Abstract
Fish schools and bird flocks exhibit complex collective dynamics whose self-organization principles are largely unknown. The influence of hydrodynamics on such collectives has been relatively unexplored theoretically, in part due to the difficulty in modeling the temporally long-lived hydrodynamic interactions between many dynamic bodies. We address this through a novel discrete-time dynamical system (iterated map) that describes the hydrodynamic interactions between flapping swimmers arranged in one- and two-dimensional lattice formations. Our 1D results exhibit good agreement with previously published experimental data, in particular predicting the bistability of schooling states and new instabilities that can be probed in experimental settings. For 2D lattices, we determine the formations for which swimmers optimally benefit from hydrodynamic interactions. We thus obtain the following hierarchy: while a side-by-side single-row “phalanx” formation offers a small improvement over a solitary swimmer, 1D in-line and 2D rectangular lattice formations exhibit substantial improvements, with the 2D diamond lattice offering the largest hydrodynamic benefit. Generally, our self-consistent modeling framework may be broadly applicable to active systems in which the collective dynamics is primarily driven by a fluid-mediated memory.
- Received 10 April 2019
- Revised 13 August 2019
- Corrected 15 November 2019
DOI:https://doi.org/10.1103/PhysRevX.9.041024
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)
Corrections
15 November 2019
Correction: The omission of a support statement in the Acknowledgments section has been remedied.
Popular Summary
The beautiful displays exhibited by schools of fish and flocks of birds have long fascinated scientists, but the role of their complex behavior remains largely unknown. In particular, the influence of hydrodynamic interactions on schooling and flocking has been the subject of intense debate in the scientific literature. Here, we present a model for flapping wings in orderly formations and determine the formations for which swimmers optimally benefit from hydrodynamic interactions. While a side-by-side single-row “phalanx” formation offers a small improvement over a solitary swimmer, 1D in-line and 2D rectangular lattice formations exhibit substantial improvements, with the 2D diamond lattice offering the largest hydrodynamic benefit.
Our model is posed as a discrete-time dynamical system: the swimmers shed vortices during each stroke and are, in turn, propelled by the hydrodynamic force due to their collective vortices. The solutions to our model exhibit good agreement with previously published experimental data on freely swimming wings in a water tank. Specifically, our model predicts a multistability of schooling states, wherein low- and high-velocity states coexist for identical kinematic parameters. The model also predicts oscillatory instabilities of schooling wings, which may be probed in experimental settings.
Our work offers a physical picture for the influence of fluid-mediated memory on schools and flocks, as the flapping wings interact through persistent flow vortices. We thus expect that our modeling framework may be broadly applied to physical systems governed by temporally long-lived hydrodynamic interactions.
