Periodic bursting cycles on the edge to square duct turbulence

Abstract

The long-time behaviour of trajectories embedded in the edge manifold between the laminar and turbulent basins of attraction is investigated in order to identify simple invariant solutions to the Navier-Stokes equations in straight ducts with square cross-section. With the aid of the iterative bisection procedure of Toh and Itano (J. Fluid Mech., vol. 481, 2003, pp. 67-76), we detect three travelling waves that represent attracting ‘edge states’ within the separatrix and saddles w.r.t. the full state space. All three travelling waves are presumably connected to solutions known in the community. In other situations, the edge states to which trajectories are attracted seem to be invariant periodic cycles that feature a rich dynamics: Along each cycle, the flow oscillates periodically between two flow states, in each of which streamwise vorticity is predominantly residing near one set of parallel walls, while the vortical activity associated to the other pair of opposing walls is markedly reduced. This ‘switching’ dynamics is accompanied by intermittent bursting-like high-dissipation excursions. While we cannot rigorously prove that the identified periodic cycle is a periodic orbit rather than a finite-precision approximation to a heteroclinic cycle, sound arguments for the former supposition are given. Interestingly, the different phases of the periodic cycle qualitatively resemble a similar ‘switching’ behaviour observed in (chaotic) marginal turbulence, which suggests that the found periodic cycles could help understanding the characteristic dynamics of duct turbulence at low Reynolds numbers.