Coexistence of Active and Hydrodynamic Turbulence in Two-Dimensional Active Nematics

C. Rorai, F. Toschi, and I. Pagonabarraga

Phys. Rev. Lett. 129, 218001 – Published 18 November 2022

Abstract

In active nematic liquid crystals, activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flow develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parameter leads to a broadening of the elastic energy spectrum that promotes a growth of kinetic energy able to trigger an inverse energy cascade.

Received 16 May 2022
Revised 29 September 2022
Accepted 25 October 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.218001

Physics Subject Headings (PhySH)

Turbulence

Active nematics Complex fluids

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. Rorai¹, F. Toschi ^2,3, and I. Pagonabarraga ^1,4,5

¹CECAM, Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne (EPFL), Batochime, Avenue Forel 2, 1015 Lausanne, Switzerland
²Department of Applied Physics, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, Netherlands
³CNR-IAC, I-00185 Rome, Italy
⁴Departament de Física de la Matèria Condensada, Universitat de Barcelona, C. Martí i Franquès 1, 08028 Barcelona, Spain
⁵University of Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, 08028 Barcelona, Spain

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Issue

Vol. 129, Iss. 21 — 18 November 2022

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Images

Figure 1
Snapshots of the magnitude of the velocity field rescaled by the root-mean-square velocity over the entire system (same scale) for the contractile (a) and extensile (b) nematic of group $F$ at time $T_{fin}$ , see Table 1. See the SM for visualizations of the vorticity and scalar order parameter. (c) Magnitude of the tensor order parameter (in color) and velocity field (black arrows) for the last time step of the run of group $E$ with $α > 0$ , $l_{a} = 0.5$ and $ξ = 2$ . Unlike contractile nematics (a), the velocity field for extensile nematics (b) and (c) generates large scale structures that span all the system size. See movies and SM. The white segment in the three panels is of 50 unit length.
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Figure 2
(a) Isotropic kinetic energy spectra rescaled by the kinetic energy at the active wave number $k_{a}$ identified by the intersection of the spectra with the dotted line. Black lines identify the $k^{- 4}$ (solid) and $k^{- 1}$ (dashed) scaling. The curves correspond to simulations of group $E$ and $D$ (Table 1). (b) Kinetic energy spectra and total kinetic energy density vs time (inset) in simulation units for systems of groups $E$ and $F$ characterized by the coexistence of active and inertial turbulence. Note the large Re numbers and $ξ$ and the different large scale scaling compared to the runs of panel (a) where inertial effects are negligible. See SM for the definitions of the isotropic kinetic, $E_{kin} (k)$ , and elastic, $E_{el} (k)$ , energy spectra.
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Figure 3
Logarithm of the total kinetic (a) and elastic (b) energy vs $A_{eff}$ , that compares the characteristic size of the largest flow structures ${\tilde{L}}_{I}$ with the effective active length scale $\tilde{l_{a}}$ at which energy is injected into the system, for the simulations of Table 1. The different colors refer to the different types of steady states: nonturbulent (blue), active turbulent (green), coupled active and inertial turbulence (orange). The size of the marker is proportional to the nominal Re number, ${Re}^{*}$ , while the different symbols correspond to different resolutions $L = 560$ (circle), $L = 2240$ (star), $L = 6144$ (diamond). The linewidth of the black symbols edge is proportional to $ξ$ : $ξ = 0$ (no edge), $ξ = 1$ (thin edge) $ξ = 2$ (thick edge). (c) Ratio between the kinetic and elastic energy. Runs where active and hydrodynamic turbulence coexist are present only in the positive semiplane.
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Figure 4
Elastic (a),(c) and kinetic (b),(d) energy spectra in simulation units for simulations of group $D$ with $l_{a} = - 0.5$ (a),(b) and $l_{a} = 0.5$ (c),(d) and different values of the flow-aligning parameter $ξ$ as reported in the legend of panel (b) and (d). The same color code is used in panels (a),(b) and (c),(d). The effective active wave number $k_{a} = 2 π / \tilde{l_{a}}$ is marked by black triangles while the wave number corresponding to the integral length scale, ${\tilde{L}}_{I}$ , is marked by a star in panel (b) and (d). The inset of panel (a) shows how the effective active length, $\tilde{l_{a}}$ , varies with $ξ$ for contractile and extensile nematics whose spectra are shown in this image. The inset of panel (c) shows how $\tilde{l_{a}}$ scaled by $l_{a}$ varies with $l_{a}$ for group $A$ excluding those in a NT state. The increase of $ξ$ affects the spectra of a contractile and extensile nematics differently.
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