Exact coherent states and connections to turbulent dynamics in minimal channel flow

Abstract

Several new families of nonlinear three-dimensional travelling wave solutions to the Navier–Stokes equation, also known as exact coherent states, are computed for Newtonian plane Poiseuille flow. The symmetries and streak/vortex structures are reported and their possible connections to critical layer dynamics are examined. While some of the solutions clearly display fluctuations that are localized around the critical layer (the surface on which the streamwise velocity matches the wave speed of the solution), for others this connection is not as clear. Dynamical trajectories along unstable directions of the solutions are computed. Over certain ranges of Reynolds number, two solution families are shown to lie on the basin boundary between laminar and turbulent flow. Direct comparison of nonlinear travelling wave solutions to turbulent flow in the same channel is presented. The state-space dynamics of the turbulent flow is organized around one of the newly identified travelling wave families, and in particular the lower-branch solutions of this family are closely approached during transient excursions away from the dominant behaviour. These observations provide a firm dynamical-systems foundation for prior observations that minimal channel turbulence displays time intervals of ‘active’ turbulence punctuated by brief periods of ‘hibernation’ (see, e.g., Xi & Graham, Phys. Rev. Lett., vol. 104, 2010, 218301). The hibernating intervals are approaches to lower-branch nonlinear travelling waves. Representing these solutions on a Prandtl–von Kármán plot illustrates how their bulk flow properties are related to those of Newtonian turbulence as well as the universal asymptotic state called maximum drag reduction (MDR) found in viscoelastic turbulent flow. In particular, the lower- and upper-branch solutions of the family around which the minimal channel dynamics is organized appear to approach the MDR asymptote and the classical Newtonian result respectively, in terms of both bulk velocity and mean velocity profile.