5.4.1 Element-Based Gradients 基于单元的梯度[cfd-5-4-1]

A common feature of this type of gradient computation is the necessity to store either information about the grid elements or some coefficients related to the geometry of the elements. Hence, we have to extend the data structure beyond the face-/edge-based formulation presented earlier for the convective fluxes. Below, we discuss three well-established methods for the cell-centred and the median-dual discretisation.

One possible way of evaluating the gradients at a face of the control volume is to define an auxiliary control volume centred at the face and to employ the Green-Gauss theorem. We already discussed this approach in Section 4.4 in the framework of the structured finite volume discretisation. For example, in the case of the median-dual scheme we can compute the gradient at the edge-midpoint as the volume average of gradients for all elements which share the edge [70]. The element-based gradients are evaluated according to Eq. (5.49) by looping over all grid cells and accumulating the gradients at the edges. The values of \(U\) at the cell-faces are obtained by averaging the nodal values, in a manner similar to Eq. (5.53). This approach is relatively costly in terms of the memory and the number of operations. However, it can be implemented for any mix of grid elements.

Another methodology, which is applicable to the median-dual scheme, was derived from the Galerkin finite element method [31]. Basically speaking, the approach transforms the integration of gradients over the surface of the control volume into an evaluation of the Hessian matrix (second derivatives) at the central node. The viscous terms then follow the differential form of the Navier-Stokes equations in Cartesian coordinates (Eq. (A.4) with \(\xi = x\), \(\eta = y\), \(\zeta = z\) and \(J^{-1} = 1\)), which contains terms such as

with \(\partial_m(\cdot) = \partial(\cdot)/\partial m\). Hence, no further integration of the viscous fluxes over the faces of the control volume is required.

The original scheme was formulated for purely triangular/tetrahedral grids. It employs a union of all elements that contain the particular node. In order to simplify the implementation, the dynamic viscosity coefficient is averaged from the nodal values, which is a difference to the Galerkin method. Then, the second derivatives can be evaluated at node \(i\) as follows [71]

The volume \(\Omega'\) contains all tetrahedra which share the node \(i\). The coefficient matrix \(\alpha\) in Eq. (5.71) is symmetric about the diagonal [71], i.e., \(\alpha_{xy} = \alpha_{yx}\), \(\alpha_{xz} = \alpha_{zx}\), and \(\alpha_{yz} = \alpha_{zy}\), respectively. Thus, it is necessary to store only six coefficients for each edge. The coefficients are given by [71]

where \(n\), \(k\) denote the \(x\), \(y\), \(z\) subscripts, and \((\vec{S}_e^{i})_k\), \((\vec{S}_e^{j})_n\) represent components of the outer face vectors \(\vec{S}_e^{i}\), \(\vec{S}_e^{j}\) displayed in Fig. 5.18. The summation is carried out over all tetrahedra (with particular volumes \(\Omega_e\)) which share the edge \(ij\). If the grid is stationary, the coefficients can be computed in a pre-processing step. A desirable feature is that the same edge-based data structure can be employed as for the convective fluxes.

The disadvantage of this approach is however that the full viscous terms are retained only on triangular or tetrahedral grids. For elements like prisms or hexahedra, this technique simplifies to a TSL-like approximation of the Navier-Stokes equations in all three coordinate directions [8]. An extension to non-simplex elements which conserves the full viscous terms was presented in [72]. But the efficient edge-based data structure can then no longer be used since the extended stencil involves nodes not connected by an edge to point \(i\).

This scheme is intended for the cell-centred type of control volume and purely tetrahedral grids. It employs a modified version [4] of the stencil introduced in [61] to evaluate the gradients at the cell faces. The approach is based on an average of the values at the three nodes which define the cell face, combined with the quantities at the cell centroids. The first derivatives at a cell face result from the solution of the linear system of equations [4]

Referring to Fig. 5.19, the subscripts \(I\), \(J\) denote the cell-centroids, and the subscripts \(1\), \(2\), \(3\) stand for the nodes \(P_1\), \(P_2\) and \(P_3\), respectively. Flow variables at the nodes can be determined either from the inverse-distance weighting (Eq. (5.45)) or by the pseudo-Laplacian weighting [2] (similar to Eq. (5.24)).