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The application of floating breakwaters in British Columbia Byres, Ronald David 1988

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T H E A P P L I C A T I O N O F F L O A T I N G B R E A K W A T E R S I N B R I T I S H C O L U M B I A Ronald David Byres B. A . Sc. (Civil Engineering) University of British Columbia, 1985 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 1988 © Ronald David Byres, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada Date: Abstract The nature of British Columbia coastal and inland waterways affords many locations where floating breakwaters are or could be used to protect small-craft harbours from wave action. A field survey of many of the current breakwater sites is undertaken in or-der to establish qualitative performance criteria of various designs. A two-dimensional numerical model is developed to predict the oscillatory response and wave transmission characteristics for a number of common breakwater designs. Finally, experiments with two configurations of breakwater models were carried out in the Hydraulics Laboratory at the University of British Columbia. The experiments were designed to validate the numerical model and to estimate viscous damping coefficients required in the numerical solution. i Acknowledgement Many people have contributed in various ways to this thesis, and although the complete list is too lengthy to list here, a few individuals deserve special recognition. The author would like to express his appreciation to Dr. Michael de St. Q. Isaacson for his advice and guidance during the preparation of this thesis. Thanks are also due to graduate students Carol Mihelcic for assistance in the labo-ratory, and to Mark Mattila for collaborating on the field work. The support of family and friends is also appreciated, in particular Adele Inouye and Doug Rowland for their timely encouragement and cynicism respectively. A special acknowledgement is made to Bill Wolferstan of the Ministry of the Envi-ronment and to Western Canada Hydraulics Laboratories for making their photographs, time, and other information freely available. Finally, the financial support of a Research Assistantship from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. ii Table of Contents Acknowledgement ii 1 I N T R O D U C T I O N 1 1.1 General 1 1.2 Applications in British Columbia 6 1.3 Literature Survey 11 2 F I E L D S U R V E Y 15 2.1 Categories of Breakwaters 15 2.2 Concrete Caissons 16 2.3 Log Bundles and Rafts 19 2.4 A-Frames 20 2.5 Other Types 21 3 N U M E R I C A L M O D E L 23 3.1 Mathematical Treatment 24 3.1.1 Wave Diffraction 24 3.1.2 Green's Function Solution 28 3.1.3 Hydrodynamic Analysis 31 3.1.4 Transmission and Reflection Coefficients 32 3.1.5 Equations of Motion 33 3.2 Numerical Analysis 35 3.2.1 Singularities 37 iii 4 P H Y S I C A L M O D E L L I N G 40 4.1 Experimental Facilities 40 4.2 Dimensional Analysis 41 4.3 Experimental Procedure 42 4.3.1 Wave Heights 44 4.3.2 Breakwater Motions 44 5 R E S U L T S 47 5.1 Field Survey 47 5.2 Field Data 48 5.3 Numerical and Experimental Results 49 5.3.1 Caisson Breakwaters 49 5.3.2 A-frame Breakwater 53 5.3.3 Summary 56 6 D I S C U S S I O N A N D C O N C L U S I O N S 57 6.1 Field Data 57 6.2 Numerical Results 58 6.3 Experimental Results 59 Bibliography 62 A Site photographs and marine charts 66 B Experimental Facilities 86 iv List of Figures 1.1 Wave attenuation by a floating breakwater 3 1.2 Several floating breakwater designs 4 1.3 Comparison of wave climates for "offshore" and "sheltered" B.C. waters 7 1.4 Floating Breakwater Locations in British Columbia 8 3.5 Definition sketch for floating cylinder 25 3.6 Definition sketch for boundary integral 28 3.7 Boundary reflected about seabed 30 3.8 Discretization of breakwater section 36 3.9 Discretization of A-frame plate . 38 4.10 Experimental setup (not to scale) 43 4.11 Measurement of breakwater motions 45 5.12 Transmission coefficients for rectangular caisson breakwater 50 5.13 Sway response for caisson breakwater 50 5.14 Heave response for caisson breakwater 51 5.15 Roll response for caisson breakwater 51 5.16 Response Amplitude Operators (RAO's) for caisson breakwater 52 5.17 Transmission coefficients for A-frame breakwater 54 5.18 Sway response for A-frame breakwater 54 5.19 Heave response for A-frame breakwater 55 5.20 Roll response for A-frame breakwater 55 v A.21 Traditional rubble-mound breakwater at Westview 67 A.22 Former A-Frame breakwater at Lund 68 A.23 Concrete caisson at Deep Cove 69 A.24 Caisson at Horseshoe Bay 70 A.25 Detached caissons at Lund 71 A.26 Caisson at Nanaimo Yacht Club 72 A.27 Caisson at Maple Bay 73 A.28 Log bundle at Reed Point Marina 74 A.29 Log bundle at Fanny Bay 75 A.30 Log raft at Ford Cove, Hornby Island 76 A.31 Log raft at Prince Rupert 77 A.32 Pontoon/tire breakwater at Eagle Harbour 78 A.33 Barge breakwater at Sunset Marina, Howe Sound 79 A.34 Breakwater of scrapped ship hulls at Powell River 80 A.35 Floating railway tank cars at Brown Bay 81 A.36 Log breakwaters at Northwest Bay and Becher Bay 82 A.37 Log bundles at Pender Island 83 A.38 Breakwaters at Sooke Basin and Esquimault 84 A. 39 Log bundles at Kelowna and Tahsis 85 B. 40 Wave basin experiments 86 B.41 Experimental data collection 87 vi List of Tables Summary of floating breakwater installations in British Columb vii Chapter 1 I N T R O D U C T I O N 1.1 General The term "floating breakwater" can be applied to any floating or semi-floating structure intended to attenuate wind- or ship-generated water waves. Such structures have found widespread application in inland and coastal waters in many parts of the world, and have met with varying degrees of success. The primary limitation to most floating breakwater designs is the fact that they lose their wave-attenuating characteristics as the length of the incident waves becomes large. Since the longest (and largest) waves at a given site generally coincide with the most severe storms, the effectiveness of the breakwater can potentially be lost when it is needed the most. As a result, the application of floating breakwater systems is generally confined to relatively sheltered fetch-limited bodies of water, where incident wavelengths have a fairly well-defined upper bound. Despite the limitations however, careful selection of design criteria can produce a breakwater which satisfies wave attenuation requirements and provides shore or harbour protection in instances where traditional breakwaters (such as rubble-mound or other bottom-resting designs) would be impractical to construct. Whereas the cost of rubble-mound structures is proportional to their volume (which increases as the square of the water depth), the cost of floating breakwaters is relatively independent of depth since only the length of the mooring lines must be altered. This makes a floating 1 Chapter 1. INTRODUCTION 2 structure particularly attractive in deep-water, sheltered areas, or in areas where only temporary protection is required. Similarly, in areas where the seafloor is soft and cannot support the weight of large amounts of rock fill, a floating structure might be sucessfully employed. Other advantages of floating systems include the provision of better water quality through unimpeded circulation, reduction or elimination of sedimentation problems, improved aesthetics from low-profile designs (particularly in areas of large tidal ranges) and the ease of relocating or removing the breakwater when it is no longer required. In addition, many designs can be fabricated at one location and towed to the required site, making their installation in remote or aggregate-poor areas easier than for rubble-mounds. A typical rubble-mound breakwater is shown in Figure A.21. Compare the scale of this installation with that the floating structures at Lund (Figures A.22 and A.25) which has a similar (though slightly less severe) wave climate. The nature of some floating breakwater designs makes them amenable to secondary uses. Concrete caissons in particular are often used for additional boat moorage. Many installations include finger wharves on the leeward side of the caisson to maximize the use of the system. The fiat deck and low-profile make caissons useful for pedestrian walkways, fuel barges, and access platforms for aquaculture. Although the designs of various floating breakwater systems vary considerably, there are essentially only two mechanisms by which a floating body can attenuate wave energy: reflection and dissipation. Wave energy which is not reflected or dissipated by the structure is transmitted to the leeward side as shown in Figure 1.1. Figure 1.2 shows several of the designs in current use. The caisson types rely on large mass to provide a relatively immovable surface from which incident waves are reflected. The vertical wall of the A-frame centreboard type also reflects wave energy, but relies on the pontoons and support frame to provide lateral and rotational stability. Dissipative Chapter 1. INTRODUCTION 3 Figure 1.1: Wave attenuation by a floating breakwater Chapter 1. INTRODUCTION Rectangular Caisson (Wave Ref lect ion) A-Frame Section (Reflection) -cooccoa Log Bundles or Rafts (Ref l ec t ion /Di s s ipa t ion) Scrap Tires Laced Together (Diss ipation) (NB: Not to Scale) Mooring lines not shown for clarity FLOATING BREAKWATER T Y P E S Used in British Columbia (Primary attenuation mechanism(s) as indicated) Figure 1.2: Several floating breakwater designs Chapter 1. INTRODUCTION 5 structures such as the floating tire breakwaters extract energy from waves by converting it into turbulence and mechanical energy. Although most systems are designed to work primarily either by reflection or dissipation, in practice most designs use a combination of the two. None of the current designs are completely effective at attenuating any but the highest frequency waves. Although most of the energy in a deep-water wave is con-centrated near the surface, some of it is contained in the water at depth. Breakwaters of practical dimensions can therefore intercept only the surface portion of this energy, transmitting the remainder. In addition, the motions of the breakwater itself generates waves which propagate outward, contributing to the transmitted wave heights. Under resonant conditions the breakwater is less effective at attenuating waves, and the large motions can be damaging to the structure, its connections, or the mooring system. For these reasons the resonant condition should (if possible) be avoided in breakwater design. This implies that a knowledge not only of the expected design wave conditions is required, but also an understanding of the response of the floating body under that sea state. Current design procedures are often based on experience with existing designs. The large number of variables involved and the variety of existing breakwaters has made it difficult for empirical relationships to be derived. For most large scale applications, it has therefore been necessary to resort to site-specific physical model tests before a particular breakwater design is adopted. Only recently have there been attempts to develop numerical design methods, and even these are used in conjunction with physical model studies. The eventual goal is to develop a numerical method sufficiently accurate over a wide range of configurations and conditions such that routine physical model studies become unneccessary. Chapter 1. INTRODUCTION 6 1.2 Applications in British Columbia The presence of Vancouver Island on the B.C. coast acts essentially like an enormous "fixed" breakwater, preventing the long-period ocean swells generated by Pacific storms from reaching the inner coast. Figures 1.3 (a) and (b) show wave scattergrams which illustrate the differences in typical wave climate as measured off the west coast of Vancouver Island (near Tofino) and within the Strait of Georgia (at Lund) respectively [11]. As is readily apparent from the scattergrams, both the wave heights and periods are considerably higher on the exposed coast. It is also interesting to note that while the "occurrence of calm" on the open coast was zero, for the inner coast a prodigious 75 % of the 1142 observations were classified as "calm". The Strait of Georgia, as well as many of the fjords and inlets on the coast and Puget Sound to the south, are therefore considered "protected" bodies of water with fetch-limited wave heights and periods. Because of the fjordal nature of many B.C. inlets, water depths drop off very rapidly as one moves away from shore. In addition, British Columbia is subject to large tidal fluctuations, exceeding five metres in some areas. The combination of all these factors provides many sites where the floating breakwaters are or could be used to advantage. Currently in British Columbia there exist more than 30 harbours and marinas which use "significant" floating breakwater systems. The locations of many of these sites are shown in Figure 1.4 and are listed in Table 1.1 (Although reasonable diligence was exercised when cataloguing the sites, it is quite possible that some were missed since many are not listed in any other publication). A number of other places use small-scale systems such as log booms to provide nominal wave protection. Most of the sites are found on coastal waters in the lower mainland area, however some are located as far north as the Queen Charlotte Islands Chapter 1. INTRODUCTION 7 10 -i STATION 103 T0FIN0 B C JANUARY 1.1981 TO DECEM8ER 31.1981 NUMBER OF OBSERVATIONS 2707 OCCURRENCES OF CALM 0 7 -1 1 1 4 S 4 11 9 13 8 13 15 16 11 16 20 8 14 20 17 36 38 26 3 1 S 8 15 39 34 32 57 60 3 14 11 21 55 54 72 51 58 8 25 56 80 122 68 77 55 54 14 55 97 135127 49 29 30 39 5 36 45 43 33 10 5 18 22 6 2 2 3 8 7 5 11 26 30 19 21 49 53 -r 1 1 2 1 4 6 2 3 10 11 10 16 45 60 " l 1 1 1 1 1 1 1 1 3 * S 6 7 8 9 10 11 12 13 14 IS 16 17 20 PEAK PERIOD IN SECONDS 5.0 n 4.5 -4.0 !E3' o 0 -2.S -_ 2 . 0 t 1 -5 " ~ 1 z 1 .0 -0.5 -0.0 STATION 117 LUNO (HAVER IDER1.B.C. OCT 12.1977 TO MARCH 25.1978 NUMBER OF OBSERVATIONS 1142 OCCURRENCES OF CALM 860 I 3 1 t 18 9 2 32 62 33 6 5B 39 16 1 Z 3 4 5 6 7 B 9 10 11 12 13 14 IS 16 17 PEAK PERI00 IN SECONDS Figure 1.3: Comparison of wave climates for "offshore" and "sheltered" B.C. waters Figure 1.4: Floating Breakwater Locations in British Columbia Chapter 1. INTRODUCTION 9 Table 1.1: Summary of floating breakwater installations in British Columbia No." Location Type Chart Ref* Year Fetch' Direction 8 Lundd A-Frame 3538 1963 29.6 SW 27 Q C C e A-Frame 3890 1967 2.4 1 Richmond Caisson 3324 1979 1.7 w 3 Deep Cove YC Caisson 3495 1976 5.5 NE 4 Burrard YC Caisson/barge 3482 1977 3.1 E 6 Horseshoe Bay Caisson/ship 3534 9.0 N 8 Lund Caisson 3538 1987 29.6 SW 15 Nanaimo Caisson 3457 1974 5.3 SE 16 Nanaimo Caisson 3457 ?? SE 19 Maple Bay Caisson 3470 1977 4.3 NE 20 Victoria Hbr Caisson 3423 0.5 S 21 Esquimault Caisson 3417 1.8 SW 29 Nakusp Caisson N/A 1986 2 Port Moody Log Bundle 3495 1976 0.8 NE 10 Fanny Bay Log Bundle 3527 2.6 NE 11 Deep Bay Log Bundle 3527 2.2 W 12 Ford Cove Log Raft 3527 6.5 NW 13 Northwest Bay' Log/styrofoam 3459 1975 37 14 Nanaimo Log 3459 0.2 E 17 Pt Browning Log 3477 2.1 SE 18 Bedwell Hr Log 3477 3.0 NW 22 Becher Bay Log 3430 1.5 SE 23 Sooke Basin Log 3430 1.9 SW 24 Prince Rupert Log 3955 25 Prince Rupert Log 3955 26 QCC Log 3890 4.4 28 Kelowna Log Bundle 3052 1978 W 30 Tahsis Log 3665 2.8 SE 4 Burrard Y C Scrap Tire 3482 1977 3.1 E 5 Eagle Harbour Pontoon/tire 3534 1977 39.1 SW 7 Powell River Ship Hull 3536 1930+ 28.7 SW 7a Howe Sound Barge 3311 12.8 NNE 9 Brown Bay Tank car 3312 1983 2.8 E "refers to numbering on Figure 1.4 6denotes Canadian Hydrographic Services Chart number cefective fetch may be less dremoved 1987 e K QCC" denotes Queen Charlotte City 1 destroyed ca. 1983 "destroyed 1980 Performance of most breakwater installations rated as "satisfactory" by owners except for sites 5 and 15, which experienced failures. Chapter 1. INTRODUCTION 10 and in lakes in the interior. With the expected increase in recreational boat traffic expected in B.C. in future decades, it will be necessary to develop more small-craft marinas. Many such sites will require wave protection and would benefit from well-designed floating breakwater systems. In addition, the recent growth of the aquaculture industry in the province has exhausted most of the natural sites having the necessary water quality and shelter characteristics. Some existing aquaculture sites have experienced negative reactions from area residents who find the farms unaesthetic and who are concerned with water quality problems from fish fecal matter. Expanding aquaculture sites into more exposed locations using floating breakwaters for wave protection could serve the dual purpose of providing improved water circulation and avoiding some of the conflicting land use problems currently being experienced. In an attempt to improve design process of floating breakwaters, a numerical model is developed to predict the wave transmission characteristics and response for a variety of breakwater shapes. In particular, the model is currently configured for breakwaters having rectangular and semi-circular cross-sections, as well as for an "A-frame" cross section consisting of a vertical plate or centreboard and two "outrigger" pontoons. Con-sideration is given to the mooring forces since past experience has shown that moorings and inter-module connections are the most likely to cause operational problems. Experiments designed to measure transmission characteristics and structure mo-tions for rectangular and A-frame type breakwaters were carried out in the Hydraulics Laboratory at the University of British Columbia. The results were used to estimate viscous damping coefficients and to validate the numerical model. Chapter 1. INTRODUCTION 11 1.3 Literature Survey Much has been written on the subject of floating breakwaters, with references in the literature dating back to 1842. Western Canada Hydraulics Laboratory (WCHL) [48] compiled an extensive bibliography with 266 entries, covering topics ranging from an-alytical formulations to in-situ experiences with particular breakwater designs. The large variety both of specific breakwater designs and of the various areas of interest have contributed to this wealth of information. Most of these papers are not cited here; for more information the reader is referred to WCHL [48] or to Hales [18]. The number of different designs that have been conceived or tested in the laboratory is almost as large and diverse as the number of papers on the subject, but a few of the more effective designs are discussed by Silvester [43], McCartney [29] discusses several additional designs as well as various mooring systems and anchorage methods. Many of the analytic techniques employed today originate from the disciplines of marine architecture and ship hydrodynamics. Several authors (for example Ursell [44], Porter [39], Frank [14], Kim [24], Vugts [46], Ijima [21], Bai [3]) have treated the case of two-dimensional wave interaction with cylinders on the free surface. Most of these papers develop expressions to solve for the velocity potential of the flow field in the vicinity of the breakwater, and to calculate the hydrodynamic coefficients neccesary to determine the fluid forces on (and motions of) the body. The case of oblique wave interaction with cylinders has been investigated by Mac-Camy [28], Levine [25], Lebreton and Margnac [26], Black and Mei [6], Garrison [16,17], Evans and Morris [13], Bolton and Ursell [7], Bai [4], and Isaacson and Nwogu [23]. Leonard et al [27] extended this approach for multiple cylinders. Field studies using full-scale prototype breakwaters are comparatively uncommon, largely due to the difficulty and expense of field instrumentation. A number of authors Chapter 1. INTRODUCTION 12 have however published field data, including Nelson et al [35], Nece and Skjelbreia [33], and Miller and Christensen [30]. The latter included a formulation for the rigid body motions of a caisson-type breakwater undergoing motions in five degrees of freedom. Comparison of these field data with numerical models indictates that the response of breakwaters can be modelled quite well with frequency-domain methods, provided that the wave frequencies modelled are not near the roll resonance frequency. For the latter case, the presence of additional damping effects in the real breakwaters provides for a response much lower than would would be indicated by the inclusion of radiation damping alone. Experience has shown that by arbitrarily doubling the calculated hydrodynamic damping, better agreement is produced [2], although Miller et al. used a square-law roll-damping formulation to quantify damping more precisely. Papers have also been presented in related areas, such as wave drift forces on floating bodies, and on static and dynamic analysis of mooring line systems. See for example work done by Dean [10], Berteaux [5], and Irvine [22]. Several authors have developed numerical models in an attempt to provide a con-venient method for preliminary design calculations. A completely rigorous treatment of the problem is difficult due to the complexity of the mathematics describing the flow field and the dynamics of a floating body in three dimensions. Strictly speak-ing, a complete model would include the effects of non-linear waves, viscous damping effects, random or spectral sea-state, directional waves, the effect of mooring lines, structure deformations, and the motions of a floating body with six or more degrees of freedom. Routine calculations of this sort in the preliminary design phases would be time-consuming and expensive, requiring the computing power of large main-frame computers1. Fortunately, many simplifying assumptions can generally be made without 1 although recent advances in microprocessor technology are rapidly making such large-scale comput-ing feasible even on desk-top microcomputers. Chapter 1. INTRODUCTION 13 severely reducing the accuracy of the results. For example, small-amplitude (linear) wave theory is often used to make the diffraction calculations more manageable. For the case of incident waves aproaching with crests parallel to the breakwater, exciting forces and motions are in-phase along the length of the section. The problem can therefore be treated as a two-dimensional one, with forces and motions in the third direction being uniform. This reduces the body degrees of freedom from six to three and simplifies the equations of motion. Fraser [15] showed that although mooring line restraints can significantly effect the oscillatory body motions under resonant condi-tions, the effect can be neglected for most of the incident wave frequencies. The body motions can therefore be calculated independantly of mooring line effects for most wave frequencies, simplifying the analysis. Niwinski [36] investigated non-linear wave effects and found there to be reasonable agreement with linear theory calculations. Thus the hydrodynamic analysis can be confined to two-dimensional sections undergoing unre-strained motions under linear waves. In the static analysis of the mooring system for a floating body, anchor line tensions and geometry are calculated by considering the effect of currents, wave drift force, and possibly the effect of wind drag. The result yields the steady state or equilibrium position of the breakwater and the average tensions in the mooring lines. The calculated dynamic motions of the (unrestrained) body can then be applied as boundary conditions to the equlibrium mooring system state to calculate maximum (dynamic) mooring line tensions and anchor loadings. The present thesis examines experience with floating breakwaters currently in use in British Columbia. A numerical method developed by Nwogu [23,37] is used to predict the response of two types of floating breakwaters. The method uses a Green's function approach to relate values of the velocity potential within the fluid domain to the known values of the potential and its derivatives on the control surfaces. The method is Chapter 1. INTRODUCTION 14 modified to account for viscous damping effects and to incorporate the geometry of plate-like structures such as found on A-frame type breakwaters. Viscous damping is included by allowing the user to specify an arbitrary amount of critical damping in each of three modes of motion. The results are then compared with physical model studies and to full-scale prototype data. Chapter 2 F I E L D S U R V E Y 2.1 Categories of Breakwaters The floating breakwaters currently in use in British Columbia mainly fall into three categories: Concrete caisson types, having generally rectangular cross sections; floating log bundles , having generally circular or raft-like cross sections; and A-frame breakwa-ters having a central vertical plate and two pontoons. All of these designs rely primarily on wave reflection to reduce the height of transmitted waves. The locations of major floating breakwater installations within the Province are shown in Table 1.1 and in Figure 1.4. Photographs and marine chartlets of selected breakwaters are presented in Appendix A. In the case of the concrete caissons, the large mass of the section provides a rela-tively stable vertical barrier able to resist the motions of waves by virtue of its inertia. Log bundles attenuate waves in the same way, though generally to a lesser degree due to narrower sections and/or mass. Log rafts have virtually their entire bulk concen-trated at the water surface, and in addition to turbulent losses associated with wave breaking are able to reflect wave energy by providing a rigid mat through which surface undulations cannot easily propagate. The widely spaced pontoons of A-frame sections provide roll-stability to a vertical plate which in turn reflects the waves. These types of sections are less resistant to heave motions due to a lighter mass per unit length, although this is generally not a 15 Chapter 2. FIELD SURVEY 16 problem since the sections typically have greater draft than other breakwater types. 2.2 Concrete Caissons Concrete caisson breakwaters are located at the Richmond Seaplane Base, Deep Cove Yacht Club (Deep Cove), Burrard Yacht Club (North Vancouver), Horseshoe Bay, Nanaimo, Maple Bay, Canadian Forces Sailing Association (Esquimault), Victoria Har-bour, Lower Arrow Lake (Nakusp, B.C.), as well as a new breakwater installed at Lund during June 1987. The latter breakwater replaces an older A-Frame breakwater which had been at that location since 1963. Most caisson designs are essentially similar in construction, consisting of a reinforced concrete box divided into internal sections by means of integral partitions, much like an inverted ice-cube tray. The intersticial spaces are sometimes left empty, allowing trapped air to provide buoyancy. More commonly the space is filled with closed-cell plastic foam to provide positive bouyancy in the event of a structural failure in the caisson wall. Most designs have a smooth, flat deck surface, making them amenable to ancillary uses such as pedestrian walkways and boat moorage. Unusual design features or site locations are described for a few cases below. The caisson at Deep Cove Yacht Club (Figure A.23) has a beam of 5.3 metres and draft of 1.25 metres, and protects yachts from fetches of 3.0 km to the northeast. The design conditions at this site indicated a wave height of 1.2 metres with a period of 3.4 seconds [48]. Although fairly well protected from the prevailing storm directions, the site is subject to occasional winter outflow conditions providing quite vigorous winds from the northeast. Numerous finger wharves are built into the leeward side of the breakwater providing moorage for approximately thirty boats. The windward side of the breakwater is often Chapter 2. FIELD SURVEY 17 used for temporary moorage for boats visiting in fair weather, although at times this has proved risky — the marina caretaker recounts that on several occasions small boats moored on the exposed side were lifted out of the water and deposited none too gently on the breakwater deck by a sudden squall. The topography in Indian Arm is fjord-like, accounting for the rather deep water at the site of the breakwater — divers carrying out routine inspection of the anchors and mooring lines are reportedly required to undergo decompression procedures when returning to the surface. Marine charts of the area indicate the breakwater is moored in approximately 20 metres of water (Lowest Normal Water) but the anchors may be located in water up to 45 metres deep. Sewells Marina in Horseshoe Bay (Figure A.24) uses an unusual caisson, which is a disused section from the Hood Canal floating bridge in Washington State. Because of its original purpose the breakwater is considerably wider than most caisson designs at more than 20 metres. In addition to a fetch of 9.0 km to the north (the direction of the prevailing winter outflow or "Squamish" winds), the breakwater protects the marina from ship wakes at one of the busiest ferry terminals in the province. Located directly to the west of the caisson is a scrapped hull from an ore carrier. The hull is ballasted to the deck level with seawater and serves as additional protection for the moorage. Although no structural problems have been experienced by the breakwater, the owners indicated that the mooring lines been difficult to maintain due to corrosion of chain or ultraviolet degradation of polypropylene rope. Nanaimo Yacht Club in Newcastle Island Channel (Figure A.26) uses a caisson to protect members boats. The maximum fetch distance (which coincides with the direction of the prevailing winter storms) is 5.4 kilometres, although the design wave conditions are not available. The sections are moored to pile dolphins on their windward side, and overall the owners are pleased with the breakwater's performance. There was concern at this site that a bottom-resting structure would cause sedimentation Chapter 2. FIELD SURVEY 18 problems in the already constricted channel, and thus a floating structure which would not impede existing circulation was prefered. The breakwater at Maple Bay (Figure A.27) consists of seven units, each with dimensions of 22.3 m x 4.6 m x 1.2 m and having a draft of 0.58 m. The sections are connected to provide a continuous breakwater, and positive foam flotation is provided. Integral finger wharves are built into the leeward side of the breakwater and provide moorage for thirty-six vessels in addition to the rest of the marina. Maximum fetch is to the Northeast (4.3 km), and the design five-year wave had a significant height of 0.6 m and period of three seconds. The specified performance criterion was to provide approximately 50% reduction in wave height under these conditions [48]. With the exception of minor surface repairs and occaisional tightening of the connections, the breakwater has required little maintenance and has provided satisfactory service. In May 1987 a new three-module caisson breakwater was installed at Lund (Figure A.25), replacing an earlier A-Frame (Figure A.22) which had served at that location for 22 years. The site is exposed to the southwest and northwest with fetches of 29.6 and 12.1 km respectively. The effective fetch to the southwest is actually somewhat less since seas approching from this direction must pass over shoals in Manson Passage (between Hernando and Savary Islands) only 9.6 km from the harbour. At higher high water however, this shoal is covered to depths of about 6.4 metres, which is approaching the deep-water limit for waves of period five to six seconds [45], typically among the largest waves along this part of the coast. This means that at high tide and with a storm approaching from the southwest, Lund is perhaps subject to the most severe wave climate of any caisson breakwater in the Strait of Georgia. The layout of the Lund breakwater is somewhat unusual in that the caisson sections are not connected to each other. The modules are staggered in the horizontal plane in an effort to avoid problems with interconnections and structural problems arising from Chapter 2. FIELD SURVEY 19 collisions between units. The disadvantages of this arrangement are a reduced length of protection because of the overlap of the sections, wave diffraction through the gaps, and a more complicated mooring line arrangement. In addition, this breakwater is in a detached position in the harbour making ancillary uses somewhat difficult. In 1985 B.C. Hydro installed a caisson breakwater on Upper Arrow Lake at Nakusp. This installation was part of a flood mitigation package dating back more than twenty years to construction of the Hugh Keenleyside dam. Because the lake is a power-generation reservoir, it is subject to large annual fluctuations (up to 25 m) in the water level. The site also experiences ice-up during winter months, complicating the design which must account for freeze-thaw cycles in the concrete as well as ice expansion pressures. 2.3 Log Bundles and Rafts Floating log bundles are used extensively throughout the province, although in their simplest application, a mere boom of single logs, they are not effective against any but the shortest period waves. Larger breakwaters consisting of booms of six or eight logs are found at Browning and Bedwell Harbours (North Pender Island), Fairview Bay and Rushbrook (Prince Rupert), Reed Point Marina (Port Moody), Fanny and Mud Bays, (Vancouver Island), Ford Cove (Hornby Island), Tahsis, Queen Charlotte City, and in Okanagon Lake (Kelowna). Fetches and location references for these sites are given in Table 1.1. The photograph in Figure A.28 clearly shows wave reflection from and diffraction around the end of the log bundle breakwater. The waves in this case were caused by the passage of a marine tug, and have an estimated wave height of 0.5 metres and period of three seconds (the scale can be inferred from the figure standing in the left Chapter 2. FIELD SURVEY 20 background). Note the relative calm in the lee of the logs even though the wavelength appears to be significantly greater than the beam dimension of the bundle. Although the relatively small waves might not be a cause for engineering concern (few structural problems would be associated with waves of this size), they are significant from an operational point of view. The marina provides berthage space to a large number of pleasure craft, and even slight wave-induced oscillations can cause abrasion to mooring lines and fibreglass hulls. The log raft at Ford Cove (Figure A.30) compliments a rubble mound breakwater to extend the area of coverage. It also provides convenient additional mooring space for visiting boats when the public wharf is full. 2.4 A-Frames An A-frame breakwater installed in Queen Charlotte City in 1967 continues to provide satisfactory service, while the one recently removed from Lund was suffering badly from corrosion (and is currently being repaired after being sold to an unspecified private party). Both of these breakwaters were based on the same design, having steel pontoons of diameter 0.76 metres, beam of 7.6 metres, and draft of 3.7 metres [48] (Figure A.22). The timber centreboard is connected to the pontoons with a steel space-frame, and extends upward from the sea surface approximately 2.0 metres (An alternate design for A-frame breakwaters has the space-frame located below the water surface, allowing the centreboard to be cut off at the surface and providing a convenient flat deck which can be covered by planking for pedestrian use.). Chapter 2. FIELD SURVEY 21 2.5 Other Types Only one BC breakwater uses energy dissipation as its primary wave reduction method. This breakwater is located at Eagle Harbour in West Vancouver, (see Figure A.32) and consists of two rows of cylindrical steel pontoons between which scrap tires are strung on conveyor belting. Although some wave reflection occurs from the surface of the pontoons, each pontoon is free to move independently of the other. This means that the effective beam for reflection is the diameter of an individual pontoon, resulting in reduced reflecting effectiveness as the wave period (and length) increases. Most of the wave reduction occurs through turbulent losses associated with the movement of water in and around the submerged tires. Another breakwater built largely of scrap tires had been in use at Burrard Yacht Club in North Vancouver, but was destroyed in 1980 after only one year of service. Although an extreme case, this appears consistent with the performance experienced with rubber-tire breakwaters elsewhere in North America, where an initially low capital cost (scrap tires being readilly available) is offset by higher maintenance and lower life expectancy. Sunset Marina in Howe Sound (approximately 3.6 kilometres north of Horseshoe Bay) uses an old steel barge to protect small craft from the often considerable winds in Howe sound (Figure A.33). The maximum fetch is 12.8 kilometres to the north-northwest, and the site is also exposed to the wash of freighter traffic travelling to Squamish. Details Details of the barge itself are unavailable, but the owners are re-portedly satisfied with the degree of wave attenuation provided. MacMillan Bloedel uses a unique breakwater of scrapped ship hulls to protect their log-booming grounds from a southwest exposure (Figure A.34). There are currently ten old concrete-hulled ships which comprise the breakwater, which was begun before Chapter 2. FIELD SURVEY 22 1930 and has gradually grown as more ships hulls were aquired. Many of the vessels were built in the U.S.A. under wartime measure and have been purchased since 1947. The ship lengths range from 102 to 128 metres (336 to 420 feet) in length and once had displacements of up to 5450 tonnes (6000 tons). The ships are anchored with eight to ten concrete anchors, each weighing up to 14.5 tonnes, using anchor chains weighing 800 N/m (55 lb/ft). Another unique breakwater is located at Brown Bay Marina (north of Campbell River), and consists of a number of disused railway tanker cars (Figure A.35) strung together with heavy chain and cable (the wheel assemblies were removed before the cars were launched). The topography of the area allows the mooring lines to be made fast not to anchors but to the surrounding cliffs, thus allowing for more convenient inspection. One of the units developed a hole from rubbing against rocks at low tide, and is currently partly submerged since no positive bouyancy was provided. Although the fetch at the location is rather small, the breakwater protects the marina from the wash of frequent ocean-going ships travelling through the Inside Passage. The marina is located just north of infamous Seymour Narrows, and is thus subject to tidal currents of several knots each day. As the breakwater is alligned parallel to the main channel, it also serves to deflect floating logs and other debris from entering the harbour. Chapter 3 N U M E R I C A L M O D E L A numerical model based on linear diffraction theory was developed in 1985 to cal-culate the response and effectiveness of breakwaters of rectangular and circular cross section subjected to waves with any specified direction of propagation (Nwogu [37], and Isaacson and Nwogu [23]). This model uses a Green's Function solution to the two dimensional problem of a floating body undergoing motions in three modes, namely sway, heave and roll. The model does not include mooring line forces in determining the oscillatory response since previous work has indicated that these effects are small except under conditions approaching resonance (Fraser [15]). Rather, the mooring lines affect the low-frequency drift motions and the location of the breakwater, and can be neglected in determining the wave transmission characteristics and oscillatory behaviour of the section. The present thesis revises this model to include breakwaters of A-frame cross sec-tion, and to allow the inclusion of viscous damping coefficients. Viscous damping forces are important terms affecting the response of floating structures, particularly in the roll mode of motion near the natural frequency. These coefficients are estimated from em-pirical measurements of model breakwaters as well as from what full scale prototype data are available. The calculated motions are used as boundary conditions in a second numerical model which calculates the forces in a three dimensional mooring system. These forces are of interest in the design of mooring systems since experience in other parts of North America has shown that moorings and inter-module connections are 23 Chapter 3. NUMERICAL MODEL 24 more likely to fail than the breakwater itself. 3.1 Mathematical Treatment 3.1.1 Wave Diffraction The linear analysis of a floating body in waves may be divided into two parts for convenience: first, the evaluation of the forces on a fixed cylinder (in this case of arbitrary section) due to a diffracted incident wave train of unit amplitude; and second, determining the resulting wave pattern caused by the oscillation of the cylinder in otherwise calm water. The breakwater being considered is assumed to be large enough to diffract the incoming wave train. This assumption implies that flow separation will not occur in the structure vicinity and that the flow field can be analysed using diffraction theory. Clearly this assumption can only be made for breakwater sections which rely primarily upon reflection for wave attenuation, since dissipative mechanisms use flow separation and the resulting turbulence to attenuate wave energy. In the case of the A-frame sections, the flow separation around the "sharp" edge at the bottom of the plate is initially neglected in the analysis and accounted for by the use of empirical viscous damping coefficients. The coordinate system is right-handed and Cartesian and is defined such that the ar-axis is in the beam-wise direction of the section, the y-axis is along the longitudinal axis of the breakwater, and the z-axis is measured upward from the still water level as shown in Figure 3.5. The fluid is initially assumed to be inviscid (viscous effects are included later in the equations of motion) and incompressible, and the flow to be irrotational. Thus the flow can be described in terms of a velocity potential $ , which must satisfy the Laplace Chapter 3. NUMERICAL MODEL 25 Figure 3.5: Definition sketch for floating cylinder equation V 2 $ = 0 (3.1) and is subject to the usual linearized boundary conditions. Small amplitude regular waves of height H and angular frequency OJ are assumed to propagate in water of constant depth d past the arbitrarily-shaped cylinder. The breakwater is assumed to oscillate periodically in each of its three modes of motion. For the case of long-crested waves approaching perpendicular to the long axis of the breakwater, the motion at one section of the breakwater will be in phase with that at any other section. For the case of obliquely incident waves, the motion is assumed to be periodic along the length of the breakwater, in other words the breakwater is flexible. For a two dimensional section the motion in the j th mode can be expressed as B,- = 5R[^e'(fcwsin^a'0] j = 1,2,3 (3.2) Chapter 3. NUMERICAL MODEL 26 where £y is the amplitude of motion (generally complex), j — 1,2,3 corresponds to motion in the sway, heave, and roll modes respectively, 9i denotes the real part of the complex expression, and (3 is the approach angle of the waves, defined as the angle between the wave orthogonal and a line normal to the long axis of the breakwater. Since the wave heights are small the boundary condition on the body surface is linearized and is given as ~=Vn onSB (3.3) where Vn represents the velocity of the body in the direction of a unit vector n normal to the body surface at that point. The velocity has components derived from each of the modes of motion, thus Vn = J2-i»tinie~iut (3-4) where ni = nx, n-i — nz, and nz = [z — e)nx — xnz, nx and nz are the direction cosines of the normal vector n, and e is the z-coordinate of the point about which roll motion is defined. The complete velocity potential $ for a floating body oscillating in waves can be considered to be made up of components associated with the incident waves (j>o, the diffracted or scattered waves <£4, and waves generated (radiated) by each of the three modes of motion of the breakwater, <j>\,<t>2, and <f>3, corresponding to sway, heave, and roll respectively. The latter potentials each are proportional to the amplitude of motion £j in that mode. Thus, the total flow potential can be expressed as 2 0t'(A:y sin /8-oit) (3.5) where A; is the wave number given by the linear dispersion relation u2 /etanh(fcd) = — (3.6) Chapter 3. NUMERICAL MODEL 27 and the incident potential <j>0 is given by linear wave theory as - jp f -»9ff cosh[fc(g + rf)] i { k v s i n P . u t ) ' 4>o = 3* (3.7) 2u> cosh(A;rf) Substitution of equation 3.5 into equation 3.1 gives the governing differential equa-tion in the fluid domain as V24>k{x,z)-u2<j>k = Q k = 1,2,3,4 where u = A:sin/?. The body surface boundary condition can be expressed as (3.8) d<t>k dn for k = 1,2,3 for k = 4 (3.9) The potentials <j>k are each subject to additional kinematic and dynamic boundary conditions as follows: d<t>k w2 " a - = —0* an g d<j>k dn d<t>k dn = 0 = ik cos /3<f>4 at the free surface at the seabed at +/ - SR (3.10) (3.11) (3.12) The radiation surfaces +/ — are located at some distance from the body where the evanescent modes are assumed to be negligible and where the far-field diffracted potential can be evaluated. The task is now to find the solution to the boundary value problem of the diffracted and forced motion potentials, subject to the conditions above. Once these potentials are known, the exciting forces and wave transmission characteristics for the breakwater can be determined from linear wave theory. Chapter 3. NUMERICAL MODEL 28 3.1.2 Green's Function Solution A boundary element method based on Green's theorem is used to obtain a solution for the potentials <f>k (k=l,2,3,4). Green's theorem relates the values of the potential within the fluid region or on the boundary to the values of the potential and its derivatives over some closed surface S. This theorem is expressed as where G(x, f) is the Green's function satisfying the requisite boundary conditions, x is the point located at (x,z) where the potential is to be evaluated, and £ is the point located at (£, f) which is on the closed surface and contributes to the potential at x. For convenience the closed surface is divided into sections consisting of the free surface Sp , the radiation surface SR , the seabed Sp , and the body surface SQ. This geometry is shown in Figure 3.6 . (3.13) Free Breakwater section I (arbitrary shape) /— Radiation Surface Seabed So Figure 3.6: Definition sketch for boundary integral Chapter 3. NUMERICAL MODEL 29 The Green's function which satisfies equation (3.13) is G(x; I) = -K0{ur) (3.14) where KQ denotes the modified Bessel function of zero order and r is the distance between the points x and £, given by the expression r = \t-x\ = yj(t-xy + (s-z)* (3.15) —* This Green's function is singular at the point x = f and so special consideration must be given to evaluating the integrand at that point. In this solution the seabed is assumed to be horizontal over the region of interest. Provided that the condition of no flow normal to the seabed is enforced, a second, more efficient Green's function can be defined which regards the seabed as a plane of symmetry. That is, the calculation proceeds by excluding Sp and including reflected versions of Sf , SR , and SB, using the Green's function G(x; I) = -[K0(i/r) + K0(ur')\ (3.16) where r' is the distance between x and the reflection of about the seabed £' = ( £ , - ( £ +2d)), given by: r' = \§ - x\ = v / ( e -x ) 2 + («r + 2d + z) 2 (3.17) (see Figure 3.7). The integral equation (3.13) is then evaluated using a boundary element procedure which discretizes the closed surface 5. This procedure is described in greater detail in section 3.2. Figure 3.7: Alternative boundary geometry which makes use of symmetry about the Chapter 3. NUMERICAL MODEL 31 3.1.3 Hydrodynamic Analysis Once the flow potentials have been obtained over the desired domain, the hydrodynamic pressure acting on the body can be determined from linear wave theory as P=-P-^ (3.18) Integrating this pressure over the body surface yields the forces and moments per unit length of the structure. The exciting forces due to the incident and diffracted wave trains are given as Fj{y, t) = ^{Cje^-^} (3.19) where Cy is the complex amplitude of the exciting force defined as Cy = \pgH f {4>0 + <f>4)njdS (3.20) * J Sg In addition, the motion of the breakwater in each of its respective modes provides additional forces which interact due to coupling terms in the equations of motion: L JSg where is the itH force component due to the jth component of motion. This force can alternately be expressed in terms of components in phase respectively with the velocity and acceleration of the body: Fn = 5R[(wV,-yf; + ioj^^e^-^} (3.22) The coefficients fiij are defined as added mass coefficients, while the AtJ- are defined as damping coefficients. These terms are so-named because of the roles they play in the equations of motion, and are given in non-dimensional form by £ = fadS] (3.23) pam JSg Chapter 3. NUMERICAL MODEL 32 and (3-24) The exponent m takes on different values according to the dimensions of the component motions, with m = 2 for pure sway or heave motions, m = 3 for coupled sway-roll or heave-roll, and m — 4 for roll motion only. 3.1.4 Transmission and Reflection Coefficients The primary quantity which describes the effectiveness of a breakwater section is the transmission coefficient Ct, which relates the heights of the incident waves to those transmitted by the breakwater and is defined as Ct = f£ (3.25) where the subscripts t and t refer to the transmitted and incident waves respectively. This quantity can be determined by evaluating the complex amplitude of the waves at the leeward radiation surface SR. The wave amplitude is composed of component parts from both the transmission of waves past a fixed cylinder as well as waves generated from the motion of the cylinder oscillating in still water. The wave amplitude n associated with a particular velocity potential is given by " = <3-2 6> Substitution of the various potentials (f>k,{k = 1,2,3,4) into this equation and super-posing the results yields the required wave amplitude, which when normalized by the incident wave amplitude gives the desired transmission coefficient. The reflection co-efficient C r can similarly be determined by evaluating the various potentials at the seaward radiation surface. Chapter 3. NUMERICAL MODEL 33 The wave energy not transmitted to the leeward side of the breakwater or reflected from the section is dissipated in heat, noise, turbulence, and through any work extracted through structural deformation. Traditional diffraction solutions neglect any losses since it is assumed that if the structure is large enough to diffract the incident wave field, flow separation does not occur and turbulent losses are not present. However, in the presence of viscous damping forces included in this model, some portion of the wave energy must be attributable to losses. By including viscous damping in the equations of motion, the calculated response is smaller than would otherwise be the case, particularly near resonant frequencies. Since part of the transmitted and reflected wave heights are due to the motions of the breakwater, Ct and CT are both lower when viscous effects are included. It is therefore necessary to define an additional coefficient d, which can be attributed to wave dissipation, and which is defined as the ratio of energy dissipated to the incident wave energy. The total energy in the system must still be conserved, and since wave energy is proportional to the square of wave height 3.1.5 Equations of Motion The generalized dynamic equations of motion for a body with multiple degrees of freedom can be expressed (in matrix form) as C] + C? + Cd = l (3.27) [M](5) + [A](H) + [C](H) = (Ft) (3.28) Chapter 3. NUMERICAL MODEL 34 where [M], [A] and [C] represent mass, damping and stiffness matrices respectively, and (Ft) is some time-dependent forcing function, (H) is a general displacement coor-dinate vector, and the dots denote first and second time derivatives of displacement. Recognizing that the motion of the breakwater is periodic and recalling the relation between the derivatives of H, the equations of motion for the floating cylinder can be expressed as 3 ]T[-w2(m,-y + ^.) _ + rp{j) + Cij]= Fi (3.29) t=i where the m,y and c,-y are the actual mass and hydrostatic stiffness coefficients defined by Nwogu (1985) as m 0 —TTIZQ rriij = 0 m 0 —THZQ 0 Io Cij = 0 0 0 0 pgB pgBxf 0 pgBxf c 3 3 (3.30) where m is the mass per unit length of the section, ZQ is the the z coordinate of the centre of gravity, Io is the polar moment of inertia per unit length of the section about the y-axis, B is the beam of the section, and xj is the location of the centroid of the waterplane line with respect to the z-axis 1 . The term c 3 3 in the stiffness matrix is given as c33 = pgV[-^- +zB- zG] (3.31) where V is the displaced volume per unit length of the section, zB is the z-coordinate of the centre of buoyancy, and S n is the waterplane moment of inertia per unit length of the section about the z-axis. The stiffness coefficients here do not include any terms involving the first mode (sway) since in the absence of mooring line forces (neglected in this analysis) there is no restoring force for horizontal translation. 1 equal to zero for bodies symmetrical about the z-axis Chapter 3. NUMERICAL MODEL 35 The terms V«i m equation(3.29) are the terms associated with viscous damping forces on the structure. These are empirical values supplied as percentages of critical damping in each mode, that is Ai = C.yAcnt.. (3.32) Critical damping is defined here as A c r , t . y = 2m,jWn = 2 v/m, ic, y (3.33) Since in this case the stiffness c and therefore critical damping in sway are identically zero, the sway damping has been defined in terms of the heave stiffness c^. Solving equation (3.29) for yields the amplitudes of motion in the respective modes. These motions or responses can be nondimensionalized with respect to the amplitude of the incident waves, yielding the Response Amplitude Operators (RAO's) defined as RAOj = ^ (3.34) 2 where H is the incident wave height and the amplitude of motion for roll is first di-mensionalized by the half-beam of the breakwater. 3.2 Numerical Analysis In order to evaluate the integral equation (3.13), the closed boundary S is discretized into N segments, over which the values of the potential 4>k and its normal derivative d<j>k/dn are taken to be constant and equal to the values at the midpoint of the segment (Figure 3.8). The continuous integral of equation (3.13) is therefore replaced by the discrete summation k = 1,2,3,4 (3.35) Chapter 3. NUMERICAL MODEL -< Beam-a >• Draft Depth Free Surface Pontoon Radiation Surface Centre point / /Edge -• 1 Double row of coincident points for A-frame plate K - A S - M Typical Boundary element Figure 3.8: Discretization of breakwater section This summation can also be written as = 0 k = 1,2,3,4 where 8ij is the Kronecker delta function defined by N £ 3=1 gAk) fa + + by-XL-on 1 for i = j 0 otherwise The coefficients a,-,- are defined as while the 6,-,- are defined as where the r,-y and rj- are given as Chapter 3. NUMERICAL MODEL 37 r'ij = yJixj-Xiy + izj+^d + Zi)2 (3.41) and the vectors xt and £j are taken at the centre of each segment. As the 8S in equations (3.38) and (3.39) are generally small, the integrals in these equations can be approximated by assuming a constant value for the Green's function over the segment, taken to be the value at the midpoint. For t ^ j, the expressions for dij and bij therefore become [(ij - Xi)Azj - {ZJ - Zi)Axj] K [ur1 1 u ^ IJ \{XJ - Xi)Azj - {ZJ + 2d + Zi)Axj] (3.42) and where by = —[Koivry) + Koiur'^AS (3.43) AZJ = Zj+i — Zj Ax, = XJ+I — Xj (3-44) ASj = ^{AZJY + (Ax,-)» and Ki is the modified Bessel function of order one. 3.2.1 Singularities The Green's function used in (3.13) is singular at the point x = £ and thus special consideration is given to these points. Physically, the singularities occur in the bound-ary intregal process when the point of interest coincides with the point over which the integration is performed. In the case of both rectangular and semi-circular breakwater sections, these points occur in the numerical algorithm only when t = j. Furthermore, there is only one such point for each j in equation (3.35). Chapter 3. NUMERICAL MODEL 38 In the case of the A-frame section however, the central plate is modelled as an impermeable vertical wall of negligible thickness, as shown in Figure 3.9. As the wall i - 2 <> <> i+5 i - 1 ii «i i+4 E n l a r g e d sec t iona l view of A - F r a m e plate e l e m e n t i *> <> i+3 i+1 ( , i+2 — H + — Figure 3.9: Discretization of A-frame plate thickness t is allowed to go to zero, the points on one "side" of the plate become coincident with the corresponding points on the other side. Thus when the points j in (3.35) are located on either side of the vertical plate, there is an additional singularity which occurs when the point of interest is directly opposite the point of summation. For the example shown in Figure 3.9, there are singularities at i both when j = i and j = i + 3 . For both types of singularities the numerical difficulty arises from the fact that the the Bessel function KQ(O) approaches infinity as a approaches zero. Both can therefore be treated in the same manner using an approximate small-argument expression for K0 as given by Abramowitz and Stegun(l964): Hm K0(ur) ~ -[In ( y ) + 7] (3.45) where 7 is Euler's constant. This provides expressions for a,-,- and 6,-,- in equations (3.42) Chapter 3. NUMERICAL MODEL 39 and (3.43) as follows: an = -Ki\2u(zi + d)]Axi (3.46) l n [ ^ ] + 7 - 1 - KQ[2v(zi + rf)]] (3.47) t AS,-Oti = 7T The expressions involving Ko(ur') do not present difficulty since r' is never zero. Once the atJ- and 6tJ- are determined, iV equations can be written relating the N values each of <f> and | £ at the facet centres along the surface. Applying the appropriate boundary conditions over the various surfaces thus leads to a total of 2N equations needed to solve for the 2JV" values of 6 and 1^  . ~ on The resulting equations are solved using a complex matrix inversion technique to determine the values of the <f>j. Chapter 4 P H Y S I C A L M O D E L L I N G In order to confirm the performance and to obtain empirical damping coefficients for the numerical model, two configurations of breakwaters were constructed and tested in the Civil Engineering Hydraulics Laboratory at the University of British Columbia. Measurements were made to determine the motions and wave transmission character-istics of the models. The experimental results are compared to those predicted by the numerical model and to full scale prototype data collected in the Puget Sound area (see [33,35]). 4.1 Experimental Facilities The wave basin at the University of British Columbia measures 13.7 metres by 4.9 me-tres, and can accomodate working water depths of up to 0.48 metres. A variable speed, electrically driven flap-type wave generator at one end of the basin provides uniform long-crested waves in periods ranging from 0.4 seconds to 2.2 seconds. The amplitude of the waves can be adjusted by altering the stroke of the wave paddle. A sloping beach consisting of an aluminum and timber frame covered with artificial hair matting is lo-cated at the opposite end of the basin and serves to reduce reflections. Wave probes used were of the capacitance type, and were driven using Hewlett Packard DC voltage supplies. Probes were calibrated prior to and after measurements by immersing the wires at specific intervals and recording the resulting voltage. Calibration curves were linear with an average correlation coefficient of 0.997. The probe output signals were 40 Chapter 4. PHYSICAL MODELLING 41 connected both to an oscilloscope and to a Hewlett Packard strip-chart recorder. 4.2 Dimensional Analysis The wave transmission characteristics (Ct) and oscillatory response (£) of a floating breakwater can be characterized by the following independent parameters: the inci-dent wave height Hi and wave length L, the water depth d, the water density p and acceleration due to gravity g, as well as two characteristic breakwater dimensions, taken to be the half-beam a and the draft h. An application of dimensional analysis provides _ £ , , a H d h. , . C"W2 = f i L - L ' L ^ <4-48> For experiments carried out in water of constant depth and with breakwater models of constant draft, the latter term drops out, leaving the beam parameter a/L, the wave steepness parameter H/L, and the water depth parameter d/L. These can be com-bined to provide an additional breakwater "shape" parameter, a/h, which conveniently describes the "fullness" of the section and its stability in roll. With this definition of shape, a section with with a/h = 0.5 is just as wide as it is deep. From previous experience the beam parameter a/L is known to be the most im-portant in characterizing the response of a floating breakwater, and the laboratory program was designed with this in mind. The wavenumber k = lis/L was chosen to represent the wavelength, and thus the actual beam parameter used in plotting the results was ka (Some investigators have used a similar beam parameter B/L, where B is the beam of the breakwater. This parameter is dimensionally equivalent to ka, differing only by a factor of IT) . The motions of the breakwater are presented in dimensionless form as the Response Amplitude Operators (RAO's), defined as the ratio of the amplitude of motion to the amplitude of the incident waves. For the case of sway and heave motion, which have Chapter 4. PHYSICAL MODELLING 42 dimensions of length, the RAO'i s are given as RA0li2 = £1,2 (4.49) H/2 Since roll motion is defined as angular displacement, the amplitude is first dimension-alized by the half-beam of the section, thus The experiments were carried out using waves of both "high" and "low" steepness even though the numerical model is based on linear diffraction theory which takes no account of the steepness effect. The results for both steepnesses are plotted together on the various curves in Chapter 5. 4.3 Experimental Procedure Two configurations of breakwaters were tested, an A-frame type and a rectangular-section caisson. The A-Frame breakwater was modelled after the prototype installations at Lund and Queen Charlotte City, which have a beam of 7.6 metres and draft of 3.7 metres. The model was constructed at a scale factor of ^ and consisted of a plywood centreboard and two rectangular pontoons partially filled with foam. The pontoons were joined to the centreboard by means of an aluminum space frame, which was constructed in such a manner as to allow the pontoon spacing to be altered. Tests on the A-frame were carried out at a number of pontoon spacings, providing values of a/h ranging between 0.86 and 1.29. The design of the caisson model was based on the concrete caisson employed in field tests by Nece et al [33] and Nelson et al [35]. The prototype caisson had an overall length of 22.9 metres, beam of 4.88 metres, and draft of 1.07 metres (75' by 16' by 3.5' respectively). The model was constructed out of plywood to a scale of ^ , and was RAO3 = H/2 (4.50) Chapter 4. PHYSICAL MODELLING 43 partially open at the bottom to admit water into the interior. The required draft was achieved by adjusting foam flotation and lead weights as necessary. Since the draft of the section was not altered during the tests, the caisson data represent a shape parameter of a/h = 0.22. The effect on the breakwater motions of the free surface within the caisson was min-imized by dividing the interior into sections with the foam floatation. The "sloshing" frequency of the water within the compartments was observed to be of higher order than that of the breakwater motions, and so was considered insignificant. For all cases tested, a vertical wall was installed in the basin to prevent waves reflected from the model from interfering with the incident wave signal. The general test arrangement is depicted in Figure 4.10. WAVE BASIN (Plan View) (13.7 m X 4.9 m) u o <-> «f u V a O V > wave travel • Anchors •3 Wall a Incident Wave Probe Mooring lines Breakwater Direction of wave travel • •3-S Transmitted Wave probe -cm o ed a; m Theodolite station | ^ Figure 4.10: Experimental setup (not to scale) Chapter 4. PHYSICAL MODELLING 44 4.3.1 Wave Heights Incident wave heights were measured in an area of the basin separated from the test section by a vertical wall as indicated in Figure 4.10. Artificial hair matting placed around the perimeter of the basin and along one side of the wall served to reduce reflected waves. The transmitted wave probe was located leeward of the breakwater and opposite the incident probe. Although some reflections were evident off the rear basin wall, placing the two probes an equal distance from the wave paddle is thought to have reduced the resulting error. The incident and transmitted wave heights were recorded over several wavelengths and average heights were used in subsequent calculations. 4.3.2 Breakwater Motions Model motions were measured optically by means of a vertical bar and targets attached to the model. Three reference markers at known intervals were located vertically above the breakwater centre of gravity. The amplitudes of motion of the markers were mea-sured by sighting through a theodolite at the targets, behind which was placed a scale. By aligning the crosshairs in turn upon each target at its maximum horizontal or vertical excursion and noting the resulting position on the scale, the motions were ob-tained. These motions were related to the motions of the breakwater center of gravity as described below. Consider the geometry of the model as shown in Figure 4.11. The sway, heave and roll motions of the model are denoted by f i , ^ and £ 3 respectively. The horizontal motion of points A, B, and C are denoted by £XB> and £ 1 C . Likewise the vertical motion of A is denoted £25 • Since points A, B and C are located vertically above the centre of gravity, their vertical motion is in phase with and is identical to that of f2-Chapter 4. PHYSICAL MODELLING 45 Measurement of Breakwater Motions Marker pins— ^ 2 A Incident Waves Anchor Lines-Figure 4.11: Measurement of breakwater motions The horizontal motion of point A can be described by £ I A C O S ( U ; < - <f>lA) = £ x cos(w* - <£i) + zA£z cos(ut - fa) (4.51) where the <j> represent phase shifts of the respective motions. If wt is defined as 0 and 4>x is set arbitrarily to zero, equation (4.51) becomes iiA cos 0 = £i cos 6 + Z A 6 COS(0 - (j>3) (4.52) This can be expressed in the form: & = + 2 £ i &*A cos <f>3 + zAZ\ (4.53) Similar expressions can be written for £ i B and £ic-CIB = £i + 2 f I 6 zB cos fa + zB i\ i\c = £i + 2 6 izzc cos <j>3 + zc £ (4.54) (4.55) Chapter 4. PHYSICAL MODELLING 46 Since it is the motions of the centre of gravity which are desired, equations (4.51) through (4.55) are solved for £i and f3 as follows: t2 _ ZAC{$.IA ~ £ I B ) ~ ZAB{€IA ~ €IB) 5g\ 3 ZAC{Z\ ~ 4 ) - ZAB{ZA ~ z2c) g = ZA#B - ZB£\A - [ZAZI - ZBZA)Zl ^ 5 7 ^ ZA — ZB and £ 2 is known directly from the measurement of £2A-Using these expressions the component motions of the centre of gravity of the break-water can be determined by measuring the vertical amplitude of motion for the top pin, & M ) a n d the horizontal amplitudes of all three pins, £1A, £ I . B , and Chapter 5 R E S U L T S 5.1 Field Survey An inspection of the breakwater systems currently in use in the province indicates that the majority are providing satisfactory service for the owners. This is particularly true for the concrete caissons, which tend to be more durable than the other designs and are more amenable to secondary uses. Although having an initially higher capital cost, the concrete breakwater systems have a lower incidence of structural damage and provide a greater degree of wave attenuation than either tire breakwaters or log bundles and rafts. In most cases marina operators have few complaints about their breakwaters, except of course in the few cases where failures have occurred. Breakwaters that are well-built and maintained should have a useful design life of 15 - 30 years, and should provide acceptable wave protection for all except extreme wave states. This does not mean, however, that failures of well-designed systems will not occur, but rather the choice of a floating breakwater system over a traditional rubble-mound must be made on the basis of cost effectiveness. It is necessary to weigh on one hand the lower capital cost, deep-water adaptability, and other benefits against lower design life, higher risk, and the amount of damage incurred in the event of failure. Annual maintenance of most breakwaters includes an inspection of the mooring and anchoring systems, typically the first component of the systems to show signs of failure. 47 Chapter 5. RESULTS 48 Of the marinas visited, only one (Horseshoe Bay) had had appreciable difficulty with mooring lines. This was associated with the corrosion of steel chain and of ultraviolet degradation of polypropylene lines. 5.2 Field Data Although full-scale field data concerning floating breakwaters are relatively rare, a number of studies ([33,35,8]) have been carried out in Washington State in this area. The data collected, however, are somewhat limited due to the difficulties in maintain-ing a program of field research. Nonetheless they are presented here along with the numerical and experimental results for comparison purposes. Nece et a/[33] carried out wave attenuation measurements using a US Coast Guard vessel to generate the incident waves. They found that there was no discernable differ-ence in the transmission characteristics for waves approaching at a variety of incident angles. Although precise measurements of the incident angle were not made, it was estimated to vary between -5 and 36 degrees (with an angle of "zero" defined as oc-curring when the wave crest was parallel to the long axis of the breakwater). These results are plotted along with numerical and model data in Figure 5.12. For shorter wave lengths the results were found to agree well with two-dimensional model studies carried out by Carver [8]. For longer wave lengths the results were inconclusive, with transmissions measured for one type of ship wave being higher than predicted, while for another ship the transmissions were lower. Nece et al. cite differences in mooring line pre-tensions as a possible cause for this discrepancy. In general their results show higher transmission coefficients than indicated either by the present numerical or experimental data. Nelson and Broderic (1984) report wave transmission data obtained on the same Chapter 5. RESULTS 49 prototype breakwater but from wind generated waves. The wave crests were reported to be essentially parallel to the breakwater axis, with wave periods taken to be that of the spectral energy peak obtained from spectral analysis of 8.5 minute wave records. The wave height was taken to be four times the standard deviation of the record (an approximate result based on the assumed Rayleigh distribution [45]). Their results are well within those predicted by the present numerical and experimental results. 5.3 Numerical and Experimental Results 5.3.1 Caisson Breakwaters Comparisons of experimental data with numerical results are shown in Figures 5.12 through 5.15. In general the numerical results model the data trends observed in the laboratory, but with mixed results. Agreement between numerical and measured values is reasonably good for both transmission coefficients (Figure 5.12) and heave response (Figure 5.14 ), but inconclusive for the sway and roll response (Figures 5.13 and 5.15 respectively). For the former the agreement is best at the higher wave frequencies (ka > 0.8), while the latter data show an almost frequency-independent behaviour (One exception to this is the high measured sway response under low-amplitude waves at ka = 0.97 (Figure 5.13) — however, it is unclear whether this is a spurious data value associated with errors in the motion measurements). For both the high- and low-amplitude waves, the transmission coefficients show a rapid decline for frequencies above ka > 0.7 1. As expected, the breakwater is more effective at attenuating the higher frequency waves. In general the breakwater effectiveness is lost as the frequency of incident waves decreases much below ka = 1. 1Some of the low-frequency C% data collected are greater than unity, which is clearly an unreasonable result. This effect is probably due to limitations of the wave basin and measurement devices, and is discussed further in Chapter 6 Chapter 5. RESULTS Transmission Coefficients for Caisson C o m p a r i s o n of N u m e r i c a l , E x p e r i m e n t a l , and F ie ld data 1.2 -i 1.1 -1 -0.9 -a 0.8 -o s 0.7 -u o 0.6 -o a o 0.5 -• tn n s 0.4 -in a « 0.3 -0.2 -0.1 -0.2 N u m e r i c a l resul t s + Measured (High) o Measured (Low) a F i e ld (Nece et al) x F i e l d (Nelson et al) 1 1.4 1.8 2.2 Frequency Parameter (ka) 3.4 Figure 5.12: Transmission coefficients for rectangular caisson breakwater Sway Response for Caisson Breakwater o. o 3 c o a 4.5 4 3.5 H 3 2.5 -2 1.5 -1 0.5 0 0.2 Numerical and Experimental Results N u m e r i c a l Results (Undamped and SX d a m p i n g ahown) E x p e r i m e n t a l Results X High Amplitude V Low Amplitude 0.8 1.4 — I — 1.8 2.2 2.6 Frequency Parameter (ka) Figure 5.13: Sway response for caisson breakwater Chapter 5. RESULTS 51 Figure 5.14: Heave response for caisson breakwater Figure 5.15: Roll response for caisson breakwater Chapter 5. RESULTS Rectangular Caisson Breakwater 52 Numerical Results o at •o G e 1 1.4 1.0 Frequency Parameter (ka) 2.6 Figure 5.16: Response Amplitude Operators (RAO's) for caisson breakwater Since a is the half-beam of the breakwater and k = the approximate region of effectiveness for a floating breakwater is when ^ > £ . Thus the width of the breakwater must be at least one third of the incident wavelength for any reasonable attenuation, irrespective of the motions of the breakwater. There appears to be local minima in Ct in the vicinity of ka = 1.0 for both the numerical and experimental results. Inspection of Figure 5.16 shows that these minima occur approximately midway between the resonant peaks associated with heave motions (ka « 0.8) and with sway and roll motions (ka « 1.1). Furthermore, there is a rise in the Ct curves in the vicinity of 0.9 < ka < 1.4 associated with resonant roll and sway motions. Since the transmitted wave is composed partly of waves generated by the oscillations of the breakwater, it is apparent that these "self-generated" waves follow a transfer-function relationship associated with the dynamic behaviour of the body. This means that the design of a given breakwater section should avoid any resonant conditions not only from structural damage considerations, but also from a Chapter 5. RESULTS 53 wave-attenuation viewpoint. That is, designing the breakwater with resonant periods sufficiently longer than that of the design waves will not only avoid large breakwater motions (and the asssociated structural damage), but will result in smaller transmitted wave heights as well. Calculated roll and heave motions are very sensitive to the value of viscous damping coefficient supplied, while sway motions are only slightly affected2. At only 5% viscous damping, the resonant peaks associated with sway, heave and roll motions are all but removed. This is consistent with other published results [see Salveson,1970; Ochi,1976; or Miller and Christensen, 1984] which state that damping near resonant frequencies, particularly in roll, is very important in determining the response of floating bodies. This is also reflected in the experimental results since once the resonant peaks are removed, calculated roll and sway motions are essentially frequency independent. 5.3.2 A-frame Breakwater The experimental results for the A-frame breakwater are shown in Figures 5.17 through 5.20 and indicate patterns similar to those observed for the caisson section. The trans-mission coefficient shows a strong frequency dependence over most of the range tested, as well as local maxima associated with resonant motions of the breakwater. The results are shown for various pontoon widths and indicate that the ka parameter properly ac-counts for the wavelength-breakwater size relationship in that no significant differences in the (dimensionless) response are apparent for the various cases studied. The application of the Green's Function approach was not successful in solving for the flow potential for the A-frame breakwater. The boundary-element approach described in Chapter 3 was found to lead to inconsistent numerical results in the vicinity 2except near resonance, 1.1 < ka < 1.3 Chapter 5. RESULTS 54 A-Frame Breakwater (Low Waves) T r a n s m i s s i o n Coeff ic ient vs Frequency  1.5 -o cient Ct 1.4 -1.3 -1.2 -1.1 -• A * + + + m • + o A a / h a / h a / h a / h = 1.29 = 1.15 = 1.00 = 0.86 lission Coeffii 1 -0.9 -0.8 -0.7 -0.6 -o • * * • • + fc VI • 0.5 -Trail 0.4 -0.3 -0.2 -% o " o • + • O A • 0.1 -0 - 1 • -I • • i , 1 1 1 r-0 2 4 6 Frequency Parameter (ka) Figure 5.17: Transmission coefficients for A-frame breakwater Q. O 3 C o a 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A - F r a m e Breakwater Sway Response — Low Waves  • a / h = 1.29 + a / h = 1.15 o a / h = 1.00 & a / h = 0.86 A0 o 2 4 Frequency Parameter (ka) Figure 5.18: Sway response for A-frame breakwater Chapter 5. RESULTS 55 a, o 3 a o a. 1.5 1.4 -1.3 1.2 1.1 -1 -0.9 -0.8 -0.7 -0.6 0.5 0.4 -0.3 -0.2 -0.1 0 A - F r a m e Breakwater Heave Response for Low Waves • a / h = 1.29 •* a / h = 1.15 o a / h = 1.00 * a / h = 0.86 2 4 Frequency Parameter (ka) Figure 5.19: Heave response for A-frame breakwater o, o •a 3 a a a. 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A—Frame Breakwater Roll Response for Low Waves - a / h = 1.29 + a / h = 1.15 o a / h = 1.00 ^ a / h = 0.86 2 4 Frequency Parameter (ka) Figure 5.20: Roll response for A-frame breakwater Chapter 5. RESULTS 56 of the vertical plate, and the difficulty was not resolved. Despite the inclusion of terms to deal with the numerical singularities associated with the plate, the flow field was incorrectly reproduced and apparently predicted an identical velocity potential on both sides of the plate. Although the plate was modelled as having negligible thickness, it nonetheless should act as a physical barrier to the transmission of wave energy since the boundary conditions state that there can be no water-particle movement passing through the plane of the plate. The result was two points in space having the same potential but different boundary conditions (due to two different outward-facing normal vectors), a result for which no solution could be found. This difficulty has not been resolved, but a more careful analysis should be able to provide more appropriate results. 5.3.3 Summary From comparisons of the experimental and numerical results, it appears that the effec-tive viscous damping coefficient for the caisson section studied is in the order of 2 — 5% of critical damping (similar estimates can not be made for the A-frame section since the numerical results are not available). Although this is low in terms of typical structural dynamics [9], it is clear that even a small amount of viscous damping dramatically alters the predicted response of the breakwater. Viscous effects should therefore be included in the motion analysis of floating breakwaters, particularly when the wave frequencies are near the resonant frequencies of the section. Chapter 6 D I S C U S S I O N A N D C O N C L U S I O N S 6.1 Field Data The limited field data presented in [33,35] show that better agreement with theory is obtained for the wind-wave data than for the ship waves. Although the data are not conclusive, it may be that the wind waves were more representative of steady-state conditions than that of a transient ship wake, and as such are more representative of the conditions modelled by the theory. A typical ship-wake wave trace presented in [33] shows a wave envelope which builds from ambient conditions, peaks, and returns to ambient waves within the space of six or seven waves and over a period of less than fifteen seconds. From observations in the laboratory and of other field situations (such as a boat wake incident upon a floating wharf), it is clear that the first few waves in a wave train tend to break over the relatively motionless body. Since the body has inertia it resists the initial forced motion, and accelerating the body extracts a higher-than-usual proportion of the wave energy. It is only after a few wave cycles that the floating structure starts to reach a uniform motion and the transmitted waves become representative of steady state. Since the entire wave envelope of the tests described by [33] has passed within a few cycles, it is unlikely that the measured response accurately reflected the transmission characteristics of the breakwater. It may be for this reason that the results presented in [33], show poorer agreement with theory than do the steady-state wind data reported 57 Chapter 6. DISCUSSION AND CONCLUSIONS 58 in [35]. 6.2 Numerical Results The numerical results for transmission coefficients show that the calculated wave trans-mission is not dependent on the amount of viscous damping present. Part of the transmitted wave height is due to "self-generated" waves due to the oscillations of the breakwater, and although the presence of damping results in reduced motion am-plitudes (and therefore smaller generated waves) no calculation of the viscous work extracted is made here. Since transmission was calculated on the basis of ideal-fluid behaviour evaluated at the radiation boundary, the wave attenuation due to viscous losses is not entirely accounted for. Clearly in a real situation such losses are impor-tant, since a reduction in the wave height will be proportional to the square-root of the energy dissipated through viscous work1. The numerical approach used was not successful in predicting the flow field in the vicinity of the plate on the A-frame breakwater. As discussed in Chapter 5, an inconsis-tent result was obtained where two points in space had the same velocity potential but algebraically opposite boundary conditions. As a result of this condition, the boundary intregal equation (3.9) applied over the windward and leeward sides of the plate invari-ably evaluated to zero. An attempt to minimize this effect by decreasing the element size was not successful, probably because for all practical element sizes, the plate width is always an order of magnitude (or more) smaller, leading to the same result. This effect has been bothersome in the past where the distributed wave-source method is known to give rise to inaccuracies as the ratio of element size to structure dimension became large. The difficulty is that while the source method predicts zero 1 There is also energy dissipation through turbulence, but this was not considered in the model. Chapter 6. DISCUSSION AND CONCLUSIONS 59 velocites on one side of a boundary, they are generally non-zero on the other side. This is not a problem if the "other side" happens to be within the structure ( it on the side of no hydrodynamic interest), but can cause problems in such "thin" structures as the plate breakwater. An alternate approach using a distributed series of wave doublets has been used by some authors (for example, Naftzger and Chakrabarti [32] ) with rather good results and might be more appropriate for the case considered here. The approach is essentially similar as for the source method except that the doublet does not exhibit the velocity discontinuity across a thin shell. Although diffraction theory has in the past been used with remarkable success for large, "blunt" bodies, the geometry of the A-frame breakwater differs significantly from that of caissons and like bodies. The application of diffracted-potential theory to a situation where there is almost certainly flow-separation may therefore be inappropri-ate. Flow-visualisation in the laboratory revealed that there are vortices shed around the sides and bottom of the A-frame plate with each wave cycle, particularly at the moderate wave frequencies, 0.5 < ka < 1.4. Thus although the Keulegan-Carpenter number K = for the breakwater section as a whole is within the diffraction range, the presence of sharp edges on the plate results in rather high values of K for which flow-separation is important. 6.3 Experimental Results Part of the scatter observed in the experimental results is likely due to the limitations of the measurement techniques and of the laboratory facilities. For example, Figure 5.12 indicates that transmission coefficients greater than unity were observed. Clearly this is impossible since transmitted waves greater than incident waves would require a Chapter 6. DISCUSSION AND CONCLUSIONS 60 net input of energy from the breakwater. In this case the transmitted waves were likely contaminated by waves reflected off the sides and back of the wave basin. A series of tests designed to evaluate the influence of basin reflections was carried out. These tests involved traversing a wave probe from one end of the basin to the other for a variety of wave frequencies, and comparing the recorded heights to those measured by another, stationary probe. The results indicated that there were indeed spatial variations in the apparent wave heights, but these were difficult to isolate since a time-varying response was also observed in the stationary probe. This indicated that some sort of "seiching" was taking place in addition to other basin reflections, particularly at the lower wave frequencies. As a result, the wave heights measured in the experiment, especially at low frequency, were subject to random error on the order of five to ten percent. This is reflected in the low-frequency transmission coefficients greater than unity. The use of the optical motion-measuring system described in Chapter 4 did not allow the isolation of individual breakwater oscillations, but provided amplitudes av-eraged over several cycles. This made the measurements subject to random errors introduced by wave basin reflections and low-frequency oscillations of the breakwater model. The paths of the marker pins were on some occaisions observed to describe a "figure-8" or "S-shaped" pattern rather than the expected elliptical orbits. Conse-quently there sometimes appeared to be two distinct sway or roll amplitudes at the same wave frequency. For these cases the outermost excursions of the markers were chosen to represent the motion. This may be the reason why measured motions ap-pear to be somewhat higher that calculated motions over much of the frequency range studied. A more accurate measurement technique would be to use a video tape of discrete breakwater oscillations, and to digitize the observed marker excursions. Alternately, Chapter 6. DISCUSSION AND CONCLUSIONS 61 a Light Emitting Diode (LED) and camera system could be used in a similar manner to to provide single-cycle motion amplitudes. The use of displacement transducers or accelerometers might also provide better instantaneous measurements. Repeating such tests a number of times would serve to eliminate any random error associated with spurious basin reflections. Bibliography Abramowitz, M. and LA. Stegun 1964. Handbook of Mathematical Functions. Dover Publications, New York. Adee, B.H. 1976. Floating breakwater performance. Proc. Coastal Engineering 1976, pp.2777-2791. Bai, K.J . 1972. A variational method in potential flows with a free surface. Rep. NA72-2,College of Engineering, University of California (Berkeley). Bai, K.J . 1975. Diffraction of oblique waves by an infinite cylinder. Journal of Fluid Mechanics 68, pp. 513-535. Berteaux, H.O. 1976. Buoy Engineering. Wiley Publications, New York. Black, J.L. and C.C. Mei. 1970. Scattering and radiation of water waves. Rept. No. 121, Water Resources and Hydrodynamics Laboratory, Dept. of Civil Engineering, MIT, Cambridge, Mass. Bolton, W.E. and F. Ursell. 1973. The wave force on an infinitely long circular cylinder in an oblique sea. J. Fluid Mech. 1973, Vol. 57, pp. 241-256. Carver, R.D., 1979. Floating breakwater wave-attenuation tests for East Bay Ma-rina, Olympia Harbour, Washington. Tech. Rept. HL-79-13, U.S. Army Waterways Experiment Station, Vicksburg, Mississippi, Aug.,1979. Clough, R.W. and J. Penzien. 1975. Dynamics of Structures. McGraw-Hill Book Co. New York. Dean, D.L. 1962. Static and dynamic analysis of guy cables. Transactions of the ASCE, Vol. 127, Part II, pp. 382-402. Department of Fisheries and Oceans. Waves recorded off Tofino (file 103).1981. And Waves recorded off Lund, B.C.(file 117-2M).1978. Marine Environmental Data Service (MEDS) report. Ottawa, Ontario. Department of Fisheries and Oceans. 1987. 1988 Canadian Tide and Current Ta-bles / Tables des marees et courants du Canada. Information and Publications Branch, DFO, Ottawa. Evans, D.V. and Morris, C.A.N. 1972. The effects of a fixed vertical barrier on obliquely incident surface waves in deep water. J. Inst. Math. Appl. Vol. 9, p. 198-213?. Frank, W. Oscillations of cylinders in or below the free surface of deep fluids. Rept. No. 2375, Naval Ship Research and Dev. Center, Washington, p. 45. 62 Bibliography 63 [15] Fraser, G. 1979. Dynamic Response of Moored Floating Breakwaters. M.A.Sc. The-sis, Dept. of Civil Engineering, University of British Columbia, Vancouver, B.C. [16] Garrison, C.J. 1969. On the interaction of an infinite shallow-draft cylinder oscil-lating at the free surface with a train of oblique waves. Journal of Fluid Mechanics, Vol. 39, pp. 227-255. [17] Garrison, C.J. 1984. Interaction of oblique waves with an infinite cylinder. Applied Ocean Research Vol. 6 No. 1, pp. 4-15. [18] Hales, L.Z., 1981. Floating breakwater: state-of-the-art literature review. 7V. 81-1, U.S. Army Coastal Engineering Research Center, C E , Fort Belvoir, Va. Oct. 1981. [19] Hogben, N., J. Osborne, and R.G. Standing. 1974. Wave loading on offshore struc-tures - theory and experiment. Proc. Symp. Ocean Eng. National Physical Labo-ratory, London, RINA, pp. 19-36. [20] Hsu, F.H. and K.A. Blenkarn. 1970. Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas. Proc. Offshore Technology Conference, Houston, Paper No. OTC 1159, Vol. I, pp. 135-146. [21] Ijima, T.,C.R. Chou and A. Yoshida. 1976. Method of analysis for two-dimensional water wave problems.Proceedings, Coastal Engineering 1976. Honolulu, pp. 2717-2736. [22] Irvine, H.M. 1981. Cable Structures. MIT Press, Cambridge, Mass. [23] Isaacson, M. , and O.U. Nwogu. 1987. Wave loads and motions of long structures in directional seas. Journal of Offshore Mechanics and Arctic Engineering. Vol. 109, No. 2, pp. 126-132. [24] Kim, W.D. 1965. On the harmonic oscillations of a rigid body on a free surface. Journ. Fluid Mech. Vol. 21, pp. 427-451. [25] Levine, H. 1965. Scattering of surface waves by by a submerged cylinder. J. Math. Phys. Vol. 6, 1231. [26] Lebreton, J.C. and A. Margnac. 1966. Traitement sur ordinateur de quelques prob-lemes concernant Taction de la houle sur les corps flottants en theorie bidimen-sionnelle. Bull. Du Centre de Reherches et D'essais de Chatou. No. 18. [27] Leonard, J.W., M.-C. Huang, and R.T. Hudspeth. 1983. Hydrodynamic interfer-ence between floating cylinders in oblique seas. Applied Ocean Research. Vol. 5, No. 3, pp. 158-167. [28] MacCamy, R.C. 1964. The motions of cylinders of shallow draft. J. Ship Research. 7(3), pp. 1-11. [29] McCartney, B.L. 1985. Floating breakwater design.Journ. Waterway Port Coastal and Ocean Eng. Vol. I l l , No. 2. March 1985. pp. 304-318. Bibliography 64 [30] Miller, R.W., and D.R. Christensen 1984. Rigid body motion of a floating break-water. Proc. Coastal Engineering 1984- pp.2663-2679. [31] Ministry of Tourism. Highway on the sea. Beautiful British Columbia Magazine Special Publication. (Undated) Victoria, B.C. [32] Naftzger, R.A., and Chakrabarti, S.K. 1975. Wave forces on a submerged hemi-spherical shell. Proc. Civil Engineering in the Oceans III, ASCE, Univ. of Delaware, pp. 959-978. [33] Nece, R.E.,and N.K. Skjelbreia 1984. Ship-wave attenuation tests of a prototype floating breakwater.Proc. Coastal Engineering 1984, pp.2515-2529. [34] Nelson, E.E. , D.R. Christensen, and A.D. Schuldt. 1983. Floating breakwater pro-totype test program. Proc. Coastal Structures '83. ASCE, Mar. 1983, pp. 433-446. [35] Nelson, E.E. , and L.L. Broderick. 1984. Floating breakwater prototype test pro-gram. Proc. 41st Meeting of the Coastal Engineering Research Board, U.S. Army Coastal Engineering Research Center, May 1984. [36] Niwinski, C T . 1982.Nonlinear Wave Forces on Floating Breakwaters. M.A.Sc. Thesis, Dept. of Civil Engineering, University of British Columbia, Vancouver, B.C. [37] Nwogu, O.U. 1985. Wave Loads and Motions of Long Structures in Directional Seas. M.A.Sc. Thesis, Dept. of Civil Engineering, University of British Columbia, Vancouver, B.C. [38] Pattison, K. 1978. Milestones on Vancouver Island. Milestone Publications, Vic-toria, B.C. [39] Porter, W.R. 1960. Pressure distribution, added-mass and damping coefficients for cylinders oscillating in a free surface. Contract No. N-ONR-222(30), Series No. 82, Issue No. 16, Inst. Eng. Research, Univ. of California (Berkeley), California. [40] Rowland, D.E. 1985. Component Yaw-Normalized Incident Celerities in Seismic Sea Measurements. Unpubl. Rep., Dept. Fisheries and Oceans. Vancouver, B.C. [41] Sarpkaya and Isaacson 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold Co. Ltd. New York NY. [42] Sorenson, R.M. 1978. Basic Coastal Engineering. Wiley-Interscience. New York. [43] Silvester, R. 1974. Coastal Engineering I. Generation, Propagation, and Influence of Waves. American Elsevier Public Commission Inc. New York, NY. [44] Ursell, F. 1949. On the heaving motion of a circular cylinder on the surface of a liquid. Quart. J. Mech. Appl. Math. £,pp. 218-231. [45] US Army Corps of Engineers. 1984.Shore Protection Manual, 4^ Ed. Coastal Engineering Research Center, Vicksburg, Mississippi. Bibliography 65 [46] Vugts, J.H. 1968. The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. Netherlands Ship Research Centre TNO, Rept. No. 112S, May 1968. [47] Watmough, D. 1984. Pacific Yachting's Cruising Guide to British Columbia, Vol IV. Special Interest Publications, Maclean-Hunter Ltd., Vancouver, B.C. [48] Western Canada Hydraulics Laboratory. 1981. Development of a Manual for the Design of Floating Breakwaters. Can.MS Rep.Fish.Aquat.Sci. 1629: 228pp. [49] Wolferstan, B. 1987. Cruising Guide to British Columbia Vol. I : Gulf Islands. Whitecap Books Ltd. North Vancouver, B.C. Canada. Appendix A Site photographs and marine charts The following appendix contains site photographs and corresponding marine chart rep-resentations for selected floating breakwater sites in British Columbia. The source of the charts is indicated as the Canadian Hydrographic Services Chart Number (CHSC nnnn), where nnnn is the chart number. Except where indicated, North is toward the top of the chartlet. The maximum fetch and its direction is given in Table 1.1. Where the caption states "View to dir", it is an indication of the direction dir in which the photo (not the chartlet) was taken. 66 Appendix A. Site photographs and marine charts Figure A.21: Traditional rubble-mound breakwater at Westview (View to N) (Note the large-profile design, particularly at low tide) Appendix A. Site photographs and marine charts Figure A.22: Former A-frame breakwater at Lund, removed in 1987 (View to (Photograph courtesy of [48]) Appendix A. Site photographs and marine charts 69 Figure A.23: Concrete caisson at Deep Cove (View to SW) Appendix A. Site photographs and marine charts Figure A.24: Caisson at Horseshoe Bay (View to SE) Appendix A. Site photographs and marine charts Figure A.25: Detached caissons at Lund (View to West) Figure A.26: Caisson at Nanaimo Yacht Club (View to East) Appendix A. Site photographs and marine charts Figure A.27: Caisson at Maple Bay (View to SE) Appendix A. Site photographs and marine charts 74 Figure A.28: Floating log bundle at Reed Point Marina (view to N W ) . (Note ship wave diffraction/reflection at right) Appendix A. Site photographs and marine charts Figure A.29: Log Bundle at Fanny Bay (view to NE) Appendix A. Site photographs and marine charts Figure A.30: Log raft at Ford Cove, Hornby Island (view to NE) (Photograph from [49]) Appendix A. Site photographs and marine charts Figure A.31: Log raft at Prince Rupert (view to N) (photograph from [31]) Figure A.32: Pontoon/tire breakwater at Eagle Harbour (view to SW) (Tires submerged beneath view) Appendix A. Site photographs and marine charts Figure A.33: Barge breakwater at Sunset Marina, Howe Sound (view to W) Appendix A. Site photographs and marine charts Figure A.34: Breakwater of scrapped ship hulls at Powell River (view to Appendix A. Site photographs and marine charts Figure A.35: Floating railway tank cars at Brown Bay (view to W) Figure A.36: Log breakwaters at Northwest Bay (top)and Becher Bay (bottom) Figure A.37: Log bundles at Pt. Browning and Bedwell Harbour, Pender Island Appendix A. Site photographs and marine charts Figure A.38: Log breakwater at Sooke Basin (top) Caisson at CFSA, Esquimault (bottom) Figure A.39: Log bundle breakwaters at Kelowna (top)and Tahsis (bottom) Appendix B Experimental Facilities Figure B.40: Wave basin showing A-frame model tests. Vertical wall serves to isolate transmitted wave probe from breakwater reflections. (Direction of wave travel is into page) 86 Figure B.41: Experimental data collection instrumentation Oscilloscope shows incident and transmitted wave profiles respectively. 

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