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On the interaction of a moving hollow vortex with an aerofoil, with application to sound generation

Published online by Cambridge University Press:  25 August 1997

F. G. LEPPINGTON
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, UK
R. A. SISSON
Affiliation:
Surpac Software International, PO Box 233, Belmont, Western Australia 6104

Abstract

A hollow vortex in the form of a straight tube, parallel to the z-axis, and of radius a, moves in a uniform stream of fluid with velocity U in the x-direction, with U small compared with the sound speed c. This steady flow is disturbed by the presence of a thin symmetric fixed aerofoil. With a change of x-coordinate, the problem is equivalent to that of a moving aerofoil cutting through an initially fixed vortex in still fluid. The aim of this work is to determine the resulting perturbed flow, and to estimate the distant sound field. A detailed calculation is given for the perturbed velocity potential in the incompressible flow case, for the linearized equations in the limit of small aerofoil thickness. A formally exact solution involves a four-fold integral and an infinite sum over all mode numbers. For the important special case where the vortex tube has small radius a compared with the aerofoil width, the deformed vortex is characterized by a hypothetical vortex filament located at the ‘mean centre’ (z, t), (z, t) of the tube. Explicit results are given for (z, t), (z, t) for the case where the aerofoil has the elementary rectangular profile; results can then be obtained for more general and realistic cylindrical aerofoils by a single integral weighted with the derivative of the aerofoil thickness function. Finally the distant sound field is estimated, representing the aerofoil by a distribution of moving monopole sources and representing the effect of the deformed vortex in terms of compressible dipoles along the mean centre of the vortex.

Type
Research Article
Copyright
© 1997 Cambridge University Press

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