[Fluent Inc. Logo] return to home search
next up previous contents index

24.2.9 Evaluation of Derivatives

The derivative $\nabla \phi$ of a given variable $\phi$ is used to discretize the convection and diffusion terms of the equations of motion. The gradient is computed using the Green-Gauss theorem as


$\displaystyle (\nabla \phi)_{c0}$ $\textstyle =$ $\displaystyle \frac{1}{\cal V} \sum_f \overline{\phi}_f \; \vec A_f$ (24.2-22)

where $\phi_f$ is the value of $\phi$ at the cell face centroid, and the summation is over all the faces enclosing the cell.

Cell-Based Derivative Evaluation

By default, the face value, $\overline{\phi}_f$, in Equation  24.2-22 is taken from the arithmetic average of the values at the neighboring cell centers, i.e.,


\begin{displaymath} \overline{\phi}_f = \frac{\phi_{c0} + \phi_{c1}}{2} \end{displaymath} (24.2-23)

To use this option, select Cell-Based under Gradient Option in the Solver panel.

Node-Based Derivative Evaluation

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


\begin{displaymath} \overline{\phi}_f = \frac{1}{N_{f}} \sum_n^{N_{f}} \overline{\phi}_n \end{displaymath} (24.2-24)

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

The nodal values, $\overline{\phi}_n$ in Equation  24.2-24, are constructed from the weighted average of the cell values surrounding the nodes, following the approach originally proposed by Holmes and Connel[ 105] and Rauch et al.[ 207]. 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 averaging scheme is known to be more accurate than the default cell-based scheme for unstructured meshes, most notably for triangular and tetrahedral meshes.

To use this option, select Node-Based under Gradient Option in the Solver panel.


next up previous contents index Previous: 24.2.8 Temporal Discretization
Up: 24.2 Discretization
Next: 24.3 The Segregated Solver
© Fluent Inc. 2003-01-25