Stability of fluid-loaded structures
It is known theoretically that infinitely long fluid loaded plates in mean flow exhibit a range of unusual phenomena in the 'long time' limit. These include convective instability, absolute instability and negative energy waves which are destabilized by dissipation. However, structures are necessarily of finite length and may have discontinuities. We have undertaken an analytical and computational study to investigate the response of finite plates, and of plates with local inhomogeneities, to ascertain if these unusual effects might be realized in practice. Analytically, we adopt Crighton & Oswell’s (1991) structural acoustics approach and take a "wave scattering" --as opposed to a "modal superposition"-- view of the fluttering plate problem. First, we derive the energy balance relations for the extended plate (i.e., plate with two sided flow, spring foundation and plate pretension) and define a generalized wave impedance valid for both positive energy waves (PEW) and negative energy waves (NEW). Next, we solve for the scattering coefficients of localized plate discontinuities using a multipole source approach. Our solutions are exact and include the nearfields due to fluid-loading effects. We introduce the concept of power normalized scattering coefficients, and show that overall power is conserved during the scattering process if the sign of the wave energy is preserved. We argue that energy conservation, combined with the presence of NEWs on the plate, are responsible for the phenomenon of over-scattering, or of amplified reflection/transmission. These are scattering processes that draw energy from the mean flow into the plate. Next, we use the Wiener-Hopf technique to solve for the scattering coefficients of a variety of plate leading and trailing edge conditions -- including the flag like configuration of a free trailing edge with wake. We find that the edges are over-reflective in the frequency range where NEWs are present. The exception is a free trailing edge with wake where, remarkably, the wake is found to absorb almost all of the incident wave energy. We use combinations of these upstream and downstream edge reflection matrices to solve for the complex resonance frequencies of long, finite plates immersed in mean flow. Finally, we construct the response of a finite plate by a superposition of infinite plate propagating waves continuously scattering off the plate ends. We solve for the unstable resonance frequencies and temporal growth rates for long plates. We derive upper and lower bounds on the unstable growth rates of finite plates with given edge conditions. We find that a flag-like configuration of a clamped leading edge and a free trailing edge with wake is destabilized for sub-critical flow speeds only for very long plate lengths and only in the presence of convectively unstable waves. We present a comparison between direct computational results and the infinite plate theory. In particular, the resonance response of a moderately sized plate is shown to be in excellent agreement with the long plate analytical predictions.