Recent experimental measurements have reported that the outer peak of the streamwise
wavenumber spectra of the streamwise velocity depends on the Reynolds number. Starting from
this puzzling observation, here it is proposed that the wall-parallel velocity components of each
of the energy-containing motions in the form of Towsnend’s attached eddies exhibit inner-scaling
nature in the region close to the wall. Some compelling evidence on this proposition has been presented
with a careful inspection of scaling of velocity spectra from DNS, a linear analysis with an
eddy viscosity, and the recently computed statistical structure of the self-similar energy-containing
motions in the logarithmic region. This observation suggests that the viscous wall effect would
not be negligible at least below the peak wall-normal location of each of the energy-containing
motions in the logarithmic and outer regions, reminiscent of the concept of the ‘mesolayer’ previously
observed in the mean momentum balance. It is shown that this behavior emerges due to a
minimal form of scale interaction, modeled by the eddy viscosity in the linear theory, and enables
one to explain the Reynolds-number-dependent behavior of the outer peak as well as the near-wall
penetration of the large-scale outer structures in a consistent manner. Incorporation of this viscous
wall effect to Townsend’s attached eddies, which were originally built with an inviscid approximation
at the wall, also reveals that the self-similarity of the wall-parallel velocity components of the
energy-containing motions would be theoretically broken in the region close to the wall.