Fluid loading with mean flow. I. Response of an elastic plate localized excitation

Abstract

The response to localized forcing of a fluid-loaded elastic plate is studied in the case when there is uniform incompressible flow over the plate. Absolute instability of the fluid-plate system is found when the dimensionless mean velocity U exceeds a threshold U c which is found exactly. For U < U c the system is convectively unstable for 0 < ω <ω s ( U ), neutrally stable, with anomalous features, for ω s ( U ) < ω < ω p ( U ) and stable, with conventional features, for ω > ω p (U), ω being the excitation frequency: here asymptotic expressions are found for the frequencies and for the wavenumbers and amplitudes of the waves found upstream and downstream of the excitation. A significant feature is that Re A 0 < 0 throughout 0 < ω < ω p , A 0 being the drive admittance (velocity at the point of application of the force); this means that throughout the convectively unstable and the anomalous neutral frequency ranges, the exciting force must absorb energy. An exact energy equation is derived, and shown to require the introduction of a new fluid-plate interaction flux UnO t , where O is the fluid potential and n the plate deflexion. The energy equation is used to illuminate properties of the convectively unstable and neutral waves, to verify the property Re A 0 < 0 and to trace the waves responsible for this. Anomalous features in the frequency range ω s (U) < ω < ω p ( U ) are investigated further from the viewpoint of the theory of negative energy waves, and it is found that not only can some wave modes in this frequency range have negative energy, but also group velocity in an inward direction (towards the excitation). It is argued that this does not contradict the outward group velocity ‘radiation condition’ of M. J. Lighthill, because that condition refers expressly to circumstances in which the excitation is the sole source of all the wave energy, whereas here the excitation acts also as a scatterer, transferring energy from the mean flow to the wave field.