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

Abstract

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 Poincaré 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. Poincaré 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.