PULSATING SLURRY ATOMIZATION, FILM THICKNESS, AND AZIMUTHAL INSTABILITIES

Abstract

A detailed numerical study on a transonic self-sustaining pulsatile three-stream coaxial airblast injector provided new insight on turbulent pulsations that affected atomization. Unique to this investigation, slurry viscosity, slurry annular thickness, and how the annular thickness interacts with inner nozzle retraction (prefilming distance) were found to be paramount to atomizer performance. Narrower annular slurry passageways yielded a thinner slurry sheet and increased injector throughput, but the resulting droplets were unexpectedly larger. As anticipated, a lower slurry viscosity resulted in smaller droplets. Both the incremental impact of viscosity and the computed slurry droplet length scale matched open literature values. The use of a partial azimuthal model produced a circumferentially periodic outer sheath of pulsing spray ligaments, whereas modeling the full domain showed a highly randomized and broken outer band of ligaments. However, quantitatively the results between the two azimuthal constructs were similar, especially farther from the injector; therefore, it was proved that modeling a wedge with periodic circumferential boundaries can be used for screening exercises. Additionally, velocity point correlations revealed that an inertial subrange was difficult to find in any of the model permutations and that droplet length scales correlated with radial velocities. Lastly, droplet size and turbulence scale predictions for two literature cases were presented for the first time using computational fluid dynamics.

Full text

Pulsating Slurry Atomization, Film Thickness, and Azimuthal Instabilities
W. Strasser*2 and F. Battaglia§
*Eastman Chemical Company, Kingsport, TN, 37660, strasser@eastman.com
§Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY 14260
Abstract
A detailed numerical study on a transonic self-sustaining pulsatile three-stream coaxial
airblast injector provided new insight on turbulent pulsations that affected atomization. Unique
to this investigation, slurry viscosity, slurry annular thickness, and how the annular thickness
interacts with inner nozzle retraction (prefilming distance) were found to be paramount to
atomizer performance. Narrower annular slurry passageways yielded a thinner slurry sheet and
increased injector throughput, but the resulting droplets were unexpectedly larger. As
anticipated, a lower slurry viscosity resulted in smaller droplets. Both the incremental impact of
viscosity and the computed slurry droplet length scale matched open literature values. The use
of a partial azimuthal model produced a circumferentially periodic outer sheath of pulsing spray
ligaments, whereas, modeling the full domain showed a highly randomized and broken outer
band of ligaments. However, quantitatively the results between the two azimuthal constructs
were similar, especially farther from the injector; therefore, it was proved that modeling a wedge
with periodic circumferential boundaries can be used for screening exercises. Additionally,
velocity point correlations revealed that an inertial subrange was difficult to find in any of the
model permutations and that droplet length scales correlated with radial velocities. Lastly,
droplet size and turbulence scale predictions for two literature cases were presented for the first
time using CFD.
2 Corresponding author: strasser@eastman.com
1

Keywords: Compressible Flow, Prefilming, VOF, Acoustics
2

Nomenclature
a Speed of sound
COV Coefficient of variation
D32 Sauter mean droplet length scale
DINozzle innermost diameter
DMNozzle intermediate diameter
DONozzle outermost diameter
E Total energy
F Surface tension body force
IG Inner gas stream
k Turbulent kinetic energy
La Laplace number = Re2/We
LAG Outer annular gap
LCCharacteristic length scale
LRI Inner retraction length
LSShoulder length, or spray lift-off
M Gas/liquid momentum ratio = (U2)G/(U2)L
Ma Mach number
OG Outer gas stream
Oh Ohnesorge number =
√
We
/Re = 1/
√
La
P Pressure
Pr Liquid phase Prandtl number
Q Viscous capillary length
Re Reynolds number = UD/
S Velocity ratio, UG/UL
St Strouhal number
SMD Sauter mean diameter (“D32”)
tLI Thickness of inner lip
tLO Thickness of outer lip
tSLAN Thickness of slurry annulus
T Static temperature
u Velocity component
U Velocity magnitude
y+uty/
We Weber number = U2LC/
Wo Womersley number =
√
2πReSt
Z Density ratio, ρG/ρL
Greek
Phase volume fraction
3

Turbulence dissipation rate
 Surface tension
Density
Specific dissipation rate
Stress tensor
γ Outer gas/liquid annular approach angle
μ Molecular viscosity
ζ Molecular thermal conductivity
Subscripts and Superscripts
i Summation index
L Liquid
G Gas
t Turbulent
ref Reference condition
I Inner gas
O Outer gas
4

1. Introduction and Objective
Atomization processes within the aerospace, agricultural, chemical, food, fire protection, and
energy-production industries, have been studied for nearly two centuries (Lefebvre 1988,
Beheshti and McIntosh 2007, Liu et al. 2015, Pathania et al. 2016, Fu et al. 2017). Specifically,
controlling liquid droplet sizing and trajectories can be paramount for high yields, prolonged life
(reliability and maintenance), and productivity. The initial generation of liquid parcels or
“droplets” from a continuous liquid stream, which are often initially non-spherical (Kourmatzis
and Masri 2015), is referred to as “primary atomization”. These droplets form when there is an
imbalance between local cohesive forces within the droplet and destructive forces within and
around the droplet. “Secondary atomization” implies existing droplets breaking up into smaller
droplets, which are much more likely to be spherical as curvature increases and surface tension
forces become more significant. In most spray nozzle configurations these processes occur, to
some extent, simultaneously.
Airblast atomization, which occurs when a high-velocity gas stream is used to facilitate
liquid breakup, and the resulting liquid surface instabilities have been extensively reviewed by
Faeth et al. (1995), Engelbert et al. (1995), Lasheras and Hopfinger (2000), and Dumouchel
(2008), while a thorough practical guide to injectors and sprays is given in Lefebvre (1988).
Relationships between factors such as air velocity, liquid density, surface tension, phase
momentum ratio, and liquid sheet thickness on droplet size have been established for many types
of atomizers. Kihm and Chigier (1991), for example, showed a mixed or weak droplet size effect
with changes in initial liquid sheet thickness. Another recent study (Shao et al. 2017) found that
both the maximum negative axial velocity and highest vorticity occurs where sheet break up
begins, but it is not clear whether the vorticity is a cause or effect. In the regions trailing the
droplets, they found that the wakes induced isotropic turbulence modulation and a clear inertial
5

subrange. On the other hand, the gas phase structures were anisotropic and more characteristic
of near-wall turbulence near the continuous liquid sheet. The reader is left considering the
interesting possibility of whether or not the droplet wake turbulence feeds back onto the sheets to
further enhance disintegration.
Despite staggering volumes of literature on primary and secondary atomization, however,
there is nothing known that quantifies intricate details of primary atomization within a bulk-
pulsating three-stream injector. Generally speaking, computational studies of high Mach number
primary atomization are also scarce. The established foundations for the present effort are
Strasser (2011), Strasser and Battaglia (Strasser and Battaglia 2016a, b, 2017b, a). These
experimental and computational studies dealt with ascertaining acoustic signatures and primary
atomization droplet size distributions within a three-stream self-sustaining industrial-scale
pulsating injector shown in Fig. 1. The three constant feed streams are designated as inner gas
(IG), outer gas (OG) and an intermediate liquid (slurry) stream, all generally flowing from the
top to the bottom. Interfacial instabilities created by the interactions between these streams when
they meet naturally produce bulk pulsations and spray bursting in a highly dynamic manner.
Our prior work was built upon assessing the effects of numerical assumptions, various
geometric configurations, and feed stream flow rates on acoustics and atomization properties.
The motivation of this new effort, however, is to assess for the first time, the effects of slurry
annular dimension (film thickness), azimuthal modeling angle, slurry viscosity, and mesh
resolution in conjunction with gas feed rate and viscosity. Additionally, a new numerical solver
evaluation will be executed, along with probing modeled domain length and exploring
turbulence length/time scales, axial velocity reversal, and how both interact with atomization.
Moreover, slurry backflow, as a surrogate measure of injector exit erosion, and two independent
injection systems will be evaluated in light of this new information. Substantially more
6

information on relevant external literature and how it applies to the present work will be included
throughout this document.
Fig. 1. Geometry for three-stream injector (from previous AAS paper??)
2. Computational Modeling
2.1 Scope
The injector inlets shown in Fig. 1 are relatively long and tortuous and extend well above
what is depicted. The naturally developing pulsations are strong enough to produce large
fluctuations in mass flow rate and vibrate the feed piping systems. “Flushed” refers to a value of
LRI = 0 and implies that there is no “prefilming” region. Thirteen newly devised simulations,
eleven of which are outlined in Table 1, will be discussed in this work. The eleven cases
involved the three-stream injector, while the other two simulations modeled two-stream
experimental studies (Mansour and Chigier 1995, Zhao et al. 2012). Geometry ("Geo."), inner
gas flow rate (“IG Flow”), mesh resolution, and azimuthal dimension (“In. An.”) will be
considered. It is worth clarifying that, although gas feed rate and retraction have been
7

considered before, the combined effects of these with azimuthal angle and slurry film thickness
are distinctively addressed here. Horizontal lines separate the six geometries considered. The
capital letters under the heading “Case” refer to either groups of geometry or inner gas flow rate.
Six geometric configurations will be evaluated as shown in Table 2 (see also Fig. 1). When
scaled by the outermost diameter (DO), outer gas annular gap (LAG), inner nozzle diameter (DI),
intermediate nozzle diameter (DM), and outer lip thickness (tLO) remain 0.097, 0.36, 0.83, and
0.014, respectively, for all simulations. The outer gas angle  and DO are purposely undisclosed
due to proprietary restrictions. Dimensionless inner nozzle retraction (LRI) ranges include 0.0 to
0.6 as shown in Table 2. The retraction could not be varied independently from the inner lip
thickness, tLI, because of the nozzle configuration and the geometric constraints of the other
process equipment. Note that higher retractions had larger inner lip thicknesses.
Dimensionless slurry annulus tSLAN ranges from 0.053 to 0.24, as shown in Table 2. Also
tabulated is the percent of the original slurry annular area retained in the new geometry. A value
of 100% implies that the design is the same as the base case. Two slurry annular dimension
changes are tested, one in which 63% of the original area is retained (37% reduction) and one in
which 31% of the original area is retained. Smaller slurry annular dimensions are expected to
thin the initial sheet meeting the inner gas core. However, the farther upstream the sheet is from
the nozzle outlet (where the slurry and inner gas meet the outer gas), the more likely it is to
recover its thickness as it radially expands. In other words, it could be postulated that the impact
of slurry annular dimension should be more exaggerated at lower retractions because less pre-
filming is available for sheet expansion.
Table 1: Simulation Matrix
Case Geo. IG Flow Mesh In. An. Purpose
A24 1 Low Base 360 360°
8

A25 1 Low Base 11.25 Solver
A26 1 Low Base 360 High Viscosity
A27 1 Low 8X 360 Mesh Resolution
C18 1High Base 360 Base
C19 1High Exp. 11.25 Expanded Domain
P 12 Low Base 11.25 Slurry Annulus
Q 13 Low Base 11.25 Slurry Annulus
R 14 Low Base 11.25 Slurry Annulus
T 16 Low Base 11.25 Slurry Annulus
U 17 Low Base 11.25 Slurry Annulus
Table 2: Geometries Simulated (LRI, tLI, tSLAN normalized by DO)
Geo. Cases LRI tLI tSLAN % Area Ret.
1A,C
0.6
0
0.038 0.238 100
12 P
0.6
0
0.15 0.129 63
13 Q
0.0
0
0.12 0.113 63
14 R
0.0
0
0.17 0.0600 31
16 T
0.3
0
0.12 0.113 63
17 U
0.3
0
0.18 0.0529 31
9

2.2 Governing Equations
A transient Eulerian framework was prescribed where computation cells are smaller than the
liquid region length scales. The continuity equation governing the mass balance of each phase
was
   
0
tu
 
� ��
�
(1)
where α is the slurry volume fraction,

is the density and u is the velocity vector. The phase-
averaged Reynolds-averaged linear momentum balance was
     
 
t ref
p
tu u u g F
     
�      �� � �� �
�
(2)
where τ is the stress tensor, p is pressure, g is gravity and F is surface tension body force. The gas
and slurry shared a common momentum field, and properties were phase-averaged. Similarly,
the phase-averaged energy equation was
     
Pr
t
t
t
E E p T
tu u

    
� �
� �
�     �� �� � �
� � � �
� �
� �
�� �
� �
(3)
where E is total energy,  is the molecular thermal conductivity, µ is the molecular viscosity, T is
the temperature, Pr is the Prandtl number, and the subscript t denote turbulence quantities. With
this method, film formation, ligament production, and droplet onset, as well as turbulence are
considered. The gradient diffusion hypothesis was used to separate the molecular and turbulent
effects in Eqs. 2 and 3. The gas density was assumed to vary as that of an ideal gas, and slurry
compressibility is ignored. The local Mach number could be as high as 0.6 at any given time. In
Eq. 3, kinetic energy, viscous heating, and pressure-work terms were included. Slurry droplet
evaporation due to gas humidity effects was ignored. Surface energy effects were treated via the
continuum surface force method of Brackbill et al. (1992). The pressure jump considers the
normal component only. Lastly, we ignore Marangoni effects caused by surface tension
gradients due to a concentration or temperature gradient.
10

The homogeneous shear stress transport (SST) two-equation linear eddy viscosity model
of Menter (1994) was used for computing the turbulent contributions to momentum and energy
transport for nearly all of the cases presented herein. A homogenous approach was considered in
that only one turbulence field was computed for both phases. In the SST model, additional
consideration is given to the transport of the principal turbulent shear stress via 1) an eddy
viscosity limiting function and 2) a cross diffusion term in the transport equation for specific
dissipation rate, ω. Also, there is a turbulence production limiter, preventing the artificial build-
up of fluctuating velocity in regions of irrotational strain. "Scalable" wall functions are an
alternative to standard wall functions and have the advantage of being less sensitive to variation
in near-wall grid resolution throughout the domain.
Stabilizing effects of compressibility on turbulence is explored by Pirozzoli and Grasso
(2004) and Gatski and Bonnet (2013) who found a reduction in the growth of shear layers due to
compressibility. The typical preferential alignment between the vorticity vector and the velocity
vector is lost with compressibility; however, the ratio of “pancake” to “cigar” flow structure
shapes is preserved. In both types of flows, enstrophy evolves in two stages, growing due to
vortex stretching up until a time scale on the order of an eddy turnover time and then decreasing.
The turnover time scale is inversely proportional to turbulent Mach number, so enstrophy
production is effectively decreased in compressible flow. Three mechanisms control dissipation
growth within incompressible flow: self-interaction (+), vortex stretching (-), and viscous action
(-). For compressible flow, shocklets significantly contribute to the increase in dissipation.
Similar findings were presented by Yoshizawa et al. (1997), who constructed a three-equation
eddy viscosity approach which takes compressibility into account. Due to other formidable
complexities embodied within this work, a compact method described in ANSYS (2013) and in
11

Strasser and Battaglia (2016a) is used to include additional compressibility-turbulence
interactions herein. Particularly, more relaxed eddy communication and damped instabilities
result. Limitations of the SST are well known, some of which are documented in the particle-
turbulence coupling study of Strasser (2008) and the cyclone studies of Strasser (2009) and
Dhakal et al. (2014) . SST improvements were proposed in Dhakal and Walters (2011) based on
simplification of a Reynolds stress model. Nonetheless, past validation work (Strasser 2011,
Strasser and Battaglia 2016a) provides sufficient evidence that the SST is valuable for this study.
Equations (1-3) were solved in ANSYS Fluent using the segregated double precision commercial
cell-centered solver 13.0 with Service Pack 2 (13SP2) for all cases except A25, A26, and A27
(Fluent version 14.5.0).
2.3 Numerics
The volume-of-fluid (VOF) method is a subset of the Eulerian-Eulerian mixture method and is
one of the various options for seeking the definition of the gas-slurry interface (Menard et al.
2007). Assuming that the interface is not always aligned with computational element faces, the
issue of differentiating between the slurry and gas phase at a length scale less than the cell
dimension (sub-grid scale, or “SGS”) is paramount for this work. The explicit piecewise linear
interface calculation (PLIC) scheme by Youngs (1982) was employed for all cases and assumed
that the interface takes on the shape of a line in 2-D and a plane in 3-D. An unsplit flux
methodology with no smoothing and a proprietary gradient method other than least-squares was
incorporated. Much more detail on VOF and the associated risks and evaluations are presented
in the previously mentioned foundational works.
12

Pressure-velocity coupling was coordinated via the Pressure Implicit with the Splitting of
Operators (PISO) scheme with both skewness and neighbor corrections. A Green-Gauss node-
based gradient method was used for discretizing derivatives and is more rigorous than a simple
arithmetical grid cell center average. The pressure field was treated with a body-force weighted
approach to assist with body force numerics for all simulations. Second-order upwinding was
used for advection terms, and first-order upwinding was used for turbulence quantities, which
were dominated by source terms. The transient term was also discretized using first-order
upwinding, but this was an ANSYS limitation. It can be noted, however, that with very small
timesteps, this should not pose any numerical problems; each time step represented only about
1/10,000th of a normal spray pulsing event. A typical time step of 5.0 ×10-7 seconds with 10
inner loops was required for the residuals to stop changing significantly with additional inner
loop steps. The results were found to be independent of timestep size in Strasser and Battaglia
(2016a) . Typical globally averaged Courant number (CN) based on fluid velocity remained
below 0.5 throughout the simulations. A typical total run time was about 2 to 3 weeks on eight
(8) Intel Xeon E5-2643 3.3GHz Intel cores for the base mesh 11.25° 3-D models to achieve
sufficient time-averages, which necessitated about 100 convective time (CT) scales. A CT was
defined as the length of time it takes for the fastest droplets to travel through the modeled
domain beyond the injector exit, which surveyed approximately 80 injector pulses. Naturally,
360° models would require 32 times the “2 to 3 weeks” mentioned for the 11.25° models, and
then 256 times that for the 8X 360° mesh to achieve the same CT.
13

2.4 Mesh and Boundary Conditions
The mesh was designed using multi-block concepts from Strasser et al. (2004) and contained
about 32,000 elements in a cross-section. The slurry sheet was resolved with approximately 30
cells. To produce a 3-D mesh, a half-cross-section face was swept some distance in the
azimuthal direction, producing a wedge shape, and periodic boundary conditions were applied.
The azimuthal included angles considered in this study are 11.25° and 360°. It was found in a
prior study (Strasser and Battaglia 2017b) that included angle did not affect critical results
significantly for angles up to 45°, but nothing was attempted above 45°. Turbulence intensity
was set to 5% of the inlet area-averaged velocity, while the integral length scale was set to 20%
of the feed nozzle length scales. These were not critical since the turbulence field develops
throughout the long inlets based on pressure gradient, boundary layer development, etc. The fed
turbulence field was completely transformed by the tortuous inlet passages before the flows
reached the prefilming zones. Typical area-averaged y+ values on all exposed inner walls were
near 10, and those values near the water annular gap were 3 times that.
Table 3 outlines important dimensionless quantities for the “high” and “low” IG flows,
including the gas-to-liquid density ratio (Z), the viscous capillary length (Q), the Reynolds
number (Re), the Ohnesorge number (Oh) and the Strouhal number (St). The Womersley
number (Wo), is >> 1 for all phases, indicating that the three feed passage velocity profiles did
not have time to develop between bulk pulses. Also note that the Weber number (We) is the
average, but the instantaneous values of the Weber number can be very different from the
average and can cause the liquid to span multiple breakup regimes. For example, Kourmatzis
and Masri (2015) stated that if the turbulence intensity is greater than 21%, the Weber number
for their work spanned 4 regimes. The liquid phase was a non-Newtonian slurry, while the inner
14

and outer gas streams were fed at an undisclosed pressure many multiples of ambient pressure.
The slurry viscosity responded strongly to local shear rate and temperature (using a proprietary
recipe), and that was programmed via a “user-defined function” (UDF). A constant high-shear
value for viscosity (Tsai et al. 1991, Aliseda et al. 2008) was used for the dimensionless groups.
The walls of the injector were hot, typically > 500°C. The temperature was a function of
position and was also implemented using a UDF. All fluids left the domain at the bottom, where
pressure was specified. Other non-wall model boundaries were treated as “openings”, through
which flow could move into or out of the modeled domain.
Table 3: Dimensionless parameters: Z = 0.082 for both IG feed rates, and Re = 180, Q = 0.053, Oh = 1.5, St = 5.4,
We = 6.8x104, and Wo = 78, all on a slurry basis. The IG Wo = 3.1x103 and OG Wo = 1.7x103 for all flow rates.
Inner Gas
Outer Gas
IG
M
S
Re/106
St
Ma
M
S
Re/106
St
Ma
Low
0.79
3.1
3.5
0.43
0.11
13
13
7.7
0.058
0.43
High 2.8 5.8 6.6 0.23 0.20 11 12 7.1 0.063 0.40
3. Results
3.1 Metrics
Multiple measures were considered throughout these evaluations, including pressure, droplet
size and spray angle. Though most of these measures are the same measures as those in our prior
work, they are used here again to reveal unique findings regarding slurry film thickness,
azimuthal modeling angle, modeled domain length, and slurry viscosity. The first metric is
pressure drop through the inner and outer gas feed passages, loosely referred to as “pressure” and
is normalized by the injector discharge pressure. Due to the intense gas stream pulsations that
affect not only the pre-filming section but also the inner feed passages, their transient signals
have been processed. They are averaged in time (“mean”), and then the standard deviation is
computed. The standard deviation divided by the mean is the coefficient of variance (“COV”).
15

Also, Fast Fourier Transform (FFT) analyses were performed for each signal, and two measures
reported include the dominant frequency (“tone”) and the FFT magnitude (“mag.”), which
signify how focused the spectrum is at the dominant frequency. The statistical Sauter mean
diameter (“D32”) of the primary liquid parcels was computed on 10 axial volume cuts using a
UDF discussed and validated in Strasser (2011) . The time-average (mean) and temporal
variability (COV) of the D32 at each axial measurement location were consistently important
measures, as they gave insights into both the droplet size and the consistency of that droplet size
versus time. Point-wise velocity sampling was carried out at 2 points in order to correlate
velocities with important variables. Lastly, the flux of slurry backflow towards the injector exit
through a plane 0.12 DO downstream of the injector was computed by a UDF. Backflow is
important in that it is a potential indicator for injector exit erosion by the slurry and has not been
previously investigated for these injectors.
3.2 Mesh-Independent Droplet Size Analysis
Various validation efforts have been discussed in prior works work (Strasser 2011, Strasser
and Battaglia 2016a); however, with updated solution methods and conditions, it was desired to
seek a more accurate droplet size in comparison to a recently discovered external correlation.
For case A27 (360° mesh at higher slurry viscosity), all computational mesh element length
scales (MLS) were halved in all three dimensions, producing a grid containing 8 times the
element count. Before halving the MLS, the typical ratio of droplet size to MLS was 5. Prior
work (Strasser and Battaglia 2017b) showed that mesh-independent results were obtained
between 2X and 8X of the base mesh, so it was known that 8X was more than sufficient. The
simulation time was a few months on 32 cores just to get stable time-averaged D32 information
after reaching a new quasi-steady behavior. It was only run half the time (order of 25 CT times)
16

as its counterpart, A26, which was run ~50 CT. The relative comparison of the droplet size
statistics is shown in Fig. 2. The ratios of the coarser mesh value to the more refined mesh value
for the D32 mean and COV at each spatial sampling location are presented.
Fig. 2. Effects of mesh resolution on D32 statistics (mean and COV) versus distance from injector; ratio of base
mesh values (denominator) to refined mesh values
The ratio of the temporal mean varies from nearly unity (data approaches ~1.1 at the exit)
near the injector and then levels out to about 2.0 in the far field. In other words, the mesh-
independent result ranges from about the same near the injector exit to about half of what has
been reported at the far-field. It is expected that, if the base mesh is not mesh-independent, its
resulting droplet sizes will be different due either to the interfacial reconstruction method
limitations and/or any changes in hydrodynamics created by improved shear layer resolution.
Various numerical issues are at play to create two scenarios to demonstrate the effect of a too-
coarse mesh on droplet size. The droplets can either be artificially large or small, depending on
the curvature of the droplets. For droplets that have low curvature and are mostly aligned with
the computational cell faces, such as the elongated slurry structures in the present primary
atomization work and noted in Kourmatzis and Masri (2015) , too-large cells simply don’t have
the spatial resolution to detect the sub-grid scale (SGS) interface. More resolution should allow
the quantification of smaller droplets. For highly curved droplets, such as spheres sitting in
17

relatively large mesh cells, premature breakup can occur. Liovic and Lakehal (2007) discussed
this as “flotsam and jetsam”, and it is sometimes referred to as “false surface tension”. The
planar interfacial reconstructions are too discontinuous from cell face to cell face. In order to
conserve mass (volume fraction), a false flatting of the interface occurs, as to be expected with
increased surface tension. In this situation, more cells should produce larger droplets.
Ling et al. (2017) demonstrated that mesh resolution effects are quite difficult to quantify due
to unpredictable non-linear offsetting effects. Mesh coarseness caused premature crest breakage
into droplets, premature hole formation, and the inability to resolve small droplets produced in
regions of high liquid curvature. The first two effects create numerical droplets smaller than they
should be, while the last one tends to only identify larger droplets. While the droplet size
distribution broadens and increases in convexity with increased grid resolution, the mean droplet
size might not be greatly impacted, and the droplets at either extreme often comprise a relatively
small mass fraction.
A mixed response for the effect of mesh resolution is not uncommon. Bagué et al. (2010)
and Wang et al. (2010) described how grid coarseness and increased surface tension reduced
amplitudes of growth eigenfunctions. Growth time-scale is linear in viscosity but depends on the
square root of surface tension. As a result, the effects are not quantitatively the same. Likewise,
Chesnel et al. (2011) showed that grid coarseness produced a delayed breakup. Boeck et al.
(2007), on the other hand, found that a coarser mesh led to an earlier breakup, i.e., grid
coarseness strengthened the instability and prevented ligaments from thinning adequately before
breakup.
Given that the current situation has more elongated droplets than spheres, consistent with
high Oh systems that tend to form elongated threads instead of distinct droplets as shown in
Hsiang and Faeth (1992), the former scenario should be dominant as observed in Fig. 2. The
coarser mesh with effectively half the droplet resolution capability produced droplets about twice
18

the size of the more resolved mesh. The ratio is not exactly 2.0, so there are other forces at play.
The relative COV values indicate that the coarser mesh creates less droplet size temporal
variability in the mid-field and more in the far-field. Some of the COV difference may be due to
the relatively low run time of the higher resolution case.
The final droplet size was 2.76 mm. Using the techniques presented in Liu et al. (2006) for
an experimental study of a similar three-stream injector and a non-Newtonian slurry, and
modified for the exact conditions and geometry of the present work, the predicted droplet size is
2.80 mm.
3.3 Typical 3-D Flow Features (Case C18)
An instantaneous snapshot from case C18 is shown in Fig. 3. The 3-D structure represents a
surface which bounds the slurry volume below the spray hardware (not shown) and is colored by
velocity. Blue is for a Mach number of 0, while red is for Mach equal to or above 0.3. The local
Mach number can be transonic in areas where there is significant gas acceleration by slurry area
blockage. The inner and outer beginnings of the slurry surface are anchored between the inner
lip of the OG feed annulus and the outer lip of the IG feed tube, respectively. Between the two
gas streams is the pre-filming zone where bulk disruptions in the slurry annular region are
initiated.
Compared to prior work showing the wedge of 11.25° for 3-D surfaces (Strasser and
Battaglia 2017a), the structure resulting from the 360° mesh is very different (Fig. 3). The
typical “branch” pattern that is analogous to a conifer tree is not so obvious. Instead of a clearly
defined azimuthally intact liquid annular sheet close to the injector, the sheet distorts annularly
with sinuous instabilities and even higher modes occurring very early, rendering the newly
discovered effects of modeling the full azimuthal domain astonishing. The ligament production
is much less regular, and the spray pulses remain closer to the injector axis.
19

Fig. 3. Instantaneous iso-surface of slurry volume fraction colored by velocity; red signifies velocities at or above a
Mach number of 0.3.
Fig. 4. Instantaneous contours of slurry volume fraction (blue is for slurry) on planar cross-cuts at even increments
of 0.24DO along the atomization trajectory
Fig. 4 shows instantaneous slurry volume fraction contours on evenly spaced sampling
planes moving away from the injector exit. Blue represents slurry, while red is for gas. Each
20

plane is placed at 0.24 DO increments along the atomization trajectory (general flow direction
along the nozzle axis), and each plane shows how the instantaneous spray axially progresses.
The distortion and breakup of the outer rim of the slurry annulus is already evident at 0.24DO.
By 0.96DO, the slurry sheet no longer looks like an annulus. By 1.92DO, primarily elongated
streamwise slurry droplets are present as the central spray region has fully ruptured.
For the first time in our research effort, instantaneous turbulent kinetic energy (k) contours on
a plane through the central axis are shown in Fig. 5 (blue = 0.0 to red = 300 m2/s2). The values
are generally higher where shear is higher, i.e. where the 1) slurry annular sheet meets the inner
gas core, 2) slurry annular sheet meets the outer gas annulus (although not the highest), and 3)
outer gas is moving through the quiescent surrounding gas. The outermost area (3) has the most
energy since the shear is greatest; the other locations involve the meeting of streams already in
motion.
Pursuant to the previously mentioned discoveries made in Shao et al. (2017) regarding
atomization enhancement, Figures 6 and 7 permit the exploration of instantaneous vorticity and
axial counter-flow (negative axial velocity), respectively, along the atomization trajectory.
Expectedly, vorticity begins in thin layers (Fig. 6, 0.24DO) where the gas and slurry meet on the
inside and outside of the slurry annular sheet. Vorticity conservation causes its concentration to
reduce in the central portion of the spray as it spreads radially. By 2.16DO, there are two distinct
pulsed regions rich with vorticity filaments. Since droplet formation continues to increase with
decreasing vorticity concentration, there appears to be no obvious positive correlation.
21

Fig. 5. Instantaneous contours of turbulent kinetic energy ranged from blue = 0.0 to red = 300 m2/s2
Fig. 6. Instantaneous contours of vorticity on planar cross-cuts at even increments of 0.24DO along the atomization
trajectory
22

Fig. 7. Instantaneous contours of axial velocity reversal (moving the opposite of the primary atomization trajectory)
on planar cross-cuts at even increments of 0.24DO along the atomization trajectory
Axial counter-flow, on the other hand, begins as eclipse-like shadows on the outside OG-
slurry interface, as shown in Fig. 7. Then, the counter-flow shows a sudden increase at 1.92DO,
which is where the atomization morphology also makes a dramatic change in Fig. 4. This newly
revealed flow reversal appears to be responsible for the complete dispersal of the central slurry
region.
3.4 Effects of Solver and Axial/Radial Domain Extent
Numerical setup considerations have been addressed in prior works and will be discussed in
other sections herein, but two current issues are elucidated here. The first is the version of the
solver used in Fluent, i.e. version 13SP2 relative to 14.5.0. The droplet size statistics from A25
(Fluent 14.5.0) compared to Case A23 in Strasser and Battaglia (2017a) were indistinguishable.
Additional spectral information extracted from point-wise velocity sampling (discussed in
23

Section 3.10) was employed to confirm the two sets of simulation data are identical; therefore,
the two Fluent solvers are treated as equivalents.
The second distinctive issue involves the axial and radial extent of the computational domain.
There is concern that pressure waves reflect from the outlets and interfere with the transient
response of the system. To investigate this issue, the computational domain of case C19 was
extended by a factor of nearly 3 times the original domain, both axially and radially. The time-
averaged inner gas pressure drop is only 0.05% different than that from the base (smaller)
domain size simulation. Frequency information was not extracted, but the signal characteristics
(not shown) are highly aligned. Droplet size results are also similar, e.g., the time-averaged D32
at 1.0DO from the injector exit is only 2% different using the smaller domain. It can be
concluded then that the base domain size is sufficient.
3.5 Effects of Azimuthal Domain Extent
Obvious primary atomization asymmetries caused by modeling the entire azimuthal domain
were shown in Figs. 3-5, and those effects will be further quantified. Cases A24 (lower IG flow)
and C18 (high IG flow) are compared with their 11.25° wedge counterparts from prior work
(Strasser and Battaglia 2017a), cases A23 for low flow and C9 for high flow, respectively, and
shown in Fig. 8. For each pair, the ratio of the time-averaged D32 value in each axial sampling
volume of the smaller domain is compared to the full domain. That is, the 11.25° wedge result is
in the numerator. The trends in Fig. 8 show that nearer to the injector exit, the 360° mesh has
smaller slurry length scales due to increased azimuthal sheet instability. The curves are similar
for both IG flows, but the azimuthal sheet instability effect is more pronounced in the mid-field
for the low IG feed rate. Evidently, the full circumferential model encourages 2-D to 3-D slurry
film transition and allows more filament elongation at smaller scales, enhancing slurry surface
24

area before breakup. Since the 11.25° model predicts length scales in the same range as (average
of only 15% higher than) those using the 360° model, an 11.25° model can be used for design
scoping analyses.
Likewise, the ratio of D32 temporal COV for both pairs of cases is plotted in Fig. 9. There is
more differentiation in the D32 COV results between the two domain models. The COV statistics
confirm that the 3-D surfaces pulse in time (refer to the discussion of Fig. 3). The 11.25° model
shows that there is an in-tact passing of sheets instead of droplets. The fluctuation in droplet
length scales is more distinguishable than those from the 360° model. Although the trends are
similar for both IG flows, the COV ratio is higher at higher IG flow. It is apparent that the
retarding of atomization caused by the reduced-order model enhances droplet size variability.
Fig. 8. Lateral time-average Sauter mean diameter versus distance; ratio of the 11.25° result (numerator) to the 360°
result for both inner gas flows
25

Fig. 9. Droplet size temporal COV versus distance from exit; ratio of the 11.25° result (numerator) to the 360° result
for both inner gas flows
3.6 Effects of Slurry Annular Opening Dimension (Low IG)
Despite the obvious effect of computational domain on the symmetry of the ligament/droplet
production process, a pivotal focus of this effort is to screen the design space for the effect of
slurry annular dimension (film thickness) at various retraction settings. The 11.25° model takes
1/32nd of the computational time of the 360° model, so it was not possible to explore retraction
and slurry annular dimension (resulting in a large design matrix) with the full 360°
computational domain.
The axial development of slurry droplet/ligament length scales is shown in Fig. 10, and the
time-averaged value in each axial sampling volume is plotted as a function of distance from the
injector exit scaled by DO. The Case P curve looks exactly like its base slurry annular
counterpart; therefore, slurry annular acceleration has little impact at 0.6 retraction. The other
two retractions (0.0 for Cases Q, R and 0.3 for Cases T, U), however, show a dramatic impact.
Every increase in slurry sheet acceleration results in a commensurate reduction in length scale
near the injector and then an increase away from the injector. These effects are greater at lower
retractions than at higher retractions due to the fact that less sheet recovery space is available at
26

lower retractions. The droplet production mechanism becomes less efficient for thinner sheets
and higher lip thicknesses (thicker gas boundary layer). It is theorized that the thinner sheets will
result in smaller droplets if the lip thickness effect is prevented.
Our present analysis continues with Fig. 11, which depicts the temporal COV of the droplet
size at each measurement location away from the nozzle as a function of distance from the
nozzle. At the highest retraction (Case P), the temporal variability begins at a lower value than
the 100% slurry annular area counterpart (not shown) but ends at the same value. At the
intermediate retraction (Cases T and U), the result is similar, where more slurry acceleration
leads to less variability near the injector (due to the metal lip thickness impact on the interphase
momentum exchange) and then more away from the injector. The trends at the lowest retraction
(Cases Q and R) also show less variability at the injector and then more downstream. The effect
is much less distinguishable at the lowest retraction due to the fact that less pre-filming is
available for instabilities to develop. Note, however, there are some large unexplained
differences between cases Q, R, and U from about 1.0 to 2.0 diameters from the injector exit.
Fig. 10. Lateral time-averaged Sauter mean diameter versus distance from exit showing effects of slurry annular
dimension.
27

Fig. 11. Droplet size temporal COV versus distance from exit showing effects of slurry annular dimension
As with Fig. 3, an instantaneous slurry surface colored by Mach number (blue = 0, red  0.3)
is provided in Fig. 12 for Case U. A previously shed annular slurry ring is shown at the bottom
of the figure. The slurry ringed layers for Case U look similar to those of its full slurry annulus
counterpart (Case S), but the outer rings are more ligament-like instead of containing distinct
droplets. A quite unusual phenomenon, however, is observed for this model that persists
throughout time sampling and was not detected using AS models (Strasser and Battaglia 2016a).
Liquid bridging in the AS models was responsible for bulk pulsations and was shown to sling
liquid upstream in the form of a “fountain”. Although fountains are present in the full annulus
model, they did not necessarily contribute to the bulk pulsations. Here, unexpectedly, this
particular combination of IG flow, retraction, and slurry annular constriction for case U gives the
bridging slurry enough momentum to eject a fountain up into the inner gas feed stream as was
observed with the earlier AS models.
28

Fig. 12. Instantaneous iso-surface of slurry volume fraction colored by velocity for case U (middle retraction,
minimum slurry annulus); red signifies velocities at or above a Mach number of 0.3.
Fig. 13. Instantaneous iso-surface of slurry volume fraction colored by velocity for case R (zero retraction,
minimum slurry annulus); red signifies velocities at or above a Mach number of 0.3.
29

The strikingly different pulsation events resulting from high amounts of slurry acceleration at
zero retraction can be seen in Fig. 13 showing an instantaneous slurry surface for Case R (lowest
retraction, smallest slurry annular dimension). Instead of wide-spread rings of sheets and
droplets, elongated ligaments develop very close to the injector axis in a regular manner. It is
evident that, like the axial D32 curves (Fig. 10), the sheet starts thin, but droplet production is
nearly nonexistent. The dramatic change in spray pattern caused by small slurry annulus was
only detected in the flushed design.
To summarize unexpected effects of slurry annular constriction on the primary
droplet/ligament size evolution, Fig. 14 shows the time-averaged D32 values at the end of the
modeled domain for all cases. The convoluting effects of retraction and slurry annular
dimension are obvious in how the different retractions respond to slurry annular changes. Zero
retraction cases show the steepest response in the D32 value, with more slurry constriction and
thicker “splitter plate” producing approximately 75% larger droplets. Even though the geometry
of Aliseda et al. (2008) was much simpler than that of the current work, their methods applied to
this geometry also predict a substantial increase in droplet size with a thinner slurry annulus. A
similar trend is seen for the 0.3 retraction cases, but the effect is not as strong. At 0.6 retraction
(with only two slurry annular areas tested) there was a minor reduction in droplet size by the
apparent offsetting of thinning the slurry sheet on the thicker lip.
Similarly, Fig. 15 shows the overall spatially averaged D32 COV versus slurry annular area.
For the two higher retraction cases, there is an increase in droplet size temporal variability with
increasing slurry sheet area. Apparently weak instabilities caused by thicker gas boundary layers
produce less repeatable droplet size distributions as was discussed in Fuster et al. (2013) . The
flushed case (zero retraction), however, shows a mixed response.
30

Fig. 14. Lateral time-averaged Sauter mean diameter at the end of the modeled domain versus slurry annular area
showing the coupling effect of inner nozzle retraction
Fig. 15. Overall average droplet size temporal COV versus slurry annular area showing the coupling effect of inner
nozzle retraction
Fig. 16. Dimensionless inner gas differential pressure versus slurry annular area
31

Time-averaged IG pressure drop is yet another important atomization consideration. The
effect that the slurry annular constriction has on pressure losses is shown in Fig. 16. Linear
increases at all retractions tested are shown. The nearly identical slopes for a least-squares fit are
0.00036, 0.00039 and 0.00037 for the 0.0, 0.3, and 0.6 retraction cases, respectively. As the
slurry annular sheet is thinned, and the metal lip separating the slurry sheet and inner gas core is
thickened, there is less interfacial shear between the gas and slurry. It is only logical that this
would equate to less viscous loss. The pressure drop for the base case slurry annular areas is
approximately 50% larger than the 31% slurry open area for the flushed and 0.3 retraction cases.
For the outer gas differential pressure, only less than 7% of a change was noted between the
various cases.
The average backflow of slurry onto the injector exit is affected by slurry annular dimension.
Fig. 17 shows the effects using normalized backflow by slurry film constriction. The strong
coupling between retraction and slurry annulus makes interpretation difficult. Two ordinates are
used due to the large difference in numbers (see the legend). Although the reported values are
very different, backflow is higher for the two lowest retractions when the slurry is constricted.
At 0.6 retraction, there is a measurable decrease in backflow when the slurry is constricted to
66%. One might conclude that certain slurry annular sizes are favorable depending on the level
of retraction.
32

Fig. 17. Dimensionless slurry backflow versus slurry annular open area
Breakup length of a central liquid stream is a typical reported value for coaxial primary
breakup, but it is not directly applicable here due to the fact there is no central liquid core in the
current geometry. However, breakup length from a twin-fluid design might be inversely related
to backflow from a three-stream design; slurry breaking closer to the injector and the resulting
radial spreading of ligaments should lead to a higher likelihood of slurry transferred upstream.
Zhao et al. (2012) studied the breakup length of a central non-Newtonian slurry stream.
Applying their developed correlation with the outer gas stream of the current work (since it
surrounds the liquid), a downward trend in backflow can be expected for increasing slurry
annular area. This is consistent with the current findings at lower retraction.
3.7 Effect of Slurry Viscosity on Base Case Injector Geometry
In cases A26 and A27 (low IG flow, base geometry, 360° domain), the viscosity is increased
to a multiple of 1.55 of the base value uniquely for this work, and the associated changes to
dimensionless parameters in Table 3 will result. A strong effect of viscosity is expected since the
Weber number is so high (Senecal et al. 1999), and the droplet length scale increases, as shown
in Fig. 18. The ratio of the high viscosity value to the low viscosity value ranges from 1.4 near
the injector exit, to 1.8 at 0.5 diameters from the exit, to 1.2 at the end. Tsai et al. (1991)
evaluated the effect of viscosity for a non-Newtonian slurry and found the droplet size to be
related to the Ohnesorge number (linear in viscosity) to the 0.3 to 0.6 power, which would imply
a value of 1.1 to 1.3 for a viscosity multiple (1.55 viscosity ratio raised to those powers). The
value of 1.2 fits well into that range. Also, Liu et al. (2006) used a viscous wave growth analysis
to find the most excited frequency to explain experimental measurements. Using their analysis
33

and the resulting viscosity functionality, a value of 1.2, again, was computed. Additionally, the
methods shown in Aliseda et al. (2008) imply a value closer to 1.3.
Also included in Fig. 18 is the temporal D32 variability ratio, ranging from 0.7 near the
injector to 1.2 at the end of the domain. Near the injector, the fluctuating slurry ligament length
scales have less variability at the higher viscosity implying that the viscous annular sheet is more
difficult to deform. Away from the injector, the ranges of length scales passing through the
measurement volumes are more erratic. This concurs with Chen and Li (1999) pertaining to
highly viscous liquids showing more three-dimensional mode destabilization (n  1, where n=0
is for varicose mode and n=1 is for sinuous mode disturbances).
Fig. 18. Effects of slurry viscosity on D32 statistics (mean and COV) versus distance from injector; ratio of high
viscosity values (denominator) to low viscosity values
3.8 More Insights on Instability Driving Force
For non-pulsating coaxial primary slurry atomization systems, the dominant instigator of the
initial liquid sheet instability mechanism is the gas phase turbulence (Fuster et al. 2013, Xiao et
al. 2014); however, it can be shown that pulsating systems are augmented by fluctuating
streamwise pressure gradients. In Strasser and Battaglia (2016a) liquid bridging was shown to
be responsible for bulk pulsations, and here a closer advanced look at the slurry sheet in the full
360° model is presented in Fig. 19. Three random, uncorrelated time samplings of pressure are
34

shown for Case P. The pressure range is arbitrary, but blue represents low pressure while red
represents high pressure. Superimposed on the pressure contours are solid black lines outlining
the local slurry surface. The top picture is annotated with the IG stream, slurry stream, OG
stream, and the injector axis (dash/dot line). Bulk flow is from left to right. Notice that only a
very small subset of the 3-D results is shown. The solid arrows show the component flow
directions, while the dashed arrows show the directions parallel with the local pressure gradients
in a few regions. No liquid bridging is seen at these time samplings for Case P. It is seen that
sometimes the pressure gradients are normal to the interface, which show that more than just
small-scale gas phase turbulence is responsible for initiating slurry wave growth. The work of
Chen and Li (1999) confirms that for compressible flow, instability is augmented by gas phase
normal stress variations. Even without liquid bridging, the large radial movement of the annular
sheet produces ligaments that direct the gas pressure gradient normal to the ligaments.
Fig. 19. Instantaneous pressure contours for Case P on a sampling plane showing only half the model colored by an
arbitrary pressure range (blue = low, red = high) and with superimposed slurry 2-D surfaces (black) at three random
instants in time; only the top picture is annotated with boundary flows.
35

Similarly, Fig. 20 presents the flow for case C18 at two random, uncorrelated sampling times.
The slurry interface is marked by 3-D grey surfaces superimposed on pressure contours plotted
on the centerplane. Flow is generally left to right and slightly into the page. In a few regions,
the pressure gradient normal to the interface is marked with a dashed arrow. Again, bulk
pressure gradients are normal to the slurry/gas interface and contribute to slurry disintegration;
however, liquid bridging is observed. The left-most arrow in the bottom picture for the gas
pressure gradient illustrates the effect of the bridge. The inside of the slurry surface cannot be
seen, but it is characterized as a high-pressure region. The gas just above the arrow is a low-
pressure region blue). Therefore, the gradient is from the inside out, making the gas buffet the
liquid in an orthogonal direction.
Fig. 20. Instantaneous pressure contours on the centerplane colored by an arbitrary pressure range (blue = low, red =
high) with superimposed slurry surfaces (grey) at two random instants in time
36

3.9 Length and Time Scales of Motion
Unlike previous publications, important length and time scales are examined. Fig. 21 shows
randomly sampled instantaneous local (laterally averaged) turbulent Reynolds number (k2/,
where  = kinematic viscosity and  = dissipation rate) for low IG flow (A24) and high IG flow
(C18), both involving 360° models with the base geometry and slurry viscosity. The values
range from 5,000 to 11,300. Regardless of the random snapshot in time or the IG flow rate, the
behavior is similar. The values are low near the injector, increase to a peak in the mid-field and
then return to low values near the end.
In Fig. 22, the ratio of the local, laterally averaged, turbulent integral time scale (ITS) to
general pulsation time (using 1000 Hz as a dominant pulsation frequency) versus distance is
considered. These instantaneous data are for Case C18, but the data were not found to strongly
depend on time or IG flow. The ratios monotonically increase from 0.2 to near unity across the
atomization domain, which indicates that most of the large turbulent structures do not survive
between major bursting events. Liquid breakup occurs on timescales greater than the local ITS
because it takes time for the instabilities to be communicated throughout the liquid surface
(Kourmatzis and Masri 2015). At higher turbulence intensities and, therefore, fluctuating We,
ITS is the controlling factor for the primary instability.
Fig. 21. Instantaneous turbulent Reynolds number versus distance from injector exit (Cases A24 and C18)
37

Fig. 22. Instantaneous ratio of turbulent integral time scale to pulsation time scale versus distance from injector exit
Fig. 23 illustrates the time-averaged, laterally averaged, ratio of droplet length scale to ILS
for the four cases in this work that had the time-averaged ILS data available. The run time
requirement for ILS quantification is substantially higher than that necessary to assess D32
statistics, so only three data sets are available. In addition to the three-stream pulsating models,
transient 360° CFD models were built to mimic the experimental primary atomization non-
Newtonian two-stream work of Zhao et al. (2012) and Mansour and Chigier (1995). Much more
detail on these is discussed in a companion paper (Strasser et al. 2015). The ILS alone (not
shown) is fairly constant across the three-stream atomization space, so the dominant player in the
shape of the curves in Fig. 23 is the descending droplet size. The ratio varies from 10,000 to
about 30 for the three-stream pulsating injector cases (A24, A25, and U), while the curve took on
a similar shape (though offset on the high side by about one order of magnitude) for the two-
stream models (Mansour and Chigier 1995, Zhao et al. 2012)). Notice that the 11.25° model
(A25) shows ratios much higher at the beginning and end of the domain than did its 360°
counterpart (A24). When comparing A25 (base geometry) with U (less retraction and more
slurry constriction), both using the 11.25° wedge domain, minimal difference is detected.
38

Fig. 23. Time-averaged and laterally-averaged ratio of droplet length scale to turbulent integral lengths scale versus
distance from injector exit for two- and three-stream injectors
If we assume that about 80% of the turbulent kinetic energy is contained between 1/6 and 6
times the size of the ILS (from 0.17 to 6 on the ordinate), we can conclude that larger eddies
throughout the primary three-stream or two-stream atomization domain are too small to directly
deform a droplet near the injector exit. From Crowe (2000), the gas phase turbulence is
augmented as much as 350% for ratios up to unity; therefore, the gas phase turbulence must be
strongly influenced by the emerging slurry structures.
A similar analysis is performed for the same two- and three-stream cases but this time
considering the ratio of ILS to MLS. The ratio is fairly constant for the three-stream cases
(plateaus at 0.06), but a steady increase across the atomization trajectory is seen for the two-
stream cases (Mansour and Chigier 1995, Zhao et al. 2012) in Fig. 24. All models consistently
begin very low, and only the two-stream cases with the larger computational domains exhibited
values above 1.0. Since the three-stream ILS values are 1/50th to 1/10th of the MLS, an LES
approach for those models would require substantial mesh refinement; the two-stream models
would not require as much for the far-field, which would make hybrid RANS-LES discussed in
Strasser et al. (2015) more attractive.
39

Fig. 24. Time-averaged and laterally-averaged ratio of turbulent integral lengths scale to mesh element length scale
versus distance from injector exit for two- and three-stream injectors
Though it is not plotted here, when the newly determined droplet length scales are
normalized by the distance from the liquid orifice, the results from our CFD models of the
experimental work (Mansour and Chigier 1995, Zhao et al. 2012) for droplet scale versus
distance from the orifice take on similar forms as all of our pulsating three-stream work. These
results from our model of Zhao et al. (2012) are fit well by the curve 0.67x1.7, while those from
our CFD model of the experimentation by Mansour and Chigier (1995) are fit well by the curve
0.90x1.4. For comparative reference, the three-stream injector data (and our CFD model of the
experimental work of Aliseda et al. (2008) ) yield curve fit coefficients and exponents that are
typically lower and closer to 0.40x1.2 when the gas phase is treated as incompressible; when
compressibility is present, instability increases, and the three-stream model predicts closer to
0.12x1.5 (Strasser and Battaglia 2017a).
3.10 Point-Wise Velocity Sampling
A number of interesting relations are now made available that can connect various velocity
statistics to design changes. Farago and Chigier (1992), for example, discussed how radial
velocity components were linked with atomization efficiency. In Figs. 25 and 26, relations
40

between radial velocities at two measurement points discussed in Strasser and Battaglia (2017a)
are explored for the various designs in which velocity data were collected. The radial
components are normalized by the mean outer gas velocity since that was the source of the shear
layer energy in which the point resides. It is apparent that outward velocities close to the nozzle
(NE) led to lower slurry length scales, while inward velocities away from the nozzle (HA) led to
lower slurry length scales. Interestingly, both retractions lie on consistent slopes on both plots.
A connection between the COV of the r-component of velocity at point “NE” is made in Fig.
277. As expected, more fluctuation energy (temporal variability) in radial velocity near the
nozzle is consistent with situations which led to smaller slurry length scales. Again, both
retractions follow a similar exponential relationship.
Fig. 255. D32 versus normalized r-component of velocity at point “NE”
Fig. 266. D32 versus normalized r-component of velocity at point “HA”
41

Fig. 277. D32 versus COV of r-component of velocity at point “NE”
Fig.28. D32 versus normalized x-component of velocity at point “NE”
The normalized axial component at point “NE” versus D32 is given in Fig.28. Both
retractions show a similar exponential increase in droplet length scale with increasing axial
velocity. This adds credence to the hypothesis that when the velocity is aimed in a radial
direction, more energy is available for droplet breakup.
Lastly, a series of cross-correlations are explored between the mean, COV, and spectral slope
among the two points and two velocity components for all low IG simulations in this work and
some from prior work where a counterpart simulation was required. Unique to this work, a
linear relationship is found between 19 combinations that showed an R 2 of at least 0.90. For
example, a flushed design yields the correlation:
42

COV of X-HA = -530(Mean of R-HA) + 42.5, R2 = 1.0 (4)
Simply stated, the temporal variability of the axial component of velocity at point “HA” is
linearly related to the temporal mean of the normalized radial component at the same point for
cases with zero retraction.
4. Conclusions
A computational study involving 11 previously undisclosed transient compressible 3-D
models has been executed in order to assess the effects of slurry film thickness, nozzle geometry,
non-Newtonian slurry viscosity, model domain length, slurry backflow, and azimuthal model
angle on the self-generating pulsatile spray produced by an industrial scale three-stream coaxial
airblast reactor injector. These simulations involved two main categories of model geometries.
The first was an 11.25° wedge used for screening purposes, while the second was a full 360°
domain used to study the primary atomization processes and turbulence length scales more
closely. Two additional transient 360° models were run to mimic two-stream (non-pulsating)
injector experimental slurry primary atomization studies in the open literature, revealing their
turbulence and droplet size characteristics for the first time.
The slurry annular dimension was found to have a major impact on primary atomization of a
three-stream pulsating injector. Due to geometry constraints within the injector (defined by the
surrounding process), the slurry annular dimension was not able to be modified independently of
the thickness of the metal lip separating the inner gas and the slurry annular sheet in the pre-
filming region; more constricted slurry sheets also mandated thicker lips. Like a splitter plate,
the thicker lip promoted less interfacial shear. As result, the slurry length scales were smaller
closer to the injector where the slurry sheet was thinned, but the length scales were larger away
from the nozzle where the integrated effects of reduced atomization efficiency were felt. Inner
43

gas pressure drop was reduced for more slurry construction, as less shear converted static
pressure to velocity fluctuations. Moreover, droplet length scale temporal variability was mainly
reduced for smaller slurry openings. Depending on the droplet length scale goals, the lower
pressure drop and reduced variability of the thinner sheet cases may offer a design advantage.
Point-wise velocity spectra sampling was considered for some of the designs. Radial
velocity was correlated with slurry annular constriction for two retractions. The effect of radial
velocity on droplet length scale was analyzed. Near the nozzle, designs which promoted radially
outward velocities contributed to lower slurry length scales; the opposite was true farther away
from the nozzle. The effects were very similar at both retractions.
Full 360° models showed very different qualitative behavior than wedge models. The
pulsations were much less defined with a significantly less ringed spray pattern. Due to a lack of
azimuthal annular sheet instability, the slurry length scales were larger for the wedge models
closer to the injector but then were mostly similar at the end of the domain. As expected, higher
slurry viscosities showed larger slurry length scales, and those length scales matched nearly
exactly those predicted by an external experimental study. In addition, the incremental effect of
viscosity matched what is shown in the literature. These more advanced models allowed the
further exploration of the pulsation mechanism. It was revealed that strong radial slurry ligament
positional fluctuation set conditions such that gas pressure gradients were sometimes normal to
slurry ligaments. Liquid bridging was present but was not solely responsible for setting up these
gradients. Those gradients provided a disintegrating force on top of the typical instability
mechanisms present in coaxial gas-liquid flow.
Characteristic length scale analyses indicated that larger turbulent structures do not survive
between pulsation events for the three-stream models. Additionally, these structures were too
small to directly deform slurry droplets for all (three-stream and two-stream) models near the
injector exit. Integral length scales relative to mesh length scales instructed that mesh refinement
44

would be required in order to perform LES or hybrid modeling near the injector, especially for
the three-stream systems.
Acknowledgment
The support of a multitude of Eastman Chemical Company personnel is greatly appreciated.
Specifically, George Chamoun, Josh Earley, Paul Fanning, Moises Figueroa-Contreras, Jason
Goepel, Steve Hrivnak, Meredith Jack, Kristi Jones, Rick McGill, Wayne Ollis, Sam Perkins,
Megan Salvato, Glenn Shoaf, Andrew Stefan, Bill Trapp, and Kevin White were supporters of
this effort. George Chamoun deserves special recognition for constructing UDFs and processing
transient signal data sets using various methods. Additionally, discussions with Mihai Mihaescu
from Royal Institute of Technology (KTH), Marcus Herrmann from Arizona State University,
David Schmidt from the University of Massachusetts, Mario Trujillo from University of
Wisconsin–Madison, Daniel Fuster of Institut Jean Le Rond D'Alembert UPMC, and Christophe
Dumouchel of Université et INSA de Rouen were extremely beneficial. Finally, Special Effects
Artist Tyler Strasser provided post-processing assistance.
45

References
Aliseda, A., Hopfinger, E. J., Lasheras, J. C., Kremer, D. M., Berchielli, A. and Connolly, E. K.,
Atomization of viscous and non-newtonian liquids by a coaxial, high-speed gas jet. Experiments
and droplet size modeling, International Journal of Multiphase Flow, vol. 34, no. 2, pp. 161-175,
2008.
ANSYS, Solver Documentation, vol., no., 2013.
Bagué, A., Fuster, D., Popinet, S., Scardovelli, R. and Zaleski, S., Instability growth rate of two-
phase mixing layers from a linear eigenvalue problem and an initial-value problem, Physics of
Fluids (1994-present), vol. 22, no. 9, pp. 092104, 2010.
Beheshti, N. and McIntosh, A. C., The bombardier beetle and its use of a pressure relief valve
system to deliver a periodic pulsed spray, Bioinspir. Biomim., vol. 2, no. 4, pp. 57-64, 2007.
Boeck, T., Li, J., López-Pagés, E., Yecko, P. and Zaleski, S., Ligament formation in sheared
liquid–gas layers, Theoretical and Computational Fluid Dynamics, vol. 21, no. 1, pp. 59-76,
2007.
Brackbill, J., Kothe, D. B. and Zemach, C., A continuum method for modeling surface tension, J.
Comput. Phys., vol. 100, no. 2, pp. 335-354, 1992.
Chen, T. and Li, X., Liquid jet atomization in a compressible gas stream, Journal of propulsion
and power, vol. 15, no. 3, pp. 369-376, 1999.
Chesnel, J., Menard, T., Reveillon, J. and Demoulin, F.-X., Subgrid analysis of liquid jet
atomization, Atomization and Sprays, vol. 21, no. 1, 2011.
Chigier, N. and Farago, Z., Morphological Classification of Disintegration of Round Liquid Jets
in a Coaxial Air Stream, vol. 2, no. 2, pp. 137-153, 1992.
Crowe, C. T., On models for turbulence modulation in fluid–particle flows, International
Journal of Multiphase Flow, vol. 26, no. 5, pp. 719-727, 2000.
Dhakal, T. P. and Walters, D. K., A Three-Equation Variant of the SST k-ω Model Sensitized to
Rotation and Curvature Effects, Journal of Fluids Engineering, vol. 133, no. 11, pp. 111201,
2011.
Dhakal, T. P., Walters, D. K. and Strasser, W., Numerical study of gas-cyclone airflow: an
investigation of turbulence modelling approaches, International Journal of Computational Fluid
Dynamics, vol. 28, no. 1-2, pp. 1-15, 2014.
46

Dumouchel, C., On the experimental investigation on primary atomization of liquid streams,
Experiments in fluids, vol. 45, no. 3, pp. 371-422, 2008.
Engelbert, C., Hardalupas, Y. and Whitelaw, J., Breakup phenomena in coaxial airblast
atomizers, Proceedings of the Royal Society of London. Series A: Mathematical and Physical
Sciences, vol. 451, no. 1941, pp. 189-229, 1995.
Faeth, G., Hsiang, L.-P. and Wu, P.-K., Structure and breakup properties of sprays, International
Journal of Multiphase Flow, vol. 21, no., pp. 99-127, 1995.
Fu, Q., Mo, C. and Yang, L., Numerical Study of a Self-pulsation Injector, Atomization and
Sprays, vol. 27, no. 2, pp. 139-149, 2017.
Fuster, D., Matas, J. P., Marty, S., Popinet, S., Hoepffner, J., Cartellier, A. and Zaleski, S.,
Instability regimes in the primary breakup region of planar coflowing sheets, Journal of Fluid
Mechanics, vol. 736, no. 12, pp. 150-176, 2013.
Gatski, T. B. and Bonnet, J.-P. Compressibility, turbulence and high speed flow, Academic Press,
2013.
Hsiang, L. P. and Faeth, G. M., Near-limit drop deformation and secondary breakup,
International Journal of Multiphase Flow, vol. 18, no. 5, pp. 635-652, 1992.
Kihm, K. D. and Chigier, N., Effect of Shock Waves on Liquid Atomization of a Two-
Dimensional Airblast Atomizer, Atomization and Sprays, vol. 1, no. 1, pp. 113-136, 1991.
Kourmatzis, A. and Masri, A., Air-assisted atomization of liquid jets in varying levels of
turbulence, Journal of Fluid Mechanics, vol. 764, no. 5, pp. 95-132, 2015.
Lasheras, J. and Hopfinger, E., Liquid jet instability and atomization in a coaxial gas stream,
Annual Review of Fluid Mechanics, vol. 32, no. 1, pp. 275-308, 2000.
Lefebvre, A. Atomization and sprays, Taylor and Francis, 1988.
Ling, Y., Fuster, D., Zaleski, S. and Tryggvason, G., Spray formation in a quasiplanar gas-liquid
mixing layer at moderate density ratios: A numerical closeup, Physical Review Fluids, vol. 2, no.
1, pp. 014005, 2017.
Liovic, P. and Lakehal, D., Multi-physics treatment in the vicinity of arbitrarily deformable gas-
liquid interfaces, Journal of Computational Physics, vol. 222, no. 2, pp. 504-535, 2007.
47

Liu, H.-F., Gong, X., Li, W.-F., Wang, F.-C. and Yu, Z.-H., Prediction of droplet size distribution
in sprays of prefilming air-blast atomizers, Chemical Engineering Science, vol. 61, no. 6, pp.
1741-1747, 2006.
Liu, S., Li, G., Shi, H., Huang, Z., Tian, S. and Yang, R., Dynamics of Cleaning by Hydraulic
Pulsed Jet, Atomization and Sprays, vol. 25, no. 11, pp. 1013-1024, 2015.
Mansour, A. and Chigier, N., Air-blast atomization of non-Newtonian liquids, Journal of Non-
Newtonian Fluid Mechanics, vol. 58, no. 2, pp. 161-194, 1995.
Menard, T., Tanguy, S. and Berlemont, A., Coupling level set/VOF/ghost fluid methods:
Validation and application to 3D simulation of the primary break-up of a liquid jet, International
Journal of Multiphase Flow, vol. 33, no. 5, pp. 510-524, 2007.
Menter, F. R., Two-equation eddy-viscosity turbulence models for engineering applications,
AIAA Journal, vol. 32, no. 8, pp. 1598-1605, 1994.
Pathania, R. S., Chakravarthy, S. R. and Mehta, P. S., Time-resolved Characterization of Low-
pressure Pulsed Injector, Atomization and Sprays, vol. 26, no. 8, pp. 755-773, 2016.
Pirozzoli, S. and Grasso, F., Direct numerical simulations of isotropic compressible turbulence:
Influence of compressibility on dynamics and structures, Physics of Fluids, vol. 16, no. 12, pp.
4386-4407, 2004.
Senecal, P. K., Schmidt, D. P., Nouar, I., Rutland, C. J., Reitz, R. D. and Corradini, M. L.,
Modeling high-speed viscous liquid sheet atomization, International Journal of Multiphase
Flow, vol. 25, no. 6-7, pp. 1073-1097, 1999.
Shao, C., Luo, K., Yang, Y. and Fan, J., Detailed numerical simulation of swirling primary
atomization using a mass conservative level set method, International Journal of Multiphase
Flow, vol. 89, no. Supplement C, pp. 57-68, 2017.
Strasser, W., Discrete particle study of turbulence coupling in a confined jet gas-liquid separator,
ASME J. Fluids Eng., vol. 130, no. 1, pp. 011101, 2008.
Strasser, W., Cyclone-ejector coupling and optimisation, Progress in Computational Fluid
Dynamics, an International Journal, vol. 10, no. 1, pp. 19-31, 2009.
Strasser, W., Towards the optimization of a pulsatile three-stream coaxial airblast injector,
International Journal of Multiphase Flow, vol. 37, no. 7, pp. 831-844, 2011.
Strasser, W. and Battaglia, F., Identification of Pulsation Mechanism in a Transonic Three-Stream
Airblast Injector, ASME J. Fluids Eng., vol. 138, no. 11, pp. 111303, 2016a.
48

Strasser, W. and Battaglia, F., The Influence of Retraction on Three-Stream Injector Pulsatile
Atomization for Air-Water Systems, ASME J. Fluids Eng., vol. 138, no. 11, pp. 111302, 2016b.
Strasser, W. and Battaglia, F., The Effects of Prefilming Length and Feed Rate on Compressible
Flow in a Self-pulsating Injector, Atomization and Sprays, vol. 27, no. 11, pp. 929-947, 2017a.
Strasser, W. and Battaglia, F., The effects of pulsation and retraction on non-Newtonian flows in
three-stream injector atomization systems, Chemical Engineering Journal, vol. 309, no. 1, pp.
532-544, 2017b.
Strasser, W., Battaglia, F. and Walters, D., Application of hybrid RANS-LES CFD methodology
to primary atomization in a coaxial injector, American Society of Mechanical Engineers,
Proceedings of the International Mechanical Engineering Congress and Exposition, Paper #
IMECE2015-53028, 2015.
Strasser, W. S., Feldman, G. M., Wilkins, F. C. and Leylek, J. H., Transonic Passage Turbine
Blade Tip Clearance With Scalloped Shroud: Part II—Losses With and Without Scrubbing
Effects in Engine Configuration, American Society of Mechanical Engineers, ASME 2004
International Mechanical Engineering Congress and Exposition, 2004.
Tsai, S. C., Ghazimorad, K. and Viers, B., Airblast Atomization of Micronized Coal Slurries
Using a Twin-Fluid Jet Atomizer, Fuel, vol. 70, no. 4, pp. 483-490, 1991.
Wang, L., Ye, W. and Li, Y., Combined effect of the density and velocity gradients in the
combination of Kelvin–Helmholtz and Rayleigh–Taylor instabilities, Physics of Plasmas (1994-
present), vol. 17, no. 4, pp. 042103, 2010.
Xiao, F., Dianat, M. and McGuirk, J. J., LES of turbulent liquid jet primary breakup in turbulent
coaxial air flow, International Journal of Multiphase Flow, vol. 60, no. 4, pp. 103-118, 2014.
Yoshizawa, A., Liou, W. W., Yokoi, N. and Shih, T. H., Modeling of compressible effects on the
Reynolds stress using a Markovianized two-scale method, Physics of Fluids, vol. 9, no. 10, pp.
3024-3036, 1997.
Youngs, D. L., Time-dependent multi-material flow with large fluid distortion, Numerical
Methods for Fluid Dynamics, vol. 24, no. 2, pp. 273-285, 1982.
Zhao, H., Liu, H.-F., Xu, J.-L., Li, W.-F. and Cheng, W., Breakup and atomization of a round coal
water slurry jet by an annular air jet, Chemical Engineering Science, vol. 78, no. 8, pp. 63-74,
2012.
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