Kolmogorovian Active Turbulence of a Sparse Assembly of Interacting Marangoni Surfers

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Kolmogorovian Active Turbulence of a Sparse Assembly of Interacting Marangoni Surfers

Mickael Bourgoin, Ronan Kervil, Cecile Cottin-Bizonne, Florence Raynal, Romain Volk, and Christophe Ybert

Phys. Rev. X 10, 021065 – Published 22 June 2020

See synopsis: Active Matter that Mimics Turbulence in Space and Time

Abstract

Active matter, composed of self-propelled entities, forms a wide class of out-of-equilibrium systems that display striking collective behaviors, among which, the so-called active turbulence where spatially and time-disordered flow patterns spontaneously arise in a variety of active systems. De facto, the active turbulence naming suggests a connection with a second seminal class of out-of-equilibrium systems, inertial turbulence, even though the latter is of very different nature with energy injected at global system scale rather than at the elementary scale of single constituents. Indeed, the existence of a possible strong tie between active and canonical turbulence remains an open question and a field of profuse research. Using an assembly of self-propelled interfacial particles, we show experimentally that the statistical properties of particles’ velocities display a turbulentlike behavior, as described by the celebrated 1941 phenomenology of Kolmogorov. Moreover, the analogy between the dynamics of the self-propelled particles and inertial turbulence is observed to hold consistently both in the Eulerian and Lagrangian frameworks. Unlike the swimmers’ velocities distribution, the subsurface fluid flow is found not turbulent, thus making Marangoni surfers’ assemblies different from other active systems generating turbulence, such as living matter. Identifying an active system in the universality class of inertial turbulence not only benefits its future development but may also provide new insights into the long-standing description of turbulent flows, arguably one of the biggest remaining mysteries in classical physics.

Received 13 November 2019
Revised 25 April 2020
Accepted 4 May 2020

DOI:https://doi.org/10.1103/PhysRevX.10.021065

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Collective behavior Interfacial flows Interparticle interactions Turbulence

Living matter & active matter

Polymers & Soft MatterFluid DynamicsNonlinear Dynamics

synopsis

Active Matter that Mimics Turbulence in Space and Time

Published 22 June 2020

Despite being driven by a different process, a system of self-propelling particles can evolve over time in a similar way to a turbulent fluid.

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Authors & Affiliations

Mickael Bourgoin ¹, Ronan Kervil², Cecile Cottin-Bizonne ², Florence Raynal ³, Romain Volk ¹, and Christophe Ybert ^2,*

¹Laboratoire de Physique, Universite Lyon, ENS de Lyon, Universite Lyon 1, CNRS, F-69342 Lyon, France
²ILM, Universite Lyon, Universite Lyon 1, CNRS, F-69622 Villeurbanne CEDEX, France
³LMFA, Université Lyon, Centrale Lyon, INSA Lyon, Université Lyon 1, CNRS, F-69134 Écully, France

^*Corresponding author. christophe.ybert@univ-lyon1.fr

Popular Summary

Active matter, composed of self-propelled entities, forms a wide class of out-of-equilibrium systems that display striking collective behaviors. One of these behaviors is active turbulence, where spatial- and time-disordered flow patterns spontaneously arise in a variety of active systems. Despite its name, active turbulence differs from canonical fluid turbulence with regard to how energy is injected into the system, however, a possible connection between the two remains an open question and a field of profuse research. Using an assembly of self-propelled interfacial particles, we show experimentally that this active system shares remarkable quantitative similarities with canonical fluid turbulence.

Our synthetic swimmers are based on the idea of “camphor boats,” small floating objects that use a surfactant as a means of propulsion. Our “boats” are 5-mm-wide agar gel disks loaded with camphor, which locally lowers the surface tension of the surrounding water. An instability then causes the disk to move (and the water to flow) in some direction. When placed in a Petri dish filled with water, the disks self-propel at about $10 m m s^{- 1}$ . We record the motion of 30 of these disks, trace their trajectories, and compute various statistics. Our analysis reveals that the particles exhibit turbulencelike behavior.

Our results draw a clear connection between active systems and fluid turbulence, which not only benefits future development of active matter but may also provide new insights for the long-standing description of turbulent flows, arguably one of the biggest remaining mysteries in classical physics.

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Vol. 10, Iss. 2 — April - June 2020

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Figure 1
(a) Top view of the experiment. Thirty self-propelled camphor disks move at an air-water interface. Arrows represent the instantaneous velocity direction, colored according to the velocity magnitude. A video is available as online Supplemental Material [23]. (b) Superposition of the trajectories of the 30 particles over a 5-min recording.
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Figure 2
(a) Normalized mean-square displacements ( $σ_{MSD}^{2} / 2 ⟨ r^{2} ⟩$ , where $⟨ r^{2} ⟩$ is the average square radial position) of the camphor boats for experiments with different number of particles $N_{p}$ . At large times, the MSD oscillates around and eventually saturates to 1, due to the finite size of the experiment. For small numbers of particles (typically $N_{p} < 20$ ), the MSDs before saturation exhibit a dominant trivial ballistic motion (where the MSD grows as the square of time). For larger numbers of particles, an intermediate diffusivelike regime (where the MSD grows linearly with time) appears, revealing a nontrivial random dynamics with a finite correlation timescale. (b) Probability density functions of velocity and acceleration components. Because of the boundedness of the system, the mean velocity and acceleration components are zero. The standard deviation of the velocity components is $σ_{v_{x}} = σ_{v_{y}} = {⟨ v_{i}^{2} ⟩}_{i = x, y}^{1 / 2} = 12.1 {mm s}^{- 1}$ (corresponding to a root-mean-square velocity amplitude $σ_{| v |} = 17.1 {mm s}^{- 1}$ ); the standard deviation of acceleration components is $σ_{a_{x}} = σ_{a_{y}} = {⟨ a_{i}^{2} ⟩}_{i = x, y}^{1 / 2} = 33.2 {mm s}^{- 2}$ (corresponding to a root-mean-square acceleration amplitude $σ_{| a |} = 46.9 {mm s}^{- 2}$ ).
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Figure 3
Single particle—two-times Lagrangian statistics. (a) Lagrangian autocorrelation function $R_{ν ν}$ of velocity component $v_{x}$ . The light red colored area represents the estimate of the Lagrangian correlation timescale $T_{L}$ . (b) Lagrangian autocorrelation function $R_{a a}$ of acceleration component $a_{x}$ . $t_{0}$ corresponds to the shortest time such as $R_{a a} (t_{0}) = 0$ . The light red colored area represents the estimate of the Lagrangian dissipative timescale $τ_{η}^{L}$ .
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Figure 4
Two particles—single-time Eulerian statistics. (a) ( $□$ ) Second-order longitudinal structure function $S_{2}^{∥}$ as a function of interparticle separation $r$ . The dashed line represents the K41 prediction for inertial turbulence $S_{2}^{∥} \propto r^{2 / 3}$ . ( $\circ$ ) $S_{2}^{∥} (r)$ for the fluid flow at the surface computed from the velocity field measured by particle tracking velocimetry (PTV) using $200 − μ m$ floating tracers. (b) Power spectral density obtained as the Fourier transform of the Eulerian correlation function of the camphor boats’ velocity. The dashed line represents the K41 prediction for inertial turbulence. The light red area on the right represents wave numbers corresponding to scales smaller that the particles’ diameter. The Kolmogorov spectrum extends over almost one decade of scales, down to scales of the order of the particles’ diameter. (c) High-order longitudinal structure functions $S_{n}^{∥}$ (for $n \leq 5$ ). Dashed lines indicate the corresponding nonintermittent K41 predictions ( $S_{n}^{∥} \propto r^{n / 3}$ ). The light gray area qualitatively indicates the corresponding inertial range for which K41 scalings hold. (d) Crossed acceleration-velocity Eulerian structure function, whose amplitude in inertial turbulence is twice the energy flux, and the sign indicates the direction (direct or inverse) of the energy cascade: The negative sign here points toward a direct (from large- to small-scale) scenario.
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