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Identification of Pulsation Mechanism in a Transonic Three-

Stream Airblast Injector

Wayne Strasser

Eastman Chemical Company

Kingsport, TN, 37660

strasser@eastman.com

ASME Fellow

Francine Battaglia

Department of Mechanical Engineering

Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

fbattaglia@vt.edu

ASME Fellow

Abstract

Acoustics and ligament formation within a self-generating and self-sustaining pulsating

three-stream injector are analyzed and discussed due the importance of breakup and atomization

of jets for agricultural, chemical, and energy-production industries. An extensive parametric

study was carried out to evaluate the effects of simulation numerics and boundary conditions

using various comparative metrics. Numerical considerations and boundary conditions made

quite significant differences in some parameters, which stress the importance of using

documented and consistent numerical discretization recipes when comparing various flow

conditions and geometries. Validation exercises confirmed that correct droplet sizes could be

produced computationally, the Sauter mean diameter of droplets/ligaments could be quantified,

and the trajectory of a droplet intersecting a shock wave could be accurately tracked. Swirl had a

minor impact by slightly moving the ligaments away from the nozzle outlet and changing the

spray to a hollow cone shape. Often, metrics were synchronized for a given simulation,

indicating that a common driving mechanism was responsible for all global instabilities, namely

2

liquid bridging and fountain production with shocklet-like structures. Interestingly, both CFD

and the experimental non-Newtonian primary droplet size results, when normalized by distance

from the injector, showed an inversely proportional relationship with injector distance. Another

important outcome was the ability to apply the models developed to other nozzle geometries,

liquid properties, and flow conditions or to other industrial applications.

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1. Introduction

1.1. Background

The breakup and atomization of jets have been of direct importance to the agricultural,

chemical, food, fire protection, and energy-production industries. Controlling the droplet

characteristics and trajectories can be critical for high yields and productivity of reactive process

equipment. Despite the fact that process fluid properties often depart significantly from air and

water at atmospheric conditions, air/water test stands (AWTS) are used as effective testing

means for industrial injectors. A vast expatiation on the design of injectors and sprays is given in

[1]. Generally speaking, droplet size decreases with increasing air velocity and liquid density, it

increases with increasing liquid viscosity and surface tension, and it shows a mixed or weak

effect with changes in initial liquid sheet thickness [2]. Important insights into the mechanisms

of coaxial primary atomization are elucidated by [3, 4]. Nourgaliev et al. [3] discussed how

fundamental continuum microscale fluid forces can be linked to Kelvin-Helmholtz instabilities

(KHI) and Rayleigh-Taylor instabilities (RTI). They note the transition from linear to nonlinear

wave growth in KHI for which turbulence is not explicitly stated to be a prerequisite. According

to [4] which is more applicable to shear-dominated flows, in the case of laminar liquid feeds, gas

phase turbulent structures deform the liquid phase interface. Small perturbations are quickly

restored by surface tension due to sharp interfacial curvature. The competition between gas

phase turbulent structures and the liquid interface joins the indigenous KHI, which eventually

result in major distortion of the contiguous liquid to form waves. Waves provide openings for

gas phase interactions with the waves; form drag creates high pressure and low pressure regions

on the windward and leeward boundaries of the waves. The light gas accelerating the heavy

liquid creates RTI until liquid droplets are pinched off. When the liquid feed is turbulent, the

4

initial perturbation is from competition between the liquid phase turbulent structures, instead of

those in the gas, and its surface tension. The remainder of the process is similar.

High velocity gradients present at the gas-liquid interface create computational

difficulties. Consequently, experimental studies are common, such as in [5] and [6]. A typical

computational approach is to study the flowfield of the gas and liquid droplets after primary

atomization. An embedded rigorous mathematical routine is used to produce a distribution of

droplets, and then that distribution is allowed to slip and interact with the gas stream and local

geometry [7, 8]. The alternative to this approach is to use an interface tracking scheme to

explicitly compute the shapes and sizes of the first ligaments and droplets during primary

atomization [9].

As noted by Lefebvre [1], airblast (implying relatively high gas feed rates) atomization

has been studied for decades. Three-stream atomizers, a subset of airblast atomizers, incorporate

two gas streams, each surrounding the annular liquid feed. When operated under certain

conditions, including pulsation-free boundary conditions, a three-stream atomization system

becomes globally unstable. Strong pulsations are generated and self-sustained [10] by the

primary atomization process, linking acoustics with primary atomization. The system behaves

much like a periodic “stable limit cycle” (Strogatz, [11]) in which various constant feed

conditions create a particular response frequency, but each cycle is not a perfect repeat of the

previous as will be shown here. One might characterize the slightly aperiodic behavior as

“chaos”, but as will be discussed in the “Numerically-Induced Instabilities” section, there is no

dependency on initial conditions, preventing it from being classified as chaos. Despite the recent

advances in computational methodologies and the proliferation of interface tracking methods,

three-stream injection systems remain relatively computationally unexplored. Pulsating sprays

5

pre-date the industrial world by thousands of years, as can be seen in the life and intricate design

of the Bombardier Beetle found mainly in Africa and Asia [12], and have been found relevant to

the food industry [13], chemical process industry [14], and heat transfer control [15]. All coaxial

atomization may fundamentally be considered pulsatile in that, from a fixed reference frame

watching the passing spray, there are temporal oscillations in liquid volume fraction and spray

droplet number density. (Plus, two fluids merging at different velocities encourage fluctuations

at some scale.) Farago and Chigier [16] discussed various breakup regimes ranging from

Rayleigh-type (droplet diameter on the order of the liquid orifice diameter) to fiber-type (droplet

diameter much smaller than the liquid orifice diameter) based on the ratio of the gas-liquid

relative velocity to the liquid surface tension. The more energetic regime was called

“superpulsating” and was observed to have bulk passing pockets of various sized droplet clouds.

Here, we define “pulsations” to imply those which are driven by, or effectively drive, pure feed

stream oscillations.

Not only are the self-sustaining global pulsations and acoustics germane to primary

atomization in a three-stream injector, but also to very high speed systems where the gas

becomes transonic inside or outside the nozzle. One of the most comprehensive works on

compressibility effects on turbulent flow is that of Gatski and Bonnet [17]. In general,

compressibility dampens turbulence by relaxing eddy communication. In addition, shocks

enhance turbulence and alter the anisotropy, wall reflections contribute to viscous heating, and

thermodynamic quantities (and their interactions with turbulence) evolve at higher frequencies in

compressible flow. These issues are more pronounced in 3-D systems. The following section

identifies the issues to be examined in this study.

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1.2. Objectives and Scope

The self-generating and self-sustaining pulsating nature of a three-stream injector with

pulsation-free boundary conditions has received surprisingly little attention in the open literature

even though its spray pattern is linked to the energy content of its pulsatile flow. It will be

shown what mechanisms are responsible for primary atomization and for bulk pulsations. The

foundation of the work here is that of Strasser [10] in which various ratios of gas and liquid feed

rates were considered with the aim first to characterize and then to optimize the operation of a

pulsatile injector in a predominantly axis-symmetric (AS) modeling framework; animations and

experimental videos were produced to illustrate various effects. However, an astounding

quantity of issues lacking in the literature remain unaddressed, including the mechanism

responsible for the self-sustaining bulk pulsations, which we will uncover. To do so, an extensive

study is conducted to investigate the effects of numerical models and provide recommendations,

strengths and drawbacks based on the parametric analysis. The outcome will be the first

comprehensive numerical study for the pulsatile injector. Lastly, a new function for a

dimensionless droplet size, when scaled by the distance from the injector, will be proposed.

As part of the computational modeling, numerical considerations, such as turbulence

model, time-step size, volume of fluid (VOF) method, inlet boundary conditions, and

discretization are investigated in the 11 modeled cases shown in Table 1. There is apparently

nothing in the open literature by other research groups regarding issues of numerics, boundary

conditions, and swirl on the flow field for a transonic three-stream injector using a VOF

computational approach even though VOF is quite commonly [18-23] used for multiphase flows.

Strasser and Wonders [24] discussed how a cancellation of errors in the momentum balance can

still produce a reasonable result, so various combinations of approaches will be explored. More

7

information will be provided for the details in Table 1 as the need arises. In addition to these 11

parametric simulations, validation simulations and experimental results are presented herein.

2. Model Formulation

2.1. Governing Equations and Models

A steady simulation will not always produce the same result as the time-average of a

transient simulation [25]. As a result, a transient solution is sought in an Eulerian framework

using ANSYS. The continuity equation governing the mass balance of each phase is:

() ( )

0u

t

αρ αρ

∂+∇⋅ =

∂

(1)

The phase-averaged Reynolds-averaged (RANS) linear momentum balance is:

() () ( )

()

tref

uu u p gF

t

ρρττρρ

∂+⋅∇ =∇⋅ + −∇+ − +

∂

(2)

Processes from transforming the multi-fluid Navier-Stokes equations at the continuum

microscale into phase-averaged RANS equations are not trivial, and details are provided in [26].

The risks are especially relevant when capturing the momentum balance at the interfacial jump

conditions as is discussed in [27]. Proper jump conditions capturing in DNS are difficult, and

work on closing the loop in current cutting-edge research are far from conquered yet; therefore,

commercial RANS solvers are lacking. The simplified mixture treatment applies mainly to cells

that contain either all gas or all liquid (and α is volume fraction). Where phases meet and cells

contain both fluids, the differentiated face fluxes of each phase within a cell are computed by the

VOF method in order to ensure mass balance. Consequently, the shape and location of the

interface is reconstructed in space and time as will be described in the “Numerics” section.

Droplet/particle tracking methods [28] are not used, as computational cells are smaller than the

droplets. Properties are mass-averaged among the phase volume fractions present in a cell. With

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this method, film formation, ligament production, and droplet onset are explicitly taken into

account by considering sub-grid scale (SGS) information, i.e., how the liquid interface looks

within a computational cell.

Equation 2 shows that a single momentum field is computed, making this a

“homogeneous” method. Since the three co-flowing immiscible phases have very different feed

velocities (cause of the instabilities) in airblast atomization, the use of a single momentum field

seems counterintuitive and only works if grid cells are smaller than droplets. Very far from the

phase interface, the phases act as a continuum requiring only a single velocity field. This

includes the liquid and gas inlet streams. At the interface, the no-slip condition requires that the

liquid and gas be at the same velocity. In between, a velocity gradient is established.

Similarly, the phase-averaged energy equation is:

() ( ) ()

Pr

tt

t

EuEp Tu

t

μ

ρρ ζ

ττ

∂+∇⋅+=∇⋅ + ∇ + ⋅ +

∂

(3)

Again, properties are mass-averaged among the phases, which differ from Favre averaging by

the inclusion of the volume fraction. For example, E = αiρiEi/ρMIX, where Ei is the total energy

for phase i and ρMIX = αiρi. The gradient diffusion hypothesis has been used to separate the

molecular and turbulent effects in equations (2) and (3). The air density is assumed to vary as

that of an ideal gas, and liquid compressibility is ignored. The local air Mach number can reach

higher than 1.0 at any given time. As a result, kinetic energy, viscous heating, and pressure-

work terms are included in Equation (3). Some ways which flows can be classified as

“compressible” are described in Gatski and Bonnet [17] and include the effects of high speeds,

molecular diffusion, viscous heating, unsteadiness, and body forces. Various dimensionless

parameters were tested, and it was found that only the traditional high-speed considerations

9

render the present system compressible. In other words, density changes due to entropy

associated with dissipative heating or molecular diffusion are insignificant. Liquid droplet

evaporation due to gas humidity effects has been ignored. Surface energy effects are treated via

the continuum surface force method of Brackbill et al. [29].

The homogeneous shear stress transport (SST) two-equation linear eddy viscosity model

of Menter [30] is used for computing the turbulent contributions to momentum and energy

transport for nearly all of the cases presented herein; realizable k-ε (RKE) of Shih et al. [31] was

employed only once as a test and was later referred to as case A13. 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 ω. Also, there is a turbulence

production limiter, as discussed in ANSYS [32], preventing the artificial build-up of fluctuating

velocity in regions of irrotational strain. "Scalable" wall functions, discussed in ANSYS [32],

are an alternative to standard wall functions. They have the advantage of being less sensitive to

variation in near-wall grid resolution throughout the domain. The distance of a given

computational cell center from the wall is computed via ∇2φ = −1, where φ is the wall distance

quantity, with the boundary condition φ = 0 at the wall, zero flux on other boundaries, and only

keeping positive roots. Additional compressibility effects are taken into account as described in

ANSYS [32]; the typical constant known as “beta”, which is used in the SST method [30] for the

computation of the turbulent viscosity, production of turbulence kinetic energy, and the

dissipation of turbulence kinetic energy, is adjusted for turbulent Mach number (2k/a2) when

turbulent Mach number exceeds 0.25. Velocity gradients at gas-liquid interfaces can induce

modeled turbulence that is actually not present in the real flow and requires a modeled damping

10

function (Hansch et al. [33], Ergorov [34], and Deendarlianto et al. [35]); a damping factor of 10

was employed in this work.

It is well-known that eddy-viscosity turbulence computations are wanting. Limitations

of the SST are known, some of which are documented in the particle-turbulence coupling study

[28] and cyclone studies [36, 37]. SST improvements were proposed [38] based on

simplification of a Reynolds stress model. The advantages of using the differential Reynolds

stress model (RSM) approach include natural realizability and the ability to capture the effects of

streamline curvature, turbulence anisotropy, and rapid changes in strain rate. Turbulence

anisotropy can be substantial in jet flows; quantification of the departure from isotropic

turbulence is discussed in Strasser [10]. With the RSM, production and molecular diffusion need

no modeling, but other terms do. The Launder, Reece and Rodi-Isotropization of Production

(LRR-IP) approach for modeling Reynolds stresses [39] is used some throughout this work, and

it caused computational run time to increase by about 10%. The turbulent diffusive stress

transport is treated with the gradient-diffusion hypothesis, and the dissipation is modeled

assuming isotropy. Pressure strain is well known to be one of the most important terms; it has

zero trace and serves to redistribute the stresses and move toward the isotropic state. In the

present work, the pressure-strain model incorporates a linear return-to-isotropy term or “slow”, a

rapid pressure-strain term, and a wall reflection term. A nonequilibrium wall function approach

was used for the near-wall cells as discussed in ANSYS. This method involves typical wall

functions that have been modified to relax the production matches dissipation assumption. Wall

reflection effects are included, and the wall boundary conditions for the stresses are derived from

the near-wall value of k. As with most Reynolds-averaged turbulence models, the boundary

layers are considered to be everywhere turbulent.

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2.2. Numerics

Computational algorithms offer many degrees of freedom. A typical approach is to pick

a method, justify it using (sometimes incomplete) information from other sources, and proceed in

a study. Often, the ramifications of these decisions are not fully explored. Here, we seek to shed

new light on the effects of these freedoms and choices for transonic pulsatile injectors.

Equations 1-3 are solved in ANSYS Fluent 13.0 (with service pack # 2) segregated double

precision commercial cell-centered solver.

A mass-conserving way of solving Equation (1) and modeling the interactions between

two immiscible phases is the VOF method. Using Fluent's VOF framework to study droplets is

not rare [40, 41] and can be used to define the gas-liquid interface [42] when the droplets are

larger than the computational grid scale. The issue of differentiating between the liquid and gas

phase at a length scale less than the cell dimension is paramount for this work; the fact that the

liquid and gas have substantially different densities further complicates interfacial detection,

along with the basic advection calculations (see [4, 43] and many others cited herein). The

explicit piecewise linear geometric reconstruction scheme from 1982 by Youngs [44] (often

referred to as “PLIC”) is used as the time-marching scheme to solve Equation (1) for cell face

fluxes and for computing the location of the SGS liquid-gas interface. To clarify, VOF does not

resolve SGS droplets (where the entire droplet fully fits within a mesh cell) but only interfacial

segments. VOF is accurate when a droplet is resolved by many cells. The PLIC method

assumes that the interface takes on the shape of a line in 2-D and a plane in 3-D. An unsplit flux

methodology is used for the advection terms after interfacial reconstruction. No smoothing is

applied to the volume fraction during interfacial reconstruction, and an unpublished gradient

method other than least-squares was utilized. Cummins et al. [45] describe three common

12

techniques for estimating interfacial curvature, including convolved VOF, distance function, and

height function, noting that the first two are more robust for unstructured meshes and/or

situations where curvature is ill-resolved. Fluent incorporates the convolved VOF method in

which a smoothing kernel is applied to the volume fraction for the calculation of a smoothed

volume fraction gradient in order to reduce high-frequency aliasing errors. The smoothing is

node-based, such that a volumetric average of node neighbors of a given cell is computed. The

history, development, and order of accuracy of modern PLIC implementations, along with

highly-detailed surface normal reconstruction, flux splitting, smoothing, and pressure

discretization information, are provided in Rider and Kothe [46] and Gueyffier et al. [47]. An

alternative is to begin with the linear interface assumption and then decompose the line (2-D)

into fragments and adjust their locations in order to maintain connectivity between the interfacial

edge cuts between adjacent cells while ensuring mass conservation, producing SGS curvature as

is noted in Liovic [48]; scope for reduced interfacial error beyond PLIC-VOF is substantial,

although the algorithms to realize these reductions are still in their infancy. More information on

SGS influences can be found [27, 33, 49-52]. A validation of the PLIC method for evaporating

droplets can be found in [53]. According to ANSYS [32] and Liovic [48], the geometric

reconstruction scheme is the most accurate method of interface capturing currently implemented

in ANSYS Fluent, so this was used for most modeling herein

A typical Courant number (CN) for the internal VOF calculations at the sub-loop level is

0.25, but the effect of reducing this to a value of 0.125 was considered (case A11). A

combination of VOF and Level-Set discussed in ANSYS [32], Anumolu and Trujillo [43], and

Xiao et al. [4] is evaluated (case A15). More computational details on the Level-Set method can

be found in Menard et al. [42]. VOF is known for its strong mass conservation at the cell level,

13

but it has difficulty reconstructing the interfacial curvature and surface normals. As a result,

spurious velocities sometimes result at the VOF-computed interface [54] due to the VOF data

supplied to the surface tension model. Level-Set has the opposite challenges and often loses

mass during the re-initialization step. Anumolu and Trujillo propose a fix for this issue in [43].

Ideally, a code would utilize the strengths of them both. Note, specifically, the downside to any

VOF (or single momentum field) formulation; that is, phase equilibrium is assumed at the cell

level. In other words, when droplets travel much faster than the surrounding gas, or vice versa,

the slippage and shear layer generation between the two is ignored. This is a reasonable

assumption for a fine grid in which there are multiple cells per droplet, but would not be the case

in which there are multiple droplets per cell (Eulerian-Lagrangian).

Pressure-velocity coupling is coordinated via the Pressure Implicit with the Splitting of

Operators (PISO) scheme with skewness and neighbor corrections. A Green-Gauss node-based

gradient method is used for discretizing derivatives and is more rigorous than a simple

arithmetical grid cell center average. The pressure field is treated with a body-force weighted

approach to assist with body force numerics for all simulations, except case A12 which

employed the PREssure STaggering Option (PRESTO!) scheme. PRESTO! uses mass balances

to obtain face pressures, instead of a traditional second-order upwinding scheme, which

incorporates geometric interpolation to obtain face pressures from cell-centered pressures. This

could be beneficial in areas of high pressure or volume fraction gradient, but has been found

from internal testing to be substantially more numerically stiff. Second-order upwinding is used

for advection terms, and first-order upwinding is used for turbulence quantities which are

dominated by source terms. The QUICK scheme, which is formally higher than 2nd order

accuracy on structured meshes, was used on the advection terms for A12. Details for the

14

discretization schemes can be found in Strasser [55] and in ANSYS [32]. The transient term is

also discretized using first-order upwinding, but this is 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. Among those who have

successfully used similar numerical considerations (geometric reconstruction, PISO, and

QUICK) for liquid break-up VOF studies is Ng and Sallam [56]. Two total variation

diminishing (TVD) slope limiters (not to be confused with flux limiters) are considered for the

second order advection schemes. The typical is the “standard” approach from Barth and

Jespersen [57], but the “multidimensional” method (more rigorous and less diffusive) of Kim et

al. [58] is investigated for A14. The advantages of this scheme are discussed more in Poe and

Walters [59].

Convergence and time-averaging were discussed in detail in Strasser [10] but it will be

repeated here that a typical time step of 5.0x10-7 seconds with 10 inner loops was required for the

residuals to stop changing significantly with additional inner loop steps. Typical globally

averaged CN based on fluid velocity remained below 0.5 throughout the simulations. In fact,

there are various bases for quantifying CN according to Menter [60], most of which are less

conservative for this gas-liquid interface-driven system. For case A9, however, a CN of 1.5 was

tested. Additionally, it can be noted that the maximum capillary time-step size from Brackbill et

al. [29] is 2 to 4 orders larger than what is used in these simulations. Total simulation time

includes reaching quasi-steady behavior (as shown in [10]) and then time-averaging data

collection. Sufficient data had to be collected to produce clean time-averages, which

necessitated many convective time (CT) scales. A CT is the length of time it takes for the fastest

droplets to travel through the modeled domain downstream of the nozzle. A typical simulation

15

time for AS models is approximately one week on 4 Intel Xeon X5690 processor cores for the

1000 CT required for the spray angle measure to stabilize.

Three proprietary subroutines (UDF) were written for this study. One UDF is required to

establish a temperature profile on the outer walls of the injector as dictated by reactor conditions.

Another UDF defines the local temperature and shear-dependent viscosity of the non-Newtonian

slurry throughout the domain during computations. A consequence of a local viscosity

calculation is a “coupled” approach to fluid properties. Cell-centered velocity gradients and

temperature set the local viscosity at the beginning of a timestep. As the loops continue

throughout the timestep, the velocity gradient is affected by the viscosity and so on. Viscous

heating adds further coupling. A third UDF is used to compute real-time droplet size statistics

for 3-D models. Though not a UDF per se, Matlab was used to compute Fast Fourier Transforms

(FFT) on various transient signals in order to assess spectral information. Lastly, video analysis

routines were written in LabVIEW.

2.3. Mesh and Boundary Conditions

The validated mesh in Figure 1 contains about 32,000 elements per cross-section with

approximately 30 cells spanning the liquid annular gap. It was constructed carefully in order to

avoid having any triangular-faced cells due to their known inferiority [55, 61]. Known

temperatures, pulsation-free flow rates, and properties from the AWTS are supplied at the three

independent computational stream inlets. Turbulence quantities were not available in the

experiment but were specified as follows: Turbulence intensity was set to 5% of the inlet area-

averaged velocity, while the integral length scale was set to a relative amount, say 20%, of the

feed nozzle length scales. These are not critical since the turbulence field will develop

throughout the long inlets based on pressure gradient, boundary layer development, etc. The

16

feed turbulence field will be completely transformed by the tortuous inlet passages through shear

and vorticity before the flows reach the retracted zone. Additionally, bulk feed pulsations with

frequencies over approximately 100 Hz (for this system) will not create any modulation or

resonance in the turbulence of the feed passages [62]. The choice of feed turbulence quantities,

therefore, is not pertinent. 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. All fluids leave the domain at

the bottom and sides, which are pressure-outlets. Outlets are treated as “openings”, at which

flow can move into or out of the modeled domain.

The three injector streams are designated as inner gas (IG), outer gas (OG) and an

intermediate liquid as shown in Figure 1. For the purposes of testing numerics in this study, only

AS modeling frameworks are considered. The significant AS disadvantage, obviously, is that

any shed droplets are not accurately modeled as “droplets” per se. Since the geometry wraps

uniformly in the azimuthal dimension, each droplet remains a uniform torus shape. In reality,

sheet perforations (caused by velocity curl and turbulence) and radial sheet thinning cause the

ligaments to break up azimuthally. However, a 3-D analysis is not required for establishing how

numerics, boundary conditions, and swirl affect acoustics and ligament formation in the retracted

(also known as “pre-filming” where the inner nozzle is recessed upstream of outer nozzle)

section and how far away from the injector face these ligaments tend to project; these elements

are the focus of this present work and are primarily 2-D in nature (see [10]), Additionally,

discrete droplet size production and analysis has not yet been established for this system, which

is part of this paper. All references to things like “spray angle” will have limited power in that

they will be useful for comparisons among cases instead of absolute application. As will be

17

shown, the axisymmetric analysis is more than adequate to identify the strengths and weaknesses

of the computational methodologies.

The dimensionless groups which govern the flowfield are shown in Table 2. The flow

rate of inner gas (IG) replicates “FC3” from Strasser [10]. Strouhal numbers in Table 2 are

based on a single dominant tone of 200 Hz. As will be evident, not all simulations show the

same dominant tone, but a general basis is desired for establishment of the Strouhal number. The

lack of specificity around Strouhal number is the reason it is only assigned 1 significant figure.

For all simulations, Z is 0.0012, while the liquid-based Re, Q, Oh, St, and We are 1.3×105,

1.1×10-8, 6.8×10-4, 3.3, and 0.0076, respectively. Note that the Weber number 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 [6] stated

that if the turbulence intensity is greater than 21%, the Weber number for their work spanned 4

regimes.

The base case geometry will be the focus of these evaluations (see Figure 1). Most

injector dimensions are normalized by the outer diameter, DO. Rather than disclosing the actual

dimensions, for proprietary issues, the angle γ and outer diameter itself are normalized by a

maximum value. In dimensionless terms, LAG, LRI, tLI, tLO, γ, DI, DM, and DO are 0.097, 0.60,

0.038, 0.0144, 0.60, 0.36, 0.83, and 0.95, respectively.

3. Terminology and Metrics

In general, parametric study metrics entail pressure signals, spray angle, and data streams

from video analyses (see Strasser [10]). The LabVIEW-based video analysis steps include

frame-by-frame examination of CFD animations, extracting binarized regions of interest, taking

measurements in these regions based on volume fraction gradients, and then exporting

18

measurement results in text format. Specifically, the shoulder distance, or the distance the

ligaments lift off the injector lower-most face for AS models, is an important consideration in

this effort. The results for all metrics for each simulation can be found in Table 3 through Table

5. A brief overview and definition of the table information will be offered here before detailed

information is given by subject category in sub-sections.

Table 3 provides all IG and OG pressure results, Table 4 involves all video analysis

shoulder distance results and spectral alignment ratios between the pressure and shoulder data,

and Table 5 shows time-averaged spray angle measurements for the simulations that involved a

large enough run time in order to assess the angle. The shoulder distances have been normalized

by the nozzle outer diameter. Any pressures are actually pressure drops throughout the entire

feed sections of the inner and outer gas passages and have been normalized by the nozzle

discharge pressure.

Definitions of the quantities are given here, while more on each of these metrics can be

found in Strasser [10]. Due to intense gas stream pulsations and the fact that they 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 “COV” and is a measure of the fluctuation

energy. The “Tone” is in units of Hz and refers to the dominant frequency, which is not

necessarily the fundamental frequency (lowest detectable frequency peak); the dominant tone is

sometimes a higher harmonic multiple. “Mag.” refers to the FFT magnitude and indicates how

focused the frequency spectrum is at the dominant tone. A more focused spectrum will contain a

larger fraction of its fluctuating energy at the dominant tone, producing a higher magnitude for

that tone. Both the tone and mag. are quantified using an FFT algorithm in Matlab. Spectral

19

alignment refers to the particular signal tone and magnitude compared with those of another

signal. Tones and magnitudes of two signals are compared in ratio form; a value of unity

indicates alignment. The premise is that if the source(s) and mechanism(s) of two instabilities

are the same, then the transient signature measures should be similar. The extent to which

various metrics in this work are tied to the pressure fluctuations, i.e., how much control one has

of these by manipulating the pressure response, is of industrial value. For all signal ratio

tabulated values (Table 4), the inner (i) and outer (o) pressure components are used in the

numerator and designates the subscript. Λ is the ratio of the pressure tone to shoulder tone, and

Γ is the ratio of pressure magnitude to shoulder magnitude.

Spray angles are given in Table 5. All spray angle conclusions must be qualified by the

fact that the current modeling framework is AS. As a result, these are not sprays in an absolute

sense, and all comparisons are relative. Spray angle is explored by leveraging two different

measures. The first is the angle containing 50% of the liquid phase by mass, and the second is

the total liquid collected in the volume enclosed by the nozzle centerline ± 30%. Both are

measured approximately 2.5DO from the injector outlet. The spray measures should be expected

to trend opposite to one another; a higher spray angle typically creates a situation in which less

liquid is collected near the centerline. Both metrics are included in that each is more sensitive to

certain situations. In the industrial system, there is no obvious benefit to a design which

produces a high or low spray angle. Typically, one would expect a wider angle, if the space is

fully utilized by droplets, to produce more interfacial area. However, the area encompassed by

the spray gives no indication of droplet size distribution. Additionally, a larger spray angle could

be an indication that the spray droplets are hitting the vessel periphery instead of being

centralized in the reaction zone.

20

4. Validation

4.1. Comparison with Air-Water Experiments

Although the validation of CFD with an AWTS was presented in Strasser [10], additional

data have been collected in order to add confidence to that endeavor. The first point involves a

data stream from a microphone set up just outside the AWTS when running with conditions like

those in this paper. The signal showed harmonics at 225 Hz, confirming that internal acoustics

were transmitting to the external atmosphere at similar frequencies as those measured in CFD.

The next set of experimental data came from a pressure transducer with an AWTS configuration

like that studied here, but with a slightly different geometry, where DO in this test is about 5%

larger than that of the base case. The AWTS IG tone and COV were 230 Hz and 12%,

respectively, compared to 210 Hz and 16% from CFD. The OG stream showed a tone and COV

of 460 Hz and 2.9% compared to CFD values of 210 Hz and 1.7%. While the AWTS OG result

is picking up the 2nd harmonic instead of the dominant tone, the results are otherwise reasonably

similar. The final set of experimental data relates to the shoulder distances coming from video

analysis of the AWTS configured both like the CFD base case and the aforementioned geometry

with a slightly larger outlet. The experimental shoulder tones for these two cases were 250 and

230, respectively, compared to 210 and 220 from CFD. The CFD magnitudes were within 21%

of the AWTS values, and the mean distances were within 26%. Lastly, the work of Tian et al.

[63] also shows about 200 Hz when framed with the conditions of the simulations for this paper.

4.2. Effects of Numerics (Cases A5, A9-A15)

Numerical considerations for such complex physics as those exhibited in this injection

system require deliberate and judicious usage in order to prevent one from arriving at incorrect

conclusions regarding design changes. While some aspects of various modeling frameworks are

21

shown to be advantageous in certain situations, the effects of these on this particular system are

not known. Specifically, during evaluations aimed at scoping the operating space, there is a risk

from a manufacturing/cost standpoint of not detecting the ideal design or flow scenario among

those tested and, even worse, mistakenly viewing a disadvantaged design as advantaged. To this

end, a number of modeling degrees of freedom will be approached through a study of run A5

(base settings) and A9-A15. There is no flawless numerical approximation; with a reasonable

foundation, one can hopefully confidently investigate, at least, directional changes among design

permutations. The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)

method was employed in Case A10. From countless proprietary industrial partner evaluations

(not shown here), CICSAM was found to be substantially more diffusive and exponentially more

computationally forgiving compared to PLIC. The purpose for testing it in the present work is to

determine whether or not it can be used as a viable substitute for PLIC. Examining Table 3, it is

seen that there are some noteworthy effects, but none are astounding. If no large shifts were

detected, they will not be discussed here. The use of PRESTO! and QUICK (A12) produces the

lowest IG tone for this geometry and AW flow. While the use of these numerical techniques is

expected to be superior, the tone drops below the experimental value. Cases A12-A14 all invoke

PRESTO! and QUICK discretization, but A13 employs RKE and A14 uses the multi-

dimensional TVD scheme. The employment of RKE (A13) does not change much compared to

A12 except a minor reduction in IG COV and magnitude and the increase of the OG COV to the

highest value in this comparative group. Application of the multi-dimensional TVD scheme

(A14) had some interesting effects. The IG COV fell to its lowest, while the magnitude rose to

the highest in the group. The IG tone recovers close to the base case (A5), as it is 50 Hz higher

than A13 and A12. The OG tone became double the IG tone, whereas normally it is either equal

22

to the IG tone or three times the IG tone. CICSAM (A10) raised the OG magnitude to the

highest in the group. Surprisingly, tripling the time step size (A9) did not affect much, indicating

that the current time step size is sufficiently small. In other words, using a Courant Number

(CN) threshold to set the time step for a PLIC reconstruction would be more than sufficient

within the context of RANS modeling. The halving of the VOF sub-loop CN (A11 vs. A5) and

the inclusion of the Level-Set computations (A15 vs. A5) affected nothing as well. It makes

sense that the level-set technique did not control the outcome substantially given that the Weber

number is so high and surface tension effects are small. Importantly, this also confirms the lack

of influence of potential spurious artificial velocity components at the VOF-calculated interface,

which are mitigated by coupling level-set to VOF [64]. The shoulder distance metric (Table 4) is

affected somewhat by the numerics. Almost all tests (except the triple time step, A9, and level-

set, A15) lowered the shoulder distance. Particularly the multi-dimensional TVD (A14) and

CICSAM (A10) simulations had the lowest distances. The CICSAM result stands out from the

other tests, both in terms of the mean and tone (peak tone is 4 times the fundamental). This is

one indication that the CICSAM method is not a viable substitute for PLIC. All methods, except

A9, produce lower means and magnitudes than the base case (A5). It should be noted here that

the similarity in the results of A5 and A9 indicates that the current solution and video analysis

methods are repeatable. Case A14 showed the highest shoulder COV. As with the pressures, the

use of PRESTO! and QUICK (A12) lowered the dominant shoulder tones. RKE (A13) showed

the most diffused spectrum (lowest mag.) in this comparative group.

In terms of spectral alignment, the four right-hand columns of Table 4, cases A12-A14

differ from the base case (A5) similarly. They have more tone alignment, but less so with the

magnitude. CICSAM (A10) stands out as vastly different from the others and lacks alignment on

23

all fronts. The uniqueness of the CICSAM result is further evident in Table 5; it had the highest

spray angle, by far, of this comparison group. Figure 2 shows the spray profile comparing the

base case (A5) to the CICSAM model (A10), among others. The A10 profile is less centrally

peaked and wider than that of A5, which is precipitous of the increased interfacial diffusion.

Interestingly, the TVD run (A14) was the most centrally peaked of all. Again, spray profiles are

generated in an AS framework, so all comparisons are relative.

The effects of using CICSAM are further elucidated in Figure 3 showing a typical

instantaneous contour plot of water volume fraction (blue). The white boxes identify areas of

diffused volume fraction gradient that are not present in any of the PLIC simulations.

Additionally, the black box demarks the wider concentration of liquid moving away from the

nozzle, which manifests in an increasing spray angle in Table 5 and lower central peak in Figure

3.

Table 6 summarizes the findings on numerics tests. Here, the three main categories of

metrics are compared to the values from Strasser [10]. A indicates that the results are

reasonably close, typically within 15%, while an “X” indicates that the values deviate

substantially from what is expected from the base case calibration work. In short, it is clear that

CICSAM and auto-refine produce unacceptable results. Likewise, the employment of PRESTO!,

QUICK, and/or the modified TVD scheme create significant shifts in measures.

4.3. Numerically-Induced Instabilities

There is always a concern that flow instabilities can be caused by numerics. For

example, central-difference convective schemes are sometimes needed for adequate momentum

stimulation in scale-adaptive simulations (Gritskevich et al. [65]). However, there are several

issues which, in conjunction, provide sufficient evidence that numerical instabilities are not the

24

root cause of computed injector pulsations. First, flows of two fluids (or the same fluid in two

different streams at different velocities) next to one another are known to produce an unstable

situation. In other words, instabilities are physically correct. Second and related to this, the

actual air-water system is highly pulsatile with a “Christmas tree” type fluctuating spray pattern,

as is well known to the industrial partner and was discussed in Strasser [10]. Third, experimental

and computational pulsations are global and not local. In other words, any spurious pulsations

would have to be significant enough to dramatically affect the feed conditions of the model.

Fourth, the computational results have been validated against the experimental results, especially

in regard to the dominant system response frequency. Fifth, when introducing an ongoing large

modulation at a random time to the inner gas stream, effectively resetting its pulses and changing

the initial conditions, the new bulk pulsations responded in a similar manner. Sixth, a model was

run in which the inner gas stream flow rate was set to zero. The bulk pulsations were eradicated

as was noted in Strasser [10]. Seventh, the coupling of level-set with VOF did not show any

detectable difference, which adds merit to the view that spurious currents are not responsible for

the near interface shear field.

4.4. Droplet Size Quantification

In preparation for 3-D studies at some point in the future, two questions for any droplet

size analysis method are posed. The first question is, “can the Sauter Mean Diameter (SMD) in a

complicated computational domain be accurately computed?” One issue is that the first

“droplets” are just ligaments that are elongated in the axial dimension, i.e., not spheres. To that

end, tests were carried out in which various shapes, sizes, placements, and orientations were

placed in a 11.25° (1/32nd) sliver of the full injector domain. Figure 4 illustrates the

computational domain for tests 3 – 6, and the test details are in Table 7. The small cube is

25

partially obscured and is therefore annotated, while the rest of the particles are various sized

spheres and a cylinder. To ensure a correct mimic of the more complex model situation, the last

sphere to the right is actually split across the periodic planes (translucent grey bounds). Dashed

black lines indicate the 5 separate volume “buckets” within which statistics are gathered.

Table 7 results are organized as follows. The vertical grouping represents different

shapes and meshes, while the horizontal groupings delineate the 5 buckets of results. For

example, Test 1 (top group) involved a relatively coarse tetrahedral mesh. The first bucket

(farthest upstream) is liquid-full, while the second contains two 0.2″ diameter spheres. The third

bucket included no liquid, while the fourth and fifth buckets had three 0.1″ spheres and one 0.4″

sphere, respectively. An SMD algorithm developed as a user defined function (UDF) computed

liquid volume and interfacial area and then returned the SMD. The “error” row shows how the

algorithm differed from the theoretical SMD. Errors as high as 6% exist. Test 2 involved nearly

what is shown in Figure 4 but without the periodically broken sphere. Errors improve with finer

tetrahedral mesh elements. Test 3 involves all shapes arranged as shown in Figure 4 but with a

slightly different meshing method and the last sphere is periodic. Test 4 uses a finer mesh than

Test 3, and the errors are lower. Test 5 is the same system as Test 4, but the mesh has been

converted directly to polyhedral elements. The errors fall significantly in each measured

volume. Lastly, Test 6 incorporates only a single measurement bucket so that all shapes are

included in a single computed SMD with a final error of about 2%. It is concluded that the

algorithm works.

4.5. Droplet Production

Now that it is has been confirmed that droplet sizes can be quantified, the question

remains as to whether or not the computational method can generate correctly-sized droplets.

26

The experimental work of Aliseda et al. [66] is duplicated in an unsteady 3-D computational

framework for validation purposes. Only their most viscous non-Newtonian liquid was

considered, and the time-averaged results are shown in Figure 5. The complex shear and time-

dependent nature of non-Newtonian materials gives them processing advantages, but the

treatment of their viscosities for simulations is not so obvious. Though Tavangar et al. [64]

discuss the complexities in assigning an effective viscosity to non-Newtonian materials when

modeling breakup processes, others [66-68] show how the viscosity can be approximated to be a

constant high-shear value even for extremely complex materials such as coal-water slurry. The

experimental error was not discussed in the original manuscript [66], but personal

communication with the lead author indicated that the expected error could be ± 15% + 2.5

microns. Their domain extent was quite different than that of the present test. They wanted to

study farther away from the liquid orifice; however, mesh and domain size limitations prevented

the inclusion of such distances in this CFD effort. SMD data are all normalized by the distance

from the nozzle, and the distance from the nozzle has been normalized by the liquid orifice

diameter. The vertical bars are temporal sampling variability (COV) of the CFD data. In the

region of overlap, between 10 to 20 liquid orifice diameters, there is strong agreement. A curve

fit of the form y = 0.36x-1.3 includes both data sets well. While various mechanisms are

competing to produce a wide variety of droplet sizes and shapes in the early stages of

atomization, it appears that there is a generally monotonic trend for non-Newtonian viscous

materials.

4.6. Shock-induced shattering

Based on the aforementioned references [2, 69-74], the translational acceleration of an

initially stationary droplet interacting with a shock wave was studied. A lone droplet of water

27

with an initial droplet size, dl.0 =1 mm was suspended in air in front of a diaphragm (vertical

wall) restraining 1.0x107 Pa (shock tube pressure ratio of about 100). The diaphragm ruptured

(wall disappears instantly) at t=0. The implicit time step size was 1.0x10-8 seconds for a CN

peak of unity. The Weber number was 2.0x104, Reynolds number = 1.0x105, Ohnesorge number

= 4.0x10-3, and aerodynamic time scale

()

()

,lo l g

BdU

ρρ

= was 24 microseconds. The peak

Mach number at the droplet was about 2.2. A sample flowfield superimposed with Mach

number of the air interacting with the droplet is shown in Figure 6 at 0.23 dimensionless times

(t/B) after the shock has passed the droplet (flow is from left to right). The air vectors are shown

scaled from blue for Mach =0.0 to red for Mach = 2.2. The bow shock is clearly seen upstream

from the front of the droplet, and the shock layer is thickening with time. Not shown in this plot

is the sonic line, which is moving leeward around the face of the droplet in the shock layer. The

droplet surface instabilities are present and are growing in time. The shattering of, and mist

formation within, the thin droplet rim is seen as the droplet major axis (perpendicular to the

direction of flow) is increasing. The dimensionless trajectory of the droplet showing its

acceleration is provided in Figure 7. The ordinate is the travel distance normalized by the

original droplet diameter, while the abscissa is dimensionless time squared. For the CFD work,

the mass centroid of the droplet is the metric; for the experimental works, the authors used

photographs to assess the droplet position. These two measures are not exactly the same, but

should be reasonably similar. From those experimental studies, the literature teaches that a

straight line on this plot should have a slope of 0.7 to 1.1. The CFD value of 0.9 lies in this

range.

28

5. Results and Discussion

5.1. General Flow Features

Sample instantanous volume fraction fields from a non-swirling (left, case A19) and

swirling (right, case A7) flow are shown in Figure 8. Bulk flow is from top to bottom, and the

contour colors of blue and red represent the water phase and gas phase, respectively. Since the

two fluids are represented here, the interface would be the thin region between the two fluid

colors. Ringed layers of water are shown being shed from the pulsing field, progressively

widening in the axial direction, and are denoted as a “Christmas tree” pattern in [10] (shown in

AS and 3-D simulations, as well as AWTS video footage) due to the wide flair as water flows

downward. Notice that the liquid annulus merges near the center forming a liquid bridge in the

non-swirling case, while the liquid does not form a bridge for the swirling case. More on the

effects of swirl will be presented in a later section.

A close examination of liquid bridging and the resulting inertial inner fountain is offered

in Figure 9. Sequential, but not necessarily evenly spaced in time, instantaneous contours of

liquid volume fraction are included beginning at the top-left and moving to the bottom-right of

the figure. The number at the top of each frame is the approximate nondimensional time in a

given cycle. At t/H = 0, where H is the approximate cycle time, the primary bridge is formed

across the liquid annulus just above and behind the text box (shown as a continuous blue color

from left to right in the figure), while the fountain is seen ejecting back into the flow stream

farther than is visible and is denoted with a yellow arrow. A new series of waves develop on the

inner edge of the liquid annulus (t/H = 0.3), marked using a white circle, and reveals the breaking

of those new waves and the Rayleigh-Taylor disintegration of the primary fountain. Next the

new waves merge with the fountain to form the secondary bridge (t/H = 0.4). Remnants of the

29

primary fountain are still moving upstream. Then, the secondary bridge grows (t/H = 0.5) while

the primary fountain stops rising vertically due to the balance of gravity and inertia. At t/H = 0.6

is the onset of a secondary upstream-facing ejection, which will be a new fountain. The old

fountain remnant moves downstream and is denoted by the yellow downward pointing arrow. In

the final frame, the two fountains merge. The shed rings of the Christmas tree pattern can also

be observed moving downward once the fountains merge. These processes repeat in a nearly

regular manner to produce a rich spectrum of pressure pulses with a dominant tone. Upon close

inspections of movies (many available in the online version of Strasser [10]), one will find that

the overall spray pulsation is more closely tied to the inner gas stream dynamics which is

supported by this analysis.

Figure 10 offers further exposition. Three contour plots from a single random time

sample are provided. The cycle time is near t/H = 0.3 (see Figure 3) just before the bridge is

formed upstream of the nozzle outer face. In the retracted section, the pressure front mimics the

movement of volume fraction front for the most part. Although there is no liquid bridge at this

time instant, pressure builds across the volume fraction annular region forming a pressure bridge

(white oval). Local areas of Mach number greater than unity are seen near the gas-liquid

interface (near the aforementioned pressure bridge) and ligament formation area farther

downstream. Local regions of relatively high Mach number are also present in the outer gas

shear layer, but these regions are not supersonic

Three additional random sample contours of Mach number can be seen at other instants

in time in Figure 11 with the same Mach color scaling. These represent a cycle time near t/H =

0.3 (bridge forming inside nozzle) taken at approximately the same time for three uncorrelated

cycles to elucidate the rapid changes in flow and corresponding Mach numbers. The dotted

30

horizontal line in Figure 10 is a reference for the narrow point in the liquid to show the (slight)

time progression of the flow; the cycles are not perfectly repeatable as would be expected in

complex systems. Focused regions of high Mach number are persistent, and there are local

Mach depressions within some of them. For the time being, these local accelerations will be

referred to as “shocklets”. The term “shocklet” is aimed more towards micro-scaled fluid

structures, likely smaller than those being resolved in the present work, and is discussed in

Gatski and Bonnet [17], Pirozzoli and Grasso[75], and Freund et al. [76]. Shocklets were found

to form more easily in 2-D turbulence than in 3-D. The exact definition is ambiguous. Pirozzoli

and Grasso [75] used a ratio based on the first invariant of the deformation tensor to locate

shocklets. Their results showed that shocklets occupied a very small fraction of the fluid volume

even up to turbulent Mach numbers of 0.8 and that they contributed approximately 20% of the

dilatational dissipation. Of those shocklets, about 80% of their volume was compressed

convergent and shear zones, while 20% was compressed eddies. Freund et al. [76] claimed

shocklets are defined by areas of negative dilatation. They computed and compared local

negative dilatation rates to local vorticity magnitudes. The vorticity could be a result of the pre-

existing shear layers in the boundary conditions, vortex stretching regions far from the shocklets,

or that resulting from curved shock fronts or local compressible flow accelerations as dictated by

Crocco’s relationship (Batley, McIntosh and Brindley [77]). In [77], shocklets formed in areas

where this ratio was as low as 0.25. Even the authors could not differentiate between shocklets

and weak oblique shocks.

The dilatation-vorticity ratio is computed in the present work for a random time sample,

different than the times sampled in prior figures, as is shown in Figure 12. On the left is a

contour plot of Mach number, and on the right is a contour plot of the ratio of negative dilatation

31

to vorticity magnitude scaled from 0.0 (blue) to 1.0 (red). They are taken at the same time

sample. The local regions of high Mach number (left) do not necessarily coincide with the areas

of high dilatation ratio (right), although the main spotty high Mach number front about 0.75

diameters away from the nozzle outlet does have a corresponding band of high dilatation ratio on

the right (comparison made in white outline). Additionally, it should be noted that the turbulent

Mach number (not shown) exceeds 0.25 primarily in these spots but that the values typically

exceed 4 in isolated regions at this instant in time and others. There are areas of high dilatation

which are even more prominent than those outlined, including some relatively macro-scale

regions in the comparatively un-sheared retracted section, and are obviously not shocklets.

Regardless of the definition of shocklet, it is well known [2, 69-74] that shock fronts passing

over droplets can, on time scales on the order of microseconds, cause aerodynamic shattering

such that the secondary droplet diameters are 2 to 3 orders of magnitude smaller than that of the

initial droplet. Olson and Cook [78] describe how in Rayleigh-Taylor instability-driven systems,

shocklets can merge to form a defined shock wave and that the resulting 3-D shocks are stronger

than their 2-D counterparts. Apparently there is not enough coherency in the shocklet-like

structures in the present work (due to liquid pulse disruptions) to allow large-scale shock wave

formation. This conclusion must be qualified by the fact that the current work is limited to an

AS modeling framework.

5.2. Effects of Swirl and Inlet Boundary Conditions (Cases A5, A7, and A17-19)

Swirling feeds offer advantages in process equipment ([79]). Of primary interest is to

find realistic and amenable swirl-inducing mechanisms inside the two gas channels and liquid

channel and evaluate their effects on atomization and acoustics. Conjoined with this swirl

consideration is the need to consider the effect of the length of the modeled inlet passages and

32

incompressibility; the reasons will soon be apparent. Figure 13 shows some of the many

examples of what was considered to induce swirling flow. The ability to create strong swirl was

capped by limitations with what could physically be installed in the reactor system. As

discussed, swirl is commonly used in reaction systems, but its effects are not known for this

injector.

In the upper right-hand corner of Figure 13 is a twisted-tape type IG mechanism, while

the lower image shows a similar liquid mechanism colored by pressure (red is high, while blue is

low). A sample mesh from a portion of the liquid mechanism is shown in the upper left-hand

corner. These tests (proprietary and not outlined here) were modeled as steady-state, single

phase, incompressible and with truncated inlets (cut down near the retracted section). Each feed

stream was isolated from the others. The purpose was to find the highest swirling velocity

profiles exiting the modeled sections. The final, optimized profiles were then fed to the inlets of

the compressible, multiphase, unsteady full domain models which comprise the bulk of this

work; the cost of addressing all physics and mechanisms simultaneously in the swirl profile

optimization studies was prohibitive. Due to the fact that the inlet profiles to the current models

were generated with truncated inlets, a truncated baseline of compressible, multiphase, unsteady

models had to be run in order to obtain a direct comparison of the swirling effect.

Firstly, the effect of inlet truncation in a compressible solver is exposed (refer to Table

3). Case A19, relative to A5, shows a significantly reduced IG mean (consistent with a shorter

passage) and magnitude, a dramatically increased IG COV (less cushion) and an OG dominant

tone lowered to its fundamental. It is surprising that the OG did not respond much to inlet

truncation. Apparently the OG dynamics are not controlled by resonance in the inlet plenum.

The shoulder distance metric response in Table 4 indicates that the main effect is a much more

33

diffused shoulder spectrum (lower mag.) and a mild reduction in the tone. As shown in Table 4,

the alignment of the spectral tones does not change much, but the magnitudes depart

substantially as a result of truncation. Case A19 has a much larger spray angle than A5 in Table

5 and is illustrated in Figure 12, where the A19 center peak is lower and spray profile is more

diffuse than A5. It was quite unexpected that the length of the inlets would have such a dramatic

effect on the spray shape but not the shoulder liftoff.

For the effect of incompressibility with a truncated geometry, A17 (incomp.) is compared

to A19 (comp.). Though the real system is compressible, the 3-D work presented in Strasser

[10] was limited to incompressible and truncated simulations; therefore, this work will quantify

the effects of those limitations and establish needs for additional studies. Table 3shows an

increase in IG mean pressure and pressure COV (due to less cushion) and a decrease in IG tone

and magnitude when compressibility is removed. The OG mean pressure falls, but the COV

significantly increases (again, due to less cushion). The OG tone fell like that of the IG, but the

magnitude increased. Additionally, the lack of compressibility significantly raised the shoulder

distance and reduced its COV, and the magnitude nearly doubled (Table 4). The spectral

alignment improved without compressibility. As seen in Table 5, removing compressibility

substantially reduced the spray angle, mostly offsetting the effect of inlet truncation.

What is very interesting to note is that the truncated and incompressible case (A17) is not

terribly different than the base case (A5) in terms of acoustics (Table 3). This is consistent with

the spray angle measurements, showing the offsetting effect of inlet truncation and

incompressibility. Obviously, there are temporal pressure variability differences, but the mean

pressures and FFT results are not in great contrast. The IG dominant tones are reasonably close,

and the OG tones are nearly multiples of one another. Given that experience has shown that the

34

shifting around of the dominant tone among its harmonics is very sensitive, there is similarity.

This implies that the overall pulsating dynamics of this system are more governed by the liquid-

gas interactions than they are the pressure communicating time scales of the feed passages.

Finally, the direct comparison for the swirling effect on the metrics comes from

simulations A7 (swirling feed profiles, compressible, and truncated) and A19 (non-swirling,

compressible, and truncated). In terms of pressure measures, there is little change with the

addition of swirl. Shoulder distance metrics did change: the mean rose, the COV fell, the tone

fell, and the spectrum became much more focused (higher mag.). The tones became less aligned,

but the magnitudes improved. The spray angle metrics did not change noticeably, but the spray

character did change slightly. With swirl, the profile was less centrally peaked with closer to a

hollow cone style profile as shown in Figure 12. The cause of the hollow cone-like profile is

shown in Table 1Figure 3.1, where volume fraction contours reveal an open central area with no

liquid bridge. The effect of swirl on reattachment length after a sudden expansion is investigated

by Vanierschot and Van den Bulck [80]. They found a reduction in reattachment distance with

increasing swirl, up to the point of diminishing returns. The implication with the present work

would be a widening of the spray angle or perhaps a reduction in shoulder distance. Neither of

those is seen here for the levels of swirl tested. Swirling effects may have been more evident if

the strength of the swirl was not fixed by limitations related to mechanical installations. In other

words, enough swirling could not be induced to achieve the desired effect.

6. Conclusions

Eleven unsteady transonic injector models, plus validation simulations and experimental

work, have been executed in order to discover the mechanism for bulk pulsations and to study

the effects of numerical methodologies and swirl on the self-generating pulsatile spray produced

35

by an industrial scale three-stream coaxial airblast reactor injector. With a focus on ligament

formation in the retracted section, measurements included inner and outer gas stream pressure

pulsations, transient shoulder distance, time-averaged spray angle, and spectral alignment. Since

often various metrics were synchronized for a given simulation, a common driving mechanism

for all global instabilities is inferred. That mechanism responsible for bulk pulsations is liquid

bridging with the production of a liquid fountain and shocklet-like structures in the retracted

zone. Swirl had a minor impact by slightly moving the ligament shoulder away from the nozzle

outlet and changing the spray to a hollow cone shape; the effects of swirling may have been

more pronounced if the capability to produce strong swirl existed in the mechanical bounds of

the system.

Numerical considerations made noticeable differences in some parameters, which stress

the importance of using documented and consistent recipes when comparing various flow

conditions and geometries. CICSAM was found not to be a drop-in replacement for the much

less forgiving PLIC scheme. Using a CN threshold to set the time step for a PLIC approach

should be sufficient for temporal resolution in a RANS framework.

Although overall system pulsations appear to be driven more by interfacial phenomena

than by feed piping acoustical time-scales, the length of feed piping is critically important to

develop accurate spray profiles. However, there is a partially offsetting effect of

incompressibility on inlet truncation. Outer gas dynamics are not controlled by resonance in the

inlet plenum but by local feedback from the fluctuating gas-liquid interface.

Three tiers of validation exercises confirmed that correct droplet sizes could be produced

computationally, the Sauter mean diameter of droplets/ligaments could be quantified, and the

trajectory of a droplet intersecting a shock wave could be accurately tracked. Both CFD and the

36

experimental non-Newtonian primary atomization droplet size results, when scaled by distance

from the injector outlet, showed an inversely proportional to injector distance. Additionally,

more data has become available from our air-water apparatus to add weight to the validity of the

overall computational approach.

One valuable outcome of this effort is the ability to apply the models developed herein to

more widely varying nozzle geometries, liquid properties, and flow conditions or to other

industrial applications. In regards to new nozzle geometries, more aggressive outer stream

meeting angles, variable retractions, larger inner stream meeting angles, and smaller diameters

(higher shear) may be considered. Similarly, higher Mach numbers might be utilized to create

shock-augmented droplet shattering. Other industrial applications might include the

consideration of instabilities within polymer spinning or extrusion operations.

Acknowledgments

The support of a multitude of Eastman Chemical Company (named 2016 ENERGY

STAR® Partner of the Year - Sustained Excellence as well as a 2015 World’s Most Ethical

Company® by the Ethisphere Institute) 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 Steffan, Bill Trapp, and Kevin White were supporters of this effort. George

Chamoun and Jason Goepel deserve special recognition for developing video analysis tools,

constructing UDFs, and processing on the order of 1000 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

37

Jean Le Rond D'Alembert UPMC, and Christophe Dumouchel of Université et INSA de Rouen

were extremely beneficial.

Nomenclature

a Speed of sound

AWTS Air/water test stand

B Aerodynamic response time =

()

,0llg

dU

ρρ

COV Coefficient of variation = standard deviation/mean×100%

dl,0 Initial droplet size

DI Nozzle innermost diameter normalized by nozzle outer diameter

DM Nozzle intermediate diameter normalized by nozzle outer diameter

DO Nozzle outermost diameter normalized by the maximum outer diameter

E Total energy

F Surface tension body force

H Injector pulse cycle time

IG Inner gas stream

k Turbulent kinetic energy

La Laplace number = Re2/We (also known as the Suratman number)

LAG Outer annular gap normalized by nozzle outer diameter

LC Characteristic length scale

LRI Inner retraction length normalized by nozzle outer diameter

LS Shoulder length, or spray lift-off, normalized by outer diameter

M Gas/liquid momentum ratio = (ρU2)G/(ρU2)L

Ma Mach number

OG Outer gas stream

Oh Ohnesorge number = We /Re = 1/

L

a

P Pressure; tabulated pressure drop normalized by outlet pressure

Pr Liquid phase Prandtl number

Q Viscous capillary length

Re Reynolds number = ρUD/μ

S Velocity ratio, UG/UL

St Strouhal number

SMD Sauter mean diameter (“D32”)

tLI Thickness of inner lip normalized by nozzle outer diameter

tLO Thickness of outer lip normalized by nozzle outer diameter

T Static temperature

u Velocity component

U Velocity magnitude

y+ ρuty/μ

We Weber number = ρU2LC/σ

Z Density ratio, ρG/ρL

38

Greek

α Phase volume fraction

ε Turbulence dissipation rate

σ Surface tension

φ Wall distance

ρ Density

ω Specific dissipation rate

τ Stress tensor

γ Outer gas/liquid annular approach angle normalized by maximum value

μ Molecular viscosity

ζ Molecular thermal conductivity

Λ Ratio of pressure tone to shoulder tone

Γ Ratio of pressure magnitude to shoulder mag.

ϑ Ratio of pressure tone to droplet length scale tone

Ξ Ratio of pressure magnitude to droplet length scale magnitude

Subscripts and Superscripts

i Summation index

L Liquid

G Gas

t Turbulent

ref Reference condition

I Inner gas

O Outer gas

39

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46

List of Tables

Table 1. Matrix of all axisymmetric air-water simulations for numerics and swirl simulations

Table 2. Dimensionless groups

Table 3. Inner and outer gas pressure drop statistics from axisymmetric air-water simulations for

all numerics and swirl simulations

Table 4. Shoulder distance metric and ratio of pressure drop signal statistics to those of the

shoulder distance metric from axisymmetric air-water simulations

Table 5. Time-averaged spray angle measures from axisymmetric air-water simulations for

numerics and swirl simulations

Table 6. Summary of numerical studies from axisymmetric air-water simulations

Table 7. SMD, computed versus theoretical, for various shapes and sizes in test injector domain.

47

Table 1. Matrix of all axisymmetric air-water simulations for numerics and swirl simulations

Case Purpose

A5 Base Case

A7 Trunc./Swirl

A9 3xTS

A10 CICSAM

A11 VOF CN

A12 Discret.

A13 RKE

A14 TVD

A15 Level-Set

A17 Trunc./Incomp.

A19 Trunc.

48

Table 2. Dimensionless groups

Inner Gas Outer Gas

M S Re/100 St Ma M S Re/100 St Ma

0.080 8.1 480 0.1 0.11 1.3 33 1000 0.01 0.44

49

Table 3. Inner and outer gas pressure drop statistics from axisymmetric air-water simulations for all

numerics and swirl simulations

Inner Gas Outer Gas

Case Mean COV Tone Mag. Mean COV Tone Mag.

A5 0.27 16 200 0.49 0.33 4.4 610 0.63

A7 0.20 130 220 0.46 0.32 4.2 220 0.35

A9 0.26 15 200 0.37 0.32 3.7 600 0.84

A10 0.26 13 200 0.48 0.33 4.1 600 1.0

A11 0.27 16 200 0.51 0.33 4.4 610 0.67

A12 0.24 18 140 0.58 0.31 5.5 140 0.63

A13 0.24 12 140 0.28 0.31 5.8 130 0.65

A14 0.26 7.7 190 1.14 0.32 5.7 380 0.37

A15 0.27 16 200 0.48 0.33 4.3 610 0.73

A17 0.28 190 170 0.39 0.27 24 170 0.82

A19 0.20 120 210 0.68 0.31 4.5 210 0.56

50

Table 4. Shoulder distance metric and ratio of pressure drop signal statistics to those of the shoulder distance

metric from axisymmetric air-water simulations

Case Mean COV Tone Mag. ΛI ΛΟ ΓI ΓΟ

A5 0.45 43 210 0.44 1.0 3.0 1.1 1.4

A7 0.54 34 110 0.15 2.0 2.0 3.1 2.4

A9 0.45 48 210 0.44 1.0 2.9 0.85 1.9

A10 0.22 49 800 0.19 0.25 0.75 2.6 5.5

A11 0.39 48 210 0.33 1.0 3.0 1.6 2.0

A12 0.39 54 140 0.14 1.0 1.0 4.2 4.5

A13 0.37 56 130 0.12 1.1 1.0 2.3 5.3

A14 0.30 68 190 0.25 1.0 2.0 4.6 1.5

A15 0.39 46 210 0.30 1.0 3.0 1.6 2.4

A17 0.64 24 170 0.027 1.0 1.0 14 30

A19 0.45 44 170 0.016 1.3 1.3 42 34

51

Table 5. Time-averaged spray angle measures from axisymmetric air-water simulations for numerics and

swirl simulations

50% +/- 30%

Case Angle Liquid

A5 13 70

A7 33 52

A9 13 71

A10 31 54

A11 12 71

A12 14 80

A13 13 82

A14 12 85

A15 12 70

A17 11 95

A19 31 56

52

Table 6. Summary of numerical studies from axisymmetric air-water simulations

Case Pressure LS Angle

A5

A9

A10 X X

A11

A12 X X X

A13 X X X

A14 X X X

A15

53

Table 7. SMD, computed versus theoretical, for various shapes and sizes in test injector domain.

Measurement

Block 1 2 3 4 5

Test 1: coarse tetrahedral mesh elements

Configuration liquid full 2 spheres (0.2″) empty 3 spheres (0.1″) 1 sphere (0.4″)

Theoretical D32 ∞ 5080 ∞ 2540 10160

CFD ∞ 4883 ∞ 2420 9559

D32 Error N/A -3.90% N/A -4.70% -5.90%

Test 2: finer tetrahedral mesh elements, more shapes

Configuration cube (0.1″) 2 spheres (0.2″) cylinder

(0.8″× 0.3″)

3 spheres (0.1″)

1 sphere (0.4″) 1 sphere (0.4″)

Theoretical D32 2540 5080 9625 8957 10160

CFD 2559 4949 9456 8701 10992

D32 Error 0.70% -2.60% -1.80% -2.90% 8.20%

Test 3: last sphere periodic, slightly different meshing

Configuration cube (0.1″) 2 spheres (0.2″) cylinder

(0.8″× 0.3″)

3 spheres (0.1″)

1 sphere (0.4″)

periodic sphere

(0.4″)

CFD 2623 4923 9384 8683 9689

D32 Error 3.30% -3.10% -2.50% -3.10% -4.60%

Test 4: configuration same as Test 3, but finer mesh

CFD 2594 4915 9361 8679 9720

D32 Error 2.10% -3.20% -2.70% -3.10% -4.30%

Test 5: configuration same as Test 4, but polyhedral mesh elements

CFD 2572 5068 9633 8959 10143

D32 Error 1.30% -0.20% 0.10% 0.00% -0.20%

Test 6: configuration same as Test 5, but only one sampling volume for all shapes

Theoretical D32 9109

CFD 8892

D32 Error -2.40%

54

List of Figures

Figure 1. Geometry and mesh for three-stream injector

Figure 2. Time-averaged spray profiles from axi-symmetric air-water simulations.

Figure 3. Sample instantaneous volume fraction contours CICSAM Case A10; blue represents

water, while red represents gas

Figure 4. Injector sliver model for testing shape length scale quantification

Figure 5. CFD results compared to experimental results of non-Newtonian primary atomization

from Aliseda et al [66].

Figure 6. Vector plot colored by Mach number of a normal shock wave in air just having passed

over a single droplet of water

Figure 7. Dimensionless lateral trajectory of a droplet having been exposed to a shock wave

Figure 8. Sample instantaneous contours from non-swirl (left, Case A19) and swirl (right; Case

A7) flows showing the swirl opening of the spray from axisymmetric air-water simulations.

Figure 9. Typical time sequence using volume fraction contours from axisymmetric air-water

simulations (Case A5), starting from top-left and proceeding to bottom-right. The number near

the top of each frame represents the approximate time (t/H) that frame captures in a given cycle,

starting with 0 for the upper left-hand frame.

Figure 10. Instantaneous contours at t/H = 0.3 of volume fraction, pressure front and Mach

number from

axisymmetric air-water simulations (Case A5). The cycle time is just before the bridge forms.

The left image is volume fraction (blue = liquid, red = gas), the middle shows the resulting

pressure front (purposely undisclosed, red = high, blue = low), and the right image provides

Mach number contours (blue = 0, while red designates ≥ 1).

Figure 11. Mach number contours at time samples from three uncorrelated cycles from

axisymmetric air-water simulations (Case A5). The cycle time is close to t/H = 0.3 just before

the bridge forms upstream of the nozzle outer face. The dotted line shows a very slight cycle

time progression from left to right.

Figure 12. Instantaneous contours showing Mach number (left) scaled from blue = 0.0 to red ≥

1.0 and the ratio of negative dilatation to vorticity magnitude (right, same time sample) scaled

from blue = 0.0 to red = 1.0 at the same time instant from axisymmetric air-water simulations

(Case A5).

Figure 13. Swirl-inducing mechanism examples; lower figure is colored by undisclosed pressure

from low (blue) to red (high)

Figure 1.

G

G

eometr

y

and

55

mesh for thr

e

e

e-stream in

je

e

ctor

56

Figure 2. Time-averaged spray profiles from axi-symmetric air-water simulations.

0

1

2

3

4

5

6

7

-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75

Water Vol ume [%]

Normalized Distance

A5

A7

A10

A12

A14

A19

Figure 3

.

.

Sample inst

a

a

ntaneous volume fraction

red

r

57

contours CI

C

r

epresents

g

a

s

C

SAM Case A

1

s

1

0; blue repr

e

e

sents water,

w

w

hile

Fi

g

ur

e

e

4. In

j

ector

s

s

liver model f

o

58

o

r testin

g

sha

p

p

e len

g

th scal

e

e

quantificati

o

o

n

59

Figure 5. CFD results compared to experimental results of non-Newtonian primary atomization from Aliseda

et al [66].

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 20406080100

Normalized Droplet Length Scale

Normalized Distance From Orifice

CFD

Experiment

Exp. Error

Figure 6.

Vector plot c

o

o

lored b

y

Ma

c

c

h number of

dr

o

60

a normal sho

c

o

plet of water

c

k wave in ai

r

r

j

ust havin

g

p

p

assed over a sin

g

le

61

Figure 7. Dimensionless lateral trajectory of a droplet having been exposed to a shock wave

0

4

8

12

16

0 4 8 12 16

x/d

l,0

(t/B)

2

CFD

Linear Fit

Figure

8

8

. Sample ins

t

showin

g

t

antaneous co

n

the swirl ope

n

n

tours from

n

n

in

g

of the sp

r

62

n

on-swirl (left

,

r

a

y

from axis

y

,

Case A19) a

n

y

mmetric air-

w

n

d swirl (ri

g

h

water simula

t

t; Case A7) fl

o

t

ions.

o

ws

Figure

9

(Case A

5

represen

t

9

. T

y

pical ti

m

5

), startin

g

fr

o

t

s the approxi

m

e sequence u

s

o

m top-left a

n

mate time (t/

H

in

g

volume f

r

n

d proceedin

g

H

) that frame

h

a

63

r

action conto

u

to bottom-ri

g

captures in a

a

nd frame.

u

rs from axis

y

g

ht. The nu

m

g

iven c

y

cle, s

t

mmetric air-

w

m

ber near the

t

artin

g

with

0

w

ater simulat

i

top of each fr

0

for the uppe

r

i

ons

r

ame

r

left-

Figure

1

axis

y

mm

e

is vol

u

undisclos

e

1

0. Instantan

e

e

tric air-wate

r

u

me fraction

(

e

d, red = hi

g

h

e

ous contours

r

simulations

(

blue = liquid

,

, blue = low),

at t/H = 0.3 o

f

(Case A5). T

h

,

red =

g

as), t

h

and the ri

g

ht

de

s

64

f

volume frac

t

h

e c

y

cle time

i

h

e middle sho

w

ima

g

e provid

s

i

g

nates ≥ 1).

t

ion, pressure

i

s

j

ust before

w

s the resulti

n

d

es Mach nu

m

e

front and M

a

the brid

g

e fo

r

ng

pressure f

r

m

ber contours

a

ch number f

r

r

ms. The left

i

r

ont (purpose

l

(blue = 0, wh

i

r

om

i

ma

g

e

ly

i

le red

Figure

1

water si

m

n

o

1

1. Mach nu

m

m

ulations (Ca

s

o

zzle outer fa

c

m

ber contours

s

e A5). The c

y

c

e. The dotte

d

at time samp

l

y

cle time is cl

o

d

line shows a

65

l

es from thre

e

o

se to t/H = 0.

3

ver

y

sli

g

ht c

y

e

uncorrelate

d

3

j

ust before t

y

cle time pro

gr

d

c

y

cles from

a

t

he brid

g

e for

m

r

ession from

l

a

xis

y

mmetric

m

s upstream

o

l

eft to ri

g

ht.

air-

o

f the

Figure

1

ratio of

n

2. Instantan

e

n

e

g

ative dilat

a

1.0 at th

e

e

ous contours

a

tion to vorti

c

e

same time i

n

showin

g

Mac

c

it

y

ma

g

nitud

e

n

stant from a

x

66

h number (le

f

e

(ri

g

ht, same

x

is

y

mmetric a

f

t) scaled fro

m

e

time sample)

a

ir-water sim

u

m

blue = 0.0 t

o

)

scaled from

b

u

lations (Case

o

red ≥ 1.0 an

d

b

lue = 0.0 to

r

A5).

d

the

r

ed =

Figure

1

1

3. Swirl-ind

u

u

cin

g

mechan

i

i

sm examples

;

(blu

e

67

;

lower fi

g

ure

e

) to red (hi

g

h

)

is colored b

y

)

undisclosed

p

p

ressure from low