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The Rise and Fall of Banana Puree: Non-Newtonian Annular Wave Formation by Transonic Self-Pulsating Flow

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We reveal mechanisms driving pre-filming wave formation of non-Newtonian banana puree inside a twin-fluid atomizer at a steam-puree mass ratio of 2.7%. Waves with a high blockage ratio form periodically at a frequency of 1000 Hz, where the collapse of one wave corresponds to the formation of another (i.e., no wave train). Wave formation and collapse occur at very regular intervals, while instabilities result in distinctly unique waves each cycle. The average wave angle and wavelength are 50{degree sign} and 0.7 nozzle diameters, respectively. Kelvin-Helmholtz instability (KHI) dominates during wave formation, while pressure effects dominate during wave collapse. Annular injection of the puree into the steam channel provides a wave pool, allowing KHI to deform the surface; then, steam shear and acceleration from decreased flow area lift the newly formed wave. The onset of flow separation appears to occur as the waves' rounded geometry transitions to a more pointed shape. Steam compression caused by wave sheltering increases pressure and temperature on the windward side of the wave, forcing both pressure and temperature to cycle with wave frequency. Wave growth peaks at the nozzle exit, at which point the pressure build-up overcomes inertia and surface tension to collapse and disintegrate the wave. Truncation of wave life by pressure build-up and shear-induced puree viscosity reduction is a prominent feature of the system, and steam turbulence does not contribute significantly to wave formation. The wave birth-death process creates bulk system pulsation, which in turn affects wave formation.
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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
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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
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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
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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 HerschelBulkley 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-nmodel; 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
QSS WT
TA WT
Base
0.3
AMD EPYC 7742
1
117/120
11
14
Ref-1
2.1
AMD EPYC 7763
1
120/120
99
37
Ref-2-45
8.3
AMD EPYC 7742
1
117/120
4.4
Ref-2
17
AMD EPYC 7742
4
118/120
24
7.5
Ref-3-AMR
~28
AMD EPYC 7763
12
64/120
8.6
5.4
Ref-3
132
AMD EPYC 7763
16
120/120
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
1 Nick Pizzo, Luc Deike and Alex Ayet, "How does the wind generate waves?" Physics Today 74 (11), 38-43 (2021). 649
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2 Harold Jeffreys, "On the Formation of Water Waves by Wind," Proceedings of the Royal Society of London. Series A, 650
Containing Papers of a Mathematical and Physical Character 107 (742), 189-206 (1925). 651
3 John W. Miles, "On the generation of surface waves by shear flows," Journal of Fluid Mechanics 3 (2), 185-204 (1957). 652
4 O. M. Phillips, "On the generation of waves by turbulent wind," Journal of Fluid Mechanics 2 (5), 417-445 (1957). 653
5 Erin L. Hult, Cary D. Troy and Jeffrey R. Koseff, "Laboratory images of breaking internal waves," Physics of Fluids 18 (9), 654
091107 (2006). 655
6 G. K. Pedersen et al., "Runup and boundary layers on sloping beaches," Physics of Fluids 25 (1), 012102 (2013). 656
7 Markus Dauth and Nuri Aksel, "Breaking of waves on thin films over topographies," Physics of Fluids 30 (8), 082113 657
(2018). 658
8 Kuang-An Chang and Philip L. -F Liu, "Experimental investigation of turbulence generated by breaking waves in water of 659
intermediate depth," Physics of Fluids 11 (11), 3390-3400 (1999). 660
9 Gang Chen et al., "Two-dimensional NavierStokes simulation of breaking waves," Physics of Fluids 11 (1), 121-133 661
(1999). 662
10 Chang (林呈) Lin et al., "Novel similarities in the free-surface profiles and velocities of solitary waves traveling over a very 663
steep beach," Physics of Fluids 32 (8), 083601 (2020). 664
11 S. Millet et al., "Wave celerity on a shear-thinning fluid film flowing down an incline," Physics of Fluids 20 (3), 031701 665
(2008). 666
12 N. J. Balmforth, R. V. Craster and C. Toniolo, "Interfacial instability in non-Newtonian fluid layers," Physics of Fluids 15 667
(11), 3370-3384 (2003). 668
13 E. Mogilevskiy, "Stability of a non-Newtonian falling film due to three-dimensional disturbances," Physics of Fluids 32 (7), 669
073101 (2020). 670
14 Sumit Tripathi et al., "Lubricated Transport of Highly Viscous Non-newtonian Fluid as Core-annular Flow: A CFD Study," 671
Procedia IUTAM 15, 278-285 (2015). 672
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24
15 Sergey V. Alekseenko et al., "Study of formation and development of disturbance waves in annular gasliquid flow," 673
International Journal of Multiphase Flow 77, 65-75 (2015). 674
16 Mikhail V. Cherdantsev et al., "Development and interaction of disturbance waves in downward annular gas-liquid flow," 675
International Journal of Multiphase Flow 138, 103614 (2021). 676
17 Wayne Strasser, "Towards Atomization for Green Energy: Viscous Slurry Core Disruption by Feed Inversion," AAS 31 (6), 677
23-43 (2021). 678
18 Daniel M. Wilson and Wayne Strasser, "Smart Atomization: Implementation of PID Control in Biosludge Atomizer," 5-6th 679
Thermal and Fluids Engineering Conference , (2021). 680
19 D. M. Wilson and W. Strasser, "A Spray of Puree: Wave-Augmented Transonic Airblast Non-Newtonian Atomization," 681
Physics of Fluids (in revision) , (2022). 682
20 D. M. Wilson, W. Strasser and R. Prichard, "Spatiotemporal Characterization of Non-Newtonian Wave-Augmented 683
Varicose Explosions (WAVE) in Transonic Airblast Atomizer," Physics of Fluids (submitted) , (2022). 684
21 M. Lagha et al., "A numerical study of compressible turbulent boundary layers," Physics of Fluids 23 (1), 015106 (2011). 685
22 Akanksha Baranwal, Diego A. Donzis and Rodney D. W. Bowersox, "Asymptotic behaviour at the wall in compressible 686
turbulent channels," Journal of Fluid Mechanics 933, (2022). 687
23 Suhas V. Patankar, Numerical Heat Transfer and Fluid Flow, edited by Anonymous 1st ed. (CRC Press, London, 1980), .
688
24 Wayne Strasser and Francine Battaglia, "Identification of Pulsation Mechanism in a Transonic Three-Stream Airblast 689
Injector," Journal of Fluids Engineering 138 (11), (2016). 690
25 David Youngs, Time-Dependent Multi-material Flow with Large Fluid Distortion, in Numerical Methods in Fluid
691
Dynamics, edited by K. W. Morton and M. J. Baines, (Academic Press, 1982), pp. 273-285.
692
26 Wayne Strasser and Francine Battaglia, "The effects of pulsation and retraction on non-Newtonian flows in three-stream 693
injector atomization systems," Chemical Engineering Journal 309, 532-544 (2017). 694
27 Wayne Strasser and Francine Battaglia, "The Influence of Retraction on Three-Stream Injector Pulsatile Atomization for 695
AirWater Systems," Journal of Fluids Engineering 138 (11), (2016). 696
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25
28 Wayne Strasser, "Towards the optimization of a pulsatile three-stream coaxial airblast injector," International Journal of 697
Multiphase Flow 37 (7), 831-844 (2011). 698
29 Wayne Strasser, "Oxidation-assisted pulsating three-stream non-Newtonian slurry atomization for energy production," 699
Chemical Engineering Science 196, 214-224 (2019). 700
30 Wayne Strasser, Francine Battaglia and Keith Walters, in , Anonymous (American Society of Mechanical Engineers Digital 701
Collection, 2016/03/07) . 702
31 Wayne Strasser and Francine Battaglia, "The Effects of Prefilming Length and Feed Rate on Compressible Flow in a Self-703
Pulsating Injector ," AAS 27 (11), (2017). 704
32 W. Strasser and F. Battaglia, "Pulsating Slurry Atomization, Film Thickness, and Azimuthal Instabilities," Atomization and 705
Sprays 28 (7), (2018). 706
33 Wayne Strasser, "The war on liquids: Disintegration and reaction by enhanced pulsed blasting," Chemical Engineering 707
Science 216, 115458 (2020). 708
34 Alan George Wonders, Howard Wood Jenkins Jr, Lee Reynolds Partin, Wayne Scott Strasser, Marcel De Vreede, US Patent 709
No. 7589231 2009). 710
35 Santanu Kumar Sahoo and Hrishikesh Gadgil, "Large scale unsteadiness during self-pulsation regime in a swirl coaxial 711
injector and its influence on the downstream spray statistics," International journal of multiphase flow 149, (2022). 712
36 Manisha B. Padwal and D. P. Mishra, "Effect of Air Injection Configuration on the Atomization of Gelled Jet A1 Fuel in an 713
Air-Assist Internally Mixed Atomizer ," Atomization and Sprays 23 (4), 327-341 (2013). 714
37 Cynthia Ditchfield et al., "Rheological Properties of Banana Puree at High Temperatures," International Journal of Food 715
Properties 7 (3), 571-584 (2004). 716
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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... Given the low , the model was treated as isothermal at ambient conditions to match the biofuel 205 experiment. These methods have been extensively validated and used in the past [7,21,[47][48][49][50] . ...
... A strong measure of how the TA is converging is the cumulative TA versus post-QSS data collection 377 time as discussed in [48]. Without computing the cumulative value, there is no way to know how much data 378 collection time is "enough". ...
... At even higher flow rates, waves might break and generate a dispersed flow in which water drops form inside the oil layer and/or oil drops form inside the water layer [1,2]. Note that even though our analysis is focused on oil-water flows assuming Newtonian rheology, waves can also be generated in non-Newtonian flows characterized very different densities and viscosities [3]. ...
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We use pseudo-spectral Direct Numerical Simulation (DNS), coupled with a Phase Field Method (PFM), to investigate the turbulent Poiseuille flow of two immiscible liquid layers inside a channel. The two liquid layers, which have the same thickness (h1 = h2 = h), are characterized by the same density (ρ1 = ρ2 = ρ) but different viscosities (η1 ≠ η2), so to mimick a stratified oil-water flow. This setting gives the possibility to study the interplay between inertial, viscous and surface tension forces to be studied in the absence of gravity. We focus on the role of turbulence in initially deforming the interface and on the subsequent growth of capillary waves. After an initial transient, we observe the emergence of a stationary capillary wave regime. Capillary wave propagation and interaction is studied by means of a spatiotemporal spectral analysis and compared with previous theoretical and experimental results. The computed power spectra of wave elevation are in line with previous experimental findings and can be explained in the frame of the weak wave turbulence theory. At wave scales larger than the turbulent forcing range the observed scaling of the one-dimensional wavenumber spectrum suggests an energy equipartition regime (k−1), which is predicted by theory and has been recently observed in experiments with capillary wave turbulence in microgravity. At wave scales directly forced by hydrodynamic turbulence, an initially milder slope (k−4) of the wavenumber spectrum is followed by a sharper decay (k−6) of wave energy at larger wavenumbers, with the transition taking place near the Kolmogorov-Hinze critical scale, where surface tension forces and turbulent inertial forces are balanced.
... Our recent study 18 revealed the atomization mechanism of the OIL injector: as the liquid stream enters the mixing chamber at a lower pressure drop, the liquid smoothly attaches to the chamber wall so that there is a gas core in the central part surrounded by a liquid film in the mixing chamber. Once the annular liquid film is ejected from the injector through the convergence section, it breaks into droplets by the synergetic effect of Kelvin-Helmholtz (K-H) and Rayleigh-Taylor (R-T) instabilities, [19][20][21] which produces droplets of diameter proportional to the instability wavelength. 22 In light of these hydrodynamic instabilities, the breakup process of the liquid film can be passively controlled by adjusting the orifice shape of the injector to promote the atomization of the liquid 23,24 and the associated spatial distribution of the resultant spray. ...
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Passive control of twin-fluid atomization can be achieved by changing the orifice shape of the injector. In this study, the characteristics of twin-fluid atomization in the outside-in-liquid injector with circular, square, and rectangular orifices at various aspect ratios were investigated experimentally and computationally. The morphology of the spray was captured by shadowgraph, the diameter and velocity of the droplets were measured by the phase Doppler particle analyzer, and numerical simulations were performed for the central gaseous core. Comparing the sprays with square and circular orifices, droplets from the non-circular orifice are generally smaller with less disparities in droplet sizes due to the more intensive turbulent disturbances and corner effect. Furthermore, the non-circular orifice also results in better spatial distribution of the spray. The droplet diameters of the spray with a square orifice do not satisfy the log-normal distribution near the orifice along the centerline of the spray, which may be attributed to the different entrainment of spray droplets by the central gas flow for the sprays with circular and non-circular orifices. The twin-fluid sprays produced by the rectangular orifice also exhibit the same axial switching effect as in the high-pressure gaseous jet flow, in which the spray diffusion in the minor axis is more extensive than that in the major axis. Moreover, the droplets' Sauter mean diameter produced by the rectangular orifice is more sensitive to the size in the minor axis of the orifice and decreases as the aspect ratio of the orifice increases given the same cross-sectional area.
... A prime example is our daily experience with toothpaste, which only flows when enough force is applied. Yield stress is also seen in numerous food products [344][345][346]. Think about mayonnaise, where ridges and peaks on the surface show the existence of a critical stress above which it flows [347][348][349]. ...
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Innovations in fluid mechanics have been leading to better food since ancient history, while creativity in cooking has inspired fundamental breakthroughs in science. This review addresses how recent advances in hydrodynamics are changing food science and the culinary arts and, reciprocally, how the surprising phenomena that arise in the kitchen are leading to new discoveries across the disciplines, including molecular gastronomy, rheology, soft matter, biophysics, medicine, and nanotechnology. This review is structured like a menu, where each course highlights different aspects of culinary fluid mechanics. Our main themes include multiphase flows, complex fluids, thermal convection, hydrodynamic instabilities, viscous flows, granular matter, porous media, percolation, chaotic advection, interfacial phenomena, and turbulence. For every topic, an introduction and its connections to food are provided, followed by a discussion of how science could be made more accessible and inclusive. The state-of-the-art knowledge is then assessed, the open problems, along with the likely directions for future research and indeed future dishes. New ideas in science and gastronomy are growing rapidly side by side.
... Numerical simulations reveal three atomization mechanisms directly resulting from wave formation: wave impact momentum, pressure buildup, and droplet breakaway, which together contribute to the "wave-augmented atomization" process. Wilson and Strasser 26 investigated wave formation inside a twin-fluid atomizer nozzle using non-Newtonian fluids numerically. It was shown that Kelvin-Helmholtz instability and pressure effects dominate the wave formation and collapse process. ...
Article
Full-text available
An annular liquid sheet sheared by a coaxial supersonic gas stream with a swirling effect is investigated using Large Eddy Simulation. Despite its wide applications in aerospace and medical devices, the instability and spatial characters have been barely investigated due to the high complexity under supersonic condition. Unlike the conventional use of the temporal dynamic mode decomposition (DMD), DMD is applied in the axial direction to evaluate the transient convective instability. The high-velocity cases show significantly stronger instability in the nozzle near-field. However, swirling has only marginal effects on the convective instability. In addition, proper orthogonal decomposition (POD) extracts the essential spatial topology of velocity, momentum, and pressure fields. Pulsatile and flapping instabilities are observed in the gas flow, where liquid flow demonstrates the schrink/expansion as well as the flapping instabilities. In addition, all POD modes of the pressure field take the form of coherent wavepacket structures, and their wavelength and spatial forms of the wavepackets are dependent on the gas flow speed rather than the swirling. Time coefficients of the leading POD modes of momentum and pressure fields show an interesting correlation. Hence, the causal–effect relationship between these leading modes of momentum and pressure field is quantified via transfer entropy from the information theory. The transfer entropy from the pressure field to the momentum field is generally higher than vice versa, and this trend is enhanced by the swirling in the low-velocity condition.
... They examine numerically the industrially relevant process of annular injection of the banana puree into a transonic pipe flow of steam in a twin-fluid atomizer using the ANSYS Fluent CFD solver. The first paper 30 focuses on the numerical analysis of the instability of the injected stream. When the puree meets the rapid gas flow, they observe the formation of annular periodic waves at the interface and their subsequent growth driven by the Kelvin-Helmholtz instability. ...
Article
Full-text available
Characterization of viscous, non-Newtonian atomization by means of internal waves is presented for a twin-fluid injector. Atomization of such fluids is challenging, especially at low gas-liquid mass ratios (GLRs). This paper details mechanisms that enhance their disintegration in a "wave-augmented atomization" process. The working fluid, banana puree, is shear-thinning and described by the Herschel-Bulkley model. Unlike a conventional airblast injector, an annular flow of banana puree is injected into a core steam flow, encouraging regular puree waves to form inside the nozzle. A pulsing flow develops with three distinct stages: stretch, bulge, and burst leading to an annular puree sheet stretching down from the nozzle exit. Rayleigh-Taylor instabilities and viscosity gradients destabilize the surface. During wave collapse, the puree sheet bulges radially outward and ruptures violently in a radial burst. Near-nozzle dynamics propagate axially as periodic fluctuations in Sauter mean diameter occur in a wave pattern. Numerical simulations reveal three atomization mechanisms that are a direct result of wave formation: 1) wave impact momentum, 2) pressure buildup, and 3) droplet breakaway. The first two are the forces that exploit puree sheet irregularities to drive rupture. The third occurs as rising waves penetrate the central steam flow; steam shear strips droplets off, and more droplets break away as the wave collapses and partially disintegrates. Waves collapse into the puree sheet with a radial momentum flux of 1.7 × 10 ⁵ kg/m-s ² , and wave-induced pressure buildup creates a large pressure gradient across the puree sheet prior to bursting.
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Full-text available
Characterization of viscous, non-Newtonian atomization by means of internal waves is presented for a twin-fluid injector. Atomization of such fluids is challenging, especially at low gas-liquid mass ratios (GLRs). This paper details mechanisms that enhance their disintegration in a "wave-augmented atomization" process. The working fluid, banana puree, is shear-thinning and described by the Herschel-Bulkley model. Unlike a conventional airblast injector, an annular flow of banana puree is injected into a core steam flow, encouraging regular puree waves to form inside the nozzle. A pulsing flow develops with three distinct stages: stretch, bulge, and burst leading to an annular puree sheet stretching down from the nozzle exit. Rayleigh-Taylor instabilities and viscosity gradients destabilize the surface. During wave collapse, the puree sheet bulges radially outward and ruptures violently in a radial burst. Near-nozzle dynamics propagate axially as periodic fluctuations in Sauter mean diameter occur in a wave pattern. Numerical simulations reveal three atomization mechanisms that are a direct result of wave formation: 1) wave impact momentum, 2) pressure buildup, and 3) droplet breakaway. The first two are the forces that exploit puree sheet irregularities to drive rupture. The third occurs as rising waves penetrate the central steam flow; steam shear strips droplets off, and more droplets break away as the wave collapses and partially disintegrates. Waves collapse into the puree sheet with a radial momentum flux of 1.7 × 10 ⁵ kg/m-s ² , and wave-induced pressure buildup creates a large pressure gradient across the puree sheet prior to bursting.
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We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number ( Re , wave inertia relative to viscous effects) and Bond number ( Bo , wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at high Re and Bo , and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulent Re up to a limiting value Reλ100Re_\lambda \sim 100 , consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deike et al. ( J. Fluid Mech. , vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale, rHr_H . We confirm two regimes in the bubble size distribution, N(r/rH)(r/rH)10/3N(r/r_H)\propto (r/r_H)^{-10/3} for r>rHr>r_H , and (r/rH)3/2\propto (r/r_H)^{-3/2} for r.Bubblesareresolveduptooneorderofmagnitudebelowr . Bubbles are resolved up to one order of magnitude below r_H , and we observe a good collapse of the numerical data compared to laboratory breaking waves (Deane & Stokes, Nature , vol. 418 (6900), 2002, pp. 839–844). We resolve droplet statistics at high Bo in good agreement with recent experiments (Erinin et al. , Geophys. Res. Lett. , vol. 46 (14), 2019, pp. 8244–8251), with a distribution shape close to N_d(r_d)\propto r_d^{-2}$ . The evolution of the droplet statistics appears controlled by the details of the impact process and subsequent splash-up. We discuss velocity distributions for the droplets, finding ejection velocities up to four times the phase speed of the wave, which are produced during the most intense splashing events of the breaking process.
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The asymptotic behaviour of Reynolds stresses close to walls is well established in incompressible flows owing to the constraint imposed by the solenoidal nature of the velocity field. For compressible flows, thus, one may expect a different asymptotic behaviour, which has indeed been noted in the literature. However, the transition from incompressible to compressible scaling, as well as the limiting behaviour for the latter, is largely unknown. Thus, we investigate the effects of compressibility on the near-wall, asymptotic behaviour of turbulent fluxes using a large direct numerical simulation (DNS) database of turbulent channel flow at higher than usual wall-normal resolutions. We vary the Mach number at a constant friction Reynolds number to directly assess compressibility effects. We observe that the near-wall asymptotic behaviour for compressible turbulent flow is different from the corresponding incompressible flow even if the mean density variations are taken into account and semi-local scalings are used. For Mach numbers near the incompressible regimes, the near-wall asymptotic behaviour follows the well-known theoretical behaviour. When the Mach number is increased, turbulent fluxes containing wall-normal components show a decrease in the slope owing to increased dilatation effects. We observe that RvvR_{vv} approaches its high-Mach-number asymptote at a lower Mach number than that required for the other fluxes. We also introduce a transition distance from the wall at which turbulent fluxes exhibit a change in scaling exponents. Implications for wall models are briefly presented.
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Although the question is a classical problem, the details of how wind transfers energy to waves at the ocean surface remain elusive.
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Drop deformation in fluid flows is investigated as an exchange between the kinetic energy of the fluid and the surface energy of the drop. We show analytically that this energetic exchange is controlled only by the stretching (or compression) of the drop surface by the rate-of-strain tensor. This mechanism is analogous to the stretching of the vorticity field in turbulence. By leveraging the non-local nature of turbulence dynamics, we introduce a new decomposition that isolates the energetic exchange arising from local drop-induced surface effects from the non-local action of turbulent fluctuations. We perform direct numerical simulations of single inertial drops in isotropic turbulence and show that an important contribution to the increments of the surface energy arises from the non-local stretching of the fluid–fluid interface by eddies far from the drop surface (outer eddies). We report that this mechanism is dominant and independent of surface dynamics in a range of Weber numbers in which drop breakup occurs. These findings shed new light on drop deformation and breakup in turbulent flows, and opens the possibility for the improvement and simplification of breakup models.
Article
The focus of the present paper is to understand the large scale unsteadiness in the atomization of swirl coaxial jets and correlate it with the downstream spray statistics. A gas-centered swirl coaxial atomizer, in which a gaseous jet fragments an annular swirling liquid sheet, displays pulsating flow at certain momentum flux ratios (MFRs). These conditions were chosen to study the unsteady dynamics in the spray formation. Various zones of the spray such as near orifice region, primary atomization zone and far field spray were diagnosed using high-speed shadowgraphy technique. Proper orthogonal decomposition was employed on the time-resolved spray images to understand various unsteady modes. Three modes observed prominently were identified as large scale unsteady structures viz. axisymmetric pulsating mode, explosive mode and swirling mode. The pulsating mode was found to be more dominant in the pulsation regime, whereas the other modes gained significance at higher MFRs when the pulsation regime ceases to exist. Fourier analysis of temporal coefficients pertaining to pulsating mode showed a definitive frequency. The analysis of liquid shedding rate was found to be in synchronization with the pulsating mode which shows the upstream influence on periodic atomization. Spatio-temporal measurements of droplet sizes were carried out far from the atomizer. The temporal variation in the droplet number density and Sauter Mean Diameter depicted unsteady behavior; however, there is no preferred frequency in the power spectrum. This clearly indicates that the far field spray loses its memory of upstream periodic atomization in the pulsation regime of swirl coaxial atomizer.
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