Question about the non-orthogonal term in surface normal gradients (FVM)
 

In FVM, the surface normal velocity gradient is used when discretizing the diffusion term.

According to Dr. Jasak thesis (setion 3.3.1.3) the gradient of the face normal is decomposed into orthogonal and non-orthogonal components.

While the orthogonal component is assessed implicitly, the non-orthogonal counterpart is explicitly evaluated as a source term, leveraging the known velocity field.

I'm curious about why the non-orthogonal aspect isn't handled implicitly. Are there any advantages to its explicit implementation?

Thank you so much for your time! :)

In more general terms it really is the gradient part and the non-gradient part, whose coefficients can be adjusted to make them the non-orthogonal and orthogonal ones. But other popular options exist (like making the gradient part the smallest possible).

The non gradient part only involves face adjcent cell center values and is easily and robustly discretized implicitly. The gradient part, instead, involves further apart cells (matrix band grows), possibly with non helpful coefficients.

Long story short: memory wise is a blood bath (also, different gradient methods might have different stencils), and doesn't even really help that much stability. Moreover, for the most common non linear problems, making everything implicit is not a silver bullet as for linear problems.