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Time-domain second-order wave interactions with floating offshore structures Ng, Joseph Y. 1993

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TIME-DOMAIN SECOND-ORDER WAVE INTERACTIONS WITHFLOATING OFFSHORE STRUCTURESbyJoseph Yee-Tak NgB.E.Sc., University of Western Ontario, 1986M.E.Sc., University of Western Ontario, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1993© Joseph Yee-Tak Ng, 1993In presenting this thesis in partial fulfillment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available forreference and study. I further agree that permission for extensive copying of this thesis forscholarly purposes may be granted by the head of the department or by his or herrepresentatives. It is understood that copying or publication of this thesis for financial gainshall not be allowed without my written permission.Department of Civil EngineeringThe University of British Columbia2324 Main MallVancouver, B.C.Canada V6T 1Z4Date :AbstractOver the past decade, numerical modelling of nonlinear wave-structure interaction problemshas been an important topic of ocean engineering research, with practical applications relatingto load and response predictions for large floating structures subjected to steep waves. Withthe increasing need to incorporate more accurate analyses of wave effects into existing offshoredesign procedures, the emphasis of recent hydrodynamic research has been on thedevelopment of numerical solutions to a wide variety of nonlinear wave-structure interactionproblems. In particular, the present theoretical work considers the second-order waveradiation and combined diffraction-radiation problems in both two and three dimensions for thecase of large floating offshore structures of arbitrary shape. The mathematical formulation ofthe corresponding initial-boundary value problem can be derived on the basis of potential flowtheory.A recent time-domain approach is extended to simulate second-order hydrodynamic effectsinvolving large floating structures which cannot be obtained by the conventional linearhydrodynamic theory. The method involves the application of Taylor series expansions andthe use of Stokes perturbation procedure to establish the corresponding first- and second-orderboundary value problems with respect to a time-independent fluid domain. A time-steppingscheme together with a suitable iterative procedure are used to solve the coupled fluid-structuregoverning equations, and an integral equation method based on Green's theorem is used toobtain the wave field at each time step. In relation to conventional nonlinear methods, thepresent method provides a relatively algebraically straightforward and computationally effectivenumerical algorithm for treating the second-order free surface flow problems.The method is used to study the second-order wave radiation problem in two and threedimensions, in which surface-piercing structures of arbitrary shape undergo forced sinusoidalmotions in otherwise still water. The method is illustrated by numerical results obtained fromtwo semi-submerged circular and rectangular cylinder sections and a truncated surface-piercingcircular cylinder. The second-order oscillatory force component due to second-order wavepotential, for both the two-dimensional vertical plane case and the general three-dimensionalcase, contributes significantly in the evaluation of the total hydrodynamic forces.Although the second-order problems have been studied quite extensively in the context ofwave diffraction and radiation separately, results for the nonlinear diffraction-radiation problemare rather scarce. This combined problem involves the equations of motion of the structure aswell as nonlinear interactions between the incident, diffracted and radiated wave components.The present method is subsequently extended to simulate second-order wave interactions withlarge floating structures in two and three dimensions, and is illustrated by applying respectivelyto the cases of a semi-submerged circular cylinder and of a floating truncated circular cylinder.Numerical computations presented relate to the transient motion of a freely-floating cylinderwith a specified initial vertical displacement, and the diffraction-radiation of Stokes second-order waves by a moored floating cylinder. In general, numerical results indicate significantsecond-order hydrodynamic effects in the forces and motions of large floating structuressubjected to regular waves, as well as in the corresponding free surface profiles and waveamplitudes.iiiX VTable of ContentsAbstractList of TablesList of FiguresList of SymbolsAcknowledgements1. INTRODUCTION 11.1 Background ^ 11.2 Literature Review 41.2.1 Frequency-Domain Methods ^ 51.2.2 Time-Stepping Methods 81.2.3 Second-Order Hydrodynamic Effects ^ 101.3 Scope of Present Study ^ 132 . MATHEMATICAL FORMULATION 162.1 Fully Nonlinear Problem ^ 162.2 Stokes Second-Order Method 202.2.1 Second-Order Expansion ^ 202.2.2 Three-Dimensional Problem 232.2.3 Two-Dimensional Problem ^ 272.2.4 Boundary Integral Equation 292.3 Free Surface Elevation ^ 332.4 Hydrodynamic Forces and Moments ^ 34iv3 . NUMERICAL PROCEDURE^ 433.1 Field Solution ^  433.2 Time-Stepping Scheme ^  463.2.1 Radiation Condition  473.2.2 Free Surface Boundary Conditions ^  484. RESULTS AND DISCUSSION^ 514.1^Introduction  ^514.2 Computational Considerations ^  524.2.1 Surface Discretization  524.2.2 Initial Conditions ^  544.2.3 Numerical Accuracy  554.3 Radiation Problems ^  574.3.1 Two-Dimensional Radiation Problem ^  584.3.2 Three-Dimensional Radiation Problem  624.4 Wave-Structure Interaction Problems ^  654.4.1 Transient Motion of Semi-Submerged Circular Cylinder ^ 654.4.2 Two-Dimensional Diffraction-Radiation Problem ^ 664.4.3 Transient Motion of Truncated Circular Cylinder  704.4.4 Three-Dimensional Diffraction-Radiation Problem ^ 715. SUMMARY AND CONCLUSIONS^ 755.1 Second-Order Time-Domain Solution  755.2 Numerical Results ^  784.4 Recommendations for Further Study ^  81References^ 83List of Tables1 Block diagram of the time-stepping scheme ^  892. Hydrodynamic force components in the z direction as functions of Nb for a semi-submerged circular cylinder in heave motion, to2a /g = 0.6, A/a = 0.1 and deepwater ^  903. Hydrodynamic force components in the z direction as functions of Nb for a semi-submerged rectangular cylinder in heave motion, web /g = 0.6, A/b = 0.1 and deepwater ^4. Hydrodynamic force components in the z direction as -functions of Nbl for atruncated circular cylinder in heave motion, ka =1.0, d/a = 1.5, h/a = 0.5 and A/a. =0.19091viList of Figures1. Definition sketch. (a) three-dimensional problem. (b) two-dimensional problem ... 922. Body geometries of illustrative computations. (a) semi-submerged circularcylinder. (b) semi-submerged rectangular cylinder. (c) truncated surface-piercingcircular cylinder  ^943. Examples of surface discretization in two dimensions. (a) semi-submerged circularcylinder Nb = 20 and NT = 138. (b) semi-submerged rectangular cylinder Nb = 30and NT = 150 ^  954. Examples of surface discretization of a truncated surface-piercing circular cylinderin three dimensions. (a) Nb = 378 and NT = 2298. (b) Nb = 378 and NT = 2108 .. 965. Development with time of vertical displacement of a semi-submerged circularcylinder, free surface elevation on body surface and vertical hydrodynamic forcecomponents for o.)2a/g = 1.6, A/a = 0.1 and deep water ^ 976. Development with time of free surface profiles for a semi-submerged circularcylinder undergoing forced heave motion, co2a/g = 1.6, A/a = 0.1 and deep water ^ 987. Amplitude and phase angle of first-order oscillatory force in the z direction on asemi-submerged circular cylinder in heave motion as functions of excitationfrequency for deep water ^  998. Magnitude of second-order steady force in the z direction on a semi-submergedcircular cylinder in heave motion as functions of excitation frequency for deepwater ^  1009. Amplitude of second-order oscillatory force in the z direction on a semi-submergedcircular cylinder in heave motion as functions of excitation frequency for deepwater ^  10110. Phase angle of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in heave motion as functions of excitation frequencyfor deep water ^  10111. Composition of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in heave motion as fiinctions of excitation frequencyfor deep water ^  10212. Amplitude and phase angle of first-order oscillatory force in the x direction on asemi-submerged circular cylinder in sway motion as functions of excitationfrequency for deep water ^  10313. Magnitude of second-order steady force in the z direction on a semi-submergedcircular cylinder in sway motion as functions of excitation frequency for deepwater   104vii14. Amplitude of second-order oscillatory force in the z direction on a semi-submergedcircular cylinder in sway motion as functions of excitation frequency for deepwater ^  10515. Phase angle of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in sway motion as functions of excitation frequency fordeep water ^  10516. Composition of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in sway motion as functions of excitation frequency fordeep ater   106- 17. Magnitude of second-order steady force and composition of second-orderoscillatory force in the z direction on a semi-submerged rectangular cylinder inheave motion as functions of excitation frequency for deep water   10618. Magnitude of second-order steady force and composition of second-orderoscillatory force in the z direction on a semi-submerged rectangular cylinder insway motion as functions of excitation frequency for deep water ^ 10719. Magnitude of second-order steady force and composition of second-orderoscillatory force in the z direction on a semi-submerged rectangular cylinder in rollmotion as functions of excitation frequency for deep water ^ 10720. Phase angle of second-order oscillatory force in the z direction on a semi-submerged rectangular cylinder as functions of excitation frequency for deep water. 10821. Development with time of vertical displacement of a truncated circular cylinder andvertical hydrodynamic force components for ka = 1.6, d/a = 1.5, h/a = 0.5 and A/a= 0.1 ^  10822. Oblique views of the free surface to second order for ka = 1.6, d/a = 1.5, h/a = 0.5and t = 3.5T. (a) heave motion. (b) surge motion ^  10923. Amplitude and phase angle of first-order oscillatory force in the z direction on atruncated circular cylinder in heave motion as functions of ka for dia. = 1.5, h/a =0.5 ^  11024. Magnitude of second-order steady force and composition of second-orderoscillatory force in the z direction on a truncated circular cylinder in heave motion asfunctions of ka for d/a = 1.5, h/a = 0.5 ^  11125. Phase angle of second-order oscillatory force in the z direction on a truncatedcircular cylinder in heave motion as functions of ka for d/a = 1.5, h/a = 0.5^. . . . 11126. Amplitude and phase angle of first-order oscillatory force in the x direction andmoment about the y-axis on a truncated circular cylinder in surge motion asfunctions of ka for d/a = 1.5, h/a = 0.5 ^  11227. Magnitude of second-order steady force and composition of second-orderoscillatory force in the z direction on a truncated circular cylinder in surge motion asfunctions of ka for d/a = 1.5, h/a = 0.5 ^  113viii28. Phase angle of second-order oscillatory force in the z direction on a truncatedcircular cylinder in surge motion as functions of ka for dia. = 1.5, h/a = 0.5 ^29. Development with time of vertical displacement and vertical hydrodynamic forcecomponents for a semi-submerged circular cylinder undergoing transient heavemotions with IAI/a = 0.1 and d/a = 10.0 ^30. Development with time of free surface profiles to second order for a semi-submerged circular cylinder undergoing transient heave motions with IAI/a = 0.1and d/a = 10.0. Successive profiles are at times 0.4T apart     11531. Development with time of the motion response and hydrodynamic forcecomponents for a moored semi-submerged circular cylinder in waves for 0) 2a/g =0.4, A/a = 0.3 and deep water ^  11632. Development with time of free surface profiles for a moored semi-submergedcircular cylinder in waves for w2a/g = 0.4, A/a = 0.3 and deep water ^ 11733. Amplitude and phase angle of first-order oscillatory forces on the cylinder asfunctions of 0.)2a/g for deep water ^  11834. Magnitude of second-order drift forces on the cylinder as functions of c02a/g fordeep water ^  11935. Amplitude and phase angle of second-order oscillatory forces on the cylinder asfunctions of (02a/g for deep water ^  12036. Amplitude and phase angle of first-order oscillatory motion response of the cylinderas functions of (02a/g for deep water  12137. Magnitude of second-order drift motion response of the cylinder as functions of(D2a/g for deep water ^  12238. Amplitude and phase an of second-order oscillatory motion response of thecylinder as functions of aiza/g for deep water ^  12339. Development with time of vertical displacement and vertical hydrodynamic forcecomponents for a floating truncated circular cylinder undergoing transient heavemotions with IAI/a = 0.1, d/a = 3.0 and h/a = 0.5 ^  12440. Oblique views of free surface profiles to second order for a floating truncatedcircular cylinder undergoing transient heave motions with IAI/a = 0.1, dia. = 3.0 andh/a = 0.5. (a) t = 7.73,1 a/g. (b) t = 13.92-Talg. (c) t = 20.101raTg. (d) t =26. 291Thig ^41. Development with time of the motion response and hydrodynamic forcecomponents for a moored truncated circular cylinder in waves for ka = 1.0, d/a =2.0, h/a = 0.5 and A/a = 0.2 ^42. Oblique views of free surface profiles to second order for a moored truncatedcircular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 and A/a = 0.2. (a) t =0.2T. (b) t =0.7T  113114125126127ix43. Perspective and contour plots of first-order wave amplitude of a moored truncatedcircular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 and A/a = 0.2.(Contours shown are normalized with respect to incident wave amplitude.) ^ 12844. Perspective and contour plots of wave crest amplitude to second order of a mooredtruncated circular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 and A/a =0.2. (Contours shown are normalized with respect to incident wave amplitude.) ^ 12945. Perspective and contour plots of second-order mean water surface elevation of amoored truncated circular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 andA/a = 0.2. (Contours shown are normalized with respect to incident waveamplitude.) ^  13046. Perspective and contour plots of second-order oscillatory wave amplitude of amoored truncated circular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 andA/a = 0.2. (Contours shown are normalized with respect to incident waveamplitude.) ^  131List of SymbolsThe following symbols are used in this thesis:a = radius of circular cylinder,A = amplitude of forced body motion; and incident wave amplitude;1A1 = initial vertical displacementAw = mean waterplane area;Aij, Bij = influence matrix coefficients;b = half beam length;cn = celerity in normal direction n;d = water depth (see Fig. 1);D = fluid domain bounded by Sb, So, and Sc;= fluid domain bounded by Sw,, Sf, and Sc;F = force vector;Fm = modulation function, (defined in Eqn 4.1);g = gravitational constant;G = moment vector,G(x, x') = Green's function;h = draft of circular cylinder,H = incident wave height;Hb = breaking wave height;H = matrix containing quadratic products of first-order rotationalmotions, (defined in Eqn 2.21);j, k = unit vector in x, y and z direction respectively;I = inertia matrix;k = wave number;K44 , K4a , K a4, Kota = structural stiffness sub-matrices;xiL = wavelength;1-Tin = waterplane area moment of inertia;Lb = length of two-dimensional computational domain;M = mass matrix;n = distance in direction of normal n;n = unit normal vector in equilibrium position;N = distance in direction of normal N;N = instantaneous unit normal vector,NT = total number of facets;Nb = number of facets on body surface;p = fluid pressure;r = radial distance between x and x';R = rank of matrix equation;s = tangential direction on mean body surface;S' = integration surface composed of Sb, So, and S o;Sb = body surface below still water level (see Fig. 1);So = control surface (see Fig. 1);Sf = instantaneous free surface (see Fig. 1);So ;=-- still water surface (see Fig. 1);Sw = instantaneous wetted body surface (see Fig. 1);t = time;T = wave period;Tm = modulation time;u = fluid velocity;U = Ursell parameter,wo = waterline contour,xb, yb, zb = location of centre of buoyancy;xiixf, yf = longitudinal and transverse coordinates of centre of floatation;x = position vector of field point;x' = position vector of source point;X = inertial coordinate system (see Fig. 1);X = body-fixed coordinate system (see Fig. 1);X g = position vector of centre of mass;An = characteristic facet length in normal direction n;At = time step size;V = displaced volume of body;= rotational vector;5 = Courant number,Sij = Kronecker delta function;e = perturbation parameter,cl) = velocity potential;rl = free surface elevation (see Fig. 1);0 = azimuthal angle measured from positive x-axis;p = fluid density;= angular frequency;co = angular velocity vector, and4 = translational vector.The following subscripts are used in this thesis:0 = steady component;1 = component at first order;2 = component at second order;k = index for image source point; and order of problem; andx, y, z = directions of axis.The following superscripts are used in this thesis:(1) = component due to first-order potential;(2) = component due to second-order potential;B = wave disturbance component; andW = incident wave component.AcknowledgementsI would like to express my sincere thanks and gratitude to my supervisor, Dr. MichaelIsaacson, for his advice and guidance during this study. His introduction of the intriguingsubject of wave hydrodynamics to a novice is greatly appreciated.Many thanks are extended to the past and present fellow graduate students of the Coastaland Ocean Engineering Group for their many inspiring discussions. Their criticisms have beenpart of my education experience.A special note of appreciation is due to Mr. Tom Nicol of the University ComputingServices for lending his expertise on numerically intensive computation.I owe a great deal of thanks to my parents for their unfailing support and encouragementthroughout my venture of overseas study. They are the important driving force behind such anundertaking. This thesis is dedicated to them.I am deeply indebted to my girl friend, Phyllis, for her support, understanding and patienceduring the long period of my absence. Her consideration has made this dilemma much morebearable.Finally, financial support for this study was provided by the Izaak Walton Killam Trust andthe University of British Columbia through their graduate fellowships. The benevolence ofthese institutions is gratefully acknowledged.xvCHAPTER 1INTRODUCTION1.1 BackgroundPredictions of wave loads and motion responses for large floating structures have generallybeen obtained on the basis of linear diffraction-radiation theory which is formally valid for thecase of small amplitude sinusoidal waves. More specifically, the mathematical formulationcan be derived with respect to the separate wave diffraction and radiation problems whichcorrespond respectively to the scattering of incident waves by a fixed structure, and to thewaves generated by the prescribed rigid body harmonic motions in otherwise still water. In alinear hydrodynamic analysis, the potential flow solutions to these two complementaryboundary value problems can be superposed to represent the resulting fluid motions around afloating structure in regular incident waves, commonly referred to as the combined problem ofwave diffraction-radiation. Analytical methods for these linear problems are well established,and detail reviews of the available approaches have been given in the literature (e.g.Wehausen, 1971; Mei, 1978; Sarpkaya and Isaacson, 1981; and Yeung, 1982a).The offshore structures considered in this thesis fall into the category of large floatingstructures which span a significant portion of a wavelength. Large floating structures whichhave been commonly used in offshore exploration and production include semi-submersibles,tension leg platforms, ship and barges, as well as gravity platforms and similar structuresprior to installation. With respect to coastal engineering applications, floating breakwatersused to provide wave protection to coastal regions and harbours may also be considered.In estimating wave-structure interaction effects in the design of such marine structures, amodification to the incident wave field associated with wave diffraction and radiation becomesan important consideration. In this context, the present study treats the second-order wave1radiation and combined diffraction-radiation problems in both two and three dimensions forthe case of large floating structures of arbitrary shape on the basis of a time-domain method.Second-order wave effects of the corresponding wave radiation and diffraction-radiationproblems are predicted respectively for forced sinusoidal rigid body motions in otherwise stillwater, and for uni-directional, regular Stokes second-order waves. In order to achieve moreprecise descriptions of realistic sea states, it would be necessary to extend the simplified waveconditions considered here to account for the various complications such as irregular waves,short-crested waves, combined waves and currents, wave groups and breaking waves.Due to the increasing demand from the offshore industry to refine existing designprocedures, the emphasis of recent hydrodynamic research has been on the development ofanalytical solutions which account for various nonlinear effects of steep waves encountered inengineering applications. In the light of rapid advances and innovative developments in thefields of computational mechanics and computer technology during the past decade, the use ofnumerical simulation in treating nonlinear wave-structure interaction problems has recentlybeen made possible, and has been widely accepted as a powerful tool in the design andanalysis of advanced offshore structures.Fully nonlinear solutions are needed to study several categories of free surfacehydrodynamic problems associated with highly nonlinear wave effects. Examples related tooffshore engineering in which wave nonlinearities are an important concern include largeamplitude motions of ships in extreme wave conditions, liquid sloshing in storage tanks, bowand stern flows and nonlinear ship wave resistance. In solving these nonlinear waveproblems, numerical computations in the time domain have been performed based on anumber of time-stepping methods. On the other hand, weakly nonlinear wave problems canbe pursued by the Stokes perturbation analysis up to second order in terms of a wavesteepness parameter. Most previous studies have formulated second-order problems usingmore conventional frequency-domain methods in which steady state hydrodynamic forces andmotions are treated as harmonic in time.2For the case of monochromatic regular waves, second-order wave forces contain a steadyforce component, generally known as the wave drift force, and a bi-harmonic wave force attwice the incident wave frequency. In the presence of irregular waves, the interactions of apair of regular wave components in a wave group will give rise to second-order wave forces atsum and difference frequencies of the waves. Although the magnitudes of these second-orderforces, proportional to the square of the incident wave amplitude, are in general smallcompared to the first-order loads at wave frequencies, they become important with respect tothe motions and mooring loads of a floating structure when the frequencies of such excitationsfall close to the natural periods of the body motions, or when restoring and damping forces aresmall. For slowly varying drift forces (difference-frequency excitations), typical examples .include the horizontal excursions of moored ships and the vertical plane motions of smallwaterplane area vessels such as semi-submersibles and small waterplane area twin hull(SWATH) ships. In addition, the high frequency forces (sum-frequency excitations) areassociated with the "ringing" of tension leg platforms and the "springing" vibrations of shiphulls. Due to the various nonlinearities arising in wave induced forces and motion inducedreactions, the linear potential flow solution alone, in most cases, leads to significant errors inthe predictions of these second-order hydrodynamic effects which are outside the spectralrange of first-order wave frequencies.In general, calculations of sum- and difference-frequency excitation forces on largeoffshore structures require the solution of the first-order problem relating to the wave field andbody motions; as well as the corresponding second-order wave potential. While the formercontribution can be obtained by using various linearized potential flow treatments, the majordifficulty of the latter is the solution of the second-order wave-structure interaction problem.Owing to the immense difficulties of solving the second-order problem, most existinganalytical models are invariably restricted to simple two or three-dimensional body geometries,and the validity and accuracy of these are still a major source of discussion in the literature.3Despite considerable progress having been made on wave-induced force and responsepredictions, the understanding on various aspects of nonlinear wave effects is far fromcomplete. To the author's knowledge, a satisfactory general nonlinear solution of complexfloating structures such as semi-submersibles and tension leg platforms has not yet beenreported. As a step towards this direction, the present study extends a time-domain second-order method recently developed by Isaacson and Cheung (1991, 1992) in the context of wavediffraction, which relates to wave interactions with fixed structures, to study nonlinearhydrodynamic problems involving two- and three-dimensional large floating structures.1.2 Literature ReviewTheoretical calculations of steep wave effects on large floating structures generally require thesolution to a fully nonlinear boundary value problem. Complications of this nonlinearproblem are that the solution is required to satisfy the two nonlinear free surface boundaryconditions and the kinematic body surface boundary condition at the free surface and thewetted body surface respectively, both of which are unknown a priori, and the formulationshould include a correct treatment of the radiation (far-field) condition. A survey of previouswork indicates that there are two categories of method available to treat the nonlinear wave-structure interaction problem. One approach is the second-order frequency-domain solutionbased on the Stokes perturbation procedure, and the other is the fully nonlinear solution to theresulting wave field which involves a time-stepping procedure.In the past few decades, second-order hydrodynamic phenomena which are especiallyimportant for floating structure applications have received considerable attention. Earlystudies were motivated by the technological need to include these effects into the design ofmooring systems for floating vessels. By simplifying the hydrodynamic problem, a numberof approximate methods which originated from heuristic arguments are used in engineeringcalculations to estimate the slow drift and sum-frequency excitation forces. Many of thesesolutions neglect the contributions of second-order wave components. With the recent4success in the development of accurate second-order solutions (e.g. Kim and Yue, 1989b;Chau and Eatock Taylor, 1992; and Isaacson and Cheung, 1992), the focus of research effortin the foreseeable future will be on the modifications and refinements of these computationalmethods and analytical solutions to incorporate various practical considerations such ascomplex structure configurations, wave directionality and irregular wave excitation.1.2.1 Frequency-Domain MethodsIn the perturbation methods, the fully nonlinear problem is reduced to a sequence of linearboundary value problems, one at each perturbation order, with respect to a time-independentfluid domain, by making use of the Stokes expansion procedure. Most frequency-domainmethods have formulated the nonlinear problem up to second order, and the study of wavediffraction by fixed structures has attracted particular attention. These frequency-domainsolutions require a number of elaborate computational considerations. In general, suchmethods enforce a weak far-field condition and are considered to be algebraically complicated.In his seminal work, Molin (1979) proposed that the second-order solution contains bothforced wave motions (phase-locked waves) associated with the quadratic forcing terms in thefree surface boundary conditions, and free wave motions (free waves) associated with theinteraction of the forced waves with the structure. If primary interest is restricted tohydrodynamic loads and not the detailed flow kinematics, including the free surface profileand wave runup, the second-order forces can be derived without explicitly solving the second-order potential problem. By utilizing the Haskind reciprocal relationship and the asymptoticbehaviour of the second-order wave components, second-order forces due to the second-orderpotential are thereby expressed in terms of free surface and body surface integrals involvingfirst-order quantities and associated radiation potentials oscillating at twice the incident wavefrequency. This indirect approach was first presented by Lighthill (1979) for infinite waterdepth, and by Molin (1979) for three-dimensional bodies in arbitrary depths. Molin and5Marion (1986) modified the method in order to calculate second-order wave forces for afloating axisymmetric body and limited numerical results are presented.Eatock Taylor and Hung (1987) adopted the theory of Molin to investigate the second-orderdiffraction problem. The major part of their work has been focussed on the asymptoticbehaviour of the free surface integral and the integral on the control surface at far-field whichare required to calculate the second-order wave forces. Their analytical solution for a verticalcircular cylinder was subsequently extended by Eatock Taylor et al. (1989) to calculate thesecond-order local pressure on the submerged body surface. More recently, Chau and EatockTaylor (1992) developed a more general semi-analytical solution for the distribution ofpotential and the associated flow kinematics within the fluid region. Abul-Azum and Williams(1988) expressed the linearized radiation potentials as eigenfunction expansions, andnumerical results were presented for truncated vertical circular cylinders. A subsequentextension of this formulation was made by Ghalayini and Williams (1989) in order to calculatesecond-order wave diffraction around a vertical cylinder of arbitrary cross section.A direct approach involves a complete solution to the second-order boundary value problemby employing an integral equation method based on Green's theorem. The wave diffractionproblem for the case of two-dimensional cylinders of arbitrary shape submerged in water ofinfinite depth has been investigated by such frequency-domain formulations (e.g. Kyozuka,1980; and Vada, 1987). The agreement between Vada's second-order results and theexperimental measurements of Chaplin (1984) is excellent. In addition, experimental resultsof the nonlinear wave forces acting on three types of horizontally submerged cylinders werereported by Inoue and Kyozuka (1988).With respect to the more general three-dimensional problems, direct second-order wavediffraction solutions are, in most cases, restricted to the simple geometry of a bottom-mounted, surface-piercing, vertical circular cylinder. Hunt and Baddour (1981) obtained ananalytical solution for the second-order velocity potential based on a Fourier-Bessel integralmethod for a vertical circular cylinder in deep water condition. Chen and Hudspeth (1982)6presented a second-order diffraction solution based on an eigenfunction expansion for Green'sfunction. The problem was decomposed into two separate linear boundary value problemshaving only one inhomogeneous boundary condition (either free surface or body surface) ineach case. Garrison (1984) proposed a solution method for the second-order velocitypotential based on a source distribution method using Green's functions. This approach issimilar to that of the first-order problem except that there is contribution from quadratic forcingterms of the free surface integral, and the Green's function for the second-order scatteredwave potential is assumed to satisfy the homogeneous free surface boundary condition attwice the incident wave frequency and Sommerfeld radiation condition. Sclavounos (1988)presented a second-order Green's function method to treat the second-order diffraction-radiation problem for three-dimensional bodies of arbitrary shape in deep water. However, nocomputational results were given to illustrate the numerical procedure.Recently, Kim and Yue (1989b) presented an elaborate second-order diffraction solutionfor axisymmetric structures by using a ring source integral equation method. Extensivenumerical results were obtained for a vertical circular cylinder and a truncated cone. Theirtechnique was subsequently extended to a consideration of bichromatic incident waves, aswell as body motions (Kim and Yue, 1990). By considering the far-field behaviour of thefree surface quadratic forcings, Newman (1990) derived an approximate solution for thesecond-order wave potential at large depths of submergence. Only a few preliminary resultswere compared with published data for some simple body shapes. Based on an eigenfunctionexpansion, Kriebel (1990) presented an analytical wave diffraction solution by a verticalcircular cylinder, which revealed physical features of nonlinear wave components.The related nonlinear wave radiation problem involving a rigid body undergoing forcedoscillations in otherwise still water has important applications in naval hydrodynamics and hasalso been studied extensively. Most previous studies have formulated the second-orderproblem using frequency-domain approaches, and only the case of two-dimensional bodieshas been considered. Parissis (1966) first used an integral equation method to calculate the .7second-order hydrodynamic forces on a horizontal circular cylinder heaving in the freesurface. Lee (1968) presented a second-order potential theory based on the multipoleexpansions method for both two-dimensional horizontal circular and U-shaped cylindersundergoing forced heave motions.Potash (1971) developed a second-order solution for the cases of sway, heave, roll andtheir coupling motions by an integral equation method, and presented numerical results forthree surface-piercing cylinder sections. SOding (1976) expressed second-order forces due tothe second-order radiation potential in terms of free surface and body surface integralsinvolving only first-order quantities. Yamashita (1977) presented an approximate third-orderwave radiation solution without considering the nonlinear free surface boundary conditionsand the velocity squared term of the Bernoulli equation in the wave force calculations, andobtained experimental results of several semi-submerged cylinders heaving in the free surface.Papanikolaou and Nowacki (1980) presented an integral equation method to study thesecond-order wave radiation problem in two dimensions, and numerical results for differentcylinder shapes were provided. Kyozuka (1981) studied the second-order wave radiationproblem in two dimensions by an integral equation method. Kyozuka (1982) conductedexperiments in wave radiation problem for a circular cylinder subject to heave and swayoscillations and comparisons were made with theoretical results.1.2.2 Time-Stepping MethodsIn time-stepping methods, the fully nonlinear problem is directly formulated with respect to atime-dependent fluid domain. With the nonlinear free surface boundary conditions applied atthe instantaneous free surface, a new system of linear algebraic equations is assembled andsolved at each time step as the free surface moves to a new position. Computationalconsiderations associated with the treatment of the body and free surface intersection line, thestability of the time-integration scheme, and the implementation of the far-field closure maylimit the numerical accuracy and duration of simulation time. In general, such methods8demand substantial computing resources and involve special numerical techniques such asartificial smoothing and regriding algorithms.Longuet-Higgins and Cokelet (1976) first introduced the mixed Eulerian-Lagrangianmethod to study two-dimensional steep waves as well as plunging wave breakers. Faltinsen(1977) adopted a related approach to treat nonlinear problems involving an oscillating body indeep water. The novel approach of Longuet-Higgins and Cokelet was subsequently modifiedto a consideration of floating bodies in finite water depth (e.g. Vinje and Brevig, 1981; andGreenhow et al., 1982). This solution scheme calculates the fluid motion in the context ofcomplex potential based on Cauchy's integral theorem, and applications of the method arerestricted to two-dimensional free surface flow problems.Since then, considerable progress has been made on the accurate simulation of a variety oftwo-dimensional flows. Lin et al. (1984) investigated breaking waves generated by a wave-maker with an emphasis on resolving the numerical treatment of the body and free surfaceintersection problem and matching boundary conditions best known as corner problem. Thisparticular problem can be circumvented by specifying both the potential and stream functionon the body surface portion of the corner. Dommermuth et al. (1988) presented experimentaland theoretical results for steep and overturning waves produced by a piston wave-maker in awave flume. Grosenbaugh and Yeung (1989) treated the nonlinear wave motions of severaltwo-dimensional, translating bows by improving Vinje and Brevig's method. Recently, time-stepping methods have also been applied to studies of wave diffraction by two-dimensionalsubmerged bodies (e.g. Jagannathan, 1988; and Cointe, 1989).By incorporating an extensive numerical approach, Dommermuth and Yue (1987) extendedthe mixed Eulerian-Lagrangian method to treat axisymmetric wave-structure interactionproblems using a ring source distribution method. For general three-dimensional nonlinearproblems involving fixed and floating bodies, Isaacson (1982) presented a time-steppingmethod by using an integral equation method based on Green's theorem. This approachinvolves a lengthy computational time, and the possible numerical instabilities associated with9the treatment of far field closure may limit the useful length of simulation time. With theapplication of the Orlanski radiation condition, Yang and Ertekin (1992) adopted an integralequation method to treat the nonlinear wave diffraction of solitary waves and Stokes waves bya vertical circular cylinder. Recently, a boundary-fitted coordinate transformation techniquewas employed by Wang et al. (1992) to solve the generalized Boussinesq model governingthree-dimensional wave diffraction in shallow water.An alternative time-domain approach, which may be considered as a hybrid of thefrequency-domain and the time-stepping methods, has recently been developed by Isaacsonand Cheung (1991, 1992) to treat wave diffraction by structures for the two-dimensionalvertical plane case and the general three-dimensional case respectively. In this approach, theapplication of Taylor series expansions and the use of Stokes perturbation procedure reducethe fully nonlinear problem to first- and second-order boundary value problems with respect toa time-independent computational domain. A time-stepping scheme is used to treat the freesurface boundary conditions and radiation condition, and the wave field is then evaluated ateach time step by an integral equation method based on Green's theorem. Since the matrixcoefficients in the discretized boundary integral equation are functions of body geometry only,the linear system of simultaneous equations is solved only once, resulting in a reduction incomputational effort in comparison with most conventional nonlinear methods.1.2.3 Second-Order Hydrodynamic EffectsWith increasing engineering importance of second-order hydrodynamic effects in theapplications of floating structures, the study of slowly varying wave drift forces (difference-frequency excitations) and high frequency wave forces (sum-frequency excitations) hasattracted considerable research interest. These second-order forces give rise to large resonantresponses which may be many times larger than the first-order wave frequency response,since compliant offshore structures and moored vessels usually have small restoring anddamping forces. Many of the theoretical and experimental investigations were initiated by the10need to predict these nonlinear forces and motions of floating structures in regular andirregular waves.A comprehensive review of available theoretical approaches for predicting second-ordereffects on floating offshore structures has been given by Ogilvie (1983). In general, second-order forces acting on fixed or floating bodies have been evaluated by two theoreticalapproaches. The far-field method (e.g. Maruo, 1960; and Faltinsen and Michelsen, 1974)equates the mean second-order forces to the total change of fluid momentum flux. Thisapproach is restricted to the mean components only, and the computation of the vertical planeforces and moments involves a number of complications. The alternative near-field approach(e.g. Pinkster, 1980; and Standing and Dacunha, 1982) involves a direct integration ofhydrodynamic pressure contributions over the instantaneous wetted body surface, and generalexpressions can be derived for all hydrodynamic force components to second order. Thismethod provides physical insight into the resulting forces and indicates the relativesignificance of each component.For the case of regular waves, the wave drift forces can simply be obtained from second-order contributions of the first-order potential and body motions. Maruo (1960) derived anexpression to calculate the horizontal drift force for a ship in beam seas based on momentumconsiderations, but his work does not treat the case of oblique seas. Newman (1967) adopteda different approach to calculate the mean drift yaw moment, and rederived the force results ofMaruo. The analytical results are based on the assumptions of slender body theory togetherwith long wavelength. Faltinsen and Michelsen (1974) modified Newman's method tocalculate steady drift forces on three-dimensional structures in finite water depth using thesource distribution method. Salvesen (1974) showed that the second-order steady drift forcesand moments on a ship advancing in oblique regular waves can be expressed in terms of theinteractions of the body generated waves and incident waves. His numerical results werepresented within the framework of the ship motion strip theory. Molin and Hairault (1983)11compared the near-field and far-field methods of vertical wave drift force calculations withexperimental results for the case of three-dimensional axisymmetric bodies in regular waves.Hsu and Blenkarn (1970) represented the drift forces by dividing the irregular wave systeminto approximate regular wave portions. The structure is assumed to experience a steadyhorizontal force and yaw moment in each regular wave. Remery and Hermans (1971)conducted model tests with a rectangular barge to measure the drift forces in regular andgrouped waves. Their experimental results were compared with the calculated drift forcesusing Hsu and Blenkarn's method of analysis. Pijfers and Brink (1977) calculated the driftforces on two semi-submersibles due to different current and mass transport velocities ofwaves using the Stokes second-order waves and structural motions.By assuming a narrow banded wave spectrum, Newman (1974) presented a simplifiedapproach in which the quadratic transfer function of low frequency forces can beapproximated from the mean second-order wave forces in regular waves. From a number ofnumerical simulations for two-dimensional bodies, Faltinsen and Loken (1979) confirmed thatNewman's approximate approach, in most cases, is a satisfactory engineering method forcalculating drift forces in irregular waves. Pinkster (1980) applied the direct integrationmethod to calculate the second-order force quadratic transfer function for floating bodies ofarbitrary shape, and the linear flow problems were solved by three-dimensional potentialtheory. By refuting Pinkster's method, similar computations have been made by Standing andDacunha (1982). Recently, Kim and Yue (1989a) studied the effects of wave directionality onthe slowly varying drift forces using Newman's approximation. For sum-frequencyexcitations, approximate solutions were presented by Herfjord and Nielsen (1986) on acylinder in irregular waves and by Petrauskas and Liu (1987) on a tension leg platform. Dueto the immense difficulties of solving the second-order problem, these methods ignored orprovided only approximations for contributions due to the second-order wave potential. Untilrecently when accurate solutions of second-order wave diffraction have become viable, theseanalyses were subsequently extended to compute slowly varying and sum-frequency12hydrodynamic forces on various three-dimensional structures (e.g. Matsui, 1989; and Kimand Yue, 1991).Traditionally, slow drift motions of the floating structure have been analyzed by a linearresponse model and the second-order drift excitation forces based on frequency-domainformulation have been calculated without accounting for the effects of slow drift motions. Afloating structure will experience additional damping known as wave drift damping due to theslowly varying velocity of the structure and the subject was studied by Wichers and van Sluijs(1979). Triantafyllou (1982) proposed that the large amplitude slow drift motions can bedecoupled from the small amplitude relatively high frequency motions by using a multiple timescale expansion. This idea was subsequently extended by Agnon and Mei (1985) toinvestigate the slow drift oscillations of a two-dimensional sliding block. Faltinsen and Zhao(1989) studied the effects of wave interactions with the quasi-steady two-dimensional flowdue to the slow drift velocity of a moored cylinder, and the resulting wave drift damping canbe evaluated from the added wave resistance for steady forward velocity in a wave train.1.3 Scope of Present StudyA survey of previous work indicates that the emphasis of recent hydrodynamic research hasbeen on the development of analytical solutions and numerical methods to treat the nonlinearinteractions of free surface waves with large offshore structures. As a particular thrust of thisresearch, the present theoretical work considers the wave radiation and combined diffraction-radiation problems to second order in both two and three dimensions for the case of largefloating offshore structures of arbitrary shape. Although the second-order diffraction andradiation problems have been pursued separately quite extensively, results for the nonlineardiffraction-radiation problem are rather scarce. This combined problem involves the equationsof motion of the structure as well as nonlinear interactions between the incident, diffracted andradiated wave components. A recently developed second-order time-domain approach is13adopted to simulate second-order hydrodynamic effects involving large floating structureswhich cannot be predicted by the conventional linear hydrodynamic theory.On the basis of potential flow, the mathematical formulation of second-order boundaryvalue problem related to wave-structure interactions is described in Chapter 2. In the second-order analysis, Taylor series expansions are applied to the body surface boundary conditionand the free surface boundary conditions, and the Stokes perturbation procedure is then usedto establish the corresponding boundary value problems at first and second order with respectto a time-independent fluid domain. The numerical implementation of the method is outlinedin Chapter 3. The method involves a time-stepping scheme together with a suitable iterativeprocedure to solve the coupled fluid-structure governing equations, and an integral equationmethod based on Green's theorem is used to obtain the wave field at each time step. Thepresent method provides a relatively algebraically straightforward and computationallyeffective numerical algorithm for treating the second-order free surface flow problems inrelation to conventional frequency-domain methods.In Chapter 4, second-order hydrodynamic results are presented to illustrate and examine thepresent method. For the two- and three-dimensional wave radiation problems involvingsurface-piercing structures undergo forced sinusoidal motions in otherwise still water,numerical results are given respectively for two-dimensional semi-submerged circular andrectangular cylinder sections and a three-dimensional truncated surface-piercing circularcylinder. The present method is subsequently extended to simulate second-order waveinteractions with large floating bodies in both two and three dimensions, and is appliedrespectively to the cases of a semi-submerged circular cylinder and of a floating truncatedcircular cylinder. Numerical computations presented relate to the transient motion of a freely-floating cylinder with a specified initial vertical displacement, and the diffraction-radiation ofStokes second-order waves by a moored floating cylinder. The numerical results demonstratethe importance of second-order hydrodynamic effects in the forces and motions of largefloating structures subjected to waves, as well as in the corresponding free surface profiles14and wave amplitudes. Finally, Chapter 5 presents the main conclusions of the study, andprovides suggestions for further studies and research. As a result of the recent development intime-domain simulation, it is suggested that the present method can be extended to calculatethe second-order forces and motions of floating structures of arbitrary shape in bichromatic orirregular wave excitations.It is hoped that this thesis will provide some contribution towards a better understanding ofthe outstanding problems associated with the nonlinear interactions of free surface waves withlarge floating structures.15CHAPTER 2MATHEMATICAL FORMULATION2.1 Fully Nonlinear ProblemThe governing boundary value problem defming nonlinear wave interactions with a large,three-dimensional floating structure is formulated. As illustrated in Fig. 1(a), the generalthree-dimensional problem is defined with respect to two right-handed Cartesian coordinatesystems: one is the inertial (space-fixed) coordinate system X = (x, y, z), in which x and yare measured horizontally and z is measured vertically upwards from the still water level; thesecond is the body-fixed coordinate system X = Y,^in which the origin is located at thecentre of mass of the rigid body. When the body is in its equilibrium position, the two sets ofcoordinate systems are parallel and the centre of mass is located at X g = (xg , yg , zg). Theseabed is assumed impermeable and horizontal along the plane z = —d. For the simplified caseof two-dimensional flow as shown in Fig. 1(b), the corresponding fully nonlinear problemcan be formulated with respect to the vertical x-z plane on the same basis.Let t denote time and tl the free surface elevation above the still water level. Assuming ahomogeneous, inviscid and incompressible fluid, and an irrotational flow, a velocity potential4) can be defined in the fluid domain such thatu=0 .^ in .0"^ (2.1)where u is the fluid velocity. The velocity potential must satisfy the continuity equation or theLaplace equation within the fluid domaina24)^a24,^a24)v24) ax2 aye az ^u in 9J^ (2.2)16The potential $4) is subject to boundary conditions on the seabed, the instantaneous wetted bodysurface SW and the free surface Sf, given respectively as0as0az = 0= v.NaNan &Oar! a4anaz — "X" — WiFca4)^1^12+^+^V(1) = 0at z = —d^ (2.3)on Sw^(2.4)on Sf^ (2.5)on Sf^ (2.6)Here g is the gravitational constant; V is the velocity vector of the body surface; and N denotesdistance in the direction of the unit normal vector N = (Ni , Ny , NZ) at the instantaneous bodysurface Sw pointing outward from the fluid region. Finally, for a well-posed boundary valueproblem, a suitable boundary condition is applied on a control surface S c located at asufficiently far distance from the body in order to ensure that all wave disturbances due to thepresence of the structure propagate away from the fluid domain. The major difficultiesassociated with the fully nonlinear problem lie in the two nonlinear free surface boundaryconditions at the free surface and the body surface boundary condition at the instantaneouswetted body surface, both of which are unknown a priori. The problem is further complicatedby a proper specification of the radiation condition on the control surface of the computationaldomain.For a three-dimensional body, the rigid body motions about its centre of mass can bedefined in terms of a translational vector 4= (4x , 4y, whose components denote thesurge, sway and heave displacements respectively, and a rotational vector a = (ax, ay, az),whose components denote the angles of roll, pitch and yaw respectively. In studying themotion response of floating structures, the hydrodynamic loads induced by the time-dependentflow field associated with wave disturbances give rise to the structure motions. The17interaction involves the kinematic body surface boundary condition relating the fluid velocitiesand structure motions as well as the equations of motion of the structure. The latter are givenfor the six modes of motion as:M + Kit + Kta a = F* (2.7)hi) + Kat + Kaa a = G* (2.8)where M is the mass matrix; I is the inertia matrix; Ktt, Kt a, Kat and Kaa are structuralstiffness sub-matrices which are associated with the mooring system; F * = (Fr , Fy, Fz) is avector containing the total hydrodynamic forces, with F r , Fy and Fz denoting the forcecomponents in the x, y and z directions respectively; G * = (Gr , Gy , Gz) is a vector containingthe corresponding moment components; and co = ((ix , ciy, az) is the angular velocity vector,where an over-dot indicates a time derivative. With the origin of the body-fixed axes chosento coincide with the centre of mass of the body, the matrix M is diagonal, with elements madeup of the mass of the structure; whereas the elements of matrix I contain moments andproducts of inertia with respect to the body-fixed coordinate system.In accordance with linear hydrodynamic theory, the source of damping for a floating bodyin waves is solely attributed to wave radiation effects. It is well known that damping due toviscous effects, such as those relating to flow separation, vortex-shedding and drag-inducedforces, is one possible cause of the discrepancies between theoretical predictions and dataobtained from full-scale and laboratory model measurements. In most practical engineeringapplications, it is convenient to include also the viscous induced damping associated with rolland pitch motions on the basis of an assumed ratio of the critical damping of the floatingstructure. Within the context of potential flow, viscous damping is not considered in thepresent study. In general, viscous effects can be considered to be minimal for the case ofKeulegan-Carpenter numbers less than unity, although flow separation may then occur atlocalized regions near the structure adjacent to any sharp corners that may be present. In thepresent time-domain formulation and in contrast to most frequency-domain methods,18contributions of the hydrodynamic added mass and damping, associated with the motions ofthe structure, as well as the hydrostatic stiffness are all included within the forcing vector onthe right-hand side of the equations of motion.As mentioned previously, a numerical solution of the fully nonlinear problem can beobtained by using various time-stepping methods, and a large number of two-dimensionalproblems have been studied extensively. In these computational models, the temporal wavemotions are calculated by tracking the free surface particles in time, and the boundary integralequation for the governing fluid equations is based on either Cauchy's integral theorem orGreen's second identity. In the former case, the use of the complex potential in themathematical formulation is applicable only to two-dimensional flow simulations.Furthermore, major difficulties common to most nonlinear methods such as highcomputational costs associated with solving a large system of linear algebraic equations at eachtime instant, the complications related to the growth of numerical instabilities and theunresolved difficulties of far-field closure have limited their widespread application to thegeneral three-dimensional case.Alternatively, in the weakly nonlinear wave regime it is justifiable to seek second-ordercontributions in terms of incident wave height and the resulting body motions by applying theStokes perturbation procedure. In the second-order expansion, the first- and second-ordermathematical problems are defined with respect to the mean positions of the fluid domain andcomputational efforts required is significantly reduced. This theoretical development has madethe three-dimensional calculations based on boundary integral equation formulation and time-stepping method well within the resources of the available state-of-the-art computertechnology.192.2 Stokes Second-Order Method2.2.1 Second-Order ExpansionIt is assumed that the amplitudes of body motion are small compared with a principal bodydimension, the wave height is moderate, and the water depth is not small compared with atypical wavelength. Specifically, in order for the second-order Stokes theory to be valid, theUrsell parameter U = HL2/d3 , which indicates the magnitude of the second-order termsrelative to the first-order terms, must not exceed a value of about 26 (e.g. Dean andDalrymple, 1984), where H and L denote respectively the height and wavelength of theincident waves. Furthermore, the wave conditions are considered to be non-breaking. Thebreaking wave height limit Hb for a progressive wave over a variety of water depths andwavelengths may be approximated as:0.8dHb = minimum 1 0.142L tanh (kd)^ (2.9)where k = 27r/L is the wave number. Due to the combined effects of the incident and reflectedwaves in the presence of a vertical wall-sided structure, the maximum wave height for the caseof standing waves may be approximated as:Hb = 0.109L tanh (kd)^ (2.10)On the basis of the above assumption, it is possible to apply Taylor series expansions totransform the body surface and free surface boundary conditions, originally defined on theinstantaneous surfaces, to conditions evaluated at the corresponding mean positions (Ogilvie,1983).Cod +^- Xg^)•0(04)) +^).N = 17 -1S1^ on Sb (2.11)ta4) an 4.41i^arh^a do an ao an an7)T^FE +TINkE—Ft - FEE - Jir)^= 0 on So (2.12)20l+gri+11V01 2) +^IV4)12) +... = 0^on So (2.13)where X denotes the exact instantaneous position vector. The boundary conditions maythereby be applied on the surface of a time-independent fluid domain which includes the meanbody surface Sb and the still water surface So . Using the Stokes expansion procedure,quantities at first and second order are separated by introducing perturbation series for 4, a, 4)and II:= Eti +^+a = e a l + e2 a2 +4)= e (01^e2 4)2 +1 = Th^C2 712(2.14)(2.15)(2.16)(2.17)where e is a perturbation parameter related to the wave steepness which is small, and thesubscripts 1 and 2 indicate respectively components at first and second order. For relativelysmall amplitude rigid body motions considered here, the exact instantaneous position vector Xand the unit normal vector N at each point on the body surface can be given by the followingtransformation relationships which are valid up to second order:X = Xg +51 +(4 +axR)+ HX^ (2.18)N=n+axn+Hn^ (2.19)The velocity of each point on the body surface can be obtained by a straightforward timedifferentiation of equation (2.18) asV= (t+axX)+ HX^ (2.20)21where n = (ny, n)72 nz ) denotes the unit normal vector directed outward from the fluid regionwhen the body is in its equilibrium position. As pointed out by Ogilvie (1983), the form ofmatrix H depends on the sequence of rotational motions. By taking these in the order roll,pitch and yaw, the matrix H is given by:H = 1-ay + oq[o^°—2a a y a+ a z2^0x^x2—2a x a z —2ay az a x2 + a yY(2.21)Note that elements of H are second-order quantities which are associated with the quadraticproducts of the first-order rotational motions and can be interpreted as correction terms whenthe body surface velocity are calculated to second order.The first- and second-order wave potentials and the free surface elevation are furtherdecomposed into incident wave components and wave disturbance components due tocombined wave diffraction-radiation effects:(1) = e($41 + ) e2(4 + (02 ) (2.22)tl = + ) e2 (t1 1 +Tel ) (2.23)The first-order potential O F: corresponds simply to a superposition of the separate lineardiffraction and radiation wave potentials which are associated respectively with the scatteringof the linear incident waves by a fixed structure and with the forced oscillatory motion of arigid body in otherwise still water. Likewise, the situation is somewhat similar but muchmore complicated at second order. The total second-order potential (0 2 comprises of nonlinearwave components due to self- and cross-interactions of the first-order incident and disturbedwave potentials, with the second-order incident wave component corresponds to the nonlinearself-interaction of first-order incident waves. In the conventional frequency-domainformulations, these second-order wave potentials would give rise to a number of sub-22boundary value problems which are both algebraically tedious and computationally difficult tobe solved. In the present approach, all the second-order wave components due to wavedisturbances are collectively represented by the wave potential 4.By substituting the Stokes perturbation expansions for 4, a, 4 and ri into the governingLaplace equation and the corresponding boundary conditions expanded about their meanpositions, and retaining terms to second order, the corresponding boundary value problemsfor the e and e2 terms in the power series expansions may be developed. It is noted that thegoverning boundary value problem at each order is now linear and is formulated with respectto a time-independent fluid domain D bounded by the seabed, the wetted body surface in itsequilibrium position Sb, the still water surface S o and the control surface Sc.It is noteworthy that Taylor series expansions and the Stokes perturbation proceduredescribed above in the context of the combined wave diffraction-radiation problem may bemodified to derive the separate wave diffraction and radiation problems in a relativelystraightforward manner. In the former case, the structure is stationary and the wave potentialcontains both the incident and diffracted wave components, whereas in the latter case theforced body motions are considered to be first-order quantities and the wave potential containsthe radiated wave component only.2.2.2 Three -Dimensional ProblemIn this section, the three-dimensional wave diffraction-radiation problem to second order isconsidered. In the k-th order boundary value problem (with k = 1, 2 in turn), the disturbancepotential satisfies the Laplace equation in DV24 = 0 in D (2.24)and is subject to the boundary conditions applied on the seabed, the mean body surface andthe still water surface, given respectively as23asOwl^• -^-_^+ (ti +. x x)• n421— an=f2 + ( t2 +^x ) • n(2.29a)a4 0az -aekfka4): ari)Bcaz - atasiBcat + =atz=—d^ (2.25)on Sb^(2.26)on So^(2.27)on So^(2.28)The terms fk, fk and f;', are given respectively as+ (li^n —^+^x )•V (V4• 1)]- n+ (a i x n ) • [(^x^— Vi(1)1]fl = 0f2 (^arr aril; \ ac t ant aot ant a2otaz2 - at ax ax — 11 az2f; = 0— — eft + g11; — II04)112 —(2.29b)(2.30a)(2.30b)(2.31a)(2.31b)Here n denotes distance in the direction of the unit normal vector n. At first order, theboundary value problem corresponds to the classical linear diffraction-radiation problem.Each quadratic forcing term in the second-order body surface boundary condition and freesurface boundary conditions can in principle be evaluated once the first-order potential andbody motions have been solved.As indicated in equation (2.29b), the kinematic body surface boundary condition at secondorder consists of five terms. The first and second terms are analogous to those of the24corresponding first-order problem. The third term is associated with the nonlinear effects inthe calculation of body surface velocity. The fourth term is the correction for the conditionapplied on the mean position of the wetted body surface instead of its instantaneous position.The last term accounts for the change of direction of the unit normal due to rotational motions.Theoretically, the second-order potential solution can be interpreted as being composed offorced wave and free wave components. The forced waves are associated with the quadraticforcing terms in the free surface boundary conditions which correspond to an oscillatorypressure field acting on the entire still water surface at twice the excitation frequency. Byimposing the second-order body surface boundary condition, the free waves which satisfy thehomogeneous second-order free surface boundary conditions account for the nonlinearinteraction of the floating structure and the forced waves. The forced waves are phase-lockedwith the first-order wave system and the corresponding celerities do not satisfy a simpledispersion relation, whereas the free waves propagate independently of the first-order systemand the corresponding celerities now satisfy the linear dispersion relation at twice the wavefrequency.In most free surface flow problems, it is often necessary to introduce an artificiallytruncated boundary known as a control surface S c, such that flow simulations can be carriedout in a reasonably sized fluid domain. From the viewpoint of theoretical computations, asuitable boundary condition is required on the control surface to ensure a unique solution. Atpresent, a generalized mathematical treatment of perfectly transparent open boundary (far-field) condition for nonlinear wave propagation problems is unavailable. The Sommerfeldradiation condition, derived on the basis of spatially and temporally periodic conditions of apotential function, can be approximated as the boundary condition on the control surface(Orlanski, 1976):a4): at + cn an _0 on Sc^(2.32)25where cn is the time-dependent celerity of the radiated waves at the control surface in thedirection of the unit normal vector n. Even though the applicability of Orlanski's approach hasnot been proved analytically, his method has extensively been adopted as a pragmatic choiceof radiation condition in various three-dimensional nonlinear wave problems formulated in thetime domain. Examples of such recent attempts include Thou and Gu (1990) for nonlinearwave-body interactions; Yang and Ertekin (1992) for nonlinear diffraction in shallow water;and Isaacson and Cheung (1992) for second-order diffraction. All of these studies havesuggested that the Orlanski condition is found to be effective and no significant distortion ofsolutions in the interior wave field has been reported.For the three-dimensional case considered here, the amplitudes of the second-order wavedisturbance components are expected to decay with increasing radial distance from the bodydenoted by r. These nonlinear wave effects become negligible when the control surface islocated sufficiently far away from the structure, and the performance of the radiation conditionis then less critical. According to the theoretical analysis of Isaacson and Cheung (1992) inthe context of the second-order wave diffraction in three dimensions, the magnitude of thecorrection term which has not been included in the Orlanski condition has been shown to be oforder 1/(kr) smaller than the existing terms in equation (2.32), and becomes negligibleprovided that the control surfaces are chosen to be at least one wavelength from the body.Finally, the equations of motion given in equations (2.7) and (2.8) may be decomposedinto equations for the first- and second-order motion components. Thus the equations for thefirst-order motions are:M 41 + Ktt1 + K4cc al = (2.33a)^+ Kat t i + Kati a l = G *1^(2.33b)In addition, the associated equations for the second -order motions are given by:M 42 + 1/44 42 + 1( cc a2^F; (2.34a)26+ co l x I co l + Kat 42 + Kaa a2 = GI^ (2.34b)where 01 = (ax l, ayl , az1) and 632 = ( ax2 + azl ayl , 6.3/2 - dtzl axl, az2 + 6y1 axl )denote the first- and second-order angular velocity vectors respectively; and in the doubleindex subscript notation, the first denotes the axis and the second the order of the motion.With the equations of motion in the present form, the force vector F* and the moment vectorG * have been decomposed into first-order components (Fr , G i* ) and second-ordercomponents (Ft , G 2). The latter represents the sum of the steady drift component and theoscillatory component at twice the first-order incident wave frequency.2.2.3 Two -Dimensional ProblemIn two-dimensional flow, the treatment of the problem corresponds to the fluid motion in thevertical x-z plane past an infinite horizontal cylinder whose axis is parallel to the y-axis, andthe normal incident waves propagate in the positive x direction. Since the two-dimensionalproblem requires substantially less algebraic and computational effort when compared with thethree-dimensional counterpart, this idealization of the problem, although with more limitedpractical applications, has generally been treated as a preliminary test case in the developmentand validation of the suitable numerical procedure.In the two-dimensional case, the rigid body motions of the floating body are therebydescribed in terms of the translational motions in the horizontal and vertical directions and therotational motion about the centre of mass, namely the sway, heave and roll motions which aredenoted by 4, C and a respectively; and the unit normal vector n = (ny, nz) with respect to thex-z plane. By eliminating all the terms involving the y coordinate in the three-dimensionalformulation, the potential flow problem can be simplified to the corresponding two-dimensional vertical plane case. The governing Laplace equation as well as the boundaryconditions applied on the seabed and the still water surface of the corresponding first- andsecond-order boundary value problems are basically similar to those of the general three-27dimensional problem and they are not repeated upon here. With respect to the kinematic bodysurface boundary condition, equation (2.26) reduces to the form:aoai a01  + (4 1 + a i z) nx + (t i — x) nz^(2.35a)anaos^a■12' + (42 + a2^+ ( 2 - a2 nz—(a2(1)1^a201^_ ,a2o1^a261- (41 + a z)^nx asan nz) (C1 - x) lanr2 nz—,as n•^ao i^aoi^•^a(1)1+ ai nz (4 1 -^nx -^n) - a i nx(Ci -^nz +^nx) (2.35b)where s is the tangential direction on the mean body surface as illustrated in Fig. 1(b). Toaccount for the structure motions, the three degrees of freedom equations of motion at eachorder (with k = 1, 2 in turn) may be expressed in matrix form asmtk K4k (2.36)Here tk = (4k, Cie ak) represents the displacement vector of the two-dimensional floatingbody, and K is the structural stiffness matrix. Since the origin of the body-fixed axes iscoincident with the centre of mass of the body, the mass matrix in the equations of motiondenoted by M is diagonal, with the first and second terms given as the mass of the structureand the third term given by the mass moment of inertia about the y-axis. The associatedhydrodynamic force vector denoted by ft corresponds to force components in the x and zdirections and the moment component in the y direction.For the two-dimensional case, the nonlinear wave disturbances do not attenuate in the far-field due to the absence of directional wave spreading, and therefore a satisfactoryimplementation of the far-field closure is crucial to the long time simulation of the problem.Early work has resolved this problem based on the assumption of spatial periodicity, and theexterior computational boundaries are simply eliminated by a suitable folding (e.g. Longuet-28Higgins and Cokelet, 1976; Vinje and Brevig, 1981; and Greenhow et al., 1982). However,the assumption is considered to be unrealistic in the presence of obstacle in the fluid flow.Dommermuth and Yue (1987) suggested that a closure by matching the nonlinear innersolution to a linear wave field outside can be employed without appreciable wave reflection forthe case of three-dimensional flow in which the energy density of outgoing waves decays withincreasing radial distance from the body. The persistence of nonlinear wave effects in the farfield prohibits the use of such matching scheme in the two-dimensional flow problem. As inthe three-dimensional case, the Orlanski condition which is originally devised for a hyperbolicflow equation has subsequently been applied to a variety of two-dimensional free surfaceflows in which the Laplace equation is considered as an elliptic partial differential equation(e.g. Lee and Leonard, 1987; Jagannathan, 1988; and Isaacson and Cheung, 1991).Recently, Isaacson and Cheung (1991) conducted numerical tests to confirm theeffectiveness of the Orlanski condition in treating the second-order wave diffraction problemin two dimensions. This involved tests with computational domains of different size. Theirresults have indicated that the second-order solution for durations over which any reflectedwaves would reach the structure, and obtained using the time-dependent celerity and a limitedcomputational domain, gives excellent agreement with the solution obtained with a largercomputational domain for durations before any reflected waves would reach the structure. Itis suggested that the effects of the outer domain on the inner solution diminish as the size ofthe artificially truncated fluid domain becomes sufficiently large, say greater than about fourwavelengths. For two-dimensional problems, the control surfaces have to be positioned atleast three times the water depth away from the structure in order to exclude the effects fromevanescent modes or local disturbances (Lee and Leonard, 1987).2.2.4 Boundary Integral EquationThe posed boundary value problems governing second-order wave-structure interactions maybe solved by an integral equation method derived from Green's theorem. Bai and Yeung29(1974) pioneered the development of the method to free surface flows with an emphasis on thecalculation of hydrodynamic coefficients in the two-dimensional wave radiation problem.Using such an integral equation formulation, it is then possible to transform the governingfluid equations defined within the domain to one on the boundary surface alone, andconsequently the dimensionality of the problem considered is reduced by one.Unlike the conventional wave source distribution method in which the wave potential isrepresented by a complicated kernel function known as the Green's function (e.g. Wehausenand Laitone, 1960), the integral equation method adopted here employs only the fundamentalsingularity (simple source) such that the numerical procedure requires the discretization of theboundary surface of the fluid region rather than the submerged portion of the body surface. Inaddition, the solution of the integral equation is expressed in terms of an unknown velocitypotential and its normal derivative directly instead of the source distribution strength. Thisdirect formulation of the potential problem leads to the mixed Fredholm integral equations ofthe first and second kind with Dirichlet conditions on the free surface and control surface andNeumann conditions on the body surface, from which the remaining unknown boundaryparameters can be solved. The integral equation method is well suited to solve potential flowproblems involving nonlinear free surface and body surface boundary conditions for the casesof two- and three-dimensional floating bodies of arbitrary shape which are the main objectivesof the present study. Furthermore, the method has the flexibility of analyzing the effects ofvariable water depths as well as rigid wall boundaries as in the case of a narrow wave channel.Three-Dimensional ProblemFor the general three-dimensional problem, the k-th order potential at a field point x = (x,y, z) on the smooth surface of the fluid domain may be expressed by the following surfaceintegral equation based on Green's second identity:1^rG(x013:(x) =^f I.a-g1(^B aG2n^' xi) an x(x')-a-w(x, x') _,OSs'(2.37)30where x' = (x', y', z') is a source point on the boundary surface S' over which surfaceintegration is performed, G is a Green's function, and n is measured from the point x'. Thesurface S' consists of the mean wetted body surface Sb below the still water level, the stillwater surface So, the control surface Sc and the seabed (see Fig. 1(a)). In the present form,the integral equation can be interpreted physically as a mixed distribution of single-layersources and double-layer normal dipoles. The potential for a point x located inside the fluiddomain can be calculated using integral equation (2.37) with the factor 1/(211) replaced by1/(41t) .For three-dimensional problems in which the body and the simulated flow are symmetricabout the x-z plane and the seabed is horizontal, the Green's function can be chosen as aRankine singularity function and its images satisfying the Laplace equation and the seabedboundary condition. This leads to a significant reduction in the computational effort required.With these simplifications, only one half of the computational domain is considered and theseabed can be excluded from S'. This three-dimensional Green's function is given as:4G(x, x') = Eik(2.38)where rk is the distance between the points x and 4. The source points x'k are defined suchthat xi = (x', y', z') is a point on S'; x2 = (x', —y', z') is the image of x i about the x-zplane; x'3 = (x', —y', —(z'+2d)) is the image of Yq about the seabed; and x4 = (x', y',—(z'+2d)) is the image of about the seabed. Thus, ric is given byr1 = [(x' — x)2 + (3/' — Y)2 + (2 — z )2]"2^(2.39a)r2 = [(x' — x )2 + (y 1 + y )2 + (z' _ z )21 1/2 (2.39b)r3 = [(x' — x )2 +^+ y + (z t + 2d + z )2] la^(2.39c)r4 = [( - x )2 + (Y t Y )2 + (z' + 2d + z )211/2^(2.39d)31Here the subscript k denotes the four quadrants of the doubly symmetric geometry.The above situation applies to a structure with at least one plane of symmetry (the x-zplane), such as a ship, and for an incident wave direction which is coincident with the plane ofsymmetry (the x direction). However, even for structures with one or two planes ofsymmetry but with an incident wave direction which is arbitrary, the above simplificationinvolving half the fluid domain cannot be made. Instead, the full fluid domain should beconsidered and the Green's function given by equation (2.38) will include two terms insteadof four.Two-Dimensional ProblemIn analogy to equation (2.37), the k-th order potential of the two-dimensional vertical planeproblem may be expressed by the following line integral equation:4Tcoo =^f[G(x x) a^B, , an^— (1) k Maa7{? X , V)] S7CS(2.40)where x = (x, z) and x' = (x', z') represent respectively a field point and a source point on thesmooth boundary surface S' over which contour integration is performed, and n is measuredfrom the point x'. In two-dimensional flow, the closed contour S' consists of the meanwetted body surface Sb below the still water level, the still water surface S o, the controlsurface So and the seabed (see Fig. 1(b)). The potential for a point x located inside the fluiddomain can be calculated using integral equation (2.40) with the factor —1/(it) replaced by—1/(2x).Once again, making use of the constant water depth assumption, the seabed can beremoved from S' by choosing a Green's function corresponding to the logarithmic singularityof the Laplace equation and its image about the seabed. The two-dimensional Green'sfunction is given by322G(x,^= E In (rk)^ (2.41)k=1where rk is the distance between the points x and x 'k. The source points xi are defined suchthat xi = (x', z') is a point on S'; and 34 = (x', -(z'+2d)) is the image of xi about thehorizontal seabed. Thus, rk is given byr1 = [(x' — x )2 + (z' — z )21 1 /2 (2.42a)r2 = [(x' — x )2 + (z' + 2d + z )21 1/2 (2.42b)Here the subscript k denotes the boundary surface and its image about the seabed.After solving for the first- and second-order velocity potentials by a suitable numericalprocedure, all physical quantities of significance including the pressure, fluid velocity, freesurface elevation as well as hydrodynamic forces can be derived.2.3 Free Surface ElevationWithin the context of a detailed second-order analysis, prediction of the resulting wave fieldand the wave runup has been one of the main themes in a few recent studies of wavediffraction by a vertical circular cylinder (e.g. Kriebel, 1990; Chau and Eatock Taylor, 1992;and Isaacson and Cheung, 1992). These results indicate the importance of second-ordereffects on the wave envelope around the structure under steeper wave conditions.The free surface elevation to second order is calculated explicitly by the dynamic freesurface boundary condition given by equation (2.28). For the case of regular wave excitation,the free surface elevation rl to second order may be expressed as a sum of three components:tl = r11 + no + 12 (2.43)where n i , n o and r12 are respectively the first-order oscillatory component at the excitationfrequency, the second-order steady component, and the second-order oscillatory component at33twice the excitation frequency. Expressions for Il i , rio and 112, which derive from equations(2.28), are given respectively as:1 Ali) \111 = — ik at /1 1 j^h2'no = — i<2- 1v4)112 + Tha >azat1 tajtz^1^2^a2 (1)1 1^12 = - i l at + f IV011 + Illazati^lbon So^(2.44)on So^(2.45)on So^(2.46)where < > denotes a time average. The components 1 0, 112 are associated respectively with asteady set-up or set-down of the mean water level due to the non-zero mean of the free surfaceboundary conditions at second order, and an oscillatory variation of the free surface elevationat second order due to both forced and free wave components. With steady state results of thefirst- and second-order free surface elevations over one complete wave cycle known, the waveamplitudes to second order can be calculated on the basis of their corresponding maximum andminimum values.2.4 Hydrodynamic Forces and MomentsIn practice, the hydrodynamic forces and moments acting on the body surface are of particularinterest and can be calculated by carrying out a direct integration of the dynamic pressure overthe instantaneous wetted body surface Sw. The method has commonly been referred to in theliterature as the near-field approach. The dynamic pressure in the fluid can be determined bythe unsteady Bernoulli equation:a(I)^1^11-7,_ 12I) = — 1)^— 2- p 1 vyi — pgz (2.47)where p is the fluid density. In order to account for the small amplitude body motions, thepressure acting on the instantaneous body surface can consistently be expanded about themean body surface by a Taylor series expansion34(p)s„ = (p +^— Xg —X) .pp + )s b^(2.48)Substituting the perturbation expansions of 4, a and J into equation (2.48) and retainingterms to second order, the pressure is given asp = —p/ — 24) IV 0 1 1 2 pgz Pg(4z1 axl Y - ay1 x)Pg(4z2 ax2 - CCy2 - pgHX — P ( ti ai x )*V ( 1 )^on Sb (2.49)Thus, the hydrodynamic force and moment vectors are given respectively by the followingintegralsF =p JpNdS^ (2.50)s wG=p fp [(X — X g ) x^dS^ (2.51)swIn the present application, the moment is taken about the centre of mass of the body. Thepressure integration over the exact wetted surface of the body Sw can be expressed as anintegral over the wetted body surface below the still water level together with a correctionintegral defined at the still waterline contour wo.For a floating structure in regular waves, the hydrodynamic forces to second order Fcontains three force components:F = F1 + F0 + F2^ (2.52)where F1, Fo and F2 are respectively the first-order oscillatory force at the excitationfrequency, the second-order steady force, and the second-order oscillatory force at twice theexcitation frequency. By a direct integration method, these may be expressed as:35aoiF, = -p f wl n a - pgAw(zi + axi yf -ay i xf) k^ (2.53)sb1Fo = --2)< f IV$112 n dS>sb- p < f[(t 1 + a 1 x i).V(41- )] n dS>sb_ p < f av (. 1 . n) dS>sb+ 1P g < f oil - z1 - a x 1 Y + a y 1 TC)2 n dw>wo- pgAw < azi (axl xf + ayl Yf ) + 1 ( ax21 + ay21 ) Zg> k^(2.54)ao2 dSF2 = -p i -Fn , - pgAw(4z2+ ax2 Yf - ay2 Xf) ksb1^r— IP JIV0112 n dSsb- p f [(t i + a i x 31)-V(1)] n dSsbfao i- P J -F(cti xsbn) dS1Pg is Oil — 4z1— ax1 + a y i FE) 2+ . ^nW o1^2^2- pgAw [ azi ( axi xf + ayi Yf) + 1- ( axi + ayi ) zgi k - Fo^(2.55)dw36where Aw is the mean waterplane area; xf and yf denote respectively the longitudinal andtransverse coordinates of the centre of floatation, which represents the centroid of thewaterplane area when the body is at rest; and k is the unit vector in the z direction. For a bodygeometry which is not vertical at the still waterline, an additional factor 1/-V 1 — n 2 isrequired for the waterline contour integral of equations (2.54) and (2.55) [see Garrison (1984)for details].The second term in equation (2.53) represents the first-order hydrostatic restoring forcesand only has a component in the z direction. Note that the second-order steady forces can becalculated from the quadratic products of the first-order quantities such as wave potential andbody motions. As shown in equation (2.55), the F2 component is made up of seven terms.The first term is the contribution due to the second-order potential. The second termcorresponds to the vertical hydrostatic restoring force component due to second-order motionswhich has the same form as the corresponding first-order component. The third term isassociated with the velocity squared term of the Bernoulli equation. The fourth term accountsfor the change in pressure due to first-order body motions. The fifth term accounts for thechange in directions of the forces due to first-order rotation and is sometimes expressed as theproduct of first-order rotational motions and hydrodynamic force vector. The sixth term isrelated to the correction for the hydrodynamic forces acting on the instantaneous watersurface. The last term accounts for the second-order hydrostatic forces associated with thedisplacements of the body from its equilibrium position.For the sake of completeness, the corresponding moment components G1, Go and G2 canbe obtained in the similar way and are given respectively as:^G 1 = —p^x n) d Ssb—pg { 4z, Aw yf + axi [V(zb — zg) + Lyy] aylLxY aziVxb—pgf - 4zi Aw xf + ayi [V(zb — zg) + 1,] — axiL„y — aziVYb .i^(2.56)37Go =^p< fri70 1 1 2 (i x n) as>2P < f [(ti^>.< X)-v( 1 )] (17 x n) as>Sb- p<^[41 x n + ga l x (JC-- x n)] dS>sb1Pg < f^- axi Y a y l X ) 2^x n) dw>wo—rgAw zg < ocL + ay2i > ( yf i —xf j ) (2.57)G2 —p f (k x -n ) dSsb- pgt tz2 Aw yf + axe [V(zb — zg) +^Ory21-1Cy aZ2VXb }- pgt -4z2 Aw xf + ay2 [V(zb — zg) + Lzx] — ax2Lxy — az2VYb1- P f^(37 x n) d SSb- P^ [(4 1 + ct i x 31).v(a )] (5c x n) dSSb— p^xa(Pi r t n+ a 1 x (i x n)] d SSbf —4z1 — ax1 Y+ a yl ) 2^x n) d wwo2gAw zg ax2i ay21 )( Yfi — xf i ) — Go (2.58)38where V is the displaced volume of the body at its mean position; (xb, yb, zb) is the location ofthe centre of buoyancy; i and j denote unit vectors in x and y directions respectively; and Limnis the waterplane area moment of inertia with respect to the m and n axes indicated by thedouble index subscript notation when the body is in its equilibrium position. As indicated inequation (2.56), the second and third terms correspond to the first-order hydrostatic restoringmoments in both the x and y directions respectively. In addition, the physical interpretationsof the various second-order moment components are somewhat analogous to thecorresponding second-order force components. It is mentioned that the general expressionsfor hydrodynamic forces and moments to second order given here have been derived byfollowing closely the approach described in detail by Ogilvie (1983).The hydrodynamic force components to second order derived originally for the generalthree-dimensional case, given in equations (2.53) to (2.55) respectively, can be simplified tothe case of two-dimensional vertical plane flow, and these may be expressed as:F1^ a4)].= -p f^ndS — pgAw (C i — a ixf) ksbF0^1 < f iV4)112 n dS>sb- P < f [(41 +^a24)1sb^anat nx^n dS>- P < f^alie24)1a24) ^) 1^nz^rix, _ n dS>sb—P f a oci(n z i — n x k) dS>sb+ 1 pg < f ( -1 1 — C 1 + a i x)2 n dw>wo(2.59)391— -2- pgAw zg<cci> ka2F2 = — p^n dS — pgAw(C2— oc2xf )ksbJ—^j iv 4, 1 1 2 nsb— a 20iP b [(41 +^anat nx Tainz)] dSsp^[(C1^ra2oi^a201x Xanat nz^fl7c)1 n dSsbI a(1)1- P J^cc i(nz i — n x k) dSsb+ Pg J 01 1^+ a^n dwwo—1 pgAw zg ai k — Fo(2.60)(2.61)Similarly, the remaining moment components for the two-dimensional case are givenrespectively as:a(01 -GI = -p r^(z nx - nz) dS jsb—pg [al(V(zb — zg) + I-xx) — b1Aw j^ (2.62)Go = 1 p < f IV•0 1 1 2 (znx — nz) dS> jsb,a2th,- P < f [ (41 + aizAan5; nx as.ralt no] nx — x nz) dS> jsb40ta1- p < i [(Ci - a ix )kan at n zsba201 \ 1_ a^sat nx" (z nx - x nz) dS> jrach-p<i -5--t (Cinx-tisbnz ) dS> j14- -2-pg< f oh — ci + a1x-)2 (inx - inz)dw>iwo1+ 1 pgAw xf zg<cei> j(1)2G2 = - p f --ai (i n x - i nz) dS j - pg [a2(d(zb - zg) + 1,xx) - C2Aw xd isb1- y p f ivo l l 2 (z n x - R n z) dS jsbf k i + ail)(a2—(121n ,_ a 2- P^01 vianat x • asat nz)i (i nx - 5i nz) dS jsb- P .1[(c,— a ii)(?-ariat nz a2(01t nxi\ i1Sb ^aSa^(i nx — i nz) dS jr0- P J aWf1-" gi nx - 41 nz) as jsb1+ f, Pg f(1li—ci 4- aii)2 (znx -X nz)dw jw 01+ --f pgAw Xf Zg 04 i — Go(2.63)(2.64)In the three-dimensional problems, each of the force and moment components at first andsecond order may be split further into components in the x, y and z directions, whereas in thetwo-dimensional problems the corresponding force components may be expressed in terms ofcomponents in the horizontal and vertical directions. Note that alternative expressions for the41force and moment components may be found in the literature, and their differences are mainlyassociated with the coordinate systems used and the assumed location of the centre of mass.42CHAPTER 3NUMERICAL PROCEDURE3.1 Field SolutionThe boundary integral equations of the first- and second-order potential flow problemsdescribed in Chapter 2 may be solved numerically by the boundary element method which iscommonly referred to in the literature as a panel method. In the numerical implementationscheme, the integral equation is approximated by discretizing all the boundaries enclosing thefluid domain into a finite number of facets over which simple sources and normal dipoles aredistributed. By assuming a prescribed variation of boundary variables over each facet, thediscretized integral equation is enforced at a set of discrete points to obtain a system of linearalgebraic equations. For a well-posed boundary value problem, the resulting matrix equationcan be solved by conventional methods of linear algebra to obtain the unknown boundaryvariables, provided that either the potential or its normal derivative on each portion of theboundary is given from the corresponding boundary conditions.On the basis of recent developments of the boundary element technique, it is possible torepresent the geometry as well as the boundary variables over each facet using higher-orderinterpolation functions (e.g. quadratic and cubic) resembling those adopted in the finiteelement analysis. However, this would significantly complicate the numerical algorithm,especially in the treatment of intersection points of two portions of the boundary, on each ofwhich a different boundary condition is imposed (Lin et al., 1984). In order to maintain theoverall simplicity of the numerical procedure, which is a major consideration in the successfuldevelopment of associated computer programs, the use of higher-order boundary elements isnot considered in the present integral equation formulation.43The surfaces Sb, Sc and So are discretized into finite numbers of facets. These facets aremade up respectively of planar quadrilateral facets for three-dimensional problems, and ofstraight-line segments for two-dimensional problems. For simplicity, the correspondingboundary values of ek and a441an are assumed piecewise constant over each facet and appliedat the facet centroid. By applying these collocation points to the boundary integral equation,this leads to the following set of linear algebraic equations for the unknown boundary valuesof ek and .34 /an:Nb^B^NT^kB Nb^/0-14):^NTAjAki +^B ii k'aii-).)^• Bij^- > A-- di •j.i^J=Nb+1 J.1(i = 1, ...., NT)^(3.1)in which NT is the total number of facets in the formulation and Nb is the number of facets onSb. Note that the surfaces are treated in the order of Sb, Sc and So as the index j increasesfrom 1 to NT. The influence coefficients Bij and Aij of the matrix equation correspondrespectively to integrals of the Green's function and its normal derivative over area of the j-thfacet and are expressed as:a GAij = y f^(xi, xi) dS + Oij^ (3.2)AS iBii = -y f G(x^dSAs j(3.3)where ASj is the area of the j-th facet, the geometric factor y corresponds to 1/(27r) for three-dimensional problems and -1/it for two-dimensional problems, and Sij is the Kronecker deltafunction, equal to 1 if i = j and 0 if i j. When the field point coincides with the sourcepoints, the influence coefficients are evaluated by a closed-form integration. Analyticalexpressions for the integral of the fundamental singularity and its normal derivative over anypolygon facet shape have been given in the literature (e.g. Hess and Smith, 1964; and Hogbenand Standing, 1974). When the field point does not coincide with the source point, the44integration is performed numerically by using variable order Gaussian quadratures. Forquadrilateral facets of arbitrary shape, the numerical integration can be facilitated by employinga bilinear transformation which maps individual facet onto a unit square. The full expressionsused in the calculations of the influence matrix coefficients Aij and Bij have been given byIsaacson and Cheung (1991, 1992) and therefore these are not elaborated upon here.In the surface discretization procedure, the vertices of each facet are generated for the entireboundary surfaces of a given computational domain considered. Based on these geometricaldata, the corresponding facet area ASj, centroid and unit normal vector can then be calculated.The computational effort of numerical discretization is expected to be directly proportional tothe number of facets used. With the problem now formulated as a matrix equation (3.1),influence matrix coefficients which are functions of the time-independent geometry only areevaluated for all possible combinations of source and field points, and assembled into the left-and right-hand side matrices. These matrices are fully populated and non-symmetric, andrelate the input and output vectors containing the time-dependent boundary variables. In thepresent Stokes second-order method, a matrix inversion and multiplication is required onlyonce in the simulation to obtain the solution to the system of linear algebraic equations, andsubsequently a multiplication of the resulting matrix and the input vector at each time step thengives the resulting field solution. The input vector contains 4 1:/an on the body surface Sb,and 443, on the control surface S c and the still water surface S o .The potential on the control surface S c and the still water surface So is given by time-stepping procedures applied to the radiation condition on S c and the free surface boundaryconditions on So . Depending on the category of the wave-structure interaction problemconsidered in this thesis, the normal derivative of the potential on the body surface Sb isevaluated in a different manner. In the wave radiation problem, the normal derivative isexplicitly known from the body surface boundary condition. In studying motions of a floatingstructure, the normal derivative is calculated by the body surface boundary condition appliedin conjunction with the equations of motion.45In general, the numerical errors introduced in the matrix solution play a crucial role in theaccuracy of the flow simulation, and this aspect has been a major concern, especially for thethree-dimensional formulation. Since a Rankine source and its images are used as the Green'sfunction in this case, the resulting matrices of the field equation are dominated by elementsalong and near the diagonal and are regarded as well-conditioned. The condition number usedto indicate the magnitude of the round-off error in the matrix inversion operation is generallyfound to be very small in compared to the rank of the matrix, and therefore good numericalaccuracy of the matrix solution can confidently be maintained even for a large system ofsimultaneous equations. It is mentioned that the computational effort associated with theevaluations of matrix coefficients and the matrix inversion are proportional to R2 and R3respectively, where R is the rank of the matrix equation. Due to the simplicity of the Green'sfunction and its normal derivative, the computational burden in the field solution is dominatedby the matrix operations involving inversion and multiplication.3.2 Time-Stepping SchemeBased on the time-stepping method, the solution of the flow development may be obtained ateach discrete time step in terms of the known solution at previous time steps. In order tosummarize all the required programming tasks of the computational sequence, Table 1presents a block diagram of the time-stepping scheme in the context of the wave radiation andcombined diffraction-radiation problems. The radiation condition and the free surfaceboundary conditions are numerically integrated in time to obtain the first- and second-orderpotentials and the relevant boundary variables on the control surface S c and the still watersurface So. The number of floating point operations of the time-stepping scheme is expectedto increase approximately with the square of the total number of facets and the number of timesteps in the simulation.463.2.1 Radiation ConditionIn order to perform a sufficient length of flow simulation in a reasonably sized computationaldomain, the application of a radiation condition on the control surface requires specialattention. As mentioned previously, the Sornmerfeld radiation condition in equation (2.32)together with a time-dependent celerity, best known as the Orlanski condition, has recentlybeen applied by a number of researchers as a radiation (far-field) condition in the simulation ofnonlinear wave-structure interaction problems; although this is, in fact, not a mathematicallysatisfactory answer.By expanding the radiation condition with a central difference in time and a leap-frogdifference in space, the potential 4k on each point x on the control surface at time step (t+At)may be expressed as (Orlanski, 1976):gOk(x,t+At) = 1-13  4 k(x,t—At) + 213 413(x—nAn,t)^(k = 1, 2)^(3.4)1+13^1+0in whichAn cn^(3.5)where x is a point on the control surface; At is a time step size; An is a small distance in theorder of a characteristic facet diameter; n is the normal vector at x; and (3 is known as theCourant number. In practice, the Courant number sets an upper bound for the fastest wavefor a given spatial and temporal grid size.For most unsteady wave motions, the dispersion characteristics as well as thecorresponding celerity cn are not well defined. With spatial and temporal variations of c nbeing small, Orlanski (1976) proposed that the celerity on the control surface at a time t can beapproximated by its value at the previous time step at locations which are just within thecontrol surface. The numerical value of c n at time (t—At) and (x—nAn) can be expressed as:47cn — An ^4)k (x—nAn,t) — 4)k (x—nAn,t-2At) (k = 1, 2)^(3.6)At 4)k (x—nAn,t) + (x—nAn^t) — 20k (x-2nAn,t—At)In the numerical implementation of equation (3.6), it is indicated that the calculated celerity isundefined when the denominator is zero. In order to overcome this difficulty, the followingnumerical scheme is carried out. When the absolute values of both aoZiat and a4)13/an are lessthan certain prescribed values, the potential at the advanced time oz(x,t+At) is simply given by41k(x,t). This is equivalent to taking c n = 0, and is used because 4): changes very little in onetime step on account of ask/at being small. For numerical stability considerations, themaximum value of the celerity calculated is limited to have a value of An/At.3.2.2 Free Surface Boundary ConditionsAmong other available explicit or implicit time-stepping formulae, the evolution of the freesurface elevation and the potential on the still water surface are given by a time-steppingprocedure which incorporates the free surface boundary conditions to second-order. The freesurface elevation rlk at a new time step (t+At) is first evaluated in terms of the known solutionup to time t by the first-order Adams Bashforth equation as13TI:(t+At) = TilBc(t) + at [3(1t-9t -^ (k = 1, 2)^(3.7)The dynamic free surface boundary condition (DFSBC), equation (2.28), can then be used toprovide as:1)13, /at at time (t+At). Subsequently, the potential 4): may then be obtained by thefirst-order Adams Moulton equation:4(t+At) =^+^+ (IL]^(k = 1, 2)^(3.8)Finally, the normal derivative a443jan on the equilibrium body surface Sb is given by the bodysurface boundary condition equation (2.26) in terms of the body displacements and velocities.These are explicitly known in the wave radiation problem. However, in the floating body48problem the kinematics of the body motions are governed by the equations of motion, givenby equations (2.33) and (2.34) for the three-dimensional case and by equation (2.36) for thetwo-dimensional case, which can be solved by employing a standard fourth-order Runge-Kutta method.By using the normal derivative of the potential on Sb known from the body surfaceboundary condition and the potentials on S c and S. evaluated from the time-steppingequations, the wave field at the advanced time step (t+At) can be obtained by solving thediscretized boundary integral equation (3.1). The output vector gives ask/an on So at theadvanced time step (t+At). The kinematic free surface boundary condition (KFSBC),equation (2.27), can now be used to provide the corresponding values of ari Ekl/at at theadvanced time step (t+At). This completes the initial calculation of all the boundary variables attime step (t+At) and the refined values of the free surface elevation at advanced time areobtained by using the first-order Adams Moulton equation instead of equation (3.7)TbB,(t+At) = 4(0+ + (a,11;z ut t ot - t-hAti (k = 1, 2) (3.9)The successive steps can then be repeated identically which lead to more accurate values of Okon Sb for hydrodynamic force calculations, with the numerical values of &vat and V4 1 areobtained by applying a central difference scheme to (1) in time and a spatial interpolationinvolving neighbouring facets respectively. Previous computational experience indicates thatonly a few iterations are required to achieve rapid numerical convergence and the computationcan then proceed to the next time step.Since the free surface elevation and the potential on the still water surface are time-steppedin the normally vertical direction, the temporal solution of the resulting flow satisfies the freesurface boundary conditions to second order. Despite the fact that such second-order solutionis restricted to wave fields of weak nonlinearity, more importantly the present algorithm basedon an Eulerian time-stepping procedure greatly reduces the computational efforts in solving49nonlinear wave-structure interaction problems and is numerically stable. Although notperformed in the present study, the numerical stability of the time-stepping formulas given inequations (3.7) to (3.9) can be assessed by a simplified von Neumann analysis (Yeung,1982a).50CHAPTER 4RESULTS AND DISCUSSION4.1 IntroductionBased on the time-domain method, vectorized computer codes in FORTRAN have beendeveloped to study the second-order wave radiation and combined diffraction-radiationproblems. In this approach, a direct integral equation method applicable to both two- andthree-dimensional formulations together with a boundary element technique are used to solvethe field equation of the governing boundary value problems, and a time-integration procedureis used to obtain the temporal solution of the corresponding boundary variables.Basically, the major computational tasks involved in the hydrodynamic analysis are thediscretization of the boundaries of the computational domain, the evaluation of the matrixcoefficients in the system of linear algebraic equations, and the solution of the wave field ateach time step in order to calculate physical quantities of interest, such as hydrodynamicforces, body motions as well as the free surface elevations. A number of numericalconsiderations relating to the surface discretization scheme, the time-stepping algorithm andthe solution of a full matrix equation have been implemented in the development of thecomputational model in order to improve the computational efficiency and numerical accuracyof the solution.In order to illustrate and examine the present method, numerical results are presented forseveral well defined body geometries as shown in Fig. 2. For the two-dimensional case, themethod is applied to a semi-submerged circular cylinder (radius a) in deep water. In addition,a semi-submerged rectangular cylinder (half beam length b, draft h = b) is also employed inorder to simulate the second-order hydrodynamic effects associated with sinusoidal rollmotions in otherwise still water. For the three-dimensional case, the method is applied to a51truncated surface-piercing circular cylinder (radius a, draft h) in finite water depth. In order tocheck the accuracy of the numerical method, convergence tests are performed for the two- andthree-dimensional wave radiation problems. Although second-order results of thecorresponding wave radiation and diffraction-radiation problems are given respectively forforced sinusoidal rigid body motions in otherwise still water and regular Stokes second-orderwaves, it is suggested that with suitable modification to the theoretical formulation the methodcan be extended to tackle the more general cases of arbitrary body motions and irregular waveexcitation.4.2 Computational ConsiderationsNumerical computations employed in this study were performed on IBM 3090/150S andRS/6000 560 computers at the University of British Columbia, and double precision is usedthroughout the computation. The execution of the computer programs has been facilitated bythe use of these virtual memory machines in which the out of core memory storage requiredfor large systems of equations is swapped automatically in and out of the core memory storagein an efficient manner. The solution of the linear algebraic equations can be obtained directlyusing subroutines available in mathematical libraries such as ESSL (Engineering and ScientificSubroutine Library) and IMSL (International Mathematical and Statistical Library) which arehighly optimized to run on specific machine architecture. The stability of the numericalsimulation is customarily assessed in terms of the facet size chosen for a given time step size.The Courant criterion which describes the numerical stability is adopted to determine themaximum time step size of a given spatial discretization and wave frequency.4.2.1 Surface DiscretizationThe geometrical representation of the computational domain through numerical discretizationhas been a common feature to many practical computational methods, and there are nostringent guidelines governing the choice of discretization scheme. In most cases, the decision52concerning the shape, size and distribution of the computational grid used is a matter ofexperience and judgement. In the application of fluid flow problems using the boundaryelement technique, a smaller sized facet is generally distributed on boundary surfaces withlarge curvature and adjacent to the fluid region where the flow pattern changes rapidly.Substantial computational effort can be saved in terms of the required CPU time and theallocation of virtual memory storage by utilizing planes of symmetry in the body geometry andthe simulated flow field. In practice, this particular aspect of computational considerationbecomes particularly important in the three-dimensional flow analysis of structures withcomplicated body geometry which may lead to a large number of facets in discretizationprocedure, and the resulting solution of the system of equations is costly. Note that adaptivemesh refinement techniques are generally employed by some computer codes in order toimprove the accuracy of the numerical results obtained with the original mesh throughsuccessive refinement of the grid size and the number of facets used and the application ofhigher-order interpolation functions. Such approaches demand substantial programmingefforts and computational resources, and are not considered in the present study.Two-Dimensional ProblemThe numerical simulation of two-dimensional wave motions corresponds to a two-dimensional fluid domain with a surface-piercing horizontal cylinder section located at thecentre. About 400 to 600 facets made up of straight-line segments are generally used for thediscretization of the body surface, the control surface and the still water surface. Theinfluence matrix coefficients Aij and Bij in the discretized integral equation are evaluated by a4-point Gaussian quadrature. Note that the choice of Green's function incorporating thesymmetry about the seabed allows the seabed to be excluded from the discretization scheme.Fig. 3 shows the surface discretization of the numerical model for both the semi-submergedcircular and rectangular cylinders for cases corresponding to d/a = d/b = 4.6 and Lh/d = 4.0,where Lh is the horizontal length of the computational domain. In order to describe the morerapid variations of the potential near the body surface and to speed up the convergence of the53numerical calculations without increasing the facet density everywhere, the facet sizes used onthe body surface and on the nearby free surface are smaller than those further away from thebody surface.Three-Dimensional ProblemThe numerical simulation of the three-dimensional flow problem corresponds to arectangular domain of constant water depth with the floating cylinder positioned at the centre.Typically, about 1800-2400 planar quadrilateral facets are generated to represent one half ofthe boundaries of the domain which include the body surface, the control surface and the stillwater surface. The influence matrix coefficients Aij and Bii in the discretized integral equationare evaluated by a 16-point Gaussian quadrature. Since the Green's function is chosen toaccount for the double symmetry, the seabed and half of the total surface can be excluded fromthe numerical discretization.Fig. 4 shows examples of the numerical discretization scheme for a truncated circularcylinder for cases corresponding respectively to d/a = 1.5 and h/a = 0.5; and to d/a = 2.0 andh/a = 0.5. Note that the horizontal dimensions of the computational domain are chosen suchthat the minimum distance between the surface of the cylinder and the control surface is L,where L denotes the wavelength of the first-order waves. In the present application, smallerfacets are used on the body surface and the neighbouring portion of the free surface in order todescribe the more rapid variations of the potential function near the body and to improve thenumerical convergence of the solution, whereas a uniform grid is used on the outer region ofthe free surface. Furthermore, the facet lengths in the vertical direction are chosen to increaselinearly with submergence.4.2.2 Initial ConditionsIn order to minimize adverse transient effects related to the abrupt initial condition and to allowa gradual development of the simulated fluid motion, the initial flow development is modulated54by the following ramp function which increases gradually from zero to unity over a specifiedmodulation time (e.g. Isaacson and Cheung, 1992; and Yang and Ertekin, 1992):1 1— [1 - cos ( 1,7ct )1Fm . 2^-1for t < Tmfor t Tm(4.1)where Tm is a modulation time. For the numerical results relating to wave radiation anddiffraction-radiation problems, a modulation time Tm = 2T, where T denotes the wave period,is adopted to provide a stable and smooth development of the steady state solution obtainedimmediately after the duration of the modulation.For each of the initial-boundary value problems considered, it is necessary to prescribe aset of initial conditions to start the computation, and consequently the solution is advanced tosatisfy the corresponding boundary conditions at successive time steps. In the case of thewave radiation problem, the body surface boundary condition is gradually imposed bymultiplying the body displacement vector by the modulation function. Thus, the initialcondition corresponds to a zero flow potential everywhere and the cylinder is about to startoscillating from its initial state of rest position. In studying wave interactions with a floatingstructure, the normal derivative of the incident wave potential of the body surface boundarycondition, and the hydrodynamic pressure forces exerted on the body surface are multiplied bythe modulation function. With a Stokes second-order wave field deployed inside thecomputational domain, the immersed body surface gradually materializes and the dynamicpressure of the surrounding fluid is slowly established over the modulation time. It ismentioned that the initial conditions used for the two-dimensional problems are essentially thesame as those in the corresponding three-dimensional cases.4.2.3. Numerical AccuracyThe numerical errors of the present solution technique mainly depend on the accuracy of thefield solution of the Laplace equation, which involves a geometrical discretization of the55computational domain boundary, and the time-integrations of the free surface and radiationboundary conditions which are used to provide the temporal solution for the flowdevelopment In particular, the numerical convergence of the present method is illustrated bythe computed steady state hydrodynamic forces with increasing number of facets used torepresent the body surface, and two- and three-dimensional wave radiation problems areconsidered here.Two-Dimensional Radiation ProblemThe particular cases of the semi-submerged circular and rectangular cylinders subject toforced sinusoidal heave motions in deep water with 0)24 = 0.6 and co2b/g = 0.6 respectivelyare examined here, where 0) is the excitation frequency. The convergence of thehydrodynamic force components at first and second order in dimensionless forms are given inTables 2 and 3. It is seen that the first-order force component (denoted by Fi r) achieves veryaccurate results at relatively low values of Nb. The second-order steady force components(denoted by For) and the second-order oscillatory force components due to $ti i (denoted by(i)F2r) converge relatively fast in relation to the second-order oscillatory force component due to4)2 (denoted by F2r). In general, the convergence of the wave force components is more rapidfor the circular cylinder than for the rectangular cylinder. This may be due to the presence ofthe corners in the rectangular geometry.Three-Dimensional Radiation ProblemFor the case of the truncated circular cylinder subject to forced sinusoidal heave motionwith ka = 1.0, d/a = 1.5, h/a = 0.5 and A/a = 0.1, Table 4 shows the vertical hydrodynamicforce components at first and second order in dimensionless forms as functions of Nbl, whereNbl represents the number of facets used in the discretization for one half of the bottomportion of the cylinder. Since heave motions are considered, only the discretization of thecylinder bottom is varied, even though this may not represent an entirely systematic approach.It is found that the first-order force component converges to very accurate results at relatively56low values of Nbl. The second-order steady force components and the second-orderoscillatory force components due to 0 1 , both of which are evaluated from the quadraticproducts of the first-order solution, converge quite rapidly. However, due to the increasingnumerical errors associated with the second-order solution, the convergence of the second-order oscillatory force component due to 0 2 is seen to be comparatively slow in relation to thecorresponding linear calculations, and consequently a finer facet size is used on the bodysurface and the neighbouring portion of the free surface in order to achieve accurate results.4.3 Radiation ProblemsWithin the scope of the present investigation, the second-order wave radiation probleminvolving forced sinusoidal motions of a rigid, surface-piercing body in otherwise still wateris considered for both the two- and three-dimensional cases. Traditionally, the first-orderoscillatory force excluding hydrostatic stiffness components associated with wave radiationproblem is presented in terms of hydrodynamic added mass and damping coefficients, whichrepresent components in phase with the acceleration and the velocity of the body motionrespectively. Based on the frequency-domain approach, these hydrodynamic coefficientstogether with the first-order wave exciting (diffracted) forces can then be used to predictmotion responses of a floating body as functions of wave frequency generally known as lineartransfer function.In order to present steady state results, the first- and second-order hydrodynamic force andmoment components reported here may explicitly be described in terms of their amplitudesnormalized as force coefficients and phase angles. For example, the total hydrodynamic forcein the j direction is expressed asFj = Foj + Fij sin(cot + 61j) + F2j sin(2cot + 82j) (4.2)where Foj is the second-order steady force; Fij and F2j are the first- and second-orderoscillatory forces respectively; and 81 j and 82j denote the phase angles of the first- and57second-order oscillatory force respectively. Note that the hydrostatic restoring force isincluded inside the first- and second-order oscillatory force components.Unlike frequency-domain methods in which the body motions are restricted to be eithermonochromatic or bichromatic, the present method can be extended to treat an arbitrary bodymotion which is considered as first order, and the first-order radiation solution containscomponents with a range of frequencies and amplitudes. Each of the components and theirinteraction terms are collectively refined to a second-order approximation. If the major interestis limited to the overall response of the system rather than the individual frequencycomponents, the present approach appears to be more efficient and effective than the moreconventional frequency-domain methods.4.3.1 Two-Dimensional Radiation ProblemIn the two-dimensional wave radiation problem, numerical results are given for two semi-submerged circular and rectangular cylinder sections in deep water. For the circular andrectangular cylinders in heave and sway motion, the centre of mass is assumed to be at theintersection of the still water level and the vertical plane of symmetry. For the rectangularcylinder in roll motion, the centre of mass is located at an elevation of h/6 below the still waterlevel. In order to validate the numerical method, comparisons of the computed hydrodynamicforces at first and second order for the case of the circular cylinder are made with previoustheoretical and experimental results and a favourable agreement is indicated.Fig. 5 shows the development with time of the specified vertical body motion, the freesurface elevation adjacent to the body surface and the vertical hydrodynamic force for thesemi-submerged circular cylinder with co2a/g = 1.6, A/a = 0.1 and deep water, where A is theamplitude of the forced sinusoidal motion. The time variations of the first- and second-ordercomponents of the free surface elevation and the vertical hydrodynamic force are included inthe figure. It is clear that a steady state solution is developed soon after the end of the secondcycle, when the specified body motion attains its full amplitude. For the relatively high58excitation frequency considered, the second-order force components are seen to affectsignificantly the total hydrodynamic force, particularly its maxima. This example highlightsthe importance of the second-order hydrodynamic effects associated with the forced motionsof a large structure. However, the second-order component of the free surface elevation isseen to be relatively small compared to the first-order component.The development of the free surface profiles with time to first and second order for thesame conditions as to those of Fig. 5 is shown in Fig. 6. In the figure, x is measured fromone edge of the computational domain, and the cylinder axis is located at x/L = 2 which is atthe centre of the domain's horizontal length of 4L. It is seen that the flow immediatelyadjacent to the body appears to repeat itself after the second cycle, while the flow further fromthe body takes somewhat longer to reach a steady state. The radiated waves at first andsecond order propagate steadily away from the body at their corresponding celerities. It isexpected that there are two systems of second-order waves. The second-order forced wavesare phased-locked with the first-order wave systems to produce an overall wave profile withsteeper crests and flatter troughs; while the second-order free waves at twice the excitationfrequency propagate independently at a celerity obtained from the linear dispersionrelationship. In this particular case, the free wave system is found to dominate the totalsecond-order wave component.Fig. 7 shows the amplitude and phase angle of the first-order oscillatory force in the zdirection on a semi-submerged circular cylinder in heave motion. The corresponding second-order steady force is presented in Fig. 8. The figures show comparisons of the present resultswith the theoretical results of Potash (1971) and indicate excellent agreement. The amplitudeand phase angle of the second-order oscillatory force in the z direction for the same conditionsare presented respectively in Figs. 9 and 10. These show comparisons of the present resultswith the previous theoretical results of Yamashita (1977) and Papanikolaou (1984) and alsowith experimental results of Yamashita (1977). Fig. 9 shows the comparison of the forceamplitude and exhibits favourable agreement; whereas the comparison in Fig. 10 for the phase.59angle indicates that the present results give higher values than the previous studies. It is ofinterest to note that the second-order oscillatory force increases with increasing excitationfrequency while the first-order oscillatory force decreases with the frequency. At highfrequencies, second-order effects are expected to become relatively important in thehydrodynamic force calculations. Fig. 11 shows the amplitudes of the second-order forcecomponents associated with 4) 1 and 4)2 , and both the force components due to 4) 1 and 4)2increase steadily with frequency.Fig. 12 shows the amplitude and phase angle of the first-order oscillatory force in the xdirection on a semi-submerged circular cylinder in sway motion. Its amplitude increases withincreasing frequency and approaches a constant value at high frequencies. The agreementbetween the present solution and the theoretical results of Potash (1971) is excellent. Fig. 13shows the magnitude of the second-order steady force in the z direction compared with thetheoretical and experimental studies of Kyozuka (1982). Excellent agreement is indicatedbetween the present results and Kyozuka's theoretical results. On the other hand, Kyozuka'sexperimental results are found to exhibit some scatter. Figs. 14 and 15 show the amplitudeand phase angle of the second-order oscillatory force in the z direction. The second-orderoscillatory force amplitude, shown in Fig. 14, appears to increase steadily at high frequenciesand Kyozuka's theoretical solution tends to give higher predictions than the present results,while Kyozuka's experimental results exhibit considerable scatter. Fig. 15, which shows thesecond-order oscillatory force phase angle, indicates that the present results have the sametrend as Kyozuka's theoretical and experimental results. As with Fig. 11, Fig. 16 shows thecomposition of the second-order oscillatory force, and indicates that the force component dueto 4)2 is the major contribution to the resultant second-order force amplitude. For asymmetrical cylinder section undergoing pure sway and roll motions in deep water, it isexpected that the second-order forces give rise to components only in the z direction, so thatthe corresponding force components in the x direction are not shown.60Numerical computations have also been carried out for a semi-submerged rectangularcylinder in deep water with a half beam length to draft ratio b/h = 1.0. Although not shown,the corresponding first-order results for the heave, sway and roll motions of the rectangularcylinder give excellent agreement with the previous first-order results presented in terms of theadded masses and damping coefficients (e.g. Vugts, 1968; and Nestegard and Sclavounos,1984).The magnitude of the second-order steady force, the amplitude of the second-orderoscillatory force and the amplitudes of the second-order force components due to 4) 1 and 4)2,all in the z direction, are presented in Fig. 17 for a semi-submerged rectangular cylinder inheave motion. It is seen that the second-order oscillatory force components increase steadilywith frequency. The magnitude of the steady second-order force is found to be considerablysmaller than the second-order oscillatory force components. The four corresponding second-order force components in the z direction for a semi-submerged rectangular cylinderundergoing sway motion are shown in Fig. 18. Note that the errors caused by neglecting thecomponent due to 4)2 in the second-order oscillatory force calculations are observed to besignificant at high frequencies. The second-order steady force exhibits the same trend over thefrequency range of interest as for the case of a circular cylinder in sway motion.In contrary to the case of a circular cylinder, the rectangular cylinder undergoing forced rollmotion gives rise to significant forces. As mentioned previously, corresponding computationswith the semi-submerged rectangular geometry have taken the centre of mass to lie at adistance of h/6 below the still water level. Fig. 19 shows the corresponding second-ordersteady force and the amplitudes of the various second-order oscillatory force components inthe z direction. As indicated in the figure, the second-order potential plays a dominant role inthe contributions towards the total second-order force. The phase angles of the second-orderoscillatory force in the z direction due to heave, sway and roll motions of the rectangularcylinder are presented in Fig. 20. A similar trend can be observed for the case of therectangular cylinder in heave and roll motions.61For the case of two-dimensional semi-submerged cylinder sections, the numerical resultsindicate that steady state solutions are achieved over a reasonably short duration, and thesecond-order components become more important in the evaluation of the total hydrodynamicforce at higher excitation frequencies. In general, the second-order oscillatory forcecomponent due to 02 contributes significantly in the evaluation of the second-order oscillatoryforce. The present results have confirmed that the second-order force contributionsapproximated by quadratic products of first-order results alone may lead to significant errors.4.3.2 Three-Dimensional Radiation ProblemIn the three-dimensional wave radiation problem, numerical results including thehydrodynamic force components and the free surface profiles to second order are given for thecase of a truncated surface-piercing circular cylinder with d/a = 1.5 and h/a = 0.5. In theresults presented here, the forced sinusoidal body oscillations of the circular cylinder arerestricted only to the surge and heave motions about the centre of mass. The present linearresults give excellent agreement with the previous theoretical data presented in terms of theadded masses and damping coefficients (e.g. Garrison, 1974). More recently, Sabuncu andCalisal (1981) presented the hydrodynamic coefficients for truncated vertical circular cylinderswith different depth to radius and draft to radius ratios.Fig. 21 shows the time histories of the specified vertical body motion, and of the verticalhydrodynamic force components on the truncated circular cylinder for the case ka = 1.6. Asteady state solution is gradually achieved at the end of the specified second cycle when thebody motion attains its full amplitude. At the relatively high excitation frequency examinedhere, the contribution of second-order force components to the total hydrodynamic force isfound to be significant. Indeed, this particular case illustrates the presence of strong second-order hydrodynamic effects due to the nonlinear interaction of the forced body motions withthe radiated wave field.62Fig. 22 shows oblique views at a particular instant t = 3.5T of the free surfaces to secondorder for a truncated circular cylinder undergoing sinusoidal heave and surge motions in turnwith ka = 1.6 and A/a = 0.1. The resulting radiated waves are generated by the forced bodymotion of the cylinder, somewhat analogous to a three-dimensional wave-maker. It is clearthat the wave runup and the resulting free surface for the heave motion are axisymmetric andthe radiated waves at first and second order propagate away from cylinder concentrically attheir corresponding celerities. On the other hand, the refinement of the free surface resultingfrom the surge motion, which is asymmetric about the y-z plane for the linear radiated waves,is associated with the second-order radiated wave components. It is emphasized that themaximum wave runup and rundown for surge motion are located at opposite sides of thecylinder surface along the x direction.Based on equations (2.45) and (2.46), the second-order free surface elevation can berepresented as due to contributions from the second-order potential and from the quadraticproducts of the first-order quantities. The second-order forced waves are phased-locked withthe first-order wave system to produce an overall wave profile with steeper crests and flattertroughs, these being responsible for the increase of the runup profile around the cylinder. Inthe present case, the free waves at twice the excitation frequency, which propagateindependently of the first-order wave system at a constant celerity obtained from lineardispersion relation, has been found to be the dominant second-order wave component.Fig. 23 shows the amplitude and phase angle of the first-order oscillatory force in the zdirection as functions of ka for a truncated circular cylinder (d/a =1.5, h/a = 0.5) in heavemotion. It is seen that the amplitude of the first-order force decreases with increasingexcitation frequency. Fig. 24 shows the corresponding second-order steady force and theamplitudes of the various second-order oscillatory force components again as functions of ka.The second-order steady force is found to be relatively small compared with the second-orderoscillatory components. In addition, the amplitudes of the second-order oscillatory forcecomponents increase steadily with excitation frequency. This feature is associated primarily63with the representation of the dimensionless force used and corresponds to a steepening ofradiated waves at higher ka values. Over the frequency range of interest, the majorcontribution to the total second-order oscillatory force comes from the second-order oscillatoryforce component due to 4)2. Fig. 25 shows the corresponding phase angle of the total second-order oscillatory force as functions of ka.Fig. 26 shows the amplitude and phase angle of the first-order oscillatory force in the xdirection and moment about the y-axis as functions of ka for a truncated circular cylinder (d/a=1.5, h/a = 0.5) in surge motion. In the figure, both first-order force and momentcomponents increase with increasing excitation frequency. Fig. 27 shows the correspondingsecond-order steady force and the composition of the second-order oscillatory vertical forcecomponents. It can be shown that there is no second-order horizontal force for a symmetricbody undergoing surge motions (e.g. Kyozuka, 1982), and hence the vertical force is shownin the figure. Note that the second-order steady force always acts in the downward directionover the range of ka values of interest. It is indicated that the amplitude of second-orderoscillatory force component due to 02 is similar to that of the second-order componentcalculated from first-order solution. The amplitude of the total second-order oscillatory forceis found to increase gradually to the maximum value and drop off steadily at large values ofka. Fig. 28 presents the corresponding phase angle of the total second-order oscillatory forceas functions of ka.It has been shown that the extension of the present method to the general three-dimensionalcase of second-order wave radiation problem is fairly straightforward although the efforts ofalgorithm development and the computational costs inevitably increase manifold. A properspecification of the radiation boundary condition enables flow simulation to be performed overa long duration of time in a reasonably sized computational domain. As in the related case oftwo-dimensional wave radiation, the contribution of the second-order force component due to4)2 is found to be important to the total hydrodynamic force calculations, especially at higherexcitation frequencies.644.4 Wave-Structure Interaction ProblemsAs mentioned previously, one of the main themes of this thesis has been the development ofsecond-order solutions for wave interactions with two- and three-dimensional large floatingstructures by employing the time-domain method. Numerical examples presented include thetransient motion of a freely-floating cylinder with a specified initial vertical displacement inotherwise still water, and the diffraction-radiation of Stokes waves by a moored floatingcylinder. In the latter case, the first- and second-order incident wave potentials for the case ofregular waves with height H, length L and period T are given respectively as:w^nH cosh [k(z+d)] sin (kx—cot)kT sinh (kd) (4.3)w^3 itH RH\ cosh [2k(z+d)] sin 1-200t—tot)]^ (4.4)4)2 — 8 kT L^sinh4(kd)^""where k is the wave number and 0) is the angular frequency. In order to validate the numericalalgorithm used for solving the equations of motion which may be expressed as a system ofcoupled second-order ordinary differential equations, the transient motion problem extended tosecond order has been investigated as a prelude to treating the more general case of a floatingbody subjected to regular waves.4.4.1 Transient Motion of Semi-Submerged Circular CylinderThe unsteady free surface flow problem relating to the transient heave motion of a freely-floating two-dimensional cylinder with a specified initial displacement has been the subject ofa number of analytical investigations based on a linearized potential flow treatment (e.g.Maskell and Urse11,1970; Yeung, 1982b; and Lee and Leonard, 1987). Since the radiatedwaves generated by the free body motions are unsteady, the celerity of the outwardpropagating waves at the control surface is determined numerically at each time step.Furthermore, for this problem the modulation function is not needed, and the initial condition65is taken to correspond to zero velocity potential everywhere, a specified initial vertical bodydisplacement, and zero initial velocity.Fig. 29 shows calculated time histories of the heave motion and the vertical hydrodynamicforce components on the cylinder for case corresponding to IAI/a = 0.1, d/a = 10.0 and Lh/d =4.0, where IAI is the initial vertical displacement of the cylinder. Note that the cylinder axis issituated at x/I, = 2.0. For convenience, time t is normalized with respect to T = 7-4Wrg whichis associated with the natural period of the cylinder in heave. The present first-order solutionof the displacement gives excellent agreement with the corresponding results of Maskell andUrsell (1970). Note that the time between each successive zero uperossing of the heavemotion and the corresponding hydrodynamic force varies slightly, in accordance with theunsteady features of the motion. The figure also indicates that differences between the first-and second-order solutions are hardly distinguishable for this case, so that errors caused byneglecting the second-order solution to the initial-boundary value problem would be minimal.The development with time of the corresponding free surface profiles to second order isshown in Fig. 30. The unsteady radiated wave motions continuously transmit energy awayfrom the cylinder and in turn damp out the harmonic body motions. In contrast to the resultsfor the case of a cylinder undergoing forced sinusoidal motions, in which substantial second-order effects can be observed, the contribution of the second-order wave components to thefree surface elevation, in the present case, is found to be insignificant.4.4.2 Two-Dimensional Diffraction-Radiation ProblemIn this section, numerical results including hydrodynamic forces and motion responses tosecond order are presented for a moored semi-submerged circular cylinder subjected to deepwater regular waves. Note that the centre of mass of the cylinder is taken to be at theintersection of the still water level and the vertical plane of symmetry. For simplicity, thecylinder motion is constrained in the x direction by a pair of linear horizontal springs located atthe still water level and with a combined equivalent stiffness of (rt/2)pga. With this particular66configuration, rotational motions induced by the moment components vanish and are thereforenot considered in the computation. The incident waves correspond to a Stokes second-orderwave train inside the computational domain, and the modulation function is used to providethe initial conditions and the initial development of the flow.Fig. 31 shows the development with time of the horizontal and vertical motions (denotedby 4 and C respectively) and the corresponding hydrodynamic force components (denoted byFx and Fz respectively) to first and second order for the same conditions, &a/g = 0.4, A/a =0.3 and deep water, where A is the amplitude of the first-order incident wave train. Asindicated in the figure, the body motions and the hydrodynamic forces are modulated tosuppress the transient effects associated with the initial conditions, and a second-order steadystate solution is gradually developed over a reasonably short duration after the full impositionof the body surface boundary condition. In this particular case, the second-order componentscontribute significantly to the total vertical hydrodynamic force and the corresponding heavemotion. On the other hand, the amplitudes of the second-order force components in the xdirection and the resulting sway response are relatively small compared with those of the first-order solution. It is seen that the horizontal drift force and the resulting steady motion are inthe direction of wave propagation as expected, while their vertical counterparts are in thenegative z direction.Fig. 32 shows the development with time of the free surface profiles to first and secondorder for co2a/g = 0.4, A/a = 0.3 and deep water. The cylinder is located at the centre of thedomain with horizontal length of 4L, where L is the wavelength of the incident wave train. Inthe figure, a steady state is attained for the fully developed wave field a few cycles after themodulation time, and the second-order wave components associated with the forced and freewaves are observed to be concentrated near the cylinder. In addition, the upwave region inwhich incident and perturbed waves propagate in opposite directions is characterized by thepresence of strong second-order wave effects. Since the incident and perturbed waves in thedownwave region are out of phase and propagate in the same direction, the resulting wave67amplitude and the related second-order components are relatively small. In the deep watercase considered here, the free wave system is observed to be the dominant second-order wavecomponent.Fig. 33 shows the amplitude and phase angle of the first-order oscillatory force in thehorizontal and vertical directions as functions of dimensionless wave frequency co 2a/g for asemi-submerged circular cylinder in deep water. In order to confirm these results,corresponding first-order results have also been obtained from a frequency-domain solution ofthe equations of motion using added masses, damping coefficients and wave exciting forcesobtained from the literature (e.g. Dean and Ursell, 1959; Vugts, 1968; and Nestegard andSciavounos, 1984). These results are identical to those of Fig. 33 and hence are not includedin the figure. The first-order horizontal hydrodynamic force is seen to approach zero when thedimensionless frequency aPaig = 1.0, corresponding to the surge natural frequency (definedwith the added mass excluded). The corresponding phase angle then undergoes a shift of 7E.These features may readily be explained by considering the equation of motion in thehorizontal direction, bearing in mind that wave radiation damping is included in thehydrodynamic force and that the structural stiffness is provided solely by the horizontalmooring lines. In a similar manner, the first-order vertical hydrodynamic force approaches thehydrostatic restoring force at the heave natural frequency. For small values of co2a/g, it isindicated that the first-order vertical hydrodynamic force is out of phase with the free surfaceelevation.Fig. 34 presents the second-order horizontal and vertical drift forces as functions of &a/gfor a semi-submerged circular cylinder in deep water. It is evident that the horizontal driftforce always acts in the direction of wave propagation. Furthermore, it has first been shownby Maruo (1960) that the steady force component of a two-dimensional floating structure isdirectly proportional to the square of the disturbed wave amplitude (or reflection coefficient) inthe upwave region. At the high wave frequency limit, the incident waves are fully reflected bythe surface-piercing cylinder and the horizontal drift force approaches an asymptotic value of68(1/2)pgA2. For the floating cylinder considered here, the vertical drift force acts downwardsand is primarily due to negative dynamic pressure associated with the velocity squared term inthe Bernoulli equation. Furthermore, its magnitude increases rapidly with wave frequency toa peak and then reduces to a fairly modest value at high wave frequencies. Note that thehorizontal and vertical drift force components are counteracted respectively by the mooringline restoring force and the second-order force due to the hydrostatic stiffness, and thesetogether give rise to the corresponding steady drift motions.The amplitude and phase angle of the corresponding second-order oscillatory force in thehorizontal and vertical directions as functions of co2a/g are presented in Fig. 35. In this case,the horizontal second-order oscillatory force approaches zero at co 2a/g = 0.25, correspondingto a wave frequency of half the natural frequency, and associated with the second-orderforcing occurring at twice the incident wave frequency. For higher wave frequencies, both thehorizontal and vertical second-order oscillatory components increase steadily with co 2a/g.Corresponding results relating to the cylinder motions are shown in Figs. 36 to 38. Fig.36 shows the amplitude and phase angle of the first-order oscillatory sway and heave responseas functions of (.02a/g. The heave motion exhibits virtually no dynamic amplification at smallvalues of co2a/g and is then in phase with the incident wave component, as expected. Asindicated in the figure, the amplitudes of the sway and heave motions, both of which exhibitsimilar trend, increase with increasing values of co2a/g to different peak values and drop offsteadily at high wave frequencies. Once again, corresponding results have been obtained onthe basis of a frequency-domain approach. These are identical with those of Fig. 36 andhence are not included in the figure.Fig. 37 shows the corresponding magnitude of the second-order drift sway and heaveresponse as functions of co2a/g. The results indicate that the second-order drift motionsexhibit similar trends as those of the corresponding drift force components, as expected. Themean sway response is caused by the horizontal drift force and corresponds to a constantoffset in the positive x direction, whereas the mean heave response is induced by the vertical69drift force and corresponds by a steady set-down of the cylinder. Fig. 38 shows theamplitude and phase angle of the corresponding second-order oscillatory sway and heaveresponse. In the figure, the amplitudes of the second-order oscillatory sway and heaveresponse exhibit considerable variations over the frequency range of interest. Furthermore,the trend of the second-harmonic heave response appears to be similar to that of verticalsecond-order oscillatory force for relatively low wave frequencies.Following the development of second-order wave radiation solution, the method has beenextended to solve the more complicated problem of combined diffraction-radiation by two-dimensional floating bodies. Unlike the previous case of transient heave motion of a freely-floating body in which no substantial second-order wave effects is present in the predicted freebody motions and free surface profiles, in the present case of a moored floating cylindersubjected to regular waves second-order force and motion components are found to besignificant at certain wave frequencies.4.4.3 Transient Motion of Truncated Circular CylinderLike the corresponding two-dimensional case, the linearized potential flow problem related tothe transient heave motion of a freely-floating three-dimensional body with a specified initialdisplacement has previously been studied by various time-domain formulations (e.g.Newman, 1985; and Beck and Liapis, 1987). For this transient problem the modulationfunction is not needed, and the initial condition is taken to correspond to a zero velocitypotential everywhere, a specified initial vertical body displacement, and a zero initial velocityof the body.Fig. 39 shows calculated time histories of the heave motion and the corresponding verticalhydrodynamic force components on the truncated circular cylinder for the case correspondingto IAI/a = 0.1, d/a = 3.0 and h/a = 0.5. Note that in this case a computational domain withhorizontal dimensions 4d x 4d has been used. As indicated in the figure, the present first-order solution of the displacement gives excellent agreement with the numerical results of70Newman (1985). The hydrodynamic force is seen to be out of phase with the resultingtransient heave motion by a phase angle of 71 corresponding to the effect of a positive addedmass. For this case, second-order effects associated with the heave motion and thehydrodynamic forces are hardly noticeable, so that satisfactory results can simply be obtainedfrom a linearized solution without any major loss of accuracy. When compared to thetransient motion of a two-dimensional semi-submerged circular cylinder, the oscillatorymotion damps out relatively slowly in the present case, implying that wave damping effectsare much weaker in this three-dimensional case.Fig. 40 shows the oblique views of the corresponding free surface profiles to second orderat several time instants -during the free body motion. It is evident that the resulting free surfaceprofiles generated by the transient motion are axisyrnmetric, and that the unsteady wavemotion continuously transmits energy concentrically away from the cylinder. Although notevident in the figure, the contribution of second-order wave components to the free surfaceelevation is found to be insignificant.4.4.4 Three-Dimensional Diffraction -Radiation ProblemIn this section, numerical results are presented for a moored floating truncated circular cylinderwith d/a = 2.0 and h/a = 0.5 subjected to sinusoidal incident waves corresponding to ka = 1.0and A/a = 0.2. Such results include motion responses and hydrodynamic forces, as well asthe corresponding free surface profiles and wave amplitudes to second order. In the particularcase considered, the body motion is constrained in the surge direction by a pair of linearhorizontal springs located at an elevation of a/4 above the still water level with a combinedequivalent stiffness of Epga2, and the radius of gyration of the cylinder about the y-axis istaken to be 0.8a. Since the cylinder is symmetric about the x-z and the y-z planes, only thesurge, heave and pitch modes of motion are non-zero.The incident waves correspond to an undisturbed Stokes second-order wave train inside thecomputational domain, and the modulation function is used to provide the initial conditions71and the initial development of the flow. For the geometry considered here, the wave radiationdamping associated with pitch motions is minimal, and consequently transient motionsassociated with the initial conditions do not attenuate within a reasonable duration. In order tofacilitate a more rapid decay of the initial transient motions and accelerate the development ofthe steady state solution, an artificial damping term is introduced for a limited duration into thepitch mode of the equation of motion. Numerical experiments suggest that stable steady statepitch motions can be attained by modulating the initial damping ratio of 0.5 gradually to zeroover a duration of 5T. Indeed, in a practical application of the method, it would in any case benecessary to include additional damping to account for viscous effects.Fig. 41 shows the development with time of the motion response and the correspondinghydrodynamic force and moment components to first and second order. It is seen that asecond-order steady state solution is gradually developed over a reasonably short durationafter the full imposition of the body surface boundary condition and the removal of theadditional pitch damping. In this particular example, the total second-order hydrodynamicmoment (pitch) is found to be of the same order as the first-order component. On the otherhand, the amplitudes of the second-order force components in the x and z directions arerelatively small compared with those of the first-order solution, although not negligible. It isseen that the dominant motion responses are associated with the wave exciting frequency, sothat the cylinder undergoes primarily sinusoidal motions. Although not evident in the figure,the drift motions to the corresponding steady second-order force and moment components arealso present.In order to visualize the characteristics of the calculated three-dimensional wave field, Fig.42 shows oblique views of the free surface profiles to second order at two instantscorresponding to t = 02T and t = 0.7T respectively, for a moored floating cylinder in wavesfor the same case, ka = 1.0 and A/a = 0.2. As indicated from these perspective plots, second-order wave disturbance components due to diffraction-radiation considerably affect theresulting free surface profiles. In accordance with second-order theory, these second-order72waves contain forced and free wave components, which are respectively associated withquadratic forcings in the free surface boundary conditions and with the interactions of theseforced waves with the floating cylinder. In this case, the complicated diffracted and radiatedwave motions generated by the free waves are particularly noticeable. In fact, they propagateaway from the cylinder in all azimuthal directions at a constant celerity which satisfies thewave dispersion relation, and which is independent of the first-order wave system. Thispronounced feature is markedly different from the case of wave diffraction by a verticalcircular cylinder in which second-order free waves are predominant only in the downwaveregion (e.g. Kriebel, 1990; and Isaacson and Cheung, 1992). On the other hand, the forcedwaves which phased-locked with the first-order wave system give rise to the feature of steepercrests and flatter troughs in the overall wave profile.Fig. 43 shows the perspective and contour plots of the first-order wave amplitude for thesame case. Note that the contours shown are normalized with respect to incident waveamplitude. Since all the quadratic forcing terms are proportional to the square of first-orderwave amplitude, the spatial variations of the second-order forced waves are directly related tothe first-order wave envelope shown in Fig. 43. Since the diffracted and radiated wavemotions propagate in the opposite direction to the incident waves in the upwave region, apartial standing wave system may be observed in this region. In the downwave region, theincident and disturbance components propagate in the same direction. There is a substantialvariation of wave amplitude around the cylinder. It is seen that the maximum and minimumwave amplitudes are located respectively on the downwave side of the cylinder (0 = 0) and onthe mid-section of the cylinder (0 = ±7./2), where 0 denotes the azimuthal angle. In contrastto this, in the case of a bottom mounted circular cylinder, the corresponding maximum waveamplitude is on the upwave side of the cylinder (0 =Fig. 44 shows the perspective and contour plots of the wave crest amplitudes to secondorder for the same case. In general, the second-order wave system substantially refines theresulting wave amplitudes from those of the first-order solution. The features of first- and73second-order partial standing wave systems can sketchily be observed in the upwave region.In the downwave region, the wave amplitudes are significantly amplified, with localized areasof high amplitude. This particular example illustrates the significant contributions of second-order wave components in the predictions of the free surface elevations.Fig. 45 shows the corresponding second-order mean free surface elevations. This second-order component corresponds to a steady set-up or set-down of the mean water level due tothe non-zero mean of the free surface boundary conditions at second order. In the upwaveregion, the partial standing wave system gives rise to the corrugated or spatially varyingpattern of the mean water level. A maximum set-up related to the strong interactions ofincident wave and wave disturbance components is seen on the rear side of the cylinder (0 =0) and drops off sharply away from the cylinder. In addition, the minimum set-down near 0= ±it/2 is associated with the localized low pressure due to the first-order wave amplitude.Fig. 46 shows the corresponding second-order oscillatory wave amplitudes. Thiscomponent originates from the oscillatory variation of the free surface elevation at secondorder due to both forced and free waves. As indicated in the figure, there is a second-orderpartial standing wave system formed in the upwave region. Substantial spatial variations ofwave amplitude are seen in the downwave region, and the corresponding maximum values arenear the lee quarters of the cylinder.In this particular case examined, it is evident that the second-order effects relating to theinteraction of Stokes regular waves with a moored floating truncated cylinder are morepronounced than the former case of three-dimensional transient heave motion.74CHAPTER 5SUMMARY AND CONCLUSIONS5.1 Second-Order Time-Domain SolutionA time-domain method has been extended to study nonlinear wave radiation and combineddiffraction-radiation problems in both two and three dimensions for large floating structures ofarbitrary shape. For weakly nonlinear wave propagation, it is possible to develop nonlinearsolutions up to second order described in terms of wave steepness and body motions by usingthe Stokes perturbation procedure together with an Eulerian time-stepping scheme. Withfurther development in this direction, it is expected that the method can be generalized byconsidering the case of bichromatic or irregular wave excitation to calculate the resultingsecond-order sum-frequency and difference-frequency forces and motions.The mathematical formulation defining the nonlinear interaction of free surface waves witha floating structure is derived by invoking the potential flow assumptions. The major obstacleof solving the fully nonlinear problem is that an exact solution is required to satisfy the twononlinear free surface boundary conditions at the free surface and the body surface boundarycondition at the instantaneous wetted body surface, both of which are unknown a priori.Another well-known computational difficulty is related to the suitable treatment of the far-fieldclosure which ensures that all wave disturbances due to the presence of the structure propagateaway from the computational domain without reflection. In addition to these intriguingproblems, high computational costs and large memory storage requirements resulting fromsolving a large size matrix equation at each time instant have impeded the extension of fullynonlinear simulation to the more general three-dimensional problems.Following second-order analysis, the application of Taylor series expansions allows thebody surface boundary condition and the two nonlinear free surface boundary conditions to be75expanded about their corresponding mean positions, and the Stokes second-order perturbationprocedure is then used to establish the corresponding first- and second-order boundary valueproblems defined with respect to a time-independent fluid domain. For floating bodiesconsidered in this thesis, the equations of motion for the first- and second-order responsecomponents used to describe the dynamics of the structural system are coupled with thecorresponding body surface boundary conditions.Unlike the linear hydrodynamic theory in which the solution can simply be obtained bysuperposition of incident, diffracted and radiated potentials; the second-order solution containsmore complicated nonlinear wave components due to self- and cross-interactions of the first-order incident and disturbed wave potentials. Conceptually, these forced and free wavecomponents are associated with quadratic forcings in the second-order free surface boundaryconditions and with the interactions of these forced waves with the structure by enforcing thesecond-order body surface boundary condition respectively. Due to the inhomogeneity of theboundary conditions at second-order, it is necessary for most frequency-domain formulationsto obtain the second-order potential by solving a number of algebraically tedious sub-boundary value problems. On the contrary, the present approach makes use of a single wavedisturbance potential to account for the second-order combined diffraction-radiation effects.With the fundamental singularity and its image about the seabed chosen as the Green'sfunction, an integral equation method based on Green's theorem applicable to both two- andthree-dimensional problems can be used to obtain the first- and second-order field solutions ateach time step. One distinctive feature of the direct formulation is that the influence matrixcoefficients are dependent on spatial geometry of the boundary surface only. Thus, thesolution of linear system of algebraic equations is required once leading to a substantialreduction in computational efforts. A time-stepping scheme together with a suitable iterativeprocedure are used to obtain the resulting flow development as well as motion responses. Ingeneral, the present method is found to be algebraically straightforward, computationallyeffective and numerically stable and appears to be the best alternative to the existing fully .76nonlinear methods. Furthermore, the time-domain analysis has the advantage of includingnonlinear effects such as those originated from viscous-induced damping and structuralstiffness of the mooring system without resorting to the method of equivalent linearization.Steady state first- and second-order solutions are developed over a reasonably shortduration by a gradual imposition of the body surface boundary condition over the modulationtime. In the present application, the prescribed set of initial conditions for radiation problemsis somewhat different from those for diffraction-radiation problems. The former correspondsto a zero flow potential everywhere and the cylinder is about to start oscillating from its stateof rest position, while the latter corresponds to a gradual materialization of the wetted bodysurface and a smooth establishment of dynamic pressure of the surrounding fluid with theintroduction of a Stokes second-order wave field inside the fluid domain.With suitable treatment of the radiation condition, it is possible to perform simulation over along period of time in a reasonably sized computational domain. In the absence of amathematically satisfactory answer, Sommerfeld radiation condition with time-dependentcelerity (Orlanski condition) is approximated as the radiation condition to treat the case ofunsteady wave propagation. By applying a time-stepping procedure to the radiation condition,the potentials on the control surface can be obtained by a numerical extrapolation based onvalues of potentials near and inside the boundary of the computational domain at previous timesteps. Based on the computational results of the examples studied here, the performance ofOrlanski condition is found to be effective, and no significant contamination of solutions hasbeen observed interior of the fluid domain.In order to facilitate the computational efficiency and numerical accuracy of the presentnumerical method, a number of numerical considerations relating to the surface discretizationscheme, the time-stepping algorithm and the solution of a full matrix equation have beenconsidered. Numerical convergence in terms of number of facets used in the surfacediscretization for the two- and three-dimensional wave radiation problems have beenexamined. The development of fully vectorized computer codes to run on high speed virtual77memory machines (e.g. vector supercomputers and massively parallel processors) has playedan important role in accomplishing fast and efficient computational performance. Previouscomputational experience has suggested that matrices assembled into the integral equation arevery well-conditioned and the condition number associated with the matrix solution operationsindicates high degree of accuracy.5.2 Numerical ResultsApplications of the time-domain method to second-order hydrodynamic problems involvinglarge floating structures are illustrated by carrying out numerical computations for someelementary body shapes. Examples investigated include: (a) the forced sinusoidal motions ofa surface-piercing cylinder in otherwise still water; (b) the transient motion of a freely-floatingcylinder with a specified initial vertical displacement; and (c) the diffraction-radiation of Stokeswaves by a moored floating cylinder. Numerical results are given for a semi-submergedcircular cylinder in deep water and a truncated surface-piercing circular cylinder in finite waterdepth. In addition, the case of a semi-submerged rectangular cylinder section undergoingforced roll motions is investigated. Steady state solutions of the hydrodynamic forcecomponents and motion responses at first and second order are expressed in terms of theiramplitudes and phase angles.With respect to the two-dimensional wave radiation problem, the predominant nonlinearwave motions are seen to be the free wave component generated by the quadratic forcings inthe second-order body surface boundary condition. For the special case of a symmetricalcylinder section considered here, the second-order hydrodynamic forces give rise to second-order components only in the vertical direction. As indicated from these results, theimportance of second-order components in the total hydrodynamic force calculations increaseswith excitation frequencies. In general, the second-order oscillatory force component due tosecond-order wave potential contributes significantly to the second-order oscillatory force.There are significant second-order forces developed due to the forced roll motion of a78rectangular cylinder. Comparisons of the computed first- and second-order hydrodynamicforces have been made with previous theoretical and experimental results for the case of asemi-submerged circular cylinder and good agreement is indicated.The extension of the method to the three-dimensional radiation problem has been achievedon the same basis of the related problem in two dimensions. For the case of a truncatedcircular cylinder undergoing forced heave motion, the second-order steady force iscomparatively smaller than the second-order oscillatory components. The correspondingsecond-order oscillatory force components are found to increase steadily with excitationfrequency. On the other hand, forced surge motion of the cylinder gives rise to the second-order steady sinkage force. The corresponding total second-order oscillatory force appears toincrease gradually and drop off steadily at higher excitation frequencies. In general, thecontribution due to second-order potential to the total second-order oscillatory force is foundto be significant.In order to validate the numerical technique used for solving structure motions of floatingbodies, the two- and three-dimensional problems relating to the transient motions of freely-floating cylinders have been extended to second order. In effect, the time-dependent radiatedwave motions continuously propagate energy away from the cylinder to damp out theharmonic free body motions. The present first-order solutions are compared with previoustheoretical results and indicate excellent agreement. It is observed that there is no substantialsecond-order effects present in the predicted transient heave motions and the correspondingfree surface profiles.With respect to the diffraction-radiation by a semi-submerged circular cylinder, the second-order wave components associated with the forced and free waves are seen to be concentratednear the floating cylinder. There is a strong presence of second-order wave effects in theupwave region where the incident and disturbed waves propagate in opposite directions. Inthe downwave region, the resulting wave amplitude and the related second-order componentsare found to be relatively small. The horizontal drift force always acts in the direction of wave79propagation as derived from the conservation of momentum principle, while the vertical driftforce acts in the negative z direction which is primarily due to negative dynamic pressureassociated with the velocity squared term in the Bernoulli equation. It is indicated thatsignificant second-order force and motion components are developed at certain wavefrequencies.Based on the development of the related two-dimensional problem, the method issubsequently extended to simulate the second-order diffraction-radiation by a truncatedcircular cylinder. The numerical results reveals the pronounced second-order hydrodynamiceffects associated with the hydrodynamic forces and motions, as well as corresponding freesurface profiles and wave amplitudes. The second-order free wave component is seen topropagate away from the cylinder in all azimuthal directions. This particular feature issomewhat different from the case of wave diffraction in which second-order free waves arepredominant only in the downwave region. Due to the strong interactions between thenonlinear wave systems with the floating cylinder, it is suggested that the contribution to thetotal second-order wave component is dominated by the free wave component. In general,the second-order wave system is seen to substantially affect the resulting wave amplitudes,and localized areas of high amplitude are observed.The simulation of second-order hydrodynamic effects involving large floating structures,which cannot be predicted by a conventional linear hydrodynamic analysis, has been the majorapplication of the time-domain method described in this thesis. Typical examples of suchfloating structures include semi-submersibles, tension leg platforms, ship and barges. Ingeneral, the wave force and motion calculations are made on the basis of potential flow, andviscous effects are considered to be negligible when the ratio between the wave height and acharacteristic dimension of the structure is small, say less than about unity. However, flowseparation effects and viscous induced damping may become important considerations forsharp edged structures. As a general guideline, the second-order Stokes theory is applicablefor the non-breaking wave conditions when the Ursell parameter is less than about 26. It is80mentioned that the present method is restricted to analyze the basic case corresponding to thediffraction-radiation of Stokes regular waves, in which the resulting second-orderhydrodynamic forces and motions contain a drift component and an oscillatory component attwice the wave frequency. The more general case of bichromatic or irregular wave excitation,which gives rise to second-order sum-frequency and difference-frequency wave forces, hasnot been investigated in the present study.5.3 Recommendations for Further StudyThe present research project has focussed on the development of second-order potentialsolutions to account for the nonlinear hydrodynamic effects associated with wave interactionswith large floating structures. For illustration, preliminary computations are presented forsimple geometrical forms. It is anticipated that the future prospect of the present method willbe considerably dependent upon the advances of tomorrow's large-scale computationaltechnology including significant improvements of computing power and increase widespreadavailability of advanced computing systems to engineering practice.Using the methodology described in this thesis, numerical simulations of nonlineardiffraction-radiation of regular waves by practical floating structures with complicatedconfigurations can readily be made on the same basis. Likewise, in the context of practicalcoastal engineering applications, nonlinear wave effects of floating breakwaters can also beassessed by the time-domain method. With straightforward modification to the theoreticalformulation and computational model, the effects of a number of complications encountered inpractice such as arbitrary and/or coupled forced body motions in otherwise still water,reflection of lateral wall boundaries, variable water depths, interactions of multiple bodies, andnonlinearity of mooring systems can be investigated, without any major difficulties beinganticipated.The reliable prediction of slowly varying drift forces and motions of a moored orunrestrained floating vessel in an irregular seaway has widely been considered as one of the81most challenging research topics to be tackled in the field of ocean engineering. In somephysical circumstances, nonlinear viscous forces due to flow separation can be important.Due to its immense practical significance, this problem is expected to receive even moreattention in the future research and development of deep water floating production systems.For many years, second-order sum-frequency and difference-frequency excitation forces havebeen interpreted to evolve from the interactions of all possible frequency pairs in wavespectrum characterizing the sea state. The evaluation of the second-order force spectrumbased on the use of quadratic transfer functions for difference-frequency excitation forcesoften ignores the effect of slow drift motions which is referred to in the literature as wave driftdamping. In fact, this second-order damping force due to the slowly varying velocitycomponent can be explained by the interaction of incident waves with a quasi-steady speed ofthe structure.In principle, the first-order wave frequency motions cannot be decoupled from the largeamplitude slow drift motions, and an accurate representation of the quasi-steady motions isparticular difficult to be accommodated into the traditional hydrodynamic analysis. Followingthe multiple time scale expansion technique (Triantafyllou, 1982; and Agnon and Mei, 1985),the second-order problem with encounter frequency dependence can then be separated fromthe first-order wave-current interaction problem. Quite apart from the established school ofthought in frequency domain, the present time-domain approach may be employed to performdeterministic flow simulations with respect to the two decomposed first- and second-orderhydrodynamic problems, and this will hopefully throw some light on the furtherunderstanding of this subject matter. However, it is suggested that the major difficulty in thiscase arises from a quasi-steady change in the wave heading as well as the encounter frequencybetween the waves and the structure caused respectively by the low frequency rotational andtranslational body motions.82ReferencesAbul-Azm, A.G. and Williams, A.N. 1988. Second-order diffraction loads on truncatedcylinders. Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE, 114(4),436-454.Agnon, Y. and Mei, C.C. 1985. Slow-drift motion of a two-dimensional block in beamseas. Journal of Fluid Mechanics,151, 279-294.Bai, K.J. and Yeung, R.W. 1974. 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Block diagram of the time-stepping scheme.89Nb Fiz/pgaA FoJpgA2 F2/pgA2 F2/pgA2 F2/pgA210 1.5698 -0.0438 0.4761 0.5929 0.533015 1.5675 -0.0410 0.4801 0.5867 0.562520 1.5668 -0.0410 0.4815 0.5812 0.567025 1.5664 -0.0409 0.4823 0.5831 0.580730 1.5662 -0.0410 0.4826 0.5782 0.579435 1.5660 -0.0410 0.4828 0.5785 0.585640 1.5659 -0.0410 0.4830 0.5802 0.5918Table 2. Hydrodynamic force components in the z direction as functions of Nb for a semi-submerged circular cylinder in heave motion, (u 2a /g = 0.6, A/a = 0.1 and deepwater.Nb F1jpgbA Fo7JPgA2 F(21),/pgA2 F2 /pgA2 F2jpgA215 1.0357 0.1726 1.0229 1.6225 0.628520 1.0504 0.1498 1.0448 1.8152 0.794225 1.0582 0.1342 1.0599 1.9625 0.922630 1.0625 0.1251 1.0684 2.0391 0.990735 1.0662 0.1145 1.0786 2.1522 1.097240 1.0687 0.1061 1.0867 2.2488 1.180245 1.0706 0.0991 1.0933 2.3335 1.257550 1.0719 0.0942 1.0980 2.3835 1.3027Table 3. Hydrodynamic force components in the z direction as functions of Nb for a semi-submerged rectangular cylinder in heave motion, web /g = 0.6, A/b = 0.1 and deepwater.90Nbl^Fidpga2A Fo7JpgaA2 F2 /pgaA2 F2 fpgaA2 F27/pgaA296 1.7369 0.4710 2.3494 3.6427 1.3199130 1.7402 0.4477 2.3792 3.2525 0.8974192 1.7506 0.4162 2.4137 3.6654 1.2690266 1.7538 0.3998 2.4323 3.4327 1.0258352 1.7563 0.3877 2.4458 3.2933 0.8737Table 4. Hydrodynamic force components in the z direction as functions of Nbl for atruncated circular cylinder in heave motion, ka = 1.0, d/a = 1.5, h/a = 0.5 and A/a =0.1.91S c(a)Figure 1. Definition sketch. (a) three-dimensional problem. (b) two-dimensional problem.92r-S c S cd'//1/////////////7////////////////////////////////(b)Figure 1. Continued.93Figure 2.^Body geometries of illustrative computations. (a) semi-submerged circularcylinder. (b) semi-submerged rectangular cylinder. (c) truncated surface-piercingcircular cylinder.94(a)(b)Figure 3. Examples of surface discretization in two dimensions. (a) semi-submerged circularcylinder Nb = 20 and NT =138. (b) semi-submerged rectangular cylinder Nb = 30and NT = 150.95(b)Figure 4. Examples of surface discretization of a truncated surface-piercing circular cylinderin three dimensions. (a) Nb = 378 and NT = 2298. (b) Nb = 378 and NT = 2108.962.00.0—2.0 ^0.02.00.0 ......^......^......^•- ... ......... -•^........ .^......... ..1.0 2.0 3.0 4.0 5.01.0 2.0 3.0 4.0 5.01.0^2.0^3.0^4.0^5.0VTFigure 5. Development with time of vertical displacement of a semi-submerged circularcylinder, free surface elevation on body surface and vertical hydrodynamic forcecomponents for co2a/g =1.6, A/a = 0.1 and deep water. ----,  solution to first order, , solution to second order,  , second-order component.972.0■ 0.0- 2.02.00.0Ps- 2.02.00.0g's- 2.02.00.0- 2.04.04.04.0I I0.0 1.0 2.0 3.0..------\.................j1 It.„---'-'I0.0 1.0 2.0 3.0-1. T =0.0 1.0 2.0 3.02.00.0-2.00.0^1.0^2.0^3.0^4.00.0^1.0^2.0^3.0^4.0x/LFigure 6. Development with time of free surface profiles for a semi-submerged circularcylinder undergoing forced heave motion, (02a/g = 1.6, A/a = 0.1 and deep water.- - - -, solution to first order; , solution to second order.982.01.5<0a)ca. toaL.0.50.00.0^0.4^0.8^1.2^1.6^2.0co2q/g0.4^0.8^1.2^1.6^2.0co2a/gFigure 7. Amplitude and phase angle of first-order oscillatory force in the z direction on asemi-submerged circular cylinder in heave motion as functions of excitationfrequency for deep water. A, present study; , Potash's theory (1971).991I^I^.^I 0.4 0.8 1.2co 2a/g 1.6^2.00.00.20.0N. eeY. ) —0.2Pc.1 •L...—0.4—0.6Figure 8. Magnitude of second-order steady force in the z direction on a semi-submergedcircular cylinder in heave motion as functions of excitation frequency for deepwater. A, present study; , Potash's theory (1971).1002.50.50.00.8^t2c2a/g0.0 0.4 1.6 2.0300.0250.050.00.0^0.4^0.8^1.22G) algFigure 9. Amplitude of second-order oscillatory force in the z direction on a semi-submergedcircular cylinder in heave motion as functions of excitation frequency for deepwater. A, present study; -- - - , Papanikolaou's theory (1984); , Yamashita'stheory (1977); A, Yamashita's experiment (1977).Figure 10. Phase angle of second-order oscillatory force in the z direction on a semi-submergedcircular cylinder in heave motion as functions of excitation frequency for deepwater. A, present study;  , Papanikolaou's theory (1984); •, Yamashita'stheory (1977); 0, Yamashita's experiment (1974).1.6^2.01013.02.0N.<ChQ.NcvLL1.00.00.0^0.4^0.8^1.2^1.6^2.0W2a/gFigure 11. Composition of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in heave motion as functions of excitation frequencyfor deep water. D, component due to c1; 0, component due to 02; •, resultant forceamplitude.1020.0^0.4^0.8^1.2w2a/g 1.6 2.01.61.2<00)Q. 0.8L.L.0.40.00.4^0.8^1.2C.J 2a/gFigure 12. Amplitude and phase angle of first-order oscillatory force in the x direction on asemi-submerged circular cylinder in sway motion as functions of excitationfrequency for deep water. A, present study; , Potash's theory (1971).103AAAAAAAAI^ 1^,^1 0.4 0.8 1.2W2a/g1.6^2.00.010.20.0'.1,ct —0.2Cr)..i.)0L., —0.4—0.6—0.8Figure 13. Magnitude of second-order steady force in the z direction on a semi-submergedcircular cylinderin sway motion as functions of excitation frequency for deep water.A, present study;  , Kyozuka's theory (1982); A, Kyozuka's experiment(1982).1041.6 2.00.0^0.4^0.8^1.22 iCA) cvg0.0-50.0-200.0N -150.0%14.03.0N-t<alcl- 2.0NLL1.00.0Figure 14. Amplitude of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in sway motion as functions of excitation frequencyfor deep water. A, present study; , Kyozuka's theory (1982); A, Kyozuka'sexperiment (1982).0.4^0.8^1.2^1.6^2.0CO2a/gFigure 15.^Phase angle of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in sway motion as functions of excitation frequencyfor deep water. A, present study; , Kyozuka's theory (1982); A, Kyozuka'sexperiment (1982).1052.52.0N.< 1.5D)Q.,„.NL 1.00.50.00.0^0.4^0.8^1.2^1.6^2.0CO2a/gFigure 16. Composition of second-order oscillatory force in the z direction on a semi-submerged circular cylinder in sway motion as functions of excitation frequencyfor deep water. ^, component due to 01; 0, component due to 02; •, resultantforce amplitude.10.08.0N.<Q.Cr)^6.0...›..1(.4Li_• r.6i^4.0L.L.2.00.00.0^0.4^0.8^1.2^1.6^2.0CO2bigFigure 17. Magnitude of second-order steady force and composition of second-order oscil-latory force in the z direction on a semi-submerged rectangular cylinder in heavemotion as functions of excitation frequency for deep water. A, steady component;^, oscillatory component due to 0 1 ; 0, oscillatory component due to 4)2; •,resultant oscillatory force amplitude.10610.08.02.00.00.4^0.8^1.2^1.6^2.0a?LAFigure 18. Magnitude of second-order steady force and composition of second-order oscil-latory force in the z direction on a semi-submerged rectangular cylinder in swaymotion as functions of excitation frequency for deep water. A, steady component;^, oscillatory component due to 41; 0, oscillatory component due to 02; •,resultant oscillatory force amplitude.0.0^0.4^0.8^1.2^1.6^2.06)20Figure 19. Magnitude of second-order steady force and composition of second-order oscil-latory force in the z direction on a semi-submerged rectangular cylinder in rollmotion as functions of excitation frequency for deep water. A, steady component;0, oscillatory component due to 01; 0, oscillatory component due to (02; •,resultant oscillatory force amplitude.107—2.00.0^1.0^2.0^3.0 4.0 5.0160.0120.0c0LLI^80.040.00.00.0^0.4^0.8^1.2^1.6^2.0CJ2b/gFigure 20. Phase angle of second-order oscillatory force in the z direction on a semi-submerged rectangular cylinder as functions of excitation frequency for deepwater. El, heave motion; 0, sway motion; •, roll motion.2.00.01.0 2.0 3.0 4.0 5.0t/TFigure 21. Development with time of vertical displacement of a truncated circular cylinderand vertical hydrodynamic force components for ka =1.6, d/a =1.5, h/a = 0.5 andA/a =0.1. ----,  solution to first order, , solution to second order,  second-order component.108Figure 22.^Oblique views of the free surface to second order for ka = 1.6, d/a =1.5, h/a =0.5 and t = 3.5T. (a) heave motion. (b) surge motion.1093.02.52.0cv<co1.5...,NLi.1.00.5Figure 23. Amplitude and phase angle of first-order oscillatory force in the z direction on atruncated circular cylinder in heave motion as functions of ka for d/a = 1.5, h/a =0.5.11010.0 ^8.0 -6.0 -4.0 -2.0 -0.00.2A^A0.6^1.0^1.4^1.8 22kaFigure 24. Magnitude of second-order steady force and composition of second-order oscil-latory force in the z direction on a truncated circular cylinder in heave motion asfunctions of ka for d/a = 1.5, h/a = 0.5. e, steady component; ^, oscillatorycomponent due to clh ; 0, oscillatory component due to 02; •, resultant oscillatoryforce amplitude.160120• a0=80vN‘,040002^0.6^1.0^14^1.8^22kaFigure 25.^Phase angle of second-order oscillatory force in the z direction on a truncatedcircular cylinder in heave motion as functions of ka for d/a = 1.5, h/a = 0.5.1111.4 1.81.0 2.20.690I 1.60.0 ^0.2ka180-1800.2 0.6^1.0ka1.4^1.8^2.2Figure 26. Amplitude and phase angle of first-order oscillatory force in the x direction andmoment about the y-axis on a truncated circular cylinder in surge motion asfunctions of ka for d/a = 1.5, h/a = 0.5. •, force component; 0, momentcomponent.112Figure 27. Magnitude of second-order steady force and composition of second-order oscil-latory force in the z direction on a truncated circular cylinder in surge motion asfunctions of ka for d/a = 1.5, h/a = 0.5. A, steady component; ^, oscillatorycomponent due to ch ; 0, oscillatory component due to 02; •, resultant oscillatoryforce amplitude.10075U) 50LiiL.L.ICC0u..i^25aN o,0-25-6002 0.6^1.0^1.4^1.8^22kaFigure 28.^Phase angle of second-order oscillatory force in the z direction on a truncatedcircular cylinder in surge motion as functions of ka for d/a = 1.5, h/a = 0.5.113,.......""'".-'---.--..'""'`......----...-.....-""'I^,^(^,^I^,^I^.^I^,1.0< 0.0.3....,1.0^2.0^3.0^4.0^5.0^6.0Figure 29. Development with time of vertical displacement and vertical hydrodynamic forcecomponents for a semi-submerged circular cylinder undergoing transient heavemotions with i A i /a = 0.1 and d/a =10.0. - - - -, solution to first order,  ,solution to second order; ^ , second-order component.114t/T = 7.4 t/T = 6.6t/T = 5.8t/T = 5.0t/T = 4.2t/T = 3.4t/T = 2.6=1.8t/T = 1.0t/T = 0.20^1^2^3^4x/dFigure 30. Development with time of free surface profiles to second order for a semi-submerged circular cylinder undergoing transient heave motions with IA I /a = 0.1and d/a = 10.0. Successive profiles are at times 0.4T apart.1153.04 0' 0-3.00.02.0< 0.0'%-.J..,-2.00.0< 3.0C 0a)0.0...,xLi- -3.00.0< 2.0coCO'z2 0.0....NU- -2.0 ^0.0 2.02.02.02.04.0t/T4.04.04.06.06.08.06.08.08.08.08.0Figure 31.^Development with time of the motion response and hydrodynamic force compo-nents for a moored semi-submerged circular cylinder in waves for (nig = 0.4,A/a = 0.3 and deep water. ----, solution to first order; ^, solution to secondorder; ^ , second-order component.1163.0.0.. 0.0P-3.00.03.0-4g.. 0.0-3.00.03.0'0. 0.0P-3.00.0^3.0 ^4 o.o --3.0 ^0.03.00.0P-3.00.01.0^2.0^3.0^4.0I^ I^ I 1.0 2.0 3.0^4.0x/Lt/T =- 21.0 2.0 3.0 4.01.0 2.0 3.0 4.01.0^2.0^3.0^4.0t/T =Figure 32. Development with time of free surface profiles for a moored semi-submergedcircular cylinder in waves for co 2a/g = 0.4, A/a = 0.3 and deep water. - - - -,solution to first order;  , solution to second order.117- 0 0 0 0 0 0 0 0 0 000^• • •^•^•^•^•0 0^oo^0o 0^0- •• ••••••• • • • • •12700.4^0.8^1.2^1.6^2.0CA)2aigFigure 33. Amplitude and phase angle of first-order oscillatory forces on the cylinder asfunctions of co2a/g for deep water. •, horizontal component; 0, verticalcomponent.1800-900.01180.8^122 ,C.J a/gFigure 34. Magnitude of second-order drift forces on the cylinder as functions of o.)2a/g fordeep water. •, horizontal component; 0, vertical component.1.6^2.01191^o^I I••• o0^• • ••••• •000^0^ 00 00 • o•S o•^•0 o•• 00I^I^i• •^8 8 80^090-901800.4 0.8^12^1.6^2.0C.J2a/gFigure 35.^Amplitude and phase angle of second-order oscillatory forces on the cylinder asfunctions of &a/g for deep water. I, horizontal component; 0, verticalcomponent.-1800.01201_ ••• •••10 o^•• • • • • • •^•^•^•0^ 00 0^00- 0000008go•0 01800.4^0.8^1.2^1.6^2.0CA)2a/gFigure 36. Amplitude and phase angle of first-order oscillatory motion response of thecylinder as functions of (02a/g for deep water. •, sway; 0, heave.90-90-1800.01210.8^1220.) amFigure 37. Magnitude of second-order drift motion response of the cylinder as functions of(.02a/g for deep water. •, sway; 0, heave.0.4 1.6^2.0122• 0I-1800.0 0.4 0.8^1.2C4.)2a/g2.01.6180 1•090 0• o o• o-90O• 0^• • Iii•••• •—• 0^••^80• o^ 000 o^o^o0•• ••00eFigure 38. Amplitude and phase angle of second-order oscillatory motion response of thecylinder as functions of co 2a/g for deep water. •, sway; 0, heave.123-2.00.0^5.0^10.0^15.0^20.0t-VaTa.25.0 30.01.0N''■•^0.04.kp►^1^►^1^►5.0^10.0^15.0^20.0^25.0^30.0Figure 39. Development with time of vertical displacement and vertical hydrodynamic forcecomponents for a floating truncated circular cylinder undergoing transient heavemotions with 1 A 1 /a = 0.1, dia. = 3.0 and h/a = 0.5. - - - -, solution to first order; , solution to second order,  , second-order component; •, Newman(1985).124Figure 40. Oblique views of free surface profiles to second order for a floating truncatedcircular cylinder undergoing transient heave motions with IAI/a = 0.1, d/a = 3.0 andh/a = 0.5. (a) t = 7.73 g. (b) t = 13.924g. (c) t = 20.10 -qa/g. (d) t = 26.294a/g.1252.0• 0.0-2.00.03.00.0-3.00.02.0CD 0.0-2.00.04.0etica0.0XLL. -4.00.0"^6.04. co0.0LLN -6.00.00.5c.,<CcoQ.• 0.02>. -0.50.0 2.02.02.02.02.02.04.04.04.04.04.04.06.0t/T6.06.06.06.06.08.08.08.08.08.08.010.010.010.010.010.010.012.012.012.012.012.012.0Figure 41.^Development with time of the motion response and hydrodynamic force compo-nents for a moored truncated circular cylinder in waves for ka = 1.0, d/a = 2.0,h/a =0.5 and A/a = 0.2. - - - -, solution to first order; ^, solution to secondorder, ^ , second-order component.126Figure 42. Oblique views of free surface profiles to second order for a moored truncatedcircular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 and A/a = 0.2. (a) t =0.2T. (b) t = 0.7T.1270.0^0.4^0.8^1.2x/LFigure 43. Perspective and contour plots of first-order wave amplitude of a moored truncatedcircular cylinder in waves for ka = 1.0, d/a = 2.0, h/a = 0.5 and A/a = 0.2. (Contoursshown are normalized with respect to incident wave amplitude.)1281.20.80.4...J■. 0.0>.•-0.4-0.8-1.2-1.2^-0.8^-0.4^0.0^0.4^0.8^1.2x/LFigure 44. Perspective and contourplots of wave crest amplitude to second order of a mooredtruncated circular cylinder in waves for ka =1.0, d/a = 2.0, h/a = 0.5 and A/a = 0.2.(Contours shown are normalized with respect to incident wave amplitude.)1291300.4^0.8 1.2-0.4^0.0x/LFigure 45. Perspective and contour plots of second-order mean water surface elevation of amoored truncated circular cylinder in waves for ka =1.0, d/a = 2.0, h/a = 0.5 andA/a = 0.2. (Contours shown are normalized with respect to incident waveamplitude.)-1.2^-0.8131-1.2^-0.8 -0.4w* cp -9pt>o 09 • 00.40.0-0.4-0.8-1.20.0x/LFigure 46. Perspective and contour plots of second-order oscillatory wave amplitude of amoored truncated circular cylinder in waves for ka =1.0, d/a = 2.0, h/a = 0.5 andA/a = 0.2. (Contours shown are normalized with respect to incident waveamplitude.)0 00. °Z0.fr■ 0.4^0.8^1.21.20.8

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