 Kolmogorov equation for structure functions

For unforced homogeneous turbulence with isotropic turbulence statistics, put $F=u^{2}f=u^2-S_2/2$ and $S_3=6u^{3}K$. The longitudinal Kármán-Howarth equation is $F_t=r^{-4}{\partial }_r[r^4(S_3/6-\nu S_2^{\prime })]$. Since ${\partial }_t{u}^2=-2ϵ/3$, regularity at zero givesNeglecting the last term in a locally equilibrated small-scale range gives the displayed equation. The Kolmogorov four-fifths law follows where the viscous term is negligible as well.