Correspondences between self-similar mean dynamics and streamwise velocity behaviors in the inertial region of the turbulent boundary layer

Self-similar mean dynamics are analytically known to exist over a well-defined inertial domain of turbulent wall-flows [Klewicki 2013, \textit{J. Fluid Mech}. \textbf{718}, 596]. Well-resolved streamwise velocity measurements up to $\delta^{+} = $ 20,000 are used to investigate three measures of self-similarity in turbulent boundary layers, and compare their behaviors with those determined via analysis of the mean momentum equation. The measures include the Kullback-Leibler divergence (KLD) [Tsuji et al. 2005, \textit{Fluid Dyn. Res}. \textbf{37}, 293], the logarithmic decrease of even statistical moments [Meneveau {\&} Marusic 2013, J. Fluid Mech. \textbf{719}, R1], and the so-called diagnostic plot [Alfredsson {\&} Orlu 2010, \textit{Euro. J. Mech. B/Fluids} \textbf{42}, 403]. The present findings indicate that the approximately constant KLD profiles and the approximately logarithmic moment profiles follow the same scaling but reside interior to the bounds of the self-similar inertial domain associated with the mean dynamics. Conversely, the bounds of the self-similar region on the diagnostic plot correspond closely to the theoretically estimated bounds. A self-consistent physical interpretation is briefly discussed.