Kolmogorov four-fifths law

Remarks

The derivation starts from the Kármán–Howarth equation, an exact consequence of the Navier–Stokes equations for the two-point velocity correlation of homogeneous isotropic turbulence. In a stationary state it relates $\langle \delta u^3 \rangle$, a viscous term proportional to $\nu \, d\langle \delta u^2 \rangle / dr$, the forcing and $\varepsilon$. For $r \ll L$ the forcing term is negligible, and as $\nu \to 0$ the viscous term vanishes at fixed $r$, leaving the four-fifths law. Kolmogorov derived it in 1941.

It is the main exact result of the statistical theory of turbulence, with no adjustable constant. The minus sign says that the velocity increments are skewed towards negative values, the signature of energy transfer to small scales in the energy cascade. It agrees with the 1941 similarity theory for $p = 3$, but it holds independently of the similarity hypotheses. The hypothesis that $\varepsilon$ stays positive as $\nu \to 0$, the dissipation anomaly, is supported by experiments and simulations but is not proved for the Navier–Stokes equations.

Going deeper

For a student

Unlike most scaling laws in turbulence, the four-fifths law is derived from the equations of motion. Its assumptions are homogeneity, isotropy, stationarity and a finite dissipation rate in the inviscid limit. It is used to measure $\varepsilon$ in experiments and to check numerical simulations. There is a related four-thirds law for the mixed third-order structure function $\langle \delta u \, \abs{\delta \mathbf{u}}^2 \rangle = -\frac43 \varepsilon r$.