Turbulence closure

A turbulence closure approximates the unclosed higher-order moments in evolution equations for lower-order turbulent statistics. For example, an equation for the velocity correlation tensor involves triple correlations, whose evolution involves fourth-order correlations and pressure. A closure must be checked against conservation laws, realizability and boundary conditions; matching an energy spectrum alone does not establish all these properties.

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.

EDQNM closure models the evolution of the turbulent energy spectrum through spectral triad interactions. Fourth-order correlations are approximated using products of second-order correlations, an eddy damping limits the resulting triple correlations, and their memory integrals are replaced by expressions using current spectra. The modeled damping times are additional physical input. This allows evolving spectra and scale-dependent transfer, but does not automatically recover coherent structures or internal intermittency. Bertoglio, Squires and Ferziger's simulation comparison explains the damping assumptions and tests their limitations.