Identification of Pulsation Mechanism in a Transonic Three-
Stream Airblast Injector
Eastman Chemical Company
Kingsport, TN, 37660
Department of Mechanical Engineering
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061
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
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.
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
. 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 . Important insights into the mechanisms
of coaxial primary atomization are elucidated by [3, 4]. Nourgaliev et al.  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  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
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  and . 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
As noted by Lefebvre , 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  by the
primary atomization process, linking acoustics with primary atomization. The system behaves
much like a periodic “stable limit cycle” (Strogatz, ) 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
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 , and have been found relevant to
the food industry , chemical process industry , and heat transfer control . 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  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
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 . 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.
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  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  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
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 . 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:
() ( )
The phase-averaged Reynolds-averaged (RANS) linear momentum balance is:
() () ( )
uu u p gF
∂+⋅∇ =∇⋅ + −∇+ − +
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 .
The risks are especially relevant when capturing the momentum balance at the interfacial jump
conditions as is discussed in . 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  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
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:
() ( ) ()
∂+∇⋅+=∇⋅ + ∇ + ⋅ +
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  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
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. .
The homogeneous shear stress transport (SST) two-equation linear eddy viscosity model
of Menter  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.  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 , preventing the artificial build-up of fluctuating
velocity in regions of irrotational strain. "Scalable" wall functions, discussed in ANSYS ,
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 ; the typical constant known as “beta”, which is used in the SST method  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
function (Hansch et al. , Ergorov , and Deendarlianto et al. ); 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
 and cyclone studies [36, 37]. SST improvements were proposed  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 . 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  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.
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  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  (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.  describe three common
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  and Gueyffier et al. . 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 ; 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 . According to ANSYS  and Liovic , 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 , Anumolu and Trujillo , and
Xiao et al.  is evaluated (case A15). More computational details on the Level-Set method can
be found in Menard et al. . VOF is known for its strong mass conservation at the cell level,
but it has difficulty reconstructing the interfacial curvature and surface normals. As a result,
spurious velocities sometimes result at the VOF-computed interface  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 .
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
discretization schemes can be found in Strasser  and in ANSYS . 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 . 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 , but the “multidimensional” method (more rigorous and less diffusive) of Kim et
al.  is investigated for A14. The advantages of this scheme are discussed more in Poe and
Convergence and time-averaging were discussed in detail in Strasser  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 , 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.  is 2 to 4 orders larger than what is used in these simulations. Total simulation time
includes reaching quasi-steady behavior (as shown in ) 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
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
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 . 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 ), 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
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 . 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  stated
that if the turbulence intensity is greater than 21%, the Weber number for their work spanned 4
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 ). 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
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
Definitions of the quantities are given here, while more on each of these metrics can be
found in Strasser . 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
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.
4.1. Comparison with Air-Water Experiments
Although the validation of CFD with an AWTS was presented in Strasser , 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.
 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
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
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 . 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
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
Table 6 summarizes the findings on numerics tests. Here, the three main categories of
metrics are compared to the values from Strasser . 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. ). However, there are several
issues which, in conjunction, provide sufficient evidence that numerical instabilities are not the
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 . 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 . 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
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
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.
The experimental work of Aliseda et al.  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. 
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 , 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
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
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
= 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
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  (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
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 ), 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
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 , Pirozzoli and Grasso, and Freund et al. . Shocklets were found
to form more easily in 2-D turbulence than in 3-D. The exact definition is ambiguous. Pirozzoli
and Grasso  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.  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 ). In , 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
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  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 (). 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
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
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
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
 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
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 . 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.
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
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
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
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.
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
Jean Le Rond D'Alembert UPMC, and Christophe Dumouchel of Université et INSA de Rouen
were extremely beneficial.
a Speed of sound
AWTS Air/water test stand
B Aerodynamic response time =
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/
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
We Weber number = ρU2LC/σ
Z Density ratio, ρG/ρL
α Phase volume fraction
ε Turbulence dissipation rate
σ Surface tension
φ Wall distance
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
ref Reference condition
I Inner gas
O Outer gas
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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.
Table 1. Matrix of all axisymmetric air-water simulations for numerics and swirl simulations
A5 Base Case
A11 VOF CN
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
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
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
Table 5. Time-averaged spray angle measures from axisymmetric air-water simulations for numerics and
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
Table 6. Summary of numerical studies from axisymmetric air-water simulations
Case Pressure LS Angle
A10 X X
A12 X X X
A13 X X X
A14 X X X
Table 7. SMD, computed versus theoretical, for various shapes and sizes in test injector domain.
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
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
3 spheres (0.1″)
1 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
D32 Error -2.40%
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 .
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
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
Figure 13. Swirl-inducing mechanism examples; lower figure is colored by undisclosed pressure
from low (blue) to red (high)
mesh for thr
Figure 2. Time-averaged spray profiles from axi-symmetric air-water simulations.
-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75
Water Vol ume [%]
ntaneous volume fraction
SAM Case A
0; blue repr
liver model f
Figure 5. CFD results compared to experimental results of non-Newtonian primary atomization from Aliseda
et al .
Normalized Droplet Length Scale
Normalized Distance From Orifice
Vector plot c
h number of
a normal sho
plet of water
k wave in ai
assed over a sin
Figure 7. Dimensionless lateral trajectory of a droplet having been exposed to a shock wave
0 4 8 12 16
. Sample ins
the swirl ope
of the sp
Case A19) a
d swirl (ri
t; Case A7) fl
s the approxi
e sequence u
m top-left a
mate time (t/
) that frame
captures in a
rs from axis
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top of each fr
for the uppe
d, red = hi
blue = liquid
, blue = low),
at t/H = 0.3 o
(Case A5). T
and the ri
e middle sho
nates ≥ 1).
s the resulti
es Mach nu
front and M
ch number f
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(blue = 0, wh
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e. The dotte
at time samp
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se to t/H = 0.
ust before t
cle time pro
eft to ri
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same time i
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lue = 0.0 to
) to red (hi
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