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 $x\to \infty$.
Categories: physics.flu-dyn