Approximate derivation of the power law for the mean streamwise velocity in a turbulent boundary layer under zero-pressure gradient

Using a Reyolds shear stress model in the boundary layer equations for flow over a flat plate, the streamwise velocity ($U$) is related to the wall-normal velocity ($V$). The empirical power-law $U\ensuremath{\sim}{\ensuremath{\eta}}^{1/n}$ for the streamwise velocity is derived theoretically, followed by a power-law for $V$.