buffer layer. (Here the $x$ denotes the streamwise direction.) We will take $I_b$ to be zero outside the buffer layer.

The splitting of the near-wall region based on different scaling of the spectrum was proposed by Perry and Chong [@PC82] who used it build an interpolation model for MVP and the variation, this model was improved in Ref. [@Va15].

The generalised log-law {#sec:fluct}
=======================

In this section, we will give a simple derivation of the log-law for the mean-square velocity profile that holds in the limit of large Reynolds number. In the following section we derive the general form of the variation that is not equally transparent.

We will generalize the derivation of the MVP in Ref. [@GGGC10], by adding a fluctuation to the mean velocity. We let the velocity along the wall be $$v_1=u+v_1-u=u+w,$$ where $u$ is the mean velocity obtained by averaging $v_1$ over time, and $w$ is the fluctuation. The same derivations as in Ref. [@GGGC10] give the following equations for a dominant eddy of radius $s=y$, if we include the velocity fluctuations. In Ref. [@GGGC10] the shear stress at the distance $y$ from the wall is given by the formula ${\bar \tau_t} = \kappa_\tau \rho y 