## Abstract

Kolmogorov flow in two dimensions - the two-dimensional (2D) Navier-Stokes equations with a sinusoidal body force - is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimics the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynamics is observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) partial differential equations (PDEs) based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions - a kink and antikink - which connect two steady one-dimensional (1D) flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

Original language | English |
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Pages (from-to) | 518-554 |

Number of pages | 37 |

Journal | Journal of Fluid Mechanics |

Volume | 750 |

DOIs | |

Publication status | Published - Jul 2014 |

## Keywords

- instability
- nonlinear dynamical systems
- pattern formation
- PLANE COUETTE TURBULENCE
- HIGH REYNOLDS-NUMBER
- PIPE-FLOW
- LABORATORY SIMULATION
- PERIODIC FLOWS
- INSTABILITY
- MOTION
- BIFURCATIONS
- STABILITY
- MODEL