Stability of a flexible insert in one wall of an inviscid channel flow

Abstract

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.

Asymptotic stability and transient growth in pulsatile Poiseuille flow through a compliant channel

“…We remark that for their figure 7, Davies & Carpenter (1997a) used a tensioned membrane instead of a spring-backed plate and this resulted in a very flexible channel that is more prone to wall-based instabilities. For the spring-backed flexible plate used in our system, an equivalent critical Reynolds number predicted for the onset of two-dimensional divergence instability for steady inviscid fluid flow in a compliant channel (Burke et al 2014) is Re D ≈ 1.25 × 10 4 , having chosen blood for the fluid and characteristic dimension typical of a small blood vessel. This value falls within the range of Reynolds numbers in figure 15(a).…”

An experimental investigation to model wheezing in lungs

“…Theoretical work has gradually increased the complexity of the models under consideration, starting with a two-dimensional analogue of the Starling resistor. Both linear stability [21][22][23][24][25][26][27][28][29][30] and direct numerical simulation [31][32][33][34][35][36] have been considered. In a similar way to the lumped parameter models, the two-dimensional models produce interesting phenomenology, but cannot be used quantitatively.…”

Flow-induced surface instabilities in a flexible-walled channel with a heavy wall

“…Conversely, SD is a class A instability, observed in the experiments of Gad-El-Hak, Blackwelder & Riley (1984), taking the form of a stationary or very slowly propagating mode (Carpenter & Garrad 1986); the mechanism of instability is not clearly understood, but non-zero wall damping is essential for destabilisation in an infinitely long system (Carpenter & Garrad 1986), although the story is more complicated when the compliant panel is of finite length (Lucey & Carpenter 1992, 1993; Peake 2004; Burke et al. 2014). In this study we consider the limit of large wall damping to elucidate the mechanism of SD in a long compliant channel.…”

“…TWF is a class B instability, observed in the experiments of Gaster (1988), which takes the form of a travelling wave propagating with wavespeed close to that of the unloaded elastic wall (Carpenter & Garrad 1986); the mechanism of instability is similar to that of wind generating water waves, sustained by viscous effects within narrow viscous layers across the flow (Benjamin 1959;Carpenter & Gajjar 1990). Conversely, SD is a class A instability, observed in the experiments of Gad-El-Hak, Blackwelder & Riley (1984), taking the form of a stationary or very slowly propagating mode (Carpenter & Garrad 1986); the mechanism of instability is not clearly understood, but non-zero wall damping is essential for destabilisation in an infinitely long system (Carpenter & Garrad 1986), although the story is more complicated when the compliant panel is of finite length (Lucey & Carpenter 1992Peake 2004;Burke et al 2014). In this study we consider the limit of large wall damping to elucidate the mechanism of SD in a long compliant channel.…”