Theoretical and Computational Fluid Dynamics

, Volume 27, Issue 6, pp 843–864 | Cite as

An unsteady airfoil theory applied to pitching motions validated against experiment and computation

  • Kiran RameshEmail author
  • Ashok Gopalarathnam
  • Jack R. Edwards
  • Michael V. Ol
  • Kenneth Granlund
Original Article


An inviscid theoretical method that is applicable to non-periodic motions and that accounts for large amplitudes and non-planar wakes (large-angle unsteady thin airfoil theory) is developed. A pitch-up, hold, pitch-down motion for a flat plate at Reynolds number 10,000 is studied using this theoretical method and also using computational (immersed boundary method) and experimental (water tunnel) methods. Results from theory are compared against those from computation and experiment which are also compared with each other. The variation of circulatory and apparent-mass loads as a function of pivot location for this motion is examined. The flow phenomena leading up to leading-edge vortex shedding and the limit of validity of the inviscid theory in the face of vortex-dominated flows are investigated. Also, the effect of pitch amplitude on leading-edge vortex shedding is examined, and two distinctly different vortex-dominated flows are studied using dye flow visualizations from experiment and vorticity plots from computation.


Unsteady aerodynamics Airfoil aerodynamics Leading-edge vortex 



Angle between the airfoil and inertial horizontal


Pressure difference over airfoil


Pitch rate


Plunge rate


Variation of camber along airfoil

γ(θ, t)

Chordwise distribution of bound vorticity on airfoil


Bound circulation of airfoil at time i


Strength of wake vortex shed at time k − 1


Velocity potential


Velocity potential from bound circulation


Velocity potential from wake circulation


Air density


Variable of transformation of chordwise distance


Rate of rotation of the body frame


Unit vector normal to camberline in body frame


Position vector of a point in the body frame


Velocity of the body frame with respect to the inertial frame


Pivot location on the airfoil from 0 to 1 (x/c)

A0, A1, A2....

Fourier coefficients


Body frame


Airfoil chord


Drag coefficient


Lift coefficient


Normal force coefficient


Leading edge suction force coefficient


Normal force on airfoil


Leading edge suction force


Plunge displacement in the inertial Z direction


Non-dimensional pitch rate = \({\dot{\alpha}c/(2U)}\)




Inertial frame


Pressure distribution over airfoil


Freestream pressure




Non-dimensional time = tU/c


Freestream velocity


Local downwash


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  1. 1.
    Ansari S.,  Zbikowski R., Knowles K.: Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 1: methodology and analysis. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 220(2), 61–83 (2006)CrossRefGoogle Scholar
  2. 2.
    Ansari S.,  Zbikowski R., Knowles K.: Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2: implementation and validation. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 220(3), 169–186 (2006)CrossRefGoogle Scholar
  3. 3.
    Carr L.: Progress in analysis and prediction of dynamic stall. J. Aircraft 25, 6–17 (1988)CrossRefGoogle Scholar
  4. 4.
    Carr L.W., Platzer M.F., Chandrasekhara M.S., Ekaterinaris J.: Experimental and computational studies of dynamic stall. In: Cebeci, T. (ed.) Numerical and Physical Aspects of Aerodynamic Flows IV, pp. 239–256. Springer, Berlin (1990)Google Scholar
  5. 5.
    Cassidy D.A., Edwards J.R., Tian M.: An investigation of interface-sharpening schemes for multi-phase mixture flows. J. Comput. Phys. 228(16), 5628–5649 (2009). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choi J., Oberoi R., Edwards J., Rosati J.: An immersed boundary method for complex incompressible flows. J. Comput. Phys. 224(2), 757–784 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choi J.I., Edwards J.R.: Large eddy simulation and zonal modeling of human-induced contaminant transport. Indoor air 18(3), 233–249 (2008)CrossRefGoogle Scholar
  8. 8.
    Eldredge, J.D., Wang, C.: High-fidelity simulations and low-order modeling of a rapidly pitching plate. AIAA paper 2010-4281 (2010)Google Scholar
  9. 9.
    Eldredge, J.D., Wang, C., Ol, M.V.: A computational study of a canonical pitch-up, pitch-down wing maneuver. AIAA paper 2009-3687 (2009)Google Scholar
  10. 10.
    Ellington, C.: Unsteady aerodynamics of insect flight. In: Symposia of the Society for Experimental Biology, vol. 49, pp. 109–129 (1995)Google Scholar
  11. 11.
    Garrick, I.: Propulsion of a flapping and oscillating airfoil. NACA Rept. 567 (1937)Google Scholar
  12. 12.
    Granlund, K., Ol, M.V., Bernal, L.: Experiments on pitching plates : force and flowfield measurements at low Reynolds Numbers. AIAA Paper 2011-0872 (2011)Google Scholar
  13. 13.
    Granlund, K., Ol, M.V., Garmann, D.J., Visbal, M.R., Bernal, L.: Experiments and computations on abstractions of perching. AIAA paper 2010-4943 (2012)Google Scholar
  14. 14.
    Johnston, C.: Review, extension and application of unsteady thin airfoil theory. CIMSS Report 04-101, Virginia Polytechnic Institute and State University (2004).
  15. 15.
    Kármán T., Sears W.: Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5(10), 379–390 (1938)CrossRefzbMATHGoogle Scholar
  16. 16.
    Katz J.: Discrete vortex method for the non-steady separated flow over an airfoil. J. Fluid Mech. 102, 315–328 (1981)CrossRefzbMATHGoogle Scholar
  17. 17.
    Katz, J., Plotkin, A.: Low-Speed Aerodynamics. Cambridge Aerospace Series (2000)Google Scholar
  18. 18.
    Leishman, J.G.: Principles of Helicopter Aerodynamics. Cambridge Aerospace Series (2002)Google Scholar
  19. 19.
    McCroskey, W.: Unsteady airfoils. Annu. Rev. Fluid Mech. 14, 285–311 (1982). doi: 10.1146/annurev.fl.14.010182.001441
  20. 20.
    McCroskey, W.J.: The phenomenon of dynamic stall. NASA TM 81264 (1981)Google Scholar
  21. 21.
    McCune J., Lam C., Scott M.: Nonlinear aerodynamics of two-dimensional airfoils in severe maneuver. AIAA J. 28(3), 385–393 (1990)CrossRefGoogle Scholar
  22. 22.
    McGowan, G., Gopalarathnam, A., Ol, M.V., Edwards, J.R., Fredberg, D.: Computation vs. experiment for high-frequency low-Reynolds number airfoil pitch and plunge. AIAA Paper 2008-0653 (2008)Google Scholar
  23. 23.
    McGowan G.Z., Granlund K., Ol M.V., Gopalarathnam A., Edwards J.R.: Investigations of lift-based pitch-plunge equivalence for airfoils at low Reynolds Numbers. AIAA J. 49(7), 1511–1524 (2011)CrossRefGoogle Scholar
  24. 24.
    Mueller, T.J. (ed.): Fixed and flapping wing aerodynamics for micro air vehicle applications. In: Progress in Astronautics and Aeronautics, vol. 195. AIAA Inc., Virginia (2001)Google Scholar
  25. 25.
    Neaves M.D., Edwards J.R.: All-speed time-accurate underwater projectile calculations using a preconditioning algorithm. J. Fluids Eng. 128(2), 284–296 (2006). doi: 10.1115/1.2169816 CrossRefGoogle Scholar
  26. 26.
    Ol, M., Bernal L., Kang C., Shyy W.: Shallow and deep dynamic stall for flapping low Reynolds number airfoils. Exp. Fluids 46(5), 883–901 (2009)Google Scholar
  27. 27.
    Ol, M., McAuliffe, B., Hanff, E., Scholz, U., Kaehler, C.: Comparison of laminar separation bubble measurements on a low Reynolds number airfoil in three facilities. AIAA paper 2005-5149 (2005)Google Scholar
  28. 28.
    Ol, M.V., Altman, A., Eldredge, J.D., Garmann, D.J., Lian, Y.: Résumé of the AIAA FDTC low Reynolds Number Discussion Group’s Canonical Cases. AIAA paper 2010-1085 (2010)Google Scholar
  29. 29.
    Ol M.V., Reeder M., Fredberg D., McGowan G.Z., Gopalarathnam A., Edwards J.R.: Computation vs experiment for high-frequency low-Reynolds Number airfoil plunge. Int. J. Micro Air Veh. 1(2), 99–119 (2009)CrossRefGoogle Scholar
  30. 30.
    Peters D.: Two-dimensional incompressible unsteady airfoil theory-an overview. J. Fluids Struct. 24(3), 295–312 (2008)CrossRefGoogle Scholar
  31. 31.
    Sane S.: The aerodynamics of insect flight. J. Exp. Biol. 206(23), 4191–4208 (2003)CrossRefGoogle Scholar
  32. 32.
    Sarpkaya T.: An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate. J. Fluid Mech. 68(01), 109–128 (1975)CrossRefzbMATHGoogle Scholar
  33. 33.
    Spalart P.: Vortex methods for separated flows. VKI Comput. Fluid Dyn. 1, 64 (1988)Google Scholar
  34. 34.
    Spalart P., Allmaras S.: A one-equation turbulence model for aerodynamic flows. La Recherche Aerospatiale 1, 5–21 (1994)Google Scholar
  35. 35.
    Theodorsen, T.: General theory of aerodynamic instability and the mechanism of flutter. NACA Rept. 496 (1935)Google Scholar
  36. 36.
    Theodorsen, T.: On some reciprocal relations in the theory of nonstationary flows. NACA Rept. 629 (1938)Google Scholar
  37. 37.
    Wagner H.: Über die Entstehung des dynamischen Auftriebes von Tragflügeln. ZaMM 5(1), 17–35 (1925)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, C., Eldredge, J.D.: Low-order phenomenological modeling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 1–22 (2012). doi: 10.1007/s00162-012-0279-5
  39. 39.
    Wang Z.: Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kiran Ramesh
    • 1
    Email author
  • Ashok Gopalarathnam
    • 1
  • Jack R. Edwards
    • 1
  • Michael V. Ol
    • 2
  • Kenneth Granlund
    • 2
  1. 1.Department of Mechanical and Aerospace EngineeringNorth Carolina State UniversityRaleighUSA
  2. 2.U.S. Air Force Research LaboratoryAir Vehicles Directorate, WPAFBDaytonUSA

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