Numerical and theoretical studies of water waves and floating bodies

Numerical and Theoretical Studies of

Water Waves and Floating Bodies

Daalen, Edwin Frank George van

Numerical and theoretical studies of water waves and floating bodies Edwin Frank George van Daalen. - [S.L : s.n.]. - Ill., fig., tab. Proefscbrift Enschede. - Met index, lit. opg. - Met samenvatting in

het Nederlands. ISBN 90-9005656-4

Numerical and Theoretical Studies of

Water Waves and Floating Bodies

Proefschrift

ter verkrij ging van

de graad van doctor ann de Universiteit Twente,

op gezag van de rector magnificus

prof.dr. Th.J.A. Popma,

volgens beshiit van het College van Dekanen

in het openbaar te verdedigen op vrijdag 8 januari 1993 te 16.45 nur

door

Edwin FrankGeorge van Daalen geboren op 13 juni 1965 te Amsterdam

PROF.DR.IR. P.J. ZANDBERGEN

Contents

Contents

V

Preface

ix

The Argonautica

xi

i

Introduction

i

V 1.1 Historical background 2

1.2 The investigations: past and present 3

1.3 Dissertation outline 4

1.4 Suggested references 6

1.5 Bibliography 7

I Mathematical Formulation and Numerical Algorithm

11

2

Mathematical Statement of the Problem

13

2.1 Introduction 14

2.2 Force mechanisms in fluid flow 14

2.3 Potential flow 15

2.4 Boundary conditions 19

2.4.1 Impermeable fixed boundaries 20

2.4.2 Impermeable moving boundaries 20

2.4.3 Free boundaries 21

2.4.4 Open boundaries 23

2.5 Wave-body formulations 24

2.5.1 Ship wave-making and wave-resistance 24

2.5.2 Linearized oscillatory motion 25

2.5.3 Forward speed radiation-diffraction 27

2.5.4 Nonlinear motion at zero forward speed 30

2.6 Problem defliiltions 31

3 Boundary Integral Equation Formulations

37

3.1 Introduction 38

3.2 Integral equations in potential theory 39

3.3 Well-posedness: existence and uniqueness 44

3.4 Choice of integral equation method 47

3.5 Bibliography 49

4 Algorithm for Wave-Body Simulations

51

4.1 Introduction 52

4.2 Review of methods for nonlinear ship motions 52 4.3 Basic algorithm for nonlinear water waves 55

4.4 Extension to nonlinear ship motions 58

4.5 Bibliography 63

II

Numerical Results

67

5

Impulsive Wavemaker Motion

69

5.1 Introduction 70

5.2 Nonlinear analysis 72

5.2.1 Governing equations and small time expansions 72

5.2.2 Leading order solutions 74

5.2.3 Initial and small time behaviour 76

5.3 Numerical results 79

5.3.1 Initial behaviour on the wavem.aker 80

5.3.2 Initial behaviour on the bottom 81

5.3.3 Initial behaviour on the free surface 81

5.3.4 Free surface elevation for small time 81

5.4 Concluding remarks 94

5.5 Bibliography 94

6 Hydrodynamic Mass and Damping

97

6.1 Introduction 98

6.2 Mathematical model for two-dimensional motion 99

6.3 Numerical results 102

6.3.1 Circular cylinder in heaving motion 102

6.3.2 Circular cylinder in swaying motion 105

6.4 Concluding remarks 112

6.5 Bibliography 112

7 Cylinders in Free Motion

115

7.1 Introduction 116

7.2 Circular heaving cylinder 116

7.5 Concluding remarks 128

7.6 Bibliography 129

III

Variational Principles and Hamiltonian Formulations

131

8 Lagrangian and Hamiltonian Formulations

133

8.1 Introduction 134

8.2 Luke's variation principle for water waves 135 8.3 Zaicharov & Broer's Harniltonian formulation 139

8.4 Wave-body formulations 145

8.4.1 A variation principle for wave-body interactions 146

8.4.2 A Hamiltonian formulation for wave-body interactions . 150

8.5 Bibliography 156

9 Symmetries and Conservation Laws

161

9.1 Introduction 162

9.2 Conserved densities for water waves 164

9.3 Conservation laws for the wave-body problem 169

9.3.1 Invariants for the two-dimensional case 169

9.3.2 Invariants for the three-dimensional case 177

9.4 The vinal and conservation law no.8 180

9.4.1 The vinal connection 180

9.4.2 The circulation alternative 185

9.4.3 Fluid-filled deformable bodies 186

9.4.4 The 'broken symmetry' argument 187

9.5 Symmetry-breaking boundaries 188

9.6 Numerical validation 191

9.7 Bibliography 198

10 Radiation Boundary Conditions

201

10.1 Introduction 201

10.2 Theory for continuous systems 202

10.3 Application to one-dimensional wave equations 211

10.4 Application to the water-wave problem 230

10.4.1 The Lagnangian-Hanniltonian approach 230

10.4.2 The Eulerian approach 235

10.5 Bibliography 238

IV Appendices

243

A An Expression for

on the Bo4y

B A Body Surface integral Condition

C The Angular Body Momentùm

D. rrogith, Structure

Index

Index, öf Symbols

List of 'igures

List of Tables

AbStrat

Samenvatting(ábstract in Dutch)

About the..author

um vitae in Dutch)

245 251 255 257 259 267 273 277 279

28i

Preface

This thesis was written within the framework of a research project entitled Further development of a three-dimensional model for nonlinear

sur-face waves, with respect to the interaction with floating objects and the influence of boundaries.

As already indicated by the title, one of the main objectives was the extension of

an existing method - developed by Itomate' - to an algorithm for the

numer-ical simulation of noniinear free surface waves in hydrodynamic interaction with

floating bodies; this particular part of the investigations is described here. In the development of the numerical model and the computer code, many problems of both theoretical and practical nature were encountered. From time to time the daily search for 'engineering' solutions was interrupted by periods in which these problems were considered from a more fundamental point of view. This theoretical research - also recorded here - resulted in some basic changes

in the numerical model. In addition, a number of theories concerning water waves and floating bodies was developed.

The investigations were financially supported by the Netherlands Technology Foundation (STW) - a subdivision of the Dutch Foundation for Scientific Re-search (NWO) - under grants TW188.1460 and TWI8O.1460.

Most of the computations were done on supercomputers; from 1989 to 1990 on a CRAY-XMP, and from 1991 on a CILAY-YMP. Computation time was granted by the Dutch Foundation for Supercomputer Facilities (NCF). Tecbnical support

from the Stichting Academisch Rekencentrum Amsterdam (SARA) is

acknowl-edged.

A committee of future users of the computer code accompanied the project.

The committee members were from the Maritime Research Institute

Nether-lands (MARIN), Deift Hydraulics, the Royal Dutch Shell Laboratory Amsterdam (KSLA), the University of Twente, and the Netherlands Tecbnology Foundation.

'Romate, J.E. 1989. The Numerical Simulation of Nonlinear Gravity Waves in Three Di-mensiona using a Higher Order Panel Method. Ph.D. thesis, University of Twente. Enschede, The Netherlands.

I would like to thank the board of directors of MARIN for the unique opportunity

to do this work in such an inspiring research institute.

Thanks go to Hans P. van der Kam (Automation and Instrumentation De-partment) and Hans Zeller (Audio-Visual DeDe-partment) for their enthusiasm and assistance in visualization and video-editing, and to Henk Luisman (Offshore Research Department) for miking the illustrations.

Special thanks go to Ir. René H.M. Huijsmans (Offshore Research Depart-ment) and Ir. Hoyte C. Raven (Ship Research DepartDepart-ment) for introducing me

to the field of ship hydrodynamics.

Also I want to thank Dr. Huib J. de Vriend (Deift Hydraulics) for his interest and participation in this project.

Very special thanks go to Ir. Jan Broeze for his good-fellowship during the past

four years, and for our fruitful cooperation which made our investigations so suc-cessful.

Next, I would like to thank Dr. Johan E. Romate (KSLA) for his invaluable

contributions to this research, both in the past and in the present.

Furthermore, I would like to thank some members of the staff of the Faculty of Applied Mathematics of the tlniversity of Twente.

I am grateful to Dr. Douwe Dijkstra for his helpful comments in 'numerical matters'.

I feel very much indebted to professor van Groesen for his interest and active

participation in this research; our frequent discussions culmnated in four joint papers, which in turn formed a sound basis to part m of this thesis.

It is a great pleasure to express my deep gratitude to professor Zandbergen for his

criticism, advice, and encouragement; it has been a privilege to do this research

under his supervision.

I thank my parents and my brother for their stimulation in my study and work.

Finally, I would like to express my sincere love to my wife Sonja; in the end, her

enormous support and endless patience have helped me to complete this work.

Devenier, The Netherlands ED VAN DAALEN

The Argonaut ica

A narrow escape: the Argo passes through the Syniplegades.

Turning over the pages of this thesis, the interested reader inevitably will en-counter several passages from the Argonauiica, an ancient Greek tale written by

Apolloxüus Rhodius. To those who are familiar with the mathematics and physics described here, each selected fragment may come as a pleasant break while reading

consecutive chapters. Perhaps even more important is the fact that this disser-tation, despite its technical nature, still offers something worthwhile to read to those who are noi familiar with exact sciences.

Several motives induced me to lard this thesis with a selection of passages from the Argonauiica. First of all, the tale of the Argonauts has a pronounced nautical character. In this sense, the Argonautica is in close harmony with the

investigations described here. The second reason is that ever since my first lesson

in classical languages I have felt a natural interest in Greek and Roman litera-ture. This particular part of the literary classics is full of fetching parables and moraiizing legends; immortal gods and mortal heroes are involved in an eternal

struggle for the sympathy of the reader. Finally, in my opinion the Argonautica has unjustly been eclipsed by Homer's well-known Iliad and Odyssee this is the

third motive for this personal mixture of literature and exact sciences.

Cited from the jacket of the Loeb Classical Library publication, with an English translation by R.C. Seaton:

'Apollonius 'of Rhodes' was a Greek grirnmarian and epic poet of

Alexandria in Egypt and lived late in the third century and early in

the second century B.C. While still young he composed his extant epic

poem of four books on the story of the Argonauts. When this work

failed to win acceptance he went to Rhodes where he not only did well

as a rhetorician but also made a success of his epic in a revised form,

for which the R.hodians gave him the 'freedom' of their city; hence his

surname. On returning to Alexandria he recited his poem again, with

applause. In 196 B.C. Ptolemy Epiphanes made him the librarian of the Museum (the University) at Alexandria. His Argonautica is

one of the better minor epics, remarkable for originaiity, powers of

observation, sincere feeling, and depiction of romantic love. His Jason

and Medea are natural and interesting, and did much to inspire Virgil (in a very different setting) in the fourth book of the Aeneid.'

The motive of the voyage of the band of Greek heroes (named Argonauts after their ship, the Argo) is the command of Pellas, king of lolcus, to bring back the 'golden fleece' from Colchis. This command is based on Pellas' desire to destroy the hero Jason, while the divine aid given to Jason results from the intention of the goddess Hera to punish Pelias for bis neglect of the honour due to her.

The first and second books describe the history of the voyage to Coichis, where Apollonius interweaves with his narrative legends, accounts of local customs. This

part of the Argonautica is filled with many exciting episodes, such as the rape of Hylas, the boxing match between Polydeuces and Amycus, the prophecies and

counsels of Phineus, and the passing through the Symplegades. The third book is occupied for the greater part by the episode of the love of Jason and Medea2, and

the accomplishment of Jason's task with her aid. The fourth book, describing

the return voyage, is invaluable for its amazing geography, such as the supposed

junction of the Rhine, Rhone, and Po rivers, the Libyan desert, and the so-called Tritonian Lúe. Each book has its own specific topic and character, and the unity of the legend is that of the voyage itself.

May it please the reader!

Chapter 1

Tnt ro

ion

Beginning with thee, O Phoebus1, I will recount the famous deeds of men of old, who, at the behest of King Pelias, down through the mouth of Pontui and between the Cyanean rocl&, sped well-benched Argo in quest of the golden fleece.

Such was the oracle thai Pelias heard, that a hateful doom awaited

him - to be slain at the prompting of the man whom he could see

coming forth from the people with but one sandal. And no long time after, in accordance with that true report, Jason crossed the stream of wintry Anaurus on foot, and saved one sandal from the mire, but the other he left in the depths held back by the flood. And straightway he

came to Pelias to share the banquet which the king was offering to his

father Poseidon and the rest of the gods, though he paid no honour io Pelasgian Hera. Quickly the king saw him and pondered, and devised

for him the toil of a troublous voyage, in order that on the sea or among

strangers he might lose his home-return.

The ship, as former bards relate, Argus wrought by the guidance of Athena. But now I will tell the lineage and the names of the heroes, and

of the long sea-paths and the deeds they wrought in their wanderings; may the Muses be the inspirers of my song!

Argonautica, Book I, Verses 1-22.

1i.e. Apollo 2i.e. theBlackSea

i. e. the Symplegades

2 _{CHAPTER 1. INTRODUCTION}

1.1

Historical background

Over the past three centuries, surface wave problems have interested a

consider-able number of mathematicians, beginning in the early eighteenth century with Euler and the Bernoullis in Switzerland, and continuing in the late eighteenth and the early nineteenth century with Lagrange, Cauchy, Navier, and Poisson in France. Later the British school of mathematical physicists paid attention to these problems, and notable contributions were made by Airy, Stokes, Rayleigh,

Kelvin, Michell, and Lamb. In the latter part of the nineteenth century the

French once more took up the subject; the work done by Boussinesq in this field

is historic, as well as the Dutch contributions from Korteweg and de Vries. Later, Poincar made excellent contributions with regard to figures of equilibrium of

ro-tating and graviro-tating liquids. One of the most outstanding accomplishments from the purely mathematical point of view - the proof of the existence of pro-gressing waves of finite amplitude - was made by Levi-Civita; the extension of

this proof to waves in a canal of finite depth was accomplished by Struik.4

The subject of surface gravity waves covers a wide range, whether regarded from the viewpoint of physical problems which occur, or from the point of view of the mathematical ideas and methods to solve these problems. The physical problems vary from wave motion over sloping beaches to flood waves in rivers,

the motion of ships in a sea-way, free oscifiations of enclosed bodies of water such

as lakes and harbors, to mention just a few. The mathematical tools employed

comprise the whole of the methods developed in the classical linear mathematical

physics concerned with partial differential equations, as well as a good part of what has been learned about the nonlinear problems of mathematical physics. Thus potential theory and the theories of linear and nonlinear wave equations, together with tools such as conformal mapping and complex variable methods in general, the Laplace and Fourier transform techniques, methods employing a Green's function, integral equations, etc. are used. The nonlinear problems are of both elliptic and hyperbolic type.

Nowadays, the evolution of nonlinear gravity driven water waves interacting with fixed or freely floating objects is an important field of research in ocean

engineer-ing. For large objects with characteristic dimensions of the order of the wave length, viscous effects of the fluid flow can be neglected, as well as effects of compressibility and surface tension. This is the so-called diffraction regime of fluid-structure interaction. Under the additional assumption of irrotational fluid flow, a velocity potential can be introduced to describe the flow characteristics. This potential satisfies a linear field equation, namely Laplace's equation, by continuity. The free surface boundary conditions render the problem nonlinear.

In many cases it is sufficient to linearize the free surface conditions and solve

40f course this historical survey is incomplete; for the classical publications mentioned here the interested reader is referred to the bibliography at the end of this chapter.

the linear problem. However, this applies only to waves with small amplitude compared with the wavelength and the mean water depth. For steep waves this linearization procedure can not be justified and other techniques must be devel-op ed. In this case the mutual interaction of waves due to the nonlinearity can be very strong; this may result, for instance, in the overturning of waves. JI a

floating body is involved, nonlinear effects may have strong influence on the wave

evolution and the body motion; in such a case a linearized approximation would provide inadequate results. In many of these problems higher order approxima-tions to the nonlinear equaapproxima-tions are also inadequate. Typical examples where approximations to the nonlinear equations give unsatisfactory results are:

wave slamming on fixed and floating structures, wave rim-up on slopes,

problems in which a part of the structure is submerged, causing the waves on top of it to break or nearly break.

Of course the assumptions of potential flow are not valid at all stages of the

physical process under consideration, but in such a case the potential solution can be very useful as input for a more general model.

1.2

The investigations: past and present

The main objective of these investigations is the extension of a higher order panel method for nonlinear gravity wave simulations to water waves in hydrodynaxnic interaction with floating bodies.

The major difficulty in solving the nonlinear potential model for this particular problem is the presence of a moving free surface and a freely floating body. Due

to the time dependent free surface conditions both the potential at, and the

position of the free surface are unknown variables, which have to be determined as part of the solution. Similarly, due to the equations of motion for the freely floating body, the potential on the wetted body surface and the body position and orientation are unknown; these variables have to be determined as part of the solution too. The difficulty in solving the nonlinear potential model is also indicated by the fact that this particular problem bas both elliptic and hyperbolic

properties, through the field equation and the free surface conditions respectively.

In the past decades numerous attempts have been made to obtain analytical solutions for general free surface wave problems. Substitution of perturbation

series for the variables into the governing equations and the use of expansions to

sometimes very high order provided many nonlinear approximations. However, analytical solutions of the fully nonlinear equations have not been found so far.

Therefore, it seems that the development of numerical methods is a more promis-ing way towards the solution of these problems.

4 CHAPTER 1. INTRODUCTION

Romate(1989) developed a very efficient boundary element method for

three-dimensional nonlinear gravity wave simulations, using a Green's formulation for

the velocity potential. Higher order approximtions for the singularity dlistribu-tions and the geometry, combined with a robust time stepping scheme, ensured

the accuracy and stability of the aigorithm.. By the end of1988,his panel method

gave excellent results for linear and weakly nonlinear waves.

From 1989 on, manpower was doubled; Broeze and van Daalen continued

Romate's research, aiming to fit his method for highly nonlinear waves and wave

interactions with fixed and freely floating objects. As a first step towards these

goais, a simplified panel method for two-dimensional nonlinear gravity wave

sim-ulations was derived from R.omate's original method. With a number of basic modifications this method was made suitable for highly nonlinear waves and waves interacting with fixed structures. After a period of dose cooperation, it

was decided that Broeze focused his attention on highly nonlinear waves in three dimensions. The extension of the panel method to two-dimensional waves in

hy-drodynamic interaction with freely floating bodies was entrusted to van Daalen; the results of these investigations cover the larger part of this thesis.

During the investigations many engineering problems were encountered,

rang-ing from grid motion control to the development of effective radiation boundary

conditions. If it was felt that these problems called for a more fundamental approach, a reasonable amount of time was spent on gaining insight through

the-oretical considerations. Sometimes these views induced significant adaptations to the numerical algorithm; at other times, ideas were profoundly worked out to

novel theories. The findings of these investigations are embodied in the remaining

part of this thesis.

1.3

Dissertation outline

This thesis comprises a number of numerical and theoretical studies of gravity

driven water waves and the interaction with fixed structures and floating bodies. In spite of the diversity of the material, it is not merely a collection of disconnected

topics, lacking unity and coherence. Considerable effort was made to supply the fundamental background in hydrodynamics - and also some of the mathematics

needed - and to plan this dissertation sudi that a self-contained and readable

whole was arrived at.

This work is split up into four main parts:

Mathematical Formulation and Numerical Algorithm (ch. 2-4)

Numerical Results (ch. 5-7)

TU. Variational Principles and Hamiltonian Formulations (ch. 8-lo)

IV. Appendices (A-D)

Each part (excepting part TV) has been written such that to some extent

separate chapters; an outline will be given next.

Parts I and II reflect the investigations with respect to the development of the mathematical model and the numerical (boundary element) algorithm for water

waves and floating bodies.

In chapter 2 the governing equations for the nonlinear water-wave problem, including the interaction with fixed objects and floating bodies, are presented. The transition from this set of governing equations to a boundary integral equa-tion formulaequa-tion is described in chapter 3. In chapter 4 the numerical algorithm for water waves and floating bodies is outlined, with a short discussion of the discrete (boundary element) approximation of the problem.

Numerical results - obtained with our computer code TJPHYS5

- for an

impulsively started wavemaker are discussed in chapter 5. The numerically com-puted hydrodynaxnic mass and damping coefficients for two-dimensional cylinders in forced harmonic motion have been compared to experimental data and

analyt-ical solutions; the findings of this study are reported in chapter 6. In chapter 7 another validation study for two-dimensional cylinders - but now in free motion

- is presented.

Part ifi is concerned with special descriptions of the hydrodynamic problems considered here. Chapter 8 reviews so-called variation principles and

Hamilto-than formulations for the dassical water-wave problem, novel extensions of these theories to water waves interacting with floating bodies are presented. Chapter 9 is devoted to invariants and conservation laws for wave-body problems. A theory

for radiation boundary conditions - for wave problems that are governed by a Lagrangian principle - that conserve a characteristic density (for instance, the energy density) is presented in chapter 10.

Concluding remarks and recommendations for future research are given in chap-ter 11.

Part IV consists of four appendices containing extensive derivations

(Appen-dices A-C) and a flow-chart of the TIPHYS-code (Appendix D).

The bibliographical footnotes have been borrowed from the New Webster's

Dic-tionary and Thesaurus of the English Language (1991 edition, Lexicon Publica-tions), and from C.B. Boyer's A History of Mathematics (1985, Princeton

Uni-versity Press).

Finally, it is noted that the equations have not been made dimensionless, unless

stated otherwise; ail variables are expressed in SI-units.

5TIPHYS: a ime doman panel method for nonlinear gravity waves and ßoating bodies;

6 CHAPTER 1. INTRODUCTION

1.4

Suggested references

For a general introduction to water waves and wave-body interactions, the reader

is referred to the many books and (review) articles on these subjects; a number

of them will be mentioned hereafter.

All of the basics of fluid dynamics used in this thesis is covered by the works of

Batchelor (1967) and Milne-Thomson (1968). Valuable information regarding the field of hydrodynamics is gathered in the historic work of Lamb (1932), of which almost a third is concerned with surface gravity waves. A mathematically oriented treatment of water waves is given by Stoker (1957). Another important source of information on wave theories is the work of Wehausen and Laitone (1960).

Wind-generated water waves are discussed extensively by Kinsman (1965). Whitham

(1974) discusses waves in a more general context; his work includes a number of chapters on water waves and the use of variational principles. An introduction to the science of wave motions in fluids, with a chapter devoted to water waves only, is given by Lighthill (1978). A very readable text on the dynamics of ocean surface

waves is due to Mei (1983). An introductory presentation of the basic theories of water waves, using direct mathematical techniques, is given by Crapper (1984).

The following reviews on various wave types should also be brought to the

attention of the reader: solitary waves, by Mlles (1980); trapped waves, by Mysak

(1980); wave instabifities, by Yuen and Lake (1980); strongly nonlinear waves,

by Schwartz and Fenton (1982); breaking waves, by Peregrine (1983) and Battjes (1988); tsunamis (i e huge waves caused by large submarine earthquakes or land-slides), by Voit (1987). Numerical methods in free surface flows are reviewed by

Mei (1978) and Yeung (1982). The use of boundary integral equation methods

in inviscid fluid mechanics is discussed by Hess (1990). A survey of Hamiltonian formulations in fluid mechanics is due to Salmon (1988).

Mathematical treatments of the motion of bodies in waves have been given by Landweber (1961) and Wehausen (1971). Newman's (1977) textbook on the hy-drodynamics of marine objects is recommendable. Wave-structure interactions

have been dealt with by Sarpkaya and Isaacson (1981); more recent contributions

with regard to this subject are due to Faltinsen (1990ab). An update of the

theory of floating bodies since John (1949, 1950) is given by Kleinman (1982). Timmin, Hermans, and Hsiao (1985) have written a very readable textbook on

water waves and ship hydrodynamics

For a survey of the different numerical methods used in the solution of free sur-face flow problems, with and without the presence of solid structures or floating

bodies, the proceedings of the International Conferences on Numerical Ship Hy-drodynamics, the proceedings of the Symposia on Naval HyHy-drodynamics, and the

abstracts of the International Workshops on Water Waves and Floating Bodies

concerning fluid-structure interaction, such as potential flows, nonlinear waves, lifting bodies, vortex flows, cavitation, ship wave-making and wave-resistance,

propulsion, boundary layers, and viscous flows.

1.5

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1975...

Proceedings of the Symposia on Naval Hydrodynamics. 1956...

RAYLEIGH, LORD. 1876. On periodical irrotational waves at the surface of deep water. Philosophical Magazine, Series 5, 1:381-389.

ROMATE, J.E. 1989. The Numerical Simulation of Nonlinear Gravity Waves in

Three Dimensions using a Higher Order Panel Metho& Ph.D. thesis, University of Twente. Enschede, The Netherlands.

SARPKAYA, T., AND ISAACSON, M. DE ST. Q. 1981. Mechanics of Wave

Forces on Offshore Structures. Van Nostrand Reinhold.

SALMON, R. 1988. Hamiltonian fluid mechanics. Annual Review of Fluid Me-chanics 20:225-256.

SCHWARTZ, L.W., AND FENTON, J.D. 1982. Strongly nonlinear waves. Annual Review of Fluid Mechanics 14:39-60.

STOKER, J.J. 1957. Water Waves. The Mathematical Theory with Applications. Interscience Publishers.

STOKES, SIR G.G. 1847. On the theory of oscillatory waves. Transactions of the Cambridge Philosophical Society 8(4):441-455.

STRUIK, D.J. 1926. Détermination rigoreuse des ondes irrotationelles périodiques dans un canal à profondeur finie. Mathematische Annalen 95:595-634.

TIMMAN, R., HERMANS, A.J., AND HsIAo, G.C. 1985. Water Waves and

Ship Hydrodynamics. An Introduction. Kiuwer Academic Publishers.

VOIT, S.S. 1987. Tsunamis. Annual Review of Fluid Mechanics 19:217-236. WEHAUSEN, J.V., AND LAITONE, E.V. 1960. Surface Waves. In Handbook of Physics, Volume IX, pp.445-778. Springer-Verlag.

WEHAUSEN, J.V. 1971. The motion of floating bodies. Annual Review of Fluid Mechanics 3:237-268.

WHITHAM, G.B. 1974. Linear and Nonlinear Waves. John Wiley and Sons.

YEUNG, R.W. 1982. Numerical methods in free surface flows. Annual Review

of Fluid Mechanics 14:395-442.

YUEN, H.C., AND LAKE, B.M. 1980. Instabilities of waves on deep water.

10 CHAPTER 1. INTRODUCTION

First then let us name Orpheus whom once Calliope bare, it is said,

wedded to Thracian Oeagrus, near the Pimpleian height. Men say that

he by the music of his songs charmed the stubborn rocks upon the

moun-tains and the course of rivers

Tiphys, son of Hagnias, left the Siphaean people of the Thespians, well skilled to foretell the rising wave on the broad sea, and well skilled

to infer from sun and star the stormy winds and the time for sailing. Tritonian Athena herself urged him to join the band of chiefs, and he came among them a welcome comrade. She herself too fashioned the swift ship; and with her Argus, son of Arestor, wrought it by her

counsels. Wherefore it proved the most excellent of all ships that have

made trial of the sea with oars

Next to him came a scion of the race of divine Danaus, Nauplius. He was the son of Clytonaeus son of Naubolus; Naubolus was son of Lernus; Lernus we know was the son of Proetus son of Natzplius; and once Amymone daughter of Danaus, wedded to Poseidon, bare

Nauplius, who surpassed all men in naval skill

After them from Taenarus came Euphemus whom, most swift-footed

of men, Europe, daughter of mighty Tuyos, bare to Poseidon. He was wont to skim the swell of the grey sea, and wetted not his swift feet,

but just dipping the tips of his toes was borne on the watery path Yea, and two other sans of Poseidon came; one Erginus, who left

the citadel of glorious Miletus, the other proud Ancaeus, who left

Par-thenia, the seat of Imbrasion Hera; both boasted their skill in sea-craft and in war

So many then were the helpers who assembled to join the son of Aeson6. All the chiefs the dwellers thereabout called Minyae, for the

most and the bravest avowed that they were sprung from the blood of the daughters of Minyas; thus Jason himself was the son of Alcimede

who was born of Clymene the daughter of Minyas7.

Now when all things had been made ready by the thralls, all things that fully-equipped ships are furnished withal when men's business leads

them to voyage across the sea, then the heroes took their way through the city to the ship where it lay on the strand that men call Magnesian

Pagasae

Argonautica, Book I, Fragments from Verses 23-238.

6i.e. Jason

TMinyas was a son of Acolus, who was a son of Zeus; hence, Jason was a descendant of the

Part I

Mathematical Formulation

and Numerical Algorithm

Chapter 2

Mathematical Statement of

the Problem

. .And they heaped their garments, one upon the other, on a smooth

stone, which the sea did not strike with its waves, but the stormy surge

had cleansed it long before. First of all, by the command of Argus,

they strongly girded the ship with a rope well twisted within, stretching

it tight on each side, in order that the planks might be well compacted by the bolts and might withstand the opposing force of the surge. And they quickly dug a trench as wide as the space the ship covered, and at

the prow as far into the sea as it would run when drawn down by their

hands. And they ever dug deeper in front of the stem, and in the furrow

laid polished rollers; and inclined the ship down upon the first rollers, that so she might glide and be borne on by them. And above, on both sides, reversing the oars, they fastened them round the thole-pins, so as to project a cubit's space. And the heroes themselves stood on both

sides ai the oars in a row, and pushed forward with chest and hand at

once. And then Tiphys leapt on board to urge the youths to push at the right moment; and calling on them he shouted loudly; and they at once,

leaning with all their strength, with one push started the ship from her

place, and strained with their feet, forcing her onward; and Pelian Argo followed swiftly; and they on each side shouted as they rushed on. And then the rollers groaned under the sturdy keel as they were chafed, and

round them rose up a dark smoke owing to the weight, and she glided

into the sea; but the heroes stood there and kept dragging her back as she sped onward. And round the thole-pins they fitted the oars, and in the ship they placed the mast and the well-made sails and the stores.

Argonautica, Book I, Verses 364-393.

2.1

Introduction

In this chapter the complete set of governing equations for the motion of an ideal

fluid with a free surface is presented. These equations include the field equation for the interior fluid flow and the conditions on all physical and artificial bound-aries. The additional equations describing the hydrodynamic interaction with floating bodies, either partially or totally submerged, are also presented. These

equations comprise the appropriate boundary conditions on the body surface and

the equations of motion for the body.

In order to give full account for assumptions that would have been made

tacitly otherwise, the present derivation of the governing equations is rather ex-tensive. In section 2.2 we discuss assumptions with respect to various types of

forces which are active in fluid flow. Then, the set of equations describing poten-tial flow is derived in section 2.3. In section 2.4 the conditions for several types of

boundaries are presented. The governing equations for a number of formulations

describing three-dimensional wave-body interactions are discussed in section 2.5. Finally, formal definitions of the particular water-wave and wave-body problems

treated in this thesis are given in section 2.6.

2.2

Force mechanisms in fluid flow

The motion of a fluid, like the motion of rigid bodies, is governed by opposing actions of different forces. In fluid dynamics, these forces are distributed con-tinuously throughout a volume filled with infinitesimal fluid particles. One may

distinguish force mechanisms associated with the fluid inertia, the fluid weight, the fluid viscosity, and other (secondary) effects such as surface tension.1 In order to analyze the three principal force mechanisms _{- inertial, gravitational,} and viscous - it is useful to estimate the orders of magnitude. Suppose that the

problem under consideration is characterized by a physical length L, a velocity U,

a fluid density p, a gravitational acceleration g, and a dynamic fluid viscosity u. An order estimation of the force magnitudes is given in the table below.

Table 2.1: Order estimation of force magnitudes.

'In section 8.2 we show that surface tension effects are negligible for ali_{waves but the shortest}

or rippIe waves.

Type of Force Symbol Order of Magnitude

Inertial Gravitational Viscous F, F9 Pv pU2L2 pgL3 iUL

2.3. POTENTL4L FLOW 15

These order estimates merely indicate how changes in any of the physical

param-eters L, U, p, g, or ¡L affect the balance of the various force mechanisms. For

instance, doubling the length scale L corresponds to multiplicative factors of 22,

2, and 21 for the inertial, gravitational, and viscous forces respectively. The

above considerations are useful not only in predicting full-scale phenomena from

tests with a scale model, but they also indicate which effects can be neglected in the mathematical model of the problem under consideration.

In order to gain insight in the relative magnitudes of the various forces, three nondimensional parameters are introduced to describe the fluid flow:

F

pU2L2

= U2/gL

-

pgL3

= pUL/,

(2.2)

= pgL/jLU . (2.3)

From any two of these ratios the third can be calculated. Hence, any pair of these

ratios determines the balance of forces in the fluid motion. Usually, the first two are used to define the characteristic numbers

Fr

U (Froude number) , (2.4)

(gL)

pUL UL

Re =

= - (Reynolds number) ,

(2.5)

where w = ¡1/pis the kinematic fluid viscosity; common values of w are 10_6 m2/s

for water and 1.5 X 10 m2/s for air. For typical values of the characteristic

velocity and the length scale, say U = i rn/s and L = 10 ni, this implies that the Reynolds number Re will be large, and hence that viscous forces will be small compared to inertial forces. Therefore, effects of viscosity can be neglected for the bulk of the fluid. However, in special regions - such as the boundary layer2 very close to a body - viscosity should be induded.

2.3

Potential flow

Assuming the laws of dassical mechanics and thermodynamics to apply, the mo-tion of a fluid can be described by a set of partial differential equamo-tions expressing conservation of mass, conservation of momentum, and conservation of energy per unit volume of the fluid. If necessary, this set of equations is complemented with

an equation of state.

(2.1)

2For a detailed discussion of skip boundary layers we refer to the review article written by Landweber and Patel (1979).

Let {,

e} denote a Cartesian3 coordinate system fixed in space, with cor-responding spatial coordinates (z, y, z), and let i = (u, y, w)T denote the fluid velocity field. Following the motion of an infinitesimal control volume 6V, the equation of mass conservation - or continuity equation - reads

D6V

[p5V]=5V+p

_Dt

=0,

where D/Dt denotes the material derivative (i.e. the substantial derivative or the total time derivative) and p

_{= p}

(z, y, z; t) is the local time dependent fluid

density.

The control volume is subject to changes due to strain only; hence, in terms

of the velocity field the material derivative of the control volume is given by

D6V IOu 0v Ow\

Dt

=++---)6V=(V.v')6V

where V represents the three-dimensional gradient-operator:

(o O

0\T

-

_{uz ay az}

With (2.7) the equation of mass conservation (2.6) can be simplified to

+ p(V . 11) = O

Then, assuming the fluid to be absolutely incompressible and homogeneous, we

have p = constant throughout the fluid dontiin, and the continuity equation

reduces to

Vii=O or

diviT=0 , (2.10)

expressing that the velocity field is free of strain (or divergence-free).

Next, for the control volume 5V, the equation of momentum conservation reads

[pii] = P+ P9 + P (2.11)

where we have dropped the incompressibility assumption. This equation ex-presses the change of momentum due to forces acting on the control volume.

These forces are, in general, inertial (pressure) forces (Í), gravity forces (1),

(2.6)

3Descartes, René (1596-1650), French philosopher, physicist and mathematician. He founded

the science of analytical geometry ('la Géometrie', 1637) and discovered the laws of geometric optics. In 'Discours de la Méthode' (1637) he divests himself of all previously held beliefs, to rebuild on his own basis of certitude, i.e. the fact of his self-conscious existence: 'dubito ergo cogito: cogito ergo sum' (I doubt, therefore I think: I think, therefore I am).

(2.7)

(2.8)

2.3. POTENTIAL FLOW 17

and frictional forces due to viscosity (x,). Substitution of explicit expressions for

these different forces yields the well-known Navier-Stokes equations: 0V . Dp Du Op + = +

(vU

+

)

, (2.12)

0V.il\

Dp Dv + = +

(vv

+

)

, (2.13) Dp Dw Op 1 2 8V.iT\

-W + p-- = -- + ¡L ,V w +

₎

pg , (2.14)

where p denotes the pressure, ji is the uniform dynamic fluid viscosity, g is the gravitational acceleration (acting downwards along the z-axis), and V2 is the

Laplace-operator:4

2==

_(++I)

Under the assumption of incompressibility, the above set of equations reduces to

the Navier-Stokes equations for incompressible fluid flow:

Du Ou

lOp

₂

= - + v Vu = --- + vV u ,

(2.16) Dt Ot

pôz

=

+ il Vv = -

_{+ z'V2v,} (2.17) Dt

at

_pay

Dw aw

l0

2

-m- =

-- + y .

Vw

= --j--

+ ¡'V w - g,

(2.18)

where z' = /1/p denotes the kinematic viscosity of the fluid. If z' 0, these equations simplify to the Euler5 equations for incompressible and inviscid fluid flow:

Dii Oil 1

(2.19)

In general, a fluid flow problem is characterized by a pressure p, a density p, a

temperature T, a viscosity ¡z, and a velocity ii = (u, y, w). For an incompressible,

homogeneous, and inviscid fluid with uniform temperature, p and T are constant

and ,i equals zero, and only four equations are needed to solve the flow problem.

Summarizing, it can be stated that the motion of an ideal and isothermal fluid is governed by the continuity equation (2.10) and the three Euler equations (2.19).

(2.15)

4Laplace, Pierre Simon, Marquis de (1749-1827), French physicist and astronomer. With Lavoisier he made the first determination of the coefficient of expansion of a metal rod, and initiated the study of thermochemistry.

5Euler, Leonhard (1707-1783), Swiss mathematician. He was responsible for the revision of nearly ail the branches of puie mathematics then known and the foundation of new methods of analysis. The mathematical symbols , e, and i were introduced by Euler; his celebrated

equality e'T + 1 = O combines the five most important numbers in mathematics in a single

Next, consider the fluid vorticity

(8w 6v Ou Ou, 6v 8u\

c=Vxii or

(c1,w,w)=

. (2.20)

Assuming that the fluid flow is irrotational at some time t = to, then it follows

from the curl of equation (2.19) that the flow wifi remain irrotational at all times:

= for t to . (2.21)

This condusion is important because an irrotational vector field can be

repre-sented as the gradient of a scalar field.6 Thus we are allowed to introduce a

potential for the velocity field:

i= Vq

. (2.22)

With (2.22) the continuity equation (2.10) becomes

V2q=0

+(VV)+gz= 7','°+B(t)

(2.23)

i.e. Laplace's equation for the velocity potential _4.

With (2.22) the Euler equations (2.19) can be integrated to a general form of

Bernoulli's7 equation:

(2.24)

where Po IS a constant reference pressure and B (t) is an arbitrary function of time. When, as in the cases which will be considered, the boundary conditions are kinematical, the solution process consists in finding a harmonic potential satisfying (2.24) and the prescribed boundary conditions. The pressure p is then indeterminate to the extent of an additive function of t. It becomes determinate when the value of p at some point of the fluid is given for all values of t. Since the term B (t) has no influence on resultant pressures, it is frequently omitted.8

6This is a direct result of Hdmholts' theorem in vector analysis, which states thatany

continuous and finite vector field can be expressed as the sum of the gradient of a scalar function and the curl of a sero-divergence vector. The divergence-free vector vanishes if the original vector

field is irrotationaL The proof of this theorem can be found in Morse and Feshbach (1953). 7Beznoulli, Swiss family of Dutch extraction, several members of which made distinguished contributions to physics and mathematics. Jacques (1654-1705) worked on analytical geometry, his brother Jean (1667-1748) discovered the exponential calculus and a method of integrating rational functions, and Jean's son Daniel (1700-1782) developed the kinetic theory of gases.

8Suppose, for instance, that a solid body moves through a fluid completely endosed by fixed boundaries, and that it is possible - say, by means of a piston to apply an arbitrary pressure at some point of the boundary. Whatever variations arc made in the magnitude of the force applied to the piston, the motion of both the fluid and the solid will be absolutely unaffected, since at all points the pressure will instantaneously rise or fail by equal amounts. Physically, the origin of this paradox is that the fluid is treated as absolutely incompressible. In actual fluids changes of pressure propagate with very great, but not infinite, velocity.

2.4. BOUNDARY CONDITIONS 19 Thus it has been shown that under the assumptions of an incompressible, ho-mogeneous, and inviscid (i.e. ideal) fluid and in the absence of vorticity (i.e. an irrotational flow), the motion of the fluid is governed by Laplace's equation for the velocity potential and the nonlinear Bernoulli equation. JI the potential is known throughout the fluid domain, the velocity field ii and the pressure field p

are easily obtained from (2.22) and (2.24) respectively. From these quantities the

potential and kinetic energies can be determined, due to the absence of friction.

2.4

Boundary conditions

To complete the set of governing equations, initial conditions are needed, as well as boundary conditions for the elliptic field equation (2.23). The initial conditions

depend on the specific problem under consideration, and are incorporated in the

final problem specifications. Obviously, the initial conditions must be compatible

with the governing equations, in order to obtain a well-posed problem.

Figure 2.1: Fluid domain 1 and bounding surfaces.

The boundary conditions depend on the type of boundary under consideration. In general, we have a certain amount of fluid occupying a simply connected9

transient domain 1 (t), see Figure 2.1. The problem of gravity driven water waves

9A region such that any simple closed curve can be shrunk to a point without leaving the region is called simply (or singly) connected.

introduces a free surface F, which is one part of the domain boundary OfI (t).

Other parts are, for instance, the bottom B and the hull of a ship S. These

and other types of boundaries and the corresponding conditions are discussed

hereafter.

2.4.1

Impermeable fixed boundaries

In most practical con.figurations, at least one fixed boundary is present; the

bot-tom - denoted by B in Figure 2.1 - which is not necessarily even. If such a

fixed boundary is impermeable, then fluid partides can not penetrate it; hence, the normal component of the fluid velocity must vanish there:

vn = ii. O (2.25)

Here is the unit normal vector along Ofl J B, pointing outwards.

Under the assumptions of potential flow, this so-called zero-flux condition can

be reformulated to

= Vç5.= O

On

stating that the normal derivative of the velocity potential vanishes.

(2.26)

2.4.2

Impermeable moving boundaries

The motion of fluid particles in the neighbourhood of impermeable moving

bound-aries - such as a wavemaker or the hull of a ship - is influenced by the boundary velocity. Effects of viscosity can be taken into account by introducing a

bound-ary layer, wherein stress forces ixLfluence the fluid motion. However, assuming

the fluid to be inviscid, the motion of a fluid partide on an impermeable moving

boundary with velocity V satisfies

v=ti.ñ=V.il

, (2.27)

expressing that the normal velocity of the fluid particle and the normal boundary velocity are equal.

In case of potential flow, this condition reads

(2.28)

Of course condition (2.26) for impermeable fixed boundaries is a special case of

2.4. BOUNDARY CONDITIONS 21

2.4.3

Free boundaries

The free surface - denoted by F in Figure 2.1 - differs from other physical

boundaries since it is a free boundary indeed; not only the potential must be determined there, but also the position of the free surface itseif. Therefore, it is obvious that two boundary conditions are needed for a proper mathematical

description of the free surface.

In the absence of surface tension, the pressure at the free surface must equal

the atmospheric pressure Po; the first condition is then obtained from Bernoulli's

equation (2.24) by substitution of p = Po = O and by the choice B (t) = O:

-

= _{+ (V}

_{. V) + gz = O ,}

_(2.29)

where z = O corresponds to the mean water level. Condition (2.29) is known as

the dynamic free surface condition, since it has been deduced from a momentum equation, i.e. a balance of forces.

In Lagrangian notation (2.29) reads

(Vç.Vq)gz

Dt - 2

(2.30)

The second condition concerns the free surface velocity; fluid particles at the free

surface bave the property that they remain part of the free surface until they

encounter a boundary of another type. Following such a free surface particle, this implies that its transient position = is determined by the Lagrangian

expression

DF

-.

(2.31)

which is known as the kinematic free surface condition, since it is concerned with the velocity field only.

In many applications the free surface elevation can be assumed to be a

single-valued function of the horizontal coordinates and the time. In these cases the free surface is described in terms of its elevation i by

F (a;, y, z; t) z - i (z, y; t) = O . (2.32)

The requirement that the material derivative of z

-

vanishes on the free surface then gives

077 077 077

+u+vw

Ot 0e Oy

or, in terms of the elevation i and the potential :

Oli 8T8Ç6 ₊

Condition (2.31) is usually preferred to (2.34), since the latter condition exdudes phenomena as overturning waves.

For waves with small amplitude A compared to the wavelength ) and the mean water depth h, the free surface conditions (2.29) and (2.34) can be linearized about the mean water level:

çbt = _at

z=O,

11t = z ₎ w ej - = (g/k)"2 dw 1 _1/2 1

c9=(g/k) =cj.

(2.35)

where a suffix denotes partial differentiation, i.e. gg = Oç/8t etc. The second

approximate boundary condition states that the vertical velocity of a free surface particle coincides with the vertical velocity of the free surface itself, thus ignoring the small horizontal deviations.

In case of a horizontal bottom, the zero-flux condition (2.26) gives

= O at

z = h .

(2.36)

A two-dimensional periodic solution satisfying Laplace's equation (2.23), the

lin-earized free surface conditions (2.35), and the bottom condition (2.36) is given by the velocity potential

w coshk(z+h)

qi (z, z; t) = A

k kh cos (kz - wt) , (2.37)

and the corresponding free surface elevation

, (z; t) = A sin (ka - wt) , (2.38)

where w is the wave frequency and k = 2ir/) is the wave number.

Substitution of (2.37-2.38) into (2.35) yields the first order dispersion relation for Stokes waves:

w2 = gktanhkh . (2.39)

For finite-amplitude waves on deep water we have kh -* 00 and consequently

w2 = gk. Hence, for deep water waves the phase velocity ej and the group

velocity Cg are given by

(2.40)

For finite-amplitude waves on shallow water we have kh -. O, and as a result w2 = ghk2. It follows that for shallow water waves e1 and Cg are equal and

independent of the wave number k:

2.4. BOUNDARY CONDITIONS 23

2.4.4

Open boundaries

In general, the fluid domAin f extends to infinity in the horizontal directions.

lu this case a so-called radiation condition at infinity is required to obtain a

uniquely solvable problem. This condition - based on conservation of energy - states that the waves should behave at infinity like progressing waves moving

away from the source of the disturbance. In problems concerning electromagnetic wave propagation this condition is known as the Sommerfeld radiation condition, see Sommerfeld (1964) or Stoker (1957).

With regard to the water-wave problem, this radiation condition states that the solution corresponds to outgoing waves only. However, for computational

reasons (limited cpu-time and computer memory) it will be necessary to truncate

the fluid domain at some distance from the area of interest. The artificial (i.e.

non-physical) boundaries thus introduced are part of the bounding surface Oíl. To obtain a well-posed problem, appropriate conditions are needed on these bound-aries.

There are two main criteria for a radiation condition on an artificial boundary. First of all, the condition should yield a well-posed problem. This boils down to the requirement that the solution to the problem exists, is unique, and depends continuously upon the initial and boundary conditions, i.e. well-posed in the

sense of Hadamard (1923). Secondly, the radiation condition should simulate the behaviour of the exduded domain as well as possible; the solution should

approx-imate the solution that one would have obtained if the boundaries would have

been chosen at infinity. With regard to the free surface wave problem, this comes

down to the requirement that radiated surface waves approaching an artificial boundary should be fully transmitted or 'absorbed'. In this sense, the radiation condition should provide an opefl boundary.

Romate (1992) has reviewed methods for the numerical simulation of open boundaries for linear and nonlinear water waves. On the basis of his literature survey, he decided to use Higdon's (1987) first- and second-order partial differ-ential equations as absorbing boundary conditions for the linearized model; the

question of well-posedness was addressed too. The numerical implementation in his panel method for linear free surface wave simulations and the stability of these

first- and second-order absorbing boundary conditions have been discussed - in a subsequent paper - by Broeze and Romate (1992).

More recently, van Daalen, Broeze, and van Groesen (1992) introduced an analytical technique for the development of radiation boundary conditions for general wave systems. At the outset of a variational principle for the wave prob-lem under consideration, additional boundary conditions are derived such that a characteristic density (for instance, the energy density) is conserved. In chapter 10 this theory is presented, and applications to a number of wave problems -including the three-dimensional nonlinear water-wave problem - are discussed.

2.5

Wave-body formulations

The presence of a floating body, either partly or totally submerged, implies that extra equations are needed for a correct mathematical description of the

fluid-body interaction. Usually, a second coordinate system {i', f2', e31} is introduced,

which moves with the body, see Figure 2.2. The origin of this coordinate system

is located at some fixed point inside the body - for example, the centre of mass G

- and the unit vectors are chosen along the body principal axes of inertia. The wetted (submerged) part of the body surface is denoted by S.

Figure 2.2: Free surface with a floating body.

Generally, different assumptions on the motion of the body and the free surface lead to different sets of governing equations. To illustrate this dependence, four

frequently used wave-body formulations are discussed next.

2.5.1

Ship wave-making and wave-resistance

Ever since Kelvin (1886) analyzed the potential waves generated by a pressure

point moving with constant velocity in otherwise calm water and Michell (1898)

derived the potential wave-resistance of a thin ship, much interest has been paid to theoretical (i.e. analytical) ship wave-resistance calculations. With the

devel-opment of fast computers this problem has also been addressed numerically.

Consider a body moving at a constant speed in otherwise calm water. For this particular system a stationary problem can be formulated, using a moving frame of reference fixed to the body. Assuming that the uniform iiiflow velocity

is given by

2.5. WAVE-BODY FORMULATIONS 25

the following steady-state formulation is obtained:

V2q5=O in ( , (2.43)

(Vt. Vq5) + gz = U2 on

F ,

(2.44)

= O on

F and

S ,

(2.45)

Vq5 = for

r

(2

+

- oo

(2.46)

Condition (2.44) is the dynamic free surface condition, and is easily obtained by application of Bernoulli's theorem to a free surface streamline extending to infinity in horizontal direction. The kinematic condition (2.45) expresses that both the free surface flow and the flow around the body are stationary.

Linearization of the free surface conditions about the mean water level z = O

and about the uniform flow Vq5 = i results in the so-called Neumann-Kelvin

problem. This problem is usually solved with integral equation techniques; in the

'Havelock-source' (or 'Kelvin-source') approach the Green's function satisfies ail boundary conditions except the hull boundary condition, while in the 'Rankine-source' approach the Green's function satisfies none of the boundary conditions. Another solution procedure is the linearization about the flow at zero Froude number. Successful computations for this so-called slow ship wave-making and

wave-resistance problem have been initiated by Gadd (1976) and Dawson (1977),

followed by Sciavounos and Nakos (1988), R.aven (1988), and many others. In recent literature a shift towards solution methods for the nonlinear problem is observed, see Ni (1987), Jensen, Bertram, and Söding (1989), Campana, Lau,

and Bulgarelli (1989), and Raven (1992).

2.5.2

Linearized oscillatory motion

If the body motion is time-harmonic and non-translatory, the amplitude of the motion can be used to linearize the governing equations. This has the important

advantage that the free surface grid is fixed in time, thus simplifying a numerical

solution procedure considerably. Introducing the body frequency a by

ç(z,y,z;t) = Re [(z,y,z)e_1t]

, (2.47)

V,, (z, y, z; t) = Re [V,. (a, y, z) e1't] , (2.48)

where the body velocity V,, is imposed, the governing equations can be Fourier'0

transformed in time to obtain a much simpler frequency domain problem:

'°Fourier, Jean Baptiste Joseph, Baron de (1768-1830), French physicist, mathematician and politician, important for his theorem that any periodic function may be resolved into sine and

V2ji=0

in (1 , (2.49)

on

P

, (2.50)

on , (2.51)

where P and denote the mean positions of the free surface and the wetted body

surface respectively. Condition (2.50) is obtained by substitution of (2.47-2.48)

into the (combined) linearized free surface conditions (2.35).

In order to render this problem uniquely solvable, Sommerfeld's radiation

condition at infinity must be imposed:

11m (kr)hI4'2

- ik

= o ,

r =

(z2+ , (2.52)

rtoc

\Or

_j

where k is the wave number. Mel (1978) reviews two approaches towards the solu-tion of this so-called harmonic radiasolu-tion problem: hybrid (coupled finite element

and boundary element) methods and integral equation methods.

The use of integral equation methods in the study of wave fields goes back to Lamb (1932), who examined the scattering of linear waves by a surface piercing body. The potential of the wave field can be described in terms of a source

distri-bution on the mean wetted part of the body surface, where the Green's function

satisfies Laplace's equation, the linearized free surface conditions, and the

radia-tion condiradia-tion at infinity. Green's funcradia-tions of this type are rather complicated and have been developed in the fifties by John (1950) and Stoker (1957); for a

more recent discussion, see Noblesse (1983). The corresponding integral equation

methods have been employed by, for instance, Brown, Eatock Taylor, and Patel (1983), and Nestegard and Sclavounos (1984) in the frequency domain, and by

Adachi and Ohmatsu (1980), Beck and Liapis (1987), and Newman (1985) in the time domain

Unfortunately, this method fails to give a unique solution at the so-called

irregular frequencies - which are the eigen.&equencies of the interior Diricblet11

problem of the body - even though the solution of the original boundary value

problem is unique. Much effort has been spent to solve this problem; Sciavounos and Lee (1985) give a short survey of methods for the removal of irregular

frequen-cies. For further results on this (non-physical) phenomenon, see Ursell (1981),

Martin (1981), Hulme (1983), and Forbes (1984). The question of well-posedness

for this specific problem was addressed by John (1950); more general existence and uniqueness proofs are given by Lenoir and Martin (1981), and Simon and

Ursell (1984).

"Dirichiet, Peter Gustav Lejeune (1805-1859), German mathematician who contributed to the theory of numbers and to the establishment of Fourier's theorem.

2.5. WAVE-BODY FORMTJLATIONS 27 Another way to treat this problem is to employ a simpler Green's function, which

only satisfies the boundary condition on the bottom. In this case singularity

distributions are needed on all other boundaries. Angeli, Hsiao, and Kleinmrn (1986) show that for a restricted class of threedimensional body geometries

-this formulation has no irregular frequencies, provided that the original boundary

value problem is uniquely solvable. Liu (1991) proved in a different setting -that the same integral equation for the two-dimensional problem does not suffer

from irregular frequencies either.

2.5.3

Forward speed radiation-diffraction

Another important problem in offshore technology is the slow drift motion of

a floating marine structure, such as a moored ship or a moored oil platform.

The motion is generated by resonance between the moored structure and slowly oscillating waves, and may result in very large horizontal displacements. The viscous damping and the wave (radiation) damping are often small; in many sea states the so-called wave drift damping - defined as the increase in wave drift forces due to a small forward velocity of a body moving in waves - may be the

dominant damping effect.

The concept of wave drift damping has been introduced by Wichers and Sluijs (1979). It has been discussed further by Wichers and Huijsmans (1984), and

Wichers (1988). An extensive study - among many other contributions - of

wave drift forces was published by Pinkster (1980).

Consider a body moving horizontally with constant forward speed U in response

to incoming regular waves with small amplitude-wavelength ratio A/A. Let the reference frame be fixed to the body, with the undisturbed free surface in the

(z, y)-plane, the z-axis in the direction of forward motion, and the z-axis vertically

upwards. In this frame the body performs small oscillations due to the incoming

waves and is embedded in a uniform current with speed U along the z-axis.

Assuming the finid to be ideal and of infinite extent in the lower half.space, and the flow to be free of vorticity, and neglecting effects of viscosity and surface tension, then there exists a velocity potential . that satisfies Laplace's equation:

= O . (2.53)

This potential can be split up as follows:

(2.54)

The first part 4i8 represents the steady flow and is independent of time. The

unsteady parts ç6 and ç axe time-harmonic with frequency of encounter o,

determined by the angle of incidence /3 and the orbital frequency c of the incoming waves: