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Directed percolation and puff jamming near the transition to pipe turbulence

Nature Physics volume 20, pages 1339–1345 (2024)Cite this article

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Abstract

The onset of turbulence in pipe flow has defied detailed understanding ever since the first observations of the spatially heterogeneous nature of the transition. Recent theoretical studies and experiments in simpler, shear-driven flows suggest that the onset of turbulence is a directed-percolation non-equilibrium phase transition, but whether these findings are generic and also apply to open or pressure-driven flows is unknown. In pipe flow, the extremely long time scales near the transition make direct observations of critical behaviour virtually impossible. Here we find a technical solution to that limitation and show that the universality class of the transition is directed percolation, from which a jammed phase of puffs emerges above the critical point. Our method is to experimentally characterize all pairwise interactions between localized patches of turbulence puffs and use these interactions as input for renormalization group and computer simulations of minimal models that extrapolate to long length and time scales. The strong interactions in the jamming regime enable us to explicitly measure the turbulent fraction and confirm model predictions. Our work shows that directed-percolation scaling applies beyond simple closed shear flows and underscores how statistical mechanics can lead to profound, quantitative and predictive insights on turbulent flows and their phases.

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Fig. 1: Illustration of the bottom-up multiscale approach to the laminar–turbulent transition in pipe flow used in this study.
Fig. 2: Puff dynamics as a function of puff-to-puff separation and Re.
Fig. 3: Scaling relationships obtained by simulating equation (3) using the experimentally obtained puff interactions.
Fig. 4: Puff jamming in pipe flow represented by a local Lindemann criterion.
Fig. 5: Puff density as a function of the rates at which puffs split (1/τs) and decay (1/τd).

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Data availability

All experimental data presented in this article are available online in the Zenodo repository at https://doi.org/10.5281/zenodo.10308791 (ref. 37).

Code availability

All computational data presented in this paper, the codes used to generate and process those data and the codes and scripts used to generate the figures are available online in the Zenodo repository at https://doi.org/10.5281/zenodo.10308791 (ref. 37).

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Acknowledgements

We gratefully acknowledge the assistance of J. M. Lopez with DNSs at an early stage of this work. This work was partially supported by two grants from the Simons Foundation (grant nos. 662985 (N.G.) and 662960 (B.H.)) and by Ministry of Science and Technology, Taiwan (grant nos. MOST 109-2112-M-001-017-MY3 and MOST 111-2112-M-001-027-MY3 (H.-Y.S.)). Part of this work was performed using computing resources of CRIANN (Normandy, France).

Author information

Authors and Affiliations

  1. LOMC, CNRS UMR 6294, Université Le Havre Normandie, Le Havre, France

    Grégoire Lemoult

  2. Institute of Science and Technology Austria, Klosterneuburg, Austria

    Grégoire Lemoult, Vasudevan Mukund & Björn Hof

  3. Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, Urbana, IL, USA

    Hong-Yan Shih & Nigel Goldenfeld

  4. Institute of Physics, Academia Sinica, Taipei, Taiwan

    Hong-Yan Shih

  5. Physics Division, National Center for Theoretical Sciences, Taipei, Taiwan

    Hong-Yan Shih

  6. Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    Gaute Linga & Joachim Mathiesen

  7. PoreLab, The Njord Centre, Departments of Physics and Geosciences, University of Oslo, Oslo, Norway

    Gaute Linga

  8. Department of Physics, University of California San Diego, La Jolla, CA, USA

    Nigel Goldenfeld

Authors
  1. Grégoire Lemoult
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  2. Vasudevan Mukund
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  3. Hong-Yan Shih
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  4. Gaute Linga
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Contributions

B.H., N.G. and J.M. designed the project. B.H., G. Lemoult and V.M. performed experiments and computer simulations. H.-Y.S., G. Linga, J.M. and N.G. performed theoretical calculations and computer simulations. All authors contributed to the interpretation of the data and the writing of the paper.

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Correspondence to Nigel Goldenfeld or Björn Hof.

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Nature Physics thanks Ron Shnapp and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Information containing details of the experiments as well as theoretical and numerical calculations of models of puff interactions near the laminar–turbulent transition in pipe flow, and Supplementary Figs. 1–14 and Tables 1–3.

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Lemoult, G., Mukund, V., Shih, HY. et al. Directed percolation and puff jamming near the transition to pipe turbulence. Nat. Phys. 20, 1339–1345 (2024). https://doi.org/10.1038/s41567-024-02513-0

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