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Modeling flying formations as flow-mediated matter

Christiana Mavroyiakoumou*, Jiajie Wu, and Leif Ristroph†

  • Courant Institute, Applied Math Lab, New York University, New York, New York 10012, USA

  • *Contact author: mavroyiakoum@maths.ox.ac.uk
  • †Contact author: ristroph@cims.nyu.edu
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Phys. Rev. Fluids 11, 063103 – Published 18 June, 2026

DOI: https://doi.org/10.1103/tp8s-76vr

Abstract

Collective locomotion of swimming and flying animals is fascinating in terms of individual-level fluid mechanics and group-level structure and dynamics. Here we bridge and relate these scales through a model of formation flight that views the collective as a material whose properties arise from the flow-mediated interactions among its members. We build on and revise an aerodynamic model describing how flapping flyers produce vortex wakes and how they are forced by others' wakes. While simplistic, the model faithfully reproduces a series of physical experiments carried out over the past decade on pairwise interactions of flapping foils. By studying longer in-line arrays, we show that the group behaves as a soft “crystal” with regularly spaced member “atoms” whose positioning is, however, susceptible to deformations and dynamical instabilities. Poking or wiggling a member excites longitudinal waves (flow-mediated phonons, or “flonons”) that pass down the group while growing in amplitude, and indeed the internal excitation from flapping is sufficient to trigger instabilities. Linear analysis of the model explains the aerodynamic origin of the lattice spacing, the springiness of the “bonds” between flyers, and the tendency for disturbances to resonantly amplify. Other properties, such as the timescales for instability growth and wave propagation, seem to involve the full nonlinear behavior. These findings suggest intriguing analogies with physical materials that could be generally useful for understanding and analyzing animal groups. Several properties displayed by our system seem particularly relevant to biological collectives, namely, group cohesion and organization, sensitive detection of and response to perturbations, and transmission of information through traveling waves.

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References (74)

  1. D. J. T. Sumpter, The principles of collective animal behaviour, Phil. Trans. R. Soc. B 361, 5 (2006).
  2. A. Cavagna and I. Giardina, Bird flocks as condensed matter, Annu. Rev. Condens. Matter Phys. 5, 183 (2014).
  3. C. K. Hemelrijk and H. Hildenbrandt, Schools of fish and flocks of birds: Their shape and internal structure by self-organization, Interface focus, 2, 726 (2012).
  4. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
  5. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  6. T. Mora, A. M. Walczak, L. Del Castello, F. Ginelli, S. Melillo, L. Parisi, M. Viale, A. Cavagna, and I. Giardina, Local equilibrium in bird flocks, Nat. Phys. 12, 1153 (2016).
  7. J. J. L. Higdon and S. Corrsin, Induced drag of a bird flock, Am. Nat. 112, 727 (1978).
  8. D. Weihs, Hydromechanics of fish schooling, Nature (London) 241, 290 (1973).
  9. D. Weihs, Some hydrodynamical aspects of fish schooling, Swim. Fly. Nat. 2, 703 (1975).
  10. P. B. S. Lissaman and C. A. Shollenberger, Formation flight of birds, Science 168, 1003 (1970).
  11. D. Hummel, Aerodynamic aspects of formation flight in birds, J. Theor. Biol. 104, 321 (1983).
  12. J. C. Liao, A review of fish swimming mechanics and behaviour in altered flows, Philos. Trans. R. Soc. B 362, 1973 (2007).
  13. B. L. Partridge and T. J. Pitcher, Evidence against a hydrodynamic function for fish schools, Nature (London) 279, 418 (1979).
  14. D. S. Pavlov and A. O. Kasumyan, Patterns and mechanisms of schooling behaviour in fish: a review, J. Ichthyol 40, S163 (2000).
  15. J. P. Badgerow and F. R. Hainsworth, Energy savings through formation flight? A re-examination of the vee formation, J. Theor. Biol. 93, 41 (1981).
  16. M. V. Abrahams and P. W. Colgan, Fish schools and their hydrodynamic function: A reanalysis, Environ. Biol. Fishes 20, 79 (1987).
  17. M. Larsson, Why do fish school? Curr. Zool. 58, 116 (2012).
  18. Sir J. Lighthill, Mathematical Biofluiddynamics (SIAM, Philadelphia, PA, 1975).
  19. S. Ramananarivo, F. Fang, A. Oza, J. Zhang, and L. Ristroph, Flow interactions lead to orderly formations of flapping wings in forward flight, Phys. Rev. Fluids 1, 071201(R) (2016).
  20. X. Zhu, G. He, and X. Zhang, Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration, Phys. Rev. Lett. 113, 238105 (2014).
  21. A. D. Becker, H. Masoud, J. W. Newbolt, M. Shelley, and L. Ristroph, Hydrodynamic schooling of flapping swimmers, Nat. Commun. 6, 8514 (2015).
  22. L. Dai, G. He, X. Zhang, and X. Zhang, Stable formations of self-propelled fish-like swimmers induced by hydrodynamic interactions, J. R. Soc. Interface. 15, 20180490 (2018).
  23. Z.-R. Peng, H. Huang, and X.-Y. Lu, Hydrodynamic schooling of multiple self-propelled flapping plates, J. Fluid Mech. 853, 587 (2018).
  24. J. W. Newbolt, J. Zhang, and L. Ristroph, Flow interactions between uncoordinated flapping swimmers give rise to group cohesion, Proc. Natl. Acad. Sci. USA 116, 2419 (2019).
  25. J. W. Newbolt, N. Lewis, M. Bleu, J. Wu, C. Mavroyiakoumou, S. Ramananarivo, and L. Ristroph, Flow interactions lead to self-organized flight formations disrupted by self-amplifying waves, Nat. Commun. 15, 3462 (2024).
  26. A. U. Oza, L. Ristroph, and M. J. Shelley, Lattices of hydrodynamically interacting flapping swimmers, Phys. Rev. X 9, 041024 (2019).
  27. M. Saadat, F. Berlinger, A. Sheshmani, R. Nagpal, G. V. Lauder, and H. Haj-Hariri, Hydrodynamic advantages of in-line schooling, Bioinspir. Biomim. 16, 046002 (2021).
  28. S. Heydari and E. Kanso, School cohesion, speed and efficiency are modulated by the swimmers flapping motion, J. Fluid Mech. 922, A27 (2021).
  29. M. Nitsche, A. U. Oza, and M. Siegel, On the stability of an in-line formation of hydrodynamically interacting flapping plates, J. Fluid Mech. 1013, A14 (2025).
  30. G. S. Triantafyllou, M. S. Triantafyllou, and M. A. Grosenbaugh, Optimal thrust development in oscillating foils with application to fish propulsion, J. Fluids Struct. 7, 205 (1993).
  31. A. Andersen, T. Bohr, T. Schnipper, and J. H. Walther, Wake structure and thrust generation of a flapping foil in two-dimensional flow, J. Fluid Mech. 812, R4 (2017).
  32. B. M. Boschitsch, P. A. Dewey, and A. J. Smits, Propulsive performance of unsteady tandem hydrofoils in an in-line configuration, Phys. Fluids 26, 051901 (2014).
  33. D. Rival, G. Hass, and C. Tropea, Recovery of energy from leading-and trailing-edge vortices in tandem-airfoil configurations, J. Aircr. 48, 203 (2011).
  34. N. Vandenberghe, J. Zhang, and S. Childress, Symmetry breaking leads to forward flapping flight, J. Fluid Mech. 506, 147 (2004).
  35. S. Alben and M. Shelley, Coherent locomotion as an attracting state for a free flapping body, Proc. Natl. Acad. Sci. USA 102, 11163 (2005).
  36. N. Vandenberghe, S. Childress, and J. Zhang, On unidirectional flight of a free flapping wing, Phys. Fluids 18, 014102 (2006).
  37. S. E. Spagnolie, L. Moret, M. J. Shelley, and J. Zhang, Surprising behaviors in flapping locomotion with passive pitching, Phys. Fluids 22, 041903 (2010).
  38. T. Han, A. Mivehchi, S. S. Sarraf, and K. W. Moored, Tailoring formations of self-organising hydrofoil schools towards high-efficiency, J. Fluid Mech. 1012, A26 (2025).
  39. D. Floryan, T. Van Buren, C. W. Rowley, and A. J. Smits, Scaling the propulsive performance of heaving and pitching foils, J. Fluid Mech. 822, 386 (2017).
  40. A. J. Smits, Undulatory and oscillatory swimming, J. Fluid Mech. 874, P1 (2019).
  41. M. Gazzola, M. Argentina, and L. Mahadevan, Scaling macroscopic aquatic locomotion, Nat. Phys. 10, 758 (2014).
  42. D. J. Tritton, Physical Fluid Dynamics (Springer Science & Business Media, Cham, 2012).
  43. H. Schlichting and K. Gersten, Boundary-Layer Theory (Springer, Berlin, 2016).
  44. The MathWorks Inc. Matlab version: R2023a (2023).
  45. F. E. Fish, J. F. Fegely, and C. J. Xanthopoulos, Burst-and-coast swimming in schooling fish (Notemigonus crysoleucas) with implications for energy economy, Comp. Biochem. Physiol. A: Physiol. 100, 633 (1991).
  46. O. K. G. Tietjens and L. Prandtl, Applied Hydro- and Aeromechanics: Based on Lectures of L. Prandtl (Courier Corporation, North Chelmsford, MA, 1957), Vol. 2.
  47. F. M. White, Fluid Mechanics (McGraw HillH, Columbus, OH, 1966).
  48. F. Fang, C. Mavroyiakoumou, L. Ristroph, and M. J. Shelley, Flow interactions and forward flight dynamics of tandem flapping wings, J. Fluid Mech. 1034, A51 (2026).
  49. K. W. Moored and D. B. Quinn, Inviscid scaling laws of a self-propelled pitching airfoil, AIAA J. 57, 3686 (2019).
  50. M. J. Lighthill, Large-amplitude elongated-body theory of fish locomotion, Proc. Roy. Soc. London B: Biol. Sci. 179, 125 (1971).
  51. C. Eloy, Optimal Strouhal number for swimming animals, J. Fluids Struct. 30, 205 (2012).
  52. M. Sfakiotakis, D. M. Lane, and J. B. C. Davies, Review of fish swimming modes for aquatic locomotion, IEEE J. Oceanic Eng. 24, 237 (1999).
  53. G. K. Taylor, R. L. Nudds, and A. L. R. Thomas, Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency, Nature (London) 425, 707 (2003).
  54. T. Beatus, T. Tlusty, and R. Bar-Ziv, Phonons in a one-dimensional microfluidic crystal, Nat. Phys. 2, 743 (2006).
  55. P. M. Chaikin, T. C. Lubensky, and T. A. Witten, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, UK, 1995), Vol. 10.
  56. A. Andersen, U. Pesavento, and Z. J. Wang, Analysis of transitions between fluttering, tumbling and steady descent of falling cards, J. Fluid Mech. 541, 91 (2005).
  57. A. Andersen, U. Pesavento, and Z. J. Wang, Unsteady aerodynamics of fluttering and tumbling plates, J. Fluid Mech. 541, 65 (2005).
  58. L. I. Sedov, Two-Dimensional Problems of Hydrodynamics and Aerodynamics (Interscience Publishers, New York, NY, 1965).
  59. P. J. Baddoo, N. J. Moore, A. U. Oza, and D. G. Crowdy, Generalization of waving-plate theory to multiple interacting swimmers, Commun. Pure Appl. Math. 76, 3811 (2023).
  60. C. H. Greenewalt, Dimensional relationships for flying animals, Smithsonian Miscellaneous Collections (Smithsonian Libraries, Washington, D.C., 1962).
  61. G. Viscor and J. F. Fuster, Relationships between morphological parameters in birds with different flying habits, Comp. Biochem. Physiol. Part A: Physiology 87, 231 (1987).
  62. G. Dimitriadis, J. D. Gardiner, P. G. Tickle, J. Codd, and R. L. Nudds, Experimental and numerical study of the flight of geese, Aeronaut. J. 119, 803 (2015).
  63. C. J. Pennycuick, Predicting wingbeat frequency and wavelength of birds, J. Exp. Biol. 150, 171 (1990).
  64. S. Dhawan, Bird flight, Sadhana, 16, 275 (1991).
  65. J. M. V. Rayner, A new approach to animal flight mechanics, J. Exp. Biol. 80, 17 (1979).
  66. K. Krishnan, B. Garde, A. Bennison, N. C. Cole, E.-L. Cole, J. Darby, K. H. Elliott, A. Fell, A. Gómez-Laich, S. De Grissac, et al., The role of wingbeat frequency and amplitude in flight power, J. R. Soc. Interface. 19, 20220168 (2022).
  67. L. A. Taylor, G. K. Taylor, B. Lambert, J. A. Walker, D. Biro, and S. J. Portugal, Birds invest wingbeats to keep a steady head and reap the ultimate benefits of flying together, PLoS Biol 17, e3000299 (2019).
  68. I. Maggini, L. V. Kennedy, A. Macmillan, K. H. Elliott, K. Dean, and C. G. Guglielmo, Light oiling of feathers increases flight energy expenditure in a migratory shorebird, J. Exp. Biol. 220, 2372 (2017).
  69. A. Nafi, H. Ben-Gida, C. G. Guglielmo, and R. Gurka, Aerodynamic forces acting on birds during flight: A comparative study of a shorebird, songbird and a strigiform, Exp. Therm Fluid Sci. 113, 110018 (2020).
  70. J. M. Segreto, A. J. Kirchhefer, E. E. Hackett, C. G. Guglielmo, G. A. Kopp, and R. Gurka, Experimental study of the unsteady aerodynamics during flapping flight of birds: European starling, Western sandpiper and American robin, in Proceedings of the 10th Pacific Symposium on Flow Visualization and Image Processing (Naples, Italy, 2015).
  71. S. Heydari, H. Hang, and E. Kanso, Mapping spatial patterns to energetic benefits in groups of flow-coupled swimmers, eLife, 13, RP96129, (2024).
  72. M. Brandenbourger, X. Locsin, E. Lerner, and C. Coulais, Non-reciprocal robotic metamaterials, Nat. Commun. 10, 4608 (2019).
  73. J. A. Carrillo, Y.-P. Choi, and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collect. Dynam. Bacteria Crowds 1 (2014).
  74. M. Bruna, M. Burger, and O. de Wit, Lane formation and aggregation spots in a model of ants, SIAM J. Appl. Dyn. Syst. 24, 675 (2025).

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Phys. Rev. Fluids 11, 063103– Published 18 June, 2026
  • Received 16 June 2025
  • Accepted 30 April 2026
Article part of CHORUS
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