4.4.2 Cell-Vertex Scheme 单元顶点格式[cfd-4-4-2]

如前所述，两种单元顶点格式的黏性通量离散都借助对偶控制体(4.2.3小节)。于是问题是如何在该控制体的面上计算一阶导数。考虑图4.12b，一种可能的做法是先在网格单元上积分求出单元中心处的梯度，这在任意拉伸网格上具有一阶精度;下一步再像文献[31]、[100]那样，把基于单元的梯度在控制体\(\Omega\)的面上平均。然而，这种做法无法防止解的奇偶失联。en

As already mentioned, both types of cell-vertex schemes resort to the dual control volume (Subsection 4.2.3) for the discretisation of the viscous fluxes. Hence, the question is how to evaluate the first derivatives at the faces of this control volume. Considering Fig. 4.12b, one possible alternative is to calculate the gradients at the cell centres first by integrating over the grid cells, which yields first-order accuracy on arbitrarily stretched grids. In a next step, the cell-based gradients are averaged at the faces of the control volume \(\Omega\) like in Refs. [31], [100]. However, this approach cannot prevent an odd-even decoupling of the solution.

另一种与单元中心格式类似的做法，是通过连接定义相邻网格单元的各边的中点，围绕该面构造辅助控制体[101]、[102]，如图4.12b所示。一阶差分的计算与单元中心格式的讨论相同，必要时采用平均量。注意，这一做法在形式上与有限差分近似完全相同[96]-[98]。该格式在任意拉伸网格上给出黏性通量的一阶精度离散，在光滑网格上达到二阶精度[96]、[98]。另一个优点是计算模板很小:二维只有9个节点，三维15个节点。en

Another possibility, similar to the cell-centred scheme, is to construct an auxiliary control volume around the face by connecting the midpoints of the edges defining adjacent grid cells [101], [102]. This is depicted in Fig. 4.12b. The evaluation of the first differences proceeds along the same lines as discussed for the cell-centred scheme, with averaged quantities where necessary. It should be noted that this approach is formally identical to the finite difference approximation [96]-[98]. This scheme leads to first-order accurate discretisation of the viscous fluxes on arbitrarily stretched grids and to second-order accuracy on smooth grids [96], [98]. Another positive feature is that the computational stencil is confined to only nine nodes in two dimensions and to 15 nodes in three dimensions.

最后，还应提到另一种方法，它选择了更复杂的积分路径，平均时纳入所有相邻节点[20]。该格式的一个严重缺点是，即使在二维也包含25点模板，通常比紧凑模板引入更多数值扩散;此外，若时间积分采用隐式格式，通量Jacobian的带宽将变得过大而无法接受。关于梯度计算各种方法的详细讨论还可参见文献[103]。en

Finally, one further approach should be mentioned, where a more complex integration path was chosen, with averaging incorporating all neighbouring nodes [20]. A serious disadvantage of this scheme is that it encompasses a 25-point stencil even in two dimensions, which adds in general more numerical diffusion than compact stencils. Furthermore, if an implicit scheme would be envisioned for the time integration, the bandwidth of the flux Jacobian would become prohibitively large. A detailed discussion of various methodologies for the gradient evaluation can also be found in Ref. [103].