A uniform momentum zone–vortical fissure model of the turbulent boundary layer

Abstract

Recent studies reveal that at large friction Reynolds number $\unicode[STIX]{x1D6FF}^{+}$ the inertially dominated region of the turbulent boundary layer is composed of large-scale zones of nearly uniform momentum segregated by narrow fissures of concentrated vorticity. Experiments show that, when scaled by the boundary-layer thickness, the fissure thickness is $\mathit{O}(1/\sqrt{\unicode[STIX]{x1D6FF}^{+}})$, while the dimensional jump in streamwise velocity across each fissure scales in proportion to the friction velocity $u_{\unicode[STIX]{x1D70F}}$. A simple model that exploits these essential elements of the turbulent boundary-layer structure at large $\unicode[STIX]{x1D6FF}^{+}$ is developed. First, a master wall-normal profile of streamwise velocity is constructed by placing a discrete number of fissures across the boundary layer. The number of fissures and their wall-normal locations follow scalings informed by analysis of the mean momentum equation. The fissures are then randomly displaced in the wall-normal direction, exchanging momentum as they move, to create an instantaneous velocity profile. This process is repeated to generate ensembles of streamwise velocity profiles from which statistical moments are computed. The modelled statistical profiles are shown to agree remarkably well with those acquired from direct numerical simulations of turbulent channel flow at large $\unicode[STIX]{x1D6FF}^{+}$. In particular, the model robustly reproduces the empirically observed sub-Gaussian behaviour for the skewness and kurtosis profiles over a large range of input parameters.