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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.

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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 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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