Constant-skewness turbulence closure ($S_3(r)=-4S_2(r{)}^{3/2}/(5{\beta }^{3/2})$)

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.

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.