Ship wave patterns and motions by a three dimensional Rankine Panel Method

EiO3iTDiJ ER/EF

d.d.

van

DOCTOR OF PHILOSOPHY

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 1990

C) Massachusetts Institute of Technology 1990

Department of Ocean Engineering

"ne 1990

Certified by

Paul D. Sclavounos Associate Professor of Naval Architecture Thesis Supervisor

Accepted by

Douglas -ii ichael

Professor of Power Engineering

Chairman, Department Committee on Graduate Students

SHIP WAVE PATTERNS AND MOTIONS

BY A THREE DIMENSIONAL RANKINE PANEL METHOD

TECHN1SCHE UNIVERSITEIT Laboratorium vox

by _{Scheepshydromechanka}

Dimitris E. Nakos Archlef

Meitelvireg 2, 26:19 CO Delft Diploma, Naval Architecture and Marine EngindMi115.786873- Falc 015.781838

National Technical University of Athens, Greece, 1985

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF

INGEKOMEN

1 9 JUNI 1990

SHIP WAVE PATTERNS AND MOTIONS

BY A THREE DIMENSIONAL RANKINE PANEL METHOD

by

Dimitris E. Nakos

Submitted to the Department of Ocean Engineering of the Massachusetts Institute of Technology

in partial fulfillment of the requirements for the degree of Doctor of Philosophy

ABSTRACT

The steady and unsteady ship wave problems are studied. A quasi-linear formula-tion of the underlying Boundary Value Problem is derived, valid over a wide range of Froude numbers, wave frequencies and ship hull geometries.

The numerical solution of the resulting free-surface boundary value problem is ob-tained by a three dimensional panel method employing the Rankine source as the Green function. A theoretical framework is developed for the rational study of the order, numerical dispersion, dissipation and stability of Rankine Panel Methods

for three-dimensional free surface flows, as well as for. the efficient numerical im-plementation of the proper radiation conditions. This new methodology is applied to the selection of a numerical solution scheme of cubic order, which is free of

nu-merical dissipation and wave reflections from the boundaries of the computational domain. The solution algorithm involves the discretization of the ship hull and part of the free surface by panels, over which a bi-quadratic spline approximation of the unknown potential is employed.

The effectiveness of the proposed solution scheme is demonstrated in the solution of the Wave Resistance and Seakeeping problems around a number of realistic hull shapes. The hydrodynamic forces, exerted on the ship, and the resulting motions are

found in remarkable agreement with related experimental measurements, demon-strating that important three-dimensional features of the flow are properly modelled by the solution scheme. Moreover, the display of computed steady and unsteady wave patterns illustrate the effectiveness of the solution algorithm in resolving all

ACKNOWLEDGMENTS

I am deeply thankful to my family and friends for making my stay in MIT possible, challenging, fruitful, joyful and rewarding. Lefteris, Irini and Angelos manage to keep the links with my roots alive, despite the pain of the long separation. Noela selflessly helped in balancing my life as well as my temper; the challenge is now on my shoulders to help her navigate through the ocean of her own dreams. This thesis is dedicated to my grandfather, the late Dimitris Nakos Sr., for offering me not only his name and his deep unconditional love, but also his spirit and mentality. I wish to thank Professor Paul D. Sclavounos for watching over my solder all these

years, not only as a research advisor but also as a. dear friend. I will never forget his

ability to transform my frustration into constructive optimism. Special thanks are also directed to Professor J. N. Newman for advising, motivating and approving my

efforts; to Professor D. K. Yue for his optimistic upbeat spirit and continuous search for scientific challenges; and to Professors J. E. Kerwin, T. F. Ogilvie and H. Schmidt

for serving as members of my thesis Committee. Professors C. Chryssostomidis and N. M. Partikalakis, Dr. M. S. Drooker and Mr. S. Tuohy are also acknowledged for making available to me the computer equipment of the Design Laboratory of the Ocean Engineering Department, as well as their expertize for the graphical illustration of this thesis.

It was a joy having the opportunity to work and socialize within the Computational Hydrodynamics Group of the Ocean Engineering Department in MIT. I can only hope that the group spirit will stay alive, integrating complex research efforts with simple human needs. Dear friends and colleagues Thanasis Dimas and Kostas

Ster-giakis deserve special notice for sharing with me happy as well as difficult moments.

Financial support, during the course of this research, was provided by the Applied Hydromechanics Research Program administered by the Office of Naval Research and the David Taylor Research Center (Contract number N00167-86-K-0010) and by A. S. Veritas Research of Norway. Computations were performed on the

VAX-11/750 of the Computational Hydrodynamics Facility supported by MIT and NSF

and on the Cray YMP of the Pittsburg Supercomputer Center (Grant number 0CE880003P).

A 4.7

TABLE OF CONTENTS

INTRODUCTION * "! A A. A A *

Id. Background

L2. Overview

IL ,STATEMENT OF THE MATHEMATICAL PROBLEM ILL The exact formulation

11.2', Linearization of the free surface boundary condition 11.3. linearization of the body boundary condition

11.4. Frequency-domain formulation of the unsteady probirrn IL5. The resultant force and the equatiOili of motion,

11.6. Integral formulation of the boundary value problem ILA-. Evaluation of the influence of the m-terms

III. DESIGN OF THE NUMERICAL SOLUTION SCHEME

...44

111.1. Error analysis of Rankine panel methods in the F01:flierspace

111.2. Design of the numerical scheme - The basis functicas. 111.3. The continuous dispersion relation

111.4. The discrete dispersion relation

111.5. Stability of the discretization scheme

111.6. Truncation of the free surface - The radiation concition, The discrete formulation in Fourier space

III.B. Asymptotic expansion of E,, m=1,2,3

,

11

IV. APPLICATION OF THE SOLUTION SCHEME 94

IV.1. The grid on the free surface and the hull Formulation of the linear system of equations

Evaluation of the wave elevation and the pressure field The computer code SWAN

V. NUMERICAL CONFIRMATION OF CONVERGENCE 106

Steady flow around elementary singularities Unsteady flow around elementary singularities

VI. NUMERICAL RESULTS : THE STEADY WAVE FLOW 121

Steady wave patterns

Steady wave forces Steady sinkage and trim

VII. NUMERICAL RESULTS : THE UNSTEADY WAVE FLOW 143

VII.1. Numerical convergence of the hydrodynamic coefficients VII.2. Prediction of ship motions in head waves

VII.3. Unsteady wave patterns

VIII. CONCLUSIONS AND RECOMMENDATIONS 163

LIST OF REFERENCES 166

VI.1.

VI.2'.

1.1. BACKGROUND

More than a century ago, Froude and Krylov modelled the ship motions by bal-ancing only the inertial and restoring forces. No attempt was made to analyze the hydrodynamic disturbance due to the presence of the vessel.

The first significant steps to account for the hydrodynamic effects dueto shiplike

ves-sels were associated with the solution of the steady wave resistance problem, where the vessel is assumed to advance steadily in an otherwise calm sea. Michell(1898) developed the thin-ship theory for the solution of this problem, under the

assump-tions of small beam, relative to the ship length and draft. In the twentieth century,

Havelock rederived and generalized Michell's wave resistance theory, which remains

popular even nowadays.

In an attempt to replace the cost-inefficient experimental techniques, several

m.athe-maticians and engineers devoted their efforts to the improvement of wave resistance

theory. Slender-body theory of aerodynamics was transferred into the field of free surface hydrodynamics. only to be proven as a special case of Michell's thin-ship theory. Peters and Stoker(1957) noted that the related boundary value problems can be also consistently linearized under the assumption of a fiat-ship, leading to a formulation analogous to the lifting-surface theory of aerodynamics.

In order to treat the case of large scale, full-shaped vessels, which greatly violate the

assumptions of geometrical slenderness, alternative theories were presented based on the assumption of slow forward motion. Ogilvie(1968) showed first that the problem can indeed be consistently linearized under the assumption of small Froude

number. Since then, several slow-ship theories have appeared in the literature,

which when coupled with the use of an efficient numerical solution scheme provide significant improvements over the classical theories. A more thorough

review of the existing slow-ship theories is given in Section 11.2. In parallel to

slow-ship theories, several solutions of the wave resistance problem were reported, based on the so-called Neumann-Kelvin formulation. This intuitive yet inconsistent approach attempts to improve upon thin-ship theory by satisfying more accurately the body boundary condition.

The Boundary Element Method forms the basis of the majority of the computational algorithms for the numerical solution of the linearized _{wave resistance problem.} These solutions schemes may be classified in two categories, based on their choice of the elementary singularity.

The Kelvin wave source _{serves as the elementary singularity of the schemes}

belong-ing in the first class. The original formulation is due to Brard(1972) and it is, 'by construction, tied to the Neumann-Kelvin free surface boundary condition. The major advantages of such schemes are the elimination of the free surface from the domain over which the resulting integral equation is defined, and the automatic satisfaction of the radiation condition. _{The success of such numerical schemes} hinges upon the accurate and efficient evaluation of the Kelvin source potential as well as the proper treatment of the so-called waterline integral. Both of _these issues are the subject of studies by Newrnan(1987), Doctors and Beck(1987a,b) and

Clarisse(1989).

The pioneering work of Gadd(1976) and Dawson(1977) ignited the second class of numerical schemes, which utilize the Rankine source as the elementary singularity.

This choice provides flexibility in satisfying free surface conditions with variable

-aiming at the decrease of the numerical dispersion and dissipation have been

pro-posed by Piers(1983), Jensen et al.(1986) and Sclavounos and Nakos(1988), while the effective enforcement of the radiation condition is the subject of studies by

Jensen(1987), Raven(1988) and Sclavounos and Nakos(1988).

The wave resistance problem has also attracted some studies of its fully nonlinear form. Most developments are related to two-dimensional problems (see eg. Von Kerzeck and Salvesen(1974) and Campana et al.(1989)), although some success has

been reported in three-dimensions by Ni(1987) and Jensen(1988).

The two International Workshops on Wave Resistance(1979,1983) and the more

recent Kelvin Wake Survey(1988) summarize the state of art in wave resistance computations. The relative merits of the different linearizations are concealed by the significant numerical difficulties, which appear to prevent convergence of the numerical schemes and cause numerical errors which in most cases are greater than the errors associated with the linearization assumptions.

Over the past forty years attention has been also focused on the ship motion or

seakeeping problem, which considers the unsteady forward motion of a vessel in sea waves. This problem was first studied by Weinblum and St.Denis(1950) and it was greatly benefited by the introduction of the spectral description of a sea state, by St.Denis and Pierson(1953). The latter development allows the formulation of a deterministic problem in regular waves, whose solution is the transfer function needed in the stochastic description of the resulting motion.

In general, the assumptions employed in the development of approximate seakeeping

theories are directly analogous to the ones used for the wave resistance problem. In particular, the thin-ship theory was extended to include unsteady motions by

Haskind(1946) and Peters and Stoker(1957) and refined by Newman(1961). Slender-body theory forms the basis of many ship motion theories that were developed in the

4

-50's and 60's. Different assumptions about the order of magnitude of the

character-istic wavelengths, or equivalently the frequency of oscillation, led to several

approxi-mate theories. The slender-body theory of Newman(1964) and Maruo(1967) _{is built} upon the long-wavelength assumption and fails to predict the resonant behavior of the vessel's motion. The strip theory of Korvin-Kroukovsky(1955), Gerritsma and Beukelman(1967) and Salvesen et al.(1970) utilized the _{available two-dimensional} solutions by properly integrating them along the length of the vessel, and it is valid in the short-wavelength regime. A rational analysis of strip theory was carried out by Ogilvie and Tuck(1969) and it resulted into some modifications that account for the forward translation in a more consistent manner. In order to bridge the gap

between the slender-body and strip theories, Newman(1978) _{and Sclavounos(1984)} developed the unified theory, which accounts for the _{three-dimensionality of the}

flow in a more consistent manner.

A common characteristic of all these slender-body _{theories is that they require the} solution of only two-dimensional boundary value problems, rendering them

corn-putationally efficient. On the other hand, the limitation _{of their applicability on} simple, geometrically slender configurations, travelling at low speed is considered

as their most serious deficiency.

With the increase in the availability _{of computer resources-, the first numerical}

solu-tions of the three-dimensional seakeeping problem_{appeared. All existing algorithms}

are based on the Neumann-Kelvin formulation, which _{- as in the steady problem} - may be considered to be an inconsistent extension of thin-ship theory. Related success was first reported by Chang(1977). During the past thirteen _{years related} studies have been published by Inglis and Price(1981), Guevel and Bougis(1982), Iwashita and Okhusu(1989) and King et al.(1988). The last addresses the transient motion, while all the rest concentrate on the steady-state limit.

forward extensions of the corresponding schemes for the steady wave resistance problem, sharing with them merits and disadvantages. Further developments of the unsteady scheme stumble, however, on the additional analytical and numerical difficulties in the evaluation of the unsteady Kelvin source potential, which serves

as the Green function.

The skepticism about the ability of a discretized free surface to model the propaga-tion of the very complicated unsteady ship wave pattern postponed the employment of Rankine Panel Methods in the numerical solution of the seakeeping problem. Re-cently, Nakos and Sclavounos(1990) proposed a free surface discretization scheme that is free of numerical damping and introduces small numerical dispersion, while providing flexibility in the choice of the linearization of the free surface boundary

conditions. Related success is also reported by Bertram(1990).

Discussions about the feasibility of numerical solutions for the fully nonlinear ship motion problem were triggered by the pioneering work of Longuet-Higgins and

Cokelet(1976). Their mixed Eulerian-Langrangian scheme appears relatively

promis-ing, but related developments are currently restricted to two-dimensions. Moreover, the implementation and use of the related numerical solution schemes still remain an art, rather than science, and are heavily dependent on the availability of

super-computing power.

In conclusion, the wave resistance and the seakeeping problem belong at the top of the list of classic problems in marine hydrodynamics. Both the academic and practical interests in these problems were the major driving force in the course of the research that led to the present dissertation.

6

-1.2. OVERVIEW

This thesis is concerned with the hydrodynamic problem which arises when a marine

vessel advances on or close to the sea surface. Both the steady wave resistance and the unsteady ship motion problems are addressed.

The ultimate objective of this study is the development ofa computational algorithm that will accept as input the configuration of the vessel and the sea-state, and will produce as output the wave pattern, which can be thought of as the footprint of the vessel, and all seakeeping characteristics of interest. The latter include the oscillatory motions and accelerations of the vessel, the relative motion between the vessel and the free surface with direct _{consequences on deck wetness, bow slamming} and propeller clearance, as well as the distribution of wave induced shearing forces

and bending moments along the hull.

One basic underlying assumption is that both the steady and unsteady free surface

flows are independently linearizable. The 'smallness' of these disturbancewave flows

may follow as a result of a combination of geometric slenderness and small forward speed, without, however, being excusively tied to one of those. The solution of the free surface flow around a full-shaped vessel advancing at high speed is beyond the scope of the present study.

A Rankine Panel Method is designed to meet the requirements. of accuracy and efficiency, and it is based on a bi-quadratic spline representation of the unknown

algorithm may easily be modified to accommodate virtually all possible linearized or quasi-linearized free surface conditions, thus it establishes a sound platform for the numerical investigation of the validity of alternative linearization assumptions.

In Chapter II the potential flow problem is

_{formulated and linearized. The} lin-earization of the free surface boundary condition is not necessarily based on either geometrical slenderness or low forward speed, and it bridges the gap between the slender/thin-ship and slow-ship theories. The body boundary condition is also lin-earized, in a consistent manner, and the importance of the so-called rn-terms is discussed. An expression for the linearized pressure on the hull is given, which results in the steady wave resistance, sinkage force and trim moment, as well as the added mass, damping and restoring coefficients _{related to the unsteady motion} of the vessel. Finally, integral formulations of the corresponding boundary value

problems are derived by means of Green's theorem with the Rankine source as the

elementary singularity.

Although the problems the solution of which we seek are linear, they can be hardly considered as easy. Analytical solution tools _{are exhausted very soon and the}

nu-merical methods arise as the lone alternative. _{A carefully designed and analyzed} numerical algorithm is presented in Chapter III. The _{numerical solution may be} classified as a Rankine Pane/ Method, since it is a Boundary Element Method us-ing the Rankine source as the elementary sus-ingularity. Of particular _{importance is} the derivation of a discrete dispersion relation which governs the wave propagation over the discretized free surface. The discretization errors may be quantified by comparing the location and gradients of the discrete and continuous _dispersion re-lations. The study of the discrete dispersion relation also results in the derivation of a Stability criterion, which restricts the choice of the grid parameters so that the discrete problem is well-posed and the discrete solution physically acceptable. In order to conclude the design of the solution scheme, the effect of the truncated free surface domain is studied and appropriate end-conditions are derived, for the

numerical implementation of the radiation condition and the minimization of wave reflection due to the truncation error.

In Chapter rv the numerical algorithm is tailored for the solution of steady and

unsteady flows about realistic ship hulls. The grid generation and the discretization of the integral equations into a system of simultaneous linear equations is discussed. Subsequently, the algorithm for the numerical evaluation of the wave elevation and

pressure fields is described. The computer code SWAN, which executes the proposed

solution scheme, is also briefly presented.

Verification of the convergence of the numerical solution algorithm is presented

in Chapter V. The flows around a translating/pulsating singular source and a

thin strut in forward translation are chosen as test cases. Closed-form analytical solutions exist for all those elementary flows. The numerical solutions are compared to their analytical counterparts, in order to illustrate the efficiency and accuracy of the proposed scheme. Moreover, the proper selection of test cases reveals the most significant features of the resulting free surface flows and illustrates the way in which they are handled by the discrete model.

Numerical results for the solution of the steady wave flow around realistic ship hulls

are presented in Chapter VI. The wave patterns due to the Wigley model and a

Series-60, ct, =0.6, vessel are predicted for a broad range of Froude numbers. Results

based on the linearization of the free surface boundary condition about the uniform incoming flow (Neumann-Kelvin) and the double-body flow are compared to each other, in order to demonstrate the effects of the linearization scheme. The discrep-ancies are generated in the near field of the vessel, but propagate far downstream

rendering in some cases significantly disparate wave patterns. The evaluation of the steady wave forces, acting on the vessel, by pressure integration over the

wave resistance. Balancing of these wave forces by hydrostatic effects _{results in the}

prediction of the vessel's equibrium position, under steady forward translation.

Chapter VII discusses and illustrates the

_{prediction of unsteady forces and} mo-tions of vessels advancing through regular_{waves. The convergence of the numerical} results for the hydrodynamic coefficients is _{studied, in order to determine the} nec-essary grid density, which provides adequate accuracy, as a function of the Froude number and frequency of oscillation. The calculated unsteady forces and motions, based on both the Neumann-Kelvin and _{double-body linearizations, are compared} with experimental data and strip theory _{calculations. The double-body} lineariza-tion, which is derived in this Thesis, is clearly favored providing robust and

accu-rate numerical predictions of the seakeeping behavior,_{over a broad range of Froude}

numbers.

Finally, some conclusions and recommendations _{for future research are listed in}

Chapter VIII Of particular interest

_{are the steps that need to be taken, in order} to develop the solution algorithm for the ship wave problem, introduced in this thesis, into a professional design tool.

-AM;fikt,

ta'NF:;-....IN.,.

II. STATEMENT OF THE MATHEMATICAL PROBLEM

a

ILL THE EXACT

_FORMULATION

The problem of determining _{the flow around a freely floating vessel, which advances} through sea waves, is considered.

(FS)

12

Figure 11.1 _{: The two coordinate}

Two different frames of reference are utilized for the study of the wave field, as

illustrated in Figure ILL The Cartesian coordinate

_{system io =}

, yo , zo) is

fixed in space with the positive x0 axis pointing upstream and_{the positive zo axis} upwards. In this frame the vessel translates at _{a constant forward speed U, while} undergoing an oscillatory motion about its mean position. The coordinate system = (x, y,z) has the same orientation as the system

but it translates with

respect to it at the mean forward speed. In the frame , the physical problem

may be, alternatively, defined as the oscillation of the vessel in the presence of an incoming uniform flow of velocity U in the direction of the negative x axis. The

Boundary Value Problem will be formulated with

_{respect to the frame E}

, while

restricted use of the frame Eo will be made where appropriate.

The physical problem is approximated under the assumptions of an inviscid and incompressible fluid and an irrotational flow. The _{potential function k ("±-,}

_{t) that}

describes the resulting velocity field, is _{a harmonic function throughout the fluid} domain, a property that follows from the satisfaction of the mass conservation

principle,

t) = 0

,

in the fluid domain

_(II.1.1)

The velocity field 17(f, t) _{may therefore be expressed as the gradient of the potential,}

(Z, t) = VW(Z,t) , (II.1.2)

(20

p pci =

p(W,1VW

VW + gz) .

1

2 2

14

(II.1.3)

Here p is the fluid density, g is the gravitational acceleration and _{pa is the} atnao-spheric pressure which will be taken as the reference pressure.

On the submerged portion of the vessel surface (B), the component of the fluid velocity in the direction n, normal to (B), is equal to the component of the vessel's velocity V3 in the same direction,

an

(i,t) =

(VB

fi)(i,t)

, for E E (B) . (11.1.4)

Since equation (II.1.4) is expressed in the frame the velocity of the vessel 173 is

due only to its oscillatory motion. The direction of the normal vector ñ. is taken by convention to point outside the fluid domain and inside the vessel.

The fluid domain is also bounded by the free surface, that is defined by its elevation z _{c(z, y, t). Wave overturning is not allowed to occur and therefore the elevation} c can be cleaned as a single-valued function of the horizontal coordinates z and y, as well as time t. The free surface is a 'material surface', obeying the kinematic free surface boundary condition,

(--k9 + VW

V)

(z

(z,y,t))

= 0 on z = c(x,y,t) . (II.1.5)

Moreover the pressure on the free surface is prescribed and the application of Bernoulli's equation (11.1.3) results in the dynamic free surface condition. Here-after, unless otherwise explicitly noted, the pressure on the free surface is taken to be equal to the reference pressure pa. For the length scales of interest to this study

the effects of surface tension can be neglected and the resulting condition reduces

to

C(x,Y,t) = --1_g

+ VW

U2)

2 2 _x= (11.1.6)

By taking the substantial derivative of (II.1.6) and making use of (11.1.5), the wave

elevation c may be eliminated, and the condition for klf reads

1

Witt +217* V111, +

2VW VW) +gTz

= 0

on

z =

.

If the fluid domain is bounded by _{no other boundary surfaces the additional} con-dition to be satisfied is that at finite times the disturbance velocity decays _{at large} distances from the vessel rendering a far field which asymptotes to the undisturbed flow.

The response of the physical system governed by the equations derived above, should

be causal. Two initial conditions are in principle necessary in order to complete the mathematical formulation. In the following Sections the Boundary Value Problem will be linearized and thus a steady-state of its response can be safely assumed to exist. In the steady-state limit, the causality condition and the requirement that the system response is stable are engraved in the radiation condition. The radiation condition may be interpreted as the requirement that the wave energy flux carried

by the disturbance due to the vessel, should propagate away from the vessel at infinity. Further discussion of the radiation condition can be found in Chapter III of this thesis. A more detailed derivation of the exact Boundary Value Problem is

-11.2.. LINEARIZATION OF THE FREE SURFACE BOUNDARY _CONDITION

The Boundary Value Problem formulated in Section 11.1 is exact within, the lim-itations of potential flow. The presence of the free surface, however renders it highly nonlinear not only because of the nonlinear form of the conditions, (II.L.6)

and (II.1.7), but also because of the fact that these conditions _{are applied over an_} unknown surface which is part of the solution.

Attempts to linearize the ship wave problem span almost a century, and they are of various degrees of sophistication. Some linearizations have been proven to be more successful and convenient, from the computational standpoint and they have been extensively utilized. The majority of the linearization studies have been carried Out in connection with the steady ship wave problem, the so called wave resistance

problem.. Analogous approximations have been also introduced in the. unsteady ship wave problem, also referred as the ship motion or ,seakeeping problem.,

The assumption of small beam-to-length ratio led to the very wellknown thin.

-ship approximation. This theory was developed by Michell in. 1898 and 'it is Still extensively used for the solution of the Wave Resistance problem. The free surface boundary conditions (II.1.16) and (II.1.7) are linearized about the uniform incoming flow under the assumption that the total disturbance due tal the presence Of the

vessel, wave-like or not is small

of the same order as the beam. In addition,

the body boundary condition 'is collapsed onto the centerplane of the Vessel, in the same fashion as in the linearized wing theory. The resulting free surface boundary condition, widely known as the Neumann-Kelvin condition, takes the form

,

-4,,a.thartfacClb5.011,-_ 411(E, t) +10(i,t)

(a

a

ao

+ gaz = o

, on t =JO

The steady condition is recovered if the time derivatives are dropped. The Neumann=

Kelvin. condition is one of constant coefficients and it is applied on the whole. plane

z 0. Fourier Transform and/or Green Function techniques can be used to

explic-itly express the solution in integral form.,

If both the beam and the draft are assumed to be small compared to the length of the 'vessel, an alternative approximation. follows along the lines of slender-body theory. The resulting solution can be interpreted as a special case- of the thin-ship approxi-mation and, for the steady problem, it was studied by Maruo(1962), Vossers(1962)1 and Tuck(1963) and reviewed by Newman(1976): Slender-body theory has also been extensively utilized for the solution of the ship motion problem. with consid-erable success. Various degrees of sophistication and different assumptions about the order of magnitude of the frequency of oscillation led to the Istrip-theories (eg. K.orvin-Kroukovski(1955), Gerritsma and. Beukelman(1967), Salvesen et al.(1970),

Ogilvie and Tuck(1969)), the composite theory of Maruo and Tokura(1974) and the

unified theory of Newman (1978) and 'Sclavounos(1984).

Geometrical slenderness does not always characterize the designs preferred by naval

Architects. In particula.r., the bluff bulbous bows and the transom. stern Violate

the. assumptions of all the aforementioned theories. The relaxation of the slen-derness/thinness, requirement was first attempted in the context of the so called Neumann-Kelvin formulation, in both its steady and unsteady Versions. The free surface boundary conditions are kept linearized about the uniform incoming flow,,

.17

bodies, but it appears to be inconsistent for the _{more interesting case of}

surface-piercing vessels.

Nevertheless, the Neumann-Kelvin formulation has emerged as one of the primarily used bases for the numerical prediction of the _{wave resistance and it is generally}

ac-cepted as an improvement _{over slender/thin-ship theories. In the field of ship motion}

predictions the Neumann-Kelvin formulation was first introduced by Chang(1977).

Unlike the wave resistance problem, the numerical _{solution of the ship motion}

prob-lem based on the Neumann-Kelvin formulation is in the development stage and has not yet replaced two-dimensional theories, in practice. That is also because of the numerical complexity of the method, due_{to the cumbersome time-consuming}

evalu-ation of the wave source potential, which consists the _{kernel of the integral equation}

that needs to be solved.

An alternative approach to the linearization of the ship _{wave problem follows from} the assumption of small forward speed U. _{Slow-ship theories, otherwise known} as low-Froude-number approximations, regard the double-body flow as the zero'th order solution and superpose the _{wave effects upon this. Linearized free surface} boundary conditions for the steady problem, based on the low-Froude-number as-sumption, have been derived by Ogilvie(1968) (in two dimensions), Keller(1974), Baba and Takekuma(1975), Newman(1976) and Maruo(1980). Although there

ex-ist differences between the abovementioned independent studies, the resulting free

surface conditions are, in general, quite similar. Sakamoto and Baba(1986) _extended

the application of the low-Froude-number approximation to the unsteady ship wave problem, deriving an unsteady free surface boundary condition of a form identical to the steady one with the mere addition of terms due to the unsteadiness. On

physical grounds, slow-ship conditions _{can be interpreted as conditions that model ,}

the convection of short waves by a slowly-varying velocity field the double-body

flow. They all read as follows

2

+ v(1).

v)

±

gad)

az

1,111ft,*,:i. 4+4

W(E,t) =

+ ck (5)

for the steady problem,

0,

for the unsteady problem.

where cl) is the double-body flow and is the wave disturbance. Condition (II.2.2b) is inhomogeneous only for the steady problem in which

case the forcing Dfil

depends on gradients of the double-body flow.

A mathematically less rigorous linearization of the free surface boundary condition was proposed by Dawson in 1977, in connection with the wave resistance problem. His arguments are not, based on _{proper ordering of the different terms but rather} on physical, intuition as well as computational ,convenience. Dawson's condition superposes the wave effects upon the double-body flow and it reads

kl/(E,t) = 1)(4 7f-,0(i) , (II.2.3a)

(11,21 01)1i + go. 4,12 (Da

on z =10 , (11.2.3b)

where the subscript t denotes differentiation along the streamlines of the double-body flow.. The popularity and extensive use that Dawson's 'condition has been enjoying over the last 13 years is striking and must be attributed to the develop-. =lent of a powerful numerical method for its impleMentation, which generated very promising- numerical resufts. Dawson's condition is most often viewed as a low-,

Froude-number approximation because it considers the double-body flow ,as the basis flow upon which, the wave effects are superposed.

It is true however, that

(11.2'.2a.)

on z

0

(11.2.2b)

D(±) ,

-5;

condition is transferred on the z = 0 plane without formal justification. The im-portance of the terms, dropped by Dawson, has been the subject of several studies which propose alternative forms of the free surface condition, along similar lines as (11.2.3) (see eg. Nakatake et al.(1979), Baba(1979), Eggers(1981), Brandsraa and

Hermans (1985), Raven (1988 ,1989) )

Physical intuition suggests that the linearization of the boundary value problem may be justified only when the disturbance due to the presence of the body is in some sense small. This 'smallness' can be a result of geometrical slenderness, slow forward translation, or a combination of the above. A full-shaped vessel travelling at high speed will definitely cause large disturbance, which may not be subject to linearization. Large disturbances are closely related to big losses of energy and

functionality as well as discomfort, and consequently they are undesirable in practice

of naval architecture.

The rest of this section is devoted to the derivation of a linearized free surface boundary condition, based on the assumption that the disturbance due to the pres-ence of the body is small and valid uniformly for thin/slender ships advancing at finite forward speed and full-shaped vessels at a slow forward rate of translation. The steady and unsteady conditions are treated simultaneously, until the very last step where the steady and unsteady terms are balanced separately. The precise quantities that are assumed small and the appropriate justifications/motivations

are given as the analysis carries on.

The total flow field P(E,t) is decomposed into the double-body flow 1)(4, the steady

wave flow q5(4, and the unsteady wave flow (p(i,t) if unsteadiness is present

A W(E,t) = (1.(4 + (1)(±1

co(,t)

. (11.2.4)

The separation of the double-body flow (1)) from the wave flow (0 ± cp) is suggested

by the body boundary condition, which is discussed in the following section. The

tiga0.44,-En.

rigid wall condition,

cD, 0 , on

z =

0 (11.2.5)

that is imposed. on cl) is arbitrarily chosen, and it is mainly motivated by the. sim-plifications that it provides to the ensuing analysis..

The wave flows q5 and yo are both assumed small and linearizable, as compared to the double-body flow 4), which is taken to be the basis 0(1) flow. The ,smallness of co can be easily justified by assuming unsteady excitation of small amplitudes..

The justification of small steady wave disturbance q5 can be justified by loosely considering the product of the slenderness times the square of the Froude number 11 (F2) and assuming that it is small.,

Plugging the decomposition (11.2.4) into the_ conditions (II.1.6) and (II.1.7) and

neglecting of higher order terms in

and p leads to

cots + 2V(1) Vcot Vec1)- V (Vil) V(0+ So)) ₊

_{(VI) VI') V(Cb}

+ g (0, +

=

940, :V(V(1)- VI)) -

on z = s-

_(11.2.6)

where the wave 'elevation is given by:

1 1

_g

(77(1).

v(D'.

U2.)

3: g

1 ,

lCot V(I) Vy0).= (11.2.7) ,

+

The total wave elevation can then be approximated as follows

(cfl

_(az

{Iv(D v(p})

(v(1. V0).__- - (co, + VI) Vcp).=17.

az 2g _ g g

z=r

s) = (V(I) VO) + (Cot + 774) ' V40))

g 71) V(Dz

with errors of 0(152, q5CP).

Taylor expanding of (11.2.6) about z = and dropping terms of order higher than

0(q5,cp) leads to +2V(1).V, +VcDV (Vc1, 77(0 + (p)) + -2-1V (V.V cD).\ 7 (0+(p) + g(0.-H)

3 {17

7

v(vci). VcD)gCD.

} (Pt ± 77431 V(g5

8z2

g VcD. = 1 =

--2g (VI) VcI) U2)z=s

-1

-2\7(V(1, VI)) , on

z =

Condition (II.2.10) is linear with respect to the wave flow potentials 0 and co and it is imposed on the surface z = , which is known after the double-body flow is

solved. A similar condition was derived by Newman(1978) for the linearization of

the unsteady flow about the steady one. In Newman's condition 11. is the total steady

disturbance flow and T. is the total steady wave elevation. In contrast, condition (11.2.10) does not model nonlinear interactions between the steady and unsteady wave disturbances, because gh and cp have been assumed to be comparably small.

22

(11.2.8)

(11.2.9)

Although the free surface boundary conditions appear to have been sufficiently' simplified into (II.2.10), one additional linearization step will be. taken, which is expected to reduce significantly the computational effort associated with the solu-tion for both and (p. Examination of the relation (11.2.8') for the double-body 'elevation' reveals that 0(c-F2 1 and consequently it is of magnitude comparable.

to q5.. A Taylor expansion of (11.2.10) about z = 0, correct up to linear order in 4),

co and leads to

VGI)- V (Vit. VO) i1\7(\71, VcD) Vc6 gO, (V(I) - VO)

1 1

2'7(\71) VI)) 17(1)

i(U2

VcI)- \711.)t. on = 0 (11.2.1 1) for the steady flow, and

(Pst ± 2 \74:1> Vrpt

VI) V(VVp)

ilV(V4Y Veto) Vcp, Tic),

(1)z, (co, + \71), Vp) on

z =

(11.2.12)

fOr the unsteady flow. The tvo)oconditions, that are given above are derived by balancing the steady and unsteady terms of (11.2.10) ,separately,, and also making use of the following two identities

VI+VcD, = 0,

on z = 0 (11.2.13a) a

{V(V(1), V(D).-Vc1)} = 0

on z = 0 (11.2.13b) az. `.44111.

-+

+

=

z + +

+

= 0

, 0 , ,

:1. s-(z,Y) = 24 ---.-...'" 1 1 1

-- Vc1)

_g(

V(1) + VI) C70 ) 2 2

/0

1 S(i, Y, t) =

--

(p, + V,D g

for the steady and unsteady wave flows, respectively.

The conditions (11.2.11) and (11.2.12) are proposed as uniformly valid alternatives to the Neumann-Kelvin and low-Froude-number conditions, since they bridge the gap in between them. At the slender/thin-ship limit, the double-body flow (1) can be replaced by the uniform incoming flow U s and the conditions (11.2.11-12) reduce to the Neumann-Kelvin conditions. On the other hand, if the assumption changes to small forward speed and short wavelengths, (11.2.11-12) reduce to the conditions

of slow-ship theory.

Furthermore, the terms retained by Dawson's condition are also a subset of the terms that constitute condition (11.2.11). Of particular interest are the terms

asso-ciated with (1) which are missing in Dawson's condition. The importance of such terms has been studied extensively and has been proved significant when the vessel is not wall-sided (Baba(1979)).

In conclusion, the linearization of the free surface conditions into equations (II.2.11)

and (11.2.12) is primarily based on physical intuition, although it is also supported by mathematical arguments. The weakest link of this study is associated with the linearization of the steady wave flow, and the resulting elimination of the nonlinear interaction between the steady and unsteady wave flows. The condition proposed by Newman(1978) for the unsteady problem takes proper care of this interaction, but its implementation stumbles upon the lack of an accurate and efficient solution of the nonlinear steady wave problem. It appears that further analytical studies

(11.2.14)

as well as systematic numerical experimentations are necessary. In this spirit, the

numerical solution scheme sketched in the following Chapter of this thesis has been

designed so that several theoretical enhancements can be easily implemented and numerically analyzed.

11.3. LINEARIZATION OF THE BODY BOUNDARY CONDITION

The body boundary condition (II.1.4) should also be interpreted in terms of the decomposition (11.24), that was introduced for the total flow potential P.

The steady forcing of the problem is the presence of the uniform incoming flow U x. On the mean position of the vessel (B), this induces a normal velocity that is balanced by the non-wavelike disturbance, rendering the body boundary condition for the double-body flow

It should be pointed out that VI) _{UT in the far-field, which is the inhomogeneity} that forces the double-body flow problem.

The steady wave flow satisfies the homogeneous condition

an

ao

leaving the right-hand-side of (11.2.1 1) as the only forcing of the steady wave

prob-lem. This forcing can be interpreted as an artificial pressure distribution _{on the free} surface due to the presence of the double-body flow.

The unsteady forcing due to the oscillatory motion of the vessel is balanced by the

unsteady wave flow cp. If Ei is the oscillatory displacement vector measured from

the mean position of the vessel (P), it follows that aco acy = Ti V(1) +0) on (B) . (11.3.3) an at 26 on (B) , _(11.3.2)

an

= 0

, on (73-) . _(II.3.1) : .

The magnitude of the displacement vector Ei, and consequently the distance between

(B) and (B), is assumed to be small

of the same order as p. Based on this

_

assumption,, Tim.man and Newman(1962) derived, a condition to be applied on the mean position of the vessel,. They Taylor expanded (11.3.3) and dropped terms of order higher than linear, to get :n

(.(Zi

'7)V + (74)

(ii) _(11.34)

The last term of (11.3.4) accounts for the interaction between the steady and un-steady disturbances in a manner which is consistent with the assumptions of Section 11.2., An alternative form of (11.3.4) can be derived if the local oscillatory displace-ment of the vessel is expressed in terms of the six rigid-body global displacedisplace-ments,

(+

x (11.3.5)

where (denotes the unsteady translation and fl the unsteady rotation of the vessel. From the use of (11.3.5) in (11.3.4), and after some vector manipulations), it follows

that :

ao

ari

(xxn)

+

c[(n-V)74} +

Vt.)] on (B)

A. significantly more compact,, as well as suggestive in the following solution steps, expression for the statement (11.3.6) ,can be derived if the. following notations are employed a el) (II.3,.7a) (e4 v

4)

(11.3.7b) (n _{n2 n-3) = 21,} (11.3.7c) =

at

. = a = (11.3.6) = , ,

an

0

aso

(rn,,ms in 6))

The notation (11.3.7e-f) was introduced by Ogilvie and Tuck(1969) and they are referred to as the m-terrns:. In terms of the quantities defined by (11.3.7), the body boundary condition. for the unsteady wave flow reduces to

aef

_{n +}

,e;

mi)

at ;

If the basis flow is approximated by the uniform incoming flow the only non-zero

m-terms are ms = Un3 and m6 = .Un2

, which merely account for the 'angle-of attack' due to yaw and pitch.. This approximation of the m-terms is employed

_in.!

most previous studies of the ship motion problem, based on an 'extrapolation' of 1 the assumption used in the derivation of the Neumann-Kelvin free surface boundary condition. The validity of this assumption, even within the limits of the Neumann Kelvin formulation,. is highly questionable and it is the subject of some analytical and numerical investigation, which are presented in Section 11.6 and in Chapter VII of the present thesis

28

,

_(V)(' .x VI))

(11.3.70 on (B) (11.3.8) ^ , = . i= 1 6

11.4. FREQUENCY-DOMAIN FORMULATION OF THE UNSTEADY PROBLEM

The principal objective of the present work is to study the steady-state limit of the unsteady response of the physical system discussed in the preceding Sections. The unsteady excitation is assumed to be due to an incident monochromatic wave

train. The frequency of the incident wave, as viewed from the stationary frame Eo, is taken to be cuo. _{In the moving reference frame E, the incident wave arrives at}

the frequency of encounter U./. _{If 13 is the angle between the phase velocity of the} incident wave and the forward velocity of the vessel, co is defined as follows

co2

= IwoU2cosi31

Therefore, in the frame , the incident wave potential of unit amplitude in deep

water is given, in complex form, by

n

(Po (±",t) = _wo (. ix cot iy sin $) eita t

(11.4.1)

(11.4.2)

The incident waves get diffracted by the hull, and they also induce an oscillatory

motion of the vessel. The underlying physical system has been linearized in Sections

11.2-3 and it can be shown to be stable. Consequently, the transient component of the response will be decaying in time rendering a time-monochromatic steady-state response at the frequency of encounter cu. Moreover, the linearity of the Boundary Value Problem that governs the physical system along with the form of the body

boundary condition (11.3.8) suggest the decomposition of the resulting wave field as

follows

Here (co;,

j = 1,

6) are the radiation, potentials due to motion of the_{vessel in each}

of the six rigid-body degrees of freedom with unit amplitude and (ea,

j

1,_....,6)

are the corresponding amplitudes of motion. Finally, car is the 'diffraction potential and A. is the amplitude of the incident wave train.,

After plugging (11.4.3) into the boundary conditions (11.2.12) and (11.3.8) _{and 8ep-,}

arate. balancing of Each 'mode', the following Boundary Value Problems are

estab-lished for .(pi

j = 1,...,7

= 0

in the fluid, domain

a (47

,==, + co2 2iciA74, Vps

az

v4).. _{(VI) - V(P)4}

for the, radiation problems, and

+ VcD Vpi)

on

z =0

an iwn ±, tn., on (T3) Radiation Condition 30 1 2\7077(D VcD) Vf,o.

j =

6, kIT.4.4) ..., , = , ,

-,

V2co, = 0

,

in the fluid domain

(9407 (9400

az

=

az + w2 ((po + yo7) 2ic,./V(1) V(cp, + (,o7)

V (VI' V(go, + ;07)) iV(VcD V(D) V(cpo + yo7)

a,p7 &pc,

an an

Radiation Condition

(11.4.5)

for the diffraction problem.

The free surface conditions in (11.4.4) and (11.4.5) are written in a form suggestive of the solution scheme that will follow. In general, they may be classified as mixed

boundary conditions, which specify the normal (to the boundary) derivative of the potential in terms of its value and tangential derivatives.

The inhomogeneity of the radiation problems appears only on the hull surface of the vessel, while the diffraction problem is also forced by an artificial pressure distribution over the free surface. The latter pressure is due to the fact that the incident waves, having no way to anticipate the presence of the vessel, do not satisfy the free surface condition (11.2.12). This additional inhomogeneity of the diffraction

problem is eliminated if the uniform incoming flow Ux is taken to be the basis flow 1), and for that reason it is not present in previous studies of the ship motion

problem. The practical significance of the free surface forcing on the exciting forces,

(iw(rpo +,,o7) + (1) v(00 ±c07))

on (73-)

--11.5. THE RESULTANT FORCE AND THE EQUATIONS OF MOTION

The potential flow is determined as the solution of the Boundary Value Problems that have been formulated in the preceding Sections'. ,Subsequently, the pressure field may be obtained from Bernoulli's ,equation (II.1.3). Of particular interest is the pressure on the vessel's wetted surface, whose integrated effect determines the force and moment acting on the vessel.

Utilizing the decomposition (11.2.3) of the total potential W, the pressure on the

hull is expressed as :,

1

PPaA

(

(Pt + 117(11 + p)

+ + yo)

₂

gz)

(11.5.1)

2 _zeun

Taylor expansion of (II.5.1) about the mean position of the vessel (33), correct up to terms linear in q5 and (p, leads to

It should be pointed out that the square of the velocity due to the steady wave disturbance,, Vck Vci) , is preserved in the first term, of (11.5.2), although it has

originally 'been assumed to be of higher order. That was done in order tocomply with the traditionally employed expression for the pressure that gives rise to the

Wave resistance. 32 V(1) + (/))1 Wt. + ch) U2, -i-gz) 2

'737)

P (Pt +VIP - V(p)zE(-Erl p.i@ct V)(-1V(1) VI) +gz)) fE(T) (11.5.2) =

The steady part of the pressure field can be extracted from (11.5.2),

Po

"75.6%._-- -6_

_g(v(cDiro)'' Vr('D 4" 0) U2 )11Et-B-1 _(Pgz)zE(r4

2

Assuming that the vessel's centerplane is a plane of symmetry, the steady pressure (11.5.3) will give rise to the wave resistance force R in

the xdirection, the steady

lift (or sinkage) force L in the zdirectioti and a steady trim moment T, as follows

(t) =

X,

ff

(P. )

1:17)

where n3 and n5 are the components of the normal vector to the hull, as they

are defined by (11.3.7).

The wave resistance force R is' balanced by the propulsive force, while the steady lift force and trim moment are balanced by the weight and its moment about the origin of the axes, defining this way the equilibrium position of the vessel.

The unsteady component of the pressure is examined under the assumptions of Section (11.4) of small monochromatic motions at frequency The components of the unsteady force and moment on the vessel,

(F F2, F,)

and ./Cf.

(F,,F5, F6) , due to the oscillatory motion are expressed as follows

n3 1 i-iwki (11.5.3) (11.5.4)

...,6

. (11.5.5)

The added mass and damping coefficients aii and b, are associated with the second term in the right-hand side of (1E5.2) and depend on both the forward speed and

7014firliraVe {(t) 6 + c,5) ds = = 1, .

-

=

pressure and of the steady dynamic pressure field. It follows from straightforward combination of (1I.5.2) and (11.5.5) that

2 ff(hop; + Vcli V(p,) r2, ds _(11.5.6)

(73-)

ff

i(ic.upa + Vito, Vyoi)

(T)

P ((Jr"V

1

(-2V4D +

gz))

ds

Due to the oscillatory rotation of the vessel the _{weight of the vessel Induces an} ad-ditional restoring moment in roll and pitch. These _{effects are traditionally included.} in the coefficients ciy. The zero-forward-speed restoring coefficients are given by Newman(1977).- The evaluation of the speed-dependent_{part of cif involves Second,}

derivatives of the double-body potential 10, On the hull and it is expected to depend critically on the accuracy of the numerical scheme. The related discussion is post-poned until the Chapter VII of the present thesis. The speed-dependent portion of the restoring coefficients is ignored by _{all previous studies of the ship motion.,} problem_ Such a simplification, however, is justified _{only at stnall forward speed} while it may be shown. to be of great importance _{at. high Froude numbers.}

The rest of the force and moment /cf. is due to the incident and diffracted wave.

potentials, (po and co, and it is referred toas the exciting force. In view of 1(11.5.2),

the components of the exciting force are 'expressed as follows

ff

V(Po, co7))n,ds (-1) 34

-..nraner7r411:::414:4

, (11.5.7) (II.5.8)t (11.5.9) 1. .1,i'lf-T=4.514;:, V =

=

ds = (B) +

+

.

The equations of motion that govern the steady-state time-harmonic response of the vessel follow from the application of Newton's second law. Using the definitions

(11.5.5) of the resultant force, it follows

6

E

iwbo. + col

= X,

I = 1,...,6

, (11.5.10)

= 1

where rnti is the mass matrix. Newman(1977) gives a general expression for the mass

matrix. The same reference also discusses the expected behavior of the Response Amplitude Operator (RAO) e., , j = 1, ...6 at small and large frequencies as well

11.6. INTEGRAL FORMULATION OF THE BOUNDARY VALUE PROBLEM

Green's second identity is applied for the unknown potentials, cl:), cl) and j

=-1, ...,7, and the Rankine source potential,

The fluid domain is bounded by the hull surface (P), the free surface (FS) and a

cylindrical 'control' surface (S). The resulting integral equation reads as follows

15(±1

-

f f

(FS) 1 G(i; = 27- Et! ao(±-) ac(E;±--)

-cizr +

0(x') clil

G(x;x)

az' an (F S)u(157) (II.6.1)

ff8w')-

an G(i; il)d±" E (FS) U (13-) . (11.6.2)

Here q5 stands for either one of (1), 0 or cp., , j = 1,...,7, which are introduced in

the preceding Sections. The surface integrals over the control surface (S) can be

shown to vanish if (S) is removed to infinity and 1±-1 remains finite.

The derivatives of cl), (15 and c,c), in the direction normal to the surfaces (ES) and (B)

are subsequently replaced in terms of their value and tangential derivatives, as well as known 'forcing' quantities, by using the related free surface and body boundary

conditions.

36

-Of particular interest is the integral over the hull that represents the part of the forcing of the unsteady radiation problems, which is associated with the gradient of the steady disturbance:

ffrn; G(;') di'

,

j = 1,...,6

. (11.6.3)

The evaluation of the na-terms by straightforward use of their definition (11.3.7) involves the computation of second order derivatives of the double-body potential (I) on the hull. The panel methods, that will be proposed for the numerical solution

for (1,, are known to suffer numerical difficulties and significant loss ofaccuracy due to such operations. Nestegard(1984) and Zhao and Faltinsen(1989) illustrate the numerical deficiencies of the straightforward numerical evaluation of the m-terms.

An alternative approach to the evaluation of the integral (11.6.3) follows from the application of Stokes' theorem on it. It is shown in Appendix ILA, that if the basis flow (I. satisfies the rigid body condition on the hull surface, the following relation

holds

ff rn;

= (V43.(i') G(±"; i1)) n; di' ,

j =

(i) (T)

(11.6.4)

The right-hand-side of (11.6.4) involves only first derivatives of (1) on the hull, and consequently it is clearly superior, from the computational standpoint. The price to be paid for avoiding the direct evaluation of the m-terms is associated with the need to compute the influence of a tangential dipole distribution over the hull. Despite the fact that the related computational burden is significant, the use of (11.6.4) is definitely preffered.

the forcing of the integral equation is dropped. It appears that the approximation

V(I) = U: results in smaller errors if it is employed on the expression of the

right-hand-side of (11.6.4). Such an approximation preserves the tangential dipole distribution, although if affects its strength.

Alternative integral equations for the unknown potentials can be derived if a differ-ent Green function is considered and/or the source methodis used instead of Green's

method. Virtually all combinations of the above are reported in the literature in

connection with the wave resistance problem (see for example the proceedingsof the

First and Second International Workshops on Wave Resistance (1979, 1983)). As far as the ship motion problem is concerned, all previous three-dimensional studies

are based on the use of unsteady Neumann-Kelvin source potential as the Green

function. If the Neumann-Kelvin. condition is to be satisfied on the free surface, this

particular Green function satisfies the condition (II.2.1b) and allows, through the use of Stokes' theorem, the reduction of the integrals over the free surface to line integrals over the waterline of the vessel. The reduction of the size of the domain over which the resulting integral equation is defined, as well as the 'automatic' sat-isfaction of the proper radiation condition are considered to be the most significant advantages of this approach. On the other hand, the complicated expression for the chosen Green function shifts the computational burden towards the evaluation of the kernel of the integral equation and away from the solution of the system of linear equations which follows from the discretization of the problem.

The choice of the integral formulations to be used in the present study is based on two strategic decisions. The first, and most important one, is related to the

maxi-mization of the flexibility of the solution scheme as far as the linearization of the free

surface condition is concerned. Tailor-made Green functions can be derived only for free surface conditions with constant coefficients. On the contrary, the use of the Rankine source as the Green function allows the design of integral formulations ba.sed on any possible linearization of the free surface condition. The second

sion is related to the computer architecture that appears to prevail nowadays. The pipeline architecture of vector-supercomputers diminishes the computational cost associated with the highly-vectorizable process of solving systems of linear equa-tions, while it does not benefit equally much the efficiency of intrinsically complex evaluation of the specialized Green functions.

Err-APPENDIX II.A. EVALUATION OF THE INFLUENCE OF THE M-TERMS

Proposition

If is the potential of the double-body flow , = (ni n3) is the unit vector

normal to the hull (-1-3), and

(m1,m2,m3) =

V)Vcr) (m4,n2,5,m6) = V)(Ex ViI))

1 1

G(i;Zi)

Then, the following relation holds, for any point E E (B)

ff

(x-') G(E; dZi

f f

(V (1.(;') V.-, G(E; x-;)) n ,

(Ti) (I)

(II.A.1)

Proof

Using two different variations of Stokes' theorem, Ogilvie and Tuck(1969) showed that for any differentiable scalar function f(I)

40

If [ f (;') ( 7-1 ) ;1f7 ( x-; ) ri ( al. ( x-; ) * V 7(ZI)] d-xl

=

cil x f (x-; ) T,t7- . (Z )

_I-f [ _{f (-xl) (ri V)(i x W- ( - -'))}

(i x) (.

_{) - V f(Zi)]}

de =

(II.A.2)

tS)

= cil x f ( ;) t'fi (;') _{I x Z'}

(8S)

under the conditions that the _{vector field W. is irrotational, has zero divergence and}

also satisfies the condition W-. -ff._{= 0 on the surface (S). The surface (S) is assumed}

continuous and its boundary (8S) is _{described in a counterclockwise manner.}

Apply the relations (II.A.2) using the surface (B)D, as (S), where D, is the

neighborhood of any point E E (B)

. The Rankine source potential G(±; z') _is

differentiable on this surface, thus it can replace f (Zi), while the double-body flow

velocity Vc1) satisfies all the conditions for W.

The boundary line of (B)D, consists of the

_{waterline of the hull and a closed}

path C, that surrounds D. On the waterline it holds that

d-1 11 71.

If the neighborhood of the point E is let to shrink onto _{it can be considered to} be a planar circular disk of center and radius E 0, as illustrated in Figure 11.2. At this limit the following relations hold

G()V(1)(Z1)

=

=

L

( Cr 1 X G(E;;')V(1)(;')) X I-W L

d-1

* di

(Tsin 0 ± "COS 6' _{-I- IC 0)} {

VI) --+ _{V(1)(E) (Tcos 00 ± :/sin00 +1; 0)} x' (II.A.3) 2 Tr f d-.1 X G(Z;;')\ 71)(Zi) _Fc1V(1)(41f

_{de cos(0 +0,) = 0}

c 21r

(Cii X G(EjZ)V(D(.;')) X X 41774)(41

f

_{de cos(0}

+6Y = 0

Finally, as the radius of the disk E

_{0, the surface (T3)D, reduces to (ig) and}

from the use of (II.A.3-4) in (II..A.2) it follows :

[f (

;)(7-1. V)) ;)

V f (Z)] dZi = 0

(I)

(II.A.5)

If[f (Z1)(ii. V) x

(E x ri)((z-')

f (z-')] dz

= 0

(T)

which are unified into the relation (II.A.1) if the definition of the rn-terms is utilized.

- 11

44

111.1. ERROR ANALYSIS OF RANKINE PANEL METHODS IN FOURIER SPACE

This Section outlines a systematic methodology for the study of the discretization errors of Rankine panel methods for free surface flows. The error analysis is carried out in the Fourier space, rather than in the physical one. The basic definitions and theorems are included herein, while their application on the design and analysis of numerical schemes for the solution of the wave resistance and ship motion problems follows in the next four Sections.

The major skepticism about numerical schemes that require the discretization of the free surface, relates to their ability to model properly the generation and propa-gation of the fairly complicated ship wave patterns. Therefore, attention is focused on the errors due to the discretization of the infinitely extended free surface. The analysis of numerical errors due to the truncation of the free surface domain, as well as due to the discretization of the hull, may be studied separately and, subsequently, be coupled with the results of the present interior error analysis.

In general terms, the problem which needs to be solved may be expressed as follows

"Wc6(x,y) = 2(x, y) ,

where 14.) is a linear integrodifferential operator of convolution type, acting on the

solution q5. The right hand side of (III.1.1), referred to as the forcing, includes all

integrals over the body surface, as well as any other present forcing terms. The formulation (III.1.1) is defined over the two-dimensional infinite domain of the free

surface.

The Fourier Transform of the function 0(x, y) and its inverse are defined by the

following pair of equations

c13-(u,v)

=

di dy 4)(x,y) e-i("÷")

(III.1.2)

1

46(x, Y) =

(2702

where u and v are the Fourier wavenumbers in the x and _{y directions respectively.}

The translational nature of 14) allows the transformation of (III.1.1) into the Fourier

space,

c-i)(u,v) = 3- (u,v) , (III.1.3)

where the complicated operation of convolution is replaced by the very straightfor-ward multiplication. The function li.)(u, v) denotes the spectrum of the operator

.

The first step of every algorithm towards the numerical solution of (III.1.1) is the approximation of the solution 0(i, y) in terms of a finite number of degrees of freedom a5, by being expressed as a linear superposition of some basis functions B,( ,y) oo -00

f f

du dv (u, v) 46 CC

where the summation is carried over the vector index j =(j', jw) , and B.7"s'n) is

the two-dimensional basis function defined as the product of the basis functions in x and ydirections. The superscripts (m, ri) denote the order of the basis function and they will be discussed in more detail in the following Section.

If the collocation method is employed, the approximation (III.1.4) is plugged in the continuous formulation (III.1.1) and after the latter is satisfied at a finite set of

i(us-i-vy)

(z, = y) , (III.1.4)

ff

1-4)(u,v)

collocation points, (x,,yi), the discrete formulation is rendered

WOh(zi,tli) = R(1,,Yi)

The discrete formulation is a system of simultaneous linear equations, but for the purposes of this analysis it will be considered as the operation of the discrete op-erator W on the discrete function Oh.

It is very important to point out that the

translational type of the continuous operator "A) is inherited by its discrete counter-part W.

In analogy to the continuous case, the discrete formulation (III.1.5) is translated to the Fourier space. This step requires the introduction of the semi-Discrete Fourier

Transform. Let

(Xk YiZ) (Ci5h )k.i 7 (I1I.1.6)

be a sequence, or in other words a discrete function, defined over the two-dimensional discrete space of infinite extent and uniform spacings (hx,hy)

= lchx ,

y, = lhy

, (k, 1) E .Z2 . (III.1.7)

The semi-Discrete Fourier Transform of the discrete function Oh IS defined by

+co

Oh (14, v; hz, hy)

hhy

k,I= - co

The transform (jih is a continuous function of the wavenumbers (u, v). This property distinguishes it from the well-known Discrete Fourier Transform (otherwise known as Fourier Series), and it is a direct consequence of the fact that the grid, over which Oh IS defined, is of infinite extent. At this note, the prefix semi- may be eliminated since there is no danger for confusion.

(111.1.5)

The inverse of the transform (III.1.8) is defined as

(Z,k, yt) = (Oh ),k,i

w/h. w/hy

1

(27)2

f du f

dv cih(u,v) ei(u=.-r=1,0 . (III1.9)

Unlike the continuous inverse Fourier Transform, the integration in (III.1.9) is_over the finite domain [ir/hz,7r/hzlx [--7r/h, 7r/h1 which is referred to as the prineipa/

wavenumber domain. This is a manifestation of the fact that_{wave components with} wavelengths shorter than (2hz ,2h) cannot be resolved on the grid with spacings

(hz,h).

Trefethen(1988) discusses the properties of the semi-Discrete Fourier Transform. Two of those properties are critical in the present analysis and they will be

subse-quently stated without proof.

Discrete Convolution

Let fi,; and g,n; be two discrete functions defined over _{an infinite grid with}

uniform spacings (hz,hy). Their discrete convolution 42 _{is defined by}

If f(u,v) and"j(u,v) are the semi-Discrete Fourier Transforms of f and g1

re-spectively, then

ci(u,v)

= f(u,v)

"j(u,v) . (111.1.1 1)

Poisson Summation formula Aliasing Theorem

Let f (x,y) be a function with Fourier Transform f(u,v). If (A), is a sampling of f (x,y) on the grid with spacings (hz, hy), then the semi-Discrete Fourier Transform

fk,1

48 k,I=