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Hello to all,


I am new o CFD in general and just started studying the SIMPLE algorithm (in simpleFoam).

The continuity equation for incompressible flows is:



And the momentum equation in a semi-discritezed manner is:



or alternatively



We can now write an expression for as:



To derive the pressure correction equation we must replace the into the continuity equation. Which results in:



We must also compute the face fluxes. In this case:




In simpleFOAM:


is defined as:

    volScalarField rAU(1.0/UEqn.A());

    volVectorField HbyA(constrainHbyA(rAU*UEqn.H(), U, p));

surfaceScalarField phiHbyA("phiHbyA", fvc::flux(HbyA));

Pressure Equation as:

       
 fvScalarMatrix pEqn
        (
            fvm::laplacian(rAtU(), p) == fvc::div(phiHbyA)
        );

rAtU() is equal to rAU for the SIMPLE algorithm

The velocity is reconstructed as:

    U = HbyA - rAtU()*fvc::grad(p);

My doubt is in the update of the fluxes.


In the code, we have :

phi = phiHbyA - pEqn.flux();

Is this equivalent to having (?):



In addition, why does the equation change in a compressible case (rhoSimpleFoam) to:

phi = phiHbyA + pEqn.flux();

Sorry in advance if this is too basic. I am still "breaking stone". I would also appreciate recommendations for literature showing how these coupling algorithms are implemented in OpenFoam