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:

I don't understand what e is/where it comes from. A book I have just calls it a "coefficient with large values." Is it some kind of approximation for P? Is the method I currently use--calculating P first and then getting a velocity profile from that--better?

Could it be that the penalty formulation stuff might be better suited to single-phase flow? Or flows with no gas (compressible)?

I guess what I'm getting at is, what ARE the advantages of the penalty method? I see that someone I respect has used it and I'm wondering why he would have chosen it over the pressure calculation method. Any advice would be very much appreciated. Thanks.

remove the ellipticity of the equations, thus allowing the use of explicit schemes.
the e is a large number, it takes ﬁnite values which can be determined through numerical experimentation.