Hi everyone:

I'm interested in getting into some fluid flow modeling (e.g., advection-diffusion). I'm a newbie, so sorry in advance for any silly questions.

Today I came across the "penalty formulation" and I'd really appreciate any basic help or directions to online references in understanding (1) how it works and (2) what the advantages/disadvantages are compared to the method I'm familiar with--using conservation of fluid mass to derive an expression for the fluid pressure p and then getting the velocity profile from that.

From what I've read, using the penalty method allows elimination of pressure from the momentum equation so that the mass and momentum equations can be combined. The result is an equation (convection-diffusion) where velocity can be calculated directly.

This seems rather strange though. The first step in using the penalty method is to replace the continuity equation by: