Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-17T01:25:54.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Energetics and optimum motion of oscillating lifting surfaces of finite span

Published online by Cambridge University Press:  21 April 2006

Ali R. Ahmadi  and
Sheila E. Widnall
Ali R. Ahmadi
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Present address: Aerospace Engineering Department, California State Polytechnic University, Pomona, California 91768.
Sheila E. Widnall
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

The energetics of an unswept wing of finite span oscillating harmonically in combined pitch and heave in in viscid incompressible flow are determined in closed form. The calculations are based on a recently developed low-frequency unsteady lifting-line theory. The energetic calculations for the wing consist of sectional and total values of thrust, leading-edge suction force, power required to maintain the wing oscillations, and energy-loss rate due to vortex shedding in the wake, where the latter quantity is only defined for the entire wing. These results are used to analyse the optimum motion of a wing oscillating harmonically: optimum motion minimizes the power input for fixed average total thrust. The optimum solution is found to be unique (at least for low reduced frequencies), in contrast to the two-dimensional optimum, which is non-unique. Numerical results are presented for the energetics and optimum motion of an elliptic wing.

To understand better the structure of the known solution for the optimum motion of an oscillating two-dimensional airfoil, the solution is recast in terms of the normal modes of the energy-loss-rate matrix. It is found that one of the modes, termed here the ‘invisible mode’, plays a central role in the optimum solution and is responsible for its non-uniqueness. The three-dimensional optimum, which is unique, does not have an invisible mode.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 162 , January 1986 , pp. 261 - 282
Copyright
© 1986 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, M.I.T., also as NASA CR-165619, 1981.
Ahmadi, A. R. & Widnall, S. E. 1982 Unsteady lifting-line theory with applications. A1AA Paper No. 82–0354, presented at AIAA 20th Aerospace Sciences Conference, Orlando, FL.
Ahmadi, A. R. & Widnall, S. E. 1985 Unsteady lifting-line theory as a singular perturbation problem. J. Fluid Mech. 153, 59–81.Google Scholar
Archer, R. D., Sapuppo, J. & Betteridge, D. S. 1979 Propulsive characteristics of flapping wings. Aero. J. 355–371.
Bennet, A. G. 1970 A preliminary study of ornithopter aerodynamics. Ph.D. thesis, University of Illinois, Urbana.
Betteridge, D. S. & Archer, R. D. 1974 A study of the mechanics of flapping wings. Aer. Quart.
Chopra, M. G. 1974 Hydrodynamics of lunate-tail swimming propulsion. J. Fluid Mech. 64, 375–391.Google Scholar
Chopra, M. G. 1976 Large amplitude lunate-tail theory of fish locomotion. J. Fluid. Mech. 74, 161–182.Google Scholar
Chopra, M. G. & Kambe, T. 1977 Hydrodynamics of lunate-tail swimming propulsion. Part 2. J. Fluid Mech. 79, 49–69.Google Scholar
Garrick, I. E. 1936 Propulsion of a flapping and oscillating airfoil. NACA Rep. No. 567.
Lan, C. E. 1979 The unsteady quasi-vortex-lattice method with applications to animal propulsion. J. Fluid Mech. 93, 747–765.Google Scholar
Lighthill, M. J. 1970 Aquatic animal propulsion of high hydrodynamic efficiency. J. Fluid Mech. 44, 265–301.Google Scholar
Mangler, K. W. 1951 Improper integrals in theoretical aerodynamics. ARC R & M 2424.Google Scholar
Reissner, E. 1947 Effect of finite span on the airload distributions for oscillating wings, I - Aerodynamic theory of oscillating wings of finite span. NACA TN No. 1194.
Sears, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical application. J. Aero. Sci. 8, 104–108.Google Scholar
Siekmann, J. 1962 Theoretical studies of sea animal locomotion, part I. Ing. Arch. 31, 214–228.Google Scholar
Siekmann, J. 1963 Theoretical studies of sea animal locomotion, part II. Ing. Arch. 32, 40–50.Google Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA TR-496.
von Kármán, T. & Burgers, J. M. 1935 General aerodynamic theory - perfect fluids. In Aerodynamic Theory (ed. W. F. Durand), pp. 301–310. Julius Springer.
Wagner, S. 1969 On the singularity method of subsonic lifting-surface theory. J. Aircraft 6, 549–558.Google Scholar
Wu, T. Y. 1961 Swimming of a waving plate. J. Fluid Mech. 10, 321–344.Google Scholar
Wu, T. Y. 1971a Hydrodynamics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable foward 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