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This turbulence closure assumes the velocity increment skewness is the same throughout the equilibrium range, with its value fixed by the Kolmogorov two-thirds law and Kolmogorov four-fifths law. It yields a simple crossover between differentiable and inertial-scale increments, but suppresses the scale dependence of skewness and has no independent spectral-transfer dynamics.

The equilibrium range consists of scales below the energy-containing motions whose statistics can adjust rapidly to the local energy supply. It includes the inertial range and dissipative scales. Approximate equilibrium concerns mean transfer and viscous dissipation, not absence of intermittent spatial fluctuations.

The integral scale of turbulence characterizes energy-containing velocity correlations; in isotropic turbulence one definition is $\ell ={\int }_0^{\infty }f(r)dr$. Its turnover time is of order $\ell /u$. Large-scale intensity fluctuations can change the energy supplied to the equilibrium range.

In the locally isotropic equilibrium range, the small-scale velocity increment statistics depend on the separation, mean viscous dissipation and kinematic viscosity, independently of large-scale forcing details. Dimensional analysis leaves the Kolmogorov microscales as normalization scales and gives the displayed second-order longitudinal structure function. Universality of $F$ is part of the hypothesis, not something proved by dimensional analysis.

External intermittency is the alternation of turbulent and non-turbulent fluid at a point as a fluctuating interface passes it. Internal intermittency in the equilibrium range instead describes uneven, bursty concentrations of small-scale velocity gradients and viscous dissipation within turbulent fluid. Integral-scale intermittency is modulation of the energy-containing motions and their energy supply. It is non-universal because it retains information about forcing, boundary conditions and the history of the large-scale flow.

The Landau intermittency counterexample challenges the unconditional universality asserted by Kolmogorov 1941 theory. Suppose turbulent populations have different conditional mean viscous dissipation rates ${ϵ}_a$. Even if each obeys the same conditional scaling, averaging over populations giveswhich generally differs from ${\beta }_P{r}^{P/3}⟨{ϵ}_a{⟩}^{P/3}$. Thus a coefficient normalized by the global mean depends on the large-scale intensity distribution. The third-order case escapes this particular objection because the power of ${ϵ}_a$ is one.

The Kolmogorov refined similarity hypothesis replaces global mean viscous dissipation by coarse-grained energy dissipation ${ϵ}_{\mathrm{AV}}(r)$. It assumes thathas universal conditional statistics at sufficiently large local Reynolds number. If the conditional $P$th moment is ${\beta }_P$, the law of total expectation givesThe refined similarity hypothesis does not itself prescribe the distribution of coarse-grained energy dissipation. Scale dependence of those moments can change the predicted exponents. As before, noninteger orders require absolute velocity increments.

The first similarity hypothesis of Kolmogorov 1941 theory states that at sufficiently large Reynolds number, the small-scale statistics in the equilibrium range are locally universal and determined by the mean turbulent kinetic energy dissipation rate $ϵ$ and kinematic viscosity $\nu$, together with the separation or wavenumber being measured. Large-scale forcing details are excluded only at scales well below the integral scale of turbulence.

These parameters define the Kolmogorov microscalesDimensional analysis gives universal functions $F$ and $G$ withThe dimensional factors follow because $S_2$ has units of velocity squared, whereas $E$ has units of velocity squared times length. The universal equilibrium forms include viscous effects; they do not impose a power law through the dissipation range.

The second similarity hypothesis asserts independence of kinematic viscosity at scales large compared with $\eta$ but still small compared with the integral scale. In this inertial range, $S_2$ can depend only on $ϵ$ and $r$. Balancing time and length dimensions forces powers $2/3$ and $2/3$. The same calculation for $E$, using $k$ instead of $r$, forces powers $2/3$ and $-5/3$. HenceThe constants $C_2$ and $C_K$ are universal within the hypotheses and the stated normalization. They are related by the exact structure-function spectral filter, rather than by assigning all energy to one sharp scale.