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

UFL/COEL-TRI121

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 First-Order 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 Semi-Analytic Technique (SAT) ......................................... 64
4.3. Semi-Analytic Technique. Determination of the Unknown Coefficients ................. 70
4.3.1. Boundary Condition on Sb .................................................. ..................71
4.3.2. Conformal M apping ................................ .......... ................................... 71
4.3.3. Left-Hand Side of the Boundary Condition on Sb ........................................ 77
4.3.4. Right-Hand Side of the Boundary Condition on Sb ........................................ 87
4.3.5. Fourier Expansion of LHS and RHS. Solution for the Unknown
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 Set-up ............................................... 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, Far-field wave amplitude
a+ = A / V Far-field amplitude. Dimension time.
a, b Vertical, and horizontal semi-axes of the elliptical
cylinder
a, Power series coefficient of the 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 Keulegan-Carpenter number
k Wave number of the incident wave
L Wave length of the incident wave
[M] Mass matrix of the floating body
n, Component of unit normal to Sb vector in a direction
Pa Atmospheric pressure
p Pressure
R Radius of the circular cylinder
Re Reynolds number
SA Part of the water surface cut out by Sb
Sb Instantaneous wetted body surface
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. Time-amplitude 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 Time-amplitude 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) Time-amplitude of the water elevation
2 Damping
U Added mass
v Kinematic viscosity
P Water density
co Circular frequency of the incident wave
a Time-amplitude of Xa
T Stream function
VTime-amplitude 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 semi-analytical method for computing the dynamics of free-floating, horizontal

cylinders, subjected to ocean waves has been developed. The bodies analyzed in this

analysis are cylinders with circular and elliptical cross sections and variable still water

drafts. The motions considered are heave, surge and pitch. The technique computes the

added mass and damping coefficients using linearized radiation theory. The efficiency of

the numerical scheme is due to the simplicity of the mathematical scheme used a

combination of original holomorphic functions, convergent infinite power series, and

conformal mapping. An object-oriented approach was used for the computational aspects

of the problem using the programming language C++.

Physical experiments in a wave tank were conducted with circular and elliptical

horizontal cylinders in order to verify the method. The cylinders were positioned at the

water surface and forced to oscillate sinusoidally first in surge and later in heave motion.

Different still water drafts and oscillation frequencies were used in the experiments. The

far field waves produced by the oscillating cylinder were measured using capacitance wave

gauges. The damping and added mass coefficients were computed from the measured

wave data. There is a good agreement between the predicted and measured coefficients.

CHAPTER 1
MOTIVATION

In XV-century Europe, the brilliant engineer, scientist and artist, Leonardo da

Vinci recorded for the first time an engineering application of a free floating buoy. The

buoy was used to measure the water velocity in streams and consisted of a weighted rod

and flotation bladder. It was released in the water flow and its downstream travel

measured after a given period of time and the average velocity computed as the distance

divided by the elapsed time. Since that time drifter buoys have proved to be very effective

in increasing the level of our understanding of the ocean environment and in improving

weather analysis and forecasting. Scientific investigations using drifting buoy systems

include measurements of atmospheric pressure, air and sea-surface temperature, solar

radiation, air humidity, wind shear, wave evolution, wave noise, light penetration, oceanic

current speeds, and water temperature and salinity. A variety of drifter buoy systems,

deployed in the world's ocean, are used to monitor the spatial and temporal distribution of

the above mentioned environmental variables. With modern satellite technology, which

provides both buoy positioning information and a means of receiving and re-transmitting

data from the buoys, the accuracy and reliability of these measurements has been greatly

improved. Measurements can be recorded with data acquisition systems mounted on the

buoys or transmitted to land based stations via satellite (Berteaux, H.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 wind-wave

parameters would allow buoy designs to be optimized for the sea state conditions in which

they are to be deployed. Therefore the buoy hull must be designed in accordance with

certain stability design criteria for different ocean and atmosphere conditions.

The purpose of the following research is to investigate how variations of draft, size

and shape of drifting buoy systems influence fluid-body dynamics and to create a

computer-based model. In the focus of the research are floating horizontal cylinders of

circular and elliptical cross sections with variable still water drafts. The computer-based

model is intended to provide designers with estimates of the dynamic response

characteristics, in terms of response amplitude operators or frequency response functions,

of relatively small buoys.

The dissertation is comprised of six chapters. Chapter 2 presents a brief historical

retrospective of floating body studies, followed by classifications that introduce criteria for

large and small bodies, and deterministic and statistical approaches. Next, the concept of

added mass is introduced, as well as classifications of damping and the most frequently

used numerical methods for the determination of these hydrodynamic properties.

Furthermore, the most significant advancements in floating body studies are presented,

followed by a review of recent scientific works on the related subjects. This chapter

concludes with an attempt to classify the current state of knowledge about floating bodies

as one going simultaneously in three main directions. As a result of the well-known

conservation laws of mass, linear, and angular momentum, chapter 3 formulates the

floating body dynamics as a system of linear second-order differential equations with

boundary conditions of Neumann and Dirichlet-Neumann type. After introducing the

generally accepted simplifications for incompressible, inviscid fluid and irrotational flow,

the linearized radiation theory is deduced. Furthermore the hydrodynamics is

decomposed from the body dynamics, and some hydrodynamic theorems are given with

purpose to express all unknown variables of the floating body dynamics as functions of the

far-field amplitudes. Chapter 4 describes an exact analytical asymptotic solution of the

radiation problem, which derives the far-field amplitudes as functions of the wetted surface

of the floating body, and the circular frequency of the incoming harmonic wave. Chapter

5 describes the experiments conducted in heave and surge motion, which main purpose is

to obtain data for the far-field wave amplitudes and thus to verify the analytic solution,

introduced in chapter 4. These experiments were carried out in a wave tank at the Coastal

Engineering Laboratory at the University of Florida. Chapter 6 presents the analysis of

the results from the experiments and comparison with the numerical solution. The

conclusion is given at the end of this chapter.

CHAPTER 2
INTRODUCTION

2.1. Historical Retrospective of Floating Body Studies

Known since the ancient civilizations, the ship and boat transportation had

naturally attracted the attention of the universal minds of the 18th century and became the

first theoretically investigated floating bodies. Following Vugts' historical survey (1971),

the great mathematician Leonhard Euler was the first who studied in a typical

mathematical framework with lemmas, corollaries and propositions the motions of ships in

still water. In 1749 his work "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 Froude-Kriloff hypothesis and dominated almost all floating

body studies up to 1953. Kriloff computed the wave exciting forces and the restoring

forces and included the hydrostatic-coupling effects between heave and pitch. In an

additional estimated term, he included a resistance to motions, thus introducing the

concept of damping into the floating body studies. Speaking in modern terms 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 Korvin-Kroukovski and Jacobs introduced a strip theory that builds on

two-dimensional solutions to get three-dimensional effects for elongated floating bodies.

At that revolutionary time, 18 years after the invention of the greatest scientific tool of the

20th century the computer whose father was the American from Bulgarian descent

Dr. John Atanasoff, all the basic elements were available to obtain engineering solutions

for a coupled hydrodynamic motion, at least for heave and pitch.

2.2. Classification

2.2.1. Large and Small Bodies

There are at least three relevant length scales in the interaction between waves and

a fixed or floating body. They are the characteristic body dimension B, defined as the

predominant horizontal size of the body projection onto the vertical plane of the wave

front, the wavelength L=2 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 Keulegan-Carpenter 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 Keulegan-Carpenter number (more easily seen for the case of

deep water: tanh(kh)=l) is the ratio between the circumference of the fluid-surface-

particle-orbital motion and the characteristic body dimension. Speaking in Navier-Stokes

equation (see equation 3.2) terms, the Keulegan-Carpenter number is the ratio between

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 -Keulegan-Carpenter number
and kB-diffraction parameter

As given by (2.3), the greater the wave steepness (H/L) the larger the Keulegan-Carpenter

number becomes. An approximation of the maximum wave steepness is given as (Patel,

1989)

(H =0.14 tanh(kh) (2.4)
L max

Therefore, from (2.4) and (2.3) the relationship between the largest Keulegan-Carpenter

number and the diffraction parameter, shown in Figure 2.1, is given by

0.44
KC = (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 = 10-6[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 large-body-diffraction-regime

B/L>0.2 (or B/L>0.1) may be analyzed by the potential theory, since no flow separation

occurs KC<2, and since the viscous effects are negligible 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 Pierson-Moskowitz, Bretschneider, JONSWAP and other energy

density spectra. The Pierson-Moskowitz spectrum is controlled by a single parameter -

significant wave height and represents fully developed seas. The Bretschneider spectrum

is controlled by the significant wave height and a modal wave period and can be used for

fully and partially developed wind generated seas. The JONSWAP spectrum is controlled

by the significant wave height, fetch-length, and shape parameter, and is used for partially

developed seas. Therefore the implementation of the energy distribution concept into a

computer model is a powerful tool for analyzing the buoy behavior in real seas, knowing

only the buoy response to a single harmonic wave with small amplitude.

2.2.3. The Concept of Added Mass

The concept of hydrodynamic added mass arises from the fact that a body having

an accelerated motion in or on the surface of the water experiences a force that is greater

than the mass of the body times the acceleration. Since this increment of force can be

defined as the multiplication of the body acceleration and a quantity having the same

dimension as the mass, it is termed added mass. The added mass is not a finite amount of

water, which oscillates rigidly connected to the body. The whole fluid will oscillate with

different fluid particle amplitudes throughout the fluid. As the linear oscillation is

associated with forces and the rotational oscillations with moments, the added-mass may

have dimensions of mass, mass multiplied by length, and even inertia moment. The

concept of hydrodynamic added mass should be understood only in terms of generalized

force on the body induced by the hydrodynamic pressure, and therefore it will depend on

the wave frequency and the wetted body surface. Inherited from the added-mass concept

is the concept of virtual mass, defined as the sum of the added mass and the mass of the

floating body.

2.2.4. Classification of Damping

By definition, damping is the ability of a structure to dissipate energy. There are

three major kinds of damping for a fixed or floating body in water: structural, material,

and fluid. Structural damping is due to friction among different parts of a structure.

Material damping is energy dissipation within the material of the body, being more

significant in materials like rubber. Fluid damping is the result of energy dissipation, as the

fluid moves relative to the vibrating body. The fluid damping can be classified further into

a damping due to wave generation and a damping due to viscous effects. The wave

generation damping or simply the wave damping dissipates the energy of the vibrating

body into the fluid, thus causing waves. The viscous effects damping can be subdivided

into skin friction effects and viscous effects due to the pressure distribution around the

body. The latter is associated with separation and formation of eddies and is usually

known as eddy-making damping in the literature (Faltinsen, 1990). The separation

changes the flow pattern about the body to a certain extent so that in may be felt in both

the damping and added mass. The skin friction effects on damping are due to shear

stresses acting tangentially on the boundary surface between the fluid and the body and are

proportional to a velocity gradient.

Since it is possible to obtain the response of a floating body in irregular seas by

linearly superimposing body responses from harmonic wave components, it is sufficient

from a hydrodynamical point of view to analyze a floating body in incident regular

sinusoidal waves of small steepness. The basic laws of physics governing the motion of

floating bodies are well known in their linearized version (Mei, 1989), and with the drag

force proportional to the square of the floating body speed (Berteaux, 1991). The

difficulty in predicting the response of a floating body to a harmonic wave arises in the

determination of the hydrodynamic properties, namely added mass and damping. Due to

the effects of water-air interface, these hydrodynamic properties depend on the wave

frequency, water depth, and the wetted body surface, which change with time. There has

been a significant amount of research on these subjects in recent years but they remain the

most difficult aspects of floating structure response prediction. Thanks to the significant

computer advancement in achieving higher computational speed and larger memory

capacity, numerical methods and techniques have been increasing their role in calculating

the dynamics of fluid-structure interaction. The differences between computed and

measured hydrodynamic forces revealed that some important phenomena are either not

well understood or the existing combinations of theories and numerical techniques can not

explain them. On the other hand the agreement between computed and measured

quantities in other particular regions, confirmed the generalization that most engineering

approaches in the floating body studies are only valid in a certain range of practical

interest, thus stimulating more investigations. The notable interplay of theories, numerical

methods, and experiments has been very fruitful for the advancements in floating body

studies.

2.2.5. Numerical Methods Classification

As classified by Yeung (1985), Mei (1989), and Vantorre (1990), several

numerical methods have been proposed for calculating potential functions in free-surface

hydrodynamics:

2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method.

2.2.5.2. Singularity distributions on the floating body surface, which leads to an integral-
equation formulation based on Green'sfunctions.

2.2.5.3. Method of finite-differences. Boundary-fitted coordinates.

2.2.5.4. Finite element method. Hybrid element method.

2.2.5.5. Boundary integral equation methods (BIEM's) based on a distribution of
"simple sources" over the total fluid domain boundary.

2.2.5.6. Methods making use of eigenfunction matching.

All these numerical methods will be explained in the frequency domain, because as

it will become evident from the linearized combined kinematic-dynamic free surface

boundary condition (3.34b), the time-domain and frequency domain solutions are simply

related.

2.2.5.1. Multipole expansion, often combined with conformal mapping methods, or with
BIE-BMP method

Generalizing the heaving motion solution for a semi-immersed circle Ursell (1949),

and its extension to a semi-immersed sphere Havelock (1955), Taylor and Hu (1991)

developed a systematic multiple expansion technique for submerged and floating

horizontal circular cylinders in infinite water depth. Considering the two-dimensional case

of a partially or totally submerged circle, the main idea is to place a set of easy-to-evaluate

elementary functions multipoles which satisfy the Laplace equation, on the level of the

center of the circle. The combined kinematic-dynamic free surface boundary condition is

satisfied with a proper combination of the multipoles and their mirror images with respect

to the mean water surface, which also satisfy the Laplace equation. In order to represent

the outgoing waves at infinity, additional potential is introduced, that has different forms

for symmetric (heave) and anti-symmetric (surge) motions respectively. With the help of

unknown coefficients, both wave and local-disturbance multipoles are combined as an

infinite sum into a velocity potential, which automatically satisfies the Laplace equation,

the free surface boundary condition, and the radiation condition. The wetted body surface

boundary condition is used to determine the unknown coefficients, after truncating the

infinite series at a finite number of terms, and using the point collocation method. A

limitation of the 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 semi-submerged circle in the region kB<1.5 (Ursell,

1949). Numerically, the convergence was found to vary with the depth of submergence,

with the fastest convergence for a fully submerged circular cylinder. The most demanding

case was found to be when the circle was just piercing the surface. In that case the body

intersects the free surface at an angle of zero degrees, which makes the linearized

boundary value problem mathematically unstable.

The multiple expansion can be applied to the analysis of more general bodies

through a coupled method, called the BIE-BMP method by Taylor and Hu, 1991. To

solve for the velocity potential, one can use a boundary integral expression in a domain

close to the body and match it with the multiple expansion outside this domain. Using

three-dimensional multipoles, Taylor and Hu (1991) outlined the same procedure for the

case of a floating or submerged sphere. A complete multiple expansion solution of a

heaving semi-immersed 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 semi-immersed sphere with an open

bottom.

2.2.5.2. Singularity distributions on the floating body surface, which leads to an integral-
equation formulation based on Green's function

The method of integral equations via Green's function, as explained by Mei

(1989), is based on applying Green's theorem on the radiation velocity potential and a

Green's function defined to be the potential at any field point due to an oscillating source

of unit strength at a particular point inside the fluid domain. Knowing the prescribed

normal velocity boundary condition, the velocity potential at any point on the wetted body

surface is a Fredholm integral equation of the second kind. By dividing the wetted body

surface into discrete panels and approximating the velocity potential in each panel by a

constant, one can obtain a system of algebraic equations for these constant values after

carrying out the integration. Solving for these constant velocity potentials on the panels

will help to express the velocity potential anywhere in the fluid domain. An advantage of

this method is the relatively small number of unknowns, while getting the matrix

coefficients is "a laborious task both for the worker and for the computer." Another

drawback of the Fredholm integral equation is the so-called irregular frequencies and

nontrivial eigensolutions in the case of a floating body on the water surface. Without

going into details the irregular frequencies are the eigenfrequencies of a fictitious interior

for the wetted body surface problem with the Dirichlet condition on the water-body

boundary. As a result the source distribution is not unique, the approximate matrix

equation becomes ill-conditioned; hence the integral equation must fail. In order to avoid

the irregular frequencies, one must introduce additional artificial unknowns and more

conditions to improve the matrix equations, thus making the methodology less

advantageous.

A similar technique was used by Martin and Farina (1997) to solve the radiation

problem of a heaving submerged horizontal disc, where the boundary integral equation is

reduced to a one-dimensional Fredholm integral equation of the second kind.

2.2.5.3. Method of finite-differences. Boundary-fitted coordinates

The classical finite-difference method is based on generating a mesh around the

floating body and using a variety of difference-schemes to express the Laplacian operator

at a particular node with the help of information from some neighboring nodes and as a

function of distance, h, between these nodes. Higher-order schemes increase their

accuracy by involving more nodes, usually at the expense of more complicated algorithms.

On the other hand, the accuracy loss due to an increase in truncation errors, expressed as a

power of the distance h, can lead to physically unacceptable solutions (Yeung, 1985).

While conveniently suited for interior nodes of the fluid domain, the difference schemes

are not easily applicable on curved boundaries. That is why the grid generation process

serves two purposes: first it produces a set of curvilinear coordinates that are specifically

adapted to the geometry in question; and second it provides a crucial numerical

transformation that allows difference-schemes to be applied in a more geometrically simple

computational domain. The boundary curves from the original physical domain have to be

transformed into coordinate lines in a logical domain of mapped variables, which facilitates

the implementation of Neumann type boundary conditions. As pointed out by Yeung

(1985), the coordinate transformation and the physical solution may be solved

concurrently in the same "sweep" with a proper adjustment for the boundary conditions.

Once the general algorithm is developed, it can be applied to different floating body

geometries with a change of boundary coordinates. The change of boundary coordinates

is much more involved for three-dimensional bodies.

2.2.5.4. Finite and hybrid element methods

The strength of this class of methods is its ability to handle curved boundaries.

The main idea is to map isoparametric boundary surface elements into local squares,

triangles, etc., on which one can calculate every element's contribution to the field and

boundary properties. The unknown function consists of a set of nodal values and a set of

predefined "shape functions," chosen to satisfy certain continuity requirements across the

elements. The requirements depend on the differential order, and the boundary conditions

(Yeung, 1985). The determination of the nodal unknowns relies on a global, integral

criterion. A brief description of the hybrid element methods, as given by Mei (1989), will

be given as a generalization of the finite element methods of Newton (1974, 1975). The

main idea of the hybrid element method is to employ a finite-element approximation in an

imaginary cylinder, which extends from the sea bottom to the water surface and surrounds

the body, with an analytical representation outside of the cylinder. Thus the finite-element

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 two-dimensional rectangular fluid domain. The domain has the following

boundaries: the vertical axis of symmetry of the body, half of the vertical cross section of

the wetted body surface, the vertical cylinder wall, the flat bottom, and fluid surface

between the body and the cylinder. In order to justify the confinement of the problem into

the cylinder, and thus the boundary element method, a modified radiation condition has to

be satisfied on the vertical cylindrical wall. Now the whole boundary is divided into

discrete panels, on the center of which the velocity potential is considered constant. As a

consequence of Green's theorem a potential on a point on the boundary is expressed as an

integral of the velocity potential and its normal derivative over the whole boundary.

Eventually the problem is reduced to the solution of set of N linear algebraic equations,

where Nis the total number of panels.

Some of the boundary integral methods are very effective for first-order diffraction

and radiation problems. However, these methods are not suitable for second-order

diffraction and radiation problems (Taylor and Hung, 1987). In this case the integrand is a

functional of the first-order potential and the Green's function. Because of the slow

convergence of the integrals, a large number of values of the first-order potential must be

evaluated, and this is not easily achieved using boundary integral methods.

2.2.5.6. Methods making use of eigenfunction matching

Considering the hydrodynamic problem of a floating vertical circular cylinder in

finite-depth water, Yeung (1981) gave an example of the eigenfunction matching methods

with treatment of interior and exterior problems. In the interior problem, the vertical

circular cylinder is considered a fictitious interior domain filled with the same fluid, where

the velocity potential is uniquely determined by solving the Laplace equation with

Dirichlet type boundary conditions. The exterior problem respectively is a Neumann type

problem and can be thought of as one driven by a flux emitted by the interior region, with

a solution written in terms of an eigen expansion with unknown coefficients. Both

problems are treated as if the conditions at the common boundary were known. By

matching both velocity potentials at.the common boundary, the problem is reduced to

solving an infinite system of linear equations. This infinite system is claimed to have

excellent truncation characteristics requiring rarely more than 20 equations to achieve an

accuracy of 1%. It should be noted that the eigen expansion in this solution was possible

because of the convenient presentation of the cylinder boundary in a cylindrical coordinate

system, and because of the finite-depth water. As Taylor and Hu (1991) point out when

the water depth increases toward infinity, the eigenvalues tend to pack together, and the

eigenfunctions become undistinguishable. Consequently, the number of terms required in

the eigen-series expression becomes unrealistically high.

2.3. Advancements in Floating Body Studies

As stated earlier, the most basic problem to solve is that of the frequency response

of the floating body when subjected to simple sinusoidal wave excitation. In 1944

Haskind introduced a way to decouple the hydrodynamics from the body dynamics and to

further decouple the hydrodynamics problem into diffraction and radiation components.

Taking advantage of the linearity of the Laplace operator and the combined Neumann and

Dirichlet-Neumann boundary conditions for the velocity potential Haskind presented the

velocity potential as a sum of diffraction and radiation potentials. Speaking in physical

terms, two hydrodynamical sub-problems were formulated. The first sub-problem

assumes the body to be rigid and restrained from any oscillation in the presence of incident

regular waves. The hydrodynamic loads are called wave excitation loads and composed

of Froude-Kriloff and diffraction forces and moments. The second sub-problem forces

the rigid body to oscillate in any motion mode with the wave excitation frequency in the

absence of incident waves. The hydrodynamic loads are identified as added mass,

damping and restoring terms. Since the body oscillates in otherwise calm water, it

radiates waves, thus the term radiation. In 1949 John (see Mei, 1989) introduced in a

systematical formal approach the complete first order wave-structure theory and showed

how higher-order extensions can be made. Following Haskind (1973), Newman (1976),

and Mei (1989), several remarkable general identities have been introduced. These

identities relate different hydrodynamic quantities and have proved to be extremely useful

in increasing the theoretical understanding of physical phenomena. Moreover, they

provide necessary checks for analytical theories or numerical methods, and minimize the

25

computing time of those quantities related by them. Based on the mathematical definition

of the incident, radiation, and diffraction wave potentials as harmonic functions, and on

the Green's theorem, the first identity relates, in an elegant way, two radiation problems.

A computationally efficient result is that the restoring force, the added mass and the

damping matrices are diagonally symmetric, which decreases significantly the number of

unknowns. Another consequence is the convenient relationship between the damping and

the asymptotic behavior of the radiation velocity potential. In physical terms the energy

given up by the oscillating body is transported by the waves propagating away from the

body. The knowledge (about damping) gained this way, can be used to find the added

mass, using the so-called Kramers-Kronig relations. Derived in a universal way by Ogilvie

(1964), these relations simply exploit the fact that damping and added mass are the real

and imaginary parts of the same function. Therefore knowledge of the damping

coefficients is sufficient to determine the corresponding added mass coefficients and vice

versa. Newman (1976) discovered that the damping matrix is singular for the case of

particular bodies of revolution where the exciting force is independent of the angle of

wave incidence. The second identity relates two diffraction problems corresponding to

different angles of incidence. As two-dimensional results, there are several relationships

between the amplitudes and the phase angles of the transmitted and reflected waves. The

three dimensional result is that the amplitude of the first scattered wave toward the second

incident wave is equal to the amplitude of the second scattered wave toward the first

incident wave. The third identity relates the diffraction potential to the radiation potential,

thanks to a brilliant Haskind theorem (1957). It expresses a generalized component of the

exciting force (due to fixed body diffraction from an incident wave moving in a particular

direction) in terms of the radiation potential and its normal derivative (due to body

oscillation in the same direction, in otherwise calm water). An elegant consequence from

the Haskind theorem is that the exciting force is linearly proportional to the far-field wave

amplitude. The very practical meaning of the Haskind theorem is that an actual solution of

the diffraction problem can be avoided and that the exciting forces are simply related to

the damping coefficients. Newman elaborated on this further and found an explicit

relationship between the magnitudes of the exciting force and the damping coefficients,

which is often used as a check on the magnitudes. An even less obvious identity between

radiation and diffraction problems was discovered by Bessho (1967) for two dimensions

and extended for three dimensions by Newman (1975,1976). It relates the far-field

radiation and diffraction wave amplitudes, and reveals the significant result that the phases

of surge and pitch antisymmetric modes differ with r .

2.4. Presentation of the Results of Investigations

In light of the above-mentioned theoretical advancements, it is instructive to give

some examples of the theory-experiment interplay. In 1965 Kim determined the added

mass and damping for a semi-ellipsoidal body with its origin on the free surface of water

of infinite depth. In order to check the validity of potential flow theory predictions Frank

(1967) conducted a series of experiments to determine the added-mass and damping

coefficients for two-dimensional bodies at the free surface.

Using Ursell's 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 two-dimensional floating cylinders. Conducting experiments with five

different cylinder cross-sections in heave, sway, and roll oscillatory motion, he has

compared the experimental results with the predictions of the linear potential theory and

particularly the 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 low-frequency 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 half-immersed floating sphere with a

bottom opening, Evans and Mclver (1984) have used a semi-analytical solution based on

an extension of the method of 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 off-diagonal values are in good agreement with the semi-analytical

solution. Moreover the numerical method showed slight differences between the off-

diagonal added mass and damping coefficients, which theoretically must be equal in

accordance with the symmetry-identity classified above. The singularity of the damping

matrix has been used as a partial check for the correctness of the semi-analytical solution.

In a technical note to the Evans and Mclver (1984) work, Falnes (1984) proposed some

empirical formulas for the added mass and damping for the semi-submerged sphere in

heave motion.

Utilizing a numerical scheme that is similar to one used by Nestegard and

Sclavounos (1984) for the method of multiple expansion combined with BIE-BMP

matching classified in 2.2.5.1, Taylor and Hu (1991) obtained added mass and damping for

floating and submerged circular cylinders. For the submerged cylinder, the diagonal added

mass and damping coefficients in sway have been confirmed to be equal to those in heave.

While damping is always positive, negative added mass has been discovered for the

case when the submerged cylinder is close to the surface. Negative added mass has been

also observed for a cylinder floating on the surface in sway motion when the cylinder is

more than three-quarters immersed. For the frequencies of negative added mass in heave

the sway added mass was positive. It was concluded that "at these frequencies the forced

heave oscillation does not transfer energy to the fluid and hence does not generate waves."

Lastly Taylor and Hu have found that in the low frequency range the added mass changes

from finite to infinite values as the cylinder emerges from below the free surface. The

incorrect conclusions, made by Taylor and Hu, show the important role of experiments as

the only tool for verification of the working capabilities of theories and numerical

techniques.

Exploring surge oscillatory motion of a single tethered half-submerged spherical

float, Vethamony et al. (1992) have computed added mass and damping from the motion

generated velocity potential, using potential flow theory. They solved a boundary value

problem with Green's function method by dividing the cross-section of the float into equal

segments, and conducted experiments to compare the results. Considering the float size

small compared to the wave-length the diffraction potential has been neglected and added

mass and damping computed from the motion generated velocity potential. With added

mass in phase with the surge motion and damping coefficient out of phase with the

motion, both were found to be frequency dependent, and to increase gradually with

respect to size of the float. It was noted that added mass and damping do not change with

water depth. In light of the general identities discussed earlier, Vethamony et al. found

that wave excitation forces calculated from incident wave potential are in good agreement

with excitation forces calculated from damping coefficients.

It was observed that nonlinear effects become important in fluid-structure

interaction when waves become steeper, or when the oscillation amplitude of the floating

body can no longer be assumed small. These natural phenomena have been stimulating the

extension of the linearized radiation problem and linearized diffraction theory which are

the first terms in Stokes perturbation expansion into higher order terms, where the

perturbation parameter is related to the wave steepness. As a result of this expansion

procedure the full nonlinear diffraction and/or radiation problems are replaced by a

sequence of linear boundary-value problems formulated for each order of perturbation.

Most difficulties of the second order problem arise from the requirement that the velocity

potential satisfies a nonhomogeneous boundary condition on the free surface of the fluid.

According to Molin (1979), the nature of second-order diffracted waves consists of two

components. One is free-waves travelling independently of the first-order wave system

and the other is phase-locked-waves accompanying the first-order waves (Chakrabarti,

31

1987). Since in many applications the hydrodynamic loads, not the flow kinematics, are of

primary interest, an indirect method can be used to determine wave-induced loads to the

second order without the explicit calculation of the second order potential. The technique

involves an application of Green's second identity and requires the solution of associated

linearized radiation problems corresponding to prescribed oscillation of the structure at

twice the first-order wave frequency. A modification that utilizes an axisymmetric Green's

function solution for the linear problem has been used by Molin and Marion (1985) to

calculate the second-order wave induced loads and associated motions for a floating body.

Taylor and Hung (1987) have performed detailed analysis of the asymptotic behavior of

the troublesome free-surface integral and have presented a computational method for its

calculation on the far field in the case of a fixed vertical circular cylinder. Adopting an

interior region that includes the fixed vertical cylinder, and an exterior region that extends

to infinity in the horizontal plane, Ghalayini and Williams (1991) presented a solution to

the free-surface integral. The interior region solution was performed numerically by

utilizing the asymptotic forms of the potentials, while the exterior region solution was

carried out analytically in terms of Fresnel functions. Inspired by Soding's work,

Vantorre (1986) developed a computational procedure for calculating hydrodynamic

forces up to the third order for the case of floating axisymmetric bodies in a symmetric

heave oscillatory motion. Soding (1976) proved that the knowledge of the first-order

potential for the exciting frequency and for twice that frequency on the mean body surface

and on the mean free surface within some distance from the body is all the information

required to compute second-order forces for oscillating cylinders. In a similar way,

Vantorre computed third-order forces by means of the first-order and second-order

32

potentials for the exciting frequency, and the first-order potential for three times the

exciting frequency. Calculation of the latter potential is not necessary if the first harmonic

of the third-order forces is the only primary interest. The computational procedure used

by Vantorre is classified in 2.2.5.4. as boundary integral equation method (BIEM) based

on a "simple sources" distribution over the total fluid domain boundary. Two experiments

have been conducted, one with a floating cone and a second with a submerged vertical

cylinder. In both cases the third harmonic was impossible to measure. It is obvious from

the experimental results for the floating cone, that the third-order theory somehow

underestimates the added mass over the entire frequency domain. This is more significant

for larger heave amplitudes. Being acceptable for the region (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 second-order forces. Using

again the boundary integral equation method (BIEM) for a floating cone in heave

oscillatory motion, Vantorre (1990) determined the influence of small draft variations on

hydrodynamic first-order forces. After deriving some second-order and third-order force

components, he used a modified BIEM to derive first and second derivatives of heave

added-mass and damping with respect to draft.

The nonlinear effects of the motion of fluid around a body can be dealt with

directly from the fundamental conservation laws of mass and momentum (Equations 3.2

and 3.3). Due to the increasing capacity of computers in the recent years, three groups of

methods for numerical treatment of flow nonlinearity around cylinders have been

developed. Following the comprehensive survey of Sumer and Fredsoe (1997), the first

group consists of methods involving the direct solution of the Navier-Stokes equations.

When the flow around a cylinder is two-dimensional and the Reynolds number is small, i.e.

Re<200, a direct solution with finite-differences yields results for the gross-flow

parameters that are in reasonable agreement with measurements, while the lift force is

grossly overestimated (Sumer and Fredsoe, 1997). For larger Reynolds numbers, vortex

shedding occurs in cells and therefore the flow becomes three-dimensional (3D). When

300
further, 3D-turbulence begins to spread into the boundary layer, and direct numerical

simulation of the Navier-Stokes equations is not feasible, because of the scales of the

dissipative part of the turbulent motion. The oscillatory flow is solved using the same

vorticity transport equation (3.7) and Poisson equation (3.8). The major source of

difficulties is the number of grid points (for finite-difference and finite-element methods)

and the corresponding number of computations required to obtain a solution increases

with increasing Reynolds number, and may become prohibitive as mentioned earlier at

large Reynolds number. The second group consists of discrete vortex methods, developed

as grid-free numerical methods. The idea is to solve the vorticity transport equation

(Equation 3.7) through a numerical simulation of convective diffusion of discrete vortices

generated on the cylinder boundary. The advantages summarized by Sumer and Fredsoe,

(1997) are (1) the inviscid theory can be employed, (2) numerical diffusion problems

associated with the vorticity gradient terms in Eulerian schemes are, to a large degree,

avoided, (3) there are no zone assumptions to require matching of an outer to an inner

flow, and (4) the method is relatively stable and well suited to vectorization on

supercomputers (Stansby and Isaacson, 1987). For a detailed review of the vortex

methods see Sarpkaya (1989). For oscillatory flow and waves the vortex methods fail to

agree with experiments within the Keulegan-Carpenter range between 1 and 2.5. The

third group of numerical methods is.based on the hydrodynamic stability approach, in

which the formation of vortex shedding is viewed as an instability of the flow in the wake.

As seen in Section 2.2.1, the oscillatory flow becomes unstable above a critical KC

number due to spanwise-periodic vortices the so-called Honji instability. After modeling

the phenomenon numerically, Zhang and Dalton (1995) have obtained a definite 3D

behavior regarding the variation 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 Navier-Stokes

equations with massive computing power. With the help of supercomputers and

mature numerically-stable methods, the existing mathematical models have succeeded

in increasing our appreciation and understanding of some natural phenomena, in

particular ranges of Reynolds and Keulegan-Carpenter numbers. Other Re and KC

ranges of practical interest require different or improved mathematical and theoretical

modeling. It is evident that oscillatory and wave-flows bring additional challenges for

the direct approach to the nonlinear problem.

* The second approach is to utilize a Stokes perturbation expansion, which restructures

the Navier-Stokes equations into a sequence of linear boundary-value problems

formulated for each order of perturbation in terms of the power of a small parameter,

for example wave-steepness. With gradually increasing complexity, any-order solution

can be obtained, provided the solutions of previous-order problems are known.

Predominantly second-order and third-order solutions of the fluid-body interaction

have been investigated, with the noticeably preferred simplifications of axisymmetrical

bodies and symmetrical heaving motion for the case of third-order problems. To

explain this preference it suffices to recall that even for the second-order problems, a

great effort is required to deal with the nonhomogeneous boundary condition on the

free surface of the fluid, as discussed earlier. Nevertheless, the role of perturbation

analysis in fluid-structure interaction studies is evident in providing valuable

information about the significance, contribution, and limits of the higher-order

nonlinear effects, as compared to the first-order solution.

The third approach is to work with the linearized diffraction and radiation theories and

with the powerful principle of superposition, which provides insight into the fluid-

structure interaction problem in irregular seas. Naturally semi-analytical solutions are

much faster than the numerical ones because of the smaller number of computations

needed. Moreover, because of the exact boundary conditions semi-analytical solutions

are principally more accurate as compared to the approximate boundary conditions in

numerical solutions. Of course there are greater restrictions on body-shapes with

semi-analytical than there is for numerical solutions. One of the main advantages of

the linearized theory is that one can explore in real time the influence of variation of

different parameters like draft, shape, size on the hydrodynamic properties, forces, and

hence fluid-body dynamics. Another advantage is that the linearized theory provides

the easiest way to analyze and comprehend the physical concepts, and meaning and

significance of observed natural phenomena. Ones seeing the big picture, further

improvements can be made, if desired and feasible, with the help of the other two

approaches.

In a historical retrospective, it is true that the three main approaches have had

great impact on the floating-body studies. Having a common goal, they have influenced

and stimulated each other's evolution. All three approaches have their own unique

advantages and disadvantages and areas of applicability.

In the present work, a new semi-analytical method is proposed for solving the

dynamics of free floating two-dimensional horizontal cylinders, of various shapes in heave,

surge and pitch. The method is based on linear radiation theory and is intended to be the

first step in the development of a mathematical model and computer program for

predicting the response of an axisymmetric, three dimensional, free floating drifter buoy

subjected to waves.

CHAPTER 3
FORMULATION OF THE PROBLEM

3.1. General Description of the Problem and its Simplifications

The most general formulation of the problem of the dynamic response of a free

floating body subjected to waves is to pose a dynamic equilibrium of forces and moments

in and on an elastic body freely moving in the air-water interface. The focus of the present

work will be on the floating body motions due to external loads, which act on the

underwater part of the body. Therefore two restrictions will be made right from the

beginning: first the body is considered rigid; and second the direct influence of the air

environment on the body is negligible. As long as no structural or vibrational problems

are to be dealt with, the first restriction can be made without any hesitation. The second

restriction is based on the fact that the density of air is roughly one one-thousandth of the

density of water. Provided that the above-water part of the floating body is close to the

water surface, and excluding strong winds during adverse weather, it is clear that for most

practical problems aerodynamic forces may be neglected with respect to hydrodynamic

forces.

3.1.1. Incompressible Fluid Assumption

A coordinate system Oxyz or simply x=(x,y,z) which is fixed in space will be used

in this analysis. The Oxy plane coincides with the still water surface, and the vertical axis

Oz is positive upward. The origin 0 is the intersection of the centerline of the floating-

body-section and the still water surface. In a wide variety of gravity wave problems, the

variation of water density is insignificant over the temporal and spatial scales of

engineering interest. The relationship between water density and pressure is given by

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 Navier-Stokes 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 Navier-Stokes equations is that the rate of change of

vorticity is due to stretching and twisting of vortex lines and to viscous diffusion.

-+ u. 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 three-dimensional compressible

flow (Dean and Dalrymple, 1991). A Streamfunction exists in two-dimensional or

axisymmetric incompressible flow and is defined by

-- = u and -= v in Oxy-plane,

or (3.6)

-- = u and = -w in Oxz-plane
9z &x

For two-dimensional incompressible flow, equations (3.4), (3.5), and (3.6) reduce to the

2D vorticity-transport 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 height-to-diameter
ratio and diffraction parameter

As observed from Figure 3.1, Hooft (1982, Figure 4.1.6) has shown that

for a fixed vertical cylinder and relatively steep waves, with ratio between wave

height and wave length: H/L=0. 1, viscous effects become important when H/B

>15 and kB < 0.06, where k is the wave number. Practically this means that if the

characteristic dimension of the fixed vertical cylinder B=1 [m], the viscous effects

become important for wave heights H>15[m], and wave periods T>9[sec]. When

B=2 [m], the viscous effects become important for wave heights H>30[m], which

in reality can be referred to as extreme wave conditions. Intuitively, it is reasonable

to expect that for a free-floating body the velocity gradient will be significantly less

than for a fixed vertical cylinder. Therefore the fluid can be considered inviscid,

and the Navier-Stokes equations (3.2) are transformed into Euler equations:

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

two-dimensional and three-dimensional 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 air-water interface is that it can not

support pressure variations and hence must respond appropriately to maintain the pressure

uniform. For the practical interest, the wave length is so long that the surface tension is

unimportant; the pressure just beneath the surface must be equal the atmospheric pressure

Pa above. Therefore the DFSBC is defined as a uniform pressure along the wave form on

the free surface. Applying Bernoulli's equation (3.12) on the free surface, we have

+ 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 (air-water) 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 kinematic-dynamic free surface boundary condition (CFSBC).

3.1.7. Sea Bottom Boundary Condition (SBBC)

On the sea bottom, denoted with Bo, (3.14) becomes

F(x,y,z,t) = z +h(x,y) = 0

where h is the water depth. Applying (3.15) on the above equation, the SBBC becomes

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

(3.19)

Using the same procedure as in 3.1.6, we state the continuity of the normal velocity with

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 first-order

theory and shows how higher-order extensions can be made. For small-amplitude motion,

we expand in powers of the wave slope s = A / L, which is the small parameter in the

perturbation analysis

z= f(0)(x,y) + 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)

= x-s[x +) O(1) x (-X(0) +0O(E2)

Y = X[- 6X(1) +8(Z Z(O)_ )(Y 102)]
= x- X(') + z-Z ( y )
(3.26)
y=y-EY )+7x-X(Oz_ o- )
z=z- EZ() +a(y-Y(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( y-Y(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 (Z-Z()) t(Y- Y(o))
-fo)[(1) )+t(x-X()) -a (z-Z())] (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, (x-X(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 zero-order portion of (3.36) is

0 = f(-pgf(O))nds Mgk (3.37)
So

while the first-order 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 zeroth-order

Mg = pg V() (3.39)

The first-order 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 (y-y('))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

(x-X)x dm = (x- X)x Pnds + (xc- X) x (-Mgk) (3.43)
Vb Sb

which following Mei's procedure leads to

x-component:
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))

y-component:
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

z-component:

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 cross-section in the Oxz-plane with incident waves travelling

along the Ox-axis, the motion can be described working with a unit-cylinder length in the

y-direction. For this two-dimensional case the displacement vector is

{X}= (X(),Z(',)fr (3.46)

The normal to the wetted body surface vector is

n} = n,,n,,n, z-Z('))- 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 = x-X ( 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 left-hand side are inertial

and buoyancy forces with mass and restoring matrices, known from the hydrostatic

equilibrium in the absence of waves. These forces are proportional to the unknown

generalized body displacements, and are balanced on the right hand side of (3.45) by

hydrodynamic forces, which are functions of the unknown velocity potential. The

unknown velocity potential can be determined from the hydrodynamic system (3.34), only

if and when the generalized body displacements are known (see (3.34d)). One way to

resolve this problem is to decouple the hydrodynamics from the body dynamics with the

following decomposition (Haskind, 1944):

dX = Re(Vae-i"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 3D-space, and 3 in 2D-space. (, has the dimensions of [Length], and

stands for the velocity potential caused by a body oscillatory motion with unit velocity in

the a-direction: for example in heave, surge, or pitch.

(b) diffraction 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 2D-space.

Sommerfeld had introduced a similar lateral boundary condition, in an analogy with an

outgoing wave system. The incident velocity potential is given by

-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 3D-space using double integration

over the wetted body surface. In 2D-space the same properties can be expressed with a

single integration over the wetted body surface. Working with the hydrodynamic pressure

of the linearized Bernoulli equation (3.12c), and with the decomposition made before, the

generalized hydrodynamic force on the body in the a-direction is decomposed into its

diffraction and radiation components:

F, = f Pnlads = If-P )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 twice-differentiable 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 left-hand side of (3.61) becomes

zero due to Laplace equation. By virtue of the boundary conditions (3.54b), and (3.54c),

neither the free surface S, nor the bottom Bo contributes to the surface integral thus

reducing the right-hand 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 = 2-aXP 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 two-dimensional 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 thefar-field 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 non-negative, the damping matrix is positive semidefinite. As a

corollary, all diagonal terms of the damping matrix are non-negative, since

,+ = pgc,(a-2 +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 so-called restoring force

matrix. This means that knowledge of the damping coefficients is sufficient to determine

the corresponding added mass coefficients and vice versa. Such relations, known as

Kramers-Kronig relations, are valid for all modes of motion regardless of forward speed

(currents, ships). Having been derived in a universal way by Ogilvie, they read:

0

Another way to find the added mass, knowing the damping as a function of the circular

frequency, is Hooft's approach (1982) of using the so-called Bode relations, which for

water waves correspond to the Kramers-Kronig relations.
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 so-called Bode relations, which for

water waves correspond to the Kramers-Kronig 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 far-field amplitudes, (3.74, and 3.78).

As mentioned in section 2.3, a remarkable Haskind theorem relates the exciting

force to the far-field amplitudes in the following manner

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 far-field 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][)-[a-j. (3.84)

3.6. Algorithm for the Solution of the Problem

In summary, the asymptotic solution of the radiation problem gives the far-field

amplitudes. The added mass matrix [u], the radiation-damping matrix [A] (3.73), and the

exciting force (3.81) can be found from these far-field amplitudes. Therefore the

asymptotic solution of the radiation problem alone will be sufficient for analyzing the body

dynamics (3.84). Practically this means that the solution of the diffraction problem can be

avoided. Having the dynamics and hydrodynamics (radiation) problems formulated, and

all the necessary relationships for their solutions, the following algorithm will be used.

First, the asymptotic solution of the radiation problem, in terms of the far field amplitudes,

will be found in heave, surge, and pitch. Second, the hydrodynamic added mass, damping

coefficients, and exciting forces will be computed as functions of the far-field amplitudes;

consecutively the body dynamics will be computed for a particular wave frequency in

terms of H, or RAO. Third, using the RAO (3.83), the spectral analysis can be used to

find the body response spectrum Syy(co) due to the incident wave spectrum S,(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 cross-section in the Oxz-plane is

oscillating on the free water surface in otherwise calm water. Part of it is below the water

surface; the other part is in the air. The rigid cylinder is oscillating with unit velocity in

the generalized O-direction with the circular frequency of the incident waves co (in the

absence of the incident waves), thus generating outgoing waves. Taking a unit-cylinder

length in the y-direction, the problem is considered two-dimensional, and the motion can

be described in the cross-sectional Oxz-plane (Figure 4.1).
Z

-TB __ -

II CB II
CG II
II < ^ II
Switch
II II
II eII
surge
heave

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

Formulated in (3.54), the radiation problem is stated as a two-dimensional Laplace

equation with a complete set of boundary conditions:

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 e-j'') (4.2)

With the imaginary unit j = I--, the wave profile is

S= A cos(kx wt) = Re(Aej(x-t)) = Re([Aej ]e-j) = 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 two-dimensional case, the physical meaning of the

Sommerfeld radiation condition (4. le) is that the generated waves are propagating

outward from both sides of the oscillating body. Therefore, the asymptotic expression of

the wave profile at infinity must be:

lim = A_,eJk-x) = (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

far-field amplitudes introduced in (3.73).

4.2. Main Idea Behind the Semi-Analytic Technique (SAT)

Inspired from a Haskind idea (1973), a new analytical solution of(4.1) will be

given for the case of partially submerged floating circular and elliptical cylinders with

varying equilibrium drafts. As will be shown below, the analytical solution is in terms of

infinite power series. Therefore, a finite number of terms must be taken to solve it

numerically thus the prefix "semi-" and the name Semi-analytic technique (SAT). For

the sake of clarity and completeness, some mathematical definitions and formulations used

in the analysis are presented first. Considering a two-dimensional (2D) wave motion in a

right-hand complex plane Oxz, the real axis Ox is defined to represent the still water

surface and the imaginary axis Oz to be perpendicular to the still water surface and

positive upward (Figure 4.1). Let y be the complex variable, which corresponds to the

complex plane Oxz, and defined as

y =x+iz (4.5)

where i = --T is the complex plane imaginary unit, which for the sake of convenience is

different from the time imaginary unit j = 1-l In the two-dimensional 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 well-known complex velocity

potential

w(y) = 0((x,z)+ i y(x,z) (4.6)

and are related to each other as conjugate functions through the Cauchy-Riemann

conditions

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 Cauchy-Riemann conditions.

Therefore by definitions (a,b,c) the complex velocity potential is a holomorphic function

(Milne-Thomson, 1950). Utilizing the property of holomorphic functions that a

combination of derivatives and integrals of holomorphic functions is also holomorphic, the

following holomorphic function is introduced.

f(y) = 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 air-water 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 air-water

interface (4.9), it is possible to continuef(y) in the upper half-plane. As a result of the

analytical continuation the values off(y) in the upper half-plane will be conjugate

imaginary of the mirror off(y) values in the lower half-plane. In this mannerf(y) will be

holomorphic in the whole complex plane except for the points on the wetted buoy

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 half-plane 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 well-known 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) = e-ikoY[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 half-plane. Consequently, when y = (x, z) -> (0, z),

the asymptotic expressions of the complex velocity potential are

limw(y) = lim(q + i/) = (A, +iA2)e-iky = (A, +iA2)e-kox+koz
y-+O y-.+m (4.14)
limw(y) = lim(q + iy) = (B, +iB,)e-i'o = (B, +iB2)e-ikx+koz
y---oo y-m--

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)
y-4-
From the linearized free surface dynamic boundary condition at z=0

1 = jco =17eJ-t
= 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 deep-water (k = k0 = r-2) relationship:
g

lim 7 = j V A coskox+A2 sinkox) = Aekx = (Va)(coskox+ jsinkox)

lim7= j V (B1 coskox+B2 sinkx) = Ae-kx =(Va)(coskox-jsinkox)

Therefore the integration constants are

A =-j-a+, A=-a+, A +iA2 =g--a+ (-), (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'(i-j)- 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. Semi-Analytic Technique: Determination of the Unknown Coefficients

The procedure for determining the unknown coefficients in (4.12) consists of

4.3.1. Constructing a boundary condition for the holomorphic function through

integration of (4.8a) over the wetted body surface Sb, so that the left-hand side

contains all unknown coefficients, and the right-hand side contains all terms

derived from the complex fluid velocity.

4.3.2. Conformal mapping of Sb and its mirror image ,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 left-hand side, so that all multipliers of the unknown coefficients

are trigonometric functions of the polar angle of the unit circle 0.

4.3.4. Determining the right-hand side, so that all term are trigonometric functions of the

polar angle of the unit circle 0. Determining the complex fluid velocity with a

generalization of a procedure outlined in Milne-Thomson (1950).

4.3.5. Determining the unknown coefficients by solving a linear system of equations,

derived through the Fourier expansion of both left- and right-hand sides of the

boundary condition.

4.3.1. Boundary Condition on 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 (1-iky)dy (4.23)

Eliminating ikw in (4.23) and (4.8a), gives a boundary condition whose right-hand side

dw
(RHS) is a function of the complex velocity and complex variable,
dy

Yr dvw -Ydw
Af (y)dy- yf ()+ikAWA = -y- + -(1-iky)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 half-submerged circle, then the

analytic presentation (4.12) in the left-hand side (LHS) of the boundary condition (4.24)

can be used. In this case the unknown coefficients can be determined from those

corresponding to the generalized motion RHS. If the wetted cross-section of the floating

buoy is different than the half-submerged circle, then Sb can be mapped into a half-

submerged circle C. Denote the analytic conformal mapping by

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 half-submerged circle

If R is the radius of the circle and h is the distance between the center of the circle

and the still water level (Figure 4.3), then the wetted body surface and its mirror image

can be written as

y = -ih + Re'"',

y = ih + Re"' and

, S:b E[P [-pA,2+t+(A],

S:, E(P[-p7Ar + (PA.

;z Sb

Fig. 4.3 Conformal mapping of more than half-submerged 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 half-submerged 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 half-submerged ellipse

If a and b are the vertical and horizontal semi-axes of the ellipse and h is the

distance between the center of the ellipse and the still water level as shown in Figure 4.4,

then the wetted body surface and its mirror image can bewritten as:

x+iz z y=x+iz

Sb \ Sb// Sb
x- A

bS

Fig. 4.4 Conformal mapping of more than half-submerged 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 = (b-a)/(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 = (b-a)/(b+a), (4.55)

a, = ce3iA (4.56)

a2 = Cle-3ir/A (4.57)

7= 2'r+ 37 +3hO, and (4.58)

k +2+EA (4.59)

4.3.2.4. Conformal mapping of exactly half-submerged ellipse

This is a particular case of 4.3.2.3, with h= 0, 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. Left-Hand Side of the Boundary Condition on Sh

To find an explicit form of the left-hand 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)= e-i'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 half-submerged 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~'n-d = i[e +nI ,,+ (4.70)

The first integral can be expressed with a complex exponential-integral 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 half-submerged 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)
n-I

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)" n-l-sh2dt (ika)s 1 (4.78)
I. =-f eikat-2 dt = -O ts--d (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 S-n+n (4.79)
ik n=1 I =, s! s2 -n

Substituting (4.79) into (4.66) results in

-i e-khj-'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 half-submerged 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 half-submerged 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 e-kh an,, ,(4.87)
+00 n=1 co n=1

1 1
I = eik(a +a2)" h2 (a a2r-h)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. Far-field wave amplitudes

(4.90)

S= f (y)eksdy
2 g s, -J i=T-j

(4.91)

= aD (n)
n=1

D+(n) has to be expressed from (4.65) and (4.41b).

4.3.3.1.2.1. Far-field wave amplitudes: Exactly half-submerged circle

Denote the integral (4.41b), upon which the far-field amplitudes depend with I.

The use of conformal mapping (4.40), (4.41) results in

y 21r ikRet2 i(n-1)"
I= ff(y)ekYdy= R a. eckR 'd= RlaneikRe e-ie(n-)idO ,
Sb + n=1 C:1=1 n=1 0

(4.92)

(4.93)

I= RZ an e(i[s-(n-1)]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 wn-wR (ikR)"'
D, (n) = 2ni =
2g ( (n 1g! ^ g ( -1)!---

(4.94)

82

4.3.3.1.2.2. Far-field wave amplitudes: More than half-submerged circle

The use of conformal mapping (4.30), (4.31) results in

I = f (y)elkody = aj ekih+a ah2nd= ah2e-kh ~ 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(h3-n+')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. Far-field wave amplitudes: Exactly half-submerged 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 C-C=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 + s-1)! = s!(n+s-1)! s!(s+n-)!

2s+n-1
G, = 2mn (-12 (-I)(kciVf) = 2ni. (ni)i2J (2k e,)
s=O s!(s+n-1)!

,when A > 0

(4.102)

where J,_-(.) denotes the Bessel function of the first kind,

= ----) 2s+n-l
G,= 2 n-"(-)-n2) -k ) =2"()-(2kclf /-)
,0 s!(s+n-1)!

,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. Far-field wave amplitudes: More than half-submerged ellipse

Using the conformal mapping (4.44), (4.45) results in

I= f(y)e'kody e-kh 'a,, '' -n 3 -a2 d-h' )d= e-k 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= s-0 i-=o ( 1-s)h,-n

and

Sb inik( aIeo+a2e dO (ika 2) (ika e-' -('-)-"]" e-2i(1-s)h-"]"r (4.109)
I-,=i se-=e-( ^)id=l (l-s)h-- -,- (4.109)
2 s= o s! = l1 (I- s)h,2-n

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 left-hand side of the boundary condition on Sh

Upon substituting (4.25), and (4.27) into (4.24) the left-hand side of the boundary

condition becomes

LHS =f (y)dy-yf(y)+ikWA = aL, (4.111)
n=1

4.3.3.2.1. Left-hand side: Exactly half-submerged circle

Using the conformal mapping (4.40), (4.41) results in

L, = RA-n"d R('-') +ikyw, (4.112)
A4.12

1- -i(n-1)0
L, = R e--id- Re-(-+ikxA = R e Re-' ) +ikxAw, (4.113)
n-1

and

L, = + ik R n-- e-i-. (4.114)
n-1 n-1

The real part of(4.114) is given as follows:

R n
Re(L,) = R Im(w) R -n-cos(n 1)0. (4.115)
n-1 n-1

4.3.3.2.2. Left-hand side: More than half-submerged circle

Using the conformal mapping (4.30), (4.31) results in

L, = h2af' d-(ih, +a4k)g"+ikyAW (4.116)

L,, = ha ei(h-'+0id 8-ihe-o ae-'('-o +ikAW (4.117)

L, = h2a e(-)-- O aie--i(n-)o + ikxAw and (4.118)
n-h

L = ikXAWn + -i(na -ihe-ink (4.119)
n-h2 n-h2

The real part of (4.119) is given as follows:

Re(L,) = lo + i cos(n-h2) +2 sin(n-h)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)
n-h2

11 nRea (4.122)
n-h,

n Ima
2-= and (4.123)
n h2

13= -h. (4.124)

4.3.3.2.3. Left-hand side. Exactly half-submerged ellipse

Using the conformal mapping (4.61), (4.62) results in

L,= cici c-n i-iC + +ikyA ,, (4.125)

C[\ p-1)- in+_ ( 1)1 ]
L = c [e -A -e-i(n-1)o '-i(n+l)O +ikx AW (4.126)
n-1 n+l1

L, = c 1 -- A n e i(n-)o nA e-i(n+)o + and (4.127a)
[ -
n-1 n+1 n-1 n+c

LI =cC I -1-+iO-Ae-2iO + Aiwl. (4.127b)
2 2

The real part of (4.127) is given as follows:

Re()= c,[ A -] Im(w)- ncl cos(n-1)0-nC+ cos(n+1)0 (4.128a)
Re(L,)= n1 n+1I nn-1 n+l

Re(L,) = c,-1- -A m(w1)- cos(20) (4.128b)

4.3.3.2.4. Left-hand side. More than half-submerged ellipse

Using the conformal mapping (4.44), (4.45) results in

L, =h2 a-"(a( -a )d- (ih, +a1h< + a24-h +kAWn,, (4.129)

-i(n-h2)8 -i(n+h)0
i1-e --(-h) -inO i(n-k)O -i(n+h2)0
L, = ih- e -ae -'("-e) c
n h2 n+h2

(4.130)

L _____h2 __1_h2 2- na1 -i(n-h2)9 na2 -i(n+h2)(
h2al -ia2 +ikYA -ih" n- e-(-) e- (4.131)
n-h2 n+h, n-h 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. Right-Hand Side of the Boundary Condition on Sh

To find an explicit form of the right-hand side of the boundary condition (4.24)

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 Cauchy-Riemann 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, -z-Z X() dx- and (4.139)
as On s ds ds V, ds \ I ds

as ds 2 L '

As described in Milne-Thomson (1950), the stream function can be presented as a

difference of a complex function g(y,y) and its complex conjugate counterpart g(y,y),

2iYR = g(y,y)-g(y,y), (4.141)

g(y,) = Ve-y + -[ y- ) ) +c, (4.142)

g(y, y) = Ve y _- [)yy-I) y(O)- + and (4.143)

2iR = Ve- y-Ve +iVyYy-Y ()--- 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 half-submerged circle

Applying the conformal mapping (4.40) in (4.144) results in

y = R"-', (4.150)

yf = R2 (4.151)

B,() = -Ve'f(R-1) +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 half-submerged 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)

MILLISECOND	CLASS.METHOD	MESSAGE
0	sobekcm_page_globals.constructor
0	sobekcm_page_globals.constructor	Application State validated or built
0	sobekcm_database.verify_item_lookup_object
0	sobekcm_page_globals.constructor	Navigation Object created from URI query string
0	sobekcm_database.verify_item_lookup_object
0	sobekcm_page_globals.display_item	Retrieving item or group information
0	sobekcm_page_globals.get_entire_collection_hierarchy	Retrieving hierarchy information
0	sobekcm_assistant.get_entire_collection_hierarchy
0	cached_data_manager.retrieve_item_aggregation
0	cached_data_manager.retrieve_item_aggregation	Found item aggregation on local cache
0	item_aggregation_builder.get_item_aggregation	Found 'all' item aggregation in cache
0	system.web.ui.page.page_load (ufdc.page_load)
0	sobekcm_page_globals.constructor.on_page_load
0	html_echo_mainwriter.add_style_references	Adding style references to HTML
0	html_echo_mainwriter.add_text_to_page	Reading the text from the file and echoing back to the output stream
114	html_echo_mainwriter.add_text_to_page	Finished reading and writing the file