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In the inertial range, velocity increment statistics also become independent of kinematic viscosity. Combined with the Kolmogorov first similarity hypothesis, this gives $S_p(r)={\beta }_p(ϵr{)}^{p/3}$ by dimensional analysis. Signed integer moments and absolute noninteger moments must be distinguished; the signed third-order coefficient is negative in a forward energy cascade.

Local isotropy of turbulence means that joint small-scale velocity increment statistics are approximately unchanged by rotations, even when the large-scale flow is anisotropic. The universal equilibrium range contains separations much smaller than the integral scale of turbulence: their adjustment times are short compared with the outer eddy turnover time, and their normalized statistics are assumed independent of the detailed large-scale forcing. It includes the inertial range and the dissipation range.

The Kolmogorov first similarity hypothesis says that these locally isotropic statistics depend only on $r$, mean viscous dissipation $ϵ$ and kinematic viscosity $\nu$. Dimensional analysis therefore expresses the second-order longitudinal structure function in terms of the Kolmogorov microscales:The universality of $F$ is an assumption of the theory, not a consequence of dimensional analysis alone.

The Kolmogorov second similarity hypothesis removes dependence on kinematic viscosity when $\eta \ll r\ll \ell$. Only $ϵ$ and $r$ remain, so the Kolmogorov two-thirds law isThe same reasoning gives the predicted order-$P$ longitudinal structure function:These are signed moments for integer $P$; for noninteger orders, use $⟨|\Delta v{|}^P⟩$. In particular, the signed third-order coefficient is negative, ${\beta }_3=-4/5$, in the Kolmogorov four-fifths law.