finite volume discretizing on non-conservative eq.
 

Hello.

I am now working on the discretizing formulation to solve incompressible flow by pseudo-compressibility method in a velocity-pressure coupled way. From my experience, we need to linearize the eq before discretization using Jacobin matrix.

for example:
suppose U is matrix of primitive variables
we often change ("dd" means 1st order partial differencing here)

ddU/ddt+ddE/ddX+ddF/ddY=RHS (1) --conservative

into

ddU/ddt+A(ddU/ddX)+B(ddU/ddY)=RHS (2) (in which A=ddE/ddU, B=ddF/ddU) --non-conservative

Theoretically, conservative form is needed before applying the FVM. I tried to discretize (1) without knowing E, F as functions of U first, and then applied E=U(ddE/ddU) and F=U(ddF/ddU) for the facial flux expression. I found that the final formulation will be the same with discretizing (2). Is this correct or wrong?

I did this because I've to handle the pressure term in solving N-S eq. I want to include the pressure term into E and F, rather than leaving it as a separated term. I think this way may make it simpler and doesn't require other tricks to discretize the pressure term. Is my idea correct? Can it work?

Thank you very much.

I heard that some solvers use a technique named "momentum interpolation" to compute the pressure. Very few CFD books talk about this. Not sure if it will work as in a coupled solver using pseudo-compressibility.

could be right
 

Hi,

What you do could be right since E=AUand F=BU are exact relations for the Euler equations: a flux vector is exactly equal to the product of the Jacobian and the state vector.

I remember that the Steger-Warming flux vector splitting scheme are constructed by using these relations.

gory