freeflow/navierstokes/staggered/fluxvariables.hh

Go to the documentation of this file.

// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
// vi: set et ts=4 sw=4 sts=4:
/*****************************************************************************
 *   See the file COPYING for full copying permissions.                      *
 *                                                                           *
 *   This program 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.                                     *
 *                                                                           *
 *   This program 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 this program.  If not, see <http://www.gnu.org/licenses/>.   *
 *****************************************************************************/
#ifndef DUMUX_NAVIERSTOKES_STAGGERED_FLUXVARIABLES_HH
#define DUMUX_NAVIERSTOKES_STAGGERED_FLUXVARIABLES_HH
 
#include <array>
#include <optional>
 
#include <dumux/common/math.hh>
#include <dumux/common/exceptions.hh>
#include <dumux/common/parameters.hh>
#include <dumux/common/properties.hh>
 
#include <dumux/flux/fluxvariablesbase.hh>
#include <dumux/discretization/method.hh>
 
 
#include "staggeredupwindfluxvariables.hh"
#include "velocitygradients.hh"
 
namespace Dumux {
 
// forward declaration
template<class TypeTag, DiscretizationMethod discMethod>
class NavierStokesFluxVariablesImpl;

template<class TypeTag>
class NavierStokesFluxVariablesImpl<TypeTag, DiscretizationMethod::staggered>
: public FluxVariablesBase<GetPropType<TypeTag, Properties::Problem>,
                           typename GetPropType<TypeTag, Properties::GridGeometry>::LocalView,
                           typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView,
                           typename GetPropType<TypeTag, Properties::GridFluxVariablesCache>::LocalView>
{
    using GridVariables = GetPropType<TypeTag, Properties::GridVariables>;
 
    using GridVolumeVariables = typename GridVariables::GridVolumeVariables;
    using ElementVolumeVariables = typename GridVolumeVariables::LocalView;
    using VolumeVariables = typename GridVolumeVariables::VolumeVariables;
 
    using GridFluxVariablesCache = typename GridVariables::GridFluxVariablesCache;
    using FluxVariablesCache = typename GridFluxVariablesCache::FluxVariablesCache;
 
    using GridFaceVariables = typename GridVariables::GridFaceVariables;
    using ElementFaceVariables = typename GridFaceVariables::LocalView;
    using FaceVariables = typename GridFaceVariables::FaceVariables;
 
    using Problem = GetPropType<TypeTag, Properties::Problem>;
    using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
    using FVElementGeometry = typename GridGeometry::LocalView;
    using GridView = typename GridGeometry::GridView;
    using Element = typename GridView::template Codim<0>::Entity;
    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
    using ModelTraits = GetPropType<TypeTag, Properties::ModelTraits>;
    using Indices = typename ModelTraits::Indices;
    using SubControlVolumeFace = typename FVElementGeometry::SubControlVolumeFace;
    using CellCenterPrimaryVariables = GetPropType<TypeTag, Properties::CellCenterPrimaryVariables>;
    using FacePrimaryVariables = GetPropType<TypeTag, Properties::FacePrimaryVariables>;
    using BoundaryTypes = GetPropType<TypeTag, Properties::BoundaryTypes>;
    using VelocityGradients = StaggeredVelocityGradients<Scalar, GridGeometry, BoundaryTypes, Indices>;
 
    static constexpr bool normalizePressure = getPropValue<TypeTag, Properties::NormalizePressure>();
 
    using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
 
    static constexpr auto cellCenterIdx = GridGeometry::cellCenterIdx();
    static constexpr auto faceIdx = GridGeometry::faceIdx();
 
    static constexpr int upwindSchemeOrder = getPropValue<TypeTag, Properties::UpwindSchemeOrder>();
    static constexpr bool useHigherOrder = upwindSchemeOrder > 1;
 
public:
 
    using HeatConductionType = GetPropType<TypeTag, Properties::HeatConductionType>;

    template<class UpwindTerm>
    static Scalar advectiveFluxForCellCenter(const Problem& problem,
                                             const ElementVolumeVariables& elemVolVars,
                                             const ElementFaceVariables& elemFaceVars,
                                             const SubControlVolumeFace &scvf,
                                             UpwindTerm upwindTerm)
    {
        const Scalar velocity = elemFaceVars[scvf].velocitySelf();
        const bool insideIsUpstream = scvf.directionSign() == sign(velocity);
        static const Scalar upwindWeight = getParamFromGroup<Scalar>(problem.paramGroup(), "Flux.UpwindWeight");
 
        const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
        const auto& outsideVolVars = elemVolVars[scvf.outsideScvIdx()];
 
        const auto& upstreamVolVars = insideIsUpstream ? insideVolVars : outsideVolVars;
        const auto& downstreamVolVars = insideIsUpstream ? outsideVolVars : insideVolVars;
 
        const Scalar flux = (upwindWeight * upwindTerm(upstreamVolVars) +
                            (1.0 - upwindWeight) * upwindTerm(downstreamVolVars))
                            * velocity * scvf.area() * scvf.directionSign();
 
        return flux * extrusionFactor_(elemVolVars, scvf);
    }

    CellCenterPrimaryVariables computeMassFlux(const Problem& problem,
                                               const Element& element,
                                               const FVElementGeometry& fvGeometry,
                                               const ElementVolumeVariables& elemVolVars,
                                               const ElementFaceVariables& elemFaceVars,
                                               const SubControlVolumeFace& scvf,
                                               const FluxVariablesCache& fluxVarsCache)
    {
        // The advectively transported quantity (i.e density for a single-phase system).
        auto upwindTerm = [](const auto& volVars) { return volVars.density(); };
 
        // Call the generic flux function.
        CellCenterPrimaryVariables result(0.0);
        result[Indices::conti0EqIdx - ModelTraits::dim()] = advectiveFluxForCellCenter(problem, elemVolVars, elemFaceVars, scvf, upwindTerm);
 
        return result;
    }

    FacePrimaryVariables computeMomentumFlux(const Problem& problem,
                                             const Element& element,
                                             const SubControlVolumeFace& scvf,
                                             const FVElementGeometry& fvGeometry,
                                             const ElementVolumeVariables& elemVolVars,
                                             const ElementFaceVariables& elemFaceVars,
                                             const GridFluxVariablesCache& gridFluxVarsCache)
    {
        return computeFrontalMomentumFlux(problem, element, scvf, fvGeometry, elemVolVars, elemFaceVars, gridFluxVarsCache) +
               computeLateralMomentumFlux(problem, element, scvf, fvGeometry, elemVolVars, elemFaceVars, gridFluxVarsCache);
    }

    FacePrimaryVariables computeFrontalMomentumFlux(const Problem& problem,
                                                    const Element& element,
                                                    const SubControlVolumeFace& scvf,
                                                    const FVElementGeometry& fvGeometry,
                                                    const ElementVolumeVariables& elemVolVars,
                                                    const ElementFaceVariables& elemFaceVars,
                                                    const GridFluxVariablesCache& gridFluxVarsCache)
    {
        FacePrimaryVariables frontalFlux(0.0);
 
        // The velocities of the dof at interest and the one of the opposite scvf.
        const Scalar velocitySelf = elemFaceVars[scvf].velocitySelf();
        const Scalar velocityOpposite = elemFaceVars[scvf].velocityOpposite();
 
        // Advective flux.
        if (problem.enableInertiaTerms())
        {
            // Get the average velocity at the center of the element (i.e. the location of the staggered face).
            const Scalar transportingVelocity = (velocitySelf + velocityOpposite) * 0.5;
 
            frontalFlux += StaggeredUpwindFluxVariables<TypeTag, upwindSchemeOrder>::computeUpwindedFrontalMomentum(scvf, elemFaceVars, elemVolVars, gridFluxVarsCache, transportingVelocity)
                           * transportingVelocity * -1.0 * scvf.directionSign();
        }
 
        // The volume variables within the current element. We only require those (and none of neighboring elements)
        // because the fluxes are calculated over the staggered face at the center of the element.
        const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
 
        // Diffusive flux.
        const Scalar velocityGrad_ii = VelocityGradients::velocityGradII(scvf, elemFaceVars[scvf]) * scvf.directionSign();
 
        static const bool enableUnsymmetrizedVelocityGradient
            = getParamFromGroup<bool>(problem.paramGroup(), "FreeFlow.EnableUnsymmetrizedVelocityGradient", false);
        const Scalar factor = enableUnsymmetrizedVelocityGradient ? 1.0 : 2.0;
        frontalFlux -= factor * insideVolVars.effectiveViscosity() * velocityGrad_ii;
 
        // The pressure term.
        // If specified, the pressure can be normalized using the initial value on the scfv of interest.
        // The scvf is used to normalize by the same value from the left and right side.
        // Can potentially help to improve the condition number of the system matrix.
        const Scalar pressure = normalizePressure ?
                                insideVolVars.pressure() - problem.initial(scvf)[Indices::pressureIdx]
                              : insideVolVars.pressure();
 
        // Account for the orientation of the staggered face's normal outer normal vector
        // (pointing in opposite direction of the scvf's one).
        frontalFlux += pressure * -1.0 * scvf.directionSign();
 
        // Account for the staggered face's area. For rectangular elements, this equals the area of the scvf
        // our velocity dof of interest lives on.
        return frontalFlux * scvf.area() * insideVolVars.extrusionFactor();
   }

    FacePrimaryVariables computeLateralMomentumFlux(const Problem& problem,
                                                    const Element& element,
                                                    const SubControlVolumeFace& scvf,
                                                    const FVElementGeometry& fvGeometry,
                                                    const ElementVolumeVariables& elemVolVars,
                                                    const ElementFaceVariables& elemFaceVars,
                                                    const GridFluxVariablesCache& gridFluxVarsCache)
    {
        FacePrimaryVariables lateralFlux(0.0);
        const auto& faceVars = elemFaceVars[scvf];
        const std::size_t numSubFaces = scvf.pairData().size();
 
        // If the current scvf is on a boundary, check if there is a Neumann BC for the stress in tangential direction.
        // Create a boundaryTypes object (will be empty if not at a boundary).
        std::optional<BoundaryTypes> currentScvfBoundaryTypes;
        if (scvf.boundary())
            currentScvfBoundaryTypes.emplace(problem.boundaryTypes(element, scvf));
 
        // Account for all sub faces.
        for (int localSubFaceIdx = 0; localSubFaceIdx < numSubFaces; ++localSubFaceIdx)
        {
            const auto eIdx = scvf.insideScvIdx();
            // Get the face normal to the face the dof lives on. The staggered sub face conincides with half of this lateral face.
            const auto& lateralScvf = fvGeometry.scvf(eIdx, scvf.pairData(localSubFaceIdx).localLateralFaceIdx);
 
            // Create a boundaryTypes object (will be empty if not at a boundary).
            std::optional<BoundaryTypes> lateralFaceBoundaryTypes;
 
            // Check if there is face/element parallel to our face of interest where the dof lives on. If there is no parallel neighbor,
            // we are on a boundary where we have to check for boundary conditions.
            if (lateralScvf.boundary())
            {
                // Retrieve the boundary types that correspond to the center of the lateral scvf. As a convention, we always query
                // the type of BCs at the center of the element's "actual" lateral scvf (not the face of the staggered control volume).
                // The value of the BC will be evaluated at the center of the staggered face.
                //     --------T######V                 || frontal face of staggered half-control-volume
                //     |      ||      | current scvf    #  lateral staggered face of interest (may lie on a boundary)
                //     |      ||      |                 x  dof position
                //     |      ||      x~~~~> vel.Self   -- element boundaries
                //     |      ||      |                 T  position at which the type of boundary conditions will be evaluated
                //     |      ||      |                    (center of lateral scvf)
                //     ----------------                 V  position at which the value of the boundary conditions will be evaluated
                //                                         (center of the staggered lateral face)
                lateralFaceBoundaryTypes.emplace(problem.boundaryTypes(element, lateralScvf));
            }
 
            // If the current scvf is on a bounary and if there is a Neumann or Beavers-Joseph-(Saffmann) BC for the stress in tangential direction,
            // assign this value for the lateral momentum flux. No further calculations are required. We assume that all lateral faces
            // have the same type of BC (Neumann or Beavers-Joseph-(Saffmann)), but we sample the value at their actual positions.
            if (currentScvfBoundaryTypes)
            {
                // Handle Neumann BCs.
                if (currentScvfBoundaryTypes->isNeumann(Indices::velocity(lateralScvf.directionIndex())))
                {
                    // Get the location of the lateral staggered face's center.
                    const auto& lateralStaggeredFaceCenter = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
                    lateralFlux += problem.neumann(element, fvGeometry, elemVolVars, elemFaceVars,
                        scvf.makeBoundaryFace(lateralStaggeredFaceCenter))[Indices::velocity(lateralScvf.directionIndex())]
                        * extrusionFactor_(elemVolVars, lateralScvf) * lateralScvf.area() * 0.5
                        * lateralScvf.directionSign();
                        continue;
                }
 
                // Treat the edge case when our current scvf has a BJ condition while the lateral face has a Neumann BC.
                // It is not clear how to evaluate the BJ condition here.
                // For symmetry reasons, our own scvf should then have the same Neumann flux as the lateral face.
                // TODO: We should clarify if this is the correct approach.
                if (currentScvfBoundaryTypes->isBeaversJoseph(Indices::velocity(lateralScvf.directionIndex())) && lateralFaceBoundaryTypes &&
                    lateralFaceBoundaryTypes->isNeumann(Indices::velocity(scvf.directionIndex())))
                {
                    const auto& lateralStaggeredFaceCenter = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
                    lateralFlux += problem.neumann(element, fvGeometry, elemVolVars, elemFaceVars,
                        lateralScvf.makeBoundaryFace(lateralStaggeredFaceCenter))[Indices::velocity(scvf.directionIndex())]
                        * extrusionFactor_(elemVolVars, lateralScvf) * lateralScvf.area() * 0.5
                        * lateralScvf.directionSign();
                        continue;
                }
            }
 
            // Check if the lateral face (perpendicular to our current scvf) lies on a boundary. If yes, boundary conditions might need to be treated
            // and further calculations can be skipped.
            if (lateralScvf.boundary())
            {
                // Check if we have a symmetry boundary condition. If yes, the tangential part of the momentum flux can be neglected
                // and we may skip any further calculations for the given sub face.
                if (lateralFaceBoundaryTypes->isSymmetry())
                    continue;
 
                // Handle Neumann boundary conditions. No further calculations are then required for the given sub face.
                if (lateralFaceBoundaryTypes->isNeumann(Indices::velocity(scvf.directionIndex())))
                {
                    // Construct a temporary scvf which corresponds to the staggered sub face, featuring the location
                    // the staggered faces's center.
                    const auto& lateralStaggeredFaceCenter = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
                    const auto lateralBoundaryFace = lateralScvf.makeBoundaryFace(lateralStaggeredFaceCenter);
                    lateralFlux += problem.neumann(element, fvGeometry, elemVolVars, elemFaceVars, lateralBoundaryFace)[Indices::velocity(scvf.directionIndex())]
                                                  * elemVolVars[lateralScvf.insideScvIdx()].extrusionFactor() * lateralScvf.area() * 0.5;
                    continue;
                }
 
                // Handle wall-function fluxes (only required for RANS models)
                if (incorporateWallFunction_(lateralFlux, problem, element, fvGeometry, scvf, elemVolVars, elemFaceVars, localSubFaceIdx))
                    continue;
            }
 
            // Check the consistency of the boundary conditions, exactly one of the following must be set
            if (lateralFaceBoundaryTypes)
            {
                std::bitset<3> admittableBcTypes;
                admittableBcTypes.set(0, lateralFaceBoundaryTypes->isDirichlet(Indices::pressureIdx));
                admittableBcTypes.set(1, lateralFaceBoundaryTypes->isDirichlet(Indices::velocity(scvf.directionIndex())));
                admittableBcTypes.set(2, lateralFaceBoundaryTypes->isBeaversJoseph(Indices::velocity(scvf.directionIndex())));
                if (admittableBcTypes.count() != 1)
                {
                    DUNE_THROW(Dune::InvalidStateException, "Invalid boundary conditions for lateral scvf "
                    "for the momentum equations at global position " << lateralStaggeredFaceCenter_(scvf, localSubFaceIdx)
                    << ", current scvf global position " << scvf.center());
                }
            }
 
            // If none of the above boundary conditions apply for the given sub face, proceed to calculate the tangential momentum flux.
            if (problem.enableInertiaTerms())
                lateralFlux += computeAdvectivePartOfLateralMomentumFlux_(problem, fvGeometry, element,
                                                                          scvf, elemVolVars, faceVars,
                                                                          gridFluxVarsCache,
                                                                          currentScvfBoundaryTypes, lateralFaceBoundaryTypes,
                                                                          localSubFaceIdx);
 
            lateralFlux += computeDiffusivePartOfLateralMomentumFlux_(problem, fvGeometry, element,
                                                                      scvf, elemVolVars, faceVars,
                                                                      currentScvfBoundaryTypes, lateralFaceBoundaryTypes,
                                                                      localSubFaceIdx);
        }
        return lateralFlux;
    }

    FacePrimaryVariables inflowOutflowBoundaryFlux(const Problem& problem,
                                                   const Element& element,
                                                   const SubControlVolumeFace& scvf,
                                                   const ElementVolumeVariables& elemVolVars,
                                                   const ElementFaceVariables& elemFaceVars) const
    {
        FacePrimaryVariables inOrOutflow(0.0);
        const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
 
        // Advective momentum flux.
        if (problem.enableInertiaTerms())
        {
            const Scalar velocitySelf = elemFaceVars[scvf].velocitySelf();
            const auto& outsideVolVars = elemVolVars[scvf.outsideScvIdx()];
            const auto& upVolVars = (scvf.directionSign() == sign(velocitySelf)) ?
                                    insideVolVars : outsideVolVars;
 
            inOrOutflow += velocitySelf * velocitySelf * upVolVars.density();
        }
 
        // Apply a pressure at the boundary.
        const Scalar boundaryPressure = normalizePressure
                                        ? (problem.dirichlet(element, scvf)[Indices::pressureIdx] -
                                           problem.initial(scvf)[Indices::pressureIdx])
                                        : problem.dirichlet(element, scvf)[Indices::pressureIdx];
        inOrOutflow += boundaryPressure;
 
        // Account for the orientation of the face at the boundary,
        return inOrOutflow * scvf.directionSign() * scvf.area() * insideVolVars.extrusionFactor();
    }
 
private:

    FacePrimaryVariables computeAdvectivePartOfLateralMomentumFlux_(const Problem& problem,
                                                                    const FVElementGeometry& fvGeometry,
                                                                    const Element& element,
                                                                    const SubControlVolumeFace& scvf,
                                                                    const ElementVolumeVariables& elemVolVars,
                                                                    const FaceVariables& faceVars,
                                                                    const GridFluxVariablesCache& gridFluxVarsCache,
                                                                    const std::optional<BoundaryTypes>& currentScvfBoundaryTypes,
                                                                    const std::optional<BoundaryTypes>& lateralFaceBoundaryTypes,
                                                                    const int localSubFaceIdx)
    {
        const auto eIdx = scvf.insideScvIdx();
        const auto& lateralFace = fvGeometry.scvf(eIdx, scvf.pairData(localSubFaceIdx).localLateralFaceIdx);
 
        // Get the transporting velocity, located at the scvf perpendicular to the current scvf where the dof
        // of interest is located.
        const Scalar transportingVelocity = [&]()
        {
            if (!scvf.boundary())
                return faceVars.velocityLateralInside(localSubFaceIdx);
            else
            {
                // Create a boundaryTypes object. Get the boundary conditions. We sample the type of BC at the center of the current scvf.
                const BoundaryTypes bcTypes = problem.boundaryTypes(element, scvf);
 
                if (bcTypes.isDirichlet(Indices::velocity(lateralFace.directionIndex())))
                {
                    // Construct a temporary scvf which corresponds to the staggered sub face, featuring the location
                    // the staggered faces's center.
                    const auto& lateralBoundaryFacePos = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
                    return problem.dirichlet(element, scvf.makeBoundaryFace(lateralBoundaryFacePos))[Indices::velocity(lateralFace.directionIndex())];
                }
                else if (bcTypes.isBeaversJoseph(Indices::velocity(lateralFace.directionIndex())))
                {
                    return VelocityGradients::beaversJosephVelocityAtCurrentScvf(problem, element, fvGeometry, scvf,  faceVars,
                                                                                 currentScvfBoundaryTypes, lateralFaceBoundaryTypes, localSubFaceIdx);
                }
                else
                    return faceVars.velocityLateralInside(localSubFaceIdx);
            }
        }();
 
        return StaggeredUpwindFluxVariables<TypeTag, upwindSchemeOrder>::computeUpwindedLateralMomentum(problem, fvGeometry, element, scvf, elemVolVars, faceVars,
                                                                     gridFluxVarsCache, localSubFaceIdx, currentScvfBoundaryTypes, lateralFaceBoundaryTypes)
               * transportingVelocity * lateralFace.directionSign() * lateralFace.area() * 0.5 * extrusionFactor_(elemVolVars, lateralFace);
    }

    FacePrimaryVariables computeDiffusivePartOfLateralMomentumFlux_(const Problem& problem,
                                                                    const FVElementGeometry& fvGeometry,
                                                                    const Element& element,
                                                                    const SubControlVolumeFace& scvf,
                                                                    const ElementVolumeVariables& elemVolVars,
                                                                    const FaceVariables& faceVars,
                                                                    const std::optional<BoundaryTypes>& currentScvfBoundaryTypes,
                                                                    const std::optional<BoundaryTypes>& lateralFaceBoundaryTypes,
                                                                    const int localSubFaceIdx)
    {
        const auto eIdx = scvf.insideScvIdx();
        const auto& lateralFace = fvGeometry.scvf(eIdx, scvf.pairData(localSubFaceIdx).localLateralFaceIdx);
 
        FacePrimaryVariables lateralDiffusiveFlux(0.0);
 
        static const bool enableUnsymmetrizedVelocityGradient
            = getParamFromGroup<bool>(problem.paramGroup(), "FreeFlow.EnableUnsymmetrizedVelocityGradient", false);
 
        // Get the volume variables of the own and the neighboring element. The neighboring
        // element is adjacent to the staggered face normal to the current scvf
        // where the dof of interest is located.
        const auto& insideVolVars = elemVolVars[lateralFace.insideScvIdx()];
        const auto& outsideVolVars = elemVolVars[lateralFace.outsideScvIdx()];
 
        // Get the averaged viscosity at the staggered face normal to the current scvf.
        const Scalar muAvg = lateralFace.boundary()
                             ? insideVolVars.effectiveViscosity()
                             : (insideVolVars.effectiveViscosity() + outsideVolVars.effectiveViscosity()) * 0.5;
 
        // Consider the shear stress caused by the gradient of the velocities normal to our face of interest.
        if (!enableUnsymmetrizedVelocityGradient)
        {
            if (!scvf.boundary() ||
                currentScvfBoundaryTypes->isDirichlet(Indices::velocity(lateralFace.directionIndex())) ||
                currentScvfBoundaryTypes->isBeaversJoseph(Indices::velocity(lateralFace.directionIndex())))
            {
                const Scalar velocityGrad_ji = VelocityGradients::velocityGradJI(problem, element, fvGeometry, scvf, faceVars, currentScvfBoundaryTypes, lateralFaceBoundaryTypes, localSubFaceIdx);
                // Account for the orientation of the staggered normal face's outer normal vector.
                lateralDiffusiveFlux -= muAvg * velocityGrad_ji * lateralFace.directionSign();
            }
        }
 
        // Consider the shear stress caused by the gradient of the velocities parallel to our face of interest.
        // If we have a Dirichlet condition for the pressure at the lateral face we assume to have a zero velocityGrad_ij velocity gradient
        // so we can skip the computation.
        if (!lateralFace.boundary() || !lateralFaceBoundaryTypes->isDirichlet(Indices::pressureIdx))
        {
            const Scalar velocityGrad_ij = VelocityGradients::velocityGradIJ(problem, element, fvGeometry, scvf, faceVars, currentScvfBoundaryTypes, lateralFaceBoundaryTypes, localSubFaceIdx);
            lateralDiffusiveFlux -= muAvg * velocityGrad_ij * lateralFace.directionSign();
        }
 
        // Account for the area of the staggered lateral face (0.5 of the coinciding scfv).
        return lateralDiffusiveFlux * lateralFace.area() * 0.5 * extrusionFactor_(elemVolVars, lateralFace);
    }

    const GlobalPosition& lateralStaggeredFaceCenter_(const SubControlVolumeFace& scvf, const int localSubFaceIdx) const
    {
        return scvf.pairData(localSubFaceIdx).lateralStaggeredFaceCenter;
    };

    static Scalar extrusionFactor_(const ElementVolumeVariables& elemVolVars, const SubControlVolumeFace& scvf)
    {
        const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
        const auto& outsideVolVars = elemVolVars[scvf.outsideScvIdx()];
        return harmonicMean(insideVolVars.extrusionFactor(), outsideVolVars.extrusionFactor());
    }

    template<class ...Args, bool turbulenceModel = ModelTraits::usesTurbulenceModel(), std::enable_if_t<!turbulenceModel, int> = 0>
    bool incorporateWallFunction_(Args&&... args) const
    { return false; }

    template<bool turbulenceModel = ModelTraits::usesTurbulenceModel(), std::enable_if_t<turbulenceModel, int> = 0>
    bool incorporateWallFunction_(FacePrimaryVariables& lateralFlux,
                                  const Problem& problem,
                                  const Element& element,
                                  const FVElementGeometry& fvGeometry,
                                  const SubControlVolumeFace& scvf,
                                  const ElementVolumeVariables& elemVolVars,
                                  const ElementFaceVariables& elemFaceVars,
                                  const std::size_t localSubFaceIdx) const
    {
        const auto eIdx = scvf.insideScvIdx();
        const auto& lateralScvf = fvGeometry.scvf(eIdx, scvf.pairData(localSubFaceIdx).localLateralFaceIdx);
 
        if (problem.useWallFunction(element, lateralScvf, Indices::velocity(scvf.directionIndex())))
        {
            const auto& lateralStaggeredFaceCenter = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
            const auto lateralBoundaryFace = lateralScvf.makeBoundaryFace(lateralStaggeredFaceCenter);
            lateralFlux += problem.wallFunction(element, fvGeometry, elemVolVars, elemFaceVars, scvf, lateralBoundaryFace)[Indices::velocity(scvf.directionIndex())]
                                               * extrusionFactor_(elemVolVars, lateralScvf) * lateralScvf.area() * 0.5;
            return true;
        }
        else
            return false;
    }
};
 
} // end namespace Dumux
 
#endif // DUMUX_NAVIERSTOKES_STAGGERED_FLUXVARIABLES_HH