From d55e34fd74897b1bb9b54d5899c13ca8eebe482e Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 16:46:31 +0200 Subject: [PATCH 01/15] fix indices --- .../boundary/open_boundary/mirroring.jl | 32 ++++++++++++------- 1 file changed, 20 insertions(+), 12 deletions(-) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 52d4d5c0d..11f7d4737 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -86,6 +86,8 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, grad_kernel = smoothing_kernel_grad(fluid_system, pos_diff, distance, particle) + # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper, + # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle L, R = correction_arrays(kernel_value, grad_kernel, pos_diff, rho_b, m_b) correction_matrix[] += L @@ -164,9 +166,12 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, end function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{3}, rho_b, m_b) - x_ab = pos_diff[1] - y_ab = pos_diff[2] - z_ab = pos_diff[3] + # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper (Tafuni et al. 2018), + # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle. + # Note that in eq. (16) and (18) the indices are swapped, i.e. `x_{kl}` is used instead of `x_{lk}`. + x_ba = -pos_diff[1] + y_ba = -pos_diff[2] + z_ba = -pos_diff[3] grad_W_ab_x = grad_W_ab[1] grad_W_ab_y = grad_W_ab[2] @@ -174,10 +179,10 @@ function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{3}, rho_b, m_b) V_b = m_b / rho_b - M = @SMatrix [W_ab W_ab*x_ab W_ab*y_ab W_ab*z_ab; - grad_W_ab_x grad_W_ab_x*x_ab grad_W_ab_x*y_ab grad_W_ab_x*z_ab; - grad_W_ab_y grad_W_ab_y*x_ab grad_W_ab_y*y_ab grad_W_ab_y*z_ab; - grad_W_ab_z grad_W_ab_z*x_ab grad_W_ab_z*y_ab grad_W_ab_z*z_ab] + M = @SMatrix [W_ab W_ab*x_ba W_ab*y_ba W_ab*z_ba; + grad_W_ab_x grad_W_ab_x*x_ba grad_W_ab_x*y_ba grad_W_ab_x*z_ba; + grad_W_ab_y grad_W_ab_y*x_ba grad_W_ab_y*y_ba grad_W_ab_y*z_ba; + grad_W_ab_z grad_W_ab_z*x_ba grad_W_ab_z*y_ba grad_W_ab_z*z_ba] L = V_b * M @@ -187,17 +192,20 @@ function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{3}, rho_b, m_b) end function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{2}, rho_b, m_b) - x_ab = pos_diff[1] - y_ab = pos_diff[2] + # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper, + # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle. + # Note that in eq. (16) and (18) the indices are swapped, i.e. `x_{kl}` is used instead of `x_{lk}`. + x_ba = -pos_diff[1] + y_ba = -pos_diff[2] grad_W_ab_x = grad_W_ab[1] grad_W_ab_y = grad_W_ab[2] V_b = m_b / rho_b - M = @SMatrix [W_ab W_ab*x_ab W_ab*y_ab; - grad_W_ab_x grad_W_ab_x*x_ab grad_W_ab_x*y_ab; - grad_W_ab_y grad_W_ab_y*x_ab grad_W_ab_y*y_ab] + M = @SMatrix [W_ab W_ab*x_ba W_ab*y_ba; + grad_W_ab_x grad_W_ab_x*x_ba grad_W_ab_x*y_ba; + grad_W_ab_y grad_W_ab_y*x_ba grad_W_ab_y*y_ba] L = V_b * M R = V_b * SVector(W_ab, grad_W_ab_x, grad_W_ab_y) From 3f21208ab9549f9fbd4b82e876ddf6b6a5f52a45 Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 19:21:24 +0200 Subject: [PATCH 02/15] fix tests --- .../data/open_boundary_extrapolated_2d.csv | 402 +- .../data/open_boundary_extrapolated_3d.csv | 8002 +++++------ .../open_boundary_extrapolated_skew_2d.csv | 582 +- .../open_boundary_extrapolated_skew_3d.csv | 11602 ++++++++-------- 4 files changed, 10294 insertions(+), 10294 deletions(-) diff --git a/test/schemes/boundary/open_boundary/data/open_boundary_extrapolated_2d.csv b/test/schemes/boundary/open_boundary/data/open_boundary_extrapolated_2d.csv index ad7d187e9..7cf6e7df7 100644 --- a/test/schemes/boundary/open_boundary/data/open_boundary_extrapolated_2d.csv +++ b/test/schemes/boundary/open_boundary/data/open_boundary_extrapolated_2d.csv @@ -1,201 +1,201 @@ -"v:0","v:1","p","Points:0","Points:1","Points:2" --0.13938,0,-0.40802,-0.025,0.025,0 --0.40145,0,-0.30006,-0.025,0.075,0 --0.62417,0,-0.16753,-0.025,0.125,0 --0.7865,0,-0.037593,-0.025,0.175,0 --0.87184,0,0.042713,-0.025,0.225,0 --0.87184,0,0.042713,-0.025,0.275,0 --0.7865,0,-0.037593,-0.025,0.325,0 --0.62417,0,-0.16753,-0.025,0.375,0 --0.40074,0,-0.29747,-0.025,0.425,0 --0.13809,0,-0.37778,-0.025,0.475,0 -0.13809,0,-0.37778,-0.025,0.525,0 -0.40074,0,-0.29747,-0.025,0.575,0 -0.62417,0,-0.16753,-0.025,0.625,0 -0.7865,0,-0.037593,-0.025,0.675,0 -0.87184,0,0.042713,-0.025,0.725,0 -0.87184,0,0.042713,-0.025,0.775,0 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+1000,0,-0.33699,0.33699,-1.9624,0.975,1.0689,0.41342 +1000,0,-0.24073,0.24073,-1.5287,0.975,1.1034,0.44787 +1000,0,-0.13372,0.13372,-0.85307,0.975,1.1378,0.48232 +1000,0,-0.022309,0.022309,-0.083282,0.975,1.1723,0.51677 +1000,0,0.092251,-0.092251,0.68136,0.975,1.2067,0.55123 +1000,0,0.2028,-0.2028,1.2631,0.975,1.2412,0.58568 +1000,0,0.30153,-0.30153,1.5633,0.975,1.2756,0.62013 +1000,0,0.38778,-0.38778,1.5462,0.975,1.3101,0.65458 From 96fc751a36e66d9f53422577072e786e47b8302f Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 20:17:28 +0200 Subject: [PATCH 03/15] add PST in pipe flow example --- examples/fluid/pipe_flow_2d.jl | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/examples/fluid/pipe_flow_2d.jl b/examples/fluid/pipe_flow_2d.jl index 3a65f4b81..8feb80f30 100644 --- a/examples/fluid/pipe_flow_2d.jl +++ b/examples/fluid/pipe_flow_2d.jl @@ -155,7 +155,8 @@ saving_callback = SolutionSavingCallback(dt=0.02, prefix="") extra_callback = nothing -callbacks = CallbackSet(info_callback, saving_callback, UpdateCallback(), extra_callback) +callbacks = CallbackSet(info_callback, saving_callback, UpdateCallback(), + ParticleShiftingCallback(), extra_callback) sol = solve(ode, RDPK3SpFSAL35(), abstol=1e-5, # Default abstol is 1e-6 (may need to be tuned to prevent boundary penetration) From e08c9ee18a92d5ca140c47f5ef7dd6eed62aaea7 Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 23:16:29 +0200 Subject: [PATCH 04/15] fix write pressure and density --- src/io/write_vtk.jl | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/src/io/write_vtk.jl b/src/io/write_vtk.jl index 0e5aaf4c1..dc3f4ada9 100644 --- a/src/io/write_vtk.jl +++ b/src/io/write_vtk.jl @@ -401,8 +401,10 @@ end function write2vtk!(vtk, v, u, t, system::OpenBoundarySPHSystem; write_meta_data=true) vtk["velocity"] = [current_velocity(v, system, particle) for particle in active_particles(system)] - vtk["density"] = current_density(v, system) - vtk["pressure"] = current_pressure(v, system) + vtk["density"] = [current_density(v, system, particle) + for particle in active_particles(system)] + vtk["pressure"] = [current_pressure(v, system, particle) + for particle in active_particles(system)] if write_meta_data vtk["boundary_zone"] = type2string(first(typeof(system.boundary_zone).parameters)) From 805109e139383895b1f109c3eb187d73df4e9e1f Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 23:16:45 +0200 Subject: [PATCH 05/15] remove velocity projection --- src/schemes/boundary/open_boundary/mirroring.jl | 6 ------ 1 file changed, 6 deletions(-) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 11f7d4737..0fde1ba2e 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -76,12 +76,6 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, pressure_b = current_pressure(v_fluid, fluid_system, neighbor) v_b = current_velocity(v_fluid, fluid_system, neighbor) - # Project `v_b` on the normal direction of the boundary zone - # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: - # "Because flow from the inlet interface occurs perpendicular to the boundary, - # only this component of interpolated velocity is kept [...]" - v_b = dot(v_b, boundary_zone.plane_normal) * boundary_zone.plane_normal - kernel_value = smoothing_kernel(fluid_system, distance, particle) grad_kernel = smoothing_kernel_grad(fluid_system, pos_diff, distance, particle) From 161665a8a2ad2f8d269ccf8a5d2a681709a3ca61 Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 23:22:48 +0200 Subject: [PATCH 06/15] Revert "remove velocity projection" This reverts commit 805109e139383895b1f109c3eb187d73df4e9e1f. --- src/schemes/boundary/open_boundary/mirroring.jl | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 0fde1ba2e..11f7d4737 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -76,6 +76,12 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, pressure_b = current_pressure(v_fluid, fluid_system, neighbor) v_b = current_velocity(v_fluid, fluid_system, neighbor) + # Project `v_b` on the normal direction of the boundary zone + # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: + # "Because flow from the inlet interface occurs perpendicular to the boundary, + # only this component of interpolated velocity is kept [...]" + v_b = dot(v_b, boundary_zone.plane_normal) * boundary_zone.plane_normal + kernel_value = smoothing_kernel(fluid_system, distance, particle) grad_kernel = smoothing_kernel_grad(fluid_system, pos_diff, distance, particle) From 4b960a4aef04f9a0f70adedae428dab7a06e34a8 Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 11 Jun 2025 23:51:48 +0200 Subject: [PATCH 07/15] project v only for inflow --- src/schemes/boundary/open_boundary/mirroring.jl | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 11f7d4737..f9e0bc0f4 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -30,7 +30,7 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, semi, t; prescribed_density=false, prescribed_pressure=false, prescribed_velocity=false) (; pressure, density, boundary_zone, reference_density, - reference_velocity, reference_pressure) = system + reference_velocity, reference_pressure) = system fluid_system = corresponding_fluid_system(system, semi) @@ -76,11 +76,13 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, pressure_b = current_pressure(v_fluid, fluid_system, neighbor) v_b = current_velocity(v_fluid, fluid_system, neighbor) - # Project `v_b` on the normal direction of the boundary zone - # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: - # "Because flow from the inlet interface occurs perpendicular to the boundary, - # only this component of interpolated velocity is kept [...]" - v_b = dot(v_b, boundary_zone.plane_normal) * boundary_zone.plane_normal + if boundary_zone.boundary_type isa InFlow + # Project `v_b` on the normal direction of the boundary zone + # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: + # "Because flow from the inlet interface occurs perpendicular to the boundary, + # only this component of interpolated velocity is kept [...]" + v_b = dot(v_b, boundary_zone.plane_normal) * boundary_zone.plane_normal + end kernel_value = smoothing_kernel(fluid_system, distance, particle) grad_kernel = smoothing_kernel_grad(fluid_system, pos_diff, distance, From 1693c0d2355376c05b3be927aa2f582021ef851f Mon Sep 17 00:00:00 2001 From: LasNikas Date: Thu, 12 Jun 2025 18:31:54 +0200 Subject: [PATCH 08/15] project only for inflow --- .../boundary/open_boundary/mirroring.jl | 28 ++++++++++++------- 1 file changed, 18 insertions(+), 10 deletions(-) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index f9e0bc0f4..8676939a6 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -30,7 +30,7 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, semi, t; prescribed_density=false, prescribed_pressure=false, prescribed_velocity=false) (; pressure, density, boundary_zone, reference_density, - reference_velocity, reference_pressure) = system + reference_velocity, reference_pressure) = system fluid_system = corresponding_fluid_system(system, semi) @@ -74,15 +74,13 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, m_b = hydrodynamic_mass(fluid_system, neighbor) rho_b = current_density(v_fluid, fluid_system, neighbor) pressure_b = current_pressure(v_fluid, fluid_system, neighbor) - v_b = current_velocity(v_fluid, fluid_system, neighbor) - - if boundary_zone.boundary_type isa InFlow - # Project `v_b` on the normal direction of the boundary zone - # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: - # "Because flow from the inlet interface occurs perpendicular to the boundary, - # only this component of interpolated velocity is kept [...]" - v_b = dot(v_b, boundary_zone.plane_normal) * boundary_zone.plane_normal - end + v_b_ = current_velocity(v_fluid, fluid_system, neighbor) + + # Project `v_b_` on the normal direction of the boundary zone (only for inflow boundaries). + # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: + # "Because flow from the inlet interface occurs perpendicular to the boundary, + # only this component of interpolated velocity is kept [...]" + v_b = project_velocity_on_plane_normal(v_b_, boundary_zone) kernel_value = smoothing_kernel(fluid_system, distance, particle) grad_kernel = smoothing_kernel_grad(fluid_system, pos_diff, distance, @@ -221,3 +219,13 @@ function mirror_position(particle_coords, boundary_zone) return particle_coords - 2 * dist * boundary_zone.plane_normal end + +project_velocity_on_plane_normal(vel, boundary_zone) = vel + +function project_velocity_on_plane_normal(vel, boundary_zone::BoundaryZone{InFlow}) + # Project `v_b` on the normal direction of the boundary zone + # See https://doi.org/10.1016/j.jcp.2020.110029 Section 3.3.: + # "Because flow from the inlet interface occurs perpendicular to the boundary, + # only this component of interpolated velocity is kept [...]" + return dot(vel, boundary_zone.plane_normal) * boundary_zone.plane_normal +end From 5ce6cca74396f496b522ab8293625bd85fb61b06 Mon Sep 17 00:00:00 2001 From: LasNikas Date: Thu, 12 Jun 2025 18:49:49 +0200 Subject: [PATCH 09/15] fix tests --- test/examples/examples_fluid.jl | 10 +++++----- test/schemes/boundary/open_boundary/mirroring.jl | 4 ++-- 2 files changed, 7 insertions(+), 7 deletions(-) diff --git a/test/examples/examples_fluid.jl b/test/examples/examples_fluid.jl index 4c557ee46..45bf2ece1 100644 --- a/test/examples/examples_fluid.jl +++ b/test/examples/examples_fluid.jl @@ -339,7 +339,8 @@ end @trixi_testset "fluid/pipe_flow_2d.jl - steady state reached (`dt`)" begin - steady_state_reached = SteadyStateReachedCallback(; dt=0.002, interval_size=10) + steady_state_reached = SteadyStateReachedCallback(; dt=0.002, interval_size=10, + reltol=1e-3) @trixi_test_nowarn trixi_include(@__MODULE__, joinpath(examples_dir(), "fluid", @@ -353,13 +354,12 @@ end @trixi_testset "fluid/pipe_flow_2d.jl - steady state reached (`interval`)" begin - steady_state_reached = SteadyStateReachedCallback(; interval=1, - interval_size=10, - abstol=1.0e-5, reltol=1.0e-4) + steady_state_reached = SteadyStateReachedCallback(; interval=1, interval_size=10, + reltol=1e-3) @trixi_test_nowarn trixi_include(@__MODULE__, joinpath(examples_dir(), "fluid", "pipe_flow_2d.jl"), - extra_callback=steady_state_reached, + extra_callback=steady_state_reached, dtmax=2e-3, tspan=(0.0, 1.5), viscosity_boundary=nothing) # Make sure that the simulation is terminated after a reasonable amount of time diff --git a/test/schemes/boundary/open_boundary/mirroring.jl b/test/schemes/boundary/open_boundary/mirroring.jl index 0865a440a..7892db837 100644 --- a/test/schemes/boundary/open_boundary/mirroring.jl +++ b/test/schemes/boundary/open_boundary/mirroring.jl @@ -55,7 +55,7 @@ fluid_system.cache.density .= domain_fluid.density @testset verbose=true "plane normal $i" for i in eachindex(files) - inflow = BoundaryZone(; plane=plane_boundary[i], + inflow = BoundaryZone(; plane=plane_boundary[i], boundary_type=InFlow(), plane_normal=plane_boundary_normal[i], open_boundary_layers=10, density=1000.0, particle_spacing) @@ -150,7 +150,7 @@ fluid_system.cache.density .= domain_fluid.density @testset verbose=true "plane normal $i" for i in eachindex(files) - inflow = BoundaryZone(; plane=plane_boundary[i], + inflow = BoundaryZone(; plane=plane_boundary[i], boundary_type=InFlow(), plane_normal=plane_boundary_normal[i], open_boundary_layers=10, density=1000.0, particle_spacing) From 09399534dc1270ba744fb94cea1066b2a134716b Mon Sep 17 00:00:00 2001 From: LasNikas Date: Thu, 12 Jun 2025 18:56:51 +0200 Subject: [PATCH 10/15] fix comments --- src/schemes/boundary/open_boundary/mirroring.jl | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 8676939a6..b7f92787a 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -86,7 +86,7 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, grad_kernel = smoothing_kernel_grad(fluid_system, pos_diff, distance, particle) - # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper, + # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper (Tafuni et al., 2018), # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle L, R = correction_arrays(kernel_value, grad_kernel, pos_diff, rho_b, m_b) @@ -166,9 +166,9 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, end function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{3}, rho_b, m_b) - # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper (Tafuni et al. 2018), - # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle. - # Note that in eq. (16) and (18) the indices are swapped, i.e. `x_{kl}` is used instead of `x_{lk}`. + # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper (Tafuni et al., 2018), + # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle + # Note that in eq. (16) and (17) the indices are swapped, i.e. `x_{lk}` is used instead of `x_{kl}`. x_ba = -pos_diff[1] y_ba = -pos_diff[2] z_ba = -pos_diff[3] @@ -192,9 +192,9 @@ function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{3}, rho_b, m_b) end function correction_arrays(W_ab, grad_W_ab, pos_diff::SVector{2}, rho_b, m_b) - # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper, - # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle. - # Note that in eq. (16) and (18) the indices are swapped, i.e. `x_{kl}` is used instead of `x_{lk}`. + # `pos_diff` corresponds to `x_{kl} = x_k - x_l` in the paper (Tafuni et al., 2018), + # where `x_k` is the position of the ghost node and `x_l` is the position of the neighbor particle + # Note that in eq. (16) and (17) the indices are swapped, i.e. `x_{lk}` is used instead of `x_{kl}`. x_ba = -pos_diff[1] y_ba = -pos_diff[2] From c8833aa3ca839c67dd2ad090aa1ef45157c9368c Mon Sep 17 00:00:00 2001 From: LasNikas Date: Thu, 12 Jun 2025 21:36:29 +0200 Subject: [PATCH 11/15] check velocity direction --- .../boundary/open_boundary/mirroring.jl | 26 +++++++++++++++++++ 1 file changed, 26 insertions(+) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index b7f92787a..9274ba4d8 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -135,6 +135,9 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, v_open_boundary[dim, particle] = f_v[1] + dot(pos_diff, df_v) end + + # In the inflow zone, the velocity must be oriented in the same direction as the prescribed flow direction. + check_velocity_direction!(v_open_boundary, system, particle, boundary_zone) end if prescribed_density @@ -229,3 +232,26 @@ function project_velocity_on_plane_normal(vel, boundary_zone::BoundaryZone{InFlo # only this component of interpolated velocity is kept [...]" return dot(vel, boundary_zone.plane_normal) * boundary_zone.plane_normal end + +function check_velocity_direction!(v_open_boundary, system, particle, boundary_zone) + return v_open_boundary +end + +# In the inflow zone, the velocity must be oriented in the same direction as the prescribed flow direction. +function check_velocity_direction!(v_open_boundary, system, particle, + boundary_zone::BoundaryZone{InFlow}) + (; flow_direction) = boundary_zone + + v_particle = current_velocity(v_open_boundary, system, particle) + + if dot(v_particle, flow_direction) < 0 + # Reflect the velocity vector across the flow direction + v_reflected = v_particle - 2 * dot(v_particle, flow_direction) * flow_direction + + @inbounds for dim in eachindex(v_particle) + v_open_boundary[dim, particle] = v_reflected[dim] + end + end + + return v_open_boundary +end From f3340c81627ee49048f6ca1640dd78d89463465f Mon Sep 17 00:00:00 2001 From: LasNikas Date: Thu, 12 Jun 2025 23:01:18 +0200 Subject: [PATCH 12/15] Revert "check velocity direction" This reverts commit c8833aa3ca839c67dd2ad090aa1ef45157c9368c. --- .../boundary/open_boundary/mirroring.jl | 26 ------------------- 1 file changed, 26 deletions(-) diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 9274ba4d8..b7f92787a 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -135,9 +135,6 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, v_open_boundary[dim, particle] = f_v[1] + dot(pos_diff, df_v) end - - # In the inflow zone, the velocity must be oriented in the same direction as the prescribed flow direction. - check_velocity_direction!(v_open_boundary, system, particle, boundary_zone) end if prescribed_density @@ -232,26 +229,3 @@ function project_velocity_on_plane_normal(vel, boundary_zone::BoundaryZone{InFlo # only this component of interpolated velocity is kept [...]" return dot(vel, boundary_zone.plane_normal) * boundary_zone.plane_normal end - -function check_velocity_direction!(v_open_boundary, system, particle, boundary_zone) - return v_open_boundary -end - -# In the inflow zone, the velocity must be oriented in the same direction as the prescribed flow direction. -function check_velocity_direction!(v_open_boundary, system, particle, - boundary_zone::BoundaryZone{InFlow}) - (; flow_direction) = boundary_zone - - v_particle = current_velocity(v_open_boundary, system, particle) - - if dot(v_particle, flow_direction) < 0 - # Reflect the velocity vector across the flow direction - v_reflected = v_particle - 2 * dot(v_particle, flow_direction) * flow_direction - - @inbounds for dim in eachindex(v_particle) - v_open_boundary[dim, particle] = v_reflected[dim] - end - end - - return v_open_boundary -end From a958907f7a6bbdf1be47f3716186ef8d0f67e3cb Mon Sep 17 00:00:00 2001 From: LasNikas Date: Sun, 15 Jun 2025 14:47:31 +0200 Subject: [PATCH 13/15] averaging inflow velocity --- examples/fluid/pipe_flow_2d.jl | 14 +++-- .../method_of_characteristics.jl | 15 ++++-- .../boundary/open_boundary/mirroring.jl | 51 ++++++++++++++++++- 3 files changed, 70 insertions(+), 10 deletions(-) diff --git a/examples/fluid/pipe_flow_2d.jl b/examples/fluid/pipe_flow_2d.jl index 3a65f4b81..dd8d3d77c 100644 --- a/examples/fluid/pipe_flow_2d.jl +++ b/examples/fluid/pipe_flow_2d.jl @@ -49,7 +49,7 @@ pipe.boundary.coordinates[1, :] .-= particle_spacing * open_boundary_layers NDIMS = ndims(pipe.fluid) -n_buffer_particles = 4 * pipe.n_particles_per_dimension[2]^(NDIMS - 1) +n_buffer_particles = 100 * pipe.n_particles_per_dimension[2]^(NDIMS - 1) # ========================================================================================== # ==== Fluid @@ -98,6 +98,7 @@ boundary_type_in = InFlow() plane_in = ([0.0, 0.0], [0.0, domain_size[2]]) inflow = BoundaryZone(; plane=plane_in, plane_normal=flow_direction, open_boundary_layers, density=fluid_density, particle_spacing, + # shift_particles=false, boundary_coordinates=pipe.boundary.coordinates, boundary_type=boundary_type_in) reference_velocity_in = velocity_function2d @@ -131,15 +132,15 @@ viscosity_boundary = ViscosityAdami(nu=1e-4) boundary_model = BoundaryModelDummyParticles(pipe.boundary.density, pipe.boundary.mass, AdamiPressureExtrapolation(), state_equation=state_equation, - viscosity=viscosity_boundary, + # viscosity=viscosity_boundary, smoothing_kernel, smoothing_length) boundary_system = BoundarySPHSystem(pipe.boundary, boundary_model) # ========================================================================================== # ==== Simulation -min_corner = minimum(pipe.boundary.coordinates .- particle_spacing, dims=2) -max_corner = maximum(pipe.boundary.coordinates .+ particle_spacing, dims=2) +min_corner = minimum(pipe.boundary.coordinates .- 100 * particle_spacing, dims=2) +max_corner = maximum(pipe.boundary.coordinates .+ 100 * particle_spacing, dims=2) nhs = GridNeighborhoodSearch{NDIMS}(; cell_list=FullGridCellList(; min_corner, max_corner), update_strategy=ParallelUpdate()) @@ -153,9 +154,12 @@ ode = semidiscretize(semi, tspan) info_callback = InfoCallback(interval=100) saving_callback = SolutionSavingCallback(dt=0.02, prefix="") +particle_shifting = nothing + extra_callback = nothing -callbacks = CallbackSet(info_callback, saving_callback, UpdateCallback(), extra_callback) +callbacks = CallbackSet(info_callback, saving_callback, UpdateCallback(), + particle_shifting, extra_callback) sol = solve(ode, RDPK3SpFSAL35(), abstol=1e-5, # Default abstol is 1e-6 (may need to be tuned to prevent boundary penetration) diff --git a/src/schemes/boundary/open_boundary/method_of_characteristics.jl b/src/schemes/boundary/open_boundary/method_of_characteristics.jl index 8dacc446e..5df4f42fe 100644 --- a/src/schemes/boundary/open_boundary/method_of_characteristics.jl +++ b/src/schemes/boundary/open_boundary/method_of_characteristics.jl @@ -13,11 +13,20 @@ For more information about the method see [description below](@ref method_of_cha the reference values are not known a priori. **Note:** This feature is experimental and has not been fully validated yet. As of now, we are not aware of any published literature supporting its use. +- `average_inflow_velocity=true`: (Only in combination with `extrapolate_reference_values=true`.) +If `true`, the extrapolated inflow velocity is averaged to impose a uniform inflow profile. +When no velocity is prescribed at the inflow and `extrapolate_reference_values=true`, +the velocity is extrapolated from the fluid domain. +Thus, turbulent flows near the inflow can lead to non-uniform buffer-particles distribution, +resulting in a potential numerical instability. Averaging mitigates these effects. """ struct BoundaryModelLastiwka - extrapolate_reference_values::Bool - function BoundaryModelLastiwka(; extrapolate_reference_values::Bool=false) - return new{}(extrapolate_reference_values) + extrapolate_reference_values :: Bool + average_inflow_velocity :: Bool + + function BoundaryModelLastiwka(; extrapolate_reference_values::Bool=false, + average_inflow_velocity::Bool=true) + return new{}(extrapolate_reference_values, average_inflow_velocity) end end diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index 52d4d5c0d..f23996b54 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -1,13 +1,25 @@ @doc raw""" - BoundaryModelTafuni() + BoundaryModelTafuni(; average_inflow_velocity::Bool=true) Boundary model for the `OpenBoundarySPHSystem`. This model implements the method of [Tafuni et al. (2018)](@cite Tafuni2018) to extrapolate the properties from the fluid domain to the buffer zones (inflow and outflow) using ghost nodes. The position of the ghost nodes is obtained by mirroring the boundary particles into the fluid along a direction that is normal to the open boundary. + +# Keywords +- `average_inflow_velocity=true`: If `true`, the extrapolated inflow velocity is averaged to impose a uniform inflow profile. +When no velocity is prescribed at the inflow, the velocity is extrapolated from the fluid domain. +Thus, turbulent flows near the inflow can lead to non-uniform buffer-particles distribution, +resulting in a potential numerical instability. Averaging mitigates these effects. """ -struct BoundaryModelTafuni end +struct BoundaryModelTafuni + average_inflow_velocity::Bool + + function BoundaryModelTafuni(; average_inflow_velocity::Bool=true) + return new{}(average_inflow_velocity) + end +end function update_boundary_quantities!(system, ::BoundaryModelTafuni, v, u, v_ode, u_ode, semi, t) @@ -160,6 +172,13 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, end end + if !(prescribed_velocity) && system.boundary_model.average_inflow_velocity + # When no velocity is prescribed at the inflow, the velocity is extrapolated from the fluid domain. + # Thus, turbulent flows near the inflow can lead to non-uniform buffer-particles distribution, + # resulting in a potential numerical instability. Averaging mitigates these effects. + average_velocity!(v_open_boundary, u_open_boundary, system, boundary_zone, semi) + end + return system end @@ -211,3 +230,31 @@ function mirror_position(particle_coords, boundary_zone) return particle_coords - 2 * dist * boundary_zone.plane_normal end + +average_velocity!(v, u, system, boundary_zone, semi) = v + +function average_velocity!(v, u, system, boundary_zone::BoundaryZone{InFlow}, semi) + (; spanning_set, zone_origin) = boundary_zone + + # We only use the extrapolated velocity in the vicinity of the transition region. + # Otherwise, if the boundary zone is too large, averaging would be excessively influenced + # by the fluid velocity farther away from the boundary. + max_dist2 = (3 / 2 * compact_support(system.fluid_system, system))^2 + + candidates = findall(x -> dot(x - zone_origin, spanning_set[1]) <= max_dist2, + reinterpret(reshape, SVector{ndims(system), eltype(u)}, + active_coordinates(u, system))) + + avg_velocity = sum(candidates) do particle + return current_velocity(v, system, particle) / length(candidates) + end + + @threaded semi for particle in each_moving_particle(system) + # Set the velocity of the ghost node to the average velocity of the fluid domain + @inbounds for dim in eachindex(avg_velocity) + v[dim, particle] = avg_velocity[dim] + end + end + + return v +end From 848a18cf72f600e6662534fa6f5a2ba24a52d8a1 Mon Sep 17 00:00:00 2001 From: LasNikas Date: Sun, 15 Jun 2025 15:11:42 +0200 Subject: [PATCH 14/15] move kword to boundary zone --- .../boundary/open_boundary/boundary_zones.jl | 29 +++++++++++++------ .../method_of_characteristics.jl | 12 ++------ .../boundary/open_boundary/mirroring.jl | 24 ++++----------- 3 files changed, 28 insertions(+), 37 deletions(-) diff --git a/src/schemes/boundary/open_boundary/boundary_zones.jl b/src/schemes/boundary/open_boundary/boundary_zones.jl index 6739f80f9..5c7916d58 100644 --- a/src/schemes/boundary/open_boundary/boundary_zones.jl +++ b/src/schemes/boundary/open_boundary/boundary_zones.jl @@ -54,6 +54,14 @@ There are three ways to specify the actual shape of the boundary zone: - `extrude_geometry=nothing`: 1D shape in 2D or 2D shape in 3D, which lies on the plane and is extruded upstream to obtain the inflow particles. See point 2 above for more details. +- `average_inflow_velocity=false`: If `true`, the extrapolated inflow velocity is averaged + to impose a uniform inflow profile. + When no velocity is prescribed at the inflow, + the velocity is extrapolated from the fluid domain. + Thus, turbulent flows near the inflow can lead to + anisotropic buffer-particles distribution, + resulting in a potential numerical instability. + Averaging mitigates these effects. # Examples ```julia @@ -82,18 +90,20 @@ bidirectional_flow = BoundaryZone(; plane=plane_points, plane_normal, particle_s This is an experimental feature and may change in any future releases. """ struct BoundaryZone{BT, IC, S, ZO, ZW, FD, PN} - initial_condition :: IC - spanning_set :: S - zone_origin :: ZO - zone_width :: ZW - flow_direction :: FD - plane_normal :: PN - boundary_type :: BT + initial_condition :: IC + spanning_set :: S + zone_origin :: ZO + zone_width :: ZW + flow_direction :: FD + plane_normal :: PN + boundary_type :: BT + average_inflow_velocity :: Bool end function BoundaryZone(; plane, plane_normal, density, particle_spacing, initial_condition=nothing, extrude_geometry=nothing, - open_boundary_layers::Integer, boundary_type=BidirectionalFlow()) + open_boundary_layers::Integer, boundary_type=BidirectionalFlow(), + average_inflow_velocity=false) if open_boundary_layers <= 0 throw(ArgumentError("`open_boundary_layers` must be positive and greater than zero")) end @@ -120,7 +130,8 @@ function BoundaryZone(; plane, plane_normal, density, particle_spacing, boundary_type=boundary_type) return BoundaryZone(ic, spanning_set_, zone_origin, zone_width, - flow_direction, plane_normal_, boundary_type) + flow_direction, plane_normal_, boundary_type, + average_inflow_velocity) end function set_up_boundary_zone(plane, plane_normal, flow_direction, density, diff --git a/src/schemes/boundary/open_boundary/method_of_characteristics.jl b/src/schemes/boundary/open_boundary/method_of_characteristics.jl index 5df4f42fe..a2a00b9cb 100644 --- a/src/schemes/boundary/open_boundary/method_of_characteristics.jl +++ b/src/schemes/boundary/open_boundary/method_of_characteristics.jl @@ -13,19 +13,11 @@ For more information about the method see [description below](@ref method_of_cha the reference values are not known a priori. **Note:** This feature is experimental and has not been fully validated yet. As of now, we are not aware of any published literature supporting its use. -- `average_inflow_velocity=true`: (Only in combination with `extrapolate_reference_values=true`.) -If `true`, the extrapolated inflow velocity is averaged to impose a uniform inflow profile. -When no velocity is prescribed at the inflow and `extrapolate_reference_values=true`, -the velocity is extrapolated from the fluid domain. -Thus, turbulent flows near the inflow can lead to non-uniform buffer-particles distribution, -resulting in a potential numerical instability. Averaging mitigates these effects. """ struct BoundaryModelLastiwka - extrapolate_reference_values :: Bool - average_inflow_velocity :: Bool + extrapolate_reference_values::Bool - function BoundaryModelLastiwka(; extrapolate_reference_values::Bool=false, - average_inflow_velocity::Bool=true) + function BoundaryModelLastiwka(; extrapolate_reference_values::Bool=false) return new{}(extrapolate_reference_values, average_inflow_velocity) end end diff --git a/src/schemes/boundary/open_boundary/mirroring.jl b/src/schemes/boundary/open_boundary/mirroring.jl index f23996b54..d0d983a2a 100644 --- a/src/schemes/boundary/open_boundary/mirroring.jl +++ b/src/schemes/boundary/open_boundary/mirroring.jl @@ -1,25 +1,13 @@ @doc raw""" - BoundaryModelTafuni(; average_inflow_velocity::Bool=true) + BoundaryModelTafuni()) Boundary model for the `OpenBoundarySPHSystem`. This model implements the method of [Tafuni et al. (2018)](@cite Tafuni2018) to extrapolate the properties from the fluid domain to the buffer zones (inflow and outflow) using ghost nodes. The position of the ghost nodes is obtained by mirroring the boundary particles into the fluid along a direction that is normal to the open boundary. - -# Keywords -- `average_inflow_velocity=true`: If `true`, the extrapolated inflow velocity is averaged to impose a uniform inflow profile. -When no velocity is prescribed at the inflow, the velocity is extrapolated from the fluid domain. -Thus, turbulent flows near the inflow can lead to non-uniform buffer-particles distribution, -resulting in a potential numerical instability. Averaging mitigates these effects. """ -struct BoundaryModelTafuni - average_inflow_velocity::Bool - - function BoundaryModelTafuni(; average_inflow_velocity::Bool=true) - return new{}(average_inflow_velocity) - end -end +struct BoundaryModelTafuni end function update_boundary_quantities!(system, ::BoundaryModelTafuni, v, u, v_ode, u_ode, semi, t) @@ -172,7 +160,7 @@ function extrapolate_values!(system, v_open_boundary, v_fluid, u_open_boundary, end end - if !(prescribed_velocity) && system.boundary_model.average_inflow_velocity + if !(prescribed_velocity) && boundary_zone.average_inflow_velocity # When no velocity is prescribed at the inflow, the velocity is extrapolated from the fluid domain. # Thus, turbulent flows near the inflow can lead to non-uniform buffer-particles distribution, # resulting in a potential numerical instability. Averaging mitigates these effects. @@ -234,14 +222,14 @@ end average_velocity!(v, u, system, boundary_zone, semi) = v function average_velocity!(v, u, system, boundary_zone::BoundaryZone{InFlow}, semi) - (; spanning_set, zone_origin) = boundary_zone + (; plane_normal, zone_origin, initial_condition) = boundary_zone # We only use the extrapolated velocity in the vicinity of the transition region. # Otherwise, if the boundary zone is too large, averaging would be excessively influenced # by the fluid velocity farther away from the boundary. - max_dist2 = (3 / 2 * compact_support(system.fluid_system, system))^2 + max_dist = initial_condition.particle_spacing - candidates = findall(x -> dot(x - zone_origin, spanning_set[1]) <= max_dist2, + candidates = findall(x -> dot(x - zone_origin, -plane_normal) <= max_dist, reinterpret(reshape, SVector{ndims(system), eltype(u)}, active_coordinates(u, system))) From 5eaa5281fa4531c1887d956903750f4a7aad910b Mon Sep 17 00:00:00 2001 From: LasNikas Date: Wed, 18 Jun 2025 18:52:06 +0200 Subject: [PATCH 15/15] add tests --- .../boundary/open_boundary/boundary_zones.jl | 7 ++- .../boundary/open_boundary/mirroring.jl | 61 +++++++++++++++++++ 2 files changed, 65 insertions(+), 3 deletions(-) diff --git a/src/schemes/boundary/open_boundary/boundary_zones.jl b/src/schemes/boundary/open_boundary/boundary_zones.jl index 5c7916d58..30e589e1c 100644 --- a/src/schemes/boundary/open_boundary/boundary_zones.jl +++ b/src/schemes/boundary/open_boundary/boundary_zones.jl @@ -7,7 +7,8 @@ struct OutFlow end @doc raw""" BoundaryZone(; plane, plane_normal, density, particle_spacing, initial_condition=nothing, extrude_geometry=nothing, - open_boundary_layers::Integer, boundary_type=BidirectionalFlow()) + open_boundary_layers::Integer, boundary_type=BidirectionalFlow(), + average_inflow_velocity=true) Boundary zone for [`OpenBoundarySPHSystem`](@ref). @@ -54,7 +55,7 @@ There are three ways to specify the actual shape of the boundary zone: - `extrude_geometry=nothing`: 1D shape in 2D or 2D shape in 3D, which lies on the plane and is extruded upstream to obtain the inflow particles. See point 2 above for more details. -- `average_inflow_velocity=false`: If `true`, the extrapolated inflow velocity is averaged +- `average_inflow_velocity=true`: If `true`, the extrapolated inflow velocity is averaged to impose a uniform inflow profile. When no velocity is prescribed at the inflow, the velocity is extrapolated from the fluid domain. @@ -103,7 +104,7 @@ end function BoundaryZone(; plane, plane_normal, density, particle_spacing, initial_condition=nothing, extrude_geometry=nothing, open_boundary_layers::Integer, boundary_type=BidirectionalFlow(), - average_inflow_velocity=false) + average_inflow_velocity=true) if open_boundary_layers <= 0 throw(ArgumentError("`open_boundary_layers` must be positive and greater than zero")) end diff --git a/test/schemes/boundary/open_boundary/mirroring.jl b/test/schemes/boundary/open_boundary/mirroring.jl index 7892db837..9a81f2234 100644 --- a/test/schemes/boundary/open_boundary/mirroring.jl +++ b/test/schemes/boundary/open_boundary/mirroring.jl @@ -57,6 +57,7 @@ @testset verbose=true "plane normal $i" for i in eachindex(files) inflow = BoundaryZone(; plane=plane_boundary[i], boundary_type=InFlow(), plane_normal=plane_boundary_normal[i], + average_inflow_velocity=false, open_boundary_layers=10, density=1000.0, particle_spacing) open_boundary = OpenBoundarySPHSystem(inflow; fluid_system, @@ -152,6 +153,7 @@ @testset verbose=true "plane normal $i" for i in eachindex(files) inflow = BoundaryZone(; plane=plane_boundary[i], boundary_type=InFlow(), plane_normal=plane_boundary_normal[i], + average_inflow_velocity=false, open_boundary_layers=10, density=1000.0, particle_spacing) open_boundary = OpenBoundarySPHSystem(inflow; fluid_system, @@ -190,4 +192,63 @@ @test isapprox(open_boundary.pressure, expected_pressure, atol=1e-2) end end + + @testset verbose=true "Average Inflow Velocity $i-D" for i in (2, 3) + particle_spacing = 0.05 + domain_length = 1.0 + open_boundary_layers = 40 + + n_particles_xy = round(Int, domain_length / particle_spacing) + + if i == 2 + domain_fluid = RectangularShape(particle_spacing, (2, 1) .* n_particles_xy, + (0.0, 0.0), density=1000.0, + velocity=x -> SVector{2}(x[1], 0.0)) + + else + domain_fluid = RectangularShape(particle_spacing, (2, 1, 1) .* n_particles_xy, + (0.0, 0.0, 0.0), density=1000.0, + velocity=x -> SVector{3}(x[1], 0.0, 0.0)) + end + + smoothing_length = 1.5 * particle_spacing + smoothing_kernel = WendlandC2Kernel{ndims(domain_fluid)}() + fluid_system = EntropicallyDampedSPHSystem(domain_fluid, smoothing_kernel, + smoothing_length, 1.0) + fluid_system.cache.density .= 1000.0 + + if i == 2 + plane_in = ([0.0, 0.0], [0.0, domain_length]) + else + plane_in = ([0.0, 0.0, 0.0], [0.0, domain_length, 0.0], + [0.0, 0.0, domain_length]) + end + + inflow = BoundaryZone(; plane=plane_in, boundary_type=InFlow(), + plane_normal=(i == 2 ? [1.0, 0.0] : [1.0, 0.0, 0.0]), + open_boundary_layers=open_boundary_layers, density=1000.0, + particle_spacing, average_inflow_velocity=true) + open_boundary_in = OpenBoundarySPHSystem(inflow; fluid_system, + boundary_model=BoundaryModelTafuni(), + buffer_size=0) + + semi = Semidiscretization(fluid_system, open_boundary_in) + TrixiParticles.initialize_neighborhood_searches!(semi) + + v_open_boundary = zero(inflow.initial_condition.velocity) + v_fluid = vcat(domain_fluid.velocity, domain_fluid.pressure') + + TrixiParticles.set_zero!(open_boundary_in.pressure) + + TrixiParticles.extrapolate_values!(open_boundary_in, v_open_boundary, v_fluid, + inflow.initial_condition.coordinates, + domain_fluid.coordinates, semi, 0.0) + + # Since the velocity profile increases linearly in positive x-direction, + # we can use the first velocity entry as a representative value. + v_x_fluid_first = v_fluid[1, 1] + + @test length(unique(v_open_boundary[1, :])) == 1 + @test isapprox(-v_x_fluid_first, first(unique(v_open_boundary[1, :]))) + end end