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Dynamical System Analyses of Aperiodic Flow-Induced Oscillations of a Collapsible Tube
C. Bertram
To cite this version:
C. Bertram. Dynamical System Analyses of Aperiodic Flow-Induced Oscillations of a Collapsible Tube. Journal de Physique III, EDP Sciences, 1995, 5 (12), pp.2101-2116. �10.1051/jp3:1995248�.
�jpa-00249440�
Classification
Phj.sics Abstracts
87.22As 87.45Hw 87.45-k
Dynamical System Analyses of Aperiodic Flow-Induced
Oscillations of a Collapsible Tube(*)
C-D- Bertram
Graduate School of Biomedical Engineering, University of New South Wales, Sydney,
2052 Australia
(Received .3 April 1995, revised 27 July1995, accepted 28 August 1995)
Abstract, The paper briefly reviews dynamical systems analysis concepts and techniques
which were applied to gain knowledge of a particular biomechanical problem, that of self-excited oscillation of flow through the collapsed flexible tubes found in many different parts of the body.
The tubes investigated had
a relatively thick wall and the through-flow was turbulent. The methods deployed did not suffice to indicate unequivocally the presence of a low-dimensional chaotic attractor amid this high-dimensional or random influence. Chaos has recently been
demonstrated in the
same system when upstream forcing interacts with otherwise periodic os-
cillations.
1. Introduction
This paper reviews efforts to demonstrate the presence of a chaotic attractor underlying exper-
imental observations of aperiodic oscillations from a collapsed tube through which fluid flows, and illustrates the various analytical approaches from the field of nonlinear dynamics which my associates and I have deployed. An outline will be given of each of the techniques deployed, and an indication of the extent to which each has been successful. Where possible reference is made to previously published papers in which further detail can be found.
The collapsible-tube system has obvious importance in the human body, and its self-excited oscillation has been implicated in lung wheezing, and in Korotkov sounds. With geometrical modifications, it is also a model for the lips and mouthpiece of a brass musical instrument
player, or for the vocal cords, which would themselves more aptly be named vocal lips in view of their anatomic shape. In concentrating on the aperiodic oscillations of such systems one
seeks understanding of the dynamics and thereby of the physics of the oscillations.
In ways which will be detailed below, the collapsible-tube oscillator was far from being an
ideal experimental system for dynamical systems analysis. Various factors such as tube aging
(*) Presented in part at the 10~~~ CongrAs Frangais de Mdcanique, Paris Campus Jussieu, 2.6
September 1991.
are impossible to control, and turbulence added a randomizing element. In consequence the
experiments and analysis were viewed as being to some extent a test of the abilities of dynamical systems tools when applied to a real as opposed to a carefully chosen 'toy' problem.
2. Methods and results
2.I. CONTROL SPACE DIAGRAMS. First, the various behaviours of the system observed
experimentally were categorized as functions of the control parameters, namely the upstream pressure pu causing aqueous flow through the silicone rubber tube, and the external pressure p~ forcing it to collapse. The relation between the tube transmural pressure and the lumen
cross-sectional area is highly nonlinear, as shown in Figure I, and includes a region between first buckling and first opposite-wall contact where very little pressure change can cause major changes in area, and thus tube configuration. Strong coupling between the fluid dynamics and the flow boundary position therefore occurs.
Oscillatory modes were plotted as different regions on control-space diagrams, of which an example is shown in Figure 2. In fact the chosen axes are not precisely those which would be expected on a strict control-space diagram, where the control parameters pu and pe would be used. Such a diagram would have disadvantages here. First, the whole region of dynamical
interest would be only a diagonal strip, because the oscillatory behaviours of interest occur at values of pu and pe that ascend roughly in parallel. This could be solved by using a linear
transformation, but instead the dependent variable jJ~~ = pe jJ~ was preferred, where p2(t) is the pressure at th~ exit of the tube. This has the qualitative advantage that it allows forbidden
zones of diveig~iit instability to appear on the diagram. These correspond to values of jl~~
which cannot be stably attained by p~-variation; each such zone corresponds to a transient
change in the mode of tube behaviour such that fl~ varies discontinuously (p2(t) itself is, of course, continuous). The mode change may involve a transition to a different waveform, or the transition from open tube to collapsed, changing the resistance to flow and thereby fl~.
Such control-space diagrams were constructed for three different values of resistance to flow downstream of the tube and four different values of tube length iii. As exemplified in Figure 2, there were several different oscillatory modes, differentiated into sharply distinct regions of low-, intermediate- and high-frequency oscillation, and in the case of low-frequency oscillations, into two modes of differing waveform shape. In all cases, the waveform was a large-amplitude, highly nonlinear limit-cycle, or relaxation oscillation, such as is shown in Figure 3. However
on the boundaries of these regions were recorded aperiodic oscillations.
Currently, about the most advanced analytical model of flow through a collapsed tube is the one-dimensional theory of Jensen [2]. His model predicts the existence of multiple oscillatory regions, with frequency of oscillation increasing discontinuously at region boundaries in the direction of increasing p~. The model also predicts the behaviour to be expected at interfaces between regions: both hysteresis, where behaviour depends on history, and quasi-periodicity
are seen numerically. Hysteresis with respect to behaviour in a region has been observed [3];
such regions display either the oscillation characterising the region above (higher pe) or that below, depending on the route of approach. Apparently quasi-periodic oscillation has also been observed on boundaries between regions of periodic oscillation; an example is shown in
Figure 4. The example shown showed no sign of mode-locking at an integer ratio of high- to
low-frequency cycles; the frequency ratio is approximately 4.44.
2.2. ANALYSIS OF APERIODIC OSCILLATIONS. Bertram et al. [I] analysed the most in-
teresting of these aperiodic oscillations by means of phase plane plots, Poincar4 sections, and return maps, as well as by Fourier spectral analysis. The processes of plotting two-dimensional
Tube constitutive relation
Transmural ~~
Pressure (kPa) PK
area
12
L/Di «17.4 R2 -'8'
+
+
H§ +
~ c nf
f ~ ~ ~ ~ ~
g
+ + + + ~
ll.2 11.7
~ ~
ii,s iji
4,44 4.70
+ + + ° + + +
~0 100
200 Pu (kPal
Fig. 2. The different modes of oscillation are indicated by regions on a 'control-space' diagram.
Operating points investigated are marked with the symbol (+) and when appropriate the frequency of self-excited oscillation in Hz. The regions are identified by the notations (o): steady flow, open tube, (c): steady flow, collapsed tube, (nf): noise-like fluctuations of pressure, etc. at high frequency and small amplitude relating to tube wall flutter, UN: unattainable zone as described in the text, LU and LD: low-frequency oscillations of two different waveshapes see ref. [3j, 1: intermediate frequency and H: high frequency. The diagram describes explicitly what happens as p~ and thereby p~~ are varied, I-e-
movement vertically; the experiments did not cover horizontal movement (variation of pu at constant
Pe2).
0.80
~f 0.60
,
3 0.40
(
~ ~'~~
0 0.3 0.6 0.9 1.2 1-S
TI34E (seconds)
Fig. 3. An example of the large-amplitude, highly nonlinear repetitive waveforms observed during limit-cycle oscillation. The signal transduced here is the tube internal cross-sectional area (normalised
as in Fig. I) at the most vigorously oscillating point, always located at the downstream end where the transmural pressure is most negative. The tube typically collapses during much less than half the cycle.
i'o«
loco
~ >900
d
~ >coo
nor
1600
TIME(seconds)
Fig. 4. Quasi-periodic oscillation has been observed experimentally. To the limit of resolution of the data, this example consists predominantly of two superimposed single-frequency oscillations with
a
frequency ratio of about 4.44. The transduced variable is again area as in Figure 3, but here expressed
in arbitrary units. There are also signs of
a secondary tube wall flutter of smaller amplitude at the
moments of maximum collapse.
aperiodicity, and resist analysis. The other difficulty is that the data are inevitably corrupted by noise, from measurement and other sources. This both obscures the interpretation of spectra and prevents the demonstration of fractal behaviour. However, the Poincar6 sections and return
maps did demonstrate unambiguously the existence of dynamics with toroidal topology.
The time-series appearance of such signals is what in radio engineering would,be called
amplitude modulation of one pure (sinusoidal) signal by another, and such signals have been recorded from collapsible tubes also by Yahia [7]. Experimentally the components were not pure sinusoids, because the oscillation was nonlinearly limited, and there was simultaneous
modulation of mean (d.c.) signal level. The amplitude-modulation components were sub- sequently separated by homomorphic filtering [8], and short-segment autoregressive spectral analysis was used to show that there was also simultaneous frequency modulation of the higher
of the two frequencies.
The problem of noise has led to the use of various tools which will 'look past the noise' in ways more sophisticated than a low-pass filter. In the collapsible-tube system, this problem
is especially severe, and in a sense fundamental. The experiments were run in silicone rubber tubes with considerable wall thickness (0.37 of inside radius), in order that the predominant
elastic restoring force be circumferential tube wall bending, instead of the longitudinal tension which dominated experiments elsewhere. A consequence of this and the need to use tubes of sufficient internal diameter (13 mm) to allow the use of internal measurement probes was that
self-excited oscillations did not occur until flow-rates well into the turbulent regime (Reynolds
number 5000 or more for the values of downstream resistance investigated). Turbulent flow in- troduced a source of random noise that was intrinsic, rather than extrinsic as in a measurement system. Furthermore it opened the possibility (which could not ultimately be excluded, despite
the techniques deployed) that manifestly aperiodic behaviour at certain operating points might
be the consequence of sensitive amplification of this random source under particular conditions.
2.3. SINGULAR VALUE DECOMPOSITION. One tool much favoured for noise reduction in
dynamical systems analysis is singular value decomposition. As implemented here, one time-
varying signal was used, namely p2(t), although records were also taken of pressure at the entrance, entrance and exit flow-rate, and lumen cross-sectional area at the site of greatest oscillation, near the downstream end. The exit pressure data points were used in groups of m points, for instance 1,2,3, 2,3,4, 3,4,5, 4,5,6 etc. if m
= 3. All N recorded points, where N
might be in the thousands or tens of thousands, were used in the construction of an m x m covariance matrix, as outlined by Broomhead and King [9]. Usually m was bet~A~een 3 and 50.
The matrix was diagonalised, and its eigenvalues used to determine the matrix rank. r. This indicates the maximum number of principal axes that can be discerned in the data above the noise floor. A total of r new time series was then formed by projecting the data onto these
principal axes. Since the method uses time delays to form the data vectors, each one of these
principal directions was associated with a particular frequency component in the data. Thus if, say, the first two series are subsequently used to reconstruct a two-dimensional projection, the data are also in effect subjected to an adaptive low-pass filtering. Subsequent analysis was
done on such reconstructions of the data iii.
For instance, the dimension of the system can be investigated by measuring the rate at which
new points are taken into an r-dimensional sphere as its radius increases. Care must be taken:
the technique is valid only in a small region of the attractor, and can therefore only be applied where there is sufficient data density [10]. The collapsible-tube experiments were difficult to conduct in conditions of complete stationarity (more details below), and consequently some of the most interesting time-series observed were available in shorter data segments than desirable
for this purpose. The results were accordingly ambiguous: local dimension ranged from 2,
implying a ribbon-like structure, to possibly 5 iii.
2.4. PERIODIC POINTS. In a further application of the singular value decomposition, using
the first three principal axes as input, two-dimensional PoincarA sections were taken as shown in Figure 5. The section was then searched for evidence of periodic points, that is to say, points
which next intersect the section close to their original position, one or more orbits previously.
Such points are evidence of unstable periodic orbits, which should occur in any chaotic attractor as an organising principle ill]. The closer such an orbit is approached, through chance hazard of the particular data window recorded, the longer the flow would be expected to spend in its vicinity. Searches for such unstable periodic orbits are a preoccupation of many dynamical
systems laboratories today, as they offer one of the most promising pathways to analysis of a
dynamical system of unknown dimension [12].
There are various computational ways to search for near returns, but one of the most ap-
pealing visually is shown in Figure 6. A small group of points on the section was marked off for investigation (Fig. 6a), choosing usually a region which appeared dense to the eye, because
that clearly could arise through the proximity of an unstable periodic orbit. Then the next intersection with the section plane of each of the orbits in question was plotted (Fig. 6b).
Equally one can plot the previous intersection of these orbits with the section plane (Fig. 6c).
But even if at first sight the results are not very promising, in that the tight group of points is
becoming scattered as well as relocated, one should persist at least as far as the next-but-one
return or the last-but-one return, because the periodic orbit sought may be complex, involving
more than one passage (in the same direction) through the section plane before returning to its starting point as defined by the choice of the group of points.
The aperiodic oscillation subjected to this analysis was one which appeared structurally stable, and of which
a long period of recording was available. Figure 7 shows an extract of
p2(t) for such an operating point. Plotted as a phase-plane portrait of p2(t) vs. p2(t + T)
with an appropriately chosen (e.g. first autocorrelation zero) delay T, the data eventually fill the plane completely, and no structure is visible. However, a lesser number of points displays
some signs of orderly, if ragged, structure (Fig. 8a). A useful tool here is real-time three- dimensional reconstruction with arbitrary rotation on the computer screen; there is still no
algorithmic substitute for the human eye in pattern recognition tasks. Such structure, once
recognised, can be clarified using software to sort through the dataset for pairs of points which
ld2 (104 points) s-v-1 « 25
x
-4 3 2 0 1 2
X ci ~)
'-' -"'
" '
i
' ,'
,
.
' '
'
. .' ~
f. -
"o~' ,
ajj,'~ ) 'i""'
'it§~'i/Tjl,')[f~i 'tS6]141"~'i
~' ''+iiliiiif""'i
"fi", ,'
""" '
b)
Fig. 5. a) Projecting the original time-series onto the first three principal axes, one effects a three- dimensional SVD reconstruction, of which a two-dimensional projection is seen here. The almost horizontal line defined by the segments outside the axes shows where an optimally chosen planar
Poincard section will cut the plane of the projection. b) The Poincard section plane, showing where the orbital strands pass through. The original uses colour to distinguish between passages up through
and down through.
occupy small neighbourhoods, as explained by the sketch in Figure 8b. The results of such a
scan are presented as a histogram in Figure 8c, showing on a logarithmic scale the number of
occurrences of an orbit of given period (in terms of number of points, n) which satisfies the
return criterion. The largest peak is for the trivial n
= 2. Peaks at high n are likely to be the result of multiple orbitting and are not significant. The important ones are the numbered
a) b)
c)
Fig. 6. A graphical procedure for testing for possibly periodic points, involving looking for evidence
of return to the same local neighbourhood in the attractor on contiguous orbits. a) A small group of
points is chosen in a dense region from the section of Figure 5b. b) The position of the next intersection
point (in the same sense) of each of the orbits in question with the section plane. c) The positions of the intersection points preceding the grouping of Figure 6a.
peaks at n = lo, 20/22, and 43. The orbital shape corresponding to each of these peaks is distinct, as can be seen in Figures 8d-f, even though the process does not perfectly separate the square perimeter march of the n = 20 orbits from the period-2 orbit of n
= 43, which avoids the upper left corner. Enough examples of each of these three orbital types is found in a dataset of limited length to make clear that the categorisation is a useful measure of the
underlying dynamical order.
Related computational techniques with less visual appeal involve simply defining an r-dimensional radius for a small neighbourhood around each point in turn and looking for returns that fall within that neighbourhood. This sort of search can be extended to periodic
orbits involving arbitrarily high numbers of passages through the section plane ill]. With an appropriate and robust algorithm, this approach can reliably pick out similar orbits in the
singular-value-decomposition reconstruction, and these in turn correspond to similar waveform shapes in the original data, as shown in Figure 9. Note that since this is a geometric analysis
