Kolmogorov 1941 theory

Remarks

The proof is dimensional analysis. By hypothesis 3, $S_p(r)$ depends only on $\varepsilon$, with units $\mathrm{m^2/s^3}$, and $r$, with units $\mathrm{m}$. The only combination of them with units of $(\mathrm{m/s})^p$ is $(\varepsilon r)^{p/3}$. The spectrum follows in the same way, since $E(k)$ has units $\mathrm{m^3/s^2}$. The conclusions are only as good as the hypotheses, which are physical assumptions, not consequences of the Navier–Stokes equations.

For $p = 2$ the law $S_2(r) \propto (\varepsilon r)^{2/3}$, the two-thirds law, and the $-5/3$ spectrum agree well with measurements in many flows. For $p = 3$ the prediction is consistent with the exact four-fifths law, which gives $C_3 = -4/5$. For larger $p$ measured exponents fall below $p/3$, an effect of intermittency that the 1941 theory ignores; Kolmogorov and Obukhov proposed a refined theory in 1962 to account for it.

Going deeper

For a student

The theory is the backbone of the phenomenology of turbulence. It gives the scalings used everywhere: the Kolmogorov microscales, the $\mathrm{Re}^{9/4}$ cost of direct simulation, and the $-5/3$ spectrum against which data and models are checked. It is not a theorem about the Navier–Stokes equations; deriving it, or even the existence of a dissipation rate independent of $\nu$, from the equations is a major open problem. Its one exact consequence is the four-fifths law for $S_3$, which Kolmogorov derived from the Kármán–Howarth equation. Deviations at higher order, called anomalous scaling, are the subject of the theory of intermittency.