  The spectral model of Perry, Henbest & Chong (1986) predicts that the
integral length-scale varies very slowly with distance to the wall in the
intermediate layer. The only way for the integral length scale's variation to
be more realistic while keeping with the Townsend-Perry attached eddy spectrum
is to add a new wavenumber range to the model at wavenumbers smaller than that
spectrum. This necessary addition can also account for the high Reynolds number
outer peak of the turbulent kinetic energy in the intermediate layer. An
analytic expression is obtained for this outer peak in agreement with extremely
high Reynolds number data by Hultmark, Vallikivi, Bailey & Smits (2012, 2013).
The finding of Dallas, Vassilicos & Hewitt (2009) that it is the eddy turnover
time and not the mean flow gradient which scales with distance to the wall and
skin friction velocity in the intermediate layer implies, when combined with
Townsend's (1976) production-dissipation balance, that the mean flow gradient
has an outer peak at the same location as the turbulent kinetic energy. This is
seen in the data of Hultmark, Vallikivi, Bailey & Smits (2012, 2013). The same
approach also predicts that the mean flow gradient has a logarithmic decay at
distances to the wall larger than the position of the outer peak, a qualitative
prediction which the aforementioned data also support.
