We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller periodically continued domains, by increasing the length to obtain streamwise localisation and then by increasing the width to achieve spanwise localisation. The states maintain the travelling wave structure of the extended states, which is then modulated by a localised envelope on larger scales. In the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number Re, but the downstream exponent is essentially independent of Re. In the spanwise direction the decay is compatible with a power-law localisation. As the width increases the localised state undergoes further bifurcations which add additional unstable directions, so that the edge state, the relative attractor on the boundary between the laminar and turbulent motions, in the system becomes chaotic.

Abstract. We investigate the laminar-turbulent boundary in plane Poiseuille flow by the method of edge tracking. In short and narrow computational domains we find for a wide range of Reynolds numbers that all states in the boundary converge to a periodic orbit with a period of the order of 10 3 time units. The attracting states in these small domains are periodically extended in the spanwise and streamwise direction, but always localized to one side of the channel in the normal direction. In wider domains the edge states are localised in the spanwise direction as well. The periodic motion found in the small domains then induces a large variety of dynamical activity that is similar to one found in the asymptotic suction boundary layer.

Many shear flows follow a route to turbulence that has striking similarities to bifurcation scenarios in lowdimensional dynamical systems. Among the bifurcations that appear, crisis bifurcations are important because they cause global transitions between open and closed attractors, or indicate drastic increases in the range of the state space that is covered by the dynamics. We here study exterior and interior crisis bifurcations in direct numerical simulations of transitional plane Poiseuille flow in a mirror-symmetric subspace. We trace the state space dynamics from the appearance of the first three-dimensional exact coherent structures to the transition from an attractor to a chaotic saddle in an exterior crisis. For intermediate Reynolds numbers, the attractor undergoes several interior crises, in which new states appear and intermittent behavior can be observed. The bifurcations contribute to increasing the complexity of the dynamics and to a more dense coverage of state space. Numerical and experimental studies of pipe and plane Couette flow have demonstrated the significance of exact coherent structures and their bifurcations for the transition to turbulence [1][2][3][4]. Typically, these states appear in saddle-node bifurcations and then undergo further bifurcations. Initially, most of their complexity lies in the temporal dynamics, so that they are better characterized as chaotic rather than turbulent. With increasing Reynolds number, more temporal and spatial degrees of freedom are activated, until the complexity of a turbulent flow is established. Parallel to the increase in complexity comes a growth of the parts of state space that participate in the chaotic and turbulent dynamics. Studies of low-dimensional dynamical systems have revealed many routes to this increased complexity [5][6][7][8]. Several of them have already been discussed in the context of high-dimensional fluid systems, e.g., in the cases of plane Couette flow [2] or pipe flow [4,9]. One contribution of the present study is to document similar phenomenology in another canonical fluid system, plane Poiseuille flow (PPF). A second one is the demonstration of interior crisis and their contribution to increasing the complexity of the attractor and of the state space region covered by it.PPF is the pressure driven flow between two parallel plates and differs from plane Couette flow and pipe flow because of the presence of a linear instability to transverse vortices, the so-called Tollmien-Schlichting modes [10][11][12]. It occurs at a critical Reynolds number of 5772.22 for a streamwise wave number α of 1.020 56 (based on the center-line velocity and half the gap width), as determined by Orszag [13]. The bifurcation is subcritical, and reaches down to about Re ≈ 2700 [14,15] (for different wavelength). However, several experiments and numerical simulations show that turbulence occurs already at Reynolds numbers around 1000 [16][17][18], and hence well below the onset of Tollmien-Schlichting modes. Thus, the linear instability cannot explai...