File:VortexStreetAnimation DifferentShapes.gif

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VortexStreetAnimation_DifferentShapes.gif(768 × 444 pixels, file size: 9.27 MB, MIME type: image/gif, looped, 81 frames, 8.1 s)

Captions

Captions

When a fluid flows slowly enough it can smoothly move around an obstacle, but when the speed increases the flow becomes unstable.

Summary[edit]

Description
English: When a fluid flows slowly enough it can smoothly move around an obstacle, but when the speed increases the flow becomes turbulent. How fast you can go before you get turbulences, and how severe they are depends a lot on the shape of the obstacle. Color is modulus of the velocity, arrows show direction.
Date
Source https://twitter.com/j_bertolotti/status/1244226965508407296
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 12.0 code[edit]

(*Basic code from https : // www.wolfram.com/language/12/nonlinear-finite-elements/transient-navier-stokes.html*)
w = 2.2; h = 0.41; (*Sizes*)
geometry1 = RegionDifference[Rectangle[{0, 0}, {w, h}], Disk[{2/5, 1/5}, 1/20]];
BoundaryDiscretizeRegion[geometry1]

eq = {
        \[Rho] 
      
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, 
              y] + \[Rho] {u[t, x, y], v[t, x, y]}.Inactive[Grad][
                u[t, x, y], {x, y}] + 
          
     Inactive[Div][(-\[Mu] Inactive[Grad][u[t, x, y], {x, y}]), {x, 
              y}] + 
     
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, y], \[Rho] 
      
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, 
              y] + \[Rho] {u[t, x, y], v[t, x, y]}.Inactive[Grad][
                v[t, x, y], {x, y}] + 
          
     Inactive[Div][(-\[Mu] Inactive[Grad][v[t, x, y], {x, y}]), {x, 
              y}] + 
     
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, y], 
    
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, y] + 
     
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, y]} /. {\[Mu] -> 10^-3, \[Rho] -> 1};

tmax = 12; (*maximum time for the simulation*)
flow[t_] := 1/(1 + Exp[-1.6 (t - 5.5)]); (*how fast the input velocity rises*)

(*boundary conditions*)
inflowBC = DirichletCondition[{u[t, x, y] == flow[t]*4*1.5*y*(h - y)/h^2, v[t, x, y] == 0}, x == 0];
outflowBC = DirichletCondition[p[t, x, y] == 0., x == w];
wallBC = DirichletCondition[{u[t, x, y] == 0, v[t, x, y] == 0}, 0 < x < w];
bcs = {inflowBC, outflowBC, wallBC};
ic = {u[0, x, y] == 0, v[0, x, y] == 0, p[0, x, y] == 0};
(*Solve*)
Monitor[AbsoluteTiming[{xVel1, yVel1, pressure1} = NDSolveValue[{eq == {0, 0, 0}, bcs, ic}, {u, v, p}, {x, y} \[Element] geometry1, {t, 0, tmax}, Method -> {"PDEDiscretization" -> {"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement", "InterpolationOrder" -> {u -> 2, v -> 2, p -> 1}, "MeshOptions" -> {"MaxCellMeasure" -> 0.0005}}}}, EvaluationMonitor :> (currentTime = Row[{"t = ", CForm[t]}])];], currentTime]

centre = 1/5; l = 1/20;
geometry2 = RegionDifference @@ (BoundaryDiscretizeRegion /@ {Rectangle[{0, 0}, {w, h}], Rectangle[{2 centre - l, centre - l}, {2 centre + l, centre + l}] })
Monitor[AbsoluteTiming[{xVel2, yVel2, pressure2} = NDSolveValue[{eq == {0, 0, 0}, bcs, ic}, {u, v, p}, {x, y} \[Element] geometry2, {t, 0, tmax}, Method -> {"PDEDiscretization" -> {"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement", "InterpolationOrder" -> {u -> 2, v -> 2, p -> 1}, "MeshOptions" -> {"MaxCellMeasure" -> 0.0005}}}}, EvaluationMonitor :> (currentTime = Row[{"t = ", CForm[t]}])];], currentTime]

geometry3 = RegionDifference @@ (BoundaryDiscretizeRegion /@ {Rectangle[{0, 0}, {w, h}], ParametricRegion[0.065 {r Cos[t], r Sin[t] Sin[t/2]^1} + {0.415, 1/5}, {{t, 0, 2 \[Pi]}, {r, 0, 1}}]})
Monitor[AbsoluteTiming[{xVel3, yVel3, pressure3} = NDSolveValue[{eq == {0, 0, 0}, bcs, ic}, {u, v, p}, {x, y} \[Element] geometry3, {t, 0, tmax}, Method -> {"PDEDiscretization" -> {"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement", "InterpolationOrder" -> {u -> 2, v -> 2, p -> 1}, "MeshOptions" -> {"MaxCellMeasure" -> 0.0005}}}}, EvaluationMonitor :> (currentTime = Row[{"t = ", CForm[t]}])];], currentTime]

p0 = Table[
   GraphicsColumn[{
     Show[
      DensityPlot[Norm[{xVel2[t, x, y], yVel2[t, x, y]}]/2, {x, 0, 2.2}, {y, 0, 0.41}, PlotPoints -> 50, PlotRange -> {0, 2.1}, AspectRatio -> Automatic, Frame -> None, ColorFunction -> "TemperatureMap", ColorFunctionScaling -> False]
      ,
      VectorPlot[{xVel2[t, x, y], yVel2[t, x, y]}, {x, 0.05, 2.15}, {y, 0.02, 0.4}, AspectRatio -> Automatic, Frame -> None, VectorStyle -> Black]
      ,
      Graphics[{White, Rectangle[{2 centre - l, centre - l}, {2 centre + l, centre + l}] }]
      ]
     ,
     Show[
      DensityPlot[Norm[{xVel1[t, x, y], yVel1[t, x, y]}]/2, {x, 0, 2.2}, {y, 0, 0.41}, PlotPoints -> 50, PlotRange -> {0, 2.1}, AspectRatio -> Automatic, Frame -> None, ColorFunction -> "TemperatureMap", 
ColorFunctionScaling -> False]
      ,
      VectorPlot[{xVel1[t, x, y], yVel1[t, x, y]}, {x, 0.05, 2.15}, {y, 0.02, 0.4}, AspectRatio -> Automatic, Frame -> None, VectorStyle -> Black]
      ,
      Graphics[{White, Disk[{2/5, 1/5}, 1/20], Black, Circle[{2/5, 1/5}, 1/20]}]
      ]
     ,
     Show[
      DensityPlot[Norm[{xVel3[t, x, y], yVel3[t, x, y]}]/2, {x, 0, 2.2}, {y, 0, 0.41}, PlotPoints -> 50, PlotRange -> {0, 2.1}, AspectRatio -> Automatic, Frame -> None, ColorFunction -> "TemperatureMap", 
ColorFunctionScaling -> False]
      ,
      VectorPlot[{xVel3[t, x, y], yVel3[t, x, y]}, {x, 0.05, 2.15}, {y, 0.02, 0.4}, AspectRatio -> Automatic, Frame -> None]
      ,
      ParametricPlot[0.065 {r Cos[\[Tau]], r Sin[\[Tau]] Sin[\[Tau]/2]^1} + {0.415, 1/5}, {\[Tau], 0, 2 \[Pi]}, {r, 0, 1}, Frame -> None, Background -> None, Axes -> False, PlotStyle -> {Directive[White, Opacity[1]]}, Mesh -> None, Epilog -> {White, Thick, Line[{{0.4, 1/5}, {0.479, 1/5}}]}]
      ]
     }, ImageSize -> Large]
   , {t, 3, 11, 0.1}];
ListAnimate[p0]

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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Date/TimeThumbnailDimensionsUserComment
current12:58, 30 March 2020Thumbnail for version as of 12:58, 30 March 2020768 × 444 (9.27 MB)Berto (talk | contribs)Uploaded own work with UploadWizard

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