Theoretical and experimental hydroelastic analysis of high speed vessels

OLE ANDREAS HERMUNDSTAD

THEORETICAL AND EXPERIMENTAL

HYDROELASTIC ANALYSIS OF

HIGH SPEED VESSELS j

TECHNISCHE BNIVERSITEIT

Scheepshydromechanica

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UNIVERSITETET I TRONDHEIM

NTH.Tryld( 1996

THEORETICAL AND EXPERIMENTAL

HYDROELASTIC ANALYSIS

OF

HIGH SPEED VESSELS

Dr.Ing. Thesis

by

Ole Andreas Hermundstad

Department of Marine Structures

ABSTRACT

A method for hydroelastic analysis of high speed monohulls and catamarans, subjected

to continuous wave loads, is presented. The flexibility of the vessel is taken into account by using a number of eigenmodes. in addition to the six rigid body modes, asgeneralized

degrees of freedom. These eigenmodes are obtained from a free vibration analysis of the "dry" structure: based on a finite element formulation.

The fluid forces are calculated by generalizing the linear 21-dimensional method of

Faltinsen and Zhao (1991b), so that the additional modes can be included.

The

2-dimensional formulation is computationally efficient, and it incorporates the important wave systems that are generated by high speed vessels.

Theoretical predictions are compared with experimental results for a flexible catamaran

model in regular and irregular longcrested waves. Head, bow and beam seas are studied.

Vertical shear forces and vertical bending moments in the hulls and the cross structure, in

addition to heave, pitch and roll motions, are investigated.

The comparisons indicate that a modal approach is well suited for the prediction of wave-induced structural responses in high speed vessels. Both quasi-static responses and

resonant vibrations (springing) were generally quite wellpredicted by the theory. Resonant

heave and pitch motions however, were overpredicted. The disturbance of the steady flow

around the ship is neglected in the unsteady problem, and this seems to be an important reason for these overpredictions. Some discrepancies may also have occurred due to hy-drodynamic interactions between the two hulls and three-dimensional effects at the stern; two other phenomena that are not included in the theory. The experiments demonstrate that the structural responses may be nonlinear even if the ship motions behave linearly.

ACKNOWLEDGEMENTS

This work has been carried out under the supervision of Professor Torgeir Moan at the Department of Marine Structures, the Norwegian Institute of Technology. His guidance and support are gratefully acknowledged.

I am indebted to Dr. MingKang Wu for his many productive ideas and helpful advice during the theoretical part of this study. The experiments were skillfully administered by Dr. Jan Vidar Aarsnes (MARINTEK). Working with him has been very inspiring and

instructive.

Advice and comments from Professor Odd M. Faltinsen (Dept. Marine Hydrodynamics)

are deeply appreciated. I am also grateful for the fruitful discussions with my friends and colleagues at the Departments of Marine Structures and Marine Hydrodynamics, and at MARINTEK. Special thanks to Dr. Tore Ulstein (MARINTEK) for valuable comments

on the manuscript, and to Dr. Jan Roger Hoff for his expert assistance with various

computer-related matters.

The initial part of this work was carried out at the Department of Naval Architecture and Offshore Engineering at the University of California at Berkeley. I wish to thank the faculty and staff at the NAOE department, Professor William C. Webster in particular, for a stimulating and memorable stay.

Sincere thanks to Professor Jorgen Juncher Jensen and Dr. Arne Nesteeard for their willingness to serve as members of the doctoral committee.

This work was made possible by a research grant from the Research Council of Norway.

Det Norske Veritas Research provided financial support during my stay at UC Berkeley,

while MARINTEK and the Faculty of Marine Technology have funded the final part of my studies. All parties are sincerely acknowledged.

Contents

ABSTRACT

ACKNOWLEDGEMENTS

NOMENCLATURE ix General Roman symbols

Greek symbols xii

Mathematical operators and other symbols xiii

Abbreviations xiii

1

2

INTRODUCTION

1.1 Background and objective

1.9 Scope and limitations

1.3 Previous work and present method

1.4 Organization

THEORY

2.1 Introduction

2.9 Equations of motion for the ship structure

2.9.1 Governing field equations

2.2.2 Finite element formulation

9.9.3 Modal analysis

9.9.4 Internal hull damping

2.2.3 Rigid body modes and coordinate system

2.2.6 External forces

2.3 Formulation of the hydrodynamic boundary value problem

2.3.1 Governing field equations

2.3.2 General hydroelastic formulation

2.3.3 Simplified formulation for slender hulls

2.4 Solution of the hydrodynamic boundary value problem

2.4.1 The steady problem

2.4.2 The unsteady problem

1 1 2 3 5 7 7 8 8 9 10 11 19 13 13 14 19 23 23 25 . . . . . .

. .... .

. .... . . -. . . .

VI CONTENTS

2,.5 Generalized fluid and gravity forces .. 26

2.5.1 Generalized fluid forces . . 26

2.5.2 Generalized gravity forces' 28

2.6 Response calculations I 30

2.6.1 Equations of steady and unsteady motion, 30

2.6.2 Total responses 4 .0, d 31

2.71 Practical implementation of the method

H,

33

2.7.1 Overview of computer programs 33

2.7.2 Generalized hydrodynamic calculations . 33

2.7.3 Simplified representation of mode shapes 35

2.8 Verification of the computer programs . F 38

3 EXPERIMENTS

39

3.1 Introduction . . ,.. . ,.

..,..

_. a.- TN, :, .4 r 4 :, 1.i ... r 2 39

3.2 Physical model . . .. .... . ,. . 1)..I . C. 1.4,. , ,,,n A 40

3.2.1 Model type . . . . V 40

3.2.2 Model description . , ... 4T

3.3

Test program ...

.. a n.

, ,

3! , 45

3.3.1 'Tests in air .. , 45

3.3.2 Tests in water . 467

4 RESULTS AND DISCUSSION

491

4.1 Introduction n. o o 01.. %v " 49

4.2 Numerical models 6.n. Li , 491

4.2.1 Finite element model . _'7:','

..1 49

4.2.2 Hydrodynamic grid .. 52

4.3 Convergence studies .= , :Th 54

4.3.1 Parameters and response quantities 54

4.3.2 Convergence with respect to modes . . 56

4.3.3 Convergence with respect to stations . 57

4.4 Sensitivity studies , 58

4.4.1 Parameters and response quantities . .. . , ., . ,,... .. , .. 58

4.4.2 Draft and trim 59

4.4.3 Center of gravity and moments of inertia '601

4.4.4 Structural damping 4. 61

4.4.5 Speed

, -

A, t 63

4.4.6 Heading . ., .

, , ,

64

4.4.7 Gauge position . .. 65

4.4.8 Structural model . t: 68

4.5 Tests in calm water . . . .. 70

4.5.1 Draft and trim . . .. ,. ., ., . .

..

4.5.2 Forces and moments . , .. . .. . r , r,7,2 .-' 7 . 74

CONTENTS vii

4.6 Tests in regular waves 75

4.6.1 Response quantities 75

4.6.2 Nonlinear effects 76

4.6.3 Measured RAOs 77

4.6,4 Comparisons of theory and measurements 80

4.6.5 Experimental error sources 88

4.6.6 Theoretical error sources 93

4.6.7 Concluding remarks 101

4.7 Tests in irregular waves 101

4.7.1 Motivation 101

4.7.2 The wave spectrum 102

4.7.3 Measured response processes 102

4.7.4 Trends in experimental response spectra 120

4.7.5 Comparisons of theory and measurements 120

4.7.6 Concluding remarks 125

5 CONCLUSIONS AND RECOMMENDATIONS

127

5.1 Conclusions 127

5.2 Recommendations for future work 129

REFERENCES

131

A Free surface stepping procedure

141

A.1 Introduction 141

A.2 The stepping procedure 141

B Generalized gravity forces

145

B.1 Introduction 145

B.2 Steady generalized gravity force 145

B.3 Generalized gravity matrix 146

C The physical catamaran model

149

C.1 Introduction 149 C.2 Stiffness distribution 149 C.3 Mass distribution 149 . . . . . .

...

. . . .

.

. . . .

....

. . . . . . n. v:9 .

NOMENCLATURE

General

Symbols are generally defined where they appear in the text for the first time. Matrices and vectors are represented by bold face characters.

Overdots signify differentiation with respect to the time parameter. Primes signify differentiation with respect to the x-coordinate.

Subindexes i and j generally refer to axial directions, where i, j = 1.2 and 3 for the

x- y- and z-axis, respectively. Subindexes x, y and z are sometimes used when a specific axis is referred to.

Subindexes k and r refer to mode numbers (except in equation (2.4)).

Index .s (sub or super) indicate that a quantity is steady: i.e. it is not varying with

time.

Roman symbols,

Ilk, Generalized added mass matrix

Vector tangential to body surface: instantaneous: steady

Waterline area

Strength of singularities distributed along the ship's axis-Discrete values of A(s) and B(x) for each station Generalized hydrodynamic damping matrix

Vector tangential to body surface: instantaneous: steady Total beam of catamarans

Beam of the hull

Transverse distance from the ship: see Figure 2.2.

Ckk Generalized structural damping matrix

CF Submatrix in C for the flexible modes Structural damping matrix

ix a,a, A(x), B b,b, b(x)

NOMENCLATURE

C(t) Constant in the Bernoulli equation that only depends on time

CB Block coefficient

Integration contours used for solving the two-dimensional,

CFO, Coo hydrodynamic boundary value problem; see Figure 2.2

d,dAp, fIFP Draft; aft perpendicular; fore perpendicular

dS Infinitesimal surface segment

dV Infinitesimal volume segment

Sum of squares defined by equation (2.118)

Fk Generalized external force vector; in mode k

FE, Fr

Subvector in F for the rigid modes; flexible modes

FsE Generalized steady excitation force vector

FD,FFR Generalized unsteady excitation force vectors: Diffraction; Froude-Krylov

External force vector

Entry in the mass force vector

f r(x) Probability density function for quantity x

Generalized fluid force in mode lc; steady; unsteady

Fif,, Fr

Generalized gravity force in mode k; steady; unsteady

Fe

Generalized diffraction force in mode It

Generalized radiation force in mode It

FirK Generalized Froude Krylov force in mode k

Froude number. 727u

F,(x

Cumulative probability distribution function Flow quantity in a region near the hull

G kr Generalized gravity matrix

GR _{Submatrix in G for the rigid modes}

Mass force vector due to gravity: instantaneous; steady Lame constant (Shear modulus or viscosity)

Acceleration of gravity Significant wave-height

r(4,)

Complex transfer function for the response x Unit vector in the x-direction

r Moments of inertia with respect to roll, pitch and yaw axes

Imaginary unit. Subindex for axial direction or for station

Subindex for axial direction

K, Kkk, Generalized structural stiffness matrix

KF Submatrix in K for the flexible modes. Structural stiffness matrix

KG Vertical distance from keel to center of gravity

Subindex for node number

Factors used for rudder-control in equation (3.1)( Vessel's length over all

CB,CFI, F.

F Fkf

, G, H k2

ROMAN SYMBOLS xi

Vessel's length between perpendiculars

LC G Horizontal distance from AP to center of gravity

Index for nodes in the hull cross section (equation (2.118))

M, Af lit Generalized structural mass matrix

MR, MP _{,Submatrix in M for the rigid modes; flexible modes}

Consistent structural mass matrix Total structural mass of the vessel

Number of modes

Flyn nth spectral moment for quantity x

N(77,) As Ns, but also defined for CFI, CFO and Ceci

Ns, Projection of ns in the transverse plane n, Surface normal vector: instantaneous; steady

Component of n, in the x-direction

Number of nodes in a hull cross section (equation (2.118))

NB Number of segments on the submerged transverse hull section

NF Number of segments on the free surface on each side of the hull

NT Total number of maxima in a record

JVXA Number of maxima above level xA in a record

it Number of degrees of freedom

P, Ps Vector of generalized coordinates; steady

pR, pF _{Subvector in p for the rigid modes; flexible modes}

pr(t) Generalized coordinate for mode r

p , Amplitude of the unsteady part of p(t); steady part

Pl, P2

Matching points used in the two-dimensional problem

pi Po Pressure; atmospheric pressure

R., Ric, The generalized restoring matrix

RR Submatrix in R for the rigid modes

rk. Deformation vector; mode r; mode k

Distance in the transverse plane., see Figure 2.2 Subindex for mode number

r44, rss, C66 Radius of gyration: roll; pitch; yaw

S, S So Wetted hull surface: instantaneous; steady; steady below z=0.

'(Se is used for the steady surface of the entire ship structure

in Appendix B)

(w) Wave spectrum

Response spectrum for quantity r

Stress tensor Surface traction

Tkr Entries in the generalized hydrodynamic force matrix

T, Peak period

Zero-crossing period

Time parameter

uttijuk,

Displacement vector; mode r; mode k;

xii NOMENCLATURE

U,Uk Displacement in the x-direction; in mode k

ui,U3 Displacement in the direction of axis i or j

Forward speed of the vessel

v, V1 Displacement in the y-direction; in mode k

Vb,V, Volume of the vessel's structure: instantaneous; steady,

wk Displacement in the z-direction: in mode k

xi Space coordinate. i = 1,2 and 3 for x, y and z, respectively Response in quantity x for mode r

x(t) Total response in quantity x

x(we) Complex amplitude of the response in quantity x

Y,Z Coordinates of a point in the submerged cross section of the hull

ZB, ZG z-coordinate of the vessel's center of buoyancy: gravity

Greek symbols

a> 137 415,1-3$ cti, az 7,-7x1 7x2 ak f5R 6=,6,,ö, (5i3, 8k3 C, Cu Cr Ci Cr 77, 77x, 71Y ^ 77: A Ap Aw

Vectors tangential to the hull surface: instantaneous; steady

Coefficients of proportionality in Rayleigh damping model

Wave direction relative to the ship. i3 = 00 is head seas

Modified CT (equation (2.67)) Skewness for quantity x

Kurtosis for quantity x

Deviation from desired heading (equation (3.1)) Logarithmic decrement in mode k

Rudder angle (equation (3.1))

Displacements of a point in the submerged cross section

Kronecker delta function Hull slenderness parameter

Entries in Cauchy's infinitesimal strain tensor Total water surface elevation; steady; unsteady

Amplitude of incident waves Amplitude of waves at station 7

Amplitude of waves that are generated by the ship oscillating in mode r

Coordinates for integration contour; see Figure 2.2 Rotations of a point in the submerged cross section Rotation vector; in mode r

Lame constant

Wavelength corresponding to the peak period Length of transverse waves generated by the ship

Viscosity

MATHEMATICAL OPERATORS AND OTHER SYMBOLS xiii

Arn nth central moment in the probability density function for:

V k Modal damping ratio in mode k

nxm matrix of the in lowest eigenvectors

7r The constant 3.14159....

19, Pb Mass density; of the ship structure

Standard deviation

Nondimensional parameter,

4). 6, O. Total velocity potential; steady; unsteady

OD, Or, OR Velocity potential: diffraction; incident waves: radiation

Velocity potential for station i

Radiation velocity potential for mode r

Modified OD (equation (2.66))

71)1 r Modified Or (equation (2.66))

Frequency of the waves in an earth-fixed coordinate system

We Encounter frequency of incident waves relative to the vessel

Eigenfrequency in mode k Nondimensional frequency. wiry'

Mathematical operators and other symbols

Transpose of a vector or matrix Gradient (Hamilton) operator Laplace operator

Summation

Absolute value (modulus) Displaced volume

Abbreviations

AP Aft perpendicular

FP Fore perpendicular

FE Finite element

LCG Longitudinal center of gravity

RAO Response amplitude operator (see equation (2.105))

VCG Vertical center of gravity

VBM Vertical bending moment

VSF Vertical shear force

Or

V

Chapter 1

INTRODUCTION

1.1

Background and objective

Marine vehicles with speeds above 25-30 knots have until recently been used primarily for passenger transport and naval applications. A considerable number of high speed monohulls and catamarans, with lengths between 20 and 40 meters. have been put into

service worldwide.

High forward speeds. complex structural outlines and the use of novel materials are factors which make the analysis of these vessels more challenging than for conventional ships. However, shipbuilders and classification societies have gradually gained substantial

experience with these vessels, and it turns out that if hull plates and scantlings are designed

to sustain local loads, the overall structure will in most cases safely sustain the global hull girder loads. Local loads, such as slamming pressures, can be obtained from various

empirical formulas.

During the last few years, there has been an increasing interest in the use of high speed ships for cargo transportation. As a result, the size of the vessels has increased significantly.

Catamarans of more than 120 meters length are being built (e.g. Svensen and Valsgird 1993), and 220 meter monohulls with a speed range of 35 to 40 knots have been designed (e.g. Jullumstro et al. 1993, Levander 1993). Several vehicles in the 50-100 meter range are already operating.

When the high speed vessel concepts are extrapolated to larger lengths, one moves into an area where new problems arise and previous experience is scarce. A new problem is the

increased importance of global wave loads. The lack of adequate experience and the costs of model tests intensify the need for practical and reliable theoretical calculation methods. The global hydrodynamic loads will depend not only on the ship's rigid body motions, but also on its elastic deformations. These hydroelastic effects can be of particular

impor-tance for large high speed vessels.

A hydroelastic phenomenon known as springing appeared with the advent of container ships with relatively high speeds, and large bulk carriers with relatively low rigidity in the 1960s (e.g. Taylor and Bell 1966, Mathews 1967, Little and Lewis 1967). Springing

CHAPTER I. INTRODUCTION

occurred when encounter frequencies of the waves synchronized with the resonant vibration

frequency of the mode for 2-noded vertical bending of the hull girder.

Theoretically, springing can occur for all types of vessels. For most conventional ships

however, it is not a serious problem, since the waves which cause resonance are very short

and contain little energy. As the ship's speed increases, high encounter frequencies are experienced for larger waves. Similarly, a reduction in the ship's rigidity will also transfer

the springing phenomenon to waves with more energy.

It is seen that high speed and low rigidity are two features of a ship which generally increase the springing problems. The abovementioned container ships possessed the first feature, while the bulk carriers possessed the other. Large high speed vessels possess the first feature to a considerably larger extent than the container ships. Moreover, the utilization of materials with a low stiffness to strength ratio, such as aluminum and high strength steel, will probably make them possessors of the second feature as well.

The springing phenomenon may not influence the extreme structural responses. but the fatigue damage could be significantly increased (e.g. Friis-Hansen et al. 1995). Most

of the fatigue damage will occur in low or moderate seas, and the problems are intensified by the use of aluminum and high strength steel in the hull structure.

Hydroelasticity can be of significance not only for steady state responses. but also for those of the transient type. The flexibility of the structure in the area of slamming impact can influence the total slamming load (e.g. Kvalsvold 1994), and the overall stiffness of the vessel's structure is important for the global (whipping) responses. However, transient phenomena will not be considered in the present work.

In most of today's calculation methods, the ship is treated as a rigid body when the fluid loads are calculated. Hydroelastic effects are therefore not included. Methods that include hydroelasticity have been presented, but the hydrodynamic calculations are either based on conventional strip theories (e.g. Bishop and Price 1979) or three-dimensional hydrodynamic theories (e.g. Wu 1984). Strip theories account for the effect of forward speed in a simplistic way, and three-dimensional methods, with an appropriate treatment of the forward speed, are computationally very demanding. Hence, there is a need for a

practical method which is applicable to flexible high speed vessels subjected to continuous

wave loading. The development and validation of such a method is the objective of the

present work.

1.2

Scope and limitations

The present study will focus on monohulls and catamarans with a high forward speed, but

the vessels should remain in the displacement or semi-displacement mode. Foils. aircushions

or other devices for static or dynamic lift will not be considered: neither will ships with

unconventional hull forms, like e.g. SWATHs.

Emphasis is put on structural responses, but since these responses are often dependent

on the ship motions. focus must also be directed towards motion predictions. The study will concentrate on global wave loads and the corresponding responses. Loads and responses in

1.3. PREVIOUS WORK AND PRESENT METHOD 3

the vertical planes are given most of the attention. Typical structural responses are vertical

shear forces and vertical bending moments, and the corresponding stresses. Such responses

are important for the hull structure (longitudinal vertical responses), and for catamarans. they are also important for the cross structure between the two hulls (transverse vertical

responses).

1.3

Previous work and present method

A fluid-structure interaction problem is approached by making three basic decisions. Firstly,

a method for the formulation and solution of the fluid problem must be selected.

Sec-ondly, a method for the structure problem must be selected; and finally, a method for

coupling the two problems must be selected. Hence, different combinations of methods for

fluid/coupling/structure will result in different hydroelastic formulations.

Early work on springing (e.g. Belgova 1962, Maximadji 1967a.b, Mathews 1967,

Good-man 1971, van Gunsteren 1970) utilized the two-dimensional strip theories for the fluid

(e.g. Korvin-Kroukovsky and Jacobs 1957, Gerritsma and Beukelman 1964. 1967), and

beam theories for the structure. Coupling was provided by a modal approach, and interest focused primarily on the 2-noded vertical bending mode.

In the 1970s. Bishop, Price and their co-workers (e.g. Bishop 1971, Bishop et al. 1977,

Bishop and Price 1979) advocated a more general modal analysis. They included the rigid

body modes as well as additional flexible modes in the formulation. The flexible modes

were obtained from an eigenvalue analysis of the "dry" ship (Bishop and Price 1974). Hence, the fluid/coupling/structure combination can be summarized as.

Two-dimensional/modal/beam

where "Two-dimensional" relates to the two-dimensional strip theories that are used for

the fluid.

In the 1980s, alternative combinations have been presented, where the strip theories are replaced by three-dimensional theories with forward speed effects, and the beam rep-resentation of the structure is replaced by a general finite element (FE) formulation (e.g. Wu 1984. Bishop et al. 1986). In terms of fluid/coupling/structure combination, these

methods feature.

Three-dimensional/modal/FE

Other authors have replaced the conventional strip theories by nonlinear extensions, so that

springing due to higher harmonics (e.g. Kumai 1974. Socling 1975, Jensen and Pedersen 1979, 1981) as well as whipping responses (e.g. Friis-Hansen et al. 1994. 1995) can be studied.

A comprehensive treatment of linear two-dimensional hydroelastic analyses can be found in Bishop and Price (1979). Methods in springing and whipping of ships are re-viewed by Jensen (1980), while Wu (1984) gives an overview of hydroelastic approaches

CHAPTER' INTRODUCTION

in the analysis of general floating bodies. Reviews of ship hydrodynamics and springing analyses can also be found in the triannual ITTC conferences and ISSC congresses.

Strip theories (e.g. Salvesen et al. 1970) apply a two-dimensional boundary condition

on the free surface and, except for the Doppler shift of the oscillation frequency, no effects of

the ship's forward speed are retained in this boundary condition. The theory will therefore.

only predict waves which propagate outwards, perpendicular to the ship's longitudinal axis.

For high speed ships moving in waves, the wedge-shaped diverging wave system will dominate (e.g. Ohkusu and Faltinsen 1990), and the conventional strip theories can not reproduce such waves. These waves can be included if a three-dimensional free surface boundary condition is used. This is done in the three-dimensional theories (e.g. Chang 1977), but the computer costs rise dramatically.

Faltinsen and Zhao (1991a,b) have presented a modified linear strip theory, where the three-dimensional free surface boundary condition is utilized to interrelate the

two-dimensional problems at the various strips. The formulation is commonly referred to as

4-dimensional, and the problem is solved by a stepping procedure which starts at the

bow. The method originates via works of Cummins (1956), Ogilvie (1972), Hirata (1972,

1975), Daod (1975), Chapman (1975, 1976), Faltinsen (1977, 1983), Yeung and Kim(1981,

1984) and Ohkusu and Faltinsen (1990).

Now, the diverging waves are included with only a small increase 'in thecomputational

costs, and the method is therefore practical for high speed vessels. Since the objective of.

the present work is to develop a hydroelastic method for these vessels, the 4-dimensional approach will be selected for the hydrodynamic calculations.

By adopting Faltinsen and Zhao's method for the fluid problem, one also adopts a

few basic assumptions in addition to those inherent in potentialflow theory. Firstly,

the method is linear, and responses excited by higher harmonics can therefore not be

predicted. Nonlinear effects are intensified not only by an increased wave steepness, but also if the higher harmonics are close to the resonance frequency of one of the ship's

modes. Depending on the vessel's flexibility one may therefore observenonlinear structural

responses also in relatively moderate seas.

Secondly, even if a three-dimensional free surface boundary condition is utilized,the theory still uses a two-dimensional field equation. The method will therefore give in-correct predictions in the vicinity of the ship's ends, and for incident waves which are short compared to the ship. The latter difficulty could be approached by reformulating the diffraction problem in a manner which replaces the two-dimensional Laplace equation with a two-dimensional Helmholtz equation (e.g. Skjordal and Faltinsen 1980, Sclavounos

1984). Or one could possibly circumvent the diffraction problem by somehowutilizing the

Haskind-Hanaoka-Newman relations (e.g. Ogilvie 1973, Ertekin et al. 1995).

Finally, the 4-dimensional method handles the forward speed effects moreproperly

than conventional strip theories, but there are still important simplifications. Interactions between the local steady and the unsteady flow are neglected, and these effects can be important for high speed vessels (e.g. Newman 1991, Zhao 1994). Faltinsen and Zhao (1991a,b) also presented a 4-dimensional formulation where these interactions were

1.4. ORGANIZATION

[counted for, but numerical problems were encountered at the intersection between the body

and the free surface.

It is not the purpose of the present work to improve or extend the

*dimensional

hydrodynamic theory beyond the modifications that are necessary to includethe flexibility

of the vessel. The present approach will therefore contain no nonlinear effects, no three-dimensional effects and no interactions between the steady and the unsteady flow fields.

By keeping these simplifications, one also retains the full computational efficiency of the method.

In the practical implementation of Faltinsen and Zhao's method (FASTSEA 1991), it is assumed that, due to the high forward speed, hydrodynamic interactions between the two hulls of a catamaran can be neglected. This assumptionwill also be adopted herein.

It should also be noted that the 21-dimensional formulation can be justified theoretically

only for Froude numbers higher than about 0.4; see Section 2.3.3. Its practical application

is therefore limited to vessels with a relatively high speed.

For the structure. a general numerical procedure is needed, since the present method should be applicable to non-beam-like structures, such as catamarans. Hence, the finite

element method is a logical choice.

Since this study is concerned with global wave loads and responses, it is reasonable to

assume that the relevant deformation patterns of the ship structure can be represented by a number of predefined global shape functions. Hence. the commonly used "dry" modal approach will be adopted. By utilizing modes or shape functions, one also maintains a high flexibility with respect to the discretization of the fluid and the structure. The only

requirement is that the mesh of each domain is sufficiently refined to represent the various mode shapes.

Alternatively one could use the structure's degrees of freedom directly, but it appears that this would produce an unnecessarily large equation system. The direct method will show to greater advantage when the structure is very flexible (e.g. Lee and Lou 1989) or when local high-frequency modes are of interest (e.g. Armand and Orsero1979).

In brief, the present method provides a new fluid/coupling/structure combination.

namely,

21-dimensional/modal/FE

which is believed to be practical for hydroelastic analysis of high speed vessels. Experiments

with a flexible high speed catamaran will be carried out to validate the method.

1.4

Organization

The mathematical details of the present approach are included in Chapter 2. together with an outline of its practical implementation. Chapter 3gives a description of the experiments

6 CHAPTER I. INTRODUCTION

presented in Chapter 4. This chapter also contains a discussion of the results and the valid-ity of assumptions upon which the theory rests. Finally, conclusions and recommendations for future work are given in Chapter 5.

Overviews of the present theory, or parts of it, have been published during the course of the work (Wu et al. 1993. Hermundstad et al. 1994, Hermundstad et al. 1995). In

Hermundstad et al. (1994), the method has been applied to a fairly realistic high speed

Chapter 2

THEORY

2:1

Introduction

The purpose of this chapter is to describe the present method in more detail. Derivations start at a very fundamental level, and this is done in order toillustrate how the governing equations for the structure and the fluid originate from the same set of fundamental axioms'.

Hence, the contents of Sections 2.2 and 2.3.1 can be found in textbooks onsolid mechanics!

(e.g. Fung 1965), structural dynamics (e.g. Clough and Penzien 1975), finite element methods (e.g. Zienkiewicz and Taylor 1989, 1991) and fluid dynamics (e.g. Newman

1977).

One is concerned with the structural responses and the motions of the ship. Thefluid

domain is. only interesting to the extent that it provides a pressure on parts of the ship's outer surface. The derivation of the method therefore starts by establishing the governing

equations for the ship structure, where fluid actions enter only as an external force vector

(Section 2.2).

In order to find these external fluid forces, a hydrodynamic boundary value problem 'must be solved, and the formulation of this problem is described in Section2.3. It will be demonstrated how the basic assumptions reduce the three-dimensional equations to the particular 2 -dimensional formulation.

The solution procedure for the hydrodynamic problem is described in Section 2.4.

Sec-tion 2.5 shows how the external fluid and gravity forces are obtained once the boundary value problem has been solved. The equation system that was established in Section 2.2 is then complete, and its solution provides the various response quantities, as described in

Section 2.6.

Section 2.7 describes some aspects of the practical implementation of the method and gives an overview of the resulting computer programs Verification of the programs is

Ii

8 CHAPTER 2. THEORY

"2.2

Equations of motion for the ship structure

.2.2.1

Governing field equations

By applying the axiom of conservation of mass together with the transport theorem to an arbitrary material volume, the continuity equation is obtained,

op apu,

K2.4

19i + ox,

A rectangular Cartesian coordinate system is used, and subscript i is the index of the three

axial directions. Hence, xl = x, x2 = y and x3 = z. Moreover, ui is the ith entry in the

displacement vector, [u, v, wIT, and p is the mass density. Overdots signify differentiation with respect to time 2, and repeated indexes mean summation.

If the axioms of conservation of momentum together with Cauchy's stress theorem

'Gauss' theorem and the transport theorem are employed to the material volume, and

equation (2.1) is utilized, the following equations can be derived:

'au, au, 1 az, 1

at

Ox j

pax;

p

These are the Eulerian equations of motion for a continuum. Here, f, is the ith entry in the mass force vector, and Tij are entries in the symmetric stress tensor.

The equations of continuity and motion constitute four equations for ten unknown functions of time and position. Hence, relationships between displacements and strains (kinematic compatibility), and between strains and stresses '(constitutive equations) must be introduced.

Assuming small deformations, the kinematic compatibility- can be represented by Cauchy'S,

infinitesimal strain tensor,

au, au,,,

- taxi+ or,

4(2.3)

For an isotropic and linearly elastic material, the constitutive equations are given by'

Hooke's law,

TE; = 2Gfi5 + Ae,*(52; (2.4)

where G and A are the Lame constants', and is the Kronecker delta function. G

is commonly known as the shear modulus in solid mechanics and the viscosity in fluid

mechanics (symbol p)F. Thermal effects are ignored.

2.2.2

Finite element formulation

The problem described by equations (2.2)-(2.4) can be cast into a finite element form by applying e.g. Hamilton's principle. For the entire structure, the discretized equations of

motion now read,.

(2.2)

= +

2.2. EQUATIONS OF MOTION FOR THE SHIP STRUCTURE'

+ cr + kr = f 1(2.3)

where in is the consistent structural mass matrix, c is the structural damping matrix, k is the structural stiffness matrix, f is the vector of external forces and r is the vector of unknown displacements u and rotations 61. The matrix dimension is nxn, where it is the

number of degrees of freedom. A linear viscous damping model has been introduced in equation (2.5) to account for energy dissipation.

2.2.3

Modal analysis.

For forced motions (i.e. f 0), any deformation of the structure can be expressed as a, linear combination of its eigenvectors,

r(t) = E rrpr(t)

(2.6)

r=1

The unknown coefficients pr(t) are usually referred to as principal or generalized

coor-dinates. Eigenvectors r, are obtained from a free vibration analysis of the undamped

structure.

In most cases, only the in lowest eigenvectors are needed to obtain the necessary accu-racy. Hence,

r(4) E rrmt)

Ep(t) (2.7)

r=/

where E is an TLX712 matrix where the in eigenvectors constitute the columns. The vector

p contains the generalized coordinates. The necessary number of modes will depend on

the response type that is being studied. Responses which are derived from the spatial

derivatives of the displacements, like strains and stresses, will generally require more modes

than for example motions and accelerations. Moreover, responses in local details of the structure will usually require more modes than global responses.

Inserting (2.7) into (2.5) yields,

m'Eji + c2-2p± kEp f (2.8)

It is easily demonstrated that the eigenmodes possess the following orthogonality

proper-ties,

crrnr,. = k

r

(2.9)

rrIcr41.= 0 k 7- (2.10)

It will be assumed that the orthogonality applies also to the damping matrix,,

rrcr,. = k r (2.11)

9

=

0

10 CHAPTER 9. THEORY

Premultiplying equation (2.8) by the transpose of the kth eigenvector and utilizing the

orthogonality properties yields the following set of uncoupled equations,

Mkkiik Ckkk Kkkpk Fk Jr= 1-777. (2.12)

where Mkk = rTmrk, Ckk= rTcrk and Kkk = rTkrk are entries in the generalized mass-,

damping- and stiffness matrices. respectively. Moreover, Fk = lakl is the kth element of the generalized external force vector, F.

It is common practice to normalize the eigenvectors so that ../Vikk = 1 and hence, Kkk =

wZ, where wk is the kth eigenfrequency. The resulting eigenvectors are often referred to as normal modes.

2.2.4

Internal hull clamping

For a ship hull, the main sources of damping are material hysteresis and friction in cargo, equipment and bolted/riveted connections. No tractable theoretical procedure for the calculation of hull damping exists. and empirical methods are therefore used (Betts et al.

1977. Jensen and Madsen 1977). It is generally assumed in ship vibration analyses that a linear viscous damping model is adequate, and that the orthogonality conditions (2.11)

apply to the damping matrix. The assumptions made in the present work are therefore in

conformity with common practice.

Instead of considering the damping matrix c. one will usually try to determine the

nonzero entries Ckk, in the generalized damping matrix directly. Based on full scale mea-surements, various formulas have been established for the logarithmic decrement 6k, in different vibration modes; see Betts et al. (1977) and Jensen and Madsen (1977). How-ever, damping measurements on high speed vessels and on ships made of materials other than steel appear to be scarce.

When has been estimated, the entries in the generalized damping matrix can be

obtained as.

MkkWk6k n

Ckk = = (2.13)

where vk =is the modal damping ratio.

The different empirical formulas for the logarithmic decrement give widely scattered values (Betts et al. 1977). Hence, in a finite element analysis of a ship, one will often resolve to the simpler Rayleigh damping model.

Now the generalized damping matrix is expressed as a linear combination of the generalized mass- and stiffness matrices,

Ckk = aiMkk Cx2Kkk (2.14)

The relationship between the coefficients al, a2 and the logarithmic decrement is given by,

bk al

= Ce-,Wk

rr Wk

2.2. EQUATIONS OF MOTION FOR THE SHIP STRUCTURE 11

An estimate of al; and az tan thereby be obtained if,the logarithmic decrement is known

for two modes.

Both modal damping and Rayleigh damping are implemented in the present method, but only modal damping will be used when calculations are comparedwith experiments/ results in Chapter 4.

Z2.5

Rigid body modes and coordinate system

in a unified hydroelastic analysis of ships, the six rigid body modes, surge, sway, heave, roll, pitch and yaw must also be taken into account. It is customary to include them as the first six modes in equation (2.7). The ship will be assumed to be freely floating, and the rigid body modes have therefore no associated structural damping or stiffness. The equations of motion 2.12) can now' be written.

Ma

.0 I 13R Ho ,o { isn{

a

a_ { FR I

0 NV'

V 1"

0 cP

pF

0 KF

pF pF

(216)

Superindexes R and F are used for quantities corresponding to rigid and flexible modes, respectively. The null matrix is denoted by 0. It is seen that thereis no coupling between

the two types of modes on the left hand side ofequation (2.16). However, it will be

demonstrated in the subsequent sections that the fluid forces included on the right hand side, will provide a strong coupling between rigid andflexible modes.

The uncoupling on the left hand side occurs because the orthogonality conditions

(2.9)-(2.11) apply also in the case where one of the modes is rigid and the other is flexible. But

if both modes are rigid, they are not generally valid (Wu 1984). Hence, the 6x6 rigid body

mass matrix MR is not diagonal in practical ship analyses; see e.g. Bishop et al. (1986). The present analysis will be restricted to ships with port-starboard symmetry. A

co-ordinate system is chosen that is fixed with respect to the meanoscillatory position of

the vessel. The xy-plane coincides with the undisturbed water surface, the x-axis points

towards the stern and the z-axis is directed upwards through the centerof gravity; see

Figure 2.1.

Rigid body modes are defined as unit translations along (surge, sway, heave), and unit rotations about (roll, pitch, yaw) the x-, y- and z-axis, respectively. In the present coordinate system,, the corresponding displacements can therefore be written,

=

and the rotations

1 0' [ 0

=

0 01

a

U2=I are,, Oli={ 0 1 0 o 1 0 3= 0 1

[

, 1 1 101 a o O 11 Y

j

1 Di 0 I 075,= z!!

x

0 1 0 .u6= 06= I x 0 ( 2.17) 1 (2.18) 0 1 R 04= 0 0

12 CHAPTER 2. THEORY

Seen from astern

MR

tz

Figure 2.1: Coordinate system for monoluills and catamarans.

The rigid body mass matrix now takes the form.

M 0 0 0 M zG 0

Here, zG is the z-coordinate of the center of gravity, and If, are moments and products

of inertia with respect to the roll, pitch and yaw axes. Since 164 = /46 (e.g. Faltinsen

1990), this matrix is symmetric. M is the total structural mass of the ship. Note that.

according to the definition following equation (2.12), the Ms on the diagonal have been multiplied twice with a unit translation. The offdiagona1 Ms have been multiplied with a unit translation only once. Hence, the diagonal and offdiagonal Ms have the same value but different units. All entries in the generalized mass matrix have the unit [kgm2] when standard SI units are used.

2.2.6

External forces

The vector F = [FR, FFiT[ contains all the generalized forces applied to the structure.

In addition to gravity, only steady and steady-state time-harmonic fluid forces will be

considered in the present work. By introducing the mass force vector due to gravity

= [0,0, pbgl, the generalized gravity force in the kth mode can be written,

Ff =

uk = pbgwkdV

vo vb

The integral should be taken over the instantaneous volume. 1//), of that have an associated mass density pb and a vertical displacement The acceleration of gravity is denoted by g, and subindex s has been that the gravity vector is constant (steady).

The generalized fluid force in the kth mode is given by,

(2.20)

all parts of the ship Wk in the kth mode. introduced to signify 0 M 0

MzG

0 0 0 C M 0 0 0 0 MaG

MzG

0 0 0 144 0 0 155

164

0 0 0 0 146 0 166 (2.19)

2.3. FORMULATION OF THE HYDRODYNAMIC BOUNDARY VALUE PROBLEM

la

Fki =

uk npdS

(2.21)

where p is the fluid pressure acting on the wetted part S, of the ship, and n is the unit

normal vector of S; positive into the fluid domain. The vector uk(x,y,z) contains the

three local displacement components of the kth eigenvector. The fluid pressure will be

found by formulating a boundary value problem for the fluid domain, and then solving it by numerical means. The next two sections are devoted to these tasks.

2.3

Formulation of the hydrodynamic boundary value

problem

2.3.1

Governing field equations

The governing field equations presented in Section 2.2.1 are also valid for the fluid do-main. For the fluid however, some additional assumptions will be made that simplify the

expressions further.

Firstly, it will be assumed that the fluid is incompressible with constant density. This simplification is generally done in seakeeping analyses, and the continuity equation (2.1)

now simplifies to,

0 (2.22)

ox,

Secondly, the viscosity of the fluid is disregarded. Hence, G = 0 and equation (2.4) can now be written,

= P8ij (2.23)

Thirdly, the fluid motion is regarded as irrotational. This is another frequently applied assumption in ship motion analyses, and it can now be shown that the fluid velocity vector can be represented by the gradient of a scalar potential (I),

24=

7-- Or

= 74) (2.24)

ax,

Inserting this into the continuity equation (2.22) yields,

524, = o axiax, or 524) 52 + E-v24) = 5r2 5y2 5,2 (2.25) (2.26)

14 CHAPTER 2. THEORY

which is the well known Laplace equation. It constitutes the governing field equation for the fluid domain. In equation (2.26) and in the subsequent derivation, the notation x, y and z has been used instead of xl, x2 and x3.

An expression for the fluid pressure in terms of the scalar potential is obtained by

inserting equations (2.23) and (2.24) into the Euler equations (2.2), and performing a few simple operations,

a4) 1

p

p(at

+ et) VI. + gz)+ C(t) (2.27)

This is the Bernoulli equation; another well known formula in potential flow theory. C(t) is a constant that is independent of position. The coordinate system defined in Section 2.2.5 has been introduced.

2.3.2

General hydroelastic formulation

In the present coordinate system. the ship's forward speed is represented by an incident

free flow field with velocity U in the positive x-direction. According to equation (2.24), the

potential of this free field can be written Us. It will now be assumed that the amplitude of the incident waves is small relative to the transverse dimensions of the ship hull, and that the ship-fluid system is stable. The oscillatory motions of the ship and the fluid can then be regarded as small, and the total velocity potential can be written.

1)(x, y, x.t) = Us + y, z) + (,75.(s, y, z, t) (2.28)

Here. os is the time-independent potential of the flow field that is generated by the ship when it moves steadily with constant speed in calm water. The unsteady potential Ou represents the incident waves and their interaction with the oscillating ship.

The unknown potentials Os and are determined by requiring the total potential (I) to satisfy the governing field equation (2.26) together with a set of boundary conditions.

Dynamic free surface boundary condition

On the free surface = ((x, y, t), the fluid pressure must equal theatmospheric pressure po

Surface tension is ignored. Inserting equation (2.28) into equation (2.27) this requirement gives, after separating position-dependent and position-independent quantities,

1 C(t) = Po + 5pU2

throughout the fluid, and

athu

ao,

acsi, 1 1

at

+u +

az

- + -vo,-vos+ -you

2 9

+ vos vo.+ gz = 0

(2.30)

on z = ((x,y.t). The total surface elevation ( is

composed of the steady elevation (s

and the unsteady elevation Cu. Since is small, the quantities in equation (2.30) can be

(2.29) z

2.3. FORMULATION OF THE HYDRODYNAMIC BOUNDARY VALUE PROBLEM 15

expanded in a Taylor series and evaluated at z = (5(x, y). Products of unsteady quantities are small and will therefore be neglected. Since the condition now applies on a steady surface, time-dependent and time-independent terms can be separated, and the following

expressions result,

aOs acbs 2 asbs 2

4s 2

U-57 +,iftrx-) +(--8)1

(

7)

gZ = 0

on z = (,kx, y) for the steady problem, and,

.90.

It

5208 iaq5u 056.c 0205

o& a,

4,820,

+ cuu-

79. .+ ' -aT +

cu(

az azaz

+ ay ayaz+ Br r-2 )

az, az,,

az,

8&Ox

ax

ay ay

arkt,

+ +

_{+ie z-}

+

=0

y

z a

on z = (;(2, y) for the unsteady problem..

Kinematic free surface boundary condition

The kinematic free surface boundary condition is obtained by requiringthat the normal

velocities of the fluid and the free surface are equal on the boundary. Hence,the substantial

derivative of the quantity C z, must be zero for z = ((x, y, t). This yields,

8(

8(

az, ac

amnia(

0,,8(

86,8(

az,

azu

+u +

-r 0

at

ax

ex ax

Ox Ox ay ay ay ay az az (2.33)

z = ((x, y, 't). As for the dynamic condition, terms can be expanded about cs, and

separated into time-independent and time-dependent parts. Hence ac, aza ac,

az,

,/

U = 11

Or

al ax

ay ay

Or

on z = C.,(x,y) for the steady problem, and,

±

Tract, az, act, ac, &Auks

azu k,

et

ax

ax ax ± ay ay

az ax ±

ay ay

azos acs 0263 azos athu

+ct(az axaz -4- ay ayaz

8z2)

az CI

on z = C,(x,,y) for the unsteady problem.

(2.31) (2.32) (2.34) (2.35) on

+

16 'CHAPTER 2. THEORY

!Kinematic body boundary condition.

The kinematic boundary condition for an impermeable body surface is obtained by requir-ing that the normal velocities of the fluid and the body surface are equal on their common boundary. The normal velocity of the fluid is V4) n while that of the body is given by U n. With (I) expressed by equation (2.28) this yields,

'Uin+V&n+V.n=ün

(2.36)

on the instantaneous wetted surface S of the body. The symbol ii denotes the unit, vector

in the positive x-direction.

Since the unsteady fluid and body motions are small, the terms in equation (2.36) can be expanded about the steady position S, of the body below the steady surface elevation.

As for the free surface boundary conditions, products of time-dependent quantities will be neglected. Hence, the steady disturbed fluid velocity, Vth takes the form Vth, + u V(V0s) while the other velocity vectors remain the same after the expansion. (Except that they are now evaluated on S, instead of 5).

Care should be taken when expanding the surface normal vector n, since it depends not

only on the position of the instantaneous body surface, but also on the surface deformation in terms of rotations and strains. The normal vector can be expanded by first defining two vectors as(x,y,z) and bs(x, y, z) that are tangential to the steady body surface S, at position (x, y, z), such that n, = a, x b where n, is the normal vector of the steady body surface (Wu et al. 1993). Expressing the instantaneous values of as and b, in afirst degree

Taylor series yields,,

a = as + u Vas,

(2.37)

and.

b = bs + u Vb,

(2.38)

The normal vector on the instantaneous position of the body surface can therefore be

written,

h = (a, + u "ca.) x (b, + u Vbs)

+ [a, x (13, V) bs x (a, - V)Itt (2.39)

Using the expanded terms and separating time-independent and time-dependent quanti-ties, equation (2.36) now yields the following conditions on the mean oscillatory (steady) position of the body surface below the steady wave elevation (3,

(Eh + Vths) = 0 (2.40)

for the steady problem, and

V(Vth,)] ns + (Ui + Vth,)

[a, x '(b, V)

bs. x (as V1u = 0 (2.41)

=

2.3. FORMULATION OF THE HYDRODYNAMIC BOUNDARY VALUE PROBLEM 17

for the unsteady problem.

Analogous to equation (2.7) the displacement vector of the body surface is written as an aggregate of the corresponding vectors in each eigenmode,

u(t)

E urpr(t)

(2.42)

r=1

The generalized coordinates will consist of a steady part ps, and an unsteady part pr.'. For regular waves that encounter the ship with frequency we, the unsteady part can be written

pre"4. Hence,

Mi) = p + rCu'l

(2.43)

Time-differentiation of unsteady quantities will therefore imply multiplication with iwe, where i is the imaginary unit.

In the present linearized unsteady analysis, the potential 0u. can bedecomposed into,

Ou(x z ,t) = 01(x, y, + D(x , y z)eiw`t + R(x , y , (2.44)

where Of is the potential of the undisturbed incident waves, OD is the potential of the waves

diffracted by the ship when there is no oscillatory ship motion, and OR is the potential of

the waves that radiate from the ship when it oscillates about its steadily advancing position

in otherwise calm water.

Infinite water depth will be assumed, and Or can then be written (e.g. Newman 1977),

0 1(x, y, z) _9(1 ekzik. cosiicysin,3 (2.45)

where C/ is the incident wave amplitude, and denotes the angle between the wave prop-agation direction and the positive x-axis. Hence, 3 = 00 is head seas and 3 = 90° is beam

seas from port side. Moreover. 4,.? is the frequencyof the incident waves in an earth-fixed

coordinate system so that,

Uw2

We = -

COS

In accordance with equation (2.42), the total radiation potential can be decomposed into

its modal contributions thy,

caR(X1 Y, z) E or(s, y,z)p, (2.47)

r.1

Inserting equations (2.42), (2.44) and (2.47) into equation (2.41) and dividing through by

e'""' yields,

(VOr+V(i)D) ns + DIV*,

iweur + V(V&)] - no

r=1

(Ui-E7c63)

x (bsV)b, x (as V)Jurlp, =0

(2.48)

y.

(2.46>

18 CHAPTER 2. THEORY

on the mean wetted body surface S,. This equation must be satisfied by any arbitrary

combination of the quantities pr. This is obtained if it is satisfied for each pr separately so that,

90D.961

_{_}

an,

an,

on S, for the diffraction problem, and aor

wen, n,

(lii + V5) [as x (b, V)

bs x (as V)]ur

Ur V(V09) n, (2.50)

an,

on S. for each of the in radiation problems. The relation VO n = ath has been utilized_an

above.

Both ODand each of the m radiation potentials cb,. must separately satisfy the Laplace

equation (2.26) and the unsteady free surface boundary conditions (2.32) and (2.35). Equation (2.50) is a generalization of the kinematic body boundary condition that was

derived by Timman and Newman in 1962. A simplified version of the formula, applicable to

slender hulls, has been presented by Wu et al. (1993). The above condition is comparable to the one presented by Wu (1984). Price and Wu (1985) and Bishop et al. (1986). but equation (2.50) contains an additional term that accounts for strain deformation. This additional term should not be neglected in a consistent derivation. It can be shown that equation (2.50) reduces to that of Wu (1984) when the local body surface undergoes a

pure rotation, and it further reduces to that of Timman and Newman (1962) when the

ship moves as a rigid body. According to Xia (1994), an expression similar to equation (2.50) was presented by Xia and Wu (1993).

Conditions at infinity

As the z-coordinate approaches minus infinity, the potentials .955 and with all their components must vanish. A radiation condition must also be imposed which ensures that the disturbance of the ship is directed away from it.

Summary and remarks

A solution to this general hydroelastic problem is obtained by first solving for the steady potential Os, which is governed by equations (2.26), (2.31), (2.34) and (2.40) in addition to the conditions at infinity. When 0, and its derivatives have been calculated, the in + 1 unsteady potentials OD and Or, r = 1, m, can be determined. They are governed by equations (2.26), (2.32), (2.35) and the infinity conditions. In addition, OD must satisfy equation (2.49) while each Or must satisfy equation (2.50). The total potential is obtained from equation (2.28) with given by equations (2.44), (2.45) and (2.47).

No explicit restrictions have been imposed on the body geometry and its forward speed

in the above derivation. Hence, the formulation is fully three-dimensional with a complete

2.3. FORMULATION OF THE HYDRODYNAMIC BOUNDARY VALUE PROBLEM 19

interaction between the steady flow fields and the linearized unsteady flow. It is also seen that the steady problem is nonlinear. An approach like this would be computationally very

demanding. and no attempts will be made to solve the problem as presently stated. Instead,

the geometrical slenderness of the ship hull will be utilized to simplify the formulation.

2.3.3

Simplified formulation for slender hulls

In order to simplify the hydrodynamic boundary value problem stated in the previous subsection, estimates of the relative magnitude of the different terms will be needed. It

will be useful in such a process to apply the concepts of perturbation theory (e.g. van Dyke 1964), in which the order of magnitude of the various quantities are consistently evaluated

in terms of a small parameter. Hence, the whole theory is developed in an asymptotic sense, so that the approximations will be more nearly valid as the parameter becomes smaller and smaller. One can seldom tell from the theory alone whether a finite value of the parameter is really small enough for the approximation to be useful in that particular case. and some other basis for judging the validity of the approximation is needed. Such a basis could be provided by a special case that can be solved exactly. by a higher-order

calculation or by experiments: see Chapters 3 and 4.

It is common practice in slender ship analysis, to let the hull slenderness parameter serve as the small perturbation quantity. This slenderness parameter is defined as the ratio of a typical transverse dimension of the submerged portion of the ship hull to its length,

L. Such a transverse dimension could be the beam b. or the draft d. Hence, L = 0(1)

while b and d are 0(f). where 0( ) means order of magnitude.

The subsequent assumptions are similar to those made by Faltinsen and Zhao (1991b). In line with conventional strip theory approaches,, it is assumed that the ship speed U is 0(1) and that the oscillation frequencyWe is 0(E-7). The x-component n31, of the surface

normal vector is taken as OW, while its two other components are 0(1).

A further supposition will be made regarding the derivatives of a flow quantity f in a region close to the ship; namely,

a f

0(f

x: f and L

0(f ci)

af

0(f ci

(2.52) ay az

Terms that are 0() relative to the other terms will be neglected.

With this set of assumptions, the equations that govern the flow can be simplified in a region close to the ship hull. Hence, the Laplace equation (2.26) takes a two-dimensional form in this near-field,

a24, a24, + = 0 ay2 az2 (2.51) (2.53) ,

20 CHAPTER 2. THEORY

and the boundary conditions on the body and the near-field free surface simplify

accord-ingly:

aN,

where N, is the projection of n, in the transverse plane. It is seen that Os is O(2), and

the dynamic free surface boundary condition (2.31) reduces to.

g

__cs

U

Hence, the steady wave elevation Cs is 0(d). Since Cs now is 0(d), all quantities can be evaluated at = 0 instead of z = (,(x.y). Equation (2.55) therefore applies at z = Oi and equation (2.54) applies to the steady position of the body surface below z = 0. This

surface will hereafter be denoted So.

Similarly, the kinematic free surface boundary condition (2.34) now reads,,

ac, 1 50,

P.56)

ax az

on z = 0. When equations (2.55) and (2.36) are combined, the conventional linear three-dimensional steady free surface boundary condition results,

The steady problem

The body boundary condition for the steady problem. equation (2.40), becomes,

Do,

Unsi

on z = 0.

The unsteady problem

The body boundary condition for the diffraction problem, equation (2.49). takes the form,

athp _(2.58)

aN,

an,

and for each of the rn radiation problems, equation (2.50) now reads

Nor

= n, [as x V) bs x (a, 71)]u,

ON,

-Both conditions apply to So.

The unsteady dynamic free surface boundary condition (2.32) simplifies to,

ao,

iwe

grr

Ox u (Pr (2.54) (2.55) (2.57) (2.59) (2.60) z aos ax2

1.3. FORMULATION OF THE If BOUNDARY VALUE PROBLEM 21

on z = I0 for r = 1, m. Here is the amplitude of waves that care caused by &he ship,

when it oscillates in mode number r.

Similarly, the kinematic free surface boundary condition (2.35)can be written.

0(, Lae 1 80,.

a,

u 'sr± az

on z = 0 for r= 1, m.

Both these free surface conditions also apply to the diffraction potential OD.

When equations (2.60) and (2.61) are combined, the conventional linear three-dimensional

'unsteady free surface boundary condition results,

2 T,Othr ,

00r

aor or +2iwe. -r-

1- 9 =

aX 8,2 a, z = 0..

Remarks

In the body boundary conditions for the unsteady problems, equations (2.58) and (2.59), some terms have been retained' that should have been neglected in a consistent first order

approach.

Firstly, Pit could have been replaced with since the x-component of n, is OW

relative to the two other components. For the same reason, the ur n.,), component of the

first term on the right hand side of equation (2.59) could have been neglected. Both terms have been retained because they are easily available.

The second and third terms in equation (2.50) constitute a generalization of what is commonly referred to as the m-terms in ship motion theories. In equation (2.59) the second expression on the right hand side represents a simplified m-term, but the analysis

'is not strictly consistent. This simplified m-term can be seen to be 0(E) relative to the first term on the right hand side, and it could therefore have been neglected. Moreover,

terms containing the second derivatives of 46, with respect to y and have been disregarded

in equation (2.59), even if they are of the same order of magnitude as the simplified m-term.

Since the steady problem is being solved prior to the unsteady problem, the 05-derivatives could in principle be made available. However, the complete m-terms have not been used due to the numerical problems that were reported by Faltinsen and Zhao (1991a1b).

The reason for not discarding the simplified m-terms is partly the fact that they are easily available, and partly the fact they are included in the conventional low-speed strip theories. Their importance for the hydrodynamic coefficients is recognized, and it is there, fore reasonable to include these forward speed effects in the present high speed theory as

well.

The assumption (2.51) is what makes the present approach a so-called 21-dimensional method in contrast to the conventional two-dimensional theories. If the x-derivative of f is taken as 0( f), the U-terms would disappear from the unsteady free surface condi-tion (2.62). Hence, a flexible body generalizacondi-tion of the convencondi-tional strip theories (e.g.

(2.61)

(2.62)

0 on

Salvesen et at 1970) emerges. This is the approach that was pursued by e.g. Bishop and

Price (1979). If in addition, the "inconsistent terms" are removed from equations (2.58)

and (2.59), the result is a generalization of the lowest order rational strip theory of Ogilvie

and Tuck (1969).

The free surface boundary condition for the steady problem, equation (2.57), reduces

to the conventional rigid-wall condition, at = 0 on z = 0, if the x-derivative of f is

OCT). Together with equations (2.53) and (2.54) this condition would constitute a two-dimensional Neumann problem in the near-field. An arbitrary function of x can therefore be added to its solution. This function must be determined by matching the near-field solution with a fax-field solution that satisfies the radiation condition. By this matching

process, it can be shown (Ogilvie 1977) that the unknown function contains a representation of transverse waves.

The present steady problem however, governed by equations (2.53), (2.54) and (2.57), is not purely two-dimensional, and it can be identified as that of a Cauchy-Poisson type (Tuck 1972). No arbitrary function of x can be added to the near-field solution, and a

representation of transverse waves can therefore not be included. The steady diverging wave

system on the other hand, is properly incorporated (Ogilvie 1977). One would therefore

expect the present formulation to be valid in a region close to the bow which is small

compared to the length of the transverse waves. Relative to the ship length, this wavelength is given by (e.g. Newman 1977),

.Aw U2

97r = 27:1,2 (2.63)

gL

where Et is the Froude number. It is evident that Aw is larger than L for Froude numbers larger than 0.4, and as the speed increases beyond this value, the method could be applica-ble to the entire ship. The present approach is therefore seen to be a high-Froude-number theory, unless it is being matched with some other formulation downstream from the bow.

Such a matching is not pursued in this work, and the high-Fronde-number restriction

therefore applies.

The situation is similar for the unsteady problem. It will be demonstrated in the next section that the unsteady problem can be rewritten in a form that makes the free surface

boundary condition similar to that for the steady problem. Hence, only the unsteady

diverging waves are included in the solution; see Faltinsen (1983), and it is not possible to match the near-field solution with a far-field representation of transverse waves, like in strip theory.

For sway, roll and yaw, no transverse wave systems are generated. For these lateral motions, the method could also be applicable for lower Froude numbers (Faltinsen 1994). Otherwise, a strip theory formulation should be used for low speed vessels. Or, as men-tioned by Ogilvie (1977), a switch to a strip theory formulation could possibly be used at a distance from the bow where the local Froude number gets too low.

2.4. SOLUTION OF THE HYDRODYNAMIC BOUNDARY VALUE PROBLEM 23

2.4

Solution of the hydrodynamic boundary value

prob-lem

2.4.1

The steady problem

Since the present steady problem formulation is exactly the same as that of Faltinsen and Zhao (1991b), their method can be applied without modifications. The situation is similar for the unsteady problem, where the only difference is the body boundary condition (2.59) for the radiation problems.

Forerunners of Faltinsen and Zhao's procedure were presented by Faltinsen (1977, 1983),

the overall strategy is presented in Faltinsen and Zhao (1991a), and a slightly modified version is described by Ohkusu and Wen (1993). For completeness, an overview of the method is included in the present section.

The vessel will be required to have port-starboard symmetry. For catamarans, only one hull needs to be studied, since hydrodynamic interactions between the hulls are neglected.

Waves generated by the ship will decay as one moves away from hull in transverse directions. Hence, in a transverse plane. one can identify an inner region where waves are present, and an outer region where there are no waves; see Figure 2.2. The free surface boundary condition

= 0 on

= 0 can therefore be used in the outer region, while

equation (2.57) applies in the inner region.

Outside a distance b(x) from the ship, the velocity potential will be written,

yz

ths(x,y,z) = A(s)y2 _{+ 22} B(x)_{(y2 +2)2} (2.64)

This means that the ship is represented by a distribution of two-dimensional vertical dipoles

and y-derivatives of such dipoles, with strengths A(x) and B(s), along the x-axis. This is a valid approximation, provided b(x) is large compared to the transverse dimensions of

the hull. In the steady problem, B(x) will be zero, unless the hull is asymmetric about its local centerline. This may be the case for catamarans.

It is assumed that conditions (2.51) and (2.52) apply in the entire inner region. One can then divide the hull into NS stations (transverse planes) along its longitudinal axis, and solve the problem at each station by a two-dimensional boundary integral method. Hence, A(x) and B(x) reduce to a set of unknown constants, Ai and Bi for each station, i. By utilizing Green's second identity, the velocity potential at a point (y, z) in the fluid

can be expressed by a distribution of two-dimensional sources In R and normal dipoles °InnON

along the boundary of the fluid domain.

alnR

27r43(Y,z)= ic'c

ON(77.

ln R

aN(?),)ids

(2.65)

Here, R = \Ay (z and NO, fl is the two-dimensional normal vector of the

boundary; positive into the fluid. The boundary C consists of the body contour CB, the inner free surface contour CFI, the outer free surface contour CFO and the contour Cro far