Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes

Abstract

We present a theoretical description of flow-induced self-excited oscillations in the Starling resistor-a pre-stretched thin-walled elastic tube that is mounted on two rigid tubes and enclosed in a pressure chamber. Assuming that the flow through the elastic tube is driven by imposing the flow rate at the downstream end, we study the development of small-amplitude long-wavelength high-frequency oscillations, combining the results of two previous studies in which we analysed the fluid and solid mechanics of the problem in isolation. We derive a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillations, and determine the slow growth or decay of the normal modes by considering the system's energy budget. We compare the theoretical predictions for the mode shapes, frequencies and growth rates with the results of direct numerical simulations, based on the solution of the three-dimensional Navier-Stokes equations, coupled to the equations of shell theory, and find good agreement between the results. Our results provide the first asymptotic predictions for the onset of self-excited oscillations in three-dimensional collapsible tube flows.

Instabilities in flexible channel flow with large external pressure

“…However, in the present case the dominant source of viscous dissipation arises around the point of greatest constriction of the static membrane, whereas for 'sloshing' modes predicted in using an analogue of our 1D model the dominant dissipation arises over entire channel (Stewart et al 2009). (Of course, in the full 2D and 3D representations of 'sloshing' the dominant dissipation is confined to narrow Stokes layers on the channel walls (Jensen & Heil 2003;Whittaker et al 2010).) In addition, in 'sloshing' the mean flow rate along the channel is increased by the oscillation, whereas here the mean flow must remain constant (due to the choice of upstream boundary condition) and instead the channel walls are driven apart over a period, releasing energy which drives the oscillation.…”

“…The oscillation profile of the primary global instability has a single extremum across the channel length, so is termed mode-1. This asymptotic mechanism can be generalised to three-dimensional flow through a collapsible tube with elliptical cross-section (Whittaker et al 2010;Whittaker 2015).…”

The energetics of flow through a rapidly oscillating tube with slowly varying amplitude

“…This current paper provides key results that are needed to build a model of selfexcited oscillations in a collapsible-tube system. The details and solution of this model are reported elsewhere [14], but, briefly, we combine the fluid mechanics solutions from the current paper with a description of the tube-wall mechanics from Whittaker et al [16]. The two are coupled via the kinematic and dynamic boundary conditions on the tube wall, and we solve the combined system to find the normal modes of oscillation.…”

“…The results derived here are used by Whittaker et al [14] in the manner described above to analyse the full stability problem for self-excited oscillations.…”

“…The frequency is determined by a balance between fluid inertia and the restoring force from the walls [14], and so is undetermined here (as the wall mechanics are not specified). This turns out not to matter, as we can scale all times on the time scale T for the oscillation period, and absorb this scale into the other dimensionless parameters in the system.…”