Exact coherent states in channel flow

Abstract

Three spatially extended travelling wave exact coherent states, together with one spanwise-localised state, are presented for channel flow. Two of the extended flows are derived by homotopy from solutions to the problem of channel flow subject to a spanwise rotation investigated by Wall & Nagata (J. Fluid Mech., vol. 727, 2013, pp. 523–581). Both these flows are asymmetric with respect to the channel centreplane, and feature streaky structures in streamwise velocity flanked by staggered vortical structures. One of these flows features two streak/vortex systems per spanwise wavelength, while the other features one such system. The former substantially reduces the value of the lowest Reynolds number at which channel flow solutions ,other than the basic flow, are known to exist down to 665. The third flow has, in contrast, half-turn rotational symmetry about a streamwise axis through a point on the channel centreplane, and is found to be the flow from which one of the asymmetric flows bifurcates in a symmetry-breaking bifurcation. This flow is found to exist on an isolated bifurcation branch, whose upper and lower branches both lie on the boundary basin separating initial conditions that lead to turbulent events, and those that directly decay back to laminar flow. The structure of this flow, in which the disturbance to the basic flow is concentrated in a core region in a spanwise period, allowed the derivation of a corresponding spanwise-localised flow, which is also discussed.