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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:

To learn how to apply the various gradients, see this section in the separate User's Guide.

Green-Gauss Theorem

When the Green-Gauss theorem is used to compute the gradient of the scalar $\phi$ at the cell center $c0$, 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.

Green-Gauss Cell-Based Gradient Evaluation

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.,

 \overline{\phi}_f = \frac{\phi_{c0} + \phi_{c1}}{2} (18.3-23)

Green-Gauss Node-Based Gradient Evaluation

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)

where $N_{f}$ is the number of nodes on the face.

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.


The node-based gradient method is not available with polyhedral meshes.

Least Squares Cell-Based Gradient Evaluation

In this method the solution is assumed to vary linearly. In Figure  18.3.4, the change in cell values between cell $c0$ and $ci$ along the vector $\delta r_i$ from the centroid of cell $c0$ to cell $ci$, can be expressed as

 ({\nabla \phi})_{c0} \cdot \Delta r_i= (\phi_{ci} - \phi_{c0}) (18.3-25)

Figure 18.3.4: Cell Centroid Evaluation

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

[J ]({\nabla\phi})_{c0} = \Delta\phi (18.3-26)

Where [J] is the coefficient matrix which is purely a function of geometry.

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.

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