221. Turbulence: Kolmogorov law derived in one line
Save This Post as PDFAs you surely understood from the previous post, today is hydrodynamics and turbulence Sunday on NEQNET. If so, let me still 5 minutes of your precious time discussing the physics of developed turbulence.
The latter is very complicated. And today is Sunday. Do I really have to overload your brain on Sunday with exceedingly technical discussion? No necessarily – fortunately the only result in turbulence which is really worth mentioning during holidays is extremely simple to understand. Moreover, it can be derived in just one line of calculation. What I am talking about is the spectrum of developed turbulence first derived by famous Russian matematician Andrey Kolmogorov.
Imagine that you pump more and more energy into the fluid flow (say, simple water that flows from the tap in your kitchen). Increasing the energy (that is, turning the tap), eventually you will see that the smooth flow becomes unstable, fragmented – for example, small drops of liquid will get off the main stream. The moment when the flow becomes unstable is determined by so called Reynolds number of the fluid
,
where 
is the velocity of the flow, 
is viscosity of the fluid and 
is a characteristic linear scale of the flow (for example, for the flow in the pipe is given by the diameter of the pipe).
The flow becomes unstable or turbulent for Reynolds numbers of the order 100.
One can ask what is the spectrum of the velocity field 
or energy of the flow 
in this situation. Naively, one may think that viscosity should play an important role in the regime of developed turbulence. However, high Reynolds numbers correspond effectively to the case of low viscosities, so viscosity cannot be important for the spectrum of turbulence. The latter should actually only depend on geometry of the flow, that is, the scale and maybe energy that we pumped into the flow. The dependence of on 
turns out to be self-similar.
It is actually hard to visually recognize this self-similarity in the turbulent flow: for example, take a look at the picture below
- it is actually a 3d simulation of Kolmogorov self-similar turbulence. Self-similarity of the flow becomes clearer if you plot the spectrum of excitations itself:
To derive the functional dependence on 
at intermediate scales, let us first understand the physical picture of developed turbulence. The energy is pumped at very large scales (corresponding to small ’s) of the order of the size of the system. Let us suppose that amount of energy 
which is pumped per unit time remains constant (this is reasonable – you opened the tap and went cooking 
).
Clearly, the same amount of energy should be dissipated during the same time (otherwise, the overall energy of the flow will grow without bound, and your tap will break down with some unpleasant consequences). But dissipation cannot happen at intermediate length scales (intermediate ) – effectively, Reynolds numbers are high there, which corresponds to low effective viscosity. The energy should be transferred first to small scales (large ’s), where effective viscosity is large and Reynolds numbers are small.
In fact, that’s enough for us to derive the self-similarity of the flow. Let us define as the amount of energy dissipated in the unit volume per unit time. The dimension of this quantity is energy/length
/time.
If 
is the kinetic energy of the flow stored in the modes with wave numbers 
, the only way to construct as the combination of and is
.
That’s it – this is famous Kolmogorov law derived by using just dimension estimations 
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Do you remember the mysterious figure of 3 in Feynman’s discussions of turbulence? Is that the same thing that appears in your exponents? If so, that’s cool!
I actually don’t, where was it – in Feynman’s lectures of physics?
Hi,
what does similarity exactly mean in turbulent flow?
habib
Dear Habib,
it means Kolmogorov’s power law scaling of the correlation functions of the energy density in the flow as well as the velocity of the flow.
Cheers,
Dmitry.
I wish it is so simple. More and more evidence is out there that the picture is not as straightforward as drawn here – viscosity turns to be important at any Reynolds number and it affects the flow from smallest through the inertial (intermediate) to upper scales, etc.
Yet, the combination of physicists, especially with a non-equilibrium tools and experimentalists (engineers) with simple measurement tools could be necessary in order to tackle this >> 1Mn problem.
Hi Alex
My impression is that Kolmogorov scaling is good first approximation into the problem. There are corrections to scaling that come from many other effects such as intermittency (please see my further posts on the problem – for example, this one as well as discussion in comments), but there is no question that Kolmogorov scaling is there.
Cheers,
Dmitry.
A corollary of the derivation of the Kolmogorov spectrum by a purely dimensional argument is that any well-dimensioned model will give the same result, so an ability to predict the Kolmogorov spectrum is no test of the correctness of a model. The log law for the velocity profile is similarly “insensitive to derivation”.
Another feature of the dimensional derivation is that it works in terms of the transfer of turbulence energy, without gains or losses, from production scale to dissipation scale. Not all turbulence energy need be in the form of turbulence kinetic energy. For example, in high-Rayleigh-number flows there can be so much energy in the pressure fluctuations that the -5/3 law becomes very approximate.