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The question is difficult to understand unless I explain why I am asking it.

I would not really be interested in Fluid Dynamics if the transition to turbulent flow were not, at least approximately, described by the Directed Percolation Universality Classes. However, it looks like it may very well be the case (see, for reference, paywalled):

https://www.nature.com/collections/rxsztdqblr/

The Directed Percolation Universality Classes seem also to be related to Regge-Theory (see, for example):

https://sunclipse.org/wp-content/downloads/2013/04/cardy-etal1980.pdf

Regge-Theory is, IMO, as close as we get to non-perturbative QCD (in its different "field versions" like the BFKL-Pomeron, for example, and leaving lattice QCD aside).

This has puzzled me a lot because the symmetry group of the incompressible Navier-Stokes equations are an enlarged version of the Galilean group and are, apparently, unrelated to Poincare's group (the only relevant group in relativistic particle interactions). How could this happen if they are linked by the same Universality Classes?

I have played (and failed) with the idea of "sound velocity being constant" replacing "light velocity being constant" to connect both dynamic systems (they should be if they belonged to the same Universality Class). However, no "pseudo-relativistic" corrections play any role in the incompressible fluid road to turbulence (or, at least, I am unable to see them).

So maybe I should give up this idea. However in real (compressible) fluids this idea does come up:

"Physique theorique" (Mecanique des fluides), Tome 6, L. Landau and E. Lifchitz, MIR 1989, Chapter IX, sections 83 and 84 (pages 448-457). (I do not have the English book, but I guess that the chapters and sections should not differ too much).

So, I would be very grateful if somebody could help me out:

What is the relevant feature of incompressible N-S transition to turbulence?

Viscosity not being able to dissipate vorticity? What is the residual viscosity role? Does it determine the minimum eddies size? Does sound speed play any role in this transition?

Why do I not see any mechanism being able to propagate perturbations (constant density)? I do not get it. How is this possible?

I know there are three question marks, but this just reflects my puzzlement and my lack of understanding of this "road" to turbulence, and that is all I mean by these three particular questions. The important question is in the title but it is far too general to be meaninful.

Thank you very much in advance.

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    $\begingroup$ I don't understand most of this, but in an incompressible fluid (which doesn't exist in real life, though it is a convenient approximation in some situations) the "sound velocity" is infinite (i.e. pressure disturbances affect the whole fluid region instantaneously) - so be careful about what you are really assuming in your math. $\endgroup$ – alephzero Oct 13 '18 at 18:57
  • $\begingroup$ OK, but then how can you study the propagation on sound waves in an incompressible fluid. I suppose that you make a perturbative analysis, something like: $\rho(\vec{x},t)=\rho_0+\frac{d \rho((\vec{x},t))}{dp}|_0\Delta p((\vec{x},t))$ and, somehow you manage to relate the small perturbations. How do you end up with perturbations being able to propagate at the speed of sound? Is this the only propagation mechanism there is? I am sorry for bothering you with so many question. I'm very interested in this subject. $\endgroup$ – Carlos L. Janer Oct 13 '18 at 19:31
  • $\begingroup$ You are basically asking and answer for the unsolved physical problem; en.wikipedia.org/wiki/… I have found some solutions to this, you can check them here; researchgate.net/project/Turbulence-22 $\endgroup$ – Jokela Oct 15 '18 at 8:34
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Transition to turbulence is observed with increasing external pressure, when the real root becomes imaginary and a turbulent complex solution begins.

Brief summary of scientific direction: Using complex values of velocity and coordinates when solving nonlinear partial differential equations

Just as the square equation has complex roots, the nonlinear partial differential equations have complex solutions. It turns out that the complex solution is probabilistic. The physical meaning of the real part is the average value of the solution, and the imaginary part means the standard deviation. The nonlinear Navier-Stokes equation is reduced to an infinite system of ordinary differential equations of the first order. The complex coordinates of the equilibrium position describe the turbulent solution. Problems arise when recalculating the imaginary part of a complex solution into a real solution. But in the attached articles, for which the abstract describes the solution to these problems. For different types of roughness, the solution to these problems is different.

YAKUBOVSKIY, EG. "STUDY OF NAVIER-STOKES EQUATION SOLUTION I. THE GENERAL SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 60-66. https://world-science.ru/pdf/2016/3/14.pdf

YAKUBOVSKIY, EG. "STUDY OF NAVIER-STOKES EQUATION SOLUTION II. THE USE OF LAMINAR SOLUTIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 67-83.https://world-science.ru/pdf/2016/3/15.pdf

YAKUBOVSKIY, E. G. "STUDY OF NAVIER–STOKES EQUATION SOLUTION III. THE PHYSICAL SENSE OF THE COMPLEX VELOCITY AND CONCLUSIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 84-87. https://www.world-science.ru/pdf/2016/3/16.pdf

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