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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

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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.

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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

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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 0.67x1.7, while those from

our CFD model of the experimentation by Mansour and Chigier (1995) are fit well by the curve

0.90x1.4. 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 0.40x1.2 when the gas phase is treated as incompressible; when

compressibility is present, instability increases, and the three-stream model predicts closer to

0.12x1.5 (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

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