ABSTRACT
Title of dissertation: AIR ENTRAINMENT IN
THE TURBULENT SHIP
HULL BOUNDARY LAYER
Nathan Washuta, Doctor of Philosophy, 2016
Dissertation directed by: Professor James Duncan
Department of Mechanical Engineering
Turbulent fluctuations in the vicinity of the water free surface along a flat, ver-
tically oriented surface-piercing plate are studied experimentally using a laboratory-
scale experiment. In this experiment, a meter-wide stainless steel belt travels hori-
zontally in a loop around two rollers with vertically oriented axes, which are sepa-
rated by 7.5 meters. This belt device is mounted inside a large water tank with the
water level set just below the top edge of the belt. The belt, rollers, and supporting
frame are contained within a sheet metal box to keep the device dry except for one
6-meter-long straight test section between rollers. The belt is launched from rest
with an acceleration of up to 3-g in order to quickly reach steady state velocity.
This creates a temporally evolving boundary layer analogous to the spatially evolv-
ing boundary layer created along a flat-sided ship moving at the same velocity, with
a length equivalent to the length of belt that has passed the measurement region
since the belt motion began.
Surface profile measurements in planes normal to the belt surface are con-
ducted using cinematic Laser Induced Fluorescence and quantitative surface pro-
files are extracted at each instant in time. Using these measurements, free surface
fluctuations are examined and the propagation behavior of these free surface ripples
is studied. It is found that free surface fluctuations are generated in a region close
to the belt surface, where sub-surface velocity fluctuations influence the behavior
of these free surface features. These rapidly-changing surface features close to the
belt appear to lead to the generation of freely-propagating waves far from the belt,
outside the influence of the boundary layer.
Sub-surface PIV measurements are performed in order to study the modifi-
cation of the boundary layer flow field due to the effects of the water free surface.
Cinematic planar PIV measurements are performed in horizontal planes parallel to
the free surface by imaging the flow from underneath the tank, providing streamwise
and wall-normal velocity fields. Additional planar PIV experiments are performed
in vertical planes parallel to the belt surface in order to study the bahvior of stream-
wise and vertical velocity fields. It is found that the boundary layer grows rapidly
near the free surface, leading to an overall thicker boundary layer close to the sur-
face. This rapid boundary layer growth appears to be linked to a process of free
surface bursting, the sudden onset of free surface fluctuations.
Cinematic white light movies are recorded from beneath the water surface in
order to determine the onset location of air entrainment. In addition, qualitative
observations of these processes are made in order to determine the mechanisms
leading to air entrainment present in this flow.
AIR ENTRAINMENT IN THE TURBULENT SHIP HULL
BOUNDARY LAYER
by
Nathan Washuta
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
2016
Advisory Committee:
Dr. James H. Duncan, Chair/Advisor
Dr. James Baeder, Dean’s Representative
Dr. Kenneth Kiger
Dr. Johan Larsson
Dr. Amir Riaz
c© Copyright by
Nathan Washuta
2016
Acknowledgments
First and foremost, I would like to thank my advisor, Dr. James Duncan, for
all that he has taught me over the years. Without his advice and guidance these
past six years, I would not have become half the student, engineer, or researcher I
am today. I will always strive to match the example he sets.
I would also like to thank my labmates, without whom I would never have been
able to accomplish anything worthwhile in graduate school. To Naeem Masnadi,
An Wang, Ren Liu, Martin Erinin, Christine Ikeda, Kyle Corfman, Jayson Geiser,
Rahul Mulinti, Dana Ehyaei, and so many others, for being so smart, hard-working,
generous, and kind. I wish them all the success they so clearly deserve.
I also want to thank Dr. Ken Kiger, Dr. Amir Riaz, Dr. Johan Larsson, and
Dr. James Baeder for generously donating their valuable time and agreeing to serve
on my dissertation committee and for the many valuable and productive discus-
sions. Special thanks go to Dr. Kiger for giving me my first crack at research as an
undergraduate.
My deepest thanks go to Jennifer Dahne, who endured my forever open-ended
quest for a Ph.D. Without her love and support, I would never have dared to attempt
anything so great.
Thank you again to Kyle Corfman and to Rosie Myers, for taking me in and
for being great friends.
And last, but certainly not least, I want to thank my family. The love and
caring they have always given to me inspire me every day to want to be better.
ii
Contents
List of Figures iv
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Prior Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Research Considerations for a Laboratory-Scale Experiment . . . . . 12
2 Experimental Details 15
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Experimental Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Water Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Ship Boundary Layer Device . . . . . . . . . . . . . . . . . . . 20
2.3 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Surface Profile Measurements . . . . . . . . . . . . . . . . . . 32
2.3.2 Fluid Velocity Measurements . . . . . . . . . . . . . . . . . . 38
2.3.3 Air Entrainment Measurements . . . . . . . . . . . . . . . . . 59
3 Results and Discussion 63
3.1 Surface Profile Measurements . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.2 Free Surface Bursting . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.3 Surface Fluctuations . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Flow Field Measurements . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.1 PIV measurements with a horizontal light sheet . . . . . . . . 79
3.2.2 PIV Measurements with a vertical light sheet . . . . . . . . . 99
3.3 Air Entrainment Events . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.3.1 Event Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 122
3.5 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 125
iii
List of Figures
1.1 Photograph of naval combatant ship showing zone of white water next
to the hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Drawing from Stern (1986) defining the various flow regions in a ship
hull boundary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Drawing from Grega et al. (1995) hypothesizing the secondary flow
motions that exist in the mixed boundary corner. . . . . . . . . . . . 5
1.4 A series of plots from Sreedhar and Stern (1998) comparing the sec-
ondary flow motions found under a variety of boundary conditions. . 7
1.5 Regions of various types of surface motions for free surface turbu-
lence with velocity fluctuation magnitude q (vertical axis) and length
scale L (horizontal axis), from Brocchini and Peregrine (2001). Air
entrainment and spray production occur in the upper region, above
the uppermost curved line. The three data points are values obtained
for the turbulent boundary layer on a flat plate with q taken as the
rms of the spanwise (which is vertical for the boundary layer along a
ship hull) velocity fluctuation (w′) and L taken as the boundary layer
thickness (δ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Two plots from Kim et al. (2014), showing wall-normal distributions
of turbulent velocity fluctuations in the (a) streamwise and (b) ver-
tical (−) and wall-normal (−·) directions. . . . . . . . . . . . . . . . . 12
1.7 Two figures from Kim et al. (2013), showing a visualization of air
entrainment in a turbulent Couette flow from (a) a side view and (b)
an end view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 A photo of the open-surface water tank, showing the transparent
acrylic walls and steel support frame. . . . . . . . . . . . . . . . . . . 16
2.2 A top view photo of one of the seams between acrylic panels that
form the walls of the water tank. Each gap between panels is first
filled with flexible Tygon tubing to reduce the thickness of the silicone
caulk layer, allowing for a fully-cured caulk layer. . . . . . . . . . . . 17
2.3 A photo of the skimmer at one end of the tank. Its height is adjustable
so that it can be set just below the desired water level in the tank. . . 18
2.4 Top view schematic of the SBL device and the tank where experiments
are performed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 A photo of one of the rollers during construction of the SBL. . . . . . 21
iv
2.6 A photo of the belt construction in progress, showing major com-
ponents of the device, including the rollers, steel support frame, hy-
draulic motors, and stainless steel belt. . . . . . . . . . . . . . . . . . 22
2.7 A photo of one end of the SBL, showing the stainless steel sheet metal
dry box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 A photo of the overflow drain piping that allows the SBL dry box to
drain in the event of a sump pump failure. . . . . . . . . . . . . . . . 24
2.9 A top view schematic of the entrance fairing, designed to create
a smooth transition over the backwards-facing step at the location
where the belt exits the SBL dry box. . . . . . . . . . . . . . . . . . . 25
2.10 Two photos of the hydraulic system showing (a) the Hydraulic Power
Unit (HPU) and (b) hydraulic fluid accumulators. . . . . . . . . . . . 26
2.11 Two photos of the crane system used to move the SBL device in and
out of the tank. Image (a) shows one of the cranes that is used and
(b) shows a photo of the process of lifting the device. . . . . . . . . . 27
2.12 Measurements of belt trajectory for 5 repeated runs at each belt
speed. The left column shows the belt velocity, U , versus time rel-
ative to the trigger signal that comes from the belt control system.
The right column shows the amount of belt travel, x, versus time
relative to trigger. Belt speeds of 3 m/s are shown in (a) and (d),
4 m/s are shown in (b) and (e), and 5 m/s are shown in (c) and (f). . 29
2.13 A comparison of the average belt launch trajectory for each belt
speed. Plot (a) shows the belt velocity U versus time, while plot
(b) shows the belt velocity U versus x, the amount of belt travel. . . 30
2.14 Schematic drawing showing the set up for the cinematic LIF mea-
surements of the free surface shape from (a) a side view and (b) an
end view perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.15 One sample frame from an LIF movie during a belt launch to 5 m/s.
Image (a) shows a raw image from the movie camera and image (b)
shows that same image with the result of the image processing plotted
on top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.16 Sample images from intermediate steps of the processing alogirthm
used to extract instantaneous surface profiles from LIF images . . . . 35
2.17 A side view schematic of the horizontal PIV setup. . . . . . . . . . . 39
2.18 A photo of the horizontal PIV setup made necessary by space con-
straints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.19 A photo of the Type 309-15 calibration target used to correct for
optical distortions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.20 Two images of the periscope used to create a light sheet below the wa-
ter surface showing (a) an overall schematic and (b) the configuration
used in the horizontal light sheet experiments. . . . . . . . . . . . . . 42
2.21 Two images of the vertical PIV setup showing (a) a side-view schematic
and (b) a photo of the actual setup through the wall of the tank,
showing the orientation of the light sheet. . . . . . . . . . . . . . . . 44
2.22 Particle frequency response for a range of particle diameters. . . . . . 49
v
2.23 Four photos of the particle seeding system. Image (a) shows the
basin in which high-concentration particle solution is mixed. Image
(b) shows the pump that draws from this basin and pumps to (c) the
particle seeder. Image (d) is a photo from beneath the water of an
initial test run of the particle seeder using fluorescent dye rather than
particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.24 Two images of the results from the PIV experiments with a horizontal
sheet far from the free surface during a launch to 4 m/s at x = 4.2 m.
Image (a) shows a sample raw PIV image and image (b) shows the
processed vector field with the vector spacing reduced by 1/2 and the
vector length increased by 10 times for clarity. The background is
colored by vector length with a scale running from 0 to 2.5 m/s. . . . 54
2.25 Two images of the results from the PIV experiments with a vertical
sheet close to the belt surface during a launch to 3 m/s at x = 5.9 m.
Image (a) shows a sample raw PIV image and image (b) shows the
processed vector field with the vector spacing reduced by 1/2 and the
vector length increased by 10 times for clarity. The background is
colored by vector length with a scale running from 0 to 1.0 m/s. . . . 55
2.26 A plot showing the viscous sublayer thickness based on the Schultz-
Grunow formula for skin friction, taken from Schlichting (1979) . . . 57
2.27 Measurements of the y-direction belt position throughout each run
with each plot showing the results of 20 repeated runs at (a) 3 m/s,
(b) 4 m/s, and (c) 5 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.28 Measurements of the RMS velocity fluctuations of the belt, normal-
ized by belt speed throughout each run. . . . . . . . . . . . . . . . . . 59
2.29 A photo of the sub-surface planar bubble measurement system, illu-
minated using two 650 watt flood lamps. . . . . . . . . . . . . . . . . 60
2.30 A schematic of the underwater stereo bubble measurement system . . 61
3.1 A sequence of five images from a high speed movie of the free surface
during a belt launch to 5 m/s. These images are taken at equivalent
belt lengths of (a) 0 m (b) 1 m (c) 2 m (d) 3 m (e) 4 m and (f)
5 m from the bow of the ship. The high reflectivity of the stainless
steel belt makes it appear as a symmetry plane on the left side of the
images. The horizontal field of view for these images is approximately
31 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 A sequence of five images from a high speed movie of the free surface
during a belt launch to 5 m/s. These images are taken at equivalent
belt lengths of (a) 0 m (b) 5 m (c) 10 m (d) 15 m and (e) 20 m from
the bow of the ship. The high reflectivity of the stainless steel belt
makes it appear as a symmetry plane on the left side of the images.
The horizontal field of view for these images is approximately 31 cm. 65
vi
3.3 Sequence of profiles of the water surface during belt launch to 3 m/s
starting from x = 21 m. The time between profiles is 4 ms and each
profile is shifted up 2 mm from the previous profile to reduce overlap
and show propagation of surface features throughout time. The belt
is positioned at the left edge of the image (y = 0). . . . . . . . . . . . 67
3.4 Cross correlation maps for profiles from x = 0 to 5 m. The left column
shows cross correlation maps averaged over a region close to the belt
and those in the right column are averaged over a region far from the
belt. The three rows from top to bottom contain plots from U = 3,
4, and 5 m/s, respectively. The black line in each plot indicates the
∆x location of the maximum correlation for each ∆t slice. . . . . . . 69
3.5 Cross correlation maps for profiles from a launch to 3 m/s. Images
(a) and (c) show cross correlation maps averaged over a region close
to the belt and (b) and (d) are averaged over a region far from the
belt. Images (a) and (b) are averaged over x positions from 5.85 m
to 17.85 m and images (c) and (d) are averaged over x positions from
17.85 m to 29.85 m. The black line in each plot indicates the ∆x
location of the maximum correlation for each ∆t slice. . . . . . . . . . 71
3.6 Cross correlation maps for profiles from a launch to 4 m/s. Images
(a) and (c) show cross correlation maps averaged over a region close
to the belt and (b) and (d) are averaged over a region far from the
belt. Images (a) and (b) are averaged over x positions from 5.85 m
to 17.85 m and images (c) and (d) are averaged over x positions from
17.85 m to 29.85 m. The black line in each plot indicates the ∆x
location of the maximum correlation for each ∆t slice. . . . . . . . . . 72
3.7 Cross correlation maps for profiles from a launch to 5 m/s. Images
(a) and (c) show cross correlation maps averaged over a region close
to the belt and (b) and (d) are averaged over a region far from the
belt. Images (a) and (b) are averaged over x positions from 5.85 m
to 17.85 m and images (c) and (d) are averaged over x positions from
17.85 m to 29.85 m. The black line in each plot indicates the ∆x
location of the maximum correlation for each ∆t slice. . . . . . . . . . 73
3.8 Wave propagation speed far from the belt averaged over 5.85 m <
x <17.85 m versus belt speed. . . . . . . . . . . . . . . . . . . . . . . 74
3.9 A sequence of six images from a high speed movie of the free surface
during a belt launch to 3 m/s. These images are cropped to have
a horizontal field of view of approximately 7 cm. Each image from
(a) to (f) is spaced by 20 ms, corresponding to 6 cm of belt travel
between images.The first image occurs at x = 2.45 m . . . . . . . . . 75
3.10 Two plots showing (a) the average x location of the onset of free
surface bursting versus belt speed and (b) the associated Reynolds
number based on x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.11 RMS surface height fluctuation averaged over y and x versus belt
speed for an equivalent ship length of 30 m. . . . . . . . . . . . . . . 77
vii
3.12 RMS surface height averaged over repeated experimental runs versus
distance from the belt for three belt speeds. . . . . . . . . . . . . . . 79
3.13 Mean streamwise velocity profiles at a belt speed of U = 3 m/s for
(a),(c) D = 14 cm and (b),(d) D = 2.5 cm. Images (a) and (b) each
plot a profile at each 0.5 m of belt travel from x = 0 to 5 m, while
(c) and (d) each plot a profile at each 5 m of belt travel from x = 5
to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.14 Mean streamwise velocity profiles at a belt speed of U = 4 m/s for
(a),(c) D = 14 cm and (b),(d) D = 2.5 cm. Images (a) and (b) each
plot a profile at each 0.5 m of belt travel from x = 0 to 5 m, while
(c) and (d) each plot a profile at each 5 m of belt travel from x = 5
to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.15 Mean streamwise velocity profiles at a belt speed of U = 5 m/s for
(a),(c) D = 14 cm and (b),(d) D = 2.5 cm. Images (a) and (b) each
plot a profile at each 0.5 m of belt travel from x = 0 to 5 m, while
(c) and (d) each plot a profile at each 5 m of belt travel from x = 5
to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.16 Plots showing (a-c) momentum thickness and (d-f) displacement thick-
ness versus x calculated from measured boundary layer profiles for
belt speeds of (a, d) 3 m/s, (b, e) 4 m/s, and (c, f) 5 m/s. . . . . . . 84
3.17 Plots showing shape factor, H, as a function of x for belt speeds of
(a) 3 m/s, (b) 4 m/s, and (c) 5 m/s. . . . . . . . . . . . . . . . . . . 86
3.18 Plots showing different scaling parameters for bursting onset, cal-
culated using measured velocity profile data. In all plots, blue dots
indicate velocity from D = 14 cm and red dots indicate velocity closer
to the surface at D = 2.5 cm. . . . . . . . . . . . . . . . . . . . . . . 88
3.19 Streamwise rms velocity profiles at a belt speed of U = 3 m/s for (a),
(c) D = 14 cm and (b), (d) D = 2.5 cm. These profiles are plotted
in (a) and (b) at each 1 m of belt travel from x = 1 to 5 m and in
(c) and (d) at each 5 m of belt travel from x = 5 to 30 m. . . . . . . 90
3.20 Streamwise rms velocity profiles at a belt speed of U = 4 m/s for (a),
(c) D = 14 cm and (b), (d) D = 2.5 cm. These profiles are plotted
in (a) and (b) at each 1 m of belt travel from x = 1 to 5 m and in
(c) and (d) at each 5 m of belt travel from x = 5 to 30 m. . . . . . . 91
3.21 Streamwise rms velocity profiles at a belt speed of U = 5 m/s for (a),
(c) D = 14 cm and (b), (d) D = 2.5 cm. These profiles are plotted
in (a) and (b) at each 1 m of belt travel from x = 1 to 5 m and in
(c) and (d) at each 5 m of belt travel from x = 5 to 30 m. . . . . . . 92
3.22 Streamwise profiles comparing the velocity profiles at different dis-
tances from the free surface. The left column shows plots of U and
the right column shows plots of urms. Each column contains plots at
each 1 m from x = 1 to 5 m. The top-most plots correspond to x =
1 m, while the bottom plots correspond to x = 5 m. . . . . . . . . . . 93
viii
3.23 Plot of x locations of four different events that indicate transition to
turbulence. Points in blue represent data from measurements at D
= 14 cm and points in red represent data from measurements at D
= 2.5 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.24 Wall-normal rms velocity profiles at a belt speed of U = 3 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. In plots (a) and (b),
profiles are plotted at each 0.5 m of belt travel from x = 0 to 5 m,
while in plots (c) and (d), profiles are plotted at each 5 m of belt
travel from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . 95
3.25 Wall-normal rms velocity profiles at a belt speed of U = 4 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. In plots (a) and (b),
profiles are plotted at each 0.5 m of belt travel from x = 0 to 5 m,
while in plots (c) and (d), profiles are plotted at each 5 m of belt
travel from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . 96
3.26 Wall-normal rms velocity profiles at a belt speed of U = 5 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. In plots (a) and (b),
profiles are plotted at each 0.5 m of belt travel from x = 0 to 5 m,
while in plots (c) and (d), profiles are plotted at each 5 m of belt
travel from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . 97
3.27 Wall-normal rms velocity profiles for belt speeds of (a) 3, (b) 4, and
(c) 5 m/s at D = 14 cm. . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.28 Streamwise mean velocity profiles at a belt speed of U = 3 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are
from early in the run, with profiles plotted every 0.5 m of belt travel
from x = 0 to 5 m. Plots (c) and (d) contain profiles plotted every
5 m from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.29 Streamwise mean velocity profiles at a belt speed of U = 4 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are
from early in the run, with profiles plotted every 0.5 m of belt travel
from x = 0 to 5 m. Plots (c) and (d) contain profiles plotted every
5 m from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.30 Streamwise mean velocity profiles at a belt speed of U = 5 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are
from early in the run, with profiles plotted every 0.5 m of belt travel
from x = 0 to 5 m. Plots (c) and (d) contain profiles plotted every
5 m from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.31 Streamwise rms velocity profiles at a belt speed of U = 3 m/s for (a),
(c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from
early in the run, with profiles plotted every 0.5 m of belt travel from
x = 0 to 5 m. Plots (c) and (d) contain profiles plotted every 5 m
from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
ix
3.32 Streamwise rms velocity profiles at a belt speed of U = 4 m/s for (a),
(c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from
early in the run, with profiles plotted every 0.5 m of belt travel from
x = 0 to 5 m. Plots (c) and (d) contain profiles plotted every 5 m
from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.33 Streamwise rms velocity profiles at a belt speed of U = 5 m/s for (a),
(c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from
early in the run, with profiles plotted every 0.5 m of belt travel from
x = 0 to 5 m. Plots (c) and (d) contain profiles plotted every 5 m
from x = 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.34 Vertical rms velocity profiles at a belt speed of U = 3 m/s for (a), (c)
y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early
in the run, with profiles plotted every 0.5 m of belt travel from x =
0 to 5 m. Plots (c) and (d) contain profiles plotted every 5 m from x
= 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.35 Vertical rms velocity profiles at a belt speed of U = 4 m/s for (a), (c)
y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early
in the run, with profiles plotted every 0.5 m of belt travel from x =
0 to 5 m. Plots (c) and (d) contain profiles plotted every 5 m from x
= 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.36 Vertical rms velocity profiles at a belt speed of U = 5 m/s for (a), (c)
y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early
in the run, with profiles plotted every 0.5 m of belt travel from x =
0 to 5 m. Plots (c) and (d) contain profiles plotted every 5 m from x
= 5 to 30 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.37 Three sequences of images from underwater white light movies at
three different belt speeds. The three columns contain frames from
belt launches to 3, 4, and 5 m/s respectively. Each row contains
frames from each 5 m of belt travel from x = 0 to 30 m. . . . . . . . 112
3.38 Two sequences of images from LIF movies depicting potential air
entraining events. Images in the left column depict a trench closure
event and images in the right column depict a wave breaking event.
Both sets of images are from a belt launch to 5.0 m/s. . . . . . . . . . 113
3.39 Two sequences of images from underwater white light movies of the
same air entrainment event. The belt is moving from left to right.
Images from the left column were captured with the left camera and
images from the right camera were captured with the right camera.
This event occured during a launch to 4.0 m/s. Because of the lighting
and imaging configuration shown above in Figure 2.30, each image
contains both the bubble of interest and its shadow on the belt. . . . 116
3.40 Two sequences of images from underwater white light movies of the
same air entrainment event. The belt is moving from left to right.
Images from the left column were captured with the left camera and
images from the right camera were captured with the right camera.
This event occured during a launch to 5.0 m/s. . . . . . . . . . . . . 117
x
3.41 A set of timelines depicting the x location of various events described
throughout this dissertation. . . . . . . . . . . . . . . . . . . . . . . . 119
3.42 Plots showing (a) θ vs x and (b) urms vs x for the initial 5 m of belt
travel at each belt speed. . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.43 Plots showing Weber number vs Froude number, calculated using θ
as a length scale and a velocity scale of (a) U or (b) urms at y =
3.04 mm, the location of the secondary peak in streamwise velocity
fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.44 A figure from Brocchini and Peregrine (2001) depicting critical regions
for air entrainment based on length and velocity scales, with recorded
data of θ and urms plotted on top. . . . . . . . . . . . . . . . . . . . . 122
xi
Chapter 1: Introduction
1.1 Background and Motivation
It has long been observed that a layer of white water is present along the hulls
of large naval ships. This white water indicates that air entrainment is occurring,
but it is not clear whether these bubbles are entrapped by breaking bow waves
and swept downstream into the boundary layer or if turbulent fluctuations in the
boundary layer are strong enough to entrain air. Although the mechanism of this
entrainment is still unclear, the result is that bubbles formed during this process are
swept into the wake of the ship, where they can persist for over a mile. This bubbly
wake can be detected by sonar, thus enhancing the remotely-detected signature of
naval combatant ships. While the air entrainment caused by breaking bow waves
has been studied previously, the possibility of bubble generation due to near-surface
velocity fluctuations in the turbulent boundary layer has received little attention.
1.2 Prior Research
Research into boundary layer turbulence in the vicinity of a free surface first
began in the 1980s when Hotta and Hatano (1983) considered the turbulence quan-
1
Figure 1.1: Photograph of naval combatant ship showing zone of white water next
to the hull.
tities in the near wake of a double tanker ship hull model, which is symmetric about
a horizontal center plane. In one condition, this model is fully submerged and in
another case, the model is set to pierce the free surface where the air-water interface
coincides with the model symmetry plane. In this way, the authors make compar-
isons to the flow around the ship hull model with and without a free surface. These
experiments were performed in a circulating water channel and velocity was mea-
sured in both cases using a hot wire probe at depths of 0.4d, 0.58d, and d, where d is
the distance from the horizontal symmetry plane to the bottom of the model. They
found that in the case where a free surface is introduced, the turbulence intensities
associated with vertical velocity fluctuations are greatly reduced at shallow depths.
2
In an attempt to isolate the turbulent boundary layer flow in the vicinity of
a free surface from the effects of hull shape, theoretical and numerical work by
Stern (1986) and the corresponding experimental research by Stern et al. (1989)
looked at the influence of a small-amplitude, two-dimensional Stokes wave on the
boundary layer of a surface-piercing flat plate. The first of these studies focused
on using scaling arguments to determine important flow field scaling parameters for
regions influenced by the solid wall and free surface boundary conditions, as shown
in Figure 1.2. The authors focused on flow in the region where the free surface
boundary condition had the largest influence (Region II in Figure 1.2), the region
where effect of the ship hull boundary layer was dominant (Region III), and the
region where both of these boundary conditions had a significant effect (Region IV).
It was found that wave steepness was the primary parameter that had a significant
effect on the boundary layer near the free surface, resulting in large variations in
boundary layer thickness. The wave effects present near the free surface in Regions
II and IV were found to have a significant influence at depths up to half of the
wavelength of the imposed surface wave.
Following this research, two primary groups of researchers began attempting
to isolate the effects of boundary layer turbulence from the flow produced by the
shape of the ship hull by studying what was referred to as a ”mixed-boundary
corner flow.” This type of flow is characterized by flow in the vicinity of a corner
formed by a vertical, stationary wall and a horizontal, flat, free-shear boundary.
The first of these studies, performed by Grega et al. (1995), consisted of an experi-
mental and numerical investigation of flow near a surface-piercing flat plate. In this
3
Figure 1.2: Drawing from Stern (1986) defining the various flow regions in a ship
hull boundary layer.
work, turbulent structures and free-surface effects were studied. Experiments were
performed along a 4.5 meter-long flat plate that pierced the free surface in an open-
surface water tunnel. In these experiments the Froude number, which describes the
ratio of turbulent kinetic energy to gravitational potential energy, was limited to
0.003, effectively eliminating the effects of waves and surface deformations from this
flow. The numerical study emulates this condition using a flat, free-shear boundary
condition at the free surface. The experiments were performed using both Laser
Induced Fluorescence (LIF) and shadowgraph for qualitative flow visualization and
Laser Doppler Velocimetry (LDV) to record single-component point velocity mea-
surements. Experiments were performed with free-stream velocities up to 21 cm/s.
From flow visualization studies, the authors found that vortex lines that connected
to the free surface caused localized depressions in the free surface, which act as a sig-
nificant source of transport for surfactant from the free surface into the bulk. Both
numerical and experimental results indicated the presence of a streamwise vortex
4
Figure 1.3: Drawing from Grega et al. (1995) hypothesizing the secondary flow
motions that exist in the mixed boundary corner.
near the intersection of the flat plate and the free surface. Additional secondary flow
features were found, but these results differed between experiments and simulations.
The authors hypothesized a secondary flow motion shown in Figure 1.3.
Longo et al. (1998) performed an experimental investigation of a towed, vertically-
oriented, surface-piercing flat plate. In this study, the flat plate was towed at low
speed (corresponding to a low Froude number) to minimize free surface disturbances.
To measure the flow in the vicinity of the plate, three-component point-velocity
measurements beneath the free surface were performed using a Laser Doppler Ve-
locimetry (LDV) system. It was found that a streamwise vortex was formed near
the solid/free-surface junction, while a second streamwise vortex rotating in the op-
posite direction was formed beneath the free surface in the outer boundary layer.
5
These vortices were found to have the effect of increasing turbulent mixing between
the boundary layer and outer flow region, leading to a thickening of the flat plate
boundary layer near the free surface.
Sreedhar and Stern (1998) performed a similar numerical study using Large
Eddy Simulation (LES) on a temporally evolving junction flow, incorporating a rigid
lid as an analog for the free surface. The rigid lid boundary condition is cited as
an approximation for a flat free surface, where vertical velocity as well as gradients
of the horizontal velocity components in the vertical direction are set to zero. A
comparison was made between this solid/rigid-lid and a solid/solid corner flow. The
mean cross-stream profiles appeared to be quite similar between these two cases,
forming an inner streamwise vortex with an outer flow that moved up along the
vertical wall and out along the horizontal boundary. The presence of streamwise
vortices at the solid/rigid-lid junction and reduction in vertical velocity fluctuations
found by the authors was consistent with prior research studies and their results are
shown in Figure 1.4. Additionally, it was found that despite a reduction in vertical
velocity fluctuations, an overall increase in turbulent kinetic energy near the free
surface was observed due to redistribution of vertical velocity fluctuations to the
two horizontal components.
Hsu et al. (2000) performed three-component LDV and horizontal planar Par-
ticle Image Velocimetry (PIV) measurments near a surface-piercing wall in an open-
surface water tunnel, extending the work from Grega et al. (1995). Measurements
were again performed at low speed in order to minimize free surface deformations.
In this research, the measurements of Longo et al. (1998) were called into question,
6
Figure 1.4: A series of plots from Sreedhar and Stern (1998) comparing the sec-
ondary flow motions found under a variety of boundary conditions.
as the flow was thought to be transient at the time of measurment. Additionally,
the computations of the solid-solid corner flow presented in Sreedhar and Stern
(1998) were questioned due to the lack of symmetry along a line that bisects the
corner. The subgrid turbulence model was also called into question, as it assumes
isotropy of turbulent fluctuations at small scales, which is considered by the authors
to be an invalid assumption. In their own study, the authors found that near the
solid boundary, streamwise velocity fluctuations are suppressed as the free surface
is approached, while far from the solid wall, distance from the free surface has little
effect on this quantity. Vertical velocity fluctuations are seen to decrease near the
free surface at almost all wall-normal distances measured. A monotonic decrease
in horizontal, wall-normal velocity fluctuations as the free-surface is approached is
seen, although far from the belt, wall-normal fluctuations peak immediately at the
7
free surface. Additionally, it is found that turbulent kinetic energy production and
dissipation are greatly reduced near the free surface, resulting in a redistribution of
turbulent fluctuations to surface-parallel components.
Grega et al. (2002) performed PIV in a cross-stream plane in order to con-
clusively determine the nature of secondary flows in the mixed boundary corner,
finding a slowly rotating streamwise vortex and wall-normal outflow from the wall
near the free surface., similar to the hypothesized flow shown in Figure 1.3.
Broglia et al. (2003) performed LES of an open-surface duct flow, utilizing
LES to simulate this flow at three different Reynolds numbers. This work attempts
to resolve discrepencies between prior research studies of a mixed boundary corner.
The authors find similar weak secondary flow motions as have been seen in previous
research, but also indicate the presence of a second outer vortex below the inner
streamwise vortex, which is assumed to be caused by the presence of the solid-
solid corner boundary at the bottom of the duct. The authors also examine the
budgets of the streamwise vorticity equation, examining the contribution of each
term to the mean secondary motion. These secondary motions are found to be
caused by anisotropy of Reynolds stresses, with the production and pressure-strain
redistribution terms having the largest effect.
To this point, boundary layer turbulence in the vicinity of a free surface has
only been studied at relatively low ship speed conditions where the free surface is
approximately flat or disturbed only by imposed gravity waves. In order to study
the physics of how turbulence interacts with the air-water interface, Shen and Yue
(2001) performed DNS of a turbulent shear flow in the presence of a deformable free
8
surface. It was found that the energy cascade to small scales desreases near the free
surface, while the stress becomes highly anisotropic due to the free-shear boundary
condition. Additionally, it was found that vortex lines connect to the free surface,
leading to the creation of regions where fluid is convected towards and away from
the free surface, referred to as ’splats’ and ’antisplats’ respectively. The authors
utilize these findings to develop more phyically-relevant sub-grid scale models for
LES simulations.
Teixeira and Belcher (2006) performed a theoretical study of the process by
which pressure fluctuations generated in a turbulent shear flow drive free surface
height fluctuations. It is found that surface fluctuations grow rapidly during the
initial development of the boundary layer, but this growth slows considerably as
time progresses. This is due to the development of long streaks in the flow, which
work to shift the turbulent forcing to larger scales. It is also found that under
certain conditions waves tend to grow through resonance, where the length and
velocity scales of the turbulent forcing matches that of the free surface waves. This
tends to cause preferential growth for freely-propagating waves. Additionally, this
resonant forcing corresponds to a time scale, which affects the decorrelation time of
the surface features.
The flow conditions leading to air entrainment have long been an object of
interest in a wide variety of flows. Chanson (1996) reviews a number of different
types of high velocity, air-entraining flows and discusses the commonalities between
them. The author finds that air entrainment in high-velocity flows can be caused
by breaking waves, droplets or jets impinging onto the free surface, by the action of
9
vorticies near the air-water interface, or by free-surface instabilities and turbulent
fluctuations near the free-surface. In reviewing previous studies, the author finds a
wide range of onset conditions for air entrainment specific to each flow geometry.
The author states that this variety of air entrainment processes are each dependent
in some way on gravity effects, surface tension effects, or viscous effects, which are
characterized by the Froude, Weber, and Reynolds numbers. Therefore, the author
argues, experimental investigations of these phenomena should only be performed
at full scale in order to retain the pyhsics of air entrainment that occur at full scale.
In order to better understand the conditions under which air entrainment oc-
curs, Brocchini and Peregrine (2001) utilize scaling arguments for critical Froude and
Weber numbers in order to determine the dimensional regimes for turbulent velocity
(q) and length (L) scales that correspond to air entrainment. The authors review a
wide range of flows in which turbulence interacts with a free surface and combine
this with scaling arguments to characterize surface features for different regions of
the q-L plane. A plot from this paper showing different free surface behaviors in
the q-L plane is shown in Figure 1.5, in which the upper region corresponds to the
conditions necessary for air entrinament. Three data points have been included in
this graph as estimates of q versus L in the ship boundary layer, based on classical
correlations for flat plate boundary layers for a plate (ship) speed of 15 m/s at three
streamise (X) locations. Because these estimates lie in the uppermost region of the
plot, air entrainment is predicted.
Only in recent years has air entrainment in wall-bounded turbulent flows be-
come the focus of some research studies. In research presented by Kim et al. (2013),
10
X = 0.5 m
X = 10.0 m
X = 100.0 m
Ballistic
Figure 1.5: Regions of various types of surface motions for free surface turbulence
with velocity fluctuation magnitude q (vertical axis) and length scale L (horizontal
axis), from Brocchini and Peregrine (2001). Air entrainment and spray production
occur in the upper region, above the uppermost curved line. The three data points
are values obtained for the turbulent boundary layer on a flat plate with q taken as
the rms of the spanwise (which is vertical for the boundary layer along a ship hull)
velocity fluctuation (w′) and L taken as the boundary layer thickness (δ).
direct numerical simulations (DNS) were performed of a turbulent two-phase couette
flow. In this study, two parallel, vertical walls move in opposite directions to create
a shear flow in the vicinity of a free surface. It is found that turbulent fluctuations
reach a maximum at a small distance from each sidewall, as shown in Figure 1.6.
Additionally, small air bubbles tend to accumulate near the sidewalls, while larger
bubbles are more highly concentrated near the center of the channel. A visualization
of the computed domain is shown in Figure 1.7. The very large concentration of
bubbles reported in this study seems unphysical and appears to be contradicted by
11
Figure 1.6: Two plots from Kim et al. (2014), showing wall-normal distributions
of turbulent velocity fluctuations in the (a) streamwise and (b) vertical (−) and
wall-normal (−·) directions.
experiments performed by Washuta et al. (2014), although the periodic nature of
the computational domain would limit the outflow of bubbles to only occur at the
free surface. This would essentially create a couette flow channel of infinite length,
as opposed to the finite length of the experimental domain. Kim et al. (2014) later
indicated that increasing the depth of the computational domain had a large effect
of reducing air entrainment, which could also contribute to the differences seen here.
1.3 Research Considerations for a Laboratory-Scale Experiment
In the proposed study, the turbulent free-surface boundary layer adjacent to
a ship hull is explored via a novel laboratory-based experimental setup. The diffi-
culty of designing laboratory scale experiments on ship boundary layers, including
12
Figure 1.7: Two figures from Kim et al. (2013), showing a visualization of air
entrainment in a turbulent Couette flow from (a) a side view and (b) an end view.
air entrainment due to the near-surface turbulence, centers around overcoming the
effects of gravity and surface tension. Consider that in the free surface boundary
layer, the air entrainment process is controlled by the ratios of the turbulent ki-
netic energy to the gravitational potential energy and the turbulent kinetic energy
to the surface tension eenergy. These ratios are given by the square of the Froude
number (Fr2 = q
2
gL
) and the Weber number (We = ρq
2L
σ
), where g is the gravita-
tional acceleration, ρ is the density of water, σ is the surface tension of water, q
is the characteristic magnitude of the turbulent velocity fluctuations and L is the
length scale of this turbulence. In order to retain the wave and bubble behavior
that control the air entrainment process at full scale, the Froude and Weber num-
bers must be matched in model scale. Because these experiments will be performed
on earth and in water, the gravity, density, and surface tension remain constant be-
tween scales. Therefore, in order to retain both Froude and Weber number scaling,
the full-scale length and velocity scales of naval combatant ships must be retained.
All previous studies of this penomenon have been performed near the limit of zero
13
Froude number with a nearly flat free surface, leading to a large gap in the present
knowledge. While some previous research has attempted to create a general frame-
work for understanding air entrainment processes in turbulent flow, these criteria
and mechanisms have yet to be applied to this problem. Therefore, this proposed
research will attempt to study the mechanisms by which turbulent boundary layer
fluctuations are capable of entraining air as is seen in real world flows by performing
experiments at full length and velocity scales of large naval ships.
In order to achieve this, these experiments will be performed using a flat-sided,
surface-piercing belt which will launch impulsively from rest and travel along its
length until it quickly reaches a constant speed. The development of the boundary
layer along the belt in time will be equivalent to the boundary layer development
along the length of a ship, where the total distance traveled by the belt is equivalent
to the length along the hull of a flat-sided ship moving at constant velocity, making
no bow waves. By using this impulsively started belt rather than a towed flat plate,
there is no leading edge, meaning that the air entrainment produced along the flat
belt can be isolated from other flow effects seen along the flat mid-hull section of a
ship.
In the following section, further details of the experimental apparatus will be
presented, along with techniques for measuring water surface profiles, sub-surface
velocity fields, and mechanisms for air entrainment. The final section will contain re-
sults of these measurements and a discussion of the findings, followed by conclusions
and directions for future work.
14
Chapter 2: Experimental Details
2.1 Overview
The present research study is an experimental investigation of the boundary
layer along a surface-piercing flat wall moving horizontally at full-scale ship speeds.
To create this moving wall in a laboratory setting, a stainless steel belt is driven in
a loop around two vertically-oriented rollers. The belt is started suddently from rest
to create a temporally-evolving boundary layer analagous to the spatially-evolving
boundary layer developed along a flat ship hull. The interaction of the boundary
layer turbulence and the free surface is investigated using a range of experimental
techniques. Water surface profiles are recorded in order to characterize height fluc-
tuations and wave propagation behavior at the free surface, while particle image
velocimetry measurements are performed beneath the free surface to determine the
interaction of turbulence with the free surface under a range of conditions. Addi-
tionally, white light movies from beneath the water surface are used to characterize
the air entrainment behavior at each flow condition.
15
2.2 Experimental Facilities
2.2.1 Water Tank
Experiments are performed in the large, open-surface water tank in the Hy-
drodynamics Laboratory at the University of Maryland, as shown in the photo in
Figure 2.1. This tank is approximately 13.4 m long, 2.4 m wide, and 1.3 m deep.
The walls and floor of the tank are made of 3.18 cm thick transparent, abrasion-
resistant acrylic sheets to allow for optical access from the bottom and on all four
sides. The supporting structure of the tank is composed of steel beams that support
Figure 2.1: A photo of the open-surface water tank, showing the transparent
acrylic walls and steel support frame.
16
the acrylic walls with minimal deflection under the pressure of the water. By mini-
mizing the deflection of the tank walls, the optical distortion when looking through
the polycarbonate is limited.
Figure 2.2: A top view photo of one of the seams between acrylic panels that form
the walls of the water tank. Each gap between panels is first filled with flexible
Tygon tubing to reduce the thickness of the silicone caulk layer, allowing for a
fully-cured caulk layer.
Because this tank is made of 25 separate acrylic panels, the seams between
panels had to be sealed with silicone caulk to prevent leaks. This silicone caulk needs
to be exposed to air in order to cure and therefore can only be applied in layers up
to 3/8” thick. Because the gaps between panels were as deep as the thickness of the
panels (3.18 cm), filling the entire gap with silicone would prevent the entire depth
17
of the seam from fully curing. Therefore, each seam between panels was first filled
with flexible Tygon tubing that was forced into the gaps, as shown in Figure 2.2,
before being covered with silicone sealant.
Figure 2.3: A photo of the skimmer at one end of the tank. Its height is adjustable
so that it can be set just below the desired water level in the tank.
A skimmer is fixed at one end of the tank, see Figure 2.3, with adjustable
height so that it can be used for a range of tank water levels. This skimmer is made
of 1 cm-thick acrylic and is sealed with silicone sealant to prevent water from leaking
into the box. This skimmer box is designed to be set just below the desired water
level of the tank so that the top layer of water pours over the edge of the box, forming
a small waterfall that contains any debris or surfactants that have accumulated on
the water surface. The skimmer is connected to the plumbing system of the tank
through a hose that connects a bulkhead fitting at the bottom of the skimmer to one
of two bulkhead fittings through the floor of the water tank. Using this connection,
the water that enters the skimmer can be either sent directly to the drain or pumped
18
through a diatomaceous earth filter and back into the opposite end of the tank to
keep the water level in the tank constant. Additionally, water from the bulk of the
tank (rather than from the skimmer) may be circulated through the filter. The
tank is typically filtered continuously when experiments are not being performed to
ensure that the water remains clean of particulate matter. This circulation pump is
turned off while experiments are performed to minimize disturbances in the tank.
Additionally, before experiments are performed, the skimmer is opened to the drain
and water is continuously added to the tank so that any surfactant or debris floating
on the water surface is skimmed to the drain.
Two stainless steel feet are set into the concrete floor of the lab and pass
through the bottom of the tank. These feet act as a mounting surface to support
the above-mentioned moving belt device, herein called the Ship Boundary Layer
(SBL) device, which is discussed below. These feet were made to be adjustable in
the horizontal plane before being permanently fixed so that the distance between
them could be set to match the distance between the feet of the SBL device. These
feet were positioned and fixed to the floor of the lab prior to the installation of the
acrylic walls of the tank. When the SBL device is in the tank, two large threaded
rods pass through the frame of the device and thread into these feet, securing the
SBL in place.
19
2.2.2 Ship Boundary Layer Device
In this experiment, a temporally-evolving boundary layer, which simulates
the boundary layer on the flat portion of the hull of a ship, is created using a
hydraulically-powered belt device. A top view schematic of the SBL device can
be seen in Figure 2.4. This device is made up of two hydraulically-driven rollers
with vertically-oriented axes that are separated by 7.5 m and that are attached to
a steel support frame. A stainless steel belt is driven in a loop around these two
rollers and one of the two straight sections of belt between rollers forms a vertical,
surface-piercing wall that translates along its length.
measurement 
region
water
entrance
fairing
2.4 m
13.4 m
6 m
7.5 m
Figure 2.4: Top view schematic of the SBL device and the tank where experiments
are performed.
One of these rollers that makes up this device can be seen laying on its side
in Figure 2.5, a photo taken before the device was assembled. Each roller is 0.46 m
in diamater with circumferential grooves designed to prevent hydroplaning, similar
to those on a rain tire. On either end of the roller, the axle of the roller passes
20
through a roller bearing, which allows the roller to rotate about its axis. On the
top of each roller, two hydraulic motors are arranged in opposite orientations, with
each motor driving a pulley attached to the axis of the roller, providing torque to
drive the roller.
Figure 2.5: A photo of one of the rollers during construction of the SBL.
To assemble the SBL device, the two rollers are attached at either end of a
steel support frame, as shown in Figure 2.6, a photo of the construction of the belt
device. One roller is directly fixed to one end of the SBL support frame and the
second roller is attached to the other end of the SBL support frame at the top
and bottom via two horizontally oriented hydraulic pistons. These pistons push in
and out to provide tension to the belt and to tilt the second roller relative to the
21
first in order to dynamically maintain the vertical position of the belt. When fully
assembled, a 0.8 mm-thick stainless steel belt is looped around these two rollers.
Figure 2.6: A photo of the belt construction in progress, showing major compo-
nents of the device, including the rollers, steel support frame, hydraulic motors, and
stainless steel belt.
The majority of the SBL device is enclosed in a stainless steel sheet metal box,
as shown in Figure 2.7, in order to keep the support frame and driving mechanism
dry and prevent the belt from hydroplaning off of the rollers. A schematic of this
setup can be seen in Figure 2.4. One of the two straight sections of belt between the
rollers exits the dry box and is exposed to the water in the tank for approximately
6 m before re-entering the box. The outer surface of this section of the belt forms
the surface of the simulated ship hull. The water level in the tank is set to a height
such that the belt pierces the free surface.
22
Figure 2.7: A photo of one end of the SBL, showing the stainless steel sheet metal
dry box.
At the locations where the belt enters and exits the sheet metal box, phenolic
laminate slats press against the belt to act as scrapers in order to keep the air con-
tained inside the box from leaking out into the test section and vice versa. Because
these slats and various other seals are imperfect, two sump pumps are positioned at
the bottom of the dry box at each end so that any water leaking into the box from
the tank is returned into the tank to keep the overall tank water level constant.
Each of these sump pumps is capable of pumping 4000 gallons per hour and has a
built in float switch so that they do not remain on when the box is completely dry.
The outlet of these pumps is positioned behind the SBL dry box so that this outflow
23
does not effect the flow field of the experiment. An overflow drain was plumbed into
the side of the SBL box at a height of approximately 5 cm from the bottom of the
sheet metal dry box, as shown in Figure 2.8. In the event that the pumps fail, this
drain allows excess water in the dry box to flow directly to the drain instead of
remaining in the dry box.
Figure 2.8: A photo of the overflow drain piping that allows the SBL dry box to
drain in the event of a sump pump failure.
Because of the thickness of the laminate scrapers and their mounting hardware,
the shape of the dry box at the location where the belt enters the water forms a
backwards-facing step. In order to reduce the effect of the flow separation caused by
this step, a stainless steel entrance fairing is attached to the dry box, as can be seen
24
in Figure 2.9. This fairing is designed with a length of 38 cm and a maximum angle
of 15 degrees in order to turn the backwards-facing step into a smooth, gradual
transition. The opposite problem occurs where the belt re-enters the box, so a
simpler fairing is placed at this location to reduce the splash caused by this high-
pressure region in front of the forward-facing step. This second fairing is located
significantly downstream of the measurement location and has no noticeable effect
on the experiment.
15.0°
38.00
Figure 2.9: A top view schematic of the entrance fairing, designed to create a
smooth transition over the backwards-facing step at the location where the belt
exits the SBL dry box.
A stationary, vertically-oriented, acrylic wall is fixed in the tank parallel to
the belt surface at a distance of approximately 1.17 m. The wall is 4.87 m long and
begins approximately 0.5 m downstream of the fairing exit. This wall is far enough
away from the belt to where it appears to have no effect on the flow field or waves
produced during a run. The area between this stationary wall and the tank wall
opposite the belt surface provides a region for return flow from the end of the tank.
This wall is also able to be moved closer to the belt for couette flow experiments
not discussed in this dissertation.
The SBL device is powered by a 60 horsepower hydraulic power unit (HPU), as
shown in Figure 2.10(a). This HPU contains a 250 gallon reservoir of hydraulic fluid
25
with two pumps that can achieve hydraulic fluid pressures up to 3000 psi. When
launching the belt, there is a large demand for high-pressure hydraulic fluid. In
order to meet this demand, the pressure line of the hydraulic system contains three
acumulators, two of which are shown in Figure 2.10(b). Each of these accumulators
is precharged with nitrogen gas to 1400 psi and contains a piston that separates
this gas from the hydraluic fluid. When entering high-pressure mode, the HPU
pumps hydraulic fluid up to 3000 psi, which compresses the nitrogen and fills the
the accumulators with high-pressure hydraulic fluid. This large volume of high-
pressure hydraulic fluid is then available to drive the hydraulic motors during the
launch of the belt.
(a) (b)
Figure 2.10: Two photos of the hydraulic system showing (a) the Hydraulic Power
Unit (HPU) and (b) hydraulic fluid accumulators.
The hydraulic system is designed to be used to power both the hydraulic
system as well as a high-speed towing carriage in the same water tank. The hydraulic
lines connected to the SBL device utilize quick disconnect fittings so that, when
26
transitioning to another experiment, these hoses can be disconnected and the SBL
device can be removed from the tank. Because this device weighs approximately
12,000 lbs, this is accomplished using two 3-ton cranes attached to horizontal ceiling
beams that run over the tank, as shown in Figure 2.11(a). In order to transition
between experiments, the hydraulic hoses are disconnected, the threaded rods that
secure the SBL device to the floor of the tank are removed, and two crane operators
simultaneously lift the device in the air, as shown in Figure 2.11(b). These cranes
are then able to translate along the ceiling beams to which they are attached. When
passing over the side wall of the tank, the SBL device has approximately 1 inch of
clearance, making this process difficult to coordinate. The device is then lowered
onto the floor outside of the tank, where it remains when not in operation.
(a) (b)
Figure 2.11: Two photos of the crane system used to move the SBL device in and
out of the tank. Image (a) shows one of the cranes that is used and (b) shows a
photo of the process of lifting the device.
When performing experiments, the belt launches from rest and is capable of
accelerating at up to 3 times gravitational acceleration until reaching constant belt
speeds of up to 15 m/s. In order to attain this rapid acceleration, the system needs
27
to utilize the power of all four hydraulic motors during launch. This four-motor
operation can only be sustained for very short periods of time before the demand
for high-pressure hydraulic fluid exceeds the ability of the HPU and accumulators
to supply this fluid. The hydraulic control system contains an option to switch from
four-motor operation to single-motor operation in the middle of a run, as a single
motor is capable of sustaining the belt motion at constant speed. This switch from
four motors to one creates a difficult challenge from a controls perspective, as that
single motor must supply four times the torque it had been supplying, while also
suddenly receiving all of the hydraulic fluid that had been going to all four motors.
Quite often, before the single motor can reach a new equilibrium, the velocity of
the belt experiences a large peak in velocity just after the switch. At the lower belt
speeds used in this set of experiments, this peak was found to reach approximately
1.5 m/s above the desired belt speed. To avoid this large discontinuity, it was
determined that using a single motor throughout the entire run was more desirable.
Due to this, the acceleration that the system is able to achieve is lower and the
launch time is increased, but the belt motion is more repeatable and overshoot is
greatly reduced.
The belt speeds used in these experiments are 3, 4, and 5 m/s. These speeds
seem to span a range of air entrainment conditions, where little to no air is entrained
at 3 m/s, some air is entrained at 4 m/s, and a large amount of air is entrained at
5 m/s. Further details of this air entrainment will be discussed later. For each
experimental setup, the belt launch reamins the same for a given belt speed. By
taking white-light video of the belt and cross-correlating successive images, it was
28
Time relative to trigger (s)
0 0.5 1 1.5
U
(m
/s
)
0
0.5
1
1.5
2
2.5
3
3.5
run 1
run 2
run 3
run 4
run 5
Time relative to trigger (s)
-0.5 0 0.5 1 1.5
x
(m
)
0
1
2
3
4
5
run 1
run 2
run 3
run 4
run 5
(a) (d)
Time relative to trigger (s)
0 0.5 1 1.5
U
(m
/s
)
0
1
2
3
4
5
run 1
run 2
run 3
run 4
run 5
Time relative to trigger (s)
-0.5 0 0.5 1 1.5
x
(m
)
0
1
2
3
4
5
6
7
run 1
run 2
run 3
run 4
run 5
(b) (e)
Time relative to trigger (s)
0 0.5 1 1.5
U
(m
/s
)
0
1
2
3
4
5
6
run 1
run 2
run 3
run 4
run 5
Time relative to trigger (s)
-0.5 0 0.5 1 1.5
x
(m
)
0
1
2
3
4
5
6
7
run 1
run 2
run 3
run 4
run 5
(c) (f)
Figure 2.12: Measurements of belt trajectory for 5 repeated runs at each belt
speed. The left column shows the belt velocity, U , versus time relative to the
trigger signal that comes from the belt control system. The right column shows the
amount of belt travel, x, versus time relative to trigger. Belt speeds of 3 m/s are
shown in (a) and (d), 4 m/s are shown in (b) and (e), and 5 m/s are shown in (c)
and (f).
29
determined that the belt launch is repeatable to within a 5 to 10 cm of belt travel
over the length of comparable runs, as can be seen in Figure 2.12.
Time relative to trigger (s)
0 0.5 1 1.5 2
U
(m
/s
)
0
1
2
3
4
5
6
U = 3 m/s
U = 4 m/s
U = 5 m/s
X (m)
0 1 2 3 4 5
U
(m
/s
)
0
1
2
3
4
5
6
V = 3 m/s
V = 4 m/s
V = 5 m/s
(a) (b)
Figure 2.13: A comparison of the average belt launch trajectory for each belt
speed. Plot (a) shows the belt velocity U versus time, while plot (b) shows the belt
velocity U versus x, the amount of belt travel.
Figure 2.13 shows a comparison between the belt launch trajectory for each of
the three belt speeds. As you can see in (a), each launch accelerates at approximately
the same rate, so at higher belt speeds, the belt travels farther before reaching full
speed. This effect is captured in (b), which shows the belt speed U versus the length
of belt travel, x. From these plots, it is found that during belt launches to 3, 4, and
5 m/s, the belt travels 0.85 m, 1.45 m, and 2.29 m before reaching 95% of the final
belt speed. These measurements of belt travel during launch will be disucssed later
in relation to the timing of other interesting events in these experiments.
When performing experiments, the high-speed cameras are triggered with a
circuit closure signal which is output by the hydraulic control system. In this way,
the timing of different runs can be matched to compare results to each other and
to the timing of the launch process. For a majority of the experiments discussed
30
herein, the length of belt travel studied is limited by the memory capacity of the
high-speed cameras. This typically limits the length of each run to ten seconds.
Therefore, at a belt speed of 3 m/s, the length of belt travel would be approximately
30 m, while at 5 m/s the belt would travel approximately 50 m. In order to make
equivalent comparisons between belt speeds, most of the results presented below are
only discussed for 30 m of belt travel. It is assumed that the length of belt that
travels past a fixed point can be considered equivalent to the distance along the
length of a ship hull. Therefore, the temporally-evolving boundary layer created
along the belt is considered comparable to the spatially-evolving boundary layer
along a flat-sided ship that makes no bow waves.
In a perfect experimental setup using this configuration, the rollers would be
so far apart that any fluid particle to pass through the measurement region during
a run would have experienced the same effect of the shear induced by the belt for
the entire experiment up to that point. In considering the effect of the finite length
of the belt test section, this assumption holds true for the beginning part of each
experiment in which the belt that passes through the measurement region has been
in contact with water for the entire run to that point. Because the section of belt
exposed to water at a given time is approximately 6 m, measurements are performed
4.75 m downstream of the end of the fairing and approximately 1 m upstream of
the downstream fairing. In this way, the measurement region is located as far
downstream as possible to maintain this assumption while still remaining outside
of the region of influence of the downstream fairing. Therefore, for the first 4.75 m
of each experiment, this assumption of fluid particles feeling the influence of the
31
belt for the entire experiment is valid. After this point, some fluid particles in the
flow are entrained from the region outside the influence of the belt and must be
accelerated into the boundary layer and it is unclear what influence this has on the
flow field.
2.3 Experimental Techniques
2.3.1 Surface Profile Measurements
Belt motion
Free surface
Light sheet from Argon CW laser
Digital movie camera
300 pps
Transparent tank bottom
Fluorescing dye
Water mixed with
uorescein dye
Side View
Belt moving normal
to page
Light sheet for LIF
Argon CW laser
Stainless steel belt
Water surface
End View
(a) (b)
Figure 2.14: Schematic drawing showing the set up for the cinematic LIF measure-
ments of the free surface shape from (a) a side view and (b) an end view perspective.
To study the water surface motion, a cinematic Laser Induced Fluorescence
(LIF) technique is utilized, as shown in Figure 2.14. In this technique, a continuous-
wave Argon Ion laser beam is converted to a thin sheet using a system of spherical
and cylindrical lenses. This sheet is projected vertically down onto the water surface
with the plane of the light sheet oriented normal to the plane of the belt. This
laser emits light primarily at wavelengths of 488 nm and 512 nm. During these
32
experiments, the water in the tank is mixed with fluorescein dye at a concentration of
about 5 ppm and dye within the light sheet is excited by the laser light, fluorescing at
a longer wavelength than the incident laser light. Two cameras view the intersection
of the light sheet and the water surface from both upstream and downstream with
viewing angles of approximately 20 degrees from horizontal. The cameras (Phantom
V641 by Vision Research, Inc.) capture 4-Mpixel, 12-bit black-and-whtie images at
frame rates of 1000 Hz. Utilizing 200 mm Nikon lenses, the images captured by the
cameras are each 2560 x 600 pixels in size and have a resolution of 7.96 pixels/mm.
This reduced vertical frame size allows the camera to store over twice as many
images in memory as with the full frame size, allowing the camera to record at this
high frame rate for the entire duration of a run.
A long-wavelength-pass 550 nm optical filter is placed in front of each camera
lens. These filters block out direct reflections of the laser light and transmit the
light from the fluorescing dye, thus preventing specular reflections of the laser light
from the water surface or belt surface from entering the camera lenses. The use of
two cameras allows for more accurate surface measurement in the event that the
intersection of the light sheet and the water surface is blocked by large deformations
of the free surface between the plane of the light sheet and either camera, although
this was unnecessary for the conditions presented in this research.
A sample LIF image from a belt launch to 5 m/s is shown in Figure 2.15(a).
The sharp boundary between the upper dark and lower bright region of each image
is the intersection of the light sheet and the water surface. The lower portion
of each image contains light that has been refracted when the light sheet passes
33
(a)
(b)
Figure 2.15: One sample frame from an LIF movie during a belt launch to 5 m/s.
Image (a) shows a raw image from the movie camera and image (b) shows that same
image with the result of the image processing plotted on top.
through the free surface and again when the fluorescing light passes through the
surface towards the camera, and is therefore not quantitative. The high-contrast
boundary where the light sheet intersects the free surface is analyzed quantitatively
to extract instantaneous surface profiles. Because these images were recorded from
the downstream camera, the belt surface appears on the right side of each image and
the belt moves out of the page. Because of the reflectivity of the belt, the intensity
pattern to the right of the belt location is a reflection of the light pattern on the left
due to the high reflectivity of the smooth surface of the belt. This line of symmetry
gives a good indication of the position of the belt in each image.
In order to extract these surface profiles, a novel image processing algorithm
was developed and applied to each frame of the LIF movies. In the first step of this
algorithm, this image is first broken into 64-pixel-wide windows. Each of these win-
34
(a)
(b)
(c)
(d)
(e)
Figure 2.16: Sample images from intermediate steps of the processing alogirthm
used to extract instantaneous surface profiles from LIF images
35
dows is then thresholded and recombined into one image, as shown in Figure 2.16(a),
in order to account for horizontal variations in light sheet intensity. In this image,
regions of connected white pixels above the free surface represent illuminated fea-
tures that are behind the plane of the light sheet. Any such regions not connected to
the surface are eliminated, as shown in image (b). The top-most white pixel in each
column of this image is used as the initial guess for the profile extraction algorithm.
This algorithm, discussed in more detail below, requires the input of a matrix that
contains a ridge of high pixel intensities that can be traced to form the surface. Due
to the many unique characteristics of the LIF images, a combination of a number
of different image transformations is used to create this matrix. The first operation
performed on the raw image is to determine its gradient in the horizontal and verti-
cal directions. Due to the nature of the free surface shape, most surface features can
be captured simply by using the gradient in the vertical direction. However, when
surface features develop a large slope, as is often the case at the meniscus, where the
water contacts the belt surface, this vertical gradient fails. Therefore, these individ-
ual gradients in each direction are combined into the total gradient image shown in
Figure 2.16(c), which is capable of capturing vertical or nearly vertical free surface
shapes. While the total gradient image shows many small edges, particularly below
the free surface, the intersection of the water free surface and the light sheet results
in a large band of high contrast. In order to emphasize this band, a separate op-
eration is performed to calculate the large-scale vertical intensity gradient. In this
operation, for each pixel in the image, the average of the intensity of the 5 pixels
above this location is subtracted from the average intensity of 5 pixels below, as
36
shown in (d). These images are then combined using a weighted average and the
resulting intensity map is again broken into 64-pixel-wide windows to normalize the
intensity of each part of the image.
In order to extract the free surface shape from this intensity image, a version
of the snake method, presented by Kaas et al. (1988), using dynamic programing,
as discussed by Amini et al. (1990) is utilized. In this method, the free surface is
defined as a snake, which is as an energy-minimizing spline that is guided by internal
spline forces and external image forces, defined below as:
Esnake =
∫ 1
0
Eint(v(s)) + Eimage(v(s))ds
Eint = (α(s)|vs(s)|2 + β(s)|vss(s)|2)/2
The first equation defines the energy of the snake as the sum of Eint, the in-
ternal energy of the snake, and Eimage, the energy of the image, which is defined by
the intensity map discussed in the previous section. The second equation defines the
internal spline energy, where v(s) is the position of the snake and α and β are ad-
justable parameters used to weight the relative importance of these two terms. The
first term in the internal spline energy equation is designed to penalize sharp spatial
changes in the shape of the snake, allowing it to act like a membrane, while the
second term penalizes the curvature of the snake, allowing it to act like a thin plate.
Therefore, the snake is attracted to high intensity regions of the prescribed intensity
map, Eimage, but penalized for discontinuous or non-smooth shapes. Through trial
37
and error, it was determined that values of α and β of 0.02 and 0, respectively, gave
good performance of this algorithm. Because of the rapid changes in curvature of
the free surface, use of a non-zero coefficient for the curvature term gave poor per-
formance. Typically this algorithm is capable of achieving consistent results within
± 1 pixel of the surface location judged by eye. Because of the complex optical
phenomena often present on the free surface, occasional errors were caused by the
presence of bubbles or by refracted light patterns below the light sheet intersection
line. These profiles were verified by eye and, if necessary, corrected using simpler
thresholding or sobel edge detection algorithms.
The horizontal location of the belt surface in each image was determined after
calculating these surface profiles. Because of the reflectivity of the belt, the image
processing algorithm was able to extract the surface shape even in the reflection of
the belt. Therefore, located towards the end of each profile, the portion of the profile
next to the belt surface was reflected as a mirror image of itself. To determine the
belt location, a one-dimensional cross-correlation was performed between a section
of the extracted profile and it’s mirror image. Because of this mirroring effect, the
location of the highest correlation value was taken to be the location of the belt
surface.
2.3.2 Fluid Velocity Measurements
In order to fully examine the complex interaction between boundary layer tur-
bulence and the free surface, it is desirable to study the turbulent velocity field
38
beneath the free surface. To accomplish this, two configurations of cinematic planar
Particle Image Velocimetry (PIV) measurements are utilized. In this technique, a
high-speed digital movie camera (Phantom V641, by Vision Research, Inc.) records
image pairs of tracer particles, illuminated by a laser light sheet, as they are con-
vected by the flow. These image pairs are recorded at a frame rate of 250 Hz, with
a frame size of 2560 x 1600 pixels. By cross-correlating the intensity map of small
interrogation windows between the frames of each image pair, a velocity vector field
is derived. Using this technique, the first set of measurements is performed by form-
ing a horizontal light sheet at varying distances from the free surface. In this way, it
is possible to study the impact of the free surface boundary condition on streamwise
and wall-normal velocity components.
Belt Surface
Water Free Surface
Horizontal Light Sheet
Transparent Tank Floor
High-Speed
Movie Camera
Figure 2.17: A side view schematic of the horizontal PIV setup.
39
Figure 2.18: A photo of the horizontal PIV setup made necessary by space con-
straints.
A schematic of the setup used for the horizontal PIV experiment is shown in
Figure 2.17. In this setup, a high-speed movie camera beneath the water tank views
a horizontally oriented light sheet located just beneath the free surface. Because of
limitations on space, the camera faces horizontally, with a large, flat mirror posi-
tioned at a 45 degree angle in front of it, as can be seen in the photo in Figure 2.18.
This mirror directs the camera’s view upward so that the illuminated horizontal
plane can be viewed. Using this orientation, the camera can look normally through
the floor of the tank, eliminating any optical aberrations that are caused by oblique
viewing.
40
Figure 2.19: A photo of the Type 309-15 calibration target used to correct for
optical distortions.
Calibration of the setup is performed prior to experiments using LaVision’s
DaVis imaging software and a Type 309-15 calibration target provided by LaVision,
as shown in Figure 2.19. This target is 309 mm by 309 mm with 3 mm diameter white
dots on a black background that are each spaced 15 mm apart. This calibration
target is used to transform the coordinates of the raw camera images into real-world
coordinates using a 3rd-order polynomial. This calibration procedure corrects any
distortions in the optical setup.
A horizontal light sheet is created using a high-repetition rate, double-pulsed
Nd:YLF laser which is directed above the tank using a series of laser-line mirrors,
and a set of two spherical convex lenses are used to prevent the beam from expanding
41
(a) (b)
Figure 2.20: Two images of the periscope used to create a light sheet below the
water surface showing (a) an overall schematic and (b) the configuration used in the
horizontal light sheet experiments.
as it travels. The effective focal length of a two-lens system is given by the equation:
1
f
=
1
f1
+
1
f2
− d
f1f2
In this equation, f is the effective focal length of the set of two lenses, f1 and f2 are
the focal lengths of each individual lens, and d is the distance between these lenses.
When d = f1 + f2, the terms on the right hand side cancel out and the effective
focal length goes to infinity. Assuming the incoming beam is collimated, the beam
that exits this two-lens system would also be collimated. Next, consider the case
when d = f1 + f2 - , where  is some small distance. In this case, the effective focal
42
length of the system would become:
f =
f1f2

Again, if  is small, this would result in the lens system having a long focal length,
potentially much longer than either of the individual lenses, and with a focal point
tunable by changing the distance between lenses. Therefore, when setting up the
optical system to create a light sheet, two spherical lenses with focal lengths of 10
and 15 cm were chosen simply because they were surplus equipment from a previous
experiment and the approximate combined distance between them of 25 cm was
convenient for the geometry of the optical system. First, these lenses were aligned
with the path of the beam that was set to travel up to the ceiling and above the
tank and then the distance between them was tuned so that the beam focused to a
point at the approximate distance from the laser that the light sheet will be located.
This is used as a sort of coarse focus for the light sheet.
From above the tank, this thin beam is directed downward and passed through
a piece of equipment similar to a periscope, as shown in Figure 2.20. This periscope
is made up of a tube with focusing optics and a pill-shaped head that contains a set
of laser-line mirrors and a cylindrical lens. This periscope assembly is air-tight so
that the pill-shaped head can be submerged in water while the optics remain dry.
Using this periscope, a laser beam is introduced from above the water surface into
the open top end of the tube and a light sheet is projected from the device beneath
the water surface. Because of this, the water free surface does not refract the beam
43
as it passes through the surface. In the horizontal PIV setup, the periscope head is
positioned approximately 80 cm from the belt surface and the light sheet is projected
directly towards the belt surface. In doing this, the distance of particle-filled water
through which the beam passes is reduced when compared to introducing the light
sheet from the far side of the tank. This is preferrable because the scattering of
light by particles outside of the measurement region reduces the intensity of light
that would eventually reach the region of interest. This periscope also contains a
set of focusing optics to allow for fine tuning of the light sheet thickness in the
measurement region.
Stainless steel belt
Moving normal to page 
Laser light sheet
High-speed 
movie camera
Water
surface
(a) (b)
Figure 2.21: Two images of the vertical PIV setup showing (a) a side-view
schematic and (b) a photo of the actual setup through the wall of the tank, showing
the orientation of the light sheet.
The second set of measurements is performed using a vertical light sheet par-
allel to the belt surface. This configuration provides a more detailed picture of the
variation of the streamwise and vertical velocity components as the free surface is ap-
proached. A schematic of this second PIV configuration is shown in Figure 2.21(a).
In this setup, a vertical light sheet is projected parallel to the belt surface using the
same periscope device discussed in the previous section. This periscope is positioned
44
downstream of the measurement region with the pill-shaped head reoriented so that
the light sheet is projected upstream and vertically up at an angle, as can be seen in
the photo in Figure 2.21(b). By positioning the head of the periscope downstream
and below the edge of the belt, its effect on the upstream flow is minimized. A high-
speed movie camera is mounted outside of the sidewall of the tank and views the
vertically-oriented light sheet straight on, without the plate mirror that is needed
in the horizontal PIV configuration. A similar calibration procedure as previously
mentioned is used to correct for optical abberations that are introduced by looking
through the tank wall.
When passing the laser beam into the top of the periscope tube, the periscope
is initially moved to its highest point to ensure that the beam is centered with respect
to the tube. Next, the tube is lowered to its eventual location and alignment with
the periscope tube is performed. Using this procedure, the beam is aligned with the
periscope tube so that there are no reflections from the reflective interior of the metal
tube. Positioning of the light sheet for each set of experiments is performed using
a ruler with a resolution of 1 mm. For the horizontal sheet, the water level is first
set by positioning the skimmer at the desired water level in the tank and a small,
continuous stream of water is added to the tank so that the skimmer continuously
overflows to the drain. This keeps the tank water level constant. Next, the light
sheet is focused using the initial position of the optics and periscope so that a thin
light sheet is formed near the belt surface. A clear ruler is then positioned in front
of the light sheet just at the location where the light sheet exits the pill-shaped head
of the periscope in order to give an initial measurement of the distance of the light
45
sheet from the free surface. Then, a measurement is performed near the belt surface,
far from the periscope. This should be the same as the previous measurement in
order to ensure that the light sheet is perfectly horizontal. The angle of the periscope
head is adjusted and these measurements are repeated iteratively until the distance
of the light sheet from the free surface remains constant over its entire path toward
the belt. Next, the vertical position of the light sheet is adjusted by translating the
periscope tube vertically until the desired light sheet position with respect to the
free surface is reached. When aligning the vertical light sheet, a similar procedure
is followed, with ruler measurements performed with respect to the belt surface
rather than the free surface. It should be noted that the focusing optics in the
periscope tube can cause some eccentric movement of the light sheet when adjusted,
changing the distance of the light sheet from the free surface or belt surface, so this
is performed prior to any light sheet alignment.
The flow is seeded with fluorescent tracer particles which are selected in or-
der to provide a sufficiently small settling velocity and sufficiently high frequency
response to faithfully follow the flow. As with the LIF experiments discussed above,
an optical long-wavelength-pass filter is placed in front of the camera in order to
block direct scattering of laser light by bubbles, the free surface, and the belt surface
and allow only fluorescent light from the particles to pass through to the camera sen-
sor. These particles are manufactured in the laboratory using the process described
by Pedocchi et al. (2008). In this process, a mixture of Rhodamine WT in epoxy is
created by first mixing 200 mL of MAS epoxy resin with 100 mL of slow-hardening
MAS epoxy hardener in a shallow pot or cake pan. Due to the exothermic curing
46
of the epoxy, these epoxy disks should remain 1 to 1.5 cm thick to ensure full cure
and this amount of epoxy is typically suitable for a 9-inch diameter pan. These
two components of epoxy should be mixed thoroughly for approximately 90 seconds
to ensure that the a proper homogeneous mixture of resin to hardener is main-
tained. After this mixing, 5 mL of Keyacid Rhodamine WT Liquid (703-010-27)
from Dyes.com is added to the epoxy mixture and is further mixed for 90 seconds, or
until the mixture appears to become completely homogeneous. This epoxy mixture
should then be allowed to cure for 24 hours to ensure that the mixture has fully
cured to become a hardened disk. This disk is then sanded iwith a belt sander to
produce a wide range of polydispersed particles. A sanding belt with 220 grit is
typically suitable to obtain a range of particles in desirable size ranges. This sand-
ing is typically performed in a ventilation hood with the operator of the belt sander
completely protected from inhaling these particles by using a respirator with N100
or P100 cartridges, which are rated to filter over 99.97% of particles. The operator
should also wear a disposable painter suit, rubber gloves that cover the wrists, and
goggles in order to prevent particles from getting onto their skin or clothes. This
sanding should be performed within a box in the vent hood in order to contain the
majority of particles that are created during this process. These particles can then
be easily collected using a dedicated vacuum with a 3-layer pleated filter, rated to
capture 99.5% of particles that are 0.5 microns or larger. By sifting these particles
through sieves of different mesh sizes, specific size ranges of particles can be created.
The vast majority of particles created by this process lie in the size ranges of 25 µm
to 53 µm and 53 µm to 106 µm.
47
In order to be considered suitable as tracers, these particles must have a set-
tling velocity that is negligible in comparison to flow velocities of interest. These
particles also should have a small enough settling velocity that the particles do
not settle out of the region of interest prior to or during the experiment. Using
Stokes’ Law as a simplified analysis of the particle settling velocity, it is found that
the smaller of these size ranges has a settling velocity between 0.044 mm/s and
0.20 mm/s, while the larger size range contains settling velocities up to 0.80 mm/s.
An additional consideration is that the tracer particles must have a sufficient fre-
quency response in order to represent the frequencies of fluid motion in the flow
field. Following the work of Mei (1996), an analysis of particle frequency response
can be performed using the following relation:
vˆp(ω)e
iωt = Hp(ω) · u(ω)eiωt
In this equation, u(ω)eiωt is the equation for a sinusiodal oscillation of a fluid
particle, vˆp(ω)e
iωt is the particle response function, and Hp(ω) represents the transfer
function between them. Via the analysis in Mei (1996), an equation for the square
of this transfer function, called the energy transfer function can be devised using
asymptotic analysis, taking the following form:
|Hp(ω)|2 = |Hp()|2 = (1 + )
2 + (+ 2
3
2)2
(1 + )2 + [+ 2
3
2 + 4
9
(ρ− 1)2]2
48
In this equation, ρ is the density ratio of the particle to the fluid and  is a Stokes
number, defined as:
 =
√
ωa2/2ν
In this equation, ω is the frequency of fluid oscillation in the flow, a is the particle
radius, and ν is the kinematic viscocity of the fluid. By substituting the particle sizes
in the ranges previously discussed, as well as the viscocity of water and the density
ratio of the particles of 1.13, according to information from the epoxy company, this
energy transfer function can be calculated for each particle size over a large range
of frequencies, as shown in Figure 2.22.
Frequency (Hz)
100 101 102 103 104 105 106 107 108
|H p
|2
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
25 µm particles
53 µm particles
106 µm particles
Figure 2.22: Particle frequency response for a range of particle diameters.
It can be seen from this analysis that any of the particles in these size ranges
retain over 84 percent of the energy of the forcing fluid flow at all frequencies.
Additionally, if you were to define an acceptable energy level for this energy transfer
function, say 90%, and evaluate each particle size at that energy level, the largest
49
particles would reach approximately 1.5 kHz, the mid-sized particles would reach
approximately 6 kHz, and the smallest particles would reach approximately 30 kHz.
Therefore, the smaller size range provides a considerably higher frequency range
over which the majority of energy in the flow is well-represented by the particle
response.
Finally, to be suitable for use as PIV tracers, images of these particles must be
a sufficient size and brightness and with a high enough seeding density in order to
perform proper cross correlation between image pairs. By performing preliminary
PIV tests with both size ranges of particles, it was determined that the particles
with diameters between 25 µm and 53 µm were the most appropriate for these
experiments. While the larger particles appeared brighter, the smaller particles
provided a better settling velocity and higher frequency response while maintaining
an acceptable level of brightness.
To introduce these particles into the region of interest of the flow, a local
seeding system is used rather than filling the entire tank with a homogeneous mixture
of particles. The concentration of particles that is required in the flow for properly-
seeded PIV images is high enough that a homogeneous mixture would cause the
water to appear cloudy, greatly reducing visibility as the camera has to look through
a large distance of water before seeing the region illuminated by the laser light sheet.
In addition, the amount of particles required to properly seed the approximately
8000 gallons of water in the tank would be prohibitive. Additionally, these particles
would settle out of the flow between runs and remixing the entire tank would be
difficult.
50
(a) (b)
(c) (d)
Figure 2.23: Four photos of the particle seeding system. Image (a) shows the
basin in which high-concentration particle solution is mixed. Image (b) shows the
pump that draws from this basin and pumps to (c) the particle seeder. Image (d)
is a photo from beneath the water of an initial test run of the particle seeder using
fluorescent dye rather than particles.
51
To create a local seeding system, a separate basin of approximately 85 gallons
is filled with a 50 mg/L mixture of particles in water, as shown in figure 2.23(a).
This mixture is then pumped from a bulkhead at the bottom of the basin into the
flow upstream of the measurement region using a 1/2 horsepower pump, shown in
image (b). To introduce these particles into the correct region of the flow, a length
of cylindrical PVC pipe with a long, narrow slot was created to evenly discharge the
particles along the length of the pipe, shown in images (c) and (d). The pump feeds
directly from the seeding basin into this pipe, which is positioned either horizontally
or vertically, depending on the orientation of the measurement plane in the PIV
setup. This pipe is positioned approximately 5 meters upstream and displaced
significantly from the measurement plane to avoid any disturbances in the wake of
the pipe from influencing the flow. Before performing experiments, the pump is
started and the belt is turned on at low speed (typically 1 m/s) for 30 seconds to
introduce some particles into the measurement region and then the belt is turned off
for 60 seconds to allow the water to calm down before an experimental run begins.
The particle seeder pumps out particles with a velocity of approximately 10 cm/s
and does not appear to have a significant effect on the flow.
Planar velocity fields consisting of streamwise and wall-normal velocity compo-
nents were measured in horizontal planes oriented perpendicular to the belt surface
and aligned with the streamwise flow direction. In these experiments, two separate
measurement planes were considered at distances below the undisturbed free sur-
face, D, of 14 cm and 2.5 cm. The location of the first of these planes, positioned
at a depth of 14 cm, was chosen to be far enough below the free surface to not be
52
influenced by any free surface waves or by the effect of the free surface boundary
condition. The second measurement location, positioned at a depth of 2.5 cm, was
chosen in order to be below the majority of bubbles and free surface depressions
that would disrupt the measurement technique, while still capturing effects of the
free surface on the boundary layer flow. These measurement locations allow for
comparison of the flow fields in order to study the modified physics in the vicinity
of the free surface.
Separate experiments were performed in which planar velocity fields consist-
ing of streamwise and vertical velocity components were measured in vertical planes
parallel to the belt surface. In these experiments, two separate planes were consid-
ered at distances from the belt surface, y, of 1.9 cm and 3.8 cm. These measurement
locations were chosen in order to study the primary region in which wave breaking
appears to occur.
In all experiments, images were recorded with a high-speed digital movie cam-
era (Phantom V641, by Vision Research, Inc.) using a Canon 180 mm lens with
remote focus and aperature control, allowing for adjustments to be made while the
camera was positioned under the tank. Image pairs were recorded at a frame rate of
250 Hz, with a frame size of 2560 x 1600 pixels and a long-wavelength-pass 550 nm
optical filter was placed in front of the lens to filter out any specular reflections from
the belt surface or bubbles in the flow. The images collected for the 14-cm-deep PIV
experiments have a resolution of 23.4 pixels/mm, with a corresponding field of view
of 10.9 cm x 6.8 cm. In moving the second sheet closer to the free surface and
farther from the camera for the D = 2.5 cm case, the field of view was increased to
53
(a)
(b)
Figure 2.24: Two images of the results from the PIV experiments with a horizontal
sheet far from the free surface during a launch to 4 m/s at x = 4.2 m. Image (a)
shows a sample raw PIV image and image (b) shows the processed vector field with
the vector spacing reduced by 1/2 and the vector length increased by 10 times for
clarity. The background is colored by vector length with a scale running from 0 to
2.5 m/s.
54
(a)
(b)
Figure 2.25: Two images of the results from the PIV experiments with a vertical
sheet close to the belt surface during a launch to 3 m/s at x = 5.9 m. Image (a)
shows a sample raw PIV image and image (b) shows the processed vector field with
the vector spacing reduced by 1/2 and the vector length increased by 10 times for
clarity. The background is colored by vector length with a scale running from 0 to
1.0 m/s.
55
12.1 cm x 7.6 cm, with a corresponding pixel resolution of 21.1 pixels/mm. In the
vertical-plane PIV experiments, the images of the light sheet closer to the belt have
a resolution of 14.6 pixels/mm and a field of view of 17.8 cm x 11.1 cm, while the y
= 3.8 cm condition yields images with a resolution of 14.8 pixels/mm and a field of
view of 17.2 cm x 10.8 cm. These PIV images were collected using a double-frame
mode, resulting in image pairs, which were processed using LaVision’s DaVis imag-
ing software. To process these images, a multi-pass PIV processing algorithm with
decreasing window size was used with a final interrogation window size of 32 x 32
pixels, using a 2 x 1 elliptical weighting function and a 50% overlap. The result-
ing vector fields were post-processed using median filtering and bad vectors were
replaced with values interpolated from surrounding vectors. Some sample images
from both the horizontal and vertical PIV setups can be seen in Figures 2.24 and
2.25. Each figure shows a typical raw PIV image on the left and its corresponding
velocity vector field on the right using this processing algorithm. This PIV process-
ing resulted in a vector spacing of 0.68 mm and 0.76 mm for the horizontal PIV
measurements at D = 14 cm and 2.5 cm, respectively. By calculating the viscous
sublayer thickness using the Schultz-Grunow formula from Schlichting (1979), as
shown in Figure 2.26, this vector spacing results in the first vector laying well out-
side of the viscous sublayer at around 60-120 viscous wall units. While it would be
preferrable to resolve this region, LIF movies show that air entrainment tends to
occur at larger scales. Vector spacings of 1.09 mm and 1.08 mm were attained for
the vertical-plane PIV measurements at y = 1.9 cm and 3.8 cm, respectively.
56
x (m)
0 10 20 30 40 50 60
δ ν
(μ
m
)
0
10
20
30
40
50
60
U = 3 m/s
U = 4 m/s
U = 5 m/s
Figure 2.26: A plot showing the viscous sublayer thickness based on the Schultz-
Grunow formula for skin friction, taken from Schlichting (1979)
Because there is water on both sides of the portion of belt that is exposed to
water, the belt is able to move in the wall-normal (y) direction under the influence
of the water. When designing the SBL device, it was assumed that the large surface
area of the belt would result in a large added mass, preventing it from moving
significantly. When performing PIV measurements, the belt was observed to move
in the y direction throughout each run. Therefore, the belt was tracked from frame to
frame using the raw PIV images in each movie. In this method, 25 pixel by 100 pixel
interrogation windows in the vicinity of the belt surface were cross correlated with
their mirror images in the y direction. This utilized the reflectivity of the belt surface
to determine the belt position in each frame and this was used as the reference from
which the PIV data was measured. The results of these measurements are shown in
Figure 2.27, which compares the belt position for 20 runs at each speed. It can be
seen that the belt moves outward by approximately 1 cm throughout each run with
57
x (m)
0 5 10 15 20 25 30
y
(m
m
)
0
2
4
6
8
10
12
(a)
x (m)
0 5 10 15 20 25 30
y
(m
m
)
0
2
4
6
8
10
12
(b)
x (m)
0 5 10 15 20 25 30
y
(m
m
)
0
2
4
6
8
10
12
(c)
Figure 2.27: Measurements of the y-direction belt position throughout each run
with each plot showing the results of 20 repeated runs at (a) 3 m/s, (b) 4 m/s, and
(c) 5 m/s.
58
x (m)
0 5 10 15 20 25 30
v r
m
s
U
×10-3
0
1
2
3
4
5
U = 3 m/s
U = 4 m/s
U = 5 m/s
Figure 2.28: Measurements of the RMS velocity fluctuations of the belt, normal-
ized by belt speed throughout each run.
a fairly repeatable motion. Using central difference to calculate the y-direction belt
velocity from this position data, the RMS velocity fluctuation of the belt can be
determined for each belt speed, as shown in Figure 2.28. It is unclear whether the
high-frequency content of these measurements is caused by actual belt vibrations or
by measurement error in using this technique. Due to the large surface area of the
belt, high-frequency vibrations seem to be a less likely cause of these fluctuations.
In either case, it is clear that these wall-normal belt fluctuations are very small
compared to the belt speed. As will be shown later, these values are about a factor
of 20 smaller than the wall-normal velocity fluctuations of the fluid near the wall.
2.3.3 Air Entrainment Measurements
In addition to the above-described surface profile and velocity field measure-
ments, observations of the onset of air entrainment are also desirable. While some
59
Figure 2.29: A photo of the sub-surface planar bubble measurement system, illu-
minated using two 650 watt flood lamps.
indications of surface breaking that might typically be associated with air entrain-
ment can be seen from surface profile measurements, this above-water perspective
cannot be used to definitively say whether a given event leads to air entrainment.
Therefore, to record these events, cinematic white light movies are recorded beneath
the water free surface, shown in Figure 2.29. The camera view used in this exper-
imental setup is similar to that described above in the vertical-sheet PIV section.
In these experiments, a high-speed camera is positioned to look directly at the belt
surface from outside the side wall of the tank. This region is illuminated using
two 650 watt flood lamps. Because the belt surface is partially reflective, this ar-
rangement results in any surface depressions or bubbles appearing as dark features
on a bright background. The resulting movies can be examined qualitatively to
determine the time at which air entrainment occurs for a given belt speed.
60
Belt Surface
Flood light
High-speed 
movie camera
Belt Motion
Water
WaterWater
Acrylic tank wall 
(3.18 cm thick)
1.
83
 m
Glass walls
Figure 2.30: A schematic of the underwater stereo bubble measurement system
A second configuration for this experiment is shown in Figure 2.30. This
configuration uses a cinematic stereo camera system in which two cameras view the
flow just under the water free surface adjacent to the belt. The cameras are set up
along the side wall of the tank that is parallel to the belt with horizontal viewing
directions and lines of sight rotated ±30◦ from a vertical plane that is normal to the
belt surface. A photo flood light with a translucent screen in front of it is placed next
to each camera. Since the belt surface is moderately reflective, each light provides
the illumination for the opposite camera. The cameras (lights) view (illuminate)
the scene through water-filled prism boxes that are set to create a straight line of
sight that is perpendicular to the side wall of the prism tank. Each camera is also
equipped with a Scheimpflug lens mount, which inclines the sensor plane relative to
the lens plane so that each camera may focus on a plane parallel to the belt despite
61
viewing from a displaced angle. Using two cameras at offset angles allows for a
more accurate characterization of the non-axisymmetric free surface features found
during entrainment events. This gives a better insight into the physical processes of
air entrainment that are occurring.
62
Chapter 3: Results and Discussion
3.1 Surface Profile Measurements
Surface profile measurements were performed at belt speeds, U , of 3, 4, and
5 m/s. Through initial trials, it was determined that a frame rate of 1000 fps was
necessary to provide a sufficient temporal resolution so that surface features could
be identified and tracked smoothly in successive frames. LIF images of the water
free surface next to the belt for an experimental run with U = 5 m/s are shown in
Figures 3.1 and 3.2. Though the camera recorded at a frame rate of 1,000 frames
per second, the five images in the figure are spaced out equally by distance of belt
travel, with the first image (a) in each figure taken at 0.0 s, the time when the
belt first starts to move. The instantaneous belt speed from the beginning of belt
motion through the acceleration portion until the belt reaches constant speed has
been measured separately and is used to correlate the time of each frame to the belt
travel distance. Here and in the following, rather than refer to images and data by
the time after the belt started moving, we refer to them by the distance, x, from the
leading edge of an equivalent flat plate, which is also the distance that the belt has
traveled. Thus, the images in Figure 3.1, showing the beginning portion of the run,
were captured at (a) 0 s, (b) 0.5 s, (c) 0.76 s, (d) 0.96 s, (e) 1.16 s, and (f) 1.35 s,
63
Belt Surface
Contact point
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.1: A sequence of five images from a high speed movie of the free surface
during a belt launch to 5 m/s. These images are taken at equivalent belt lengths
of (a) 0 m (b) 1 m (c) 2 m (d) 3 m (e) 4 m and (f) 5 m from the bow of the
ship. The high reflectivity of the stainless steel belt makes it appear as a symmetry
plane on the left side of the images. The horizontal field of view for these images is
approximately 31 cm.
64
Belt Surface
Contact point
(a)
(b)
(c)
(d)
(e)
Figure 3.2: A sequence of five images from a high speed movie of the free surface
during a belt launch to 5 m/s. These images are taken at equivalent belt lengths of
(a) 0 m (b) 5 m (c) 10 m (d) 15 m and (e) 20 m from the bow of the ship. The high
reflectivity of the stainless steel belt makes it appear as a symmetry plane on the left
side of the images. The horizontal field of view for these images is approximately
31 cm.
65
corresponding to x = 0.0, 1.0, 2.0, 3.0, 4.0, and 5.0 m, respectively. Similarly, images
from Figure 3.2 depict the later portion of the run, with images (a), (b), (c), (d) and
(e) captured at 0 s, 1.35 s, 2.35 s, 3.35 s, and 4.35 s, respectively, corresponding to
x = 0.0, 5.0, 10.0, 15.0 and 20.0 m. As discussed in the previous section, the plane
of the vertical light sheet is oriented normal to the belt surface and the cameras
look parallel to the belt surface and down at the water surface at a small angle from
horizontal. The images in Figure 3.1 are from the downstream camera, and these
images have been flipped horizontally for convenie so that the belt is near the left
side of each image and is moving out of the page, which will match the coordinate
system of later plots. The position of the belt is marked on the left side of image (a)
and the intensity pattern to the left of this location is a reflection of the light pattern
on the right due to the high reflectivity of the smooth surface of the belt. This line
of symmetry gives a good indication of the position of the belt in each image. The
sharp boundary between the upper dark and lower bright region of each image is the
intersection of the light sheet and the water surface. The upper regions of the later
images contain light scattered from roughness features on the water surface behind
the light sheet. These roughness features include bubbles that appear to be floating
on the water surface and moving primarily in the direction of the belt motion.
An example series of water surface profiles from a run with U = 3.0 m/s over
a range of x from 21 m to 21.3 m is shown in Figure 3.3. The horizontal axis
in the plot is horizontal distance, y, from the belt surface and the vertical axis is
water surface height above the mean water level. The profiles are equally spaced
in x by 1.2 cm and each successive profile is plotted 1.5 mm above the previous
66
0 50 100 150 200 250 300
0
50
100
150
y (mm)
z
(m
m
)
Figure 3.3: Sequence of profiles of the water surface during belt launch to 3 m/s
starting from x = 21 m. The time between profiles is 4 ms and each profile is
shifted up 2 mm from the previous profile to reduce overlap and show propagation
of surface features throughout time. The belt is positioned at the left edge of the
image (y = 0).
profile so that overlap is reduced and the evolution of surface features can be seen.
Surface features like ripple crests can be tracked over a number of successive profiles
and the slopes of imaginary lines connecting these features are an indication of their
horizontal speed away from the belt surface. It is clear that close to the belt surface,
the ripple features last for only a few profiles and their paths do not contain a single
dominant propagation speed. This region appears to be more chaotic than the
region to the right. In this outer region, surface features remain visible over many
frames giving support to the idea that they are freely propagating ripples, and their
velocity remains fairly constant. The red line in the figure, which was drawn by
67
eye to approximate that slope of the imaginary lines connection the ripple crests
in the outer region, corresponds to a velocity of about 34 cm/s. It should be kept
in mind that this is only the y-component of the ripple phase speed. Information
about the component of the phase speed in the x direction is unavailable from these
experiments, but could be obtained from experiments with the LIF system oriented
to record water surface profiles in planes parallel to the belt surface.
3.1.1 Wave Propagation
In order to study the differences in the propagation of free surface ripples in a
more quantitative way, we can perform cross correlation between profiles both close
to the belt (0.6 cm > y > 6.9 cm) and far away (21.4 cm > y > 27.6 cm), as shown
in Figures 3.4, 3.5, 3.6, and 3.7. The cross correlation function R is defined as:
R(∆y,∆t) =
∑
(Z1(y, t)− Z¯1)(Z2(y + ∆y, t+ ∆t)− Z¯2)√∑
(Z1(y, t)− Z¯1)2
√∑
(Z2(y + ∆y, t+ ∆t)− Z¯2)2
In this case, y is the horizontal, wall-normal coordinate, t is the time at which each
profile is measured, ∆y and ∆t are defined as the spatial and temporal shift between
correlated profiles, and Z1 and Z2 are the two profiles which are being correlated.
This cross correlation is performed for all profiles over lengths of belt travel from
x = 0 to 30 m, and the resulting correlation maps are summed into three groups
from x = 0 to 5 m, 5.85 m to 17.85 m, and 17.85 m to 29.85 m. This procedure is
repeated for belt speeds of 3, 4, and 5 m/s.
68
(a) (b)
(c) (d)
(e) (f)
Figure 3.4: Cross correlation maps for profiles from x = 0 to 5 m. The left column
shows cross correlation maps averaged over a region close to the belt and those in
the right column are averaged over a region far from the belt. The three rows from
top to bottom contain plots from U = 3, 4, and 5 m/s, respectively. The black line
in each plot indicates the ∆x location of the maximum correlation for each ∆t slice.
69
The first set of correlation maps in Figure 3.4 contains plots from early in each
run. The left column contains cross correlation maps from close to the belt and the
right column contains cross correlation maps far from the belt. The three rows from
top to bottom contain results from belt speeds of 3, 4, and 5 m/s, respectively. It
is clear that at this early stage, no waves have reached the region far from the belt,
so only a flat surface is being correlated. Close to the belt, a well-defined ridge is
beginning to develop, with correlations lasting for fairly long ∆t. Still, the ridge in
each correlation map is fairly broad, indicating that free surface features are moving
with a fairly wide range of speeds.
It is clear when comparing the results from the mid- and late- run cross corre-
lation maps that the correlation drops off very quickly close to the belt and that it is
difficult to pick out a single dominant velocity component. Far from the belt, how-
ever, the correlation lasts for long ∆t and has a clearly defined ridge, indicating a
single dominant velocity, particularly in the range of x from 5.85 m to 17.85 m. This
behavior could indicate that ripples close to the belt are heavily influenced by tur-
bulent velocity fluctuations beneath the surface and change in form rapidly. These
nonpropagating surface fluctuations are most likely the generators for the freely
propagating waves in the outer region, which retain their form for longer periods of
time. Alternatively, this could simply be an indication that these surface features
are primarily convected in the x direction and have a short enough wavelength in
that direction to become quickly uncorrelated. In addition, the correlation maps far
from the belt appear to weaken later in each run, which could be an indication that
70
(a) (b)
(c) (d)
Figure 3.5: Cross correlation maps for profiles from a launch to 3 m/s. Images (a)
and (c) show cross correlation maps averaged over a region close to the belt and (b)
and (d) are averaged over a region far from the belt. Images (a) and (b) are averaged
over x positions from 5.85 m to 17.85 m and images (c) and (d) are averaged over
x positions from 17.85 m to 29.85 m. The black line in each plot indicates the ∆x
location of the maximum correlation for each ∆t slice.
71
(a) (b)
(c) (d)
Figure 3.6: Cross correlation maps for profiles from a launch to 4 m/s. Images (a)
and (c) show cross correlation maps averaged over a region close to the belt and (b)
and (d) are averaged over a region far from the belt. Images (a) and (b) are averaged
over x positions from 5.85 m to 17.85 m and images (c) and (d) are averaged over
x positions from 17.85 m to 29.85 m. The black line in each plot indicates the ∆x
location of the maximum correlation for each ∆t slice.
72
(a) (b)
(c) (d)
Figure 3.7: Cross correlation maps for profiles from a launch to 5 m/s. Images (a)
and (c) show cross correlation maps averaged over a region close to the belt and (b)
and (d) are averaged over a region far from the belt. Images (a) and (b) are averaged
over x positions from 5.85 m to 17.85 m and images (c) and (d) are averaged over
x positions from 17.85 m to 29.85 m. The black line in each plot indicates the ∆x
location of the maximum correlation for each ∆t slice.
73
the growing boundary layer has a larger effect on modifying the free surface features
farther from the belt.
U (m/s)
2.5 3 3.5 4 4.5 5 5.5
c p
y
(c
m
/
s)
28
30
32
34
36
Figure 3.8: Wave propagation speed far from the belt averaged over 5.85 m <
x <17.85 m versus belt speed.
Looking at the correlation maps far from the belt during the middle portion
of each run, as can be seen from Figures 3.5(b), 3.6(b), and 3.7(b), the slope of the
ridge in the cross correlation maps appears to increase as the belt speed increases.
The black line in each image shows the ∆x location of the maximum correlation
for each ∆t slice. Using this, the slope of the ridge can be tracked, showing the
propagation speed of surface features, shown in Figure 3.8. This propagation speed
in the wall-normal direction increases with belt speed. Further information about
the propagation in the x direction is needed to more fully explore the significance
of this behavior.
74
(a) (d)
(b) (e)
(c) (f)
Figure 3.9: A sequence of six images from a high speed movie of the free surface
during a belt launch to 3 m/s. These images are cropped to have a horizontal field
of view of approximately 7 cm. Each image from (a) to (f) is spaced by 20 ms,
corresponding to 6 cm of belt travel between images.The first image occurs at x =
2.45 m
75
3.1.2 Free Surface Bursting
Another interesting phenomenon that occurs during the launch of the belt
that can be seen from these LIF images is that the surface appears to burst with
activity shortly after reaching constant velocity. Initially during launch, the belt
appears to have no noticeable effect on the free surface and appears to simply slip
past without creating waves. Shortly after reaching constant belt speed, the water
free surface near the belt appears to burst very suddenly, as you can see from the
sequence of images in Figure 3.9. Before this bursting occurs, the surface looks very
similar to image (a) for the entire acceleration period. After this point, free surface
fluctuations are continually generated close to the belt and this generation region
grows in time. This appears to be an indication of transition to turbulence, but this
transition point could be a moderated turbulence signal that has been filtered by
surface tension and gravity, which would act to smooth out free surface motions.
U (m/s)
2.5 3 3.5 4 4.5 5 5.5
x
on
se
t
(m
)
0
0.5
1
1.5
2
2.5
U (m/s)
2.5 3 3.5 4 4.5 5 5.5
R
e x
×106
0
2
4
6
8
10
(a) (b)
Figure 3.10: Two plots showing (a) the average x location of the onset of free
surface bursting versus belt speed and (b) the associated Reynolds number based
on x.
76
Therefore, the turbulence intensities may have to reach a critical threshhold in order
to mobilize the air-water interface. This will be further discussed in the following
sections. The x location of the onset of bursting in each run can be easily seen
from the LIF movies and additional repeated runs were performed to increase the
statistical significance of this measurement. Figure 3.10(a) shows that the average
onset location decreases with increasing belt speed, averaged over 20 runs at each
belt speed. The average Reynolds number based on x location of onset, shown in
(b) appears fairly consistent, but more belt speeds would be required to see if this
trend continues to hold true. Other potential scaling parameters for this bursting
onset will be discussed in more detail below.
3.1.3 Surface Fluctuations
2.5 3 3.5 4 4.5 5 5.5
0
0.5
1
1.5
2
2.5
3
3.5
4
U (m/s)
H
r
m
s
(m
m
)
Figure 3.11: RMS surface height fluctuation averaged over y and x versus belt
speed for an equivalent ship length of 30 m.
77
In order to characterize these free surface motions, the instantaneous surface
profiles extracted from the LIF movies are analyzed to determine RMS surface
height fluctuations, Hrms. A mean profile is first determined by averaging the time
series of surface height data for each wall-normal position throughout a given run.
This average surface height profile is then subtracted from each individual profile
and RMS fluctuations about the mean are calculated from the resulting data. This
data is averaged over five repeated runs for belt speeds of 3 and 4 m/s, while ten
repeated runs are utilized for the belt speed of 5 m/s due to its increased surface
fluctuation and wave breaking behavior. The RMS surface height is first averaged
across both wall-normal position and equivalent ship length to quantify the overall
mean fluctuation amplitude for each belt velocity, as shown in Figure 3.11. Because
these profiles are averaged across every surface height measured in both x and y for
each speed, the RMS values are the most statistically significant of the quantities
presented here. The RMS height fluctuations increase monotonically with increasing
belt speed, but a larger range of belt speeds would be required to gain an insight
into whether this trend is linear or if it follows some higher order behavior.
More detail about the RMS height fluctuations can be found by examining
their distribution in y. By averaging the RMS height fluctuation over the time series
of points for each wall-normal position, the average RMS surface height fluctuation
versus distance from the belt can be determined, as shown in Figure 3.12. These
profiles for each speed are averaged over the same equivalent ship length of 30 m.
While the surface fluctuations increase steadily from one speed to the next at all
values of y, some differences can be seen in the shape of the distributions. Each
78
y (cm)
0 5 10 15 20 25 30
H
r
m
s
(m
m
)
0
1
2
3
4
5
6
U = 3 m/s
U = 4 m/s
U = 5 m/s
Figure 3.12: RMS surface height averaged over repeated experimental runs versus
distance from the belt for three belt speeds.
surface height fluctuation profile reaches a maximum at the belt, but in the highest
speed case the values decrease more gradually with increasing distance from the belt
than in the lower speed cases.
3.2 Flow Field Measurements
3.2.1 PIV measurements with a horizontal light sheet
Because the boundary layer evolves temporally along the entire belt at the
same time, the processed PIV vector fields were able to be averaged in the streamwsie
direction for greater statistical convergence. Experiments at each light sheet location
and belt speed were also repeated 20 times for the purpose of ensemble averaging.
In addition, some averaging in x by ± 8 frames was performed for each condition,
which corresponds to 9.6 cm, 12.8 cm, and 16 cm of belt travel for belt speeds of 3, 4,
and 5 m/s, respectively. Velocity profiles, u(y), averaged in this way for belt speeds
79
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.13: Mean streamwise velocity profiles at a belt speed of U = 3 m/s for
(a),(c) D = 14 cm and (b),(d) D = 2.5 cm. Images (a) and (b) each plot a profile
at each 0.5 m of belt travel from x = 0 to 5 m, while (c) and (d) each plot a profile
at each 5 m of belt travel from x = 5 to 30 m.
80
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.14: Mean streamwise velocity profiles at a belt speed of U = 4 m/s for
(a),(c) D = 14 cm and (b),(d) D = 2.5 cm. Images (a) and (b) each plot a profile
at each 0.5 m of belt travel from x = 0 to 5 m, while (c) and (d) each plot a profile
at each 5 m of belt travel from x = 5 to 30 m.
81
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.15: Mean streamwise velocity profiles at a belt speed of U = 5 m/s for
(a),(c) D = 14 cm and (b),(d) D = 2.5 cm. Images (a) and (b) each plot a profile
at each 0.5 m of belt travel from x = 0 to 5 m, while (c) and (d) each plot a profile
at each 5 m of belt travel from x = 5 to 30 m.
82
of 3, 4, and 5 m/s can be seen in Figures 3.13, 3.14, and 3.15, respectively. The plots
in the left column of each figure contain profiles from the D = 14 cm condition, while
the plots in the right-hand column contain profiles from the D = 2.5 cm condition.
The plots in the top row depict profiles from each 0.5 m of belt travel from x = 0
to 5 m, while the bottom row contains profiles at each 5 m from x = 5 to 30 m. It
can be seen by comparing the plots in the top row that the boundary layer growth
appears quite different between these different conditions. Far from the surface, the
profiles from x = 0 to 3 m are very closely clustered together before the growth rate
suddenly accelerates after 2 to 3 m of travel. Near the surface, this burst happens
much more quickly, with the growth rate accelerating after only about 1 m of belt
travel. This burst appears to be an indication of transition to turbulence and the
onset location of this mean burst will be discussed later in relation to other events
that occur in the flow.
These mean velocity profiles can be used to calculate both the displacement
thickness and the momentum thickness up until the x location where the bound-
ary layer grows beyond the frame of the measurement region. The displacement
thickness is the distance that the surface would have to be displaced in order for a
uniform free stream velocity profile to retain the same flow rate and is defined using
the following equation:
δ∗ =
∫ ∞
0
(1− u(y)
U
)dy
The momentum thickness is the distance that the surface would have to be displaced
in order for a uniform free stream velocity profile to retain the same total momentum
83
x (m)
0.25 1 2 3 4 5
θ
(m
m
)
0
1
2
3
4
D = 14 cm
D = 2.5 cm
x (m)
0.25 1 2 3 4 5
δ
∗
(m
m
)
0
1
2
3
4
5
D = 14 cm
D = 2.5 cm
(a) (d)
x (m)
0.25 1 2 3 4 5
θ
(m
m
)
0
1
2
3
4
D = 14 cm
D = 2.5 cm
x (m)
0.25 1 2 3 4 5
δ
∗
(m
m
)
0
1
2
3
4
5
D = 14 cm
D = 2.5 cm
(b) (e)
x (m)
0.25 1 2 3 4 5
θ
(m
m
)
0
1
2
3
4
D = 14 cm
D = 2.5 cm
x (m)
0.25 1 2 3 4 5
δ
∗
(m
m
)
0
1
2
3
4
5
D = 14 cm
D = 2.5 cm
(c) (f)
Figure 3.16: Plots showing (a-c) momentum thickness and (d-f) displacement
thickness versus x calculated from measured boundary layer profiles for belt speeds
of (a, d) 3 m/s, (b, e) 4 m/s, and (c, f) 5 m/s.
84
and is defined using the following equation:
θ =
∫ ∞
0
u(y)
U
(1− u(y)
U
)dy
These relations can be applied to the velocity profiles at each x location for both
horizontal planes, as shown in Figure 3.16. These plots confirm that the boundary
layer grows faster closer to the surface, with both momentum thickness and dis-
placement thickness both indicating a thicker boundary layer at every x location.
Additionally, it can be seen in each momentum thickness profile that after an initial
period of slow boundary layer growth, a kink in the profile occurs, followed by a
higher boundary layer growth rate. This kink is an indicator of transition to turbu-
lence and again appears to occur earlier in the plane closer to the surface. As with
the x location of bursting seen in mean velocity profiles, the x location of this kink
can be measured from these plots and will be discussed in more detail later.
From the calculations of displacement thickness and momentum thickness, the
shape factor H can be defined as the ratio of displacement thickness to momentum
thickness. For a laminar Blasius boundary layer, this typically has a value of 2.59 and
for a turbulent boundary layer, a value between 1.3 and 1.4 is typical. Figure 3.17
shows the shape factor as a function of x for each belt speed and light sheet location.
These plots show a value of H between 2 and 2.25 that gradually increases during
the initial portion of each launch, except for the plot for U = 3 m/s at the deeper
measurement location. At this speed, in the time between the belt reaching full
speed (around 0.85 m) and transition to turbulence, the shape factor grows to a
85
x (m)
0.25 1 2 3 4 5
H
1
1.5
2
2.5
3
D = 14 cm
D = 2.5 cm
(a)
x (m)
0.25 1 2 3 4 5
H
1
1.5
2
2.5
3
D = 14 cm
D = 2.5 cm
(b)
x (m)
0.25 1 2 3 4 5
H
1
1.5
2
2.5
3
D = 14 cm
D = 2.5 cm
(c)
Figure 3.17: Plots showing shape factor, H, as a function of x for belt speeds of
(a) 3 m/s, (b) 4 m/s, and (c) 5 m/s.
86
value of around 2.9. Because the belt reaches full speed in this condition well before
transition, this could be an indication that the acceleration portion of the belt launch
is initially suppressing the shape factor. Alternatively, this higher peak could be due
to the relatively higher level of overshoot that occurs at this belt speed, as shown
in Figure 2.13, which could lead to an adverse pressure gradient. In each plot,
the initial value of shape factor forms a kink and transitions toward a value more
typical of a turbulent boundary layer, just below 1.4. As with momentum thickness,
this kink produces another indication of transition to turbulence. These numerous
indications of transition will be discussed in further detail below.
These momentum thickness calculations can also be performed at the x loca-
tions of free surface bursting presented earlier, as shown in Figure 3.18 (a). These
plots use blue dots to indicate velocity measurements from the deeper horizontal
plane, D = 14 cm and red dots indicate velocity measurements from the shallow
plane, D = 2.5 cm. Plot (a) shows the same plot for the x location of onset shown in
Figure 3.10(a). Figure 3.18 (b) shows the displacement thickness calculated at those
locations, while plot (c) shows the momentum thickness at each of those locations.
Plots (d), (e), and (f) use these calculated momentum thickness values as a length
scale in order to calculate Reynolds number, Froude number, and Weber number,
respectively. It can be seen in these plots that both the displacement thickness and
momentum thickness are greater closer to the surface, which is in agreement with
the idea that the boundary layer grows faster near the free surface. The Froude
and Weber number at bursting onset both appear to show a monotonic increase
87
U (m/s)
2.5 3 3.5 4 4.5 5 5.5
x
on
se
t
(m
)
0
0.5
1
1.5
2
2.5
U (m/s)
3 4 5
R
e θ
0
1000
2000
3000
4000 D = 14 cm
D = 2.5 cm
(a) (d)
U (m/s)
3 4 5
δ
∗ on
se
t
(m
m
)
0
0.5
1
1.5
2
2.5
D = 14 cm
D = 2.5 cm
U (m/s)
3 4 5
F
r θ
0
20
40
60
80
100
120
D = 14 cm
D = 2.5 cm
(b) (e)
U (m/s)
3 4 5
θ
on
se
t
(m
m
)
0
0.2
0.4
0.6
0.8
1
1.2
D = 14 cm
D = 2.5 cm
U (m/s)
3 4 5
W
e θ
0
50
100
150
200
250
D = 14 cm
D = 2.5 cm
(c) (f)
Figure 3.18: Plots showing different scaling parameters for bursting onset, calcu-
lated using measured velocity profile data. In all plots, blue dots indicate velocity
from D = 14 cm and red dots indicate velocity closer to the surface at D = 2.5 cm.
88
with belt speed, while Reynolds number based on momentum thickness appears to
provide the most consistent indicator of bursting onset.
In addition to mean velocity profiles, additional information can be gained
from looking at RMS velocity fluctuations. The RMS velocity at each location is
defined as the RMS fluctuation about a mean velocity profile at that x location. At
belt speeds of 3, 4, and 5 m/s, Figures 3.19, 3.20, and 3.21 each show a range of
streamwise RMS velocity fluctuations at a range of x values. In each plot, the top
row of plots contains profiles from every 1 m of belt travel from x = 1 to 5 m and
the bottom row contains profiles at each 5 m of belt travel from x = 5 to 30 m.
In each figure, the left column of plots again shows profiles for the condition D =
14 cm, while the right column of plots shows profiles for D = 2.5 cm. Far from the
surface, early profiles have a sharp peak near the belt and decrease away from the
belt. After about 2 to 3 m of belt travel, a second peak in this distribution begins to
grow around y = 3 to 4 mm. Close to the surface, a similar sudden burst of activity
occurs after only about 1 m of belt travel. The appearance of this second peak in
the urms distribution again seems to coincide with transition to turbulence. After
5 m of belt travel, this second peak flattens out and spreads quickly away from the
belt surface.
In order to better compare the streamwise flow at these two measurement
depths, Figure 3.22 shows plots of profiles at the same condition for both light sheet
depths, D = 2.5 and 14 cm. In these plots, mean velocity profiles are shown in the
left columnn and streamwise RMS velocity profiles in the right column. Each row
contains plots at each 1 m of belt travel from x = 1 to 5 m. These plots clearly
89
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 1.0 m
X = 2.0 m
X = 3.0 m
X = 4.0 m
X = 5.0 m
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 1.0 m
X = 2.0 m
X = 3.0 m
X = 4.0 m
X = 5.0 m
(a) (b)
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.19: Streamwise rms velocity profiles at a belt speed of U = 3 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. These profiles are plotted in (a) and
(b) at each 1 m of belt travel from x = 1 to 5 m and in (c) and (d) at each 5 m of
belt travel from x = 5 to 30 m.
90
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 1.0 m
X = 2.0 m
X = 3.0 m
X = 4.0 m
X = 5.0 m
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 1.0 m
X = 2.0 m
X = 3.0 m
X = 4.0 m
X = 5.0 m
(a) (b)
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.20: Streamwise rms velocity profiles at a belt speed of U = 4 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. These profiles are plotted in (a) and
(b) at each 1 m of belt travel from x = 1 to 5 m and in (c) and (d) at each 5 m of
belt travel from x = 5 to 30 m.
91
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 1.0 m
X = 2.0 m
X = 3.0 m
X = 4.0 m
X = 5.0 m
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 1.0 m
X = 2.0 m
X = 3.0 m
X = 4.0 m
X = 5.0 m
(a) (b)
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
urms(y)
U
0 0.1 0.2 0.3 0.4
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.21: Streamwise rms velocity profiles at a belt speed of U = 5 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. These profiles are plotted in (a) and
(b) at each 1 m of belt travel from x = 1 to 5 m and in (c) and (d) at each 5 m of
belt travel from x = 5 to 30 m.
92
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
urms(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
urms(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
urms(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
urms(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
U−u(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
urms(y)
U
0 0.5 1
y
(m
m
)
0
10
20
30 D = 14 cm
D = 2.5 cm
Figure 3.22: Streamwise profiles comparing the velocity profiles at different dis-
tances from the free surface. The left column shows plots of U and the right column
shows plots of urms. Each column contains plots at each 1 m from x = 1 to 5 m.
The top-most plots correspond to x = 1 m, while the bottom plots correspond to x
= 5 m. 93
depict the burst process observed in the earlier plots, as well as an overall faster
growth of the boundary layer near the free surface. The numerous indications of
transition to turbulence discussed within this section can be compared, as shown
in Figure 3.23. It is clear that each of these events can be used as an indication of
transition to turbulence and this plot gives an indication of the types of events that
this transition sets into motion. This x location of transition will later be used as a
reference point in discussing the timing of other events recorded throughout these
experiments.
U (m/s)
3 4 5
x
tr
a
n
si
ti
on
(m
)
0
0.5
1
1.5
2
2.5
3
3.5
4
Momentum Thickness Kink
RMS Knee
Mean Burst
Shape Factor Kink
Momentum Thickness Kink
RMS Knee
Shape Factor Kink
Shape Factor Kink
Figure 3.23: Plot of x locations of four different events that indicate transition to
turbulence. Points in blue represent data from measurements at D = 14 cm and
points in red represent data from measurements at D = 2.5 cm.
In addition to the streamwise velocity component, it is also useful to examine
the behavior of wall-normal velocity fluctuations, as shown in Figures 3.24, 3.25,
and 3.26. Each figure contains plots of profiles at each 0.5 m from x = 0 to 5 m in
94
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.24: Wall-normal rms velocity profiles at a belt speed of U = 3 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. In plots (a) and (b), profiles are
plotted at each 0.5 m of belt travel from x = 0 to 5 m, while in plots (c) and (d),
profiles are plotted at each 5 m of belt travel from x = 5 to 30 m.
95
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.25: Wall-normal rms velocity profiles at a belt speed of U = 4 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. In plots (a) and (b), profiles are
plotted at each 0.5 m of belt travel from x = 0 to 5 m, while in plots (c) and (d),
profiles are plotted at each 5 m of belt travel from x = 5 to 30 m.
96
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.26: Wall-normal rms velocity profiles at a belt speed of U = 5 m/s for
(a), (c) D = 14 cm and (b), (d) D = 2.5 cm. In plots (a) and (b), profiles are
plotted at each 0.5 m of belt travel from x = 0 to 5 m, while in plots (c) and (d),
profiles are plotted at each 5 m of belt travel from x = 5 to 30 m.
97
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
vrms(y)
U
0 0.05 0.1 0.15
y
(m
m
)
0
10
20
30
40
50
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b) (c)
Figure 3.27: Wall-normal rms velocity profiles for belt speeds of (a) 3, (b) 4, and
(c) 5 m/s at D = 14 cm.
the top row and plots at each 5 m of belt travel from x = 5 to 30 m in the bottom
row. Plots in the left column are far from the free surface and plots in the right
column are close to the free surface. In each figure, plot (a) shows that far from the
surface, these wall-normal velocity fluctuations contain a peak slightly away from
the wall that grows and moves farther from the wall with increasing x. Close to the
surface, however, these vrms profiles contain a sharp peak close to the belt surface
with a plateau farther from the belt. These profiles also appear to show overall larger
fluctuations in wall-normal velocity at all wall-normal positions. In comparing these
profiles for the three belt speeds in the D = 14 cm case, see Figure 3.27, it can be
seen that this near-wall peak seen in the near-surface conditions also appears to grow
with increasing belt speed in the deep-water condition. Plot (a) shows a peak in the
range of y = 3 to 5 mm for the lowest-speed case, while (b) shows a plateau of this
quanitity near the wall, and (c) shows a large, sharp peak near the wall. It is possible
98
that this region exists at low speed, with a layer that is simply too thin to resolve
with the present resolution. Additionally, this near-wall peak could be influenced
slightly by the y direction belt fluctuations, as shown earlier in Figure 2.27.
3.2.2 PIV Measurements with a vertical light sheet
While the velocity profiles extracted from the horizontal velocity field measure-
ments show that the flow is altered in the vicinity of the free surface, vertical velocity
fields are necessary to quantify this trend. In these vertical velocity fields, averaging
at each belt speed is cut off near the surface at a level below the deepest free sur-
face depressions, as determined from raw images. In this way, any cross-correlations
performed in regions without particles that may bias statistics are avoided. This
distance was found to be 8 mm, 17 mm, and 21 mm for belt speeds of 3, 4, and
5 m/s, respectively. Figures 3.28, 3.29, and 3.30 show streamwise mean velocity
profiles plotted at a range of x locations, with the top row of plots in each figure
containing profiles from every 0.5 m of belt travel from x = 0 to 5 m and the bottom
row containing profiles at every 5 m of belt travel from x = 5 to 30 m. The plots
in the left column contain profiles from close to the belt (y = 1.9 cm) and plots
in the right column contain profiles farther from the belt (y = 3.8 cm). It is clear
in all cases that these velocity profiles increase significantly as the free surface is
approached, particularly at low speed where statistics are valid for shallower depths
and at greater x locations when the boundary layer has grown significantly. This
behavior confirms the trends seen in the horizontal PIV measurements that indicate
99
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.28: Streamwise mean velocity profiles at a belt speed of U = 3 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the
run, with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c)
and (d) contain profiles plotted every 5 m from x = 5 to 30 m.
100
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.29: Streamwise mean velocity profiles at a belt speed of U = 4 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the
run, with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c)
and (d) contain profiles plotted every 5 m from x = 5 to 30 m.
101
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
u(y)
U
0 0.2 0.4 0.6 0.8 1
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.30: Streamwise mean velocity profiles at a belt speed of U = 5 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the
run, with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c)
and (d) contain profiles plotted every 5 m from x = 5 to 30 m.
102
a thicker boundary layer near the free surface. This depth of influence of the free
surface appears to become thicker at higher belt speeds, reaching depths of up to
5 cm at U = 5 m/s.
Additional information about the effect of the free surface boundary condi-
tion can be gained from examining streamwise velocity fluctuations, as shown in
Figures 3.31, 3.32, and 3.33, which each contain urms profiles plotted at a range of
x locations, with the top row of plots in each figure containing profiles from every
0.5 m of belt travel from x = 0 to 5 m and the bottom row containing profiles at
every 5 m of belt travel from x = 5 to 30 m. The plots in the left column contain
profiles from close to the belt (y = 1.9 cm) and plots in the right column contain
profiles farther from the belt (y = 3.8 cm). It can be seen in Figure 3.31, and
particularly in plots (a) and (b) from early in the run, that these streamwise rms
velocity profiles increase close to the surface, with the strongest effect appearing to
be limited to within 20 cm of the surface. In plot (a) of each of these figures, a
slight inclination of these profiles also appears to exist during the development of
the boundary layer close to the belt. Overall, the level of noise present in these
profiles prevents any conclusive statements about the behavior of the streamwise
rms velocity fluctuations from being made.
In addition to streamwise velocity profiles, vertical velocity fluctuations can
provide a valuable insight into the effects of the free surface boundary condition.
Figures 3.34, 3.35, and 3.36 each contain wrms profiles plotted at a range of x
locations, with the top row of plots in each figure containing profiles from every
0.5 m of belt travel from x = 0 to 5 m and the bottom row containing profiles
103
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.31: Streamwise rms velocity profiles at a belt speed of U = 3 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the
run, with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c)
and (d) contain profiles plotted every 5 m from x = 5 to 30 m.
104
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.32: Streamwise rms velocity profiles at a belt speed of U = 4 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the
run, with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c)
and (d) contain profiles plotted every 5 m from x = 5 to 30 m.
105
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
urms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.33: Streamwise rms velocity profiles at a belt speed of U = 5 m/s for
(a), (c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the
run, with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c)
and (d) contain profiles plotted every 5 m from x = 5 to 30 m.
106
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.34: Vertical rms velocity profiles at a belt speed of U = 3 m/s for (a),
(c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the run,
with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c) and (d)
contain profiles plotted every 5 m from x = 5 to 30 m.
107
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.35: Vertical rms velocity profiles at a belt speed of U = 4 m/s for (a),
(c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the run,
with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c) and (d)
contain profiles plotted every 5 m from x = 5 to 30 m.
108
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 0.0 m
X = 0.5 m
X = 1.0 m
X = 1.5 m
X = 2.0 m
X = 2.5 m
X = 3.0 m
X = 3.5 m
X = 4.0 m
X = 4.5 m
X = 5.0 m
(a) (b)
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
wrms(y)
U
0 0.05 0.1 0.15 0.2
z
(m
m
)
0
20
40
60
80
X = 5.0 m
X = 10.0 m
X = 15.0 m
X = 20.0 m
X = 25.0 m
X = 30.0 m
(c) (d)
Figure 3.36: Vertical rms velocity profiles at a belt speed of U = 5 m/s for (a),
(c) y = 1.9 cm and (b), (d) y = 3.8 cm. Plots (a) and (b) are from early in the run,
with profiles plotted every 0.5 m of belt travel from x = 0 to 5 m. Plots (c) and (d)
contain profiles plotted every 5 m from x = 5 to 30 m.
109
at every 5 m of belt travel from x = 5 to 30 m. The plots in the left column
contain profiles from close to the belt (y = 1.9 cm) and plots in the right column
contain profiles farther from the belt (y = 3.8 cm). In examining these profiles, it
can be seen that, as with vertical profiles of urms, the largest effect is seen within
20 mm of the undisturbed free surface at U = 3 m/s where data from this region
is considered valid. These plots appear to show an increase in vertical velocity
fluctuations close to the surface early in the run. Close to the belt, in plot (a) of
each figure, a slight, gradual increase in vertical velocity fluctuations can be seen as
the free surface is approached. This trend appears to reverse later in each run, with
a gradual decrease in vertical fluctuations close to the free surface as the boundary
layer becomes thicker. This behavior seems to show that the thicker boundary layer
leads to a quick increase of turbulent fluctuations close to the surface early on, but
as the boundary layer develops, the constraining effect of the free surface has greater
influence and works to reduce these fluctuations. As with the streamwise velocity
fluctuations, the level of noise present in these profiles prohibits firm conclusions
about these behaviors.
3.3 Air Entrainment Events
In order to study the air entrainment behavior in this boundary layer flow, it is
necessary to record movies from below the free surface. Using the same camera view
as was used in the vertical plane PIV experiments, white light movies were captured
in order to determine the onset location of air entrainment. Frames captured from
110
movies of belt launches to 3, 4, and 5 m/s at every 5 m of belt travel from x = 0 to
30 m are shown in Figure 3.37. It is clear from watching these movies that almost
no air entrainment occurs at U = 3 m/s. While a few bubbles are occasionally
entrained, this belt speed does not exhibit sustained air entrainment as the higher
speeds do. It can also be seen that the volume and quantity of bubbles entrained
at U = 5 m/s appear to be greater than during a launch to 4 m/s.
From watching these movies, air entrainment can be observed and recorded in
each movie. One method of judging the onset of air entrainment is to record the
first instance of air being entrained within the field of view. This method yields x
onset locations of 3.61 m and 2.66 m for belt speeds of 4 and 5 m/s, respectively.
Another method is to record the location of the first bubble to enter the field of
view of the camera. This yields air entrainment onset locations of x = 2.51 and
2.31 m for belt speeds of 4 and 5 m/s, respectively. While the first method gives
an exact measurement of when air is entrained in a given field of view, this method
appears to be somewhat more random and less consistent because it relies more on
local flow conditions to generate air entrainment. In addition, it is clear that air
entrainment often begins earlier than this recorded onset location because of the
presence of bubbles passing into the field of view of the camera. In either case,
no consistent air entrainment can be deduced for belt speeds of 3 m/s. These two
measurements of air entrainment onset location will later be discussed in relation to
the flow statistics presented previously.
From both surface profile movies and underwater white light movies, a vari-
ety of air entrainment events can be seen. Perhaps the most prominent of these
111
Figure 3.37: Three sequences of images from underwater white light movies at
three different belt speeds. The three columns contain frames from belt launches to
3, 4, and 5 m/s respectively. Each row contains frames from each 5 m of belt travel
from x = 0 to 30 m.
112
(a) (f)
(b) (g)
(c) (h)
(d) (i)
(e) (j)
Figure 3.38: Two sequences of images from LIF movies depicting potential air
entraining events. Images in the left column depict a trench closure event and
images in the right column depict a wave breaking event. Both sets of images are
from a belt launch to 5.0 m/s.
113
mechanisms begins with the development of deep narrow troughs that are elongated
in a quasi-streamwise direction. Some such troughs become so deep that the sides
collapse together. When the sides of the trough meet, air is often entrapped beneath
the free surface and bubbles begin to pinch off and convect downstream. Another
air entrainment mechanism is caused by ejections from the free surface. These ejec-
tions can fold over the surface and trap a pocket of air, similar to the process of air
entrainment in plunging breaking waves.
From the LIF movies, some evidence of both types of air entrainment events
can be seen. Figure 3.38 contains two sets of images to help illustrate the mechanisms
for air entrainment as seen from above the free surface. In the left column, a trough
collapse is depicted. In images (a) and (b), a deep narrow trough can be seen clearly
toward the left side of the image. In these first images, the left side of the trough is
beginning to turn over. In image (c), the two sides of the trough meet and a droplet
is ejected into the air. Presumably, this process entraps a pocket of air beneath
the free surface. In the right column, the second type of surface breaking event can
be seen. In the first image, (f), a jet can be seen forming near the contact point
of the free surface with the belt, toward the left side of the image. This jet ejects
from the surface and becomes nearly horizontal by image (h). In image (j), the
jet makes contact with the free surface, potentially trapping any air that remained
below it during the formation of the jet. Since the camera is above the free surface
and because of blockage of the camera line of sight by other free surface features
between the camera and the light sheet, the LIF images do not give us a clear
114
indication of whether or not air was actually entrained by either of these particular
events.
In order to be able to determine if air is entrained by surface breaking events,
another way to visualize the events at the air-water interface is by recording white
light movies beneath the free surface. This arrangement reduces line of sight block-
age by other surface features and allows for a more detailed study of how the air is
entrained rather than just showing the surface features that are likely to produce
bubbles. Figure 3.39 shows two sets of images from two views of the same event.
Each image contains both the feature of interest and its shadow projected onto the
belt by the light source for each camera. This can be seen especially well in im-
age pair (d)-(j). In image (d), the camera is on the left side, with the light source
coming from the right side, so that the actual trough marked as object 2, casts a
shadow shown in object 1. In image (j), the camera is on the right side with the
light source to the left, so the same trough is shown as object 4 with its shadow
projected to the right, shown as object 5. The relative horizontal distance between
and object and its shadow is an indication of the distance of the object from the
belt, where the smaller the distance, the closer the object is to the belt surface. In
the first three sets of images, the trough can be seen to become deep and narrow.
In the subsequent images, the left side of the trough can be seen turning over and
contacting the opposite side of the trough, as it pinches off and becomes a large
bubble in the final image pair. These events, where one side turns over completely,
seem to primarily form large bubbles when entraining pockets of air.
115
(a) (g)
(b) (h)
(c) (i)
1 2 3
5 64
(d) (j)
(e) (k)
(f) (l)
Figure 3.39: Two sequences of images from underwater white light movies of the
same air entrainment event. The belt is moving from left to right. Images from the
left column were captured with the left camera and images from the right camera
were captured with the right camera. This event occured during a launch to 4.0 m/s.
Because of the lighting and imaging configuration shown above in Figure 2.30, each
image contains both the bubble of interest and its shadow on the belt.
116
(a) (g)
(b) (h)
(c) (i)
(d) (j)
(e) (k)
(f) (l)
Figure 3.40: Two sequences of images from underwater white light movies of the
same air entrainment event. The belt is moving from left to right. Images from the
left column were captured with the left camera and images from the right camera
were captured with the right camera. This event occured during a launch to 5.0 m/s.
117
A second type of air entrainment event can be seen in Figure 3.40. As with the
previous Figure, this event is shown from two camera views in two columns. This
event begins, as shown in image (g), with the two sides of the trough closest and
farthest from the right camera meeting at a single point in the middle of the trough
and forming a round hole in the trough on the top left of the image. Following
this event, the hole spreads rapidly, propagating outward from the meeting point
and forming a large ligament of air that can be seen in image (h). In the following
images, this ligament can be seen retracting towards the surface as it deposits a
series of small bubbles into the flow.
3.3.1 Event Timing
In an attempt to study the influence of flow field parameters on free sur-
face bursting and air entrainment, the timing of these events can be compared.
Figure 3.41 shows a set of timelines that depict the timing of a variety of events dis-
cussed previously at each belt speed, with flow field events taken from near-surface
measurements. In these timelines, colored lines connect the same event between belt
speeds. It an be seen that the numerous events indicating transition to turbulence,
as discussed earlier and plotted in red, occur at a similar x location. The onset of
free surface bursting occurs closer to this transition point with increasing belt speed.
At 5 m/s, it appears that bursting occurs a short time after transition. In all cases,
bursting occurs before the secondary peak in streamwise RMS reaches a maximum.
The rise of this second peak away from the wall could contribute to the onset of
118
bursting and will be discussed in more detail below. Both measurements of air en-
trainment, presented in green, occur well after bursting and after the streamwise
velocity fluctuations away from the wall reach a maximum and begin to decrease.
x (m)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
U
(m
/s
)
3
4
5
Full Speed
Mean Transition
Momentum Transition
Shape Factor Transition
RMS Transition
Max RMS Peak
Free Surface Bursting
First Bubble
Air Entrainment
Figure 3.41: A set of timelines depicting the x location of various events described
throughout this dissertation.
The time series of velocity data collected for each speed can be used to calculate
the momentum thickness, θ, and urms at the location of the secondary peak (y =
3.04 mm) versus x throughout each run, as shown in Figure 3.42. Using these values
as length and velocity scales, a variety of scaling parameters can be calculated for
each x value in an attempt to determine the important physical processes influencing
free surface fluctuations and air entrainment. While the length scale θ increases
steadily throughout the run, this velocity scale reaches a peak and then decreases
fairly steadily after that point. This could indicate that air entrainment processes
are strongest in this region of rapid boundary layer growth and are reduced later in
each run.
From these velocity and length scales, the Weber and Froude numbers can
be calculated at each x location and these parameters can be compared across belt
119
x (m)
1 2 3 4 5
θ
(m
m
)
0
1
2
3
4
U = 3 m/s
U = 4 m/s
U = 5 m/s
x (m)
0 1 2 3 4 5
u
rm
s
(m
/
s)
0
0.1
0.2
0.3
0.4
0.5
U = 3 m/s
U = 4 m/s
U = 5 m/s
(a) (b)
Figure 3.42: Plots showing (a) θ vs x and (b) urms vs x for the initial 5 m of belt
travel at each belt speed.
speeds for both bursting and air entrainment. Figure 3.43 shows plots of Weber
number vs Froude number at each speed, with the values at bursting and air en-
trainment marked with connected lines, as in Figure 3.41. Plot (a) shows Weber
number and Froude number calculated using θ as a length scale and U as a velocity
scale. Using this scaling, both bursting and air entrainment appear to depend on
reaching a critical Weber number in order for onset to occur. While this appears
to be a reasonable conclusion for free surface bursting, this may not be true for air
entrainment. Because the momentum thicnkess length scale should continue to grow
beyond what has been calculated here and the velocity scale remains constant, the
Weber number at each belt speed will continue to grow. Therefore, even at a belt
speed of 3 m/s, the critical value of Weber number for air entrainment at higher
speeds will be surpassed. Therefore, perhaps a more appropriate velocity scale is
the urms of the secondary peak, discussed above. A plot of Weber vs Froude number
using both θ as a length scale and urms as a velocity scale are shown in Figure 3.43
(b). Using this scaling, it is unclear how free surface bursting depends on Weber or
120
Froude number, but air entrainment appears to scale better using these length and
velocity scales. From both measurements of air entrainment, it appears that a criti-
cal Weber number of approximately 3 to 3.5 must be reached before air entrainment
occurs. While this threshhold is reached at 3 m/s, it appears to only occur briefly,
while at higher speeds, the Weber number increases sharply after this point.
Weθ,U
0 500 1000 1500
F
r θ
,U
0
20
40
60
80
100
U = 3 m/s
U = 4 m/s
U = 5 m/s
Bursting
First Bubble
Air Entrainment
Weθ,urms
0 2 4 6 8 10
F
r θ
,u
rm
s
0
1
2
3
4
5
U = 3 m/s
U = 4 m/s
U = 5 m/s
Bursting
First Bubble
Air Entrainment
(a) (b)
Figure 3.43: Plots showing Weber number vs Froude number, calculated using
θ as a length scale and a velocity scale of (a) U or (b) urms at y = 3.04 mm, the
location of the secondary peak in streamwise velocity fluctuations.
An alternative explanation to the idea of a single critical Weber or Froude
number is that there is some combined effect of both scaling parameters. This idea
is expressed in Brocchini and Peregrine (2001), as shown in Figure 1.5. The recorded
data discussed above can be plotted on top of this figure, as shown in Figure 3.44.
While the boundary for air entrainment onset does not appear to be appropriate for
the length and velocity scales used here, the curves for higher belt speeds do appear
to reach further into the upper air entrainment region of the plot, indicating that
this figure could be describing the physics correctly.
121
Figure 3.44: A figure from Brocchini and Peregrine (2001) depicting critical regions
for air entrainment based on length and velocity scales, with recorded data of θ and
urms plotted on top.
3.4 Summary and Conclusions
The entrainment of air due to turbulent fluctuations in the boundary layers
of large naval ships is an important problem for the detectability of these ships.
Due to the large range of scales important to the physics of this problem, research
on this topic using traditional means, both experimental and computational, has
been limited to this point. A review of the literature found only research that
is tangentially related to this problem. In this dissertation, a novel laboratory-
scale device was created in order to experimentally study the interaction of the
turbulent boundary layer with a free surface. This device utilizes a stainless steel
belt, driven by hydraulic motors around two rollers as a surface piercing vertical
wall of infinite length. This belt accelerates in under 0.5 seconds to constant speed
122
in an effort to mimick the sudden passage of a flat-sided ship. Utilizing the full
length and velocity scales of large naval ships, this device creates a temporally-
evolving boundary layer analagous to the spatially-evolving boundary layer along
the length of a ship. Water surface profiles were recorded with a cinematic LIF
system to study the generation of surface height fluctuations by the sub-surface
turbulence. Sub-surface velocity fields were recorded using a cinematic planar PIV
system in order to study the modification of the flow field in the vicinity of the
free surface. Underwater white-light movies were recorded to determine the onset
location of air entrainment as well as mechanisms for this entrainment. It was found
that the free surface remains calm during the acceleration portion of the belt launch
before bursting with activity close to the belt. This burst location was seen to vary
from 1 to 2 m, depending on belt speed. Reynolds number based on x appear to
show a consistent indicator of bursting around Rex = 6 ×106. After this point, the
free surface fluctuates and changes form rapidly near the belt surface where wave
breaking begins to occur. These free surface height fluctuations are seen to peak
directly at the belt surface and decrease gradually away from the belt. At all wall-
normal positions, these fluctuations increase with belt speed. This Hrms peak near
the belt reaches a value approximately 50% higher than in the far field. The free
surface ripples created near the belt surface appear to lead to the generation of freely-
propagating waves far from the belt surface. The speed of these waves normal to the
belt surface increases with increasing belt speed, with waves traveling approximately
25% faster at U = 5 m/s when compared to those at U = 3 m/s. In studying
sub-surface velocity fields, it was found that the boundary layer exhibits similar
123
bursting to what is seen in free surface profiles, with an accompanying secondary
peak in streamwise velocity fluctuations. This second peak appears to occur at
a y location of approximately 3 mm from the belt surface and evidence of this
second peak is found in the literature for high Reynolds number boundary layer
flows. The rapid boundary layer growth seen in these velocity profiles happens
earlier in planes closer to the surface, leading to an overall thicker boundary layer
in the vicinity of the free surface. By analyzing these mean streamwise velocity
profiles and comparing the results to the x locations of free surface bursting, it is
found that both displacement thickness and momentum thickness at the location of
bursting decreases monotonically with belt speed, while Reynolds number based on
momentum thickness provides a fairly consistent indicator of free surface bursting
at Reθ of approximately 2300 based on deep velocity profiles and 4400 based on
shallow velocity profiles. Additionally, an overall increase in wall-normal velocity
fluctuations is seen close to the surface early in each run, with unusual behavior
occurring near the belt surface. While the near-surface velocity profiles always
show a near-wall peak in the vrms profiles, this peak is reduced at U = 4 m/s
and completely eliminated at 3 m/s when far from the surface. In vertical velocity
fields, boundary layer growth of up to 30% is found near the free surface and this
effect is apparent for depths of up to 5 cm. Free-surface normal fluctuations appear
to show a small increase toward the surface during the period of rapid boundary
layer growth, which then reverses and becomes a small, gradual decrease as the
free surface is approached later in the run. This again lends credence to the idea
that the boundary layer grows faster near the surface, while later in the run, the
124
restraining effects of surface tension and gravity become more dominant forces. From
underwater white-light movies, it is found that no sustained air entrainment occurs
at U = 3 m/s, while some air entrainment occurs at 4 m/s, and a large amount of
air entrainment occurs at 5 m/s. Additionally, different types of air entrainment
events are detected at small distances away from the belt, which entrain a variety
of bubble sizes. It is apparent that deep, quasi-streamwise troughs form close to
the belt and may become unstable before collapsing. In separate events, plunging
wave breaking appears to overturn the surface and entrap pockets of air. In all
cases, these air entrainment events appear to occur a small distance away from the
belt surface, appearing to coincide with location of the growing secondary peak in
steamwise velocity fluctuations. Onset of both bursting and air entrainment appear
to depend in some way on Weber number. Free surface bursting appears to scale
well with Weber number of approximately 200 based on momentum thickness and
belt velocity, while air entrainment appears to occur after reaching a Weber number
of approximately 3 to 3.5 based on momentum thickness and streamwise velocity
fluctuations. These scaling parameters appear to agree with the behavior of the
scaling arguments presented in Brocchini and Peregrine (2001), if not the exact
values for this air entrainment boundary.
3.5 Directions for Future Work
In future work, further exploration of vertical velocity fluctuations near the
surface could lead to a greater understanding of the coupling between sub-surface ve-
125
locity fluctuations and free surface deformations. An increase in repeated runs with
a vertically-oriented light sheet in order to increase statistical significance would be
the first step towards improving our understanding of this behavior. Alternatively,
while the present study measured these separately, a measurement campaign that
combined LIF with vertical-plane PIV could allow for measurements to more closely
approach the free surface and give insight into the flow field near particular events.
Additionally, LIF in planes parallel to the belt surface would lead to a greater un-
derstanding of the complete wave propagation behavior, particularly when close to
the belt. Finally, it would be desireable to study the mechanisms and location of
air entrainment in more detail, perhaps by extracting quantitative information from
stereo white light movies of bubbles, perhaps combining this with LIF in order to
direcly match wave breaking to air entrainment.
126
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