Periodic orbits near onset of chaos in plane Couette flow

Abstract

We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo a periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show that the ones that exist fall into the universal sequence described by Metropolis, Stein and Stein for unimodal maps. The periodic orbits cover much of the turbulent dynamics in that their temporal evolution overlaps with turbulent motions when projected onto a plane spanned by energy production and dissipation.

Secondary instability and tertiary states in rotating plane Couette flow

“…Flowfield depictions of a stable oWVF state at two points during its oscillation are shown in figure 9. Its structure is much simpler than the periodic orbits found in non-rotating subcritical plane Couette flow by, for example, Kawahara & Kida (2001) and Viswanath (2007), though its E cf evolution is reminiscent of the orbit denoted 1 in symbolic dynamics by Kreilos & Eckhardt (2012). oWVF holds the same symmetries as WVF and wTWI: shift-reflection S and shift-rotation Ω.…”

“…A body of work has grown, detailing the properties of the structures in plane Couette flow: Nagata (1990Nagata ( , 1997, Waleffe (1997Waleffe ( , 1998Waleffe ( , 2003, Clever & Busse (1997), Viswanath (2007), Gibson et al (2008Gibson et al ( , 2009), Itano & Generalis (2009), and in pipe flow: Faisst & Eckhardt (2003), Wang et al (2007), Wedin & Kerswell (2004), Duguet et al (2008), Pringle & Kerswell (2007). Exact coherent structures have been shown to support chaotic dynamics (Gibson et al (2009), Kreilos & Eckhardt (2012)), and also act as edge states between laminar and turbulent regimes (Schneider et al (2008)). Therefore, exact coherent structures play a key role in transition of the subcritical shear flows.…”

Transition to turbulence in shear flows

“…3, for the case of plane Couette flow [50]. Initially, below the formation of the structures, all initial conditions return to the laminar profile indicated by the blue circle.…”

“…[50]): in the blue regions all initial conditions return to the laminar profile. In the red/brown area they are attracted to the nodal state.…”

Doubly localized states in plane Couette flow

“…For good measure, then also show what happens when the structure is perturbed: as befits a state that is intermediate between laminar and turbulent, some perturbations of the state increase and become turbulent (at least for a while), others return to the laminar state more directly. This is in many ways reminiscent of the crisis bifurcations that turn the stable attractor into an open saddle in small domains (Kreilos & Eckhardt 2012). Just how this bifurcation changes when there are other neutral directions remains to be seen.…”