Skin Friction in Simple Wall - Bounded Shear Flows in Large Reynolds Number Limit

Abstract

A global approach to analysis of fully developed turbulent flows in pipes/channels and zero pressure gradient boundary layers is proposed. A new dynamic definition of the boundary layer thickness $\delta(x)$, where $x$ is the distance to the plate origin, is proposed. The Coles - Fernholtz empirical correlation for skin friction $\lambda=\frac{2\tau_{w}}{\rho U_{0}^{2}}\propto 1/\ln^{2}\delta(x)$ and $\delta(x)\propto x/\ln^{2}(\frac{x}{x_{0}})$ are derived from the Navier-Stokes equations in the limit $Re_{x}\to \infty$. 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 $\frac{d\delta(x)}{dx}\to 0$ in the limit $x\to \infty$.