alphaEqn.H

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{
    word alphaScheme("div(phi,alpha)");
    word alpharScheme("div(phirb,alpha)");

    tmp<fv::ddtScheme<scalar>> ddtAlpha
    (
        fv::ddtScheme<scalar>::New
        (
            mesh,
            mesh.ddtScheme("ddt(alpha)")
        )
    );

    // Set the off-centering coefficient according to ddt scheme
    scalar ocCoeff = 0;
    if
    (
        isType<fv::EulerDdtScheme<scalar>>(ddtAlpha())
     || isType<fv::localEulerDdtScheme<scalar>>(ddtAlpha())
    )
    {
        ocCoeff = 0;
    }
    else if (isType<fv::CrankNicolsonDdtScheme<scalar>>(ddtAlpha()))
    {
        if (nAlphaSubCycles > 1)
        {
            FatalErrorInFunction
                << "Sub-cycling is not supported "
                   "with the CrankNicolson ddt scheme"
                << exit(FatalError);
        }

        ocCoeff =
            refCast<const fv::CrankNicolsonDdtScheme<scalar>>(ddtAlpha())
           .ocCoeff();
    }
    else
    {
        FatalErrorInFunction
            << "Only Euler and CrankNicolson ddt schemes are supported"
            << exit(FatalError);
    }

    scalar cnCoeff = 1.0/(1.0 + ocCoeff);

    // Standard face-flux compression coefficient
    surfaceScalarField phic(mixture.cAlpha()*mag(phi/mesh.magSf()));

    // Add the optional isotropic compression contribution
    if (icAlpha > 0)
    {
        phic *= (1.0 - icAlpha);
        phic += (mixture.cAlpha()*icAlpha)*fvc::interpolate(mag(U));
    }

    surfaceScalarField::Boundary& phicBf =
        phic.boundaryFieldRef();

    // Do not compress interface at non-coupled boundary faces
    // (inlets, outlets etc.)
    forAll(phic.boundaryField(), patchi)
    {
        fvsPatchScalarField& phicp = phicBf[patchi];

        if (!phicp.coupled())
        {
            phicp == 0;
        }
    }

    tmp<surfaceScalarField> phiCN(phi);

    // Calculate the Crank-Nicolson off-centred volumetric flux
    if (ocCoeff > 0)
    {
        phiCN = cnCoeff*phi + (1.0 - cnCoeff)*phi.oldTime();
    }

    if (MULESCorr)
    {
        fvScalarMatrix alpha1Eqn
        (
            (
                LTS
              ? fv::localEulerDdtScheme<scalar>(mesh).fvmDdt(alpha1)
              : fv::EulerDdtScheme<scalar>(mesh).fvmDdt(alpha1)
            )
          + fv::gaussConvectionScheme<scalar>
            (
                mesh,
                phiCN,
                upwind<scalar>(mesh, phiCN)
            ).fvmDiv(phiCN, alpha1)
        );

        alpha1Eqn.solve();

        Info<< "Phase-1 volume fraction = "
            << alpha1.weightedAverage(mesh.Vsc()).value()
            << "  Min(" << alpha1.name() << ") = " << min(alpha1).value()
            << "  Max(" << alpha1.name() << ") = " << max(alpha1).value()
            << endl;

        tmp<surfaceScalarField> talphaPhiUD(alpha1Eqn.flux());
        alphaPhi = talphaPhiUD();

        if (alphaApplyPrevCorr && talphaPhiCorr0.valid())
        {
            Info<< "Applying the previous iteration compression flux" << endl;
            MULES::correct(alpha1, alphaPhi, talphaPhiCorr0.ref(), 1, 0);

            alphaPhi += talphaPhiCorr0();
        }

        // Cache the upwind-flux
        talphaPhiCorr0 = talphaPhiUD;

        alpha2 = 1.0 - alpha1;

        mixture.correct();
    }


    for (int aCorr=0; aCorr<nAlphaCorr; aCorr++)
    {
        surfaceScalarField phir(phic*mixture.nHatf());

        tmp<surfaceScalarField> talphaPhiUn
        (
            fvc::flux
            (
                phi,
                alpha1,
                alphaScheme
            )
          + fvc::flux
            (
               -fvc::flux(-phir, alpha2, alpharScheme),
                alpha1,
                alpharScheme
            )
        );

        // Calculate the Crank-Nicolson off-centred alpha flux
        if (ocCoeff > 0)
        {
            talphaPhiUn =
                cnCoeff*talphaPhiUn + (1.0 - cnCoeff)*alphaPhi.oldTime();
        }

        if (MULESCorr)
        {
            tmp<surfaceScalarField> talphaPhiCorr(talphaPhiUn() - alphaPhi);
            volScalarField alpha10("alpha10", alpha1);

            MULES::correct(alpha1, talphaPhiUn(), talphaPhiCorr.ref(), 1, 0);

            // Under-relax the correction for all but the 1st corrector
            if (aCorr == 0)
            {
                alphaPhi += talphaPhiCorr();
            }
            else
            {
                alpha1 = 0.5*alpha1 + 0.5*alpha10;
                alphaPhi += 0.5*talphaPhiCorr();
            }
        }
        else
        {
            alphaPhi = talphaPhiUn;

            MULES::explicitSolve(alpha1, phiCN, alphaPhi, 1, 0);
        }

        alpha2 = 1.0 - alpha1;

        mixture.correct();
    }

    if (alphaApplyPrevCorr && MULESCorr)
    {
        talphaPhiCorr0 = alphaPhi - talphaPhiCorr0;
    }

    if
    (
        word(mesh.ddtScheme("ddt(rho,U)"))
     == fv::EulerDdtScheme<vector>::typeName
    )
    {
        rhoPhi = alphaPhi*(rho1 - rho2) + phiCN*rho2;
    }
    else
    {
        if (ocCoeff > 0)
        {
            // Calculate the end-of-time-step alpha flux
            alphaPhi = (alphaPhi - (1.0 - cnCoeff)*alphaPhi.oldTime())/cnCoeff;
        }

        // Calculate the end-of-time-step mass flux
        rhoPhi = alphaPhi*(rho1 - rho2) + phi*rho2;
    }

    Info<< "Phase-1 volume fraction = "
        << alpha1.weightedAverage(mesh.Vsc()).value()
        << "  Min(" << alpha1.name() << ") = " << min(alpha1).value()
        << "  Max(" << alpha1.name() << ") = " << max(alpha1).value()
        << endl;
}