DASpalartAllmarasFv3.C

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/*---------------------------------------------------------------------------*\
 
    DAFoam  : Discrete Adjoint with OpenFOAM
    Version : v3
 
    This file is modified from OpenFOAM's source code
    src/TurbulenceModels/turbulenceModels/RAS/SpalartAllmaras/SpalartAllmaras.C
 
    OpenFOAM: The Open Source CFD Toolbox
 
    Copyright (C): 2011-2016 OpenFOAM Foundation
 
    OpenFOAM License:
 
        OpenFOAM is free software: you can redistribute it and/or modify it
        under the terms of the GNU General Public License as published by
        the Free Software Foundation, either version 3 of the License, or
        (at your option) any later version.
    
        OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
        ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
        FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
        for more details.
    
        You should have received a copy of the GNU General Public License
        along with OpenFOAM.  If not, see <http://www.gnu.org/licenses/>.
 
\*---------------------------------------------------------------------------*/
 
#include "DASpalartAllmarasFv3.H"
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
namespace Foam
{
 
defineTypeNameAndDebug(DASpalartAllmarasFv3, 0);
addToRunTimeSelectionTable(DATurbulenceModel, DASpalartAllmarasFv3, dictionary);
// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //
 
DASpalartAllmarasFv3::DASpalartAllmarasFv3(
    const word modelType,
    const fvMesh& mesh,
    const DAOption& daOption)
    : DATurbulenceModel(modelType, mesh, daOption),
      // SA parameters
      sigmaNut_(dimensioned<scalar>::lookupOrAddToDict(
          "sigmaNut",
          this->coeffDict_,
          0.66666)),
      kappa_(dimensioned<scalar>::lookupOrAddToDict(
          "kappa",
          this->coeffDict_,
          0.41)),
 
      Cb1_(dimensioned<scalar>::lookupOrAddToDict(
          "Cb1",
          this->coeffDict_,
          0.1355)),
      Cb2_(dimensioned<scalar>::lookupOrAddToDict(
          "Cb2",
          this->coeffDict_,
          0.622)),
      Cw1_(Cb1_ / sqr(kappa_) + (1.0 + Cb2_) / sigmaNut_),
      Cw2_(dimensioned<scalar>::lookupOrAddToDict(
          "Cw2",
          this->coeffDict_,
          0.3)),
      Cw3_(dimensioned<scalar>::lookupOrAddToDict(
          "Cw3",
          this->coeffDict_,
          2.0)),
      Cv1_(dimensioned<scalar>::lookupOrAddToDict(
          "Cv1",
          this->coeffDict_,
          7.1)),
      Cv2_(dimensioned<scalar>::lookupOrAddToDict(
          "Cv2",
          this->coeffDict_,
          5.0)),
      // Augmented variables
      nuTilda_(const_cast<volScalarField&>(
          mesh.thisDb().lookupObject<volScalarField>("nuTilda"))),
      nuTildaRes_(
          IOobject(
              "nuTildaRes",
              mesh.time().timeName(),
              mesh,
              IOobject::NO_READ,
              IOobject::NO_WRITE),
          mesh,
#ifdef CompressibleFlow
          dimensionedScalar("nuTildaRes", dimensionSet(1, -1, -2, 0, 0, 0, 0), 0.0),
#endif
#ifdef IncompressibleFlow
          dimensionedScalar("nuTildaRes", dimensionSet(0, 2, -2, 0, 0, 0, 0), 0.0),
#endif
          zeroGradientFvPatchField<scalar>::typeName),
      // pseudoNuTilda_ and pseudoNuTildaEqn_ for solving adjoint equation
      pseudoNuTilda_(
          IOobject(
              "pseudoNuTilda",
              mesh.time().timeName(),
              mesh,
              IOobject::NO_READ,
              IOobject::NO_WRITE),
          nuTilda_),
      pseudoNuTildaEqn_(fvm::div(phi_, pseudoNuTilda_, "div(phi,nuTilda)")),
      y_(mesh.thisDb().lookupObject<volScalarField>("yWall"))
{
 
    // initialize printInterval_ we need to check whether it is a steady state
    // or unsteady primal solver
    IOdictionary fvSchemes(
        IOobject(
            "fvSchemes",
            mesh.time().system(),
            mesh,
            IOobject::MUST_READ,
            IOobject::NO_WRITE,
            false));
    word ddtScheme = word(fvSchemes.subDict("ddtSchemes").lookup("default"));
    if (ddtScheme == "steadyState")
    {
        printInterval_ =
            daOption.getAllOptions().lookupOrDefault<label>("printInterval", 100);
    }
    else
    {
        printInterval_ =
            daOption.getAllOptions().lookupOrDefault<label>("printIntervalUnsteady", 500);
    }
 
    // get fvSolution and fvSchemes info for fixed-point adjoint
    const fvSolution& myFvSolution = mesh.thisDb().lookupObject<fvSolution>("fvSolution");
    solverDictNuTilda_ = myFvSolution.subDict("solvers").subDict("nuTilda");
    if (myFvSolution.found("relaxationFactors"))
    {
        if (myFvSolution.subDict("relaxationFactors").found("equations"))
        {
            if (myFvSolution.subDict("relaxationFactors").subDict("equations").found("nuTilda"))
            {
                relaxNuTildaEqn_ = myFvSolution.subDict("relaxationFactors").subDict("equations").getScalar("nuTilda");
            }
        }
    }
}
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
// SA member functions. these functions are copied from
tmp<volScalarField> DASpalartAllmarasFv3::chi() const
{
    return nuTilda_ / this->nu();
}
 
tmp<volScalarField> DASpalartAllmarasFv3::fv1(
    const volScalarField& chi) const
{
    const volScalarField chi3(pow3(chi));
    return chi3 / (chi3 + pow3(Cv1_));
}
 
tmp<volScalarField> DASpalartAllmarasFv3::fv2(
    const volScalarField& chi,
    const volScalarField& fv1) const
{
    return 1.0 / pow3(scalar(1) + chi / Cv2_);
}
 
tmp<volScalarField> DASpalartAllmarasFv3::fv3(
    const volScalarField& chi,
    const volScalarField& fv1) const
{
 
    const volScalarField chiByCv2((1 / Cv2_) * chi);
 
    return (scalar(1) + chi * fv1)
        * (1 / Cv2_)
        * (3 * (scalar(1) + chiByCv2) + sqr(chiByCv2))
        / pow3(scalar(1) + chiByCv2);
}
 
tmp<volScalarField> DASpalartAllmarasFv3::fw(
    const volScalarField& Stilda) const
{
    volScalarField r(
        min(
            nuTilda_
                / (max(
                       Stilda,
                       dimensionedScalar("SMALL", Stilda.dimensions(), SMALL))
                   * sqr(kappa_ * y_)),
            scalar(10.0)));
    r.boundaryFieldRef() == 0.0;
 
    const volScalarField g(r + Cw2_ * (pow6(r) - r));
 
    return g * pow((1.0 + pow6(Cw3_)) / (pow6(g) + pow6(Cw3_)), 1.0 / 6.0);
}
 
tmp<volScalarField> DASpalartAllmarasFv3::DnuTildaEff() const
{
    return tmp<volScalarField>(
        new volScalarField("DnuTildaEff", (nuTilda_ + this->nu()) / sigmaNut_));
}
 
// Augmented functions
void DASpalartAllmarasFv3::correctModelStates(wordList& modelStates) const
{
    /*
    Description:
        Update the name in modelStates based on the selected physical model at runtime
 
    Example:
        In DAStateInfo, if the modelStates reads:
        
        modelStates = {"nut"}
        
        then for the SA model, calling correctModelStates(modelStates) will give:
    
        modelStates={"nuTilda"}
        
        while calling correctModelStates(modelStates) for the SST model will give 
        
        modelStates={"k","omega"}
        
        We don't udpate the names for the radiation model becasue users are 
        supposed to set modelStates={"G"}
    */
 
    // replace nut with nuTilda
    forAll(modelStates, idxI)
    {
        word stateName = modelStates[idxI];
        if (stateName == "nut")
        {
            modelStates[idxI] = "nuTilda";
        }
    }
}
 
void DASpalartAllmarasFv3::correctNut()
{
    /*
    Description:
        Update nut based on other turbulence variables and update the BCs
        Also update alphat if is present
    */
 
    const volScalarField chi(this->chi());
    const volScalarField fv1(this->fv1(chi));
    nut_ = nuTilda_ * fv1;
 
    nut_.correctBoundaryConditions();
 
    // this is basically BasicTurbulenceModel::correctNut();
    this->correctAlphat();
 
    return;
}
 
void DASpalartAllmarasFv3::correctBoundaryConditions()
{
    /*
    Description:
        Update turbulence variable boundary values
    */
 
    // correct the BCs for the perturbed fields
    nuTilda_.correctBoundaryConditions();
}
 
void DASpalartAllmarasFv3::updateIntermediateVariables()
{
    /*
    Description:
        Update nut based on nuTilda. Note: we need to update nut and its BC since we 
        may have perturbed other turbulence vars that affect the nut values
    */
 
    this->correctNut();
}
 
void DASpalartAllmarasFv3::correctStateResidualModelCon(List<List<word>>& stateCon) const
{
    /*
    Description:
        Update the original variable connectivity for the adjoint state 
        residuals in stateCon. Basically, we modify/add state variables based on the
        original model variables defined in stateCon.
 
    Input:
    
        stateResCon: the connectivity levels for a state residual, defined in Foam::DAJacCon
 
    Example:
        If stateCon reads:
        stateCon=
        {
            {"U", "p", "nut"},
            {"p"}
        }
    
        For the SA turbulence model, calling this function for will get a new stateCon
        stateCon=
        {
            {"U", "p", "nuTilda"},
            {"p"}
        }
    
        For the SST turbulence model, calling this function will give
        stateCon=
        {
            {"U", "p", "k", "omega"},
            {"p", "U"}
        }
        ***NOTE***: we add a extra level of U connectivity because nut is 
        related to grad(U), k, and omega in SST!
    */
 
    forAll(stateCon, idxI)
    {
        forAll(stateCon[idxI], idxJ)
        {
            word conStateName = stateCon[idxI][idxJ];
            if (conStateName == "nut")
            {
                stateCon[idxI][idxJ] = "nuTilda";
            }
        }
    }
}
 
void DASpalartAllmarasFv3::addModelResidualCon(HashTable<List<List<word>>>& allCon) const
{
    /*
    Description:
        Add the connectivity levels for all physical model residuals to allCon
 
    Input:
        allCon: the connectivity levels for all state residual, defined in DAJacCon
 
    Example:
        If stateCon reads:
        allCon=
        {
            "URes":
            {
               {"U", "p", "nut"},
               {"p"}
            }
        }
    
        For the SA turbulence model, calling this function for will get a new stateCon,
        something like this:
        allCon=
        {
            "URes":
            {
               {"U", "p", "nuTilda"},
               {"p"}
            },
            "nuTildaRes": 
            {
                {"U", "phi", "nuTilda"},
                {"U"}
            }
        }
 
    */
 
    word pName;
 
    if (mesh_.thisDb().foundObject<volScalarField>("p"))
    {
        pName = "p";
    }
    else if (mesh_.thisDb().foundObject<volScalarField>("p_rgh"))
    {
        pName = "p_rgh";
    }
    else
    {
        FatalErrorIn(
            "Neither p nor p_rgh was found in mesh.thisDb()!"
            "addModelResidualCon failed to setup turbulence residuals!")
            << exit(FatalError);
    }
 
    // NOTE: for compressible flow, it depends on rho so we need to add T and p
#ifdef IncompressibleFlow
    allCon.set(
        "nuTildaRes",
        {
            {"U", "nuTilda", "phi"}, // lv0
            {"U", "nuTilda"}, // lv1
            {"nuTilda"} // lv2
        });
#endif
 
#ifdef CompressibleFlow
    allCon.set(
        "nuTildaRes",
        {
            {"U", "T", pName, "nuTilda", "phi"}, // lv0
            {"U", "T", pName, "nuTilda"}, // lv1
            {"T", pName, "nuTilda"} // lv2
        });
#endif
}
 
void DASpalartAllmarasFv3::correct()
{
    /*
    Descroption:
        Solve the residual equations and update the state. This function will be called 
        by the DASolver. It is needed because we want to control the output frequency
        of the residual convergence every 100 steps. If using the correct from turbulence
        it will output residual every step which will be too much of information.
    */
 
    // We set the flag solveTurbState_ to 1 such that in the calcResiduals function
    // we will solve and update nuTilda
    solveTurbState_ = 1;
    dictionary dummyOptions;
    this->calcResiduals(dummyOptions);
    // after it, we reset solveTurbState_ = 0 such that calcResiduals will not
    // update nuTilda when calling from the adjoint class, i.e., solveAdjoint from DASolver.
    solveTurbState_ = 0;
}
 
void DASpalartAllmarasFv3::calcResiduals(const dictionary& options)
{
    /*
    Descroption:
        If solveTurbState_ == 1, this function solve and update nuTilda, and 
        is the same as calling turbulence.correct(). If solveTurbState_ == 0,
        this function compute residuals for turbulence variables, e.g., nuTildaRes_
 
    Input:
        options.isPC: 1 means computing residuals for preconditioner matrix.
        This essentially use the first order scheme for div(phi,nuTilda)
 
        p_, U_, phi_, etc: State variables in OpenFOAM
    
    Output:
        nuTildaRes_: If solveTurbState_ == 0, update the residual field variable
 
        nuTilda_: If solveTurbState_ == 1, update nuTilda
    */
 
    // Copy and modify based on the "correct" function
 
    label printToScreen = this->isPrintTime(mesh_.time(), printInterval_);
 
    word divNuTildaScheme = "div(phi,nuTilda)";
 
    label isPC = 0;
 
    if (!solveTurbState_)
    {
        isPC = options.getLabel("isPC");
 
        if (isPC)
        {
            divNuTildaScheme = "div(pc)";
        }
    }
 
    const volScalarField chi(this->chi());
    const volScalarField fv1(this->fv1(chi));
 
    const volScalarField Stilda(
        this->fv3(chi, fv1) * ::sqrt(2.0) * mag(skew(fvc::grad(U_)))
        + this->fv2(chi, fv1) * nuTilda_ / sqr(kappa_ * y_));
 
    tmp<fvScalarMatrix> nuTildaEqn(
        fvm::ddt(phase_, rho_, nuTilda_)
            + fvm::div(phaseRhoPhi_, nuTilda_, divNuTildaScheme)
            - fvm::laplacian(phase_ * rho_ * DnuTildaEff(), nuTilda_)
            - Cb2_ / sigmaNut_ * phase_ * rho_ * magSqr(fvc::grad(nuTilda_))
        == Cb1_ * phase_ * rho_ * Stilda * nuTilda_
            - fvm::Sp(Cw1_ * phase_ * rho_ * fw(Stilda) * nuTilda_ / sqr(y_), nuTilda_));
 
    nuTildaEqn.ref().relax();
 
    if (solveTurbState_)
    {
 
        // get the solver performance info such as initial
        // and final residuals
        SolverPerformance<scalar> solverNuTilda = solve(nuTildaEqn);
        if (printToScreen)
        {
            Info << "nuTilda Initial residual: " << solverNuTilda.initialResidual() << endl
                 << "          Final residual: " << solverNuTilda.finalResidual() << endl;
        }
 
        DAUtility::boundVar(allOptions_, nuTilda_, printToScreen);
        nuTilda_.correctBoundaryConditions();
 
        // ***************** NOTE*****************
        // In the original SA, it is correctNut(fv1) and fv1 is not
        // updated based on the latest nuTilda. We use correctNut which
        // recompute fv1 with the latest nuTilda
        this->correctNut();
    }
    else
    {
        // calculate residuals
        nuTildaRes_ = nuTildaEqn.ref() & nuTilda_;
        // need to normalize residuals
        normalizeResiduals(nuTildaRes);
    }
 
    return;
}
 
void DASpalartAllmarasFv3::invTranProdNuTildaEqn(
    const volScalarField& mySource,
    volScalarField& pseudoNuTilda)
{
    /*
    Description:
        Inverse transpose product, M_nuTilda^(-T)
        Based on inverseProduct_nuTildaEqn from simpleFoamSAPrimal, but swaping upper() and lower()
        We won't ADR this function, so we can treat most of the arguments as const
    */
 
    // Make sure pseudoNuTilda = nuTilda;
    //if (pseudoNuTildaEqnInitialized_ == 0)
    //{
    /*
    pseudoNuTilda_ = nuTilda_;
 
    const volScalarField chi(this->chi());
    const volScalarField fv1(this->fv1(chi));
 
    // Get myStilda
    const volScalarField Stilda(
        this->fv3(chi, fv1) * ::sqrt(2.0) * mag(skew(fvc::grad(U_)))
        + this->fv2(chi, fv1) * nuTilda_ / sqr(kappa_ * y_));
 
    // Get the pseudoNuTildaEqn,
    // the most important thing here is to make sure the l.h.s. mathces that of nuTildaEqn.
    // Some explicit terms that only contributes to the r.h.s. are diabled
    pseudoNuTildaEqn_ =
        //fvm::ddt(pseudoNuTilda_)
        fvm::div(phi_, pseudoNuTilda_, "div(phi,nuTilda)")
        - fvm::laplacian(DnuTildaEff(), pseudoNuTilda_)
        + fvm::Sp(Cw1_ * fw(Stilda) * pseudoNuTilda_ / sqr(y_), pseudoNuTilda_);
    pseudoNuTildaEqn_.relax(relaxNuTildaEqn_);
 
    // Swap upper() and lower()
    List<scalar> temp = pseudoNuTildaEqn_.upper();
    pseudoNuTildaEqn_.upper() = pseudoNuTildaEqn_.lower();
    pseudoNuTildaEqn_.lower() = temp;
 
    // mark it as initialized
    //    pseudoNuTildaEqnInitialized_ = 1;
    //}
 
    // Overwrite the r.h.s.
    pseudoNuTildaEqn_.source() = mySource.primitiveField();
 
    // Make sure that boundary contribution to source is zero,
    // Alternatively, we can deduct source by boundary contribution, so that it would cancel out during solve.
    forAll(pseudoNuTilda_.boundaryField(), patchI)
    {
        const fvPatch& pp = pseudoNuTilda_.boundaryField()[patchI].patch();
        forAll(pp, faceI)
        {
            label cellI = pp.faceCells()[faceI];
            pseudoNuTildaEqn_.source()[cellI] -= pseudoNuTildaEqn_.boundaryCoeffs()[patchI][faceI];
        }
    }
 
    // Before solve, force xxEqn.psi to be solved into all zero
    // This ensures the zero (internal) initial guess
    forAll(pseudoNuTilda_.primitiveFieldRef(), cellI)
    {
        pseudoNuTilda_.primitiveFieldRef()[cellI] = 0;
    }
    // Solve using the zero (internal) initial guess
    pseudoNuTildaEqn_.solve(solverDictNuTilda_);
 
    forAll(pseudoNuTilda, cellI)
    {
        pseudoNuTilda[cellI] = pseudoNuTilda_[cellI];
    }
*/
}
 
void DASpalartAllmarasFv3::constructPseudoNuTildaEqn()
{
    /*
    Description:
        construct the pseudo nuTildaEqn, 
        which is nuTildaEqn with the lhs upper and lower arrays swapped,
        we also don't care about the rhs of pseudo nuTildaEqn.
    */
 
    // Make sure pseudoNuTilda is indeed identical to nuTilda
    pseudoNuTilda_ = nuTilda_;
    pseudoNuTilda_.correctBoundaryConditions();
 
    const volScalarField chi(this->chi());
    const volScalarField fv1(this->fv1(chi));
 
    // Get myStilda
    const volScalarField Stilda(
        this->fv3(chi, fv1) * ::sqrt(2.0) * mag(skew(fvc::grad(U_)))
        + this->fv2(chi, fv1) * nuTilda_ / sqr(kappa_ * y_));
 
    // Get the pseudoNuTildaEqn,
    // the most important thing here is to make sure the l.h.s. mathces that of nuTildaEqn.
    // Some explicit terms that only contributes to the r.h.s. are diabled
    pseudoNuTildaEqn_ =
        //fvm::ddt(pseudoNuTilda_)
        fvm::div(phi_, pseudoNuTilda_, "div(phi,nuTilda)")
        - fvm::laplacian(DnuTildaEff(), pseudoNuTilda_)
        + fvm::Sp(Cw1_ * fw(Stilda) * pseudoNuTilda_ / sqr(y_), pseudoNuTilda_);
    pseudoNuTildaEqn_.relax(relaxNuTildaEqn_);
 
    // Swap upper() and lower()
    // We will use the swap function once it's being moved to the DAUtility
    List<scalar> temp = pseudoNuTildaEqn_.upper();
    pseudoNuTildaEqn_.upper() = pseudoNuTildaEqn_.lower();
    pseudoNuTildaEqn_.lower() = temp;
}
 
void DASpalartAllmarasFv3::rhsSolvePseudoNuTildaEqn(const volScalarField& nuTildaSource)
{
    /*
    Description:
        solve the pseudo nuTildaEqn with a overwritten rhs
    */
 
    // Overwrite the r.h.s.
    pseudoNuTildaEqn_.source() = nuTildaSource.primitiveField();
 
    // Make sure that boundary contribution to source is zero,
    // Alternatively, we can deduct source by boundary contribution, so that it would cancel out during solve.
    forAll(pseudoNuTilda_.boundaryField(), patchI)
    {
        const fvPatch& pp = pseudoNuTilda_.boundaryField()[patchI].patch();
        forAll(pp, faceI)
        {
            label cellI = pp.faceCells()[faceI];
            pseudoNuTildaEqn_.source()[cellI] -= pseudoNuTildaEqn_.boundaryCoeffs()[patchI][faceI];
        }
    }
 
    // Before solve, force xxEqn.psi to be solved into all zero
    // This ensures the zero (internal) initial guess
    forAll(pseudoNuTilda_.primitiveFieldRef(), cellI)
    {
        pseudoNuTilda_.primitiveFieldRef()[cellI] = 0;
    }
    // Solve using the zero (internal) initial guess
    pseudoNuTildaEqn_.solve(solverDictNuTilda_);
}
 
void DASpalartAllmarasFv3::calcLduResidualTurb(volScalarField& nuTildaRes)
{
    /*
    Description:
        calculate the turbulence residual using LDU matrix
    */
 
    const volScalarField chi(this->chi());
    const volScalarField fv1(this->fv1(chi));
 
    // Get myStilda
    const volScalarField Stilda(
        this->fv3(chi, fv1) * ::sqrt(2.0) * mag(skew(fvc::grad(U_)))
        + this->fv2(chi, fv1) * nuTilda_ / sqr(kappa_ * y_));
 
    // Construct nuTildaEqn using our own SA implementation
    fvScalarMatrix nuTildaEqn(
        fvm::div(phi_, nuTilda_, "div(phi,nuTilda)")
            - fvm::laplacian(DnuTildaEff(), nuTilda_)
            - Cb2_ / sigmaNut_ * magSqr(fvc::grad(nuTilda_))
        == Cb1_ * Stilda * nuTilda_
            - fvm::Sp(Cw1_ * fw(Stilda) * nuTilda_ / sqr(y_), nuTilda_));
 
    List<scalar>& nuTildaSource = nuTildaEqn.source();
    List<scalar>& nuTildaDiag = nuTildaEqn.diag();
 
    // Initiate nuTildaRes, with no boundary contribution
    for (label i = 0; i < nuTilda_.size(); i++)
    {
        nuTildaRes[i] = nuTildaDiag[i] * nuTilda_[i] - nuTildaSource[i];
    }
    nuTildaRes.primitiveFieldRef() -= nuTildaEqn.lduMatrix::H(nuTilda_);
 
    // Boundary correction
    forAll(nuTilda_.boundaryField(), patchI)
    {
        const fvPatch& pp = nuTilda_.boundaryField()[patchI].patch();
        forAll(pp, faceI)
        {
            // Both ways of getting cellI work
            // Below is the previous way of getting the address
            label cellI = pp.faceCells()[faceI];
            // Below is using lduAddr().patchAddr(patchi)
            //label cellI = nuTildaEqn.lduAddr().patchAddr(patchI)[faceI];
            //myDiag[cellI] += TEqn.internalCoeffs()[patchI][faceI];
            nuTildaRes[cellI] += nuTildaEqn.internalCoeffs()[patchI][faceI] * nuTilda_[cellI];
            nuTildaRes[cellI] -= nuTildaEqn.boundaryCoeffs()[patchI][faceI];
        }
    }
 
    // Below is not necessary, but it doesn't hurt
    nuTildaRes.correctBoundaryConditions();
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
} // End namespace Foam
 
// ************************************************************************* //