ILASS Americas, 20th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 2007

A Model for Deformation of Liquid Jets and Droplets Subjected to Gaseous Flows

A. Mashayek

1

, A. Jafari

2

and N. Ashgriz

2*

1

Department of Mechanical and Aerospace Engineering

University of California San Diego, La Jolla, CA 92093-0411, USA

2

Department of Mechanical and Industrial Engineering

University of Toronto, 5 King’s College Road, Toronto, ON, M2M 2G5 Canada

Abstract An analytical model is used to calculate the deformation and spreading of the two-dimensional and axisymmetric

liquid drops in a gas stream. The model is based on the expansion of the Navier-Stokes equations in a series of small

parameters. The zeroth and first order terms correspond to the deceleration and deformation of the liquid body re-

spectively. The axisymmetric formulation is used for calculating the spreading of the spherical droplets while the

two-dimensional formulation is used for predicting the spreading of the liquid jet’s cross section in cross flows.

There is good agreement between the present calculations and experimental data.

*

Corresponding author

Introduction

Understanding of mechanism of motion and de-

formation of liquid droplets in gaseous flow is of great

importance for description and modeling of sprays and

other dispersed systems. These types of flows are ubiq-

uitously found in the nature as well as industries. Ex-

amples include fuel injection in combustion chamber of

aerial and automotive engines [1], rain properties [2],

and interaction between aircraft and raindrops [3],

among others. In addition to the spherical droplets, liq-

uid jet in gaseous cross flow (JICF) systems also have

applications in fuel injection systems such as gas tur-

bines, afterburners, augmenters and ramjet/scramjet

combustors. This type of injection into a gaseous cross

flow improves fuel atomization and vaporization char-

acteristics and is commonly used in lean premixed pre-

vaporized combustion systems. The deformation and

atomization of such liquid jets has important effects on

the flow and combustion characteristics. The deforma-

tion of liquid jet and the blockage it imposes on the gas

may change the flow field significantly. The breakup

regimes involved in this type of atomization depend

significantly on the aerodynamic forces which affect

the jet deformation and spreading, its deflection, and

the rate of mass stripping from it. Especially, at flows

involving weaker aerodynamic conditions where the

dominant breakup regimes are column and bag breakup,

the jet’s deformation at early stages plays a governing

role in its deflection and breakup process.

Jet in cross flow atomization problem involves

very complex flow physics such as strong vortical

structures (created by liquid jet blockage against the gas

flow), small scale wave formation, stripping of small

droplets from the jet surface, and formation of differ-

ently sized ligaments and droplets. Therefore, the com-

plete numerical simulation of such a problem, resolving

most important flow scales on the Eulerian frame, is not

still feasible, especially for industrial applications.

These issues signal the demand for some simpler, yet

reliable, models that can be used for industrial design

purposes and can take into account important parame-

ters such as flow conditions and physical properties of

the liquid and gas phases. A computationally affordable

solution, at the moment, is to combine some simple

models, which calculate the jet shape and trajectory,

with an improved Lagrangian droplet tracking scheme

which accounts for the secondary breakup of the drop-

lets.

To reproduce the complex vortical structures on the

leeward of the liquid jet, one can construct an obstacle

of a shape similar to the continuous liquid body in the

flow field. Droplets can be injected from different loca-

tions along the obstacle’s trajectory to model the mass

stripping from the liquid column and tracked by La-

grangian schemes. The presence of the jet-like obstacle

helps develop a realistic flow field. This procedure can

be used to predict the droplet size, velocity distribution

and mass flux for different JICF conditions.

So far, most of the correlations that predict the de-

formation and penetration of a liquid jet in a cross flow

has been based on experimental studies focusing on a

certain range of flow parameters and fluid properties.

The constants in these correlations were tuned based on

the specific experimental data which in turn questions

their applicability to a wider range of parameters. Ma-

zallon et al. [4] experimentally investigated the primary

breakup of laminar round liquid jets in gaseous cross-

flows. They suggested that there are qualitative simi-

larities between the primary breakup of nonturbulent

round liquid jets in crossflows and the secondary

breakup of droplets. This similarity has been reported in

a number of other articles. We use this similarity in the

present model and simulate the 2D droplet deformation

in order to gain an insight into the JICF problem.

Here, we start from the Navier-Stokes equations

and, with some simplifications, attempt at solving these

equations for the cross section of a liquid drop (axi-

symmetric formulation) or a liquid jet (two-dimensional

formulation). Currently, the model assumes there is no

mass loss from the body in the process of deformation.

Thus, it is suited for studying the deformation, shape

evolution, and spreading of liquid droplets and jets be-

fore mass shedding or liquid body disintegration occurs.

Let us consider the deformation of a spherical

droplet as an example. The aerodynamic pressure force

on a droplet traveling through the gas stream is bal-

anced against its inertia. This produces a non-uniform

pressure along the droplet interface which is accommo-

dated by surface tension and variations in the surface

curvature. As a result, the droplet gets distorted and the

aerodynamic force will increase upon the droplet distor-

tion. This makes the droplet more oblate and increases

its frontal area. When the conditions are severe enough,

it will breakup, mostly as determined by the Weber

number. [3]

In the absence of heat and mass exchange between

the droplet and the surrounding gas, the droplet condi-

tions are adequately described by the Weber number

We, which represents the ratio of the gas inertial force

to the surface tension force, and Ohnesorge number Oh,

which quantifies the effect of liquid internal viscous

forces relative to the surface tension forces.

VUP

VU

0

0

2

,

DOhDUWe

L

LG (1)

where the subscripts L and G represent the liquid and

gas phases, respectively. Furthermore U, ȡ, µ, ı and D0

are the relative velocity between droplet and gas

stream, the density, the dynamic viscosity, the surface

tension and the initial droplet diameter respectively.

Some studies consider the Weber and Reynolds as the

non-dimensional parameters. It should be noted that the

definition of Weber, Reynolds and drag coefficient are

normally based on the volume equivalent diameter,

which is assumed to be constant if no stripping or

breakup occurs. In JICF applications, another important

parameter called momentum ratio, q, is used exten-

sively. It compares the liquid momentum to that of gas.

2

2

GG

LL

UUq

UU

(2)

Here, we briefly review the most relevant works on

the field of droplet deformation in gas streams. Some

other experimental investigations are reviewed by

Wierzba [5]. Based on Weber and Ohnesorge number,

Hsiang and Faeth [6] specify the transition between

different deformation and breakup regimes. Aalburg et al. [7,8], simulated the low to moderate deformation of

2D and axisymmetric droplets under shock conditions

using level set method. However, even for 2D simula-

tions, the simulation of high density ratios as occurs in

real droplet systems needed considerable computational

resources.

Fuchs [9] performed accurate simulations of de-

forming droplets. He found out that for the deforming

droplets, the drag coefficient cannot be characterized

solely based on Reynolds number. Surface tension is a

very important factor which has to be considered too.

Thus, for a droplet of certain Reynolds number but with

different Weber numbers, drag coefficient changes sig-

nificantly. For example according to Fuchs’ computa-

tions, at Re=100, for solid particle (We=0), CD=1.103,

for slightly deformable droplet (We=0.1), CD=1.33, and

for a more deformable droplet (We=1), CD=1.73 [9].

The majority of spray studies involving Lagrangian

tracking schemes assumes spherical droplets and use

drag coefficient of spherical droplets. In many condi-

tions, the droplets are far from the spherical shape.

Strongly deformed droplets are observed during the

disintegration of liquid jets, sheets and ligaments. Dur-

ing the droplet transport, they may be deformed due to

the surrounding gas flow or due to collisions with a

wall or other droplets. In our method, based on the ve-

locity and the pressure at the interface, the drag coeffi-

cient can be calculated with good accuracy which is

another advantage of using such method.

There are a number of studies on prediction of the

droplet and jet cross section shapes. Some of them have

assumed that the droplet deforms into an ellipse or el-

lipsoid which is valid for low deformations but not for

large deformations since the non-symmetric flow ef-

fects come into the picture.

Clark [10] derived a model for small deformations

of a droplet by calculating the linearized terms for the

viscous, interfacial tension and inertial forces based on

the analogy between an oscillating two-dimensional

droplet and a forced mass-spring system.

Mashayek et al. [11] extended Clark’s model for

nonlinear terms, along with improved drag coefficients

[12] and obtained trajectories in good agreement with

the experimental data. While, this method was appro-

priate for determining the jet trajectory, the predicted

shape of cross section for large deformations was not

realistic. This is because the jet cross section shape

changes to kidney in large deformation cases, while the

model only predicts a ellipse of large aspect ratio.

For spherical droplets, at higher Weber numbers,

the aerodynamic shear breakup becomes important and

the periphery of the droplet converts to long ligaments

which is out of the scopes of the present work.

Based on the similarity suggested by Mazallon [4] and

Hsiang and Faeth [6] between JICF cross section and

2D drop, we have utilized a theoretical model for calcu-

lating the deformation of a 2D or axisymmetric liquid

droplet. The axisymmetric equations are solved for

studying spherical droplet deformation, while 2D equa-

tions govern the deformation of the cross section of the

liquid jet. Details of the theoretical model for both axi-

symmetric and 2D cases are provided and the results for

different cases at various test conditions are presented

and discussed.

Part A: Axisymmetric droplet The model is based on the droplet deformation and

spreading formulation originally developed by Gonor

and Zolotova [13] which allows for small to relatively

large deformations of axisymmetric liquid droplets sub-

jected to gaseous flows. The formulation is then modi-

fied for 2D droplets. Comparison between the 2D and

axisymmetric cases is followed by application of the 2D

formulation to prediction of the behavior of liquid jets

subjected to gas cross flows.

Assuming an axisymmetric droplet subjected to

gaseous flow as depicted in Fig. 1, the system of equa-

tions governing the motion of the droplet is [13]

¸̧¹

·¨̈©

§ww

�ww

�ww

�ww

�

ww

�ww

�ww

ru

rru

zu

zp

k

ruu

zuu

tu

zzz

zr

zz

z

1

Re

11

2

2

2

2

(3)

¸̧¹

·¨̈©

§�

ww

�ww

�ww

�ww

�

ww

�ww

�ww

22

2

2

2

1

Re

11

ru

ru

rru

zu

rp

k

ruu

zuu

tu

rrrr

rr

rr

r

(4)

0 �ww

�ww

ru

ru

zu rrz

(5)

where k is the density ratio ȡ/ȡG and Re is the Reynolds

number defined base on the gas density ȡG, droplet di-

ameter d, gas dynamic viscosity µG, and gas velocity uG,

which is in the Z direction. The involving parameters

are non-dimensionalized using

2

00

,,,

GGG

G

upp

rrr

uuu

rutt

U c c c c (6)

where r0 is the initial droplet radius. The boundary con-

ditions are discussed in details in Gonor and Zolotova

[13]. The pressure coefficients over the surface of a

sphere and a 2D droplet located in a potential flow are

plotted in Fig. 2. At relatively high Reynolds numbers,

the flow separation occurs which affects the pressure

distribution on the droplet surface. Thus, the pressure

coefficient in real condition does not follow that of the

potential flow. Figure 3 plots the pressure coefficient

for solid spheres and 2D cylinders at various Reynolds

number. [12,14]. To impose a more realistic pressure

boundary condition on the droplet surface rather than

that of the potential flow, Gonor and Zolotova applied a

smoothing function to produce the nearly flat profile on

the leeward of the droplet as proposed by Fig. 3. In or-

der to calculate the pressure distribution around a de-

forming droplet in the gas flow, the pressure is ex-

panded in the form of

...210

��� pppp (7)

where p0 is the pressure distribution on a sphere subject

to a potential flow in the Z direction [13]. The velocity

components ur and uș are respectively expanded in the

form of

...,

...,

310

310

���

���

rrrr

zzzz

uuuu

uuuu (8)

where , are also obtained from the solution of

the potential flow past a sphere.

0zu

0ru

Defining the gas velocity relative to the droplet as

, the potential and the corresponding

pressure distribution are [15]

)(10

tuU z�

)

2

1)((3

3

0

rrCosUr �� TM (9)

))(sin

4

9

1(

2

122

0

TU �� f Upp (10)

Evaluating Eq. (3),(4) for the potential flow and

noting that , we obtain 00

ru

³ ww

� dtxp

ktuz

0

1

)(0

(11)

Thus, the velocity of the solid sphere becomes [13]

kl

ltlttuz

8

9

,

1

)(0

�

(12)

To calculate the velocity on the surface of the de-

forming droplet, and have to be calculated. Sub-

stituting Eqs. (7) and (8) into (3-5) and moving to a

non-inertial frame attached to the moving center of

mass of the droplet, yields [13]

1zu

1ru

¸¸¹

·¨¨©

§

w

w�

w

w�

w

w�

ww

� w

w

ru

rru

zu

zp

ktu zzzz

1111

1

Re

11

2

2

2

2

(13)

¸¸¹

·¨¨©

§�

w

w�

w

w�

w

w�

ww

� w

w2

1

2

2

2

2

1111

1

Re

11

ru

ru

rru

zu

rp

ktu rrrrr

(14)

0111 �

w

w�

w

w

ru

ru

zu rrz

(15)

Gonor and Zolotova [13] expanded the solution of

this linear system in the form of the following double

series

jiij

ji

jiijz yxayxau 2

0,

2

1

{ ¦ ¦f

f

(16)

121

0,12

1

��f

f

��� ¦¦ ji

ijji

r yxaj

iuK

(17)

After ample calculations, they worked out the fol-

lowing identity equations to calculate the aij coefficients

,...1,0,1]

2)12(

)1)(2)(3)(4(

)1)(2(2

)12)(22[(

Re

1

2)12(

)1)(2(

)(

1,4

,2

1,

1,2

t��

�����

���

����

c����

� c

��

�

�

��

ijajjiiii

aii

ajj

ajj

iita

ji

ji

ji

jiij

K

K

K

(18)

and,

,...1,0,1

])1()1()1(2)1(2

)1(

)22)(12(

)3)(2)(1(

)1(2

)1(

)12(

)1)(2(

[

Re

1

)12(

)1(

)22)(12(

)1()1(),,(1

222

0

12112

22

2

122

0

0

t

�������

�������

���

�����

���

�c�

���

�c��

ww

� c

��

���

�

��

ijxxaiixaiixa

xxajjiiiixxija

xxaj

iiij

xxaijjxxaii

xtyxp

kxa

jiij

ii

iij

jiij

jiij

jiij

jiij

jiiji

i

K

K

KK

K

(19)

where

> @

.]

[

1

Re

2

424422424

22222222

2222222

42

ijji

ijji

ijji

ij

jiij

jiij

jiij

jiij

ijji

ijji

ijji

ij

AyxGyxQyxI

yxFyxEyxHyxDWe

ayxNyxMyxKk

xxxxp

������

����

����

���

����

���

��� ww GJED

(20)

The parameters introduced in (20) are described

fully in Gonor and Zolotova [13].

Considering the Eqs. (18) and (19) without the

presence of the viscous terms, they degenerate into a

system of differential equations for the coefficients

21020121114030201000

21020121114030201000

,,,,,,,,,

,,,,,,,,,,

AAAAAAAAAAaaaaaaaaaa

cccccccccccccccccccc

(21)

where Aij=a’ij. This system can be solved using numeri-

cal integration from t=0 up to the desired time. The

velocities on the interface of the droplet can be calcu-

lated by plugging back the integrated values into Eqs.

(16) and (17) and finally into (8). This task is worked

out by differentiating (21) once and integrating the co-

efficients from 0 to t using 4

th

order Runge Kutta

method. The initial value of the aij terms are zero at t=0 and the initial values of a’ij terms can be calculated by

solving the system of linear equations of Eq. (21) with-

out the surface tension forces, i.e., without the presence

of the Aij terms. Finally, to calculate the deformation of

the droplet at time t, the following equations can be

added to the system of differential equations and solved

for simultaneously

³ t

z dttrzuz0

),,(1

(22)

³ t

r dttrzur0

),,(1

(23)

where and, are defined in Eqs.(16) and (17). 1

zu1ru

Results and Discussion (Part A) As the first test case, we consider the deformation

of a spherical water droplet in air flow with We § 33

and Re §4000. Figure 4 compares the deformed droplet

shapes at non-dimensional times from 0 to 80 with the

experimental data of Gelfand et al.[16]. As the figure

shows, the windward side of the droplet flattens at early

stages. As the deformation grows, the droplet begins to

spread and the leeward side takes a flat shape. The

spreading will continue to form a bag shape which in

reality, will be followed by bag and shear breakup and

mass stripping from the droplet at this Weber number

as reported by Hirahara and Kawahashi [17].

As the second case, the deformation of the droplet

at different times for Re=40, We=40, and k=50 is

shown in Fig. 5. It is clearly observed that the droplet

deforms and flattens by t=15. The spreading radius a in

the figure is used to define a/r0 as a measure of the ra-

dial spreading of the droplet. To compare this case with

the results of Quan and Schmidt [18], the deformation

a/r0 is compared with their results at exactly the same

time and flow conditions in Fig. 6. The good agreement

between the results validate the ability of the present

model to predict the deformation of the droplet before it

reaches critical shapes and stripping or breakup re-

gimes. It can be clearly observed that the rate of radial

spreading of the droplet is slower at the early stages

(t§0-5) and increases from that point on. This is mainly

due to initial dominance of the surface tension forces at

early stages because of larger curvatures. As the droplet

deforms, its local Weber number, We=ȡGU2(2a)/ı, and frontal area, ʌa2

, increase and so does its local drag

coefficient which lead to faster deformation [11,18].

Figure 7 shows the interface velocity of droplet for the

same case of Fig. 5. Top row shows the velocities

measured in an inertial frame. It is observed that the

velocity profile on the surface of the droplet reaches

almost a uniform profile by t=15. The bottom row

shows velocities measured in the non-inertial frame

attached to the center of mass of the droplet. From the

left to right, it can be observed that the windward side

of the droplet is moving toward the center of the droplet

due to direct contact with the upstream gas flow while

the leeward side of the drop is moving backward, with

respect to the center of mass of the droplet, due to resis-

tance of the gas behind it. Focusing on the velocity pro-

file on the leeward side of the droplets in the second

row, one can see that at early stages (first 4 frames), the

velocity vectors point in the direction opposite to the air

flow. However, as time increases (the last three

frames), the vectors in the upper half of the droplet

point to the upper left direction and the vectors in the

lower half point to the lower left direction. This can be

justified by considering the evolution of the recircula-

tion region and double vortices that form behind an

axisymmetric or 2D deformed droplet at low Reynolds

numbers as reported widely in literature such as

[11,18].

As the last case in this section, we investigate the

effect of liquid to gas density ratio, k¸ on the droplet

deformation. Figure 8 demonstrates the deformation of

droplets in time as the density ratio changes from nearly

590 in the top row to almost 30 in the last row. It should

be noted that the third row is the same as that of Fig. 5.

As the density ratio decreases, the rate of momentum

exchange between the gas stream and the droplet in-

creases leading to considerably faster deformations

which can be clearly observed by comparing the first

and last rows.

Part B: 2D droplet and application in JICF

We now turn our attention to modeling the spread-

ing of a 2D droplet and its application in modeling the

deformation of a liquid jet in a gas cross flow. Equa-

tions (13-15) can be rewritten in the form of

¸̧¹

·¨̈©

§ww

�ww

�ww

� ww

�ww

�ww

2

2

2

2

Re

11

yu

xu

xp

kyuv

xuu

tu

(24)

¸̧¹

·¨̈©

§ww

�ww

�ww

� ww

�ww

�ww

2

2

2

2

Re

11

yv

xv

yp

kyvv

xvu

tv

(25)

0 ww

�ww

yv

xu

(26)

where u denotes the velocity in the stream-wise direc-

tion and v is the velocity in the lateral direction. Fol-

lowing the same procedure of part A, we can write

...,

...,

...

210

210

210

��� ��� ���

vvvvuuuu

pppp (27)

The zeroth order terms p0 and u0 can be obtained

using the potential flow past a circular cylinder with the

potential of [19]

))((

2

0

rrrCosug � TM (28)

It should be noted that v0 is zero due to symmetry. Proceeding as the axisymmetric case of Wang and Jo-

seph [15], it can be shown that p0 takes the form of

))(sin41(

2

122

0

TU �� f Upp (29)

Similar to the previous section, we get

kl

ltlttu 2

,

1

)(0

�

(30)

The rest of the procedure in calculating the spread-

ing of a 2D droplet is exactly the same as part A. Equa-

tions (18-20) can be still used to calculate the coeffi-

cients of the velocity series by setting Ș=0. Mashayek et al. [11] studied the deformation and

deflection of a liquid jet in cross flow using a 2D ellip-

tic model which allowed for deformation of the jet

cross section from an initial circle to elongated ellipses.

They assumed that the jet velocity is constant along its

trajectory up to the breakup location and calculated the

trajectory using a force balance. To calculate the dis-

placement of the droplet, they balanced the forces in the

cross sectional plane using the local drag coefficient

and the frontal area of the cross sectional element with

unit thickness (similar to that shown in Fig. 9). The

local drag coefficient was obtained using interpolation

between some numerical-based correlations for drag

coefficients of elliptic cylinders at various Reynolds

numbers[12]. We shall apply a similar approach here

by replacing the elliptic model employed there by the

present model to predict the deformations of the jet

cross section. Since the displacement of the droplet is

already calculated in this model, no further force bal-

ance in the cross sectional plane is needed. This consid-

erably simplifies the process of mapping the 2D drop

deformation into a jet by skipping the approximate drag

coefficient interpolation since the drag force is already

calculated in the present model. Other details like cal-

culations of the deflection angle and the jet penetration

are very similar to Mashayek et al. [11] and are not

repeated here. One key point in this process is to project

the gas stream velocity on to the plane of the cross sec-

tion at each instant as depicted in Fig. 9. Neglecting this

step leads to unrealistically large local Weber numbers

which cause unwanted large deformations in the jet

cross section[11].

Results and Discussion (Part B)

To illustrate the substantial difference between the

spreading of axisymmetric and 2D droplets, Fig. 10

shows the deformation of a 2D droplet as well as an

axisymmetric droplet at equal flow conditions. The

Weber number is 35 and for the sake of comparison, it

is assumed that no mass is being stripped from the

droplets even though it is expected at this Weber num-

ber. The considerably larger deformations of the 2D

droplet with respect to the corresponding axisymmetric

case are evident in the figure. For the case of the axi-

symmetric droplet, the gas flow can travel around the

droplet in the radial direction while this cannot happen

for the 2D droplet due to its two dimensional nature.

Thus, the gas can only travel in the transverse direction

in the droplet plane which rapidly stretches the droplet

in the cross stream wise direction. As shown in the last

frames, the large shear force exerted at the tips of the

deformed droplet by the gas stream elongates the drop-

let in the direction of gas flow. This phenomenon is

clearly illustrated in Fig 10. It is known that the shear

around the droplet is higher and surface tension is lower

for 2D droplets compared to that of spherical droplets.

In 2D droplet, the curvature is only present at one plane

while for spherical droplet curvature is present at two

perpendicular planes and allows for the droplet to de-

form spatially rather than a two dimensional deforma-

tion.

Next, we consider the application of a 2D droplet

model in JICF problem. Figure 11 compares the defor-

mation and penetration of two liquid jets with Weber

number of We=10 and momentum ratios of q=9.54 and

q=1.94 with results of Madabhushi et al. [20]. The gen-

eral shape of the jets and the amount they penetrate into

the gas stream are in good agreement with the experi-

mental images especially for case (a) (q=9.54). It can be

seen that the liquid column becomes thinner in the side

view shortly after it exits the nozzle due to lateral

spreading of its cross section (2D drop). The side view

shows the bouncing back of the cross section as the jet

moves on which makes the jet look circular again.

However this bounce back seems to be more severe for

the calculated jet shape with respect to the experimental

image. This difference is even more apparent in case (b)

where the liquid jet has a smaller velocity and thus,

more oscillations of the jet cross section can be ob-

served within the limits of the image. The main reason

behind this difference between the present results and

the experiments might be the absence of the viscous

forces in the present model which leads to more defor-

mation in the cross section. Also, the cross section

would bounce back to the initial circular shape in ab-

sence of viscosity which is clearly not the case in the

experimental images. Focusing on the case (b)

(q=1.94), it is observed that the cross section does not

bounce back to the initial circular shape in the real jet

and, thus, is more flattened along its path comparing to

the calculated jet. The less flat and more circular shape

of the calculated jet leads to a smaller drag force which

reduces the aerodynamic force and makes it penetrate

more into the gas stream comparing to the experimental

figure. Although the calculated jet oscillations do not

completely match with the experimental figures, they

suggest the existence of a close relation between the jet

oscillation and the deformation pattern shown in the

experimental figures of Madabhushi et al. [20]. This

confirms the suggested similarities between the defor-

mation and breakup characteristics of liquid droplets

and jets in cross flows as reported previously by various

studies [4, 6].

To better demonstrate the retraction of the liquid

column upon deformation, Fig 12 shows the side and

35

o

rotated views of a deformed liquid jet at Weber

number of 15. Figure 12(c) shows the velocity on the

jet surface at several locations from the non-inertial

frame attached to the center of mass of its cross section.

As it can be seen from the velocity vectors, the jet flat-

tens after it exits the nozzle due to the aerodynamic

force. The surface tension forces the cross section to

bounce back in the absence of viscosity. It should be

noted again that were the viscous forces present, it

would not bounce back to the same circular jet as it

does in the figure. Figure 12(d) shows the velocity on

the jet surface at several locations from the inertial

frame.

At higher Weber numbers, the aerodynamic force

dominates the surface tension force and there will be no

bounce back in the liquid column as shown in Fig. 13

for a case with Weber of 35. To understand the reason

behind the curly shape observed on the edged of the

flattened jet in Figs. 13(a) and 13(b), Fig. (13c) shows

the deformation of jet cross section over time. The back

and forth movement of the tip of the cross section leads

to the wave like shape formed on jet surface. At this

level of deformation and with increase in the Weber

number, mass will start to strip off the liquid column

[4] especially since the stretched ligaments can be sepa-

rated and broken into smaller ligaments and child drop-

lets.

As the last case in the JICF section we investigate

the effect of changing the momentum ratio and Weber

number on the jets deformation and penetration. Fig.

14(a) and 14(b) show the side and 35

o

rotated view of 5

water jets in air flow with common jet velocity of 4.7

m/s and various air velocities. The Weber number

changes from zero on the first frame to 20 on the last

frame while the momentum ratio q, changes from infin-

ity on the first frame to 6 on the last one. It is clear that

as the momentum ratio decreases (from left to right),

the jet penetrates less into the cross stream as reported

by all the previous studies on the JICF problem [11,

21]. Again, it is evident that with increase in the Weber

number (from left to right), the deformations in the

cross section grow larger. Larger Weber numbers lead

to irreversible deformations as shown by Fig. (13).

Conclusion

The deformation of axisymmetric droplets was

studied employing the model proposed by Gonor and

Zoltova with some modifications in the general process

of solving for the series coefficients. Results were in

good agreement with experimental and numerical litera-

ture used for comparison.

The model was modified for 2D droplets and the

results showed considerable difference between the

behaviors of a 2D drop in the gas flow with that of an

axisymmetric flow at the same flow conditions.

The 2D droplet spreading was mapped into three

dimensions to simulate the deformation of the cross

section of a liquid jet in a gaseous cross flow. The re-

sults were in relatively good agreement with the ex-

perimental results while the absence of viscosity in the

present model was observable in the calculated jet de-

formations. The calculations suggest close similarities

between the deformation of a liquid column in the cross

flow and spreading of a 2D droplet as reported in vari-

ous available literature.

The more realistic prediction of the deformation of

the jet cross section with respect to models that allowed

for elliptic cross sectional deformations, takes into ac-

count many parameters that were missing in the theo-

retical models available for predicting the jet in cross

flow atomization.

The significance of the present model is its extreme

speed (less than 15 seconds for each case on a regular

PC) which makes it a proper option for parametric stud-

ies. Its superior speed over Eulerian simulations such as

volume of fluid and level set method, along with its

accurate results make it an excellent choice for model-

ing and industrial applications. One other key advan-

tage of this model is its independence of any tunable

parameter and works properly for various flow condi-

tions, within the scope of the model of course, without

any fine-tuning.

Nomenclature a series coefficient, spreading radius

A series coefficient

D diameter

k density ratio

p pressure

q momentum ratio

r radial coordinate, radius

t time

U velocity

z flow direction

We Weber number

Re Reynolds number

CD drag coefficient

Cp pressure coefficient

U density µ viscosity

ı surface tension

ș circumferential angle

Į jet bending angle

Subscripts

G gas L liquid

0 initial

� free stream

ij series coefficient subscript

Superscripts

‘ time derivative

References

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1998.

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K.A., Atomization and Sprays, 15(3):271-294

(2005).

9. Fuchs L., SIAMUF Seminar, Glumslöv, Sweden,

October 2006.

10. Clark, M.M., Chemical Engineering Science,

43(3):671–679, (1988).

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12. Mashayek, A., Jafari, A., and Ashgriz, N., submitted

to Phys. Fluids, (2007).

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17. Hirahara, H, and Kawahashi, M., Exp. Fluids

13:423-428 (1992).

18. Quan, S., and Schmidt, D.P., Phys. Fluids 18:

102103, (2006).

19. Pozrikidis, C., Fluid Dynamics, Theory, computa-

tion, and Numerical simulation, Kluwer (Springer),

2001.

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Brown, C.T., and McDonell, V.G., 19th Annual Con-ference on Liquid Atomization and Spray Systems,

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A.S., J. Propulsion and Power, 13(1):64-73, (1997).

Figure 1. Schematic of gas flow over a liquid droplet.

Figure 2. Pressure coefficient over the surface of a solid sphere located in 2D and 3D potential flows.

Figure 3. Pressure coefficient on a solid spheres at (a)Re=10,000, (b)Re=1,140,000, (c)Re=100,000 and on a circu-

lar cylinder at (d)Re=500, (e)Re=2,000, (f)Re=8,000. ((a)-(c) from calculations of Constantinescu and Squires [14]

and (d)-(f) from calculations of Mashayek et al. [12]).

Figure 4. Droplet shapes at different times from left to right: t=0, 23, 40, 72, and 78. Top row, experimental images

by Gelfand et al. [16]; bottom row, present calculations.

Figure 5. Shape evolution of spherical droplet (Re=40, We=40, and k=50) at non-dimensional times t= 0.0, 2.5, 5.0,

7.0, 10.0, 12.0, and 15.0.

Figure 6. Comparison of droplet deformations of Fig. (5) with the computational results of Quan and Schmidt [18]

Figure 7. The interface velocity for (Re=40, We=40, and k=50) case. Top row, velocities measured in an inertial

frame. Bottom row, velocities measured in the non-inertial frame.

Figure 8. Deformation of water droplet in air stream at We=40 at t=0, 5, 10, 15 (from left to right). (a)K=590; (b) K=83; (c) K=50; (d) K=29.

Figure 9. Projecting the gas velocity on the cross sectional plane in the application of the 2D drop in JICF.

Figure 10. Comparison between the deformation of an axisymmetric water drop (top row) with that of a 2D water

drop (bottom row) at air with We=35. Times from left to right: t=0, 10, 20, 30, 40, 50, 55, and 58.

(a)

(b)

Figure 11. Comparison between present calculations (right) and the experimental results reproduced from Mad-

habushi et al.[20] by permission(left); (a) q=9.54, Re=1007; (b) q=1.94, Re=454.

Figure 12. Deformation of a water jet in air flow at We=15. (a) Side view; (b) 35

o

view; (c) Velocity vectors in a

non-inertial frame; (d) Velocity vectors in an inertial frame.

(a) (b)

(c)

Figure 13. Deformation of a water liquid jet in air stream at We=35. (a) Side view; (b) 35

0

view; (c) Evolution of

the jet’s cross section from the nozzle to tip.

Figure 14. Deformation and penetration of water jets in air stream. Side view on the top row and 35

o

on the second

row. (a) We=0, q=�; (b) We=5, q=30; (c) We=10, q=15; (d) We=15, q=10; (e) We=20, q=7.5.