The Influence of Density Ratio on the Primary Atomization of a Turbulent LiquidJet in Crossflow

M. Herrmann

School of Mechanical, Aerospace, Chemical, and Materials EngineeringArizona State University, Tempe, AZ 85287-6106, USA.

Abstract

In this paper we study the impact of density ratio on turbulent liquid jet in crossflow penetration and atomization

if all other characteristic parameters, i.e. momentum flux ratio, jet and crossflow Weber and Reynolds numbers, are

maintained constant. We perform detailed simulations of the primary atomization region using the refined level set

grid method to track the motion of the liquid/gas phase interface. We employ a balanced force, interface projected

curvature method to ensure high accuracy of the surface tension forces, use a multi-scale approach to transfer broken

off, small scale nearly spherical drops into a Lagrangian point particle description allowing for full two-way coupling

and continued secondary atomization, and employ a dynamic Smagorinsky large eddy simulation approach in the

single phase regions of the flow to describe turbulence. We compare simulation results obtained previously using a

liquid to gas density ratio of 10 for a momentum flux ratio 6.6, Weber number 330, and Reynolds number 14000

liquid jet injected into a Reynolds number 740,000 gaseous crossflow to those at a density ratio of 100, a value typical

for gas turbine combustors. The results show that the increase in density ratio results in a noticeable increase in jet

penetration, change in drop size distribution resulting from primary atomization, change in drop velocities generated

by primary atomization in the crossflow and jet direction, but virtually no change in drop velocities in the transverse

direction.

Keywords: atomization, level set, jet in crossflow, jet penetration, drop sizes

1. Introduction

The atomization of turbulent liquid jets injected into fast moving, subsonic gaseous crossflows is an important

application for example in gas turbines, ramjets, and augmentors. It is a highly complex process, that has been

extensively studied experimentally over the past decades. Early studies of the atomization of non-turbulent liquid

jets in crossflows have recently been reviewed in [1], whereas newer studies of this case include the work reported in

Preprint submitted to Elsevier May 27, 2010

[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Most experimental work has focused on jet penetration, including both the column

trajectory and the resulting spray penetration under ambient, atmospheric conditions. Some recent jet penetration

correlations under these conditions can be found in [3, 10, 11, 12]. Under non-atmospheric conditions, jet penetration

correlations have been derived in [2, 13]. The identified characteristic parameters determining jet penetration from

these studies are the momentum flux ratio [12], momentum flux ratio and crossflow Weber number [11], momentum

flux ratio and water to injected liquid viscosity ratio [3], momentum flux ratio, crossflow Weber number, and water to

injected liquid viscosity ratio [10], momentum flux ratio, crossflow Weber and Reynolds numbers [2], and momentum

flux ratio, crossflow Weber number, and channel pressure [13]. Since gaseous density is proportional to channel

pressure, only the latter correlation contains the impact of density ratio on liquid jet penetration directly. However,

although momentum flux ratio and crossflow and jet Weber numbers were approximately held constant, at least one

of the Reynolds numbers could not have been maintained, thus potentially overlapping two separate effects. It should

be pointed out that some other studies exist that vary the channel pressure from its atmospheric value, however,

these studies consider modified configurations, for example airblast assisted atomization [14], or did not result in

modifications to the liquid jet core penetration correlations [15].

All the above studies at non-atmospheric conditions use as a starting point correlations obtained at atmospheric

conditions, since there is an abundance of data there. Density ratios found in applications like gas turbine combustors

are thus approached from above, i.e. atmospheric conditions result in too large density ratios and non-atmospheric

experiments analyze the effect of increasing the crossflow channel pressure and thereby reducing the density ratio in

steps to those values representative of gas turbine combustors [13].

Detailed numerical simulations of turbulent atomization processes, on the other hand, are more easily performed

at density ratios smaller than those found in gas turbine combustors due to often times encountered numerical

instabilities associated with the density discontinuity on highly deformed interfaces. Yet how representative are

these lower density ratio simulations of the actual application in gas turbine combustors? We aim to address this

question by comparing detailed simulation results of the primary atomization region at two different density ratios,

one lower, the other representative of gas turbine injectors, under the constraint that all characteristic numbers, i.e.

momentum flux ratio, crossflow and jet Weber numbers, and crossflow and jet Reynolds numbers are kept constant.

This effectively isolates the impact of density ratio, a feature not yet reproduced experimentally.

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Besides jet penetration, a change in density ratio, while maintaining all other characteristic numbers, might also

impact the primary atomization mechanism of the jet, and thus the resulting primary atomization drop size and

velocity distributions.

The outline of this paper is the following: after summarizing the governing equations, the numerical methods

employed to solve them are briefly outlined. Then, simulation results are presented and the impact of density ratio

on the atomization process is discussed.

2. Governing equations

The equations governing the motion of an unsteady, incompressible, immiscible, two-fluid system are the Navier-

Stokes equations,

∂u

∂t+ u · ∇u = −1

ρ∇p+ 1

ρ∇ ·

(µ(∇u +∇Tu

))+

1

ρT σ , (1)

where u is the velocity, ρ the density, p the pressure, µ the dynamic viscosity, and T σ the surface tension force

which is non-zero only at the location of the phase interface xf . Furthermore, the continuity equation results in a

divergence-free constraint on the velocity field, ∇ · u = 0. The phase interface location xf between the two fluids is

described by a level set scalar G, with G(xf , t) = 0 at the interface, G(x, t) > 0 in fluid 1, and G(x, t) < 0 in fluid

2. Differentiating this with respect to time yields,

∂G

∂t+ u · ∇G = 0 . (2)

Assuming material properties to be constant within each fluid, they can be calculated at any point x from

α(x) = H(G)α1 + (1−H(G))α2 (3)

where α is either density or viscosity, indices 1 and 2 denote values in fluid 1, respectively 2, and H is the Heaviside

function. The surface tension force T σ can be expressed as

T σ(x) = σκδ(x− xf )n = σκδ(G)|∇G|n , (4)

with σ the surface tension coefficient and interface normal n and curvature κ directly calculated from the level set

scalar.

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3. Numerical methods

To keep track of the position and motion of the phase interface, we solve all level set related equations using the

Refined Level Set Grid (RLSG) on a separate, equidistant Cartesian grid using a dual-narrow band methodology for

efficiency. This so-called G-grid is overlaid onto the flow solver grid on which the Navier-Stokes equations are solved

and can be independently refined, providing high resolution of the tracked phase interface geometry. Details of the

resulting method and extensive verification results are reported in [16].

The flow solver used to solve the incompressible two-phase Navier-Stokes equations on unstructured grids using

a finite volume balanced force algorithm is Cascade Technologies’ CDP. In the single phase regions, the employed

scheme conserves the kinetic energy discretely and turbulence is modeled using a dynamic Smagorinsky LES model.

However, none of the terms arising from filtering the phase interface, like subfilter surface tension or subfilter liquid

volume fraction transport, are modeled. The approach instead relies on resolving all relevant scales at the phase

interface, thus ideally reverting to a DNS at the phase interface. It is for this reason that the highest grid resolutions

in the simulations reported here are provided in the vicinity of the phase interface. As such, the current simulation

results are a combination of LES in the single phase regions and an under-resolved DNS on the two-dimensional

subset of the phase interface [17]. The flow solver CDP and the RLSG interface tracking software are coupled using

the parallel multi-code coupling library CHIMPS [16, 18].

Finally, resolving the entire phase interface geometry by tracking the phase interface associated with each at-

omized drop quickly becomes prohibitively expensive. Instead, we follow a multi-scale coupled Eulerian/Lagrangian

procedure in that we track the phase interface by the Eulerian level set method in the near injector primary atomiza-

tion region and transfer broken-off, nearly spherical liquid structures into a Lagrangian point particle description [19].

To this end, the level set solution field is searched at the end of every time step to identify all unique G > 0 volumes.

If the volume is smaller than a given threshold value, it’s eccentricity measure defined as the maximum distance

from the structure’s surface to its center of mass scaled by the radius of the equivalent volume sphere is calculated.

If the eccentricity measure is smaller than a given threshold value, the liquid structure is removed from the level set

representation, by simply switching the sign of the level set values within the structure, thereby resetting the volume

fraction and densities of the structure’s occupied volume to the gaseous values. A Lagrangian particle is added to

the flow solver at the structure’s center of mass with the volume averaged velocity, thereby preserving the removed

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structure’s mass and momentum. In the Lagrangian description, full two-way momentum coupling between the drop

and continuous phase is used [20], including a stochastic secondary atomization model [21]. However, the cell volume

occupied by the Lagrangian drops is not explicitly taken into account and neither drop/drop nor drop/tracked phase

interface collisions are modeled. But, as long as liquid structures have not been transferred into the Lagrangian

description, i.e. they are still tracked by the level set scalar, cell volume effects and all collisions are fully captured.

4. Computational Domain and Operating Conditions

The case analyzed in this paper is based on one studied experimentally by Brown & McDonell [4]. Table 1

summarizes the operating conditions and resulting characteristic numbers. Results obtained for the density ratio 10

case were reported previously in [17].

The computational domain (−25D . . . 50D × 0 . . . 25D × −10D . . . 10D) is smaller than the channel used in the

experiment (−77D . . . 127D × 0 . . . 54D ×−27D . . . 27D). However, simulations using the full experimental channel

geometry were conducted to verify that the reduced computational domain does not impact the reported results.

Furthermore, the details of the injector geometry are taken into account by performing at a constant momentum

flux ratio a turbulent single phase LES of the full geometry and storing the resulting velocities in a database for use

as the injector exit-plane inflow boundary condition in the atomization simulations [17].

We previously reported on a grid refinement study for the ρj/ρc = 10 case using three different grid resolutions

[17]. The obtained results indicate that jet penetration can be well predicted even on the coarsest grid, column and

surface breakup modes can be observed on all three grids, and that grid independent drops resulting from primary

atomization can be obtained for drops resolved by at least 2 grid nodes of the RLSG grid. Unless otherwise indicated,

all results in the following are obtained using the medium resolution grid c12 of [17] as he base grid. It resolves the

injector diameter by 32 control volumes using a static mesh refinement strategy in the injector region in the flow

solver and 64 grid nodes in the RLSG solver. The resulting mesh sizes are 21 million control volumes for the flow

solver and 13 million active RLSG grid nodes. At t = 0, the liquid jet is initialized in the computational domain by

a small cylindrical section of length D capped by a half-sphere, protruding into the crossflow channel. The threshold

volume used to transfer drops from the level set to the Lagrangian description is 0.14µl and the chosen threshold

eccentricity measure is 1.5.

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In the following, unless otherwise indicated, results are reported in dimensionless quantities, with uj and D being

the reference velocity and length. Thus, non-dimensional coordinates are the same in both density cases, whereas

non-dimensional times and velocities are different by a factor 3.16. After an initial startup phase, the atomizing jets

reach a statistical steady state of the level set tracked liquid mass at around t = 20 in the low density ratio case and

t = 10 in the high density ratio case. All results reported in the following are obtained only after this initial startup

phase.

5. Results and Discussion

Figure 1 shows instantaneous snapshots of the surface geometry of the two cases. Jet penetration appears different,

with the higher density case being less bent in the crossflow direction, and the liquid core not extending as far in

that direction.

To allow for a detailed quantitative analysis of the mean jet penetration, the following procedure is adopted to

extract such data from time snapshots of the level set solution field. Since the interest here is on time averaged data,

it should be pointed out that a simple time averaging of the level set scalar is doomed to fail and give inaccurate

results, since such an approach violates the inherent symmetries of the level set equation [22]. Instead, we base

our averaging tool on the spatial filters for level set scalars proposed in [23], applied to temporal filtering instead of

spatial filtering. The idea in this approach is to perform the averaging on a Heaviside transform of the level set scalar

instead of on the level set scalar itself. This, in essence, results in a scalar that describes the threshold probability

of finding liquid at a given location (the probability is x% or greater). Additionally, this scalar is consistent with

a mean liquid volume fraction. We define the 50% probability iso-surface to be the location of the mean surface,

although, in principle, any other value could be used.

Figure 2 presents the position of the liquid core’s mean center of mass and windward and lee-side mean interface

position in the injector’s center plane. The center of mass position was determined by an iterative procedure evalu-

ating the center of mass in planes normal to its trajectory. Also shown are power law fits, where y/D = 3.3(x/D)0.33

and y/D = 3.8(x/D)0.40 fit the center of mass trajectory in the low and high density ratio case, respectively. The

50% probability leading edge can be fitted by y/D = 3.18(x/D)0.49, respectively y/D = 3.4(x/D)0.45. The power

law fits obtained from experiments are y/D = 3.5(x/D)0.50 [12] and y/D = 3.6(x/D)0.39 [24].

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It should, be pointed out, that the pre-factor and exponent in the power-law fit are very sensitive to the chosen

value of the probability iso-line. Choosing values ranging from 10% up to 90% results in power law fit pre-factors

ranging from 2.74 to 3.50 and exponents from 0.39 to 0.52 in the low density ratio case, and 3.1 to 3.7 with exponents

from 0.36 to 0.48 in the high density ratio case. This significant spread in parameters highlights a potential weakness

in the way windward edge trajectories, and therefore, jet penetration is experimentally measured. The results indicate

that the power law parameters are a strong function of the threshold value used to identify the windward edge. A

more objective measure for jet penetration would be the jet’s center of mass, which is, however, hard to measure

experimentally. Nonetheless the obtained leading edge correlations are within the range determined experimentally,

thus providing some level of validation of the simulation data. To verify the numerical solution, Fig. ?? shows the

result of a grid dependency study using a factor 2 coarser grid in each direction compared to the base grid. Windward

and lee side mean interface position show practically no change between the two grids, thus verifying the presented

solutions.

Figure 3 shows the discrete number and volume frequency distributions of drops generated by primary atomization

in the two density ratio cases. Since there is no precise definition of the demarcation between primary and secondary

atomization, we define primary atomization as all topology change events resulting in liquid structures smaller than

a threshold volume and nearly spherical in shape as defined by the eccentricity measure previously stated, i.e. drops

triggering a transfer from the level set tracked representatioan to the Lagrangian description. The threshold volume

used in the simulations corresponds to a maximum spherical drop size of d = 1/2, thereby allowing for the transfer of

very large structures. The used eccentricity threshold however prevents the transfer of ligament or sheet structures

even if they are small in volume and broken off from the main liquid column. Instead, the drops resulting from the

breakup of these highly non-spherical structures are mostly spherical and thus are considered to be the result of

primary atomization since they typically trigger a transfer.

The discrete frequency distribution is obtained by binning the drop sizes using 20 bins of equal logarithmic drop

size between the smallest drop size resolved by at least 2 G-grid points and d = 1/4 which is larger than the largest

transferred drop throughout either simulation. The distribution is normalized by the total number of drops in this

size range, 24,745 drops in the low density ratio case and 22,233 drops in the high density ratio case.

It should be pointed out that it is practically impossible to obtain a fully grid independent number frequency

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distribution using fixed grid interface capturing/tracking schemes like the Volume-of-Fluid or the level set method.

This is due to the fact that in these methods, topology change events are automatically initiated if two opposing

front segments enter into the same computational grid cell. This automatic topology change event is always grid

dependent and the local mesh size acts as an inherent topology change parameter, unless a dedicated topology

change model based on micro-scale physics like Van der Waals forces would be employed [25]. Although the exact

moment of topology change might not be predicted accurately without such a micro-scale model, nevertheless the

error in volume of the broken-off liquid structure is at most of the order of the local mesh volume and thus small

compared to drops resolved by several grid points pre-transfer. Indeed grid independence of such larger drops has

been demonstrated in the past, see [17, 26] for examples. We thus consider only drops resolved by at least 2 G-grid

points in all of the presented results. Finally, to ensure that results of the two density ratios are comparable, we

necessarily have to use the same inherent topology change model in each case, i.e. the same grid resolution, as is the

case in the present study.

The discrete frequency distributions show that changing the density ratio does have a noticeable impact on

drop sizes resulting from primary atomization. Increasing the density ratio decreases the number of larger drops

significantly and yields a somewhat narrower distribution. This observation is consistent with the Sauter mean and

standard deviation calculated from the binned data giving d32 = 0.0762 and d32 = 0.0575 with σ = 8.21 · 10−3 and

σ = 7.02 · 10−3 in the low respective high density ratio case. Note that these values are calculated using the discrete

number frequency distribution in the given size range only, so only the trend should be considered, not the actual

quantitative values.

At first glance the observed trend that an increase in density ratio should result in a decrease in the mean drop size

is somewhat puzzling. One would expect that an increase in density ratio should reduce the impact of the gas phase

aerodynamic effect, thereby producing larger mean drops instead of smaller ones. This line of thinking however

neglects the fact that the simulations were carried out maintaining as constant all other characteristic numbers,

thereby isolating the effect of density ratio only. To keep for example the momentum flux ratio constant, we have to

decrease the jet velocity if we increase the jet density to increase the density ratio, see Tab. 1. This will necessarily

change the relative velocity between the jet and the crossflow in such a way that a rough estimate of the most

unstable wave length of a Kelvin-Helmholtz type instability would in fact indicate that the wave length decreases

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when increasing the density ratio from 10 to 100, thus resulting in potentially smaller drops if one were to assume a

Kelvin-Helmholtz type instability as the dominant instability mode leading to breakup of the liquid column [27].

Figure 4 shows the location at which drops are generated by primary atomization, i.e. the location of transfer

from the level set to the Lagrangian representation. In the high density ratio case, atomization occurs significantly

closer to the injector, and the region of generation is significantly less bend in the crossflow direction. This can

be explained by the different trajectories of the jets center of mass in the two density ratio cases. Theprimary

atomization is also more compact in the transverse direction in the higher density ratio case. Although primary

atomization seems to occur earlier, the transverse spread of the resulting spray is increased when the density ratio

increases as can be seen in Fig. 1 showing top view snapshots of the developing spray. This trend is consistent with

experimental observations showing an increase in spray area and plume width with increasing density ratios [2].

Finally, Figs. 5-7 show the discrete number frequency distribution of primary atomization drop velocities at

their moment of transfer from the level set to the Lagrangian representation. Data like this is of importance to

modeling efforts of the primary atomization process that inject drops along projected mean liquid core paths with

a given velocity [28]. The distributions have been generated by first binning drops into 6 size classes between

2∆xG < d < 0.25 of equal logarithmic size named d0 to d5. Within each size class, the velocity data in each direction

is binned into 20 equally sized bins between the minimum and maximum velocities in the respective directions. Total

number of drops used in each size class for the low density ratio case are 15,594 for d0, 645 for d3, and 139 for d4,

and for the high density case 14,225 for d0, 178 for d3, and 7 for d4.

Figure 5 shows that density ratio has a significant impact on the crossflow direction drop velocity. Here scaling is

performed with the gaseous crossflow velocity as the relevant velocity scale for this direction. In each case, velocities

for large drops converge to a uniform velocity, consistent with experimental observations [12]. The value however is

significantly different in the two cases. In the low density ratio case, the asymptotic velocity is close to 1, whereas in

the high density ratio case the value is roughly 0.25. A similar trend in drop velocity has been reported by Elshamy

[13], although measured significantly further downstream of the injector at x/D = 40. The observed difference is

easily explained by the lower relative Stokes number of the liquid in the low density ratio case.

Small drops, on the other hand exhibit a broad range of crossflow velocities, with a considerable number moving

upstream at the moment of primary atomization. This is due to the fact that the liquid jet generates significant

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recirculation regions in its wake and small drops, due to their low Stokes number will follow the vortices there more

easily resulting in potential upstream movement. This behavior is of significance, since residence times of drops

caught in the recirculation region of the wake will be increased and combined with the fact that these drops tend to

be small, they might provide a gaseous fuel source near the injector in combustion applications due to evaporation.

Furthermore, the variance of the crossflow velocity distribution is noticeably larger in the low density ratio case,

explainable again by the difference in Stokes numbers.

Figure 6 shows on the top the jet direction velocities scaled with the liquid jet velocity as the seemingly relevant

velocity scale in this direction. Density ratio appears to again have a significant impact with the high density

ratio case showing a larger mean value, again due to the relative larger Stokes number, but also a much broader

distribution. This might be due to the fact that the high density ratio case better retains the initial range of velocity

fluctuations from the turbulent flow of the injector. Interestingly, however, if one scales the jet direction velocity

with the cross flow velocity as shown in the bottom of Fig. 6, the two distributions nearly collapse. This indicates a

scaling with the square root of the density ratio, since the momentum flux ratio is maintained constant. Furthermore,

the distributions show a considerable number of drops moving in the opposite direction of the injected jet at their

moment of primary atomization, again explainable by the turbulent wake region behind the injected jet.

Finally Fig. 7 shows the transverse direction velocities scaled with the crossflow velocity as the relevant velocity

scale. As expected, the distributions are symmetric, and appear not to be predominantly a function of density

ratio. Larger drops seem to exhibit a more uniform velocity, however this trend is much less pronounced than in the

crossflow direction.

6. Conclusions

Detailed simulation results of the primary atomization of a turbulent liquid jet injected into a subsonic gaseous

crossflow have been presented, studying the impact of density ratio on the primary atomization region. Results were

obtained using the same numerical topology change model given by the same local grid spacing in both cases. The

simulation results indicate that the density ratio has a noticeable effect on primary atomization, even if all other

relevant characteristic numbers are being kept constant. Mean jet penetration, location of primary atomization, drop

sizes, crossflow and jet direction drop velocities at the moment of primary atomization are impacted by the change

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in density ratio. The simulation results, however, show that transverse direction drop velocities are not significantly

changed.

The obtained results can form a starting point for improving modeling attempts of jet in crossflow atomization

based on the idea of injecting drops along projected mean liquid core paths with given velocity distributions [28].

Acknowledgments

This work was supported in part by CASCADE Technologies Inc. under NavAir SBIR N07-046. The author would

like to thank M. Arienti and M. Soteriou from United Technologies Research Center for many helpful discussions

and participation in the NavAir SBIR as well as S. Hajiloo and F. Ham from Cascade Technologies Inc.

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sim. 1 sim. 2

density ratio R = ρj/ρc 10 100

jet exit diameter D [mm] 1.3 1.3

crossflow density ρc [kg/m3] 1.225 1.225

jet density ρj [kg/m3] 12.25 122.25

crossflow velocity uc [m/s] 120.4 120.4

jet velocity uj [m/s] 97.84 30.94

crossflow viscosity µc [kg/ms] 1.82 e-5 1.82e-5

jet viscosity µj [kg/ms] 1.11e-4 3.5e-4

surface tension coeff. σ [N/m] 0.07 0.07

momentum flux ratio q 6.6 6.6

crossflow Weber number Wec 330 330

jet Weber number Wej 2178 2178

crossflow Reynolds number Rec 5.7e5 5.7e5

jet Reynolds number Rej 14079 14079

Table 1: Operating conditions and characteristic numbers.

14

Figure 1: Side (top) and top (bottom) view of jet in crossflow atomization at t=30, ρj/ρc = 10 (left) and 100 (right).

y

0

1

2

3

4

-1 0 1 2 3

y

0

1

2

3

4

-1 0 1 2 3

y

0

1

2

3

4

-1 0 1 2 3

y

x0

1

2

3

4

-1 0 1 2 3x

Figure 2: Top: center of mass, windward and lee side 50% probability isoline (lines) and power law fits (symbols) in the injector midplane.

Bottom: Impact of grid resolution: base grid (lines), coarse grid (symbols). ρj/ρc = 10 (left) and 100 (right).

N/Nsum

10-5

0.0001

0.001

0.01

0.1

1

0.03 0.05 0.1 0.2 0.3d

10-4

D3 N/Nsum

10-7

10-6

10-5

0.03 0.05 0.1 0.2 0.3d

Figure 3: Discrete number (left) and volume (right) frequency distributions of primary atomization drops, ρj/ρc = 10 (circles, black)

and 100 (squares, gray).

15

Figure 4: Location of drop generation side view (center), top view (right), ρj/ρc = 10 (circles, black) and 100 (squares, gray).

0

0.1

0.2

0.3

0.4

0.5

-1 -0.5 0 0.5 1 1.5up /uc up/uc

0

0.1

0.2

0.3

0.4

0.5

-1 -0.5 0 0.5 1 1.5

N/Nsum

Figure 5: Primary atomization crossflow direction drop velocities for drop size classes d0 (left) and d4 (right) for ρj/ρc = 10 (black) and

100 (gray).

N/Nsum

N/Nsum

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5vp /ujvp /uj

vp /ucvp /uc

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

-0.5 0 0.5 1

Figure 6: Primary atomization jet direction drop velocities scaled by jet (top) and crossflow velocity (bottom) for drop size classes d0

(left) and d3 (right) for ρj/ρc = 10 (black) and 100 (gray).

16

N/Nsum

0

0.05

0.1

0.15

0.2

0.25

0.3

-1 -0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

-1 -0.5 0 0.5 1wp/ucwp/uc

Figure 7: Primary atomization transverse direction drop velocities for drop size classes d0 (left) and d3 (right) for ρj/ρc = 10 (black)

and 100 (gray).

17

List of Tables

1 Operating conditions and characteristic numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

18

List of Figures

1 Side (top) and top (bottom) view of jet in crossflow atomization at t=30, ρj/ρc = 10 (left) and 100

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Top: center of mass, windward and lee side 50% probability isoline (lines) and power law fits (symbols)

in the injector midplane. Bottom: Impact of grid resolution: base grid (lines), coarse grid (symbols).

ρj/ρc = 10 (left) and 100 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Discrete number (left) and volume (right) frequency distributions of primary atomization drops,

ρj/ρc = 10 (circles, black) and 100 (squares, gray). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Location of drop generation side view (center), top view (right), ρj/ρc = 10 (circles, black) and 100

(squares, gray). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Primary atomization crossflow direction drop velocities for drop size classes d0 (left) and d4 (right)

for ρj/ρc = 10 (black) and 100 (gray). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Primary atomization jet direction drop velocities scaled by jet (top) and crossflow velocity (bottom)

for drop size classes d0 (left) and d3 (right) for ρj/ρc = 10 (black) and 100 (gray). . . . . . . . . . . . 16

7 Primary atomization transverse direction drop velocities for drop size classes d0 (left) and d3 (right)

for ρj/ρc = 10 (black) and 100 (gray). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

19