### Abstract

Original language | English |
---|---|

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

### Cite this

**Turbulent transition in a truncated one-dimensional model for shear flow.** / Dawes, Jonathan H P; Giles, W J.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences*, vol. 467, no. 2135, pp. 3066-3087. https://doi.org/10.1098/rspa.2011.0225

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TY - JOUR

T1 - Turbulent transition in a truncated one-dimensional model for shear flow

AU - Dawes, Jonathan H P

AU - Giles, W J

PY - 2011/11/8

Y1 - 2011/11/8

N2 - 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.

AB - 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.

KW - dynamical systems

KW - fluid flow

KW - turbulence

UR - http://www.scopus.com/inward/record.url?scp=80755175242&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1107.0580v1

UR - http://dx.doi.org/10.1098/rspa.2011.0225

U2 - 10.1098/rspa.2011.0225

DO - 10.1098/rspa.2011.0225

M3 - Article

VL - 467

SP - 3066

EP - 3087

JO - Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences

SN - 1364-503X

IS - 2135

ER -