Skip to main content Accessibility help

Effects of flexibility on the aerodynamic performance of flapping wings

  • C.-K. Kang (a1), H. Aono (a1), C. E. S. Cesnik (a1) and W. Shyy (a1) (a2)


Effects of chordwise, spanwise, and isotropic flexibility on the force generation and propulsive efficiency of flapping wings are elucidated. For a moving body immersed in viscous fluid, different types of forces, as a function of the Reynolds number, reduced frequency (k), and Strouhal number (St), acting on the moving body are identified based on a scaling argument. In particular, at the Reynolds number regime of and the reduced frequency of , the added mass force, related to the acceleration of the wing, is important. Based on the order of magnitude and energy balance arguments, a relationship between the propulsive force and the maximum relative wing-tip deformation parameter () is established. The parameter depends on the density ratio, St, k, natural and flapping frequency ratio, and flapping amplitude. The lift generation, and the propulsive efficiency can be deduced by the same scaling procedures. It seems that the maximum propulsive force is obtained when flapping near the resonance, whereas the optimal propulsive efficiency is reached when flapping at about half of the natural frequency; both are supported by the reported studies. The established scaling relationships can offer direct guidance for micro air vehicle design and performance analysis.


Corresponding author

Email address for correspondence:


Hide All
1. Aono, H., Chimakurthi, S. K., Wu, P., Sällström, E., Stanford, B. K., Cesnik, C. E. S., Ifju, P., Ukeiley, L. & Shyy, W. 2010 A computational and experimental study of flexible flapping wing aerodynamics. AIAA Paper 2010–554.
2. Azuma, A. 2006 The Biokinetics of Flying and Swimming, 2nd edn. AIAA.
3. Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. & Zhang, H. 2010 PETSc users manual. Tech. Rep. Argonne National Laboratory.
4. Balay, S., Brown, J., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. & Zhang, H. 2011 PETSc web page.
5. Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. 1997 Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools for Scientific Computing, pp. 163202. Birkhäuser.
6. Barenblatt, G. I. 2003 Scaling. Cambridge University Press.
7. Biewener, A. A. 2003 Animal Locomotion. Oxford University Press.
8. Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 1996 Aeroelasticity. Dover.
9. de Boer, A., van der Schoot, M. S. & Bijl, H. 2007 Mesh deformation based on radial basis function interpolation. Comput. Struct. 85 (11–14), 784795.
10. Bos, F. M., Lentink, D., van Oudheusden, B. W. & Bijl, H. 2008 Influence of wing kinematics on aerodynamic performance in hovering insect flight. J. Fluid Mech. 594, 341368.
11. Buchwald, R. & Dudley, R. 2010 Limits to vertical force and power production in bumblebees (Hymenoptera: Bombus impatiens). J. Expl Biol. 213, 426432.
12. Chen, J.-S., Chen, J.-Y. & Chou, Y.-F. 2008 On the natural frequencies and mode shapes of dragonfly wings. J. Sound Vib. 313, 643654.
13. Chimakurthi, S. K., Cesnik, C. E. S. & Stanford, B. 2011 Flapping-wing structural dynamics formulation based on a corotational shell finite element. AIAA J. 49 (1), 128142.
14. Chimakurthi, S. K., Tang, J., Palacios, R., Cesnik, C. E. S. & Shyy, W. 2009 Computational aeroelasticity framework for analysing flapping wing micro air vehicles. AIAA J. 47, 18651878.
15. Combes, S. A. & Daniel, T. L. 2003a Flexural stiffness in insect wings. Part I. Scaling and the influence of wing venation. J. Expl Biol. 206 (17), 29792987.
16. Combes, S. A. & Daniel, T. L. 2003b Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta. J. Expl Biol. 206 (17), 29993006.
17. Daniel, T. L. & Combes, S. A. 2002 Flexible wings and fins: bending by inertial or fluid-dynamic forces? Integr. Compar. Biol. 42 (5), 10441049.
18. Dudley, R. 2002 The Biomechanics of Insect Flight: Form, Function, Evolution. Princeton University Press.
19. Falgout, R. & Yang, U. 2002 hypre: a library of high performance preconditioners. In Computational Science ICCS 2002 (ed. Sloot, P., Hoekstra, A., Tan, C. & Dongarra, J. ), pp. 632641. Springer.
20. Ferreira de Sousa, P. J. S. A. & Allen, J. J. 2011 Thrust efficiency of harmonically oscillating flexible flat plates. J. Fluid Mech. 674, 4366.
21. Garrick, I. E. 1937 Propulsion of a flapping and oscillating aerofoil. NACA report 567, 419–427.
22. Gogulapati, A. & Friedmann, P. 2011 Approximate aerodynamic and aeroelastic modeling of flapping wings in hover and forward flight. AIAA Paper 2011–2008.
23. Gordnier, R. E., Attar, P. J., Chimakurthi, S. K. & Cesnik, C. E. S. 2010 Implicit LES simulations of a flexible flapping wing. AIAA Paper 20102960.
24. Graebel, W. 2007 Advanced Fluid Mechanics. Academic Press.
25. Heathcote, S. & Gursul, I. 2007 Flexible flapping aerofoil propulsion at low Reynolds numbers. AIAA J. 45 (5), 10661079.
26. Heathcote, S., Wang, Z. & Gursul, I. 2008 Effect of spanwise flexibility on flapping wing propulsion. J. Fluids Struct. 24 (2), 183199.
27. Ishihara, D., Horie, T. & Denda, M. 2009a A two-dimensional computational study on the fluid-structure interaction cause of wing pitch changes in dipteran flapping flight. J. Expl Biol. 212 (1), 110.
28. Ishihara, D., Yamashita, Y., Horie, T., Yoshida, S. & Niho, T. 2009b Passive maintenance of high angle of attack and its generation during flapping translation in crane fly wing. J. Expl Biol. 212, 38823891.
29. Kamakoti, R. & Shyy, W. 2004 Evaluation of geometric conservation law using pressure-based fluid solver and moving grid technique. Intl J. Heat Fluid Flow 14 (7), 851865.
30. Kamakoti, R., Thakur, S., Wright, J. & Shyy, W. 2006 Validation of a new parallel all-speed CFD code in a rule-based framework for multidisciplinary applications. AIAA Paper 2006–3063.
31. Kang, C.-K., Aono, H., Cesnik, C. E. S. & Shyy, W. 2011 A scaling parameter for thrust generation of flapping flexible wings. AIAA Paper 2011–1313.
32. Katz, J. & Plotkin, A. 2001 Low-Speed Aerodynamics. Cambridge University Press.
33. Khosravi, P., Ganesan, R. & Sedaghati, R. 2007 Corotational nonlinear analysis of thin plates and shells using a new shell element. Intl J. Numer. Meth. Engng 69 (4), 859885.
34. Kim, D. & Gharib, M. 2011 Flexibility effects on vortex formation of translating plates. J. Fluid Mech. 677, 255271.
35. Küttler, U. & Wall, W. A. 2008 Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput. Mech. 43, 6172.
36. Luke, E. A. & George, T. 2005 Loci: a rule-based framework for parallel multi-disciplinary simulation synthesis. J. Funct. Program 15 (03), 477502.
37. Masoud, H. & Alexeev, A. 2010 Resonance of flexible flapping wings at low Reynolds number. Phys. Rev. E 81, 056304.
38. Maxworthy, T. 1981 The fluid dynamics of insect flight. Annu. Rev. Fluid. Mech. 13, 329350.
39. Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21, 071902.
40. Noca, F. 1997 On the evaluation of time-dependent fluid-dynamic forces on bluff bodies. PhD thesis, California Institute of Technology.
41. Norberg, U. M. 1990 Vertebrate Flight: Mechanics, Physiology, Morphology, Ecology, and Evolution. Springer.
42. Pennycuick, C. J. 1996 Wingbeat frequency of birds in steady cruising flight: new data and improved predictions. J. Expl Biol. 199, 16131618.
43. Ramananarivo, S., Godoy-Diana, R. & Thiria, B. 2011 Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance. Proc. Natl. Acad. Sci. USA 108 (15), 59645969.
44. Saffman, P. G. 1995 Vortex Dynamics. Cambridge University Press.
45. Sane, S. P 2003 The aerodynamics of insect flight. J. Expl Biol. 206, 41914208.
46. Shevtsova, E., Hansson, C., Janzen, D. H. & Kjaerandsen, J. 2011 Stable structural colour patterns displayed on transparent insect wings. Proc. Natl. Acad. Sci. USA 108 (213), 668673.
47. Shyy, W., Aono, H., Chimakurthi, S. K., Trizila, P., Kang, C.-K., Cesnik, C. E. S. & Liu, H. 2010 Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46 (7), 284327.
48. Shyy, W., Berg, M. & Ljungqvist, D. 1999 Flapping and flexible wings for biological and micro air vehicles. Prog. Aerosp. Sci. 35 (5), 455505.
49. Shyy, W., Lian, L., Tang, J., Liu, H., Trizila, P., Stanford, B., Bernal, L. P., Cesnik, C. E. S., Friedmann, P. & Ifju, P. 2008a Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications. Acta Mechanica Sin. 24, 351373.
50. Shyy, W., Lian, Y., Tang, J., Viieru, D. & Liu, H. 2008b Aerodynamics of Low Reynolds Number Flyers. Cambridge University Press.
51. Shyy, W., Trizila, P., Kang, C.-K. & Aono, H. 2009 Can tip vortices enhance lift of a flapping wing? AIAA 47, 289293.
52. Smith, R. W. & Wright, J. A. 2003 An implicit edge-based ale method for the incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 43, 253279.
53. Spagnolie, S. E., Moret, L., Shelley, M. J. & Zhang, J. 2010 Surprising behaviours in flapping locomotion with passive pitching. Phys. Fluids 22, 041903.
54. Sunada, S., Zeng, L. & Kawachi, K. 1998 The relationship between dragonfly wing structure and torsional deformation. J. Theor. Biol. 193, 3945.
55. Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. Tech. Rep. NACA report.
56. Thiria, B. & Godoy-Diana, R. 2010 How wing compliance drives the efficiency of self-propelled flapping flyers. Phys. Rev. E 82 (1), 015303.
57. Thomas, P. D. & Lombard, C. K. 1979 Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17 (10), 10301037.
58. Timoshenko, S., Young, D. H. & Weaver, J. R. W. 1974 Vibration Problems in Engineering. Wiley.
59. Tobalske, B. W. 2007 Biomechanics of bird flight. J. Expl Biol. 210, 31353146.
60. Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid. Mech. 32, 3353.
61. Trizila, P., Kang, C.-K., Aono, H., Visbal, M. & Shyy, W. 2011 Low-Reynolds-number aerodynamics of a flapping rigid flat plate. AIAA J. 49 (4), 806823.
62. Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E. & Balachandran, B. 2009 Influence of flexibility on the aerodynamic performance of a hovering wing. J. Expl Biol. 212, 95105.
63. Visbal, M. R., Gordnier, R. E. & Galbraith, M. C. 2009 High-fidelity simulations of moving and flexible aerofoils at low Reynolds numbers. Exp. Fluids 46, 903922.
64. Vogel, S. 1966 Flight in Drosophila. Part I. Flight performance of tethered flies. J. Expl Biol. 44, 567578.
65. Wang, Z. J. 2008 Aerodynamic efficiency of flapping flight: analysis of a two-stroke model. J. Expl Biol. 211, 234238.
66. Whitney, J. P. & Wood, R. J. 2010 Aeromechanics of passive rotation in flapping flight. J. Fluid Mech. 660, 197220.
67. Willmott, A. P. & Ellington, C. P. 1997a The mechanics of flight in the hawkmoth Manduca sexta. Part I. Kinematics of hovering and forward flight. J. Expl Biol. 200, 27052722.
68. Willmott, A. P. & Ellington, C. P. 1997b The mechanics of flight in the hawkmoth Manduca sexta. Part II. Aerodynamic consequences of kinematic and morphological variation. J. Expl Biol. 200, 27232745.
69. Wright, J. A. & Smith, R. W 2001 An edge-based method for the incompressible Navier–Stokes equations on polygonal meshes. J. Comput. Phys. 169, 2443.
70. Wu, P., Ijfu, P. & Stanford, B. 2010 Flapping wing structural deformation and thrust correlation study with flexible membrane wings. AIAA J. 48 (9), 21112122.
71. Yin, B. & Luo, H. 2010 Effect of wing inertia on hovering performance of flexible flapping wings. Phys. Fluids 22, 111902.
72. Zhang, J., Liu, N.-S. & Lu, X.-Y. 2010 Locomotion of a passively flapping flat plate. J. Fluid Mech. 659, 4368.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed