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: [email protected]
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)