twoPhaseEulerFoam: Add partial-elimination to phase flux and velocity correction

Improves stability and convergence of systems in which drag dominates
e.g. small particles in high-speed gas flow.

Additionally a new ddtPhiCorr strategy is included in which correction
is applied only where the phases are nearly pure.  This reduces
staggering patters near the free-surface of bubble-column simulations.

applications/solvers/multiphase/twoPhaseEulerFoam/pEqn.H

+// --- Pressure corrector loop
+while (pimple.correct())
 {
     surfaceScalarField alpha1f("alpha1f", fvc::interpolate(alpha1));
     surfaceScalarField alpha2f("alpha2f", scalar(1) - alpha1f);

applications/solvers/multiphase/twoPhaseEulerFoam/pEqnElim.H

+surfaceScalarField alpha1f("alpha1f", fvc::interpolate(alpha1));
+surfaceScalarField alpha2f("alpha2f", scalar(1) - alpha1f);
+
+rAU1 = 1.0/U1Eqn.A();
+rAU2 = 1.0/U2Eqn.A();
+
+surfaceScalarField rAlphaAU1f(fvc::interpolate(alpha1*rho1*rAU1));
+surfaceScalarField rAlphaAU2f(fvc::interpolate(alpha2*rho2*rAU2));
+
+// --- Pressure corrector loop
+while (pimple.correct())
+{
+    volVectorField HbyA1
+    (
+        IOobject::groupName("HbyA", phase1.name()),
+        U1
+    );
+    HbyA1 = rAU1*U1Eqn.H();
+
+    volVectorField HbyA2
+    (
+        IOobject::groupName("HbyA", phase2.name()),
+        U2
+    );
+    HbyA2 = rAU2*U2Eqn.H();
+
+    // Phase pressure flux (e.g. due to particle-particle pressure)
+    surfaceScalarField phiF1
+    (
+        "phiF1",
+        fvc::interpolate(rAU1*phase1.turbulence().pPrime())
+       *fvc::snGrad(alpha1)*mesh.magSf()
+    );
+
+    // Phase-2 pressure flux (e.g. due to particle-particle pressure)
+    surfaceScalarField phiF2
+    (
+        "phiF2",
+        fvc::interpolate(rAU2*phase2.turbulence().pPrime())
+       *fvc::snGrad(alpha2)*mesh.magSf()
+    );
+
+    // Mean density for buoyancy force and p_rgh -> p
+    volScalarField rho("rho", fluid.rho());
+
+    // Add Buoyancy force
+    {
+        surfaceScalarField ghSnGradRho(ghf*fvc::snGrad(rho)*mesh.magSf());
+
+        phiF1 +=
+            rAlphaAU1f
+           *(
+                ghSnGradRho
+              - alpha2f*fvc::interpolate(rho1 - rho2)*(g & mesh.Sf())
+            )/fvc::interpolate(rho1);
+
+        phiF2 +=
+            rAlphaAU2f
+           *(
+                ghSnGradRho
+              - alpha1f*fvc::interpolate(rho2 - rho1)*(g & mesh.Sf())
+            )/fvc::interpolate(rho2);
+    }
+
+    // ddtPhiCorr filter -- only apply in pure(ish) phases
+    surfaceScalarField alpha1fBar(fvc::interpolate(fvc::average(alpha1f)));
+    surfaceScalarField phiCorrCoeff1(pos(alpha1fBar - 0.99));
+    surfaceScalarField phiCorrCoeff2(pos(0.01 - alpha1fBar));
+
+    // Set ddtPhiCorr to 0 on non-coupled boundaries
+    forAll(mesh.boundary(), patchi)
+    {
+        if
+        (
+            !mesh.boundary()[patchi].coupled()
+         || isA<cyclicAMIFvPatch>(mesh.boundary()[patchi])
+        )
+        {
+            phiCorrCoeff1.boundaryField()[patchi] = 0;
+            phiCorrCoeff2.boundaryField()[patchi] = 0;
+        }
+    }
+
+    // Phase-1 predicted flux
+    surfaceScalarField phiHbyA1
+    (
+        IOobject::groupName("phiHbyA", phase1.name()),
+        (fvc::interpolate(HbyA1) & mesh.Sf())
+      + phiCorrCoeff1*rAlphaAU1f
+       *(
+            mrfZones.absolute(phi1.oldTime())
+          - (fvc::interpolate(U1.oldTime()) & mesh.Sf())
+        )/runTime.deltaT()
+      - phiF1
+    );
+
+    // Phase-2 predicted flux
+    surfaceScalarField phiHbyA2
+    (
+        IOobject::groupName("phiHbyA", phase2.name()),
+        (fvc::interpolate(HbyA2) & mesh.Sf())
+      + phiCorrCoeff2*rAlphaAU2f
+       *(
+            mrfZones.absolute(phi2.oldTime())
+          - (fvc::interpolate(U2.oldTime()) & mesh.Sf())
+        )/runTime.deltaT()
+      - phiF2
+    );
+
+    // Face-drag coefficients
+    surfaceScalarField D1f(fvc::interpolate(rAU1*dragCoeff));
+    surfaceScalarField D2f(fvc::interpolate(rAU2*dragCoeff));
+
+    // Construct the mean predicted flux
+    // including explicit drag contributions based on absolute fluxes
+    surfaceScalarField phiHbyA
+    (
+        "phiHbyA",
+        alpha1f*(phiHbyA1 + D1f*mrfZones.absolute(phi2))
+      + alpha2f*(phiHbyA2 + D2f*mrfZones.absolute(phi1))
+    );
+    mrfZones.makeRelative(phiHbyA);
+
+    // Construct pressure "diffusivity"
+    surfaceScalarField rAUf
+    (
+        "rAUf",
+        mag
+        (
+            alpha1f*rAlphaAU1f/fvc::interpolate(rho1)
+          + alpha2f*rAlphaAU2f/fvc::interpolate(rho2)
+        )
+    );
+
+    // Update the fixedFluxPressure BCs to ensure flux consistency
+    setSnGrad<fixedFluxPressureFvPatchScalarField>
+    (
+        p_rgh.boundaryField(),
+        (
+            phiHbyA.boundaryField()
+          - mrfZones.relative
+            (
+                alpha1f.boundaryField()
+               *(mesh.Sf().boundaryField() & U1.boundaryField())
+              + alpha2f.boundaryField()
+               *(mesh.Sf().boundaryField() & U2.boundaryField())
+            )
+        )/(mesh.magSf().boundaryField()*rAUf.boundaryField())
+    );
+
+    tmp<fvScalarMatrix> pEqnComp1;
+    tmp<fvScalarMatrix> pEqnComp2;
+
+    // Construct the compressibility parts of the pressure equation
+    if (pimple.transonic())
+    {
+        surfaceScalarField phid1
+        (
+            IOobject::groupName("phid", phase1.name()),
+            fvc::interpolate(psi1)*phi1
+        );
+        surfaceScalarField phid2
+        (
+            IOobject::groupName("phid", phase2.name()),
+            fvc::interpolate(psi2)*phi2
+        );
+
+        pEqnComp1 =
+            (
+                contErr1
+              - fvc::Sp(fvc::ddt(alpha1) + fvc::div(alphaPhi1), rho1)
+            )/rho1
+          + (alpha1/rho1)*correction
+            (
+                psi1*fvm::ddt(p_rgh)
+              + fvm::div(phid1, p_rgh) - fvm::Sp(fvc::div(phid1), p_rgh)
+            );
+        deleteDemandDrivenData(pEqnComp1().faceFluxCorrectionPtr());
+        pEqnComp1().relax();
+
+        pEqnComp2 =
+            (
+                contErr2
+              - fvc::Sp(fvc::ddt(alpha2) + fvc::div(alphaPhi2), rho2)
+            )/rho2
+          + (alpha2/rho2)*correction
+            (
+                psi2*fvm::ddt(p_rgh)
+              + fvm::div(phid2, p_rgh) - fvm::Sp(fvc::div(phid2), p_rgh)
+            );
+        deleteDemandDrivenData(pEqnComp2().faceFluxCorrectionPtr());
+        pEqnComp2().relax();
+    }
+    else
+    {
+        pEqnComp1 =
+            (
+                contErr1
+              - fvc::Sp(fvc::ddt(alpha1) + fvc::div(alphaPhi1), rho1)
+            )/rho1
+          + (alpha1*psi1/rho1)*correction(fvm::ddt(p_rgh));
+
+        pEqnComp2 =
+            (
+                contErr2
+              - fvc::Sp(fvc::ddt(alpha2) + fvc::div(alphaPhi2), rho2)
+            )/rho2
+          + (alpha2*psi2/rho2)*correction(fvm::ddt(p_rgh));
+    }
+
+    // Cache p prior to solve for density update
+    volScalarField p_rgh_0(p_rgh);
+
+    // Iterate over the pressure equation to correct for non-orthogonality
+    while (pimple.correctNonOrthogonal())
+    {
+        // Construct the transport part of the pressure equation
+        fvScalarMatrix pEqnIncomp
+        (
+            fvc::div(phiHbyA)
+          - fvm::laplacian(rAUf, p_rgh)
+        );
+
+        solve
+        (
+            pEqnComp1() + pEqnComp2() + pEqnIncomp,
+            mesh.solver(p_rgh.select(pimple.finalInnerIter()))
+        );
+
+        // Correct fluxes and velocities on last non-orthogonal iteration
+        if (pimple.finalNonOrthogonalIter())
+        {
+            phi = phiHbyA + pEqnIncomp.flux();
+
+            surfaceScalarField mSfGradp("mSfGradp", pEqnIncomp.flux()/rAUf);
+
+            // Partial-elimination phase-flux corrector
+            {
+                surfaceScalarField phi1s
+                (
+                    phiHbyA1 + rAlphaAU1f*mSfGradp/fvc::interpolate(rho1)
+                );
+
+                surfaceScalarField phi2s
+                (
+                    phiHbyA2 + rAlphaAU2f*mSfGradp/fvc::interpolate(rho2)
+                );
+
+                surfaceScalarField phir
+                (
+                    ((phi1s + D1f*phi2s) - (phi2s + D2f*phi1s))/(1 - D1f*D2f)
+                );
+
+                phi1.boundaryField() ==
+                    mrfZones.relative
+                    (
+                        mesh.Sf().boundaryField() & U1.boundaryField()
+                    );
+                phi1 = phi + alpha2f*phir;
+
+                phi2.boundaryField() ==
+                    mrfZones.relative
+                    (
+                        mesh.Sf().boundaryField() & U2.boundaryField()
+                    );
+                phi2 = phi - alpha1f*phir;
+            }
+
+            // Compressibility correction for phase-fraction equations
+            fluid.dgdt() =
+            (
+                alpha1*(pEqnComp2 & p_rgh)
+              - alpha2*(pEqnComp1 & p_rgh)
+            );
+
+            // Optionally relax pressure for velocity correction
+            p_rgh.relax();
+
+            // Update the static pressure
+            p = max(p_rgh + rho*gh, pMin);
+            p_rgh = p - rho*gh;
+
+            mSfGradp = pEqnIncomp.flux()/rAUf;
+
+            // Partial-elimination phase-velocity corrector
+            {
+                volVectorField U1s
+                (
+                    HbyA1
+                  + fvc::reconstruct
+                    (
+                        rAlphaAU1f*mSfGradp/fvc::interpolate(rho1)
+                      - phiF1
+                    )
+                );
+
+                volVectorField U2s
+                (
+                    HbyA2
+                  + fvc::reconstruct
+                    (
+                        rAlphaAU2f*mSfGradp/fvc::interpolate(rho2)
+                      - phiF2
+                    )
+                );
+
+                volScalarField D1(rAU1*dragCoeff);
+                volScalarField D2(rAU2*dragCoeff);
+
+                U = alpha1*(U1s + D1*U2) + alpha2*(U2s + D2*U1);
+                volVectorField Ur(((1 - D2)*U1s - (1 - D1)*U2s)/(1 - D1*D2));
+
+                U1 = U + alpha2*Ur;
+                U1.correctBoundaryConditions();
+                fvOptions.correct(U1);
+
+                U2 = U - alpha1*Ur;
+                U2.correctBoundaryConditions();
+                fvOptions.correct(U2);
+
+                U = fluid.U();
+            }
+        }
+    }
+
+    // Update densities from change in p
+    rho1 += psi1*(p_rgh - p_rgh_0);
+    rho2 += psi2*(p_rgh - p_rgh_0);
+
+    // Update the phase kinetic energies
+    K1 = 0.5*magSqr(U1);
+    K2 = 0.5*magSqr(U2);
+
+    // Update the pressure time-derivative if required
+    if (thermo1.dpdt() || thermo2.dpdt())
+    {
+        dpdt = fvc::ddt(p);
+    }
+}

applications/solvers/multiphase/twoPhaseEulerFoam/twoPhaseEulerFoam.C

@@ -90,16 +90,10 @@ int main(int argc, char *argv[])
               - (fvOptions(alpha2, rho2)&rho2)
             );
 
-
             #include "UEqns.H"
             #include "EEqns.H"
-
-            // --- Pressure corrector loop
-            while (pimple.correct())
-            {
-                #include "pEqn.H"
-            }
-
+            //#include "pEqn.H"
+            #include "pEqnElim.H"
             #include "DDtU.H"
 
             if (pimple.turbCorr())

src/finiteVolume/cfdTools/general/MRF/MRFZoneList.C

@@ -2,7 +2,7 @@
   =========                 |
   \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
    \\    /   O peration     |
-    \\  /    A nd           | Copyright (C) 2012-2014 OpenFOAM Foundation
+    \\  /    A nd           | Copyright (C) 2012-2015 OpenFOAM Foundation
      \\/     M anipulation  |
 -------------------------------------------------------------------------------
 License
@@ -218,6 +218,17 @@ void Foam::MRFZoneList::makeRelative(surfaceScalarField& phi) const
 }
 
 
+Foam::tmp<Foam::surfaceScalarField> Foam::MRFZoneList::relative
+(
+    const tmp<surfaceScalarField>& phi
+) const
+{
+    tmp<surfaceScalarField> rphi(phi.ptr());
+    makeRelative(rphi());
+    return rphi;
+}
+
+
 Foam::tmp<Foam::FieldField<Foam::fvsPatchField, Foam::scalar> >
 Foam::MRFZoneList::relative
 (
@@ -266,6 +277,17 @@ void Foam::MRFZoneList::makeAbsolute(surfaceScalarField& phi) const
 }
 
 
+Foam::tmp<Foam::surfaceScalarField> Foam::MRFZoneList::absolute
+(
+    const tmp<surfaceScalarField>& phi
+) const
+{
+    tmp<surfaceScalarField> rphi(phi.ptr());
+    makeAbsolute(rphi());
+    return rphi;
+}
+
+
 void Foam::MRFZoneList::makeAbsolute
 (
     const surfaceScalarField& rho,

src/finiteVolume/cfdTools/general/MRF/MRFZoneList.H

@@ -2,7 +2,7 @@
   =========                 |
   \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
    \\    /   O peration     |
-    \\  /    A nd           | Copyright (C) 2012-2014 OpenFOAM Foundation
+    \\  /    A nd           | Copyright (C) 2012-2015 OpenFOAM Foundation
      \\/     M anipulation  |
 -------------------------------------------------------------------------------
 License
@@ -118,12 +118,15 @@ public:
         //- Make the given absolute velocity relative within the MRF region
         void makeRelative(volVectorField& U) const;
 
-        //- Make the given relative velocity absolute within the MRF region
-        void makeAbsolute(volVectorField& U) const;
-
         //- Make the given absolute flux relative within the MRF region
         void makeRelative(surfaceScalarField& phi) const;
 
+        //- Return the given absolute flux relative within the MRF region
+        tmp<surfaceScalarField> relative
+        (
+            const tmp<surfaceScalarField>& phi
+        ) const;
+
         //- Return the given absolute boundary flux relative within
         //  the MRF region
         tmp<FieldField<fvsPatchField, scalar> > relative
@@ -138,9 +141,18 @@ public:
             surfaceScalarField& phi
         ) const;
 
+        //- Make the given relative velocity absolute within the MRF region
+        void makeAbsolute(volVectorField& U) const;
+
         //- Make the given relative flux absolute within the MRF region
         void makeAbsolute(surfaceScalarField& phi) const;
 
+        //- Return the given relative flux absolute within the MRF region
+        tmp<surfaceScalarField> absolute
+        (
+            const tmp<surfaceScalarField>& phi
+        ) const;
+
         //- Make the given relative mass-flux absolute within the MRF region
         void makeAbsolute
         (