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Airfoil leading-edge suction and energy conservation for compressible flow

Published online by Cambridge University Press:  26 April 2006

R. K. Amiet
Affiliation:
3306 Dover Road, Wooster, OH 44691, USA

Abstract

When a flat-plate airfoil at zero angle of attack encounters a vertical gust in an otherwise uniform flow, it experiences a force along the chord. This leading-edge suction force is examined for compressible flow with a time-dependent gust. A simple derivation of the thrust force is based on the fact that the leading edge is a singular point so that the flow here is dominated by the leading-edge dipole strength. From the viewpoint of a fluid-fixed observer the fluid does work on the airfoil, and this energy must come from the incident gust. Demonstrating energy conservation is not surprising, but it gives a better understanding of the relationship between the individual energy terms. The derivation shows that the acoustic energy can be calculated using compact assumptions at low frequency, but that it must be calculated non-compactly at high frequency. For a general gust the work done on the airfoil is shown to equal the energy taken from the fluid, the energy transfer occurring at the leading edge. For a sinusoidal gust the energy contained in the incident gust is shown to equal the sum of the energy remaining in the wake, the work done on the airfoil and the acoustic energy radiated away. The relative proportions of the incident energy going to these three energy types depends on the gust frequency, the acoustic radiation becoming more efficient as the frequency increases. For a fixed gust frequency, the thrust force goes to zero at a Mach number of one, and for an incident gust consisting of vorticity on the airfoil axis, the entire energy of the gust is radiated as acoustic energy at this Mach number.

Type
Research Article
Copyright
© 1995 Cambridge University Press

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References

Amiet, R. K. 1974 Compressibility effects in unsteady thin-airfoil theory. AIAA J. 12, 252255.Google Scholar
Amiet, R. K. 1976 High frequency thin-airfoil theory for subsonic flow. AIAA J. 14, 10761082.Google Scholar
Amiet, R. K. 1990 Gust response for flat-plate airfoils and the Kutta condition. AIAA J. 28, 17181727.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Char, B. W., Geddes, K. O., Gonnett, G. H. et al. 1991 Maple V Language Reference Manual. Springer.
Ffowcs Williams, J. E. & Guo, Y. P. 1988 Sound generated from the interruption of a steady flow by a supersonically moving aerofoil. J. Fluid Mech. 195, 113135.Google Scholar
Fung, Y. C. 1969 An Introduction to the Theory of Aeroelasticity. Dover.
Garrick, I. E. 1936 Propulsion of a flapping and oscillating airfoil. NACA Rep. 567, pp. 419427.
Garrick, I. E. 1957 Nonsteady wing characteristics. Aerodynamic Components of Aircraft at High Speeds, Vol. VII (ed. A. F. Donovan & H. R. Lawrence). Princeton University Press.
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Tables of Integrals Series and Products. Academic.
Guo, Y. P. 1989 A note on sound from interruption of cylindrical flow by a semi-infinite aerofoil of subsonic speed. J. Sound Vib. 128, 275286.Google Scholar
Guo, Y. P. 1991a Sound diffraction and dissipation at a sharp trailing edge in a supersonic flow. J. Sound Vib. 145, 179193.Google Scholar
Guo, Y. P. 1991b Energetics of sound radiation from flow-aerofoil interaction. J. Sound Vib. 151, 247262.Google Scholar
Jones, R. T. 1950 Leading edge singularities in thin-airfoil theory. J. Aeronaut. Sci. 17, 307310.Google Scholar
Kármán, T. von & Burgers, J. M. 1935 General aerodynamic theory – perfect fluids. In Aerodynamic Theory (ed. W. F. Durand), vol. II. Dover.
Katzmayr, R. 1922 Effects of periodic changes in angle of attack on behavior of airfoils. Z. Flugtechnik Motorluftschiffahrt, March 31 and April 13, 80–82 and 95–101; see also NACA TM-147.
Landahl, M. T. 1961 Unsteady Transonic Flow. Pergamon.
Levine, H. 1991 On the energetics of a baffled piston with time-varying velocity. J. Acoustique 4, 1320.Google Scholar
Lighthill, M. J. 1951 A new approach to thin-aerofoil theory. Aeronaut. Q. 3, 193210.Google Scholar
Lighthill, J. 1975 Mathematical Biofluiddynamics. Society for Industrial and Applied Mathematics, Philadelphia.
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics. McGraw Hill.
Pierce, A. D. 1981 Acoustics, An Introduction to Its Physical Principles and Applications. McGraw Hill.
Ribner, H. S. 1993 Thrust imparted to an airfoil by passage through a sinusoidal upwash field. AIAA J. 31, 18631868.Google Scholar
Robinson, A. & Laurmann, J. A. 1956 Wing Theory. Cambridge University Press.