ABSTRACT
Title of dissertation: THE DYNAMICS OF PLUNGING BREAKERS
AND THE GENERATION OF SPRAY DROPLETS
Martin A. Erinin
Doctor of Philosophy, 2020
Dissertation directed by: Professor James H. Duncan
Department of Mechanical Engineering
In this dissertation, drop generation by plunging breaking waves is studied in
laboratory-scale experiments. The breaking waves are generated by a programmable
wavemaker that is set to produce a dispersively focused wave packet. Breaker profile
and drop measurements are obtained for three breaking waves of increasing inten-
sity, called herein the, weak, moderate, and strong plunging breakers, respectively.
Drops, with radius ? 50 ?m, are measured using a cinematic in-line holographic sys-
tem positioned at 28 streamwise measurement locations, arranged in a horizontal
plane, called herein the measurement plane, positioned 1 cm above the maximum
wave crest height, during many repeated breaking events. From the holograms, the
radius, three-dimensional location, and velocity of the drops is determined. The
evolution of the breaker profile is measured at the center plane of the tank using
a cinematic laser-induced fluorescence (LIF) technique. Drop and breaker profile
movies are taken simultaneously and recorded at 650 frames per second.
The breaker profile and drop measurements from the weak breaker are used to
identify and quantify spray generation processes in plunging breakers. Two spatially
and temporally separated regions of drop production are found. The first region (I)
of drop production is associated with the active phase of wave breaking and begins
with jet impact. In this region, drops are produced by jet impact, large bubble
bursting events, and splashing. The second region (II) of drop production occurs
after the active phase of wave breaking, approximately one wave period after jet
impact. In this region, drops are generated by small air bubbles, initially entrained
by the breaker, that rise to the free-surface and burst.
Various features of the breaker profiles and drop production are compared
and contrasted between the three waves. The temporal evolution of the breaker
profile is measured in 10 unique realizations for each of the three breakers. At every
instant in time, the phase averaged mean breaker profile and the distribution of
standard deviation (SD) in breaker height along the streamwise direction of the
mean profile is computed. Using the mean profiles, the three breakers are physically
characterized based on their wave crest and jet impact speed. When aligned to
the plunging jet impact location in space and time, the breaker profiles are found
to be highly repeatable throughout the non-linear wave breaking process. Profile
regions with high SD in height correspond to regions of high drop production. The
number of drops generated per breaking event is found to scale exponentially with
jet impact speed. The drop probability distribution from each of the three waves
follows a power law scaling where large and small drop regions obey power laws with
different coefficients.
THE DYNAMICS OF PLUNGING BREAKERS
AND THE GENERATION OF SPRAY DROPLETS
by
Martin A. Erinin
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2020
Advisory Committee:
Professor James H. Duncan, Chair/Advisor
Professor Kenneth Kiger
Associate Professor Johan Larsson
Emeritus Professor Peter Bernard
Associate Professor Anya Jones, Dean?s Representative
?c Copyright by
Martin A. Erinin
2020
Acknowledgments
This section is dedicated to the people who helped me throughout my PhD.
First and foremost, I would like to thank my advisor, Professor James Duncan, for
giving me the opportunity to pursuit a PhD and being a great mentor. You are
truly the embodiment of a rigorous scientist, the lessons I have learned from you
will be with me for the rest of my life.
I would also like to thank my lab mates and Professor Liu from whom I have
learned much from and without whom I would not have been able to achieve nearly
as much in my academic career. To Nate, Naeem, and An, who have taught me how
to get around the lab from the early days when I was an undergraduate, I would
like to give my sincerest gratitude.
I also want to thank all the undergraduates who have worked with me over
the years. It has been a pleasure to work with you such young and talented people
and I wish you all success in the future. I would especially like to thank Benjamin
Schaefer for helping me with the tedious task of processing countless wave surface
profiles during the last few months of my PhD.
Thank you to my committee members, Professor Kiger, Professor Larsson,
Professor Bernard, and Professor Jones for generously giving their time, attention,
and insightful comments on this dissertation.
Last but not least, I would like to thank my close family and friends. The
love, care, and support I have received from you has made this journey easier.
ii
Contents
Acknowledgements ii
List of Tables v
List of Figures vi
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of Previous Research . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Review of Some Laboratory and Field Experiments on Drop
Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Review of Recent Numerical Simulations of Bubbles and Drop
Generated by Breaking Waves . . . . . . . . . . . . . . . . . . 10
2 Experimental Details 15
2.1 Overview and Research Objective . . . . . . . . . . . . . . . . . . . . 15
2.2 Experimental Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Water Wave Tank . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Wave Generation Method . . . . . . . . . . . . . . . . . . . . 19
2.3 Experimental Measurement Techniques . . . . . . . . . . . . . . . . . 23
2.3.1 Drop Measurements Using In-line Holography . . . . . . . . . 23
2.3.1.1 Winter 2012 Collimated Laser Beam Optical Setup . 25
2.3.1.2 Summer 2019 Collimated Laser Beam Optical Setup 25
2.3.1.3 Camera Setup Used to Record Holograms . . . . . . 28
2.3.1.4 Hologram Image Processing . . . . . . . . . . . . . . 30
2.3.1.5 Drop Radius Measurement . . . . . . . . . . . . . . . 36
2.3.1.6 In-line Holography Calibration . . . . . . . . . . . . 38
2.3.2 Drop Tracking Code . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.3 Surface Profile Measurements . . . . . . . . . . . . . . . . . . 45
2.3.4 Humidity and Drop Evaporation . . . . . . . . . . . . . . . . 52
2.3.5 Surface Tension Measurements . . . . . . . . . . . . . . . . . . 54
3 Results and Discussion 58
3.1 Spray Generation Mechanisms by a Plunging Breaking Wave . . . . . 59
3.2 Comparison of Spray Generated by a Weak, Moderate, and Strong
Intensity Plunging Breakers . . . . . . . . . . . . . . . . . . . . . . . 78
iii
3.2.1 Surface Profile Measurements . . . . . . . . . . . . . . . . . . 79
3.2.2 Drop Measurements . . . . . . . . . . . . . . . . . . . . . . . 90
4 Summary and Conclusions 105
4.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
iv
List of Tables
2.1 Table of wavemaker parameters for the three waves studied in this
dissertation, the weak, moderate and strong plunging breakers. All
three waves have the same average wave packet frequency of f0 =
1.15 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 List of parameters used as input to the model by Andreas [1989],
which simulates drop evaporation. . . . . . . . . . . . . . . . . . . . . 53
3.1 The values for the regression coefficients, p1, p2, and p3, used for the
second order polynomial fit to xc(t), the position of the wave crest
point, for the weak, moderate, and strong breakers. The second order
polynomial has the following form xc(t) = p
2
1t +p2t+p3. The function
fit xc(t) is used to obtain the velocity and acceleration of the wave
crest during the jet formation phase. . . . . . . . . . . . . . . . . . . 82
3.2 The values for the regression coefficients, xp1, xp2, xp3, yp1, yp2,
and yp3, from second order polynomials which are fitted to the xj(t)
and yj(t) motion of the jet tip point. The second order polynomials
have the following form; xj(t) = xp1t
2 + xp2t + xp3 and yj(t) =
yp1t
2 + yp2t+ yp3 for x and y motion, respectively. . . . . . . . . . . 86
3.3 Table comparing physical wave parameters for the weak, moderate,
and strong plunging breakers. The average phase speed of the wave c?
is measured just before the jet starts to form on the wave, see Figure
3.9. The average jet impact speed, v?jet, is measured just before jet
impact (see Figure 3.10) and the total number of drops indicates the
number of drops produced per breaking event per crest width whose
D > 100 ?m, measured in section 3.2.2. . . . . . . . . . . . . . . . . . 90
3.4 Table comparing the number drops produced per breaking event per
crest width (whose d > 100 ?m) during the different stages of break-
ing for the weak, moderate, and strong plunging breakers. The jet
impact and large bubble bursting regions are defined spatially and
temporally (see text for full description) while the late bubble burst-
ing region is defined from 1 < t < 2 seconds. . . . . . . . . . . . . . . 101
3.5 The fit parameters to drop size distributions from Figures 3.19 for
the three waves. Where ? and ? are the slope of the line fits for the
small and large drops respectively and ri identifies the drop radius
(in ?m) where the two lines intersect. . . . . . . . . . . . . . . . . . . 104
v
List of Figures
1.1 Drop production mechanisms as envisioned by an artist in Veron
[2015] (originally Figure 1 in that review article). Spume drops, which
are globules of water sheared off from the wave crest, can be seen on
the top right of the image. Entrained bubbles can be seen behind the
wave crest as they rise to the free surface and pop, creating film and
jet drops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Drop size distributions from laboratory generated wind-waves at var-
ious wind speeds (vertical plots) and height above water surface (hor-
izontal plots) from Wu [1973]. . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Drop size distributions 13 cm above the water surface at various wind
speeds (left plot, where U = 6.4 m/s (open circles), U = 7.5 m/s
(filled circles), U = 8.0 m/s (solid circles)) and at the same wind speed
(U = 7.5 m/s) but different height above water surface(z = 13 cm
(open circles) and z = 18 cm (solid circles)) from Wu et al. [1984]. . . 9
1.4 Spume drop size distributions from laboratory generated wind-waves
at various wind speeds (U = 31.3 m/s, 41.2 m/s, 47.1 m/s, solid black
dots) from Veron et al. [2012] plotted along with models from Fairall
et al. [1995] and Mueller and Veron [2009] also shown. . . . . . . . . . 11
1.5 A side view of a numerical simulation from Wang et al. [2016] of a
breaking wave just after jet impact. An air cavity and bubbles are
present below the free surface, while drops can be seen suspended up
above. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 A visualization of the air-water interface from direct numerical sim-
ulations just after wave breaking from Deike et al. [2016]. Numerous
drops and bubbles are visible in the water and air. . . . . . . . . . . . 12
1.7 A series of snapshots from a numerical simulation of breaking waves
with wind above from Tang et al. [2017]. . . . . . . . . . . . . . . . . 13
vi
2.1 A side view schematic of the wave tank. The tank is 14.8 m long,
1.15 m wide, and has a water depth of 0.91 m. The wedge (seen on
the right) is part of the wavemaker control system and is typically
partially submerged in the water. Most laser, optical, and camera
equipment is attached to the instrument carriage. The instrument
carriage is shown here configured for wave profile measurements. At
the end opposite to the wavemaker is an artificial beach, a sloped
panel of plexiglas spanning the width of the tank, used to dissipate
wave energy after the experiment has been conducted. . . . . . . . . . 19
2.2 The wave maker motion (position vs. time) for the weak, moderate,
and strong breakers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 An end (left) and plan (right) schematic drawings of the wave wave
tank with the drop and surface profile measurement system compo-
nents shown. The end view schematic shows the laser light sheet,
used to illuminate the water surface and measure breaker profiles,
attached to the instrument carriage. Also attached to the instrument
carriage are two high speed cameras for the drop measurements (one
behind the other, bottom camera in the end view) and a camera used
to measure the free surface profiles (top camera in the end view). The
plan view (right schematic) shows the two cameras used for drop mea-
surements along with two separated collimated laser beams directed
into the two cameras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 The optical setup used to generate a collimated laser beam approxi-
mately 50 mm in diameter used to collect drop in-line holographic
data for the weak and strong plunging breakers. A high-energy
Nd:YLF laser (50 mJ/pulse) is used as the light source. The light
coming out of the laser is polarized and most of the energy from the
pulse is dumped. The light is then passed through a spatial filter,
collimated, and expanded through a series of two achromatic lenses
(one converging one diverging) are used to create the collimated laser
beam. The beam is then directed horizontally across the width of the
tank (1.15 m) and into a camera on the other side of the tank. . . . . 26
2.5 The new and improved optical setup used to generate a collimated
laser beam approximately 50 mm in diameter that was used to collect
drop in-line holographic data for the moderate plunging breakers. A
low-powered Nd:YLF laser (50 ?J/pulse) is used as the light source.
The light is passed through a spatial filter and a neutral density filter,
and is then collimated by a single achromatic converging lens. The
collimated laser beam is then directed horizontally across the width
of the tank (1.15 m) and into a camera on the other side of the tank. 29
vii
2.6 Two sample images of drop holograms where (a) was recorded with
the hologram illumination system used for the weak and strong break-
ers collected in the winter 2012, while the hologram in (b) was recorded
with an improved illumination system used for the moderate breaker.
Holograms of drops show up as either black dots surrounded by in-
terference rings or just as interference rings. Both images are approx-
imately 25.6 mm x 16 mm with spatial resolution of 10 ?m per pixel
and are able to reconstruct drops with diameter of d > 50?m over
the entire width of the tank, 1150 mm. The red box is referenced in
the next section and in Figure 2.7. . . . . . . . . . . . . . . . . . . . 30
2.7 Original recorded hologram is shown in (a) (originally cropped from
the red box from the hologram in Figure 2.6 (a)). Three distinct
interference patterns, indicating that three drops are present, can be
seen, labeled by the letters A, B, and C. In (b), the original image in
(a) is reconstructed at a distance of z = 133 mm away from the focal
plane where drop B is in focus. Image (c) shows the raw hologram in
(a) reconstructed at z = 268 mm where drop C comes into focus. Fi-
nally, image (d) shows the image in (a) reconstructed at z = 331 mm
where drop A comes into focus. . . . . . . . . . . . . . . . . . . . . . 33
2.8 Series of images showing the first stage of the reconstruction process,
where drop location and approximate radius is measured. The origi-
nal recorded hologram is shown in (a). A time-averaged background is
subtracted from (a) and is shown in (b). This background subtracted
hologram is reconstructed every 5 mm in depth and the resulting 3D
reconstructed hologram volume is collapsed by taking the minimum
intensity of each pixel in depth. The resulting collapsed hologram
is shown in (c). Finally, the drops? location and approximate size
are measured by thresholding the collapsed hologram, with the result
shown in (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Series of images showing the second stage of the reconstruction pro-
cess, in which the plane of best focus is calculated for each drop.
The raw cropped hologram of a drop is shown in (a). The sharpness
criteria, based on the Tenegrad operator, at a single plane is shown
in (b). The mean drop sharpness vs. depth is shown in (c) with a
defined peak near the plane of best focus for the drop. Finally, the
reconstructed drop at the plane of best focus is shown in (d). . . . . . 37
2.10 The inverse of the intensity map of a 1000 micron dot from the cal-
ibration reticle shown in Figure 2.11 (a) is shown on the left. The
image on the right shows the hyperbolic tangent function fit of the
same drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
viii
2.11 A calibration reticle with 14 chrome sputter deposited circles ranging
in size from d = 3000 ?m down to d = 30 ?m. Image (a) shows the
calibration target imaged by the digital in-line holographic system
at the focal point of the camera lens. Hologram (b) shows the same
calibration target imaged at the far sidewall of the tank (605 mm
away from the focal plane); interference patterns can be seen in the
hologram. Image (c) is the digital reconstruction of image (b) at
z = 605 mm. Note that the original image is 2560 x 1600 pixels in
size and that the d = 100 ?m dot (seventh dot from the left in the
bottom row) is visible and in focus in the reconstructed hologram. . . 41
2.12 A sample plot of drop trajectories from one run spanning approxi-
mately 1300 images (2 seconds) with spatial dimensions of 25.6 by
16 mm. The relative drop diameters are indicated by the diameters of
the markers shown in the plot. The drop positions are not necessarily
equally spaced in time. Because holographic reconstructions of drops
near the exterior boundaries of the image are more inaccurate, only
drops 200 pixels (2 mm) inside the measurement window (the white
region outlined by solid and dashed red lines) are reconstructed and
their radius is measured. The measurement plane is indicated by a
solid red line 200 pixels above the bottom edge of the frame. . . . . . 45
2.13 A sample sequence of LIF images of a breaking wave. The wave is
moving from right to left with time advancing from image (a) to (c).
The free surface is initially smooth in image (a), before the wave has
broken. In image (b) the wave is breaking and surface roughness can
be seen in the middle of the image. The part of the wave closest to the
camera can be seen in the foreground of image (b). Finally, in image
(c) the wave has passed. The surface profile in (c) looks less uniform
due to a number of reasons, such as free surface roughness (causing
water to act like a lens) or free surface features in the foreground of
the image between the camera and lens. . . . . . . . . . . . . . . . . 47
2.14 Three time-synchronized surface profile images just after jet impact
and at the onset of the initial splash up. The three cameras have
slightly overlapping fields of view and the wave is moving right to left. 48
2.15 Images of the same calibration checkerboard target from the three
cameras with slightly overlapping fields of view of view. The image
points, (Xi, Yi), are detecetd by an image processing code and used
to convert from image coordinates to physical coordinates and trans-
form the points to an orthogonal coordinate system. Image (a) is
the left most camera, (b) is the middle camera, and (c) is the right
camera, with the direction of the wave propagating from right to left.
The yellow points seen in (a-c) are shared common points where the
camera fields of view overlap, which are used to stitch the images
together. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
ix
2.16 Stitched images from the three camera fields of view, covering a span-
wise distance approximately 1300 cm at a spatial resolution of 180
?m/pixel, of the calibration board shown in Figure 2.15 and surface
profile images shown in Figure 2.14. . . . . . . . . . . . . . . . . . . . 51
2.17 Evaporation model for d = 100 micron (red), d = 200 micron (blue),
and d = 400 micron (green) for fresh water and 50 percent relative
humidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.18 Surface tension isotherms of wave tank water from seven measure-
ments from different days. The five red curves show the surface ten-
sion isotherms that was deemed dirty because of a significant drop in
surface tension after about 70% compression. The blue curves show
the water quality in the same wave tank after the filtration system
was cleaned. The quality of the water was deemed important for
the repeatability of the experiment. Therefore, the surface tension
isotherm for the water used in the moderate breaker droplet mea-
surements and all profile measurements was maintained as close as
possible to the blue curves. . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Water surface profiles (a) and images (b) from an LIF movie of the
breaker used in this study. Each profile in (a) is obtained from one
image in the LIF movie. The movie was taken at 650 pps (?t =
0.00154 s) but only every fourth profile (?t = 0.0062) is shown in (a).
Each profile is plotted 8 mm above the previous profile for clarity. The
first profile in the sequence was recorded 0.0077 s after the plunging
jet hit the front face of the wave, t = 0 s, and the last profile was
taken at t = 0.6154 s. The red filled triangles mark the position of
the highest point on each profile. The sequence of bold profiles in
(a) corresponds to the sequence of images in (b). This figure was
originally published in Erinin et al. [2019]. . . . . . . . . . . . . . . . 63
3.2 Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest
length, plotted on an x-t plane. The black line is drawn with a slope
of 1.7 m/s. The bolded free surface profiles and images in Figure 3.1
(a & b) correspond to the red lines on the lower right corner of the
contour plot. The magenta line corresponds to the maximum wave
crest height from Figure 3.1. The plot on the right is Nx, the number
of drops generated per unit time per crest length, vs time. The plot
on top is Nt, the number of drops generated per mm in the streamwise
direction per unit crest length, vs steamwise position. This figure was
originally published in Erinin et al. [2019]. . . . . . . . . . . . . . . . 66
x
3.3 (a). Plot of the number of drops generated per breaking event (Nx(t))
per time interval ?t = 0.025 s, per tank width ?y = 1.05 m. (b).
Plot of the number of drops generated per breaking event (Nt(x))
per distance interval ?x = 21.6 mm, per tank width ?y = 1.05 m.
These two plots are the data in Figure 3.2 integrated over x and t,
respectively to yield a total of 539 drops per breaking event. (c).
Images (a-c) show the free surface from t = 0.1 s to 0.3 s with ?t =
0.1 s between each image. Images (d-f) show the free surface from t
= 0.334 s to 0.435 s with ?t = 0.051 s between successive images.
Images (g-i) show traces of bubbles on the free surface from t = 1.5 s
to 1.7 s with ?t = 0.1 s between successive images. . . . . . . . . . . . 68
3.4 Mean drop speed as a function of drop radius plotted on a logarith-
mically spaced bin ranging from r = 50 to 1500 ?m with a total of 32
bins. The mean drop speed is calculated from the x and y velocity
components as the drop passes through the measurement plane. . . . 69
3.5 Histograms of drop angles (?, from 0 to ?, where 0 is horizontal in
the direction of wave motion) for all drops at various stages during
the breaking processes as they pass through the plane 1.2 cm above
the maximum wave crest height. (a) shows the distribution of drop
angles for the whole measurement region. (b) show the distribution
of drop angles from 0 ? x ? 12.5 cm, 0.325 ? t ? 0.5 s.
(c) show drop angles from 0 ? t ? 0.8s excluding drops from
0 ? x ? 12.5 cm, 0.325 ? t ? 0.5 s. Finally, (d) shows all drop
angles after t > 0.8s. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6 Profile history of a breaking event with drop generation locations in
time and space. In (a), the dots are located at the x and t where the
drops passed through the horizontal plane located 1.2 cm above the
highest point reached by the free surface. In (b), the dots are located
at the x and t where the drops are calculated to have been generated.
This calculation is performed with the data on the drop motion while
it is in the measurement window and a ballistic trajectory model for
the drop motion. ? 800 ? d, ? 300 ? d < 800, ? 200 ? d < 300, ?
125 ? d < 200, ? d < 125 ?m. . . . . . . . . . . . . . . . . . . . . . . 73
3.7 Drop radius probability distributions. Vertical error bars show ?1
standard deviation, while horizontal error bars show radius measure-
ment error obtained from a calibration process. The main plot is for
all of the drops produced by the breaker while the sub plot contains
separate data sets for the drops produced during the jet impact phase
(Region I, 0 < t < 0.8 s, ?) and during the bubble ascend and popping
phase (Region II, 0.8 < t < 2 s, ?). A Hinze-scale-like break in slope
is identified in all three data sets. The slopes of the distributions for
the larger and smaller bubbles in each data set are identified by ?
and ?, respectively. For all the drops, break at r = 273 ?m. This
figure was originally published in Erinin et al. [2019]. . . . . . . . . . 76
xi
3.8 Sample measurements of wave crest point, (green square) and jet tip
location (red triangle) from one run of the moderate breaker. Each
surface profile, shown in blue, is obtained from one LIF image, similar
to the ones shown in Figure 2.13. Each successive profile is separated
by a time interval of ?t = 0.0123 s and plotted 10 mm above the
previous for clarity. The wave crest and jet tip trajectories form 10
runs for each of the three breakers are shown in Figures 3.9 and 3.10,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.9 The x and y positions vs. time of the wave crest point during the
plunging jet formation from 10 runs for the weak, moderate, and
strong plunging breakers. The vertical dashed lines indicated the
average time of jet impact from the 10 runs for the three waves. The
weak breaker is shown in red, the moderate in blue, and the strong
in green. The maximum wave crest height from 10 runs for xc(t) and
yc(t) are shown in (a) and (c) respectively, while (b) and (d) show the
averaged position of the wave crest point xc(t) and yc(t) respectively.
The vertical error bars in (b) and (d) show ?1 standard deviation of
the xc and yc over the 10 runs (which are too small to see in (b)).
The wave crest speed reported in Table 3.3 is measured by taking the
derivative of xc(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.10 Jet tip trajectories for the three plunging breakers measured from 10
individual runs. The time interval between each measurement point
is ?t = 0.0015 s (1/650 s). The wave is moving from right to left and
the first point is measured just as the jet tip starts to form from the
wave crest and the last point is measured when the jet impacts the free
surface. The jet trajectories are not adjusted spatially or temporally,
and therefore indicate the repeatability of the experiment from run
to run for the three waves. . . . . . . . . . . . . . . . . . . . . . . . . 87
3.11 Mean surface profiles for the weak breaker (f0 = 1.15 Hz, A =
0.070?0) averaged over 10 unique runs. The range of color along
each profile indicates the standard deviation of surface height at each
streamwise position over the 10 runs. Each successive profile is plot-
ted 10 mm above the previous with a temporal separation of ?t =
0.0123 s between each profile, with breaker profile (i) occurring at
t = 0.0031 s. The surface profiles are shown in a reference frame
fixed to the wave crest measured before the jet is formed (c = 1330
mm/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xii
3.12 Mean surface profiles for the moderate breaker (f0 = 1.15 Hz, A =
0.074?0) averaged over 10 unique runs. The range of color along
each profile indicates the standard deviation of surface height at each
streamwise position over the 10 runs. Each successive profile is plot-
ted 10 mm above the previous with a temporal separation of ?t =
0.0123 s between each profile, with breaker profile (i) occurring at
t = 0.0031 s. The surface profiles are shown in a reference frame
fixed to the wave crest measured before the jet is formed (c = 1330
mm/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.13 Mean surface profiles for the strong breaker (f0 = 1.15 Hz, A =
0.076?0) averaged over 10 unique runs. The range of color along
each profile indicates the standard deviation of surface height at each
streamwise position over the 10 runs. Each successive profile is plot-
ted 10 mm above the previous with a temporal separation of ?t =
0.0123 s between each profile, with breaker profile (i) occurring at
t = 0.0031 s. The surface profiles are shown in a reference frame
fixed to the wave crest measured before the jet is formed (c = 1360
mm/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.14 Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest
length, plotted on an x-t plane for the weak breaker. The black line
is drawn with a slope of 1.7 m/s, indicating the drop ejection front.
This plot is identical to Figure 3.2 except that it is plotted using the
same colorbar as Figures 3.15 and 3.16. . . . . . . . . . . . . . . . . . 95
3.15 Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest
length, plotted on an x-t plane for the moderate breaker. The black
line is drawn with a slope of 1.7 m/s, indicating the drop ejection front. 96
3.16 Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest
length, plotted on an x-t plane for the strong breaker. The black
line is drawn with a slope of 1.7 m/s, indicating the drop ejection front. 97
3.17 A comparison of the number of drops generated per time interval
per crest length vs. time, Nx(t) for the weak (in red), moderate (in
blue), and strong breaker (in green). Three pronounced peaks are
present in the curves from all three waves. The first associated with
jet impact, the second with the large bubble popping region and the
third with the smaller bubbles coming to the free surface later in time.
The three curves were generated by spatially integrating the data in
Figures 3.14, 3.15, and 3.16. . . . . . . . . . . . . . . . . . . . . . . . 99
xiii
3.18 A comparison of the number of drops generated per mm in the stream-
wise direction per crest length vs. streamwise position, Nt(x), for the
weak (red circles), moderate (blue triangles), and strong breakers
(green squares). The weak breaker has a simple spatial distribu-
tion of drops that increases at the jet impact location and gradually
decreases. In contrast, the moderate and strong breakers have pro-
nounced peaks and features at the locations of jet impact and large
bubble popping. These three curves were generated by temporally
integrating N(x, t), shown in Figures 3.14, 3.15, and 3.16. . . . . . . . 100
3.19 Drop probability distributions for weak (red circles), moderate (blue
triangles), and strong breakers (green squares) for drops generated
through out the entire breaking event 0 < t < 2 seconds. A break in
slope is identified in all three data sets and function fitting parameters
and break in slope radii are shown in Table 3.5. . . . . . . . . . . . . 103
xiv
Chapter 1: Introduction
1.1 Background and Motivation
Sea spray consists of water drops that are typically generated by breaking wind
water waves. The transfer of mass, momentum, and heat between the ocean and
the atmosphere is greatly influenced by sea spray [Veron, 2015, Wu, 1973, Veron
et al., 2012, O?Dowd and de Leeuw, 2007, Andreas, 2011, Bao et al., 2011, Bianco
et al., 2011, Mueller and Veron, 2014a,b, Anguelova et al., 1999a, Koga, 1981]. Sea
spray has the potential to dramatically increase the intensity of tropical cyclones,
see Andreas and Emanuel [2001] and Wang et al. [2001]. Sea spray is also known to
enhance some chemical reactions that occur near the air-sea interface. Finally, sea
spray aerosol can act as reactive medium in the atmosphere, transforming chemical
reactions [Blanchard and Syzdek, 1972, Ravishankara, 1997].
Extensive efforts have been devoted to studying drop generation by breaking
waves. Previous investigations have reported measurements of the temporally aver-
aged droplet size distributions and fluxes at fixed heights in wind-wave systems at
sea and in laboratory-scale experiments. It is commonly accepted that droplets are
primarily produced by three breaking subprocesses: wind shear (primarily at the
wave crest), splashing, and the popping of breaker-entrained air bubbles that rise
1
to the free surface [Veron, 2015]. However, there is little experimental data that
quantitatively link the relationship between breaking events and droplet production
by these subprocesses.
1.2 Overview of Previous Research
Comprehensive reviews of spray generated by breaking waves have been pub-
lished recently by Veron [2015] and de Leeuw et al. [2011]. According to the review
article by Veron [2015] there are two dominant mechanism by which drops are gen-
erated by a breaking wave. The first drop production mechanism is by bubbles,
initially entrained by the breaking wave, rising to the free surface and bursting.
When a bubble bursts it creates two types of drops, film drops, generated by the
rupturing of the bubble film, and jet drops, formed by the collapse of the bubble
cavity. The second primary mechanism of drop production identified by Veron [2015]
is when the wind shear stress is sufficiently large to tear off globules of water from
the wave crest. Both of these drop generation mechanisms are shown in an artistic
rendering in Veron [2015]?s Figure 1, which is reproduced here as Figure 1.1. Recent
studies have been carried out in the field (see for example, Monahan et al. [1986],
Wu et al. [1984], de Leeuw [1986], Smith et al. [1993], and Reid et al. [2001]) and in
the lab (see for example, Wu [1973] and Veron et al. [2012]), which are discussed in
more detail in the sections below. In these studies drop production is parameterized
with wind speed rather than the sea state characteristics. It should be noted that
our knowledge of the relationship between breaking events and drop production is
2
Figure 1.1: Drop production mechanisms as envisioned by an artist in
Veron [2015] (originally Figure 1 in that review article). Spume drops,
which are globules of water sheared off from the wave crest, can be seen
on the top right of the image. Entrained bubbles can be seen behind the
wave crest as they rise to the free surface and pop, creating film and jet
drops.
still quite limited [Veron, 2015, de Leeuw et al., 2011].
In an effort to study the relationship between breaking and drop production,
some investigators have explored drop production by bubble bursting. As mentioned
previously, when a bubble bursts two types of drops are produced, film drops, which
tend to be relatively small (drop radius r between 0.01 ?m and 5 ?m) and jet drops,
which can be considerably larger, r u 400 ?m [Veron, 2015, Andreas, 1998]. One of
the earliest photographic measurements on the jet drops were described in Kientzler
et al. [1954]. Other notable works on bubble popping include Spiel [1995, 1998],
Resch [1986], Resch and Afeti [1991] and Cipriano and Blanchard [1981]. Recently,
the size and ejection speed of jet drops as a function of the bubble radius have
been extensively studied by a number of investigators using theory and numerical
simulation (see Gan?a?n Calvo [2017], Deike et al. [2018] and Brasz et al. [2018]).
3
These recent studies have attempted to establish some universal scaling laws for
drops produced by bursting bubbles. Additionally, the role of surfactants have
been studied in laboratory experiments by Sellegri et al. [2006] where the effects
of surfactants (sodium dodecyl sulphate, SDS) on the production of sub-micron
marine aerosols by bubble bursting were investigated. The authors showed that
in the presence of SDS, the size distribution of drops shifts toward smaller sizes
and the relative importance of larger drops increases dramatically. In a laboratory
breaking wave analog, Modini et al. [2013] and Callaghan et al. [2014] demonstrated
the effects of soluble surfactants on the persistence of bubbles and the production
of aerosol particles upon bursting in salt water.
Many of the above-mentioned studies were conducted under ideal conditions
with a calm free surface and single bubbles, which is presumably different from
the surface conditions found in a breaking wave. In plunging breakers, the surface
experiences a dramatic topographical change as the jet is formed at the wave crest
and plunges onto the front of the wave face. A large pocket of air is trapped during
a breaking event as the jet impact takes place. Additionally, after jet impact there
is more air entrainment due to the violent splash of the free surface caused by the jet
impact. Both laboratory experiments by Lamarre and Melville [1991], Deane and
Stokes [2002] and numerical simulations by Deike et al. [2016], Lubin and Glockner
[2015] have shown that the total volume of air entrained and the size distribution
of bubbles can be physically scaled to the energy dissipated due to breaking and to
the characteristic wave slope at breaking.
Various studies have shown that salt and surfactants strongly affect the pro-
4
duction of drops by bursting bubbles. However, the precise connections between
surfactant concentration, water salinity, and drop production is poorly understood.
At moderate wind speeds, air entrainment and drop production are probably associ-
ated with whitecaps. However, our knowledge of the relationship between breaking
events and drop production is limited, Veron [2015] and de Leeuw et al. [2011]. The
processes of drop generation by breaking waves are also modified by the wind. At
high wind speeds, spume drops are produced as foam is ripped off the wave crests
by the wind. At moderate wind speeds, air entrainment and drop production are
probably associated with whitecaps.
A number of studies have attempted to model the drop production in the
ocean. A model accounting for the variation of the drop production rate with wind
speed was first proposed by Monahan et al. [1986]. In this time-dependent model
the aerosol population is modeled in the open-ocean in terms of the drop radius.
The model accounts for aerosol production coming from the bursting of bubbles
from whitecaps at moderate wind speed close to the water surface. A number of
refinements to this model have been proposed, first by de Leeuw [1986] who modeled
the distribution of particle sizes in the region from the free surface to a height of 11 m.
Wu [1990] combined data and models from previous researchers and found that drop
concentration increases exponentially with wind velocity. Additional refinements to
Monahan et al. [1986]?s model were proposed by Andreas [1992] and Andreas et al.
[1995]. Despite the extensive efforts to model the sea spray in the ocean, a review
by Andreas [1998] points out that estimates of the production rate of drops can vary
over six orders of magnitude.
5
1.2.1 Review of Some Laboratory and Field Experiments on Drop
Generation
A number of laboratory and field experiments have attempted to investigate
the relationship between wave breaking and sea spray generation. Some of these
studies were summarized briefly in this paragraph, while a few selected studies are
reported in the preceding two paragraphs. Lai and Shemdin [1974] measured the
wave height distribution and the vertical distribution of spray in a wind-wave tank.
Monahan et al. [1982, 1983] correlated the decay of whitecaps with the number
of drops in both a laboratory tank and the field. Koga [1981, 1984] measured the
vertical distribution and velocity (along the wave profile) of giant drops (diameter >
0.81 mm) produced directly from breaking wind waves. Anguelova et al. [1999b] and
Veron et al. [2012] measured spume drops, which are generated by wind tearing, in
the laboratory while Fairall et al. [2009] investigated large spume drops production
by breaking wind waves in laboratory. Stokes et al. [2013] built a reference tank
system as a breaking wave analogue to study the production of foam and sea-spray
aerosols. Near-surface measurements of sea-spray aerosol production by individual
breaking wave events in the open ocean have been given by Norris et al. [2013]. In
this work, an aerosol spectrometer probe was placed approximately 1 m above the
breaker whitecaps.
Wu [1973] used a laser and a photoresistor to measure drop size distribution in
a laboratory wind wave tank. The measurements were taken at three wind speeds
(U = 10.72 m/s, 12.62 m/s, 13.43 m/s) and two heights above the water surface. One
6
of his primary results are shown in Figure 1.2 where he shows the drop diameter
distributions at the different wind speeds and different heights above the water
surface. Wu [1973] observed a power-law relationship in the drop size distribution
and a notable difference in the slope between the small and large drops occurring
at a drop of diameter D > 200 ?m.
Wu et al. [1984] conducted field measurements using a similar measurement
technique described in the previous paragraph (from Wu [1973]). Drop size distri-
bution data was collected at different wind speeds and at different height above the
water surface. Figure 1.3 shows two drop size distributions, the left subplot shows
data at three wind speeds (U = 6.4 m/s, 7.5 m/s, 8.0 m/s) while the right subplot
shows data at the same wind speed (U0 = 7.5 m/s) at two different heights, z =
13 cm and 18 cm. Again, Wu et al. [1984] observed a power-law relationship in the
distribution of drop sizes and a significant change in slope between large and small
drops occurring between 200 ?m < D < 400 ?m.
Veron et al. [2012] conducted experiments in a wind-wave tank at high wind
speeds (up to U = 47.1 m/s) to obtain spume drop size distributions and com-
pared his results to modern spray distribution models by Mueller and Veron [2009]
and Fairall et al. [1995]. He used a shadowgraph technique to image drops (with
D > 140 ?m) and measured their radius using image processing techniques. Figure
1.4 shows the spume drop size distributions at three different wind speeds (U =
31.3 m/s, 41.2 m/s, 47.1 m/s). Similar to the lab and field experiments of Wu
[1973] and Wu et al. [1984], a power-law relationship is apparent in the drop size
distribution with a difference in slope between large and small drops with a break
7
Figure 1.2: Drop size distributions from laboratory generated wind-
waves at various wind speeds (vertical plots) and height above water
surface (horizontal plots) from Wu [1973].
8
Figure 1.3: Drop size distributions 13 cm above the water surface at
various wind speeds (left plot, where U = 6.4 m/s (open circles), U =
7.5 m/s (filled circles), U = 8.0 m/s (solid circles)) and at the same wind
speed (U = 7.5 m/s) but different height above water surface(z = 13 cm
(open circles) and z = 18 cm (solid circles)) from Wu et al. [1984].
9
in slope around D = 1000 ?m.
1.2.2 Review of Recent Numerical Simulations of Bubbles and Drop
Generated by Breaking Waves
Numerical simulations of drops generated by breaking waves have recently
become available through improvements in numerical schemes and increases in com-
putational capacity. By using a high resolution simulation, Wang et al. [2016] in-
vestigated the bubbles and drops generated by plunging breakers using a direct
numerical simulation (DNS). Figure 1.5 shows a side view of a breaking wave from
Wang et al. [2016] just after the jet impact, where the air-water interface is iden-
tified by the blue colored isosurface . A tube of entrained air can be seen in the
middle of image, while bubbles are seen below the free surface and drops are seen
above the air-water interface. A power law scaling is measured for the large drops
with a slope of -4.5. It should be noted that these simulations do not resolve small
drops, therefore the difference in power law scaling found in field measurements and
experiments is not observed in the DNS.
Deike et al. [2016] studied bubble entrainment during wave breaking using
direct numerical simulations. For bubbles smaller than the Hinze scale, they de-
veloped an improved scaling model for the bubble size distributions which is given
by:
N(r, t) = B(V0/2?)((t???)Wg)r?10/3r?2/3m
10
Figure 1.4: Spume drop size distributions from laboratory generated
wind-waves at various wind speeds (U = 31.3 m/s, 41.2 m/s, 47.1 m/s,
solid black dots) from Veron et al. [2012] plotted along with models from
Fairall et al. [1995] and Mueller and Veron [2009] also shown.
11
Figure 1.5: A side view of a numerical simulation from Wang et al. [2016]
of a breaking wave just after jet impact. An air cavity and bubbles are
present below the free surface, while drops can be seen suspended up
above.
Figure 1.6: A visualization of the air-water interface from direct nu-
merical simulations just after wave breaking from Deike et al. [2016].
Numerous drops and bubbles are visible in the water and air.
12
Figure 1.7: A series of snapshots from a numerical simulation of breaking
waves with wind above from Tang et al. [2017].
where (t ? ??) is the time-dependent turbulent dissipation rate, ?? is the time
scale of the collapse of the air pocket initially entrained by the wave, W is a weighted
velocity for the raising bubbles, rm is the maximum bubble radius, g gravity, V0 is
the initial volume of entrained air, B is a dimensionless constant, and r the bubble
radius. Additionally, they found scaling laws for the total volume of entrained air
during a breaking event. Figure 1.6 shows a visualization of the air-water interface
after the wave breaking where bubbles and drops can be seen.
13
Tang et al. [2017] developed a direct numerical simulation scheme to study the
generation and transportation of spume drops by wind blowing over breaking waves.
They found that drops are generated near the wave crest for young wave ages and
near and behind the wave crest for old wave ages. Additionally, they found that
plunging breakers generated more drops than spilling breakers. Figure 1.7 shows
four snapshots from their direct numerical simulations where drops are shown in
white and time advances from (a) to (d).
14
Chapter 2: Experimental Details
This chapter describes the facilities and methods used in the present exper-
iments to study drop production mechanisms in breaking waves. In these experi-
ments, three mechanically generated breakers, all with the same average wave packet
frequency but with varying breaking intensity, were selected for study. In this disser-
tation, the three breakers are referred to as weak, moderate, and strong. Simultane-
ous measurements of drops, at many streamwise positions above each breaker, and
surface profile histories were conducted during many repeated breaking events for
each wave condition. In the following chapter, the first section is a brief overview of
the research objectives of these experiments. The second section discusses the exper-
imental facilities and methods used to generate the breakers, while the third section
discusses the techniques used to measure the drops and breaker surface profiles.
2.1 Overview and Research Objective
The previous chapter highlighted recent field and laboratory experiments that
have studied spray generation by breaking water waves. However, the precise phys-
ical mechanisms that generated drops during wave breaking are still poorly under-
stood. The objective of the experimental study in this dissertation is to explore
15
the physics of the wave breaking and drop generation processes and determine how
they relate to breaker intensity. Breaker profile histories are used to measure wave
parameters, such as jet impact speed, that can be used as quantitative measured
of breaker intensity. Measurements of drops generated by a breaking wave are used
to obtain drop statistics like the number, size, and spatio-temporal distribution of
drops.
2.2 Experimental Facilities
All of the data analyzed in this dissertation was collected in the wind-wave
facility in the Hydrodynamics Laboratory at the University of Maryland. This
includes droplet measurements for the weak and strong plunging breakers, which
were originally performed by Sophie D. Wang Wang [2012], and similar data for
the droplet measurements for the moderate plunging breaker and surface profile
measurements for weak, moderate, and strong breakers, which were performed by
the author. The wind-wave facility and wave generation technique is described in
this section.
2.2.1 Water Wave Tank
The following description of the wave tank, wave maker, and wave maker mo-
tion applies to the experiments for all three breakers. Experiments are performed
in a wind-wave tank that is 14.8 m long, 1.15 m wide, 2.2 m tall, with and a wa-
ter depth of 0.91 m, as shown in Figure 2.1. The tank includes a programmable
16
wavemaker, a beach located on the opposite end of the tank from the wave maker, a
programmable instrument carriage, an open return wind tunnel, and a water filtra-
tion system. Finally, a beach, which dampens the wave energy after the experiment
has concluded, is located on the opposite end of the tank from the programmable
wave make.
The programmable wavemaker consists of a wedge spanning the width of the
tank and is located at one end. The side of the wedge closest to the end of the
tank is vertical and the opposite side (facing the tank) is inclined at 30? from the
vertical. The wedge is driven by an electric servo motor via a ball screw and rides
on two parallel vertically oriented linear bearings. A feedback control system is used
in conjunction with a position sensor and tachometer to control the motion of the
wedge. Repeated measurements indicate that the RMS variation of the position of
the wedge at the time of the peak amplitude of the wave maker motion is only 0.5
%.
The programmable instrument carriage runs along the length of the wave tank
and is supported by two linear tracks that are mounted on top of the tank. The
carriage rides on hydraulic oil bearings, which provide low vibration during carriage
motion, and is driven by an electric motor. A feedback control system, working in
tandem with a position sensor, is used to create highly repeatable carriage motions.
Most of the optical and camera systems used in this study are directly attached
to the instrument carriage. With this mounting system, the optical measurement
equipment can be moved to various locations along the tank without realignment.
Using the carriage control system, it can be positioned with and accuracy of ?
17
0.025 mm.
The wind tunnel is powered by two 7.5 HP fans and is open to the atmosphere
at the inlet and outlet, see Figure 2.1. Air enters the wind tunnel through the fans
and the flow coming from the fan outlets passes through a system of turning vanes,
screens, and flow straighteners in order to condition the flow. The outlet spans the
1.15 m width of the tank and is 0.76 m high. The bottom of the outlet is 5 cm
from the still water level when the water depth in the tank is 0.91 m. The wind
tunnel is capable of generating wind speeds up to 10 m/s. Waves in the tank can be
generated by the wind tunnel alone, by the mechanical wave maker or both systems
can be used in conjunction. In the present study, waves were generated by the wave
maker alone and the wind tunnel was only used to clean the water surface between
measurement runs, see below.
A water skimmer and filtration system is used to fill the tank and periodically
clean the water in the tank. The filling procedure for the tank start by filling a
holding tank with tap water, which is filtered through a series of two 20, 10, and
5 ?m filters. Hypochlorite is added and mixed to the water in the holding tank
at a concentration of 10 ppm in order to neutralize and prevent organic material
from growing in the water. The hypochlorinated water is transferred to the wave
tank and is filtered through a diatomaceous earth filter for two days. Just before
the breaking wave experiments the free chlorine level in the tank is reduced by the
addition of Hydrogen Peroxide (H2O2) and a sufficient amount of Fluorescein dye is
added and mixed into the tank for wave profile measurements, elaborated upon in a
later section. The reduction of free chlorine is necessary since chlorine bleaches the
18
Figure 2.1: A side view schematic of the wave tank. The tank is 14.8 m
long, 1.15 m wide, and has a water depth of 0.91 m. The wedge (seen
on the right) is part of the wavemaker control system and is typically
partially submerged in the water. Most laser, optical, and camera equip-
ment is attached to the instrument carriage. The instrument carriage is
shown here configured for wave profile measurements. At the end oppo-
site to the wavemaker is an artificial beach, a sloped panel of plexiglas
spanning the width of the tank, used to dissipate wave energy after the
experiment has been conducted.
Flurescein dye. Between each experimental run, the water skimming, wind tunnel,
and filtration systems are turned on periodically to maintain a low level of naturally
occurring surfactants on the water surface.
2.2.2 Wave Generation Method
The waves in this study are generated via a dispersively focused wave packet
technique as described in Duncan et al. [1999] and Wang et al. [2018]. The wave
generation method, first proposed by Rapp and Melville [1990] and Longuet-Higgins
[1976], produces a packet of linear deep-water waves with varying frequencies such
that the packet converges as it travels down the length of the tank. The convergence
19
of the packet causes the amplitudes of each component to constructively interfere.
If the initial amplitudes of the waves are sufficiently large, the largest wave in the
converged packet will reach a critical amplitude and form a highly nonlinear breaking
wave.
Linear deep-water wave theory is used to compute a suitable motion for the
wavemaker. The packet consists of N sinusoidal components where the vertical
wavemaker motion is given by:
?N2? 1 ( (? ) )i ?
zw = w(t) cos xb ? ki ? ?it+ (2.1)
N k c? 2
i=1 i
where A is the overall wavemaker amplitude, N is the number of wave components,
ki and ?i are the wave number and frequency of each wave component, c? is the
average wave group velocity of all N components (c?i = 1/2(?i/ki)). The frequencies
of each wave component are equally space, ?i+1 = ?i + ?? where ?? is a constant.
The variable xb determines the breaking position, measured from the back of the
wedge, along the length of the tank as predicted by linear wave theory. The window
function w(t) gives the wedge zero motion when the sum of all N components results
in very small wavemaker motion and is chosen as:
1( )( )
w(t) = tanh(???(t? t1)) + 1 1? tanh(???(t? t2)) (2.2)
4
?
where ? determines the rate rise of the window function and ? = 1/N i=Ni=1 ?i.
Except for the rise and fall time regions close to t1 and t2, the window function is
equal to 1 for the time interval t1 < t < t2 and zero at all other times. The times t1
20
Table 2.1: Table of wavemaker parameters for the three waves studied in this dis-
sertation, the weak, moderate and strong plunging breakers. All three waves have
the same average wave packet frequency of f0 = 1.15 Hz.
xb/?0 A/?0 Breaker Type Date collected
6.20 0.076 Strong Plunging Breaker Winter 2012
6.20 0.074 Moderate Plunging Breaker Summer 2019 (new holographic setup)
6.17 0.070 Weak Plunging Breaker Winter 2012
and t2 were chosen to allow the highest and lowest frequency wave components to
be generated and travel to xb, the breaking position, and are given by:
1 1
t1 = xb( ? ), (2.3)
c? cN
1 1
t2 = xb( ? ) + 10. (2.4)
c? c1
Three plunging breaking waves of different intensities, a weak, moderate, and
strong plunging breaker, were generated using the technique described above. For
all three breakers the wave generation parameters were N = 32, h/?0 = 0.35792
(where h is the distance between the mean water level and the vertex of the wedge
and ?0 = 2?g/??
2, where g is the gravitational acceleration constant), H/?0 = 0.7458
(where H is the mean water depth in the tank), N??/?? = 0.77, and f0 = ??/2? =
1.15 Hz (where f0 is the average wave packet frequency in hertz). The wave maker
motions for the three breakers differ in the values chosen for the breaking distance,
xb/?0, and the overall wavemaker amplitude, A/?0. Important wave generating
parameters for the three breakers are given in Table 2.1. Plots of the wedge motion
(height verses time) for the three breakers are shown in Figure 2.2.
21
5
0
-5
0 5 10 15 20
5
0
-5
0 5 10 15 20
5
0
-5
0 5 10 15 20
Figure 2.2: The wave maker motion (position vs. time) for the weak,
moderate, and strong breakers.
22
2.3 Experimental Measurement Techniques
In this section, the measurement techniques used to obtain drop and breaker
profile measurements are presented and discussed. In subsection 2.3.1, the in-line
holographic system, which is used to measure drops, and an associated custom drop
tracking code are discussed. Then, in subsection 2.3.3, the laser induced fluorescence
technique, which is used to measure the breaker surface profiles, is described. Finally,
techniques for measuring the humidity and surface tension, which are monitored
during the experiments, are discussed.
2.3.1 Drop Measurements Using In-line Holography
In-line holography uses a coherent and collimated laser beam as a light source
to illuminate a camera whose line of sight is coincident with the optical axis of the
collimated laser beam. An object, in the case of these experiments a drop, that is
in the path of the collimated laser beam is recorded as a hologram by the camera
sensor. A hologram is a record of the interference formed by the collimated and
coherent laser beam and the light diffracted by the drop. The recorded interference
pattern contains the phase and amplitude of the diffracted waves [Katz and Sheng,
2010]. This information contained in the interference pattern is used to reconstruct
the hologram via image processing and computational techniques.
The holographic image sequence for the weak and strong intensity plunging
breakers were collected in the Winter of 2012 by Sophie D. Wang, who used a
different illumination and optical setup than was used in the measurements for the
23
Mirror
Instrument carriage
Light sheet
from Argon Expanded low-power
ion laser beams from Nd:YLF 
Expanded low-power High-speed pulsed lasers (holographic
beam from Nd:YLF digital movie measurement system)
pulsed laser (holographic cameras
Long-distance measurement system)
microscope
lens
Focal plane 
of microscope
lens
High-speed Light sheet
Long-distance
digital movie from Argon
microscope lens
cameras ion laser
1.15 m 1.15 m
Figure 2.3: An end (left) and plan (right) schematic drawings of the wave
wave tank with the drop and surface profile measurement system com-
ponents shown. The end view schematic shows the laser light sheet, used
to illuminate the water surface and measure breaker profiles, attached
to the instrument carriage. Also attached to the instrument carriage
are two high speed cameras for the drop measurements (one behind the
other, bottom camera in the end view) and a camera used to measure
the free surface profiles (top camera in the end view). The plan view
(right schematic) shows the two cameras used for drop measurements
along with two separated collimated laser beams directed into the two
cameras.
24
moderate intensity breaker (see Table 2.1). The two optical setups and illumination
light sources are described in two separate paragraphs below.
2.3.1.1 Winter 2012 Collimated Laser Beam Optical Setup
The illumination in the holographic system employed to capture drop data
for the weak and strong breakers is provided by a high-energy pulsed Nd:YLF laser
(Photonics Industries, DM50-527, 50 mJ/pulse at repetition rates of up to 1 kHz,
pulse width 120 ns). Most of the beam power is dumped using a series of half-wave
plates, thin-film beam splitters, and beam dumps. The remainder of the beam (on
the order of about 40 ?Joules/pulse) is spatially filtered and expanded to a diameter
of about 50 mm. In the final conditioning stage, the beam is split and projected
horizontally across the 1.15 m width of the wave tank. This optical setup is depicted
in Figure 2.4.
2.3.1.2 Summer 2019 Collimated Laser Beam Optical Setup
The optical system employed to generate the collimated laser beam for the
weak and strong breakers used a laser that was intended for more demanding ap-
plications (such as high speed PIV). The laser was too powerful (up to 99.9 percent
of the laser power was dumped), had a high pulse to pulse intensity variation, and
was large and cumbersome. Therefore, a simpler optical system for the in-line holo-
graphic drop measurements was designed and used to record drop data from the
moderate breaker.
25
Thin-film 
polarizer
Nd:YLF pulsed laser
Thin-film 
polarizer Spatial filter
Polarizer Polarizer (with pinhole)
High-speed camera lens f = 200mm
Beam dump Beam dump
K-2 microscopic lens
Acromatic 
Focal Plane converging lens Acromatic 
diverging lens
Measurement Region
(Wave Tank)
Figure 2.4: The optical setup used to generate a collimated laser beam
approximately 50 mm in diameter used to collect drop in-line holographic
data for the weak and strong plunging breakers. A high-energy Nd:YLF
laser (50 mJ/pulse) is used as the light source. The light coming out of
the laser is polarized and most of the energy from the pulse is dumped.
The light is then passed through a spatial filter, collimated, and ex-
panded through a series of two achromatic lenses (one converging one
diverging) are used to create the collimated laser beam. The beam is
then directed horizontally across the width of the tank (1.15 m) and into
a camera on the other side of the tank.
26
The new system, shown in Figure 2.5, uses a low-powered Nd:YLF laser (Crys-
talLaser, QL527-200, 80 ?J/pulse at repetition rates of 1 kHz, with a pulse width of
20-30 ns) as the illumination source. The beam is spatially filtered to improve beam
quality, passed through neutral density (ND) filters to reduce the intensity output
by approximately half, and collimated by a single 50.8 mm achromatic converging
lens, whose focal point is located at the pinhole. The collimation of the resulting
50 mm diameter laser beam is checked with a shear plate interferometer (Newport
Optics, model 20QS20). The collimated laser beam is projected horizontally across
the 1.15 m width of the wave tank.
The spatial filter consists of a pinhole, objective, and a position platform.
Selection of the optimal pinhole (and objective) is dependent on input beam param-
eters and is given by the equation:
fefl?
Dopt = , (2.5)
ain
where Dopt is the optimal pinhole diameter, f is the effective focal length of the
microscope objective, ? is the wavelength of the laser, and ain is the radius of the
input beam. A 20x objective (Newport Optics, model M-20X) with an effective
focal length f = 9 mm was chosen. The wavelength of the laser is ? = 527 nm
and the input beam radius is measured to be 0.77 mm. The input beam radius can
be measured by using the divergence angle of the beam (3 mrad) and the radius
of the beam at a known distance (0.54 mm in radius 25.4 cm away from laser).
Plugging these number into Equation 2.5, the optimal pinhole diameter was found
27
to be approximately 5 ?m.
After the beam passes through the spatial filter it has divergence angle given
by:
( D )in
? = tan?1 , (2.6)
2fefl
where ? is the divergence angle measured from the centerline of the beam, Din is the
diameter of the input beam, and fefl is the effective focal length of the microscope
objective used. Using Din = 1.54 mm and fefl = 9 mm the divergence angle is
found to be 9.71 deg. The resulting collimated laser beam diameter is calculated to
be 64 mm when using a 200 mm collimating lens.
The new optical setup is significantly better than the one used for the strong
and weak breakers because it uses a low-powered laser, which has a lower signal
to noise ratio, smaller pulse to pulse energy variation, and lower pulse duration.
Additionally, the low laser power results in a much simpler optical setup, as evident
by comparing Figures 2.4 and 2.5, and a more compact optical system that can
be easily moved to a new experimental setup. A comparison of typical recorded
holograms of drops using the old ((a), Winter 2012) and new ((b), Summer 2019)
collimated laser beam optical setups is shown in Figure 2.6.
2.3.1.3 Camera Setup Used to Record Holograms
On the opposite side of the tank from the light source, a high-speed digital
movie camera (Phantom V640, 4 Megapixel 12 bit images) fitted with a long-distance
28
High-speed Camera
K-2 microscopic lens Acromatic Spatial filter
Focal Plane converging lens (with pinhole)
Nd:YLF pulsed laser
Measurement Region
(Wave Tank) ND  Filter
Figure 2.5: The new and improved optical setup used to generate a
collimated laser beam approximately 50 mm in diameter that was used to
collect drop in-line holographic data for the moderate plunging breakers.
A low-powered Nd:YLF laser (50 ?J/pulse) is used as the light source.
The light is passed through a spatial filter and a neutral density filter,
and is then collimated by a single achromatic converging lens. The
collimated laser beam is then directed horizontally across the width of
the tank (1.15 m) and into a camera on the other side of the tank.
microscope lens (model K2 by Infinity, Inc.) is aligned so that the face of the sensor
and the optical axis of the lens is coincident with the collimated laser beam. The
laser pulse and camera are synchronized to take holographic image sequences at a
rate of 650 Hz with the bottom edge of the images located 12 mm above the max-
imum wave crest height. The horizontal surface 12 mm above the maximum wave
crest height is referred to as the measurement plane in this dissertation. Holographic
image sequences are taken at 28 streamwise locations above the breaking wave and
the results are interpolated to cover regions between adjacent locations where there
are no measurements. The streamwise drop measurement locations cover a stream-
wise region from just before the jet impact to approximately 1 meter downstream
from the (farther from the wave maker) jet impact site. At each location, at least
29
(a) (b)
Figure 2.6: Two sample images of drop holograms where (a) was recorded with the
hologram illumination system used for the weak and strong breakers collected in the
winter 2012, while the hologram in (b) was recorded with an improved illumination
system used for the moderate breaker. Holograms of drops show up as either black
dots surrounded by interference rings or just as interference rings. Both images are
approximately 25.6 mm x 16 mm with spatial resolution of 10 ?m per pixel and are
able to reconstruct drops with diameter of d > 50?m over the entire width of the
tank, 1150 mm. The red box is referenced in the next section and in Figure 2.7.
10 experimental runs were performed for a total of at least 140 unique realizations
of each breaking wave.
2.3.1.4 Hologram Image Processing
Holography is a three-dimensional (3D) imaging technique where interference
patterns formed by the interaction of a reference beam and light diffracted at object
boundaries are recorded by a camera. With recent advanced in digital cameras
and computing power, holography has been widely adapted to measure the 3D
position, tracks, and size of bubbles and drops Katz and Sheng [2010]. In the
present experiments a form of holography in which the reference beam and the
recorded image are parallel, called inline holography, is used.
The basic principles of hologram reconstruction is shown in Figure 2.7. The
image shown in Figure 2.7 (a) is a cropped image inside the red box in the hologram
30
shown in Figure 2.6 (a). In this cropped region, the holograms of three water drops,
labeled A, B, and C, generated by a breaking water wave, can be seen. The holo-
gram of each drop consists of a characteristic circular pattern caused interference
of the reference beam and diffracted light from the drop. This image can be digi-
tally reconstructed at any position in the depth direction, z, along the axis of the
collimated beam (where z = 0 is the location of the focal plane of the camera lens)
using a model for the propagation of diffracted light Katz and Sheng [2010]. When
the hologram is reconstructed at the actual location of the drop, the reconstructed
image of the drop is typically a solid black circle with a sharp edge. As the loca-
tion of the reconstruction is moved away from the z location of the drop, circular
interference patterns start to appear inside and around the edges of the drop.
Figure 2.7 (b) shows the recorded hologram in (a) reconstructed at a depth
of z = 133 mm away from the focal plane. In this image drop B becomes sharp
and in focus while drops A and C are not. Because B is sharp in Figure 2.7 (b),
it indicates that the plane of best focus for drop B is close to z = 133 mm, which
is also close to its physical position in depth. Similarly, Figure 2.7 (c) and (d)
show the hologram reconstructed at z = 268 mm and z = 331 mm away from the
focal plane, where drops C and A come into focus, respectively. Figure 2.7 (d)
specifically highlights the power of holography in measuring the size and position of
a drop. In the original recorded hologram, drop A appeared only as a faint, barely
distinguishable interference pattern, yet in Figure 2.7 (d) the same drop appears
sharp and in focus.
A recently developed GPU compatible MATLAB hologram reconstruction al-
31
gorithm, used in the image processing of holograms, was provided to us by Professor
Joseph Katz form Johns Hopkins University. We gratefully acknowledge Professor
Katz and his research group, because this reconstruction algorithm significantly
decreased the computational time required to reconstruct holograms.
The image processing and the determination of the plane of best focus code
for drop hologram processing was developed in-house and is loosely based on the
Hybrid method proposed by Guildenbecher et al. [2013]. The process involves the
following image processing steps for each hologram image:
1. Pre-process hologram image by subtracting background (Figure 2.8 (b)).
2. Reconstruct hologram at a coarsely spaced array in the depth direction, z,
over the entire width of the wave tank (Figure 2.8 (c)).
3. Collapse the reconstructed image array into a single 2D (x-y) image and detect
the drops by thresholding (Figure 2.8 (d)).
4. Reconstruct the holographic image in the depth direction, z, spaced every
500 ?m in the vicinity of each detected drop. (Figure 2.9 (a)).
? Calculate a mean sharpness criteria at each reconstructed plane (Figure
2.9 (b)).
5. Find plane of best focus by calculating a mean drop sharpness in the depth
direction, z (Figure 2.9 (c)).
6. Reconstruct the drop at the plane of best focus (Figure 2.9 (d)).
32
(a) (b)
(c) (d)
Figure 2.7: Original recorded hologram is shown in (a) (originally
cropped from the red box from the hologram in Figure 2.6 (a)). Three
distinct interference patterns, indicating that three drops are present,
can be seen, labeled by the letters A, B, and C. In (b), the original im-
age in (a) is reconstructed at a distance of z = 133 mm away from the
focal plane where drop B is in focus. Image (c) shows the raw hologram
in (a) reconstructed at z = 268 mm where drop C comes into focus.
Finally, image (d) shows the image in (a) reconstructed at z = 331 mm
where drop A comes into focus.
33
The results of the above hologram image processing steps are depicted in
Figures 2.8 and 2.9. Figure 2.8 (a) shows the original recorded hologram as it
was captured by the laser-optical-camera system (hence forth referred to as the
hologram). The first processing step is to homogenize the intensity distribution of
each hologram and remove any drops that may be on the tank wall or imperfections
from the manufacturing process of the tank?s plastic walls. This is accomplished by
calculating a time series (typically 50 images) averaged hologram from holograms
recorded before drops are generated by the breaking wave. The original hologram
is subtracted from the background averaged hologram and referred to herein as the
enhanced hologram, seen in Figure 2.8 (b). The only pronounced features in the
enhanced hologram are interference patterns of drops generated by the breaking
wave. Note that there are many interference patterns in Figure 2.8 (a) that do not
change in time and are filtered out by this pre-processing.
In step 2, the background-subtracted hologram, shown in Figure 2.8 (b), is
digitally reconstructed every 5 mm in the depth (z) direction. The 3D reconstructed
volume is collapsed into a single image containing the minimum intensity at each
x-y image location overall all the images in the depth direction, referred to as the
collapsed hologram and shown in Figure 2.8 (c). The approximate location and size
of a drop is measured by thresholding the collapsed hologram, shown in Figure 2.8
(d).
The steps outlined in the previous two paragraphs were solely for the purpose
of detecting the approximate 3D location and approximate size of drops in each
hologram image. The plane of best focus cannot be obtained accurately from the
34
(a) (b)
(c) (d)
Figure 2.8: Series of images showing the first stage of the reconstruction process,
where drop location and approximate radius is measured. The original recorded
hologram is shown in (a). A time-averaged background is subtracted from (a) and
is shown in (b). This background subtracted hologram is reconstructed every 5 mm
in depth and the resulting 3D reconstructed hologram volume is collapsed by taking
the minimum intensity of each pixel in depth. The resulting collapsed hologram is
shown in (c). Finally, the drops? location and approximate size are measured by
thresholding the collapsed hologram, with the result shown in (d).
35
coarse hologram reconstruction. In order to obtain an accurate measurement of the
plane of best focus for each drop, a small 200 by 200 pixel window around each
drop is cropped from the raw hologram, shown in 2.9 (a). The extracted hologram
is reconstructed using the above described algorithm, at an array z locations spaced
every 500 ?m around the estimated z location of the drop. A sharpness criteria,
based on the Tenengrad operator (see Guildenbecher et al. [2013]), is calculated
at each reconstructed plane, see Figure 2.9 (b). The idea behind calculating a
sharpness criteria at each plane stems from the idea that the drop image will be
sharpest when it is in focus. A mean value for the drop sharpness is calculated at
each reconstructed plane and a plot of mean drop sharpness vs. depth is obtained,
see for example Figure 2.9 (c). The mean sharpness has a peak at about 74 mm
from the lens focal plane for this particular drop, which indicates the plane of best
focus. When the hologram is reconstructed at this depth location the image appears
sharp and in focus, see 2.9 (d). This process is repeated for all detected drops in
each hologram.
2.3.1.5 Drop Radius Measurement
The drop radius is measured by fitting a hyperbolic tangent function by least-
squares linear regression to the inverse intensity map of the drop when it is recon-
structed in the plane of best focus. The fitting function has the following form:
[ ( )]
? = ?f 1? tanh ? ((x ? x )2f i,j f + (yi,j ? y 2f ) ? rf ) , (2.7)
36
(a) (b)
(c) (d)
Figure 2.9: Series of images showing the second stage of the reconstruction process,
in which the plane of best focus is calculated for each drop. The raw cropped
hologram of a drop is shown in (a). The sharpness criteria, based on the Tenegrad
operator, at a single plane is shown in (b). The mean drop sharpness vs. depth is
shown in (c) with a defined peak near the plane of best focus for the drop. Finally,
the reconstructed drop at the plane of best focus is shown in (d).
37
(a) (b)
Figure 2.10: The inverse of the intensity map of a 1000 micron dot from
the calibration reticle shown in Figure 2.11 (a) is shown on the left. The
image on the right shows the hyperbolic tangent function fit of the same
drop.
where ? is the fitted inverse intensity profile, ?f is the maximum value of the inverse
intensity of the drop, ?f is a parameter that characterizes the slope of the function
at the boundary between the intensity of the drop and the background, xf and yf
are the fitted x and y positions of the drop and rf is the fitted drop radius. The
initial guess for the parameters in Equation 2.7 used for least-square fitting comes
from the estimate of the position and radius obtained in the above-described coarse
analysis, and radius guess from hologram image processing described above. Figure
2.10 shows the inverse intensity map for a 1000 ?m circle from the calibration target
(see following paragraph) and the resulting hyperbolic tangent fit of the same drop.
2.3.1.6 In-line Holography Calibration
The in-line holographic system is calibrated with a custom calibration target
(referred to as the reticle) in order to ensure 1:1 magnification, determine measure-
38
ment region limits, and quantify drop radius measurement error. The reticle consists
of a glass slide with 14 chrome sputter deposited circles with diameters ranging from
3000 ?m to 30 ?m. In order to ensure 1:1 magnification, the calibration target is
placed in the focal plane of the K2 microscope lens (the focal plane is located near
the center plane of the tank width). The magnification on the microscope lens is
adjusted until it is close to 1:1 or 10 ?m/pixel. The recorded hologram of the reticle
(positioned near the focal plane of the lens) with the lens set to 1:1 magnification
is shown in Figure 2.11 (a).
The second purpose of the calibration reticle is to ensure that the smallest drop
considered in this dissertation (d = 100 ?m) can be measured at the extreme ends
of the measurement volume (near either side of the tank wall). In order perform this
verification, the calibration target is placed in the furthest in-tank position from the
focal plane of the microscopic lens, i.e. next to either tank wall. The hologram of
the reticle is recorded and reconstructed at the z location of the reticle to ensure
that the drop of d = 100 ?m can still be measured accurately. The imaged and
reconstructed holograms of the reticle near the tank wall are shown in Figure 2.11
(b) and (c). Note that in the reconstructed image the d = 100 ?m drop appears
sharp and in focus.
The final purpose of the calibration reticle is to quantify the drop radius mea-
surement error. In order to achieve this the reticle is attached to a motorized linear
traverser (NEAT 310M Programmable Stepping Motor Controller). The reticle is
traversed and imaged across the measurement volume at known positions away from
the focal plane. The recorded hologram at each known position from the focal plane
39
is reconstructed and the radius of each of the dots is measured, using the inverse
hyperbolic tangent fit described in the previous section. From this data the drop
radius measurement error is assessed. This procedure is also used to ensure the
expanded laser beam used by in-line holographic system is collimated. The mea-
surement error in drop radius is assessed to be no more than 3 percent of the drop
radius over the entire width of the tank.
2.3.2 Drop Tracking Code
A custom MATLAB drop tracking code was written in-house using ideas from
existing tracking algorithms and new ideas designed to overcome specific challenges
related to tracking the drops generated by the breaking waves. One drop tracking
challenge is due to the large variation of drop speeds in hologram movies. For
example, in a series of five images there may be drops that are reaching the peak
of their trajectories, thereby displacing only 100 pixels in the five frames, while
simultaneously there may be a drop traveling much faster in the same set of images
and displacing a total of 1600 pixels between the five frames. This variation in
speeds makes it difficult to track drops using traditional tracking algorithms like
nearest neighbor search.
Having determined the 3D location and size of all drops in each image for
each run, the trajectory of the drops, ?i(x, y, z, r, t), is determined using a drop
tracking algorithms. The tracking algorithm used in this study is composed of two
sub routines. The first subroutine of the algorithm is a nearest-neighbor search using
40
(a)
(b) (c)
Figure 2.11: A calibration reticle with 14 chrome sputter deposited cir-
cles ranging in size from d = 3000 ?m down to d = 30 ?m. Image (a)
shows the calibration target imaged by the digital in-line holographic sys-
tem at the focal point of the camera lens. Hologram (b) shows the same
calibration target imaged at the far sidewall of the tank (605 mm away
from the focal plane); interference patterns can be seen in the hologram.
Image (c) is the digital reconstruction of image (b) at z = 605 mm. Note
that the original image is 2560 x 1600 pixels in size and that the d =
100 ?m dot (seventh dot from the left in the bottom row) is visible and
in focus in the reconstructed hologram.
41
the drop?s 3D location and size. The second subroutine of the tracking algorithm
was developed in-house to uniquely solve the problem of the wide range of drop
velocities at each instant. The subroutine uses the tracking algorithm described
above to iteratively track drops of increasing speeds, starting with the slowest drops.
The drop tracking algorithm used in this study is based on a modified nearest-
neighbor algorithm developed for Brownian particle motion by Crocker and Grier
[1996]. The modified code tracks particles based on their 3D position and radius
between two consecutive frames by voting on the most likely track link from a set of
drop candidates in the next frame. The set of candidates is selected based on the n
drops in the next frame nearest to the drop being tracked from the previous frame,
henceforth referred to as the tracked drop (typically, n = 5). The distance between
the tracked and candidate drop is taken to be the 2D euclidean distance between the
two based on x and y positions. If a candidate drop is greater than dmax away from
the tracked drop, it is discarded from the candidate list. The remaining drops in the
candidate list are then ranked from least to most likely matching candidate based
on the relative x, y, z positions and radius r. Finally, the most likely candidate
is required to pass a set of constraints based on the difference in size between the
tracked and candidate drop, as well as the displacement in the z direction. Because
the measurement of drop position in the z direction is not as accurate as the x and
y position, this constraint is applied with loose tolerance.
In this tracking algorithm, linking drop positions into trajectories is only feasi-
ble if the typical drop displacement from frame to frame is smaller than the typical
drop spacing in each of the same two frames. In other words, if two consecutive
42
frames contain a dense population of drops, and some of these drops are displacing
by a distance greater than the typical drop spacing in one frame, then this tracking
algorithm will not be able to link the tracks of the fast moving drops, which have
the largest displacements from frame to frame. In order to address this issue an
iterative subroutine that tracks slow moving drops first and faster drops later was
developed in-house. The subroutine consists of the following steps:
1. Drops are initially tracked using the nearest neighbor algorithm described
above with a very conservative maximum drop linking distance, dmax.
2. Drop tracks with fewer than three drop links are unlinked.
3. For each trajectory (resulting from the initial tracking algorithm) that is longer
than three drop links:
(a) The drop trajectory is predicted back in time by fitting a first order
polynomial to the x and y position vs. time for the three nearest drop
positions in time.
(b) Any unlinked drop in the previous time step that is sufficiently close to
the x and y predicted positions (within a given distance threshold) are
assigned to the track existing track.
(c) If no new drop is assigned to the existing track, the previous step is
repeated for an additional previous time step (up to 7 times, then the
code stops looking for candidate drops)
(d) If a new drop is assigned to the track, the search is advanced one time
43
step back in time and step (a) is repeated with the a new set of three
nearest drop positions in time.
4. The drops search process above is repeated forward in time.
5. All drops tracks with three or more drop links are removed from the data set
of candidate drops for linking
6. The subroutine is repeated for the left over drops using a larger value for dmax.
At this point, all the drops with tracks are removed from further tracking con-
sideration and the drops tracking algorithm is used iteratively with ever increasing
maximum drops linking distance dmax until no drops are left to be tracked. The sub-
routine introduces physics of the pseudo-ballistic trajectories of the spray drops into
the drop tracking method since it tries to predict future and past drops positions
based on known information about the drop track.
The trajectory data for each drop is fitted to third order polynomials for
xd(t), yd(t) and zd(t). The polynomials are then used to find the time t0, x0 and
z0 position, and velocity of each drop as it crosses the measurement plane, which
is 1.2 cm above the maximum wave crest height. Only drops that are moving up
through the measurement plane are included in the final data set. Drops that are
moving downward at the bottom or top of the image space or through the side walls
of the image space are assumed to have been accounted for at other measurement
positions. In order to be tracked successfully, a drop must appear three times in
a sequence of images. This means that the maximum vertical velocity of a drop
44
Figure 2.12: A sample plot of drop trajectories from one run spanning
approximately 1300 images (2 seconds) with spatial dimensions of 25.6
by 16 mm. The relative drop diameters are indicated by the diameters
of the markers shown in the plot. The drop positions are not necessarily
equally spaced in time. Because holographic reconstructions of drops
near the exterior boundaries of the image are more inaccurate, only
drops 200 pixels (2 mm) inside the measurement window (the white
region outlined by solid and dashed red lines) are reconstructed and
their radius is measured. The measurement plane is indicated by a solid
red line 200 pixels above the bottom edge of the frame.
must be below 2.6 m/s. A sample plot of drop trajectories from one run is shown
in Figure 2.12.
2.3.3 Surface Profile Measurements
Surface profiles of the waves were measured with a cinematic Laser Induced
Fluorescence (LIF) technique that has been used extensively in the Hydrodynamics
Laboratory (see for example Duncan et al. [1999] and Wang et al. [2018]). The
45
method uses a thin (?1 mm in thickness) vertically oriented light sheet from a 7-
Watt Argon ion laser, with the majority of power coming from wavelengths of 488
and 524 nm. The laser light sheet illuminates the water in the longitudinal center
plane of the wave tank. The water is mixed with a low concentration of Fluorescein
dye which has strong excitation wavelengths of 475 and 490 nm and strong emission
wavelengths near 510 and 520 nm. When the laser light sheet from the Argon ion
laser (with peak power at ? = 488 nm, blue-green in color) illuminates the dyed
water, it fluoresces a neon-green color (? = 510? 520 nm).
The laser is located on the downstream end of the wave tank. Several mirrors
are used to reflect the laser beam so that it is positioned above and along the center
plane of the wave tank. The laser beam is directed from the downstream end to
the upstream end of the wave tank on top of the instrument carriage (discussed in
section 2.2.1). A mirror is used to direct the beam vertically downward from the
carriage and through two convex lenses, which are used to focus the laser beam on
the water surface. Finally, a mirror directs the beam at an angle of 30 deg up
from the horizontal to a rotating mirror that generates a thin laser light sheet that
intersects the water surface. The light sheet thickness at the calm water surface is
measured to be approximately 1 mm.
This LIF technique is used along with three time-synchronized high-speed
cameras with slightly overlapping fields of view to capture a time history of the
surface profile evolution during the active phase of wave breaking. Each high-speed
digital movie camera (Phantom V640 or V9) views the intersection of the light sheet
and free surface looking down at a shallow angle. Any residual laser light is filtered
46
(a)
(b)
(c)
Figure 2.13: A sample sequence of LIF images of a breaking wave. The
wave is moving from right to left with time advancing from image (a)
to (c). The free surface is initially smooth in image (a), before the wave
has broken. In image (b) the wave is breaking and surface roughness can
be seen in the middle of the image. The part of the wave closest to the
camera can be seen in the foreground of image (b). Finally, in image (c)
the wave has passed. The surface profile in (c) looks less uniform due
to a number of reasons, such as free surface roughness (causing water to
act like a lens) or free surface features in the foreground of the image
between the camera and lens.
47
(a) (b) (c)
Figure 2.14: Three time-synchronized surface profile images just after jet impact and
at the onset of the initial splash up. The three cameras have slightly overlapping
fields of view and the wave is moving right to left.
by a long-wavelength-pass optical filter in front of the camera lens. A sample series
of images from a breaking wave are shown in Figure 2.13. The cameras view the
intersection of the light sheet and the water surface from a point just above the
water level and off to the side of the tank. Movies are recorded at 650 images per
second and the cameras have a sensor size of 2560 x 1600, with a combined field of
view that covers approximately 1300 cm in the horizontal direction and 300 cm in
the vertical direction with a spatial resolution of approximately 180 ?m/pixel. The
three cameras cover the plunging jet formation, jet impact, and subsequent splash
up, capturing a total of 2000 images with a recording time of just over 3 seconds.
Sample images recorded by the three cameras are shown in Figure 2.14.
The shape of the water surface at any instant is determined from the images by
edge detection techniques implemented in MATLAB. The water surface is located
to an accuracy of about ?1 pixel or approximately ?180 ?m. Each set of three LIF
images is captured simultaneously with the two holographic images. It should be
kept in mind that the LIF image is captured over an interval of about 1/650 s while
the holographic image is captured over the duration of the Nd:YLF laser light pulse,
48
which is on the order of 100 to 20 ns. Also, the drops are measured over the entire
width of the tank while the water surface profile is measured over the thickness of
the light sheet, about 1 mm, in the center-plane of the tank.
The image processing to determine the surface profile shape along the inter-
section of the laser light sheet and the water surface is done by a combination of
automatic and manual image processing techniques based on gradient edge detec-
tion. First, an automatic edge detection MATLAB code is used to measure the
surface profile shape from one side of the image to the other. If a surface profile is
not detected from one side of the image to the other by the automatic processing
code it is tagged and processed manually with the use of a touch sensitive tablet-pen
system (Wacom, model Cintiq 21ux).
Once the surface profiles are processed, they need to be rectified and calibrated
since the optical axes of the camera lenses are not perpendicular to the plane of the
light sheet. Additionally, a camera calibration process can correct for any spherical
abberation from the camera lens, convert the profiles from image coordinates to
physical coordinates, and allow the surface profiles from three adjacent cameras to be
stitched together. The calibration process is accomplished by placing a calibration
grid, which spans the combined width of the fields of view covered by the three
cameras, in the plane of the laser light sheet. The calibration board consists of a
checkerboard pattern with squares that are 25.4 mm x 25.4 mm attached rigidly to
an aluminum frame. The calibration process is carried out after the surface profile
experiments were finished. In total, the experiments and calibration process were
conducted over three days.
49
(a) (b) (c)
Figure 2.15: Images of the same calibration checkerboard target from the three
cameras with slightly overlapping fields of view of view. The image points, (Xi, Yi),
are detecetd by an image processing code and used to convert from image coordinates
to physical coordinates and transform the points to an orthogonal coordinate system.
Image (a) is the left most camera, (b) is the middle camera, and (c) is the right
camera, with the direction of the wave propagating from right to left. The yellow
points seen in (a-c) are shared common points where the camera fields of view
overlap, which are used to stitch the images together.
The calibration process for each individual camera is similar to the one de-
scribed by Wang [2017]. Image processing is used to locate the intersection points
between four squares on the checkerboard; these intersection points make up the set
of points referred to as the image points (Xi, Yi). Because the coordinates of these
points in physical space is known (Xp, Yp), two quadvariate quartic functions can
be fitted to obtain Xp and Yp from a set of two image points. This method is used
to convert the images and surface profiles from pixel to physical coordinates, rectify
the images to an orthogonal coordinate system, and correct for any spherical ab-
berations. A sample of the detected image points for the three cameras is shown in
Figure 2.15 and the rectified images resulting from this calibration processes shown
in Figure 2.16 (a).
As mentioned previously, the three cameras have slightly overlapping fields of
view. The image points shared between each pair of side-by-side cameras is shown
in yellow in Figure 2.15. Using the calibration process described in the previous
paragraph, the shared image points can be used to stitch the three camera fields of
50
(a)
(b)
Figure 2.16: Stitched images from the three camera fields of view, covering a span-
wise distance approximately 1300 cm at a spatial resolution of 180 ?m/pixel, of the
calibration board shown in Figure 2.15 and surface profile images shown in Figure
2.14.
view together and convert the stitched image to physical coordinates. Figure 2.16
shows stitched images of the calibration board shown in Figure 2.15 and surface
profile images shown in Figure 2.14. This calibration and stitching process can
be applied to both surface profiles (lines) or images. It should be noted that the
stitching process does not always produce a smooth transition from one image or
surface profile to another. This is especially noticeable when the free surface becomes
rough and 3D structures appear in the foreground. In this case, since the two
adjacent cameras have slightly different perspectives, the measured surface profile
can create discontinuities in the detected stitched surface profile due to different
line of sight blocking in the camera?s field of view. Another reason could be errors
in making the calibration board parallel to the laser lightsheet, thereby causing a
mismatch between image and physical coordinates for adjacent cameras.
51
2.3.4 Humidity and Drop Evaporation
The relative humidity (measured approximately 25 cm from the water surface)
and the ambient temperature in the lab were measured in order to ensure that the
smallest measurable drop (d = 100 ?m) does not evaporate by a significant amount
by the time it reaches the measurement plane. The humidity and temperature
measurements were used as input into a drop evaporation model to ensure that drop
radius remains relatively constant. It was found that the humidity in the air over
the water was no lower than 55 %, with an ambient temperature of approximately
20 ?C ? 1 ?C.
These measurements were used as input into the drop evaporation model for
fresh water that was presented in Andreas [1989]. Figure 2.17 shows the normalized
drop radius vs. time on a semi-log plot for three drops of different initial diameter.
The red curve is for a drop of d = 100 ?m, blue for d = 200 ?m, and green d =
400 ?m. The plot shows that after 10 seconds, the smallest drop, with an initial
diameter of d = 100 ?m, decreases about 0.05 percent of its original radius due to
evaporation. Therefore it was deemed that the smallest measurable drop, which
have a typical speed of 0.65 m/s and reach the measurement plane in 50-90 ms, do
not evaporate significantly by the time they reach the measurement plane. Table
2.2 shows the physical parameter inputs used to simulate the drop evaporation.
52
Table 2.2: List of parameters used as input to the model by Andreas [1989], which
simulates drop evaporation.
Parameter Value Unit
Water Temperature ?C 18
Air Temperature ?C 20
Atmospheric Pressure hPa 1013.25
Relative Humidity - 50
Molecular Weight of Water kg/mol 18.0160e-3
Molecular Weight of Air kg/mol 28.9644e-3
Specific Heat of Air (const. pressure) J/kg/?C 1.006e3
1
0.9999
0.9998
0.9997
0.9996
0.9995
-2 -1 0 1
10 10 10 10
Figure 2.17: Evaporation model for d = 100 micron (red), d = 200 micron
(blue), and d = 400 micron (green) for fresh water and 50 percent relative
humidity.
53
2.3.5 Surface Tension Measurements
The static surface tension for the weak and strong plunging breakers were
originally measured by Dan Wang in 2012. The static surface tension measurement
process is reported in her proposal Wang [2012], but is briefly repeated in this section
for completeness. The static surface tension for the weak and strong experiments
was measured using a Wilhelmy Plate Tensiometer from NIMA with a 3 L water
sample taken from the tank just before the experiment was conducted. Before each
measurement the beaker is cleaned and filled with water from the tank. The beaker
is skimmed for about 15 minutes by continuously overflowing the beaker with water
siphoned from the tank. Just before the measurement, the siphon is stopped and
the static surface tension is measured for about 30 minutes.
For the moderate breaker the surface tension isotherm was measured using
a Langmuir trough (KSV Nima). Just after the first and last run of each day, a
sample of water is collected from the wave tank in a 500 mL beaker and transferred
into the Teflon trough. The Teflon trough and barriers were previously thoroughly
cleaned with soap and tap water and rinsed with tap water. As a final step to the
cleaning process, the trough and barriers are rinsed in the wave tank just before
the surface tension isotherm is measured. The surface tension is measured by a
platinum Wilhelmy plate that makes contact with the water sample (forming a
meniscus) and is suspended on an electronic balance that measures the downward
pull (surface tension) of the water sample. The isotherm is measured when two
Teflon barriers compress the water surface, thereby causing any surface surfactants
54
in the water sample to be compressed, in turn lowering the surface tension. A
sample plot of seven different surface tension vs. surface compression measurements
is shown in Figure 2.18.
The red curves in Figure 2.18 shown the surface tension isotherm for tank
water that was not properly filtered (hence forth referred to dirty water) and the
blue curves are for properly filtered tank water (referred to as clean water). The
static surface tension (the surface tension at 0% compression) for both samples is
close to the surface tension of clean water. However, as the surface is compressed,
the surface tension for the red curves decreases significantly, up to a 16 mN/m
difference, at about 70% compression while the blue curves change at most 1 mN/m
at 90% compression. The blue surface tension isotherms are from water samples
collected from the wave tank after the water filtration systems (incoming water
filters and diatomaceous earth filter) were cleaned.
It was found that certain wave features like wave crest speed, jet impact speed,
and jet impact location, for breakers generated in dirty water were not consistent
from run to run. Therefore, having relatively clean water was deemed important to
experimental repeatability. For the droplet measurements for the moderate breaker
and the profile measurements for the three waves, the water quality was maintained
as close as possible to the blue curves in Figure 2.18. Although the surface ten-
sion isotherms were not measured for the droplet measurements in the weak and
strong breakers, the water quality and filtration systems were maintained in a way
that?s consistent with the clean water quality obtained for the moderate breaker.
Additionally, wave features like wave crest speed, jet impact speed, and jet impact
55
location, from the weak and strong breakers are consistent from run to run and con-
sistent with experiments of the same waves conducted just before the experiments
for the moderate breaker. Therefore, it was deemed that the water surface tension
quality between the two experiments was similar.
56
76
74
72
70
68
66
64
62
60
58
56
0 20 40 60 80 100
Figure 2.18: Surface tension isotherms of wave tank water from seven
measurements from different days. The five red curves show the surface
tension isotherms that was deemed dirty because of a significant drop
in surface tension after about 70% compression. The blue curves show
the water quality in the same wave tank after the filtration system was
cleaned. The quality of the water was deemed important for the repeata-
bility of the experiment. Therefore, the surface tension isotherm for the
water used in the moderate breaker droplet measurements and all profile
measurements was maintained as close as possible to the blue curves.
57
Chapter 3: Results and Discussion
In this chapter, experimental results of spray produced by mechanically gener-
ated breaking waves in laboratory scaled experiments are presented. Simultaneous
spatially and temporally resolved measurements of the breaker crest profile evolu-
tion and similarly resolved measurements of drops ranging in radius down to 50
?m are reported and discussed for plunging breakers of weak, moderate, and strong
intensities.
In the first section of this chapter (section 3.1), spray generation mechanisms
by the weak plunging breaker are reported in great detail and linked to wave breaking
sub-processes in three distinct spatial and temporal zones of drop production. In
the second section (section 3.2.2), drop statistics and mean breaker profiles are
compared for the three breakers. The mean surface profiles are found to be highly
repeatable and share common features between the three breakers. The relative
importance of the three spray generation mechanisms identified in section 3.1 are
reported for all three breakers. The droplet probability distribution for the three
waves follows a separate power law scaling for large and small drops and varies with
breaker intensity.
58
3.1 Spray Generation Mechanisms by a Plunging Breaking Wave
In this subsection, breaking profile and drop production measurements for
the weak breaker are presented and discussed in detail. The measurements are
taken simultaneously during many repeated breaking events, and the time, location,
and drop characteristics of the various drop generation events during the breaking
process are identified from the data. Some of the work presented in this section
was published (in a slightly modified form) in Geophysical Research Letters [Erinin
et al., 2019]. Figures and text from the paper are cited where appropriate.
During a breaking event, drops are generated by small-scale interfacial mo-
tions with length scales on the order of the typical drop size. These motions are
in turn generated by the large-scale motions of breaking, for example jet impact,
with length scales on the order of the wave height. The waves studied in this re-
search are highly repeatable and nearly two-dimensional up until the instant that
the plunging jet collides with the front face of the wave. After this point in time,
the flow becomes turbulent and, while the large-scale features found in the profile
evolution averaged over an ensemble of repeated breaking events is repeatable and
probably two-dimensional, the small-scale features of the flow are three dimensional
and vary significantly from run to run. The interfacial features produced during the
breaking process that are relevant to drop generation are discussed in the following
paragraphs.
The evolution of the breaking process and the small- and large-scale motions
are exhibited in the water surface profile sequence and LIF images presented in
59
Figure 3.1(a) and (b), respectively. The profiles and images were taken from the
same high-speed LIF movie and the bold profiles in (a) were obtained from the
corresponding images in (b). The images show some drops; however, it is important
to keep in mind that for a drops to be visible as a bright spot in the LIF images, it
must be relatively large and be positioned within the 1-mm-thick laser light sheet.
Instead, drops are measured using in-line holography, which can measure drops with
positions across the entire width of the tank (1.15 m wide) and with radii down to
50 ?m. Thus, the number of drops seen in the typical holographic image is much
larger than the number visible in the 1-mm-thick laser light sheet region of the LIF
images coincident with the hologram location.
The surface profiles are first used to characterize the breaker through the
measurements of the geometry and speeds of the crest region during jet formation
and through the measurements of plunging jet speed and geometry during the time
just before jet impact. One significant time during this incipient breaking period
is the moment when the plunging jet begins to form. At this point in time, a
small region near the crest on the front face of the wave becomes vertical. When
this condition is reached, the measured wave crest speed (the horizontal speed of the
highest point on the crest region) is 1.37 m/s and the the maximum wave crest height
is 10.28 cm above the still water level. A second significant time is the moment just
before the plunging jet collides with the front face of the wave. At this moment, the
wave crest speed has dropped to 1.29 m/s and the maximum wave crest height has
risen to 12.2 cm above the free surface. Also, the jet tip speed is 1.63 m/s just before
impact and the location of the jet tip impact point is 4.3 cm horizontally ahead and
60
5.67 cm vertically down from the wave crest point. Qualitatively speaking, this is
a weak energetic plunging breaker since the jet impact point is only about half the
distance vertically down from the crest to the still water level. Regarding the small
decrease in crest speed while the jet is plunging, it is important to keep in mind that
the crest region has a relatively large radius of curvature and is constantly changing
in shape. These factors make it difficult to define, measure, and interpret the phase
speed.
The sequences of profiles and images in Figure 3.1 (a) and (b), respectively,
depict the breaking event as follows. Image (i) was taken a short time after the
plunging jet collided with the water surface upstream of the crest and a region of
splash has formed upstream (to the left) of the jet impact point. During the period
between the plunging jet impact (the first profile in (a)) and the first bolded profile
corresponding to image (i)) the leading edge of the splash region advances with
a speed of 1.86 m/s, as estimated from the profile sequence. At about the time
between images (i) and (ii), the speed of the leading edge of the first splash slows
down to 1.07 m/s. As the splash region continues to evolve, the sharp indent at the
location of the boundary between the upper surface of the jet and the back surface
of the splash remains distinct. In images taken with the camera placed to the right
of the camera position for the images shown in Figure 3.1(b), one can see that this
indent is actually a narrow crater that extends below the apparent free surface seen
in images (ii) and (iii). In Figure 3.1(a), the location of the point of maximum
height on each profile is marked by a red triangle. This point is initially the wave
crest, but later transfers to the first splash. The speed of the highest profile point is
61
0.944 m/s when it is on the wave crest during the jet plunging process but is 0.896
m/s when it is on the first splash. Between the images (ii) and (iii), the leading edge
of the splash first slows down and then speeds up. During the speed up period, a
second splash springs forward and between images (iii) and (iv), the highest point
on the profiles transfers to this second splash. The motion of the highest point
among the various features of the breaking event is consistent with the idea of the
wave continuing to propagate through the breaker-generated turbulent vortical flow
field. This patch of turbulent flow moves with the surrounding water flow field,
which typically has speeds on the order of the wave slope times the phase speed.
Following the indent between the upper surface of the plunging jet and the
back face of the first splash, one can see an eruption at about the time of image (iii).
By the time of image (iv), many ligaments and drops appear to issue from this point
in the images. These processes are visible in the surface profile plot as well. This
eruption involves the appearance of large bubbles and the rupture of their surface
films. The process of the crater collapse and film popping is, of course, related to the
canonical experiments of single bubbles popping at the surface of a quiescent pool
but is much more complicated due to the non-axisymmetric geometry and three-
dimensional turbulent free surface flow that accompanies the process [Erinin et al.,
2019].
A primary statistic indicating the drop production during the above-described
breaking process is N(x, t), the number of drops moving up across the measurement
plane per unit time (?t) per unit streamwise distance (?x) per meter of crest
length per breaking event. This function is measured experimentally and shown as
62
(a) (b)
(vi)
(v)
(iv)
(iii)
(ii)
(i)
Figure 3.1: Water surface profiles (a) and images (b) from an LIF movie of the
breaker used in this study. Each profile in (a) is obtained from one image in the
LIF movie. The movie was taken at 650 pps (?t = 0.00154 s) but only every fourth
profile (?t = 0.0062) is shown in (a). Each profile is plotted 8 mm above the
previous profile for clarity. The first profile in the sequence was recorded 0.0077 s
after the plunging jet hit the front face of the wave, t = 0 s, and the last profile was
taken at t = 0.6154 s. The red filled triangles mark the position of the highest point
on each profile. The sequence of bold profiles in (a) corresponds to the sequence of
images in (b). This figure was originally published in Erinin et al. [2019].
63
a contour plot on an x-t plane in Figure 3.2. Interpolating the drop data spatially
and temporally from at least ten experimental runs at each measurement location,
it is estimated that 653 drops per breaking event are produced. There are two large
regions of drop production, measuring about 600 mm long and 0.3 s high, (labeled
region I and region II). Since the measurement plane is 1.2 cm above the maximum
wave crest height, there is a time delay between the moment that a drops is generated
and when it is measured. Using the synchronized drop and profile measurements
along with a simple drag model for a solid sphere moving in stationary air [Cheng,
2009], the average time of flight between the surface and the measurement plane is
estimated to be between 40 and 70 ms. The effect of the air flow induced by the
breaker is estimated to be large (particularly for the smaller drops), but, since air
flow measurements were not attempted, this effect is not included in the model at
this time. The drop radius decrease due to evaporation during the time of flight
was estimated via the theory of H.R. Pruppacher [1978]. Using a relative humidity
of 50% (estimated as a minimum value from later measurements), it was found that
the decrease in drop diameter would be less than 4% after one second for a drop
with an initial radius of r = 50?m, and therefore insignificant for the 70-ms flight
time found here.
In region I of the contour plot of N(x, t) in Figure 3.2, there are three promi-
nent local maxima. The first, which is the most intense, is located in the vicinity
of the point (x, t) = (140 mm, 0.18 s). By examining the corresponding region
below profile (ii) in the profile history plotted in Figure 3.1 (a), it can be seen
that this region of high drop number production is associated with the initial jet
64
impact and resulting splash. The second local maximum occurs at approximately
(x, t) = (325 mm, 0.275 s) and this point is in the region of the second jet impact
and splash, roughly along profile (iii) in Figure 3.1 (a). The peak value of N at
this location is about 46 percent of that at the first local maximum. The third
local maximum occurs in the vicinity of (x, t) =(275 mm, 0.38 s) and is associated
with closing of the impact crater and the sudden eruption of large air bubbles, be-
tween profiles (iii) & (iv). The large bubbles that come to the surface and pop
are probably from the air entrapped under the plunging jet during impact and as
the impact crater closes. The drops generated in Region II occur after the main
breaking process has subsided. From the LIF profile movies it appears that these
drops are generated by small bubbles that rise to the surface and eventually pop
[Erinin et al., 2019].
In order to analyze the drop generation data further, the measured function
N(x, t) was integrated over all x to obtain the number of drops generated per unit
time per unit crest length, Nx(t), and integrated over all t to obtain the number of
drops generated per unit x per unit crest length, Nt(x):
? 1.05 m
Nx(t) = ? N(x, t)dx, (3.1)02.0 s
Nt(x) = N(x, t)dt. (3.2)
0
A plot of Nx(t) is presented in Figure 3.3(a) and to the right of the contour plot in
Figure 3.2. This function has three local maxima, found at approximately t = 0.18,
65
1.5
1
0.5
0
2
N(t,x)
0 2 4 6 8
Region II
1.5
1
0.5
Region I
0
1000 800 600 400 200 0 0 1 2
Streamwise position above wave (mm) Nx(t) 10
-3
Figure 3.2: Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest length, plotted
on an x-t plane. The black line is drawn with a slope of 1.7 m/s. The bolded
free surface profiles and images in Figure 3.1 (a & b) correspond to the red lines
on the lower right corner of the contour plot. The magenta line corresponds to
the maximum wave crest height from Figure 3.1. The plot on the right is Nx, the
number of drops generated per unit time per crest length, vs time. The plot on top
is Nt, the number of drops generated per mm in the streamwise direction per unit
crest length, vs steamwise position. This figure was originally published in Erinin
et al. [2019].
66
Time (s) Nt(x)
0.38 and 1.40 s after initial jet impact. The three LIF images in Figure 3.3 of the
breaker in each of the columns, (A), (D) and (G), were taken around the times of
these local maxima, respectively. From the images, one can see that the first local
maximum occurs at the time of the growth of the first splash ahead of the plunging
jet impact point. The number of drops produced in the first peak accounts for about
22% of the total number of drops measured. The second local maximum occurs at
about the time when the top surface of the jet collides with the back surface of
the first splash and results in the impact-crater eruption (from 0.25 < t ? 1.00s),
this region accounts for 44% of total drops. Finally, the third maximum occurs at
later time when the surface is relatively free of breaking crest features and entrained
bubbles appear to come to the surface and pop. The number of drops produced in
the third peak accounts for 34% of the total drops.
The plot of Nt(x) in Figure 3.3(b) and on top of the contour plot in Figure 3.2,
has a simpler structure with one maximum at about x = 0.2 m. The position
of the maximum corresponds to the general location of the splash and the large
bubble eruption along the impact crater. The fact that Nt(x) goes to zero on both
ends of the plot is consistent with the observation during the experiments that no
drops reach the measurement plane upstream or downstream of this 1.0-meter-long
measurement zone.
Figure 3.4 shows the mean drop speed vs. drop radius. The speed of each
drop is measured from the x and y velocity components, u(y) and v(y), as the drop
passes through the measurement plane. The components u and v are obtained by
?
fitting a second order polynomial to drop tracks where the speed s = u2 + v2. In
67
(a) (b)
1.4
1800
1600 1.2
1400
1
1200
0.8
1000
800 0.6
600
0.4
400
0.2
200
0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1000 900 800 700 600 500 400 300 200 100 0
Time (s) Streamwise position above wave (mm)
Figure 3.3: (a). Plot of the number of drops generated per breaking event (Nx(t))
per time interval ?t = 0.025 s, per tank width ?y = 1.05 m. (b). Plot of the number
of drops generated per breaking event (Nt(x)) per distance interval ?x = 21.6 mm,
per tank width ?y = 1.05 m. These two plots are the data in Figure 3.2 integrated
over x and t, respectively to yield a total of 539 drops per breaking event. (c).
Images (a-c) show the free surface from t = 0.1 s to 0.3 s with ?t = 0.1 s between
each image. Images (d-f) show the free surface from t = 0.334 s to 0.435 s with ?t
= 0.051 s between successive images. Images (g-i) show traces of bubbles on the
free surface from t = 1.5 s to 1.7 s with ?t = 0.1 s between successive images.
68
Nx(t) per time interval per crest length
Nt(x) per distance per tank width
2
1.8
1.6
1.4
1.2
1
102
0.8
0.6 101
0.4
100
0.2
101 102
0
0 200 400 600 800 1000 1200
Figure 3.4: Mean drop speed as a function of drop radius plotted on a logarithmically
spaced bin ranging from r = 50 to 1500 ?m with a total of 32 bins. The mean drop
speed is calculated from the x and y velocity components as the drop passes through
the measurement plane.
the plot, a logarithmic spacing for dr is used with a total of 32 bins ranging from
r = 50 to 1500 ?m.
It was found that the cross stream velocity component, v, was typically v ? 15
percent of the drop speed. The mean drop speed increases from about s = 0.8 m/s
for r = 50 ?m to about 1.4 m/s at r = 250 ?m; from there to higher radii the local
average remains fairly constant. The increased variation in mean speed at larger
radii is at least partly due the small number of large drops, as mentioned previously.
The average speed of all drops is 1.07 m/s.
Figure 3.5 shows the distribution of drop trajectory angles (?) as measured in
69
the streamwise-vertical (x? y) plane when a drop passes through the measurement
plane. In the following, an angle of 0? indicates a drop is moving horizontally
downstream, while an angle of 90? indicates a drop is moving vertically up. Subplot
(a) contains the data for all drops measured during the entire breaking event. The
mean drop angle is 73.9?, i.e., approximately 26? downstream from vertical, with a
standard deviation of 36.4?.
Subplot (b) contains data from drops passing through the measurement plane
in the time range 0.325 < t < 0.5 s and the position range 0 < x < 0.33 m. This
region is intended to capture the drops that are generated by the collapse of the
crater between the back face of the first splash and the upper surface of the plunging
jet. The average velocity is essentially vertical, ? = 87.9? with a standard deviation
of 41.3?.
The drops considered in subplot (c) are the difference between the drops mea-
sured from all streamwise positions during the period from 0 < t < 0.8 s and the
drops from 0 < x < 0.33 m and 0.325 < t < 0.5 s. This region is intended to capture
the drops generated by the turbulent region generated by the plunging jet impact.
The average drop velocity was 23.7? downstream from vertical, ? = 76.3?, with a
standard deviation of 39?.
Finally, subplot (d) includes drops from all streamwise positions appearing
after t > 0.8 s. This region is intended to capture the drops resulting from the
third peak in the drop production curve shown in figure 3.3(a), which is believed to
result from late time bubble popping at the free surface. The drop velocities in this
region have a mean angle of 71.7? with a standard deviation of 29.6?. It was initially
70
Figure 3.5: Histograms of drop angles (?, from 0 to ?, where 0 is hori-
zontal in the direction of wave motion) for all drops at various stages dur-
ing the breaking processes as they pass through the plane 1.2 cm above the
maximum wave crest height. (a) shows the distribution of drop angles for
the whole measurement region. (b) show the distribution of drop angles from
0 ? x ? 12.5 cm, 0.325 ? t ? 0.5 s. (c) show drop angles from 0 ? t ? 0.8s
excluding drops from 0 ? x ? 12.5 cm, 0.325 ? t ? 0.5 s. Finally, (d) shows
all drop angles after t > 0.8s.
thought that these drop velocities would be evenly distributed about vertical since
their source appears to be small popping bubbles in the latter phase of breaking
where the water surface is much smoother than in the earlier phases. The measured
angle indicates that motion of the free surface and/or the underling flow must play
a role in determining this angle.
Further identification of the breaker features associated with drop production
is provided by the plots in Figure 3.6. The plot in subfigure (a) shows the profile
of the breaker crest as it evolves in time (the same data shown in Figure 3.1) with
71
colored dots plotted at the measured position and time when a drop crosses the
measurement plane. It should be kept in mind that these drops were measured
over a total of 160 unique realizations of the breaker, while the profiles are from
a single run, i.e., a single breaking event. The vertical bands with no drops in
Figure 3.6(a) appear at locations where drop data was not collected. The plotted
drops in subfigure (a) are concentrated in two regions. The first is near the rough
region created by the jet impact, advancing from the right to the left and running
diagonally on the plot. The second concentration of drops is located in the vicinity
of the impact of the back face of the first splash with the upper surface of the
plunging jet. As discussed above, drop velocities are primarily up and with a smaller
component in the upstream or downstream direction. Thus, the positions and times
of the generation of the drops are not those marked in Figure 3.6(a). The distance
and direction of the drop generation point on the water surface relative to the marked
position of each dot on the plot should vary from drop to drop.
In an effort to get an approximate x?t position of the generation point of each
drop, a model of the drop motion was used along with the measured drop velocities
and positions at the measurement plane. In this model, the forces of gravity and
drag relative to still air are considered and the equations of motion are:
( )?( )2 ( )
d2
2
Xd 1 dXp
md = Cd ?a ? ? ?
dXp ?dYp+ (3.3)
dt2 2 ( dt ) ( dt ) ( dt)
d2
2 2
Yd 1 ?dYp ?dXp dYpmd = Cd ?a + ? +mdg (3.4)
dt2 2 dt dt dt
72
(a) (b)
Figure 3.6: Profile history of a breaking event with drop generation locations in
time and space. In (a), the dots are located at the x and t where the drops passed
through the horizontal plane located 1.2 cm above the highest point reached by
the free surface. In (b), the dots are located at the x and t where the drops are
calculated to have been generated. This calculation is performed with the data on
the drop motion while it is in the measurement window and a ballistic trajectory
model for the drop motion. ? 800 ? d, ? 300 ? d < 800, ? 200 ? d < 300, ? 125
? d < 200, ? d < 125 ?m.
73
where
?
d u2 2
Re = rel
+ vrel , (3.5)
?a
24 [ ]
Cd = (1 + 0.27Re)
0.43 + 0.47 1? exp(?0.04Re0.38) , (3.6)
Re
urel and vrel are x and y components, respectively, of drop velocity relative to the
air, d is the drop diameter, ?a is the kinematic viscosity of air, Re is the Reynolds
number of the drop as seen in equation 3.5, Cd is the coefficient of drag (in this case
given by Cheng [2009] for a solid spherical particle), ?a is the density of the air,
md is the mass of the drop, and g is the gravitational constant. The assumption of
stationary air was used because there were no measurements or calculations of this
flow field. This approximation probably causes larger errors for the smaller drops.
In the calculations, the drop position is simulated backward in time until it hits
the temporally and spatially varying wave profile, which we point out again was
measured in a single breaking event and at the center plane of the tank. In spite of
the inaccuracies, it seems that the presentation of the results at this stage may be
useful. The results are shown in Figure 3.6(b) where the water surface profiles in
(a) are repeated but the drop positions are as computed by the backward tracking
model. From these simple simulations, it appears that drops are primarily generated
from the turbulent region created by the jet impact and from the impact between
the back face of the first splash and the top surface of the plunging jet.
The turbulent front region, created by the jet impact and propagating from
74
right to left, seems to be a major source of drop production since many of the sim-
ulated drops project back to the rough surface features created by the jet impact.
Meanwhile, the cloud of drops around the impact of the turbulent front region and
back face of the wave in subfigure (a) seems to be more concentrated towards the
rough surface features when simulated back in time, shown in subfigure (b). As
noted above, there are locations in Figure 3.6(a) where drop data was not collected
(indicated by the vertical bands where no drops appear). These empty regions un-
doubtedly have an influence the appearance of Figure 3.6(b). This may be corrected
in future work.
The normalized distribution of drop radii for all drops that pass through the
measurement plane during a breaking event is shown on a log-log plot in Figure
3.7. The probability distribution was calculated from the data obtained from 10
runs at each of the 28 streamwise measurement locations. In order to obtain this
distribution, radius probability distributions are first generated at all 28 measure-
ment locations using the same set of logarithmically spaced radius bins ranging from
r = 50 ?m to r = 1400 ?m with 26 bins. The number of observed drops in each bin
at each location is divided by the number of runs, the width of the bin in microns,
the streamwise width of the holographic camera?s field of view, and the width of
the tank (1.15 m) to obtain the number of drops generated per breaking event per
micron of radius per meter of crest length versus streamwise position, x. This func-
tion for each bin is then numerically integrated using the trapezoidal rule over the
range of x for which measurements were taken. In this way, data at locations where
no measurements were taken are approximated by linear interpolation. The verti-
75
102
 = -2.3
101
100
102  = -4.9
 = -2.0
10-1 101 I
100  = -2.5  = -4.8II I
10-1  = -5.0
-2 II10
10-2
2 3
10-3
10 10
102 103
Radius (r) ( m)
Figure 3.7: Drop radius probability distributions. Vertical error bars show ?1 stan-
dard deviation, while horizontal error bars show radius measurement error obtained
from a calibration process. The main plot is for all of the drops produced by the
breaker while the sub plot contains separate data sets for the drops produced during
the jet impact phase (Region I, 0 < t < 0.8 s, ?) and during the bubble ascend and
popping phase (Region II, 0.8 < t < 2 s, ?). A Hinze-scale-like break in slope is
identified in all three data sets. The slopes of the distributions for the larger and
smaller bubbles in each data set are identified by ? and ?, respectively. For all the
drops, break at r = 273 ?m. This figure was originally published in Erinin et al.
[2019].
76
n(r) (per m radius increment per m crest width)
cal error bars show ?1 standard deviation, while horizontal error bars show radius
measurement error obtained from a calibration process. The four largest radius bins
do not have error bars because only one drop was measured in each bin.
This calculation yields an estimate for the total number drops observed per
breaking event per micron of radius per meter of crest length in each bin and in
Figure 3.7 the result is plotted versus the mean radius for each bin. The probability
distribution from all drops is shown in the main plot, while in the inset, separate
distributions are plotted for the first (t < 0.9 s) and second (t ? 0.9 s) drops
production regions in Figure 3.2. In all three data sets, separate straight lines were
fitted by least square error minimization to the smaller diameter and larger diameter
drop data.
The boundary between the larger and smaller radius groups was determined
by an iterative bisection-like routine outlined as follows: first, an initial guess for the
break in slope, r0, is estimated, the data is split into two distinct sets and a power
law, of the form described above, is fitted to each set. The radius where the two
lines intersect, ri, is found. If the difference between r0 and ri does not fall within
a specific tolerance (in this case rtol = 1 ?m), a new guess for r0 half way between
the previous values of r0 and ri is assigned and the processes is repeated until the
tolerance is reached.
The break in slope occurred approximately at the Hinze scale Hinze [1955] that
is computed using the turbulent dissipation rate to determine the velocity scale in
the wind field. This indicates that the wind plays an important role during the drop
generation, which is quite different from the drop generation mechanisms shown
77
herein with a mechanically generated breaking wave in the absence of wind.
It should be noted that this determination of the ?Hinze? radius is inherently
inaccurate because there are only a few drops in each bin at the larger radii. This is a
classic problem in bubble and drop measurements and to emphasize this inadequacy,
the line for the fit to the drops with larger radii is drawn as a dashed line. In a
laboratory wind wave system, Wu et al. [1984] measured the drops size distribution
at several heights above the mean water level. The measurements were made over
long periods of time and no correlation with breaking events was attempted. The
distribution has a shape that is qualitatively similar to the one shown here Erinin
et al. [2019].
3.2 Comparison of Spray Generated by a Weak, Moderate, and Strong
Intensity Plunging Breakers
In this section, mean breaking profiles and drops characteristics generated by
the three plunging breaking waves are presented and discussed. For each of the
three breakers, the mean breaker profiles are calculated from 10 breaker realizations
at each instant in time. The three breakers are characterized by their mean ge-
ometric features, such as wave crest and jet impact speed (see Table 2.1). When
aligned to the plunging jet impact location in space and time, the breaker profiles are
found to be highly repeatable and share common features throughout the non-linear
wave breaking process. Drop statistics, including the spatio-temporal distribution
of drops, number of drops produced per breaking event, and drop probability distri-
78
butions are reported and discussed for the three breakers. Surface profile and drop
data presented in this section are shown in red for the weak, blue for the moderate,
and green for the strong breaker. The weak breaker drop data presented in this
section is the same as the one reported in the previous section.
3.2.1 Surface Profile Measurements
The surface profile histories are first used to characterize the three breakers as
they approach the point of jet impact. To this end, two quantities were extracted
from the profiles: the horizontal speed of the highest point on the wave crest profile
at the time when the plunging jet begins to form and the speed of the plunging jet
just before impact, denoted as c and Vjet, respectively. These quantities are taken
from the derivatives of the trajectories of the vertical and horizontal positions of the
highest point on the crest (called the crest point), as determined quantitatively from
the profiles, and of the jet tip location, as determined by eye from the profiles and
images. A sample measurement of the wave crest point (marked by green square)
and jet tip location (marked by red triangle) from a single realization of the moderate
breaker is shown in Figure 3.8.
The trajectories of the horizontal and vertical coordinates of the crest points,
denoted as xc and yc, were obtained from the 10 runs for each of the three breakers
over the time of jet formation. The individual crest point trajectories (xc(t) and
yc(t)) from the 10 runs are shown in Figure 3.9 (a) and (c), respectively, and the
trajectories of the average values (xc(t) and yc(t)) are shown in Figure 3.9 (b) and
79
Figure 3.8: Sample measurements of wave crest point, (green square) and jet tip
location (red triangle) from one run of the moderate breaker. Each surface profile,
shown in blue, is obtained from one LIF image, similar to the ones shown in Figure
2.13. Each successive profile is separated by a time interval of ?t = 0.0123 s and
plotted 10 mm above the previous for clarity. The wave crest and jet tip trajectories
form 10 runs for each of the three breakers are shown in Figures 3.9 and 3.10,
respectively.
80
(d). The colored vertical dashed lines in (a-d) were obtained by averaging the time
of jet impact over the 10 runs for each breaker, respectively; while vertical error
bars in subplots (b) and (d) show ?1 standard deviation (in mm) of xc and yc from
run to run, respectively. The wave crest point trajectories were collected over the
time period starting just before the jet starts to form, t ? 0.32 s, and ending shortly
after jet impact. This time period is referred to as the jet formation phase in the
subsequent paragraphs.
The slope of the trajectory of xc(t) remains nearly constant during the jet
formation phase for all three waves, indicating the horizontal speed of the wave crest
point (c) is constant. In each time interval, the average and standard deviation is
computed over the 10 runs. Averaging these quantities over the 24 time intervals in
the plots, yields an average horizontal displacement and standard deviation of xc(t)
16.6 ? 0.309, 16.4 ? 0.426, 16.3 ? 0.596 mm for the weak, moderate, and strong
breakers, respectively.
The motion of the wave crest during the jet formation phase is assessed in detail
by fitting a second order polynomial to xc(t) using least-squares linear regression.
The second order polynomial has the form: xc(t) = p t
2
1 +p2t+p3 where p1, p2, and p3
are the regression coefficients determined by the fitting. The speed of the wave crest
point at time t is given by the first derivative of xc(t), c(t) = d(xc(t))/dt = 2p1t+p2
and the acceleration (assumed to be constant) by the second derivative of c(t),
ac = d
2(xc(t))/dt
2 = 2p1. The regression coefficient fits of xc(t) for the three
breakers are given in Table 3.1.
The average speed of the wave crest point and standard deviation during the
81
Table 3.1: The values for the regression coefficients, p1, p2, and p3, used for the
second order polynomial fit to xc(t), the position of the wave crest point, for the
weak, moderate, and strong breakers. The second order polynomial has the following
form xc(t) = p
2
1t + p2t + p3. The function fit xc(t) is used to obtain the velocity
and acceleration of the wave crest during the jet formation phase.
Fit coefficient Weak Moderate Strong
p1 445 348 325
p2 -1174 -1658 -1635
p3 887 835 811
jet formation phase is reported in Table 3.3 and was nearly the same for all three
waves, measuring at 1.33 ? 0.003 m/s, 1.33 ? 0.002 m/s, and 1.36 ? 0.002 m/s for
the weak, moderate, and strong breakers respectively. These measurements can be
compared to the wave phase speed predicted by linear deep water wave theory:
g
cp = (3.7)
2?f
where cp is the wave phase speed, g is the gravitational acceleration constant, and
f is the wave frequency. Taking g = 9.81 m/s2, f = f0 = 1.15 Hz the wave phase
speed is calculated to be cp = 1.357 m/s. The measured value of the crest point
speed (see Table 3.3) are lower than the value obtained by linear wave theory. The
measured and theoretical values are not expected to match since Equations 3.7 is
from a linear theory for a deep water infinitely long wave train of permanent shape,
where as the experiments involve a highly nonlinear temporally evolving wave packet
with 32 wave components.
The slope of the trajectory of yc(t), shown in Figure 3.9 (d), is initially constant
and upwards from t = 0.33 to 0.43 s, displacing vertically by approximately 8 mm for
all three waves. During this phase, the speed of yc(t), measured by the same method
82
used to measure cp, is 0.103 m/s, 0.107 m/s, and 0.110 m/s for the weak, moderate,
and strong breakers, respectively. Shortly after the start of the formation of the jet,
at t ? 0.37 s, yc(t) reaches a local maxima for all three breakers at ? 0.43 s. After
reaching the local maxima, yc(t) starts to decrease, eventually exhibiting irregular
motion shortly after jet impact. The irregular vertical motion of the wave crest point
can partly be explained by the fact that shortly after jet impact, the wave motion
begins a rapid loss of energy due to energy transfer to turbulent fluid motions.
The trajectories of the horizontal and vertical coordinates of the jet tip point,
denoted as xj and yj, were obtained from 10 runs of each breaker from moment of jet
formation until jet impact. The jet initially forms from a small region on the front
face of the wave near the crest as the face there becomes vertical. This condition
can be seen in Figure 3.8 just before the first measurement of the jet tip location
(red triangle). The time elapsed from jet formation, tj,0 ? 0.37 s, to jet impact,
tj,i ? 0.53 s is ? 0.16 s. The individual positions of the jet tip are determined by
eye from the images and measurements of the breaker profiles. Jet tip trajectories,
?(xj(t), yj(t)), for from the 10 runs are given in Figure 3.10. It should also be
noted that the jet trajectories shown in Figure 3.10 are not adjusted spatially or
temporally, and thus the variability of the profiles indicates the repeatability of
the breaking event from run to run. The standard deviation of the streamwise jet
impact location, where the mean streamwise jet impact location is located at x =
0 mm, monotonically increases for the three breakers, from 5.05 mm, 5.63 mm, and
6.66 mm for the weak, moderate, and strong breakers, respectively.
The average horizontal and vertical displacement of the jet tip from jet forma-
83
(a) (b)
(c) (d)
Figure 3.9: The x and y positions vs. time of the wave crest point during
the plunging jet formation from 10 runs for the weak, moderate, and
strong plunging breakers. The vertical dashed lines indicated the average
time of jet impact from the 10 runs for the three waves. The weak
breaker is shown in red, the moderate in blue, and the strong in green.
The maximum wave crest height from 10 runs for xc(t) and yc(t) are
shown in (a) and (c) respectively, while (b) and (d) show the averaged
position of the wave crest point xc(t) and yc(t) respectively. The vertical
error bars in (b) and (d) show ?1 standard deviation of the xc and yc
over the 10 runs (which are too small to see in (b)). The wave crest speed
reported in Table 3.3 is measured by taking the derivative of xc(t).
84
tion to jet impact can be measured from the jet tip trajectories. The total horizontal
displacement of the jet tip point is 197 mm, 217 mm, and 257 mm for the weak,
moderate, and strong breakers, respectively; while vertical displacement of the jets
during the same interval of time is 50 mm, 58 mm, and 61 mm for the weak, moder-
ate, and strong breakers, respectively. During jet impact, a tube of air is entrained
by the plunging jet and this tube of air is broken up by the underlying fluid motion,
creating bubbles that eventually reach the free surface, pop, and generate drops.
The volume of the entrained air cavity is determined by geometric parameters of
the jet as described above. Qualitatively speaking, these numbers are consistent
with the idea that as the breaker height increases, the jet travels a longer horizontal
and vertical distance before impact and entrains a larger cavity of air.
Similar to the analysis carried out for the trajectory of the wave crest point,
the motion of the jet tip is assessed by fitting a second order polynomial to the
average jet tip position, xj and yj vs. time using least-squares linear regression. The
polynomial has a form xj(t) = xp
2
1t + xp2t + xp3 and yj(t) = yp1t
2 + yp2t + yp3
for the x and y jet tip trajectories, where xp1, xp2, xp3, yp1, yp2, and yp3 are the
regression coefficients determined by fitting, shown in Table 3.2. The velocity, uj(t)
and vj(t), and accelerations ax and ay, are given by the first and second derivative
of xj(t) and yj(t) with respect to time, respectively.
The horizontal speed of the jet tip, uj, moments after the jet begins to form,
uj(0.33 < t < 0.36s), are 1.36 m/s, 1.38 m/s, and 1.34 m/s for the weak, moderate,
and strong breakers respectively. The horizontal and vertical components of the jet
tip velocity, (uimpact,vimpact), at the moment of jet impact are (1.38, 0.77) m/s, (1.49,
85
Table 3.2: The values for the regression coefficients, xp1, xp2, xp3, yp1, yp2, and
yp3, from second order polynomials which are fitted to the xj(t) and yj(t) motion
of the jet tip point. The second order polynomials have the following form; xj(t) =
xp1t
2 + xp2t+ xp3 and y (t) = yp t
2
j 1 + yp2t+ yp3 for x and y motion, respectively.
Fit coefficient Weak Moderate Strong
xp1 -104 -311 -508
xp2 -1313 -1182 -1028
xp3 775 727 675
yp1 -3177 -3367 -2976
yp2 2655 2617 2223
yp3 -506 -451 -353
0.84) m/s, and (1.52, 0.82) m/s for the weak, moderate, and stron?g breakers respec-
tively. The corresponding jet impact speed, taken to be Vjet = u2 2impact + vimpact,
monotonically increases with breaker intensity from 1.60, 1.69, 1.75 m/s for the
three breakers respectively. The average wave crest speed and jet impact speeds are
reported in Table 3.3.
The vertical component of the jet tip acceleration, ay, (assuming constant
acceleration) is given by the second derivative of yj(t), ay = d
2(yj(t))/dt
2 = 2yp1.
The x component of acceleration of the jet tip is measured to be 6.5 m/s2, 6.7 m/s2
and 5.95 m/s2 for the weak, moderate, and strong breakers respectively. These
accelerations are about 20% lower than the constant acceleration due to gravity,
g = 9.81 m/s2, and could indicate the effects of surface tension or aerodynamic
forces influence the motion of the jet just before impact significantly. Similarly, the
horizontal component of acceleration for the jet tip is given by ax = d
2(xj(t))/dt
2 =
2xp1. The horizontal acceleration, ay, for the weak, moderate and strong breakers
are 0.2 m/s2, 0.6 m/s2, 1.0 m/s2. The horizontal component of acceleration is found
to increase with breaker intensity.
86
Weak Breaker
100
80
60
40
20
0
0 50 100 150 200 250
Moderate Breaker
100
80
60
40
20
0
0 50 100 150 200 250
Strong Breaker
100
80
60
40
20
0
0 50 100 150 200 250
Streamwise Position (mm)
Figure 3.10: Jet tip trajectories for the three plunging breakers measured from 10
individual runs. The time interval between each measurement point is ?t = 0.0015 s
(1/650 s). The wave is moving from right to left and the first point is measured just
as the jet tip starts to form from the wave crest and the last point is measured when
the jet impacts the free surface. The jet trajectories are not adjusted spatially or
temporally, and therefore indicate the repeatability of the experiment from run to
run for the three waves.
87
Height above calm flat surface (mm)
Figures 3.11, 3.12, and 3.13 (referred to as the three figures in the subsequent
paragraphs) show the surface profile evolution during the active phase of wave break-
ing for the weak, moderate, and strong plunging breakers respectively. The surface
profiles are phase averaged over 10 repeated runs of the same breaker and are ad-
justed spatially (in x) and temporally (in t) so that the point of jet impact occurs
at the same x and t for all three breakers. The surface profiles are plotted in the ref-
erence frame of the wave crest speed as reported in Table 3.3. The gradient of color
along each profile indicates the value of the standard deviation of surface height at
each streamwise position over the 10 runs. Each profile is plotted 10 mm above the
previous and is separated in time by ?t = 0.0123 s. Because the profiles are plotted
in the reference frame of the wave crest, features that move to the left are moving
faster than the wave crest speed and features moving to the right are moving slower
than the wave crest speed.
Despite the fact that wave breaking is a highly non-linear process, common
small-scale and large-scale features are present in the three figures. The first of these
features is the motion of the leading edge of the splash region, which is initially
formed by the jet impact. Moments before the formation of the splash region, the
breaker profile is characterized by high variation in surface height associated with the
time of jet impact, seen along surface profile (i) between x+ c ? t = 50 to 100 (mm).
After jet impact, the leading edge of the splash region accelerates in all three figures
and goes through cycles of accelerating and decelerating between profiles (ii) and
(iv). The speed of the front splash region is close to the inital phase speed of the
wave, between surface profile (iv) and (v), and eventually decelerates by the time it
88
reaches surface profile (vi).
Another common feature between the three waves is the sharp indent that
forms between the initial jet splash-up and the upper surface of the jet, appearing
just before surface profile (ii) at around x = 50 mm in the three figures. In images
taken with the camera positioned to examine the crater in more detail, one can see
this indent is actually a narrow crater that extends below the apparent free surface
that closes off rapidly, ejecting a wall of drops into the air. Following the indent in
time, it begins to slow down after surface profile (ii), just as the wave crest passes
through the feature. Between surface profiles (iii) and (iv) the feature appears to
diminish, when a violent air bubble explosion occurs just after surface profile (iv)
and x = 250 mm. The violent bubble bursting was identified as one of the primary
drop producing regions in Section 3.1.
The initial indent seems to form a boundary between the irrotational water
fluid motion to the right of the indent and the vortical region on the left generated
by the splash up from the jet impact. From a qualitative analysis, the jet impact and
subsequent splash could generate a vortex pair, one vortex to the right of the indent
and another to the left, with both vortices rotating counter clockwise. A similar
vortex pair was observed just after jet impact in 2D simulations of wave breaking
by Iafrati [2009], Figure 2 (d). The induced motion of the vortex pair would be in
the direction of the wave crest travel and the vortex decay and diffusion could be
responsible for the diminishing of the intent feature.
Similar secondary and tertiary indents form just before surface profile (iv)
at x = 70 mm and surface profile (v) at x = 80 mm, respectively. The standard
89
Table 3.3: Table comparing physical wave parameters for the weak, moderate, and
strong plunging breakers. The average phase speed of the wave c? is measured just
before the jet starts to form on the wave, see Figure 3.9. The average jet impact
speed, v?jet, is measured just before jet impact (see Figure 3.10) and the total number
of drops indicates the number of drops produced per breaking event per crest width
whose D > 100 ?m, measured in section 3.2.2.
Plunging Breaker Type A/?0 c? (m/s) V?jet (m/s) # of drops
Weak 0.070 1.33 ? 0.003 1.60 ? 0.022 696
Moderate 0.074 1.33 ? 0.002 1.69 ? 0.018 860
Strong 0.076 1.36 ? 0.002 1.75 ? 0.016 1173
deviation of the free surface height is higher in these two regions, similar to the initial
indent. The time difference between the formation of the primary and secondary
and secondary and tertiary indents are 0.22 s and 0.15 s respectively. One possible
formation mechanism for the secondary and tertiary indents could be the rebounding
of the jet.
3.2.2 Drop Measurements
The spatio-temporal distribution of drops, introduced in Section 3.1 as N(x, t),
the number of drops moving up across the measurement plane per crest width per
breaking event per unit time (?t) per unit streamwise direction (?x), is measured
experimentally and shown in Figures 3.14, 3.15, and 3.16 for the weak, moderate, and
strong plunging breakers respectively. Figures 3.14, 3.15, and 3.16 are collectively
referred to as the three figures in the following paragraphs. Figure 3.14 is the same
as Figure 3.3 except the color scale for the number of drops in Figure 3.14 matches
that of 3.15 and 3.16.
In the three figures, there are two large spatio-temporal regions of drop pro-
90
Figure 3.11: Mean surface profiles for the weak breaker (f0 = 1.15 Hz, A = 0.070?0)
averaged over 10 unique runs. The range of color along each profile indicates the
standard deviation of surface height at each streamwise position over the 10 runs.
Each successive profile is plotted 10 mm above the previous with a temporal sepa-
ration of ?t = 0.0123 s between each profile, with breaker profile (i) occurring at
t = 0.0031 s. The surface profiles are shown in a reference frame fixed to the wave
crest measured before the jet is formed (c = 1330 mm/s).
91
Figure 3.12: Mean surface profiles for the moderate breaker (f0 = 1.15 Hz, A =
0.074?0) averaged over 10 unique runs. The range of color along each profile indicates
the standard deviation of surface height at each streamwise position over the 10
runs. Each successive profile is plotted 10 mm above the previous with a temporal
separation of ?t = 0.0123 s between each profile, with breaker profile (i) occurring
at t = 0.0031 s. The surface profiles are shown in a reference frame fixed to the
wave crest measured before the jet is formed (c = 1330 mm/s).
92
Figure 3.13: Mean surface profiles for the strong breaker (f0 = 1.15 Hz, A = 0.076?0)
averaged over 10 unique runs. The range of color along each profile indicates the
standard deviation of surface height at each streamwise position over the 10 runs.
Each successive profile is plotted 10 mm above the previous with a temporal sepa-
ration of ?t = 0.0123 s between each profile, with breaker profile (i) occurring at
t = 0.0031 s. The surface profiles are shown in a reference frame fixed to the wave
crest measured before the jet is formed (c = 1360 mm/s).
93
duction measuring about 600-800 mm long and 0.6 s high. These regions were
previously identified as Region I and II in Section 3.1. In region I for the weakest
breaker, three prominent features, identified by a local maxima in N(x, t), were
identified in Section 3.1. The first of these prominent features is a local maxima
around (x, t) = (140 mm,0.18 s) and corresponds to the approximate location and
time of jet impact. The second maxima is located around (x, t) = (325 mm,0.275 s)
and is associated with the second jet impact and splashing. Finally, the third local
maxima is located at (x, t) = (275 mm,0.38 s) and is associated with the bursting
of large air bubbles, initially entrained by the plunging jet.
In contrast, region I for the moderate breaker contains only two prominent
local maxima. The first maxima is located around (x, t) = (190 mm,0.20 s) and is
associated with the region of the indent formed by the jet impact and first splash
region, between breaker profile (ii) and (iii) in Figure 3.12. Drops produced in this
region are confined to a narrow spatial and temporal location. The second local
maxima occurs in the vicinity of (x, t) = (390 mm,0.30 s) and is associated with
diminishing of the impact crater and the sudden eruption of large air bubbles, just
before profile (iv) in Figure 3.12.
Similarly, the spatio-temporal distribution of N in region I for the strong
breaker has two pronounced local maxima at similar spatial and temporal locations.
The first local maxima is located at (x, t) = (200 mm,0.30 s) and is associated with
the location of the indent formed by the jet impact and front splash region, between
breaker profile (iii) and (iv) in Figure 3.13. The second local maxima is located at
(x, t) = (420 mm,0.40 s) is associated with large bubble bursting event, seen just
94
2
N(x,t)
1.8
0 10 20 30
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1000 800 600 400 200 0
Streamwise position (mm)
Figure 3.14: Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest length, plotted
on an x-t plane for the weak breaker. The black line is drawn with a slope of 1.7
m/s, indicating the drop ejection front. This plot is identical to Figure 3.2 except
that it is plotted using the same colorbar as Figures 3.15 and 3.16.
after breaker profile (iv) in Figure 3.13. The second local maxima in region I for
the weak breaker (Figure 3.14), identified as the region of the second jet impact
and splashing, is no longer noticeable in the plots of N(x, t) for the moderate, and
strong breakers.
Similar to the analysis carried out in Section 3.1, the measured function N(x, t)
was spatially and temporally integrated separately over all x and t to obtain number
of droplets generated per unit time per unit crest length, Nx(t), and the number of
95
Time (s)
2
N(x,t)
1.8
0 10 20 30
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1000 800 600 400 200 0
Streamwise position (mm)
Figure 3.15: Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest length, plotted
on an x-t plane for the moderate breaker. The black line is drawn with a slope
of 1.7 m/s, indicating the drop ejection front.
96
Time (s)
2
N(x,t)
1.8
0 10 20 30
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1000 800 600 400 200 0
Streamwise position (mm)
Figure 3.16: Contours of N(x, t), the number of drops with d ? 100 ?m generated
per breaking event, per ?t = 0.025 s, per ?x = 13.02 mm, per crest length, plotted
on an x-t plane for the strong breaker. The black line is drawn with a slope of
1.7 m/s, indicating the drop ejection front.
97
Time (s)
droplets generated per unit x per unit crest length, Nt(x), respectively. A plot of
Nx(t) for the weak, moderate, and strong breakers is presented in Figure 3.17. The
three functions share similar features and local maxima, found at approximately
t = 0.20, 0.40, and 1.50 s after initial jet impact. The first local maxima occurs
between t = 0.15 and 0.30 s and was previously identified as the approximate time
of jet impact. Comparing the maximum value of the first peak between the weak
and moderate and the weak and strong breakers, it was found that the first peak is
about 2.5 times larger for the moderate and 3.7 times larger for the strong breaker.
The number of drops measured during the jet impact phase is shown in Table 3.4
and accounts for 14%, 19%, and 23% of the total number of drops produced in the
weak, moderate, and strong breakers.
Shortly after the first peak, a second local maxima, associated with the large
bubble popping region, occurs between t = 0.30 and 0.40 s. The number of drops
measured during the large bubble bursting phase is 356, 456, 735 accounting for
52%, 53%, and 62% of the total number of drops produced between the weak,
moderate, and strong breakers, respectively. The total number and proportion of
drops generated in region I increases with breaker intensity.
For the weak breaker, the second local maxima is slightly larger, measuring
at Nx(0.175) = 1374 for the first peak and Nx(0.375) = 1812 for the second peak.
However, in the case of the moderate and strong breakers the first peak is more
intense than the second peak. The third and final local maxima is associated with
small bubbles rising to the free surface and popping, from about t = 1 to 2 seconds.
The temporal range and maximum peak of the third local maxima is similar between
98
6000
Weak
Moderate
5000 Strong
4000
3000
2000
1000
0
0 0.5 1 1.5 2
Time (s)
Figure 3.17: A comparison of the number of drops generated per time interval per
crest length vs. time, Nx(t) for the weak (in red), moderate (in blue), and strong
breaker (in green). Three pronounced peaks are present in the curves from all
three waves. The first associated with jet impact, the second with the large bubble
popping region and the third with the smaller bubbles coming to the free surface
later in time. The three curves were generated by spatially integrating the data in
Figures 3.14, 3.15, and 3.16.
the three breakers.
Figure 3.18 shows a plot of Nt(x), the number of drops generated per unit x per
unit crest length, for the three breakers. Each data point (circle, triangle, or square)
indicates the center location of a drop measurement window. The curve for the weak
breaker (red circles) has a simple structure with a single local maxima at about
x = 200 mm, which approximately corresponds to the jet impact location. Local
features and maxima emerge in the curves of the moderate and strong breakers,
becoming more pronounced with wave intensity. The first local maxima in the
99
N (t)
x
5
Weak
Moderate
Strong
4
3
2
1
0
1000 800 600 400 200 0
Streamwise position (mm)
Figure 3.18: A comparison of the number of drops generated per mm in the stream-
wise direction per crest length vs. streamwise position, Nt(x), for the weak (red
circles), moderate (blue triangles), and strong breakers (green squares). The weak
breaker has a simple spatial distribution of drops that increases at the jet impact
location and gradually decreases. In contrast, the moderate and strong breakers
have pronounced peaks and features at the locations of jet impact and large bub-
ble popping. These three curves were generated by temporally integrating N(x, t),
shown in Figures 3.14, 3.15, and 3.16.
moderate and strong breakers occurs at x = 200 mm and is associated with the
approximate location of jet impact. A second local maxima is present at x = 400 mm
for the moderate and x = 450 mm for the strong breaker associated with the large
bubble bursting event.
Table 3.4 compares the number of drops measured per breaking event per unit
crest length for the three breakers during the different stages of breaking. The total
number of measured drops over all t and x, reported in the last column, is 696,
100
N (x)
t
Table 3.4: Table comparing the number drops produced per breaking event per
crest width (whose d > 100 ?m) during the different stages of breaking for the
weak, moderate, and strong plunging breakers. The jet impact and large bubble
bursting regions are defined spatially and temporally (see text for full description)
while the late bubble bursting region is defined from 1 < t < 2 seconds.
Region Weak Breaker Moderate Breaker Strong Breaker
Jet Impact 100 (14%) 166 (19%) 265 (23%)
Large Bubble Burst 356 (52%) 456 (53%) 735 (62%)
Late Time Bubbles 240 (34%) 238 (28%) 172 (15%)
Total 696 860 1173
860, and 1173 drops for the weak, moderate, and strong breakers respectively. The
number of drops generated during jet impact is also reported. The jet impact region
is defined as (x, t) = (0 - 200 mm, 0.00 - 0.30 s), (x, t) = (0 - 250 mm, 0.00 - 0.30 s),
and (x, t) = (0 - 250 mm, 0.00 - 0.35 s) for the weak, moderate, and strong breakers,
respectively. During this phase of wave breaking the number of drops generated
were 100, 166, and 265 accounting for 14%, 19%, and 23% of the total drops for the
weak, moderate and strong breakers.
The drops attributed to large bubble bursting region are taken as those passing
through the measurement plane before t < 1 s, excluding drops generated by the
jet impact in the spatio-temporal region above. The number of drops generated
during this phase of wave breaking are 356, 456, and 735 accounting for 52%, 53%,
and 62% of the total drops generated by the three breakers. Finally, the number
of drops generated in the late time bubble popping, after the active phase of wave
breaking, are 240, 238, 172 drops and accounting for 34%, 28%, and 15% for the
three breakers. The number of drops produced during the late time bubble popping
is similar for the three breakers, although slightly lower for the stronger breaker.
101
A comparison of the normalized distributions of drop radii from the three
breakers is shown on a log-log plot in Figure 3.19. The probability distributions
were obtained in the same manner as for the distributions in Figure 3.7, see section
3.1 for details, except, in the present case, a total of 32 logarithmically spaced
bins ranging from r = 50 to 1500 ?m are used. The break in slope is calculated
using the same bisection method described in section 3.1. Separate straight lines
were fit, by least squares error minimization, to the smaller and larger diameter
data. The slopes of the line fits for the small and large drops are represented by
? and ?, respectively. It should be noted that the determination of ri and ? is
inherently inaccurate because there are only a few drops in each bin for the larger
radii. This is a classic problem in bubble and drop measurements, and to emphasize
this inadequacy, dashed lines are used to represent the fitted functions in the larger
radii region of each plot. The ? and ? slope coefficents and ri for the three waves
are shown in Table 3.5.
For the large drops, ? = -5.4 to -6.1 with no noticeable trend as the intensity
of the wave increases. From Figure 3.19, it is evident that the bins containing large
drop data from the weak breaker is noisy, with some bins containing only a single
drop. Comparatively, the moderate and strong breaker data is progressively less
noisy since more large drops are produced by these breakers; therefore, each bin
is averaged over more drops. It is possible that the bimodal nature of the radius
probability distribution is an artifact of the lack of bin convergence for the large
drops.
The break in slope, ri, grows as the intensity of the wave increase from ri =
102
2 Weak
10
Moderate
Strong
0
10
-2
10
50 100 200 500 1000
r ( m)
Figure 3.19: Drop probability distributions for weak (red circles), moderate (blue
triangles), and strong breakers (green squares) for drops generated through out the
entire breaking event 0 < t < 2 seconds. A break in slope is identified in all three
data sets and function fitting parameters and break in slope radii are shown in Table
3.5.
460 to 570 to 740 ?m between the three breakers. Using the computed values of ri,
it is possible to calculate a corresponding Weber number, We 2i = vj ri/? where vj is
the impact speed of the jet, and ? = 73 dynes/cm. The calculated values of We are
16.1, 22.3, and 31.4 respectively. For the small drops, ? decrease with wave intensity
from -2.4 to -2.1 to -1.9, indicating that as breaker intensity increases, the average
drop radius increases. This result is consistent with the idea and the observed trend
that as more large drops are produced, the break in slope ri would shift to the right.
103
n(r) ( m-1 m-1)
Table 3.5: The fit parameters to drop size distributions from Figures 3.19 for the
three waves. Where ? and ? are the slope of the line fits for the small and large
drops respectively and ri identifies the drop radius (in ?m) where the two lines
intersect.
Variable Weak Breaker Moderate Breaker Strong Breaker
? -2.4 -2.1 -1.9
? -5.4 -6.1 -5.8
ri (?m) 460 570 740
104
Chapter 4: Summary and Conclusions
Drop generation by breaking water waves is an important problem which has
a significant impact on air-sea interaction. The mechanisms by which drops are
generated are poorly understood. Previous studies of drop generation have mainly
focused on average distributions and fluxes of drops in wind-wave tanks, wind-wave
systems in the field, and more recently, numerical simulations. Though there is
a qualitative understanding of how droplets are produced during a wave breaking
event, there is little quantitative data to back up these assertions.
In this dissertation, the generation and dynamics of drops produced by break-
ing water waves were studied in laboratory-scale experiments. The breakers were
generated by a programmable wavemaker using a dispersively focused wave packet
technique. The wave maker motions for the three breakers differ in the values chosen
for the breaking distance from the wavemaker, xb/?0, and the overall wavemaker
amplitude, A/?0. The three plunging breakers are characterized (and referred to)
as weak, moderate, and strong, based on their overall wavemaker amplitude, where
the weak breaker corresponds to the smallest, the moderate breaker to the middle,
and strongest breaker to the largest A/?0.
For each of the three breakers, the breaker profile and the spatio-temporal
105
distribution of drops are measured using highly resolved spatial and temporal mea-
surement techniques. The two-dimensional profile evolution at the center plane
of the wave tank is measured using a laser induced fluorescence technique. Drop
radii, trajectories, and the spatio-temporal distribution of drop numbers are mea-
sured using in-line holography. The two measurements are taken simultaneously at
650 frames per second. As discussed in more detail below, the measurements of
the breaker profile evolution and the spatio-temporal distribution of drop genera-
tion are used to identify spray generation mechanisms during wave breaking. Also,
the breaker profile measurements are used to explore the dynamics of the break-
ing process through examination of ensemble averaged breaker profiles for the three
breakers. Finally, the number and size distribution of drops generated by the three
breakers are related to the previously identified spray generation mechanism and
breaker characteristics such as jet impact speed.
The evolution of breaker profiles were measured over 10 unique realizations
for each of the three breakers. Using the data from these 10 runs, the mean profile
and the standard deviation of the breaker profiles were computed at each instant
in time. The mean breaker profiles were used to measure a set of pre-breaking
geometrical parameters of the wave crest shape and plunging jet with the purpose
of determining independent variables to characterize the breaker. The speed of the
wave crest point, c, was defined as the highest point on the wave crest profile and is
measured during the formation of the plunging jet. It was found that c was nearly
the same for the three waves during the jet formation phase, and was measured to
be 1.33, 1.33, and 1.36 m/s for the three breakers. These values are close to the
106
phase speed of a linear uniform wave train with a frequency equal to the average
frequency of the wave packet used to generate the breakers (f0 = 1.15 Hz). The
jet impact speed, V jet, was defined as the jet speed just before jet impact with the
front face of the wave. The measured values of V jet are 1.60, 1.69, and 1.75 m/s
for the three breakers, where V jet increases with breaker strength. The difference
in average jet impact speed between the weak and strong plunging breakers is 9.3%
while the difference in maximum wave crest height, ymax, is 15% (ymax = 26.5 mm
for the weak and 30.5 mm for the strong breaker).
When the breaker profiles are aligned horizontally relative to their streamwise
jet impact locations and averaged over 10 realization, the mean breaker profiles
produce many common surface features between the three breakers. Despite the
fact that wave breaking is a highly non-linear event, large-scale and small-scale
breaker features are preserved between the three breakers. These common features
take the form of ripples with rounded crests and sharp troughs that slow down and,
in the reference frame of the breaking crest, propagate over the crest of the wave.
The first of these features to appear is a sharp trough, or indent, formed between
the initial jet splash-up and the upper surface of the plunging jet and referred to
herein as the impact crater. The impact crater slows down relative to the wave crest
speed and diminishes in time; eventually, a large bubble bursting event, takes place
just upstream of the impact crater. The standard deviation of surface profile height
is higher in the region of the large bubble bursting event and increases with breaker
intensity.
The results of the spatio-temporal distribution of drops indicate that for me-
107
chanically generated plunging breakers, drop generation occurs in two regions of
space and time. The first region (I) is associated with the active phase of wave
breaking and begins with the plunging jet impact. In this region, drops are gener-
ated by the jet impact and splashing events, and by the bursting of the large air
bubbles that are entrained during the initial jet impact. The second region (II) is
separated from region (I) and occurs after the active phase of wave breaking follow-
ing jet impact. In this region, drops are generated by small air bubbles, initially
entrained by the plunging breaker, which rise to the free surface and burst.
Breaker intensity is related to drop production from the two breaking regions
identified in the previous paragraph as regions (I) and (II) and their associated sub-
processes. In region (I), drops are produced by two sub-processes, jet impact and
large bubble bursting and splashing. During these two sub-processes, the average
number of drops increases with breaker intensity and range from 100 in the weak
breaker to 265 in the strong breaker for jet impact and 456 for the weak breaker
to 735 for the large breaker drops for the large bubble bursting. In region (II)
the number of drops measured per breaking event decreases from 240 for the weak
breaker 172 for the strong breaker. When plotted on a semi-log plot, the total
number of drops generated during wave breaking and the number of drops generated
by jet impact is found to scale exponentially with jet impact speed.
The drop radius probability distribution is reported for each the three breakers.
When plotted on a log-log plot, the small and large radius data is found to follow
separate inverse power law scalings, with power law exponents represented by ? and
?, respectively. The small drop scaling exponent is found to decrease with wave
108
intensity from -2.4 for the weak to -1.9 for the strong breaker. The large drop power
law exponent, ?, does not follow a trend with increasing breaker intensity. It should
be noted that determination of ? is inherently inaccurate because there are few
drops in each radius bin at larger radii. The break in slope, ri is determined by
the intersection point of the two straight lines fits by an iterative bisection method.
As breaker intensity increases, ri increases monotonically from 460 ?m for the weak
breaker to 740 ?m for the strong breaker. The increase in ri indicates that more
large drops are produced as breaker intensity increases. Drop radii distributions
with two linear regions have been previously found in natural wind-wave system (at
U10m ? 10 m/s) by Wu et al. [1984] and in laboratory wind-wave experiments (at
U10m > 30 m/s) by Veron et al. [2012].
4.1 Future Directions
In future work, the effects of wavelength, surfactants, and wind on droplet
characteristics and breaking sub-processes can be studied in a set of experiments
similar to the ones presented in this dissertation. Studying the effects of wave-
length on drop generation can aid in the development of scaling laws that extend
the results presented in this dissertation. Particularly, parameterizing droplet pro-
duction to wavelength may be of practical interest since the wavelength is an easier
wave characteristic to measure in the field than jet impact speed or volume of air
entrained.
Surfactants are ubiquitous on natural water surfaces and their concentration
109
and properties vary broadly. Previously, the effects of surfactants on drops produced
by bursting bubbles has been studied in laboratory scale experiments where the
droplet size distribution was found to shift to smaller sizes when compared to the
clean water case [Sellegri et al., 2006] (see Chapter 1). However, the effects of
surfactants on drop production by plunging breaking waves has not been related to
the breaking sub-processes identified in this dissertation.
Finally, it would be desirable to study how low to moderate wind speed condi-
tions influence drop generation and transport. This problem is of particular interest
since wind is often present in the field and is one of the primary mechanisms by
which water waves are generated. Studying how wind modifies the spatio-temporal
distribution of drops for a range of wind speeds would lead to a better understating
of how droplets are transported into the marine atmospheric boundary layer, which
would lead to improved parameterization of air-sea fluxes.
110
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