shallowwaterviscousflux.hh

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#ifndef DUMUX_FLUX_SHALLOW_WATER_VISCOUS_FLUX_HH
#define DUMUX_FLUX_SHALLOW_WATER_VISCOUS_FLUX_HH
 
#include <cmath>
#include <algorithm>
#include <utility>
#include <type_traits>
#include <array>
 
#include <dune/common/std/type_traits.hh>
#include <dune/common/exceptions.hh>
 
#include <dumux/common/parameters.hh>
#include <dumux/flux/fluxvariablescaching.hh>
#include <dumux/flux/shallowwater/fluxlimiterlet.hh>
 
namespace Dumux {
 
#ifndef DOXYGEN
namespace Detail {
// helper struct detecting if the user-defined spatial params class has a frictionLaw function
template <typename T, typename ...Ts>
using FrictionLawDetector = decltype(std::declval<T>().frictionLaw(std::declval<Ts>()...));
 
template<class T, typename ...Args>
static constexpr bool implementsFrictionLaw()
{ return Dune::Std::is_detected<FrictionLawDetector, T, Args...>::value; }
} // end namespace Detail
#endif

template<class NumEqVector, typename std::enable_if_t<NumEqVector::size() == 3, int> = 0>
class ShallowWaterViscousFlux
{
 
public:
 
    using Cache = FluxVariablesCaching::EmptyDiffusionCache;
    using CacheFiller = FluxVariablesCaching::EmptyCacheFiller;
    template<class Problem, class FVElementGeometry, class ElementVolumeVariables>
    static NumEqVector flux(const Problem& problem,
                            const typename FVElementGeometry::GridGeometry::GridView::template Codim<0>::Entity& element,
                            const FVElementGeometry& fvGeometry,
                            const ElementVolumeVariables& elemVolVars,
                            const typename FVElementGeometry::SubControlVolumeFace& scvf)
    {
        using Scalar = typename NumEqVector::value_type;
        using FluidSystem = typename ElementVolumeVariables::VolumeVariables::FluidSystem;
 
        NumEqVector localFlux(0.0);
 
        // Get the inside and outside volume variables
        const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
        const auto& outsideVolVars = elemVolVars[scvf.outsideScvIdx()];
 
        const auto& insideScv = fvGeometry.scv(scvf.insideScvIdx());
        const auto& outsideScv = fvGeometry.scv(scvf.outsideScvIdx());
 
        const auto gradVelocity = [&]()
        {
            // The left (inside) and right (outside) states
            const auto velocityXLeft   = insideVolVars.velocity(0);
            const auto velocityYLeft   = insideVolVars.velocity(1);
            const auto velocityXRight  = outsideVolVars.velocity(0);
            const auto velocityYRight  = outsideVolVars.velocity(1);
 
            // Compute the velocity gradients normal to the face
            // Factor that takes the direction of the unit vector into account
            const auto& cellCenterToCellCenter = outsideScv.center() - insideScv.center();
            const auto distance = cellCenterToCellCenter.two_norm();
            const auto& unitNormal = scvf.unitOuterNormal();
            const auto direction = (unitNormal*cellCenterToCellCenter)/distance;
            return std::array<Scalar, 2>{
                (velocityXRight-velocityXLeft)*direction/distance,
                (velocityYRight-velocityYLeft)*direction/distance
            };
        }();
 
        // Use a harmonic average of the depth at the interface.
        const auto waterDepthLeft  = insideVolVars.waterDepth();
        const auto waterDepthRight = outsideVolVars.waterDepth();
        const auto averageDepth = 2.0*(waterDepthLeft*waterDepthRight)/(waterDepthLeft + waterDepthRight);
 
        // compute the turbulent viscosity contribution
        const Scalar turbViscosity = [&]()
        {
            // The (constant) background turbulent viscosity
            static const auto turbBGViscosity = getParamFromGroup<Scalar>(problem.paramGroup(), "ShallowWater.TurbulentViscosity", 1.0e-6);
 
            // Check whether the mixing-length turbulence model is used
            static const auto useMixingLengthTurbulenceModel = getParamFromGroup<bool>(problem.paramGroup(), "ShallowWater.UseMixingLengthTurbulenceModel", false);
 
            // constant eddy viscosity equal to the prescribed background eddy viscosity
            if (!useMixingLengthTurbulenceModel)
                return turbBGViscosity;
 
            using SpatialParams = typename Problem::SpatialParams;
            using E = typename FVElementGeometry::GridGeometry::GridView::template Codim<0>::Entity;
            using SCV = typename FVElementGeometry::SubControlVolume;
            if constexpr (!Detail::implementsFrictionLaw<SpatialParams, E, SCV>())
                DUNE_THROW(Dune::IOError, "Mixing length turbulence model enabled but spatial parameters do not implement the frictionLaw interface!");
            else
            {
                // turbulence model based on mixing length
                // Compute the turbulent viscosity using a combined horizontal/vertical mixing length approach
                // Turbulence coefficients: vertical (Elder like) and horizontal (Smagorinsky like)
                static const auto turbConstV = getParamFromGroup<Scalar>(problem.paramGroup(), "ShallowWater.VerticalCoefficientOfMixingLengthModel", 1.0);
                static const auto turbConstH = getParamFromGroup<Scalar>(problem.paramGroup(), "ShallowWater.HorizontalCoefficientOfMixingLengthModel", 0.1);

                constexpr Scalar kappa = 0.41;
                // Compute the average shear velocity on the face
                const Scalar ustar = [&]()
                {
                    // Get the bottom shear stress in the two adjacent cells
                    // Note that element is not needed in spatialParams().frictionLaw (should be removed). For now we simply pass the same element twice
                    const auto& bottomShearStressInside = problem.spatialParams().frictionLaw(element, insideScv).bottomShearStress(insideVolVars);
                    const auto& bottomShearStressOutside = problem.spatialParams().frictionLaw(element, outsideScv).bottomShearStress(outsideVolVars);
                    const auto bottomShearStressInsideMag = bottomShearStressInside.two_norm();
                    const auto bottomShearStressOutsideMag = bottomShearStressOutside.two_norm();
 
                    // Use a harmonic average of the viscosity and the depth at the interface.
                    using std::max;
                    const auto averageBottomShearStress = 2.0*(bottomShearStressInsideMag*bottomShearStressOutsideMag)
                            / max(1.0e-8,bottomShearStressInsideMag + bottomShearStressOutsideMag);
 
                    // Shear velocity possibly needed in mixing-length turbulence model in the computation of the diffusion flux
                    using std::sqrt;
                    static_assert(!FluidSystem::isCompressible(0),
                        "The shallow water model assumes incompressible fluids"
                    );
                    // Since the shallow water equations are incompressible, the density is constant.
                    // Therefore, insideVolVars.density() is equal to outsideVolVars.density().
                    return sqrt(averageBottomShearStress/insideVolVars.density());
                }();
 
                const auto turbViscosityV = turbConstV * (kappa/6.0) * ustar * averageDepth;

                using std::sqrt;
                const auto [gradU, gradV] = gradVelocity;
                const auto mixingLengthSquared = turbConstH * turbConstH * averageDepth * averageDepth;
                const auto turbViscosityH = mixingLengthSquared * sqrt(2.0*gradU*gradU + 2.0*gradV*gradV);
 
                // Total turbulent viscosity
                return turbBGViscosity + sqrt(turbViscosityV*turbViscosityV + turbViscosityH*turbViscosityH);
            }
        }();
 
        // Compute the viscous momentum fluxes with connecting water depth at the interface
        using std::min;
        using std::max;
        const auto freeSurfaceInside = insideVolVars.waterDepth() + insideVolVars.bedSurface();
        const auto freeSurfaceOutside = outsideVolVars.waterDepth() + outsideVolVars.bedSurface();
        const auto interfaceWaterDepth = max(min(freeSurfaceInside , freeSurfaceOutside) - max(insideVolVars.bedSurface(),outsideVolVars.bedSurface()),0.0);
        const auto& [gradU, gradV] = gradVelocity;
        const auto uViscousFlux = turbViscosity * interfaceWaterDepth * gradU;
        const auto vViscousFlux = turbViscosity * interfaceWaterDepth * gradV;
 
        localFlux[0] = 0.0;
        localFlux[1] = -uViscousFlux * scvf.area();
        localFlux[2] = -vViscousFlux * scvf.area();
 
        return localFlux;
    }
};
 
} // end namespace Dumux
 
#endif