the streamwise fluctuations in the wall shear stress, and remove the assumption made in Refs. [@BC16] and [@KBK19], and mentioned above. Using theory-informed by data analysis, we can construct the Townsend-Perry constants and the generalised Townsend-Perry constants. In Section \[sec:BW\], we extend the formulas for the mean square fluctuations to the buffer layer and the energetic wake. In Section \[sec:data\], we compare the predicted MVP and mean-square velocity profile from this spectral theory to experimental data. In Section \[sec:summary\], we conclude with a discussion on the proposed spectral theory and the role that Townsend’s attached eddies play in it.

The Spectral Theory {#sec:MVP}
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The typical velocity of an inertial eddy of size $s$ can be obtained by integrating out the kinetic energy contained in all eddies of sizes up to $s$ as in Ref. [@GGGC10] $$\label{eq:vs}
v_s^2 = \int_{1/s}^\infty E(k)dk,$$ where kinetic energy spectrum follows the Kolmogorov-Obukhov scaling with cutoffs in the injection scale and viscous scales, $E(k) =c_d(\eta k) \frac{2}{3}(\kappa_\epsilon \epsilon )^{2/3}k^{-5/3} c_e(Rk)$, with $\frac{2}{3}(\kappa_\epsilon \epsilon)^{2/3}k^{-5/3}$ being the Kolmogorov-Obukhov spectrum and $c_d (\eta k)$ and $c_e(R k)$ the phenomenological dimensionless corrections functions in the dissipative (set by the Kolmogorov scale $\eta$) and energetic range (set by the system 