The trinity of so-called “canonical” wall-bounded turbulent flows, comprising the zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL, turbulent pipe flow, and channel/duct flows has continued to receive intense attention as new and more reliable experimental data have become available. Nevertheless, the debate on whether the logarithmic part of the mean velocity profile, in particular the Kármán constant $\kappa$, is identical for these three canonical flows or flow-dependent is still ongoing. In this paper, the asymptotic matching requirement of equal $\kappa$ in the logarithmic overlap layer, which links the inner and outer flow regions, and in the expression for the centerline/free-stream velocity is reiterated and shown to preclude a universal logarithmic overlap layer in the three canonical flows. However, the majority of pipe and channel flow studies at friction Reynolds numbers ${\mathrm{Re}}_{\tau }$ below $\approx {10}^4$ extract from near-wall profiles the same $\kappa$ of 0.38–0.39 as in the ZPG TBL. This apparent contradiction is resolved by a careful reanalysis of high-quality mean velocity profiles in the Princeton “Superpipe” and other pipes, channels, and ducts, which shows that the mean velocity in a near-wall region extending to around 700 “+” units in channels and ducts and 500 “+” units in pipes is the same as in the ZPG TBL. In other words, all the “canonical” flow profiles contain the lower end of the ZPG TBL log-region, which starts at a wall distance of $150–200$ “+” units with a universal $\kappa$ of ${\kappa }_{\mathrm{ZPG}}\approx 0.384$. This interior log-region is followed by a second logarithmic region with a flow specific $\kappa >{\kappa }_{\mathrm{ZPG}}$, which increases monotonically with pressure gradient. This second, exterior log-layer is the actual overlap layer matching up to the outer expansion, which implies equality of the exterior $\kappa$ and ${\kappa }_{\mathrm{CL}}$ obtained from the evolution of the respective centerline velocity with Reynolds number. The location of the switch-over point implies furthermore that this second log-layer only becomes clearly identifiable, i.e., separated from the wake region, for ${\mathrm{Re}}_{\tau }$ well beyond ${10}^4$ (see Fig. 1). This explains the discrepancies between the Kármán constants of 0.38–0.39, extracted from near-wall pipe profiles below ${\mathrm{Re}}_{\tau }\approx {10}^4$ and the $\kappa$'s obtained from the evolution of the centerline velocity with Reynolds number. The same analysis is successfully applied to velocity profiles in channels and ducts even though experiments and numerical simulations have not yet reached Reynolds numbers where the different layers have even started to clearly separate.

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• Structure & turbulence of boundary layers