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  • Perspective
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Directed percolation and the transition to turbulence

Nature Reviews Physics volume 5, pages 62–72 (2023)Cite this article

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Abstract

It may come as a surprise that a phenomenon as ubiquitous and prominent as the transition from laminar to turbulent flow has resisted combined efforts by physicists, engineers and mathematicians, and remained unresolved for almost one and a half centuries. In recent years, various studies have proposed analogies to directed percolation, a well-known universality class in statistical mechanics, which describes a non-equilibrium phase transition from a fluctuating active phase into an absorbing state. It is this unlikely relation between the multiscale, high-dimensional dynamics that signify the transition process in virtually all flows of practical relevance, and the arguably most basic non-equilibrium phase transition, that so far has mainly been the subject of model studies, which I review in this Perspective.

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Fig. 1: Analogy between directed percolation and proliferation of turbulent puffs.
Fig. 2: Laminar–turbulent co-existence patterns.
Fig. 3: Median lifetimes (left branch) and splitting times (right branch) of isolated turbulent puffs in pipe flow.
Fig. 4: Definitions of active sites and the order parameter.
Fig. 5: Comparison of measurements of critical exponents from recent years.
Fig. 6: Dangers in disregarding non-zero data points.

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Acknowledgements

The author thanks M. Avila and M. Vasudevan for their comments on an earlier version of this article.

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    Björn Hof

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Hof, B. Directed percolation and the transition to turbulence. Nat Rev Phys 5, 62–72 (2023). https://doi.org/10.1038/s42254-022-00539-y

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