# Kolmogorov’s Route Part I – Universality Assumptions

“An individual tends to absorb the surrounding spirit and to radiate the acquired lifestyle and worldview to anyone around, not just to a select friend.”

Andrei Kolmogorov contribution to the field of turbulence was not less than astonishing.  The original Kolmogorov  papers were difficult to read as some of the assumptions made are not implicitly stated and the resulting results do not clearly follow from the assumptions as they are simply highly nontrivial. On the other hand, these results have been validated under the correct assumptions.

A good starting point would be to remember the symmetries found at initial low Reynolds numbers and which are broke as the turbulent phase approaches but still posses certain symmetries in the statistical sense. This assumption stands in the basics of Kolmogorv’s Universality Assumptions (there are actually two of them):

### Kolmogorov’s  First Universality Assumption

At very high, but not infinite, Reynolds number, all of the small-scale statistical properties are uniquely and universally determined by the length scale ℓ, the mean dissipation rate (per unit mass) ε and the viscosity ν.

### Kolmogorov’s  Second Universality Assumption

In the limit of infinite Reynolds number, all small-scale statistical properties are uniquely and universally determined by the length scale ℓ and the mean dissipation rate ε.

It is important to note that both of these statements concern “small-scale” statistical properties. The actual fact is that it is needed first to indicate what Kolomogorov actually meant by small scale. These small scales are certainly not the  fluctuating quantities in a Reynolds Averaging Decomposition (although even one equate them to those hell doesn’t brake loose and the essential validity still stands albeit not in the classical Kolmogorov meaning).The more precise way is to affiliate those fluctuating quantities to a high-pass filter part of Navier-Stokes solution, meaning the high wave number components. This is most easily explained in Hilbert space, but it’s enough to say at this point that these small scales are associated with length scales that are far smaller than the integral scale.

Kolmogorov was had a mastery at the essential use of dimensional analysis and by that related certain parameters according to which he found would be most influential. This simple yet ingenious thinking let him to results that due to the “cold war” were not available to the west until quite some time elapsed untill Uriel Frisch astonishing work translated them to the building blocks of turbulence.

Part II shall be a dive into Frisch interpretation of Kolmogrov’s universality laws. It might look at the beginning as a tough complication understandable approach, but the kind blogger shall walk you through the tough part until we get to the essentials: The K41 Theory”, “The 2/3 Law” and fa and foremost ” The k-3/5 Law energy spectrum.

This is turbulent charm at its best!