On the mixing length eddies and logarithmic mean velocity profile in wall turbulence

TL;DR

This paper links classical turbulent eddies in wall turbulence to observable uniform momentum zones, explaining the origin of the logarithmic velocity profile and confirming the theory across various flow conditions.

Contribution

It provides evidence connecting mixing length eddies to uniform momentum zones, enhancing understanding of the mean velocity profile in turbulent boundary layers.

Findings

UMZs correspond to mixing length eddies

Instantaneous velocity profiles are step-like due to UMZs

Mean velocity profile scales with friction velocity and wall-normal distance

Abstract

Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterise surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer.