microscopic flow characteristics and relates to generic features such as space dimensionality. The distance to the wall $y$ and the mean fluid velocity $u$ along the wall, are typically expressed in the “wall units” determined by the wall shear stress $\tau_0$. This is because $\tau_0$ is an important theoretical concept that is also experimentally measurable. The friction velocity $u_\tau = \sqrt{\langle \tau_0 \rangle/\rho}$ which is set by the wall shear stress $\tau_0$ and the kinematic viscosity $\nu$, and enters in the unit rescalings as $\tilde u= u/u_\tau$ and $\tilde y = yu_\tau/ \nu$. The constant fluid density is $\rho$ and the $B$ is a dimensionless constant that is fitted to experimental data, e.g. [@P53].

![Theoretical predictions from the spectral theory for the MVP $\langle u \rangle$ and mean square velocity fluctuations $\langle w^2\rangle$ (dimensionless variables in wall units). []{data-label="fig:wx1"}](Figure_0.pdf){width=".4\textwidth"}

A log-law of the wall was also derived from the “attached eddy hypothesis” by Townsend [@T76]. Townsend showed that the velocity fluctuations, $\tilde w =w/u_\tau$, $\tilde u= \langle \tilde u \rangle + \tilde w$, also follow the log-law of the wall in its second moment, namely $$\label{eq:l-lfluct}
{\langle \tilde w^2 \rangle} = - A_1 \log(\tilde y) + B_1,$$ where the coefficients 