Asymptotic behavior of velocity spectra and variances in wall-bounded turbulence*

We analyze the behavior of the streamwise velocity spectra in wall-bounded turbulent flow using DNS data from channel and pipe flows up to Re_τ ≈12000. The data consistently indicate the presence of a near-wall layer where the velocity spectra exhibit a power-law behavior, distinctly different from the k^-1 predicted from the attached-eddy model. This deviation is attributed to the formation of turbulent Stokes layers underneath the attached eddies. The key implication is that the near-wall footprint of superstructures gradually diminishes with increasing Reynolds number, and the streamwise velocity variance follows a defect power law, with universal wall scaling being restored in the infinite Reynolds number limit.