Velocity gradient statistics in turbulent shear flow: an extension of Kolmogorov's local equilibrium theory

Abstract

This paper presents an extension of Kolmogorov's local similarity hypotheses of turbulence to include the influence of mean shear on the statistics of the fluctuating velocity in the dissipation range of turbulent shear flow. According to the extension, the moments of the fluctuating velocity gradients are determined by the local mean rate of the turbulent energy dissipation $\left \langle \epsilon \right \rangle$ per unit mass, kinematic viscosity $\nu$ and parameter $\gamma \equiv S (\nu /\left \langle \epsilon \right \rangle )^{1/2}$ , provided that $\gamma$ is small in an appropriate sense, where $S$ is an appropriate norm of the local gradients of the mean flow. The statistics of the moments are nearly isotropic for sufficiently small $\gamma$ , and the anisotropy of moments decreases approximately in proportion to $\gamma$ . This paper also presents a report on the second-order moments of the fluctuating velocity gradients in direct numerical simulations (DNSs) of turbulent channel flow (TCF) with the friction Reynolds number $Re_\tau$ up to $\approx 8000$ . In the TCF, there is a range $y$ where $\gamma$ scales approximately $\propto y^ {-1/2}$ , and the anisotropy of the moments of the gradients decreases with $y$ nearly in proportion to $y^ {-1/2}$ , where $y$ is the distance from the wall. The theoretical conjectures proposed in the first part are in good agreement with the DNS results.