Reynolds number scaling and inner-outer overlap of stream-wise Reynolds stress in wall turbulence

The scaling of Reynolds stresses in turbulent wall-bounded flows is the subject of a long running debate. In the near-wall ``inner'' region, a sizeable group, inspired by the ``attached eddy model'', has advocated the unlimited growth of $\langle uu\rangle^+$ and in particular of its inner peak at $y^+\approxeq 15$, with $\ln\Reytau$ \citep[see e.g.][and references therein]{smitsetal2021}. Only recently, \citet{chen_sreeni2021,chen_sreeni2022} have argued on the basis of bounded dissipation, that $\langle uu\rangle^+$ remains finite in the inner near-wall region for $\Reytau\rightarrow\infty$, with finite Reynolds number corrections of order $\Reytau^{-1/4}$. In this paper, the overlap between the two-term inner expansion $f_0(y^+) + f_1(y^+)/\Reytau^{1/4}$ of \citet{monkewitz22} and the leading order outer expansion for $\langle uu\rangle^+$ is shown to be of the form $C_0 + C_1\,(y^+/\Reytau)^{1/4}$. With a new indicator function, overlaps of this form are reliably identified in $\langle uu\rangle^+$ profiles for channels and pipes, while the situation in boundary layers requires further clarification. On the other hand, the standard logarithmic indicator function, evaluated for the same data, shows no sign of a logarithmic law to connect an inner expansion of $\langle uu\rangle^+$ growing as $\ln{\Reytau}$ to an outer expansion of order unity.