A hybrid of computational and theoretical methods is extended and used to investigate the instabilities of a flexible surface inserted into one wall of an otherwise rigid channel conveying an inviscid flow. The computational aspects of the modelling combine finite-difference and boundary-element methods for structural and fluid elements respectively. The resulting equations are coupled in state-space form to yield an eigenvalue problem for the fluid-structure system. In tandem, the governing equations are solved to yield an analytical solution applicable to inserts of infinite length as an approximation for modes of deformation that are very much shorter than the overall length of the insert. A comprehensive investigation of different types of inserts-elastic plate, damped flexible plate, tensioned membrane and spring-backed flexible plate-is conducted and the effect of the proximity of the upper channel wall on stability characteristics is quantified. Results show that the presence of the upper-channel wall does not significantly modify the solution morphology that characterises the corresponding open-flow configuration, i.e. in the absence of the rigid upper channel wall. However, decreasing the channel height is shown to have a very significant effect on instability-onset flow speeds and flutter frequencies, both of which are reduced. The channel height above which channel-confinement effects are negligible is shown to be of the order of the wavelength of the critical mode at instability onset. For spring-backed flexible plates the wavelength of the critical mode is much shorter than the insert length and we show very good agreement between the predictions of the analytical and the state-space solutions developed in this paper. The small discrepancies that do exist are shown to be caused by an amplitude modulation of the critical mode on an insert of finite length that is unaccounted for in the travelling-wave assumption of the analytical model. Overall, the key contribution of this paper is the quantification of the stability bounds of a fundamental fluid-structure interaction (FSI) system which has hitherto remained largely unexplored.

The stability of a finite flexible wall occupying part of a rigid wall that separates two inviscid channel flows is investigated. The two-dimensional system is solved using a boundary-element method coupled with a finite-difference method. The motion of the wall is driven by the transmural pressure while the no-flux condition at the wall provides the kinematic boundary condition for each of the flows. Flows and structure are fully coupled to yield a system equation that is then transformed into state-space form so that its eigenvalues can be analysed. The flow velocities at which divergence and modal-coalescence flutter of the flexible wall occur are then determined as are mode shapes. We show that decreasing the channel heights and increasing the fluid density causes instabilities to occur at lower flow velocities. When the channels flow in opposite directions it is possible to suppress modal-coalescence of the first two modes.

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