experiments on droplet collisions, …engineering.uci.edu/~melissao/journal/coll_review.pdfprogress...

Progress in Energy and Combustion Science Vol. 23, pp. 65-79, 1997 1

EXPERIMENTS ON DROPLET COLLISIONS, BOUNCE, COALESCENCE

AND DISRUPTION

by Melissa Orme

ABSTRACT

There has been a significant effort to understand the events that occur when two or more droplets

collide. The majority of the early work, which dates from the 1960’s, focused on droplet growth relating

to precipitation and was thus limited to water droplet collision phenomena. Recently, studies have

sought information on the collision behavior of fuel droplets for application to combustion spray systems.

Researchers have found distinct differences between the collision behavior of water droplets and fuel

droplets. The extensive experimental data on the droplet collision process is reviewed and presented

here for both water and fuel droplet collisions. Collision outcomes of bounce, temporary coalescence

followed by separation or catastrophic fragmentation, and permanent coalescence are examined and an

effort is made to relate the existing findings to a unified description of collisional droplet behavior.

keywords: droplet collisions, coalescence, disruption

1. INTRODUCTION

Some of the earliest work in droplet collisions dates back to Lord Rayleigh [1] who noted that small

rain droplets bounce upon collision with a larger pool of water. In 1897 he asserted that the failure to

achieve coalescence in such circumstances was caused by a layer of air trapped between the two

colliding surfaces which prevented true contact.


Later, in the period of time beginning approximately in 1960 and extending over two decades,

many investigations involving water droplets of different sizes traveling at their terminal velocities were

conducted in order to promote the understanding of the natural process of the formation of precipitation.

Inertial capture, in which droplets with larger terminal velocities overtake and capture smaller droplets in

their path is an important mechanism in precipitation. The terminal velocity is that constant speed that the

particle will move after the force of gravity which produces its motion is exactly balanced by the drag

forces that resist the motion. Hence, differences in terminal velocity of a droplet pair traveling in the

same medium could occur only from differences in droplet fluid densities or droplet diameters. Many of

the early investigations are limited to collisions which occur due to differences in their terminal velocities.

Detailed descriptions of theses experiments are reviewed in the article by Abbott [2].

Recently, attention has been focused on the collision behavior of hydrocarbon droplets for

application to the study of spray combustion. Since the conditions encountered in spray combustion are

notably different than those occurring in the natural atmosphere (e.g., gas, pressure, liquid), the earlier

results obtained in an effort to promote an understanding of raindrop formation cannot be directly

applied to combustion studies. This paper will attempt to detail the differences encountered in the

collision behavior of hydrocarbon droplets and water droplets.

Figure 1 illustrates the nomenclature adopted in this paper which is common to both water

droplet collisions and hydrocarbon droplet collisions. For generality, the droplets are assumed to be

different sizes with radii rL and rS denoting the large and small droplet radii respectively. The droplet

size ratio, R, is equal to rL/rS. The velocities of the large and small droplets are denoted by VL and VS.

The impact parameter, b, is defined as the distance from the center of one droplet to the relative


velocity vector placed on the center of the other droplet. The relative velocity U of the interacting

droplets is given by:

U V V V VL S L S2 2 2

1 22= + − +cos( )θ θ (1)

where θ1 and θ2 are the trajectory angles of the large and small droplet respectively. The collision angle

ψ is the angle formed by the line connecting the centers of the droplets at the instant of contact and the

relative velocity vector.

Many investigators have shown that it is convenient to characterize the collision process in terms

of the Weber number, We, and the impact parameter, b. The Weber number is the ratio of the inertial

force to the surface force and is defined as:

We U DS=ρ σ2 / (2)

where ρ is the droplet density, DS is the diameter of the smaller droplet equal to 2rS and σ is the surface

tension of the droplet fluid.

It is a widely accepted fact that when two droplets interact during flight several events may

occur. Generally speaking, the droplets may experience bounce, stable coalescence, temporary

coalescence followed by disruption, or temporary coalescence followed by fragmentation depending on

the value of We and b and the fluid property. Examples of these events are sketched in Figure 2.

Droplet bounce will occur if the surfaces of the droplets do not make contact due to the

presence of a thin intervening gas film. In this case the droplet’s surfaces undergo a flattening

deformation, but the surfaces do not make contact since the collisional kinetic energy, CKE, is


insufficient to expel the intervening layer of gas. Using the notation of Low and List [3], the collisional

kinetic energy CKE of the droplet pair (assuming similar droplet fluids) is given by:

CKED D

D DV VL S

L SL S=

+

−

ρπ12

3 3

3 3

2( ) (3)

Equation (3) can be rewritten as:

CKED

RV VL

L S=+

−

ρπ12 1

3

32( ) (4)

where R is the droplet size ratio rL/rS.

Droplet coalescence, in which a post-collision droplet is formed whose mass is equal to the sum

of the masses of the pre-collision droplets, follows droplet contact. The colliding droplets will coalesce

when the air film thickness reaches a critical value. This critical value was shown by Mackay and

Mason [4] to be typically of the order of 102 Å. The droplets may coalesce temporarily or permanently,

depending on the CKE and b. Temporary coalescence occurs when the collisional kinetic energy

exceeds the value for stable coalescence and will result in either disruption or fragmentation. In

disruption, the droplet’s surfaces have made contact, and the droplets temporarily coalesce. However

the increase in collisional kinetic energy was too severe and the coalescence is followed by disruption.

Disruption is that case when the collision product separates into the same number of droplets which

existed prior to the collision. A collision resulting in bounce may be difficult to distinguish from the case

when the temporary coalescence followed by disruption results in two droplets with masses equal to the

pre-collision droplet masses. The last event sketched is fragmentation which occurs in collisions with


excessive kinetic energies. In fragmentation, the coalesced droplet undergoes catastrophic break-up

into numerous small droplets.

Figure 3 illustrates influence of the impact parameter, b, on the post-collision product. For

convenience both droplet streams in Figure 3a-c are pictured the same size although this need not be the

case. Here, VL and VS are the velocities of the two droplets with radii rL and rS respectively. In each

example the z axis is aligned with the gravity vector and θ1 and θ2 are the trajectory angles measured in

the y-z plane. The impact parameter b is measured in the x-y plane as shown. The “phasing

parameter” is measured in the y-z plane. For head-on collisions (b=0) and relatively low high impact

energies the droplets will experience stable coalescence in which the post-collision droplet has the mass

of the two pre-collision droplets as indicated in Figure 3a. The coalesced droplet oscillates for a few

wavelengths (depending on fluid properties such as viscosity and surface tension) until it relaxes into a

stable spherical form.

Head-on collisions at high impact velocities result in a post-collision mass which has spread to a

flat disk as if they were impinging on a flat surface. Depending on the collisional kinetic energy of the

interaction, the disk will either relax back into a spherical form, or, at sufficiently high values of CKE,

grow until it becomes unstable and disrupts into myriad of small droplets emanating radially from the

disk’s center. Figure 3b illustrates a head on collision of relatively high CKE which results in the

formation of flat disks. In this case, however, the “phasing parameter” measured in the y-z plane is

non-zero (note that the droplets emanating from the right stream are slightly ahead of the droplets from

the left stream). Such a collision results in the formation of flat disks as shown edge-on in the figure.

The orientation of the disks is a function of the phasing parameter. If the phasing parameter were zero,


the thin disks would be aligned with the z axis, and the disks would interfere with each other. The disks

grow until they become unstable and disrupt. This collision event has been termed a “reflexive

separation” and will be discussed in detail in the next section.

Figure 3c illustrates a high energy grazing collision, where the impact parameter, b, measured in

the x-y plane is nearly rL+rS. In order for collision to occur, the impact parameter, b, must lie between

0 and ± (rL+rS). In the collision with a non-zero impact parameter, part of the kinetic energy is

transferred into rotational energy. The grazing collision results in a thin ligament connecting two large

fluid masses which ultimately disrupts into smaller satellite droplets as shown in the figure. This collision

event has been termed a “stretching separation.”

The ability to understand and predict the characteristics of droplet collision outcomes is of

primary importance for the application of the spray processes associated with combustion, materials

synthesis, painting, insecticide distribution, and the growth of cloud and water droplets. This paper will

attempt to detail the collision outcomes of water and hydrocarbon droplets, to explain discrepancies

found by different investigators, and to present a unified explanation of the collision outcomes.

2. WATER DROPLET COLLISIONS

2.1. Water Droplets Colliding with a Fixed Hemispherical Water Surface

Some of the earliest work on droplet collisions and growth is that of Shotland [5], Jayaratne and Mason

[6], List and Whelpdale [7], and Whelpdale and List [8] who studied the interactions between small

droplets with a large supported water surface in an effort to understand the mechanisms governing cloud


and raindrop growth. Their qualitative work showed that coalescence was dependent on the angle of

incidence of the droplet, θi, (as defined in Figure 4) and the Weber number, We, as defined by equation

(2) in which U is the normal velocity component of the incident droplet.

The above investigators [5-8] reached the consensus that low speed collisions had the

propensity to experience bounce and high speed collisions tended to favor coalescence. More

specifically, Shotland [5] showed that bounce occurs when We<3.0, and List and Whelpdale [7]

showed that bounce occurs when We<3.6. In order for coalescence to occur, the thin air film between

the liquid surfaces must be expelled by the action of dynamical forces. Jayaratne and Mason [6]

showed that the contact time of the colliding droplet with the target hemisphere is proportional to the

velocity of impact. Thus, it can be reasoned that high speed collisions experience a longer contact time,

and hence a longer opportunity to expel the air film which acts as a barrier to coalescence.

It was also found that the impact angle, θi, was an important parameter in determining the

characteristics of the collision outcome. Experimental results showed that, for a fixed incident droplet

speed, near head-on collisions generally favored coalescence and grazing collisions generally favored

bounce. An explanation for the observed behavior can be postulated with the following physical model

[6]. It can be reasoned that the kinetic energy of the impinging droplet is transformed into surface

energy of the surface depression which results due to the collision. The impinging droplet decelerates as

it contacts the surface, and at some point is brought to rest. In the following instant, the surface relaxes

upward toward its equilibrium position and repels the droplet in a manner analogous to a membrane on

a drum. The energy stored in the surface is partly converted into kinetic energy of the rebounding

droplet and partly into the energy of forming surface waves on the large hemispherical surface. Hence,


droplets impinging at nearly grazing incidence penetrate less than those impinging at normal incidence

and produce only shallow valleys in the water surface. Thus, these droplets tend to skim across the

surface and fail to coalesce. Also, droplets colliding at low speeds penetrate the surface less than

droplets colliding at high speeds and produce only shallow valleys in the water surface. Hence,

collisions at low speeds have the same effect as collisions at grazing incidence and favor bounce.

Jayaratne and Mason [6] introduced the method of displaying results of a collision where the

action of the initial kinetic energy lost by a droplet on impact is expressed as:

( ) /V V Vi b i2 2 2− (5)

where Vi is the impact velocity and Vb is the rebounding velocity as defined in Figure 4. Equation (5)

represents the fraction of kinetic energy lost by a droplet on impact provided that the droplet mass

remains constant. Experimental results show that the fractional energy loss increases with increasing

angle of incidence until the angle of incidence approaches 90o. For the case of nearly normal incidence,

the fractional energy loss tends to a limiting value of 0.95 and is independent of droplet diameter which

means that about 95% of the kinetic energy of the impacting droplet is being absorbed by the

hemispherical target. For nearly normal incidence, Vb/ Vi is equal to 0.22. Commensurate with the

physical model discussed earlier, the fractional energy loss is less for small angles of impact because less

energy is dissipated in deforming the surface and in producing surface waves.

Jayaratne and Mason [6], appear to be the only investigators to investigate the influence of

droplet diameter on the collision outcome for the case when smaller droplets impinge on a fixed

hemispherical surface. They found that the critical velocity and the critical impact angle required for

coalescence both decrease as the droplet diameter increases, hence, there exists a broader range of


conditions for which large drops coalesce compared to small drops. They also found that the behavior

of drops of radius greater than about 140 µm is notably different from that of smaller droplets. This fact

is evident in their studies on the relationship between the critical value of velocity and the critical value of

impact parameter required for coalescence. For drops with a radii smaller than about 140 µm, there

are well defined zones (ranges of θi) in which coalescence and bouncing occur. For larger drops, the

critical value of impact velocity decreases monotonically with increasing θi. In the work of Shotland [5],

300 µm diameter droplets were made to impinge on an 8mm hemispherical water target. He varied the

height at which the droplets were released, and found that the critical height required for droplet

coalescence decreased monotonically with increasing θi. He did not report the zones of bounce and

coalescence, which is likely due to the fact that, as confirmed in the later work of Jayaratne and Mason

[6], such zones do not occur for droplets of radii larger than about 140 µm.

2.2. Binary Water Drop Collisions

The drive for understanding the growth of raindrops led to the extension of previous studies, in

which small droplets collided with an unnaturally constrained larger droplet, to the collisions of two

water droplets flying in air. These studies can generally be classified in two groups: (1) those that

suspended the larger droplet in a vertically oriented wind-tunnel and injected smaller droplets into the

flow resulting in forced collisions, and (2) those studies in which both large and small droplets are falling

freely in still air. Since these studies were conducted in an effort to simulate the conditions found in

natural precipitation, most collisions occurred between large and small droplets of radii and speeds rL,

VL and rS, VS respectively. In addition to the experimental works cited in this paper, there has also

been a significant number of publications which focused on the theoretical or numerical aspects of binary


droplet collisions (e.g., see references [9-13]). This review will focus on the experimental investigations

of droplet collisions.

Droplet Collisions with an Aerodynamically Supported Drop

Much of the early work on droplet collisions [14-18] focused on droplets suspended in a

vertically oriented wind-tunnel and injected droplets to collide with the suspended droplet. In these

studies, it has been found that when larger droplets are injected downward toward the suspended

droplet, the low velocity in the wake of the lower droplet causes the droplet above it to accelerate when

it enters the wake and to coalesce with the suspended drop. Alternatively, when smaller drops are

injected into the airflow from below the suspended drop, the smaller droplet will rise to collide with the

under-surface of the larger drop. The collision outcome can be either bounce, permanent coalescence

or temporary coalescence. Woods and Mason [16] found that bounce always occurred when the

droplet diameter of the injected droplet was 70 µm or less. They attributed this finding to the fact that

the wake of the small droplet was insufficient to accelerate the large droplet towards it. Droplets with

diameters greater than 70 µm frequently resulted in coalescence, and it was postulated that this

occurred due to a well developed wake of the larger droplets.

Consistent with this finding, Montgomery [17] photographed collisions of relatively large

droplets with sizes in the range 0.25 ≤ rS ≤ 1.25 mm with an aerodynamically suspended droplet

roughly twice the size of the injected droplet. They reported collision outcomes of stable or unstable

coalescence, and not droplet bounce. Although their quantitative results are limited, they showed that

when the average relative velocity of the collision was approximately 40% higher than the terminal

velocity, coalescence occurred but was always temporary. When the relative velocity was


approximately 50% lower than the terminal velocity, the collision product was found to have a higher

likelihood for stability.

Spengler and Gokhale [18] also studied droplet collisions by suspending large diameter droplets

(>4mm) in a vertical wind tunnel and found that the collision outcome never resulted in bounce. The

droplets were impacted by a range of smaller droplets ranging in diameter from 500 µm to 4 mm. They

reported that the two influential factors leading to temporary or permanent coalescence were the

collisional kinetic energy and the impact parameter. They also noted that while keeping CKE constant,

the type of disruption (reported as filament pull-out, splash, or crown) depends on the impact

parameter. Filament pull-out occurs when the smaller droplet impacts near the edge of the large

droplet, the symmetrical crown action occurs when the smaller droplet impinges at small impact

parameters b, and splash occurs at intermediate impact parameters.

The application of collision results which rely on injected droplets impinging on an

aerodynamically supported target drop to the study of precipitation is not straightforward since artificial

flow patterns are used to obtain droplet stability, and the relative velocities used were not always

realistic. This fact caused others to formulate experiments which more closely resembled conditions

leading to precipitation growth [19-29].

Droplet Collisions Between Two Droplets Traveling in Still Air

Adam et al. [19] conducted experiments with water droplet collisions and paid careful attention

to the aerodynamic environment that surrounded the event. They asserted that using colliding streams of

droplets (as opposed to single droplets) has the disadvantage that the droplet stream drags along

ambient air which may produce extraneous aerodynamic forces. To overcome the problem, they


applied an electrostatic charge on a free-flying droplet stream and were able to isolate single droplets by

pulsing them out of the stream. They conducted experiments for equally sized water droplets of 120 µm

and 600 µm in diameter. Even for collisions between the smaller droplets, they did not experience

bounce as a collision outcome, but rather only found stable or unstable coalescence. This finding is

consistent with those of Woods and Mason [16] who found that droplets less than 70 µm in diameter

resulted in bounce and those greater than 70 µm in diameter frequently resulted in coalescence. Thus,

even though the aerodynamic effects pertaining to their experiments are vastly different [16, 19], their

results are in good qualitative agreement, leaving open the possibility that bounce experienced by small

droplets (<70 µm in diameter) is not entirely due to wake effects, but rather due to insufficient CKE (as

manifested through small droplet diameters or low relative velocities) to expel the intervening layer of

gas between the droplets. In the work of Adam et al. [19], a small positive charge of approximately

10-14C was applied to each droplet. This caused them to find that the impact parameter required for

coalescence was a function of the charge for impact velocities less than 1m/s.

Droplet Bounce

The physical phenomena which causes bounce or coalescence when small droplets strike a

supported hemispherical liquid surface as described in section 2.1 also apply to droplet collisions in

flight. That is, the thin air film which is the barrier to coalescence during flight must be expelled before

the two interacting droplets can make contact and coalesce. Researchers [20-22] have shown that the

critical factors which result in droplet bounce for collisions between two free-flying drops in still air are

the same for collisions between a droplet and a supported liquid hemisphere as discussed in the


previous section. Hence, in both cases, collisions at low speeds and/or grazing incidence and favor

bounce.

Brazier-Smith, et al. [20], examined the collisions between streams of water droplets traveling

in still air. Droplet radii were produced in the range between 150 and 750 µm and collisions were

produced with 1 ≤ R ≤ 2.5, with 0.3 ≤ U ≤ 3.0 m/s. They were able produce streams of droplets with

separations about 10 radii which caused the droplets to exert negligible influence on each other.

Although their work was directed primarily at investigating those situations which led to coalescence,

they photographed bouncing collisions between equally sized water droplets. They found that bouncing

occurred for low velocity collisions (assumed to be ≤ 0.3m/s, though not explicitly stated) with values of

b very close to rL+rS. Gunn [21] examined the collisions of large water drops (2mm) traveling freely in

air and found that those colliding at low relative velocities (0.4 m/s) tended to bounce. Park [22] also

produced collisions between streams of water droplets traveling freely in still air and showed pictorially

that near head-on collisions between pairs of equally sized drops with 700 µm diameters and a relative

velocity of 0.45 m/s resulted in stable coalescence and collisions at the same relative velocity but at high

impact parameters (θi≥50o) resulted in bounce.

Gunn [21] includes photographs of 1 mm water droplet interactions which include bounce and

near miss. He reported that coalescence could occur if the kinetic energy involved in the interaction

exceeded a certain minimum value. This minimum kinetic energy was assumed to have been exceeded

when the relative velocity of the droplets was greater than (4σ/ρr)1/2, that is, he found that bounce

occurs for We<8, which is a higher cut-off Weber number than reported by Shotland [5] and List and

Whelpdale [7] for droplets colliding with a supported droplet. Sartor and Abbott [23] arranged


bouncing collisions between droplets of nearly equal radii of 390 µm using a relative velocity of 0.3 m/s.

Hence, they found that bounce occurs for We=.96 which agrees qualitatively with the earlier results of

Shotland [5], List and Whelpdale [7] and those of Gunn [21]. Arkhipov et al. [24] studied collisions of

water droplets with R=1.9. They found that droplet bounce occurred for 0.7<We<1.5. Unfortunately

their quantitative results are lean as they did not report the impact parameter b in their studies.

Coalescence Efficiency

If the collisional kinetic energy is sufficient to expel the intervening air film between the two

droplets, coalescence will occur. The probability that a collision event will result in permanent

coalescence as sketched in Figure 2b and 3a is given by the coalescence efficiency (Brazier-Smith et

al. [20]):

Eb

r rcoalC

L S

=+

2

(6)

where bc is the critical value of the impact parameter above which separation occurs, and below which

the coalescence is permanent. Beard et al. [25] established that the coalescence efficiency is 45% for

water droplet collisional pairs with diameters of 81 and 20 µm falling at their terminal velocities.

McTaggart-Cowan and List [26], and Low and List [3] found that droplets greater in diameter than

200 µm possess sufficient collisional kinetic energy when traveling at their terminal velocities to

overcome any intervening air which would cause bouncing. Thus for conditions concerning rainfall

where droplets are often smaller in diameter than 200µm, bouncing is a likely event following the

collision interaction and the coalescence efficiency is likely to be less than 1.0.


Referring to equation (4) we see that when the size ratio R becomes large, the collisional kinetic

energy becomes small and we approach the case of small droplets impinging on a hemispherical surface

as discussed in the previous section. When the droplets coalesce, the newly formed droplet has a

smaller surface area (assuming a spherically shaped droplet) than the sum of the two pre-collision

droplets. The decrease in surface energy ∆Sσ is given by [3]:

∆S S ST Cσ = − (7)

where ST is the total surface energy of the pre-collision droplets:

S D DT L S= +πσ( )2 2 (8)

and DL and DS are the large and small droplet diameters respectively. SC is the surface energy of the

spherical equivalent post-collision droplet:

S D DC L S= +πσ( )3 3 23 . (9)

In order for coalescence to occur, both CKE and ∆Sσ must be dissipated through work done during

the action of expelling the air barrier between the droplet surfaces and through oscillations and

deformations of the coalesced droplet. The ability of the coalesced droplet to dissipate the total

collisional energy ET (sum of CKE and ∆Sσ) would establish the coalescence efficiency Ecoal.

Low and List [3] also conducted experiments with unequally sized colliding water droplets

which were traveling at their terminal velocity. Coalescence or breakup was determined based on the

number of post-collision particles. Breakup was defined as any collision which produced more than

two droplets. Their data indicate a general increase in Ecoal with a reduction in both droplet diameters.

That is, decreasing the collisional kinetic energy increases the probability for stable coalescence, and


increasing the CKE increases the probability for separation. They formulated an empirical relation for

Ecoal:

ED

D

E

Scoals

L

T

C

= +

−

−

αβσ

12 2

exp for ET <5.0 µJ (10)

Ecoal = 0 , otherwise (11)

Where α and β are constants equal to 0.778 and 2.61×106 J-2m2 respectively, and σ is the surface

tension of the water. These results approximate the results of Whelpdale and List [8] at the lower end

of the size spectrum. The results of Low and List [3] show that raindrop growth through coalescence is

significant (with Ecoal>0.5) only when small droplets (rS<300 µm) are collided with larger droplets.

Collisions with rS>300 µm generally resulted in breakup. They found that droplets did not coalesce for

ET>5.0 µJ and therefore did not examine these cases. They point out that equations (10) and (11) are

fits to the data and there is no justification to expect that the empirical relations should hold for other

conditions as well.

Brazier-Smith et al. [20] developed the following relation for the coalescence efficiency Ecoal:

EU r

fcoal =

240 2 1.

σρ

, (12)

where

fR R R

R R1

2 3 23 3 11

3

6 2

4 8 1 1 11

= + − + ++

. [ ( ) ]( )( )

(13)

They showed that the above relationship is in excellent agreement with their experimental results as well

as with those of Adam et al. [19]. The parameters used in their study were droplet diameters ranging


from 150 to 750 µm with relative speeds of 0.3 to 3 m/s. They cast their results in terms of Ecoal which

is measured as a function of the parameters b, DS, DL, R, and relative velocity U. They found that high

impact velocity collisions led to either permanent coalescence, coalescence followed by disruption, or

coalescence followed by fragmentation depending on the value of the impact parameter b.

Fragmentation is droplet separation which gives rise to an assemblage of small droplets often termed

satellite droplets as sketched in Figure 2. Satellite droplets are droplets which are much smaller in

diameter than the primary droplets. Their results showed that Ecoal decreased from 1.0 towards

zero as the Weber number increased from 6.4 to infinity. They also examined Ecoal as a function of the

droplet size ratio R, and found that Ecoal increases as R increases since less angular momentum is

available to pull the two droplets apart. This is corroborated by the results of Low and List [3] who

found that reducing the CKE increases Ecoal, and examination of eq. (6) shows that increasing R

reduces CKE.

Equations (12) and (13), combined with equation (6) lead to the following relation for the

critical value of the collision angle ψcr,

ψ cr

f RWe

=

arcsin( )1

12

(14)

where collision angle ψ is the angle formed by the line connecting the centers of the two droplets at the

moment of contact and the relative velocity vector as illustrated in Figure 1. Angles less than ψcr result

in stable coalescence. The relation was obtained by comparing the kinetic energy of rotation of the

droplet formed as a result of coalescence with the surface energies of the original droplets.


Bradley and Stow [27, 28] also engineered and conducted experiments in which pairs of

unequally sized water droplets collided in flight at their terminal velocities in still air with values which

approximate natural rainfall. The droplet pairs were ejected at constant rates and depending on droplet

diameter, the droplet spacing varied from 10 to 75 droplet diameters making it possible to ignore wake

interaction effects. Their qualitative results showed that for small values of the impact parameter b,

(near head-on collisions) the coalesced droplet oscillated and rotation was minimal. For intermediate

values of b, rotation occurred with and without breakup. For high values of b (grazing incidence), either

droplet rotation or shear generally resulted in breakup. Bounce was not a reported collision outcome.

Effect of Impact Parameter and We

Adam et al. [19] were among the first to study quantitatively the effects of impact parameter

and impact velocity on the stability of the droplet collision. Figure 5 shows two plots which represent

their results for water droplet collisions of 120 µm diameter (5a) and 600 µm (5b). Their data has been

recast in terms of Weber number in order to compare it with results from other studies. Since the data

for different droplet diameters does not collapse onto a single curve, it is asserted that the collision

process is more complicated than a function of Weber number and impact parameter alone. However,

this work was the first to show that the transition between stable coalescence and separation is

monotonic in the (b, We) plane for moderate We. For 120 µm water droplets impinging at low Weber

numbers (We<7.6) and high impact parameters (b>0.8) no coalescence occurs. At We > 7.6 and at

high values of b, the colliding droplets coalesce and disrupt. As We increases, the impact parameter for

disruption decreases. An increase in We implies an increase in the ratio of kinetic energy to surface

energy. Hence while keeping the impact parameter constant and increasing We, at some value of We


the kinetic energy will overcome the surface energy. If the impact parameter is high, part of the kinetic

energy is transferred to rotational energy. If the rotational energy is high enough the droplets may

separate. For this reason, the stable region decreases with increasing impact parameter. If the impact

parameter is low, a smaller amount of kinetic energy is transferred to rotational energy. As We

increases, the kinetic energy of the original droplets may be sufficient to cause disruption of the

coalesced droplet.

As can be seen from their data for 120 µm droplets, the We range 60 ≤ We ≤ 105 provides

two values of the impact parameter for disruption. For a head-on collision with an impact parameter of

b=0, the impact velocity associated with We=60 provides the minimum kinetic energy required to

overcome the surface tension force and causes the droplet to disrupt. If the impact parameter is

increased while holding We fixed, the coalesced droplet is stable and the CKE is dissipated through the

spinning and oscillating motion of the droplet. If the impact parameter is increased still further, excess

kinetic energy is transferred into rotational energy and the coalesced droplets pull apart due to the

centrifugal force on them. A similar plot is shown in Figure 5b for larger droplet collisions (600 µm

diameter). The same trends in Weber number and impact parameter hold for the larger droplet

collisions as for the smaller droplets. It can be seen that for identical Weber numbers, the larger

droplets remain more stable for a larger range of impact parameter. Another finding in their study was

that in the unstable region, there is a tendency for the number of fragments to increase and their size to

decrease as the impact velocity is increased for a given impact parameter.

The region of unstable coalescence at low impact parameters and high We has been termed

“reflexive” separation (termed so since it is the reflexive action of the surface tension forces which


causes separation), and the region of unstable coalescence at high impact parameters has been termed

“stretching” separation by Ashgriz and Poo [29]. A sketch of a head-on reflexive separation collision

for two equal sized drops is provided in Figure 6 (adapted from [29] and [30]). In the reflexive

separation collision, the collision product spreads to a shape which approximates a flat vertically

oriented disk. Surface tension causes the fluid near the circumference of the disk which is associated

with a high curvature to contract back towards the center of the droplet mass, forming a horizontally

oriented disk as shown. At the critical We, the disk will break into two or more droplets, where the

number of droplets is dependent on the value of We as documented by Ashgriz and Poo [29]. At

values of We less than the critical We, the disk will oscillate for a few periods until a spherical droplet is

formed (stable coalescence).

Stretching separation occurs at higher impact parameters. The reason for this is when two

droplets collide at high impact parameters, only a portion of the droplet fluid will interact, leaving the

remaining fluid not in the interaction region to travel toward its original trajectory. Thus, for stretching

separation there is a competition between the surface energy associated with the interaction region

which acts to hold the droplet together and the CKE associated with the non-interacting portion of the

pre-collision droplets which acts to separate the merged mass. For a given We, increasing b from the

value denoting the transition between coalescence and stretching separation to some critical value of b

causes an increase in the number of satellites.

Recently, Ashgriz and Poo [29] presented a detailed study of binary collision dynamics of water

droplets for various values of R, We and b. Their efforts were concentrated on coalescence and

separating collisions, and hence selected their experimental parameters to avoid bounce. Their results


which include the regions of coalescence and separation have been reproduced in Figure 7, and agree

qualitatively with the results of Adam et al. [19], and quantitatively with the work of Park [22]. It

should be noted that Figure 7 demarcates regions of coalescence stability or instability as defined by

their numerous experimental data points which are not present in Figure 7. In disagreement with the

work of Adam et al. [19], Ashgriz and Poo [29] and Park [20] found the We demarcating the onset

of reflexive separation to be independent of the droplet diameter but dependent on the drop size ratio

R. The experimental data in [29] defines the regions shown in Figure 7 for two size ratios R formed with

droplets emanating from nozzle diameters of 100, 200, 300, 400, and 500µm. Data for all droplet

diameters were combined onto single plots without distinguishing droplet diameter. Ashgriz and Poo

[29] found that the transition between stable coalescence and reflexive separation for head-on collisions

occurs at We=19 for R=1, and We=35 for R=2 which is in excellent agreement with the work of Park

[22]. In the previous work, Park measured the transition impact velocity for two independent sets of

equally sized droplets (R=1) with diameters of 200 µm and 700 µm Extracting the Weber number from

his data shows that the transition between stable coalescence and reflexive separation occurs at a

Weber number of approximately 19 for both cases. Similar measurements were also undertaken for

two independent sets of droplets with DS=200 µm and 450 µm and R=2 and it is found that transition

occurs at approximately We=37 for both cases, again in excellent agreement with the work of Ashgriz

and Poo [29].

Ashgriz and Poo [29] showed that keeping the impact parameter fixed at b=0 (head-on

collisions) and increasing We caused the number of satellite droplets resulting from separation to

increase. Their experimental results also showed that while holding We constant in the region of


reflexive separation and increasing b from 0 to the value where stable coalescence occurs causes the

total number of droplets produced by the reflexive collision to reduce. For example, for b=0 and

We=19 (onset of reflexive separation) their experimental results showed that the coalesced droplet will

separate reflexively into two droplets. Holding We constant at this value and increasing b to a non-zero

value will result in stable coalescence, i.e., the collision products have been reduced from two to one by

increasing the impact parameter while holding We fixed. As another example, reflexive interactions due

to head-on collisions at a higher value of We may cause three droplets to be formed. Increasing the

impact parameter while holding We constant at this higher value will cause two droplets to be formed.

Increasing the impact parameter further will lead to stable coalescence. Hence increasing the impact

parameter for fixed We (in the region of reflexive separation) has the effect of reducing the number of

droplets produced.

As indicated previously, it was found that the We which determines the onset of reflexive

separation depends on R [22, 29]. For larger R (or smaller DS), there is a larger region of impact

parameter for stable coalescence which is in agreement with the work of others [3, 20]. Also, for larger

R, stretching separation occurs at higher impact parameters for a given We. This is due to the fact that

the momentum of the smaller droplet is not sufficient to overcome the inertia and surface forces in the

fluid of the region of interaction. Unlike collisions involving equally sized droplets, as soon as the

droplets make contact, the higher pressure of the smaller droplet causes fluid to flow from the small

droplet to the large droplet.

An unexplained outcome of this survey is the lack of quantitative agreement between the results

of Adam et al. [19] and those of Park [22] and Ashgriz and Poo [29]; the only investigations which


quantified their results in a format that is conducive to direct quantitative comparisons for water droplet

interactions. Although they each showed that the transition between stable coalescence and separation is

monotonic in the (b,We) plane, the value of the transition Weber number between coalescence and

separation for head-on collisions differs in the work of Adam et al. [19]. One noted difference in the

aforementioned experiments is that the droplets carried a small electrostatic charge in the work of Adam

et al. [19] and were uncharged in the works of Park [22] and Ashgriz and Poo [29]. In the former

work [19], measurements were made of the impact parameter required for collision as a function of

droplet charge for droplets 120 µm in diameter and a relative velocity of 1 m/s. While these

measurements showed no indication of electrostatic interactions for this particular condition, the

possibility of electrostatic interaction effects as a function of droplet size (due to variations in charge-to-

mass ratio) and impact velocity has not been ruled out.

Other factors which could cause discrepancies in the data are the background gas density, gas

viscosity, humidity, and whether or not the droplet fluid is “clean”. Shotland [5] showed that the

propensity for permanent coalescence was increased when the density of the gas was reduced. This

finding has recently been corroborated by Qian and Law [30], who also showed that increased gas

viscosity inhibits coalescence. Also, it should be noted here that in all of the literature cited in this

review, there has been no mention of whether or not the stream surface is “clean” i.e., free of surface

tension degrading contaminants such as surfactants. Non-clean surfaces will have markedly different

collisional behavior since the fundamental surface properties defining the interface which must be formed

for coalescence will be dramatically altered.


3. COLLISIONS WITH DROPLET FLUIDS OTHER THAN WATER

Studies of spray combustion by Faeth [31] and O’Rourke and Bracco [32] have emphasized

the importance of understanding the phenomena occurring within the dense spray. The previous studies

discussed in this paper, which were largely motivated by meteorological interests, have often been used

for the understanding of phenomena associated with spray combustion. The application of the results

obtained from the study of rainfall to the combustion process is not entirely valid since the conditions

characterizing the two phenomena are vastly different in terms of background pressures, gases and

droplet fluid. Despite this fact, the literature base is relatively lean on the subject of droplet collisions

formed with droplets which are not composed of water.

There are five significant studies in which collisions of fuel droplets have been investigated for

direct application to spray combustion processes. In chronological order these studies include those of

Brenn and Frohn [33], who first studied the collision and merging of two droplets of propanol-2, and

later studied other liquids such as water, and n-Hexadecane [34]. Ashgriz and Givi [35] studied the

collisions of n-Hexane fuel droplets in burning or non-burning environments. The results reported in

these studies [33 - 35], although they were break-throughs in character since they were among the first

which dealt the dynamics of fuel droplet collisions, are generally observational in nature. Jiang et al.

[36] provided the most complete quantitative assessment of droplet collisions of hydrocarbon droplets

(heptane, decane, dodecane, tetradecane and hexadecane), and later Qian and Law [30] extended

those results to include the effects of background pressure, density, and viscosity. In each of the above

studies [30, 33 - 36], monodisperse streams of droplets were made to collide at various angles,

resulting in periodic collisions with identical parameters as sketched in Figure 3. The sequences of


collisions were photographed and studied. Additionally, Podvysotsky et al. [37] studied the collisions

between pairs of droplets of water-glycerin solutions and oil droplet pairs. They studied the cases when

1.9≤R≤12 and when there was always sufficient collisional kinetic energy to avoid bounce.

Unfortunately their data is cast in terms of parameters which make it impossible to extract the Weber

number for comparison to other studies and to identify which of their results are from oil and which are

from water-glycerin. Generally however, they did not find a sharp boundary between regions of

coalescence and break-up as observed by others for water [19, 22, 29, 30, 35, 36].

It has been shown by Jiang et al. [36] and Qian and Law [30] that the collisional dynamics for

hydrocarbon droplets is significantly different from those for water droplets. Their studies constitute the

most complete investigation of the collision dynamics of various fluids to date. They studied collisions of

equally sized droplets (150 µm in diameter) for a range of We from 0-100 for water, heptane (C7H16),

decane (C10H22), dodecane (C12H26), tetradecane (C14H30) and hexadecane (C16H34). Additionally,

they found that the transition between stable coalescence and stretching separation for hydrocarbon

droplet collisions is far from being monotonic in the (b, We) plane as was observed for water droplet

collisions by others for water [19, 22, 29] and shown on Figures 5 and 7. Thus, the collision behavior

embodies far more complicated phenomena than is apparent from the results of water droplet collisions.

The qualitative behavior of hydrocarbon droplet collisions is described in the conceptual sketch of

Figure 8 which is based on reported experimental observations [36].

A large discrepancy in the collision behavior between water droplets and hydrocarbon droplets

lies in the fact that at one atmosphere pressure, bounce is not observed in water drops for head-on

collisions. As discussed in the previous section, bounce has been observed in collisions between free-


flying water droplets with low impact velocity and high impact parameter [20-22], and in small droplets

(<70µm) for droplets colliding with an aerodynamically suspended droplet [16]. The criterion which

determines droplet bounce or coalescence in water droplet collisions or hydrocarbon droplet collisions

is whether or not the intervening layer of gas between the droplets which is the barrier to coalescence

can be expelled. In the bouncing regime (regime II in the figure), high pressure is built up in the gap

between the droplets as they approach each other. The pressure buildup in the gap causes the droplets

to deform by flattening as they approach each other. When this occurs, the a portion kinetic energy is

converted into surface energy. Bounce occurs when the droplets lose all of their impact kinetic energy

and are unable to expel the intervening layer of gas causing them to avoid coalescence and bounce

apart. Therefore, droplet bounce should depend not only on We, b, R, and droplet diameter, as

determined by the previous studies which focused on water collisions, but should also depend on the

viscosity and surface tension of the fluid as well as the viscosity and density of the surrounding gas [30].

It is thought that for this reason, the collision behavior of water droplets is different from the collision

behavior of hydrocarbon droplets.

As shown by Jiang et al. [36], the behavior of hydrocarbon droplet collisions is classified into

regimes I - V, where Wea, Web, and Wec denote the Weber numbers of transition between regime I

and II, II and III, and III and IV respectively for head-on collisions. Though each hydrocarbon droplet

collision behavior can be described qualitatively by the plot shown in Figure 8, the values of Wea, Web,

and Wec are different for each hydrocarbon studied (see Table 1).

For head-on collisions with low collisional kinetic energy (regime I), the collisions of

hydrocarbon droplets result in stable coalescence and not bounce. It is reasoned that this is because as


the droplets approach each other at low speeds, there is sufficient time to expel the intervening gas layer

which is the barrier to coalescence. Increasing We (regime II) causes the collision to be sufficiently

energetic for droplet deformation to occur, however merging does not occur in this regime and the

droplets experience bounce as discussed earlier. A further increase in We implies an increase in

collisional kinetic energy which can be sufficient to force the gas out of the gap between the approaching

drops. Hence, droplet collisions in regime III coalesce permanently. The coalesced droplet deforms

into a disk, and oscillates until it relaxes into a sphere. At the transition to regime IV, the collision

energy overcomes surface energy and oscillation causes the droplet to separate reflexively into two

droplets. Increasing the We in regime IV causes the formation of a ligament which connects the two

droplet masses and ultimately breaks to form one or more satellite droplets.

In the aforementioned study [36], head-on collisions were examined in the range of We which

was thought to result in stable coalescence (We<40) for water droplets as evidenced by others [19,

20]. However Park [22] and Ashgriz and Poo [29] found reflexive separation to occur for head-on

collisions at We ≈ 19 for equally sized water droplets. As can be seen from the sketch of Figure 8, the

collision behavior is markedly different at off-center collisions for hydrocarbon droplets than for water

droplets. That is, the transition between permanent coalescence and stretching separation does not

occur monotonically in the (We, b) plane.

Stretching separation occurs in high energy, high impact parameter collisions (regime V). The

qualitative behavior of the non-zero impact parameter collision described by regimes I-IV is similar to

the head-on collisions except that a rotational energy is imparted to the droplets upon contact. For high


impact parameter collisions (regime V), the strong centrifugal force causes the droplet to stretch and

break-up.

Jiang et al. [36] developed a phenomenological analysis from which they examined the

deformation and viscous dissipation of the coalesced droplets for regimes III and IV. From

experiments involving the collisions of various hydrocarbon fluids with differing viscosities, they found

that the surface area of the spheroid at maximum deformation increased linearly with We at a rate which

depends on the energy dissipation coefficient. The data for all collisions of different hydrocarbon fluids

used collapsed around a single line indicating that the energy dissipation coefficient is independent of

viscosity. Further, results from their phenomenological model indicate that the amount of viscous energy

dissipated is independent of the viscosity which is corroborated by the experiments.

Evaluation of the characteristics of the transition Weber numbers Wea, Web, and Wec,

demonstrated that Wec varies linearly with µ/σ, the ratio of the dynamic viscosity to the surface tension

of the droplet fluid. Their phenomenological model led to the judgment that the droplet energy

dissipation is considered to occur in two phases. The first phase is independent of viscosity and is

characterized by large amplitude deformation which is damped out quickly. The second phase occurs

during the small amplitude free oscillations through surface tension forces. Experimental results also

indicate that Web increases linearly and Wea decreases linearly with µ/σ. Tabulated values for the

critical Weber numbers are provided in Table 1 below:

C7H16 C10H22 C12H26 C14H30 C16H34

Wea 2.9 2.5 2.5 2.5 1.2

Web 5.0 5.5 10.0 14.0 15.5


Wec N.R.* 23.5 28.5 35.5 42.0

µ × 103 (Ns/m2) 0.4 0.9 1.5 2.3 3.4

σ × 103 (N/m) 21.0 24.0 25.0 26.0 27.0

Table 1: Experimental values for critical wavenumbers for different hydrocarbons [36]

(*NR indicates no record)

In earlier studies, Brenn and Frohn [33] investigated the dynamics of collisions similar to that

shown in Figure 3 for droplet pairs with diameters ranging from 70 to 200 µm and impacting speeds

from 2.8 to 11.7 m/s. They found that for head-on collisions with collisional kinetic energies at the lower

end of their range results in a coalesced droplet which oscillated before relaxing into a spherical form

similar to that shown in Figure 3a (Regime III). Increasing the collisional kinetic energy caused disks to

form as sketched in Figure 3b. In their experiments however the phasing parameter was zero (unlike

those in Figure 3b where the left stream lags the right resulting in the formation of angled disks), and the

disks were in alignment with the z axis, causing the disks to interact with each other, ultimately

connecting to form a single jet which later disintegrated. For droplet collisions which were generated

with a non-zero b, coalescence occurred when the collisional kinetic energy was not very high (still in

Regime III). For larger impact parameter b, the stream characteristics resembled those shown in Figure

3c (stretching separation, Regime V). For collisional kinetic energies commensurate with low We in

Regime V, the thin ligament which was formed as the collision product stretched apart was always

found to disintegrate into an odd number of satellite droplets. The reason why they found only an odd

number of satellite droplets in this range in unexplained, and not in agreement with the findings of others


[36]. Increasing the collisional kinetic energy in Regime V caused the ligament to grow, i.e., caused

more stretching to occur prior to disintegration.

Collisions of droplets of different sizes were also studied [33, 34] by directing two streams of

droplets of different sizes but identical separations to collide. For these conditions, the collision product

loses its symmetry about the z axis and will be skewed at an angle toward the original trajectory of the

larger droplet. The generation of similar structures as those attained with equal droplet sizes were

observed, one difference being that the structures were skewed at an angle. Additionally, if the

collision resulted in the generation of satellite droplets, these droplets have the potential to recombine

with the larger droplet which was not the case with equal sized collisions.

Brenn and Frohn also sought to study collisions of propanol-2, water, and n-hexadecane

droplets in which regular disintegration occurs [34]. They studied collisions of unequally sized droplets

with size ratios R=1.4 and 1.17. For each droplet fluid considered, they found that increasing the

collisional kinetic energy caused the limiting impact parameter for which stable coalescence to occur (bc)

to decrease which is in agreement with work of others which concentrated on water droplet collisions

[3, 20, 22, 29] and fuel droplet collisions [30, 36]. They found that increasing the size ratio results in a

larger range of b for a given collisional kinetic energy for stable coalescence (or a larger range in

collisional kinetic energy for a given b) which is in qualitative agreement with the earlier reported results

of Ashgriz and Poo [29] and Park [22] for unequal sized water droplets. Unfortunately, no quantitative

conclusion regarding the differences in the collisional behavior in the different fluid systems could be

drawn from their results.


Ashgriz and Givi [35] also studied the coalescence efficiency Ecoal of n-Hexane droplets in

unequally sized binary collisions and expanded the previous studies to include the effects of burning

droplets. From results of non-burning droplets which are presented in tabular form, it was reported that

as We increased from 1 to 38, Ecoal increased from 0.0 to unity, and then monotonically decreased to

0.0. These results were explained in the following way. As demonstrated by others, bounce occurs for

small We giving an Ecoal =0.0. Increasing We (which corresponds to increasing the collisional kinetic

energy relative to the surface energy) increases the probability that the air film can be expelled between

the two droplets resulting in collision until ultimately Ecoal =1.0. If the value of b is large however, the

coalesced droplets may separate due to the rotational energy imparted due to the geometry of the

collision. A further increase in We causes Ecoal to decrease monotonically since the high kinetic energy

causes separation or fragmentation.

To examine the effects of combustion, the two streams were ignited prior to collision [35].

They found that the collisions of burning droplets had a greater likelihood of leading to coalescence than

non-burning droplets for the low and intermediate range of We (that range of We in which Ecoal

increased from 0.0 to 1.0 for non-burning droplets). The reason for the decrease in Ecoal was

attributed to the reduction in surface tension directly as a result of increasing their temperature and

indirectly by creating fuel vapor on the surface of the droplets. The reduction in surface tension increases

the capacity of the droplet to penetrate into the surface of the other droplet for a given collisional kinetic

energy, and hence collisions with lower We will also coalesce. Qian and Law [30] also showed that

non-burning fuel droplet collisions which occur in an environment containing fuel vapor have a greater

propensity to experience stable coalescence. However, as We increased above a critical value, the


high temperature environment has a reverse effect on the collision in that the efficiencies are less than

those for non-burning droplets. This decrease of Ecoal is attributed again to the decrease surface

tension due to the fuel vapor in the environment, and in this case, the decrease in surface tension results

in a lowering of surface energy of the post collision droplet so that the kinetic energy associated with the

high We dominates the collision interaction, acting to tear them apart.

In summary, the investigations of [30, 33 - 36] and most notably [30, 36] have shown that

collisional behavior for non-water droplets is markedly different from that of water droplets as can be

seen by comparing Figures 5 and 7 (for water) with Figure 8 (for hydrocarbons). The difference in

behavior is believed to be caused by the difference in properties of the fluids such as surface tension and

viscosity.

In corroboration with the above conjecture, Arkhipov et al. [38] developed the following

relationship between R and ψcr based on experiments of a water-glycerin solution:

ψ cr

f RWe

=

arcsin( )2 (15)

fR

R R2

3 136

3 2

2 3 11

= ++

. ( )( )

(16)

The differences in equation (14) developed for water found by Brazier-Smith et al. [20] and equation

(15) for water-glycerin yields substantially different results, indicating that the collision mechanism

depends on the fluid properties which cannot be accounted for in We, b and R alone.


4. SUMMARY

The behavior of droplet collisions has been reviewed and several consistent trends have been

presented. For water droplets collisions, it has been found that regions of stable coalescence, reflexive

separation, and stretching separation are dependent on We and b, and more specifically, the transition

between stable coalescence and separation is monotonic in the (We, b) plane. The value of the critical

We and b which defines the transition between stable coalescence and separation was found to be

consistent between two independent investigations [22, 29], but a third was in disagreement [19].

Reasons for discrepancies have been discussed and include: effects droplet charging, gas pressure, and

humidity. The author of this review also puts forth the conjecture that the differences found in the critical

We and b for like fluids could be due to the fact that the “cleanliness” of the surface was not considered.

In the works reviewed, there has been little or no discussion regarding the definition of the interfacial

conditions. It has been shown [39] that the presence of surface contaminants such as surfactants can

significantly affect the collision behavior of the interacting droplets. In order to obtain a true comparison

between different investigations on similar droplet fluids, it is necessary that the droplet fluid be cleansed

with a cleaning device such as that used by Hodgson and Woods [40] in their experiments investigating

the collision of droplets suspended in an aqueous solution. Indirectly corroborating this conjecture,

Ashgriz and Givi [35] identified that burning droplets exhibit significantly different collision characteristics

since the increase in temperature caused by droplet burning results in a reduction in surface tension.

The behavior of fuel droplet collisions has been found to be notably different from the behavior

of water droplet collisions. For phenomenological reasons discussed in the text, the behavior of droplet

collisions is found to be dependent on the surface tension and viscosity of the droplet fluid as well as the


gas density, pressure, and viscosity, and hence, the collision behavior of water droplets and fuel droplets

are expected to experience differences. Hence, the value of the critical We and b which define the

transitions between regions between stable coalescence, bounce, reflexive separation, and stretching

separation will vary with droplet fluid and background gas conditions. Additionally, unlike water droplet

collisions, the transition between stable coalescence and separation for fuel droplet collisions is far from

monotonic in the (We, b) plane.


REFERENCES

[1] Lord Rayleigh in The Theory of Sound, vol II. (Dover publications, New York, 2nd ed., 1945)

[2] C.E. Abbott, “A Survey of Waterdrop Interaction Experiments,” Reviews of Geophysics and Space physics, 15, 3, 363 - 374, 1977

[3] T.B. Low and R. List “Collision, Coalescence and Breakup of Raindrops. Part I: Experimentally Established Coalescence Efficiencies and Fragment Size Distributions in Breakup,” J. Atmospheric Sciences, 39, 1591 - 1606, 1982

[4] G.D.M. Mackay and S.G. Mason, “The Gravity Approach and Coalescence of Fluid Droplets and Liquid Interfaces, Can. J. Chem. Engng, 41, 203, 1963

[5] R. M. Shotland, “Experimental Results Relating to the Coalescence of water droplets with water surfaces” Disc. Faraday. Soc. 30, 72-77, 1960

[6] O.W. Jayaratne and B.J. Mason, “The Coalescence and bouncing of water droplets at an air/water interface,” Proc. Roy. Soc. (London), A280, 545-565, 1964

[7] R. List and D.M. Whelpdale, “A preliminary investigation of factors affecting the coalescence of colliding water droplets,” J. Atmospheric. Sci. 26, 305 - 308, 1969

[8] D.M. Whelpdale and R. List, “The coalescence process in raindrop growth,” J. Geophys. Res., 76, 2836-2856, 1971

[9] U. Shafrir and T. Gal-Chen, “A Numerical Study of Collision Efficiencies and Coalescence Parameters for Droplet Pairs with Radii up to 300 Microns,” J. Atmos. Sci., 28, 741 - 751, 1971

[10] L.D. Reed and F.A. Morrison, Jr., “The Slow Motion of Two Touching Fluid Spheres Along Their Line of Centers,” Int. J. Multiphase Flow, 1, 573 - 584, 1974

[11] N. Carjan, “Effect of Dissipation on Ternary Fission in Very Heavy Nuclear Systems,” Nuclear Physics, A452, 381 - 397, 1986

[12] G. Tryggvason, D. Juric, M.H.R. Nobari, and S. Nas, “Computations of Drop Collision and Coalescence,” NASA Conference Publication, Issue # 3276, Second Microgravity Fluid Physics Conference, 135 - 140, 1995

[13] A. Menchaca-Rocha, “From Nuclei to Liquid Drops” Proceedings of the annual meeting of the American Chemical Soc., Nuclear Chemistry award symposium, Anaheim, CA, World Scientific Publishing, Singapore.

[14] D.C. Blanchard “The behavior of water drops at terminal velocity in air,” Trans. Am. Geophys. 31, 836-842, 1950


[15] J.W. Telford, N.S. Thorndike and E.G. Bowen, “The coalescence between small water drops,” Quart. J. Roy. Meteor. Soc. 81, 241-250, 1955

[16] J.D. Woods and B.H. Mason, “The wake capture of water drops in air, “ Quart. J. Roy. Meteor. Soc. 91, 35 - 43 , 1965

[17] D.N. Montgomery, “Collision and Coalescence of Water Droplets,” J. Atmospheric Science, 28, 292 - 293, 1971

[18] J.D. Spengler and N.R. Gokhale, “Droplet Impactions,” Journal of Applied Meteorology, 12, 316 - 321, 1973

[19] J.R. Adam, N.R. Lindblad, and C.D. Hendricks, “The Collision, Coalescence ,and Disruption of Water Droplets,” Journal of Applied Physics, 39, 11, 5173 - 5180, 1968

[20] P.R. Brazier-Smith, S.G. Jennings and J. Latham, “The interaction of falling water droplets: Coalescence,” Proc. R. Soc. Lond. A, 326, 393-408, 1972

[21] K. Gunn, “Collision characteristics of freely falling water droplets, SCIENCE, 150, 888-889, 1965a

[22] R. W. Park, “Behavior of Water Drops Colliding in Humid Nitrogen,” Ph.D. dissertation, Univ. of Wis., Madison, 1970.

[23] J.D. Sartor and C.E. Abbott, “Charge transfer between uncharged water droplets in free fall in an electric field,” J. Geophysical Research, 73, 20, 6415 - 6423, 1968

[24] V. Arkhipov, G. Ratanov, and V. Trofimov, “Experimental Investigation of the Interactions of Colliding Droplets,” J. Appl. Mech. Tech. Phys. 2, 19, 201 - 204, 1978

[25] K.V. Beard, H.T. Ochs, and T.S. Tung, “A Measurement of the Efficiency for Collection between Cloud Droplets” J. Atmospheric Sciences, 36, 2479 - 2483, 1979

[26] J.D. McTaggart-Cowan and R. List, “Collision and Breakup of Water Droplets at Terminal Velocity,” J. Atmospheric Science, 1401 - 1411, 1975

[27] S.G. Bradley and C.D. Stow, “Collisions between Liquid Droplets,” Phil. Trans. R. Soc. London., A 287, 635 - 675, 1978

[28] S.G. Bradley and C.D. Stow, “On the Production of Satellite Droplets During Collisions Between Water Droplets Falling in Still Air, “Journal of Atmospheric Sciences, 36, 494 - 500, 1979

[29] N. Ashgriz and J.Y. Poo, “Coalescence and Separation in Binary Collisions of Liquid Droplets” J. Fluid Mech., 221, 183 - 204, 1990


[30] J. Qian and C.K. Law, “Regimes of Coalescence and Separation of Droplet Collision,” J. Fluid Mech., (in print)

[31] G.M. Faeth, “Current Status of Droplet and Liquid Combustion,” Prog. Energy Combust. Sci. 3, 191, 1977

[32] P.J. O’Rourke and F.V. Bracco, “Modeling of Droplet Interactions in Thick Sprays and a Comparison with Experiments,” Stratified Charge Auto. Engng Conf., 101-115 Inst. Mech. Engng Pub. ISBN 0-85298-4693

[33] G. Brenn and A. Frohn, “Collision and Merging of Two Equal Droplets of Propanol,” Expts in Fluids, 7, 441-446, 1989

[34] G. Brenn and A Frohn, “Collision and Coalescence of Droplets of Various Liquids, “J. Aerosol Sci., 20, 8, 1027-1030, 1989

[35] N. Ashgriz and P. Givi, “Coalescence Efficiencies of Fuel Droplets in Binary Collisions, “Int. Comm. Heat and Mass Transfer” 16, 11-20, 1989

[36] Y.J. Jiang, A. Umemura and C.K. Law, “An experimental investigation on the collision behavior of hydrocarbon droplets,” J. Fluid Mech., 234, 171 - 190, 1990

[37] A.M. Podvysotsky A., and A. Shraiber, “Coalescence and Break-up of Droplets in Two Phase Flows,” Int. J. Multiphase Flow, 10, 2, 195 - 209, 1984

[38] V. Arkhipov, G. Ratanov, and V. Trofimov, “Experimental Study of Droplets Interaction at Collisions,” J. Appl. Mech. Tech. Phys. 2, 73-77, 1978

[39] W. Rommel, W. Meon, E. Blass, “Hydrodynamic Modeling of Droplet Coalescence at Liquid-Liquid Interfaces”, Separation Science and Technology, 27, 2, 129 - 159, 1992

[40] T.D. Hodgson , D.R. Woods, “The Effect of Surfactants on the Coalescence of a Droplet at an Interface. II”, J. Colloid Interface Sci., 30, 4, 429 - 446, 1969


FIGURE CAPTIONS

Figure 1: Nomenclature of droplet collision process. b is the impact parameter, ψ is the collision angle, VL and VS are the velocities of the large and small drop respectively, U is the relative velocity, rL and rS are the radii of the large and small drop respectively, and θ1 and θ2 are the trajectory angles measured from the reference of the gravitational vector.

Figure 2: Terminology of possible droplet collision outcomes (adapted from Abbott [2])

Figure 3: Examples of binary droplet collisions illustrating the effect of impact velocity and impact angle b. (a) low impact speed collision resulting stable coalescence with zero impact and phasing parameter, (b) high impact speed collision resulting in unstable coalescence leading to fragmentation with zero impact parameter and finite phasing parameter, (c) high impact speed collision resulting in unstable coalescence leading to fragmentation with zero phasing parameter and finite impact parameter (grazing collision).

Figure 4: Nomenclature for rebounding droplet trajectories (adapted from Jayaratne and Mason [6])

Figure 5: Data from Adam et al. [19] for water droplet collisions which shows the regions of stable and unstable collisions. (a) droplet diameters are 120 µm, (b) droplet diameters are 600 µm.

Figure 6: Schematic of a head-on collision leading to reflexive separation.

Figure 7: Regions of Coalescence, reflexive separation, and stretching separation for water droplets formed from 100, 200, 300, 400, and 500 µm nozzles (Adapted from Ashgriz and Poo [29]). (a) droplet size ratio R=1.0, (b) droplet size ratio R=2.0.

Figure 8: Regions of experimental observations of Jiang et al. [36]. In Regime I collisions result in coalescence, Regime II bounce, Regime stable coalescence, Regime IV reflexive separation, and Regime V stretching separation. Shaded regions are characterized by stable coalescence.


b

ψ

VL

SVθ2

rS

U

U

SV

1θ

VL

Lr

Figure 1: Nomenclature of droplet collision process. b is the impact parameter, ψψ is the collision angle, VL and VS are the velocities of the large and small drop respectively, U is the relative velocity, rL and rS are the radii of the large and small drop respectively, and θθ 11 and θθ 22 are the trajectory angles measured from the reference of the gravitational vector.


coalescence

bounce

fragmentation

disruption

Figure 2: Terminology of possible droplet collision outcomes (adapted from Abbott [2])

LVVs

θ21θ

)a( )c()b(

bU

y

x

z

x

y

Figure 3: Examples of binary droplet collisions illustrating the effect of impact velocity and impact angle b. (a) low impact speed collision resulting stable coalescence with zero impact and phasing parameter, (b) high impact speed collision resulting in unstable coalescence leading to fragmentation with zero impact parameter and finite phasing parameter, (c) high impact speed collision resulting in unstable coalescence leading to fragmentation with zero phasing parameter and finite impact parameter (grazing collision).


drops

water

air

Vi

ιθ

Vb

bθ

Figure 4: Nomenclature for rebounding droplet trajectories (Adapted from Jayaratne and Mason [6])


unstable

stable

(a)

b/D

0.0

0.2

0.4

0.6

0.8

1.0

0.0 20.0 40.0 60.0 80.0 100.0 120.0

Weber Number

stable

(b)

b/D

unstable

0.0 100.0 200.0 300.0 400.0

Weber Number

500.00.0

0.2

0.4

0.6

0.8

1.0

Figure 5: Data from Adam et al. [19] for water droplet collisions which shows the regions of stable and unstable collisions. (a) droplet diameters are 120 µµm, (b) droplet diameters are 600 µµm.


Figure 6: Schematic of a head-on collision leading to reflexive separation.


Stretching Separation

Reflexive Separation

Coalescence

b/D

(a)

0 20 40 60 80 100

W e

0.0

0.2

0.4

0.6

0.8

1.0

Coalescence

Reflexive Separation0 20 40 60 80 100

W e

Stretching Separation

0.0

0.2

0.4

0.6

0.8

1.0

b/D

(b)

Figure 7: Regions of Coalescence, reflexive separation, and stretching separation for water droplets formed from 100, 200, 300, 400, and 500 µµm nozzles (Adapted from Ashgriz and Poo [29]). (a) droplet size ratio R=1.0, (b) droplet size ratio R=2.0.


We

V

III

0

a bWe We

I II

b/D

1

cWe

IV

Figure 8: Regions of experimental observations of Jiang et al. [36]. In regime I collisions result in coalescence, Regime II bounce, Regime III stable coalescence, regime IV separation due to high impact energies, and regime V separation due to shear. Shaded regions are characterized by stable coalescence.

experiments on droplet collisions, …engineering.uci.edu/~melissao/journal/coll_review.pdfprogress...

Documents