v_y u'$ where $u'$ denotes the $y$ derivative of the velocity $u$ along the wall, and the overline indicates a not-fluctuating quantity. When velocity fluctuations are included the shear stress becomes: $$\label{eq:shear-stress}
\tau_t = \kappa_\tau \rho y v_y (u'+w'),$$ where $\rho$ is the density $v_y$ is the (rotational) velocity of an eddy a distance $y$ from the wall and $\kappa_\tau$ is the dimensionless proportionality factor. The energy dissipation rate is related to the wall shear stress as ${\bar \epsilon} = \tau_t u'/\rho$ [@GGGC10] , and including the fluctuations, this becomes $$\label{eq:energy}
\epsilon = \tau_t(u'+w')/\rho.$$ The eddy velocity for an eddy with radius $s=y$ at the distance $y$ from the wall is the same as in Ref. [@GGGC10], and as discussed above, $$\label{eq:e-viscosity}
v_y= (\kappa_\epsilon \epsilon y)^{1/3} \sqrt{I},$$ where $I$ is the integral from Eq. (\[eq:spectcont\]) and $\kappa_\epsilon$ is a dimensionless proportionality factor. In the inertial layer $I=1$ and $\kappa_\epsilon = 4/5$ according to Kolmogorov’s $4/5$ law.

Eliminating $\epsilon$ and $v_y$ from the three equations above, we obtain $$\label{eq:shear-stress_1}
\tau_t= (\kappa_\epsilon \kappa_\tau^3)^{1/2} \rho y^2 (u'+w')^2 I^{3/4}.$$ The viscous shear stress is $\rho \nu (u'+w')$ so the total shear stress, including the contribution from the fluctuation is [@T76] $$\tau_t + \rho \nu 