On the causal behaviour of flow over an elastic wall

In this paper we consider the causal response of the inviscid shear-layer f low over an elastic surface to excitation by a time-harmonic line force. In the case of uniform flow, Brazier-Smith & Scott (1984) and Crighton & Oswe ll (1991) have analysed the long-time limit of the response. They find that the system is absolutely unstable for sufficiently high flow speeds, and t hat at lower speeds there exist certain anomalous neutral modes with group velocity directed towards the driver (in contradiction of the usual radiati on condition of out-going disturbances). Our aim in this paper is to repeat their analysis for more realistic shear profiles, and in particular to det ermine 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 lo ading, denoted epsilon, becomes small. For a simple broken-line linear shea r profile we find that the results are qualitatively similar to those for u niform flow. However, for the more realistic Blasius profile very significa nt differences arise, essentially due to the presence of the critical layer . In particular, we find that as epsilon --> 0 the minimum flow speed requi red for absolute instability is pushed to considerably higher values than w as found for uniform flow, leading us to conclude that the uniform-flow pro blem is an unattainable singular limit of our more general problem. In cont rast, we find that the uniform-flow anomalous modes (written as exp (ikx - l omega t), say) do persist for non-zero shear over a wide range of epsilon , although now becoming non-neutral. Unlike the case of uniform flow, howev er, the k-loci of these modes can now change direction more than once as th e imaginary part of omega is increased, and we describe the connection betw een this behaviour and local properties of the dispersion function. Finally , in order to investigate whether or not these anomalous modes might be rea lizable at a finite time after the driver is switched on, we evaluate the d ouble Fourier inversion integrals for the unsteady flow numerically. We fin d that the anomalous mode is indeed present at finite time, once initial tr ansients have propagated away, not only for impulsive start-up but also whe n the forcing amplitude is allowed to grow slowly from a small value at som e initial instant. This behaviour has significant implications for the appl ication of standard radiation conditions in wave problems with mean flow.