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The trinity of so-called "canonical" wall-bounded turbulent flows, comprisingthe zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL,turbulent pipe flow and channel/duct flows has continued to receive intenseattention as new and more reliable experimental data have become available.Nevertheless, the debate on whether the logarithmic part of the mean velocityprofile, in particular the K\'arm\'an constant $\kappa$, is identical for thesethree canonical flows or flow-dependent is still ongoing. In this paper, whichexpands upon Monkewitz and Nagib (24th ICTAM Conf., Montreal, 2016), theasymptotic matching requirement of equal $\kappa$ in the log-law and in theexpression for the centerline/free-stream velocity is reiterated and shown topreclude a single universal log-law in the three canonical flows or at leastmake it very unlikely. The current re-analysis of high quality mean velocityprofiles in ZPG TBL's, the Princeton "Superpipe" and in channels and ductsleads to a coherent description of (almost) all seemingly contradictory datainterpretations in terms of TWO logarithmic regions in pipes and channels: Auniversal interior, near-wall logarithmic region with the same parameters as inthe ZPG TBL, in particular $\kappa_{\mathrm{wall}} \cong 0.384$, but onlyextending from around $150$ to around $10^3$ wall units, and shrinking withincreasing pressure gradient, followed by an exterior logarithmic region with aflow specific $\kappa$ matching the logarithmic slope of the respectivefree-stream or centerline velocity. The log-law parameters of the exteriorlogarithmic region in channels and pipes are shown to depend monotonically onthe pressure gradient.

Revisiting the quest for a universal log-law and the role of pressure gradient in "canonical" wall-bounded turbulent flows