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How can I write down two-dimensional Navier-Stokes equations for a simple rigid object immersed in a flow and freely falling due to gravity? I'm trying to view the vorticity that's induced by the fluid-solid interaction.

I'm using Matlab for the simulations.

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  • $\begingroup$ You can't have still background flow at the same time as an object moving through your medium. The two are incompatible with each other. $\endgroup$
    – Wolfgang Bangerth
    Commented Apr 18, 2020 at 2:03

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The challenge is not the visualization, but writing the Navier-Stokes solver. That's because not only is that one difficult equation to solve to begin with, but you are now also using a domain that changes with time and, on top, how exactly it changes (i.e., how the ellipse falls) depends on the details of the flow.

The area of scientific computing that deals with this is called "fluid-structure interaction". For your particular case, the typical method people use is called the "immersed boundary method" and you will find quite a number of publications about more or less the problem you care about. A reasonable starting point are the papers by Luca Heltai (https://scholar.google.it/citations?user=9TUDvmgAAAAJ&hl=en) or the man himself, Charles Peskin (https://scholar.google.com/scholar?hl=en&as_sdt=0%2C6&q=charles+peskin&btnG=&oq=charles+pesk). You might also be interested in examples of implementation, for example this tutorial program: https://www.dealii.org/developer/doxygen/deal.II/step_60.html

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  • $\begingroup$ Is the immersed boundary method significantly easier to understand and implement, if the body is rigid? The IB method seems to be accounting for thin elastic bodies. $\endgroup$
    – user35678
    Commented Apr 20, 2020 at 3:47
  • $\begingroup$ @user35678 You don't need fluid-structure interaction or IB to simulate flow around a falling rigid object. You can think of it as your rigid object is fixed and just flow around it moves and that would reduce your problem to something like flow around a sphere or cylinder or whatever is the shape of rigid object. By the way, you formulated your question wrongly I think. When your object is completely rigid, there is no fluid-structure interaction. There would be only some disturbance of flow because of that rigid body motion which could be captured as I described here without IB. $\endgroup$
    – Mithridates the Great
    Commented Apr 20, 2020 at 6:42
  • $\begingroup$ @user35678 No, there is no MATLAB command to do this. You need to formulate your problem in terms of solving Navier-Stokes or Stoke's equation if it is a creepy flow, and then solve it with a proper numerical tool like FEM. It needs more efforts. $\endgroup$
    – Mithridates the Great
    Commented Apr 20, 2020 at 15:02
  • $\begingroup$ @AloneProgrammer -- there is only no fluid-structure interaction if the object is falling down in a straight line and does not rotate. That would only be the case for a sphere in a Stokes fluid of infinite extension But in practice, falling objects do rotate (which changes their shape too, unless they are spheres) and they move sideways in response to the flow field. $\endgroup$
    – Wolfgang Bangerth
    Commented Apr 20, 2020 at 16:38
  • $\begingroup$ @user35678: Yes, the formulation becomes easier if your object is rigid. In that case, you can parameterize the solid body by just 5 numbers: The position, and the angles of rotation (or, equivalently, by the velocity and the angular velocity). $\endgroup$
    – Wolfgang Bangerth
    Commented Apr 20, 2020 at 16:40

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