Statistical Estimates For Channel Flows Driven By A Pressure Gradient

TL;DR

This paper provides rigorous statistical estimates for physical quantities in channel flows driven by pressure gradients, including bounds on skin friction, energy dissipation, and velocity, using the concept of stationary statistical solutions.

Contribution

It introduces new bounds for key flow quantities in channel flows, improving previous estimates and connecting them to the Reynolds number and energy dissipation laws.

Findings

Lower bound for mean skin friction coefficient.

Improved upper bound for energy dissipation rate.

Upper bound on scale-by-scale energy injection.

Abstract

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 solution, which is related to the notion of ensemble average 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 in terms of the Reynolds number. We also present lower and upper bound estimates for the mean rate of energy dissipation, the mean longitudinal velocity (in the direction of the pressure gradient), and the mean kinetic energy. 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 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.