5.2 General Discretisation Methodologies 一般离散化方法[cfd-5-2]

We speak of a cell-centred scheme if the control volumes are identical with the grid cells and if the flow variables are associated with their centroids, as it is sketched in Fig. 5.6. When we evaluate the discretised flow equations (5.2), we have to supply the convective and the viscous fluxes at the midpoints of the faces of the control volume, which is sufficient for a second-order accurate discretisation on smooth grids (averaged normal vector is employed for quadrilateral faces). The fluxes can be approximated in one of three 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 unit normal vector (generally applied only to the convective fluxes);
• by using an average of variables associated with the centroids of the grid cells adjacent to the left and to the right side of the cell face;
• by computing the fluxes from flow quantities reconstructed separately on both sides of the cell face from values in the surrounding cells (employed only for the convective fluxes).

Thus, considering for example the cell face with the unit normal vector \(\vec{n}_{01}\) in Fig. 5.6, the first approach - average of fluxes - reads in two dimensions

with the face area \(\Delta S_{01}\) computed from Eqs. (5.6) and (5.7).

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

where the conservative/dependent variables at the face with the unit normal vector \(\vec{n}_{01}\) are defined as the arithmetic average of values at the two adjacent cells. Thus,

The flux vector \(\vec{F}\) in Eq. (5.19) represents either the convective or the viscous fluxes.

The third methodology starts with an interpolation of flow quantities (usually velocity components, pressure, density and total enthalpy) separately to both sides of the cell face. The reconstructed quantities - termed the left and the right state (see Subsection 5.3.3) - differ in general. The fluxes through the cell face are then evaluated from the difference of the left and right state using an appropriate non-linear function. Hence,

where

represent the reconstructed states.

Of course, similar relations hold for the other control volume faces as well. The above approximations can be employed in the same way in three dimensions. The face vector \(\vec{S}\) is then evaluated using the formulae (5.11) or (5.13), respectively.

As we already stated in the introduction to this section, the basic data structure which describes the elements has to be extended in an appropriate way to support the discretisation methodology. It is obvious from the previous discussion that numerical operations are carried out using mainly the faces of the elements (control volumes) together with values at the centres of the adjacent cells. It is therefore quite natural to employ a face-based data structure for the spatial discretisation. Such data structure stores for each particular face in the grid (see Fig. 5.6):

• pointers to the two cells which share the respective face - this allows us to access the flow variables associated with the two cells (\(C_0\), \(C_1\));
• the face vector (\(\vec{S}_{01} = \vec{n}_{01}\Delta S_{01}\)) - must point consistently either outwards or inwards;
• two vectors from the centroid each cell to the midpoint of the face \(M_{01}\), i.e., (\(C_0-M_{01}\)), (\(C_1-M_{01}\)) - required for an accurate interpolation of flow variables to the face. This is not necessary for purely tetrahedral grids, where a simple extrapolation formula can be used [26], [2] (cf. Eq. (5.44)).

Hence, the integration of the fluxes (e.g., according to Eq. (5.19)) would be implemented as a loop over all (i.e., internal and boundary) faces contained in the grid:

After the loop is completed and the source term \(\vec{Q}_I\Omega_I\) is added, we obtain the final residuals (\(\vec{R}\)) in all cells. A less efficient approach would be to loop over elements because the face vectors would have to be stored twice and the fluxes would be computed twice (with the exception of the boundaries). Furthermore, because we use exactly the same face vector \(\vec{S}_{IJ}\) in order to to evaluate the partial fluxes into the volumes \(\Omega_I\) and \(\Omega_J\), the conservation properties of the governing equations are automatically retained.

Within the cell-vertex scheme, the flow variables are associated with the grid nodes (vertices). Median-dual control volumes are formed by connecting the centroids, face- and edge-midpoints of all cells sharing the particular node. This is depicted in Fig. 5.7a for a tetrahedron and in Fig. 5.7b for a hexahedron. The definition of a median-dual control volume results in a polyhedral hull around each grid node, as it is sketched in Fig. 5.8 for a 2-D mixed grid. This polyhedra can be viewed as a dual grid - hence the name of the scheme. It is interesting to note that the median-dual finite volume discretisation is equivalent to the Galerkin finite element scheme with linear elements (see, e.g., [36]).

In order to evaluate the discretised flow equations (5.2), we have to integrate the convective and viscous fluxes over the surface of the control volume. Hence, we would have to compute the fluxes for each partial face (e.g., \(F_1-M_{13}-F_2-C\) in Fig. 5.7a) separately. However, this is only required for a third- or higher-order accurate discretisations [37], [38]. In the case of a second-order scheme, which is most frequently employed, we may assume the flow variables to be constant for all faces grouped around a particular edge. The fluxes are then evaluated at the midpoint of the edge using the variables and the gradients from both nodes. This approach allows us to define a mean unit normal vector and a total face area associated with each edge. Thus referring to Fig. 5.8, the mean normal vector, e.g., for the edge \(P_0-P_1\) becomes

and the total face area is given by: \(\Delta S_{01} = \Delta S_L + \Delta S_R\). The same applies also in 3D, where the mean normal vector results from a sum over all partial faces having the particular edge-midpoint in common, as it is rendered in Fig. 5.9. The face vector (\(\vec{S} = \vec{n}\Delta S\)) is computed in 2D from Eq. (5.6). In three dimensions, where the partial faces are always quadrilaterals, we can either divide them into triangles and use Eq. (5.11), or we can employ a simplified treatment due to Eq. (5.13), which is sufficient for smooth grids. The element centroids and the face centres are obtained by the formulae (5.17), (5.8), or (5.9), respectively.

The fluxes can then be evaluated according to one of the three following methodologies:

• by the average of fluxes computed from values at both nodes of an edge, but using the same mean unit normal vector (generally applied only to the convective fluxes);
• by using an average of variables stored at the two nodes of an edge;
• by computing the fluxes from flow quantities reconstructed separately on both sides of the face of the control volume from values at the surrounding nodes (employed only for the convective fluxes).

The computation of fluxes follows formally the same approaches as for the cell-centred scheme. Thus, the formulae (5.18)-(5.22) are applicable also in the case of the median-dual scheme. If we utilise the above approach which associates each edge with a mean unit normal, the most efficient methodology is to employ an edge-based data structure for the spatial discretisation. The edge-based data structure stores for each particular edge in the grid (cf. Fig. 5.9):

• pointers to the two nodes which define the edge - this allows us to access the flow variables associated with the two control volumes \(\Omega_i\) and \(\Omega_j\);
• the face vector (\(\vec{S}_{ij} = \vec{n}_{ij}\Delta S_{ij}\)) - must point consistently either outwards or inwards;
• the edge vector from node \(i\) to node \(j\) - required for the interpolation of flow variables to the face (solution reconstruction). Alternatively, the edge vector can be computed on the fly from coordinates of the nodes. This is not required for the standard central scheme with artificial dissipation (Subsection 5.3.1).

With this, the integration of the fluxes (e.g., according to Eq. (5.19)) would be implemented as a loop over all edges in the grid:

After the loop is completed and the source term \(\vec{Q}_i\Omega_i\) is added, we obtain the final residuals (\(\vec{R}\)) in all nodes. This approach is significantly more efficient than summing up the fluxes over each control volume separately, because we store each mean face vector only once and we also visit each edge only once instead of twice. Furthermore, since we use exactly the same mean face vector \(\vec{S}_{ij}\) in order to to evaluate the partial fluxes into the volumes \(\Omega_i\) and \(\Omega_j\), the mass, momentum and energy remain exactly conserved.

The relative advantages and disadvantages of the cell-centred and the median-dual scheme are the subject of controversial debates. The main reason is the lack of fair comparisons of the two methodologies with respect to accuracy, computational time and memory for realistic configurations. Our intention here is to collect the most important arguments for and against each of the approaches regarding:

• accuracy,
• computational work,
• memory requirements, and
• flexibility.

This should lead to a greater understanding of the problems inherent to each scheme and should be of help in selecting the most suitable scheme for the intended applications.

A cell-centred scheme on a triangular/tetrahedral grid leads to about twice/six times as many control volumes and hence degrees of freedom as a median-dual scheme [36]. On typical mixed grids, which consist of tetrahedra and prisms, a cell-centred scheme gives roughly three times more unknowns than a median-dual scheme. This suggests that cell-centred schemes are more accurate than cell-vertex discretisations on an identical grid. However, the residual of a cell-centred scheme results from a much smaller number of fluxes as compared to a median-dual scheme (three versus approximately seven on a tetrahedral grid), which may impair the accuracy. Thus, there is no clear evidence about which scheme might be superior.

The median-dual scheme suffers from a particular problem on stretched triangular and tetrahedral grids. Consider, for example, Fig. 5.10, which shows a tessellation composed of right triangles, as it is often employed near solid walls for viscous flows. We can see in Fig. 5.10a that the face \(\Delta S_{ij}\) becomes highly skewed with respect to the edge \(ij\). However, spatial discretisation schemes mostly assume fluxes to be orthogonal to a face (especially Riemann solvers). Thus, an error is introduced which is particularly significant for a first-order scheme [38]. The situation can be improved using the so-called containment-dual control volume [39]. As depicted in Fig. 5.11, the containment-dual approach employs the centres of the minimum spanning circles/spheres instead of the cell centroids to define the faces. This leads to control volumes identical to those on quadrilateral grids (Fig. 5.10b). Notice that there is no face area associated with diagonal edges like \(ij'\). An additional effort is required for pre-processing, but the solution accuracy can be improved noticeably [40]. Of course, another possibility is to employ directly quadrilateral or hexahedral

cells within the boundary layers. Further discussion of grid-induced errors can be found in Ref. [22] and [23].

Another problem inherent to the median-dual scheme is the discretisation at boundaries of the physical domain. What happens is that there is only about one half of the control volume left at the boundary (cf. Fig. 4.6). The integration of fluxes around the faces results in a residual located inside - ideally at the centre - of the control volume. However, the residual is associated with the node, residing directly on the boundary. This mismatch leads to increased discretisation error in comparison to the cell-centred scheme, which is particularly undesirable on solid walls. The definition of the dual control volume causes also problems at sharp corners (like trailing edges), which show up as unphysical peaks in pressure or density. Further complications arise at periodic boundaries (see Chapter 8.8), where the fluxes from both parts of the control volume have to be summed up correctly.

The mismatch between the centroid of the control volume and the node where the residual is stored has also a further implication for the median-dual scheme. It arises as the mass matrix in the case of unsteady flows. We discussed this point already at the beginning of Section 3.2. The advantage of the cell-centred scheme is that the mass matrix can be eliminated from the equations, without compromising the solution accuracy. By contrast, the median-dual scheme requires a special treatment of the mass matrix [41], [42].

In order to judge the computational effort required for both schemes, we have to consider primarily the integration of the fluxes. We know from the previous discussion that the cell-centred scheme uses a loop over cell faces whereas the median-dual scheme loops over the edges. Since the evaluation of the fluxes at an interface is quite similar for both schemes, the ratio of the number of cell faces to the number of edges gives the ratio of the computational work. Thus, on a tetrahedral grid, where the number cell faces (if counted only once for every two cells) is approximately two times larger than the number of edges, the cell-centred scheme is computationally twice as much expensive as the median-dual scheme on an identical grid [36]. The cell-centred approach becomes however more competitive on mixed grids containing prismatic elements. Apart from boundary treatment, both methods are computationally equivalent on hexahedral grids, where the number of faces equals the number of edges.

Considering the memory requirements, the cell-centred scheme has to store about six times more flow variables on tetrahedral and about three times more variables on usual mixed grids as compared to the median-dual scheme. Furthermore, as we saw, both schemes require to store two integers and three reals (pointers and face vector) per cell face or edge, respectively. Additionally, the cell-centred scheme has to keep two vectors to the face-midpoint - 6 reals - per cell face in memory. On the contrary, the median-dual scheme can work with the node coordinates only, which are considerably fewer values. Thus in summary, the cell-centred scheme needs, on average, more than twice as much computer memory as the median-dual method.

One significant advantage of the cell-centred scheme appears in the case of non-conforming cell interfaces, like those at the letter "F" in Fig. 3.4. In contrast to the median-dual methodology, no special and expensive procedure is required for the computation of the fluxes at the interface. This allows for an increased flexibility in the grid generation and also in the grid adaptation.