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| 1 | // -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- | ||
| 2 | // vi: set et ts=4 sw=4 sts=4: | ||
| 3 | // | ||
| 4 | // SPDX-FileCopyrightInfo: Copyright © DuMux Project contributors, see AUTHORS.md in root folder | ||
| 5 | // SPDX-License-Identifier: GPL-3.0-or-later | ||
| 6 | // | ||
| 7 | |||
| 8 | #ifndef DUMUX_POISEUILLE_FLOW_TEST_PROBLEM_HH | ||
| 9 | #define DUMUX_POISEUILLE_FLOW_TEST_PROBLEM_HH | ||
| 10 | |||
| 11 | #include <algorithm> | ||
| 12 | #include <cctype> | ||
| 13 | |||
| 14 | #include <dumux/common/properties.hh> | ||
| 15 | #include <dumux/common/parameters.hh> | ||
| 16 | #include <dumux/common/numeqvector.hh> | ||
| 17 | |||
| 18 | #include <dumux/freeflow/shallowwater/problem.hh> | ||
| 19 | #include <dumux/freeflow/shallowwater/boundaryfluxes.hh> | ||
| 20 | |||
| 21 | namespace Dumux { | ||
| 22 | |||
| 23 | /*! | ||
| 24 | * \ingroup ShallowWaterTests | ||
| 25 | * \brief A simple test for the 2D flow in a channel with rough side walls (Poiseuille flow). | ||
| 26 | * | ||
| 27 | * The initial water depth corresponds to the analytical solution: | ||
| 28 | * having a slope equal to \f$ ib = d \Theta / L \f$ , where \f$ d \Theta \f$ is the water level difference between upstream and downstream and \f$ L \f$ is the domain length. | ||
| 29 | * At the west/left (inflow) boundary a discharge is prescribed of \f$ Q_{in} \f$ or \f$ q_{in} \f$ per meter. | ||
| 30 | * At the east/right (outflow) boundary a fixed water level is prescribed of \f$ \Theta \f$. | ||
| 31 | * The south and north boundaries are set to roughwall type boundaries, | ||
| 32 | * with a coefficient alphaWall = 1.0, where: | ||
| 33 | * alphaWall = 0.0: full slip (smooth wall) | ||
| 34 | * 0.0 <alphaWall < 1.0: partial slip (partially-rough wall) | ||
| 35 | * alphaWall = 1.0: no slip (fully-rough wall) | ||
| 36 | * Additionally these (south and north) boundaries are (automatically) set to no-flow boundaries. | ||
| 37 | |||
| 38 | * The flow in the channel experiences two forces: | ||
| 39 | * 1) the pressure gradient, driving the flow downstream | ||
| 40 | * 2) the wall roughness, | ||
| 41 | |||
| 42 | * It can be verified that this force balance reduces the momentum equation in X-direction to: | ||
| 43 | * | ||
| 44 | \f[ | ||
| 45 | \frac{\partial p}{\partial x} = \nu_T \frac{\partial^2 u}{\partial y^2} | ||
| 46 | \f] | ||
| 47 | * where \f$ \nu_T \f$ is the turbulent viscosity. | ||
| 48 | * | ||
| 49 | * This ordinary differential equation can be solved (applying the boundary conditions to obtain the integration constants), | ||
| 50 | * resulting in a parabolic velocity profile in lateral direction (over the width of the channel): | ||
| 51 | * | ||
| 52 | \f[ | ||
| 53 | u(y) = \frac{g d \Theta}{8 \nu_T L} \left(4y^2 - W^2\right) | ||
| 54 | \f] | ||
| 55 | * | ||
| 56 | * where y the coordinate in lateral direction. | ||
| 57 | * The velocity has a maximum value at y = 0: | ||
| 58 | \f[ | ||
| 59 | u_{max} = \frac{g d \Theta W^2}{8 \nu_T L} | ||
| 60 | \f] | ||
| 61 | * | ||
| 62 | The formula for \f$ u(y) \f$ is also used to calculate the analytic solution. | ||
| 63 | * It should be noted that u = f(x) and that v = 0 in the whole domain (in the final steady state). | ||
| 64 | * Therefore the momentum advection/convection terms should be zero for this test, as: | ||
| 65 | * \f$ u \partial u / \partial x = v \partial u / \partial y = u \partial v / \partial x = v \partial v / \partial y = 0 \f$ | ||
| 66 | * | ||
| 67 | * This problem uses the \ref ShallowWaterModel | ||
| 68 | */ | ||
| 69 | template<class TypeTag> | ||
| 70 | class PoiseuilleFlowProblem | ||
| 71 | : public ShallowWaterProblem<TypeTag> | ||
| 72 | { | ||
| 73 | using ParentType = ShallowWaterProblem<TypeTag>; | ||
| 74 | using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>; | ||
| 75 | using BoundaryTypes = Dumux::BoundaryTypes<GetPropType<TypeTag, Properties::ModelTraits>::numEq()>; | ||
| 76 | using Scalar = GetPropType<TypeTag, Properties::Scalar>; | ||
| 77 | using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices; | ||
| 78 | using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>; | ||
| 79 | using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView; | ||
| 80 | using GridVariables = GetPropType<TypeTag, Properties::GridVariables>; | ||
| 81 | using ElementFluxVariablesCache = typename GridVariables::GridFluxVariablesCache::LocalView; | ||
| 82 | using VolumeVariables = typename ElementVolumeVariables::VolumeVariables; | ||
| 83 | using FVElementGeometry = typename GetPropType<TypeTag, Properties::GridGeometry>::LocalView; | ||
| 84 | using SubControlVolumeFace = typename FVElementGeometry::SubControlVolumeFace; | ||
| 85 | using GridView = typename GetPropType<TypeTag, Properties::GridGeometry>::GridView; | ||
| 86 | using Element = typename GridView::template Codim<0>::Entity; | ||
| 87 | using GlobalPosition = typename Element::Geometry::GlobalCoordinate; | ||
| 88 | using NumEqVector = Dumux::NumEqVector<PrimaryVariables>; | ||
| 89 | using NeumannFluxes = NumEqVector; | ||
| 90 | using SubControlVolume = typename FVElementGeometry::SubControlVolume; | ||
| 91 | |||
| 92 | public: | ||
| 93 | 6 | PoiseuilleFlowProblem(std::shared_ptr<const GridGeometry> gridGeometry) | |
| 94 |
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18 | : ParentType(gridGeometry) |
| 95 | { | ||
| 96 |
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6 | name_ = getParam<std::string>("Problem.Name"); |
| 97 |
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12 | exactWaterDepth_.resize(gridGeometry->numDofs(), 0.0); |
| 98 |
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12 | exactVelocityX_.resize(gridGeometry->numDofs(), 0.0); |
| 99 |
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12 | exactVelocityY_.resize(gridGeometry->numDofs(), 0.0); |
| 100 |
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6 | bedSlope_ = getParam<Scalar>("Problem.BedSlope"); |
| 101 |
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6 | discharge_ = getParam<Scalar>("Problem.Discharge"); |
| 102 |
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6 | hBoundary_ = getParam<Scalar>("Problem.HBoundary"); |
| 103 |
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6 | alphaWall_ = getParam<Scalar>("Problem.AlphaWall"); |
| 104 |
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6 | turbViscosity_ = getParam<Scalar>("ShallowWater.TurbulentViscosity"); |
| 105 |
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6 | ksWall_ = getParam<Scalar>("Problem.KsWall"); |
| 106 |
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6 | wallFrictionLawType_ = getParam<std::string>("Problem.WallFrictionLaw"); |
| 107 | // Make the wallFrictionLawType_ lower case | ||
| 108 | 60 | std::transform(wallFrictionLawType_.begin(), wallFrictionLawType_.end(), wallFrictionLawType_.begin(), [](unsigned char c){ return std::tolower(c); }); | |
| 109 | 6 | } | |
| 110 | |||
| 111 | //! Get the analytical water depth | ||
| 112 | const std::vector<Scalar>& getExactWaterDepth() const | ||
| 113 |
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6 | { return exactWaterDepth_; } |
| 114 | |||
| 115 | //! Get the analytical U-velocity | ||
| 116 | const std::vector<Scalar>& getExactVelocityX() const | ||
| 117 |
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6 | { return exactVelocityX_; } |
| 118 | |||
| 119 | //! Get the analytical V-velocity | ||
| 120 | const std::vector<Scalar>& getExactVelocityY() const | ||
| 121 |
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6 | { return exactVelocityY_; } |
| 122 | |||
| 123 | //! Update the analytical solution | ||
| 124 | 219 | void updateAnalyticalSolution() | |
| 125 | { | ||
| 126 |
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149789 | for (const auto& element : elements(this->gridGeometry().gridView())) |
| 127 | { | ||
| 128 | 223698 | const auto eIdx = this->gridGeometry().elementMapper().index(element); | |
| 129 | 118892 | const auto& globalPos = element.geometry().center(); | |
| 130 | 104806 | const auto gravity = this->spatialParams().gravity(globalPos); | |
| 131 | 104806 | const Scalar y = globalPos[1]; | |
| 132 | 477636 | const Scalar width = this->gridGeometry().bBoxMax()[1] - this->gridGeometry().bBoxMin()[1]; | |
| 133 | 74566 | const Scalar h = hBoundary_; | |
| 134 | 74566 | const Scalar u = -(gravity*bedSlope_/(8.0*turbViscosity_)) * (4.0*y*y - width*width); | |
| 135 | 104806 | exactWaterDepth_[eIdx] = h; | |
| 136 | 104806 | exactVelocityX_[eIdx] = u; | |
| 137 | 149132 | exactVelocityY_[eIdx] = 0.0; | |
| 138 | } | ||
| 139 | 219 | } | |
| 140 | |||
| 141 | const std::string& name() const | ||
| 142 |
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6 | { return name_; } |
| 143 | |||
| 144 | /*! | ||
| 145 | * \name Boundary conditions | ||
| 146 | */ | ||
| 147 | // \{ | ||
| 148 | |||
| 149 | /*! | ||
| 150 | * \brief Specifies which kind of boundary condition should be | ||
| 151 | * used for which equation on a given boundary segment. | ||
| 152 | * | ||
| 153 | * \param globalPos The position for which the boundary type is set | ||
| 154 | */ | ||
| 155 | ✗ | BoundaryTypes boundaryTypesAtPos(const GlobalPosition &globalPos) const | |
| 156 | { | ||
| 157 | 148431 | BoundaryTypes bcTypes; | |
| 158 |
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148431 | bcTypes.setAllNeumann(); |
| 159 | ✗ | return bcTypes; | |
| 160 | } | ||
| 161 | |||
| 162 | /*! | ||
| 163 | * \brief Specifies the neumann boundary | ||
| 164 | * | ||
| 165 | * We need the Riemann invariants to compute the values depending of the boundary type. | ||
| 166 | * Since we use a weak imposition we do not have a dirichlet value. We impose fluxes | ||
| 167 | * based on q, h, etc. computed with the Riemann invariants | ||
| 168 | */ | ||
| 169 | 116671 | NeumannFluxes neumann(const Element& element, | |
| 170 | const FVElementGeometry& fvGeometry, | ||
| 171 | const ElementVolumeVariables& elemVolVars, | ||
| 172 | const ElementFluxVariablesCache& elemFluxVarsCache, | ||
| 173 | const SubControlVolumeFace& scvf) const | ||
| 174 | { | ||
| 175 | 116671 | NeumannFluxes values(0.0); | |
| 176 | |||
| 177 | 233342 | const auto& insideScv = fvGeometry.scv(scvf.insideScvIdx()); | |
| 178 | 116671 | const auto& insideVolVars = elemVolVars[insideScv]; | |
| 179 |
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116671 | const auto& unitNormal = scvf.unitOuterNormal(); |
| 180 |
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233342 | const auto gravity = this->spatialParams().gravity(scvf.center()); |
| 181 | std::array<Scalar, 3> boundaryStateVariables; | ||
| 182 | |||
| 183 | // impose discharge at the left side | ||
| 184 |
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700026 | if (scvf.center()[0] < this->gridGeometry().bBoxMin()[0] + eps_) |
| 185 | { | ||
| 186 | // Prescribe the exact q-distribution long the inflow boundaryStateVariables | ||
| 187 | // based on the parabolic u-profile: | ||
| 188 | // q(y) = (g*H*ib/(8*nu_T)) * (4*y^2^-W^2) | ||
| 189 | // Note the opposite sign to the velocity u, due to the fact that it is an inflow discharge | ||
| 190 | 36384 | const auto y = scvf.center()[1]; | |
| 191 | 109152 | const auto width = this->gridGeometry().bBoxMax()[1] - this->gridGeometry().bBoxMin()[1]; | |
| 192 | 18192 | const auto h = hBoundary_; | |
| 193 | // Now compute a weighted average between the constant q (from input) and the parabolic q from the analytical solution | ||
| 194 | // based on the prescribed alphaWall | ||
| 195 | 18192 | const auto q_in0 = (gravity*h*bedSlope_/(8.0*turbViscosity_)) * (4.0*y*y - width*width); | |
| 196 | 18192 | const auto q_in1 = discharge_; | |
| 197 | 18192 | const auto q_in = (1.0-alphaWall_)*q_in1 + alphaWall_*q_in0; | |
| 198 | 72768 | boundaryStateVariables = ShallowWater::fixedDischargeBoundary(q_in, | |
| 199 | insideVolVars.waterDepth(), | ||
| 200 | insideVolVars.velocity(0), | ||
| 201 | insideVolVars.velocity(1), | ||
| 202 | gravity, | ||
| 203 | unitNormal); | ||
| 204 | } | ||
| 205 | // impose water depth at the right side | ||
| 206 |
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590874 | else if (scvf.center()[0] > this->gridGeometry().bBoxMax()[0] - eps_) |
| 207 | { | ||
| 208 | 73224 | boundaryStateVariables = ShallowWater::fixedWaterDepthBoundary(hBoundary_, | |
| 209 | insideVolVars.waterDepth(), | ||
| 210 | insideVolVars.velocity(0), | ||
| 211 | insideVolVars.velocity(1), | ||
| 212 | gravity, | ||
| 213 | unitNormal); | ||
| 214 | } | ||
| 215 | // no flow boundary | ||
| 216 | else | ||
| 217 | { | ||
| 218 | // For the rough side walls of type no-slip we prescribe a slip condition based on alphaWall | ||
| 219 | // for smooth closed walls we prescribed a full-slip condition (zero-roughness) | ||
| 220 | |||
| 221 | // Get inside velocity components in cell face coordinates (t,n) using normal vector unitNormal | ||
| 222 | // Note that the first component is the normal component | ||
| 223 | // since we are rotating to the normal vector coordinate system | ||
| 224 |
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320692 | const Scalar insideVelocityNWall = insideVolVars.velocity(0)*unitNormal[0] + insideVolVars.velocity(1)*unitNormal[1]; |
| 225 |
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320692 | const Scalar insideVelocityTWall = -insideVolVars.velocity(0)*unitNormal[1] + insideVolVars.velocity(1)*unitNormal[0]; |
| 226 | |||
| 227 | // Initialisation of outside velocities | ||
| 228 | 80173 | auto outsideVelocityNWall = insideVelocityNWall; | |
| 229 | 80173 | auto outsideVelocityTWall = insideVelocityTWall; | |
| 230 | |||
| 231 | // Now set the outside (ghost cell) velocities based on the chosen slip condition | ||
| 232 |
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481038 | if ((scvf.center()[1] < this->gridGeometry().bBoxMin()[1] + eps_ || scvf.center()[1] > this->gridGeometry().bBoxMax()[1] - eps_) && (wallFrictionLawType_ == "noslip" || wallFrictionLawType_ == "nikuradse")) |
| 233 | { | ||
| 234 | // Set the outside state using the no-slip wall roughness conditions based on alphaWall | ||
| 235 | // alphaWall = 0.0: full slip (wall tangential velocity equals inside tangential velocity) | ||
| 236 | // alphaWall = 1.0: no slip (wall tangential velocity=0.0: point mirroring of the velocity) | ||
| 237 | // 0.0 < alphaWall < 1.0: 'partial' slip. | ||
| 238 | // e.g. for alphaWall = 0.5 the wall tangential velocity is half the inside tangential velocity. | ||
| 239 | 80173 | outsideVelocityNWall = -insideVelocityNWall; | |
| 240 | 80173 | outsideVelocityTWall = (1.0 - 2.0*alphaWall_)*insideVelocityTWall; | |
| 241 | } | ||
| 242 | else | ||
| 243 | { | ||
| 244 | // Set the outside state using the full-slip wall roughness conditions (line mirroring in the boundary face) | ||
| 245 | // Only mirror the normal component | ||
| 246 | ✗ | outsideVelocityNWall = -insideVelocityNWall; | |
| 247 | ✗ | outsideVelocityTWall = insideVelocityTWall; | |
| 248 | } | ||
| 249 | // Rotate back to cartesian coordinate system | ||
| 250 | 160346 | const Scalar outsideVelocityXWall = outsideVelocityNWall*unitNormal[0] - outsideVelocityTWall*unitNormal[1]; | |
| 251 | 160346 | const Scalar outsideVelocityYWall = outsideVelocityNWall*unitNormal[1] + outsideVelocityTWall*unitNormal[0]; | |
| 252 | |||
| 253 | 160346 | boundaryStateVariables[0] = insideVolVars.waterDepth(); | |
| 254 | 80173 | boundaryStateVariables[1] = outsideVelocityXWall; | |
| 255 | 160346 | boundaryStateVariables[2] = outsideVelocityYWall; | |
| 256 | |||
| 257 | } | ||
| 258 | |||
| 259 | 116671 | auto riemannFlux = ShallowWater::riemannProblem(insideVolVars.waterDepth(), | |
| 260 | 233342 | boundaryStateVariables[0], | |
| 261 | insideVolVars.velocity(0), | ||
| 262 | 233342 | boundaryStateVariables[1], | |
| 263 | insideVolVars.velocity(1), | ||
| 264 | 233342 | boundaryStateVariables[2], | |
| 265 | insideVolVars.bedSurface(), | ||
| 266 | insideVolVars.bedSurface(), | ||
| 267 | gravity, | ||
| 268 | unitNormal); | ||
| 269 | |||
| 270 |
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233342 | values[Indices::massBalanceIdx] = riemannFlux[0]; |
| 271 |
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233342 | values[Indices::velocityXIdx] = riemannFlux[1]; |
| 272 |
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233342 | values[Indices::velocityYIdx] = riemannFlux[2]; |
| 273 | |||
| 274 | // Addition of viscosity/diffusive flux rough wall boundaries | ||
| 275 | // No-slip wall (with coefficient alphaWall): | ||
| 276 | // Compute wall shear stress using turbulent viscosity and local velocity gradient | ||
| 277 | // Assume velocity gradient (in cell adjacent to wall) equal to alphaWall*(0 - u_c) | ||
| 278 | 116671 | std::array<Scalar, 3> roughWallFlux{}; | |
| 279 |
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700026 | if (scvf.center()[1] < this->gridGeometry().bBoxMin()[1] + eps_ || scvf.center()[1] > this->gridGeometry().bBoxMax()[1] - eps_) |
| 280 | { | ||
| 281 | // Distance vector between the inside cell center and the boundary face center | ||
| 282 | 240519 | const auto& cellCenterToBoundaryFaceCenter = scvf.center() - insideScv.center(); | |
| 283 | |||
| 284 | // The left (inside) state vector | ||
| 285 | 80173 | const auto& leftState = insideVolVars.priVars(); | |
| 286 | |||
| 287 |
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80173 | if (wallFrictionLawType_ == "noslip") |
| 288 | { | ||
| 289 | 80173 | roughWallFlux = ShallowWater::noslipWallBoundary(alphaWall_, | |
| 290 | 80173 | turbViscosity_, | |
| 291 | leftState, | ||
| 292 | cellCenterToBoundaryFaceCenter, | ||
| 293 | unitNormal); | ||
| 294 | } | ||
| 295 | ✗ | else if (wallFrictionLawType_ == "nikuradse") | |
| 296 | { | ||
| 297 | ✗ | roughWallFlux = ShallowWater::nikuradseWallBoundary(ksWall_, | |
| 298 | leftState, | ||
| 299 | cellCenterToBoundaryFaceCenter, | ||
| 300 | unitNormal); | ||
| 301 | } | ||
| 302 | } | ||
| 303 | |||
| 304 | 350013 | values[Indices::massBalanceIdx] += roughWallFlux[0]; | |
| 305 | 350013 | values[Indices::velocityXIdx] += roughWallFlux[1]; | |
| 306 | 350013 | values[Indices::velocityYIdx] += roughWallFlux[2]; | |
| 307 | |||
| 308 | 116671 | return values; | |
| 309 | } | ||
| 310 | |||
| 311 | // \} | ||
| 312 | |||
| 313 | /*! | ||
| 314 | * \brief Evaluate the initial values for a control volume. | ||
| 315 | * \param globalPos The position for which the boundary type is set | ||
| 316 | */ | ||
| 317 | 2038 | PrimaryVariables initialAtPos(const GlobalPosition &globalPos) const | |
| 318 | { | ||
| 319 | 2038 | PrimaryVariables values(0.0); | |
| 320 | |||
| 321 | // Set the initial values to the analytical solution | ||
| 322 | 4076 | const auto gravity = this->spatialParams().gravity(globalPos); | |
| 323 | 14266 | const auto width = this->gridGeometry().bBoxMax()[1] - this->gridGeometry().bBoxMin()[1]; | |
| 324 | |||
| 325 | 4076 | values[0] = hBoundary_; | |
| 326 | 8152 | values[1] = -(gravity*bedSlope_/(8.0*turbViscosity_)) * (4.0*globalPos[1]*globalPos[1] - width*width); | |
| 327 | 4076 | values[2] = 0.0; | |
| 328 | |||
| 329 | 2038 | return values; | |
| 330 | }; | ||
| 331 | |||
| 332 | private: | ||
| 333 | |||
| 334 | std::vector<Scalar> exactWaterDepth_; | ||
| 335 | std::vector<Scalar> exactVelocityX_; | ||
| 336 | std::vector<Scalar> exactVelocityY_; | ||
| 337 | |||
| 338 | Scalar hBoundary_; // water level at the outflow boundary | ||
| 339 | Scalar bedSlope_; // bed slope (positive downwards) | ||
| 340 | Scalar discharge_; // discharge at the inflow boundary | ||
| 341 | Scalar alphaWall_; // wall roughness coefficient for no-slip type wall roughness | ||
| 342 | Scalar ksWall_; // Nikuradse wall roughness height for Nikuradse type wall roughness | ||
| 343 | Scalar turbViscosity_; // turbulent viscosity | ||
| 344 | std::string wallFrictionLawType_; // wall friction law type | ||
| 345 | |||
| 346 | static constexpr Scalar eps_ = 1.0e-6; | ||
| 347 | std::string name_; | ||
| 348 | }; | ||
| 349 | |||
| 350 | } //end namespace Dumux | ||
| 351 | |||
| 352 | #endif | ||
| 353 |