Discretization of boundary conditions

As described in [1], Dirichlet conditions for the velocity are implemented by setting the nodal value of the normal velocity component or by linear interpolation for the tangential velocity components. The homogeneous boundary Neumann conditions for the pressure are discretized by, e.g.

$$p_{0,j,k}=-p_{1,j,k}~.$$

In case of convex edges some special treatment is required. Consider the situation shown in Figure .

Figure: convex corner

If $\Gamma_1$ denotes the face between the cells $_{i,j}$ and $_{i,j+1}$ and $\Gamma_2$ the face between the cells $_{i,j}$ and $_{i+1,j}$, then the homogeneous Neumann conditions are discretized by

$$p_{i,j}=-\frac{\vert\Gamma_1\vert p_{i,j+1}+\vert\Gamma_2\vert p_{i+1,j}}{\vert\Gamma_1\vert+\vert\Gamma_2\vert} \quad.$$