312. Turbulence. Statistical approach 2
Save This Post as PDFLet us continue our brief discussion of stochastic approach to description of developed turbulence.
3. Kolmogorov scaling
One of the most important and beautiful results of stochastic approach is Kolmogorov scaling. Earlier, I have already discussed Kolmogorov’s turbulence on the blog in details, let me ramble about it today a little bit more.
The base of the Kolmogorov’s theory is the fact that Navier-Stokes equation describing a viscous fluid
is scale-invariant – i.e., invariant w.r.t. the transformation 
, 
, 
, where 
is scaling exponent. It is postulated that
a) scale invariance holds only in statistical sense – i.e., only averaged quantities are scale invariant,
b) there is finite energy flux 
from large scales to small ones,
c) the flux 
at scale 
may only depend on quantities defined at the same scale (in particular, on the scale itself and characteristic velocity 
of vortices with characteristic linear scale ). The last postulate is the most restrictive one, and weak deviations from scaling observed in Nature are mostly related to its violation.
Since has a dimension of mass divided by time, we find
from simple dimensional analysis. Together with scaling relations above we find 
, so scale invariance implies that the scaling exponent 
.
Scale invariance only holds at intermediate scales (in inertial range) small compared to the scale of the flow itself (say, for the flow in a pipe this large scale is diameter of the pipe) and large compared to dissipative scale
,
where viscosity becomes important (I’ve explained why it is expected to be only important as short scales in my earlier posts).
There are many technical consequences of the Kolmogorov scaling. For example, using scaling we can immediately derive an expression for the velocity correlation function of arbitrary order 
:
(1)
or expression for the turbulent viscosity
.
That’s more or less how Kolmogorov spectra look like in experiment.
It is established experimentally that Kolmogorov turbulence holds for an arbitrary simple homogeneous and isotropic medium, whenever an internal scale is absent (deviations from the Kolmogorov scaling law 
in the inertial range are about 2%). In more complicated media, for example, in plasma, or in anysotropic problems such as turbulent flow in the presence of gravity there are strong deviations from the Kolmogorov law – this is of no surprise if we look again at the Kolmogorov’s postulates. In every particular case, it is rather clear which one breaks down.
4. Intermittency
What is more surprising is that experimentally Kolmogorov scaling often holds even in the regime where it is not supposed to hold: for example, at intermediate Reynolds numbers, when the flow cannot be considered homogeneous and isotropic in average. This probably means that naive derivation of Kolmogorov law does not take into account all important features of Navier-Stokes turbulence.
Still, in the case of homogeneous and isotropic turbulence, there exists one important deviation from the Kolmogorov law. Namely, as you remember, Kolmogorov postulates that – the energy flux from larger to smaller scales (coinciding with the dissipation rate) – is the only important demensionful parameter in the theory. As it turns out, when itself strongly fluctuates, statistical parameters of its fluctuations become also important. Experimentally, it is established that the main contribution to the expectation values of (and its correlation functions) comes from strong rare fluctuations.
Strong fluctuations in the flow (seen as filaments on the Fig. above) mean intermittency. Image by LBNL visualization group.
Attempts to take this feature into account resulted in construction of fractal models of developed turbulence. The physical idea behind fractal theory of turbulence is the following. When we consider energy transfer from larger vortices to smaller ones (well, it is better to say “eddies”, but I like “vortices” more since it gives an impression that Navier-Stokes turbulence is related to Eulerian flow), only relatively small part of the large vortex participates in the energy transfer. We define how large this part is by introducing a coefficient 
equal to the ratio of the volume of created smaller vortices with the scale 
to the volume of initial vortex with characteristic linear scale 
. Correspondingly, the expression for the velocity correlation function (1) is changed and now reads
, (2)
where
.
Although we have another parameter 
to fit observed turbulent spectra, the fractal model is not able to survive – it predicts a very particular dependence of correlation functions (2) on , contradicting to experiment.
One can fight with this fact by introducing additional fitting parameters (such as in so called multifractal model of turbulence), but the sense that we have a better understanding of turbulence is clearly lost.
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Hi Dmitry
“The Navier-Stokes is scale-invariant.” and the all the physics of the fluid mechanics is captured in Navier-Stokes eq, how can the solution be not scale invariant?
Also, I wonder if simulation agrees well with experiment. If they do, I think we at least know all the physics.
Keith
Hi Keith,
Lubos has pretty much outlined the answer to your question, but I would also advice you to check out what is spontaneous symmetry breaking.
Regarding simulations – they actually pretty much do, so yes, there is a strong impression that Navier-Stokes is a very good approximation for description of fluids.
Cheers,
Dmitry.
Keith: isn’t it like asking why isn’t your body rotationally invariant under rotations around the z axis even though the laws of Nature are invariant?
Is it really news that the laws invariant under symmetries still have most solutions being non-invariant under the symmetry?
Sorry for asking about obvious questions (as I often do).
“Is it really news that the laws invariant under symmetries still have most solutions being non-invariant under the symmetry?”
Probably I could only solve spherical cows and forget most of the cows are not spherical symmetric. In terms of solving differential equations, probably most of the time the symmetries are broken by initial/boundary conditions.
It is interesting to note that the concept of spontaneous symmetry breaking is so ubiquitous and powerful; not just useful in high energy physics.
Keith
Hi Keith,
You don’t need to be sorry. Many really good physicists I knew (like Ginzburg) were never afraid to ask stupid questions, that was one of the reasons they knew so much.
Cheers,
Dmitry.