Oscillations in Collapsible Tubes

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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; Heil & Hazel 2011).

Experimental studies of flow in elastic-walled 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).

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.

My Research

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 kinetic energy 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.

Prescribed Oscillations

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.

A single time-frame from a simulation of flow in a tube with prescribed wall oscillations. Upper image: full flow. Lower image: magnified perturbation to steady 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 (Whittaker et al. 2010a). The theoretical predictions are found to be in good agreement with numerical simulations (Whittaker et al. 2010b).

Graphs showing good agreement between the theoretical predictions and numerical simulations. Left: Energy flux E transmitted to the wall against strength of mean flow St-1. Right: Critical mean flow Stc-1 against tube length l.

Wall Mechanics

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 (Whittaker et al. 2010b).

Graph showing good agreement between the predictions from the theoretical tube law (lines) and the results of numerical simulations (points) for an initially uniform elliptical tube. The graph shows the relative change in cross-sectional area, α = (A-A0)/A0, against dimensionless axial position z, for two different dimensionless axial tensions F. The different point styles correspond to different ways in which the flexible tube is attached to the rigid supports at the ends.

Self-Excited Oscillations

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 (Whittaker et al. 2010d; Whittaker et al.2011).

Graphs showing good agreement between the theoretical predictions and numerical simulations. We plot (a) the dimensionless oscillation frequency ω, and (b) the dimensionless growth rate Λ, as functions of the mean flow Reynolds number Re. Results for three different dimensionless tube lengths l=10 (squares), l=15 (circles), and l=20 (triangles).

Wall Inertia

Ongoing work by my PhD student is examining the effect of inertia in the tube wall on the stability and growth of self-excited oscillations. Since the wall inertia is associated with the weaker cross-sectional flow, rather than the primary axial sloshing flow, the additional terms from wall inertia appear in a different place in the equations from the terms describing the fluid inertia.

Tube-end boundary layers in the wall mechanics

The tube law described above is only second-order in the axial coordinate z, and hence solutions are incapable of satisfying the full set of 'clamped' boundary conditions that should be applied where an elastic tube is joined to a rigid section. Higher axial derivatives were neglected during the derivation as they were asymptotically small on the axial length-scale of the whole system. However, we would expect these terms to re-enter on shorter axial length-scales, and so give rise to boundary layers close to the ends of the elastic tube. Work by myself and my PhD Student Martin Walters has revealed a rich variety of such boundary layers, involving in-plane shearing, axial bending, and transverse shearing. Which boundary layers are needed, and their thicknesses depend on the key ratios of the wall thickness to tube diameter δ, and the axial tension to bending stiffness F.

The initial tube-law did not even allow the canonical 'pinned' boundary conditions to be satisfied at the tube ends. This was found to cause discrepancies between the model and numerical simulations when δ2F was small. The discrepancies were found to be caused by the neglect of in-plane shear forces, rather than axial bending. A shear-relaxation boundary layer was found and solved asymptotically to allow the pinned boundary conditions to be satisfied. (Whittaker 2015).

Work by Martin Walters has examined the boundary layers required to allow the full set of clamped conditions to be satisfied at the tube ends. Various regimes have been discovered with either axial bending or normal-shear effects included.

Non-linear effect of base-state axial curvature

The work above has used a tube law to describe the elastic response of the tube wall, which has been linearised for small-amplitude deformations about an axially uniform initial state. Even if the oscillations take place about a slightly non-uniform mean state, the linearisation means that certain effects are neglected. I am currently working with a postdoctoral research assistant, Dr Thomas Ward, on an EPSRC-funded project to consider the inclusion of these effects.

Specifically, we are investigating what happens in the case of O(ε) amplitude oscillations about an O(δ) steady deformation, in the case where ε ≪ δ ≪ 1. Here additional terms appear in the tube law due to axial stretching arising from the curvature of the mean state.

For a 2D channel, the extra term in the tube law is a dimensionless parameter K, times an integral I times the dimensionless axial curvature perturbation ηzz. The integral I is the product of the dimensionless base-state axial curvature and the dimensionless normal displacements, which represents the increase in axial tension due changes in axial extension. The dimensionless parameter K gives the size of this non-linear effect relative to the other restoring forces in the tube law, and involves the mean-deformation amplitude δ, the extensional-to-bending-stiffness ratio K/D=h2/12 of the wall, and the aspect ratio =L/a of the channel.

For sufficiently large K, the effect of the additional term in the tube law is to force the oscillatory mode to alter its shape so that I=O(K-1). For larger K, this forces the oscillatory mode to be almost orthogonal to the base-state curvature. In the case where the base-state curvature has an axial mode-1 shape, this forces the primary (lowest frequency) oscillatory mode towards an axial mode-2 shape, and the secondary oscillatory mode towards an axial mode-3 shape. This in turn affects the stability properties of the modes, and we have found a region of parameter space in which the secondary oscillatory mode is more unstable than the primary oscillatory mode.

Peer Reviewed Journal Articles

The Energetics of Flow through a Rapidly Oscillating Tube. Part 1. General Theory
Robert J. Whittaker, Sarah L. Waters, Oliver E. Jensen, Jonathan Boyle & Matthias Heil, 2010.
Journal of Fluid Mechanics 648, 83–121.
The Energetics of Flow through a Rapidly Oscillating Tube. Part 2. Application to an Elliptical Tube
Robert J. Whittaker, Matthias Heil, Jonathan Boyle, Oliver E. Jensen & Sarah L. Waters, 2010.
Journal of Fluid Mechanics 648, 123–153.
A Rational Derivation of a Tube Law from Shell Theory
Robert J. Whittaker, Matthias Heil, Oliver E. Jensen & Sarah L. Waters, 2010.
Quarterly Journal of Mechanics and Applied Mathematics 63 (4), 465–496.
Predicting the Onset of High-Frequency Self-Excited Oscillations in Elastic-Walled Tubes
Robert J. Whittaker, Matthias Heil, Oliver E. Jensen & Sarah L. Waters, 2010.
Proceedings of the Royal Society A 466 (2124), 3635–3657.
The Energetics of Flow through a Rapidly Oscillating Tube with Slowly Varying Amplitude
Robert J. Whittaker, Matthias Heil & Sarah L. Waters, 2011.
Phil. Trans. Roy. Soc. A 369 (1947), 2989–3006.
A Shear-Induced Boundary Layer near the Pinned Ends of a Buckled Elastic-Walled Tube
Robert J. Whittaker, 2015.
IMA Journal of Applied Mathematics (advanced online access).

Conference Posters

Fluid–Structure Interaction: Shell Theory and 3D Flow Instability in Collapsible Tubes
Robert J. Whittaker, Sarah L. Waters, Oliver E. Jensen & Matthias Heil, 2007.
Conference Poster for the Fourth Physiological Flow Network Meeting, University of Manchester.
High-Frequency Self-Excited Oscillations in Collapsible Tube Flows
Robert J. Whittaker, Sarah L. Waters, Oliver E. Jensen, Jonathan Boyle & Matthias Heil, 2009.
Poster at the Cardiac Physiome Project Meeting, Isaac Newton Institute, University of Cambridge.

Articles in Preparation and Under Review

The Effect of Wall Inertia on High-Frequency Self-Excited Oscillations in an Elastic-Walled Tube
Martin C. Walters & Robert J. Whittaker
In preparation.
Bending boundary layers at the Clamped Ends of a Buckled Elastic-Walled Tube
Martin C. Walters & Robert J. Whittaker
In preparation.
The Non-Linear Effect of Base-State Curvature on Self-Excited Oscillations in an Elastic-Walled Channel
Thomas Ward & Robert J. Whittaker
In preparation.