In this paper we consider the causal response of the inviscid shear-layer flow over
an elastic surface to excitation by a time-harmonic line force. In the case of uniform
flow, Brazier-Smith & Scott (1984) and Crighton & Oswell (1991) have analysed the
long-time limit of the response. They find that the system is absolutely unstable for
sufficiently high flow speeds, and that at lower speeds there exist certain anomalous
neutral modes with group velocity directed towards the driver (in contradiction of the
usual radiation condition of out-going disturbances). Our aim in this paper is to repeat
their analysis for more realistic shear profiles, and in particular to determine whether
or not the uniform-flow results can be regained in the limit in which the shear-layer
thickness on a length scale based on the fluid loading, denoted ε, becomes small.
For a simple broken-line linear shear profile we find that the results are qualitatively
similar to those for uniform flow. However, for the more realistic Blasius profile very
significant differences arise, essentially due to the presence of the critical layer. In
particular, we find that as ε → 0 the minimum flow speed required for absolute
instability is pushed to considerably higher values than was found for uniform flow,
leading us to conclude that the uniform-flow problem is an unattainable singular limit
of our more general problem. In contrast, we find that the uniform-flow anomalous
modes (written as exp (ikx − iωt), say) do persist
for non-zero shear over a wide range
of ε, although now becoming non-neutral. Unlike the case of uniform flow, however,
the k-loci of these modes can now change direction more than once as the imaginary
part of ω is increased, and we describe the connection between this behaviour and
local properties of the dispersion function. Finally, in order to investigate whether
or not these anomalous modes might be realizable at a finite time after the driver is
switched on, we evaluate the double Fourier inversion integrals for the unsteady flow
numerically. We find that the anomalous mode is indeed present at finite time, once
initial transients have propagated away, not only for impulsive start-up but also when
the forcing amplitude is allowed to grow slowly from a small value at some initial
instant. This behaviour has significant implications for the application of standard
radiation conditions in wave problems with mean flow.
Wave Forcing of Zonal Mean Angular Momentum in Various Coordinate Systems
