How to compute gradient of a cell having a boundary face?

Question

In many situations in unstructured mesh solvers, one needs to compute gradient of arbitrary variable $phi$ such as temperature or velocity at face centers (one of such situations is correction for mesh non-orthogonality), which can be approximated as:
$$ nabla phi_f = (g_C  nabla phi_C) + (g_F  nabla phi_F)$$
Where $C$ & $F$ are two adjacent cells sharing the face $f$ and $g_C$ & $g_F$ are interpolation weight factors.
Now, $nabla phi_C$ and $nabla phi_F$ can be approximated as the average gradient across the cell, using Gauss-Green theorem as:
$$ nabla phi_C = frac{1}{V_C} sum_{faces} phi_f mathbf{S_f} $$
where $mathbf{S_f}$ is the face normal vector at face $f$, and $V_C$ is the cell volume.
The $phi_f$ values for interior faces can be easily known.

What if the face is a boundary face and insulated? how to get $phi_f$ in such case?
What if the face has a fixed boundary condition $phi_{f} =
    phi_{specified}$ (meaning that face has only one adjacent cell, so first interpolation equation cannot be used), how to compute the gradient at the face in such
case?

Algo · Accepted Answer

Disclaimer: I'm not 100% sure but I thought that I should provide my working solution to the above problem, for any future visitor that might have the same questions. The answer is for a steady temperature diffusion problem.

What if the face is a boundary face and insulated? how to get $phi_f$ in such case?

Since a boundary face is in contact with only one cell, we cannot use interpolation, but since an insulated boundary will have a zero gradient $nabla phi=0$ across all its faces, and by assuming linear variation between cell centroid and face centroid, thus for an arbitrary cell with a face boundary one can write:
$$ phi_C = phi_f + (r_C - r_f).nabla phi_f implies phi_C = phi_f$$

What if the face has a fixed boundary condition $phi_{f} =
phi_{specified}$ (meaning that face has only one adjacent cell, so first interpolation equation cannot be used), how to compute the gradient at the face in such
case?

Again, since a linear profile is assumed:
$$ nabla phi_f = frac{phi_{specified} - phi_C}{|r_f - r_C|} hat{e}$$
where $hat{e} = frac{r_f - r_C}{|r_f - r_C|}$
I tried this on an unstructured non-orthogonal mesh and the solution finally converged, having a good agreement with laplacianFoam (I still didn't implement correction for skewness, which justifies the deviation from OpenFOAM (and not enough iterations), and used simple Gauss-Seidel solver), and this is the center line temperature distribution

How to compute gradient of a cell having a boundary face?

One Answer

Add your own answers!

Ask a Question