NAVORD REPORT 4 15

UNDERWATER EXPLOSION PHENOMENA:THE PARAMETERS OF MIGRATING BUBBLES

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12 OCTOBER 1962

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U. S. NAVAL ORDNANCE LABORATOR'

WHITE OAK, MARYLANI:r-i

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IWVORD "185

UNDERWATER EXPLOS ION PHENOMENA: THE PARAMETERSOF MIGRATING BUBBLES

byHans G. Sr&-.

41' ABSTRACT: The migration of underwater explosion bubbles caused by y

'14• Cf •oyancy) affects the energ'y of the pulsation as well as per!-S .r mumradius in the second aad subsequent cycles. Experimental data on I ofexplosions in various depths are analyzed, and the bubble energy is Ined

for five cycles of the oscillation as a function of the strength of tion.The result is given in dimension]ass form which permits the calcula f

e-periods, maximum radii, and migration for a wide range of conditior•An pnergy balance shows the surising resui• that the bubble enerE he

"second cycle increases with increasing intensity of migration until imum

is reached at a condition of strong migiation. Beyond this point, ergyC , decreases again. This gain is found to be due to the decrease of t rgy

radiated by the bubble pulse.

PUBLISHED DECEMBER 1962

EXPLOSIONS RESEARCH DEPARTMNU. S. NAVAL ORDNANCE LABORATORY

WHITE OAK, YARYLAND

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KAVORD 4185

NAVORD REPORT 4185 12

UNDMMTER FUOSION PENOM -: THE P.A A4-MiS OF MIGRATING BUBB

In this report the attempt is made ti further the understanding ohydrodynamic processes associated with migrating explosion bubbleresults are obtained from an evaluation of experimental obse.-vati

y the use of involved theoretical calculations. The ir-port- . tcMuM r alculations is a graph showing the bubble energy ratios fo.i

of cycles of the osci'U.I-ion. Combined with simple equations this i

Ined permits the calculation of the bubble parameters for almost all p

tion. impjrtant conditions of common explosives detonated under water.f coming NOLTR 62-184 utilizes the results of this paper and presen

method for expedient calculations,

he Preliminary results of this study have been reported in previous

inum Although this final version shows litule differences, the previouergy the bubble energy ratios must be considered to be superseded.

rgy This report is part of a comprehensive study on the behavior of ebubbles. The work has been carried out umder Task No. 301-664/43and RUM-3-E-OOO/212-1/wFCo8-i0-00o4 PA OO2.

R. E. ODENING

Captain, USNCommander

J. ARONSONBy directton

ii

NAVORD 41b5

NAVORD WORT 4185 12 October 1962

UNDERWkTM EXPLOSION PHENOMENA: TIE PUWWIWS OF MIMRATING B" jB'.sF

"t *his report the attempt is made to Pwther the understand. ng of the various"rodynsmic processes associated with migrating explosion bubbles. Revealing

results are obtained from an evaluation of A-xperimental observations withoutthe use of involved theoretical calculations. The important outcome of thecalculations i-7 R graph showing the bubble energy ratios for the first fivecycl 5 of the oscillation. Combired with -JiMple equations thIs informationpermits the calculation of the bubDle parameters for almost all practicallyimportant condi-.ons of cmmon exulosives detonated under water. The forth-com.ýng NOLTR 62-134 utilizes the results of this paper and presents a simplemethod for expedient calculations.

celiminary results of this study have been reported in previous papers.",hough th s final version shows little differences, the previous data on

",he bubble ener *j ratios must be considered to be superseded.

This report is part of a comprehetisive study on the behavior of explosionbubbles. The work ! is been carried out under Task No. 301-664/43003/01040and RUME-3-E-0O0/212-l/WFCOWl.O-O04 PA 002.

R. E. ODENING

Captain, USNCcmmander

1. J. ARONSONBy direction

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CONTEWS

Page

I.* INTRODUCTION... ..0C....**. .~oo.......*.* o.o. soe. .... *- . see..o.o, 1II. EQUATION FOR THE BUBBLE ENERGY RATIO..... . ........ 31II. THE SURFACE CORRECTION TERM,, ........ ,, o.,o.,.,•o,,.* 6IV* THE MIGRATION COEFFICIEN2..o,° ........ ... e.o...ec... *.o...e .. 7V. EXPERWN]AL IP..T... ......... .... ,. ,..,s,....o , ... 12VI. THE ENERGY OF MIGRATING BUBBLES F ....... ............... ...... 13

LIST OF SY i-Oo.s... o ...................................... 23LIST OF REF CE. ......................... ......... o ........ 25

ILLUSTRATIONS

Figure Title Page

1 Evidence on Bubble Migration ...... .................. 152 Surface Correction..o.. oo.. .oe.*. 0....o...o.o.o. o....•,, 163 Positions and Radii of the Bubble Maxima....... ............

Figure 3a Firing Depth 130 ft. .......................... 17

Figure 3b Firing Depth 140 ft ........................... 18Figure 3c Firing Depth 150 ft.... ......... ...e . .... 19

4 Period Ratio T /1 .................................... 205 Period Ratio ?-Ti# O...~@ess~ses~s*O~s~ 21

6 Calculated iuble-Energy Ratios* .... 22

ACKNOWLEDGEMENT

The author wishes to express his appreciation for the helpful assistance ofMiss R. V. Tipton in the computational efforts of this project.

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NAVoRD 4185

I. ImTROIDCTION

There is an extensive literature wi the p'nsation .- lmbb..es produced byunderwater explosions, listed in references (a) and (1 ,-. Almost all of thiswork refers to non-migrating bubbles, i.e. it is assu ed that -he center ofthe bubble remains stationary. This assumption is valid for small chargesexploded in great depth, e.g. 1 lb in 500 ft. The bubbles of larg-r explosionsmove upward because of their buoyancy and thereby change the characteristicsof the pulsation narameters. Little is known today about the fundamentalbehavicr of such vubbles in the secre, and the following cycles.

A theoretical treatment of this problem is difficult for two reasons:(a) Migrating bubbles change their shape. When contracting, the lower inter-face of the originally spherical bubble moves faster invard than the upperinterface. At an intermediate moment, the cross section of the bubbleresembles that of a kidney. Later, the upper and lower interfaces collideand the bubble becomes a torus. Upon re-expansion, the spherical shape isroughly restored, but energy has been dissipated by the impact of the inter-faces. (b) Near the bubble minimum, the gas-water interface becanes unstable.It tends to dissolve into a water spray vhich is proJected into the interiorof the bubble. This brings forth a cooling of the explosion gases and, thus,again a dissipation of energy.

Both of these dissipative processes are difficult to account for theoret-ically. A further portion of the bubble energy is radiated by the bubblepulse. These energies are not available for the subsequent cycles of thebubble pul tion. A knowledge of the remaining energy is of prime importancein any quantitative calculation of the later bubble phenonena.

A rather crude approach is used in this paper to find the bubble energyin each cycle cf the oscillation. The analysis is based on the change ofthe bubble period caused by the various degrees of migration. Strong migrationcarries the bubble into a shallower depth. As a consequence the period of thenext oscillation is increased whea compared with that of a non-migrating bubble.To carry out the analysis, the magnitude of the bubble migration must be known.This process has been experimentally studied by means of acoustic ranging of

* The list of references is found at the end of thiz report.

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the bubble minimum. Further evidence is obtained in this paper from theobservation --f the time at which the bubble breaks the water surface. Oncea bubble migrates into the proximity of the water surface, the emission ofthe bubble pulse ceases. The evidence whether or not a bubble pulse isobserved before the breakthrough yields information on the dr Gh of thepreceding bubble max•mum.

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II. EQUATION FOR TIE BUBBLE FEERGY RATIO

The period of the bubble oscillation in the n-t1- -y* e is given by

(I) Tn 0.374 t (rnQD)13 -z 5

The symbols are explained In the list at the end of this report. The firstportion of (l) ib given in reference (b), .here rQ is denoted by e. The lastterm in the parenthesis accounts for the effect of the water surface whichshortens the period This term will be discussed In paragraph III. The effectof the bottom of the sea is not included in (i), because the tests vhich wewill analyze were made in water sufficiently deep as to make this effectnegligibly small.

Forming the ratio of the n-th to the first period, we have

Tn rn-1/3 ( Z -.5/6 1 -tAyDn(2) ý r '' 1 A , D

In this equation, the dimensionless period t has been cancelled ihich amountsto the tacit assumption that it is the same for all cycles of the pulsation.On the basis of the classic bubble theory, this is valid if the adiabaticexponent gamma of the explosion gases is betweeL 1.2 and 1.3 or so. Then, tis essentia-Ily constant and independent of the amplitude of the pulsation, asseen in Figure 2 of reference (b). The gamma of most of the comn explosivesis within this range. For A-1,er values of gamma, the above assumption is onlyapproximately valid.

,The maximm bubble radius in the n-th cycle is, reference (b):

(rnQW >1/3

(3) An - 1 *733 aM nr 1n

A surface correction is not necessary fcr the maximum radius, reference (a)Chapter 8. Assuming that aM is the same for all cycles, we can eliminateSin (2) by means of

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NAVORD 4185

A,, A41( 1/3 z 1/3 D,

n 1 1 U f

The assumption of an equal aM is not as good an apprc jatioa as thatof equal t. Since the significance of the maximn radii in the later pulsesis less critical than that of the periods, the accuracy is acceptable fo.our analysis.

Re-arrangement of (2) yields the foJ ..ng quadratic for the cube rootof the bubble energy ',.tio (rn/rl)l/3:

(5)

2/3 11/3 /33 516

171 an n-I 1

Mn order to find expressions for the ratics of the hylrost.-ic head Zn/•and the depth Dn/]Dl, the migration of the bubble must, be knovn. We referD. to the center of the n-th bubble maximum and we set for the rise betweentwo such points

D -Dn÷ AI21/2Dn -n +l &Zn.C12AMn(l-a ) I' ray -(6) zn 2n U= Z• (- a, -- 2 ,•/"J r

n Znn ZnaD

This formula is cnmmonly used for the migration between the point of explosionand the first mini-un. It represents an approximation of Taylor's fomuladerived in reference (c).

If the assumption is nade that the bubble remains spherical, theoreticalcalculations yield for the coefficient C* the value 132. (For the r sebetween two successive maxim, C* in (6) would be about twice as large.)Coarmaison of the theoretical value with that obtained from measured V-ubblemigrations shows the former about 40% too high. Figure 1 illustrates thisdiscrepancy and also illustrates the great scatter of the data.

The magnitude lf'ZIz2 in (6) is rela.ed to the Froude Number F whichfor our purposes is best defined by

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NAVoRD 41B5

AM am Z4/3 z4 1 1/3

(7) p T- -77i97 -W-7 tc. w 2

Thus

(8) 6 -

For the use in ccnjunction vith the quadratic (5), (6) can be broughtinto the form

Di - n+l "( --1/ AM.-i Z, [1(9) z 2 Z1, 1 Z. n Dn 2 in\

ThenZ n-i D - D.+

(10) z- xzJul

andD U-1 5. - Dj+l Z,

(11) 'Dý " 1-X Z, DJul 1

The computation of the reduced migration in the n-th cycle requiresonly the knowledge of A•/Z and D /Z,. This provides all relationsnecessery for the evaluation of (51. The energy ratios rn/r1 can be cwztedif the foll ving information is given:

(a) numercal values foi 0 and a

Tn A• D1

(b) Ix as a function of ths andaDlT 1 1, 1

The next paragmphs deal vith these data.

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III. THE SURFACE OORRECTION TERM

The surface correction term in (1) is given in the literature in variousforms. Reference (d) lists the same form as (1), but footnote 12, page 342,of reference (a) shows the surface correction as

(Reference (a) erroneousl7 shows a 5/6 power for Z.) For the case of a freewater surface and infinitely deep water, 1 0(x) is unity. With the introduc-tion of (3) and aM 0.92 (which is appropriate for our conditions), oneobtains o - 0.215.

Essentially the saxe value is obtained for the coefficient occurring inthe surface correction term for the migration (6) for which we have used thesame syubol a. From equation (8.67) of reference (a), this magnitude is foundto be 0.2. (There are two misprints in reference (a) at this place: Inequation (8.67) the factor 0.2 is omitted. In the preceding equation, thefactor should read 0.4 and not 0.2). Reference (d) quotes : - 1/5 for thesurface c3rrection of the migration.

These values as well as the form of the surface correction term are firstorder approximations. Figure 2 shows the period constants versus depth ofexplosion corrected with the use of various coefficients a. These data arefrom a test series which contributed the most important values to our evalua-tion. It is seen that a - 0.1 makes the period constant independent of depth,i.e. eliminates the surface effect, whereas a = 0.2 "overcorrects". Actually,otis not a constant, but depends on AM/D as well as A./Z. Figure 8.21 ofreference (a) shows fair agreement for 300 lb TNT charges and Figure 8.20 goodagreement fa 0.66 lb tetryl charges, if the periods are corrected with a - 0.2.But, a closer inspection again reveals an overcorrection, if shallow firingcorditions are excluded. It s. ms that a - 0.1 is appropriate for such caseswhere the bubble is not too close to the surface, as in Figure 2, where Af/D< 0.5. For larger values of AM/D, i.e. for shallower explosions, a value ofa larger than 0.1 is a medei or couplete correction. This shows that thesimple form of the surface correction term in (1) is not sufficient for aprecise description of this effect. For the purpose tf ý.nis paper the ocmpli.cations of a more elaborate relationship would not be worthwhile. In themajority of cases which are of interest here, the bubble is so far awy fromthe water surface, that the value f - .l is appll-able. For the instancesdiscussed in paragraýhM, v.ere the bW ,Ie break-through is considered, itwas found that a had little effect on the location of the bubble ma-id andon the times of the break-through. Thus, it seems safe to use a - 0,1 for thepurposes of this paper.

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NAVUIRD 4185

IV. THE MIGRATION COEFFICIENT

In Figure 1 experimental data on the bubble migration up to the firstbubble jinimum are compiled. The reduced migration LYZ/Z is 1l-)tted versus100 WuI//Z 2 " The data stem from various experimental serk', vizh, a varietyof charge weights and explosives. Most of the experJmer ii points %mre takenfrom a compilation in reference (f). Only typical poin, 3 are shown in Figure 1in order to avoid an overloading of the graph. All values are reduced to anexplosive hrving the properties of TNT with the use of the appropriate equiva-lent factors. Furthermore, all data are corrected to free field conditions,i.e. an infinite medium. The correction was made as described in Paragraph IIIof this paper usirg the factor a = 0.3. The z-esults of the 290 lb TNT testsare not very certain, because they were carried ou; in water oi Limited depth.The method used to coy-ect for the effect of the bottom consisted of including

the term - a A Z/H2 into the parentheses of (6), where H is the distance be-Mtween bottom and point of explosion. This correction is probably not veryaccurate, but the only one known today.

The migration was measured by the sound ranging metunod. The point fromwhich the bubble pulse is emitted was located by means of triangulation froma vertical string carrying several pressure gages. One of the difficultiesof this method arises from the fact that the pulse is not emitted from a mathe-matical point, but from the bubble surface which at the moment of the minimumstill has a considerable size. The great scatter of data shown in thsq figureillustrates the difficulties of this measurement and, therefore, the ratherapproximate nature of the information.

Figure 1 also shows the result of numerical calculations using Taylor'smigration fo iula, reference (c). The points shown have been obtained by ratherlaborious computations made during World War II by various British agenciesquoted in reference(W). ie. irence (d) stated that on this basis the rise isproportional to wll/ 4/Zll/6. Kennard, reference (e),has noticed that the linedrawn through these theoretical points can be equally well represented by asimple relationship which in our notation corresponds to

(13) (ýZ-h =ý 132 W1Z ýh z 2"

The experimentally observed migratcn (.n then be represented by

(14) i W/2

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AVORD 4185

where C* is between 80 and 90 depending on the weight one desires to gire themigration at shallow or greater depths of explosion respectively. ActuallyFigure 1 suggests a slightly different functional dependency of W and Z fromthat of the above equation. It is dubious whether or not the ,peri-mentalevidence is sufficientl. accurate to establish such a funct1'-..l rc:lationship.In view of the scatter and the difficulties of measuring mi' ration, the form(14) appears to be adequate. Also, it seems that either of 'he above quotedvalues of the C* fits the data equally well and can be used with equal con-fidence.

For the pr'-tpose c- the present analysis the migration between successivebubble maxima is needed and not that betwee.. the point of explosion and thebubble minimum vhich is shown in Figure 1. The theory of reference (g)(which deals with spherical. bubbles) gives the following picture about thesemigration terms:

Migration between

(a) point of explosion and first maximrim

(15) D o gTI, [n 2l/2]

(b) point of explosion and first minimum

(16) Do - D•1 in -4- c

(c) first minimum and second maximmn2

(17) D*l -D2 g 9 [ln2 1/2+in4c]23 -

(d) first maximum and second maximum

(18) D -D - [T10ln 4 c +.r 2 in 4c1 2 3 li 1 2 2

+ (T22 T T1' )(ln 2 - 1/22-)j

The magnitude c depends on the ratio of maximum to minion bubble radius:

(19) c .2 (AM/A) 3 _ 1.

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NAVORD 4185

As before, the subscripts 1 and 2 refer to the first and second cycle of thebubble pulsation respectively.

On the basis of this theory, the migration between the two maxima is justtwice the migration between the point of explosion and týie irst minimum, ifthe period and the coefficient c are equal in bobh cyclea- At'tbough thisresult gives a valuable hint, the approximations made 1 (10), (Li), and (6)are apparent: (a) Migration betwe m the point of exploLion and the firstbubble maximum is neglected in (10) and (11) for simplicity. Relation (15)which is probably rather accurate, since all bubbles remain spherical duringthe first expansion, indicates that this migration term can indeed be neglected.It is sma!l in cor-qarison with the other tcrms if the firing conditions aresuch tha;. sevez a! cycles of bubble pult ation occur. It is a poor approxima-tion for large charges exploded so shallow that the breakthrough at the watersurface occurs after ,he first minimum. But, such conditions are uot thesubject of our study. (b) Since neither T nor c are consistently equal forsuccessive cycles, the factor 2 is not necessarily applicable in (6). .But,nei-ther is any constant factor, since the migration between two maxima depends onthe parameters of both cycles. It is the simplicity of the enalysis as wellas the present lack of any better information which justifies approximation(6). In view of these discussions, it appears to be desirable to obtain anindependent check for the migration formula (6) and the factor t.C

Such an additional evidence of the bubble migration can be obtainedusing the analysis developed in this paper and considering the time at whichthe migrating bubble breaks through the water surface. This event can beconveniently recorded by means of photography of the surface phenomena. Apertinent result is listed in Table I.

TABLE !

Charge We.ght: 1590 lb TNT Equivalent

Firing Depth 130 140 150 ft

Time of Bubble Break-through Observed 2.50 3.24 3.25 secKineratographically

Time of Latest 2.55 3.32 2.96 secBubble PulseObserved 3rd pulse 4th pulse 4th pulse

Time counts from moment of detonation.

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The slight discrepancies between the observed bubble pulses and thetimes of t?- bubble breakthrough are probably experimental errors. Oneshould expect the time of breakthrough to be somewhat larger than the timeof the pulse. For our purpose the magnitude of the time intervalr observedis not important, but only the correlation of these events.

Figures 3a to 3c show the position and size of the bubb] Mm calcu-lated with the coefficients C - 3.7, 3.5, and 3.2 for the coLtitions ofTable I. Explosion bubbles are spherical up to the first maxim= only. Thecircles referring to the subsequent cycles in Figures 3a to c must be under-stood as idealized measures of size. Such bubble shapes are not well definedand can be considered spherical in a crude way only. The numbers shown nearthe bubble refer to *e cycle of the puls'ation.

At 130 ft firing depth, the center of the fourth bubble maximum occursaccording to these calcuiations either above or so close to the water suriacethat a fourth bubble pulse could not have. been emitted. This is in agreementwith the experimental evidence vhich shows that the breakthrough must havqoccurred iniediately after the 3rd bubble minimum. It must be visualizedthat the bubble center Jumps at this manent raral!y from the position "3" totile position "4". Figure 3a shows that for C - 3.7 and 3.5 a considerablepart of the fourth bubble maximum is above the water surface. This wouldresult in the observed surface disturbances. On the other hand the bubbleis too deep at the third cycle for C - :.2 in ordLr to produce such surfaceeffects. This shows that C must be larger than 3.o.

At 140 ft firing depth, a fourth bubble pulse was observed. For C = 3.7the bubble is too shallow in the fourth cycle in order to produce a pulse.It is generally assumed that ince the depth of the bubble center is less than90% of the maximum radius no bubble pulse is emitted. (This depth is calledthe venting depth. We prefer the term "blow-in".) Although this evidenceis establishel for tae first bubble maximum only, it is, at least approxi-mately, applicable to the later cycles also. It turns out that in this caseC = 3.5 is the largest value fo which a fourth bubble pulse can be expected.We have here a rather sensitive criterion for the migration coefficient C.

Figure 3c shows the case of 150 ft firing depth. The calculations usingany of the three values for C are compatible with the experimental evidence.One may argue that C - 3.2 again appears low in view of the evidence that thebreakthrough essentially coincided with the fourth bubble miniml .

It is obvious that the data available are not sufficient for an unmbiguousdetermination of the ngration peramete" However, it is significant that thevalue C - 3.5 'which is the largest value consistent with the evidence of i30 ft

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firing depth and which holds for the third cycle is compatible with theacou•stcally measured migrations shown in Figure 1 which hold for the firstcycle. C* and C are interrelated by

(13) c* - 3/2 c.

On this basis C* - 80 corresponds to C - 1.785. Th. s refers ýo migrationsup to the minimum. The value needed here would be about twice as large, hence3.57.

For the practical calculations it wa decided to use the rounded value

C - 3°5.

It is realized tha; this value cannot claim a high degree of accuracy. It isconsistent with the experimental evidence available today, but because of thegreat scatter of the data a considerable uncertainty remain .

In the following analysis, the value of C is sonevhat less critical thanit might be expected. The bubble energies in the various cyclAes of thepulsation are, of zourse, dependent on C. But, when these bubble energies arelater used to calculate bubble parameters, some of the uncertainties connectedwith C will be eliminated. For instance, the periods calculated by this methodwill almost exactly reflect the input periods. The migration of the bubbleand the position of the bubble center of the various bubble maxina a- moresensitive to the migration parameter. But, these represent the best informa-tion available today.

A sensitive method to check the validity of our method is the observa-tion of the migrating bubble in a high gravity tank. A preliminary studyresulte, in excellent agreement. A comprehensive test series aimed at athorough check of the migration parameter ib planned.

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V. ExpmERD nAL INP

in addition to the surface correction factor a and the bubble migrationcoefficient C, information is needed on the period ratios Tnt1. The form ofequation (5) does not require the knowledge of the periods by %b ,delves. Also,the specific firing conditions aie not needed for the evaluatl , oz c•h.s equa-tion. It is sufficient to know the ratio AWDI as well as i .e ratio 'i/Dl.The nature of this input makes it possible to utilize test reiults frout explo-sive charges having different explosive material, different charge weights, anddifferent firing conditions. The bulk of the information stems from a testseries listed in reference (h). (Denoted as M-series in Figures 4 and 5.)Other data (C-series) are from the files of the E Department of the Naval Ord-nance Laboratory.

Figures 4 and 5 show the period ratios T2 /T1 and T3 /T_ plotted versusAMI/ZI. The correspond.iag values of Z /D are given in Figure 4 as curves.The limiting values for great depth 0 -- O) are T2 /T1 - 0.70 and T3/T 1 -

0.565.

Information on the fourth bubble period is sparse and uncertain. Thecurve in lg ire 6 is based on the following three values:

TABLE II

1/Z 1 D1/Z1l

- 0 1.0 0,51

0.15 0.81 1.17

o.166 0.82 1.o3

The period ratios of non-migrating bubbles (A.,/Z 1 0) show a slightvariatioi for different explosives, reference (J). "fThe v-ues chosen refer toexplosives which mosT closely resemble those employed in the M- and C--eries:minol for the second and torpex for the third cycle. The ratio for The fourthcycle is estimated from the TNT result, reference (i), on the basis of thetrend exhibited by the different explosives in the preceding cycles.

A cr• e estimate of r 5 /r 4 can be based on the period ratio of non-migratingTNT-bubbles, Tr/T - '%927, reference Wi'. Di absence of any ttter informationthe constant value r 5 /r 4 - 0.80 might be used as a rough approximation.

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VI. TH E XEGY OF MIGRATING BUBBLES

The result of the calculations outlined above is shoir in Figurc 6, wherethe energy fractions rn/rn.1 are plotted versus the reduc . mijcration. Theenergy rn QW refers to the bubble energy at the instan4 ef ti- i-th bubble

maximum. It is well known that the major portion of i ,is energy is potentialenergy stored in the water during the bubble expansion. The rmainder is theinternal energy of the gas. Kinetic energy and the energy of migration are

negligibly small at this moment.

At each bub.'t minimum a reduction of the bubble energy takes place.The enerst, vbich is loot at this poin. essentially comprises two terms:(a) enema acoustically radiated by the bubble pulse and (b) dissipatedenergy. Both of theue energy terms depend on the degree of bubble migration,Strong migration reduces the amplitude of the bubble pulse, thus reduces theenergy loss due to the acoustic radiation.

For non-migrating bubbles the energy dissipation --t thc. bubble uinimumis probably a consequence of the Taylor instability of the bubble interface.With increasing migration and with the consequent decrease of the excesspressure in the bubble near its minimum, the instability decreases. Thus,the intensity of the spray prvjected into the bubble interior is also de-creased and so is the energy dissipation. However, migration causes theinversion of the bubble and an impinging of the upper and lower bubble inter-faces. The water hamher and the spray formation connected with it areprobably the alternative mechanisos of energy dissipation. These take overwith increasing intensity as the effects of the Taylor instability decrease.

These vonsiderations illustrate the important role of migration in thepartition of these energy t--ms. In fact, our analysis clearly shows thatthe bubble energy fraction de;ýends on the strength of the migration. It alsosuggests the term which is used in Figure 6 as abb,,issa.

Figure 6 shows the energy balance for the second cycle at the left hanmside. The bubble energy ratio r /r is plotted as obtained from our analysisand the input discussed in Pararpi V. This ratio refers to the energy ofthe bubble at the second maximm. Since this energy is supplied from thebubble energy of the first cycle, tle term 1 - r /r represents the energydecrease which occurs at the first .ab.:2.e minium. kThe enerk- of Ghe pulewhiých is emitted at the end of the firzit and the beginning of the second cycle

is shwn as a dashed curve. (This curve is a rather crude estimate made for

the ssake of illustration.) The two curves divide the plot into three bandsthe he.tghtsof which represent (a) the bubble energy, (b) dissipated energy, and

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NAVORD 4185

(c) the pulse energy. In the dimensionless diagram shown, these three energiesadd up to unity which is to say that. the bubble energy of the first cycle splitsinto these 'xree terms.

An interesting result is that the bubble energy of the second "yc-leinitially increases with increasing migration, if the other parar .,era obfthe explosion are held constant. In view of the preceding coeu r.s, -h.9is to be expected. It is also seen that the dissipated energ is roughlyconstant over a considerable range of v igration intensity. M3.s is surprising,since two different mechanisms of energy dissipation are probably involved,both of which depend on migration. Although one of them decreases and theother increases with increasing migration, a roughly constant sum of thesefactors, as suggested by Figure 6, was not necessarily to be expected.

For very strong migration, the bubble inversion and the consequentenergy dissipation appear to be so violent that a decrease of the bubbleenergy in the second cycle results. For such conditions the energy radiatedby the bubble pulse is practically negligible.

The most important outcome of these calculations is that the energy ratiosgiven in Figure 6 permit the calculation of the parameters of Nigrmt~ng bubblesfor almost all explosion conditions of practical interest. (Nuclear explosionsare, of course, not included*) The calculations can be extended with reasonableconfidence up to the end of the fourth cycle. Estimates for the fifth cycle arepossible with the use of the approximate energy ratio r 5 /r 4 ' 0.8.

The conditions at the bubble minimum, its size and shape as well as itsenergy of translation are not covered in this enalysis. However, it ispossible to calculate the position of the bubble minimum simply by assumingit halfgy between successive bubble maxia.

According to the present state of knowledge the energy curves are appli-cable to most explosives. Some deviations may be expected for deep explosions,where only slight migration takes place. The evidence of non-migrating bubbles,reference (J), shows a slight tren- in the period ratios for explosives ofdifferent compositions. But, the results of the M- and C-series give no indica.-tion of such a treud. If desired the effect of different e:losives can beaccounted for in Figure 6 by constructing a new curve Which passes thrc-gh theappropriate value at AM/Z = 0 and subsequently merges into the old curve. But,in most cases, such a precaution will not be necessary.

The practical application of the information obtained here, is discussedin NOLTR 62-184. In this paper graphs are oresented which permit a convenientreading of tbe bubble parameters for a widt ran•e of conditions.

14

NAVORD REPORT 9 35

U,w U) cJ WCD ~.-

< wD W

-i ~ > > (

D ~ . 0 0o 0 00(a -w i Z.J4 -< I- IaZ > z z z w

W~0 Z W.2 W: FL OR .

Lii - CL 0 Irc a 0C u-

W 00

00

0

TTr

_ _ * _ _ _ _ _ C-j

0zU.0

Sw

4 w____ N

ZW~ ___ 0

CD om

I 0

0 6 U)N

15

NAVORD REPORT 4185

1.0 4- • .

Z 1.02

Z CORRECTED0 PERIOD0 .10 CONSTANTS

a 1.002frwaX. 0

> 0.98

0 6w - OBSERVED" ,,PERIOD

0.9. )CONSTANTS0.96.

FI0.94

6

50 100 150FIRING DEPTH IFT)

FIG.2 SURFACE CORRECTION OF THE FIRST BUBBLE PERIOD.DATA ARE FROM CHARGES OF 1580LB TNT EQUIVALENT.THE DASHED LINE BETWEEN THE POINTS " INDICATESTHE UNCERTAINTY IN THE READING OF THE PERIOD.

16

NAVORD REPORT 4185

WATER SURFACE

0- ("4

4420-

40-o- 3

IL60-

X 80-

wo100-

120-

C=5.2 C=3.5 C=3.7

FIG. 30 POSITIONS AND RADII OF THE BUBBLE MAXIMA CALCULATED FORTHREE VALUES OF THE MIGRATION PARAMET ER C_ CHARGE WEIGHT1580 LB -TNT EQUIVALENT, FIRING DEPTH 130 FEET.

17

NAVORD REPORT 4185

WATER SURFACE

0 4

20-

S60- 3 3

" 80-

4gwI-l I00 .- 2 .

120 -

140 1

C 3.2 C= 3.5 C =3.7

FIG. 3b POSITIONS AND RADII OF THE BUBBLE MAXIMA CALCULATED FORTHREE VALUES OF THE AIGRATION PARAMETER C. CHARGE WEIGHT1580 LB -TNT EQUIVALENT, FIRING DEPTH 140 FEET.,

18

NAVORD REPORT 4185

WATER SURFACE

0440- 4

18 3

3

(Ll

2 2 2120-

140-

0=3.2 C=3.5 C=3.7

FIG. 3c POSITIONS AND RI AU OF THE BUBBLE MAXIMA CALCULATED FORTHREE VALUES OF THE MIGRATION PARAMETER C. CHARGE WEIGHT1580 LB-TNT EQUiVALENT, FIRING DEPTH 150 FEET.

19

NAVORD REPORT 4185

1.2

X C- SERIES, M- SERIES

0I.9

0.

z 4.0

HEAD ABCSAI S YR.TA; EDRDCDB H

0.8_ _ C3 __ 0 _ _

0.7• 'Z -,° o

0.6:3 4 6 8 1 0 15 20 30 40K 60 80

REDUCED HYDROSTATIC HEAD ZI/AM,

FIG 4 PERIOD RATIO T2 /1'1 AND RATIO OF FIRING DEPTH TO HYDROSTATICHEAD. ABSCISSA IS THE HYDRCrTAT;C HEAD REDUCED BY THE

FIRSI MAXIMUM BUBBLE RADIUS.

20

NAVO-(D REPORT 4185

I.I

1.0 __I

0.9 -

: 0.8 - __ _ "

0

0.6' •

0.5

3 4 6 10 15 20 30 40 60 80

REDUCED HYDROSTATIC HEAD Z I/Am,

FIG. 5 PERIOD RATIO T3/TI . THIý MAGNITUDES USED AS ABSCISSA REFERTO THE FIRST CYCLE OF .IULS•ATION

21

NAVORD 4185

LIST OF SYMBOLS

AM Maximum Bubble Radius (ft)

Am = Minimum Bubble Radius (ft)

NH = Reduced Maximum Bubble Radius (dim ,ionless)

c = See equation (19)

C* = Migration Coefficient (ft /lb /2)

C Migration Ccefcin. ( . dimensionles)

D = Depth of Bubble Center at Bubble Maximum (ft)

Do =Firing Depth (ft)

F Froude Number (Dimensionless)

F'(x) = Surface Correction Function (dimensionless)

g = Acceleration of Gravity (ft/sec2 )

H = Distance to Bottom (ft)

J = Radius Constant (ft4/3/lb1/3)

K = Period Constant (sec ft, 5/lb /3)

n = Subscript Referring to Cycle of Pulsation

Q = Chemical Energy per Unit Mass of Explosive (cal/ilb)

r = Energy Fraction Referring to Bubble (dimensionless)

rQW = Bubble Energy (cal)

T = Period (sec)

t = Reduced Period ýdir..,mnsionless)

W = Charge Weight (!b)

x = (D-HO/(D+H) (dimensionless)

23

NAVORI 4185

Z = Total Hydrostatic Head = D + 33 ft for sea water

Zo = Total Hydrostatic Head at firing depth (ft)

0! M Surface Correction Factor (dimensionless)

e = rQ

24

NaVOjRI 4185

REFD.MNCS

(a) "U~nderwiate Explobions", R. H. Cole, Princeton University Press, 19I48.

(b) "Underwater Explosion Phenomena: The Parameters 6f - oa-1-MieratingBubble Oscillating in an Incctpreasible Medium", H. Go Sm~y andE. A. Christian, IIAVORD Report 2437, 1952. Unclass..tied.

(c) "Vertical Motion of a Spherical Bubble and the Pressure Surrounding it",Go 1. Tayrlor, Underwater ExplosionkResearch II, OUR 1950p page 131-144f,Unclassified.

(d) "The Behavior of an Underwater Expl~ouion Bubble", A. R. Bryant, UnderwaterExplosion Research II, OUR 1950, page 163-182 and 321-328, Unclassified.

(e) "Migration of Underwater Gas Globes Due to Gravity and NeighboringSurfaces", E. H. Keznnard, Underwater Explosion Research II, OUR 1950,page 377-414, Unclassified.

(f) "Note on the Migration of Explosion Bubbles'", J. C. Decius and M. Arsove,NPAVORD Report 419, 1949, Confidential.

(g) "Migration of Explosion Bubbles in a Rotating Test T&Ak", H. Go Snayr,NoLTR 61,-1.45, 1962, Unclassified.

(h) "Measurement of Bubble Pulse Phenomena", C. P. Slichter, W. G. Schneider,and R. H. Cole. OSRD Report 6442, 1946, confidential.

(i) "Measurement of Bubble Pulse Phenaiena XV: Press'-ýe-Time Measurementsin Free Water", A. B. A :-ns, J. P. Slifkio, A. Ho Carter, Do R, Yennie,and C. L. Wingate, NAVORD Report W08, 19J47, Unclassified.

(j) "Measuremr-.nt of Bubble Pulse Phenomena, III. Radius and Period Studies",E. Swift, Jr. and J. C. Decius, Underwater Explosion Research 11, OUR

1950 Page 553-599, 1947, Uaclassified.

25

NAVORD 4185

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