A simple Starling resistor setup. A flexible section of tube is contained within a pressure chamber, allowing the external pressure to be controlled. Upstream and downstream pressures inside rigid tube extensions control the axial flow rate.
Flow-induced oscillations of fluid-conveying elastic vessels arise in many engineering and biomechanical systems. Examples include pipe flutter, wheezing during forced expiration from the pulmonary airways, and the development of Korotkoff sounds during blood pressure measurement by sphygmomanometry (Heil & Jensen 2003; Grotberg & Jensen 2004).
Experimental studies of flow in collapsible tubes are typically performed with a Starling resistor. A finite-length elastic tube is mounted on two rigid tubes and flow is driven through the system either by imposing the flow rate (using a volumetric pump) or by applying a fixed pressure drop between the far-upstream and far-downstream ends of the two rigid tubes. The collapsible segment is contained inside a pressure chamber which allows the external pressure acting on the elastic tube to be controlled independently of the fluid pressure. If the transmural (external minus internal) pressure becomes sufficiently large, the elastic tube buckles non-axisymmetrically. Once buckled, the tube is very flexible and, as a result, small changes in transmural pressure suffice to induce large changes in the tube shape, resulting in strong fluid-structure interaction. Experiments show that the elastic tube segment has a propensity to develop large-amplitude self-excited oscillations of great complexity when the flow rate is increased beyond a certain value (Bertram 2003).
I am currently studying a particular class of high-frequency instability of flow through a collapsible tube. The basic setup comprises an elastic-walled tube along which a steady axial flow is driven either by a prescribed pressure drop between the ends, or by a flux condition at one end. The aim is then to consider self-excited oscillations of this system involving bother the fluid and the tube walls. The mechanism I am looking at was first proposed by Jensen & Heil (2003) for 2D channel flows, and I have shown that it persists in 3D tubes. (The mechanism involves the extraction of KE from the mean flow via a larger oscillatory velocity perturbation at the upstream end than at the downstream end.)
I have been carrying out theoretical work in collaboration with Sarah Waters at Oxford and Oliver Jensen at Nottingham. Another team in Manchester, lead by Matthias Heil have been carrying out numerical simulations with which we compare our results.
As a first step in analysing the instability, I have used asymptotic analysis to derive leading order solutions for the fluid flow in response to prescribed oscillations of the tube wall. Changes in cross-sectional area result in an axial sloshing flow in addition to the mean flow.
Asymptotic analysis of the fluid problem reveals the energy budget for the system and shows that the instability mechanism seen in 2D channel flows persists in 3D. A general stability criterion is derived in terms of the oscillation mode shape, original cross-section, and the physical parameters of the system. The theoretical predictions are found to be in good agreement with numerical simulations.
The next stage in the work is to develop a simple model to capture the wall mechanics. I have derived a so-called tube law linking the cross-sectional area changes to the transmural pressure for small-amplitude long-wavelength deformations. We start by deriving a rational approximation to a full shell-theory description, which consists of an infinite set of coupled ODEs in an infinite set of axially varying amplitude parameters bn(z). Truncating after the first term yields a simple tube law which is a good approximation to the full solution.
Finally, we combine the solid and fluid mechanics descriptions to consider the full fluid–structure interaction problem. We determine the shapes of the normal modes of oscillation, and the growth or decay rate of each mode as a function of the problem parameters. Our theoretical predictions are in good agreement with numerical simulations conducted using the oomph-lib C++ library.
