### Abstract

We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation, while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localized in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localized initial conditions, the subcritical transition to a 'turbulent' state (i) takes place more abruptly, with a boundary between laminar and turbulent flows that appears to be much less 'structured' and (ii) results in a spatio-temporally chaotic regime within which the lifetimes of spatio-temporally complicated transients are longer, and are even more sensitive to initial conditions. The minimum initial energy E(0) required for a spanwise-localized initial perturbation to excite a chaotic transient has a power-law scaling with the Reynolds number E(0) similar to Re(p) with p approximate to -4.3. The exponent p depends only weakly on the width of the localized perturbation and is lower than that commonly observed in previous low-dimensional models where typically p approximate to -2. The distributions of lifetimes of chaotic transients at the fixed Reynolds number are found to be consistent with exponential distributions.

Original language | English |
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Pages (from-to) | 3066-3087 |

Number of pages | 22 |

Journal | Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences |

Volume | 467 |

Issue number | 2135 |

DOIs | |

Publication status | Published - 8 Nov 2011 |

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### Keywords

- dynamical systems
- fluid flow
- turbulence