Origin of the scaling laws of developing turbulent boundary layers

Abstract

In this Perspective article, we seek the origin of the scaling laws of developing turbulent boundary layers over a flat plate from the perspective of the phenomenological theory of turbulence. The scaling laws of the boundary-layer thickness and the boundary shear stress in rough and smooth boundary-layer flows are established. In a rough boundary-layer flow, the boundary-layer thickness (scaled with the boundary roughness) and the boundary shear stress (scaled with the dynamic pressure) obey the “2/(1− σ)” and “(1+ σ)/(1− σ)” scaling laws, respectively, with the streamwise distance (scaled with the boundary roughness). Here, σ is the spectral exponent. In a smooth boundary-layer flow, the boundary-layer thickness (scaled with the viscous length scale) and the boundary shear stress (scaled with the dynamic pressure) obey the “8/(5 − 3 σ)” and “3(1+ σ)/(5 − 3 σ)” scaling laws, respectively, with the Reynolds number characterized by the streamwise distance.