We were talking about Navier-Stokes 3d turbulence of chemically homogeneous liquid flow – the most common example of turbulence in nature. Sure, there are many other examples of turbulence not described by Kolmogorov scaling – such as weak turbulence of surface waves, where exponents are different, and of course, turbulent flow in the presence of polymers. So what? You are talking about different physical problem.

On the other hand, if you are saying that there are strong deviations from Kolmogorov scaling in 3d N.-S. turbulence, I think, you need to provide at least some evidence in favor of so out-of-mainstream statement, a reference to a research (experimental?) paper will do.

Cheers,
Dmitry.

I am honored to see you at NEQNET!

Interesting take, though it is not true that viscosity does not play any role in the inertial scale?it does indirectly by serving as a sink for energy and enstrophy. Perhaps, you meant no direct role.

Sure, but that’s why I said – “in inertial interval”, at length scales much larger than the sink scale.

I actually had the same picture with turbulent viscosity and reconnections of vortices in my mind. What’s not clear to me in this picture is the following.

Suppose that we are in the regime of developed turbulence, where everything is more or less self-similar according to Kolmogorov (at least approximately as you pointed out, but the approximation is good). There are probably lots of small vortices (by “small” I mean small lengths L of closed vortex lines) which reconnect rapidly with each other. Of course, they also easily reconnect to themselves.

Since there is approximate Kolmogorov self-similarity, larger vortices also reconnect but just at longer time scales (which can be estimated by self-similarity).

So, doesn’t self-similarity really mean that there can be no relevant length scale where reconnection can be neglected?

Also, could you present your views on the mechanism of breakdown of self-similarity?

Cheers,
Dmitry.

I don’t question relevance of attractors – I don’t see how to estimate the number of hierarchy levels that you mention.

Also, I think, the behavior of velocity correlators in the vicinity of transition to turbulence might be not self-similar.

Cheers,
Dmitry.

thanks for your very nice comment. My impression is that the Claymath problem is related to turbulence in two respects:

a) whether the general solution of the N.-S. equation exists will probably show you whether the Kolmogorov attractor is universal (i.e., if a general solution exists, then Kolmogorov attractor is universal, probably not vice versa).

b) how much is this solution (if it exists) smooth will show you how viscosity is relevant.

If not I think that the turbulence looks like a more important question (ok, maybe not availabe to a precise mathemathical formulation) and there should be a similar prize about it?s resolution, what you think?

Sure, turbulence is more general problem, but then again, mathematicians like nicely and precisely formulated problems. How would you formulate “the problem of turbulence”?

Cheers,
Dmitry.

However, experiments by Couder et al in France with bubbles that helps in turbulent flow visualization show that dissipation is hardly uniform in space and time. There are particular regions of large dissipation; people believe this is where vortex reconnection process is important–this kind of non uniformity in the dissipation parameter epsilon is not taken into consideration in Kolmogorov theory–not surprisingly people see deviations from the scaling laws.

The vortex reconnection process is believed to be the signature of vortex lines in 3-D Euler approximation forming kinks as they come together and touch. So, going into the question of when Euler will approximation will be valid for large Reynolds number, it will be in regions of space and time when vortex reconnection is not important. Unfortunately, because of the complications of actual numerical computation for large enough Reynolds number, let alone a mathematical proof, this is all a conjecture.

If you doubted that fractals are relevant for turbulence, see e.g.

http://journals.cambridge.org/.....aid=392671
http://scholar.google.com/scho.....turbulence

Two summers ago I assisted to a seminar about fluid dynamis in the UCM. The semianr was clerarly addressed towards the Claymath problem on Navier-Stokes and, consequently, they talked mainly about evolution of solutions along charazterisitics and wether of not there was a blowing up of them.

The lecturer (sorry, I don?t remember his name) stated that the common belief among mathemathicians was that viscosity seemed almost irrelevant for that question and that the Euler equaion contained the difficoult part. In fact it looked as if adding viscosity the questions were a litle bit easiests. The actual results were centered in the two dimensional equation where the availability of the vortational formulation simplified the things a lot, and consequantly, it was not obviouws how to extend results to the thre dimensional case.

The questions is that I don?t see very clear wether or not that claymath problem and the question or turbulence dynamics are equivalent (or even relateed in a more or least direct way).

If not I think that the turbulence looks like a more important question (ok, maybe not availabe to a precise mathemathical formulation) and there should be a similar prize about it?s resolution, what you think?

I think I’ve got your idea, but, as I understand, typically in turbulence there are no distinct levels of hierarchy like, say, in Sierpinski triangle.

Sometimes (like in Taylor-Quette) you can separate vortex dynamics and dynamics of sound waves along them. First, chaos gets realized for interacting sound waves and then for vortex degrees of freedom. You don’t see strict self-similarity though when approach to turbulence (self-similarity appears only after the developed turbulence is established). For example, Taylor vortices have some very definite size depending on sizes and angular velocities of cylinders.

Cheers,
Dmitry.

let me offer a draft answer to your 1st question. (In fact, the critical Re number is often as high as 10,000.) Turbulence generates self-similarities. So the typical length scales produce geometric series.

Imagine that you need K – of order one, like 2.pi or 6 – levels of the hierarchy for the turbulence to propagate everywhere inside. And the ratio of length scales in the “fractal” is M – another number of order one, also around 6.

So the length scale, in natural units, needed for turbulence is of order K^M, and 6^6 is around 50,000, more than needed. Recall that the length scale in natural units is nothing else than the Reynolds number.

So morally speaking, I reduce your first question to the question why there is self-similarity.

Best
Lubos