Skip to main content Accessibility help

Unsteady force generation and vortex dynamics of pitching and plunging aerofoils

  • Yeon Sik Baik (a1), Luis P. Bernal (a1), Kenneth Granlund (a2) and Michael V. Ol (a2)


Experimental studies of the flow topology, leading-edge vortex dynamics and unsteady force produced by pitching and plunging flat-plate aerofoils in forward flight at Reynolds numbers in the range 5000–20 000 are described. We consider the effects of varying frequency and plunge amplitude for the same effective angle-of-attack time history. The effective angle-of-attack history is a sinusoidal oscillation in the range to with mean of and amplitude of . The reduced frequency is varied in the range 0.314–1.0 and the Strouhal number range is 0.10–0.48. Results show that for constant effective angle of attack, the flow evolution is independent of Strouhal number, and as the reduced frequency is increased the leading-edge vortex (LEV) separates later in phase during the downstroke. The LEV trajectory, circulation and area are reported. It is shown that the effective angle of attack and reduced frequency determine the flow evolution, and the Strouhal number is the main parameter determining the aerodynamic force acting on the aerofoil. At low Strouhal numbers, the lift coefficient is proportional to the effective angle of attack, indicating the validity of the quasi-steady approximation. Large values of force coefficients () are measured at high Strouhal number. The measurement results are compared with linear potential flow theory and found to be in reasonable agreement. During the downstroke, when the LEV is present, better agreement is found when the wake effect is ignored for both the lift and drag coefficients.


Corresponding author

Email address for correspondence:


Hide All
1. Anderson, J. M., Streitlein, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 4172.
2. Baik, Y. 2011 Unsteady force generation and vortex dynamics of pitching and plunging aerofoils at low Reynolds number. PhD thesis, University of Michigan.
3. Baik, Y., Rausch, J. M., Bernal, L. P., Shyy, W. & Ol, M. 2010 Experimental study of governing parameters in pitching and plunging aerofoil at low Reynolds number. AIAA Paper 2010-0388.
4. Birch, J. M., Dickson, W. B. & Dickinson, M. H. 2004 Force production and flow structure of the leading edge vortex on flapping wings. J. Expl Biol. 207, 10631072.
5. Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 1996 Aeroelasticity. Dover.
6. Chakraborty, P., Balachandar, S. & Adrian, R. J. 2005 On the relationship between local vortex identification schemes. J. Fluid Mech. 535, 189214.
7. Dabiri, J. O. 2009 Optimal vortex formation as a unifying principle in biological propulsion. Annu. Rev. Fluid Mech. 41, 1733.
8. Dickinson, M. H. & Gotz, K. G. 1993 Unsteady aerodynamic performance on model wings at low Reynolds numbers. J. Expl Biol. 174, 4564.
9. von Ellenrieder, K. D. & Posthos, S. 2008 PIV measurements of the asymmetric wake of a two-dimensional heaving hydrofoil. Exp. Fluids 44, 733745.
10. Ellington, C. P., van den Berg, C., Willmott, A. P. & Thomas, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626630.
11. Garrick, I. E. 1936 Propulsion of a flapping and oscillating aerofoil. NASA Tech. Rep. 567.
12. Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121140.
13. Godoy-Diana, R., Aider, J. L. & Wesfried, J. E. 2009 A model for the symmetry breaking of the reverse Bénard–von Kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 2332.
14. Graftieaux, L., Michard, M. & Grosjean, N. 2001 Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows. Meas. Sci. Tech. 12, 14221429.
15. Granlund, K., Ol, M. & Bernal, L. 2011 Experiments on pitching plates: force and flow field measurements at low Reynolds numbers. AIAA Paper 2011-0872.
16. Hover, F. S., Haugsdal, O. & Triantafyllou, M. S. 2004 Forces on oscillating foils for propulsion and maneuvering. J. Fluid Struct. 19, 3747.
17. Jones, A. R. & Babinsky, H. 2010 Unsteady lift generation on rotating wings at low Reynolds numbers. J. Aircraft 47, 10131021.
18. Kang, C.-K., Baik, Y., Bernal, L. P., Ol, M. V. & Shyy, W. 2009 Fluid dynamics of pitching and plunging aerofoils of Reynolds number between and . AIAA Paper 2009-536.
19. von Kármán, T. & Sears, W. R. 1938 Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5, 379390.
20. Koochesfahani, M. M. 1989 Vortical patterns in the wake of an oscillating aerofoil. AIAA J. 27, 12001205.
21. Krueger, P., Dabiri, J. O. & Gharib, M. 2006 The formation number of vortex rings formed in uniform background co-flow. J. Fluid Mech. 556, 147166.
22. Lai, J. C. S. & Platzer, M. 1999 Jet characteristics of a plunging aerofoil. AIAA J. 37, 15291537.
23. Lighthill, M. J. 1969 Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1, 413446.
24. Lua, K. B., Lim, T. T., Yeo, K. S. & Oo, G. Y. 2007 Wake-structure formation of a heaving two-dimensional elliptic aerofoil. AIAA J. 45, 15711583.
25. Maxworthy, T. 1981 The fluid dynamics of insect flight. Annu. Rev. Fluid Mech. 13, 329350.
26. McCroskey, W. J. 1981 The phenomenon of dynamic stall. Tech. Rep. 81264. NASA Tech. Mem.
27. McCroskey, W. J. 1982 Unsteady aerofoils. Annu. Rev. Fluid Mech 14, 285311.
28. McGowan, G. Z., Granlund, K., Ol, M. V., Gopalarathnam, A. & Edwards, J. R. 2011 Investigations of lift-based pitch–plunge equivalence for aerofoils at low Reynolds numbers. AIAA J. 49, 15111524.
29. Milano, M. & Gharib, M. 2005 Uncovering the physics of flapping flat plates with artificial evolution. J. Fluid Mech. 534, 403409.
30. Ohmi, K., Coutanceau, M., Daube, O. & Loc, T. P. 1991 Further experiments on vortex formation around an oscillating and translating aerofoil at large incidences. J. Fluid Mech. 225, 607630.
31. Ohmi, K., Coutanceau, M., Loc, T. P. & Dulieu, A. 1990 Vortex formation around an oscillating and translating aerofoil at large incidences. J. Fluid Mech. 211, 3760.
32. Ol, M., Bernal, L. P., Kang, C. & Shyy, W. 2009 Shallow and deep dynamic stall for flapping low Reynolds number aerofoils. Exp. Fluids 46, 883901.
33. Ol, M., McAuliffe, B. R., Hanff, E. S., Scholz, U. & Kaehler, Ch. 2005 Comparison of laminar separation bubble measurements on a low Reynolds number aerofoil in three facilities. AIAA Paper 2005-5149.
34. Platzer, M., Jones, K., Young, J. & Lai, J. 2008 Flapping wing aerodynamics: progress and challenges. AIAA J. 46, 21362149.
35. Polhamus, E. C. 1966 A concept of the vortex lift of sharp-edge delta wings based on a leading-edge-suction analogy. Tech. Rep. NASA Technical Note.
36. Read, D. A., Hover, F. S. & Triantafyllou, M. S. 2003 Effect of angle of attack profiles in flapping foil propulsion. J. Fluid Struct. 17, 163183.
37. Ringuette, M., Michelle, M. & Gharib, M. 2007 Role of the tip vortex in the force generation of low-aspect-ratio normal flat plates. J. Fluid Mech. 581, 453468.
38. Rival, D., Prangemeier, T. & Tropea, C. 2009 The influence of aerofoil kinematics on the formation of leading-edge vortices in bio-inspired flight. Exp. Fluids 46, 823833.
39. Rival, D. & Tropea, C. 2010 Characteristics of pitching and plunging aerofoils under dynamic-stall conditions. J. Aircraft 47, 8086.
40. Sane, S. P. 2003 The aerodynamics of insect flight. J. Expl Biol. 206, 41954208.
41. Shyy, W., Lian, Y., Viieru, D. & Liu, H. 2008 Aerodynamics of Low Reynolds Number Flyers. Cambridge University Press.
42. Shyy, W. & Liu, H. 2007 Flapping wings and aerodynamics lift: the role of leading-edge vortices. AIAA J. 45, 28172819.
43. Strickland, J. H. & Graham, G. M. 1987 Force coefficients for a NACA-0015 aerofoil undergoing constant pitch rate motions. AIAA J. 25, 622624.
44. Taylor, G. K., Nudds, R. L. & Thomas, A. L. R. 2004 Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. J. Fluid Struct. 19, 3747.
45. Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Tech. Rep. 496.
46. Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1992 Optimal thrust development in oscillating foils with applications to fish propulsion. J. Fluid Struct. 7, 205224.
47. Visbal, M. R. & Shang, J. S. 1989 Investigation of the flow structure around a rapidly pitching aerofoil. AIAA J. 27, 10441051.
48. Young, J. & Lai, J. C. S. 2004 Oscillation frequency and amplitude effects on the wake of a punging aerofoil. AIAA J. 42, 20422052.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Unsteady force generation and vortex dynamics of pitching and plunging aerofoils

  • Yeon Sik Baik (a1), Luis P. Bernal (a1), Kenneth Granlund (a2) and Michael V. Ol (a2)


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