Locality appears to be an omnipresent feature of physics, including within turbulent fluid dynamics.

In 1941 Kolmogorov postulated that the turbulent energy cascade is driven by scale-local interactions, where the eddies at a given scale solely interact with eddies of nearby scales. Hence they transfer kinetic energy in a waterfall-like "cascade" from the large integral scale all the way down to the dissipative ones, wherein the energy is converted into heat.

As far as I understand, much numerical investigation has been done on this, and this scale-local nature of interactions has largely been confirmed to be true. See e.g. (DOI) 10.1103/PhysRevFluids.3.084601 or 10.1063/1.3266883 (arXiv).

However, I cannot find much on the scale-locality of correlations. It seems that the only correlations that are measured in turbulence research is the correlations between eddies in real-space, i.e. in terms of their physical seperation distance. Why has nobody investigated the correlations across scale? Does the scale-locality of interactions imply that also the correlations between eddies are scale-local?


1 Answer 1


The study of interscale interactions from a scale perspective is investigated using what has come to be called triadic interaction analysis. The idea is to convert the equations to their Fourier representation and then look at which wave numbers interact with each other and by how much. A good representative paper from early in the development of the method is Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interactions.

The studies reveal that there are local, non-local, and distant interactions between scales. Under high Reynolds number, homogeneous conditions these interactions are dominated by local behavior as Kolmogorov hypothesized. However, under more realistic conditions there are other effects that become important, such as non-local interactions where small scales transfer energy to large ones in an inverse cascade (also know as back-scatter). It's also much more complex for compressible turbulence, where the interactions happen amongst 4 wave-number pairs instead of 3. These are all active areas of investigation.

  • $\begingroup$ Thanks! But two things: (1) I'm specifically curious about the correlations. I.e., what is the characteristic correlation length between the eddies in scale space? In your example for instance, assuming homogeneity and high Re, are also the correlations scale-local? (2) Also, how "high" is high Re? Presumably, what was high Re in 1994 is different to what is considered high Re now. I think these two papers seem to argue strongly for a universally scale-local view of turbulence: aip.scitation.org/doi/pdf/10.1063/1.3266883, aip.scitation.org/doi/pdf/10.1063/1.3266948. $\endgroup$
    – Niko
    Commented Jan 4, 2021 at 13:54
  • $\begingroup$ @NikGourianov You'll have to be more precise in what you are calling correlation lengths... Things like the integral, Taylor, or Kolomogorov length scales are all "correlation lengths" and they all have direct analogues in wave number space. For some other correlation lengths, you'll have to be more precise about the definitions of them, but generally speaking you can take the Fourier transform of the expression to find a corresponding wave number correlation "length." $\endgroup$
    – tpg2114
    Commented Jan 4, 2021 at 14:02
  • $\begingroup$ As for how "high" is "high", that's one of the great fiats of Kolmogorov's theory -- if a flow doesn't follow the hypothesis, then it's just not high enough Reynolds number. The theories are all based on the assumption of "For high enough Reynolds number..." and so there really isn't a definition for how high is high enough. $\endgroup$
    – tpg2114
    Commented Jan 4, 2021 at 14:03
  • $\begingroup$ Lastly -- the paper you linked to is saying the same thing I am, which is that scale-local interactions dominate generic turbulent flows. However, compressibility (due to high speed, or combustion, or density inhomogeneity like in the atmosphere) and the presence of boundaries like walls lead to things like inverse cascades even at high Re, which are non-local in nature in general (although how much and under what conditions is an active research area). $\endgroup$
    – tpg2114
    Commented Jan 4, 2021 at 14:06
  • $\begingroup$ (1/2) thanks a lot @tpg2114! With "characteristic correlation length in scale space", I meant some characteristic scale-seperation under which eddies can be expected to be correlated with each other. In e.g. science.sciencemag.org/content/357/6353/782.abstract, they isolate eddies at a given scale "delta" using a filter, and then they calculate a sort of correlation between eddies at scales delta, delta/2, delta/4, etc. I suppose one could also look at the correlation between two points in wavenumber space, as I guess the physics don't depend on how one views scale? $\endgroup$
    – Niko
    Commented Jan 4, 2021 at 15:44

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