twoPhaseEulerFoam: Added experimental face-based momentum equation formulation

This formulation provides C-grid like pressure-flux staggering on an
unstructured mesh which is hugely beneficial for Euler-Euler multiphase
equations as it allows for all forces to be treated in a consistent
manner on the cell-faces which provides better balance, stability and
accuracy.  However, to achieve face-force consistency the momentum
transport terms must be interpolated to the faces reducing accuracy of
this part of the system but this is offset by the increase in accuracy
of the force-balance.

Currently it is not clear if this face-based momentum equation
formulation is preferable for all Euler-Euler simulations so I have
included it on a switch to allow evaluation and comparison with the
previous cell-based formulation.  To try the new algorithm simply switch
it on, e.g.:

PIMPLE
{
    nOuterCorrectors 3;
    nCorrectors      1;
    nNonOrthogonalCorrectors 0;
    faceMomentum     yes;
}

It is proving particularly good for bubbly flows, eliminating the
staggering patterns often seen in the air velocity field with the
previous algorithm, removing other spurious numerical artifacts in the
velocity fields and improving stability and allowing larger time-steps
For particle-gas flows the advantage is noticeable but not nearly as
pronounced as in the bubbly flow cases.

Please test the new algorithm on your cases and provide feedback.

Henry G. Weller
CFD Direct

applications/solvers/multiphase/twoPhaseEulerFoam/twoPhaseSystem/twoPhaseSystem.H

@@ -40,13 +40,13 @@ SourceFiles
 #include "orderedPhasePair.H"
 #include "volFields.H"
 #include "surfaceFields.H"
+#include "dragModel.H"
 
 // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
 namespace Foam
 {
 
-class dragModel;
 class virtualMassModel;
 class heatTransferModel;
 class liftModel;
@@ -113,6 +113,9 @@ class twoPhaseSystem
         autoPtr<BlendedInterfacialModel<turbulentDispersionModel> >
             turbulentDispersion_;
 
+        //-
+        static dimensionedScalar zeroResidualAlpha_;
+
 
     // Private member functions
 
@@ -141,26 +144,29 @@ public:
         tmp<volVectorField> U() const;
 
         //- Return the drag coefficient
-        tmp<volScalarField> dragCoeff() const;
+        tmp<volScalarField> Kd() const;
+
+        //- Return the face drag coefficient
+        tmp<surfaceScalarField> Kdf() const;
 
         //- Return the virtual mass coefficient
-        tmp<volScalarField> virtualMassCoeff() const;
+        tmp<volScalarField> Vm() const;
+
+        //- Return the face virtual mass coefficient
+        tmp<surfaceScalarField> Vmf() const;
 
         //- Return the heat transfer coefficient
-        tmp<volScalarField> heatTransferCoeff() const;
+        tmp<volScalarField> Kh() const;
 
-        //- Return the lift force
-        tmp<volVectorField> liftForce() const;
+        //- Return the combined force (lift + wall-lubrication)
+        tmp<volVectorField> F() const;
 
-        //- Return the wall lubrication force
-        tmp<volVectorField> wallLubricationForce() const;
+        //- Return the combined face-force (lift + wall-lubrication)
+        tmp<surfaceScalarField> Ff() const;
 
         //- Return the turbulent diffusivity
         //  Multiplies the phase-fraction gradient
-        tmp<volScalarField> turbulentDiffusivity() const;
-
-        //- Return the turbulent dispersion force
-        tmp<volVectorField> turbulentDispersionForce() const;
+        tmp<volScalarField> D() const;
 
         //- Solve for the two-phase-fractions
         void solve();
@@ -176,10 +182,17 @@ public:
 
         // Access
 
-            //- Return the drag model for the supplied phase
+            //- Return the residual phase-fraction for given phase
+            //  Used to stabilize the phase momentum as the phase-fraction -> 0
+            const dimensionedScalar& residualAlpha
+            (
+                const phaseModel& phase
+            ) const;
+
+            //- Return the drag model for the given phase
             const dragModel& drag(const phaseModel& phase) const;
 
-            //- Return the virtual mass model for the supplied phase
+            //- Return the virtual mass model for the given phase
             const virtualMassModel& virtualMass(const phaseModel& phase) const;
 
             //- Return the surface tension coefficient