312. Turbulence. Statistical approach 2

Let 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.