Journal of Fluids and Structures (1990) 4, 125-153

MAPPING OF INSTABILITIES FOR FLOW THROUGH COLLAPSED TUBES OF

DIFFERING LENGTH

C. D. BERTRAM AND C. J. RAYMOND

Centre for Biomedical Engineering, University of New South Wales, Kensington, Australia 2033

AND

T. J. PEDLEY

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, U.K.

(Received 22 January 1988 and in revised form 10 May 1989)

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.

1. I N T R O D U C T I O N

FLUID FLOWS IN THE HUMAN BOOr mostly Occur in elastic tubes which can be deformed significantly by hydrodynamic variations in fluid pressure, and in particular can collapse if the internal pressure falls below the external pressure. While flow and wave propagation in distended tubes is largely well understood (e.g., Lighthill 1975), our understanding of flows in collapsed vessels does not yet permit good comparisons between theory and experiment.

0889-9746/90/020125 + 29 $03.00 ~) 1990 Academic Press Limited

126 C. D. BERTRAM E T AL.

We consider here a collapsible tube which is connected at each end to a rigid circular pipe of similar internal diameter (see Figure 1). The tube is pressurized externally, uniformly over its entire length. At Reynolds numbers in excess of about 100, two qualitatively novel types of behaviour arise which are not observed in distended tubes. The better known is the appearance of self-excited oscillations, as studied by Conrad (1969), who also showed that, at a given parameter combination, all possible operating points lie on a curved surface in three-dimensional variable-space. Bonis (1979) was apparently the first to point out the other novel behaviour, namely that some operating points (some parts of the surface) are not attainable experimen- tally. An operating point is defined by the values of two variables, here chosen to be the flow-rate through and the pressure drop down the collapsed tube; this choice corresponds to that made by previous workers (e.g. Brower & Noordergraaf 1973; Brower & Scholten 1975; Bonis 1979), and arises naturally in theoretical studies (e.g. Jensen & Pedley 1989). The flow-rate depends on the pressure drop and also on the external pressure (relative to internal pressure at a particular point in the tube). Thus, in the three-dimensional space defined by the pressure drop, the transmural (internal minus external) pressure at, say, the downstream end of the collapsible segment, and the flow-rate, only coordinates lying on a unique surface are possible. Other coordinates lie on other surfaces, corresponding to different settings of the adjustable parameters. These parameters include the longitudinal tension, the tube radius, length, wall thickness and elastic modulus, the fluid density and viscosity, and, in the up- and downstream apparatus, the resistance to flow and the fluid inertia.

The purpose of this paper is to show how both types of experimentally observed instability are affected by changes in two of the adjustable parameters. Those chosen were resistance downstream of the collapsed tube, and tube length. For the systematic mapping of the various possible types of behaviour, we present methods which divide the operating point space into well-defined regions, each exhibiting a particular behaviour. It is shown that, in some circumstances, the existence of self-excited oscillations depends on the path by which the operating point is reached. We also point out that high-frequency oscillatory modes sometimes occur at moderate pressure drop and flow-rate, conditions normally conducive only to low-frequency modes.

The mechanism of a given experimentally observed oscillation remains conjectural at this stage. Part of the motivation behind this work was to provide evidence for or against some of the possible mechanisms, by finding out whether the length of the tube controls the oscillation frequency. The experiments reveal only a weak dependence of minimum frequency on tube length, suggesting that end-to-end wave propagation is not central to the mechanism of the low-frequency oscillations. They do not rule out the possibility that such waves govern the higher-frequency oscillations seen as the tube is compressed to greater degrees of collapse.

The results of these experiments demonstrate that the finite collapsible tube represents a dynamical system of great complexity, with a rich structure of bifurcations between steady, oscillatory and possibly chaotic types of behaviour. The mathematical modelling of this system is a substantial challenge for future research.

2. METHODS

The term Starling resistor describes concisely the tube set-up, which is shown in Figure 1. Since these experiments are an extension of work reported previously (Bertram 1986a), only a brief description of the methodology will be given here. The silicone rubber tubes used (Young's modulus 3-8 MPa) have a thick wall [(thickness)/(mid-wall

MAPPING COLLAPSED TUBE INSTABILITIES 127

1 ~ / ~ . Catheter

°2['n-J[ Doubl~.~ ~ Adjustable Pl ~ DomiCile ~ ~ . e s i s t a n c e

¢ Pressure ' ~ "~ / /%. "o transducer a2,t 2 " ~ .

nstant-head Constant-pressure supply eeservoir air supply

Figure 1. Schematic diagram of the apparatus, showing the symbols used as abbreviations for the measured quantities.

radius) = 0-30], which has important consequences. Firstly, the ratio of the elastic restoring force to the viscous force generated by the flow is large, and longitudinal tube motion is therefore very slight. Elastic forces due to tube wall bending are also large in relation to hydrostatic effects. Hence inclination of the tube axis does not cause significant differences in the tube configuration. The tubes are also largely self- supporting; the cross-section at zero transmural pressure is very close to circular, and only the longest of the four tubes required intermediate supports to avoid sag. In terms of inside diameter, the distance between the rigid pipes to which the tubes were clamped at each end was 4.0, 8.2, 17-4 and 34.2 units for the four tubes. Thus each successive tube had approximately twice the length of the previous one. They will be identified henceforth as the 4D tube, the 8D tube, etc.

The quasistatic rheology of the tubing has been described previously (Bertram 1987). All the results described here were obtained with approximately the same longitudinal tension, which corresponded to the axial force provided by a suspended mass of 0.32 kg. The behaviour of the 17D tube has been described (Bertram 1986a), but our techniques for mapping the zones of unattainable operating points have subsequently been refined, allowing the presentation here of much more detail.

A further consequence of using relatively thick-walled tubes is that the range of driving pressure head needed for an adequate exploration of all types of behaviour is greater than can be conveniently provided by an elevated reservoir. A system was therefore used which involves a compressed-air reservoir to which the working fluid (essentially water) is returned by pumps (Bertram 1986b). All pressures are referenced to atmosphere at the downstream exit, beyond the calibrated downstream flow resistance and inertance. Although hydrostatic head differences of a few centimetres are relatively insignificant here, care was taken that the exit elevation equalled the tube elevation; flow exited as a horizontal jet into a closed, vented tank above the main collecting reservoir.

The behaviour of each tube was characterized at three different values of the downstream resistance, R2. The high, medium and low R2-values will be referred to as R2 h, R~' and Rt2, respectively. These resistances correspond to pressure-flow curves 1, 3 and 8, respectively in Bertram's (1986a) figure 3. The components making up the downstream resistance included orifice plates and stiff polyethylene tubing, while the variable component was a modified commercial valve. R~' and R~ correspond to settings used previously (Bertram 1986a); R h is less than the maximum resistance used previously, which had always yielded steady flow. A relation of the form Ap = kQ m, where Ap = pressure drop and Q = flow-rate, was fitted to the steady flow measure- ments of each R2. The fits yielded exponents m of 2.2 for R h, and 2-1 for RT and Rt~. The exponent for the upstream resistance R1 was 1.6. If m is constrained to equal 2, k

128 c D. BERTRAM E T AL.

has the units Pa (m 3 s-1) -2. The best-fit measured values of k with m = 2 are given below as dimensionless quantities.

The four tube lengths and three downstream resistances R2 comprise a total of 12 locations on a plane through a higher-dimensional parameter space. Each of these 12 locations has been investigated at the same seven constant values of upstream driving pressure Pu, and the same five values of transmural pressure at the downstream end of the tube, Ap2e =P2--Pc, where P2 is the internal pressure there and Pe is the external pressure in the chamber. Where the operating point gave oscillations, the time-mean P2 was employed. Additional values of A p 2 e w e r e investigated as necessary to map the boundaries of behaviour zones or to document behaviour which changed qualitatively in a small range of Ap2e. The actual measured pressures were pl (at the upstream end of the collapsible segment), P2 and Pe. Ap2e o r Aft2 e was continuously monitored when changing Pc. The pressure Pe did not vary significantly when the tube oscillated between open and collapsed configurations, because the chamber was relatively large. It had inside diameter 140 mm (to be compared with a nominal tube outside diameter of 17.5 mm), and was 128 mm longer than the unsupported segment of the collapsible tube.

Data were recorded (i) by noting values, (ii) on an analog multi-channel chart recorder with frequency response flat to 100 Hz, and (iii) digitally, using a 12-bit a.d.c., at sample frequencies up to 5000 s -1 per channel. Six quantities were continuously recorded (see Figure 6 for example): Pl, P2 and pc, the flow-rates Q1 and Q2 (into and out of the collapsible segment), and the cross-sectional area A at the point near the downstream end where collapse is most severe (the 'throat'). See Bertram (1987) for a description of the area measurement technique. The time-averaged flow-rates Q1 and Q2 are identical, but instantaneously Q1 differed markedly from Q2 in both phase and amplitude. For the pressure signals, the digitised least-significant-bit resolution was 100Pa. The waveforms of Pl and P2 in Figure 4 to be discussed later, were first captured on a digital storage oscilloscope then transferred to a pen recorder using the oscilloscope's digital-to-analog converter.

The results are presented largely as curves of Ap12=P1-P2 (or A/~2) versus flow-rate Q (or ~?) at constant Apze (see Figure 2), to conform with existing practice (Brower & Noordergraaf 1973; Brower & Scholten 1975; Bonis 1979). Pressures Pl and P2 (and hence Ap12 and Ape2), and the flow-rate Q, are all dependent on the values of Pu and Pe. In a forthcoming paper, a way will be proposed of presenting the results that reflects this situation.

Operating points yielding oscillations are identified by the fundamental frequency of oscillation in Hz next to the data symbol. Operating points exhibiting noise-like fluctuations of small amplitude in all measured variables are identified as 'nf' ('hf' in Bertram 1986a). The symbol 'c' means that the flow was steady and the tube collapsed.

Unattainable operating points are mapped in two ways. First, the Ap12 - Q curves at Ap2~ = -20 , -40, -60, -80, and -100 kPa are located, using both stable (steady flow) and oscillatory operating points. The data for all three values of R2 are combined, since the position (not the stability) of operating points plotted in this way is independent of conditions outside the collapsible tube (Brower & Noordergraaf 1973). Unattainable points on a given curve are then located (Bertram 1986a) by interpolating linearly between the nearest attainable points having the same values of p , and R2 to find the intersection with the curve.

In this paper the boundaries of unattainable zones are also located, in terms of their position along these lines of constant Pu (see the broken line in Figure 5 in this report,

MAPPING COLLAPSED TUBE INSTABILITIES 129

or figure 11 in Bertram 1986a). With p , constant, Pe is slowly varied until the abrupt transition across the unattainable zone occurs. The value of me2 e immediately prior to the transition is plotted on the constant-p, line, at a point corresponding to its 'distance' in pressure from each of the appropriate neighbouring constant-Ap2e curves. Both the upper and lower boundaries of the zone, where they cross the line of constant Pu, are found in this way. Since there is hysteresis in the transition process, the boundaries eventually plotted by joining the points found in this way on various constant-pu curves are inner boundaries. The values of Apze immediately after the transition would yield an outer boundary, dependent on the dynamics of the apparatus up- and downstream of the collapsible tube, which does not merit plotting.

Owing to the large number of adjustable parameters that affect the dynamics, quantitative comparison with the unsteady results obtained by other experimenters in this field is not possible. In most cases insufficient information on the system surrounding the collapsible tube is reported for even a crude estimate of the appropriate values. Even if the information were available, each independent laboratory would be found to have investigated unique coordinates in the multidimen- sional parameter space. For this reason, little is gained experimentally by presenting data in nondimensional form. However, the data in this paper can readily be converted to dimensionless form for comparison with theory using values (Bertram 1987) for the tube inside diameter (13.5 mm), flexural rigidity pressure unit (10-9kPa--see the nomenclature appendix), and fluid density (103kgm-3). Using this scheme, the dimensionless values of k with m = 2 were 172.6 for R2 h, 26.9 for R~', 11.53 for R~, and 5-99 for R1. The pulsatile flow measurements, with a mean and an almost sinusoidal component, yielded dimensionless k-values for R h of from 156-7 at Womersley number (defined in the Appendix) o:w = 19.4, to 187-3 at trw = 25.8. For R~ the pulsatile k-values ranged from 10.81 (a~w = 17.1) to 11-38 (a~w = 24.2). For R1 they ranged from 4.89 (trw= 17.0) to 5-04 (tew =24.1). Similarly, the dimensionless inertances were measured as 363 to 427 upstream, and 322 to 374 downstream. [Note that slightly different values for the diameter and the flexural rigidity of the tube were adopted in earlier work (Bertram 1986a), leading to different dimensionless values for resistance and inertance.]

3. RESULTS

3.1. OSCILLATIONS

The transmural pressure at the downstream end of the tube is conventionally defined as Ap2e =Pa-Pe, but it will be more convenient here to use Ape2=Pe-P2 to avoid negative numbers. The four panels of Figure 2 shows operating point curves for the four different tube lengths. Such curves are intersections in (Q, Ap12, Ape2) variable-space of constant-Ape2 planes with the operating-point surface. In each panel filled, unfilled and half-filled symbols are used to distinguish Rtz, R'~, and R h, respectively. Symbol shape distinguishes the constant-Apea curve to which the point belongs. The time-mean Reynolds number based on tube diameter is in every case slightly less than 100-times the flow-rate in ml s -1, and hence ranged from about 400 to almost 50,000. Either the Strouhal or the Womersley number can be used to describe the degree of fluid mechanical unsteadiness associated with an oscillatory frequency. The Womersley number based on tube radius ranged from 25 to over 300. Some previous results for the 17D tube have been published (Bertram 1986a), and likewise a few earlier results for the 4D and 8D tubes have been presented elsewhere (Bertram

130 c . D . BERTRAM E T A L .

0

100[

100 Mean f l o w - r a t e ( m i l s )

200 300 400 I I

(a) Length = 4 .0D

I I 500

fl_

v

I

o .

O_

80

6 0

40

20

C I

nf

c ~

nf

342

235

3-881200

nf

i r r e g . ~ 1 ~ l 898

~ 2 " 3 nf -~

irreg.

6 ; ' 3 65"3

" . ; 80

irreg,

5.71

irreg.

5 .97/hf

APe 2 R; R 7 R I

100 • o • 80 [] [ ] • 60 A & • 40 ~' V •

20 <~ 0 •

20 CC 54-3

(10/)

Figure 2. Equilibrium curves for collapsible tubes of four different lengths, at three different values of downstream resistance. The length is given relative to internal diameter in each panel. Resistance values: low = filled symbols, medium = unfilled, high = half-filled (see text for numerical values). Symbol shape denotes the value of Ape 2 pertaining to the curve: diamond = 20 kPa, inverted triangle = 40, non-inverted triangle = 60, square = 80, circle = 100. Where appropriate, a number accompanying the symbol denotes oscillation frequency in Hz. Two frequencies, e.g. 3-02/237, indicate that the high frequency is modulated by the low one. The letter c denotes steady flow through a collapsed tube, and nf denotes small-amplitude,

broad-band, noise-like fluctuations. At ~Pe2 = 20 kPa the tube was circular.

1986c). The more recent results do not completely correspond everywhere, and therefore the earlier results are referred to where there is a discrepancy.

Various broad trends can be discerned from Figure 2. Repetitive oscillations are most prevalent at low Rz (and persist to extremely low R2 values not presented here). At the intermediate value (RT), respectively few points appear, because the cor- responding operating points were often 'unattainable'. At R h many operating points are stable, and at a yet higher resistance setting, all operating points investigated in the 17D tube were stable (Bertram 1986a). Longer tubes appear predisposed to repetitive oscillation of low frequency, although high-frequency repetitive modes can be excited at sufficiently high Pu and Ape 2. AS the tube length decreases, low-frequency modes

MAPPING COLLAPSED TUBE INSTABILITIES 131

Mean f low ra te (m i l s )

0 100 200 300 400 500 I I 1 I

(b) Length = 8 - 2 D

100 7 1 - 6 pe_~2 =lOOkP a

nf

80 . ¢ ~ f

/ /u3301 nf _. 47,4 / . . . . . . . nr _...,-.a~ 80

I¢ C 14 60 y ~ _ nf

nf g"

4-97 I / 343

2O

I M-~ ic c ~ 5 .31

149" 2v9° 20

O" _ i I (101

Figure 2, (Continued.)

appear at fewer operating points; the rest exhibit either high-frequency repetitive modes or instabilities of the 'nf' type.

At a given location in the parameter space (i.e. with R2 and length specified), the frequency of oscillation generally increases with Ape2 and Q. For each tube the minimum frequency of oscillation is seen at or near Ape2 = 40 kPa (at Ap~2 = 20 kPa the tube is not collapsed), at the minimum Q at which the equilibrium is unstable. This minimum frequency was 4-3 Hz at length 4D (cf. 4-5 Hz in Bertram 1986c), 2-9 Hz at length 8D (cf. 3-2 Hz in Bertram 1986c), 2-5 Hz at length 17D (cf. 2.7 Hz in Bertram 1986a) and 2.0 Hz at length 34D. In every case the minimum frequency occurred at R~, although frequencies only slightly higher were seen at R~'. Only at length 34D were low frequencies seen at Rh2; there the minimum observed (in a later series of experiments) was 2-9 Hz, at p , = 166 and Ap~2 = 62 kPa.

The maximum observable frequency is less clear-cut, for two reasons. Firstly, it appears that ever-higher-frequency modes of repetitive oscillation can be excited as Ape2 is increased [e.g. 625 Hz at 120 kPa on figure 2(a) in Bertram 1986c]. Although the results suggest that all repetitive modes at a given flow-rate eventually give way to

132

n- v

I

t l o

P

n

0

100

8 0

6 0

4 0

2 0

C. D. BERTRAM ET AL.

M e a n f l o w - r a t e ( m i l s )

100 2 0 0 3 0 0 1 I I

4 0 0 5 0 0

(c ) L e n g t h = 17.4D

n f

aft,_. C ~ " n f

~f

653 ~ / ~ 6 ~ . o n f . ~ 2 2 . 3 / i r r e g .

60 .2 ~ 8 0

62.4 [] ~ • n f [] • ~ 11.7

i r r e g .

"1%.~4

4.37 irreg.~slow 1 6 0 1 9 . 4 / 3 2 9 ~

" ZX 11-5 4.17 4-24

3.13 3-26

126 3 .02 /237 • 4.44

4.32

3,11

4 0

c 20

Figure 2. (Continued.)

(101)

'nf' modes a s Ape 2 increases, it has not been possible to verify this, given the constraints on maximum pressure and flow-rate in this apparatus. Secondly, the orderly progression towards higher frequencies with increases in Q and Ape2 is interrupted by 'islands' of high-frequency modes at moderate Q and Ape2. These islands can be seen on Figure 2; notice the points marked as having frequencies in excess of 50 Hz on the curves for Ape2 = 40 and 60 kPa [similar results appear also on figure 2(a) in Bertram 1986c]. This behaviour was missed previously (Bertram 1986a), because the boundaries between regions were not explored in detail. Qualitatively similar 'islands' have now been observed in all four different-length tubes, although at length 34D the behaviour is not readily reproducible. However, one can legitimately compare the maximum frequency observed up to p , = 200 and Ape 2 = 100 kPa in each tube, excluding results from these aberrant islands. On this basis, a maximum frequency of repetitive oscillation of 85.8 Hz is seen at these p,, Ape 2 values at length 4D (cf. 95.5 Hz in Bertram 1986c). At length 8D the corresponding operating point reaches 71.6 Hz ('nf' in Bertram 1986c). At length 17D, it reached 68.0 Hz (cf. 62.7 Hz at a neighbouring point in Bertram 1986a), and at length 34D, 10.0 Hz.

Q.

I

ic£ (3- o

"o

D_

0

100

80

60

40

20

MAPPING COLLAPSED TUBE INSTABILITIES

Mean f l o w - r a t e ( m i l s )

100 200 300 400 I

500

(d) Length= 34'2D

3.91

_ t •

3.95

3.88 0

10.08 f ie- P2 =100 kPa

9"32 ,, 80

nf

3-80

= 3"78

3.94 ZZ

Z~ 3.40 3'52

3.96

2'77

4.74

2'59 ~ 2.81 2.33

133 2-56

40

20

133

Figure 2. (Continued.)

(6 / )

The magnitude of the oscillations is portrayed in Figure 3, where two operating points have each been shown both as the time-averaged point and as a dynamic orbit of instantaneous Ap12 and Q values visited during a few cycles of oscillation. The frequencies of oscillation differ by a factor of 17, yet the magnitudes of the Ap12 and Q excursions during the cycles are comparable. Similarly large magnitudes, relative to a family of operating point curves, have been reported by Bonis (1979). Still-higher- frequency modes are necessarily of smaller magnitude, and the 'nf' modes are relatively insignificant.

Figure 4 shows complex behaviour in the islands of exceptionally high frequency, here in the 4D tube. At R2 h, almost sinusoidal waveforms at a frequency of 70-72 Hz are seen [Figure 4(a)]. The frequency decreases as Pe is increased. In Figure 4(a), Pl and Pz move roughly in antiphase, as might be expected: collapse reduces Pz and increases Pl- Oscillations at R~ are shown in parts (b) and (c). Starting at Apez above 50kPa and decreasing Pe, oscillations are first seen in the range l12-130Hz [Figure 4(b)], then a t A p e 2 = 47 kPa, steady flow is restored. Higher frequency (160-165 Hz) oscillations with a very different waveform shape break out again at 43 kPa [part (c)]o

134 C. D. B E R T R A M ET AL.

0 200 --

1 0 0

8 0

v

ID'~ 6 0 i

L "o

~, 4 0

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100 o 0 •

a o r l [ ] i 6 0 A Z~ • 4 0 • V •

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200 300

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f I

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',Ill 6 8

/'~ il ~r~r k

:!:Ii it~ I ~

-~/ i I I i i J t

t ~ l I j ~ d

t l ~ ,, ,,', i i I i i I i d

i i r i i I i i 1 t t}

LI r

1 ! " l o • ~ ~ I ; ~ l I

( 1O I )

Figure 3. The dynamic orbits of two oscillatory operating points on axes of instantaneous pressure drop Ap12 and downstream flow-rate Q2, superimposed on the equil ibrium curves for the ]7D tube to show the magnitude of the oscillations in relation to the distance between operating points plotted in Figure 2. The position on the equilibrium diagram of each of the two operating points is shown, together with the oscillation frequency in Hz. In both cases R~; p , = ]66 and Ape 2 = 44 kPa for the 4.] Hz oscillation, whereas p . = 200 and Ape 2 = 100 kPa for the 68 Hz. Note the phase change: the low-frequency oscillation circulates anticlockwise around its main loop, whereas the fast oscillation apparently circulates clockwise; this is a

consequence of the limited frequency response of the flowmeter.

M A P P I N G C O L L A P S E D T U B E I N S T A B I L I T I E S 1 3 5

(a)

( b )

Pe - P 2 = 4 0 k P a 2 6 . 2 8

2 O

(d)

u}

n

3 0 -- Pe-P2 = 5 0 k P a 2 6 " 2

~ P2

I I 5 m s

3 0

0 --

Pa - P 2 = 5 0 k P a

- 2 6 - 1 6

P1

P2

(e)

30 Pe - P c = 4 6 k P a

2 6 " 6

/ • P 2 o- 1 I

5ms

(c) 30 P e - P2 = 4 2 k P a 2 6 " 2 2

~ P 2

0 -

I t 10 ms

( f ) 3 0

O.

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-- Pe - P2 = 4 2 k P a 2 6 " B

::'., > : . . . . ., , , .

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I t.~.!.~' ' I~I ,h.h,l~,f ' ,hi

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• . . ,~.', ". ".'. k , '~" P2

i ,,*'it l =:llpII

o - I I 2 0 0 m s

Figure 4. High-frequency oscillations at moderate flow-rate and pressure drop mpl2, in a tube of length 4D at p, = 32 kPa. ,, (a) R~ and Ape2 = 40kPa; f = 72Hz; (b) R 2 and Ape2= 50 kPa; f = ll6Hz;

m 1 (c) R2 and Ape2 = 42 kPa; f = 165 Hz; (d) R2 and Ape2 = 50 kPa; f = 210 Hz; 1 1 (e) R 2 and Ape2 = 46 kPa; f = 194 Hz; (f) R2 and Ap,2 = 42 kPa; f = 4.6/195 Hz;

(note that the regularity of the high-frequency mode in panel (t') is obscured by aliasing). Frequencies quoted here do not correspond exactly to those shown in Figure 2(a) because the results here derive from an earlier study which was undertaken to examine in more detail the corresponding island of high-frequency oscillation

seen in Bertram (1986c) figure 2a.

Figure 4(d), (e) and (f) show oscillations at R/ . The frequency range is 195 -210 Hz [(d)], but now increasing with Pe. The magnitude of the p l -excurs ion is here at least as large as the p2-excursion (unlike any m o d e s reported in Bertram 1986a), and Pl and P2 are now in phase. A s Pe is reduced (and P2 accordingly increases with Q) the waveforms acquire a more complex shape [part (e)] , and at Ape2= 4 2 k P a are combined with a low-frequency (5.2 Hz) m o d e [part (f)]. The high-frequency oscilla- tion then appears only during that phase of the low-frequency cycle when the oppos i te walls o f the tube at the throat c o m e together. H o w e v e r , depending on h o w the

136 C. D. BERTRAM E T AL.

operating point is approached, one can obtain only the 200 Hz mode at the same Ape2, with the waveform shape resembling that in Figure 4(e).

The frequencies in the islands can range even higher. The 4D and 8D tubes both reach 342 Hz at Apez--57, p, = 66 kPa for R/, and at the same settings the 17D tube vibrates at 330 Hz (with a pronounced irregular modulation of amplitude at about 19 Hz), as shown in Figure 8(b)--this figure is discussed in detail later in this report. In the experiments reported here the highest frequency from the 34D tube was 133 Hz at Pu = 33, Ape2 = 40 kPa with R~', as shown in Figure 2, but this mode could not be consistently reproduced.

3.2. STABILITY MAPPING

Besides unexpectedly high-frequency oscillations, the other notable characteristic of the region bordering the nf (noisy) or c (steady) modes is a large hysteresis in the dynamic behaviour. The behaviour depends not only on the existing values of Pu and Ape2 but also on whether that value of Ape 2 has been approached from a lesser or a greater value. We have been able to map the boundaries of this 'overlap' region by a similar approach to that described in the methods section for mapping the boundaries of 'unattainable' zones. It is convenient to present the results of both procedures on the same diagram. When considering oscillation frequencies alone, it is possible to present results for different settings of the R2 parameter on the one graph as in Figure 2, but the zone diagrams lose all legibility if superimposed. With a total of twelve different parameter combinations to consider, it is therefore necessary to select certain diagrams as illustrations and summarise the results of the rest in tabular form and in the text. The diagrams all exhibit broadly the same features, with behaviour varying gradually and systematically with changes in tube length and Rz. The important exceptions to such trends are noted in what follows.

3.2.1. 34D tube: R~

This parameter combination, shown in Figure 5, produces one of the simpler diagrams. Curves of constant APe2 are taken from the corresponding panel in Figure 2. The superimposed unattainable zones (diagonal-shaded) and overlap regions (speckled) yield a complete prediction of behaviour anywhere on the operating point surface up to the maximum Pu employed (200 kPa, shown as a dashed line). Above that p,, no data at RE are available, but one can validly extend the curves of constant Ape 2 according to the Rtz results, on the basis that the curves are independent of R E. However, one cannot predict the stability behaviour in that region.

The diagonal-shaded areas are zones of unattainability. The lower one, overlying the Ape2=40kPa curve, will be generically identified below as the collapse-onset unattainable zone. It lies above a region where there is steady flow through a noncoUapsed tube, and below one where the collapsed tube undergoes repetitive oscillations at low frequency. This latter region is labelled LU (low; up) to distinguish the mode of oscilllation from that in the region labelled LD (low; down) above the second, smaller unattainable zone. L indicates frequency; U or D distinguishes the waveform shapes on the basis of whether the area trace spends more time above or below a horizontal line drawn midway between the maxima and minima of the trace. Physically this means that the tube is open for the majority of the cycle during LU oscillations, and collapsed for most of it during LD oscillations. This distinction is illustrated in Figure 6, which shows an example of pressure, flow-rate and area

MAPPING COLLAPSED TUBE INSTABILITIES 137

140,

120

100

' 8 0 IQT 13_ o "{3

o

~ 60

Q_

40

t t

t

:i~: o :: ::.;.'.i: i .

6--b%= 100 kPa

60

20

0 100 200 300 400 5-00

Mean flow ra te (mils)

Figure 5. Equilibrium stability map for the 34D tube at R~ n, showing curves of constant Ape 2 and open symbols denoting stable or oscillatory operating points at R~ from Figure 2(d), diagonal-shaded zones of unattainability, and (speckle-shaded) the zone of 'overlap' (see text) between small nf fluctuations and large-amplitude low-frequency repetitive oscillations. The boundary between these two is completed by a very small zone of unattainability (shown chequer-shaded) at low Q and Apl 2. Small filled triangles denote the actual locations of the experimentally-measured boundary points; the triangles indicate the direction of ApeE-change at the transition. See the text for the definition of (and Figure 6 for an illustration of) the LU/LD classification of oscillations. LD oscillations occur everywhere in the unshaded region between the overlap region and the upper unattainable zone. LU oscillations occur everywhere in the unshaded region between the two unattainable zones, and below and to the left of the upper unattainable zone. The behaviour termed nf (see text, and caption to Figure 2) was found to the left of the overlap region, dying away to leave steady flow at low flow-rates as shown in figure 2(d). The broken line joins points where

p . = 200 kPa (see text).

waveforms for each type. Both the LU and the LD oscillations are of large amplitude, as exemplified in Figure 3. Thus their dynamic orbits are not confined within the regions where their time-averaged operating points are located in Figure 5. This is true also of the intermediate-frequency (I) and high-frequency (H) oscillation modes to be presented below.

138 C. D, BERTRAM ET AL.

175 (a)

1 6 0

Q'~ 145

130 l • " t i -~

1 6 0 ~

8 0 '

o

- 8 0

(~ 275

250 350

250

c7

150

°ioOo o 0.60

~ o. . . . . . . o

Time (s)

175 (b)

- 8 0 - - - - ~ '

2 6 5

E 245

f f 2 2 5

2 0 0

t004- - ~ - ,

222222 0.71

~. 0"53

' ~034

0.15 ~ ~ ~ ~ ~ - - ~ 0'00 030 0'60 0'90 1'20 1'50 Time (s)

Figure 6. Waveforms of pl , P2, Q1, Q2 and A (scaled with undeformed area Ao) from the 34D tube at R~ and p~ = 166 kPa, illustrating by example the definition of LU and LD oscillations, as used in Figures 5 and 7. Note different ordinate scales. (a) LU (up) denotes oscillations where the measured cross-sectional area trace (and the p2-trace) spend the greater part of each cycle above a line midway between the extremes of the waveform. Pressure Ape 2 = 60 kPa; f = 4.09 Hz. (b) LD (down)--tbe area trace spends more time below the horizontal halfway between the extremes. The frequency (4.02 Hz) is almost exactly as in (a), but here

Ape2 = 100 kPa.

MAPPING COLLAPSED TUBE INSTABILITIES 139

The LD waveforms often bear some similarity to that termed '2-out-of-3-beats' in earlier work (Bertram t986a, figure 10). The '2-out-of-3' type can be defined as follows: the tube collapses briefly twice per repeated cycle, but collapse is not usually as severe the second time, and these two collapse events occupy about two-thirds of the total cycle. That mode is actually a waveform of LU type. Moreover, with further slight increase in Pe it always changed frequency discontinuously (to a '3-out-of-3' rhythm relative to the '2-out-of-3', or in other words to intermediate frequencies, roughly three times higher than the low ones).

The LD waveform instead pauses at small area after the rapid collapse. The pressure P2, which is increasing relatively slowly after descending to a minimum very close to the instant of maximal collapse, may redescend somewhat, but there is usually insufficient area recovery after the first collapse to define a second one. With a further increase in Pc, the LD waveform does not lead to a discontinuity in frequency equivalent to the LU-to-I transition described below.

At low flow-rates, the LU region merges smoothly into the LD region by the gradual appearance of a hesitation at small area after collapse. The unattainable zone between the two regions indicates that at greater flow-rates the mode changes abruptly; P2 changes discontinuously even though P2 is continuous, crossing the zone in the process. The transition is relatively nonhysteretic compared to that defining the lower unattainable zone.

In the large speckled region above and to the left of the LD region, the behaviour depends on whether an operating point has been approached from below the lower boundary of the region or from above the upper boundary. Thus the speckled region displays hysteresis, or 'overlap' between two different behaviours. Slow oscillations of the LU or LD type can be maintained as far as the upper margin, before giving way to the 'nf' mode in the unshaded region above and to the left of the speckled region. Conversely, the 'nf' mode alone can persist down to the lower margin of the overlap region before slow repetitive oscillations break out. Because the nf fluctuations are of small amplitude, the overlap regioncan also be regarded simply as an area of hysteresis for the appearance and disappearance of slow oscillations. The unshaded nf region itself merges smoothly to the left, i.e. at lower flow-rates, into a region of steady flow through a collapsed tube; the nf mode gradually dies away. Finally, intersecting the Apez = 40kPa curve near Q = 100ml s -1 is a small unattainable zone (chequered) between the LU and nf regions.

3.2.2. 34D tube: R~2

With low downstream resistance, the same tube exhibits more complicated behaviour, as shown in Figure 7. Although most of the additional zone boundaries occur to the right of those flow-rates accessible at R~, the placement of the three major shaded zones seen at R~' is also different at R~.

A new region labelled 'I', of self-excited oscillation at intermediate frequency (between 9.3 and 10.3 Hz), is cordoned off by dot-array-shaded zones of hysteresis (or overlap) from the LU region below and the nf region above. In both this figure and Figure 5, small triangular symbols show the transition points investigated experimen- tally. The triangles point along a line of constant p~ (not marked), showing the direction of change of Ape2 when the transition occurred. Thus, bounding an unattainable zone the triangles point inwards, whereas bounding an overlap region they point outwards.

140 C. D. BERTRAM ET AL.

120

100

13.

C~ 0 L

2

80

60

40

R'

O0 kPa

LU

20

20

0 v = I , I ! 100 3 0 0 4 0 0 500

Mean f l o w r a t e ( m i l s ) "

Figure 7. Operating point stability map for the 34D tube at R~. Refer to the caption of Figure 5 for explanation of symbols and shading. Additional overlap zones at this R2, shown by dot-array shading, separate regions of regular oscillation at low (LU) and medium (I) frequency and of small-amplitude

noise-like fluctuation.

Two downward-pointing triangles appear in the middle of the I region, indicating a further complication of behaviour in that region. If pe was increased rapidly so that Ap~2 rose quickly through the LU-to-I hysteresis region, LD-type oscillations could be produced in the upper half of the I region, above the isolated triangles. The result of arriving at the same location either from higher Ap~2, or from lower Ap~2 slowly, was I-mode oscillation.

3.2.3. Other tube lengths: all three Re values

To summarize the results of similar studies at the other parameter combinations, Table 1 shows for each tube length and R2 the number of operating points which were unattainable, i.e. which were in diagonal-shaded zones. Both those in the zone where the tube collapsed abruptly, and those in the zone where the oscillatory mode changed from LU to LD, are counted. Each number in the table is out of a possible maximum of 28, this being the number of collapsed-tube (p, , &pe2)-combinations investigated

MAPPING COLLAPSED TUBE INSTABILITIES

TABLE 1

Comparison of the extent of the unattainable zones at all values of Rz and tube length investigated. Tube length is here given in terms of diameters. Each location in para- meter space was investigated at the 28 collapsed-tube variable-space coordinates defined by p, = 13, 33, 67, 100, 133, 167, 200; Ape2=40, 60, 80, 100kPa. The extent of the zones is approximated by counting the number of such operating points in the zones, i.e. that have to be located by interpolation as described in the text. Of the points counted, all but three are due to the collapse transition; the exceptions are (i) at R~ n in the 34D tube, due to the LU/LD transition, (ii) at R~' in the 17D tube, on the H-to-nf boundary, (iii) at R~ in the 17D tube, on the

LU-to-H boundary.

Length/D

R2 4.0 8-2 17.4 34.2

l 4 4 2 2 m 8 8 5 5 h 7 5 6 3

141

(see the methods section). For instance, at R~ in the 4D tube, only four points were unattainable, all the 24 others being either oscillatory or stable. The table shows that the 'static instability' (Bonis, 1979) manifested experimental ly as an unattainable operating point becomes more prevalent as tube length is reduced. The table also shows that this type of instability is most prevalent at the intermediate resistance Rg'. At lesser or greater downst ream resistance, either oscillatory instability or stable flow, respectively, is more prevalent .

The reproducibility of these results is not as high as one would like. For comparison, Table 2 shows the corresponding numbers of unattainable points found in earlier studies on tubes of approximately the same 4D, 8D and 17D lengths. (The numbers in the table correspond to data in figure 12 of Ber t ram 1986a for the 17D tube, in figure l(b) of Ber t ram 1986c for the 8D tube, and in figure 2(b) of Ber t ram 1986c for the 4D tube.) Fewer unattainable points occur in the more recent studies at all equivalent parameter locations except R~ in the 4D tube and R2 h in the 8D tube.

TABLE 2

As Table 1, but for the corresponding data at length 17D in Bertram (1986a), and at lengths 3-9D and 8.2D in Bertram (1986c). Comparable data at length 17D with R2 h are not available because a yet higher resistance

value was used instead in Bertram (1986a).

Length/D

R2 3-9 8"2 17"0

1 5 5 1 m 10 12 9 h 5 5 (na)

142 c . D . BERTRAM ET AL.

3.2.4. 17D tube

Figures 5 and 7, and Tables 1 and 2, do not detail all observed behaviour in the four tubes. Some additional complications for the 17D tube were presented in earlier work (Bertram 1986a). These include, at R~2, discontinuous changes between regions of low (2.5 to 5-3Hz), intermediate ( l l - f - l l - T H z ) and high frequency (68 and 81Hz at p , = 200 kPa for A pe 2 = 100 and 120 kPa, respectively--aU the frequencies quoted here relate to the present study as in Figure 2). The transition from low to intermediate frequency is usually a simple frequency-doubling, but exceptionally may involve a stable intermediate '2-out-of-3' mode, as at Pu =166, APe2=60kPa. Here the sequence with increase in Ape 2 is 'l-out-of-3' (4.1Hz), 2-out-of-3 (4-3Hz), then 3-out-of-3, i.e. 1-out-of-l, since all three cycles are identical ( l l -2Hz) . The same '2-out-of-3' mode was observed at a slightly different operating point in A as an intermediate state between 'l-out-of-2' and 'l-out-of-l ' modes, i.e. during a frequency- doubling, rather than frequency-tripling, transition. However, the frequency jumps in these two types of transition are more similar than this description would suggest. For example the 'frequency-tripling' event at p , = 166 kPa actually involves the frequency ratio 2.75, while the neighbouring 'frequency-doubling' (with no intermediate state) had the ratio 2.36.

The 17D tube displays less extended overlap regions than the 34D tube. However, numerous exceptional oscillatory modes populate the boundary between collapsed 'c'

5 5 Q.

5O

~ 35

Q'~ -5 - 2 5 2oe~

-E 195

0

d ¢ L

2oo AAAAA;175 150 ~- ~ ~ A L ~ _ ~ •

0 . 6 5

0 3 5

0.20 OOO 0.30 0 ' 6 0 0 . 9 0 1.20 1'50

T i m e (s)

Figure 8. Waveforms from the 17D tube at Rt2, illustrating the hysteresis of the dynamics in the overlap region. In both panels (a) and (b), p , = 67 and Ape 2 = 60 kPa, but (a) shows repetitive oscillations at 4-15 Hz resulting from approaching the operating point (by adjustment of Pe) from a Ape z below the overlap region. High frequencies in the range 250-400 Hz occur during one phase of the cycle. In contrast, when approached from Ap~ 2 above the overlap region (b), the operating point shows irregular oscillations at about 19 Hz, plus almost continuous vibration in a range centred on 330 Hz. Note that the frequency response of the flow-rate

and area measurement systems is insufficient to represent the high frequencies.

MAPPING COLLAPSED TUBE INSTABILITIES 143

5 8 - 5 ~

A 57.0 £

~ 555

54.0 _ _ 40

~ 20

~ 10

0

86"5

--~ 184"5

~-1825 180-5 ~- I I I

2oo I

180

031 ~ ~

0 2 8 ~ I 0"00 0'25 )'50 0"75 1"00 1-25 Time (s)

Figure 8. (Continued.)

(at low flow-rates) or noisy 'nf' (at higher flow-rates) states on the one hand, and low, intermediate or high frequency repetitive oscillatory states on the other. Figure 8 shows an example. Approached from lower Ape2, this operating point exhibits a conventional low-frequency oscillation [Figure 8(a)], whereas when approached from higher Apez, i.e. from the c region, the mode shown in Figure 8(b) was found. Note that these exceptional modes are clearly distinguished from the nf mode, which is of much smaller amplitude. The mode in Figure 8(b), found on the boundary, is not representative of the behaviour at higher Ape2, which is non-oscillatory. At higher flow-rates, further examples of irregular, i.e. nonrepetitive, oscillatory states of substantial amplitude are found on the same boundary, between nf and I regions. At both low and high flow-rates (p, = 33 and 200 kPa), amplitude modulation at low frequencies of high or very high frequency vibrations occurs on or just below the boundary. All these operating points are shown on Figure 2(c), identified by frequency and filled symbols.

At R~, the 17D tube displays no intermediate, I, region. An irregular slow oscillation is seen on the boundary between the high frequency H region (frequencies between 60 and 65 Hz) and the collapse-onset unattainable zone, which borders the H region directly. At R2 h, there are no large regions of repetitive oscillation, but all the operating points on the boundary between the collapse-onset unattainable zone and the large 'c' region (stable collapsed-tube flow) display high-frequency oscillations, as at Rt2.

3.2.5. 8D tube

Intermediate frequencies are not seen in the 8D or 4D tubes. At Rt2 the 8D tube H region, as at R~ in the 17D tube, abuts the collapse-onset unattainable zone directly~

144 c, D. BERTRAM E T AL.

(a)

46 " ~

~ 26" "% Q" 10

-8

240 d 230

225

20O

045

• - • I " -

Time (s) (b) 2 13o ~~j

113 C

95

~ 5 4

-16 3 8 0 ~

"~ 365 350

335 4 0 0 ~

350

o 300

0.70

0'00 0'40 0-80 1-20 1'60 2'00 Time (S)

Figure 9. Four examples of 'multiple collapse' waveforms from within the region of low-frequency oscillations observed at R~ in the 8D tube. (a) p . = 65 and Ap~2 = 40 kPa; [ = 5.00 Hz; (b) p . -- 135 and Ape 2 = 45 kPa; f = 4.48 Hz; (c) p . = 102 and Ape 2 = 49 kPa; f = 4.98 Hz; (d) p . = 135 and Ape 2 = 60 kPa;

f = 4-97 Hz.

MAPPING COLLAPSED TUBE INSTABILITIES 145

95 (c) 9 o ~ ~ 85

75 70

40

"~ 20 Q.

o I ' I I I I I I I -20 - - ~

275 v

25C ' '

~ 30O ~ 250

200 o-65

~" 0.45

< 0.35

0"25 0"00 0.40 0.8Q 1,20 1"60 2.00 Time (s)

(d)

113 ~T

95

~ 50

1o

-30

3 5 0 ~

_~ 335

6 320

305

"~ 35O

300 o

250 ,' ~ "J " ~ ~ 0 . 6 5 ~

,qO 0-55 '~ 0-45

035

0.25 ~ " ' ' ' 0-00 0.40 0"80 1-20 1.60 2 O0 Time (s)

Figure 9. (Continued.)

146 C. D. B E R T R A M ET AL.

with irregular oscillations on the margin. This region includes frequencies from 47 to 105 Hz, increasing systematically with Ape2. The large L region includes an inner region of 'multiple collapse' modes, as exemplified in Figure 9, which include both U and D types as defined earlier. The U/D categorization is less useful here, and no discontinuous transition giving rise to an unattainable zone has been observed. Isolated very-high-frequency repetitive oscillatory states again inhabit the upper boundary of the L region, beyond which only stable or nf states are observed.

At R~, the nf region abuts the collapse-onset unattainable zone. The L region is confined to lower flow-rates, while the H region appears only above p~ = 166 kPa. At Ape2 = p , = 100kPa, far above the Apez value at which the nf region started, the tube vibrated at the very high frequency of 330 Hz. The map at R h resembled that described above for R~ in the 17D tube.

3.2.6. 4D tube

At R~, the 4D tube exhibits a large region within which only irregular, non-repetitive oscillations are seen. This region is bounded below by a region of low frequencies and above at lower flow-rates by an nf region and at higher flow-rates by H oscillations. L oscillations range in frequency from 3.88 to 5 .97Hz, increasing with flow-rate. H oscillations, observed at Pu = 200 kPa from Ape2= 100 to &Pc2 = 160 kPa, stay in a narrow frequency range (86-90 Hz) before giving way to the nf mode at higher Ape2. As before, much higher frequencies still are seen near the L-to-nf boundary. This is also true at R~, where irregular oscillations are absent. Here , a low frequency occurs only once, at the highest flow-rate reached, modulating a high-frequency oscillation on the boundary between the H region and the collapse-onset unattainable zone.

The map at R h generally resembles those at the same downstream resistance in the 8D and 17D tubes. Two 'islands' of aberrantly high frequency are observed along the upper border of the collapse-onset unattainable zone, one involving frequencies from 64 to 88 Hz, the other from 230 to 235 Hz. Stable collapsed-tube flows occur in the region above and separate these islands. Repetitive oscillation at 617 Hz was observed at Pu = 200 and Ap~ e = 120 kPa, on the outer margin of states observable with the apparatus.

Table 3 shows how many of the standard 28 operating points for each tube length and R2 value were oscillatory. The table shows that longer tubes in general display more widespread oscillatory instabilities. (An exception occurs at R2 h, where in the shortest tube three of the designated p,, Pez intersections coincide with aberrant

TABLE 3 Following the same procedure as used to count unat- tainable operating points for Table 1, the number of operating points (out of the total of 28 defined by p, = 13, 33, 67, 100, 133, 167, 200 and Ape 2 = 40, 60, 80, 100 kPa) that are oscillatory is here tabulated. Oscillatory here

excludes nf modes.

Length/D

Rz 4.0 8.2 17.4 34.2

l 9 10 14 16 m 4 3 9 13 h 2 - - - - - -

MAPPING COLLAPSED TUBE INSTABILITIES 147

oscillations along the border between the 'c' region and the collapse-onset unattainable zone.) Excepting R h, the total prevalence to instability, whether oscillatory or divergent, is greatest in long tubes, as can be seen by adding the corresponding numbers in Tables 1 and 3.

4. DISCUSSION

All the types of behaviour described were found by first setting a constant value of p , then varying Pe SO as to attain desired values of Ape2. Although the results at a given location in parameter-space range over the whole surface of operating points, only specific constant-pu curves on that surface have been investigated. Hence, qualitatively distinct behaviour confined to small regions may have been missed. More detailed mapping might show the behaviour of some of the observed operating points to be more systematic than has been realized.

We have demonstrated that operating point behaviour can depend on direction of approach along a curve of constant Pu- Approaching from different directions, for example varying pu while Ape 2 is held constant, might reveal further types of behaviour, but this is not possible with the present apparatus. Furthermore, in at least one instance (the 34D tube in the I region at R~), the behaviour of the tube has been shown to depend on rate of approach; such dependence is well-known in the context of Taylor vortices (Fenstermacher et al. 1979).

It is important to ask whether the aberrant types of behaviour described in the results section in fact reflect a dependence of the dynamics on as yet unrecognised variables (such as rate of change of Ape2). This cannot be ruled out, but other explanations are possible. Considering in particular the 'islands' of exceptionally high frequency and irregular oscillation found along the lower boundary of the nf region, it would appear that in this area of the operating point surface, the stability map genuinely presents a much finer structure than elsewhere. Such 'fine structure' in certain regions is found in much simpler systems than that studied here, for example in solutions of Duffing's equation for a forced oscillator (Ueda 1980).

Discrepancies between Tables 1 and 2, and other differences between the present results and comparable results in earlier papers (Bertram 1986a, c) were pointed out in the previous section. We cannot explain the disparity in size of the collapse-onset unattainable zones in equivalent experiments on the shorter tubes reported here and in Bertram's (1986c) paper. The cumulative effect of small differences in many variables between the two series of experiments are presumed to be responsible. Such differences may readily be imagined in each of the adjustable parameters explicitly mentioned in the introduction. One additional factor likely to play an important role is the fixing of the collapsible tube at each end. To withstand the considerable pressures involved, each end of the tube is clamped on to the rigid pipe using two 'Denilok' cable ties and a tensioning gun. Even with the greatest care it is impossible to avoid slight asymmetries and torsion, which would assume greater significance in the shorter tubes. Another effect which may play a role is the aging of the silicone rubber tubing itself, aspects of which have been documented already (Bertram 1987).

Nevertheless, a reasonably complete characterization of the dynamics of the 'Starling resistor' with changes in two of the adjustable parameters has been achieved, in the process providing a considerable augmentation of the experimental data available for comparison with theory. To our knowledge the dependence of the dynamics on route of approach has not been described previously, nor has the occurrence of very high frequencies of oscillation at moderate flow-rate and pressure drop.

148 c .D. BERTRAM ET AL.

4.1. DIFFERENT TYPES OF OSCILLATION AT SIMILAR FREQUENCIES

Of particular note in the results of Figure 4 are the phase changes (p~ and P2 in antiphase at 5 Hz [Figure 4(0] and 72 Hz, but in phase at 210 Hz), and the momentary fall of the pressure drop Ap12 in part (a) to approximately zero as p~ passes through its minimum and P2 its maximum. In contrast the pressure drop stays continuously positive during the cycle at 68 Hz in Figure 3. Thus these two operating points of similar frequency (68 and 72 Hz) are differentiated both by position on the diagram and by the mechanics of events in the tube.

The 68 Hz oscillation derives from a 'conventional' position involving high Ape 2 and the forcing of a large volume flow-rate at high fluid velocity through a 17D tube that is highly collapsed at the downstream end. In consequence the viscous pressure drop in the tube is large enough to outweigh at all times in the cycle the inertial component, which must swing positive and negative as fluid in the tube is accelerated or, at the time of collapse, retarded. Measurements of Ap12 may at times understate the positive component of this swing, since there is certainly some pressure recovery at the divergent section downstream of the tube 'throat' or narrowest point, but are unlikely to understate the magnitude of the negative swing, since there is no corresponding phase of reverse flow (see Figure 3).

In contrast the cycle in Figure 4(a) involves a tube that is only lightly compressed beyond the initial collapse. The mean flow-rate is only 15% of that in the 'conventional' cycle, so that with more tube cross-sectional area available the fluid velocities and the viscous pressure drop are much lower. Inertial effects within the tube thus appear relatively larger. All this is strongly suggestive that different physical mechanisms are at work in the two types of oscillation. If so, this would partly explain the confinement of the Figure 4 modes to 'islands' far from their expected position on the characteristics in terms of frequency.

4.2. THE RELEVANCE OF CHOKING

Oscillatory operating points are commonly attributed to the mechanism of 'choking' in the tube throat. Following a compressible gas dynamics analogy (Shapiro 1977), sufficiently high flow-rates generate enough viscous drag that the internal pressure and tube cross-sectional area fall sharply along the tube, bringing the fluid velocity up to the decreased wave-speed. With the aid of, for example, an externally applied constriction, transition through to supercritical flow velocities can be achieved in steady flow, subcritical flow being restored at a shock-like transition where the tube widens to meet the downstream rigid pipe. In the absence of some such artifice, a sub- to supercritical flow site cannot be maintained in a uniform tube with steady flow. The flow becomes unsteady (oscillation) or diverges to a sustainable operating point, the 'choice' depending on the global stability of the system including surrounding resistances and inertances.

While it has not been shown that supercritical flow velocities are attained during an oscillation cycle, there is strong indirect evidence (Brower & Scholten 1975) that this is the case for thinwalled tubes. By definition supercritical flow prevents the upstream transmission of pressure wave information originating at the throat, although long- itudinal tension in the tube wall can still offer a mechanism for upstream transmission of short wavelength waves (Cancelli & Pedley 1985).

Even when the flow is subcritical, the effective distance from the throat to the upstream end of the tube in terms of wavelengths is increased by the downstream

MAPPING COLLAPSED TUBE INSTABILITIES 149

advection of pressure waves at the fluid velocity. Thus, considerable attenuation of pressure excursions initiated at the throat is to be expected at the upstream end of the tube. Despite this, and the proximity of the throat to the downstream end, the magnitude of the excursion of pl during a cycle in Figure 4(d) is greater than that ofpz. In Figure 4(a), (c), (e) and (f) the magnitude of the prexcursion at high frequency is comparable to that of Pz, whereas Figures 6, 8 and 9 (and all the illustrations of waveforms in (Bertram 1986a) depict modes where the pl-excursion is relatively small. These qualitative discrepancies are suggestive of a different mechanism at work in the Figure 4 modes.

A satisfactory explanation of the phase changes in Figure 4 eludes us. In Figure 4(d), Pl is in phase with P2 presumably because the delay in transmitting the pressure wave from the throat to the upstream end of the tube is one or more whole cycles (4.8 ms) longer than that to the downstream end. This could be both because the throat is nearer the downstream end than the upstream and because pressure waves are advected with the flow. However, this would again suggest that the pz-excursion should be larger than that of pl.

Evidence on whether critical flow velocities are attained during the oscillation cycles in Figure 4 must unfortunately be based on rather indirect measurements. Spatially- averaged instantaneous flow velocity is calculated from the measured Qa and tube cross-sectional area A at the throat, while pressure-wave velocity is derived from the known relations between transmural pressure PrM and A as modified by axial tension and tube profile (Bertram 1987). Under the static conditions in which such relations have been measured so far, dA/dprM becomes so large as the tube collapses that the calculated pressure-wave velocity reaches very low values (1-2 m s -1 at the centre of a 17D tube at the axial strain used here). On this basis critical velocities are almost certainly attained even at the moderate flow-rates of the modes in Figure 4. However, dynamic elastic properties as the tube collapses have not been measured. The stiffness of the tube during high-frequency oscillation may be much greater, as a result of tube profile differences and visco-elasticity. Were this the case the wave-speed could remain above the flow velocity throughout the cycle.

If the mechanism of 'choking' underlies the instabilities at the majority of operating points, the tendency shown in Table 1 for a smaller unattainable zone in a longer tube is apparently unexpected. In general there is more opportunity for choking in a long than a short tube, in that the reduction of internal pressure by viscous friction and, once taper starts, by conversion to kinetic energy, will be greater. However, a different picture emerges when the oscillatory instabilities are considered. As noted in Section 3, the total prevalence of oscillatory and divergent instability was greatest in long tubes.

4.3. POSSIBLE MECHANISMS FOR DIFFERENT MODES

The discontinuous changes in oscillation type, between regions of different frequency band, or between regions of different waveform shape as in Figure 6, pose questions about what is occurring at each such step. It was suggested (Bertram 1986a) that regions of increased frequency represent harmonics of a fundamental, albeit in such a nonlinear system that frequency ratios become inexact. This idea remains feasible for the transition from low to intermediate frequencies. However, as on a wind musical instrument, the spacing between such harmonics should progressively decrease with frequency, whereas the gap between the intermediate and the high frequencies is larger than that between the low and the intermediate ones. It is therefore concluded (as in

150 C, D. BERTRAM ET AL.

Bertram, 1986a) that the mechanism of the high-frequency oscillations may be different from that of the lower-frequency ones.

Weaver & Pffidoussis (1977) observed two different modes of oscillation in a thin-walled tube of length 12.7 diameters, one in the range 70-95 Hz and the other, starting at 155 Hz, taking over at a well-defined transition point as flow-rate was increased. However, in their case the higher-frequency (and larger amplitude) mode consisted of alternate collapse half-cycles in orthogonal directions, the major axis of the collapsing cross-section in one half-cycle becoming the minor axis in the next. No such mode has been observed in the tubes used here. At Ape 2 values above those where large-amplitude periodic oscillations occurred, again in thin-walled tubes, Bonis (1979) observed 'vibrations' near the tube exit, of small amplitude and 'much higher frequency'. These he attributed to vortices engendered by separation downstream of the tube throat.

The isolated examples of very high-frequency (>500 Hz) oscillation seen at the outer limits of pressure and flow-rate may not be very important. Presumably, during flow in such highly compressed tubes there are opportunities for extremely localized vibration of the pressed-together tube walls against each other. Such local modes would not necessarily depend on any global parameter such as tube length or diameter.

The present results do not support an oscillation mechanism in which the time taken for waves to propagate to the upstream end of the tube and back is what governs the frequency. If this were the case, one would expect to see a systematic variation in frequency with the inverse of tube length, over an eight-fold range for the four tubes used here. The actual range of minimum low frequencies, from 2-0 to 4.3 Hz, (see Section 3), while systematic, is closer to a two-fold spread. This may be explicable solely on the basis of the increased difficulty of collapsing a short tube: axial arc length must increase relatively more in a short than in a long tube for a given degree of central collapse. However, higher-frequency modes show no greater dependence on tube length. Instead, tube length seems to determine whether a mode of a particular frequency will occur, but has little effect on the frequency itself. This finding is in accord with theoretical expectation: the mechanism proposed by Cancelli & Pedley (1985), who tentatively identified the oscillations in their model (based on hysteresis in the process of unsteady flow separation) with an observed high-frequency repetitive mode (Bertram, 1982), does not involve dependence of frequency on tube length.

A large number of points could not be assigned a particular oscillation frequency. Those of 'nf' type, where the noise-like fluctuations extend to high frequency, are attributed to turbulence (see also Bertram 1986a). [A second potential source of noise is cavitation. The minimum Pz reached during low-frequency oscillations is subatmos- pheric (see Figures 6, 8(a) and 9). The exiting fluid has been observed to become slightly 'misty' once per cycle, and this may be attributed to air coming out of solution. This may explain the noisiness of the P2 traces at P2 < 0.]

Irregular oscillations of larger amplitude, such as that shown in Figure 8(b), have not been reported previously. One might attribute these modes to a defect of the experiment whereby conditions are not being held sufficiently constant for the true repetitive behaviour to be manifest. However, increasing experience and much re-examination of the apparatus has led to the realization that these modes are sufficiently stable and prevalent to warrant analysis in their own right as possible examples of chaotic dynamics (Thompson & Stewart 1986). The results of such analysis will be presented in a forthcoming paper.

Even excluding the >500 Hz modes from consideration, the co-existence of so many

MAPPING C O L L A P S E D TUBE INSTABILITIES 151

modes of oscillation in a relatively simple experimental system represents a theoretical challenge. The difficulty of showing that a particular theoretical model of collapsible tube oscillation, of which there are now several, applies to a particular experiment, has been noted (Bertram & Pedley 1982). Up to now each independent laboratory has usually described oscillations in a single region of variable space, as revealed by continuous dependence of frequency on, for instance, flow-rate. (There is a notably large discrepancy in the reported magnitudes of these oscillations, from small perturbations up to violent limit cycles as here, even amongst those working with thin-walled tubes.) Similarly, in general, each theory has predicted one mode of oscillation, without discontinuities of frequency. Short of using a different theory to explain each region, it will be of interest to see what factors need to be included in a model to reproduce such discontinuities. The coexistence of different types of behaviour at a single operating point, prominent in these results, would also seem to demand fresh thinking. Theoretical modelling of all the complex behaviour seen here is a considerable challenge, but it would dearly be desirable to identify a single model which will link up all well-documented experiments in the field.

5. CONCLUSIONS

The findings in this paper are here summarised. The frequency of self-excited oscillation in a Starling resistor is not strongly dependent on tube length, at either the low frequencies where the most non-sinusoidal and largest magnitude oscillations occur, or the higher frequencies. Oscillations with an identifiable repetition frequency occur in discrete frequency bands corresponding to different regions on the pressure- flow characteristics. Within each such region, frequency increases systematically with flow-rate and with increasing tube compression by negative transmural pressure as measured at the downstream end. Large-magnitude oscillations occupy these frequency bands. The intermediate band is reached from the low-frequency band by a frequency-doubling or -tripling event that resembles a jump to a higher harmonic. The high-frequency band is further in frequency ratio from the intermediate band and probably arises from a different mechanism. The predominant effect of tube length is to predispose the system to a particular mode of oscillation. Intermediate frequencies are absent in the shortest tubes while they replace high frequencies in the longest. Short tubes also display a greater tendency to static instability, i.e. operating points that are unattainable by variation of external pressure. Such points may be rendered stable by high downstream flow resistance, and oscillatory by low resistance. Regions of steady flow with the tube collapsed or open, of oscillatory instability, and of static instability, may be mapped on the pressure-flow characteristics for a given set of parameter values. Regions of hysteresis, where behaviour depends on the direction of approach, are also found. Aberrantly high frequencies are observed along one such boundary at moderate pressure drop and flow. These modes are considered to arise from a different mechanism again.

ACKNOWLEDGEMENTS

We thank the Australian Research Grants Committee for their support of this work. The paper was drafted while C.D.B. was the recipient of a Visiting Fellowship from the Science and Engineering Research Council of Great Britain. Valuable discussions with and suggestions from Professor R. D. Kamm are gratefully acknowledged.

152 C. D. BERTRAM E T AL.

REFERENCES

BERTRAM, C. D. 1982 Two modes of instability in a thick-walled collapsible tube conveying a flow. Journal of Biomechanics 15, 223-224.

BERTRAM, C. D. 1986a Unstable equilibrium behaviour in collapsible tubes. Journal of Biomechanics 19, 61-69.

BERTRAM, C. D. 1986b An adjustable hydrostatic-head source using compressed air. Journal of Physics E: Scientific Instruments 19, 201-202.

BERTRAM, C. D. 1986c Collapsed tube instability during flow: effects of tube length. Fifth International Conference on Mechanics in Medicine and Biology, Bologna, July 1-5, 1986, pp. 91-94.

BERTRAM, C. D. 1987 The effects of wall thickness, axial strain and end proximity on the pressure-area relation of collapsible tubes. Journal of Biomechanics 20, 863-876.

BERTRAM, C. O. & PEDLEY, T. J. 1982 A mathematical model of unsteady collapsible tube behaviour. Journal of Biomechanics 15, 39-50.

BONIS, M. 1979 Ecoulement visqueux permanent dans un tube collabable elliptique. Th~se de Doctorat d'Etat, Universit6 de Technologie de Compi~gne.

BROWER, R. W. & NOORDERGRAAF, A. 1973 Pressure-flow characteristics of collapsible tubes: a reconciliation of seemingly contradictory results. Annals of Biomedical Engineering 1, 333-355.

BROWER, R. W. • SCHOLTEN, C. 1975 Experimental evidence on the mechanism for the instability of flow in collapsible vessels. Medical and Biological Engineering and Computing 13, 839-845.

CANCELLI, C. ~£ PEDLEY, T. J. 1985 A separated-flow model for collapsible tube oscillation. Journal of Fluid Mechanics 157, 375-404.

CONRAD, W. A. 1969 Pressure-flow relationships in collapsible tubes. IEEE Transactions on Biomedical Engineering 16, 284-295.

FENSTERMACHER, P. R., SWINNEY, H. L. & GOLLUB, J. P. 1979 Dynamic instabilities and the transition to chaotic Taylor vortex flow. Journal of Fluid Mechanics 94, 103-128.

JENSEN, O. E. ~£ PEDLEY, T. J. 1989 The existence of steady flow in a collapsible tube. Journal of Fluid Mechanics 206, 339-374.

LIGHTHILL, M. J. 1975 Mathematical Biofluiddynamics. Philadelphia: Society for Industrial and Applied Mathematics.

SHAPIRO, A. H. 1977 Steady flow in collapsible tubes. ASME Journal of Biomechanical Engineering 99, 126-147.

THOMPSON, J. M. T. & STEWART, H. B. 1986 Nonlinear Dynamics and Chaos. Chichester: John Wiley & Sons.

UEDA, Y. 1980 Steady motions exhibited by Duffing's equation: a picture book of regular and chaotic motions. New Approaches to Nonlinear Problems in Dynamics (ed. P. J. Holmes), pp. 311-322. Philadelphia: Society for Industrial and Applied Mathematics.

WEAVER, D. S. & PAiDOUSSIS, M. P. 1977 On collapse and flutter phenomena in thin tubes conveying fluid. Journal of Sound and Vibration 50, 117-132.

A P P E N D I X : N O M E N C L A T U R E

VARIABLES AND PARAMETERS P

Ap a A R nD k

m ~w

pressure (kPa) time-averaged (pressure) pressure difference volume flow-rate (ml/s) internal cross-sectional area resistance to fluid flow of length n inside diameters constant relating Ap and Q", defining the magnitude of a non-linear resistance; the units are kPa/(ml/s) z when m = 2 exponent describing the power-law relation between Ap and Q in a resistance the Womersley number (ratio of tube radius to oscillatory boundary layer thickness) = a~[(to/v), where a = radius, to = radian frequency and v = momentum diffusivity.

MAPPING COLLAPSED TUBE INSTABILITIES 153

SUBSCRIPTS 1 upstream of the collapsed tube 2 downstream of the collapsed tube e external (pressure), outside the tube u far upstream (pressure), the driving head

SUPERSCRIPTS high, medium and low (downstream resistance) h, m, l

MODES nf C L I H U D

low-amplitude broad-band (pressure) fluctuations, tube collapsed steady flow, tube collapsed low-frequency oscillation intermediate-frequency oscillation high-frequency oscillation oscillation with collapse during less than half the cycle oscillation with collapse during more than half the cycle

NONDIMENSIONALIZATION

The pressure unit (PK) is F/(r3), where F is the flexural rigidity of the tube wall and r is the logarithmic mean radius of the tube (see Bertram, 1987). The other two fundamental units are the tube internal diameter (D) and the fluid density (p). From these are derived a unit of velocity u (= P~/P--~r/~) and a unit of flow-rate Q (= ~D2u/4). The unit of resistance magnitude k is then pK/(Q2), or equivalently (4[~)2p/D 4. The unit of inertance is (4/:r)p/D; that of time is