interFoam family: Add support for MULES-bounded Crank-Nicolson 2nd-order ddt(alpha)

This is an experimental feature demonstrating the potential of MULES to
create bounded solution which are 2nd-order in time AND space.

Crank-Nicolson may be selected on U and/or alpha but will only be fully
2nd-order if used on both within the PIMPLE-loop to converge the
interaction between the flux and phase-fraction.  Note also that
Crank-Nicolson may not be used with sub-cycling but all the features of
semi-implicit MULES are available in particular MULESCorr and
alphaApplyPrevCorr.

Examples of ddt specification:

ddtSchemes
{
    default         Euler;
}

ddtSchemes
{
    default         CrankNicolson 0.9;
}

ddtSchemes
{
    default         none;
    ddt(alpha)      CrankNicolson 0.9;
    ddt(rho,U)      CrankNicolson 0.9;
}

ddtSchemes
{
    default         none;
    ddt(alpha)      Euler;
    ddt(rho,U)      CrankNicolson 0.9;
}

ddtSchemes
{
    default         none;
    ddt(alpha)      CrankNicolson 0.9;
    ddt(rho,U)      Euler;
}

In these examples a small amount of off-centering in used to stabilize
the Crank-Nicolson scheme.  Also the specification for alpha1 is via the
generic phase-fraction name to ensure in multiphase solvers (when
Crank-Nicolson support is added) the scheme is identical for all phase
fractions.

applications/solvers/multiphase/interFoam/LTSInterFoam/LTSInterFoam.C

@@ -39,6 +39,8 @@ Description
 
 #include "fvCFD.H"
 #include "CMULES.H"
+#include "EulerDdtScheme.H"
+#include "CrankNicolsonDdtScheme.H"
 #include "subCycle.H"
 #include "immiscibleIncompressibleTwoPhaseMixture.H"
 #include "turbulentTransportModel.H"

applications/solvers/multiphase/interFoam/alphaEqn.H

@@ -2,6 +2,43 @@
     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()))
+    {
+        ocCoeff = 0;
+    }
+    else if (isType<fv::CrankNicolsonDdtScheme<scalar> >(ddtAlpha()))
+    {
+        if (nAlphaSubCycles > 1)
+        {
+            FatalErrorIn(args.executable())
+                << "Sub-cycling is not supported "
+                   "with the CrankNicolson ddt scheme"
+                << exit(FatalError);
+        }
+
+        ocCoeff =
+            refCast<fv::CrankNicolsonDdtScheme<scalar> >(ddtAlpha()).ocCoeff();
+    }
+    else
+    {
+        FatalErrorIn(args.executable())
+            << "Only Euler and CrankNicolson ddt schemes are supported"
+            << exit(FatalError);
+    }
+
+    scalar cnCoeff = ocCoeff/2.0;
+
     // Standard face-flux compression coefficient
     surfaceScalarField phic(mixture.cAlpha()*mag(phi/mesh.magSf()));
 
@@ -24,10 +61,16 @@
         }
     }
 
-    tmp<surfaceScalarField> tphiAlpha;
-
     if (MULESCorr)
     {
+        tmp<surfaceScalarField> phiCN(phi);
+
+        // Calculate the Crank-Nicolson off-centred volumetric flux
+        if (ocCoeff > 0)
+        {
+            phiCN = (1.0 - cnCoeff)*phi + cnCoeff*phi.oldTime();
+        }
+
         fvScalarMatrix alpha1Eqn
         (
             #ifdef LTSSOLVE
@@ -38,9 +81,9 @@
           + fv::gaussConvectionScheme<scalar>
             (
                 mesh,
-                phi,
-                upwind<scalar>(mesh, phi)
-            ).fvmDiv(phi, alpha1)
+                phiCN,
+                upwind<scalar>(mesh, phiCN)
+            ).fvmDiv(phiCN, alpha1)
         );
 
         alpha1Eqn.solve();
@@ -52,21 +95,18 @@
             << endl;
 
         tmp<surfaceScalarField> tphiAlphaUD(alpha1Eqn.flux());
-        tphiAlpha = tmp<surfaceScalarField>
-        (
-            new surfaceScalarField(tphiAlphaUD())
-        );
+        phiAlpha = tphiAlphaUD();
 
         if (alphaApplyPrevCorr && tphiAlphaCorr0.valid())
         {
             Info<< "Applying the previous iteration compression flux" << endl;
             #ifdef LTSSOLVE
-            MULES::LTScorrect(alpha1, tphiAlpha(), tphiAlphaCorr0(), 1, 0);
+            MULES::LTScorrect(alpha1, phiAlpha, tphiAlphaCorr0(), 1, 0);
             #else
-            MULES::correct(alpha1, tphiAlpha(), tphiAlphaCorr0(), 1, 0);
+            MULES::correct(alpha1, phiAlpha, tphiAlphaCorr0(), 1, 0);
             #endif
 
-            tphiAlpha() += tphiAlphaCorr0();
+            phiAlpha += tphiAlphaCorr0();
         }
 
         // Cache the upwind-flux
@@ -77,6 +117,7 @@
         mixture.correct();
     }
 
+
     for (int aCorr=0; aCorr<nAlphaCorr; aCorr++)
     {
         surfaceScalarField phir(phic*mixture.nHatf());
@@ -97,9 +138,17 @@
             )
         );
 
+        // Calculate the Crank-Nicolson off-centred alpha flux
+        if (ocCoeff > 0)
+        {
+            tphiAlphaUn =
+                (1.0 - cnCoeff)*tphiAlphaUn
+              + cnCoeff*phiAlpha.oldTime();
+        }
+
         if (MULESCorr)
         {
-            tmp<surfaceScalarField> tphiAlphaCorr(tphiAlphaUn() - tphiAlpha());
+            tmp<surfaceScalarField> tphiAlphaCorr(tphiAlphaUn() - phiAlpha);
             volScalarField alpha10("alpha10", alpha1);
 
             #ifdef LTSSOLVE
@@ -111,22 +160,22 @@
             // Under-relax the correction for all but the 1st corrector
             if (aCorr == 0)
             {
-                tphiAlpha() += tphiAlphaCorr();
+                phiAlpha += tphiAlphaCorr();
             }
             else
             {
                 alpha1 = 0.5*alpha1 + 0.5*alpha10;
-                tphiAlpha() += 0.5*tphiAlphaCorr();
+                phiAlpha += 0.5*tphiAlphaCorr();
             }
         }
         else
         {
-            tphiAlpha = tphiAlphaUn;
+            phiAlpha = tphiAlphaUn;
 
             #ifdef LTSSOLVE
-            MULES::explicitLTSSolve(alpha1, phi, tphiAlpha(), 1, 0);
+            MULES::explicitLTSSolve(alpha1, phi, phiAlpha, 1, 0);
             #else
-            MULES::explicitSolve(alpha1, phi, tphiAlpha(), 1, 0);
+            MULES::explicitSolve(alpha1, phi, phiAlpha, 1, 0);
             #endif
         }
 
@@ -135,11 +184,47 @@
         mixture.correct();
     }
 
-    rhoPhi = tphiAlpha()*(rho1 - rho2) + phi*rho2;
-
     if (alphaApplyPrevCorr && MULESCorr)
     {
-        tphiAlphaCorr0 = tphiAlpha() - tphiAlphaCorr0;
+        tphiAlphaCorr0 = phiAlpha - tphiAlphaCorr0;
+    }
+
+    if
+    (
+        word(mesh.ddtScheme("ddt(rho,U)"))
+     == fv::EulerDdtScheme<vector>::typeName
+    )
+    {
+        if (ocCoeff > 0)
+        {
+            // Calculate the Crank-Nicolson off-centred volumetric flux
+            surfaceScalarField phiCN
+            (
+                "phiCN",
+                (1.0 - cnCoeff)*phi + cnCoeff*phi.oldTime()
+            );
+
+            Info<< "Calculate the Crank-Nicolson off-centred mass flux" << endl;
+            // Calculate the Crank-Nicolson off-centred mass flux
+            rhoPhi = phiAlpha*(rho1 - rho2) + phiCN*rho2;
+        }
+        else
+        {
+            // Calculate the end-of-time-step mass flux
+            rhoPhi = phiAlpha*(rho1 - rho2) + phi*rho2;
+        }
+    }
+    else
+    {
+        if (ocCoeff > 0)
+        {
+            Info<< "Calculate the end-of-time-step alpha flux" << endl;
+            // Calculate the end-of-time-step alpha flux
+            phiAlpha = (phiAlpha - cnCoeff*phiAlpha.oldTime())/(1.0 - cnCoeff);
+        }
+
+        // Calculate the end-of-time-step mass flux
+        rhoPhi = phiAlpha*(rho1 - rho2) + phi*rho2;
     }
 
     Info<< "Phase-1 volume fraction = "

applications/solvers/multiphase/interFoam/createFields.H

@@ -78,16 +78,6 @@
 
     #include "readGravitationalAcceleration.H"
 
-    /*
-    dimensionedVector g0(g);
-
-    // Read the data file and initialise the interpolation table
-    interpolationTable<vector> timeSeriesAcceleration
-    (
-        runTime.path()/runTime.caseConstant()/"acceleration.dat"
-    );
-    */
-
     Info<< "Calculating field g.h\n" << endl;
     volScalarField gh("gh", g & mesh.C());
     surfaceScalarField ghf("ghf", g & mesh.Cf());
@@ -127,6 +117,19 @@
         p_rgh = p - rho*gh;
     }
 
+    // MULES flux from previous time-step
+    surfaceScalarField phiAlpha
+    (
+        IOobject
+        (
+            "phiAlpha",
+            runTime.timeName(),
+            mesh,
+            IOobject::READ_IF_PRESENT,
+            IOobject::AUTO_WRITE
+        ),
+        phi*fvc::interpolate(alpha1)
+    );
 
     // MULES Correction
     tmp<surfaceScalarField> tphiAlphaCorr0;

applications/solvers/multiphase/interFoam/interDyMFoam/interDyMFoam.C

@@ -35,6 +35,8 @@ Description
 #include "fvCFD.H"
 #include "dynamicFvMesh.H"
 #include "CMULES.H"
+#include "EulerDdtScheme.H"
+#include "CrankNicolsonDdtScheme.H"
 #include "subCycle.H"
 #include "immiscibleIncompressibleTwoPhaseMixture.H"
 #include "turbulentTransportModel.H"

applications/solvers/multiphase/interFoam/interFoam.C

@@ -39,6 +39,8 @@ Description
 
 #include "fvCFD.H"
 #include "CMULES.H"
+#include "EulerDdtScheme.H"
+#include "CrankNicolsonDdtScheme.H"
 #include "subCycle.H"
 #include "immiscibleIncompressibleTwoPhaseMixture.H"
 #include "turbulentTransportModel.H"