Physical Explanation for “Kolmogorov -5/3 spectrum” in Fluid Mechanics

Question

According to Kolmogorov, the energy spectrum function of a turbulent fluid is given as,

$E(k)=C\epsilon^{\frac{2}{3}}k^{\frac{-5}{3}}$

where $\epsilon$ is the energy flux and $k=\frac{2\pi}{r}$ where $r$ is the length scale.

The normal explanation I see in most physics texts and articles is that the -5/3 exponent is found purely through dimensional analysis. Obviously dimensional analysis is very useful, but often times there is a more physical explanation as well. Does anyone have any insights?

Answer 1

The standard explanation is that there is a constant flux of energy from large eddies to smaller eddies. The time scale for an eddy of scale $r$ to turn over is $\tau \sim \frac{r}{v(r)}\sim \frac{1}{kv}$, the energy density for scale $r$ is $\sim v(r)^2$, so you get an energy flux rate $\epsilon \sim v^2/\tau \sim v^3k$ which is assumed constant. You then estimate she spectral energy density as $E \sim v^2/k \sim \epsilon^{2/3}k^{-2/3 -1} = \epsilon^{2/3} k^{-5/3}$. I hope that is a little better than pure dimensional analysis.