Lehigh University
Lehigh Preserve
Theses and Dissertations
1998
Free-surface wave interaction with a horizontal
cylinder
Peter M. Oshkai
Lehigh University
Follow this and additional works at: http://preserve.lehigh.edu/etd
Recommended Citation
Oshkai, Peter M., "Free-surface wave interaction with a horizontal cylinder" (1998). Theses and Dissertations. Paper 566.
This Thesis is brought to you for free and open access by Lehigh Preserve. It has been accepted for inclusion in Theses and Dissertations by an
authorized administrator of Lehigh Preserve. For more information, please contact [email protected].
Oshkai, Peter M.
Free-Surface
Wave Interaction
with a Horizontal
Cylinder
January 10,1999
Free-surface Wave Interaction
With a Horizontal Cylinder
by
Peter M. Oshkai
A Thesis
Presented to the Graduate and Research Corinnittee
of Lehigh University
in Candidacy for the Degree of
Master of Science
in
Applied Mathematics
Department of Mechanical Engineering and Mechanics
Lehigh University
November, 1998
ACKNOWLEDGEMENTS
I would like to extend my thanks to my advisor Dr. Donald Rockwell for creating
a great environment for study and research at the Fluids lab at Lehigh. I sincerely
appreciate his very valuable and professional advice in all aspects of my research and
academic life.
I am grateful to all people from the Fluids labs with whom I worked during this
program. I want to thank Dr. Jung-Chang Lin for sharing his expertise in experimental
techniques, Dr. Fevzi Unal for his invaluable help and advice on the force calculation
methods, and Ms. Oksan Cetiner for valuable ideas and being a great lab partner. I want
to express special thanks to Dr. Peter Vorobieff for his help and professional advice, for
influencing my view of academic research through his dedication and excitement, and,
most of all, for being a true friend.
I would like to thank my parents for their support and encouragement and my wife.
Svetlana for always being by my side and for her patience, support and help in absolutely
everything I do.
iii
TABLE OF CONTENTS
TITLE
i
CERTIFICATE OF APPROVAL
ii
ACKNOWLEDGEMENTS
.iii
TABLE OF CONTENTS
iv
LIST OF FIGURES
vi
ABSTRACT
1
r-
1. INTRODUCTION
1.1 Circular Cylinder in Oscillatory Flow
3
1.2 Circular Cylinder in Orbital Flow/Orbital Motion
6
1.3 Circular Cylinder Beneath a Nominally-Stationary Free-Surface
7
1.4 Unresolved Issues
8
2. EXPERIMENTAL SYSTEM AND TECHNIQUES
3.
3
10
2.1 Wave Tank
10
2.2 Experimental Apparatus
11
2.3 Strain Gauge Assembly
13
2.4 Laser Scanning and Image Acquisition Techniques
15
2.5 Image Processing and Interrogation
17
WAVB INTERACTION WITH A WELL-SUBMERGED CYLINDER:
PATTERNS OF VORTICITY AND VELOCITY-STREAMLINE TOPOLOGY
19
4. RELATION OF INSTANTANEOUS LOADING OF CYLINDER TO PATTERNS
OF VORTICITY
24
5. IDENTIFICATION OF FORCE CONTRIBUTIONS DUE TO INDIVIDUAL
VORTICES
28
6. EFFECTS OF DEGREE OF SUBMERGENCE OF CYLINDER: ..I
31
6.1 Initial Stage of Development of Patterns of Vorticity
31
6.2 Intermediate Stage of Development of Patterns of Vorticity
32
6.3 Final Stage of Development of Patterns of Vorticity
33
iv
7. CONCLUDIN"G REMARKS
REFERENCES
34
;
38
APPENDIX A: DEFINITION OF KEULEGAN-CARPENTER NUMBER AND
RELATED ISSUES
59
APPENDIX B: TIME SEQUENCES OF VORTICITY DISTRIBUTIONS
63
Case 1: Depth of Submergence hID = 1.57
63
Case 2: Depth of Submergence hID = 0.55
67
Case 3: Depth of Submergence hID = 0
71
APPENDIX C: TIME SEQUENCES OF MOMENTS OF VORTICITY
DISTRIBUTIONS
75
X-Moments of Vorticity
75
Y-Moments of Vorticity
92
APPENDIX D: Oscillations of the Cylinder in an Incident Wave: Effects of Phase Shift...
................................................................................................................................. 110
APPENDIX E: Additional Images
VITA
116
:
v
119
LIST OF FIGURES
Figure la: Configuration of experimental system.
43
Figure lb: Closeup of the test section.
44
Figure Ie: Overview of experimental arrangement and principal parameters for wavecylinder interaction.
45
Figure ld: Top view of the test section.
46
Figure Ie: Detailed schematic of the test cylinder.
47
Figure If: General procedure for image processing.
48
Figure 2a: Time history of patterns of positive (solid line) and negative (dashed line)
concentrations of vorticity and corresponding velocity field and streamline topology for
the case of relatively deep submergence of the horizontal cylinder beneath the freesurface wave. Frame number of cinema sequence is designated by N. Nominal
submergence of the cylinder is hID = 1.57. Minimum and incremental values of vorticity
are COmin = Aoo = 3 sec-I.
49
Figure 2b: Time history of patterns of positive (solid line) and negative (dashed line)
concentrations of vorticity and zoomed-in representations of corresponding velocity fields
associated with vortex pairs for the case of relatively deep submergence of the horizontal
cylinder beneath the free-surface wave. Frame number of cinema sequence is designated
by N. Nominal submergence of the cylinder is hID = 1.57. Minimum and incremental
values of vorticity are COmin = Aoo= 3 sec-I.
50
Figure 3: Dlustration of the principal types of trajectories of positive and· negative
vorticity concentrations. Level of submergence of cylinder beneath the free-surface is
hID =1.57.
51
Figure 4: Comparison of selected patterns of instantaneous vorticity with time histories
of horizontal Cx and vertical Cy force coefficients for relatively deep submergence of
cylinder hID =1.57.
52
Figure 5a: Calculated moments of vorticity (Mro)x corresponding to the horizontal (x)
direction force, in which hollow and filled square symbols represent respectively
contributions from positive and negative vorticity concentrations. Circular symbol
represents net sum of positive and negative contributions. Solid line corresponds to the
measured time integral of the horizontal force coefficient, ICxdt. Each frame number N
corresponds to an instantaneous PN image. Level of submergence of cylinder is
hID = 1.57.
53
Figure 5b: Calculated moments of vorticity corresponding to the vertical (y) direction of
force, in which hollow and filled square symbols represent respectively contributions
from positive and negative vorticity concentrations. Circular symbol represents net sum
vi
of positive and negative contributions. Solid line corresponds to the measured time
integral of the vertical force coefficient, ICydt. Each frame number N represents an
instantaneous PlY image. Level of submergence of cylinder is hID = 1.57.
54
Figure 6: Contributions of major concentrations of vorticity A, B, A', B',e and D to the
moments of vorticity corresponding to the x and y direction forces. Level of submergence
of cylinder beneath the free-surface is hID = 1.57.
55
Figure 7a: Comparison of instantaneous patterns of vorticity at the same frame numbers
N =7 and 9 for varying levels of submergence hID = 1.57, 0.55, and 0 of the cylinder
beneath the free-surface. Frame N =7 represents the instant where the crest of the freesurface wave is above the cylinder. Minimum and incremental values of vorticity are
COmin =3 sec- l and Am =3 sec-I.
56
Figure 7b: Comparison of instantaneous patterns or vorticity at the same frame numbers
N = 11 and 15 for varying levels of submergence hID = 1.57,0.55, and 0 of the cylinder
beneath the free-surface. Frame N = 15 represents the instant at which the trough of the
free-surface wave is above the cylinder. Minimum and incremental values of vorticity are
l
COmin = 3 sec- l and Am =3 sec· .
57
Figure 7c: Comparison of instantaneous patterns of vorticity at the same frame numbers
N =17 and 19 for varying levels of submergence hID =1.57, 0.55, and 0 of the cylinder
beneath the free-surface. Minimum and incremental values of vorticity are COmin =3 sec·I
and Am =3 sec· l .
58
Figure A-I: Characteristic velocity calculation.
61
Figure A-2: Mean velocity vectors (upstream region) for the stationary cylinder case.
62
Figure B-1: Time sequence of contours of constant positive (solid lines) and negative
(dashed lines) vorticity. Stationary cylinder case. Depth of submergence h =20 mm.
64
Figure B-2: Time sequence of contours of constant positive (solid lines) and negative
(dashed lines) vorticity. Stationary cylinder case. Depth ofsubmergence h =7 mm.
68
Figure B-3: Time sequence of contours of constant positive (solid lines) and negative
(dashed lines) vorticity. Stationary cylinder case. Depth of submergence h =0 mm.
72
Figure C-l: Time sequence of contours of constant x-moments of vorticity «Mro)y ) in a
propagating wave past a stationary cylinder at various depths of cylinder submergence.
76
Vll
Figure C-2: Time sequence of contours of constant y-moments of vorticity ((Mro)x ) in a
propagating wave past a stationary cylinder at various depths of cylinder submergence.
93
Figure D-l: Oscillations of a cyli~der in an incident wave: effects of phase shift. 113
Figure D-2: Images for maximum Fx case. Oscillating cylinder. Olmin = 3 sec· l and Am =3
sec· l ~
.
114
Figure D-3: Images for minimum Fx case. Oscillating cylinder. Olmin = 3 sec· I and Am =3
. 115
sec· l .
Figure E-l: Velocity fields and streamline topology at two instants of time for two
different depths of submergence.
117
Figure E-2: Images corresponding to frame number 9. Stationary cylinder. Various
depths of cylinder submergence.
118
viii
ABSTRACT
Classes of vortex formation from a horizontal cylinder adjacent to an undulating freesurface wave are characterized using high-image-density particle image velocimetry.
Instantaneous representations of the velocity field, streamline topology and vorticity
patterns yield insight into the origin of unsteady loading of the cylinder.
For sufficiently deep submergence of the cylinder, the orbital nature of the wave motion
results in multiple sites of vortex development, Le., onset of vorticity concentrations,
along the surface of the cylinder, followed by distinctive types of shedding from the
cylinder. All of these concentrations of vorticity then exhibit orbital motion about the
cylinder. Their contributions to the instantaneous values of the force coeffiCients are
assessed by calculating moments of vorticity. It is shown that large contributions to the
moments and their rate of change with time can occur for those vorticity concentrations
having relatively small amplitude orbital trajectories. In a limiting case, collision with the
surface of the cylinder can occur. Such vortex-cylinder interactions exhibit abrupt
changes in the streamline topology during the wave cycle, including abrupt switching of
the location of saddle points in the wave.
The effect of nominal depth of submergence of the cylinder is characterized in terms of
the time history of patterns of vorticity generated from the cylinder and the free-surface.
1
Generally speaking, generic types of vorticity concentrations are fonned from the cylinder
during the cycle of the wave motion for all values of submergence. The proximity of the
free-surface, however, can exert a remarkable influence on: the initial fonnation; the
eventual strength; and the subsequent motion of concentrations of vorticity.
For
sufficiently shallow submergence, large-scale vortex fonnation from the upper surface of
the cylinder is inhibited and, in contrast, that from the lower surface of the cylinder is
intensified. Moreover, decreasing the depth of submergence retards the orbital migration
of previously-shed concentrations of vorticity about the cylinder.
2
1. INTRODUCTION
The interaction of waves with structures in the ocean environment is a major source of
unsteady loading and vibration of a variety of structural components including long
cables, components of offshore platforms, risers, as well as a number of other related
geometrical configurations. In recent decades, substantial efforts have been devoted to
enhancing our understanding of this class of flow-structure interaction. In the present
investigation, the focus is on wave interaction with a horizontal cylinder.
Previous
related studies may be categorized according to: a circular cylinder in oscillatory flow; a
cylinder in orbital flow/orbital motion; and a cylinder beneath a nominally stationary freesurface. Summaries of these efforts are provided in the following.
1.1 Circular Cylinder in Oscillatory Flow
A variety of theoretical, numerical and experimental investigations have addressed the
case of unidirectional, oscillatory flow past a circular cylinder. Sarpkaya & Isaacson
(1981) summarize early studies, extending over a range of Keulegan-Carpenter number
Kc, where Kc = UrnTID = 21tA/D, in which Urn is the maximum velocity of the flow, Tis
the period of the wave motion, D is the cylinder diameter, and A is aniplitude. Recent
investigations include those of Singh (1979), Honji (1981), Bearman et.
at.
(1981),
Sarpkaya & Isaacson (1981), Ikeda & Yamamoto (1981), Iwagaki et
at.
(1983),
Williamson (1985), Sarpkaya (1986), Obasaju et. ai. (1988), and Tatsuno & Bearman
3
(1990).
These investigations provide insight into, among other features, the vortex
patterns that can arise over various ranges of Kc.
From a theoretical perspective, description of the relationship between point
vortices and loading on a cylinder was formulated by Sarpkaya (1963, 1968, 1969).
Using this type of approach, Maull & Milliner (1978) qualitatively related the movement
of visualized vortices about the cylinder to variations of the measured forces. Ikeda &
Yamamoto (1981) approximated the variation of the lift force by tracking vortices and
estimating their circulation using particle streak-marker visualization. Lin & Rockwell
(1996) employed quantitative images of the patterns of distributed vorticity generated by
an oscillating cylinder, and the lift force was approximated using the Maull-Milliner
(1978) approach whereby each vortex was represented as an equivalent point vortex of
given circulation. In addition, the vorticity moment approach of Lighthill (1986) was
employed to account for the distributed nature of the instantaneous patterns of vorticity.
Noca et. al. (1997) extended this technique to account for vorticity outside the field of
view of the near-wake of the cylinder. Unal et. al. (1997) and Noca (1997) invoked yet
another class of approaches for evaluating the forces. These methods, which are based on
instantaneous patterns of velocity obtained by quantitative imaging, involve momentumbased approaches that employ a control volume about the cylinder.
Numerical approaches to this class of flows include a variety of techniques.
Discrete vortex approaches have been employed and are -reviewed by Sarpkaya (1989)
and Lin et. ai. (1996). Finite-difference solutions of the Navier-Stokes equations were
used by Baba & Miyata (1987), Wang & Dalton (1991), Justesen (1991) and Miyata &
4
Lee (1990). Solution of the Navier-Stokes equations has also been approached by
Anagnostopoulos et. ai. using a finite-element analysis and by Sun & Dalton (1996) via a
method of large eddy simulation.
Insight into the decomposition of forces on cylinders in small-amplitude
oscillatory flows is provided by Bearman et ai. (1985a). The total force was considered
to be composed of three parts: inertia of the accelerating flow; effects of viscous
boundary layers, and shedding of vortices. Since this investigation focused on low values
of Kc, the dominant part of the force is the inviscid inertia force.
Onset of vortex
shedding from a circular· cylinder and a substantial contribution to the force occurs at a
value of Kc of the order of 5.
The rich variety of vortex patterns that are generated at larger values of flow (or
cylinder) oscillation amplitude, Le., Kc
= 21tAID,
are summarized, for example, by
Sarpkaya & Isaacson (1981), Williamson (1985) and Obasaju et. ai. (1988). The
interrelationship. between vortex patterns and the induced forces on the cylinder were
addressed by Obasaju et. ai. (1988) for values of Kc up to 55. Over this range, the
patterns of vortex formation were classified as asymmetric, transverse, diagonal and
quasi-steady, which, in turn, have important consequences for the forces on the cylinder.
Lin and Rockwell (1997) employed a technique of high-image-density particle image
ve10cimetry to determine the instantaneous patterns of vorticity and velocity about the
cylinder at a value of Kc =10. The physics of vortex formation were shown to involve a
complex series of events, described by vorticity and streamline topology. Moreover, in
presence of a free-surface, the nature of vortex formation was found to be a strong
.5
function of the degree of submergence of the cylinder. These features were related to
signatures of instantaneous lift.
1.2 Circular Cylinder in Orbital Flow/Orbital Motion
In the event that orbital, as opposed to unidirectional,.Jflow interacts with the cylinder, or
equivalently, orbital versus unidirectional motion of the cylinder occurs in quiescent
fluid, the processes of vortex formation and the associated loading are expected to be
altered accordingly.
Due to the orbital particle trajectories inherent to a wave, the
velocity vectors of the water particles rotate, in contrast to the horizontally-directed
velocities of a unidirectional, oscillatory flow.
This distinction was emphasized by
Ramberg & Niedzwecki (1979), who addressed the differences of the inertia Cm and drag
Cd coefficients of Morison's equation for planar oscillatory flow versus orbital flow.
Chaplin (1981a) calculated the limiting case of irrotational flow around a
horizontal cylinder under long-crested waves, using Milne-Thompson's circle theorem.
The presence of the cylinder was found to significantly distort the wave flow. Fully
accounting for viscous effects, Borthwick (1986)..performed a finite-difference solution of
the Navier-Stokes equations for the case of flow due to orbital motion of a cylinder.
Further numerical studies involving orbital wave motion past a circular cylinder were
performed by Miyata & Lee (1990) and Chaplin (1992, 1993). These studies show the
substantial influence of viscous effects and, at sufficiently high values of Kc, the
possibility of vortex formation, which significantly influences the flow patterns and the
corresponding forces on cylinders.in wave-induced flows.
6
Corresponding
experimental
investigations
have
focused
primarily
on
measurement of the loading on a cylinder. The forces on bodies in orbital motion were
measured by Holmes & Chaplin (1978), Chaplin (1981b), Sarpkaya (1984), Grass et. ai.
(1985), Borthwick (1987), and Chaplin (1988). Williamson et. ai. (1998) characterized
the fluid loading and vortex dynamics for a cylinder moving in elliptic orbits. It was
found that as the ellipticity of the orbit increases, there is a significant reduction in the
drag force. Moreover, for circular orbits, the mean radial force is directed inwards relative
to the orbit, due to the background circulation set up by repeated orbital cycles for the
case of orbital wave motion past a stationary cylinder. For the case of wave motion past a
stationary cylinder, Chaplin (1984) determined the forces on the cylinder for relatively
low Kc < 3. For Kc < 2, the inertial force provides the dominant contribution to the
loading. For Kc > 2, nonlinear contributions to the loading were found to be substantial.
Forces on horizontal cylinders under periodic and random waves, at Keulegan-Carpenter
numbers up to 20, were measured by Bearman et ai. (1985b). The force coefficients
were found to be similar for regular and random waves. ill random waves, the vortex
shedding is triggered when Kc exceeds a value of 7.
1.3 Circular Cylinder Beneath a Nominally-Stationary Free-Surface
ill contrast to the foregoing cases of oscillatory or orbital flow, or the corresponding
motion of a cylinder in quiescent fluid, where presence of the free-surface has not b~en an
issue, recent investigations have addressed the consequence of proximity of a free-surface
on the patterns of vortices formed both from the cylinder and the free-surface. .
7
Miyata et. al. (1990) studied the steady translation of a cylinder beneath a freesurface via qualitative visualization and numerical simulation. Using particle-streaks and
focussing on the case where the free-surface exhibited negligible distortion, they show
interesting patterns of vortices, which were interpreted with corresponding force
measurements. Sheridan et. al. (1997) undertook a quantitative investigation of the
nominally steady flow past a cylinder close to a free-surface, using the technique of highimage-density particle image velocimetry. The value of Froude number was sufficiently
high, such that substantial distortion of the free-surface could occur. Patterns of
instantaneous velocity and vorticity exhibit a variety of fundamental mechanisms of
generation of vorticity layers from the cylinder and the free-surface. The development and
interaction of these layers suggest important consequences for the loading of the cylinder.
In the event that a cylinder penetrates the free-surface and is subjected to oscillation in the
vertical direction, while the Froude number is maintained sufficiently low to avoid
significant free-surface distortions, still additional classes of vorticity concentrations can
develop in the near-wake of the cylinder, as characterized by Lin, Sheridan and Rockwell
(1996). The foregoing features may be present, in analogous forms, for wave motion past
a cylinder located close to a free-surface; this aspect has remained, however,
uninvestigated.
1.4 Unresolved Issues
The foregoing studies have provided considerable insight into the flow patterns and
loading due to oscillatory or orbital flow past a cylinder, or the equivalent motion of a
8
cylinder in quiescent fluid. To date, an experimentally-based investigation that provides
representations of the instantaneous, quantitative flow patterns about the cylinder,
acquired simultaneously with the measured loading, has not been undertaken.
At a
sufficiently high value of Keulegan-Carpenter number Kc, it is well known that shedding
of vortices will occur, but the manner in which these vortices are initially formed and
shed from the cylinder has not been 'addressed. Moreover, the nature of their orbital
migration about the cylinder is unclear. An intriguing possibility is the formation of
vortices from multiple sites about the surface of the cylinder during the imposed orbital
motion of the wave cycle, in contrast to what is known for the case of unidirectional
oscillatory flow or, for that matter, for the well known case of Karnuln vortex formation
from a cylinder in a uniform stream. Moreover, the eventual shedding of a number of
vortices from the cylinder during an orbital cycle of the wave motion is expected to
produce patterns of multiple, interacting vortices; the issue of whether the trajectories of
these vortices follow closely the imposed orbital motion of the wave, or are primarily
influenced by mutual induction effects, has not been characterized.
An adequate
understanding of all of these complex features requires space-time characterization of the
instantaneous patterns of vorticity and streamline topology, in conjunction with
instantaneous force measurements. The present investigation employs this approach.
9
2. EXPERIMENTAL SYSTEM AND TECHNIQUES
2.1 Wave Tank
Waves were generated with a paddle-type wavemaker having actively-controlled force
feedback to compensate for wave reflections. The total height of the wavelllaker was
0.90 m, relative to the depth 0.75 m of the quiescent water. At the opposite end of the
wave tank of length 9 m, a wedge-type beach constructed of absorbent foam material
provided additional attenuation of reflected waves. The motivation of the present
investigation is the particularly severe loading of structures occurring in a system of
incident-reflected wave systems having orbital particle trajectories of the type defined in
the visualization of Van Dyke (1982). The limiting case is a pure standing wave; such
wave cylinder interactions are described for a number of practical scenarios by
Naudascher & Rockwell (1994). A major criterion of the present experiments was to
retain the orbital character of the local wave motion past the cylinder, while attaining
essentiall'y phase-locked, spanwise coherent vortex formation. Preliminary experiments
revealed that wave motion having circular orbits, I.e., particle trajectories, yielded
degeneration of phase-repetitive vortex shedding and intermittent loss of quasidimensionality of the shed vortices, relative to waves having at least 2: 1 elliptical orbits.
These considerations led to generation of an orbital particle trajectory of the wave motion
by adjustment of the beach. The orbital trajectory had the following characteristics: ratio
of major to minor axis of 3: 1; angle of inclination eof approximately 41 0 with respect to
the horizontal; and ratio of major axis to the cylinder diameter of 2.2: 1, corresponding to
10
an effective Keulegan-Carpenter number Kc
=6.96 based on the amplitude of the major
axis. These parameters were determined at a location 2.35 D to the left of the center of
the cylinder, and at a constant depth of 1.57 D beneath the quiescent free-surface. At this
reference location, a total of 225 instantaneous velocity vectors were spatially averaged
over a square domain 0.97 D by 0.97 D, in order to obtain a single, representative
instantaneous vector of the wave motion. The maximum vector magnitudes of the wave
motion occurred in the first quadrant (8 = 44°, IVI = 0.044 m/sec) at instants
corresponding to image frame numbers (defined subsequently) N
third quadrant (8 = 224°, IVI = 0.034 m/sec) at N = 11.
=4 and 20, and in the
At the deepest value of
submergence, hID = 1.57, distortion of the wave motion at the reference location due to
presence of the cylinder is minim31, except for that portion of the wave cycle for which
the vorticity moves towards the reference location. At shallower depths of submergence,
especially for the surface piercing case, hID
= 0,
significant distortions of the wave
system occur in the near-field of the cylinder, Le., at the reference location.
2.2 Experimental Apparatus
A horizontal cylinder extending along the entire width of the wave tank was located a
distance of approximately 7.7 m from the wave paddle. The support system allowed
adjustment of the vertical position of the cylinder to various depths of submergence h, as
defined in Figure 1c. Figures 1d and Ie provide schematics of the test cylinder and
I
"
experimental system. The cylinder had a diameter D
=1.27 cm and a length of 36.8 cm.
It was mounted on a 3.2 mm2 sting with an arrangement of 8 strain gauges to measure the
11
in-line and transverse components of the fluid loading. End plates were employed at
either end of the cylinder. The end plate was 3 mm-thick Plexiglas, which allowed use of
particle image velocimetry. The gap between the end of the cylinder and this end plate
was kept at 0.7 mm. On the far side of the cylinder, the end plate had a thickness of 6.4
mm; it was beveled at an angle of 30°. This end plate was mounted in such a fashion that
it did not come in contact with the cylinder or the support sting. In order to allow
penetration of the sheet of laser light through the cylinder with minimum refraction, a
12 mm-wide thin-wall window was machined in the cylinder. The center of this window
was located a distance of 12 mm from the end of the cylinder, which was within the
region of spanwise coherent vortex formation, as visualized with a scanning laser sheet
across the tank. The window was filled with distilled water.
The entire cylinder apparatus can undergo arbitrary motion provided by high
resolution stepper motors (Parker Compumotor AX57-102) which have an angular
resolution of 25,000 steps per revolution. The motors are controlled by a microcomputer
via Parker PC-23 indexers. The motor displacement for each discrete time step is
specified by the Stream Function Generator (SFG) software (Magness, 1990). The
software Automated Laboratory Technician (ALT) (Magness and Troiano, 1991) executes
the command sequences produced by the SFG software for up to three stepper motors and
allows for simultaneous data acquisition through a Data Translation (DT 2801 series)
NO converter. Figure la shows a diagram of the experimental system.
During the first three experiments, the cylinder was kept stationary at the submergence
depths of 20 mm, 7 mm and 0 mm respectively. During two additional experiments, the
12
cylinder was undergoing a small-amplitude orbital motion obtained by programming a
sinusoidal and a cosinusoidal motions in horizontal and vertical directions respectively.
The experiments involving moving cylinder are discussed in appendix D.
2.3 Strain Gauge Assembly
The cantilevered strain gauge sting was constructed at ATLSS Instrumentation by
Edward Tomlinson. All details of the strain gauge design are documented in Strain Gage
Stings (Tomlinson, 1996). The sting is constructed of a 3.175 mm brass stock. A strain
gauge (Measurements Group, Inc.; EA-13-125BT-120) is mounted on each face of the
stock in order to measure the lift and drag components of the fluid loading. The strain
gauge sting has tree layers of waterproof and protective coating, which enables operation
of the sting under the water.
The same ALT program that was used for control of stepper motors was used for
force data acquisition. The lift and drag signals were amplified by Clip AE 301 S6
measurement amplifiers and then passed through analog filters (Krohn-Hite 3750) which
operated in the low-pass mode with a cut-off frequency of 20 Hz. The filter gain was set
at 20 db and the slope was 24 db per octave. The ALT software acquired the force data
through a Data Translation DT-2801 series AID converter. For each of the force signals
(lift and drag), a total of 4096 data points were acquired at a Nyquist frequency of 454.5
Hz. The gain of the AID converter was set at 8.
Several tests were developed to ensure proper operation of the strain gauge sting
during the experiment. The zero level of the output voltage was carefully adjusted and
13
monitored before and immediately after each experiment. In addition, known in-line and
transverse forces are required for suitable calibration of the strain gauge sting. The
cylinder was oscillated at the Keulegan-Carpenter numbers of 1 and 10, and the rms of
the force coefficient CPnns was measured for both in-line and transverse motions of the
cylinder, then compared with the results of Bearman et ai. (1985). Herein, instantaneous
force coefficients Cx and Cyare employed. They are defined as Cx = Fx/[~pU2Df] andCy
= FyI[Y2pU2Df], where Fx and Fy are the horizontal and vertical dimensional forces, U is
the maximum velocity of the orbital motion of the wave, and D and f are the cylinder
diameter and length respectively.
To obtain the expression for the force coefficient Cf of the form
Cf=k
E
1
zPU
2
where E is the voltage output of the strain gauge, p is the density of the water, U is the
maximum velocity of the flow, and k is the calibration coefficient, a static calibration was
performed for both vertical and horizontal directions by hanging weights from the end of
the cylinder. The calibration coefficient k was found to be equal 0.813 for the vertical
direction and 0.794 for the horizontal direction.
For the strain gauge configuration in the present apparatus, the polarity of the lift
and drag forces are positive meaning that an upward force results in a positive voltage,
and a drag force in the downstream direction results in the positive voltage.
14
2.4 Laser Scanning and Image Acquisition Techniques
Quantitative flow visualization was provided by a laser scanning version of
particle image velocimetry (PN). The ability of obtaining two-dimensional instantaneous
velocity fields with high spatial resolution makes PN a powerful tool for investigating
separated, unsteady flows. A two-dimensional velocity field can be used for calculation of
other flow properties such as vorticity distribution and streamline pattern, as well as other
information. Comprehensive reviews of the PN technique are presented by Adrian
(1991), Landreth and Adrian (1988), Rockwell et ai. (1993).
PN tracks the motion of neutrally-buoyant particles suspended in the fluid to
determine the velocity. In the present experiments, a mixture of metallic-coated glass
particles with an average diameter of 14
diameter of 10
~m
~m
and hollow glass spheres with an average
manufactured by Potters Industries Inc. was illuminated by a laser
sheet, approximately 1 mm in thickness. The laser sheet is created using the laser-optical
system described by Corcoran (1992). A continuous Argon-ion laser (Coherent Innova
series) with a maximum power of 30 watts produces a multi-band beam which passes
through a series of optical lenses. The beam is focused on a 72-facet rotating mirror
manufactured by Lincoln Laser Corp. As the mirror rotates, reflections of the laser beam
from its facets create a continuous vertical light sheet. In the case when the cylinder was
moving with the zero phase shift with respect to the propagating wave, the rotating mirror
was driven by a variable frequency motor at a frequency fm = 1.9 Hz to produce an
effective scanning frequency fsc =72fm = 136.8 Hz. In all other cases, the rotating mirror
was driven at a frequency fm = 2.9 Hz to produce an effective scanning frequency fsc =
15
208.8 Hz. The values of scanning frequency were chosen to optimize the distance
between particle images on" the PN image. To minimize the unsteadiness in the laser
scanning rate at lower rotational speeds, an external square function generator (Heath Co.
SG-1274) was used to provide a high precision square wave to drive the motor.
The entire optical system is located under the wave tank (Figure 1c). The steering
mirrors and the focusing lenses are positioned on a table that can be moved along the
length of the tank. In addition, the entire system can be translated along the width of the
wave tank to allow adjustment of the plane of interest. The circular support of the rotating
mirror allows the laser sheet to be oriented at any angle with respect to the direction of
wave propagation.
A Canon EOS-1 N RS camera with a 100 mm telephoto lens was used to capture
the illuminated flow field on 35 mm Kodak TMAX 400 film. In the case of the cylinder
moving with zero phase shift with respect to the propagating wave, a shutter speed of
1/40 sec and an f-number of 6.3 were used. In all the other cases, a shutter speed of 1/50
sec and an f-number of 4.0 were used. The maximum framing rate of the camera
(approximately 8 frameslsec for these shutter speeds) allowed to take approximately 16
frames during one cycle of wave motion (T
=2 sec) providing a time resolution AtIT =
0.0625.
The circular motion of the water in a propagating wave and large regions of the
separated flow in the plane of interest causes directional ambiguity in the velocity field
(Adrian, 1986). This ambiguity is eliminated by adding an artificial bias velocity to each
image. The bias velocity also decreases the dynamic range to permit better interrogation
16
of the image. The bias is added by using a 45 0 rotating mirror between the camera and the
laser sheet. The frequency of the bias mirror oscillation was equal 10Hz for ·all the
experiments. The peak to peak amplitude of the ramp signal for the stationary cylinder
cases was equal 0.432 V (depths of the cylinder submergence h
=20 mrn and h =7 mm)
and 0.480 V (h = 0 mm). In the case of the cylinder moving with zero phase shift with
respect to the propagating wave, the amplitude was equal to 0.272 V. In case of 1t radians
phase shift, the amplitude was equal to 0.340 V. The actual bias velocity added to the
flow ranged approximately from 2 to 6 times the velocity of the flow and was oriented in
the direction of the wave propagation.
The use of a bias mirror to produce image shifting may give rise to systematic
errors discussed by Raffel and Kompenhans (1995). This error is significant with
sufficiently large rotation angles of the mirror, when the mirror is located close to the
laser sheet, and with large magnification factors. In the present experiments, the bias
mirror was located at a maximum distance from the laser sheet and was rotated much less
than 10 • It has been assumed that the error associated with the use of a bias mirror is
negligible with respect to other uncertainties involved in the PIV technique.
The proper choice of the parameters described in this section allows acquisition of
multiple images of individual particles on the film negative. The spacing between the
particles is proportional to the vector sum of .the local and bias velocities.
2.5 Image Processing and Interrogation
The photographic negatives, having a resolution of 300 lines/mrn, are digitized
17
using a Nikon LS-351OAF 35 mm film scanner at a s~anning resolution of 125
pixels/mm. Each·image is saved as a Transferablehnage File (TIF) (Figure It).
The software PIV3 (Seke, 1993) uses the TIF files as an input and evaluates the
distances between particle images to calculate displacement vectors. The software uses a
single-frame cross-correlation method involving the application of two successive fast
Fourier transforms to a given window of the total image. An average displacement vector
is calculated for each window and is assumed to be located at the center of the window.
For the present experiments, a window size of 90 x 90 pixels (0.72 mm x 0.72 mm) with
50% overlap in both vertical and horizontal directions was used, resulting in a grid size of
0.36 mm x 0.36 mm on the film. In order to satisfy the high-image-density criterion, each
interrogation window contained a minimum of 50 multiply-exposed particle images. The
software TRACETIF (hnage Alchemy, 1992) and TWOBOUN (Lin, 1996) were used to
trace the contours of the free-surface and the cylinder in the TIF file.
The software V3 (Robinson, 1992) is used to display and evaluate the preliminary
flow field of displacement vectors. V3 enables the user to identify and remove
inappropriate vectors which are the result of shadows, reflections or laser sheet
distortions in the flow field.
The software NFILV (Lin, 1994) applies a bilinear interpolation algorithm to the
areas of the flow field for which the displacement vectors have been removed. NFILV
uses the magnification factor, laser scanning rate, and bias velocity as an I input to
calculate actual velocity vectors from the particle displacement vectors. The interpolated
velocity field is smoothed by a Gaussian weighted average technique described by
18
Landreth and Adrian (1989), with a smoothing parameter of 1.3, which provides minimal
distortion of the velocity field. , which provides minimal distortion of the velocity field.
The out-of-plane component of vorticity can be calculated from the two-dimensional
velocity field using the software SURFER (Golden Software Inc., 1989).
3.
WAVE INTERACTION WITH A WELL-SUBMERGED CYLINDER:
PATTERNS OF VORTICITY AND VELOCITY-STREAMLINE TOPOLOGY
Figures 2a and b show the time evolution of patterns of vorticity. In addition, Figure 2a
exhibits corresponding patterns of instantaneous velocity and superposed streamlines, and
Figure 2b gives close-ups of the patterns of velocity vectors near key vorticity
concentrations. Contours of positive and negative vorticity are designated respectively by
solid and dashed lines. The time sequence of images is represented by excerpts from the
sequence of frames N = 7 through 19, which extend over nearly one complete cycle of the
wave motion. At the instants corresponding approximately to frames N = 7 and 15, the
free-surface attains its maximum and minimum elevations respectively.
Frame N =7 shows three major concentrations of vorticity. Negative vortex A is
shed from the top of the cylinder, while positive vortices B and D develop respectively
along the right and bottom sides of the cylinder. Positive vortex C was generated during
the previous cycle of the wave motion and is located above the cylinder.
corresponding image of the velocity field at N
The
=7 shows that the instantaneous direction
of the wave is essentially from left to right, Le., the velocity vectors in the region of the
19
undistorted wave away from the cylinder are horizontal and essentially parallel to the
free-surface. The locus of vortex A is clearly identifiable in the instantaneous streamline
pattern, which takes the form of a limit cycle. A saddle point (locus of intersecting
streamlines) is located immediately to the right of this limit cycle. Moreover, an alleyway
flow occurs in the region between: the right half of the cylinder; and the streamline that
leads to the saddle point. A localized separation bubble is discernible along the right
surface of the cylinder; it corresponds to the vorticity concentration.B along that surface.
At a later instant, corresponding to frame N
=9, further development of vortices A
through D is indicated. Vortex A has approximately attained its maximum displacement
from the surface of the cylinder, vortices B and Dhave moved clockwise around the
cylinder and vortex C has translated horizontally to the right. At this instant, the velocity
field of the undisturbed portion of the wave is oriented in the downward, nearly vertical
direction. The large-scale negative vortex A is again clearly defined by the pattern of
streamlines. It exhibits an outward-spiral from its center, corresponding to an unstable
focus. In contrast to the streamline topology at N
=7, the saddle point has now switched
to the left side of vortex A. An alleyway flow exists between the top surface of the
cylinder and the streamline connected to the saddle point. This .alleyway flow is oriented
in the leftward direction, which is compatible with the outward-spiraling motion of the
velocity field and streamline pattern associated with vortex A.
At the instant corresponding to the frame N
= 11,
vortex A has moved back
towards the cylinder and is about to collide with its surface and undergo severe distention.
The vorticity concentration B indicated in frame N =9 has further moved in the clockwise
20
direction about the cylinder, separated from its surface. and joined with the vorticity
concentration D immediately adjacent to the left side of the cylinder. The consequence is
formation of a new, two-part vorticity concentration, henceforth designated as B,D.
Vortex C has moved still further to the right, and its peak vorticity level has decreased.
The corresponding velocity-streamline pattern indicates that the direction of the
undisturbed portion of the wave motion is downward to the left. The identities of vortex
A and all remaining concentrations of vorticity are not indicated by the streamline
topology.
Further development ofthe pattern of vorticity at frame N
= 13 (not shown herein)
involves interaction of vortex A with the cylinder and its severe distension about the
cylinder surface. At a substantially later instant of time, represented by frame N
=15 (top
of Figure 2b), a remnant of vortex A is evident, but the most predominant features are the
large-scale vortex B,D that has separated from the surface of the cylinder and,
simultaneously, onset of a vorticity concentration A' immediately adjacent to the left side
of the cylinder. Vortex C has continued its orbital-like motion about the cylinder and
translated downward relative to its position in frame N = 11. At this instant, N
=15, the
free-surface attains its minimum elevation corresponding to the trough of the free-surface
wave. The corresponding image of the velocity pattern shows an enlarged view of the jetlike flow between the counterrotating vortex system B,D and A ~
In frame N = 17, the large-scale positive vortex B,D has translated up and to the
left of the cylinder. Vortex A' has continued to increase in scale while continuing to
move in the clockwise direction about the surface of the cylinder and vortex C exhibits a
21
further decrease in maximum vorticity level. The corresponding image of the velocity
field again focuses on the jet-like flow associated with the system B,D and A' .
Finally, frame N =19 shows the continued upward movement of vorticity
concentration B, along with the matured development of vortex A', which has migrated
further in the clockwise direction about the surface of the cylinder. Furthermore, a new
positive vorticity concentration B' develops along the lower right surface of the cylinder.
The zoomed-in view of the velocity field reveals the large velocity gradient associated
with the development of vortex A' in the near-wake of the cylinder. This region of high
gradient is coincident with the extremum of vorticity concentration A ~
The trajectories of the major vorticity concentrations are shown on the xy plane in
Figure 3. These diagrams were obtained from the entire succession of PN images over
approximately one cycle of the wave motion. The position of each vorticity concentration
was determined by locating the coordinates of the maximum absolute value of vorticity.
This extremum is not necessarily coincident with the'centroid; moreover, the shape of the
vorticity concentration distorts with time. As a consequence, the trajectories of Figure 3
must be viewed as approximate.
Nevertheless, the generally orbital trajectories of
positive vorticity concentrations are evident in the left diagram and negative
concentrations in the right diagram. The direction of the vortex motion is indicated by
the arrows.
I
'
.
Since the flow pattern was generally repeatable from cycle to cycle, the paths of
positive vorticity concentrations B~ B and E shown in the left diagram essentially
represent the continuous trajectory of the vorticity concentration that is initially formed
22
along the right side of the cylinder, moves along the cylinder surface, is shed from the
bottom of the cylinder surface, and follows its orbit about the cylinder. Likewise,
concentrations D and C represent the trajectory of the concentration that is originally
formed along, then shed from the left side of the cylinder. Finally, the paths of the
negative concentrations of vorticity A and A' given in the right diagram represent the
trajectory of the concentration that is formed along the left side of the cylinder, moves
along the cylinder surface, then eventually separates from the upper surface of the
cylinder; its small orbital trajectory leads to eventual collision with the surface of the
cylinder. Viewing all of the foregoing processes together, it is evident that, during a
single cycle of the wave motion, vortex formation originates at three distinct sites,
followed by eventual shedding from three locations along the surface of the cylinder, i.e.
approximately on the upper, lower, and left surfaces.
The orbits of the positive vorticity concentrations shown in the left diagram of
Figure 3 have ratios of major to minor axes of roughly 1.6: 1 and 1.5: 1 and are inclined at
an angle of about 36 0 to the horizontal. These values compare with those of the wave
orbit in the vicinity of the cylinder of 2.2: 1 and 41 0 • No doubt, mutual induction and
image effects influence the paths of the vortical structures. Yet the reasonable
correspondence between the wave and vortex orbits suggests that the wave motion exerts
a predominant influence.
23
4. RELATION OF INSTANTANEOUS LOADING OF CYLINDER TO
PATTERNS OF VORTICITY
The time histories of the instantaneous horizontal Fx and vertical Fy forces were acquired
simultaneously with the instantaneous images of the flow patterns described in the
previous section. The force coefficients Cx and Cy are shown in Figure 4 as a function of
frame number N and the corresponding elapsed time t. The force coefficient is defined as
Cx =Fx![1I2pU2D£], in which p is the fluid density, U is the characteristic velocity of the
flow, £ is the length of the cylinder, D is the cylinder diameter, and similarly for Cy•
Patterns of instantaileous vorticity, taken from the sequence of images described in the
previous section, are related to the occurrence of minima and maxima of the force
coefficients Cx and Cy in Figure 4.
The maximum-negative value of Cx occurs in the vicinity of N
= 12.
At this
instant, the large-scale negative vortex A is on the return portion of its trajectory towards
the cyiinder and is undergoing initial stages of severe distortion during its interaction with
the cylinder. The maximum positive value of ex, which occurs (initially) at approximately
N = 18, coincides with the continued development of the large-scale negative vortex A'
along the top surface of the cylinder, as it moves in the clockwise direction above the
surface of the cylinder. The remaining concentrations of vorticity, which, of course,
contribute to Cx, develop in the fashion described in the previous section.
Concerning the vertical force coefficient Cy, the maximum-negative value
coincides with movement of the: vorticity concentration A up and away from the surface
of the cylinder. On the other hand, the maximum-positive value of Cy occurs when
24
vorticity concentration A' attains its position at the top of the cylinder during its clockwise
migration about the cylinder surface, while the remaining vorticity concentrations develop
as previously described.
The relationship between the space-time evolution of the instantaneous velocity
field about the cylinder and the effective force
E acting on the cylinder can be described
using a concept described by Wu (1981) and Lighthill (1986). In essence, the vector
force acting on a body can be expressed as the time derivative of the spatial integral of the
moment of vorticity according to:
F= -'![pJrxoxlV]
dt
--
(la)
-
in which p is the fluid density,
(0
is vorticity, and f LX (0 dV is the integral of moments of
vorticity with respect to the center of the body over the entire flow field of volume V.
Only the shed vorticity is considered, so forces due to added mass and Froude-Krylov
effects are not included in equation (1 a). This concept is addressed by Wu (1981) and
Unal (1996); additionally, it has been verified by the discrete vortex calculations of Unal
(1996). In the case of quasi-two-dimenslonal flow, the force per unit length on the
cylinder can be evaluated from the vorticity field according to the relationship:
F =-'![l.pJrxoxlA]
R.
dt
2
(lb)
--
in which R. is the cylinder length and f r x oxlA is the integral of the moment of vorticity
with respect to the center of the cylinder over the domain of interest, and
Therefore, the horizontal force coefficient can be expressed as ex
25
(0
= kroz.
= d[(Mco)xl/dt, where
(Mro)x = -IYOlzdA/U
ne is the normlliized integrlli of the y moment of vorticity that acts in
2
the x direction, 0 is the cylinder diameter, and U is the maximum velocity of the flow.
Similarly, the verticlli force coefficient is Cy = d[(Mro)y]/dt, where (Mro)y = IXOlzdA/U20.e
is the normlliized integral of the x moment of vorticity.
For the case of deepest nominlli submergence of the cylinder, hID = 1.57, excerpts
of which are shown in Figures 2a and b, the free-surface remains sufficiently far above
the surface of the cylinder,
~and
interactions of vorticity concentrations with the free-
surface are minimlli, thereby providing the opportunity for application of equation (16).
According to equation (lb), the time derivative of the moment of vorticity is required to
determine the instantaneous force and, thereby the corresponding force coefficients.
Since the temporlli resolution of the acquired images is inadequate to llilow accurate
evlliuation of the time derivative, an approach that emphasizes identification of the
relative contribution of each of the vorticity concentrations to the induced force
coefficients, Cx and Cy, is employed. For the horizontlli force coefficient Cx, for example,
the time integrlli of cx, Le., ICxdt, is evlliuated from the force transducer data. This
measured integrlli is then compared with the instantaneous moment of vorticity (Mro)x. A
similar procedure is followed for the verticlli force coefficient Cy. This approach was first
introduced by Zhu et.
at. (1998).
Figure Sa exhibits comparison of: measured ICxdt, which is designated by the
continuous solid line, and moments of vorticity. The top plot of Figure Sa emphasizes the
separate contributions of the moments (Mro)x corresponding to positive (hollow square
26
symbol) and negative (filled square symbol) vorticity contributions. They exhibit a
similar period, but different amplitude and phase shift relative to the measured ICxdt. The
bottom plot of Figure 5a focuses on the net moment (Mco)x obtained by summing the
positive and negative vorticity contributions to the moment given in the top plot. The
variation of the calculated moment generally has the same form as the measured ICxdt.
The discrepancy is no doubt due to a combination of several factors, the most important
of which is the existence of low-level vorticity concentrations outside the field of view of
the PIV images, excerpts of which are shown in Figures 2a and b.
Figure 5b provides a similar comparison of the integral of the vertical force
coefficient ICydt and the calculated moments (Mco)y. The top plot again indicates the
separate contributions of the moments (Mco)y due to positive and negative vorticity. The
overall form of each moment distribution is essentially an inversion of the other, though
their amplitudes are different. As a consequence, the variation of the net sum of positive
and negative moments, shown in the bottom diagram of· Figure 6b, exhibits small
deviations about a nominal value of approximately -1. The lack of agreement between
the measured ICydt and calculated net moment (Mco)y is most likely due to the failure to
account for significant moments arising from large moment arms. Such moments can
arise from large values of x, in combination with low level concentrations of vorticity
outside the field of view. This situation contrasts with the foregoing case of y moment
arms employed to calculate (Mco)x, where the value of y is bounded by the presence of the
free-surface.
27
5. IDENTIFICATION OF FORCE CONTRIBUTIONS DUE TO INDIVIDUAL
VORTICES
Each of the major concentrations of vorticity exhibited in Figures 2a and b will make
different contributions to the time integrals of the horizontal and vertical force
coefficients, JCxdt and JCydt, depending upon the variation of their circulation and
position with time during the wave cycle.
Contributions of the principal vorticity
concentrations A, B, C, D, A', .and B' to the x and y components of the vorticity moments,
Le., (Mro)x and (Mro)y, are given in Figure 6.
The top plot of Figure 6 gives the variation of (Mro)x as a function of frame
number N. Generally speaking, the major concentrations A and A' produce the largest
moments. Moreover, the change of the vorticity moment with time, represented by the
slope (8.(Mro)x /m) has markedly larger values for concentrations A and A' than those
due to the remaining vorticity concentrations B, B ' , and C. Comparison with the image
sequence of Figures 2a and b shows the physical basis for large changes of the moments
due to A and A'. First, consider the rapid movement of vorticity concentration A back
towards the cylinder, represented by frames N = 9 and 11 in Figure 2; in fact this
downward movement continues at least through frame N = 12, as shown in the top left
image of Figure 4. As a consequence, the values of moment arm y of the vorticity
concentration A decrease, producing decreases in the value of (Mro)x. This observation is
in accord with the pronounced negative v~ue of the slope (8.(Mro)x 1m) between frames
N
= 9 and
12 in the upper plot of Figure 6. Moreover, the actual value of the force
coefficient Cx shown in Figure 4 attains its largest negative values over N =11 to 13.
28
On the other hand, movement of vorticity concentration A' in the upward direction
and its continuing increase in scale, evident in frames N
=17 and 19 in Figure 2b, as well
as in images N = 20 to 23 (not shown herein) promote an increase in the value of the
moment arm y and thereby moment (Mco)x of concentration A' with increasing N; this
corresponds to a large positive slope A(Mco)xldN over N = 17 to 23 in Figure 6. In tum,
this slope corresponds to the positive peak of
ex over N = 18 to 23 in the top trace of
Figure 4. To be sure, the concentration of vorticity B,D makes significant, but generally
smaller, contributions, as indicated in the top plot of Figure 6.
A similar assessment cali be
undert~en
for the individual contributions to the
moment (Mco)y as shown in the bottom plot of Figure 6. The largest contributions to
(Mco)y generally are from: concentrations A and A' over the same portions of the
oscillation cycle as for the (Mco)x contributions of the top plot of Figure 6; and
concentration B,D during the initial part of its trajectory about the cylinder. The major
vorticity concentrations A and A' produce large positive values of moment (Mco)y at small
and large values of N; likewise, concentration B,D yields large moments at intermediate
values of N. Again, Figure 2 shows the underlying physics. For vorticity concentration
A, frames 7, 9 and 11 suggest decreases in the positive value of (Mco)y , due both to a
decrease in the peak value of vorticity and a decrease in the x moment arm. This negative
slope ,1(Mco)yldN produces, in accord with equation (1), a negative peak in the vertical
force coefficient Cy indicated in the bottom plot of Figure 4. Similarly, as indicated in the
bottom plot of Figure 6, at large values of N = 23 and 24 (not shown), vorticity
29
concentration A' generates both large positive values of moment (Mro)y and slope of
moment il(Mro)y/AN, relative to those of the other major vortices B and B~
Correspondingly, the plot of Figure 4 shows the onset of a positive peak of Cy• These
observations are due to the movement of the concentration A' away from the surface of
the cylinder; although frames N
= 23 and 24 are not shown in the image sequence of
Figure 2, the initial launching of concentration A' is evident in frame N =19.
Finally, the bottom plot of Figure 6 also shows that vorticity concentration B,D
makes substantial contributions to the moment (Mro)y over the range of intermediate
values of N, say N = 14 to 19. Attainment of the maximum positive value of moment
(Mro)y at
N = 16 is due to movement of the large-scale concentration B,D to the left of
the cylinder, as shown in Figure 2b. At this instant, the slope of the moment il(Mro)ylAN
is approximately zero, and the value of Cy in Figure 4 is small, Le., slightly positive.
On the basis of the foregoing observations of the dominant contributions of
negative vortices A, A' and the positive vortex B,D to both (Mro)x and (Mro)y, we conclude
that such contributions occur in the relatively early stages of development of these
vorticity concentrations, corresponding to small displacements away from the surface of
the cylinder. This effect, involving a relatively short moment arm and large circulation of
the vorticity concentration, appears to be more prevalent than the large moment arm and
reduced value of circulation of the vorticity concentrations associated with large orbital
trajectories.
30
6. EFFECTS OF DEGREE OF SUBMERGENCE OF CYLINDER:
SPACE-TIME EVOLUTION OF VORTICITY PATTERNS
Figure 7 shows a time sequence of images of the instantaneous vorticity distribution for
three depths of nominal submergence of the cylinder: relatively deep submergence
(hID = 1.57), corresponding to the first row. of images; intermediate submergence
(hID = 0.55), shown in the second row; and surface-piercing (hID = 0), given in the third
row of images. Each time sequence consists of representative images over the span of
frame numbers N = 7 through 19. At a given value of N, the phase of the undulating freesurface relative to the cylinder is the same.
6.1 Initial Stage of Development of Patterns of Vorticity
In the layout of Figure 7a, corresponding to frames N =7 and 9, analogous concentrations
of vorticity A, Band C are identifiable for all levels of submergence hID. Their form,
circulation and position, however, vary substantially. At the deepest submergence (top
row of images), vorticity concentration A takes an approximately circular form, whereas
at the intermediate depth of submergence (middle row), concentration A is substantially
elongated and eventually takes the form of a sequence of small-scale concentrations of
vorticity embedded hi. a vorticity layer. For the shallowest submergence (bottom row),
corresponding to surface piercing during a subsequent instant of the wave cycle, vorticity
concentration A is dramatically reduced in both the circulation and spatial extent. It
initially appears on the upper surface of the cylinder, then migrates in the clockwise
direction and, in conjunction with interaction of vorticity concentration B with the
31
cylinder, it eventually takes the form of two distinct vorticity concentrations. Taking an
overview of the· evolution of vorticity concentration B for varying degrees of .
submergence, its peak level of vorticity increases with decreasing hID; at the shallowest
submergence, Le., smallest value of hID, it takes on a highly concentrated form. Finally,
vorticity concentration C appears above the cylinder at the deep and intermediate values
of submergence, but is located adjacent to the lower surface of the cylinder at the
shallowest submergence. Generally speaking, the effect of decreasing submergence is to
retard the orbital motion of vorticity concentration C in the clockwise direction.
6.2 Intermediate Stage of Development of Patterns of Vorticity
The next sequence of events is represented in Figure 7b by frames N = 11 and 15. For
deep submergence (top row), vorticity concentration A is on the return part of its
trajectory, about to collide with the cylinder at N = 11, and only a small residual of the
post-collision process is evident at N
= 15.
On the other hand, the major vortex A at
intermediate submergence (middle row) does not directly encounter the cylinder and
remains intact at N
= 15.
At the shallowest submergence (bottom row), generation of
vortex A is retarded relative to deeper values of submergence, and it occurs from the
bottom surface of the cylinder at N
= 11.
At N
= 15, the separated vortex A is located
well away from the surface of the cylinder. At all levels of submergence, for the images at
N
= 15, an additional concentration of vorticity A' is generated from the left side of the
cylinder; it is the sequel to the originally generated vortex A from an earlier portion of the
32
wave cycle. The degree of concentration of vorticity. A' appears to Increase with
decreasing submergence.
Moreover, all levels of submergence of the cylinder exhibit formation ofvorticity
concentrations B and D. The sequence of events leading to their initial development at
N = 11, as well as their subsequent evolution at N =15, is similar for deep (top row) and
intermediate (middle row) values of submergence. At shallow submergence (bottom row
of images), however, vortex B separates early from the surface of the cylinder, and at
N = 11, appears immediately beneath extended vorticity concentration A. At a later time,
N = 15, concentration B moves well to the left of the cylinder.
Further, vorticity concentration C continues its orbital trajectory about the cylinder
at deep submergence in images N
=11 and 15, collides with the free-surface at N = 1 for
intermediate submergence, and remains well submerged beneath the free-surface at the
shallowest submergence, both in images N =11 and 15.
A distinctive feature of the vorticity field development at the shallowest
submergence of the cylinder is generation of an additional concentration of vorticity E
from the free-surface at N
= 15. It is induced by vortex D. All of these events at the
shallowest submergence produce two sets of triplet vortices at N =15; together, they form
an array of six counter-rotating vortices.
6.3 Final Stage of Development of Patterns of Vorticity
The final stage of development is indicated in Figure 7c. At relatively deep submergence
(top row of images), vortices A', B and D migrate in the clockwise direction in accord
33
with the wave trajectory, and vortex B' appears on the lower surface of the cylinder at
N= 19.
For intermediate submergence (middle row), vorticity concentra.tion A, which,
unlike the case of deepest submergence, did not collide with the cylinder, is still
identifiable. Development of concentration A' in images N = 17 and 19 is closely similar
to its development at the deepest submergence. Vorticity pattern B,D retains its identity
of two concentrations at N
=17; they appear distinctly separated at N =19.
At the shallowest submergence, the array of six counterrotating concentrations of
vorticity is still identifiable at N
= 17, though the peak vorticity levels· of each of the
concentrations has generally decreased. At N
= 19, onset of vorticity concentration B'
along the right surface of the cylinder occurs in a fashion analogous to .that at
intermediate and deep levels of submergence.
7. CONCLUDING REMARKS
Interaction of a free-surface wave with a horizontal cylinder is examined ata sufficiently
high value of Keulegan-Carpenter number such that well-defined concentrations of
vorticity are generated during a typical wave cycle. The mutual interaction of these
vorticity concentrations at varying degrees of submergence beneath the free-surface
provides a rich array of space-time patterns of the instantaneous vorticity field.
Irrespective of the particular pattern,however, central elements of the vortex formation
process appear to persist. In essence, due to orbital motion of the wave, the evolution of a
34
given pattern of vorticity follows a well-defined sequence: (i) growth of a pronounced
region of vorticity having an identifiable extremum on the surface of the cylinder; (ii) .
migration of this vorticity concentration about the surface of the cylinder in a direction
compatible with the orbital motion of the wave; and (iii) separation of the vorticity
concentration from the surface of the cylinder. This process of growth and migration of a
vorticity concentration can originate from three sites, i.e. approximately on the upper,
right, and lower surfaces of the cylinder, during a single oscillation cycle.
The foregoing process appears to be largely controlled by the orbital motion of the
wave. In fact, the form and orientation of the orbital trajectories of the eventually-formed
concentrations of vorticity resemble those of the particle trajectories of the wave.
Basically, two types of orbits of the vorticity concentrations are discernible. The first is a
relatively small amplitude orbit; in the limiting case of such an orbit, abrupt reversal in
direction of movement of the vorticity concentration occurs, fo.1lowed by collision of the
concentration with the surface of the cylinder for sufficiently deep submergence. This
type of small amplitude orbital motion is shown to actually give rise to relatively large
moments of vorticity and large contributions to the instantaneous forces on the cylinder,
as deduced by application of a concept described by Lighthill (1986). The second type of
orbital motion of the vorticity concentrations exhibits a large amplitude trajectory. At a
sufficiently large value of nominal submergence, the vorticity concentration navigates one
complete loop about the cylinder. This type of large amplitude orbit, however, generally
gives rise to smaller moments of vorticity. This is due to the fact that the large amplitude
35
orbit involves a relatively long time scale, and thereby the opportunity for substantial
decay of circulation of the vortex.
The nominal submergence of the cylinder beneath the wave has a pronounced
effect on the pattern of vorticity concentrations. It has been demonstrated that generic
concentrations of vorticity are common to all patterns of vorticity at various depths of
submergence.
For relatively deep submergence, as previously noted, the separated
vorticity concentrations exhibit large-scale orbital motion about the cylinder. For
interrnediate submergence, however, the presence of the free-surface exerts a significant
influence, and this large-scale orbital motion is substantially hindered. Moreover, the
distorted velocity field of the wave, due to proximity of the free-surface of the cylinder,
promotes formation of elongated vorticity concentrations from the top of the cylinder and,
in addition, significantly enhances the vorticity. level and scale of the vorticity
concentrations from the bottom surface of the cylinder. In the extreme case of shallow
submergence, whereby the cylinder pierces the free-surface during the wave cycle, the
presence of the free-surface induces a substantial lag of the process of vortex formation
from the surface of the cylinder. In addition, vortex formation is induced from the freesurface during a portion of the wave cycle.
These fascinating processes of vortex
formation and interaction yield an ordered array of as many as six counterrotating vortices
immediately beneath the free-surface.
Finally, it has been demonstrated that, at relatively large submergence, the spacetime evolution of the vorticity field is associated with identifiable topological features of
the instantaneous streamline patterns. Changes in the predominant orientation of the
36
velocity field of the undisturbed portion of the free-surface wave, in conjunction with the
defined patterns of positive and negative vorticity concentrations about the cylinder, can
induce well-defined singularities in the form of foci and saddle points. Even though the
pattern of vorticity concentrations undergoes a very mild change from one instant to the
next during the wave motion, nearly discontinuous changes in the streamline topology
can occur. Such changes include, for example, abrupt switching of the location of a
saddle point from one region of the flow to another. These changes are apparently due to
the fact that the predominant direction of the velocity field of the wave varies
substantially during the wave cycle.
37
REFERENCES
Anagnostopoulos, P., lliadis, G., and Ganoulis, J. 1995 Flow ancl Response Parameters of
a Circular Cylinder Vibrating In-Line with the oscillating stream. In Flow-Induced
Vibration (ed. Bearman), pp. 167-179.
Baba, N., and Miyata, H. 1987 Higher-Order Accurate Difference Solutions of Vortex
Generation from a Circular Cylinder in an Oscillatory Flow. Journal of Computational
Physics, 69,362-396.
Bearman, P.W. 1985 Vortex trajectories in oscillatory flow. In Proceedings of
International Symposium on Separated Flow around Marine Structures. Norwegian
. Institute of Technology, Trondheim, Norway. 133-153. NOT IN TEXT
Bearman, P.W., Downie, MJ., Graham, J.M.R., and Obasaju, F.D. 1985a Forces on
Cylinders in Viscous Oscillatory Flow at Low Keulegan-Carpenter Numbers. Journal of
FluidMechanics, 154,337-356.
Bearman, P.W., Chaplin, J.R., Graham, J.M.R., Kostense, J.K., Hall, P.E, and Klopman,
G. 1985b The loading on the cylinder in post-critical flow beneath periodic and random
waves. In Proceedings of4th Conference on Behaviour of Offshore Structures. Delft. 2,
213.
Bearman, P.W., Graham, J.M.R., Naylor, P., and Obasaju, E.D. 1981 The Role of
Vortices in Oscillatory Flow about Bluff Bodies. In Proceedings of International
Symposium on Hydrodynamics and Ocean Engineering (The Norwegian Institute of
Technology),621-644.
Borthwick, A. 1986.·Orbital Flow Past a Cylinder: a Numerical Approach. International
Journalfor Numerical Methods in Fluids, 6,677-713.
Borthwick, A. 1987 Orbital Flow Past a Cylinder: ExperimentalWork. In Proceedings
of the 5th Conference of Offshore Mechanics and Arctic Engineering, 123.
Chaplin, J.R. 1981a On the Irrotational Flow around a Horizontal Cylinder in Waves.
Journal ofApplied Mechanics, 48, 689-694.
Chaplin, J.R. 1981b Boundary Layer Separat~on from a Cylinder in Waves. In
Proceedings of International Symposium on Hydrodynamics in Ocean Engineering.
Trondheim, Norway. 1,645-666.
38
Chaplin~ J.R. 1984 Nonlinear Forces on a Horizontal Cylinder Beneath Waves. Journal of
FluidMechanics, 147,449-464.
Chaplin, J.R. 1988 Loading on a cylinder in oscillatory flow: Elliptical orbital flow.
Applied Ocean Research. 10, 199. Chaplin, lR. 1988 Loading on a cylinder in oscillatory
flow: Elliptical orbital flow. Applied Ocean Research. 10, 199.
Chaplin, J.R. 1992 Orbital Flow Around a Circular Cylinder. Part 1. Steady Streaming in
Non-Uniform Conditions. Journal ofFluid Mechanics, 237,395-411.
Chaplin, J.R. 1993 Orbital Flow Around a Circular Cylinder. Part 2. Attached Flow at
Larger Amplitudes. Journal ofFluid Mechanics, 246, 397-418.
Grass, A. J., Simons, R. R. and Cavanaugh, N. J. 1985 Fluid Loading on Horizontal
Cylinders in Wave Type Orbital Flow. ill Proceedings of 4th Conference of Offshore
Mechanics an Arctic Engineering, Dallas, Texas, 1,576.
Holmes, P., Chaplin, lR. 1978 Wave loads on horizontal cylinders. ill Proceedings of
16th International Conference on Coastal Engineering. Hamburg. 3,2449.
Honji, H. 1981 Streaked Flow Around an Oscillating Circular Cylinder. Journal of Fluid
Mechanics, 107,509-520.
Ikeda, S., and Yamamoto, Y. 1981 Lift Force on Cylinders in Oscillatory Flows. ill
Report ofDepartment ofFound. Engineering & Const. Engineering (Saitama University),
10.
Iwagaki, Y., Asano, T., and Nagai, F. 1983 Hydrodynamic Forces on a Circular Cylinder
Placed in Wave-Current Co-Existing Fields. ill Memo Faculty of Engineering. (Kyoto
University, Japan), 45, pp. 11-23.
Justesen, P. 1991 A Numerical Study of Oscillating Flow around a Circular Cylinder.
Journal ofFluid Mechanics, 222, 157-196.
Koterayama, W. 1979 Wave forces exerted on submerged circular cylinders fixed in deep
waves. In Reports of Research Institute for Applied Mechanics. Kyushu University, 27,
(No. 84), 25.
Lighthill, J. 1986 Fundamentals Concerning Wave Loading on Offshore Structures.
Journal ofFluid Mechanics, 173,667-681.
Lin, X.W., Bearman, P.W., and Graham, J.M.R. 1996 A Numerical Study of Oscillatory
Flow about a Circular Cylinder for Low Values of Beta Parameter. Journal of Fluids and
Structures, 10,501-526.
39
Lin, J.-C., and Rockwell, D. 1996 Force Identification By Vorticity Fields: Techniques
Based on Flow Imaging. Journal ofFluids and Structures, 10, 663-668.
Lin, J.-C., and Rockwell, D. Horizontal Oscillations of a Cylinder beneath a FreeSurface: Vortex Formation and Loading. Journal ofFluid Mechanics (in press).
Lin, J.-C., Sheridan, J. and Rockwell, D. 1996 "Near-Wake of a Perturbed, Horizontal
Cylinder at a Free-Surface", Physics ofFluids, 8 (8), 2107-2116.
Maull, DJ., and Milliner, M.G. 1978 Sinusoidal Flow Past a Circular Cylinder. Coastal
.
Engineering, 2, 149-168.
Miyata, H., and Lee, Y.-G. 1990 Vortex Motions about a Horizontal Cylinder in Waves.
Ocean Engineering, 17, 279-305.
Miyata, H., Shikazono, N., Kanai, M. 1990 Forces on a Circular Cylinder Advancing
Steadily Beneath the Free-Surface. Ocean Engineering, 17, 81-104.
Naudascher, E., Rockwell D. Flow-Induced Vibrations: An Engineering Guide, Balkema
Press, Rotterdam.
Noca, F. 1997 On the Evaluation of Time-Dependent Fluid Dynamic Forces on Bluff
Bodies. Ph.D. Dissertation, California Institute of Technology.
Noca, F., Shiels, D., and Jeon, D. 1997 Measuring Instantaneous Fluid Dynamic Forces
on Bodies, Using Only Velocity Field and Their Derivatives. Journal of Fluids and
. Structures, 11,345-350.
Obasaju, E.D., Bearman, P.W., Graham, J.M.R. 1988 A Study of Forces, Circulation and
Vortex Patterns Around a Circular Cylinder in Oscillating Flow. Journal of Fluid
Mechanics, 196,467-494.
Ramberg, S.E., Niedzwecki, J.M. 1979 Some uncertainties and errors in wave force
computations. In Proceedings of 11th Offshore Technology Conference. Houston, TX. 3,
2091-2101.
Rockwell, D., Magness, C., Towfighi, J., Akin, O. and Corcoran, T. 1993 "High ImageDensity Particle Image Velocimetry Using Laser Scanning Techniques", Experiments in
Fluids, 14, 181-192.
Sarpkaya, T. 1963 Lift, Drag, and Added-Mass Coefficients for a Circular Cylinder
Immersed in a Time-Dependent Flow. ASME Journal ofApplied Mechanics, 85, 13-15.
40
Sarpkaya, T. 1968 An Analytical Study of Separated Flow About Circular Cylinders.
ASME Journal ofBasic Engineering, 90, 511-520.
.
Sarpkaya, T. 1969 Analytical Study of Separated Flow About Circular Cylinders. Physics
ofFluids, 12, Supplement II, p.145.
Sarpkaya, T. 1977 In-line and transverse forces on cylinders in oscillatory flow at high
Reynolds numbers. Journal ofShip Research 21, 200.
Sarpkaya, T. 1984 Discussion of paper by Stansby, P. K. et. al., Quasi 2D Forces on a
Vertical Cylinder in Waves", Journal of Waterway, Port, Coastal and Ocean
Engineering, 110, 120-124.
Sarpkaya, T. 1986 Force on a Circular Cylinder in Viscous Oscillatory Flow at Low
Keulegan-Carpenter Numbers. Journal ofFluid Mechanics, 165, 61-71.
Sarpkaya, T. 1989 Computation Methods with Vortices - The 1988 Freeman Scholar
Lecture. Transactions ofASME, Journal ofFluids Engineering, 111,5-52.
Sarpkaya, T., Isaacson, M. Mechanics ofWave Forces on Offshore Structures, New York,
Van Nostrand Reinhold.
Sheridan, J., Lin, l-C., and Rockwell, D. 1997 Flow Past a Cylinder Close to a FreeSurface. Journal ofFluid Mechanics, 330, 1-30.
Singh, S. 1979 Forces on Bodies in Oscillatory Flow. Ph.D. Thesis,
London.
University of
Sun, X., and Dalton, C. 1996 Application of the LES Method to the Oscillating Flow Past
a Circular Cylinder. Journal ofFluids and Structures, 10,851-872.
Tatsuno,.M., and Bearman, P.W. 1990 A Visual Study of the Flow around an Oscillating
Circular Cylinder at low Keulegan-Carpenter Numbers and Low Stokes Numbers.
Journal ofFluid Mechanics, 211, 157-182.
Unal, M.P. 1996 Force Calculation by Control Volume and Vortex Methods. Lecture
Series, Department of Mechanical Engineering and Mechanics, Lehigh University.
Unal, M.P., Lin, l-C., and Rockwell, D. Force Prediction via PIV: A Momentum-Based
Approach. Journal ofFluids and Structures (in press).
Van Dyke, M. 1982 An Album of Fluid Motion, Parabolic Press, Stanford, CA, 110.
41
Wang, X., Dalton, C. 1991 Oscillating Flow Past a Rigid Circular Cylinder: A Finite
Difference Calculation. Journal ofFluids Engineering, 13, 377-383.
Williamson, C.H.K. 1985 Sinusoidal Flow Relative to Circular Cylinders. Journal of
Fluid Mechanics, 155, 141-174.
Williamson, C.H.K., Hess, P., Peter, M., Govardhan, R. 1998 Fluid Loading and Vortex
Dynamics for a Body in Elliptic Orbits. Proceedings of 1998 ASME Fluids Engineering
Division Summer Meeting.
Wu, J.C. 1981 Theory for Aerodynamic Force in Viscous Flows. AIAA Journal, 19 (4),
432-441. .
Zhu, Q., Lin, J.-C., Unal, M.E, Rockwell, D. 1998 Motion of a Cylinder Adjacent to a
Free-Surface: Flow Patterns and Loading, submitted to Experiments in Fluids.
42
c
~
}J2
~
h
/
.j:>.
,
w
Wavemaklng system: 11-_.,
actively controlled
i
with feedback
I Arbitrary motion
of cylinder
Laser
illumination
system
Image (PIV)
acquisition
systems
Figure la. Configutation of experimental system
Foree
acquisition
system
c
I
lc_L
1J2
---------~-----------------
<I>
~
~
~-
- --
---
,..../~'\ \
\ ~d/~-1-------jr---------I-
-I
;'
I
\
" .....
',
......,-.,.-----
---
-
"'(
' . . . _ . . ",/
I
Figurelb. Closeup of the test section
AJd = 0.5, hid = 1.6, Ajd = 0.5, IJd = 494.5
I':!.
L
Free-surface
Laser sheet
~
VI
/
laser
---H
Opflcal1Tain
y_
,f~)"\
0 ~ steering
~
Rota1ing :::s
Figure Ie. Overview of experimental arrangement and principal parameters formteraction ofa progressive wave with the cylinder.
Compumotors
Cylinder ---1~~1
~ Laser sheet
1-0
PlY camera
~
~Bias mirror
Figure ld. Top view of test section
46
23mm
~-
LASER SHEET
Q,7mm
12.5 mm
265mm
.J::>..
--.l
O.7mm
'---WINDOW
3mm
'-J
430 mm
Figure Ie. Detailed schematic of the test cylinder
"
Film
Negative
Digitizer
TIF File
Interrogation
(PIV3)
Raw Velocity Data
Vector
Validation (V3)
Interpolationl
Smoothing
(NFILVB)
~~
Vorticity
Velocity
dJ
Surfer
!
!
Streamlines
Vorticity Contours
Figure If. General Procedure for image processing
48
Figure 2a. Time history of patterns of positive (thick line) and negative (thin line) concentrations of
vorticity and corresponding velocity field and streamline topology for the case of relatively deep
submergence ofthe horizontal cylinder beneath the free-surface wave. Frame number ofcinema sequence
is designated by N. Nominal submergence ofthe cylinder is bID = 1.57. Minimum and incremental values
ofvorticity are mmi. =11m =3sec·l •
49
INTENTIONAL SECOND EXPOSURE
1=::----=
i __
L.,<~~:~'~·~~~~~"~·~·-··~·-··---"---
Figure 2a. Tinle histor;.'- of patterns of positive (thick lfH-e}-u4'!d negative (thin line-') cOElcentratlons
~lnd corresponding velocity field and streamline topology for the case of
"llbl::c':-~:e:1Ccof the horIzontal cylinder beneath the free-surE1ce \\·3\"e. FraIne nurnber ofcinen1~1.sequence
ul',;::,:::a:c0 by N. Norninzll submergence of the cylinder is h/D = 1.57. \linimun1 and
values
are
=60)=3
49
Figure 2b: Time history ofpattems ofpositive (thick line) and negative (thin line) concentrations of
vorticity and zoomed-in representations ofcorresponding velocity fields associated with vortex pairs
for the case ofrelatively deep submergence ofthe horizontal cylinder beneath the free-surface wave.
Frame number ofcinema sequence is designated by N. Nominal submergence ofthe cylinder is hID =
1.57. Minimum and incremental values ofvorticity are romin = ~ro =3 sec· l •
50
INTENTIONAL SECOND EXPOSURE
10 'l
•
..,,,
"t:"Illlf:"
_6'li'S"1f:::C
Figure 2b: TiIne
of patterns of positive (thick line) Zlnd negative t, thin line) concentr'j.ll(1[l'
representations of corresponding
fields assoc13ted \\-lth '-UrIe:\. PJ.I~S
for :h:: ,,::::'<:: ofrerati"'/cI~,>· Je·:;:p submergence of the horizor1t~l1 cylinder beneath the fr;:;;:e-surf:lce \\J,\C.
0=
f L1Flc nLtl11Der of cine-rna sequence is designated by' :-.:. ?\orninzd subrnergence of the .,,;; -";'"
= ~(.U =3 sec·
~lnd zoorned-lD
50
NEGATIVE VORTICITY
POSITIVE VORTICITY
y
(mm)
-
40
30
Of
~(
(~o
20
-
10
0
20
30
40
50
I
I
I
I
20
30
40
50
X (mm)
X (mm)
0
0
0
B'
t::.
A
'V
C
0
B
0
p:
0
E
Figure 3: Illustration ofthe principal types oftrajectories ofthe separatedvorticity concentrations
due to positive and negative vorticity concentrations. Level of submergence of cylinder beneath
the free-surface is hID = 1.57.
51
ex
3
2
0
-1
-2
-3
-4
5
0.0
15
10
0.5
1.0
20
1.5
2.0
25
2.5
30
3.0
N
3.5
t (sec)
Figure 4: Comparison ofselected patterns ofinstantaneous vorticity with time histories ofhorizontal
Cx and vertical Cy force coefficients for relatively deep submergence ofcylinderhID =1.57.
52
ex
3
2
0
-1
-2
-3
-.:.,;.
5
10
15
10
15
30
25
20
N
-2
---------------~-~-
-4
u.u
0.5
1.0
1.5
2,0
30
25
20
2,5
3,0
Figure 4: Companson of selected patterns of instantaneous vorticity with time historle'$
C and vertical force coefficients for relatively deep s~;JJJller:;-enceofcylinder hiD = !
N
t (sec)
(MJx
2
'---'---'--'----'-""T--r---'--'--'--'--'---'--.--T----r-....---r----'--'--'--'--'---.--.--T----r-..,.---,---.--.,--,
(sec)
5
20
10
25
35
30
N
o
2
3
t (sec)
(Moot due to positive vorticity contribution
• (Moo)x due to negative vorticity contribution
o
o
Net
(Moo)x
- - Measured ICxdt
Figure 5a: Calculated moments of vorticity (M)x corresponding to the horizontal (x) direction force, in
which hollow and filled square symbols represent respectively contributions from positive and negative
vorticity concentrations. Circular symbol represents net sum of positive and negative contributions. Solid
line corresponds to the measured
time integral ofthe horizontal force coefficient, ICxdt. Each frame number N
I
corresponds to an instantaneous PlY image. Level ofsubmergence ofcylinderis hID = 1.57.
53
(M",)y
(sec)
0
-1
.,~
~
~
,,11-11 ~o</
\.---./.~
•
-2
10
5
15
20
25
30
35
N
(M",)y
(sec)
0
/0,
/0,
-1
'0
o-D--d
'(yO-O,
";'2
-3
5
10
15
20
25
30
35
N
o
2
3
t
o
(M",)y due to positive vorticity contribution
•
(M",)y due to negative vorticity contribution
°
Net
(sec)
(M.,)y
Measured fe,dt
Figure 5b: Calculated moments ofvorticity corresponding to the vertical (y) direction of force, in which
hollow and filled square symbols represent respectively contributions from positive and negative vorticity
concentrations. Circular symbol represents net sum of positive and negative contributions. Solid line
corresponds to the measured time integral of the vertical force coefficient, ICydt. Each frame niunber N
represents an instantaneous PlY image. Level ofsubmergence ofcylinderis hID = 1.57.
54
(McJx
2
(sec)
1
o
-1
-(Mlll)y
15
10
20
N
1
(sec)
o
-1
-2
'---'------'-_-'----1---'_--'------'-_-'----1---'_--'------'-_-'----1----''-----'------'-------'
15
10
20
N
OA
•
0.5
TA'
\lC
BOB'
-0
1.0
1.5
2.0
2.5
t
(sec)
Figure 6: Contributions of major concentrations of vorticity A, B, A', B', C and D to the moments of
vorticity corresponding to the x and y direction forces. Level ofsubmergence ofcylinderbeneath the freesurface is hID = 1.57.
ss
Figure 7a: Comparison of instantaneous patterns of vorticity at the same frame numbers N = 7 and 9 for
varying levels ofsubmergence hID = 1.57,0.55, and 0ofthe cylinderbeneath the free-si.uface. Frame N = 7
represents the instant where the crest of the free-surface wave is above the cylinder. Minimum and
incremental values ofvorticity are romin =3 sec·! andAro =3sec·l •
56
Figure 7b: Comparison ofinstantaneous patterns ofvorticity at the same frame numbers N = 11 and 15 for
varying levels ofsubmergence hID = 1.57,0.55, and 0 ofthe cylinder beneath the free-surface. Frame N =
15 represents the instant at which the trough ofthe free-surface wave is above the cylinder. Minimum and
incremental values ofvorticity are mmin,.; 3sec'! and Am =3 sec'!.
57
INTENTIONAL SECOND EXPOSURE
Figure 7b:
of IriSIJ.ntaneous patterns of vorticity at the same frame numbers ?'\ = 1 2nd 15 for
ie '-C'ls of submergence h,iD = 1.57.0,55. and 0 of the cylinder beneath the free-surfJi'-.>:. Fr:uTloe 0 =
5 r~r:T~s~nts the Instant :t \' hieD the trough
the free-surface \V3.ve is above the
.\Ei,ir;lur.n.2nd
\z~~nI1g
or
=~.. sec' 2.I1d ~Uj
=3 sec
57
Figure 7c: Comparison of instantaneous patterns ofvorticity at the same frame numbers N = 17 and
19 for varying levels of submergence hID = 1.57, 0.55, and 0 of the cylinder beneath the free-surface.
Minimum and incremental values ofvorticity are OOmin = 3 sec·· and ~OO =3 sec·l •
58
EXPOSUR
Figure 7c:
r~,Jr
OElpJ.flson
ie\:eis of
or~
cns,tarlta'n,,()us patterns of vorticity' at the saIne frarne numbers
h.,/D = 1.57. 0.55. Gnd 0 of the cy-tinder beneath th,c rne,,-s,urr:lce.
58
APPENDIX A: DEFINITION OF KEULEGAN-CARPENTER NUMBER AND
RELATED ISSUES
In general, Keulegan-Carpenter number is defined by Kc =UmTID, where Umis the
maximum velocity of the flow, T is the pe~od of the wave motion, and D is the cylinder
diameter. For unidirectional, sinusoidal motion of the fluid past the cylinder, this
definition reduces to Kc =2rr.a/D, where a is the amplitude of the motion. It can be shown
that for a small-amplitude wave of. wavelength 21t1k,
U =2M[coshk(d-ho)]
m
T
. sinhkd
where A is the wave amplitude, T is the wave period, d is the water depth and ho is the
depth of submergence below the still water level. Another definition of the KeuleganCarpenter number for a small-amplitude deep-water wave, where the orbital motion is
circular, is given by Chaplin (1984):
This formula can be extended to cases where the trajectory of a fluid particle motion is
elliptical. In these cases, the amplitude of the wave is multiplied by the orbital parameter
n,
which is defined as the ratio of the major to the minor axes of the ellipse. From
Lighthill (1978), it can be shown that n =cosh[k(d-hJ]lsinh[k(d-hJ].
In these experiments, the general definition of Keulegan-Carpenter number Kc =
UmTID was used, where maximum flow velocity Umwas measured experimentally (figure
59
A-I). Velocity magnitude was obtained from each PIV image of instantaneous velocity
field by calculating mean values of u and v velocity components from a small undisturbed
region of the flow field. Um was found as a.maximum of calculated velocity magnitudes.
For all three depths of submergence (ho = 20 mm, ho = 7 mm, and ho = 0 mm) Um was
found to be equal 0.04 m/sec. For the wave period T
=2 sec and diameter of the cylinder
D = 0.0127 m, Keulegan-Carpenter number was found to be equal 6.96..
Figure A-2 shows plots of sequences of space-averaged velocity vectors for the
various depths of cylinder submergence. The vectors were determined at a reference
location 2.35 D to the left of the center of the cylinder, and at a constant depth of 1.57 D
beneath the quiescent free-surface. The labels of the vectors indicate the corresponding
frame numbers. Figure A-2 also shows superimposed vectors for different depths of
submergence for frames number 7,9; and 11.
60
f11f.plt
0\
(U m )lI=(u_mean'+v_mean')1/2 = 34.1102 mm/sec
Frame No. 11
Urn = max {
(U~)41
... ; (U rn )3 s} = 0.04 m/sec
Figure A-I :Characteristic velocity calculation
18
19
19
20
21
16
6
22
.."
'"
14
13
13
12
12
.."
11
h/D=1.57
11
h/D=Q.55
10
7
/0
_ _~>~o--JO"'O.55
"'--1.57
17
20
9
21
(i
o 1.57
0.55
10
h/D=Q
Figure A-2. Mean velocity vectors (upstream region) for the stationary cylinder case.
62
APPENDIX B: TIME SEQUENCES OF VORTICITY DISTRIBUTION
The following images (figures B-1, B-2, and B-3) are the raw data in the form of
time sequences of contours of constant vorticity. In all three cases the cylinder was
stationary, and the wave had a period of 2 sec and an amplitude of 6 mm. Depth of the
cylinder submergence h was measured from the top of the cylinder to the still water level.
For the cases 1 (figure B-1), 2 (figure B-2), and 3 (figure B-3), h was equal 20 mm, 7
mm, and 0 mm respectively. The time resolution of the images is /),.tlT
= 0.0625. The
contours of constant positive and negative vorticity are represented by solid and dashed
1
lines respectively. The minimum vorticity contour in all the cases is at 0) = 3 sec· and /),.0)
l
= 3 sec· • The images were not edited, and the small-scale structures were not filtered out. .
Case 1: Depth of Submergence hID = 1.57
Figure B-1 is a time sequence of contours of constant vorticity in a wave past a
stationary cylinder at depth of submergence of 20 mm. The time sequence consists of
frames number 7 through 23.
63
VORTICITY
TestS Roll13
10/30/96
d = 20mm
ftl, d=20mm. dw=3
f10, d=20mm, dw=3
f8, d=20mm. dw=3
f 11
o
o
o
e::;,
Q
0
f12
f9, d=20mm. dw=3
o
C)
0
o
o
6
Figure B-1: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergence h =20mm.
64
VORTICITY
Tests RaU13
10/30/96
d= 20mm
f13
f16, d=20mm, dw=3
o
o
o
f14, d=20mm, dw=3
f17, d=20mm, dw=3
f15, d=20mm, dw=3
f18
o
J
\!
0
o
(:
{)
Figure B-1: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergenceh =20 mm.
65
VORTICITY
TestS Roll13
10/30/96
d = 20mm
f19
f22. d=20mm. dw=3
C:)
o
o
f20
f23. d=20mm. dw=3
o
f21. d=20mm, dw=3
o
~.
•o
o
r
I
~)
o
r-
1....
o
Figure B-1: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergence h =20 mm.
66
Case 2: Depth of Submergence bID = 0.55
Figure B-2 is a time sequence of contours of constant vorticity in a wave past a
stationary cylinder at depth of submergence of 7 rom. The time sequence consists of
frames number 7 through 23.
67
VORTICITY
TestS Roll23
10/30/96
d=7mm
f7, d=7mm, dw=3
f10, d=7mm. dw=3
f8, d=7mm. dw=3
f11. d=7mm, dw=3
f9. d=7mm, dw=3
f12. d=7mm, dw=3
'0
o
o
Figure B-2: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergenceh = 7 mID.
68
VORTICllY
d=7mm
Test5 Roll23
10/30/96
f13, d=7mm, dw=3
f16, d=7mm. dw=3
f14, d=7mm, dw=3
f17, d=7mm, dw=3
f15, d=7mm, dw=3
f18, d=7mm. dw=3
Q
o
o
o
Figure B-2: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergence h =7mm.
69
VORTICITY
d=7mm
TestS Roll23
10/30/96
f19, d=7mm, dw=3
f22, d=7mm, dw=3
f20, d=7mm, dw=3
f23, d=7mm, dw=3
f21, d=7mm, dw=3
Figure B-2: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergenceh =7mm.
70
Case 3: Depth of Submergence hID
=0
Figure B-3 is a time sequence of contours of constant vorticity in a wave past a
stationary cylinder at depth of submergence of 0 rom. The time sequence consists of
frames number 7 through 23.
71
TestS RoU33
10/30/96
VORTICITY
f7, d=O, dw=3
d=Omm
f10, d=O, dw=3
-,
~
O{]
0
~:J 0
.'.
.
(?
()
Q
0
o
f8, d=O, dw=3
f11, d=O, dw=3
f9, d=O, dw=3
f12, d=O, dw=3
--...,....
o
Figure B-3: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergenceh =0 mID.
72
VORTICITY
d=Omm
f13, d=O, dw=3
Test5 Roll33
10/30/96
f16, d=O, dw=3
o
o
f14, d=O, dw=3
f17, d=O, dw=3
f15, d=O, dw=3
f18, d=O, dw=3
Figure B-3: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergence h= 0mm.
.
73
VORTICITY
TestS Roll33
10/30/96
d=Omm
f19, d=O, dw=3
f22, d=O, dw=3
f20, d=O, dw=3
f23, d=O, dw=3
f21, d=O, dw=3
o
Figure B-3: Time sequence of contours of constant positive (solid lines) and negative (dashed lines)
vorticity. Stationary cylinder case. Depth ofsubmergenceh =0mm.
74
APPENDIX C: TIME SEQUENCES OF MOMENTS OF VORTICITY
DISTRIBUTION
The following images (figures eland C2) are the raw data in the form of time
sequences of contours of constant moments of vorticity. The cylinder was stationary, and
the propagating wave had a period of 2 sec and an amplitude of 6 mm. Depth of the
cylinder submergence h was equal to 20 mm (top row of images), 7 mm (middle row of
images), and 0 mm (bottom row of images). The time resolution of the images is
/).tfl' =
0.0625. The contours of constant positive and negative moments of vorticity are
represented by solid and dashed lines respectively. The minimum contour is at Moo =
0.0001 sec and /).q>oo
= 0.002
sec. The images were not edited, and the small-scale
structures were not filtered out.
X-Moments of Vorticity.
Figure C1 is a time sequence of contours of constant x-moments of vorticity (
(Moo)y ) in a wave past a stationary cylinder at various depths of cylinder submergence.
The time sequence consists of frames number 7 through 22.
75
X-MOMENTS OF VORTICITY
MOl
d
AMOl
20mm
=
f7cl5, m=O.0001, delta=O.002
f7ldw. m=O.0001, dw=O.002
D
~.
d= 7mm
f7ldw, d=7mm. m=O.0001. delta=O.002
f7c13, d=7mm. m=O.0001, delta=O.002
...-:],.:~__~
D
@
o
o
d=Omm
flcl3. d=O, m=O.0001, delta=O.002
f7ldw, d=O. m=O.0001. delta=O.002
Figure C-l: Time sequence of contours of constant x-moments ofvorticity ({Mro)y) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
76
X-MOMENTS OF VORTICllY
Mcu
d = 20mm
f8ldw. m=O.0001, dw=O.002
f8c15, m=O.0001. delta=O.002
o
d=7mm
f8ldw, d=7mm, m=O.0001, delta=O.002
f8c13, d=7mm,
o
@
.'
d=Omm
f8c13, d=O. m=O.0001. delta=O.002
f81dw, d=O. m=O.0001. delta=O.002
~0~
;lj
,~~
'" \®J@)
0
. o:tl.'@
Q~
)'..........,
.Jot
\..".-
Ge
a
0
~
~
~ ~)
c:,
0
'0
Figure C-l: Time sequence of contours of constant x-moments of vorticity ( (MID)y) in a propagating
wave past a stationary cylinder at various depths ofcylinder submergence.
77
X-MOMENTS OF VORTICllY
d
=
20mm
Mc.u
LlMc.u
f9cl5, m=O.0001, delta=O.002
f91dw, m=O.0001, dw=O.002
•
~@
@ .....,
f.·-~1
0)
(g~~v
/.-r-."
":=::.::::F"
v.:,.
e
~t:1;f?
"i
~
d=7mm
f9c13, d=7mm, m=O.0001, delta=O.002
f91dw, d=7mm, m=O.0001, delta=O.002
d=Omm
f9c13, d=O, m=O.0001, delta=O.002
f91dw, d=O, m=O.0001, delta=O.002
Figure C-l: Time sequence of contours of constant x-moments of vorticity ( (Mro)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
78
d
=
20mm
Mcu
f10elS, m=O.0001, delta=O.002
f10ldw. m=O.0001, dw=O.002
d=7mm
flOel3, d=7mm, m=O.0001, delta=O.002
f1 Oldw, d= 7mm. m=O.OOO 1. delta=O.002
d=Omm
f10e13, d=O, m=O.0001, delta=O.002
f10ldw, d=O, m=O.0001, delta=O.002
@
Figure C-l: Time sequence of contours of constant x-moments ofvorticity ({Mro)y) in a propagating
.wave past a stationary cylinder at various depths ofcylinder submergence.
79
d
=
~Mcu
Mcu
20mm
f11 c15. m=O.0001. delta=O.002
f11ldw, m=O.0001. dw=O.002
t1
G
~:f)
d= 7mm
f11ldw. d=7mm. m=O.0001. delta=O.002
f11c13. d=7mm. m=O.0001. delta=O.002
@
0=-0~
d=Omm
f11 c13. d=O. m=O.0001, delta=O.002
f11ldw. d=O. m=O.0001. delta=O.002
Figure C-l: Time sequence of contours of constant x-moments ofvorticity «Mro)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
80
d
=
20mm
Mcu
f12c15, m=O.0001, delta=O.002
f12ldw, m=O.0001, dw=O.002
d= 7mm
f12ldw, d=7mm, m=O.OOOl, delta=O.002
f12c13, d=7mm, m=O.0001, delta=O.002
d=Omm
f12c13, d=O, m=O.0001, delta=O.002
f12ldw, d=O, m=O.0001, delta=O.002
-./'
Figure C-l: Time sequence of contours of constant x-moments of vorticity «Mco)) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
81
d
=
20mm
Mcu
. f13ldw. m=O,OOO 1, dw=O,002
f13c15. m=O,0001. delta=O,002
0
.~
o
,,,@
9
d= 7mm
f13ldw. d=7mm. m=O.0001, delta=O,002
f13c13. d=7mm. m=O,0001. delta=O,002
o
@
d=Omm
f13c13. d=O. m=O.0001. delta=O.002
f13ldw. d=O, m=O.0001. delta=O.002
o
Figure C-l: Time sequence of contours of constant x-moments of vorticity «Mro)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
82
d
=
20mm
f14c15, m=O.0001, delta=O.002
f141dw, m=O.0001, dw=O.002
0
o
~
~
c::.
tJ:,~1o
@
<s:~
d= 7mm
f14c13, d=7mm, m=O.0001, delta=O.002
f141dw, d=7mm, m=O.OOOl, delta=O.002
d=Omm
f14cl3, d=O, m=O.OOOl, delta=O.002
f141dw, d=O, m=O.0001, delta=O.002
Figure C-l: Time sequence of contours of constant x-moments of vorticity ( (Moo)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
83
d = 20mm
Mw
f15ldw, m=O.0001, dw=O.002
f 15c15, m=O.OOO 1, delta=O.002
d= 7mm
f151dw, d=7mm, m=O.0001, delta=O.002
f15c13, d=7mm, m=O.0001, delta=O.002
d=Omm
f15ldw, d=O, m=O.0001, delta=O.002
f15c13, d=O, m=O.0001, delta=O.002
o
Figure C-l: Time sequence of contours of constant x-moments of vorticity «Mro)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
84
d
=
Mcu
20mm
f16c15. m=O.0001. delta=O.002
f16ldw. m=O.0001. dw=O.002
0
~
~ .r;..1>;"::::"7>o
t
i
~t(.vff
ll; •. ~
(:
0
0
~
®
...
~ C)
<.
()
J~
C.i
J
Q:.7
~
@~
ii:i;O
-----d= 7mm
f16ldw, d=7mm. m=O.0001, delta=O.002
f16c13, d=7mm. m=O.OOO1. delta=O.002
d=Omm
f16c13. d=O. m=O.0001, delta=O.002
f16ldw, d=O. m=O.0001, delta=O.002
o
Figure C-l: Time sequence of contours of constant x-moments of vorticity «Mro)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
85
d = 20mm
Mcu
f17ldw, m=O.0001. dw=O.002
f17c15, m=O.0001, delta=O.002
o
d=7mm
f17ldw. d=7mm, m=O.0001, delta=O.002
f17c13, d=7mm, m=O.0001, delta=O.002
d=Omm
f17c13, d=O, m=O.0001. delta=O.002
f17ldw. d=O, m=O.0001. delta=O.002
o
Figure C-l: Time sequence of contours of constant x-moments ofvorticity ( (Mro)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
86
d
=
Mcu
20mm
f18cl5, m=O.0001, delta=O.002
f18ldw, m=O.0001, dw=O.002
~~C:QJ
@
d=7mm
f18c13, d=7mm, m=O.0001, delta=O.002
f181dw, d=7mm, m=O.0001, delta=O.002
d=Omm
f18c13, d=O, m=O.OOOl, delta=O.002
f181dw, d=O, m=O.OOOl, delta=O.002
o
Figure C-l: Time sequence of contours of constant x-moments of vorticity ( (MID)y) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
87
d =·20mm
Mcu
f19c15. m=O.0001. delta=O.002
f191dw, m=O.0001, dw=O.002
.<>
~j
d=7mm
f191dw, d=7mm, m=O.0001, delta=O.002
f19c13, d=7mm. m=O.0001. delta=O.002
o
d=Omm
f19c13. d=O. m=O.0001. delta=O.002
f19ldw, d=O, m=O.0001, delta=O.002
c'
o
o
Figure C-l: Time sequence of contours of constant x-moments of vorticity ( (Moo)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
88
d
=
20mm
~Mw
Mw
f20c15, m=O.0001, delta=O.002
f20ldw, m=O.0001, dw=O.002
d= 7mm
f20ldw, d=7mm, m=O.0001, delta=O.002
f20c13, d=7mm, m=O.0001, delta=O.002
o
o
d=Omm
f20c13, d=O, m=O.b001, delta=O.002
f20ldw, d=O, m=O.0001, delta=O.002
o
Go
Figure C-l: Time sequence ofcontours ofconstant x-moments of vorticity ( (Mco)y) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
89
d
=
Mw
20mm
~Mw
f21ldw. m=O.0001. dw=O.002
f21 c15. m=O.0001. delta=O.002
o
d= 7mm
f21ldw. d=7mm. m=O.0001. delta=O.002
f21 c13, d= 7mm, m=O.0001. delta=O.002
o
d=Omm
f21 c13, d=O, m=O.OOO 1, delta=O.002
f211dw, d=O, m=O.0001, delta=O.002
@
Figure C-l: Time sequence of contours ofconstant x-moments of vorticity «Mro)y) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
90
d
=
20mm
Mw
f22c15, m=O.0001, delta=O.002
f22ldw, m=O.0001, dw=O.002
d=7mm
f22c13, d=7mm. m=O.0001, delta=O.002
f22ldw. d=7mm. m=O.OOOl, delta=O.002
o
d=Omm
f22c13. d=O, m=O.OOOl, delta=O.002
f22ldw. d=O. m=O.0001. delta=O.002
6
tt..'
@
Figure C-l: Time sequence of contours of constant x-moments ofvorticity ( (Moo)y) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
91
V-Moments of Vorticity.
Figure C2 is a time sequence of contours of constant y-moments of vorticity (
(Mro)x ) in a wave past a stationary cylinder at various depths of cylinder submergence.
The time sequence consists of frames number 7 through 23.
92
d
=
20mm
Mw
f7cd5, m=O.0001, delta=O.002
f7ddw, m=O.0001, delta=O.002
d=7mm
f7cd3, m=O.0001, delta=O.002
f7ddw, m=O.0001, delta=O.002
'"I"v;;
"--.,'
00
o
d=Omm
f7cd3, m=O.0001, delta=O.002
f7ddw, m=O.0001, delta=O.002
o
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
93
d
=
20mm
~Mcu
Mcu
f8edS, m=O.0001. delta=O.002
f8ddw. m=O.0001. delta=O.002
o
d= 7mm
f8ddw, m=O.0001. delta=O.002
f8ed3, m=O.0001. delta=O.002
o
()
@
d=Omm
f8ed3. m=O.0001. delta=O.002
60~
0
CJ
0
0
fBddw, m=O.0001, delta=O.002
0
~o
0
Figure C-2: Time sequence of contours of constant y-moments ofvorticity ( (Moo).) in a propagating
wave past a· stationary cylinder at various depths of cylinder submergence.
94
d == 20mm
Mcu
f9cd5. m=O.0001, delta=O.002
f9ddw. m=O.0001, delta=O.002
o
d=7mm
f9ddw. m=O.0001. delta=O.002
f9cd3, m=O.0001, delta=O.002
d;"'Omm
f9cd3. m=O.0001, delta=O.002
f9ddw, m=O.0001, delta=O.002
o
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
95
d
=
Mw
20mm
f10ddw, m=O.0001, delta=O.002
f10cd5, m=O.0001, delta=O.002
o
o
o
d=7mm
f10ddw, m=O.0001, delta=O.002
f10cd3, m=O.0001, delta::::O.002
o
o
d=Omm
fl0cd3, m=O.0001, delta=O.002
f10ddw, m=O.0001, delta=O.002
C:J
'0CJ
(i
@
@@
-==:>
@
0@
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
96
d = 20mm
Mw
f11 cdS, m=O.0001, delta=O.002
f11 ddw, m=O.0001, delta=O.002
o
d= 7mm
f11 ddw, m=O.0001, delta=O.002
f11 cd3, m=O.0001, delta=O.002
,r.....
o
o
~o
0
o
d=Omm
fl1 cd3, m=O.OOOl, delta=O.002
fl1ddw, m=O.0001, delta=O.002
o
o
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo).) in a propagating
wave past a stationary cylinder at various depths ofcylinder submergence.
97
d
=
20mm
LlMcu
Mcu
f12cd5, m=O.0001, delta=O.002
f12ddw, m=O.OOO( delta=O.002
d=7mm
f12ddw, m=O.0001, delta=O.002
f12cd3, m=O.OOO 1, delta=O.002
o
\) 0
o
0
o
c,
d=Omm
f12cd3, m=O.OOOl, delta=O.002
f12ddw, m=O.0001, delta=O.002
o
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo)x) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
98
d
=
20mm
Mcu
f13cd5, m=O.0001, delta=O.002
f13ddw, m=O.0001, delta=O.002
•
o
(';:;.
0
o
o
d=7mm
f13ddw, m=O.0001, delta=O.002
f13cd3, m=O.0001, delta=O.002
o
@
d=Omm
f13cd3, m=O.OOOl, delta=O.002
f13ddw, m=O.0001, delta=O.002
0
U
~.
.:....-E)
~
@{{P
@
*
Figure C-2: Time sequence ofcontours of constant y-moments ofvorticity ( (MOl).) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
99
d
=
Mw
20mm
f14cd5, m=O.0001, delta=O.002
f14ddw, m=O.0001, delta=O.002
0 ..'
". .V:
<..~
~S>o
<0
O.~,
r:.......
~
\)
\)
0
()
d=7mm
f14cd3, m=O.0001, delta=O.002
(J
0
c:CJ
\)
f14ddw, m=O.OOO 1, delta=O.002
()
00
<:;J0
V'
(;;)0
~
0
~
0
00
0
d=Omm
f14cd3, m=O.0001, delta=O.002
f14ddw, m=O.0001, delta=O.002
o
Figure C-2: Time sequence ofcontours of constant y-moments of vorticity ( (Mro)x) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
100
d
=
20mm
Mcu
f15cd5, m=O.0001, delta=O.002
f15ddw, m=O.0001, delta=O.002
o
d=7mm
f15ddw, m=O.0001, d,elta=O.002
f15cd3, m=O.0001, delta=O.002
o
,.........
0>
o
\ ..(
o
d=Omm
f15cd3, m=O.0001, delta=O.002
f15ddw, m=O.OOO 1, delta=O.002
Figure C-2: Time sequence of contours of constant y-moments ofvorticity ( (Moo).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
101
d
=
20mm
Mcu
LlMcu
f16cd5, m=O.0001, delta=O.002
~0(Q)
<';)
@)
(j
f16ddw. m=O.0001, delta=O.002
~
v·
a
\)
aD
0
0
0
d= 7mm
f16cd3, m=O.0001, delta=O.002
f16ddw. m=O.0001, delta=O.002
.0
o
0'
d=Omm
f16d3. m=O.0001, delta=O.002
f16ddw. m=O.0001. delta=O.002
Figure C-2: Time sequence ofcontours ofconstant y-moments ofvorticity ( (Moo).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
102
d
=
20mm
Mcu
f17ddw, m=O.0001, delta=O.002
f17cd5, m=O.0001, delta=O.002
o
o
d=7mm
f17ddw, m=O.0001, delta=O.002
f17cd3, .m=O.0001, delta=O.002
o·
o
d=Omm
f17d3, m=O.0001, delta=O.002
f17ddw, m=O.OOOl, delta=O.002
~.
Figure C-2: Time sequence ofcontours of constant y-moments of vorticity ( (MOl).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
103
d = 20mm
Mw
f18cd5, m=O.0001, delta=O.002
f18ddw. m=O.0001. delta=O.002
,,../()"';\(
'. ,
°0 _
0--'/
d=7mm
f18ddw. m=O.0001. delta=O.002
f18cd3, m=O.0001, delta=O.002
o
o
d=Omm
f18d3, m=O.0001, delta=O.002
f18ddw. m=O.0001. delta=O.002
o
Figure C-2: Time sequence of contours of constant y-moments ofvorticity ( (MID).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
104
d = 20mm
Mc.u
LlMc.u
f19cd5. m=O.0001. delta=O.002
f19ddw. m=O.0001. delta=O.002
o
d= 7mm
f19ddw. m=O.0001. delta=O.002
f19cd3. m=O.0001. delta=O.002
o
o
o
o
d=Omm
f19d3. m=O.0001. delta=O.002
f19ddw. m=O.0001. delta=O.002
o
o
Figure C-2: Time sequence of contours of constant y-moments ofvorticity ( (MID).) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
105
d
=
20mm
Mcu
.6Mcu
f20cd5, m=O.0001, delta=O.002
f20ddw, m=O.0001, delta=O.002
o
Cf
d=7mm
f20ddw, m=O.0001. delta=O.002
f20cd3, m=O.0001, delta=O.002
o
d=Omm
f20d3, m=O.0001, delta=O.002.
f20ddw, m=O.0001, delta=O.002
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo)x) in apropagating
wave past a stationary cylinder at various depths of cylinder submergence.
106
d
=
20mm
Mw
L\Mw
f21 cdS, m=O.OOO 1, delta=O.002
f21 ddw, m=O.OOO 1, delta=O.002
(J
d=7mm
f21 ddw, m=O.OOOl, delta=O.002
f21 cd3, m=O.OOOl, delta=O.002
d=Omm
f21 cd3, m=O.OOOl, delta=O.002
f21 ddw, m=O.OOOl, delta=O.002
o
o
ed o
o 0
o
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (Moo)x) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
107
d
=
Mc.u
20mm
LlMc.u
f22cd5, m=O.0001, delta=O.002
f22ddw, m=O.0001, delta=O.002
o
d= 7mm
f22ddw. m=O.0001.
f22cd3. m=O.0001, delta=O.002
o
.d
= Omm
f22cd3, m=O.0001, delta=O.002
f22ddw, m=O.0001, delta=O.002
o
Q
6
o C?~
0
s;;;j
Figure C-2: Time sequence of contours of constant y-moments of vorticity ( (MID)x) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
108
d = 20mm
L\Mw
Mw
f23cd5, m=O.0001, delta=O.002
d= 7mm
f23cd3, m=O.0001,
d=Omm
f23cd3. m=O.0001, delta=O.002
f23md.cdr
Figure C-2: Time sequence of contours ofconstant y-moments ofvorticity ( (MID).) in a propagating
wave past a stationary cylinder at various depths of cylinder submergence.
109
APPENDIX D: OSCILLATIONS OF THE CYLINDER IN AN INCIDENT WAVE:
EFFECTS OF PHASE SHIFT.
During two additional experiments, the cylinder underwent a small-amplitude
orbital motion obtained by programming sinusoidal and cosinusoidal motions in the
horizontal and vertical directions respectively. The period of the orbital motion of the
cylinder was 2 sec, and the diameter of the orbit was 0.127 m. The depth of submergence
of the cylinder at the highest point of its trajectory was 20 rom for both experiments. The
phase shift between the cylinder motion and the wave motion was approximately zero
radians in the first case and approximately 1t radians in the second case. The parameters
of the motion were chosen such as to match the Keulegan-Carpenter number of the
cylinder motion, given by
(
)
KCcyl
21tA
=n
to the Keulegan-Carpenter number of the wave motion, given by (Kc)wave = UrnTID. Here
A is the amplitude of the cylinder motion, D is the cylinder diameter, Urn is the maximum
velocity of the flow, and T is the wave period. Definition of the Keulegan-Carpenter
number and the procedure of measurement of Urn are discussed in Appendix A. The value
of (Kc)wave was found to be equal to 6.96 for all three values of the cylinder submergence.
Two cases were considered. During the first experiment, the phase shift between
the cylinder motion and the wave motion was approximately 1t radians,'meaning that a
trough of the propagating wave was directly above the cylinder when the cylinder was at
the highest point of its trajectory. The top graph of Figure D-l shows the trace of
110
measured horizontal force with respect to time. In this case (so-called maximum Fx case),
the peak-to-peak amplitude of the signal is approximately 0.3 Volts, and the. period is
equal to 2 sec (same as the period of the wave motion and the period of the cylinder
motion).
During the second experiment, the cylinder was moving in-phase with the
propagating wave, meaning that a crest of the wave was directly above the cylinder when
it was at its highest position. In this case (so-called minimum Fx case), the value of the
measured horizontal force is almost zero throughout the whole wave cycle, as shown on
the bottom graph of Figure D-l.
Figure D-2 shows images of the flow field in the maximum Fx case. The left
column of images represents the instant tl corresponding to the maximum negative value
of the F x ' The right column represents the instant t2 corresponding to the maximum
positive value of the Fx'
Top images show the instantaneous vorticity distributions in the flow field.
Contours of constant positive and negative vorticity are represented by the solid and
dashed lines respectively. There are well-defined concentrations of vorticity with high
values of circulation at both instants of time. Positive and negative vortices develop on
the surface of the cylinder and interact with the vortices shed during previous wave
cycles.
Lower images show instantaneous velocity fields and streamline patterns. At both
instants, the flow velocity experiences significant changes in direction and magnitude
111
throughout the field of view. The singular points in streamline patterns indicate positions
of the vortices.
The left image in Figure D-3 shows an instantaneous vorticity distribution for the
minimum Fx case. Vortices that are developing on the surface of the cylinder have low
values of circulation. There are no vortices shed during previous wave cycles.
The right image shows the corresponding velocity field and streamline pattern.
The velocity field is very uniform, and the streamline pattern is not distorted and has no
singular points.
Hence, moving the cylinder in-phase with the propagating wave significantly.
reduces the magnitude of the horizontal force acting on the cylinder and reduces the
vortex shedding.
112
Fx
~.jt.,
'_I
(Volts)
0.2
0.1 pt ~ V'v
II""
~~
0.0
f-
-0.1
f-
-0.2
I-
"-
Fx
~
\
~
I~
-0.3
o0
rv IAKI MU M
05
'1 0
1I
~
I
15
V
.t'
r--
~ ro"" ,.., \'toll JI-' ~I\
\
25
20
1\
f\.
~ V'
30
I
V
/
1I
4
35
o
sec)
Fx
~-
(Volts)
I
,_I,
0.2
0.1
I-
Mn.
..J1l
0.0
l-
I
I
.....
r"
~
.1
A
lJ..
,.1.,1,
I
M
III Ni MU
:
-0.1
-0.3
o0
-
x
-0.2
I
J. 5
-
,
1.(
15
_
,
~. 0
I
I
.
.)
3.0
35
( sec)
10
20
30
N (Frame No.)
Figure D-l: Oscillations of a cylinder in an incident wave: effects of phase shift.
113
t
Figure D-2: Images for maximumFx case. Oscillating cylinder. romio =3 S·I, I:!..ro =3 S·I.
114
iNTEN
I
'
i i! ,
i J~:
I
~
I
T
I
1
I
I.
I
t
Figure D-3: Images forminimumFx case. Oscillating cylinder.mmin =3 S·I, Am =3 S·l
115
t
115
APPENDIX E: ADDITIONAL IMAGES
Figure E-1 shows velocity fields and streamline topology for two different depths
of submergence of the stationary cylinder. The top row of images corresponds to hID =
1.57. Frames 9 and 15 correspond to the crest and the trough of the propagating wave
respectively. The bottom row of images shows the same phases of the wave motion (the
crest and the trough) for hID = 0.55. Streamline patterns corresponding to the shallow
submergence case exhibit additional features (saddle points and foci) while having the
same general direction of the flow as in the case of deep cylinder submergence.
Figure E-2 shows images of instantaneous vorticity distribution (left column) and
velocity field with superimposed streamlines (right column) corresponding to frame 9 of
the stationary cylinder case for three different depths of submergence. Contours of
positive and negative vorticity are represented by solid and dashed lines respectively.
Streamline patterns exhibit singular points corresponding to the major vorticity
concentrations for all three submergence levels.
116
Figure E-l. Velocity fields and streamline topology at two instants oftime for two different depths of
submergence.
117
Figure E-2. Images corresponding to frame number 9. Stationary cylinder. Various depths of cylinder
submergence.
118
EXPOSURE
-~-----~
Figure £-2.
j
18
. _ - ~ ­
VITA
Peter Oshkai was born to Tamara and Mikhail Oshkai on March 16, 1974 in
Voronezh, Russia.
Peter Oshkai completed a specialized program in mathematics and physics during
his last two years in high school (College N° 1 of Voronezh) and was accepted as a
student to the Department of Applied Mathematics and Mechanics of Voronezh State
University without entrance examinations as a winner of 1991 Regional Contest in
Mathematics and Physics. After three years of studies he was selected to participate in a
student exchange program between the universities of Russia and United States. He
completed the fourth year of studies at Lehigh University in Bethlehem, Pennsylvania,
earning the degree of Bachelor of Arts in Mathematics in May 1995. At the same time,
Peter Oshkai was accepted to the graduate school of Lehigh University and was awarded
Baldwin Fellowship to begin his Master's program at the Department of Mechanical
Engineering and Mechanics in September 1995.
During his work towards the Master's degree, Peter Oshkai was awarded several
teaching and research assistantships. Current research resulted .in the following
publications and presentations:
"Free-Surface Wave Interaction with a Horizontal Cylinder", Submitted to Journal of
Fluids and Structures.
"A Free-Surface Wave past an Oscillating Cylinder". Technical note. In progress.
119
"A Free-Surface Wave past an Oscillating Cylinder: Control Concepts", American
Physical Society's 49th Meeting of the Division of Fluid Dynamics, Syracuse, NY,
November, 1996
"Wave-Induced Flow past a Cylinder in Orbital Motion", Lehigh University Graduate
Student Symposium. 1997
"Wave Interaction with the Horizontal Cylinder". To be presented at American Physical
Society's 5Ft Meeting of the Division of Fluid Dynamics, Philadelphia, PA, November,
1998.
120
END
OF
TITLE