UFL/COELTRI121
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
ACKN OW LED GM EN TS ............................................................................................ iii
K EY TO SY M B OLS................................................ ........ ..............vi
A B ST R A C T ........................................................... .... .................. .................... viii
CHAPTERS
1 M O T IV A T IO N ....................................................................... ..................... 1
2 IN TR O D U CTIO N ....................................................................... ..................... 4
2.1. Historical Retrospective of Floating Body Studies...................................................4
2.2. C classification. ................................................... ............................................7
2.2.1. Large and Sm all Bodies... ............................. ............................................7
2.2.2. Deterministical and Statistical Approaches ........................... ............ 11
2.2.3. The Concept of A dded M ass....................................................................... 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 A ssum ption. ........................................................................... 39
3.1.4. Irrotational Flow A ssum ption.......................................................................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 FirstOrder Theory..........................................................44
3.2. Floating Body Dynamics ......................... ..................................... ........................ 47
3.2.1. Conservation of Linear Momentum............................................................47
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 SemiAnalytic Technique (SAT) ......................................... 64
4.3. SemiAnalytic Technique. Determination of the Unknown Coefficients ................. 70
4.3.1. Boundary Condition on Sb .................................................. ..................71
4.3.2. Conformal M apping ................................ .......... ................................... 71
4.3.3. LeftHand Side of the Boundary Condition on Sb ........................................ 77
4.3.4. RightHand Side of the Boundary Condition on Sb ........................................ 87
4.3.5. Fourier Expansion of LHS and RHS. Solution for the Unknown
C o effi cients........................... ............................................................................ 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 Setup ............................................... 106
5.2.1. C cylinders ........... .................... ........ ................................ ...... 108
5.2.2. Wave Absorption at the Ends of the Tank. ........................ ................... 110
5.2.3. W ave G auges ................................................................ ................... 110
5.2.4. Surface Tension ............................... ......................... 111
5.3. Wave Absorption and Reflection Analysis................... .................................. 112
5.4. M odel 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 M ode O scillations ................................................................ ................. 130
6.2. Heave Mode Oscillations .............................................. .. ..................... 140
6.3. Damping, Added Mass, and Frequency Response Function ................................. 149
6.4. N um erical Convergence................................................................ .................. 164
6 .5 C o n clu sio n s .................... ..................................... .................... ............... .... 16 6
A PPE N D IX .......................... .......................... .. ............ ..... .............. .... ..... 168
LIST OF REFERENCES............................................................ ................... 175
BIOGRAPHICAL SKETCH ....................................................................... 181
KEY TO SYMBOLS
Symbol Description
A Amplitude of the incident wave
A, Farfield wave amplitude
a+ = A / V Farfield amplitude. Dimension time.
a, b Vertical, and horizontal semiaxes of the elliptical
cylinder
a, Power series coefficient of the nh 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 =a V Total derivative in space and time
() = (.) + u V(.)
Dt at
E Water bulk modulus
{FD} Exciting force vector due to diffraction
g Gravity acceleration
H Height of the incident wave
H(o) = RAO Transfer function
h Water depth
i imaginary unit
KC KeuleganCarpenter 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
b Mirror image of Sb in the air
Sm Lateral boundary at infinity
S. (C) Incident wave spectrum
S, (o) 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
V. Timeamplitude of generalized velocity
= dW / dt Energy flux
Xa Generalized displacement in a direction
x = (x, c,z ) Coordinates of the center of mass
y = x+ i z Complex variable
6 Small parameter
cD Velocity potential
0 Timeamplitude of the velocity potential
1D, R Diffraction, radiation velocity potential
ba Radiation velocity potential due to unit velocity in
generalized a direction
l(x,y) Timeamplitude of the water elevation
2 Damping
U Added mass
v Kinematic viscosity
P Water density
co Circular frequency of the incident wave
a Timeamplitude of Xa
T Stream function
VTimeamplitude of the stream function
4(x, 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 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 semianalytical method for computing the dynamics of freefloating, 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 objectoriented 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
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 XVcentury 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 drifting buoy systems
include measurements of atmospheric pressure, air and seasurface 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 retransmitting
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
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 windwave
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 fluidbody dynamics and to create a
computerbased model. In the focus of the research are floating horizontal cylinders of
circular and elliptical cross sections with variable still water drafts. The computerbased
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 wellknown
conservation laws of mass, linear, and angular momentum, chapter 3 formulates the
floating body dynamics as a system of linear secondorder differential equations with
boundary conditions of Neumann and DirichletNeumann 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
farfield amplitudes. Chapter 4 describes an exact analytical asymptotic solution of the
radiation problem, which derives the farfield 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 farfield 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 "Sciertia 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 Non
Uniform 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 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 FroudeKriloff hypothesis and dominated almost all floating
body studies up to 1953. Kriloff computed the wave exciting forces and the restoring
forces and included the hydrostaticcoupling 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 Kriloffleft
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 KorvinKroukovski and Jacobs introduced a strip theory that builds on
twodimensional solutions to get threedimensional 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 n7/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 2 nr to the KeuleganCarpenter number, KC, defined as
u
KC Um.T . (2.1)
B a u
at
where the wave period is T, and the magnitude of the horizontal velocity of a harmonic
progressive wave is
U. = (2.2)
tanh(kh)2)
h is the water depth, and Co is the circular frequency. Now with (2.2), the Keulegan
Carpenter number is
H
KC 27A L (2.3)
B tanh(kh) B tanh(kh)
L
The physical meaning of the KeuleganCarpenter number (more easily seen for the case of
deep water: tanh(kh)=l) is the ratio between the circumference of the fluidsurface
particleorbital motion and the characteristic body dimension. Speaking in NavierStokes
equation (see equation 3.2) terms, the KeuleganCarpenter number is the ratio between
Du Bu
spatial u and temporal 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<<1; 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 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.
0! 0.2 03
B/L
0 0.5 1.0
kB
Fig.2.1 Wave force regimes (Sarpkaya and Isaacson, 1981). Importance of
diffraction and flow separation as functions ofKC KeuleganCarpenter number
and kBdiffraction parameter
As given by (2.3), the greater the wave steepness (H/L) the larger the KeuleganCarpenter
number becomes. An approximation of the maximum wave steepness is given as (Patel,
1989)
(H =0.14 tanh(kh) (2.4)
L max
Therefore, from (2.4) and (2.3) the relationship between the largest KeuleganCarpenter
number and the diffraction parameter, shown in Figure 2.1, is given by
0.44
KC = (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 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
BU, B2
Re= KC (2.6)
v vT
where the kinematic viscosity is v = 106[m2 / sec]. When for example Be[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 largebodydiffractionregime
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 Re>>O(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
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
2 5 10 15 20 25 30 35
0
20
5 10 15 20 25 30 35
20
20
5 10 15 20 25 30 35
20
20
5 10 15 20 25 30 35
0
50 5 10 15 20 25 30 35
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 PiersonMoskowitz, Bretschneider, JONSWAP and other energy
density spectra. The PiersonMoskowitz 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, fetchlength, 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 addedmass 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 addedmass 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 eddymaking 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 waterair 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 fluidstructure 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 freesurface
hydrodynamics:
2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIEBMP method.
2.2.5.2. Singularity distributions on the floating body surface, which leads to an integral
equation formulation based on Green'sfunctions.
2.2.5.3. Method of finitedifferences. Boundaryfitted 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 kinematicdynamic free surface
boundary condition (3.34b), the timedomain and frequency domain solutions are simply
related.
2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIEBMP method
Generalizing the heaving motion solution for a semiimmersed circle Ursell (1949),
and its extension to a semiimmersed sphere Havelock (1955), Taylor and Hu (1991)
developed a systematic multiple expansion technique for submerged and floating
horizontal circular cylinders in infinite water depth. Considering the twodimensional case
of a partially or totally submerged circle, the main idea is to place a set of easytoevaluate
elementary functions multipoles which satisfy the Laplace equation, on the level of the
center of the circle. The combined kinematicdynamic 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 antisymmetric (surge) motions respectively. With the help of
unknown coefficients, both wave and localdisturbance 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 multiple 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 semisubmerged 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 multiple expansion can be applied to the analysis of more general bodies
through a coupled method, called the BIEBMP 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 multiple expansion outside this domain. Using
threedimensional multipoles, Taylor and Hu (1991) outlined the same procedure for the
case of a floating or submerged sphere. A complete multiple expansion solution of a
heaving semiimmersed sphere was given by Hulme (1982), who simplified Havelock's
solution by making certain explicit integration. This method was developed further by
Evans and Mclver (1984) for the case of a heaving semiimmersed 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 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 socalled 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 waterbody
boundary. As a result the source distribution is not unique, the approximate matrix
equation becomes illconditioned; 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 onedimensional Fredholm integral equation of the second kind.
2.2.5.3. Method of finitedifferences. Boundaryfitted coordinates
The classical finitedifference method is based on generating a mesh around the
floating body and using a variety of differenceschemes 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. Higherorder 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 differenceschemes 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 threedimensional 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 finiteelement 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 finiteelement
I
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 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
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 twodimensional 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 firstorder diffraction
and radiation problems. However, these methods are not suitable for secondorder
diffraction and radiation problems (Taylor and Hung, 1987). In this case the integrand is a
functional of the firstorder potential and the Green's function. Because of the slow
convergence of the integrals, a large number of values of the firstorder 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
finitedepth 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 finitedepth 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 eigenseries 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
DirichletNeumann 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 subproblems were formulated. The first subproblem
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 FroudeKriloff and diffraction forces and moments. The second subproblem 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 wavestructure theory and showed
how higherorder 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 socalled KramersKronig 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 twodimensional 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 farfield 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 farfield
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 abovementioned theoretical advancements, it is instructive to give
some examples of the theoryexperiment interplay. In 1965 Kim determined the added
mass and damping for a semiellipsoidal 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 addedmass and damping
coefficients for twodimensional bodies at the free surface.
Using Ursell's multiple 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 twodimensional floating cylinders. Conducting experiments with five
different cylinder crosssections in heave, sway, and roll oscillatory motion, he has
compared the experimental results with the predictions of the linear potential theory and
particularly the multiple 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 lowfrequency range, o B / (2g) < 0.33, the multiple 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 < coB / (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 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 halfimmersed floating sphere with a
bottom opening, Evans and Mclver (1984) have used a semianalytical solution based on
an extension of the method of multiple 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 offdiagonal values are in good agreement with the semianalytical
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 symmetryidentity classified above. The singularity of the damping
matrix has been used as a partial check for the correctness of the semianalytical 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 semisubmerged sphere in
heave motion.
Utilizing a numerical scheme that is similar to one used by Nestegard and
Sclavounos (1984) for the method of multiple expansion combined with BIEBMP
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 threequarters 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 halfsubmerged 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 crosssection of the float into equal
segments, and conducted experiments to compare the results. Considering the float size
small compared to the wavelength 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 fluidstructure
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 boundaryvalue 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 secondorder diffracted waves consists of two
components. One is freewaves travelling independently of the firstorder wave system
and the other is phaselockedwaves accompanying the firstorder 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 waveinduced 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 firstorder 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 secondorder 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 freesurface 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 freesurface 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 firstorder
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 secondorder forces for oscillating cylinders. In a similar way,
Vantorre computed thirdorder forces by means of the firstorder and secondorder
32
potentials for the exciting frequency, and the firstorder potential for three times the
exciting frequency. Calculation of the latter potential is not necessary if the first harmonic
of the thirdorder 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 thirdorder theory somehow
underestimates the added mass over the entire frequency domain. This is more significant
for larger heave amplitudes. Being acceptable for the region (coB / 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 secondorder 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 firstorder forces. After deriving some secondorder and thirdorder force
components, he used a modified BIEM to derive first and second derivatives of heave
addedmass 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 NavierStokes equations.
When the flow around a cylinder is twodimensional and the Reynolds number is small, i.e.
Re<200, a direct solution with finitedifferences yields results for the grossflow
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 threedimensional (3D). When
300
further, 3Dturbulence begins to spread into the boundary layer, and direct numerical
simulation of the NavierStokes 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 finitedifference and finiteelement 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 gridfree 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
agree with experiments within the KeuleganCarpenter 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 spanwiseperiodic vortices the socalled Honji instability. After modeling
the phenomenon numerically, Zhang and Dalton (1995) have obtained a definite 3D
behavior regarding the variation ofvorticity, 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 NavierStokes
equations with massive computing power. With the help of supercomputers and
mature numericallystable methods, the existing mathematical models have succeeded
in increasing our appreciation and understanding of some natural phenomena, in
particular ranges of Reynolds and KeuleganCarpenter numbers. Other Re and KC
ranges of practical interest require different or improved mathematical and theoretical
modeling. It is evident that oscillatory and waveflows bring additional challenges for
the direct approach to the nonlinear problem.
* The second approach is to utilize a Stokes perturbation expansion, which restructures
the NavierStokes equations into a sequence of linear boundaryvalue problems
formulated for each order of perturbation in terms of the power of a small parameter,
for example wavesteepness. With gradually increasing complexity, anyorder solution
can be obtained, provided the solutions of previousorder problems are known.
Predominantly secondorder and thirdorder solutions of the fluidbody interaction
have been investigated, with the noticeably preferred simplifications of axisymmetrical
bodies and symmetrical heaving motion for the case of thirdorder problems. To
explain this preference it suffices to recall that even for the secondorder 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 fluidstructure interaction studies is evident in providing valuable
information about the significance, contribution, and limits of the higherorder
nonlinear effects, as compared to the firstorder 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 semianalytical solutions are
much faster than the numerical ones because of the smaller number of computations
needed. Moreover, because of the exact boundary conditions semianalytical solutions
are principally more accurate as compared to the approximate boundary conditions in
numerical solutions. Of course there are greater restrictions on bodyshapes with
semianalytical 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 fluidbody 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 floatingbody 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 semianalytical method is proposed for solving the
dynamics of free floating twodimensional 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 airwater 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 onethousandth of the
density of water. Provided that the abovewater 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
Oz is positive upward. The origin 0 is the intersection of the centerline of the floating
bodysection 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
SDp_ DP (3.1)
p Dt E Dt
D a
where (.) = (.) + 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 NavierStokes equations
+ u.Vu= V V+gz +vV2u (3.2)
at P
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
f(x,) = V x u(x,t) (3.4)
an important deduction from the NavierStokes equations is that the rate of change of
vorticity is due to stretching and twisting of vortex lines and to viscous diffusion.
+ u. V)2 = .Vu +vV2 (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 threedimensional compressible
flow (Dean and Dalrymple, 1991). A Streamfunction exists in twodimensional or
axisymmetric incompressible flow and is defined by
 = u and = v in Oxyplane,
or (3.6)
 = u and = w in Oxzplane
9z &x
For twodimensional incompressible flow, equations (3.4), (3.5), and (3.6) reduce to the
2D vorticitytransport equation
+ u. V = vV2 (3.7)
and the Poisson equation
+ a (3.8)
9x2 Dy2
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.
LR B
Viscous effects
become important
Line of constant wave
steepness H/L=0/1
H/2B
Potential effects
are dominant
kB
Fig. 3.1 Wave force regimes (Hooft, 1982). Importance of
viscous and potential effects as functions of wave heighttodiameter
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 freefloating body the velocity gradient will be significantly less
than for a fixed vertical cylinder. Therefore the fluid can be considered inviscid,
and the NavierStokes equations (3.2) are transformed into Euler equations:
a+ u. Vu = V +gz (3.9)
at ( P
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
twodimensional and threedimensional irrotational flows
u= VD (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
Vr +' I 2 VO 2 = v[+ gz (3.12a)
LOt 2 p
Integrating (3.12a) with respect to the space variables away from the body, we derive
Bernoulli's equation
1 P
S +IjV +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 airwater 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
+ 1 V 2 + P on(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 (x,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 (airwater) across which there is no flow, a particle
remains and moves only tangentially on that material surface when
DF u. V1 =0 (3.15)
Dt 8t )
which gives the KFSBC
a4 a9I9ag ac _a4
at + n a 
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
DP,
Dt p
a'(u2 C 2
+ +g++u.Vu =0
5t2 9z at 2
,on z=",
(3.17)
which is the combined kinematicdynamic 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
a0 ah 9D h _a
ax + y az
8x 8x Sy ay 8z
, 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)= zf(x,y,t)= 0
(3.19)
Using the same procedure as in 3.1.6, we state the continuity of the normal velocity with
af a0 af ac f Oaf
+ += ,on z= f(x,y,t). (3.20)
at ox Ox ay ay az
(3.18)
3.1.9. Linearization to First Order Theory
Following Mei (1989), the derivation below leads to the complete firstorder
theory and shows how higherorder extensions can be made. For smallamplitude 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) + f(1)(x,y, t) + 2 f(2)(x,y, t)+... (3.21)
where f(0)(x,y) represents the wetted body surface rest position, that is S(). The
velocity potential can be expanded in the same manner
D = ~(l) +e2 (2)+... (3.22)
Considering small body motion, any function evaluated on Sb may be expanded about
S~): z = f0)(x,y). To the order O(e), equation (3.20) can be written as
l)f () + ()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(0) + EX()(t) + X(2)(t)+... ,X= (XY,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 i = (x, y, z) be the coordinate system fixed with the body in a way that I x
when the body is at its rest position. Denoting the angular displacement of the body with
e()(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 = Y+ X(,) + 0(1) x(Y X()] +O(g2) (3.25)
= xs[x +) O(1) x (X(0) +0O(E2)
Y = X[ 6X(1) +8(Z Z(O)_ )(Y 102)]
= x X(') + zZ ( y )
(3.26)
y=yEY )+7xX(Oz_ o )
z=z EZ() +a(yY(O)) f xX(O))]
When the body is at its rest position, then i x and
= f ()(,.y) (3.27)
Substituting (3.26) into (3.27), expanding about So), and comparing with (3.21), results
in
f(l)z '+ a( yY(o)) (x X()) fo)[x ) +p Z(o) r( y(o)]
(3.28)
fO)[y() + x 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
(1fxO) 1)f()+ (I1) = f(O)[xt') +Jt (ZZ()) t(Y Y(o))
fo)[(1) )+t(xX()) a (zZ())] (3.29)
+Z) + a,(y Y(o)) p(x X(o))
The unit normal vector n directed into the body becomes
n = (f) jo) (),i)[ + (fj)) +(fo) ] (3.30)
Equation (3.29) can be rewritten as
n [x( +0) X.(Xo)].n= (3.31)
where
{Xa} = {X(1), = oX(1), Y(1), a,P, Y} (3.32)
n}= in, (xX(o) x n = n,,n2 ,[n(z Z(O)) n3(Y o))]
(3.33)
[3(x ()) n(z Z())],[(y y) n2(x X0) )
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)
+g = 0 on the free surface (3.34b)
9t2 az
 0 on the sea bottom (3.34c)
an
a(e 6 dX
a= d a, on the wetted body surface (3.34d)
an i dt
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 = (xy ,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
Mx, = Pnds Mgk (3.35)
Sb
where k is unit vector of Oz axis. Using the linearized Bernoulli equation
P = pgf ep (') + O(82) (3.12c)
and (3.26), (3.35) can be written as
d&wX4) + 00) x (v X()] = f (pgf sp'))nds Mgk + 0(e2) (3.36)
Sb
The zeroorder portion of (3.36) is
0 = f(pgf(O))nds Mgk (3.37)
So
while the firstorder portion of (3.36) is
M[X + )+ x (i X())] = J(p() p?)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(), and let
A(') be the area of S) Following the procedure shown in Mei (1989) results in
Archimedes' law for the zerothorder
Mg = pg V() (3.39)
The firstorder equations are
M[Z~ ) + att(y Y(O)) ,( _ Xo))] = p ds pg)na I'+ ZA('0)A
s)
M[XI)+ f( Z()) )tt "())] = p\ al'nds (3.40b)
sbo)
M[4,1) + r7(c X(o)) at ( Z())]= pff 1)n2ds (3.40c)
IA = f(x X())dxdy I = f (yy('))ddy (3.41)
s!o) s!o
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' = (xW,yC,z'), such as fffJx xdm Mxc, the conservation of angular momentum
Vb
requires
Sxx dm= f J xx Pnds+ x (Mgk) (3.42)
v, s,
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
(xX)x dm = (x X)x Pnds + (xc X) x (Mgk) (3.43)
Vb Sb
which following Mei's procedure leads to
xcomponent:
ib(1 1by) +( I)a  II/ _3 = Pr\ _P (Ol)ds
22 t 33a +t tt 1 t n4ds
os (3.44a)
pg[z)IM + a(I + 1) ] + Mg[a(r Z(o)) y(c X(o))
ycomponent:
ib ()I ) + (133 + Ijb) b tt att = P a)(I')ds
S"X tt') 132 tt t n,
sbO) (3.44b)
+pg[z()IA + a/ fl(IA + 4') + 4I] + Mg[3(y Z(o)) y(y y(O))]
+pg 1 + 21 1 +Z 3'Y(0)1
zcomponent:
Iltt ytt 1) 1 + I,)y, I3a, I 13 = P 1)n6ds (3.44c)
where the first and second moments of inertia are defined as follows
S= fI(x X(O))dm M(x X())
IP = X (0) )2 dm
V2= X
Ib2 = J(x y Y())dm
v
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 d"(X fd(1)
M] d + [C]{X} pI t n}d (3.45)
where [M] is the mass matrix and [C] is the buoyancy restoring force matrix. For a long
horizontal cylinder with its crosssection in the Oxzplane with incident waves travelling
along the Oxaxis, the motion can be described working with a unitcylinder length in the
ydirection. For this twodimensional case the displacement vector is
{X}= (X(),Z(',)fr (3.46)
The normal to the wetted body surface vector is
n} = n,,n,,n, zZ(')) n(x X(O))} (3.47)
(note that nxds=dz, nzds= dc) (3.47a)
The mass matrix is
M [ M[o Z(O)
[M] = 0M M( X()) (3.48)
M( Z(0)) M^ _ X(o) )+
and the buoyancy restoring force matrix is
0 0 0
[C]= 0 pgA pgjA (3.49)
0 pgIA pg(I +I) Mg Z(
where
IA = xX ( r (x X())dx I = (z Z())dxdz
so) s ) v(o)
S= JJ(x X())dm Ii = x 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 lefthand 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(Vaei"o) (3.51a)
dt
(D = Re(e'') (3.51b)
0 = iR(adiaion) + iD(ffraction)
(3.51c)
where V, is the time amplitude of the generalized body velocity; 5 is the time amplitude
of the velocity potential; co is the wave circular frequency; and the imaginary unit is
defined as i = V. 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 3.51c). 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 R = Va ,, where the summation is over the elementary components of the
body velocity 6 in 3Dspace, and 3 in 2Dspace. (, has the dimensions of [Length], and
stands for the velocity potential caused by a body oscillatory motion with unit velocity in
the adirection: 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 5', 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
for the diffraction subproblem as "D = 0, + 's. The dimensions ofb', Is, and bD are
[Length2/Time].
Thus the necessary decomposition of the time amplitude of the velocity potential is given
by
=D i"+R + = (' +S)+"VY'A (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)
D 2 D = 0 on the free surface SF (3.52b)
az g
D = 0 on the sea bottom Bo (3.52c)
9z
= 0 on the wetted body surface Sb (3.52d)
cn
lim Tiks\ = 0 waves outgoing at infinity (3.52e)
kx>+ [ &
The only missing lateral boundary condition in (3.34) is now given by (3.52e) in 2Dspace.
Sommerfeld had introduced a similar lateral boundary condition, in an analogy with an
outgoing wave system. The incident velocity potential is given by
igA coshk(z +h) + (5)
co coshkh
In a similar way, the radiation subproblem is formulated as
V20. = 0 in the fluid domain (3.54a)
a~ =0 on the free surface SF (3.54b)
8z g
ao" = 0 on the sea bottom Bo (3.54c)
az
 n on the wetted body surface Sb (3.54d)
an "
lirm T ik, = 0 waves outgoing at infinity. (3.54e)
kx[m Q 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 3Dspace using double integration
over the wetted body surface. In 2Dspace 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 adirection is decomposed into its
diffraction and radiation components:
F, = f Pnlads = IfP )nads = Re [icopfJ (jD + )ds e''4E = F
Sb S L at s I
where
F, = Re{Fe' }, FD = icopJf n~ds (3.55)
Sb
FR = ReCFRe'}, FhR = ijop) Rnds = IVpfp f<=io pff fpnnds
Fe, =Ret }, F 6= 6B I =fJJVa
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
[pu]: pf i = Re pJ nds = Im(fa,) and (3.56)
[A]: A = Im pBn,ds = Re(f). (3.57)
The index notation /pa denotes the added mass, which causes a force in direction / due
to acceleration in direction a. The index notation A 6 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
j6 dt2 8Pa dx (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] + [i,]) iw[A]]j} = (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 twicedifferentiable functions O,, and #j, the Green's
theorem states:
f(',V jj V2 i)dn= S S (3.61)
where Q is a closed volume with boundary 802 consisting of the wetted body surface Sb,
the free surface S,, the bottom Bo, and a vertical circular cylinder with an arbitrary large
radius S, If 4,, and Oj, are two velocity potentials, the lefthand 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 S, nor the bottom Bo contributes to the surface integral thus
reducing the righthand side to:
i bi jb JdS = 0. (3.62)
Sb +S.
If 0 = 0,, and 0, = 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
Jfs ()d =O, or (3.63)
SJ a dS = ~ fJq n 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
tpa = tap, and A2, = 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 = Wawy ,where (3.66)
7 1t+T
Wv= Wdt. (3.67)
T
From (3.58), Wbuoy becomes
Wbuoy = F' = l Xj + ,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
a 6 i + 'U0a X P fia u0
1 X
%.. PfaXpX =2 aXiXt+a, 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 = 2aXP X a = Z, . (3.69)
a / a f
Next, the energy flux far away from the buoy can be expressed as
awav = Reds = [R Rc R (3.70)
[2 Su C J 4i St 6i J I
Moreover, with the help of (3.62), (3.70) can be transformed to
Waway 4= iJ [5R IR R Rds (3.71)
4 && I n
When R = Va y a and the twodimensional asymptotic behavior of 0,
lim 0 iga coshk(z+h) (
im _a) e (3.72)
x )co cosh kh
are substituted in (3.71), and (3.71) equalized to (3.69), the law of conservation of energy
flux expresses damping in terms of ad
ga = pg (aa +aa;) (3.73)
where Cg is the group velocity, and (.) denotes the complex conjugate. a0 will be
referred to as thefarfield amplitude, that has dimension of time since a = A+ / Va.
Since the average rate of work done by the body on the fluid (energy transmitted to the
fluid by the body) is nonnegative, the damping matrix is positive semidefinite. As a
corollary, all diagonal terms of the damping matrix are nonnegative, since
,+ = pgc,(a2 +a2) 0. (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 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 socalled 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
KramersKronig 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:
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 socalled Bode relations, which for
water waves correspond to the KramersKronig relations.
Rp,(t) 2a(O)cos(t)d (3.74)
Aja(t) =R A (c0) =osfo2d 2 (3.75)
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 socalled Bode relations, which for
water waves correspond to the KramersKronig relations.
,() fJ (co)cos( 0)dco c (3.77)
01' 0
~
a(c) P(  Rq(t) sin(ct)dt (3.78)
R,, (t) is called the retardation function, and is obtained through a Fourier transform of
Aq (c) Therefore the added mass is related to the farfield amplitudes, (3.74, and 3.78).
As mentioned in section 2.3, a remarkable Haskind theorem relates the exciting
force to the farfield amplitudes in the following manner
FaD= ff pfds=iw p ff( +' S)n ds= opffI( + )S ds
(3.79)
sb awp $ On sn
where (3.63) has been applied. Since  = is true on Sb, then (3.79) becomes
On On
FD = j a opf('ds. (3.80)
Next, substituting the asymptotic forms (3.72) into (3.80) results in
FaD = 2pgCAaa, (3.81)
thus expressing the exciting force in terms of the farfield amplitudes. Upon substituting
(3.81) into (3.59), the matrix form of the dynamics equations become
[[C] C([M]+ [ i])i ]]{[ } = 2pgCA{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 RAO,(c) Ha(wco) (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] ([M][)[aj. (3.84)
3.6. Algorithm for the Solution of the Problem
In summary, the asymptotic solution of the radiation problem gives the farfield
amplitudes. The added mass matrix [u], the radiationdamping matrix [A] (3.73), and the
exciting force (3.81) can be found from these farfield 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 farfield 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,(o) for the
entire frequency domain of interest, namely
S,(co) = H(o)2S,(m). (3.85)
CHAPTER 4
RADIATION PROBLEM SOLUTION
4.1. Problem Statement and Definitions
A long horizontal rigid cylinder with its crosssection in the Oxzplane 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 Odirection with the circular frequency of the incident waves co (in the
absence of the incident waves), thus generating outgoing waves. Taking a unitcylinder
length in the ydirection, the problem is considered twodimensional, and the motion can
be described in the crosssectional Oxzplane (Figure 4.1).
Z
TB __ 
II CB II
CG II
II < ^ II
Switch
II II
II eII
surge
heave
Fig. 4.1 Crosssection of the floating body. CB, CG centers of buoyancy and gravity.
Fig. 4.1 Crosssection of the floating body. CB, CG centers of buoyancy and gravity.
Formulated in (3.54), the radiation problem is stated as a twodimensional Laplace
equation with a complete set of boundary conditions:
2 + 0a2 =o in the fluid domain (4.1a)
9x2 6z2
S Ca 0 = 0 on the free surface S (4.lb)
az g
0a 0 on the sea bottom Bo (4. c)
0#
az
oa = n on the wetted body surface Sb (4.1 d)
an "
lim o  jk = 0 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
{OD,X,)T = Re({O,V, r1}T ej'') (4.2)
With the imaginary unit j = I, the wave profile is
S= A cos(kx wt) = Re(Aej(xt)) = Re([Aej ]ej) = Re([7]e'"') (4.3a)
r= 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 twodimensional 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 = A_,eJkx) = (Ae'k")e = lim 77e', (4.4a)
lim = Aej = (Vaa)ekx (4.4b)
x*>+co
where A+ and A. are the asymptotic expressions of the wave amplitudes, and ad are the
farfield amplitudes introduced in (3.73).
4.2. Main Idea Behind the SemiAnalytic 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 Semianalytic technique (SAT). For
the sake of clarity and completeness, some mathematical definitions and formulations used
in the analysis are presented first. Considering a twodimensional (2D) wave motion in a
righthand 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 = 1l In the twodimensional irrotational flow
of an ideal fluid, both the velocity potential q(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 O(x,z) and stream function i/(x,z) are harmonic functions. Combined in
the complex plane these two harmonic functions define the wellknown 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 CauchyRiemann
conditions
x 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 CauchyRiemann conditions.
Therefore by definitions (a,b,c) the complex velocity potential is a holomorphic function
(MilneThomson, 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) = w( 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
f(y)= ) ikw(y)= ko ( ) _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 airwater interface can be written as:
Im {f(y)} = 0 ., onz=0 (4.9)
By virtue of the Schwarz Reflection Principle and the boundary condition on the airwater
interface (4.9), it is possible to continuef(y) in the upper halfplane. As a result of the
analytical continuation the values off(y) in the upper halfplane will be conjugate
imaginary of the mirror off(y) values in the lower halfplane. In this mannerf(y) will be
holomorphic in the whole complex plane except for the points on the wetted buoy
surface Sb and their mirror images denoted by S (Figure 4.2) Thus, on an abstract
mathematical ground, the problem has been extended from the lower halfplane to the
entire complex plane.
iz
Y 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"kY[A, +iA, + Jf(y)eikoYdy] (4.10)
domain
where At 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 wellknown mathematical technique
is to represent the still unknown holomorphic function by power series, which has the
general form
f(y) = a, +a,(y yo)+a2,(y y) +...+a,(y y,)+... (4.11)
The power series (4.11) is convergent within a circle y yo < R around the fixed point yo
of radius R = lim a ,and it can be divergent outside that circle ly yo > R # 0 (see
n"^ an+1
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
f/")(yo)
coefficients are uniquely determined as Taylor's series coefficients a = 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(y) = + + + 4 +... (4.12)
yF y 2 y 3 y
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
w(y) = eikoY[A, +iA2 + Jf ()ek'gd4] (4.13)
+W0
where Ai+iA2 is an integration constant, is a dummy variable, and the integration is
taken over a curve lying in the lower halfplane. Consequently, when y = (x, z) > (0, z),
the asymptotic expressions of the complex velocity potential are
limw(y) = lim(q + i/) = (A, +iA2)eiky = (A, +iA2)ekox+koz
y+O y.+m (4.14)
limw(y) = lim(q + iy) = (B, +iB,)ei'o = (B, +iB2)eikx+koz
yoo ym
where
B, +iB2 = A, +iA2 + f(y)e'kdy (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)
y (4.16)
lim (y) = ekoz(B, coskx + B2 sin kx)
y4
From the linearized free surface dynamic boundary condition at z=0
1 = jco =17eJt
= z=o g = l
(4.17)
17 = V) V _=
Substituting (4.16) into (4.17) and comparing with the asymptotic wave profile (4.4),
results in the following deepwater (k = k0 = r2) relationship:
g
lim 7 = j V A coskox+A2 sinkox) = Aekx = (Va)(coskox+ jsinkox)
lim7= j V (B1 coskox+B2 sinkx) = Aekx =(Va)(coskoxjsinkox)
Therefore the integration constants are
A =ja+, A=a+, A +iA2 =ga+ (), (4.18a)
O) O) (0CO
Substituting the expressions for A, +iA2 and B, +iB, into (4.15), results in
0+00
9a(j+i)= a'(ij) f (y)eik'dy (4.18c)
Upon substituting i = j and i = j into (4.18c) the far field wave amplitudes become:
a = f (y)erdy (4.19)
2 g_. Ji=T
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 S, in the counter clockwise direction. The proof, based on a lemma due to Jordan
(Solomentsev, 1988), is given in Doynov (1992).
a= ff (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. SemiAnalytic 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 lefthand side
contains all unknown coefficients, and the righthand side contains all terms
derived from the complex fluid velocity.
4.3.2. Conformal mapping of Sb and its mirror image ,b 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 lefthand 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 righthand 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 MilneThomson (1950).
4.3.5. Determining the unknown coefficients by solving a linear system of equations,
derived through the Fourier expansion of both left and righthand sides of the
boundary condition.
4.3.1. Boundary Condition on Sh
Integrating (4.8a) over the wetted body surface Sb in the clockwise direction from
point A to point Y (Figure 4.2), results in
f f(y)dy= [dw()) dy (4.21)
Substituting (4.22)
J wdy = wy w yA f ydy (4.22)
A4 fAdy
into (4.21) results in
f (y)dy = ikyw +(ikw)y+ y (1iky)dy (4.23)
Eliminating ikw in (4.23) and (4.8a), gives a boundary condition whose righthand side
dw
(RHS) is a function of the complex velocity and complex variable,
dy
Yr dvw Ydw
Af (y)dy yf ()+ikAWA = y + (1iky)dy (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 halfsubmerged circle, then the
analytic presentation (4.12) in the lefthand 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 crosssection of the floating
buoy is different than the halfsubmerged circle, then Sb can be mapped into a half
submerged 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
Sf(( () = d4' d +ikw() (4.26)
The conformal mapping properties are:
dA)
a) Angles between vectors are preserved as long as 0
dr
A
d f ()
b) A vector is dilated by = d and
c) A vector is rotated by arg df,)
dr
At the infinite point of the transformed plane Orq the following analytic presentation can
be used,
f()= (4.27)
n=l i
4.3.2.1. Conformal mapping of more than halfsubmerged 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
, S:b E[P [pA,2+t+(A],
S:, E(P[p7Ar + (PA.
;z Sb
Fig. 4.3 Conformal mapping of more than halfsubmerged circle
If the conformal mapping
y= f() = ih, + agh
(4.30)
dy d() a h = a, 0, (4.31)
d4 d4
where hi 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.e', where 0 e [,0], then
p = 2r+ (p +h20, (4.32)
n + +2PA (4.33)
a = Re'A and (4.34)
(4.28)
(4.29)
A 
h = h. (4.35)
If Sb (4.29) is transformed into the upper half of the unit circle C: = .e'o, where
09 [2r,n], then
p=21r+3, +h29, (4.36)
a = Re3'p, and (4.36)
h=h (4.38)
4.3.2.2. Conformal mapping of exactly halfsubmerged circle
This is a particular case of 4.3.2.1, with h= 0, 9A = 0, h = 0, h, =0, and a= 1.
The result is
y= Re"' Sb:( ~ [0,27r] (4.39)
The conformal mapping which transforms Sb (4.39) into the lower half of the unit circle
C: 1= 1.e'6, where 0 e[c,0] is
y = f()= R( ,and (4.40)
dy df() R O (4.41)
d4 de
4.3.2.3. Conformal mapping of more than halfsubmerged ellipse
If a and b are the vertical and horizontal semiaxes 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:
x+iz z y=x+iz
Sb \ Sb// Sb
x A
bS
Fig. 4.4 Conformal mapping of more than halfsubmerged ellipse
y = ih +bcos(7) + iasin(q7) S: 77 [n r1A ,2nz + 17A (4.42)
y=ih+bcos(rq)+iasin(r{) S,.r E[77AI7+ A]. (4.43)
If the conformal mapping
y = f() = ih, + a,1 + a2 and (4.44)
d = d = a.~) = h2(a 1 ) 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,0], then
h, = h (4.46)
c = (b+a)/2, (4.47)
A = (ba)/(b+a), (4.48)
a, = cle'A (4.49)
a2 Ce'iA (4.50)
77= 2+ 77 +h2, and (4.51)
h= +27A^ (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 = (ba)/(b+a), (4.55)
a, = ce3iA (4.56)
a2 = Cle3ir/A (4.57)
7= 2'r+ 37 +3hO, and (4.58)
k +2+EA (4.59)
4.3.2.4. Conformal mapping of exactly halfsubmerged ellipse
This is a particular case of 4.3.2.3, with h= 0, r7A =0, h1= 0, and h2= 0, and
results in
y = bcos(77) + iasin(77) 77 e[0,27r]. (4.60)
The conformal mapping which transforms Sb (4.60) into the lower half of the unit circle
C: = 1.e'", where 0 e[7,0] is
y= ) =f c, ~ ( and (4.61)
A
dy _d f1() ;1 0. (4.62)
d4 d4 2
4.3.3. LeftHand Side of the Boundary Condition on Sh
To find an explicit form of the lefthand side of the boundary condition (4.24) A,,
the complex velocity potential at point A must be found (Figure 4.2).
4.3.3.1. Complex velocity potential at point A
Substituting yA fory in (4.13) will give the following expression for w,
= W(= (A)= ei'A[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, = jga+, A,=ga+,
equation (4.36) takes the following form:
wA = [A, os(xA) + A2 sin(kA) +i(A2 cos(A) A sin(kxA))]+ e'Y 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,
a = aD+ (n) and (4.65)
n=1
YA 00
e' f f(y)e'dy = a,(P, +iQ), (4.66)
+.0 n=1
thus giving wA, expressed as a series of the unknown a, coefficients
wA = nw, (4.67)
n=l
where
w, =[j g eJ^D+(n)+P, +i eJ~AD+(n)+Q (4.68)
4.3.3.1.1. Determination of P,+iQ_
P, +iQ, (4.66) can be expressed in the following manner
YA
e f (y)e'Ydy= a,(P, +iQ,).
+00 n=1
4.3.3.1.1.1. Determination of P+iO Exactly halfsubmerged circle
Using the conformal mapping (4.40), (4.41) results in
YA .0 1 W
I = Jf(y)e'"dy = R ajfei'c nd'= RyanI (4.69)
+00 n=1 n=1
where I, can be determined knowing I, and the following recurrent formula
= Jei~'nd = i[e +nI ,,+ (4.70)
The first integral can be expressed with a complex exponentialintegral function
1 ikR( kR it
1 = re d= dt = E,(ikR), (Gradshteyn and Ryzhik, 1980) (4.71)
a, t
79
ikR.I e' and (4.72)
n
P, +iQ, = e 'RI,. (4.73)
4.3.3.1.1.2 Determination of P,+iQO More than halfsubmerged circle
The use of conformal mapping (4.30), (4.31) results in
n=l n=1 e>
kh o 1 kh I ika 4 1 1
= = a+ ndeika2  +nJ e ka ("+)d (4.75)
k n=l o n=1 oo
kh oo
Za [eika +n], (4.76)
nI
where I, is
I. f= ("')eika2 d. (4.77)
After changing the variable of integration 4 with t = and correspondingly
d4'= dt, I, becomes
S(ika)" nlsh2dt (ika)s 1 (4.78)
I. =f eikat2 dt = O tsd (4.78)
0 S o s! 0 S 0 s! s4 n
Substituting (4.78) into (4.75) gives
kh a (ika)( 1
I= e a, ek Sn+n (4.79)
ik n=1 I =, s! s2 n
Substituting (4.79) into (4.66) results in
i ekhj'ikA i (ika)s 1
P +iQ = ek +n (k (4.80)
S k s= s! sh,n
4.3.3.1.1.3. Determination of P,+iO Exactly halfsubmerged ellipse
The use of conformal mapping (4.61), (4.62) results in
YA co C
I = f (y)edy = cJanei'^k(+ n 2) c, a, [I (4.81)
+co n=l oo n=l
I,= i ik(. (Ik ,,, (4.82)
oo s .=0
I,, J= eik("+)d = [e +(n+s)I++], (4.83)
n .i kcc
1 ikcl k kl it
I, = d=I dt = Ei(kc), (4.84)
ikcl' e'ik
In+.= and (4.85)
n+s
P, +iQ. = e"^c,[I, M+2]. (4.86)
4.3.3.1.1.4. Determination of P,+iQ More than halfsubmerged ellipse
The use of conformal mapping (4.44), (4.45) results in
I = f (y )e dy = ek f eik(ah +ah2i "h2 ( ah, h4 )di ekh an,, ,(4.87)
+00 n=1 co n=1
1 1
I = eik(a +a2)" h2 (a a2rh)d= ik 1 dek +a (4.88)
00 c0
I = e(,+2 + f, )' +a2 )()n+I ik(+a) eik t+ nldt (4.89)
k L+ e lk
ik Wik0
P, + iQ, = e"A k I.
4.3.3.1.2. Farfield wave amplitudes
(4.90)
S= f (y)eksdy
2 g s, J i=Tj
(4.91)
= aD (n)
n=1
D+(n) has to be expressed from (4.65) and (4.41b).
4.3.3.1.2.1. Farfield wave amplitudes: Exactly halfsubmerged circle
Denote the integral (4.41b), upon which the farfield amplitudes depend with I.
The use of conformal mapping (4.40), (4.41) results in
y 21r ikRet2 i(n1)"
I= ff(y)ekYdy= R a. eckR 'd= RlaneikRe eie(n)idO ,
Sb + n=1 C:1=1 n=1 0
(4.92)
(4.93)
I= RZ an e(i[s(n1)]oidO = R a,, 2n;i
n=1 =0 SO 2r I n=1 (n
From (4.91) and (4.93) it can be concluded that
=/giR (ikR)n wnwR (ikR)"'
D, (n) = 2ni =
2g ( (n 1g! ^ g ( 1)!
(4.94)
82
4.3.3.1.2.2. Farfield wave amplitudes: More than halfsubmerged circle
The use of conformal mapping (4.30), (4.31) results in
I = f (y)elkody = aj ekih+a ah2nd= ah2ekh ~ a (4.95)
Sb+b n=1 C n=l
where the I, integral is to be calculated over the unit circle (Figure 4.3)
I,= fe ka'h"d (4.96)
c:(1=1
In a cylindrical coordinate system I, can be expressed as
I = ik l"i(h3n+')idO= (ika) e(idO = (ika 0 ,i[(s+l)n] (4.97)
i e'e Ie'Oiid9= d( (4.97)
Se 2a + 3=0 s 2r h2(s+1)n
.as+le kll (1 e '[ (s+1)ni)], + I as+le+k (1 e [h2(s+1)"nir)'Ti[h2(s+1)n] 1
n= s=O 2(s+1)n
(4.98)
From (4.91) and (4.98) it can be concluded that
9k_ ( a s+ le kh 1 e'[hk(s+1)n] T) + [a s+le+kh i e [2(s+ )n] t)'ih (s+1)n] 1
Dn (n) = (
2g = s! h(s+l)n
(4.99)
4.3.3.1.2.3. Farfield wave amplitudes: Exactly halfsubmerged ellipse
The use of conformal mapping (4.61), (4.62) results in
I = f(y)e'iody = c, a, ei'kc( l')(f 2)d= cia[Gn AG,+], (4.100)
Sb+Sb n=1 CC=1 n=1
(4.101)
G= ei( ) = (ikc)s fe'kc"(n+s)d.
c:k1=1 s=O c.=1
Applying the same technique as in (4.92), (4.93), gives
W(ikcc (ikc)) ' (ikc)2 =s+n i (l)s(kc )2s+n s
G= = ni 1)! 2nil)! = 2 n I ,l) and
s=0 st (n + s1)! = s!(n+s1)! s!(s+n)!
2s+n1
G, = 2mn (12 (I)(kciVf) = 2ni. (ni)i2J (2k e,)
s=O s!(s+n1)!
,when A > 0
(4.102)
where J,_(.) denotes the Bessel function of the first kind,
= ) 2s+nl
G,= 2 n"()n2) k ) =2"()(2kclf /)
,0 s!(s+n1)!
,when A, <0
(4.103)
where I,_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,, )
(4.104)
4.3.3.1.2.4. Farfield wave amplitudes: More than halfsubmerged ellipse
Using the conformal mapping (4.44), (4.45) results in
I= f(y)e'kody ekh 'a,, '' n 3 a2 dh' )d= ek n
Sb +Sb n=1 C:=l n=1
(4.105)
In = e ikah.,; n aI h3 a2h )d4 1 j. nodeke +a(4.106)
C: =1 2rk
n [eik(al+a2) in k( a+e" +azetk
k (4.107)
in ikre (aje' +aze"A) eik( " +ia2e")2 + Is]
ik L b
iS =i e"oeik (e'+ae (ik 2) (ika)'1 e'[(l)"]'n (4.108)
S= s0 i=o ( 1s)h,n
and
Sb inik( aIeo+a2e dO (ika 2) (ika e' (')"]" e2i(1s)h"]"r (4.109)
I,=i se=e( ^)id=l (ls)h , (4.109)
2 s= o s! = l1 (I s)h,2n
From (4.91), (4.107), (4.108) and (4.109)
D (n) = Ie .I (4.110)
2g
4.3.3.2. Real part of the lefthand side of the boundary condition on Sh
Upon substituting (4.25), and (4.27) into (4.24) the lefthand side of the boundary
condition becomes
LHS =f (y)dyyf(y)+ikWA = aL, (4.111)
n=1
4.3.3.2.1. Lefthand side: Exactly halfsubmerged circle
Using the conformal mapping (4.40), (4.41) results in
L, = RAn"d R('') +ikyw, (4.112)
A4.12
1 i(n1)0
L, = R eid Re(+ikxA = R e Re' ) +ikxAw, (4.113)
n1
and
L, = + ik R n ei. (4.114)
n1 n1
The real part of(4.114) is given as follows:
R n
Re(L,) = R Im(w) R ncos(n 1)0. (4.115)
n1 n1
4.3.3.2.2. Lefthand side: More than halfsubmerged circle
Using the conformal mapping (4.30), (4.31) results in
L, = h2af' d(ih, +a4k)g"+ikyAW (4.116)
L,, = ha ei(h'+0id 8iheo ae'('o +ikAW (4.117)
L, = h2a e() O aiei(n)o + ikxAw and (4.118)
nh
L = ikXAWn + i(na iheink (4.119)
nh2 nh2
The real part of (4.119) is given as follows:
Re(L,) = lo + i cos(nh2) +2 sin(nh)0+13 sin(n0). (4.120)
The real coefficients ii e Re,(i = 0,1,2,3) are
S= kA Im(w,) + Rea (4.121)
nh2
11 nRea (4.122)
nh,
n Ima
2= and (4.123)
n h2
13= h. (4.124)
4.3.3.2.3. Lefthand side. Exactly halfsubmerged ellipse
Using the conformal mapping (4.61), (4.62) results in
L,= cici cn iiC + +ikyA ,, (4.125)
C[\ p1) in+_ ( 1)1 ]
L = c [e A ei(n1)o 'i(n+l)O +ikx AW (4.126)
n1 n+l1
L, = c 1  A n e i(n)o nA ei(n+)o + and (4.127a)
[ 
n1 n+1 n1 n+c
LI =cC I 1+iOAe2iO + Aiwl. (4.127b)
2 2
The real part of (4.127) is given as follows:
Re()= c,[ A ] Im(w) ncl cos(n1)0nC+ cos(n+1)0 (4.128a)
Re(L,)= n1 n+1I nn1 n+l
Re(L,) = c,1 A m(w1) cos(20) (4.128b)
4.3.3.2.4. Lefthand side. More than halfsubmerged ellipse
Using the conformal mapping (4.44), (4.45) results in
L, =h2 a"(a( a )d (ih, +a1h< + a24h +kAWn,, (4.129)
i(nh2)8 i(n+h)0
i1e (h) inO i(nk)O i(n+h2)0
L, = ih e ae '("e) c
n h2 n+h2
(4.130)
L _____h2 __1_h2 2 na1 i(nh2)9 na2 i(n+h2)(
h2al ia2 +ikYA ih" n e() e (4.131)
nh2 n+h, nh n+h,
Re(L,) = Re a n+h2, 1 n n [Re(a,) cos(n h)+ Im() sin(n h,)]
n [Re(a) cos(n+hk)0+ Im(a2) sin(n+h)0] h, sin(n0)
n+h,
(4.132)
4.3.4. RightHand Side of the Boundary Condition on Sh
To find an explicit form of the righthand side of the boundary condition (4.24)
w the complex velocity around the boundary Sb, 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)
0R Vhh A + V, + V and (4.133)
R Vh + V Vo +Vp, v, +V +Vhnz +VP[nz. Z n(x X(o)]. (4.134)
The components of the unit normal vector arena
The components of the unit normal vector are
dz _dx
nX = n T (4.135)
ds ds
and ds denotes the elementary increment along the wetted boundary Sb. Making use of
the CauchyRiemann conditions, which relate the velocity potential with the stream
function and denoting
V = Vcosp (4.136)
Vh = V sinf and (4.137)
S= tan'(Vh /V), (4.138)
results in
a /R 8R dz d + x
R vv, zZ X() dx and (4.139)
as On s ds ds V, ds \ I ds
as ds 2 L '
As described in MilneThomson (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),
2iYR = g(y,y)g(y,y), (4.141)
g(y,) = Vey + [ y ) ) +c, (4.142)
g(y, y) = Ve y _ [)yyI) y(O) + and (4.143)
2iR = Ve yVe +iVyYyY () Ty + 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 = ) and recognizing that
substituting the conformal mapping into a unit circle y = f() and recognizing that
= 1 (4.145)
an expression for the stream function (4.144) in Or77 can be obtained
2iVfR = B +B,(1)+B2(B ). (4.146)
B0 is a constant, B,(4) contains all the negative powers of 4, and B2( ) contains all the
positive powers of 4. Using the following relationship, which is proven below
B2(0) = ((4.147)
results in
2iR = Bo + B() B,() = wR R and (4.148)
R = 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 halfsubmerged circle
Applying the conformal mapping (4.40) in (4.144) results in
y = R"', (4.150)
yf = R2 (4.151)
B,() = Ve'f(R1) +iV,[Y(o)RC ], and (4.152)
B2() = Ve' (R4)+iV,[Y (0)Rj. (4.153)
Obviously (4.147) is justified. Making use of(4.152) and (4.149) results in
wR = +B,()=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' (R2) +VP[iRY()2 (4.155)
From the decomposition (3.5 Id), (4.50) of the radiation potential, the complex velocity
should be
dwRv dw, Vhdwh+ dw
=, +V^ +V (4.156)
d, d h d de'
which means
dw iR
d _2 surge mode (4.157)
h = heave mode and (4.158)
dwp iRY()
dw RY(, pitch mode. (4.159)
d(" 2
4.3.4.1.2. Complex velocity around the boundary S. More than halfsubmerged circle
Applying the conformal mapping (4.30) in (4.144) results in
j7 = ih, + (4.160)
yy = h + 6cz ihc a + iiaS h2 (4.161)
B,() = Ve' )+iV[(ih, Y(O))+] and (4.162)
B() = Ve(a()+iVp(ih, [o)a ] (4.163)
It is obvious from (4.162) and (4.163) that
B2)= _ 1) (4.164)
