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Rogue wave: stable fluids algorithm for air/water interface simulation

POSTED BY: Alexander Trounev
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@Alexander, first of all, this is a wonderful work, thank you very much for sharing! I gave a shout-out on LinkedIN and the feedback was great:

https://www.linkedin.com/feed/update/urn:li:activity:6866071848321368064

I am curious about stable fluid algorithm FDM you mentioned. Is there a general reference to it or it was developed in your report? I know NDSolve does use FDM

but I assume your FDM is a very custom thing.
POSTED BY: Vitaliy Kaurov

Dear Vitaliy, thank you very much for links to NDSolve tutorial and LinkedIn. Actually I have used NDSolve with MOL, DAE and FEM solver as well in different applications. Stably fluids algorithm is very specific for implementation with Mathematica. Every step 1-9 discussed above can be realized with NDSolve, but we prefer Module, Do and Compile with option CompilationTarget -> C to speedup simulation. First time this algorithm has been developed by Jos Stam in the last century for computer graphics applications. For our problem we rebuild this algorithm to make it more stable.

POSTED BY: Alexander Trounev
POSTED BY: Paul Abbott

Thank you very much for reference link. Note, that in our code we have to implement the stable fluid algorithm, therefore we use FDM, not FEM. As nice example of this implementation we can visualize flow around square cylinder
enter image description here

POSTED BY: Alexander Trounev

Next step is to animate wave transformation using interface function. We can show velocity field over density as follows

Show[ContourPlot[1 - rh[sm][x, y]/rhoWater20C, {x, 0, 1}, {y, 0, 1}, 
  PlotRange -> All, ColorFunction -> "BlueGreenYellow", Contours -> 8,
   ContourStyle -> Yellow, Frame -> False, PlotLegends -> Automatic], 
 StreamPlot[{Uvel[sm][x, y], Vvel[sm][x, y]}, {x, 0, 1}, {y, 0, 1}, 
  PlotRange -> All, StreamColorFunction -> None, VectorPoints -> Fine,
   VectorColorFunction -> Hue, PlotLegends -> Automatic, 
  StreamColorFunctionScaling -> True, StreamStyle -> LightGray]]

Figure 3

We can also compute frames for animation with using interface function 

frames=Table[ContourPlot[1 - rh[i][x, y], {x, 0, 1}, {y, 0, 1}, 
  ColorFunction -> "BlueGreenYellow", Frame -> False, 
  PlotRange -> All, ImageSize -> Small, MaxRecursion -> 2, 
  Contours -> 8, ContourStyle -> Yellow, PlotLabel -> i], {i, 20, sm, 
  20}]; Animate[frames]
POSTED BY: Alexander Trounev

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