  This paper addresses the integral energy fluxes in natural and controlled
turbulent channel flows, where active skin-friction drag reduction techniques
allow a more efficient use of the available power. We study whether the
increased efficiency shows any general trend in how energy is dissipated by the
mean velocity field (mean dissipation) and by the fluctuating velocity field
(turbulent dissipation).
  Direct Numerical Simulations (DNS) of different control strategies are
performed at Constant Power Input (CPI), so that at statistical equilibrium
each flow (either uncontrolled or controlled by different means) has the same
power input, hence the same global energy flux and, by definition, the same
total energy dissipation rate. The simulations reveal that changes in mean and
turbulent energy dissipation rates can be of either sign in a successfully
controlled flow.
  A quantitative description of these changes is made possible by a new
decomposition of the total dissipation, stemming from an extended Reynolds
decomposition, where the mean velocity is split into a laminar component and a
deviation from it. Thanks to the analytical expressions of the laminar
quantities, exact relationships are derived that link the achieved flow rate
increase and all energy fluxes in the flow system with two wall-normal
integrals of the Reynolds shear stress and the Reynolds number. The dependence
of the energy fluxes on the Reynolds number is elucidated with a simple model
in which the control-dependent changes of the Reynolds shear stress are
accounted for via a modification of the mean velocity profile. The physical
meaning of the energy fluxes stemming from the new decomposition unveils their
inter-relations and connection to flow control, so that a clear target for flow
control can be identified.
