An experimental–theoretical framework for reducing scatter in the log-law parameters κ and B of wall-bounded shear flows

This paper revisits the classical logarithmic velocity profile of wall-bounded turbulent shear flows, with particular emphasis on the persistent scatter in its defining parameters—the von Kármán constant (κ) and the additive constant (B). A combined experimental–theoretical framework is developed to extract these parameters across the three canonical flows: plane channel, circular pipe, and zero-pressure-gradient (ZPG) boundary layer. The authors' derivations show that three different partial differential equations governing these flows in the fully developed state behave similarly as Reτ→∞, particularly within the inertial sublayer, thereby supporting the universality of the log-law, where Reτ denotes the friction Reynolds number. The theoretical approach, based on the slope of the mean velocity profile in logarithmic coordinates, is validated using new hot-wire measurements together with literature data, accounting for wall-curvature effects in pipe flow and flow development in ZPG boundary layers. The analysis indicates that, at sufficiently high Reynolds numbers, the inertial sublayer is governed by the log-law, with an estimated von Kármán constant of κ≈1/e≈0.367879⋯. This value is supported by renormalization group theory and exponential self-similarity arguments, reinforcing that κ is not merely a fitting parameter but a consequence of turbulence dynamics. Furthermore, the present study demonstrates that κ and B cannot be specified independently, as multiple κ–B pairs yield equally valid log-law predictions. A linear κB–B relation, κB = 0.4582 B − 0.3134, is established, reducing data scatter and clarifying the cross-dependence of the two parameters. The present findings highlight the robustness of the logarithmic law and the ongoing challenge of defining its universal constants in wall-bounded turbulent flows, while providing a physical framework for developing and calibrating turbulence models.