Skin friction in zero-pressure-gradient boundary layers

TL;DR

This paper presents a global solution to the Navier-Stokes-Prandtl equations for zero-pressure-gradient boundary layers, revealing how boundary layer thickness and skin friction scale with Reynolds number as it becomes very large.

Contribution

It introduces a self-consistent theoretical framework for boundary layer behavior at high Reynolds numbers, deriving explicit scaling laws for thickness and skin friction.

Findings

Boundary layer thickness scales as x/ln^2(Re_x)

Skin friction is proportional to 1/ln^2(delta)

Theory applies as Reynolds number approaches infinity

Abstract

A global approach leading to a self-consistent solution to the Navier-Stokes-Prandtl equations for zero-pressure-gradient boundary layers is presented. It is shown that as $Re_{\delta}\rightarrow \infty$, the dynamically defined boundary layer thickness $\delta(x)\propto x/\ln^{2}Re_{x}$ and the skin friction $\lambda=\frac{2\tau_{w}}{\rho U_{0}^{2}}\propto 1/\ln^{2}\delta(x)$. Here $\tau_{w}$ and $U_{0}$ are the wall shear stress and free stream velocity, respectively. The theory is formulated as an expansion in powers of a small dimensionless parameter $\frac{d\delta(x)}{dx}\rightarrow 0$ in the limit $x\rightarrow \infty$.