Nonlinear stability of a fluid-loaded elastic plate with mean flow

Abstract

It has been known for some time that the unsteady interaction between a simple elastic plate and a mean flow has a number of interesting features, which include, but are not limited to, the existence of negative-energy waves (NEWs) which are destabilized by the introduction of dashpot dissipation, and convective instabilities associated with the flow–surface interaction. In this paper we consider the nonlinear evolution of these two types of waves in uniform mean flow. It is shown that the NEW can become saturated at weakly nonlinear amplitude. For general parameter values this saturation can be achieved for wavenumber k corresponding to low-frequency oscillations, but in the realistic case in which the coefficient of the nonlinear tension term (in our normalization proportional to the square of the solid–fluid density ratio) is large, saturation is achieved for all k in the NEW range. In both cases the nonlinearities act so as increase the restorative stiffness in the plate, the oscillation frequency of the dashpots driving the NEW instability decreases, and the system approaches a state of static deflection (in agreement with the results of the numerical simulations of Lucey et al. 1997). With regard to the marginal convective instability, we show that the wave-train evolution is described by the defocusing form of the nonlinear Schrödinger (NLS) equation, suggesting (at least for wave trains with compact support) that in the long-time limit the marginal convective instability decays to zero. In contrast, expansion about a range of other points on the neutral curve yields the focusing form of the NLS equation, allowing the existence of isolated soliton solutions, whose amplitude is shown to be potentially significant for realistic parameter values. Moreover, when slow spanwise modulation is included, it turns out that even the marginal convective instability can exhibit solitary-wave behaviour for modulation directions lying outside broad wedges about the flow direction.