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Plane Couette flow transitions to turbulence for Re~325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are quite distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. Computing such connections is essential for understanding the global dynamics of spatially localized structures that occur in transitional plane Couette flow. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.
Equilibrium, traveling wave, and periodic orbit solutions of pipe, channel, and plane Couette flows can now be computed precisely at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of wall-bounde
Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical,
We consider a 9-PDE (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localisation in detail. Of the many steady Eckhaus (spanwise
Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, stead
We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders structure stri