Cell Velocity Gradient

When the Smagorisnky horizontal mixing method is utilized, the required velocity gradients are computed at cells and then interpolated to faces. The cell velocity gradients are computed using Gauss’ divergence theorem. The velocity gradients for the velocity in the x-direction may be written as:

1)	$\begin{array}{l}\displaystyle \nabla u_i = \frac{1}{A_i} \int _A \nabla u dA = \frac{1}{A_i} \sum _{k \in \mathcal{K} (i) } (u_N)_k \textbf{n}_{ik} L_k\end{array}$

where $\begin{array}{l}L_k\end{array}$ is the length of face k, and $\begin{array}{l}A_i\end{array}$ is the area of cell i. A similar equation may be written for the velocity in the y-direction.