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Ecological collapse and the emergence of travelling waves at the onset of shear turbulence

Nature Physics volume 12, pages 245248 (2016) | Download Citation

Abstract

The mechanisms and universality class underlying the remarkable phenomena at the transition to turbulence remain a puzzle 130 years after their discovery1. Near the onset to turbulence in pipes1, plane Poiseuille flow2 and Taylor–Couette flow3, transient turbulent regions decay either directly4 or through splitting5,6,7,8, with characteristic timescales that exhibit a super-exponential dependence on Reynolds number9,10. The statistical behaviour is thought to be related to directed percolation (DP; refs 6,11,12,13). Attempts to understand transitional turbulence dynamically invoke periodic orbits and streamwise vortices14,15,16,17,18,19, the dynamics of long-lived chaotic transients20, and model equations based on analogies to excitable media21. Here we report direct numerical simulations of transitional pipe flow, showing that a zonal flow emerges at large scales, activated by anisotropic turbulent fluctuations; in turn, the zonal flow suppresses the small-scale turbulence leading to stochastic predator–prey dynamics. We show that this ecological model of transitional turbulence, which is asymptotically equivalent to DP at the transition22, reproduces the lifetime statistics and phenomenology of pipe flow experiments. Our work demonstrates that a fluid on the edge of turbulence exhibits the same transitional scaling behaviour as a predator–prey ecosystem on the edge of extinction, and establishes a precise connection with the DP universality class.

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Acknowledgements

We gratefully acknowledge helpful discussions with Y. Duguet and Z. Goldenfeld. We especially thank A. Willis for permission to use his code ‘Open Pipe Flow’23. This work was partially supported by the National Science Foundation through grant NSF-DMR-1044901.

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Affiliations

  1. Loomis Laboratory of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street Urbana, Illinois 61801, USA

    • Hong-Yan Shih
    • , Tsung-Lin Hsieh
    •  & Nigel Goldenfeld

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Contributions

H.-Y.S. and N.G. designed the project. Computer simulations of pipe turbulence were performed by T.-L.H. Computer simulations of stochastic predator–prey dynamics were performed by H.-Y.S. All authors contributed to the interpretation of the data and the writing of the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Nigel Goldenfeld.

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https://doi.org/10.1038/nphys3548

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