$A_1$   $B_1$
  -------------- -------- ------- --------
  6000            9.449   0.730   9.373
  10,000         15.628   1.207   13.073
  14,500         15.500   1.197   13.573
  20,000         14.994   1.158   13.673

  : Here, the approximate $A_1$ value is computed from $C_1$ using the proportionality factor $A_1=C_1/(K|y^*|^{\zeta_1})=C_1/12.952$.

![Sketch of the instantaneous streaks, in the streamwise direction, and the wall-attached eddies, in the spanwise direction.[]{data-label="fig:wx1"}](Figure_4.pdf){width=".4\textwidth"}

Discussion {#sec:summary}
==========

We used the spectral theory of the MVP and the variation profile to represent both, and compare with experiment [@Sa18] for a range of Reynolds numbers. Assuming that the wall shear stress is a fluctuating quantity, we can derive that log-law for the variation (\[eq:l-lfluct\]) that was proposed by Townsend and measured by Perry and Chong. This law involves the Townsend-Perry constants. This was first done in the large Reynolds number limit and then for general Reynolds numbers. The Reynolds number dependence of the Townsend-Perry constants is determined by the stochastic closure theory [@BC16], [@KBK19]. We derive the log-law for the higher moments of the fluctuations and the Generalized Townsend-Perry constants based on the functional form of the variation and use the stochastic closure theory to express them in terms of the Kolmogorov-Obukhov coefficients of the structure functions of turbulence [@KBBS17]. This confirms 