The late start of the mean velocity overlap layer at $y^+=\mathcal{O}(10^3)$ -- A generic feature of ducted turbulent flows

Abstract

One of the key observations in the Princeton Superpipe was the late start of the logarithmic mean velocity overlap layer at a wall distance of the order of $10^3$ inner units. Between $y^+\approx 150$, the start of the overlap layer in zero pressure gradient turbulent boundary layers, and $y^+\approx 10^3$, the Superpipe profile is modeled equally well by a power law or a log-law with a larger slope than in the overlap layer. In this paper it is shown that the mean velocity profile in turbulent plane channel flow exhibits analogous characteristics, namely a sudden decrease of logarithmic slope (increase of $κ$) at a $y^+_{\mathrm{break}}\approx 600$, which marks the start of the actual overlap layer. This demonstration results from the first construction of the complete inner and outer asymptotic expansions up to order $\mathcal{O}(\mathrm{Re}_τ)^{-1}$ from mean velocity profiles of direct numerical simulations (DNS) at moderate Reynolds numbers. A preliminary analysis of a Couette flow DNS, on the other hand, yields an increase of logarithmic slope (decrease of $κ$) at a $y^+_{\mathrm{break}}\approx 400$. The correlation between the sign of the slope change and the flow symmetry motivates the hypothesis that the breakpoint between the possibly universal short inner logarithmic region and the actual overlap log-law corresponds to the penetration depth of large-scale turbulent structures originating from the opposite wall.