Spatiotemporal Characterization of Wave-Augmented Varicose Explosions
D. M. Wilson1, W. Strasser1, R. Prichard1
1School of Engineering, Liberty University, Lynchburg, VA 24515, USA
The onset of three-dimensional instabilities during -(WAVE) liquid
disintegration is characterized for an annular flow of a shear-thinning slurry that is fed into a central transonic
steam flow. Droplet production inside the nozzle is enhanced by ligaments radially flicking up from the slurry
wave into the steam flow with radial:axial velocity ratios exceeding 0.5. The wave also leaves residual
ligaments in its wake, which facilitate further disintegration. After birth, a wave spends 80% of the wave cycle
period building up to peak height at the nozzle exit. Two effervescent mechanisms are provided as 1) steam
penetrates the rising wave and surface deformation allows steam fingers to force through, and 2) the wave
collapses on itself, trapping steam. Baroclinic torque drives the development of Rayleigh-Taylor (RT)
instabilities and reaches values on the order of 1/s2. Both RT and Kelvin-Helmholtz instabilities are
self-amplified in a viscosity-shear-temperature instability cycle because the slurry is non-Newtonian.
Velocities inside the nozzle (wave formation region) are generally azimuthally similar (two-dimensional), but
those outside the nozzle (radial bursting region) are azimuthally uncorrelated (three-dimensional). Inter-
variable correlations show significant decoupling of quantities beyond the nozzle exit, and local strain rate
fluctuations were found to correlate particularly well with bulk system pulsation. Although adaptive mesh
refinement (AMR) can provide computationally efficient resolution of gas-liquid interfaces, this technique
produced different results than an equivalent non-dynamic mesh when modeling WAVE. Gradients were
particularly affected by AMR, and turbulent kinetic energy showed differences greater than 150% outside the
Keywords: CFD, multiphase, transonic, instability, adaptive mesh refinement
The author to whom correspondence may be addressed: firstname.lastname@example.org@liberty.edu
The relevance of atomization for modern life cannot be understated. Those atomization processes found in nature
designs have been successfully exploited for applications in many industries, such as automotive, aerospace,
chemical, and agricultural. Though widely employed, atomization is a complex and diverse process that continues
to be studied with great vigor. Various fluid instabilities lead to atomization, and the nature of droplet formation
makes atomization an inherently three-dimensional (3D) process. The Kelvin-Helmholtz instability (KHI),
induced by velocity gradients, is often a contributor to atomization but is only two-dimensional (2D). For a moving
surface, KHI operates in the longitudinal direction to excite the interface, creating perforations, tongues, and
waves. Variation in the transverse direction can arise from Rayleigh-Taylor instabilities (RTI), which are induced
by a misalignment between the density and pressure gradient vectors; RTI arise from the extra term
in the vorticity equation, where is density, is pressure, and is the direction vector. Additionally, surface
curvature can induce Rayleigh-Plateau capillary instabilities (RPI). KHI, RTI, and RPI can be important and
simultaneous players in the disintegration of liquids.
Controlled atomization processes utilize a range of methods. Some common types of atomization-enhancing
techniques include swirl,1-3 effervescence,4, 5 and an assisting gas stream.6-8 Twin-fluid atomization utilizes a high-
speed co-lowing gas stream to disrupt the liquid. Typical designs involve a central liquid stream surrounded by a
co-axial gas flow. The gas-liquid shear encourages destabilization of the bulk liquid for disintegration and droplet
production. Gas-Centered Swirl Coaxial (GCSC) injector designs have been used for propellant atomization with
inverted the feeds.9 A central gas flow is then surrounded by a co-axial liquid flow, and pulsing characteristics can
develop at certain gas-to-liquid momentum ratios.10 Alekseenko et al. studied disturbance waves in vertical
annular liquid flow around a central gas stream, but these waves are small relative to the nozzle.11
A Newtonian fluid is typical for atomization processes, and non-Newtonian fluids will alter atomization
characteristics. Although KHI, RTI, and RPI are, fundamentally, inviscid instabilities, variations in viscosity can
both 1) modify the growth of these perturbations and 2) further destabilize the interface. Non-Newtonian fluid
layers can be prone to instability even when operating at low Reynolds numbers.12 When studying a shear-thinning
fluid on a slope, Millet et al. reported increased celerity and likelihood of instability compared to the Newtonian
counterpart.13 A unique breakup mode for secondary atomization was discovered in the study of non-Newtonian
coal water slurries.14 Guo et al. report that the central gas flow promotes instability of high-speed annular power-
law fuel jets and that a thinner fuel film and higher gas density also encourage breakup.15 Both inverted feeds and
an ing viscous, non-
Newtonian waste slurries.16 Applications could include waste-to-energy conver
technology17 and gelled propellant atomization.18-21
It has been shown that a certain inverted-feed, forced interaction atomizer design produces periodic, high-
blockage-ratio waves inside the nozzle when a viscous, shear-thinning fluid present.22 Two past numerical
studies have investigated aspects of this phenomenon with hot, subsonic steam as the assisting gas and banana
puree as the working fluid.22, 23 Banana puree is both viscous and shear-thinning, with well-recorded viscosity
data in the literature.24 WAVE atomization could be instrumental for manure slurries (energy reclamation) and
gelled propellants. A predominantly 2D analysis revealed basic mechanisms driving liquid wave formation
and collapse.22 An investigation of atomization downstream of the nozzle highlighted the importance of waves
for transonic disintegration of puree via periodic radial bursting.23 Since the characteristic wave cycling leads
to bursting (that in turn enhances atomization), the process has been termed -Augmented Varicose
However, major gaps in understanding the WAVE phenomenon remain. Most
importantly, the 3D nature of the interior waves and related instability cycle are almost entirely unexplored.
Among the many questions that arise are the following: Where and how do RTI form in the wave prior to
rupture, and is enough time in a wave cycle allotted for RTI to develop? To what degree are waves azimuthally
uniform? If not, where does the uniformity break down? What mechanism drives droplet breakaway inside the
nozzle before the radial burst is it wave stripping or wave flicking? Do quantities at all locations fluctuate
with wave cycles? Do flow metrics correlate with one another at various times and locations?
Adaptive mesh refinement (AMR) was considered as a numerical technique, but initial results showed AMR
to be an outlier to numerical trends.22 Because of the complexity and small scales in these atomization
phenomena, it can be computationally expensive to resolve the explicitly gas-liquid interface using CFD. To
increase computational efficiency, AMR has been used been used to model both waves and atomization.25, 26
Rather than refine all cells within the computational domain, AMR locally and dynamically refines the mesh only
around the interface as the simulation proceeds. Despite the benefits of AMR, the aforementioned results indicate
a need to further understand how AMR affects simulations. It remains unclear how AMR differs qualitatively and
which quantities are most affected by AMR.
In this paper, a numerical study is presented to reveal for the first time the spatiotemporal aspects of WAVE;
azimuthal instabilities are a primary focus. Careful evaluation of adaptive mesh refinement (AMR) and how it
affects the modeling of banana puree slurry atomization is also an important contribution. Our new findings are
communicated in four distinct sections, each of which seeks to address unanswered questions from previous
studies. Section 3.1 includes cartesian (unraveled from cylindrical) pictures and animations of the wave (so that
the azimuthal variation and onset of RTI is evident), as well as calculations of baroclinic torque and discussion of
the liquid viscosity-shear-temperature instability cycle. Section 3.2 presents contours of radial versus axial
velocity to elucidate mechanisms for droplet breakaway from the liquid wave inside the nozzle. Section 3.3
provides extensive frequency and correlation analyses across 100 spatially diverse signals, including
demonstration of Fast Fourier Transform (FFT) convergence. An analysis of wave cycle timing is also included,
showing leading and lagging responses. Finally, Section 3.4 presents a qualitative and quantitative assessment to
clarify the effect of AMR on WAVE simulations.
2.1 Computational Methods
The governing Navier-Stokes equations, formulated for multiphase flow in vector notation, are presented in
Equations 1-3. Symbols in Equations 1-3 are defined as follows: is time, is the velocity vector, is the static
temperature, is phase volume fraction, is density, is constant pressure heat capacity, is laminar
conductivity, is the turbulent viscosity, is the turbulent Prantdl number, is pressure, is gravity, is the
surface tension force vector, is the laminar shear stress tensor, and is turbulent shear stress tensor. Properties
are arithmetically phase-averaged, the banana puree slurry is modeled as incompressible, and the ideal gas equation
of state is used to compute steam density.
The compressible Reynolds-Averaged Navier-Stokes and volume-of-fluid (VOF) equations were discretized and
solved using double precision segregated ANSYS Fluent 2020R1 software. The gas-liquid interface was
reconstructed explicitly by means of the geometric reconstruction technique (also known as piecewise linear
interface capturing or PLIC).27, 28 Turbulent effects were included via a homogeneous shear stress transport (SST)
k-model. SIMPLE (Semi-IMplicit Pressure Linked Equations) was used for pressure-velocity coupling, and
mostly second order discretization stencils were employed. With the segregated approach of SIMPLE, pressure
and velocity are updated sequentially instead of simultaneously.29 Time step was varied to preserve a Courant
number of 1 throughout the entirety of the simulation. For the finest mesh, the time step was generally on the
order of one-hundred-thousandth of a wave/pulsation cycle (1×10-8 s).
The Herschel-Bulkley model is used to describe the shear-thinning (beyond a yield stress) and temperature-
thinning nature of the slurry. Banana puree viscosity is modeled as a function of both strain rate and temperature
according to the data of Ditchfield et al.24 A user define function (UDF) in Fluent computes viscosity () according
to Equations 4-7 from strain rate magnitude () and temperature ( in °C) values. In Equation 4, is the yield
stress, and are calculated according to Equations 6 and 7,
is the lower strain rate bound, and
is the corresponding upper viscosity bound. Although our implementation method used herein matches that
of the validated method used in Ref. 30,30 we sought to further verify our calculations. Instantaneous cell-
centered values of strain rate and temperature were collected at five locations (to be discussed more in a future
section) within the model for a single moment in time. The viscosity observed in Fluent matched excellently
that calculated by hand with Equations 4-7 (less than 0.005% difference), verifying correct implementation of
the viscosity UDF.
Extensive validation of the methods here employed for transonic wave formation and atomization has already
been conducted over the course of the last 10 years.27, 31-38 The SST k-turbulence model, which is employed
here, was sufficient to reveal important physical mechanisms. Experimental results were reproduced both
quantitatively and qualitatively, and the primary validation exercise are summarized as follows. First,
computations revealed the globally pulsing nature of an industrial three-stream air-water atomizer as
qualitatively observed in experiments. Second, the experimental acoustic signature of pulsations and primary
atomization ligament wave positions were quantitatively reproduced with numerical simulations. Third, the
axial droplet size distribution in a non-Newtonian injector from experiments aligned quantitatively with
numerical results. Additionally, assessment of droplet size distribution did not significantly alter with changing
azimuthal angle, even for 1/32nd of a full 360° azimuth. Fourth, the numerical trajectory of a disintegrating
droplet (after exposure to a normal shock wave) matched the analytical trajectory for said droplet. Furthermore,
broadly similar atomization systems have been studied (recessed with a high gas-to-liquid momentum ratio
and an inverted feed), and we find that the results presented here correspond to the globally varicose (but
locally sinuous from the Lagrangian perspective of the flapping annular waves) pulsing nature expected of
Models were considered to be at quasi steady state (QSS) when various point monitor signals (discussed later)
were statistically stationary. Unless otherwise noted, all results presented in this paper are using QSS data, and
quantities were only time-averaged across QSS data. Since the system experiences bulk pulsation, flow time is
normalized by the time elapsed between pulses. This is referred to as or, equivalently, as a
. Waves and the general pulsing phenomenon are operating together in the same cyclic pattern.
The model used for the majority of results, Ref-3, was run for 12 PTs of QSS data and includes 8.5 PTs of time-
averaged data. A recently developed methodology was used to optimize the hardware utilization.39
2.2 Mesh and Boundary Conditions
A summary of the geometry, mesh, and boundary conditions is provided here. Rather than the typical twin-fluid
atomizer design with a central liquid flow surrounded by a coaxial gas flow, the streams are reversed. An outer
slurry annulus surrounds a central steam flow. The slurry pool is exposed to the hot subsonic center steam flow
before the nozzle exit, and the nozzle orifice is extended to encourage significant steam-slurry interaction before
exiting the nozzle. The full 360° domain is simulated by a 90° azimuthal slice bookended by periodic boundary
conditions. Changing the angle from 45° to 90° produced no noticeable change in the axial droplet size profile,23
and the sufficiency of the 90° azimuth will be further discussed among the results presented here. The downstream
atomization domain spans 1.5 nozzle diameters in the radial direction and extends 2 nozzle diameters in the axial
direction. Constant mass flow rates of 0.021 kg/s for steam and 0.79 kg/s for the slurry resulted in a gas-liquid
mass ratio of 2.7%. Inlet temperatures for the slurry and steam were set to 304 K and 393 K, respectively. The
steam inlet feed turbulence is set by defining a turbulent kinetic energy () and specific dissipation rate (), where
= 48 m2/s2 and = 2.5×105 1/s. Both and are spatially constant and were chosen arbitrarily. All results
presented in this paper should be understood with these inlet conditions in mind, and other inlet conditions could
be studied to determine the influence of GLR and turbulence feed conditions on wave formation and atomization.
Figure 1 provides an overview of the geometry and shows an example of the Ref-3-AMR mesh.
represent the radial, axial, and azimuthal directions, respectively. In total, six distinct meshes were evaluated. A
Base mesh was refined n times to produce Ref-1, Ref-2, and Ref-3 meshes. The total element count increases by
a factor of nearly eight during refinement, as each cell length is cut in half. Note that the Base mesh was not refined
farther back in the slurry annulus and steam pipe, but all regions of gas-liquid interaction were included in the
refinements. The additional two meshes are Ref-2-45, which replaces the 90° azimuth with a 45° azimuth, and
Ref-3-AMR, which utilizes adaptive mesh refinement (AMR). AMR provides the same refinement level as Ref-3
but increases computational efficiency by refining dynamically only around the gas-liquid interface while the rest
of the domain remains at a Ref-2 refinement level. To closely track the gas-liquid interface, the mesh is refined
every 5 time steps, and a stringent VOF gradient criterion is used. Ref-3-AMR maintained around 28 million
elements compared to the 132 million elements in the Ref-3 mesh.
The Ref-3 mesh was found to be reasonably mesh independent across a range of metrics, including turbulent
kinetic energy, velocity, and slurry volume fraction at the nozzle exit.22 Furthermore, wave physics comparable to
those produced by Ref-3 (pulsation frequency and wavelength, for example) were revealed by Ref-1, two
refinement levels below Ref-3. We acknowledge that all relevant length scales are not resolved, but Ref-3
maintains the requirement of at least four cells across each droplet,23 which was shown to be sufficient for viscous
slurry atomization using our computational methods.40 Strasser and Battaglia provide more discussion on
implications of mesh resolution for atomization.31 The trend of numerical results versus mesh size showed Ref-3-
AMR to be an outlier. For these reasons, Ref-3 is used for all results presented in this paper unless otherwise noted.
The difference in numerical output from the Ref-3 and Ref-3-AMR meshes is an important point; a comparison
of the two meshes will be presented later. Figure 2 presents a side view of the pre-filming (wave formation) region
of the Ref-3 mesh. The region where slurry (this is where slurry
waves rise into the steam flow) and the extension of the nozzle orifice
before exiting the nozzle.
Figure 1 Oblique view of a representative Ref-3-AMR surface mesh, illustrating the nature of adaptive mesh
refinement (AMR). Starting with a Ref-2 mesh, the cells around the gas-liquid interface are dynamically
refined every 5 time steps. Through a given pulsing cycle, Ref-3-AMR fluctuates between ~26 and ~30 million
elements compared to the 132 million Ref-3 mesh. Both Ref-3 and Ref-3-AMR use a 90° azimuth with periodic
boundary conditions on either side. The radial, axial, and
Figure 2 Side view of the 132 million element Ref-3 mesh for the pre-filming (wave formation) region of the
atomizer. A central steam flow is interrupted by the injection of the slurry, forcing the two phases to interact
before exiting the nozzle. slurry and steam meet) and travel across
3. RESULTS AND DISCUSSION
3.1 3D Wave Cycle
The inverted-feed, forced interaction nozzle design leads to highly regular bulk pulsations in the system. These
pulsations may be looked at from two perspectives: inside and outside the nozzle. We will examine both in
this paper. Inside the nozzle, annular waves form at regular intervals (corresponding to bulk system pulsation),
rising out of the wave pool (labeled in Figure 2). One wave forms
toward the center of the steam flow and collapses while exiting the nozzle. Previous studies have mostly
considered 2D aspects of these waves (KHI, which is integral to the wave formation process, is 2D),22 leaving
important 3D characteristics uninvestigated (RTI, for example, manifests in the azimuthal dimension). We
seek to clarify how azimuthal variation develops in the wave and leads to atomization downstream.
Outside the nozzle, regular radial bursting of slurry is observed. This bursting manifests as a three-part
sequence: 1) stretch, 2) bulge, and 3) burst. While a wave is forming inside the nozzle, an annular sheet of
slurry stretches from the nozzle exit. As the wave collapses, the windward pressure build-up propagates
through the wave, and the slurry sheet bulges radially. Finally, the slurry sheet ruptures altogether. Both wave
cycling and radial bursting occur at a frequency around 1000 Hz (more on this later). The wave formation is a
sinuous instability manifestation (from a Lagrangian perspective of an observer moving with the wave leaving
the nozzle), but the bulk pulsation outside of the nozzle (radial bursting) is a varicose manifestation. Wave
formation and collapse are integral -by forcing the wave to crash
into itself and enhance the disintegration of viscous, non-Newtonian fluids with radial explosions. The Weber
(), Reynolds ( ), and Strouhal ( ) numbers characterizing the wave are
2.0×104, 1.2×104, and 0.08, respectively. Here, is wavelength, is wave speed, is wave frequency, is the
steam-slurry bulk velocity, is slurry density, is surface tension, and is the lower end of slurry viscosity.
To better understand cross-wave variation during the 3D wave cycle, Figures 3-6 present a unique view of the
annular wave: the 90° azimuthal slice has been unraveled to (rather than annular)
visualization of the slurry interfacial motion with a superimposed contour in the background. The result is
more comparable to what ocean waves on a beach look like and provides insight into 3D aspects of the wave
physics. Wave formation is illustrated in frames 1-3 and wave collapse in frames 4-6. Figure 3 (Multimedia
view) shows the unraveled slurry surface colored by viscosity, and the background contour is colored by strain
rate. This view is from the perspective of inside the nozzle to observe wave formation. High strain-rate regions
appear as the hot steam contacts the slurry both at the wave and with droplets in the free stream. Viscosity
reduces according to the shear-thinning nature of the slurry where strain rate is highest and the temperature-
thinning effect by the hotter steam. Viscosity reduction causes ligaments to flick up into the steam,
compounding the shear-thinning effect. Some azimuthal variation in viscosity is evident, although spatial
variability fluctuates throughout the wave cycle. Azimuthal variation in the slurry is present, though minimal,
in frames 1-3 but becomes very pronounced as the wave crashes in frames 4-6. Frames 4 and 5 show irregular
valleys and ridges forming across the azimuth. The RTI time scale, as approximated with a low-Re method41
for the low end of the slurry viscosity spectrum, is an order of magnitude lower than the wave time scale. This
suggests that sufficient time is available for RTI development.
Figure 3 Sequential views of unraveled wave with the slurry surface colored by viscosity and contours of
strain rate in the background. Pictures are at equally spaced flow time intervals through one representative
wave cycle. The view is from the inside of the nozzle looking down on the wave. The shear- and temperature-
thinning nature of the slurry is highlighted by viscosity reduction in response to high strain rate as the wave
penetrates into the hot steam. Azimuthal variation is present as the wave forms (frames 1-3) but increases
significantly as the wave collapses (frames 4-6). (Multimedia view).
Figure 4 (Multimedia view) presents a similar visualization as that in Figure 3 but viewed from outside the
atomizer (looking underneath the wave). This perspective shows the annular slurry sheet (now flattened)
stretching out from the nozzle as a wave rises. In this view, the wave is rising underneath the slurry sheet. The
point of maximal extension before rupture (frame 4) corresponds to the highest viscosity of the slurry sheet,
as it experiences minimal shear outside the nozzle radius. The sheet is destabilized by both azimuthal variation
from RTI (heavier slurry being accelerated by the lighter steam) and azimuthal variation in viscosity. Frame 4
reveals significant and azimuthally variant bulging underneath the wave, which corresponds to valleys in the
wave surface. Clearly, the wave is not impacting the slurry sheet uniformly. We also note that the multiple
bulges across this 90° azimuthal slice indicates that 90° is a sufficient angle to capture variation in the azimuth.
Parts of the wave contact local portions of the slurry sheet before others-
slurry sheet. Frame 5 marks the pre-rupture of the slurry sheet, where small portions begin to break apart. The
collapse of the wave and the instability of the slurry sheet both likely contribute to pre-rupturing. The further
destabilized slurry sheet completely ruptures in frame 6 in a violent radial burst.
Figure 4 Sequential views of unraveled wave with the slurry surface colored by viscosity and contours of
strain rate in the background. Pictures are at equally spaced flow time intervals through one representative
pulsing sequence. The view is from the outside of the nozzle (looking up from underneath the wave) to
illustrate the radial bursting phenomenon. As the slurry stretches out in an annular (flattened for these views)
sheet, RTI - as the collapsing
wave starts to contact the slurry sheet, followed by complete rupture in frame 6. (Multimedia view).
Waves have a high blockage ratio that significantly affects the steam pressure and velocity via local
acceleration; both fluctuate periodically with the pulsing cycles. Figure 5 (Multimedia view) illustrates these
effects by showing the unraveled wave surface colored by pressure with a Mach number contour in the
background. As the wave rises, it shelters its leeward side from the oncoming steam flow, thereby reducing
the flow area for the steam exiting the nozzle. Steam is then accelerated to transonic velocities. Though not
explicitly shown in Figure 5, small regions are supersonic. Consequently, steam compresses on the windward
side of the wave to build up pressure, and the steam accelerates above the wave crest through the reduced-area
opening. Both phenomena have implications for wave formation. The windward high-pressure zone exploits
irregularities in the slurry surface caused by RTI and viscosity gradients. The transition from wave formation
to wave collapse (frame 3 to frame 4) is significant: pressure increases sufficiently to overcome inertia and
surface tension, and the increase in azimuthal variability is marked. We observe in frame 5 the propagation of
the windward high-pressure zone axially as the wave collapses, which is a driving force in the radial bursting.
Figure 5 Sequential views of unraveled wave with the slurry surface colored by pressure with contours of
Mach number in the background. Pictures are at equally spaced flow time intervals through one representative
wave cycle. The view is from the inside of the nozzle looking down on the wave. The high blockage ratio of
the wave leads to pressure build up on its windward side and reduces the exit area for the steam to accelerate
it above the wave crest. Steam clearly reaches transonic speeds. Pressure buildup drives wave collapse and
exploits azimuthal variations in the slurry surface (caused by RTI and viscosity gradients) for disintegration.
Figure 6 (Multimedia view) provides a side view to of the wave cycle (gray interface) with a contour of
temperature in the background. Much like pressure, temperature cycles as the steam periodically compresses
on the windward side of the wave. Highest steam temperatures occur in frames 4 and 5 and will contribute to
surface destabilization and wave disintegration by reducing slurry viscosity, although shear is undoubtedly the
dominant driving force behind viscosity changes. The wave appears to be largely 2D as it rises and approaches
the nozzle exit. As it reaches the nozzle exit in frame 4, the wave is transitioning to a more 3D surface. This
corresponds to the dramatic increase in azimuthal irregularity as pressure exacerbates existing surface
variations and temperature thins the slurry.
Figure 6 Sequential views of unraveled wave (gray) with contours of temperature in the background. Pictures
are at equally spaced flow time intervals through one representative wave cycle. The view is directly from the
side of the wave, effectively outing the wave profile. Wave blockage causes the steam to compress, cycling
the temperature with wave formation and collapse. The wave is largely 2D until around the nozzle exit, where
it collapses, and azimuthal variations are exacerbated for a strongly 3D surface. (Multimedia view).
It has already been noted that RTI is driven by the baroclinic torque term
in the vorticity equation.
Density-based torque on the interface creates vorticity that will tend to increase the misalignment of pressure and
density gradient vectors. This in turn creates additional vorticity, leading to further misalignment. Figure 7
(Multimedia view) presents contours of baroclinic torque. The highest values are on the order of 1/s2,
indicating significant misalignment of gradient vectors and strong RTI activity in the wave. Baroclinic torque is
greatest at the steam-slurry interface, where density gradients are highest. Aside from directly at the interface,
baroclinic torque is highest in the regions where droplets are being stripped off the wave and interacting with the
steam. This indicates that as the shear layer forms, RTI becomes important very early in the wave life.
Figure 7 Sequential side contours of baroclinic torque through one representative wave cycle. Rayleigh-Taylor
instabilities (RTI) arise from this baroclinic torque term in the vorticity equation. Baroclinic torque is highest
at the interface, where the density gradient is highest. The largest values shown are on the order of
1/s2, indicating significant RTI activity present in the wave from its birth. (Multimedia view).
RTI and KHI are present early in the wave and lead to minute interfacial deformation. As the approaching
steam navigates the newly roughened surface, spatial variations in temperature and strain rate occur across the
wave. The shear- and temperature-dependent slurry responds accordingly, producing local variations in
viscosity across the surface. Because of viscosity variation, the steam-slurry interface develops axial and
azimuthal wavelength spatial variability. Also, due to viscosity variation, the interfacial stress develops spatial
variation. Finally, the process is repeated as the surface deformation excites RTI and KHI. Figure 8 illustrates
the cycle, where T is temperature, SR is strain rate, and WL is wavelength. Note that we are not utilizing linear
instability analysis but rather the local momentum balance to reveal these instabilities.
Figure 8 Instability cycle, where the Kelvin-Helmholtz instability (KHI) and Rayleigh-Taylor instability (RTI)
cause surface variations that in turn excite the instabilities. Here, T is temperature, SR is strain rate, and WL
3.2 Droplet Production
In a previous study, the effect of the wave on droplet production was largely addressed from the perspective
of bursting outside the nozzle,23 but questions remain regarding the role of 3D surface instabilities in droplet
production inside the nozzle. It has been noted that small droplets break away from the wave inside the nozzle,
and this was primarily attributed to steam shear (and later the collapse of the wave).23 A more thorough
investigation of this phenomenon is presented here, but we will first put the atomizer and its droplet production
mechanisms in context.
The Wave-Augmented Varicose Explosions (WAVE) design is essentially a combination of a Gas-Centered
Swirling Coaxial (GCSC) and Effervescent atomizer which capitalizes on the advantages of both and
incorporates an additional element of radial momentum generation. Effervescence is introduced when the wave
crashes onto the annular slurry sheet at regular intervals. Steam is sandwiched between the wave and sheet,
introducing bubbles into the liquid phase. The Gas-Centered feature of GCSC is preserved, though swirl is not
included in the current design. The lack of swirl necessity is a benefit of the current WAVE design because
swirl increases internal geometric complexity and could require maintenance of fouled swirl elements. Besides,
a previous study showed the impact of swirl in a multi-stream non-Newtonian atomizer to be minimal.33 Future
research may include revisiting this idea.
three mechanisms. 1) The momentum of the crashing wave creates radial bulges in the liquid film, bursting
droplets outward at regular intervals. 2) The high blockage ratio of the wave causes an intense pressure increase
behind the wave that assists the wave momentum in driving liquid film rupture and slurry disintegration. 3)
The extension of the slurry as a wave into the steam flow increases the interfacial area, providing more space
for the steam to peel off droplets. 42 refers to the fact that, from a fixed external observatory frame,
refers to the radial blasting of droplets caused by wave crashing.
The first two mechanisms have been thoroughly investigated.23 It is the third mechanism that will be explored
here. Droplets clearly break away from the wave as it penetrates into the steam flow inside the nozzle (before
the wave completely collapses), but what factors contribute to this outcome? It remains unclear whether the
steam is merely stripping droplets off an axially moving wave surface or if slurry ligaments are being flicked
up into the steam with significant radial velocity. To address this question, we present contours of radial
velocity divided by the absolute value of axial velocity in Figure 9 and Figure 10 (Multimedia view). The
outline of the steam-slurry interface is marked in black. Blue represents movement inward towards the central
steam flow, and red represents movement outward toward the beach and beyond. For the purposes of this
discussion (and from the perspective of Figure 9 and Figure 10), we will refer to blue regions as moving
The time sequence in Figure 9 reveals the temporal development of radial versus axial velocity through a given
wave cycle. By frame 3, the wave is penetrating significantly into the steam flow. Consequently, the wave
develops dark blue on its back, corresponding to the deflection of steam upward. This deflection
is significant close to the wave but less so further upstream. Just behind the wave jet, much of the slurry is
directed axially (white), but the wave jet itself has significant radial thrust pushing upward into the steam flow.
Radial velocity is at least 50% of the axial velocity in dark blue regions, which is the limit of the scale. By
frame 4, where the wave height peaks, much of the wave base is moving in the axial direction. However, we
still observe strong upward movement at the top of the wave, and some ligaments are present. In other words,
we observe the slinging of ligaments upward as the main body of the wave is moving forward. Steam around
the wave tip and ligaments also has strong upward motion. Non-axial flow in an atomizer has been found to
correlate with higher droplet production efficiency.31 As an aside, the beginning of the radial bulge is quite
evident as the red patch just at the end of the beach in frame 4. The flicking up of ligaments in frame 4 helps
explain what we term . For example, there is a nearly vertical ligament around the
nozzle exit in frame 5, which is unexpected because of the high-velocity steam directed toward it. This appears
to be a residual ligament derived from the windward ligament in frame 4. We call the windward ligament a
primary wave tip. By frame 5, the ligament is moving downward, but
it is an enduring form of the upward moving ligament in frame 4, and its position enables steam to disintegrate
the slurry more effectively. Curiously, the steam on the leeward side of the residual ligament is moving upward,
opposite the direction of the ligament and most of the surrounding slurry. This upward moving steam is likely
helping strip away droplets via shear. By the time the wave has collapsed (frame 6), a clear divide is observed
around the middle of the beach. To the left, slurry is moving upwards, and to the right, slurry is moving
downwards. The primary exception to this trend is the downward moving slurry in the sheltered region on the
leeward side of the newly formed wave.
Figure 9 The ratio of radial velocity to the absolute value of axial velocity through one representative wave
cycle. Contours are on a plane corresponding to a 30° azimuthal angle. As the wave peaks in frame 4, much
of the wave base is moving in the axial direction (white). The wave tip and associated ligaments, however, are
still slinging upward (negative radial direction) into the steam flow, which will enhance droplet production.
Figure 10 shows contours at the moment in time roughly where the wave peaks in height (equivalent to frame
4 in Figure 9) at 6 azimuthal angles. The base of the wave is generally uniform across the various azimuthal
slices, but the wave tips differ significantly across angles, even at 6° increments. What is consistent across
angles is this: while the wave base is moving with a largely axial velocity, the wave tip has a significant radial
velocity component. In frames 1 and 6 especially, we see a distinct secondary ligament form behind the wave
tip, which also has a high radial velocity. Most frames include some level of a steam gap between the wave tip
and a secondary ligament or wave base. The resulting wave tips can take on something of a hammerhead
shape before breaking away (for example, frames 2, 4, and 6) as the slurry connecting the tip to the base is
thinned. We direct the reader to notice particularly the penetration of steam into the wave as the tip rises. In
frames 2-4, the steam is pushing down into the wave, creating significant necking to break off the tip. Frame
5 shows smaller pockets of steam inside the wave, where steam fingers have forced through the deformed
surface. The upward movement of the wave, then, provides an effervescence mechanism in addition to the
trapping of steam as the wave crashes.
Figure 10 The ratio of radial velocity to the absolute value of axial velocity at a fixed time, roughly where the
wave peaks (equivalent to frame 4 in Figure 9). Contours are on planes corresponding to azimuthal angles of
30°, 36°, 42°, 48°, 54°, and 60°. The wave base consistently moves largely in the axial direction (white), while
the wave tip generally includes a significant upward velocity component. Steam penetrates the wave to break
off the tip (frames 2-4) and as smaller bubbles (frame 5), providing a mechanism for effervescence.
It is clear that flicking is an important mechanism for droplet production in the nozzle, but is stripping also
important? While flicking involves ligaments rising perpendicular to the steam flow, by stripping, we mean
droplets breaking away from the slurry surface or a ligament stretching in the direction of steam flow (parallel).
A close look at Figure 9 seems to show stripping towards the beginning of wave growth, but this is made
clearer in the animation for Figure 10. This animation shows stripping early on where droplets break free from
both the surface and parallel ligaments. As the wave matures, the primary droplet production mechanism
transitions to flicking. Finally, outside the nozzle, effervescence and the collapse of the wave lead to its
complete disintegration. In summary, droplet production inside the nozzle is dominated by stripping in the
early wave and later by flicking as the wave crests.
To conclude our discussion, we note several consequences (and benefits) of slurry ligaments flicking radially
up into the gas stream. Effervescence is introduced via a second mechanism before the wave crashes and
sandwiches steam. While the slurry wave jet is accelerated by the steam and thinned by shear, causing the jet
to buckle upwards and sling the liquid tips25 up into the gas stream as ligaments, the gas infiltrates the liquid
as bubbles. The thin fingers from the liquid sheet make it easier for the steam to peel off droplets,43 and this
avoids the need for a forcibly thinned sheet using a thin slurry annulus with high pressure drop and risk of
plugging. Additionally, the shearing and thinning of the fingers lowers the timescale for RTI to take effect,
making RTI more active and sooner.
3.3 Point Monitors
In a previous study, only two quantities were tracked at a single point monitor.22 We greatly expand point
monitor analysis here to include 100 signals for the purpose of understanding the spatial and temporal variation
of quantities, both azimuthally and along the slurry flow path. Velocity magnitude, slurry volume fraction
(VF), strain rate, and turbulent kinetic energy (TKE) were tracked at five point monitors placed roughly along
the trajectory of the slurry interface as it interacts with the steam. Each point monitor is present at five different
azimuthal angles (4.5°, 13.5°, 22.5°, 31.5°, and 40.5°), spaced evenly within a 45° azimuth, which is half of
the total 90° azimuth. In total then, 100 individual signals were recorded (4 quantities across 25 points) for the
entirety of the QSS window. Due to the extensive nature of this temporal data collection, we provide only
select plots and summary statistics. The locations of the five point monitors in a given azimuthal slice are
shown in Figure 11. The labels of Inner, Middle, Exit, Tip, and Outer, will continue to be used in reference to
these points, along with their azimuthal angles. The Inner, Middle, Exit, Tip, and Outer points are located 1,
0.75, 0.67, 1.25, and 1.5 nozzle radii from the nozzle axis (bottom edge of Figure 11), respectively. The Inner
point is centered axially on the wave pool. The Middle and Exit points are positioned axially at the start and
end of the beach. The Tip and Outer points are located 0.25 and 1.25 nozzle radii downstream of the nozzle
exit, respectively. The output data monitored at these five points will be discussed throughout the remainder
of the paper, revealing the spatiotemporal characteristics of the system.
Figure 11 Locations of five points monitors (roughly along the slurry interface trajectory) where certain data
quantities are collected. Each point monitor exists at 5 azimuthal angles within a 45° azimuth (half of the total
90° azimuth): 4.5°, 13.5°, 22.5°, 31.5°, and 40.5°.
Comparing the Exit point monitors across azimuthal angles reveals aspects of the wave interface as it exits the
nozzle. Figure 12 shows a moving average time series for (a) velocity and (b) slurry volume fraction through
12.5 pulsing cycles. Velocity signals are fairly uniform across angles and from wave to wave. In other words,
the general motion of the wave seems to vary minimally as it exits the nozzle, both azimuthally and temporally.
However, slurry volume varies significantly in the azimuthal and temporal dimensions. We attribute this
variability largely to the work of RTI, which is exacerbated by windward pressure buildup. We note again that
it is around the nozzle exit that the wave collapses, marking a transition from a somewhat 2D wave to a more
azimuthally diverse 3D wave. Table I provides the mean and coefficient of variation (COV) for each signal in
Figure 12 as well as the other quantities that are not displayed graphically. Note that this is a small subset of
all point monitor data. The COV is a normalized standard deviation, where the standard deviation is divided
by the mean and converted to a percentage.
Figure 12 Moving average time series of (a) velocity and (b) slurry volume fraction at the Exit point monitors
across all five azimuthal angles. The velocity signals are fairly uniform, both azimuthally and temporally. RTI
causes slurry volume fraction, however, to vary significantly in the azimuthal and temporal dimensions.
Table I Mean and coefficient of variation (COV) for signals across all five azimuthal angles at the Exit point.
COV is calculated by dividing the standard deviation by the mean and converting to a percentage.
Strain Rate [1/s]
On the other hand, we can reveal the axial (roughly) development of quantities by examining all five point
monitors at a given azimuthal angle. Figure 13 shows the moving average time series for (a) velocity and (b)
strain rate for all point monitors at the 22.5° azimuthal angle. Velocity magnitudes increase significantly as
the wave moves from the pool to the nozzle exit but decreases at the Tip and Outer point monitors. The slurry
is disintegrating at these last two points, so the monitors are picking up both slurry droplets and relatively
stagnant steam. Periodicity is much more evident inside the nozzle than outside the nozzle. This is indicative
of the relatively uniform wave motion inside the nozzle and the chaotic rupture outside the nozzle. Strain rate
follows a similar pattern through the slurry motion, increasing to the nozzle exit and then decreasing outside
the nozzle. However, the difference between the Middle/Exit points and the Tip/Outer points is less marked
than velocity. Compared to velocity, the temporal periodicity of strain rate is less pronounced for the middle
Middle/Exit points and more pronounced for the Tip/Outer points. In other words, velocity gradients are
fluctuating more consistently than velocity magnitude outside the nozzle. Strain rate also shows more wave-
to-wave variation than velocity within the nozzle. Table II provides the mean and COV for each signal in
Figure 13 as well as the other quantities that are not displayed graphically.
Figure 13 Moving average time series of (a) velocity and (b) strain rate at all five point monitors at the 22.5°
angle, illustrating variation of these quantities spatiotemporally as slurry moves through the system. The
velocity increases up to the nozzle exit as waves form and is much more periodic inside the nozzle than outside
the nozzle, showing the contrast between the more ordered wave formation and the more chaotic radial
bursting. Inside the nozzle, strain rate shows more wave-to-wave variation than velocity, and the Tip and Outer
points show more periodicity.
Table II Mean and coefficient of variation (COV) for signals across all point monitors at the 22.5° azimuthal
angle. COV is calculated by dividing the standard deviation by the mean and converting to a percentage.
Strain Rate [1/s]
3.3.2 Frequency Analysis
FFTs were performed to determine the dominate frequencies in the atomizer at various locations. Because of
the pulsing nature of the system, we expect most quantities to cycle at a consistent overall pulsing frequency
(frequency of wave formation). Velocity magnitude, strain rate, slurry volume fraction, and turbulent kinetic
energy were tracked at all 25 point monitor locations. FFTs reveal frequencies around 1000 Hz for the vast
majority of these quantities and point monitors. 83% of signals show a 1068 Hz peak frequency (note that
slurry volume does not fluctuate at any Inner point locations). 8% show a 2060 Hz peak frequency, and one
shows 3128 Hz, illustrating the higher mode harmonics. Strain rate is the most consistent quantity: all 25 point
monitors show a dominate frequency of 1068 Hz. The most prominent frequencies across point monitors for
the Ref-1 and Ref-2 meshes are 963 Hz and 992 Hz, respectively. Frequencies vary wildly for the Base mesh,
and no prominent frequency is evident. This corresponds to previous findings: major wave characteristics are
present, and consistent, with progressively increased mesh resolution, beginning with the resolution of Ref-
1.22 The characteristic pulsing of the system is largely absent from the Base case, and the Ref-3 mesh is two
refinement levels above Ref-1.
To understand how FFT peak frequencies vary across the azimuth, the COV was computed, which shows
azimuthal point-to-point variation as a percentage. For a given point location, such as Inner, a single peak
frequency was computed at each azimuthal angle, and the COV was computed from these 5 frequencies. Figure
14 shows the azimuthal frequency COV from the Inner to the Outer points (effectively along the interfacial
trajectory). Most frequencies are consistent inside the nozzle, but the variation increases significantly outside
the nozzle. The only quantity that continues to fluctuate uniformly in the azimuth outside the nozzle is strain
rate. Figure 14 illustrates again the consistency of strain rate as a pulsating quantity.
Figure 14 Azimuthal coefficient of variation (COV) for frequency at various points along the slurry interface
flow path. COV represents the percent of variation in the azimuth at each point. VF is volume fraction, and
TKE is turbulent kinetic energy. The peak frequencies were determined by FFTs. Frequencies are generally
less consistent outside of the nozzle as the slurry disintegrates. Strain rate stands out, fluctuating at consistent
frequencies at all locations (inside the nozzle, outside the nozzle, and across the azimuth).
FFTs are meaningless unless the resulting peak frequencies have converged over the course of the simulation.
For those signals that showed a peak frequency of 1068 Hz (the prominent frequency among all signals
evaluated), the FFTs generally converged within 4 pulsing cycles after the flow was already at QSS. This was
not necessarily the case for less periodic signals like velocity at the Tip and Outer points (see Figure 13).
Figure 15 provides a sample FFT peak frequency convergence plot for strain rate at the Exit point in the 22.5°
plane (see Figure 13 for time series). The vertical dashed line is the 4 pulsing cycles mark, and the green star
is the final value. Zero-padding was employed, resulting in the discrete step values towards convergence.
Figure 15 Convergence of FFT peak frequency for the strain rate signal at the Exit point monitor in the 22.5°
plane. FFTs included zero-padding, which produces the discrete steps towards convergence. The vertical
dashed line is at 4 pulsing times, and the green star represents the final frequency value of 1068 Hz. Signals
generally converged to this value within 4 pulsing cycles.
Figure 16 presents two FFT examples, one for velocity at the Inner point (left) and one for strain rate at the
Exit point (right). Both are at a 22.5° azimuthal angle, and the time series were shown in Figure 13. The Inner
point is located at the surface of the wave pool, where waves are being produced. The wave pool surface, then,
has a fluctuating velocity at 1068 Hz, which sets the pace for bulk pulsation in the system. Geometric
parameters might be varied to determine their influence on pulsing frequency, but that is beyond the scope of
this study. The Inner velocity FFT also shows harmonics at roughly 2060 Hz and 3110 Hz, which correspond
to the peak frequencies for a minority of signals. The Exit strain rate FFT shows the same peak frequency as
the Inner velocity, although it is less pronounced, and no harmonics are evident.
Figure 16 FFTs with peak frequency labeled for (a) velocity at the Inner point and (b) strain rate at the Exit
point. Both points are at an azimuthal angle of 22.5°. The velocity at the Inner point (where waves are formed)
shows a peak frequency of 1068 Hz and clear harmonics at around 2060 Hz and 3110 Hz. Strain rate at the
Exit point has the same peak frequency, but it is less distinct and with no harmonics.
As shown, the preponderance of frequencies are close to, or multiples of, approximately 1000 Hz, which results
from the wave pool generation process (i.e. KHI working with Bernoulli to amplify surface disturbances).
Therefore, the wave-generation process sets up an absolute instability in the system that is likely to be
unaffected by upstream turbulence effects. A future study could include evaluations of this.
The spread of information in the azimuthal and radial directions can be assessed by determining the cross-
correlation between signals; a normalized cross-correlation of 1 indicates that two time-series signals are
perfectly correlated and implies 2D (axisymmetric) motion. The time difference between two given locations
was accounted for by time-shifting the signals to align them before calculating the normalized cross-
correlation. Cross-correlations (calculated using the NumPy package in Python) were normalized by
subtracting the means from the signals and dividing the cross-correlation by the number of data points and the
standard deviations of the two signals. Figure 17 shows the normalized cross-correlation between velocity
signals both azimuthally and along the slurry interface trajectory. Correlation is calculated between the velocity
signal at a given azimuthal angle or point and the first angle or point. In essence, we are estimating how much
the motion at one place in the flow field might be related to the motion at another place in the flow field.
The motion at the inner point monitor appears to be uniform across all angles (indicating its 2D nature), as the
signals show almost perfect correlation with the first angle (4.5°). The Middle and Exit points show relatively
high correlation, and the correlation remains largely the same across angles. Outside of the nozzle, where the
wave and slurry sheet are being ruptured, the tip and outer point monitors show low correlation between angles.
The Tip point monitor, which is closer to the nozzle (around the bulging and bursting), shows slightly higher
correlation. These results suggest that fluid motion inside the nozzle is generally azimuthally similar (2D), but
the fluid motion outside the nozzle as the slurry bursts is azimuthally unrelated (3D). Furthermore, since the
correlation, for a given point monitor, is quite consistent across 45°, we conclude that a 90° mesh is more than
sufficient to capture azimuthal variation. The Inner point, which is at the surface of the wave pool, marks the
location of wave generation. The Middle and Exit points are well-correlated with the Inner point, but the Tip
and Outer points are not. In other words, the velocities at Tip and Outer do not show much relation to wave
Figure 17 Normalized cross-correlation between transient velocity magnitude signals from point monitors at
five points along the slurry interfacial flow and five azimuthal angles. A given normalized cross-correlation
was calculated after the two signals were time-shifted to align. A value of 1 indicates perfect correlation
between signals. The correlation between a given angle or point with the first angle (4.5°) or point (Inner) is
computed. High correlation is observed both azimuthally and along the slurry flow inside the nozzle but is
greatly diminished outside the nozzle. The lack of azimuthal variation in correlation across 45° demonstrates
the sufficiency of a 90° mesh to capture variations in the azimuth.
When signals are reasonably correlated, a time lag shows how much the motions are temporally offset. The
time lag is calculated as the amount the signals must be time-shifted to produce the maximum cross-correlation.
In Figure 18, time lags are normalized by the pulsing time, so a given value represents the fraction of a pulse
cycle by which the signals are offset. Time lag is meaningless if the signals are not well-correlated, so the Tip
and Outer points have been removed from the time lag plot. Figure 18 displays the extent to which the Middle
and Exit velocity signals lag the Inner velocity signal. These results are communicating where the Middle and
Exit points are, temporally, within the wave cycle (which repeats regularly every 1 PT). Velocity fluctuations
at the wave pool (Inner point) reach the Middle and Exit points 0.7 and 0.8 PTs, respectively, after the initial
wave pool motion. Fluid motion at the nozzle exit then lags the Middle point by 0.1 PT. Velocity spikes again
in the wave pool 0.2 PTs after a velocity increase at the Exit point. This cycle is illustrated in Figure 18, which
shows the percentage of a pulse cycle for velocity fluctuations at Inner (I) to reach Middle (M) and then Exit
(E) and then start again at Inner. The majority of a given pulse cycle (70%) involves the wave growing out of
the wave pool and reaching the beach. After reaching the beach, the wave travels more rapidly to the nozzle
exit and beyond.
Figure 18 Transient velocity signal time lags from the Inner point at five azimuthal angles. The time lag
corresponds to the signal offset that produces the maximum cross-correlation and has been normalized by the
pulsing time. Wave pool velocity fluctuations (Inner point, I) take 0.7 and 0.8 pulsing times to reach the Middle
(M) and Exit (E) points, respectively. Initial growth of a wave consumes the majority (70%) of a given pulse
In addition to inter-point correlations for velocity, correlations between different quantities were calculated at
all 25 points. Three contour plots in Figure 19 summarize these data and show the normalized cross-correlation
between 1) velocity magnitude and slurry VF (far left), 2) TKE and strain rate (middle), and 3) velocity
magnitude and strain rate (far right). The slurry volume fraction maintains a value of 1 at the Inner points,
making any normalized cross-correlation value meaningless. For this reason, the inner points were excluded
on the leftmost plot in Figure 19. The trend across all 3 sets of correlations is that any two quantities are most
highly correlated at the Inner point, and correlation decreases downstream. Correlation between quantities is
also fairly azimuthally uniform, particularly inside the nozzle. Velocity and slurry VF are the least well-
correlated overall. The middle plot indicates that, after the Exit point, TKE and strain rate are decoupled; thus,
TKE downstream must have been produced by some earlier shear. In summary, we observe the decoupling of
quantities exiting the nozzle: strong correlations inside the nozzle and very weak correlations outside the
Figure 19 Contours across all 25 points for the normalized cross-correlations between 1) velocity magnitude
and slurry volume fraction (left), 2) turbulent kinetic energy strain rate (middle), and 3) velocity magnitude
and strain rate (right). The Inner points have been excluded on the leftmost plot because the slurry volume
fraction maintains a value of 1. Correlations are generally strong inside the nozzle but very weak outside the
nozzle, showing a decoupling of quantities exiting the nozzle.
3.4 Adaptive Mesh Refinement
Up to this point, the Ref-3 mesh has been used entirely for visualization and analysis. It has been noted that
Ref-3-AMR, though a more cost-efficient alternative, is the outlier to the numerical trends of two point monitor
signals,22 but no visual comparison was provided, and a more rigorous comparison is lacking. A qualitative
and quantitative comparison of Ref-3 and Ref-3-AMR is here provided to clarify the differences between the
two models. We emphasize again that Ref-3 and Ref-3-AMR have the exact same mesh resolution at the slurry-
steam interface. A rigorous VOF gradient criterion and adaption frequency were used, but it should be noted that
there are other criteria that can determine the regions of adaption for AMR. It is possible that a different criterion
would positively affect AMR results, though we find this doubtful based on the extensive AMR refinement in this
study. We also acknowledge that any conclusions about AMR as a technique are limited to ANSYS Fluent and its
underlying algorithms. All validation efforts were conducted using meshes equivalent to Ref-3 (mostly
hexahedral elements swept in the flow direction), and AMR has not been validated experimentally for this
work. The differences between Ref-3 and Ref-3 AMR indicate problems with the communication of
information through split cells, which are constantly created by AMR.
Figure 20, which shows the slurry surface (coloured by slurry viscosity) as it exits the nozzle (flow is generally
downward), reveals qualitative differences. The leftmost set of images in Figure 20 show that Ref-3-AMR
produces a significantly more rippled surface, and slurry viscosity is generally higher. A higher viscosity
indicates lower strain rate, perhaps suggesting that the mesh gradients around the slurry-steam interface are
affecting velocity gradients. We remind the reader that RTI and viscosity gradients serve to destabilize the
annular slurry sheet, priming it for rupture and atomization as the wave crashes into it. The middle and
rightmost images in Figure 20 reveal Ref-3 rupturing more readily than Ref-3-AMR. In the middle images,
little pre-rupture (localized bursting events) in the annular slurry sheet is observed for Ref-3-AMR. Much more
can be seen for Ref-3 (corresponding to the preliminary rupture for Ref-3 in frame 5 of Figure 4). In the
rightmost images, the radial burst of slurry is more violent for Ref-3 than Ref-3-AMR, and Ref-3 is clearly
producing smaller droplets at this stage of the pulsing sequence. Interestingly, Ref-3-AMR was found to have
slightly larger droplets throughout the domain past the nozzle exit.23
Quantitative discrepancies between Ref-3 and Ref-3-AMR are summarized in Figure 21, which shows percent
differences between azimuthally-averaged quantities. TKE shows by far the greatest difference between the
models, especially outside the nozzle, where a 160% difference is observed. In general, the difference between
model outputs is higher outside (shown in Figure 20) than inside the nozzle. This follows the trend of an
increasing presence of mesh element size gradients as the slurry disintegrates outside the nozzle. Interestingly,
the % difference for strain rate (velocity gradients) increases outside the nozzle, but that for velocity does not.
This observation indicates that gradients are more strongly affected by AMR. We note also that TKE
production is driven by velocity gradients. Our conclusion: the AMR technique within ANSYS Fluent 2020R1
is not sufficient to accurately model non-Newtonian wave-augmented atomization in the present system.
Figure 20 Comparison of slurry surface as computed by the Ref-3 (top row) and Ref-3-AMR (bottom row)
meshes at three points in the wave cycle. Both models represent a 90° azimuthal slice with periodic boundary
conditions. The top three images are at roughly the same stages within a given pulsing cycle as the bottom
three images. Ref-3-AMR produces a noticeably more rippled surface than Ref-3. In the Ref-3-AMR case, the
slurry sheet stretching down from the nozzle maintains a higher viscosity and does not rupture as quickly as
Figure 21 The percent difference between azimuthally-averaged quantities at various points along the slurry
interface flow path for Ref-3 and Ref-3-AMR. VF is volume fraction, and TKE is turbulent kinetic energy.
TKE shows the greatest difference between the models, with % differences above 150% outside the nozzle.
The general trend is a greater divergence in model outputs outside than inside the nozzle.
Using a computational framework which has been validated and used extensively over the course of the last
decade, we have analyzed the spatiotemporal characteristics of the novel WAVE process (Wave-Augmented
Varicose Explosions) in which non-Newtonian waves facilitate disintegration in an inverted feed twin-fluid
atomizer. Annular slurry hot steam flow, creating a secondary nozzle effect for the
steam, as an annular slurry sheet stretches from the nozzle. Waves then collapse as they exit the nozzle,
crashing into the slurry sheet in a violent radial burst to enhance droplet formation. The wave birth-death cycle
is part of a general bulk system pulsation phenomenon, causing many quantities to fluctuate periodically.
Important knowledge gaps from previous studies have been filled to provide a complete understanding of this
efficient atomization phenomenon. A summary of new contributions to the literature is provided in the
We began by presenting a unique visual perspective: unraveled wave views to elucidate characteristics of the
3D wave cycle. An estimate of RTI time scale showed sufficient time for RTI development, and baroclinic
torque on the order of 1/s2 indicates strong RTI activity in the wave. KHI and RTI cause surface
variations in the non-Newtonian slurry that then excite the instabilities in a self-amplification cycle. During
the early stages of wave growth, stripping is a dominant mechanism for droplet production. Later, as the wave
crests, ligaments flicking up into the steam flow at the wave tip facilitate droplet production inside the nozzle.
Meanwhile, the radial thrust of the wave allows for steam penetration to increase effervescence and sometimes
break off the wave tip. Further disintegration is encouraged as the wave leaves residual ligaments in its wake.
An evaluation of velocity, slurry volume fraction, strain rate, and turbulent kinetic energy at 25 axially and
azimuthally spaced points revealed a loss of consistent fluctuation frequency outside of the nozzle. Strain rate,
however, cycles with the dominant system frequency at all points. FFT analysis was an important component
of this study, and peak frequencies generally converged within 4 pulsing cycles. The nozzle exit marks a
significant increase in azimuthal variation of velocities. Velocities show strong azimuthal correlation (2D) in
the wave formation region inside the nozzle but are azimuthally unrelated (3D) outside the nozzle, where radial
bursting is occurring. Outside the nozzle, fluid motion did not show strong correlation with wave pool motion.
Correlations between quantities, though strong in the wave formation region, showed a consistent trend of
significant decoupling outside the nozzle. Time lags revealed that, for a given pulse cycle, the wave rising out
of the pool to reach the beach takes 70% of the total pulse time.
Azimuthal correlations of velocity demonstrate that a 90° angle is sufficient for to capture azimuthal
variability, which is important for atomization. Evaluation of Ref-3-AMR, both qualitatively and
quantitatively, showed that this particular AMR technique is insufficient for accurately modeling non-
Newtonian, wave-augmented atomization in the present system, despite the allure of computational savings.
Velocity gradients were more affected by AMR than velocity magnitude. Turbulent kinetic energy differed
most drastically between Ref-3 and Ref-3-AMR, particularly outside the nozzle. Future research could
evaluate how geometry changes affect the spatiotemporal characteristics of the atomization process, such as
The authors thank Valda Rowe and Dr. Mark Horstemeyer for their administrative support.
The data that support the findings of this study are available from the corresponding author upon reasonable
CONFLICT OF INTEREST
The authors have no conflicts to disclose.
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23 D. M. Wilson and W. Strasser, "A Spray of Puree: Wave-Augmented Transonic Airblast Non-Newtonian Atomization,"
Physics of Fluids 34 (7), (2022).
24 Cynthia Ditchfield et al., "Rheological Properties of Banana Puree at High Temperatures," International Journal of Food
Properties 7 (3), 571-584 (2004).
25 W. Mostert, S. Popinet and L. Deike, "High-resolution direct simulation of deep water breaking waves: transition to
turbulence, bubbles and droplets production," Journal of fluid mechanics 942 (A27), (2022).
26 Chenwei (张宸玮) Zhang et al., "Atomization of misaligned impinging liquid jets," Physics of Fluids 33 (9), 093311
27 Wayne Strasser and Francine Battaglia, "The effects of pulsation and retraction on non-Newtonian flows in three-stream
injector atomization systems," Chemical Engineering Journal 309, 532-544 (2017).
28 David Youngs, Time-Dependent Multi-material Flow with Large Fluid Distortion, in Numerical Methods in Fluid
Dynamics, edited by K. W. Morton and M. J. Baines, (Academic Press, 1982), pp. 273-285.
29 Suhas V. Patankar, Numerical Heat Transfer and Fluid Flow, edited by Anonymous 1st ed. (CRC Press, London, 1980), .
30 Subramanian Easwaran Iyer et al, US Patent No. 8734909 (May 27, 2014).
31 W. Strasser and F. Battaglia, "Pulsating Slurry Atomization, Film Thickness, and Azimuthal Instabilities," Atomization and
Sprays 28 (7), 643-672 (2018).
32 Wayne Strasser, "Towards the optimization of a pulsatile three-stream coaxial airblast injector," International Journal of
Multiphase Flow 37 (7), 831-844 (2011).
33 Wayne Strasser and Francine Battaglia, "Identification of Pulsation Mechanism in a Transonic Three-Stream Airblast
Injector," Journal of Fluids Engineering 138 (11), (2016).
34 Wayne Strasser and Francine Battaglia, "The Influence of Retraction on Three-Stream Injector Pulsatile Atomization for
AirWater Systems," Journal of Fluids Engineering 138 (11), (2016).
35 Wayne Strasser, Francine Battaglia and Keith Walters, "Application of a Hybrid RANS-LES CFD Methodology to Primary
Atomization in a Coaxial Injector," Volume 7A: Fluids Engineering Systems and Technologies , (2015).
36 Wayne Strasser and Francine Battaglia, "The Effects of Prefilming Length and Feed Rate on Compressible Flow in a Self-
Pulsating Injector ," AAS 27 (11), (2017).
37 Wayne Strasser, "Oxidation-assisted pulsating three-stream non-Newtonian slurry atomization for energy production,"
Chemical Engineering Science 196, 214-224 (2019).
38 Wayne Strasser, "The war on liquids: Disintegration and reaction by enhanced pulsed blasting," Chemical Engineering
Science 216, 115458 (2020).
39 Reid Prichard and Wayne Strasser, "Optimizing Selection and Allocation of High-Performance Computing Resources for
Computational Fluid Dynamics," 7th Thermal and Fluids Engineering Conference (under review) , (2022).
40 Wayne Strasser, "Toward Atomization for Green Energy: Viscous Slurry Core Disruption by Feed Inversion," AAS 31 (6),
Taylor Instabilities," Journal of geophysical research. Solid earth 123 (5), 3593-3607 (2018).
42 P. K. Senecal et al., "Modeling high-speed viscous liquid sheet atomization," International journal of multiphase flow 25
(6), 1073-1097 (1999).
43 Yue Ling et al., "Spray formation in a quasiplanar gas-liquid mixing layer at moderate density ratios: A numerical closeup,"
Physical review fluids 2 (1), (2017).