Chaotic bubblingD. J. Tritton and C. Egdell

Citation: Physics of Fluids A: Fluid Dynamics (1989-1993) 5, 503 (1993); doi: 10.1063/1.858874 View online: http://dx.doi.org/10.1063/1.858874 View Table of Contents: http://scitation.aip.org/content/aip/journal/pofa/5/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Chemical oscillations in bubbles: Resolving the mystery of chaotic sonoluminescence J. Acoust. Soc. Am. 115, 2381 (2004); 10.1121/1.4780285 Experimental Chaotic Bubbling AIP Conf. Proc. 676, 213 (2003); 10.1063/1.1612215 Noise induced stabilization of chaotic oscillations of cavitation bubbles AIP Conf. Proc. 524, 519 (2000); 10.1063/1.1309277 When bubbles go bad…. Chaotic shape oscillations of air bubbles in water J. Acoust. Soc. Am. 87, S55 (1990); 10.1121/1.2028273 Regular and chaotic bubble oscillations in periodically driven pressure fields Phys. Fluids 30, 3342 (1987); 10.1063/1.866466

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

213.246.241.238 On: Sat, 10 May 2014 17:26:12

Chaotic bubbling D. J. Trittona) and C. Egdell Department of Physics, University of Newcastle upon Tyne, NE1 7R V, England

(Received 10 June 1992; accepted 2 September 1992)

A short set of experiments on bubbling from a submerged orifice has been carried out to investigate whether the fact that bubbling is sometimes regular and sometimes irregular may be related to modern concepts of chaotic dynamics. A period-doubling sequence leading to chaotic behavior has been observed, followed by further developments in which intermittency of varying intermittency factor and “period-3” patterns are prominent.

Dripping from a faucet or tube is often cited as a conceptually simple system that exemplifies chaotic dy- namics.lT3 A general, although not detailed, analogy sug- gested to us that bubbling from a submerged orifice might exhibit similar behavior and thus provide another example with the additional advantage of greater technological rel- evance. Also there have been observations4” of bubbles occurring in twofold or fourfold patterns that, in the light of present knowledge, are suggestive of a period-doubling sequence. (One set of observations4 influenced the design of our own experiment-in particular with regard to the choice of liquid-but a similar behavior has been observed in markedly different circumstances.‘) We have carried out a short set of experiments which have confirmed this in- terpretation and shown interesting subsequent develop- ments.

The liquid used was a water-glycerol mixture of 24% by volume of water. Nitrogen from a gas cylinder was allowed to bubble into a 5 liter beaker of the liquid from an upwardly pointing vertical tube of internal diameter 1.2 mm, remote from the walls of the beaker. The pressure drop in the flow line was large compared with likely pres- sure fluctuations at the nozzle during the bubble formation process; hence, the bubbling was of the constant flow rate type.‘16 The flow rate was maintained constant during ob- servations of each bubbling mode but changed to find dif- ferent modes. It was measured using two “rotameters” (flow meters with a lifting float in a tapered tube) covering different ranges; their calibrations were checked by timing the filling of measuring cylinders. The bubbling induced some stirring of the liquid; observation of small suspended bubbles indicated that the resulting liquid velocity in the vicinity of the nozzle was small compared with the velocity directly associated with bubble formation, except possibly at the highest bubbling rates.

Evidence for the bubbling behavior is given below in terms of the output of a transducer. We also observed and photographed the bubbling in both steady and stroboscopic light. These visual observations were essential in providing a check on the interpretation of the transducer output (and a check that the presence of the probe did not significantly modify the bubbling).

The transducer was a hot-film anemometer (Dantec type 55R32) placed typically about 3 mm above the nozzle and 4 mm away from the axis of symmetry. It was ensured that the bubbles did not intersect the probe, which there- fore responded to movement of the liquid produced by

formation and release of the bubbles. This is a nonstandard way of using a hot-tilm anemometer and it is uncertain exactly what is being measured. However, this does not matter; one needs only a signal that shows the character of the process: periodic, period doubled, chaotic, etc. The an- emometer output was interpreted both by direct inspection of oscilldgrams and by digitizing it, using a 1Zbit A/D converter sampling at intervals of 286 psec, for further analysis.

Figures 1 and 2 show examples of the observations. In addition to AH,+, vs AH, plots like those in Fig. 2, plots ofHn.1 vs H, (notation in caption to Fig. 2) and similar ones of time intervals between maxima contributed to the interpretation. The AH,,+ l vs AH, plots are displayed here as the ones that most immediately illustrate the discussion below.

Table I lists the bubbling modes observed. [The flow rate and the bubbling rate have been nondimensionalized in the simplest available way, rather than in the more com-

(b)

FIG. 1. Examples of output of hot-film anemometer. Flow rates Q/(&g)ln (cf. Table I): (a) 37; (b) 60; (c) 83; (d) 110; (e) 130; (f) 165; (g) 205; (h) 225; (i) 345; and (j) 405. Time span is 1.43 set for (e) and (g); 0.43 set for all other cases. In traces (d), (h), and (j) the section shown has been selected to be within an ordered interval of intermittent behavior. Relative amplitudes of different traces are of no significance.

503 Phys. Fluids A 5 (2), February 1993 0899-8213/93/020503-03$06.00 @ 1993 American Institute of Physics 503 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

213.246.241.238 On: Sat, 10 May 2014 17:26:12

(a)

. . ‘i

(9) . .

. . .> - \ a&+ ‘: . .

*. :. . . . g.1 . . . . . , .\:- -_ -z

--I-.

Y&f :; .i . .,,;;.* . . _ .,Iy$.”

_x-

3::. 1 (bl

. . . I . . . 1 -y$./. . .

. .

-7 *& I;: ._-a : ,.., a&+&j -: - .*-

- .; . . . I (h)

. . .*.. . . A+-,.$:.: :

.

.:* . . . . . :

.- . . t . . . .( . . .: . . . . . _. . . . . _ +

%&+.’ f

*.‘. -& . .

b . . . . . L---L-

:- : . *. 9 * . . _.. . . ; z.. ;.

: k.-* :3.,; . .

. . if) ‘3.Y.

. . .

+

*. . . . . * *- .

. .+,&*

__.- -_- * .I. (i) . . ‘~~~-: . .~ ;: _ . :. : . . : $ . i .--::: ,;&+ i

:I *. . :. *:. .*. . . . . . $#.::>; . .

FIG. 2 Examples of analysis of hot-film anemometer output, based on maxima of records such as those in Fig. 1. Defining H, as the height of the nth maximum and AH, as (H,,--HE-,), each plot shows AH,,, vertically against AH, horizontally for observations over 6.6 sec. (The extent of each axis corresponds to the largest value of 1 AHH, 1 occurring during this time.) Flow rates Q/(&g)” (cf. Table I): (a) 77; (b) 95; (c) 130; (d) 155; (e) 205; (f) 225; (g) 270; (h) 365; and (i) 405.

plicated ways sometimes used6 on the basis of supplemen- tary physical considerations. Information in the caption to Table I enables conversion to any preferred form.) At small flow rates, the bubbling is periodic and the bubbles are widely spaced. As the flow rate is increased, the spacing decreases; the formation of each bubble may be influenced by the presence of the previous one. The sequence in Table

I then occurs. Jargon in this table requiring explanation is discussed below. The number of bubbles per unit time is taken as the number of maxima per unit time in the ane- mometer output; for the regular bubbling modes (or inter- vals) this identification was verified by the visual observa- tions, and it seems reasonable to extend it, at least as a good approximation, to the chaotic ones.

TABLE I. Summary of the observations. Values of other nondimensional parameters (based on tabulated data for water-glycerol mixtures): rr/pgd2=3.8; $/pdgd3=0.10. (Notation: Q=volume flow rate; n=number of bubbles per unit time; d=tube internal diameter; g=acceleration due to gravity; p=liquid density; a=surface tension; p=liquid viscosity.)

Bubbling rate ,,d’“,g’/’

Range Flow rate Q/C&)“’ Character

Overall average

Ordered intervals*

045 periodic 45-85 period doubling 85-105 limited chaos

105-145 intermittency (2) 145-200 large-scale chaos 200-320 intermittency (3 fast) 320-385 “two-level” chaos 38%> 420 intermittency (3 slow)

O-0.30 0.3CO.44 0.44-0.50 0.51 ztO.Olb 0.50-0.55 0.61-0.65 0.66-0.76 0.74-0.83’ 0.75-0.62d 0.61 AO.Olb 0.60*0.01b~e

‘When intermittent. bLittle variation with flow rate. ‘Systematically larger than overall average. dDecreasing with increasing flow rate. eNot significantly different from overall average.

504 Phys. Fluids A, Vol. 5, No. 2, February 1993 Brief Communications 504

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

IP: 213.246.241.238 On: Sat, 10 May 2014 17:26:12

The occurrence of the period-doubling sequence, as the way in which the first departure from periodic production of identical bubbles occurs, provides the most direct evi- dence for the relevance of concepts from nonlinear dynam- ics. In this sequence, the stages period 1, period 2, and period 4 have been clearly identified [Figs. 1 (a)-1 (c) and 2(a)], period 8 more tentatively so. The sequence leads, as to be expected, to chaotic behavior in which the pattern of fluctuations retains much of the structure of the sequence; e.g., the basic alternation of period 2 is still present [Fig. 2(b)]. In Table I this behavior is classified as “limited chaos,” using the nomenclature proposed in Table 2-1 of Ref. 7.

As with many systems exhibiting a transition to cha- otic behavior, subsequent developments [Table I) are quite complicated and show ranges of more ordered behavior between ones of chaotic. We have not, however, observed further ranges of complete order without intermittent bursts of chaotic fluctuations.

The first development after the period-doubling se- quence is a transition from “limited” to “large-scale” chaos (again using the nomenclature of Ref. 7). This occurs by the large-scale chaos appearing in intermittent bursts [Figs. 1 (e) and 2 (c)] and the intermittency factor (the fraction of the time that the bubbling is chaotic) increasing with increasing flow rate. We summarize this range in Table I as “Intermittency (2),” the 2 indicating that the most prom- inent feature of the ordered intervals is a period-Ltype alternation. It does not, however, imply that it is exactly period 2; indeed, the intermittency may be between limited chaos and large-scale chaos. Speculatively, this is type III intermittency in the usual classification,* with the super- critical bifurcation giving rise to the period-doubling se- quence being followed by a subcritical one.

At the top of this range, one has the least structured bubbling of any that we have observed [Figs. l(f) and 2(d)]. [There is a curious jump in the average bubbling rate (Table I), although in other respects the chaotic fluc- tuations occurring continuously in range 5 of Table I ap- pear similar to those occurring intermittently in range 4.1

Further developments involve particularly two pat- terns of ordered behavior with predominantly period-3 structures. One appears in the anemometer output as a sequence of a high maximum and two low ones [Fig. 1 (h)], the other as a sequence of two high and one low [Fig. 1 (j)]. (Approximately they correspond, respectively, to a big bubble and two small ones and to two big and one small; but there are significant variations in bubble shape as well as size.) They are also distinguished by the fact that the second type involves a lower bubbling rate even though it occurs at higher flow rates (Table I). We therefore denote the two modes as “3 fast” and “3 slow.”

The fast pattern appears intermittently [Figs. 1 (g) and 2(e)], with the intermittency factor decreasing with in- creasing flow rate to a minimum of about 0.14, so that one has predominantly ordered bubbling with only occasional bursts of large-scale chaos [Fig. 2(f)]. However, the inter- mittency factor then increases again [Fig. 2(g)]. Although

“3 fast” has a predominant period-3 structure, it is not pure period 3. In particular, in the vicinity of the minimum in the intermittency factor there is a clear period-6 struc- ture [Fig. l(h); note particularly the alternations in the details of the two small peaks]. Its association with period 3 suggests that the intermittency in this range may be of type I. Speculatively,- in the light of what happens next (ranges 7 and 8 of Table I), its failure to reach zero inter- mittency factor may be due to a growing influence of the “3 slow” mode as an alternative attractor.

Range 7 is designated as “two-level chaos,” because the maxima in the anemometer output are mostly large or small, with few of intermediate size [Fig. 1 (i)]; a histogram of the frequency with which H, falls in various ranges has two peaks. The chaotic aspect is primarily in the sequence of large and small. This range may be seen as a transition between ranges 6 and 8; a pattern resembling Fig. 2(h) would be given by superimposing Figs. 2(g) and 2(i).

We have examined ranges 7 and 8 more briefly than the other ranges, because the stirring of the liquid by the bubbling may now be significantly affecting the bubble for- mation process. Similarly we have not extended the obser- vations to higher flow rates. There is, of course, no reason to suppose that there are not further developments beyond the range of Table I.

The purpose of this Brief Communication has been primarily to note the existence of “chaotic bubbling.” A much more extensive program of experiments would be needed to know what generality our observations have be- yond indicating the relevance of a modern dynamical sys- tems approach; there is a wide variety of regimes of bub- bling associated with the different proportions in which surface tension forces, viscous forces, and inertia balance the buoyancy of a bubble and with the range of conditions of which constant flow rate and constant pressure are lim- , iting cases.6

ACKNOWLEDGMENTS

We are most grateful to Dr. L. Molyneux for the loan of the A/D converter and associated equipment and for providing programs for and advice on the data processing; and to Mr. G. Robb for extensive technical assistance.

aJCorrespondence to the first author at: Institut de Mecanique de Greno- ble, B.P. 53X, 38041 Grenoble Cedex, France.

‘P. Martien, S. C. Pope, P. L. Scott, and R. S. Shaw, “The chaotic behavior of the leaky faucet,” Phvs. Lett. A 110, 399 -( 19851. -I.

*D. J. Tritton, Physical Fluid Dykzmics, 2nd ed. (Clarendon, Oxford, 1988), pp. 408-410.

“X. Wu and Z. A. Schelly, “The effects of surface tension and tempera- ture on the nonlinear dynamics of the dripping faucet,“- Physica D 40, 433 (1989).

4F. W. Helsby and K. R. Tuson, “Behavior of air bubbles in aqueous solutions,” Research 8, 270 ( 1955).

‘J. F. Davidson and B. 0. G. Schtiler, “Bubble formation at an orifice in an inviscid liquid,” Trans. Inst. Chem. Eng. 38, 335 (1960).

‘R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978), pp. 322-330.

‘F. C!. Moon, Chaotic Vibrations (Wiley, New York, 1987), p. 39. “P. Berg& Y. Pomeau, and C.. Vidal, Order within Chnos (Hermann, Paris, and Wiley, New York, 1984), pp. 109 and 223-258.

505 Phys. Fluids A, Vol. 5, No. 2, February 1993 Brief Communications 505

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

213.246.241.238 On: Sat, 10 May 2014 17:26:12