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Spatiotemporal Characterization of Wave-Augmented Varicose Explosions

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D. M. Wilson1, W. Strasser1, R. Prichard1

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1School of Engineering, Liberty University, Lynchburg, VA 24515, USA

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ABSTRACT

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The onset of three-dimensional instabilities during -(WAVE) liquid

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disintegration is characterized for an annular flow of a shear-thinning slurry that is fed into a central transonic

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steam flow. Droplet production inside the nozzle is enhanced by ligaments radially flicking up from the slurry

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wave into the steam flow with radial:axial velocity ratios exceeding 0.5. The wave also leaves residual

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ligaments in its wake, which facilitate further disintegration. After birth, a wave spends 80% of the wave cycle

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period building up to peak height at the nozzle exit. Two effervescent mechanisms are provided as 1) steam

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penetrates the rising wave and surface deformation allows steam fingers to force through, and 2) the wave

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collapses on itself, trapping steam. Baroclinic torque drives the development of Rayleigh-Taylor (RT)

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instabilities and reaches values on the order of 1/s2. Both RT and Kelvin-Helmholtz instabilities are

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self-amplified in a viscosity-shear-temperature instability cycle because the slurry is non-Newtonian.

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Velocities inside the nozzle (wave formation region) are generally azimuthally similar (two-dimensional), but

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those outside the nozzle (radial bursting region) are azimuthally uncorrelated (three-dimensional). Inter-

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variable correlations show significant decoupling of quantities beyond the nozzle exit, and local strain rate

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fluctuations were found to correlate particularly well with bulk system pulsation. Although adaptive mesh

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refinement (AMR) can provide computationally efficient resolution of gas-liquid interfaces, this technique

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produced different results than an equivalent non-dynamic mesh when modeling WAVE. Gradients were

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particularly affected by AMR, and turbulent kinetic energy showed differences greater than 150% outside the

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nozzle.

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Keywords: CFD, multiphase, transonic, instability, adaptive mesh refinement

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The author to whom correspondence may be addressed: wstrasser@liberty.edu@liberty.edu

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1. INTRODUCTION

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The relevance of atomization for modern life cannot be understated. Those atomization processes found in nature

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designs have been successfully exploited for applications in many industries, such as automotive, aerospace,

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chemical, and agricultural. Though widely employed, atomization is a complex and diverse process that continues

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to be studied with great vigor. Various fluid instabilities lead to atomization, and the nature of droplet formation

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makes atomization an inherently three-dimensional (3D) process. The Kelvin-Helmholtz instability (KHI),

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induced by velocity gradients, is often a contributor to atomization but is only two-dimensional (2D). For a moving

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surface, KHI operates in the longitudinal direction to excite the interface, creating perforations, tongues, and

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waves. Variation in the transverse direction can arise from Rayleigh-Taylor instabilities (RTI), which are induced

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by a misalignment between the density and pressure gradient vectors; RTI arise from the extra term

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in the vorticity equation, where is density, is pressure, and is the direction vector. Additionally, surface

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curvature can induce Rayleigh-Plateau capillary instabilities (RPI). KHI, RTI, and RPI can be important and

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simultaneous players in the disintegration of liquids.

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Controlled atomization processes utilize a range of methods. Some common types of atomization-enhancing

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techniques include swirl,1-3 effervescence,4, 5 and an assisting gas stream.6-8 Twin-fluid atomization utilizes a high-

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speed co-lowing gas stream to disrupt the liquid. Typical designs involve a central liquid stream surrounded by a

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co-axial gas flow. The gas-liquid shear encourages destabilization of the bulk liquid for disintegration and droplet

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production. Gas-Centered Swirl Coaxial (GCSC) injector designs have been used for propellant atomization with

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inverted the feeds.9 A central gas flow is then surrounded by a co-axial liquid flow, and pulsing characteristics can

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develop at certain gas-to-liquid momentum ratios.10 Alekseenko et al. studied disturbance waves in vertical

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annular liquid flow around a central gas stream, but these waves are small relative to the nozzle.11

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A Newtonian fluid is typical for atomization processes, and non-Newtonian fluids will alter atomization

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characteristics. Although KHI, RTI, and RPI are, fundamentally, inviscid instabilities, variations in viscosity can

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both 1) modify the growth of these perturbations and 2) further destabilize the interface. Non-Newtonian fluid

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layers can be prone to instability even when operating at low Reynolds numbers.12 When studying a shear-thinning

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fluid on a slope, Millet et al. reported increased celerity and likelihood of instability compared to the Newtonian

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counterpart.13 A unique breakup mode for secondary atomization was discovered in the study of non-Newtonian

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coal water slurries.14 Guo et al. report that the central gas flow promotes instability of high-speed annular power-

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law fuel jets and that a thinner fuel film and higher gas density also encourage breakup.15 Both inverted feeds and

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an ing viscous, non-

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Newtonian waste slurries.16 Applications could include waste-to-energy conver

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technology17 and gelled propellant atomization.18-21

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It has been shown that a certain inverted-feed, forced interaction atomizer design produces periodic, high-

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blockage-ratio waves inside the nozzle when a viscous, shear-thinning fluid present.22 Two past numerical

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studies have investigated aspects of this phenomenon with hot, subsonic steam as the assisting gas and banana

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puree as the working fluid.22, 23 Banana puree is both viscous and shear-thinning, with well-recorded viscosity

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data in the literature.24 WAVE atomization could be instrumental for manure slurries (energy reclamation) and

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gelled propellants. A predominantly 2D analysis revealed basic mechanisms driving liquid wave formation

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and collapse.22 An investigation of atomization downstream of the nozzle highlighted the importance of waves

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for transonic disintegration of puree via periodic radial bursting.23 Since the characteristic wave cycling leads

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to bursting (that in turn enhances atomization), the process has been termed -Augmented Varicose

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However, major gaps in understanding the WAVE phenomenon remain. Most

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importantly, the 3D nature of the interior waves and related instability cycle are almost entirely unexplored.

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Among the many questions that arise are the following: Where and how do RTI form in the wave prior to

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rupture, and is enough time in a wave cycle allotted for RTI to develop? To what degree are waves azimuthally

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uniform? If not, where does the uniformity break down? What mechanism drives droplet breakaway inside the

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nozzle before the radial burst is it wave stripping or wave flicking? Do quantities at all locations fluctuate

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with wave cycles? Do flow metrics correlate with one another at various times and locations?

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Adaptive mesh refinement (AMR) was considered as a numerical technique, but initial results showed AMR

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to be an outlier to numerical trends.22 Because of the complexity and small scales in these atomization

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phenomena, it can be computationally expensive to resolve the explicitly gas-liquid interface using CFD. To

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increase computational efficiency, AMR has been used been used to model both waves and atomization.25, 26

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Rather than refine all cells within the computational domain, AMR locally and dynamically refines the mesh only

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around the interface as the simulation proceeds. Despite the benefits of AMR, the aforementioned results indicate

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a need to further understand how AMR affects simulations. It remains unclear how AMR differs qualitatively and

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which quantities are most affected by AMR.

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In this paper, a numerical study is presented to reveal for the first time the spatiotemporal aspects of WAVE;

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azimuthal instabilities are a primary focus. Careful evaluation of adaptive mesh refinement (AMR) and how it

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affects the modeling of banana puree slurry atomization is also an important contribution. Our new findings are

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communicated in four distinct sections, each of which seeks to address unanswered questions from previous

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studies. Section 3.1 includes cartesian (unraveled from cylindrical) pictures and animations of the wave (so that

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the azimuthal variation and onset of RTI is evident), as well as calculations of baroclinic torque and discussion of

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the liquid viscosity-shear-temperature instability cycle. Section 3.2 presents contours of radial versus axial

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velocity to elucidate mechanisms for droplet breakaway from the liquid wave inside the nozzle. Section 3.3

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provides extensive frequency and correlation analyses across 100 spatially diverse signals, including

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demonstration of Fast Fourier Transform (FFT) convergence. An analysis of wave cycle timing is also included,

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showing leading and lagging responses. Finally, Section 3.4 presents a qualitative and quantitative assessment to

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clarify the effect of AMR on WAVE simulations.

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2. METHODS

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2.1 Computational Methods

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The governing Navier-Stokes equations, formulated for multiphase flow in vector notation, are presented in

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Equations 1-3. Symbols in Equations 1-3 are defined as follows: is time, is the velocity vector, is the static

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temperature, is phase volume fraction, is density, is constant pressure heat capacity, is laminar

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conductivity, is the turbulent viscosity, is the turbulent Prantdl number, is pressure, is gravity, is the

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surface tension force vector, is the laminar shear stress tensor, and is turbulent shear stress tensor. Properties

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are arithmetically phase-averaged, the banana puree slurry is modeled as incompressible, and the ideal gas equation

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of state is used to compute steam density.

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(1)

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(2)

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(3)

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The compressible Reynolds-Averaged Navier-Stokes and volume-of-fluid (VOF) equations were discretized and

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solved using double precision segregated ANSYS Fluent 2020R1 software. The gas-liquid interface was

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reconstructed explicitly by means of the geometric reconstruction technique (also known as piecewise linear

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interface capturing or PLIC).27, 28 Turbulent effects were included via a homogeneous shear stress transport (SST)

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k-model. SIMPLE (Semi-IMplicit Pressure Linked Equations) was used for pressure-velocity coupling, and

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mostly second order discretization stencils were employed. With the segregated approach of SIMPLE, pressure

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and velocity are updated sequentially instead of simultaneously.29 Time step was varied to preserve a Courant

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number of 1 throughout the entirety of the simulation. For the finest mesh, the time step was generally on the

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order of one-hundred-thousandth of a wave/pulsation cycle (1×10-8 s).

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The Herschel-Bulkley model is used to describe the shear-thinning (beyond a yield stress) and temperature-

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thinning nature of the slurry. Banana puree viscosity is modeled as a function of both strain rate and temperature

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according to the data of Ditchfield et al.24 A user define function (UDF) in Fluent computes viscosity () according

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to Equations 4-7 from strain rate magnitude () and temperature ( in °C) values. In Equation 4, is the yield

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stress, and are calculated according to Equations 6 and 7,

is the lower strain rate bound, and

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is the corresponding upper viscosity bound. Although our implementation method used herein matches that

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of the validated method used in Ref. 30,30 we sought to further verify our calculations. Instantaneous cell-

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centered values of strain rate and temperature were collected at five locations (to be discussed more in a future

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section) within the model for a single moment in time. The viscosity observed in Fluent matched excellently

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that calculated by hand with Equations 4-7 (less than 0.005% difference), verifying correct implementation of

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the viscosity UDF.

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(4)

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(5)

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(6)

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(7)

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Extensive validation of the methods here employed for transonic wave formation and atomization has already

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been conducted over the course of the last 10 years.27, 31-38 The SST k-turbulence model, which is employed

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here, was sufficient to reveal important physical mechanisms. Experimental results were reproduced both

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quantitatively and qualitatively, and the primary validation exercise are summarized as follows. First,

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computations revealed the globally pulsing nature of an industrial three-stream air-water atomizer as

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qualitatively observed in experiments. Second, the experimental acoustic signature of pulsations and primary

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atomization ligament wave positions were quantitatively reproduced with numerical simulations. Third, the

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axial droplet size distribution in a non-Newtonian injector from experiments aligned quantitatively with

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numerical results. Additionally, assessment of droplet size distribution did not significantly alter with changing

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azimuthal angle, even for 1/32nd of a full 360° azimuth. Fourth, the numerical trajectory of a disintegrating

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droplet (after exposure to a normal shock wave) matched the analytical trajectory for said droplet. Furthermore,

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broadly similar atomization systems have been studied (recessed with a high gas-to-liquid momentum ratio

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and an inverted feed), and we find that the results presented here correspond to the globally varicose (but

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locally sinuous from the Lagrangian perspective of the flapping annular waves) pulsing nature expected of

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such systems.10

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Models were considered to be at quasi steady state (QSS) when various point monitor signals (discussed later)

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were statistically stationary. Unless otherwise noted, all results presented in this paper are using QSS data, and

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quantities were only time-averaged across QSS data. Since the system experiences bulk pulsation, flow time is

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normalized by the time elapsed between pulses. This is referred to as or, equivalently, as a

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. Waves and the general pulsing phenomenon are operating together in the same cyclic pattern.

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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-

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averaged data. A recently developed methodology was used to optimize the hardware utilization.39

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2.2 Mesh and Boundary Conditions

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A summary of the geometry, mesh, and boundary conditions is provided here. Rather than the typical twin-fluid

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atomizer design with a central liquid flow surrounded by a coaxial gas flow, the streams are reversed. An outer

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slurry annulus surrounds a central steam flow. The slurry pool is exposed to the hot subsonic center steam flow

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before the nozzle exit, and the nozzle orifice is extended to encourage significant steam-slurry interaction before

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exiting the nozzle. The full 360° domain is simulated by a 90° azimuthal slice bookended by periodic boundary

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conditions. Changing the angle from 45° to 90° produced no noticeable change in the axial droplet size profile,23

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and the sufficiency of the 90° azimuth will be further discussed among the results presented here. The downstream

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atomization domain spans 1.5 nozzle diameters in the radial direction and extends 2 nozzle diameters in the axial

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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

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mass ratio of 2.7%. Inlet temperatures for the slurry and steam were set to 304 K and 393 K, respectively. The

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steam inlet feed turbulence is set by defining a turbulent kinetic energy () and specific dissipation rate (), where

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= 48 m2/s2 and = 2.5×105 1/s. Both and are spatially constant and were chosen arbitrarily. All results

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presented in this paper should be understood with these inlet conditions in mind, and other inlet conditions could

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be studied to determine the influence of GLR and turbulence feed conditions on wave formation and atomization.

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Figure 1 provides an overview of the geometry and shows an example of the Ref-3-AMR mesh.

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represent the radial, axial, and azimuthal directions, respectively. In total, six distinct meshes were evaluated. A

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Base mesh was refined n times to produce Ref-1, Ref-2, and Ref-3 meshes. The total element count increases by

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a factor of nearly eight during refinement, as each cell length is cut in half. Note that the Base mesh was not refined

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farther back in the slurry annulus and steam pipe, but all regions of gas-liquid interaction were included in the

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refinements. The additional two meshes are Ref-2-45, which replaces the 90° azimuth with a 45° azimuth, and

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Ref-3-AMR, which utilizes adaptive mesh refinement (AMR). AMR provides the same refinement level as Ref-3

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but increases computational efficiency by refining dynamically only around the gas-liquid interface while the rest

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of the domain remains at a Ref-2 refinement level. To closely track the gas-liquid interface, the mesh is refined

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every 5 time steps, and a stringent VOF gradient criterion is used. Ref-3-AMR maintained around 28 million

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elements compared to the 132 million elements in the Ref-3 mesh.

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The Ref-3 mesh was found to be reasonably mesh independent across a range of metrics, including turbulent

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kinetic energy, velocity, and slurry volume fraction at the nozzle exit.22 Furthermore, wave physics comparable to

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those produced by Ref-3 (pulsation frequency and wavelength, for example) were revealed by Ref-1, two

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refinement levels below Ref-3. We acknowledge that all relevant length scales are not resolved, but Ref-3

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maintains the requirement of at least four cells across each droplet,23 which was shown to be sufficient for viscous

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slurry atomization using our computational methods.40 Strasser and Battaglia provide more discussion on

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implications of mesh resolution for atomization.31 The trend of numerical results versus mesh size showed Ref-3-

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AMR to be an outlier. For these reasons, Ref-3 is used for all results presented in this paper unless otherwise noted.

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The difference in numerical output from the Ref-3 and Ref-3-AMR meshes is an important point; a comparison

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of the two meshes will be presented later. Figure 2 presents a side view of the pre-filming (wave formation) region

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of the Ref-3 mesh. The region where slurry (this is where slurry

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waves rise into the steam flow) and the extension of the nozzle orifice

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before exiting the nozzle.

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Figure 1 Oblique view of a representative Ref-3-AMR surface mesh, illustrating the nature of adaptive mesh

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refinement (AMR). Starting with a Ref-2 mesh, the cells around the gas-liquid interface are dynamically

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refined every 5 time steps. Through a given pulsing cycle, Ref-3-AMR fluctuates between ~26 and ~30 million

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elements compared to the 132 million Ref-3 mesh. Both Ref-3 and Ref-3-AMR use a 90° azimuth with periodic

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boundary conditions on either side. The radial, axial, and

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respectively.

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Figure 2 Side view of the 132 million element Ref-3 mesh for the pre-filming (wave formation) region of the

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atomizer. A central steam flow is interrupted by the injection of the slurry, forcing the two phases to interact

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before exiting the nozzle. slurry and steam meet) and travel across

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3. RESULTS AND DISCUSSION

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3.1 3D Wave Cycle

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The inverted-feed, forced interaction nozzle design leads to highly regular bulk pulsations in the system. These

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pulsations may be looked at from two perspectives: inside and outside the nozzle. We will examine both in

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this paper. Inside the nozzle, annular waves form at regular intervals (corresponding to bulk system pulsation),

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rising out of the wave pool (labeled in Figure 2). One wave forms

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toward the center of the steam flow and collapses while exiting the nozzle. Previous studies have mostly

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considered 2D aspects of these waves (KHI, which is integral to the wave formation process, is 2D),22 leaving

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important 3D characteristics uninvestigated (RTI, for example, manifests in the azimuthal dimension). We

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seek to clarify how azimuthal variation develops in the wave and leads to atomization downstream.

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Outside the nozzle, regular radial bursting of slurry is observed. This bursting manifests as a three-part

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sequence: 1) stretch, 2) bulge, and 3) burst. While a wave is forming inside the nozzle, an annular sheet of

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slurry stretches from the nozzle exit. As the wave collapses, the windward pressure build-up propagates

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through the wave, and the slurry sheet bulges radially. Finally, the slurry sheet ruptures altogether. Both wave

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cycling and radial bursting occur at a frequency around 1000 Hz (more on this later). The wave formation is a

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sinuous instability manifestation (from a Lagrangian perspective of an observer moving with the wave leaving

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the nozzle), but the bulk pulsation outside of the nozzle (radial bursting) is a varicose manifestation. Wave

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formation and collapse are integral -by forcing the wave to crash

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into itself and enhance the disintegration of viscous, non-Newtonian fluids with radial explosions. The Weber

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(), Reynolds ( ), and Strouhal ( ) numbers characterizing the wave are

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2.0×104, 1.2×104, and 0.08, respectively. Here, is wavelength, is wave speed, is wave frequency, is the

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steam-slurry bulk velocity, is slurry density, is surface tension, and is the lower end of slurry viscosity.

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To better understand cross-wave variation during the 3D wave cycle, Figures 3-6 present a unique view of the

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annular wave: the 90° azimuthal slice has been unraveled to (rather than annular)

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visualization of the slurry interfacial motion with a superimposed contour in the background. The result is

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more comparable to what ocean waves on a beach look like and provides insight into 3D aspects of the wave

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physics. Wave formation is illustrated in frames 1-3 and wave collapse in frames 4-6. Figure 3 (Multimedia

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view) shows the unraveled slurry surface colored by viscosity, and the background contour is colored by strain

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rate. This view is from the perspective of inside the nozzle to observe wave formation. High strain-rate regions

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appear as the hot steam contacts the slurry both at the wave and with droplets in the free stream. Viscosity

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reduces according to the shear-thinning nature of the slurry where strain rate is highest and the temperature-

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thinning effect by the hotter steam. Viscosity reduction causes ligaments to flick up into the steam,

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compounding the shear-thinning effect. Some azimuthal variation in viscosity is evident, although spatial

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variability fluctuates throughout the wave cycle. Azimuthal variation in the slurry is present, though minimal,

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in frames 1-3 but becomes very pronounced as the wave crashes in frames 4-6. Frames 4 and 5 show irregular

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valleys and ridges forming across the azimuth. The RTI time scale, as approximated with a low-Re method41

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for the low end of the slurry viscosity spectrum, is an order of magnitude lower than the wave time scale. This

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suggests that sufficient time is available for RTI development.

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Figure 3 Sequential views of unraveled wave with the slurry surface colored by viscosity and contours of

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strain rate in the background. Pictures are at equally spaced flow time intervals through one representative

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wave cycle. The view is from the inside of the nozzle looking down on the wave. The shear- and temperature-

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thinning nature of the slurry is highlighted by viscosity reduction in response to high strain rate as the wave

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penetrates into the hot steam. Azimuthal variation is present as the wave forms (frames 1-3) but increases

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significantly as the wave collapses (frames 4-6). (Multimedia view).

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Figure 4 (Multimedia view) presents a similar visualization as that in Figure 3 but viewed from outside the

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atomizer (looking underneath the wave). This perspective shows the annular slurry sheet (now flattened)

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stretching out from the nozzle as a wave rises. In this view, the wave is rising underneath the slurry sheet. The

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point of maximal extension before rupture (frame 4) corresponds to the highest viscosity of the slurry sheet,

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as it experiences minimal shear outside the nozzle radius. The sheet is destabilized by both azimuthal variation

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from RTI (heavier slurry being accelerated by the lighter steam) and azimuthal variation in viscosity. Frame 4

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reveals significant and azimuthally variant bulging underneath the wave, which corresponds to valleys in the

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wave surface. Clearly, the wave is not impacting the slurry sheet uniformly. We also note that the multiple

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bulges across this 90° azimuthal slice indicates that 90° is a sufficient angle to capture variation in the azimuth.

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Parts of the wave contact local portions of the slurry sheet before others-

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slurry sheet. Frame 5 marks the pre-rupture of the slurry sheet, where small portions begin to break apart. The

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collapse of the wave and the instability of the slurry sheet both likely contribute to pre-rupturing. The further

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destabilized slurry sheet completely ruptures in frame 6 in a violent radial burst.

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Figure 4 Sequential views of unraveled wave with the slurry surface colored by viscosity and contours of

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strain rate in the background. Pictures are at equally spaced flow time intervals through one representative

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pulsing sequence. The view is from the outside of the nozzle (looking up from underneath the wave) to

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illustrate the radial bursting phenomenon. As the slurry stretches out in an annular (flattened for these views)

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sheet, RTI - as the collapsing

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wave starts to contact the slurry sheet, followed by complete rupture in frame 6. (Multimedia view).

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Waves have a high blockage ratio that significantly affects the steam pressure and velocity via local

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acceleration; both fluctuate periodically with the pulsing cycles. Figure 5 (Multimedia view) illustrates these

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effects by showing the unraveled wave surface colored by pressure with a Mach number contour in the

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background. As the wave rises, it shelters its leeward side from the oncoming steam flow, thereby reducing

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the flow area for the steam exiting the nozzle. Steam is then accelerated to transonic velocities. Though not

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explicitly shown in Figure 5, small regions are supersonic. Consequently, steam compresses on the windward

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side of the wave to build up pressure, and the steam accelerates above the wave crest through the reduced-area

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opening. Both phenomena have implications for wave formation. The windward high-pressure zone exploits

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irregularities in the slurry surface caused by RTI and viscosity gradients. The transition from wave formation

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to wave collapse (frame 3 to frame 4) is significant: pressure increases sufficiently to overcome inertia and

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surface tension, and the increase in azimuthal variability is marked. We observe in frame 5 the propagation of

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the windward high-pressure zone axially as the wave collapses, which is a driving force in the radial bursting.

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Figure 5 Sequential views of unraveled wave with the slurry surface colored by pressure with contours of

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Mach number in the background. Pictures are at equally spaced flow time intervals through one representative

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wave cycle. The view is from the inside of the nozzle looking down on the wave. The high blockage ratio of

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the wave leads to pressure build up on its windward side and reduces the exit area for the steam to accelerate

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it above the wave crest. Steam clearly reaches transonic speeds. Pressure buildup drives wave collapse and

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exploits azimuthal variations in the slurry surface (caused by RTI and viscosity gradients) for disintegration.

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(Multimedia view).

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Figure 6 (Multimedia view) provides a side view to of the wave cycle (gray interface) with a contour of

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temperature in the background. Much like pressure, temperature cycles as the steam periodically compresses

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on the windward side of the wave. Highest steam temperatures occur in frames 4 and 5 and will contribute to

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surface destabilization and wave disintegration by reducing slurry viscosity, although shear is undoubtedly the

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dominant driving force behind viscosity changes. The wave appears to be largely 2D as it rises and approaches

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the nozzle exit. As it reaches the nozzle exit in frame 4, the wave is transitioning to a more 3D surface. This

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corresponds to the dramatic increase in azimuthal irregularity as pressure exacerbates existing surface

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variations and temperature thins the slurry.

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Figure 6 Sequential views of unraveled wave (gray) with contours of temperature in the background. Pictures

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are at equally spaced flow time intervals through one representative wave cycle. The view is directly from the

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side of the wave, effectively outing the wave profile. Wave blockage causes the steam to compress, cycling

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the temperature with wave formation and collapse. The wave is largely 2D until around the nozzle exit, where

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it collapses, and azimuthal variations are exacerbated for a strongly 3D surface. (Multimedia view).

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It has already been noted that RTI is driven by the baroclinic torque term

in the vorticity equation.

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Density-based torque on the interface creates vorticity that will tend to increase the misalignment of pressure and

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density gradient vectors. This in turn creates additional vorticity, leading to further misalignment. Figure 7

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(Multimedia view) presents contours of baroclinic torque. The highest values are on the order of 1/s2,

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indicating significant misalignment of gradient vectors and strong RTI activity in the wave. Baroclinic torque is

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greatest at the steam-slurry interface, where density gradients are highest. Aside from directly at the interface,

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baroclinic torque is highest in the regions where droplets are being stripped off the wave and interacting with the

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steam. This indicates that as the shear layer forms, RTI becomes important very early in the wave life.

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Figure 7 Sequential side contours of baroclinic torque through one representative wave cycle. Rayleigh-Taylor

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instabilities (RTI) arise from this baroclinic torque term in the vorticity equation. Baroclinic torque is highest

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at the interface, where the density gradient is highest. The largest values shown are on the order of

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1/s2, indicating significant RTI activity present in the wave from its birth. (Multimedia view).

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RTI and KHI are present early in the wave and lead to minute interfacial deformation. As the approaching

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steam navigates the newly roughened surface, spatial variations in temperature and strain rate occur across the

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wave. The shear- and temperature-dependent slurry responds accordingly, producing local variations in

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viscosity across the surface. Because of viscosity variation, the steam-slurry interface develops axial and

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azimuthal wavelength spatial variability. Also, due to viscosity variation, the interfacial stress develops spatial

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variation. Finally, the process is repeated as the surface deformation excites RTI and KHI. Figure 8 illustrates

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the cycle, where T is temperature, SR is strain rate, and WL is wavelength. Note that we are not utilizing linear

367

instability analysis but rather the local momentum balance to reveal these instabilities.

368

369

370

Figure 8 Instability cycle, where the Kelvin-Helmholtz instability (KHI) and Rayleigh-Taylor instability (RTI)

371

cause surface variations that in turn excite the instabilities. Here, T is temperature, SR is strain rate, and WL

372

is wavelength.

373

374

375

3.2 Droplet Production

376

In a previous study, the effect of the wave on droplet production was largely addressed from the perspective

377

of bursting outside the nozzle,23 but questions remain regarding the role of 3D surface instabilities in droplet

378

production inside the nozzle. It has been noted that small droplets break away from the wave inside the nozzle,

379

and this was primarily attributed to steam shear (and later the collapse of the wave).23 A more thorough

380

investigation of this phenomenon is presented here, but we will first put the atomizer and its droplet production

381

mechanisms in context.

382

383

The Wave-Augmented Varicose Explosions (WAVE) design is essentially a combination of a Gas-Centered

384

Swirling Coaxial (GCSC) and Effervescent atomizer which capitalizes on the advantages of both and

385

incorporates an additional element of radial momentum generation. Effervescence is introduced when the wave

386

crashes onto the annular slurry sheet at regular intervals. Steam is sandwiched between the wave and sheet,

387

introducing bubbles into the liquid phase. The Gas-Centered feature of GCSC is preserved, though swirl is not

388

included in the current design. The lack of swirl necessity is a benefit of the current WAVE design because

389

swirl increases internal geometric complexity and could require maintenance of fouled swirl elements. Besides,

390

a previous study showed the impact of swirl in a multi-stream non-Newtonian atomizer to be minimal.33 Future

391

research may include revisiting this idea.

392

17

393

Tat least

394

three mechanisms. 1) The momentum of the crashing wave creates radial bulges in the liquid film, bursting

395

droplets outward at regular intervals. 2) The high blockage ratio of the wave causes an intense pressure increase

396

behind the wave that assists the wave momentum in driving liquid film rupture and slurry disintegration. 3)

397

The extension of the slurry as a wave into the steam flow increases the interfacial area, providing more space

398

for the steam to peel off droplets. 42 refers to the fact that, from a fixed external observatory frame,

399

400

refers to the radial blasting of droplets caused by wave crashing.

401

402

The first two mechanisms have been thoroughly investigated.23 It is the third mechanism that will be explored

403

here. Droplets clearly break away from the wave as it penetrates into the steam flow inside the nozzle (before

404

the wave completely collapses), but what factors contribute to this outcome? It remains unclear whether the

405

steam is merely stripping droplets off an axially moving wave surface or if slurry ligaments are being flicked

406

up into the steam with significant radial velocity. To address this question, we present contours of radial

407

velocity divided by the absolute value of axial velocity in Figure 9 and Figure 10 (Multimedia view). The

408

outline of the steam-slurry interface is marked in black. Blue represents movement inward towards the central

409

steam flow, and red represents movement outward toward the beach and beyond. For the purposes of this

410

discussion (and from the perspective of Figure 9 and Figure 10), we will refer to blue regions as moving

411

412

413

The time sequence in Figure 9 reveals the temporal development of radial versus axial velocity through a given

414

wave cycle. By frame 3, the wave is penetrating significantly into the steam flow. Consequently, the wave

415

develops dark blue on its back, corresponding to the deflection of steam upward. This deflection

416

is significant close to the wave but less so further upstream. Just behind the wave jet, much of the slurry is

417

directed axially (white), but the wave jet itself has significant radial thrust pushing upward into the steam flow.

418

Radial velocity is at least 50% of the axial velocity in dark blue regions, which is the limit of the scale. By

419

18

frame 4, where the wave height peaks, much of the wave base is moving in the axial direction. However, we

420

still observe strong upward movement at the top of the wave, and some ligaments are present. In other words,

421

we observe the slinging of ligaments upward as the main body of the wave is moving forward. Steam around

422

the wave tip and ligaments also has strong upward motion. Non-axial flow in an atomizer has been found to

423

correlate with higher droplet production efficiency.31 As an aside, the beginning of the radial bulge is quite

424

evident as the red patch just at the end of the beach in frame 4. The flicking up of ligaments in frame 4 helps

425

explain what we term . For example, there is a nearly vertical ligament around the

426

nozzle exit in frame 5, which is unexpected because of the high-velocity steam directed toward it. This appears

427

to be a residual ligament derived from the windward ligament in frame 4. We call the windward ligament a

428

primary wave tip. By frame 5, the ligament is moving downward, but

429

it is an enduring form of the upward moving ligament in frame 4, and its position enables steam to disintegrate

430

the slurry more effectively. Curiously, the steam on the leeward side of the residual ligament is moving upward,

431

opposite the direction of the ligament and most of the surrounding slurry. This upward moving steam is likely

432

helping strip away droplets via shear. By the time the wave has collapsed (frame 6), a clear divide is observed

433

around the middle of the beach. To the left, slurry is moving upwards, and to the right, slurry is moving

434

downwards. The primary exception to this trend is the downward moving slurry in the sheltered region on the

435

leeward side of the newly formed wave.

436

437

438

19

439

Figure 9 The ratio of radial velocity to the absolute value of axial velocity through one representative wave

440

cycle. Contours are on a plane corresponding to a 30° azimuthal angle. As the wave peaks in frame 4, much

441

of the wave base is moving in the axial direction (white). The wave tip and associated ligaments, however, are

442

still slinging upward (negative radial direction) into the steam flow, which will enhance droplet production.

443

444

Figure 10 shows contours at the moment in time roughly where the wave peaks in height (equivalent to frame

445

4 in Figure 9) at 6 azimuthal angles. The base of the wave is generally uniform across the various azimuthal

446

slices, but the wave tips differ significantly across angles, even at 6° increments. What is consistent across

447

angles is this: while the wave base is moving with a largely axial velocity, the wave tip has a significant radial

448

velocity component. In frames 1 and 6 especially, we see a distinct secondary ligament form behind the wave

449

tip, which also has a high radial velocity. Most frames include some level of a steam gap between the wave tip

450

and a secondary ligament or wave base. The resulting wave tips can take on something of a hammerhead

451

shape before breaking away (for example, frames 2, 4, and 6) as the slurry connecting the tip to the base is

452

thinned. We direct the reader to notice particularly the penetration of steam into the wave as the tip rises. In

453

frames 2-4, the steam is pushing down into the wave, creating significant necking to break off the tip. Frame

454

5 shows smaller pockets of steam inside the wave, where steam fingers have forced through the deformed

455

surface. The upward movement of the wave, then, provides an effervescence mechanism in addition to the

456

trapping of steam as the wave crashes.

457

458

459

460

20

Figure 10 The ratio of radial velocity to the absolute value of axial velocity at a fixed time, roughly where the

461

wave peaks (equivalent to frame 4 in Figure 9). Contours are on planes corresponding to azimuthal angles of

462

30°, 36°, 42°, 48°, 54°, and 60°. The wave base consistently moves largely in the axial direction (white), while

463

the wave tip generally includes a significant upward velocity component. Steam penetrates the wave to break

464

off the tip (frames 2-4) and as smaller bubbles (frame 5), providing a mechanism for effervescence.

465

(Multimedia view).

466

467

It is clear that flicking is an important mechanism for droplet production in the nozzle, but is stripping also

468

important? While flicking involves ligaments rising perpendicular to the steam flow, by stripping, we mean

469

droplets breaking away from the slurry surface or a ligament stretching in the direction of steam flow (parallel).

470

A close look at Figure 9 seems to show stripping towards the beginning of wave growth, but this is made

471

clearer in the animation for Figure 10. This animation shows stripping early on where droplets break free from

472

both the surface and parallel ligaments. As the wave matures, the primary droplet production mechanism

473

transitions to flicking. Finally, outside the nozzle, effervescence and the collapse of the wave lead to its

474

complete disintegration. In summary, droplet production inside the nozzle is dominated by stripping in the

475

early wave and later by flicking as the wave crests.

476

477

To conclude our discussion, we note several consequences (and benefits) of slurry ligaments flicking radially

478

up into the gas stream. Effervescence is introduced via a second mechanism before the wave crashes and

479

sandwiches steam. While the slurry wave jet is accelerated by the steam and thinned by shear, causing the jet

480

to buckle upwards and sling the liquid tips25 up into the gas stream as ligaments, the gas infiltrates the liquid

481

as bubbles. The thin fingers from the liquid sheet make it easier for the steam to peel off droplets,43 and this

482

avoids the need for a forcibly thinned sheet using a thin slurry annulus with high pressure drop and risk of

483

plugging. Additionally, the shearing and thinning of the fingers lowers the timescale for RTI to take effect,

484

making RTI more active and sooner.

485

486

3.3 Point Monitors

487

3.3.1 Overview

488

In a previous study, only two quantities were tracked at a single point monitor.22 We greatly expand point

489

monitor analysis here to include 100 signals for the purpose of understanding the spatial and temporal variation

490

21

of quantities, both azimuthally and along the slurry flow path. Velocity magnitude, slurry volume fraction

491

(VF), strain rate, and turbulent kinetic energy (TKE) were tracked at five point monitors placed roughly along

492

the trajectory of the slurry interface as it interacts with the steam. Each point monitor is present at five different

493

azimuthal angles (4.5°, 13.5°, 22.5°, 31.5°, and 40.5°), spaced evenly within a 45° azimuth, which is half of

494

the total 90° azimuth. In total then, 100 individual signals were recorded (4 quantities across 25 points) for the

495

entirety of the QSS window. Due to the extensive nature of this temporal data collection, we provide only

496

select plots and summary statistics. The locations of the five point monitors in a given azimuthal slice are

497

shown in Figure 11. The labels of Inner, Middle, Exit, Tip, and Outer, will continue to be used in reference to

498

these points, along with their azimuthal angles. The Inner, Middle, Exit, Tip, and Outer points are located 1,

499

0.75, 0.67, 1.25, and 1.5 nozzle radii from the nozzle axis (bottom edge of Figure 11), respectively. The Inner

500

point is centered axially on the wave pool. The Middle and Exit points are positioned axially at the start and

501

end of the beach. The Tip and Outer points are located 0.25 and 1.25 nozzle radii downstream of the nozzle

502

exit, respectively. The output data monitored at these five points will be discussed throughout the remainder

503

of the paper, revealing the spatiotemporal characteristics of the system.

504

505

506

507

Figure 11 Locations of five points monitors (roughly along the slurry interface trajectory) where certain data

508

quantities are collected. Each point monitor exists at 5 azimuthal angles within a 45° azimuth (half of the total

509

90° azimuth): 4.5°, 13.5°, 22.5°, 31.5°, and 40.5°.

510

511

Comparing the Exit point monitors across azimuthal angles reveals aspects of the wave interface as it exits the

512

nozzle. Figure 12 shows a moving average time series for (a) velocity and (b) slurry volume fraction through

513

22

12.5 pulsing cycles. Velocity signals are fairly uniform across angles and from wave to wave. In other words,

514

the general motion of the wave seems to vary minimally as it exits the nozzle, both azimuthally and temporally.

515

However, slurry volume varies significantly in the azimuthal and temporal dimensions. We attribute this

516

variability largely to the work of RTI, which is exacerbated by windward pressure buildup. We note again that

517

it is around the nozzle exit that the wave collapses, marking a transition from a somewhat 2D wave to a more

518

azimuthally diverse 3D wave. Table I provides the mean and coefficient of variation (COV) for each signal in

519

Figure 12 as well as the other quantities that are not displayed graphically. Note that this is a small subset of

520

all point monitor data. The COV is a normalized standard deviation, where the standard deviation is divided

521

by the mean and converted to a percentage.

522

523

524

Figure 12 Moving average time series of (a) velocity and (b) slurry volume fraction at the Exit point monitors

525

across all five azimuthal angles. The velocity signals are fairly uniform, both azimuthally and temporally. RTI

526

causes slurry volume fraction, however, to vary significantly in the azimuthal and temporal dimensions.

527

528

529

Table I Mean and coefficient of variation (COV) for signals across all five azimuthal angles at the Exit point.

530

COV is calculated by dividing the standard deviation by the mean and converting to a percentage.

531

532

Velocity [m/s]

Slurry VF

Strain Rate [1/s]

TKE [m2/s2]

Angle

Mean

COV

Mean

COV

Mean

COV

Mean

COV

4.5

142

94

0.17

219

4.5×105

156

412

142

13.5

157

83

0.09

312

4.5×105

141

414

129

22.5

161

83

0.09

311

3.6×105

152

345

131

31.5

167

82

0.10

301

3.0×105

170

325

144

40.5

171

79

0.10

299

2.9×105

166

312

148

533

23

On the other hand, we can reveal the axial (roughly) development of quantities by examining all five point

534

monitors at a given azimuthal angle. Figure 13 shows the moving average time series for (a) velocity and (b)

535

strain rate for all point monitors at the 22.5° azimuthal angle. Velocity magnitudes increase significantly as

536

the wave moves from the pool to the nozzle exit but decreases at the Tip and Outer point monitors. The slurry

537

is disintegrating at these last two points, so the monitors are picking up both slurry droplets and relatively

538

stagnant steam. Periodicity is much more evident inside the nozzle than outside the nozzle. This is indicative

539

of the relatively uniform wave motion inside the nozzle and the chaotic rupture outside the nozzle. Strain rate

540

follows a similar pattern through the slurry motion, increasing to the nozzle exit and then decreasing outside

541

the nozzle. However, the difference between the Middle/Exit points and the Tip/Outer points is less marked

542

than velocity. Compared to velocity, the temporal periodicity of strain rate is less pronounced for the middle

543

Middle/Exit points and more pronounced for the Tip/Outer points. In other words, velocity gradients are

544

fluctuating more consistently than velocity magnitude outside the nozzle. Strain rate also shows more wave-

545

to-wave variation than velocity within the nozzle. Table II provides the mean and COV for each signal in

546

Figure 13 as well as the other quantities that are not displayed graphically.

547

548

549

Figure 13 Moving average time series of (a) velocity and (b) strain rate at all five point monitors at the 22.5°

550

angle, illustrating variation of these quantities spatiotemporally as slurry moves through the system. The

551

velocity increases up to the nozzle exit as waves form and is much more periodic inside the nozzle than outside

552

the nozzle, showing the contrast between the more ordered wave formation and the more chaotic radial

553

bursting. Inside the nozzle, strain rate shows more wave-to-wave variation than velocity, and the Tip and Outer

554

points show more periodicity.

555

556

557

558

24

Table II Mean and coefficient of variation (COV) for signals across all point monitors at the 22.5° azimuthal

559

angle. COV is calculated by dividing the standard deviation by the mean and converting to a percentage.

560

561

Velocity [m/s]

Slurry VF

Strain Rate [1/s]

TKE [m2/s2]

Point

Mean

COV

Mean

COV

Mean

COV

Mean

COV

Inner

4.2

18

1.0

0.0

1.9×103

54

1.1

60

Middle

98

86

0.241

176

2.2×105

136

255

147

Exit

161

83

0.090

311

3.6×105

152

345

131

Tip

31

67

0.064

367

6.8×104

257

28

326

Outer

30

46

0.026

550

7.7×104

137

14

320

562

563

3.3.2 Frequency Analysis

564

FFTs were performed to determine the dominate frequencies in the atomizer at various locations. Because of

565

the pulsing nature of the system, we expect most quantities to cycle at a consistent overall pulsing frequency

566

(frequency of wave formation). Velocity magnitude, strain rate, slurry volume fraction, and turbulent kinetic

567

energy were tracked at all 25 point monitor locations. FFTs reveal frequencies around 1000 Hz for the vast

568

majority of these quantities and point monitors. 83% of signals show a 1068 Hz peak frequency (note that

569

slurry volume does not fluctuate at any Inner point locations). 8% show a 2060 Hz peak frequency, and one

570

shows 3128 Hz, illustrating the higher mode harmonics. Strain rate is the most consistent quantity: all 25 point

571

monitors show a dominate frequency of 1068 Hz. The most prominent frequencies across point monitors for

572

the Ref-1 and Ref-2 meshes are 963 Hz and 992 Hz, respectively. Frequencies vary wildly for the Base mesh,

573

and no prominent frequency is evident. This corresponds to previous findings: major wave characteristics are

574

present, and consistent, with progressively increased mesh resolution, beginning with the resolution of Ref-

575

1.22 The characteristic pulsing of the system is largely absent from the Base case, and the Ref-3 mesh is two

576

refinement levels above Ref-1.

577

578

To understand how FFT peak frequencies vary across the azimuth, the COV was computed, which shows

579

azimuthal point-to-point variation as a percentage. For a given point location, such as Inner, a single peak

580

frequency was computed at each azimuthal angle, and the COV was computed from these 5 frequencies. Figure

581

14 shows the azimuthal frequency COV from the Inner to the Outer points (effectively along the interfacial

582

trajectory). Most frequencies are consistent inside the nozzle, but the variation increases significantly outside

583

25

the nozzle. The only quantity that continues to fluctuate uniformly in the azimuth outside the nozzle is strain

584

rate. Figure 14 illustrates again the consistency of strain rate as a pulsating quantity.

585

586

587

Figure 14 Azimuthal coefficient of variation (COV) for frequency at various points along the slurry interface

588

flow path. COV represents the percent of variation in the azimuth at each point. VF is volume fraction, and

589

TKE is turbulent kinetic energy. The peak frequencies were determined by FFTs. Frequencies are generally

590

less consistent outside of the nozzle as the slurry disintegrates. Strain rate stands out, fluctuating at consistent

591

frequencies at all locations (inside the nozzle, outside the nozzle, and across the azimuth).

592

593

FFTs are meaningless unless the resulting peak frequencies have converged over the course of the simulation.

594

For those signals that showed a peak frequency of 1068 Hz (the prominent frequency among all signals

595

evaluated), the FFTs generally converged within 4 pulsing cycles after the flow was already at QSS. This was

596

not necessarily the case for less periodic signals like velocity at the Tip and Outer points (see Figure 13).

597

Figure 15 provides a sample FFT peak frequency convergence plot for strain rate at the Exit point in the 22.5°

598

plane (see Figure 13 for time series). The vertical dashed line is the 4 pulsing cycles mark, and the green star

599

is the final value. Zero-padding was employed, resulting in the discrete step values towards convergence.

600

601

26

602

603

Figure 15 Convergence of FFT peak frequency for the strain rate signal at the Exit point monitor in the 22.5°

604

plane. FFTs included zero-padding, which produces the discrete steps towards convergence. The vertical

605

dashed line is at 4 pulsing times, and the green star represents the final frequency value of 1068 Hz. Signals

606

generally converged to this value within 4 pulsing cycles.

607

608

Figure 16 presents two FFT examples, one for velocity at the Inner point (left) and one for strain rate at the

609

Exit point (right). Both are at a 22.5° azimuthal angle, and the time series were shown in Figure 13. The Inner

610

point is located at the surface of the wave pool, where waves are being produced. The wave pool surface, then,

611

has a fluctuating velocity at 1068 Hz, which sets the pace for bulk pulsation in the system. Geometric

612

parameters might be varied to determine their influence on pulsing frequency, but that is beyond the scope of

613

this study. The Inner velocity FFT also shows harmonics at roughly 2060 Hz and 3110 Hz, which correspond

614

to the peak frequencies for a minority of signals. The Exit strain rate FFT shows the same peak frequency as

615

the Inner velocity, although it is less pronounced, and no harmonics are evident.

616

617

618

27

Figure 16 FFTs with peak frequency labeled for (a) velocity at the Inner point and (b) strain rate at the Exit

619

point. Both points are at an azimuthal angle of 22.5°. The velocity at the Inner point (where waves are formed)

620

shows a peak frequency of 1068 Hz and clear harmonics at around 2060 Hz and 3110 Hz. Strain rate at the

621

Exit point has the same peak frequency, but it is less distinct and with no harmonics.

622

623

As shown, the preponderance of frequencies are close to, or multiples of, approximately 1000 Hz, which results

624

from the wave pool generation process (i.e. KHI working with Bernoulli to amplify surface disturbances).

625

Therefore, the wave-generation process sets up an absolute instability in the system that is likely to be

626

unaffected by upstream turbulence effects. A future study could include evaluations of this.

627

628

3.3.3 Cross-Correlations

629

The spread of information in the azimuthal and radial directions can be assessed by determining the cross-

630

correlation between signals; a normalized cross-correlation of 1 indicates that two time-series signals are

631

perfectly correlated and implies 2D (axisymmetric) motion. The time difference between two given locations

632

was accounted for by time-shifting the signals to align them before calculating the normalized cross-

633

correlation. Cross-correlations (calculated using the NumPy package in Python) were normalized by

634

subtracting the means from the signals and dividing the cross-correlation by the number of data points and the

635

standard deviations of the two signals. Figure 17 shows the normalized cross-correlation between velocity

636

signals both azimuthally and along the slurry interface trajectory. Correlation is calculated between the velocity

637

signal at a given azimuthal angle or point and the first angle or point. In essence, we are estimating how much

638

the motion at one place in the flow field might be related to the motion at another place in the flow field.

639

640

The motion at the inner point monitor appears to be uniform across all angles (indicating its 2D nature), as the

641

signals show almost perfect correlation with the first angle (4.5°). The Middle and Exit points show relatively

642

high correlation, and the correlation remains largely the same across angles. Outside of the nozzle, where the

643

wave and slurry sheet are being ruptured, the tip and outer point monitors show low correlation between angles.

644

The Tip point monitor, which is closer to the nozzle (around the bulging and bursting), shows slightly higher

645

correlation. These results suggest that fluid motion inside the nozzle is generally azimuthally similar (2D), but

646

the fluid motion outside the nozzle as the slurry bursts is azimuthally unrelated (3D). Furthermore, since the

647

28

correlation, for a given point monitor, is quite consistent across 45°, we conclude that a 90° mesh is more than

648

sufficient to capture azimuthal variation. The Inner point, which is at the surface of the wave pool, marks the

649

location of wave generation. The Middle and Exit points are well-correlated with the Inner point, but the Tip

650

and Outer points are not. In other words, the velocities at Tip and Outer do not show much relation to wave

651

pool motion.

652

653

654

655

Figure 17 Normalized cross-correlation between transient velocity magnitude signals from point monitors at

656

five points along the slurry interfacial flow and five azimuthal angles. A given normalized cross-correlation

657

was calculated after the two signals were time-shifted to align. A value of 1 indicates perfect correlation

658

between signals. The correlation between a given angle or point with the first angle (4.5°) or point (Inner) is

659

computed. High correlation is observed both azimuthally and along the slurry flow inside the nozzle but is

660

greatly diminished outside the nozzle. The lack of azimuthal variation in correlation across 45° demonstrates

661

the sufficiency of a 90° mesh to capture variations in the azimuth.

662

663

When signals are reasonably correlated, a time lag shows how much the motions are temporally offset. The

664

time lag is calculated as the amount the signals must be time-shifted to produce the maximum cross-correlation.

665

In Figure 18, time lags are normalized by the pulsing time, so a given value represents the fraction of a pulse

666

cycle by which the signals are offset. Time lag is meaningless if the signals are not well-correlated, so the Tip

667

and Outer points have been removed from the time lag plot. Figure 18 displays the extent to which the Middle

668

and Exit velocity signals lag the Inner velocity signal. These results are communicating where the Middle and

669

Exit points are, temporally, within the wave cycle (which repeats regularly every 1 PT). Velocity fluctuations

670

at the wave pool (Inner point) reach the Middle and Exit points 0.7 and 0.8 PTs, respectively, after the initial

671

wave pool motion. Fluid motion at the nozzle exit then lags the Middle point by 0.1 PT. Velocity spikes again

672

29

in the wave pool 0.2 PTs after a velocity increase at the Exit point. This cycle is illustrated in Figure 18, which

673

shows the percentage of a pulse cycle for velocity fluctuations at Inner (I) to reach Middle (M) and then Exit

674

(E) and then start again at Inner. The majority of a given pulse cycle (70%) involves the wave growing out of

675

the wave pool and reaching the beach. After reaching the beach, the wave travels more rapidly to the nozzle

676

exit and beyond.

677

678

Figure 18 Transient velocity signal time lags from the Inner point at five azimuthal angles. The time lag

679

corresponds to the signal offset that produces the maximum cross-correlation and has been normalized by the

680

pulsing time. Wave pool velocity fluctuations (Inner point, I) take 0.7 and 0.8 pulsing times to reach the Middle

681

(M) and Exit (E) points, respectively. Initial growth of a wave consumes the majority (70%) of a given pulse

682

cycle.

683

684

In addition to inter-point correlations for velocity, correlations between different quantities were calculated at

685

all 25 points. Three contour plots in Figure 19 summarize these data and show the normalized cross-correlation

686

between 1) velocity magnitude and slurry VF (far left), 2) TKE and strain rate (middle), and 3) velocity

687

magnitude and strain rate (far right). The slurry volume fraction maintains a value of 1 at the Inner points,

688

making any normalized cross-correlation value meaningless. For this reason, the inner points were excluded

689

on the leftmost plot in Figure 19. The trend across all 3 sets of correlations is that any two quantities are most

690

highly correlated at the Inner point, and correlation decreases downstream. Correlation between quantities is

691

also fairly azimuthally uniform, particularly inside the nozzle. Velocity and slurry VF are the least well-

692

correlated overall. The middle plot indicates that, after the Exit point, TKE and strain rate are decoupled; thus,

693

TKE downstream must have been produced by some earlier shear. In summary, we observe the decoupling of

694

quantities exiting the nozzle: strong correlations inside the nozzle and very weak correlations outside the

695

nozzle.

696

30

697

698

Figure 19 Contours across all 25 points for the normalized cross-correlations between 1) velocity magnitude

699

and slurry volume fraction (left), 2) turbulent kinetic energy strain rate (middle), and 3) velocity magnitude

700

and strain rate (right). The Inner points have been excluded on the leftmost plot because the slurry volume

701

fraction maintains a value of 1. Correlations are generally strong inside the nozzle but very weak outside the

702

nozzle, showing a decoupling of quantities exiting the nozzle.

703

704

705

3.4 Adaptive Mesh Refinement

706

Up to this point, the Ref-3 mesh has been used entirely for visualization and analysis. It has been noted that

707

Ref-3-AMR, though a more cost-efficient alternative, is the outlier to the numerical trends of two point monitor

708

signals,22 but no visual comparison was provided, and a more rigorous comparison is lacking. A qualitative

709

and quantitative comparison of Ref-3 and Ref-3-AMR is here provided to clarify the differences between the

710

two models. We emphasize again that Ref-3 and Ref-3-AMR have the exact same mesh resolution at the slurry-

711

steam interface. A rigorous VOF gradient criterion and adaption frequency were used, but it should be noted that

712

there are other criteria that can determine the regions of adaption for AMR. It is possible that a different criterion

713

would positively affect AMR results, though we find this doubtful based on the extensive AMR refinement in this

714

study. We also acknowledge that any conclusions about AMR as a technique are limited to ANSYS Fluent and its

715

underlying algorithms. All validation efforts were conducted using meshes equivalent to Ref-3 (mostly

716

hexahedral elements swept in the flow direction), and AMR has not been validated experimentally for this

717

work. The differences between Ref-3 and Ref-3 AMR indicate problems with the communication of

718

information through split cells, which are constantly created by AMR.

719

720

Figure 20, which shows the slurry surface (coloured by slurry viscosity) as it exits the nozzle (flow is generally

721

downward), reveals qualitative differences. The leftmost set of images in Figure 20 show that Ref-3-AMR

722

31

produces a significantly more rippled surface, and slurry viscosity is generally higher. A higher viscosity

723

indicates lower strain rate, perhaps suggesting that the mesh gradients around the slurry-steam interface are

724

affecting velocity gradients. We remind the reader that RTI and viscosity gradients serve to destabilize the

725

annular slurry sheet, priming it for rupture and atomization as the wave crashes into it. The middle and

726

rightmost images in Figure 20 reveal Ref-3 rupturing more readily than Ref-3-AMR. In the middle images,

727

little pre-rupture (localized bursting events) in the annular slurry sheet is observed for Ref-3-AMR. Much more

728

can be seen for Ref-3 (corresponding to the preliminary rupture for Ref-3 in frame 5 of Figure 4). In the

729

rightmost images, the radial burst of slurry is more violent for Ref-3 than Ref-3-AMR, and Ref-3 is clearly

730

producing smaller droplets at this stage of the pulsing sequence. Interestingly, Ref-3-AMR was found to have

731

slightly larger droplets throughout the domain past the nozzle exit.23

732

733

Quantitative discrepancies between Ref-3 and Ref-3-AMR are summarized in Figure 21, which shows percent

734

differences between azimuthally-averaged quantities. TKE shows by far the greatest difference between the

735

models, especially outside the nozzle, where a 160% difference is observed. In general, the difference between

736

model outputs is higher outside (shown in Figure 20) than inside the nozzle. This follows the trend of an

737

increasing presence of mesh element size gradients as the slurry disintegrates outside the nozzle. Interestingly,

738

the % difference for strain rate (velocity gradients) increases outside the nozzle, but that for velocity does not.

739

This observation indicates that gradients are more strongly affected by AMR. We note also that TKE

740

production is driven by velocity gradients. Our conclusion: the AMR technique within ANSYS Fluent 2020R1

741

is not sufficient to accurately model non-Newtonian wave-augmented atomization in the present system.

742

743

32

744

745

Figure 20 Comparison of slurry surface as computed by the Ref-3 (top row) and Ref-3-AMR (bottom row)

746

meshes at three points in the wave cycle. Both models represent a 90° azimuthal slice with periodic boundary

747

conditions. The top three images are at roughly the same stages within a given pulsing cycle as the bottom

748

three images. Ref-3-AMR produces a noticeably more rippled surface than Ref-3. In the Ref-3-AMR case, the

749

slurry sheet stretching down from the nozzle maintains a higher viscosity and does not rupture as quickly as

750

Ref-3.

751

752

753

Figure 21 The percent difference between azimuthally-averaged quantities at various points along the slurry

754

interface flow path for Ref-3 and Ref-3-AMR. VF is volume fraction, and TKE is turbulent kinetic energy.

755

TKE shows the greatest difference between the models, with % differences above 150% outside the nozzle.

756

The general trend is a greater divergence in model outputs outside than inside the nozzle.

757

758

759

760

33

4. CONCLUSION

761

Using a computational framework which has been validated and used extensively over the course of the last

762

decade, we have analyzed the spatiotemporal characteristics of the novel WAVE process (Wave-Augmented

763

Varicose Explosions) in which non-Newtonian waves facilitate disintegration in an inverted feed twin-fluid

764

atomizer. Annular slurry hot steam flow, creating a secondary nozzle effect for the

765

steam, as an annular slurry sheet stretches from the nozzle. Waves then collapse as they exit the nozzle,

766

crashing into the slurry sheet in a violent radial burst to enhance droplet formation. The wave birth-death cycle

767

is part of a general bulk system pulsation phenomenon, causing many quantities to fluctuate periodically.

768

Important knowledge gaps from previous studies have been filled to provide a complete understanding of this

769

efficient atomization phenomenon. A summary of new contributions to the literature is provided in the

770

following paragraphs.

771

772

We began by presenting a unique visual perspective: unraveled wave views to elucidate characteristics of the

773

3D wave cycle. An estimate of RTI time scale showed sufficient time for RTI development, and baroclinic

774

torque on the order of 1/s2 indicates strong RTI activity in the wave. KHI and RTI cause surface

775

variations in the non-Newtonian slurry that then excite the instabilities in a self-amplification cycle. During

776

the early stages of wave growth, stripping is a dominant mechanism for droplet production. Later, as the wave

777

crests, ligaments flicking up into the steam flow at the wave tip facilitate droplet production inside the nozzle.

778

Meanwhile, the radial thrust of the wave allows for steam penetration to increase effervescence and sometimes

779

break off the wave tip. Further disintegration is encouraged as the wave leaves residual ligaments in its wake.

780

781

An evaluation of velocity, slurry volume fraction, strain rate, and turbulent kinetic energy at 25 axially and

782

azimuthally spaced points revealed a loss of consistent fluctuation frequency outside of the nozzle. Strain rate,

783

however, cycles with the dominant system frequency at all points. FFT analysis was an important component

784

of this study, and peak frequencies generally converged within 4 pulsing cycles. The nozzle exit marks a

785

significant increase in azimuthal variation of velocities. Velocities show strong azimuthal correlation (2D) in

786

the wave formation region inside the nozzle but are azimuthally unrelated (3D) outside the nozzle, where radial

787

34

bursting is occurring. Outside the nozzle, fluid motion did not show strong correlation with wave pool motion.

788

Correlations between quantities, though strong in the wave formation region, showed a consistent trend of

789

significant decoupling outside the nozzle. Time lags revealed that, for a given pulse cycle, the wave rising out

790

of the pool to reach the beach takes 70% of the total pulse time.

791

792

Azimuthal correlations of velocity demonstrate that a 90° angle is sufficient for to capture azimuthal

793

variability, which is important for atomization. Evaluation of Ref-3-AMR, both qualitatively and

794

quantitatively, showed that this particular AMR technique is insufficient for accurately modeling non-

795

Newtonian, wave-augmented atomization in the present system, despite the allure of computational savings.

796

Velocity gradients were more affected by AMR than velocity magnitude. Turbulent kinetic energy differed

797

most drastically between Ref-3 and Ref-3-AMR, particularly outside the nozzle. Future research could

798

evaluate how geometry changes affect the spatiotemporal characteristics of the atomization process, such as

799

pulsing frequency.

800

801

ACKNOWLEDGEMENT

802

The authors thank Valda Rowe and Dr. Mark Horstemeyer for their administrative support.

803

804

DATA AVAILABILITY

805

The data that support the findings of this study are available from the corresponding author upon reasonable

806

request.

807

808

CONFLICT OF INTEREST

809

The authors have no conflicts to disclose.

810

811

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