Fluid-loaded elastic plate with mean flow: point forcing and three-dimensional effects

Abstract

The unsteady behaviour of an infinitely long fluid-loaded elastic plate subject to a single-frequency line forcing in the presence of a uniform mean flow is known to exhibit a number of interesting phenomena. These include the onset of absolute instability for non-dimensional flow speeds U in excess of some critical speed Uc, and various interesting propagation effects when U < Uc. In the latter respect Crighton & Oswell (1991) have shown that over a particular frequency range there exists an anomalous neutral mode with group velocity directed towards the driver, in violation of the usual Lighthill outgoing radiation condition. Similar results have been found by Peake (1997) when transverse curvature effects are included. In this paper we seek to extend these results and consider the substantially harder problem of a fluid-loaded elastic plate with uniform mean flow which is subject to a point forcing, thereby resulting in a two-dimensional structural problem. A systematic method for determining the absolute instability threshold is developed, and it is shown that the flow is absolutely unstable for flow speeds U > Uc, where Uc is the one-dimensional value found by Crighton & Oswell. At flow speeds U < Uc the flow is marginally stable and convective growth is found to occur downstream of the driver, over a particular frequency range depending on the transverse Fourier wavenumber ky, within a wedge-shaped region. Outside this wedge-shaped region there is only neutral mode behaviour. Asymptotic forms are found for the dominant large-distance causal flexion response downstream of the driver inside and outside the wedge region, and the appropriate critical angle for the wedge region is identified. Within the convective instability wedge the flexion and critical angle take two different forms depending on whether the frequency ω is greater or less than U2/√5. In addition to this interesting behaviour, the flow also exhibits the usual anomalous neutral mode behaviour and, as with Peake's problem, we also find an extra stability (hoop) resulting in neutral mode behaviour over a small frequency range. Asymptotic forms are also found for the threshold frequencies which divide up the various regions of stability of the system (neutral, neutral anomalous, convectively unstable), as a function of ky, and are compared with the results of both Crighton & Oswell and Peake.