  Data available in literature from direct numerical simulations of
two-dimensional turbulent channels by Lee & Moser (2015), Bernardini et al.
(2014), Yamamoto and Tsuji (2018) and Orlandi et al. (2015) in a large range of
Reynolds number have been used to find that shear parameter the ratio between
the eddy turnover time and the time scale of the mean deformation (1/S), scales
very well with the Reynolds number in the near-wall region. The good scaling is
due to the eddy turnover time, although the turbulent kinetic energy and the
rate of isotropic dissipation show a Reynolds dependence near the wall. the
shear parameter is linked to the flow structures, as well as the second
invariant, and also this quantity presents a good scaling. It has been found
that the maximum of turbulent kinetic energy production occurs in the layer
with the second invariant approximately zero, that is where the unstable
sheet-like structures roll-up to become rods. The decomposition of production
in the contribution of elongational and compressive strain demonstrates that
the two contribution present a good scaling. The perfect scaling however holds
when the near-wall and the outer structures are separated. The same statistics
have been evaluated by direct simulations of turbulent channels with different
type of corrugations on both walls. The flow physics in the layer near the
plane of the crests is strongly linked to the shape of the surface and it has
been demonstrated that the normal to the wall velocity fluctuations are
responsible for the modification of the flow structures, for the increase of
the resistance and of the turbulent kinetic energy production. These
simulations at intermediate Reynolds number indicated that in the outer region
the Townsend similarity hypothesis holds.
