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

## TL;DR

This paper develops a theoretical framework for turbulent wall-bounded flows at high Reynolds numbers, deriving new relations for boundary layer thickness and skin friction based on Navier-Stokes equations.

## Contribution

It introduces a new dynamic boundary layer thickness definition and derives empirical correlations for skin friction and boundary layer growth at large Reynolds numbers.

## Key findings

- Derived a new correlation for skin friction proportional to 1/ln^2(δ(x))
- Proposed a dynamic boundary layer thickness δ(x) proportional to x/ln^2(x/x_0)
- Formulated the theory as an expansion in a small parameter as x approaches infinity

## 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 $x\to \infty$.