315. Turbulence: order and disorder in turbulent flow
Save This Post as PDFLet us continue our short excursion into physics of developed turbulence (I hope that you don’t mind, if you do – please let me know 
) Last time we have discussed technicalities related to averaging and statistics of the turbulent flow, today I would like to get back to foundations and discuss a bit various structures typically seen in the turbulent flow. This way we will understand better which features our future complete theory of turbulence will have to explain
I’ll start by discussing the so called Richardson model. This is a very old model (created back in 1920s – even before Kolmogorov) qualitatively explaining the structure of the turbulent flow.
(Richardson, a rather interesting fellow, was the first (I think) who proposed to solve differential equations in order to predict weather There were no computers at that epoch yet, and once he tried to solve corresp. differential equations by bare hands. Failed. Anyway, let me get back to his model )
As you may remember, one of the most important parameters in the physics of turbulence is Reynolds number. We also know that the energy cascade in the flow is directed from larger scales towards smaller scales. These two observations are enough to qualitatively understand the Richardson model.
Let us start from the largest possible scale in te flow, he said. Reynolds number 
is large there, so the flow is unstable. The instability develops with time, and this leads to breakdown of larger vortices (eddies) into smaller ones (with characteristic linear scale 
and velocity 
). The Reynolds number characterizing motion of the fluid in these smaller vortices is somewhat smaller:
,
but still large enough for the flow in smaller vortices to be also unstable. Therefore, smaller vortices will also break down into even smaller ones etc. etc. until the effective Reynolds number won't drop below the critical value. At this point, smallest vortices will become dynamically stable, albeit dissipating due to viscosity (see an article on Wikipedia about Taylor-Green vortex – not too lengthy discussion, though – or any of the textbooks I mention at the bottom of the post).
Although, I think, this picture is ultimately correct, Nature turns out to be far more rich than that.
For example, when you warm up the bottom layer of the fluid, it will lead to convection and eventually to turbulence, if you heat your tea pot too much.
Rayleigh-Benard convection at Rayleigh number about 108. Image by www.lbmethod.org.
Another example of the turbulent flow is Taylor-Quette flow (a flow between two rotating cylinders – see the Fig. below). Generally, as it turns out, ordered structures may appear spontaneously and even form various lattices in turbulent flows. Topology of these structures does depend on geometry of the flow.
Increasing instability of the flow (by, say, increasing amount of pumped energy) leads to both increase of complexity in these structures and appearance of defects. Complexity in the flow typically increases in several discrete steps or bifurcations (compare it to the discussion of increasing complexity in the Taylor-Quette flow). It so happens that usually individual cells in the structures are so stable, that it is possible to identify them both in the transitive and strongly turbulent regimes. In such situations, space-time disorder in the flow is related to chaotic motion of defects along strongly ordered lattice.
While the large scale structure of the turbulent flow can be very well ordered, its structure at smaller scales is usually so much more complicated that eye cannot pick any correlation at all. But then again – sometimes it can, such as when turbulent flow is strongly anisotropic and/or inhomogeneous. In this case, dynamical and kinematical constraints strongly affect topology of structures in the flow at all scales – the flow becomes effectively low dimensional, and low dimensional turbulence is a completely different story (I would like to talk about it later as well).
For example, so called vortex dynamo mechanism may lead to explicit formation of ordered large scale structure of the flow from completely disordered low scale structure. This process is characterized by the inverse cascade – transfer of energy from smaller scales to larger scales.
So, I guess, if we will want to ultimately understand turbulence, apart from answering the questions I mentioned before we will have to understand
1. how geometry of the turbulent flow affects topology of ordered structures (or the possibility of their appearance if you want a more general question) and
2. how dimensionality affects the problem
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Reading about your posts on developed turbulence, I can’t help but think about Feigenbaum’s constant.
Surely someone has tried before to reconcile the energy cascade and onset of turbulence with this remarkable result.
It seems to me that it should not only explain turbulence, but intermittency as well. I would be shocked if it didn’t.
There must be some analytical description of the mechanism by which the energy is dissipated that is subject to this onset of chaos.
Surely the release of energy isn’t continuous and subject to some kind of hysteresis, or at least some oscillation.
Has this been investigated before ?
Hi J.D.
Feigenbaum constant is a “feature” of systems where transition to chaos happens via period doubling. There was long time belief (that goes back to Landau 1944) that in Navier-Stokes turbulence transition to chaos happens also through period doubling. For N.-S., such regime was later found to be unstable but the doubling story does not seem to be completely finished yet – I was going to talk about it more in one of the next posts (this week, probably).
Cheers,
Dmitry.