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Physical models and vortex dynamics of swimming and flying: a review

  • Review and Perspective in Mechanics (by invitation only)
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Abstract

The swimming of aquatic animals and flying of insects and birds have fascinated physicists and biologists for more than a century. In this regard, great efforts have been made to develop new features and promote their applications in underwater and air propulsion. However, many challenges remain in understanding these forms of physical processes. Five key physical models are summarized to show how researchers use numerical and experimental methods to understand physiology, movement ecology and evolution from the viewpoint of fluid mechanics. They are morphological model, flexibility model, kinematics model, tethered/free model and force measurement model. Then, the latest progresses on the vortex dynamics of some simplified models and even high-fidelity models are presented, including the forming, growth, interaction, role and influence factors of the vortical structures. Some other aspects in swimming and flying, including stability, manoeuvrability and acoustics, are also briefly reviewed. Finally, the major challenges and several open issues in this field are highlighted.

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References

  1. Salazar, R., Fuentes, V., Abdelkefi, A.: Classification of biological and bioinspired aquatic systems: a review. Ocean Eng. 148, 75–114 (2018)

    Article  Google Scholar 

  2. Liu, H., Ravi, S., Kolomenskiy, D., Tanaka, H.: Biomechanics and biomimetics in insect-inspired flight systems. Philos. Trans. R. Soc. B: Biol. Sci. 371(1704), 20150390 (2016)

    Article  Google Scholar 

  3. Gray, J.: Animal Locomotion. Weidenfeld & Nicolson, London (1968)

    Google Scholar 

  4. Videler, J.J.: Fish Swimming, vol. 10. Springer, Chapman and Hall (1993)

    Book  Google Scholar 

  5. Lauder, G.V.: Fish locomotion: recent advances and new directions. Annu. Rev. Mar. Sci. 7, 521–545 (2015)

    Article  Google Scholar 

  6. Triantafyllou, M.S., Triantafyllou, G., Yue, D.: Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32(1), 33–53 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chin, D.D., Lentink, D.: Flapping wing aerodynamics: from insects to vertebrates. J. Exp. Biol. 219(7), 920–932 (2016)

    Article  Google Scholar 

  8. Wang, Z.J.: Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, H., Nakata, T., Li, G., Kolomenskiy, D.: Biomechanics and biomimetics in flying and swimming. In: Industrial Biomimetics, pp. 29–80. Jenny Stanford Publishing, Boca Raton (2019)

  10. Sane, S.P.: The aerodynamics of insect flight. J. Exp. Biol. 206(23), 4191–4208 (2003)

    Article  Google Scholar 

  11. Bhat, S., Zhao, J., Sheridan, J., Hourigan, K., Thompson, M.: Aspect ratio studies on insect wings. Phys. Fluids 31(12), 121301 (2019)

    Article  Google Scholar 

  12. Shyy, W., Aono, H., Chimakurthi, S.K., Trizila, P., Kang, C.-K., Cesnik, C.E., Liu, H.: Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46(7), 284–327 (2010)

    Article  Google Scholar 

  13. Smits, A.J.: Undulatory and oscillatory swimming. J. Fluid Mech. 874 (2019)

  14. Wu, T.Y.: Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech. 43, 25–58 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Triantafyllou, M.S., Weymouth, G.D., Miao, J.: Biomimetic survival hydrodynamics and flow sensing. Annu. Rev. Fluid Mech. 48, 1–24 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shelley, M.J., Zhang, J.: Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43, 449–465 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fish, F., Lauder, G.V.: Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193–224 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fish, F.E.: The myth and reality of gray’s paradox: implication of dolphin drag reduction for technology. Bioinspir. Biomimetics 1(2), 17 (2006)

    Article  Google Scholar 

  19. Costello, J.H., Colin, S.P., Dabiri, J.O., Gemmell, B.J., Lucas, K.N., Sutherland, K.R.: The hydrodynamics of jellyfish swimming. Annu. Rev. Mar. Sci. 13, 375–396 (2021)

    Article  Google Scholar 

  20. Eldredge, J.D., Jones, A.R.: Leading-edge vortices: mechanics and modeling. Annu. Rev. Fluid Mech. 51, 75–104 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Izraelevitz, J.S., Kotidis, M., Triantafyllou, M.S.: Optimized kinematics enable both aerial and aquatic propulsion from a single three-dimensional flapping wing. Phys. Rev. Fluids 3(7), 073102 (2018)

  22. Webb, P.W.: The biology of fish swimming. Mechanics and physiology of animal swimming 4562 (1994)

  23. Lauder, G.V., Madden, P.G.: Fish locomotion: kinematics and hydrodynamics of flexible foil-like fins. Exp. Fluids 43(5), 641–653 (2007)

    Article  Google Scholar 

  24. Triantafyllou, G.S., Triantafyllou, M.S., Grosenbaugh, M.A.: Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7(2), 205–224 (1993)

    Article  Google Scholar 

  25. Floryan, D., Van Buren, T., Rowley, C.W., Smits, A.J.: Scaling the propulsive performance of heaving and pitching foils. J. Fluid Mech. 822, 386–397 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chao, L.-M., Pan, G., Zhang, D., Yan, G.-X.: Numerical investigations on the force generation and wake structures of a nonsinusoidal pitching foil. J. Fluids Struct. 85, 27–39 (2019)

    Article  Google Scholar 

  27. Zhang, D., Pan, G., Chao, L., Yan, G.: Mechanisms influencing the efficiency of aquatic locomotion. Mod. Phys. Lett. B 32(25), 1850299 (2018)

    Article  Google Scholar 

  28. Zhang, D., Chao, L., Pan, G.: Ground effect on a self-propelled undulatory foil. Mod. Phys. Lett. B 32(11), 1850135 (2018)

    Article  Google Scholar 

  29. Lin, X., Wu, J., Zhang, T.: Self-directed propulsion of an unconstrained flapping swimmer at low reynolds number: hydrodynamic behaviour and scaling laws. J. Fluid Mech. 907 (2021)

  30. Deng, H.-B., Xu, Y.-Q., Chen, D.-D., Dai, H., Wu, J., Tian, F.-B.: On numerical modeling of animal swimming and flight. Comput. Mech. 52(6), 1221–1242 (2013)

    Article  MathSciNet  Google Scholar 

  31. Danos, N., Lauder, G.V.: Challenging zebrafish escape responses by increasing water viscosity. J. Exp. Biol. 215(11), 1854–1862 (2012)

    Article  Google Scholar 

  32. Breder, C.M., Jr.: The locomotion of fishes. Zoologica 4, 159–291 (1926)

    Google Scholar 

  33. Videler, J., Stamhuis, E., Povel, G.: Leading-edge vortex lifts swifts. Science 306(5703), 1960–1962 (2004)

    Article  Google Scholar 

  34. Thielicke, W., Stamhuis, E.J.: The influence of wing morphology on the three-dimensional flow patterns of a flapping wing at bird scale. J. Fluid Mech. 768, 240–260 (2015)

    Article  Google Scholar 

  35. Oh, S., Lee, B., Park, H., Choi, H., Kim, S.-T.: A numerical and theoretical study of the aerodynamic performance of a hovering rhinoceros beetle (trypoxylus dichotomus). J. Fluid Mech. 885 (2020)

  36. Shen, C., Sun, M.: Wing and body kinematics measurement and force analyses of landing in fruit flies. Bioinspiration & biomimetics 13(1), 016004 (2017)

  37. Bode-Oke, A.T., Zeyghami, S., Dong, H.: Aerodynamics and flow features of a damselfly in takeoff flight. Bioinspir. Biomimetics 12(5), 056006 (2017)

  38. Flammang, B.E., Lauder, G.V.: Pectoral fins aid in navigation of a complex environment by bluegill sunfish under sensory deprivation conditions. J. Exp. Biol. 216(16), 3084–3089 (2013)

    Article  Google Scholar 

  39. Wainwright, D.K., Lauder, G.V.: Tunas as a high-performance fish platform for inspiring the next generation of autonomous underwater vehicles. Bioinspir. Biomimetics 15(3), 035007 (2020)

  40. Zhang, J., Childress, S., Libchaber, A., Shelley, M.: Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408(6814), 835–839 (2000)

    Article  Google Scholar 

  41. Uddin, E., Huang, W.-X., Sung, H.J.: Interaction modes of multiple flexible flags in a uniform flow. J. Fluid Mech. 729, 563–583 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Uddin, E., Huang, W.-X., Sung, H.J.: Actively flapping tandem flexible flags in a viscous flow. J. Fluid Mech. 780, 120–142 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lim, J.L., Lauder, G.V.: Mechanisms of anguilliform locomotion in fishes studied using simple three-dimensional physical models. Bioinspir. Biomimetics 11(4), 046006 (2016)

  44. Zurman-Nasution, A., Ganapathisubramani, B., Weymouth, G.: Influence of three-dimensionality on propulsive flapping. J. Fluid Mech. 886 (2020)

  45. Zhang, D., Pan, G., Chao, L., Zhang, Y.: Effects of reynolds number and thickness on an undulatory self-propelled foil. Phys. Fluids 30(7), 071902 (2018)

    Article  Google Scholar 

  46. Zhang, C., Huang, H., Lu, X.-Y.: Effect of trailing-edge shape on the self-propulsive performance of heaving flexible plates. J. Fluid Mech. 887 (2020)

  47. Koehler, C., Liang, Z., Gaston, Z., Wan, H., Dong, H.: 3d reconstruction and analysis of wing deformation in free-flying dragonflies. J. Exp. Biol. 215(17), 3018–3027 (2012)

    Google Scholar 

  48. Kern, S., Koumoutsakos, P.: Simulations of optimized anguilliform swimming. J. Exp. Biol. 209(24), 4841–4857 (2006)

    Article  Google Scholar 

  49. Zhu, Q., Wolfgang, M., Yue, D., Triantafyllou, M.: Three-dimensional flow structures and vorticity control in fish-like swimming. J. Fluid Mech. 468, 1–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tanaka, H., Li, G., Uchida, Y., Nakamura, M., Ikeda, T., Liu, H.: Measurement of time-varying kinematics of a dolphin in burst accelerating swimming. PLoS ONE 14(1), 0210860 (2019)

    Article  Google Scholar 

  51. Huang, Q., Zhang, D., Pan, G.: Computational model construction and analysis of the hydrodynamics of a rhinoptera javanica. IEEE Access 8, 30410–30420 (2020)

    Article  Google Scholar 

  52. Zou, P.-Y., Lai, Y.-H., Yang, J.-T.: Effects of phase lag on the hovering flight of damselfly and dragonfly. Phys. Rev. E 100(6), 063102 (2019)

    Article  Google Scholar 

  53. Xu, R., Zhang, X., Liu, H.: Effects of wing-to-body mass ratio on insect flapping flights. Phys. Fluids 33(2), 021902 (2021)

    Article  Google Scholar 

  54. Nakata, T., Liu, H.: Aerodynamic performance of a hovering hawkmoth with flexible wings: a computational approach. Proc. R. Soc. B: Biol. Sci. 279(1729), 722–731 (2012)

    Article  Google Scholar 

  55. Aono, H., Liang, F., Liu, H.: Near-and far-field aerodynamics in insect hovering flight: an integrated computational study. J. Exp. Biol. 211(2), 239–257 (2008)

    Article  Google Scholar 

  56. Liu, L.-G., Du, G., Sun, M.: Aerodynamic-force production mechanisms in hovering mosquitoes. J. Fluid Mech. 898 (2020)

  57. Watts, P., Mitchell, E.J., Swartz, S.M.: A computational model for estimating the mechanics of horizontal flapping flight in bats: model description and validation. J. Exp. Biol. 204(16), 2873–2898 (2001)

    Article  Google Scholar 

  58. Wang, S., Zhang, X., He, G., Liu, T.: Numerical simulation of unsteady flows over a slow-flying bat. Theor. Appl. Mech. Lett. 5(1), 5–8 (2015)

    Article  Google Scholar 

  59. Wang, S., Zhang, X., He, G., Liu, T.: Lift enhancement by bats’ dynamically changing wingspan. J. R. Soc. Interface 12(113), 20150821 (2015)

    Article  Google Scholar 

  60. Maeda, M., Nakata, T., Kitamura, I., Tanaka, H., Liu, H.: Quantifying the dynamic wing morphing of hovering hummingbird. R. Soc. Open Sci. 4(9), 170307 (2017)

  61. Luo, H., Dai, H., Das, S.S., Song, J., Doyle, J.: Toward high-fidelity modeling of the fluid-structure interaction for insect wings. In: 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, p. 1212 (2012)

  62. Chang, E., Matloff, L.Y., Stowers, A.K., Lentink, D.: Soft biohybrid morphing wings with feathers underactuated by wrist and finger motion. Sci. Robotics 5(38) (2020)

  63. Huang, W.-X., Sung, H.J.: Three-dimensional simulation of a flapping flag in a uniform flow. J. Fluid Mech. 653, 301–336 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  64. Huang, W.-X., Shin, S.J., Sung, H.J.: Simulation of flexible filaments in a uniform flow by the immersed boundary method. J. Comput. Phys. 226(2), 2206–2228 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  65. Huang, W.-X., Sung, H.J.: An immersed boundary method for fluid-flexible structure interaction. Comput. Methods Appl. Mech. Eng. 198(33–36), 2650–2661 (2009)

    Article  MATH  Google Scholar 

  66. Shahzad, A., Tian, F.-B., Young, J., Lai, J.C.: Effects of flexibility on the hovering performance of flapping wings with different shapes and aspect ratios. J. Fluids Struct. 81, 69–96 (2018)

    Article  Google Scholar 

  67. Tian, F.-B., Luo, H., Song, J., Lu, X.-Y.: Force production and asymmetric deformation of a flexible flapping wing in forward flight. J. Fluids Struct. 36, 149–161 (2013)

    Article  Google Scholar 

  68. Wang, C., Ren, F., Tang, H.: Enhancing propulsion performance of a flexible heaving foil through dynamically adjusting its flexibility. Bioinspir. Biomimetics 14(6), 064002 (2019)

  69. Ryu, J., Park, S.G., Huang, W.-X., Sung, H.J.: Hydrodynamics of a three-dimensional self-propelled flexible plate. Phys. Fluids 31(2), 021902 (2019)

    Article  Google Scholar 

  70. Wang, S., Ryu, J., He, G.-Q., Qin, F., Sung, H.J.: A self-propelled flexible plate with a navier slip surface. Phys. Fluids 32(2), 021906 (2020)

    Article  Google Scholar 

  71. Tian, F.-B.: A numerical study of linear and nonlinear kinematic models in fish swimming with the dsd/sst method. Comput. Mech. 55(3), 469–477 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  72. Chen, Y., Gravish, N., Desbiens, A.L., Malka, R., Wood, R.J.: Experimental and computational studies of the aerodynamic performance of a flapping and passively rotating insect wing. J. Fluid Mech. 791, 1–33 (2016)

    Article  MathSciNet  Google Scholar 

  73. Lei, M., Li, C.: The aerodynamic performance of passive wing pitch in hovering flight. Phys. Fluids 32(5), 051902 (2020)

    Article  Google Scholar 

  74. Ishihara, D., Horie, T.: Passive mechanism of pitch recoil in flapping insect wings. Bioinspir. Biomimetics 12(1), 016008 (2016)

  75. Zeyghami, S., Zhong, Q., Liu, G., Dong, H.: Passive pitching of a flapping wing in turning flight. AIAA J. 57(9), 3744–3752 (2019)

    Article  Google Scholar 

  76. Zhao, F., Qadri, M.M., Wang, Z., Tang, H.: Flow-energy harvesting using a fully passive flapping foil: a guideline on design and operation. Int. J. Mech. Sci. 197, 106323 (2021)

  77. Qadri, M.M., Zhao, F., Tang, H.: Fluid-structure interaction of a fully passive flapping foil for flow energy extraction. Int. J. Mech. Sci. 177, 105587 (2020)

    Article  Google Scholar 

  78. Qadri, M.M., Shahzad, A., Zhao, F., Tang, H., et al.: An experimental investigation of a passively flapping foil in energy harvesting mode (2019)

  79. Tytell, E.D., Hsu, C.-Y., Williams, T.L., Cohen, A.H., Fauci, L.J.: Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl. Acad. Sci. 107(46), 19832–19837 (2010)

    Article  Google Scholar 

  80. Combes, S., Daniel, T.: Flexural stiffness in insect wings ii. spatial distribution and dynamic wing bending. J. Exp. Biol. 206(17), 2989–2997 (2003)

  81. Shahzad, A., Tian, F.-B., Young, J., Lai, J.C.: Effects of hawkmoth-like flexibility on the aerodynamic performance of flapping wings with different shapes and aspect ratios. Phys. Fluids 30(9), 091902 (2018)

    Article  Google Scholar 

  82. Webb, P.W.: Form and function in fish swimming. Sci. Am. 251(1), 72–83 (1984)

    Article  Google Scholar 

  83. Khalid, M.S.U., Wang, J., Dong, H., Liu, M.: Flow transitions and mapping for undulating swimmers. Phys. Rev. Fluids 5(6), 063104 (2020)

  84. Tytell, E.D., Lauder, G.V.: The hydrodynamics of eel swimming: I. Wake structure. J. Exp. Biol. 207(11), 1825–1841 (2004)

  85. Borazjani, I., Sotiropoulos, F.: On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming. J. Exp. Biol. 213(1), 89–107 (2010)

    Article  Google Scholar 

  86. Borazjani, I., Sotiropoulos, F.: Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 211(10), 1541–1558 (2008)

    Article  Google Scholar 

  87. Zhang, J.-D., Sung, H.J., Huang, W.-X.: Specialization of tuna: a numerical study on the function of caudal keels. Phys. Fluids 32(11), 111902 (2020)

  88. Senturk, U., Smits, A.J.: Numerical simulations of the flow around a square pitching panel. J. Fluids Struct. 76, 454–468 (2018)

    Article  Google Scholar 

  89. Senturk, U., Smits, A.J.: Reynolds number scaling of the propulsive performance of a pitching airfoil. AIAA J. 57(7), 2663–2669 (2019)

    Article  Google Scholar 

  90. Van Buren, T., Floryan, D., Smits, A.J.: Scaling and performance of simultaneously heaving and pitching foils. AIAA J. 57(9), 3666–3677 (2019)

    Article  MATH  Google Scholar 

  91. Floryan, D., Van Buren, T., Smits, A.J.: Large-amplitude oscillations of foils for efficient propulsion. Phys. Rev. Fluids 4(9), 093102 (2019)

  92. Schouveiler, L., Hover, F., Triantafyllou, M.: Performance of flapping foil propulsion. J. Fluids Struct. 20(7), 949–959 (2005)

    Article  Google Scholar 

  93. Dong, H., Mittal, R., Najjar, F.: Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils. J. Fluid Mech. 566, 309–343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  94. Zhang, D., Huang, Q.-G., Pan, G., Yang, L.-M., Huang, W.-X.: Vortex dynamics and hydrodynamic performance enhancement mechanism in batoid fish oscillatory swimming. J. Fluid Mech. 930 (2022)

  95. Thekkethil, N., Sharma, A., Agrawal, A.: Three-dimensional biological hydrodynamics study on various types of batoid fishlike locomotion. Phys. Rev. Fluids 5(2), 023101 (2020)

  96. Bottom Ii, R., Borazjani, I., Blevins, E., Lauder, G.: Hydrodynamics of swimming in stingrays: numerical simulations and the role of the leading-edge vortex. J. Fluid Mech. 788, 407–443 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  97. Blevins, E.L., Lauder, G.V.: Rajiform locomotion: three-dimensional kinematics of the pectoral fin surface during swimming in the freshwater stingray potamotrygon orbignyi. J. Exp. Biol. 215(18), 3231–3241 (2012)

    Google Scholar 

  98. Su, G., Shen, H., Li, N., Zhu, Y., Su, Y.: Numerical investigation of the hydrodynamics of stingray swimming under self-propulsion. J. Fluids Struct. 106, 103383 (2021)

    Article  Google Scholar 

  99. Standen, E., Lauder, G.V.: Dorsal and anal fin function in bluegill sunfish lepomis macrochirus: three-dimensional kinematics during propulsion and maneuvering. J. Exp. Biol. 208(14), 2753–2763 (2005)

    Article  Google Scholar 

  100. Bozkurttas, M., Mittal, R., Dong, H., Lauder, G., Madden, P.: Low-dimensional models and performance scaling of a highly deformable fish pectoral fin. J. Fluid Mech. 631, 311–342 (2009)

    Article  MATH  Google Scholar 

  101. Wang, J., Ren, Y., Li, C., Dong, H.: Computational investigation of wing-body interaction and its lift enhancement effect in hummingbird forward flight. Bioinspir. biomimetics 14(4), 046010 (2019)

  102. Liu, Y., Sun, M.: Wing kinematics measurement and aerodynamics of hovering droneflies. J. Exp. Biol. 211(13), 2014–2025 (2008)

    Article  Google Scholar 

  103. Wan, H., Dong, H., Gai, K.: Computational investigation of cicada aerodynamics in forward flight. J. R. Soc. Interface 12(102), 20141116 (2015)

    Article  Google Scholar 

  104. Song, J., Tobalske, B.W., Powers, D.R., Hedrick, T.L., Luo, H.: Three-dimensional simulation for fast forward flight of a calliope hummingbird. R. Soc. Open Sci. 3(6), 160230 (2016)

  105. Li, C., Dong, H.: Wing kinematics measurement and aerodynamics of a dragonfly in turning flight. Bioinspir. Biomimetics 12(2), 026001 (2017)

  106. Bode-Oke, A.T., Zeyghami, S., Dong, H.: Flying in reverse: kinematics and aerodynamics of a dragonfly in backward free flight. J. R. Soc. Interface 15(143), 20180102 (2018)

    Article  Google Scholar 

  107. Bode-Oke, A.T., Dong, H.: The reverse flight of a monarch butterfly (danaus plexippus) is characterized by a weight-supporting upstroke and postural changes. J. R. Soc. Interface 17(167), 20200268 (2020)

    Article  Google Scholar 

  108. Ellington, C.P., Van Den Berg, C., Willmott, A.P., Thomas, A.L.: Leading-edge vortices in insect flight. Nature 384(6610), 626–630 (1996)

    Article  Google Scholar 

  109. Fuchiwaki, M., Kuroki, T., Tanaka, K., Tababa, T.: Dynamic behavior of the vortex ring formed on a butterfly wing. Exp. Fluids 54(1), 1–12 (2013)

    Article  Google Scholar 

  110. Henningsson, P., Michaelis, D., Nakata, T., Schanz, D., Geisler, R., Schröder, A., Bomphrey, R.J.: The complex aerodynamic footprint of desert locusts revealed by large-volume tomographic particle image velocimetry. J. R. Soc. Interface 12(108), 20150119 (2015)

    Article  Google Scholar 

  111. Thomas, A.L., Taylor, G.K., Srygley, R.B., Nudds, R.L., Bomphrey, R.J.: 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(24), 4299–4323 (2004)

    Article  Google Scholar 

  112. Bomphrey, R.J., Taylor, G.K., Thomas, A.L.: Smoke visualization of free-flying bumblebees indicates independent leading-edge vortices on each wing pair. In: Animal Locomotion, pp. 249–259. Springer, Berlin (2010)

  113. Hubel, T.Y., Riskin, D.K., Swartz, S.M., Breuer, K.S.: Wake structure and wing kinematics: the flight of the lesser dog-faced fruit bat, cynopterus brachyotis. J. Exp. Biol. 213(20), 3427–3440 (2010)

    Article  Google Scholar 

  114. Pournazeri, S., Segre, P.S., Princevac, M., Altshuler, D.L.: Hummingbirds generate bilateral vortex loops during hovering: evidence from flow visualization. Exp. Fluids 54(1), 1–11 (2013)

    Article  Google Scholar 

  115. Flammang, B.E., Lauder, G.V., Troolin, D.R., Strand, T.: Volumetric imaging of shark tail hydrodynamics reveals a three-dimensional dual-ring vortex wake structure. Proc. R. Soc. B: Biol. Sci. 278(1725), 3670–3678 (2011)

    Article  Google Scholar 

  116. Flammang, B.E., Lauder, G.V., Troolin, D.R., Strand, T.E.: Volumetric imaging of fish locomotion. Biol. Let. 7(5), 695–698 (2011)

    Article  Google Scholar 

  117. Tytell, E.D., Lauder, G.V.: Hydrodynamics of the escape response in bluegill sunfish, lepomis macrochirus. J. Exp. Biol. 211(21), 3359–3369 (2008)

    Article  Google Scholar 

  118. Van Buren, T., Floryan, D., Wei, N., Smits, A.J.: Flow speed has little impact on propulsive characteristics of oscillating foils. Phys. Rev. Fluids 3(1), 013103 (2018)

  119. Lighthill, M.: Note on the swimming of slender fish. J. Fluid Mech. 9(2), 305–317 (1960)

    Article  MathSciNet  Google Scholar 

  120. Liu, G., Ren, Y., Dong, H., Akanyeti, O., Liao, J.C., Lauder, G.V.: Computational analysis of vortex dynamics and performance enhancement due to body-fin and fin-fin interactions in fish-like locomotion. J. Fluid Mech. 829, 65–88 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  121. Han, P., Lauder, G.V., Dong, H.: Hydrodynamics of median-fin interactions in fish-like locomotion: Effects of fin shape and movement. Phys. Fluids 32(1), 011902 (2020)

    Article  Google Scholar 

  122. Borazjani, I., Sotiropoulos, F.: Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 212(4), 576–592 (2009)

    Article  Google Scholar 

  123. Fish, F.E., Schreiber, C.M., Moored, K.W., Liu, G., Dong, H., Bart-Smith, H.: Hydrodynamic performance of aquatic flapping: efficiency of underwater flight in the manta. Aerospace 3(3), 20 (2016)

    Article  Google Scholar 

  124. Xiao, Q., Hu, J., Liu, H.: Effect of torsional stiffness and inertia on the dynamics of low aspect ratio flapping wings. Bioinspir. Biomimetics 9(1), 016008 (2014)

  125. Bale, R., Hao, M., Bhalla, A.P.S., Patel, N., Patankar, N.A.: Gray’s paradox: a fluid mechanical perspective. Sci. Rep. 4(1), 1–5 (2014)

    Google Scholar 

  126. Lighthill, M.J.: Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. Ser. B. Biol. Sci. 179(1055), 125–138 (1971)

  127. Cheng, J.-Y., Blickhan, R.: Note on the calculation of propeller efficiency using elongated body theory. J. Exp. Biol. 192(1), 169–177 (1994)

    Article  Google Scholar 

  128. Maertens, A., Triantafyllou, M.S., Yue, D.K.: Efficiency of fish propulsion. Bioinspir. Biomimetics 10(4), 046013 (2015)

  129. Fish, F.E., Rohr, J.: Review of dolphin hydrodynamics and swimming performance (1999)

  130. Bale, R., Hao, M., Bhalla, A.P.S., Patankar, N.A.: Energy efficiency and allometry of movement of swimming and flying animals. Proc. Natl. Acad. Sci. 111(21), 7517–7521 (2014)

    Article  Google Scholar 

  131. Liu, H., Kolomenskiy, D., Nakata, T., Li, G.: Unsteady bio-fluid dynamics in flying and swimming. Acta. Mech. Sin. 33(4), 663–684 (2017)

    Article  Google Scholar 

  132. Muijres, F.T., Johansson, L.C., Hedenström, A.: Leading edge vortex in a slow-flying passerine. Biol. Let. 8(4), 554–557 (2012)

    Article  Google Scholar 

  133. Muijres, F., Johansson, L.C., Barfield, R., Wolf, M., Spedding, G., Hedenström, A.: Leading-edge vortex improves lift in slow-flying bats. Science 319(5867), 1250–1253 (2008)

    Article  Google Scholar 

  134. Windes, P., Fan, X., Bender, M., Tafti, D.K., Müller, R.: A computational investigation of lift generation and power expenditure of pratt’s roundleaf bat (hipposideros pratti) in forward flight. PLoS ONE 13(11), 0207613 (2018)

    Article  Google Scholar 

  135. Windes, P., Tafti, D.K., Müller, R.: Determination of spatial fidelity required to accurately mimic the flight dynamics of a bat. Bioinspir. Biomimetics 14(6), 066011 (2019)

  136. Lentink, D., Dickinson, M.H.: Rotational accelerations stabilize leading edge vortices on revolving fly wings. J. Exp. Biol. 212(16), 2705–2719 (2009)

    Article  Google Scholar 

  137. Johansson, L.C., Norberg, R.Å.: Delta-wing function of webbed feet gives hydrodynamic lift for swimming propulsion in birds. Nature 424(6944), 65–68 (2003)

    Article  Google Scholar 

  138. Sun, M.: Insect flight dynamics: stability and control. Rev. Mod. Phys. 86(2), 615 (2014)

    Article  Google Scholar 

  139. Wu, J.C.: Theory for aerodynamic force and moment in viscous flows. AIAA J. 19(4), 432–441 (1981)

    Article  MATH  Google Scholar 

  140. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2007)

    Google Scholar 

  141. Wu, J.-Z., Lu, X.-Y., Zhuang, L.-X.: Integral force acting on a body due to local flow structures. J. Fluid Mech. 576, 265–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  142. Kang, L., Liu, L., Su, W., Wu, J.: Minimum-domain impulse theory for unsteady aerodynamic force. Phys. Fluids 30(1), 016107 (2018)

    Article  Google Scholar 

  143. Werner, N.H., Chung, H., Wang, J., Liu, G., Cimbala, J.M., Dong, H., Cheng, B.: Radial planetary vorticity tilting in the leading-edge vortex of revolving wings. Phys. Fluids 31(4), 041902 (2019)

    Article  Google Scholar 

  144. Maxworthy, T.: The formation and maintenance of a leading-edge vortex during the forward motion of an animal wing. J. Fluid Mech. 587, 471–475 (2007)

    Article  MATH  Google Scholar 

  145. Cheng, B., Sane, S.P., Barbera, G., Troolin, D.R., Strand, T., Deng, X.: Three-dimensional flow visualization and vorticity dynamics in revolving wings. Exp. Fluids 54(1), 1423 (2013)

    Article  Google Scholar 

  146. Wojcik, C.J., Buchholz, J.H.: Vorticity transport in the leading-edge vortex on a rotating blade. J. Fluid Mech. 743, 249–261 (2014)

    Article  Google Scholar 

  147. Garmann, D., Visbal, M.: Dynamics of revolving wings for various aspect ratios. J. Fluid Mech. 748, 932–956 (2014)

    Article  Google Scholar 

  148. Jardin, T., David, L.: Coriolis effects enhance lift on revolving wings. Phys. Rev. E 91(3), 031001 (2015)

    Article  Google Scholar 

  149. Phillips, N., Knowles, K., Bomphrey, R.J.: The effect of aspect ratio on the leading-edge vortex over an insect-like flapping wing. Bioinspir. Biomimetics 10(5), 056020 (2015)

  150. Phillips, N., Knowles, K., Bomphrey, R.J.: Petiolate wings: effects on the leading-edge vortex in flapping flight. Interface Focus 7(1), 20160084 (2017)

    Article  Google Scholar 

  151. Du, G., Sun, M.: Effects of wing deformation on aerodynamic forces in hovering hoverflies. J. Exp. Biol. 213(13), 2273–2283 (2010)

    Article  Google Scholar 

  152. Engels, T., Kolomenskiy, D., Schneider, K., Farge, M., Lehmann, F.-O., Sesterhenn, J.: Helical vortices generated by flapping wings of bumblebees. Fluid Dyn. Res. 50(1), 011419 (2018)

    Article  MathSciNet  Google Scholar 

  153. Srygley, R., Thomas, A.: Unconventional lift-generating mechanisms in free-flying butterflies. Nature 420(6916), 660–664 (2002)

    Article  Google Scholar 

  154. Lin, Y.-J., Chang, S.-K., Lai, Y.-H., Yang, J.-T.: Beneficial wake-capture effect for forward propulsion with a restrained wing-pitch motion of a butterfly. R. Soc. Open Sci. 8(8), 202172 (2021)

  155. Liu, Y., Lozano, A.D., Hedrick, T.L., Li, C.: Comparison of experimental and numerical studies on the flow structures of hovering hawkmoths. J. Fluids Struct. 107, 103405 (2021)

    Article  Google Scholar 

  156. Zhang, J.-D., Huang, W.-X.: On the role of vortical structures in aerodynamic performance of a hovering mosquito. Phys. Fluids 31(5), 051906 (2019)

    Article  Google Scholar 

  157. Linehan, T., Mohseni, K.: On the maintenance of an attached leading-edge vortex via model bird alula. J. Fluid Mech. 897 (2020)

  158. Lentink, D., Dickson, W.B., Van Leeuwen, J.L., Dickinson, M.H.: Leading-edge vortices elevate lift of autorotating plant seeds. Science 324(5933), 1438–1440 (2009)

    Article  Google Scholar 

  159. Seale, M., Nakayama, N.: From passive to informed: mechanical mechanisms of seed dispersal. New Phytol. 225(2), 653–658 (2020)

    Article  Google Scholar 

  160. NORBERG, R.Å.: Autorotation, self-stability, and structure of single-winged fruits and seeds (samaras) with comparative remarks on animal flight. Biol. Rev. 48(4), 561–596 (1973)

  161. Salcedo, E., Treviño, C., Vargas, R.O., Martínez-Suástegui, L.: Stereoscopic particle image velocimetry measurements of the three-dimensional flow field of a descending autorotating mahogany seed (swietenia macrophylla). J. Exp. Biol. 216(11), 2017–2030 (2013)

    Google Scholar 

  162. Lee, I., Choi, H.: Flight of a falling maple seed. Phys. Rev. Fluids 2(9), 090511 (2017)

  163. Lee, I., Choi, H.: Scaling law for the lift force of autorotating falling seeds at terminal velocity. J. Fluid Mech. 835, 406–420 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  164. Borazjani, I., Daghooghi, M.: The fish tail motion forms an attached leading edge vortex. Proc. R. Soc. B: Biol. Sci. 280(1756), 20122071 (2013)

    Article  Google Scholar 

  165. Zhu, J., White, C., Wainwright, D.K., Di Santo, V., Lauder, G.V., Bart-Smith, H.: Tuna robotics: A high-frequency experimental platform exploring the performance space of swimming fishes. Sci. Robotics 4(34) (2019)

  166. Jones, A., Ford, C.P., Babinsky, H.: Three-dimensional effects on sliding and waving wings. J. Aircr. 48(2), 633–644 (2011)

    Article  Google Scholar 

  167. Dickinson, M.H., Lehmann, F.-O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science 284(5422), 1954–1960 (1999)

    Article  Google Scholar 

  168. Jones, A., Babinsky, H.: Unsteady lift generation on rotating wings at low Reynolds numbers. J. Aircr. 47(3), 1013–1021 (2010)

    Article  Google Scholar 

  169. Ohmi, K., Coutanceau, M., Loc, T.P., Dulieu, A.: Vortex formation around an oscillating and translating airfoil at large incidences. J. Fluid Mech. 211, 37–60 (1990)

    Article  Google Scholar 

  170. Panda, J., Zaman, K.B.: Experimental investigation of the flow field of an oscillating airfoil and estimation of lift from wake surveys. J. Fluid Mech. 265, 65–95 (1994)

    Article  Google Scholar 

  171. Bomphrey, R.J., Nakata, T., Phillips, N., Walker, S.M.: Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight. Nature 544(7648), 92–95 (2017)

    Article  Google Scholar 

  172. Zhu, H.J., Sun, M.: Unsteady aerodynamic force mechanisms of a hoverfly hovering with a short stroke-amplitude. Phys. Fluids 29(8), 081901 (2017)

  173. Shekarriz, A., Fu, T., Katz, J., Huang, T.: Near-field behavior of a tip vortex. AIAA J. 31(1), 112–118 (1993)

    Article  Google Scholar 

  174. Spalart, P.R.: Airplane trailing vortices. Annu. Rev. Fluid Mech. 30(1), 107–138 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  175. Gerz, T., Holzäpfel, F., Darracq, D.: Commercial aircraft wake vortices. Prog. Aerosp. Sci. 38(3), 181–208 (2002)

    Article  Google Scholar 

  176. Leete, K.M., Gee, K.L., Liu, J., Wall, A.T.: Coherence analysis of the noise from a simulated highly heated laboratory-scale jet. AIAA J. 58(8), 3426–3435 (2020)

    Article  Google Scholar 

  177. Gerz, T., Holzapfel, F.: Wing-tip vortices, turbulence, and the distribution of emissions. AIAA J. 37(10), 1270–1276 (1999)

    Article  Google Scholar 

  178. Birch, D., Lee, T., Mokhtarian, F., Kafyeke, F.: Structure and induced drag of a tip vortex. J. Aircr. 41(5), 1138–1145 (2004)

    Article  Google Scholar 

  179. Ringuette, M.J., Milano, M., Gharib, M.: Role of the tip vortex in the force generation of low-aspect-ratio normal flat plates. J. Fluid Mech. 581, 453–468 (2007)

    Article  MATH  Google Scholar 

  180. Kim, D., Gharib, M.: Experimental study of three-dimensional vortex structures in translating and rotating plates. Exp. Fluids 49(1), 329–339 (2010)

    Article  Google Scholar 

  181. Carr, Z., Chen, C., Ringuette, M.: Finite-span rotating wings: three-dimensional vortex formation and variations with aspect ratio. Exp. Fluids 54(2), 1–26 (2013)

    Article  Google Scholar 

  182. Carr, Z.R., DeVoria, A.C., Ringuette, M.J.: Aspect-ratio effects on rotating wings: circulation and forces. J. Fluid Mech. 767, 497–525 (2015)

    Article  Google Scholar 

  183. Birch, D., Lee, T.: Investigation of the near-field tip vortex behind an oscillating wing. J. Fluid Mech. 544, 201–241 (2005)

    Article  MATH  Google Scholar 

  184. Li, C., Dong, H., Cheng, B.: Tip vortices formation and evolution of rotating wings at low Reynolds numbers. Phys. Fluids 32(2), 021905 (2020)

  185. Poelma, C., Dickson, W., Dickinson, M.: Time-resolved reconstruction of the full velocity field around a dynamically-scaled flapping wing. Exp. Fluids 41(2), 213–225 (2006)

    Article  Google Scholar 

  186. Hartloper, C., Kinzel, M., Rival, D.E.: On the competition between leading-edge and tip-vortex growth for a pitching plate. Exp. Fluids 54(1), 1–11 (2013)

    Article  Google Scholar 

  187. Yilmaz, T.O., Rockwell, D.: Flow structure on finite-span wings due to pitch-up motion. J. Fluid Mech. 691, 518–545 (2012)

    Article  MATH  Google Scholar 

  188. Dong, H., Bozkurttas, M., Mittal, R., Madden, P., Lauder, G.: Computational modelling and analysis of the hydrodynamics of a highly deformable fish pectoral fin. J. Fluid Mech. 645, 345–373 (2010)

    Article  MATH  Google Scholar 

  189. Liu, G., Dong, H., Li, C.: Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight. J. Fluid Mech. 795, 634–651 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  190. Shyy, W., Trizila, P., Kang, C.-K., Aono, H.: Can tip vortices enhance lift of a flapping wing? AIAA J. 47(2), 289–293 (2009)

    Article  Google Scholar 

  191. Newbolt, J.W., Zhang, J., Ristroph, L.: Flow interactions between uncoordinated flapping swimmers give rise to group cohesion. Proc. Natl. Acad. Sci. 116(7), 2419–2424 (2019)

    Article  Google Scholar 

  192. Lissaman, P., Shollenberger, C.A.: Formation flight of birds. Science 168(3934), 1003–1005 (1970)

    Article  Google Scholar 

  193. Weihs, D.: Hydromechanics of fish schooling. Nature 241(5387), 290–291 (1973)

    Article  Google Scholar 

  194. Nakata, T., Henningsson, P., Lin, H.-T., Bomphrey, R.J.: Recent progress on the flight of dragonflies and damselflies. Int. J. Odonatol. 23(1), 41–49 (2020)

    Article  Google Scholar 

  195. Peng, L., Zheng, M., Pan, T., Su, G., Li, Q.: Tandem-wing interactions on aerodynamic performance inspired by dragonfly hovering. R. Soc. Open Sci. 8(8), 202275 (2021)

  196. Matthews, D., Lauder, G.V.: Fin-fin interactions during locomotion in a simplified biomimetic fish model. Bioinspir. Biomimetics (2021)

  197. Zhu, X., He, G., Zhang, X.: Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration. Phys. Rev. Lett. 113(23), 238105 (2014)

  198. Portugal, S.J., Hubel, T.Y., Fritz, J., Heese, S., Trobe, D., Voelkl, B., Hailes, S., Wilson, A.M., Usherwood, J.R.: Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature 505(7483), 399–402 (2014)

    Article  Google Scholar 

  199. Alaminos-Quesada, J., Fernandez-Feria, R.: Aerodynamics of heaving and pitching foils in tandem from linear potential theory. AIAA J. 58(1), 37–52 (2020)

    Article  Google Scholar 

  200. Alaminos-Quesada, J., Fernandez-Feria, R.: Propulsion performance of tandem flapping foils with chordwise prescribed deflection from linear potential theory. Phys. Rev. Fluids 6(1), 013102 (2021)

  201. Becker, A.D., Masoud, H., Newbolt, J.W., Shelley, M., Ristroph, L.: Hydrodynamic schooling of flapping swimmers. Nat. Commun. 6(1), 1–8 (2015)

    Article  Google Scholar 

  202. Ramananarivo, S., Fang, F., Oza, A., Zhang, J., Ristroph, L.: Flow interactions lead to orderly formations of flapping wings in forward flight. Phys. Rev. Fluids 1(7), 071201 (2016)

  203. Lagopoulos, N., Weymouth, G., Ganapathisubramani, B.: Deflected wake interaction of tandem flapping foils. J. Fluid Mech. 903 (2020)

  204. Akhtar, I., Mittal, R., Lauder, G.V., Drucker, E.: Hydrodynamics of a biologically inspired tandem flapping foil configuration. Theor. Comput. Fluid Dyn. 21(3), 155–170 (2007)

    Article  MATH  Google Scholar 

  205. Zheng, Y., Wu, Y., Tang, H.: An experimental study on the forewing-hindwing interactions in hovering and forward flights. Int. J. Heat Fluid Flow 59, 62–73 (2016)

    Article  Google Scholar 

  206. Jia, L.-B., Yin, X.-Z.: Passive oscillations of two tandem flexible filaments in a flowing soap film. Phys. Rev. Lett. 100(22), 228104 (2008)

  207. Jia, L.-B., Yin, X.-Z.: Response modes of a flexible filament in the wake of a cylinder in a flowing soap film. Phys. Fluids 21(10), 101704 (2009)

  208. Mazharmanesh, S., Young, J., Tian, F.-B., Lai, J.C.: Energy harvesting of two inverted piezoelectric flags in tandem, side-by-side and staggered arrangements. Int. J. Heat Fluid Flow 83, 108589 (2020)

  209. Streitlien, K., Triantafyllou, G.S., Triantafyllou, M.S.: Efficient foil propulsion through vortex control. AIAA J. 34(11), 2315–2319 (1996)

    Article  MATH  Google Scholar 

  210. Alben, S.: On the swimming of a flexible body in a vortex street. J. Fluid Mech. 635, 27–45 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  211. Wang, W., Huang, H., Lu, X.-Y.: Self-propelled plate in wakes behind tandem cylinders. Phys. Rev. E 100(3), 033114 (2019)

  212. Maybury, W.J., Lehmann, F.-O.: The fluid dynamics of flight control by kinematic phase lag variation between two robotic insect wings. J. Exp. Biol. 207(26), 4707–4726 (2004)

    Article  Google Scholar 

  213. Wang, Z.J., Russell, D.: Effect of forewing and hindwing interactions on aerodynamic forces and power in hovering dragonfly flight. Phys. Rev. Lett. 99(14), 148101 (2007)

  214. Xie, C.-M., Huang, W.-X.: Vortex interactions between forewing and hindwing of dragonfly in hovering flight. Theor. Appl. Mech. Lett. 5(1), 24–29 (2015)

    Article  Google Scholar 

  215. Ristroph, L., Zhang, J.: Anomalous hydrodynamic drafting of interacting flapping flags. Phys. Rev. Lett. 101(19), 194502 (2008)

  216. Alben, S.: Wake-mediated synchronization and drafting in coupled flags. J. Fluid Mech. 641, 489–496 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  217. Kim, S., Huang, W.-X., Sung, H.J.: Constructive and destructive interaction modes between two tandem flexible flags in viscous flow. J. Fluid Mech. 661, 511–521 (2010)

    Article  MATH  Google Scholar 

  218. Boschitsch, B.M., Dewey, P.A., Smits, A.J.: Propulsive performance of unsteady tandem hydrofoils in an in-line configuration. Phys. Fluids 26(5), 051901 (2014)

  219. Kurt, M., Moored, K.W.: Flow interactions of two-and three-dimensional networked bio-inspired control elements in an in-line arrangement. Bioinspir. Biomimetics 13(4), 045002 (2018)

  220. Han, P., Pan, Y., Liu, G., Dong, H.: Propulsive performance and vortex wakes of multiple tandem foils pitching in-line. J. Fluids Struct. 108, 103422 (2022)

    Article  Google Scholar 

  221. Peng, Z.-R., Huang, H., Lu, X.-Y.: Hydrodynamic schooling of multiple self-propelled flapping plates. J. Fluid Mech. 853, 587–600 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  222. Lin, X., Wu, J., Zhang, T., Yang, L.: Self-organization of multiple self-propelling flapping foils: energy saving and increased speed. J. Fluid Mech. 884 (2020)

  223. Zhang, J.-D., Huang, W.-X.: Numerical model and hydrodynamic performance of tuna finlets. Theor. Appl. Mech. Lett. 100322 (2022)

  224. Liao, J.C., Beal, D.N., Lauder, G.V., Triantafyllou, M.S.: The kármán gait: novel body kinematics of rainbow trout swimming in a vortex street. J. Exp. Biol. 206(6), 1059–1073 (2003)

    Article  Google Scholar 

  225. Taguchi, M., Liao, J.C.: Rainbow trout consume less oxygen in turbulence: the energetics of swimming behaviors at different speeds. J. Exp. Biol. 214(9), 1428–1436 (2011)

    Article  Google Scholar 

  226. Drucker, E.G., Lauder, G.V.: Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish. J. Exp. Biol. 204(17), 2943–2958 (2001)

    Article  Google Scholar 

  227. Drucker, E.G., Lauder, G.V.: Locomotor function of the dorsal fin in rainbow trout: kinematic patterns and hydrodynamic forces. J. Exp. Biol. 208(23), 4479–4494 (2005)

    Article  Google Scholar 

  228. Standen, E., Lauder, G.V.: Hydrodynamic function of dorsal and anal fins in brook trout (salvelinus fontinalis). J. Exp. Biol. 210(2), 325–339 (2007)

    Article  Google Scholar 

  229. Wang, J., Wainwright, D.K., Lindengren, R.E., Lauder, G.V., Dong, H.: Tuna locomotion: a computational hydrodynamic analysis of finlet function. J. R. Soc. Interface 17(165), 20190590 (2020)

    Article  Google Scholar 

  230. Gazzola, M., Argentina, M., Mahadevan, L.: Scaling macroscopic aquatic locomotion. Nat. Phys. 10(10), 758–761 (2014)

    Article  Google Scholar 

  231. Muller, U.K., van den Boogaart, J.G., van Leeuwen, J.L.: Flow patterns of larval fish: undulatory swimming in the intermediate flow regime. J. Exp. Biol. 211(2), 196–205 (2008)

    Article  Google Scholar 

  232. Cai, Y., Bi, S., Zheng, L.: Design and experiments of a robotic fish imitating cow-nosed ray. J. Bionic Eng. 7(2), 120–126 (2010)

    Article  Google Scholar 

  233. Clark, R.P., Smits, A.J.: Thrust production and wake structure of a batoid-inspired oscillating fin. J. Fluid Mech. 562, 415–429 (2006)

    Article  MATH  Google Scholar 

  234. Liu, G., Ren, Y., Zhu, J., Bart-Smith, H., Dong, H.: Thrust producing mechanisms in ray-inspired underwater vehicle propulsion. Theor. Appl. Mech. Lett. 5(1), 54–57 (2015)

    Article  Google Scholar 

  235. Moored, K.W., Smith, W., Hester, J., Chang, W., Bart-Smith, H.: Investigating the thrust production of a myliobatoid-inspired oscillating wing. In: Advances in Science and Technology, vol. 58, pp. 25–30 (2008). Trans Tech Publ

  236. Azarsina, F.: Designing a hydrodynamic shape and thrust mechanism for a batoid underwater robot. Mar. Technol. Soc. J. 50(5), 45–58 (2016)

    Article  Google Scholar 

  237. Zhao, J., Mao, Q., Pan, G., Huang, Q., Sung, H.J.: Hydrodynamic benefit of cephalic fins in a self-propelled flexible manta ray. Phys. Fluids 33(8), 081906 (2021)

  238. Mittal, R., Dong, H., Bozkurttas, M., Lauder, G., Madden, P.: Locomotion with flexible propulsors: Ii. computational modeling of pectoral fin swimming in sunfish. Bioinspir. Biomimetics 1(4), 35 (2006)

  239. Tytell, E.D.: Kinematics and hydrodynamics of linear acceleration in eels, anguilla rostrata. Proc. R. Soc. Lond. Ser. B: Biol. Sci. 271(1557), 2535–2540 (2004)

  240. Novati, G., Verma, S., Alexeev, D., Rossinelli, D., Van Rees, W.M., Koumoutsakos, P.: Synchronisation through learning for two self-propelled swimmers. Bioinspir. Biomimetics 12(3), 036001 (2017)

  241. Khalid, M.S.U., Akhtar, I., Dong, H.: Hydrodynamics of a tandem fish school with asynchronous undulation of individuals. J. Fluids Struct. 66, 19–35 (2016)

    Article  Google Scholar 

  242. Barrett, D., Triantafyllou, M., Yue, D., Grosenbaugh, M., Wolfgang, M.: Drag reduction in fish-like locomotion. J. Fluid Mech. 392, 183–212 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  243. Khalid, M.S.U., Wang, J., Akhtar, I., Dong, H., Liu, M.: Modal decompositions of the kinematics of crevalle jack and the fluid-caudal fin interaction. Bioinspir. Biomimetics 16(1), 016018 (2020)

  244. White, C.H., Lauder, G.V., Bart-Smith, H.: Tunabot flex: a tuna-inspired robot with body flexibility improves high-performance swimming. Bioinspir. Biomimetics 16(2), 026019 (2021)

  245. Wang, S., Zhang, X., He, G.: Numerical simulation of a three-dimensional fish-like body swimming with finlets. Commun. Comput. Phys. 11(4), 1323–1333 (2012)

    Article  Google Scholar 

  246. Zhong, Q., Dong, H., Quinn, D.B.: How dorsal fin sharpness affects swimming speed and economy. J. Fluid Mech. 878, 370–385 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  247. Wang, J., Tran, H., Christino, M., White, C., Zhu, J., Lauder, G., Bart-Smith, H., Dong, H.: Hydrodynamics and flow characterization of tuna-inspired propulsion in forward swimming. In: Fluids Engineering Division Summer Meeting, vol. 59025, pp. 001–01025 (2019). American Society of Mechanical Engineers

  248. Liu, H., Kawachi, K.: A numerical study of undulatory swimming. J. Comput. Phys. 155(2), 223–247 (1999)

    Article  MATH  Google Scholar 

  249. Von Ellenrieder, K.D., Parker, K., Soria, J.: Flow structures behind a heaving and pitching finite-span wing. J. Fluid Mech. 490, 129–138 (2003)

    Article  MATH  Google Scholar 

  250. Anderson, J.M., Streitlien, K., Barrett, D., Triantafyllou, M.S.: Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  251. Koochesfahani, M.M.: Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27(9), 1200–1205 (1989)

    Article  Google Scholar 

  252. Jones, K., Dohring, C., Platzer, M.: Experimental and computational investigation of the knoller-betz effect. AIAA J. 36(7), 1240–1246 (1998)

    Article  Google Scholar 

  253. Isogai, K., Shinmoto, Y., Watanabe, Y.: Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil. AIAA J. 37(10), 1145–1151 (1999)

    Article  Google Scholar 

  254. Tuncer, I.H., Platzer, M.F.: Computational study of flapping airfoil aerodynamics. J. Aircr. 37(3), 514–520 (2000)

    Article  Google Scholar 

  255. Mittal, R., Utturkar, Y., Udaykumar, H.: Computational modeling and analysis of biomimetic flight mechanisms. In: 40th AIAA Aerospace Sciences Meeting & Exhibit, p. 865 (2002)

  256. Lewin, G.C., Haj-Hariri, H.: Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. J. Fluid Mech. 492, 339–362 (2003)

    Article  MATH  Google Scholar 

  257. Guglielmini, L., Blondeaux, P.: Propulsive efficiency of oscillating foils. Eur. J. Mech.-B/Fluids 23(2), 255–278 (2004)

    Article  MATH  Google Scholar 

  258. Godoy-Diana, R., Marais, C., Aider, J.-L., Wesfreid, J.E.: A model for the symmetry breaking of the reverse bénard-von kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 23–32 (2009)

    Article  MATH  Google Scholar 

  259. Wong, J., Rival, D., et al.: Flow separation on flapping and rotating profiles with spanwise gradients. Bioinspir. Biomimetics 12(2), 026008 (2017)

  260. Mackowski, A., Williamson, C.: Direct measurement of thrust and efficiency of an airfoil undergoing pure pitching. J. Fluid Mech. 765, 524–543 (2015)

    Article  Google Scholar 

  261. Menon, K., Mittal, R.: Dynamic mode decomposition based analysis of flow over a sinusoidally pitching airfoil. J. Fluids Struct. 94, 102886 (2020)

  262. Zhu, Q.: Energy harvesting by a purely passive flapping foil from shear flows. J. Fluids Struct. 34, 157–169 (2012)

    Article  Google Scholar 

  263. Muscutt, L., Weymouth, G., Ganapathisubramani, B.: Performance augmentation mechanism of in-line tandem flapping foils. J. Fluid Mech. 827, 484–505 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  264. Chen, D., Kolomenskiy, D., Liu, H.: Closed-form solution for the edge vortex of a revolving plate. J. Fluid Mech. 821, 200–218 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  265. Hoover, A.P., Cortez, R., Tytell, E.D., Fauci, L.J.: Swimming performance, resonance and shape evolution in heaving flexible panels. J. Fluid Mech. 847, 386–416 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  266. Li, C., Dong, H.: Three-dimensional wake topology and propulsive performance of low-aspect-ratio pitching-rolling plates. Phys. Fluids 28(7), 071901 (2016)

  267. Han, J.-S., Chang, J.-W., Kim, S.-T.: Reynolds number dependency of an insect-based flapping wing. Bioinspir. Biomimetics 9(4), 046012 (2014)

  268. Han, J.-S., Chang, J.W., Han, J.-H.: The advance ratio effect on the lift augmentations of an insect-like flapping wing in forward flight. J. Fluid Mech. 808, 485–510 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  269. Green, M.A., Rowley, C.W., Smits, A.J.: The unsteady three-dimensional wake produced by a trapezoidal pitching panel. J. Fluid Mech. 685, 117–145 (2011)

    Article  MATH  Google Scholar 

  270. Shelton, R.M., Thornycroft, P.J., Lauder, G.V.: Undulatory locomotion of flexible foils as biomimetic models for understanding fish propulsion. J. Exp. Biol. 217(12), 2110–2120 (2014)

    Google Scholar 

  271. Quinn, D.B., Lauder, G.V., Smits, A.J.: Maximizing the efficiency of a flexible propulsor using experimental optimization. J. Fluid Mech. 767, 430–448 (2015)

    Article  Google Scholar 

  272. Lehn, A.M., Thornycroft, P.J., Lauder, G.V., Leftwich, M.C.: Effect of input perturbation on the performance and wake dynamics of aquatic propulsion in heaving flexible foils. Phys. Rev. Fluids 2(2), 023101 (2017)

  273. Alben, S., Witt, C., Baker, T.V., Anderson, E., Lauder, G.V.: Dynamics of freely swimming flexible foils. Phys. Fluids 24(5), 051901 (2012)

  274. Blevins, E., Lauder, G.V.: Swimming near the substrate: a simple robotic model of stingray locomotion. Bioinspir. Biomimetics 8(1), 016005 (2013)

  275. Lucas, K.N., Thornycroft, P.J., Gemmell, B.J., Colin, S.P., Costello, J.H., Lauder, G.V.: Effects of non-uniform stiffness on the swimming performance of a passively-flexing, fish-like foil model. Bioinspir. Biomimetics 10(5), 056019 (2015)

  276. Jia, K., Fang, L., Huang, W.-X.: Coupled states of dual side-by-side inverted flags in a uniform flow. J. Fluids Struct. 91, 102768 (2019)

  277. Park, S.G., Sung, H.J.: Vortex interaction between two tandem flexible propulsors with a paddling-based locomotion. J. Fluid Mech. 793, 612–632 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  278. Park, S.G., Sung, H.J.: Hydrodynamics of flexible fins propelled in tandem, diagonal, triangular and diamond configurations. J. Fluid Mech. 840, 154–189 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  279. Van Dyke, M., Van Dyke, M.: An Album of Fluid Motion, vol. 176. Parabolic Press, Stanford (1982)

    MATH  Google Scholar 

  280. Jantzen, R.T., Taira, K., Granlund, K.O., Ol, M.V.: Vortex dynamics around pitching plates. Phys. Fluids 26(5), 053606 (2014)

  281. Shyy, W., Liu, H.: Flapping wings and aerodynamic lift: the role of leading-edge vortices. AIAA J. 45(12), 2817–2819 (2007)

    Article  Google Scholar 

  282. Harbig, R.R., Sheridan, J., Thompson, M.C.: Reynolds number and aspect ratio effects on the leading-edge vortex for rotating insect wing planforms. J. Fluid Mech. 717, 166–192 (2013)

    Article  MATH  Google Scholar 

  283. Chen, D., Kolomenskiy, D., Nakata, T., Liu, H.: Forewings match the formation of leading-edge vortices and dominate aerodynamic force production in revolving insect wings. Bioinspir. Biomimetics 13(1), 016009 (2017)

  284. Ma, M., Huang, W.X., Xu, C.X.: A dynamic wall model for large eddy simulation of turbulent flow over complex/moving boundaries based on the immersed boundary method. Phys. Fluids 31(11), 115101 (2019)

  285. Ma, M., Huang, W.X., Xu, C.X., Cui, G.X.: A hybrid immersed boundary/wall-model approach for large-eddy simulation of high-reynolds-number turbulent flows. Int. J. Heat Fluid Flow 88, 108769 (2021)

  286. Triantafyllou, M., Triantafyllou, G., Gopalkrishnan, R.: Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3(12), 2835–2837 (1991)

    Article  Google Scholar 

  287. Saadat, M., Fish, F.E., Domel, A., Di Santo, V., Lauder, G., Haj-Hariri, H.: On the rules for aquatic locomotion. Phys. Rev. Fluids 2(8), 083102 (2017)

  288. Rohr, J.J., Fish, F.E.: Strouhal numbers and optimization of swimming by odontocete cetaceans. J. Exp. Biol. 207(10), 1633–1642 (2004)

    Article  Google Scholar 

  289. Floryan, D., Van Buren, T., Smits, A.J.: Efficient cruising for swimming and flying animals is dictated by fluid drag. Proc. Natl. Acad. Sci. 115(32), 8116–8118 (2018)

    Article  Google Scholar 

  290. Sfakiotakis, M., Lane, D.M., Davies, J.B.C.: Review of fish swimming modes for aquatic locomotion. IEEE J. Oceanic Eng. 24(2), 237–252 (1999)

    Article  Google Scholar 

  291. Fish, F.E., Lauder, G.V.: Control surfaces of aquatic vertebrates: active and passive design and function. J. Exp. Biol. 220(23), 4351–4363 (2017)

    Article  Google Scholar 

  292. Walker, J.A.: Does a rigid body limit maneuverability? J. Exp. Biol. 203(22), 3391–3396 (2000)

    Article  Google Scholar 

  293. Hove, J., O’Bryan, L., Gordon, M.S., Webb, P.W., Weihs, D.: Boxfishes (teleostei: Ostraciidae) as a model system for fishes swimming with many fins: kinematics. J. Exp. Biol. 204(8), 1459–1471 (2001)

    Article  Google Scholar 

  294. Bartol, I.K., Gharib, M., Weihs, D., Webb, P.W., Hove, J.R., Gordon, M.S.: Hydrodynamic stability of swimming in ostraciid fishes: role of the carapace in the smooth trunkfish lactophrys triqueter (teleostei: Ostraciidae). J. Exp. Biol. 206(4), 725–744 (2003)

    Article  Google Scholar 

  295. Bartol, I.K., Gharib, M., Webb, P.W., Weihs, D., Gordon, M.S.: Bodyinduced vortical flows: a common mechanism for self-corrective trimming control in boxfishes. J. Exp. Biol. 208(2), 327–344 (2005)

  296. Bartol, I.K., Gordon, M.S., Webb, P., Weihs, D., Gharib, M.: Evidence of self-correcting spiral flows in swimming boxfishes. Bioinspir. Biomimetics 3(1), 014001 (2008)

  297. Blake, R.: On ostraciiform locomotion. J. Mar. Biol. Assoc. UK 57(4), 1047–1055 (1977)

    Article  Google Scholar 

  298. Van Wassenbergh, S., van Manen, K., Marcroft, T.A., Alfaro, M.E., Stamhuis, E.J.: Boxfish swimming paradox resolved: forces by the flow of water around the body promote manoeuvrability. J. R. Soc. Interface 12(103), 20141146 (2015)

    Article  Google Scholar 

  299. Mistick, E.A., Mountcastle, A.M., Combes, S.A.: Wing flexibility improves bumblebee flight stability. J. Exp. Biol. 219(21), 3384–3390 (2016)

    Google Scholar 

  300. Zhu, H.J., Meng, X.G., Sun, M.: Forward flight stability in a drone-fly. Sci. Rep. 10(1), 1–12 (2020)

    Google Scholar 

  301. Sarradj, E., Fritzsche, C., Geyer, T.: Silent owl flight: bird flyover noise measurements. AIAA J. 49(4), 769–779 (2011)

    Article  Google Scholar 

  302. Jackson, J.C., Robert, D.: Nonlinear auditory mechanism enhances female sounds for male mosquitoes. Proc. Natl. Acad. Sci. 103(45), 16734–16739 (2006)

    Article  Google Scholar 

  303. Srygley, R.B.: Evolution of the wave: aerodynamic and aposematic functions of butterfly wing motion. Proc. R. Soc. B: Biol. Sci. 274(1612), 913–917 (2007)

    Article  Google Scholar 

  304. Jaworski, J.W., Peake, N.: Aeroacoustics of silent owl flight. Annu. Rev. Fluid Mech. 52, 395–420 (2020)

    Article  MATH  Google Scholar 

  305. Ji, X., Wang, L., Ravi, S., Tian, F.-B., Young, J., Lai, J.C.: Influences of serrated trailing edge on the aerodynamic and aeroacoustic performance of a flapping wing during hovering flight. Phys. Fluids 34(1), 011902 (2022)

  306. Wang, L., Tian, F.-B.: Numerical study of flexible flapping wings with an immersed boundary method: Fluid-structure-acoustics interaction. J. Fluids Struct. 90, 396–409 (2019)

    Article  Google Scholar 

  307. Wang, L., Tian, F.-B.: Numerical study of sound generation by three-dimensional flexible flapping wings during hovering flight. J. Fluids Struct. 99, 103165 (2020)

  308. Geng, B., Xue, Q., Zheng, X., Liu, G., Ren, Y., Dong, H.: The effect of wing flexibility on sound generation of flapping wings. Bioinspir. Biomimetics 13(1), 016010 (2017)

  309. Seo, J.-H., Hedrick, T.L., Mittal, R.: Mechanism and scaling of wing tone generation in mosquitoes. Bioinspir. Biomimetics 15(1), 016008 (2019)

  310. Seo, J.-H., Hedrick, T.L., Mittal, R.: Mosquitoes buzz and fruit flies don’ta comparative aeroacoustic analysis of wing-tone generation. Bioinspir. Biomimetics 16(4), 046019 (2021)

  311. Wagenhoffer, N., Moored, K.W., Jaworski, J.W.: Unsteady propulsion and the acoustic signature of undulatory swimmers in and out of ground effect. Phys. Rev. Fluids 6(3), 033101 (2021)

  312. Zhu, Y., Tian, F.-B., Young, J., Liao, J.C., Lai, J.C.: A numerical study of fish adaption behaviors in complex environments with a deep reinforcement learning and immersed boundary-lattice boltzmann method. Sci. Rep. 11(1), 1–20 (2021)

    Google Scholar 

  313. Zhu, Y., Pang, J.-H., Tian, F.-B.: Stable schooling formations emerge from the combined effect of the active control and passive self-organization. Fluids 7(1), 41 (2022)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772172 and 12102227) and the Institute for Guo Qiang of Tsinghua University (Grant No. 2019GQG1012.).

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  1. AML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Dong Zhang, Jun-Duo Zhang & Wei-Xi Huang

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Zhang, D., Zhang, JD. & Huang, WX. Physical models and vortex dynamics of swimming and flying: a review. Acta Mech 233, 1249–1288 (2022). https://doi.org/10.1007/s00707-022-03192-9

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