Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-19T01:04:45.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Unsteady lifting-line theory as a singular perturbation problem

Published online by Cambridge University Press:  20 April 2006

Ali R. Ahmadi  and
Sheila E. Widnall
Ali R. Ahmadi
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Present address: Bolt Beranek and Newman Inc., Cambridge, Massachusetts 02238.
Sheila E. Widnall
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting-line theory of James (1975), and points out a limitation in that of Van Holten (1975, 1976, 1977).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 153 , April 1985 , pp. 59 - 81
Copyright
© 1985 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

Ahmadi, A. R. 1980 An asymptotic unsteady lifting-line theory with energetics and optimum motion of thrust-producing lifting surfaces. Ph.D. thesis, MIT, also as NASA CR-165679 (1981).
Ahmadi, A. R. & Widnall, S. E. 1983 Energetics and optimum motion of oscillating lifting surfaces. AIAA Paper 83-1710. presented at AIAA 16th Fluid and Plasma Dynamics Conf., Danvers, MA, July.
Ashley, H. & Landahl, M. T. 1965 Aerodynamics of Wings and Bodies. Addison-Wesley.
Cheng, H. K. 1975 On lifting-line theory in unsteady aerodynamics. In Unsteady Aerodynamics: Proc. Symp. at University of Arizona, 18–20 March (ed. R. B. Kinney), pp. 719–739.
Cheng, H. K. & Meng, S. Y. 1980 The oblique wing as a lifting-line problem in transonic flow. J. Fluid Mech. 97, 531–556.Google Scholar
Cheng, H. K. & Murillo, L. E. 1984 Lunate-tail swimmming propulsion as a problem of curved lifting line in unsteady flow. Part 1. Asymptotic theory. J. Fluid Mech. 143, 327–350.Google Scholar
Erdélyi, A. (ed.) 1953 Higher Transcendental Functions. Bateman Manuscript Project, vol. 11, McGraw Hill.
Gröbner, W. & Hofreiter, N. 1961 Integraltafel, Zweiter Teil, Bestimmte Integrale. Springer.
Guiraud, J. P. & Slama, G. 1981 Lifting line asymptotic theory in incompressible oscillating flow. Rech. Aerospat. 1.Google Scholar
James, E. C. 1975 Lifting-line theory for an unsteady wing as a singular perturbation problem. J. Fluid Mech. 70, 753–771.Google Scholar
Jordan, P. F. 1971a The parabolic wing tip in subsonic flow. AIAA Paper 71-10.
Jordan, P. F. 1971b Span loading and formation of wake. In Aircraft Wake Turbulence and Its Detection (ed. J. H. Olsen, A. Goldberg & M. Rogers), pp. 207–227. Plenum.
Kinner, W. 1937 Die kreisförmige Tragfläche auf potentialtheoretischer Grundlage. Ing. Archiv. 8, 47–80.Google Scholar
Küssner, H. G. 1941 General airfoil theory. NACA TM 979.
Landahl, M. T. 1968 Pressure loading functions for oscillating wings with control surfaces. AIAA J. 6, 345–348.Google Scholar
Mangler, K. W. 1951 Improper integrals in theoretical aerodynamics. ARC R & M 2424.
Murillo, L. E. 1979 Hydromechanical performance of lunate tails analyzed as a lifting-line problem in unsteady flow. Ph.D. thesis, USC.
Reissner, E. 1944 On the general theory of thin airfoils for nonuniform motion. NACA TM 946.
Schade, T. & Krienes, K. 1947 The oscillating circular airfoil on the basis of potential theory. NACA TM 1098.
Sears, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical application. J. Aero. Sci. 8, 104–108.Google Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA TR 496.
Van Dyke, M. 1963 Lifting-line theory as a singular-perturbation problem. SUDAER 165 (August).
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics, annotated edn. Parabolic.
Van Holten, T. 1975 The computation of aerodynamic loads on helicopter blades in forward flight, using the method of the acceleration potential. Delft Univ. Tech., Dept Aerosp. Engng Rep. VTH-189.Google Scholar
Van Holten, T. 1976 Some notes on unsteady lifting-line theory. J. Fluid Mech. 77, 561–579.Google Scholar
Van Holten, T. 1977 On the validity of lifting line concepts in rotor analysis. Vertica 1, 239–254.Google Scholar
Watkins, C. E., Woolston, D. S. & Cunningham, H. J. 1959 A systematic kernel function procedure for determining aerodynamic forces on oscillating or steady finite wings at subsonic speeds. NASA TR R-48.
Widnall, S. E. 1964 Unsteady loads on hydrofoils including free surface effects and cavitation. Sc.D. thesis, MIT.
Wu, T. Y. 1971a Hydrodynamics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speed in an inviscid fluid. J. Fluid Mech. 46, 337–355.Google Scholar
Wu, T. Y. 1971b Hydrodynamics of swimming propulsion. Part 2. Some optimum shape problems. J. Fluid Mech. 46, 521–544.Google Scholar