Understanding the origin of large-scale structures in high Reynolds number wall turbulence
has been a central issue over a number of years. Recently, Rawat et al. (J.
Fluid Mech., 2015, 782, p515) have computed invariant solutions for the large-scale
structures in turbulent Couette flow at Reτ ≃ 128 using an over-damped LES with the
Smagorinsky model to account for the effect of the surrounding small-scale motions.
Here, we extend this approach to an order of magnitude higher Reynolds numbers in
turbulent channel flow, towards the regime where the large-scale structures in the form
of very-large-scale motions (long streaky motions) and large-scale motions (short vortical
structures) energetically emerge. We demonstrate that a set of invariant solutions can
be computed from simulations of the self-sustaining large-scale structures in the minimal
unit (domain of size Lx = 3.0h streamwise and Lz = 1.5h spanwise) with midplane
reflection symmetry at least up to Reτ ≃ 1000. By approximating the surrounding small
scales with an artificially elevated Smagorinsky constant, a set of equilibrium states are
found, labelled upper- and lower-branch according to their associated drag. It is shown
that the upper-branch equilibrium state is a reasonable proxy for the spatial structure
and the turbulent statistics of the self-sustaining large-scale structures.