Why is there a breakdown of Kolmogorov scaling in turbulence? What causes intermittency?


The basic Kolomogorov theory is a mean-field theory -- dissipation rate is considered to be constant over the whole volume of liquid. While the dissipation rate should depend on position -- that is where intermittency comes from.

Somewhere in 60's Kolmogorov and Obukhov attempted to account for this effect. But the problem is that there is the "Landau objection" -- the impossibility to devise universal formula, independent on the geometry of the problem (with scale $l$) for relatively small scales (which are $ \ll l$):

... It may be thought that the possibility exists in principle of obtaining a universal formula, applicable to any turbulent flow, which should give $B_{rr}$ and $B_{tt}$ for all distances $r$ that are small compared with $l$. In fact, however, there can be no such formula, as we see from the following argument. The instantaneous value of $(v_{2i}-v_{1i})(v_{2k}-v_{1k})$ might in principle be expressed as a universal function of the energy dissipation $\epsilon$ at the instant considered. When we average these expressions, however, an important part will be played by the manner of variation of $\epsilon$ over times of the order of the periods of large eddies (with size $\sim l$) and this variation is different for different flows. The result of averaging therefore cannot be universal.

(Fluid Mechanics, Landau and Lifshitz, chapter 34.)

  • $\begingroup$ Great question and a great answer @Kostya +1 $\endgroup$ – user346 Feb 24 '11 at 13:16

Another way of seeing this has recently come out of the Functional Renormalisation Group (FRG) method.

Usually Renormalisation Group (RG) flow equations are local in momentum. This means that the correlation functions of two theories that are coarse grained at two neighbouring (momentum) scales $k$ and $k+dk$ only differ for momenta close to $k$. It was found here that this decoupling property (momenta that are not close to $k$ decouple from the rest of the RG flow) is not satisfied for the problem of turbulence. This property was associated with the presence of a cascade of energy here.

Then there is indeed no chance for universality to occur since large scale properties affect the RG flow of correlation functions at all momentum scales. Moreover, this makes it possible to consider multi-scaling within the RG framework since the RG flow can never truly reach an Infra-Red fixed point.


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