Asymptotics of streamwise Reynolds stress in wall turbulence

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