The Characteristic Length Scale of the Intermediate Structure in Zero-Pressure-Gradient Boundary Layer Flow

TL;DR

This work challenges the classical wake-region model for zero-pressure-gradient turbulent boundary layers by proposing an intermediate structure comprised of two self-similar layers separated by a sharp interface. By defining two characteristic length scales, the wall-region thickness $\lambda$ and a Reynolds-number-based scale $\Lambda$ via $Re=U\Lambda/\nu$, the authors show that the first intermediate layer adheres to a pipe-flow-like scaling form when the Reynolds number is determined with $\Lambda$, i.e., $\phi=(\tfrac{1}{\sqrt{3}}\ln Re+\tfrac{5}{2})\eta^{\tfrac{3}{2\ln Re}}$. They derive the intersection condition $\eta_*=(A/B)^{1/(\beta-\alpha)}$ giving $\lambda=(A/B)^{1/(\beta-\alpha)}(\nu/u_*)$ and relate $\Lambda$ to $\lambda$ via $\Lambda/\lambda=(u_*/U)Re\,\eta_*^{-1}$. Re-analysis of Österlund’s data and other zero-$Pg$ experiments reveals a consistent two-layer structure with a sharp boundary and finds $\Lambda$ close to $\lambda$ (within experimental uncertainty), contradicting the wake-region model and supporting a potentially universal scaling framework for wall-bounded shear flows at large $Re$. This has implications for cross-flow scaling and the interpretation of near-wall turbulence across different geometries.

Abstract

In a turbulent boundary layer over a smooth flat plate with zero pressure gradient, the intermediate structure between the viscous sublayer and the free stream consists of two layers: one adjacent to the viscous sublayer and one adjacent to the free stream. When the level of turbulence in the free stream is low, the boundary between the two layers is sharp and both have a self-similar structure described by Reynolds-number-dependent scaling (power) laws. This structure introduces two length scales: one --- the wall region thickness --- determined by the sharp boundary between the two intermediate layers, the second determined by the condition that the velocity distribution in the first intermediate layer be the one common to all wall-bounded flows, and in particular coincide with the scaling law previously determined for pipe flows. Using recent experimental data we determine both these length scales and show that they are close. Our results disagree with the classical model of the "wake region".