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A dynamic response model for free floating horizontal cylinders subjected to waves

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Title:
A dynamic response model for free floating horizontal cylinders subjected to waves
Series Title:
UFLCOEL-TR
Creator:
Doynov, Krassimir I., 1963-
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Language:
English
Physical Description:
ix, 182 leaves : ill. ; 29 cm.

Thesis/Dissertation Information

Degree:
Ph.D.
Degree Grantor:
University of Florida
Degree Disciplines:
Coastal and Oceanographic Engineering
Committee Chair:
Sheppard, Max
Committee Members:
Dean, Robert
Hanes, Daniel
Kurzweg, Ulrich H.
Ochi, Michel
Sheng, Peter
Thieke, Robert

Subjects

Subjects / Keywords:
Amplitude ( jstor )
Boundary conditions ( jstor )
Buoyancy ( jstor )
Cylinders ( jstor )
Damping ( jstor )
Elliptical cylinders ( jstor )
Hydrodynamics ( jstor )
Velocity ( jstor )
Wave diffraction ( jstor )
Waves ( jstor )
Coastal and Oceanographic Engineering thesis, Ph. D ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
A semi-analytic method for computing the dynamics of free-floating, horizontal cylinders, subjected to ocean waves has been developed. The bodies analyzed in this analysis are cylinders with circular and elliptical cross sections and variable still water drafts. The motions considered are heave, surge, and pitch. the technique computes the added mass and damping coefficients using linearized radiation theory. The efficiency of the numerical scheme is due to the simplicity of the mathematical scheme used - a combination of original holomorphic functions, convergent infinite power series, and conformal mapping. An object-oriented approach was sued for the computational aspects of the problem using the programming language C++. Physical experiments in a wave tank were conducted with circular and elliptical horizontal cylinders in order to verify the method. The cylinders were positioned at the water surface and forced to oscillate sinusoidally first in surge and later in heave motion. Different still water drafts and oscillation frequencies were used in the experiments. The far field waves produced by the oscillating cylinder were measured using capacitance wave gauges. the damping and added mass coefficients were computed from the measured wave data. there is a good agreement between the predicted and measured coefficients.
Thesis:
Thesis (Ph.D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 175-180).
General Note:
Typescript.
General Note:
Vita.
Funding:
Technical report (University of Florida. Coastal and Oceanographic Engineering Dept.) ;
Statement of Responsibility:
by Krassimir I. Doynov.

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UFL/COEL-TR/121

A DYNAMIC RESPONSE MODEL FOR FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES by
Krassimir I. Doynov

Dissertation

1998




A DYNAMIC RESPONSE MODEL FOR
FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES
By
Krassimir I. Doynov

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

1998




To Boris and Galina




ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my advisor Dr. Max Sheppard for his guidance, technical, and moral support during my doctoral research. Being chairman of my graduate committee, he provided me with his insight and perspective and gave me the freedom to pursue my research interests. Being a noble soul, he granted me his friendship and moral help during the difficult moments I had to go through as an international student. I would also like to thank the members of my committee: Dr. Robert Dean, Dr. Michel Ochi, and Dr. Ulrich Kurzweg for their time and advice, Dr. Peter Sheng, Dr. Robert Thieke, and Dr. Daniel Hanes for reviewing this work.
For the clarity of all drawings in my dissertation, for her help, trust, inspiration, and love, I am forever grateful to my wife, Galina.
For their constant support, encouragement, and inspiration, I am deeply grateful to my parents, lordan Doynov and Nadejda Doynova, and to my brother Ivan.
Additional thanks for making my time here enjoyable go to my fellow students Wayne Walker, USA; Thanasis Pritsivelis, Greece; Roberto Liotta, Italy; Emre Otay, Turkey; Ahmed Omar, Egypt; Kerry Anne Donohue, USA; Wendy Smith, USA; and Matthew Henderson, USA.
Finally, words cannot express my love to my son Boris, whose presence and love make my life a real adventure.




TABLE OF CONTENTS
pNge
ACKNOWLEDGMENTS ..................................................................11.1i
KEY TO SYMBOLS........................................................................... vi
ABSTRACT................................................................................. viii
CHAPTERS
1 MOTIVATION ............................................................................ 1
2 INTRODUCTION.......................................................................... 4
2. 1. Historical Retrospective of Floating Body Studies....................................... 4
2.2. Classification................................................................................ 7
2.2. 1. Large and Small Bodies ............................................................. 7
2.2.2. Deterministical and Statistical Approaches ....................................... 11
2.2.3. The Concept of Added Mass ...................................................... 13
2.2.4. Classification of Damping.......................................................... 14
2.2.5. Numerical Methods Classification ................................................. 16
2.3. Advancements in Floating Body Studies ................................................ 24
2.4. Presentation of the Results of Investigations............................................ 26
2.5. Some Thoughts about the Current State of Knowledge................................ 34
3 FORMULATION OF THE PROBLEM ................................................. 37
3. 1. General Description of the Problem and its Simplifications ........................... 37
3. 1.1. Incompressible Fluid Assumption.................................................. 37
3.1.2. Governing Equations and Definitions ............................................. 38
3.1.3. Inviscid Fluid Assumption.......................................................... 39
3.1.4. Irrotational Flow Assumption ..................................................... 41
3.1.5. Dynamic Free Surface Boundary Condition (DFSBC) ......................... 42
3.1.6. Kinematic Free Surface B'oundary Condition (KFSBC)......................... 42
3.1.7. Sea Bottom Boundary Condition (SBBC)........................................ 43
3.1.8. Wetted Body Surface Boundary Condition (Sb) .................................. 43
3.1.9. Linearization to First-Order Theory............................................... 44
3.2. Floating Body Dynamics ................................................................. 47
3.2. 1. Conservation of Linear Momentum............................................... 47




3.2.2. Conservation of Angular M om entum .............................................................. 48
3.2.3. M atrix Form of the Dynamics Equations ....................................................... 50
3.3 Decomposition and Separation of the Hydrodynamics from the Body Dynamics ....... 51 3.4. Hydrodynamic Properties and Forces ...................................................................... 54
3.5. Hydrodynamic Relationships, Identities, and D efinitions .......................................... 56
3.6. Algorithm for the Solution of the Problem .............................................................. 61
4 RADIATION PROBLEM SOLU TION ................................................................... 62
4. 1. Problem Statem ent and D efinitions .......................................................................... 62
4.2. M ain Idea behind the Semi-Analytic Technique (SAT) ............................................ 64
4.3. Semi-Analytic Technique. Determination of the Unknown Coefficients ................... 70
4.3. 1. Boundary Condition on Sb .............................................................................. 71
4.3.2. Conform al M apping ....................................................................................... 71
4.3.3. Left-H and Side of the Boundary Condition on Sb ........................................... 77
4.3.4. Right-H and Side of the Boundary Condition on Sb ......................................... 87
4.3.5. Fourier Expansion of LHS and RHS. Solution for the Unknown
Coeffi cients .............................................................................................................. 97
4.3.6. Discussion of the Uniqueness of the Solution ................................................ 101
5 EXPERIM EN TS ................................................................................................... 106
5. 1. Purpose of the Experim ents .................................................................................. 106
5.2. General Set-up ...................................................................................................... 106
5.2. 1. Cylinders ...................................................................................................... 108
5.2.2. W ave Absorption at the Ends of the Tank ................................................... 110
5.2.3. W ave Gauges ............................................................................................... 110
5.2.4. Surface Tension ........................................................................................... III
5.3. W ave Absorption and Reflection Analysis ............................................................. 112
5.4. M odel Scale Selection: Froude Scaling .................................................................. 117
5.5. Discussion of the Experim ental Accuracy .............................................................. 118
5.6. Discussion of the Experim ental Procedure ............................................................. 118
6 ANALY SIS OF THE RESULTS ........................................................................... 128
6. 1. Surge M ode Oscillations ....................................................................................... 130
6.2. H eave M ode Oscillations ...................................................................................... 140
6.3. D amping, Added M ass, and Frequency Response Function ................................... 149
6.4. Num erical Convergence ........................................................................................ 164
6.5. Conclusions .......................................................................................................... 166
APPEND IX ................................................................................................................. 168
LIST OF REFEREN CES ............................................................................................. 175
BIOGRAPHICAL SKETCH ....................................................................................... 181




KEY TO SYMBOLS

Symbol Description
A Amplitude of the incident wave
A, Far-field wave amplitude
a+ =>A/ V Far-field amplitude. Dimension time.
a, b Vertical, and horizontal semi-axes of the elliptical
cylinder
a,, Power series coefficient of the nth term
B Breadth of the waterline section of Sb
B0 Sea bottom boundary
CZ Group velocity
[C] Buoyancy restoring force matrix of the floating body
D _0 a V(a) Total derivative in space and time
Dt ait
E Water bulk modulus
{FD} Exciting force vector due to diffraction
g Gravity acceleration
H Height of the incident wave
pH(co)= RAO Transfer function
h Water depth
i imaginary unit
KC Keulegan-Carpenter number
k Wave number of the incident wave
L Wave length of the incident wave
[M] Mass matrix of the floating body
n,, Component of unit normal to Sb vector in a direction
Pa Atmospheric pressure
p Pressure
R Radius of the circular cylinder
Re Reynolds number
SA Part of the water surface cut out by Sb
Sb Instantaneous wetted body surface
Mirror image of Sb in the air
S. Lateral boundary at infinity
S.(CO) Incident wave spectrum




S, (CO) Response spectrum of the floating body
T Period of the incident wave
U. Magnitude of the horizontal velocity of the incident wave
u = (u,v,w) Fluid particle velocity
V Time-amplitude of generalized velocity
= dW / dt Energy flux
Xa Generalized displacement in a direction
x = (xc, yc, z) Coordinates of the center of mass
y = x+ i z Complex variable
8 Small parameter
(D Velocity potential
0 Time-amplitude of the velocity potential
0 D, oR Diffraction, radiation velocity potential
b. Radiation velocity potential due to unit velocity in
generalized a direction
70,y) Time-amplitude of the water elevation
2 Damping
/' Added mass
v Kinematic viscosity
P Water density
co Circular frequency of the incident wave
1 Time-amplitude of X,
T Stream function
VTime-amplitude of the stream function 4(xy, t) Water elevation




Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
A DYNAMvIC RESPONSE MODEL
FOR
FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES By
Krassimir I. Doynov
December, 1998
Chairman: D. Max Sheppard
Major Department: Coastal & Oceanographic Engineering
A semi-analytical method for computing the dynamics of free-floating, horizontal cylinders, subjected to ocean waves has been developed. The bodies analyzed in this analysis are cylinders with circular and elliptical cross sections and variable still water drafts. The motions considered are heave, surge and pitch. The technique computes the added mass and damping coefficients using linearized radiation theory. The efficiency of the numerical scheme is due to the simplicity of the mathematical scheme used a combination of original holomorphic functions, convergent infinite power series, and conformal mapping. An object-oriented approach was used for the computational aspects of the problem using the programming language C++.
Physical experiments in a wave tank were conducted with circular and elliptical horizontal cylinders in order to verifyr the method. The cylinders were positioned at the




water surface and forced to oscillate sinusoidally first in surge and later in heave motion. Different still water drafts and oscillation frequencies were used in the experiments. The far field waves produced by the oscillating cylinder were measured using capacitance wave gauges. The damping and added mass coefficients were computed from the measured wave data. There is a good agreement between the predicted and measured coefficients.




CHAPTER 1
MOTIVATION
In XV-century Europe, the brilliant engineer, scientist and artist, Leonardo da Vinci recorded for the first time an engineering application of a free floating buoy. The buoy was used to measure the water velocity in streams and consisted of a weighted rod and flotation bladder. It was released in the water flow and its downstream travel measured after a given period of time and the average velocity computed as the distance divided by the elapsed time. Since that time drifter buoys have proved to be very effective in increasing the level of our understanding of the ocean environment and in improving weather analysis and forecasting. Scientific investigations using drifing buoy systems include measurements of atmospheric pressure, air and sea-surface temperature, solar radiation, air humidity, wind shear, wave evolution, wave noise, light penetration, oceanic current speeds, and water temperature and salinity. A variety of drifter buoy systems, deployed in the world's ocean, are used to monitor the spatial and temporal distribution of the above mentioned environmental variables. With modern satellite technology, which provides both buoy positioning information and a means of receiving and re-transmitting data from the buoys, the accuracy and reliability of these measurements has been greatly improved. Measurements can be recorded with data acquisition systems mounted on the buoys or transmitted to land based stations via satellite (Berteaux, H.O0., 199 1). Data from these buoys are vital to weather organizations for early detection of storms and for daily weather predictions. This information is essential for those organizations and




companies involved in offshore oil exploration and production, marine transportation, commercial and recreational fishing and boating, and military operations. For reliable data transmission to the monitoring satellite, it is crucial that the surface floats provide a stable platform under a variety of wave, current, and wind conditions. Since it is the surface water motion and properties that are of interest, the question becomes how well do these buoys track the currents and how do they respond to the wave motion. How a buoy responds to currents and waves depends on a number of quantities including the buoy size, shape, mass and mass distribution. Knowledge of the response characteristics of a buoy for a given set of wind and wave conditions as a function of their structure and wind-wave parameters would allow buoy designs to be optimized for the sea state conditions in which they are to be deployed. Therefore the buoy hull must be designed in accordance with certain stability design criteria for different ocean and atmosphere conditions.
The purpose of the following research is to investigate how variations of draft, size and shape of drifting buoy systems influence fluid-body dynamics and to create a computer-based model. In the focus of the research are floating horizontal cylinders of circular and elliptical cross sections with variable still water drafts. The computer-based model is intended to provide designers with estimates of the dynamic response characteristics, in terms of response amplitude operators or frequency response functions, of relatively small buoys.
The dissertation is comprised of six chapters. Chapter 2 presents a brief historical retrospective of floating body studies, followed by classifications that introduce criteria for large and small bodies, and deterministic and statistical approaches. Next, the concept of added mass is introduced, as well as classifications of damping and the most frequently




used numerical methods for the determination of these hydrodynamic properties. Furthermore, the most significant advancements in floating body studies are presented, followed by a review of recent scientific works on the related subjects. This chapter concludes with an attempt to classify the current state of knowledge about floating bodies as one going simultaneously in three main directions. As a result of the well-known conservation laws of mass, linear, and angular momentum, chapter 3 formulates the floating body dynamics as a system of linear second-order differential equations with boundary conditions of Neumann and Dirichiet-Neumann type. After introducing the generally accepted simplifications for incompressible, inviscid fluid and irrotational flow, the linearized radiation theory is deduced. Furthermore the hydrodynamics is decomposed from the body dynamics, and some hydrodynamic theorems are given with purpose to express all unknown variables of the floating body dynamics as functions of the far-field amplitudes. Chapter 4 describes an exact analytical asymptotic solution of the radiation problem, which derives the far-field amplitudes as functions of the wetted surface of the floating body, and the circular frequency of the incoming harmonic wave. Chapter 5 describes the experiments conducted in heave and surge motion, which main purpose is to obtain data for the far-field wave amplitudes and thus to verify the analytic solution, introduced in chapter 4. These experiments were carried out in a wave tank at the Coastal Engineering Laboratory at the University of Florida. Chapter 6 presents the analysis of the results from the experiments and comparison with the numerical solution. The conclusion is given at the end of this chapter.




CHAPTER 2
INTRODUCTION
2. 1. Historical Retrospective of Floating Body Studies
Known since the ancient civilizations, the ship and boat transportation had
naturally attracted the attention of the universal minds of the 18th century and became the first theoretically investigated floating bodies. Following Vugts' historical survey (1971), the great mathematician Leonhard Euler was the first who studied in a typical mathematical framework with lemmas, corollaries and propositions the motions of ships in still water. In 1749 his work "Scientia Navalis" was edited in two volumes and published in Latin in St. Petersburg, Russia. In 1746 the French scientist Bouguer published a similar work and noted that he was familiar with the fact that Euler had been working on the subject but that he had not yet been able to lay hands on his results. Daniel Bernoulli was the first who examined ship motions in waves, and won the prize of the French Academy of Sciences for his work in 1757. Considering the resonance phenomenon, he examined forced oscillations of ships in waves. Having wrong ideas about the wave motion, Bernoulli did not arrive at correct conclusions. Nevertheless his work was considered as classic for a long time. In 1861 William Froude published his paper "On the Rolling of Ships" where the ship was assumed sailing broadside to the waves and had to follow the wave slope and the orbital motion of the wave particles. Practically Froude's study dealt with a range of very low frequency motions, thus originating the generalization




that most engineering approaches in floating body studies are only valid in a certain range of practical interest. Developing further Froude's idea with a paper in 1896 "The NonUniform Rolling of Ships" William Froude's son, R. E. Froude, added the forcing of the rolling motion for regular and irregular waves. Significant advancements in floating body studies were made by Kriloff (1896, 1898), who considered for the first time the threedimensional problem, working with six degrees of freedom. Kriloff introduced the hypothesis of approximating the actual pressure on the floating body surface by the corresponding pressure in the wave structure, not being disturbed by the presence of the floating body. As pointed out by Vugts (1971), the same hypothesis had been implicitly made by William Froude in his earlier and more restricted work. Since then this hypothesis has been known as Froude-Kriloff hypothesis and dominated almost all floating body studies up to 1953. Kriloff computed the wave exciting forces and the restoring forces and included the hydrostatic-coupling effects between heave and pitch. In an additional estimated term, he included a resistance to motions, thus introducing the concept of damping into the floating body studies. Speaking in modern terms Kriloff left out the hydrodynamic mass and the hydrodynamic coupling among the various motions. The hydrodynamic mass became known earlier. William Froude had realized that for a longer natural roll period an apparent increase in mass was necessary. Because of the presence of the free surface of the fluid and the determination of the hydrodynamic mass, respectively, the added mass becomes much more complicated due to the wave frequency dependence. In the first half of the 20th century, some attempts were made to get more information on the added mass and damping, particularly for ships and ships' sections. With an exception for the limiting case of infinitely high motion frequencies, no major




success has been achieved in increasing the knowledge of these hydrodynamic properties, as evidenced in a comprehensive survey of the available knowledge made by Weinblum and St. Denis in 1950. Despite the insufficient understanding of the effects of hydrodynamic mass and damping and the omitting of hydrodynamic coupling, the greatest limitation they felt at that time was the restriction to regular waves. Weinblum and St. Denis quote a saying of Lord Rayleigh: "The basic law of the seaway is the apparent lack of any law." Only three years later St. Denis and Pierson (1953) wrote in a paper that "In a broad sense the laws of nature are Gaussian," thus introducing the statistical description of sea waves and body motions to the floating body studies. The concept of spectral analysis of sea waves was defined by assuming the wave elevation as the sum of a large number of simple sinusoidal waves, each having its own length, height and direction. Consequently, the floating body motion is considered the sum of its responses to each individual wave component. Since the phases of those wave components are randomly distributed only statistical information can be obtained. The superposition of harmonic waves and harmonic body responses meant enormous stimulation for the floating body studies. In 1957 Korvin-Kroukovski and Jacobs introduced a strip theory that builds on two-dimensional solutions to get three-dimensional effects for elongated floating bodies. At that revolutionary time, 18 years after the invention of the greatest scientific tool of the 20th century the computer whose father was the American from Bulgarian descent Dr. John Atanasoff, all the basic elements were available to obtain engineering solutions for a coupled hydrodynamic motion, at least for heave and pitch.




2.2. Classification
2.2. 1. Large and Small Bodies
There are at least three relevant length scales in the interaction between waves and a fixed or floating body. They are the characteristic body dimension B, defined as the predominant horizontal size of the body projection onto the vertical plane of the wave front, the wavelength L=2 if/k (where k is the wave number), and the wave amplitude A. If B, for a fixed vertical cylinder, becomes relatively large then the presence of the cylinder will disturb the incident wave pattern. As the incident waves impinge on the cylinder, reflected waves move outward. In the shadow zone, on the sheltered side of the cylinder, wave fronts are bent around the cylinder and thus form diffracted waves. The combination of reflected and diffracted waves is usually called scattered waves, but the process itself is generally termed diffraction. The three scales, B, L, and A, may form two physically meaningful ratios. The first one is the diffraction parameter B/L an important dimensionless variable relating to the intensity of the scattered waves. Often used with the same purpose is kB derived from the multiplication of the diffraction parameter with the constant 2 if. The second ratio is A/B, which in deep water is proportional with the same constant 2 if to the Keulegan-Carpenter number, KC, defined as
U UKC= .__ (2.1)
B aut
at
where the wave period is T, and the magnitude of the horizontal velocity of a harmonic progressive wave is




U.- oA(2)
Un -tanh(kh) (2.2)
h is the water depth, and (o is the circular frequency. Now with (2.2), the KeuleganCarpenter number is
H
KC= 2 r L (2.3)
B tanh(kh) B tanh(kh)
L
The physical meaning of the Keulegan-Carpenter number (more easily seen for the case of deep water: tanh(kh)=1) is the ratio between the circumference of the fluid-surfaceparticle-orbital motion and the characteristic body dimension. Speaking in Navier-Stokes equation (see equation 3.2) terms, the Keulegan-Carpenter number is the ratio between spatial u and temporal u accelerations. In accordance with Mei's definitions (1989),
ax at
a body is regarded as large when kB 0(1); its presence can significantly alter the pattern of wave propagation, produce wave diffraction, and the disturbance can propagate in a much wider area far away from the large body. On the contrary, a body is regarded as small when kB<_ 0(1) the local velocity gradient near the small body augments the effect of viscosity and induces flow separation and vortex shedding, leading to so calledform drag. It should be noted that the associated viscous forces are not mean shear forces, but pressure forces due to separated flow. The influence of a small body is usually bounded to a comparatively narrow area. A more precise classification is given in Sarpkaya and Isaacson (1981), where Isaacson has presented a convenient means of indicating the conditions under which the diffraction,




flow separation and nonlinear effects become important for the case of a fixed vertical circular cylinder, as seen in Figure 2.1.

kB
Fig.2.1 Wave force regimes (Sarpkaya and Isaacson, 1981). Importance of
diffraction and flow separation as functions ofKC -Keulegan-Carpenter number
and kB-diffraction parameter
As given by (2.3), the greater the wave steepness (H/L) the larger the Keulegan-Carpenter number becomes. An approximation of the maximum wave steepness is given as (Patel, 1989)




H) = 0.14 tanh(kh) (2.4)
Therefore, from (2.4) and (2.3) the relationship between the largest Keulegan-Carpenter number and the diffraction parameter, shown in Figure 2.1, is given by
0.44
KC = 0 (2.5)
B L
and is a simple hyperbolic curve. Isaacson pointed out that the critical value of the diffraction parameter that roughly separates large from small bodies is BIL=0.2, because the curve of the largest KC (without wave breaking) does not exceed 2 for the range B/L>O. On the other side, flow separation should be more important than the diffraction when KC>2, which according to Figure 2.1 happens when B/L>0.2. At the same time the condition for diffraction B/L> 0.2 and (2.4) imply that H/B0.2) bodies. As seen in Figure 2.1, for a wave with steepness one half of the maximum steepness (0.5H/L), KC does not exceed 2 for the region B/L>O.1. This fact suggests that the "generally accepted" (Sawaragi, 1995; Sumer and Fredsoe, 1997) critical value of the diffraction parameter B/L=0.2 is not a fixed value it may vary even for rounded bodies. It is well known that the flow regimes about a fixed vertical cylinder depend not only on the KC number but also on the Reynolds number defined as the ratio between inertia and viscous forces




BU B2
Re= =KC (2.6)
v vT
where the kinematic viscosity is v = 10-6[m2 / sec]. When for example B[1,3][m], TE [2,20][sec], and tanh(kh)=1 for the case of deep water, then the Reynolds number becomes large Re>>O(103). As evidenced in Sumer and Fredsoe, 1997 with Figures 3.15,
3.2, and 3.16, when KC<2 and Re>>O(103), the flow will not be separated; when KC approaches 2, there will be separation, but not very extensive. This analysis suggests that the flow about a fixed vertical circular cylinder in the large-body-diffraction-regime B/L>0.2 (or B/L>O. 1) may be analyzed by the potential theory, since no flow separation occurs KC<2, and since the viscous effects are negligible Re>>O( 10).
In the case of a floating body, there is another hydrodynamic force due to the
body motion in addition to the wave forces exerted on the body under the fixed condition. According to Sawaragi (1995), the generation mechanism of hydrodynamic forces due to body motion can also be classified briefly into two regimes in the same way as the case of the fixed body.
2.2.2. Deterministical and Statistical Approaches
A phenomenon, which is changing with time, can be described deterministically or statistically. In the deterministic approach all the variables are functions of time and known at any moment of time, usually after solving differential and integral equations. In the statistical approach the explicit time dependence is not considered. A variable is usually known as an average or as a probability of occurrence. The time history is unknown and therefore the variable is unknown at any moment of time. The problem is




formulated as a distribution of the relevant quantities over the independent variables. An
excellent example found in nature, which explains the two formulations and their
relationship, is the irregular sea. Obviously the time history is very difficult to obtain and
is not important. In order to obtain statistical estimates, the linear theory simulates the
irregular sea as a superposition of linear harmonic waves. In the case of a floating body, if
its response to a harmonic wave is solved deterministically, it will help to find statistically
its response to the irregular sea. One of the most generally used ways to describe and to
work practically with a random sea is to consider the distribution of its energy content as a
function of wave frequency (sea spectrum). The concept is to sum a large number of
sinusoids with small amplitudes, different frequencies and phases, with some waves adding
to build up larger ones and others canceling each other, thus forming an irregular profile
with no set pattern as to amplitude or periodicity (Figure 2.2.).
Irregular profile as a sum of four harmonic waves 20 0
-20
5 10 15 20 25 30 35
20
0
5 10 15 20 25 30 35
20 0
-20
5 10 15 20 25 30 35
20 0
-20
50 5 10 15 20 25 30 35
0
-50

5 10 15 20 25 30 35
Time in seconds
Fig. 2.2. Irregular wave as a sum of four harmonic waves with different amplitudes and circular frequencies




From here it follows that the energy content of the random sea irregular profile can be presented as a sum of the energy of all the component waves irrespective of their phases. The concept of sea spectrum had been justified theoretically and experimentally for 40 years by utilizing the Pierson-Moskowitz, Bretschneider, TONS WAP and other energy density spectra. The Pierson-Moskowitz spectrum is controlled by a single parameter significant wave height and represents fully developed seas. The Bretschneider spectrum is controlled by the significant wave height and a modal wave period and can be used for fully and partially developed wind generated seas. The JONSWAP spectrum is controlled by the significant wave height, fetch~length, and shape parameter, and is used for partially developed seas. Therefore the implementation of the energy distribution concept into a computer model is a powerful tool for analyzing the buoy behavior in real seas, knowing only the buoy response to a single harmonic wave with small amplitude.
2.2.3. The Concept of Added Mass
The concept of hydrodynamic added mass arises from the fact that a body having an accelerated motion in or on the surface of the water experiences a force that is greater than the mass of the body times the acceleration. Since this increment of force can be defined as the multiplication of the body acceleration and a quantity having the same dimension as the mass, it is termed added mass The added mass is not a finite amount of water, which oscillates rigidly connected to the body. The whole fluid will oscillate with different fluid particle amplitudes throughout the fluid. As the linear oscillation is associated with forces and the rotational oscillations with moments, the added-mass may have dimensions of mass, mass multiplied by length, and even inertia moment. The




concept of hydrodynamic added mass should be understood only in terms of generalized force on the body induced by the hydrodynamic pressure, and therefore it will depend on the wave frequency and the wetted body surface. Inherited from the added-mass concept is the concept of virtual mass, defined as the sum of the added mass and the mass of the floating body.
2.2.4. Classification of Dampin
By definition, damping is the ability of a structure to dissipate energy. There are three major kinds of damping for a fixed or floating body in water: structural, material, and fluid. Structural damping is due to friction among different parts of a structure. Material damping is energy dissipation within the material of the body, being more significant in materials like rubber. Fluid damping is the result of energy dissipation, as the fluid moves relative to the vibrating body. The fluid damping can be classified further into a damping due to wave generation and a damping due to viscous effects. The wave generation damping or simply the wave damping dissipates the energy of the vibrating body into the fluid, thus causing waves. The viscous effects damping can be subdivided into skin friction effects and viscous effects due to the pressure distribution around the body. The latter is associated with separation and formation of eddies and is usually known as eddy-making damping in the literature (Faltinsen, 1990). The separation changes the flow pattern about the body to a certain extent so that in may be felt in both the damping and added mass. The skin friction effects on damping are due to shear stresses acting tangentially on the boundary surface between the fluid and the body and are proportional to a velocity gradient.




Since it is possible to obtain the response of a floating body in irregular seas by linearly superimposing body responses from harmonic wave components, it is sufficient from a hydrodynamical point of view to analyze a floating body in incident regular sinusoidal waves of small steepness. The basic laws of physics governing the motion of floating bodies are well known in their linearized version (Mei, 1989), and with the drag force proportional to the square of the floating body speed (Berteaux, 1991). The difficulty in predicting the response of a floating body to a harmonic wave arises in the determination of the hydrodynamic properties, namely added mass and damping. Due to the effects of water-air interface, these hydrodynamic properties depend on the wave frequency, water depth, and the wetted body surface, which change with time. There has been a significant amount of research on these subjects in recent years but they remain the most difficult aspects of floating str cture response prediction. Thanks to the significant computer advancement in achieving higher computational speed and larger memory capacity, numerical methods and techniques have been increasing their role in calculating the dynamics of fluid-structure interaction. The differences between computed and measured hydrodynamic forces revealed that some important phenomena are either not well understood or the existing combinations of theories and numerical techniques can not explain them. On the other hand the agreement between computed and measured quantities in other particular regions, confirmed the generalization that most engineering approaches in the floating body studies are only valid in a certain range of practical interest, thus stimulating more investigations. The notable interplay of theories, numerical methods, and experiments has been very fruitful for the advancements in floating body studies.




2.2.5. Numerical Methods Classification
As classified by Yeung (1985), Mei (1989), and Vantorre (1990), several
numerical methods have been proposed for calculating potential functions in free-surface hydrodynamics:
2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method
2.2.5.2. Singularity distributions on the floating body surface, which leads to an integralequation formulation based on Green's functions.
2.2.5.3. Method of finite-differences. Boundary-fitted coordinates.
2.2.5.4. Finite element method Hybrid element method
2.2.5.5. Boundary integral equation methods (BIEM's) based on a distribution of
"simple sources" over the total fluid domain boundary.
2.2.5.6. Methods making use of eigenfunction matching.
All these numerical methods will be explained in the frequency domain, because as it will become evident from the linearized combined kinematic-dynamic free surface boundary condition (3.34b), the time-domain and frequency domain solutions are simply related.
2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method
Generalizing the heaving motion solution for a semi-immersed circle Ursell (1949), and its extension to a semi-immersed sphere Havelock (1955), Taylor and Hu (1991) developed a systematic multipole expansion technique for submerged and floating horizontal circular cylinders in infinite water depth. Considering the two-dimensional case of a partially or totally submerged circle, the main idea is to place a set of easy-to-evaluate




elementary functions multipoles which satisfy the Laplace equation, on the level of the center of the circle. The combined kinematic-dynamic free surface boundary condition is satisfied with a proper combination of the multipoles and their mirror images with respect to the mean water surface, which also satisfy the Laplace equation. In order to represent the outgoing waves at infinity, additional potential is introduced, that has different forms for symmetric (heave) and anti-symmetric (surge) motions respectively. With the help of unknown coefficients, both wave and local-disturbance multipoles are combined as an infinite sum into a velocity potential, which automatically satisfies the Laplace equation, the free surface boundary condition, and the radiation condition. The wetted body surface boundary condition is used to determine the unknown coefficients, after truncating the infinite series at a finite number of terms, and using the point collocation method. A limitation of the multipole expansion technique is that the general existence of the expansion is very difficult to justify, or in other words the convergence of that expansion has been proved only in the case of a semi-submerged circle in the region kB<]. S (Ursell, 1949). Numerically, the convergence was found to vary with the depth of submergence, with the fastest convergence for a fully submerged circular cylinder. The most demanding case was found to be when the circle was just piercing the surface. In that case the body intersects the free surface at an angle of zero degrees, which makes the linearized boundary value problem mathematically unstable.
The multipole expansion can be applied to the analysis of more general bodies through a coupled method, called the BIE-BMP method by Taylor and Hu, 199 1. To solve for the velocity potential, one can use a boundary integral expression in a domain close to the body and match it with the multipole expansion outside this domain. Using




three-dimensional multipoles, Taylor and Hu (1991) outlined the same procedure for the case of a floating or submerged sphere. A complete multipole expansion solution of a heaving semi-immersed sphere was given by Hulme (1982), who simplified Havelock's solution by making certain explicit integrations. This method was developed further by Evans and Mclver (1984) for the case of a heaving semi-immersed sphere with an open bottom.
2.2.5.2. Singularity distributions on the floating body surface, which leads to an integralequation formulation based on Green's function
The method of integral equations via Green's function, as explained by Mei
(1989), is based on applying Green's theorem on the radiation velocity potential and a Green's function defined to be the potential at any field point due to an oscillating source of unit strength at a particular point inside the fluid domain. Knowing the prescribed normal velocity boundary condition, the velocity potential at any point on the wetted body surface is a Fredholm integral equation of the second kind. By dividing the wetted body surface into discrete panels and approximating the velocity potential in each panel by a constant, one can obtain a system of algebraic equations for these constant values after carrying out the integration. Solving for these constant velocity potentials on the panels will help to express the velocity potential anywhere in the fluid domain. An advantage of this method is the relatively small number of unknowns, while getting the matrix coefficients is "a laborious task both for the worker and for the computer." Another drawback of the Fredholm integral equation is the so-called irregular frequencies and nontrivial eigensolutions in the case of a floating body on the water surface. Without




going into details the irregular frequencies are the eigenfrequencies of a fictitious interior for the wetted body surface problem with the Dirichlet condition on the water-body boundary. As a result the source distribution is not unique, the approximate matrix equation becomes ill-conditioned; hence the integral equation must fail. In order to avoid the irregular frequencies, one must introduce additional artificial unknowns and more conditions to improve the matrix equations, thus making the methodology less advantageous.
A similar technique was used by Martin and Farina (1997) to solve the radiation problem of a heaving submerged horizontal disc, where the boundary integral equation is reduced to a one-dimensional Fredholm integral equation of the second kind.
2.2.5.3. Method of finite-differences. Boundary-fitted coordinates
The classical finite-difference method is based on generating a mesh around the floating body and using a variety of difference-schemes to express the Laplacian operator at a particular node with the help of information from some neighboring nodes and as a function of distance, h, between these nodes. ligher-order schemes increase their accuracy by involving more nodes, usually at the expense of more complicated algorithms. On the other hand, the accuracy loss due to an increase in truncation errors, expressed as a power of the distance h, can lead to physically unacceptable solutions (Yeung, 1985). While conveniently suited for interior nodes of the fluid domain, the difference schemes are not easily applicable on curved boundaries. That is why the grid generation process serves two purposes: first it produces a set of curvilinear coordinates that are specifically adapted to the geometry in question; and second it provides a crucial numerical




transformation that allows difference-schemes to be applied in a more geometrically simple computational domain. The boundary curves from the original physical domain have to be transformed into coordinate lines in a logical domain of mapped variables, which facilitates the implementation of Neumann type boundary conditions. As pointed out by Yeung (198 5), the coordinate transformation and the physical solution may be solved concurrently in the same "sweep" with a proper adjustment for the boundary conditions. Once the general algorithm is developed, it can be applied to different floating body geometries with a change of boundary coordinates. The change of boundary coordinates is much more involved for three-dimensional bodies.
2.2.5.4. Finite and hybrid element methods
The strength of this class of methods is its ability to handle curved boundaries. The main idea is to map isoparametric boundary surface elements into local squares, triangles, etc., on which one can calculate every element's contribution to the field and boundary properties. The unknown function consists of a set of nodal values and a set of predefined "shape functions," chosen to satisfy certain continuity requirements across the elements. The requirements depend on the differential order, and the boundary conditions (Yeung, 1985). The determination of the nodal unknowns relies on a global, integral criterion. A brief description of the hybrid element methods, as given by Mei (1989), will be given as a generalization of the finite element methods of Newton (1974, 1975). The main idea of the hybrid element method is to employ a finite-element approximation in an imaginary cylinder, which extends from the sea bottom to the water surface and surrounds the body, with an analytical representation outside of the cylinder. Thus the finite-element




region can serve as a transition zone that transforms a geometry of higher complexity into a simpler cylindrical geometry where the singular kernels (associated with integralequation methods) can be more conveniently calculated. Variational principles can be utilized to formulate the radiation problem and to seek an approximate solution for the velocity potential in the inner domain surrounding the body. The strength of the variational principles is that they permit an exact coupling of the approximate interior solution with the analytical solution for the velocity potential in the outer domain. The outer domain solution is usually presented in one of the following two ways. The first one is to use Green's function and express the velocity potential as a superposition of sources of unknown strength on the boundary between the inner and outer domains, while the second way is to use eigenfunction expansions with unknown coefficients. In the case of infinite water depth, the eigenfunction expansion was found to be inefficient, and the Green's function approach was recommended instead. By obtaining two different answers for two different grids for the velocity potential at a particular point, it was proven that the general identities and the energy conservation between rate of work done by the body and the rate of energy flux through the boundary between the inner and outer domains are necessary but not sufficient conditions to guarantee an accurate solution. Mei generalized that similar caution was warranted in other numerical methods.




2.2.5.5. Boundary integral equation methods (BIEM's) based on a "simple sources"
distribution over the total fluid domain boundary
Developed for the numerical calculation of linear potential functions for heaving axisymmietric bodies by Ferdinande and Kritis (see Vantorre, 1986, 1990) the philosophy of this method is to confine the problem into a finite cylinder, which surrounds the heaving body. Since both the body and the motion are axisymmetrical, the problem is reduced and solved in a two-dimensional rectangular fluid domain. The domain has the following boundaries: the vertical axis of symmetry of the body, half of the vertical cross section of the wetted body surface, the vertical cylinder wall, the flat bottom, and fluid surface between the body and the cylinder. In order to justify the confinement of the problem into the cylinder, and thus the boundary element method, a modified radiation condition has to be satisfied on the vertical cylindrical wall. Now the whole boundary is divided into discrete panels, on the center of which the velocity potential is considered constant. As a consequence of Green's theorem a potential on a point on the boundary is expressed as an integral of the velocity potential and its normal derivative over the whole boundary. Eventually the problem is reduced to the solution of set of N linear algebraic equations, where Nis the total number of panels.
Some of the boundary integral methods are very effective for first-order diffraction and radiation problems. However, these methods are not suitable for second-order diffraction and radiation problems (Taylor and Hung, 1987). In this case the integrand is a functional of the first-order potential and the Green's function. Because of the slow convergence of the integrals, a large number of values of the first-order potential must be evaluated, and this is not easily achieved using boundary integral methods.




2.2.5.6. Methods making use of eigenfunction matching
Considering the hydrodynamic problem of a floating vertical circular cylinder in
finite-depth water, Yeung (1981) gave an example of the eigenfunction matching methods with treatment of interior and exterior problems. In the interior problem, the vertical circular cylinder is considered a fictitious interior domain filled with the same fluid, where the velocity potential is uniquely determined by solving the Laplace equation with Dirichlet type boundary conditions. The exterior problem respectively is a Neumann type problem and can be thought of as one driven by a flux emitted by the interior region, with a solution written in terms of an eigen expansion with unknown coefficients. Both problems are treated as if the conditions at the common boundary were known. By matching both velocity potentials at.the common boundary, the problem is reduced to solving an infinite system of linear equations. This infinite system is claimed to have excellent truncation characteristics requiring rarely more than 20 equations to achieve an accuracy of 1%. It should be noted that the eigen expansion in this solution was possible because of the convenient presentation of the cylinder boundary in a cylindrical coordinate system, and because of the finite-depth water. As Taylor and Hu (1991) point out when the water depth increases toward infinity, the eigenvalues tend to pack together, and the eigenfunctions become undistinguishable. Consequently, the number of terms required in the eigen-series expression becomes unrealistically high.




2.3. Advancements in Floating Body Studies
As stated earlier, the most basic problem to solve is that of the frequency response of the floating body when subjected to simple sinusoidal wave excitation. In 1944 Haskind introduced a way to decouple the hydrodynamics from the body dynamics and to further decouple the hydrodynamics problem into diffraction and radiation components. Taking advantage of the linearity of the Laplace operator and the combined Neumann and Dirichlet-Neumann boundary conditions for the velocity potential Haskind presented the velocity potential as a sum of diffraction and radiation potentials. Speaking in physical terms, two hydrodynamical sub-problems were formulated. The first sub-problem assumes the body to be rigid and restrained from any oscillation in the presence of incident regular waves. The hydrodynamic loads are called wave excitation loads and composed of Froude-Kriloff and diffraction forces and moments. The second sub-problem forces the rigid body to oscillate in any motion mode with the wave excitation frequency in the absence of incident waves. The hydrodynamic loads are identified as added mass, damping and restoring terms. Since the body oscillates in otherwise calm water, it radiates waves, thus the term radiation. In 1949 John (see Mei, 1989) introduced in a systematical formal approach the complete first order wave-structure theory and showed how higher-order extensions can be made. Following Haskind (1973), Newman (1976), and Mei (1989), several remarkable general identities have been introduced. These identities relate different hydrodynamic quantities and have proved to be extremely useful in increasing the theoretical understanding of physical phenomena. Moreover, they provide necessary checks for analytical theories or numerical methods, and minimize the




25
computing time of those quantities related by them. Based on the mathematical definition of the incident, radiation, and diffraction wave potentials as harmonic functions, and on the Green's theorem, the first identity relates, in an elegant way, two radiation problems. A computationally efficient result is that the restoring force, the added mass and the damping matrices are diagonally symmetric, which decreases significantly the number of unknowns. Another consequence is the convenient relationship between the damping and the asymptotic behavior of the radiation velocity potential. In physical terms the energy given up by the oscillating body is transported by the waves propagating away from the body. The knowledge (about damping) gained this way, can be used to find the added mass, using the so-called Kramers-Kronig relations. Derived in a universal way by Ogilvie (1964), these relations simply exploit the fact that damping and added mass are the real and imaginary parts of the same function. Therefore knowledge of the damping coefficients is sufficient to determine the corresponding added mass coefficients and vice versa. Newman (1976) discovered that the damping matrix is singular for the case of particular bodies of revolution where the exciting force is independent of the angle of wave incidence. The second identity relates two diffraction problems corresponding to different angles of incidence. As two-dimensional results, there are several relationships between the amplitudes and the phase angles of the transmitted and reflected waves. The three dimensional result is that the amplitude of the first scattered wave toward the second incident wave is equal to the amplitude of the second scattered wave toward the first incident wave. The third identity relates the diffraction potential to the radiation potential, thanks to a brilliant Haskind theorem (1957). It expresses a generalized component of the exciting force (due to fixed body diffraction from an incident wave moving in a particular




direction) in terms of the radiation potential and its normal derivative (due to body oscillation in the same direction, in otherwise calm water). An elegant consequence from the Haskind theorem is that the exciting force is linearly proportional to the far-field wave amplitude. The very practical meaning of the Haskind theorem is that an actual solution of the diffraction problem can be avoided and that the exciting forces are simply related to the damping coefficients. Newman elaborated on this further and found an explicit relationship between the magnitudes of the exciting force and the damping coefficients, which is often used as a check on the magnitudes. An even less obvious identity between radiation and diffraction problems was discovered by Bessho (1967) for two dimensions and extended for three dimensions by Newman (1975,1976). It relates the far-field radiation and diffraction wave amplitudes, and reveals the significant result that the phases of surge and pitch antisymmetric modes differ with ;r.
2.4. Presentation of the Results of Investigations
In light of the above-mentioned theoretical advancements, it is instructive to give some examples of the theory-experiment interplay. In 1965 Kim determined the added mass and damping for a semi-ellipsoidal body with its origin on the free surface of water of infinite depth. In order to check the validity of potential flow theory predictions Frank (1967) conducted a series of experiments to determine the added-mass and damping coefficients for two-dimensional bodies at the free surface.
Using Ursell's multipole expansion method and De Jong's extension to sway and roll motions of arbitrary shaped cylinders, Vugts (1968) solved the linear radiation




27
problem for two-dimensional floating cylinders. Conducting experiments with five different cylinder cross-sections in heave, sway, and roll oscillatory motion, he has compared the experimental results with the predictions of the linear potential theory and particularly the multipole expansion solution. With the influence of viscosity negligible, Vugts has found good agreement between calculated (with the potential theory) and measured added mass and damping coefficients in heave for the middle and high frequency range. However, in the low-frequency range, coVB / (2g) < 0.33, the multipole expansion method significantly overestimates the experimentally obtained added mass. In sway, the calculated added mass is in good agreement with measured values for most of the frequency range: 1.125 < o B / (2g) < 1.25. In the higher frequency range the damping was underestimated. For the case of coupling of sway and roll the differences between the measured and calculated added mass and damping coefficients are higher. Viscous effects in terms of separation and eddy making have been observed in sway motion for the relatively sharp edged sections. With predominant wave damping, the viscous damping contributes from 10% at low frequencies to 40% at high frequencies for sharp edged sections. Obviously the energy loss due to eddy formation has been one of the reasons for these differences in damping. Interestingly enough it appears that eddy formation does not seriously affect the total pressure distribution in phase with the body acceleration since the added mass has been predicted relatively well. For rolling oscillatory motion the calculated added mass moment of inertia overestimates the measured one, while for the damping coefficient the observations were the same as in the case of sway. Since the wave damping part for the sections considered is of an order smaller than for swaying, the




viscous effects become much more important. That is why the damping in roll motion is significantly underestimated compared to the damping in sway motion. The exciting forces, calculated with Newman's explicit relationship of the magnitudes of the exciting force and damping coefficients, have been found to be in a relatively good agreement with the measured ones.
Haskind (1973) developed a techniqu e for computing added mass and damping for horizontal and vertical plates floating at the water surface, as a function of body geometry, water depth and frequency. Newton et al. (1974) and Newton (1975) have developed a finite element model (FEM) for computing these coefficients for ship hull forms. Using potential flow theory, Chung (1976, 1977) presented added mass and damping as a function of frequency, direction of oscillation and depth of submergence. Hanif (1983) determined these hydrodynamic coefficients using FEM and compared his results with other investigators.
While investigating the heave motion of a half-immersed floating sphere with a
bottom opening, Evans and Mclver (1984) have used a semi-analytical solution based on an extension of the method of multipole expansion, classified in 2.2.5. 1. The added mass and damping coefficients calculated with this method were compared with their values obtained via the numerical singularity distribution method classified in 2.2.5.2. They found that the numerical method tends to overestimate the diagonal hydrodynamic coefficients, while their off-diagonal values are in good agreement with the semi-analytical solution. Moreover the numerical method showed slight differences between the offdiagonal added mass and damping coefficients, which theoretically must be equal in accordance with the symmetry-identity classified above. The singularity of the damping




matrix has been used as a partial check for the correctness of the semi-analytical solution. In a technical note to the Evans and Mclver (1984) work, Falnes (1984) proposed some empirical formulas for the added mass and damping for the semi-submerged sphere in heave motion.
Utilizing a numerical scheme that is similar to one used by Nestegard and
Sclavounos (1984) for the method of multipole expansion combined with BIE-BMP matching classified in 2.2.5.1, Taylor and Hu (1991) obtained added mass and damping for floating and submerged circular cylinders. For the submerged cylinder, the diagonal added mass and damping coefficients in sway have been confirmed to be equal to those in heave. While damping is always positive, negative added mass has been discovered for the case when the submerged cylinder is close to the surface. Negative added mass has been also observed for a cylinder floating on the surface in sway motion when the cylinder is more than three-quarters immersed. For the frequencies of negative added mass in heave the sway added mass was positive. It was concluded that "at these frequencies the forced heave oscillation does not transfer energy to the fluid and hence does not generate waves." Lastly Taylor and Hu have found that in the low frequency range the added mass changes from finite to infinite values as the cylinder emerges from below the free surface. The incorrect conclusions, made by Taylor and Hu, show the important role of experiments as the only tool for verification of the working capabilities of theories and numerical techniques.
Exploring surge oscillatory motion of a single tethered half-submerged spherical float, Vethamony et al. (1992) have computed added mass and damping from the motion generated velocity potential, using potential flow theory. They solved a boundary value




problem with Green's function method by dividing the cross-section of the float into equal segments, and conducted experiments to compare the results. Considering the float size small compared to the wave-length the diffraction potential has been neglected and added mass and damping computed from the motion generated velocity potential. With added mass in phase with the surge motion and damping coefficient out of phase with the motion, both were found to be frequency dependent, and to increase gradually with respect to size of the float. It was noted that added mass and damping do not change with water depth. In light of the general identities discussed earlier, Vethamony et at. found that wave excitation forces calculated from incident wave potential are in good agreement with excitation forces calculated from damping coefficients.
It was observed that nonlinear effects become important in fluid-structure
interaction when waves become steeper, or when the oscillation amplitude of the floating body can no longer be assumed small. These natural phenomena have been stimulating the extension of the linearized radiation problem and linearized diffraction theory which are the first terms in Stokes perturbation expansion into higher order terms, where the perturbation parameter is related to the wave steepness. As a result of this expansion procedure the full nonlinear diffraction and/or radiation problems are replaced by a sequence of linear boundary-value problems formulated for each order of perturbation. Most difficulties of the second order problem arise from the requirement that the velocity potential satisfies a nonhomogeneous boundary condition on the free surface of the fluid. According to Molin (1979), the nature of second-order diffracted waves consists of two components. One is free-waves travelling independently of the first-order wave system and the other is phase-locked-waves accompanying the first-order waves (Chakrabarti,




31
1987). Since in many applications the hydrodynamic loads, not the flow kinematics, are of primary interest, an indirect method can be used to determine wave-induced loads to the second order without the explicit calculation of the second order potential. The technique involves an application of Green's second identity and requires the solution of associated linearized radiation problems corresponding to prescribed oscillation of the structure at twice the first-order wave frequency. A modification that utilizes an axisymmetric Green's function solution for the linear problem has been used by Molin and Marion (1985) to calculate the second-order wave induced loads and associated motions for a floating body. Taylor and Hung (1987) have performed detailed analysis of the asymptotic behavior of the troublesome free-surface integral and have presented a computational method for its calculation on the far field in the case of a fixed vertical circular cylinder. Adopting an interior region that includes the fixed vertical cylinder, and an exterior region that extends to infinity in the horizontal plane, Ghalayini and Williams (199 1) presented a solution to the free-surface integral. The interior region solution was performed numerically by utilizing the asymptotic forms of the potentials, while the exterior region solution was carried out analytically in terms of Fresnel functions. Inspired by Soding' s work, Vantorre (1986) developed a computational procedure for calculating hydrodynamic forces up to the third order for the case of floating axisymmetric bodies in a symmetric heave oscillatory motion. Soding (1976) proved that the knowledge of the first-order potential for the exciting frequency and for twice that frequency on the mean body surface and on the mean free surface within some distance from the body is all the information required to compute second-order forces for oscillating cylinders. In a similar way, Vantorre computed third-order forces by means of the first-order and second-order




32
potentials for the exciting frequency, and the first-order potential for three times the exciting frequency. Calculation of the latter potential is not necessary if the first harmonic of the third-order forces is the only primary interest. The computational procedure used by Vantorre is classified in 2.2.5.4. as boundary integral equation method (BIIEM) based on a "simple sources" distribution over the total fluid domain boundary. Two experiments have been conducted, one with a floating cone and a second with a submerged vertical cylinder. In both cases the third harmonic was impossible to measure. It is obvious from the experimental results for the floating cone, that the third-order theory somehow underestimates the added mass over the entire frequency domain. This is more significant for larger heave amplitudes. Being acceptable for the region (w02B / 2g) < 1, the thirdorder theory overestimates the experimental heave damping in the higher frequency range. The same frequency relationships can be observed for the second-order forces. Using again the boundary integral equation method (BIEM) for a floating cone in heave oscillatory motion, Vantorre (1990) determined the influence of small draft variations on hydrodynamic first-order forces. After deriving some second-order and third-order force components, he used a modified BIEM to derive first and second derivatives of heave added-mass and damping with respect to draft.
The nonlinear effects of the motion of fluid around a body can be dealt with
directly from the fundamental conservation laws of mass and momentum (Equations 3.2 and 3.3). Due to the increasing capacity of computers in the recent years, three groups of methods for numerical treatment of flow nonlinearity around cylinders have been developed. Following the comprehensive survey of Sumer and Fredsoe (1997), the first group consists of methods involving the direct solution of the Navier-Stokes equations.




When the flow around a cylinder is two-dimensional and the Reynolds number is small, i.e. Re<200, a direct solution with finite-differences yields results for the gross-flow parameters that are in reasonable agreement with measurements, while the lift force is grossly overestimated (Sumer and Fredsoe, 1997). For larger Reynolds numbers, vortex shedding occurs in cells and therefore the flow becomes three-dimensional (3D). When 300



agree with experiments within the Keulegan-Carpenter range between 1 and 2.5. The third group of numerical methods is.based on the hydrodynamic stability approach, in which the formation of vortex shedding is viewed as an instability of the flow in the wake. As seen in Section 2.2.1, the oscillatory flow becomes unstable above a'critical KC number due to spanwise-periodic vortices the so-called Honji instability. After modeling the phenomenon numerically, Zhang and Dalton (1995) have obtained a definite 3D behavior regarding the variation of vorticity, and found that the sectional lift coefficient has a strong spanwise variation.
2.5. Some Thoughts About the Current State of Knowledge
For a long time scientists and engineers have tried to approximate natural
phenomena with different mathematical formulations, brought into life with a variety of numerical techniques, validated or rejected through experiments. Keeping in mind Poincare's words: "Mathematics can never tell what is, it can only say what would be if', the current state of knowledge about floating bodies is a direct result of all these trials and errors, and can be classified as going simultaneously in three main directions:
* The first approach is to directly solve for the nonlinearities of the Navier-Stokes
equations with massive computing power. With the help of supercomputers and
mature numerically-stable methods, the existing mathematical models have succeeded
in increasing our appreciation and understanding of some natural phenomena, in
particular ranges of Reynolds and Keulegan-Carpenter numbers. Other Re and KC
ranges of practical interest require different or improVed mathematical and theoretical




modeling. It is evident that oscillatory and wave-flows bring additional challenges for the direct approach to the nonlinear problem.
*The second approach is to utilize a Stokes perturbation expansion, which restructures the Navier-Stokes equations into a sequence of linear boundary-value problems formulated for each order of perturbation in terms of the power of a small parameter, for example wave-steepness. With gradually increasing complexity, any-order solution can be obtained, provided the solutions of previous-order problems are known.
Predominantly second-order and third-order solutions of the fluid-body interaction
have been investigated, with the noticeably preferred simp lifications of axisymmetrical
bodies and symmetrical heaving motion for the case of third-order problems. To
explain this preference it suffices to recall that even for the second-order problems, a great effort is required to deal with the nonhomogeneous boundary condition on the free surface of the fluid, as discussed earlier. Nevertheless, the role of perturbation
analysis in fluid-structure interaction studies is evident in providing valuable
information about the significance, contribution, and limits of the higher-order
nonlinear effects, as compared to the first-order solution.
*The third approach is to work with the linearized diffraction and radiation theories and
with the powerful principle of superposition, which provides insight into the fluidstructure interaction problem in irregular seas. Naturally semi-analytical solutions are
much faster than the numerical ones because of the smaller number of computations
needed. Moreover, because of the exact boundary conditions semi-analytical solutions
are principally more accurate as compared to the approximate boundary conditions in
numerical solutions. Of course there are greater restrictions on body-shapes with




semni-analytical than there is for numerical solutions. One of the main advantages of the linearized theory is that one can explore in real time the influence of variation of
different parameters like draft, shape, size on the hydrodynamic properties, forces, and
hence fluid-body dynamics. Another advantage is that the linearized theory provides
the easiest way to analyze and comprehend the physical concepts, and meaning and
significance of observed natural phenomena. Ones seeing the big picture, further improvements can be made, if desired and feasible, with the help of the other two
approaches.
In a historical retrospective, it is true that the three main approaches have had
great impact on the floating-body studies. Having a common goal, they have influenced and stimulated each other's evolution. All three approaches have their own unique advantages and disadvantages and areas of applicability.
In the present work, a new semi-analytical method is proposed for solving the
dynamics of free floating two-dimensional horizontal cylinders, of various shapes in heave, surge and pitch. The method is based on linear radiation theory and is intended to be the first step in the development of a mathematical model and computer program for predicting the response of an axisynimetric, three dimensional, free floating drifter buoy subjected to waves.




CHAPTER 3
FORMULATION OF THE PROBLEM
3. 1. General Description of the Problem and its Simnplifications
The most general formulation of the problem of the dynamic response of a free
floating body subjected to waves is to pose a dynamic equilibrium of forces and moments in and on an elastic body freely moving in the air-water interface. The focus of the present work will be on the floating body motions due to external loads, which act on the underwater part of the body. Therefore two restrictions will be made right from the beginning: first the body is considered rigid; and second the direct influence of the air environment on the body is negligible. As long as no structural or vibrational problems are to be dealt with, the first restriction can be made without any hesitation. The second restriction is based on the fact that the density of air is roughly one one-thousandth of the density of water. Provided that the above-water part of the floating body is close to the water surface, and excluding strong winds during adverse weather, it is clear that for most practical problems aerodynamic forces may be neglected with respect to hydrodynamic forces.
3. 1.1. Incompressible Fluid Assumption
A coordinate system Oxyz or simply x=xyz) which is fixed in space will be used in this analysis. The Oxy plane coincides with the still water surface, and the vertical axis




Oz is positive upward. The origin 0 is the intersection of the centerline of the floatingbody-section and the still water surface. In a wide variety of gravity wave problems, the variation of water density is insignificant over the temporal and spatial scales of engineering interest. The relationship between water density and pressure is given by I Dp_ I DP (3.1)
p Dt E Dt
where D(.) = .) + u. V(.) is the total derivative in space and time, u = (u,v,w) is the
Dt at
fluid particle velocity, P is the pressure, p is the water density, and E 2.07 x 109 [Pa] is the water bulk modulus. Since an increase in pressure of 1 MPA results in 0.05% change in water density, for practical purposes the fluid can be considered incompressible.
3.1.2. Governing Equations and Definitions
Using Mei's notation (1989), the motion of fluid around a body is governed by the fundamental conservation of momentum law or the Navier-Stokes equations a+u.Vu =-v P+gz)+vV2u (3.2)
and the conservation of mass law or continuity equation V. u = 0 (3.3)
where v = 0. 01 [cm2/sec] is the kinematic viscosity, and g=9.814 [m/sec2] is the acceleration of gravity. Dots represent the scalar multiplication of two vector quantities (Batchelor, 1967). Defining the vorticity vector as the curl of velocity vector Q(x,t) = V x u(x,t) (3.4)




an important deduction from the Navier-Stokes equations is that the rate of change of vorticity is due to stretching and twisting of vortex lines and to viscous diffusion. a u.V 2 = n. Vu + vV2E (3.5)
Defining a streamline as a line tangent everywhere to the velocity vector, it is obvious that the physical concept of streamlines must exist in a general three-dimensional compressible flow (Dean and Dalrymple, 1991). A Streamfunction exists in two-dimensional or axisymmetric incompressible flow and is defined by
= u and -= v in Oxy-plane,
or (3.6)
= u and = -w in Oxz-plane
8z x
For two-dimensional incompressible flow, equations (3.4), (3.5), and (3.6) reduce to the 2D vorticity-transport equation
+ u. Vf2 = vV20 (3.7)
and the Poisson equation
82T a2 T
2 = -a2 (3.8)
8x2 y2
3.1.3. Inviscid Fluid Assumption
In water the kinematic viscosity is small: v = 0.01 [cm2/sec], which means that the last term of equation (3.5) is negligible, except in regions'of large velocity gradient and




strong vorticity as mentioned earlier.
L B
Viscous effects
S become important
Line of constant wave
steepness HIL=OI1
steepness HL =0/1
potential effects
0r0 0 .t
are dominant
kB
Fig. 3.1 Wave force regimes (Hooft, 1982). Importance of
viscous and potential effects as functions of wave height-to-diameter
ratio and diffraction parameter
As observed from Figure 3.1, Hooft (1982, Figure 4.1.6) has shown that
for a fixed vertical cylinder and relatively steep waves, with ratio between wave height and wave length: H/L=0. 1, viscous effects become important when H/B
>15 and kB < 0.06, where k is the wave number. Practically this means that if the characteristic dimension of the fixed vertical cylinder B=1 [m], the viscous effects become important for wave heights H>15[m], and wave periods T>9[sec]. When B=2 [m], the viscous effects become important for wave heights H>30[m], which in reality can be referred to as extreme wave conditions. Intuitively, it is reasonable to expect that for a free-floating body the velocity gradient will be significantly less
than for a fixed vertical cylinder. Therefore the fluid can be considered inviscid,




and the Navier-Stokes equations (3.2) are transformed into Euler equations:
+ u. V u = -V + g (3.9)
3.1.4. Irrotational Flow Assumption
For an inviscid and incompressible fluid, where the Euler equations are valid, there are only normal stresses acting on the surface of a fluid particle; since shear stresses are zero, there are no stresses to impart a rotation on a fluid particle. Therefore any nonrotating particle remains nonrotating, but if initial vorticity exists, vorticity remains constant (Dean and Dalrymple, 1991). An important class of problems is one where f 0 and is called irrotationalflow. For an inviscid irrotational flow, the velocity u is usually expressed as the gradient of a scalar function called velocity potential. It exists in two-dimensional and three-dimensional irrotational flows u= V@ (3.10)
Conservation of mass (3.3) requires that the velocity potential satisfies Laplace's equation V2 = 0 (3.11)
while conservation of momentum (3.2) transforms into V[ +I = IVt2 ] v[ + v[ gz] (3.12a)
L t 2 p
Integrating (3.12a) with respect to the space variables away from the body, we derive Bernoulli's equation
2 P
- +- Iv +gz= --+C(t) (3.12b)
at 2 p




with a temporal constant C(t) which can be omitted by redefining the velocity potential without affecting the velocity field.
3.1.5. Dynamic Free Surface Boundary Condition (DFSBC)
A characteristic feature of the "free" surface of air-water interface is that it can not support pressure variations and hence must respond appropriately to maintain the pressure uniform. For the practical interest, the wave length is so long that the surface tension is unimportant; the pressure just beneath the surface must be equal the atmospheric pressure Pa above. Therefore the DFSBC is defined as a uniform pressure along the wave form on the free surface. Applying Bernoulli's equation (3.12) on the free surface, we have
I P (3.13)
at 2 p
3.1.6. Kinematic Free Surface Boundary Condition (KFSBC)
The instantaneous free surface of a wave can be described with the equation
F(x,y,z,t) = z-4(x,y,t) = 0 (3.14)
where 4(x,y, t) is the displacement of the free surface about the horizontal plane z=O. Defining "material" surface as surface (air-water) across which there is no flow, a particle remains and moves only tangentially on that material surface when DF -0 +u.V1F 0 (3.15)
Dt tt

which gives the KFSBC




at a yay =z
at ax ax ay ay az

,on z = (

(3.16)

Taking the total derivative of (3.13), the two surface boundary conditions (3.13) and (3.16) may be combined in terms of the velocity potential

D P,
Dt p

S +g-+-+-u.Vu 0
5t2 8z at 2

, on z=",

(3.17)

which is the combined kinematic-dynamic free surface boundary condition (CFSBC).
3.1.7. Sea Bottom Boundary Condition (SBBC)
On the sea bottom, denoted with Bo, (3.14) becomes F(x,y,z,t) = z + h(x,y) = 0 where h is the water depth. Applying (3.15) on the above equation, the SBBC becomes

-0 aha I a ax ay y az

, on z =-h(x,y)

3.1.8. Wetted Body Surface Boundary Condition (Sb)

Let the instantaneous position of the surface of Sb be described by the equation

F(x,y,z,t)=z-f(x,y,t)= 0O

(3.19)

Using the same procedure as in 3.1.6, we state the continuity of the normal velocity with af --= ,on z= f(x,y,t). (3.20)
at ax ax By az

(3.18)




3.1.9. Linearization to First- Order Theory
Following Mei (1989), the derivation below leads to the complete first-order
theory and shows how higher-order extensions can be made. For small-amplitude motion, we expand in powers of the wave slope s = A / L, which is the small parameter in the perturbation analysis
z f(0)(x,y) + Ef(1)(x,y, t) + E2f (2)(x,y, t)+... (3.21)
where fC()(x,y) represents the wetted body surface rest position, that is S() The velocity potential can be expanded in the same manner (D= 6I0) + E2 ((2) +... (3.22)
Considering small body motion, any function evaluated on Sb may be expanded about b:z = f)(x,y). To the order O(), equation (3.20) can be written as )f(0) + ()fy() +f(l) ,on z f()(x,y) (3.23)
It is necessary to find f('). Let the center of rotation of the rigid body be Q, which has the following moving coordinate:
X(t) = X() + EX0)(t) + XX(2)(t) ,X=(X,,Y,Z) (3.24)
where X() is the rest position of Q independent of time. In some cases for example a moored buoy the center of rotation may not coincide with the center of gravity, denoted by C. Let Y = (Y, j, Y) be the coordinate system fixed with the body in a way that Y x when the body is at its rest position. Denoting the angular displacement of the body with 600)(t) = e(a,3, y) with rotational components about axes x, y, and z, the two coordinate systems are related to the first order by




x = X + X(,) + (1) x (Y- x(o)] +O(s2) (3.25)
Y = X EX( ) +E(1) x (x- X(o +O(62) = x- X(') + z (_o -_ Yo)] x= z EZ({) + a(z Y(a)) -# X(O))] When the body is at its rest position, then i x and Y = f (o)(~,y) (3.27)
Substituting (3.26) into (3.27), expanding about Slo), and comparing with (3.21), results in
f ()= Z 1 + a(y y(o) f(x X(O)) fo) [X() +(z Z(o) r(Y y(O))] (3.28) (3.28)
-f(O) [y() + X(O)) a(z -Z(O))]
Substituting (3.23) into (3.28), results in the first order kinematic boundary condition on the wetted body surface
-()(1fx(o) (1)f(o) + i 1) -(O[ () Jt" (+z-Z( ))- t(Y- (o))]
-fo) [Yt)+ t(x-X( ))-a, (z-Z())] (3.29)
+Z + a, y Y()) -tp(x X(o) The unit normal vector n directed into the body becomes n = -f (o) o),i)[i + (O))2 +(fo ) )2-1/2 (3.30)
Equation (3.29) can be rewritten as
= [x= +0()xx o.n (3.31)
an .1. dt (




where
{XJ = {X(1), E)(1)} I ={X(1), Y(1), Z() a,18, r} (3.32)
= in, (X X(O)) x n} = {n,n2, n3,[n (z Z(O)) n3 (Y o))](3.33)
-[n(X- X(O)) n(z Z(o))], -[1(y y( )- n2(x X(3. )33
The physical meaning of the boundary condition (3.31) is that at any point on the wetted body surface, the normal component of the velocity of the body should be equal to the normal component of the velocity of the fluid at that point. Finally, assuming that the atmospheric pressure Pa is constant, which is true over large sea areas, and applying the known expansions into (3.17), (3.18), the hydrodynamic problem is completely linearized V2( = 0 in the fluid domain (3.34a)
2 g- = 0 on the free surface (3.34b)
at2 az
an 0 on the sea bottom (3.34c)
an
aeD 6 dX
a -- dt a on the wetted body surface (3.34d)
with the exception of a boundary condition on the lateral boundaries, which will be added later. Thanks to the linearity of the Laplace equation and the boundary conditions in Equation (3.34), the problem of the response of a floating buoy to irregular waves can be reduced greatly with the principle of linear superposition of motions. What actually remains is to study the oscillations of a rigid floating body subjected to a simple harmonic excitation due to a train of surface gravity waves.




3.2. Floating Body Dynamics
3.2.1. Conservation of Linear Momentum
Let the entire mass of the floating body be denoted withM, part of which is above the free surface, and let the center of mass be denoted with C: xc = (x, C yzc). Integrating the linearized version of Euler equations (3.9), i.e. without the nonlinear term in the left hand side, along the wetted body surface, the conservation of linear momentum states
Mxc = Pnds MgK (3.35)
Sb
where k is unit vector of Oz axis. Using the linearized Bernoulli equation P = -pg ep') + O(82) (3.12c)
and (3.26), (3.35) can be written as &wMx4) + 00) x (v X = f (-pgf sp@ '))nds Mgk + 0(62) (3.36)
Sb
The zero-order portion of (3.36) is
0 = ff(-pgf())nds Mgk (3.37)
So
while the first-order portion of (3.36) is M[X() + E() x (ic X(o))] = (-pe(1) pe') nds (3.38)
Sb
Considering the buoyancy term -pgf of (3.37), and having that on the instantaneous body surface Sb
nds = (- f ,- f, ,1)dxdy




we can replace the domain of integration Sb with the part of the water surface cut out by Sb, denoted with SA. With an error of O(1) we can replace the integration over Sb and SA with integration over S(o) and S() when the body is in its rest position. Lets denote the instantaneous volume of the displaced water with V and in rest position with V(0), and let A(0) be the area of S(). Following the procedure shown in Mei (1989) results in Archimedes' law for the zeroth-order
Mg = pg V(0) (3.39)
The first-order equations are
M[Z) + att(y (o) #(c X(O))] -p (1)n3ds -pg(j JA +Z- + x + (0)
Sb
M[X )+ f,(ic Z(o))- rtt(c- y(0))]= _p (ll)nds (3.40b)
So)
M[Y() + rt(5c X(o)) at(zc Z(o))]= -p(f1f)n2ds (3.40c)
s40)
IA = (x X(o))dxdy I = (y-y(0))dvedy (3.41)
3.2.2. Conservation of Angular Momentum
If Vb represents the volume of the whole body including the part above the free surface, dm is the body mass per unit volume, and the center of mass is C: x' = (x,y, zc), such as fff x xdm = Mxc, the conservation of angular momentum
b

requires




d2x
fff xx dm= ff xx Pnds+ x x (-Mgk) (3.42)
Vb Sb
Taking the cross product of (3.35) with X, and subtracting the result from (3.42), results in the conservation of angular momentum with respect to the center of rotation Q
(x-X)x d 2X dm= f (x- X)x Pnds + (xc- X) x (-Mgk) (3.43)
Vb d2 Sb
which following Mei's procedure leads to x-component:
ibZ(1)-_ /byla) + (Ib2 + Ib 3)att -- IIb. r 7 =t _Pl (O)nd I2 33 tt I 3 tf t n4ds
o (3.44a)
-pg[Z)I + a(I4 +4I))- + Mg[a(Yc Z(o)) y(c X(o))]
-pg1 Z + a(I22 1 1 Zo X10I'
y-component:
ib ) -IZ) + (I33 + Ib att =2 -P a()
"tt 1 32 2ttt ,d
SO) (3.44b)
+pg[Z()IA + a2 fl(IA + ) + ]+ Mg[3( o (o)) y( y(O))]
z-component:
[lb y(1) [byX(t) +(b b_-Iba,- 2~l pff~~~s 34c
It 2)I tX +(I22 tt f3 (1)n6ds (3.44c)
where the first and second moments of inertia are defined as follows
Il = f(x X(O))dm =- M(Yc- X(o)
I = ff ( X (0) 2 dm
Vb
I12 = Jf(x X())(y- Y())dm
Vb




3.2.3 Matrix Form of the Dynamics Equations
The linear system of equations (3.40) and (3.44) can be written in matrix form as
M, ~ ~ "( f d Xl ds(1
[M] + [C]{X} = -p n}d (3.45)
(0)
where [M] is the mass matrix and [C] is the buoyancy restoring force matrix. For a long horizontal cylinder with its cross-section in the Oxz-plane with incident waves travelling along the Ox-axis, the motion can be described working with a unit-cylinder length in the y-direction. For this two-dimensional case the displacement vector is {X} = X(),Z(',f8} (3.46)
The normal to the wetted body surface vector is {n} = n,n,nx(z- Z())-n(x X(O))} (3.47)
(note that nxds= dz, nzds= -d) (3.47a)
The mass matrix is
M 0 M Fc Z(a)
[M] 0 M -M(5 X(0) (3.48)
M Fe Z(0) -M~, X(o)) +
and the buoyancy restoring force matrix is
0 0 0
[C]= 0 pgA -pgIA (3.49)
0 -pgIA -pg(I V +If) Mg Z(o))
where




IA = f x Xfx- X (0) ) dx 3 =(z Z())dxdz
So) () V(O)
I = (x X())dm Ii=Jx- X(0) 2dm (3.50)
Vb Vb
3.3. Decomposition and Separation of the Hydrodynamics from the Body Dynamics
The dynamics of a freely floating body subjected to a train of harmonic incident
waves is described by (3.45). This matrix equation is a second order differential equation and represents a dynamic balance of forces. The forces on the left-hand side are inertial and buoyancy forces with mass and restoring matrices, known from the hydrostatic equilibrium in the absence of waves. These forces are proportional to the unknown generalized body displacements, and are balanced on the right hand side of (3.45) by hydrodynamic forces, which are functions of the unknown velocity potential. The unknown velocity potential can be determined from the hydrodynamic system (3.34), only if and when the generalized body displacements are known (see (3.34d)). One way to resolve this problem is to decouple the hydrodynamics from the body dynamics with the following decomposition (Haskind, 1944): dX = Re(Va e_"t) (3.51a)
dt
(D = Re(oe-'") (3.5 lb)

S= R(ad~iafion) + rD(ifraction)

(3.51c)




where V is the time amplitude of the generalized body velocity; 5 is the time amplitude of the velocity potential; o is the wave circular frequency; and the imaginary unit is defined as i = V--i. At this point both the velocity potential (D and its time amplitude J have the same dimensions, which is [Length2/Time]. For the sake of brevity the notations for the real parts will be omitted and only the complex forms will be used instead, but only the real parts have physical meaning. Due to the linearity of the system (3.34) two contributions of a different nature can be separated entirely (see 3.51 c). Therefore two subproblems arise, each of which is more tractable than the complete problem:
(a) radiation subproblem: the rigid body oscillates harmonically in an
otherwise undisturbed body of water, thus generating waves which propagate or radiate away from the body. Physically the corresponding radiation velocity potential OR is a direct result of the motion of the body and should be proportional to this motion. Therefore OR = EV,,,,, where the summation is over the elementary components of the body velocity 6 in 3D-space, and 3 in 2D-space. 0, has the dimensions of [Length], and stands for the velocity potential caused by a body oscillatory motion with unit velocity in the oa-direction: for example in heave, surge, or pitch.
(b) diffraction subproblem: the forces on the rigid body, fixed in space, are caused by a train of harmonic incident waves. The velocity potential for the incident waves is ', while the velocity potential for the scattered waves (defined in Section 2.2.1. as sum of reflected and diffracted waves) is denoted by 0'. Define the velocity potential




for the diffraction subproblem as OD = + s. The dimensions of 0b s, and bD are [Length2/Time].
Thus the necessary decomposition of the time amplitude of the velocity potential is given by
O=OD +OR = ( +ps)+"V,'0" (3.51d)
Next the complete hydrodynamics problem, (3.34), is reformulated in terms of time amplitudes of the diffraction velocity potential with (3.51): V2oD = 0 in the fluid domain (3.52a)
SD 2 D = 0 on the free surface SF (3.52b)
az g
aD = 0 ,on the sea bottom Bo (3.52c)
8z
SD = 0 on the wetted body surface- Sb (3.52d)
-In
lim s Tiks = 0 waves outgoing at infinity (3.52e)
kx-*+co [
The only missing lateral boundary condition in (3.34) is now given by (3.52e) in 2D-space. Sommerfeld had introduced a similar lateral boundary condition, in an analogy with an outgoing wave system. The incident velocity potential is given by
0 -igA coshk(z + h) e+ikx (3.53)
co coshkh
In a similar way, the radiation subproblem is formulated as V2. = 0 in the fluid domain (3.54a)




a C 2 =0 on the free surface SF (3.54b)
8z g
a" = 0 on the sea bottom Bo (3.54c)
az
" n on the wetted body surface- Sb (3.54d)
an"
rlim "- Tik, = 0 waves outgoing at infinity. (3.54e)
kx-- m L &J
In a summary, the hydrodynamics (3.52d)+(3.54d) has been decoupled from the body dynamics (3.34d) with the help of decomposition (3.51). Now all that is needed is to solve the hydrodynamics (3.52)+(3.54) first, and then deal with the body dynamics (3.45).
3.4. Hydrodynamic Properties and Forces
The hydrodynamic properties will be derived in 3D-space using double integration over the wetted body surface. In 2D-space the same properties can be expressed with a single integration over the wetted body surface. Working with the hydrodynamic pressure of the linearized Bernoulli equation (3.12c), and with the decomposition made before, the generalized hydrodynamic force on the body in the a-direction is decomposed into its diffraction and radiation components:
F, = ff Pnads= f-(P c)nads = Re iCopf 01 +( OR)nads]e-' = F +
sb tL ( atsb

where




F D = Re{F e-' FD = Dicopf a~nds (3.55)
Sb
F, = RetFje-'}, FR=6.. j Sb 6Sb
The diffraction force is Ff Its time amplitude FD is known in the literature as the exciting force on a stationary body due to diffraction. The radiation component, the matrix [fp ], is known in the literature as the restoring force matrix, and FR as the restoring force. The radiation component (Mei, 1989) can be expanded further by defining the added mass and radiation damping matrices, namely [p]: pf,a = Re pf 0,ncds = Im(fra,), and (3.56)
[A]: Aflj = Im Pm pnds= Re fp) (3.57)
b,]:
The index notation p,a denotes the added mass, which causes a force in direction 8 due to acceleration in direction a. The index notation A6a denotes the damping, which cause a force in direction f due to velocity in direction a. In terms of these matrices the restoring force is expressed as
j dt2 X dXt (3.58)
Finally, with the velocity potential decomposition and the hydrodynamic property definitions, the dynamics of the floating rigid body (3.45) may be rewritten as [C] 2([M]+ [,u]) iwco[Aj]]{} = {FD} (3.59)




where {J} is the time amplitude of the generalized displacements {X}: {X} = Re({}e-) (3.60)
3.5. Hydrodynamic Relationships, Identities, and Definitions
As introduced and explained in Section 2.3, all necessary relationships and
identities among the hydrodynamic properties and forces will be summarized briefly for further use, without detailed derivation. For a complete description the reader is referred to Mei (1989). For any two twice-differentiable functions Oj, and 0j, the Green's theorem states:
ff(i1jOVO~n f q5i a~ A0 (3.61)
where Q is a closed volume with boundary Mi consisting of the wetted body surface Sb, the free surface SF, the bottom B., and a vertical circular cylinder with an arbitrary large radius S, If ,, and 01, are two velocity potentials, the left-hand side of (3.61) becomes zero due to Laplace equation. By virtue of the boundary conditions (3.54b), and (3.54c), neither the free surface SF nor the bottom Bo contributes to the surface integral thus reducing the right-hand side to:
s i aoj JdS = 0. (3.62)
Sb +S.




If a = 0,, and = Oj are two radiation velocity potentials, than the surface integral at the lateral boundaries vanishes due to the boundary condition (3.54e). This results in f (fl d =O 0, or (3.63)
S 8 n an
ff J; af dS = ff dS. (3.64)
Sb Sb
Therefore, the added mass (3.56), and damping matrices (3.57) must all be symmetric, regardless of whether the body is symmetrical or not, due to (3.64), namely PtS6 =/u and 2A,, = A, (3.65)
The law of conservation of energy flux requires that the average rate of work done by the oscillating buoy on the fluid over a period, denoted with Wbuoy, should be equal to the energy flux far away from the buoy denoted with Waway Wbuoy = away ,where (3.66)
St+T
W= f Wdt. (3.67)
T t
From (3.58), Wboy becomes
Wbuoy =-TFXa = 1 Pa X, X + A X, X (3.68)
Because of the symmetry (3.65) the first term of(3.68) can be written as
Because of the symmetry (3.65) the first term of (3.68) can be written as




58
1d X X,
-**i --*=I YI ''8x x =**
11 PrfaXX =2 "XX,+p. aX) 2 fl /3a dt
and vanishes due to periodicity. This means that the average rate of work done by the oscillating buoy on the fluid over a period is Wbuoy = a a2 6 a = VZ aP6 (3.69)
a /3 a f?
Next, the energy flux far away from the buoy can be expressed as
awav = -Re r Rds = dsR (3.70)
2 &bC- 4i St &'
Moreover, with the help of (3.62), (3.70) can be transformed to Waway = R R R C R ds. (3.71)
4fS.[ h
When = V a and the two-dimensional asymptotic behavior of 0, lim -iga coshk(z+h)
im = e (3.72)
x+ co coshkh
are substituted in (3.71), and (3.71) equalized to (3.69), the law of conservation of energy flux expresses damping in terms of a 1, = pcg(aa-- +a:a) (3.73)
where Cg is the group velocity, and (.) denotes the complex conjugate. a will be referred to as thefar-field amplitude, that has dimension of time since a = 4 / Va.




Since the average rate of work done by the body on the fluid (energy transmitted to the fluid by the body) is non-negative, the damping matrix is positive semidefinite. As a corollary, all diagonal terms of the damping matrix are non-negative, since = pgcg(_a 12+aI)o. (3.73b)
It is noteworthy that the same as (3.72) asymptotic behavior is valid for 0s, in accordance with the lateral boundary condition (3.52d), except that a will be replaced by another term with a dimension of length. It is also noteworthy that the damping and added mass matrices are proportional to the real and imaginary parts of the so-called restoring force matrix. This means that knowledge of the damping coefficients is sufficient to determine the corresponding added mass coefficients and vice versa. Such relations, known as Kramers-Kronig relations, are valid for all modes of motion regardless of forward speed (currents, ships). Having been derived in a universal way by Ogilvie, they read:
2 -2 A'( >) Af(l.), (3.74)
= a~T~i(C 2~(3.75)
p[.ia ) =- 7ta,(c2)]d_ = 0 (3.76)
0
Another way to find the added mass, knowing the damping as a function of the circular frequency, is Hooft's approach (1982) of using the so-called Bode relations, which for water waves correspond to the Kramers-Kronig relations. Rpa(t) = 2A ()cos(ot)dof (3.77)
Ir0C




oo
P,(co)- P(o)- 1- f Ry(t)sin(cmt)dt (3.78)
00
R,6, (t) is called the retardation function, and is obtained through a Fourier transform of A2p (co). Therefore the added mass is related to the far-field amplitudes, (3.74, and 3.78).
As mentioned in section 2.3, a remarkable Haskind theorem relates the exciting force to the far-field amplitudes in the following manner FD= ff pads (opf (' + 0 S)nd s copf ( +I S) ds
Sb b OSb O(3.79)
sb an sn
where (3.63) has been applied. Since 0- 0 is true on Sb, then (3.79) becomes On On
F D I a opff ds. (3.80)
Next, substituting the asymptotic forms (3.72) into (3.80) results in FD = -2pgCg Aa, (3.81)
thus expressing the exciting force in terms of the far-field amplitudes. Upon substituting (3.81) into (3.59), the matrix form of the dynamics equations become [[c] C2([M]+ [P])- ioA]]{} = -2pgCgA{a }. (3.82)
Defined as a ratio between the amplitude of displacement in the generalized direction a and the amplitude of the incident harmonic wave A, the frequency response function and the corresponding response amplitude operator are H,,() = / A, and RAO1(co) Ha(o) 2. (3.83)




The purpose of the frequency response function and the response amplitude operator are to characterize the dynamic response of the floating body as a function of the circular frequency of the incident wave. The vector form of the frequency response function is [H) -~~ (-2pgCg)[[C] _c2([M][J])-i)[A]][aj. (3.84)
3.6. Algorithm for the Solution of the Problem
In summary, the asymptotic solution of the radiation problem gives the far-field
amplitudes. The added mass matrix [u], the radiation-damping matrix [A] (3.73), and the exciting force (3.81) can be found from these far-field amplitudes. Therefore the asymptotic solution of the radiation problem alone will be sufficient for analyzing the body dynamics (3.84). Practically this means that the solution of the diffraction problem can be avoided. Having the dynamics and hydrodynamics (radiation) problems formulated, and all the necessary relationships for their solutions, the following algorithm will be used. First, the asymptotic solution of the radiation problem, in terms of the far field amplitudes, will be found in heave, surge, and pitch. Second, the hydrodynamic added mass, damping coefficients, and exciting forces will be computed as functions of the far-field amplitudes; consecutively the body dynamics will be computed for a particular wave frequency in terms of H, or RAO. Third, using the RAO (3.83), the spectral analysis can be used to find the body response spectrum Syy(co) due to the incident wave spectrum S.(co) for the entire frequency domain of interest, namely
S,(co) = H(o) 2S.(ao). (3.85)




CHAPTER 4
RADIATION PROBLEM SOLUTION
4.1. Problem Statement and Definitions
A long horizontal rigid cylinder with its cross-section in the Oxz-plane is
oscillating on the free water surface in otherwise calm water. Part of it is below the water surface; the other part is in the air. The rigid cylinder is oscillating with unit velocity in the generalized E -direction with the circular frequency of the incident waves co (in the absence of the incident waves), thus generating outgoing waves. Taking a unit-cylinder length in the y-direction, the problem is considered two-dimensional, and the motion can be described in the cross-sectional Oxz-plane (Figure 4.1).
Z
-T CB __II CB II
CG I
1I I pitch
surge
IIheave /-777 1 r77n f 7 7 7b
Fig. 4.1 Cross-section of the floating body. CB, CG centers of buoyancy and gravity.




Formulated in (3.54), the radiation problem is stated as a two-dimensional Laplace equation with a complete set of boundary conditions:
+ -2a= 0 in the fluid domain (4.1 a)
8x2 az2
8 a 02 =0 on the free surface SF (4. lb)
az g
S 0 on the sea bottom Bo (4.1c)
az
aa =n on the wetted body surface- Sb (4.1 d)
an"
lim jk = 0 waves outgoing at infinity. (4.1 e)
x-_+[ Dx J
As explained at the end of chapter 3, the practical mathematical problem to solve will be to find the asymptotic solution of (4.1), particularly the far field amplitudes. For the convenience of mathematical manipulation, time is removed from the problem by using the exponential time dependence (3.51), which restated is {D,X,}")T = Re({O,V, 7} e-j ') (4.2)
With the imaginary unit j= fi-, the wave profile is
= A cos(kx ot) = Re(Aej(a-t)) = Re([Ae'j ]e-j) = Re([-7]e-j') (4.3a)
7= Aej (4.3b)
As stated before, for brevity the sign Re (the real part of) will be omitted, but accounted for in the final results. In the two-dimensional case, the physical meaning of the Sommerfeld radiation condition (4. le) is that the generated waves are propagating outward from both sides of the oscillating body. Therefore, the asymptotic expression of




the wave profile at infinity must be:
lim 4 = AeJ( Ae) = ( )e-j = lim re-j', (4.4a)
x-.+ooD x--+-)._o
lim = = (va )ek (4.4b)
where A+ and A. are the asymptotic expressions of the wave amplitudes, and a2 are the far-field amplitudes introduced in (3.73).
4.2. Main Idea Behind the Semi-Analytic Technique (SAT)
Inspired from a Haskind idea (1973), a new analytical solution of(4. 1) will be given for the case of partially submerged floating circular and elliptical cylinders with varying equilibrium drafts. As will be shown below, the analytical solution is in terms of infinite power series. Therefore, a finite number of terms must be taken to solve it numerically thus the prefix "semi-" and the name Semi-analytic technique (SAT). For the sake of clarity and completeness, some mathematical definitions and formulations used in the analysis are presented first. Considering a two-dimensional (2D) wave motion in a right-hand complex plane Oxz, the real axis Ox is defined to represent the still water surface and the imaginary axis Oz to be perpendicular to the still water surface and positive upward (Figure 4.1). Let y be the complex variable, which corresponds to the complex plane Oxz, and defined as
y = x+iz (4.5)
where i = ,]-T is the complex plane imaginary unit, which for the sake of convenience is different from the time imaginary unit j = .-17. In the two-dimensional irrotational flow




of an ideal fluid, both the velocity potential (x,z) and stream function l(x,z) exist. In the whole fluid domain they have continuous first and second partial derivatives and satisfy the Laplace differential equation. Therefore by definition (Solomentsev, 1988) both velocity potential O(x,z) and stream function V(x,z) are harmonic functions. Combined in the complex plane these two harmonic functions define the well-known complex velocity potential
w(y) = 0 (x,z)+ i y (x,z) (4.6)
and are related to each other as conjugate functions through the Cauchy-Riemann conditions
k 01z (4.7)
At any point inside the fluid domain the complex velocity potential:
(a) has unique value
(b) has continuous derivatives of all orders
(c) satisfies the Cauchy-Riemann conditions. Therefore by definitions (a,b,c) the complex velocity potential is a holomorphic function (Milne-Thomson, 1950). Utilizing the property of holomorphic functions that a combination of derivatives and integrals of holomorphic functions is also holomorphic, the following holomorphic function is introduced. f (Y) = dw(y) + ikow(y) ,in the fluid domain without Sb (4.8a)
dy

In terms of its real and imaginary parts, f(y) can be written as




_ dy + ikow(y) = -kovi+(kj _--koo) (4.8b)
dy (Y
As pointed out in Haskind (1973), the combination (4.8a) was introduced for the first time by Keldysh in 1935. Comparing the imaginary parts of(4.8b) and (4. 1b), the boundary condition on the air-water interface can be written as: Im {f(y)}= 0, on z=O (4.9)
By virtue of the Schwarz Reflection Principle and the boundary condition on the air-water interface (4.9), it is possible to continuef(y) in the upper half-plane. As a result of the analytical continuation the values off(y) in the upper half-plane will be conjugate imaginary of the mirror off(y) values in the lower half-plane. In this mannerf(y) will be holomorphic in the whole complex plane except for the points on the wetted buoy surfaceSb and their mirror images denoted bySY (Figure 4.2) Thus, on an abstract mathematical ground, the problem has been extended from the lower half-plane to the entire complex plane.
iz
B A
K 5
Fig. 4.2 Wetted body surface and its mirror image Iff(y) were known, then the complex velocity potential would simply be the solution to




the ordinary differential equation (4.8a), namely
fluid
w(y) = e-k Y[A +iA2 + Jf(y)eikYdy] (4.10)
domain
where A1 and A2 are constants. Therefore the key to the solution of problem stated in equation (4.1) is to find a convenient form forf(y). A well-known mathematical technique is to represent the still unknown holomorphic function by apower series, which has the general form
f (y) = a. +at(y- y.)+a(y- _yo)+...+a,,(y- yo)+... (4.11)
The power series (4.11) is convergent within a circle ly -yo < R around the fixed point yo of radius R = lim an ,and it can be divergent outside that circle Iy -yo > R # 0 (see Solomentsev, 1988). The power series derivatives and integrals of any order have the same radius of convergence. Within the circle of convergence the power series coefficients are uniquely determined as Taylor's series coefficients an By definition an analytic function is defined as a power series, which within its circle of convergence is uniquely determined as a Taylor's series and possesses derivatives of all orders (Taylor and Mann, 1983; Solomentsev, 1988). Thanks to the similarity in their definitions, the analytic function will be the answer for the convenient presentation of (4.8) holomorphic function in (4.10). At infinity, the analytic function (4.11) can be written as
f = + 1'2 + Y3 + Y4 +... (4.12)
y I -- y y3 y4
From (4.9) it follows that all coefficients yi, (i = 1,2,3,...) are real. Substituting (4.12)




into the solution of the complex velocity potential (4.10) results in
y
w(y) = e-ikoY[A, +iA2 + f f ()e'kgdf] (4.13)
+o0
where A1+iA2 is an integration constant, 4 is a dummy variable, and the integration is taken over a curve lying in the lower half-plane. Consequently, when y = (x, z) -> (, z), the asymptotic expressions of the complex velocity potential are limw(y) = lim(q5 +iV) = (A1 +iA2)e-iko'y = (A1 +iA2)e-kox+koz
y-+wO y-4+.o (4.14)
limw(y) = lim(q +iy) = (BI +iB2)e-'oy = (BI +iB2)e-ikox+koz
where
-00
B, +iB2 = A, +iA2 + f (y)e'kYdy (4.15)
+W0
Taking the real part of (4.14), the corresponding asymptotic expressions of the velocity potential are
lim (y) = ekoz(A, coskox + A2 sinkox)
.. (4.16)
limo(y) = ekoz(B, cosk0x + B2 sinkox)
y-4-c-o
From the linearized free surface dynamic boundary condition at z=O
= OD jco < = e-j
=0 (4.17)
r7 =ji V) = =
Substituting (4.16) into (4.17) and comparing with the asymptotic wave profile (4.4),
2
results in the following deep-water (k = k0 = O2 ) relationship:
g




lim7 = jC V A, coskox+A2 sinkox) = Aejkox = (Va)(coskox + jsinkox) lim77= j V (B1 coskox+B2 sink0x)= Ae-jkkox (Va-)(coskox jsinkox) Therefore the integration constants are
Al =-j-a+, A2= -a+, Al +iA2 = -a+ (i- j), (4.18a)
B, =-jga-, B2 = -ga-, and B, +iB2 = -g-a-(j+i). (4.18b)
O) ) (0
Substituting the expressions for A, + iA2 and B, + iB2 into (4.15), results in
-ga-(j+i)= -a+ (i-j)- f (y)eiko'dy (4.18c)
Upon substituting i = j and i = -j into (4.18c) the far field wave amplitudes become: a = f (y)e'kYdy (4.19)
2 g _. i=TWhen the buoy oscillations are the source of wave generation then the integration in (4.19) can be replaced with an integration over the wetted buoy surface Sb and its mirror image S, in the counter clockwise direction. The proof, based on a lemma due to Jordan (Solomentsev, 1988), is given in Doynov (1992). a=[ -i m f (y)ekoYdy (4.20)
Therefore, in order to find a-, the unknown coefficients yi, (i = 1,2,3,...) of the analytic functionf(y) given in (4.12) must be determined.




4.3. Semi-Analytic Technique: Determination of the Unknown Coefficients
The procedure for determining the unknown coefficients in (4.12) consists of
4.3.1. Constructing a boundary condition for the holomorphic function through
integration of (4.8a) over the wetted body surface S, so that the left-hand side
contains all unknown coefficients, and the right-hand side contains all terms
derived from the complex fluid velocity.
4.3.2. Conformal mapping of Sb and its mirror image Yb into a unit circle, which permits
expressing the mapped holomorphic function with an analytic function of type
(4.12), convergent outside the unit circle
4.3.3. Determining the left-hand side, so that all multipliers of the unknown coefficients
are trigonometric functions of the polar angle of the unit circle 0.
4.3.4. Determining the right-hand side, so that all term are trigonometric functions of the
polar angle of the unit circle 0. Determining the complex fluid velocity with a
generalization of a procedure outlined in Milne-Thomson (1950).
4.3.5. Determining the unknown coefficients by solving a linear system of equations,
derived through the Fourier expansion of both left- and right-hand sides of the
boundary condition.
4.3.1. Boundary Condition on SIntegrating (4.8a) over the wetted body surface Sb in the clockwise direction from point A to point Y (Figure 4.2), results in




Sf (y)dy= [ dw(y) w(y)dy (4.21)
IL dy
Substituting (4.22)
J wdy = wy WAYA d ydy (4.22)
A4 fAdy
into (4.21) results in
~f(y)dy = -ikyAwA +(ikw)y+ I-(1-iky)dy (4.23)
f, (ydy _'YAWA .('W~y+ A dy
Eliminating ikw in (4.23) and (4.8a), gives a boundary condition whose right-hand side aw
(RHS) is a function of the complex velocity and complex variable y, dy
r dw Ydw
Sf (y)dy- yf (y) + ikyAW A = -y- + f- (1-iky)dy (4.24)
dy Ad
As stated before, only the real part of (4.24) will matter after deriving its explicit form. Some thoughts about the uniqueness of the solution of the boundary condition on Sb are given in section 4.3.6.
4.3.2. Conformal Mapping
If the wetted cross section of the floating buoy is a half-submerged circle, then the analytic presentation (4.12) in the left-hand side (LHS) of the boundary condition (4.24) can be used. In this case the unknown coefficients can be determined from those corresponding to the generalized motion RHS. If the wetted cross-section of the floating buoy is different than the half-submerged circle, then Sb can be mapped into a halfsubmerged circle C. Denote the analytic conformal mapping by




y = f() (4.25)
It transforms the holomorphic functionf(y) from the original plane Oxz:(y = x + iz = re*) into a holomorphic function f(() in the transformed plane Or:( = +iq = pe'o). This can be written as
dw (() /d f (
f(y) = f f() = () = d4' d +ikw() (4.26)
The conformal mapping properties are: dAf(()
a) Angles between vectors are preserved as long as d 0 d(
A
dy d df()
b) A vector is dilated by = d ,and
c) A vector is rotated by arg[ d(
At the infinite point of the transformed plane Orq the following analytic presentation can be used,
(.)= (4.27)
4.3.2.1. Conformal mapping of more than half-submerged circle
If R is the radius of the circle and h is the distance between the center of the circle and the still water level (Figure 4.3), then the wetted body surface and its mirror image can be written as




y = -ih + Re*, y = ih + Re"' and

, Sb:P G[- A,2"+PA],. $:p ( P A 91 +(A].

z Sb

Fig. 4.3 Conformal mapping of more than half-submerged circle If the conformal mapping

A= = ih +
y= f()= ihl + a4-k

(4.30)

dy d(4) ah-l = ah2,, 0, (4.31)
d4 d4
where h; and h2 are real and a is complex, is used to transform Sb (4.28) into the lower half of the unit circle C: 4 = 1.ei'o, where 0 e [-,0], then p = 2; + pA +h20, (4.32)
k + 2PA, (4.33)
a = Re'A and (4.34)

(4.28) (4.29) +AL+J




-h. (4.35)
If Sb (4.29) is transformed into the upper half of the unit circle C: e" where
0 [-2 ,-n],then
p=21r+3(p, +h20, (4.36)
a = Re 3"'A and (4.36)
h=h (4.38)
4.3.2.2. Conformal mapping of exactly half-submerged circle
This is a particular case of 4.3.2.1, with h = 0, 9A =0, h =0, h2 =0, and a =1. The result is
y= Re', S:qp ([0,27r] (4.39)
The conformal mapping which transforms Sb (4.39) into the lower half of the unit circle C: "= 1. e', where 0 c[--,0] is
y =f()=R and (4.40)
A
dy _df() -R O. (4.41)
d4 d =(
4.3.2.3. Conformal mapping of more than half-submerged ellipse
If a and b are the vertical and horizontal semi-axes of the ellipse and h is the
distance between the center of the ellipse and the still water level as shown in Figure 4.4, then the wetted body surface and its mirror image can bewritten as:




iz iz,
x- A
/c
SSb b
,kit, \\a C = e
Fig. 4.4 Conformal mapping of more than half-submerged ellipse
y= -ih+bcos(rq)+iasin(q) Sb: re [7 A,2r + 1A] (4.42)
y=ih+bcos(rq)+iasin(rq) q.r:q[-7AI7+ 7A]. (4.43)
If the conformal mapping
A
y= f() =ih1 + a1( + a2- and (4.44)
- h) 2 I =a2--') = h(a4 2 )- 0 (4.45)
d" d(
is used to transform Sb (4.42) into the lower half of the unit circle C: = 1.elo, where
0 e[-Ir,O], then
h; = -h, (4.46)
c; = (b+a)/2, (4.47)
A = (b-a)/(b+a), (4.48)
a, = cze'"A, (4.49)
a2 = ce-A (4.50)




r/= 2-+A, +h20O, and (4.51)
g +2r7A (4.52)
7Z*
If Sb (4.43) is transformed into the upper half of the unit circle C: = 1.e'", where
0 e[-27r,-7r], then
h = h, (4.53)
c, = (b+a)/2, (4.54)
A = (b-a)/(b+a), (4.55)
a, = ce3 (4.56)
a2 = C/e-3ir'a (4.57)
7= 2'r+3r7A +hO, and (4.58)
k =+2A (4.59)
4.3.2.4. Conformal mapping of exactly half-submerged ellipse
This is a particular case of 4.3.2.3, with h= 0, = 0, h = 0, and h2 = 0, and results in
y = b cos(7) + iasin(q) iE [0,2r]. (4.60)
The conformal mapping which transforms Sb (4.60) into the lower half of the unit circle C: = 1.e, where 0 e[-r,O] is
y= f) = c, + and (4.61)




A
dy _d f(2) -c, "- 0. (4.62)
d4 d4 2
4.3.3. Left-Hand Side of the Boundary Condition on Sb
To find an explicit form of the left-hand side of the boundary condition (4.24) WA, the complex velocity potential at point A must be found (Figure 4.2).
4.3.3.1. Complex velocity potential at point A
Substituting y.4 for y in (4.13) will give the following expression for w, YA
W.4 = W(yA) = e-ikA[A +iA2 + f (y)e~dy] (4.63)
+M0
Since the constants A1, A2 are proportional to the far field wave amplitudes in (4.18a) A, = -j-a, A2 =ga+,
equation (4.36) takes the following form: wA = [A1 cos(kxA) + A2 sin(kXA) +i(A2 Cos(kXA) A sin(kxA))] + e-'kYA f (y)ekydy (4.64) By virtue of the conformal mapping (4.26, 4.27), which will be proven below in 4.3.3.1.1 and 4.3.3.1.2, both terms in (4.64) can be expressed in terms of the unknown coefficients a,
al = aD+ (n) and (4.65)
n=1




YA 00
e'YA f (y)e'Ydy= "a,(P +iQn), (4.66)
+.0 n=1
thus giving wA, expressed as a series of the unknown an coefficients
cc
wA = a nw (4.67)
n=1
where
w, =[j ge AD+(n)+P ]+i KeJAD+(n)+Qn (4.68)
4.3.3.1.1. Determination of P,+iQP, +iQ, (4.66) can be expressed in the following manner
YA
e- 'kyAf f(y)e'Ydy= an(P +iQ,).
+00 n=1
4.3.3.1.1.1. Determination of P+iO_ Exactly half-submerged circle
Using the conformal mapping (4.40), (4.41) results in
YA .0 1 Wo
I = f (y)e'kdy = R" aje'cR "-nd4"= R al (4.69)
+o n=1 n=1
where I, can be determined knowing I, and the following recurrent formula
I 1= e~ik-"d4 = i[e +nI,,+ (4.70)
The first integral can be expressed with a complex exponential-integral function
I ikR kR it
1 f e d(= e dt = -E(-ikR), (Gradshteyn and Ryzhik, 1980) (4.71)
a, ;




79
n= ikR.I e' and (4.72)
n
P, +iQ, = e' AR, (4.73)
4.3.3.1.1.2 Determination ofP,+iQ More than half-submerged circle
The use of conformal mapping (4.30), (4.31) results in
YAM I I
y 1 1 Q
I= f (y)e'Jdy = f ek "i+ ]a '- "d= e-kh n -n (e'k' a2 h3 )d(, (4.74) +,0 n=1 W n=1 em
-kh 1 kh ika4 1 1
S= a~ -ndeika=2 n +n e (ka -+')d (4.75)
i n=l ik n=1 oo
-kh w
= Za [eka +nI, (4.76)
ik ,=,
n=1
where In is
1
I = f (" +')eika d(. (4.77)
1
After changing the variable of integration 4 with t = and correspondingly
1
di= -dt, In becomes
t2
-ika-tn- (ika)s 1 -sh2dt (ika)" 1 (4.78)
In = -eikatn1dt = -- [ t"-s! sdtn (4.78)
o S== o S! 0 '-0 S! s/ n
Substituting (4.78) into (4.75) gives
-kh oo a s (ika) 1
I = a, ek +nj (4.79)
ik n=1 I =o s! s/ n




Substituting (4.79) into (4.66) results in
-ie -kht-'1Aikes (ika)s I
P, +iQ kak e +nz (ika) (4.80)
k S=0 s! sh -n"
4.3.3.1.1.3. Determination of P,+iOQ Exactly half-submerged ellipse
The use of conformal mapping (4.61), (4.62) results in
YA co C 0
I = f (y)edy = c an eik -2d=c, a, [I, -,,+: (4.81)
+c0 n=1 0 n=1
= c (ilc,.)"
I, ike (+AC _"d(= (ikcA)' 1,, (4.82)
oo sS=0
1
I',, = f eikqc(-(n+s)d= ~ e' +(n+s)I+,,+1], (4.83)
n .ikc
S ikq( kc it
I, = f d= t dt = Ei(kc, (4.84)
ikcl' -eikq
I = "n* and (4.85)
nn+s
P, +iQ = e-" 'A c1 [I, -Mn+2] (4.86)
4.3.3.1.1.4. Determination of P,+iQ, More than half-submerged ellipse
The use of conformal mapping (4.44), (4.45) results in
I = f (y)e-dy =- n ik ah 4 1h -h4d= e-kh a ,, ,(4.87)
+00 n=1 co n=1
11
I, = ek(' +a,;"r -n h 2 -h4)d = -",deika +a.,( ,;-) (4.88)
In f eka1+2) n( ~ a2 -h4)dc =jJ~dl~i ik2 f (4.88)
cc c




ik(a+a) f ]k(a4 +a2 h2) 1 1 k( cz+a, (,k( t-h+.2t fl (l
[=-e' +nI e' f )-~~tdI(.9
ik L ikL 0 1

P, +iQ, = e-i"A- k' In .
4.3.3.1.2. Far-field wave amplitudes

(4.90)

S= f (y)ekody
2 g se _- i=T-j

(4.91)

= Za, D (n)
n=1

D+(n) has to be expressed from (4.65) and (4.41b).
4.3.3.1.2.1. Far-field wave amplitudes: Exactly half-submerged circle
Denote the integral (4.4 1b), upon which the far-field amplitudes depend with I. The use of conformal mapping (4.40), (4.41) results in

y 21r ikRet i(n-1)"
I= f(y)ekoYdy= R an e(-'d = RiLan e kRe (nl)idO ,
Sb + n=1 C11=1 n=a1 0

(4.92)

(4.93)

I= RZ an ei[s-(n-1)]oidO = R a,, 2i
n=1 Ls=O s! -2r I n=1 (n
From (4.91) and (4.93) it can be concluded that
2g (n-1)! g (n-1)!

(4.94)




82
4.3.3.1.2.2. Far-field wave amplitudes: More than half-submerged circle
The use of conformal mapping (4.30), (4.31) results in
I = f (y)e koYdy= a ik:ih +a h2-nd= ah2e-k anl n (4.95)
Sb+b n=1 C n=1
where the I, integral is to be calculated over the unit circle (Figure 4.3) In= e ika4 "h3nd (4.96)
C:(=1
In a cylindrical coordinate system I can be expressed as
2=er i(h3-n+1)d il(s+n ik k) e [h(s+l)-n]o (4.97)
i e e=0 si' -2x so S d (4.97)
0a -- s+=s+1 h(s+l)-n
': 2 .a0 ik's [as+'e- (1- e-'[lh(s+1)-n]i)] +[a '+' e+kk(1 e-' [h(s+1)-]ir)e-i[h2(s+1)-n]"1 I= I an
n -=0 S! (s+1)-n
(4.98)
From (4.91) and (4.98) it can be concluded that
9k_ (k) "[as+ le-kh -i hz(s+1)-n;r a s+ l+kht ( i e I )h(s+1-n]7 -i h(s+1)-n] A D.+ (n) = 1
2g s=0 s! (s+1)-n
(4.99)
4.3.3.1.2.3. Far-field wave amplitudes: Exactly half-submerged ellipse
The use of conformal mapping (4.61), (4.62) results in I = f(y)ekoYdy = c1Za ekt-,, 4e' -2)d; = ci a[Gn -AG,+2], (4.100)
sb +9 n=1 C:=1-l n=1




(4.101)

Gn= eikc(+A nd = (ikc1)S eikc (n+s)d.
c:11=1 s= C =1

Applying the same technique as in (4.92), (4.93), gives
(ikc-c (ikc) (ikc)2s+n-2 S (-)(kc')2s+n- as
G == 2ni 2ni = 2nI and
s=0 s! (n+s-1)! =0 s!(n+s-1)! = s!(s+n-1)!

2s+n-1
G, = 29i"(n-1)2: =(-l)(k 2 2 ,i (n-1)i2j n_1 (2k-y5 )
s=O s!(s+n-1)!

,when A > 0

(4.102)

where J, (.) denotes the Bessel function of the first kind,
= ( ( -- 2s+n-1
= 2n(A)-(n-1)/2 c = 2n(A)-(n-/ 2,_z(2kcIn
0 s!(s+n-1)

,when A <0

(4.103)

where ,_1(.) denotes the modified or hyperbolic Bessel function of the first kind. From (4.91), (4.100), (4.102) and (4.103) it can be concluded that

D (n) = [-i(G, G)

(4.104)

4.3.3.1.2.4. Far-field wave amplitudes: More than half-submerged ellipse
Using the conformal mapping (4.44), (4.45) results in
I = f(y)e'ikoYdy =e-kh a,, i + -n ah3 -a2z-h' )d= e- an
Sb- +b n=1 C:11=1 n=1

(4.105)




I, = e ik(. +a2h )4,-h (ah3 -a2h )d4 e- nodekae +ae (4.106)
C: =1 -2r
In = [e ik(al+a2) in ik a +nI +
k n ] (4.107)
ins ikae-ukr +aze")-e ik(.,e-"k +a2e2 2 + Is]
ik L
I S = i 0e-"oe ik age k+azek )dO = (ika2 [ -i 1-sh- (4.108)
S-0 S! 0 1 (1-s)h,-n
and
I b e inOeik a eIo+ae2dO (ika2) (ikal) e-' [(-s)-"]" e-2i(1-s)h2-n]. (4.109)
-i s= o S t (l-s)h2-n
-2 Yr S=O =0
From (4.91), (4.107), (4.108) and (4.109) D+ (n) = e I,, (4.110)
2g
4.3.3.2. Real part of the left-hand side of the boundary condition on Sb
Upon substituting (4.25), and (4.27) into (4.24) the left-hand side of the boundary condition becomes
00
LHS = f(y)dy yf(y)+ikAWA L (4.111)
n=1l
4.3.3.2.1. Left-hand side: Exactly half-submerged circle
Using the conformal mapping (4.40), (4.41) results in
Ln = Rf'd(- R('-') +ikyaw, (4.112)




1- -i(n-1)0
L, = RJ e-'-lid- Re-(n-1o+ik xAw = R 1- e Re o +ikAW (4.113)
n-1
and
L R nw ~ -)
L, = -R +ikxAw R n e-n-. (4.114)
n-i n-1
The real part of(4.114) is given as follows:
R n
Re(L,) R kXAIm(w,) R ncos(n 1)0. (4.115)
n-1 n-1
4.3.3.2.2. Left-hand side: More than half-submerged circle
Using the conformal mapping (4.30), (4.31) results in
L = h2af' "d(-(ih +af' )" +ikyAW (4.116)
(h3n+)9 in -i(n-h2)O
4~ -ae A(4.117)
L,, = h2 i(k-n+idO ie- ae-en-ho +ikX AWn (4.117)
L- = h2a -( ih e-no ae-i(n-)0 + ikxAWn, and (4.118)
n-hk
L = ikXw + a -i(n-) ihe-inkeo. (4.119)
n-h2 n-h2
The real part of (4.119) is given as follows: Re(L,) = 10 +1 cos(n-h2)O +2 sin(n-kh )0+13sin(n9). (4.120)
The real coefficients 1i c Re,(i= 0,1,2,3) are 10 = -kXA Im(w,) + hk Rea (4.121)
n- h2
/1 nRea (4.122)
n-h




n Im a
12 n- and (4.123)
n h2
13 = -h1. (4.124)
4.3.3.2.3. Left-hand side. Exactly half-submerged ellipse
Using the conformal mapping (4.61), (4.62) results in
L = [ci f f c ( +)" +ikyAw ] (4.125)
CL [ c_.__-(n-1) 1 e-i(n+1)(
S= c[-e e e- ein-) i(n 1) +ikxAw,, (4.126)
n-1 n+1
L. = c, 1 A n e-(_n-)o nA e-i(n+1)o + ikxAwn and (4.127a)
In-1 n+1 n-1 n+1
L1 = c -1- +il- e-2O+ikxAwl. (4.127b)
2 2
The real part of (4.127) is given as follows:
Re(L,)= 1 -kx lm(w) nc' cos(n- 1)0nC cos(n+1)0 (4.128a)
n-1 n+n-1 n+1
Re(L,)= c,[ -kXA In(w1)- Ccos(20) (4.128b)
4.3.3.2.4. Left-hand side. More than half-submerged ellipse
Using the conformal mapping (4.44), (4.45) results in
L, =h2 n1' -a2 )d(-(ih +al



L t i 1-e -h2)9 e- 1-e ( -inO i(n-h)O i(n+h2)0
, ih e haa e--h) a2-a2e A
n-h2 n h2
(4.130)
L ___=_h2_1_h2 2nna1 -i(n-hs)9 na2 -i(n+h2)(
h2al -ka2 +ikYAW,-iI e" h e e (4.131)
n-h2 n+h2 n-k n+h
Re(L) = Re ha h2 A n h2 [Re(az) cos(n h2)O +Im(al) sin(n h,)O]
(n-k n+k ) n-h2
n- [Re(a2)cos(n+hk)O+ Im(a2)sin(n+hk)O] -hl sin(n9)
n + h2
(4.132)
4.3.4. Right-Hand Side of the Boundary Condition on Sb
To find an explicit form of the right-hand side of the boundary condition (4.24) dw the complex velocity around the boundary Sb, must be determined. dy
4.3.4.1. Complex velocity around the boundary Sb. Surge, heave, and pitch mode
The procedure given below, is a generalization of a procedure outlined in MilneThomson (1950). Consider again the radiation velocity potential and its normal (to the wetted body surface) derivative for heave, surge, and pitch (4. 1d) 0R Vhh + V + Vp and (4.133)
R Vh + +V o +V P = Vnx +Vhnz +VP [nz-Z(-n(x X(o))]. (4.134) Tn c pn o n an
The components of the unit normal vector are




dz dx
n = nz = (4.135)
nxds ds
and ds denotes the elementary increment along the wetted boundary Sb. Making use of the Cauchy-Riemann conditions, which relate the velocity potential with the stream function and denoting
V =Vcosf (4.136)
Vh = V sinf and (4.137)
f = tan-'(V V/ S), (4.138)
results in
ay 8/ 0 OR dz x Z (O dz + 0 x
R V, +V z(z + x-X and (4.139)
as On ds h dsd
IR =d Vdz Vhdx +V d[(x- X(0) 2 + z- Z(O))2} (4.140)
as ds 2 L 'J
As described in Milne-Thomson (1950), the stream function can be presented as a difference of a complex function g(y,y) and its complex conjugate counterpart g(y,y), 2ifR = g(y,y)- g(y,y), (4.141)
g(y,) = Ve y +i --[y- (O)y (O)y +c, (4.142)
2
g(y, Y) = Ve y )7 0 _y y(O)y + J and (4.143)
2iy /R = Ve- Veify +iV y-Y(O) _y()y + c' (4.144)
where c, and c' are constants. The notation (.) denotes the complex conjugate. Upon substituting the conformal mapping into a unit circle y = f(4) and recognizing that




= 1 (4.145)
an expression for the stream function (4.144) in O r can be obtained 2iVfR = B +B,()+B2(). (4.146)
B0 is a constant, B1(4) contains all the negative powers of 4, and B2(4) contains all the positive powers of 4. Using the following relationship, which is proven below B2()= (-) (4.147)
results in
2iyR = Bo + B,()- B() = wR WR and (4.148)
wR = c"+B,(() (4.149)
where c"is a constant. Leaving the generalities, lets look for particular conformal mapping implementations.
4.3.4.1.1. Complex velocity around the boundary Sb: Exactly half-submerged circle
Applying the conformal mapping (4.40) in (4.144) results in
T = RC-', (4.150)
yf = R2, (4.151)
BI() = -Ve'fl (R4-1) +iV[-Y()R;-1 ], and (4.152)
B2(()= Ve-'"(R()+iV, -Y(0)R(j. (4.153)
Obviously (4.147) is justified. Making use of(4.152) and (4.149) results in wR c"+B(4)= -Ve'"(R4-')+iV[-Y(O)R+,'] +c", (4.154)




where c" is a constant. Therefore the complex velocity on the boundary C in Orl is dw= Ve' (R-2) +V[iRY(0) 2] (4.155)
d4" e~R2
From the decomposition (3.51 d), (4.50) of the radiation potential, the complex velocity should be
dw v dw +Vh dwh+ dwP (4.156)
= V,1 h +VP+ (4.156)
d, d h d d'
which means
dw R
dW _- surge mode (4.157)
dw iR
wh iR heave mode and (4.158)
dwp iRY (0)
dw- iRY- pitch mode. (4.159)
d( -2
4.3.4.1.2. Complex velocity around the boundary S. More than half-submerged circle
Applying the conformal mapping (4.30) in (4.144) results in
j7 = -ih, + a- (4.160)
yy =h + h i14' +i1i5-h (4.161)
B() = -Vef h )+iVp[(ihi Y(O)) and (4.162)
B() = Ve-(( )+iVp[(-ih- -o))a2 (4.163)
It is obvious from (4.162) and (4.163) that B2( )= -1 (4.164)




Full Text
93
and
(4.184)
B2{£) = V{-a2eip + ale~ifi)Cht +iVp axa2Ch2 -(y(0)cc2 + Y{0)ax +ihl(al a2))^
(4.185)
It is obvious from (4.184) and (4.185) that B2(= ~B,(£ ') and
w* V(a2eip ajPyr" +iv\axa2^ +W]a2-ihx{ax a2))c*
+ c".
(4.186)
The complex velocity on the boundary C in 0%tj is
dwR
~dQ
= V(-a2e-if> + aleif,)h1fh'-x +iVp\-2h2a1a2C2hl'' +h2(r{0)a\ +Ywa2 -ihx{ax-a2))Ch'
Compare (4.187) with (4.156) the complex velocity should be
dws (ax-a2)h2
dcr p
dwh i(ax + a2)h2
~dC~ ?
, surge mode
, heave mode and
(4.187)
(4.188)
(4.189)
dwp h2(iY{0)al+iY{0)a2+hx{al-a2))
dC
ch
c
h4 + hi
, pitch mode. (4.190)
4,3,4.2. Real part of the right-hand side of the boundary condition on Sh
In O^Tj the complex velocity potential and the right-hand side of the boundary
condition (4.24) are


115
Ka = To illustrate the effectiveness of the wave absorber at the North model boundary, the
dimensionless absorption coefficient for a typical set of experiments is plotted on Figure
5.5 versus the dimensionless water depth to wave length variable kh.
File= e2v52n.txt; Absorption Coefficient at the North End
Fig. 5.5 Absorption coefficient at the North end
With the very effective absorption at the North end, the reflected waves from the North
model boundary could be neglected and therefore the waves between the wave maker and
the South end are not contaminated by wave reflections from the North end. In order to


165
to choose nMax=24. As a characteristic of the speed of convergence, a partial sum is
defined as
ZkA()|
5(A0 = ia1 (6-14)
I I nMax
where l criterion (6.13) are illustrated on the following figure, in an example of a SAT numerical
solution for a particular circular frequency. As can be seen, four terms of the series are
sufficient to provide the necessary convergence. Moreover all the terms satisfy the
Cauchy-Hadamard convergence criterion, with q=0.3< 1.
Convergence of the far-field amplitude (FFA)
Fig. 6.51 Convergence of the far-field amplitude series. Cauchy-Hadamard criterion.


94
dw = dw{£) ,df(£)
dy dC d£
and
(4.191)
rhs = -y^f-+- ity)dy = -/(£)
dy iA dy
M£) ,df{c), ?M£)
dC'' d£
+
j:
d(
l-ikf(()W.
(4.192)
4,3.4,2,1, Real part of the right-hand side: Exactly half-submerged circle
It can be easily observed that (4.157), (4.158), and (4.159) have the same form,
namely
_ co
dC ?
(4.193)
and differ only by a constant c0 in the numerator. Therefore from (4.191) the RHS is
RHS = +[*r(l- ikR()dC = -f+Slip-ikRfp (4 194)
In the cylindrical coordinate system= l.e,e the RHS (4.194) transforms to
e
RHS = -c0e~w + J d(-c0e~w + kRco0) = c0- 2 cpid + kRc0O and
0
Re(RHS) = Re(c0 2coe~10 + kRc06).
(4.195)
(4.196)
4.3,4,2,2, Real part of the right-hand side: More than half-submerged circle
It can be easily observed that (4.169), (4.170), and (4.171) have the same form,
namely


i
69
lim/7 =
yH-co
lim^ =
y-+-co
v)j (.Bx eosk0x + B2 sinA:0x) = A_e~jkoX = {VCl~Ycosk0x jsmk0x)
( vi A
J^V
V S )
Therefore the integration constants are
Ax=-jZ-a<
co
A=-a+,
CO
Ax+iA2=-^a+(i-j),
co
(4.18a)
Bx =-jCT,
CO
B2 = ~a~, and Bx+iB2=-^-a-(j+i)- (4.18b)
co co
Substituting the expressions for Ax +iA^ and Bx +iB2 into (4.15), results in
+00
-a~(j+0 = f f(yykaydy
co CO L
(4.18c)
Upon substituting / = j and i = -j into (4.18c) the far field wave amplitudes become:
a- =
/ CO
2 g:
t-oo
\f(y)e**cty
(4.19)
Jl=+J
When the buoy oscillations are the source of wave generation then the integration in
(4.19) can be replaced with an integration over the wetted buoy surface Sb and its mirror
image Sb in the counter clockwise direction. The proof, based on a lemma due to Jordan
(Solomentsev, 1988), is given in Doynov (1992).
a- =
--
1 £ Sb+s
(4.20)
Ji=+;
Therefore, in order to find Or, the unknown coefficients /,, (/ = 1,2,3,...) of the analytic
function f(y) given in (4.12) must be determined.


98
Applying the same Fourier series expansion for each term n=l,2,3,... of the left-hand side
(4.120) of the boundary condition (4.24), results in
R<4) = 2>, coiW). (4.217)
/=0
where n=1,2,3,...
c, = \L+l.cos(n-h2)0 + l2sin(n-h2)0 + l2sin(n0)]cos(l0)d0 (4.218)
when 1=1,2,3,...; and for the case of 1=0 the coefficient is
c0n = f \lo+l^cos(n-h2)0+l2sm(n-h2)0+l3sm(n0))d0. (4.219)
The coefficients bo, b¡, cn0, and c,-can be determined using the formulae given below,
where v is real [v e Re]; k,l,m are integers \k = 1,2,3, ]
2 2 m
J sin(/w#) cos(I0)d0 =
n l2 -m2
0
when m +1 = 2k +1
when m + l = 2k
(4.220)
[sin (m0)d0
TT *
11
n m
0
2
u
l
[ cos v0) cos(l0)d0 =
J TT
when m-2k + \
when m = 2k
sin( v + t)n ^ sin( v t)n
v+l
v-l
? 0 -j
[ sin( v0) codl0)d0 =
7tJ n
n
2
U
J_
n
-l + COs(v-l-/)^^ 1 + COs( V 1^71
v+l
v-l
(4.221)
(4.222)
(4.223)
0
\
1
'vr
o
o
1
H
+
'rf
o
o
1
sin(v + /)^
si n{v-)n
I (/ vUS vKJ J vUS t C/ JCl (/ JL
-K
n2(v + l)2 tt2(v-1)2
7r(v + l)
7t{v-)
0
l-cos(v'^-) sin(v7r)
-7T
n2 v1 nv
(4.224)


80
Substituting (4.79) into (4.66) results in
Pn+iQn =
-ie
-khl~ikxA
Aka
A (ika)s 1
s\ sh^-n
(4.80)
4,3.3,1,1.3, Determination of P+iOv. Exactly half-submerged ellipse
The use of conformal mapping (4.61), (4.62) results in
yA co 1/ _j\ co
I = = C(1"K'1 )d£ = ct Z a[7 ~^+2] >
-f CO W=1 CO
w
Jkc^XC1)^ (ikCxX)
s=0
s!
/, = j -L [eto' +( + *)/,],
CO *"1
L Pikc£ k<2 e
h = J^ = J y = Ei{K)>
co oO
1 rt+.+l
n + s
and
(4.81)
(4.82)
(4.83)
(4.84)
(4.85)
(4.86)
4,3,3,1,1.4, Determination of P+iO, More than half-submerged ellipse
The use of conformal mapping (4.44), (4.45) results in
/ = ]f(y)e,kydy = e~*aHj-a2CK)dC = ,(4.87)
4-00 W = 1 CO
W
¡fc(a1^+ct2^)/,_
C'A!(a1f>-a!rK= j/C*'
1 f ikla^+arf-*1)
(4.88)


171
O 0.2 0.4 0.6 0.8 1 1.2 1.4
dimensionless circular frequency
Fig. 6.44 Added mass in pitch. Circular cylinder. Draft variations.


ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to my advisor Dr. Max Sheppard for
his guidance, technical, and moral support during my doctoral research. Being chairman
of my graduate committee, he provided me with his insight and perspective and gave me
the freedom to pursue my research interests. Being a noble soul, he granted me his
friendship and moral help during the difficult moments I had to go through as an
international student. I would also like to thank the members of my committee:
Dr. Robert Dean, Dr. Michel Ochi, and Dr. Ulrich Kurzweg for their time and advice, Dr.
Peter Sheng, Dr. Robert Thieke, and Dr. Daniel Hanes for reviewing this work.
For the clarity of all drawings in my dissertation, for her help, trust, inspiration,
and love, I am forever grateful to my wife, Galina.
For their constant support, encouragement, and inspiration, I am deeply grateful to
my parents, Iordan Doynov and Nadejda Doynova, and to my brother Ivan.
Additional thanks for making my time here enjoyable go to my fellow students
Wayne Walker, USA; Thanasis Pritsivelis, Greece; Roberto Liotta, Italy; Emre Otay,
Turkey; Ahmed Omar, Egypt; Kerry Anne Donohue, USA; Wendy Smith, USA; and
Matthew Henderson, USA.
Finally, words cannot express my love to my son Boris, whose presence and love
make my life a real adventure.


Fig. 5.2 Wave absorption at the North end
SCREEN 3
FIG. 5.2.b
SECTION B-B
WAVE ABSORPTION AT THE NORTH END
EXPERIMENT: A DYNAMIC RESPONSE MODEL
FOR FREE FLOATING BUOYS SUBJECTED TO WAVES
57n
5.?.h
O
VO


48
we can replace the domain of integration Sb with the part of the water surface cut out by
Sb, denoted with SA. With an error of 0((?) we can replace the integration over Sb and SA
with integration over S0) and when the body is in its rest position. Lets denote the
instantaneous volume of the displaced water with V, and in rest position with V{0), and let
A{0> be the area of S^. Following the procedure shown in Mei (1989) results in
Archimedes law for the zeroth-order
Mg = pgV{0) (3.39)
The first-order equations are
m[z+a(r r,0>) p(r A'"1)]=-pjf 4>i Vr- (/> I,Afi+Z< V0))
M
Af+fi,(r -zm)-r,{r- f4'1)]=-pJJ fi'S*
.,(o)
- <*(? Z<0))1=-pSI 9'^ds
?<)
If = \\[x-X{0))dxdy
s<>
I = li(y-ym)dxPy
.?()
(3.40b)
(3.40c)
(3.41)
3.2.2. Conservation of Angular Momentum
If Vb represents the volume of the whole body including the part above the free
surface, dm is the body mass per unit volume, and the center of mass is C:
xc =(xc,yc,zc), suchas JJJx xdm f Mxc, the conservation of angular momentum
v>
requires


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i
117
5.4. Model Scale Selection: Froude Scaling
Using an analogy of a rectangular barge in heave motion, a proper qualitative
relationship between wave height H and displacement S is
p2 V
v^2a
B
(5.8)
where B is the characteristic body dimension, introduced in chapter 2 L is the wave
length, and F is the Froude number defined as
V2 (tfA)2
F =
(5.9)
gB gB
with V, the floating body velocity, and co, the circular wave frequency. Therefore
proper dynamic and geometric scaling requires
(*),=(n,. (5l)
f B^\
vA,
rB2
2| and
a-di
(5.11)
(5.12)
wherep, m subscripts denote prototype and model respectively. Using the dimensions of
the wave tank (Figure 5.1), and a geometrically undistorted model results in a length scale
of 1:20, and a corresponding temporal scale of 1:4.5.


164
It can be seen from Figure 6.41, that the natural frequency decreases when the draft
increases. This is true despite the fact that added mass, fj., decreases with draft because
an increase in draft means an increase in mass, M, and eventually means an increase of the
denominator in (6.10). The lower the natural frequency, the lower the denominator in
(6.9), and the larger FRF(a>n) becomes. Figure 6.42 presents heave damping, added
mass, and FRF for a half-submerged elliptical cylinder. .
6,4. Numerical Convergence
In order to prove the numerical convergence of the far-field amplitude series,
(6.11)
it is sufficient to provide its absolute convergence (Taylor and Mann, 1983), namely
(6.12)
In accordance with Cauchy's root test, also known as the Cauchy-Hadamard convergence
criterion (Solomentsev, 1988), the series (6.12) is absolutely convergent if there is a
positive number q< 1, such that
n
(6.13)
for all sufficiently large values of n. For the numerical solution, it was practically sufficient


dimensionless damping o\ dimensionless damping
131
dimensionless circular frequency
1 Damping in surge. Half-submerged circular cylinder D=0.114 m. Stroke variations.
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
dimensionless circular frequency
Fig. 6.2 Damping in surge. Half-submerged circular cylinder D^O. 102m. Stroke variations.


55
F=RejXV"},
FaD = imp
JJ nads
(3.55)
F*=R Fa = ivpjj fnads = £ Vpfpa fpa = G>p\\ p^js
The diffraction force is Its time amplitude Ff is known in the literature as the
exciting force on a stationary body due to diffraction. The radiation component, the
matrix \fpa ], is known in the literature as the restoring force matrix, and F* as the
restoring force. The radiation component (Mei, 1989) can be expanded further by
defining the added mass and radiation damping matrices, namely
I ) 1
[p\ Fpa = Re PH pnadS = -lm{ffia) > ^d
V s
(3.56)
[A]. Xpa = Im
Pjj pnads =-Re(/^)
Vs )
(3.57)
The index notation ppa denotes the added mass, which causes a force in direction /? due
to acceleration in direction a The index notation Xpa denotes the damping, which cause
a force in direction /? due to velocity in direction a In terms of these matrices the
restoring force is expressed as
^ d2Xfi dXB
F ~~^pa~di2 ^Pa~dT
Finally, with the velocity potential decomposition and the hydrodynamic property
definitions, the dynamics of the floating rigid body (3.45) may be rewritten as
{[C]-m\[M] + [p])-im[X\]{^{FD}
(3.58)
(3.59)


162


40
strong vorticity as mentioned earlier.
Fig. 3.1 Wave force regimes (Hooft, 1982). Importance of
viscous and potential effects as functions of wave height-to-diameter
ratio and diffraction parameter
As observed from Figure 3.1, Hooft (1982, Figure 4.1.6) has shown that
for a fixed vertical cylinder and relatively steep waves, with ratio between wave
height and wave length: H/L=0.1, viscous effects become important when H/B
>15 and kB < 0.06, where k is the wave number. Practically this means that if the
characteristic dimension of the fixed vertical cylinder B=1 [m], the viscous effects
become important for wave heights H>15[mJ, and wave periods T>9[sec]. When
B=2 [m], the viscous effects become important for wave heights H>30[m], which
in reality can be referred to as extreme wave conditions. Intuitively, it is reasonable
to expect that for a free-floating body the velocity gradient will be significantly less
than for a fixed vertical cylinder. Therefore the fluid can be considered inviscid,


44
3.1.9. Linearization to First- Order Theory
Following Mei (1989), the derivation below leads to the complete first-order
theory and shows how higher-order extensions can be made. For small-amplitude motion,
we expand/in powers of the wave slope e- A! L, which is the small parameter in the
perturbation analysis
z = f(Q\x,y) + ef(x){x,y,t) + e2 f(2\x,y,t)+... (3.21)
where f(0\x,y) represents the wetted body surface rest position, that is The
velocity potential can be expanded in the same manner
0 = £(I)+,/ Considering small body motion, any function evaluated on Sb may be expanded about
:z = f(0\x,y). To the order 0(e), equation (3.20) can be written as
L1)/i0) + + /(1) = on z = f(0]{x,y) (3.23)
It is necessary to find Let the center of rotation of the rigid body be Q, which has
the following moving coordinate:
X(t) = X{0]+eX{l)(t) + s2X{2)(t)+..'. ,X=(X,Y,Z) (3.24)
where X(0) is the rest position of 0 independent of time. In some cases for example a
moored buoy the center of rotation may not coincide with the center of gravity, denoted
by C. Let x = (x,y,z) be the coordinate system fixed with the body in a way that x = x
when the body is at its rest position. Denoting the angular displacement of the body with
£n)(t) = s(a,P,y) with rotational components about axes x, y, and z, the two coordinate
systems are related to the first order by


170
the damping. Figure 6.48 shows that the added mass in pitch motion increases with
increasing draft. Using the same flap-type wavemaker analogy, the greater the draft, the
greater the local pressure gradient, the greater the underwater surface area on which the
local pressure gradient acts, the greater the added mass. Figure 6.49 presents the FRF in
pitch motion for the three drafts. The units of FRF are degree per meter. The peak values
of FRF have the same character as given in Bhattacharyya (1978). The greater the draft,
the lower the dimensionless frequency of the peak value of FRF. The differences between
the peak values of Figures 6.45, and 6.49 are due to the asymmetry of the elliptical shape
in pitch mode ocsillations.


178
Mei, C.C. (1989). The applied dynamics of ocean surface waves. World Scientific,
Englewood Cliffs, NJ.
Milne-Thomson, L.M. (1950). Theoretical hydrodynamics. The Macmillan Company, New
York.
Molin, B (1979). Second-order diffraction loads upon three-dimensional bodies. Applied
Ocean Research, Vol. 1,No.4, pp. 197-202.
Molin, B. and Marion, A. (1985). Second-order loads and motions for floating bodies in
regular waves. Proceedings 5th International Offshore Mechanics and Arctic Engineering
Conference, pp. 353-360. Tokyo.
Nestegard, A. and Sclavounos, P.D. (1984). A numerical solution of two-dimensional
deep water wave-body problems. Journal of Ship Research, Vol. 28, pp.48-54.
Newman, J.N. (1975). Interaction of waves with two dimensional obstacles: a relation
between the radiation and scattering problems. Journal of Fluid Mechanics. Vol. 71,
pp.273-282
Newman, J.N. (1976). The interaction of stationary vessels with regular waves.
Proceedings of 11th Symposium in Naval Hydrodynamics. Office of Naval Research, pp.
491-502. Washington.
Newton, R.E. (1975). Finite elements in fluids. Gallagher et al., Vol.l, pp.219-232
Newton, R.E., Chenault, D.W. and Smith, D.A. (1974). Finite element solution for added
mass and damping. Proceedings of the International Symposium on Finite Element
Methods in Flow Problems, Swansee, U.K.
Ogilvie, T.F. (1964). Recent progress toward the understanding and predictions of ship
motions. Proceedings of the fifth Symposium of Naval Hydrodynamics, Bergen, Norway.
pp.3-80
Patel, M.H. (1989). Dynamics of offshore structures. Butterworths. London.
Rahman, M. (1994). Ocean waves engineering. Computational Mechanics Publications.
Sahin, I. (1985). Motion analysis of floating structures by a surface singularity panel
method." Proceedings, pp. 1071-1076.
Sarpkaya, T. and Isaacson, M. (1981). Mechanics of wave forces on offshore structures.
Van Nostrand Reinhold Company. New York.


47
3.2. Floating Body Dynamics
3.2.1, Conservation of Linear Momentum
Let the entire mass of the floating body be denoted withM, part of which is above
the free surface, and let the center of mass be denoted with C: xc = (xc,yc,zc).
Integrating the linearized version of Euler equations (3.9), i.e. without the nonlinear term
in the left hand side, along the wetted body surface, the conservation of linear momentum
states
Mxctt = JJ Pnds Mgk (3.35)
where k is unit vector of Oz axis. Using the linearized Bernoulli equation
P = ~pgf +0(s2)
and (3.26), (3.35) can be written as
(3.12c)
eM
+ 0^ x (xc X(0)= JJ(-pgf ep&^nds- Mgk + 0[e2)
(3.36)
The zero-order portion of (3.36) is
0 = JJ (-pgf^nds Mgk
St
while the first-order portion of (3.36) is
M[X(1) + S x (xc X(0))] = JJ [~pgf[l) pounds
(3.37)
(3.38)
Considering the buoyancy term -pgf of (3.37), and having that on the instantaneous body
surface Sb
nds-[-fx ,-fy,\)dxdy


J
143
Figures 6.16 and 6.17 show the influence of draft variations on damping caused by heave
oscillations for both circular cylinders. The measurements of damping are shown on these
two figures for the draft-to-radius ratios 1, 1.2, and 1.4. Theoretically obtained damping
curves are in good agreement with the measured ones, for draft-to-radius ratios of 1.2,
and 1.4, but for 1 the theoretical damping curve overestimates the measured one. It
should be noted that this overestimation is due to the 39 mm data set (Figure 6.17) and 52
mm data set (Figure 6.16). It can be seen in Figures 6.15 and 6.14 the other data sets
produced by different strokes are closer to the theoretical solutions than both of these 39
mm and 52 mm sets. Both theoretical and measured damping curves on Figures 6.16 and
6.17 suggest that damping in heave decreases with the draft. The plots in the preceding
Figures 6.14, 6.15, 6.16, and 6.17 suggest that damping coefficients in heave increase with
frequency until they reach a maximum, after which they decrease with further increment of
the circular frequency. Figure 6.18 presents heave damping for half-submerged elliptical
cylinder with a horizontal semi major axis. Figure 6.19 shows heave damping for half-
submerged elliptical cylinder with a vertical semi major axis. In both cases the theoretical
damping curves are in good agreement with the measured ones. The measurements for
the case of the horizontal semi major axis show that heave damping slightly increases with
the magnitude of the stroke. This observation is almost completely reversed for the case
of the vertically oriented elliptical cylinder. It is noteworthy that the heave damping
curves for the elliptical cylinder with a horizontal semi major axis are steeper than those
for the cylinder with a vertical semi major axis, when plotted versus increasing circular
frequency.


149
6.3. Damping. Added Mass, and Frequency Response Function
As defined at the end of section 3.5 and summarized in section 3.6, the dynamic
response of a floating body (3.82) is characterized by its frequency response function FRF,
and the response amplitude operator (3.83), (3.84). The mass and restoring force matrices
are determined from the static equilibrium state (3.48) and (3.49), and therefore are
independent of the frequency of the incident wave. On the other hand, the added mass,
damping, and exciting force are functions of the circular frequency of the incident wave.
Consequently, it is instructive to show how the frequency, shape and draft dependence of
the added mass and damping influence the FRF and the dynamic response of the floating
body. To illustrate this frequency, shape, and draft dependence, the following two
numerical examples are considered utilizing the semi-analytic technique. In the first
example, a circular cylinder of radius R=0.5 m is investigated for three draft-to-radius
ratios of 1, 1.2, and 1.4. In all three cases the center of gravity is placed at a fixed distance
of 0.3 m from the lowest point on the cylinder, as shown on Figure 6.25. The breadth of
the waterline section is decreasing with increasing draft. In the second example, shown on
Figure 6.26, an elliptical cylinder with fixed horizontal semi-axis =0.5 m, and fixed
waterline breadth B=2b= 1 m is investigated for three semi vertical to semi horizontal axis
ratios of a/b-1.01, 1.1, and 0.9. Again, in all three cases the center of gravity is placed at
the same fixed distance from the lowest point on the cylinder. The breadth of the
waterline section is constant B-2b= 1 m in all three cases.


132
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
dimensionless circular frequency
Fig.6.3 Damping in surge. Circular cylinder D=0.114 m. Variations in draft.
0 I I i X 1 1 1 1 i i_l
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
dimensionless circular frequency
Fig.6.4 Damping in surge. Circular cylinder D=0.102 m. Variations in draft.


72
y=m (4-25)
It transforms the holomorphic function f(y) from the original plane Oxz:[y = x + iz = re"p)
into a holomorphic function f(£) in the transformed plane O^tj:= <^+irj = pe,e). This
can be written as
/W=A Ad) =/(?)=^/
The conformal mapping properties are:
df(C)
a) Angles between vectors are preserved as long as ^ 0
UCy
(4.26)
b) A vector is dilated by
dy
dC
dt;
and
c) A vector is rotated by arg
r a
dm
dC
v y
At the infinite point of the transformed plane 0%r¡ the following analytic presentation can
be used,
/( n= 1 b
(4.27)
4,3,2.1, Conformal mapping of more than half-submerged circle
If R is the radius of the circle and h is the distance between the center of the circle
and the still water level (Figure 4.3), then the wetted body surface and its mirror image
can be written as


89
6T=i (4145>
an expression for the stream function (4.144) in 0£r¡ can be obtained
2iy,R=B0+Bx(£) + B2{£). (4146)
B0 is a constant, Bx{£) contains all the negative powers of £, and B2(£) contains all the
positive powers of £. Using the following relationship, which is proven below
fi,(f) = -S,(rl) (4.147)
results in
2/ y/R = BQ + B, (£) Bx{£) = wR-wR and (4.148)
wR=c"+Bl(£) (4.149)
where c" is a constant. Leaving the generalities, lets look for particular conformal
mapping implementations.
4,3.4.1.1. Complex velocity around the boundary S Exactly half-submerged circle
Applying the conformal mapping (4.40) in (4.144) results in
y = RC,
yy-R2,
Bx{£) = -Veip{RCx) + iVp\-Y{(>)RC'\and
B2{£) = Ve^{R£) + iVp
Y{0)R£
(4.150)
(4.151)
(4.152)
(4.153)
Obviously (4.147) is justified. Making use of (4.152) and (4.149) results in
wR =c"+Bx{£) = -Vei,}(R£-x) + iVp[-Y{0)RC-x] + c\ (4.154)


127
Fig. 5.14 Side view of the wave gauges.


156
Fig. 6.31 Damping in heave. Circular cylinder. Draft variations.


The RHS of (4.241) is a function of the constants rA, and y/ A and a function of the
known normal velocity along the wetted body surface Sb. The harmonic function
r(x,z) = r(p,6) is analytic in the lower half-plane z < 0, vanishes at infinity (4.242), its
normal derivative on the free surface z = 0 is zero (4.243), and along the wetted body
surface Sb satisfies (4.241). Now, consider two functions rx, and r2 which are harmonic
in the half-plane z < 0, and satisfy (4.241), (4.242), and (4.243). For two twice-
differentiable functions rx, and r2, Green's theorem is
(4.244)
where the free surface and the infinite surface are shown on Figure 4.5, and do not
contribute to the integral (4.244) because of (4.242) and (4.243). Therefore the only
surface integral that remains is the one on Sb.
(4.245)
If the right hand side of (4.241) is denoted with RHS, then both harmonic functions rx and
r2 must satisfy
(4.246)
(4.247)
From here it follows that on the wetted boundary surface


S
Response spectrum of the floating body
T
Period of the incident wave
um
Magnitude of the horizontal velocity of the incident wave
u = (u,v,w)
Fluid particle velocity
K
Time-amplitude of generalized velocity
W = dW / di
Energy flux
xa
Generalized displacement in a direction
i'=(x-ys)
Coordinates of the center of mass
y =x+i z
Complex variable
s
Small parameter
G>
Velocity potential
<¡>
Time-amplitude of the velocity potential
tD,R
Diffraction, radiation velocity potential
a
Radiation velocity potential due to unit velocity in
generalized a direction
*Kx>y)
Time-amplitude of the water elevation
A
Damping
V
Added mass
V
Kinematic viscosity
P
Water density
(0
Circular frequency of the incident wave
Time-amplitude of Xa
Stream function
Â¥
Time-amplitude of the stream function
C(x,y,t)
Water elevation
Vil


151
Since pitch mode experiments were not conducted and since stability is strongly
dependent on the metacentric height (usually small in pitch), the added mass, damping,
and FRF in pitch mode for these two examples, are given in the appendix. It is assumed
that keeping the center of gravity below the instantaneous center of buoyancy provides
stability. This is usually achieved by placing ballast at the bottom of the floating body.
Nevertheless, the coupling between surge and pitch is mentioned in the following
discussion. The circular frequency, damping, and added mass are made again non-
dimensional with the help of the breath of the section at the waterline (6.1), (6.2), and
(6.3).
With reference to the first numerical example (Figure 6.25), Figures 6.27, 6.28,
and 6.29 present the damping, added mass, and FRF in surge motion of a circular cylinder
with varying drafts. It can be seen from Figure 6.27 that in general, damping in surge
increases with increasing draft for dimensionless frequencies up to 0.95, For higher
dimensionless frequency above 0.95, or periods below 1.55 seconds, the increment of the
draft-to-radius ratio from 1.2 to 1.4 results in less damping in surge motion. An intuitive
explanation for this phenomenon can be offered from the wavemaker-theory point of view.
First consider the limiting case of substituting the underwater shapes with the same draft
vertically oriented plates with part of them above the water. If all three drafts are
oscillated with a constant surge amplitude, the greater the draft, the greater the displaced
water volume, and the greater the propagating wave amplitude and thus the greater the
damping. While this analogy works in the practical frequency range, it is not entirely
applicable for higher frequencies in the ratio transition from 1.2 to 1.4. In the particular


179
Sarpkaya,T. (1989). Computational methods with vortices The Freeman Scholar
Lecture. Journal of Fluids Engineering. Vol.165, pp. 61-71
Sawaragi, T. (1995). Coastal Engineering waves, beaches, wave-structure interactions.
Elsevier. Netherlands.
Soding, H. (1976). Second-order forces on oscillating cylinders in waves. Schiffstechnik.
Vol.23, pp.205-209.
Solomentsev, E.D. (1988). Functions of complex variables and their applications (in
Russian). Izdatelstvo Vysshaia Shkola. Moscow. Russia.
St. Denis, M. and Pierson, W.J. (1953). On the motions of ships in confused seas. Trans.
Soc. ofNaval Architecture and Marine Engineering. Vol.61, pp.280-357. New York.
Stansby, P.K. and Isaacson, M. (1987). Recent developments in offshore hydrodynamics:
workshop report. Applied Ocean Research, Vol.9 (3), pp. 118-127
Sumer, B.M. and Fredsoe, J. (1997). Hydrodynamics around cylindrical structures. World
Scientific, New Jersey.
Taylor, A.E. and Mann, W.R. (1983). Advanced calculus. John Wiley & Sons. New
York.
Taylor, R.E. and Hu, C.S, (1991). Multipole expansions for wave diffraction and radiation
in deep water. Ocean Engineering, Vol.18, No.3, pp.191-224.
Taylor, R.E. and Hung, S.M. (1987). Second-order diffraction forces on a vertical
cylinder in regular waves. Applied Ocean Research, Vol.9, pp. 19-30
Ursell, F (1949). On the heaving motion of a circular cylinder on the surface of a fluid.
Quarterly Journal of Mechanics and Applied Mathematics, Vol. 2, pp.218-231
Vantorre, M. (1986). Third-order theory for determining the hydrodynamic forces on
axisymmetric floating or submerged bodies in oscillatory heaving motion.
Ocean Engineering, Vol. 13, No.4, pp.339-371.
Vantorre, M. (1990). Influence of draft variation on heave added-mass and hydrodynamic
damping coefficients of floating bodies. Journal of Ship Research, Vol.34, No.3, pp. 172-
178.
Vethamony, P., Chandramohan, P., Sastry, J.S., and Narasimhan, S. (1992). Estimation of
added-mass and damping coefficients of a tethered spherical float using potential flow
theory. Ocean Engineering, Vol. 19, No.5, pp.427-436. '


41
and the Navier-Stokes equations (3.2) are transformed into Euler equations:
f- u. V u = -V
dt
P
+ gz
VP ,
(3.9)
3.1.4. Irrotational Flow Assumption
For an inviscid and incompressible fluid, where the Euler equations are valid, there
are only normal stresses acting on the surface of a fluid particle; since shear stresses are
zero, there are no stresses to impart a rotation on a fluid particle. Therefore any
nonrotating particle remains nonrotating, but if initial vorticity exists, vorticity remains
constant (Dean and Dalrymple, 1991). An important class of problems is one where
QsO and is called irrotational flow. For an inviscid irrotational flow, the velocity u is
usually expressed as the gradient of a scalar function called velocity potential. It exists in
two-dimensional and three-dimensional irrotational flows
u = VO (3.10)
Conservation of mass (3.3) requires that the velocity potential satisfies Laplaces equation
V2O = 0 (3.11)
while conservation of momentum (3-.2) transforms into
50 1 1^,2
+ VO
dt 2' 1
(3.12a)
Integrating (3.12a) with respect to the space variables away from the body, we derive
Bernoullis equation
50
+|VO|2 +gz = ~+C(t)
dt 21 1 p W
(3.12b)


To Boris and Galina


29
matrix has been used as a partial check for the correctness of the semi-analytical solution.
In a technical note to the Evans and Mclver (1984) work, Falnes (1984) proposed some
empirical formulas for the added mass and damping for the semi-submerged sphere in
heave motion.
Utilizing a numerical scheme that is similar to one used by Nestegard and
Sclavounos (1984) for the method of multipole expansion combined with BIE-BMP
matching classified in 2.2.5.1, Taylor and Hu (1991) obtained added mass and damping for
floating and submerged circular cylinders. For the submerged cylinder, the diagonal added
mass and damping coefficients in sway have been confirmed to be equal to those in heave.
While damping is always positive, negative added mass has been discovered for the
case when the submerged cylinder is close to the surface. Negative added mass has been
also observed for a cylinder floating on the surface in sway motion when the cylinder is
more than three-quarters immersed. For the frequencies of negative added mass in heave
the sway added mass was positive. It was concluded that at these frequencies the forced
heave oscillation does not transfer energy to the fluid and hence does not generate waves.
Lastly Taylor and Hu have found that in the low frequency range the added mass changes
from finite to infinite values as the cylinder emerges from below the free surface. The
incorrect conclusions, made by Taylor and Hu, show the important role of experiments as
the only tool for verification of the working capabilities of theories and numerical
techniques.
Exploring surge oscillatory motion of a single tethered half-submerged spherical
float, Vethamony et al. (1992) have computed added mass and damping from the motion
generated velocity potential, using potential flow theory. They solved a boundary value


30
problem with Greens function method by dividing the cross-section of the float into equal
segments, and conducted experiments to compare the results. Considering the float size
small compared to the wave-length the diffraction potential has been neglected and added
mass and damping computed from the motion generated velocity potential. With added
mass in phase with the surge motion and damping coefficient out of phase with the
motion, both were found to be frequency dependent, and to increase gradually with
respect to size of the float. It was noted that added mass and damping do not change with
water depth. In light of the general identities discussed earlier, Vethamony et al. found
that wave excitation forces calculated from incident wave potential are in good agreement
with excitation forces calculated from damping coefficients.
It was observed that nonlinear effects become important in fluid-structure
interaction when waves become steeper, or when the oscillation amplitude of the floating
body can no longer be assumed small. These natural phenomena have been stimulating the
extension of the linearized radiation problem and linearized diffraction theory which are
the first terms in Stokes perturbation expansion into higher order terms, where the
perturbation parameter is related to the wave steepness. As a result of this expansion
procedure the full nonlinear diffraction and/or radiation problems are replaced by a
sequence of linear boundary-value problems formulated for each order of perturbation.
Most difficulties of the second order problem arise from the requirement that the velocity
potential satisfies a nonhomogeneous boundary condition on the free surface of the fluid.
According to Molin (1979), the nature of second-order diffracted waves consists of two
components. One is free-waves travelling independently of the first-order wave system
and the other is phase-locked-waves accompanying the first-order waves (Chakrabarti,


133
Figures 6.3 and 6.4 show the influence of draft variations on damping caused by surge
oscillations for both circular cylinders. With a stroke of 63 mm for the large cylinder and
52 mm for the small cylinder, the measurements of damping are shown for three draft-to-
radius ratios 1, 1.2, and 1.4. On both figures, there is little scatter among the measured
points for the three variations in draft. Theoretically obtained damping curves for the
small cylinder, slightly overestimate the measured ones. Both theoretical and measured
damping curves suggest that damping in surge increases with the draft. There is a
noticeable branching among the different draft sets with surge damping increasing with
draft. The same tendency of branching can be observed from the theoretical curves in
both Figures 6.3 and 6.4 for draft-to-radius ratios of 1, 1.2, and 1.4. This branching also
suggests that it would be more appropriate to non-dimensionalize surge damping and
circular frequency with draft instead of breadth of the section at the waterline. Figure 6.5
presents surge damping for a horizontally oriented elliptical cylinder. Its semi major axis
coincides with the still water surface while the semi minor axis is normal to the still water
surface. Figure 6.6 shows surge damping for a vertically oriented elliptical cylinder. This
time its semi minor axis coincides with the still water. In both Figures 6.5 and 6.6, the
measured results for the larger strokes are slightly above those with the smaller strokes.
In both cases, the theoretical damping curves are in good agreement with the measured
ones.


67
the ordinary differential equation (4.8a), namely
fluid
w(y) = e-'k[Al+iA1+ J / (y)eiky dy] (4.10)
domain
where Aj and A2 are constants. Therefore the key to the solution of problem stated in
equation (4.1) is to find a convenient form for f(y). A well-known mathematical technique
is to represent the still unknown holomorphic function by a power series, which has the
general form
f(y) = 0 + Y+ + (y To)"+ (4-11)
The power series (4.11) is convergent within a circle [y-y0| < R around the fixed pointya
of radius R = lim
M co
a.
a.
n+l
, and it can be divergent outside that circle \y -y0\ > R 0 (see
Solomentsev, 1988). The power series derivatives and integrals of any order have the
same radius of convergence. Within the circle of convergence the power series
/"O')
coefficients are uniquely determined as Taylors series coefficients an =
n\
By
definition an analytic function is defined as a power series, which within its circle of
convergence is uniquely determined as a Taylors series and possesses derivatives of all
orders (Taylor and Mann, 1983; Solomentsev, 1988). Thanks to the similarity in their
definitions, the analytic function will be the answer for the convenient presentation of (4.8)
holomorphic function in (4.10). At infinity, the analytic function (4.11) can be written as
Ay) = LT+Lt+ZT+lT+- (4.12)
y y y y
From (4.9) it follows that all coefficients y,,(i = 1,2,3,...) are real. Substituting (4.12)


I
31
1987). Since in many applications the hydrodynamic loads, not the flow kinematics, are of
primary interest, an indirect method can be used to determine wave-induced loads to the
second order without the explicit calculation of the second order potential. The technique
involves an application of Greens second identity and requires the solution of associated
linearized radiation problems corresponding to prescribed oscillation of the structure at
twice the first-order wave frequency. A modification that utilizes an axisymmetric Greens
function solution for the linear problem has been used by Molin and Marion (1985) to
calculate the second-order wave induced loads and associated motions for a floating body.
Taylor and Hung (1987) have performed detailed analysis of the asymptotic behavior of
the troublesome free-surface integral and have presented a computational method for its
calculation on the far field in the case of a fixed vertical circular cylinder. Adopting an
interior region that includes the fixed vertical cylinder, and an exterior region that extends
to infinity in the horizontal plane, Ghalayini and Williams (1991) presented a solution to
the free-surface integral. The interior region solution was performed numerically by
utilizing the asymptotic forms of the potentials, while the exterior region solution was
carried out analytically in terms of Fresnel functions. Inspired by Sodings work,
Vantorre (1986) developed a computational procedure for calculating hydrodynamic
forces up to the third order for the case of floating axisymmetric bodies in a symmetric
heave oscillatory motion. Soding (1976) proved that the knowledge of the first-order
potential for the exciting frequency and for twice that frequency on the mean body surface
and on the mean free surface within some distance from the body is all the information
required to compute second-order forces for oscillating cylinders. In a similar way,
Vantorre computed third-order forces by means of the first-order and second-order


3.2.2. Conservation of Angular Momentum 48
3.2.3. Matrix Form of the Dynamics Equations 50
3.3 Decomposition and Separation of the Hydrodynamics from the Body Dynamics 51
3.4. Hydrodynamic Properties and Forces 54
3.5. Hydrodynamic Relationships, Identities, and Definitions 56
3.6. Algorithm for the Solution of the Problem 61
4 RADIATION PROBLEM SOLUTION 62
4.1. Problem Statement and Definitions 62
4.2. Main Idea behind the Semi-Analytic Technique (SAT) 64
4.3. Semi-Analytic Technique. Determination of the Unknown Coefficients 70
4.3.1. Boundary Condition on St, 71
4.3.2. Conformal Mapping 71
4.3.3. Left-Hand Side of the Boundary Condition on Sb 77
4.3.4. Right-Hand Side of the Boundary Condition on Sb 87
4.3.5. Fourier Expansion of LHS and RHS. Solution for the Unknown
Coefficients 97
4.3.6. Discussion of the Uniqueness of the Solution 101
5 EXPERIMENTS 106
5.1. Purpose of the Experiments 106
5.2. General Set-up 106
5.2.1. Cylinders 108
5.2.2. Wave Absorption at the Ends of the Tank 110
5.2.3. Wave Gauges 110
5.2.4. Surface Tension Ill
5.3. Wave Absorption and Reflection Analysis 112
5.4. Model Scale Selection: Froude Scaling 117
5.5. Discussion of the Experimental Accuracy 118
5.6. Discussion of the Experimental Procedure 118
6 ANALYSIS OF THE RESULTS 128
6.1. Surge Mode Oscillations 130
6.2. Heave Mode Oscillations 140
6.3. Damping, Added Mass, and Frequency Response Function 149
6.4. Numerical Convergence 164
6.5. Conclusions 166
APPENDIX 168
LIST OF REFERENCES 175
BIOGRAPHICAL SKETCH 181
v


28
viscous effects become much more important. That is why the damping in roll motion is
significantly underestimated compared to the damping in sway motion. The exciting
forces, calculated with Newmans explicit relationship of the magnitudes of the exciting
force and damping coefficients, have been found to be in a relatively good agreement with
the measured ones.
Haskind (1973) developed a technique for computing added mass and damping for
horizontal and vertical plates floating at the water surface, as a function of body geometry,
water depth and frequency. Newton et al. (1974) and Newton (1975) have developed a
finite element model (FEM) for computing these coefficients for ship hull forms. Using
potential flow theory, Chung (1976, 1977) presented added mass and damping as a
function of frequency, direction of oscillation and depth of submergence. Hanif (1983)
determined these hydrodynamic coefficients using FEM and compared his results with
other investigators.
While investigating the heave motion of a half-immersed floating sphere with a
bottom opening, Evans and Mclver (1984) have used a semi-analytical solution based on
an extension of the method of multipole expansion, classified in 2.2.5.1. The added mass
and damping coefficients calculated with this method were compared with their values
obtained via the numerical singularity distribution method classified in 2.2.5.2. They
found that the numerical method tends to overestimate the diagonal hydrodynamic
coefficients, while their off-diagonal values are in good agreement with the semi-analytical
solution. Moreover the numerical method showed slight differences between the off-
diagonal added mass and damping coefficients, which theoretically must be equal in
accordance with the symmetry-identity classified above. The singularity of the damping


136
Figures 6.7 and 6.8 present the influence of draft variations on damping caused by surge
oscillations of an elliptical cylinder with horizontally and vertically oriented semi major
axis. In these figures the measured damping coefficients are shown for the same three
draft-to-vertical semi-ratios of 1, 1.2 and 1.4, and a stroke of 52 mm. Again, there is little
scatter among the measured points for all draft variations. As in the case of a circular
cylinder with varying draft, the same branching tendency among the different draft sets can
be observed (Figures 6.7 and 6.8) with surge damping increasing with draft. In Figure
6.7, the theoretical solution slightly overestimates the measured damping for ratio 1.2. In
Figure 6.8, There is good agreement between the theoretical and measured values. It is
noteworthy that the damping curves for the elliptical cylinder with a vertical semi major
axis are steeper than those for the elliptical cylinder with a horizontal semi major axis,
when plotted versus increasing circular frequency. Despite the same volume of water
displacement, the draft increment engenders greater damping in surge mode oscillations.
The amplitudes of the outgoing waves in surge are presented in Figures 6.9
through 6.13. They are shown in their non-dimensional form of amplitude-to-stroke ratio
versus the dimensionless circular frequency, introduced in (6.1). Figure 6.9 presents the
influence of the size of semi-submerged circular cylinders on the non-dimensional
amplitudes of the outgoing waves. It can be seen that wave amplitudes increase with the
characteristic size, since the draft increases with the size. Figure 6.10 reveals how draft
variations of a circular cylinder influence the outgoing wave amplitudes. It is clear that the
amplitudes increase with the draft. The same is true for the case of a vertical ellipse with
varying draft, shown on Figure 6.12. The most evident proof that surge-generated
amplitudes increase with draft is illustrated on Figure 6.11 for half-submerged elliptical


46
where
{jr}={x(,),0"} = {xi'\r,z<-'\a,p,r}
{} = {n,(x-X(0)) xnj = [n,, n2,,-[,(? -Zi,)-n,(y-i'v'1)
-[,(* -r(0))'-n,(z Z(0))],-[,(> F(0>)-2(x X<">)]}
(3.32)
(3.33)
The physical meaning of the boundary condition (3.31) is that at any point on the wetted
body surface, the normal component of the velocity of the body should be equal to the
normal component of the velocity of the fluid at that point. Finally, assuming that the
atmospheric pressure Pa is constant, which is true over large sea areas, and applying the
known expansions into (3.17), (3.18), the hydrodynamic problem is completely linearized
V20 = 0 in the fluid domain (3.34a)
520
r+£ =
dr
ao
dn
dz
= 0
on the free surface
on the sea bottom
ao dXa
= / ^n
Dn di a
, on the wetted body surface
(3.34b)
(3.34c)
(3.34d)
with the exception of a boundary condition on the lateral boundaries, which will be added
later. Thanks to the linearity of the Laplace equation and the boundary conditions in
Equation (3.34), the problem of the response of a floating buoy to irregular waves can be
reduced greatly with the principle of linear superposition of motions. What actually
remains is to study the oscillations of a rigid floating body subjected to a simple harmonic
excitation due to a train of surface gravity waves.


19
going into details the irregular frequencies are the eigenfrequencies of a fictitious interior
for the wetted body surface problem with the Dirichlet condition on the water-body
boundary. As a result the source distribution is not unique, the approximate matrix
equation becomes ill-conditioned; hence the integral equation must fail. In order to avoid
the irregular frequencies, one must introduce additional artificial unknowns and more
conditions to improve the matrix equations, thus making the methodology less
advantageous.
A similar technique was used by Martin and Farina (1997) to solve the radiation
problem of a heaving submerged horizontal disc, where the boundary integral equation is
reduced to a one-dimensional Fredholm integral equation of the second kind.
2,2.5.3. Method of finite-differences. Boundary-fitted coordinates
The classical finite-difference method is based on generating a mesh around the
floating body and using a variety of difference-schemes to express the Laplacian operator
at a particular node with the help of information from some neighboring nodes and as a
function of distance, h, between these nodes. Higher-order schemes increase their
accuracy by involving more nodes, usually at the expense of more complicated algorithms.
On the other hand, the accuracy loss due to an increase in truncation errors, expressed as a
power of the distance h, can lead to physically unacceptable solutions (Yeung, 1985).
While conveniently suited for interior nodes of the fluid domain, the difference schemes
are not easily applicable on curved boundaries. That is why the grid generation process
serves two purposes: first it produces a set of curvilinear coordinates that are specifically
adapted to the geometry in question; and second it provides a crucial numerical


35
modeling. It is evident that oscillatory and wave-flows bring additional challenges for
the direct approach to the nonlinear problem.
The second approach is to utilize a Stokes perturbation expansion, which restructures
the Navier-Stokes equations into a sequence of linear boundary-value problems
formulated for each order of perturbation in terms of the power of a small parameter,
for example wave-steepness. With gradually increasing complexity, any-order solution
can be obtained, provided the solutions of previous-order problems are known.
Predominantly second-order and third-order solutions of the fluid-body interaction
have been investigated, with the noticeably preferred simplifications of axisymmetrical
bodies and symmetrical heaving motion for the case of third-order problems. To
explain this preference it suffices to recall that even for the second-order problems, a
great effort is required to deal with the nonhomogeneous boundary condition on the
free surface of the fluid, as discussed earlier. Nevertheless, the role of perturbation
analysis in fluid-structure interaction studies is evident in providing valuable
information about the significance, contribution, and limits of the higher-order
nonlinear effects, as compared to the first-order solution.
The third approach is to work with the linearized diffraction and radiation theories and
with the powerful principle of superposition, which provides insight into the fluid-
structure interaction problem in irregular seas. Naturally semi-analytical solutions are
much faster than the numerical ones because of the smaller number of computations
needed. Moreover, because of the exact boundary conditions semi-analytical solutions
are principally more accurate as compared to the approximate boundary conditions in
numerical solutions. Of course there are greater restrictions on body-shapes with


137
cylinders. When there is no motion, the cylinders with a vertical and a horizontal semi
major axes displace the same volume of water. When these elliptical cylinders oscillate
with a particular frequency and constant stroke, the greater draft of the vertical semi major
axes displaces a greater amount of water than the horizontal one, and therefore generates
waves with higher amplitudes in surge motion. This argument is in perfect synchrony with
the conservation of mass. It is noteworthy that surge-generated amplitudes increase with
the circular frequency in the frequency range of the experiment, but this is not necessarily
true outside this range.
Fig.6.9 Wave amplitudes. Circular cylinders in surge. Draft= D/2.


dimensionless damping P' dimensionless damping
142
16 Damping in heave. Circular cylinder D=0.114 m. Draft variations.
Fig. 6.17 Damping in heave. Circular cylinder D=0.102 m. Draft variations.


110
5.2.2. Wave Absorption at the Ends of the Tank
The design of the wave absorption at both ends of the tank followed the main idea
of the experiments, which was to obtain the far-field wave amplitudes before and without
any contamination from reflected waves. With a greater distance between the South end
and the wave maker, the wave absorption at the South end consisted of constructing a
steep porous beach with a screen in front of it. With a smaller distance between the North
end and the wave maker, the wave absorption at the North end consisted of placing a
group of three inclined screens in front of the model boundary (Figure 5.2) the purpose of
which was to absorb the energy of the waves. Mesh screen porosity decreased toward the
rear of the North end wave absorber. The wave energy dissipation was designed in
accordance with Jamieson and Mansard's (1987) recommendations. The high porosity
front screen works best for absorbing energy from steep waves, while the rear low
porosity screen absorbs energy from the waves with low steepness. The length of the
wave absorber at the North end was 3.56 m, chosen in accordance with Jamieson and
Mansard's (1987) recommendations for 35%-100% of maximum wavelength. The front
screen had increasing porosity with water depth. The supporting framework of the
screens was designed with minimum frontal area in order to reduce wave reflection.
5.2.3, Wave gauges
The active elements of the capacitance wave gauges are two thin insulated
vertically oriented wires held taut by a supporting rod. The rod was constructed of
stainless steel with a minimum cross-section to reduce flow disturbance. The capacitance
between the two wires varies with the level of submergence. The electronic circuit was


112
depths are less then 2 cm. At these small parameter values, the restoring force of surface
tension becomes significant and the model will experience wave motion damping that does
not occur in the prototype. Since the particular model had water depths above 90 cm and
wave periods above 0.8 sec, surface tension did not affect the experiment results.
5.3, Wave Absorption and Reflection Analysis
A portion of the wave energy was absorbed at the North boundary and a portion
was reflected. As a result of the superposition of the incident and reflected waves, a
partially standing wave is formed which can be presented as
r¡[t) = 4 cos(£c-cot+ £,) + Ar cos(Ax + cot + sr). (5.1)
As will be shown below, having time series of the water elevation collected by two gauges
situated at distances Xj ('¡=3,4) (see Figure 5.1) is sufficient to determine the amplitudes
and phases of both incident and reflected waves travelling between the wave maker and
the end of the tank. Assuming the wave time series was a linear superposition of many
sinusoidal components, the wave data was analyzed in the frequency domain using spectral
analysis. With 212 data points in a record and a Fast Fourier Transform, the energy
spectrum was estimated and the primary harmonics filtered out. It was found that the
principle of linear superposition was well justified over the entire frequency range by using
the first two or three harmonics (co,2a>,3co) to match the wave time series. This fact is
illustrated in Figure 5.3, which shows a typical stationary-mode-wave-time series and
power density spectrum for one of the experiments. Figure 5.4 shows an almost perfect fit
between


100
The number of terms N nedded is practically determined from the far-field amplitude
oo
power series (4.91), namely = £aD(n). Knowing D(n) exactly from (4.99), N
n~ 1
can be determined by specifying a desired accuracy in advance, thus cutting off all terms
whose absolute value is less than the chosen accuracy. For example if the desired accuracy
is defined as 1CT16, then only the first A terms n=l,2, ...,N such that
\anD(n)\>\Q-16 ,n=l,2,...,N (4.232)
are needed to be kept, and all other terms n=N+l,N+2,N+3, ...can be omitted
\anD(n)\ < 10-16 n=N+l,N+2,N+ 3,... (4.233)
In summary, solving the linear system (4.231) gives the unknown coefficients an,
which when substituted in (4.91) produce the far-field wave amplitudes which is the main
goal of this semi-analytic technique. As explained at the end of chapter 3, knowing the far-
field amplitudes is sufficient to find the hydrodynamic added mass, damping coefficients,
and the exciting forces, and consequently to solve the floating buoy dynamics problem.


60
1 ^
Upa H ftp* H = - J Rpa{t) Sin {C0t)dt
(3.78)
Rpa{t) is called the retardation function, and is obtained through a Fourier transform of
Xpa(co). Therefore the added mass is related to the far-field amplitudes, (3.74, and 3.78).
As mentioned in section 2.3, a remarkable Haskind theorem relates the exciting
force to the far-field amplitudes in the following manner
(3.79)
where (3.63) has been applied. Since = is true on Sb, then (3.79) becomes
dn dn
Fa -ia>P
(3.80)
Next, substituting the asymptotic forms (3.72) into (3.80) results in
Ff = -2pgC ACl~,
(3.81)
thus expressing the exciting force in terms of the far-field amplitudes. Upon substituting
(3.81) into (3.59), the matrix form of the dynamics equations become
[[C]- co\[M]+[/,])-; Defined as a ratio between the amplitude £,a of displacement in the generalized direction
a and the amplitude of the incident harmonic wave A, the frequency response function
and the corresponding response amplitude operator are
Ha{a>) = SJA, and RAOa{co) = \Ha{(of. (3.83)


34
agree with experiments within the Keulegan-Carpenter range between 1 and 2.5. The
third group of numerical methods is.based on the hydrodynamic stability approach, in
which the formation of vortex shedding is viewed as an instability of the flow in the wake.
As seen in Section 2.2.1, the oscillatory flow becomes unstable above a critical KC
number due to spanwise-periodic vortices the so-called Honji instability. After modeling
the phenomenon numerically, Zhang and Dalton (1995) have obtained a definite 3D
behavior regarding the variation of vorticity, and found that the sectional lift coefficient
has a strong spanwise variation.
2.5, Some Thoughts About the Current State of Knowledge
For a long time scientists and engineers have tried to approximate natural
phenomena with different mathematical formulations, brought into life with a variety of
numerical techniques, validated or rejected through experiments. Keeping in mind
Poincares words: Mathematics can never tell what is, it can only say what would be if,
the current state of knowledge about floating bodies is a direct result of all these trials and
errors, and can be classified as going simultaneously in three main directions:
The first approach is to directly solve for the nonlinearities of the Navier-Stokes
equations with massive computing power. With the help of supercomputers and
mature numerically-stable methods, the existing mathematical models have succeeded
in increasing our appreciation and understanding of some natural phenomena, in
particular ranges of Reynolds and Keulegan-Carpenter numbers. Other Re and KC
ranges of practical interest require different or improved mathematical and theoretical


dimensionless damping
144
Fig.6.18 Damping in heave. Elliptical cylinder with horizontal major axis.Stroke variations
Fig. 6.19 Damping in heave. Elliptical cylinder with vertical major axis. Stroke variations.


57
If (j)a =(j)n and p = j are two radiation velocity potentials, than the surface integral at
the lateral boundaries vanishes due to the boundary condition (3.54e). This results in
JJ
d ^ 8n dn
iS = 0, or
(3.63)
St,
(3.64)
Therefore, the added mass (3.56), and damping matrices (3.57) must all be symmetric,
regardless of whether the body is symmetrical or not, due to (3.64), namely
M/Sa^api an^ ^-pa=^ap (3-65)
The law of conservation of energy flux requires that the average rate of work done
by the oscillating buoy on the fluid over a period, denoted with Wbuoy, should be equal to
the energy flux far away from the buoy denoted with W,
away
buoy IT away
T 2 <+.r .
W = j Wdt.
where
(3.66)
(3.67)
From (3.58), Wbuoy becomes
(
= -£ F* X. = Y, I /V. xf x.+V x, x.
a a p V 7
(3.68)
Because of the symmetry (3.65) the first term of (3.68) can be written as


96
r3 = kRe(ac0),
(4.206)
(2 + ^)lm(c0)
K
and
r5 = Im
K
f
v
(4.207)
(4.208)
4.3.4,2.3. Real part of the right-hand side: Exactly half-submerged ellipse
While surge and heave (4.179), (4.180) have the already investigated form,
= the pitch mode's (4.181) complex velocity has an additional term proportional
dC C
to
dwp cx
-3
(4.209)
where c¡ is a constant. The contribution of (4.209) to the RHS is
= c,
01
+ l) r? 1 ikcx ikcxA
' f-ic +H?r_lrr_?_
K
(4.210)
In the cylindrical coordinate system E, = l.e (4.210) transforms to
RHS c,
01
(ew+Ae-w) { e-** ikcAe
-r+ \d\ +ikcxe,g +1
el'e-Xe'9 2 1 3
3iO \
(4.211)
RHS c.
01
(e + Ae->) + +ifc| _!)+_,)
1,9
Aew
(4.212)
and


167
(6.16)
where
/l(3) = pgCg{craCrp + <2*Qp j is the solution given by (3.73), and the addition due to
higher harmonics is
cu [a (lC0)a U<) + a (ico)af ch]
(6.17)
In (6.17) c¡j are the integration constants derived from (3.71). As explained in section
3.5, the diagonal terms of the damping matrix are non-negative, which means that
- > 0
(6.18)
The addition due to higher harmonics is positive. Since the theoretical model, which is
based on linearized radiation theory can not predict these higher harmonics, it is
understandable that the experimentally measured damping is less than that computed by
the model, because of (6.18). Nevertheless, the general agreement between the measured
and computed hydrodynamic coefficients is sufficient for using the following engineering
approach when investigating floating body dynamics:
a) Use the linearized radiation theory to obtain the hydrodynamic damping and
added mass as functions of wave frequency and body geometry and then
b) Use the derived functions of the hydrodynamic coefficients to analyze higher
order nonlinear effects on design forces and allowable amplitudes of
oscillation.


7
2.2. Classification
2.2.1, Large and Small Bodies
There are at least three relevant length scales in the interaction between waves and
a fixed or floating body. They are the characteristic body dimension B, defined as the
predominant horizontal size of the body projection onto the vertical plane of the wave
front, the wavelength L=2 n/k (where k is the wave number), and the wave amplitude A.
If B, for a fixed vertical cylinder, becomes relatively large then the presence of the cylinder
will disturb the incident wave pattern. As the incident waves impinge on the cylinder,
reflected waves move outward. In the shadow zone, on the sheltered side of the cylinder,
wave fronts are bent around the cylinder and thus form diffracted waves. The combination
of reflected and diffracted waves is usually called scattered waves, but the process itself is
generally termed diffraction. The three scales, B, L, and A, may form two physically
meaningful ratios. The first one is the diffraction parameter B/L an important
dimensionless variable relating to the intensity of the scattered waves. Often used with the
same purpose is kB derived from the multiplication of the diffraction parameter with the
constant 2 n. The second ratio is A/B, which in deep water is proportional with the same
constant 2n to the Keulegan-Carpenter number, KC, defined as
du
(2.1)
KC =
UJ_ _Udx
B
du
dt
where the wave period is T, and the magnitude of the horizontal velocity of a harmonic
progressive wave is


Fig.5.1 Longitudinal and transverse sections.
SURGIR
V '
-4>a
0.1 90m
0.0994
Section A-A
O 15m
FIG. 5.1.e
Steel
Section A-A
o
o
FIG. 5.1 .c
Side View
0.568m
o
o
to
3
-7Â¥
EXPERIMENT: A DYNAMIC RESPONSE MODEL
FOR FREE FLOATING BUOYS SUBJECTED TO WAVES
Gauges. Longitudinal section.
Gauges. Transverse section.
Cylinder. Side view,
Aluminurn cylinder (cicular). Section A-A.
Steel cylinder (^circular). Section AA.
Wood cylinder TelipticaQ. Section AA.
Wood cylinder (elipticall. section A-A,
JL.L
-5-1-f
-5-JU
107


125
Fig. 5.12 Side view of the oscillating cylinder and its end plates.


66
(4.8b)
As pointed out in Haskind (1973), the combination (4.8a) was introduced for the first time
by Keldysh in 1935. Comparing the imaginary parts of (4.8b) and (4. lb), the boundary
condition on the air-water interface can be written as:
(4.9)
.,on 2=0
Im {f(y)} = 0
By virtue of the Schwarz Reflection Principle and the boundary condition on the air-water
interface (4.9), it is possible to continue f(y) in the upper half-plane. As a result of the
analytical continuation the values off(y) in the upper half-plane will be conjugate
imaginary of the mirror off(y) values in the lower half-plane. In this manner f(y) will be
holomorphic in the whole complex plane except for the points on the wetted buoy
surface^ and their mirror images denoted by.V (Figure 4.2). Thus, on an abstract
mathematical ground, the problem has been extended from the lower half-plane to the
entire complex plane.
Fig. 4.2 Wetted body surface and its mirror image
Iff(y) were known, then the complex velocity potential would simply be the solution to


22
2.2,5.5. Boundary integral equation methods fBIEMs) based on a simple sources
distribution over the total fluid domain boundary
Developed for the numerical calculation of linear potential functions for heaving
axisymmetric bodies by Ferdinande and Kritis (see Vantorre, 1986, 1990) the philosophy
of this method is to confine the problem into a finite cylinder, which surrounds the heaving
body. Since both the body and the motion are axisymmetrical, the problem is reduced and
solved in a two-dimensional rectangular fluid domain. The domain has the following
boundaries: the vertical axis of symmetry of the body, half of the vertical cross section of
the wetted body surface, the vertical cylinder wall, the flat bottom, and fluid surface
between the body and the cylinder. In order to justify the confinement of the problem into
the cylinder, and thus the boundary element method, a modified radiation condition has to
be satisfied on the vertical cylindrical wall. Now the whole boundary is divided into
discrete panels, on the center of which the velocity potential is considered constant. As a
consequence of Greens theorem a potential on a point on the boundary is expressed as an
integral of the velocity potential and its normal derivative over the whole boundary.
Eventually the problem is reduced to the solution of set of N linear algebraic equations,
where N is the total number of panels.
Some of the boundary integral methods are very effective for first-order diffraction
and radiation problems. However, these methods are not suitable for second-order
diffraction and radiation problems (Taylor and Hung, 1987). In this case the integrand is a
functional of the first-order potential and the Greens function. Because of the slow
convergence of the integrals, a large number of values of the first-order potential must be
evaluated, and this is not easily achieved using boundary integral methods.


11
Re = ^L = KC (2.6)
v vT
where the kinematic viscosity is v = 10~6[m2 / sec]. When for example Be[1,37[m], Te
[2,20]\sec], and tanh(kh)=l for the case of deep water, then the Reynolds number
becomes large ReO(103). As evidenced in Sumer and Fredsoe, 1997 with Figures 3.15,
3.2, and 3.16, when KC<2 and ReO(103), the flow will not be separated; when KC
approaches 2, there will be separation, but not very extensive. This analysis suggests that
the flow about a fixed vertical circular cylinder in the large-body-diffraction-regime
B/L>0.2 (or B/L>0.1) may be analyzed by the potential theory, since no flow separation
occurs KC<2, and since the viscous effects are negligible ReO(103).
In the case of a floating body, there is another hydrodynamic force due to the
body motion in addition to the wave forces exerted on the body under the fixed condition.
According to Sawaragi (1995), the generation mechanism of hydrodynamic forces due to
body motion can also be classified briefly into two regimes in the same way as the case of
the fixed body.
2.2.2. Deterministical and Statistical Approaches
A phenomenon, which is changing with time, can be described deterministically or
statistically. In the deterministic approach all the variables are functions of time and
known at any moment of time, usually after solving differential and integral equations. In
the statistical approach the explicit time dependence is not considered. A variable is
usually known as an average or as a probability of occurrence. The time history is
unknown and therefore the variable is unknown at any moment of time. The problem is


63
Formulated in (3.54), the radiation problem is stated as a two-dimensional Laplace
equation with a complete set of boundary conditions:
d\-+8\=0
etc2 dz2
, in the fluid domain
(4.1a)
dK %=o
dz g
, on the free surface SF
(4.1b)
^-=0
dz
, on the sea bottom Ba
(4.1c)
dta
It
on
, on the wetted body surface- Sb
(4-Id)
hm{ tc
( dx J
, waves outgoing at infinity.
(4.1 e)
As explained at the end of chapter 3, the practical mathematical problem to solve will be
to find the asymptotic solution of (4.1), particularly the far field amplitudes. For the
convenience of mathematical manipulation, time is removed from the problem by using the
exponential time dependence (3.51), which restated is
{,X,QT = R e({L,77}V;c") (4.2)
With the imaginary unit j = V-1, the wave profile is
£=A cos(kx at) = Re(^e>(fcf- jj=AeJla (4.3b)
As stated before, for brevity the sign Re (the real part of) will be omitted, but accounted
for iathe final results. In the two-dimensional case, the physical meaning of the
Sommerfeld radiation condition (4.1e) is that the generated waves are propagating
outward from both sides of the oscillating body. Therefore, the asymptotic expression of


145
Fig.6.20 Damping in heave. Elliptical cylinder with horizontal major axis. Draft variations.
Fig. 6.21 Damping in heave. Elliptical cylinder with vertical major axis. Draft variations.


APPENDIX
PITCH MODE OSCILLATIONS
Since pitch mode experiments were not conducted and since stability is strongly
dependent on the metacentric height (usually small in pitch), the added mass, damping,
and FRF in pitch mode for these two examples, are given in the appendix. It is assumed
that keeping the center of gravity below the instantaneous center of buoyancy provides
stability. This is usually achieved by placing ballast at the bottom of the floating body.
The circular frequency, damping, and added mass are made again non-dimensional with
the help of the breath of the section at the waterline (6.1), (6.2), and (6.3).
With reference to the first numerical example (Figure 6.25), Figures 6.43, 6.44,
and 6.45 present the damping, added mass and FRF in pitch motion for a circular cylinder
with various drafts. It can be seen from Figure 6.43, that damping in pitch increases when
the draft-to-radius ratio increases. An intuitive explanation for this phenomenon can be
offered from the wavemaker-theory point of view. Consider the limiting case of
substituting the underwater shapes with a flap-type wavemaker in the form of a vertical
plate with the same draft, extended above the water surface, and hinged at the center of
gravity of the floating body. Since the center of gravity is at a fixed distance from the
bottom point, an increment in draft means an increment of the segment of the plate
extending from the center of gravity to the waterline. As long as there is no mooring line,
the center of rotation coincides with the center of gravity. If all three draft variations are
168


155
With reference to the first numerical example (Figure 6.25), Figures 6.31, 6.32,
and 6.33 present the damping, added mass, and FRF in heave motion for a circular
cylinder with varying draft. It can be seen from Figure 6.31, that damping in heave
decreases with increasing draft. An intuitive explanation for this phenomenon can be
offered from a wavemaker-theory point of view. The greater the draft, the less the
waterline breadth B, the deeper the location of the displacement that results from an
increment in vertical motion. If all three draft variations are to oscillate with constant
heave amplitude, the less the waterline breadth, the less the displaced water volume, the
less energy input is required, and the less the damping, since damping characterizes the
energy given up from the body (3.69). The closer the underwater shape is to the limiting
case of a horizontal plate, the closer the displacement is to the surface where the
propagating waves are formed, the greater the displaced water volume, the more energy
input is required, and the greater the damping becomes. Figure 6.32 shows that the added
mass in heave motion decreases with increasing draft. As explained for the heave
damping, the greater the draft is, the less the displaced water volume is, the less the local
pressure gradient is, the less the added mass becomes. Figure 6.33 shows that the peak
value of FRF in heave motion increases when the draft increases. To explain this fact, it
suffices to consider the heave-only form of the FRF from (3.84), namely
(6.9)
where C is the spring constant, and con is the natural circular frequency, defined as


10
rH_
U
max
= 0.14tanh(£/i)
(2.4)
Therefore, from (2.4) and (2.3) the relationship between the largest Keulegan-Carpenter
number and the diffraction parameter, shown in Figure 2.1, is given by
(2.5)
L
and is a simple hyperbolic curve. Isaacson pointed out that the critical value of the
diffraction parameter that roughly separates large from small bodies is B/L=0.2, because
the curve of the largest KC (without wave breaking) does not exceed 2 for the range
B/L>0. On the other side, flow separation should be more important than the diffraction
when KC>2, which according to Figure 2.1 happens when B/L>0.2. At the same time
the condition for diffraction B/L>0.2 and (2.4) imply that H/B<1 and that the drag forces
will be small since wave amplitude is less than the body dimension. It should be noted that
the Isaacson criterion is only true for a fixed vertical circular cylinder; for any rectangular
cylinder flow separation inevitably occurs and its effect might not be negligible for large
(B/L>0.2) bodies. As seen in Figure 2.1, for a wave with steepness one half of the
maximum steepness (0.5H/L), KC does not exceed 2 for the region B/L>0.1. This fact
suggests that the generally accepted (Sawaragi, 1995; Sumer and Fredsoe, 1997) critical
value of the diffraction parameter B/L=0.2 is not a fixed value it may vary even for
rounded bodies. It is well known that the flow regimes about a fixed vertical cylinder
depend not only on the KC number but also on the Reynolds number defined as the ratio
between inertia and viscous forces


154
O)
OI i 1 1 1 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
dimensionless circular frequency
Fig. 6.30 Damping, added mass, and FRF in surge. Circular cylinder.


68
into the solution of the complex velocity potential (4.10) results in
w(y) = +4, + J/(Oe^dCI
(4.13)
where Ai+iA2 is an integration constant, C, is a dummy variable, and the integration is
taken over a curve lying in the lower half-plane. Consequently, when y = (x,z) > (oo,z),
the asymptotic expressions of the complex velocity potential are
limw(^) = lim(^ + />0 = {A + iA2)e~,ky = + iA2)e~'kx+k,>z
y>+oo y>+co
limvv(^) = lim {+iy/) = (Bx +iB2)e~,ky = (2?, +iB2)e~'k,,x+k,,z
(4.14)
y->- co
y->-
where
Bx + iB2 = A +iA2 + Jf(y)e'kydy (4.15)
Taking the real part of (4.14), the corresponding asymptotic expressions of the velocity
potential are
lim^(^) = ekz(A, cosk0x + A2 sin&0x)
_y>+co
Ym\(f)(y) = ek2{Bl cos k0x + B2 sin^0x)
(4.16)
y-y~ oo
From the linearized free surface dynamic boundary condition at z=0
<=~
1 c
g a
z=0
g
(D
= 7¡e~JWI
J=0
77 =
( Vi A
HOLy
\g )

(4.17)
Substituting (4.16) into (4.17) and comparing with the asymptotic wave profile (4.4),
co1
results in the following deep-water (k = k0 = ) relationship:
g


Since the average rate of work done by the body on the fluid (energy transmitted to the
fluid by the body) is non-negative, the damping matrix is positive semidefinite. As a
corollary, all diagonal terms of the damping matrix are non-negative, since
(3.73b)
It is noteworthy that the same as (3.72) asymptotic behavior is valid for s, in accordance
with the lateral boundary condition (3.52d), except that <2* will be replaced by another
term with a dimension of length. It is also noteworthy that the damping and added mass
matrices are proportional to the real and imaginary parts of the so-called restoring force
matrix. This means that knowledge of the damping coefficients is sufficient to determine
the corresponding added mass coefficients and vice versa. Such relations, known as
Kramers-Kronig relations, are valid for all modes of motion regardless of forward speed
(currents, ships). Having been derived in a universal way by Ogilvie, they read:
2 rV(^)-VH
(3.74)
(3.75)
oo
(3.76)
Another way to find the added mass, knowing the damping as a function of the circular
frequency, is Hoofts approach (1982) of using the so-called Bode relations, which for
water waves correspond to the Kramers-Kronig relations.


6
success has been achieved in increasing the knowledge of these hydrodynamic properties,
as evidenced in a comprehensive survey of the available knowledge made by Weinblum
and St. Denis in 1950. Despite the insufficient understanding of the effects of
hydrodynamic mass and damping and the omitting of hydrodynamic coupling, the greatest
limitation they felt at that time was the restriction to regular waves. Weinblum and St.
Denis quote a saying of Lord Rayleigh: The basic law of the seaway is the apparent lack
of any law. Only three years later St. Denis and Pierson (1953) wrote in a paper that In
a broad sense the laws of nature are Gaussian, thus introducing the statistical description
of sea waves and body motions to the floating body studies. The concept of spectral
analysis of sea waves was defined by assuming the wave elevation as the sum of a large
number of simple sinusoidal waves, each having its own length, height and direction.
Consequently, the floating body motion is considered the sum of its responses to each
individual wave component. Since the phases of those wave components are randomly
distributed only statistical information can be obtained. The superposition of harmonic
waves and harmonic body responses meant enormous stimulation for the floating body
studies. In 1957 Korvin-Kroukovski and Jacobs introduced a strip theory that builds on
two-dimensional solutions to get three-dimensional effects for elongated floating bodies.
At that revolutionary time, 18 years after the invention of the greatest scientific tool of the
20th century the computer whose father was the American from Bulgarian descent
Dr. John Atanasoff, all the basic elements were available to obtain engineering solutions
for a coupled hydrodynamic motion, at least for heave and pitch.


dimensionless damping 'as dimensionless damping
134
5 Damping in surge. Elliptical cylinder with horizontal major axis. Stroke variations.
Fig.6.6 Damping in surge. Elliptical cylinder with vertical major axis. Stroke variations.


119
x = A sin(atf).
As a direct solution of this simplified system the added mass coefficient is
F cos(>)
(5.14)
F =
co1 A
-M
(5.15)
and the damping coefficient is
F sin(£')
A = -
coA
(5.16)
In the present experiment, higher harmonics 2 Figure 5.3. This fact suggests that regardless of whether the force required to sustain the
motion includes higher harmonics or not, the motion of the cylinder should account for
these higher harmonics. Consequently, ones the motion (5.14) or the forcing in the right-
hand side of (5.13) is changed with an addition of higher harmonics, (5.15) and (5.16) can
no longer be used to determine accurately the added mass and damping coefficients. In
summary, this experimental procedure (5.13-5.16) is not capable of detecting the presence
of higher harmonics, and therefore gives an approximate estimate of the added mass and
damping coefficients.
The procedure used in the present experiment to obtain data for the far-field wave
amplitudes again gives an approximate solution for the added mass and damping
coefficients as it will be proven in the next chapter 6. The only difference from the Vugts'
procedure (5.13-5.16) is that it detects the higher harmonics, and that it gives an idea for
the degree at which the nonlinearities are involved in this fluid-structure interaction.
Pictures of the experimental set-up are given on Figures 5.7 through 5.14.


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Fig. 5.10 A closer side view of the Scotch Yoke motion mechanism.


36
semi-analytical than there is for numerical solutions. One of the main advantages of
the linearized theory is that one can explore in real time the influence of variation of
different parameters like draft, shape, size on the hydrodynamic properties, forces, and
hence fluid-body dynamics. Another advantage is that the linearized theory provides
the easiest way to analyze and comprehend the physical concepts, and meaning and
significance of observed natural phenomena. Ones seeing the big picture, further
improvements can be made, if desired and feasible, with the help of the other two
approaches.
In a historical retrospective, it is true that the three main approaches have had
great impact on the floating-body studies. Having a common goal, they have influenced
and stimulated each others evolution. All three approaches have their own unique
advantages and disadvantages and areas of applicability.
In the present work, a new semi-analytical method is proposed for solving the
dynamics of free floating two-dimensional horizontal cylinders, of various shapes in heave,
surge and pitch. The method is based on linear radiation theory and is intended to be the
first step in the development of a mathematical model and computer program for
predicting the response of an axisymmetric, three dimensional, free floating drifter buoy
subjected to waves.


122
Fig. 5.9 Side view of the Scotch Yoke
motion mechanism.


146
Figures 6.20 and 6.21 present the influence of draft variations on damping caused by
heave oscillations of the elliptical cylinder with horizontally and vertically oriented semi
major axes. In these figures, the measured damping coefficients are shown for the same
three draft-to-vertical semi-axis ratios of 1, 1.2, and 1.4, and strokes of 27 and 39 mm.
Again, there is some scatter among the measured points for all draft variations. In Figure
6.20, the theoretical solution slightly overestimates the measured damping for draft-to-
vertical semi axis ratios of 1 and 1.2 in the middle of frequency range, while there is good
agreement for draft-to-vertical semi axis ratios of 1.4. In Figure 6.21, there is good
agreement between the theory and measurements for all three draft-to-vertical semi-axis
ratios. As in the case of a circular cylinder with varying draft, the heave damping
decreases with draft for elliptical cylinders with both horizontally and vertically oriented
semi major axes. Again, the heave damping curves are steeper for the horizontal than for
the vertical semi major axis, when plotted versus increasing circular frequency.
The amplitudes of the outgoing waves generated by heave motion are presented in
Figures 6.22, 6.23, and 6.24. They are shown in their non-dimensional form of amplitude-
to-stroke ratio versus the dimensionless circular frequency, introduced in (6.1). Figure
6.22 reveals how draft variations of a circular cylinder influence the outgoing wave
amplitudes. It is clear that the amplitudes decrease with increasing draft. The same is true
for the cases of elliptical cylinders with vertical and horizontal semi major axes with
varying draft, shown on Figures 6.23 and 6.24. Since the waterline section decreases
when the draft increases, the volume of water activated by heave oscillations also
decreases which leads to smaller wave amplitudes. A clear evidence for this explanation is
that the elliptical cylinder with horizontal semi major axis produces higher amplitudes than


38
Oz is positive upward. The origin O is the intersection of the centerline of the floating-
body-section and the still water surface. In a wide variety of gravity wave problems, the
variation of water density is insignificant over the temporal and spatial scales of
engineering interest. The relationship between water density and pressure is given by
1 Dp _1 DP
p Dt E Dt
(3.1)
where (.) = (.)+ u.V(.)
Dr dr w
is the total derivative in space and time, u = (u,v,w) is the
fluid particle velocity, P is the pressure, p is the water density, and E = 2.07 x 109 [Pa] is
the water bulk modulus. Since an increase in pressure of 1 MPA results in 0.05% change
in water density, for practical purposes the fluid can be considered incompressible.
3.1.2. Governing Equations and Definitions
Using Meis notation (1989), the motion of fluid around a body is governed by the
fundamental conservation of momentum law or the Navier-Stokes equations
(d ^
(P \
4-u.V u = -V
+ gz
{dt )
l p ;
+ vV2u
(3.2)
and the conservation of mass law or continuity equation
V.u = 0 (3.3)
where v= 0.01 [cm2/sec] is the kinematic viscosity, and g=9.814 [m/sec2] is the
acceleration of gravity. Dots represent the scalar multiplication of two vector quantities
(Batchelor, 1967). Defining the vorticity vector as the curl of velocity vector
fi(x,/) = V x u(x,f) (3.4)


150
Fig. 6.25 First example. Circular cylinder. Variations in draft. Center of gravity
- placed at a fixed distance from the lowest point on the cylinder.
Fig. 6.26 Second example. Elliptical cylinders. Constant waterline breadth.
Variations in the vertical semi-axis. Center of gravity
- placed at a fixed distance from the lowest point on the cylinder.


17
elementary functions multipoles which satisfy the Laplace equation, on the level of the
center of the circle. The combined kinematic-dynamic free surface boundary condition is
satisfied with a proper combination of the multipoles and their mirror images with respect
to the mean water surface, which also satisfy the Laplace equation. In order to represent
the outgoing waves at infinity, additional potential is introduced, that has different forms
for symmetric (heave) and anti-symmetric (surge) motions respectively. With the help of
unknown coefficients, both wave and local-disturbance multipoles are combined as an
infinite sum into a velocity potential, which automatically satisfies the Laplace equation,
the free surface boundary condition, and the radiation condition. The wetted body surface
boundary condition is used to determine the unknown coefficients, after truncating the
infinite series at a finite number of terms, and using the point collocation method. A
limitation of the multipole expansion technique is that the general existence of the
expansion is very difficult to justify, or in other words the convergence of that expansion
has been proved only in the case of a semi-submerged circle in the region kB<1.5 (Ursell,
1949). Numerically, the convergence was found to vary with the depth of submergence,
with the fastest convergence for a fully submerged circular cylinder. The most demanding
case was found to be when the circle was just piercing the surface. In that case the body
intersects the free surface at an angle of zero degrees, which makes the linearized
boundary value problem mathematically unstable.
The multipole expansion can be applied to the analysis of more general bodies
through a coupled method, called the BIE-BMP method by Taylor and Hu, 1991. To
solve for the velocity potential, one can use a boundary integral expression in a domain
close to the body and match it with the multipole expansion outside this domain. Using


65
of an ideal fluid, both the velocity potential (¡>(x,z) and stream function y/(x,z) exist. In the
whole fluid domain they have continuous first and second partial derivatives and satisfy
the Laplace differential equation. Therefore by definition (Solomentsev, 1988) both
velocity potential the complex plane these two harmonic functions define the well-known complex velocity
potential
w(y) = 4>(x,z)+ i y/(x,z) (4.6)
and are related to each other as conjugate functions through the Cauchy-Riemann
conditions
d dy/
dx dz
d dz dx
(4.7)
At any point inside the fluid domain the complex velocity potential:
(a) has unique value
(b) has continuous derivatives of all orders
(c) satisfies the Cauchy-Riemann conditions.
Therefore by definitions (a,b,c) the complex velocity potential is a holomorphic function
(Milne-Thomson, 1950). Utilizing the property of holomorphic functions that a
combination of derivatives and integrals of holomorphic functions is also holomorphic, the
following holomorphic function is introduced.
f(y) =
^+*Ay)
dy
,in the fluid domain without St (4.8a)
In terms of its real and imaginary parts, / (y) can be written as


166
For all circular frequencies of interest, the number of terms necessary to provide the
absolute convergence did not exceed 9. To be on the safe side, the usual number of terms
used in the computer program was 24.
6.5. Conclusions
In general there is good agreement between the computed and measured
experiments and results for surge and heave motion. The differences are due to
experimental error (discussed in chapter 5), to numerical error, and to the theoretical
limitations of the mathematical model. As shown in section 3.5 with equations (3.66-
3.69), the damping is associated with the energy given up by the oscillating body.
Nonlinear effects of the fluid-body interaction were observed in the results of the
experiments by the presence of the second and third harmonics, shown in Figure 5.3. By
virtue of the fact that these higher harmonics carry part of the energy given up by the
oscillating body, they unavoidably constitute a portion of the damping. This can be
qualitatively estimated by deriving the damping coefficients (3.73), from (3.71) with the
following form of the radiation velocity potential
(6.15)
a
which includes the higher harmonics. If the asymptotic solutions of the higher harmonics
a2(lco), (j)al{2co),... as well as their far-field amplitudes £72(2co), £7*3(3 known, than the damping coefficients would be


20
transformation that allows difference-schemes to be applied in a more geometrically simple
computational domain. The boundary curves from the original physical domain have to be
transformed into coordinate lines in a logical domain of mapped variables, which facilitates
the implementation of Neumann type boundary conditions. As pointed out by Yeung
(1985), the coordinate transformation and the physical solution may be solved
concurrently in the same sweep with a proper adjustment for the boundary conditions.
Once the general algorithm is developed, it can be applied to different floating body
geometries with a change of boundary coordinates. The change of boundary coordinates
is much more involved for three-dimensional bodies.
2.2,5,4, Finite and hybrid element methods
The strength of this class of methods is its ability to handle curved boundaries.
The main idea is to map isoparametric boundary surface elements into local squares,
triangles, etc., on which one can calculate every elements contribution to the field and
boundary properties. The unknown function consists of a set of nodal values and a set of
predefined shape functions, chosen to satisfy certain continuity requirements across the
elements. The requirements depend on the differential order, and the boundary conditions
(Yeung, 1985). The determination of the nodal unknowns relies on a global, integral
criterion. A brief description of the hybrid element methods, as given by Mei (1989), will
be given as a generalization of the finite element methods of Newton (1974, 1975). The
main idea of the hybrid element method is to employ a finite-element approximation in an
imaginary cylinder, which extends from the sea bottom to the water surface and surrounds
the body, with an analytical representation outside of the cylinder. Thus the finite-element


52
where Va is the time amplitude of the generalized body velocity; is the time amplitude
of the velocity potential; co is the wave circular frequency; and the imaginary unit is
defined as / = 4~T. At this point both the velocity potential O and its time amplitude
have the same dimensions, which is [Length2/Time], For the sake of brevity the notations
for the real parts will be omitted and only the complex forms will be used instead, but only
the real parts have physical meaning. Due to the linearity of the system (3.34) two
contributions of a different nature can be separated entirely (see 3.51c). Therefore two
subproblems arise, each of which is more tractable than the complete problem:
(a) radiation sabproblem. the rigid body oscillates harmonically in an
otherwise undisturbed body of water, thus generating waves which propagate or radiate
away from the body. Physically the corresponding radiation velocity potential (f>R is a
direct result of the motion of the body and should be proportional to this motion.
Therefore <¡>R = ^Va(f>a where the summation is over the elementary components of the
a
body velocity 6 in 3D-space, and 3 in 2D-space. <¡>a has the dimensions of [Length], and
stands for the velocity potential caused by a body oscillatory motion with unit velocity in
the a-direction: for example in heave, surge, or pitch.
(b) diffraction subproblenr. the forces on the rigid body, fixed in space, are
caused by a train of harmonic incident waves. The velocity potential for the incident
waves is (f)1, while the velocity potential for the scattered waves (defined in Section 2.2.1.
as sum of reflected and diffracted waves) is denoted by s. Define the velocity potential


CHAPTER 3
FORMULATION OF THE PROBLEM
3.1. General Description of the Problem and its Simplifications
The most general formulation of the problem of the dynamic response of a free
floating body subjected to waves is to pose a dynamic equilibrium of forces and moments
in and on an elastic body freely moving in the air-water interface. The focus of the present
work will be on the floating body motions due to external loads, which act on the
underwater part of the body. Therefore two restrictions will be made right from the
beginning: first the body is considered rigid; and second the direct influence of the air
environment on the body is negligible. As long as no structural or vibrational problems
are to be dealt with, the first restriction can be made without any hesitation. The second
restriction is based on the fact that the density of air is roughly one one-thousandth of the
density of water. Provided that the above-water part of the floating body is close to the
water surface, and excluding strong winds during adverse weather, it is clear that for most
practical problems aerodynamic forces may be neglected with respect to hydrodynamic
forces.
.3,1.1. Incompressible Fluid Assumption
A coordinate system Oxyz or simply x=(x,y,z) which is fixed in space will be used
in this analysis. The Oxy plane coincides with the still water surface, and the vertical axis
37


73
y = -ih + Re', Sb\

y = ih + Re' and Sb:

Fig. 4.3 Conformal mapping of more than half-submerged circle
If the conformal mapping
y = f(£) = ihl+a£hl and
dy df(0
dC d = 0 ,
(4.30)
(4.31)
where h¡ and h2 are real and a is complex, is used to transform Sb (4.28) into the lower
half of the unit circle C: C, l.e'0, where 0 e[-7r,0], then

(4.32)
x + ?- K
(4.33)
a = Re'1, and
(4.34)


CHAPTER 5
EXPERIMENTS
51. Purpose of the Experiments
The main purpose of the experiment described in this chapter was to obtain data
for the far-field wave amplitudes for partially submerged floating cylinders of circular and
elliptical shape and thus to verify the Semi-analytic technique (SAT) introduced in
Chapter 4. The necessity of these experiments is a consequence of the lack of
experimental data in the published literature for elliptical shape cylinders and for more than
one half submerged circular cylinders.
5,2, General Set-Up
The experiments were carried out in a wave tank at the Coastal Engineering
Laboratory at the University of Florida. The tank has a North-South orientation,
rectangular cross-section with glass sides and bottom, and its length, width, and depth are
27.80 m x 0.572 m x 1.22 m. Since the drafts of the cylinders for the various modes of
motion were different, the water depth varied between 0.908 m and 1.04 m. At about one
fifth of its length from the North end of the tank a bridge was constructed across the top
of the tank. On this bridge a Scotch Yoke motion mechanism (Mabie and Ocvirk, 1963)
was installed for imparting pure sinusoidal motion in heave and surge to the cylinder with
106


64
the wave profile at infinity must be:
lim £ = AejXkx-M) = (AeJtx)e'JM = lim rje^, (4.4a)
lim tj = A+elkx = (Vaal )ella (4.4b)
where A+ and A. are the asymptotic expressions of the wave amplitudes, and £7* are the
far-field amplitudes introduced in (3.73).
4.2. Main Idea Behind the Semi-Analvtic Technique (SAT)
Inspired from a Haskind idea (1973), a new analytical solution of (4.1) will be
given for the case of partially submerged floating circular and elliptical cylinders with
varying equilibrium drafts. As will be shown below, the analytical solution is in terms of
infinite power series. Therefore, a finite number of terms must be taken to solve it
numerically thus the prefix semi- and the name Semi-analytic technique (SAT). For
the sake of clarity and completeness, some mathematical definitions and formulations used
in the analysis are presented first. Considering a two-dimensional (2D) wave motion in a
right-hand complex plane Oxz, the real axis Ox is defined to represent the still water
surface and the imaginary axis Oz to be perpendicular to the still water surface and
positive upward (Figure 4.1). Let y be the complex variable, which corresponds to the
complex plane Oxz, and defined as
y=x+iz (4.5)
where i = V-T is the complex plane imaginary unit, which for the sake of convenience is
different from the time imaginary unit j = V-. In the two-dimensional irrotational flow


182
looks forward to applying the knowledge, gained in coastal and oceanographic
engineering as well as to materializing his love of computers and computer programming.


102
If the first holomorphic function of the left-hand side of (4.235) has a real part r and an
imaginary part 5, namely
F(y) = J f{y)dy = r + is, (4.236)
then from complex variable theory it follows that
r \ dr .dr
(4.237)
Now, if at the infinite point the analytic form of the holomorphic function is
n=l y
then
,as y oo
(4.238)
F(y) = J f{y)dy = a, (iff) +
,as y
oo.
(4.239)
The real part of the boundary condition (4.235) over the wetted body surface Sb can be
derived after substituting (4.236) and (4.237) into (4.235), and written as follows
dr dr}
r r. -1 x +z
dx dz
= ^aWa +Re|-T^ + /;^(1-%)^| or (4.240)
r -
+ +Re{-yF + l'(1.,mJ
(4.241)
From (4.236), and (4.239) it follows that at infinity
lim/' = 0.
p->oo
(4.242)
From (4.8b), (4.237), and (4.238) it follows that on the free surface
dr
dz
= 0.
2=0
(4.243)


UFL/COEL-TR/121
A DYNAMIC RESPONSE MODEL FOR FREE FLOATING
HORIZONTAL CYLINDERS SUBJECTED TO WAVES
by
Krassimir I. Doynov
Dissertation
1998

A DYNAMIC RESPONSE MODEL
FOR
FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES
By
Krassimir I. Doynov
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998

To Boris and Galina

ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to my advisor Dr. Max Sheppard for
his guidance, technical, and moral support during my doctoral research. Being chairman
of my graduate committee, he provided me with his insight and perspective and gave me
the freedom to pursue my research interests. Being a noble soul, he granted me his
friendship and moral help during the difficult moments I had to go through as an
international student. I would also like to thank the members of my committee:
Dr. Robert Dean, Dr. Michel Ochi, and Dr. Ulrich Kurzweg for their time and advice, Dr.
Peter Sheng, Dr. Robert Thieke, and Dr. Daniel Hanes for reviewing this work.
For the clarity of all drawings in my dissertation, for her help, trust, inspiration,
and love, I am forever grateful to my wife, Galina.
For their constant support, encouragement, and inspiration, I am deeply grateful to
my parents, Iordan Doynov and Nadejda Doynova, and to my brother Ivan.
Additional thanks for making my time here enjoyable go to my fellow students
Wayne Walker, USA; Thanasis Pritsivelis, Greece; Roberto Liotta, Italy; Emre Otay,
Turkey; Ahmed Omar, Egypt; Kerry Anne Donohue, USA; Wendy Smith, USA; and
Matthew Henderson, USA.
Finally, words cannot express my love to my son Boris, whose presence and love
make my life a real adventure.

TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
KEY TO SYMBOLS vi
ABSTRACT viii
CHAPTERS
1 MOTIVATION 1
2 INTRODUCTION 4
2.1. Historical Retrospective of Floating Body Studies 4
2.2. Classification 7
2.2.1. Large and Small Bodies 7
2.2.2. Deterministical and Statistical Approaches 11
2.2.3. The Concept of Added Mass 13
2.2.4. Classification of Damping 14
2.2.5. Numerical Methods Classification 16
2.3. Advancements in Floating Body Studies 24
2.4. Presentation of the Results of Investigations 26
2.5. Some Thoughts about the Current State of Knowledge 34
3 FORMULATION OF THE PROBLEM 37
3.1. General Description of the Problem and its Simplifications 37
3.1.1. Incompressible Fluid Assumption 37
3.1.2. Governing Equations and Definitions 38
3.1.3. Inviscid Fluid Assumption 39
3.1.4. Irrotational Flow Assumption 41
3.1.5. Dynamic Free Surface Boundary Condition (DFSBC) 42
3.1.6. Kinematic Free Surface Boundary Condition (KFSBC) 42
3.1.7. Sea Bottom Boundary Condition (SBBC) 43
3.1.8. Wetted Body Surface Boundary Condition (S) 43
3.1.9. Linearization to First-Order Theory 44
3.2. Floating Body Dynamics 47
3.2.1. Conservation of Linear Momentum 47
IV

3.2.2. Conservation of Angular Momentum 48
3.2.3. Matrix Form of the Dynamics Equations 50
3.3 Decomposition and Separation of the Hydrodynamics from the Body Dynamics 51
3.4. Hydrodynamic Properties and Forces 54
3.5. Hydrodynamic Relationships, Identities, and Definitions 56
3.6. Algorithm for the Solution of the Problem 61
4 RADIATION PROBLEM SOLUTION 62
4.1. Problem Statement and Definitions 62
4.2. Main Idea behind the Semi-Analytic Technique (SAT) 64
4.3. Semi-Analytic Technique. Determination of the Unknown Coefficients 70
4.3.1. Boundary Condition on St, 71
4.3.2. Conformal Mapping 71
4.3.3. Left-Hand Side of the Boundary Condition on Sb 77
4.3.4. Right-Hand Side of the Boundary Condition on Sb 87
4.3.5. Fourier Expansion of LHS and RHS. Solution for the Unknown
Coefficients 97
4.3.6. Discussion of the Uniqueness of the Solution 101
5 EXPERIMENTS 106
5.1. Purpose of the Experiments 106
5.2. General Set-up 106
5.2.1. Cylinders 108
5.2.2. Wave Absorption at the Ends of the Tank 110
5.2.3. Wave Gauges 110
5.2.4. Surface Tension Ill
5.3. Wave Absorption and Reflection Analysis 112
5.4. Model Scale Selection: Froude Scaling 117
5.5. Discussion of the Experimental Accuracy 118
5.6. Discussion of the Experimental Procedure 118
6 ANALYSIS OF THE RESULTS 128
6.1. Surge Mode Oscillations 130
6.2. Heave Mode Oscillations 140
6.3. Damping, Added Mass, and Frequency Response Function 149
6.4. Numerical Convergence 164
6.5. Conclusions 166
APPENDIX 168
LIST OF REFERENCES 175
BIOGRAPHICAL SKETCH 181
v

KEY TO SYMBOLS
Symbol
Description
A
Amplitude of the incident wave
A
Far-field wave amplitude
a* = A/va
Far-field amplitude. Dimension time.
a, b
Vertical, and horizontal semi-axes of the elliptical
cylinder
an
Power series coefficient of the nth term
B
Breadth of the waterline section of Sb
A
Sea bottom boundary
ce
Group velocity
rci
Buoyancy restoring force matrix of the floating body
^(.) = |(.) + u.V(.)
Total derivative in space and time
E
Water bulk modulus
i^}
Exciting force vector due to diffraction
8
Gravity acceleration
H
Height of the incident wave
\H(co)\ = RAO
Transfer function
h
Water depth
i
imaginary unit
KC
Keulegan-Carpenter number
k
Wave number of the incident wave
L
Wave length of the incident wave
\M\
Mass matrix of the floating body
"a
Component of unit normal to Sb vector in a direction
Pa
Atmospheric pressure
P
Pressure
R
Radius of the circular cylinder
Re
Reynolds number
SA
Part of the water surface cut out by Sb
sb
Instantaneous wetted body surface
sb
Mirror image of Sb in the air
Lateral boundary at infinity
SJ>)
Incident wave spectrum
vi

S
Response spectrum of the floating body
T
Period of the incident wave
um
Magnitude of the horizontal velocity of the incident wave
u = (u,v,w)
Fluid particle velocity
K
Time-amplitude of generalized velocity
W = dW / di
Energy flux
xa
Generalized displacement in a direction
i'=(x-ys)
Coordinates of the center of mass
y =x+i z
Complex variable
s
Small parameter
G>
Velocity potential
<¡>
Time-amplitude of the velocity potential
tD,R
Diffraction, radiation velocity potential
a
Radiation velocity potential due to unit velocity in
generalized a direction
*Kx>y)
Time-amplitude of the water elevation
A
Damping
V
Added mass
V
Kinematic viscosity
P
Water density
(0
Circular frequency of the incident wave
Time-amplitude of Xa
Stream function
Â¥
Time-amplitude of the stream function
C(x,y,t)
Water elevation
Vil

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
A DYNAMIC RESPONSE MODEL
FOR
FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES
By
Krassimir I. Doynov
December, 1998
Chairman: D. Max Sheppard
Major Department: Coastal & Oceanographic Engineering
A semi-analytical method for computing the dynamics of free-floating, horizontal
cylinders, subjected to ocean waves has been developed. The bodies analyzed in this
analysis are cylinders with circular and elliptical cross sections and variable still water
drafts. The motions considered are heave, surge and pitch. The technique computes the
added mass and damping coefficients using linearized radiation theory. The efficiency of
the numerical scheme is due to the simplicity of the mathematical scheme used a
combination of original holomorphic functions, convergent infinite power series, and
conformal mapping. An object-oriented approach was used for the computational aspects
of the problem using the programming language C++.
Physical experiments in a wave tank were conducted with circular and elliptical
horizontal cylinders in order to verify the method. The cylinders were positioned at the
viii

water surface and forced to oscillate sinusoidally first in surge and later in heave motion.
Different still water drafts and oscillation frequencies were used in the experiments. The
far field waves produced by the oscillating cylinder were measured using capacitance wave
gauges. The damping and added mass coefficients were computed from the measured
wave data. There is a good agreement between the predicted and measured coefficients.
IX

CHAPTER 1
MOTIVATION
In XV-century Europe, the brilliant engineer, scientist and artist, Leonardo da
Vinci recorded for the first time an ngineering application of a free floating buoy. The
buoy was used to measure the water velocity in streams and consisted of a weighted rod
and flotation bladder. It was released in the water flow and its downstream travel
measured after a given period of time and the average velocity computed as the distance
divided by the elapsed time. Since that time drifter buoys have proved to be very effective
in increasing the level of our understanding of the ocean environment and in improving
weather analysis and forecasting. Scientific investigations using drifting buoy systems
include measurements of atmospheric pressure, air and sea-surface temperature, solar
radiation, air humidity, wind shear, wave evolution, wave noise, light penetration, oceanic
current speeds, and water temperature and salinity. A variety of drifter buoy systems,
deployed in the worlds ocean, are used to monitor the spatial and temporal distribution of
the above mentioned environmental variables. With modern satellite technology, which
provides both buoy positioning information and a means of receiving and re-transmitting
data from the buoys, the accuracy and reliability of these measurements has been greatly
improved. Measurements can be recorded with data acquisition systems mounted on the
buoys or transmitted to land based stations via satellite (Berteaux, H.O., 1991). Data
from these buoys are vital to weather organizations for early detection of storms and for
daily weather predictions. This information is essential for those organizations and
1

2
companies involved in offshore oil exploration and production, marine transportation,
commercial and recreational fishing and boating, and military operations. For reliable data
transmission to the monitoring satellite, it is crucial that the surface floats provide a stable
platform under a variety of wave, current, and wind conditions. Since it is the surface
water motion and properties that are of interest, the question becomes how well do these
buoys track the currents and how do they respond to the wave motion. How a buoy
responds to currents and waves depends on a number of quantities including the buoy size,
shape, mass and mass distribution. Knowledge of the response characteristics of a buoy
for a given set of wind and wave conditions as a function of their structure and wind-wave
parameters would allow buoy designs to be optimized for the sea state conditions in which
they are to be deployed. Therefore the buoy hull must be designed in accordance with
certain stability design criteria for different ocean and atmosphere conditions.
The purpose of the following research is to investigate how variations of draft, size
and shape of drifting buoy systems influence fluid-body dynamics and to create a
computer-based model. In the focus of the research are floating horizontal cylinders of
circular and elliptical cross sections with variable still water drafts. The computer-based
model is intended to provide designers with estimates of the dynamic response
characteristics, in terms of response amplitude operators or frequency response functions,
of relatively small buoys.
The dissertation is comprised of six chapters. Chapter 2 presents a brief historical
retrospective of floating body studies, followed by classifications that introduce criteria for
large and small bodies, and deterministic and statistical approaches. Next, the concept of
added mass is introduced, as well as classifications of damping and the most frequently

3
used numerical methods for the determination of these hydrodynamic properties.
Furthermore, the most significant advancements in floating body studies are presented,
followed by a review of recent scientific works on the related subjects. This chapter
concludes with an attempt to classify the current state of knowledge about floating bodies
as one going simultaneously in three main directions. As a result of the well-known
conservation laws of mass, linear, and angular momentum, chapter 3 formulates the
floating body dynamics as a system of linear second-order differential equations with
boundary conditions of Neumann and Dirichlet-Neumann type. After introducing the
generally accepted simplifications for incompressible, inviscid fluid and irrotational flow,
the linearized radiation theory is deduced. Furthermore the hydrodynamics is
decomposed from the body dynamics, and some hydrodynamic theorems are given with
purpose to express all unknown variables of the floating body dynamics as functions of the
far-field amplitudes. Chapter 4 describes an exact analytical asymptotic solution of the
radiation problem, which derives the far-field amplitudes as functions of the wetted surface
of the floating body, and the circular frequency of the incoming harmonic wave. Chapter
5 describes the experiments conducted in heave and surge motion, which main purpose is
to obtain data for the far-field wave amplitudes and thus to verify the analytic solution,
introduced in chapter 4. These experiments were carried out in a wave tank at the Coastal
Engineering Laboratory at the University of Florida. Chapter 6 presents the analysis of
the results from the experiments and comparison with the numerical solution. The
conclusion is given at the end of this chapter.

CHAPTER 2
INTRODUCTION
2.1. Historical Retrospective of Floating Body Studies
Known since the ancient civilizations, the ship and boat transportation had
naturally attracted the attention of the universal minds of the 18th century and became the
first theoretically investigated floating bodies. Following Vugts historical survey (1971),
the great mathematician Leonhard Euler was the first who studied in a typical
mathematical framework with lemmas, corollaries and propositions the motions of ships in
still water. In 1749 his work Scieritia Navalis was edited in two volumes and published
in Latin in St. Petersburg, Russia. In 1746 the French scientist Bouguer published a
similar work and noted that he was familiar with the fact that Euler had been working on
the subject but that he had not yet been able to lay hands on his results. Daniel Bernoulli
was the first who examined ship motions in waves, and won the prize of the French
Academy of Sciences for his work in 1757. Considering the resonance phenomenon, he
examined forced oscillations of ships in waves. Having wrong ideas about the wave
motion, Bernoulli did not arrive at correct conclusions. Nevertheless his work was
considered as classic for a long time. In 1861 William Froude published his paper On
the Rolling of Ships where the ship was assumed sailing broadside to the waves and had
to follow the wave slope and the orbital motion of the wave particles. Practically Froudes
study dealt with a range of very low frequency motions, thus originating the generalization
4

5
that most engineering approaches in floating body studies are only valid in a certain range
of practical interest. Developing further Froudes idea with a paper in 1896 The Non-
Uniform Rolling of Ships William Froudes son, R. E. Froude, added the forcing of the
rolling motion for regular and irregular waves. Significant advancements in floating body
studies were made by Kriloff (1896, 1898), who considered for the first time the three-
dimensional problem, working with six degrees of freedom. Kriloff introduced the
hypothesis of approximating the actual pressure on the floating body surface by the
corresponding pressure in the wave structure, not being disturbed by the presence of the
floating body. As pointed out by Vugts (1971), the same hypothesis had been implicitly
made by William Froude in his earlier and more restricted work. Since then this
hypothesis has been known as Froude-Kriloff hypothesis and dominated almost all floating
body studies up to 1953. Kriloff computed the wave exciting forces and the restoring
forces and included the hydrostatic-coupling effects between heave and pitch. In an
additional estimated term, he included a resistance to motions, thus introducing the
concept of damping into the floating body studies. Speaking in modern terms Kriloff left
out the hydrodynamic mass and the hydrodynamic coupling among the various motions.
The hydrodynamic mass became known earlier. William Froude had realized that for a
longer natural roll period an apparent increase in mass was necessary. Because of the
presence of the free surface of the fluid and the determination of the hydrodynamic mass,
respectively, the added mass becomes much more complicated due to the wave frequency
dependence. In the first half of the 20th century, some attempts were made to get more
information on the added mass and damping, particularly for ships and ships sections.
With an exception for the limiting case of infinitely high motion frequencies, no major

6
success has been achieved in increasing the knowledge of these hydrodynamic properties,
as evidenced in a comprehensive survey of the available knowledge made by Weinblum
and St. Denis in 1950. Despite the insufficient understanding of the effects of
hydrodynamic mass and damping and the omitting of hydrodynamic coupling, the greatest
limitation they felt at that time was the restriction to regular waves. Weinblum and St.
Denis quote a saying of Lord Rayleigh: The basic law of the seaway is the apparent lack
of any law. Only three years later St. Denis and Pierson (1953) wrote in a paper that In
a broad sense the laws of nature are Gaussian, thus introducing the statistical description
of sea waves and body motions to the floating body studies. The concept of spectral
analysis of sea waves was defined by assuming the wave elevation as the sum of a large
number of simple sinusoidal waves, each having its own length, height and direction.
Consequently, the floating body motion is considered the sum of its responses to each
individual wave component. Since the phases of those wave components are randomly
distributed only statistical information can be obtained. The superposition of harmonic
waves and harmonic body responses meant enormous stimulation for the floating body
studies. In 1957 Korvin-Kroukovski and Jacobs introduced a strip theory that builds on
two-dimensional solutions to get three-dimensional effects for elongated floating bodies.
At that revolutionary time, 18 years after the invention of the greatest scientific tool of the
20th century the computer whose father was the American from Bulgarian descent
Dr. John Atanasoff, all the basic elements were available to obtain engineering solutions
for a coupled hydrodynamic motion, at least for heave and pitch.

7
2.2. Classification
2.2.1, Large and Small Bodies
There are at least three relevant length scales in the interaction between waves and
a fixed or floating body. They are the characteristic body dimension B, defined as the
predominant horizontal size of the body projection onto the vertical plane of the wave
front, the wavelength L=2 n/k (where k is the wave number), and the wave amplitude A.
If B, for a fixed vertical cylinder, becomes relatively large then the presence of the cylinder
will disturb the incident wave pattern. As the incident waves impinge on the cylinder,
reflected waves move outward. In the shadow zone, on the sheltered side of the cylinder,
wave fronts are bent around the cylinder and thus form diffracted waves. The combination
of reflected and diffracted waves is usually called scattered waves, but the process itself is
generally termed diffraction. The three scales, B, L, and A, may form two physically
meaningful ratios. The first one is the diffraction parameter B/L an important
dimensionless variable relating to the intensity of the scattered waves. Often used with the
same purpose is kB derived from the multiplication of the diffraction parameter with the
constant 2 n. The second ratio is A/B, which in deep water is proportional with the same
constant 2n to the Keulegan-Carpenter number, KC, defined as
du
(2.1)
KC =
UJ_ _Udx
B
du
dt
where the wave period is T, and the magnitude of the horizontal velocity of a harmonic
progressive wave is

8
(2.2)
h is the water depth, and co is the circular frequency. Now with (2.2), the Keulegan-
Carpenter number is
H
n
2 tiA 'l
(2.3)
Btmh(kh) 5tanh(M)
L
The physical meaning of the Keulegan-Carpenter number (more easily seen for the case of
deep water: tanh(kh)=l) is the ratio between the circumference of the fluid-surface-
particle-orbital motion and the characteristic body dimension. Speaking in Navier-Stokes
equation (see equation 3.2) terms, the Keulegan-Carpenter number is the ratio between
$11
spatial u and temporal accelerations. In accordance with Meis definitions (1989),
dx dt
a body is regarded as large when kB>0(l)\ its presence can significantly alter the pattern
of wave propagation, produce wave diffraction, and the disturbance can propagate in a
much wider area far away from the large body. On the contrary, a body is regarded as
small when kBl; diffraction is of minor importance. When A/B> 0(1) the local velocity
gradient near the small body augments the effect of viscosity and induces flow separation
and vortex shedding, leading to so called form drag. It should be noted that the
associated viscous forces are not mean shear forces, but pressure forces due to separated
flow. The influence of a small body is usually bounded to a comparatively narrow area.
A more precise classification is given in Sarpkaya and Isaacson (1981), where Isaacson
has presented a convenient means of indicating the conditions under which the diffraction,

9
flow separation and nonlinear effects become important for the case of a fixed vertical
circular cylinder, as seen in Figure 2.1.
jNCReasmassr
Wotaht\
DifFSlCTiCW-
MSRcCA&tNSW:
Fig.2.1 Wave force regimes (Sarpkaya and Isaacson, 1981). Importance of
diffraction and flow separation as functions of KC -Keulegan-Carpenter number
and ^-diffraction parameter
As given by (2.3), the greater the wave steepness (H/L) the larger the Keulegan-Carpenter
number becomes. An approximation of the maximum wave steepness is given as (Patel,
1989)

10
rH_
U
max
= 0.14tanh(£/i)
(2.4)
Therefore, from (2.4) and (2.3) the relationship between the largest Keulegan-Carpenter
number and the diffraction parameter, shown in Figure 2.1, is given by
(2.5)
L
and is a simple hyperbolic curve. Isaacson pointed out that the critical value of the
diffraction parameter that roughly separates large from small bodies is B/L=0.2, because
the curve of the largest KC (without wave breaking) does not exceed 2 for the range
B/L>0. On the other side, flow separation should be more important than the diffraction
when KC>2, which according to Figure 2.1 happens when B/L>0.2. At the same time
the condition for diffraction B/L>0.2 and (2.4) imply that H/B<1 and that the drag forces
will be small since wave amplitude is less than the body dimension. It should be noted that
the Isaacson criterion is only true for a fixed vertical circular cylinder; for any rectangular
cylinder flow separation inevitably occurs and its effect might not be negligible for large
(B/L>0.2) bodies. As seen in Figure 2.1, for a wave with steepness one half of the
maximum steepness (0.5H/L), KC does not exceed 2 for the region B/L>0.1. This fact
suggests that the generally accepted (Sawaragi, 1995; Sumer and Fredsoe, 1997) critical
value of the diffraction parameter B/L=0.2 is not a fixed value it may vary even for
rounded bodies. It is well known that the flow regimes about a fixed vertical cylinder
depend not only on the KC number but also on the Reynolds number defined as the ratio
between inertia and viscous forces

11
Re = ^L = KC (2.6)
v vT
where the kinematic viscosity is v = 10~6[m2 / sec]. When for example Be[1,37[m], Te
[2,20]\sec], and tanh(kh)=l for the case of deep water, then the Reynolds number
becomes large ReO(103). As evidenced in Sumer and Fredsoe, 1997 with Figures 3.15,
3.2, and 3.16, when KC<2 and ReO(103), the flow will not be separated; when KC
approaches 2, there will be separation, but not very extensive. This analysis suggests that
the flow about a fixed vertical circular cylinder in the large-body-diffraction-regime
B/L>0.2 (or B/L>0.1) may be analyzed by the potential theory, since no flow separation
occurs KC<2, and since the viscous effects are negligible ReO(103).
In the case of a floating body, there is another hydrodynamic force due to the
body motion in addition to the wave forces exerted on the body under the fixed condition.
According to Sawaragi (1995), the generation mechanism of hydrodynamic forces due to
body motion can also be classified briefly into two regimes in the same way as the case of
the fixed body.
2.2.2. Deterministical and Statistical Approaches
A phenomenon, which is changing with time, can be described deterministically or
statistically. In the deterministic approach all the variables are functions of time and
known at any moment of time, usually after solving differential and integral equations. In
the statistical approach the explicit time dependence is not considered. A variable is
usually known as an average or as a probability of occurrence. The time history is
unknown and therefore the variable is unknown at any moment of time. The problem is

12
formulated as a distribution of the relevant quantities over the independent variables. An
excellent example found in nature, which explains the two formulations and their
relationship, is the irregular sea. Obviously the time history is very difficult to obtain and
is not important. In order to obtain statistical estimates, the linear theory simulates the
irregular sea as a superposition of linear harmonic waves. In the case of a floating body, if
its response to a harmonic wave is solved deterministically, it will help to find statistically
its response to the irregular sea. One of the most generally used ways to describe and to
work practically with a random sea is to consider the distribution of its energy content as a
function of wave frequency (sea spectrum). The concept is to sum a large number of
sinusoids with small amplitudes, different frequencies and phases, with some waves adding
to build up larger ones and others canceling each other, thus forming an irregular profile
with no set pattern as to amplitude or periodicity (Figure 2.2.).
20
o
-20
20 5 10 15 20 25 30 35
0
-20
20
0
-20
20
0
-20
50
0
-50
5 10 15 20 25 30 35
Time in seconds
Fig.2.2. Irregular wave as a sum of four harmonic waves with different
amplitudes and circular frequencies
Irregular profile as a sum of four harmonic waves

13
From here it follows that the energy content of the random sea irregular profile can be
presented as a sum of the energy of all the component waves irrespective of their phases.
The concept of sea spectrum had been justified theoretically and experimentally for 40
years by utilizing the Pierson-Moskowitz, Bretschneider, JONSWAP and other energy
density spectra. The Pierson-Moskowitz spectrum is controlled by a single parameter -
significant wave height and represents fully developed seas. The Bretschneider spectrum
is controlled by the significant wave height and a modal wave period and can be used for
fully and partially developed wind generated seas. The JONSWAP spectrum is controlled
by the significant wave height, fetch-length, and shape parameter, and is used for partially
developed seas. Therefore the implementation of the energy distribution concept into a
computer model is a powerful tool for analyzing the buoy behavior in real seas, knowing
only the buoy response to a single harmonic wave with small amplitude.
2,2,3. The Concept of Added Mass
The concept of hydrodynamic added mass arises from the fact that a body having
an accelerated motion in or on the surface of the water experiences a force that is greater
than the mass of the body times the acceleration. Since this increment of force can be
defined as the multiplication of the body acceleration and a quantity having the same
dimension as the mass, it is termed added mass. The added mass is not a finite amount of
water, which oscillates rigidly connected to the body. The whole fluid will oscillate with
different fluid particle amplitudes throughout the fluid. As the linear oscillation is
associated with forces and the rotational oscillations with moments, the added-mass may
have dimensions of mass, mass multiplied by length, and even inertia moment. The

14
concept of hydrodynamic added mass should be understood only in terms of generalized
force on the body induced by the hydrodynamic pressure, and therefore it will depend on
the wave frequency and the wetted body surface. Inherited from the added-mass concept
is the concept of virtual mass, defined as the sum of the added mass and the mass of the
floating body.
2.2.4, Classification of Damping
By definition, damping is the ability of a structure to dissipate energy. There are
three major kinds of damping for a fixed or floating body in water: structural, material,
and fluid. Structural damping is due to friction among different parts of a structure.
Material damping is energy dissipation within the material of the body, being more
significant in materials like rubber. Fluid damping is the result of energy dissipation, as the
fluid moves relative to the vibrating body. The fluid damping can be classified further into
a damping due to wave generation and a damping due to viscous effects. The wave
generation damping or simply the wave damping dissipates the energy of the vibrating
body into the fluid, thus causing waves. The viscous effects damping can be subdivided
into skin friction effects and viscous effects due to the pressure distribution around the
body. The latter is associated with separation and formation of eddies and is usually
known as eddy-making damping in the literature (Faltinsen, 1990). The separation
changes the flow pattern about the body to a certain extent so that in may be felt in both
the damping and added mass. The skin friction effects on damping are due to shear
stresses acting tangentially on the boundary surface between the fluid and the body and are
proportional to a velocity gradient.

15
Since it is possible to obtain the response of a floating body in irregular seas by
linearly superimposing body responses from harmonic wave components, it is sufficient
from a hydrodynamical point of view to analyze a floating body in incident regular
sinusoidal waves of small steepness. The basic laws of physics governing the motion of
floating bodies are well known in their linearized version (Mei, 1989), and with the drag
force proportional to the square of the floating body speed (Berteaux, 1991). The
difficulty in predicting the response of a floating body to a harmonic wave arises in the
determination of the hydrodynamic properties, namely added mass and damping. Due to
the effects of water-air interface, these hydrodynamic properties depend on the wave
frequency, water depth, and the wetted body surface, which change with time. There has
been a significant amount of research on these subjects in recent years but they remain the
most difficult aspects of floating structure response prediction. Thanks to the significant
computer advancement in achieving higher computational speed and larger memory
capacity, numerical methods and techniques have been increasing their role in calculating
the dynamics of fluid-structure interaction. The differences between computed and
measured hydrodynamic forces revealed that some important phenomena are either not
well understood or the existing combinations of theories and numerical techniques can not
explain them. On the other hand the agreement between computed and measured
quantities in other particular regions, confirmed the generalization that most engineering
approaches in the floating body studies are only valid in a certain range of practical
interest, thus stimulating more investigations. The notable interplay of theories, numerical
methods, and experiments has been very fruitful for the advancements in floating body
studies.

16
2.2.5.Numerical Methods Classification
As classified by Yeung (1985), Mei (1989), and Vantorre (1990), several
numerical methods have been proposed for calculating potential functions in free-surface
hydrodynamics:
2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method.
2.2.5.2. Singularity distributions on the floating body surface, which leads to an integral-
equation formulation based on Greens functions.
2.2.5.3. Method offinite-differences. Boundary-fitted coordinates.
2.2.5.4. Finite element method. Hybrid element method.
2.2.5.5. Boundary integral equation methods (BIEMs) based on a distribution of
simple sources over the total fluid domain boundary.
2.2.5.6. Methods making use of eigenfunction matching.
All these numerical methods will be explained in the frequency domain, because as
it will become evident from the linearized combined kinematic-dynamic free surface
boundary condition (3.34b), the time-domain and frequency domain solutions are simply
related.
2,2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method
Generalizing the heaving motion solution for a semi-immersed circle Ursell (1949),
and its extension to a semi-immersed sphere Havelock (1955), Taylor and Hu (1991)
developed a systematic multipole expansion technique for submerged and floating
horizontal circular cylinders in infinite water depth. Considering the two-dimensional case
of a partially or totally submerged circle, the main idea is to place a set of easy-to-evaluate

17
elementary functions multipoles which satisfy the Laplace equation, on the level of the
center of the circle. The combined kinematic-dynamic free surface boundary condition is
satisfied with a proper combination of the multipoles and their mirror images with respect
to the mean water surface, which also satisfy the Laplace equation. In order to represent
the outgoing waves at infinity, additional potential is introduced, that has different forms
for symmetric (heave) and anti-symmetric (surge) motions respectively. With the help of
unknown coefficients, both wave and local-disturbance multipoles are combined as an
infinite sum into a velocity potential, which automatically satisfies the Laplace equation,
the free surface boundary condition, and the radiation condition. The wetted body surface
boundary condition is used to determine the unknown coefficients, after truncating the
infinite series at a finite number of terms, and using the point collocation method. A
limitation of the multipole expansion technique is that the general existence of the
expansion is very difficult to justify, or in other words the convergence of that expansion
has been proved only in the case of a semi-submerged circle in the region kB<1.5 (Ursell,
1949). Numerically, the convergence was found to vary with the depth of submergence,
with the fastest convergence for a fully submerged circular cylinder. The most demanding
case was found to be when the circle was just piercing the surface. In that case the body
intersects the free surface at an angle of zero degrees, which makes the linearized
boundary value problem mathematically unstable.
The multipole expansion can be applied to the analysis of more general bodies
through a coupled method, called the BIE-BMP method by Taylor and Hu, 1991. To
solve for the velocity potential, one can use a boundary integral expression in a domain
close to the body and match it with the multipole expansion outside this domain. Using

18
three-dimensional multipoles, Taylor and Hu (1991) outlined the same procedure for the
case of a floating or submerged sphere. A complete multipole expansion solution of a
heaving semi-immersed sphere was given by Hulme (1982), who simplified Havelocks
solution by making certain explicit integrations. This method was developed further by
Evans and Mclver (1984) for the case of a heaving semi-immersed sphere with an open
bottom.
2,2,5,2, Singularity distributions on the floating body surface, which leads to an integral-
equation formulation based on Greens function
The method of integral equations via Greens function, as explained by Mei
(1989), is based on applying Greens theorem on the radiation velocity potential and a
Greens function defined to be the potential at any field point due to an oscillating source
of unit strength at a particular point inside the fluid domain. Knowing the prescribed
normal velocity boundary condition, the velocity potential at any point on the wetted body
surface is a Fredholm integral equation of the second kind. By dividing the wetted body
surface into discrete panels and approximating the velocity potential in each panel by a
constant, one can obtain a system of algebraic equations for these constant values after
carrying out the integration. Solving for these constant velocity potentials on the panels
will help to express the velocity potential anywhere in the fluid domain. An advantage of
this method is the relatively small number of unknowns, while getting the matrix
coefficients is a laborious task both for the worker and for the computer. Another
drawback of the Fredholm integral equation is the so-called irregular frequencies and
nontrivial eigensolutions in the case of a floating body on the water surface. Without

19
going into details the irregular frequencies are the eigenfrequencies of a fictitious interior
for the wetted body surface problem with the Dirichlet condition on the water-body
boundary. As a result the source distribution is not unique, the approximate matrix
equation becomes ill-conditioned; hence the integral equation must fail. In order to avoid
the irregular frequencies, one must introduce additional artificial unknowns and more
conditions to improve the matrix equations, thus making the methodology less
advantageous.
A similar technique was used by Martin and Farina (1997) to solve the radiation
problem of a heaving submerged horizontal disc, where the boundary integral equation is
reduced to a one-dimensional Fredholm integral equation of the second kind.
2,2.5.3. Method of finite-differences. Boundary-fitted coordinates
The classical finite-difference method is based on generating a mesh around the
floating body and using a variety of difference-schemes to express the Laplacian operator
at a particular node with the help of information from some neighboring nodes and as a
function of distance, h, between these nodes. Higher-order schemes increase their
accuracy by involving more nodes, usually at the expense of more complicated algorithms.
On the other hand, the accuracy loss due to an increase in truncation errors, expressed as a
power of the distance h, can lead to physically unacceptable solutions (Yeung, 1985).
While conveniently suited for interior nodes of the fluid domain, the difference schemes
are not easily applicable on curved boundaries. That is why the grid generation process
serves two purposes: first it produces a set of curvilinear coordinates that are specifically
adapted to the geometry in question; and second it provides a crucial numerical

20
transformation that allows difference-schemes to be applied in a more geometrically simple
computational domain. The boundary curves from the original physical domain have to be
transformed into coordinate lines in a logical domain of mapped variables, which facilitates
the implementation of Neumann type boundary conditions. As pointed out by Yeung
(1985), the coordinate transformation and the physical solution may be solved
concurrently in the same sweep with a proper adjustment for the boundary conditions.
Once the general algorithm is developed, it can be applied to different floating body
geometries with a change of boundary coordinates. The change of boundary coordinates
is much more involved for three-dimensional bodies.
2.2,5,4, Finite and hybrid element methods
The strength of this class of methods is its ability to handle curved boundaries.
The main idea is to map isoparametric boundary surface elements into local squares,
triangles, etc., on which one can calculate every elements contribution to the field and
boundary properties. The unknown function consists of a set of nodal values and a set of
predefined shape functions, chosen to satisfy certain continuity requirements across the
elements. The requirements depend on the differential order, and the boundary conditions
(Yeung, 1985). The determination of the nodal unknowns relies on a global, integral
criterion. A brief description of the hybrid element methods, as given by Mei (1989), will
be given as a generalization of the finite element methods of Newton (1974, 1975). The
main idea of the hybrid element method is to employ a finite-element approximation in an
imaginary cylinder, which extends from the sea bottom to the water surface and surrounds
the body, with an analytical representation outside of the cylinder. Thus the finite-element

21
region can serve as a transition zone that transforms a geometry of higher complexity into
a simpler cylindrical geometry where the singular kernels (associated with integral-
equation methods) can be more conveniently calculated. Variational principles can be
utilized to formulate the radiation problem and to seek an approximate solution for the
velocity potential in the inner domain surrounding the body. The strength of the
variational principles is that they permit an exact coupling of the approximate interior
solution with the analytical solution for the velocity potential in the outer domain. The
outer domain solution is usually presented in one of the following two ways. The first one
is to use Greens function and express the velocity potential as a superposition of sources
of unknown strength on the boundary between the inner and outer domains, while the
second way is to use eigenfunction expansions with unknown coefficients. In the case of
infinite water depth, the eigenfunction expansion was found to be inefficient, and the
Greens function approach was recommended instead. By obtaining two different answers
for two different grids for the velocity potential at a particular point, it was proven that the
general identities and the energy conservation between rate of work done by the body
and the rate of energy flux through the boundary between the inner and outer domains -
are necessary but not sufficient conditions to guarantee an accurate solution. Mei
generalized that similar caution was warranted in other numerical methods.

22
2.2,5.5. Boundary integral equation methods fBIEMs) based on a simple sources
distribution over the total fluid domain boundary
Developed for the numerical calculation of linear potential functions for heaving
axisymmetric bodies by Ferdinande and Kritis (see Vantorre, 1986, 1990) the philosophy
of this method is to confine the problem into a finite cylinder, which surrounds the heaving
body. Since both the body and the motion are axisymmetrical, the problem is reduced and
solved in a two-dimensional rectangular fluid domain. The domain has the following
boundaries: the vertical axis of symmetry of the body, half of the vertical cross section of
the wetted body surface, the vertical cylinder wall, the flat bottom, and fluid surface
between the body and the cylinder. In order to justify the confinement of the problem into
the cylinder, and thus the boundary element method, a modified radiation condition has to
be satisfied on the vertical cylindrical wall. Now the whole boundary is divided into
discrete panels, on the center of which the velocity potential is considered constant. As a
consequence of Greens theorem a potential on a point on the boundary is expressed as an
integral of the velocity potential and its normal derivative over the whole boundary.
Eventually the problem is reduced to the solution of set of N linear algebraic equations,
where N is the total number of panels.
Some of the boundary integral methods are very effective for first-order diffraction
and radiation problems. However, these methods are not suitable for second-order
diffraction and radiation problems (Taylor and Hung, 1987). In this case the integrand is a
functional of the first-order potential and the Greens function. Because of the slow
convergence of the integrals, a large number of values of the first-order potential must be
evaluated, and this is not easily achieved using boundary integral methods.

23
2.2.5.6, Methods making use of eigenfunction matching
Considering the hydrodynamic problem of a floating vertical circular cylinder in
finite-depth water, Yeung (1981) gave an example of the eigenfunction matching methods
with treatment of interior and exterior problems. In the interior problem, the vertical
circular cylinder is considered a fictitious interior domain filled with the same fluid, where
the velocity potential is uniquely determined by solving the Laplace equation with
Dirichlet type boundary conditions. The exterior problem respectively is a Neumann type
problem and can be thought of as one driven by a flux emitted by the interior region, with
a solution written in terms of an eigen expansion with unknown coefficients. Both
problems are treated as if the conditions at the common boundary were known. By
matching both velocity potentials at .the common boundary, the problem is reduced to
solving an infinite system of linear equations. This infinite system is claimed to have
excellent truncation characteristics requiring rarely more than 20 equations to achieve an
accuracy of 1%. It should be noted that the eigen expansion in this solution was possible
because of the convenient presentation of the cylinder boundary in a cylindrical coordinate
system, and because of the finite-depth water. As Taylor and Hu (1991) point out when
the water depth increases toward infinity, the eigenvalues tend to pack together, and the
eigenfunctions become undistinguishable. Consequently, the number of terms required in
the eigen-series expression becomes unrealistically high.

24
2.3. Advancements in Floating Body Studies
As stated earlier, the most basic problem to solve is that of the frequency response
of the floating body when subjected to simple sinusoidal wave excitation. In 1944
Haskind introduced a way to decouple the hydrodynamics from the body dynamics and to
further decouple the hydrodynamics problem into diffraction and radiation components.
Taking advantage of the linearity of the Laplace operator and the combined Neumann and
Dirichlet-Neumann boundary conditions for the velocity potential Haskind presented the
velocity potential as a sum of diffraction and radiation potentials. Speaking in physical
terms, two hydrodynamical sub-problems were formulated. The first sub-problem
assumes the body to be rigid and restrained from any oscillation in the presence of incident
regular waves. The hydrodynamic loads are called wave excitation loads and composed
of Froude-Kriloff and diffraction forces and moments. The second sub-problem forces
the rigid body to oscillate in any motion mode with the wave excitation frequency in the
absence of incident waves. The hydrodynamic loads are identified as added mass,
damping and restoring terms. Since the body oscillates in otherwise calm water, it
radiates waves, thus the term radiation. In 1949 John (see Mei, 1989) introduced in a
systematical formal approach the complete first order wave-structure theory and showed
how higher-order extensions can be made. Following Haskind (1973), Newman (1976),
and Mei (1989), several remarkable general identities have been introduced. These
identities relate different hydrodynamic quantities and have proved to be extremely useful
in increasing the theoretical understanding of physical phenomena. Moreover, they
provide necessary checks for analytical theories or numerical methods, and minimize the

1
25
computing time of those quantities related by them. Based on the mathematical definition
of the incident, radiation, and diffraction wave potentials as harmonic functions, and on
the Greens theorem, the first identity relates, in an elegant way, two radiation problems.
A computationally efficient result is that the restoring force, the added mass and the
damping matrices are diagonally symmetric, which decreases significantly the number of
unknowns. Another consequence is the convenient relationship between the damping and
the asymptotic behavior of the radiation velocity potential. In physical terms the energy
given up by the oscillating body is transported by the waves propagating away from the
body. The knowledge (about damping) gained this way, can be used to find the added
mass, using the so-called Kramers-Kronig relations. Derived in a universal way by Ogilvie
(1964), these relations simply exploit the fact that damping and added mass are the real
and imaginary parts of the same function. Therefore knowledge of the damping
coefficients is sufficient to determine the corresponding added mass coefficients and vice
versa. Newman (1976) discovered that the damping matrix is singular for the case of
particular bodies of revolution where the exciting force is independent of the angle of
wave incidence. The second identity relates two diffraction problems corresponding to
different angles of incidence. As two-dimensional results, there are several relationships
between the amplitudes and the phase angles of the transmitted and reflected waves. The
three dimensional result is that the amplitude of the first scattered wave toward the second
incident wave is equal to the amplitude of the second scattered wave toward the first
incident wave. The third identity relates the diffraction potential to the radiation potential,
thanks to a brilliant Haskind theorem (1957). It expresses a generalized component of the
exciting force (due to fixed body diffraction from an incident wave moving in a particular

26
direction) in terms of the radiation potential and its normal derivative (due to body
oscillation in the same direction, in otherwise calm water). An elegant consequence from
the Haskind theorem is that the exciting force is linearly proportional to the far-field wave
amplitude. The very practical meaning of the Haskind theorem is that an actual solution of
the diffraction problem can be avoided and that the exciting forces are simply related to
the damping coefficients. Newman elaborated on this further and found an explicit
relationship between the magnitudes of the exciting force and the damping coefficients,
which is often used as a check on the magnitudes. An even less obvious identity between
radiation and diffraction problems was discovered by Bessho (1967) for two dimensions
and extended for three dimensions by Newman (1975,1976). It relates the far-field
radiation and diffraction wave amplitudes, and reveals the significant result that the phases
of surge and pitch antisymmetric modes differ with n.
2.4. Presentation of the Results of Investigations
In light of the above-mentioned theoretical advancements, it is instructive to give
some examples of the theory-experiment interplay. In 1965 Kim determined the added
mass and damping for a semi-ellipsoidal body with its origin on the free surface of water
of infinite depth. In order to check the validity of potential flow theory predictions Frank
(1967) conducted a series of experiments to determine the added-mass and damping
coefficients for two-dimensional bodies at the free surface.
Using Ursells multipole expansion method and De Jongs extension to sway and
roll motions of arbitrary shaped cylinders, Vugts (1968) solved the linear radiation

27
problem for two-dimensional floating cylinders. Conducting experiments with five
different cylinder cross-sections in heave, sway, and roll oscillatory motion, he has
compared the experimental results with the predictions of the linear potential theory and
particularly the multipole expansion solution. With the influence of viscosity negligible,
Vugts has found good agreement between calculated (with the potential theory) and
measured added mass and damping coefficients in heave for the middle and high frequency
range. However, in the low-frequency range, co^B / (2g) < 0.33, the multipole expansion
method significantly overestimates the experimentally obtained added mass. In sway, the
calculated added mass is in good agreement with measured values for most of the
frequency range: 1.125 < co^Jb /(2g) < 1.25. In the higher frequency range the damping
was underestimated. For the case of coupling of sway and roll the differences between the
measured and calculated added mass and damping coefficients are higher. Viscous effects
in terms of separation and eddy making have been observed in sway motion for the
relatively sharp edged sections. With predominant wave damping, the viscous damping
contributes from 10% at low frequencies to 40% at high frequencies for sharp edged
sections. Obviously the energy loss due to eddy formation has been one of the reasons for
these differences in damping. Interestingly enough it appears that eddy formation does not
seriously affect the total pressure distribution in phase with the body acceleration since the
added mass has been predicted relatively well. For rolling oscillatory motion the
calculated added mass moment of inertia overestimates the measured one, while for the
damping coefficient the observations were the same as in the case of sway. Since the
wave damping part for the sections considered is of an order smaller than for swaying, the

28
viscous effects become much more important. That is why the damping in roll motion is
significantly underestimated compared to the damping in sway motion. The exciting
forces, calculated with Newmans explicit relationship of the magnitudes of the exciting
force and damping coefficients, have been found to be in a relatively good agreement with
the measured ones.
Haskind (1973) developed a technique for computing added mass and damping for
horizontal and vertical plates floating at the water surface, as a function of body geometry,
water depth and frequency. Newton et al. (1974) and Newton (1975) have developed a
finite element model (FEM) for computing these coefficients for ship hull forms. Using
potential flow theory, Chung (1976, 1977) presented added mass and damping as a
function of frequency, direction of oscillation and depth of submergence. Hanif (1983)
determined these hydrodynamic coefficients using FEM and compared his results with
other investigators.
While investigating the heave motion of a half-immersed floating sphere with a
bottom opening, Evans and Mclver (1984) have used a semi-analytical solution based on
an extension of the method of multipole expansion, classified in 2.2.5.1. The added mass
and damping coefficients calculated with this method were compared with their values
obtained via the numerical singularity distribution method classified in 2.2.5.2. They
found that the numerical method tends to overestimate the diagonal hydrodynamic
coefficients, while their off-diagonal values are in good agreement with the semi-analytical
solution. Moreover the numerical method showed slight differences between the off-
diagonal added mass and damping coefficients, which theoretically must be equal in
accordance with the symmetry-identity classified above. The singularity of the damping

29
matrix has been used as a partial check for the correctness of the semi-analytical solution.
In a technical note to the Evans and Mclver (1984) work, Falnes (1984) proposed some
empirical formulas for the added mass and damping for the semi-submerged sphere in
heave motion.
Utilizing a numerical scheme that is similar to one used by Nestegard and
Sclavounos (1984) for the method of multipole expansion combined with BIE-BMP
matching classified in 2.2.5.1, Taylor and Hu (1991) obtained added mass and damping for
floating and submerged circular cylinders. For the submerged cylinder, the diagonal added
mass and damping coefficients in sway have been confirmed to be equal to those in heave.
While damping is always positive, negative added mass has been discovered for the
case when the submerged cylinder is close to the surface. Negative added mass has been
also observed for a cylinder floating on the surface in sway motion when the cylinder is
more than three-quarters immersed. For the frequencies of negative added mass in heave
the sway added mass was positive. It was concluded that at these frequencies the forced
heave oscillation does not transfer energy to the fluid and hence does not generate waves.
Lastly Taylor and Hu have found that in the low frequency range the added mass changes
from finite to infinite values as the cylinder emerges from below the free surface. The
incorrect conclusions, made by Taylor and Hu, show the important role of experiments as
the only tool for verification of the working capabilities of theories and numerical
techniques.
Exploring surge oscillatory motion of a single tethered half-submerged spherical
float, Vethamony et al. (1992) have computed added mass and damping from the motion
generated velocity potential, using potential flow theory. They solved a boundary value

30
problem with Greens function method by dividing the cross-section of the float into equal
segments, and conducted experiments to compare the results. Considering the float size
small compared to the wave-length the diffraction potential has been neglected and added
mass and damping computed from the motion generated velocity potential. With added
mass in phase with the surge motion and damping coefficient out of phase with the
motion, both were found to be frequency dependent, and to increase gradually with
respect to size of the float. It was noted that added mass and damping do not change with
water depth. In light of the general identities discussed earlier, Vethamony et al. found
that wave excitation forces calculated from incident wave potential are in good agreement
with excitation forces calculated from damping coefficients.
It was observed that nonlinear effects become important in fluid-structure
interaction when waves become steeper, or when the oscillation amplitude of the floating
body can no longer be assumed small. These natural phenomena have been stimulating the
extension of the linearized radiation problem and linearized diffraction theory which are
the first terms in Stokes perturbation expansion into higher order terms, where the
perturbation parameter is related to the wave steepness. As a result of this expansion
procedure the full nonlinear diffraction and/or radiation problems are replaced by a
sequence of linear boundary-value problems formulated for each order of perturbation.
Most difficulties of the second order problem arise from the requirement that the velocity
potential satisfies a nonhomogeneous boundary condition on the free surface of the fluid.
According to Molin (1979), the nature of second-order diffracted waves consists of two
components. One is free-waves travelling independently of the first-order wave system
and the other is phase-locked-waves accompanying the first-order waves (Chakrabarti,

I
31
1987). Since in many applications the hydrodynamic loads, not the flow kinematics, are of
primary interest, an indirect method can be used to determine wave-induced loads to the
second order without the explicit calculation of the second order potential. The technique
involves an application of Greens second identity and requires the solution of associated
linearized radiation problems corresponding to prescribed oscillation of the structure at
twice the first-order wave frequency. A modification that utilizes an axisymmetric Greens
function solution for the linear problem has been used by Molin and Marion (1985) to
calculate the second-order wave induced loads and associated motions for a floating body.
Taylor and Hung (1987) have performed detailed analysis of the asymptotic behavior of
the troublesome free-surface integral and have presented a computational method for its
calculation on the far field in the case of a fixed vertical circular cylinder. Adopting an
interior region that includes the fixed vertical cylinder, and an exterior region that extends
to infinity in the horizontal plane, Ghalayini and Williams (1991) presented a solution to
the free-surface integral. The interior region solution was performed numerically by
utilizing the asymptotic forms of the potentials, while the exterior region solution was
carried out analytically in terms of Fresnel functions. Inspired by Sodings work,
Vantorre (1986) developed a computational procedure for calculating hydrodynamic
forces up to the third order for the case of floating axisymmetric bodies in a symmetric
heave oscillatory motion. Soding (1976) proved that the knowledge of the first-order
potential for the exciting frequency and for twice that frequency on the mean body surface
and on the mean free surface within some distance from the body is all the information
required to compute second-order forces for oscillating cylinders. In a similar way,
Vantorre computed third-order forces by means of the first-order and second-order

32
potentials for the exciting frequency, and the first-order potential for three times the
exciting frequency. Calculation of the latter potential is not necessary if the first harmonic
of the third-order forces is the only primary interest. The computational procedure used
by Vantorre is classified in 2.2.5.4. as boundary integral equation method (BIEM) based
on a simple sources distribution over the total fluid domain boundary. Two experiments
have been conducted, one with a floating cone and a second with a submerged vertical
cylinder. In both cases the third harmonic was impossible to measure. It is obvious from
the experimental results for the floating cone, that the third-order theory somehow
underestimates the added mass over the entire frequency domain. This is more significant
for larger heave amplitudes. Being acceptable for the region [co'B / 2g) < 1, the third-
order theory overestimates the experimental heave damping in the higher frequency range.
The same frequency relationships can be observed for the second-order forces. Using
again the boundary integral equation method (BIEM) for a floating cone in heave
oscillatory motion, Vantorre (1990) determined the influence of small draft variations on
hydrodynamic first-order forces. After deriving some second-order and third-order force
components, he used a modified BIEM to derive first and second derivatives of heave
added-mass and damping with respect to draft.
The nonlinear effects of the motion of fluid around a body can be dealt with
directly from the fundamental conservation laws of mass and momentum (Equations 3.2
and 3.3). Due to the increasing capacity of computers in the recent years, three groups of
methods for numerical treatment of flow nonlinearity around cylinders have been
developed. Following the comprehensive survey of Sumer and Fredsoe (1997), the first
group consists of methods involving the direct solution of the Navier-Stokes equations.

33
When the flow around a cylinder is two-dimensional and the Reynolds number is small, i.e.
Re<200, a direct solution with finite-differences yields results for the gross-flow
parameters that are in reasonable agreement with measurements, while the lift force is
grossly overestimated (Sumer and Fredsoe, 1997). For larger Reynolds numbers, vortex
shedding occurs in cells and therefore the flow becomes three-dimensional (3D). When
300 further, 3D-turbulence begins to spread into the boundary layer, and direct numerical
simulation of the Navier-Stokes equations is not feasible, because of the scales of the
dissipative part of the turbulent motion. The oscillatory flow is solved using the same
vorticity transport equation (3.7) and Poisson equation (3.8). The major source of
difficulties is the number of grid points (for finite-difference and finite-element methods)
and the corresponding number of computations required to obtain a solution increases
with increasing Reynolds number, and may become prohibitive as mentioned earlier at
large Reynolds number. The second group consists of discrete vortex methods, developed
as grid-free numerical methods. The idea is to solve the vorticity transport equation
(Equation 3.7) through a numerical simulation of convective diffusion of discrete vortices
generated on the cylinder boundary. The advantages summarized by Sumer and Fredsoe,
(1997) are (1) the inviscid theory can be employed, (2) numerical diffusion problems
associated with the vorticity gradient terms in Eulerian schemes are, to a large degree,
avoided, (3) there are no zone assumptions to require matching of an outer to an inner
flow, and (4) the method is relatively stable and well suited to vectorization on
supercomputers (Stansby and Isaacson, 1987). For a detailed review of the vortex
methods see Sarpkaya (1989). For oscillatory flow and waves the vortex methods fail to

34
agree with experiments within the Keulegan-Carpenter range between 1 and 2.5. The
third group of numerical methods is.based on the hydrodynamic stability approach, in
which the formation of vortex shedding is viewed as an instability of the flow in the wake.
As seen in Section 2.2.1, the oscillatory flow becomes unstable above a critical KC
number due to spanwise-periodic vortices the so-called Honji instability. After modeling
the phenomenon numerically, Zhang and Dalton (1995) have obtained a definite 3D
behavior regarding the variation of vorticity, and found that the sectional lift coefficient
has a strong spanwise variation.
2.5, Some Thoughts About the Current State of Knowledge
For a long time scientists and engineers have tried to approximate natural
phenomena with different mathematical formulations, brought into life with a variety of
numerical techniques, validated or rejected through experiments. Keeping in mind
Poincares words: Mathematics can never tell what is, it can only say what would be if,
the current state of knowledge about floating bodies is a direct result of all these trials and
errors, and can be classified as going simultaneously in three main directions:
The first approach is to directly solve for the nonlinearities of the Navier-Stokes
equations with massive computing power. With the help of supercomputers and
mature numerically-stable methods, the existing mathematical models have succeeded
in increasing our appreciation and understanding of some natural phenomena, in
particular ranges of Reynolds and Keulegan-Carpenter numbers. Other Re and KC
ranges of practical interest require different or improved mathematical and theoretical

35
modeling. It is evident that oscillatory and wave-flows bring additional challenges for
the direct approach to the nonlinear problem.
The second approach is to utilize a Stokes perturbation expansion, which restructures
the Navier-Stokes equations into a sequence of linear boundary-value problems
formulated for each order of perturbation in terms of the power of a small parameter,
for example wave-steepness. With gradually increasing complexity, any-order solution
can be obtained, provided the solutions of previous-order problems are known.
Predominantly second-order and third-order solutions of the fluid-body interaction
have been investigated, with the noticeably preferred simplifications of axisymmetrical
bodies and symmetrical heaving motion for the case of third-order problems. To
explain this preference it suffices to recall that even for the second-order problems, a
great effort is required to deal with the nonhomogeneous boundary condition on the
free surface of the fluid, as discussed earlier. Nevertheless, the role of perturbation
analysis in fluid-structure interaction studies is evident in providing valuable
information about the significance, contribution, and limits of the higher-order
nonlinear effects, as compared to the first-order solution.
The third approach is to work with the linearized diffraction and radiation theories and
with the powerful principle of superposition, which provides insight into the fluid-
structure interaction problem in irregular seas. Naturally semi-analytical solutions are
much faster than the numerical ones because of the smaller number of computations
needed. Moreover, because of the exact boundary conditions semi-analytical solutions
are principally more accurate as compared to the approximate boundary conditions in
numerical solutions. Of course there are greater restrictions on body-shapes with

36
semi-analytical than there is for numerical solutions. One of the main advantages of
the linearized theory is that one can explore in real time the influence of variation of
different parameters like draft, shape, size on the hydrodynamic properties, forces, and
hence fluid-body dynamics. Another advantage is that the linearized theory provides
the easiest way to analyze and comprehend the physical concepts, and meaning and
significance of observed natural phenomena. Ones seeing the big picture, further
improvements can be made, if desired and feasible, with the help of the other two
approaches.
In a historical retrospective, it is true that the three main approaches have had
great impact on the floating-body studies. Having a common goal, they have influenced
and stimulated each others evolution. All three approaches have their own unique
advantages and disadvantages and areas of applicability.
In the present work, a new semi-analytical method is proposed for solving the
dynamics of free floating two-dimensional horizontal cylinders, of various shapes in heave,
surge and pitch. The method is based on linear radiation theory and is intended to be the
first step in the development of a mathematical model and computer program for
predicting the response of an axisymmetric, three dimensional, free floating drifter buoy
subjected to waves.

CHAPTER 3
FORMULATION OF THE PROBLEM
3.1. General Description of the Problem and its Simplifications
The most general formulation of the problem of the dynamic response of a free
floating body subjected to waves is to pose a dynamic equilibrium of forces and moments
in and on an elastic body freely moving in the air-water interface. The focus of the present
work will be on the floating body motions due to external loads, which act on the
underwater part of the body. Therefore two restrictions will be made right from the
beginning: first the body is considered rigid; and second the direct influence of the air
environment on the body is negligible. As long as no structural or vibrational problems
are to be dealt with, the first restriction can be made without any hesitation. The second
restriction is based on the fact that the density of air is roughly one one-thousandth of the
density of water. Provided that the above-water part of the floating body is close to the
water surface, and excluding strong winds during adverse weather, it is clear that for most
practical problems aerodynamic forces may be neglected with respect to hydrodynamic
forces.
.3,1.1. Incompressible Fluid Assumption
A coordinate system Oxyz or simply x=(x,y,z) which is fixed in space will be used
in this analysis. The Oxy plane coincides with the still water surface, and the vertical axis
37

38
Oz is positive upward. The origin O is the intersection of the centerline of the floating-
body-section and the still water surface. In a wide variety of gravity wave problems, the
variation of water density is insignificant over the temporal and spatial scales of
engineering interest. The relationship between water density and pressure is given by
1 Dp _1 DP
p Dt E Dt
(3.1)
where (.) = (.)+ u.V(.)
Dr dr w
is the total derivative in space and time, u = (u,v,w) is the
fluid particle velocity, P is the pressure, p is the water density, and E = 2.07 x 109 [Pa] is
the water bulk modulus. Since an increase in pressure of 1 MPA results in 0.05% change
in water density, for practical purposes the fluid can be considered incompressible.
3.1.2. Governing Equations and Definitions
Using Meis notation (1989), the motion of fluid around a body is governed by the
fundamental conservation of momentum law or the Navier-Stokes equations
(d ^
(P \
4-u.V u = -V
+ gz
{dt )
l p ;
+ vV2u
(3.2)
and the conservation of mass law or continuity equation
V.u = 0 (3.3)
where v= 0.01 [cm2/sec] is the kinematic viscosity, and g=9.814 [m/sec2] is the
acceleration of gravity. Dots represent the scalar multiplication of two vector quantities
(Batchelor, 1967). Defining the vorticity vector as the curl of velocity vector
fi(x,/) = V x u(x,f) (3.4)

39
an important deduction from the Navier-Stokes equations is that the rate of change of
vorticity is due to stretching and twisting of vortex lines and to viscous diffusion.
^ + u.vja = aVu + vV2Q (3.5)
Defining a streamline as a line tangent everywhere to the velocity vector, it is obvious that
the physical concept of streamlines must exist in a general three-dimensional compressible
flow (Dean and Dalrymple, 1991). A Streamfunction exists in two-dimensional or
axisymmetric incompressible flow and is defined by
ay
dy
= u
and
ay
dx
= -v
in Oxy-plane,
or
(3.6)
= u and = -w in Oxz-plane
dz dx
For two-dimensional incompressible flow, equations (3.4), (3.5), and (3.6) reduce to the
2D vorticity-transport equation
+u.v)q = vV2H
U )
and the Poisson equation
(3.7)
a2y a2y
dx2 dy2
= -Q
(3.8)
3,1.3. Inviscid Fluid Assumption
In water the kinematic viscosity is small: v = 0.01 [cm2/sec], which means that the
last term of equation (3.5) is negligible, except in regions'of large velocity gradient and

40
strong vorticity as mentioned earlier.
Fig. 3.1 Wave force regimes (Hooft, 1982). Importance of
viscous and potential effects as functions of wave height-to-diameter
ratio and diffraction parameter
As observed from Figure 3.1, Hooft (1982, Figure 4.1.6) has shown that
for a fixed vertical cylinder and relatively steep waves, with ratio between wave
height and wave length: H/L=0.1, viscous effects become important when H/B
>15 and kB < 0.06, where k is the wave number. Practically this means that if the
characteristic dimension of the fixed vertical cylinder B=1 [m], the viscous effects
become important for wave heights H>15[mJ, and wave periods T>9[sec]. When
B=2 [m], the viscous effects become important for wave heights H>30[m], which
in reality can be referred to as extreme wave conditions. Intuitively, it is reasonable
to expect that for a free-floating body the velocity gradient will be significantly less
than for a fixed vertical cylinder. Therefore the fluid can be considered inviscid,

41
and the Navier-Stokes equations (3.2) are transformed into Euler equations:
f- u. V u = -V
dt
P
+ gz
VP ,
(3.9)
3.1.4. Irrotational Flow Assumption
For an inviscid and incompressible fluid, where the Euler equations are valid, there
are only normal stresses acting on the surface of a fluid particle; since shear stresses are
zero, there are no stresses to impart a rotation on a fluid particle. Therefore any
nonrotating particle remains nonrotating, but if initial vorticity exists, vorticity remains
constant (Dean and Dalrymple, 1991). An important class of problems is one where
QsO and is called irrotational flow. For an inviscid irrotational flow, the velocity u is
usually expressed as the gradient of a scalar function called velocity potential. It exists in
two-dimensional and three-dimensional irrotational flows
u = VO (3.10)
Conservation of mass (3.3) requires that the velocity potential satisfies Laplaces equation
V2O = 0 (3.11)
while conservation of momentum (3-.2) transforms into
50 1 1^,2
+ VO
dt 2' 1
(3.12a)
Integrating (3.12a) with respect to the space variables away from the body, we derive
Bernoullis equation
50
+|VO|2 +gz = ~+C(t)
dt 21 1 p W
(3.12b)

42
with a temporal constant C(t) which can be omitted by redefining the velocity potential
without affecting the velocity field.
3,1.5. Dynamic Free Surface Boundary Condition fDFSBO
A characteristic feature of the free surface of air-water interface is that it can not
support pressure variations and hence must respond appropriately to maintain the pressure
uniform. For the practical interest, the wave length is so long that the surface tension is
unimportant; the pressure just beneath the surface must be equal the atmospheric pressure
Pa above. Therefore the DFSBC is defined as a uniform pressure along the wave form on
the free surface. Applying Bernoullis equation (3.12) on the free surface, we have
,on: = (
(3.13)
3.1.6. Kinematic Free Surface Boundary Condition (TCFSBC)
The instantaneous free surface of a wave can be described with the equation
F{x,y,z,i) = z-fe,y,t) = 0
(3.14)
where £(x,y,t) is the displacement of the free surface about the horizontal plane z=0.
Defining material surface as surface (air-water) across which there is no flow, a particle
remains and moves only tangentially on that material surface when
(3.15)
which gives the KFSBC

i
43
d£+^K+d ,onz = (
(3.16)
dt dx dx dy dy dz
Taking the total derivative of (3.13), the two surface boundary conditions (3.13)
and (3.16) may be combined in terms of the velocity potential
Dt p
+
d20 dO du 1
dt2
g +
dz dt 2
+u.Vir
= 0
, on z = C,
(3.17)
which is the combined kinematic-dynamic free surface boundary condition (CFSBC).
3,1.7. Sea Bottom Boundary Condition (SBBC)
On the sea bottom, denoted with B0, (3.14) becomes
F(x,y,z,t) = z+h(x,y) = 0
where h is the water depth. Applying (3.15) on the above equation, the SBBC becomes
c dh dO dh 50
dx dx dy dy dz
, on z = -h(x,y)
(3.18)
3,1,8. Wetted Body Surface Boundary Condition hSV)
Let the instantaneous position of the surface of Sb be described by the equation
F(x,y,z,t) = z-f(x,y,t) = 0 (3.19)
Using the same procedure as in 3.1.6, we state the continuity of the normal velocity with
dt dx dx
+
do df
dy dy
do
dz
, on z f(x,y,t).
(3.20)

44
3.1.9. Linearization to First- Order Theory
Following Mei (1989), the derivation below leads to the complete first-order
theory and shows how higher-order extensions can be made. For small-amplitude motion,
we expand/in powers of the wave slope e- A! L, which is the small parameter in the
perturbation analysis
z = f(Q\x,y) + ef(x){x,y,t) + e2 f(2\x,y,t)+... (3.21)
where f(0\x,y) represents the wetted body surface rest position, that is The
velocity potential can be expanded in the same manner
0 = £(I)+,/ Considering small body motion, any function evaluated on Sb may be expanded about
:z = f(0\x,y). To the order 0(e), equation (3.20) can be written as
L1)/i0) + + /(1) = on z = f(0]{x,y) (3.23)
It is necessary to find Let the center of rotation of the rigid body be Q, which has
the following moving coordinate:
X(t) = X{0]+eX{l)(t) + s2X{2)(t)+..'. ,X=(X,Y,Z) (3.24)
where X(0) is the rest position of 0 independent of time. In some cases for example a
moored buoy the center of rotation may not coincide with the center of gravity, denoted
by C. Let x = (x,y,z) be the coordinate system fixed with the body in a way that x = x
when the body is at its rest position. Denoting the angular displacement of the body with
£n)(t) = s(a,P,y) with rotational components about axes x, y, and z, the two coordinate
systems are related to the first order by

45
X = X + £
X(1)+0(1)x(x-X(o))]-h3(>2)
x = x-s
X(1) + 0(1) x ^x X(0))] + 0{s2)
x = x-e
z(1)+ye(z-z{0))-r(y-Y{0)^
y = y-s
Y{1)+r(x-X{0))-a{z-Z{0))
z = z e
Z{l)+a(y-Y{0))-p(x-X{0)f
(3.25)
(3.26)
When the body is at its rest position, then x = x and
z = /<>(*, jO (3.27)
Substituting (3.26) into (3.27) expanding about S\ and comparing with (3.21), results
in
/(1) = Z(1) + a(y 7(0)) p(x X(0)) /j0) [x(1) + fi(z Z(0)) r(y ~ 7(0) )
-/;o)[7(1) +r(x- X(0)) a(z Z(0))
(3.28)
Substituting (3.23) into (3.28), results in the first order kinematic boundary condition on
the wetted body surface
-M0) -?/,'01+P,{z-zm)-r,(y-Ym)
-/."[if +r,(x-X(0|)-a,(z-Z<0))] (3.29)
+Z')+a,(y-Ym)-fil(x-Xm)
The unit normal vector it directed into the body becomes
n = (-/iV/f.lf 1 + (/i0))J +{4})2
-1-1/2
Equation (3.29) can be rewritten as
30
(i)
dn
xM+0Wx(x-X)].n = £^
J 1
(3.30)
(3.31)

46
where
{jr}={x(,),0"} = {xi'\r,z<-'\a,p,r}
{} = {n,(x-X(0)) xnj = [n,, n2,,-[,(? -Zi,)-n,(y-i'v'1)
-[,(* -r(0))'-n,(z Z(0))],-[,(> F(0>)-2(x X<">)]}
(3.32)
(3.33)
The physical meaning of the boundary condition (3.31) is that at any point on the wetted
body surface, the normal component of the velocity of the body should be equal to the
normal component of the velocity of the fluid at that point. Finally, assuming that the
atmospheric pressure Pa is constant, which is true over large sea areas, and applying the
known expansions into (3.17), (3.18), the hydrodynamic problem is completely linearized
V20 = 0 in the fluid domain (3.34a)
520
r+£ =
dr
ao
dn
dz
= 0
on the free surface
on the sea bottom
ao dXa
= / ^n
Dn di a
, on the wetted body surface
(3.34b)
(3.34c)
(3.34d)
with the exception of a boundary condition on the lateral boundaries, which will be added
later. Thanks to the linearity of the Laplace equation and the boundary conditions in
Equation (3.34), the problem of the response of a floating buoy to irregular waves can be
reduced greatly with the principle of linear superposition of motions. What actually
remains is to study the oscillations of a rigid floating body subjected to a simple harmonic
excitation due to a train of surface gravity waves.

47
3.2. Floating Body Dynamics
3.2.1, Conservation of Linear Momentum
Let the entire mass of the floating body be denoted withM, part of which is above
the free surface, and let the center of mass be denoted with C: xc = (xc,yc,zc).
Integrating the linearized version of Euler equations (3.9), i.e. without the nonlinear term
in the left hand side, along the wetted body surface, the conservation of linear momentum
states
Mxctt = JJ Pnds Mgk (3.35)
where k is unit vector of Oz axis. Using the linearized Bernoulli equation
P = ~pgf +0(s2)
and (3.26), (3.35) can be written as
(3.12c)
eM
+ 0^ x (xc X(0)= JJ(-pgf ep&^nds- Mgk + 0[e2)
(3.36)
The zero-order portion of (3.36) is
0 = JJ (-pgf^nds Mgk
St
while the first-order portion of (3.36) is
M[X(1) + S x (xc X(0))] = JJ [~pgf[l) pounds
(3.37)
(3.38)
Considering the buoyancy term -pgf of (3.37), and having that on the instantaneous body
surface Sb
nds-[-fx ,-fy,\)dxdy

48
we can replace the domain of integration Sb with the part of the water surface cut out by
Sb, denoted with SA. With an error of 0((?) we can replace the integration over Sb and SA
with integration over S0) and when the body is in its rest position. Lets denote the
instantaneous volume of the displaced water with V, and in rest position with V{0), and let
A{0> be the area of S^. Following the procedure shown in Mei (1989) results in
Archimedes law for the zeroth-order
Mg = pgV{0) (3.39)
The first-order equations are
m[z+a(r r,0>) p(r A'"1)]=-pjf 4>i Vr- (/> I,Afi+Z< V0))
M
Af+fi,(r -zm)-r,{r- f4'1)]=-pJJ fi'S*
.,(o)
- <*(? Z<0))1=-pSI 9'^ds
?<)
If = \\[x-X{0))dxdy
s<>
I = li(y-ym)dxPy
.?()
(3.40b)
(3.40c)
(3.41)
3.2.2. Conservation of Angular Momentum
If Vb represents the volume of the whole body including the part above the free
surface, dm is the body mass per unit volume, and the center of mass is C:
xc =(xc,yc,zc), suchas JJJx xdm f Mxc, the conservation of angular momentum
v>
requires

49
JJJx x JJ x x Pnds + xc x (-Mgk)
ir dt p
(3.42)
Taking the cross product of (3.35) with X, and subtracting the result from (3.42), results
in the conservation of angular momentum with respect to the center of rotation Q
JJJ (x X) x ^-j^-dm = JJ(x X) x Pnds + (xc X) x (-Mgk)
which following Meis procedure leads to
x-component:
44" 44" + (4 + 4, K 4/4 4r, = -p JJ 4'V*
sr
-pg[z(l,4 + a(42 + 4) pi* yl\] + Mg[a(r Z(0>) y{? 4>)
y-component:
44" 44"+(4= + 4)4 4r 4 = -p if 4 V*
4s
+^g[z(1>4 +<-4(4 +4,')+4]+Mg[/?(r-z(0))-r(j-401)
z-component:
44" 44" +(4i+42 )r,, 4. 4A = -/>JJ 4V*
4
where the first and second moments of inertia are defined as follows
4=JJJ(*-4>) 4,=J}\{x-X^dm
(3.43)
(3.44a)
(3.44b)
(3.44c)
Ibn=\\\{x-X^){y-Y^)dm

50
3.2.3 Matrix Form of the Dynamics Equations
The linear system of equations (3.40) and (3.44) can be written in matrix form as
M{^} + [C]W = -pJJ^{4* (3.45)
where [M] is the mass matrix and [C] is the buoyancy restoring force matrix. For a long
horizontal cylinder with its cross-section in the Oxz-plane with incident waves travelling
along the Ox-axis, the motion can be described working with a unit-cylinder length in the
^-direction. For this two-dimensional case the displacement vector is
{X} = {x[x),Z(x\p)T (3.46)
The normal to the wetted body surface vector is
{}=
nx,n2,nx(z-
Z|0))-n,(*-X(0))]
T
(3.47)
(note that
nxds =
S
1
II
^3
N
(3.47a)
The mass matrix is
-
M
0
m(zc -;
j())
M =
0
M
-m(f -
X>)
(3.48)
Ml
r z(0
) -m[xc-X{0))
Iu+I
b
33
and the buoyancy restoring force matrix
is
0
0
0
[C]=
0
PgA
-Ptft
(3.49)
0
~PgI?
Ml
o
1
N,
o
where

r
51
fix -
I\\ = f(*~ ^(0>) &
= ff (z Z^Adxdz
J \ /
c(0)
J \ }
M
J J \ /
yW
i
o
T
In=H(x-X{0))2dm
(3.50)
vb
3 3. Decomposition and Separation of the Hydrodynamics from the Body Dynamics
The dynamics of a freely floating body subjected to a train of harmonic incident
waves is described by (3.45). This matrix equation is a second order differential equation
and represents a dynamic balance of forces. The forces on the left-hand side are inertial
and buoyancy forces with mass and restoring matrices, known from the hydrostatic
equilibrium in the absence of waves. These forces are proportional to the unknown
generalized body displacements, and are balanced on the right hand side of (3.45) by
hydrodynamic forces, which are functions of the unknown velocity potential. The
unknown velocity potential can be determined from the hydrodynamic system (3.34), only
if and when the generalized body displacements are known (see (3.34d)). One way to
resolve this problem is to decouple the hydrodynamics from the body dynamics with the
following decomposition (Haskind, 1944):
dX.
dt
f- = Re(Vae~')
(3.51a)
O = Re(ife,<2")
(3.51b)
<¡> = 0
R(adiation)
+
Diffraction)
(3.51c)

52
where Va is the time amplitude of the generalized body velocity; is the time amplitude
of the velocity potential; co is the wave circular frequency; and the imaginary unit is
defined as / = 4~T. At this point both the velocity potential O and its time amplitude
have the same dimensions, which is [Length2/Time], For the sake of brevity the notations
for the real parts will be omitted and only the complex forms will be used instead, but only
the real parts have physical meaning. Due to the linearity of the system (3.34) two
contributions of a different nature can be separated entirely (see 3.51c). Therefore two
subproblems arise, each of which is more tractable than the complete problem:
(a) radiation sabproblem. the rigid body oscillates harmonically in an
otherwise undisturbed body of water, thus generating waves which propagate or radiate
away from the body. Physically the corresponding radiation velocity potential (f>R is a
direct result of the motion of the body and should be proportional to this motion.
Therefore <¡>R = ^Va(f>a where the summation is over the elementary components of the
a
body velocity 6 in 3D-space, and 3 in 2D-space. <¡>a has the dimensions of [Length], and
stands for the velocity potential caused by a body oscillatory motion with unit velocity in
the a-direction: for example in heave, surge, or pitch.
(b) diffraction subproblenr. the forces on the rigid body, fixed in space, are
caused by a train of harmonic incident waves. The velocity potential for the incident
waves is (f)1, while the velocity potential for the scattered waves (defined in Section 2.2.1.
as sum of reflected and diffracted waves) is denoted by s. Define the velocity potential

53
for the diffraction subproblem as s. The dimensions of (j)1, s, and D are
[Length2/Time],
Thus the necessary decomposition of the time amplitude of the velocity potential is given
by
*=*D+^=(*J+^)+ErA (3-51d)
Next the complete hydrodynamics problem, (3.34), is reformulated in terms of time
amplitudes of the diffraction velocity potential with (3.51):
VVD =0 in the fluid domain
d£_
dz
dx J
, on the free surface Sf
, on the sea bottom Ba
, on the wetted body surface- Sb
, waves outgoing at infinity
(3.52a)
(3.52b)
(3.52c)
(3.52d)
(3.52e)
The only missing lateral boundary condition in (3.34) is now given by (3.52e) in 2D-space.
Sommerfeld had introduced a similar lateral boundary condition, in an analogy with an
outgoing wave system. The incident velocity potential is given by
^-^cosh*(£+*) e_ (3.53)
co cosh kh
In a similar way, the radiation subproblem is formulated as
V2(j>a = 0 in the fluid domain
(3.54a)

54
dz
a=o
g
, on the free surface Sp
(3.54b)
da
dz
= 0
, on the sea bottom B0
(3.54c)
dn
=
, on the wetted body surface- Sb
(3.54d)
lim
, waves outgoing at infinity.
(3.54e)
In a summary, the hydrodynamics (3.52d)+(3.54d) has been decoupled from the body
dynamics (3.34d) with the help of decomposition (3.51). Now all that is needed is to
solve the hydrodynamics (3.52)+(3.54) first, and then deal with the body dynamics (3.45).
3,4. Hydrodynamic Properties and Forces
The hydrodynamic properties will be derived in 3D-space using double integration
over the wetted body surface. In 2D-space the same properties can be expressed with a
single integration over the wetted body surface. Working with the hydrodynamic pressure
of the linearized Bernoulli equation (3.12c), and with the decomposition made before, the
generalized hydrodynamic force on the body in the a-direction is decomposed into its
diffraction and radiation components:
K = JJ Pnads = \\{-p^nads = Re<
ia>p\\(R)nads
F + Fi
where

55
F=RejXV"},
FaD = imp
JJ nads
(3.55)
F*=R Fa = ivpjj fnads = £ Vpfpa fpa = G>p\\ p^js
The diffraction force is Its time amplitude Ff is known in the literature as the
exciting force on a stationary body due to diffraction. The radiation component, the
matrix \fpa ], is known in the literature as the restoring force matrix, and F* as the
restoring force. The radiation component (Mei, 1989) can be expanded further by
defining the added mass and radiation damping matrices, namely
I ) 1
[p\ Fpa = Re PH pnadS = -lm{ffia) > ^d
V s
(3.56)
[A]. Xpa = Im
Pjj pnads =-Re(/^)
Vs )
(3.57)
The index notation ppa denotes the added mass, which causes a force in direction /? due
to acceleration in direction a The index notation Xpa denotes the damping, which cause
a force in direction /? due to velocity in direction a In terms of these matrices the
restoring force is expressed as
^ d2Xfi dXB
F ~~^pa~di2 ^Pa~dT
Finally, with the velocity potential decomposition and the hydrodynamic property
definitions, the dynamics of the floating rigid body (3.45) may be rewritten as
{[C]-m\[M] + [p])-im[X\]{^{FD}
(3.58)
(3.59)

56
where {£} is the time amplitude of the generalized displacements {X}:
{Z) = Re({^>-'i) (3.60)
3,5, Hydrodynamic Relationships. Identities, and Definitions
As introduced and explained in Section 2.3, all necessary relationships and
identities among the hydrodynamic properties and forces will be summarized briefly for
further use, without detailed derivation. For a complete description the reader is referred
to Mei (1989). For any two twice-differentiable functions t, and ^., the Green's
theorem states:
(3.61)
where O is a closed volume with boundary 60 consisting of the wetted body surface Sb,
the free surface SF, the bottom Bo, and a vertical circular cylinder with an arbitrary large
radius Sx If i, and zero due to Laplace equation. By virtue of the boundary conditions (3.54b), and (3.54c),
neither the free surface SF nor the bottom Bo contributes to the surface integral thus
reducing the right-hand side to:
Sb+S<0
dn
dS = 0.
(3.62)

57
If (j)a =(j)n and p = j are two radiation velocity potentials, than the surface integral at
the lateral boundaries vanishes due to the boundary condition (3.54e). This results in
JJ
d ^ 8n dn
iS = 0, or
(3.63)
St,
(3.64)
Therefore, the added mass (3.56), and damping matrices (3.57) must all be symmetric,
regardless of whether the body is symmetrical or not, due to (3.64), namely
M/Sa^api an^ ^-pa=^ap (3-65)
The law of conservation of energy flux requires that the average rate of work done
by the oscillating buoy on the fluid over a period, denoted with Wbuoy, should be equal to
the energy flux far away from the buoy denoted with W,
away
buoy IT away
T 2 <+.r .
W = j Wdt.
where
(3.66)
(3.67)
From (3.58), Wbuoy becomes
(
= -£ F* X. = Y, I /V. xf x.+V x, x.
a a p V 7
(3.68)
Because of the symmetry (3.65) the first term of (3.68) can be written as

58
P
a p
II ^x,x. =1;Ii'Z\mxi,x.+m*x.xX}-'ZZ
a P
dp,
dX^
dt
= 0,
and vanishes due to periodicity. This means that the average rate of work done by the
oscillating buoy on the fluid over a period is
a P
a P
Next, the energy flux far away from the buoy can be expressed as
(3.69)
Wa
Re<
imp re ,R d(j)h
[il*
cfo
ds
sk

k eR W
dn
dn
ds.
(3.70)
Moreover, with the help of (3.62), (3.70) can be transformed to
^=Tii
ir d(j>R -ARdh
f
3i
ds.
(3.71)
When^R = %2^Va(f>a and the two-dimensional asymptotic behavior of c
lim a
X>oo
-igal cosh k(z + h)cjkx
m cosh kh
(3.72)
are substituted in (3.71), and (3.71) equalized to (3.69), the law of conservation of energy
flux expresses damping in terms of Q-l
Kp=pgCg{a~aa-+a:a;) (3.73)
where Cg is the group velocity, and (.) denotes the complex conjugate. Cl^ will be
referred to as the far-field amplitude, that has dimension of time since Cl^ =A+/Va.

Since the average rate of work done by the body on the fluid (energy transmitted to the
fluid by the body) is non-negative, the damping matrix is positive semidefinite. As a
corollary, all diagonal terms of the damping matrix are non-negative, since
(3.73b)
It is noteworthy that the same as (3.72) asymptotic behavior is valid for s, in accordance
with the lateral boundary condition (3.52d), except that <2* will be replaced by another
term with a dimension of length. It is also noteworthy that the damping and added mass
matrices are proportional to the real and imaginary parts of the so-called restoring force
matrix. This means that knowledge of the damping coefficients is sufficient to determine
the corresponding added mass coefficients and vice versa. Such relations, known as
Kramers-Kronig relations, are valid for all modes of motion regardless of forward speed
(currents, ships). Having been derived in a universal way by Ogilvie, they read:
2 rV(^)-VH
(3.74)
(3.75)
oo
(3.76)
Another way to find the added mass, knowing the damping as a function of the circular
frequency, is Hoofts approach (1982) of using the so-called Bode relations, which for
water waves correspond to the Kramers-Kronig relations.

60
1 ^
Upa H ftp* H = - J Rpa{t) Sin {C0t)dt
(3.78)
Rpa{t) is called the retardation function, and is obtained through a Fourier transform of
Xpa(co). Therefore the added mass is related to the far-field amplitudes, (3.74, and 3.78).
As mentioned in section 2.3, a remarkable Haskind theorem relates the exciting
force to the far-field amplitudes in the following manner
(3.79)
where (3.63) has been applied. Since = is true on Sb, then (3.79) becomes
dn dn
Fa -ia>P
(3.80)
Next, substituting the asymptotic forms (3.72) into (3.80) results in
Ff = -2pgC ACl~,
(3.81)
thus expressing the exciting force in terms of the far-field amplitudes. Upon substituting
(3.81) into (3.59), the matrix form of the dynamics equations become
[[C]- co\[M]+[/,])-; Defined as a ratio between the amplitude £,a of displacement in the generalized direction
a and the amplitude of the incident harmonic wave A, the frequency response function
and the corresponding response amplitude operator are
Ha{a>) = SJA, and RAOa{co) = \Ha{(of. (3.83)

61
The purpose of the frequency response function and the response amplitude operator are
to characterize the dynamic response of the floating body as a function of the circular
frequency of the incident wave. The vector form of the frequency response function is
W = ^}^PgCt)[{C]-^([M]+[f,])-im[>.]]'{a-}. (3.84)
3.6. Algorithm for the Solution of the Problem
In summary, the asymptotic solution of the radiation problem gives the far-field
amplitudes. The added mass matrix [//], the radiation-damping matrix [X] (3.73), and the
exciting force (3.81) can be found from these far-field amplitudes. Therefore the
asymptotic solution of the radiation problem alone will be sufficient for analyzing the body
dynamics (3.84). Practically this means that the solution of the diffraction problem can be
avoided. Having the dynamics and hydrodynamics (radiation) problems formulated, and
all the necessary relationships for their solutions, the following algorithm will be used.
First, the asymptotic solution of the radiation problem, in terms of the far field amplitudes,
will be found in heave, surge, and pitch. Second, the hydrodynamic added mass, damping
coefficients, and exciting forces will be computed as functions of the far-field amplitudes;
consecutively the body dynamics will be computed for a particular wave frequency in
terms of H, or RAO. Third, using the RAO (3.83), the spectral analysis can be used to
find the body response spectrum S^co) due to the incident wave spectrum S^co) for the
entire frequency domain of interest, namely
S(co) = \H(vfs(a).
(3.85)

CHAPTER 4
RADIATION PROBLEM SOLUTION
41. Problem Statement and Definitions
A long horizontal rigid cylinder with its cross-section in the Oxz-plane is
oscillating on the free water surface in otherwise calm water. Part of it is below the water
surface; the other part is in the air. The rigid cylinder is oscillating with unit velocity in
the generalized -direction with the circular frequency of the incident waves co (in the
absence of the incident waves), thus generating outgoing waves. Taking a unit-cylinder
length in the ^-direction, the problem is considered two-dimensional, and the motion can
be described in the cross-sectional Oxz-plane (Figure 4.1).
z

Fig. 4.1 Cross-section of the floating body. CB, CG centers of buoyancy and gravity.
62

63
Formulated in (3.54), the radiation problem is stated as a two-dimensional Laplace
equation with a complete set of boundary conditions:
d\-+8\=0
etc2 dz2
, in the fluid domain
(4.1a)
dK %=o
dz g
, on the free surface SF
(4.1b)
^-=0
dz
, on the sea bottom Ba
(4.1c)
dta
It
on
, on the wetted body surface- Sb
(4-Id)
hm{ tc
( dx J
, waves outgoing at infinity.
(4.1 e)
As explained at the end of chapter 3, the practical mathematical problem to solve will be
to find the asymptotic solution of (4.1), particularly the far field amplitudes. For the
convenience of mathematical manipulation, time is removed from the problem by using the
exponential time dependence (3.51), which restated is
{,X,QT = R e({L,77}V;c") (4.2)
With the imaginary unit j = V-1, the wave profile is
£=A cos(kx at) = Re(^e>(fcf- jj=AeJla (4.3b)
As stated before, for brevity the sign Re (the real part of) will be omitted, but accounted
for iathe final results. In the two-dimensional case, the physical meaning of the
Sommerfeld radiation condition (4.1e) is that the generated waves are propagating
outward from both sides of the oscillating body. Therefore, the asymptotic expression of

64
the wave profile at infinity must be:
lim £ = AejXkx-M) = (AeJtx)e'JM = lim rje^, (4.4a)
lim tj = A+elkx = (Vaal )ella (4.4b)
where A+ and A. are the asymptotic expressions of the wave amplitudes, and £7* are the
far-field amplitudes introduced in (3.73).
4.2. Main Idea Behind the Semi-Analvtic Technique (SAT)
Inspired from a Haskind idea (1973), a new analytical solution of (4.1) will be
given for the case of partially submerged floating circular and elliptical cylinders with
varying equilibrium drafts. As will be shown below, the analytical solution is in terms of
infinite power series. Therefore, a finite number of terms must be taken to solve it
numerically thus the prefix semi- and the name Semi-analytic technique (SAT). For
the sake of clarity and completeness, some mathematical definitions and formulations used
in the analysis are presented first. Considering a two-dimensional (2D) wave motion in a
right-hand complex plane Oxz, the real axis Ox is defined to represent the still water
surface and the imaginary axis Oz to be perpendicular to the still water surface and
positive upward (Figure 4.1). Let y be the complex variable, which corresponds to the
complex plane Oxz, and defined as
y=x+iz (4.5)
where i = V-T is the complex plane imaginary unit, which for the sake of convenience is
different from the time imaginary unit j = V-. In the two-dimensional irrotational flow

65
of an ideal fluid, both the velocity potential (¡>(x,z) and stream function y/(x,z) exist. In the
whole fluid domain they have continuous first and second partial derivatives and satisfy
the Laplace differential equation. Therefore by definition (Solomentsev, 1988) both
velocity potential the complex plane these two harmonic functions define the well-known complex velocity
potential
w(y) = 4>(x,z)+ i y/(x,z) (4.6)
and are related to each other as conjugate functions through the Cauchy-Riemann
conditions
d dy/
dx dz
d dz dx
(4.7)
At any point inside the fluid domain the complex velocity potential:
(a) has unique value
(b) has continuous derivatives of all orders
(c) satisfies the Cauchy-Riemann conditions.
Therefore by definitions (a,b,c) the complex velocity potential is a holomorphic function
(Milne-Thomson, 1950). Utilizing the property of holomorphic functions that a
combination of derivatives and integrals of holomorphic functions is also holomorphic, the
following holomorphic function is introduced.
f(y) =
^+*Ay)
dy
,in the fluid domain without St (4.8a)
In terms of its real and imaginary parts, / (y) can be written as

66
(4.8b)
As pointed out in Haskind (1973), the combination (4.8a) was introduced for the first time
by Keldysh in 1935. Comparing the imaginary parts of (4.8b) and (4. lb), the boundary
condition on the air-water interface can be written as:
(4.9)
.,on 2=0
Im {f(y)} = 0
By virtue of the Schwarz Reflection Principle and the boundary condition on the air-water
interface (4.9), it is possible to continue f(y) in the upper half-plane. As a result of the
analytical continuation the values off(y) in the upper half-plane will be conjugate
imaginary of the mirror off(y) values in the lower half-plane. In this manner f(y) will be
holomorphic in the whole complex plane except for the points on the wetted buoy
surface^ and their mirror images denoted by.V (Figure 4.2). Thus, on an abstract
mathematical ground, the problem has been extended from the lower half-plane to the
entire complex plane.
Fig. 4.2 Wetted body surface and its mirror image
Iff(y) were known, then the complex velocity potential would simply be the solution to

67
the ordinary differential equation (4.8a), namely
fluid
w(y) = e-'k[Al+iA1+ J / (y)eiky dy] (4.10)
domain
where Aj and A2 are constants. Therefore the key to the solution of problem stated in
equation (4.1) is to find a convenient form for f(y). A well-known mathematical technique
is to represent the still unknown holomorphic function by a power series, which has the
general form
f(y) = 0 + Y+ + (y To)"+ (4-11)
The power series (4.11) is convergent within a circle [y-y0| < R around the fixed pointya
of radius R = lim
M co
a.
a.
n+l
, and it can be divergent outside that circle \y -y0\ > R 0 (see
Solomentsev, 1988). The power series derivatives and integrals of any order have the
same radius of convergence. Within the circle of convergence the power series
/"O')
coefficients are uniquely determined as Taylors series coefficients an =
n\
By
definition an analytic function is defined as a power series, which within its circle of
convergence is uniquely determined as a Taylors series and possesses derivatives of all
orders (Taylor and Mann, 1983; Solomentsev, 1988). Thanks to the similarity in their
definitions, the analytic function will be the answer for the convenient presentation of (4.8)
holomorphic function in (4.10). At infinity, the analytic function (4.11) can be written as
Ay) = LT+Lt+ZT+lT+- (4.12)
y y y y
From (4.9) it follows that all coefficients y,,(i = 1,2,3,...) are real. Substituting (4.12)

68
into the solution of the complex velocity potential (4.10) results in
w(y) = +4, + J/(Oe^dCI
(4.13)
where Ai+iA2 is an integration constant, C, is a dummy variable, and the integration is
taken over a curve lying in the lower half-plane. Consequently, when y = (x,z) > (oo,z),
the asymptotic expressions of the complex velocity potential are
limw(^) = lim(^ + />0 = {A + iA2)e~,ky = + iA2)e~'kx+k,>z
y>+oo y>+co
limvv(^) = lim {+iy/) = (Bx +iB2)e~,ky = (2?, +iB2)e~'k,,x+k,,z
(4.14)
y->- co
y->-
where
Bx + iB2 = A +iA2 + Jf(y)e'kydy (4.15)
Taking the real part of (4.14), the corresponding asymptotic expressions of the velocity
potential are
lim^(^) = ekz(A, cosk0x + A2 sin&0x)
_y>+co
Ym\(f)(y) = ek2{Bl cos k0x + B2 sin^0x)
(4.16)
y-y~ oo
From the linearized free surface dynamic boundary condition at z=0
<=~
1 c
g a
z=0
g
(D
= 7¡e~JWI
J=0
77 =
( Vi A
HOLy
\g )

(4.17)
Substituting (4.16) into (4.17) and comparing with the asymptotic wave profile (4.4),
co1
results in the following deep-water (k = k0 = ) relationship:
g

i
69
lim/7 =
yH-co
lim^ =
y-+-co
v)j (.Bx eosk0x + B2 sinA:0x) = A_e~jkoX = {VCl~Ycosk0x jsmk0x)
( vi A
J^V
V S )
Therefore the integration constants are
Ax=-jZ-a<
co
A=-a+,
CO
Ax+iA2=-^a+(i-j),
co
(4.18a)
Bx =-jCT,
CO
B2 = ~a~, and Bx+iB2=-^-a-(j+i)- (4.18b)
co co
Substituting the expressions for Ax +iA^ and Bx +iB2 into (4.15), results in
+00
-a~(j+0 = f f(yykaydy
co CO L
(4.18c)
Upon substituting / = j and i = -j into (4.18c) the far field wave amplitudes become:
a- =
/ CO
2 g:
t-oo
\f(y)e**cty
(4.19)
Jl=+J
When the buoy oscillations are the source of wave generation then the integration in
(4.19) can be replaced with an integration over the wetted buoy surface Sb and its mirror
image Sb in the counter clockwise direction. The proof, based on a lemma due to Jordan
(Solomentsev, 1988), is given in Doynov (1992).
a- =
--
1 £ Sb+s
(4.20)
Ji=+;
Therefore, in order to find Or, the unknown coefficients /,, (/ = 1,2,3,...) of the analytic
function f(y) given in (4.12) must be determined.

70
4.3.Semi-Analvtic Technique: Determination of the Unknown Coefficients
The procedure for determining the unknown coefficients in (4.12) consists of
4.3.1. Constructing a boundary condition for the holomorphic function through
integration of (4.8a) over the wetted body surface Sb, so that the left-hand side
contains all unknown coefficients, and the right-hand side contains all terms
derived from the complex fluid velocity.
4.3.2. Conformal mapping of Sb and its mirror image Sb into a unit circle, which permits
expressing the mapped holomorphic function with an analytic function of type
(4.12), convergent outside the unit circle
4.3.3. Determining the left-hand side, so that all multipliers of the unknown coefficients
are trigonometric functions of the polar angle of the unit circle 9.
4.3.4. Determining the right-hand side, so that all term are trigonometric functions of the
polar angle of the unit circle 6. Determining the complex fluid velocity with a
generalization of a procedure outlined in Milne-Thomson (1950).
4.3.5. Determining the unknown coefficients by solving a linear system of equations,
derived through the Fourier expansion of both left- and right-hand sides of the
boundary condition.
4,3.1. Boundary Condition on Sh
Integrating (4.8a) over the wetted body surface Sfein the clockwise direction from
point A to point Y (Figure 4.2), results in

71
£>wv=j,
dw[y)
dy
+ ikw[y)
dy
(4.21)
Substituting (4.22)
fwdy = wy- wAyA fcydy
(4.22)
into (4.21) results in
jAf(y)dy = -ikyAwA + [ikw)y + £^(i- iky)dy
(4.23)
Eliminating ikw in (4.23) and (4.8a), gives a boundary condition whose right-hand side
dw
(RHS) is a function of the complex velocity and complex variable >>,
dy
\YJ(y)dy-yf(y)+ih>AwA =
(4.24)
As stated before, only the real part of (4.24) will matter after deriving its explicit form.
Some thoughts about the uniqueness of the solution of the boundary condition on Sb are
given in section 4.3.6.
4,3.2. Conformal Mapping
If the wetted cross section of the floating buoy is a half-submerged circle, then the
analytic presentation (4.12) in the left-hand side (LHS) of the boundary condition (4.24)
can be used. In this case the unknown coefficients can be determined from those
corresponding to the generalized motion RHS. If the wetted cross-section of the floating
buoy is different than the half-submerged circle, then Sb can be mapped into a half-
submerged circle C. Denote the analytic conformal mapping by

72
y=m (4-25)
It transforms the holomorphic function f(y) from the original plane Oxz:[y = x + iz = re"p)
into a holomorphic function f(£) in the transformed plane O^tj:= <^+irj = pe,e). This
can be written as
/W=A Ad) =/(?)=^/
The conformal mapping properties are:
df(C)
a) Angles between vectors are preserved as long as ^ 0
UCy
(4.26)
b) A vector is dilated by
dy
dC
dt;
and
c) A vector is rotated by arg
r a
dm
dC
v y
At the infinite point of the transformed plane 0%r¡ the following analytic presentation can
be used,
/( n= 1 b
(4.27)
4,3,2.1, Conformal mapping of more than half-submerged circle
If R is the radius of the circle and h is the distance between the center of the circle
and the still water level (Figure 4.3), then the wetted body surface and its mirror image
can be written as

73
y = -ih + Re', Sb\

y = ih + Re' and Sb:

Fig. 4.3 Conformal mapping of more than half-submerged circle
If the conformal mapping
y = f(£) = ihl+a£hl and
dy df(0
dC d = 0 ,
(4.30)
(4.31)
where h¡ and h2 are real and a is complex, is used to transform Sb (4.28) into the lower
half of the unit circle C: C, l.e'0, where 0 e[-7r,0], then

(4.32)
x + ?- K
(4.33)
a = Re'1, and
(4.34)

74
hl=-h. (4.35)
If Sb (4.29) is transformed into the upper half of the unit circle C: £= \.e'e, where
0 e[-2n~n\, then
(p-2K+2(pA+h20, (4.36)
a = Re3'^, and (4 36)
hl=h (4.38)
4.3,2.2, Conformal mapping of exactly half-submerged circle
This is a particular case of 4.3.2.1, with h- 0, The result is
y = Reip > Sb:

The conformal mapping which transforms Sb (4.39) into the lower half of the unit circle
C: £ = l.e'e, where 6 e[-^-,0] is
y=/(£) = &. >and
dy df{Q
dC
= R* 0 .
(4.40)
(4.41)
4.3.2.3, Conformal mapping of more than half-submerged ellipse
If a and b are the vertical and horizontal semi-axes of the ellipse and h is the
distance between the center of the ellipse and the still water level as shown in Figure 4.4,
then the wetted body surface and its mirror image can be-written as:

75
Fig. 4.4 Conformal mapping of more than half-submerged ellipse
y = -ih+bcos[j]) + iasin(77) Sb:rj e[7C-T¡A,27V+TjA] (4.42)
y = ih+bcos{Tj)+iasm(T}) Sb:r¡&[-7]A,7i + rjA]. (4.43)
If the conformal mapping
y = f{g) = ihx + + a2CK and (4.44)
A
% = .r*1"')=*,(,<* 0 (4.)
UL, UL,
is used to transform Sb (4.42) into the lower half of the unit circle C: £ = l.e10, where
0 e[-7T,0], then
h, = -h, (4.46)
c¡ = (b+a)/2, (4.47)
A = (b-a)/(b+a), (4.48)
, = c1e"7/1, (4.49)
a2 = ^/le'"7-4 ,
(4.50)

76
tj=2tt + tja +lt,6, and
(4.51)
(4.52)
TV
If Sb (4.43) is transformed into the upper half of the unit circle C: £ = l.e,e, where
6 e[-2 n-n], then
h¡ = h,
(4.53)
Ci = (b+a)/2,
(4.54)
A = (b-a)/(b+a),
(4.55)
ax = cle3,r,A,
(4.56)
a2 = cxAe~3,r,A,
(4.57)
7i-2tt + 3t]a Jrh19, and
(4.58)
= n+2rjA
(4.59)
n
4,3,2,4, Conformal mapping of exactly half-submerged ellipse
This is a particular case of 4.3.2.3, with h = 0, 774 = 0, \ 0, and = 0, and
results in
y = bcos^T]) +iasm(?j) ,7e[0,2^]. (4.60)
The conformal mapping which transforms Sb (4.60) into the lower half of the unit circle
C: £= l.e'9, where 0 e[-7r,0] is
y = f(C) =
ci
r
V
and
(4.61)

77
dy df{C)_
dC d£
= c,
A
*0.
(4.62)
4.3,3, Left-Hand Side of the Boundary Condition on Sb
To find an explicit form of the left-hand side of the boundary condition (4.24) wA,
the complex velocity potential at point A must be found (Figure 4.2).
4 3 3.1. Complex velocity potential at point A
Substituting^ fory in (4.13) will give the following expression for wA
wA = w(yA)
= e~,kyA
K +iA + jfiy^dy]
(4.63)
Since the constants Ax, are proportional to the far field wave amplitudes in (4.18a)
Ax=-j^a\
CO CO.
equation (4.36) takes the following form:
w
yA
= | Ax cos(Ax^) + ^2 si^Ax^)-!-/^, cos(fcr/1)- Ax sin(AxJ)] + e~,kyA J f(y)e,kydy
(4.64)
By virtue of the conformal mapping (4.26, 4.27), which will be proven below in 4.3.3.1.1
and 4.3.3.1.2, both terms in (4.64) can be expressed in terms of the unknown
coefficients an
£7* = AaA+(w) and
n= 1
(4.65)

78
-'* ] f(y)edy = a, (7> + IQ.),
(4.66)
thus giving expressed as a series of the unknown an coefficients
n=l
WA =EaW
where
(4.67)
w_ =
-jV'i.Wtf,
+/
CO
(4.68)
4.3.3,1,1. Determination ofP+/^
+/£) (4.66) can be expressed in the following manner
^]nyy-dy=fJ.(P.+/&).
+co
4.3,3.1.1.1. Determination of P+iO. Exactly half-submerged circle
Using the conformal mapping (4.40), (4.41) results in
I = J JiyYdy = Ra. j e'^CdC = K a,/, (4.69)
+00 **=1 00 W 1
where / can be determined knowing h and the following recurrent formula
/= j emCC"d( ["+/,]
(4.70)
The first integral can be expressed with a complex exponential-integral function
1 me kR it
/, = J d£ = Jdt = -Ex{-ikR), (Gradshteyn and Ryzhik, 1980) (4.71)

79
In+\ ~
ikR.J -e
m
and
n
Pn +Qn = e ikXARI
(4.72)
(4.73)
4,3,3,1,1,2 Determination of P+iO,. More than half-submerged circle
The use of conformal mapping (4.30); (4.31) results in
r = ]/(y)e* M=l
-kh oo
1 =
ik
n=l
1 jW* 00
Jka^
1 1
lev*-* = 2>,
00 ^
e
+JVteiV("+1)^
00
-kh oo
lK n=1
where I is
In=j^n+%ika^dc.
1
After changing the variable of integration £ with t = and correspondingly
1
d£=--^dt, In becomes
Substituting (4.78) into (4.75) gives
, e~kh^
!=-,r2>.
k n=l
(/far)' 1
e +2j
j=0
5! shi-n
(4.75)
(4.76)
(4.77)
(4.78)
(4.79)

80
Substituting (4.79) into (4.66) results in
Pn+iQn =
-ie
-khl~ikxA
Aka
A (ika)s 1
s\ sh^-n
(4.80)
4,3.3,1,1.3, Determination of P+iOv. Exactly half-submerged ellipse
The use of conformal mapping (4.61), (4.62) results in
yA co 1/ _j\ co
I = = C(1"K'1 )d£ = ct Z a[7 ~^+2] >
-f CO W=1 CO
w
Jkc^XC1)^ (ikCxX)
s=0
s!
/, = j -L [eto' +( + *)/,],
CO *"1
L Pikc£ k<2 e
h = J^ = J y = Ei{K)>
co oO
1 rt+.+l
n + s
and
(4.81)
(4.82)
(4.83)
(4.84)
(4.85)
(4.86)
4,3,3,1,1.4, Determination of P+iO, More than half-submerged ellipse
The use of conformal mapping (4.44), (4.45) results in
/ = ]f(y)e,kydy = e~*aHj-a2CK)dC = ,(4.87)
4-00 W = 1 CO
W
¡fc(a1^+ct2^)/,_
C'A!(a1f>-a!rK= j/C*'
1 f ikla^+arf-*1)
(4.88)

81
ik
eik[a¡+a2) jrn^eik[a^hl+a^ h)^-(n+x)^^
J_
ik
ik(ai+a2)
P.+iQ. i.
(4.89)
(4.90)
4,3.3.1.2. Far-field wave amplitudes
a* =
l CO
§f(y)eikydy
2 % Sb+St
= YkanD(n)
-**=+J
w1
D(n) has to be expressed from (4.65) and (4.41b).
(4.91)
4.3.3.1.2.1, Far-field wave amplitudes: Exactly half-submerged circle
Denote the integral (4.41b), upon which the far-field amplitudes depend with I.
The use of conformal mapping (4.40), (4.41) results in
oo ou
/= §/(y)e,,dy = R'£a, §e"(C"d(=
Sb+Sb =1 C: |i|=l
= 1
2 n
\
eikRe e-i{n-l)9id0
and
n=1
Wide
e|
5=0 -2jt
= R'Ean
n=1
2 m
(ikR)
n-1
(-!)!
From (4.91) and (4.93) it can be concluded that
D(n)
icoR
2 g
2m
. (ikR)
7
n-1 \
(n-1)!
tvcoR
>=*;
g
(ikR)
n-1
(w-1)!
(4.92)
(4.93)
(4.94)

82
4.3.3.1.2.2. Far-field wave amplitudes: More than half-submerged circle
The use of conformal mapping (4.30), (4.31) results in
/= = (4.95)
Sb+S "=1 C "=>
where the In integral is to be calculated over the unit circle (Figure 4.3)
/ = Uka C:|f|=l
In a cylindrical coordinate system In can be expressed as
ln 00 lilrrtY 0 , o (ih-r/Y . th(s+l)-n]B
ln = Je^^ide = J d __ (4.97)
s=0
-2*
j=0
5! j, A,(j + 1)-/i
co oo
=1
5*!
/^(s + l)-/?
(4.98)
From (4.91) and (4.98) it can be concluded that
Z)+()=_^M
K) 2g s\
as+xe~k\ e-,[Ms+1H*j + ^4+ig+ArA, ^
/z2(j' + l)-
(4.99)
4,3.3.1.2,3. Far-field wave amplitudes: Exactly half-submerged ellipse
The use of conformal mapping (4.61), (4.62) results in
1= §f(yykycty = can §eikCl{(+XC\n(\-C2)dC = cl'an[Gn-AGn+2), (4.100)
Si+S* n=l C:|f|=l n=1

83
G= je
c-\(\=i
^ {ikCxX) | ktfs-ln+s),
s=0
5!
e,kCl(£-(n+s>dC.
(4.101)
C:|i|=l
Applying the same technique as in (4.92), (4.93), gives
G.-Z
{ikcxX)¡
i=0
s\
2m
(ikcx)
/7+.S-1
(n + s-1)!
-(ikcx)2s+n~l Xs (-\Y(kcx)2s+n~lAs
2m^Yj- = 2mnY/ M : and
i=0
s!(w + 5 1)! ^ i!(s + -l)!
v 2+rt-l
Gn=2mnX-{n-x)/2Y< ^ = 2mnX{n-x),2Jn hkcxJx)
sl(s+n-l)¡ v
,when X > 0
(4.102)
where denotes the Bessel function of the first kind,
Gn=2m\-Xr^
2 j+w1
5) s!(s + n-1)
= 2mn(-X) 1nIn_x{2kcx4-X} ,when X < 0
(4.103)
where denotes the modified or hyperbolic Bessel function of the first kind. From
(4.91), (4.100), (4.102) and (4.103) it can be concluded that
(4.104)
4.3,3.1,2.4. Far-field wave amplitudes: More than half-submerged ellipse
Using the conformal mapping (4.44), (4.45) results in
^a^)di=e-^aJn
/= §f(yYl,dy=e-l'''La.
=1 c:|£|=l
n=1
(4.105)

84
/ = -a2r-K=T I6'"*"
C:||=l '*
-2t
(4.106)
7"=T
ik L
ik(a¡+a2) airvta,k{aie~i,ri' +a1e>*) +nJsi
-e e
+
Jsh
ik .
e,nKe
ik[aie-^,a2e^) _Jk{aie-^,a^) +njSb
-e
(4.107)
js J = jr fel te)-
j=0
5!
;=o
/!
1-e
(/ s)/^ n
(4.108)
and
* fe)' ^ fe)'
e e
-2rr
^ -s! =0 /!
(l-s)h2 -n
(4.109)
From (4.91), (4.107), (4.108) and (4.109)
n ( \ icoe~kK r
DAn) = /
(4.110)
4,3.3,2, Real part of the left-hand side of the boundary condition on Sh
Upon substituting (4.25), and (4.27) into (4.24) the left-hand side of the boundary
condition becomes
LHS = \YJ(y)dy-yf{y)+ikyAwA
= 2>4
(4.111)
4,3,3.2.1. Left-hand side: Exactly half-submerged circle
Using the conformal mapping (4.40), (4.41) results in
4=*£c"*r-sr,"','+5v*,, .
(4.112)

85
.g 1 p-i(n-l)e
L =/?f e-Kn-l)eide-Kel{n-x)e+ihcAwn = R Kq^9+ikxAwn, (4.113)
"Jo n-1
and
Ln=+ikxAwn-Re-iin-l)S. (4.114)
n-1 -1
The real part of (4.114) is given as follows:
Re(J = ^-j-cos(-l)0. (4.115)
4,3,3.2.2. Left-hand side: More than half-submerged circle
Using the conformal mapping (4.30), (4.31) results in
L, = h1afi(J>-"dC-(ihl +<-)C" <4.116)
L. =h,af eHH-"wid0-ihle-M-cte-K-V +tkx/,w (4.117)
Jo
1 _J(h2-n)0
Ln h2a ihie~in9 ae-i{n~h)8 +ikxAwn and (4.118)
n-\\
Ln = ikxAwn +-^ £Le-0-W -i^9. (4.119)
n-hz n-h2
The real part of (4.119) is given as follows:
Re(Z) = /o+/,cos(-/?2)0 + /2 sin( )<9 + /3 sin(n<9). (4.120)
The real coefficients /, sRe,(/ = 0,1,2,3) are
+ (4.121)
n-h,
j Rea
1 7
n-K
(4.122)

86
, nlma
/2= and
n-K
4 = V
4.3.3.2.3. Left-hand side. Exactly half-submerged ellipse
Using the conformal mapping (4.61), (4.62) results in
L =
C
f 4
c+-
{ o
r" +%,*'
4=4,
1 ,,-'0-1)0 \p~' (n+1)0
l~e ^ 1 e e-i(n-1)0 ^-¡(+1)0
w-1
77 + 1
+ikxAwn ^,
4 c\
1 A
n-1 H + l -l
c--(-i)g n^ c-'(+i)0
+ 1
+ ikxAwn, and
4 ci
_1 _*L+W-*Le-
2 2
+ikxAwl.
The real part of (4.127) is given as follows:
1 X
Re(4) = Ci
n-\ n+1
- kxA Im(wn) H£l- cos(n -1)0 Cos(n + 1)9
Re(4) = ci
-1-
X
- kxA Im(Wj) cos(2 9) .
4,3,3.2.4. Left-hand side. More than half-submerged ellipse
Using the conformal mapping (4.44), (4.45) results in
L =h1\[c(a£i'+ a¡<+ + a,r*)C"
(4.123)
(4.124)
(4.125)
(4.126)
(4.127a)
(4.127b)
(4.128a)
(4.128b)
(4.129)

87
t -in-hy )9 < -i(n+h¡)9
l~e '-ne ~ -a2e~,{n+hz)0 +ikyAwn
Ln~h2a\ Kai .
n-h, n + h~
-ihxe -axe
(4.130)
L =A^l__A^ + /iy w _iUe-ine -WC^e-i^e (4.131)
" n-/*2 +/>2 ^ ^ n-\ n+h2
Re(Z) = Re
h2al h2a2
n-h2 + /t.
+ ikyAwn
Re(a1)cos(-/22)6, +101(0;! )sin(A7-/z2)6)]
n h.
[Re(a2) cos( + )6 + Im(a2) sin( + )\ hx sin(n?)
n + h.
(4.132)
4,3.4. Right-Hand Side of the Boundary Condition on Sh
To find an explicit form of the right-hand side of the boundary condition (4.24)
dw
, the complex velocity around the boundary St, must be determined.
dy
4,3.4.1. Complex velocity around the boundary Sh. Surge, heave, and pitch mode
The procedure given below, is a generalization of a procedure outlined in Milne-
Thomson (1950). Consider again the radiation velocity potential and its normal (to the
wetted body surface) derivative for heave, surge, and pitch (4. Id)
R=Vh+VA + Vpp and
(4.133)
-= K^~ + Vs~+vJ?p- = Vsnx+ Vhnz + Vp\nx (z- Z(0)) nt (x -- ^f(0))]. (4.134)
dn on an on L v K /J
The components of the unit normal vector are

88
dz
dx
=
n =
(4.135)
ds z ds
and ds denotes the elementary increment along the wetted boundary V Making use of
the Cauchy-Riemann conditions, which relate the velocity potential with the stream
function and denoting
VS=V cos/3 (4.136)
Vh-V sin/? ,
/3 = txa-'{VJVs),
and
(4.137)
(4.138)
results in
dy/R d(j)R dz dx
ds dn ds h ds p
lz-Z{)\ +(x-X^)
\ / ds \ f dx
and
(4.139)
dy/R d V,
ds
V'dz-Vhdx+-^-d
(x-X(0))2+(2-Z(0))2
(4.140)
As described in Milne-Thomson (1950), the stream function can be presented as a
difference of a complex function g(y,y) and its complex conjugate counterpart g(y,y),
2 ifg =g(y,y)-g(y,y),
g(y,y) = Ve-fy+~L[jy r(0,y r(y
+ C
(4.141)
(4.142)
g(y,y) = Veipyyy-Yi)y-Yl
<)y
+ c and
(4.143)
2/ y/R = Ve~ipy Ve,py + iVp
yy-Y(0)y-Y[a)y
+ c'
(4.144)
where c, and c are constants. The notation (.) denotes the complex conjugate. Upon
' A
substituting the conformal mapping into a unit circle y-f(C) and recognizing that

89
6T=i (4145>
an expression for the stream function (4.144) in 0£r¡ can be obtained
2iy,R=B0+Bx(£) + B2{£). (4146)
B0 is a constant, Bx{£) contains all the negative powers of £, and B2(£) contains all the
positive powers of £. Using the following relationship, which is proven below
fi,(f) = -S,(rl) (4.147)
results in
2/ y/R = BQ + B, (£) Bx{£) = wR-wR and (4.148)
wR=c"+Bl(£) (4.149)
where c" is a constant. Leaving the generalities, lets look for particular conformal
mapping implementations.
4,3.4.1.1. Complex velocity around the boundary S Exactly half-submerged circle
Applying the conformal mapping (4.40) in (4.144) results in
y = RC,
yy-R2,
Bx{£) = -Veip{RCx) + iVp\-Y{(>)RC'\and
B2{£) = Ve^{R£) + iVp
Y{0)R£
(4.150)
(4.151)
(4.152)
(4.153)
Obviously (4.147) is justified. Making use of (4.152) and (4.149) results in
wR =c"+Bx{£) = -Vei,}(R£-x) + iVp[-Y{0)RC-x] + c\ (4.154)

90
where c" is a constant. Therefore the complex velocity on the boundary C in 0¡r¡ is
dwR
dQ
= Velf(R(-2) + V, [iRYmi~2 ]
(4.155)
From the decomposition (3.5 Id), (4.50) of the radiation potential, the complex velocity
should be
d (4.156)
which means
dw. R
dwh iR
~d^~~C
dwp jRY{0)
dC ?
, surge mode
(4.157)
, heave mode and
(4.158)
, pitch mode.
(4.159)
4,3.4.1.2. Complex velocity around the boundary Sh. More than half-submerged circle
Applying the conformal mapping (4.30) in (4.144) results in
y = -ih{ +a£ ^ (4.160)
yy + ad ih^a^ + ih^dC'^ (4.161)
B,{C) = ~Ve'p(d^) + [(/'/?, Y(0])dCh] and (4.162)
B2(fl = Ve-iil(aCh>)+ivU-ihl-Y{0))aCh>] (4.163)
It is obvious from (4.162) and (4.163) that
Bt(d = -Bl(C')
(4.164)

91
w* = cn+Bx{Cj = -Ve^f-1*) +iVp\(il\ Y{0))aCh2] + c"
(4.166)
where c" is a constant. Therefore the complex velocity on the boundary C in 0%r¡ is
dwR
d<;
dwR
dC
= Ve'f(!hciC''-')+Vt[h1{hi +K))aT
K-1
and
= V,(K SC1'-') + Vt(iKaC''-')+Vr\h1(h, +y(0))a^*,]
Comparing (4.168) with (4.156) the complex velocity becomes
dws h^a
d<; £*+I
dwh ih^a
~dC~^
7*2+1
dwp +/T(0))
d;
c
*2+1
, surge mode
heave mode and
, pitch mode.
(4.167)
(4.168)
(4.169)
(4.170)
(4.171)
4.3 A. 1.3, Complex velocity around the boundary Sr,: Exactly half-submerged ellipse
Applying the conformal mapping (4.61) in (4.144) results in
J = c1(C1+^),
yy c\{A^2 + AC, 2 + A2 +1) ,
A(0 = V{cxAe-ip-c^)C' +iVp\cx2AC2 + W\a)cx
, and
B2(C) = V{cxeip-cxAeip)Cx +iVp c2A? -{Y(0)cxA + Y(!>]cx)C
(4.172)
(4.173)
(4.174)
(4.175)
It is obvious from (4.174) and (4.175) that B2(£) = -Bx(£ l) and

92
= c"+Bx{C¡ = V(cxXe'ip-cxeip)Cx +iv\cx2XC2 -(l^c, +Y{\x)c1
+ c", (4.176)
where c" is a constant. Substituting (4.47), and (4.48) in (4.176) results in
wR =F(-a cos/?-/'sin/?)<£1 +Vp
+(z{0)a)cl
+ c .
(4.177)
Therefore the complex velocity on the boundary C in 0%ij is
dwR
~dC¡
= V(acosP+ibn0)¡ 2 + Vf
(4.178)
Comparing (4.178) with (4.156) the complex velocity should be
dws
a
, surge mode
(4.179)
dC,
dwh
ib
, heave mode and
(4.180)
c
dwp
, pitch mode.
(4.181)
d4
2 '
4.3,4.1 A. Complex velocity around the boundary S More than half-submerged ellipse
Applying the conformal mapping (4.44) in (4.144) results in
y = -ihx + a^ + a^, (4.182)
yy = ihx(a2-ax)^hl +ihx(a2-ax)Chl +(axa2)C2hl +{ccxa2)£2fh +axax +a2a2,
Bx(£) = V(a2e"p -axeip)Ch +iVp axa2C'2hi -(Y(0)ax+Y{0)a2-ihx(a1-a2))ch2
(4.183)

93
and
(4.184)
B2{£) = V{-a2eip + ale~ifi)Cht +iVp axa2Ch2 -(y(0)cc2 + Y{0)ax +ihl(al a2))^
(4.185)
It is obvious from (4.184) and (4.185) that B2(= ~B,(£ ') and
w* V(a2eip ajPyr" +iv\axa2^ +W]a2-ihx{ax a2))c*
+ c".
(4.186)
The complex velocity on the boundary C in 0%tj is
dwR
~dQ
= V(-a2e-if> + aleif,)h1fh'-x +iVp\-2h2a1a2C2hl'' +h2(r{0)a\ +Ywa2 -ihx{ax-a2))Ch'
Compare (4.187) with (4.156) the complex velocity should be
dws (ax-a2)h2
dcr p
dwh i(ax + a2)h2
~dC~ ?
, surge mode
, heave mode and
(4.187)
(4.188)
(4.189)
dwp h2(iY{0)al+iY{0)a2+hx{al-a2))
dC
ch
c
h4 + hi
, pitch mode. (4.190)
4,3,4.2. Real part of the right-hand side of the boundary condition on Sh
In O^Tj the complex velocity potential and the right-hand side of the boundary
condition (4.24) are

94
dw = dw{£) ,df(£)
dy dC d£
and
(4.191)
rhs = -y^f-+- ity)dy = -/(£)
dy iA dy
M£) ,df{c), ?M£)
dC'' d£
+
j:
d(
l-ikf(()W.
(4.192)
4,3.4,2,1, Real part of the right-hand side: Exactly half-submerged circle
It can be easily observed that (4.157), (4.158), and (4.159) have the same form,
namely
_ co
dC ?
(4.193)
and differ only by a constant c0 in the numerator. Therefore from (4.191) the RHS is
RHS = +[*r(l- ikR()dC = -f+Slip-ikRfp (4 194)
In the cylindrical coordinate system= l.e,e the RHS (4.194) transforms to
e
RHS = -c0e~w + J d(-c0e~w + kRco0) = c0- 2 cpid + kRc0O and
0
Re(RHS) = Re(c0 2coe~10 + kRc06).
(4.195)
(4.196)
4.3,4,2,2, Real part of the right-hand side: More than half-submerged circle
It can be easily observed that (4.169), (4.170), and (4.171) have the same form,
namely

95
d*. co
d£ ^+1
and differ only by a constant c0 in the numerator. Therefore from (4.191) the RHS is
(4.197)
RHS = -(ihx+a^)j^-^+\YA-^{\-ik(iK + a£h>))d£ and
RHS =
*Acr A
, Ay
C2ih +
f_c^
v A J
rr 1 ik(ih + a£h')
(4.197)
(4.198)
In the cylindrical coordinate system £ = l.e the RHS (4.198) transforms to
RHS =
RHS =
r AO
v Ay
e-2^ +
_ fkjg-M + Co j[(! + kh, )ie-ih>e + ka\dQ,
A
%o
v Ay
y \
e-2W +
l a;
coO+*A)
A
[e-v_i]
and
aQ+*A)s
A ,
+ (&aco)0 +
_ co(2 + ^A)V-iy + *Aco
A J
v Ay
,-2 The real part of (4.74c) is given as follows:
Rq(RHS) = r0+rx cos(h28) + r2 co^l/^O)+r30 + r4 sin(h20) + r5 sin(2h2d),
where the real coefficients rt e Re, (/ = 0,1,2,3,4,5) are
ro =
(l + ^)Re(c0)
A
r,=-
(2 + /:/?, )Re(c0)
_A
k = Re
A
A
f A"
^ ay
(4.199)
(4.200)
(4.201)
(4.202)
(4.203)
(4.204)
(4.205)

96
r3 = kRe(ac0),
(4.206)
(2 + ^)lm(c0)
K
and
r5 = Im
K
f
v
(4.207)
(4.208)
4.3.4,2.3. Real part of the right-hand side: Exactly half-submerged ellipse
While surge and heave (4.179), (4.180) have the already investigated form,
= the pitch mode's (4.181) complex velocity has an additional term proportional
dC C
to
dwp cx
-3
(4.209)
where c¡ is a constant. The contribution of (4.209) to the RHS is
= c,
01
+ l) r? 1 ikcx ikcxA
' f-ic +H?r_lrr_?_
K
(4.210)
In the cylindrical coordinate system E, = l.e (4.210) transforms to
RHS c,
01
(ew+Ae-w) { e-** ikcAe
-r+ \d\ +ikcxe,g +1
el'e-Xe'9 2 1 3
3iO \
(4.211)
RHS c.
01
(e + Ae->) + +ifc| _!)+_,)
1,9
Aew
(4.212)
and

97
(eie + Ae~i0)
e3'e Ae,a
-e~a +
2 3
(4.213)
Since
J = c, 1 - is a sum of two terms that appear in the denominator of
dC dC \ c)
A
clear in 4.3.5.
4.3,42.4, Real part of the right-hand side: More than half-submerged ellipse
The procedure is analogous to that described in 4.3.4.2.3.
4,3,5. Fourier Expansion of LHS and RHS: Solution for the Unknown Coefficients
For the sake of brevity, the solution for the unknown coefficients will be explained
for the case of the more than half-submerged circle. Having the explicit form (4.202) of
right-hand side of the boundary condition (4.24), it can be expanded it in a Fourier cosine
series in the interval 0 e[-7r,0] as follows:
oo
R Q(RHS) = ^b,cos(ie).
(4.214)
;=o
when 1= 1,2,3,...; and for the case of 1=0 the coefficient is

98
Applying the same Fourier series expansion for each term n=l,2,3,... of the left-hand side
(4.120) of the boundary condition (4.24), results in
R<4) = 2>, coiW). (4.217)
/=0
where n=1,2,3,...
c, = \L+l.cos(n-h2)0 + l2sin(n-h2)0 + l2sin(n0)]cos(l0)d0 (4.218)
when 1=1,2,3,...; and for the case of 1=0 the coefficient is
c0n = f \lo+l^cos(n-h2)0+l2sm(n-h2)0+l3sm(n0))d0. (4.219)
The coefficients bo, b¡, cn0, and c,-can be determined using the formulae given below,
where v is real [v e Re]; k,l,m are integers \k = 1,2,3, ]
2 2 m
J sin(/w#) cos(I0)d0 =
n l2 -m2
0
when m +1 = 2k +1
when m + l = 2k
(4.220)
[sin (m0)d0
TT *
11
n m
0
2
u
l
[ cos v0) cos(l0)d0 =
J TT
when m-2k + \
when m = 2k
sin( v + t)n ^ sin( v t)n
v+l
v-l
? 0 -j
[ sin( v0) codl0)d0 =
7tJ n
n
2
U
J_
n
-l + COs(v-l-/)^^ 1 + COs( V 1^71
v+l
v-l
(4.221)
(4.222)
(4.223)
0
\
1
'vr
o
o
1
H
+
'rf
o
o
1
sin(v + /)^
si n{v-)n
I (/ vUS vKJ J vUS t C/ JCl (/ JL
-K
n2(v + l)2 tt2(v-1)2
7r(v + l)
7t{v-)
0
l-cos(v'^-) sin(v7r)
-7T
n2 v1 nv
(4.224)

99
rs V
\6n{v6)co?{W)dd-7i
7T *
sin(v + /)^ ^ sin(v-/)^ cos(v' + /);t cos(v-/j
X2{v + l)2 7T2{v-l)2 K {y+l) 7i{y-l)
1 0
16 sm{vG)d6 = 7z
^ -K
1 0
OdO--, and
n * 2
sin(vTr) cos(v7r)
nv
2 2
7T V
-|0cos(/0>/0 =
_2 2_
K l2
0
when l = 2k +1
when / = 2£
Substituting (4.214) and (4.217) into the boundary condition (4.24) yields
2>.
1 = 1
2>,cos(/0)
/=0
= 6, cos(/<9).
;=o
Regrouping the terms on the LHS, results in
I
/=0
Lw=i
cos(/#) = ]>^[>,]cos(/?).
/=0
Equating the terms in the square brackets on both sides of (4.89),
c/a =bi / = 0,1,2,..,,oo ,
=1
results in the following system of linear equations
/ = 0
C01
C02
C03
C0H
/=1
cu
CU
C13
C\N
/ = 2
^21
C22
C23
C2N
= jV-1
CN-1.2
CW-1,3
CH-\,N _
n-1
n = 2
n = 3
... = #
a,
a.
a3 } = i
a
N
Ph- 1.
(4.225)
(4.226)
(4.227)
(4.228)
(4.229)
(4.230)
(4.231)

100
The number of terms N nedded is practically determined from the far-field amplitude
oo
power series (4.91), namely = £aD(n). Knowing D(n) exactly from (4.99), N
n~ 1
can be determined by specifying a desired accuracy in advance, thus cutting off all terms
whose absolute value is less than the chosen accuracy. For example if the desired accuracy
is defined as 1CT16, then only the first A terms n=l,2, ...,N such that
\anD(n)\>\Q-16 ,n=l,2,...,N (4.232)
are needed to be kept, and all other terms n=N+l,N+2,N+3, ...can be omitted
\anD(n)\ < 10-16 n=N+l,N+2,N+ 3,... (4.233)
In summary, solving the linear system (4.231) gives the unknown coefficients an,
which when substituted in (4.91) produce the far-field wave amplitudes which is the main
goal of this semi-analytic technique. As explained at the end of chapter 3, knowing the far-
field amplitudes is sufficient to find the hydrodynamic added mass, damping coefficients,
and the exciting forces, and consequently to solve the floating buoy dynamics problem.

101
4.3.6. Discussion of the Uniqueness of the Solution
A half-submerged circular cylinder of radius R is presented on Figure 4.5 in the
orthogonal coordinate system Oxz.
x
Fig. 4.5. Exactly half-submerged circular cylinder in Oxz.
A polar coordinate system Opd is related to the orthogonal system Oxz in the following
manner
x = pcosO
z = psind
(423'
dp dx dz
y -x+iz = pe'9
The boundary condition (4.24) over the wetted body surface St can be written as
(4.235)

102
If the first holomorphic function of the left-hand side of (4.235) has a real part r and an
imaginary part 5, namely
F(y) = J f{y)dy = r + is, (4.236)
then from complex variable theory it follows that
r \ dr .dr
(4.237)
Now, if at the infinite point the analytic form of the holomorphic function is
n=l y
then
,as y oo
(4.238)
F(y) = J f{y)dy = a, (iff) +
,as y
oo.
(4.239)
The real part of the boundary condition (4.235) over the wetted body surface Sb can be
derived after substituting (4.236) and (4.237) into (4.235), and written as follows
dr dr}
r r. -1 x +z
dx dz
= ^aWa +Re|-T^ + /;^(1-%)^| or (4.240)
r -
+ +Re{-yF + l'(1.,mJ
(4.241)
From (4.236), and (4.239) it follows that at infinity
lim/' = 0.
p->oo
(4.242)
From (4.8b), (4.237), and (4.238) it follows that on the free surface
dr
dz
= 0.
2=0
(4.243)

The RHS of (4.241) is a function of the constants rA, and y/ A and a function of the
known normal velocity along the wetted body surface Sb. The harmonic function
r(x,z) = r(p,6) is analytic in the lower half-plane z < 0, vanishes at infinity (4.242), its
normal derivative on the free surface z = 0 is zero (4.243), and along the wetted body
surface Sb satisfies (4.241). Now, consider two functions rx, and r2 which are harmonic
in the half-plane z < 0, and satisfy (4.241), (4.242), and (4.243). For two twice-
differentiable functions rx, and r2, Green's theorem is
(4.244)
where the free surface and the infinite surface are shown on Figure 4.5, and do not
contribute to the integral (4.244) because of (4.242) and (4.243). Therefore the only
surface integral that remains is the one on Sb.
(4.245)
If the right hand side of (4.241) is denoted with RHS, then both harmonic functions rx and
r2 must satisfy
(4.246)
(4.247)
From here it follows that on the wetted boundary surface

104
dr2 dr2
dn dp
p=R
=\(h-RHS)
(4.249)
Therefore, substituting (4.248) and (4.249) into (4.245) leads to
|J(r,VVJ-r,VV,)dV = 4j[r,(r!-WiS)-r!(r,-iWS)]ilK
= Jt\[r (4.250)
The uniqueness of the solution requires that (rx r2) = 0 in (4.250). It is obvious from
(4.250) that they can differ by a constant, namely (rx -r2) = const. The origin of this
constant is the fact that the holomorphic function f(y), (4.8a), consists of two terms
dwiy)
proportional to both complex velocity and complex velocity potential w(y).
dy
dw(y)
While the exact form of can be determined from section 4.3.4.1, the form of w(y)
dy
can be determined only approximately, with accuracy of additive and/or multiplicative
constants.
Another form of the same boundary condition (4.24) can be derived in the following
manner:
\YJ(y)dy = ^SjjLdy+itfvHfy = w-wA +ik\YA(w -wA+wA)dy,
fAf(y)dy W wA + ikwA (y-yA) + ik^ (w -wA)dy,
(4.251)
(4.252)
(4.253)

105
(4.254)
jj(y)dy = -
f(y)-
dw
dy.
+[ik{y-yA)~ l]wA + ik\YA{w WA)dy, and (4.255)
\rj(y)dy+jf{y)-[ik(y-yA)-\]wA =^-~+ik^A(w-wA)dy. (4.256)
A similar analysis (4.244 4.250) will result in the same conclusion that the solution can
differ by a constant. Since it is natural to expect a unique solution in reality, the constant
can be determined from the experiments described in the next chapter.

CHAPTER 5
EXPERIMENTS
51. Purpose of the Experiments
The main purpose of the experiment described in this chapter was to obtain data
for the far-field wave amplitudes for partially submerged floating cylinders of circular and
elliptical shape and thus to verify the Semi-analytic technique (SAT) introduced in
Chapter 4. The necessity of these experiments is a consequence of the lack of
experimental data in the published literature for elliptical shape cylinders and for more than
one half submerged circular cylinders.
5,2, General Set-Up
The experiments were carried out in a wave tank at the Coastal Engineering
Laboratory at the University of Florida. The tank has a North-South orientation,
rectangular cross-section with glass sides and bottom, and its length, width, and depth are
27.80 m x 0.572 m x 1.22 m. Since the drafts of the cylinders for the various modes of
motion were different, the water depth varied between 0.908 m and 1.04 m. At about one
fifth of its length from the North end of the tank a bridge was constructed across the top
of the tank. On this bridge a Scotch Yoke motion mechanism (Mabie and Ocvirk, 1963)
was installed for imparting pure sinusoidal motion in heave and surge to the cylinder with
106

Fig.5.1 Longitudinal and transverse sections.
SURGIR
V '
-4>a
0.1 90m
0.0994
Section A-A
O 15m
FIG. 5.1.e
Steel
Section A-A
o
o
FIG. 5.1 .c
Side View
0.568m
o
o
to
3
-7Â¥
EXPERIMENT: A DYNAMIC RESPONSE MODEL
FOR FREE FLOATING BUOYS SUBJECTED TO WAVES
Gauges. Longitudinal section.
Gauges. Transverse section.
Cylinder. Side view,
Aluminurn cylinder (cicular). Section A-A.
Steel cylinder (^circular). Section AA.
Wood cylinder TelipticaQ. Section AA.
Wood cylinder (elipticall. section A-A,
JL.L
-5-1-f
-5-JU
107

108
frequencies up to 1.2Hz. Four capacitance-type wave gauges were mounted in the tank,
two on each side of the test buoy as shown schematically in Figure 5.1, Measurements
were done for three different amplitudes of motion and a period range from 0.8 to 6.3
seconds. The surge motion amplitudes were 20.5, 26, and 31.5 mm.
5.2,1. Cylinders
As shown below in table 5.1', two circular and one elliptical cylinders were driven
by the Scotch yoke mechanism to generate waves in otherwise still water.
Table 5.1. Principal data of cylinders
Material
aluminum
steel
wood
wood
Cross section
circle
circle
horizontal ellipse
vertical ellipse
Length in [mml
568
568
568
568
Breadth in [mm]
114.3
101.6
131
99.4
Draft in [mm]
57.2-68.6-80
50.8-61-71.1
49.7-59.6-69.6
65.5-78.6-91.6
End plates diameter
190
150
190
190
thickness in [mm]
2
2
2
2
In order to provide similar surface roughness the wooden elliptical cylinder was covered
with several coats of acrylic. End plates were fixed to both ends of all cylinders to prevent
vortex formation around the edges and to reduce three-dimensional wave effects. Their
diameter was greater than 1.5 times the cylinder breadth, as shown on Figure 5.1. The
clearance between the end plates and the glass walls of the tank was 2 mm or 0.35% of the
width of the tank, thus minimizing the effects of the leakage around the end plates.

Fig. 5.2 Wave absorption at the North end
SCREEN 3
FIG. 5.2.b
SECTION B-B
WAVE ABSORPTION AT THE NORTH END
EXPERIMENT: A DYNAMIC RESPONSE MODEL
FOR FREE FLOATING BUOYS SUBJECTED TO WAVES
57n
5.?.h
O
VO

110
5.2.2. Wave Absorption at the Ends of the Tank
The design of the wave absorption at both ends of the tank followed the main idea
of the experiments, which was to obtain the far-field wave amplitudes before and without
any contamination from reflected waves. With a greater distance between the South end
and the wave maker, the wave absorption at the South end consisted of constructing a
steep porous beach with a screen in front of it. With a smaller distance between the North
end and the wave maker, the wave absorption at the North end consisted of placing a
group of three inclined screens in front of the model boundary (Figure 5.2) the purpose of
which was to absorb the energy of the waves. Mesh screen porosity decreased toward the
rear of the North end wave absorber. The wave energy dissipation was designed in
accordance with Jamieson and Mansard's (1987) recommendations. The high porosity
front screen works best for absorbing energy from steep waves, while the rear low
porosity screen absorbs energy from the waves with low steepness. The length of the
wave absorber at the North end was 3.56 m, chosen in accordance with Jamieson and
Mansard's (1987) recommendations for 35%-100% of maximum wavelength. The front
screen had increasing porosity with water depth. The supporting framework of the
screens was designed with minimum frontal area in order to reduce wave reflection.
5.2.3, Wave gauges
The active elements of the capacitance wave gauges are two thin insulated
vertically oriented wires held taut by a supporting rod. The rod was constructed of
stainless steel with a minimum cross-section to reduce flow disturbance. The capacitance
between the two wires varies with the level of submergence. The electronic circuit was

Ill
designed to detect the variation in capacitance and the analog output signal was digitized
with A to D converter and the data stored in a computer. The constant 50 Hz sampling
frequency produced data with uniform spacing between data points. An analog filter was
used to remove frequencies higher than 100 Hz. For a typical laboratory wave of 1
second a sampling frequency of 50 Hz means 50 evenly spaced samples obtained over one
wave period. This provides a more than adequate which representation of the wave form
since the commonly used "rule of thumb" is 10 evenly spaced samples per wave length
(Hughes, 1993). The capacitance type wave gauges showed good linear response and had
a resolution of about 0.1 mm. The main disadvantage was the need for frequent
calibrations, due to changing water temperature. The calibration of the gauges was
carried out statically by manually lowering and raising them at three incremental distances
of 10 mm, into the still water surface before every set of experiments. The calibration
relationships were obtained from mathematical curve-fits between the recorded gauge
outputs and the corresponding elevations at all four gauge locations. The tests showed
that the calibration characteristics were stable over a period of 3 hours the time interval
of a set of experiments provided that there was little change in water temperature and
the gauge wires were clean and free of dust and other foreign matter.
5,2.4, Surface Tension
The scale effect due to surface tension forces not being in proper proportion in a
Froude-scaled model becomes important when water waves are very short or the water
depth is very shallow. Usual "rules of thumb" (Hughes, 1993) are that the surface tension
effects must be considered when wave periods are less than 0.35 seconds and when water

112
depths are less then 2 cm. At these small parameter values, the restoring force of surface
tension becomes significant and the model will experience wave motion damping that does
not occur in the prototype. Since the particular model had water depths above 90 cm and
wave periods above 0.8 sec, surface tension did not affect the experiment results.
5.3, Wave Absorption and Reflection Analysis
A portion of the wave energy was absorbed at the North boundary and a portion
was reflected. As a result of the superposition of the incident and reflected waves, a
partially standing wave is formed which can be presented as
r¡[t) = 4 cos(£c-cot+ £,) + Ar cos(Ax + cot + sr). (5.1)
As will be shown below, having time series of the water elevation collected by two gauges
situated at distances Xj ('¡=3,4) (see Figure 5.1) is sufficient to determine the amplitudes
and phases of both incident and reflected waves travelling between the wave maker and
the end of the tank. Assuming the wave time series was a linear superposition of many
sinusoidal components, the wave data was analyzed in the frequency domain using spectral
analysis. With 212 data points in a record and a Fast Fourier Transform, the energy
spectrum was estimated and the primary harmonics filtered out. It was found that the
principle of linear superposition was well justified over the entire frequency range by using
the first two or three harmonics (co,2a>,3co) to match the wave time series. This fact is
illustrated in Figure 5.3, which shows a typical stationary-mode-wave-time series and
power density spectrum for one of the experiments. Figure 5.4 shows an almost perfect fit
between

113
F¡)e=b2h52651 dat; GAGE #2 Wave Time Series
Fig. 5.3 Measured data and its power spectrum density
File=b2h52651.dat; GAGE #2 T=0.77475[s]
81.2 81.3
81.4 81.5 81.6 81.7
81.8 81.9
Fig. 5.4 Measured data as a sum of the first three harmonics

114
gauge #2 data points and the first three harmonics after applying the smoothing numerical
algorithm.
The spectral analysis at gauge #j, (j=3,4) permits
r¡[t) = j cos(yt) + bj sin(yt) (5.2)
filtering out the first harmonic parameters Equating (5.1) and (5.2) results in
= 4 cos^Ax; +£,) + Ar cos^£c;. + and (5.3)
bj = 4 sin(kXj + £,)- Ar r\(kcJ + £,.) (5.4)
After some algebra a linear system of two equations (5.5) is obtained
4 cos^Sj) = [(c3 -c4)f + (d3 -d4)g\/ (f + g2), and (5.5a)
4 sin(i,) = [-(c3 ~c4)g+(d3-d4)f]/(f2 +g2) (5.5b)
with a known right hand side, namely
c;. = a, cos^hCj)-b} sin(x.) wherej=3,4
dj = cij sin(/ocy) + bj cos^kXj) where j=3,4
f = cos(2Ax3) cos(2/fcx4), and
g sin(2Ax3) sin(2x4).
A straightforward solution of the incident wave amplitude and phase follows from (5.5).
Knowing the incident wave the reflected wave amplitude and phase can be determined
from (5.3), and (5.4). The reflection and absorption coefficients for the model boundary
can be defined as follows:
, reflection coefficient, and
(5.6)

115
Ka = To illustrate the effectiveness of the wave absorber at the North model boundary, the
dimensionless absorption coefficient for a typical set of experiments is plotted on Figure
5.5 versus the dimensionless water depth to wave length variable kh.
File= e2v52n.txt; Absorption Coefficient at the North End
Fig. 5.5 Absorption coefficient at the North end
With the very effective absorption at the North end, the reflected waves from the North
model boundary could be neglected and therefore the waves between the wave maker and
the South end are not contaminated by wave reflections from the North end. In order to

116
insure that the waves measured were not contaminated by reflected waves from the South
boundary, the data-records started at the time of initial wave generation, and ended at the
time required for the waves to travel to the South end of the tank and back to gauge #2
(Figure 5.6).
0 5 10 15 20 25 30
"uncontaminated time" =32.1 [sec]
Fig.5.6 Uncontaminated time

i
117
5.4. Model Scale Selection: Froude Scaling
Using an analogy of a rectangular barge in heave motion, a proper qualitative
relationship between wave height H and displacement S is
p2 V
v^2a
B
(5.8)
where B is the characteristic body dimension, introduced in chapter 2 L is the wave
length, and F is the Froude number defined as
V2 (tfA)2
F =
(5.9)
gB gB
with V, the floating body velocity, and co, the circular wave frequency. Therefore
proper dynamic and geometric scaling requires
(*),=(n,. (5l)
f B^\
vA,
rB2
2| and
a-di
(5.11)
(5.12)
wherep, m subscripts denote prototype and model respectively. Using the dimensions of
the wave tank (Figure 5.1), and a geometrically undistorted model results in a length scale
of 1:20, and a corresponding temporal scale of 1:4.5.

118
5.5. Discussion of the Experimental Accuracy
In an experiment, such as the one described here, it is not possible to overcome all
sources of possible errors which influence the experimental accuracy. Even though on the
order of millimeters, the gap between the end plates of the cylinder and the glass walls of
the tank was still a source of error caused by the water leakage. The wave absorption at
the North end (Figure 5.5) was not perfect and varied with the wave frequency. The
nonlinear effects of the fluid-body interaction, evidenced by the presence of the second
and third harmonics (Figure 5.3), also contributed to the experimental error.
5,6, Discussion of the Experimental Procedure
Vugts (1968) used another experimental procedure for an approximate estimate of
the hydrodynamic damping and added mass. When the cylinder has a forced oscillation in
one mode of motion, the force required to sustain the motion can be measured. For
example, as given in Vugts (1968), the surge mode oscillation can be described by
(M + ¡u)x + Ax = F sin(cot + e) (5.13)
where M is the mass of the cylinder section, ju is the added mass, A is the damping
coefficient, F,co,s are the force amplitude circular frequency and phase angle with
respect to the motion. In surge mode, there is no restoring force component proportional
to x. The motion of the center of gravity of the cylinder is considered harmonic with
amplitude A and oscillating with the forcing circular frequency co, namely

119
x = A sin(atf).
As a direct solution of this simplified system the added mass coefficient is
F cos(>)
(5.14)
F =
co1 A
-M
(5.15)
and the damping coefficient is
F sin(£')
A = -
coA
(5.16)
In the present experiment, higher harmonics 2 Figure 5.3. This fact suggests that regardless of whether the force required to sustain the
motion includes higher harmonics or not, the motion of the cylinder should account for
these higher harmonics. Consequently, ones the motion (5.14) or the forcing in the right-
hand side of (5.13) is changed with an addition of higher harmonics, (5.15) and (5.16) can
no longer be used to determine accurately the added mass and damping coefficients. In
summary, this experimental procedure (5.13-5.16) is not capable of detecting the presence
of higher harmonics, and therefore gives an approximate estimate of the added mass and
damping coefficients.
The procedure used in the present experiment to obtain data for the far-field wave
amplitudes again gives an approximate solution for the added mass and damping
coefficients as it will be proven in the next chapter 6. The only difference from the Vugts'
procedure (5.13-5.16) is that it detects the higher harmonics, and that it gives an idea for
the degree at which the nonlinearities are involved in this fluid-structure interaction.
Pictures of the experimental set-up are given on Figures 5.7 through 5.14.

120
Fig. 5.7 Side view of the wave tank. Wave absorbers at the North end of the wave tank.

121

122
Fig. 5.9 Side view of the Scotch Yoke
motion mechanism.

Fig. 5.10 A closer side view of the Scotch Yoke motion mechanism.

Fig. 5.11 Side view of the motion mechanism and the oscillating cylinder.

125
Fig. 5.12 Side view of the oscillating cylinder and its end plates.

126
Fig. 5.13 Side view of the cylinder in motion.

127
Fig. 5.14 Side view of the wave gauges.

CHAPTER 6
ANALYSIS OF THE RESULTS
This chapter presents a comparison between the theoretically obtained
hydrodynamic damping and added mass coefficients for horizontal cylinders and those
obtained experimentally as described in chapter 5. Next, both theoretical and experimental
results are compared with experimental values obtained by Vugts (1968). Vugts'
experiments were limited to half-submerged horizontal circular cylinders. His data is
shown with square symbols on the graphs presented in this chapter. Vugts' experiments
were conducted in a 4.2 m wide tank in water depths ranging from 1.8 to 2.25 m. The
cylinder diameter and draft were 0.3 and 0.15 m respectively. The amplitudes of motion
were 0.01, 0.02, 0.03 m; and the circular frequency ranged from 1 to 12 rad/sec.
Theoretical curves of hydrodynamic and dynamic properties computed with the semi-
analytic technique, presented in chapter 4, are referred to by the abbreviation SAT. The
dimensionless hydrodynamic coefficients versus dimensionless circular frequency are given
below in the following notations, which were also used by Vugts
dimensionless circular frequency,
(6.1)
dimensionless damping, and
(6.2)
128

129
dimensionless added mass (6.3)
where X and p are the damping and added mass coefficients, Awet is the area of the
cross-section below the still water level, B is the breath of the section at the waterline, p
is the water density, and g is the acceleration of gravity. The damping coefficient is
computed from the measured far-field wave amplitude Ga (3.63) using the expression
X = 2pgCga2a (6.4)
where Cg is the group velocity. The far-field amplitude results directly from the measured
wave height H, wave circular frequency co, and amplitude of oscillations S/2
a =
a coS
From linear wave theory the wave group velocity is
,( A i \
(6.5)
c
2 2
1 +
2 kh
V
sinh(2&/j)
= tanh(/z)
1 +
2 kh
\
sinh(2 kh)
(6.6)
where k is the wave number, h is the water depth, and C is the wave celerity. The formula
for computing the wave damping coefficient from the measured wave height is
X pg2 tanh(/z)
1 +
2 kh
sinh(2A/i)
H2
3r*2
)S
(6.7)
Consequently, the dimensionless damping coefficient is given by
B _{B/2f 1
AC \ 2 S
A (
0)
tanh(kh)
1 +
2kh
sinh(2 kh)
H2
S2
(6.8)

130
6.1. Surge Mode Oscillations
The non-dimensional damping coefficients in surge (6.8) are presented in Figures
6.1 through 6.8. There is some scatter among the measured points but the consistency of
the experiments is good. The upper limit of the dimensionless frequency corresponds to
the maximum frequency of the driving motor, 1.2 Hz while the lower limit of the
- dimensionless frequency represents the physical boundary below which waves cannot be
measured with an accuracy of 0.1 mm. For a prototype buoy with a characteristic size of
2 m, length scale 1:20, and temporal scale 1:4.5, this frequency range corresponds to a
prototype period range between 3.6 and 12 sec. This means that the maximum frequency
of the driving motor imposes a limitation on wave periods below 3.6 sec. Figures 6.1 and
6.2 present the surge oscillation of two semi-submerged circular cylinders with drafts
equal to half of their corresponding diameters of 0.114 m, and 0.102 m. It can be seen in
Figure 6.1 that the sets of damping measurements for 52 and 63 mm strokes, produced by
the 0.114 m diameter cylinder surround Vugts' data when co
> 0.35 and are below
I B
Vugts' measurements when co J < 0.35. In the same figure the theoretical solution is in
almost perfect agreement with the 52 mm set of measurements. In Figure 6.2, the sets of
measurements produced by the 0.102 m diameter cylinder are slightly below Vugts' data
and the theoretical solution. In light of the dimensionless notations used the result of the
experiments presented in Figure 6.1 and 6.2 suggest that damping in surge motion
increases with the size of the buoy.

dimensionless damping o\ dimensionless damping
131
dimensionless circular frequency
1 Damping in surge. Half-submerged circular cylinder D=0.114 m. Stroke variations.
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
dimensionless circular frequency
Fig. 6.2 Damping in surge. Half-submerged circular cylinder D^O. 102m. Stroke variations.

132
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
dimensionless circular frequency
Fig.6.3 Damping in surge. Circular cylinder D=0.114 m. Variations in draft.
0 I I i X 1 1 1 1 i i_l
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
dimensionless circular frequency
Fig.6.4 Damping in surge. Circular cylinder D=0.102 m. Variations in draft.

133
Figures 6.3 and 6.4 show the influence of draft variations on damping caused by surge
oscillations for both circular cylinders. With a stroke of 63 mm for the large cylinder and
52 mm for the small cylinder, the measurements of damping are shown for three draft-to-
radius ratios 1, 1.2, and 1.4. On both figures, there is little scatter among the measured
points for the three variations in draft. Theoretically obtained damping curves for the
small cylinder, slightly overestimate the measured ones. Both theoretical and measured
damping curves suggest that damping in surge increases with the draft. There is a
noticeable branching among the different draft sets with surge damping increasing with
draft. The same tendency of branching can be observed from the theoretical curves in
both Figures 6.3 and 6.4 for draft-to-radius ratios of 1, 1.2, and 1.4. This branching also
suggests that it would be more appropriate to non-dimensionalize surge damping and
circular frequency with draft instead of breadth of the section at the waterline. Figure 6.5
presents surge damping for a horizontally oriented elliptical cylinder. Its semi major axis
coincides with the still water surface while the semi minor axis is normal to the still water
surface. Figure 6.6 shows surge damping for a vertically oriented elliptical cylinder. This
time its semi minor axis coincides with the still water. In both Figures 6.5 and 6.6, the
measured results for the larger strokes are slightly above those with the smaller strokes.
In both cases, the theoretical damping curves are in good agreement with the measured
ones.

dimensionless damping 'as dimensionless damping
134
5 Damping in surge. Elliptical cylinder with horizontal major axis. Stroke variations.
Fig.6.6 Damping in surge. Elliptical cylinder with vertical major axis. Stroke variations.

135
dimensionless circular frequency
Fig.6.7 Damping in surge. Elliptical cylinder with horizontal major axis. Draft variations.
Fig.6.8 Damping in surge. Elliptical cylinder with horizontal major axis. Draft variations.

136
Figures 6.7 and 6.8 present the influence of draft variations on damping caused by surge
oscillations of an elliptical cylinder with horizontally and vertically oriented semi major
axis. In these figures the measured damping coefficients are shown for the same three
draft-to-vertical semi-ratios of 1, 1.2 and 1.4, and a stroke of 52 mm. Again, there is little
scatter among the measured points for all draft variations. As in the case of a circular
cylinder with varying draft, the same branching tendency among the different draft sets can
be observed (Figures 6.7 and 6.8) with surge damping increasing with draft. In Figure
6.7, the theoretical solution slightly overestimates the measured damping for ratio 1.2. In
Figure 6.8, There is good agreement between the theoretical and measured values. It is
noteworthy that the damping curves for the elliptical cylinder with a vertical semi major
axis are steeper than those for the elliptical cylinder with a horizontal semi major axis,
when plotted versus increasing circular frequency. Despite the same volume of water
displacement, the draft increment engenders greater damping in surge mode oscillations.
The amplitudes of the outgoing waves in surge are presented in Figures 6.9
through 6.13. They are shown in their non-dimensional form of amplitude-to-stroke ratio
versus the dimensionless circular frequency, introduced in (6.1). Figure 6.9 presents the
influence of the size of semi-submerged circular cylinders on the non-dimensional
amplitudes of the outgoing waves. It can be seen that wave amplitudes increase with the
characteristic size, since the draft increases with the size. Figure 6.10 reveals how draft
variations of a circular cylinder influence the outgoing wave amplitudes. It is clear that the
amplitudes increase with the draft. The same is true for the case of a vertical ellipse with
varying draft, shown on Figure 6.12. The most evident proof that surge-generated
amplitudes increase with draft is illustrated on Figure 6.11 for half-submerged elliptical

137
cylinders. When there is no motion, the cylinders with a vertical and a horizontal semi
major axes displace the same volume of water. When these elliptical cylinders oscillate
with a particular frequency and constant stroke, the greater draft of the vertical semi major
axes displaces a greater amount of water than the horizontal one, and therefore generates
waves with higher amplitudes in surge motion. This argument is in perfect synchrony with
the conservation of mass. It is noteworthy that surge-generated amplitudes increase with
the circular frequency in the frequency range of the experiment, but this is not necessarily
true outside this range.
Fig.6.9 Wave amplitudes. Circular cylinders in surge. Draft= D/2.

amplitude-to-stroke ratio
138
0.6
0.5
O
to 0.4
(D
o
i
to
0.3
"D
3
E 0.2
CO
0.1 -
+ 4- draft =D/2
< < draft=1.2T>/2
0 0 draft =1.4T)/2
Stroke =63mm;D =114 mm
<
+
<
+
^M.
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
dimensionless circular frequency
Fig. 6.10 Wave amplitudes. Circular cylinder in surge.
0.4
-
0.35
+ + Stroke = 41 mm; Horizontal
<] 0 0 Stroke = 41 mm; Vertical
0.3
Stroke = 52 mm; Vertical
-
0.25
0.2
-
s
0
0.15
0
0.1
0
$
0.05
0 <1
-
. 0 < *
Si AiL. <* 1 i
1 1
0.15*
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.6
dimensionless circular frequency
Fig. 6.11 Wave amplitudes. Elliptical cylinders in surge.

amplitude-to-stroke ratio
139
t 1 r
0,6
o
2
03
O
v-


T3
3
0.5
0.4
0.3
+ 4 draft=a
0
<] <| draft=1.2*a
Q ^ draft=1.4'a
Stroke = 52 mm
-
<
<
0 +
*
<1
Q.
s
ro 0.2
0.1 -
04
15
0.2
0.25
0.3
0.35
0
+
0.4 0.45 0.5
0.55
dimensionless circular frequency
Fig. 6.12 Wave amplitudes. Vertical elliptical cylinder in surge.
0.45
1
1 1 1 1
if**
0 .
4 4 draft=a
0
0.4
<] <3 draft=.1.2*a
-
0 0 draft=1.4*a
0.35
Stroke = 52 mm
< -
0.3
1
V
o
0.25
*-
*
<
0.2
0
0.15
-
1
+
V
0.1
0
+
0.05
'
<1
+
0
&
*
_
1 1 L l
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
dimensionless circular frequency
Fig. 6.13 Wave amplitudes. Horizontal elliptical cylinder in surge.

140
6.2. Heave Mode Oscillations
The non-dimensional damping coefficients in heave mode oscillations (6.8) are
presented in Figures 6.14 through 6.21. The scatter among the measured points in heave
is greater than in surge, but the consistency of the experiments is satisfactory. As in the
surge mode, the upper limit of the dimensionless frequency corresponds to the maximum
frequency of 1.2 Hz of the driving motor. The lower limit of the dimensionless frequency,
below which waves cannot be measured for heave motion, is less than the corresponding
limit for surge motion. This fact means that there is a frequency limit, below which forced
oscillations in surge cannot transfer energy to the fluid, while forced oscillations in heave
can still transfer energy to the fluid. Again, for a characteristic prototype size of 2 m and
the length and temporal scales given in chapter 5, this frequency range corresponds to a
prototype period range between 3.6 and 30 sec. Figures 6.14 and 6.15 present the heave
oscillation of semi-submerged circular cylinders with drafts equal to half of their
corresponding diameters, 0.114 m and 0.102 m. On both Figures 6.14 and 6.15, the
theoretical solution is in good agreement with the measurements of the present experiment
as well as with Vugts' measurements. On both figures, the theoretical solution slightly
overestimates heave damping coefficients. It can be seen in Figures 6.14 and 6.15 that the
different sets of damping measurements, with strokes of 52, 39, and 27 mm, produced by
both cylinders surround Vugts' data: Thus, these measurements are consistent with Vugts'
results in spite of the scatter.

dimensionless damping dimensionless damping
141
Fig. 6.14 Damping in heave. Circular cylinder D=0.114 m. Stroke variations.
Fig. 6.15 Damping in heave. Circular cylinder D=0.102 m. Stroke variations.

dimensionless damping P' dimensionless damping
142
16 Damping in heave. Circular cylinder D=0.114 m. Draft variations.
Fig. 6.17 Damping in heave. Circular cylinder D=0.102 m. Draft variations.

J
143
Figures 6.16 and 6.17 show the influence of draft variations on damping caused by heave
oscillations for both circular cylinders. The measurements of damping are shown on these
two figures for the draft-to-radius ratios 1, 1.2, and 1.4. Theoretically obtained damping
curves are in good agreement with the measured ones, for draft-to-radius ratios of 1.2,
and 1.4, but for 1 the theoretical damping curve overestimates the measured one. It
should be noted that this overestimation is due to the 39 mm data set (Figure 6.17) and 52
mm data set (Figure 6.16). It can be seen in Figures 6.15 and 6.14 the other data sets
produced by different strokes are closer to the theoretical solutions than both of these 39
mm and 52 mm sets. Both theoretical and measured damping curves on Figures 6.16 and
6.17 suggest that damping in heave decreases with the draft. The plots in the preceding
Figures 6.14, 6.15, 6.16, and 6.17 suggest that damping coefficients in heave increase with
frequency until they reach a maximum, after which they decrease with further increment of
the circular frequency. Figure 6.18 presents heave damping for half-submerged elliptical
cylinder with a horizontal semi major axis. Figure 6.19 shows heave damping for half-
submerged elliptical cylinder with a vertical semi major axis. In both cases the theoretical
damping curves are in good agreement with the measured ones. The measurements for
the case of the horizontal semi major axis show that heave damping slightly increases with
the magnitude of the stroke. This observation is almost completely reversed for the case
of the vertically oriented elliptical cylinder. It is noteworthy that the heave damping
curves for the elliptical cylinder with a horizontal semi major axis are steeper than those
for the cylinder with a vertical semi major axis, when plotted versus increasing circular
frequency.

dimensionless damping
144
Fig.6.18 Damping in heave. Elliptical cylinder with horizontal major axis.Stroke variations
Fig. 6.19 Damping in heave. Elliptical cylinder with vertical major axis. Stroke variations.

145
Fig.6.20 Damping in heave. Elliptical cylinder with horizontal major axis. Draft variations.
Fig. 6.21 Damping in heave. Elliptical cylinder with vertical major axis. Draft variations.

146
Figures 6.20 and 6.21 present the influence of draft variations on damping caused by
heave oscillations of the elliptical cylinder with horizontally and vertically oriented semi
major axes. In these figures, the measured damping coefficients are shown for the same
three draft-to-vertical semi-axis ratios of 1, 1.2, and 1.4, and strokes of 27 and 39 mm.
Again, there is some scatter among the measured points for all draft variations. In Figure
6.20, the theoretical solution slightly overestimates the measured damping for draft-to-
vertical semi axis ratios of 1 and 1.2 in the middle of frequency range, while there is good
agreement for draft-to-vertical semi axis ratios of 1.4. In Figure 6.21, there is good
agreement between the theory and measurements for all three draft-to-vertical semi-axis
ratios. As in the case of a circular cylinder with varying draft, the heave damping
decreases with draft for elliptical cylinders with both horizontally and vertically oriented
semi major axes. Again, the heave damping curves are steeper for the horizontal than for
the vertical semi major axis, when plotted versus increasing circular frequency.
The amplitudes of the outgoing waves generated by heave motion are presented in
Figures 6.22, 6.23, and 6.24. They are shown in their non-dimensional form of amplitude-
to-stroke ratio versus the dimensionless circular frequency, introduced in (6.1). Figure
6.22 reveals how draft variations of a circular cylinder influence the outgoing wave
amplitudes. It is clear that the amplitudes decrease with increasing draft. The same is true
for the cases of elliptical cylinders with vertical and horizontal semi major axes with
varying draft, shown on Figures 6.23 and 6.24. Since the waterline section decreases
when the draft increases, the volume of water activated by heave oscillations also
decreases which leads to smaller wave amplitudes. A clear evidence for this explanation is
that the elliptical cylinder with horizontal semi major axis produces higher amplitudes than

147
the one with vertical semi major axis for the same vertical displacement, Az, despite the
equal volume displacement in calm water (Figures 6.23 and 6.24).
Fig. 6.22 Wave amplitudes of a circular cylinder in heave. Draft variations.

amplitude-to-stroke ratio
148
Fig. 6.23 Wave amplitudes of a horizontal elliptical cylinder in heave. Draft variations.
Fig. 6.24 Wave amplitudes of a vertical elliptical cylinder in heave. Draft variations.

149
6.3. Damping. Added Mass, and Frequency Response Function
As defined at the end of section 3.5 and summarized in section 3.6, the dynamic
response of a floating body (3.82) is characterized by its frequency response function FRF,
and the response amplitude operator (3.83), (3.84). The mass and restoring force matrices
are determined from the static equilibrium state (3.48) and (3.49), and therefore are
independent of the frequency of the incident wave. On the other hand, the added mass,
damping, and exciting force are functions of the circular frequency of the incident wave.
Consequently, it is instructive to show how the frequency, shape and draft dependence of
the added mass and damping influence the FRF and the dynamic response of the floating
body. To illustrate this frequency, shape, and draft dependence, the following two
numerical examples are considered utilizing the semi-analytic technique. In the first
example, a circular cylinder of radius R=0.5 m is investigated for three draft-to-radius
ratios of 1, 1.2, and 1.4. In all three cases the center of gravity is placed at a fixed distance
of 0.3 m from the lowest point on the cylinder, as shown on Figure 6.25. The breadth of
the waterline section is decreasing with increasing draft. In the second example, shown on
Figure 6.26, an elliptical cylinder with fixed horizontal semi-axis =0.5 m, and fixed
waterline breadth B=2b= 1 m is investigated for three semi vertical to semi horizontal axis
ratios of a/b-1.01, 1.1, and 0.9. Again, in all three cases the center of gravity is placed at
the same fixed distance from the lowest point on the cylinder. The breadth of the
waterline section is constant B-2b= 1 m in all three cases.

150
Fig. 6.25 First example. Circular cylinder. Variations in draft. Center of gravity
- placed at a fixed distance from the lowest point on the cylinder.
Fig. 6.26 Second example. Elliptical cylinders. Constant waterline breadth.
Variations in the vertical semi-axis. Center of gravity
- placed at a fixed distance from the lowest point on the cylinder.

151
Since pitch mode experiments were not conducted and since stability is strongly
dependent on the metacentric height (usually small in pitch), the added mass, damping,
and FRF in pitch mode for these two examples, are given in the appendix. It is assumed
that keeping the center of gravity below the instantaneous center of buoyancy provides
stability. This is usually achieved by placing ballast at the bottom of the floating body.
Nevertheless, the coupling between surge and pitch is mentioned in the following
discussion. The circular frequency, damping, and added mass are made again non-
dimensional with the help of the breath of the section at the waterline (6.1), (6.2), and
(6.3).
With reference to the first numerical example (Figure 6.25), Figures 6.27, 6.28,
and 6.29 present the damping, added mass, and FRF in surge motion of a circular cylinder
with varying drafts. It can be seen from Figure 6.27 that in general, damping in surge
increases with increasing draft for dimensionless frequencies up to 0.95, For higher
dimensionless frequency above 0.95, or periods below 1.55 seconds, the increment of the
draft-to-radius ratio from 1.2 to 1.4 results in less damping in surge motion. An intuitive
explanation for this phenomenon can be offered from the wavemaker-theory point of view.
First consider the limiting case of substituting the underwater shapes with the same draft
vertically oriented plates with part of them above the water. If all three drafts are
oscillated with a constant surge amplitude, the greater the draft, the greater the displaced
water volume, and the greater the propagating wave amplitude and thus the greater the
damping. While this analogy works in the practical frequency range, it is not entirely
applicable for higher frequencies in the ratio transition from 1.2 to 1.4. In the particular

152
case of higher frequencies, the side view of the underwater body surface can be imagined
as one composed of multiple vertical paddles in a staircase fashion along the convex
shape. While all paddles transmit constant normal velocity to the neighboring water
particles a phase lag is imposed by the paddles above the point of inflection, which
eventually results in energy transmitted from the body to the water, and consecutively less
damping. Figure 6.28 shows that the added mass in surge motion generally increases with
increasing draft. An intuitive explanation for the fact is that, the greater the draft, the
greater the underwater surface area is upon which the local pressure gradient acts, and
thus the greater the added mass. Figure 6.29 presents the FRF in surge motion for the
three drafts. A general tendency can be inferred that at low frequencies the floating bodies
are almost perfect wave followers. This behavior decreases as the circular frequency
increases. The disturbances at dimensionless frequency of 0.6-0.7 are due to the coupling
with pitch mode oscillations shown in the appendix. FRF in pitch has its maximum value
at the same dimensionless frequency, 0.6-0.7. Obviously pitch acts in an opposite to surge
direction, with a phase difference n, a result confirmed by Mei (1989).

dimensionless added mass 9s dimensionless damping
153
Fig. 6.28 Added mass in surge. Circular cylinder. Draft variations.

154
O)
OI i 1 1 1 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
dimensionless circular frequency
Fig. 6.30 Damping, added mass, and FRF in surge. Circular cylinder.

155
With reference to the first numerical example (Figure 6.25), Figures 6.31, 6.32,
and 6.33 present the damping, added mass, and FRF in heave motion for a circular
cylinder with varying draft. It can be seen from Figure 6.31, that damping in heave
decreases with increasing draft. An intuitive explanation for this phenomenon can be
offered from a wavemaker-theory point of view. The greater the draft, the less the
waterline breadth B, the deeper the location of the displacement that results from an
increment in vertical motion. If all three draft variations are to oscillate with constant
heave amplitude, the less the waterline breadth, the less the displaced water volume, the
less energy input is required, and the less the damping, since damping characterizes the
energy given up from the body (3.69). The closer the underwater shape is to the limiting
case of a horizontal plate, the closer the displacement is to the surface where the
propagating waves are formed, the greater the displaced water volume, the more energy
input is required, and the greater the damping becomes. Figure 6.32 shows that the added
mass in heave motion decreases with increasing draft. As explained for the heave
damping, the greater the draft is, the less the displaced water volume is, the less the local
pressure gradient is, the less the added mass becomes. Figure 6.33 shows that the peak
value of FRF in heave motion increases when the draft increases. To explain this fact, it
suffices to consider the heave-only form of the FRF from (3.84), namely
(6.9)
where C is the spring constant, and con is the natural circular frequency, defined as

156
Fig. 6.31 Damping in heave. Circular cylinder. Draft variations.

157
Fig. 6.33 FRF in heave. Circular cylinder. Draft variations.
05
Fig. 6.34 Damping, added mass, and FRF in heave. Half-submerged circular cylinder.

158
O). =
C
M + n
(6.10)
Obviously, the peak value of FRF in heave motion FRF{con) corresponds to oscillation of
the floating body with the natural frequency, since the denominator of (6.9) achieves its
minimum when co con. It can be seen from Figure 6.33 that the natural frequency
decreases when the draft increases. This is true despite the fact that added mass, ju ,
decreases with draft because an increment in draft means an increment in mass, M, and
eventually means an increment of the denominator in (6.10). The lower the natural
frequency, the lower the denominator (Xcon) in (6.9), and the larger FRF(con) becomes.
Figure 6.34 presents heave damping, added mass, and FRF for a half-submerged circular
cylinder.
With reference to the second numerical example (Figure 6.26), Figures 6.35, 6.36,
and 6.37 present the damping, added mass, and FRF in surge motion of an elliptical
cylinder with various drafts. It can be seen from Figure 6.35 that in general, damping in
surge increases with increasing draft. For dimensionless frequencies above 1, or periods
below 1.4 seconds, the picture is reversed. An intuitive explanation for this behavior can
be offered from the wavemaker-theory point of view. Consider the limiting case of
substituting the underwater shapes with the same draft vertically oriented plates with part
of the plate above the water surface. If all three drafts are to oscillate with a constant
surge amplitude, the greater the draft, the greater the displaced water volume, and the
greater the damping. Figure 6.36 shows that the added mass in surge motion generally
increases with increasing draft. An intuitive explanation for this fact is that the greater the
draft, the greater the underwater surface area upon which the local pressure gradient acts,

dimensionless Added mass O dimensionless damping
159
Fig. 6.36 Added mass in surge. Elliptical cylinder. Draft'variations.

FRF added mass dimensionless damping p\ Frequency response function
160
Fig. 6.38 Damping, added mass, and FRF in surge. Half-submerged elliptical cylinder.

161
and thus the greater the added mass. Figure 6.37 presents the FRF in surge motion for the
three drafts. A general tendency can be inferred that at low frequencies the floating bodies
are almost perfect wave followers, with their behavior decreasing with increased circular
frequency. The disturbances at the dimensionless frequency of 1 are due to the coupling
with pitch mode oscillations shown in the appendix. At the same dimensionless frequency
of 1 the FRF in pitch has its maximum value, which is shown in the appendix. Obviously
pitch acts in an opposite direction to surge with a phase difference of n a result
confirmed by Mei (1989).
With reference to the second numerical example (Figure 6.26), Figures 6.39, 6.40,
and 6.41 present the damping, added mass, and FRF in heave motion for an elliptical
cylinder with various drafts. It can be seen from Figure 6.39, that damping in heave
decreases with increasing draft. An intuitive explanation for this phenomenon can be
offered from the wavemaker-theory point of view. The closer the underwater shape is to
the limiting case of a horizontal plate, the greater the displaced water volume, and the
greater the wave height generated and thus the greater the damping. Figure 6.40 shows
that the added mass in heave motion decreases with increasing draft. As explained for the
heave damping, the greater the draft, the less the displaced water volume, the less the local
pressure gradient is, the less the added mass becomes. Once again this is consistent with
the experimental results of Vugts (1968) for a horizontal cylinder with a rectangular cross
section. Figure 6.41 shows that the peak value of FRF in heave motion increases when
the draft increases. Obviously, the peak value of FRF in heave motion FRF(con)
corresponds to oscillation of the floating body with the natural frequency, since the
denominator of (6.9) achieves its minimum when co = con.

162

163
Fig. 6.41 FRF in heave. Elliptical cylinder. Draft variations.
o>
Fig. 6.42 Damping, added mass, and FRF in heave. Half-submerged elliptical cylinder.

164
It can be seen from Figure 6.41, that the natural frequency decreases when the draft
increases. This is true despite the fact that added mass, fj., decreases with draft because
an increase in draft means an increase in mass, M, and eventually means an increase of the
denominator in (6.10). The lower the natural frequency, the lower the denominator in
(6.9), and the larger FRF(a>n) becomes. Figure 6.42 presents heave damping, added
mass, and FRF for a half-submerged elliptical cylinder. .
6,4. Numerical Convergence
In order to prove the numerical convergence of the far-field amplitude series,
(6.11)
it is sufficient to provide its absolute convergence (Taylor and Mann, 1983), namely
(6.12)
In accordance with Cauchy's root test, also known as the Cauchy-Hadamard convergence
criterion (Solomentsev, 1988), the series (6.12) is absolutely convergent if there is a
positive number q< 1, such that
n
(6.13)
for all sufficiently large values of n. For the numerical solution, it was practically sufficient

165
to choose nMax=24. As a characteristic of the speed of convergence, a partial sum is
defined as
ZkA()|
5(A0 = ia1 (6-14)
I I nMax
where l criterion (6.13) are illustrated on the following figure, in an example of a SAT numerical
solution for a particular circular frequency. As can be seen, four terms of the series are
sufficient to provide the necessary convergence. Moreover all the terms satisfy the
Cauchy-Hadamard convergence criterion, with q=0.3< 1.
Convergence of the far-field amplitude (FFA)
Fig. 6.51 Convergence of the far-field amplitude series. Cauchy-Hadamard criterion.

166
For all circular frequencies of interest, the number of terms necessary to provide the
absolute convergence did not exceed 9. To be on the safe side, the usual number of terms
used in the computer program was 24.
6.5. Conclusions
In general there is good agreement between the computed and measured
experiments and results for surge and heave motion. The differences are due to
experimental error (discussed in chapter 5), to numerical error, and to the theoretical
limitations of the mathematical model. As shown in section 3.5 with equations (3.66-
3.69), the damping is associated with the energy given up by the oscillating body.
Nonlinear effects of the fluid-body interaction were observed in the results of the
experiments by the presence of the second and third harmonics, shown in Figure 5.3. By
virtue of the fact that these higher harmonics carry part of the energy given up by the
oscillating body, they unavoidably constitute a portion of the damping. This can be
qualitatively estimated by deriving the damping coefficients (3.73), from (3.71) with the
following form of the radiation velocity potential
(6.15)
a
which includes the higher harmonics. If the asymptotic solutions of the higher harmonics
a2(lco), (j)al{2co),... as well as their far-field amplitudes £72(2co), £7*3(3 known, than the damping coefficients would be

167
(6.16)
where
/l(3) = pgCg{craCrp + <2*Qp j is the solution given by (3.73), and the addition due to
higher harmonics is
cu [a (lC0)a U<) + a (ico)af ch]
(6.17)
In (6.17) c¡j are the integration constants derived from (3.71). As explained in section
3.5, the diagonal terms of the damping matrix are non-negative, which means that
- > 0
(6.18)
The addition due to higher harmonics is positive. Since the theoretical model, which is
based on linearized radiation theory can not predict these higher harmonics, it is
understandable that the experimentally measured damping is less than that computed by
the model, because of (6.18). Nevertheless, the general agreement between the measured
and computed hydrodynamic coefficients is sufficient for using the following engineering
approach when investigating floating body dynamics:
a) Use the linearized radiation theory to obtain the hydrodynamic damping and
added mass as functions of wave frequency and body geometry and then
b) Use the derived functions of the hydrodynamic coefficients to analyze higher
order nonlinear effects on design forces and allowable amplitudes of
oscillation.

APPENDIX
PITCH MODE OSCILLATIONS
Since pitch mode experiments were not conducted and since stability is strongly
dependent on the metacentric height (usually small in pitch), the added mass, damping,
and FRF in pitch mode for these two examples, are given in the appendix. It is assumed
that keeping the center of gravity below the instantaneous center of buoyancy provides
stability. This is usually achieved by placing ballast at the bottom of the floating body.
The circular frequency, damping, and added mass are made again non-dimensional with
the help of the breath of the section at the waterline (6.1), (6.2), and (6.3).
With reference to the first numerical example (Figure 6.25), Figures 6.43, 6.44,
and 6.45 present the damping, added mass and FRF in pitch motion for a circular cylinder
with various drafts. It can be seen from Figure 6.43, that damping in pitch increases when
the draft-to-radius ratio increases. An intuitive explanation for this phenomenon can be
offered from the wavemaker-theory point of view. Consider the limiting case of
substituting the underwater shapes with a flap-type wavemaker in the form of a vertical
plate with the same draft, extended above the water surface, and hinged at the center of
gravity of the floating body. Since the center of gravity is at a fixed distance from the
bottom point, an increment in draft means an increment of the segment of the plate
extending from the center of gravity to the waterline. As long as there is no mooring line,
the center of rotation coincides with the center of gravity. If all three draft variations are
168

169
to oscillate with constant pitch amplitude, the greater the draft, the greater the displaced
water volume, and the greater the amplitude of the waves generated and thus the greater
the damping. Figure 6.44 shows that the added mass in pitch motion increases with
increasing draft. Using the same flap-type wavemaker analogy, as presented in chapter 6,
the greater the draft, the greater the local pressure gradient, the greater the underwater
surface area on which the local pressure gradient acts, and thus the greater the added
mass. Figure 6.45 presents the FRF in pich motion for the three drafts. The units of FRF
are degree per meter. The peak values of FRF have exactly the same character as given in
Bhattacharyya (1978). The greater the draft, the lower the dimensionless frequency of the
peak value of FRF.
With reference to the second numerical example (Figure 6.26), Figures 6.47, 6.48,
and 6.49 present the damping, added mass and FRF in pitch motion of an elliptical cylinder
with various drafts. It can be seen from Figure 6.47, that damping in pitch increases when
the draft-to-radius ratio increases. An intuitive explanation for this phenomenon can be
offered from the wavemaker-theorypoint of view. Consider the limiting case of
substituting the underwater shapes with a flap-type wavemaker in the form of a vertical
plate with the same draft, extended above the water surface, and hinged at the center of
gravity of the floating body. Since the center of gravity is at a fixed distance from the
bottom point, an increment in draft means an increment of the segment of the plate
extending from the center of gravity to the waterline. As long as there is no mooring line,
the center of rotation coincides with the center of gravity. If all three draft variations are
to oscillate with a constant pitch amplitude, the greater the draft, the greater the displaced
water volume, and the greater the amplitude of the waves generated and thus the greater

170
the damping. Figure 6.48 shows that the added mass in pitch motion increases with
increasing draft. Using the same flap-type wavemaker analogy, the greater the draft, the
greater the local pressure gradient, the greater the underwater surface area on which the
local pressure gradient acts, the greater the added mass. Figure 6.49 presents the FRF in
pitch motion for the three drafts. The units of FRF are degree per meter. The peak values
of FRF have the same character as given in Bhattacharyya (1978). The greater the draft,
the lower the dimensionless frequency of the peak value of FRF. The differences between
the peak values of Figures 6.45, and 6.49 are due to the asymmetry of the elliptical shape
in pitch mode ocsillations.

171
O 0.2 0.4 0.6 0.8 1 1.2 1.4
dimensionless circular frequency
Fig. 6.44 Added mass in pitch. Circular cylinder. Draft variations.

172
Fig. 6.45 FRF in pitch. Circular cylinder. Draft variations.
O)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
dimensionless circular frequency
Fig. 6.46 Damping, added mass, and FRF in pitch. Circular cylinder.

173
Fig. 6.47 Damping in pitch. Elliptical cylinder. Draft variations.

174
Fig. 6.50 Damping, added mass, and FRF in pitch. Elliptical cylinder.

LIST OF REFERENCES
Batchelor, G.K. (1967). An introduction to fluid dynamics. Cambridge University Press,
New York.
Bernoulli, D. (1757). Principes hydrostatiques et mechaniques, ou memoire sur la maniere
de diminuer le roulis et le tangage dun navire sans quil perde sensiblement aucune des
bonnes qualites que sa construction doit lui donner. French Academy of Sciences, Paris.
Berteaux, H.O. (1991). Coastal and oceanic buoy engineering. Woods Hole, MA: H.O.
Berteaux.
Bhattacharyya, R. (1978). Dynamics of marine vehicles. Wiley, New York.
Bitterman, D.S., Niiler, P.P., Aoustin,Y. and du Chaffaut, A. (1990). Drift buoy
intercomparison test results. NOAA Data Report ERL AOML-17, National Oceanic
and Atmospheric Administration, Environmental Research Laboratories, Silver Spring,
MD.
Bouguer, P. (1746). Traite du navire, de sa construction et de ses mouvemens. Paris.
Bronshteyn, I.N., and Semendiaev, K.A.. (1986). Handbook of mathematics for engineers
(in Russian). Izdatelstvo Nauka, Moscow.
Chakrabarti, S.K. (1987). Hydrodynamics of offshore structures. Computational
Mechanics Publication, Springer-Verlag, London.
Chung, J.S. (1976). Motion of floating structure in water of uniform depth.
Journal of Hydrodynamics, Vol.10, pp. 65-75.
Chung, J.S. (1977). Forces on submerged cylinders oscillating near a free surface. Journal
of Hydrodynamics, Vol. 11, pp. 100-106.
Dean, R.G. and Dalrymple,R.A. (1991). Water wave mechanics for engineers and
scientists. World Scientific, Englewood Cliffs, New Jersey.
Doynov, K.I. (1992). Numerical technique for wave radiation and diffraction problem
solution. Proceedings of the Jubilee Scientific-Technical Conference. Varna Technical
175

176
University, Bulgaria.
Euler, L. (1749). Scientia Navalis, sev tractatus de construendis ac dirigendis navibus
(in Latin). St. Petersburg, Russia.
Evans, D.V., and Mclver, P. (1984). Added mass and damping of a sphere section in
heave. Applied Ocean Research, Vol. 6, No.l, pp.45-53.
Falnes, J. (1984). Technical note: Comments on 'Added mass and damping of a sphere
section in heave. Applied Ocean Research, Vol. 6, No. 4, pp.229-230.
Faltinsen, O.M. (1990). Sea loads on ships and offshore structures. Cambridge University
Press, New York.
Frank, W. (1967). Oscillations of cylinders in or below the free surface of deep fluids.
Naval Ship Research and Development Center Report 2375, Maryland.
Froude, W. (1862). On the rolling of ships. Trans. Inst, of Naval Architecture. Vol.3,
pp. 1-62. London.
Froude, R.E. (1896). The non-uniform rolling of ships. Trans. Inst, of Naval Architecture.
Vol.37, pp.293-325. London.
Ghalayini, S.A., and Williams, A.N. (1991). Nonlinear wave forces on vertical cylinder
arrays. Journal of Fluids and Structures, Vol 5, pp.1-32.
Gradshteyn, I S ., and Ryzhik, I.M. (1980). Table of integrals, series and products
(translation from Russian). Academic Press, New York.
Hamilton, G.D. (1988). Guide to drifting data buoys. Intergovernmenal Oceanographic
Commission, World Meteorological Organization, Manuals and Guides, Geneva.
Hamilton, G.D. (1990). Guide to moored buoys and other ocean data acquisition systems.
Report on marine science affairs 16, WMO No.750, World Meteorological Organization,
Geneva.
Harhf, M. (1983). Analysis of heaving and swaying motion of a floating breakwater by
finite element method. Ocean Engineering, No. 3, pp. 181-190.
Haskind, M.D. (1944). The oscillations of a body immersed in a heavy fluid. Prikladnaia
Matematika i Mehanika, Vol.8, pp.287-300.
Haskind, M.D. (1957). The exciting forces and wetting of ships in waves (in Russian).
Izdatelstvo Akademii Nauk SSSR, Otdelenie Teknicheskih Nauk, Vol.7, pp. 65-79.
Moscow.

177
Haskind, M.D. (1973). Hydrodynamics theory of ship oscillations (in Russian). Izdatelstvo
Nauka, Moscow.
Havelock, T.H. (1955). Waves due to a floating hemisphere undergoing forced periodic
oscillations. Proceedings: Royal Society of London, Vol. A231, pp.1-7,
Hooft, J.P. (1982). Advanced dynamics of marine structures. Wiley-Interscience, New
York.
Huang, E.T. and Pauling, J.R. (1993). Sea loads on large buoyant cargo during ocean
transport. Ocean Engineering, Vol. 20, No.5, pp.509-527.
Hughes, S.A. (1993). Physical models and laboratory techniques in coastal engineering.
World Scientific, New Jersey.
Hulme, A. (1982). The wave forces acting on a floating hemisphere undergoing forced
periodic oscillations. Journal of Fluid Mechanics, Vol. 121, pp.443-453.
Keldysh, M.V. (1935). Technical notes on some motions of heavy fluid (in Russian).
Izdatelstvo "Tzagi", Moscow.
Kim, W.D. (1965). On the harmonic oscillations of rigid body on a free surface.
Journal of Fluid Mechanics, Vol. 21, pp.427-451.
Korvin-Kroukovski, B.V. and Jacobs, W.R. (1957). Pitching and heaving motions of a
ship in regular waves. Trans. Soc. of Naval Architecture and Marine Engineering, Vol.65,
pp.590-632. New York.
Kriloff, A. (1896). A new theory of the pitching of ships on waves, and of stresses
produced by this motion. Trans. Inst, of Naval Architecture, Vol.37, pp.326-368.
London.
Kriloff, A. (1898). A general theory of the oscillations of a ship on waves.
Trans. Inst, of Naval Architecture, Vol.40, pp.135-196. London.
Landau, L.D., and Lifshitz, E.M. (1988). Hydrodynamics (in Russian). Izdatelstvo
Nauka, Moscow, Russia.
Mabie, H.H. and Ocvirk, F.W. (1963). Mechanisms and dynamics of machinery. John
Wiley and Sons, New York.
Martin, P.A. and Farina, L. (1997). Radiation of water waves by a heaving submerged
horizontal disc. Journal of Fluid Mechanics, Vol. 337, pp.365-379.

178
Mei, C.C. (1989). The applied dynamics of ocean surface waves. World Scientific,
Englewood Cliffs, NJ.
Milne-Thomson, L.M. (1950). Theoretical hydrodynamics. The Macmillan Company, New
York.
Molin, B (1979). Second-order diffraction loads upon three-dimensional bodies. Applied
Ocean Research, Vol. 1,No.4, pp. 197-202.
Molin, B. and Marion, A. (1985). Second-order loads and motions for floating bodies in
regular waves. Proceedings 5th International Offshore Mechanics and Arctic Engineering
Conference, pp. 353-360. Tokyo.
Nestegard, A. and Sclavounos, P.D. (1984). A numerical solution of two-dimensional
deep water wave-body problems. Journal of Ship Research, Vol. 28, pp.48-54.
Newman, J.N. (1975). Interaction of waves with two dimensional obstacles: a relation
between the radiation and scattering problems. Journal of Fluid Mechanics. Vol. 71,
pp.273-282
Newman, J.N. (1976). The interaction of stationary vessels with regular waves.
Proceedings of 11th Symposium in Naval Hydrodynamics. Office of Naval Research, pp.
491-502. Washington.
Newton, R.E. (1975). Finite elements in fluids. Gallagher et al., Vol.l, pp.219-232
Newton, R.E., Chenault, D.W. and Smith, D.A. (1974). Finite element solution for added
mass and damping. Proceedings of the International Symposium on Finite Element
Methods in Flow Problems, Swansee, U.K.
Ogilvie, T.F. (1964). Recent progress toward the understanding and predictions of ship
motions. Proceedings of the fifth Symposium of Naval Hydrodynamics, Bergen, Norway.
pp.3-80
Patel, M.H. (1989). Dynamics of offshore structures. Butterworths. London.
Rahman, M. (1994). Ocean waves engineering. Computational Mechanics Publications.
Sahin, I. (1985). Motion analysis of floating structures by a surface singularity panel
method." Proceedings, pp. 1071-1076.
Sarpkaya, T. and Isaacson, M. (1981). Mechanics of wave forces on offshore structures.
Van Nostrand Reinhold Company. New York.

179
Sarpkaya,T. (1989). Computational methods with vortices The Freeman Scholar
Lecture. Journal of Fluids Engineering. Vol.165, pp. 61-71
Sawaragi, T. (1995). Coastal Engineering waves, beaches, wave-structure interactions.
Elsevier. Netherlands.
Soding, H. (1976). Second-order forces on oscillating cylinders in waves. Schiffstechnik.
Vol.23, pp.205-209.
Solomentsev, E.D. (1988). Functions of complex variables and their applications (in
Russian). Izdatelstvo Vysshaia Shkola. Moscow. Russia.
St. Denis, M. and Pierson, W.J. (1953). On the motions of ships in confused seas. Trans.
Soc. ofNaval Architecture and Marine Engineering. Vol.61, pp.280-357. New York.
Stansby, P.K. and Isaacson, M. (1987). Recent developments in offshore hydrodynamics:
workshop report. Applied Ocean Research, Vol.9 (3), pp. 118-127
Sumer, B.M. and Fredsoe, J. (1997). Hydrodynamics around cylindrical structures. World
Scientific, New Jersey.
Taylor, A.E. and Mann, W.R. (1983). Advanced calculus. John Wiley & Sons. New
York.
Taylor, R.E. and Hu, C.S, (1991). Multipole expansions for wave diffraction and radiation
in deep water. Ocean Engineering, Vol.18, No.3, pp.191-224.
Taylor, R.E. and Hung, S.M. (1987). Second-order diffraction forces on a vertical
cylinder in regular waves. Applied Ocean Research, Vol.9, pp. 19-30
Ursell, F (1949). On the heaving motion of a circular cylinder on the surface of a fluid.
Quarterly Journal of Mechanics and Applied Mathematics, Vol. 2, pp.218-231
Vantorre, M. (1986). Third-order theory for determining the hydrodynamic forces on
axisymmetric floating or submerged bodies in oscillatory heaving motion.
Ocean Engineering, Vol. 13, No.4, pp.339-371.
Vantorre, M. (1990). Influence of draft variation on heave added-mass and hydrodynamic
damping coefficients of floating bodies. Journal of Ship Research, Vol.34, No.3, pp. 172-
178.
Vethamony, P., Chandramohan, P., Sastry, J.S., and Narasimhan, S. (1992). Estimation of
added-mass and damping coefficients of a tethered spherical float using potential flow
theory. Ocean Engineering, Vol. 19, No.5, pp.427-436. '

180
Vugts, J.H. (1968). The hydrodynamic coefficients for swaying, heaving and rolling
cylinders in a free surface. Report No. 112S. Netherlands Ship Research Center TNO.
Shipbuilding Department, Delft, Netherlands.
Vugts, J.H. (1971). The hydrodynamic forces and ship motions in oblique waves.
Report No. 150S (S 2/238). Netherlands Ship Research Center TNO. Shipbuilding
Department, Delft, Netherlands.
Weinblum, G. and St. Denis, M. (1950). On the motions of ships at sea. Trans. Soc. of
Naval Architecture and Marine Engineering. Vol.58, pp. 184-248. New York.
Yeung, R.W. (1981). Added mass and damping of a vertical cylinder in finite-depth
waters. Applied Ocean Research, Vol.3, No.3, pp.l 19-133
Yeung, R.W. (1985). A comparative evaluation of numeric methods in free-surface
hydrodynamics. Hydrodynamics of ocean wave-energy utilization. IUTAM Symposium.
Lisbon.
Zhang, J. and Dalton, C. (1995). The onset of a three-dimensional wake in two
dimensional oscillatory flow past a circular cylinder. Presented at the 6th Asian Conference
on Fluid Mechanics. Singapore.

BIOGRAPHICAL SKETCH
Krassimir Doynov was bom in 1963 in the city of Iambol, Bulgaria, where the
lands of ancient Thracia still bring reminiscence of old glory. Raised in a family of
teachers, he was encouraged to develop his knowledge perpetually. His father, whose
History of the city of Iambol, was published along with other historic surveys, was
inspiration for the young Krassimir.
Being a math nerd, Krassimir was admitted to a gifted class in the school with a
priority in mathematics Lobachevski. There, he was encouraged by several exceptional
math and science teachers, and afterward, pursued a bachelors degree in civil engineering
at the University of Architecture, Civil Engineering and Geodesy in Sofia, Bulgaria. From
1988-1990 he earned his masters degree in applied mathematics and computer science at
the Technical University in Sofia.
In 1990, along with his family, he moved to the city of Varna a city of beauty and
tradition on the Black Sea coast of Bulgaria. His appreciation for the sea was affected by
his life on the shore. His work was related to the sea, also. He investigated the dynamics
of a wave energy extraction device as a graduate research assistant at the Technical
University-V arna.
In 1994 Krassimir joined the Department of Coastal & Oceanographic Engineering
at the University of Florida in Gainesville. Since that time, he has been pursuing a Ph.D.
degree under the auspices of Professor D. Max Sheppard. Upon his graduation, Krassimir
181

182
looks forward to applying the knowledge, gained in coastal and oceanographic
engineering as well as to materializing his love of computers and computer programming.



82
4.3.3.1.2.2. Far-field wave amplitudes: More than half-submerged circle
The use of conformal mapping (4.30), (4.31) results in
/= = (4.95)
Sb+S "=1 C "=>
where the In integral is to be calculated over the unit circle (Figure 4.3)
/ = Uka C:|f|=l
In a cylindrical coordinate system In can be expressed as
ln 00 lilrrtY 0 , o (ih-r/Y . th(s+l)-n]B
ln = Je^^ide = J d __ (4.97)
s=0
-2*
j=0
5! j, A,(j + 1)-/i
co oo
=1
5*!
/^(s + l)-/?
(4.98)
From (4.91) and (4.98) it can be concluded that
Z)+()=_^M
K) 2g s\
as+xe~k\ e-,[Ms+1H*j + ^4+ig+ArA, ^
/z2(j' + l)-
(4.99)
4,3.3.1.2,3. Far-field wave amplitudes: Exactly half-submerged ellipse
The use of conformal mapping (4.61), (4.62) results in
1= §f(yykycty = can §eikCl{(+XC\n(\-C2)dC = cl'an[Gn-AGn+2), (4.100)
Si+S* n=l C:|f|=l n=1


8
(2.2)
h is the water depth, and co is the circular frequency. Now with (2.2), the Keulegan-
Carpenter number is
H
n
2 tiA 'l
(2.3)
Btmh(kh) 5tanh(M)
L
The physical meaning of the Keulegan-Carpenter number (more easily seen for the case of
deep water: tanh(kh)=l) is the ratio between the circumference of the fluid-surface-
particle-orbital motion and the characteristic body dimension. Speaking in Navier-Stokes
equation (see equation 3.2) terms, the Keulegan-Carpenter number is the ratio between
$11
spatial u and temporal accelerations. In accordance with Meis definitions (1989),
dx dt
a body is regarded as large when kB>0(l)\ its presence can significantly alter the pattern
of wave propagation, produce wave diffraction, and the disturbance can propagate in a
much wider area far away from the large body. On the contrary, a body is regarded as
small when kBl; diffraction is of minor importance. When A/B> 0(1) the local velocity
gradient near the small body augments the effect of viscosity and induces flow separation
and vortex shedding, leading to so called form drag. It should be noted that the
associated viscous forces are not mean shear forces, but pressure forces due to separated
flow. The influence of a small body is usually bounded to a comparatively narrow area.
A more precise classification is given in Sarpkaya and Isaacson (1981), where Isaacson
has presented a convenient means of indicating the conditions under which the diffraction,


101
4.3.6. Discussion of the Uniqueness of the Solution
A half-submerged circular cylinder of radius R is presented on Figure 4.5 in the
orthogonal coordinate system Oxz.
x
Fig. 4.5. Exactly half-submerged circular cylinder in Oxz.
A polar coordinate system Opd is related to the orthogonal system Oxz in the following
manner
x = pcosO
z = psind
(423'
dp dx dz
y -x+iz = pe'9
The boundary condition (4.24) over the wetted body surface St can be written as
(4.235)


56
where {£} is the time amplitude of the generalized displacements {X}:
{Z) = Re({^>-'i) (3.60)
3,5, Hydrodynamic Relationships. Identities, and Definitions
As introduced and explained in Section 2.3, all necessary relationships and
identities among the hydrodynamic properties and forces will be summarized briefly for
further use, without detailed derivation. For a complete description the reader is referred
to Mei (1989). For any two twice-differentiable functions t, and ^., the Green's
theorem states:
(3.61)
where O is a closed volume with boundary 60 consisting of the wetted body surface Sb,
the free surface SF, the bottom Bo, and a vertical circular cylinder with an arbitrary large
radius Sx If i, and zero due to Laplace equation. By virtue of the boundary conditions (3.54b), and (3.54c),
neither the free surface SF nor the bottom Bo contributes to the surface integral thus
reducing the right-hand side to:
Sb+S<0
dn
dS = 0.
(3.62)


CHAPTER 1
MOTIVATION
In XV-century Europe, the brilliant engineer, scientist and artist, Leonardo da
Vinci recorded for the first time an ngineering application of a free floating buoy. The
buoy was used to measure the water velocity in streams and consisted of a weighted rod
and flotation bladder. It was released in the water flow and its downstream travel
measured after a given period of time and the average velocity computed as the distance
divided by the elapsed time. Since that time drifter buoys have proved to be very effective
in increasing the level of our understanding of the ocean environment and in improving
weather analysis and forecasting. Scientific investigations using drifting buoy systems
include measurements of atmospheric pressure, air and sea-surface temperature, solar
radiation, air humidity, wind shear, wave evolution, wave noise, light penetration, oceanic
current speeds, and water temperature and salinity. A variety of drifter buoy systems,
deployed in the worlds ocean, are used to monitor the spatial and temporal distribution of
the above mentioned environmental variables. With modern satellite technology, which
provides both buoy positioning information and a means of receiving and re-transmitting
data from the buoys, the accuracy and reliability of these measurements has been greatly
improved. Measurements can be recorded with data acquisition systems mounted on the
buoys or transmitted to land based stations via satellite (Berteaux, H.O., 1991). Data
from these buoys are vital to weather organizations for early detection of storms and for
daily weather predictions. This information is essential for those organizations and
1


58
P
a p
II ^x,x. =1;Ii'Z\mxi,x.+m*x.xX}-'ZZ
a P
dp,
dX^
dt
= 0,
and vanishes due to periodicity. This means that the average rate of work done by the
oscillating buoy on the fluid over a period is
a P
a P
Next, the energy flux far away from the buoy can be expressed as
(3.69)
Wa
Re<
imp re ,R d(j)h
[il*
cfo
ds
sk

k eR W
dn
dn
ds.
(3.70)
Moreover, with the help of (3.62), (3.70) can be transformed to
^=Tii
ir d(j>R -ARdh
f
3i
ds.
(3.71)
When^R = %2^Va(f>a and the two-dimensional asymptotic behavior of c
lim a
X>oo
-igal cosh k(z + h)cjkx
m cosh kh
(3.72)
are substituted in (3.71), and (3.71) equalized to (3.69), the law of conservation of energy
flux expresses damping in terms of Q-l
Kp=pgCg{a~aa-+a:a;) (3.73)
where Cg is the group velocity, and (.) denotes the complex conjugate. Cl^ will be
referred to as the far-field amplitude, that has dimension of time since Cl^ =A+/Va.


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
KEY TO SYMBOLS vi
ABSTRACT viii
CHAPTERS
1 MOTIVATION 1
2 INTRODUCTION 4
2.1. Historical Retrospective of Floating Body Studies 4
2.2. Classification 7
2.2.1. Large and Small Bodies 7
2.2.2. Deterministical and Statistical Approaches 11
2.2.3. The Concept of Added Mass 13
2.2.4. Classification of Damping 14
2.2.5. Numerical Methods Classification 16
2.3. Advancements in Floating Body Studies 24
2.4. Presentation of the Results of Investigations 26
2.5. Some Thoughts about the Current State of Knowledge 34
3 FORMULATION OF THE PROBLEM 37
3.1. General Description of the Problem and its Simplifications 37
3.1.1. Incompressible Fluid Assumption 37
3.1.2. Governing Equations and Definitions 38
3.1.3. Inviscid Fluid Assumption 39
3.1.4. Irrotational Flow Assumption 41
3.1.5. Dynamic Free Surface Boundary Condition (DFSBC) 42
3.1.6. Kinematic Free Surface Boundary Condition (KFSBC) 42
3.1.7. Sea Bottom Boundary Condition (SBBC) 43
3.1.8. Wetted Body Surface Boundary Condition (S) 43
3.1.9. Linearization to First-Order Theory 44
3.2. Floating Body Dynamics 47
3.2.1. Conservation of Linear Momentum 47
IV


61
The purpose of the frequency response function and the response amplitude operator are
to characterize the dynamic response of the floating body as a function of the circular
frequency of the incident wave. The vector form of the frequency response function is
W = ^}^PgCt)[{C]-^([M]+[f,])-im[>.]]'{a-}. (3.84)
3.6. Algorithm for the Solution of the Problem
In summary, the asymptotic solution of the radiation problem gives the far-field
amplitudes. The added mass matrix [//], the radiation-damping matrix [X] (3.73), and the
exciting force (3.81) can be found from these far-field amplitudes. Therefore the
asymptotic solution of the radiation problem alone will be sufficient for analyzing the body
dynamics (3.84). Practically this means that the solution of the diffraction problem can be
avoided. Having the dynamics and hydrodynamics (radiation) problems formulated, and
all the necessary relationships for their solutions, the following algorithm will be used.
First, the asymptotic solution of the radiation problem, in terms of the far field amplitudes,
will be found in heave, surge, and pitch. Second, the hydrodynamic added mass, damping
coefficients, and exciting forces will be computed as functions of the far-field amplitudes;
consecutively the body dynamics will be computed for a particular wave frequency in
terms of H, or RAO. Third, using the RAO (3.83), the spectral analysis can be used to
find the body response spectrum S^co) due to the incident wave spectrum S^co) for the
entire frequency domain of interest, namely
S(co) = \H(vfs(a).
(3.85)


86
, nlma
/2= and
n-K
4 = V
4.3.3.2.3. Left-hand side. Exactly half-submerged ellipse
Using the conformal mapping (4.61), (4.62) results in
L =
C
f 4
c+-
{ o
r" +%,*'
4=4,
1 ,,-'0-1)0 \p~' (n+1)0
l~e ^ 1 e e-i(n-1)0 ^-¡(+1)0
w-1
77 + 1
+ikxAwn ^,
4 c\
1 A
n-1 H + l -l
c--(-i)g n^ c-'(+i)0
+ 1
+ ikxAwn, and
4 ci
_1 _*L+W-*Le-
2 2
+ikxAwl.
The real part of (4.127) is given as follows:
1 X
Re(4) = Ci
n-\ n+1
- kxA Im(wn) H£l- cos(n -1)0 Cos(n + 1)9
Re(4) = ci
-1-
X
- kxA Im(Wj) cos(2 9) .
4,3,3.2.4. Left-hand side. More than half-submerged ellipse
Using the conformal mapping (4.44), (4.45) results in
L =h1\[c(a£i'+ a¡<+ + a,r*)C"
(4.123)
(4.124)
(4.125)
(4.126)
(4.127a)
(4.127b)
(4.128a)
(4.128b)
(4.129)


CHAPTER 4
RADIATION PROBLEM SOLUTION
41. Problem Statement and Definitions
A long horizontal rigid cylinder with its cross-section in the Oxz-plane is
oscillating on the free water surface in otherwise calm water. Part of it is below the water
surface; the other part is in the air. The rigid cylinder is oscillating with unit velocity in
the generalized -direction with the circular frequency of the incident waves co (in the
absence of the incident waves), thus generating outgoing waves. Taking a unit-cylinder
length in the ^-direction, the problem is considered two-dimensional, and the motion can
be described in the cross-sectional Oxz-plane (Figure 4.1).
z

Fig. 4.1 Cross-section of the floating body. CB, CG centers of buoyancy and gravity.
62


99
rs V
\6n{v6)co?{W)dd-7i
7T *
sin(v + /)^ ^ sin(v-/)^ cos(v' + /);t cos(v-/j
X2{v + l)2 7T2{v-l)2 K {y+l) 7i{y-l)
1 0
16 sm{vG)d6 = 7z
^ -K
1 0
OdO--, and
n * 2
sin(vTr) cos(v7r)
nv
2 2
7T V
-|0cos(/0>/0 =
_2 2_
K l2
0
when l = 2k +1
when / = 2£
Substituting (4.214) and (4.217) into the boundary condition (4.24) yields
2>.
1 = 1
2>,cos(/0)
/=0
= 6, cos(/<9).
;=o
Regrouping the terms on the LHS, results in
I
/=0
Lw=i
cos(/#) = ]>^[>,]cos(/?).
/=0
Equating the terms in the square brackets on both sides of (4.89),
c/a =bi / = 0,1,2,..,,oo ,
=1
results in the following system of linear equations
/ = 0
C01
C02
C03
C0H
/=1
cu
CU
C13
C\N
/ = 2
^21
C22
C23
C2N
= jV-1
CN-1.2
CW-1,3
CH-\,N _
n-1
n = 2
n = 3
... = #
a,
a.
a3 } = i
a
N
Ph- 1.
(4.225)
(4.226)
(4.227)
(4.228)
(4.229)
(4.230)
(4.231)


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
A DYNAMIC RESPONSE MODEL
FOR
FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES
By
Krassimir I. Doynov
December, 1998
Chairman: D. Max Sheppard
Major Department: Coastal & Oceanographic Engineering
A semi-analytical method for computing the dynamics of free-floating, horizontal
cylinders, subjected to ocean waves has been developed. The bodies analyzed in this
analysis are cylinders with circular and elliptical cross sections and variable still water
drafts. The motions considered are heave, surge and pitch. The technique computes the
added mass and damping coefficients using linearized radiation theory. The efficiency of
the numerical scheme is due to the simplicity of the mathematical scheme used a
combination of original holomorphic functions, convergent infinite power series, and
conformal mapping. An object-oriented approach was used for the computational aspects
of the problem using the programming language C++.
Physical experiments in a wave tank were conducted with circular and elliptical
horizontal cylinders in order to verify the method. The cylinders were positioned at the
viii


26
direction) in terms of the radiation potential and its normal derivative (due to body
oscillation in the same direction, in otherwise calm water). An elegant consequence from
the Haskind theorem is that the exciting force is linearly proportional to the far-field wave
amplitude. The very practical meaning of the Haskind theorem is that an actual solution of
the diffraction problem can be avoided and that the exciting forces are simply related to
the damping coefficients. Newman elaborated on this further and found an explicit
relationship between the magnitudes of the exciting force and the damping coefficients,
which is often used as a check on the magnitudes. An even less obvious identity between
radiation and diffraction problems was discovered by Bessho (1967) for two dimensions
and extended for three dimensions by Newman (1975,1976). It relates the far-field
radiation and diffraction wave amplitudes, and reveals the significant result that the phases
of surge and pitch antisymmetric modes differ with n.
2.4. Presentation of the Results of Investigations
In light of the above-mentioned theoretical advancements, it is instructive to give
some examples of the theory-experiment interplay. In 1965 Kim determined the added
mass and damping for a semi-ellipsoidal body with its origin on the free surface of water
of infinite depth. In order to check the validity of potential flow theory predictions Frank
(1967) conducted a series of experiments to determine the added-mass and damping
coefficients for two-dimensional bodies at the free surface.
Using Ursells multipole expansion method and De Jongs extension to sway and
roll motions of arbitrary shaped cylinders, Vugts (1968) solved the linear radiation


2
companies involved in offshore oil exploration and production, marine transportation,
commercial and recreational fishing and boating, and military operations. For reliable data
transmission to the monitoring satellite, it is crucial that the surface floats provide a stable
platform under a variety of wave, current, and wind conditions. Since it is the surface
water motion and properties that are of interest, the question becomes how well do these
buoys track the currents and how do they respond to the wave motion. How a buoy
responds to currents and waves depends on a number of quantities including the buoy size,
shape, mass and mass distribution. Knowledge of the response characteristics of a buoy
for a given set of wind and wave conditions as a function of their structure and wind-wave
parameters would allow buoy designs to be optimized for the sea state conditions in which
they are to be deployed. Therefore the buoy hull must be designed in accordance with
certain stability design criteria for different ocean and atmosphere conditions.
The purpose of the following research is to investigate how variations of draft, size
and shape of drifting buoy systems influence fluid-body dynamics and to create a
computer-based model. In the focus of the research are floating horizontal cylinders of
circular and elliptical cross sections with variable still water drafts. The computer-based
model is intended to provide designers with estimates of the dynamic response
characteristics, in terms of response amplitude operators or frequency response functions,
of relatively small buoys.
The dissertation is comprised of six chapters. Chapter 2 presents a brief historical
retrospective of floating body studies, followed by classifications that introduce criteria for
large and small bodies, and deterministic and statistical approaches. Next, the concept of
added mass is introduced, as well as classifications of damping and the most frequently


97
(eie + Ae~i0)
e3'e Ae,a
-e~a +
2 3
(4.213)
Since
J = c, 1 - is a sum of two terms that appear in the denominator of
dC dC \ c)
A
clear in 4.3.5.
4.3,42.4, Real part of the right-hand side: More than half-submerged ellipse
The procedure is analogous to that described in 4.3.4.2.3.
4,3,5. Fourier Expansion of LHS and RHS: Solution for the Unknown Coefficients
For the sake of brevity, the solution for the unknown coefficients will be explained
for the case of the more than half-submerged circle. Having the explicit form (4.202) of
right-hand side of the boundary condition (4.24), it can be expanded it in a Fourier cosine
series in the interval 0 e[-7r,0] as follows:
oo
R Q(RHS) = ^b,cos(ie).
(4.214)
;=o
when 1= 1,2,3,...; and for the case of 1=0 the coefficient is


172
Fig. 6.45 FRF in pitch. Circular cylinder. Draft variations.
O)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
dimensionless circular frequency
Fig. 6.46 Damping, added mass, and FRF in pitch. Circular cylinder.


104
dr2 dr2
dn dp
p=R
=\(h-RHS)
(4.249)
Therefore, substituting (4.248) and (4.249) into (4.245) leads to
|J(r,VVJ-r,VV,)dV = 4j[r,(r!-WiS)-r!(r,-iWS)]ilK
= Jt\[r (4.250)
The uniqueness of the solution requires that (rx r2) = 0 in (4.250). It is obvious from
(4.250) that they can differ by a constant, namely (rx -r2) = const. The origin of this
constant is the fact that the holomorphic function f(y), (4.8a), consists of two terms
dwiy)
proportional to both complex velocity and complex velocity potential w(y).
dy
dw(y)
While the exact form of can be determined from section 4.3.4.1, the form of w(y)
dy
can be determined only approximately, with accuracy of additive and/or multiplicative
constants.
Another form of the same boundary condition (4.24) can be derived in the following
manner:
\YJ(y)dy = ^SjjLdy+itfvHfy = w-wA +ik\YA(w -wA+wA)dy,
fAf(y)dy W wA + ikwA (y-yA) + ik^ (w -wA)dy,
(4.251)
(4.252)
(4.253)


130
6.1. Surge Mode Oscillations
The non-dimensional damping coefficients in surge (6.8) are presented in Figures
6.1 through 6.8. There is some scatter among the measured points but the consistency of
the experiments is good. The upper limit of the dimensionless frequency corresponds to
the maximum frequency of the driving motor, 1.2 Hz while the lower limit of the
- dimensionless frequency represents the physical boundary below which waves cannot be
measured with an accuracy of 0.1 mm. For a prototype buoy with a characteristic size of
2 m, length scale 1:20, and temporal scale 1:4.5, this frequency range corresponds to a
prototype period range between 3.6 and 12 sec. This means that the maximum frequency
of the driving motor imposes a limitation on wave periods below 3.6 sec. Figures 6.1 and
6.2 present the surge oscillation of two semi-submerged circular cylinders with drafts
equal to half of their corresponding diameters of 0.114 m, and 0.102 m. It can be seen in
Figure 6.1 that the sets of damping measurements for 52 and 63 mm strokes, produced by
the 0.114 m diameter cylinder surround Vugts' data when co
> 0.35 and are below
I B
Vugts' measurements when co J < 0.35. In the same figure the theoretical solution is in
almost perfect agreement with the 52 mm set of measurements. In Figure 6.2, the sets of
measurements produced by the 0.102 m diameter cylinder are slightly below Vugts' data
and the theoretical solution. In light of the dimensionless notations used the result of the
experiments presented in Figure 6.1 and 6.2 suggest that damping in surge motion
increases with the size of the buoy.


42
with a temporal constant C(t) which can be omitted by redefining the velocity potential
without affecting the velocity field.
3,1.5. Dynamic Free Surface Boundary Condition fDFSBO
A characteristic feature of the free surface of air-water interface is that it can not
support pressure variations and hence must respond appropriately to maintain the pressure
uniform. For the practical interest, the wave length is so long that the surface tension is
unimportant; the pressure just beneath the surface must be equal the atmospheric pressure
Pa above. Therefore the DFSBC is defined as a uniform pressure along the wave form on
the free surface. Applying Bernoullis equation (3.12) on the free surface, we have
,on: = (
(3.13)
3.1.6. Kinematic Free Surface Boundary Condition (TCFSBC)
The instantaneous free surface of a wave can be described with the equation
F{x,y,z,i) = z-fe,y,t) = 0
(3.14)
where £(x,y,t) is the displacement of the free surface about the horizontal plane z=0.
Defining material surface as surface (air-water) across which there is no flow, a particle
remains and moves only tangentially on that material surface when
(3.15)
which gives the KFSBC


LIST OF REFERENCES
Batchelor, G.K. (1967). An introduction to fluid dynamics. Cambridge University Press,
New York.
Bernoulli, D. (1757). Principes hydrostatiques et mechaniques, ou memoire sur la maniere
de diminuer le roulis et le tangage dun navire sans quil perde sensiblement aucune des
bonnes qualites que sa construction doit lui donner. French Academy of Sciences, Paris.
Berteaux, H.O. (1991). Coastal and oceanic buoy engineering. Woods Hole, MA: H.O.
Berteaux.
Bhattacharyya, R. (1978). Dynamics of marine vehicles. Wiley, New York.
Bitterman, D.S., Niiler, P.P., Aoustin,Y. and du Chaffaut, A. (1990). Drift buoy
intercomparison test results. NOAA Data Report ERL AOML-17, National Oceanic
and Atmospheric Administration, Environmental Research Laboratories, Silver Spring,
MD.
Bouguer, P. (1746). Traite du navire, de sa construction et de ses mouvemens. Paris.
Bronshteyn, I.N., and Semendiaev, K.A.. (1986). Handbook of mathematics for engineers
(in Russian). Izdatelstvo Nauka, Moscow.
Chakrabarti, S.K. (1987). Hydrodynamics of offshore structures. Computational
Mechanics Publication, Springer-Verlag, London.
Chung, J.S. (1976). Motion of floating structure in water of uniform depth.
Journal of Hydrodynamics, Vol.10, pp. 65-75.
Chung, J.S. (1977). Forces on submerged cylinders oscillating near a free surface. Journal
of Hydrodynamics, Vol. 11, pp. 100-106.
Dean, R.G. and Dalrymple,R.A. (1991). Water wave mechanics for engineers and
scientists. World Scientific, Englewood Cliffs, New Jersey.
Doynov, K.I. (1992). Numerical technique for wave radiation and diffraction problem
solution. Proceedings of the Jubilee Scientific-Technical Conference. Varna Technical
175


70
4.3.Semi-Analvtic Technique: Determination of the Unknown Coefficients
The procedure for determining the unknown coefficients in (4.12) consists of
4.3.1. Constructing a boundary condition for the holomorphic function through
integration of (4.8a) over the wetted body surface Sb, so that the left-hand side
contains all unknown coefficients, and the right-hand side contains all terms
derived from the complex fluid velocity.
4.3.2. Conformal mapping of Sb and its mirror image Sb into a unit circle, which permits
expressing the mapped holomorphic function with an analytic function of type
(4.12), convergent outside the unit circle
4.3.3. Determining the left-hand side, so that all multipliers of the unknown coefficients
are trigonometric functions of the polar angle of the unit circle 9.
4.3.4. Determining the right-hand side, so that all term are trigonometric functions of the
polar angle of the unit circle 6. Determining the complex fluid velocity with a
generalization of a procedure outlined in Milne-Thomson (1950).
4.3.5. Determining the unknown coefficients by solving a linear system of equations,
derived through the Fourier expansion of both left- and right-hand sides of the
boundary condition.
4,3.1. Boundary Condition on Sh
Integrating (4.8a) over the wetted body surface Sfein the clockwise direction from
point A to point Y (Figure 4.2), results in


83
G= je
c-\(\=i
^ {ikCxX) | ktfs-ln+s),
s=0
5!
e,kCl(£-(n+s>dC.
(4.101)
C:|i|=l
Applying the same technique as in (4.92), (4.93), gives
G.-Z
{ikcxX)¡
i=0
s\
2m
(ikcx)
/7+.S-1
(n + s-1)!
-(ikcx)2s+n~l Xs (-\Y(kcx)2s+n~lAs
2m^Yj- = 2mnY/ M : and
i=0
s!(w + 5 1)! ^ i!(s + -l)!
v 2+rt-l
Gn=2mnX-{n-x)/2Y< ^ = 2mnX{n-x),2Jn hkcxJx)
sl(s+n-l)¡ v
,when X > 0
(4.102)
where denotes the Bessel function of the first kind,
Gn=2m\-Xr^
2 j+w1
5) s!(s + n-1)
= 2mn(-X) 1nIn_x{2kcx4-X} ,when X < 0
(4.103)
where denotes the modified or hyperbolic Bessel function of the first kind. From
(4.91), (4.100), (4.102) and (4.103) it can be concluded that
(4.104)
4.3,3.1,2.4. Far-field wave amplitudes: More than half-submerged ellipse
Using the conformal mapping (4.44), (4.45) results in
^a^)di=e-^aJn
/= §f(yYl,dy=e-l'''La.
=1 c:|£|=l
n=1
(4.105)


87
t -in-hy )9 < -i(n+h¡)9
l~e '-ne ~ -a2e~,{n+hz)0 +ikyAwn
Ln~h2a\ Kai .
n-h, n + h~
-ihxe -axe
(4.130)
L =A^l__A^ + /iy w _iUe-ine -WC^e-i^e (4.131)
" n-/*2 +/>2 ^ ^ n-\ n+h2
Re(Z) = Re
h2al h2a2
n-h2 + /t.
+ ikyAwn
Re(a1)cos(-/22)6, +101(0;! )sin(A7-/z2)6)]
n h.
[Re(a2) cos( + )6 + Im(a2) sin( + )\ hx sin(n?)
n + h.
(4.132)
4,3.4. Right-Hand Side of the Boundary Condition on Sh
To find an explicit form of the right-hand side of the boundary condition (4.24)
dw
, the complex velocity around the boundary St, must be determined.
dy
4,3.4.1. Complex velocity around the boundary Sh. Surge, heave, and pitch mode
The procedure given below, is a generalization of a procedure outlined in Milne-
Thomson (1950). Consider again the radiation velocity potential and its normal (to the
wetted body surface) derivative for heave, surge, and pitch (4. Id)
R=Vh+VA + Vpp and
(4.133)
-= K^~ + Vs~+vJ?p- = Vsnx+ Vhnz + Vp\nx (z- Z(0)) nt (x -- ^f(0))]. (4.134)
dn on an on L v K /J
The components of the unit normal vector are


71
£>wv=j,
dw[y)
dy
+ ikw[y)
dy
(4.21)
Substituting (4.22)
fwdy = wy- wAyA fcydy
(4.22)
into (4.21) results in
jAf(y)dy = -ikyAwA + [ikw)y + £^(i- iky)dy
(4.23)
Eliminating ikw in (4.23) and (4.8a), gives a boundary condition whose right-hand side
dw
(RHS) is a function of the complex velocity and complex variable >>,
dy
\YJ(y)dy-yf(y)+ih>AwA =
(4.24)
As stated before, only the real part of (4.24) will matter after deriving its explicit form.
Some thoughts about the uniqueness of the solution of the boundary condition on Sb are
given in section 4.3.6.
4,3.2. Conformal Mapping
If the wetted cross section of the floating buoy is a half-submerged circle, then the
analytic presentation (4.12) in the left-hand side (LHS) of the boundary condition (4.24)
can be used. In this case the unknown coefficients can be determined from those
corresponding to the generalized motion RHS. If the wetted cross-section of the floating
buoy is different than the half-submerged circle, then Sb can be mapped into a half-
submerged circle C. Denote the analytic conformal mapping by


12
formulated as a distribution of the relevant quantities over the independent variables. An
excellent example found in nature, which explains the two formulations and their
relationship, is the irregular sea. Obviously the time history is very difficult to obtain and
is not important. In order to obtain statistical estimates, the linear theory simulates the
irregular sea as a superposition of linear harmonic waves. In the case of a floating body, if
its response to a harmonic wave is solved deterministically, it will help to find statistically
its response to the irregular sea. One of the most generally used ways to describe and to
work practically with a random sea is to consider the distribution of its energy content as a
function of wave frequency (sea spectrum). The concept is to sum a large number of
sinusoids with small amplitudes, different frequencies and phases, with some waves adding
to build up larger ones and others canceling each other, thus forming an irregular profile
with no set pattern as to amplitude or periodicity (Figure 2.2.).
20
o
-20
20 5 10 15 20 25 30 35
0
-20
20
0
-20
20
0
-20
50
0
-50
5 10 15 20 25 30 35
Time in seconds
Fig.2.2. Irregular wave as a sum of four harmonic waves with different
amplitudes and circular frequencies
Irregular profile as a sum of four harmonic waves


85
.g 1 p-i(n-l)e
L =/?f e-Kn-l)eide-Kel{n-x)e+ihcAwn = R Kq^9+ikxAwn, (4.113)
"Jo n-1
and
Ln=+ikxAwn-Re-iin-l)S. (4.114)
n-1 -1
The real part of (4.114) is given as follows:
Re(J = ^-j-cos(-l)0. (4.115)
4,3,3.2.2. Left-hand side: More than half-submerged circle
Using the conformal mapping (4.30), (4.31) results in
L, = h1afi(J>-"dC-(ihl +<-)C" <4.116)
L. =h,af eHH-"wid0-ihle-M-cte-K-V +tkx/,w (4.117)
Jo
1 _J(h2-n)0
Ln h2a ihie~in9 ae-i{n~h)8 +ikxAwn and (4.118)
n-\\
Ln = ikxAwn +-^ £Le-0-W -i^9. (4.119)
n-hz n-h2
The real part of (4.119) is given as follows:
Re(Z) = /o+/,cos(-/?2)0 + /2 sin( )<9 + /3 sin(n<9). (4.120)
The real coefficients /, sRe,(/ = 0,1,2,3) are
+ (4.121)
n-h,
j Rea
1 7
n-K
(4.122)


dimensionless added mass 9s dimensionless damping
153
Fig. 6.28 Added mass in surge. Circular cylinder. Draft variations.


90
where c" is a constant. Therefore the complex velocity on the boundary C in 0¡r¡ is
dwR
dQ
= Velf(R(-2) + V, [iRYmi~2 ]
(4.155)
From the decomposition (3.5 Id), (4.50) of the radiation potential, the complex velocity
should be
d (4.156)
which means
dw. R
dwh iR
~d^~~C
dwp jRY{0)
dC ?
, surge mode
(4.157)
, heave mode and
(4.158)
, pitch mode.
(4.159)
4,3.4.1.2. Complex velocity around the boundary Sh. More than half-submerged circle
Applying the conformal mapping (4.30) in (4.144) results in
y = -ih{ +a£ ^ (4.160)
yy + ad ih^a^ + ih^dC'^ (4.161)
B,{C) = ~Ve'p(d^) + [(/'/?, Y(0])dCh] and (4.162)
B2(fl = Ve-iil(aCh>)+ivU-ihl-Y{0))aCh>] (4.163)
It is obvious from (4.162) and (4.163) that
Bt(d = -Bl(C')
(4.164)


water surface and forced to oscillate sinusoidally first in surge and later in heave motion.
Different still water drafts and oscillation frequencies were used in the experiments. The
far field waves produced by the oscillating cylinder were measured using capacitance wave
gauges. The damping and added mass coefficients were computed from the measured
wave data. There is a good agreement between the predicted and measured coefficients.
IX


21
region can serve as a transition zone that transforms a geometry of higher complexity into
a simpler cylindrical geometry where the singular kernels (associated with integral-
equation methods) can be more conveniently calculated. Variational principles can be
utilized to formulate the radiation problem and to seek an approximate solution for the
velocity potential in the inner domain surrounding the body. The strength of the
variational principles is that they permit an exact coupling of the approximate interior
solution with the analytical solution for the velocity potential in the outer domain. The
outer domain solution is usually presented in one of the following two ways. The first one
is to use Greens function and express the velocity potential as a superposition of sources
of unknown strength on the boundary between the inner and outer domains, while the
second way is to use eigenfunction expansions with unknown coefficients. In the case of
infinite water depth, the eigenfunction expansion was found to be inefficient, and the
Greens function approach was recommended instead. By obtaining two different answers
for two different grids for the velocity potential at a particular point, it was proven that the
general identities and the energy conservation between rate of work done by the body
and the rate of energy flux through the boundary between the inner and outer domains -
are necessary but not sufficient conditions to guarantee an accurate solution. Mei
generalized that similar caution was warranted in other numerical methods.


3
used numerical methods for the determination of these hydrodynamic properties.
Furthermore, the most significant advancements in floating body studies are presented,
followed by a review of recent scientific works on the related subjects. This chapter
concludes with an attempt to classify the current state of knowledge about floating bodies
as one going simultaneously in three main directions. As a result of the well-known
conservation laws of mass, linear, and angular momentum, chapter 3 formulates the
floating body dynamics as a system of linear second-order differential equations with
boundary conditions of Neumann and Dirichlet-Neumann type. After introducing the
generally accepted simplifications for incompressible, inviscid fluid and irrotational flow,
the linearized radiation theory is deduced. Furthermore the hydrodynamics is
decomposed from the body dynamics, and some hydrodynamic theorems are given with
purpose to express all unknown variables of the floating body dynamics as functions of the
far-field amplitudes. Chapter 4 describes an exact analytical asymptotic solution of the
radiation problem, which derives the far-field amplitudes as functions of the wetted surface
of the floating body, and the circular frequency of the incoming harmonic wave. Chapter
5 describes the experiments conducted in heave and surge motion, which main purpose is
to obtain data for the far-field wave amplitudes and thus to verify the analytic solution,
introduced in chapter 4. These experiments were carried out in a wave tank at the Coastal
Engineering Laboratory at the University of Florida. Chapter 6 presents the analysis of
the results from the experiments and comparison with the numerical solution. The
conclusion is given at the end of this chapter.


116
insure that the waves measured were not contaminated by reflected waves from the South
boundary, the data-records started at the time of initial wave generation, and ended at the
time required for the waves to travel to the South end of the tank and back to gauge #2
(Figure 5.6).
0 5 10 15 20 25 30
"uncontaminated time" =32.1 [sec]
Fig.5.6 Uncontaminated time


158
O). =
C
M + n
(6.10)
Obviously, the peak value of FRF in heave motion FRF{con) corresponds to oscillation of
the floating body with the natural frequency, since the denominator of (6.9) achieves its
minimum when co con. It can be seen from Figure 6.33 that the natural frequency
decreases when the draft increases. This is true despite the fact that added mass, ju ,
decreases with draft because an increment in draft means an increment in mass, M, and
eventually means an increment of the denominator in (6.10). The lower the natural
frequency, the lower the denominator (Xcon) in (6.9), and the larger FRF(con) becomes.
Figure 6.34 presents heave damping, added mass, and FRF for a half-submerged circular
cylinder.
With reference to the second numerical example (Figure 6.26), Figures 6.35, 6.36,
and 6.37 present the damping, added mass, and FRF in surge motion of an elliptical
cylinder with various drafts. It can be seen from Figure 6.35 that in general, damping in
surge increases with increasing draft. For dimensionless frequencies above 1, or periods
below 1.4 seconds, the picture is reversed. An intuitive explanation for this behavior can
be offered from the wavemaker-theory point of view. Consider the limiting case of
substituting the underwater shapes with the same draft vertically oriented plates with part
of the plate above the water surface. If all three drafts are to oscillate with a constant
surge amplitude, the greater the draft, the greater the displaced water volume, and the
greater the damping. Figure 6.36 shows that the added mass in surge motion generally
increases with increasing draft. An intuitive explanation for this fact is that the greater the
draft, the greater the underwater surface area upon which the local pressure gradient acts,


174
Fig. 6.50 Damping, added mass, and FRF in pitch. Elliptical cylinder.


33
When the flow around a cylinder is two-dimensional and the Reynolds number is small, i.e.
Re<200, a direct solution with finite-differences yields results for the gross-flow
parameters that are in reasonable agreement with measurements, while the lift force is
grossly overestimated (Sumer and Fredsoe, 1997). For larger Reynolds numbers, vortex
shedding occurs in cells and therefore the flow becomes three-dimensional (3D). When
300 further, 3D-turbulence begins to spread into the boundary layer, and direct numerical
simulation of the Navier-Stokes equations is not feasible, because of the scales of the
dissipative part of the turbulent motion. The oscillatory flow is solved using the same
vorticity transport equation (3.7) and Poisson equation (3.8). The major source of
difficulties is the number of grid points (for finite-difference and finite-element methods)
and the corresponding number of computations required to obtain a solution increases
with increasing Reynolds number, and may become prohibitive as mentioned earlier at
large Reynolds number. The second group consists of discrete vortex methods, developed
as grid-free numerical methods. The idea is to solve the vorticity transport equation
(Equation 3.7) through a numerical simulation of convective diffusion of discrete vortices
generated on the cylinder boundary. The advantages summarized by Sumer and Fredsoe,
(1997) are (1) the inviscid theory can be employed, (2) numerical diffusion problems
associated with the vorticity gradient terms in Eulerian schemes are, to a large degree,
avoided, (3) there are no zone assumptions to require matching of an outer to an inner
flow, and (4) the method is relatively stable and well suited to vectorization on
supercomputers (Stansby and Isaacson, 1987). For a detailed review of the vortex
methods see Sarpkaya (1989). For oscillatory flow and waves the vortex methods fail to


39
an important deduction from the Navier-Stokes equations is that the rate of change of
vorticity is due to stretching and twisting of vortex lines and to viscous diffusion.
^ + u.vja = aVu + vV2Q (3.5)
Defining a streamline as a line tangent everywhere to the velocity vector, it is obvious that
the physical concept of streamlines must exist in a general three-dimensional compressible
flow (Dean and Dalrymple, 1991). A Streamfunction exists in two-dimensional or
axisymmetric incompressible flow and is defined by
ay
dy
= u
and
ay
dx
= -v
in Oxy-plane,
or
(3.6)
= u and = -w in Oxz-plane
dz dx
For two-dimensional incompressible flow, equations (3.4), (3.5), and (3.6) reduce to the
2D vorticity-transport equation
+u.v)q = vV2H
U )
and the Poisson equation
(3.7)
a2y a2y
dx2 dy2
= -Q
(3.8)
3,1.3. Inviscid Fluid Assumption
In water the kinematic viscosity is small: v = 0.01 [cm2/sec], which means that the
last term of equation (3.5) is negligible, except in regions'of large velocity gradient and


79
In+\ ~
ikR.J -e
m
and
n
Pn +Qn = e ikXARI
(4.72)
(4.73)
4,3,3,1,1,2 Determination of P+iO,. More than half-submerged circle
The use of conformal mapping (4.30); (4.31) results in
r = ]/(y)e* M=l
-kh oo
1 =
ik
n=l
1 jW* 00
Jka^
1 1
lev*-* = 2>,
00 ^
e
+JVteiV("+1)^
00
-kh oo
lK n=1
where I is
In=j^n+%ika^dc.
1
After changing the variable of integration £ with t = and correspondingly
1
d£=--^dt, In becomes
Substituting (4.78) into (4.75) gives
, e~kh^
!=-,r2>.
k n=l
(/far)' 1
e +2j
j=0
5! shi-n
(4.75)
(4.76)
(4.77)
(4.78)
(4.79)


163
Fig. 6.41 FRF in heave. Elliptical cylinder. Draft variations.
o>
Fig. 6.42 Damping, added mass, and FRF in heave. Half-submerged elliptical cylinder.


53
for the diffraction subproblem as s. The dimensions of (j)1, s, and D are
[Length2/Time],
Thus the necessary decomposition of the time amplitude of the velocity potential is given
by
*=*D+^=(*J+^)+ErA (3-51d)
Next the complete hydrodynamics problem, (3.34), is reformulated in terms of time
amplitudes of the diffraction velocity potential with (3.51):
VVD =0 in the fluid domain
d£_
dz
dx J
, on the free surface Sf
, on the sea bottom Ba
, on the wetted body surface- Sb
, waves outgoing at infinity
(3.52a)
(3.52b)
(3.52c)
(3.52d)
(3.52e)
The only missing lateral boundary condition in (3.34) is now given by (3.52e) in 2D-space.
Sommerfeld had introduced a similar lateral boundary condition, in an analogy with an
outgoing wave system. The incident velocity potential is given by
^-^cosh*(£+*) e_ (3.53)
co cosh kh
In a similar way, the radiation subproblem is formulated as
V2(j>a = 0 in the fluid domain
(3.54a)


amplitude-to-stroke ratio
138
0.6
0.5
O
to 0.4
(D
o
i
to
0.3
"D
3
E 0.2
CO
0.1 -
+ 4- draft =D/2
< < draft=1.2T>/2
0 0 draft =1.4T)/2
Stroke =63mm;D =114 mm
<
+
<
+
^M.
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
dimensionless circular frequency
Fig. 6.10 Wave amplitudes. Circular cylinder in surge.
0.4
-
0.35
+ + Stroke = 41 mm; Horizontal
<] 0 0 Stroke = 41 mm; Vertical
0.3
Stroke = 52 mm; Vertical
-
0.25
0.2
-
s
0
0.15
0
0.1
0
$
0.05
0 <1
-
. 0 < *
Si AiL. <* 1 i
1 1
0.15*
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.6
dimensionless circular frequency
Fig. 6.11 Wave amplitudes. Elliptical cylinders in surge.


76
tj=2tt + tja +lt,6, and
(4.51)
(4.52)
TV
If Sb (4.43) is transformed into the upper half of the unit circle C: £ = l.e,e, where
6 e[-2 n-n], then
h¡ = h,
(4.53)
Ci = (b+a)/2,
(4.54)
A = (b-a)/(b+a),
(4.55)
ax = cle3,r,A,
(4.56)
a2 = cxAe~3,r,A,
(4.57)
7i-2tt + 3t]a Jrh19, and
(4.58)
= n+2rjA
(4.59)
n
4,3,2,4, Conformal mapping of exactly half-submerged ellipse
This is a particular case of 4.3.2.3, with h = 0, 774 = 0, \ 0, and = 0, and
results in
y = bcos^T]) +iasm(?j) ,7e[0,2^]. (4.60)
The conformal mapping which transforms Sb (4.60) into the lower half of the unit circle
C: £= l.e'9, where 0 e[-7r,0] is
y = f(C) =
ci
r
V
and
(4.61)


A DYNAMIC RESPONSE MODEL
FOR
FREE FLOATING HORIZONTAL CYLINDERS SUBJECTED TO WAVES
By
Krassimir I. Doynov
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998


49
JJJx x JJ x x Pnds + xc x (-Mgk)
ir dt p
(3.42)
Taking the cross product of (3.35) with X, and subtracting the result from (3.42), results
in the conservation of angular momentum with respect to the center of rotation Q
JJJ (x X) x ^-j^-dm = JJ(x X) x Pnds + (xc X) x (-Mgk)
which following Meis procedure leads to
x-component:
44" 44" + (4 + 4, K 4/4 4r, = -p JJ 4'V*
sr
-pg[z(l,4 + a(42 + 4) pi* yl\] + Mg[a(r Z(0>) y{? 4>)
y-component:
44" 44"+(4= + 4)4 4r 4 = -p if 4 V*
4s
+^g[z(1>4 +<-4(4 +4,')+4]+Mg[/?(r-z(0))-r(j-401)
z-component:
44" 44" +(4i+42 )r,, 4. 4A = -/>JJ 4V*
4
where the first and second moments of inertia are defined as follows
4=JJJ(*-4>) 4,=J}\{x-X^dm
(3.43)
(3.44a)
(3.44b)
(3.44c)
Ibn=\\\{x-X^){y-Y^)dm


129
dimensionless added mass (6.3)
where X and p are the damping and added mass coefficients, Awet is the area of the
cross-section below the still water level, B is the breath of the section at the waterline, p
is the water density, and g is the acceleration of gravity. The damping coefficient is
computed from the measured far-field wave amplitude Ga (3.63) using the expression
X = 2pgCga2a (6.4)
where Cg is the group velocity. The far-field amplitude results directly from the measured
wave height H, wave circular frequency co, and amplitude of oscillations S/2
a =
a coS
From linear wave theory the wave group velocity is
,( A i \
(6.5)
c
2 2
1 +
2 kh
V
sinh(2&/j)
= tanh(/z)
1 +
2 kh
\
sinh(2 kh)
(6.6)
where k is the wave number, h is the water depth, and C is the wave celerity. The formula
for computing the wave damping coefficient from the measured wave height is
X pg2 tanh(/z)
1 +
2 kh
sinh(2A/i)
H2
3r*2
)S
(6.7)
Consequently, the dimensionless damping coefficient is given by
B _{B/2f 1
AC \ 2 S
A (
0)
tanh(kh)
1 +
2kh
sinh(2 kh)
H2
S2
(6.8)


78
-'* ] f(y)edy = a, (7> + IQ.),
(4.66)
thus giving expressed as a series of the unknown an coefficients
n=l
WA =EaW
where
(4.67)
w_ =
-jV'i.Wtf,
+/
CO
(4.68)
4.3.3,1,1. Determination ofP+/^
+/£) (4.66) can be expressed in the following manner
^]nyy-dy=fJ.(P.+/&).
+co
4.3,3.1.1.1. Determination of P+iO. Exactly half-submerged circle
Using the conformal mapping (4.40), (4.41) results in
I = J JiyYdy = Ra. j e'^CdC = K a,/, (4.69)
+00 **=1 00 W 1
where / can be determined knowing h and the following recurrent formula
/= j emCC"d( ["+/,]
(4.70)
The first integral can be expressed with a complex exponential-integral function
1 me kR it
/, = J d£ = Jdt = -Ex{-ikR), (Gradshteyn and Ryzhik, 1980) (4.71)


KEY TO SYMBOLS
Symbol
Description
A
Amplitude of the incident wave
A
Far-field wave amplitude
a* = A/va
Far-field amplitude. Dimension time.
a, b
Vertical, and horizontal semi-axes of the elliptical
cylinder
an
Power series coefficient of the nth term
B
Breadth of the waterline section of Sb
A
Sea bottom boundary
ce
Group velocity
rci
Buoyancy restoring force matrix of the floating body
^(.) = |(.) + u.V(.)
Total derivative in space and time
E
Water bulk modulus
i^}
Exciting force vector due to diffraction
8
Gravity acceleration
H
Height of the incident wave
\H(co)\ = RAO
Transfer function
h
Water depth
i
imaginary unit
KC
Keulegan-Carpenter number
k
Wave number of the incident wave
L
Wave length of the incident wave
\M\
Mass matrix of the floating body
"a
Component of unit normal to Sb vector in a direction
Pa
Atmospheric pressure
P
Pressure
R
Radius of the circular cylinder
Re
Reynolds number
SA
Part of the water surface cut out by Sb
sb
Instantaneous wetted body surface
sb
Mirror image of Sb in the air
Lateral boundary at infinity
SJ>)
Incident wave spectrum
vi


27
problem for two-dimensional floating cylinders. Conducting experiments with five
different cylinder cross-sections in heave, sway, and roll oscillatory motion, he has
compared the experimental results with the predictions of the linear potential theory and
particularly the multipole expansion solution. With the influence of viscosity negligible,
Vugts has found good agreement between calculated (with the potential theory) and
measured added mass and damping coefficients in heave for the middle and high frequency
range. However, in the low-frequency range, co^B / (2g) < 0.33, the multipole expansion
method significantly overestimates the experimentally obtained added mass. In sway, the
calculated added mass is in good agreement with measured values for most of the
frequency range: 1.125 < co^Jb /(2g) < 1.25. In the higher frequency range the damping
was underestimated. For the case of coupling of sway and roll the differences between the
measured and calculated added mass and damping coefficients are higher. Viscous effects
in terms of separation and eddy making have been observed in sway motion for the
relatively sharp edged sections. With predominant wave damping, the viscous damping
contributes from 10% at low frequencies to 40% at high frequencies for sharp edged
sections. Obviously the energy loss due to eddy formation has been one of the reasons for
these differences in damping. Interestingly enough it appears that eddy formation does not
seriously affect the total pressure distribution in phase with the body acceleration since the
added mass has been predicted relatively well. For rolling oscillatory motion the
calculated added mass moment of inertia overestimates the measured one, while for the
damping coefficient the observations were the same as in the case of sway. Since the
wave damping part for the sections considered is of an order smaller than for swaying, the


74
hl=-h. (4.35)
If Sb (4.29) is transformed into the upper half of the unit circle C: £= \.e'e, where
0 e[-2n~n\, then
(p-2K+2(pA+h20, (4.36)
a = Re3'^, and (4 36)
hl=h (4.38)
4.3,2.2, Conformal mapping of exactly half-submerged circle
This is a particular case of 4.3.2.1, with h- 0, The result is
y = Reip > Sb:

The conformal mapping which transforms Sb (4.39) into the lower half of the unit circle
C: £ = l.e'e, where 6 e[-^-,0] is
y=/(£) = &. >and
dy df{Q
dC
= R* 0 .
(4.40)
(4.41)
4.3.2.3, Conformal mapping of more than half-submerged ellipse
If a and b are the vertical and horizontal semi-axes of the ellipse and h is the
distance between the center of the ellipse and the still water level as shown in Figure 4.4,
then the wetted body surface and its mirror image can be-written as:


108
frequencies up to 1.2Hz. Four capacitance-type wave gauges were mounted in the tank,
two on each side of the test buoy as shown schematically in Figure 5.1, Measurements
were done for three different amplitudes of motion and a period range from 0.8 to 6.3
seconds. The surge motion amplitudes were 20.5, 26, and 31.5 mm.
5.2,1. Cylinders
As shown below in table 5.1', two circular and one elliptical cylinders were driven
by the Scotch yoke mechanism to generate waves in otherwise still water.
Table 5.1. Principal data of cylinders
Material
aluminum
steel
wood
wood
Cross section
circle
circle
horizontal ellipse
vertical ellipse
Length in [mml
568
568
568
568
Breadth in [mm]
114.3
101.6
131
99.4
Draft in [mm]
57.2-68.6-80
50.8-61-71.1
49.7-59.6-69.6
65.5-78.6-91.6
End plates diameter
190
150
190
190
thickness in [mm]
2
2
2
2
In order to provide similar surface roughness the wooden elliptical cylinder was covered
with several coats of acrylic. End plates were fixed to both ends of all cylinders to prevent
vortex formation around the edges and to reduce three-dimensional wave effects. Their
diameter was greater than 1.5 times the cylinder breadth, as shown on Figure 5.1. The
clearance between the end plates and the glass walls of the tank was 2 mm or 0.35% of the
width of the tank, thus minimizing the effects of the leakage around the end plates.


152
case of higher frequencies, the side view of the underwater body surface can be imagined
as one composed of multiple vertical paddles in a staircase fashion along the convex
shape. While all paddles transmit constant normal velocity to the neighboring water
particles a phase lag is imposed by the paddles above the point of inflection, which
eventually results in energy transmitted from the body to the water, and consecutively less
damping. Figure 6.28 shows that the added mass in surge motion generally increases with
increasing draft. An intuitive explanation for the fact is that, the greater the draft, the
greater the underwater surface area is upon which the local pressure gradient acts, and
thus the greater the added mass. Figure 6.29 presents the FRF in surge motion for the
three drafts. A general tendency can be inferred that at low frequencies the floating bodies
are almost perfect wave followers. This behavior decreases as the circular frequency
increases. The disturbances at dimensionless frequency of 0.6-0.7 are due to the coupling
with pitch mode oscillations shown in the appendix. FRF in pitch has its maximum value
at the same dimensionless frequency, 0.6-0.7. Obviously pitch acts in an opposite to surge
direction, with a phase difference n, a result confirmed by Mei (1989).


77
dy df{C)_
dC d£
= c,
A
*0.
(4.62)
4.3,3, Left-Hand Side of the Boundary Condition on Sb
To find an explicit form of the left-hand side of the boundary condition (4.24) wA,
the complex velocity potential at point A must be found (Figure 4.2).
4 3 3.1. Complex velocity potential at point A
Substituting^ fory in (4.13) will give the following expression for wA
wA = w(yA)
= e~,kyA
K +iA + jfiy^dy]
(4.63)
Since the constants Ax, are proportional to the far field wave amplitudes in (4.18a)
Ax=-j^a\
CO CO.
equation (4.36) takes the following form:
w
yA
= | Ax cos(Ax^) + ^2 si^Ax^)-!-/^, cos(fcr/1)- Ax sin(AxJ)] + e~,kyA J f(y)e,kydy
(4.64)
By virtue of the conformal mapping (4.26, 4.27), which will be proven below in 4.3.3.1.1
and 4.3.3.1.2, both terms in (4.64) can be expressed in terms of the unknown
coefficients an
£7* = AaA+(w) and
n= 1
(4.65)


5
that most engineering approaches in floating body studies are only valid in a certain range
of practical interest. Developing further Froudes idea with a paper in 1896 The Non-
Uniform Rolling of Ships William Froudes son, R. E. Froude, added the forcing of the
rolling motion for regular and irregular waves. Significant advancements in floating body
studies were made by Kriloff (1896, 1898), who considered for the first time the three-
dimensional problem, working with six degrees of freedom. Kriloff introduced the
hypothesis of approximating the actual pressure on the floating body surface by the
corresponding pressure in the wave structure, not being disturbed by the presence of the
floating body. As pointed out by Vugts (1971), the same hypothesis had been implicitly
made by William Froude in his earlier and more restricted work. Since then this
hypothesis has been known as Froude-Kriloff hypothesis and dominated almost all floating
body studies up to 1953. Kriloff computed the wave exciting forces and the restoring
forces and included the hydrostatic-coupling effects between heave and pitch. In an
additional estimated term, he included a resistance to motions, thus introducing the
concept of damping into the floating body studies. Speaking in modern terms Kriloff left
out the hydrodynamic mass and the hydrodynamic coupling among the various motions.
The hydrodynamic mass became known earlier. William Froude had realized that for a
longer natural roll period an apparent increase in mass was necessary. Because of the
presence of the free surface of the fluid and the determination of the hydrodynamic mass,
respectively, the added mass becomes much more complicated due to the wave frequency
dependence. In the first half of the 20th century, some attempts were made to get more
information on the added mass and damping, particularly for ships and ships sections.
With an exception for the limiting case of infinitely high motion frequencies, no major


24
2.3. Advancements in Floating Body Studies
As stated earlier, the most basic problem to solve is that of the frequency response
of the floating body when subjected to simple sinusoidal wave excitation. In 1944
Haskind introduced a way to decouple the hydrodynamics from the body dynamics and to
further decouple the hydrodynamics problem into diffraction and radiation components.
Taking advantage of the linearity of the Laplace operator and the combined Neumann and
Dirichlet-Neumann boundary conditions for the velocity potential Haskind presented the
velocity potential as a sum of diffraction and radiation potentials. Speaking in physical
terms, two hydrodynamical sub-problems were formulated. The first sub-problem
assumes the body to be rigid and restrained from any oscillation in the presence of incident
regular waves. The hydrodynamic loads are called wave excitation loads and composed
of Froude-Kriloff and diffraction forces and moments. The second sub-problem forces
the rigid body to oscillate in any motion mode with the wave excitation frequency in the
absence of incident waves. The hydrodynamic loads are identified as added mass,
damping and restoring terms. Since the body oscillates in otherwise calm water, it
radiates waves, thus the term radiation. In 1949 John (see Mei, 1989) introduced in a
systematical formal approach the complete first order wave-structure theory and showed
how higher-order extensions can be made. Following Haskind (1973), Newman (1976),
and Mei (1989), several remarkable general identities have been introduced. These
identities relate different hydrodynamic quantities and have proved to be extremely useful
in increasing the theoretical understanding of physical phenomena. Moreover, they
provide necessary checks for analytical theories or numerical methods, and minimize the


UFL/COEL-TR/121
A DYNAMIC RESPONSE MODEL FOR FREE FLOATING
HORIZONTAL CYLINDERS SUBJECTED TO WAVES
by
Krassimir I. Doynov
Dissertation
1998


114
gauge #2 data points and the first three harmonics after applying the smoothing numerical
algorithm.
The spectral analysis at gauge #j, (j=3,4) permits
r¡[t) = j cos(yt) + bj sin(yt) (5.2)
filtering out the first harmonic parameters Equating (5.1) and (5.2) results in
= 4 cos^Ax; +£,) + Ar cos^£c;. + and (5.3)
bj = 4 sin(kXj + £,)- Ar r\(kcJ + £,.) (5.4)
After some algebra a linear system of two equations (5.5) is obtained
4 cos^Sj) = [(c3 -c4)f + (d3 -d4)g\/ (f + g2), and (5.5a)
4 sin(i,) = [-(c3 ~c4)g+(d3-d4)f]/(f2 +g2) (5.5b)
with a known right hand side, namely
c;. = a, cos^hCj)-b} sin(x.) wherej=3,4
dj = cij sin(/ocy) + bj cos^kXj) where j=3,4
f = cos(2Ax3) cos(2/fcx4), and
g sin(2Ax3) sin(2x4).
A straightforward solution of the incident wave amplitude and phase follows from (5.5).
Knowing the incident wave the reflected wave amplitude and phase can be determined
from (5.3), and (5.4). The reflection and absorption coefficients for the model boundary
can be defined as follows:
, reflection coefficient, and
(5.6)


1
25
computing time of those quantities related by them. Based on the mathematical definition
of the incident, radiation, and diffraction wave potentials as harmonic functions, and on
the Greens theorem, the first identity relates, in an elegant way, two radiation problems.
A computationally efficient result is that the restoring force, the added mass and the
damping matrices are diagonally symmetric, which decreases significantly the number of
unknowns. Another consequence is the convenient relationship between the damping and
the asymptotic behavior of the radiation velocity potential. In physical terms the energy
given up by the oscillating body is transported by the waves propagating away from the
body. The knowledge (about damping) gained this way, can be used to find the added
mass, using the so-called Kramers-Kronig relations. Derived in a universal way by Ogilvie
(1964), these relations simply exploit the fact that damping and added mass are the real
and imaginary parts of the same function. Therefore knowledge of the damping
coefficients is sufficient to determine the corresponding added mass coefficients and vice
versa. Newman (1976) discovered that the damping matrix is singular for the case of
particular bodies of revolution where the exciting force is independent of the angle of
wave incidence. The second identity relates two diffraction problems corresponding to
different angles of incidence. As two-dimensional results, there are several relationships
between the amplitudes and the phase angles of the transmitted and reflected waves. The
three dimensional result is that the amplitude of the first scattered wave toward the second
incident wave is equal to the amplitude of the second scattered wave toward the first
incident wave. The third identity relates the diffraction potential to the radiation potential,
thanks to a brilliant Haskind theorem (1957). It expresses a generalized component of the
exciting force (due to fixed body diffraction from an incident wave moving in a particular


120
Fig. 5.7 Side view of the wave tank. Wave absorbers at the North end of the wave tank.


177
Haskind, M.D. (1973). Hydrodynamics theory of ship oscillations (in Russian). Izdatelstvo
Nauka, Moscow.
Havelock, T.H. (1955). Waves due to a floating hemisphere undergoing forced periodic
oscillations. Proceedings: Royal Society of London, Vol. A231, pp.1-7,
Hooft, J.P. (1982). Advanced dynamics of marine structures. Wiley-Interscience, New
York.
Huang, E.T. and Pauling, J.R. (1993). Sea loads on large buoyant cargo during ocean
transport. Ocean Engineering, Vol. 20, No.5, pp.509-527.
Hughes, S.A. (1993). Physical models and laboratory techniques in coastal engineering.
World Scientific, New Jersey.
Hulme, A. (1982). The wave forces acting on a floating hemisphere undergoing forced
periodic oscillations. Journal of Fluid Mechanics, Vol. 121, pp.443-453.
Keldysh, M.V. (1935). Technical notes on some motions of heavy fluid (in Russian).
Izdatelstvo "Tzagi", Moscow.
Kim, W.D. (1965). On the harmonic oscillations of rigid body on a free surface.
Journal of Fluid Mechanics, Vol. 21, pp.427-451.
Korvin-Kroukovski, B.V. and Jacobs, W.R. (1957). Pitching and heaving motions of a
ship in regular waves. Trans. Soc. of Naval Architecture and Marine Engineering, Vol.65,
pp.590-632. New York.
Kriloff, A. (1896). A new theory of the pitching of ships on waves, and of stresses
produced by this motion. Trans. Inst, of Naval Architecture, Vol.37, pp.326-368.
London.
Kriloff, A. (1898). A general theory of the oscillations of a ship on waves.
Trans. Inst, of Naval Architecture, Vol.40, pp.135-196. London.
Landau, L.D., and Lifshitz, E.M. (1988). Hydrodynamics (in Russian). Izdatelstvo
Nauka, Moscow, Russia.
Mabie, H.H. and Ocvirk, F.W. (1963). Mechanisms and dynamics of machinery. John
Wiley and Sons, New York.
Martin, P.A. and Farina, L. (1997). Radiation of water waves by a heaving submerged
horizontal disc. Journal of Fluid Mechanics, Vol. 337, pp.365-379.


45
X = X + £
X(1)+0(1)x(x-X(o))]-h3(>2)
x = x-s
X(1) + 0(1) x ^x X(0))] + 0{s2)
x = x-e
z(1)+ye(z-z{0))-r(y-Y{0)^
y = y-s
Y{1)+r(x-X{0))-a{z-Z{0))
z = z e
Z{l)+a(y-Y{0))-p(x-X{0)f
(3.25)
(3.26)
When the body is at its rest position, then x = x and
z = /<>(*, jO (3.27)
Substituting (3.26) into (3.27) expanding about S\ and comparing with (3.21), results
in
/(1) = Z(1) + a(y 7(0)) p(x X(0)) /j0) [x(1) + fi(z Z(0)) r(y ~ 7(0) )
-/;o)[7(1) +r(x- X(0)) a(z Z(0))
(3.28)
Substituting (3.23) into (3.28), results in the first order kinematic boundary condition on
the wetted body surface
-M0) -?/,'01+P,{z-zm)-r,(y-Ym)
-/."[if +r,(x-X(0|)-a,(z-Z<0))] (3.29)
+Z')+a,(y-Ym)-fil(x-Xm)
The unit normal vector it directed into the body becomes
n = (-/iV/f.lf 1 + (/i0))J +{4})2
-1-1/2
Equation (3.29) can be rewritten as
30
(i)
dn
xM+0Wx(x-X)].n = £^
J 1
(3.30)
(3.31)


75
Fig. 4.4 Conformal mapping of more than half-submerged ellipse
y = -ih+bcos[j]) + iasin(77) Sb:rj e[7C-T¡A,27V+TjA] (4.42)
y = ih+bcos{Tj)+iasm(T}) Sb:r¡&[-7]A,7i + rjA]. (4.43)
If the conformal mapping
y = f{g) = ihx + + a2CK and (4.44)
A
% = .r*1"')=*,(,<* 0 (4.)
UL, UL,
is used to transform Sb (4.42) into the lower half of the unit circle C: £ = l.e10, where
0 e[-7T,0], then
h, = -h, (4.46)
c¡ = (b+a)/2, (4.47)
A = (b-a)/(b+a), (4.48)
, = c1e"7/1, (4.49)
a2 = ^/le'"7-4 ,
(4.50)


105
(4.254)
jj(y)dy = -
f(y)-
dw
dy.
+[ik{y-yA)~ l]wA + ik\YA{w WA)dy, and (4.255)
\rj(y)dy+jf{y)-[ik(y-yA)-\]wA =^-~+ik^A(w-wA)dy. (4.256)
A similar analysis (4.244 4.250) will result in the same conclusion that the solution can
differ by a constant. Since it is natural to expect a unique solution in reality, the constant
can be determined from the experiments described in the next chapter.


95
d*. co
d£ ^+1
and differ only by a constant c0 in the numerator. Therefore from (4.191) the RHS is
(4.197)
RHS = -(ihx+a^)j^-^+\YA-^{\-ik(iK + a£h>))d£ and
RHS =
*Acr A
, Ay
C2ih +
f_c^
v A J
rr 1 ik(ih + a£h')
(4.197)
(4.198)
In the cylindrical coordinate system £ = l.e the RHS (4.198) transforms to
RHS =
RHS =
r AO
v Ay
e-2^ +
_ fkjg-M + Co j[(! + kh, )ie-ih>e + ka\dQ,
A
%o
v Ay
y \
e-2W +
l a;
coO+*A)
A
[e-v_i]
and
aQ+*A)s
A ,
+ (&aco)0 +
_ co(2 + ^A)V-iy + *Aco
A J
v Ay
,-2 The real part of (4.74c) is given as follows:
Rq(RHS) = r0+rx cos(h28) + r2 co^l/^O)+r30 + r4 sin(h20) + r5 sin(2h2d),
where the real coefficients rt e Re, (/ = 0,1,2,3,4,5) are
ro =
(l + ^)Re(c0)
A
r,=-
(2 + /:/?, )Re(c0)
_A
k = Re
A
A
f A"
^ ay
(4.199)
(4.200)
(4.201)
(4.202)
(4.203)
(4.204)
(4.205)


Ill
designed to detect the variation in capacitance and the analog output signal was digitized
with A to D converter and the data stored in a computer. The constant 50 Hz sampling
frequency produced data with uniform spacing between data points. An analog filter was
used to remove frequencies higher than 100 Hz. For a typical laboratory wave of 1
second a sampling frequency of 50 Hz means 50 evenly spaced samples obtained over one
wave period. This provides a more than adequate which representation of the wave form
since the commonly used "rule of thumb" is 10 evenly spaced samples per wave length
(Hughes, 1993). The capacitance type wave gauges showed good linear response and had
a resolution of about 0.1 mm. The main disadvantage was the need for frequent
calibrations, due to changing water temperature. The calibration of the gauges was
carried out statically by manually lowering and raising them at three incremental distances
of 10 mm, into the still water surface before every set of experiments. The calibration
relationships were obtained from mathematical curve-fits between the recorded gauge
outputs and the corresponding elevations at all four gauge locations. The tests showed
that the calibration characteristics were stable over a period of 3 hours the time interval
of a set of experiments provided that there was little change in water temperature and
the gauge wires were clean and free of dust and other foreign matter.
5,2.4, Surface Tension
The scale effect due to surface tension forces not being in proper proportion in a
Froude-scaled model becomes important when water waves are very short or the water
depth is very shallow. Usual "rules of thumb" (Hughes, 1993) are that the surface tension
effects must be considered when wave periods are less than 0.35 seconds and when water


91
w* = cn+Bx{Cj = -Ve^f-1*) +iVp\(il\ Y{0))aCh2] + c"
(4.166)
where c" is a constant. Therefore the complex velocity on the boundary C in 0%r¡ is
dwR
d<;
dwR
dC
= Ve'f(!hciC''-')+Vt[h1{hi +K))aT
K-1
and
= V,(K SC1'-') + Vt(iKaC''-')+Vr\h1(h, +y(0))a^*,]
Comparing (4.168) with (4.156) the complex velocity becomes
dws h^a
d<; £*+I
dwh ih^a
~dC~^
7*2+1
dwp +/T(0))
d;
c
*2+1
, surge mode
heave mode and
, pitch mode.
(4.167)
(4.168)
(4.169)
(4.170)
(4.171)
4.3 A. 1.3, Complex velocity around the boundary Sr,: Exactly half-submerged ellipse
Applying the conformal mapping (4.61) in (4.144) results in
J = c1(C1+^),
yy c\{A^2 + AC, 2 + A2 +1) ,
A(0 = V{cxAe-ip-c^)C' +iVp\cx2AC2 + W\a)cx
, and
B2(C) = V{cxeip-cxAeip)Cx +iVp c2A? -{Y(0)cxA + Y(!>]cx)C
(4.172)
(4.173)
(4.174)
(4.175)
It is obvious from (4.174) and (4.175) that B2(£) = -Bx(£ l) and


126
Fig. 5.13 Side view of the cylinder in motion.


FRF added mass dimensionless damping p\ Frequency response function
160
Fig. 6.38 Damping, added mass, and FRF in surge. Half-submerged elliptical cylinder.


169
to oscillate with constant pitch amplitude, the greater the draft, the greater the displaced
water volume, and the greater the amplitude of the waves generated and thus the greater
the damping. Figure 6.44 shows that the added mass in pitch motion increases with
increasing draft. Using the same flap-type wavemaker analogy, as presented in chapter 6,
the greater the draft, the greater the local pressure gradient, the greater the underwater
surface area on which the local pressure gradient acts, and thus the greater the added
mass. Figure 6.45 presents the FRF in pich motion for the three drafts. The units of FRF
are degree per meter. The peak values of FRF have exactly the same character as given in
Bhattacharyya (1978). The greater the draft, the lower the dimensionless frequency of the
peak value of FRF.
With reference to the second numerical example (Figure 6.26), Figures 6.47, 6.48,
and 6.49 present the damping, added mass and FRF in pitch motion of an elliptical cylinder
with various drafts. It can be seen from Figure 6.47, that damping in pitch increases when
the draft-to-radius ratio increases. An intuitive explanation for this phenomenon can be
offered from the wavemaker-theorypoint of view. Consider the limiting case of
substituting the underwater shapes with a flap-type wavemaker in the form of a vertical
plate with the same draft, extended above the water surface, and hinged at the center of
gravity of the floating body. Since the center of gravity is at a fixed distance from the
bottom point, an increment in draft means an increment of the segment of the plate
extending from the center of gravity to the waterline. As long as there is no mooring line,
the center of rotation coincides with the center of gravity. If all three draft variations are
to oscillate with a constant pitch amplitude, the greater the draft, the greater the displaced
water volume, and the greater the amplitude of the waves generated and thus the greater


i
43
d£+^K+d ,onz = (
(3.16)
dt dx dx dy dy dz
Taking the total derivative of (3.13), the two surface boundary conditions (3.13)
and (3.16) may be combined in terms of the velocity potential
Dt p
+
d20 dO du 1
dt2
g +
dz dt 2
+u.Vir
= 0
, on z = C,
(3.17)
which is the combined kinematic-dynamic free surface boundary condition (CFSBC).
3,1.7. Sea Bottom Boundary Condition (SBBC)
On the sea bottom, denoted with B0, (3.14) becomes
F(x,y,z,t) = z+h(x,y) = 0
where h is the water depth. Applying (3.15) on the above equation, the SBBC becomes
c dh dO dh 50
dx dx dy dy dz
, on z = -h(x,y)
(3.18)
3,1,8. Wetted Body Surface Boundary Condition hSV)
Let the instantaneous position of the surface of Sb be described by the equation
F(x,y,z,t) = z-f(x,y,t) = 0 (3.19)
Using the same procedure as in 3.1.6, we state the continuity of the normal velocity with
dt dx dx
+
do df
dy dy
do
dz
, on z f(x,y,t).
(3.20)


121


81
ik
eik[a¡+a2) jrn^eik[a^hl+a^ h)^-(n+x)^^
J_
ik
ik(ai+a2)
P.+iQ. i.
(4.89)
(4.90)
4,3.3.1.2. Far-field wave amplitudes
a* =
l CO
§f(y)eikydy
2 % Sb+St
= YkanD(n)
-**=+J
w1
D(n) has to be expressed from (4.65) and (4.41b).
(4.91)
4.3.3.1.2.1, Far-field wave amplitudes: Exactly half-submerged circle
Denote the integral (4.41b), upon which the far-field amplitudes depend with I.
The use of conformal mapping (4.40), (4.41) results in
oo ou
/= §/(y)e,,dy = R'£a, §e"(C"d(=
Sb+Sb =1 C: |i|=l
= 1
2 n
\
eikRe e-i{n-l)9id0
and
n=1
Wide
e|
5=0 -2jt
= R'Ean
n=1
2 m
(ikR)
n-1
(-!)!
From (4.91) and (4.93) it can be concluded that
D(n)
icoR
2 g
2m
. (ikR)
7
n-1 \
(n-1)!
tvcoR
>=*;
g
(ikR)
n-1
(w-1)!
(4.92)
(4.93)
(4.94)


54
dz
a=o
g
, on the free surface Sp
(3.54b)
da
dz
= 0
, on the sea bottom B0
(3.54c)
dn
=
, on the wetted body surface- Sb
(3.54d)
lim
, waves outgoing at infinity.
(3.54e)
In a summary, the hydrodynamics (3.52d)+(3.54d) has been decoupled from the body
dynamics (3.34d) with the help of decomposition (3.51). Now all that is needed is to
solve the hydrodynamics (3.52)+(3.54) first, and then deal with the body dynamics (3.45).
3,4. Hydrodynamic Properties and Forces
The hydrodynamic properties will be derived in 3D-space using double integration
over the wetted body surface. In 2D-space the same properties can be expressed with a
single integration over the wetted body surface. Working with the hydrodynamic pressure
of the linearized Bernoulli equation (3.12c), and with the decomposition made before, the
generalized hydrodynamic force on the body in the a-direction is decomposed into its
diffraction and radiation components:
K = JJ Pnads = \\{-p^nads = Re<
ia>p\\(R)nads
F + Fi
where


CHAPTER 6
ANALYSIS OF THE RESULTS
This chapter presents a comparison between the theoretically obtained
hydrodynamic damping and added mass coefficients for horizontal cylinders and those
obtained experimentally as described in chapter 5. Next, both theoretical and experimental
results are compared with experimental values obtained by Vugts (1968). Vugts'
experiments were limited to half-submerged horizontal circular cylinders. His data is
shown with square symbols on the graphs presented in this chapter. Vugts' experiments
were conducted in a 4.2 m wide tank in water depths ranging from 1.8 to 2.25 m. The
cylinder diameter and draft were 0.3 and 0.15 m respectively. The amplitudes of motion
were 0.01, 0.02, 0.03 m; and the circular frequency ranged from 1 to 12 rad/sec.
Theoretical curves of hydrodynamic and dynamic properties computed with the semi-
analytic technique, presented in chapter 4, are referred to by the abbreviation SAT. The
dimensionless hydrodynamic coefficients versus dimensionless circular frequency are given
below in the following notations, which were also used by Vugts
dimensionless circular frequency,
(6.1)
dimensionless damping, and
(6.2)
128


amplitude-to-stroke ratio
148
Fig. 6.23 Wave amplitudes of a horizontal elliptical cylinder in heave. Draft variations.
Fig. 6.24 Wave amplitudes of a vertical elliptical cylinder in heave. Draft variations.


18
three-dimensional multipoles, Taylor and Hu (1991) outlined the same procedure for the
case of a floating or submerged sphere. A complete multipole expansion solution of a
heaving semi-immersed sphere was given by Hulme (1982), who simplified Havelocks
solution by making certain explicit integrations. This method was developed further by
Evans and Mclver (1984) for the case of a heaving semi-immersed sphere with an open
bottom.
2,2,5,2, Singularity distributions on the floating body surface, which leads to an integral-
equation formulation based on Greens function
The method of integral equations via Greens function, as explained by Mei
(1989), is based on applying Greens theorem on the radiation velocity potential and a
Greens function defined to be the potential at any field point due to an oscillating source
of unit strength at a particular point inside the fluid domain. Knowing the prescribed
normal velocity boundary condition, the velocity potential at any point on the wetted body
surface is a Fredholm integral equation of the second kind. By dividing the wetted body
surface into discrete panels and approximating the velocity potential in each panel by a
constant, one can obtain a system of algebraic equations for these constant values after
carrying out the integration. Solving for these constant velocity potentials on the panels
will help to express the velocity potential anywhere in the fluid domain. An advantage of
this method is the relatively small number of unknowns, while getting the matrix
coefficients is a laborious task both for the worker and for the computer. Another
drawback of the Fredholm integral equation is the so-called irregular frequencies and
nontrivial eigensolutions in the case of a floating body on the water surface. Without


161
and thus the greater the added mass. Figure 6.37 presents the FRF in surge motion for the
three drafts. A general tendency can be inferred that at low frequencies the floating bodies
are almost perfect wave followers, with their behavior decreasing with increased circular
frequency. The disturbances at the dimensionless frequency of 1 are due to the coupling
with pitch mode oscillations shown in the appendix. At the same dimensionless frequency
of 1 the FRF in pitch has its maximum value, which is shown in the appendix. Obviously
pitch acts in an opposite direction to surge with a phase difference of n a result
confirmed by Mei (1989).
With reference to the second numerical example (Figure 6.26), Figures 6.39, 6.40,
and 6.41 present the damping, added mass, and FRF in heave motion for an elliptical
cylinder with various drafts. It can be seen from Figure 6.39, that damping in heave
decreases with increasing draft. An intuitive explanation for this phenomenon can be
offered from the wavemaker-theory point of view. The closer the underwater shape is to
the limiting case of a horizontal plate, the greater the displaced water volume, and the
greater the wave height generated and thus the greater the damping. Figure 6.40 shows
that the added mass in heave motion decreases with increasing draft. As explained for the
heave damping, the greater the draft, the less the displaced water volume, the less the local
pressure gradient is, the less the added mass becomes. Once again this is consistent with
the experimental results of Vugts (1968) for a horizontal cylinder with a rectangular cross
section. Figure 6.41 shows that the peak value of FRF in heave motion increases when
the draft increases. Obviously, the peak value of FRF in heave motion FRF(con)
corresponds to oscillation of the floating body with the natural frequency, since the
denominator of (6.9) achieves its minimum when co = con.


14
concept of hydrodynamic added mass should be understood only in terms of generalized
force on the body induced by the hydrodynamic pressure, and therefore it will depend on
the wave frequency and the wetted body surface. Inherited from the added-mass concept
is the concept of virtual mass, defined as the sum of the added mass and the mass of the
floating body.
2.2.4, Classification of Damping
By definition, damping is the ability of a structure to dissipate energy. There are
three major kinds of damping for a fixed or floating body in water: structural, material,
and fluid. Structural damping is due to friction among different parts of a structure.
Material damping is energy dissipation within the material of the body, being more
significant in materials like rubber. Fluid damping is the result of energy dissipation, as the
fluid moves relative to the vibrating body. The fluid damping can be classified further into
a damping due to wave generation and a damping due to viscous effects. The wave
generation damping or simply the wave damping dissipates the energy of the vibrating
body into the fluid, thus causing waves. The viscous effects damping can be subdivided
into skin friction effects and viscous effects due to the pressure distribution around the
body. The latter is associated with separation and formation of eddies and is usually
known as eddy-making damping in the literature (Faltinsen, 1990). The separation
changes the flow pattern about the body to a certain extent so that in may be felt in both
the damping and added mass. The skin friction effects on damping are due to shear
stresses acting tangentially on the boundary surface between the fluid and the body and are
proportional to a velocity gradient.


13
From here it follows that the energy content of the random sea irregular profile can be
presented as a sum of the energy of all the component waves irrespective of their phases.
The concept of sea spectrum had been justified theoretically and experimentally for 40
years by utilizing the Pierson-Moskowitz, Bretschneider, JONSWAP and other energy
density spectra. The Pierson-Moskowitz spectrum is controlled by a single parameter -
significant wave height and represents fully developed seas. The Bretschneider spectrum
is controlled by the significant wave height and a modal wave period and can be used for
fully and partially developed wind generated seas. The JONSWAP spectrum is controlled
by the significant wave height, fetch-length, and shape parameter, and is used for partially
developed seas. Therefore the implementation of the energy distribution concept into a
computer model is a powerful tool for analyzing the buoy behavior in real seas, knowing
only the buoy response to a single harmonic wave with small amplitude.
2,2,3. The Concept of Added Mass
The concept of hydrodynamic added mass arises from the fact that a body having
an accelerated motion in or on the surface of the water experiences a force that is greater
than the mass of the body times the acceleration. Since this increment of force can be
defined as the multiplication of the body acceleration and a quantity having the same
dimension as the mass, it is termed added mass. The added mass is not a finite amount of
water, which oscillates rigidly connected to the body. The whole fluid will oscillate with
different fluid particle amplitudes throughout the fluid. As the linear oscillation is
associated with forces and the rotational oscillations with moments, the added-mass may
have dimensions of mass, mass multiplied by length, and even inertia moment. The


Fig. 5.11 Side view of the motion mechanism and the oscillating cylinder.


50
3.2.3 Matrix Form of the Dynamics Equations
The linear system of equations (3.40) and (3.44) can be written in matrix form as
M{^} + [C]W = -pJJ^{4* (3.45)
where [M] is the mass matrix and [C] is the buoyancy restoring force matrix. For a long
horizontal cylinder with its cross-section in the Oxz-plane with incident waves travelling
along the Ox-axis, the motion can be described working with a unit-cylinder length in the
^-direction. For this two-dimensional case the displacement vector is
{X} = {x[x),Z(x\p)T (3.46)
The normal to the wetted body surface vector is
{}=
nx,n2,nx(z-
Z|0))-n,(*-X(0))]
T
(3.47)
(note that
nxds =
S
1
II
^3
N
(3.47a)
The mass matrix is
-
M
0
m(zc -;
j())
M =
0
M
-m(f -
X>)
(3.48)
Ml
r z(0
) -m[xc-X{0))
Iu+I
b
33
and the buoyancy restoring force matrix
is
0
0
0
[C]=
0
PgA
-Ptft
(3.49)
0
~PgI?
Ml
o
1
N,
o
where


118
5.5. Discussion of the Experimental Accuracy
In an experiment, such as the one described here, it is not possible to overcome all
sources of possible errors which influence the experimental accuracy. Even though on the
order of millimeters, the gap between the end plates of the cylinder and the glass walls of
the tank was still a source of error caused by the water leakage. The wave absorption at
the North end (Figure 5.5) was not perfect and varied with the wave frequency. The
nonlinear effects of the fluid-body interaction, evidenced by the presence of the second
and third harmonics (Figure 5.3), also contributed to the experimental error.
5,6, Discussion of the Experimental Procedure
Vugts (1968) used another experimental procedure for an approximate estimate of
the hydrodynamic damping and added mass. When the cylinder has a forced oscillation in
one mode of motion, the force required to sustain the motion can be measured. For
example, as given in Vugts (1968), the surge mode oscillation can be described by
(M + ¡u)x + Ax = F sin(cot + e) (5.13)
where M is the mass of the cylinder section, ju is the added mass, A is the damping
coefficient, F,co,s are the force amplitude circular frequency and phase angle with
respect to the motion. In surge mode, there is no restoring force component proportional
to x. The motion of the center of gravity of the cylinder is considered harmonic with
amplitude A and oscillating with the forcing circular frequency co, namely


CHAPTER 2
INTRODUCTION
2.1. Historical Retrospective of Floating Body Studies
Known since the ancient civilizations, the ship and boat transportation had
naturally attracted the attention of the universal minds of the 18th century and became the
first theoretically investigated floating bodies. Following Vugts historical survey (1971),
the great mathematician Leonhard Euler was the first who studied in a typical
mathematical framework with lemmas, corollaries and propositions the motions of ships in
still water. In 1749 his work Scieritia Navalis was edited in two volumes and published
in Latin in St. Petersburg, Russia. In 1746 the French scientist Bouguer published a
similar work and noted that he was familiar with the fact that Euler had been working on
the subject but that he had not yet been able to lay hands on his results. Daniel Bernoulli
was the first who examined ship motions in waves, and won the prize of the French
Academy of Sciences for his work in 1757. Considering the resonance phenomenon, he
examined forced oscillations of ships in waves. Having wrong ideas about the wave
motion, Bernoulli did not arrive at correct conclusions. Nevertheless his work was
considered as classic for a long time. In 1861 William Froude published his paper On
the Rolling of Ships where the ship was assumed sailing broadside to the waves and had
to follow the wave slope and the orbital motion of the wave particles. Practically Froudes
study dealt with a range of very low frequency motions, thus originating the generalization
4


113
F¡)e=b2h52651 dat; GAGE #2 Wave Time Series
Fig. 5.3 Measured data and its power spectrum density
File=b2h52651.dat; GAGE #2 T=0.77475[s]
81.2 81.3
81.4 81.5 81.6 81.7
81.8 81.9
Fig. 5.4 Measured data as a sum of the first three harmonics


32
potentials for the exciting frequency, and the first-order potential for three times the
exciting frequency. Calculation of the latter potential is not necessary if the first harmonic
of the third-order forces is the only primary interest. The computational procedure used
by Vantorre is classified in 2.2.5.4. as boundary integral equation method (BIEM) based
on a simple sources distribution over the total fluid domain boundary. Two experiments
have been conducted, one with a floating cone and a second with a submerged vertical
cylinder. In both cases the third harmonic was impossible to measure. It is obvious from
the experimental results for the floating cone, that the third-order theory somehow
underestimates the added mass over the entire frequency domain. This is more significant
for larger heave amplitudes. Being acceptable for the region [co'B / 2g) < 1, the third-
order theory overestimates the experimental heave damping in the higher frequency range.
The same frequency relationships can be observed for the second-order forces. Using
again the boundary integral equation method (BIEM) for a floating cone in heave
oscillatory motion, Vantorre (1990) determined the influence of small draft variations on
hydrodynamic first-order forces. After deriving some second-order and third-order force
components, he used a modified BIEM to derive first and second derivatives of heave
added-mass and damping with respect to draft.
The nonlinear effects of the motion of fluid around a body can be dealt with
directly from the fundamental conservation laws of mass and momentum (Equations 3.2
and 3.3). Due to the increasing capacity of computers in the recent years, three groups of
methods for numerical treatment of flow nonlinearity around cylinders have been
developed. Following the comprehensive survey of Sumer and Fredsoe (1997), the first
group consists of methods involving the direct solution of the Navier-Stokes equations.


dimensionless damping dimensionless damping
141
Fig. 6.14 Damping in heave. Circular cylinder D=0.114 m. Stroke variations.
Fig. 6.15 Damping in heave. Circular cylinder D=0.102 m. Stroke variations.


15
Since it is possible to obtain the response of a floating body in irregular seas by
linearly superimposing body responses from harmonic wave components, it is sufficient
from a hydrodynamical point of view to analyze a floating body in incident regular
sinusoidal waves of small steepness. The basic laws of physics governing the motion of
floating bodies are well known in their linearized version (Mei, 1989), and with the drag
force proportional to the square of the floating body speed (Berteaux, 1991). The
difficulty in predicting the response of a floating body to a harmonic wave arises in the
determination of the hydrodynamic properties, namely added mass and damping. Due to
the effects of water-air interface, these hydrodynamic properties depend on the wave
frequency, water depth, and the wetted body surface, which change with time. There has
been a significant amount of research on these subjects in recent years but they remain the
most difficult aspects of floating structure response prediction. Thanks to the significant
computer advancement in achieving higher computational speed and larger memory
capacity, numerical methods and techniques have been increasing their role in calculating
the dynamics of fluid-structure interaction. The differences between computed and
measured hydrodynamic forces revealed that some important phenomena are either not
well understood or the existing combinations of theories and numerical techniques can not
explain them. On the other hand the agreement between computed and measured
quantities in other particular regions, confirmed the generalization that most engineering
approaches in the floating body studies are only valid in a certain range of practical
interest, thus stimulating more investigations. The notable interplay of theories, numerical
methods, and experiments has been very fruitful for the advancements in floating body
studies.


16
2.2.5.Numerical Methods Classification
As classified by Yeung (1985), Mei (1989), and Vantorre (1990), several
numerical methods have been proposed for calculating potential functions in free-surface
hydrodynamics:
2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method.
2.2.5.2. Singularity distributions on the floating body surface, which leads to an integral-
equation formulation based on Greens functions.
2.2.5.3. Method offinite-differences. Boundary-fitted coordinates.
2.2.5.4. Finite element method. Hybrid element method.
2.2.5.5. Boundary integral equation methods (BIEMs) based on a distribution of
simple sources over the total fluid domain boundary.
2.2.5.6. Methods making use of eigenfunction matching.
All these numerical methods will be explained in the frequency domain, because as
it will become evident from the linearized combined kinematic-dynamic free surface
boundary condition (3.34b), the time-domain and frequency domain solutions are simply
related.
2,2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method
Generalizing the heaving motion solution for a semi-immersed circle Ursell (1949),
and its extension to a semi-immersed sphere Havelock (1955), Taylor and Hu (1991)
developed a systematic multipole expansion technique for submerged and floating
horizontal circular cylinders in infinite water depth. Considering the two-dimensional case
of a partially or totally submerged circle, the main idea is to place a set of easy-to-evaluate


176
University, Bulgaria.
Euler, L. (1749). Scientia Navalis, sev tractatus de construendis ac dirigendis navibus
(in Latin). St. Petersburg, Russia.
Evans, D.V., and Mclver, P. (1984). Added mass and damping of a sphere section in
heave. Applied Ocean Research, Vol. 6, No.l, pp.45-53.
Falnes, J. (1984). Technical note: Comments on 'Added mass and damping of a sphere
section in heave. Applied Ocean Research, Vol. 6, No. 4, pp.229-230.
Faltinsen, O.M. (1990). Sea loads on ships and offshore structures. Cambridge University
Press, New York.
Frank, W. (1967). Oscillations of cylinders in or below the free surface of deep fluids.
Naval Ship Research and Development Center Report 2375, Maryland.
Froude, W. (1862). On the rolling of ships. Trans. Inst, of Naval Architecture. Vol.3,
pp. 1-62. London.
Froude, R.E. (1896). The non-uniform rolling of ships. Trans. Inst, of Naval Architecture.
Vol.37, pp.293-325. London.
Ghalayini, S.A., and Williams, A.N. (1991). Nonlinear wave forces on vertical cylinder
arrays. Journal of Fluids and Structures, Vol 5, pp.1-32.
Gradshteyn, I S ., and Ryzhik, I.M. (1980). Table of integrals, series and products
(translation from Russian). Academic Press, New York.
Hamilton, G.D. (1988). Guide to drifting data buoys. Intergovernmenal Oceanographic
Commission, World Meteorological Organization, Manuals and Guides, Geneva.
Hamilton, G.D. (1990). Guide to moored buoys and other ocean data acquisition systems.
Report on marine science affairs 16, WMO No.750, World Meteorological Organization,
Geneva.
Harhf, M. (1983). Analysis of heaving and swaying motion of a floating breakwater by
finite element method. Ocean Engineering, No. 3, pp. 181-190.
Haskind, M.D. (1944). The oscillations of a body immersed in a heavy fluid. Prikladnaia
Matematika i Mehanika, Vol.8, pp.287-300.
Haskind, M.D. (1957). The exciting forces and wetting of ships in waves (in Russian).
Izdatelstvo Akademii Nauk SSSR, Otdelenie Teknicheskih Nauk, Vol.7, pp. 65-79.
Moscow.


amplitude-to-stroke ratio
139
t 1 r
0,6
o
2
03
O
v-


T3
3
0.5
0.4
0.3
+ 4 draft=a
0
<] <| draft=1.2*a
Q ^ draft=1.4'a
Stroke = 52 mm
-
<
<
0 +
*
<1
Q.
s
ro 0.2
0.1 -
04
15
0.2
0.25
0.3
0.35
0
+
0.4 0.45 0.5
0.55
dimensionless circular frequency
Fig. 6.12 Wave amplitudes. Vertical elliptical cylinder in surge.
0.45
1
1 1 1 1
if**
0 .
4 4 draft=a
0
0.4
<] <3 draft=.1.2*a
-
0 0 draft=1.4*a
0.35
Stroke = 52 mm
< -
0.3
1
V
o
0.25
*-
*
<
0.2
0
0.15
-
1
+
V
0.1
0
+
0.05
'
<1
+
0
&
*
_
1 1 L l
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
dimensionless circular frequency
Fig. 6.13 Wave amplitudes. Horizontal elliptical cylinder in surge.


dimensionless Added mass O dimensionless damping
159
Fig. 6.36 Added mass in surge. Elliptical cylinder. Draft'variations.


173
Fig. 6.47 Damping in pitch. Elliptical cylinder. Draft variations.


140
6.2. Heave Mode Oscillations
The non-dimensional damping coefficients in heave mode oscillations (6.8) are
presented in Figures 6.14 through 6.21. The scatter among the measured points in heave
is greater than in surge, but the consistency of the experiments is satisfactory. As in the
surge mode, the upper limit of the dimensionless frequency corresponds to the maximum
frequency of 1.2 Hz of the driving motor. The lower limit of the dimensionless frequency,
below which waves cannot be measured for heave motion, is less than the corresponding
limit for surge motion. This fact means that there is a frequency limit, below which forced
oscillations in surge cannot transfer energy to the fluid, while forced oscillations in heave
can still transfer energy to the fluid. Again, for a characteristic prototype size of 2 m and
the length and temporal scales given in chapter 5, this frequency range corresponds to a
prototype period range between 3.6 and 30 sec. Figures 6.14 and 6.15 present the heave
oscillation of semi-submerged circular cylinders with drafts equal to half of their
corresponding diameters, 0.114 m and 0.102 m. On both Figures 6.14 and 6.15, the
theoretical solution is in good agreement with the measurements of the present experiment
as well as with Vugts' measurements. On both figures, the theoretical solution slightly
overestimates heave damping coefficients. It can be seen in Figures 6.14 and 6.15 that the
different sets of damping measurements, with strokes of 52, 39, and 27 mm, produced by
both cylinders surround Vugts' data: Thus, these measurements are consistent with Vugts'
results in spite of the scatter.


147
the one with vertical semi major axis for the same vertical displacement, Az, despite the
equal volume displacement in calm water (Figures 6.23 and 6.24).
Fig. 6.22 Wave amplitudes of a circular cylinder in heave. Draft variations.


84
/ = -a2r-K=T I6'"*"
C:||=l '*
-2t
(4.106)
7"=T
ik L
ik(a¡+a2) airvta,k{aie~i,ri' +a1e>*) +nJsi
-e e
+
Jsh
ik .
e,nKe
ik[aie-^,a2e^) _Jk{aie-^,a^) +njSb
-e
(4.107)
js J = jr fel te)-
j=0
5!
;=o
/!
1-e
(/ s)/^ n
(4.108)
and
* fe)' ^ fe)'
e e
-2rr
^ -s! =0 /!
(l-s)h2 -n
(4.109)
From (4.91), (4.107), (4.108) and (4.109)
n ( \ icoe~kK r
DAn) = /
(4.110)
4,3.3,2, Real part of the left-hand side of the boundary condition on Sh
Upon substituting (4.25), and (4.27) into (4.24) the left-hand side of the boundary
condition becomes
LHS = \YJ(y)dy-yf{y)+ikyAwA
= 2>4
(4.111)
4,3,3.2.1. Left-hand side: Exactly half-submerged circle
Using the conformal mapping (4.40), (4.41) results in
4=*£c"*r-sr,"','+5v*,, .
(4.112)


180
Vugts, J.H. (1968). The hydrodynamic coefficients for swaying, heaving and rolling
cylinders in a free surface. Report No. 112S. Netherlands Ship Research Center TNO.
Shipbuilding Department, Delft, Netherlands.
Vugts, J.H. (1971). The hydrodynamic forces and ship motions in oblique waves.
Report No. 150S (S 2/238). Netherlands Ship Research Center TNO. Shipbuilding
Department, Delft, Netherlands.
Weinblum, G. and St. Denis, M. (1950). On the motions of ships at sea. Trans. Soc. of
Naval Architecture and Marine Engineering. Vol.58, pp. 184-248. New York.
Yeung, R.W. (1981). Added mass and damping of a vertical cylinder in finite-depth
waters. Applied Ocean Research, Vol.3, No.3, pp.l 19-133
Yeung, R.W. (1985). A comparative evaluation of numeric methods in free-surface
hydrodynamics. Hydrodynamics of ocean wave-energy utilization. IUTAM Symposium.
Lisbon.
Zhang, J. and Dalton, C. (1995). The onset of a three-dimensional wake in two
dimensional oscillatory flow past a circular cylinder. Presented at the 6th Asian Conference
on Fluid Mechanics. Singapore.


92
= c"+Bx{C¡ = V(cxXe'ip-cxeip)Cx +iv\cx2XC2 -(l^c, +Y{\x)c1
+ c", (4.176)
where c" is a constant. Substituting (4.47), and (4.48) in (4.176) results in
wR =F(-a cos/?-/'sin/?)<£1 +Vp
+(z{0)a)cl
+ c .
(4.177)
Therefore the complex velocity on the boundary C in 0%ij is
dwR
~dC¡
= V(acosP+ibn0)¡ 2 + Vf
(4.178)
Comparing (4.178) with (4.156) the complex velocity should be
dws
a
, surge mode
(4.179)
dC,
dwh
ib
, heave mode and
(4.180)
c
dwp
, pitch mode.
(4.181)
d4
2 '
4.3,4.1 A. Complex velocity around the boundary S More than half-submerged ellipse
Applying the conformal mapping (4.44) in (4.144) results in
y = -ihx + a^ + a^, (4.182)
yy = ihx(a2-ax)^hl +ihx(a2-ax)Chl +(axa2)C2hl +{ccxa2)£2fh +axax +a2a2,
Bx(£) = V(a2e"p -axeip)Ch +iVp axa2C'2hi -(Y(0)ax+Y{0)a2-ihx(a1-a2))ch2
(4.183)


r
51
fix -
I\\ = f(*~ ^(0>) &
= ff (z Z^Adxdz
J \ /
c(0)
J \ }
M
J J \ /
yW
i
o
T
In=H(x-X{0))2dm
(3.50)
vb
3 3. Decomposition and Separation of the Hydrodynamics from the Body Dynamics
The dynamics of a freely floating body subjected to a train of harmonic incident
waves is described by (3.45). This matrix equation is a second order differential equation
and represents a dynamic balance of forces. The forces on the left-hand side are inertial
and buoyancy forces with mass and restoring matrices, known from the hydrostatic
equilibrium in the absence of waves. These forces are proportional to the unknown
generalized body displacements, and are balanced on the right hand side of (3.45) by
hydrodynamic forces, which are functions of the unknown velocity potential. The
unknown velocity potential can be determined from the hydrodynamic system (3.34), only
if and when the generalized body displacements are known (see (3.34d)). One way to
resolve this problem is to decouple the hydrodynamics from the body dynamics with the
following decomposition (Haskind, 1944):
dX.
dt
f- = Re(Vae~')
(3.51a)
O = Re(ife,<2")
(3.51b)
<¡> = 0
R(adiation)
+
Diffraction)
(3.51c)


88
dz
dx
=
n =
(4.135)
ds z ds
and ds denotes the elementary increment along the wetted boundary V Making use of
the Cauchy-Riemann conditions, which relate the velocity potential with the stream
function and denoting
VS=V cos/3 (4.136)
Vh-V sin/? ,
/3 = txa-'{VJVs),
and
(4.137)
(4.138)
results in
dy/R d(j)R dz dx
ds dn ds h ds p
lz-Z{)\ +(x-X^)
\ / ds \ f dx
and
(4.139)
dy/R d V,
ds
V'dz-Vhdx+-^-d
(x-X(0))2+(2-Z(0))2
(4.140)
As described in Milne-Thomson (1950), the stream function can be presented as a
difference of a complex function g(y,y) and its complex conjugate counterpart g(y,y),
2 ifg =g(y,y)-g(y,y),
g(y,y) = Ve-fy+~L[jy r(0,y r(y
+ C
(4.141)
(4.142)
g(y,y) = Veipyyy-Yi)y-Yl
<)y
+ c and
(4.143)
2/ y/R = Ve~ipy Ve,py + iVp
yy-Y(0)y-Y[a)y
+ c'
(4.144)
where c, and c are constants. The notation (.) denotes the complex conjugate. Upon
' A
substituting the conformal mapping into a unit circle y-f(C) and recognizing that


157
Fig. 6.33 FRF in heave. Circular cylinder. Draft variations.
05
Fig. 6.34 Damping, added mass, and FRF in heave. Half-submerged circular cylinder.


135
dimensionless circular frequency
Fig.6.7 Damping in surge. Elliptical cylinder with horizontal major axis. Draft variations.
Fig.6.8 Damping in surge. Elliptical cylinder with horizontal major axis. Draft variations.


9
flow separation and nonlinear effects become important for the case of a fixed vertical
circular cylinder, as seen in Figure 2.1.
jNCReasmassr
Wotaht\
DifFSlCTiCW-
MSRcCA&tNSW:
Fig.2.1 Wave force regimes (Sarpkaya and Isaacson, 1981). Importance of
diffraction and flow separation as functions of KC -Keulegan-Carpenter number
and ^-diffraction parameter
As given by (2.3), the greater the wave steepness (H/L) the larger the Keulegan-Carpenter
number becomes. An approximation of the maximum wave steepness is given as (Patel,
1989)


BIOGRAPHICAL SKETCH
Krassimir Doynov was bom in 1963 in the city of Iambol, Bulgaria, where the
lands of ancient Thracia still bring reminiscence of old glory. Raised in a family of
teachers, he was encouraged to develop his knowledge perpetually. His father, whose
History of the city of Iambol, was published along with other historic surveys, was
inspiration for the young Krassimir.
Being a math nerd, Krassimir was admitted to a gifted class in the school with a
priority in mathematics Lobachevski. There, he was encouraged by several exceptional
math and science teachers, and afterward, pursued a bachelors degree in civil engineering
at the University of Architecture, Civil Engineering and Geodesy in Sofia, Bulgaria. From
1988-1990 he earned his masters degree in applied mathematics and computer science at
the Technical University in Sofia.
In 1990, along with his family, he moved to the city of Varna a city of beauty and
tradition on the Black Sea coast of Bulgaria. His appreciation for the sea was affected by
his life on the shore. His work was related to the sea, also. He investigated the dynamics
of a wave energy extraction device as a graduate research assistant at the Technical
University-V arna.
In 1994 Krassimir joined the Department of Coastal & Oceanographic Engineering
at the University of Florida in Gainesville. Since that time, he has been pursuing a Ph.D.
degree under the auspices of Professor D. Max Sheppard. Upon his graduation, Krassimir
181


23
2.2.5.6, Methods making use of eigenfunction matching
Considering the hydrodynamic problem of a floating vertical circular cylinder in
finite-depth water, Yeung (1981) gave an example of the eigenfunction matching methods
with treatment of interior and exterior problems. In the interior problem, the vertical
circular cylinder is considered a fictitious interior domain filled with the same fluid, where
the velocity potential is uniquely determined by solving the Laplace equation with
Dirichlet type boundary conditions. The exterior problem respectively is a Neumann type
problem and can be thought of as one driven by a flux emitted by the interior region, with
a solution written in terms of an eigen expansion with unknown coefficients. Both
problems are treated as if the conditions at the common boundary were known. By
matching both velocity potentials at .the common boundary, the problem is reduced to
solving an infinite system of linear equations. This infinite system is claimed to have
excellent truncation characteristics requiring rarely more than 20 equations to achieve an
accuracy of 1%. It should be noted that the eigen expansion in this solution was possible
because of the convenient presentation of the cylinder boundary in a cylindrical coordinate
system, and because of the finite-depth water. As Taylor and Hu (1991) point out when
the water depth increases toward infinity, the eigenvalues tend to pack together, and the
eigenfunctions become undistinguishable. Consequently, the number of terms required in
the eigen-series expression becomes unrealistically high.