的工作是把某些梯度置零，如前面第8.6节所述（参见式(8.40)）。en

The cell-centred scheme requires no special provisions at symmetry or periodic boundaries. The implementation is identical to that discussed for the fluxes in Section 8.6 or 8.8. This holds also for the median-dual scheme, if the gradients are evaluated using the least-squares approach. The only additional work required is to set certain gradients to zero as described previously in Section 8.6 (cf. Eq. (8.40)).

如果在中位对偶格式中采用Green-Gauss方法（即应用式(5.50)），就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）：做法是把面向量中所有垂直于对称面的分量置零。最后，梯度按第8.6节所述进行修正。在周期边界处，边界两侧的梯度和体积必须按第8.8节对通量的做法（式(8.43)）求和。在旋转周期性的情形下，梯度需要通过应用式(8.44)中的旋转矩阵来变换。en

If the Green-Gauss approach is employed within the median-dual scheme (i.e., if Eq. (5.50) is applied), it is necessary to correct normal vectors of those faces of the control volume, which touch the boundary (like at point $2^{*}$ in Fig. 8.4). This is done by setting all components of the face vector to zero, which are normal to the symmetry plane. Finally, the gradients are corrected as discussed in Section 8.6. At periodic boundaries, the gradients and the volumes from both sides of the boundary have to be summed up as presented in Section 8.8 for the fluxes (Eq. (8.43)). In the case of rotational periodicity, the gradients needs to be transformed by applying the rotation matrix in Eq. (8.44).