- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Invited
  • Open Access

Hydrodynamic interactions in tandem flapping wing systems

Oscar Flores

Manuel Garcia-Villalba

  • Institute of Fluid Mechanics and Heat Transfer, TU Wien, Vienna, Austria

PDF

Phys. Rev. Fluids 10, 100502 – Published 15 October, 2025

DOI: https://doi.org/10.1103/1ry9-t1rt

Abstract

We review recent studies on the hydrodynamic interactions in tandem flapping wing systems, with a focus on the interplay between vortex dynamics and structural flexibility. We examine configurations including rigid wings with prescribed motion, spanwise-flexible wings, and chordwise-flexible wings. For rigid wings, three-dimensional effects are shown to reduce thrust, but they can be mitigated through aspect ratio tuning. Flexibility introduces fluid-structure resonance phenomena that enhance aerodynamic performance. In tandem systems, performance optimization requires careful coordination: forewings benefit from resonance, while hindwings perform best with slightly stiffer configurations to effectively capture upstream wake structures. Finally, in systems of self-propelled flexible flappers, hydrodynamic interactions give rise to coordinated motion and energy savings through flow-mediated feedback.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (99)

  1. M. García-Villalba, Bioinspired fluid-structure interaction: From winged seeds to flapping wings, 77th Annual Meeting of the Division of Fluid Dynamics, Bull. Amer. Phys. Soc. G03, G03.00001 (2024).
  2. C. P. Ellington, The novel aerodynamics of insect flight: Applications to micro-air vehicles, J. Exp. Biol. 202, 3439 (1999).
  3. T. J. Mueller, Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications (AIAA, Reston, Virginia, 2001).
  4. N. Haider, A. Shahzad, M. N. M. Qadri, and S. I. A. Shah, Recent progress in flapping wings for micro aerial vehicle applications, Proc. Inst. Mech. Eng. C. J. Mech. Eng. Sci. 235, 245 (2021).
  5. D. Quinn and G. Lauder, Tunable stiffness in fish robotics: Mechanisms and advantages, Bioinspir. Biomim. 17, 011002 (2022).
  6. A. Prakash, A. R. Nair, H. Arunav, P. Rthuraj, V. M. Akhil, C. Tawk, and K. V. Shankar, Bioinspiration and biomimetics in marine robotics: A review on current applications and future trends, Bioinspir. Biomim. 19, 031002 (2024).
  7. S. P. Sane, The aerodynamics of insect flight, J. Exp. Biol. 206, 4191 (2003).
  8. Z. J. Wang, Dissecting insect flight, Annu. Rev. Fluid Mech. 37, 183 (2005).
  9. B. W. Tobalske, Biomechanics of bird flight, J. Exp. Biol. 210, 3135 (2007).
  10. W. Shyy, H. Aono, C.-K. Kang, and H. Liu, An Introduction to Flapping Wing Aerodynamics (Cambridge University Press, Cambridge, 2013), Vol. 37.
  11. D. D. Chin and D. Lentink, Flapping wing aerodynamics: From insects to vertebrates, J. Exp. Biol. 219, 920 (2016).
  12. H. Liu, S. Wang, and T. Liu, Vortices and forces in biological flight: Insects, birds, and bats, Annu. Rev. Fluid Mech. 56, 147 (2024).
  13. M. S. Triantafyllou, G. S. Triantafyllou, and D. K. P. Yue, Hydrodynamics of fishlike swimming, Annu. Rev. Fluid Mech. 32, 33 (2000).
  14. J. E. Colgate and K. M. Lynch, Mechanics and control of swimming: A review, IEEE J. Oceanic Eng. 29, 660 (2004).
  15. F. E. Fish and G. V. Lauder, Passive and active flow control by swimming fishes and mammals, Annu. Rev. Fluid Mech. 38, 193 (2006).
  16. J. C. Liao, A review of fish swimming mechanics and behavior in altered flows, Philos. Trans. R. Soc. B 362, 1973 (2007).
  17. T. Y. Wu, Fish swimming and bird/insect flight, Annu. Rev. Fluid Mech. 43, 25 (2011).
  18. G. V. Lauder, Fish locomotion: Recent advances and new directions, Annu. Rev. Mar. Sci. 7, 521 (2015).
  19. A. J. Smits, Undulatory and oscillatory swimming, J. Fluid Mech. 874, P1 (2019).
  20. R. J. Bomphrey, T. Nakata, P. Henningsson, and H.-T. Lin, Flight of the dragonflies and damselflies, Philos. Trans. R. Soc. B 371, 20150389 (2016).
  21. J. M. Wakeling and C. P. Ellington, Dragonfly flight: II. Velocities, accelerations and kinematics of flapping flight, J. Exp. Biol. 200, 557 (1997).
  22. A. L. R. Thomas, G. K. Taylor, R. B. Srygley, R. L. Nudds, and R. J. Bomphrey, Dragonfly flight: Free-flight and tethered flow visualizations reveal a diverse array of unsteady lift-generating mechanisms, controlled primarily via angle of attack, J. Exp. Biol. 207, 4299 (2004).
  23. I. Ashraf, R. Godoy-Diana, J. Halloy, B. Collignon, and B. Thiria, Synchronization and collective swimming patterns in fish (Hemigrammus bleheri), J. R. Soc. Interface. 13, 20160734 (2016).
  24. M. Ligman, J. Lund, and M. Fürth, A comprehensive review of hydrodynamic studies on fish schooling, Bioinspir. Biomim. 19, 011002 (2024).
  25. 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).
  26. M. L. Timm, R. S. Pandhare, and H. Masoud, Multi-body hydrodynamic interactions in fish-like swimming, Appl. Mech. Rev. 76, 030801 (2024).
  27. W. Shyy, H. Aono, S. Chimakurthi, P. Trizila, C.-K. Kang, C. Cesnik, and H. Liu, Recent progress in flapping wing aerodynamics and aeroelasticity, Prog. Aerosp. Sci. 46, 284 (2010).
  28. M. F. Platzer, K. D. Jones, J. Young, and J. C. S. Lai, Flapping wing aerodynamics: Progress and challenges, AIAA J. 46, 2136 (2008).
  29. 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).
  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. M. Saadat, F. E. Fish, A. G. Domel, V. Di Santo, G. V. Lauder, and H. Haj-Hariri, On the rules for aquatic locomotion, Phys. Rev. Fluids 2, 083102 (2017).
  32. D. Floryan, T. Van Buren, and A. J. Smits, Efficient cruising for swimming and flying animals is dictated by fluid drag, Proc. Natl. Acad. Sci. USA 115, 8116 (2018).
  33. K. D. von Ellenrieder, K. Parker, and J. Soria, Fluid mechanics of flapping wings, Exp. Therm Fluid Sci. 32, 1578 (2008).
  34. C. P. Ellington, C. Van Den Berg, A. P. Willmott, and A. L. R. Thomas, Leading-edge vortices in insect flight, Nature (London) 384, 626 (1996).
  35. J. D. Eldredge and A. R. Jones, Leading-edge vortices: Mechanics and modeling, Annu. Rev. Fluid Mech. 51, 75 (2019).
  36. A. Gonzalo, M. García-Villalba, and O. Flores, A quantitative analysis of leading edge vortices in flapping wing aerodynamics, arXiv:2509.05040.
  37. L. Quartapelle and M. Napolitano, Force and moment in incompressible flows, AIAA J. 21, 911 (1983).
  38. C.-c. Chang, Potential flow and forces for incompressible viscous flow, Proc. R. Soc. A 437, 517 (1992).
  39. M. Moriche, G. Sedky, A. R. Jones, O. Flores, and M. García-Villalba, Characterization of aerodynamic forces on wings in plunge maneuvers, AIAA J. 59, 751 (2021).
  40. J. M. Anderson, K. Streitlien, D. S. Barrett, and M. S. Triantafyllou, Oscillating foils of high propulsive efficiency, J. Fluid Mech. 360, 41 (1998).
  41. D. A. Read, F. S. Hover, and M. S. Triantafyllou, Forces on oscillating foils for propulsion and maneuvering, J. Fluids Struct. 17, 163 (2003).
  42. J. Young and J. C. S. Lai, Mechanisms influencing the efficiency of oscillating airfoil propulsion, AIAA J. 45, 1695 (2007).
  43. M. Moriche, O. Flores, and M. García-Villalba, On the aerodynamic forces on heaving and pitching airfoils at low Reynolds number, J. Fluid Mech. 828, 395 (2017).
  44. R. Gopalkrishnan, M. S. Triantafyllou, G. S. Triantafyllou, and D. Barrett, Active vorticity control in a shear flow using a flapping foil, J. Fluid Mech. 274, 1 (1994).
  45. T. Engels, D. Kolomenskiy, K. Schneider, F.-O. Lehmann, and J. Sesterhenn, Bumblebee flight in heavy turbulence, Phys. Rev. Lett. 116, 028103 (2016).
  46. T. Engels, D. Kolomenskiy, K. Schneider, M. Farge, F.-O. Lehmann, and J. Sesterhenn, Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments, Phys. Rev. Fluids 4, 013103 (2019).
  47. S. Olivieri, J. M. Catalán, M. Garcia-Villalba, and O. Flores, Aerodynamic characterisation of a flapping wing in turbulent free-stream conditions, J. Fluid Mech. 1018, A12 (2025).
  48. J. M. Birch and M. H. Dickinson, The influence of wing–wake interactions on the production of aerodynamic forces in flapping flight, J. Exp. Biol. 206, 2257 (2003).
  49. F.-O. Lehmann, S. P. Sane, and M. Dickinson, The aerodynamic effects of wing–wing interaction in flapping insect wings, J. Exp. Biol. 208, 3075 (2005).
  50. B. Boschitsch, P. Dewey, and A. Smits, Propulsive performance of unsteady tandem hydrofoils in an in-line configuration, Phys. Fluids 26, 051901 (2014).
  51. T. Broering and Y. Lian, Numerical study of tandem flapping wing aerodynamics in both two and three dimensions, Comput. Fluids 115, 124 (2015).
  52. K. Lua, H. Lu, X. Zhang, T. Lim, and K. Yeo, Aerodynamics of two-dimensional flapping wings in tandem configuration, Phys. Fluids 28, 121901 (2016).
  53. F. Huera-Huarte, Propulsive performance of a pair of pitching foils in staggered configurations, J. Fluids Struct. 81, 1 (2018).
  54. J. D. Eldredge, J. Toomey, and A. Medina, On the roles of chord-wise flexibility in a flapping wing with hovering kinematics, J. Fluid Mech. 659, 94 (2010).
  55. L. Zhao, Q. Huang, X. Deng, and S. P. Sane, Aerodynamic effects of flexibility in flapping wings, J. R. Soc. Interface. 7, 485 (2010).
  56. S. Heathcote, Z. Wang, and I. Gursul, Effect of spanwise flexibility on flapping wing propulsion, J. Fluids Struct. 24, 183 (2008).
  57. C.-K. Kang, H. Aono, C. E. S. Cesnik, and W. Shyy, Effects of flexibility on the aerodynamic performance of flapping wings, J. Fluid Mech. 689, 32 (2011).
  58. M. Vanella, T. Fitzgerald, S. Preidikman, E. Balaras, and B. Balachandran, Influence of flexibility on the aerodynamic performance of a hovering wing, J. Exp. Biol. 212, 95 (2009).
  59. M. Moriche, A numerical study on the aerodynamic forces and the wake stability of flapping flight at low Reynolds number, Ph.D. thesis, Universidad Carlos III de Madrid, 2017.
  60. M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys. 209, 448 (2005).
  61. N. Arora, C.-K. Kang, W. Shyy, and A. Gupta, Analysis of passive flexion in propelling a plunging plate using a torsion spring model, J. Fluid Mech. 857, 562 (2018).
  62. M. L. Felis, RBDL: An efficient rigid-body dynamics library using recursive algorithms, Auton. Robots 41, 495 (2017).
  63. G. Arranz, C. Martínez-Muriel, O. Flores, and M. García-Villalba, Fluid–structure interaction of multi-body systems: Methodology and applications, J. Fluids Struct. 110, 103519 (2022).
  64. M. Moriche, M. Raiola, S. Discetti, A. Ianiro, O. Flores, and M. Garcia-Villalba, Assessing aerodynamic force estimation with experiments and simulations of flapping-airfoil flows on the verge of three-dimensionality, Proc. Inst. Mech. Eng. G J. Aerosp. Eng. 234, 428 (2020).
  65. M. Guerrero-Hurtado, J. M. Catalán, M. Moriche, A. Gonzalo, and O. Flores, A python-based flow solver for numerical simulations using an immersed boundary method on single GPUs, Comput. Fluids 288, 106511 (2025).
  66. D. Zhang, J.-D. Zhang, and W.-X. Huang, Physical models and vortex dynamics of swimming and flying: A review, Acta Mech. 233, 1249 (2022).
  67. G. Arranz, O. Flores, and M. Garcia-Villalba, Three-dimensional effects on the aerodynamic performance of flapping wings in tandem configuration, J. Fluids Struct. 94, 102893 (2020).
  68. R. Jurado, G. Arranz, O. Flores, and M. García-Villalba, Numerical simulation of flow over flapping wings in tandem: Wingspan effects, Phys. Fluids 34, 017114 (2022).
  69. C. Martínez-Muriel, G. Arranz, M. García-Villalba, and O. Flores, Fluid–structure resonance in spanwise-flexible flapping wings, J. Fluid Mech. 964, A5 (2023).
  70. C. Martínez-Muriel, M. García-Villalba, and O. Flores, On the role of wake-capture and resonance in spanwise-flexible flapping wings in tandem, J. Fluids Struct. 130, 104175 (2024).
  71. I. Akhtar, R. Mittal, G. V. Lauder, and E. Drucker, Hydrodynamics of a biologically inspired tandem flapping foil configuration, Theor. Comput. Fluid Dyn. 21, 155 (2007).
  72. L. Muscutt, G. Weymouth, and B. Ganapathisubramani, Performance augmentation mechanism of in-line tandem flapping foils, J. Fluid Mech. 827, 484 (2017).
  73. Y. Zheng, Y. Wu, and H. Tang, An experimental study on the forewing–hindwing interactions in hovering and forward flights, Int. J. Heat Fluid Flow 59, 62 (2016).
  74. M. Kurt and K. W. Moored, Flow interactions of two- and three-dimensional networked bio-inspired control elements in an in-line arrangement, Bioinspir. Biomim. 13, 045002 (2018).
  75. N. S. Lagopoulos, G. D. Weymouth, and B. Ganapathisubramani, Effect of aspect ratio on the propulsive performance of tandem flapping foils, Flow 3, E1 (2023).
  76. J. Ortega-Casanova and R. Fernández-Feria, Maximum propulsive efficiency of two pitching and plunging plates in tandem at low Reynolds number: A numerical analysis, Int. J. Numer. Method. Heat Fluid Flow 29, 4013 (2019).
  77. H. Dong, R. Mittal, and F. M. Najjar, Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils, J. Fluid Mech. 566, 309 (2006).
  78. W. Maybury and F. Lehmann, The fluid dynamics of flight control by kinematic phase lag variation between two robotic insect wings, J. Exp. Biol. 207, 4707 (2004).
  79. H. Nagai, K. Fujita, and M. Murozono, Experimental study on forewing–hindwing phasing in hovering and forward flapping flight, AIAA J. 57, 3779 (2019).
  80. J. Buchholz and A. Smits, The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel, J. Fluid Mech. 603, 331 (2008).
  81. C. Martínez-López, Aerodynamic performance optimization of two wings in tandem configuration, B.Sc. thesis, Universidad Carlos III de Madrid, 2019.
  82. M. N. J. Moore, Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing, Phys. Fluids 27, 091701 (2015).
  83. S. Ramananarivo, R. Godoy-Diana, and B. Thiria, Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance, Proc. Natl. Acad. Sci. USA 108, 5964 (2011).
  84. G. Arranz, O. Flores, and M. García-Villalba, Flow interaction of three-dimensional self-propelled flexible plates in tandem, J. Fluid Mech. 931, A5 (2022).
  85. 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).
  86. 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).
  87. 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).
  88. 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).
  89. A. P. Maertens, A. Gao, and M. S. Triantafyllou, Optimal undulatory swimming for a single fish-like body and for a pair of interacting swimmers, J. Fluid Mech. 813, 301 (2017).
  90. Z.-R. Peng, H. Huang, and X.-Y. Lu, Collective locomotion of two closely spaced self-propelled flapping plates, J. Fluid Mech. 849, 1068 (2018).
  91. Z.-R. Peng, H. Huang, and X.-Y. Lu, Collective locomotion of two self-propelled flapping plates with different propulsive capacities, Phys. Fluids 30, 111901 (2018).
  92. X. Lin, J. Wu, T. Zhang, and L. Yang, Phase difference effect on collective locomotion of two tandem autopropelled flapping foils, Phys. Rev. Fluids 4, 054101 (2019).
  93. J. Ryu, J. Yang, S. G. Park, and H. J. Sung, Phase-mediated locomotion of two self-propelled flexible plates in a tandem arrangement, Phys. Fluids 32, 041901 (2020).
  94. S. Heydari and E. Kanso, School cohesion, speed and efficiency are modulated by the swimmers flapping motion, J. Fluid Mech. 922, A27 (2021).
  95. X. Lin, J. Wu, T. Zhang, and L. Yang, Flow-mediated organization of two freely flapping swimmers, J. Fluid Mech. 912, A37 (2021).
  96. S. Verma, G. Novati, and P. Koumoutsakos, Efficient collective swimming by harnessing vortices through deep reinforcement learning, Proc. Natl. Acad. Sci. USA 115, 5849 (2018).
  97. M. Kurt, P. A. Eslam, and K. W. Moored, Flow interactions between low aspect ratio hydrofoils in in-line and staggered arrangements, Biomimetics 5, 13 (2020).
  98. H. Dai, H. Luo, and J. F. Doyle, Dynamic pitching of an elastic rectangular wing in hovering motion, J. Fluid Mech. 693, 473 (2012).
  99. P. D. Yeh and A. Alexeev, Free swimming of an elastic plate plunging at low Reynolds number, Phys. Fluids 26, 053604 (2014).

Outline

Information

Phys. Rev. Fluids 10, 100502– Published 15 October, 2025
  • Received 17 June 2025
  • Accepted 11 August 2025
Crossmark - Check for updates button
Creative Commons Logo

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

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation