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Set $S_2=\sqrt{15}{\beta }^{3/2}{v}_{\eta }^{2}h$ and $r=(15\beta {)}^{3/4}\eta x$. The Kolmogorov equation for structure functions and the constant-skewness turbulence closure give the displayed ordinary differential equation, with $h(0)=0$. Its small-separation balance is $h\sim x^2$; its large-separation balance is $h\sim x^{2/3}$. The first matches differentiable velocity increments, and the second recovers the assumed Kolmogorov two-thirds law.

Combining the Kolmogorov two-thirds law with the Kolmogorov four-fifths law gives the velocity increment skewness in the inertial range:The constant-skewness turbulence closure assumes this same value throughout the universal equilibrium range, including the dissipative crossover. This is an extra modeling assumption; neither similarity law establishes it near $r=0$.

Write $A=\beta (15\beta {)}^{1/2}$, $B=(15\beta {)}^{3/4}$, $S_2=A{v}^{2}h$ and $r=B\eta x$. Since $\nu =v\eta$ and $ϵ\eta =v^3$, substituting $S_3=-4S_2^{3/2}/(5{\beta }^{3/2})$ into the Kolmogorov equation for structure functions givesDividing by $2B/5$ yields the normalized constant-skewness structure-function equation:At small $x$, $h\sim x^2$, recovering $S_2\sim ϵr^2/(15\nu )$ for differentiable velocity increments. At large $x$, the dominant balance gives $h\sim x^{2/3}$, recovering the assumed Kolmogorov two-thirds law.

This turbulence closure is inexpensive: a single ordinary differential equation interpolates between the two regimes while respecting the local third-order balance. Its prescribed skewness cannot independently predict scale-dependent derivative statistics, transient spectral evolution or internal intermittency.

EDQNM closure instead evolves the turbulent energy spectrum through modeled spectral triad interactions. Its quasi-normal fourth-order approximation is supplemented by eddy damping and a Markovian treatment of correlation memory. It retains more information about scale-dependent transfer and evolving spectra, at the cost of spectral integrals and modeled damping times. It is still approximate and does not automatically describe coherent structures or internal intermittency. The simple constant-skewness model is therefore useful as an equilibrium interpolation; EDQNM closure addresses a broader spectral evolution problem.