Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers

TLDR

Quadrant analysis is employed to examine how Reynolds shear stress arises in zero‑pressure‑gradient turbulent boundary layers over both rough and smooth surfaces. The authors assume that the passage time of large coherent structures equals the mean interval between threshold‑level stress events in a given quadrant. Sweeps dominate the Reynolds shear stress near rough surfaces, with their influence increasing with roughness and proximity to the wall, defining a roughness sublayer several roughness heights deep; beyond this layer the flow follows standard similarity laws, the sweep–ejection imbalance correlates with third‑order velocity moments, and the coherent‑structure passage time scales with friction velocity and boundary‑layer thickness, remaining invariant to surface roughness.

Abstract

Quadrant analysis has been used to investigate the events contributing to the Reynolds shear stress in zero-pressure-gradient turbulent boundary layers over regularly arrayed rough surfaces of several different densities, and over a smooth surface. By partitioning the stress into ejections, sweeps, and inward and outward interactions, it is shown that sweeps account for most of the stress close to rough surfaces, and that the relative magnitude of the sweep component increases both with surface roughness and with proximity to the surface. The sweep-dominated region delineates a ‘roughness sublayer’ with a depth of up to several roughness element heights, in which the turbulence characteristics depend explicitly on the roughness. In the remainder of the inner (or constant-stress) layer, and in the outer layer, the flow obeys familiar similarity laws with respect to surface roughness. The difference Δ S 0 between the fractional contributions of sweeps and ejections to the stress is shown to be well related everywhere to the third moments of the streamwise and normal velocity fluctuations. Experimental proportionalities are established between the third moments and δ S 0 , and are shown to agree with predictions made from cumulant-discard theory. The time scale for the passage of large coherent structures past a fixed point, T , is assumed proportional to the mean time between occurrences in a specified quadrant of an instantaneous stress u'w’ at least H times the local mean stress u'w’ , where H is a threshold level. For both the ejection and sweep quadrants and for any choice of H , it is found that T scales with the friction velocity u * and the boundary-layer thickness δ, such that Tu * /δ is invariant with change of surface roughness.