Skip to main content

Advertisement

Springer Nature Link
Log in

Low-order phenomenological modeling of leading-edge vortex formation

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

A low-order point vortex model for the two-dimensional unsteady aerodynamics of a flat plate wing section is developed. A vortex is released from both the trailing and leading edges of the flat plate, and the strength of each is determined by enforcing the Kutta condition at the edges. The strength of a vortex is frozen when it reaches an extremum, and a new vortex is released from the corresponding edge. The motion of variable-strength vortices is computed in one of two ways. In the first approach, the Brown–Michael equation is used in order to ensure that no spurious force is generated by the branch cut associated with each vortex. In the second approach, we propose a new evolution equation for a vortex by equating the rate of change of its impulse with that of an equivalent surrogate vortex with identical properties but constant strength. This impulse matching approach leads to a model that admits more general criteria for shedding, since the variable-strength vortex can be exchanged for its constant-strength surrogate at any instant. We show that the results of the new model, when applied to a pitching or perching plate, agree better with experiments and high-fidelity simulations than the Brown–Michael model, using fewer than ten degrees of freedom. We also assess the model performance on the impulsive start of a flat plate at various angles of attack. Current limitations of the model and extensions to more general unsteady aerodynamic problems are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Alben S., Shelley M.J.: Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett. 100, 074301 (2008)

    Article  Google Scholar 

  2. Ansari, S.A., Zbikowski, R., Knowles, K.: Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 1: methodology and analysis. Proc. IMechE Part G J. Aerosp. Eng. 220, 61–83 (2006)

  3. Ansari, S.A., Zbikowski, R., Knowles, K.: Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2: implementation and validation. Proc. IMechE Part G J. Aerosp. Eng. 220, 169–186 (2006)

  4. Berman G.J., Wang Z.J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153–168 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brown C.E., Michael W.H.: Effect of leading edge separation on the lift of a delta wing. J. Aerosp. Sci. 21, 690–694 (1954)

    MATH  Google Scholar 

  6. Chen K.K., Colonius T., Taira K.: The leading-edge vortex and quasisteady vortex shedding on an accelerating plate. Phys. Fluids 22(3), 033601 (2010)

    Article  Google Scholar 

  7. Clements R.R.: An inviscid model for two-dimensional vortex shedding. J. Fluid Mech. 57(2), 321–336 (1973)

    Article  MATH  Google Scholar 

  8. Cortelezzi L., Leonard A.: Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion. Fluid Dyn. Res. 11, 263–295 (1993)

    Article  Google Scholar 

  9. Davis, C.I.R., Shajiee, S., Mohseni, K.: Lift Enhancement in a Flapping Airfoil by an Attached Free Vortex and Sink. AIAA Paper 2009-0390 (2009)

  10. Dore, B.D.: The Unsteady Forces on Finite Wings in Transient Motion. Technical report 3456. Aeronautical Research Council (1966)

  11. Edwards R.H.: Leading edge separation from delta wings. J. Aerosp. Sci. 21, 134–135 (1954)

    Google Scholar 

  12. Eldredge, J., Wang, C.: High-Fidelity Simulations and Low-Order Modeling of a Rapidly Pitching Plate. AIAA Paper 2010-4281, 40th AIAA Fluid Dynamics Conference, Chicago (2010)

  13. Eldredge J.D.: Numerical simulation of the fluid dynamics of 2D rigid body motion with the vortex particle method. J. Comput. Phys. 221, 626–648 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eldredge J.D.: A reconciliation of viscous and inviscid approaches to computing locomotion of deforming bodies. Exp. Mech. 50(9), 1349–1353 (2009)

    Article  Google Scholar 

  15. Eldredge J.D., Toomey J., Medina A.: On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. J. Fluid Mech. 659, 94–115 (2010)

    Article  MATH  Google Scholar 

  16. 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)

  17. Gendrich C.P., Koochesfahani M.M., Visbal M.R.: Effects of initial acceleration on the flow field development around rapidly pitching airfoils. J. Fluids Eng. 117, 45–49 (1995)

    Article  Google Scholar 

  18. Graham, J.M.R.: The forces on sharp-edged cylinders in oscillatory flow at low Keulegan–Carpenter numbers. J. Fluid Mech. 97(1), 331–346 (1980)

    Google Scholar 

  19. Granlund, K., Ol, M.V., Garmann, D.J., Visbal, M.R., Bernal, L.: Experiments and Computations on Abstractions of Perching. AIAA Paper 2010-4943, 28th AIAA Applied Aerodynamic Conference, Chicago (2010)

  20. Grauer J., Ulrich E., Hubbard J., Pines D., Humbert J.S.: Testing and system identification of an ornithopter in longitudinal flight. J. Aircr. 48(2), 660–667 (2010)

    Article  Google Scholar 

  21. Jones, K.D., Platzer, M.F.: Flapping-Wing Propulsion for a Micro Air Vehicle. AIAA Paper 2000-0897 (2000)

  22. Jones M.A.: The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jones, R.T.: The Unsteady Lift of a Wing of Finite Aspect Ratio. Technical report 681, NACA (1939)

  24. Katz J., Weihs D.: Behavior of vortex wakes from oscillating airfoils. AIAA J. 15(12), 861–863 (1978)

    Google Scholar 

  25. Koochesfahani M.M., Smiljanovski V.: Initial acceleration effects on flow evolution around airfoils pitching to high angles of attack. AIAA J. 31(8), 1529–1531 (1993)

    Article  Google Scholar 

  26. Krasny, R.: Vortex sheet computations: roll-up, wake, separation. In: Lectures in Applied Mathematics, vol. 28, pp. 385–402. AMS (1991)

  27. Krechetnikov R., Marsden J.E., Nagib H.M.: A low-dimensional model of separation bubbles. Phys. D 238, 1152–1160 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. McCroskey W.J.: Unsteady airfoils. Ann. Rev. Fluid Mech. 14, 285–311 (1982)

    Article  Google Scholar 

  29. Michelin S., Llewellyn Smith S.G.: An unsteady point vortex method for coupled fluid–solid problems. Theor. Comput. Fluid Dyn. 23, 127–153 (2009)

    Article  MATH  Google Scholar 

  30. Michelin S., Llewellyn Smith S.G.: Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model. Theor. Comput. Fluid Dyn. 24, 195–200 (2010)

    Article  MATH  Google Scholar 

  31. Milano M., Gharib M.: Uncovering the physics of flapping flat plates with artificial evolution. J. Fluid Mech. 534, 403–409 (2005)

    Article  MATH  Google Scholar 

  32. Milne-Thomson L.M.: Theoretical Hydrodynamics. Dover Publications, Inc., New York (1996)

    Google Scholar 

  33. Mourtos N.J., Brooks M.: Flow past a flat plate with a vortex/sink combination. J. Appl. Mech. 63, 543–550 (1996)

    Article  MATH  Google Scholar 

  34. Nitsche M., Krasny R.: A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ol M.V., Eldredge J.D., Wang C.: High-amplitude pitch of a flat plate: an abstraction of perching and flapping. Int. J. Micro Air Veh. 1(3), 203–216 (2009)

    Article  Google Scholar 

  36. Pitt Ford, C.W., Babinsky, H.: Lift and the Leading Edge Vortex. AIAA Paper 2012-0911 (2012)

  37. Polhamus, E.C.: A Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge-Suction Analogy. Technical report TN D-3767, NASA (1966)

  38. Pullin D.I.: The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88(3), 401–430 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pullin D.I., Wang Z.J.: Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid Mech. 509, 1–21 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ramesh, K., Gopalarathnam, A., Ol, M.V., Granlund, K., Edwards, J.R.: Augmentation of Inviscid Airfoil Theory to Predict and Model 2d Unsteady Vortex Dominated Flows. AIAA Paper 2011-3578 (2011)

  41. Rossow V.J.: Lift enhancement by an externally trapped vortex. J. Aircr. 15(9), 618–625 (1978)

    Article  Google Scholar 

  42. Saffman P.G., Sheffield J.S.: Flow over a wing with an attached free vortex. Stud. Appl. Math. 57, 107–117 (1977)

    Google Scholar 

  43. Sedov L.I.: Two-Dimensional Problems in Hydrodynamics and Aerodynamics. New York, London (1965)

    MATH  Google Scholar 

  44. Shukla R.K., Eldredge J.D.: An inviscid model for vortex shedding from a deforming body. Theor. Comput. Fluid Dyn. 21, 343–368 (2007)

    Article  Google Scholar 

  45. Tchieu, A.A., Leonard, A.: A discrete-vortex model for the arbitrary motion of a thin airfoil with fluidic control. J. Fluids Struct. 27(5–6), 680–693 (2011) (submitted)

    Google Scholar 

  46. Theodorsen, T.: General Theory of Aerodynamic Instability and the Mechanism of Flutter. TR-496, NACA (1935)

  47. Visbal M.R.: Dynamic stall of a constant-rate pitching airfoil. J. Aircr. 27(5), 400–407 (1990)

    Article  Google Scholar 

  48. von Kármán T., Sears W.R.: Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5(10), 379–390 (1938)

    MATH  Google Scholar 

  49. Wagner, H.: Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Zeitschrift für Angewandte Mathematik und Mechanik 5(1), 17–35 (1925)

    Google Scholar 

  50. Wang Z.J., Birch J.M., Dickinson M.H.: Unsteady forces and flows in low Reynolds number hovering flight: two dimensional computations vs robotic wing experiments. J. Exp. Biol. 207, 449–460 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical and Aerospace Engineering Department, University of California Los Angeles, 420 Westwood Plaza, Los Angeles, CA, 90095-1597, USA

    Chengjie Wang & Jeff D. Eldredge

Authors
  1. Chengjie Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jeff D. Eldredge
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jeff D. Eldredge.

Additional information

Communicated by T. Colonius.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, C., Eldredge, J.D. Low-order phenomenological modeling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27, 577–598 (2013). https://doi.org/10.1007/s00162-012-0279-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-012-0279-5

Keywords