The Amplitude of Turbulent Shear Flow

Abstract

A new formulation of the problem of the statistical stability of fully turbulent shear flow is proposed, in which one seeks mean fields that bound the observed flow from the stable side. In the spirit of maximum transport theory, this formulation admits a larger set of “flows” than are dynamically possible. A sequence of constraints derived from the equations of motion can narrow this set, permitting at each step the determination of a “most stable” field free of any empirical elements. Turbulent channel flow is proposed as the first application and test of this quantitative theory. Past deductive theories for this flow, from “mean field” to “transport upper bounds,” are assessed. It is shown why these theories do not retain the significant destabilizing mechanisms of the actual flow. The implications for turbulent flow of recent work on the nonlinear and three-dimensional instability of laminar shearing flow are described. In first exploration of the “decoupled mean” stability theory proposed here, approximate analytical and numerical stability methods are used to find an amplitude and structure for the averaged flow propoerties. The quantitative results differ by considerably less than two from the observed values, providing an incentive for a more complete numerical study and for further constraints on the admitted class of flows. In the language now current for nonlinear stability theory, evidence is advanced here that an N-dimensional central manifold is adjacent to the realized turbulent flow, where Nhas the largest possible value compatible with the dynamical relations.