Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-14T07:40:52.872Z Has data issue: false hasContentIssue false
  • English
  • Français

On unsteady surface forces, and sound produced by the normal chopping of a rectilinear vortex

Published online by Cambridge University Press:  26 April 2006

M. S. Howe
M. S. Howe
Affiliation:
BBN Laboratories, 10 Moulton Street, Cambridge MA 02238, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

An investigation is made of the sound produced when a rectilinear vortex is cut at right angles to its axis by a non-lifting airfoil of symmetric section. The motions are at sufficiently low Mach number that the wavelength of the sound is large relative to the chord of the airfoil. In these circumstances the airfoil experiences no fluctuating lift during the interaction, and the radiation may be ascribed to an acoustic source of dipole type whose strength is equal to the unsteady drag. It is argued that previous analyses of the related problem of ‘unsteady thickness noise’ have ignored certain terms whose inclusion greatly reduces the predicted intensity of the radiation. A general formula for the surface forces (derived in an appendix) is applied to deduce that the dipole strength is proportional to the square of the circulation of the vortex, and depends on the spanwise acceleration of the vortex induced by images in the airfoil. Numerical results are presented for typical airfoil sections, and a comparison is made with the unsteady lifting noise generated when the axis of the vortex is inclined at a small angle to the normal to the median plane of the airfoil.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 206 , September 1989 , pp. 131 - 153
Copyright
© 1989 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

Amiet, R. K. 1986 Airfoil gust response and the sound produced by airfoil-vortex interaction. J. Sound Vib. 107, 487–506.Google Scholar
Atassi, H. M. 1984 The Sears problem for a lifting airfoil revisited - new results. J. Fluid Mech. 141, 109–122.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7, 83–103.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.
Curle, N. 1955 On the influence of solid boundaries upon aerodynamic sound.. Proc. R. Soc. Lond. A 231, 505–513.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Ffowcs Williams, J. E. & Hawkings, D. L. 1969 Sound generation by turbulence and surfaces in arbitrary motion.. Phil. Trans. R. Soc. Lond. A 264, 321–342.Google Scholar
Glegg, S. A. L. 1987 Significance of unsteady thickness noise sources. AIAA J. 25, 839–844.Google 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 (see also Corrigendum, 91, 1979, 788).Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals, Series, and Products (corrected edition). Academic.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Hawkings, D. L. 1978 A possible unsteady thickness noise mechanism for helicopter rotors. Paper presented at the 95th meeting of the Acoustical Society of America, Providence, RI.
Howe, M. S. 1975 Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71, 625–673.Google Scholar
Howe, M. S. 1988a Correlation of lift and thickness noise sources in vortex-airfoil interaction. J. Sound Vib. (in press).Google Scholar
Howe, M. S. 1988b Contributions to the theory of sound production by vortex-airfoil interaction, with application to vortices with finite axial velocity defect.. Proc. R. Soc. Lond. A 420, 157–182.Google Scholar
Lighthill, J. 1986 An Informal Introduction to Theoretical Fluid Mechanics. Clarendon.
Michalke, A. & Timme, A. 1967 On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, 647–666.Google Scholar
Quartapelle, L. & Napolitano, M. 1983 Force and moment in incompressible flows. AIAA J. 21, 911–913.Google Scholar
Ribner, H. S. & Tucker, M. 1953 Spectrum of turbulence in a contracting stream. NACA Tech. Rep. 1113.Google Scholar
Wu, J. C. 1981 The theory of aerodynamic force and moment in viscous flows. AIAA J. 19, 432–441.Google Scholar