4.4.1 Cell-Centred Scheme 单元中心格式[cfd-4-4-1]

为了应用把一阶导数的体积分与 $$U$$ 的面积分联系起来的Green定理ï¼必须先定义一个合适的控制体。由于方程(4.2)中的求和需要面中点处的导数ï¼我们通过连接定义相邻网格单元的各边的中点ï¼构造一个以该面为中心的辅助控制体[31]、[19]、[98]ï¼如图4.12a所示。为了计算面 $$(I+1/2)$$ 处的一阶导数——在图4.12a中以菱形符号标记——必须把相应的流动变量 $$U$$ 沿辅助控制体的边界积分(以下以上标' 表示)。例如ï¼对x方向的导数en

In order to apply Green's theorem, which relates the volume integral of the first derivative to the surface integral of $$U$$ , we have to define a suitable control volume first. Since we need the derivatives at the midpoints of the faces for the summation in Eq. (4.2), we construct an auxiliary control volume centred at the face by connecting the midpoints of the edges defining adjacent grid cells [31], [19], [98] as shown in Fig. 4.12a. In order to evaluate the first derivative at the face $$(I+1/2)$$ - marked by a diamond symbol in Fig. 4.12a - we have to integrate the corresponding flow variable $$U$$ over the boundary of the auxiliary control volume (denoted by the superscript ' in the following). Thus, e.g., for the derivative in the x-direction

\frac{\partial U}{\partial x} = \frac{1}{\Omega'}\int_{\partial\Omega'} U\,dS'_x \approx \frac{1}{\Omega'}\sum_{m=1}^{N_F} U_m\,S'_{x,m}\,, \tag{4.125}

其中 $$N_F$$ 表示面的数目(二维 $$N_F = 4$$ ï¼三维 $$N_F = 6$$ )。体积 $\Omega'$ 与面向量 $\vec{S}'_m = [S'_{x,m}, S'_{y,m}, S'_{z,m}]^T$ 的分量按4.1节中已介绍的方法计算。面值 $$U_m$$ 或直接取单元中心值(即左、右面上的 $U_{i,j}$ 与 $U_{i+1,j}$ )ï¼或在上面与下面的面上取平均ï¼例如在 $$J+1/2$$ 处en

where $$N_F$$ stands for the number of faces ( $$N_F = 4$$ in 2D and $$N_F = 6$$ in 3D). The volume $\Omega'$ and the components of the face vector $\vec{S}'_m = [S'_{x,m}, S'_{y,m}, S'_{z,m}]^T$ , respectively, are computed as already presented in Section 4.1. The face values $$U_m$$ are obtained either directly as cell-centred values (i.e., $U_{i,j}$ and $U_{i+1,j}$ on the left and the right face), or by averaging like on the upper and the lower face, e.g., at $$J+1/2$$

U_{m_{I,J+1/2}} = \frac{1}{4}(U_{I,J} + U_{I+1,J} + U_{I,J+1} + U_{I+1,J+1})\,, \text{ etc.} \tag{4.126}

同样的方法可用于三维ï¼此时同样可用四个单元中心值作平均ï¼即en

We can apply the same approach in three dimensions, where again four cell-centred values can be utilised for the averaging. Hence,

\begin{aligned} U_{m_{I,J+1/2,K}} &= \frac{1}{4}(U_{I,J,K} + U_{I+1,J,K} + U_{I,J+1,K} + U_{I+1,J+1,K})\,,\\ U_{m_{I,J+1/2,K+1/2}} &= \frac{1}{4}(U_{I,J,K} + U_{I,J,K+1} + U_{I,J+1,K} + U_{I,J+1,K+1})\,, \end{aligned} \tag{4.127}

上述格式相当紧凑ï¼计算模板在二维只覆盖9个单元ï¼三维为15个。注意ï¼这种计算一阶导数的方法无法抑制两类虚假模态(相邻单元中心处解的失联)的产生[99]、[19]:一是棋盘格模态(chequer-board mode)ï¼源于围绕控制体的积分形式;二是一对波纹(搓衣板)模态ï¼源于相邻单元值的平均。不过ï¼实践中一般不会因此遇到困难。更严重的问题会出现在先把每个单元的梯度算出(类似对流通量的做法)、再在单元面上平均的做法中;这种做法看似更有吸引力ï¼但由于会导致强烈的奇偶失联ï¼并不推荐。en

The above scheme is quite compact, with the computational stencil extending over only nine cells in two dimensions and over 15 in three dimensions. It should be noted that this approach for computing the first derivatives cannot suppress the generation of two types of spurious modes (decoupled solutions at neighbouring cell centres) [99], [19]: the chequer-board mode, arising from the form of the integral around the control volume, and a pair of corrugated or washboard modes, arising from the averaging of values in neighbouring cells. However, there are generally no difficulties with this in practice. A more serious problem would occur, if the gradients would be first evaluated for each cell (similar to the convective fluxes) and then averaged at the cell faces. Although this approach may appear more attractive than the current methodology, it is not recommended since it leads to strong odd-even decoupling.

上述格式的缺点是:当网格不均匀时精度会下降[98]、[99]。就是说ï¼对任意拉伸的网格ï¼导数近似变得不相容(辅助控制体的形心不再对应面中心)。因此ï¼只有对适度且光滑拉伸的网格ï¼黏性通量才具有二阶精度的离散。最后ï¼应当指出,Navier-Stokes方程的TSL近似(2.4.3小节)很容易实现ï¼只需在计算梯度时略去相应的贡献即可。例如ï¼若边界层沿图4.12a的 $$I$$ 方向ï¼则辅助控制体左侧 $$(I, J)$$ 与右侧 $$(I+1, J)$$ 的贡献将被略去。en

A disadvantage of above scheme is a loss of accuracy if the grid is not uniform [98], [99]. Namely, for arbitrarily stretched grids the approximation of the derivatives becomes inconsistent (the centroid of the auxiliary control volume does no longer correspond to the face centre). Thus, the viscous fluxes are discretised with second-order accuracy only for moderately and smoothly stretched grids. Finally, it should be noted that the TSL approximation of the Navier-Stokes equations (Subsection 2.4.3) can easily be realised by omitting the appropriate contributions when computing the gradients. For example, if the boundary layer would be oriented along the $$I$$ -direction in Fig. 4.12a, contributions from the left $$(I, J)$$ and the right side $$(I+1, J)$$ of the auxiliary control volume would be dropped.