T. J. Pedley

A separated-flow model for collapsible-tube oscillations

A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of ( a ) longitudinal wall tension and ( b ) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.

A mathematical model of unsteady collapsible tube behaviour

A simple, third-order lumped parameter model is presented to describe unsteady flow in a short segment of collapsible tube held between two rigid segments and contained in a pressurised chamber. Equilibrium states and their stability are analysed in detail, as is fully non-linear time dependent behaviour, including in particular the excitation and sustenance of limit--cycle oscillations. The model explicitly neglects both wave propagation (and hence the possibility of choking) and the influence on the elastic properties of the tube of longitudinal tension, but it is otherwise firmly based on fluid mechanical principles. The results emphasise the profound importance of (a) the unsteady head loss (but with some pressure recovery) in the separated flow at the oscillating throat, and (b) the mechanical properties of the parts of the system both downstream and upstream of the collapsible segment. The nature of the upstream segment in particular determines whether it is an upstream pressure head or the inflow to the collapsible segment that is held constant during oscillations. The results are discussed in the context both of other models and of experiment.

Mapping of instabilities for flow through collapsed tubes of differing length

Aqueous flow through thick-walled silicone rubber tubes held open at both ends and externally pressurized is investigated for tubes of four different lengths, each at three levels of downstream flow resistance. The tubes are compared at operating points spanning all the observed types of dynamic behaviour, where an operating point is set by adjusting driving pressure head and external pressure. It is found that longer tubes display relatively more oscillatory operating points, while shorter tubes display more divergently unstable operating points. The observed self-excited oscillations can be divided into well-separated bands of low, intermediate and high frequency, within each of which the frequency generally increases gradually with flow-rate and external pressure. In addition, in the region at high external pressure where turbulent noise dominates, isolated operating points display very-high-frequency repetitive oscillations of small amplitude. The border between the noise-dominated region and the region below, independently of whether the latter is oscillatory or divergent, displays complex behaviour. This includes aberrantly high frequency oscillation which is sometimes superimposed on a particular phase of a low-frequency oscillation, and the behaviour depends on whether the external pressure has previously been higher or lower. Whereas the regions of low, intermediate and high frequency oscillations are arranged such that in general higher flow-rates and external pressures cause transitions to higher-frequency bands, these aberrant `border oscillations yield very high frequencies at low flow-rate and external pressure. The minimum frequency of oscillation decreases in longer tubes, but the dependence is far weaker than if end-to-end wave propagation were the period-setting mechanism. Longer tubes appear predisposed to more widespread low-frequency modes, although high frequencies can be excited with sufficient flow-rate and external pressure. Few low-frequency operating points are found with short tubes. As downstream resistance is decreased, steady flow gives way to divergent operating points which in turn become oscillatory. Possible mechanisms for all these behaviours are discussed.

Application of nonlinear dynamics concepts to the analysis of self-excited oscillations of a collapsible tube conveying a fluid

Several different types of oscillation were observed during flow through a thick-walled silicone rubber tube when the external pressure was large enough to cause collapse. The Reynolds number was above 4,400. With upstream head, and transmural pressure at the downstream end of the tube, as control variables, control-space diagrams exhibited well-defined regions of low (2–6 Hz), intermediate, and high frequency (over 60 Hz) oscillation, and of small noise-like fluctuations. The data, including aperiodic oscillatory operating points which may indicate the presence of chaos, are analyzed by dynamical systems methods. Transitions between different regions of control space are discussed in terms of topological bifurcation types. Spectral analysis is used to distinguish between quasi-periodic and aperiodic waveforms. Although the dimension of the dynamical system is unknown, phase planes are plotted, both as one transduced signal versus another and as one versus itself delayed. Return maps and Poincare sections are plotted, the latter using three-dimensional phase portraits in which the third coordinate axis was produced by further delay of the one signal. Coordinates for higher-dimensional phase portraits are also defined, using the eigenvectors of covariance matrices constructed from sequences of the recorded data points for one signal. Poincare sections are plotted for such three-dimensional portraits, using the lowest-frequency-component coordinates. Singular value decomposition of “local neighbourhood” matrices is used to define the local dimension of the system in a small region of the high-dimension phase space. Despite the use of these sophisticated techniques, one cannot unequivocally conclude from these data sets that the system is chaotic. The applicability of such methods to complex experiments that yield data which are nonoptimal for these purposes are discussed.

The effect of secondary motion on axial transport in oscillatory tube flow

In oscillatory flows through systems of branched or curved tubes, Taylor dispersion is modified both by the oscillation and by the induced secondary motions. As a model for this process, We examine axial transport in an annular region containing an oscillatory axial and steady secondary (circumferential) flow. Two complementary approaches are used: an asymptotic analysis for an annulus with a narrow gap (δ) and for large values of the secondary flow Peclet number ( P ); and a numerical solution for arbitrary values of δ and P . The results exhibit a form of resonance when the secondary-flow time equals the oscillation period, giving rise to a prominent maximum in the transport rate. This observation is consistent with preliminary numerical results for oscillatory flow in a curved tube, and can be explained physically.

The cascade structure of linear instability in collapsible channel flows

This paper studies the unsteady behaviour and linear stability of the flow in a collapsible channel using a fluid–beam model. The solid mechanics is analysed in a plane strain configuration, in which the principal stretch is defined with a zero initial strain. Two approaches are employed: unsteady numerical simulations solving the nonlinear fully coupled fluid–structure interaction problem; and the corresponding linearized eigenvalue approach solving the Orr–Sommerfeld equations modified by the beam. The two approaches give good agreement with each other in predicting the frequencies and growth rates of the perturbation modes, close to the neutral curves. For a given Reynolds number in the range of 200–600, a cascade of instabilities is discovered as the wall stiffness (or effective tension) is reduced. Under small perturbation to steady solutions for the same Reynolds number, the system loses stability by passing through a succession of unstable zones, with mode number increasing as the wall stiffness is decreased. It is found that this cascade structure can, in principle, be extended to many modes, depending on the parameters. A puzzling ‘tongue’ shaped stable zone in the wall stiffness– Re space turns out to be the zone sandwiched by the mode-2 and mode-3 instabilities. Self-excited oscillations dominated by modes 2–4 are found near their corresponding neutral curves. These modes can also interact and form period-doubling oscillations. Extensive comparisons of the results with existing analytical models are made, and a physical explanation for the cascade structure is proposed.

Steady and unsteady separation in an approximately two-dimensional indented channel

Experiments are performed on steady and impulsively started flow in an approximately two-dimensional closed channel, with one wall locally indented. In plan the indentation is a long trapezium which halves the channel width: the inclination of the sloping walls is approximately 5.7°, and these tapered sections merge smoothly into the narrowest section via rounded corners. The Reynolds number

The fluid mechanics of large blood vessels

Re = a_0overline{u}_0/nu

Flow in Collapsible Tubes: A Brief Review

( a 0 = unindented channel width,

Modelling Flow and Oscillations in Collapsible Tubes

overline{u}_0