ANSYS FLUENT 12.0 Theory Guide - 18.3.3 Evaluation of Gradients and Derivatives

Gradients are needed not only for constructing values of a scalar at the cell faces, but also for computing secondary diffusion terms and velocity derivatives. The gradient $\nabla \phi$ of a given variable $\phi$ is used to discretize the convection and diffusion terms in the flow conservation equations. The gradients are computed in ANSYS FLUENT according to the following methods:

When the Green-Gauss theorem is used to compute the gradient of the scalar $\phi$ at the cell center

, the following discrete form is written as

$\displaystyle (\nabla \phi)_{c0}$

$\textstyle =$

$\displaystyle \frac{1}{\cal V} \sum_f \overline{\phi}_f \; \vec A_f$

(18.3-22)

where $\phi_f$ is the value of $\phi$ at the cell face centroid, computed as shown in the sections below. The summation is over all the faces enclosing the cell.

By default, the face value, $\overline{\phi}_f$ , in Equation 18.3-22 is taken from the arithmetic average of the values at the neighboring cell centers, i.e.,

Alternatively, $\overline{\phi}_f$ can be computed by the arithmetic average of the nodal values on the face.

$\overline{\phi}_f = \frac{1}{N_{f}} \sum_n^{N_{f}} \overline{\phi}_n$

(18.3-24)

The nodal values, $\overline{\phi}_n$ in Equation 18.3-24, are constructed from the weighted average of the cell values surrounding the nodes, following the approach originally proposed by Holmes and Connel[ 132] and Rauch et al.[ 286]. This scheme reconstructs exact values of a linear function at a node from surrounding cell-centered values on arbitrary unstructured meshes by solving a constrained minimization problem, preserving a second-order spatial accuracy.

The node-based gradient is known to be more accurate than the cell-based gradient particularly on irregular (skewed and distorted) unstructured meshes, however, it is relatively more expensive to compute than the cell-based gradient scheme.

In this method the solution is assumed to vary linearly. In Figure 18.3.4, the change in cell values between cell

and

along the vector $\delta r_i$ from the centroid of cell

to cell

, can be expressed as

If we write similar equations for each cell surrounding the cell c0, we obtain the following system written in compact form:

The objective here is to determine the cell gradient ( $\nabla \phi{_0} = \phi{_x}$ î $+ \phi{_y}$ $\textrm{\^{j\/}}$ $+ \phi{_z}$ $\textrm{\^{k\/}}$ ) by solving the minimization problem for the system of the non-square coefficient matrix in a least-squares sense.

The above linear-system of equation is over-determined and can be solved by decomposing the coefficient matrix using the Gram-Schmidt process [ 6]. This decomposition yields a matrix of weights for each cell. Thus for our cell-centered scheme this means that the three components of the weights ( ${W^x}_{i0},{W^y}_{i0},{W^z}_{i0}$ ) are produced for each of the faces of cell c0.

Therefore, the gradient at the cell center can then be computed by multiplying the weight factors by the difference vector $\Delta \phi = (\phi_{c1} - \phi_{c0})$ ,

$\displaystyle ({\phi_x})_{c0}$	$\textstyle =$	$\displaystyle \sum_{i=1}^{n}{{W^x}_{i0} \cdot (\phi_{ci} - \phi_{c0})}$	(18.3-27)
$\displaystyle ({\phi_y})_{c0}$	$\textstyle =$	$\displaystyle \sum_{i=1}^{n}{{W^y}_{i0} \cdot (\phi_{ci} - \phi_{c0})}$	(18.3-28)
$\displaystyle ({\phi_z})_{c0}$	$\textstyle =$	$\displaystyle \sum_{i=1}^{n}{{W^z}_{i0} \cdot (\phi_{ci} - \phi_{c0})}$	(18.3-29)

On irregular (skewed and distorted) unstructured meshes, the accuracy of the least-squares gradient method is comparable to that of the node-based gradient (and both are much more superior compared to the cell-based gradient). However, it is less expensive to compute the least-squares gradient than the node-based gradient. Therefore, it has been selected as the default gradient method in the ANSYS FLUENT solver.