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’
x¯(z, t), y¯(z,
t) of the tube. Explicit results are given for
x¯(z, t), y¯(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.