Asymptotics of streamwise Reynolds stress in wall turbulence

Abstract

The scaling of different features of streamwise normal stress profiles $\langle uu\rangle ^+(y^+)$ in turbulent wall-bounded flows is the subject of a long-running debate. Particular points of contention are the scaling of the ‘inner’ and ‘outer’ peaks of $\langle uu\rangle ^+$ at $y^+\approxeq ~15$ and $y^+ ={O}(10^3)$ , respectively, their infinite Reynolds number limit, and the rate of logarithmic decay in the outer part of the flow. Inspired by the thought-provoking paper of Chen & Sreenivasan (J. Fluid Mech., vol. 908, 2021, p. R3), two terms of an inner asymptotic expansion of $\langle uu\rangle ^+$ in the small parameter $Re_{\tau }^{-1/4}$ are constructed from a set of direct numerical simulations (DNS) of channel flow. This inner expansion is for the first time matched through an overlap layer to an outer expansion, which not only fits the same set of channel DNS within 1.5 % of the peak stress, but also provides a good match of laboratory data in pipes and the near-wall part of boundary layers, up to the highest $Re_{\tau }$ values of $10^5$ . The salient features of the new composite expansion are first, an inner $\langle uu\rangle ^+$ peak, which saturates at 11.3 and decreases as $Re_{\tau }^{-1/4}$ . This inner peak is followed by a short ‘wall log law’ with a slope that becomes positive for $Re_{\tau }$ beyond ${O}(10^4)$ , leading up to an outer peak, followed by the logarithmic overlap layer with a negative slope going continuously to zero for $Re_{\tau }\to \infty$ .