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Unsteady lifting-line theory by the method of matched asymptotic expansions

Published online by Cambridge University Press:  21 April 2006

P. Wilmott
P. Wilmott
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, UK
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Abstract

An unsteady lifting-line theory is presented for a general motion of a wing of high aspect ratio. Our matched-asymptotic-expansions analysis parallels that of Van Dyke (1964) in his solution for the steady lifting line, but is complicated by the shedding of transverse vortices associated with variation of circulation with time. The principal result is an expression for the downwash due to three-dimensional effects. Numerical calculations are presented for a wing of elliptic planform following a curved path.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 186 , January 1988 , pp. 303 - 320
Copyright
© 1988 Cambridge University Press

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References

Ahmadi, A. R. & Widnall, S. E. 1985 Unsteady lifting-line theory as a singular perturbation problem. J. Fluid Mech. 153, 59–81.Google Scholar
Betteridge, D. S. & Archer, R. D. 1974 A study of the mechanics of flapping flight. Aero. Q. 25, 129.Google Scholar
Cheng, H. K. 1975 On lifting-line theory in unsteady aerodynamics. In Unsteady Aerodynamics: Proc. Symp., Univ. of Arizona (ed. R. B. Kinney), pp. 719–739.
Cheng, H. K. & Meng, S. Y. 1980 The oblique wing as a lifting line in transonic flow. J. Fluid Mech. 97, 531–556.Google Scholar
Cheng, H. K. & Murillo, L. E. 1984 Lunate-tail swimming propulsion as a problem of curved lifting line in unsteady flow. Part 1. Asymptotic theory. J. Fluid Mech. 143, 327–350.Google Scholar
Crighton, D. G. 1985 The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech. 17, 411–445.Google Scholar
Dragoş, L. 1985 The theory of oscillating thick wings in subsonic flow. Lifting line theory. Acta Mech. 54, 221–238.Google Scholar
Goldstein, S. 1952 Approximate two-dimensional aerofoil theory. Curr. Pap. Aero. Res. Counc. Lond. 68–73.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1981 Tables of Integrals, Series and Products. Academic.
Isaacs, R. 1945 Airfoil theory for flow of variable velocity. J. Aero. Sci. 12, 113.Google Scholar
James, E. L. 1975 Lifting line theory for an unsteady wing as a singular perturbation problem. J. Fluid Mech. 70, 753.Google Scholar
Lagerstrom, P. A. & Casten, R. G. 1972 Basic concepts underlying singular perturbation techniques. SIAM Rev. 14, 63–120.Google Scholar
Milne-Thomson, L. M. 1938 Theoretical Hydrodynamics. Macmillan.
Nehari, Z. 1952 Conformal Mapping. McGraw-Hill.
Ockendon, J. R. & Wilmott, P. 1986 Matching and singularity distributions in inviscid flow. IMA J. Appl. Maths 37, 199–211.Google Scholar
Phlips, P. J., East, R. A. & Pratt, N. H. 1981 An unsteady lifting line theory of flapping wings with application to forward flight of birds. J. Fluid Mech. 112, 97.Google Scholar
Theodorsen, T. 1932 Theory of wing sections of arbitrary shape. Rep. Natl Adv. Comm. Aero. Washington 411.Google Scholar
Thwaites, B. 1960 Incompressible Aerodynamics. Cambridge University Press.
Van Dyke, M. D. 1964 Lifting Line theory as a singular perturbation problem. Arch. Mech. Stos. 16, 601.Google Scholar
Van Holten, Th. 1976 Some notes on unsteady lifting line theory. J. Fluid Mech. 77, 561.Google Scholar
Wilmott, P. 1985 The asymptotic solution of models in theoretical mechanics. D.Phil. thesis, University of Oxford.