= Solution
Define the <friction velocity> by $u_*=(|\tau_w|/\rho)^{1/2}$, 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 $-\overline{u'v'}\simeq u_*^2$.
Combining this <Reynolds stress> with the logarithmic shear gives $\mathcal P\simeq u_*^3/(\kappa y)$. 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. Thus
$$
\boxed{\mathcal P\simeq\epsilon=\frac{u_*^3}{\kappa y}.}
$$
This 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 $\epsilon$ diffusion term remains essential.
Back to article page
