4.2.2 Cell-Vertex Scheme: Overlapping Control Volumes 单元顶点格式:重叠控制体[cfd-4-2-2]

In a cell-vertex scheme, all flow variables are associated with the nodes of the computational grid. Within the approach based on overlapping control volumes, the grid cells still represent the control volumes, just as in the case of the cell-centred scheme. The difference is that now the residuals computed for the control volumes have to be distributed to the grid points [4], [7], [8]. The situation is sketched in Fig. 4.4.

Let us consider the control volume \(\Omega_{I,J}\) in Fig. 4.4, which is defined by the nodes

Note that the point \((i,j)\) is located at the lower left corner of \(\Omega_{I,J}\). The convective fluxes, e.g., for the face \(\Delta S_{I,J-1/2}\), which is given by the points \((i,j)\) and \((i+1,j)\), are approximated as

The variables at the midpoint of the face are evaluated using an arithmetic average of the variables at the nodes defining the face, i.e.,

The face area \(\Delta S_{I,J-1/2}\) is computed from the relations (4.6) and (4.7).

The approach remains the same in three dimensions. For example, for the face associated with the normal vector \(\vec{n}_3\) in Fig. 4.1b, the averaged variables read

If we assume the edge 1-2 being oriented in the \(i\)-direction, edge 1-5 in the \(j\)-direction, edge 1-4 in the \(k\)-direction, and if we finally associate the point \((i,j,k)\) with the corner 1 in Fig. 4.1b, the average in Eq. (4.27) can also be written as

The convective flux is then again obtained from

where the face area \(\Delta S_{I,J,K-1/2}\) results from the formulae (4.8) and (4.9). The viscous fluxes are normally computed employing the same approach as for the dual control-volume scheme [9], [10], which results in a more compact scheme (i.e., one which involves fewer nodal values).

Summing up all face contributions given by relations (4.25), (4.26) and (4.28), (4.29), respectively, we obtain intermediate residuals \(\vec{R}_{I,J,K}\) for all grid cells. In order to relate the cell-based to the node-based residuals, a further approximation is made using a residual distribution formula. It is basically a function, which evaluates the unknown node-based residual from a weighted sum of all cells having the particular grid node in common. The following distribution formulae were devised:

• volume weighted sum due to Ni [7];
• non-weighted sum due to Hall [8];
• characteristic (upwind) weighting procedure of Rossow [11], [12].

Theoretical investigations of the truncation error [4] suggest that Ni's scheme is more accurate than Hall's approach. However, in practice Ni's distribution formula leads to problems in places, where the grid is strongly distorted and stretched. For example, strong oscillations of the pressure field near the trailing edge of an airfoil were observed when using O-grids [13], [4]. Furthermore, convergence could only be achieved when the numerical viscosity was increased considerably. The underlying idea of the upwind weighting procedure [11], [12] is quite similar to that of the fluctuation-splitting schemes [14]-[17] (cf. Subsection 3.1.5), but the implementation is numerically much simpler. Basically, the residuals are sent only upstream in the characteristic direction.

Of the three approaches, Hall's distribution scheme proved to be the most robust. In Hall's scheme, the residual at a particular node results from a simple sum of all intermediate residuals \(\vec{R}_{I,J,K}\), which cells share the node. Thus, in the 2-D case rendered in Fig. 4.4, we get

In three dimensions, in total eight cell-based residuals have to be summed up in the same way. A close inspection of (4.30) reveals that \(\vec{R}_{i,j}\) is just the net flux through the boundary of the supercell

due to the fact that the fluxes across the inner faces cancel each other. The supercell also represents the "total" control volume, centred at the point \((i,j)\). In the 3-D case, the total volume consists of the cells

As it can be seen from Fig. 4.4, the control volumes overlap by at least one cell, which gave the scheme its name.

The source term is calculated using the flow variables from the corresponding grid node, i.e.,

With the above definitions, the time-stepping scheme in Eq. (4.2) becomes

This represents a system of ordinary differential equations, which has to be solved in each grid point \((i,j,k)\) in the same way as for the cell-centred scheme. Note that the total volume (4.31) or (4.32), respectively, is utilised in Eq. (4.34).