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

TL;DR

This paper proposes a scaling law for high-frequency streamwise energy spectra in turbulent boundary layers, based on viscous scales, and validates it with high-Reynolds-number data, capturing near-wall streaks.

Contribution

It introduces a new scaling relation for high-frequency spectra in turbulent boundary layers, extending the law-of-the-wall concept to spectral analysis at high Reynolds numbers.

Findings

High frequency regime starts at approximately 0.005 f+ across Reynolds numbers.

Spectral collapse observed when using the proposed scaling relation.

Captures energetic viscous-scaled motions like near-wall streaks.

Abstract

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}.