ILASS Americas 28th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2016

High Fidelity Simulation of the Impact of Density Ratio on Liquid Jet in Crossflow Atomi-

zation

Xiaoyi Li*, and Marios C. Soteriou

United Technologies Research Center

East Hartford, CT 06108, USA

Abstract

Atomization of liquid fuel jets by cross-flowing air is critical to the performance of many aerospace combustors.

Recent advances in numerical methods and increases in computational power have enabled the first principle, high

fidelity simulation of this phenomenon. In the recent past we demonstrated for the first time such simulations that

were comprehensively validated against experimental data obtained at ambient conditions. At combustor operating

conditions, however, both temperature and pressure are significantly elevated. In this work we perform a computa-

tional study of the impact of reduced liquid-gas density ratio due to increased air density associated with operating

pressure elevation on the atomization physics. A previously validated ambient condition case is used as the baseline

for comparison with three cases with decreasing density ratios. The density ratio is independently varied by adjust-

ing the gas density and velocity together so that the momentum flux ratio and Weber number are maintained con-

stant. Results indicate a significant modification of the atomization process at lower density ratios. Although the

global-scale jet penetration and trajectory are not significantly modified by the conditions, both the process of liquid

breakup and the degree of atomization are altered. The trends in the degree of atomization represented by the liquid

volume to area ratio extracted from in the simulation results agree with the observations from a recent experiment at

elevated pressure conditions. Further effort is still required to understand the detailed physical mechanisms for at-

omization at different density ratios.

*Corresponding author: lixy2@utrc.utc.com

ILASS Americas 28th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2016

Introduction

Liquid Jet atomization In a Crossflow (LJIC) using

aerodynamic forcing is a critical process occurring in

the liquid fuel injection step during the operation of

aircraft engine combustors. The increasingly strong

requirements and regulations on improving the aero-

engine combustor efficiency and reducing pollutant

emissions have driven the increases in the combustor

inlet pressure and temperature for an enhanced liquid

fuel evaporation rate and fuel-air mixing. Since the

fuel/air properties are highly sensitive to the operating

pressure and temperature, e.g. the air density being

strongly dependent on the pressure, the sensitivity of

the atomization process to the operating conditions is

strong. Thus understanding and optimizing LJIC in

such elevated conditions has become an important sub-

ject in the liquid atomization research. In the current

study we focus on the dependence of atomization pro-

cess on the liquid-gas density ratio altered by pressure

conditions.

Traditional liquid atomization research has relied

on experimental approaches that were mostly con-

strained to ambient pressure conditions due to the com-

plexity and high cost of experiments at elevated condi-

tions. Global features such as liquid jet trajectory and

penetration and far-field spray distribution were meas-

ured and reported using a variety of empirical correla-

tions [1-6]. A number of detailed experiments focusing

on near-field atomization details [7-11] shed some light

towards understanding the fundamental multiphase

breakup mechanisms, despite the fact that whether we

could extrapolate these understandings to assess LJIC at

elevated pressure condition is still questionable.

Results from only a few high pressure experiments

of LJIC were reported in the literature [12-15]. Becker

and Hassa [12] studied the breakup of kerosene jet in

crossflow with pressure up to 15 bar. They explored the

impact of pressure on liquid atomization regime, jet

penetration and lateral dispersion, and droplet size dis-

tribution. However, the effects of elevated pressure

were lumped into the effects of increased air momen-

tum flux or Weber number as a result of an increase in

air density. In fact, the density ratio and the Weber

number are two controlling parameters that can be in-

dependently varied, with the effects of the former being

rarely studied and the impact of the latter being rela-

tively well understood from the ambient condition

work. The observed changes in high pressure atomiza-

tion in terms of a reduction in jet penetration and re-

duced sensitivity of droplet sizes to Weber number var-

iations [12] can be explained by the shift into a higher

Weber numbers shear breakup regime. And such physi-

cal link has been established during the ambient condi-

tion investigations. In Bellofiore et al. [14], a large

number of flow conditions at 10 bar and 20 bar were

tested, allowing the extraction of the density ratio ef-

fects independent of the Weber number. The impact of

pressure on spray trajectory, plume width and coverage

area was reported, yet the large degree of data scatter-

ing causes difficulty in extracting the detailed impact of

density ratio, as pointed out by Herrmann et al. [16]. In

a recent LJIC experiment by Song et al. [15], the air

pressure was elevated from 2.07 to 9.65 bar, and the

impact of density ratio was independently investigated

by comparing data at fixed momentum flux ratio and

Weber number. Jet breakup regime and mean droplet

size downstream were shown to have a strong depend-

ency on the density ratio while the dependence of pene-

tration/trajectory on density ratio seemed to be weak.

At very high pressure, the reduced liquid-gas den-

sity ratio may become comparable to the density ratio

existing in many Gaseous Jets In Crossflow (GJIC)

applications. The physics of GJIC has been extensively

studied [17-20] in terms of a complex set of interacting

vortex system. While the knowledge developed from

GJIC studies may be borrowed for understanding the

large scale vortical flow structures in LJIC at high pres-

sure, the multiphase breakup phenomena unique to

LJIC have to be understood by accounting for the phase

separation caused by the presence of surface tension.

Due to the complex multiphase multiscale physics

involved in the liquid atomization process, traditional

modelling approaches have encountered severe difficul-

ties in capturing the impact of operating conditions. The

applicability of the phenomenological models [21, 22]

calibrated at ambient condition is questionable when

the operating conditions are elevated. High fidelity

simulation of liquid atomization [23-28] has emerged to

provide a very promising path for detailed investiga-

tions of the impact of operating conditions without reli-

ance on the experimental calibration due to its first-

principle nature. Yet because of the challenges of over-

coming the numerical instabilities that typically occur

when the liquid/gas density contrast is high, a number

of high-fidelity simulations of LJIC have been conduct-

ed at reduced density ratios only [25, 29], which hap-

pened to reflect the scenarios at elevated pressure con-

ditions. Numerical study of the impact of density ratio

on LJIC atomization was initiated by Herrmann and co-

workers [16] considering two density ratios rρ=10 and

rρ=100, both lower than the typical liquid-gas density

ratios at ambient condition. Reducing density ratio was

found to cause the decrease of liquid core penetration

together with an increased bending and transverse

spreading. It also increases the column wavelength and

the mean droplet size [30]. The decrease in density ratio

also leads to an increase in the normalized crossflow

droplet velocity and a decrease in the normalized trans-

verse droplet velocity, due to the increase in the relative

Stokes number controlled by size. However, in addition

to the limit on the density ratio set by the numerical

instability challenges, in Herrmann’s work [16, 30] , the

variation of density ratio was configured by altering the

liquid density, not exactly the same as an air density

change, which is the dominant response of pressure

change. To which degree such configurations represent

the high pressure condition needs further verification.

Recently, simulations of high density-ratio LJIC at

ambient condition have been performed by our team

and successfully validated against near-field experiment

[8] in terms of detailed column features and droplet

formation [28, 31]. The solver with enhanced interface

tracking capabilities allows stable simulations at density

ratios covering a broader range of pressure conditions,

thus enables a comprehensive study of the impact of

density ratio changes associated with operating pressure

variations. In this work, we present the results from

three LJIC simulation cases with increasing air density

reflecting the trends with pressure increases. For the

current investigation, all other impacts of pressure are

ignored. The cases were set up based upon a previously

validated ambient condition case. The density ratio was

independently varied while the momentum flux ratio

and the Weber number were fixed to be constants. Pre-

vious ambient condition simulation case was used as

the baseline for investigating the impact of decreasing

density ratio.

In the following, previously adopted formulation

and numerical methods are briefly highlighted. The

computational configurations of LJIC with varying den-

sity ratios are described. The impact of density ratio on

the qualitative feature and quantitative degree of atomi-

zation is presented. Finally, summary and conclusions

are provided.

Computational Approach

A. Formulation and Numerical Methods

It is assumed that the fluid properties for each

phase are spatially invariant at the specified operating

conditions and the two-phase flow of liquid and gas is

incompressible and can be represented by a single fluid

formulation. Under these assumptions the governing

mass and momentum conservation equations are:

0 u . (1)

Hpt

)2(

1DIuu

u, (2)

where p is the fluid dynamic pressure, the density,

the viscosity, I the identity tensor, D the deviatoric

strain rate tensor, the constant interfacial tension, the local curvature and H the Heaviside function de-

fined as

(gas)00

(liquid)01)(H . (3)

Here is a function that identifies the interface

location. The density and viscosity are correspondingly

defined as

HH GL 1

HH GL 1 . (4)

The motion of interface follows

0

u

t. (5)

Since the numerical methods adopted in this paper

have been comprehensively described in our previous

work [26, 28, 32], only a brief highlight is provided

here for the completeness of the paper. Our computa-

tional approach uses the Coupled Level-Set and Vol-

ume Of Fluid (CLSVOF) method [33] to capture the

liquid-gas interface. The method capitalizes on the ad-

vantages of both the accurate geometric interface repre-

sentation in level set method and the volume-preserving

properties in volume of fluid method. The Eulerian

CLSVOF interface tracking method is implemented

under the framework of a block structured adaptive

mesh refinement (AMR) [33, 34], and also coupled

with a Lagrangian droplet transformation and tracking

approach to capture the smallest spherical droplets with

significant cost-saving benefits, especially when the jet

column and dense spray region occupy only a small

part of the domain. The Eulerian to Lagrangian trans-

formation follows the algorithms similar to the imple-

mentation by Herrmann [35] and a number of criteria

(e.g. size, sphericity and diluteness criteria) are required

to be met before the transformation occurs [26, 36]. The

flow solver features a two-fluid advection approach [33,

37] to avoid artificial smearing of velocity field across

the interface, which causes poorly resolved gas velocity

gradient leading to solver divergence at high density

ratios. The pressure projection equation is solved using

a Multi-Grid Preconditioned Conjugate Gradient meth-

od (MGPCG). The method is augmented by a ghost

fluid (GF) treatment for pressure jump conditions to

achieve stable and fast pressure solution. Such a suite of

sharp interface treatments mitigate the problem of solv-

er divergence that typically occurs at high density rati-

os.

B. Computational Setup

In our previous work [28], the computational ap-

proach was validated against near-field experimental

measurements [8] for a non-turbulent water jet in steady

crossflow of air at the ambient condition. The inlet flow

turbulence was experimentally suppressed to focus the

study on liquid atomization due to aerodynamic forces.

And correspondingly plug-flow profiles were set for

both the liquid and gas inlets in the simulations. In this

work, we inherit the same computational configuration

and use one of the validated ambient cases as the base-

line for investigating the impact of density ratio.

The two-phase flow and breakup are controlled by

the competition between surface tension and aerody-

namic flow forces at the liquid-gas interface, and can be

characterized by a density ratio rρ=ρl/ρg, a momentum

flux ratio q=ρlUl2/ ρgUg

2 and gas Weber number

We=ρgUg2d0/σ. The other two independent non-

dimensional parameters are the liquid Reynolds number

Rel= ρlUld0/µl, and viscosity ratio rμ= μl/µg. In this

study, the system is maintained at ambient temperature

and the liquid evaporation is not considered. We focus

on the effects of density ratio and fix other fluid proper-

ties. The fixed fluid properties and flow parameters are

listed in Table 1. The density ratio is varied by adjust-

ing the air density. Here we make a low Mach number

assumption, and the impact of fluid dynamic pressure

on the change of air density is assumed small and ne-

glected. The air density affects multiple non-

dimensional numbers and here we fix the momentum

flux ratio at q=88.2, and Weber number at We=160, the

same as in one of the ambient condition cases [28]. We

select the case at this Weber number as the baseline due

to the predominantly high Weber number condition in

aircraft engine applications. The parameters allowed to

vary are listed in Table 2. Case 1 is the baseline case.

As the air density increases from 1.2 to 590.0 kg/m3,

the density ratio decreases from 845.0 to 1.7. To main-

tain a constant gaseous momentum flux, the gas inlet

velocity is decreased from 109.5 to 4.9 m/s. The gase-

ous Reynolds numbers are relatively large and the ef-

fects of viscosity are assumed to be secondary for all

the cases. Since the gas inlet turbulence is excluded in

the simulations, the smallest relevant flows scales are

generated at the liquid-gas interface by multiphase flow

instabilities.

In the simulations, the coordinate system has the x-

axis in the crossflow direction and the z-axis in the di-

rection of liquid injection. The computational domain is

a box of 3.0 cm × 2.0 cm × 3.0 cm. The jet orifice is

located at a coordinate of (0.2, 1.0, 0.0) with a diameter

of d0=0.8 mm. Impermeable no-slip boundary condi-

tions are imposed at the z = 0 plane, except at the jet

orifice where a liquid velocity inlet condition is im-

posed. Gas velocity inflow is imposed at the inlet

boundary located at x = 0 cm. Outflow boundary condi-

tions are imposed on the remaining boundary planes.

Three levels of AMR are used in the simulations to

refine the grid near the liquid-gas interface. The use of

AMR greatly improves the affordability of the simula-

tions. The finest grid size is set to be ∆x = 39 µm.

Smaller-scale events such as liquid pinch-off do occur

in reality, however, we postulate here that the smaller-

scale physics has little impact on the larger-scale flow.

Previous validations [28] have shown that when the

grid resolution is smaller than the ligament or droplet

size observed in the experiment [8], the simulation can

resolve physics down to the experimentally measured

scales and the under-resolved flow and pinch-off phys-

ics do not have a significant impact on the atomization

features of interest at the measurement scales. Since it

has been reported that decreasing density-ratio at higher

pressure tends to increase the mean droplet size [15,

30], the grid size as required by the baseline high densi-

ty ratio case is deemed to be sufficient for capturing the

atomization processes in other lower density ratio cases.

The time stepping for all the simulations is defined

by two criteria: CFL criterion and surface tension crite-

rion [33]

3 2

,

1 1min ,

2 2 8

l

ni j

xt x

u. (7)

For all the cases, the jet reaches full penetration within

1.2 non-dimensional flow-through time τflow= max(Lx/

Ug, Lz/Ul). Data are collected over another flow-through

time afterwards to provide reasonable flow and inter-

face statistics.

Computational Results

A. Qualitative atomization features

In Fig. 1, the qualitative LJIC atomization features

at different density ratios are illustrated using liquid

surface images rendered in three orthogonal views. Re-

sults from the previously validated ambient condition

case [28] are shown in the first column of images as the

baseline for comparison. As the jets penetrate into the

crossflow, they bend towards the direction of the cross-

flow stream. The degree of bending in the initial stage

before column fracture points does not seem to be very

sensitive to the change of density ratio (Fig. 1(a)-(d))

due to the same momentum flux ratios imposed for all

the cases. It is consistent with a similar degree of

blockage across all the cases as inferred by the similar

degree of column flattening in the transverse direction

(Fig. 1(e)-(h)).

It has been shown in the ambient condition LJIC

study that the gaseous Weber number controls the mul-

tiphase instability/breakup, and the breakup process can

be categorized by several breakup regimes such as bag,

multi-mode and shear breakup. Although the Weber

number is held constant for all the cases in this work,

significant changes in the liquid breakup details as a

result of density ratio variations are observed. As the

density ratio decreases, the characteristics of the insta-

bilities developed on the column surface is altered sig-

nificantly. The amplitude of column waves increases

with decreasing density ratio and the onset location for

column breakup is shifted towards the injection point at

lower density ratios (Fig. 1(e)-(h)). The column waves

are also observed to change their characteristics from

appearing only on the windward surface at high density

ratio (Fig. 1(a)) [28] to being present on the whole cir-

cumference of the liquid column at low density ratio

(Fig. 1(d)). This can also be observed in the comparison

of jet column shape for different conditions in Fig. 2.

As the density ratio decreases, the circumferential in-

stability becomes stronger and disturbances start at a

height closer to the injection orifice. The surface strip-

ping of droplets at the transverse edge of the column

surface as observed in the high density ratio case does

not seem to occur for the very low density ratio scenar-

io (Fig. 1(d)(h) and Fig. 2(m)-(p)). Although the devel-

opment of column waves is delayed in the higher densi-

ty ratio cases, the liquid secondary breakup proceeds at

a faster rate so that the size of droplets formed after

column fracture points is small (Fig. 1(a)-(d), (i)-(l)).

Based on side-view experimental shadowgraph im-

ages, Song et al. [15] observed an increase in surface

wavelength and wave amplitude as the pressure was

increased from 2.07 bar to 9.65 bar while the momen-

tum flux ratio and Weber number were kept constant at

q=10 and We=500. The simulations by Herrmann et al.

[16] also showed an increase in surface wavelength

with the decrease in density ratio. Such observed trends

qualitatively agree with the simulation results shown in

Fig. 1(a) to (c). Based on such observations, Song et al.

[15] also suggest that the critical Weber number for

transitioning the breakup regimes is shifted to higher

values at higher pressure (or lower density ratio) condi-

tions, e.g. at the same Weber number, a high density

ratio jet may experience shear breakup while a low den-

sity ratio jet may experience multi-mode or bag

breakup. The simulation results shown in Fig. 1, how-

ever, suggest that more complex instability transition-

ing may occur as the density ratio is varied, e.g. the

onset location of surface breakup approaches the injec-

tion orifice as the density ratio decreases.

B. Spray plume boundary and degree of atomiza-

tion

The boundaries for the spray plume were quantita-

tively extracted from the simulation data and plotted in

Fig. 3. The boundaries were defined as the minimum

and maximum locations of liquid surfaces (including

both Eulerian surface and Lagrangian droplets represen-

tation) for each x-bin. The bin size was set to be 0.2

mm. Too small bin size leads to large oscillations of

data due to the limited number of samples while too

large bin size fails to capture the detailed boundary evo-

lution. The data extracted over 20 snapshots for each

case are averaged and plotted in Fig. 3.

Although the detailed breakup process changes

with conditions, both the z and y plume boundaries

shown in Fig. 3(a) and (b) display very little sensitivity

to the variation of density ratio. While the simulation by

Herrmann et al. [16] reported a noticeable increase in

the near-field core penetration with increasing density

ratio, the experimentally observed spray trajectories

identified by the maximum Mie-scattering intensity did

not show significant dependence on the density ratio

[15] and momentum flux ratio has been confirmed to be

the most dominant factor in determining spray penetra-

tion. The data in Fig. 3 quantitatively confirm that the

changes in breakup processes at different density ratios

mainly cause local differences in liquid structures that

may alter the spray boundaries in a minor way (Fig.

3(a) and (b)), and the overall spray penetration and

spread are largely dictated by the global momentum

balance independent of the density ratio changes in the

current study.

A common way to quantify the degree by which

density ratio influences atomization is to measure or

extract the size of droplets after jet primary breakup. As

in our previous simulation work, the droplet data be-

come readily available after the Lagrangian droplet

transformation is introduced as shown in Fig. 4(a). The

transformation approach using pre-defined size and

sphericity criteria works well for the ambient condition

case. The liquid is atomized to such a large degree that

all the droplets can meet the criteria and be transformed

into the Lagrangian phase before they leave the simula-

tion domain. However, for the low density ratio case

shown in Fig. 4(c) and (d), the atomization characteris-

tics are different, and large and highly deformed liga-

ments/blobs may survive longer and persist beyond the

simulation domain. As shown in Fig. 4(a)-(d), using the

same transformation criteria, an increasing proportion

of liquid remain in the Eulerian phase as the density

ratio decreases. This presents a difficulty in accurately

extracting the droplet size distribution only based on the

Lagrangian phase data.

To characterize the averaged degree of atomization

in the above complex low density ratio scenario, we

compute an averaged liquid volume to area ratio, which

represents the effective size of the liquid structures. For

the Eulerian phase, the surface area and volume are

computed by numerical surface integration i

iS and

i

iS 3ii nx (based on divergence theorem). The

calculation of the area and volume of the Lagrangian

droplets is straightforward. The total volume and area

for both the Eulerian and the Lagrangian phases are

j

ji

il dSV 63 3ii nx (8)

j

ji

il dSA 2 (9)

where i sums over all the Eulerian surface elements and

j sums over all the Lagrangian droplets. The total vol-

ume, area and averaged volume to area ratio for all the

liquid in the domain are compared in Fig. 5(a)-(c) for

different density ratio conditions. The data represent

values averaged over 20 snapshots for each case. The

computed volume for all the liquid in the domain in

Fig. 5(a) shows a monotonic increase decreasing densi-

ty ratio. The increasing accumulation of liquid in the

domain (see Fig. 1) is due to the decreases in gas veloc-

ity to keep the gas momentum flux constant (see Table

2). The liquid surface area also shows a monotonic in-

crease with decreasing density ratio in Fig. 5(b) and a

monotonic trend in the total volume to area ratio cannot

be identified in Fig. 5(c). Although the degree of atomi-

zation seems to be higher for the high density ratio case

in terms of the size of the liquid structures downstream

after column fracture point (see Fig. 1 and 4), the more

intensive breakup of liquid column surface close to the

injection point in the low density ratio case also con-

tributes to an increase in the local volume to area ratio.

As a result, the domain-averaged degree of atomization

is comparable between cases with different density rati-

os.

The spatial variations of the atomization degree are

further investigated by computing the liquid volume

and surface area for four equal-size bins at different x

locations. The variations of bin liquid volume, area and

volume to area ratio along the crossflow x-direction are

compared in Fig. 5(d)-(f) for different conditions. As

shown in Fig. 5(d), the bin volume for the high density

ratio case decreases along the x-direction and finally

reaches a saturation value. As the LJIC process reaches

steady state, the liquid flow rate through each x plane

reaches a constant value equal to the liquid injection

flow rate. The decreases in the bin liquid volume in the

crossflow direction can be explained by an increase in

the averaged liquid x-velocity caused by the accelera-

tion due to the crossflow. As the density ratio decreas-

es, the crossflow velocity decreases and the bin liquid

volume increases since the liquid flow rate is the same

for all the conditions. It is interesting to observe a non-

monotonic change of bin liquid volume for the lower

density ratio rρ=16.9 case, which first decreases then

increases. A more significant increase in bin liquid vol-

ume is observed for the rρ=1.7 case. The cause of such

bin volume increase requires further investigation. One

possible explanation is that a larger proportion of liquid

is trapped in the low speed wake zone in the low densi-

ty ratio cases than in the high density ratio cases.

In Fig. 5(e), the bin liquid surface area shows a

monotonic decrease in the x-direction for all the condi-

tions. The degree of atomization represented by the bin

liquid volume to area ratio shows some interesting

trends in Fig. 5(f). Near the injection orifice, the effec-

tive size of liquid structures is larger in the higher den-

sity ratio case since the early breakup on the column

surface is weak for the high density ratio case and be-

comes progressively stronger as the density ratio de-

creases (see Fig. 1 and 4). The reverse trend is observed

further downstream. In the high density ratio case, the

high shear between liquid and gas due to the imposed

high gas velocity drives the acceleration and the sec-

ondary breakup of liquid ligaments/blobs into smaller

droplets. In the low density ratio case, even though the

early stage breakup near the injection orifice is strong,

the shear between liquid and gas is getting lower due to

a lower gas velocity imposed. The variation of velocity

magnitude on the Eulerian liquid surface at different

conditions is shown in Fig. 6. Compared to the liquid

injection velocity, which is held the same for all the

cases, the surface velocity progressively increases

downstream in the higher density ratio cases (Fig.

6(a)(b), but progressively decreases in the lower density

ratio cases (Fig. 6(c)(d)). The transition between accel-

eration and deceleration occurs when the imposed liq-

uid velocity being equal to the imposed gas velocity,

i.e. Ug=Ul or q=rρ, which happens in a condition be-

tween case 2 and 3 (see Table 2). Because of the pro-

gressively reduced shear in the lower density ratio cas-

es, the surface tension may act to drive a recovery of

elongated ligaments or deformed blobs into large spher-

ical droplets without further breakup. This probably

explains the increases in effective size or decreases in

the degree of atomization in Fig. 5(f) for the lower den-

sity ratio case. Note that an increase of droplet size with

decreasing density ratio was also reported in other sim-

ulation [30] and experiment work [15], although no

clear physical explanation of the phenomena has been

provided.

Summary and Conclusions

A computational investigation of the impact of

density ratio variation associated with pressure change

on the liquid jet atomization in a crossflow has been

performed. The effects were independently studied by

fixing the momentum flux ratio and Weber number to

the values specified in a previously validated high den-

sity ratio simulation case at the ambient condition. Cas-

es with three density ratios were simulated and com-

pared with the baseline ambient condition case. The

density ratio manifests its impact mostly in altering the

breakup and atomization characteristics and does not

show significant influences on the large scale spray

penetration and spreading. A new approach was devel-

oped to generically characterize the degree of atomiza-

tion when large and highly deformed liquid liga-

ments/blobs are present. The quantitative results point

to an increase in droplet size or a decrease in the degree

of atomization with decreasing density ratio, which

qualitatively agrees with conclusions from the simula-

tions by Herrmann [30] and the recent experimental

observation by Song et al. [15]. The regions with the

most intensive liquid breakup transition from column

fracture point in the ambient high density ratio case to

the point close to liquid injection in the low density

ration case. More investigation is necessary to under-

stand this transition and the change in physical mecha-

nisms of atomization due to density ratio change and

this will be the subject of future work.

Nomenclature

A = Area

d0 = injector orifice diameter

D = deviatoric strain rate tensor

H = Heaviside function

I = identity tensor

L = length scale

p = fluid dynamic pressure

q = momentum flux ratio

rμ = viscosity ratiorρ = density ratio

Re = Reynolds number

t = time

u = velocity

U = imposed velocity

V = volume

We = Weber number

x = coordinate

x, = coordinate in direction of crossflow

y, = coordinate orthogonal to x and z

z, = coordinate in direction of liquid injection

t = time step

x = grid spacing

Φ = phase indicator function

κ = curvature

= dynamic viscosity

= density

σ = surface tension

τ = Non-dimensional time

subscripts

g = gas property

l = liquid property

min = minimum

superscripts

n = step n

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ρl μl μg σ Ul q We Rel

997 0.000894 0.0000186 0.0708 35.4 88.2 160 31582.8

Table 1. Fixed fluid properties and flow parameters (SI unit).

Table 2. Fluid properties and flow parameters varying with operating pressure (SI unit).

Figure 1. Instantaneous snapshots of liquid atomization at different conditions in different views. Density ratios for

images from left to right are 845, 169, 16.9 and 1.7, respectively.

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Cases ρg Ug rρ Reg

1 1.2 109.5 845.0 5557.4

2 5.9 49.0 169.0 12424.3

3 59.0 15.5 16.9 39295.3

4 590.0 4.9 1.7 124242.6

Figure 2. Comparison of jet column shape in several x-y plane cross-sections at different conditions. For images

from left to right, z = 0.0005, 0.001, 0.0015, and 0.002 m. For images from top to bottom, rρ = 845, 169, 16.9 and

1.7.

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 3. Comparison of spray plume boundaries at different conditions.

(a) (b)

Figure 4. Illustration of Eulerian to Lagrangian transformation at different conditions. (a) rρ = 845, (b rρ = 169, (c) rρ

= 16.9, and (d) rρ = 1.7.

(a) (b)

(c) (d)

Figure 5. Comparison of liquid volume, area and volume-to-area ratio at different conditions.

845 169 16.9 1.7 rρ

(a) (b) (c)

(d) (e) (f)

845 169 16.9 1.7 rρ 845 169 16.9 1.7 rρ

Figure 6. Velocity magnitude contour on the Eulerian liquid surface at different conditions. (a) rρ = 845, (b) rρ =

169, (c) rρ = 16.9, and (d) rρ = 1.7.

(a) (b)

(c) (d)