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The Question can be Formally presented as follows;

How is the Numerical Model (CFD, Navier Stokes) of fluid dynamics for Vortexes at in it's limits?

At this Publication from 2013 at page 48 is said;

If the free surface is bent at the at the dimple, the complexity and the velocity gradient in the air core is growing fast. At this stage the actual limit is reached that numerical simulations can provide nowadays.

and continued at page 49;

An even more sophisticated step would be an adequate simulation of the air entrainment itself, which is still not possible.

QUESTION; Why it's still not possible?

There is many "exact solutions of NS-equations" for Vortexes; Ie. Burgers Vortex and Lamb-Oseen vortex

Basically Vortex has a clear construction, which can mainly be expressed mathematically very simply way. Ie Rankine Vortex An example of this construction is picture below; Vortex pattern

If you look the German-wiki from the "Lamb-Oseen vortex" you find this picture, I modified it to show the interes of my Question (green line).

Vortex velocity

I am interested about there true shape of this peak. How the numerical model's works at this peak compared to experiments. I think here lies the problem. It's clear that most of the Vortexes aren't like "Lamb-Oseen", but I doubt if they are abslutely like the "Rankine" either. So it's something like the "Burger". When i think the current calculation power, it sound not reasonable that the problem lies just on complexity, as there is even exact solutions available.

Few aspects;

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    $\begingroup$ Why must it have an exact solution if it is commonly seen? $\endgroup$ – Kyle Kanos Dec 24 '15 at 18:07
  • $\begingroup$ @KyleKanos "easily reproducible" is not "commonly seen", most "exact solution" in the provided list are also easy to verify in the lab. I do found stuff like this; cfd-online.com/Forums/cfx/21544-free-surface-vortex-flow.html and paper's like ie. Tomoyoshi Okamura 2007 stating; "The calculated streamlines and Vortex core lines are not able to be used to predict the visible vortices with much accuracy." or this kind of videos; youtube.com/watch?v=lwtGJeebwg8 which seems to be very unrealistic; like the old films with wrong Reynolds number. $\endgroup$ – Jokela Dec 24 '15 at 18:11
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    $\begingroup$ I'm not sure why "easily reproducible" is not equivalent to "commonly seen" (anyone who has tried draining a full sink will have seen it), but that's not really important. What is important is whether or not there are reproducible fluid dynamic experiments that don't have analytic solutions. If these exist, then your present thesis is unfounded. $\endgroup$ – Kyle Kanos Dec 24 '15 at 18:16
  • $\begingroup$ @KyleKanos Well, yes, if you start this with complete still stand, you don't have any analytic solution; it can rotate either way, or not at all. But if you give some initial rotation, you will always get an Vortex with air core. And I expect this vortex should have analytic solution. -But I see a flaw in my question; the start direction must be given, or othervice we are just throwing a dice. The point is that I am simply seeking a realistic CFD of an air core vortex. $\endgroup$ – Jokela Dec 24 '15 at 18:39
  • $\begingroup$ Well the NS equations do lead to the vorticity equations, so I'm not sure why you'd suspect that something that's been verified time and time again is somehow mistaken because you can't find someone who's created one particular simulation... $\endgroup$ – Kyle Kanos Dec 24 '15 at 19:46