Statistical estimates for channel flows driven by a pressure gradient

We present rigorous estimates for some physical quantities related to turbulent and non-turbulent channel flows driven by a uniform pressure gradient. Such results are based on the concept of stationary statistical solutions, which is related to the notion of ensemble averages for flows in statistical equilibrium. We provide a lower bound estimate for the mean skin friction coefficient and improve on a previous upper bound estimate for the same quantity; both estimates are derived in terms of the Reynolds number. We also present lower and upper bound estimates for the mean rate of energy dissipation, the mean longitudinal bulk velocity (in the direction of the pressure gradient), and the mean kinetic energy in terms of various physical parameters. In particular, we obtain an upper bound related to the energy dissipation law, namely that the mean rate of energy dissipation is essentially bounded by a non-dimensional universal constant times the cube of the mean longitudinal bulk velocity over a characteristic macro-scale length. Finally, we investigate the scale-by-scale energy injection due to the pressure gradient, proving an upper bound estimate for the decrease of this energy injection as the scale length decreases.