Chem. Eng. Commun. Vol. 28 pp. 355-367ooe8-6445/8 4 I 28M-A3ss$ 1 8.50/0

@ Cordon and Breach, Science Publishers, Inc., 1984Printed in the U.S.A.

PRESSURES GENERATED BY ABUBBLE CLOUD COLLAPSE

G.L. CHAHINE

Tr acor H ydr onautics, I ncorpor atedLaurel, Maryland

(Receiued March 24, 1983;in final form August 22,1983)

The collapse of a low density cloud of bubbles is studied using matched asymptotic expansions. Equationsdescribing the motion and deformation of an individual bubble accounting for its interaction with the otherbubbles in the cloud are derived. Numerical results for the case of a symmetrical cloud configuration show adramatic increase of the pressure generated at the end of the collapse due to collective effects. This explainsthe observed high erosion associated with cloud cavitation.KEYWORDS Cloud cavitation Bubble interaction Bubble dynamics Bubble collapse

Bubble cloud

INTRODUCTION

Downstream of the unsteady sheet cavity formed on a cavitating propeller blade, aregion of high population of tiny bubbles can be observed and is especially known to beassociated with erosion. These clouds are either detached from the frothy mixture at thetrailing end of the unsteady sheet, or generated downstream in a finite region of theliquid where significant fluctuating pr€ssures exist.

As the pressures generated by single bubble collapse are not strong enough toexplain the intense erosion in the subject region, and the high forces required forexample, to bend the trailing edge, cloud cavitation has been held responsible since VanManen's work.r This is supported experimentally by a very close correlation betweenthe dynamics of these clouds and the sharpest and highest pressure pulses detected onan oscillating hydrofoil.2 Similar phenomena have been observed with ultrasoniccavitation.3

To our knowledge, since the early work of Van Wijngaarden4 only a fewpublications by M/rchs-7 have dealt theoretically with the problem of "concerted

cavity clusters collapse". Van Wijngaarden4 considered a uniform, unidimensional,low void fraction layer of identical spherical bubbles on a wall, and derived theequations of motion neglecting convective and dissipative effects. Taking into accountthe individual bubble radial motion and translation, he found a considerable increaseof the pressure along the wall due to collective effects. M/rch considered a densespherical (or cylindrical) cluster, and treated the motion of the cluster surface as thepropagation of a spherical (or cylindrical) shock wave which annihilates any vaporbubbles along its path. Although very interesting in determining the global behavior ofthe cloud of bubbles, this approach is limited to high void fractions and is not capable,in its present form, of calculating accurately the pressure.field, since it does not allowthe bubbles to contain noncondensables.

3s6 G.L. CHAHINE

We suggested recently,s to extend the singular perturbation theory we havedeveloped to study the interaction of two collapsing bubblese'r0 to the collectivecollapse of a low-void fraction multibubble cloud system. In an on-going program thisis one of the approaches used to study the problem. We will briefly present here theasymptotic theory,rl and the numerical results obtained for a few particular

configurations of bubble distribution which are concerned with the individual bubbledynamics, as well as the pressures generated.

ASYMPTOTIC APPROACH

Let us consider the case of a cloud of low void fraction in an unbounded medium.

Provided that the characteristic size rro of a bubble is small compared to its char-

acteristic distance /o from its neighbors we can assume that interactions are weak

enough so that, to the first order of approximation, and in absence of relative velocity

with the surrounding fluid, each of the individual bubbles reacts to the local pressure

variations spherically, as if isolated. With this assumption we can consider, with no

further restrictions, the behavior of any given distribution of N bubbles in an initiallyknown volume; mutual bubble interactions, individual bubble motions and de-formations are taken into account in the higher orders of approximation. These orders

are studied by means of the method of matched asymptotic expansions. The "outer

problem" is that considered when the reference length is chosen to be lo. This problem

is concerned with the macrobehavior of the cloud, and the bubbles appear in it only as

singularities. The "inner problem" is that considered when the lengths are normalizedby rro.The solution of this problem applies to the microscale of the cloud, i.e., to thevicinity of an individual bubble of center Bt. The presence of the other bubbles, all

considered to be at infinity in the "inner problem", is sensed only by means of thematching condition with the "outer problem". That is to say, physically the boundaryconditions at infinity for the "inner problem" are obtained, at each order ofapproximation, by the asymptotic behavior of the outer solution in the vicinity of B'.Thus if one knows the behavior of all the bubbles except Bt, the motion, deformationand pressure field of this cavity can be determined. In first approximation (e : r6o/le) is

equal to zero, all the bubbles behave as in an infinite medium and the time dependenceof their radii, a'o!), is given by the Rayleigh-Plesset equation. This first determinationof the whole flow field sets the boundary conditions at infinity for the following order ofapproximation. The same process is then repeated for the successive higher orders.

When the liquid is assumed inviscid and incompressible, and the flow is potential, theanalytical study we have performedll shows that, up to the order e 3, concerning thedynamics of a bubble Bi, the remaining cavities can schematically be replaced by aunique equivalent bubble, Gi, whose growth rate and position are determined by thegeometrical distribution and the growth rate of these cavities. The strength of thesource representing Gt, q'f , and its distance,l'&, from Bi arc defined by:

q':l l ' f :Lqilt6,j

g-tn' q'f I (\fl ' : \

9,i ' q',/(l ' i lt,

(1 )

(2)

BUBBLE CLOUD COLLAPSE 357

where eioand eijare respectively unit vectors of the directions BiGi and B'Bi (Figure 1),andnistheorderof approximation.If ?is istheanglebetween BiGiandafield point,M,the equation of the surface of the bubble Btcan be written in the form:

R(g'r, t) : ab?) + eair(t\ + e2lal(t) + fi@.cos gtrl

+ e3[a5(r) + / ' . ( r )cos Ote + gir( t )Pr(cos gio)] + . . . , (3)

where ab\) is given by the Rayleigh-Plesset equation. The other corrections areobtained by solving differential equations which differ from those we obtained in thecase of two bubbltte'10 only by the fact that the right-hand sides (independent of thesought unknown function) are sums of N-l terms rather than a unique (j ) term. Whenall the initial radii of the bubbles in the cloud are identical, these right-hand sides areobtained by multiplying the two-bubble case right-hand sides by one of the geometricalcons tan ts c1 , c2 , c3 l

cr : I (lolld),j

cz : L4oll,i lr.cos gti/cos gir, g)

j

c: : I (rofl ' i l ' - Pr(cos oii) lrr@os gtr).

We can now compute the U.njuio, of Bi by solving the obtained differentialequations using a multi-Runge-Kutta procedure. The behavior of the whole cloud canthen be obtained. This appears at first to be a very long task. However, there exists foreach of the equations, what we will call a unit-solution from which the real solution canbe immediately deduced for any other initial bubble size. Indeed, this is evident for a'o()where one can show that if ao(r) is the nondimensional solution for a bubble of unitinit ial radius the solution e's, for a bubble of normalized init ial radius l" is such that:

a 'o(Lt) : Lao1).

PRESSURE FIELD

Once the "inner problems" are solved, the nondimensional outer potential, S, can bewritten:

O( t t , t ) : * e 2

where bars denote nondimensional "outer" quantities, and tildes nondimensional"inner" quantities.

0 : O.eTf r lo, 4 ' , : q i -Tlr io, r i : r i l lo . Q)

T is the characteristic time of the bubble collapse and rt is the distance between a fieldpoint M and Bt. The Bernoulli equation enables one to calculate P using (6). In the

(5)

-T[+ +,+ *. r(# -*,o,0") * o(. ')], (6)

G.L. CHAHINE

-o(.)()()(,o

o

d)

o

q)

c)

z.

r r )

l

I

ku o

358

BUBBLE CLOUD COLLAPSE

nondimensional form we have:

3s9

(e)

p(M,r:ry: -e#-),-tTor. (8)AP is the amplitude of the pressure driving the collapse and t : tlT, where

T : rbo\/ plLp.

In the following, we will consider as an illustration a symmetrical field of bubbles;any bubble has the same geometrical position relative to the others, and thus the samebehavior. The general expression (8) simplifies considerably to become:

p(M,t): (eQo * ezdr + ,rd, * eud)T (il

- rnfr , t f99,9"\ - . -d. l / r \ lz- " \ -'- )-'"; lv I (.i,)l

+ o(e4) (10)

In this expression, the two first summations are geometrical constants similar to c1,c2, c3. The last one is more complex, but is more easily calculated when written asfollows:

lul(i)l':+(#)'-, (11)of the direction BiM.If one knows the direction, MVo, of

rst order of approximation, and if at' is the angle BiMVo

lul(i)':(ITs)' 02)

where ei^ is the unit vectorthe velocity at M, at the fi(Figure 1), then:

EXAMPLES: SPHERICAL SHELL OF BUBBLES

We consider a distribution of bubbles centered on the surface of a sphere, and we admitthat each of the bubbles has the same position relative to the others. In this case theexpression (10) is valid, and the numerical computation time is reduced. As illustrationwe will consider the pressures generated a) in the center of the sphere; b) at the locationof the bubble Bi if it were removed and c) in a point outside the cloud at a distance rbofrom Bt. We will compare the results with the isolated bubble case.

Knowing the initial bubble configuration and thus cr, c2,andc. the relation betweenthe cloud radius, R, and /o is: R : ilo ctlcz. In the cloud center, case (a), the threesummations in (10) have respectively the values (N/R, N/R and 0). In case (b) thesevalues are (cr,cr,and c.), and at a distance rbo from Bt the values are approximated by(c, + €-r ,c2 - € t , ( r , + , - ' ) ' ) .

Bubble Dynamics

Various spherically symmetrical cloud configurations were investigated numericallyfor a sudden jump in the imposed ambient pressure. In Figure 2, the results of five

G.L. CHAHINE

or / r 8 r 3 = r ' 3 3 N V r S l O - I V N O I S N S W I O N O N

360

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BUBBLE CLOUD COLLAPSE 36T

different computations are compared expansions being conducted up to e 3. The ratio,e : r6of Io, was kept constant and at a value of 0.05. The cases of two, three and twelvebubbles are presented together with that of an isolated bubble. The fifth case is anintermediate situation between the configurations of three and twelve bubbles. Thiscase is arbitrary and is only determined by the choice of c,, c2,ltrd c.. In each case thevariation with time of the distance, BiEi,between the extreme point on a bubble Ei, andits initial center, Bi, is chosen to represent the bubble dynamics. Taking the bubblecollapse in an unbounded fluid as reference, it is easy to see from Figure 2 howincreasing the number of bubbles changes the dynamics of the one studied. We canobserve first that, during the early slow phase of the implosion process, the collapse issignificantly delayed. At any given nondimensional time the distance between Bi and Ei(and simultaneously the bubble characteristic size) is greater when the number, N, ofinteracting bubbles increases. Then, in the final phaseof the implosion the tendency isreversed: the phenomenon speeds up and, in a shorter total implosion time, the finalvelocities of the motion are higher when N increases.

Generated Pressures

To examine this let us consider the variations of the pressure generated at a distance /oby the collapse of a bubble of initial radius rro in an infinite medium. As we can see fromFigure 3, the perturbation pressure, i.e., the difference between the pressure at /o and thefar-field pressure, is negative for t < 0.75. This observation is important in the sensethat a fictitious bubble placed at the distance /o from this spherical bubble will sense aless important and more gradual increase in the surrounding pressure. In theconsidered case, instead of a sudden nondimensionaljump of the pressure from 0 to 1,P surges only to 0.84, then rises slowly, not attaining I unti l t < 0.75. This would effectthe bubble dynamics exactly as observed in Figure 2, namely a less violent start of thecollapse. As a result, we find at the end of this process a larger bubble than would beobserved in an infinite medium. This added to the fact that in the later stages (r = 0.75)the driving pressure increases up to 2.25 times the far-field pressure makes thesubsequent end of collapse much more violent.

In Figure 4, we have represented the pressure peaks generated at the end of thecollapse for the same value of N, Pno,,and e as in Figure 2.Thecurves in solid lines aresimilar to that in Figure 3 and represent the pressure generated by the cloud at thecenter of one bubble, Bi, in its absence. The cumulative effect is obvious, since thispressure increases dramatically from 0.4 AP for N : 2 to 4400 AP for N : 12. Thesenumbers show only orders of magnitude and should not be considered accurate sinceother scales for times, pressures and lengths are needed at the end of the collapse.However, it gives a good indication of how tremendous pressures can be generatedwith an increasing number of interacting bubbles. In the same Figure 4 are drawn thepressure variations with time in the cloud center and at an equivalent distance for anisolated single bubble. As expected the ratio between the two pressures becomes verylarge when N increases. The cumulative effect is even stronger when one looks at thevicinity of a given bubble at a distance rbo from its center. At the end of the collapsethe pressure generated, shown in Figure 5, are orders of magnitude higher than whenthe bubble is isolated.

G.L. CHAHINE362

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363BUBBLE CLOUD COLLAPSE

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366 G.L. CHAHINE

Figure 6, shows the importance of the gas content of the bubbles. Increasing {o from0.1 in Figure 5 to 0.2 here dramatically reduces the generated pressures. This comesmainly from the fact that the cushioning effect of the gas reduces significantly thevelocities attained at the end of the implosion.

CONCLUSIONS

This study has shown that, even for very low void fractions, collective bubble collapsecan generate pressure orders of magnitude higher than those produced by singlebubble collapse. This would tend to explain the observed high erosion intensities andthe bending of trailing edges. The cumulative effect comes from the fact that theinteraction increases the driving pressure of collapse of each individual bubble. Thisaugments the violence of its implosion and thus the interaction with the other bubbles.Thus, each bubble ends its collapse not under the effect of a pressure of the same orderas the ambient, but orders of magnitude higher. This cumulative effect would not existif the void fraction is high enough for the cloud to behave as a single bubble. This leadsus to believe that there exists a critical value for the void fraction for maximum erosion.

One major assumption of the study is that the liquid is incompressible. Thisassumption is valid as long as the fluid velocity does not exceed the speed of sound. Forsingle bubble dynamics this does not usually happen until the final phase of thecollapse. Here, however, two factors contribute to limit the validity of the assumption.First, the rate of implosion is higher and second, more important, the velocity of sounddrops considerably when the void fraction increases. This introduces a possible seriouslimitation on the values of the accepted void fraction, d, (or on e since d - €3) forthe validity of this approach. Another l imitation, of the same nature, appears in theasymptotic theory as € < l/cr. This seems to fix the domain of application of thepresent method to values of a less than 10-3. Both l imitations would become lessimportant if the model were modified to allow for a compressible behavior at themacroscale, in which case a delay in the propagation of the far-field pressure into thecloud would be accounted for by a finite speed of sound. In the same way, the numberof bubbles interacting with Btwould then be l imited to a radius determined by thespeed of sound and the period of oscillation of the bubble.

ACKNOWLEDGMENTS

This work was supported by the Naval Sea Systems Command, General Hydrome

chanics Research Program administered by the David Taylor Naval Ship Research

and Development Center under contract number N00014-82-C-0009.

REFERENCES

1. Van Manen, J.D., "Bent Trail ing Edges of Propeller Blades of High Powered Single Screw Ships",I nter nat ional Shipbuilding Progre ss, I 0( I 0 1), 3 -7 (1963).

2. Bark, G., and Van Barlekom, W.8., "Experimental Investigations of Cavitation Noise", ProceedingsI2th Symposium on Naual Hydrodynamics,4T0-493 (1979).

BUBBLE CLOUD COLLAPSE 367

3. Hansson, I., and M/rch, K.A., "The Dynamics of Cavity Clusters in Ultrasonic (Vibratory) CavitationErosion", Journal of Applied Physics, 51, 465I-4658 (I980).

4. Van Wijngaarden, L., "On the Collective Collapse ofa Large Number of Gas Bubbles in Water",Proceedings I Ith International Congress of Applied Mechanics, Springer, Berlin, 854-861 (1964).

5. M/rch, K.A., "Concerted Collapse of Cavities in Ultrasonic Cavitation", Proceedings AcousticC auitation Me eting, London, 62-7 0 (I97 7 ).

6. M/rch, K.A., "Energy Considerations on the Collapse of Cavity Clusters", Proceedings IUTAMSymposium on the Mechanics of Bubbles in Fluids, Pasadena, California, June 1981.

7. M$rch,K.A., "Cavity Cluster Dynamics and Cavitation Erosion", ASM E Cauitation and Polyphase FlowForum, l -11 (1981) .

8. Chahine, G.L., "Experimental and Asymptotic Study of Nonspherical Bubble Collapse", hoceedings,IUTAM Symposium on the Mechanics of Bubbles in Fluids, Pasadena, California, June 1981.

9. Chahine, G.L., and Bovis, A.G., "Pressure Field Generated by Nonspherical Bubble Collapse",Cauitation Erosion in Fluid Systems, ASME, New York, 27-40 (1981).

10. Bovis, A.G., and Chahine, G.L.,'Etude Asymptotique de I'Interaction d'une Bulle Oscillante avec uneSurface Libre Voisine", Journal de Mecanique,20(3),537-556 (I981).

11. Chahine, G.L., "Asymptotic Theory of Collective Bubble Growth and Collapse", Proceedings 5th Int.Symposium on Water Column Separation, Obernach, Germany, September 1981.