Universal scaling of active nematic turbulence

Abstract

A landmark of turbulence is the emergence of universal scaling laws, such as Kolmogorov’s E(q) ~ q−5∕3 scaling of the kinetic energy spectrum of inertial turbulence with the wavevector q. In recent years, active fluids have been shown to exhibit turbulent-like flows at low Reynolds number. However, the existence of universal scaling properties in these flows has remained unclear. To address this issue, here we propose a minimal defect-free hydrodynamic theory for two-dimensional active nematic fluids at vanishing Reynolds number. By means of large-scale simulations and analytical arguments, we show that the kinetic energy spectrum exhibits a universal scaling E(q) ~ q−1 at long wavelengths. We find that the energy injection due to activity has a peak at a characteristic length scale, which is selected by a nonlinear mechanism. In contrast to inertial turbulence, energy is entirely dissipated at the scale where it is injected, thus precluding energy cascades. Nevertheless, the non-local character of the Stokes flow establishes long-range velocity correlations, which lead to the scaling behaviour. We conclude that active nematic fluids define a distinct universality class of turbulence at low Reynolds number.

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Fig. 1: Stationary patterns upon the spontaneous flow instability.
Fig. 2: Disordered pattern of director domains with a characteristic wavelength.
Fig. 3: Spectra of the four contributions to the energy balance, equation (9) (Supplementary equation (32) in Supplementary Note).
Fig. 4: Universal scaling of the flow spectra at large scales.

Data availability

All the data presented in this study are available upon request.

Code availability

All the computer code used in this study is available upon request.

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Acknowledgements

We thank J. Prost for discussions. R.A. thanks A. Frishman for discussions. R.A. acknowledges support from Fundació “La Caixa” and from the Human Frontiers of Science Program (LT000475/2018-C). R.A. thanks J. Prost and acknowledges The Company of Biologists (Development Travelling Fellowship DEVTF-151206) and Fundació Universitària Agustí Pedro i Pons for supporting visits to Institut Curie. J.C. and R.A. acknowledge financial support by MINECO under project FIS2016-78507-C2-2-P and Generalitat de Catalunya under project 2017-SGR-1061. J.C. and J.-F.J. acknowledge support from the Labex Celtisphybio ANR-10-LABX-0038 part of the Idex PSL.

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J.C. conceived the research. R.A. and J.C. performed analytical calculations. R.A. performed the simulations. All authors designed the research and interpreted the results. All authors wrote the paper.

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Correspondence to Jaume Casademunt.

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Supplementary Information

Supplementary Discussion, Figs. 1–3, Descriptions of Supplementary movies 1–3 and refs. 1–5.

Supplementary Video 1

Evolution of director angle at A = 500

Supplementary Video 2

Evolution of director angle at A = 320,000

Supplementary Video 3

Evolution of stream function at A = 20,000

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Alert, R., Joanny, JF. & Casademunt, J. Universal scaling of active nematic turbulence. Nat. Phys. 16, 682–688 (2020). https://doi.org/10.1038/s41567-020-0854-4

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