[nimasam]

hi foamer
i have a term like this in my equations

del(K del (T))

K is not constant!

which one of following openFOAM expression is correct for above term?

1) fvm :: laplacian ( K,T)

2)fvm::laplacian(K , T) + (fvc::grad(T) & fvc::grad(K))

[Cyp]

Hi Nimasam !

It is the first expression which is correct otherwise it would be :

K*fvm::laplacian(T) + (fvc::grad(T) & fvc::grad(K))

because you have the following relationship:

fvm::laplacian(K,T) = K*fvm::laplacian(T) + (fvc::grad(T) & fvc::grad(K))

In the programmer guide P-38 you can notice that the laplacian term is coded with the diffusion term which allow you to use a variable diffusion.

Best,
Cyp

[nimasam]

are you sure following equation sides are equal?
fvm::laplacian(K,T) = K*fvm::laplacian(T) + (fvc::grad(T) & fvc::grad(K))
i used both of them in code and i received different answers!

[santiagomarquezd]

I think it depends on the kind of operation you want to, namely:

1. Implicit (fvm)
2. Explicit (fvc)
3. Mixed treatment

from Programmer's Guide Table 2.2 we have for laplacian:

Laplacian Implicit/Explicit \nabla^2\phi laplacian(phi) (1)
\nabla\cdot\Gamma\nabla\phi laplacian(Gamma, phi) (2)

then both in implicit and explicit treatment we have the operator with and without Gamma within the divergence. In the first case (1) constant gamma is assumed, in (2) laplacian(Gamma, phi) expands in the div-grad formulation intended for variable Gamma. So we have four cases,

a. Explicit, variable Gamma
b. Explicit, constant Gamma (no gamma within the operator)
c. Implicit, variable Gamma
d. Implicit, constant Gamma (no gamma within the operator)

a and b give geometricField as a result and c and d fvMatrix (implying an integration). Reading the code we have in gaussLaplacianScheme.C, for a, b, c

00186 template<class Type, class GType>
00187 tmp<GeometricField<Type, fvPatchField, volMesh> >
00188 gaussLaplacianScheme<Type, GType>::fvcLaplacian
00189 (
00190     const GeometricField<GType, fvsPatchField, surfaceMesh>& gamma,
00191     const GeometricField<Type, fvPatchField, volMesh>& vf
00192 )
00193 {
00194     const fvMesh& mesh = this->mesh();
00195 
00196     surfaceVectorField Sn = mesh.Sf()/mesh.magSf();
00197 
00198     surfaceVectorField SfGamma = mesh.Sf() & gamma;
00199     GeometricField<scalar, fvsPatchField, surfaceMesh> SfGammaSn = SfGamma & Sn;
00200     surfaceVectorField SfGammaCorr = SfGamma - SfGammaSn*Sn;
00201 
00202     tmp<GeometricField<Type, fvPatchField, volMesh> > tLaplacian
00203     (
00204         fvc::div
00205         (
00206             SfGammaSn*this->tsnGradScheme_().snGrad(vf)
00207           + gammaSnGradCorr(SfGammaCorr, vf)
00208         )
00209     );
00210 
00211     tLaplacian().rename("laplacian(" + gamma.name() + ',' + vf.name() + ')');
00212 
00213     return tLaplacian;
00214 }

00127 template<class Type, class GType>
00128 tmp<GeometricField<Type, fvPatchField, volMesh> >
00129 gaussLaplacianScheme<Type, GType>::fvcLaplacian
00130 (
00131     const GeometricField<Type, fvPatchField, volMesh>& vf
00132 )
00133 {
00134     const fvMesh& mesh = this->mesh();
00135 
00136     tmp<GeometricField<Type, fvPatchField, volMesh> > tLaplacian
00137     (
00138         fvc::div(this->tsnGradScheme_().snGrad(vf)*mesh.magSf())
00139     );
00140 
00141     tLaplacian().rename("laplacian(" + vf.name() + ')');
00142 
00143     return tLaplacian;
00144 }

00147 template<class Type, class GType>
00148 tmp<fvMatrix<Type> >
00149 gaussLaplacianScheme<Type, GType>::fvmLaplacian
00150 (
00151     const GeometricField<GType, fvsPatchField, surfaceMesh>& gamma,
00152     GeometricField<Type, fvPatchField, volMesh>& vf
00153 )
00154 {
00155     const fvMesh& mesh = this->mesh();
00156 
00157     surfaceVectorField Sn = mesh.Sf()/mesh.magSf();
00158 
00159     surfaceVectorField SfGamma = mesh.Sf() & gamma;
00160     GeometricField<scalar, fvsPatchField, surfaceMesh> SfGammaSn = SfGamma & Sn;
00161     surfaceVectorField SfGammaCorr = SfGamma - SfGammaSn*Sn;
00162 
00163     tmp<fvMatrix<Type> > tfvm = fvmLaplacianUncorrected(SfGammaSn, vf);
00164     fvMatrix<Type>& fvm = tfvm();
00165 
00166     tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > tfaceFluxCorrection
00167         = gammaSnGradCorr(SfGammaCorr, vf);
00168 
00169     if (this->tsnGradScheme_().corrected())
00170     {
00171         tfaceFluxCorrection() +=
00172             SfGammaSn*this->tsnGradScheme_().correction(vf);
00173     }
00174 
00175     fvm.source() -= mesh.V()*fvc::div(tfaceFluxCorrection())().internalField();
00176 
00177     if (mesh.fluxRequired(vf.name()))
00178     {
00179         fvm.faceFluxCorrectionPtr() = tfaceFluxCorrection.ptr();
00180     }
00181 
00182     return tfvm;
00183 }

and for d, from fvmLaplacian.C

00069 template<class Type>
00070 tmp<fvMatrix<Type> >
00071 laplacian
00072 (
00073     GeometricField<Type, fvPatchField, volMesh>& vf
00074 )
00075 {
00076     surfaceScalarField Gamma
00077     (
00078         IOobject
00079         (
00080             "1",
00081             vf.time().constant(),
00082             vf.mesh(),
00083             IOobject::NO_READ
00084         ),
00085         vf.mesh(),
00086         dimensionedScalar("1", dimless, 1.0)
00087     );
00088 
00089     return fvm::laplacian
00090     (
00091         Gamma,
00092         vf,
00093         "laplacian(" + vf.name() + ')'
00094     );
00095 }

from the last it is clear that d=c, with Gamma=1. Finally the correct implicit formulation in #1 is

1) fvm::laplacian ( K,T)

formulation given #3

fvm::laplacian(K,T) = K*fvm::laplacian(T) + (fvc::grad(T) & fvc::grad(K))

is a mixed one with the spatial variation of T treated implicitly and the spatial variation of K explicitly, the fvc part is put in the RHS, meanwhile fvm part contributes to the matrix, so that it is natural to hope different results (I think).

Regards.

[nimasam]

hi Santiago
thank you, for detailed response , now i find why these expressions give different numerical result
now tell me which one of them i should choose for energy equation in two phase flow?
when implicit manner is suitable and when explicit manner is useful?

[santiagomarquezd]

I think it is not very different to what is done in scalarTransportFoam solver. That includes variable scalar K (if it is tensorial things change a little bit).

Regards.

[nimasam]

now i add the energy equation into interFoam i even consider evaporation, im just in doubt which one of the above expression is suitable for conduction term?

[santiagomarquezd]

The first one.

Regards.