An approximate theory is given for the generation of internal
gravity waves in a viscous
Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder.
A
parameter λ
which is proportional to the square of the ratio of the thickness of the
oscillatory
boundary layer that surrounds the cylinder to a typical dimension
of its cross-section
is introduced. When λ[Lt ]1 (or equivalently when the Reynolds number
R[Gt ]1), the
viscous boundary condition at the surface of the cylinder may to first
order in λ be
replaced by the inviscid one. A viscous solution is proposed for the case
λ[Lt ]1 in which
the Fourier representation of the stream function found in Part 1 (Hurley
1997) is
modified by including in the integrands a factor to account for viscous
dissipation. In
the limit λ→0 the proposed solution becomes the inviscid one
at each point in the flow field.For ease of presentation the case of a circular cylinder of radius
a is considered first
and we take a to be the typical dimension of its cross-section
in
the definition of λ
above. The accuracy of the proposed approximate solution is investigated
both
analytically and numerically and it is concluded that it is accurate
throughout the flow
field if λ is sufficiently small, except in a small region near
where the characteristics touch the cylinder where viscous effects dominate.Computations indicate that the velocity on the centreline on a typical
beam of waves, at a distance s along the beam from the centre of the cylinder,
agrees, within about 1%,
with the (constant) inviscid values provided λs/a
is less than about 10−3. This result is
interpreted as indicating that those viscous effects which originate from
the
characteristics that touch the cylinder (places where the inviscid velocity
is singular)
reach the centreline of the beam when λs/a
is about 10−3. For larger values of s, viscous
effects are significant throughout the beam and the velocity profile of
the
beam changes
until it attains, within about 1% when λs/a
is about 2, the value given by the similarity
solution obtained by Thomas & Stevenson (1972). For larger values of
λs/a, their similarity solution applies.In an important paper Makarov et al. (1990) give an approximate
solution for the
circular cylinder that is very similar to ours. However, it does not reduce
to the inviscid one when the viscosity is taken to be zero.Finally it is shown that our results for a circular cylinder apply,
after small
modifications, to all elliptical cylinders.
Aeroacoustic Simulation around a Circular Cylinder by the Equations Split for Incompressible Flow Field and Acoustic Field
