Borders of disorder: in turbulent channel flow

Abstract

A quantitative theory of the average features of turbulent flow in a channel is described without the introduction of empirical parameters. The qualitative problem consists of maximizing the dissipation rate of the mean flow subject to the Rayleigh condition that the mean flow has no inflections. The quantitative features result from a boundary stability study which determines a smallest scale of motion in the transport of momentum. The velocity fields satisfying these conditions, the averaged equations and the boundary conditions uniquely determine an entire mean velocity profile at all Reynolds numbers within ten per cent of the data. The maximizing condition for the reproducibility of averages emerges from the Navier–Stokes equations as a consequence of a novel definition of nonlinear instability. The smallest scale of motion results from a theory for a time-dependent re-stabilization of the boundary layer following a disruptive instability. Computer reassessment of the several asymptotic estimates of the critical boundary eigenstructure can establish the limits of validity of the quantitative results.