  We analyse the inter-scale transfer of energy for two types of plane
Poiseuille flow: the P4U exact coherent state of Park and Graham (2015) and
turbulent flow in a minimal channel. For both flows, the dominant
energy-producing modes are streamwise-constant streaks with a spanwise spacing
of approximately 100 wall units. Since the viscous dissipation for these scales
is not sufficient to balance production, the nonlinear terms redistribute the
excess energy to other scales. Spanwise-constant scales (that is,
Tollmien-Schlichting-like modes with zero spanwise wavenumber), in particular,
account for a significant amount of net energy gain from the nonlinear terms.
We compare the energy balance to predictions from resolvent analysis and we
show that it does not model energy transfer well. Nevertheless, we find that
the energy transferred from the streamwise-constant streaks can be predicted
reasonably well by a Cess eddy viscosity profile. As such, eddy viscosity is an
effective model for the nonlinear terms in resolvent analysis and explains good
predictions for the most energetic streamwise-constant streaks. It also
improves resolvent modes as a basis for structures whose streamwise lengths are
greater than their spanwise widths by counteracting non-normality of the
resolvent operator. Eddy viscosity does not respect the conservative nature of
the nonlinear energy transfer which must sum to zero over all scales. It is
less effective, consequently, for scales which receive energy from the
nonlinear terms.
