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Accepted to Phys. Fluids 10.1063/5.0088341

1

The Rise and Fall of Banana Puree: Non-Newtonian Annular Wave Formation 1

by Transonic Self-Pulsating Flow 2

3

D. M. Wilson1, W. Strasser1 4

5

1 School of Engineering, Liberty University, Lynchburg, VA 24515, USA 6

7

8

ABSTRACT 9

We reveal mechanisms driving pre-filming wave formation of non-Newtonian banana puree inside a twin-fluid 10

atomizer at a steam-puree mass ratio of 2.7%. Waves with a high blockage ratio form periodically at a frequency 11

of 1000 Hz, where the collapse of one wave corresponds to the formation of another (i.e., no wave train). Wave 12

formation and collapse occur at very regular intervals, while instabilities result in distinctly unique waves each 13

cycle. The average wave angle and wavelength are 50° and 0.7 nozzle diameters, respectively. Kelvin-Helmholtz 14

instability (KHI) dominates during wave formation, while pressure effects dominate during wave collapse. 15

Annular injection of the puree into the steam channel provides a wave pool, allowing KHI to deform the surface; 16

then, steam shear and acceleration from decreased flow area lift the newly formed wave. The onset of flow 17

separation appears to occur as the waves’ rounded geometry transitions to a more pointed shape. Steam 18

compression caused by wave sheltering increases pressure and temperature on the windward side of the wave, 19

forcing both pressure and temperature to cycle with wave frequency. Wave growth peaks at the nozzle exit, at 20

which point the pressure build-up overcomes inertia and surface tension to collapse and disintegrate the wave. 21

Truncation of wave life by pressure build-up and shear-induced puree viscosity reduction is a prominent feature 22

of the system, and steam turbulence does not contribute significantly to wave formation. The wave birth-death 23

process creates bulk system pulsation, which in turn affects wave formation. 24

25

The author to whom correspondence may be addressed: dwilson221@liberty.edu 26

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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1. INTRODUCTION 27

Waves are one of the most widely recognizable fluid phenomena. Many peoples from time immemorial have had 28

a popular level understanding of waves, but the details of wave formation are complex.1 Lord Kelvin and Hermann 29

von Helmholtz were two key 19th-century players in the modern study of waves, providing a mechanism for wave 30

formation known as the Kelvin-Helmholtz instability (KHI). For a liquid and gas traveling at different velocities, 31

KHI will cause a sufficiently perturbed interface to become unstable via induced pressure gradients. Many other 32

figures in the 20th century built upon the work of Kelvin and Helmholtz, including Sir Harold Jeffreys, John Miles, 33

and Owen Phillips. Jeffreys introduced the concept of wave sheltering, which gives rise to flow separation.2 Miles 34

and Phillips proposed wave formation mechanisms to account for the mean wind profile and wind turbulence, 35

respectively.3, 4 36

37

More recently, in the 21st century, wave studies have continued with great vigor in the scientific community. Both 38

internal and external wave motion on sloped surfaces has been studied experimentally, including the effect of 39

undulated inclines on wave evolution.5-7 Chang and Liu characterized an order of magnitude approximation for 40

the turbulence intensity generated by breaking waves as proportional to the phase speed.8 2D simulations have 41

revealed that breaking waves induce vortex-like motion below the surface, and 80% of the wave’s energy 42

dissipates within three wave periods.9 Lin et al. described similarities in dimensionless aspects of waves travelling 43

over a slope for waves in the same length scale regime.10 They noted that at significantly smaller length scales, 44

viscous friction and surface tension play a more prominent role, reducing similarity. 45

46

Wave formation takes on different characteristics when the liquid is non-Newtonian. Compared to its Newtonian 47

counterpart, Millet et al. found a shear-thinning fluid on a slope to have a greater celerity and tend more readily 48

toward instability.11 Even at low Reynolds numbers, non-Newtonian fluid layers are prone to instability.12 Both 49

2D and 3D effects are important, but Mogilevskiy demonstrates that 2D disturbances grow at a higher rate than 50

3D disturbances for a falling non-Newtonian film, irrespective of rheological model.13 Tripathi et al. studied the 51

interaction between a highly viscous non-Newtonian oil core and injected water in a pipe (core annular flow), 52

though this does not represent multiphase wave formation.14 Disturbance waves in an annular liquid film have 53

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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been studied in vertical gas-liquid flow; such waves are on a small scale relative to the nozzle and propagate in 54

series.15, 16 55

56

Annular waves in vertical conduits, where the wave height is large relative to the conduit radius (referred to as 57

“high blockage ratio”), have the potential to differ significantly from planar waves. As a planar wave rises in the 58

absence of any adjacent phase shear, the local acceleration and deformation is controlled by windward and leeward 59

inertial effects and refractions caused by surface variations at the bottom of the liquid reservoir. When an annular 60

wave “rises” (radially inward), on the other hand, the area for flow is reduced if the wave height is large relative 61

to the conduit radius. Flow area reduction creates local acceleration approaching the crest (due to liquid phase 62

viscosity) irrespective of any gas stimulation, which in turn will create a wave height increase. This feedback 63

process continues until a new equilibrium is reached, the wave breaks, or some area change is encountered. Any 64

gas phase support for wave production would further complicate the matter. 65

66

Given the scarcity of applicable studies, this paper adds to the current body of research a numerical study of 67

viscous, non-Newtonian wave formation by annular injection of liquid into an enclosed, transonic gas flow. Our 68

working non-Newtonian fluid, banana puree, is shear-thinning beyond a yield stress. An annular flow of puree is 69

injected into a central steam pipe before exiting the atomizer nozzle. The proximity of the injection region to the 70

nozzle exit makes for a uniquely brief wave shore, and wave formation is periodic with no wave train and a high 71

blockage ratio. Prior studies do not address annular wave formation in an enclosed channel with a high blockage 72

ratio in the presence of transonic flow, and our work is further distinguished by a highly viscous, non-Newtonian 73

fluid. 74

75

The context of the overall wave generation system is a novel “core disrupting” twin-fluid atomization nozzle. Most 76

twin-fluid atomizers inject gas around a central liquid flow, but inverting these feeds disintegrates viscous slurries 77

more effectively and with lower gas flow rates.17 Such a design could be instrumental for gelled propellants or 78

energy reclamation by atomizing manure slurries with dynamically varying viscosity via smart atomization 79

technology.18 Another consequence of feed inversion for a non-Newtonian fluid is the formation of periodic waves 80

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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(object of the present study) inside the nozzle leading to bulk atomizer pulsations and improved atomization 81

downstream of the waves. We refer to the rupturing of puree in this novel atomization process as “wave-augmented 82

varicose explosions” (WAVE). We use CFD to reveal mechanisms, instabilities, and characteristics of wave 83

formation and collapse. Ref. 19 assesses the impact of wave formation on the downstream atomization.19 While 84

the computational domain is 3D, this paper focuses on the dominant 2D effects; Ref. 20 closely investigates the 85

role of 3D instabilities.20 86

87

2. METHODS 88

2.1 Computational Methods 89

The Navier-Stokes equations, which govern the present system, are presented in Equations 1-3, formulated for 90

multiphase flow in vector notation. In Equations 1-3, is phase volume fraction, is density, is time, is the 91

velocity vector, is the laminar shear stress tensor, is turbulent shear stress tensor, is pressure, is gravity, 92

is the surface tension force vector, is constant pressure heat capacity, is the static temperature, is laminar 93

conductivity, is the turbulent viscosity, and is the turbulent Prantdl number. Properties are arithmetically 94

phase-averaged, and the steam density is set by the ideal gas equation of state. The banana puree was modeled 95

as an incompressible fluid. The effects of compressibility on turbulence and all other sub-grid-scale modeled 96

quantities were ignored, as well as kinetic energy and viscous heating effects. Some authors propose that 97

supersonic/hypersonic boundary layers exhibit close similarities to incompressible boundary layers.21 Others have 98

found differences between near-wall turbulent fluxes in compressible flow and incompressible scaling laws, 99

but these differences only potentially affect the relatively short low-Mach number region upstream of wave 100

development and can be ignored here.22 101

102

(1) 103

104

(2) 105

106

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(3) 107

108

The Reynolds-Averaged Navier-Stokes and volume-of-fluid (VOF) equations were solved with commercially 109

available CFD solver ANSYS Fluent 2020 R1. The CFD model uses the shear stress transport (SST) formulation 110

of the k-ω turbulence model and the SIMPLE (Semi-IMplicit Pressure Linked Equations) pressure-velocity 111

coupling scheme. SIMPLE is a segregated approach that updates pressure and velocity in sequential steps rather 112

than simultaneously.23 The discretization schemes are as follows: PRESTO! (PREssure STaggering Option) for 113

pressure; second order upwind for momentum, energy, and density; and first order upwind for turbulence 114

quantities. PRESSTO! is a pressure staggering scheme that obtains face pressures by means of continuity balances 115

rather than central differencing.24 The VOF interface was closely resolved explicitly using the geometric 116

reconstruction technique by Youngs,25 otherwise known as “piecewise linear interface capturing” (PLIC).9, 26 The 117

PLIC approach approximates the interface as a plane in each computational cell for 3D solvers. A variable time 118

step size adjusted every 3 time steps to maintain a convective Courant number of about 1, with a maximum of 7 119

iterations per time step. The average time step size naturally varied with mesh size but stayed on the order of 120

s for the finest meshes. The consequence is approximately timesteps within the life of a wave. 121

The same numerics are employed by all models; mesh alone is the distinguishing variable in mesh resolution 122

studies. 123

124

Models were deemed at quasi steady state (QSS) when a multitude of point-monitored quantities reached 125

statistically stationary values. Time-averaged statistics were only collected after QSS was reached. Methods for 126

determining convergence of time-averaged quantities will be discussed later. In all presentations of data, flow time 127

is normalized by the wave time scale, which is defined as the time for a wave to proceed through one cycle of 128

formation and collapse (more on this later). Hereafter, flow time will be referred to in units of “wave times” (WTs) 129

rather than flow seconds. The computational techniques employed herein have been extensively validated and 130

analyzed for sensitivity in past transonic wave formation studies.24, 26-34 In particular, the SST k-ω turbulence model 131

and 90° azimuthal wedge mesh (employed in this present study) were shown to be advantageous for accurately 132

capturing the important physical mechanisms. For example, changing the azimuthal angle from 11.25° to 360° 133

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only altered the Sauter mean diameter by 10%.32 Again, each time step amounts to one-hundred-thousandth of a 134

wave cycle. 135

136

Based on the aforementioned citations, the following validation exercises have been carried out on our 137

computational methodology. 1) The experimentally determined globally pulsing nature of an industrial three-138

stream air-water atomizer was qualitatively demonstrated computationally. 2) The acoustic signature of that 139

injector’s pulsations was quantitatively matched computationally. 3) The primary atomization ligament wave 140

positions were quantitatively paralleled numerically. 4) The analytical trajectory of a droplet disintegrating after 141

being exposed to a normal shock wave was quantitively reproduced numerically. 5) The axially decaying droplet 142

size distribution from a non-Newtonian atomizer was quantitatively replicated numerically. 6) A study of 143

azimuthal angle revealed that reduced order models encompassing only 1/32nd of a full 360° azimuth do not 144

significantly degrade the assessment of axial droplet size distribution. Finally, our present study qualitatively 145

demonstrates the expected globally varicose pulsing nature of a recessed, inverted feed atomizer at high gas-liquid 146

momentum ratio.35 147

148

2.2 Mesh and Boundary Conditions 149

Figure 1 provides an overview of the geometry of the system: banana puree is injected into a central steam pipe 150

from an outer annulus, after which the two phases interact and exit the nozzle. Flow is generally from left to right. 151

Hereafter, the region where the puree meets the steam pipe will be referred to as the “wave pool” and the nozzle 152

extension just past this as the “beach” (equivalent to “shore”). We model a 90° azimuthal slice of the full 360° 153

azimuth for computational efficiency (barring one exception with a 45° azimuth). Increasing the azimuthal angle 154

from 45° to 90° was shown to have negligible effect on droplet sizes, which are dictated by wave physics.19 The 155

domain extends two nozzle diameters past the nozzle exit in the axial direction, but that is beyond the scope of the 156

present study. Both phase conduits extend upstream beyond what is shown. The important geometric and flow 157

parameters describing the nozzle are provided in Table I. 158

159

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The 90° wedge model is bookended by periodic boundary conditions. All walls are adiabatic. The steam and 160

puree inlets were set to constant mass flow rates for a resultant gas-liquid mass ratio (GLR) of 2.7%. Similar 161

GLRs have been used for reduced-GLR airblast atomization of gelled propellants.36 Effective atomization at 162

lower GLRs is desirable for improved efficiency. Inlet temperatures are 393 K and 304 K for the steam and 163

puree, respectively. Steam inlet turbulence intensity and turbulent viscosity ratio were set to 5% and 10, 164

respectively, which translates to spatially constant values of = 48 m2/s2 and = 2.5×105 1/s. It is expected 165

that and in the upstream steam pipe will influence steam boundary layer development, which might affect 166

KHI growth rate in the wave pool. Our results should, therefore, be interpreted in light of this particular choice 167

of feed turbulence conditions. Other geometries with other flow rates and/or other feed conditions (turbulence 168

quantities or otherwise) can be studied in a future effort. 169

170

The non-Newtonian nature of banana puree can be described by the Herschel–Bulkley model; beyond a yield 171

stress, the puree is shear-thinning. Depending on production methods, banana puree might take on a range of 172

properties. For the present study, banana puree is treated according to experimental data of Ditchfield et al., 173

which spans the range of inlet temperatures.37 The Herschel-Bulkley model was incorporated as a user-defined 174

function (UDF) in Fluent to make the puree viscosity a function of both strain rate and temperature. The UDF 175

follows Equations 4-7 to compute viscosity, where is puree viscosity, is strain rate magnitude, and is 176

the yield stress. and are both calculated from temperature for incorporation in Equation 4.

177

is the minimum strain rate that we have data for and is used as a lower bound to prevent the viscosity from 178

diverging at low strain rates. represents a bounding value of viscosity calculated using . 179

180

(4) 181

182

(5) 183

184

(6) 185

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186

(7) 187

188

A Base mesh with a 90° azimuth was refined three times to create a serious of four increasingly fine meshes, as 189

presented in Table II and Figure 2 (flow from left to right). Refinement cuts each cell length in half in all three 190

dimensions (within a prescribed region), effectively splitting a given cell into eight smaller cells. The refinement 191

region is a majority subset of the entire domain that includes all areas except farther back into the steam pipe and 192

puree annulus. Importantly, the refinement region includes all steam/puree interaction. All meshes are composed 193

almost entirely of hexahedral elements, with a very small fraction of triangular prims. To produce a “Ref-n” mesh, 194

the Base mesh was refined n times. Two additional meshes deviate from this simple pattern: Ref-2-45 entails a 195

45° azimuth, and Ref-3-AMR uses adaptive mesh refinement (AMR). 196

197

AMR dynamically refines the mesh only in localized areas around the gas-liquid interface and has been used 198

before in direct numerical simulations of waves.38 The obvious benefit of AMR is computational efficiency: the 199

Ref-3-AMR mesh has a Ref-3 refinement level but only as needed to adequately resolve the interface. While the 200

exact number of elements fluctuates between ~26 and ~30 million during a simulation, Ref-3-AMR consistently 201

utilizes less than a quarter of the Ref-3 cell count. Figure 3 provides a snapshot to illustrate the nature of AMR. 202

To adequately follow the constantly moving interface, including all droplets, the AMR region was updated every 203

5 time steps based on a stringent volume fraction gradient criterion. 204

205

Hardware and run rates for all models are presented in Table II. The hardware configuration (including node 206

counts and cores/node utilized) was optimized, producing the highest possible run rate, for all models except the 207

Base case, according to a novel in-house methodology.39 The influence of mesh, convergence, and sampling time 208

for time-averaged statistics will be further discussed throughout this paper. 209

210

211

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Table I Parameters describing the geometric and flow conditions of the atomizer nozzle. The steam pipe 212

diameter is equivalent to the nozzle diameter. The inner and outer angles, which are with respect to the nozzle 213

axis, describe the angled injection of the puree from the outer annulus into the central steam pipe. 214

215

Parameter

Value

Units

Nozzle Diameter

16

mm

Wave Pool Width

6.8

mm

Beach Width

3

mm

Puree Annular Gap

9.8

mm

Inner Angle

29

degrees

Outer Angle

60

degrees

Steam Mass Flow

0.021

kg/s

Puree Mass Flow

0.79

kg/s

216

Table II Mesh size, hardware, and run time. Numerics are consistent across models. The Base mesh with a 217

90° azimuth was refined n times to produce a “Ref-n” model; the two exceptions to this pattern are Ref-2-45 218

and Ref-3-AMR. All computational node configurations, except for Base, were optimized for run rate. QSS 219

refers to “quasi steady state” and TA refers to “time-averaged.” WT is the “wave time” between periodic wave 220

formations, representing how often the general system dynamics repeat themselves. 221

222

Model Name

Cell Count

[106]

Core Type

Node

Count

Core

Loading

Run Rate

[WT/day]

QSS WT

TA WT

Base

0.3

AMD EPYC 7742

1

117/120

27

11

14

Ref-1

2.1

AMD EPYC 7763

1

120/120

5.2

99

37

Ref-2-45

8.3

AMD EPYC 7742

1

117/120

0.52

4.4

Ref-2

17

AMD EPYC 7742

4

118/120

0.91

24

7.5

Ref-3-AMR

~28

AMD EPYC 7763

12

64/120

0.29

8.6

5.4

Ref-3

132

AMD EPYC 7763

16

120/120

0.18

12

8.5

223

224

225

Figure 1 Isometric view of Ref-3 surface mesh for the pre-filming (wave formation) region of our twin-fluid 226

atomizer. Unless otherwise noted, all models include a 90° slice of the azimuth, as shown here, with bookended 227

periodic boundary conditions. The full Ref-3 mesh contains 132 million elements. Banana puree flows in an 228

annulus around an inner steam pipe before the two phases are forced to interact and then exit the nozzle. 229

230

231

232

Figure 2 Side view of meshes with increasing levels of refinement. The Base mesh is refined in the entire 233

atomization domain (right side of mesh) and back into the central steam pipe and outer puree annulus (pre-234

filming region where waves live); therefore, the scope of refinement includes all puree-steam interaction. The 235

element counts from left to right are as follows: 0.3, 2.1, 17, 28 (on average), and 132 million. 236

237

238

239

Figure 3 Side view of the Ref-3-AMR mesh, illustrating the nature of adaptive mesh refinement. Rather than 240

adapting all cells in Ref-2, only those cells near the puree-steam interface were refined in all three dimensions. 241

The adaption process is updated every 5 time steps to ensure the thrice-adapted region adequately follows the 242

constantly deforming puree-steam interface. On average, Ref-3-AMR maintains around 28 million elements 243

compared to the underlying mesh size of 17 million. 244

245

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3. RESULTS AND DISCUSSION 246

3.1 Wave Cycle Overview 247

As banana puree is injected into the central steam pipe, waves develop in a cyclical pattern. An overview of a 248

single wave cycle is provided in Figure 4 (Multimedia view). No train of waves exists. Rather, a new wave 249

rises only in the wake of a dying wave. When both a newly forming wave and a dying wave can be seen 250

together, the former may be referred to as the “daughter” wave and the latter as the “parent” wave to provide 251

distinction. The life of a wave in a “wave cycle” is divided into two stages: formation and collapse. In Figure 252

4, frames 1-5 show the formation of a wave as it rises out of the relatively stagnant puree wave pool. Note that 253

the wave thickness is fairly large relative to the steam pipe radius; therefore, as the wave rises, the puree flow 254

area is reduced. The viscosity of the puree causes a velocity gradient (vorticity) to develop inside the wave, 255

and the crest is accelerated more. The shear-thinning nature of the puree reduces the viscosity approaching the 256

crest, allowing it to accelerate even more. 257

258

As the wave base approaches the beach, its base is decelerated by the no-slip boundary at the nozzle wall. This 259

produces vorticity as the wave wants to curl over. A shear stress imbalance exists where the gas is attempting 260

to accelerate on top of the wave while the beach is attempting to decelerate the wave. Frames 6-8 show the 261

wave collapse, starting roughly where the wave crest reaches the nozzle exit and attains its maximum height. 262

Synchronous with the collapse of the parent wave is the bulging of puree radially outward just past the nozzle 263

exit (starts in frame 6, but clearer in frames 7 and 8). It is evident that the collapse of a given wave corresponds 264

precisely to the formation of a new wave, thereby beginning another wave cycle. The enclosed channel 265

environment with high-speed, transonic steam truncates the natural growth and breaking of the wave and 266

accelerates disintegration (see free-surface waves simulated by Mostert et al., for example, where 267

disintegration and droplet formation largely happen after breaking with splash-up).38 More on atomization is 268

discussed in Ref. 19. It is for this reason that we describe the wave as “collapsing” rather than the conventional 269

“breaking” towards the end of its life. 270

271

272

273

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Figure 4 Sequential side views of banana puree (yellow) through one predominantly 2D wave cycle for the 274

Ref-3 mesh. Pictures are at equally spaced flow time intervals. Rather than a train of waves, only one wave 275

forms at a given time. The collapse and disintegration of one wave corresponds to the formation of a new 276

wave. This cycle is repeated at regular intervals one “wave time” apart. (Multimedia view). 277

278

Figure 5 labels major dimensions for the geometry and a given wave. Here, is the steam pipe (and nozzle) 279

radius, is the depth of puree injected, is the width of the wave pool, and is the width of the beach 280

(distance from injection zone to nozzle exit). We expect radius of atomizer curvature to influence wave 281

formation, but that is beyond the scope of this study. Because the puree injection annular walls are angled, we 282

consider to be the distance from the outer edge of the steam pipe to the radial center of the puree annulus ( 283

extends below the bottom of Figure 5). A puree wave is described by its height (), puree thickness at the 284

nozzle exit (), the wave angle (), the wavelength (), and the wave speed (). In the absence of wave trains, 285

the wavelength is considered the axial distance between a newly forming wave peak and the outermost point 286

of the radial bulge resulting from the collapse of the previous wave. The average wavelength was measured 287

for three sampled independent wave cycles to be 0.7 nozzle diameters with only a 0.57% coefficient of 288

variation (COV), demonstrating remarkable consistency. Wave angles before collapse were around 50°. The 289

measured wave speed is greater than the natural wave speed by a factor of 37. We suggest several 290

dimensionless ratios to describe the system:

,

,

,

, and 291

. Here, is the peak wave height, and is the approximate time-averaged value for all meshes except 292

the Base case. highlights the abbreviated beach, which is 0.28 times smaller than the wavelength. 293

represents the wave blockage ratio, the percentage of the steam pipe that is blocked by the wave. These ratios 294

are likely themselves functions of governing dimensionless numbers, such as Reynolds, Mach, Ohnesorge, 295

and Weber numbers, and could be especially useful when considering geometry changes and their impact on 296

wave formation. The ratio

is expected to be an important geometric ratio, and its impact on wave 297

formation is of great interest. However, effects of geometry are relegated to a future study. 298

299

These particularly well-known dimensionless numbers will be used to characterize waves in this system: 300

Weber number ( ), Reynolds number (), and Strouhal number ( ). Our 301

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characteristic length scale is . Rather than using the deep-water wave velocity for and ,40 wave speed 302

to is more appropriate for our system. is the density of puree, is surface tension, is the lowest puree 303

viscosity observed in the wave, is frequency, and is bulk velocity of the steam and puree. The wave Weber 304

number, which captures the ratio of inertial to surface tension effects, is 2.0×104. The Reynolds number, which 305

captures the ratio of inertial to viscous effects, is 1.2×104. The Strouhal number here is effectively the ratio of 306

wave velocity to bulk velocity in the channel and is 0.08. The wave and are respectively higher and 307

lower by an order of magnitude than those reported by Chan et al. for free-surface water waves.40 The waves 308

in this system have a high inertia, which overcomes surface tension to disintegrate more rapidly than is usually 309

the case for free-surface waves. A lower stems largely from the high viscosity of the banana puree. 310

311

312

Figure 5 Side view of predominantly 2D banana puree wave a) as it crests and b) after it collapses (where a 313

new wave forms) with major dimensions labelled for the Ref-3 mesh. Since no wave train exists, wavelength 314 is considered the axial distance from the peak of a wave just forming to the center of the radial bulge 315

produced by the former wave’s collapse. Wavelengths were consistently 0.7 nozzle diameters with only a 316

0.57% coefficient of variation (COV). A point monitor is located at an azimuthal angle of 22.5° within the 90° 317

computational domain. 318

319

Select quantities were collected at a point monitor in line with the nozzle exit, 1/3 of the way in from the outer 320

steam pipe radius, and at a 22.5° azimuthal angle within the 90° wedge (see Figure 5). Each wave moves 321

through this point, and the nozzle exit is of particular interest as the point of transition from wave growth to 322

collapse. This point will hereafter be referred to simply as the “point monitor.” 323

324

A study of multiple wave cycles reveals two characteristics: 1) wave formation and collapse occur at very 325

regular intervals, and 2) instabilities result in distinctly unique waves each cycle. To illustrate these dynamics, 326

the moving average of the velocity magnitude at the point monitor is presented in Figure 6. Nine wave cycles 327

are evident, repeating approximately every WT = 0.001 s. We therefore expect the dominant frequency of the 328

system to be 1/WT = 1000 Hz. 329

330

Figure 6 Moving average of velocity magnitude at the point monitor in the Ref-3 mesh (location shown in 331

Figure 5) using a 50-point moving window for data collected every 10 time steps. Two aspects of the waves 332

are highlighted: 1) wave formation and collapse constitute a distinctly periodic cycle and 2) instabilities 333

distinguish each cycle as unique. 334

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335

3.2 Mesh Independence 336

The Ref-3 mesh is sufficient for the purposes of this study based on the convergence shown in Figure 7, where 337

the abscissa is the cell count relative to the Ref-1 mesh. Both mean and standard deviation of velocity and 338

turbulent kinetic energy (TKE), as measured at the point monitor at QSS, are reasonably close to convergence 339

by Ref-3 (highest element count on the right). We conclude that the Ref-3 mesh is sufficient for identifying 340

major wave mechanisms and characteristics. Ref-3-AMR falls as an outlier to all power law trends; 341

consequently, the lines connecting the points do not include AMR results. Further investigation is needed to 342

fully assess the effects and implications of the AMR technique for wave formation, and Figure 7 suggests 343

caution. We observe 83% and 128% transient variability in velocity and TKE, respectively. Fast Fourier 344

Transforms (FFTs) reveal similar peak frequencies across all meshes, including Ref-3-AMR, except for Base. 345

Besides the Base mesh, all peak frequencies are within 7% of 1000 Hz, which corresponds to the inverse of 346

WT = 0.001 s. FFT analysis highlights an important point: the general physics of wave formation are similar 347

in meshes Ref-1 through Ref-3, but the Base mesh physics are distinctly different. In particular, very little 348

wave formation and periodicity is evident in the prohibitively coarse Base mesh. 349

350

351

Figure 7 Convergence of velocity and turbulent kinetic energy (TKE) statistics at the point monitor (location 352

shown in Figure 5) with increasing mesh element count. Ref-3-AMR values are outliers to the asymptotic 353

power law trends and are therefore excluded from them. For the Ref-3 mesh, 83% and 128% transient 354

variability is observed for velocity and TKE, respectively. The peak frequencies revealed by FFTs are 355

consistent, within 7% for all meshes except Base case. Quantities are sufficiently converged by Ref-3, which 356

represents the highest element count on the right. 357

358

The cumulative mean and standard deviation of velocity in Figure 8 (for Ref-3) have largely settled within the 359

QSS sampling period used to calculate statistics in Figure 7. Though not shown, peak frequencies from FFTs 360

also converged within the sampling period. The error associated with insufficient sampling time for a transient 361

model scales with , where T is a convective time scale.41 We thus conclude that our QSS sampling time 362

is sufficient for time-averaged quantities and statistics. 363

364

365

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Figure 8 Cumulative mean and standard deviation of velocity magnitude at the point monitor (location shown 366

in Figure 5) for the Ref-3 mesh. All data represented here are at QSS, so 0 CT on this plot is not the start of 367

the simulation. After initial oscillations, both statistics settled to roughly constant values, indicating sufficient 368

sampling time. Time-averaged values are not expected to change significantly with further sampling time. 369

370

We consider 3 WTs sufficient to produce time-averaged profiles. The series of velocity profiles in Figure 9 371

illustrate the mixture boundary layer profile at the start of the wave pool. They become reasonably invariant 372

within 3 WTs. The time-averaged WTs for all models are well above this range (see Table II). 373

374

375

Figure 9 Development of the QSS time-averaged velocity profile at the start of the wave pool (just before the 376

steam interacts with the puree). 3 WTs are sufficient for time-averaging quantities, as the time-averaged 377

velocity profile remains largely unchanged beyond this time. 378

379

An important measure is the TA radial puree thickness on the beach at the nozzle exit, which results from the 380

collective effect of the aforementioned forces over time, and the response of the puree’s viscosity, at the beach. 381

The TA puree volume fraction profiles at the nozzle exit in Figure 10 are largely similar except for the Base 382

mesh. These profiles essentially communicate the puree depth in Figure 5, and the time-averaged exit 383

velocity of the puree sheet is about 10 m/s. Wave crests produce the bulges on the left; more pronounced bulges 384

correspond to more pronounced waves. Again, we observed that the Base case’s physics are not consistent 385

with other models. The other models are reasonably uniform, and the primary differentiating factor is the wave 386

crest on the left. Interestingly, Ref-2 has a more pronounced crest than Ref-3 and Ref-3-AMR. Visual 387

inspection reveals a slightly earlier start to wave disintegration for the finer meshes. Still, from Figure 10 and 388

other metrics (pulsation frequency and wavelength, for example), we note that the essential wave physics of 389

Ref-3 were captured by Ref-1 at the first refinement level. Ref-3, then, is sufficiently close to Ref-2 and Ref-390

1 for the purposes of this study. 391

392

393

Figure 10 Time-averaged puree volume fraction profiles at the nozzle exit for all meshes. Except for the Base 394

mesh outlier, all meshes produce similar profiles, with only slight differences in the bulge on the left. 395

is approximately 0.17 for all but the Base case, where is the puree thickness at the nozzle exit and is the 396

wave pool width. This bulge stems from the wave crest and is indicative of more pronounced wave 397

development. The similarity between Ref-2 and Ref-3 profiles demonstrates the sufficiency of the Ref-3 mesh 398

to capture wave physics. 399

400

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3.3 Wave Mechanisms 401

Each wave starts forming by the KHI, which is induced by velocity gradients. Gravitational effects are 402

irrelevant for this study, as the Froude number () is 425. The two competing forces, then, are 403

surface tension (stabilizing) and the Bernoulli effect (destabilizing). It follows that there exists some steam 404

velocity below which waves will not form. Unlike free-surface waves, the mass flow of puree into the steam 405

pipe provides an additional mechanism for deforming the interface. The decrease in pressure associated with 406

the increase in velocity as the streamlines curve around the protruding puree (i.e., the Bernoulli effect), lifts 407

the puree from the wave pool into the steam flow. After initial rising, the wave continues to climb radially 408

inward due to KHI, shear lifting, puree injection, and acceleration from reduced flow area. 409

410

The mechanism of streamline deflection by the interfacial geometry, which produces flow separation and 411

sheltering, was first introduced and defended by Sir Harold Jeffreys.2 Figure 11 (Multimedia view) illustrates 412

this phenomenon in our system. The reader will notice a low-velocity region on the windward side of a newly 413

forming wave (first noticeable in frame 2), what we refer to as the “well” behind the parent wave. As the wave 414

travels to the right, it leaves a surface dip in its wake where steam decelerates (frames 2-4). Lower velocity 415

will induce higher pressure in the well, which could explain why the next wave begins to rise behind the well 416

(frame 5). While the geometry of the daughter wave is a rounded hump (frames 6-7), the flow continues to 417

decelerate on its leeward side. We hypothesize the following: flow separation from the new wave first occurs 418

in frame 8, where steam shear alters the geometry from a rounded hump to a more pointed peak. At this point, 419

the low-velocity well disappears. The effect of shear in transitioning the geometry is marked by droplets being 420

stripped off the wave (first seen in frame 8 for the new wave). Towards the later stages of wave life, the wave 421

disrupts steam flow in more complex ways as it begins to disintegrate. 422

423

424

Figure 11 Side view of puree-steam interface outline with velocity vectors and contour lines of velocity 425

magnitude through one typical wave cycle for the Ref-3 mesh. Waves start forming via the KHI. Unlike free-426

surface waves, the system here involves a mass flow of puree into the steam flow, which provides an additional 427

mechanism for deforming the surface. Streamlines are deflected as a new wave rises, leading to flow separation. 428

Through formation and collapse, waves seem to follow a roughly parabolic trajectory. (Multimedia view). 429

430

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As the wave approaches the nozzle exit (frame 5), some disintegration is observed on the on the windward 431

side of the wave crest. In frame 6 of Figure 12 (Multimedia view), the wave crests and transitions from 432

formation to collapse around the nozzle exit. At the same time, puree starts to bulge radially just outside the 433

nozzle exit. In frames 2-5, no pressure gradient exists to alter the direction of puree flow. Consequently, puree 434

stretches from the nozzle in an annular sheet. This sheet will be non-uniform from Rayleigh-Tayler instabilities 435

(RTI) induced by density gradients, but that is beyond the scope of this document (see Ref. 20 for more details). 436

The windward pressure acts normal to the surface of the wave and propagates through the wave (see frames 437

5-7), highlighting the significance of wave angle. By the time the wave reaches the nozzle exit, the pressure 438

has sufficient magnitude and acting area to move the puree radially outward (well before the wave collapses 439

into the annular puree sheet). 440

441

The difference between windward and leeward pressure (reduced by sheltering) drives the wave to collapse 442

by overcoming puree inertia. We note again that free-surface waves experience no such windward pressure 443

build-up, which is a direct result of steam compression in the enclosed space. Pressure also overcomes surface 444

tension to partially disintegrate the wave. After collapse, the pressure distributes, leaving a stark pressure 445

difference across the bulging puree sheet (frame 8). RTI, wave collapse induced imperfections, and viscosity 446

gradients create weak points where the pressure is most likely to rupture the puree sheet.19 The cycling of 447

pressure produces axial steam pressure fluctuations that most probably superimpose on other effects to 448

influence wave physics, in particular adding to the Bernoulli effect pulling on the wave pool surface. In 449

summary, a given wave starts to grow by KHI (formation stage) but is later dominated by pressure effects 450

(collapse stage). The vectors in Figure 11 show the rise and fall of a wave to follow a roughly parabolic 451

trajectory. 452

453

454

Figure 12 Sequential side contours of pressure with the puree-steam interface outlined through one typical 455

wave cycle for the Ref-3 mesh. Pressure build-up on the windward side of the wave drives wave collapse. 456

Windward pressure acts normal to the wave, causing puree to bulge radially outside the nozzle (frames 6-8). 457

(Multimedia view). 458

459

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Pressure gradient in Figure 13 (Multimedia view) reveals areas of high steam compression. Droplets stripped 460

off the wave obstruct the steam flow, locally compressing the steam. Pressure gradients behind the wave start 461

relaxing as the wave collapses (frames 8 and 1) and build back up as the next wave rises (frames 2-7). A region 462

of high pressure gradient appears in the puree on the leeward side of a newly forming wave (frames 1-4). By 463

frame 5, this region appears only on the leeward side of the next wave. 464

465

466

Figure 13 Sequential side contours of pressure gradient through one typical wave cycle for the Ref-3 mesh. 467

High pressure gradients correspond to regions where steam is highly compressed. (Multimedia view). 468

469

The change in puree surface free energy per unit area is estimated as the product of surface tension and strain rate 470

at the interface.42 Figure 14 (Multimedia view) illustrates the evolution of surface energy during wave formation 471

and collapse. As the wave crests (frame 5), there is a distinct lack of surface energy at the beginning of the 472

wave pool, where the next wave will form. A gradual increase in surface energy marks the rise of a new wave 473

(frames 6 and 7), increasing until droplets start stripping away (frame 8). The distribution of surface energy 474

up to the nozzle exit is relatively uniform in frame 8, before sheltering occurs. 475

476

Sheltered regions, particularly when the wave rises to shelter its leeward side (frames 1-6), contain less surface 477

energy. The breaking away of droplets and general disintegration of the wave occurs, as expected, in regions 478

of high surface energy. The exception to this is the windward side of the wave, where the steep slope of the 479

wave and the curvature of the streamlines leave droplets nowhere to go, despite relatively high surface energy. 480

It is only after the slope of the windward side decreases in the wake of wave collapse (frames 7-8) that droplets 481

break away from this surface. 482

483

484

Figure 14 Sequential side contours representing the surface free energy per unit area for the Ref-3 mesh. This 485

quantity is estimated as the product of surface tension and the strain rate at the interface. As a new wave forms, 486

surface energy gradually increases. Eventually, as the wave shelters its leeward side, the surface energy 487

distribution strongly favors the windward side. Droplet breakaway and wave disintegration correspond to areas 488

with higher surface energy. (Multimedia view). 489

490

We expect the non-Newtonian nature of banana puree to affect wave formation and increase fluid instability.11 491

Figure 15 (Multimedia view) shows the puree-steam interface colored by viscosity looking towards the 492

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windward side of the wave (only a 90° azimuth is shown). Significant Rayleigh-Taylor variability is evident 493

across the azimuth, which indicates that 90° is sufficient to capture azimuthal effects.20 A high-viscosity region 494

appears behind the rising wave (frames 3-7), which corresponds to the low-velocity well in Figure 11 and is a 495

low strain-rate zone. As the gentle hump of a new wave (frames 5-6) transitions to a more pointed wave with 496

a steeper slope (frames 7-8), the high-viscosity region largely disappears, corresponding to the disappearance 497

of the well in Figure 11. Viscosity variation could contribute to the rising of a new wave and the location at 498

which this occurs. The clear reduction in puree viscosity caused by upward penetration into the steam flow 499

enables increased wave speed and enhanced disintegration. The reader will notice that the wave is 500

axisymmetric (2D) through much of its life. There are obvious azimuthal instabilities and breakup which drive 501

droplet production, but those are beyond the scope of this paper; they are discussed in Ref. 19 and Ref. 20. We 502

continue to focus on the life of the contiguous wave itself. 503

504

Figure 16 (Multimedia view) shows the puree-steam interface colored by viscosity looking towards the leeward 505

side of the wave. As the wave crests, the annular puree sheet stretching past the nozzle exit downstream 506

develops significant viscosity gradients (frames 4-6, particularly frame 5). Viscosity gradients, in addition to 507

RTI, increase the instability of the annular sheet, “priming” it for disintegration upon wave collapse.19 It is 508

difficult to say to what degree shear-dependent viscosity affects wave formation, growth, and collapse without 509

a comparison to a Newtonian fluid in the same system. Such a comparison is relegated to a future study. 510

511

512

Figure 15 Windward side of the puree-steam interface colored by puree viscosity through one typical wave 513

cycle for the Ref-3 mesh. A high-viscosity region forms behind the wave in frames 3-7 but largely disappears 514

in frame 8. This high-viscosity region corresponds to a low-velocity well on the windward side of the growing 515

wave. The clear viscosity reduction as the wave penetrates into the steam facilitates wave acceleration and 516

droplet formation. (Multimedia view). 517

518

519

520

Figure 16 Leeward side of the puree-steam interface colored by puree viscosity through one typical wave 521

cycle for the Ref-3 mesh. Significant viscosity gradients develop in the annular puree sheet as it stretches away 522

from the nozzle, priming it for disintegration. (Multimedia view). 523

524

525

526

527

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3.4 Wave Feedback 528

Wave formation and collapse cause both Mach number and temperature to fluctuate periodically. The cycling 529

of these quantities is largely a consequence of wave cycling rather than a driving factor. However, both provide 530

feedback that influence the wave physics. As a wave rises, steam decelerates on its windward side and 531

accelerates through the reduced-area opening above the wave crest (Figure 17, Multimedia view). Droplet 532

shedding and wave disintegration visibly disrupt the steam flow. Though not explicit in Figure 17, small 533

portions of the flow reach Mach 1. Steam acceleration further decreases pressure above the wave crest, which 534

enhances the wave-lifting force. The transition of daughter wave geometry from rounded to pointed in frame 535

8 leads to a zone of low Mach number on the leeward side. As the parent wave is still accelerating the steam, 536

the daughter wave causes the steam to accelerate further upstream. The time-averaged Mach number contour 537

in Figure 18 confirms an average nozzle effect, and it is only because of this effect that the steam reaches 538

transonic speeds. For the current geometry, the vena contracta is located halfway across the beach; this may 539

shift as the geometry or flow conditions are altered. The Mach number profile at this location is displayed in 540

Figure 18. 541

542

543

Figure 17 Sequential side contours of steam Mach number through one typical wave cycle for the Ref-3 mesh. 544

As the wave crests, the steam flow accelerates through the reduced-area gap. Wave collapse leads to an increase 545

in Mach number further back into the steam pipe (starting in frame 8). (Multimedia view). 546

547

548

549

Figure 18 Contour of time-averaged Mach number for Ref-3. On average, the waves produce a nozzle effect, 550

accelerating the steam through the nozzle exit. The nozzle effect enables the steam to reach transonic speeds. 551

The black dashed line represents the location of the vena contracta (halfway across the beach), and the solid 552

black line is the Mach number profile at this location. 553

554

Rapid compression and decompression of steam through a wave cycle causes thermal cycling in the windward 555

side (Figure 19, Multimedia view). Wave formation leads to steam heating (frames 3-7) while collapse leads 556

to cooling (frames 8-2). Increase in puree temperature will lower its viscosity. Apparently, strain rate primarily 557

drives puree viscosity; the high-viscosity region in the well (Figure 15) does not generally correspond to any 558

low-temperature regions. However, higher temperatures would contribute to early disintegration by viscosity 559

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reduction. Sheltering by a newly forming wave reduces temperature locally (see low-temperature leeward 560

regions in frames 8 and 1). 561

562

563

Figure 19 Sequential side contours of temperature through one typical wave cycle for the Ref-3 mesh. Thermal 564

cycling is evident as the steam is compressed and decompressed with wave formation and collapse. The 565

compressive heating would contribute to reduction in puree viscosity, enhancing wave disintegration. 566

(Multimedia view). 567

568

569

3.5 Effect of Turbulence 570

The battle between surface tension (stabilizing) and turbulent forces (destabilizing) is largely won by surface 571

tension for our feed turbulence conditions. Figure 20 (Multimedia view) shows contours of the turbulent Weber 572

number (), which is the ratio of these two forces and defined in Equation 8. Here, is the density of the 573

gas phase (steam), is the rms of the velocity fluctuations, is the integral length scale, and is again the 574

surface tension of the liquid phase (puree). These approximations are used: and , where 575

and are the TKE and specific dissipation rate, respectively. 576

577

(8) 578

579

is highest on the windward side of the rising wave (frames 1, 2, and 8), just after the daughter wave 580

geometry transitions from rounded to pointed and flow separation occurs. It appears that the wave is shear 581

sheltering beginning in frame 8, stagnating TKE production and increasing . TKE production in the free 582

stream shear layer increases and thus . Since is largely less than 1, surface tension is balancing RANS 583

modelled fluctuation inertial effects, indicating that other forces besides turbulence are primarily responsible 584

for wave formation and disintegration for our turbulence feed conditions. 585

586

587

Figure 20 Sequential side contours representing turbulent Weber number () through one typical wave cycle 588

for the Ref-3 mesh. is the ratio of turbulent forces (destabilizing) to surface tension (stabilizing). In general, 589

surface tension is the dominant force, indicating that non-turbulent forces are primarily responsible for the 590

wave physics. (Multimedia view). 591

592

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Also, for our particular turbulence feed state, the integral length scale near the wave (see Figure 21) is more 593

than two orders of magnitude smaller than the wavelength. In other words, the approximately length scales of 594

the largest, anisotropic, energy-containing structures are too small to influent the birth and death of the wave. 595

From this we conclude that the effect of turbulent structures on wave formation is minimal, confirming that 596

our previously validated RANS approach is sufficient for the present study. The relevance of turbulent 597

structures is expected to depend on the GLR, so investigation at varying GLRs using large eddy simulations 598

(LES) could be useful. According to the Phillips mechanism, turbulent eddies can play a significant role in the 599

initial formation of free-surface waves,4 but we remind the reader that this system does differ from that of free-600

surface waves. 601

602

603

Figure 21 Time-averaged integral length scale () contour for Ref-3. is in meters and more than two orders 604

of magnitude smaller than the wavelength, leading us to conclude that turbulent structures do not contribute 605

significantly to wave formation. 606

607

4. CONCLUSION 608

We have performed a numerical investigation of wave formation inside a twin-fluid atomizer nozzle. The 609

context for wave formation in this study is unique in multiple ways, and to our knowledge, this work represents 610

the first study of non-Newtonian transonic periodic wave formation in an enclosed channel with a high 611

blockage ratio. With a GLR of 2.7%, injection of banana puree into a central steam pipe leads to periodic wave 612

formation, which has important implications for atomization systems. No train of waves exists; the formation 613

of one wave corresponds to the collapse of the prior wave. Though each wave is unique, wave cycle frequency, 614

angle, and wavelength are consistent at 1000 Hz, 50° and 0.7 nozzle diameters, respectively. 615

616

Wave formation is initiated by KHI, where the puree wave pool provides a vehicle for interfacial deformation. 617

In addition to steam shear and KHI, the reduction in steam flow area and reduced puree viscosity encourages 618

accelerated wave growth. Sheltering by the developing wave compresses the steam, which generates 619

considerable pressure build-up and truncates wave life. As the wave passes the nozzle exit and peaks in height, 620

pressure overcomes inertia and surface tension. The wave begins to collapse and disintegrate, and pressure 621

propagates through the wave to move puree radially outwards. The periodic compression and decompression 622

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of the steam produces pressure, temperature, and velocity cycling and feedback. Axial steam pressure 623

fluctuations most probably contribute to wave formation, adding to the Bernoulli effect, and higher 624

temperatures reduce puree viscosity. The reduction in effective orifice exit area by waves produces a nozzle 625

effect to accelerate the outgoing steam above the wave crest, decreasing pressure and increasing the lifting 626

force. Turbulence was shown to have minimal effect on wave formation, though this is undoubtedly a function 627

of GLR and feed turbulence conditions. 628

629

We have revealed for the first time what we term “wave-augmented varicose explosions” (WAVE). Now that 630

it has been demonstrated, the dependency of WAVE on various parameters can be investigated. Future efforts 631

might explore the general dependence of wave formation on GLR. The response of a Newtonian fluid in this 632

system might also be of interest to determine the precise role of viscosity in wave formation. Finally, an 633

evaluation of wave dependence on geometric parameters could enable nozzle design optimization. 634

635

ACKNOWLEDGEMENT 636

The authors thank Reid Prichard for helping to optimize computational hardware, along with Valda Rowe, 637

Eric Turman, and Dr. Mark Horstemeyer for their administrative support. 638

639

DATA AVAILABILITY 640

The data that support the findings of this study are available from the corresponding author upon reasonable 641

request. 642

643

CONFLICT OF INTEREST 644

The authors have no conflicts to disclose. 645

646

REFERENCES 647

648

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38 Wouter Mostert, Stéphane Popinet and Luc Deike, "High-resolution direct simulation of deep water breaking waves: 717

transition to turbulence, bubbles and droplet production," Journal of Fluid Mechanics (in review) , (2021). 718

39 Reid Prichard and Wayne Strasser, "Optimizing Selection and Allocation of High-Performance Computing Resources for 719

Computational Fluid Dynamics," 7th Thermal and Fluids Engineering Conference (under review) , (2022). 720

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26

40 W. H. R. Chan et al., "Formation and dynamics of bubbles in breaking waves: Part II. The evolution of the bubble size 721

distribution and breakup/coalescence statistics," Center for Turbulence Research , (2018). 722

41 Todd A. Oliver et al., "Estimating uncertainties in statistics computed from direct numerical simulation," Physics of fluids 723

(1994) 26 (3), (2014). 724

42 Alberto Vela-Martín and Marc Avila, "Deformation of drops by outer eddies in turbulence," Journal of Fluid Mechanics 725

929, (2021). 726

727

728

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Accepted to Phys. Fluids

10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids

10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids

10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids 10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341

Accepted to Phys. Fluids

10.1063/5.0088341

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Accepted to Phys. Fluids 10.1063/5.0088341

Temperature [K]

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Accepted to Phys. Fluids

10.1063/5.0088341

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0088341