Numerical study of two-airfoil arrangements by a discrete vortex method

  • Thierry M. FaureEmail author
  • Laurent Dumas
  • Olivier Montagnier
Original Article


The aerodynamic characteristics of two neighboring airfoils are greatly different from those of a single airfoil, for both attached and detached flow conditions. In order to study the features of a two-airfoil arrangement with variations in the angle of attack and distances between the airfoils, and considering possible flow detachments, an adaptation of a discrete-time vortex numerical method is conducted. It is based on the fact that for a given airfoil and Reynolds number, there is a critical value of the leading-edge suction parameter. If its instantaneous value exceeds the critical value, vortex shedding occurs at the leading edge, representing the shear layer associated with flow detachment. In addition, Kelvin’s theorem imposes for each time step that the total circulation equals zero. In the present paper, Kelvin’s theorem is extended for a two-airfoil arrangement with the initial starting flow condition. That numerical method allows to obtain instantaneous flow features and airfoil forces for different arrangements. It is validated first for unsteady motions of airfoils in oscillation and plunge. Then, steady airfoils cases are considered without plunging motion and a constant angle-of-attack. Comparisons with available experimental data are presented in terms of flow field and aerodynamic coefficients both for unsteady motions or steady airfoils. In particular, in order to show the possible improvement in the two-airfoil arrangement performance, the averaged lift coefficient is compared with the single-airfoil configuration. A discussion on the lift efficiency ratio with previous measurements and time development of the flow field permits to understand the mechanisms contributing to a positive interaction.


Discrete vortex method Aerodynamics Detached flow Airfoil interaction 



  1. 1.
    Ansari, S.A., Zbikowski, R., Knowles, K.: A nonlinear 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. 30(2), 169–186 (2006)CrossRefGoogle Scholar
  2. 2.
    Barnes, J., Hut, P.: A hierarchical O(NlogN) force-calculation algorithm. Nature 324, 446–449 (1986)CrossRefGoogle Scholar
  3. 3.
    Bentley, J.L.: Multidimensional binary search trees used for associative searching. Commun. ACM 18(9), 509–517 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Birnbaum, W.: Die tragende Wirbelfläche als Hilfsmittel zur Behandlung des ebenen Problems der Tragflügeltheorie. ZAMM J. Appl. Math. Mech. 3(4), 290–297 (1923)zbMATHCrossRefGoogle Scholar
  5. 5.
    Broering, T., Lian, Y.: The effect of phase angle and wing spacing on tandem flapping wings. Acta. Mech. Sin. 28(6), 1557–1571 (2012)CrossRefGoogle Scholar
  6. 6.
    Carrier, J., Greengard, L., Rokhlin, V.: A fast adaptative multipole algorithm for particle simulations. SIAM J. Comput. Phys. 9(4), 5628–5649 (1988)zbMATHGoogle Scholar
  7. 7.
    Clements, R.R.: An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57(2), 321–336 (1973)zbMATHCrossRefGoogle Scholar
  8. 8.
    Clements, R.R., Maull, D.J.: The representation of sheets of vorticity by discrete vortices. Prog. Aerosp. Sci. 16(2), 129–146 (1975)CrossRefGoogle Scholar
  9. 9.
    Crowdy, D.: Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions. Math. Proc. Camb. Philos. Soc. 142, 319–339 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Crowdy, D.: The Schwarz problem in multiply connected domains and the Schottky–Klein prime function. Complex Var. Elliptic Equ. 53(3), 221–236 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Crowdy, D.: A new calculus for two-dimensional vortex dynamics. Theor. Comput. Fluid Dyn. 24(1–4), 9–24 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    Crowdy, D., Marshall, J.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Crowdy, D., Surana, A.: Contour dynamics in complex domains. J. Fluid Mech. 593, 235–254 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Crowdy, D., Surana, A., Yick, K.: The irrotational motion generated by two planar stirrers in inviscid fluid. Phys. Fluids 19(1), 018103 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Darakananda, D., Eldredge, J.D.: A versatile taxonomy of low-dimensional vortex models for unsteady aerodynamics. J. Fluid Mech. 858, 917–948 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Darakananda, D., de Castro da Silva, A.F., Colonius, T., Eldredge, J.: Data-assimilated low-order vortex modeling of separated flow. Phys. Rev. Fluids 3, 124701 (2018)CrossRefGoogle Scholar
  17. 17.
    Faure, T.M., Hétru, L., Montagnier, O.: Aerodynamic features of a two-airfoil arrangement. Exp. Fluids 58(10), 146 (2017)CrossRefGoogle Scholar
  18. 18.
    Faure, T.M., Dumas, L., Drouet, V., Montagnier, O.: A modified discrete-vortex method algorithm with shedding criterion for aerodynamic coefficients prediction at high angle of attack. Appl. Math. Model. 69, 32–46 (2019)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Garrick, I.E.: Propulsion of a flapping and oscillating airfoil. Technical report, NACA TN-D-85 (1937)Google Scholar
  20. 20.
    Glauert, H.: The Elements of Aerofoil and Airscrew Theory. Cambridge University Press, Cambridge (1926)zbMATHGoogle Scholar
  21. 21.
    Graftieaux, L., Michard, M., Grosjean, N.: Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows. Meas. Sci. Technol. 12(9), 1422 (2001)CrossRefGoogle Scholar
  22. 22.
    Hammer, P., Altman, A., Eastep, F.: Validation of a discrete vortex method for low Reynolds number unsteady flow. AIAA J. 52(3), 643–649 (2014)CrossRefGoogle Scholar
  23. 23.
    Jones, R., Cleaver, D., Gursul, I.: Aerodynamics of biplane and tandem wings at low Reynolds numbers. Exp. Fluids 56(124), 1–25 (2015)Google Scholar
  24. 24.
    Katz, J.: Discrete vortex method for the non-steady separated flow over an aerofoil. J. Fluid Mech. 102, 315–328 (1981)zbMATHCrossRefGoogle Scholar
  25. 25.
    Katz, J., Plotkin, A.: Low-Speed Aerodynamics. Cambridge University Press, Cambridge (2001)zbMATHCrossRefGoogle Scholar
  26. 26.
    Kiya, M., Arie, M.: A contribution to an inviscid vortex-shedding model for an inclined flat plate in uniform flow. J. Fluid Mech. 82(2), 241–253 (1977)zbMATHCrossRefGoogle Scholar
  27. 27.
    Kuwahara, K.: Numerical study of flow past an inclined flat plate by an inviscid model. J. Phys. Soc. Jpn. 35, 1545 (1973)CrossRefGoogle Scholar
  28. 28.
    Laitone, E.V.: Wind tunnel tests of wings at Reynolds numbers below 70 000. Exp. Fluids 23(5), 405–409 (1997)CrossRefGoogle Scholar
  29. 29.
    Leonard, A.: Vortex methods for flow simulation. J. Comput. Phys. 37(3), 289–335 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lighthill, M.: On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60(1), 1–17 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Michelin, S., Smith, S.G.L.: An unsteady point vortex method for coupled fluid–solid problems. Theor. Comput. Fluid Dyn. 23(2), 127–153 (2009)zbMATHCrossRefGoogle Scholar
  32. 32.
    Mueller, T.J., DeLaurier, J.D.: Aerodynamics of small vehicles. Annu. Rev. Fluid Mech. 35(1), 89–111 (2003)zbMATHCrossRefGoogle Scholar
  33. 33.
    Munk, M.: General theory of thin wing sections. Technical report, 142 NACA (1922)Google Scholar
  34. 34.
    Ramesh, K.: Theory and low-order modeling of unsteady airfoil flows. Ph.D. thesis, North Carolina State University, Rayleigh, NC, USA (2013)Google Scholar
  35. 35.
    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. In: 41st AIAA Fluid Dynamics Conference and Exhibit, Honolulu, Hawai, USA, AIAA Paper 2011-3578 (2011)Google Scholar
  36. 36.
    Ramesh, K., Gopalarathnam, A., Edwards, J., Granlund, K., Ol, M.: Theoretical analysis of perching and hovering maneuvers. In: 31st AIAA Applied Aerodynamics Conference (2013a)Google Scholar
  37. 37.
    Ramesh, K., Gopalarathnam, A., Edwards, J.R., Ol, M.V., Granlund, K.: An unsteady airfoil theory applied to pitching motions validated against experiments and computation. Theor. Comput. Fluid Dyn. 27(6), 843–864 (2013b)CrossRefGoogle Scholar
  38. 38.
    Ramesh, K., Gopalarathnam, A., Granlund, K., Ol, M.V., Edwards, J.R.: Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding. J. Fluid Mech. 751, 500–538 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Ramesh, K., Granlund, K., Ol, M.V., Gopalarathnam, A., Edwards, J.R.: Leading-edge flow criticality as a governing factor in leading-edge vortex initiation in unsteady airfoil flows. Theor. Comput. Fluid Dyn. 32(2), 109–136 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Rival, D., Manejev, R., Tropea, C.: Measurement of parallel blade-vortex interaction at low reynolds numbers. Exp. Fluids 49, 89–99 (2010)CrossRefGoogle Scholar
  41. 41.
    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(1), 109–128 (1975)zbMATHCrossRefGoogle Scholar
  42. 42.
    Scharpf, D.F., Mueller, T.J.: Experimental study of a low Reynolds number tandem airfoil configuration. J. Aircr. 29(2), 231–236 (1992)CrossRefGoogle Scholar
  43. 43.
    Schmidt, W.: Der Wellpropeller, ein neuer Antrieb fuer Wasserland und Luftfahrzeuge. Z. Flugwiss. Weltraumforsch. 13(12), 472–479 (1965)Google Scholar
  44. 44.
    SureshBabu, A.V., Ramesh, K., Gopalarathnam, A.: Model reduction in discrete-vortex methods for 2D unsteady aerodynamic flows. In: 34th AIAA Applied Aerodynamics Conference, AIAA Paper 2016-4163 (2016)Google Scholar
  45. 45.
    Theodorsen, T.: General theory of aerodynamic instability and the mechanism of flutter. Technical report, 496 NACA (1935)Google Scholar
  46. 46.
    Vatistas, G.H., Kozel, V., Mih, W.C.: A simpler model for concentrated vortices. Exp. Fluids 11(1), 73–76 (1991)CrossRefGoogle Scholar
  47. 47.
    von Kármán, T., Sears, W.: Aerofoil theory for non-uniform motion. J. Aeronaut. Sci. 5(10), 379–390 (1938)zbMATHCrossRefGoogle Scholar
  48. 48.
    Wagner, H.: Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5(1), 17–35 (1925)zbMATHCrossRefGoogle Scholar
  49. 49.
    Wang, C., Eldredge, J.D.: Low-order phenomenological modelling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27(5), 577–598 (2013)CrossRefGoogle Scholar
  50. 50.
    Weis-Fogh, T.: Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59, 169–230 (1973)Google Scholar
  51. 51.
    Xia, X., Mohseni, K.: Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25, 091901 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Centre de Recherche de l’École de l’AirÉcole de l’AirSalon-de-ProvenceFrance

Personalised recommendations