dc.description.abstract | 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. | |