Near-wall turbulence intensity as $Re_\tau \rightarrow \infty$

In this study, asymptotic scaling of near-wall streamwise turbulence intensity $\overline{u'u'}/u_\tau^2$ ($u_\tau$ is the friction velocity) is theoretically explored. The three scalings previously proposed are first reviewed with their derivation and physical justification: 1) $\overline{u'u'}/u_\tau^2 \sim \ln Re_\tau$ ($Re_\tau$ is the friction velocity); 2) $\overline{u'u'}/u_\tau^2 \sim 1/U_\infty^+$ ($U_\infty^+$ is the inner-scaled freestream velocity in boundary layer); 3) $\overline{u'u'}/u_\tau^2 \sim Re_\tau^{-1/4}$. A new analysis is subsequently developed based on velocity spectrum, and two possible scenarios are identified based on the asymptotic behaviour of the outer-scaling part of the near-wall velocity spectrum. In the former case, the outer-scaling part of the spectrum is assumed to reach a non-zero constant as $Re_\tau \rightarrow \infty$, and it results in the scaling of $\overline{u'u'}/u_\tau^2 \sim \ln Re_\tau$, both physically and theoretically consistent with the classical attached eddy model. In the latter case, a sufficiently rapid decay of the outer-scaling part of the spectrum with $Re_\tau$ is assumed due to the effect of viscosity, such that $\overline{u'u'}/u_\tau^2 < \infty$ for all $Re_\tau$. The following analysis yields $\overline{u'u'}/u_\tau^2 \sim 1/\ln Re_\tau$, asymptotically consistent with the scaling of $\overline{u'u'}/u_\tau^2 \sim 1/U_\infty^+$. The scalings are further verified with the existing simulation and experimental data and those from a quasilinear approximation (Holford \emph{et al.}, 2023, \texttt{arXiv:2305.15043}), the spectra of which all appear to favour $\overline{u'u'}/u_\tau^2 \sim 1/\ln Re_\tau$, although new datasets for $Re_\tau \gtrsim O(10^4)$ would be necessary to conclude this issue.