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diff --git a/previews/PR126/assets/examples_generated_Actuator2D.md.CjJk3Yks.js b/previews/PR126/assets/examples_generated_Actuator2D.md.CjJk3Yks.js
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+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const k="/IncompressibleNavierStokes.jl/previews/PR126/assets/ziapkjz.D0gpz0HH.png",p="/IncompressibleNavierStokes.jl/previews/PR126/Actuator2D.mp4",l="/IncompressibleNavierStokes.jl/previews/PR126/assets/mqehavm.B0GX2W7p.png",t="/IncompressibleNavierStokes.jl/previews/PR126/assets/izkcxez.BjVPut0E.png",e="/IncompressibleNavierStokes.jl/previews/PR126/assets/ksyekmh.BQ4kmQL6.png",C=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator2D.md","filePath":"examples/generated/Actuator2D.md","lastUpdated":null}'),E={name:"examples/generated/Actuator2D.md"};function d(r,s,g,y,F,c){return h(),a("div",null,s[0]||(s[0]=[n(`Unsteady actuator case - 2D In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
Packages A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Setup A 2D grid is a Cartesian product of two vectors
julia n = 40 x = LinRange ( 0.0 , 10.0 , 5 n + 1 ), LinRange ( - 2.0 , 2.0 , 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 )))
Boundary conditions
julia inflow (dim, x, y, t) = sinpi ( sinpi (t / 6 ) / 6 + (dim == 1 ) / 2 ) boundary_conditions = (( DirichletBC (inflow), PressureBC ()), ( PressureBC (), PressureBC ())) ((DirichletBC{typeof(Main.inflow)}(Main.inflow), PressureBC()), (PressureBC(), PressureBC())) Actuator body force: A thrust coefficient Cₜ distributed over a thin rectangle
julia xc, yc = 2.0 , 0.0 # Disk center D = 1.0 # Disk diameter δ = 0.11 # Disk thickness C = 0.2 # Thrust coefficient c = C / (D * δ) # Normalize inside (x, y) = abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 bodyforce (dim, x, y, t) = - c * (dim == 1 ) * inside (x, y) bodyforce (generic function with 1 method) Build setup
julia setup = Setup (; x, Re = 100.0 , boundary_conditions, bodyforce, issteadybodyforce = true ); Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> inflow (dim, x, y, 0.0 )) 202×83×2 Array{Float64, 3}: [:, :, 1] = 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ⋮ ⋮ ⋱ ⋮ 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 [:, :, 2] = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Solve unsteady problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 12.0 ), method = RKMethods . RK44P2 (), Δt = 0.05 , processors = ( rtp = realtimeplotter (; setup, size = ( 600 , 300 ), nupdate = 5 ), log = timelogger (; nupdate = 24 ), ), ); [ Info: t = 1.2 Δt = 0.05 umax = 1.1 itertime = 0.11 [ Info: t = 2.4 Δt = 0.05 umax = 1 itertime = 0.04 [ Info: t = 3.6 Δt = 0.05 umax = 1 itertime = 0.0093 [ Info: t = 4.8 Δt = 0.05 umax = 1 itertime = 0.0087 [ Info: t = 6 Δt = 0.05 umax = 1 itertime = 0.0087 [ Info: t = 7.2 Δt = 0.05 umax = 1 itertime = 0.0086 [ Info: t = 8.4 Δt = 0.05 umax = 1 itertime = 0.0088 [ Info: t = 9.6 Δt = 0.05 umax = 1 itertime = 0.0089 [ Info: t = 10.8 Δt = 0.05 umax = 1 itertime = 0.009 [ Info: t = 12 Δt = 0.05 umax = 1 itertime = 0.0087
Post-process We create a box to visualize the actuator.
julia box = ( [xc - δ / 2 , xc - δ / 2 , xc + δ / 2 , xc + δ / 2 , xc - δ / 2 ], [yc + D / 2 , yc - D / 2 , yc - D / 2 , yc + D / 2 , yc + D / 2 ], ) ([1.945, 1.945, 2.055, 2.055, 1.945], [0.5, -0.5, -0.5, 0.5, 0.5]) Plot pressure
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! (box ... ; color = :red ) fig
Plot velocity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! (box ... ; color = :red ) fig
Plot vorticity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! (box ... ; color = :red ) fig
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes n = 40 x = LinRange ( 0.0 , 10.0 , 5 n + 1 ), LinRange ( - 2.0 , 2.0 , 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 ))) inflow (dim, x, y, t) = sinpi ( sinpi (t / 6 ) / 6 + (dim == 1 ) / 2 ) boundary_conditions = (( DirichletBC (inflow), PressureBC ()), ( PressureBC (), PressureBC ())) xc, yc = 2.0 , 0.0 # Disk center D = 1.0 # Disk diameter δ = 0.11 # Disk thickness C = 0.2 # Thrust coefficient c = C / (D * δ) # Normalize inside (x, y) = abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 bodyforce (dim, x, y, t) = - c * (dim == 1 ) * inside (x, y) setup = Setup (; x, Re = 100.0 , boundary_conditions, bodyforce, issteadybodyforce = true ); ustart = velocityfield (setup, (dim, x, y) -> inflow (dim, x, y, 0.0 )) state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 12.0 ), method = RKMethods . RK44P2 (), Δt = 0.05 , processors = ( rtp = realtimeplotter (; setup, size = ( 600 , 300 ), nupdate = 5 ), log = timelogger (; nupdate = 24 ), ), ); box = ( [xc - δ / 2 , xc - δ / 2 , xc + δ / 2 , xc + δ / 2 , xc - δ / 2 ], [yc + D / 2 , yc - D / 2 , yc - D / 2 , yc + D / 2 , yc + D / 2 ], ) fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! (box ... ; color = :red ) fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! (box ... ; color = :red ) fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! (box ... ; color = :red ) fig This page was generated using Literate.jl .
`,42)]))}const u=i(E,[["render",d]]);export{C as __pageData,u as default};
diff --git a/previews/PR126/assets/examples_generated_Actuator2D.md.CjJk3Yks.lean.js b/previews/PR126/assets/examples_generated_Actuator2D.md.CjJk3Yks.lean.js
new file mode 100644
index 00000000..450aeea2
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_Actuator2D.md.CjJk3Yks.lean.js
@@ -0,0 +1,131 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const k="/IncompressibleNavierStokes.jl/previews/PR126/assets/ziapkjz.D0gpz0HH.png",p="/IncompressibleNavierStokes.jl/previews/PR126/Actuator2D.mp4",l="/IncompressibleNavierStokes.jl/previews/PR126/assets/mqehavm.B0GX2W7p.png",t="/IncompressibleNavierStokes.jl/previews/PR126/assets/izkcxez.BjVPut0E.png",e="/IncompressibleNavierStokes.jl/previews/PR126/assets/ksyekmh.BQ4kmQL6.png",C=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator2D.md","filePath":"examples/generated/Actuator2D.md","lastUpdated":null}'),E={name:"examples/generated/Actuator2D.md"};function d(r,s,g,y,F,c){return h(),a("div",null,s[0]||(s[0]=[n(`Unsteady actuator case - 2D In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
Packages A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Setup A 2D grid is a Cartesian product of two vectors
julia n = 40 x = LinRange ( 0.0 , 10.0 , 5 n + 1 ), LinRange ( - 2.0 , 2.0 , 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 )))
Boundary conditions
julia inflow (dim, x, y, t) = sinpi ( sinpi (t / 6 ) / 6 + (dim == 1 ) / 2 ) boundary_conditions = (( DirichletBC (inflow), PressureBC ()), ( PressureBC (), PressureBC ())) ((DirichletBC{typeof(Main.inflow)}(Main.inflow), PressureBC()), (PressureBC(), PressureBC())) Actuator body force: A thrust coefficient Cₜ distributed over a thin rectangle
julia xc, yc = 2.0 , 0.0 # Disk center D = 1.0 # Disk diameter δ = 0.11 # Disk thickness C = 0.2 # Thrust coefficient c = C / (D * δ) # Normalize inside (x, y) = abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 bodyforce (dim, x, y, t) = - c * (dim == 1 ) * inside (x, y) bodyforce (generic function with 1 method) Build setup
julia setup = Setup (; x, Re = 100.0 , boundary_conditions, bodyforce, issteadybodyforce = true ); Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> inflow (dim, x, y, 0.0 )) 202×83×2 Array{Float64, 3}: [:, :, 1] = 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ⋮ ⋮ ⋱ ⋮ 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 [:, :, 2] = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Solve unsteady problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 12.0 ), method = RKMethods . RK44P2 (), Δt = 0.05 , processors = ( rtp = realtimeplotter (; setup, size = ( 600 , 300 ), nupdate = 5 ), log = timelogger (; nupdate = 24 ), ), ); [ Info: t = 1.2 Δt = 0.05 umax = 1.1 itertime = 0.11 [ Info: t = 2.4 Δt = 0.05 umax = 1 itertime = 0.04 [ Info: t = 3.6 Δt = 0.05 umax = 1 itertime = 0.0093 [ Info: t = 4.8 Δt = 0.05 umax = 1 itertime = 0.0087 [ Info: t = 6 Δt = 0.05 umax = 1 itertime = 0.0087 [ Info: t = 7.2 Δt = 0.05 umax = 1 itertime = 0.0086 [ Info: t = 8.4 Δt = 0.05 umax = 1 itertime = 0.0088 [ Info: t = 9.6 Δt = 0.05 umax = 1 itertime = 0.0089 [ Info: t = 10.8 Δt = 0.05 umax = 1 itertime = 0.009 [ Info: t = 12 Δt = 0.05 umax = 1 itertime = 0.0087
Post-process We create a box to visualize the actuator.
julia box = ( [xc - δ / 2 , xc - δ / 2 , xc + δ / 2 , xc + δ / 2 , xc - δ / 2 ], [yc + D / 2 , yc - D / 2 , yc - D / 2 , yc + D / 2 , yc + D / 2 ], ) ([1.945, 1.945, 2.055, 2.055, 1.945], [0.5, -0.5, -0.5, 0.5, 0.5]) Plot pressure
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! (box ... ; color = :red ) fig
Plot velocity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! (box ... ; color = :red ) fig
Plot vorticity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! (box ... ; color = :red ) fig
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes n = 40 x = LinRange ( 0.0 , 10.0 , 5 n + 1 ), LinRange ( - 2.0 , 2.0 , 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 ))) inflow (dim, x, y, t) = sinpi ( sinpi (t / 6 ) / 6 + (dim == 1 ) / 2 ) boundary_conditions = (( DirichletBC (inflow), PressureBC ()), ( PressureBC (), PressureBC ())) xc, yc = 2.0 , 0.0 # Disk center D = 1.0 # Disk diameter δ = 0.11 # Disk thickness C = 0.2 # Thrust coefficient c = C / (D * δ) # Normalize inside (x, y) = abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 bodyforce (dim, x, y, t) = - c * (dim == 1 ) * inside (x, y) setup = Setup (; x, Re = 100.0 , boundary_conditions, bodyforce, issteadybodyforce = true ); ustart = velocityfield (setup, (dim, x, y) -> inflow (dim, x, y, 0.0 )) state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 12.0 ), method = RKMethods . RK44P2 (), Δt = 0.05 , processors = ( rtp = realtimeplotter (; setup, size = ( 600 , 300 ), nupdate = 5 ), log = timelogger (; nupdate = 24 ), ), ); box = ( [xc - δ / 2 , xc - δ / 2 , xc + δ / 2 , xc + δ / 2 , xc - δ / 2 ], [yc + D / 2 , yc - D / 2 , yc - D / 2 , yc + D / 2 , yc + D / 2 ], ) fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! (box ... ; color = :red ) fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! (box ... ; color = :red ) fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! (box ... ; color = :red ) fig This page was generated using Literate.jl .
`,42)]))}const u=i(E,[["render",d]]);export{C as __pageData,u as default};
diff --git a/previews/PR126/assets/examples_generated_Actuator3D.md.DDNf_NQa.js b/previews/PR126/assets/examples_generated_Actuator3D.md.DDNf_NQa.js
new file mode 100644
index 00000000..acba18f2
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_Actuator3D.md.DDNf_NQa.js
@@ -0,0 +1,116 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Unsteady actuator case - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator3D.md","filePath":"examples/generated/Actuator3D.md","lastUpdated":null}'),k={name:"examples/generated/Actuator3D.md"};function t(p,s,l,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Unsteady actuator case - 3D In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a short cylinder.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "Actuator3D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 3D grid is a Cartesian product of three vectors
julia x = LinRange ( 0.0 , 6.0 , 31 ), LinRange ( - 2.0 , 2.0 , 41 ), LinRange ( - 2.0 , 2.0 , 41 ) plotgrid (x ... ) Boundary conditions: Unsteady BC requires time derivatives
julia boundary_conditions = ( # x left, x right ( DirichletBC ( (dim, x, y, z, t) -> dim == 1 ? cos ( π / 6 * sin ( π / 6 * t)) : dim == 2 ? sin ( π / 6 * sin ( π / 6 * t)) : zero (x), ), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), # z rear, z front ( PressureBC (), PressureBC ()), ) Actuator body force: A thrust coefficient Cₜ distributed over a short cylinder
julia cx, cy, cz = T ( 2 ), T ( 0 ), T ( 0 ) # Disk center D = T ( 1 ) # Disk diameter δ = T ( 0.11 ) # Disk thickness Cₜ = T ( 0.2 ) # Thrust coefficient cₜ = Cₜ / ( π * (D / 2 ) ^ 2 * δ) inside (x, y, z) = abs (x - cx) ≤ δ / 2 && (y - cy) ^ 2 + (z - cz) ^ 2 ≤ (D / 2 ) ^ 2 bodyforce (dim, x, y, z) = dim == 1 ? - cₜ * inside (x, y, z) : zero (x) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, bodyforce, backend); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y, z) -> dim == 1 ? one (x) : zero (x)); nothing #hide Solve unsteady problem
julia (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 3 )), method = RKMethods . RK44P2 (), Δt = T ( 0.05 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (V, p)
Field plot
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "Actuator3D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 100 ) x = LinRange ( 0.0 , 6.0 , 31 ), LinRange ( - 2.0 , 2.0 , 41 ), LinRange ( - 2.0 , 2.0 , 41 ) plotgrid (x ... ) boundary_conditions = ( # x left, x right ( DirichletBC ( (dim, x, y, z, t) -> dim == 1 ? cos ( π / 6 * sin ( π / 6 * t)) : dim == 2 ? sin ( π / 6 * sin ( π / 6 * t)) : zero (x), ), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), # z rear, z front ( PressureBC (), PressureBC ()), ) cx, cy, cz = T ( 2 ), T ( 0 ), T ( 0 ) # Disk center D = T ( 1 ) # Disk diameter δ = T ( 0.11 ) # Disk thickness Cₜ = T ( 0.2 ) # Thrust coefficient cₜ = Cₜ / ( π * (D / 2 ) ^ 2 * δ) inside (x, y, z) = abs (x - cx) ≤ δ / 2 && (y - cy) ^ 2 + (z - cz) ^ 2 ≤ (D / 2 ) ^ 2 bodyforce (dim, x, y, z) = dim == 1 ? - cₜ * inside (x, y, z) : zero (x) setup = Setup (; x, Re, boundary_conditions, bodyforce, backend); ustart = velocityfield (setup, (dim, x, y, z) -> dim == 1 ? one (x) : zero (x)); (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 3 )), method = RKMethods . RK44P2 (), Δt = T ( 0.05 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" )) This page was generated using Literate.jl .
`,36)]))}const y=i(k,[["render",t]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_Actuator3D.md.DDNf_NQa.lean.js b/previews/PR126/assets/examples_generated_Actuator3D.md.DDNf_NQa.lean.js
new file mode 100644
index 00000000..acba18f2
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_Actuator3D.md.DDNf_NQa.lean.js
@@ -0,0 +1,116 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Unsteady actuator case - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator3D.md","filePath":"examples/generated/Actuator3D.md","lastUpdated":null}'),k={name:"examples/generated/Actuator3D.md"};function t(p,s,l,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Unsteady actuator case - 3D In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a short cylinder.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "Actuator3D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 3D grid is a Cartesian product of three vectors
julia x = LinRange ( 0.0 , 6.0 , 31 ), LinRange ( - 2.0 , 2.0 , 41 ), LinRange ( - 2.0 , 2.0 , 41 ) plotgrid (x ... ) Boundary conditions: Unsteady BC requires time derivatives
julia boundary_conditions = ( # x left, x right ( DirichletBC ( (dim, x, y, z, t) -> dim == 1 ? cos ( π / 6 * sin ( π / 6 * t)) : dim == 2 ? sin ( π / 6 * sin ( π / 6 * t)) : zero (x), ), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), # z rear, z front ( PressureBC (), PressureBC ()), ) Actuator body force: A thrust coefficient Cₜ distributed over a short cylinder
julia cx, cy, cz = T ( 2 ), T ( 0 ), T ( 0 ) # Disk center D = T ( 1 ) # Disk diameter δ = T ( 0.11 ) # Disk thickness Cₜ = T ( 0.2 ) # Thrust coefficient cₜ = Cₜ / ( π * (D / 2 ) ^ 2 * δ) inside (x, y, z) = abs (x - cx) ≤ δ / 2 && (y - cy) ^ 2 + (z - cz) ^ 2 ≤ (D / 2 ) ^ 2 bodyforce (dim, x, y, z) = dim == 1 ? - cₜ * inside (x, y, z) : zero (x) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, bodyforce, backend); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y, z) -> dim == 1 ? one (x) : zero (x)); nothing #hide Solve unsteady problem
julia (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 3 )), method = RKMethods . RK44P2 (), Δt = T ( 0.05 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (V, p)
Field plot
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "Actuator3D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 100 ) x = LinRange ( 0.0 , 6.0 , 31 ), LinRange ( - 2.0 , 2.0 , 41 ), LinRange ( - 2.0 , 2.0 , 41 ) plotgrid (x ... ) boundary_conditions = ( # x left, x right ( DirichletBC ( (dim, x, y, z, t) -> dim == 1 ? cos ( π / 6 * sin ( π / 6 * t)) : dim == 2 ? sin ( π / 6 * sin ( π / 6 * t)) : zero (x), ), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), # z rear, z front ( PressureBC (), PressureBC ()), ) cx, cy, cz = T ( 2 ), T ( 0 ), T ( 0 ) # Disk center D = T ( 1 ) # Disk diameter δ = T ( 0.11 ) # Disk thickness Cₜ = T ( 0.2 ) # Thrust coefficient cₜ = Cₜ / ( π * (D / 2 ) ^ 2 * δ) inside (x, y, z) = abs (x - cx) ≤ δ / 2 && (y - cy) ^ 2 + (z - cz) ^ 2 ≤ (D / 2 ) ^ 2 bodyforce (dim, x, y, z) = dim == 1 ? - cₜ * inside (x, y, z) : zero (x) setup = Setup (; x, Re, boundary_conditions, bodyforce, backend); ustart = velocityfield (setup, (dim, x, y, z) -> dim == 1 ? one (x) : zero (x)); (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 3 )), method = RKMethods . RK44P2 (), Δt = T ( 0.05 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" )) This page was generated using Literate.jl .
`,36)]))}const y=i(k,[["render",t]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_BackwardFacingStep2D.md.B1L_JUM9.js b/previews/PR126/assets/examples_generated_BackwardFacingStep2D.md.B1L_JUM9.js
new file mode 100644
index 00000000..ef055957
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_BackwardFacingStep2D.md.B1L_JUM9.js
@@ -0,0 +1,96 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Backward Facing Step - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/BackwardFacingStep2D.md","filePath":"examples/generated/BackwardFacingStep2D.md","lastUpdated":null}'),t={name:"examples/generated/BackwardFacingStep2D.md"};function p(k,s,l,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Backward Facing Step - 2D In this example we consider a channel with walls at the top and bottom, and a step at the left with a parabolic inflow. Initially the velocity is an extension of the inflow, but as time passes the velocity finds a new steady state.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep2D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
Boundary conditions: steady inflow on the top half
julia U (dim, x, y, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) + randn ( typeof (x)) / 1_000 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), ) A 2D grid is a Cartesian product of two vectors. Here we refine the grid near the walls.
julia x = LinRange ( T ( 0 ), T ( 10 ), 301 ), cosine_grid ( - T ( 0.5 ), T ( 0.5 ), 51 ) plotgrid (x ... ; figure = (; size = ( 600 , 150 ))) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, backend); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, zero (x))); nothing #hide Solve steady state problem
julia # u, p = solve_steady_state(setup, u₀, p₀); nothing Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.002 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 150 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 500 ), ), ); nothing #hide Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :pressure ) Plot velocity
julia fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep2D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 3_000 ) U (dim, x, y, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) + randn ( typeof (x)) / 1_000 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), ) x = LinRange ( T ( 0 ), T ( 10 ), 301 ), cosine_grid ( - T ( 0.5 ), T ( 0.5 ), 51 ) plotgrid (x ... ; figure = (; size = ( 600 , 150 ))) setup = Setup (; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, zero (x))); # u, p = solve_steady_state(setup, u₀, p₀); nothing state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.002 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 150 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 500 ), ), ); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :pressure ) fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :velocitynorm ) fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :vorticity ) This page was generated using Literate.jl .
`,40)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_BackwardFacingStep2D.md.B1L_JUM9.lean.js b/previews/PR126/assets/examples_generated_BackwardFacingStep2D.md.B1L_JUM9.lean.js
new file mode 100644
index 00000000..ef055957
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_BackwardFacingStep2D.md.B1L_JUM9.lean.js
@@ -0,0 +1,96 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Backward Facing Step - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/BackwardFacingStep2D.md","filePath":"examples/generated/BackwardFacingStep2D.md","lastUpdated":null}'),t={name:"examples/generated/BackwardFacingStep2D.md"};function p(k,s,l,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Backward Facing Step - 2D In this example we consider a channel with walls at the top and bottom, and a step at the left with a parabolic inflow. Initially the velocity is an extension of the inflow, but as time passes the velocity finds a new steady state.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep2D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
Boundary conditions: steady inflow on the top half
julia U (dim, x, y, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) + randn ( typeof (x)) / 1_000 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), ) A 2D grid is a Cartesian product of two vectors. Here we refine the grid near the walls.
julia x = LinRange ( T ( 0 ), T ( 10 ), 301 ), cosine_grid ( - T ( 0.5 ), T ( 0.5 ), 51 ) plotgrid (x ... ; figure = (; size = ( 600 , 150 ))) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, backend); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, zero (x))); nothing #hide Solve steady state problem
julia # u, p = solve_steady_state(setup, u₀, p₀); nothing Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.002 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 150 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 500 ), ), ); nothing #hide Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :pressure ) Plot velocity
julia fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep2D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 3_000 ) U (dim, x, y, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) + randn ( typeof (x)) / 1_000 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), ) x = LinRange ( T ( 0 ), T ( 10 ), 301 ), cosine_grid ( - T ( 0.5 ), T ( 0.5 ), 51 ) plotgrid (x ... ; figure = (; size = ( 600 , 150 ))) setup = Setup (; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, zero (x))); # u, p = solve_steady_state(setup, u₀, p₀); nothing state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.002 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 150 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 500 ), ), ); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :pressure ) fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :velocitynorm ) fieldplot (state; setup, size = ( 600 , 150 ), fieldname = :vorticity ) This page was generated using Literate.jl .
`,40)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_BackwardFacingStep3D.md.BQ3_PVJu.js b/previews/PR126/assets/examples_generated_BackwardFacingStep3D.md.BQ3_PVJu.js
new file mode 100644
index 00000000..dfd69af5
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_BackwardFacingStep3D.md.BQ3_PVJu.js
@@ -0,0 +1,99 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Backward Facing Step - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/BackwardFacingStep3D.md","filePath":"examples/generated/BackwardFacingStep3D.md","lastUpdated":null}'),t={name:"examples/generated/BackwardFacingStep3D.md"};function p(l,s,k,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Backward Facing Step - 3D In this example we consider a channel with periodic side boundaries, walls at the top and bottom, and a step at the left with a parabolic inflow. Initially the velocity is an extension of the inflow, but as time passes the velocity finds a new steady state.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep3D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 3D grid is a Cartesian product of three vectors
julia x = LinRange ( T ( 0 ), T ( 10 ), 129 ), LinRange ( - T ( 0.5 ), T ( 0.5 ), 17 ), LinRange ( - T ( 0.25 ), T ( 0.25 ), 9 ) plotgrid (x ... ) Boundary conditions: steady inflow on the top half
julia U (dim, x, y, z, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, backend); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y, z) -> U (dim, x, y, z, zero (x))); nothing #hide Solve steady state problem
julia # u, p = solve_steady_state(setup, u₀, p₀); nothing Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ) Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot velocity
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep3D" ) T = Float32 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 1000 ) x = LinRange ( T ( 0 ), T ( 10 ), 129 ), LinRange ( - T ( 0.5 ), T ( 0.5 ), 17 ), LinRange ( - T ( 0.25 ), T ( 0.25 ), 9 ) plotgrid (x ... ) U (dim, x, y, z, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) setup = Setup (; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y, z) -> U (dim, x, y, z, zero (x))); # u, p = solve_steady_state(setup, u₀, p₀); nothing state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ) save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) This page was generated using Literate.jl .
`,40)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_BackwardFacingStep3D.md.BQ3_PVJu.lean.js b/previews/PR126/assets/examples_generated_BackwardFacingStep3D.md.BQ3_PVJu.lean.js
new file mode 100644
index 00000000..dfd69af5
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_BackwardFacingStep3D.md.BQ3_PVJu.lean.js
@@ -0,0 +1,99 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Backward Facing Step - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/BackwardFacingStep3D.md","filePath":"examples/generated/BackwardFacingStep3D.md","lastUpdated":null}'),t={name:"examples/generated/BackwardFacingStep3D.md"};function p(l,s,k,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Backward Facing Step - 3D In this example we consider a channel with periodic side boundaries, walls at the top and bottom, and a step at the left with a parabolic inflow. Initially the velocity is an extension of the inflow, but as time passes the velocity finds a new steady state.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep3D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 3D grid is a Cartesian product of three vectors
julia x = LinRange ( T ( 0 ), T ( 10 ), 129 ), LinRange ( - T ( 0.5 ), T ( 0.5 ), 17 ), LinRange ( - T ( 0.25 ), T ( 0.25 ), 9 ) plotgrid (x ... ) Boundary conditions: steady inflow on the top half
julia U (dim, x, y, z, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, backend); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y, z) -> U (dim, x, y, z, zero (x))); nothing #hide Solve steady state problem
julia # u, p = solve_steady_state(setup, u₀, p₀); nothing Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ) Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot velocity
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "BackwardFacingStep3D" ) T = Float32 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 1000 ) x = LinRange ( T ( 0 ), T ( 10 ), 129 ), LinRange ( - T ( 0.5 ), T ( 0.5 ), 17 ), LinRange ( - T ( 0.25 ), T ( 0.25 ), 9 ) plotgrid (x ... ) U (dim, x, y, z, t) = dim == 1 && y ≥ 0 ? 24 y * ( one (x) / 2 - y) : zero (x) boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( DirichletBC (), DirichletBC ()), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) setup = Setup (; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y, z) -> U (dim, x, y, z, zero (x))); # u, p = solve_steady_state(setup, u₀, p₀); nothing state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 7 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ) save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) This page was generated using Literate.jl .
`,40)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_DecayingTurbulence2D.md.WF0Cn1Zb.js b/previews/PR126/assets/examples_generated_DecayingTurbulence2D.md.WF0Cn1Zb.js
new file mode 100644
index 00000000..a149eee4
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_DecayingTurbulence2D.md.WF0Cn1Zb.js
@@ -0,0 +1,66 @@
+import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.BSoZtefh.js";const t="/IncompressibleNavierStokes.jl/previews/PR126/DecayingTurbulence2D.mp4",l="/IncompressibleNavierStokes.jl/previews/PR126/assets/smpfhbj.J4Xy6kEr.png",e="/IncompressibleNavierStokes.jl/previews/PR126/assets/ncjebvf.Ckr30c9N.png",h="/IncompressibleNavierStokes.jl/previews/PR126/assets/ylwwdlu.DcGjID-r.png",u=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence2D.md","filePath":"examples/generated/DecayingTurbulence2D.md","lastUpdated":null}'),k={name:"examples/generated/DecayingTurbulence2D.md"};function E(r,s,d,g,y,o){return p(),a("div",null,s[0]||(s[0]=[n(`Decaying Homogeneous Isotropic Turbulence - 2D In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1 ] and [2 ]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
Packages We just need IncompressibleNavierStokes and a Makie plotting backend.
julia using CairoMakie using IncompressibleNavierStokes Setup
julia n = 256 ax = LinRange ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (ax, ax), Re = 4e3 ); ustart = random_field (setup, 0.0 ); Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 1.0 ), processors = ( rtp = realtimeplotter (; setup, nupdate = 10 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); [ Info: t = 0.0955696 Δt = 0.00098 umax = 3.6 itertime = 0.027 [ Info: t = 0.193437 Δt = 0.0011 umax = 3.3 itertime = 0.016 [ Info: t = 0.303347 Δt = 0.0013 umax = 2.8 itertime = 0.015 [ Info: t = 0.427623 Δt = 0.0012 umax = 2.9 itertime = 0.015 [ Info: t = 0.549335 Δt = 0.0012 umax = 3 itertime = 0.015 [ Info: t = 0.669572 Δt = 0.0015 umax = 2.4 itertime = 0.015 [ Info: t = 0.81377 Δt = 0.0013 umax = 2.7 itertime = 0.015 [ Info: t = 0.935269 Δt = 0.0012 umax = 2.8 itertime = 0.015
Post-process We may visualize or export the computed fields
Energy history
Energy spectrum
Plot field
julia fieldplot (state; setup)
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes n = 256 ax = LinRange ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (ax, ax), Re = 4e3 ); ustart = random_field (setup, 0.0 ); state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 1.0 ), processors = ( rtp = realtimeplotter (; setup, nupdate = 10 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); outputs . ehist outputs . espec fieldplot (state; setup) This page was generated using Literate.jl .
`,27)]))}const F=i(k,[["render",E]]);export{u as __pageData,F as default};
diff --git a/previews/PR126/assets/examples_generated_DecayingTurbulence2D.md.WF0Cn1Zb.lean.js b/previews/PR126/assets/examples_generated_DecayingTurbulence2D.md.WF0Cn1Zb.lean.js
new file mode 100644
index 00000000..a149eee4
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_DecayingTurbulence2D.md.WF0Cn1Zb.lean.js
@@ -0,0 +1,66 @@
+import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.BSoZtefh.js";const t="/IncompressibleNavierStokes.jl/previews/PR126/DecayingTurbulence2D.mp4",l="/IncompressibleNavierStokes.jl/previews/PR126/assets/smpfhbj.J4Xy6kEr.png",e="/IncompressibleNavierStokes.jl/previews/PR126/assets/ncjebvf.Ckr30c9N.png",h="/IncompressibleNavierStokes.jl/previews/PR126/assets/ylwwdlu.DcGjID-r.png",u=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence2D.md","filePath":"examples/generated/DecayingTurbulence2D.md","lastUpdated":null}'),k={name:"examples/generated/DecayingTurbulence2D.md"};function E(r,s,d,g,y,o){return p(),a("div",null,s[0]||(s[0]=[n(`Decaying Homogeneous Isotropic Turbulence - 2D In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1 ] and [2 ]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
Packages We just need IncompressibleNavierStokes and a Makie plotting backend.
julia using CairoMakie using IncompressibleNavierStokes Setup
julia n = 256 ax = LinRange ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (ax, ax), Re = 4e3 ); ustart = random_field (setup, 0.0 ); Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 1.0 ), processors = ( rtp = realtimeplotter (; setup, nupdate = 10 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); [ Info: t = 0.0955696 Δt = 0.00098 umax = 3.6 itertime = 0.027 [ Info: t = 0.193437 Δt = 0.0011 umax = 3.3 itertime = 0.016 [ Info: t = 0.303347 Δt = 0.0013 umax = 2.8 itertime = 0.015 [ Info: t = 0.427623 Δt = 0.0012 umax = 2.9 itertime = 0.015 [ Info: t = 0.549335 Δt = 0.0012 umax = 3 itertime = 0.015 [ Info: t = 0.669572 Δt = 0.0015 umax = 2.4 itertime = 0.015 [ Info: t = 0.81377 Δt = 0.0013 umax = 2.7 itertime = 0.015 [ Info: t = 0.935269 Δt = 0.0012 umax = 2.8 itertime = 0.015
Post-process We may visualize or export the computed fields
Energy history
Energy spectrum
Plot field
julia fieldplot (state; setup)
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes n = 256 ax = LinRange ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (ax, ax), Re = 4e3 ); ustart = random_field (setup, 0.0 ); state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 1.0 ), processors = ( rtp = realtimeplotter (; setup, nupdate = 10 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); outputs . ehist outputs . espec fieldplot (state; setup) This page was generated using Literate.jl .
`,27)]))}const F=i(k,[["render",E]]);export{u as __pageData,F as default};
diff --git a/previews/PR126/assets/examples_generated_DecayingTurbulence3D.md.CPesWMyv.js b/previews/PR126/assets/examples_generated_DecayingTurbulence3D.md.CPesWMyv.js
new file mode 100644
index 00000000..a727fbde
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_DecayingTurbulence3D.md.CPesWMyv.js
@@ -0,0 +1,57 @@
+import{_ as i,c as a,a5 as n,o as t}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence3D.md","filePath":"examples/generated/DecayingTurbulence3D.md","lastUpdated":null}'),p={name:"examples/generated/DecayingTurbulence3D.md"};function l(e,s,h,k,r,E){return t(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Decaying Homogeneous Isotropic Turbulence - 3D In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1 ] and [2 ]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
julia using CairoMakie using IncompressibleNavierStokes Problem setup julia backend = CPU () # using CUDA; backend = CUDABackend() T = Float32 n = 128 ax = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (ax, ax, ax), Re = T ( 4e3 ), backend); psolver = default_psolver (setup); ustart = random_field (setup; psolver); nothing #hide Solve problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 2 )), psolver, processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 10 ), ), ); nothing #hide Post-process Field plot
julia # outputs.rtp # fieldplot(state; setup, levels = 0:5) Energy history
Energy spectrum
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes backend = CPU () # using CUDA; backend = CUDABackend() T = Float32 n = 128 ax = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (ax, ax, ax), Re = T ( 4e3 ), backend); psolver = default_psolver (setup); ustart = random_field (setup; psolver); state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 2 )), psolver, processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 10 ), ), ); # outputs.rtp # fieldplot(state; setup, levels = 0:5) # outputs.ehist outputs . espec This page was generated using Literate.jl .
`,20)]))}const o=i(p,[["render",l]]);export{g as __pageData,o as default};
diff --git a/previews/PR126/assets/examples_generated_DecayingTurbulence3D.md.CPesWMyv.lean.js b/previews/PR126/assets/examples_generated_DecayingTurbulence3D.md.CPesWMyv.lean.js
new file mode 100644
index 00000000..a727fbde
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_DecayingTurbulence3D.md.CPesWMyv.lean.js
@@ -0,0 +1,57 @@
+import{_ as i,c as a,a5 as n,o as t}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence3D.md","filePath":"examples/generated/DecayingTurbulence3D.md","lastUpdated":null}'),p={name:"examples/generated/DecayingTurbulence3D.md"};function l(e,s,h,k,r,E){return t(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Decaying Homogeneous Isotropic Turbulence - 3D In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1 ] and [2 ]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
julia using CairoMakie using IncompressibleNavierStokes Problem setup julia backend = CPU () # using CUDA; backend = CUDABackend() T = Float32 n = 128 ax = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (ax, ax, ax), Re = T ( 4e3 ), backend); psolver = default_psolver (setup); ustart = random_field (setup; psolver); nothing #hide Solve problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 2 )), psolver, processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 10 ), ), ); nothing #hide Post-process Field plot
julia # outputs.rtp # fieldplot(state; setup, levels = 0:5) Energy history
Energy spectrum
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes backend = CPU () # using CUDA; backend = CUDABackend() T = Float32 n = 128 ax = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (ax, ax, ax), Re = T ( 4e3 ), backend); psolver = default_psolver (setup); ustart = random_field (setup; psolver); state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 2 )), psolver, processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 10 ), ), ); # outputs.rtp # fieldplot(state; setup, levels = 0:5) # outputs.ehist outputs . espec This page was generated using Literate.jl .
`,20)]))}const o=i(p,[["render",l]]);export{g as __pageData,o as default};
diff --git a/previews/PR126/assets/examples_generated_Kolmogorov2D.md.C7XWVq5k.js b/previews/PR126/assets/examples_generated_Kolmogorov2D.md.C7XWVq5k.js
new file mode 100644
index 00000000..c8157a24
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_Kolmogorov2D.md.C7XWVq5k.js
@@ -0,0 +1,75 @@
+import{_ as h,c as a,j as s,a as t,a5 as n,o as l}from"./chunks/framework.BSoZtefh.js";const c=JSON.parse('{"title":"Kolmogorov flow (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Kolmogorov2D.md","filePath":"examples/generated/Kolmogorov2D.md","lastUpdated":null}'),p={name:"examples/generated/Kolmogorov2D.md"},e={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.165ex",height:"2.452ex",role:"img",focusable:"false",viewBox:"0 -833.9 4492.8 1083.9","aria-hidden":"true"},r={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.149ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.876ex",height:"5.43ex",role:"img",focusable:"false",viewBox:"0 -1450 9227.2 2400","aria-hidden":"true"};function E(g,i,o,y,T,Q){return l(),a("div",null,[i[6]||(i[6]=s("p",null,[s("em",null,"Note: Output is not generated for this example (to save resources on GitHub).")],-1)),i[7]||(i[7]=s("h1",{id:"Kolmogorov-flow-(2D)",tabindex:"-1"},[t("Kolmogorov flow (2D) "),s("a",{class:"header-anchor",href:"#Kolmogorov-flow-(2D)","aria-label":'Permalink to "Kolmogorov flow (2D) {#Kolmogorov-flow-(2D)}"'},"")],-1)),s("p",null,[i[2]||(i[2]=t("The Kolmogorov flow in a periodic box ")),s("mjx-container",e,[(l(),a("svg",k,i[0]||(i[0]=[n('where k is the wavenumber where energy is injected.
Packages We just need the IncompressibleNavierStokes and a Makie plotting package.
julia using CairoMakie using IncompressibleNavierStokes Setup Define a uniform grid with a steady body force field.
julia n = 256 axis = range ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (axis, axis), Re = 2e3 , bodyforce = (dim, x, y, t) -> (dim == 1 ) * 5 * sinpi ( 8 * y), issteadybodyforce = true , ); ustart = random_field (setup, 0.0 ; A = 1e-2 ); nothing #hide Plot body force Since the force is steady, it is just stored as a field.
julia heatmap (setup . bodyforce[:, :, 1 ]) Solve unsteady problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 2.0 ), Δt = 1e-3 , processors = ( rtp = realtimeplotter (; setup, nupdate = 100 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Field plot
Energy history
Energy spectrum
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes n = 256 axis = range ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (axis, axis), Re = 2e3 , bodyforce = (dim, x, y, t) -> (dim == 1 ) * 5 * sinpi ( 8 * y), issteadybodyforce = true , ); ustart = random_field (setup, 0.0 ; A = 1e-2 ); heatmap (setup . bodyforce[:, :, 1 ]) state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 2.0 ), Δt = 1e-3 , processors = ( rtp = realtimeplotter (; setup, nupdate = 100 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp outputs . ehist outputs . espec This page was generated using Literate.jl .
`,23))])}const u=h(p,[["render",E]]);export{c as __pageData,u as default};
diff --git a/previews/PR126/assets/examples_generated_Kolmogorov2D.md.C7XWVq5k.lean.js b/previews/PR126/assets/examples_generated_Kolmogorov2D.md.C7XWVq5k.lean.js
new file mode 100644
index 00000000..c8157a24
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_Kolmogorov2D.md.C7XWVq5k.lean.js
@@ -0,0 +1,75 @@
+import{_ as h,c as a,j as s,a as t,a5 as n,o as l}from"./chunks/framework.BSoZtefh.js";const c=JSON.parse('{"title":"Kolmogorov flow (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Kolmogorov2D.md","filePath":"examples/generated/Kolmogorov2D.md","lastUpdated":null}'),p={name:"examples/generated/Kolmogorov2D.md"},e={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.165ex",height:"2.452ex",role:"img",focusable:"false",viewBox:"0 -833.9 4492.8 1083.9","aria-hidden":"true"},r={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.149ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.876ex",height:"5.43ex",role:"img",focusable:"false",viewBox:"0 -1450 9227.2 2400","aria-hidden":"true"};function E(g,i,o,y,T,Q){return l(),a("div",null,[i[6]||(i[6]=s("p",null,[s("em",null,"Note: Output is not generated for this example (to save resources on GitHub).")],-1)),i[7]||(i[7]=s("h1",{id:"Kolmogorov-flow-(2D)",tabindex:"-1"},[t("Kolmogorov flow (2D) "),s("a",{class:"header-anchor",href:"#Kolmogorov-flow-(2D)","aria-label":'Permalink to "Kolmogorov flow (2D) {#Kolmogorov-flow-(2D)}"'},"")],-1)),s("p",null,[i[2]||(i[2]=t("The Kolmogorov flow in a periodic box ")),s("mjx-container",e,[(l(),a("svg",k,i[0]||(i[0]=[n('where k is the wavenumber where energy is injected.
Packages We just need the IncompressibleNavierStokes and a Makie plotting package.
julia using CairoMakie using IncompressibleNavierStokes Setup Define a uniform grid with a steady body force field.
julia n = 256 axis = range ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (axis, axis), Re = 2e3 , bodyforce = (dim, x, y, t) -> (dim == 1 ) * 5 * sinpi ( 8 * y), issteadybodyforce = true , ); ustart = random_field (setup, 0.0 ; A = 1e-2 ); nothing #hide Plot body force Since the force is steady, it is just stored as a field.
julia heatmap (setup . bodyforce[:, :, 1 ]) Solve unsteady problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 2.0 ), Δt = 1e-3 , processors = ( rtp = realtimeplotter (; setup, nupdate = 100 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Field plot
Energy history
Energy spectrum
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes n = 256 axis = range ( 0.0 , 1.0 , n + 1 ) setup = Setup (; x = (axis, axis), Re = 2e3 , bodyforce = (dim, x, y, t) -> (dim == 1 ) * 5 * sinpi ( 8 * y), issteadybodyforce = true , ); ustart = random_field (setup, 0.0 ; A = 1e-2 ); heatmap (setup . bodyforce[:, :, 1 ]) state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 2.0 ), Δt = 1e-3 , processors = ( rtp = realtimeplotter (; setup, nupdate = 100 ), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 , displayfig = false , ), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp outputs . ehist outputs . espec This page was generated using Literate.jl .
`,23))])}const u=h(p,[["render",E]]);export{c as __pageData,u as default};
diff --git a/previews/PR126/assets/examples_generated_LidDrivenCavity2D.md.D_iNsMnv.js b/previews/PR126/assets/examples_generated_LidDrivenCavity2D.md.D_iNsMnv.js
new file mode 100644
index 00000000..7912e64c
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_LidDrivenCavity2D.md.D_iNsMnv.js
@@ -0,0 +1,91 @@
+import{_ as i,c as a,a5 as n,o as t}from"./chunks/framework.BSoZtefh.js";const o=JSON.parse('{"title":"Tutorial: Lid-Driven Cavity - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/LidDrivenCavity2D.md","filePath":"examples/generated/LidDrivenCavity2D.md","lastUpdated":null}'),l={name:"examples/generated/LidDrivenCavity2D.md"};function p(e,s,h,k,d,r){return t(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Tutorial: Lid-Driven Cavity - 2D In this example we consider a box with a moving lid. The velocity is initially at rest. The solution should reach at steady state equilibrium after a certain time. The same steady state should be obtained when solving a steady state problem.
We start by loading packages. A Makie plotting backend is needed
for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Case name for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity2D" ) The code allows for using different floating point number types, including single precision (Float32) and double precision (Float64). On the CPU, the speed is not really different, but double precision uses twice as much memory as single precision. When running on the GPU, single precision is preferred. Half precision (Float16) is also an option, but then the values should be scaled judiciously to avoid vanishing digits when applying differential operators of the form "right minus left divided by small distance".
Note how floating point type hygiene is enforced in the following using T to avoid mixing different precisions.
T = Float64
julia T = Float32 # T = Float16 We can also choose to do the computations on a different device. By default, the computations are performed on the host (CPU). An optional backend allows for moving arrays to a different device such as a GPU.
Note: For GPUs, single precision is preferred.
julia backend = CPU () # using CUDA; backend = CUDABackend() Here we choose a moderate Reynolds number. Note how we pass the floating point type.
Non-zero Dirichlet boundary conditions are specified as plain Julia functions.
julia U = ( T ( 1 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y bottom, y top ( DirichletBC (), DirichletBC (U)), ) We create a two-dimensional domain with a box of size [1, 1]. The grid is created as a Cartesian product between two vectors. We add a refinement near the walls.
julia n = 32 lims = T ( 0 ), T ( 1 ) x = cosine_grid (lims ... , n), cosine_grid (lims ... , n) plotgrid (x ... ) We can now build the setup and assemble operators. A 3D setup is built if we also provide a vector of z-coordinates.
julia setup = Setup (; x, boundary_conditions, Re, backend); nothing #hide The initial conditions are provided in function. The value dim() determines the velocity component.
julia ustart = velocityfield (setup, (dim, x, y) -> zero (x)); nothing #hide Iteration processors are called after every nupdate time steps. This can be useful for logging, plotting, or saving results. Their respective outputs are later returned by solve_unsteady.
julia processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1000 ), ); nothing #hide By default, a standard fourth order Runge-Kutta method is used. If we don't provide the time step explicitly, an adaptive time step is used.
julia tlims = ( T ( 0 ), T ( 10 )) state, outputs = solve_unsteady (; setup, ustart, tlims, Δt = T ( 1e-3 ), processors); nothing #hide Post-process We may visualize or export the computed fields
Export fields to VTK. The file outdir/solution.vti may be opened for visualization in ParaView . This is particularly useful for inspecting results from 3D simulations.
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot velocity. Note the time stamp used for computing boundary conditions, if any.
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) In addition, the named tuple outputs contains quantities from our processors. The logger returns nothing.
julia # outputs.rtp # outputs.ehist # outputs.espec # outputs.anim # outputs.vtk # outputs.field outputs . log Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity2D" ) T = Float32 # T = Float16 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 1_000 ) U = ( T ( 1 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y bottom, y top ( DirichletBC (), DirichletBC (U)), ) n = 32 lims = T ( 0 ), T ( 1 ) x = cosine_grid (lims ... , n), cosine_grid (lims ... , n) plotgrid (x ... ) setup = Setup (; x, boundary_conditions, Re, backend); ustart = velocityfield (setup, (dim, x, y) -> zero (x)); processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1000 ), ); tlims = ( T ( 0 ), T ( 10 )) state, outputs = solve_unsteady (; setup, ustart, tlims, Δt = T ( 1e-3 ), processors); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) # outputs.rtp # outputs.ehist # outputs.espec # outputs.anim # outputs.vtk # outputs.field outputs . log This page was generated using Literate.jl .
`,46)]))}const g=i(l,[["render",p]]);export{o as __pageData,g as default};
diff --git a/previews/PR126/assets/examples_generated_LidDrivenCavity2D.md.D_iNsMnv.lean.js b/previews/PR126/assets/examples_generated_LidDrivenCavity2D.md.D_iNsMnv.lean.js
new file mode 100644
index 00000000..7912e64c
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_LidDrivenCavity2D.md.D_iNsMnv.lean.js
@@ -0,0 +1,91 @@
+import{_ as i,c as a,a5 as n,o as t}from"./chunks/framework.BSoZtefh.js";const o=JSON.parse('{"title":"Tutorial: Lid-Driven Cavity - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/LidDrivenCavity2D.md","filePath":"examples/generated/LidDrivenCavity2D.md","lastUpdated":null}'),l={name:"examples/generated/LidDrivenCavity2D.md"};function p(e,s,h,k,d,r){return t(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Tutorial: Lid-Driven Cavity - 2D In this example we consider a box with a moving lid. The velocity is initially at rest. The solution should reach at steady state equilibrium after a certain time. The same steady state should be obtained when solving a steady state problem.
We start by loading packages. A Makie plotting backend is needed
for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Case name for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity2D" ) The code allows for using different floating point number types, including single precision (Float32) and double precision (Float64). On the CPU, the speed is not really different, but double precision uses twice as much memory as single precision. When running on the GPU, single precision is preferred. Half precision (Float16) is also an option, but then the values should be scaled judiciously to avoid vanishing digits when applying differential operators of the form "right minus left divided by small distance".
Note how floating point type hygiene is enforced in the following using T to avoid mixing different precisions.
T = Float64
julia T = Float32 # T = Float16 We can also choose to do the computations on a different device. By default, the computations are performed on the host (CPU). An optional backend allows for moving arrays to a different device such as a GPU.
Note: For GPUs, single precision is preferred.
julia backend = CPU () # using CUDA; backend = CUDABackend() Here we choose a moderate Reynolds number. Note how we pass the floating point type.
Non-zero Dirichlet boundary conditions are specified as plain Julia functions.
julia U = ( T ( 1 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y bottom, y top ( DirichletBC (), DirichletBC (U)), ) We create a two-dimensional domain with a box of size [1, 1]. The grid is created as a Cartesian product between two vectors. We add a refinement near the walls.
julia n = 32 lims = T ( 0 ), T ( 1 ) x = cosine_grid (lims ... , n), cosine_grid (lims ... , n) plotgrid (x ... ) We can now build the setup and assemble operators. A 3D setup is built if we also provide a vector of z-coordinates.
julia setup = Setup (; x, boundary_conditions, Re, backend); nothing #hide The initial conditions are provided in function. The value dim() determines the velocity component.
julia ustart = velocityfield (setup, (dim, x, y) -> zero (x)); nothing #hide Iteration processors are called after every nupdate time steps. This can be useful for logging, plotting, or saving results. Their respective outputs are later returned by solve_unsteady.
julia processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1000 ), ); nothing #hide By default, a standard fourth order Runge-Kutta method is used. If we don't provide the time step explicitly, an adaptive time step is used.
julia tlims = ( T ( 0 ), T ( 10 )) state, outputs = solve_unsteady (; setup, ustart, tlims, Δt = T ( 1e-3 ), processors); nothing #hide Post-process We may visualize or export the computed fields
Export fields to VTK. The file outdir/solution.vti may be opened for visualization in ParaView . This is particularly useful for inspecting results from 3D simulations.
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot velocity. Note the time stamp used for computing boundary conditions, if any.
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) In addition, the named tuple outputs contains quantities from our processors. The logger returns nothing.
julia # outputs.rtp # outputs.ehist # outputs.espec # outputs.anim # outputs.vtk # outputs.field outputs . log Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity2D" ) T = Float32 # T = Float16 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 1_000 ) U = ( T ( 1 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y bottom, y top ( DirichletBC (), DirichletBC (U)), ) n = 32 lims = T ( 0 ), T ( 1 ) x = cosine_grid (lims ... , n), cosine_grid (lims ... , n) plotgrid (x ... ) setup = Setup (; x, boundary_conditions, Re, backend); ustart = velocityfield (setup, (dim, x, y) -> zero (x)); processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 10), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 1000 ), ); tlims = ( T ( 0 ), T ( 10 )) state, outputs = solve_unsteady (; setup, ustart, tlims, Δt = T ( 1e-3 ), processors); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) # outputs.rtp # outputs.ehist # outputs.espec # outputs.anim # outputs.vtk # outputs.field outputs . log This page was generated using Literate.jl .
`,46)]))}const g=i(l,[["render",p]]);export{o as __pageData,g as default};
diff --git a/previews/PR126/assets/examples_generated_LidDrivenCavity3D.md.CXnvIfHv.js b/previews/PR126/assets/examples_generated_LidDrivenCavity3D.md.CXnvIfHv.js
new file mode 100644
index 00000000..fd58d55b
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_LidDrivenCavity3D.md.CXnvIfHv.js
@@ -0,0 +1,79 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Lid-Driven Cavity - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/LidDrivenCavity3D.md","filePath":"examples/generated/LidDrivenCavity3D.md","lastUpdated":null}'),t={name:"examples/generated/LidDrivenCavity3D.md"};function p(l,s,k,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Lid-Driven Cavity - 3D In this example we consider a box with a moving lid. The velocity is initially at rest. The solution should reach at steady state equilibrium after a certain time. The same steady state should be obtained when solving a steady state problem.
julia using CairoMakie using IncompressibleNavierStokes Case name for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity3D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 3D grid is a Cartesian product of three vectors. Here we refine the grid near the walls.
julia x = cosine_grid ( T ( 0 ), T ( 1 ), 25 ), cosine_grid ( T ( 0 ), T ( 1 ), 25 ), LinRange ( - T ( 0.2 ), T ( 0.2 ), 11 ) plotgrid (x ... ) Boundary conditions: horizontal movement of the top lid
julia U = ( T ( 1 ), T ( 1 / 5 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y rear, y front ( DirichletBC (), DirichletBC (U)), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, backend); nothing #hide Initial conditions
julia ustart = velocityfield (setup, (dim, x, y, z) -> zero (x)) Solve unsteady problem
julia (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 0.2 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 ), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 20 ), ), ); nothing #hide Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Energy history
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity3D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 1_000 ) x = cosine_grid ( T ( 0 ), T ( 1 ), 25 ), cosine_grid ( T ( 0 ), T ( 1 ), 25 ), LinRange ( - T ( 0.2 ), T ( 0.2 ), 11 ) plotgrid (x ... ) U = ( T ( 1 ), T ( 1 / 5 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y rear, y front ( DirichletBC (), DirichletBC (U)), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) setup = Setup (; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y, z) -> zero (x)) (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 0.2 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 ), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 20 ), ), ); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) outputs . ehist This page was generated using Literate.jl .
`,33)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_LidDrivenCavity3D.md.CXnvIfHv.lean.js b/previews/PR126/assets/examples_generated_LidDrivenCavity3D.md.CXnvIfHv.lean.js
new file mode 100644
index 00000000..fd58d55b
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_LidDrivenCavity3D.md.CXnvIfHv.lean.js
@@ -0,0 +1,79 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Lid-Driven Cavity - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/LidDrivenCavity3D.md","filePath":"examples/generated/LidDrivenCavity3D.md","lastUpdated":null}'),t={name:"examples/generated/LidDrivenCavity3D.md"};function p(l,s,k,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Lid-Driven Cavity - 3D In this example we consider a box with a moving lid. The velocity is initially at rest. The solution should reach at steady state equilibrium after a certain time. The same steady state should be obtained when solving a steady state problem.
julia using CairoMakie using IncompressibleNavierStokes Case name for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity3D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 3D grid is a Cartesian product of three vectors. Here we refine the grid near the walls.
julia x = cosine_grid ( T ( 0 ), T ( 1 ), 25 ), cosine_grid ( T ( 0 ), T ( 1 ), 25 ), LinRange ( - T ( 0.2 ), T ( 0.2 ), 11 ) plotgrid (x ... ) Boundary conditions: horizontal movement of the top lid
julia U = ( T ( 1 ), T ( 1 / 5 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y rear, y front ( DirichletBC (), DirichletBC (U)), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions, backend); nothing #hide Initial conditions
julia ustart = velocityfield (setup, (dim, x, y, z) -> zero (x)) Solve unsteady problem
julia (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 0.2 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 ), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 20 ), ), ); nothing #hide Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Energy history
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "LidDrivenCavity3D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 1_000 ) x = cosine_grid ( T ( 0 ), T ( 1 ), 25 ), cosine_grid ( T ( 0 ), T ( 1 ), 25 ), LinRange ( - T ( 0.2 ), T ( 0.2 ), 11 ) plotgrid (x ... ) U = ( T ( 1 ), T ( 1 / 5 ), T ( 0 )) boundary_conditions = ( # x left, x right ( DirichletBC (), DirichletBC ()), # y rear, y front ( DirichletBC (), DirichletBC (U)), # z bottom, z top ( PeriodicBC (), PeriodicBC ()), ) setup = Setup (; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y, z) -> zero (x)) (; u, t), outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 0.2 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 50), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 10 ), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 10), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 100, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 20 ), ), ); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) outputs . ehist This page was generated using Literate.jl .
`,33)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_MultiActuator.md.DeF4XHSW.js b/previews/PR126/assets/examples_generated_MultiActuator.md.DeF4XHSW.js
new file mode 100644
index 00000000..c9244c80
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_MultiActuator.md.DeF4XHSW.js
@@ -0,0 +1,193 @@
+import{_ as i,c as a,a5 as h,o as n}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/MultiActuator.md","filePath":"examples/generated/MultiActuator.md","lastUpdated":null}'),k={name:"examples/generated/MultiActuator.md"};function p(l,s,t,e,E,d){return n(),a("div",null,s[0]||(s[0]=[h(`Note: Output is not generated for this example (to save resources on GitHub).
Unsteady actuator case - 2D In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
julia using CairoMakie using IncompressibleNavierStokes using Random Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "MultiActuator" ) Floating point precision
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Boundary conditions
julia boundary_conditions = ( # x left, x right ( DirichletBC ((dim, x, y, t) -> sinpi ( sinpi (t / 6 ) / 6 + one (x) / 2 * (dim == 1 ))), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), ) Actuator body force: A thrust coefficient Cₜ distributed over a thin rectangle
julia create_bodyforce (; xc, yc, D, δ, C) = (dim, x, y, t) -> dim == 1 && abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 ? - C / (D * δ) : zero (x) create_manyforce (forces ... ) = function (dim, x, y, t) out = zero (x) for f in forces out += f (dim, x, y, t) end out end disk = (; D = T ( 1 ), δ = T ( 0.11 ), C = T ( 0.2 )) bodyforce = create_manyforce ( create_bodyforce (; xc = T ( 2 ), yc = T ( 0 ), disk ... ), create_bodyforce (; xc = T ( 4 ), yc = T ( 0.7 ), disk ... ), create_bodyforce (; xc = T ( 6.4 ), yc = T ( - 1 ), disk ... ), ) A 2D grid is a Cartesian product of two vectors
julia n = 50 x = LinRange ( T ( 0 ), T ( 10 ), 5 n + 1 ), LinRange ( - T ( 2 ), T ( 2 ), 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 ))) Build setup and assemble operators
julia setup = Setup (; x, Re = T ( 1000 ), boundary_conditions, bodyforce, issteadybodyforce = true , backend, ); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? one (x) : zero (x)); t = T ( 0 ) boxes = map (bodyforce . forces) do (; xc, yc, D, δ) [ Point2f (xc - δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc + D / 2 ), ] end box = boxes[ 1 ] Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 12 )), method = RKMethods . RK44P2 (), processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # fieldname = :velocitynorm, # fieldname = :pressure, size = ( 600 , 300 ), nupdate = 1 , ), boxplotter = processor () do state for box in boxes lines! ( current_axis (), box; color = :red ) end end , # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 1), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 1), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (u, p).
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! .(boxes; color = :red ); fig Plot velocity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! .(boxes; color = :red ); fig Plot vorticity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! .(boxes; color = :red ); fig Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes using Random outdir = joinpath ( @__DIR__ , "output" , "MultiActuator" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() boundary_conditions = ( # x left, x right ( DirichletBC ((dim, x, y, t) -> sinpi ( sinpi (t / 6 ) / 6 + one (x) / 2 * (dim == 1 ))), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), ) create_bodyforce (; xc, yc, D, δ, C) = (dim, x, y, t) -> dim == 1 && abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 ? - C / (D * δ) : zero (x) create_manyforce (forces ... ) = function (dim, x, y, t) out = zero (x) for f in forces out += f (dim, x, y, t) end out end disk = (; D = T ( 1 ), δ = T ( 0.11 ), C = T ( 0.2 )) bodyforce = create_manyforce ( create_bodyforce (; xc = T ( 2 ), yc = T ( 0 ), disk ... ), create_bodyforce (; xc = T ( 4 ), yc = T ( 0.7 ), disk ... ), create_bodyforce (; xc = T ( 6.4 ), yc = T ( - 1 ), disk ... ), ) n = 50 x = LinRange ( T ( 0 ), T ( 10 ), 5 n + 1 ), LinRange ( - T ( 2 ), T ( 2 ), 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 ))) setup = Setup (; x, Re = T ( 1000 ), boundary_conditions, bodyforce, issteadybodyforce = true , backend, ); ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? one (x) : zero (x)); t = T ( 0 ) boxes = map (bodyforce . forces) do (; xc, yc, D, δ) [ Point2f (xc - δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc + D / 2 ), ] end box = boxes[ 1 ] state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 12 )), method = RKMethods . RK44P2 (), processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # fieldname = :velocitynorm, # fieldname = :pressure, size = ( 600 , 300 ), nupdate = 1 , ), boxplotter = processor () do state for box in boxes lines! ( current_axis (), box; color = :red ) end end , # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 1), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 1), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! .(boxes; color = :red ); fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! .(boxes; color = :red ); fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! .(boxes; color = :red ); fig This page was generated using Literate.jl .
`,37)]))}const y=i(k,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_MultiActuator.md.DeF4XHSW.lean.js b/previews/PR126/assets/examples_generated_MultiActuator.md.DeF4XHSW.lean.js
new file mode 100644
index 00000000..c9244c80
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_MultiActuator.md.DeF4XHSW.lean.js
@@ -0,0 +1,193 @@
+import{_ as i,c as a,a5 as h,o as n}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/MultiActuator.md","filePath":"examples/generated/MultiActuator.md","lastUpdated":null}'),k={name:"examples/generated/MultiActuator.md"};function p(l,s,t,e,E,d){return n(),a("div",null,s[0]||(s[0]=[h(`Note: Output is not generated for this example (to save resources on GitHub).
Unsteady actuator case - 2D In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
julia using CairoMakie using IncompressibleNavierStokes using Random Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "MultiActuator" ) Floating point precision
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Boundary conditions
julia boundary_conditions = ( # x left, x right ( DirichletBC ((dim, x, y, t) -> sinpi ( sinpi (t / 6 ) / 6 + one (x) / 2 * (dim == 1 ))), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), ) Actuator body force: A thrust coefficient Cₜ distributed over a thin rectangle
julia create_bodyforce (; xc, yc, D, δ, C) = (dim, x, y, t) -> dim == 1 && abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 ? - C / (D * δ) : zero (x) create_manyforce (forces ... ) = function (dim, x, y, t) out = zero (x) for f in forces out += f (dim, x, y, t) end out end disk = (; D = T ( 1 ), δ = T ( 0.11 ), C = T ( 0.2 )) bodyforce = create_manyforce ( create_bodyforce (; xc = T ( 2 ), yc = T ( 0 ), disk ... ), create_bodyforce (; xc = T ( 4 ), yc = T ( 0.7 ), disk ... ), create_bodyforce (; xc = T ( 6.4 ), yc = T ( - 1 ), disk ... ), ) A 2D grid is a Cartesian product of two vectors
julia n = 50 x = LinRange ( T ( 0 ), T ( 10 ), 5 n + 1 ), LinRange ( - T ( 2 ), T ( 2 ), 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 ))) Build setup and assemble operators
julia setup = Setup (; x, Re = T ( 1000 ), boundary_conditions, bodyforce, issteadybodyforce = true , backend, ); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? one (x) : zero (x)); t = T ( 0 ) boxes = map (bodyforce . forces) do (; xc, yc, D, δ) [ Point2f (xc - δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc + D / 2 ), ] end box = boxes[ 1 ] Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 12 )), method = RKMethods . RK44P2 (), processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # fieldname = :velocitynorm, # fieldname = :pressure, size = ( 600 , 300 ), nupdate = 1 , ), boxplotter = processor () do state for box in boxes lines! ( current_axis (), box; color = :red ) end end , # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 1), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 1), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (u, p).
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! .(boxes; color = :red ); fig Plot velocity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! .(boxes; color = :red ); fig Plot vorticity
julia fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! .(boxes; color = :red ); fig Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes using Random outdir = joinpath ( @__DIR__ , "output" , "MultiActuator" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() boundary_conditions = ( # x left, x right ( DirichletBC ((dim, x, y, t) -> sinpi ( sinpi (t / 6 ) / 6 + one (x) / 2 * (dim == 1 ))), PressureBC (), ), # y rear, y front ( PressureBC (), PressureBC ()), ) create_bodyforce (; xc, yc, D, δ, C) = (dim, x, y, t) -> dim == 1 && abs (x - xc) ≤ δ / 2 && abs (y - yc) ≤ D / 2 ? - C / (D * δ) : zero (x) create_manyforce (forces ... ) = function (dim, x, y, t) out = zero (x) for f in forces out += f (dim, x, y, t) end out end disk = (; D = T ( 1 ), δ = T ( 0.11 ), C = T ( 0.2 )) bodyforce = create_manyforce ( create_bodyforce (; xc = T ( 2 ), yc = T ( 0 ), disk ... ), create_bodyforce (; xc = T ( 4 ), yc = T ( 0.7 ), disk ... ), create_bodyforce (; xc = T ( 6.4 ), yc = T ( - 1 ), disk ... ), ) n = 50 x = LinRange ( T ( 0 ), T ( 10 ), 5 n + 1 ), LinRange ( - T ( 2 ), T ( 2 ), 2 n + 1 ) plotgrid (x ... ; figure = (; size = ( 600 , 300 ))) setup = Setup (; x, Re = T ( 1000 ), boundary_conditions, bodyforce, issteadybodyforce = true , backend, ); ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? one (x) : zero (x)); t = T ( 0 ) boxes = map (bodyforce . forces) do (; xc, yc, D, δ) [ Point2f (xc - δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc - D / 2 ), Point2f (xc + δ / 2 , yc + D / 2 ), Point2f (xc - δ / 2 , yc + D / 2 ), ] end box = boxes[ 1 ] state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 12 )), method = RKMethods . RK44P2 (), processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # fieldname = :velocitynorm, # fieldname = :pressure, size = ( 600 , 300 ), nupdate = 1 , ), boxplotter = processor () do state for box in boxes lines! ( current_axis (), box; color = :red ) end end , # ehist = realtimeplotter(; setup, plot = energy_history_plot, nupdate = 1), # espec = realtimeplotter(; setup, plot = energy_spectrum_plot, nupdate = 1), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = "$outdir", filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :pressure ) lines! .(boxes; color = :red ); fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :velocitynorm ) lines! .(boxes; color = :red ); fig fig = fieldplot (state; setup, size = ( 600 , 300 ), fieldname = :vorticity ) lines! .(boxes; color = :red ); fig This page was generated using Literate.jl .
`,37)]))}const y=i(k,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_PlanarMixing2D.md.DlU0y6-e.js b/previews/PR126/assets/examples_generated_PlanarMixing2D.md.DlU0y6-e.js
new file mode 100644
index 00000000..225afe90
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_PlanarMixing2D.md.DlU0y6-e.js
@@ -0,0 +1,106 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Planar mixing - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/PlanarMixing2D.md","filePath":"examples/generated/PlanarMixing2D.md","lastUpdated":null}'),k={name:"examples/generated/PlanarMixing2D.md"};function p(l,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Planar mixing - 2D Planar mixing example, as presented in [3 ].
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "PlanarMixing2D" ) Viscosity model
Boundary conditions
julia ΔU = 1.0 Ubar = 1.0 ϵ = ( 0.082 Ubar, 0.012 Ubar) n = ( 0.4π , 0.3π ) ω = ( 0.22 , 0.11 ) U (dim, x, y, t) = dim == 1 ? 1.0 + ΔU / 2 * tanh ( 2 y) + sum ( @. ϵ * ( 1 - tanh (y / 2 ) ^ 2 ) * cos (n * y) * sin (ω * t)) : 0.0 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( PressureBC (), PressureBC ()), ) A 2D grid is a Cartesian product of two vectors
julia n = 64 # n = 256 x = LinRange ( 0.0 , 256.0 , 4 n), LinRange ( - 32.0 , 32.0 , n) plotgrid (x ... ; figure = (; size = ( 600 , 250 ))) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions); psolver = psolver_direct (setup); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, 0.0 ); psolver); nothing #hide Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 100.0 ), psolver, method = RKMethods . RK44P2 (), # Δt = 0.1, processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 250 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (u, p)
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" ), psolver) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "PlanarMixing2D" ) Re = 500.0 ΔU = 1.0 Ubar = 1.0 ϵ = ( 0.082 Ubar, 0.012 Ubar) n = ( 0.4π , 0.3π ) ω = ( 0.22 , 0.11 ) U (dim, x, y, t) = dim == 1 ? 1.0 + ΔU / 2 * tanh ( 2 y) + sum ( @. ϵ * ( 1 - tanh (y / 2 ) ^ 2 ) * cos (n * y) * sin (ω * t)) : 0.0 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( PressureBC (), PressureBC ()), ) n = 64 # n = 256 x = LinRange ( 0.0 , 256.0 , 4 n), LinRange ( - 32.0 , 32.0 , n) plotgrid (x ... ; figure = (; size = ( 600 , 250 ))) setup = Setup (; x, Re, boundary_conditions); psolver = psolver_direct (setup); ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, 0.0 ); psolver); state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 100.0 ), psolver, method = RKMethods . RK44P2 (), # Δt = 0.1, processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 250 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" ), psolver) This page was generated using Literate.jl .
`,29)]))}const y=i(k,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_PlanarMixing2D.md.DlU0y6-e.lean.js b/previews/PR126/assets/examples_generated_PlanarMixing2D.md.DlU0y6-e.lean.js
new file mode 100644
index 00000000..225afe90
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_PlanarMixing2D.md.DlU0y6-e.lean.js
@@ -0,0 +1,106 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Planar mixing - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/PlanarMixing2D.md","filePath":"examples/generated/PlanarMixing2D.md","lastUpdated":null}'),k={name:"examples/generated/PlanarMixing2D.md"};function p(l,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Planar mixing - 2D Planar mixing example, as presented in [3 ].
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "PlanarMixing2D" ) Viscosity model
Boundary conditions
julia ΔU = 1.0 Ubar = 1.0 ϵ = ( 0.082 Ubar, 0.012 Ubar) n = ( 0.4π , 0.3π ) ω = ( 0.22 , 0.11 ) U (dim, x, y, t) = dim == 1 ? 1.0 + ΔU / 2 * tanh ( 2 y) + sum ( @. ϵ * ( 1 - tanh (y / 2 ) ^ 2 ) * cos (n * y) * sin (ω * t)) : 0.0 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( PressureBC (), PressureBC ()), ) A 2D grid is a Cartesian product of two vectors
julia n = 64 # n = 256 x = LinRange ( 0.0 , 256.0 , 4 n), LinRange ( - 32.0 , 32.0 , n) plotgrid (x ... ; figure = (; size = ( 600 , 250 ))) Build setup and assemble operators
julia setup = Setup (; x, Re, boundary_conditions); psolver = psolver_direct (setup); nothing #hide Initial conditions (extend inflow)
julia ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, 0.0 ); psolver); nothing #hide Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 100.0 ), psolver, method = RKMethods . RK44P2 (), # Δt = 0.1, processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 250 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (u, p)
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" ), psolver) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "PlanarMixing2D" ) Re = 500.0 ΔU = 1.0 Ubar = 1.0 ϵ = ( 0.082 Ubar, 0.012 Ubar) n = ( 0.4π , 0.3π ) ω = ( 0.22 , 0.11 ) U (dim, x, y, t) = dim == 1 ? 1.0 + ΔU / 2 * tanh ( 2 y) + sum ( @. ϵ * ( 1 - tanh (y / 2 ) ^ 2 ) * cos (n * y) * sin (ω * t)) : 0.0 boundary_conditions = ( # x left, x right ( DirichletBC (U), PressureBC ()), # y rear, y front ( PressureBC (), PressureBC ()), ) n = 64 # n = 256 x = LinRange ( 0.0 , 256.0 , 4 n), LinRange ( - 32.0 , 32.0 , n) plotgrid (x ... ; figure = (; size = ( 600 , 250 ))) setup = Setup (; x, Re, boundary_conditions); psolver = psolver_direct (setup); ustart = velocityfield (setup, (dim, x, y) -> U (dim, x, y, 0.0 ); psolver); state, outputs = solve_unsteady (; setup, ustart, tlims = ( 0.0 , 100.0 ), psolver, method = RKMethods . RK44P2 (), # Δt = 0.1, processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, docolorbar = false , size = ( 600 , 250 ), nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" ), psolver) This page was generated using Literate.jl .
`,29)]))}const y=i(k,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_PlaneJets2D.md.Cxpw6Mt8.js b/previews/PR126/assets/examples_generated_PlaneJets2D.md.Cxpw6Mt8.js
new file mode 100644
index 00000000..b5f02751
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_PlaneJets2D.md.Cxpw6Mt8.js
@@ -0,0 +1,273 @@
+import{_ as i,c as a,a5 as h,o as k}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Plane jets - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/PlaneJets2D.md","filePath":"examples/generated/PlaneJets2D.md","lastUpdated":null}'),n={name:"examples/generated/PlaneJets2D.md"};function l(p,s,t,e,E,d){return k(),a("div",null,s[0]||(s[0]=[h(`Note: Output is not generated for this example (to save resources on GitHub).
Plane jets - 2D Plane jets example, as presented in [4 ]. Note that the original formulation is in 3D.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using FFTW using CairoMakie using IncompressibleNavierStokes using LaTeXStrings Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "PlaneJets2D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
Test cases (A, B, C, D; in order)
julia # V() = sqrt(T(467.4)) V () = T ( 21.619435700313733 ) U_A (y) = V () / 2 * ( tanh ((y + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - T ( 0.5 )) / T ( 0.1 ))) U_B (y) = V () / 2 * ( tanh ((y + 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y + 1 - T ( 0.5 )) / T ( 0.1 ))) + V () / 2 * ( tanh ((y - 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - 1 - T ( 0.5 )) / T ( 0.1 ))) U_C (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) + V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) U_D (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) - V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) # U(y) = U_A(y) # U(y) = U_B(y) U (y) = U_C (y) # U(y) = U_D(y) Random noise to stimulate turbulence
julia U (x, y) = ( 1 + T ( 0.1 ) * ( rand (T) - T ( 0.5 ))) * U (y) # boundary_conditions = ( # (PeriodicBC(), PeriodicBC()), # (PressureBC(), PressureBC()) # ) A 2D grid is a Cartesian product of two vectors
julia n = 64 # n = 128 # n = 256 x = LinRange ( T ( 0 ), T ( 16 ), 4 n + 1 ), LinRange ( - T ( 10 ), T ( 10 ), 5 n + 1 ) plotgrid (x ... ) Build setup and assemble operators
julia setup = Setup (x, Re, backend); # setup = Setup(; x, Re, boundary_conditions, backend); Initial conditions
julia ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U (x, y) : zero (x)); nothing #hide Real time plot: Streamwise average and spectrum
julia function meanplot (state; setup) (; xp, Iu, Ip, Nu, N) = setup . grid umean = lift (state) do (; u, p, t) reshape ( sum (u[ 1 ][Iu[ 1 ]]; dims = 1 ), :) ./ Nu[ 1 ][ 1 ] ./ V () end K = Nu[ 1 ][ 2 ] ÷ 2 k = 1 : (K - 1 ) # Find energy spectrum where y = 0 n₀ = findmin (abs, xp[ 2 ])[ 2 ] E₀ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₀] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₀ = xp[ 2 ][n₀] # Find energy spectrum where y = 1 n₁ = findmin (y -> abs (y - 1 ), xp[ 2 ])[ 2 ] E₁ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₁] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₁ = xp[ 2 ][n₁] fig = Figure () ax = Axis ( fig[ 1 , 1 ]; title = "Mean streamwise flow" , xlabel = "y" , ylabel = L " \\l angle u \\r angle / U_0" , ) lines! (ax, xp[ 2 ][ 2 : end - 1 ], umean) ax = Axis ( fig[ 1 , 2 ]; title = "Streamwise energy spectrum" , xscale = log10, yscale = log10, xlabel = L "k_x" , ylabel = L " \\h at{U}_{cl} / U_0" , ) # ylims!(ax, (10^(0.0), 10^4.0)) ksub = k[ 10 : end ] # lines!(ax, ksub, 1000 .* ksub .^ (-5 / 3); label = L"k^{-5/3}") lines! (ax, ksub, 1e7 .* ksub .^ - 3 ; label = L "k^{-3}" ) scatter! (ax, k, E₀; label = "y = $y₀ " ) scatter! (ax, k, E₁; label = "y = $y₁ " ) axislegend (ax; position = :lb ) # on(_ -> autolimits!(ax), E₁) fig end Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1 )), method = RKMethods . RK44P2 (), Δt = 0.001 , processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, plot = meanplot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 4), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (u, p)
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot initial velocity
julia fieldplot ((; u = u₀, p = p₀, t = T ( 0 )); setup, fieldname = :velocitynorm ) Plot final velocity
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using FFTW using GLMakie using IncompressibleNavierStokes using LaTeXStrings outdir = joinpath ( @__DIR__ , "output" , "PlaneJets2D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 6_000 ) # V() = sqrt(T(467.4)) V () = T ( 21.619435700313733 ) U_A (y) = V () / 2 * ( tanh ((y + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - T ( 0.5 )) / T ( 0.1 ))) U_B (y) = V () / 2 * ( tanh ((y + 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y + 1 - T ( 0.5 )) / T ( 0.1 ))) + V () / 2 * ( tanh ((y - 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - 1 - T ( 0.5 )) / T ( 0.1 ))) U_C (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) + V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) U_D (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) - V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) # U(y) = U_A(y) # U(y) = U_B(y) U (y) = U_C (y) # U(y) = U_D(y) U (x, y) = ( 1 + T ( 0.1 ) * ( rand (T) - T ( 0.5 ))) * U (y) # boundary_conditions = ( # (PeriodicBC(), PeriodicBC()), # (PressureBC(), PressureBC()) # ) n = 64 # n = 128 # n = 256 x = LinRange ( T ( 0 ), T ( 16 ), 4 n + 1 ), LinRange ( - T ( 10 ), T ( 10 ), 5 n + 1 ) plotgrid (x ... ) setup = Setup (x, Re, backend); # setup = Setup(; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U (x, y) : zero (x)); function meanplot (state; setup) (; xp, Iu, Ip, Nu, N) = setup . grid umean = lift (state) do (; u, p, t) reshape ( sum (u[ 1 ][Iu[ 1 ]]; dims = 1 ), :) ./ Nu[ 1 ][ 1 ] ./ V () end K = Nu[ 1 ][ 2 ] ÷ 2 k = 1 : (K - 1 ) # Find energy spectrum where y = 0 n₀ = findmin (abs, xp[ 2 ])[ 2 ] E₀ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₀] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₀ = xp[ 2 ][n₀] # Find energy spectrum where y = 1 n₁ = findmin (y -> abs (y - 1 ), xp[ 2 ])[ 2 ] E₁ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₁] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₁ = xp[ 2 ][n₁] fig = Figure () ax = Axis ( fig[ 1 , 1 ]; title = "Mean streamwise flow" , xlabel = "y" , ylabel = L " \\l angle u \\r angle / U_0" , ) lines! (ax, xp[ 2 ][ 2 : end - 1 ], umean) ax = Axis ( fig[ 1 , 2 ]; title = "Streamwise energy spectrum" , xscale = log10, yscale = log10, xlabel = L "k_x" , ylabel = L " \\h at{U}_{cl} / U_0" , ) # ylims!(ax, (10^(0.0), 10^4.0)) ksub = k[ 10 : end ] # lines!(ax, ksub, 1000 .* ksub .^ (-5 / 3); label = L"k^{-5/3}") lines! (ax, ksub, 1e7 .* ksub .^ - 3 ; label = L "k^{-3}" ) scatter! (ax, k, E₀; label = "y = $y₀ " ) scatter! (ax, k, E₁; label = "y = $y₁ " ) axislegend (ax; position = :lb ) # on(_ -> autolimits!(ax), E₁) fig end state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1 )), method = RKMethods . RK44P2 (), Δt = 0.001 , processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, plot = meanplot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 4), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot ((; u = u₀, p = p₀, t = T ( 0 )); setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) This page was generated using Literate.jl .
`,45)]))}const y=i(n,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_PlaneJets2D.md.Cxpw6Mt8.lean.js b/previews/PR126/assets/examples_generated_PlaneJets2D.md.Cxpw6Mt8.lean.js
new file mode 100644
index 00000000..b5f02751
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_PlaneJets2D.md.Cxpw6Mt8.lean.js
@@ -0,0 +1,273 @@
+import{_ as i,c as a,a5 as h,o as k}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Plane jets - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/PlaneJets2D.md","filePath":"examples/generated/PlaneJets2D.md","lastUpdated":null}'),n={name:"examples/generated/PlaneJets2D.md"};function l(p,s,t,e,E,d){return k(),a("div",null,s[0]||(s[0]=[h(`Note: Output is not generated for this example (to save resources on GitHub).
Plane jets - 2D Plane jets example, as presented in [4 ]. Note that the original formulation is in 3D.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using FFTW using CairoMakie using IncompressibleNavierStokes using LaTeXStrings Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "PlaneJets2D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
Test cases (A, B, C, D; in order)
julia # V() = sqrt(T(467.4)) V () = T ( 21.619435700313733 ) U_A (y) = V () / 2 * ( tanh ((y + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - T ( 0.5 )) / T ( 0.1 ))) U_B (y) = V () / 2 * ( tanh ((y + 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y + 1 - T ( 0.5 )) / T ( 0.1 ))) + V () / 2 * ( tanh ((y - 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - 1 - T ( 0.5 )) / T ( 0.1 ))) U_C (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) + V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) U_D (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) - V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) # U(y) = U_A(y) # U(y) = U_B(y) U (y) = U_C (y) # U(y) = U_D(y) Random noise to stimulate turbulence
julia U (x, y) = ( 1 + T ( 0.1 ) * ( rand (T) - T ( 0.5 ))) * U (y) # boundary_conditions = ( # (PeriodicBC(), PeriodicBC()), # (PressureBC(), PressureBC()) # ) A 2D grid is a Cartesian product of two vectors
julia n = 64 # n = 128 # n = 256 x = LinRange ( T ( 0 ), T ( 16 ), 4 n + 1 ), LinRange ( - T ( 10 ), T ( 10 ), 5 n + 1 ) plotgrid (x ... ) Build setup and assemble operators
julia setup = Setup (x, Re, backend); # setup = Setup(; x, Re, boundary_conditions, backend); Initial conditions
julia ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U (x, y) : zero (x)); nothing #hide Real time plot: Streamwise average and spectrum
julia function meanplot (state; setup) (; xp, Iu, Ip, Nu, N) = setup . grid umean = lift (state) do (; u, p, t) reshape ( sum (u[ 1 ][Iu[ 1 ]]; dims = 1 ), :) ./ Nu[ 1 ][ 1 ] ./ V () end K = Nu[ 1 ][ 2 ] ÷ 2 k = 1 : (K - 1 ) # Find energy spectrum where y = 0 n₀ = findmin (abs, xp[ 2 ])[ 2 ] E₀ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₀] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₀ = xp[ 2 ][n₀] # Find energy spectrum where y = 1 n₁ = findmin (y -> abs (y - 1 ), xp[ 2 ])[ 2 ] E₁ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₁] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₁ = xp[ 2 ][n₁] fig = Figure () ax = Axis ( fig[ 1 , 1 ]; title = "Mean streamwise flow" , xlabel = "y" , ylabel = L " \\l angle u \\r angle / U_0" , ) lines! (ax, xp[ 2 ][ 2 : end - 1 ], umean) ax = Axis ( fig[ 1 , 2 ]; title = "Streamwise energy spectrum" , xscale = log10, yscale = log10, xlabel = L "k_x" , ylabel = L " \\h at{U}_{cl} / U_0" , ) # ylims!(ax, (10^(0.0), 10^4.0)) ksub = k[ 10 : end ] # lines!(ax, ksub, 1000 .* ksub .^ (-5 / 3); label = L"k^{-5/3}") lines! (ax, ksub, 1e7 .* ksub .^ - 3 ; label = L "k^{-3}" ) scatter! (ax, k, E₀; label = "y = $y₀ " ) scatter! (ax, k, E₁; label = "y = $y₁ " ) axislegend (ax; position = :lb ) # on(_ -> autolimits!(ax), E₁) fig end Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1 )), method = RKMethods . RK44P2 (), Δt = 0.001 , processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, plot = meanplot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 4), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields (u, p)
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot initial velocity
julia fieldplot ((; u = u₀, p = p₀, t = T ( 0 )); setup, fieldname = :velocitynorm ) Plot final velocity
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using FFTW using GLMakie using IncompressibleNavierStokes using LaTeXStrings outdir = joinpath ( @__DIR__ , "output" , "PlaneJets2D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 6_000 ) # V() = sqrt(T(467.4)) V () = T ( 21.619435700313733 ) U_A (y) = V () / 2 * ( tanh ((y + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - T ( 0.5 )) / T ( 0.1 ))) U_B (y) = V () / 2 * ( tanh ((y + 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y + 1 - T ( 0.5 )) / T ( 0.1 ))) + V () / 2 * ( tanh ((y - 1 + T ( 0.5 )) / T ( 0.1 )) - tanh ((y - 1 - T ( 0.5 )) / T ( 0.1 ))) U_C (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) + V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) U_D (y) = V () / 2 * ( tanh (((y + T ( 1.0 )) / 1 + T ( 0.5 )) / T ( 0.1 )) - tanh (((y + T ( 1.0 )) / 1 - T ( 0.5 )) / T ( 0.1 )) ) - V () / 4 * ( tanh (((y - T ( 1.5 )) / 2 + T ( 0.5 )) / T ( 0.2 )) - tanh (((y - T ( 1.5 )) / 2 - T ( 0.5 )) / T ( 0.2 )) ) # U(y) = U_A(y) # U(y) = U_B(y) U (y) = U_C (y) # U(y) = U_D(y) U (x, y) = ( 1 + T ( 0.1 ) * ( rand (T) - T ( 0.5 ))) * U (y) # boundary_conditions = ( # (PeriodicBC(), PeriodicBC()), # (PressureBC(), PressureBC()) # ) n = 64 # n = 128 # n = 256 x = LinRange ( T ( 0 ), T ( 16 ), 4 n + 1 ), LinRange ( - T ( 10 ), T ( 10 ), 5 n + 1 ) plotgrid (x ... ) setup = Setup (x, Re, backend); # setup = Setup(; x, Re, boundary_conditions, backend); ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U (x, y) : zero (x)); function meanplot (state; setup) (; xp, Iu, Ip, Nu, N) = setup . grid umean = lift (state) do (; u, p, t) reshape ( sum (u[ 1 ][Iu[ 1 ]]; dims = 1 ), :) ./ Nu[ 1 ][ 1 ] ./ V () end K = Nu[ 1 ][ 2 ] ÷ 2 k = 1 : (K - 1 ) # Find energy spectrum where y = 0 n₀ = findmin (abs, xp[ 2 ])[ 2 ] E₀ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₀] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₀ = xp[ 2 ][n₀] # Find energy spectrum where y = 1 n₁ = findmin (y -> abs (y - 1 ), xp[ 2 ])[ 2 ] E₁ = lift (state) do (; u, p, t) u_y = u[ 1 ][:, n₁] abs .( fft (u_y .^ 2 ))[k .+ 1 ] end y₁ = xp[ 2 ][n₁] fig = Figure () ax = Axis ( fig[ 1 , 1 ]; title = "Mean streamwise flow" , xlabel = "y" , ylabel = L " \\l angle u \\r angle / U_0" , ) lines! (ax, xp[ 2 ][ 2 : end - 1 ], umean) ax = Axis ( fig[ 1 , 2 ]; title = "Streamwise energy spectrum" , xscale = log10, yscale = log10, xlabel = L "k_x" , ylabel = L " \\h at{U}_{cl} / U_0" , ) # ylims!(ax, (10^(0.0), 10^4.0)) ksub = k[ 10 : end ] # lines!(ax, ksub, 1000 .* ksub .^ (-5 / 3); label = L"k^{-5/3}") lines! (ax, ksub, 1e7 .* ksub .^ - 3 ; label = L "k^{-3}" ) scatter! (ax, k, E₀; label = "y = $y₀ " ) scatter! (ax, k, E₁; label = "y = $y₁ " ) axislegend (ax; position = :lb ) # on(_ -> autolimits!(ax), E₁) fig end state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1 )), method = RKMethods . RK44P2 (), Δt = 0.001 , processors = ( rtp = realtimeplotter (; setup, # plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, plot = meanplot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 4), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot ((; u = u₀, p = p₀, t = T ( 0 )); setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) This page was generated using Literate.jl .
`,45)]))}const y=i(n,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighBenard2D.md.DlQrJFaM.js b/previews/PR126/assets/examples_generated_RayleighBenard2D.md.DlQrJFaM.js
new file mode 100644
index 00000000..8bc9b654
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_RayleighBenard2D.md.DlQrJFaM.js
@@ -0,0 +1,222 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const t="/IncompressibleNavierStokes.jl/previews/PR126/assets/cuibxhd.BxypEGK2.png",l="/IncompressibleNavierStokes.jl/previews/PR126/RayleighBenard2D.mp4",k="/IncompressibleNavierStokes.jl/previews/PR126/assets/xsmunma.C3CW_rGA.png",p="/IncompressibleNavierStokes.jl/previews/PR126/assets/jqxyinv.D8geYuhY.png",C=JSON.parse('{"title":"Rayleigh-Bénard convection (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard2D.md","filePath":"examples/generated/RayleighBenard2D.md","lastUpdated":null}'),e={name:"examples/generated/RayleighBenard2D.md"};function E(r,s,d,g,y,F){return h(),a("div",null,s[0]||(s[0]=[n(`Rayleigh-Bénard convection (2D) A hot and a cold plate generate a convection cell in a box.
julia using CairoMakie using IncompressibleNavierStokes Output directory for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard2D" ) "/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighBenard2D" Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Define observer function to track Nusselt numbers on top and bottom plates.
julia function nusseltplot (state; setup) state isa Observable || (state = Observable (state)) (; Δ, Δu) = setup . grid Δy1 = Δu[ 2 ][ 1 : 1 ] |> sum Δy2 = Δu[ 2 ][ end - 1 : end - 1 ] |> sum # Observe Nusselt numbers Nu1 = Observable (Point2f[]) Nu2 = Observable (Point2f[]) on (state) do (; temp, t) dTdy = @. (temp[:, 2 ] - temp[:, 1 ]) / Δy1 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu1[], Point2f (t, Nu)) dTdy = @. (temp[:, end - 1 ] - temp[:, end - 2 ]) / Δy2 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu2[], Point2f (t, Nu)) (Nu1, Nu2) .|> notify ## Update plot end # Plot Nu history fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Nusselt number" , xlabel = "t" , ylabel = "Nu" ) lines! (ax, Nu1; label = "Lower plate" ) lines! (ax, Nu2; label = "Upper plate" ) axislegend (ax) on (_ -> autolimits! (ax), Nu2) fig end nusseltplot (generic function with 1 method) Define observer function to track average temperature.
julia function averagetemp (state; setup) state isa Observable || (state = Observable (state)) (; xp, Δ, Ip) = setup . grid ix = Ip . indices[ 1 ] Ty = lift (state) do (; temp) Ty = sum (temp[ix, :] .* Δ[ 1 ][ix]; dims = 1 ) ./ sum (Δ[ 1 ][ix]) Array (Ty)[:] end Ty0 = copy (Ty[]) yy = Array (xp[ 2 ]) fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Average temperature" , xlabel = "T" , ylabel = "y" ) lines! (ax, Ty0, yy; label = "t = 0" ) lines! (ax, Ty, yy; label = "t = t" ) axislegend (ax) on (_ -> autolimits! (ax), Ty) fig end averagetemp (generic function with 1 method) Instabilities should depend on the floating point precision. Try both Float32 and Float64.
Temperature equation setup.
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 2 , nondim_type = 1 , ) (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2) Grid
julia n = 100 x = tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )) plotgrid (x ... )
Setup
julia setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, backend, ); Initial conditions
julia ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (y) / 2 + max ( sinpi ( 20 * x) / 100 , 0 )); Processors
julia processors = (; rtp = realtimeplotter (; setup, fieldname = :temperature , colorrange = ( T ( 0 ), T ( 1 )), size = ( 600 , 350 ), colormap = :seaborn_icefire_gradient , nupdate = 20 , ), nusselt = realtimeplotter (; setup, plot = nusseltplot, nupdate = 20 , ), avg = realtimeplotter (; setup, plot = averagetemp, nupdate = 50 , ), log = timelogger (; nupdate = 1000 ), ) (rtp = (initialize = IncompressibleNavierStokesMakieExt.var"#10#12"{@NamedTuple{grid::@NamedTuple{xlims::Tuple{Tuple{Float32, Float32}, Tuple{Float32, Float32}}, dimension::IncompressibleNavierStokes.Dimension{2}, N::Tuple{Int64, Int64}, Nu::Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}, Np::Tuple{Int64, Int64}, Iu::Tuple{CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}, Ip::CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, x::Tuple{Vector{Float32}, Vector{Float32}}, xu::Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, xp::Tuple{Vector{Float32}, Vector{Float32}}, Δ::Tuple{Vector{Float32}, Vector{Float32}}, Δu::Tuple{Vector{Float32}, Vector{Float32}}, A::Tuple{Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}}}, boundary_conditions::Tuple{Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}, Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}}, Re::Float32, bodyforce::Nothing, issteadybodyforce::Bool, closure_model::Nothing, backend::CPU, workgroupsize::Int64, temperature::@NamedTuple{α1::Float32, α2::Float32, α3::Float32, α4::Float32, γ::Float32, dodissipation::Bool, boundary_conditions::Tuple{Tuple{SymmetricBC, SymmetricBC}, Tuple{DirichletBC{Float32}, DirichletBC{Float32}}}, gdir::Int64}}, typeof(fieldplot), Int64, Bool, Nothing, Bool, Nothing, Base.Pairs{Symbol, Any, NTuple{4, Symbol}, @NamedTuple{fieldname::Symbol, colorrange::Tuple{Float32, Float32}, size::Tuple{Int64, Int64}, colormap::Symbol}}}((grid = (xlims = ((0.0f0, 2.0f0), (0.0f0, 1.0f0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (202, 102), Nu = ((199, 100), (200, 99)), Np = (200, 100), Iu = (CartesianIndices((2:200, 2:101)), CartesianIndices((2:201, 2:100))), Ip = CartesianIndices((2:201, 2:101)), x = (Float32[0.0, 0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0]), xu = ((Float32[0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393, 0.043240547 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679, 0.047299176 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0])), xp = (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), Δ = (Float32[1.1920929f-7, 0.0044347644, 0.0045239925, 0.0046145916, 0.004706502, 0.004799843, 0.004894674, 0.004990816, 0.00508821, 0.005187154 … 0.005187154, 0.00508821, 0.004990816, 0.0048947334, 0.004799843, 0.0047063828, 0.0046145916, 0.0045239925, 0.004434824, 1.1920929f-7], Float32[1.1920929f-7, 0.0044793785, 0.004660547, 0.0048472583, 0.005039513, 0.0052372515, 0.0054405034, 0.005649179, 0.00586316, 0.006082386 … 0.006082356, 0.00586313, 0.005649209, 0.0054404736, 0.0052372813, 0.005039513, 0.004847288, 0.004660487, 0.0044794083, 1.1920929f-7]), Δu = (Float32[0.0022173822, 0.0044793785, 0.004569292, 0.004660547, 0.0047531724, 0.0048472583, 0.004942745, 0.005039513, 0.005137682, 0.0052372515 … 0.005137682, 0.0050395727, 0.004942775, 0.004847288, 0.004753113, 0.004660487, 0.004569292, 0.0044794083, 0.002217412, 1.1920929f-7], Float32[0.0022396892, 0.0045699626, 0.0047539026, 0.0049433857, 0.0051383823, 0.0053388774, 0.0055448413, 0.0057561696, 0.005972773, 0.0061945915 … 0.005972743, 0.0057561994, 0.0055448413, 0.0053389072, 0.005138397, 0.004943371, 0.0047538877, 0.0045699477, 0.0022397041, 1.1920929f-7]), A = (((Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), (Float32[1.0, 1.0, 0.5099108, 0.50981885, 0.5097228, 0.50962067, 0.50951755, 0.5094086, 0.50929356, 0.509176 … 0.49093604, 0.49081892, 0.4907065, 0.49058872, 0.4904881, 0.49038374, 0.49028164, 0.49017644, 0.49009407, 0.0], Float32[0.0, 0.49008918, 0.49018115, 0.49027717, 0.49037933, 0.49048245, 0.49059144, 0.49070647, 0.49082395, 0.49094325 … 0.5091811, 0.5092935, 0.5094113, 0.5095119, 0.50961626, 0.50971836, 0.50982356, 0.50990593, 1.0, 1.0])), ((Float32[1.0, 1.0, 0.50497997, 0.50495696, 0.50493026, 0.5049094, 0.5048909, 0.5048628, 0.50483155, 0.5048146 … 0.49520862, 0.4951738, 0.4951626, 0.49514025, 0.49510598, 0.4950843, 0.49507612, 0.49504304, 0.49502343, 0.0], Float32[0.0, 0.49502006, 0.49504304, 0.49506977, 0.4950906, 0.49510905, 0.4951372, 0.49516848, 0.4951854, 0.49521717 … 0.5048262, 0.5048374, 0.50485975, 0.504894, 0.5049157, 0.5049239, 0.50495696, 0.5049766, 1.0, 1.0]), (Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing)), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 3752.933f0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)), IncompressibleNavierStokes.fieldplot, 20, true, nothing, false, nothing, Base.Pairs{Symbol, Any, NTuple{4, Symbol}, @NamedTuple{fieldname::Symbol, colorrange::Tuple{Float32, Float32}, size::Tuple{Int64, Int64}, colormap::Symbol}}(:fieldname => :temperature, :colorrange => (0.0f0, 1.0f0), :size => (600, 350), :colormap => :seaborn_icefire_gradient)), finalize = IncompressibleNavierStokes.var"#295#296"()), nusselt = (initialize = IncompressibleNavierStokesMakieExt.var"#10#12"{@NamedTuple{grid::@NamedTuple{xlims::Tuple{Tuple{Float32, Float32}, Tuple{Float32, Float32}}, dimension::IncompressibleNavierStokes.Dimension{2}, N::Tuple{Int64, Int64}, Nu::Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}, Np::Tuple{Int64, Int64}, Iu::Tuple{CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}, Ip::CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, x::Tuple{Vector{Float32}, Vector{Float32}}, xu::Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, xp::Tuple{Vector{Float32}, Vector{Float32}}, Δ::Tuple{Vector{Float32}, Vector{Float32}}, Δu::Tuple{Vector{Float32}, Vector{Float32}}, A::Tuple{Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}}}, boundary_conditions::Tuple{Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}, Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}}, Re::Float32, bodyforce::Nothing, issteadybodyforce::Bool, closure_model::Nothing, backend::CPU, workgroupsize::Int64, temperature::@NamedTuple{α1::Float32, α2::Float32, α3::Float32, α4::Float32, γ::Float32, dodissipation::Bool, boundary_conditions::Tuple{Tuple{SymmetricBC, SymmetricBC}, Tuple{DirichletBC{Float32}, DirichletBC{Float32}}}, gdir::Int64}}, typeof(Main.nusseltplot), Int64, Bool, Nothing, Bool, Nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}((grid = (xlims = ((0.0f0, 2.0f0), (0.0f0, 1.0f0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (202, 102), Nu = ((199, 100), (200, 99)), Np = (200, 100), Iu = (CartesianIndices((2:200, 2:101)), CartesianIndices((2:201, 2:100))), Ip = CartesianIndices((2:201, 2:101)), x = (Float32[0.0, 0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0]), xu = ((Float32[0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393, 0.043240547 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679, 0.047299176 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0])), xp = (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), Δ = (Float32[1.1920929f-7, 0.0044347644, 0.0045239925, 0.0046145916, 0.004706502, 0.004799843, 0.004894674, 0.004990816, 0.00508821, 0.005187154 … 0.005187154, 0.00508821, 0.004990816, 0.0048947334, 0.004799843, 0.0047063828, 0.0046145916, 0.0045239925, 0.004434824, 1.1920929f-7], Float32[1.1920929f-7, 0.0044793785, 0.004660547, 0.0048472583, 0.005039513, 0.0052372515, 0.0054405034, 0.005649179, 0.00586316, 0.006082386 … 0.006082356, 0.00586313, 0.005649209, 0.0054404736, 0.0052372813, 0.005039513, 0.004847288, 0.004660487, 0.0044794083, 1.1920929f-7]), Δu = (Float32[0.0022173822, 0.0044793785, 0.004569292, 0.004660547, 0.0047531724, 0.0048472583, 0.004942745, 0.005039513, 0.005137682, 0.0052372515 … 0.005137682, 0.0050395727, 0.004942775, 0.004847288, 0.004753113, 0.004660487, 0.004569292, 0.0044794083, 0.002217412, 1.1920929f-7], Float32[0.0022396892, 0.0045699626, 0.0047539026, 0.0049433857, 0.0051383823, 0.0053388774, 0.0055448413, 0.0057561696, 0.005972773, 0.0061945915 … 0.005972743, 0.0057561994, 0.0055448413, 0.0053389072, 0.005138397, 0.004943371, 0.0047538877, 0.0045699477, 0.0022397041, 1.1920929f-7]), A = (((Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), (Float32[1.0, 1.0, 0.5099108, 0.50981885, 0.5097228, 0.50962067, 0.50951755, 0.5094086, 0.50929356, 0.509176 … 0.49093604, 0.49081892, 0.4907065, 0.49058872, 0.4904881, 0.49038374, 0.49028164, 0.49017644, 0.49009407, 0.0], Float32[0.0, 0.49008918, 0.49018115, 0.49027717, 0.49037933, 0.49048245, 0.49059144, 0.49070647, 0.49082395, 0.49094325 … 0.5091811, 0.5092935, 0.5094113, 0.5095119, 0.50961626, 0.50971836, 0.50982356, 0.50990593, 1.0, 1.0])), ((Float32[1.0, 1.0, 0.50497997, 0.50495696, 0.50493026, 0.5049094, 0.5048909, 0.5048628, 0.50483155, 0.5048146 … 0.49520862, 0.4951738, 0.4951626, 0.49514025, 0.49510598, 0.4950843, 0.49507612, 0.49504304, 0.49502343, 0.0], Float32[0.0, 0.49502006, 0.49504304, 0.49506977, 0.4950906, 0.49510905, 0.4951372, 0.49516848, 0.4951854, 0.49521717 … 0.5048262, 0.5048374, 0.50485975, 0.504894, 0.5049157, 0.5049239, 0.50495696, 0.5049766, 1.0, 1.0]), (Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing)), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 3752.933f0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)), Main.nusseltplot, 20, true, nothing, false, nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}()), finalize = IncompressibleNavierStokes.var"#295#296"()), avg = (initialize = IncompressibleNavierStokesMakieExt.var"#10#12"{@NamedTuple{grid::@NamedTuple{xlims::Tuple{Tuple{Float32, Float32}, Tuple{Float32, Float32}}, dimension::IncompressibleNavierStokes.Dimension{2}, N::Tuple{Int64, Int64}, Nu::Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}, Np::Tuple{Int64, Int64}, Iu::Tuple{CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}, Ip::CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, x::Tuple{Vector{Float32}, Vector{Float32}}, xu::Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, xp::Tuple{Vector{Float32}, Vector{Float32}}, Δ::Tuple{Vector{Float32}, Vector{Float32}}, Δu::Tuple{Vector{Float32}, Vector{Float32}}, A::Tuple{Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}}}, boundary_conditions::Tuple{Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}, Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}}, Re::Float32, bodyforce::Nothing, issteadybodyforce::Bool, closure_model::Nothing, backend::CPU, workgroupsize::Int64, temperature::@NamedTuple{α1::Float32, α2::Float32, α3::Float32, α4::Float32, γ::Float32, dodissipation::Bool, boundary_conditions::Tuple{Tuple{SymmetricBC, SymmetricBC}, Tuple{DirichletBC{Float32}, DirichletBC{Float32}}}, gdir::Int64}}, typeof(Main.averagetemp), Int64, Bool, Nothing, Bool, Nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}((grid = (xlims = ((0.0f0, 2.0f0), (0.0f0, 1.0f0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (202, 102), Nu = ((199, 100), (200, 99)), Np = (200, 100), Iu = (CartesianIndices((2:200, 2:101)), CartesianIndices((2:201, 2:100))), Ip = CartesianIndices((2:201, 2:101)), x = (Float32[0.0, 0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0]), xu = ((Float32[0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393, 0.043240547 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679, 0.047299176 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0])), xp = (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), Δ = (Float32[1.1920929f-7, 0.0044347644, 0.0045239925, 0.0046145916, 0.004706502, 0.004799843, 0.004894674, 0.004990816, 0.00508821, 0.005187154 … 0.005187154, 0.00508821, 0.004990816, 0.0048947334, 0.004799843, 0.0047063828, 0.0046145916, 0.0045239925, 0.004434824, 1.1920929f-7], Float32[1.1920929f-7, 0.0044793785, 0.004660547, 0.0048472583, 0.005039513, 0.0052372515, 0.0054405034, 0.005649179, 0.00586316, 0.006082386 … 0.006082356, 0.00586313, 0.005649209, 0.0054404736, 0.0052372813, 0.005039513, 0.004847288, 0.004660487, 0.0044794083, 1.1920929f-7]), Δu = (Float32[0.0022173822, 0.0044793785, 0.004569292, 0.004660547, 0.0047531724, 0.0048472583, 0.004942745, 0.005039513, 0.005137682, 0.0052372515 … 0.005137682, 0.0050395727, 0.004942775, 0.004847288, 0.004753113, 0.004660487, 0.004569292, 0.0044794083, 0.002217412, 1.1920929f-7], Float32[0.0022396892, 0.0045699626, 0.0047539026, 0.0049433857, 0.0051383823, 0.0053388774, 0.0055448413, 0.0057561696, 0.005972773, 0.0061945915 … 0.005972743, 0.0057561994, 0.0055448413, 0.0053389072, 0.005138397, 0.004943371, 0.0047538877, 0.0045699477, 0.0022397041, 1.1920929f-7]), A = (((Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), (Float32[1.0, 1.0, 0.5099108, 0.50981885, 0.5097228, 0.50962067, 0.50951755, 0.5094086, 0.50929356, 0.509176 … 0.49093604, 0.49081892, 0.4907065, 0.49058872, 0.4904881, 0.49038374, 0.49028164, 0.49017644, 0.49009407, 0.0], Float32[0.0, 0.49008918, 0.49018115, 0.49027717, 0.49037933, 0.49048245, 0.49059144, 0.49070647, 0.49082395, 0.49094325 … 0.5091811, 0.5092935, 0.5094113, 0.5095119, 0.50961626, 0.50971836, 0.50982356, 0.50990593, 1.0, 1.0])), ((Float32[1.0, 1.0, 0.50497997, 0.50495696, 0.50493026, 0.5049094, 0.5048909, 0.5048628, 0.50483155, 0.5048146 … 0.49520862, 0.4951738, 0.4951626, 0.49514025, 0.49510598, 0.4950843, 0.49507612, 0.49504304, 0.49502343, 0.0], Float32[0.0, 0.49502006, 0.49504304, 0.49506977, 0.4950906, 0.49510905, 0.4951372, 0.49516848, 0.4951854, 0.49521717 … 0.5048262, 0.5048374, 0.50485975, 0.504894, 0.5049157, 0.5049239, 0.50495696, 0.5049766, 1.0, 1.0]), (Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing)), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 3752.933f0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)), Main.averagetemp, 50, true, nothing, false, nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}()), finalize = IncompressibleNavierStokes.var"#295#296"()), log = (initialize = IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}(false, true, true, true, true, 1000), finalize = IncompressibleNavierStokes.var"#295#296"())) Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 20 )), Δt = T ( 1e-2 ), processors, ); [ Info: t = 10.0001 Δt = 0.01 umax = 0.24 itertime = 0.014 [ Info: t = 20.0004 Δt = 0.01 umax = 0.55 itertime = 0.012
Nusselt numbers
Average temperature
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard2D" ) backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() function nusseltplot (state; setup) state isa Observable || (state = Observable (state)) (; Δ, Δu) = setup . grid Δy1 = Δu[ 2 ][ 1 : 1 ] |> sum Δy2 = Δu[ 2 ][ end - 1 : end - 1 ] |> sum # Observe Nusselt numbers Nu1 = Observable (Point2f[]) Nu2 = Observable (Point2f[]) on (state) do (; temp, t) dTdy = @. (temp[:, 2 ] - temp[:, 1 ]) / Δy1 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu1[], Point2f (t, Nu)) dTdy = @. (temp[:, end - 1 ] - temp[:, end - 2 ]) / Δy2 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu2[], Point2f (t, Nu)) (Nu1, Nu2) .|> notify ## Update plot end # Plot Nu history fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Nusselt number" , xlabel = "t" , ylabel = "Nu" ) lines! (ax, Nu1; label = "Lower plate" ) lines! (ax, Nu2; label = "Upper plate" ) axislegend (ax) on (_ -> autolimits! (ax), Nu2) fig end function averagetemp (state; setup) state isa Observable || (state = Observable (state)) (; xp, Δ, Ip) = setup . grid ix = Ip . indices[ 1 ] Ty = lift (state) do (; temp) Ty = sum (temp[ix, :] .* Δ[ 1 ][ix]; dims = 1 ) ./ sum (Δ[ 1 ][ix]) Array (Ty)[:] end Ty0 = copy (Ty[]) yy = Array (xp[ 2 ]) fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Average temperature" , xlabel = "T" , ylabel = "y" ) lines! (ax, Ty0, yy; label = "t = 0" ) lines! (ax, Ty, yy; label = "t = t" ) axislegend (ax) on (_ -> autolimits! (ax), Ty) fig end T = Float32 temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 2 , nondim_type = 1 , ) n = 100 x = tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )) plotgrid (x ... ) setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, backend, ); ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (y) / 2 + max ( sinpi ( 20 * x) / 100 , 0 )); GLMakie . closeall () processors = (; rtp = realtimeplotter (; screen = GLMakie . Screen (), setup, fieldname = :temperature , colorrange = ( T ( 0 ), T ( 1 )), size = ( 600 , 350 ), colormap = :seaborn_icefire_gradient , nupdate = 20 , ), nusselt = realtimeplotter (; screen = GLMakie . Screen (), setup, plot = nusseltplot, nupdate = 20 , ), avg = realtimeplotter (; screen = GLMakie . Screen (), setup, plot = averagetemp, nupdate = 50 , ), log = timelogger (; nupdate = 1000 ), ) state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 20 )), Δt = T ( 1e-2 ), processors, ); outputs . nusselt outputs . avg This page was generated using Literate.jl .
`,46)]))}const c=i(e,[["render",E]]);export{C as __pageData,c as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighBenard2D.md.DlQrJFaM.lean.js b/previews/PR126/assets/examples_generated_RayleighBenard2D.md.DlQrJFaM.lean.js
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+++ b/previews/PR126/assets/examples_generated_RayleighBenard2D.md.DlQrJFaM.lean.js
@@ -0,0 +1,222 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const t="/IncompressibleNavierStokes.jl/previews/PR126/assets/cuibxhd.BxypEGK2.png",l="/IncompressibleNavierStokes.jl/previews/PR126/RayleighBenard2D.mp4",k="/IncompressibleNavierStokes.jl/previews/PR126/assets/xsmunma.C3CW_rGA.png",p="/IncompressibleNavierStokes.jl/previews/PR126/assets/jqxyinv.D8geYuhY.png",C=JSON.parse('{"title":"Rayleigh-Bénard convection (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard2D.md","filePath":"examples/generated/RayleighBenard2D.md","lastUpdated":null}'),e={name:"examples/generated/RayleighBenard2D.md"};function E(r,s,d,g,y,F){return h(),a("div",null,s[0]||(s[0]=[n(`Rayleigh-Bénard convection (2D) A hot and a cold plate generate a convection cell in a box.
julia using CairoMakie using IncompressibleNavierStokes Output directory for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard2D" ) "/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighBenard2D" Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Define observer function to track Nusselt numbers on top and bottom plates.
julia function nusseltplot (state; setup) state isa Observable || (state = Observable (state)) (; Δ, Δu) = setup . grid Δy1 = Δu[ 2 ][ 1 : 1 ] |> sum Δy2 = Δu[ 2 ][ end - 1 : end - 1 ] |> sum # Observe Nusselt numbers Nu1 = Observable (Point2f[]) Nu2 = Observable (Point2f[]) on (state) do (; temp, t) dTdy = @. (temp[:, 2 ] - temp[:, 1 ]) / Δy1 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu1[], Point2f (t, Nu)) dTdy = @. (temp[:, end - 1 ] - temp[:, end - 2 ]) / Δy2 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu2[], Point2f (t, Nu)) (Nu1, Nu2) .|> notify ## Update plot end # Plot Nu history fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Nusselt number" , xlabel = "t" , ylabel = "Nu" ) lines! (ax, Nu1; label = "Lower plate" ) lines! (ax, Nu2; label = "Upper plate" ) axislegend (ax) on (_ -> autolimits! (ax), Nu2) fig end nusseltplot (generic function with 1 method) Define observer function to track average temperature.
julia function averagetemp (state; setup) state isa Observable || (state = Observable (state)) (; xp, Δ, Ip) = setup . grid ix = Ip . indices[ 1 ] Ty = lift (state) do (; temp) Ty = sum (temp[ix, :] .* Δ[ 1 ][ix]; dims = 1 ) ./ sum (Δ[ 1 ][ix]) Array (Ty)[:] end Ty0 = copy (Ty[]) yy = Array (xp[ 2 ]) fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Average temperature" , xlabel = "T" , ylabel = "y" ) lines! (ax, Ty0, yy; label = "t = 0" ) lines! (ax, Ty, yy; label = "t = t" ) axislegend (ax) on (_ -> autolimits! (ax), Ty) fig end averagetemp (generic function with 1 method) Instabilities should depend on the floating point precision. Try both Float32 and Float64.
Temperature equation setup.
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 2 , nondim_type = 1 , ) (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2) Grid
julia n = 100 x = tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )) plotgrid (x ... )
Setup
julia setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, backend, ); Initial conditions
julia ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (y) / 2 + max ( sinpi ( 20 * x) / 100 , 0 )); Processors
julia processors = (; rtp = realtimeplotter (; setup, fieldname = :temperature , colorrange = ( T ( 0 ), T ( 1 )), size = ( 600 , 350 ), colormap = :seaborn_icefire_gradient , nupdate = 20 , ), nusselt = realtimeplotter (; setup, plot = nusseltplot, nupdate = 20 , ), avg = realtimeplotter (; setup, plot = averagetemp, nupdate = 50 , ), log = timelogger (; nupdate = 1000 ), ) (rtp = (initialize = IncompressibleNavierStokesMakieExt.var"#10#12"{@NamedTuple{grid::@NamedTuple{xlims::Tuple{Tuple{Float32, Float32}, Tuple{Float32, Float32}}, dimension::IncompressibleNavierStokes.Dimension{2}, N::Tuple{Int64, Int64}, Nu::Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}, Np::Tuple{Int64, Int64}, Iu::Tuple{CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}, Ip::CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, x::Tuple{Vector{Float32}, Vector{Float32}}, xu::Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, xp::Tuple{Vector{Float32}, Vector{Float32}}, Δ::Tuple{Vector{Float32}, Vector{Float32}}, Δu::Tuple{Vector{Float32}, Vector{Float32}}, A::Tuple{Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}}}, boundary_conditions::Tuple{Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}, Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}}, Re::Float32, bodyforce::Nothing, issteadybodyforce::Bool, closure_model::Nothing, backend::CPU, workgroupsize::Int64, temperature::@NamedTuple{α1::Float32, α2::Float32, α3::Float32, α4::Float32, γ::Float32, dodissipation::Bool, boundary_conditions::Tuple{Tuple{SymmetricBC, SymmetricBC}, Tuple{DirichletBC{Float32}, DirichletBC{Float32}}}, gdir::Int64}}, typeof(fieldplot), Int64, Bool, Nothing, Bool, Nothing, Base.Pairs{Symbol, Any, NTuple{4, Symbol}, @NamedTuple{fieldname::Symbol, colorrange::Tuple{Float32, Float32}, size::Tuple{Int64, Int64}, colormap::Symbol}}}((grid = (xlims = ((0.0f0, 2.0f0), (0.0f0, 1.0f0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (202, 102), Nu = ((199, 100), (200, 99)), Np = (200, 100), Iu = (CartesianIndices((2:200, 2:101)), CartesianIndices((2:201, 2:100))), Ip = CartesianIndices((2:201, 2:101)), x = (Float32[0.0, 0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0]), xu = ((Float32[0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393, 0.043240547 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679, 0.047299176 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0])), xp = (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), Δ = (Float32[1.1920929f-7, 0.0044347644, 0.0045239925, 0.0046145916, 0.004706502, 0.004799843, 0.004894674, 0.004990816, 0.00508821, 0.005187154 … 0.005187154, 0.00508821, 0.004990816, 0.0048947334, 0.004799843, 0.0047063828, 0.0046145916, 0.0045239925, 0.004434824, 1.1920929f-7], Float32[1.1920929f-7, 0.0044793785, 0.004660547, 0.0048472583, 0.005039513, 0.0052372515, 0.0054405034, 0.005649179, 0.00586316, 0.006082386 … 0.006082356, 0.00586313, 0.005649209, 0.0054404736, 0.0052372813, 0.005039513, 0.004847288, 0.004660487, 0.0044794083, 1.1920929f-7]), Δu = (Float32[0.0022173822, 0.0044793785, 0.004569292, 0.004660547, 0.0047531724, 0.0048472583, 0.004942745, 0.005039513, 0.005137682, 0.0052372515 … 0.005137682, 0.0050395727, 0.004942775, 0.004847288, 0.004753113, 0.004660487, 0.004569292, 0.0044794083, 0.002217412, 1.1920929f-7], Float32[0.0022396892, 0.0045699626, 0.0047539026, 0.0049433857, 0.0051383823, 0.0053388774, 0.0055448413, 0.0057561696, 0.005972773, 0.0061945915 … 0.005972743, 0.0057561994, 0.0055448413, 0.0053389072, 0.005138397, 0.004943371, 0.0047538877, 0.0045699477, 0.0022397041, 1.1920929f-7]), A = (((Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), (Float32[1.0, 1.0, 0.5099108, 0.50981885, 0.5097228, 0.50962067, 0.50951755, 0.5094086, 0.50929356, 0.509176 … 0.49093604, 0.49081892, 0.4907065, 0.49058872, 0.4904881, 0.49038374, 0.49028164, 0.49017644, 0.49009407, 0.0], Float32[0.0, 0.49008918, 0.49018115, 0.49027717, 0.49037933, 0.49048245, 0.49059144, 0.49070647, 0.49082395, 0.49094325 … 0.5091811, 0.5092935, 0.5094113, 0.5095119, 0.50961626, 0.50971836, 0.50982356, 0.50990593, 1.0, 1.0])), ((Float32[1.0, 1.0, 0.50497997, 0.50495696, 0.50493026, 0.5049094, 0.5048909, 0.5048628, 0.50483155, 0.5048146 … 0.49520862, 0.4951738, 0.4951626, 0.49514025, 0.49510598, 0.4950843, 0.49507612, 0.49504304, 0.49502343, 0.0], Float32[0.0, 0.49502006, 0.49504304, 0.49506977, 0.4950906, 0.49510905, 0.4951372, 0.49516848, 0.4951854, 0.49521717 … 0.5048262, 0.5048374, 0.50485975, 0.504894, 0.5049157, 0.5049239, 0.50495696, 0.5049766, 1.0, 1.0]), (Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing)), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 3752.933f0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)), IncompressibleNavierStokes.fieldplot, 20, true, nothing, false, nothing, Base.Pairs{Symbol, Any, NTuple{4, Symbol}, @NamedTuple{fieldname::Symbol, colorrange::Tuple{Float32, Float32}, size::Tuple{Int64, Int64}, colormap::Symbol}}(:fieldname => :temperature, :colorrange => (0.0f0, 1.0f0), :size => (600, 350), :colormap => :seaborn_icefire_gradient)), finalize = IncompressibleNavierStokes.var"#295#296"()), nusselt = (initialize = IncompressibleNavierStokesMakieExt.var"#10#12"{@NamedTuple{grid::@NamedTuple{xlims::Tuple{Tuple{Float32, Float32}, Tuple{Float32, Float32}}, dimension::IncompressibleNavierStokes.Dimension{2}, N::Tuple{Int64, Int64}, Nu::Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}, Np::Tuple{Int64, Int64}, Iu::Tuple{CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}, Ip::CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, x::Tuple{Vector{Float32}, Vector{Float32}}, xu::Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, xp::Tuple{Vector{Float32}, Vector{Float32}}, Δ::Tuple{Vector{Float32}, Vector{Float32}}, Δu::Tuple{Vector{Float32}, Vector{Float32}}, A::Tuple{Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}}}, boundary_conditions::Tuple{Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}, Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}}, Re::Float32, bodyforce::Nothing, issteadybodyforce::Bool, closure_model::Nothing, backend::CPU, workgroupsize::Int64, temperature::@NamedTuple{α1::Float32, α2::Float32, α3::Float32, α4::Float32, γ::Float32, dodissipation::Bool, boundary_conditions::Tuple{Tuple{SymmetricBC, SymmetricBC}, Tuple{DirichletBC{Float32}, DirichletBC{Float32}}}, gdir::Int64}}, typeof(Main.nusseltplot), Int64, Bool, Nothing, Bool, Nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}((grid = (xlims = ((0.0f0, 2.0f0), (0.0f0, 1.0f0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (202, 102), Nu = ((199, 100), (200, 99)), Np = (200, 100), Iu = (CartesianIndices((2:200, 2:101)), CartesianIndices((2:201, 2:100))), Ip = CartesianIndices((2:201, 2:101)), x = (Float32[0.0, 0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0]), xu = ((Float32[0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393, 0.043240547 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679, 0.047299176 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0])), xp = (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), Δ = (Float32[1.1920929f-7, 0.0044347644, 0.0045239925, 0.0046145916, 0.004706502, 0.004799843, 0.004894674, 0.004990816, 0.00508821, 0.005187154 … 0.005187154, 0.00508821, 0.004990816, 0.0048947334, 0.004799843, 0.0047063828, 0.0046145916, 0.0045239925, 0.004434824, 1.1920929f-7], Float32[1.1920929f-7, 0.0044793785, 0.004660547, 0.0048472583, 0.005039513, 0.0052372515, 0.0054405034, 0.005649179, 0.00586316, 0.006082386 … 0.006082356, 0.00586313, 0.005649209, 0.0054404736, 0.0052372813, 0.005039513, 0.004847288, 0.004660487, 0.0044794083, 1.1920929f-7]), Δu = (Float32[0.0022173822, 0.0044793785, 0.004569292, 0.004660547, 0.0047531724, 0.0048472583, 0.004942745, 0.005039513, 0.005137682, 0.0052372515 … 0.005137682, 0.0050395727, 0.004942775, 0.004847288, 0.004753113, 0.004660487, 0.004569292, 0.0044794083, 0.002217412, 1.1920929f-7], Float32[0.0022396892, 0.0045699626, 0.0047539026, 0.0049433857, 0.0051383823, 0.0053388774, 0.0055448413, 0.0057561696, 0.005972773, 0.0061945915 … 0.005972743, 0.0057561994, 0.0055448413, 0.0053389072, 0.005138397, 0.004943371, 0.0047538877, 0.0045699477, 0.0022397041, 1.1920929f-7]), A = (((Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), (Float32[1.0, 1.0, 0.5099108, 0.50981885, 0.5097228, 0.50962067, 0.50951755, 0.5094086, 0.50929356, 0.509176 … 0.49093604, 0.49081892, 0.4907065, 0.49058872, 0.4904881, 0.49038374, 0.49028164, 0.49017644, 0.49009407, 0.0], Float32[0.0, 0.49008918, 0.49018115, 0.49027717, 0.49037933, 0.49048245, 0.49059144, 0.49070647, 0.49082395, 0.49094325 … 0.5091811, 0.5092935, 0.5094113, 0.5095119, 0.50961626, 0.50971836, 0.50982356, 0.50990593, 1.0, 1.0])), ((Float32[1.0, 1.0, 0.50497997, 0.50495696, 0.50493026, 0.5049094, 0.5048909, 0.5048628, 0.50483155, 0.5048146 … 0.49520862, 0.4951738, 0.4951626, 0.49514025, 0.49510598, 0.4950843, 0.49507612, 0.49504304, 0.49502343, 0.0], Float32[0.0, 0.49502006, 0.49504304, 0.49506977, 0.4950906, 0.49510905, 0.4951372, 0.49516848, 0.4951854, 0.49521717 … 0.5048262, 0.5048374, 0.50485975, 0.504894, 0.5049157, 0.5049239, 0.50495696, 0.5049766, 1.0, 1.0]), (Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing)), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 3752.933f0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)), Main.nusseltplot, 20, true, nothing, false, nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}()), finalize = IncompressibleNavierStokes.var"#295#296"()), avg = (initialize = IncompressibleNavierStokesMakieExt.var"#10#12"{@NamedTuple{grid::@NamedTuple{xlims::Tuple{Tuple{Float32, Float32}, Tuple{Float32, Float32}}, dimension::IncompressibleNavierStokes.Dimension{2}, N::Tuple{Int64, Int64}, Nu::Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}, Np::Tuple{Int64, Int64}, Iu::Tuple{CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}, Ip::CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}, x::Tuple{Vector{Float32}, Vector{Float32}}, xu::Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, xp::Tuple{Vector{Float32}, Vector{Float32}}, Δ::Tuple{Vector{Float32}, Vector{Float32}}, Δu::Tuple{Vector{Float32}, Vector{Float32}}, A::Tuple{Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}, Tuple{Tuple{Vector{Float32}, Vector{Float32}}, Tuple{Vector{Float32}, Vector{Float32}}}}}, boundary_conditions::Tuple{Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}, Tuple{DirichletBC{Nothing}, DirichletBC{Nothing}}}, Re::Float32, bodyforce::Nothing, issteadybodyforce::Bool, closure_model::Nothing, backend::CPU, workgroupsize::Int64, temperature::@NamedTuple{α1::Float32, α2::Float32, α3::Float32, α4::Float32, γ::Float32, dodissipation::Bool, boundary_conditions::Tuple{Tuple{SymmetricBC, SymmetricBC}, Tuple{DirichletBC{Float32}, DirichletBC{Float32}}}, gdir::Int64}}, typeof(Main.averagetemp), Int64, Bool, Nothing, Bool, Nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}((grid = (xlims = ((0.0f0, 2.0f0), (0.0f0, 1.0f0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (202, 102), Nu = ((199, 100), (200, 99)), Np = (200, 100), Iu = (CartesianIndices((2:200, 2:101)), CartesianIndices((2:201, 2:100))), Ip = CartesianIndices((2:201, 2:101)), x = (Float32[0.0, 0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0]), xu = ((Float32[0.0, 0.0044347644, 0.008958757, 0.0135733485, 0.01827985, 0.023079693, 0.027974367, 0.032965183, 0.038053393, 0.043240547 … 1.9619466, 1.9670348, 1.9720256, 1.9769204, 1.9817202, 1.9864266, 1.9910412, 1.9955652, 2.0, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0044793785, 0.009139925, 0.013987184, 0.019026697, 0.024263948, 0.029704452, 0.03535363, 0.04121679, 0.047299176 … 0.9587832, 0.96464634, 0.97029555, 0.975736, 0.9809733, 0.9860128, 0.9908601, 0.9955206, 1.0, 1.0])), xp = (Float32[0.0, 0.0022173822, 0.0066967607, 0.011266053, 0.0159266, 0.020679772, 0.02552703, 0.030469775, 0.03550929, 0.04064697 … 1.959353, 1.9644907, 1.9695302, 1.974473, 1.9793203, 1.9840734, 1.9887339, 1.9933032, 1.9977826, 2.0], Float32[0.0, 0.0022396892, 0.006809652, 0.011563554, 0.01650694, 0.021645322, 0.0269842, 0.03252904, 0.03828521, 0.044257984 … 0.955742, 0.96171474, 0.96747094, 0.9730158, 0.9783547, 0.9834931, 0.98843646, 0.99319035, 0.9977603, 1.0]), Δ = (Float32[1.1920929f-7, 0.0044347644, 0.0045239925, 0.0046145916, 0.004706502, 0.004799843, 0.004894674, 0.004990816, 0.00508821, 0.005187154 … 0.005187154, 0.00508821, 0.004990816, 0.0048947334, 0.004799843, 0.0047063828, 0.0046145916, 0.0045239925, 0.004434824, 1.1920929f-7], Float32[1.1920929f-7, 0.0044793785, 0.004660547, 0.0048472583, 0.005039513, 0.0052372515, 0.0054405034, 0.005649179, 0.00586316, 0.006082386 … 0.006082356, 0.00586313, 0.005649209, 0.0054404736, 0.0052372813, 0.005039513, 0.004847288, 0.004660487, 0.0044794083, 1.1920929f-7]), Δu = (Float32[0.0022173822, 0.0044793785, 0.004569292, 0.004660547, 0.0047531724, 0.0048472583, 0.004942745, 0.005039513, 0.005137682, 0.0052372515 … 0.005137682, 0.0050395727, 0.004942775, 0.004847288, 0.004753113, 0.004660487, 0.004569292, 0.0044794083, 0.002217412, 1.1920929f-7], Float32[0.0022396892, 0.0045699626, 0.0047539026, 0.0049433857, 0.0051383823, 0.0053388774, 0.0055448413, 0.0057561696, 0.005972773, 0.0061945915 … 0.005972743, 0.0057561994, 0.0055448413, 0.0053389072, 0.005138397, 0.004943371, 0.0047538877, 0.0045699477, 0.0022397041, 1.1920929f-7]), A = (((Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), (Float32[1.0, 1.0, 0.5099108, 0.50981885, 0.5097228, 0.50962067, 0.50951755, 0.5094086, 0.50929356, 0.509176 … 0.49093604, 0.49081892, 0.4907065, 0.49058872, 0.4904881, 0.49038374, 0.49028164, 0.49017644, 0.49009407, 0.0], Float32[0.0, 0.49008918, 0.49018115, 0.49027717, 0.49037933, 0.49048245, 0.49059144, 0.49070647, 0.49082395, 0.49094325 … 0.5091811, 0.5092935, 0.5094113, 0.5095119, 0.50961626, 0.50971836, 0.50982356, 0.50990593, 1.0, 1.0])), ((Float32[1.0, 1.0, 0.50497997, 0.50495696, 0.50493026, 0.5049094, 0.5048909, 0.5048628, 0.50483155, 0.5048146 … 0.49520862, 0.4951738, 0.4951626, 0.49514025, 0.49510598, 0.4950843, 0.49507612, 0.49504304, 0.49502343, 0.0], Float32[0.0, 0.49502006, 0.49504304, 0.49506977, 0.4950906, 0.49510905, 0.4951372, 0.49516848, 0.4951854, 0.49521717 … 0.5048262, 0.5048374, 0.50485975, 0.504894, 0.5049157, 0.5049239, 0.50495696, 0.5049766, 1.0, 1.0]), (Float32[1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], Float32[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing)), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 3752.933f0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)), Main.averagetemp, 50, true, nothing, false, nothing, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}()), finalize = IncompressibleNavierStokes.var"#295#296"()), log = (initialize = IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}(false, true, true, true, true, 1000), finalize = IncompressibleNavierStokes.var"#295#296"())) Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 20 )), Δt = T ( 1e-2 ), processors, ); [ Info: t = 10.0001 Δt = 0.01 umax = 0.24 itertime = 0.014 [ Info: t = 20.0004 Δt = 0.01 umax = 0.55 itertime = 0.012
Nusselt numbers
Average temperature
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard2D" ) backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() function nusseltplot (state; setup) state isa Observable || (state = Observable (state)) (; Δ, Δu) = setup . grid Δy1 = Δu[ 2 ][ 1 : 1 ] |> sum Δy2 = Δu[ 2 ][ end - 1 : end - 1 ] |> sum # Observe Nusselt numbers Nu1 = Observable (Point2f[]) Nu2 = Observable (Point2f[]) on (state) do (; temp, t) dTdy = @. (temp[:, 2 ] - temp[:, 1 ]) / Δy1 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu1[], Point2f (t, Nu)) dTdy = @. (temp[:, end - 1 ] - temp[:, end - 2 ]) / Δy2 Nu = sum (( .- dTdy .* Δ[ 1 ])[ 2 : end - 1 ]) push! (Nu2[], Point2f (t, Nu)) (Nu1, Nu2) .|> notify ## Update plot end # Plot Nu history fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Nusselt number" , xlabel = "t" , ylabel = "Nu" ) lines! (ax, Nu1; label = "Lower plate" ) lines! (ax, Nu2; label = "Upper plate" ) axislegend (ax) on (_ -> autolimits! (ax), Nu2) fig end function averagetemp (state; setup) state isa Observable || (state = Observable (state)) (; xp, Δ, Ip) = setup . grid ix = Ip . indices[ 1 ] Ty = lift (state) do (; temp) Ty = sum (temp[ix, :] .* Δ[ 1 ][ix]; dims = 1 ) ./ sum (Δ[ 1 ][ix]) Array (Ty)[:] end Ty0 = copy (Ty[]) yy = Array (xp[ 2 ]) fig = Figure () ax = Axis (fig[ 1 , 1 ]; title = "Average temperature" , xlabel = "T" , ylabel = "y" ) lines! (ax, Ty0, yy; label = "t = 0" ) lines! (ax, Ty, yy; label = "t = t" ) axislegend (ax) on (_ -> autolimits! (ax), Ty) fig end T = Float32 temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 2 , nondim_type = 1 , ) n = 100 x = tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )) plotgrid (x ... ) setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, backend, ); ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (y) / 2 + max ( sinpi ( 20 * x) / 100 , 0 )); GLMakie . closeall () processors = (; rtp = realtimeplotter (; screen = GLMakie . Screen (), setup, fieldname = :temperature , colorrange = ( T ( 0 ), T ( 1 )), size = ( 600 , 350 ), colormap = :seaborn_icefire_gradient , nupdate = 20 , ), nusselt = realtimeplotter (; screen = GLMakie . Screen (), setup, plot = nusseltplot, nupdate = 20 , ), avg = realtimeplotter (; screen = GLMakie . Screen (), setup, plot = averagetemp, nupdate = 50 , ), log = timelogger (; nupdate = 1000 ), ) state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 20 )), Δt = T ( 1e-2 ), processors, ); outputs . nusselt outputs . avg This page was generated using Literate.jl .
`,46)]))}const c=i(e,[["render",E]]);export{C as __pageData,c as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighBenard3D.md.BFDE04bU.js b/previews/PR126/assets/examples_generated_RayleighBenard3D.md.BFDE04bU.js
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+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Rayleigh-Bénard convection (3D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard3D.md","filePath":"examples/generated/RayleighBenard3D.md","lastUpdated":null}'),k={name:"examples/generated/RayleighBenard3D.md"};function l(p,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Rayleigh-Bénard convection (3D) A hot and a cold plate generate a convection cell in a box.
julia using CairoMakie using IncompressibleNavierStokes Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Precision
Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard3D" ) Temperature equation
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 3 , nondim_type = 1 , ) Setup
julia n = 60 x = ( LinRange ( T ( 0 ), T ( π ), 2 n), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), ) setup = Setup (; x, boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); plotgrid (x[ 1 ], x[ 2 ]) plotgrid (x[ 2 ], x[ 3 ]) This will factorize the Laplace matrix
julia @time psolver = psolver_direct (setup) Initial conditions
julia ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> one (x) / 2 + sin ( 20 * x) * sinpi ( 20 * y) / 100 ); nothing #hide Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), method = RKMethods . RK33C2 (; T), # Δt = T(1e-2), psolver, processors = (; # anim = animator(; # path = "$outdir/RB3D.mp4", rtp = realtimeplotter (; setup, nupdate = 1 , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), fieldname = :eig2field , # levels = LinRange{T}(0.01, 0.99, 10), # fieldname = :temperature, ), # vtk = vtk_writer(; # setup, # dir = joinpath(outdir, "RB3D_$n"), # nupdate = 20, # ## fieldnames = (:velocity, :temperature, :eig2field) # fieldnames = (:temperature,), # ), log = timelogger (; nupdate = 100 ), ), ); field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) fieldplot ( # (; u = ustart, temp = tempstart, t = T(0)); state; setup, levels = LinRange{T} ( 0.5 , 2 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() T = Float32 outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard3D" ) temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 3 , nondim_type = 1 , ) n = 60 x = ( LinRange ( T ( 0 ), T ( π ), 2 n), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), ) setup = Setup (; x, boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); plotgrid (x[ 1 ], x[ 2 ]) plotgrid (x[ 2 ], x[ 3 ]) @time psolver = psolver_direct (setup) ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> one (x) / 2 + sin ( 20 * x) * sinpi ( 20 * y) / 100 ); state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), method = RKMethods . RK33C2 (; T), # Δt = T(1e-2), psolver, processors = (; # anim = animator(; # path = "$outdir/RB3D.mp4", rtp = realtimeplotter (; setup, nupdate = 1 , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), fieldname = :eig2field , # levels = LinRange{T}(0.01, 0.99, 10), # fieldname = :temperature, ), # vtk = vtk_writer(; # setup, # dir = joinpath(outdir, "RB3D_$n"), # nupdate = 20, # ## fieldnames = (:velocity, :temperature, :eig2field) # fieldnames = (:temperature,), # ), log = timelogger (; nupdate = 100 ), ), ); field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) fieldplot ( # (; u = ustart, temp = tempstart, t = T(0)); state; setup, levels = LinRange{T} ( 0.5 , 2 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , ) This page was generated using Literate.jl .
`,25)]))}const y=i(k,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighBenard3D.md.BFDE04bU.lean.js b/previews/PR126/assets/examples_generated_RayleighBenard3D.md.BFDE04bU.lean.js
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+++ b/previews/PR126/assets/examples_generated_RayleighBenard3D.md.BFDE04bU.lean.js
@@ -0,0 +1,174 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Rayleigh-Bénard convection (3D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard3D.md","filePath":"examples/generated/RayleighBenard3D.md","lastUpdated":null}'),k={name:"examples/generated/RayleighBenard3D.md"};function l(p,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Rayleigh-Bénard convection (3D) A hot and a cold plate generate a convection cell in a box.
julia using CairoMakie using IncompressibleNavierStokes Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Precision
Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard3D" ) Temperature equation
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 3 , nondim_type = 1 , ) Setup
julia n = 60 x = ( LinRange ( T ( 0 ), T ( π ), 2 n), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), ) setup = Setup (; x, boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); plotgrid (x[ 1 ], x[ 2 ]) plotgrid (x[ 2 ], x[ 3 ]) This will factorize the Laplace matrix
julia @time psolver = psolver_direct (setup) Initial conditions
julia ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> one (x) / 2 + sin ( 20 * x) * sinpi ( 20 * y) / 100 ); nothing #hide Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), method = RKMethods . RK33C2 (; T), # Δt = T(1e-2), psolver, processors = (; # anim = animator(; # path = "$outdir/RB3D.mp4", rtp = realtimeplotter (; setup, nupdate = 1 , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), fieldname = :eig2field , # levels = LinRange{T}(0.01, 0.99, 10), # fieldname = :temperature, ), # vtk = vtk_writer(; # setup, # dir = joinpath(outdir, "RB3D_$n"), # nupdate = 20, # ## fieldnames = (:velocity, :temperature, :eig2field) # fieldnames = (:temperature,), # ), log = timelogger (; nupdate = 100 ), ), ); field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) fieldplot ( # (; u = ustart, temp = tempstart, t = T(0)); state; setup, levels = LinRange{T} ( 0.5 , 2 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() T = Float32 outdir = joinpath ( @__DIR__ , "output" , "RayleighBenard3D" ) temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e7 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( SymmetricBC (), SymmetricBC ()), ( DirichletBC ( T ( 1 )), DirichletBC ( T ( 0 ))), ), gdir = 3 , nondim_type = 1 , ) n = 60 x = ( LinRange ( T ( 0 ), T ( π ), 2 n), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.2 )), ) setup = Setup (; x, boundary_conditions = ( ( PeriodicBC (), PeriodicBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); plotgrid (x[ 1 ], x[ 2 ]) plotgrid (x[ 2 ], x[ 3 ]) @time psolver = psolver_direct (setup) ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> one (x) / 2 + sin ( 20 * x) * sinpi ( 20 * y) / 100 ); state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), method = RKMethods . RK33C2 (; T), # Δt = T(1e-2), psolver, processors = (; # anim = animator(; # path = "$outdir/RB3D.mp4", rtp = realtimeplotter (; setup, nupdate = 1 , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), fieldname = :eig2field , # levels = LinRange{T}(0.01, 0.99, 10), # fieldname = :temperature, ), # vtk = vtk_writer(; # setup, # dir = joinpath(outdir, "RB3D_$n"), # nupdate = 20, # ## fieldnames = (:velocity, :temperature, :eig2field) # fieldnames = (:temperature,), # ), log = timelogger (; nupdate = 100 ), ), ); field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) fieldplot ( # (; u = ustart, temp = tempstart, t = T(0)); state; setup, levels = LinRange{T} ( 0.5 , 2 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , ) This page was generated using Literate.jl .
`,25)]))}const y=i(k,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighTaylor2D.md.BS44XuaT.js b/previews/PR126/assets/examples_generated_RayleighTaylor2D.md.BS44XuaT.js
new file mode 100644
index 00000000..01eedddf
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_RayleighTaylor2D.md.BS44XuaT.js
@@ -0,0 +1,96 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const k="/IncompressibleNavierStokes.jl/previews/PR126/assets/kktjebp.C2NI-x6W.png",t="/IncompressibleNavierStokes.jl/previews/PR126/RayleighTaylor2D.mp4",F=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor2D.md","filePath":"examples/generated/RayleighTaylor2D.md","lastUpdated":null}'),p={name:"examples/generated/RayleighTaylor2D.md"};function l(e,s,E,r,d,g){return h(),a("div",null,s[0]||(s[0]=[n(`Rayleigh-Taylor instability in 2D Two fluids with different temperatures start mixing.
julia using CairoMakie using IncompressibleNavierStokes Output directory for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor2D" ) "/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighTaylor2D" Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Precision
Grid
julia n = 50 x = tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.5 )), tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.5 )) plotgrid (x ... ; figure = (; size = ( 300 , 600 )))
Setup
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = (( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ())), gdir = 2 , nondim_type = 1 , ) setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, ); Initial conditions
julia ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (x) * ( 1 + sinpi (x) / 50 > y)); Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), Δt = T ( 5e-3 ), processors = (; rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :temperature , size = ( 400 , 600 ), ), log = timelogger (; nupdate = 200 ), ), ); [ Info: t = 1 Δt = 0.005 umax = 0.1 itertime = 0.011 [ Info: t = 2 Δt = 0.005 umax = 0.32 itertime = 0.0032 [ Info: t = 3 Δt = 0.005 umax = 0.78 itertime = 0.0032 [ Info: t = 4 Δt = 0.005 umax = 0.75 itertime = 0.0033 [ Info: t = 5 Δt = 0.005 umax = 0.78 itertime = 0.0032 [ Info: t = 6 Δt = 0.005 umax = 0.84 itertime = 0.0032 [ Info: t = 7 Δt = 0.005 umax = 0.84 itertime = 0.0032 [ Info: t = 8 Δt = 0.005 umax = 0.89 itertime = 0.0032 [ Info: t = 9 Δt = 0.005 umax = 1.2 itertime = 0.0033 [ Info: t = 10 Δt = 0.005 umax = 0.88 itertime = 0.0033
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor2D" ) backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() T = Float64 n = 50 x = tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.5 )), tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.5 )) plotgrid (x ... ; figure = (; size = ( 300 , 600 ))) temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = (( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ())), gdir = 2 , nondim_type = 1 , ) setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, ); ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (x) * ( 1 + sinpi (x) / 50 > y)); state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), Δt = T ( 5e-3 ), processors = (; rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :temperature , size = ( 400 , 600 ), ), log = timelogger (; nupdate = 200 ), ), ); This page was generated using Literate.jl .
`,28)]))}const C=i(p,[["render",l]]);export{F as __pageData,C as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighTaylor2D.md.BS44XuaT.lean.js b/previews/PR126/assets/examples_generated_RayleighTaylor2D.md.BS44XuaT.lean.js
new file mode 100644
index 00000000..01eedddf
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_RayleighTaylor2D.md.BS44XuaT.lean.js
@@ -0,0 +1,96 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const k="/IncompressibleNavierStokes.jl/previews/PR126/assets/kktjebp.C2NI-x6W.png",t="/IncompressibleNavierStokes.jl/previews/PR126/RayleighTaylor2D.mp4",F=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor2D.md","filePath":"examples/generated/RayleighTaylor2D.md","lastUpdated":null}'),p={name:"examples/generated/RayleighTaylor2D.md"};function l(e,s,E,r,d,g){return h(),a("div",null,s[0]||(s[0]=[n(`Rayleigh-Taylor instability in 2D Two fluids with different temperatures start mixing.
julia using CairoMakie using IncompressibleNavierStokes Output directory for saving results
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor2D" ) "/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighTaylor2D" Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Precision
Grid
julia n = 50 x = tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.5 )), tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.5 )) plotgrid (x ... ; figure = (; size = ( 300 , 600 )))
Setup
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = (( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ())), gdir = 2 , nondim_type = 1 , ) setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, ); Initial conditions
julia ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (x) * ( 1 + sinpi (x) / 50 > y)); Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), Δt = T ( 5e-3 ), processors = (; rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :temperature , size = ( 400 , 600 ), ), log = timelogger (; nupdate = 200 ), ), ); [ Info: t = 1 Δt = 0.005 umax = 0.1 itertime = 0.011 [ Info: t = 2 Δt = 0.005 umax = 0.32 itertime = 0.0032 [ Info: t = 3 Δt = 0.005 umax = 0.78 itertime = 0.0032 [ Info: t = 4 Δt = 0.005 umax = 0.75 itertime = 0.0033 [ Info: t = 5 Δt = 0.005 umax = 0.78 itertime = 0.0032 [ Info: t = 6 Δt = 0.005 umax = 0.84 itertime = 0.0032 [ Info: t = 7 Δt = 0.005 umax = 0.84 itertime = 0.0032 [ Info: t = 8 Δt = 0.005 umax = 0.89 itertime = 0.0032 [ Info: t = 9 Δt = 0.005 umax = 1.2 itertime = 0.0033 [ Info: t = 10 Δt = 0.005 umax = 0.88 itertime = 0.0033
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor2D" ) backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() T = Float64 n = 50 x = tanh_grid ( T ( 0 ), T ( 1 ), n, T ( 1.5 )), tanh_grid ( T ( 0 ), T ( 2 ), 2 n, T ( 1.5 )) plotgrid (x ... ; figure = (; size = ( 300 , 600 ))) temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = (( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ())), gdir = 2 , nondim_type = 1 , ) setup = Setup (; x, boundary_conditions = (( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ())), Re = 1 / temperature . α1, temperature, ); ustart = velocityfield (setup, (dim, x, y) -> zero (x)); tempstart = temperaturefield (setup, (x, y) -> one (x) * ( 1 + sinpi (x) / 50 > y)); state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 10 )), Δt = T ( 5e-3 ), processors = (; rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :temperature , size = ( 400 , 600 ), ), log = timelogger (; nupdate = 200 ), ), ); This page was generated using Literate.jl .
`,28)]))}const C=i(p,[["render",l]]);export{F as __pageData,C as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighTaylor3D.md.DNAhoUfj.js b/previews/PR126/assets/examples_generated_RayleighTaylor3D.md.DNAhoUfj.js
new file mode 100644
index 00000000..ce56f9c6
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_RayleighTaylor3D.md.DNAhoUfj.js
@@ -0,0 +1,161 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor3D.md","filePath":"examples/generated/RayleighTaylor3D.md","lastUpdated":null}'),k={name:"examples/generated/RayleighTaylor3D.md"};function l(p,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Rayleigh-Taylor instability in 2D Two fluids with different temperatures start mixing.
julia using CairoMakie using IncompressibleNavierStokes Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Precision
Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor3D" ) Temperature equation
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ), gdir = 3 , nondim_type = 1 , ) Setup
julia n = 80 x = LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 2 ), 2 n + 1 ) setup = Setup (; x, boundary_conditions = ( ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); nothing #hide This will factorize the Laplace matrix
julia @time psolver = psolver_direct (setup) Initial conditions
julia ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> ( 1 + sinpi (x / 20 ) * sinpi (y) > z)); fieldplot ( (; u = ustart, temp = tempstart, t = T ( 0 )); # state; setup, levels = LinRange{T} ( 0.8 , 1 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , size = ( 400 , 600 ), ) Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 40 )), Δt = T ( 1e-2 ), psolver, processors = (; # anim = animator(; # path = "$outdir/RT3D.mp4", rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :eig2field , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), # fieldname = :temperature, # levels = LinRange{T}(0, 1, 10), size = ( 400 , 600 ), ), # vtk = vtk_writer(; # setup, # nupdate = 10, # dir = outdir, # fieldnames = (:velocity, :pressure, :temperature), # psolver, # ), log = timelogger (; nupdate = 400 ), ), ); nothing #hide Check distribution of vortex structures for choosing plot levels
julia field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) Plot temperature field
julia fieldplot (state; setup, fieldname = :temperature ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() T = Float64 outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor3D" ) temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ), gdir = 3 , nondim_type = 1 , ) n = 80 x = LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 2 ), 2 n + 1 ) setup = Setup (; x, boundary_conditions = ( ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); @time psolver = psolver_direct (setup) ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> ( 1 + sinpi (x / 20 ) * sinpi (y) > z)); fieldplot ( (; u = ustart, temp = tempstart, t = T ( 0 )); # state; setup, levels = LinRange{T} ( 0.8 , 1 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , size = ( 400 , 600 ), ) state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 40 )), Δt = T ( 1e-2 ), psolver, processors = (; # anim = animator(; # path = "$outdir/RT3D.mp4", rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :eig2field , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), # fieldname = :temperature, # levels = LinRange{T}(0, 1, 10), size = ( 400 , 600 ), ), # vtk = vtk_writer(; # setup, # nupdate = 10, # dir = outdir, # fieldnames = (:velocity, :pressure, :temperature), # psolver, # ), log = timelogger (; nupdate = 400 ), ), ); field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) fieldplot (state; setup, fieldname = :temperature ) This page was generated using Literate.jl .
`,29)]))}const y=i(k,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_RayleighTaylor3D.md.DNAhoUfj.lean.js b/previews/PR126/assets/examples_generated_RayleighTaylor3D.md.DNAhoUfj.lean.js
new file mode 100644
index 00000000..ce56f9c6
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_RayleighTaylor3D.md.DNAhoUfj.lean.js
@@ -0,0 +1,161 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor3D.md","filePath":"examples/generated/RayleighTaylor3D.md","lastUpdated":null}'),k={name:"examples/generated/RayleighTaylor3D.md"};function l(p,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Rayleigh-Taylor instability in 2D Two fluids with different temperatures start mixing.
julia using CairoMakie using IncompressibleNavierStokes Hardware
julia backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() Precision
Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor3D" ) Temperature equation
julia temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ), gdir = 3 , nondim_type = 1 , ) Setup
julia n = 80 x = LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 2 ), 2 n + 1 ) setup = Setup (; x, boundary_conditions = ( ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); nothing #hide This will factorize the Laplace matrix
julia @time psolver = psolver_direct (setup) Initial conditions
julia ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> ( 1 + sinpi (x / 20 ) * sinpi (y) > z)); fieldplot ( (; u = ustart, temp = tempstart, t = T ( 0 )); # state; setup, levels = LinRange{T} ( 0.8 , 1 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , size = ( 400 , 600 ), ) Solve equation
julia state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 40 )), Δt = T ( 1e-2 ), psolver, processors = (; # anim = animator(; # path = "$outdir/RT3D.mp4", rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :eig2field , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), # fieldname = :temperature, # levels = LinRange{T}(0, 1, 10), size = ( 400 , 600 ), ), # vtk = vtk_writer(; # setup, # nupdate = 10, # dir = outdir, # fieldnames = (:velocity, :pressure, :temperature), # psolver, # ), log = timelogger (; nupdate = 400 ), ), ); nothing #hide Check distribution of vortex structures for choosing plot levels
julia field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) Plot temperature field
julia fieldplot (state; setup, fieldname = :temperature ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes backend = CPU () # using CUDA, CUDSS # backend = CUDABackend() T = Float64 outdir = joinpath ( @__DIR__ , "output" , "RayleighTaylor3D" ) temperature = temperature_equation (; Pr = T ( 0.71 ), Ra = T ( 1e6 ), Ge = T ( 1.0 ), dodissipation = true , boundary_conditions = ( ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ( SymmetricBC (), SymmetricBC ()), ), gdir = 3 , nondim_type = 1 , ) n = 80 x = LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 1 ), n + 1 ), LinRange ( T ( 0 ), T ( 2 ), 2 n + 1 ) setup = Setup (; x, boundary_conditions = ( ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ( DirichletBC (), DirichletBC ()), ), Re = 1 / temperature . α1, temperature, backend, ); @time psolver = psolver_direct (setup) ustart = velocityfield (setup, (dim, x, y, z) -> zero (x); psolver); tempstart = temperaturefield (setup, (x, y, z) -> ( 1 + sinpi (x / 20 ) * sinpi (y) > z)); fieldplot ( (; u = ustart, temp = tempstart, t = T ( 0 )); # state; setup, levels = LinRange{T} ( 0.8 , 1 , 5 ), # levels = LinRange(-T(5), T(1), 10), # fieldname = :eig2field, fieldname = :temperature , size = ( 400 , 600 ), ) state, outputs = solve_unsteady (; setup, ustart, tempstart, tlims = ( T ( 0 ), T ( 40 )), Δt = T ( 1e-2 ), psolver, processors = (; # anim = animator(; # path = "$outdir/RT3D.mp4", rtp = realtimeplotter (; setup, nupdate = 20 , fieldname = :eig2field , levels = LinRange ( - T ( 5 ), T ( 1 ), 10 ), # fieldname = :temperature, # levels = LinRange{T}(0, 1, 10), size = ( 400 , 600 ), ), # vtk = vtk_writer(; # setup, # nupdate = 10, # dir = outdir, # fieldnames = (:velocity, :pressure, :temperature), # psolver, # ), log = timelogger (; nupdate = 400 ), ), ); field = IncompressibleNavierStokes . eig2field (state . u, setup)[setup . grid . Ip] hist ( vec ( Array ( log .( max .( eps (T), .- field))))) fieldplot (state; setup, fieldname = :temperature ) This page was generated using Literate.jl .
`,29)]))}const y=i(k,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_ShearLayer2D.md.DGqD8BRv.js b/previews/PR126/assets/examples_generated_ShearLayer2D.md.DGqD8BRv.js
new file mode 100644
index 00000000..5007ce07
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_ShearLayer2D.md.DGqD8BRv.js
@@ -0,0 +1,84 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Shear layer - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/ShearLayer2D.md","filePath":"examples/generated/ShearLayer2D.md","lastUpdated":null}'),t={name:"examples/generated/ShearLayer2D.md"};function p(l,s,k,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Shear layer - 2D Shear layer example.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "ShearLayer2D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 2D grid is a Cartesian product of two vectors
julia n = 128 lims = T ( 0 ), T ( 2π ) x = LinRange (lims ... , n + 1 ), LinRange (lims ... , n + 1 ) plotgrid (x ... ) Build setup and assemble operators
julia setup = Setup (; x, Re, backend); nothing #hide Initial conditions: We add 1 to u in order to make global momentum conservation less trivial
julia d = T ( π / 15 ) e = T ( 0.05 ) U1 (y) = y ≤ π ? tanh ((y - T ( π / 2 )) / d) : tanh (( T ( 3π / 2 ) - y) / d) # U1(y) = T(1) + (y ≤ π ? tanh((y - T(π / 2)) / d) : tanh((T(3π / 2) - y) / d)) ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U1 (y) : e * sin (x)); nothing #hide Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 8 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot velocity
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "ShearLayer2D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 2000 ) n = 128 lims = T ( 0 ), T ( 2π ) x = LinRange (lims ... , n + 1 ), LinRange (lims ... , n + 1 ) plotgrid (x ... ) setup = Setup (; x, Re, backend); d = T ( π / 15 ) e = T ( 0.05 ) U1 (y) = y ≤ π ? tanh ((y - T ( π / 2 )) / d) : tanh (( T ( 3π / 2 ) - y) / d) # U1(y) = T(1) + (y ≤ π ? tanh((y - T(π / 2)) / d) : tanh((T(3π / 2) - y) / d)) ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U1 (y) : e * sin (x)); state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 8 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) This page was generated using Literate.jl .
`,37)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_ShearLayer2D.md.DGqD8BRv.lean.js b/previews/PR126/assets/examples_generated_ShearLayer2D.md.DGqD8BRv.lean.js
new file mode 100644
index 00000000..5007ce07
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_ShearLayer2D.md.DGqD8BRv.lean.js
@@ -0,0 +1,84 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Shear layer - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/ShearLayer2D.md","filePath":"examples/generated/ShearLayer2D.md","lastUpdated":null}'),t={name:"examples/generated/ShearLayer2D.md"};function p(l,s,k,e,E,d){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Shear layer - 2D Shear layer example.
We start by loading packages. A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia using CairoMakie using IncompressibleNavierStokes Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "ShearLayer2D" ) Floating point type
Backend
julia backend = CPU () # using CUDA; backend = CUDABackend() Reynolds number
A 2D grid is a Cartesian product of two vectors
julia n = 128 lims = T ( 0 ), T ( 2π ) x = LinRange (lims ... , n + 1 ), LinRange (lims ... , n + 1 ) plotgrid (x ... ) Build setup and assemble operators
julia setup = Setup (; x, Re, backend); nothing #hide Initial conditions: We add 1 to u in order to make global momentum conservation less trivial
julia d = T ( π / 15 ) e = T ( 0.05 ) U1 (y) = y ≤ π ? tanh ((y - T ( π / 2 )) / d) : tanh (( T ( 3π / 2 ) - y) / d) # U1(y) = T(1) + (y ≤ π ? tanh((y - T(π / 2)) / d) : tanh((T(3π / 2) - y) / d)) ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U1 (y) : e * sin (x)); nothing #hide Solve unsteady problem
julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 8 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); nothing #hide Post-process We may visualize or export the computed fields
Export to VTK
julia save_vtk (state; setup, filename = joinpath (outdir, "solution" )) Plot pressure
julia fieldplot (state; setup, fieldname = :pressure ) Plot velocity
julia fieldplot (state; setup, fieldname = :velocitynorm ) Plot vorticity
julia fieldplot (state; setup, fieldname = :vorticity ) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes outdir = joinpath ( @__DIR__ , "output" , "ShearLayer2D" ) T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() Re = T ( 2000 ) n = 128 lims = T ( 0 ), T ( 2π ) x = LinRange (lims ... , n + 1 ), LinRange (lims ... , n + 1 ) plotgrid (x ... ) setup = Setup (; x, Re, backend); d = T ( π / 15 ) e = T ( 0.05 ) U1 (y) = y ≤ π ? tanh ((y - T ( π / 2 )) / d) : tanh (( T ( 3π / 2 ) - y) / d) # U1(y) = T(1) + (y ≤ π ? tanh((y - T(π / 2)) / d) : tanh((T(3π / 2) - y) / d)) ustart = velocityfield (setup, (dim, x, y) -> dim == 1 ? U1 (y) : e * sin (x)); state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 8 )), Δt = T ( 0.01 ), processors = ( rtp = realtimeplotter (; setup, plot = fieldplot, # plot = energy_history_plot, # plot = energy_spectrum_plot, nupdate = 1 , ), # anim = animator(; setup, path = "$outdir/vorticity.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), # field = fieldsaver(; setup, nupdate = 10), log = timelogger (; nupdate = 100 ), ), ); outputs . rtp save_vtk (state; setup, filename = joinpath (outdir, "solution" )) fieldplot (state; setup, fieldname = :pressure ) fieldplot (state; setup, fieldname = :velocitynorm ) fieldplot (state; setup, fieldname = :vorticity ) This page was generated using Literate.jl .
`,37)]))}const y=i(t,[["render",p]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_TaylorGreenVortex2D.md.ByImGK6X.js b/previews/PR126/assets/examples_generated_TaylorGreenVortex2D.md.ByImGK6X.js
new file mode 100644
index 00000000..ea945e28
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_TaylorGreenVortex2D.md.ByImGK6X.js
@@ -0,0 +1,135 @@
+import{_ as l,c as a,j as s,a as h,a5 as t,o as n}from"./chunks/framework.BSoZtefh.js";const p="/IncompressibleNavierStokes.jl/previews/PR126/assets/cssxjov.CGB02zQg.png",m=JSON.parse('{"title":"Convergence study: Taylor-Green vortex (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex2D.md","filePath":"examples/generated/TaylorGreenVortex2D.md","lastUpdated":null}'),k={name:"examples/generated/TaylorGreenVortex2D.md"},e={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.474ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.107ex",height:"6.078ex",role:"img",focusable:"false",viewBox:"0 -1593.3 14633.5 2686.7","aria-hidden":"true"};function r(E,i,T,Q,g,y){return n(),a("div",null,[i[2]||(i[2]=s("h1",{id:"Convergence-study:-Taylor-Green-vortex-(2D)",tabindex:"-1"},[h("Convergence study: Taylor-Green vortex (2D) "),s("a",{class:"header-anchor",href:"#Convergence-study:-Taylor-Green-vortex-(2D)","aria-label":'Permalink to "Convergence study: Taylor-Green vortex (2D) {#Convergence-study:-Taylor-Green-vortex-(2D)}"'},"")],-1)),i[3]||(i[3]=s("p",null,"In this example we consider the Taylor-Green vortex. In 2D, it has an analytical solution, given by",-1)),s("mjx-container",e,[(n(),a("svg",d,i[0]||(i[0]=[t('This allows us to test the convergence of our solver.
julia using CairoMakie using IncompressibleNavierStokes using LinearAlgebra Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "TaylorGreenVortex2D" ) ispath (outdir) || mkpath (outdir) "/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/TaylorGreenVortex2D" Convergence
julia """ Compare numerical solution with analytical solution at final time. """ function compute_convergence (; D, nlist, lims, Re, tlims, Δt, uref, backend = CPU ()) T = typeof (lims[ 1 ]) e = zeros (T, length (nlist)) for (i, n) in enumerate (nlist) @info "Computing error for n = $n " x = ntuple (α -> LinRange (lims ... , n + 1 ), D) setup = Setup (; x, Re, backend) psolver = psolver_spectral (setup) ustart = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 1 ]), tlims[ 1 ]; psolver, ) ut = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 2 ]), tlims[ 2 ]; psolver, doproject = false , ) (; u, t), outputs = solve_unsteady (; setup, ustart, tlims, Δt, psolver) (; Ip) = setup . grid a = sum (abs2, u[Ip, :] - ut[Ip, :]) b = sum (abs2, ut[Ip, :]) e[i] = sqrt (a) / sqrt (b) end e end Analytical solution for 2D Taylor-Green vortex
julia solution (Re) = (dim, x, y, t) -> (dim == 1 ? - sin (x) * cos (y) : cos (x) * sin (y)) * exp ( - 2 t / Re) solution (generic function with 1 method) Compute error for different resolutions
julia Re = 2.0e3 nlist = [ 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ] e = compute_convergence (; D = 2 , nlist, lims = ( 0.0 , 2π ), Re, tlims = ( 0.0 , 2.0 ), Δt = 0.01 , uref = solution (Re), ) 8-element Vector{Float64}: 0.4242656052086717 0.000378932838635171 0.0001007226650833037 2.5570665390686788e-5 6.417292500615618e-6 1.6058662108882434e-6 4.01563057410408e-7 1.003967967340085e-7 Plot convergence
julia fig = Figure () ax = Axis ( fig[ 1 , 1 ]; xscale = log10, yscale = log10, xticks = nlist, xlabel = "n" , title = "Relative error" , ) scatterlines! (ax, nlist, e; label = "Data" ) lines! (ax, collect ( extrema (nlist)), n -> n ^- 2.0 ; linestyle = :dash , label = "n^-2" ) axislegend (ax) fig
Save figure
julia save ( joinpath (outdir, "convergence.png" ), fig) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes using LinearAlgebra outdir = joinpath ( @__DIR__ , "output" , "TaylorGreenVortex2D" ) ispath (outdir) || mkpath (outdir) """ Compare numerical solution with analytical solution at final time. """ function compute_convergence (; D, nlist, lims, Re, tlims, Δt, uref, backend = CPU ()) T = typeof (lims[ 1 ]) e = zeros (T, length (nlist)) for (i, n) in enumerate (nlist) @info "Computing error for n = $n " x = ntuple (α -> LinRange (lims ... , n + 1 ), D) setup = Setup (; x, Re, backend) psolver = psolver_spectral (setup) ustart = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 1 ]), tlims[ 1 ]; psolver, ) ut = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 2 ]), tlims[ 2 ]; psolver, doproject = false , ) (; u, t), outputs = solve_unsteady (; setup, ustart, tlims, Δt, psolver) (; Ip) = setup . grid a = sum (abs2, u[Ip, :] - ut[Ip, :]) b = sum (abs2, ut[Ip, :]) e[i] = sqrt (a) / sqrt (b) end e end solution (Re) = (dim, x, y, t) -> (dim == 1 ? - sin (x) * cos (y) : cos (x) * sin (y)) * exp ( - 2 t / Re) Re = 2.0e3 nlist = [ 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ] e = compute_convergence (; D = 2 , nlist, lims = ( 0.0 , 2π ), Re, tlims = ( 0.0 , 2.0 ), Δt = 0.01 , uref = solution (Re), ) fig = Figure () ax = Axis ( fig[ 1 , 1 ]; xscale = log10, yscale = log10, xticks = nlist, xlabel = "n" , title = "Relative error" , ) scatterlines! (ax, nlist, e; label = "Data" ) lines! (ax, collect ( extrema (nlist)), n -> n ^- 2.0 ; linestyle = :dash , label = "n^-2" ) axislegend (ax) fig save ( joinpath (outdir, "convergence.png" ), fig) This page was generated using Literate.jl .
`,25))])}const F=l(k,[["render",r]]);export{m as __pageData,F as default};
diff --git a/previews/PR126/assets/examples_generated_TaylorGreenVortex2D.md.ByImGK6X.lean.js b/previews/PR126/assets/examples_generated_TaylorGreenVortex2D.md.ByImGK6X.lean.js
new file mode 100644
index 00000000..ea945e28
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_TaylorGreenVortex2D.md.ByImGK6X.lean.js
@@ -0,0 +1,135 @@
+import{_ as l,c as a,j as s,a as h,a5 as t,o as n}from"./chunks/framework.BSoZtefh.js";const p="/IncompressibleNavierStokes.jl/previews/PR126/assets/cssxjov.CGB02zQg.png",m=JSON.parse('{"title":"Convergence study: Taylor-Green vortex (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex2D.md","filePath":"examples/generated/TaylorGreenVortex2D.md","lastUpdated":null}'),k={name:"examples/generated/TaylorGreenVortex2D.md"},e={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.474ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.107ex",height:"6.078ex",role:"img",focusable:"false",viewBox:"0 -1593.3 14633.5 2686.7","aria-hidden":"true"};function r(E,i,T,Q,g,y){return n(),a("div",null,[i[2]||(i[2]=s("h1",{id:"Convergence-study:-Taylor-Green-vortex-(2D)",tabindex:"-1"},[h("Convergence study: Taylor-Green vortex (2D) "),s("a",{class:"header-anchor",href:"#Convergence-study:-Taylor-Green-vortex-(2D)","aria-label":'Permalink to "Convergence study: Taylor-Green vortex (2D) {#Convergence-study:-Taylor-Green-vortex-(2D)}"'},"")],-1)),i[3]||(i[3]=s("p",null,"In this example we consider the Taylor-Green vortex. In 2D, it has an analytical solution, given by",-1)),s("mjx-container",e,[(n(),a("svg",d,i[0]||(i[0]=[t('This allows us to test the convergence of our solver.
julia using CairoMakie using IncompressibleNavierStokes using LinearAlgebra Output directory
julia outdir = joinpath ( @__DIR__ , "output" , "TaylorGreenVortex2D" ) ispath (outdir) || mkpath (outdir) "/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/TaylorGreenVortex2D" Convergence
julia """ Compare numerical solution with analytical solution at final time. """ function compute_convergence (; D, nlist, lims, Re, tlims, Δt, uref, backend = CPU ()) T = typeof (lims[ 1 ]) e = zeros (T, length (nlist)) for (i, n) in enumerate (nlist) @info "Computing error for n = $n " x = ntuple (α -> LinRange (lims ... , n + 1 ), D) setup = Setup (; x, Re, backend) psolver = psolver_spectral (setup) ustart = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 1 ]), tlims[ 1 ]; psolver, ) ut = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 2 ]), tlims[ 2 ]; psolver, doproject = false , ) (; u, t), outputs = solve_unsteady (; setup, ustart, tlims, Δt, psolver) (; Ip) = setup . grid a = sum (abs2, u[Ip, :] - ut[Ip, :]) b = sum (abs2, ut[Ip, :]) e[i] = sqrt (a) / sqrt (b) end e end Analytical solution for 2D Taylor-Green vortex
julia solution (Re) = (dim, x, y, t) -> (dim == 1 ? - sin (x) * cos (y) : cos (x) * sin (y)) * exp ( - 2 t / Re) solution (generic function with 1 method) Compute error for different resolutions
julia Re = 2.0e3 nlist = [ 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ] e = compute_convergence (; D = 2 , nlist, lims = ( 0.0 , 2π ), Re, tlims = ( 0.0 , 2.0 ), Δt = 0.01 , uref = solution (Re), ) 8-element Vector{Float64}: 0.4242656052086717 0.000378932838635171 0.0001007226650833037 2.5570665390686788e-5 6.417292500615618e-6 1.6058662108882434e-6 4.01563057410408e-7 1.003967967340085e-7 Plot convergence
julia fig = Figure () ax = Axis ( fig[ 1 , 1 ]; xscale = log10, yscale = log10, xticks = nlist, xlabel = "n" , title = "Relative error" , ) scatterlines! (ax, nlist, e; label = "Data" ) lines! (ax, collect ( extrema (nlist)), n -> n ^- 2.0 ; linestyle = :dash , label = "n^-2" ) axislegend (ax) fig
Save figure
julia save ( joinpath (outdir, "convergence.png" ), fig) Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes using LinearAlgebra outdir = joinpath ( @__DIR__ , "output" , "TaylorGreenVortex2D" ) ispath (outdir) || mkpath (outdir) """ Compare numerical solution with analytical solution at final time. """ function compute_convergence (; D, nlist, lims, Re, tlims, Δt, uref, backend = CPU ()) T = typeof (lims[ 1 ]) e = zeros (T, length (nlist)) for (i, n) in enumerate (nlist) @info "Computing error for n = $n " x = ntuple (α -> LinRange (lims ... , n + 1 ), D) setup = Setup (; x, Re, backend) psolver = psolver_spectral (setup) ustart = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 1 ]), tlims[ 1 ]; psolver, ) ut = velocityfield ( setup, (dim, x ... ) -> uref (dim, x ... , tlims[ 2 ]), tlims[ 2 ]; psolver, doproject = false , ) (; u, t), outputs = solve_unsteady (; setup, ustart, tlims, Δt, psolver) (; Ip) = setup . grid a = sum (abs2, u[Ip, :] - ut[Ip, :]) b = sum (abs2, ut[Ip, :]) e[i] = sqrt (a) / sqrt (b) end e end solution (Re) = (dim, x, y, t) -> (dim == 1 ? - sin (x) * cos (y) : cos (x) * sin (y)) * exp ( - 2 t / Re) Re = 2.0e3 nlist = [ 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ] e = compute_convergence (; D = 2 , nlist, lims = ( 0.0 , 2π ), Re, tlims = ( 0.0 , 2.0 ), Δt = 0.01 , uref = solution (Re), ) fig = Figure () ax = Axis ( fig[ 1 , 1 ]; xscale = log10, yscale = log10, xticks = nlist, xlabel = "n" , title = "Relative error" , ) scatterlines! (ax, nlist, e; label = "Data" ) lines! (ax, collect ( extrema (nlist)), n -> n ^- 2.0 ; linestyle = :dash , label = "n^-2" ) axislegend (ax) fig save ( joinpath (outdir, "convergence.png" ), fig) This page was generated using Literate.jl .
`,25))])}const F=l(k,[["render",r]]);export{m as __pageData,F as default};
diff --git a/previews/PR126/assets/examples_generated_TaylorGreenVortex3D.md.CgnLg1YH.js b/previews/PR126/assets/examples_generated_TaylorGreenVortex3D.md.CgnLg1YH.js
new file mode 100644
index 00000000..8f19a554
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_TaylorGreenVortex3D.md.CgnLg1YH.js
@@ -0,0 +1,82 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Taylor-Green vortex - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex3D.md","filePath":"examples/generated/TaylorGreenVortex3D.md","lastUpdated":null}'),p={name:"examples/generated/TaylorGreenVortex3D.md"};function l(k,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Taylor-Green vortex - 3D In this example we consider the Taylor-Green vortex.
julia using CairoMakie using IncompressibleNavierStokes Floating point precision
Backend Running in 3D is heavier than in 2D. If you are running this on a CPU, consider using multiple threads by starting Julia with julia -t auto, or add "julia.NumThreads": "auto" to the settings in VSCode.
julia backend = CPU () # using CUDA; backend = CUDABackend() Setup julia n = 32 r = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (r, r, r), Re = T ( 1e3 ), backend); psolver = psolver_spectral (setup); nothing #hide Initial conditions
julia U (dim, x, y, z) = if dim == 1 sinpi ( 2 x) * cospi ( 2 y) * sinpi ( 2 z) / 2 elseif dim == 2 - cospi ( 2 x) * sinpi ( 2 y) * sinpi ( 2 z) / 2 else zero (x) end ustart = velocityfield (setup, U, psolver); nothing #hide Solve unsteady problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1.0 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 1 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 100 ), ), psolver, ); nothing #hide Post-process Energy history
Energy spectrum
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() n = 32 r = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (r, r, r), Re = T ( 1e3 ), backend); psolver = psolver_spectral (setup); U (dim, x, y, z) = if dim == 1 sinpi ( 2 x) * cospi ( 2 y) * sinpi ( 2 z) / 2 elseif dim == 2 - cospi ( 2 x) * sinpi ( 2 y) * sinpi ( 2 z) / 2 else zero (x) end ustart = velocityfield (setup, U, psolver); state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1.0 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 1 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 100 ), ), psolver, ); outputs . ehist outputs . espec This page was generated using Literate.jl .
`,25)]))}const y=i(p,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_generated_TaylorGreenVortex3D.md.CgnLg1YH.lean.js b/previews/PR126/assets/examples_generated_TaylorGreenVortex3D.md.CgnLg1YH.lean.js
new file mode 100644
index 00000000..8f19a554
--- /dev/null
+++ b/previews/PR126/assets/examples_generated_TaylorGreenVortex3D.md.CgnLg1YH.lean.js
@@ -0,0 +1,82 @@
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Taylor-Green vortex - 3D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex3D.md","filePath":"examples/generated/TaylorGreenVortex3D.md","lastUpdated":null}'),p={name:"examples/generated/TaylorGreenVortex3D.md"};function l(k,s,t,e,E,r){return h(),a("div",null,s[0]||(s[0]=[n(`Note: Output is not generated for this example (to save resources on GitHub).
Taylor-Green vortex - 3D In this example we consider the Taylor-Green vortex.
julia using CairoMakie using IncompressibleNavierStokes Floating point precision
Backend Running in 3D is heavier than in 2D. If you are running this on a CPU, consider using multiple threads by starting Julia with julia -t auto, or add "julia.NumThreads": "auto" to the settings in VSCode.
julia backend = CPU () # using CUDA; backend = CUDABackend() Setup julia n = 32 r = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (r, r, r), Re = T ( 1e3 ), backend); psolver = psolver_spectral (setup); nothing #hide Initial conditions
julia U (dim, x, y, z) = if dim == 1 sinpi ( 2 x) * cospi ( 2 y) * sinpi ( 2 z) / 2 elseif dim == 2 - cospi ( 2 x) * sinpi ( 2 y) * sinpi ( 2 z) / 2 else zero (x) end ustart = velocityfield (setup, U, psolver); nothing #hide Solve unsteady problem julia state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1.0 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 1 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 100 ), ), psolver, ); nothing #hide Post-process Energy history
Energy spectrum
Copy-pasteable code Below is the full code for this example stripped of comments and output.
julia using GLMakie using IncompressibleNavierStokes T = Float64 backend = CPU () # using CUDA; backend = CUDABackend() n = 32 r = range ( T ( 0 ), T ( 1 ), n + 1 ) setup = Setup (; x = (r, r, r), Re = T ( 1e3 ), backend); psolver = psolver_spectral (setup); U (dim, x, y, z) = if dim == 1 sinpi ( 2 x) * cospi ( 2 y) * sinpi ( 2 z) / 2 elseif dim == 2 - cospi ( 2 x) * sinpi ( 2 y) * sinpi ( 2 z) / 2 else zero (x) end ustart = velocityfield (setup, U, psolver); state, outputs = solve_unsteady (; setup, ustart, tlims = ( T ( 0 ), T ( 1.0 )), Δt = T ( 1e-3 ), processors = ( # rtp = realtimeplotter(; setup, plot = fieldplot, nupdate = 10), ehist = realtimeplotter (; setup, plot = energy_history_plot, nupdate = 1 , displayfig = false , ), espec = realtimeplotter (; setup, plot = energy_spectrum_plot, nupdate = 10 ), # anim = animator(; setup, path = "$outdir/solution.mkv", nupdate = 20), # vtk = vtk_writer(; setup, nupdate = 10, dir = outdir, filename = "solution"), log = timelogger (; nupdate = 100 ), ), psolver, ); outputs . ehist outputs . espec This page was generated using Literate.jl .
`,25)]))}const y=i(p,[["render",l]]);export{g as __pageData,y as default};
diff --git a/previews/PR126/assets/examples_index.md.Bv0KGHOS.js b/previews/PR126/assets/examples_index.md.Bv0KGHOS.js
new file mode 100644
index 00000000..60999c3c
--- /dev/null
+++ b/previews/PR126/assets/examples_index.md.Bv0KGHOS.js
@@ -0,0 +1 @@
+import{d,o as t,c as i,j as e,k as m,g as f,t as p,_ as u,F as y,C as D,b,K as w,a5 as v,G as o,a as c}from"./chunks/framework.BSoZtefh.js";const x={class:"img-box"},_=["href"],k=["src"],N={class:"transparent-box1"},T={class:"caption"},S={class:"transparent-box2"},B={class:"subcaption"},F={class:"opacity-low"},C=d({__name:"GalleryImage",props:{href:{},src:{},caption:{},desc:{}},setup(g){return(a,s)=>(t(),i("div",x,[e("a",{href:a.href},[e("img",{src:m(f)(a.src),height:"100px",alt:""},null,8,k),e("div",N,[e("div",T,[e("h2",null,p(a.caption),1)])]),e("div",S,[e("div",B,[e("p",F,p(a.desc),1)])])],8,_)]))}}),G=u(C,[["__scopeId","data-v-f778be06"]]),P={class:"gallery-image"},R=d({__name:"Gallery",props:{images:{}},setup(g){return(a,s)=>(t(),i("div",P,[(t(!0),i(y,null,D(a.images,l=>(t(),b(G,w({ref_for:!0},l),null,16))),256))]))}}),n=u(R,[["__scopeId","data-v-9f22d17b"]]),A=JSON.parse('{"title":"Examples gallery","description":"","frontmatter":{},"headers":[],"relativePath":"examples/index.md","filePath":"examples/index.md","lastUpdated":null}'),q={name:"examples/index.md"},E=d({...q,setup(g){const a=[{href:"generated/DecayingTurbulence2D",src:"../DecayingTurbulence2D.gif",caption:"Decaying turbulence (2D)",desc:"Decaying turbulence in a periodic 2D-box"},{href:"generated/DecayingTurbulence3D",src:"../DecayingTurbulence3D.png",caption:"Decaying turbulence (3D)",desc:"Decaying turbulence in a periodic 3D-box"},{href:"generated/TaylorGreenVortex2D",src:"../TaylorGreenVortex2D.png",caption:"Taylor-Green vortex (2D)",desc:"Decaying vortex structures in a periodic 2D-box"},{href:"generated/TaylorGreenVortex3D",src:"../TaylorGreenVortex3D.png",caption:"Taylor-Green vortex (3D)",desc:"Decaying vortex structures in a periodic 3D-box"},{href:"generated/Kolmogorov2D",src:"../logo.svg",caption:"Kolmogorov flow (2D)",desc:"Initiate a flow through a sinusoidal force"},{href:"generated/ShearLayer2D",src:"../logo.svg",caption:"Shear-layer (2D)",desc:"Two layers with different velocities"},{href:"generated/PlanarMixing2D",src:"../logo.svg",caption:"Planar mixing (2D)",desc:"Planar mixing layers"}],s=[{href:"generated/Actuator2D",src:"../Actuator2D.gif",caption:"Actuator (2D)",desc:"Unsteady inflow around an actuator disk"},{href:"generated/Actuator3D",src:"../Actuator3D.png",caption:"Actuator (3D)",desc:"Unsteady inflow around an actuator disk"},{href:"generated/BackwardFacingStep2D",src:"../BackwardFacingStep2D.png",caption:"Backward Facing Step (2D)",desc:"Flow past a backward facing step"},{href:"generated/BackwardFacingStep3D",src:"../BackwardFacingStep3D.png",caption:"Backward Facing Step (3D)",desc:"Flow past a backward facing step"},{href:"generated/LidDrivenCavity2D",src:"../logo.svg",caption:"Lid-driven cavity (2D)",desc:"Generate a flow caused by a moving lid"},{href:"generated/LidDrivenCavity3D",src:"../logo.svg",caption:"Lid-driven cavity (3D)",desc:"Generate a flow caused by a moving lid"},{href:"generated/MultiActuator",src:"../logo.svg",desc:"Multi-actuator (2D)",caption:"Unsteady inflow around multiple actuator disks"}],l=[{href:"generated/RayleighBenard2D",src:"../RayleighBenard2D.gif",caption:"Rayleigh-Bénard convection (2D)",desc:"Convection generated by a temperature gradient between a hot and a cold plate"},{href:"generated/RayleighBenard3D",src:"../RayleighBenard3D.gif",caption:"Rayleigh-Bénard convection (3D)",desc:"Convection generated by a temperature gradient between a hot and a cold plate"},{href:"generated/RayleighTaylor2D",src:"../RayleighTaylor2D.gif",caption:"Rayleigh-Taylor instability (2D)",desc:"Convection generated by a temperature gradient between hot and cold fluids"},{href:"generated/RayleighTaylor3D",src:"../logo.svg",caption:"Rayleigh-Taylor instability (3D)",desc:"Convection generated by a temperature gradient between hot and cold fluids"}],h=[{href:"generated/prioranalysis",src:"../logo.svg",caption:"Filter analysis",desc:"Compare discrete filters and their properties with DNS data"},{href:"generated/postanalysis",src:"../logo.svg",caption:"CNN closure models",desc:"Compare CNN closure models for different filters, grid sizes, and projection orders"},{href:"generated/symmetryanalysis",src:"../logo.svg",caption:"Symmetry-preserving closure models",desc:"Train group equivariant CNNs and compare to normal CNNs"}];return(L,r)=>(t(),i("div",null,[r[0]||(r[0]=v('Examples gallery Here is a gallery of selected commented example simulations. The outputs are generated with Literate.jl and displayed inline. Copy-pasteable code is available at the bottom of each example. The raw Julia source scripts can be found in the examples folder.
Simple flows ',3)),o(n,{images:a}),r[1]||(r[1]=e("h2",{id:"Flows-with-mixed-boundary-conditions",tabindex:"-1"},[c("Flows with mixed boundary conditions "),e("a",{class:"header-anchor",href:"#Flows-with-mixed-boundary-conditions","aria-label":'Permalink to "Flows with mixed boundary conditions {#Flows-with-mixed-boundary-conditions}"'},"")],-1)),o(n,{images:s}),r[2]||(r[2]=e("h2",{id:"With-a-temperature-equation",tabindex:"-1"},[c("With a temperature equation "),e("a",{class:"header-anchor",href:"#With-a-temperature-equation","aria-label":'Permalink to "With a temperature equation {#With-a-temperature-equation}"'},"")],-1)),o(n,{images:l}),r[3]||(r[3]=e("h2",{id:"Neural-network-closure-models",tabindex:"-1"},[c("Neural network closure models "),e("a",{class:"header-anchor",href:"#Neural-network-closure-models","aria-label":'Permalink to "Neural network closure models {#Neural-network-closure-models}"'},"")],-1)),o(n,{images:h})]))}});export{A as __pageData,E as default};
diff --git a/previews/PR126/assets/examples_index.md.Bv0KGHOS.lean.js b/previews/PR126/assets/examples_index.md.Bv0KGHOS.lean.js
new file mode 100644
index 00000000..60999c3c
--- /dev/null
+++ b/previews/PR126/assets/examples_index.md.Bv0KGHOS.lean.js
@@ -0,0 +1 @@
+import{d,o as t,c as i,j as e,k as m,g as f,t as p,_ as u,F as y,C as D,b,K as w,a5 as v,G as o,a as c}from"./chunks/framework.BSoZtefh.js";const x={class:"img-box"},_=["href"],k=["src"],N={class:"transparent-box1"},T={class:"caption"},S={class:"transparent-box2"},B={class:"subcaption"},F={class:"opacity-low"},C=d({__name:"GalleryImage",props:{href:{},src:{},caption:{},desc:{}},setup(g){return(a,s)=>(t(),i("div",x,[e("a",{href:a.href},[e("img",{src:m(f)(a.src),height:"100px",alt:""},null,8,k),e("div",N,[e("div",T,[e("h2",null,p(a.caption),1)])]),e("div",S,[e("div",B,[e("p",F,p(a.desc),1)])])],8,_)]))}}),G=u(C,[["__scopeId","data-v-f778be06"]]),P={class:"gallery-image"},R=d({__name:"Gallery",props:{images:{}},setup(g){return(a,s)=>(t(),i("div",P,[(t(!0),i(y,null,D(a.images,l=>(t(),b(G,w({ref_for:!0},l),null,16))),256))]))}}),n=u(R,[["__scopeId","data-v-9f22d17b"]]),A=JSON.parse('{"title":"Examples gallery","description":"","frontmatter":{},"headers":[],"relativePath":"examples/index.md","filePath":"examples/index.md","lastUpdated":null}'),q={name:"examples/index.md"},E=d({...q,setup(g){const a=[{href:"generated/DecayingTurbulence2D",src:"../DecayingTurbulence2D.gif",caption:"Decaying turbulence (2D)",desc:"Decaying turbulence in a periodic 2D-box"},{href:"generated/DecayingTurbulence3D",src:"../DecayingTurbulence3D.png",caption:"Decaying turbulence (3D)",desc:"Decaying turbulence in a periodic 3D-box"},{href:"generated/TaylorGreenVortex2D",src:"../TaylorGreenVortex2D.png",caption:"Taylor-Green vortex (2D)",desc:"Decaying vortex structures in a periodic 2D-box"},{href:"generated/TaylorGreenVortex3D",src:"../TaylorGreenVortex3D.png",caption:"Taylor-Green vortex (3D)",desc:"Decaying vortex structures in a periodic 3D-box"},{href:"generated/Kolmogorov2D",src:"../logo.svg",caption:"Kolmogorov flow (2D)",desc:"Initiate a flow through a sinusoidal force"},{href:"generated/ShearLayer2D",src:"../logo.svg",caption:"Shear-layer (2D)",desc:"Two layers with different velocities"},{href:"generated/PlanarMixing2D",src:"../logo.svg",caption:"Planar mixing (2D)",desc:"Planar mixing layers"}],s=[{href:"generated/Actuator2D",src:"../Actuator2D.gif",caption:"Actuator (2D)",desc:"Unsteady inflow around an actuator disk"},{href:"generated/Actuator3D",src:"../Actuator3D.png",caption:"Actuator (3D)",desc:"Unsteady inflow around an actuator disk"},{href:"generated/BackwardFacingStep2D",src:"../BackwardFacingStep2D.png",caption:"Backward Facing Step (2D)",desc:"Flow past a backward facing step"},{href:"generated/BackwardFacingStep3D",src:"../BackwardFacingStep3D.png",caption:"Backward Facing Step (3D)",desc:"Flow past a backward facing step"},{href:"generated/LidDrivenCavity2D",src:"../logo.svg",caption:"Lid-driven cavity (2D)",desc:"Generate a flow caused by a moving lid"},{href:"generated/LidDrivenCavity3D",src:"../logo.svg",caption:"Lid-driven cavity (3D)",desc:"Generate a flow caused by a moving lid"},{href:"generated/MultiActuator",src:"../logo.svg",desc:"Multi-actuator (2D)",caption:"Unsteady inflow around multiple actuator disks"}],l=[{href:"generated/RayleighBenard2D",src:"../RayleighBenard2D.gif",caption:"Rayleigh-Bénard convection (2D)",desc:"Convection generated by a temperature gradient between a hot and a cold plate"},{href:"generated/RayleighBenard3D",src:"../RayleighBenard3D.gif",caption:"Rayleigh-Bénard convection (3D)",desc:"Convection generated by a temperature gradient between a hot and a cold plate"},{href:"generated/RayleighTaylor2D",src:"../RayleighTaylor2D.gif",caption:"Rayleigh-Taylor instability (2D)",desc:"Convection generated by a temperature gradient between hot and cold fluids"},{href:"generated/RayleighTaylor3D",src:"../logo.svg",caption:"Rayleigh-Taylor instability (3D)",desc:"Convection generated by a temperature gradient between hot and cold fluids"}],h=[{href:"generated/prioranalysis",src:"../logo.svg",caption:"Filter analysis",desc:"Compare discrete filters and their properties with DNS data"},{href:"generated/postanalysis",src:"../logo.svg",caption:"CNN closure models",desc:"Compare CNN closure models for different filters, grid sizes, and projection orders"},{href:"generated/symmetryanalysis",src:"../logo.svg",caption:"Symmetry-preserving closure models",desc:"Train group equivariant CNNs and compare to normal CNNs"}];return(L,r)=>(t(),i("div",null,[r[0]||(r[0]=v('Examples gallery Here is a gallery of selected commented example simulations. The outputs are generated with Literate.jl and displayed inline. Copy-pasteable code is available at the bottom of each example. The raw Julia source scripts can be found in the examples folder.
Simple flows ',3)),o(n,{images:a}),r[1]||(r[1]=e("h2",{id:"Flows-with-mixed-boundary-conditions",tabindex:"-1"},[c("Flows with mixed boundary conditions "),e("a",{class:"header-anchor",href:"#Flows-with-mixed-boundary-conditions","aria-label":'Permalink to "Flows with mixed boundary conditions {#Flows-with-mixed-boundary-conditions}"'},"")],-1)),o(n,{images:s}),r[2]||(r[2]=e("h2",{id:"With-a-temperature-equation",tabindex:"-1"},[c("With a temperature equation "),e("a",{class:"header-anchor",href:"#With-a-temperature-equation","aria-label":'Permalink to "With a temperature equation {#With-a-temperature-equation}"'},"")],-1)),o(n,{images:l}),r[3]||(r[3]=e("h2",{id:"Neural-network-closure-models",tabindex:"-1"},[c("Neural network closure models "),e("a",{class:"header-anchor",href:"#Neural-network-closure-models","aria-label":'Permalink to "Neural network closure models {#Neural-network-closure-models}"'},"")],-1)),o(n,{images:h})]))}});export{A as __pageData,E as default};
diff --git a/previews/PR126/assets/getting_started.md.CNlPkqwq.js b/previews/PR126/assets/getting_started.md.CNlPkqwq.js
new file mode 100644
index 00000000..91bffc35
--- /dev/null
+++ b/previews/PR126/assets/getting_started.md.CNlPkqwq.js
@@ -0,0 +1 @@
+import{_ as t,c as a,a5 as s,o as i}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Getting Started","description":"","frontmatter":{},"headers":[],"relativePath":"getting_started.md","filePath":"getting_started.md","lastUpdated":null}'),n={name:"getting_started.md"};function o(r,e,l,d,p,c){return i(),a("div",null,e[0]||(e[0]=[s('Getting Started To install IncompressibleNavierStokes, open up a Julia-REPL, type ] to get into Pkg-mode, and type:
sh add IncompressibleNavierStokes which will install the package and all dependencies to your local environment. Note that IncompressibleNavierStokes requires Julia version 1.9 or above.
Check out the gallery for commented example simulations.
',5)]))}const m=t(n,[["render",o]]);export{g as __pageData,m as default};
diff --git a/previews/PR126/assets/getting_started.md.CNlPkqwq.lean.js b/previews/PR126/assets/getting_started.md.CNlPkqwq.lean.js
new file mode 100644
index 00000000..91bffc35
--- /dev/null
+++ b/previews/PR126/assets/getting_started.md.CNlPkqwq.lean.js
@@ -0,0 +1 @@
+import{_ as t,c as a,a5 as s,o as i}from"./chunks/framework.BSoZtefh.js";const g=JSON.parse('{"title":"Getting Started","description":"","frontmatter":{},"headers":[],"relativePath":"getting_started.md","filePath":"getting_started.md","lastUpdated":null}'),n={name:"getting_started.md"};function o(r,e,l,d,p,c){return i(),a("div",null,e[0]||(e[0]=[s('Getting Started To install IncompressibleNavierStokes, open up a Julia-REPL, type ] to get into Pkg-mode, and type:
sh add IncompressibleNavierStokes which will install the package and all dependencies to your local environment. Note that IncompressibleNavierStokes requires Julia version 1.9 or above.
Check out the gallery for commented example simulations.
',5)]))}const m=t(n,[["render",o]]);export{g as __pageData,m as default};
diff --git a/previews/PR126/assets/grid.BpiJkA_F.png b/previews/PR126/assets/grid.BpiJkA_F.png
new file mode 100644
index 00000000..a40dbc86
Binary files /dev/null and b/previews/PR126/assets/grid.BpiJkA_F.png differ
diff --git a/previews/PR126/assets/index.md.DfziRIh_.js b/previews/PR126/assets/index.md.DfziRIh_.js
new file mode 100644
index 00000000..5e7502d6
--- /dev/null
+++ b/previews/PR126/assets/index.md.DfziRIh_.js
@@ -0,0 +1 @@
+import{_ as e,c as t,o as a}from"./chunks/framework.BSoZtefh.js";const h=JSON.parse('{"title":"","description":"","frontmatter":{"layout":"home","hero":{"name":"Navier-Stokes","text":"Experience 21st century turbulence","tagline":"Differentiable fluid solver written in Julia","image":{"src":"/logo.svg","alt":"IncompressibleNavierStokes"},"actions":[{"theme":"brand","text":"Getting started","link":"/getting_started"},{"theme":"alt","text":"View on GitHub","link":"https://github.com/agdestein/IncompressibleNavierStokes.jl"},{"theme":"alt","text":"Gallery","link":"/examples"}]},"features":[{"icon":"Neural closure models NeuralClosure
These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:
sh git clone https://github.com/agdestein/IncompressibleNavierStokes.jl cd IncompressibleNavierStokes/lib/NeuralClosure Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by
`,3)),t("mjx-container",p,[(o(),n("svg",Q,e[0]||(e[0]=[a('Interpolate velocity components to volume centers.
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',3))]),t("details",z,[t("summary",null,[e[50]||(e[50]=t("a",{id:"NeuralClosure.decollocate-Tuple{Any}",href:"#NeuralClosure.decollocate-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.decollocate")],-1)),e[51]||(e[51]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[52]||(e[52]=a('julia decollocate (u) -> Any Interpolate closure force from volume centers to volume faces.
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`,3))]),e[169]||(e[169]=t("h2",{id:"filters",tabindex:"-1"},[s("Filters "),t("a",{class:"header-anchor",href:"#filters","aria-label":'Permalink to "Filters"'},"")],-1)),e[170]||(e[170]=t("p",null,"The following filters are available:",-1)),t("details",q,[t("summary",null,[e[56]||(e[56]=t("a",{id:"NeuralClosure.AbstractFilter",href:"#NeuralClosure.AbstractFilter"},[t("span",{class:"jlbinding"},"NeuralClosure.AbstractFilter")],-1)),e[57]||(e[57]=s()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[58]||(e[58]=a('julia abstract type AbstractFilter Discrete DNS filter.
Subtypes ConcreteFilter should implement the in-place method:
(::ConcreteFilter)(v, u, setup_les, compression) which filters the DNS field u and put result in LES field v. Then the out-of place method:
(::ConcreteFilter)(u, setup_les, compression) automatically becomes available.
Fields
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',9))]),t("details",X,[t("summary",null,[e[59]||(e[59]=t("a",{id:"NeuralClosure.FaceAverage",href:"#NeuralClosure.FaceAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.FaceAverage")],-1)),e[60]||(e[60]=s()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[61]||(e[61]=a('julia struct FaceAverage <: NeuralClosure.AbstractFilter Average fine grid velocity field over coarse volume face.
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',4))]),t("details",J,[t("summary",null,[e[62]||(e[62]=t("a",{id:"NeuralClosure.VolumeAverage",href:"#NeuralClosure.VolumeAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.VolumeAverage")],-1)),e[63]||(e[63]=s()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[64]||(e[64]=a('julia struct VolumeAverage <: NeuralClosure.AbstractFilter Average fine grid velocity field over coarse volume.
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',3))]),t("details",U,[t("summary",null,[e[68]||(e[68]=t("a",{id:"NeuralClosure.reconstruct-NTuple{4, Any}",href:"#NeuralClosure.reconstruct-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct")],-1)),e[69]||(e[69]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[70]||(e[70]=a('julia reconstruct (v, setup_dns, setup_les, comp) -> Any Reconstruct DNS velocity field. See also reconstruct!
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',3))]),e[171]||(e[171]=t("h2",{id:"training",tabindex:"-1"},[s("Training "),t("a",{class:"header-anchor",href:"#training","aria-label":'Permalink to "Training"'},"")],-1)),t("p",null,[e[75]||(e[75]=s("To improve the model parameters, we exploit exact filtered DNS data ")),t("mjx-container",W,[(o(),n("svg",$,e[71]||(e[71]=[a('julia create_callback ( err; θ, callbackstate, displayref, displayfig, displayupdates, figfile, nupdate ) Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.
If state is nonempty, it also plots previous convergence.
If not using interactive GLMakie window, set displayupdates to true.
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`,5))]),t("details",T1,[t("summary",null,[e[95]||(e[95]=t("a",{id:"NeuralClosure.create_dataloader_post-Tuple{Any}",href:"#NeuralClosure.create_dataloader_post-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_post")],-1)),e[96]||(e[96]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[97]||(e[97]=a(`julia create_dataloader_post ( trajectories; ntrajectory, nunroll, device ) Create trajectory dataloader.
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`,3))]),t("details",p1,[t("summary",null,[e[98]||(e[98]=t("a",{id:"NeuralClosure.create_dataloader_prior-Tuple{Any}",href:"#NeuralClosure.create_dataloader_prior-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_prior")],-1)),e[99]||(e[99]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[100]||(e[100]=a(`julia create_dataloader_prior ( data; batchsize, device ) -> NeuralClosure . var"#dataloader#40" {Int64} Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.
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`,3))]),t("details",Q1,[t("summary",null,[e[101]||(e[101]=t("a",{id:"NeuralClosure.create_loss_post-Tuple{}",href:"#NeuralClosure.create_loss_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_post")],-1)),e[102]||(e[102]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[103]||(e[103]=a(`julia create_loss_post ( ; setup, method, psolver, closure_model, nsubstep ) Create a-posteriori loss function.
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`,3))]),t("details",m1,[t("summary",null,[e[104]||(e[104]=t("a",{id:"NeuralClosure.create_loss_prior",href:"#NeuralClosure.create_loss_prior"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_prior")],-1)),e[105]||(e[105]=s()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[106]||(e[106]=a(`julia create_loss_prior ( f ) -> NeuralClosure . var"#loss_prior#53" {_A, NeuralClosure . var"#51#52" } where _A create_loss_prior ( f, normalize ) -> NeuralClosure . var"#loss_prior#53" Return mean squared error loss for the predictor f.
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`,3))]),t("details",h1,[t("summary",null,[e[107]||(e[107]=t("a",{id:"NeuralClosure.create_relerr_post-Tuple{}",href:"#NeuralClosure.create_relerr_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_post")],-1)),e[108]||(e[108]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[109]||(e[109]=a(`julia create_relerr_post ( ; data, setup, method, psolver, closure_model, nsubstep ) Create a-posteriori relative error.
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`,3))]),t("details",u1,[t("summary",null,[e[110]||(e[110]=t("a",{id:"NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}",href:"#NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_prior")],-1)),e[111]||(e[111]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[112]||(e[112]=a('julia create_relerr_prior (f, x, y) -> NeuralClosure . var"#54#55" Create a-priori error.
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',3))]),t("details",c1,[t("summary",null,[e[113]||(e[113]=t("a",{id:"NeuralClosure.create_relerr_symmetry_post-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_post")],-1)),e[114]||(e[114]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[115]||(e[115]=a(`julia create_relerr_symmetry_post ( ; u, setup, method, psolver, Δt, nstep, g ) Create a-posteriori symmetry error.
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`,3))]),t("details",g1,[t("summary",null,[e[116]||(e[116]=t("a",{id:"NeuralClosure.create_relerr_symmetry_prior-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_prior-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_prior")],-1)),e[117]||(e[117]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[118]||(e[118]=a('julia create_relerr_symmetry_prior (; u, setup, g) Create a-priori equivariance error.
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',3))]),t("details",k1,[t("summary",null,[e[119]||(e[119]=t("a",{id:"NeuralClosure.train-Tuple{}",href:"#NeuralClosure.train-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.train")],-1)),e[120]||(e[120]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[121]||(e[121]=a(`julia train ( ; dataloader, loss, trainstate, niter, callback, callbackstate, λ ) Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),t("details",b1,[t("summary",null,[e[122]||(e[122]=t("a",{id:"NeuralClosure.trainepoch-Tuple{}",href:"#NeuralClosure.trainepoch-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.trainepoch")],-1)),e[123]||(e[123]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[124]||(e[124]=a(`julia trainepoch ( ; dataloader, loss, trainstate, callback, callbackstate, device, noiselevel, λ ) Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
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`,4))]),e[173]||(e[173]=t("h2",{id:"Neural-architectures",tabindex:"-1"},[s("Neural architectures "),t("a",{class:"header-anchor",href:"#Neural-architectures","aria-label":'Permalink to "Neural architectures {#Neural-architectures}"'},"")],-1)),e[174]||(e[174]=t("p",null,"We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).",-1)),t("details",y1,[t("summary",null,[e[125]||(e[125]=t("a",{id:"NeuralClosure.cnn-Tuple{}",href:"#NeuralClosure.cnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.cnn")],-1)),e[126]||(e[126]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[127]||(e[127]=a(`julia cnn ( ; setup, radii, channels, activations, use_bias, channel_augmenter, rng ) Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
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`,3))]),t("details",x1,[t("summary",null,[e[128]||(e[128]=t("a",{id:"NeuralClosure.GroupConv2D",href:"#NeuralClosure.GroupConv2D"},[t("span",{class:"jlbinding"},"NeuralClosure.GroupConv2D")],-1)),e[129]||(e[129]=s()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[130]||(e[130]=a('julia struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.
The kwargs are passed to the Conv layer.
The layer has three variants:
If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).
If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).
Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).
Fields
islifting
isprojecting
cin
cout
conv
source
',8))]),t("details",E1,[t("summary",null,[e[131]||(e[131]=t("a",{id:"NeuralClosure.gcnn-Tuple{}",href:"#NeuralClosure.gcnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.gcnn")],-1)),e[132]||(e[132]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[133]||(e[133]=a('julia gcnn (; setup, radii, channels, activations, use_bias, rng) Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
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',3))]),t("details",f1,[t("summary",null,[e[134]||(e[134]=t("a",{id:"NeuralClosure.rot2",href:"#NeuralClosure.rot2"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),e[135]||(e[135]=s()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[136]||(e[136]=a('Rotate the field u by 90 degrees counter-clockwise r - 1 times.
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',3))]),t("details",H1,[t("summary",null,[e[143]||(e[143]=t("a",{id:"NeuralClosure.vecrot2-Tuple{Any, Any}",href:"#NeuralClosure.vecrot2-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.vecrot2")],-1)),e[144]||(e[144]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[145]||(e[145]=a('julia vecrot2 (u, r) -> Any Rotate vector fields [ux;;; uy]
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',3))]),t("details",j1,[t("summary",null,[e[146]||(e[146]=t("a",{id:"NeuralClosure.FourierLayer",href:"#NeuralClosure.FourierLayer"},[t("span",{class:"jlbinding"},"NeuralClosure.FourierLayer")],-1)),e[147]||(e[147]=s()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[148]||(e[148]=a('julia struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer Fourier layer operating on uniformly discretized functions.
Some important sizes:
dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).
(nx..., cin, nsample): Input size
(nx..., cout, nsample): Output size
nx = fill(n, dimension()): Number of points in each spatial dimension
n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber
kmax: Cut-off wavenumber
nsample: Number of input samples (treated independently)
Fields
dimension
kmax
cin
cout
σ
init_weight
source
',7))]),t("details",C1,[t("summary",null,[e[149]||(e[149]=t("a",{id:"NeuralClosure.fno-Tuple{}",href:"#NeuralClosure.fno-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.fno")],-1)),e[150]||(e[150]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[151]||(e[151]=a('julia fno (; setup, kmax, c, σ, ψ, rng, kwargs ... ) Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.
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`,3))]),t("details",Z1,[t("summary",null,[e[163]||(e[163]=t("a",{id:"NeuralClosure.filtersaver-NTuple{6, Any}",href:"#NeuralClosure.filtersaver-NTuple{6, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.filtersaver")],-1)),e[164]||(e[164]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[165]||(e[165]=a(`julia filtersaver ( dns, les, filters, compression, psolver_dns, psolver_les; nupdate, filenames, F, p ) Save filtered DNS data.
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diff --git a/previews/PR126/assets/manual_closure.md.DJlmWyys.lean.js b/previews/PR126/assets/manual_closure.md.DJlmWyys.lean.js
new file mode 100644
index 00000000..975ca61c
--- /dev/null
+++ b/previews/PR126/assets/manual_closure.md.DJlmWyys.lean.js
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1000","aria-hidden":"true"},N1={class:"jldocstring custom-block",open:""},Z1={class:"jldocstring custom-block",open:""};function D1(F1,e,A1,O1,S1,B1){const l=d("Badge");return o(),n("div",null,[e[166]||(e[166]=a(`Neural closure models NeuralClosure
These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:
sh git clone https://github.com/agdestein/IncompressibleNavierStokes.jl cd IncompressibleNavierStokes/lib/NeuralClosure Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by
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Subtypes ConcreteFilter should implement the in-place method:
(::ConcreteFilter)(v, u, setup_les, compression) which filters the DNS field u and put result in LES field v. Then the out-of place method:
(::ConcreteFilter)(u, setup_les, compression) automatically becomes available.
Fields
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If state is nonempty, it also plots previous convergence.
If not using interactive GLMakie window, set displayupdates to true.
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`,3))]),t("details",Q1,[t("summary",null,[e[101]||(e[101]=t("a",{id:"NeuralClosure.create_loss_post-Tuple{}",href:"#NeuralClosure.create_loss_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_post")],-1)),e[102]||(e[102]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[103]||(e[103]=a(`julia create_loss_post ( ; setup, method, psolver, closure_model, nsubstep ) Create a-posteriori loss function.
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',3))]),t("details",k1,[t("summary",null,[e[119]||(e[119]=t("a",{id:"NeuralClosure.train-Tuple{}",href:"#NeuralClosure.train-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.train")],-1)),e[120]||(e[120]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[121]||(e[121]=a(`julia train ( ; dataloader, loss, trainstate, niter, callback, callbackstate, λ ) Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
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`,4))]),t("details",b1,[t("summary",null,[e[122]||(e[122]=t("a",{id:"NeuralClosure.trainepoch-Tuple{}",href:"#NeuralClosure.trainepoch-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.trainepoch")],-1)),e[123]||(e[123]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[124]||(e[124]=a(`julia trainepoch ( ; dataloader, loss, trainstate, callback, callbackstate, device, noiselevel, λ ) Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
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`,4))]),e[173]||(e[173]=t("h2",{id:"Neural-architectures",tabindex:"-1"},[s("Neural architectures "),t("a",{class:"header-anchor",href:"#Neural-architectures","aria-label":'Permalink to "Neural architectures {#Neural-architectures}"'},"")],-1)),e[174]||(e[174]=t("p",null,"We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).",-1)),t("details",y1,[t("summary",null,[e[125]||(e[125]=t("a",{id:"NeuralClosure.cnn-Tuple{}",href:"#NeuralClosure.cnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.cnn")],-1)),e[126]||(e[126]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[127]||(e[127]=a(`julia cnn ( ; setup, radii, channels, activations, use_bias, channel_augmenter, rng ) Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
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`,3))]),t("details",x1,[t("summary",null,[e[128]||(e[128]=t("a",{id:"NeuralClosure.GroupConv2D",href:"#NeuralClosure.GroupConv2D"},[t("span",{class:"jlbinding"},"NeuralClosure.GroupConv2D")],-1)),e[129]||(e[129]=s()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[130]||(e[130]=a('julia struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.
The kwargs are passed to the Conv layer.
The layer has three variants:
If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).
If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).
Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).
Fields
islifting
isprojecting
cin
cout
conv
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',8))]),t("details",E1,[t("summary",null,[e[131]||(e[131]=t("a",{id:"NeuralClosure.gcnn-Tuple{}",href:"#NeuralClosure.gcnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.gcnn")],-1)),e[132]||(e[132]=s()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[133]||(e[133]=a('julia gcnn (; setup, radii, channels, activations, use_bias, rng) Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
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',3))]),t("details",f1,[t("summary",null,[e[134]||(e[134]=t("a",{id:"NeuralClosure.rot2",href:"#NeuralClosure.rot2"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),e[135]||(e[135]=s()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[136]||(e[136]=a('Rotate the field u by 90 degrees counter-clockwise r - 1 times.
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Some important sizes:
dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).
(nx..., cin, nsample): Input size
(nx..., cout, nsample): Output size
nx = fill(n, dimension()): Number of points in each spatial dimension
n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber
kmax: Cut-off wavenumber
nsample: Number of input samples (treated independently)
Fields
dimension
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cin
cout
σ
init_weight
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diff --git a/previews/PR126/assets/manual_differentiability.md.Do-3poq7.js b/previews/PR126/assets/manual_differentiability.md.Do-3poq7.js
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@@ -0,0 +1,93 @@
+import{_ as p,c as n,a5 as e,j as i,a,o as t}from"./chunks/framework.BSoZtefh.js";const u=JSON.parse('{"title":"Differentiating through the code","description":"","frontmatter":{},"headers":[],"relativePath":"manual/differentiability.md","filePath":"manual/differentiability.md","lastUpdated":null}'),l={name:"manual/differentiability.md"},h={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.131ex",height:"1.507ex",role:"img",focusable:"false",viewBox:"0 -666 500 666","aria-hidden":"true"};function r(d,s,o,g,c,E){return t(),n("div",null,[s[10]||(s[10]=e(`Differentiating through the code IncompressibleNavierStokes is reverse-mode differentiable , which means that you can back-propagate gradients through the code. Two AD libraries are currently supported:
Zygote.jl
Enzyme.jl
Automatic differentiation with Zygote Zygote.jl is the default choice for AD backend because it is easy to understand, compatible with most of the Julia ecosystem and good with vectorized code and BLAS. This comes at a cost however, as intermediate velocity fields need to be stored in memory for use in the backward pass. For this reason, many of the operators come in two versions: a slow differentiable allocating non-mutating variant (e.g. divergencedivergence!
Zygote limitation: array mutation
To make your code differentiable, you must use the differentiable versions of the operators (without the exclamation marks).
Example: Gradient of kinetic energy To differentiate outputs of a simulation with respect to the initial conditions, make a time stepping loop composed of differentiable operations:
julia using IncompressibleNavierStokes ax = range ( 0 , 1 , 101 ) setup = Setup (; x = (ax, ax), Re = 500.0 ) psolver = default_psolver (setup) method = RKMethods . RK44P2 () Δt = 0.01 nstep = 100 (; Iu) = setup . grid function final_energy (u) stepper = create_stepper (method; setup, psolver, u, temp = nothing , t = 0.0 ) for it = 1 : nstep stepper = timestep (method, stepper, Δt) end (; u) = stepper sum (abs2, u[Iu[ 1 ], 1 ]) / 2 + sum (abs2, u[Iu[ 2 ], 2 ]) / 2 end u = random_field (setup) using Zygote g, = Zygote . gradient (final_energy, u) @show size (u) size (g) Now g is the gradient of final_energy with respect to the initial conditions u, and consequently has the same size.
Note that every operation in the final_energy function is non-mutating and thus differentiable.
Automatic differentiation with Enzyme Enzyme.jl is highly-efficient and its ability to perform AD on optimized code allows Enzyme to meet or exceed the performance of state-of-the-art AD tools. The downside is that restricts the user's defined f function to not do things like require garbage collection or calls to BLAS/LAPACK. However, mutation is supported, meaning that in-place f with fully mutating non-allocating code will work with Enzyme and this will be the most efficient adjoint implementation.
Enzyme limitation: vector returns
Enzyme's autodiff function can only handle functions with scalar output. To implement pullbacks for array-valued functions, use a mutating function that returns nothing and stores its result in one of the arguments, which must be passed wrapped in a Duplicated. In IncompressibleNavierStokes, we provide enzyme_wrapper to automatically wrap the function and its arguments in the correct way.
Example: Gradient of the right-hand side In this example we differentiate the right-hand side of the Navier-Stokes equations with respect to the velocity field u:
julia using Enzyme ax = range ( 0 , 1 , 101 ) setup = Setup (; x = (ax, ax), Re = 500.0 ) psolver = default_psolver (setup) u = random_field (setup) dudt = similar (u) t = 0.0 f! = right_hand_side! right_hand_side! (generic function with 1 method) Notice that we are using the mutating (in-place) version of the right-hand side function. This function can not be differentiate by Zygote, which requires the slower non-mutating version of the right-hand side.
We then define the Dual part of the input and output, required to store the adjoint values:
julia ddudt = Enzyme . make_zero (dudt) .+ 1 ; du = Enzyme . make_zero (u); 102×102×2 Array{Float64, 3}: [:, :, 1] = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [:, :, 2] = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Then we pack the parameters to be passed to right_hand_side!:
julia params = [setup, psolver]; params_ref = Ref (params); Base.RefValue{Vector{Any}}(Any[(grid = (xlims = ((0.0, 1.0), (0.0, 1.0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (102, 102), Nu = ((100, 100), (100, 100)), Np = (100, 100), Iu = (CartesianIndices((2:101, 2:101)), CartesianIndices((2:101, 2:101))), Ip = CartesianIndices((2:101, 2:101)), x = ([-0.010000000000000009, 0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01], [-0.010000000000000009, 0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01]), xu = (([0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01], [-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005]), ([-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005], [0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01])), xp = ([-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005], [-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005]), Δ = ([0.010000000000000009, 0.01, 0.01, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000009, 0.009999999999999995, 0.009999999999999995 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009], [0.010000000000000009, 0.01, 0.01, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000009, 0.009999999999999995, 0.009999999999999995 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009]), Δu = ([0.010000000000000005, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000002, 0.010000000000000002, 0.010000000000000009, 0.009999999999999981, 0.010000000000000009 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.0050000000000000044], [0.010000000000000005, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000002, 0.010000000000000002, 0.010000000000000009, 0.009999999999999981, 0.010000000000000009 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.0050000000000000044]), A = ((([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 0.4999999999999998, 0.4999999999999999, 0.5, 0.5000000000000003, 0.5, 0.49999999999999967, 0.5000000000000003, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5000000000000056, 0.5, 0.5, 0.5, 0.5, 0.5, 0.49999999999999445], [0.5000000000000002, 0.5000000000000001, 0.5, 0.49999999999999967, 0.5, 0.5000000000000003, 0.49999999999999967, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.49999999999999445, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5000000000000056, 1.0])), (([1.0, 0.4999999999999998, 0.4999999999999999, 0.5, 0.5000000000000003, 0.5, 0.49999999999999967, 0.5000000000000003, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5000000000000056, 0.5, 0.5, 0.5, 0.5, 0.5, 0.49999999999999445], [0.5000000000000002, 0.5000000000000001, 0.5, 0.49999999999999967, 0.5, 0.5000000000000003, 0.49999999999999967, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.49999999999999445, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5000000000000056, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((PeriodicBC(), PeriodicBC()), (PeriodicBC(), PeriodicBC())), Re = 500.0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = nothing), IncompressibleNavierStokes.var"#psolve!#124"{FFTW.rFFTWPlan{Float64, -1, false, 2, Tuple{Int64, Int64}}, Matrix{Float64}, Matrix{ComplexF64}, Int64, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}(FFTW real-to-complex plan for 100×100 array of Float64 (rdft2-rank>=2/1 (rdft2-vrank>=1-x100/1 (rdft2-ct-dit/20 (hc2c-direct-20/76/0 "hc2cfdftv_20_avx2" (rdft2-ct-dit/2 (hc2c-direct-2/4/0 "hc2cfdftv_2_avx2" (rdft2-r2hc-direct-2 "r2cf_2") (rdft2-r2hc01-direct-2 "r2cfII_2")) (dft-direct-10 "n1fv_10_avx2_128")) (rdft2-nop)) (dft-direct-5-x10 "n1fv_5_avx2_128"))) (dft-vrank>=1-x51/1 (dft-ct-dit/10 (dftw-direct-10/6 "t3fv_10_avx2_128") (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [6.91365635410076e-310 6.9136563563142e-310 … 1.265e-321 1.75e-321; 6.9136563541229e-310 6.9136563563363e-310 … 1.265e-321 1.764e-321; … ; 6.9136563562699e-310 6.9136563584833e-310 … 1.75e-321 2.233e-321; 6.91365635629204e-310 6.91365635850546e-310 … 1.75e-321 2.25e-321], ComplexF64[0.0 + 0.0im 6.986129655e-315 + 1.27319747483e-313im … 0.0 + 0.0im 0.0 + 0.0im; 1.443e-321 + 6.98612574e-315im -9.605593679422648e85 + 1.63e-322im … 0.0 + 0.0im 1.443e-321 + 6.98652487e-315im; … ; 1.966162503567101e-236 + 1.24823282e-315im 1.443e-321 + 6.98613357e-315im … 0.0 + 0.0im 6.986528623e-315 + 8.487983164e-314im; 0.0 + 1.443e-321im 1.27319747483e-313 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698 … 3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698 … 0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))]) Now, we call the autodiff function from Enzyme:
julia Enzyme . autodiff (Enzyme . Reverse, f!, Duplicated (dudt,ddudt), Duplicated (u,du), Const (params_ref), Const (t)) ((nothing, nothing, nothing, nothing),) Since we have passed a Duplicated object, the gradient of u is stored in du.
Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:
julia f = create_right_hand_side (setup, psolver) _, zpull = Zygote . pullback (f, u, nothing , 0.0 ); @assert zpull (dudt)[ 1 ] == du Differentiating through the code IncompressibleNavierStokes is reverse-mode differentiable , which means that you can back-propagate gradients through the code. Two AD libraries are currently supported:
Zygote.jl
Enzyme.jl
Automatic differentiation with Zygote Zygote.jl is the default choice for AD backend because it is easy to understand, compatible with most of the Julia ecosystem and good with vectorized code and BLAS. This comes at a cost however, as intermediate velocity fields need to be stored in memory for use in the backward pass. For this reason, many of the operators come in two versions: a slow differentiable allocating non-mutating variant (e.g. divergencedivergence!
Zygote limitation: array mutation
To make your code differentiable, you must use the differentiable versions of the operators (without the exclamation marks).
Example: Gradient of kinetic energy To differentiate outputs of a simulation with respect to the initial conditions, make a time stepping loop composed of differentiable operations:
julia using IncompressibleNavierStokes ax = range ( 0 , 1 , 101 ) setup = Setup (; x = (ax, ax), Re = 500.0 ) psolver = default_psolver (setup) method = RKMethods . RK44P2 () Δt = 0.01 nstep = 100 (; Iu) = setup . grid function final_energy (u) stepper = create_stepper (method; setup, psolver, u, temp = nothing , t = 0.0 ) for it = 1 : nstep stepper = timestep (method, stepper, Δt) end (; u) = stepper sum (abs2, u[Iu[ 1 ], 1 ]) / 2 + sum (abs2, u[Iu[ 2 ], 2 ]) / 2 end u = random_field (setup) using Zygote g, = Zygote . gradient (final_energy, u) @show size (u) size (g) Now g is the gradient of final_energy with respect to the initial conditions u, and consequently has the same size.
Note that every operation in the final_energy function is non-mutating and thus differentiable.
Automatic differentiation with Enzyme Enzyme.jl is highly-efficient and its ability to perform AD on optimized code allows Enzyme to meet or exceed the performance of state-of-the-art AD tools. The downside is that restricts the user's defined f function to not do things like require garbage collection or calls to BLAS/LAPACK. However, mutation is supported, meaning that in-place f with fully mutating non-allocating code will work with Enzyme and this will be the most efficient adjoint implementation.
Enzyme limitation: vector returns
Enzyme's autodiff function can only handle functions with scalar output. To implement pullbacks for array-valued functions, use a mutating function that returns nothing and stores its result in one of the arguments, which must be passed wrapped in a Duplicated. In IncompressibleNavierStokes, we provide enzyme_wrapper to automatically wrap the function and its arguments in the correct way.
Example: Gradient of the right-hand side In this example we differentiate the right-hand side of the Navier-Stokes equations with respect to the velocity field u:
julia using Enzyme ax = range ( 0 , 1 , 101 ) setup = Setup (; x = (ax, ax), Re = 500.0 ) psolver = default_psolver (setup) u = random_field (setup) dudt = similar (u) t = 0.0 f! = right_hand_side! right_hand_side! (generic function with 1 method) Notice that we are using the mutating (in-place) version of the right-hand side function. This function can not be differentiate by Zygote, which requires the slower non-mutating version of the right-hand side.
We then define the Dual part of the input and output, required to store the adjoint values:
julia ddudt = Enzyme . make_zero (dudt) .+ 1 ; du = Enzyme . make_zero (u); 102×102×2 Array{Float64, 3}: [:, :, 1] = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [:, :, 2] = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Then we pack the parameters to be passed to right_hand_side!:
julia params = [setup, psolver]; params_ref = Ref (params); Base.RefValue{Vector{Any}}(Any[(grid = (xlims = ((0.0, 1.0), (0.0, 1.0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (102, 102), Nu = ((100, 100), (100, 100)), Np = (100, 100), Iu = (CartesianIndices((2:101, 2:101)), CartesianIndices((2:101, 2:101))), Ip = CartesianIndices((2:101, 2:101)), x = ([-0.010000000000000009, 0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01], [-0.010000000000000009, 0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01]), xu = (([0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01], [-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005]), ([-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005], [0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 … 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.0, 1.01])), xp = ([-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005], [-0.0050000000000000044, 0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.07500000000000001, 0.08499999999999999 … 0.915, 0.925, 0.935, 0.945, 0.955, 0.965, 0.975, 0.985, 0.995, 1.005]), Δ = ([0.010000000000000009, 0.01, 0.01, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000009, 0.009999999999999995, 0.009999999999999995 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009], [0.010000000000000009, 0.01, 0.01, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000009, 0.009999999999999995, 0.009999999999999995 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009]), Δu = ([0.010000000000000005, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000002, 0.010000000000000002, 0.010000000000000009, 0.009999999999999981, 0.010000000000000009 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.0050000000000000044], [0.010000000000000005, 0.009999999999999998, 0.010000000000000002, 0.010000000000000002, 0.009999999999999995, 0.010000000000000002, 0.010000000000000002, 0.010000000000000009, 0.009999999999999981, 0.010000000000000009 … 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.010000000000000009, 0.009999999999999898, 0.0050000000000000044]), A = ((([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 0.4999999999999998, 0.4999999999999999, 0.5, 0.5000000000000003, 0.5, 0.49999999999999967, 0.5000000000000003, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5000000000000056, 0.5, 0.5, 0.5, 0.5, 0.5, 0.49999999999999445], [0.5000000000000002, 0.5000000000000001, 0.5, 0.49999999999999967, 0.5, 0.5000000000000003, 0.49999999999999967, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.49999999999999445, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5000000000000056, 1.0])), (([1.0, 0.4999999999999998, 0.4999999999999999, 0.5, 0.5000000000000003, 0.5, 0.49999999999999967, 0.5000000000000003, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5000000000000056, 0.5, 0.5, 0.5, 0.5, 0.5, 0.49999999999999445], [0.5000000000000002, 0.5000000000000001, 0.5, 0.49999999999999967, 0.5, 0.5000000000000003, 0.49999999999999967, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.49999999999999445, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5000000000000056, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5 … 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((PeriodicBC(), PeriodicBC()), (PeriodicBC(), PeriodicBC())), Re = 500.0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = nothing), IncompressibleNavierStokes.var"#psolve!#124"{FFTW.rFFTWPlan{Float64, -1, false, 2, Tuple{Int64, Int64}}, Matrix{Float64}, Matrix{ComplexF64}, Int64, CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}}(FFTW real-to-complex plan for 100×100 array of Float64 (rdft2-rank>=2/1 (rdft2-vrank>=1-x100/1 (rdft2-ct-dit/20 (hc2c-direct-20/76/0 "hc2cfdftv_20_avx2" (rdft2-ct-dit/2 (hc2c-direct-2/4/0 "hc2cfdftv_2_avx2" (rdft2-r2hc-direct-2 "r2cf_2") (rdft2-r2hc01-direct-2 "r2cfII_2")) (dft-direct-10 "n1fv_10_avx2_128")) (rdft2-nop)) (dft-direct-5-x10 "n1fv_5_avx2_128"))) (dft-vrank>=1-x51/1 (dft-ct-dit/10 (dftw-direct-10/6 "t3fv_10_avx2_128") (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [6.91365635410076e-310 6.9136563563142e-310 … 1.265e-321 1.75e-321; 6.9136563541229e-310 6.9136563563363e-310 … 1.265e-321 1.764e-321; … ; 6.9136563562699e-310 6.9136563584833e-310 … 1.75e-321 2.233e-321; 6.91365635629204e-310 6.91365635850546e-310 … 1.75e-321 2.25e-321], ComplexF64[0.0 + 0.0im 6.986129655e-315 + 1.27319747483e-313im … 0.0 + 0.0im 0.0 + 0.0im; 1.443e-321 + 6.98612574e-315im -9.605593679422648e85 + 1.63e-322im … 0.0 + 0.0im 1.443e-321 + 6.98652487e-315im; … ; 1.966162503567101e-236 + 1.24823282e-315im 1.443e-321 + 6.98613357e-315im … 0.0 + 0.0im 6.986528623e-315 + 8.487983164e-314im; 0.0 + 1.443e-321im 1.27319747483e-313 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698 … 3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698 … 0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))]) Now, we call the autodiff function from Enzyme:
julia Enzyme . autodiff (Enzyme . Reverse, f!, Duplicated (dudt,ddudt), Duplicated (u,du), Const (params_ref), Const (t)) ((nothing, nothing, nothing, nothing),) Since we have passed a Duplicated object, the gradient of u is stored in du.
Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:
julia f = create_right_hand_side (setup, psolver) _, zpull = Zygote . pullback (f, u, nothing , 0.0 ); @assert zpull (dudt)[ 1 ] == du GPU Support IncompressibleNavierStokes supports various array types. The desired backend only has to be passed to the Setup
julia using CUDA setup = Setup (; kwargs ... , backend = CUDABackend ()) All operators have been made are backend agnostic by using KernelAbstractions.jl . Even if a GPU is not available, the operators are multithreaded if Julia is started with multiple threads (e.g. julia -t 4)
This has been tested with CUDA compatible GPUs.
This has not been tested with other GPU interfaces, such as
If they start supporting sparse matrices and fast Fourier transforms they could also be used.
psolver_direct on CUDA
To use a specialized linear solver for CUDA, make sure to install and using CUDA.jl and CUDSS.jl. Then psolver_direct will automatically use the CUDSS solver.
GPU Support IncompressibleNavierStokes supports various array types. The desired backend only has to be passed to the Setup
julia using CUDA setup = Setup (; kwargs ... , backend = CUDABackend ()) All operators have been made are backend agnostic by using KernelAbstractions.jl . Even if a GPU is not available, the operators are multithreaded if Julia is started with multiple threads (e.g. julia -t 4)
This has been tested with CUDA compatible GPUs.
This has not been tested with other GPU interfaces, such as
If they start supporting sparse matrices and fast Fourier transforms they could also be used.
psolver_direct on CUDA
To use a specialized linear solver for CUDA, make sure to install and using CUDA.jl and CUDSS.jl. Then psolver_direct will automatically use the CUDSS solver.
Large eddy simulation Depending on the problem specification, a given grid resolution may not be sufficient to resolve all spatial features of the flow. Consider the following example:
On the left, the grid spacing is too large to capt the smallest eddies in the flow. These eddies create sub-grid stresses that also affect the large scale features. The grid must be refined if we want to compute these stresses exactly.
On the right, the smallest spatial feature of the flow is fully resolved, and there are no sub-grid stresses. The equations can be solved without worrying about errors from unresolved features. This is known as Direct Numerical Simulation (DNS).
If refining the grid is too costly, a closure model can be used to predict the sub-grid stresses. The models only give an estimate for these stresses, and may need to be calibrated to the given problem. When used correctly, they can predict the evolution of the large fluid motions without computing the sub-grid motions themselves. This is known as Large Eddy Simulation (LES).
Eddy viscosity models add a local contribution to the global baseline viscosity. The baseline viscosity models transfer of energy from resolved to atomic scales. The new turbulent viscosity on the other hand, models energy transfer from resolved to unresolved scales. This non-constant field is computed from the local velocity field.
',7)]))}const p=t(r,[["render",l]]);export{f as __pageData,p as default};
diff --git a/previews/PR126/assets/manual_les.md.C92GZ80j.lean.js b/previews/PR126/assets/manual_les.md.C92GZ80j.lean.js
new file mode 100644
index 00000000..2212d9e8
--- /dev/null
+++ b/previews/PR126/assets/manual_les.md.C92GZ80j.lean.js
@@ -0,0 +1 @@
+import{_ as t,c as s,a5 as o,o as a}from"./chunks/framework.BSoZtefh.js";const i="/IncompressibleNavierStokes.jl/previews/PR126/assets/resolution.DAYTxiG0.png",f=JSON.parse('{"title":"Large eddy simulation","description":"","frontmatter":{},"headers":[],"relativePath":"manual/les.md","filePath":"manual/les.md","lastUpdated":null}'),r={name:"manual/les.md"};function l(n,e,d,c,m,u){return a(),s("div",null,e[0]||(e[0]=[o('Large eddy simulation Depending on the problem specification, a given grid resolution may not be sufficient to resolve all spatial features of the flow. Consider the following example:
On the left, the grid spacing is too large to capt the smallest eddies in the flow. These eddies create sub-grid stresses that also affect the large scale features. The grid must be refined if we want to compute these stresses exactly.
On the right, the smallest spatial feature of the flow is fully resolved, and there are no sub-grid stresses. The equations can be solved without worrying about errors from unresolved features. This is known as Direct Numerical Simulation (DNS).
If refining the grid is too costly, a closure model can be used to predict the sub-grid stresses. The models only give an estimate for these stresses, and may need to be calibrated to the given problem. When used correctly, they can predict the evolution of the large fluid motions without computing the sub-grid motions themselves. This is known as Large Eddy Simulation (LES).
Eddy viscosity models add a local contribution to the global baseline viscosity. The baseline viscosity models transfer of energy from resolved to atomic scales. The new turbulent viscosity on the other hand, models energy transfer from resolved to unresolved scales. This non-constant field is computed from the local velocity field.
',7)]))}const p=t(r,[["render",l]]);export{f as __pageData,p as default};
diff --git a/previews/PR126/assets/manual_matrices.md.CJA9PRBT.js b/previews/PR126/assets/manual_matrices.md.CJA9PRBT.js
new file mode 100644
index 00000000..a57d3d6b
--- /dev/null
+++ b/previews/PR126/assets/manual_matrices.md.CJA9PRBT.js
@@ -0,0 +1,119 @@
+import{_ as r,c as l,a5 as n,j as a,a as i,G as t,B as o,o as p}from"./chunks/framework.BSoZtefh.js";const L=JSON.parse('{"title":"Sparse matrices","description":"","frontmatter":{},"headers":[],"relativePath":"manual/matrices.md","filePath":"manual/matrices.md","lastUpdated":null}'),h={name:"manual/matrices.md"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.986ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2204 1000","aria-hidden":"true"},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},g={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.801ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2122 1000","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.53ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3328.4 1000","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.17ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 4495.2 1000","aria-hidden":"true"},b={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},F={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""};function S(B,s,A,D,I,M){const e=o("Badge");return p(),l("div",null,[s[53]||(s[53]=n(`Sparse matrices In IncompressibleNavierStokes, all operators are implemented as matrix-free kernels. However, access to the underlying matrices can sometimes be useful, for example to precompute matrix factorizations. We therefore provide sparse matrix versions of some of the linear operators (see full list below ).
Example Consider a simple setup
julia using IncompressibleNavierStokes ax = range ( 0 , 1 , 17 ); setup = Setup (; x = (ax, ax), Re = 1e3 ); (grid = (xlims = ((0.0, 1.0), (0.0, 1.0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (18, 18), Nu = ((16, 16), (16, 16)), Np = (16, 16), Iu = (CartesianIndices((2:17, 2:17)), CartesianIndices((2:17, 2:17))), Ip = CartesianIndices((2:17, 2:17)), x = ([-0.0625, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625], [-0.0625, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625]), xu = (([0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625], [-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125]), ([-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125], [0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625])), xp = ([-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125], [-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125]), Δ = ([0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625], [0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625]), Δu = ([0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125], [0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125]), A = ((([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])), (([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((PeriodicBC(), PeriodicBC()), (PeriodicBC(), PeriodicBC())), Re = 1000.0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = nothing) The matrices for the linear operators are named by appending _mat to the function name, for example: divergence, pressuregradient, and diffusion become divergence_mat, pressuregradient_mat, diffusion_mat etc.
Let's assemble some matrices:
julia divergence_mat (setup) 324×648 SparseArrays.SparseMatrixCSC{Float64, Int64} with 1024 stored entries: ⎡⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠀⠈⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⣄⠀⠀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⎦ julia pressuregradient_mat (setup) 648×324 SparseArrays.SparseMatrixCSC{Float64, Int64} with 1024 stored entries: ⎡⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠀⠀⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⎥ ⎢⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣦⡀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⢦⡀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⎦ julia diffusion_mat (setup) 648×648 SparseArrays.SparseMatrixCSC{Float64, Int64} with 2560 stored entries: ⎡⣄⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠈⠻⢷⢗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠈⠻⣿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠈⠛⣟⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣽⣤⡀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣦⡀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢵⢷⣦⡀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠙⎦ 648×648 SparseArrays.SparseMatrixCSC{Float64, Int64} with 648 stored entries: ⎡⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠀⠁⠱⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢆⢀⠀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⎦ We can verify that the diffusion matrix gives the same results as the diffusion kernel (without viscosity):
julia using Random u = randn! ( vectorfield (setup)) B = bc_u_mat (setup) D = diffusion_mat (setup) d_kernel = diffusion ( apply_bc_u (u, 0.0 , setup), setup; use_viscosity = false ) d_matrix = reshape (D * B * u[:], size (u)) maximum (abs, d_matrix - d_kernel) Boundary conditions and matrices Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:
julia setup = Setup (; x = (ax, ax), boundary_conditions = ( ( DirichletBC (( 10.0 , 0.0 )), PressureBC ()), ( DirichletBC (), DirichletBC ()), ), ) (grid = (xlims = ((0.0, 1.0), (0.0, 1.0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (18, 18), Nu = ((16, 16), (16, 15)), Np = (16, 16), Iu = (CartesianIndices((2:17, 2:17)), CartesianIndices((2:17, 2:16))), Ip = CartesianIndices((2:17, 2:17)), x = ([0.0, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0], [0.0, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0]), xu = (([0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0], [0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0]), ([0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0], [0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0])), xp = ([0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0], [0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0]), Δ = ([2.220446049250313e-16, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 2.220446049250313e-16], [2.220446049250313e-16, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 2.220446049250313e-16]), Δu = ([0.03125, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125, 2.220446049250313e-16], [0.03125, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125, 2.220446049250313e-16]), A = ((([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.0], [0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0, 1.0])), (([1.0, 1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.0], [0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Tuple{Float64, Float64}}((10.0, 0.0)), PressureBC()), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 1000.0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = nothing) We can assert that the kernel and matrix versions of the divergence give different results:
julia using Random u = randn! ( vectorfield (setup)) B = bc_u_mat (setup) M = divergence_mat (setup) div_kernel = divergence ( apply_bc_u (u, 0.0 , setup), setup) div_matrix = reshape (M * B * u[:], size (div_kernel)) maximum (abs, div_matrix - div_kernel) The solution is to create a vector containing the boundary conditions. This is done by applying the BC kernel on an empty field:
julia uzero = zero (u) yu = apply_bc_u (uzero, 0.0 , setup) yM = divergence (yu, setup) 18×18 Matrix{Float64}: 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 -160.0 -160.0 -160.0 -160.0 -160.0 -160.0 -160.0 -160.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 By adding yM, we get the equality:
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Example Consider a simple setup
julia using IncompressibleNavierStokes ax = range ( 0 , 1 , 17 ); setup = Setup (; x = (ax, ax), Re = 1e3 ); (grid = (xlims = ((0.0, 1.0), (0.0, 1.0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (18, 18), Nu = ((16, 16), (16, 16)), Np = (16, 16), Iu = (CartesianIndices((2:17, 2:17)), CartesianIndices((2:17, 2:17))), Ip = CartesianIndices((2:17, 2:17)), x = ([-0.0625, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625], [-0.0625, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625]), xu = (([0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625], [-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125]), ([-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125], [0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0625])), xp = ([-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125], [-0.03125, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.03125]), Δ = ([0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625], [0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625]), Δu = ([0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125], [0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125]), A = ((([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])), (([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((PeriodicBC(), PeriodicBC()), (PeriodicBC(), PeriodicBC())), Re = 1000.0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = nothing) The matrices for the linear operators are named by appending _mat to the function name, for example: divergence, pressuregradient, and diffusion become divergence_mat, pressuregradient_mat, diffusion_mat etc.
Let's assemble some matrices:
julia divergence_mat (setup) 324×648 SparseArrays.SparseMatrixCSC{Float64, Int64} with 1024 stored entries: ⎡⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠀⠈⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⣄⠀⠀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⎦ julia pressuregradient_mat (setup) 648×324 SparseArrays.SparseMatrixCSC{Float64, Int64} with 1024 stored entries: ⎡⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠀⠀⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⎥ ⎢⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣦⡀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⢦⡀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⎦ julia diffusion_mat (setup) 648×648 SparseArrays.SparseMatrixCSC{Float64, Int64} with 2560 stored entries: ⎡⣄⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠈⠻⢷⢗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠈⠻⣿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠈⠛⣟⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣽⣤⡀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣦⡀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢵⢷⣦⡀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠙⎦ 648×648 SparseArrays.SparseMatrixCSC{Float64, Int64} with 648 stored entries: ⎡⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤ ⎢⠀⠁⠱⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⎥ ⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢆⢀⠀⎥ ⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⎦ We can verify that the diffusion matrix gives the same results as the diffusion kernel (without viscosity):
julia using Random u = randn! ( vectorfield (setup)) B = bc_u_mat (setup) D = diffusion_mat (setup) d_kernel = diffusion ( apply_bc_u (u, 0.0 , setup), setup; use_viscosity = false ) d_matrix = reshape (D * B * u[:], size (u)) maximum (abs, d_matrix - d_kernel) Boundary conditions and matrices Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:
julia setup = Setup (; x = (ax, ax), boundary_conditions = ( ( DirichletBC (( 10.0 , 0.0 )), PressureBC ()), ( DirichletBC (), DirichletBC ()), ), ) (grid = (xlims = ((0.0, 1.0), (0.0, 1.0)), dimension = IncompressibleNavierStokes.Dimension{2}(), N = (18, 18), Nu = ((16, 16), (16, 15)), Np = (16, 16), Iu = (CartesianIndices((2:17, 2:17)), CartesianIndices((2:17, 2:16))), Ip = CartesianIndices((2:17, 2:17)), x = ([0.0, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0], [0.0, 0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0]), xu = (([0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0], [0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0]), ([0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0], [0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0, 1.0])), xp = ([0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0], [0.0, 0.03125, 0.09375, 0.15625, 0.21875, 0.28125, 0.34375, 0.40625, 0.46875, 0.53125, 0.59375, 0.65625, 0.71875, 0.78125, 0.84375, 0.90625, 0.96875, 1.0]), Δ = ([2.220446049250313e-16, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 2.220446049250313e-16], [2.220446049250313e-16, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 2.220446049250313e-16]), Δu = ([0.03125, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125, 2.220446049250313e-16], [0.03125, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.03125, 2.220446049250313e-16]), A = ((([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0]), ([1.0, 1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.0], [0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0, 1.0])), (([1.0, 1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.0], [0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0, 1.0]), ([1.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], [0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1.0])))), boundary_conditions = ((DirichletBC{Tuple{Float64, Float64}}((10.0, 0.0)), PressureBC()), (DirichletBC{Nothing}(nothing), DirichletBC{Nothing}(nothing))), Re = 1000.0, bodyforce = nothing, issteadybodyforce = false, closure_model = nothing, backend = CPU(false), workgroupsize = 64, temperature = nothing) We can assert that the kernel and matrix versions of the divergence give different results:
julia using Random u = randn! ( vectorfield (setup)) B = bc_u_mat (setup) M = divergence_mat (setup) div_kernel = divergence ( apply_bc_u (u, 0.0 , setup), setup) div_matrix = reshape (M * B * u[:], size (div_kernel)) maximum (abs, div_matrix - div_kernel) The solution is to create a vector containing the boundary conditions. This is done by applying the BC kernel on an empty field:
julia uzero = zero (u) yu = apply_bc_u (uzero, 0.0 , setup) yM = divergence (yu, setup) 18×18 Matrix{Float64}: 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 -160.0 -160.0 -160.0 -160.0 -160.0 -160.0 -160.0 -160.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 By adding yM, we get the equality:
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index 00000000..f3f50be2
--- /dev/null
+++ b/previews/PR126/assets/manual_precision.md.B1snHDv2.js
@@ -0,0 +1 @@
+import{_ as o,c as i,a5 as s,o as r}from"./chunks/framework.BSoZtefh.js";const f=JSON.parse('{"title":"Floating point precision","description":"","frontmatter":{},"headers":[],"relativePath":"manual/precision.md","filePath":"manual/precision.md","lastUpdated":null}'),a={name:"manual/precision.md"};function n(t,e,c,l,p,d){return r(),i("div",null,e[0]||(e[0]=[s('Floating point precision IncompressibleNavierStokes generates efficient code for different floating point precisions, such as
To use single or half precision, all user input floats should be converted to the desired type. Mixing different precisions causes unnecessary conversions and may break the code.
GPU precision
For GPUs, single precision is preferred. CUDA.jls cu converts to single precision.
Pressure solvers
SparseArrays.jlpsolver_directFloat64. Consider using an iterative solver such as psolver_cg
Floating point precision IncompressibleNavierStokes generates efficient code for different floating point precisions, such as
To use single or half precision, all user input floats should be converted to the desired type. Mixing different precisions causes unnecessary conversions and may break the code.
GPU precision
For GPUs, single precision is preferred. CUDA.jls cu converts to single precision.
Pressure solvers
SparseArrays.jlpsolver_directFloat64. Consider using an iterative solver such as psolver_cg
julia default_psolver ( setup ) -> Union{IncompressibleNavierStokes . var"#psolve!#101" {Bool}, IncompressibleNavierStokes . var"#psolve!#124" } Get default Poisson solver from setup.
source
`,3))]),e("details",u,[e("summary",null,[s[5]||(s[5]=e("a",{id:"IncompressibleNavierStokes.poisson!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson!")],-1)),s[6]||(s[6]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[7]||(s[7]=t('julia poisson! (psolver, f) -> Any Solve the Poisson equation for the pressure (in-place version).
source
',3))]),e("details",g,[e("summary",null,[s[8]||(s[8]=e("a",{id:"IncompressibleNavierStokes.poisson-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson")],-1)),s[9]||(s[9]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=t('julia poisson (psolver, f) -> Any Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.
Differentiable version.
source
',4))]),e("details",b,[e("summary",null,[s[11]||(s[11]=e("a",{id:"IncompressibleNavierStokes.pressure!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.pressure!-NTuple{5, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure!")],-1)),s[12]||(s[12]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=t('julia pressure! (p, u, temp, t, setup; psolver, F) Compute pressure from velocity field (in-place version).
source
',3))]),e("details",Q,[e("summary",null,[s[14]||(s[14]=e("a",{id:"IncompressibleNavierStokes.pressure-NTuple{4, Any}",href:"#IncompressibleNavierStokes.pressure-NTuple{4, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure")],-1)),s[15]||(s[15]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[16]||(s[16]=t('julia pressure (u, temp, t, setup; psolver) Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.
Differentiable version.
source
',4))]),e("details",m,[e("summary",null,[s[17]||(s[17]=e("a",{id:"IncompressibleNavierStokes.project!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project!")],-1)),s[18]||(s[18]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[19]||(s[19]=t('julia project! (u, setup; psolver, p) Project velocity field onto divergence-free space (in-place version).
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',3))]),e("details",T,[e("summary",null,[s[20]||(s[20]=e("a",{id:"IncompressibleNavierStokes.project-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project")],-1)),s[21]||(s[21]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[22]||(s[22]=t('julia project (u, setup; psolver) Project velocity field onto divergence-free space (differentiable version).
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',3))]),e("details",y,[e("summary",null,[s[23]||(s[23]=e("a",{id:"IncompressibleNavierStokes.psolver_cg-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg")],-1)),s[24]||(s[24]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[25]||(s[25]=t(`julia psolver_cg ( setup; abstol, reltol, maxiter, preconditioner ) -> IncompressibleNavierStokes . var"#psolve!#111" {_A, _B, _C, IncompressibleNavierStokes . var"#laplace_diag#109" {ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize} Conjugate gradients iterative Poisson solver.
source
`,3))]),e("details",v,[e("summary",null,[s[26]||(s[26]=e("a",{id:"IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg_matrix")],-1)),s[27]||(s[27]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[28]||(s[28]=t(`julia psolver_cg_matrix ( setup; kwargs ... ) -> IncompressibleNavierStokes . var"#psolve!#105" {Base . Pairs{Symbol, Union{}, Tuple{}, @NamedTuple {}}} Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.
source
`,3))]),e("details",E,[e("summary",null,[s[29]||(s[29]=e("a",{id:"IncompressibleNavierStokes.psolver_direct-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_direct-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_direct")],-1)),s[30]||(s[30]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[31]||(s[31]=t(`julia psolver_direct ( setup ) -> IncompressibleNavierStokes . var"#psolve!#101" {Bool} Create direct Poisson solver using an appropriate matrix decomposition.
source
`,3))]),e("details",j,[e("summary",null,[s[32]||(s[32]=e("a",{id:"IncompressibleNavierStokes.psolver_spectral-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_spectral-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_spectral")],-1)),s[33]||(s[33]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=t(`julia psolver_spectral ( setup ) -> IncompressibleNavierStokes . var"#psolve!#124" Create spectral Poisson solver from setup.
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`,3))])])}const H=r(d,[["render",f]]);export{I as __pageData,H as default};
diff --git a/previews/PR126/assets/manual_pressure.md.Bb8fJUK2.lean.js b/previews/PR126/assets/manual_pressure.md.Bb8fJUK2.lean.js
new file mode 100644
index 00000000..3060ea05
--- /dev/null
+++ b/previews/PR126/assets/manual_pressure.md.Bb8fJUK2.lean.js
@@ -0,0 +1,16 @@
+import{_ as r,c as n,j as e,a as i,a5 as t,G as l,B as o,o as p}from"./chunks/framework.BSoZtefh.js";const I=JSON.parse('{"title":"Pressure solvers","description":"","frontmatter":{},"headers":[],"relativePath":"manual/pressure.md","filePath":"manual/pressure.md","lastUpdated":null}'),d={name:"manual/pressure.md"},h={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.194ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 6715.6 1000","aria-hidden":"true"},c={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""};function f(C,s,N,F,S,x){const a=o("Badge");return p(),n("div",null,[s[35]||(s[35]=e("h1",{id:"Pressure-solvers",tabindex:"-1"},[i("Pressure solvers "),e("a",{class:"header-anchor",href:"#Pressure-solvers","aria-label":'Permalink to "Pressure solvers {#Pressure-solvers}"'},"")],-1)),s[36]||(s[36]=e("p",null,"The discrete pressure Poisson equation",-1)),e("mjx-container",h,[(p(),n("svg",k,s[0]||(s[0]=[t('julia default_psolver ( setup ) -> Union{IncompressibleNavierStokes . var"#psolve!#101" {Bool}, IncompressibleNavierStokes . var"#psolve!#124" } Get default Poisson solver from setup.
source
`,3))]),e("details",u,[e("summary",null,[s[5]||(s[5]=e("a",{id:"IncompressibleNavierStokes.poisson!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson!")],-1)),s[6]||(s[6]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[7]||(s[7]=t('julia poisson! (psolver, f) -> Any Solve the Poisson equation for the pressure (in-place version).
source
',3))]),e("details",g,[e("summary",null,[s[8]||(s[8]=e("a",{id:"IncompressibleNavierStokes.poisson-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson")],-1)),s[9]||(s[9]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=t('julia poisson (psolver, f) -> Any Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.
Differentiable version.
source
',4))]),e("details",b,[e("summary",null,[s[11]||(s[11]=e("a",{id:"IncompressibleNavierStokes.pressure!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.pressure!-NTuple{5, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure!")],-1)),s[12]||(s[12]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=t('julia pressure! (p, u, temp, t, setup; psolver, F) Compute pressure from velocity field (in-place version).
source
',3))]),e("details",Q,[e("summary",null,[s[14]||(s[14]=e("a",{id:"IncompressibleNavierStokes.pressure-NTuple{4, Any}",href:"#IncompressibleNavierStokes.pressure-NTuple{4, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure")],-1)),s[15]||(s[15]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[16]||(s[16]=t('julia pressure (u, temp, t, setup; psolver) Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.
Differentiable version.
source
',4))]),e("details",m,[e("summary",null,[s[17]||(s[17]=e("a",{id:"IncompressibleNavierStokes.project!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project!")],-1)),s[18]||(s[18]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[19]||(s[19]=t('julia project! (u, setup; psolver, p) Project velocity field onto divergence-free space (in-place version).
source
',3))]),e("details",T,[e("summary",null,[s[20]||(s[20]=e("a",{id:"IncompressibleNavierStokes.project-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project")],-1)),s[21]||(s[21]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[22]||(s[22]=t('julia project (u, setup; psolver) Project velocity field onto divergence-free space (differentiable version).
source
',3))]),e("details",y,[e("summary",null,[s[23]||(s[23]=e("a",{id:"IncompressibleNavierStokes.psolver_cg-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg")],-1)),s[24]||(s[24]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[25]||(s[25]=t(`julia psolver_cg ( setup; abstol, reltol, maxiter, preconditioner ) -> IncompressibleNavierStokes . var"#psolve!#111" {_A, _B, _C, IncompressibleNavierStokes . var"#laplace_diag#109" {ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize} Conjugate gradients iterative Poisson solver.
source
`,3))]),e("details",v,[e("summary",null,[s[26]||(s[26]=e("a",{id:"IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg_matrix")],-1)),s[27]||(s[27]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[28]||(s[28]=t(`julia psolver_cg_matrix ( setup; kwargs ... ) -> IncompressibleNavierStokes . var"#psolve!#105" {Base . Pairs{Symbol, Union{}, Tuple{}, @NamedTuple {}}} Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.
source
`,3))]),e("details",E,[e("summary",null,[s[29]||(s[29]=e("a",{id:"IncompressibleNavierStokes.psolver_direct-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_direct-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_direct")],-1)),s[30]||(s[30]=i()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[31]||(s[31]=t(`julia psolver_direct ( setup ) -> IncompressibleNavierStokes . var"#psolve!#101" {Bool} Create direct Poisson solver using an appropriate matrix decomposition.
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`,3))])])}const H=r(d,[["render",f]]);export{I as __pageData,H as default};
diff --git a/previews/PR126/assets/manual_sciml.md.CogLuKgu.js b/previews/PR126/assets/manual_sciml.md.CogLuKgu.js
new file mode 100644
index 00000000..b8c2a8b0
--- /dev/null
+++ b/previews/PR126/assets/manual_sciml.md.CogLuKgu.js
@@ -0,0 +1,61 @@
+import{_ as T,c as i,j as s,a as t,a5 as n,G as Q,B as p,o as e}from"./chunks/framework.BSoZtefh.js";const L=JSON.parse('{"title":"Using IncompressibleNavierStokes in SciML","description":"","frontmatter":{},"headers":[],"relativePath":"manual/sciml.md","filePath":"manual/sciml.md","lastUpdated":null}'),r={name:"manual/sciml.md"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},o={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.577ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.605ex",height:"4.676ex",role:"img",focusable:"false",viewBox:"0 -1370 5571.2 2067","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-3.243ex"},xmlns:"http://www.w3.org/2000/svg",width:"52.497ex",height:"7.618ex",role:"img",focusable:"false",viewBox:"0 -1933.5 23203.8 3367","aria-hidden":"true"},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function c(u,a,y,E,H,f){const l=p("Badge");return e(),i("div",null,[a[14]||(a[14]=s("h1",{id:"Using-IncompressibleNavierStokes-in-SciML",tabindex:"-1"},[t("Using IncompressibleNavierStokes in SciML "),s("a",{class:"header-anchor",href:"#Using-IncompressibleNavierStokes-in-SciML","aria-label":'Permalink to "Using IncompressibleNavierStokes in SciML {#Using-IncompressibleNavierStokes-in-SciML}"'},"")],-1)),a[15]||(a[15]=s("p",null,[t("The "),s("a",{href:"https://sciml.ai/",target:"_blank",rel:"noreferrer"},"SciML organization"),t(" is a collection of tools for solving equations and modeling systems. It has a coherent development principle, unified APIs over large collections of equation solvers, pervasive differentiability and sensitivity analysis, and features many of the highest performance and parallel implementations one can find.")],-1)),s("p",null,[a[2]||(a[2]=t("In particular, ")),a[3]||(a[3]=s("a",{href:"https://docs.sciml.ai/DiffEqDocs/stable/",target:"_blank",rel:"noreferrer"},"DifferentialEquations.jl",-1)),a[4]||(a[4]=t(" contains tools to solve differential equations defined as ")),s("mjx-container",d,[(e(),i("svg",o,a[0]||(a[0]=[n('The derivation and the drawbacks of this approach are discussed in the documentation.
This projected right-hand side can be used in the SciML solvers to solve the Navier-Stokes equations. The following example shows how to use the SciML solvers to solve the ODEs obtained from the Navier-Stokes equations.
OrdinaryDiffEqTsit5
We here use the explicit solver Tsit5 from OrdinaryDiffEqTsit5 directly to skip loading all the toolchains for implicit solvers, which takes a while to install on GitHub.
julia using OrdinaryDiffEqTsit5 using IncompressibleNavierStokes ax = range ( 0 , 1 , 101 ) setup = Setup (; x = (ax, ax), Re = 500.0 ) psolver = default_psolver (setup) f = create_right_hand_side (setup, psolver) u0 = random_field (setup) tspan = ( 0.0 , 1.0 ) # time span where to solve. problem = ODEProblem (f, u0, tspan) #SciMLBase.ODEProblem sol = solve (problem, Tsit5 (), reltol = 1e-8 , abstol = 1e-8 ) # sol: SciMLBase.ODESolution retcode: Success Interpolation: specialized 4th order "free" interpolation t: 205-element Vector{Float64}: 0.0 0.0016532604327842326 0.0024146677536511806 0.00363820027112176 0.0046728379665379825 0.005865854727739244 0.00703229270934183 0.008261979706355872 0.00950289172478397 0.010776693536653715 ⋮ 0.8941795613844775 0.9078054755801236 0.9216552716704063 0.9357343583321837 0.9500492205436287 0.9646067031383294 0.9794153561555403 0.9944855551823507 1.0 u: 205-element Vector{Array{Float64, 3}}: [-1.1497959004240497 -1.103859466061489 … -1.1497959004240497 -1.103859466061489; 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-2.0213960714842503 -1.4528396960719592 … -2.0213960714842503 -1.4528396960719592] Alternatively, it is also possible to use an in-place formulation
julia f! (du,u,p,t) = right_hand_side! (du, u, Ref ([setup, psolver]), t) u = similar (u0) du = similar (u0) p = nothing t = 0.0 f! (du,u,p,t) that is usually faster than the out-of-place formulation.
You can look here for more information on how to use the SciML solvers and all the options available.
API `,10)),s("details",g,[s("summary",null,[a[8]||(a[8]=s("a",{id:"IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_right_hand_side")],-1)),a[9]||(a[9]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),a[10]||(a[10]=n(`julia create_right_hand_side ( setup, psolver ) -> IncompressibleNavierStokes . var"#right_hand_side#343" create_right_hand_side(setup, psolver) Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.
Arguments
Returns
A function that computes the right-hand side of the Navier-Stokes equations.
source
`,8))]),s("details",k,[s("summary",null,[a[11]||(a[11]=s("a",{id:"IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.right_hand_side!")],-1)),a[12]||(a[12]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),a[13]||(a[13]=n('julia right_hand_side! (dudt, u, params_ref, t) right_hand_side!(dudt, u, params_ref, t) Computes the right-hand side of the Navier-Stokes equations in-place.
Arguments
dudt: The array to store the computed right-hand side.
u: The current velocity field.
params_ref: A reference to the parameters containing the setup and pressure solver.
t: The current time.
Returns
Nothing. The result is stored in dudt.
source
',8))])])}const v=T(r,[["render",c]]);export{L as __pageData,v as default};
diff --git a/previews/PR126/assets/manual_sciml.md.CogLuKgu.lean.js b/previews/PR126/assets/manual_sciml.md.CogLuKgu.lean.js
new file mode 100644
index 00000000..b8c2a8b0
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+++ b/previews/PR126/assets/manual_sciml.md.CogLuKgu.lean.js
@@ -0,0 +1,61 @@
+import{_ as T,c as i,j as s,a as t,a5 as n,G as Q,B as p,o as e}from"./chunks/framework.BSoZtefh.js";const L=JSON.parse('{"title":"Using IncompressibleNavierStokes in SciML","description":"","frontmatter":{},"headers":[],"relativePath":"manual/sciml.md","filePath":"manual/sciml.md","lastUpdated":null}'),r={name:"manual/sciml.md"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},o={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.577ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.605ex",height:"4.676ex",role:"img",focusable:"false",viewBox:"0 -1370 5571.2 2067","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-3.243ex"},xmlns:"http://www.w3.org/2000/svg",width:"52.497ex",height:"7.618ex",role:"img",focusable:"false",viewBox:"0 -1933.5 23203.8 3367","aria-hidden":"true"},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function c(u,a,y,E,H,f){const l=p("Badge");return e(),i("div",null,[a[14]||(a[14]=s("h1",{id:"Using-IncompressibleNavierStokes-in-SciML",tabindex:"-1"},[t("Using IncompressibleNavierStokes in SciML "),s("a",{class:"header-anchor",href:"#Using-IncompressibleNavierStokes-in-SciML","aria-label":'Permalink to "Using IncompressibleNavierStokes in SciML {#Using-IncompressibleNavierStokes-in-SciML}"'},"")],-1)),a[15]||(a[15]=s("p",null,[t("The "),s("a",{href:"https://sciml.ai/",target:"_blank",rel:"noreferrer"},"SciML organization"),t(" is a collection of tools for solving equations and modeling systems. It has a coherent development principle, unified APIs over large collections of equation solvers, pervasive differentiability and sensitivity analysis, and features many of the highest performance and parallel implementations one can find.")],-1)),s("p",null,[a[2]||(a[2]=t("In particular, ")),a[3]||(a[3]=s("a",{href:"https://docs.sciml.ai/DiffEqDocs/stable/",target:"_blank",rel:"noreferrer"},"DifferentialEquations.jl",-1)),a[4]||(a[4]=t(" contains tools to solve differential equations defined as ")),s("mjx-container",d,[(e(),i("svg",o,a[0]||(a[0]=[n('The derivation and the drawbacks of this approach are discussed in the documentation.
This projected right-hand side can be used in the SciML solvers to solve the Navier-Stokes equations. The following example shows how to use the SciML solvers to solve the ODEs obtained from the Navier-Stokes equations.
OrdinaryDiffEqTsit5
We here use the explicit solver Tsit5 from OrdinaryDiffEqTsit5 directly to skip loading all the toolchains for implicit solvers, which takes a while to install on GitHub.
julia using OrdinaryDiffEqTsit5 using IncompressibleNavierStokes ax = range ( 0 , 1 , 101 ) setup = Setup (; x = (ax, ax), Re = 500.0 ) psolver = default_psolver (setup) f = create_right_hand_side (setup, psolver) u0 = random_field (setup) tspan = ( 0.0 , 1.0 ) # time span where to solve. problem = ODEProblem (f, u0, tspan) #SciMLBase.ODEProblem sol = solve (problem, Tsit5 (), reltol = 1e-8 , abstol = 1e-8 ) # sol: SciMLBase.ODESolution retcode: Success Interpolation: specialized 4th order "free" interpolation t: 205-element Vector{Float64}: 0.0 0.0016532604327842326 0.0024146677536511806 0.00363820027112176 0.0046728379665379825 0.005865854727739244 0.00703229270934183 0.008261979706355872 0.00950289172478397 0.010776693536653715 ⋮ 0.8941795613844775 0.9078054755801236 0.9216552716704063 0.9357343583321837 0.9500492205436287 0.9646067031383294 0.9794153561555403 0.9944855551823507 1.0 u: 205-element Vector{Array{Float64, 3}}: [-1.1497959004240497 -1.103859466061489 … -1.1497959004240497 -1.103859466061489; -1.1115143411258799 -0.9429561713756396 … -1.1115143411258799 -0.9429561713756396; … ; -1.1497959004240497 -1.103859466061489 … -1.1497959004240497 -1.103859466061489; -1.1115143411258799 -0.9429561713756396 … -1.1115143411258799 -0.9429561713756396;;; 0.5780115453257968 0.5469734191664433 … 0.5780115453257968 0.5469734191664433; 0.38274219346754745 0.22183889878169927 … 0.38274219346754745 0.22183889878169927; … ; 0.5780115453257968 0.5469734191664433 … 0.5780115453257968 0.5469734191664433; 0.38274219346754745 0.22183889878169927 … 0.38274219346754745 0.22183889878169927] [-1.1346613606845686 -1.0855072360047884 … -1.1346613606845686 -1.0855072360047884; -1.085037198186501 -0.9549916902794789 … -1.1141909765562603 -0.9096660650644737; … ; -1.1346613606845686 -1.1166526851389142 … -1.1526234274797011 -1.0855072360047884; -1.085037198186501 -0.9096660650644737 … -1.085037198186501 -0.9096660650644737;;; 0.5348228779300479 0.49420120289039515 … 0.5348228779300479 0.49420120289039515; 0.36016046276699126 0.22339559857696653 … 0.38505659343640064 0.18790302303797013; … ; 0.5348228779300479 0.5391582593084667 … 0.5729023909384506 0.49420120289039515; 0.36016046276699126 0.18790302303797013 … 0.36016046276699126 0.18790302303797013] [-1.1275112161577667 -1.0766474494567906 … -1.1275112161577667 -1.0766474494567906; -1.0727913835432734 -0.9600781739950909 … -1.1150543083213575 -0.89417212831079; … ; -1.1275112161577667 -1.1220668609155946 … -1.1535842546753374 -1.0766474494567906; -1.0727913835432734 -0.89417212831079 … -1.0727913835432734 -0.89417212831079;;; 0.5153486288796626 0.47013625606217296 … 0.5153486288796626 0.47013625606217296; 0.35015697415699687 0.22415383919864973 … 0.3861425261191531 0.17253189812249553; … ; 0.5153486288796626 0.5356791510589715 … 0.5706805537830523 0.47013625606217296; 0.35015697415699687 0.17253189812249553 … 0.35015697415699687 0.17253189812249553] [-1.115791210346112 -1.0618913129242453 … -1.115791210346112 -1.0618913129242453; -1.0530471523275629 -0.9676671701020543 … -1.1159698680800068 -0.8690782171432961; … ; -1.115791210346112 -1.1301641192252425 … -1.1547077134986383 -1.0618913129242453; -1.0530471523275629 -0.8690782171432961 … -1.0530471523275629 -0.8690782171432961;;; 0.48460061794136766 0.4318044995873348 … 0.48460061794136766 0.4318044995873348; 0.3346205322468639 0.22543475374003244 … 0.3879317028632203 0.14820437517173785; … ; 0.48460061794136766 0.5302324605858955 … 0.5672680917061951 0.4318044995873348; 0.3346205322468639 0.14820437517173785 … 0.3346205322468639 0.14820437517173785] [-1.1056623764044726 -1.0489286853670308 … -1.1056623764044726 -1.0489286853670308; -1.0362876016533211 -0.973536845827031 … -1.116302725354632 -0.8476828485928333; … ; -1.1056623764044726 -1.13645399605545 … -1.15527603203835 -1.0489286853670308; -1.0362876016533211 -0.8476828485928333 … -1.0362876016533211 -0.8476828485928333;;; 0.45912379905057876 0.3997350530360164 … 0.45912379905057876 0.3997350530360164; 0.32201394196248373 0.22658484203123946 … 0.38950199225965854 0.1280170517465823; … ; 0.45912379905057876 0.5257537338406175 … 0.5645235044022956 0.3997350530360164; 0.32201394196248373 0.1280170517465823 … 0.32201394196248373 0.1280170517465823] [-1.0937360551737667 -1.033448432276319 … -1.0937360551737667 -1.033448432276319; -1.016888150998382 -0.9797010100588621 … -1.1162000839959745 -0.8228280508550142; … ; -1.0937360551737667 -1.1431022705819454 … -1.155525040969178 -1.033448432276319; -1.016888150998382 -0.8228280508550142 … -1.016888150998382 -0.8228280508550142;;; 0.4303448034117328 0.3631723200120001 … 0.4303448034117328 0.3631723200120001; 0.30810063100471446 0.22799658044988205 … 0.3913978409729652 0.10521028774297245; … ; 0.4303448034117328 0.5207197729247011 … 0.5615068658901985 0.3631723200120001; 0.30810063100471446 0.10521028774297245 … 0.30810063100471446 0.10521028774297245] [-1.0818179126332947 -1.017777477021206 … -1.0818179126332947 -1.017777477021206; -0.9978395965840717 -0.985121109614881 … -1.1156110131887644 -0.7983499821356742; … ; -1.0818179126332947 -1.1490058520154316 … -1.155373929433073 -1.017777477021206; -0.9978395965840717 -0.7983499821356742 … -0.9978395965840717 -0.7983499821356742;;; 0.4028305926913382 0.32787966255870815 … 0.4028305926913382 0.32787966255870815; 0.29516328161840344 0.22947578757224885 … 0.39336052997279874 0.08343345266475344; … ; 0.4028305926913382 0.5159196315853526 … 0.5587001992401808 0.32787966255870815; 0.29516328161840344 0.08343345266475344 … 0.29516328161840344 0.08343345266475344] [-1.0689728566019476 -1.0007005310637664 … -1.0689728566019476 -1.0007005310637664; -0.9776642956614491 -0.9902057985315252 … -1.1144827384890157 -0.7723686929371345; … ; -1.0689728566019476 -1.1546228014296884 … -1.1548192169142335 -1.0007005310637664; -0.9776642956614491 -0.7723686929371345 … -0.9776642956614491 -0.7723686929371345;;; 0.37450087975122837 0.2911900290948907 … 0.37450087975122837 0.2911900290948907; 0.2822627011814225 0.2311534258019005 … 0.3955704287000623 0.06107350259837589; … ; 0.37450087975122837 0.5109760351214467 … 0.5558845173235043 0.2911900290948907; 0.2822627011814225 0.06107350259837589 … 0.2822627011814225 0.06107350259837589] [-1.0557104502212835 -0.98290497413801 … -1.0557104502212835 -0.98290497413801; -0.9571981772852441 -0.9947020543270665 … -1.112831934231493 -0.7459792626280486; … ; -1.0557104502212835 -1.159691496764516 … -1.1538735439479935 -0.98290497413801; -0.9571981772852441 -0.7459792626280486 … -0.9571981772852441 -0.7459792626280486;;; 0.3466294362329091 0.25473497640670373 … 0.3466294362329091 0.25473497640670373; 0.2700409002155431 0.23298308009129398 … 0.39797252252874177 0.03917271924219536; … ; 0.3466294362329091 0.5060968589005629 … 0.5531844874459036 0.25473497640670373; 0.2700409002155431 0.03917271924219536 … 0.2700409002155431 0.03917271924219536] [-1.0417716988708505 -0.9640648283520794 … -1.0417716988708505 -0.9640648283520794; -0.9360670368222949 -0.9986737540219488 … -1.1106175142395858 -0.7187223936406196; … ; -1.0417716988708505 -1.164299793748669 … -1.1525237629964271 -0.9640648283520794; -0.9360670368222949 -0.7187223936406196 … -0.9360670368222949 -0.7187223936406196;;; 0.31878633596239114 0.21794448332547656 … 0.31878633596239114 0.21794448332547656; 0.2583595321941497 0.23501950287987414 … 0.4006455426065916 0.01742911897904579; … ; 0.31878633596239114 0.5011925616695612 … 0.5505552476458456 0.21794448332547656; 0.2583595321941497 0.01742911897904579 … 0.2583595321941497 0.01742911897904579] ⋮ [-0.3358150426428933 0.08514883033891986 … -0.3358150426428933 0.08514883033891986; -0.6822163312167452 -0.2538408443323854 … -0.23209785262194438 -0.14429868098101484; … ; -0.3358150426428933 -0.22512241076345169 … -0.20162486397754953 0.08514883033891986; -0.6822163312167452 -0.14429868098101484 … -0.6822163312167452 -0.14429868098101484;;; -2.413252043540796 -1.712717842276099 … -2.413252043540796 -1.712717842276099; -1.9714430928489248 -0.03294202753333163 … -0.06166046110225782 -1.3811337699727226; … ; -2.413252043540796 -0.028377068257785687 … -0.059007992026413034 -1.712717842276099; -1.9714430928489248 -1.3811337699727226 … -1.9714430928489248 -1.3811337699727226] [-0.3443703509054663 0.07804405812322168 … -0.3443703509054663 0.07804405812322168; -0.6900799622769885 -0.253560889619927 … -0.2335135045673072 -0.15045383338219104; … ; -0.3443703509054663 -0.22697545368575067 … -0.2050262150692242 0.07804405812322168; -0.6900799622769885 -0.15045383338219104 … -0.6900799622769885 -0.15045383338219104;;; -2.4127188120242553 -1.714952104037445 … -2.4127188120242553 -1.714952104037445; -1.973869748314881 -0.029852159852162042 … -0.05643759578633058 -1.38694462894733; … ; -2.4127188120242553 -0.02189198831650714 … -0.05092301185049 -1.714952104037445; -1.973869748314881 -1.38694462894733 … -1.973869748314881 -1.38694462894733] [-0.3521208550434111 0.07166696164953267 … -0.3521208550434111 0.07166696164953267; -0.6970256705507138 -0.2526021649984651 … -0.2341365128310068 -0.15581321954177493; … ; -0.3521208550434111 -0.22814552544624836 … -0.20770415442477874 0.07166696164953267; -0.6970256705507138 -0.15581321954177493 … -0.6970256705507138 -0.15581321954177493;;; -2.413653040525908 -1.7186928211396268 … -2.413653040525908 -1.7186928211396268; -1.9779073862294974 -0.028053785497743476 … -0.05251042504995188 -1.3942856234021603; … ; -2.413653040525908 -0.01669002567933338 … -0.044026891631948575 -1.7186928211396268; -1.9779073862294974 -1.3942856234021603 … -1.9779073862294974 -1.3942856234021603] [-0.3590863750002272 0.06598805856519219 … -0.3590863750002272 0.06598805856519219; -0.7031107550673309 -0.2510409346609563 … -0.23402738021341402 -0.16043991156025073; … ; -0.3590863750002272 -0.22867111893081069 … -0.20967527421369125 0.06598805856519219; -0.7031107550673309 -0.16043991156025073 … -0.7031107550673309 -0.16043991156025073;;; -2.416024534868256 -1.723872489579331 … -2.416024534868256 -1.723872489579331; -1.9834413457601845 -0.027429681496909608 … -0.04979949722704688 -1.403004376360536; … ; -2.416024534868256 -0.012727978639355316 … -0.03831240805257915 -1.723872489579331; -1.9834413457601845 -1.403004376360536 … -1.9834413457601845 -1.403004376360536] [-0.3652980205085376 0.06096950392486076 … -0.3652980205085376 0.06096950392486076; -0.7084032158463106 -0.24895698959505633 … -0.2332565481814833 -0.1644037721629653; … ; -0.3652980205085376 -0.22859911461839777 … -0.2109689121025174 0.06096950392486076; -0.7084032158463106 -0.1644037721629653 … -0.7084032158463106 -0.1644037721629653;;; -2.4197773569414047 -1.7304006749449456 … -2.4197773569414047 -1.7304006749449456; -1.9903347484325364 -0.02784369088700459 … -0.04820156586365496 -1.4129330066989474; … ; -2.4197773569414047 -0.009938183248749033 … -0.03374644742324412 -1.7304006749449456; -1.9903347484325364 -1.4129330066989474 … -1.9903347484325364 -1.4129330066989474] [-0.37079634137002254 0.056566824007777634 … -0.37079634137002254 0.056566824007777634; -0.712978650101423 -0.24643040015093676 … -0.2319006320762357 -0.16777870978576348; … ; -0.37079634137002254 -0.22798219939833356 … -0.2116246634276885 0.056566824007777634; -0.712978650101423 -0.16777870978576348 … -0.712978650101423 -0.16777870978576348;;; -2.4248325802875517 -1.738167507524028 … -2.4248325802875517 -1.738167507524028; -1.9984340450617921 -0.029146265694757346 … -0.047594466447352254 -1.4238938738403897; … ; -2.4248325802875517 -0.0082317942377285 … -0.0302723003246664 -1.738167507524028; -1.9984340450617921 -1.4238938738403897 … -1.9984340450617921 -1.4238938738403897] [-0.3756301420976148 0.05273005356731644 … -0.3756301420976148 0.05273005356731644; -0.7169178171593397 -0.24353842434524517 … -0.2300387249620156 -0.1706405509962637; … ; -0.3756301420976148 -0.22687634313714175 … -0.21168989553081277 0.05273005356731644; -0.7169178171593397 -0.1706405509962637 … -0.7169178171593397 -0.1706405509962637;;; -2.431092300940214 -1.7470485803590445 … -2.431092300940214 -1.7470485803590445; -2.0075756948585846 -0.03118012716031498 … -0.04784220836840984 -1.4357065008740522; … ; -2.431092300940214 -0.007502321042711482 … -0.02781217117877902 -1.7470485803590445; -2.0075756948585846 -1.4357065008740522 … -2.0075756948585846 -1.4357065008740522] [-0.37985447595772653 0.04940548238785496 … -0.37985447595772653 0.04940548238785496; -0.7203036773419135 -0.24035309346769054 … -0.22774905685784178 -0.17306463080541906; … ; -0.37985447595772653 -0.22533838987028504 … -0.21121702489250443 0.04940548238785496; -0.7203036773419135 -0.17306463080541906 … -0.7203036773419135 -0.17306463080541906;;; -2.4384439931530535 -1.756909669413564 … -2.4384439931530535 -1.756909669413564; -2.0175922027968425 -0.03378575200165975 … -0.04880045559905625 -1.4481932237374657; … ; -2.4384439931530535 -0.007629985783351074 … -0.02627092263190513 -1.756909669413564; -2.0175922027968425 -1.4481932237374657 … -2.0175922027968425 -1.4481932237374657] [-0.38124295443269 0.048318947816810065 … -0.38124295443269 0.048318947816810065; -0.7214075191431352 -0.23914227277597178 … -0.22683070520154086 -0.17384740055570305; … ; -0.38124295443269 -0.22468695331191318 … -0.210924277835274 0.048318947816810065; -0.7214075191431352 -0.17384740055570305 … -0.7214075191431352 -0.17384740055570305;;; -2.4413467449387483 -1.7606885966448282 … -2.4413467449387483 -1.7606885966448282; -2.0213960714842503 -0.034837801211528435 … -0.04929312067557789 -1.4528396960719592; … ; -2.4413467449387483 -0.007862325088914351 … -0.025919216403936674 -1.7606885966448282; -2.0213960714842503 -1.4528396960719592 … -2.0213960714842503 -1.4528396960719592] Alternatively, it is also possible to use an in-place formulation
julia f! (du,u,p,t) = right_hand_side! (du, u, Ref ([setup, psolver]), t) u = similar (u0) du = similar (u0) p = nothing t = 0.0 f! (du,u,p,t) that is usually faster than the out-of-place formulation.
You can look here for more information on how to use the SciML solvers and all the options available.
API `,10)),s("details",g,[s("summary",null,[a[8]||(a[8]=s("a",{id:"IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_right_hand_side")],-1)),a[9]||(a[9]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),a[10]||(a[10]=n(`julia create_right_hand_side ( setup, psolver ) -> IncompressibleNavierStokes . var"#right_hand_side#343" create_right_hand_side(setup, psolver) Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.
Arguments
Returns
A function that computes the right-hand side of the Navier-Stokes equations.
source
`,8))]),s("details",k,[s("summary",null,[a[11]||(a[11]=s("a",{id:"IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.right_hand_side!")],-1)),a[12]||(a[12]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),a[13]||(a[13]=n('julia right_hand_side! (dudt, u, params_ref, t) right_hand_side!(dudt, u, params_ref, t) Computes the right-hand side of the Navier-Stokes equations in-place.
Arguments
dudt: The array to store the computed right-hand side.
u: The current velocity field.
params_ref: A reference to the parameters containing the setup and pressure solver.
t: The current time.
Returns
Nothing. The result is stored in dudt.
source
',8))])])}const v=T(r,[["render",c]]);export{L as __pageData,v as default};
diff --git a/previews/PR126/assets/manual_setup.md.BQcEIomz.js b/previews/PR126/assets/manual_setup.md.BQcEIomz.js
new file mode 100644
index 00000000..35b8d916
--- /dev/null
+++ b/previews/PR126/assets/manual_setup.md.BQcEIomz.js
@@ -0,0 +1,31 @@
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new file mode 100644
index 00000000..35b8d916
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new file mode 100644
index 00000000..3346e971
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+++ b/previews/PR126/assets/manual_solver.md.DxVH47Jn.js
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+import{_ as p,c as n,j as s,a as t,G as l,a5 as i,B as r,o}from"./chunks/framework.BSoZtefh.js";const V=JSON.parse('{"title":"Solvers","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solver.md","filePath":"manual/solver.md","lastUpdated":null}'),d={name:"manual/solver.md"},h={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.703ex",height:"1.636ex",role:"img",focusable:"false",viewBox:"0 -683 2520.6 723","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.428ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.783ex",height:"5.507ex",role:"img",focusable:"false",viewBox:"0 -1361 11837.9 2434.1","aria-hidden":"true"},E={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},F={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},L={class:"jldocstring custom-block",open:""},S={class:"jldocstring custom-block",open:""};function N(I,e,H,M,B,A){const a=r("Badge");return o(),n("div",null,[e[55]||(e[55]=s("h1",{id:"solvers",tabindex:"-1"},[t("Solvers "),s("a",{class:"header-anchor",href:"#solvers","aria-label":'Permalink to "Solvers"'},"")],-1)),e[56]||(e[56]=s("h2",{id:"Solvers-2",tabindex:"-1"},[t("Solvers "),s("a",{class:"header-anchor",href:"#Solvers-2","aria-label":'Permalink to "Solvers {#Solvers-2}"'},"")],-1)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_cfl_timestep!")],-1)),e[1]||(e[1]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[2]||(e[2]=i('julia get_cfl_timestep! (buf, u, setup) -> Any Get proposed maximum time step for convection and diffusion terms.
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',3))]),s("details",k,[s("summary",null,[e[3]||(e[3]=s("a",{id:"IncompressibleNavierStokes.get_state-Tuple{Any}",href:"#IncompressibleNavierStokes.get_state-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_state")],-1)),e[4]||(e[4]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[5]||(e[5]=i(`julia get_state ( stepper ) -> NamedTuple{( :u , :temp , :t , :n ), <: NTuple{4, Any} } Get state (; u, temp, t, n) from stepper.
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`,3))]),s("details",c,[s("summary",null,[e[6]||(e[6]=s("a",{id:"IncompressibleNavierStokes.solve_unsteady-Tuple{}",href:"#IncompressibleNavierStokes.solve_unsteady-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.solve_unsteady")],-1)),e[7]||(e[7]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[8]||(e[8]=i(`julia solve_unsteady ( ; setup, tlims, ustart, tempstart, method, psolver, Δt, Δt_min, cfl, n_adapt_Δt, docopy, processors, θ, cache ) Solve unsteady problem using method.
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`,7))]),e[57]||(e[57]=s("h2",{id:"processors",tabindex:"-1"},[t("Processors "),s("a",{class:"header-anchor",href:"#processors","aria-label":'Permalink to "Processors"'},"")],-1)),e[58]||(e[58]=s("p",null,[t("Processors can be used to process the solution in "),s("a",{href:"/IncompressibleNavierStokes.jl/previews/PR126/manual/solver#IncompressibleNavierStokes.solve_unsteady-Tuple{}"},[s("code",null,"solve_unsteady")]),t(" after every time step.")],-1)),s("details",m,[s("summary",null,[e[9]||(e[9]=s("a",{id:"IncompressibleNavierStokes.animator",href:"#IncompressibleNavierStokes.animator"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.animator")],-1)),e[10]||(e[10]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[11]||(e[11]=i('Animate a plot of the solution every update iteration. The animation is saved to path, which should have one of the following extensions:
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',4))]),s("details",g,[s("summary",null,[e[12]||(e[12]=s("a",{id:"IncompressibleNavierStokes.energy_history_plot",href:"#IncompressibleNavierStokes.energy_history_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_history_plot")],-1)),e[13]||(e[13]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[14]||(e[14]=s("p",null,"Create energy history plot.",-1)),e[15]||(e[15]=s("p",null,[s("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/505e79d2b6bbbc243eed9b17029223fcc45d6236/src/processors.jl#L392",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",T,[s("summary",null,[e[16]||(e[16]=s("a",{id:"IncompressibleNavierStokes.energy_spectrum_plot",href:"#IncompressibleNavierStokes.energy_spectrum_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_spectrum_plot")],-1)),e[17]||(e[17]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s("p",null,[e[20]||(e[20]=t("Create energy spectrum plot. The energy at a scalar wavenumber level ")),s("mjx-container",Q,[(o(),n("svg",u,e[18]||(e[18]=[i('as in San and Staples [12 ].
Keyword arguments:
sloperange = [0.6, 0.9]: Percentage (between 0 and 1) of x-axis where the slope is plotted.
slopeoffset = 1.3: How far above the energy spectrum the inertial slope is plotted.
kwargs...: They are passed to observespectrum
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',4))]),s("details",E,[s("summary",null,[e[25]||(e[25]=s("a",{id:"IncompressibleNavierStokes.fieldplot",href:"#IncompressibleNavierStokes.fieldplot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.fieldplot")],-1)),e[26]||(e[26]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i('Plot state field in pressure points. If state is Observable, then the plot is interactive.
Available fieldnames are:
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Available plot types for 3D are:
The alpha value gets passed to contour in 3D.
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',9))]),s("details",v,[s("summary",null,[e[28]||(e[28]=s("a",{id:"IncompressibleNavierStokes.fieldsaver-Tuple{}",href:"#IncompressibleNavierStokes.fieldsaver-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.fieldsaver")],-1)),e[29]||(e[29]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[30]||(e[30]=i('julia fieldsaver (; setup, nupdate) Create processor that stores the solution and time every nupdate time step.
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',3))]),s("details",j,[s("summary",null,[e[34]||(e[34]=s("a",{id:"IncompressibleNavierStokes.observespectrum-Tuple{Any}",href:"#IncompressibleNavierStokes.observespectrum-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.observespectrum")],-1)),e[35]||(e[35]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[36]||(e[36]=i('julia observespectrum (state; setup, npoint, a) Observe energy spectrum of state.
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',3))]),s("details",F,[s("summary",null,[e[37]||(e[37]=s("a",{id:"IncompressibleNavierStokes.processor",href:"#IncompressibleNavierStokes.processor"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.processor")],-1)),e[38]||(e[38]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[39]||(e[39]=i(`julia processor ( initialize ) -> NamedTuple{( :initialize , :finalize ), <: Tuple{Any, IncompressibleNavierStokes.var"#295#296"} } processor ( initialize, finalize ) -> NamedTuple{( :initialize , :finalize ), <: Tuple{Any, Any} } Process results from time stepping. Before time stepping, the initialize function is called on an observable of the time stepper state, returning initialized. The observable is updated every time step.
After timestepping, the finalize function is called on initialized and the final state.
See the following example:
julia function initialize (state) s = 0 println ( "Let's sum up the time steps" ) on (state) do (; n, t) println ( "The summand is $n , the time is $t " ) s = s + n end s end finalize (i, state) = println ( "The final sum (at time t= $(state . t) ) is $s " ) p = processor (initialize, finalize) When solved for 6 time steps from t=0 to t=2 the displayed output is
Let's sum up the time steps The summand is 0, the time is 0.0 The summand is 1, the time is 0.4 The summand is 2, the time is 0.8 The summand is 3, the time is 1.2 The summand is 4, the time is 1.6 The summand is 5, the time is 2.0 The final sum (at time t=2.0) is 15 source
`,8))]),s("details",C,[s("summary",null,[e[40]||(e[40]=s("a",{id:"IncompressibleNavierStokes.realtimeplotter",href:"#IncompressibleNavierStokes.realtimeplotter"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.realtimeplotter")],-1)),e[41]||(e[41]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[42]||(e[42]=i('Processor for plotting the solution in real time.
Keyword arguments:
plot: Plot function.
nupdate: Show solution every nupdate time step.
displayfig: Display the figure at the start.
screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
displayupdates: Display the figure at every update (if using CairoMakie).
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',5))]),s("details",x,[s("summary",null,[e[43]||(e[43]=s("a",{id:"IncompressibleNavierStokes.save_vtk-Tuple{Any}",href:"#IncompressibleNavierStokes.save_vtk-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.save_vtk")],-1)),e[44]||(e[44]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[45]||(e[45]=i('julia save_vtk (state; setup, filename, kwargs ... ) Save fields to vtk file.
The kwargs are passed to snapshotsaver
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',4))]),s("details",w,[s("summary",null,[e[46]||(e[46]=s("a",{id:"IncompressibleNavierStokes.snapshotsaver-Tuple{Any}",href:"#IncompressibleNavierStokes.snapshotsaver-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.snapshotsaver")],-1)),e[47]||(e[47]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[48]||(e[48]=i('julia snapshotsaver (state; setup, fieldnames, psolver) In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
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',3))]),s("details",L,[s("summary",null,[e[49]||(e[49]=s("a",{id:"IncompressibleNavierStokes.timelogger-Tuple{}",href:"#IncompressibleNavierStokes.timelogger-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.timelogger")],-1)),e[50]||(e[50]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[51]||(e[51]=i(`julia timelogger ( ; showiter, showt, showdt, showmax, showspeed, nupdate ) -> @NamedTuple {initialize :: IncompressibleNavierStokes.var"#298#300" {Bool, Bool, Bool, Bool, Bool, Int64}, finalize :: IncompressibleNavierStokes.var"#295#296" } Create processor that logs time step information.
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`,3))]),s("details",S,[s("summary",null,[e[52]||(e[52]=s("a",{id:"IncompressibleNavierStokes.vtk_writer-Tuple{}",href:"#IncompressibleNavierStokes.vtk_writer-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.vtk_writer")],-1)),e[53]||(e[53]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[54]||(e[54]=i('julia vtk_writer (; setup, nupdate, dir, filename, kwargs ... ) Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver
source
',3))])])}const Z=p(d,[["render",N]]);export{V as __pageData,Z as default};
diff --git a/previews/PR126/assets/manual_solver.md.DxVH47Jn.lean.js b/previews/PR126/assets/manual_solver.md.DxVH47Jn.lean.js
new file mode 100644
index 00000000..3346e971
--- /dev/null
+++ b/previews/PR126/assets/manual_solver.md.DxVH47Jn.lean.js
@@ -0,0 +1,51 @@
+import{_ as p,c as n,j as s,a as t,G as l,a5 as i,B as r,o}from"./chunks/framework.BSoZtefh.js";const V=JSON.parse('{"title":"Solvers","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solver.md","filePath":"manual/solver.md","lastUpdated":null}'),d={name:"manual/solver.md"},h={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.703ex",height:"1.636ex",role:"img",focusable:"false",viewBox:"0 -683 2520.6 723","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.428ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.783ex",height:"5.507ex",role:"img",focusable:"false",viewBox:"0 -1361 11837.9 2434.1","aria-hidden":"true"},E={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},F={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},L={class:"jldocstring custom-block",open:""},S={class:"jldocstring custom-block",open:""};function N(I,e,H,M,B,A){const a=r("Badge");return o(),n("div",null,[e[55]||(e[55]=s("h1",{id:"solvers",tabindex:"-1"},[t("Solvers "),s("a",{class:"header-anchor",href:"#solvers","aria-label":'Permalink to "Solvers"'},"")],-1)),e[56]||(e[56]=s("h2",{id:"Solvers-2",tabindex:"-1"},[t("Solvers "),s("a",{class:"header-anchor",href:"#Solvers-2","aria-label":'Permalink to "Solvers {#Solvers-2}"'},"")],-1)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_cfl_timestep!")],-1)),e[1]||(e[1]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[2]||(e[2]=i('julia get_cfl_timestep! (buf, u, setup) -> Any Get proposed maximum time step for convection and diffusion terms.
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`,3))]),s("details",c,[s("summary",null,[e[6]||(e[6]=s("a",{id:"IncompressibleNavierStokes.solve_unsteady-Tuple{}",href:"#IncompressibleNavierStokes.solve_unsteady-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.solve_unsteady")],-1)),e[7]||(e[7]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[8]||(e[8]=i(`julia solve_unsteady ( ; setup, tlims, ustart, tempstart, method, psolver, Δt, Δt_min, cfl, n_adapt_Δt, docopy, processors, θ, cache ) Solve unsteady problem using method.
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The plot is determined by a plotter processor. Additional kwargs are passed to plot.
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',4))]),s("details",g,[s("summary",null,[e[12]||(e[12]=s("a",{id:"IncompressibleNavierStokes.energy_history_plot",href:"#IncompressibleNavierStokes.energy_history_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_history_plot")],-1)),e[13]||(e[13]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[14]||(e[14]=s("p",null,"Create energy history plot.",-1)),e[15]||(e[15]=s("p",null,[s("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/505e79d2b6bbbc243eed9b17029223fcc45d6236/src/processors.jl#L392",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",T,[s("summary",null,[e[16]||(e[16]=s("a",{id:"IncompressibleNavierStokes.energy_spectrum_plot",href:"#IncompressibleNavierStokes.energy_spectrum_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_spectrum_plot")],-1)),e[17]||(e[17]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s("p",null,[e[20]||(e[20]=t("Create energy spectrum plot. The energy at a scalar wavenumber level ")),s("mjx-container",Q,[(o(),n("svg",u,e[18]||(e[18]=[i('as in San and Staples [12 ].
Keyword arguments:
sloperange = [0.6, 0.9]: Percentage (between 0 and 1) of x-axis where the slope is plotted.
slopeoffset = 1.3: How far above the energy spectrum the inertial slope is plotted.
kwargs...: They are passed to observespectrum
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',4))]),s("details",E,[s("summary",null,[e[25]||(e[25]=s("a",{id:"IncompressibleNavierStokes.fieldplot",href:"#IncompressibleNavierStokes.fieldplot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.fieldplot")],-1)),e[26]||(e[26]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i('Plot state field in pressure points. If state is Observable, then the plot is interactive.
Available fieldnames are:
:velocity,
:vorticity,
:streamfunction,
:pressure.
Available plot types for 2D are:
heatmap (default),
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contour,
contourf.
Available plot types for 3D are:
The alpha value gets passed to contour in 3D.
source
',9))]),s("details",v,[s("summary",null,[e[28]||(e[28]=s("a",{id:"IncompressibleNavierStokes.fieldsaver-Tuple{}",href:"#IncompressibleNavierStokes.fieldsaver-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.fieldsaver")],-1)),e[29]||(e[29]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[30]||(e[30]=i('julia fieldsaver (; setup, nupdate) Create processor that stores the solution and time every nupdate time step.
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',3))]),s("details",f,[s("summary",null,[e[31]||(e[31]=s("a",{id:"IncompressibleNavierStokes.observefield-Tuple{Any}",href:"#IncompressibleNavierStokes.observefield-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.observefield")],-1)),e[32]||(e[32]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[33]||(e[33]=i('julia observefield (state; setup, fieldname, logtol, psolver) Observe field fieldname at pressure points.
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',3))]),s("details",j,[s("summary",null,[e[34]||(e[34]=s("a",{id:"IncompressibleNavierStokes.observespectrum-Tuple{Any}",href:"#IncompressibleNavierStokes.observespectrum-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.observespectrum")],-1)),e[35]||(e[35]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[36]||(e[36]=i('julia observespectrum (state; setup, npoint, a) Observe energy spectrum of state.
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',3))]),s("details",F,[s("summary",null,[e[37]||(e[37]=s("a",{id:"IncompressibleNavierStokes.processor",href:"#IncompressibleNavierStokes.processor"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.processor")],-1)),e[38]||(e[38]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[39]||(e[39]=i(`julia processor ( initialize ) -> NamedTuple{( :initialize , :finalize ), <: Tuple{Any, IncompressibleNavierStokes.var"#295#296"} } processor ( initialize, finalize ) -> NamedTuple{( :initialize , :finalize ), <: Tuple{Any, Any} } Process results from time stepping. Before time stepping, the initialize function is called on an observable of the time stepper state, returning initialized. The observable is updated every time step.
After timestepping, the finalize function is called on initialized and the final state.
See the following example:
julia function initialize (state) s = 0 println ( "Let's sum up the time steps" ) on (state) do (; n, t) println ( "The summand is $n , the time is $t " ) s = s + n end s end finalize (i, state) = println ( "The final sum (at time t= $(state . t) ) is $s " ) p = processor (initialize, finalize) When solved for 6 time steps from t=0 to t=2 the displayed output is
Let's sum up the time steps The summand is 0, the time is 0.0 The summand is 1, the time is 0.4 The summand is 2, the time is 0.8 The summand is 3, the time is 1.2 The summand is 4, the time is 1.6 The summand is 5, the time is 2.0 The final sum (at time t=2.0) is 15 source
`,8))]),s("details",C,[s("summary",null,[e[40]||(e[40]=s("a",{id:"IncompressibleNavierStokes.realtimeplotter",href:"#IncompressibleNavierStokes.realtimeplotter"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.realtimeplotter")],-1)),e[41]||(e[41]=t()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[42]||(e[42]=i('Processor for plotting the solution in real time.
Keyword arguments:
plot: Plot function.
nupdate: Show solution every nupdate time step.
displayfig: Display the figure at the start.
screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
displayupdates: Display the figure at every update (if using CairoMakie).
sleeptime: The sleeptime is slept at every update, to give Makie time to update the plot. Set this to nothing to skip sleeping.
Additional kwargs are passed to the plot function.
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The kwargs are passed to snapshotsaver
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',4))]),s("details",w,[s("summary",null,[e[46]||(e[46]=s("a",{id:"IncompressibleNavierStokes.snapshotsaver-Tuple{Any}",href:"#IncompressibleNavierStokes.snapshotsaver-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.snapshotsaver")],-1)),e[47]||(e[47]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[48]||(e[48]=i('julia snapshotsaver (state; setup, fieldnames, psolver) In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
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',3))]),s("details",L,[s("summary",null,[e[49]||(e[49]=s("a",{id:"IncompressibleNavierStokes.timelogger-Tuple{}",href:"#IncompressibleNavierStokes.timelogger-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.timelogger")],-1)),e[50]||(e[50]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[51]||(e[51]=i(`julia timelogger ( ; showiter, showt, showdt, showmax, showspeed, nupdate ) -> @NamedTuple {initialize :: IncompressibleNavierStokes.var"#298#300" {Bool, Bool, Bool, Bool, Bool, Int64}, finalize :: IncompressibleNavierStokes.var"#295#296" } Create processor that logs time step information.
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`,3))]),s("details",S,[s("summary",null,[e[52]||(e[52]=s("a",{id:"IncompressibleNavierStokes.vtk_writer-Tuple{}",href:"#IncompressibleNavierStokes.vtk_writer-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.vtk_writer")],-1)),e[53]||(e[53]=t()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[54]||(e[54]=i('julia vtk_writer (; setup, nupdate, dir, filename, kwargs ... ) Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver
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',3))])])}const Z=p(d,[["render",N]]);export{V as __pageData,Z as default};
diff --git a/previews/PR126/assets/manual_spatial.md.Cj3WsAzX.js b/previews/PR126/assets/manual_spatial.md.Cj3WsAzX.js
new file mode 100644
index 00000000..924ddbd3
--- /dev/null
+++ b/previews/PR126/assets/manual_spatial.md.Cj3WsAzX.js
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M. Kurz, P. Offenhäuser and A. Beck. Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations arXiv (2022).
B. List, L.-W. Chen and N. Thuerey. Learned Turbulence Modelling with Differentiable Fluid Solvers arxiv:2202.06988 (2022).
J. F. MacArt, J. Sirignano and J. B. Freund. Embedded training of neural-network sub-grid-scale turbulence models 2021 ).
J. Li and P. M. Carrica. A simple approach for vortex core visualization arXiv:1910.06998 [physics.flu-dyn] .
J. Jeong and F. Hussain. On the identification of a vortex J. Fluid. Mech. 285 , 69–94 (1995).
S. B. Pope. Turbulent Flows (Cambridge University Press, Cambridge, England, 2000).
B. Sanderse and F. X. Trias. Energy-consistent discretization of viscous dissipation with application to natural convection flow arXiv:2307.10874v1 [physics.flu-dyn] .
M. H. Silvis, R. A. Remmerswaal and R. Verstappen. Physical consistency of subgrid-scale models for large-eddy simulation of incompressible turbulent flows Physics of Fluids 29 (2017).
F. X. TRIAS, M. SORIA, A. OLIVA and C. D. PÉREZ-SEGARRA. Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4 . Journal of Fluid Mechanics 586 , 259–293 (2007).
P. Orlandi. Fluid flow phenomena: a numerical toolkit . Vol. 55 (Springer Science & Business Media, 2000).
O. San and A. E. Staples. High-order methods for decaying two-dimensional homogeneous isotropic turbulence Computers & Fluids 63 , 105–127 (2012).
F. H. Harlow and J. E. Welch. * Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface * . The Physics of Fluids 8 , 2182–2189 (1965), arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.1761178 .
R. W. Verstappen and A. E. Veldman. Symmetry-Preserving Discretization of Turbulent Flow J. Comput. Phys. 187 , 343–368 (2003).
B. Sanderse, R. Verstappen and B. Koren. Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations Journal of Computational Physics 257 , 1472–1505 (2014). Physics-compatible numerical methods.
R. Verstappen and A. Veldman. Direct Numerical Simulation of Turbulence at Lower Costs Journal of Engineering Mathematics 32 , 143–159 (1997).
B. Sanderse and B. Koren. Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations Journal of Computational Physics 231 , 3041–3063 (2012).
B. Sanderse. Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations Journal of Computational Physics 233 , 100–131 (2013).
References Bibliography D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner and S. Hoyer. Machine learning-accelerated computational fluid dynamics Proceedings of the National Academy of Sciences 118 (2021).
M. Kurz, P. Offenhäuser and A. Beck. Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations arXiv (2022).
B. List, L.-W. Chen and N. Thuerey. Learned Turbulence Modelling with Differentiable Fluid Solvers arxiv:2202.06988 (2022).
J. F. MacArt, J. Sirignano and J. B. Freund. Embedded training of neural-network sub-grid-scale turbulence models 2021 ).
J. Li and P. M. Carrica. A simple approach for vortex core visualization arXiv:1910.06998 [physics.flu-dyn] .
J. Jeong and F. Hussain. On the identification of a vortex J. Fluid. Mech. 285 , 69–94 (1995).
S. B. Pope. Turbulent Flows (Cambridge University Press, Cambridge, England, 2000).
B. Sanderse and F. X. Trias. Energy-consistent discretization of viscous dissipation with application to natural convection flow arXiv:2307.10874v1 [physics.flu-dyn] .
M. H. Silvis, R. A. Remmerswaal and R. Verstappen. Physical consistency of subgrid-scale models for large-eddy simulation of incompressible turbulent flows Physics of Fluids 29 (2017).
F. X. TRIAS, M. SORIA, A. OLIVA and C. D. PÉREZ-SEGARRA. Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4 . Journal of Fluid Mechanics 586 , 259–293 (2007).
P. Orlandi. Fluid flow phenomena: a numerical toolkit . Vol. 55 (Springer Science & Business Media, 2000).
O. San and A. E. Staples. High-order methods for decaying two-dimensional homogeneous isotropic turbulence Computers & Fluids 63 , 105–127 (2012).
F. H. Harlow and J. E. Welch. * Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface * . The Physics of Fluids 8 , 2182–2189 (1965), arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.1761178 .
R. W. Verstappen and A. E. Veldman. Symmetry-Preserving Discretization of Turbulent Flow J. Comput. Phys. 187 , 343–368 (2003).
B. Sanderse, R. Verstappen and B. Koren. Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations Journal of Computational Physics 257 , 1472–1505 (2014). Physics-compatible numerical methods.
R. Verstappen and A. Veldman. Direct Numerical Simulation of Turbulence at Lower Costs Journal of Engineering Mathematics 32 , 143–159 (1997).
B. Sanderse and B. Koren. Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations Journal of Computational Physics 231 , 3041–3063 (2012).
B. Sanderse. Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations Journal of Computational Physics 233 , 100–131 (2013).
Unsteady actuator case - 2D | IncompressibleNavierStokes.jl
+ Unsteady actuator case - 3D | IncompressibleNavierStokes.jl
+ Backward Facing Step - 2D | IncompressibleNavierStokes.jl
+ Backward Facing Step - 3D | IncompressibleNavierStokes.jl
+ Decaying Homogeneous Isotropic Turbulence - 2D | IncompressibleNavierStokes.jl
+ Decaying Homogeneous Isotropic Turbulence - 3D | IncompressibleNavierStokes.jl
+ Kolmogorov flow (2D) | IncompressibleNavierStokes.jl
+ Tutorial: Lid-Driven Cavity - 2D | IncompressibleNavierStokes.jl
+ Lid-Driven Cavity - 3D | IncompressibleNavierStokes.jl
+ Unsteady actuator case - 2D | IncompressibleNavierStokes.jl
+ Planar mixing - 2D | IncompressibleNavierStokes.jl
+ Plane jets - 2D | IncompressibleNavierStokes.jl
+ Rayleigh-Bénard convection (2D) | IncompressibleNavierStokes.jl
+ Rayleigh-Bénard convection (3D) | IncompressibleNavierStokes.jl
+ Rayleigh-Taylor instability in 2D | IncompressibleNavierStokes.jl
+ Rayleigh-Taylor instability in 2D | IncompressibleNavierStokes.jl
+ Shear layer - 2D | IncompressibleNavierStokes.jl
+ Convergence study: Taylor-Green vortex (2D) | IncompressibleNavierStokes.jl
+ Taylor-Green vortex - 3D | IncompressibleNavierStokes.jl
+ Examples gallery | IncompressibleNavierStokes.jl
+ Getting Started | IncompressibleNavierStokes.jl
+ Skip to content Getting Started To install IncompressibleNavierStokes, open up a Julia-REPL, type ] to get into Pkg-mode, and type:
sh add IncompressibleNavierStokes which will install the package and all dependencies to your local environment. Note that IncompressibleNavierStokes requires Julia version 1.9 or above.
Check out the gallery for commented example simulations.
IncompressibleNavierStokes.jl
+ Skip to content Navier-Stokes Experience 21st century turbulence
Differentiable fluid solver written in Julia
+
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+ N av i e r S t o k e s . j l I n c o m p r e ss i b l e
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+ Neural closure models | IncompressibleNavierStokes.jl
+ Differentiating through the code | IncompressibleNavierStokes.jl
+ GPU Support | IncompressibleNavierStokes.jl
+ Large eddy simulation | IncompressibleNavierStokes.jl
+ Sparse matrices | IncompressibleNavierStokes.jl
+ Incompressible Navier-Stokes equations | IncompressibleNavierStokes.jl
+ Operators | IncompressibleNavierStokes.jl
+ Floating point precision | IncompressibleNavierStokes.jl
+ Pressure solvers | IncompressibleNavierStokes.jl
+ Using IncompressibleNavierStokes in SciML | IncompressibleNavierStokes.jl
+ Problem setup | IncompressibleNavierStokes.jl
+ Solvers | IncompressibleNavierStokes.jl
+ Spatial discretization | IncompressibleNavierStokes.jl
+ Temperature equation | IncompressibleNavierStokes.jl
+ Time discretization | IncompressibleNavierStokes.jl
+ Utils | IncompressibleNavierStokes.jl
+ References | IncompressibleNavierStokes.jl
+ Skip to content References Bibliography D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner and S. Hoyer. Machine learning-accelerated computational fluid dynamics Proceedings of the National Academy of Sciences 118 (2021).
M. Kurz, P. Offenhäuser and A. Beck. Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations arXiv (2022).
B. List, L.-W. Chen and N. Thuerey. Learned Turbulence Modelling with Differentiable Fluid Solvers arxiv:2202.06988 (2022).
J. F. MacArt, J. Sirignano and J. B. Freund. Embedded training of neural-network sub-grid-scale turbulence models 2021 ).
J. Li and P. M. Carrica. A simple approach for vortex core visualization arXiv:1910.06998 [physics.flu-dyn] .
J. Jeong and F. Hussain. On the identification of a vortex J. Fluid. Mech. 285 , 69–94 (1995).
S. B. Pope. Turbulent Flows (Cambridge University Press, Cambridge, England, 2000).
B. Sanderse and F. X. Trias. Energy-consistent discretization of viscous dissipation with application to natural convection flow arXiv:2307.10874v1 [physics.flu-dyn] .
M. H. Silvis, R. A. Remmerswaal and R. Verstappen. Physical consistency of subgrid-scale models for large-eddy simulation of incompressible turbulent flows Physics of Fluids 29 (2017).
F. X. TRIAS, M. SORIA, A. OLIVA and C. D. PÉREZ-SEGARRA. Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4 . Journal of Fluid Mechanics 586 , 259–293 (2007).
P. Orlandi. Fluid flow phenomena: a numerical toolkit . Vol. 55 (Springer Science & Business Media, 2000).
O. San and A. E. Staples. High-order methods for decaying two-dimensional homogeneous isotropic turbulence Computers & Fluids 63 , 105–127 (2012).
F. H. Harlow and J. E. Welch. * Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface * . The Physics of Fluids 8 , 2182–2189 (1965), arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.1761178 .
R. W. Verstappen and A. E. Veldman. Symmetry-Preserving Discretization of Turbulent Flow J. Comput. Phys. 187 , 343–368 (2003).
B. Sanderse, R. Verstappen and B. Koren. Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations Journal of Computational Physics 257 , 1472–1505 (2014). Physics-compatible numerical methods.
R. Verstappen and A. Veldman. Direct Numerical Simulation of Turbulence at Lower Costs Journal of Engineering Mathematics 32 , 143–159 (1997).
B. Sanderse and B. Koren. Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations Journal of Computational Physics 231 , 3041–3063 (2012).
B. Sanderse. Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations Journal of Computational Physics 233 , 100–131 (2013).
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