  Turbulent plane Poiseuille and Couette flows share the same geometry, but
produce their flow rate owing to different external drivers, pressure gradient
and shear respectively. By looking at integral energy fluxes, we pose and
answer the question of which flow performs better at creating flow rate. We
define a flow {\em efficiency}, that quantifies the fraction of power used to
produce flow rate instead of being wasted as a turbulent overhead; {\em
effectiveness}, instead, describes the amount of flow rate produced by a given
power. The work by Gatti \emph{et al.} (\emph{J. Fluid Mech.} vol.857, 2018,
pp. 345--373), where the constant power input (CPI) concept was developed to
compare turbulent Poiseuille flows with drag reduction, is here extended to
compare different flows. By decomposing the mean velocity field into a laminar
contribution and a deviation, analytical expressions are derived which are the
energy-flux equivalents of the FIK identity. These concepts are applied to
literature data supplemented by a new set of direct numerical simulations, to
find that Couette flows are less efficient but more effective than Poiseuille
ones. The reason is traced to the more effective laminar component of Couette
flows, which compensates for their higher turbulent activity. It is also
observed that, when the fluctuating fields of the two flows are fed with the
same total power fraction, Couette flows dissipate a smaller percentage of it
via turbulent dissipation. A decomposition of the fluctuating field into large
and small scales explains this feature: Couette flows develop stronger
large-scale structures, which alter the mean flow while contributing less
significantly to dissipation.
