Structure functions and invariants of the anisotropic Reynolds stress tensor in turbulent flows on water-worked gravel beds

Abstract

In this paper, a statistical description of turbulence in flow on a water-worked gravel-bed is presented by applying the laws of turbulence in conjunction with the double-averaging methodology. To this end, a laboratory experiment was performed, in which the gravel-bed was worked by flowing water. From the Taylor frozen-in approximation, the energy spectra of streamwise velocity fluctuation reveal the existence of the inertial subrange (lying between the energy containing and the dissipation ranges), where the turbulent kinetic energy (TKE) dissipation rate is constant. It is revealed that, in this range, Kolmogorov’s 4/5-law for the spatial increments of streamwise velocity is valid, thus allowing an accurate estimation of the TKE dissipation rate. Although Monin–Yaglom’s 4/3-law for the third-order mixed structure function provides a behavior similar to that of Kolmogorov’s 4/5-law, the estimation of the TKE dissipation rate by Monin–Yaglom’s 4/3-law is not quite accurate, owing to the departure from the isotropic turbulence at large scales. Therefore, the present study demonstrates the validity of this statistical approach (Kolmogorov’s 4/5-law) to investigate the turbulence at small scales. Besides, the data plots for an anisotropy invariant map suggest that near-bed anisotropic turbulence tends to reduce to the three-dimensional isotropic turbulence with an increase in the vertical distance, indicating a relaxation of the effects of bed roughness on anisotropic turbulence toward the free surface.