Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-23T06:53:28.854Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of airfoil thickness on unsteady aerodynamic loads on pitching airfoils

Published online by Cambridge University Press:  11 June 2015

Valentina Motta ,
Alberto Guardone  and
Giuseppe Quaranta Open the ORCID record for Giuseppe Quaranta [Opens in a new window]
Valentina Motta
Affiliation:
Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa 34, 20156, Milano, Italy
Alberto Guardone
Affiliation:
Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa 34, 20156, Milano, Italy
Giuseppe Quaranta*
Affiliation:
Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa 34, 20156, Milano, Italy
*
†Email address for correspondence: giuseppe.quaranta@polimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The influence of the airfoil thickness on aerodynamic loads is investigated numerically for harmonically pitching airfoils at low incidence, under the incompressible and inviscid flow approximation. Force coefficients obtained from finite-volume unsteady simulations of symmetrical 4-digit NACA airfoils are found to depart from the linear Theodorsen model of an oscillating flat plate. In particular, the value of the reduced frequency resulting in the inversion – from clockwise to counter-clockwise – of the lift/angle-of-attack hysteresis curve is found to increase with the airfoil thickness. Both the magnitude and direction of the velocity vector due to pitching over the airfoil surface differ from their flat-plate values. During the upstroke, namely nose-up rotation, phase, this results in a decrease (increase) of the normal velocity magnitude over the upper (lower) surface of the airfoil. The opposite occurs during the downstroke phase. This is confirmed by comparing the computed pressure distribution to the flat-plate linear Küssner model. Therefore, beyond the inversion frequency, the lift coefficient of a finite-thickness airfoil is higher during upstroke and lower during downstroke than its flat-plate counterpart. A similar dependence is also found for the quarter-chord moment coefficient. Accordingly, a modification to the classical Theodorsen model is proposed to take into account the effects of the airfoil thickness on unsteady loads. The new model is found to accurately predict the unsteady aerodynamics of a thick symmetric and a slightly cambered airfoil with a maximum thickness in the range 4–24 %. The limits of the present inviscid flow analysis are assessed by means of numerical simulation of high Reynolds number ($\mathit{Re}=10^{6}$) flows.

JFM classification

Aerodynamics: Aerodynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 774 , 10 July 2015 , pp. 460 - 487
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbott, I. H. & von Doenhoff, A. E. 1949 Theory of Wing Sections. Dover.Google Scholar
Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72.CrossRefGoogle Scholar
Archibald, F. S. 1975 Unsteady Kutta condition at high values of the reduced frequency parameter. J. Aircraft 12 (6), 545–550.CrossRefGoogle Scholar
Baik, Y. S., Bernal, L. P., Granlund, K. & Ol, M. V. 2012 Unsteady force generation and vortex dynamics of pitching and plunging aerofoils. J. Fluid Mech. 709, 37–68.CrossRefGoogle Scholar
Barger, R. L.1975 Adaptation of the Theodorsen theory to the representation of an airfoil as a combination of a lifting line and a thickness distribution. NASA TN D-8117.Google Scholar
Biava, M.2007 RANS computations of rotor/fuselage unsteady interactional aerodynamics. PhD thesis, Politecnico di Milano.Google Scholar
Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 1955 Aeroelasticity. Addison-Wesley.Google Scholar
Bohl, D. G. & Koochesfahani, M. M. 2009 MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency. J. Fluid Mech. 620, 63–88.CrossRefGoogle Scholar
Cicala, P. 1936 Le azioni aerodinamiche sul profilo oscillante. L’Aerotecnica 16, 652–655.Google Scholar
Currie, I. G. 2012 Fundamental Mechanics of Fluids, 4th edn. CRC Press.Google Scholar
Drikakis, D., Zhong, B., Barakos, G., Steijl, R., Biava, M., Vigevano, L., Brocklehurst, A., Boelens, O. J., Dietz, M., Embacher, M., Khier, W. & Antoniadis, A. F. 2012 Methods against experimental flow measurements for helicopter flows. Aerosp. Sci. Technol. 19 (1), 86–100.Google Scholar
Freymuth, P. 1988 Propulsive vortical signature of plunging and pitching airfoils. AIAA J. 26 (7), 881–883.CrossRefGoogle Scholar
Fung, Y. C. 1955 Theory of Aeroelasticity. John Wiley & Sons.Google Scholar
Garrick, I. E.1936 Propulsion of a flapping and oscillating airfoil. NACA TR 567.Google Scholar
Garrick, I. E.1938 On some reciprocal relations in the theory of nonstationary flows. NACA TR 629.Google Scholar
Gennaretti, M., Testa, C. & Bernardini, G. 2013 An unsteady aerodynamic formulation for efficient rotor tonal noise prediction. J. Sound Vib. 332 (25), 6743–6754.CrossRefGoogle Scholar
Glegg, S. & Devenport, W. 2009 Unsteady loading on an airfoil of arbitrary thickness. J. Sound Vib. 319 (3–5), 1252–1270.CrossRefGoogle Scholar
Goldstein, M. E. & Atassi, H. 1976 A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust. J. Fluid Mech. 74, 741–765.CrossRefGoogle Scholar
Halfman, R. L.1952 Experimental aerodynamic derivatives of a sinusoidally oscillating airfoil in two-dimensional flow. NACA TR 1108.Google Scholar
Jameson, A., Schmidt, W. & Turkel, E.1981 Numerical solution of the Euler equations by finite volume methods schemes. AIAA 14th Fluid and Plasma Dynamic Conference, June 23–25, 1981, Palo Alto, CA, USA.CrossRefGoogle Scholar
Johnson, W. 1980 Helicopter Theory. Princeton University Press.Google Scholar
Katz, J. & Plotkin, A. 1991 Low-Speed Aerodynamics: From Wing Theory to Panel Methods. McGraw-Hill.Google Scholar
Khier, W., Vigevano, L. & Biava, M. 2012 Prediction of air flow past a full helicopter configuration. Aerosp. Sci. Technol. 19 (1), 3–18.Google Scholar
Kinzel, M. P., Maughmer, M. D. & Duque, E. P. 2010 Numerical investigation on the aerodynamics of oscillating airfoils with deployable gurney flaps. AIAA J. 48 (7), 1457–1469.CrossRefGoogle Scholar
Kurosaka, M. 1974 On the flow field of a rapidly oscillating airfoil in a supersonic flow. J. Fluid Mech. 62 (4), 811–827.CrossRefGoogle Scholar
Küssner, H. G. 1936 Zusammenfassender Bericht über den instationären Auftrieb von Flügeln. Luftfahrtforschung 13 (12), 410–424.Google Scholar
Küssner, H. G. 1960 Nonstationary theory of airfoils of finite thickness in incompressible flow. In AGARD Manual on Aeroelasticity, Part 2 (ed. Jones, W. P.), chap. 8.Google Scholar
Küssner, H. G. & Schwarz, L.1941 The oscillating wing with aerodynamically balanced elevator. NACA TM 991, translated from Luftfahrtforschung, vol. 17, 1940, pp. 337–354.Google Scholar
Lee, T. & Gerontakos, P. 2004 Investigation of flow over an oscillating airfoil. J. Fluid Mech. 512, 313–341.CrossRefGoogle Scholar
Leishman, J. G. 2006 Principles of Helicopter Aerodynamics, 2nd edn. Cambridge University Press.Google Scholar
Liebeck, R. H. 1978 Design of subsonic airfoils for high lift. J. Aircraft 15 (9), 547–561.CrossRefGoogle Scholar
Lin, P. T., Baker, T. J., Martinelli, L. & Jameson, A. 2006 Two-dimensional implicit time-dependent calculations on adaptive unstructured meshes with time evolving boundaries. Intl J. Numer. Meth. Fluids 50 (2), 199–218.CrossRefGoogle Scholar
McCroskey, W. J. 1973 Inviscid flowfield of an unsteady airfoil. AIAA J. 11 (8), 1130–1137.CrossRefGoogle Scholar
McCroskey, W. J. 1982 Unsteady airfoils. Annu. Rev. Fluid Mech. 14 (1), 285–311.CrossRefGoogle Scholar
Morino, L.1974 A general theory of compressible potential aerodynamics. NASA CR 2464.Google Scholar
Morino, L. 2003 Is there a difference between aeroacoustics and aerodynamics? An aeroelastician’s viewpoint. AIAA J. 41 (7), 1209–1223.CrossRefGoogle Scholar
Morino, L., Chen, L. T. & Suciu, E. O. 1975 Steady and oscillatory subsonic and supersonic aerodynamics around complex configurations. AIAA J. 13 (3), 368–374.CrossRefGoogle Scholar
Motta, V.2015 Computational fluid dynamic analysis of a L-shaped Gurney flap for vibration control. PhD thesis, Politecnico di Milano.Google Scholar
Motta, V. & Quaranta, G. 2015 Linear reduced-order model for unsteady aerodynamics of an L-shaped Gurney flap. J. Aircraft (in press); http://arc.aiaa.org/doi/abs/10.2514/1.C033099.CrossRefGoogle Scholar
Panda, J. & Zaman, K. B. M. Q. 1994 Experimental investigation of the flow field of an oscillating airfoil and estimation of lift from wake surveys. J. Fluid Mech. 265, 65–95.CrossRefGoogle Scholar
Rainey, A. G.1957 Measurement of aerodynamic forces for various mean angles of attack on an airfoil oscillating in pitch and on two finite-span wings oscillating in bending with enphasis on damping in the stall. NACA TR 1305.Google Scholar
Roe, P. L. 1981 Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372.CrossRefGoogle Scholar
Spalart, P. R. & Allmaras, S. R. 1994 A one-equation turbulence model for aerodynamic flows. La Recerche Aérosp. 1, 5–21.Google Scholar
Theodorsen, T.1931 Theory of wing sections of arbitrary shape. NACA Tech. Rep. 411. Langley Memorial Aeronautical Laboratory.Google Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter. NACA TR 496.Google Scholar
Uldrick, J. P. & Siekmann, J. 1964 On the swimming of a flexible plate of arbitrary finite thickness. J. Fluid Mech. 20 (1), 1–33.CrossRefGoogle Scholar
Van Dyke, M. D. 1953 On second-order supersonic flow past a slowly oscillating airfoil. J. Aeronaut. Sci. 20 (1), 61.CrossRefGoogle Scholar
Venkatakrishnan, V. & Mavriplis, D. J. 1996 Implicit method for the computation of unsteady flows on unstructured grids. J. Comput. Phys. 127 (2), 380–397.CrossRefGoogle Scholar
VV. AA. 1982 Compendium of unsteady aerodynamic measurements. Rep. 702. AGARD.Google Scholar
Wagner, H. 1925 Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5 (1), 17–35.CrossRefGoogle Scholar