TEMPORAL PROPERTIES OF DROP BREAKUP IN THE SHEAR.pdf

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Int. J. Multiphase FlowVol. 23, No. 4, pp. 651-669, 1997

1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain

PI I: S0301-9322(97)00006-2 0301-9322/97 17.00 + 0.00

T E M P O R L P R O P E R T IE S O F D R O P B R E K U P I N T H E S H E R

B R E K U P R E G I M E

W.-H. CHOU, L.-P. HSIANG and G. M. FAETHf

Depar tment of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2118, U.S.A.

Received 2 June 1996; in revised form 27 January 1997)

Abstr act--T he temporal properties of drop breakup in the shear breakup regime were studied using pulsed

shadowgraphy and holography for shock wave disturbances in air at normal temperature and pressure.

Test condi tions included Weber numbers o f 125-375, Ohnesorge numbers of 0.0034).040, liquid/gas

density ratios of 670-990 and Reynolds numbers of 3000-12000. The size distributions of drops produced

by breakup satisfied Simmons' universal root normal distribution function at each instant of time, with

Sauter mean diameters independent of surface tension that exhibited transient and quasi-steady regimes

as a function of time. The velocity distribution functions of drops produced by breakup were uniform,

with mean drop velocities somewhat larger than the velocity of the parent drop and rms drop velocity

fluctuations of 30-40% of the mean streamwise velocity of the gas relative to the parent drop, at each

instant of time. The rate of liquid removal from the parent drop was correlated reasonably well by a

clipped Gaussian function. The measurements showed that shear breakup is not a localized event; instead,

it extends over streamwise distances of 0-100 initial drop diameters, which suggests that it should be

treated as a rate process, rather than by jump conditions, in some instances. 1997 Elsevier Science Ltd.

Key Words: drop breakup, drop dynamics, pulsed holography, sprays, atomization

1. INTRODUCTION

The breakup of individual drops, which is often called secondary breakup, is an important

fundamental process o f sprays. For example, drops formed by breakup of liquid surfaces, which

is often called primary breakup, are intrinsically unstable to secondary breakup, while secondary

breakup can be the rate controlling process within dense sprays in much the same way that drop

vapor izat ion can be the rate controlling process within dilute sprays Faeth 1990, 1996; Wu e t a l

1995). Motivated by these observations, the objective of the present invest igation was to extend

earlier studies of the regimes and outcomes of secondary breakup due to shock-wave disturbances

Hsiang and Faeth 1992, 1993, 1995), to consider the evolution of breakup as a function of time

during breakup.

Earlier studies of drop breakup are discussed by Wu

e t a l

1995), Faeth 1990, 1996), Giffen

and Muraszew 1953), Hinze 1955), Clift e t a l 1979), Krzeczkowski 1980) and Wierzba and

Takayama 1987, 1988), among others. Shock-wave disturbances were considered during most

earlier studies, providing a step change of the ambient environment of the drop, similar to

conditions experienced by drops at the end of primary breakup. The main findings of this work

included the conditions required for particular deformation and breakup regimes, the times

required for the onset and end of breakup, the drag properties of deformed drops, and drop size

and velocity distributions at the end of the breakup process i.e. the jump conditions). An

interesting feature of these results is that drop breakup extended over appreciable regions of time

and space and was not properly described by jump conditions in some instances. This behavior

can be illustrated in terms of the characteristic breakup time, t*, of Ranger and Nicholls 1968),

defined as follows

1 = g o ( p L / p o ) 2 / U o , [ ]

fTo whom correspondence should be addressed.

651


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65 w H. CHOU et al

where d is the drop diameter, p is density, u is steamwise drop velocity relative to the gas, the

subscripts L and G denote liquid and gas properties, and the subscript o denotes conditions at the

start of breakup. Liang e t a l . 1988) show tha t breakup times for a wide range of conditions are

5.5 t*, which is comparable to flow residence times within the dense spray region where secondary

breakup is a dominant process Faeth 1990, 1996; Wu e t a l . 1995). Viewed another way, the original

or parent) drop moves roughly 40 initial drop diameters, while the smallest drops formed by

breakup move up to 100 initial drop diameters, during the period of breakup within the shear

breakup regime Hsiang and Faeth 1992, 1993, 1995). Such distances can represent a significant

fraction of the length of the dense spray region. These observations suggest that the time-resolved

features of secondary breakup eventually must be understood. Thus, the present study seeks to

provide some of this information by carrying out new measurements within the shear breakup

regime, where breakup proceeds by the stripping of drop liquid from the periphery of the parent

drop, because this regime tends to dominate drop breakup in practical sprays Hsiang and Faeth

1995). Phenomenological theories also were developed to help interpret and correlate the

measurements.

The present measurements were carried out using a shock tube facility, with the drop

environment approximating air at normal temperature and pressure NTP). Properties during

breakup were observed using pulsed shadowgraphy and holography. Test conditions were limited

to relatively large liquid/gas density ratios p L / p ~ > 500) in order to minimize potential

complications due to the inertia of the continuous phase. The test conditions also involved limited

ranges of Weber and Ohnesorge numbers , which Hinze 1955) has shown define the boundaries

of the shear breakup regime at large liquid/gas density ratios. The Weber number, We, is a measure

of the relative importance of drop drag and surface tension forces, and is defined as follows

We = p G d o u 2 / a , [21

where a is the surface tension of the drop liquid. The Ohnesorge number, Oh, is a measure of the

relative importance of liquid viscous forces and surface tension forces, and is defined as follows

Oh = t~L/ pLdoa) t /z , [31

where ~t is the molecular viscosity. The experiments involved relatively small initial Ohnesorge

numbers Oh < 0.04) in order to minimize potential complications due to effects of liquid viscosity.

For this range of Oh, operation in the shear breakup regime requires We > 90, in order to exceed

the mult imode/shear breakup regime transition, and We < 800, in order not to exceed the

shear/drop-piercing or shear/catastrophic) breakup regime transition Giffen and Muraszew 1953;

Hinze 1955; Reinecke and McKay 1969; Reinecke and Waldman 1970). Drop liquids included

water, ethyl alcohol and various glycerol mixtures in order to provide information about effects

of drop liquid properties.

2. EXPERIMENTAL METHODS

2 . I . A p p a r a t u s

The test apparatus will be described briefly because it was similar to earlier work Hsiang and

Faeth 1992, 1993, 1995). A shock tube with the driven section open to the atmosphere was used

for the measurements. The driven section had a rectangular cross-section 38 mm wide and 64 mm

high) and was sized to provide test times of 17-21 ms in the uniform flow region behind the incident

shock wave. The test location had quartz windows to allow observations of drop breakup.

A vibrating capillary tube drop generator, similar to the arrangement described by Dabora

1967), was used to generate a stream of drops having a constant diameter. An electrostatic drop

selection system, similar to Sangiovanni and Kestin 1977), was used to control the spacing between

drops. This drop stream passed vertically across the shock tube at the test location. The spacing

between drops was 5-7 mm while drop sizes were < 1 mm; therefore, drops always were present

within the region observed while interactions between adjacent drops during shear breakup were

negligible.


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TEMPORAL PROPERTIES OF DROP BREAKUP IN THE SHEAR BREAKUP REGIME 65

2 2 I n s t r u m e n t a t i o n

P u l s e d h o l o g r a p h y a n d s h a d o w g r a p h y w e r e us e d t o o b s e r v e t h e p ro p e r t ie s o f th e p a r e n t d r o p

a n d t h e d r o p s p r o d u c e d b y b r e a k u p a s a f u n c t i o n o f t im e d u r in g b r e a k u p . T h e h o l o c a m e r a u s e d

t w o f r e q u e n c y d o u b l e d Y A G l as er s (S p e c t ra P h y s i c s M o d e l G C R - 1 3 0 , 5 3 2 n m w a v e l en g t h , 7 n s

p u l s e d u r a t i o n , u p t o 3 0 0 m J p e r p u l s e ) w h i c h c o u l d b e f ir e d w i t h p u l s e s e p a r a t i o n s a s s m a l l a s

1 00 n s . A n o f f - a x is h o l o c a m e r a a r r a n g e m e n t w a s u s e d w i t h t h e o p ti c s p r o v i d i n g a 2 5 m m d i a m e t e r

f ie l d o f v i ew a t t h e t e s t d r o p l o c a t i o n . R e c o n s t r u c t i o n o f th e d o u b l e - p u l s e h o l o g r a m s y i e l d e d tw o

i m a g e s o f t h e s p r a y s o t h a t d r o p v e l o ci ti e s c o u l d b e f o u n d g i v e n t h e t im e o f s e p a r a t i o n b e t w e e n

t h e p u l s e s ( w h i c h w a s m e a s u r e d w i t h a d i g i t a l o s c il lo s c o p e ). T h e s e c o n d l a s e r p u ls e w a s s o m e w h a t

w e a k e r t h a n t h e f ir st , w h i c h a l l o w e d d i r e c t i o n a l a m b i g u i t y t o b e r e s o l v e d b e c a u s e s t r o n g e r p u l s e s

y i e l d e d s h a r p e r r e c o n s t r u c t e d i m a g e s . T h i s a r r a n g e m e n t a l s o p r o v i d e d s h a d o w g r a p h y b y b l o c k i n g

t h e r e f e r e n c e b e a m .

T h e h o l o g r a m r e c o n s t r u c t i o n s y s t e m w a s m o d i f i ed f r o m H s i a n g a n d F a e t h ( 19 9 2 , 1 99 3, 1 99 5).

A h e l i u m - n e o n l a s e r ( S p e c t r a P h y s i cs M o d e l 1 2 4B , c w l a se r , 35 m W o f o p t i c a l p o w e r ) w a s u s e d

t o r e c o n s t r u c t t h e i m a g e , w h i c h w a s o b s e r v e d u s i n g a C C D c a m e r a ( S o n y , M o d e l X C - 7 7 ) w i t h

o p t i c s t o y i e l d a m a g n i f i c a t i o n o f 3 0 0 :1 a n d a f ie ld o f v ie w o f t h e i m a g e ( o n t h e m o n i t o r ) o f

1 .2 1 . 4 m m . T h e o p t i c a l d a t a w a s o b t a i n e d u s i n g a f r a m e g r a b b e r ( D a t a T r a n s l a t i o n D T 2 8 5 1 )

a n d p r o c e s s e d u s i n g M e d i a C y b e r n e t i c s I m a g e - P r o P l u s s o f t w a r e . V a r i o u s l o c a t i o n s i n t h e

h o l o g r a m r e c o n s t r u c t i o n w e r e o b s e r v e d b y t r a v e r s i n g t h e h o l o g r a m i n t w o d i r e c t i o n s , a n d t h e

v i d e o c a m e r a o f th e i m a g e d i s p l a y i n t h e t h i r d d i r e c t i o n . P o s i t i o n s w e r e s e l ec t ed f o r v i e w i n g u si n g

s t e p p i n g m o t o r d r i v e n l in e a r t r a v e r s in g s y s te m s ( V e lm e x , M o d e l V P 9 0 0 0 ) h a v i n g 1 m p o s i t i o n i n g

a c c u r a c ie s . T h e c o m b i n e d h o l o c a m e r a / r e c o n s t r u c t i o n s y s t e m a l l o w e d o b j e c ts a s s m a l l a s 3 p m t o

b e o b s e r v e d a n d o b j e c t s a s sm a l l a s 5 m t o b e m e a s u r e d w i t h 5 % a c c u r a c y .

D r o p s iz e s a n d v e l o c i t ie s w e r e m e a s u r e d a s d e s c r i b e d b y H s i a n g a n d F a e t h ( 1 9 9 2 , 1 99 3, 1 99 5 ).

P r e s e n t r e s u lt s w e r e f o u n d b y s u m m i n g o v e r a t l e as t f o u r r e a li z a ti o n s , a n d c o n s i d e r i n g 5 0 - 1 0 0

l i q u id e l e m e n t s , i n o r d e r t o p r o v i d e d r o p d i a m e t e r a n d v e l o c i ty c o r r e l a t io n s . T h e s e s a m p l e s i ze s

w e r e s m a l le r th a n p a s t s t u d i es o f j u m p c o n d i t i o n s i n o r d e r t o m a i n t a i n a m a n a g e a b l e t e s t p r o g r a m

w h i l e re s o l v i n g d r o p p r o p e r t i e s a s a f u n c t i o n o f ti m e . E s t i m a t e d e x p e r i m e n t a l u n c e r t a i n t ie s ( 9 5 %

c o n f i d e n c e ) a r e le ss th a n 1 5 % f o r d r o p d i a m e t e r s a n d l es s t h a n 2 0 % f o r s t re a m w i s e m e a n d r o p

v e l o c it ie s a n d r m s d r o p v e l o c i ty f lu c t u a t io n s ; u n c e r t a i n t ie s o f c r o s s - s t re a m m e a n v e l o ci ti e s a r e

l a r g e r d u e t o t h e s m a l l e r v a l u e s o f t h e s e v e l o c i ti e s , a s d i s c u s s e d l a t e r .

2 3 T es t cond i ti ons

T h e t e s t c o n d i t i o n s a r e s u m m a r i z e d i n t ab l e 1 . T h e l i q u i d p r o p e r t i e s w e r e o b t a i n e d f r o m L a n g e

( 1 95 2 ), e x c e p t f o r t h e s u r f a c e t e n s io n s o f t h e g l y c e r o l m i x t u r e s w h i c h w e r e m e a s u r e d i n t h e s a m e

m a n n e r a s W u e t a l ( 1 9 9 1 ). S h o c k w a v e M a c h n u m b e r s i n t h e s h o c k t u b e w e r e r e la t i v e l y l o w , le s s

t h a n 1 .1 5; t h e r e f o r e , t h e p h y s i c a l p r o p e r t i e s o f th e g a s i n t h e u n i f o r m f lo w r e g io n b e h i n d t h e s h o c k

w a v e , w h e r e d r o p b r e a k u p o c c u r r e d , w e r e n e a r l y t h e s a m e a s a i r a t N T P .

T h e s p e c if ic ra n g e o f t h e p r e s e n t t es t s is il l u s tr a t ed o n t h e d r o p d e f o r m a t i o n a n d b r e a k u p r e g im e

m a p a p p e a r i n g i n f i g u r e 1 . T h i s r e g i m e m a p i s e x t e n d e d f r o m H s i a n g a n d F a e t h ( 1 9 9 5 ) t o i n c l u d e

n e w t r a n s i t i o n s m e a s u r e d d u r i n g t h e p r e s e n t i n v e s t i g a t i o n , a s f o l l o w s : t h e s h e a r / d r o p - p i e r c i n g ( o r

c a t a s t r o p h i c ) b r e a k u p r e g i m e t r a n s i t io n a t l ar g e W e , fi rs t o b s e r v e d b y R e i n e c k e a n d M c K a y ( 1 96 9 ),

a n d t h e m o r e q u a l i t a t iv e s h e a r / l o n g - l i g a m e n t b r e a k u p r e g im e t r a n s i ti o n a t l a r g e O h , t o b e d i s cu s s e d

l a te r . A s i d e f r o m e x p e r i m e n t s u s e d t o d e f i n e t h e s h e a r / d r o p - p i e r c i n g a n d s h e a r / l o n g - l i g a m e n t

Table 1. Summary of the test conditionst

pL pL X 104 O do

Liquid (kg/m3) ( k g /m s) (m N/m ) (pm) Oh x 103 Re

Wa te r 997 8 .94 70 .8 59 0- 10 00 3 .4 ~ , . 4 4930- 11150

Ethyl a lcohol 800 16.0 24 .0 78 0- 10 00 11.5-13.1 3070-5740

Glyc erol (42%):~ 1105 35.0 65.4 1000 1 3 .0 600 0-11 180

Glycerol (63%)5 1162 108.0 64.8 1000 3 9 .4 5840-11 240

tAir initially at 98.8 kPa and 298 + 2 K in the d riven section of the shock tube. Sh ock Mach

numbers in the rang e 1.01-1.15 with W e in the range 125-375. Properties of the air w ere taken

at normal tem perature and pressure: pc = 1.18 kg/m3, pG = 18.5 x 10 -4 kg/m s.

:[:Percentage gly cerin by mass.


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654

O s

W. H. CHOU et a

i I I I

I

' I l ] | ' i l ' ' ' i l l ' ' I

L I Q U I D L I Q U ID S O U R C E

I I ' I I l

n - H E P T A N E o G L Y C E R O L [ 1 2 % H A N S O N e t L 1 1 96 3 1 6

W A T E R c l G L Y C E R O L 9 7 % H I N Z E { 1 9 55 ) 6

G L Y C E R O L 2 1 % a , G L Y C E R O L g i L 5 % T L A N E 1 19 51 ) l ,

G L Y C E R O L 6 3 % ,~ M E R C U R Y @ L O P A R E V ( I J T S l 'q

G L Y C E R O L 7 5 % O 2 0 0 F L U I D ) ' K R Z E C Z K O W S K I II 9 8 5 ) - - -

G L Y C E R O L 8 4 % P R E S E N T

- p l. i P o 8 1 1 0 - 1 2 0 0 0

ira

w

E

Z

rv

I . d

m

b J

I 0 3

. P I E R C I N G B R E A K U P ~ , , . m ' ~

o l I S H E A R B R E A K U P

S .E A R .R E ~ U P I r ' O ~ I C ' E N T S f i '

I ~ ~ ~

i o =

. . . .

~ _ . _ . . . . ~ _ J . J / i ~ 0 ~ i ~ l O N

I o :

Oa~LkJ_O RZ X /

. . . . - > ; ; _ . , , ~ I . t

D E F O R M A T I O N t _ - . _ ~ j - . ] ~ , , r /

_ 2 0 %

.,,- ~

, r . ~

0 % ( D E F O R M A T I O N (

ioo ~ ~ ~ o R ~ T iO N < s ~

~, ,, ., DEF~MAr,O~' 10 . _ - - ' : .~

,o 1 6 3 1 62 1 6 i o , d , o ~

O H N E S O R G E N U M B E R

Figure I. Drop deformationand breakup regimemap for shock-wavedisturbances, extendedfrom Hsiang

and Faeth 1995).

i 0 3

breakup regime transitions, however, present measurements were confined to the cross-hatched

region which is conservatively located well within the shear breakup regime at small Ohnesorge

numbers. This involved

pL/pG

of 670--990, We o f 125-375 and Oh of 0.003-0.040. The range of

initial drop Reynolds numbers, Re, was 3000-12000, where

Re =

pGuodo/ G.

[4]

These values of Re are higher than conditions where the gas viscosity has a significant effect on

drop drag properties, e.g. the drag coefficient, C, for spheres only varies in the range 0.4-0.5 for

this Reynolds number range White 1974).

3. RESULTS AND DISCUSSION

3.1. Flow v i sual i zat ion

Pulsed shadowgraph flow visualizations provide an overview of the tempora l properties of shear

breakup. Figure 2 is an illustration of a series of these shadowgraphs for a condition toward the

lower Oh values considered during the present study, see figure 1. It should be noted that the onset

and end o f breakup where liquid removal from the parent drop begins and ends) occur at

t / t *

of roughly 1.5 and 5.5, respectively Liang

et al .

1988), for reference purposes. The shock wave,

and the flow velocities behind the shock wave, pass from the top to the bottom of the

shadowgraphs.


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TEMPORAL PROPERTIES OF DROP BREAKUP IN THE SHEAR BREAKUP REGIME 655

The shadowgraph at

t / t = 0

in figure 2 provides a size reference and illustrates the initial

spherical shape of the drop. After passage of the shock wave, the drop deforms into a flattened

shape, (see

t / t

= 1), because the liquid is drawn toward the drop periphery where the pressure

is low due to acceleration of the gas flow over the drop surface. Ligaments and drops begin to

be stripped from the drop when

t / t = 1 .5

(which is not shown in figure 2); even at

t / t

= 2,

however, there is an extensive system of ligaments protruding from the periphery of the parent

drop, with numerous individual drops present near the downst ream end of the ligaments as a result

of ligament breakup. Subsequently, the diameter and length of the ligaments, the diameters of the

drops produced by breakup of the ligaments, the number of individual drops, and the range of

streamwise distances where drops are observed, all increase as t / t increases; in contrast, the

number of ligaments and the size of the parent drop both decrease as

t / t

increases--compare

results at t / t = 2, 3, 4 and 5. Finally, the diameter of the parent drop, and the gas velocity relative

to the parent drop, become small so that drop breakup ends at

t / t = 5 .5

(just beyond the range

of photographs in figure 2). Subsequently the parent drop evolves toward a spherical shape as its

relative velocity continues to decrease.

Increasing Ohnesorge numbers cause the maximum lengths of the ligaments to become

progressively larger. This behavior follows from evolution of shear breakup toward large Oh

conditions that are dominated by drop deformation and the formation of elongated single drops

that are resistant to ligament breakup. Such configurations vastly complicate problems of

temporally resolving drop breakup; therefore, present experiments were confined to Oh < 0.1,

which was somewhat arbitrarily defined as the onset of the long ligament regime of shear breakup.

Figure 3 is an illustration of breakup behavior at the transition condition to the long ligament

regime. A series of shadowgraphs at the same values of

t / t

as figure 2 are shown for a glycerol

(75 glycerin by mass) drop having do = 1000 pro, We = 250 and Oh = 0.099. As before, both

shock and flow velocities are directed from the top to the bottom of the shadowgraphs.

The shadowgraphs illustrated in figure 3 are qualitatively similar to those of figure 2. Ligament

lengths and the extent of the region containing drops progressively increase, while the size of the

parent drop progressively decreases, as

t / t

increases. Tracking and identifying intermediate

W A T E R W e = 2 5 0 O h = 0 . 0 0 4 4

t / t = 0 = 1 = 2

t / t = 3 = 4 = 5

Figure 2. Flash s had owg ra phs of the shear brea kup o f a wate r drop as a function of t ime: do = 590 pro

We = 250 and Oh = 0.0044.


6/19

656 W.-H. CHOUetal.

G LYC ER OL (75 ) , W e=250 , Oh =0 .099

t /t = O = 1 = 2

t / t = 3 = 4 = 5

Figure 3. Flash shadowgraphs of the shear breakup of a glyceroldrop (75 glycerinby mass) as a fun ction

of tim e: d,, = 1000/~m. We = 250 and Oh = 0.099.

b r e a k u p p o i n t s a l o n g v e r y l o n g l i g a m e n t s s ee n a t t h e s e c o n d i t i o n s , e v e n w h e n u s i n g h o l o g r a p h y ,

h o w e v e r , is v e r y p r o b l e m a t i c a l . T h u s , p r e s e n t t e s t c o n d i t i o n s w e r e li m i te d t o O h < 0 .0 4 w h i c h

y i e l d e d m a n a g e a b l e l i g a m e n t l e n g t h s i n a r e g io n s o m e w h a t b e l o w t h e t r a n s i t io n t o t h e

l o n g - l i g a m e n t s h e a r b r e a k u p r e g i m e .

3 .2 . Drop s i ze s

3 . 2 . 1 . D r o p s i z e d i s t r i b u t i o n . T h e e v o l u t i o n o f d r o p s iz e d i s t r ib u t i o n s d u r i n g s h e a r b r e a k u p w a s

c o n s i d e r e d f i r s t b e c a u s e t h i s a f f e c t s t h e i n f o r m a t i o n n e e d e d t o c h a r a c t e r i z e s e c o n d a r y b r e a k u p

p r o p e r t i e s . T h e m a i n i s s u e w a s t o d e t e r m i n e w h e t h e r d r o p s iz e d i s t r ib u t i o n s v a r i e d a p p r e c i a b l y

f r o m t h e u n i v e r s a l r o o t n o r m a l d i s t r i b u t i o n w i t h M M D / S M D = 1 .2 , whe r e M M D a n d S M D a r e

t h e m a s s m e d i a n a n d S a u t e r m e a n d i a m e t e r s o f th e d i s t ri b u t i o n , r e s p e c ti v e ly , p r o p o s e d b y S i m m o n s

( 1 97 7 ), w h i c h h a s b e e n f o u n d t o b e s a t i s fa c t o r y f o r a v a r i e t y o f d r o p b r e a k u p p r o c e s s e s (F a e t h

1990 , 1996 ; W u e t al . 1995; Hs i a ng a nd Fa e th 1992 , 1993, 1995) . Se e Be lz ( 1973) f o r a d i sc u ss ion

o f th e p r o p e r t i e s o f th e r o o t n o r m a l d i s t r i b u t i o n f u n c t i o n .

T y p i c a l r e s u l ts f r o m t h e d r o p s iz e d i s tr i b u t i o n m e a s u r e m e n t s a r e i l lu s t r a te d i n f i g u re 4 . T h e

n u m b e r o f d r o p s a v a i l a b l e to d e f in e th e d r o p s iz e d i s t r i b u t io n f o r e a c h b r e a k u p c o n d i t i o n a n d t / t

i s l i m i t e d , w h i c h a c c o u n t s f o r th e s i g n i f i c a n t s c a t t e r o f d r o p s iz e d i s t r i b u t i o n p r o p e r t i e s s e e n i n

f ig u r e 4 . N e v e r t h e l e s s , w i th i n t h e s c a t t e r o f t h e d a t a , d r o p s iz es p r o d u c e d b y s e c o n d a r y b r e a k u p

a r e r e p r e s e n t e d r e a s o n a b l y w e l l b y t h e u n i v e r s a l r o o t n o r m a l s iz e d i s t r i b u t i o n f u n c t i o n w i t h

M M D / S M D = 1 .2 . T h i s b e h a v i o r is p l a u s ib l e b e c a u s e p r i m a r y a n d s e c o n d a r y b r e a k u p p r o c e s s e s ,

a s w e ll as d r o p s a t v a r i o u s p o s i t io n s w i t h i n d e n s e s p r a y s , g e n e r a l ly s a t is f y th e u n i v e r s a l r o o t n o r m a l

s i z e d i s t r i b u t i o n f u n c t i o n , a s m e n t i o n e d e a r l i e r . W i t h t h e t w o - p a r a m e t e r r o o t n o r m a l s i z e

d i s t r i b u t i o n f u n c t i o n e s t a b l is h e d f o r th e t e m p o r a l b e h a v i o r o f s e c o n d a r y d r o p b r e a k u p , d r o p s iz e

i n f o r m a t i o n c a n b e s u m m a r i z e d b y t h e S M D a l o n e ( H s i a n g a n d F a e t h 1 9 9 2 ) .

3 . 2 . 2 . T e m p o r a l e v o lu t i o n o f th e S M D . G e n e r a l d e s c r i p ti o n . C o r r e l a t i n g e x p r e s s io n s f o r t h e S M D

a s a f u n c t i o n o f ti m e d u r i n g s e c o n d a r y b r e a k u p i n t h e s h e a r b r e a k u p r e g i m e w e r e s o u g h t u s i n g

m e t h o d s s i m i l a r t o H s i a n g a n d F a e t h ( 1 9 9 2 ) a n d W u e t al . ( 1 9 9 1 ) , T h e p r e s e n t a p p r o a c h w a s

m o t i v a t e d b y t h e f l o w v i s u a l i z a t i o n s i l l u s t r a t e d i n f i g u r es 2 a n d 3 . T h e s e r es u l t s s h o w t h a t l i g a m e n t s


7/19

TEMPORAL PROPERTIES OF DROP BREAKUP IN THE SHEAR BREAKUP REGIME 6 7

w h i ch s u b s eq u en t l y b r eak u p i n t o d r o p s h av i n g co m p a r ab l e d i am e t e r s ) a r e s t r i p p ed a t t h e

p e r i p h e r y o f t h e p a r en t d r o p f r o m t h e l iq u i d - p h ase v o r ti c a l reg i o n o r b o u n d a r y l ay e r- li k e f lo w )

t h a t f o r m s a l o n g t h e l iq u i d s u rf ace o n th e u p s t r eam w i n d w ar d ) s i d e o f th e d r o p d u r i n g s eco n d a r y

b r eak u p i n t h e s h ea r b r eak u p r eg i m e . T h i s b eh av i o r a l s o w as s u g g est ed b y ea rl ie r w o r k w h i ch

s h o w ed t h a t d r o p s i z e s a f t e r s eco n d a r y b r eak u p m a i n l y d ep en d o n t h e v i s co s i t y , r a t h e r t h an t h e

sur face t ens ion , o f the liqu id phase Hs iang and Fae th 1992) . In add i t ion , in i t ia l measurem ents

dur ing the presen t inves t iga t ion a l so sugges ted a s t rong ef f ec t o f l iqu id v i scos i ty on the drop s izes

p r o d u ced b y s eco n d a r y b r eak u p . F i n a ll y , t w o b a s ic t y p es o f b eh av i o r w e r e o b s e r v ed d u r i n g p r e s en t

exper imen t s , as fo llows : 1 ) a r eg ime where there was a p rogres s ive increase of l igament d iameter s ,

an d a co r r e s p o n d i n g i n c rea s e o f t h e

SM

o f d r o p s f o r m ed f r o m t h e se l i gam en t s, a s a f u n c t i o n

of t ime du r ing b reaku p , w hich was main ly s een when the l iqu id v i scos i ty and the t ime af t e r the

s ta r t o f b re aku p were bo th sm al l th is behav ior is bes t charac te r i zed b y the flow v i sua l iza t ion o f

f ig u r e 2 ); an d 2 ) a r eg im e w h e r e t h e li g am en t d i am e t e r s, an d t h e S M D o f d r o p s f o r m ed f r o m t h e se

l igaments , we re r e la t ive ly indepe nden t o f t ime, which w as main ly s een when the l iqu id v i scos i ty

and the t ime af t e r the s t a r t o f b rea ku p were bo th l a rge th is beha vior is bes t charac te r i zed by the

f low v i sua l i za t ion of figure 3 ), Thus , l iqu id v i scos i ty had an im por ta n t e f f ect on d rop s izes even

thou gh a ll t es t con di t ions invo lved suf fi c i en t ly smal l Ohnesorge num bers so tha t var i a t ions of l iqu id

v i scos i ty d id n o t a f f ec t c r i t e r ia fo r the on se t o f b rea kup , s ee f igure 1 . Both of these behav ior s sugges t

tha t vor t i c i ty w i th in the pare n t d ro p a f f ec ts b reak up; therefore , these e ff ec ts w i ll be cons idered

s i m i la r t o p a s t s t u d y o f d r o p s iz e j u m p co n d i t io n s d u e t o H s i an g an d F ae t h 1 99 2) .

O

~

5

4

3

2

L I Q U I D W e S Y M B O L ( t/ t* )

W A T E R 1 2 5 ~ ( 2 , 2 ) 4 ' ( 3 .2 ) [ ] ( 4 . 2 ) I n ( 5 . 2 )

2 5 0 - ~ ( 2 . 2 ) < ~ ( 3 . 2 ) - ( 4 . 2 ) ~ ( 5 . 2 )

3 7 5 ( 2 . 2 ) ( 3 . 2 ) ( 4 . 2 ) - 4 ) ( 5 , 2 )

E T H Y L A L C O H O L 1 2 5

L I ( 2 . 3 ) I - 1 ( 3 . 3 ) O ( 4 . 3 )

2 5 0 & ( 2 . 3 ) [ ] ( 3 . 3 ) 4 ~ ( 4 . 3 )

3 7 5 ; I ( 2 . 3 ) [ ] ( 3 .3 ) V ( 4 . 3 )

G L Y C E R O L ( 4 2 % ) 1 2 5 X ( 3 ,0 ) . ( 4 .0 ) ~ ( 5 .0 )

2 5 0 ( 2 . 0 ) & ( 3 . 0 ) ( 5 . 0 )

3 7 5 Q ) ( 2 . 2 ) ~ ( 3 .0 ) ( 4 . 0 ) ( 5 . 0 )

G L Y C E R O L ( 6 3 % ) 1 2 5 Q ( 2 .2 ) . ( 3 .0 ) ~ ( 4 .0 ) ~ (5 . 0 )

2 5 0 A ( 2 . 2 ) A ( 3 . 0 ) ~ ( 4 . 0 ) w ( 5 . 0 )

375 3 .0 ) 4 4 .0 ) , 5 .0 )

1 . 5 M M D I S M O

1 . 2

_ > 4~ 1 . 1

O . I I I 0 5 0 8 0 7 0 g O 9 9

C U M U L A T I V E V O L U M E P E R C E N T A G E

Figure 4. D iameter distribution of d rops produced by shear breakup plotted according to the ro ot normal

distribution function.


8/19

6 5 8 w . H . C H O U e l

a /

u u , ~ p o / p O l e . u .

L.

T R A N S I E N T

Figure 5. Sk etch of the transient shear breakup mechanism at small Ohnesorge num bers

t / t c

1).


9/19

TEMPORAL PROPERTIES OF DROP BREAKUP IN THE SHEAR BREAKUP REGIME 6 9

1 . 0 0

0 . 1 0 m

0 . 0 1

0 . 0 0 0 1

|

W e 1 2 5 2 5 0 3 7 5

W TER

E T H Y L A L C O H O L O

G L Y C E R O L ( 4 2 % ) V [ ] .

G L Y C E R O L ( 6 3 % ) 4 I ~ , k

V

i

m

- - - - - Q U A S I - S T E A D Y

\ T H E O R Y -

- - ' - - T R A N S I E N T T H E O R Y

i i I I l i l l l i i i i l l l l l l i i i l l i l

0 . 0 0 1 0 0 . 0 1 0 0 0 . 1 0 0 0

V f f d

Figure 7. Temporal variation of the S M D of drops produced by shear breakup.

the drops currently being formed by shear breakup, similar to earlier considerations of jump

conditions to find drop sizes after shear breakup due to Hsiang and Faeth (1992). Then, the two

types of behavior noted earlier represent different states of the transient development of the vortical

region, which will be denoted the transient and quasi-steady shear breakup regimes in the following.

In order to fix ideas, the transition between the transient and quasi-steady shear breakup regimes

will be assumed to occur at a time tc to be quantified later.

T r a n s i e n t s h e a r b r e a k u p .

The transient breakup mechanism is illustrated in figure 5. This regime

is observed at short times after the start of breakup, particularly for liquids that have a small

viscosity so that the temporal rate of growth of the thickness of the boundary layer on the

windward side of the drop is relatively slow. Then the thickness of the boundary layer, normalized

by the init ial boundary layer thickness, can be expressed as follows (Schlichting 1975)

c ~( t ) / do = Ct ( VL t / ~ ) / 2 , t / t < 1, [5]

where VL is the kinematic viscosity of the drop liquid and C, is an empirical constant on the order

of unity. Assuming S M D ( t ) ,~ 6 ( 0 , an equation for the temporal variation of drop sizes in the

transient shear breakup regime can be obtained from [5], as follows

S M D ( t ) / d o = C s C t ( V L t / ~ ) /2 , t /t

< 1, [6]

where Cs is another empirical constant on the order of unity.

Present measurements of S M D ( t ) are correlated in terms of [6] for the transient shear breakup

regime in figure 7. At small values of V L t / ~ , the measurements exhibit an excellent correlation

IJMF 23/4 C


10/19

66 W. H. CHOU e t a l

a c c o r d i n g t o t h e t r a n s i e n t t h e o r y o f [6 ]; t h e c o r r e s p o n d i n g t h e o r e t ic a l c o r r e l a t io n , i n v o l v i n g a

l e a s t- s q u a r e s f it b a s e d o n [6 ] w h i l e m a i n t a i n i n g t h e s q u a r e r o o t d e p e n d e n c e o f

( V L t / ~ ) ,

i s a s f o l l ows

S M D ( t ) / d o = 2 . 0 ( v J / ~ ) '/2 , t / tc < 1. [7]

T h e b e s t f i t e x p r e s s i o n o f [7 ] is a l so p l o t t e d i n f ig u r e . 7 . L i m i t i n g t h e d a t a u s e d t o c o r r e l a t e [ 7] t o

v L t / ~

< 0 . 0 0 0 2 0 , y i e ld s a s t a n d a r d d e v i a t i o n o f t h e c o e f f i c ie n t o n t h e r i g h t h a n d s i d e o f [7 ] o f 1 3 ,

w i t h t h e c o r r e l a t i o n c o e f f i c ie n t o f t h e f it b e i n g 0 . 9 6 . I f t h e p o w e r o f

( v L t / ~ )

i n [ 7 ] i s f o u n d f r o m

a l e a s t - s q u a r e s f i t o f t h e s a m e d a t a s e t , a v a l u e o f 0 .5 7 w i t h a s t a n d a r d d e v i a t i o n o f 0 . 0 4 is o b t a i n e d ,

w h i c h i s n o t s t at i st i ca l ly d i f f er e n t f r o m t h e 1 /2 p o w e r b a s e d o n t h e p h e n o m e n o l o g i c a l t h e o r y u s e d

i n [ 7 ]. I n a d d i t i o n , t h e c o e f f ic i e n t o n t h e r i g h t h a n d s i d e o f [7 ] h a s a n o r d e r o f m a g n i t u d e o f u n i t y

a s a n t i c i p a t e d .

T h e e x p e r i m e n t a l r e s u l t s i l l u s t r a t e d i n f i g u r e 7 e x h i b i t a t r a n s i t i o n f r o m t h e t r a n s i e n t r e g i m e f o r

v a l u e s o f

v L t / ~

> 0 . 0 0 2 0 , w h i c h w i l l b e t a k e n t o r e p r e s e n t c o n d i t i o n s i n t h e q u a s i - s t e a d y s h e a r

b r e a k u p r e g i m e i n th e f o l l o w i n g , a s d e n o t e d o n t h e f ig u r e. T h i s t r a n s i ti o n c o m p l e t e s t h e d e f i n i t io n

o f t o, w h i c h c a n b e e x p r e s s e d a s f o l l o w s

te l l* = O.O02(pG/pL)12Uodo/VL.

[8]

B a s e d o n a p p r o x i m a t e c o n s e r v a t i o n o f m o m e n t u m s c al in g , a c h a r a c te r i s ti c in i ti a l l i q u id p h a s e

ve lo c i t y , UL o, c a n be de f ine d a s

pLU to

= p cu :,; th e r e f o r e , t h e f a c t o r o n t h e r i g h t - h a n d s i d e o f [8 ] c a n

b e r e c o g n i z e d a s a c h a r a c t e r i s t i c i n i t ia l l iq u i d p h a s e R e y n o l d s n u m b e r b a s e d o n t h i s v e l o c i t y , i .e ,

ReL =

ULodo/VL.

T h e n , n o t i n g t h a t t h e b r e a k u p p e r i o d e n d s w h e n

t / t *

= 5 .5 f r o m L i a n g

e t a l .

(1988) ,

[8 ] i m p l i e s t h a t t r a n s i e n t b e h a v i o r w il l b e o b s e r v e d f o r t h e e n t i r e b r e a k u p p r o c e s s w h e n R e L > 2 7 5 0 ,

F o r p r e s e n t t e st s, s u c h c o n d i t i o n s w e r e e n c o u n t e r e d f o r w a t e r d r o p s h a v i n g W e = 2 5 0 a n d 3 7 5.

A t t h e o t h e r e x t r e m e , p r e s e n t m e a s u r e m e n t s f o r e th y l a l c o h o l a n d g l y c e ro l d r o p s w e r e m a i n l y in

t h e q u a s i - s t e a d y s h e a r b r e a k u p r e g i m e .

T h e c o n s i s t e n c y o f [7 ] w i t h t h e e a r li e r m e a s u r e m e n t s o f H s i a n g a n d F a e t h ( 1 99 2 ) o f t h e S M D

a t t h e e n d o f s h e a r b r e a k u p ( o r th e j u m p c o n d i t i o n s ) f o r e x p e r im e n t s d o m i n a t e d b y l ar g e W e a n d

l o w v i s c o s i t y l i q u i d s ( w h i c h i m p l i e s b e h a v i o r m a i n l y i n t h e t r a n s i e n t s h e a r b r e a k u p r e g i m e ) is a l s o

o f in t e re s t . I n p a r t i c u l a r , t h is r e l a t i o n s h i p c a n b e e x a m i n e d b y a s s u m i n g t h a t d r o p s iz es a t t h e e n d

o f b r e a k u p a r e d o m i n a t e d b y t h e la r g e s t d r o p s i ze s p r o d u c e d b y b r e a k u p w h i c h a l so a r e g e n e r a t e d

a t t h e e n d o f b r e a k u p w h e r e t = tb = 5 . 5 t * . T h e n , i n t r o d u c i n g [1 ] f o r t * i n t o [ 7] y i e l d s t h e f o l l o w i n g

e x p r e s s i o n f o r

S M D e ,

t h e

S M D

f o r t h e e n t i r e s e c o n d a r y b r e a k u p p r o c e s s

SM D e/ do = 2Ce(tb /t*)L 2(pL/pG)~/ [VL/(doUo)] 1/2

[9]

w h e r e C e i s a n e m p i r i c al f a c t o r t o c o r r e c t f o r t h e f a c t t h a t j u m p c o n d i t i o n s f o r d r o p s iz es i n v o lv e

t h e e n t ir e b r e a k u p p r o c e s s , a n d t h e c o n t r i b u t i o n o f t h e r e m a i n i n g p a r e n t d r o p , r a t h e r t h a n j u s t

t h e s iz e o f d r o p s p r o d u c e d a t t h e e n d o f s h e a r b r e a k u p . N e v e r t h e l e s s, [9 ] b e c o m e s i d e n ti c a l to t h e

j u m p c o n d i t i o n s o f H s i a n g a n d F a e t h ( 1 9 9 2) i f

2Ce(tb / t*) I :2

= 6 . 2. R e c a l l i n g t h a t

t b / t *

= 5 .5 , im pl i e s

t h a t C e = 1 .3 , w h i c h i s a v a l u e o n t h e o r d e r o f u n i t y a s e x p e c t e d . T h u s , p r e s e n t f i n d i n g s f o r t h e

e v o l u t i o n o f S M D a s a f u n c t i o n o f t im e d u r i n g s h e a r b r e a k u p a r e c o n s i s t e n t w i t h t h e j u m p

c o n d i t i o n s f o u n d b y H s i a n g a n d F a e t h ( 1 99 2 ) f o r s e c o n d a r y b r e a k u p i n t h e s h e a r b r e a k u p r e g im e .

A n i n t e re s t in g f e a t u r e o f b o t h [7 ] f o r

S M D ( t )

a nd [ 9 ] f o r

S M D o

i s t h a t n e i t h e r r e s u l t d e p e n d s

o n t h e s u r f a c e te n s i o n , a n d t h u s W e , e v e n t h o u g h c o n d i t i o n s r e q u i r e d f o r t h e a p p e a r a n c e o f t h e

s h e a r b r e a k u p r e g i m e a t l o w O h a n d l a r g e

pL /pG

a r e c o n t r o l l e d b y W e , a n d t h u s a . I n a s e n s e , t h i s

b e h a v i o r i s a n a l o g o u s t o t h e r o l e o f R e y n o l d s n u m b e r s f o r t u r b u l e n t m i x i n g , w h e r e t h e p r e s e n c e

o f t u r b u l e n t m i x i n g f o r j e ts , w a k e s , e tc . d e p e n d s o n t h e R e y n o l d s n u m b e r o f t h e f lo w , ev e n t h o u g h

t h e r a t e o f m i x i n g i ts e l f is e s se n t ia l ly i n d e p e n d e n t o f th e R e y n o l d s n u m b e r o n c e t h e f l o w is

t u r b u l e n t .

S i m i l a r t o t h e c o r r e l a t i o n o f S M D e o f H s i a n g a n d F a e t h ( 19 9 2 ), [7 ] c a n b e p u t i n t o a f o r m

e m p h a s i z in g t h e W e b e r n u m b e r o f d r o p s p r o d u c e d b y s e c o n d a r y b r e a k u p , a s f o ll o w s

p c S M D ( t ) u Z o / a = 2 ( t / t * ) l / 2 W e / R e ~ '2, t /t c

< 1, [10]

w h e r e t h e l e f t - h a n d s i de o f [1 0] c a n b e r e c o g n i z e d a s t h e W e b e r n u m b e r o f d r o p s f o r m e d b y

s e c o n d a r y b r e a k u p b a s e d o n t h e

S M D

a n d t h e i n i t ia l r e l a t i v e v e l o c i t y . T h e n , s i m i l a r to p r e v i o u s

c o n s i d e r a t i o n s o f th e j u m p c o n d i t i o n s t o y i e l d S M D o ( H s i a n g a n d F a e t h 1 9 9 2 , 1 9 9 3 ) , [ 1 0 ] s h o w s


11/19

TEMPOR L PROPERTIES OF DROP BRE KUP IN THE SHE R BRE KUP REGIME 661

that the Weber number based on S M D ( t ) and Uo can exceed values of We needed to initiate

secondary breakup by shock-wave disturbances. As discussed by Hsiang and Faeth (1993, 1995),

however, subsequent tertiary breakup does not occur because these drops have had time to adjust

to the disturbance and are subject to different criteria for breakup in addition to effects of reduced

relative velocities compared to uo. Finally, even though [10] involves surface tension, it should be

recalled that the surface tension only affects requirements for the onset of secondary breakup

regimes for present conditions, while drop sizes produced by secondary shear breakup are

independent of surface tension, see [7]-[9].

Qu asi-s teady shear breakup.

The next issue tha t must be addressed with respect to the temporal

evolution of

S M D

during shear breakup involves behavior in the quasi-steady shear breakup

regime. There are two main possibilities for defining behavior in the quasi-steady shear breakup

regime, as follows: (1) stabilization of the flow within the drop at the end of the transient period

implies 6 ,-~ d, relat ively independent of properties like ReL, as illustrated in figure 6; and (2)

complete development of the boundary layer near the surface of the liquid on the windward side

of the drop yields fi proportional to the thickness of this boundary layer near the drop periphery,

along the lines of the analysis of Hsiang and Faeth (1992). The somewhat increased scatter of the

data for the quasi-steady shear breakup regime illustrated in figure 7 suggests the potential for

complications due to contributions from both these limits; nevertheless, based on this informat ion,

it is reasonable to accept the approach illustrated in figure 6 for the quasi-steady shear breakup

regime and adopt the approximation S M D ( t ) ~ d o to yield

S M O ( t ) / d o

= 0.09,

t/tc

> 1, [11]

which is illustrated on the plot. Limiting the data used to correlate [11] to

t / t c >

1, or

v L t / ~ > 0.0020, yields a standard deviat ion of the coefficient on the right hand side of [11] of 22 ,

with the correlation coefficient of the fit being 0.91. If the power of

v L t / ~

in [11] is found from

a least-squares fit using the same data set, a value of -0.06 with a standard deviation of 0.10 is

obtained, which is not statistically different from the power of zero based on the phenomenological

description of [11].

It is also of interest to compare the approximation 6 ,-~ d used to find [11] with estimates of the

thickness of the boundary layer formed near the surface of the liquid on the windward side of the

drop. For flows typical of the interior of drops this boundary layer is laminar and its thickness

was estimated as the characteristic thickness of a laminar boundary layer on a flat plate having

an ambient velocity of ULo and a length do, which implies (Schlichting 1975)

6(do)/do = 4.0/[(pc/pr) /2uodo/vr] /2. [12]

The values of

6(do)/do

computed from [12] for present test conditions are summarized in table 2.

The tabulation indicates that for measurements involving the quasi-steady shear breakup regime

(e.g. alcohol and glycerol drops),

6(do)/do

from [12] was generally larger than

S M D / d o

from [I 1]

and much more variable than the range of S M D / d o seen in figure 7 for these liquids. In contrast,

only results for water drops, which generally did not reach quasi-steady conditions, yield boundary

layer thicknesses less than the estimate of [10], which is also consistent with the behavior seen in

figure 7. Taken together, these results support the present phenomenological approach where the

transient regime ends when the thickness of the vortical region reaches a fraction of the parent

drop diameter, as a result of the confined internal flow configuration of the deformed parent drop.

Naturally, this limit does not yield formulas for the jump conditions for S M D that are consistent

with the earlier results of Hsiang and Faeth (1992), similar to the transient shear breakup regime.

T a b l e 2 . S u m m a r y o f q u a s i -s t e a d y l i qu i d b o u n d a r y l a y e r th i c k n e ss e s 6 do ) /do) t

D r o p l i q u i d

W e W a t e r E t h y l a l c o h o l G l y c e r o l ( 4 2 ) G l y c e r o l ( 6 8 )

125 0 .079 0 .137 0 .154 0 .237

250 0 .067 0 .115 0 .115 0 .200

375 0 .060 0 .104 0 .110 0 .180

t E s t i m a t e d f r o m [ 1 1 ] .


12/19

66 w m C H O U et l

A s m e n t i o n e d e a r li e r , h o w e v e r , th i s b e h a v i o r i s n o t s u r p r i s in g b e c a u s e t h e m e a s u r e m e n t s o f H s i a n g

a n d F a e t h ( 1 9 9 2 ) w e r e d o m i n a t e d b y r e s u l t s f r o m t h e t r a n s i e n t s h e a r b r e a k u p r e g i m e . A n o t h e r

f a c t o r i s t h a t d r o p s iz es t o w a r d t h e e n d o f th e t r a n s i e n t s h e a r b r e a k u p r e g i m e a re c o m p a r a b l e t o

t h o s e i n t h e q u a s i - s t e a d y b r e a k u p r e g i m e , s e e f i g u r e 7 ; t h e r e f o r e , b o t h s e t s o f r e s u l t s t e n d t o

c o r r e l a t e i n a s i m i l a r m a n n e r .

3 . 3 . P a r e n t d r o p v e l o c i t ie s

T h e t e m p o r a l b e h a v i o r o f th e v e l o c i ty d i s t r i b u t i o n s o f d r o p s p r o d u c e d b y s h e a r b r e a k u p i s c l o se l y

a s s o c i a t e d w i t h t h e t e m p o r a l b e h a v i o r o f th e v e l o c i t y o f t h e p a r e n t d r o p ; t h e r e f o r e , t h i s is s u e w i ll

b e c o n s i d e r e d f ir s t. A c o r r e l a t i n g e x p r e s s i o n f o r t h e v e l o c i t y o f t h e p a r e n t d r o p w i t h t i m e w a s b a s e d

o n t h e p h e n o m e n o l o g i c a l a n a ly s i s o f H s i a n g a n d F a e t h ( 19 93 ). T h e m a j o r a s s u m p t i o n s o f t h is

a n a l y s i s a r e a s f o l l o w s : v i r t u a l m a s s , B a s s e t t h i s t o r y a n d g r a v i t a t i o n a l f o r c es i g n o r e d ; g a s v e l o ci ti e s

a s s u m e d t o b e c o n s t a n t ; m a s s r e m o v a l f r o m t h e p a r e n t d r o p i g n o r e d ; a n d c o n s t a n t a v e r a g e d r a g

c o e f f ic i e n t a s s u m e d o v e r t h e p e r i o d o f b r e a k u p . F o r p r e s e n t c o n d i t i o n s , v i rt u a l m a s s a n d B a s s e t t

h i s t o r y f o r c e s a r e s m a l l d u e t o t h e l a r g e v a l u e s o f

p L / p ~

of t he f l ow (Fa e t h 1990) . S i mi l a rl y ,

g r a v i t a t i o n a l f o r c e s w e r e n o t a f a c t o r b e c a u s e d r o p m o t i o n w a s n e a r l y h o r i z o n t a l a n d d r a g f o r c e s

w e r e m u c h l a r g e r t h a n g r a v i t a t i o n a l f o r c e s . I n a d d i t i o n , u n i f o r m g a s p r o p e r t i e s w e r e a c o n d i t i o n

o f th e p r e s e n t e x p e r im e n t s . I n c o n t r a s t , t h e u n i f o r m p a r e n t d r o p s iz e a p p r o x i m a t i o n w a s n o t r e a l ly

j u s t if i e d f o r p r e s e n t c o n d i t i o n s b e c a u s e p a r e n t d r o p d i a m e t e r s a t t h e e n d o f b r e a k u p w e r e o n l y

1 2 - 3 0 o f t h e o r ig i n a l d r o p d i a m e t e r s a n d v a r y c o n s id e r a b l y o v e r t h e p e r io d o f b r e a k u p ( H s i a n g

a n d F a e t h 1 9 93 ). N e v e r t h e le s s , a c c o u n t i n g f o r t h e se c h a n g e s b y a d o p t i n g t h e o r i g i n a l d r o p d i a m e t e r

a n d s e l e c t i n g a m e a n d r a g c o e f f i c i e n t , C D , t o b e s t f i t t h e m e a s u r e m e n t s y i e l d e d r e a s o n a b l y g o o d

r e s u l t s i n t h e p a s t ( H s i a n g a n d F a e t h 1 9 9 3 ) , a n d w a s c o n t i n u e d d u r i n g t h e p r e s e n t s t u d y .

T h e a n a l y s i s t o f i n d p a r e n t d r o p v e l o c i t i e s i n a l a b o r a t o r y r e f e r e n c e f r a m e , u p , a s a f u n c t i o n o f

t i m e u n d e r t h e p r e c e d i n g a p p r o x i m a t i o n s i s p r e s e n t e d b y H s i a n g a n d F a e t h ( 1 9 9 3 ) . T h e s e r e s u l t s

c a n b e p l a c e d i n t h e f o l l o w i n g f o r m

( u p - U p o )/ (u G - U p o ) - -- 3 C D ( t / ( 4 t * ) ) / ( p ~ / p L ) 2 , [13]

wh e re fo r p re se n t c on di t i on s Upo = 0 a n d u~ is t he ga s ve l oc i t y in a l a b ora t o ry re fe re nc e f r a me .

Earli____er ev a lu a t io n o f par en t d ro p ve loc i t i es a t the end of s eco nd ary br eak up yie lde d a bes t f i t va lu e

of C D = 5 i n [13] (H s i a ng a n d Fa e t h 1993).

M e a s u r e m e n t s o f p a r e n t d r o p v e lo c it ie s fo r v a r io u s s e c o n d a r y b r e a k u p c o n d i t io n s , a n d t im e s

d u r i n g s e c o n d a r y b r e a k u p , w e r e o b t a i n e d f r o m b o t h t h e p r e s en t i n v e s t ig a t io n a n d f r o m t h e e ar li er

w ork of H s i a ng a nd Fa e t h (1993). T he se re su l t s a re p l o t t e d a c c o rd i n g t o [13] i n fi gure 8. A b e s t -f i t

c o r r e l a t i o n a c c o r d i n g t o [ 1 3 ] a l s o is s h o w n o n t h e fi g u re . T h e c o m p a r i s o n b e t w e e n t h e

m e a s u r e m e n t s a n d t h e c o r r e l a t io n i s s e e n t o b e q u i t e g o o d i n sp i te o f th e a p p r o x i m a t i o n s o f t h e

s i mpl i f i ed a na l ys i s . Th i s y i e l ds t he s a me be s t - fi t va l ue C D = 5 .0 , a s t he r e su l t s fou nd e a r l i e r by

H s i a n g a n d F a e t h ( 19 93 ), w i t h a n e x p e r i m e n t a l u n c e r t a i n t y ( 9 5 c o n f i d e n c e ) o f t h e f it o f 1 5 .

3 . 4 . D r o p v e l o c i t y d i s t r i b u t i o n s

M e a n v e l o c i t i e s . T h e

v e l o ci ty d i st r ib u t i o n s o f d r o p s p r o d u c e d b y s h e a r b r e a k u p w e r e m e a s u r e d

a s a f u n c t i o n o f t im e f o r a ll te s t c o n d i t i o n s . I t w a s f o u n d t h a t t h e v e l o c it ie s o f d r o p s p r o d u c e d

b y s e c o n d a r y b r e a k u p , r e l at i v e t o t h e v e l o c i ty o f t h e p a r e n t d r o p , w e r e r e l a t e d t o t h e c h a r a c t e r is t i c

ve l oc i t y o f t he l i qu i d a t e a c h i ns t a n t o f t i me , i .e . 1~

p ~ / p e ) - ( u G - u p ) . I n a d d i t i o n , i t w a s f o u n d t h a t

d r op ve l oc i ti e s we re r e l a t i ve l y i nd e pe nd e nt o f d ro p s i ze , i .e . t he d rop ve l oc i t y d i s t r i bu t i ons w e re

n e a r l y u n i f o r m . T__hus, v o l u m e - a v e r a g e d m e a n s t r ea m w i s e a n d c r o s s - s t r e a m v e l o ci ti e s f o r s h e a r

bre a ku p , UL a n d VL, nor ma l i z e d b y t he c ha ra c t e r i s t i c l iqu i d ve l oc i t y , a re p l o t t e d a s a fun c t i on o f

t i t * i n f i gure 9 . S i mi l a r t o t he d rop d i a me t e r s i l l us t r a t e d i n f i gure 7 , t he re i s a ppre c i a b l e s c a t t e r

o f t h e d r o p v e l o c i t i e s i n f i g u r e 9 . T h i s b e h a v i o r c o m e s a b o u t b e c a u s e r e l a t i v e l y f e w d r o p s a r e

a v a i l a b l e t o f i n d a p p r o p r i a t e a v e r a g e d r o p v e l o c i t i e s f o r a g i v e n b r e a k u p c o n d i t i o n a n d t i m e . I n

a d d i t i o n , r a n d o m m o t i o n s o f t h e l ig a m e n t s a n d t h e p a r e n t d r o p , s ee f i g u re s 2 a n d 3 , y i e ld

t u r b u l e n t - l i k e v e l o c i t y v a r i a t i o n s t h a t c a u s e c o r r e s p o n d i n g v a r i a t i o n s o f m e a n v e l o c i t i e s .

N e v e r t h e l e s s , i t i s e v i d e n t t h a t t h e m e a n v o l u m e - a v e r a g e d s t r e a m w i s e a n d c r o s s - s t r e a m v e l o c i t i e s


13/19

1 . 6 0

T E M P O R L P R O P E R T I E S O F D R O P B R E K U P I N T H E S H E R B R E K U P R E G I M E

i

= I = I = I = I

663

i

- W e 1 2 5 2 5 0 3 7 5

W A T E R I I

1 . 2 0 - E T H Y L A L C O H O L 0

G L Y C E R O L ( 4 2% )

[ ] ZX

_

0 8 0

=

' e t

= < > V

~ - T H E O R Y ~ 1 ~ I - l y t

o o L

0 . 0 0

0 0 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5

1 2

P G I P L ) t / t

Fi gu r e 8 . S t r e a mwi s e ve l oc i t i e s o f t he pa r e n t d r op a s a f unc t i on o f t i me du r i ng s he a r b r e a kup .

a r e r e l a t i v e ly i n d e p e n d e n t o f t im e a s w e l l a s d r o p s iz e a n d c a n b e c o r r e l a t e d r e a s o n a b l y w e ll , a s

f o l l o w s

P L / P G ) I / E u L - - U p ) / U 6 - -

Up) =

9.5

[ 1 4 ]

a n d

I

( p L / p . ) J Y L / ( U o - - U p) ~ 0

[ 5 ]

w i t h a s t a n d a r d d e v i a t io n o f t h e c o n s t a n t o n t h e r ig h t - h a n d s i d e o f [1 4] o f 2 8 . T h e c o r r e s p o n d i n g

stan_._dard de v i a t i on o f t he c on s t a n t o n t he r i g h t h a n d s i de o f [15] i s 4 . 7 ; t he re fore , t he m e a n v a l ue

of VL i s no t s t a t i s t i c a l l y d i f fe re n t f rom z e ro . A d i f f i c u l t y wi t h t he c o r re l a t i on o f s t r e a m wi se v e l oc i t y

i n [ 1 4 ] i s t h a t t h e a c t u a l v a l u e o f t h e r e l a t iv e v e l o c i ty in c r e a s e o f t h e d r o p s p r o d u c e d b y s h e a r

b r e a k u p i s n o t e a s i ly c o m p a r e d w i t h t h e r e l a ti v e v e lo c i t y o f t h e p a r e n t d r o p d u e t o t h e e f fe c t o f

t h e d e n s i t y r a t io . T h u s , c o r r e l a t i n g t h e s t r e a m w i s e v e l o c i t y d a t a d i r e c t ly i n t e r m s o f v e lo c i ti e s

r e l a ti v e t o t h e v e l o c i t y o f th e p a r e n t d r o p y i e ld s

(UL -- Up)/ UG - Up) = 0.3 7 [16]

w i t h t h e s t a n d a r d d e v i a t i o n o f th e c o n s t a n t o n t h e r i g h t - h a n d s i d e o f [1 6 ] o f 0 .0 8. T h i s r e s u l t

s u g g e s t s t h a t t h e r e is a p p r e c i a b l e a c c e l e r a t i o n o f t h e d r o p l i q u id d u r i n g b r e a k u p , m a i n l y a s a r e s u lt

o f a c c e l e r a t i o n o f t h e l i q u i d i n th e v o r t i c a l l a y e r n e a r t h e s u r f a c e o f t h e p a r e n t d r o p a s w e ll a s

a c c e l e r a t i o n o f l i q u id i n t h e l i g a m e n t s p r i o r t o f i n a l b r e a k u p i n t o d r o p s . T h e r e l a t iv e l y l a r g e

va r i a t i ons o f VL se e n i n f i gure 9 c e r t a i n l y t e n d t o sup po r t s i gn i f i c a n t e f fe c ts o f l iqu i d a c c e l e ra t i o n

i n t h e l i g a m e n t s . T h e c o r r e s p o n d i n g v a l u e s f o r VL y i e ld VL/ UG- U p ) = - - 0 . 0 1 w i t h a s t a n d a r d

d e v i a t i o n o f 0 . 15 , w h i c h i m p l i e s t h a t m e a n r a d i a l v e l o c it ie s a r e n o t s t a t i s ti c a ll y d i ff e r e n t f r o m z e r o ,

a s be fore .


14/19

664 W. H. CHOU t

a l

n

3 0

e~

=,

A

a.

, = l 2 0 - -

- I

~

ol

~ -

Q.

1

Q.

1 0

0

2 0 - -

R

1 0 - -

D.

i > 0 - -

a .

1 0

2 0

0 . 0

I I I I I

We 125 250 375

W A TE R i - ~ ~ -

ETHYL ALCO HOL

GLYCEROL (42%) V [ ] A

G L Y C E R O L , (6 3 )

I

CORRELATION

I I v

i - - ~ r I

m

D ~

l : _ ~

~ o

% I L e e ,

A I

I

I

/ S T A R T B RE AK U P I -

I ~ E N D B R E A K U P ~ I. _

CORRELATION

v \

[ ]

Z

I . e

m

/x

u

m

A ~ n

V O

I

5 . 0 6 . 0

I I I I

1 . 0 2 . 0 3 . 0 4 . 0

t / t *

Figure 9. Streamwise and cross s tream mean velocities of drops produ ced by shear bre akup as a function

of time during breakup.

T h e b e h a v i o r o f d r o p v e l o ci t ie s as t h e d r o p s a r e f o r m e d a s a f u n c t i o n o f ti m e , g i v e n b y [1 4] a n d

[1 5], is in m a r k e d c o n t r a s t t o t h e d r o p v e l o c i t y d i s t r i b u t i o n a s a f u n c t i o n o f d r o p s iz e a t t h e e n d

o f b r e a k u p t h e j u m p c o n d i t i o n s ) d i s cu s s e d b y H s i a n g a n d F a e t h 1 9 9 3 , 19 95 ). F o r t h e j u m p

c o n d i t i o n s , d r o p v e l o ci ti e s r e l at i v e to t h e g a s b e c a m e p r o g r e s s i v e ly s m a l l e r as t h e d r o p s iz es b e c o m e

s m a l le r , r a t h e r t h a n r e m a i n i n g c o n s t a n t c o m p a r e d t o t h e r e l a ti v e v e l o c i t y o f t h e g as w i t h r e s p e c t

t o t h e p a r e n t d r o p , s i m i l a r t o t h e r e s u l t s i l l u s t r a t e d i n f i g u r e 9 . T h i s b e h a v i o r c o m e s a b o u t b e c a u s e

t h e c h a r a c te r i s ti c r e l a x a t io n t i m e s o f sm a l l d r o p s a r e s m a l l e r t h a n t h o s e o f la r g e d r o p s H s i a n g a n d


15/19


~ . . L mu m W , m m x . ( w )

1 0 0 . 0 0 0

I

I : : : W A ~ I W ~ g - ~ t ( U ) m ( 4 ~ ( U )

I - m - 9 .( 'z ~ 9 ( u ) o . ( , u ) ~ k ( s . ~

/ ~ A ( u ) E l M ) e (4 .s )

. . 1 0 . 0 0 0 L - -- m . l u ) E ( M ) ~ ( ,,u p

GLYCI~ROL(42 ) I I X (~ * (4~ t~ (E.G)

I ~ z m O ( : L ~ A a J ) I ( M ) .

I ; m o ( :t .~ o l u ) T ( 4 . o ) ~ l l t m -

I - ~ . ~ n n ~ o ~ . ( m ) u s 9 ~ 9 - I M I ~ t dL O ) .O ( S .O ) -

L I m ~ & I M I 12 (4.11113(11.01

L . [ ]

~ [ ] C O R R E L A T I O N

1 0 0 0 ~ - * * m ~ ~ O * v /

o. oo r o -

o . o m . r Q j ~

]

e

0 . 0 0 1 I I I I I I t I I

0 . 0 0 .2 0 . 4 0 . 6 0 .8 1 . 0 1 . 2 1 .4 1 . 6 1 . 8 2 . 0

c U M M D

Figure 10. Streamwise and cross stream rms velocity fluctuations of drops produced by shear breakup

as a function of drop size.

F a e t h 1 99 2 ; t h e r e f o r e , s m a ll d r o p s u n d e r g o a g r e a t e r a c c e l e ra t i o n a f te r t h e y a r e f o r m e d t h a n l a rg e

d r o p s , a n d m o r e c l o s e ly a p p r o a c h t h e g a s v e lo c i t y a s a r es u lt .

Veloci tyf luctuat ions .Volume.averaged

r m s s t r e a m w i s e a n d c r o s s - s t r e a m v e l o c i t y f l u c t u a t i o n s , uL

a n d vL a r e p l o t t e d a s a f u n c t i o n o f

d / MMD ,

w i t h W e a n d

t / t

a s p a r a m e t e r s , i n f ig u r e 1 0 . I n d i v i d u a l

d a t a p o i n t s o n t h i s f ig u r e e x h i b i t s i g n i f i c a n t s c a t t e r , m a i n l y b e c a u s e e a c h t e s t c o n d i t i o n i n v o l v e s

a l i m i te d n u m b e r o f t e s t d r o p s . N e v e r t h e l e s s , e ff e ct s o f d r o p s iz e , W e a n d

t / t

a p p e a r t o b e s m a l l

o v e r t h e e n t ir e d a t a s e t , w h e n v o l u m e - a v e r a g e d f l u c t u a t i o n s a r e n o r m a l i z e d b y t h e m e a n s t r e a m w i s e


16/19

W -H CHOU

et al

v e l o c i t y o f t h e d r o p s r e l a t iv e t o t h e g a s . T h e r e s u lt i n g v o l u m e - a v e r a g e d r m s s t r ea m w i s e a n d

c r o s s - s t re a m d r o p v e l o c i ty f l u c t u a t io n s c a n b e s u m m a r i z e d , a s f o l lo w s

u~/ uc

- uL) = 0.31 [17]

a n d

v~/ u~

- uL) = 0.3 7, [18]

w h e r e t h e s t a n d a r d d e v i a t i o n s o f t h e n u m b e r s o n t h e r ig h t-_ _ ha nd s i d es o f t h e s e e q u a t i o n s a r e 2 2 .

I n v i e w of t he se u nc e r t a in t i e s , t h e m a g ni tu de s o f u~ a nd v~ a r e n o t s t a t i s t i c a l l y d if fe re n t__ .T a k i n g

t h e d a t a s e t s as a w h o l e , h o w e v e r , t h e l a r g e n u m b e r o f t o t a l s a m p l e s a v a i l a b l e t o f i n d u~ a n d v~

r e d u c e t h e e x p e r i m e n t a l u n c e r t a i n t i e s o f t h e s e e s t i m a t e s ( 9 5 c o n f i d e n c e ) t o l es s t h a n 1 0 .

3 . 5 . D r o p f o r m a t i o n r a t es

D r o p f o r m a t i o n r a t e s w e r e e s t i m a t e d u s i n g a s i m p l i f i e d a n a l y s i s . T h i s i n v o l v e d t h e f o l l o w i n g

m a j o r a s s u m p t i o n s : l iq u i d r e m o v a l r a t e s w e r e p r o p o r t i o n a l t o t h e t h i ck n e s s o f t h e b o u n d a r y l a y e r

i n t h e li q u i d o n t h e u p s t r e a m s u r f ac e o f th e d r o p ; l i q u i d r e m o v a l r a t e s w e r e p r o p o r t i o n a l t o t h e

v e l o c i t y o f t h e d r o p s f o r m e d r e l a t iv e t o t h e v e l o c i t y o f th e p a r e n t d r o p , e s t i m a t e d f r o m [1 4]; li q u id

r e m o v a l r a te s w e r e p r o p o r t i o n a l t o t h e p e r i m e t e r o f t h e d r o p a t i ts p e r ip h e r y ; a n d b r e a k u p b e g i n s

a n d e n d s a t t / t * = 1 .5 a n d 5 .5 , r e s p e c t i v e l y , a s d e t e r m i n e d b y L i a n g e t a l . ( 1 9 8 8 ) . T h e r e s u l t i n g

f o r m u l a t i o n f o r th e r a t e o f p r o d u c t i o n o f d is p e r s ed l i q u id d r o p s b y s e c o n d a r y b r e a k u p , f o r t h e

t r a n s i e n t a n d q u a s i - s t e a d y d r o p b r e a k u p r e g im e s , is re l a ti v e ly c o m p l e x . I t w a s n o t e d , h o w e v e r , t h a t

t h e a m o u n t o f li q ui d r e m o v e d f r o m t h e d r o p c o u l d b e a p p r o x i m a t e d b y a c l ip p e d G a u s s i a n f u n c ti o n

w h i c h s i m p l if ie s th e t r e a t m e n t o f t h e o n s e t a n d e n d o f s e c o n d a r y b r e a k u p . T h u s , o n l y t h e s i m p li fi ed

a p p r o a c h w i ll b e p r e s e n t e d h e r e b e c a u s e it s h o u l d b e u s e f u l f o r d e ta i l ed a n a l y s is o f d r o p b r e a k u p

p r o c e s s e s .

P r e s e n t m e a s u r e m e n t s o f t h e c u m u l a t i v e v o l u m e o f l iq u i d r e m o v e d f r o m t h e p a r e n t d r o p a s a

f u n c t i o n o f t i m e a r e p l o t t e d i n f i g u r e 1 1. T h e s e r e s u l t s i n c l u d e a ll te s t c o n d i t i o n s c o n s i d e r e d d u r i n g

t h e p r e s e n t i n v e s t i g a t i o n . T h e b e s t - fi t c o r r e l a t i o n o f t h e s e r es u l t s , a c c o r d i n g t o a c l i p p e d G a u s s i a n

f u n c t i o n , a l s o is s h o w n o n t h e p l o t. I t is e v id e n t t h a t t h e c l i p p e d G a u s s i a n f u n c t i o n p r o v i d e s a g o o d

f it o f t h e c u m u l a t i v e l o s s o f v o l u m e o f th e p a r e n t d r o p a s a f u n c t i o n o f t / t * . T h i s f o r m u l a a l s o

p r o v i d e s a r e a s o n a b l y g o o d f i t o f th e r a t e o f re m o v a l o f d r o p l iq u i d f r o m t h e p a r e n t d r o p , e x c e p t

f o r t h e s in g u l a r p o i n t s a t t h e b e g i n n i n g a n d e n d o f th e p e r i o d w h e r e d r o p m a s s i s b e i n g r e m o v e d .

T h e r e s u lt s i l lu s t r a te d i n f i g u re 11 c a n b e c o r r e l a t e d t o p r o v i d e t h e m a s s r a t e o f f o r m a t i o n o f

d i s p e r se d d r o p s d u e t o s h e a r b r e a k u p , m p , n o r m a l i z e d b y t h e i n i ti a l d r o p m a s s a n d t * , as f o l lo w s

6 m p t * / ~ p L ~ ) = 0 . 4 2 e x p { O . 8 t / t* - 3.5)2}, 1.5 _< t / t*


17/19

TEMPOR L PROPERTIES OF DROP BRE KUP IN THE SHE R BRE KUP REGIME

6 6 7

4

2

| I I | I I I

E

L I Q U I D 1 2 5 2 5 3 7 E

W TER

GLYCEROL (42%) r l 0 ~,

GLYO~OL

ca )

@ ) 0

1

ETHYL ALCOHOL

@ V

@ ~ A O @

A

I I

@

~

OORRELATION

I ' ' I I I '

~ I I I t ) I I O I O 1 10 I N ) I I I

CUMUL TIVE R E M O V E D V O L U M E PERCENT GE

F i g u r e l I D e g r e e o f m a s s r e m o v a l f r o m t h e p a r e n t d r o p a s a fu n c t i o n o f ti m e d u r i n g s h e a r b r e a k u p

pL/pc = 500. Similarly, the mean drop-containing region increases from zero at t / t = 1.5 to

roughly

x / d

= 40-100 at

t/t

= 5.5. As noted earlier, the span of the secondary breakup times, the

distance traveled by the parent drop, and the span of streamwise distances where drops are present

at the end of breakup can be significant in some instances. In such cases, the information found

during the present investigation about the temporal evolution of the sizes and velocities of drops

produced by secondary breakup, as well as the rate of liquid removal from the parent drop during

secondary breakup, should be helpful.

4. CONCLUSIONS

The properties of drop breakup in the shear breakup regime were studied as a function of time

for shock-wave disturbances in air at NTP, for the test conditions summarized in table 1. The major

conclusions of the study are as follows:

1) The maximum lengths of ligaments protruding from the periphery of the drops progressively

increase with increasing Ohnesorge number causing transition to a long-ligament shear breakup

regime at Oh ,~ 0.1; present results are limited to the conventional shear breakup regime at small

Ohnesorge numbers Oh < 0.1).

2) Drops produced by shear breakup at small Ohnesorge numbers satisfy the universal root

normal drop size distribution funct ion with

M M D / S M D

= 1.2, of Simmons 1977), at each instant

of time.

3) The S M D of drops produced by shear breakup at small Ohnesorge numbers exhibit transient

and quasi-steady regimes as a funct ion of time, based on the development of the liquid boundary

layer within the parent drop; this behavior was correlated based on a phenomenological analysis


18/19

668 W . H . C H O U et al

lO

1 2 0

8 0

4 0

I O N S E T O F

/ B R E A K U P

I B R E A K U P PERIO

: P L / P G = I O O O ( T Y P ~

', p , - 5 o o P )

/ J

/

I # o

I

/

I

I

,I

I

I O

:

I

I

I

I

R M O T O R O P

D R O P S P R A Y

R E G I O N

I

I

P A T H O F

P A R E N T D R O P

I

I

I

I

, ,, -E N D O F B R E A K U P

0 2 4 6 8 1 0

t / t

Figure 12. Growth of the spray conta ining region dur ing shear breakup.

which implied that drops produced by breakup had diameters comparable to the thickness of this

liquid viscous region.

(4) The parent drop accelerates rapidly due to the large drag coefficient caused by its

deformation; a phenomenological analysis provided an effective correlation o f the resulting parent

drop velocities.

(5) The mean velocities of drops produced by shear breakup at small Ohnesorge numbers were

relatively independent of drop size, and were somewhat larger than the velocities o f the parent drop,

at each instant of time.

(6) The rms velocity fluctuations of drops produced by shear breakup at small Ohnesorge

numbers were relatively independent of drop size, and were on the order of 30-40 of the mean

streamwise velocity of the gas relative to the parent drop, at each instant of time.

(7) The rate of liquid removal from the paren t drop could be interpreted reasonably well based

on the variations of parent drop diameter and the size and velocity of drops leaving the periphery

of the parent drop; these results were correlated concisely in terms of an empirical clipped Gaussian

function.

(8) Shear breakup at small Ohnesorge numbers extends over streamwise distances of 0-100

initial drop diameters, and 0-5.5 characteristic drop breakup times; this behavior suggests that

shear breakup should be treated as a rate process, rather than by jump conditions, in some

instances.

Acknowledgements This

research was sponsored by the Air Force Office of Scientific Research

Grant Nos. F49620-92-J-0399 and F49620-95-1-0364, under the technical management of J. M.

Tishkoff. The authors would like to thank C. W. Kauffman for the loan of the shock tube facility

and advice concerning its operation. The U.S. Government is authorized to reproduce and

distribute copies of this article for governmental purposes notwiths tanding any copyright notation

thereon.


19/19


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