  Plane Couette flow at Re=1200 (based on the channel half-height and half the
velocity difference between the top and bottom plates) is investigated with a
spatial domain designed to retain only two spanwise integral length scales. In
this system, the computation of invariant solutions that are physically
representative of the turbulent state has been understood to be challenging. To
address this challenge, our approach is to employ an accurate reduced-order
model with 600 degrees of freedom (Cavalieri & Nogueira, {Phys. Rev. Fluids},
vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal
cross-correlation of key observables, it is first demonstrated that the model
contains most of the multi-scale physical processes identified recently (Doohan
et al., J. Fluid Mech., vol. 913, 2021, A8): i.e. the large- and small-scale
self-sustaining processes, the energy cascade for turbulent dissipation, and an
energy-cascade mediated small-scale production mechanism. Invariant solutions
of the reduced-order model are subsequently computed, including 96 equilibria
and 43 periodic orbits. It is found that none of the computed equilibrium
solutions are able to reproduce an accurate energy balance associated with the
multi-scale dynamics of turbulent state. Incorporation of unsteadiness into
invariant solutions is seen to be essential for a sensible description of the
multi-scale turbulent dynamics and the related energetics, at least in this
type of flow, as periodic orbits with a sufficiently long period are mainly
able to describe the complex spatiotemporal dynamics associated with the known
multi-scale phenomena.
