  Townsend (1976) proposed a structural model for the logarithmic layer
(log-layer) of wall turbulence at high Reynolds numbers, where the dominant
momentum-carrying motions are organised into a multi-scale population of eddies
attached to the wall. In the attached eddy framework, the relevant length and
velocity scales of the wall-attached eddies are the friction velocity and the
distance to the wall. In the present work, we hypothesise that the
momentum-carrying eddies are controlled by the mean momentum flux and mean
shear with no explicit reference to the distance to the wall and propose new
characteristic velocity, length, and time scales consistent with this argument.
Our hypothesis is supported by direct numerical simulation (DNS) of turbulent
channel flows driven by non-uniform body forces and modified mean velocity
profiles, where the resulting outer-layer flow structures are substantially
altered to accommodate the new mean momentum transfer. The proposed scaling is
further corroborated by simulations where the no-slip wall is replaced by a
Robin boundary condition for the three velocity components, allowing for
substantial wall-normal transpiration at all lengths scales. We show that the
outer-layer one-point statistics and spectra of this channel with transpiration
agree quantitatively with those of its wall-bounded counterpart. The results
reveal that the wall-parallel no-slip condition is not required to recover
classic wall-bounded turbulence far from the wall and, more importantly,
neither is the impermeability condition at the wall.
