# Kolmogorov Route Employed by Uriel Frisch – Part II

(By Tomer Avraham)

U. Frisch treatment of Kolmogorov’s raw assumptions was a foundation to many research and analysis in the 80-90’s of the previous century.

The origin for this treatments are three hypothesis devised by Frisch which are not as simple and straight forward as Kolmogorov’s original assumptions:

Hypothesis 1:
In the Re → ∞ limit, all possible symmetries of the N.–S. equations, usually broken
by the (physical) mechanisms producing turbulence, are restored in a statistical sense at small scales and away from boundaries.

Hypothesis 2: Under the same assumptions as above, turbulent flow is self similar at small scales; i.e., it possesses a unique scaling exponent h such that:

with those length increments samll compared to the integral scale.

Hypothesis 3:

Again, under the same assumptions as in Hypothesis 1, turbulent flow has a finite,
non vanishing mean rate of dissipation per unit mass, ε.

Comparing Frisch hypothesis with Kolmogorov assumptions the most important thing to note is that we are dealing with length scales which are much smaller then the integral scale, hence homogeneity assumptions might be applied and that their statistics be invariant under arbitrary stationary flows.

These observations are at the basics of the 2/3 law and used to proof the 4/5 law.
Frisch does not employ the first of the Kolmogorov assumptions; in particular, all of the hypotheses used in [80] involve the Re → ∞ limit.
all three of Frisch’s hypotheses require the assumption of Navier-Stokes symmetries (in its statistical meaning), while in Kolmogorov assumptions there is no mention of that.
Finally, Hypothesis 3 coincides with experimental observations, and it also a portion in the mathematical consequences. To understand the nature of these we must remember definition of dissipation rate, given originally as:
As S is the strain rate of course.

Pay close attention: as Re → ∞ as required in all of Frisch’s hypotheses (and in Kolmogorov’s second assumption), ν → 0. Then if ε is to remain finite, it must be the case that some (first) derivatives of U become unbounded. But we emphasize that ε is a mean dissipation rate, with a spatial average constructed in Frisch’s proofs. This actually means that finite ε when there exists Re → ∞ limit is consistent with modern mathematical theories of the Navier-Stokes. equations, even though the the form of the definition of ε. As a remark it must be noted that this is true for theories which permit U to be unbound on sets of zero measure (shall be explained in part III of K41 theory), while remembering that we may ignore such sets without affecting the value of the average (and of course that ε not more then a definition),
and it does not naturally appear in the N.–S. equations. In particular, within the context So concluding our Part II of U.frisch hypothessis following Kolmogorov’s assumptions
Re → ∞ is not inconsistent with the Navier-Stokes equations if we treat the ε definition in a subtle manner.