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Old   December 8, 2017, 14:14
Default EulerD2dt2Scheme
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Lucas Ribeiro
Join Date: Aug 2016
Posts: 23
Rep Power: 8
Warlord is on a distinguished road
Hello folks,

Does anyone know how to derive the formulas implemented for the EulerD2dt2Scheme::fvmD2dt2? Specially for the first one, applied when the mesh is moving. Any reference?

template<class Type>
tmp<fvMatrix<Type> >
    const dimensionedScalar& rho,
    GeometricField<Type, fvPatchField, volMesh>& vf
    tmp<fvMatrix<Type> > tfvm
        new fvMatrix<Type>

    fvMatrix<Type>& fvm = tfvm();

    scalar deltaT = mesh().time().deltaT().value();
    scalar deltaT0 = mesh().time().deltaT0().value();

    scalar coefft   = (deltaT + deltaT0)/(2*deltaT);
    scalar coefft00 = (deltaT + deltaT0)/(2*deltaT0);

    scalar rDeltaT2 = 4.0/sqr(deltaT + deltaT0);

    if (mesh().moving())
        scalar halfRdeltaT2 = 0.5*rDeltaT2;

        scalarField VV0 = mesh().V() + mesh().V0();

        scalarField V0V00 = mesh().V0() + mesh().V00();

        fvm.diag() = rho.value()*(coefft*halfRdeltaT2)*VV0;

        fvm.source() = halfRdeltaT2*rho.value()*
            (coefft*VV0 + coefft00*V0V00)

          - (coefft00*V0V00)*vf.oldTime().oldTime().internalField()
        fvm.diag() = (coefft*rDeltaT2)*mesh().V()*rho.value();

        fvm.source() = rDeltaT2*mesh().V()*rho.value()*
            (coefft + coefft00)*vf.oldTime().internalField()
          - coefft00*vf.oldTime().oldTime().internalField()

    return tfvm;
Thank you
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euler, foam-extend-3.2, fvmd2dt2

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