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Hydrodynamic schooling of multiple self-propelled flapping plates

  • Ze-Rui Peng (a1), Haibo Huang (a1) and Xi-Yun Lu (a1)

Abstract

While hydrodynamic interactions for aggregates of swimmers have received significant attention in the low Reynolds number realm ( $Re\ll 1$ ), there has been far less work at higher Reynolds numbers, in which fluid and body inertia are involved. Here we study the collective behaviour of multiple self-propelled plates in tandem configurations, which are driven by harmonic flapping motions of identical frequency and amplitude. Both fast modes with compact configurations and slow modes with sparse configurations were observed. The Lighthill conjecture that orderly configurations may emerge passively from hydrodynamic interactions was verified on a larger scale with up to eight plates. The whole group may consist of subgroups and individuals with regular separations. Hydrodynamic forces experienced by the plates near their multiple equilibrium locations are all springlike restoring forces, which stabilize the orderly formation and maintain group cohesion. For the cruising speed of the whole group, the leading subgroup or individual plays the role of ‘leading goose’.

Copyright

Corresponding author

Email address for correspondence: xlu@ustc.edu.cn

References

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Alben, S. 2009 Wake-mediated synchronization and drafting in coupled flags. J. Fluid Mech. 641, 489496.
Alben, S. & Shelley, M. J. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. USA 102, 1116311166.
Becker, A. D., Masoud, H., Newbolt, J. W., Shelley, M. & Ristroph, L. 2015 Hydrodynamic schooling of flapping swimmers. Nat. Commun. 6, 8514.
Boschitsch, B. M., Dewey, P. A. & Smits, A. J. 2014 Propulsive performance of unsteady tandem hydrofoils in an in-line configuration. Phys. Fluids 26, 051901.
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329364.
Couzin, I. D., Krause, J., Franks, N. R. & Levin, S. A. 2005 Effective leadership and decision-making in animal groups on the move. Nature 433 (7025), 513516.
Doyle, J. F. 2001 Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability. Springer.
Godoy-Diana, R., Marais, C., Aider, J.-L. & Wesfreid, J.-E. 2009 A model for the symmetry breaking of the reverse Bénard–von Kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 2332.
Hua, R.-N., Zhu, L. & Lu, X.-Y. 2013 Locomotion of a flapping flexible plate. Phys. Fluids 25, 121901.
Hua, R.-N., Zhu, L. & Lu, X.-Y. 2014 Dynamics of fluid flow over a circular flexible plate. J. Fluid Mech. 759, 5672.
Kelley, D. H. & Ouellette, N. T. 2013 Emergent dynamics of laboratory insect swarms. Sci. Rep. 3, 1073.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.
Lighthill, M. J. 1975 Mathematical Biofluiddynamics. SIAM.
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239261.
Parrish, J. K. & Edelstein-Keshet, L. 1999 Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science 284, 99101.
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479517.
Portugal, S. J., Hubel, T. Y., Fritz, J., Heese, S., Trobe, D., Voelkl, B., Hailes, S., Wilson, A. M. & Usherwood, J. R. 2014 Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature 505 (7483), 399402.
Ramananarivo, S., Fang, F., Oza, A., Zhang, J. & Ristroph, L. 2016 Flow interactions lead to orderly formations of flapping wings in forward flight. Phys. Rev. Fluids 1 (7), 071201.
Ristroph, L. & Zhang, J. 2008 Anomalous hydrodynamic drafting of interacting flapping flags. Phys. Rev. Lett. 101 (19), 194502.
Rosellini, L. & Zhang, J.2007 The effect of geometry on the flapping flight of a simple wing. http://www.physics.nyu.edu/jz11/publications/LT10521stop.pdf.
Saintillan, D. & Shelley, M. J. 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103.
Tytell, E. D., Hsu, C.-Y., Williams, T. L., Cohen, A. H. & Fauci, L. J. 2010 Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl Acad. Sci. USA 107 (46), 1983219837.
Uddin, E., Huang, W.-X. & Sung, H. J. 2015 Actively flapping tandem flexible flags in a viscous flow. J. Fluid Mech. 780, 120142.
Vicsek, T. & Zafeiris, A. 2012 Collective motion. Phys. Rep. 517 (3), 71140.
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241, 290291.
Wu, T. Y. & Chwang, A. T. 1975 Extraction of flow energy by fish and birds in a wavy stream. In Swimming and Flying in Nature, pp. 687702. Springer.
Zhang, H.-P., Beer, A., Florin, E.-L. & Swinney, H. 2010 Collective motion and density fluctuations in bacterial colonies. Proc. Natl Acad. Sci. USA 107, 1362613630.
Zhu, X., He, G. & Zhang, X. 2014 Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration. Phys. Rev. Lett. 113, 238105.
Zou, Q. & He, X. 1997 On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9 (6), 15911598.
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JFM classification

Type Description Title
VIDEO
Movies

Peng et al. supplementary movie 1
Dynamics of the orderly formation (Fast Mode "C3+1"). The initial gap spacing is G0=1. At the equilibrium state, the whole group with N=4 consists of a compact subgroup “C3” and an individual "1" following the subgroup.

 Video (7.1 MB)
7.1 MB
VIDEO
Movies

Peng et al. supplementary movie 2
Equilibrium state of the orderly formation (Fast Mode "C3+C2+1"). The initial gap spacing is G0=1. The whole group with N=6 consists of two compact subgroups “C3”, “C2” and an individual "1" following the subgroups.

 Video (6.7 MB)
6.7 MB
VIDEO
Movies

Peng et al. supplementary movie 3
Dynamics of the orderly formation (Slow Mode "S4"). The initial gap spacing is G0=3. At the equilibrium state, the sparse configuration emerges and the four plates are equally spaced with normalized gap spacing G/λ=1.

 Video (5.2 MB)
5.2 MB
VIDEO
Movies

Peng et al. supplementary movie 4
Equilibrium state of the orderly formation (Slow Mode "S6+S2"). The initial gap spacing is G0=3. The whole group with N=8 consists of two sparse subgroups “S6” and “S2”.

 Video (5.9 MB)
5.9 MB
VIDEO
Movies

Peng et al. supplementary movie 5
Equilibrium state of the orderly formation (Slow Mode "S6+S2"). The initial gap spacing is G0=3. The whole group with N=8 consists of two sparse subgroups “S6” and “S2”.

 Video (3.3 MB)
3.3 MB
VIDEO
Movies

Peng et al. supplementary movie 6
Equilibrium state of the orderly formation (Slow Mode "S8"). The initial gap spacing is G0=5. In the sparse configuration, the eight plates are equally spaced with normalized gap spacing G/λ =1.

 Video (4.3 MB)
4.3 MB

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