Law-of-the-wall for streamwise energy spectra in high-Reynolds-number turbulent boundary layers

A scaling relation for the high frequency regime of streamwise energy spectra (in frequency domain) in the near-wall region is proposed. This is based on the dimensional analysis approach of \cite{perry1977} and \cite{Zamalloa2014} together with the hypothesis that the small-scale fluctuations in the near-wall region should only depend on the viscous scales, analogous to the Prandtl's law-of-the-wall for the mean flow. This allows us to examine the lower bound for the high frequency regime where law-of-the-wall in spectra would hold. Observations in high-Reynolds-number turbulent boundary layer data indicate that a conservative estimate for the start of this high frequency regime is $f^+$ = 0.005 (which corresponds to 200 viscous time-units) across a range of wall-normal positions and Reynolds numbers. This is sufficient to capture the energetic viscous-scaled motions such as the near-wall streaks, which has a time scale of approximately 100 viscous units. This scaling relation and the spectral collapse is consistent with the observations in internal flows at lower Reynolds numbers \cite{Zamalloa2014}.