247. Physics of turbulence: four puzzles
Save This Post as PDFBefore starting to discuss theories and models describing phenomena of weak and developed turbulence in fluids, plasmas etc., etc., let us first recall why exactly theoretical physicists were so unsuccessful so far in understanding turbulence at the quantitative level. In order not to make this post too long 
, I am going to list only four main puzzles which, I hope, will show what is the heart of the problem.
I already mentioned the first puzzle before -
1. Why critical value of Reynolds number is so high?
As you remember, the dimensionless Reynolds number made of viscosity 
, velocity of the turbulent flow 
and a characteristic linear scale of the flow 
is defined as
. (1)
The actual value of 
controls whether the flow of the fluid is turbulent or smooth, laminar, as physicists say.
Universally, if you change parameters of the flow – increase its velocity or/and somehow decrease viscosity (for example, by changing chemical composition of the fluid) in such a way that Reynolds number increases – the flow looses stability and becomes turbulent at some critical at some 
.
Typically, this critical value (where the flow starts to loose stability) is rather high:
. (2)
So, the first puzzle of turbulence is why (2) is actually so high? My point is that usually in physics a sharp transition from one regime to another corresponds to values of controlling parameter of the order 1.
2. Is viscosity really important? Vortices
As I said above, a strongly turbulent regime corresponds to high Reynolds numbers, i.e., to effectively low viscosity (note that the latter is in the denominator of (1)).
Naively, viscosity is not important. Indeed, Kolmogorov spectra of developed turbulence are universal (in the inertial interval) and don’t know anything about viscosity.
So, what if in order to treat developed turbulence we will neglect viscosity all together: that is, instead of studying very complicated Navier-Stokes equation
we study much simpler Euler equation
?
As it turns out, the life of a Eulerian flow is in a sense more complicated, not less complicated, than the life of a Navier-Stokes flow. First of all, according to Kelvin-Thompson theorem, Eulerian flow supports infinite number of non-local integrals of motion – the latter correspond to the vortex field frozen-in into the flow. Vortices strongly interact with each other and with themselves (!) – their interaction is described by Coulomb law. As it seems, this interaction may lead to appearance of finite time singularities in the velocity field (and correlation functions of velocity).
In other words, a typical solution of the Euler equation corresponding to the flow with a large number of vortices wants to blow up in finite time.
It is clear though, that divergent behavior of the velocity field can be cured by effects of viscosity that smooth out physics at short scales.
So, I would like to formulate the second puzzle as follows: is viscosity really important for turbulence in inertial interval or not?
3. Universality vs. non-universality
Once the flow is in the strong turbulent regime, spectra of excitations of the flow (let us say for definiteness – sound waves) are described by the Kolmogorov law. The latter does not know anything about microphysics of the fluid (viscosity, etc.) – it is absolutely universal – that is, all turbulent phenomena in Nature exhibit strongly similar behavior described by one and the same Kolmogorov law.
On the other hand, is the Kolmogorov attractor itself universal? If I introduce some initial conditions for the flow of the fluid and start to increase the Reynolds number, will I observe the same Kolmogorov behavior at late times for arbitrary initial conditions? Or can I instead face with something else? My example with Euler flow above shows that I probably could.
Also, as we know, the value of critical Reynolds number, where the flow becomes unstable can vary quite a bit if we vary geometry of the flow – but orders of magnitude. Why itself is not universal?
Finally, if you’ve read the previous post on turbulence, you might have noticed that transition to turbulence also depends on the geometry of the flow – it can happen either suddenly or in several steps, with subsequent increase in complexity of the flow. What does the transition scenario really depend on?
To conclude, the third puzzle is formulated as follows: what part of the physics of turbulence is really universal, and what is not? Why some things (like ) that have to be universal are not universal?
4. Intermittency
Among different scenarios of transition to turbulence one stays apart – the scenario called intermittency.
Intermittency in a few words means the following. Suppose you slowly increase the value of the Reynolds number. Of course, eventually, the flow will loose stability and become turbulent. But then, increasing even more, you again return to the laminar behavior of the flow! Increase again – and again you are in the turbulent regime, then again – in laminar etc. etc. The number of jumps from the laminar behavior to the turbulent one can be rather large or intermittency can be absent at all depending on geometry of the flow.
The fourth puzzle is what is the physics behind intermittency? how to describe it?
I hope that my short post will give you some impression why studying the physics of turbulence is so much fun.
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Privet, 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
Dear Lubos,
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.
Dear Dmitry, I don’t think that my argument requires any “distinct levels”, if I remember well that you mean that the level’s label is integer-valued or that the self-similarity becomes exact in the UV.
If you doubted that fractals are relevant for turbulence, see e.g.
http://journals.cambridge.org/.....aid=392671
http://scholar.google.com/scho.....turbulence
Dear Lubos,
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.
Hi Dmitry.
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?
Dear Javier,
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.
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.
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. In this context, the study of weak solution to 3-D Euler flows that do not preserve energy by some mathematicians is relevant.
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.
Dear Prof. Tanveer,
I am honored to see you at NEQNET!
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 suggest you to go a bit deeper into the literature before discussing turbulence on such high level. E.g. Tsinober’s book “An informal introduction to turbulence” has a whole chapter of “misconceptions” like: ’since viscosity is active on small scales, it’s not important for inertial range’ and so alike. The problem in such statement arises from misuse of the Kolmogorov’s postulates. One great example is the effect of dilute polymers: few ppms of long molecules, that are 3 order of magnitude smaller than the Kolmogorov scale of the flow, affect the flow from small, through inertial and to the large scales. Check it out. So, how can you say that viscosity is not important? How can one claim self-similarity in this case?
Dear Alex,
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.