On the unsteady motion of a long fluid-loaded elastic plate with mean flow

Abstract

In this paper we consider the dynamic behaviour of a fluid-loaded elastic plate of dimensionless length $2L$, set in a rigid baffle in the presence of uniform mean flow of dimensionless speed $U$, and including plate pre-tension (dimensionless tension $T$) and a spring foundation (dimensionless spring constant $\lambda$). Our aim is to extend previous analytical work of Crighton & Oswell (1991), who examined an infinite plate with $T\,{=}\,\lambda\,{=}\,0$. This is achieved by considering a plate which is long on the scale of typical plate bending waves, and then by expressing the unsteady motion as a superposition of infinite-plate waves which are continuously rescattered in near-field regions at the plate leading and trailing edges. We find that the finite plate possesses resonant solutions which are temporally unstable, both for parameter values for which the infinite plate is convectively unstable ($T$ and/or $\lambda$ sufficiently small) and parameter values for which it is stable ($T$ and/or $\lambda$ sufficiently large). It is shown that instability is present in the absence of structural damping on the finite plate, in agreement with numerical results of Lucey & Carpenter (1992). Neutral resonant states are also found, for which we derive a generalization of the Landahl (1962) and Benjamin (1963) concept of wave energy. Finally, we replace the linear pre-tension $T$ by the nonlinear tension induced by bending, and analyse the nonlinear evolution of the states of negative wave energy in the presence of weak structural damping. We show that the system possesses points of minimum action at non-zero frequency, which act as attractors, predicting the existence of nonlinear fluttering motion.