(u'+w') =  \tau_0(1-y/R).$$ Our assumption is that the wall shear stress $\tau_0$ is also a quantity that fluctuates about its mean value.

We change the rescaled variables in the wall units written here in terms of the friction factor $f$: $\tilde y=y Re\sqrt{f}/R$, $\tilde u = u/(U\sqrt{f})$ and $\tilde w=w/(U\sqrt{f})$ and let $f=\langle \tau_0\rangle/\rho U^2$. Then, the equation above becomes $$\label{eq:total_stress1}
{\tilde \kappa}^2 {\tilde y}^2(\tilde u'+\tilde w')^2 I^{3/4}+(\tilde u'+\tilde w') = \frac{\tau_0}{\langle \tau_0 \rangle}\left(1-\frac{\tilde y}{Re\sqrt{f}}\right).$$ If we let $\tilde y \to 0$, $\tilde w \to 0$ and integrate, we get the law of the viscous layer $$\label{eq:viscous}
\tilde u = \tilde y,$$ the laminar profile being $$\label{eq:laminar}
\tilde u = \left(\tilde y-\frac{\tilde y^2}{2Re\sqrt{f}}\right).$$ In the large Reynolds number limit, solving just for the mean velocity, we obtain the Prandtl-von Kármán law $$\label{eq:velocity}
\tilde u = \frac{1}{\tilde \kappa}\log (\tilde y)+D.$$ This is the correct leading term but the full formulas in the next section are more complicated. We now motivate the log-law for the variation. If we solve for both the mean velocity and the fluctuation in the large Reynolds number limit, we get that $$\label{eq:velocity1}
\tilde u+\tilde  w= \frac{\sqrt{\tau_0}}{\langle \tau_0\rangle^{1/2} \tilde \kappa} \log (\tilde y)+C.$$ This is consistent with the 