Face-Normal Gradient

The face-normal gradient is computed for the water surface in the pressure gradient term and the current velocities when simulating momentum diffusion with the conservative formulation. The operator is described here for the water surface elevation but is the same for the current velocity. The face-normal gradient is computed with a simple two-point stencil as

where $//<![CDATA[ \begin{array}{l}\Delta x_N\end{array} //]]>$ is the face-normal distance between the cell centers as described in the figure below. The distance  is the distance between points  and . The water surface elevation in the neighboring cells is assumed to be spatially constant.

where $//<![CDATA[ \begin{array}{l}\Delta x_N\end{array} //]]>$ is the face-normal distance between points $//<![CDATA[ \begin{array}{l}L'\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}R'\end{array} //]]>$ as described in the figure below. The scheme assumes that cell variables are piece-wise constant and does not include a non-orthogonal correction. The method is second-order for regular Cartesian cells and first-order for general polygonal cells. 

Figure 1. Cell Directional Derivatives.

Face-normal gradients at closed boundaries are set to zero. In addition, velocity face-normal gradients at wet/dry boundaries are set to zero.