diff --git a/dumux/freeflow/navierstokes/staggered/fluxvariables.hh b/dumux/freeflow/navierstokes/staggered/fluxvariables.hh
index 70b8c08024888ab2f6daf1f9db5849cfc516730e..cb1ffcb09077d7aba2613a41defd098347e19d1e 100644
--- a/dumux/freeflow/navierstokes/staggered/fluxvariables.hh
+++ b/dumux/freeflow/navierstokes/staggered/fluxvariables.hh
@@ -282,6 +282,12 @@ public:
const auto& faceVars = elemFaceVars[scvf];
const int 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).
+ Dune::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)
{
@@ -289,31 +295,39 @@ public:
// 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)
+ // If the current scvf is on a bounary and if there is a Neumann 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), but we sample the value at their actual positions.
+ if (currentScvfBoundaryTypes && 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())]
+ * elemVolVars[scvf.insideScvIdx()].extrusionFactor() * lateralScvf.area() * 0.5
+ * lateralScvf.directionSign();
+ continue;
+ }
+
+ // Create a boundaryTypes object (will be empty if not at a boundary).
Dune::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())
{
- // Construct a temporary scvf which corresponds to the staggered sub face, featuring the location
- // the sub faces's center.
- auto lateralBoundaryFaceCenter = scvf.pairData(localSubFaceIdx).lateralStaggeredFaceCenter + lateralScvf.center();
- lateralBoundaryFaceCenter *= 0.5;
-
- // ________________
- // --------####o### || frontal face of staggered half-control-volume
- // | || | current scvf # lateralBoundaryFace of interest, lies on lateral boundary
- // | || | x dof position
- // | || x~~~~> vel.Self -- element boundaries
- // | || | __ domain boundary
- // | || | o position at which the boundary conditions will be evaluated
- // ---------------- (lateralBoundaryFaceCenter)
-
- const auto lateralBoundaryFace = lateralScvf.makeBoundaryFace(lateralBoundaryFaceCenter);
-
- // Retrieve the boundary types that correspond to the sub face.
- lateralFaceBoundaryTypes.emplace(problem.boundaryTypes(element, lateralBoundaryFace));
+ // 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));
// 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.
@@ -323,13 +337,17 @@ public:
// 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, lateralBoundaryFace, elemVolVars, elemFaceVars))
+ if (incorporateWallFunction_(lateralFlux, problem, element, fvGeometry, scvf, elemVolVars, elemFaceVars, localSubFaceIdx))
continue;
// Check the consistency of the boundary conditions, only one of the following must be set
@@ -340,7 +358,7 @@ public:
if (admittableBcTypes.count() != 1)
{
DUNE_THROW(Dune::InvalidStateException, "Something went wrong with the boundary conditions "
- "for the momentum equations at global position " << lateralBoundaryFaceCenter);
+ "for the momentum equations at global position " << lateralStaggeredFaceCenter_(scvf, localSubFaceIdx));
}
}
@@ -457,9 +475,8 @@ private:
{
// Create a boundaryTypes object (will be empty if not at a boundary).
Dune::Std::optional<BoundaryTypes> bcTypes;
- const auto& boundaryFace = scvf.makeBoundaryFace(scvf.pairData(localSubFaceIdx).lateralStaggeredFaceCenter);
- // Get the boundary conditions if we are at a boundary.
+ // Get the boundary conditions if we are at a boundary. We sample the type of BC at the center of the current scvf.
if (scvf.boundary())
bcTypes.emplace(problem.boundaryTypes(element, scvf));
@@ -493,7 +510,11 @@ private:
if (!scvf.boundary())
return faceVars.velocityLateralOutside(localSubFaceIdx);
else if (bcTypes->isDirichlet(Indices::velocity(lateralFace.directionIndex())))
- return problem.dirichlet(element, boundaryFace)[Indices::velocity(lateralFace.directionIndex())];
+ {
+ // Sample the value of the Dirichlet BC at the center of the lateral face intersecting with the boundary.
+ const auto& lateralBoundaryFacePos = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
+ return problem.dirichlet(element, scvf.makeBoundaryFace(lateralBoundaryFacePos))[Indices::velocity(lateralFace.directionIndex())];
+ }
else
{
// Compute the BJS slip velocity at the boundary. Note that the relevant velocity gradient is now
@@ -529,7 +550,11 @@ private:
if (!lateralFace.boundary())
return faceVars.velocityParallel(localSubFaceIdx, 0);
else if (lateralFaceBoundaryTypes->isDirichlet(Indices::velocity(scvf.directionIndex())))
- return problem.dirichlet(element, makeParallelGhostFace_(scvf, localSubFaceIdx))[Indices::velocity(scvf.directionIndex())];
+ {
+ // Sample the value of the Dirichlet BC at the center of the staggered lateral face.
+ const auto& lateralBoundaryFacePos = lateralStaggeredFaceCenter_(scvf, localSubFaceIdx);
+ return problem.dirichlet(element, lateralFace.makeBoundaryFace(lateralBoundaryFacePos))[Indices::velocity(scvf.directionIndex())];
+ }
else
{
const auto& scv = fvGeometry.scv(scvf.insideScvIdx());
@@ -601,19 +626,23 @@ private:
private:
- //! helper function to conveniently create a ghost face which is outside the domain, parallel to the scvf of interest
- SubControlVolumeFace makeParallelGhostFace_(const SubControlVolumeFace& ownScvf, const int localSubFaceIdx) const
+ /*!
+ * \brief Get the location of the lateral staggered face's center.
+ * Only needed for boundary conditions if the current scvf or the lateral one is on a bounary.
+ *
+ * \verbatim
+ * --------#######o || 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, current scvf may lie on a boundary
+ * | || | o position at which the boundary conditions will be evaluated
+ * | || | (lateralStaggeredFaceCenter)
+ * ----------------
+ * \endverbatim
+ */
+ const GlobalPosition& lateralStaggeredFaceCenter_(const SubControlVolumeFace& scvf, const int localSubFaceIdx) const
{
- // ________________
- // --------#######o || frontal face of staggered half-control-volume
- // | || | current scvf # lateralBoundaryFace of interest, lies on lateral boundary
- // | || | x dof position
- // | || x~~~~> vel.Self -- element boundaries
- // | || | __ domain boundary
- // | || | o position at which the boundary conditions will be evaluated
- // ----------------
-
- return ownScvf.makeBoundaryFace(ownScvf.pairData(localSubFaceIdx).lateralStaggeredFaceCenter);
+ return scvf.pairData(localSubFaceIdx).lateralStaggeredFaceCenter;
};
//! helper function to get the averaged extrusion factor for a face
@@ -635,14 +664,19 @@ private:
const Element& element,
const FVElementGeometry& fvGeometry,
const SubControlVolumeFace& scvf,
- const SubControlVolumeFace& lateralBoundaryFace,
const ElementVolumeVariables& elemVolVars,
- const ElementFaceVariables& elemFaceVars) const
+ const ElementFaceVariables& elemFaceVars,
+ const std::size_t localSubFaceIdx) const
{
- if (problem.useWallFunction(element, lateralBoundaryFace, Indices::velocity(scvf.directionIndex())))
+ 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())]
- * elemVolVars[lateralBoundaryFace.insideScvIdx()].extrusionFactor() * lateralBoundaryFace.area() * 0.5;
+ * extrusionFactor_(elemVolVars, lateralScvf) * lateralScvf.area() * 0.5;
return true;
}
else
diff --git a/test/freeflow/navierstokes/channel/2d/CMakeLists.txt b/test/freeflow/navierstokes/channel/2d/CMakeLists.txt
index 8ab6ef643dec0733c1c8796876907749b9542986..120d1ee36c2e0c1e3907e647aa83ae3c3b2dfa47 100644
--- a/test/freeflow/navierstokes/channel/2d/CMakeLists.txt
+++ b/test/freeflow/navierstokes/channel/2d/CMakeLists.txt
@@ -2,7 +2,7 @@ dune_symlink_to_source_files(FILES "params.input" "params_navierstokes.input" "p
add_executable(test_ff_channel EXCLUDE_FROM_ALL main.cc)
-dune_add_test(NAME test_ff_stokes_channel
+dune_add_test(NAME test_ff_stokes_channel_outflow
TARGET test_ff_channel
LABELS freeflow navierstokes
CMAKE_GUARD HAVE_UMFPACK
@@ -10,9 +10,42 @@ dune_add_test(NAME test_ff_stokes_channel
CMD_ARGS --script fuzzy
--files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel-reference.vtu
${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel-00002.vtu
- --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_channel params.input
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_channel params.input -Problem.OutletCondition Outflow
-Problem.Name test_ff_stokes_channel")
+dune_add_test(NAME test_ff_stokes_channel_neumann_x_dirichlet_y
+ TARGET test_ff_channel
+ LABELS freeflow navierstokes
+ CMAKE_GUARD HAVE_UMFPACK
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel-00002.vtu
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_channel params.input -Problem.OutletCondition NeumannX_DirichletY
+ -Problem.Name test_ff_stokes_channel")
+
+dune_add_test(NAME test_ff_stokes_channel_neumann_x_neumann_y
+ TARGET test_ff_channel
+ LABELS freeflow navierstokes
+ CMAKE_GUARD HAVE_UMFPACK
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel_neumann-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_neumann-00002.vtu
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_channel params.input -Grid.UpperRight \"2 1\" -Grid.Cells \"50 25\"
+ -Problem.Name test_ff_stokes_channel_neumann -Problem.OutletCondition NeumannX_NeumannY -Problem.UseVelocityProfile true -Problem.OutletPressure 0")
+
+dune_add_test(NAME test_ff_stokes_channel_do_nothing
+ TARGET test_ff_channel
+ LABELS freeflow navierstokes
+ CMAKE_GUARD HAVE_UMFPACK
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel_do_nothing-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_do_nothing-00002.vtu
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_channel params.input -Problem.OutletCondition DoNothing -Grid.UpperRight \"2 1\" -Grid.Cells \"50 25\"
+ -Problem.Name test_ff_stokes_channel_do_nothing -Problem.OutletCondition DoNothing -Problem.UseVelocityProfile true -Problem.OutletPressure 0")
+
dune_add_test(NAME test_ff_navierstokes_channel
TARGET test_ff_channel
LABELS freeflow navierstokes
diff --git a/test/freeflow/navierstokes/channel/2d/params.input b/test/freeflow/navierstokes/channel/2d/params.input
index dcca38b35ae80695179bf5bb56c5156299b885a9..d2630ebbb5c361741ba88bdabf09354c2a727cba 100644
--- a/test/freeflow/navierstokes/channel/2d/params.input
+++ b/test/freeflow/navierstokes/channel/2d/params.input
@@ -11,6 +11,7 @@ Name = test_channel_stokes # name passed to the output routines
InletVelocity = 1
EnableGravity = false
EnableInertiaTerms = false
+OutletCondition = Outflow
[Component]
LiquidDensity = 1.0
diff --git a/test/freeflow/navierstokes/channel/2d/problem.hh b/test/freeflow/navierstokes/channel/2d/problem.hh
index d54983b1284829dae36ed6330c45ec52ea2672eb..0b575a3482355913cd9fb8bb34adce189551df17 100644
--- a/test/freeflow/navierstokes/channel/2d/problem.hh
+++ b/test/freeflow/navierstokes/channel/2d/problem.hh
@@ -96,23 +96,43 @@ class ChannelTestProblem : public NavierStokesProblem<TypeTag>
using BoundaryTypes = GetPropType<TypeTag, Properties::BoundaryTypes>;
using FVGridGeometry = GetPropType<TypeTag, Properties::FVGridGeometry>;
+ using FVElementGeometry = typename FVGridGeometry::LocalView;
+ using SubControlVolumeFace = typename FVGridGeometry::SubControlVolumeFace;
using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
using NumEqVector = GetPropType<TypeTag, Properties::NumEqVector>;
using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
using Scalar = GetPropType<TypeTag, Properties::Scalar>;
- static constexpr auto dimWorld = GetPropType<TypeTag, Properties::GridView>::dimensionworld;
-
using Element = typename FVGridGeometry::GridView::template Codim<0>::Entity;
using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
using TimeLoopPtr = std::shared_ptr<CheckPointTimeLoop<Scalar>>;
+ // the types of outlet boundary conditions
+ enum class OutletCondition
+ {
+ outflow, doNothing, neumannXdirichletY, neumannXneumannY
+ };
+
public:
ChannelTestProblem(std::shared_ptr<const FVGridGeometry> fvGridGeometry)
: ParentType(fvGridGeometry), eps_(1e-6)
{
inletVelocity_ = getParam<Scalar>("Problem.InletVelocity");
+ const auto tmp = getParam<std::string>("Problem.OutletCondition", "Outflow");
+ if (tmp == "Outflow")
+ outletCondition_ = OutletCondition::outflow;
+ else if (tmp == "DoNothing")
+ outletCondition_ = OutletCondition::doNothing;
+ else if (tmp == "NeumannX_DirichletY")
+ outletCondition_ = OutletCondition::neumannXdirichletY;
+ else if (tmp == "NeumannX_NeumannY")
+ outletCondition_ = OutletCondition::neumannXneumannY;
+ else
+ DUNE_THROW(Dune::InvalidStateException, tmp + " is not a valid outlet boundary condition");
+
+ useVelocityProfile_ = getParam<bool>("Problem.UseVelocityProfile", false);
+ outletPressure_ = getParam<Scalar>("Problem.OutletPressure", 1.1e5);
}
/*!
@@ -120,12 +140,6 @@ public:
*/
// \{
-
- bool shouldWriteRestartFile() const
- {
- return false;
- }
-
/*!
* \brief Returns the temperature within the domain in [K].
*
@@ -169,7 +183,19 @@ public:
}
else if(isOutlet_(globalPos))
{
- values.setDirichlet(Indices::pressureIdx);
+ if (outletCondition_ == OutletCondition::outflow)
+ values.setDirichlet(Indices::pressureIdx);
+ else if (outletCondition_ == OutletCondition::doNothing ||
+ outletCondition_ == OutletCondition::neumannXneumannY)
+ {
+ values.setNeumann(Indices::momentumXBalanceIdx);
+ values.setNeumann(Indices::momentumYBalanceIdx);
+ }
+ else // (outletCondition_ == OutletCondition::neumannXdirichletY)
+ {
+ values.setDirichlet(Indices::velocityYIdx);
+ values.setNeumann(Indices::momentumXBalanceIdx);
+ }
#if NONISOTHERMAL
values.setOutflow(Indices::energyEqIdx);
#endif
@@ -197,7 +223,10 @@ public:
if(isInlet_(globalPos))
{
- values[Indices::velocityXIdx] = inletVelocity_;
+ if (useVelocityProfile_)
+ values[Indices::velocityXIdx] = parabolicProfile(globalPos[1], inletVelocity_);
+ else
+ values[Indices::velocityXIdx] = inletVelocity_;
#if NONISOTHERMAL
// give the system some time so that the pressure can equilibrate, then start the injection of the hot liquid
if(time() >= 200.0)
@@ -208,6 +237,63 @@ public:
return values;
}
+ /*!
+ * \brief Evaluates the boundary conditions for a Neumann control volume.
+ *
+ * \param element The element for which the Neumann boundary condition is set
+ * \param fvGeometry The fvGeometry
+ * \param elemVolVars The element volume variables
+ * \param elemFaceVars The element face variables
+ * \param scvf The boundary sub control volume face
+ */
+ template<class ElementVolumeVariables, class ElementFaceVariables>
+ NumEqVector neumann(const Element& element,
+ const FVElementGeometry& fvGeometry,
+ const ElementVolumeVariables& elemVolVars,
+ const ElementFaceVariables& elemFaceVars,
+ const SubControlVolumeFace& scvf) const
+ {
+ NumEqVector values(0.0);
+
+ values[scvf.directionIndex()] = initialAtPos(scvf.center())[Indices::pressureIdx];
+
+ // make sure to normalize the pressure if the property is set true
+ if (getPropValue<TypeTag, Properties::NormalizePressure>())
+ values[scvf.directionIndex()] -= initialAtPos(scvf.center())[Indices::pressureIdx];
+
+ if (outletCondition_ != OutletCondition::doNothing)
+ values[1] = -dudy(scvf.center()[1], inletVelocity_) * elemVolVars[scvf.insideScvIdx()].viscosity();
+
+ return values;
+ }
+
+ /*!
+ * \brief A parabolic velocity profile.
+ *
+ * \param y The position where the velocity is evaluated.
+ * \param vMax The profile's maxmium velocity.
+ */
+ Scalar parabolicProfile(const Scalar y, const Scalar vMax) const
+ {
+ const Scalar yMin = this->fvGridGeometry().bBoxMin()[1];
+ const Scalar yMax = this->fvGridGeometry().bBoxMax()[1];
+ return vMax * (y - yMin)*(yMax - y) / (0.25*(yMax - yMin)*(yMax - yMin));
+ }
+
+ /*!
+ * \brief The partial dervivative of the horizontal velocity (following a parabolic profile for
+ * Stokes flow) w.r.t. to the y-coordinate (du/dy).
+ *
+ * \param y The position where the derivative is evaluated.
+ * \param vMax The profile's maxmium velocity.
+ */
+ Scalar dudy(const Scalar y, const Scalar vMax) const
+ {
+ const Scalar yMin = this->fvGridGeometry().bBoxMin()[1];
+ const Scalar yMax = this->fvGridGeometry().bBoxMax()[1];
+ return vMax * (4.0*yMin + 4*yMax - 8.0*y) / ((yMin-yMax)*(yMin-yMax));
+ }
+
// \}
/*!
@@ -223,7 +309,7 @@ public:
PrimaryVariables initialAtPos(const GlobalPosition &globalPos) const
{
PrimaryVariables values;
- values[Indices::pressureIdx] = 1.1e+5;
+ values[Indices::pressureIdx] = outletPressure_;
values[Indices::velocityXIdx] = 0.0;
values[Indices::velocityYIdx] = 0.0;
@@ -261,6 +347,9 @@ private:
Scalar eps_;
Scalar inletVelocity_;
+ Scalar outletPressure_;
+ OutletCondition outletCondition_;
+ bool useVelocityProfile_;
TimeLoopPtr timeLoop_;
};
} // end namespace Dumux
diff --git a/test/references/test_ff_stokes_channel_do_nothing-reference.vtu b/test/references/test_ff_stokes_channel_do_nothing-reference.vtu
new file mode 100644
index 0000000000000000000000000000000000000000..88b31f17c4f6b0631f8d0bb27c4ef125c3834c64
--- /dev/null
+++ b/test/references/test_ff_stokes_channel_do_nothing-reference.vtu
@@ -0,0 +1,1616 @@
+<?xml version="1.0"?>
+<VTKFile type="UnstructuredGrid" version="0.1" byte_order="LittleEndian">
+ <UnstructuredGrid>
+ <Piece NumberOfCells="1250" NumberOfPoints="1326">
+ <CellData Scalars="p" Vectors="velocity_liq (m/s)">
+ <DataArray type="Float32" Name="p" NumberOfComponents="1" format="ascii">
+ 15.2536 14.9779 14.6775 14.3668 14.0517 13.7345 13.4164 13.0977 12.7788 12.4597 12.1404 11.8211
+ 11.5018 11.1825 10.8632 10.5438 10.2245 9.9051 9.58572 9.26632 8.94689 8.62743 8.30792 7.98835
+ 7.6687 7.34898 7.02917 6.70927 6.38929 6.06927 5.74926 5.42939 5.10985 4.79093 4.47313 4.15716
+ 3.84409 3.5355 3.23372 2.94211 2.66558 2.41128 2.18974 2.01654 1.91492 1.91918 2.07692 2.43937
+ 2.99413 3.36659 15.2894 14.9843 14.6765 14.3645 14.0496 13.7329 13.4152 13.0968 12.7781 12.4592
+ 12.1401 11.8209 11.5016 11.1824 10.8631 10.5437 10.2244 9.90506 9.5857 9.26632 8.94692 8.6275
+ 8.30804 7.98854 7.669 7.34943 7.02983 6.71023 6.39065 6.07117 5.75189 5.43298 5.11469 4.7974
+ 4.48166 4.16827 3.85838 3.55361 3.25623 2.96945 2.6977 2.4471 2.22606 2.04581 1.92081 1.86769
+ 1.8999 2.01079 2.13165 2.06435 15.309 14.9935 14.6797 14.3651 14.0492 13.7323 13.4145 13.0963
+ 12.7777 12.4588 12.1398 11.8207 11.5015 11.1822 10.863 10.5437 10.2244 9.90504 9.5857 9.26635
+ 8.94698 8.6276 8.30819 7.98877 7.66934 7.34991 7.0305 6.71115 6.39191 6.07286 5.75414 5.43594
+ 5.11853 4.8023 4.48783 4.17589 3.86755 3.56431 3.26815 2.98176 2.70865 2.45337 2.22157 2.01984
+ 1.85483 1.73085 1.64421 1.57225 1.45734 1.19987 15.32 15.0009 14.6838 14.367 14.05 13.7325
+ 13.4145 13.0961 12.7774 12.4586 12.1396 11.8205 11.5013 11.1821 10.8629 10.5436 10.2243 9.90504
+ 9.58572 9.26639 8.94706 8.62771 8.30836 7.98902 7.66968 7.35037 7.03112 6.71196 6.39297 6.07423
+ 5.75588 5.43811 5.12118 4.80546 4.49147 4.17989 3.87164 3.56793 3.27031 2.98078 2.70178 2.43616
+ 2.18702 1.95714 1.74782 1.55669 1.37384 1.17666 0.925586 0.568756 15.3266 15.0062 14.6874 14.3693
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diff --git a/test/references/test_ff_stokes_channel_neumann-reference.vtu b/test/references/test_ff_stokes_channel_neumann-reference.vtu
new file mode 100644
index 0000000000000000000000000000000000000000..4470ef84f0fc66ed22359c440dd5a1e0990894a7
--- /dev/null
+++ b/test/references/test_ff_stokes_channel_neumann-reference.vtu
@@ -0,0 +1,1616 @@
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