ON THE INTERACTION OF A MOVING HOLLOW VORTEX WITH AN AEROFOIL, WITH APPLICATION TO SOUND GENERATION

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 ae rofoil. With a change of x-coordinate, the problem is equivalent to th at of a moving aerofoil cutting through an initially fixed vortex in s till fluid. The aim of this work is to determine the resulting perturb ed flow, and to estimate the distant sound field. A detailed calculati on is given for the perturbed velocity potential in the incompressible flow case, for the linearized equations in the limit of small aerofoi l thickness. A formally exact solution involves a four-fold integral a nd an infinite sum over all mode numbers. For the important special ca se 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' (x) over bar(z, t), (y) over bar (z, t) of the tube. Explicit results are given for (x) over bar(z, t), (y) over bar(z, t) for the case where the aerofoil has the elementary rectangular profile; results can then be obtained for more general an d realistic cylindrical aerofoils by a single integral weighted with t he 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.