4.2 General Discretisation Methodologies 通用离散方法学[cfd-4-2]

We speak of a cell-centred scheme if the control volumes are identical with the grid cells and if the flow variables are located at the centroids of the grid cells as indicated in Fig. 4.3. When we evaluate the discretised flow equations (4.2), we have to supply the convective and the viscous fluxes at the faces of a cell [6]. They can be approximated in one of the three following ways:

• by the average of fluxes computed from values at the centroids of the grid cells to the left and to the right of the cell face, but using the same face vector (generally applied only to the convective fluxes);
• by using an average of variables associated with the centroids of the grid cells to the left and to the right of the cell face;
• by computing the fluxes from flow quantities interpolated separately to the left and to the right side of the cell face (employed only for the convective fluxes).

Thus, taking the cell face \(\vec{n}_{I+1/2,J}\) in Fig. 4.3 as an example, the first approach - average of fluxes - reads in two dimensions

with \(\Delta S_{I+1/2,J}\) computed from Eqs. (4.6) and (4.7).

The second possible approach - average of variables - can be formulated as follows

where the conservative/dependent variables at the face \(\vec{n}_{I+1/2,J}\) of the control volume are defined as the arithmetic average of values at the two adjacent cells, i.e.,

The flux vector \(\vec{F}\) in Eq. (4.16) stands either for the convective or for the viscous fluxes.

The third methodology starts with an interpolation of flow quantities (being mostly velocity components, pressure, density and total enthalpy) separately to both sides of the cell face. The interpolated quantities - termed the left and the right state (see the begin of Section 4.3) - differ in general between both sides. The fluxes through the cell face are then evaluated from the difference of the left and right state using some non-linear function. Hence,

where

represent the interpolated states. Of course, similar relations like (4.15)–(4.19) hold also for the other cell faces.

The same approximations are employed in three dimensions. For example, at the cell face \(\vec{n}_{I+1/2,J,K}\) (e.g., identical to \(\vec{n}_2\) in Fig. 4.1b) the average of fluxes in Eq. (4.15) becomes

with \(\Delta S_{I+1/2,J,K}\) being defined correspondingly to Equations (4.8) and (4.9). The average of variables reads similarly to Eq. (4.16) as

with

The way over the interpolation of the flow variables results similarly to Eq. (4.18) in

where \(\vec{U}_L\) and \(\vec{U}_R\) are the interpolated values at the cell face.

The last term in the discretised flow equations (4.2) which remains to be evaluated is the source term \(\vec{Q}\). As we already stated in the introduction, the source term is usually supposed to be constant inside the control volume. For this reason, it is calculated using the flow variables from the corresponding cell centre. Hence, we may define

Using the above relations, the fluxes through the faces can be computed and the numerical integration over the boundary of \(\Omega_{I,J,K}\) may be performed according to (4.2). In other words, the complete residual \(\vec{R}_{I,J,K}\) is obtained. In Sections 4.3 and 4.4, we shall learn more about the details of the evaluation of the convective and viscous fluxes.

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).