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A l A A 94-0560Drop Defo rmat ion and Breakup due toShock Wave and Steady Dis turbancesL.-P. Hsiang and G. M. FaethDepartment of Aerospace EngineeringThe University of MichiganAnn Arbor, Michigan

32nd Aerospace SciencesMeeting ExhibitJanuary 10-13, 1994 / Reno, NVk6 ermissionto copy or republish,contact the A m rl c an Inot i tute of Aeronautics and Astronautics370 L'Enfan t Prom enade, S.W., Washin gto n, D.C. 20024

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DROP DEFORMATION AND BREAKUP DUE TO SHOCK WAVE ANDSTEADY DISTURBANCESL.-P. Hsiang" and G . M. FaethtDepam nent of Aerospace Engineering../ The University of M ichiganAnn Arbor. MI 48109-2118

tlslwIAn experimental study of drop deformation propertiesinduced by both shock wave and steady disturbances isdescribed. Three test facilities were used, as follows: a shocktube facility for measurements of effects of shock wavedisturbances on drops in gases, a IO m high drop tube facilityfor measurements of effects of steady disturbances on drops ingases, and a 1 m high drop tube facility for measurements ofeffects of steady disturbances on drops in liquids. Variousdispersed and continuous phase gases and liquids wereconsidered to provide disp ersd con tinu ou s phase density ratiosof 1.15-12000, Ohnesorge numbers of 0.0006-600, ebernumbers of 0.004-700nd Reynolds numbers of 0.03-16000.At low Ohnesorge numbers (< 0.1) for all types of

disturbances, significant drop deformation (ca. 5 ) began atWeber numbers of roughly unity, with the deformation regimeending due IO he onset of breakup at Weber numbe rs of 10-20.These transitions were relatively unaffected by the Ohnesorgenumber for steady disturbances, however, increasingOhnesorge numbers caused progressive increases of the W ebernumber range for both deformation and breakup regimes forshock w ave disturbances- n effect that could be explainedusing phenomenolog ical theory. Anoth er transition, betweendome- and bow l-shaped drops (related to the transition betweenbag and shear breakup), was correlated mainly in terms ofWeber and Reynolds numbers for present conditions. Dropdeformation for steady disturbances was relatively independ entof dispersdcontinuous phase density ratios but generally wassmaller than for shock wave disturbances at comparableconditions due to the absence of overshoot fmm inertial effects.In contrast, drop drag coefficients, normalized by the dragcoefficient of a solid sphere at the same Reynolds number,correlated quite well by the degree of deformation alone.

L

Nomenclaturea =d rop accelerationC,C =empirical constantsco,TD =d ro p drag coefficient, mean value of CD=d rag coefficient for equivalent sphere=d rop diameterDSPddmax dmin =max imum and minimum dimensions of a dropEo = ~ a t v ~ sumber, apdd21a or glPd-Pcld%g = acceleration of gravityK , K =emp irical constantsOh = Ohnesorge number, Pd I (Pddoa)Re = Reynolds number, PcdouolCI,t = time

112

Graduate Studen t Research Assistant ,?Professor, Fellow AIAAInstitute of A eronautics and Astronautics, Inc. with permission.- opyright 1994 by G.M. Faeth. Published by the Am erican

t b =dro p breakup timeI*U = streamwise relative velocityWeAP*P = molecular viscosityP =densitya =surface tensionTTUI = shear s u e s s

= characteristic breakup time, d,(pd I pc) l i2 I uo= Weber number ofdrop, P,doui 1a=effe ctive pressure coefficient,Eg. (14)

=characteristic deformation time at large O h,Pd I (PcU$

SubsniDtsC = continuous-phase pmpem escr =crit ical value for a transitiond =dispersed-phase pmperties0 =ini tial conditionea = local ambient condition

Processes of deformation and secondary breakup ofdrops have received significant attention as important classicalmultiphase f low phenomena with numerous practicalapplications, e.g., industrial and agricultural sprays, liquid-fueled power and propulsion systems, and rainfall, amongothers. In pmicular. recent studies suggest that secondarybreakup is a rate-controlling process in the near-injector regionof pressure-atomized sprays. through its effect on drop sizes. 1Additionally, primary breakup at the surface of nonturbulentand turbulent liquids yields drops that intrinsically are unstableto se co nd an breakuD.14 sum orti ne the classical descrintionthat atomiiation occurs h y 'p ri m G breaku near a l iquidsurface followed by secondary breakup.? Finally, highpressure combustion involves conditions where the dropsurface tension becomes small because the liquid surfa ceapproaches the thermodynamic critical point: such conditionsalso imply signif icant effects of secondary breakup.5Motivated by these observations, the objectives of the presentinvestigation were to extend earlier studies of secondary dropdeformation and breakup in this laboratory, due to Hsiang andFaeth, 6.7 emphasizing the properties of drop deformation fromboth shock wave and continuousdisturbances.Past work on drop deformation and breakup will onlybe considered bncfly. see W ienb a and Tak ay am ,* Giffen andMuraszew? Hinze,]" Krzcczkowski.1' Clift c t a]..'* andreferences cited therein, for more complete reviews. Past workgenerally has been limited to two kinds of well defineddisturbances that cause deformation and breakup of drops:shock wave disturbances that provide step changes in theambient environment of a drop, e.g., represen ting a drop at theend of rapid primary breakup; and steady disturbances, e.g.,representing freely-falling drops in a rainstorm or a mixingcolumn. Effects of shock wave disturbances have received themost attention with high-speed photography used to identifde format ion and secondary breakup regimes .5-22Measurements of transitions between breakup regimes have

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. . een limited to p > 500 and Re > 100.6-'2~ or theseconditions, Hinzeft shows that breakup regime transitionslargely are functions of the ratio of drag to surface tensionforces, represented by the W eber number, We, and the ratio ofliquid viscous to surface tension forces, represented by theOhnesorge numb er, Oh. At low Oh, drop deformationbecomes significant at We ca. 1 and drop breakup becomessignificant at We ca. 10, with bag, multimode and shearbreakup regimes o bserved at progressively larger We (see Refs.8-12 fo r photographs and detailed descriptions of the varioustypes of drop breakup). With increasing Oh, however,progressively larger We are required for the onset of dropdeformation and breakup because viscous forces inhibit dropdeformation which is the first stage of the breakup process.This behavior has been co n fm ed by several investigations,&"however, maximum values of Oh < 4 so that behavior of largeOh has not been resolved. This is unfortunate because highpressure combustion processes involve large Oh as a result ofdrop surfaces approaching the thermodynamic critical point.Thi s behavior occurs because surface tension becomes smallwhile liquid viscosity remains finite as liquid surfaces approachtheir thermodynam ic critical point.5 Thus, whether combustionat high pressures involves enhanced, or entirely suppressed,effects of secondary breakup has not yet been resolved.The time required for breakup is another aspect ofsecondary breakup that has received significant attention forshock wave disturbances at p d p C > 500. At low Oh, Liang et

aLZ6found that breakup times normalized by the characteristicbreakup time of Ranger and Nicholls,17 t*. were remarkablyindependen t of both the breakup regime and We. As might beexpected from the effect of Oh on breakup regimes, however,breakup times have been observed to increase with increasingOh.6 Additionally, processes of drop deformation, and thevariation of drop drag coefficient with time, also appear toscale systematically in terms oft' at low Oh but behavior atlarge Oh is uncertain.Finally, the outcome of secondary breakup for shock- ave disturbances at p pc > 500 and small Oh also has

distributions after secondary breakup satisfied the universalroot normal distribution of Simmons27 in all three breakupregimes, after removing the core (or drop forming) drop fromthe drop population for shear breakup. The size and velocity ofthe core drop after shear breakup then was correlatedsuccessfully based on the observation that the end of dropstripping corresponded to a constant EO.^ The relativevelocities of the drop liquid were significantly reduced aftersecondary breakup, which could be correlated successfullybased on simplified phenom enological theory.' These resultsshowed that secondary breakup processes extend over asignificant region, ca. 40 initial drop diameters. Thus,secondary breakup is not a particularly localized event whichraises con cerns about its dynamics, e.g., liquid motion duringbreakup, the distribution of drop liquid in space and time, etc.Work treating these issues, however, has not yet been reponed.

Dro p deformation and breakup for steady disturbancesalso has been studied, motivated by interest i n 2 p 3 q m t i e s ofrain and liquid-liquid exuaction equipment. The mainobjective of this work has been to develop ways to estimate thevelocity of fall of particular sized drops in gas and liquidenvironments, and to determine conditions for drop breakup. Ithas been found that We for breakupm omparable for steadyand shock wave disturbances. How ever, the properties ofbreakup for steady disturbances are not well understood due tothe intrusion of processes of drop formation. For example,whether the mechanism of breakup is due to nozzle-induceddisturbances, b ag breakup, simple splitting into a few drops, oris significantly affected by collisions, still has not beenresolved.33-34 Other areas of uncertainty involve relationshipsbetween drop deformation and drag, effects of large Oh, andthe relationships between the parameters of dispersed and

rece ived a t t e n t i ~ n . ~ , ~z4 It was found that drop size

continuou s phases and the drop shape. Since steadydisturbances provide conditions where drop deformation andbreakup response to disturbance levels are relatively simple tointerpret and to correlate, it is clear that more progress oneffects of steady disturbances must be made before the morecomplex processes involving shock wave or more generaldisturbances can be understood.The objective of the present investigation was to extendthe work of Hsian g and Faeth,6.7 in o rder to help resolve someof the issues discussed in the preceding review of the literature.Foremost among these issues is nature of the deformation andbreakup regime map for shock wave disturbances at large Oh.Other issues considered involved fac tors influencing dropdeformation and shape, emphasizing effects of thedispersedhontinuous phase density ratio, and the relationshipbetween drop drag and deformation in various environments.These problems were addressed using three test facilities, asfollows: a shock tube facility for measurem ents of effects ofshock wave disturbances on drops in gases, a 10 m high droptube facility for measureme nts of effects of steady disturbanceson drops in gases, and a 1 m high drop tube facility formeasurements of effects of steady disturbances on drops inliquids. Various dispersed and continuous phase gases andliquids were used to provide p d / p c of 1.15-12000, Oh of0.0005-600, We of 0.004-700 and Re of 0.03-16000.Phenomenological analysis was used to help interpret andcorrelate some aspects of the measurem ents.The paper begins with a discussion of experimentalmethods. Results are then considered, treating dropdeformation and breakup regimes, drop deformation and dropdrag, in turn.

-1 M e t h dl s h d a k

ADDaratus. The shock tube apparatus is illusuated inFig. 1. The arrangement involved a driven section o n to theatmosphe re, similar to earlier work in this labora tory.eJ7 Thedriven section had a rectangular cross section (38 mm wide x64mm h igh) and a length of 6.7m with the test location 4.Omfrom the downstream end. This provided test times of 17-21ms in the uniform flow region behind the incident sh ock wave.The test location had quar tz windows (25 mm high x 305 mmlong, mounted flush with the interior side walls) to allowobservation of drop breakup. Breakup was observed in airinitially at 98 kPa and 297 + 2K in the driven section of theshock tube with shock Mach numbers in the range 1.08-1.31.Instrumentation was synchronized with the passing of theshock wave using the piezoelectric pressure transducers thatmonitored the suength of the shock wave in the driven section.

Two different drop generator systems were used for theshock tube experiments. Operation at low and moderate Ohinvolved the use of vibrating capillary tube drop generator,similar to Dabora,35 which generated a sue am of drops. Thi sdrop sueam passed thmugh 6 mm diameter holes in the top andbottom of the shock tube, crossing the central plane of thedriven section at the test location. A n elecuostatic dropselection system, similar toSanjiovanni and K estin,x was usedto deflect a fraction of the drops out of the sueam . Thisyielded a drop spacing of roughly 7 mm so that drops alwayswere present in the region observed while interactions betweendrops during secondary breakup were eliminated.

Drop generation for large Oh conditions required adifferent approach. In panicular, it is very difficult to form adrop that has a large Oh; instead, such drops generally evolveto these conditions after formation at low Oh, e.g., large Ohconditions are approached during high pressure combustionbecause the drop surface eventually becomes heated toconditions near the thermody namic critical point. Th e2

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approach taken during the present investigation was to form alow Oh drop from a liquid solution, and to levitate the drop atthe test location until the solvent evaporated away, leaving asmall drop consisting of a highly viscou s liquid which providedthe desired large Oh condition. Dow Corning 200 Fluidshaving unusually large viscosities, 10,000 and 30,000mP)were used for the viscous liquids, with n-heptane used as thevolatile solvent. Original drop diameters were in the range320-700 pm , with final drop diameters of 160-350,pm, hichrequired several minutes i o complete evaporation of thesolventThe levitation system used for theevaporanon process to prepare large Oh drops is illustrated inFig. 2. The arrangemen t consists of a horn and reflector havingdiameters of 10 mm that were positioned near the top andbottom of the test section (roughly 55mm apan). Two 50mmdiameter piezoelectric ceramic elements, combined with acentral mount and a resonator, were used to power the horn.The horn operated at a frequency of roughly 47 lrHz with thetip of the horn having an amplitude of roughly 200 pm. Thepiezoelectric ceramic elements were actuated using aWilcoxan, Model PA8 -1 power supply (110 V, 8A maximumoutput conditions) operating at the required frequency. A stepup transformer w as used to increase the outpu t voltage to loo0V in order to properly drive the ceramic elements. This circuitalso incorporated proper impedance matching circuitry for theceramic elements. Th e horn driving unit was cooled with an

air blower system. The long shank of the horn also helped tominimize thermal disturbances, due to the large powerdissipation of the horn driving unit. within the test section.The d rops were placed in the aco ustic field of the hornand reflector system using a hypodermic syringe. Access to thetest section for the syringe (not shown in Fig. 2) was providedby a port that could be sealed by a cap screw, whose inner endwas flush with the inside wall of the test section.

.

J ~ QL The d eformation and breakup process_-as observed using ulsed shadowgraph motion pictures,simi lar to earlier work 7 A copper v a p r laser was used as thelight source with a 35 mm drum camera used to record theshadowgraph images at unity magnification. A functiongenerator was used to pulse the laser when the shock waveneared the drop stream locaaon, with pulse frequencies of 6-8kHz or 20 pulses. Each laser pulse duration was 30 ns, whichwas sufficient to stop the motion of the drop on the rotatingfilm drum. The drum camera recorded the images with anopen shutter within a darkened room. The time betweenshadowgraph pictures was monitored by recording the signalgenerator output using a digital oscilloscope.

The film records were analyzed using a Gould FD oImage Display. The procedure was to obtain three motionpicture shadowgraphs for a particular test condition and groupthe data to obtain statistically-signifcant results as ensembleaverages. Experimental uncertainties (95 confidence) of themeasurements reported here are as follows: initial dropdiameter and subsequent drop dimensions, less than 10 ;anddrop drag coefficients, less than 30%, limited by the accuracyof finding drop centroid motion at small times after passage ofthe shock wave.Test conditions are summarized inTable I%%%% of earlier work? test drops of water,n-heptane. mercury, two Dow Coming 200 Fluids and variousglycerol mixtures were used to provide a wide range of liquidproperties. Th e liquid properties listed in Table 1 wereobtained from L a ~ ~ g e . ~ ~xcept for the properties of the DawCorning 200 Fluids which were obtained from themanufacnuer. and the surface tensions of the glycerol mixtureswhich were m easured in the same manner as Wu et al.* Initialdrop diameters were in the range 150-1550 pm. Ranges of- ther variables are a s follows: pd/p, of 500-12ooO, h of

3

0.0006-560, e of 0.5-680 nd Re of 340-15760. The Werange includes processes from no deformation into the shearbreakup regime of interest to phenomena within dense s rays,5but does not reach the catastro hic breakup regime s d e d byReinecke and co wo rk er s . ld The Re range of thesemeasurements is higher than conditions where gas viscosityplays a strong role in drop drag properties; within the presentReynolds number range, the drag coefficient for spheres onlyvaries in the range 0.6.0.4.5.38 Shock Mach numbers weremodest so that physical properties of the gas were essentiallythe same as room tem perature air.IhuwsLs

Anoaratus. Gas-liquid and liquid-liquid drop towerswere used for tests with steady disturbances. The gas-liquiddrop tower was constructed of PVC pipe having an insidediameter of 300 nun and a height of 9.2m, that was open at thetop and the bottom. Drops were released along the axis of thetube, at its top, using a simple buret system. The drops werewidely spaced, and reached terminal velocities at roughly Sm,well before the tube exit. Measurements were made when thedrops were roughly 200 mm below the bottom of the tube.Instrumentation was synchron ized with the passing of the dropusing a simple light interception triggering system based on aHeNe laser directed across the exit of the PVC pipe. Dropswere collected in a flask at the end of their fall.The liquid-liquid drop tower was constructed ofPlexiglas to provide a 150 x 150 mm cross section and avertical height of 1.2 m. The dispersed (drop) and continuousphase liquids were immiscible, and were fully saturated withthe other liquid prior to testing. The drop liquid was releasedusing a buret discharging under the surface of the continuousphase liquid. Th e method of dro p introduction was notimportant for present results, however, because terminalvelocity conditions were reached w ell before the region wheremeasurements were made. Present test conditions did notinvolve oscillating drops. Measurements were made roughly300 nun above the bottom of the tank Drop motion for theseconditions was very slow so that it was possible to use manu alsynchronization when obtainin g test records. Drops simplycollected at the bottom of the drop tower as an immiscibleliquid layer that was removed from time to time.Insrmmentation. Drops were observed using single-and double-pulsed shadow graph photographs. Th e light sourcewas a General Radio lamp (type U-31A) with a flash durationof roughly ws. The lamp output was collimated and directedhorizontally through the axis of the drop tower. The image wasrecorded using a Graphlex camera (4 x 5 inch film format,Polaroid Type 55 film) at a magnification of 6:l. Thephotographs were obtained in a darkened room using an opencamera shutter. Th e time of separation between pulses wascontrolled by a function generator. As noted earlier, the timeof the fust photograph was controlled by a light interceptionsystem for the liquid-gas experiments, and manually for theliquid-liquid experiments. These imag es were processedsimilar to the shoc k tube measurements. Experimentaluncertainties (95% confidence) of the measurements reportedhere are as follows: drop dimensions, less than 5 ; and dropdrag coefficient, less than 10 . Both these uncertainties weredominated by fin ite sampling accuracy.The effective diameters of the drops were computedsimilar to earlier work.6.7 This involved measuring maximu mand minimum diameters through the centroid of the image;d m a and dm in.Assuming ellipsoidal shapes, the diameter wasthen taken to be the dia ter of a sphere having the samel k&hdbm Test conditions for drops falling attheir terminal velocities in air arc summarized n Table 2. Inthis case, test liquids were limited to water and various glycerol

volume; namely, d3 = d m m f m X ,

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. mixtures, with properties found as described for Table 1.Initial drop diameters were in the range 2.0-7.8mm while theranges of other variables were as follows: p I p of 845 1070,Oh of O.oO12-2.9, We of 1.2-9.8 and Re of%30-4600. in thiscase, the range of W e w as rath er n m w because smaller valuesresulted in negligible drop deformation, while larger valuescaused drop breakup, over the available range of larger Oh.Similar to the test conditions of Table 1, the present Reynoldsnumber range involves a rather modest variation of drop dragcoefficients.Test conditions for the liquid-liquid drop towerexperiments are summarized in Table 3. Test liquids for thedispersed phase included water, ethylene glycol, variousglycerol mixtures, carbon disulfide and a Dow Coming 200Fluid having a viscosity of 30,000 cP. Test liquids for thecontinu ous phase included paraffin oil and water. Liquidproperties were found as described in Table 1, except that theproperties of the paraffin oil also were obtained from themanufacturer. Initial drop diameters were in the range 0.1-58mm while ranges of other variables were as follows: p I pof 1.15-1.45, Oh of O.ooo5-215, We of 0.004-24 and d e O0.03-1190. Th e We range was limited to conditions whereinteresting effects of deformation were observed but limited bythe onset of breakup. The Re range extends from the Stokesregime to the region wher t e drag coefficient becomesrelatively independent of Re. t

ShQck Wave D&u~bmc& Results for shock wavedisturbances mainly involved extending the deformation andbreakup regime map of Ref. 6 to higher Oh, based on presentresults using the Dow Coming 200 Fluids. The resultingdeformation and breakup regime map, showing transitions asfunctions o f We and Oh sim ilar to Hinze 'O andL Krzeczkowski," is illusuated in Fig. 3. As noted in Ref. 6, thevarious breakup regimes identif ied by Hinzelo andKrzeczkowski" arc in excellent agreement with the presentmeasurements in the region where they overlap. This includesbag breakup at the onset of breakup, shear breakup thatinvolves the snipping of liquid from the periphery of the drop,and the complex multimode breakup regime between themwhich merges aspects of the tw o bounding breakup regimes.

Oh also disappear with increasing Oh, e&. oscillatorydeformation at Oh ca. 0.3 and bag breakup at Oh ca. 4.Hinze'O observed this tendency for the limited range of Ohavailable at the time (Oh < 1) and conjectured that breakupmight no onger be observed for Oh > 2. However, the largeOh behavior observed in Fig. 3does not suggest such alimitation; rather, there is a progressive (almost linear) increaseof We at the deformation and breakup transitions. withincreasing Oh. Clearly it is crucial to establish whether largeOh im lies no deformation or breakup as suggested byHinze,g or simply rather large values of We at the transitions,as suggested by the measurements illusuated in Fig. 3.Thus,phenomenological analysis is considered next in order to gainmore insight about effects of large Oh on deformation andbreakup regime transitions.Based on drop deformations results for steadydisturbances, to be discussed subsequently, it did not appearthat the liquid viscosity had a significant effect on the shapeand hydrodynam ic state of a deformed drop. Instead, he liquidviscosity for shock wave d isN ban ces appeared to be reductionof the rate of deformation of the drop. This behavior allowsmore time for drop relaxation to the local ambient velocity.tending to reduce the relative velocity, and thus the drivingpotential for drop deformation at each stage of the deformationprocess. A simple analysis incorporating these ideas wascarried out, in order to quantify the effect of liquid viscosity(represented by large Oh behavior) on deformation and

breakuo reeime transitions.-The major assumptions of regime transition analysiswere similar to earlier analysis of drop motion duringbreakup,6n1 as follows: virtual mass, Bassett history andgravitational forces were ignored; gas velocities, and otherproperties, were assumed to be constant; drop mass wasassum ed to be co nstant ; and a consta nt av erage d rag coefficientwas used over the period of interest. For present conditions,virtual mass and Bossett history forces are small because pd tkis large.5 Sim i~a r~y ,ravitational forces are not a factor becausedrop motion was nearly horizontal and drag forces were muchgreater than gravitational forces. Additionally, uniform gasvelocities, and other properties, were a condition of theexperiments. Similarly, present considerations are limited todeformation and breakup regime transitions so that there is nomass loss of the drop. Finally, although drop drag coefficientsvary considerably when drops are deformed, use of the originaldiameter and a constant average drag coefficient have beeneffective for earlier considerations of drop motion.' Based onThe transitions to the nonoscillatory and oscillatorydeformation regimes illustrated in Fig. 3 have not henreponed by others but are important because they defineconditions where drop dra e behavior deoans sienificantlv from

these assumptions, the qu at io n governing the relative velocity,U, Of the drop Can be written as follow:'

du I dt = 3 ~ ~ p ~ u '(%do) (1)that of a solid sphere. Thus, the first defokation ;egimeinvolves the maximum (cross seeam)imension (normalizedby the original drop diameter) in the range 1.05-1.10; withsubsequent deformation regimes defined by this ratio being inthe range 1.10-1.20 and greater than 1.20 ut prim to transitionto oscillatory deformation (at low Oh) or bag breakup (at largeOh). The oscillatory deformation regime is discus j Ref, 6:it is defined by conditions where the drop oscillated with aweakly dilmped amplitude (w here the secon d peak of the drop

where the initial relative velocity is equal to u and CD is anappropriate average drag coefficient. Now, previous resultsshowed that the time required for breakup, etc., of large Ohdrops could not be scaled systematically in terms of thecharacteristic low Oh breakup time, t , of Ranger andN i ~ h o l l s . ' ~hus, he more appropriate characteristic time forlarge Oh conditions defined by Hinze39 was used instead asfollows:diameter Fluctuations exceeds a diameter ratio of 1.1)Perhaps the most striking feature of Fig. 3is that whilethe We required for particular deformation and b ea ku p regimetransitions remain relatively constant for small o h (,,dues essthat 0.1), the We required for the various transitionsprogressively increase with increasing Oh at large Oh (valuesgreater than 1). Thus the onset of deformation (5-10%deformation) and breakup (bag breakup) occur at We of ca. 0.6and 13 for Oh c 0.1; however, breakup no onger is observed

deformation) disappears for a similar We range when Oh >1ooO. Other deformation and breakup regimes observed at low

Then, i t was assumed that the maximum deformationcondition, or the onset of breakup condition, occurs at a time, t-KI , where K s an empirical constant for the process beingconsidered. Thus, Completing the integration of Q. (1) fromt=O where u= u, t o t=kT where U=U, yields:(3)for We < lo00 when Oh > 10 while deformation (5-10% UdU=l+l('M(Pd do Ug)-

4

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whereK =3c~K/4 (4)

using Eq. (3), the local Weber number of the drop can beexpressed as follows:-- pCu2do a= We / (I + K'Oh(p, / pd)1'2 / We1/2)2 ( 5 )wh ere We=p, u:d, / c s the value based on initial conditions.Then it was assumed that the local Weber number must reach aparticu lar value. We,. for the regim e transition of interest tooccur. Finally, solving for the initial We t o achieve the requiredWe,,, yields:

We = (1 + (1 + (4K/We~n(pJpd)ln)2Wecr/4 (6)Values of Wecr and K were fitted to Eq. (6) to yieldbest-tit predicted transitions for the 5-IO , 10-20 and 20deformation regime transitions as well as the first breakupregime transition (typically for bag breakup). The resultingtheoretical predictions of the regime transitions are illustratedin Fig. 3.In view of the simplifications of the theory, theagreement between predicted and measured regime transitionsis seen to be reasonably good. Notably, Eq. 6) suggests thatthe transition We - Oh at large Oh rather than an ultimate l imitfor particular transitions at large Oh, as su ggested by Hinze.IoThis is a very important difference i n behavior that hassignif icant relevance for processes of high pressurecombustion, as noted earlier: therefor, the issue clearly meritsadditional study.

D The present drop towerexperiments were designed to define the properties ofdeformed drop, and did no t reveal strong effects of Oh over theavailable test range. Thus, a deformation and breakup regimemap analogous to Fig. 3for shock wave disturbances was notdeveloped for steady disturbances (although effects of We ondeformation for steady disturbances will be consideredsubsequently). How ever, a transition in drop shape, fromdome- to bowl-shaped drops, was explored which will bediscussed in the following.Dome-and bowl-shaped drops are observed for bothshock wave an d steady disturbances. Typical examples of these

deformation regimes are illustrated in Figs. 4and 5 for steadydisturbances with liquid drops in liquid continuous phases. Thewindward or forward stagnation point side of a dome-shapeddrop (Fig. 4) is flattened while the downstream side is rounded.This shape is similar to the appearance of a drop during bagbreakup for shock wave disturbances, j u s t at rhe stm of theperiod where the bag begins to grow due to deformation of thecenter of the drop in a downsueam dm ction.6 Thus, conditionsfor dome-shaped drops are somewhat analogous to conditionsfor bag breakup, and appear to involve interactions betweendrag and surface tension forces.A typical bowl-shaped drop is illustrated in fig. 5 . Inthis case, the forward stagnation region is rounded while thedownsbeam side of the drop tends to be flattened, or even cupshaped in some instances. This shape is similar to theappearance of a drop during shear breakup for shock wave

disturbances, just at the start of the period where drops aresu ip ed from the periphery of the core (or drop forming)analogous to conditions for shear breakup, and appear toinvolve interactions between drag and viscous forces.

The observations discussed in connection with Figs. 4and 5 suggest a simple means for establishmg dome-or bowl-shaped drop regimes from the ratio of shear smsses to surfacetension stresses. In doing this shear stresses shall be estimatedfor the continuous phase, exploiting the fact that shear s t r essesare continuous at the drop surface, barring significant effects of

-

drop. ; Thus, conditions for bowl-shaped drops are somewhat

surface tension gradients. Other major assumptions for theseconsiderations are as follows: shear stresses are approxlmatedby conditions for a laminar plane boundary layer, dispersedphase velocities are assumed to be small so that relativevelocities are directly related velocity of the center of mass ofthe dispersed phase, and fluid properties are assumed to beconstant. Then taking the length of the boundary layer to beproportional to the initial drop diameter, the characteristic shearstress becomes,'Tw =c kd(kcdlpouo) n (7)

where C is a constant of proportionality. The correspondingsurface tension swsss proportional to a/&, assuming thatis a reasonable measure of the curvature of the drop surface.Equating these stresses then yields an expression for thetransition Weber number between dome-and Bowl-shapeddrops, as follows:We=c 'Re ln (8)

where C is a constant of proportionalityThe dom e-to bowl-shaped drop transition expression ofEq. (8) was evaluated using available data for both steady andshock wave disturbances. The results are plotted according tothe variables of Eq. (8) in Fig. 5 .The results for steadydisturbances involve both liquid-liquid and liquid-gas systems,however, the liquid-gas systems generally involve relativelylarge Re and relatively small We so that these conditions arewell within the dome-shaped drop regime. Thus, the transitioncriterion illustrated in the figure was found using the liquid-liquid measurements, to yield the following correlation base onEq. 8)

We = 0.5 Re' n, steady disturbances ( 9 )Results for shock wave disturbances of drops in gases at lowOh also were considered. This transition was based on theobservation that dome-and bowl-shaped drops are observedwithin the multimode breakup regime, near the end of thedeformation period for values of We smaller and larger thanroughly 40, respectively. This implies, for Ob e 0.1, thefollowing relationship

w e =0.7Rein, shock wave disNbances (10)which has been entered on the plot as well.In spite of the wide range of conditions, the differentkinds of disturbances, and t h e different density ratios, thedome- to bowl-shaped drop transition illustrated in Fig. 5isreasonably consistent with Eq. (8). Funhermore, the transitionover the present test range is reasonably expressed by eitherEq. (9) or Eq. (10). Perhaps this is not surprising; in fact, anumber of investigators have suggested Eq. 8) as a criterionfor the onset of shear breakup, as discussed by Borisov et a l . 2However, it should be noted that this result generally pertainsto conditions where Re is substantially greater than the stokesregime (typically Re > 10). In contrast, othcr criteria know n forthe stokes flow regime where a somewhat similar transition(from an oblate to a prelate spheroid) has been studied for some

time, see Wellek et al.32 and references cited therein. Thus,more work is needed to reconcile these drop shape and breakupregime transitions w ithin the Sto kes and moderate Reynoldsnumber regimes.

Earlier work for shock wave disturbances found arelatively simple relationship between drop deformation andWeber number at small Oh.6 This result was based onphenomenological analysis of the interaction between surfacetension and pressure forces when a drop is drawn into a5

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SWierzba, A., and Takayama, K., "ExperimentaInvestigations on Liquid Droplet Breakup in a Gas Stream,Reo. Inst. Hieh S n e d Mech.. Tohoku Univ,, Vol. 53, No. 382,1987,pp. 1-99.9Giffen, E., and Muraszew. A., The Aromizarion ofLiquid Fuels, John Wiley Sons, New York, London, 1953.'OHinze, J.O. "Fundamentals of the HydrodynamicMechanism of Splitting in Dispersion Processes," AiChEVol. 1,1955, pp. 289-295.llKrzec zkow ski, S.A., "Measurement of Liquid DropletDisintegration Mechanisms," Int. J. Mu ltinha se Flow Vol. 6,1980, pp. 2 27-239.12C lift, R., Grace, J.R., and We ber, M.E., Bubbles,Drops and Particles, Academic Press, New York, 1978, pp. 26and 339-347.

Ind. Enm. Chem., Vol.43 , 1 951,pp . 1312-1317.'3Lane, W . R., "Shatte r of Drop s in S treams of Air,"14En.eel. O.G., "Fraementation of W aterdroos in theZone B e h i d an Air Shock," Journal of ResearLh of theVol. 60 958, pp. 245-2 80.

*5Loparev. V.P., "Experimental Investigation of theAtomization of Drops of Liquid under Co nditions of a GradualRise of the Exte rnal Forces," Aka demi i Nauk SSSR.Me kha nik a Zh idk osti i Gaz~,V%?%75, pp . 174.178.26Liang, P.Y., astes, T.W.. and Gharakhan, A.,"Computer Simulations of Drop Deformation and DropBreakup," AIAA Paper No. 88-31 42, 1988.27Simmons, H.C. "The Correlation of D r o p S i z eDisuibutions in Fuel N ozzle Sprays," J. En=. for Pow er, Vol.

99, 1 977,pp . 309-319.ZEMagono, C., On he Shape ofWater Drops Falling inStagn ant Air," I J d e t . ~ . . ol. 11,1954. pp. 77-79.29Hu, S., and Kin tner, R.C., "The Fall of Single DropsThro ugh W ater,"m. ol. 1,1955, pp, 42-48.30M agarvey , R.H.. and Taylor, B.W.. "Free FallBreakup o f Large D rops,'' J. ADDI.Phvs., Vol. 27, 1956, pp,1129- 135.3'Schro eder. R.R., and Kinme r, R.C., "Os cillation s ofDrops Falling in a Liquid Field," AIChE J., Vol. 1I 1965, pp.5-8.

15Ha nson. A.R., Dom ich, E.G.. and Adam s. H.S.,"Shock-Tube Investigation of the Breakup of Drops by AirBlasts," ,Vol. 6, 19 63, pp. 1070-1080.I6Haas. F.C., "Stability of Droplet s Suddenly Exp osedto a High V elocity Gas Stream,"m. ol. 10,1964, pp.920-924.I7Ranger. A.A., and N icholls, J.A.. "The Aerod ynam icShattering of Liquid Drops,"m. ol. 7, 1969, pp. 285-290.18Reinecke,W.G., and McK ay, W.L., "Experiments onWaterdrop Breakup Behind Mach 3 to 12 Shocks," SandiaC o p Rep t SC-CR-70.6063, 1969.'gReinecke, W.G., and Waldman, G.D., "A Study ofDrop Breakup Behind Strong Shocks with Applications toFlight." Avco Rept. AVSD -0110-70-77, 1970."'Volynskii, M.S., and Lipatov . AS. , "Deform ation andDisintegration of Liquid Drops i n a Gas Flow," J nzh eme meFizecheskii Zhurnal,Vol. 18, 1970, pp, 838-843.21H assler, G., "Un tersuchung Zu r ZerstBrung vonWassemopfen Durch Aerodynamische Krafte,"W.,Vol. 38, 1972. pp. 183-192.22Simpkins. P.G., and Bales, E.J.. "Water-DropResponse to Sudden A ccelerations," J. Fluid M ech., Vol. 5 5 ,1972, pp. 62 9-639.Z%el'fand, B.E. Gubin, S.A., and Kogarko, S.M.,"Various Form s of Drop Fractionation in Shock Waves and. .Their Special Characteristics,"pZhuol. 27, 1974 , pp. 119-126.

32Wellek, R.M., Agraw al, A.K., and S kelland , A.H.P.,"Shape of Liquid Drops Moving in Liquid Media," AIChE J.,Vol. 12,1966,pp. 854-862.33Pruppacher, H.R., and Pitter, R.L.. "A Semi-Empirical Determination of the S IDrops," J. Am os . Sciences, Vol. 28,34Ryan, R.T., "The Behavior of Large Low-Surface-Tension W ater Drops Falling at T erminal Velocity i n Air, LAool. Meteroloey Vol. 15, 1976, pp. 157-165.35Dabora, E.K.. "Production ofMonod isperse Sprays,"Rev.. ol. 38, 1967, pp. 502-506.36Saneiovanni. J.. and Kestin . A.S.. "A Theo retical and

Experimental-Investigauon of the Ignition of Fuel Droplets,"Combust..ol. 6,1977, pp. 59-70.37L ange. N.A.. H and boo k of Chemis t ry , 8th ed..Handbook P ublishers, Inc., Sandusky , Ohio, 1952, pp. 1134and 1709.38White, F.M., Viscous Fluid Flow, McGraw -Hill, NewYork, 1974.39Hinze. J.O., "Critical Speeds and Sizes of LiquidGlobules," . &&L&S _. ol. 11,19 48 , pp. 273-287.

HBorisov. A.A., Gel'fand, B.E., Natanzon, M.S., andKossov, O.M.. "Droplet Breakup Regimes and Criteria forTheir Existence," m z h u -nzh I Vol. 40,1981, pp. 64-70.

8I .

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Table 1 Summary of liquid/air shock t be test ConditimSaDispersed Pd MX X ~(Y ( a ~ 1 0 3 do We oh RePhase (kg/m3) (kg/ms) Nh) m)Watern-HeptaneMercuryDC200 FluidbDC200FluidbGlycerol 21%Glycerol 63%Glycero l 75%Glycerol 84%Glycerol 92%Glycerol 97%Glycerol WS %

997. .68313,6009809xn

124012531260

R.94~.3.9415.0100,ooO300,00016.0108.0356.01 0 0~.3270835012,500

70.820.0475.020.020.067.361.863.8~~~63.262.562.462.0

1.00.50.850.15-0.350.15-0.351.21.21.21.21.21.51.55

0.5-23614-13710-1335-100100-5808-1301-1292-1281-1271-2681-2051-612

365-555 1170-52000.007 1540-63900 . 0 3 9 0 ~ R O - M Z O~0.0990 730-62700.260 500.62101.050 530-83301.700 600-8880~~ ~~3.850 630-15,760

aAir initially at 98.8 kPa and 298 f K in the driven section of the shock tube with shock Mach numbers in the range 1.01-1.24.Roperties of air taken at normal temperatux and pressure: pc = 1.18 kg/m3,k 18.5 x l o 6 kg/ms.bDow Corning 200 Fluid.

Table 2 Summary of liquid/air drop ower test conditionsap,,xi104 a x 10 3 do We ch ReDispersed PdPhase (kg/m3) WW (N/m) (mm)

Water 997 8.94 70.8 2.0-7.8 1.2-9.8 0.0012-0.0024 830-4600Glycerol 42% 1105 35.0 65.4 2.0-7.8 1.4-9.5 0.0046-0.0091 870-4210Glycerol 63% 1162 108 64.8 2.0-7.8 1.5-9.8 0.0140-0.0275 880-4390Glycero l 84% 1219 1Mx) 63.2 2.1-7.8 1.7-9.5 0.126-0.248 940-4260Glycerol92 1240 3270 62.5 2.5-6.4 2.5-8.6 0.455-0.729 1180-350Glycerol 99.5% 1260 12,500 62.0 2.4-6.4 2.3-8.6 1.74-2.90 1060-345awops falling in air at 98.8 kPa and 297 f K. Properties of air taken at normal temperatu re and pressure: pc = 1.18 kg/m3, Pc= 18.5 x 10-6 kg/ms.

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U6

CI

IUIU L

Fig. 1 Sketch of the shock ube apparatus.W

Fig.t . , . . , I 1 . , . , . . , , ,,., I 1 1 1 1 I n

Sketch of the drop levitation system.

4

Fig. 3

Fig. 4

Drop deformation and breakup regime map forshockwave disturbances.

Photoeraoh of a twical dome-shaoed OD: free. ~.falling water drop mau, We=7 2, Re3200 andOh=O 002 .,~ .10

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Fig. 5 Photograph of a typical bowl-shaped drop: waterdrop in paraffin oil, p& = 1.15, We = 12.4, Oh =0.0007 nd Re = 39.

102k ' ' I ' ' I ' ' I ' , q

10'0.'1 10' 102 10'

Fig. 6 Dom e- and bowl-shaped d rop regime map.

F A , , , , , , ,, e , ,O L l F E R O L M U P * I Ul H O ( LOL SLIOLI .V* PIIVLltw MLCARBON 013 LnDf W A R l

1 0 ' Io- ' 100 we 10'Fig. 7 Drop deformation as a function of Weher num ber.

~

1 1

w e - 1

w e = 1 0

Fig. 8 Drop shape and pressure disuibution for steady(left-hand column) and shock wave (right-handcolumn) disturbances.

I O ' I I

Fig. 9 Drop drag coefficient as a function of deformationfor steady disturbances.

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lo .w- .W5 6.Ws6a 4.w0

2.w

0 . W 2 00 4 W 6 W 8 W 1000I I I I I2.00 4.W 6.W 8 . W 10.00d- I dmn

Fig. 1 Drop drag coefficient as a function of deformationforboth shock wave and steady disturbances.

1 2