5.4.2 Average of Gradients 梯度平均[cfd-5-4-2]

既然我们已经在每个控制体内部算出了梯度(例如用分段线性重构ï¼式(5.41)或(5.42))ï¼自然会想用简单平均来计算面中点处的梯度[73]en

Since we already computed the gradients inside each control volume (e.g., using the piecewise linear reconstruction, Eq. (5.41) or (5.42)), it would be tempting to evaluate the gradient at the face-midpoint by the simple average [73]

\overline{\nabla U}_{IJ} = \frac{1}{2}\left[\nabla U_I + \nabla U_J\right]. \tag{5.74}

这种方法特别有吸引力ï¼因为它只需要基本的基于面或边的数据结构ï¼不需要额外存储。然而ï¼正如文献[71]等指出的ï¼它导致一个权重分布不利的宽模板[49]。此外,[49]中证明ï¼该模板使解在四边形或六面体网格上可能发生失联(decoupling)。en

This approach is particularly attractive, because it requires only the basic face- or edge-based data structure and no additional storage. However, as it was pointed out, e.g., in Ref. [71], it leads to a wide stencil with an unfavourable distribution of the weights [49]. Furthermore, it was demonstrated in [49] that the stencil allows for the decoupling of the solution on quadrilateral or hexahedral grids.

利用沿单元形心连线方向的方向导数(对单元中心格式)ï¼可以改善该方法的性质ï¼尤其可以防止失联ï¼即en

The properties of the method can be improved, and particularly the decoupling can be prevented, by using the directional derivative along the connection between the cell-centroids (in the case of the cell-centred scheme), i.e.,

\left(\frac{\partial U}{\partial\ell}\right)_{IJ} \approx \frac{U_J - U_I}{\ell_{IJ}}\,, \tag{5.75}

其中 $\ell_{IJ}$ 表示两个单元形心 $$I$$ 与 $$J$$ 之间的距离(图5.19中的虚线)。对中点对偶格式也有类似表达式ï¼其中 $\vec{r}_{ij}$ 按式(5.43)定义。定义沿 $$I$$ 与 $$J$$ 连线的单位向量 $\vec{t}_{IJ}$ 为en

where $\ell_{IJ}$ represents the distance between the both cell-centroids $$I$$ and $$J$$ (dashed line in Fig. 5.19). A similar expression holds also for the median-dual scheme with $\vec{r}_{ij}$ according to Eq. (5.43). With the definition of the unit vector $\vec{t}_{IJ}$ along the line connecting $$I$$ and $$J$$ ,

\vec{t}_{IJ} = \frac{\vec{r}_{IJ}}{\ell_{IJ}}\,, \tag{5.76}

修正的平均可写为[74]、[75]en

the modified average may be written as [74], [75]

\nabla U_{IJ} = \overline{\nabla U}_{IJ} - \left[\overline{\nabla U}_{IJ}\cdot\vec{t}_{IJ} - \left(\frac{\partial U}{\partial\ell}\right)_{IJ}\right]\vec{t}_{IJ} \tag{5.77}

其中 $\overline{\nabla U}_{IJ}$ 由式(5.74)给出。这一修正在四面体以及棱柱或六面体网格上都导致强耦合的模板[49]。修正后的方法仍与基于面/边的数据结构相容ï¼且不需要额外存储。因此ï¼只要控制体内部的梯度反正要用于对流通量的计算ï¼它就比基于单元的方法更有吸引力。en

where $\overline{\nabla U}_{IJ}$ is given by Eq. (5.74). The modification leads to strongly coupled stencils on tetrahedral as well as on prismatic or hexahedral grids [49]. The modified approach is also still compatible with the face-/edge-based data structure and requires no additional storage. It is therefore more attractive than the element-based methodology, provided the gradients inside control volumes are utilised for the convective fluxes anyway.