 Solution

Define the friction velocity by $u_{\ast }=(|{\tau }_w|/\rho {)}^{1/2}$, where ${\tau }_w$ is the total tangential wall stress. It is a stress-derived velocity scale, not the local mean velocity. In the overlap region, $y$ is small relative to the channel half-width, so the total stress is approximately its wall value. Molecular stress is small there, giving $-\overline{u^{\prime }{v}^{\prime }}\simeq u_{\ast }^2$.

Combining this Reynolds stress with the logarithmic shear gives $\mathcal{P}\simeq u_{\ast }^3/(\kappa y)$. For locally equilibrated wall turbulence, energy transfer to small scales occurs over the local eddy turnover time and is balanced by molecular dissipation; energy storage and the net energy-transport divergence are small in the leading overlap-layer balance. ThusThis is the local turbulent kinetic energy balance. It is a modeling approximation rather than a general identity for all wall positions. The constant-energy solution derived next is consistent with it: its modeled $k$ diffusion term is exactly zero, while the $ϵ$ diffusion term remains essential.