= Kolmogorov equation for structure functions
{c}
{title2=$S_3-6\nu S_2'=-4\epsilon r/5$}
For unforced <homogeneous turbulence> with <isotropic turbulence> statistics, put $F=u^2f=u^2-S_2/2$ and $S_3=6u^3K$. The longitudinal <Kármán-Howarth equation> is $F_t=r^{-4}\partial_r[r^4(S_3/6-\nu S_2')]$. Since $\partial_tu^2=-2\epsilon/3$, regularity at zero gives
$$
S_3-6\nu S_2'=-\frac45\epsilon r-\frac3{r^4}\int_0^r s^4\partial_tS_2(s,t)\,ds.
$$
Neglecting 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.
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