[RodriguezFatz]

Hello to everyone!

I have a question regarding the SIMPLE-algorithm for finite volumes.
The derivation of the algorithm starts (in different books) with the discretized momentum equation, such as:

A_p * u_p + SUM (A_x * u_x) = Q_p - pressure gradient_p

I am a bit confused about this equation:
I though we need the volume integrated momentum equation to start with, with conservative integrations of all terms - because that is what we are interested in, in FV. Here, the derivation starts with something that looks like the momentum equation itself. Or I can not explain the appearance of the pressure gradient...

Can someone clear this up for me?

[Aeronautics El. K.]

If my memory is not failing me, this is derived from the integral form of the momentum equation by approximating each of its terms.

[RodriguezFatz]

Then, I am confused by the Ferziger / Peric book. They say that nearly all commercial codes use the conservative form of integrating the pressure term. Later they explain SIMPLE, SIMPLEC,... using a derivation that starts with the non conservative form.

Anyway, what you say, means that the cell volume "dV" is taken from the "pressure gradient_p" term and put into the coefficients of the other summands, correct?

[FMDenaro]

Originally Posted by RodriguezFatz

Then, I am confused by the Ferziger / Peric book. They say that nearly all commercial codes use the conservative form of integrating the pressure term. Later they explain SIMPLE, SIMPLEC,... using a derivation that starts with the non conservative form.

Anyway, what you say, means that the cell volume "dV" is taken from the "pressure gradient_p" term and put into the coefficients of the other summands, correct?

The key is that the integrals are approximated by centred second order formulas (mean value approximation). This often leads to have discrete equations that appears similar both for FV and FD.
The confusion vanishes for higher order of accuracy

[RodriguezFatz]

I was talking about conservative and non-conservative formulation. It looks like they use a non-conservative pressure integral, although they previously state that nearly everyone (commercial) uses the conservative form.

[FMDenaro]

Originally Posted by RodriguezFatz

I was talking about conservative and non-conservative formulation. It looks like they use a non-conservative pressure integral, although they previously state that nearly everyone (commercial) uses the conservative form.

The flux-integrated form of the pressure term should write as

Int [S] n.(pI) dS

but I think that by using the volume average of the pressure gradient, the second order discretization is done with the mean value formula therefore it seems a non-conservative formula ...