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A study of wave-induced forcing and damage of rock armour on rubble-mound breakwaters Cornett, Andrew Malcolm 1995

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A S T U D Y O F W A V E - I N D U C E D F O R C I N G A N D D A M A G E O F R O C K A R M O U R O N R U B B L E - M O U N D B R E A K W A T E R S by A N D R E W M . C O R N E T T B .A .Sc , Queen's University, 1983 M . A . S c , University of British Columbia, 1987 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Civ i l Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRIT ISH C O L U M B I A December 1995 © Andrew M . Cornett, 1995 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department Of C i v i l Engineering The University of B r i t i s h Columbia Vancouver, Canada Date .) C ^ ^ o r t - g Abstract This study investigates the relationship between the wave-induced forcing and the resulting damage of rock armour on rubble-mound breakwaters, and the manner in which these processes are influenced by changes in wave height, wave period, core permeability and slope angle. Experiments with physical models of various rubble-mound breakwaters have been carried out at the National Research Council in Ottawa, and at B.C. Research in Vancouver. Some of these featured simultaneous measurement of incident wave conditions, velocities on the surface of the test structure, wave-induced forces acting on sections of the armour layer, and the growth of damage to the armour. Different structures were investigated which allow for an assessment of the effects of core permeability and slope angle on the processes leading to damage. The influences of wave height and wave period are investigated using measurements obtained in a range of long-crested regular waves and irregular sea states. Results on the regular and irregular wave conditions required to initiate damage are com-pared to predictions from the design equations of Hudson and of van der Meer. Different estimates of the wave-induced forcing required to initiate damage are derived by considering the balance of driving and resisting forces on a single armour stone at the threshold of motion in five different failure modes. The relative stability of the armour in each failure mode is quantified in terms of a dimensionless failure index. These expressions of relative stability are used to compare measurements of wave-induced forcing to observations of damage in regular waves. Initiation of damage is found to be closely linked to the peak shear stress acting on the armour layer in the down-slope direction. Furthermore, the magnitude of the shear stresses ii on the armour can be related to the slope-parallel velocity using the same wave friction factor developed for oscillatory flow over rough seabeds. Several different aspects of the wave-induced forcing of the armour layer are investigated in detail. Analysis of the vertical distribution of the peak horizontal forces indicates that the most dangerous forces occur below the still waterline. The temporal variations of the forcing at this critical elevation depend strongly on the type of wave breaking that occurs on the slope. Under plunging breakers, the strongest forces result from the sudden flow reversal that occurs under the steep wave crest. Under surging breakers, the largest forces result from seepage flows that occur towards the end of the downrush phase of the surface flow cycle. Collapsing breakers are particularly damaging to the armour layer because these two forcing mechanisms tend to occur simultaneously. Armour stones tend to be more stable on structures with greater permeability because of a reduction in the shear stresses acting on the surface of the armour layer. The forces generated by the individual waves in an irregular wave train are considered and found to be highly varied. While much of this variability can be attributed to differences in the height and period of each wave, some of it is due to additional factors including differences in the shape of each wave and the sequencing of successive waves. A n analysis of the contributions due to these additional factors is presented which indicates that on milder slopes, the most dangerous irregular waves will likely feature a large upcrossing wave height and a deep trough and will likely follow a wave with a much shallower trough. From the perspective of a coastal engineer faced with the challenge of designing a rubble-mound structure, the most important conclusion of this study is that the results and analysis presented herein support the design equations recently proposed by van der Meer. These equations include the effects of wave height, wave period, slope angle, permeability and storm duration in a manner that is generally consistent with the findings of this study. This endorse-ment is based on a combined assessment of the wave-induced forcing of the armour and the damage response. However, this endorsement must be supplemented by the recommendation that van der Meer's equations should not be applied at very low damage levels. i i i Contents Abstract ii List o f Tables ix List o f Figures xi List o f Symbols xx Acknowledgment xxviii 1 Introduction 1 1.1 Literature Review 3 1.1.1 Wave-Induced Flows on Rubble-Mound Breakwaters 4 1.1.2 Forcing of Armour 7 1.1.3 Damage of Armour 8 1.2 Objectives 12 2 Experiments 15 2.1 Scale Effects . 16 2.2 Exploratory Tests 18 2.3 N R C Tests 20 2.3.1 Lay-out 20 2.3.2 Test Structures 23 2.3.3 Data Acquisition 29 2.3.4 Measurement of Waves and Waterline Motion 31 iv 2.3.5 Measurement of Velocity • • • 31 2.3.6 Measurement of Forces 33 2.3.7 Dynamometer Calibration 38 2.3.8 Resolving the Fluid Force 38 2.3.9 Separation of Hydrodynamic Forces 43 2.3.10 Measurement of Damage 47 2.3.11 Wave Synthesis 50 2.3.12 Test Conditions and Procedures 52 2.3.13 Wave Reflections 56 3 Damage of Rock Armour 58 3.1 Design Equations 59 3.1.1 Governing Variables 60 3.1.2 Design Equation of Iribarren 62 3.1.3 Design Equation of Hudson 63 3.1.4 Design Equation of Losada and Gimenez-Curto 67 3.1.5 Design Equation of Hedar 69 3.1.6 Design Equation of van der Meer 71 3.2 Initiation of Damage 74 3.2.1 Regular Waves 76 3.2.2 Irregular Waves 81 3.3 Influence of Wave Height. 87 3.4 Influence of Wave Period 90 3.5 Influence of Slope Angle 90 3.6 Influence of Core Permeability 92 3.7 Influence of Storm Duration 93 v 4 Fa i lure M e c h a n i s m s for A r m o u r Stones 96 4.1 Forces on Armour Stones 96 4.2 Failure Mechanisms for Armour Stones 99 4.2.1 Lifting Failure Mode 99 4.2.2 Hudson Failure Mode 101 4.2.3 Sliding and Rolling Failure Modes 102 4.2.4 Shields Failure Mode 106 4.3 Relation of Forces on an Armour Panel to Forces on Individual Stones 113 4.3.1 Forces 113 4.3.2 Failure Indices 115 4.4 Wave-Induced Forcing Required for Damage 119 4.4.1 Calculation of Failure Indices 119 4.4.2 Calculation of Damage Values 124 4.4.3 Failure Indices and Damage 124 5 Surface F lows and W a v e Forces 136 5.1 Wave-Induced Surface Flows 138 5.1.1 Description 139 5.1.2 Types of Wave Breaking 140 5.1.3 Effect of Breaker Type on Waterline Motions 142 5.1.4 Effect of Breaker Type on Kinematics 144 5.1.5 Peak Slope-Parallel Velocities 150 5.2 Vertical Distribution of Peak Horizontal Wave Forces 158 5.2.1 Influence of Wave Height 162 5.2.2 Influence of Wave Period 164 5.3 Temporal Character of Wave Forces 166 5.3.1 Influence of Wave Height 171 vi 5.3.2 Influence of Wave Period 174 5.3.3 Influence of Slope Angle 178 5.3.4 Influence of Core Permeability' 181 5.4 Character of the Maximum Wave Force 185 5.4.1 Magnitude of the Maximum Wave Force 186 5.4.2 Phase of the Maximum Wave Force 194 5.4.3 Direction of the Maximum Wave Force 197 6 W ave - Ind u ced Stresses 199 6.1 Temporal Variations of Velocity, Shear and Normal Stress 201 6.2 Peak Shear Stress 206 6.3 Peak Normal Stress 212 6.4 Friction Factors 216 6.4.1 Wave Friction Factor 216 6.4.2 Friction Factors for Rock Armour 219 7 I r regular W a v e Effects 229 7.1 Irregular Waves in Design Formulae 230 7.2 Wave Characteristics and Force Quantities for Individual Irregular Waves . . . . 232 7.2.1 Definition of Wave Parameters 232 7.2.2 Detection of Individual Cycles of Surface Flow 234 7.2.3 Force Quantities for Individual Waves . 234 7.3 Distributions and Extreme Values 236 7.3.1 Distribution of Wave Height 236 7.3.2 Distribution of Waterline Height 236 7.3.3 Distribution of Peak Hydrodynamic Force 241 7.3.4 Distribution of Shields Failure Index 245 7.3.5 Extreme Values 247 vii 7.4 Variability of Irregular Wave Forcing 252 7.4.1 Repeatability of Measurements 257 7.5 Indicators of Damaging Waves 258 7.5.1 Wave Potential Energy 259 7.5.2 Wave Shape and the Sequencing of Successive Waves 263 8 Conclusions 274 8.1 Surface Flows 275 8.2 Wave-Induced Forcing 276 8.3 Wave-Induced Damage 278 8.4 Irregular Wave Effects 279 8.5 Applications to Numerical Modelling 280 8.6 Summary of Principal Conclusions 281 8.7 Recommendations for Further Study 283 Bibliography 286 viii List of Tables 2.1 Characteristics of test structures E l , E2 and E3 19 2.2 Salient characteristics of test series 2 through 6 29 2.3 Coordinates of instrumented reactions for the upper and lower armour panels. . 41 2.4 Summary of test conditions 54 3.1 Parameters for the stability design equation of Hedar (1986) 70 3.2 Wave height multipliers for different damage levels (from SPM-84) 89 3.3 Influence of slope angle on initiation of damage wave height for regular waves. . 91 3.4 Influence of slope angle on initiation of damage wave height for irregular waves. 92 3.5 Influence of permeability on initiation of damage wave height for regular waves. 93 3.6 Influence of permeability on initiation of damage wave height for irregular waves. 93 5.1 Elevation parameters for test series 2, 3, 5 and 6 160 7.1 Hi/n/Have for irregular waves on the series 3, 4 and 5 rubble-mounds 249 7.2 HVsi/n/HVstave for irregular waves on the series 3, 4 and 5 rubble-mounds. . . . 249 7.3 FXjnlFave for irregular waves on the series 3, 4 and 5 rubble-mounds 250 7.4 Rs,i/n/Rs,ave for irregular waves on the series 3, 4 and 5 rubble-mounds 250 7.5 Normalized maximum values of H, HVs, F and Rs for irregular waves on the series 3, 4 and 5 rubble-mounds 250 7.6 Linear regression statistics between F and H for test series 3, 4 and 5 255 ix 7.7 Linear correlation coefficients between Rs and selected wave parameters (upper panel, test series 3, 4 and 5) 261 7.8 Linear correlation coefficients between T and selected wave parameters for indi-vidual irregular waves (upper panel, test series 3, 4 and 5) 272 x List of Figures 2-1 Typical layout for the exploratory tests 19 2-2 Basin lay-out for the N R C experiments 21 2-3 Typical test site lay-out for the N R C experiments 22 2-4 Grading curves for the armour, filter and core materials 24 2-5 Permeameter test data fitted to the Forchheimer equation 26 2-6 Permeability coefficients for the armour and permeable core as a function of hydraulic gradient 27 2-7 General view of the breakwater test sections (test series 2) 30 2-8 Sketch of the series 4 rubble-mound 32 2-9 Sketch of the series 5 rubble-mound 32 2-10 Photograph of the armour panels 34 2-11 View of the custom-built dynamometers mounted to a channel side-wall 36 2-12 Removal of inertial forces using a 10 Hz low-pass filter 39 2-13 Definition sketch for resolution of armour panel loads 39 2-14 Sketch of the fluid, hydrodynamic and buoyancy forces on an armour panel. . . . 44 2-15 Separation of hydrodynamic and buoyancy forces 45 2-16 Profiles of the series 3 rubble-mound with SA = 7.2 49 2-17 Selected J O N S W A P wave spectra 52 2-18 Reflection analysis of an irregular wave test (test series 4, Hs = 14 cm, Tp = 2 s). 57 xi 3-1 Solutions to Iribarren's design equation with K = 0.015 63 3-2 Solutions to Hudson's design equation for rock armour with KD = 2 and 4. . . . 64 3-3 Sketch of plunging, collapsing and surging breakers 68 3-4 Solutions to the design equation of Losada and Gimenez-Curto (1979) 69 3-5 Solutions to the design equation of Hedar (1986) for permeable and impermeable structures (0 = 45°, hb = 1.28i76) 71 3-6 Values of the coefficient P recommended for various structures (from van der Meer (1988)) 72 3-7 Photograph of the initiation of damage condition (5 = 1) 77 3-8 Damage curve for regular waves (test series 4, T = 2 s) 79 3-9 Regular wave height required to initiate damage (cot a = 1.75) 79 3-10 Effect of slope angle on the regular wave height required to initiate damage. . . 80 3-11 Damage curve for irregular waves (test series 3, TP = 2.5 s) 82 3-12 Growth of damage for an exploratory test (structure E2, TP — 2 s) 83 3-13 Significant wave heights required to initiate damage on permeable-core structures compared to predictions of van der Meer's design formulae with 5 = 1 84 3-14 Significant wave heights required to initiate damage on permeable-core structures compared to predictions of van der Meer's design formulae with 5 = 2 85 3-15 Significant wave heights required to initiate damage on permeable-core structures compared to predictions from Hudson's design formula extended to account for irregular waves and storm duration 86 3-16 Significant wave heights required to initiate damage on impermeable-core struc-tures compared to predictions of van der Meer's design formulae with 5 = 2. . . 87 3-17 Significant wave heights required to initiate damage on impermeable-core struc-tures compared to predictions of Hudson's design formula extended to account for irregular waves and storm duration 88 3-18 Growth of damage with storm duration for irregular waves 95 xii 4-1 Definition sketch of the forces acting on an armour stone 97 4-2 Definition sketch of forces for rolling failure 103 4-3 Contour plot of the rolling-sliding failure index (cot a = 1.75, <f> = 40°) 105 4-4 Contour plot of the rolling-sliding failure index (cot a = 3, cj> = 40°) 106 4-5 Definition sketch for Shields failure on a horizontal surface 107 4-6 Definition sketch for Shields failure on an incline 109 4-7 Variation of the Shields failure index ((f) = 40°, n = 0.4 and $ = 0.05) 112 4-8 Time series of forces and failure indices under regular waves (upper panel, test series 3, H = 13 cm, T = 2 s) 121 4-9 Time series of forces and failure indices under regular waves (upper panel, test series 4, H - 13 cm, T = 2 s) 122 4-10 Time series of forces and failure indices under regular waves (upper panel, test series 5, H = 14cm, T = 2 s) 123 4-11 Relationship between damage and Rg (upper panel, test series 4) 125 4-12 Relationship between damage and Rs (lower panel, test series 4) 126 4-13 Relationship between damage and Rs (upper panel, test series 3, 4 and 5). . . . 127 4-14 Calibration of the Rs to damage (upper panel, test series 3, 4 and 5) 129 4-15 Relationship between damage and RR (upper panel, test series 3, 4 and 5). . . . 131 4-16 Relationship between damage and RH (upper panel, test series 3, 4 and 5). . . . 132 4-17 Relationship between damage and RL (upper panel, test series 3, 4 and 5). . . . 133 4- 18 Relationship between damage and RE (upper panel, test series 3, 4 and 5). . . . 134 5- 1 Height of waterline motions in test series 3, 4 and 5 144 5-2 Waterline motion and velocities under plunging breakers 146 5-3 Waterline motion and velocities under surging breakers 146 5-4 Waterline motion and velocities under near-collapsing breakers 147 xiii 5-5 Peak slope-parallel uprush velocities at point 1 under regular waves in test series 4 and 5 151 5-6 Peak slope-parallel uprush velocities at point 2 under regular waves in test series 4 and 5 152 5-7 Influence of significant wave height on peak uprush velocities at point 1 153 5-8 Influence of irregular wave period on peak uprush velocities at point 1 154 5-9 Peak slope-parallel downrush velocities at point 1 under regular waves in test series 4 and 5 155 5-10 Peak slope-parallel downrush velocities at point 2 under regular waves in test series 4 and 5 156 5-11 Influence of significant wave height on peak downrush velocities at point 1. . . . 157 5-12 Influence of irregular wave period on peak downrush velocities at point 1 158 5-13 Influence of wave height on the vertical distribution of the peak horizontal forces (cota = 3, T = 3s) 162 5-14 Influence of wave height on the vertical distribution of the peak horizontal forces (cot a = 1.75, T = 2 s) 163 5-15 Influence of wave period on the vertical distribution of peak horizontal forces (cot a = 1.75, H = 15 cm) 165 5-16 Influence of wave period on the vertical distribution of peak horizontal forces (cot a = 3, H = 15 cm) 165 5-17 Waterline motion and hydrodynamic forces under regular waves with £ = 2.9 (test series 3, upper panel, H — 13.4 cm and T — 1.5 s) 168 5-18 Influence of wave height on the hydrodynamic forcing of the armour (lower panel, series 4, T = 2 s) 172 5-19 Influence of wave height on the hydrodynamic forcing of the armour (upper panel, series 5, T = 1.5s) 173 xiv 5-20 Waterline motion and hydrodynamic forces under regular waves with £ = 3.9 (test series 3, upper panel, H = 13.3 cm and T = 2 s) 174 5-21 Waterline motion and hydrodynamic forces under regular waves with £ = 5.9 (test series 3, upper panel, H — 13.1 cm and T = Ss) 175 5-22 Influence of wave period on the hydrodynamic forcing of the armour (upper panel, series 3, H ~ 13 cm) 177 5-23 Waterline motion and hydrodynamic forces under regular waves with £ = 2.2 (test series 5, upper panel, H = 14cm and T — 2s) 179 5-24 Influence of slope angle on the hydrodynamic forcing of the armour (upper panel, # ~ 1 4 c m , T = 2s) 180 5-25 Waterline motion and hydrodynamic forces under regular waves with £ = 4 (test series 4, upper panel, H = 12.8cm and T = 2s) 182 5-26 Influence of core permeability on the hydrodynamic forcing of the armour (upper panel, T = 2 s, H ^ 13 cm, £ = 4) 183 5-27 Influence of core permeability on the hydrodynamic forcing of the armour (lower panel, T = 2 5, H ~ 13 cm, £ = 4) 184 5-28 Influence of core permeability, wave height and period on peak hydrodynamic force (upper panel, test series 3 and 4) 187 5-29 Influence of slope angle, wave height and period on peak hydrodynamic force (upper panel, test series 3 and 5) 188 5-30 Influence of surf similarity on the relation between peak hydrodynamic force and regular wave height (upper panel, test series 3, 4 and 5) 189 5-31 Influence of significant wave height, core permeability and slope angle on F1/3 (Tp = 2 s) 190 5-32 Relationship between peak hydrodynamic force and va (upper panel, test series 3, 4 and 5) 193 xv 5-33 Relationship between peak hydrodynamic force and v (upper panel, test series 3, 4 and 5) 194 5-34 Influence of surf similarity on the phase of the peak hydrodynamic force (upper panel, test series 3, 4 and 5) 195 5-35 Phase of the peak hydrodynamic force on the upper and lower panels (test series 4) 196 5- 36 Influence of wave steepness on the direction of the peak hydrodynamic force (upper panel, test series 3, 4 and 5) 197 6- 1 Shear and normal stresses on the lower armour panel and slope-parallel velocity at point 2 under plunging breakers (test series 5, H = 16.5 cm, T = 1.5 s) 203 6-2 Shear and normal stresses on the lower armour panel and slope-parallel velocity at point 2 under surging breakers (test series 4, H = 16.6 cm, T = 3 s) 204 6-3 Shear and normal stresses on the lower armour panel and slope-parallel velocity at point 2 under near-collapsing breakers (test series 5, H = 16.6 cm, T = 3 s). . 204 6-4 Peak-to-peak shear stress from test series 3 (cot a = 1.75, impermeable core). . . 206 6-5 Peak-to-peak shear stress from test series 4 (cot a = 1.75, permeable core). . . . 207 6-6 Peak-to-peak shear stress from test series 5 (cot a = 3, impermeable core) 207 6-7 Influence of core permeability on peak down-slope shear stresses (upper panel, test series 3 and 4) 209 6-8 Influence of core permeability and slope angle on significant peak-to-peak shear stresses (upper panel, test series 3, 4 and 5) 211 6-9 Influence of surf similarity and core permeability on peak normal stress (upper panel, test series 3 and 4) 212 6-10 Influence of surf similarity and slope angle on peak normal stress (upper panel, test series 3 and 5) 213 xvi 6-11 Influence of surf similarity and elevation on peak normal stress (upper and lower panels, test series 3) 214 6-12 Influence of core permeability and slope angle on significant peak normal stresses (upper panel, test series 3, 4 and 5) 215 6-13 Comparison of predictive equations for wave friction factor 218 6-14 Orbital amplitude Reynolds number at velocity measurement point 2 221 6-15 Comparison of measured peak-to-peak friction factors and predictions of fw from Kamphuis (1975) 224 6-16 Comparison of measured uprush friction factors to predictions of fw from K a m -phuis (1975) 225 6-17 Comparison of measured downrush friction factors to predictions of fw from Kamphuis (1975) 225 6-18 Comparison of measured significant peak-to-peak friction factors to predictions of fw from Kamphuis (1975) 226 6- 19 Comparison of measured and predicted values of fmi/20 227 7- 1 Definition sketch for zero-crossing analysis of waves 232 7-2 Time series of rj (t), r)s (t), F (t) and Rs (t) in irregular waves (upper panel, test series 5, Hs = 14 cm, Tp = 2 s) 235 7-3 Comparison of measured wave heights to the Rayleigh distribution (test series 5, Hs = 14 cm, Tp = 2 s) 237 7-4 Comparison of measured wave heights and waterline heights to the Rayleigh distribution (test series 5, Hs = 14 cm, Tp = 2s) 239 7-5 Comparison of measured wave heights and waterline heights to the Rayleigh distribution (test series 3, Hs = 13 cm, Tp = 2 s) 240 7-6 Modified-Rayleigh distributions with b = 0.7, 1.0 and 1.5 .243 xvii 7-7 Hydrodynamic force peaks compared to the Rayleigh and modified-Rayleigh dis-tributions (upper panel, test series 5, Hs = 14 cm, Tp = 2s) 244 7-8 Cumulative distributions of wave heights and force peaks compared to the Rayleigh distribution (upper panel, test series 3, Hs = 13 cm, Tp = 2s) 244 7-9 Wave heights and peak values of the Shields failure index compared to the Rayleigh distribution (test series 5, upper panel, Hs = 14 cm, Tp = 2 s) 246 7-10 Wave heights and peak values of the Shields failure index compared to the Rayleigh distribution (test series 3, Hs = 13cm, Tp = 2s). . 246 7-11 Peak hydrodynamic force versus wave height for individiual irregular waves (up-per panel, test series 3) 253 7-12 Influence of surf similarity on the variation of wave forcing due to individual irregular waves (upper panel, test series 5) 256 7-13 Repeatability of irregular waves and wave-induced forcing (test series 5) 257 7-14 Peak value of the Shields failure index versus wave height for individual irregular waves (upper panel, test series 5) 260 7-15 Peak value of the Shields failure index versus potential wave energy for individual irregular waves (upper panel, test series 5) 261 7-16 Peak value of the Shields failure index versus average potential wave energy for individual irregular waves (upper panel, test series 3) 262 7-17 Peak value of the Shields failure index versus average potential wave energy for individual irregular waves (upper panel, test series 4) 263 7-18 Failure surface Rs (H,T) under regular waves (upper panel, test series 5) 265 7-19 r versus H for individual irregular waves (upper panel, test series 5) 267 7-20 r versus T for individual irregular waves (upper panel, test series 5). . . . . . . . 268 7-21 Horizontal asymmetry as an indicator of damaging irregular waves (upper panel, test series 5) 270 xvii i 7-22 Difference in upcrossing and downcrossing wave height as an indicator of dam-aging irregular waves (upper panel, test series 5) 270 7-23 Horizontal asymmetry as an indicator of damaging irregular waves (upper panel, test series 5, Hi > 15 cm) 271 7-24 Difference in upcrossing and downcrossing wave height as an indicator of dam-aging irregular waves (upper panel, test series 5, Hi > 15 cm) 272 xix List of Symbols A Ae Aa a ac ax D Dis D&5 D% Dn Dn50 E[] e e c erf[] area eroded area in cross-section accreted area in cross-section amplitude of water particle displacement amplitude of wave crest amplitude of wave trough reflection coefficient diameter particle diameter exceeded by 85 % by weight of a sample particle diameter exceeded by 15 % by weight of a sample damage as defined in the "Shore Protection Manual" (CERC, 1984) nominal diameter = (M/pa)1^3 nominal diameter = (Mso/pa) 1 ^ 3 expected value of [ ] position vector = (ex,ey,ez) elevation of the breakwater crest error function of [ ]_ F, F' xx F, F' hydrodynamic force magnitude Fb, Ft buoyancy force magnitude Ff, F< fluid force magnitude FH, F'H horizontal component of hydrodynamic force FN, F^ slope-normal component of hydrodynamic force FP, F'p slope-parallel component of hydrodynamic force Fc, F'c contact force FSw, Fsw submerged weight F F' weight Fdr, F'dr driving force F F' resisting force Fr Froude number f frequency f friction factor fd friction factor for downrush fpp friction factor for the complete flow cycle fu friction factor for uprush fw wave friction factor 9 acceleration due to gravity H wave height Have average wave height HD downcrossing wave height Hi height of an individual irregular wave HS significant wave height = #1/3 HID initiation of damage wave height H\/N average height of the highest one-n t h of all waves HVs height of waterline excursions h water depth i hydraulic gradient KD damage coefficient in the Hudson equation k coefficient of permeability for laminar internal flow k equivalent coefficient of permeability for turbulent internal flow A;s Nikuradse roughness L j reaction in the j t h flexure = (LjtX,Ljiy,LjiZ) LQ deep-water wave length I length M moment = ( M x , MY, MZ) M mass M [ ] most probable value of [ ] MA moment about point A M50 average mass of a sample of stones N number Na number of displaced stones Nw number of waves n porosity = Vv/Vr h local porosity near an armour panel P permeability coefficient in van der Meer's design equations P-w total potential wave energy qs rate of sediment transport R, R' generic failure index R'ave spatially averaged value of R' 'roll failure index for the rolling failure mode xxii failure index for the sliding failure mode RE empirical failure index RH, R'H failure index for the Hudson failure mode RL, R'L failure index for the lifting failure mode RR, R'R failure index for the rolling-sliding failure mode Rs, R'S failure index for the Shields failure mode Re Reynolds number Re* grain size Reynolds number Repp Reynolds number for flows on a rubble-mound r linear correlation coefficient rJ position vector for the jth flexure •= (^j,x,^j,y,^j,z) S damage level SA damage level from the eroded area = Ae/D^50 SN damage level from the number of displaced stones Sv damage level from visual assessment Srj wave spectrum T wave period TC period of wave crest Td downcrossing wave period Ti period of an individual irregular wave Tm average zero-crossing wave period wave period corresponding to the peak of the wave spectrum TT period of wave trough t time ta thickness of the armour layer thickness of the filter layer xxii i u component of velocity parallel to the surface u* shear velocity VT total volume W volume of voids Vw average potential wave energy v vertical velocity of waterline motion v bulk velocity of internal flow v velocity of water in a wave crest va slope-parallel velocity of waterline motion w component of velocity perpendicular to the surface wc width of breakwater crest x, y, z Cartesian coordinates Xt position of the breakwater toe xc position of the breakwater crest z(x) profile of the armour layer zc elevation of the panel centroid a angle of the seaward slope Pi wave skewness /?3 wave atiltness 7 peakedness parameter for the J O N S W A P wave spectrum 6 displacement thickness of a boundary layer 77 water surface elevation rjs elevation of the waterline on the structure surface fjs runup fjs rundown A scale factor xxiv A wave vertical asymmetry p wave horizontal asymmetry p mean value v kinematic viscosity of water £ surf similarity parameter for regular waves = tan a^JgT2/2TTH £ m surf similarity parameter for irregular waves = ta.nay/gT2l/2irHs p density of water pa density of armour stone a normal stress a standard deviation r shear stress To bed shear stress r c critical shear stress (p natural angle of repose r normalized peak failure index value for an individual irregular wave r [ ] gamma function of [ ]. A relative density = (pa — p) jp AH percentage difference in wave height = 100 • (H — iJpreutous) /H AT percentage difference in wave period = 100 • (T — T^-e^ous) jT AHu-d percentage difference in upcrossing and downcrossing height AT u_rf percentage difference in upcrossing and downcrossing period 9 [ ] direction of [ ] $ [ ] phase of [ ] ^ critical value of Shields parameter xxv General Operators A V Subscripts ave b C cr » d f ID i i max min model pp panel previous prototype Ray time-average first derivative with respect to time positive peak value negative peak value non-temporal average pertaining to wave breaking pertaining to the crest critical value pertaining to downrush final pertaining to the initiation of damage initial pertaining to an individual irregular wave maximum value minimum value pertaining to the model peak-to-peak value pertaining to the panel pertaining to the previous flow cycle pertaining to the prototype pertaining to the Rayleigh distribution xxvi pertaining to the trough pertaining to uprush significant value the average of the largest one-nth of all values Superscripts ' pertaining to an individual armour stone l/n xxvii Acknowledgment This thesis is the result of research conducted while the author was employed by the National Research Council of Canada. The author wishes to thank the Council for granting him the time, facilities and resources required to complete this study. The author also expresses his sincere thanks to the many individuals who assisted him in this work. The support of the entire staff of the NRC's Coastal Engineering Program in Ottawa is gratefully acknowledged. In particular, the author wishes to thank Dr. M . Davies, Mr. E. Funke, Dr. B.Pratte, Dr. E. Mansard and Dr. O. Nwogu for their contributions to this study, and Mr. B. Ai tk in , Mr . C. Hermann, Mr. D. Pelletier for their excellent technical support during the experimental work. The assistance of Mr. S. K i m is also acknowledged with appre-ciation. Part of this research was performed at B. C. Research in Vancouver. The author gratefully acknowledges the support of the staff at the Ocean Engineering Centre, and especially the assistance of Mr . M . Shaver, during this phase of the study. The author would like to thank the members of his supervisory committee, and in particular, Dr. M . Isaacson, for their guidance throughout this research. Finally, the author wishes to thank his family for their unwavering support and encourage-ment throughout the course of this study. xxvii i Chapter 1 Introduction A n armour layer can be described as a protective covering made up of individual units which rely on their mass and geometry for stability. Armour layers are used to protect many different types of coastal structures, including breakwaters, jetties, groins, and shoreline revetments, from destruction under wave attack. Armour layers are most commonly comprised of angular rock, but in situations where particularly large units are required, specially shaped concrete units are sometimes used. Riprap is relatively lightweight broadly graded rock armour that is typically used in situations where wave attack is not severe. The primary role of an armour layer is to dissipate large amounts of wave energy while protecting the overall integrity of a coastal structure. Wave energy is dissipated through a combination of turbulent wave breaking, flow over the rough surface of the armour layer, and internal flows between armour units. The character of the incident wave and the geometry of the structure have an important influence on the type of wave breaking that occurs, which in turn affects the flow over the surface of the armour layer and the energy dissipation. The permeability of the armour and underlying layers influences the dissipation of energy through the internal flow mechanism. Wave-induced flows on the surface and within the armour layer exert temporally and spatially varying hydrodynamic forces on the individual armour units. Damage to the armour layer occurs when the wave-induced forcing is sufficient to displace 1 armour units so that they no longer perform their protective role. Rubble-mound structures designed according to conventional practice generally feature a uniformly sloping outer surface armoured with relatively large units designed to sustain only minimal damage under very severe wave attack. The armour layer is typically at least two units thick and is placed over one or more graded filter layers. In the case of rubble-mound breakwaters, the core of the structure can be either permeable or impermeable, depending on the size and grading of the material used. Armour layers for rubble-mound structures are currently designed using a combination of semi-empirical design formulae, experience, and hydraulic model tests. Design formulae are typically used for preliminary design, including comparisons of various alternative geometries and armouring schemes. For important projects, hydraulic model tests are used to optimize and verify the performance of the favored design, and often lead to more economic design. Equations for preliminary armour design provide semi-empirical relationships between sim-ple wave characteristics such as wave height and in some cases wave period, basic properties of the structure such as the slope angle, and the size of armour units required for minimal damage. Design equations in use today are only very loosely based, if at all, on the physics behind the complex process whereby wave attack results in damage to an armour layer. Hydraulic model tests provide a means to simulate the detailed physics of this process, and are therefore strongly recommended for final design by design guides such as the Shore Protection Manual (CERC, 1984) prepared by the U.S. Army Coastal Engineering Research Center, and the Report of Working Group 12 (PIANC, 1992) prepared by the Permanent International Association of Navigation Congresses. Rubble-mound breakwaters designed according to existing practice have suffered numerous failures in recent years. Some of the most noticeable failures are discussed in P I A N C (1985) and in Bruun (1985). The prevalence of damaged armour layers and the semi-empirical nature of current design formulae suggest that there is considerable need for improved design methods that are based on the fundamental physics governing the response of armour layers to wave 2 attack. The overall process whereby wave attack causes damage to an armour layer can be separated into four sub-processes. 1. Wave propagation onto the structure and wave breaking. 2. Fluid flows on the surface and within the permeable zones of the structure. 3. Hydrodynamic forcing of the units in the armour layer driven by the wave-induced flows. 4. Displacement of armour stones due to the wave-induced forcing. Within the last 10 years, various numerical models have been developed that simulate the interaction between waves and armoured coastal structures. These numerical models use mathematical expressions to simulate wave propagation, wave breaking, flows on the surface and internal flows. In general, the accuracy of the simulation depends on the validity of the mathematical expressions adopted to model the physical processes, and on the proficiency of the numerical solution scheme. The pace of development has been rapid and if it continues unabated these numerical tools may soon be used to guide the design of armoured coastal structures. At present, approximate methods are used to simulate wave breaking and to estimate the forcing and response of armour units. Further development is hampered by a lack of quantitative information on, and physical understanding of the various sub-processes identified above. 1.1 Literature Review Historically, a considerable majority of the studies on rubble-mound structures have attempted to identify the size of armour units required to resist displacement (damage) under wave attack. These studies were used to develop semi-empirical relationships between a few of the most important parameters. While these studies provided a general qualitative understanding of the process whereby wave attack leads to armour damage, quantitative information on the details of this process are not well known. In the following, a brief review of the literature relevant to 3 this thesis is presented in three sections: the wave-induced flows on rubble-mound breakwaters; the forcing of armour due to these flows; and the damage of armour resulting from wave attack. 1.1.1 Wave-Induced Flows on Rubble -Mound Breakwaters Battjes (1974) showed that the type of wave breaking that occurs on various slopes can be described as a function of the surf similarity parameter £ = tana-y/'gT2/2-KH', which represents a ratio between the slope angle a and the square-root of wave steepness, where H is the wave height, T is the wave period and g represents the acceleration due to gravity. Plunging breakers prevail for steeper waves on milder slopes (£ < 2), while surging breakers prevail for less steep waves on steeper slopes (£ > 4). Collapsing breakers represent a transitional form of wave breaking between these two forms. Bruun and Giinbak (1978), Giinbak (1979), Sawaragi et.al.(1982, 1983) and Bruun (1985) applied the term "resonance" to describe conditions on a steep slope when the period of incident waves equals the combined duration just required for a full cycle of uprush and downrush. These authors indicate that resonance conditions tend to occur in relatively shallow water with waves for which 2 < £ < 3. Moreover, resonance conditions were associated with maximum flow velocities on the surface of the slope, suggesting that they thus represent the most dangerous conditions for armour units. Runup and rundown, denned as the maximum and minimum elevations respectively of the waterline on the surface of a slope under wave attack, are the characteristics of the surface flow that have been studied most intensively. Giinbak (1979), Losada and Gimenez-Curto (1981), Ahrens and Titus (1985), Ahrens and Heimbaugh (1988), and van der Meer (1993) have all used the surf similarity parameter to successfully describe runup and rundown on various smooth and rough slopes due to regular or irregular waves. On smooth slopes, runup is largely controlled by surf conditions, slope inclinations and wave nonlinearities. On rough permeable slopes, runup can be dramatically attenuated through the dissipation of wave energy due to friction against, and flow through the permeable armour layer. 4 Observations of flow velocities on rubble-mound slopes are less common. Sawaragi, Ryu and Iwata (1983) made Lagrangian measurements of velocity on the surface of armoured slopes under regular wave attack and found that the strongest slope-parallel velocities occurred over the range 2 < £ < 3. More recently, T0rum and van Gent (1992) and Tjzsrum (1994) recorded Eulerian velocities on the surface of a berm breakwater in regular waves and made comparisons to predictions from numerical simulations and to force measurements obtained on a single unit of rock armour. Lagrangian measurements of velocity during downrush flows on a slope armoured with Accropods (a type of concrete armour unit) are reported by Mani et. al. (1994). Experimental, numerical and field studies of the kinematics within waves breaking at constant depth or over smooth mild slopes are more common, but results from these studies cannot be directly applied to wave breaking on steep armoured slopes. A number of numerical models have been developed to simulate the interaction of waves with armoured slopes. At first, these were generally restricted to simulations of regular waves on impermeable slopes that were not too steep. More recently, simulations of irregular waves on steep permeable slopes have been obtained. One-dimensional time domain models solved by finite difference techniques have been described by Kobayashi et. al. (1986, 1988, 1990a, 1990b, 1990c, 1992), Thompson (1988), Allsop et. al. (1988), van Gent (1992), and Wurjanto and Kobayashi (1993). These models use approximate methods to treat plunging and collapsing wave breaking. Kobayashi and Otta (1987) developed a theoretical model for the hydraulic stability of armour stones in which hydrodynamic forces are represented by a combination of drag, inertia and lift forces. This theoretical model was combined with subsequent flow models by Kobayashi et. al. to provide estimates of the stability threshold for armour stones. Two-dimensional models for flows on smooth impermeable slopes are reported by Gri l l i and Svendsen (1989), Svendsen and Gri l l i (1990), and Chian and Gerritsen (1990). Two-dimensional models for flows on rough permeable slopes are described by Sakakiyama and Kaj ima (1992), Sun, Williams and Allsop (1992) and van der Meer et. al. (1992). Sakakiyama and Kaj ima (1992) presented numerical estimates of the hydrodynamic forces acting on armour units. 5 Hall and Hettiarachchi (1992) provide a review of the techniques commonly used to model flows in porous media, including internal flows within permeable rubble-mounds. Other articles that focus on internal flow processes include Hannoura and McCorquodale (1985a, 1985b), Barends (1986), Barends and Holscher (1988), Hall (1989) and van Gent (1993). These works describe the internal flows within rubble-mounds, but do not specifically address the issue of the forcing or stability of armour units on the surface of the armour layer. The model of van Gent (1992) is typical of the one-dimensional finite difference models for computing wave action on and in coastal structures. In van Gent's model, the fluid flow outside the structure is computed using a one-dimensional approach with depth-averaged ve-locities and hydrostatic pressures. A breaking wave is modelled as a bore. Flows inside the structure are computed using a similar one-dimensional depth-averaged approach developed for porous media. The internal and external flow models are coupled to each other at the sloping porous boundary that represents the surface of the structure. Special boundary conditions are developed at this interface to accommodate inflows and outflows and allow for a discontinuity in the internal and external free surfaces. The governing equations are solved with an explicit second-order method (Lax-Wendroff) using a constant grid space. At each time step, the ex-ternal flow model is computed first to obtain pressure gradients at the porous interface. The new pressure gradients are used to update the internal flow model, which responds by comput-ing a new interface discharge that is implemented in the external flow model at the next time step. A constant friction factor is used to parameterize the energy dissipation due to surface roughness. Dissipation in the internal flow model is treated using the unsteady Forchheimer equation. Although this model does not compute any vertical velocities directly; vertical veloc-ities are estimated based on the relative slopes of the free surface, the impermeable interface, the impermeable bottom and the variation of the free surface with time. In this way, a two dimensional impression of the flow is obtained from a one-dimensional model. This model was not extended to compute the forces acting on the armour or to predict the stability of armour stones. 6 1.1.2 Forcing of Armour Sigurdsson (1962) measured the forces exerted by regular waves on a single spherical armour unit placed at various locations on an idealized rubble-mound breakwater section. The idealized section was constructed entirely from identical spheres, arranged in three layers. The hydraulic forces, including contributions due to buoyancy and fluid flow, were correlated to wave char-acteristics and aspects of the breakwater geometry. Sigurdsson showed that the wave-induced forcing was very complex, and concluded that the most dangerous forces occur under the toe of an advancing breaker or when water is flowing out of the breakwater. Jensen and Juhl (1988) and Juhl and Jensen (1989,1990) measured wave forces due to regular and irregular waves on an highly idealized two-dimensional breakwater with an armour layer consisting of two rows of horizontal pipes. A simple stability model was used to express the measured forces in terms of the weight of armour that would be required to resist roll-up or roll-down. They found that the forcing most critical to stability occurred below the still waterline in waves for which £ was between 2 and 4, corresponding to the transition between plunging and surging breakers. Several studies of wave-induced forcing of concrete armour units have recently been reported. In the U.K., research has been focussed on a family of concrete armour units, collectively known as hollow block units (of which the Cob and Shed units are specific examples), that are designed to be placed in a regular close fitting single layer. Hettiarachchi and Holmes (1988) measured hydraulic forces due to regular waves on individual Cob and Shed units at small scale. Separate laboratory experiments and field studies of the hydraulic forces exerted by regular and irregular wave attack on Cob and Shed units are reported by Allsop, Herbert and Davis (1991) and by Herbert and Waldron (1992). For this type of armour, wave forces are seldom large enough to move a unit out of the armour layer, but failure can occur through breakage of the unit. Several notable breakwater failures have been caused by, or at least exacerbated by, struc-tural failures of concrete armour units (Bruun, 1985). In response to these failures, several stud-7 ies have been performed to measure stresses within individual concrete armour units. Studies with dolos (a type of concrete armour unit) at model scale are reported by Anglin et. al. (1988), Scott et.al . (1988), Anglin, Scott and Turcke (1990), Scott, Turcke and Baird (1990), and Burcharth et. al. (1990). Burger, Smidt and Partenscky (1992) measured stresses in tetrapods (another type of concrete armour unit). Stresses within instrumented dolos in a breakwater at Crescent City, California are reported by Howell (1988), Howell Burcharth and Rhee (1990) and Kendall and Melby (1992). These studies show that different mechanisms act to produce significant stresses within concrete armour units. These mechanisms include settlement, tides, fluid flows due to waves, wave impacts, and impacts against other armour units. In general, the peak stresses caused by fluid flows due to waves (without the static and impulsive compo-nents) are highly scattered, but are shown to increase with increasing wave height and reach a maximum for surf similarities between 2.5 and 4. T0rum (1994) measured the forces exerted by regular waves on a single unit of rock armour located at a single position below the still waterline on the surface of a reshaped berm break-water. The slope-parallel and slope-normal force components were analysed as Morison type forces (Morison et. al., 1950) using velocities measured in the "free" flow above the instrumented unit. Drag and inertia force coefficients were presented for several different wave conditions. T0rum concluded that the slope-parallel forces were drag dominated, and that the slope-normal forces were not lift dominated, and could not be well represented by the assumed Morison force formulation. T0rum comments that, "a Morison force formulation based on the "free" flow on a breakwater may not give a good description of the forces." No attempt was made to relate the wave-induced forcing to wave characteristics, structure geometry, or the stability of the armour. 1.1.3 Damage of Armour One advantage of rubble-mound structures is that failure of the armour layer generally occurs as a gradual process that is partial in extent and spread out over the duration of a storm. When damage does occur, the structure generally continues to function and can be repaired as 8 required after the storm abates. Raichlen (1974) provides an excellent review of earlier work on the effects of waves on rubble-mound structures. Iribarren (1938) proposed a design formula for armour stones under wave attack that includes five fundamental parameters: wave height H; water density p; slope angle a ; the density of the armour pa; and the mass of armour units M required for no damage. A coefficient K was used to account for the behaviour of different types of armour. Hudson (1958) proposed a similar formula M = ^ 3 (1.1) KD(^-l) cot a that has been, and still is extensively used in practice. Hudson's formula includes the same five parameters considered by Iribarren, as well as a damage coefficient Kr>- Numerous experiments with regular waves on various structures have been used to establish appropriate values of KQ for a wide range of conditions. References to these studies and appropriate values of Kr> for many different types of armour can be found in the "Shore Protection Manual" (CERC, 1984). Factors affecting the performance of the armour that are taken into account through the damage coefficient include: shape of armour units; thickness of the armour layer; manner of placing armour units; surface roughness and angularity of armour units; and type of wave attack (breaking or non-breaking). In spite of its extensive use in practice, the Hudson formula has a number of shortcomings. The original formula, and the variety of Kr> values recommended in the Shore Protection Manual, were developed on the basis of tests with mono-chromatic (regular) waves, whereas waves in nature are random (irregular). It also does not include the effects of wave period, storm duration, or permeability. Several different modifications have been proposed to adapt Hudson's equation for use with irregular waves. The most recent edition of the Shore Protection Manual (CERC, 1984) rec-ommends that Hi/io, equal to the average height of the highest 1 0 % of waves, be used in place 9 of H to account for the effects of irregular waves. A previous edition of the Shore Protection Manual (CERC, 1977) had recommended that the significant wave height Hs equal to the av-erage height of the highest one-third of all waves be used in place of H. Medina and McDougal (1988) suggested a modification that accounts for the combined effects of irregular waves and storm duration. Hydraulic model tests with regular waves at relatively large scale by Ahrens (1975) clearly showed a dependence between wave period and armour damage, such that minimum stability occurred in waves that formed collapsing breakers. Losada and Gimenez-Curto (1979) proposed a design formula for regular wave attack that included the influence of wave period through the surf similarity parameter. Their formula predicts minimum stability for values of £ in the range of collapsing breakers. Hedar (1986) presented design formulae for armour stability in which the permeability of the underlayer has a significant influence on the weight of armour units required to resist damage under regular wave attack. Hedar also observed that downrush flow is critical for damage on steeper slopes, while uprush flow is critical on milder slopes. Thompson and Shuttler (1976) conducted systematic experiments on the stability of riprap under irregular wave attack. They identified the significant wave height Hs, slope angle a, and storm duration (parameterized by the number of waves Nw) to be the three most significant parameters influencing the damage of a riprap armour layer. Their data was re-analysed by van der Meer (1988) who showed that wave period also had an influence on the performance of the riprap armour, and that the influence of wave period was similar to that previously obtained for rock armour under regular waves by Ahrens (1975). Van der Meer (1988) presented design equations for rock and riprap armour under irregular wave attack based on an extensive set of experiments, supplemented in part by his re-analysis of the results of Thompson and Shuttler (1976). Separate equations were developed for plunging and surging waves, which in addition to the five fundamental parameters used by Iribarren and Hudson, contain parameters to account for the effects of storm duration, damage level, permeability, and wave period. Van der Meer's 10 formulae have been recommended for initial design with irregular waves by the Permanent International Association of Navigation Congresses (PIANC, 1992). Because of the randomness of waves in nature and the complex transformations that occur through shoaling, it is often difficult to select and specify wave conditions that are most appro-priate for the design of a rubble-mound structure. Historically, random sea states have been specified by a limited number of parameters, most notably the significant wave height Hs and the peak wave period Tp, that do not provide a complete description of the sea state. Many different realizations of irregular wave trains with the same Hs and Tp are possible - and each of these will exert a different sequence of forcing on the armour layer, often resulting in markedly different damage response. Mansard (1990) reviewed some of the concepts recently developed to provide a more complete description of unidirectional random sea states, including wave grouping, wave asymmetries, and wave nonlinearities. The concept of wave grouping describes the temporal variation of wave energy within an irregular wave train. Johnson, Mansard and Ploeg (1978) found that certain sequences of high waves, such as those within well defined wave groups, can cause greater damage to rubble-mound structures than equally high individual waves dispersed throughout a wave train. Medina, Fassardi and Hudspeth (1990) concluded that wave grouping could be responsible for up to 5 0 % of the variability in damage to an armour layer resulting from different realizations of irregular waves drawn from a single spectrum. On the other hand, van der Meer (1988) concluded that wave grouping has only a minor influence on the response of an armour layer. Myrhaug and Kjeldsen (1986) defined several parameters to quantify the steepness and asymmetry of individual waves in the time domain, including the crest front steepness, the crest rear steepness, the vertical asymmetry A, and the horizontal asymmetry fi. Goda (1986) defined a parameter called wave atiltness /?3 to describe the degree of forward tilting of wave profiles. In the design formulae proposed by van der Meer (1988), the effects of irregular waves are parameterized by the significant wave height Ha and the average wave period Tm. Bruun (1990a, 11 1990b) emphasized that these parameters provide only a limited description of the wave condi-tions contained in a random sea state, and that for more detailed design, consideration should be given to the effects of special events embedded in a wave train including groups of waves in "resonance", odd wave geometries like "jumps", and "freak" waves. Qualitative descriptions of the various sequences of waves which may be particularly damaging to an armour layer were previously given by Bruun (1985). In his reply to the discussion of Bruun (1990b), van der Meer re-asserts that for preliminary design, irregular waves can be satisfactorily parameterized by Hs and Tp. Kobayashi, Wurjanto and Cox (1990c) used a limited number of numerical simulations of rock slopes under irregular wave action to examine the profiles of waves that may be most damaging to the stability of armour stones. Their results suggest that the most damaging conditions on mild slopes are caused by uprush from the wave with the maximum crest elevation, while on steeper slopes, minimum stability is caused by the downrush from waves with a high wave crest followed by a deep trough. Moreover, they found that the most damaging wave in a wave train was often not the wave with the largest height. 1.2 Objectives Armour layers for coastal rubble-mound structures are traditionally designed using semi-empirical formulae which provide very simple approximations to the complex process whereby wave at-tack causes damage to the armour. In many cases, these methods have led to unsafe or overly conservative structures. In the future, more reliable designs will likely be achieved with the aid of numerical models which more accurately reproduce the physics controlling the processes responsible for damage. Further advancement of such numerical design tools hinges on devel-oping a better understanding of the key processes that are involved, namely: the fluid flows within, and on the surface of a porous rubble-mound driven by wave attack; the forces acting on the armour units due to these flows; and the response of the armour to this forcing. The 12 main thrust of this thesis is to advance the rational and quantified understanding of these key processes. The primary objective of this study is to measure and analyze the wave-induced hydrody-namic forces acting on rock armour and identify the connections between the characteristics of the incident waves and the forcing, and between the forcing and the damage that occurs. A number of more specific secondary objectives were also established. • Develop new instrumentation to measure the hydraulic forces acting on a patch of rock armour. • Study the spatial and temporal variations of the wave-induced hydrodynamic forcing with emphasis on the critical location where damage starts. • Determine the character (magnitude, direction and phase) of the largest forces. • Develop a rational analysis of the stability of armour stones under wave attack and apply this analysis to determine the component of the hydrodynamic forcing that is responsible for initial damage to the armour layer. • Review several commonly used design equations and assess their rational basis and con-cordance with the processes responsible for wave-induced damage. • Review the characteristics of plunging, surging and collapsing breakers and investigate the influence of breaker type on the forces acting on armour stones. • Study the influences of wave height, wave period, structure slope and structure perme-ability on the surface flow kinematics, the forcing of the armour and the damage that occurs. • Determine whether the concept of wave friction factors can be applied to describe the connection between surface flows and the forcing of the armour layer. 13 • Quantify the variability of the forces due to irregular waves and determine the distribution of the forces and their extreme values. • Identify the characteristics of the most damaging waves in an irregular wave train. In this thesis, consideration is given to the wave-induced forcing and damage of rock armour on rubble-mound breakwaters of conventional design subject to normally incident wave attack. Of course, many real structures incorporate non-conventional design elements such as composite slopes, re-shaping berms or concrete armour units, and most of them are exposed to wave attack from various directions. These additional factors increase the complexity of a process that is already very complex. The fundamental case is considered in this thesis because it has a broad relevance to a wide variety of coastal structures and is slightly more simple to study. By concentrating on the fundamental case, the author does not mean to trivialize the importance of these other issues. Coastal engineers working with real structures must account for the important effects of wave direction and the performance of non-conventional design elements. Rubble-mound structures can fail through a wide range of mechanisms including displace-ments of armour units, structural failures (breakage) of armour units, geotechnical failures (slides), undermining of toe protection, subsidence of the core or foundation, and overtopping. This study is focussed on the displacement of armour units due to unidirectional (long-crested) waves that are normally incident to the trunk section of rubble-mound structures. A joint publication by the International Association for Hydraulics Research and the Per-manent International Association of Navigation Congresses ( IAHR &; P I A N C , 1987) contains a recommended list of notation for sea state parameters. Where possible, the notation of symbols used in this thesis are consistent with the list of sea state parameters proposed therein. It is the author's hope that the results and analysis presented in this thesis will advance the understanding of wave-induced forcing and damage of rock armour and contribute towards the development of more accurate and reliable tools for the design of rubble-mound coastal structures. 14 Chapter 2 Experiments Two separate sets of experiments were conducted as part of this research. The first of these was an exploratory test program conducted at the Ocean Engineering Centre (OEC) of B.C. Research in Vancouver. These tests were the first to be carried out at the O E C using a WM-15 wave machine and a data acquisition system supplied by the National Research Council (NRC). These exploratory tests provided an opportunity to examine the characteristics of regular and irregular shallow water waves incident to several different rubble-mound breakwaters and to evaluate the performance of rock armour subject to a variety of incident wave conditions. Re-sults from this test program contributed to the design of the more comprehensive test program that followed. A second set of experiments were carried out in the 14 m wide by 63 m long by 1.5 m deep Coastal Wave Basin at the National Research Council in Ottawa. In addition to the mea-surement of incident wave conditions and resulting damage, these tests featured simultaneous measurement of the forces acting on rock armour using a pair of instrumented armour panels. The majority of results presented in this thesis were obtained from the second set of exper-iments; however, these are supplemented by data from the exploratory test program where appropriate. 15 2.1 Scale Effects Froude scaling was used to construct geometrically similar models of realistic breakwater sec-tions. Froude similarity requires that the Froude number in which u is a characteristic velocity, / is a characteristic length, and g is the acceleration due to gravity, be the same in the model as in the prototype. The Froude number scale represents the ratio between the prototype and model values of Froude number, which can be written as \FT — -^prototype/.Fr-model- Froude similarity requires that A ^ R — 1 and ensures that the ratio of inertial forces to gravitational forces remains equal in the model and prototype. By setting the Froude number scale equal to unity, and conducting experiments under the same gravitational field that exists in nature, Froude similarity implies that the scales for velocity Xu and length A; are related according to (2.2) Scales for other quantities of interest can be determined as follows. time: Xt = Xi/Xu — density: Xp = 1 since the same materials are used in the model and prototype. mass: XM XpXf — Xf. force: A ^ stress: A T A f / A ? = A , . The Reynolds number ul Re = — v (2.3) 16 where u is the kinematic viscosity of the fluid, represents a ratio of inertial forces to viscous forces. A n ideal model would also preserve the similarity of this ratio; however, under Froude scaling where water is used in both the model and prototype, the Reynolds number scale is distorted according to A* e = ^ = A, 1 5 . (2.4) Under this distortion, viscous forces play a relatively greater role in the model than in the prototype. Physical models of rubble-mound breakwaters must be large enough to minimize the effects associated with improper scaling of Reynolds number. Many authors have examined scale effects on the stability of rock armour. After surveying the work of Dai and Kamel (1969), Thomsen et. al. (1972), Broderick and Ahrens (1982), Jensen and Klinting (1983), S0rensen and Jensen (1985), and Burcharth and Frigaard (1987), van der Meer (1988) concludes that the lowest Reynolds number for which scale effects become negligible can be set in the range Re = V ^ £ > n 5 ° ~ 1 x 10 4 to 4 x 10 4 , (2.5) v where Hs is the significant wave height and D^Q = (Mso/pa)1^ is the nominal diameter of armour stones with median mass M50 and density pa. Results from his own tests at small and large scales confirm this criterion. Scale effects will be most noticeable within porous cores consisting of fine material and small void spaces. As viscous forces become proportionately larger at model scale, greater energy losses will take place during flows through porous core material, and the penetration of water into the core will be reduced. For small models of rubble-mound breakwaters (Re < 1 x 104) these effects tend to reduce the stability of armour stones. Similar armour stones, with D n5o = 0.044 m and 0.042 m were used during the exploratory and N R C experiments. Setting g = 9.81 m/s 2 , u = 1.14 x 10~ 6 m 2 /s, Dn5o = 0.042m and assuming a median value for van der Meer's criterion of Re > 2.5 x 10 4 , the condition for 17 negligible scale effect can be written ) 2 Ha>- = 0.047 m. (2.6) This condition holds for all tests in both experiments, which suggests that the wave-induced forces acting on the rock armour and the armour response to those forces will not be significantly affected by scale effects. 2.2 Exploratory Tests Exploratory tests were conducted in the Coastal and Ocean Wave Basin at the O E C in Van-couver. The WM-15 wave machine was used to generate regular and irregular unidirectional waves in a 15 m wide by 30 m long flume. This flume was longitudinally subdivided by rigid vertical walls into three 5 m wide test channels. Temporary floors with plane slopes of 1:15, 1:20, and 1:25 were constructed in each channel to provide a range of shoaling bathymetries. During testing, the water depth h at the wave generator was 0.80 m. Breakwater test sections were constructed along the centre-line of each channel such that the water depth at the toe of each structure was 0.40 m. The layout for these tests is sketched in Figure 2-1. Three different rubble-mound breakwaters were studied in these exploratory tests. The structures are identified here by the labels E l , E2 and E3. Salient characteristics of these structures are summarized in Table 2.1. The armour for these structures consisted of angular, crushed granite with nominal diameter Dn^Q — 4Acm, density pa = 2680 kg/m? and grading D85/-D15 = 1.05. Individual armour stones were hand-picked and weighed to ensure conformity with a target mass distribution. A narrow grading was used to achieve a nearly homogeneous armour layer. Armour stones were individually placed by hand to simulate the construction of prototype armour by crane. Structures E l and E2 were constructed with an impermeable core covered by a 3 cm thick filter layer with Dn5o = 0.8 cm and D&5/Di5 — 2. The same filter material was used to construct a permeable core for the E3 structure. Seaward slopes of the 18 P L A N V I E W chine slope =1:25 > 5 m — i — i slope =1:20 WM slope = =1:15 30 m rubble-mound S E C T I O N Figure 2-1: Typical layout for the exploratory tests. E l E2 E3 cota 1.5 3.0 1.5 Dn5o armour (cm) 4.4 4.4 4.4 DS5/D15 armour 1.05 1.05 1.05 thickness of armour (cm) 9 9 9 Dn5o filter (cm) 0.8 0.8 -D85/D15 filter 2 2 -thickness of filter (cm) 3 3 -core impermeable impermeable permeable Drtso core (cm) - - 0.8 Dss/Dvs core - - 2 Table 2.1: Characteristics of test structures E l , E2 and E3. 19 E l , E2 and E3 breakwaters are cot a = 1.5, 3.0 and 1.5 respectively, where a is the vertical angle of the surface with respect to horizontal. Structures E l , E2 and E3 were exposed to a variety of regular and irregular waves. Regular waves with periods T = 1.5, 2.0, 2.5 s, and heights between H = 9 and 17 cm were used. Irregular waves were synthesized from J O N S W A P wave spectra using the iterative F F T method described by Funke and Mansard (1984). Peak periods Tp = 1.5, 2.0, 2.5 s and significant wave heights between Hs = 9 and 18 cm were used. During these tests, damage to the armour was measured by counting the number of displaced stones while incident and reflected waves were measured using an array of wave probes deployed in front of the structure. 2.3 N R C Tests A second, more comprehensive set of experiments were carried out at the Coastal Engineering Laboratory of the National Research Council in Ottawa. These experiments were designed to permit simultaneous measurement of incident wave conditions, armour damage, and wave-induced forcing of the armour layer. 2.3.1 Lay-out The N R C experiments were conducted in the 14 m wide by 63 m long by 1.5 m deep Coastal Wave Basin. The general lay-out of the basin is sketched in Figure 2-2. A false floor was constructed within the basin to provide deeper water at the wave machine and shallower water at the test site. The test site was located on a level section of false floor, constructed 45 cm above the original basin floor. The test section was preceded by a plane slope at 1:30 to provide a gradual transition between the two elevations. Three separate test channels, each 65 cm wide, were constructed near the centre of the raised test site. Identical two-dimensional sections of a rubble-mound were constructed in two of these three test channels; the third channel was left open to record incident wave conditions. A porous gravel beach constructed behind the test 20 P L A N V I E W false floor slope 1:30 test channels gravel beach slope 1:15 A B 14 m A jf B 63 m" S E C T I O N A - A a s S E C T I O N B - B I 0 . 8 5 - 1.0 m rubble-mound Figure 2-2: Basin lay-out for the N R C experiments. 21 P L A N V I E W Wave Probes V Armour Panels Waterline Gauge ? 0.65 m Channel 1 Gravel Beach Channel 2 1.2 m 0.65 m - h 1.2 m Gravel Beach Channel 3 0.65 m i _ SECTION, C H A N N E L 2 Window 10m Figure 2-3: Typical test site lay-out for the N R C experiments. site at a slope of 1:15 was found to provide good wave absorption. Figure 2-3 shows a sketch of the three test channels, including a typical arrangement of test structures and instrumentation. Incident waves were measured in test channel 1 using an array of capacitance wave probes. Measurements of the wave-induced forces on a portion of the armour layer were made in the central test channel (channel 2) using a pair of instrumented armour panels, described below in more detail. A second breakwater section, located in channel 3, was used to monitor damage to the armour layer. Wave probes were also located in channels 2 and 3 to monitor wave conditions in front of the two rubble-mound sections. The channel walls were constructed from concrete blocks. A large plexiglas window was installed in channel 3 to permit a side view of the wave interactions with the rubble-mound test section. Adjacent test channels were separated by a 1.3 m wide section of porous gravel beach 22 at a slope of 1:10. This separation was provided to isolate the incident wave to each channel from the effects of reflected waves radiating out of neighbouring channels. The entrance to each test channel was fitted with adjustable wave guides that were calibrated to ensure very similar wave conditions in each channel. This lay-out allows the simultaneous measurement of incident waves, wave loads and breakwater damage, with minimal contamination by reflected waves. 2.3.2 Test Structures Three different rubble-mound breakwaters were tested in the N R C experiments. These struc-tures are similar, but not identical to those used in the exploratory experiments, and are also similar to those studied by van der Meer (1988). The basic structure features an impermeable core covered by a thin filter layer and two layers of armour stones at a slope of cot a = 1.75. This slope was chosen as being representative of steeply sloping structures commonly encoun-tered in practice. The second structure is a variant of this basic structure with an identical composition at a milder slope of cot a = 3. The third structure is a different variant of the basic structure with an identical slope of cot a = 1.75, but a permeable core in place of the filter layer and impermeable core. When the permeable core was tested, the armour layer was placed directly on the core. When an impermeable core was used, the core was covered by a thin impervious rubber sheet. This simulates a core with zero permeability. These three structures offer the opportunity to assess the effects of core permeability and slope angle while representing realistic structures typically used in practice. A l l structures were protected by finely graded armour with Dnzo = 4.2 cm and D^/D\^ = 1.1, constructed using the same stones used in the exploratory tests. A l l armour stones were obtained from a single sample of crushed granite by individual weighing. Only stones with mass between 0.15 and 0 . 2 5 % were retained. Stones with highly eccentric shapes were also discarded. The density of the armour is pa = 2680 kg/m3 and the shape of individual stones can be characterized as angular. The natural angle of repose for the armour was determined by experiment to be 0 ~ 40°. This was determined by placing a sample of armour stone 23 03 00 100 90 80 70 60 2 50 c CU 40 30 20 10 0 Core Stones Armour Stones Filter stones / H 1—l-i f l l I -I 1 1 1 — l l l l H 1 1 — l l l l 10 100 1000 Mass (g) Figure 2-4: Grading curves for the armour, filter and core materials. in a shallow box and gradually increasing the slope until the armour became unstable. The filter material has Dnso = 1.9 cm and D^/D\^ = 1.3, while the permeable core material has Dnso = 2.5 cm and D^/Di^ = 1.3. Mass distribution curves for the core, filter and armour materials are presented in Figure 2-4. The permeability of the armour and the permeable core material was investigated under steady flow using a constant-head permeameter. The internal diameter of the permeameter is 0.305 m, and its internal volume is 0.064 m 3 . The armour was tested at two different packing densities, corresponding to porosities of n = 0.43 and 0.47. The lower porosity is representative of conditions for the armour layer placed on the test structures. A single sample of core material with a porosity of n = 0.47 was tested. This relatively high porosity may be due in part to the narrow gradation of the core material and in particular to the absence of fine particles (see Figure 2-4), and in part to the fact that the sample was not compacted when placed into the permeameter. The behaviour of steady internal flow within soils generally follows Darcy's law, which can 24 be written as 1 (2.7) where v is the bulk velocity of the internal flow, i is the hydraulic gradient driving the flow, and k is the coefficient of permeability for the porous medium. Darcy's law is valid for laminar internal flows which typically prevail in soils. In this regime, A; is a constant that is independent of v or i. In more permeable media, such as the armour, filter and permeable core materials used in rubble-mound breakwaters, the internal flows become turbulent and the relationship between v and i is no longer well described by the linear Darcy law. In this turbulent regime, steady flows within porous media are commonly described by the Forchheimer equation, which can be written as where a and b are coefficients that depend on properties of the material such as the porosity and the size of the void spaces. The linear term in Equation (2.8) represents laminar flow resistance while the quadratic term represents turbulent flow resistance. By analogy to Equation (2.7), an equivalent coefficient of permeability k for steady turbulent internal flows is defined by A; is a variable quantity for non-Darcy flows. Substituting the Forchheimer equation for i, the equivalent coefficient of permeability can be expressed in terms of the bulk velocity and the Forchheimer coefficients as or in terms of the hydraulic gradient as i = av + bv2 (2.8) (2.9) k(i) = 2 (2.11) a + Vo,2 + Abi 25 2.5 a CO •rH T) C O S H O u 1.5 0.5 2 - i = 18.3v • i = O.lv + 29.Ov 2 - i = 37.5v 2 o A r m o u r , n = 0.47 A A r m o u r , n = 0.43 * P e r m e a b l e c o r e / / / / A / / .-cr A' 0.05 0.1 0.15 0.2 b u l k ve loc i ty , v ( m / s ) 0.25 0.3 Figure 2-5: Permeameter test data fitted to the Forchheimer equation. Results of the permeameter tests for each sample are favorably compared in Figure 2-5 to the Forchheimer relationship. The values of the Forchheimer coefficients a and b that best describe the flow characteristics of each sample are shown in the legend to this figure. The dominance of the quadratic term of the Forchheimer equation is a clear indication that turbulent flow prevails within each of these materials for i > 0 . Equivalent permeability coefficients for the armour and permeable core material are shown in Figure 2-6 as a function of the hydraulic gradient. These curves were computed from Equation (2.11) with the Forchheimer coefficients a, b obtained from Figure 2-5. The permeable core material offers slightly more flow resistance than the densely packed armour, which is in turn less permeable than the loosely packed armour. For structures in which an impermeable core was used, k ~ 0 can be taken for the core material. The fractional difference in the permeability coefficients for two porous materials, whose 26 i% 0.8 C o 0.6 0) o 0.4 cs 0.2 a u a 0 w L v. \ \ \ V'-. A r m o u r , n = 0 . 4 7 A r m o u r , n = 0 . 4 3 P e r m e a b l e c o r e 0.25 0.5 0.75 1 hydrau l i c gradient , i ( -) 1.25 1.5 Figure 2-6: Permeability coefficients for the armour and permeable core as a function of hy-draulic gradient. flow resistance is described by a\, b\ and a-z, bi respectively, can be written jfei _ 02 + y/a^ + ^i (2.12) For the materials considered here where a <C fr, this expression can be approximated by for a -C b and i » 0 . (2.13) Substituting &i = 29.0 for the densely packed armour and &2 = 37.5 for the core material gives fci/^2 = 1.14. This result indicates that for the same hydraulic gradient, the armour is only 14 % more permeable than the core material over the range of hydraulic gradient of interest where i 0. It is likely that the permeable core material placed in the test structure had a slightly lower porosity than the sample tested in the permeameter. If so, the core material wil l be slightly 27 less permeable than indicated by the preceding analysis. The extent of this porosity effect is considered in the following. Van Gent (1993) indicates that the Forchheimer coefficient b can be obtained approximately as b = - ? - 1 - ^ (2.14) gDn50 where (3 is a dimensionless coefficient that is independent of porosity. Van Gent reports values of f3 in steady flow between 0.3 and 1.1, but recommends that a value of f3 = 0.6 be used. The rate of change in b with respect to changes in porosity can be obtained by differentiating Equation (2.14) as * _ » f i + m = $ \ . ( 2 . 1 5 ) dn gDn50 \n3 n 4 Evaluated for n = 0.45, this yields db dn = • (2-16) n=0.45 <?Ai50 This equation indicates that if the core material (with -Dn50 = 0.025 m) were placed at a porosity of n = 0.43 instead of n — 0.47 as tested in the permeameter, the Forchheimer coefficient can be expected to increase from b = 37.5 to b = 42.5. According to Equation (2.13), this corresponds to a 6 % reduction in permeability for the more highly compacted core material. Assuming that the permeable core in the test structure was placed with a porosity of n — 0.43, the permeability difference between the armour and the core increases from 14 % to 21 %. These results indicate that the test structure with a permeable core can be considered to be somewhat inhomogeneous with respect to permeability. It should be mentioned that internal flows within a rubble-mound under wave attack are unsteady, and that such unsteady turbulent porous media flows can be modelled using an extended form of the Forchheimer equation that includes an additional term proportional to the bulk fluid acceleration dv/dt. The relative permeabilities of the armour and permeable core material will not likely be significantly different under unsteady flow. 28 Test Series Series 2 Series 3 Series 4 Series 5 Series 6 h (cm) 40 55 55 55 40 cota 1.75 1.75 1.75 3.0 3.0 DnbO armour (cm) 4.2 4.2 4.2 4.2 4.2 D&5/D15 armour 1.1 1.1 1.1 1.1 1.1 thickness of armour (cm) 9 9 9 9 9 Dn5o filter (cm) 1.9 1.9 - 1.9 1.9 D85/D15 filter 1.3 1.3 - 1.3 1.3 thickness of filter (cm) 3 3 - 3 3 core imperm. imperm. perm. imperm. imperm. Dn50 core (cm) - - 2.5 - -zc upper panel (cm) 43.5 43.5 43.5 41.5 41.5 zc lower panel (cm) 31.5 31.5 31.5 33.5 33.5 Table 2.2: Salient characteristics of test series 2 through 6. Both impermeable core structures were tested at two water levels, corresponding to water depths at the structure of 55 cm and 40 cm, while the permeable core structure was tested at a single water depth of 55 cm. Thus, a total of five different combinations of water depth and test structure were investigated. These five combinations will be referred to throughout this thesis as test series 2 through 6. Salient characteristics of these test series are summarized in Table 2.2. (Test series 1 involved calibration of the incident wave conditions to each test channel with no test structures in place.) The photograph in Figure 2-7 shows a general view of the breakwater sections in channel 2 (foreground) and channel 3 (background) during test series 2. 2.3.3 Data Acquisition The data acquisition system converts analog signals from various transducers to digital form for storage in computer data files. The following transducers were sampled during either some or all experiments. • Wave probes at various locations throughout the test area. • Waterline elevation gauge on the surface of the rubble-mound in channel 2. 29 30 • Bi-directional electromagnetic velocimeters positioned above the surface of the rubble-mound in channel 2. • Load cells used within the armour panel dynamometers on the channel 2 rubble-mound. • Displacement and rotation of the electro-mechanical profiler used to measure damage on the test structure in channel 3. A l l transducers were sampled at 20 Hz using a Series 100, 15 bit analog-to-digital converter with optional signal conditioning and amplification, manufactured by N E F F Instruments. Be-fore sampling, the analog signals were low-pass filtered by 3-pole Butterworth filters with a 10 Hz cut-off frequency. Sampling was controlled using the G E D A P package of data-acquisition and experiment control software, developed at the N R C . G E D A P data files were generated and stored on a VAXstation model 3100 computer, manufactured by Digital Equipment. 2.3.4 Measurement of Waves and Waterline Motion Water surface levels were measured using capacitive-wire wave gauges developed at the N R C . These gauges feature an accuracy of ± 1 m m , and were calibrated regularly throughout the test program by changing their elevation with respect to a fixed water level. These gauges exhibited a highly linear response, with calibration errors generally less than 1 % of the 20 cm calibration range. The elevation of the free surface with respect to the still water level is denoted by r)(t). The elevation of the waterline on the surface of the rubble-mound in channel 2 was recorded using a wave gauge inclined parallel to, and located approximately 1 cm above the surface of the structure. This waterline gauge was calibrated by varying the water level within the basin. The vertical elevation of the waterline with respect to the still water level is denoted by rjs (t). 2.3.5 Measurement of Velocity During test series 5 and part of test series 4, bi-directional electromagnetic velocimeters were used to measure velocities at two locations (points 1 and 2) above the surface of the break-31 Figure 2-8: Sketch of the series 4 rubble-mound. Figure 2-9: Sketch of the series 5 rubble-mound. water section in channel 2. Sections of the series 4 and 5 breakwaters, including the locations of velocity measurement, are sketched in Figures 2-8 and 2-9 respectively. The velocimeters were located approximately 4 cm above the surface of the instrumented armour panels, at the positions shown in Figures 2-8 and 2-9. The velocimeter at position 1 was oriented to mea-sure velocity in the plane parallel to the surface of the rubble-mound, while the velocimeter at position 2 was oriented to measure velocity in the vertical plane aligned with the direction of wave propagation. Velocity components parallel and perpendicular to the surface of the rubble-mound are denoted by u(t) and w(t) respectively, defined such that u(t) is positive up-slope, and w(t) is positive away from surface of the structure. Two types of velocimeter were used; one is manufactured by Delft Hydraulics and the other 32 was developed at the N R C . Both instruments use the Faraday principle of electromagnetic induction to measure the velocity of fluid flow. These instruments feature similar performance characteristics, including repeatability of 1% over a calibrated operating range up to 5 m/s. The velocimeters were calibrated by towing through still water at various speeds. Outputs from these velocimeters were low-pass filtered with a 3 Hz cut-off to eliminate spurious high frequency fluctuations. 2.3.6 Measurement of Forces Wave-induced forces were measured on a pair of armour panels, each comprised of 50 irregularly shaped, aluminum model rocks bonded together by spot-welds into a rigid, porous, rectangular mat of armour stones. The model rocks were fabricated in various shapes to simulate the sizes and shapes of the granite armour stones used on the test structures. The model rocks in each panel were randomly placed to closely approximate the packing and porosity of the loose granite armour. Each panel represents a rectangular patch of armour stones in a single layer with approximate overall dimensions 25 cm by 63 cm by 5 cm. Figure 2-10 shows a photograph of the armour panels. The armour panels were installed as part of the surface layer of armour on the various breakwater sections tested in channel 2. They spanned the full width of the test channel, and were invariably positioned immediately adjacent to each other, centered about an elevation 40 cm vertically above the base of the structure. The locations of the armour panels for each test series are summarized in Table 2.2 in terms of the elevation of the centroid zc above the base of the rubble-mound. Four degree-of-freedom measurements of the forces acting on each armour panel were ob-tained by a pair of custom-built load cell dynamometers. Components of the dynamometers were mounted to the outer face of the vertical walls on either side of test channel 2. At this location, the channel side-walls were constructed from reinforced aluminum plate. Each ar-mour panel was connected to its dynamometer at three points by stainless steel pins inserted 33 34 through holes in the side-walls of the test channel. Two connection pins passed through one side-wall while a single pin passed through the opposite side-wall. Adjustment screws, located on the outer face of the channel side-walls, were used to fine-tune the positions of the armour panels relative to each other and to neighbouring armour stones. Five load cells were used in each dynamometer to measure the force acting on an armour panel. Shear beam load cells, manufactured by Interface Instruments, were used. These load cells contain a balanced array of strain gauges that respond to the strains induced by external forces. These load cells feature a combined non-linearity and hysteresis of 0.06 %. Forces were conveyed to the load cells through flexures (long slender steel bars with narrow sections at either end), which effectively isolate the load cells from bending moments. The load cells were protected from water damage by a flexible water-proof coating. Cabling from the load cells was protected by a sheath of plastic tubing. A photograph of the outer face of one of the test channel side-walls taken with the armour panels and dynamometers in place is presented in Figure 2-11. To obtain reliable measurements of the fluid force acting on the armour panels, it was necessary to avoid contact between the panels and the surrounding armour stones. A l l armour stones immediately adjacent to the panels were fixed in place by small quantities of marine epoxy adhesive. These neighbouring stones, including the stones in the lower layer of armour directly below the panels, were arranged so that their irregular contours were matched to the undulations of the panels. The panels were isolated from bordering materials by a thin gap that followed the irregular outline of each panel. The thickness of this gap was controlled, using adjustment screws, to be approximately 2 mm. A l l of the armour stones in the panel were located approximately 2 mm above the position they would have taken if allowed to rest on the lower layer of armour. Away from the armour panels, wire mesh was placed over the surface of the rubble-mound to restrain the motion of loose granite armour stones. This mesh was necessary to prevent stones from tumbling down the rubble-mound onto the armour panels. This set-up was adopted as a trade-off between the need to isolate the panels from surrounding breakwater materials, and the desire to minimize unrealistic distortions to the modelling of the 35 Figure 2-11: View of the custom-built dynamometers mounted to a channel side-wall. 36 rubble-mound breakwater. The effect of the thin gap below the panel on the local permeability of the armour layer can be assessed by comparing the porosity of an armour layer consisting entirely of natural rock armour to one where an armour panel is installed. The porosity of a material is the ratio of the volume of void spaces Vy to the total volume VT- In this study, armour stones were placed by hand at a density corresponding to a porosity of n = Vy/VT — 0.41. The average thickness of the armour layer was determined by profiling the structure before and after placement of the armour to be ta — 0.09 m. The thin irregular gap below the panels represents an additional void space that increases the porosity of the armour layer in the vicinity of the panels. The volume of the gap can be written as (tg/ta) Vp, where the gap thickness is tg ~ 0.002 m. The modified porosity of the armour layer where the panel is installed is f i = V v + (t9/ta)VT (1 +tg/ta) VT [ ] which reduces to h = ! L ^ ± * £ . (2.18) ta "\~ tg Substituting ta = 0.090 m, tg = 0.002 m and n = 0.410 into Equation (2.18) yields ft = 0.423, which represents a 3 % increase in porosity. This difference in porosity is within the range of porosity variations that can result naturally within an armour layer. The effect of changes in porosity on the permeability of the armour can be seen in Figure 2-6. Results from the permeameter tests of the armour at porosities of 0.43 and 0.47 indicate an approximate 25 % increase in permeability for a 9 % increase in porosity. This suggests that the thin gap below the armour panels will increase the permeability of the armour layer by approximately 8 %. Because the additional void spaces due to the gap are inter-connected and principally aligned parallel to the surface, the additional permeability will likely favor internal flow in the direction parallel to the surface of the rubble-mound. 37 2.3.7 Dynamometer Calibration Al l load cells used in the armour panel dynamometers were individually calibrated up to ±229 N using a series of static weights. In all cases, response was linear with errors less than 0.06 % of the calibration range. The performance of the armour panel dynamometers was verified after construction of each test structure using static test loads up to 196 A^, applied at different locations in various directions. Vertical test loads were applied by resting mass on the surface of the armour panel. Non-vertical loads were applied using weights suspended from a cable attached to the panel and looped over a low-friction pulley. Checks were also made by comparing the dynamometer output under submerged and non-submerged conditions. During these checks, the force components measured by the dynamometers were within 2 % of the correct values. The dynamic characteristics of the force measuring system were explored using several free vibration tests, in which a 196 N pre-load was suddenly removed from an armour panel. Load cell responses were initially sampled without filtering at 100 Hz, the maximum frequency offered by the data acquisition system. Spectral analysis of the responses to this step loading indicates fundamental response frequencies near 45 Hz for vibration modes transverse to the plane of the panel. Additional free vibration tests, sampled at 100 Hz, but low-pass filtered at 10 Hz, were conducted to verify the ability of the filter to remove inertial forces associated with free vibrations of the panel. Figure 2-12 demonstrates the effect of the analog 10 Hz filter on the dynamometer measurement by comparing the magnitude of the resultant force measured during similar free vibration tests recorded with and without filtering. Further analysis confirms that low-pass filtering at 10 Hz combined with sampling at 20 Hz effectively isolates the force measurements from inertial forces generated through vibrations of the panel. 2.3.8 Resolving the Fluid Force Figure 2-13 shows a simplified sketch of an armour panel supported at three points, A, B, C. 38 200 150 S 1 0 0 o 50 — i n o f i l t e r 10 H z f i l t e r 1 2 3 4 Time (s) Figure 2-12: Removal of inertial forces using a 10 Hz low-pass filter. -65 cm Figure 2-13: Definition sketch for resolution of armour panel loads. 39 A Cartesian coordinate system is defined at the geometric centre of the panel, point O, such that the x and z axes lie in the plane of the panel and the y axis is normal to it. When the panel is installed on the surface of a rubble-mound, the positive x axis points up-slope parallel to the surface of the structure in-line with the direction of wave propagation, the positive z axis points across-slope parallel to the surface of the structure and transverse to the direction of wave propagation, while the positive y axis points upwards and away from structure, perpendicular to its surface. The panel is supported by seven flexures, represented in Figure 2-13 by arrows numbered 1 through 7. The flexures are designed to convey tensile or compressive forces, but not moments. The external force acting on the panel is sensed by five load cells mounted in line with flexures 1-5. Flexures 6 and 7 are not instrumented with load cells and act purely as lateral restraints. Wi th this arrangement, no measurements were obtained of the forces acting transverse to the direction of wave propagation. The external force acting on the panel at a specific instant is represented by the unknown vector F = (Fx, Fy, Fz) acting through the unknown point P, which lies in the x-z plane. The position vector of P with respect to the origin is denoted by e = (ex, 0, ez). The rotational moments about the origin (point O) due to the external force are given by M = e x F = (ex, 0, ez) x (Fx , Fy , Fz) . (2.19) The reaction in the jth flexure is denoted Lj = (LjtX, LjiV, LjtZ), with j = 1 - 7 for each of the seven flexures. The lateral restraint flexures are oriented parallel to the z axis such that Li6 = (0, 0, L,6<z) and L 7 = (0, 0, £7,2). The position vector of each reaction is denoted by rj ~ (rhx-> rj,y> rj>)> with j = 1-7. For these panels, all points of support lie in the x-z plane, such that rjty = 0 for all j. The coordinates of the instrumented flexures for the two armour panels are summarized in Table 2.3. Resolving the external load involves determining the three components of force F , and the two unknown components of the position vector e. At all times, the forces and moments acting 40 Panel 3 rj,x (cm) rj,y (cm) ri,z (cm) lower 1 -7.62 0 -35.5 lower 2 -7.62 0 -35.5 lower 3 7.62 0 -35.5 lower 4 -2.54 0 35.5 lower 5 -2.54 0 35.5 upper 1 -7.62 0 -35.5 upper 2 7.62 0 -35.5 upper 3 7.62 0 -35.5 upper 4 0 0 35.5 upper 5 0 0 35.5 Table 2.3: Coordinates of instrumented reactions for the upper and lower armour panels, on the panel must sum to zero, which provides the equations F + ^ L ^ O (2.20) 3=1 7 M + Yl (rJ X Li) = °- (2-21) 3=1 Equations (2.20) and (2.21) represent a system of six equations (two for each orthogonal di -rection). However, including the unknown reactions L.6 and L 7 , there are a total of seven unknowns, which prevents a complete solution. The transverse component of the external load Fz, and the rotational moment about the y axis My, cannot be determined without knowing the reactions in the lateral restraint flexures. Solutions for the four remaining degrees-of-freedom can be readily obtained as follows. From Equation (2.20), the force component normal to the plane of the armour panel, and thus normal to the surface of the structure, is given by 7 Fv = -12LJ.v (2-22) 3=1 Fy can be determined because the lateral restraint flexures with j = 6 and 7 do not contribute 41 any force in the y direction. Similarly, the force component parallel to the plane of the armour panel (and parallel to the surface of the rubble-mound) in-line with the direction of wave propagation is given by 7 FX = -J2L3,X. (2.23) 3=1 Equation (2.21) taken in the z direction yields 7 {exFy - eyFx) + {rj,xLjiy - rjtVLjtX) = 0 . (2.24) 3=1 The x-coordinate of the point of action of the external force can be written in terms of known load cell reactions and coordinates as _ ~ Z)J=1 (R3,xLj,y — rj,yLj,x) _ Z)J=1 rj,xLj>y ex - J - — 7 — • {l.Zb) Similarly, the z-coordinate of the point of action of the external force can be written in terms of known quantities as e* - p - — ^ 7 r • Equations (2.25) and (2.26) define the eccentricity of the external force acting on the panel with respect to the geometric centre. ex and ez can only be reliably determined for |Fj,| > 0. The rotational moment about the a; axis is given by 7 MX = eYFZ - eZFY = £ rjiZLjiV , (2.27) 3=1 where ey = 0 and ez and FY are obtained from Equations (2.26) and (2.22) respectively. Like-wise, the rotational moment about the z axis can be obtained from 7 Mz = eXFY - eYFX = - ^ rj>xLj<y . (2.28) 3=1 42 where Equations (2.25) and (2.22) have been substituted for ex and Fy respectively. The equations developed above have been applied at each instant in time to resolve the x and y components of force and the x and z components of moment acting on the armour panels, and to determine the eccentricity of the external force with respect to the geometric centre. The armour panel dynamometer can be described as a two-dimensional force measurement system since it provides information on the forces acting in the two-dimensional vertical plane aligned with the direction of wave propagation. Although individual armour stones may experience some intermittent transverse forcing (in the z direction) under normally incident wave attack, the net effect when spatially integrated over an armour panel will be very small. The important forces in terms of the performance of the armour are those acting in the vertical plane aligned with the direction of wave propagation; that is, in the x and y directions. 2.3.9 Separation of Hydrodynamic Forces The time-varying vector of fluid force Ff(t) acting on an armour panel can be written Ff(t) = F(i ) + Fb(t) where F( i ) denotes the hydrodynamic force and F(,(i) is the buoyancy force. The hydrodynamic force component parallel to the x axis will be referred to as the slope-parallel force, denoted by Fp(t), while the component parallel to the y axis is denoted Fjv(t) which represents the slope-normal force. Fp(t) is defined positive up-slope, while Fpf(t) is defined positive upwards and away from the structure. The buoyancy force always acts vertically upwards, opposite to the gravitational acceleration, so that its slope-parallel and slope-normal components are given by Fb(t) sin a and Fb(t) cos a respectively, where a is the slope angle of the rubble-mound. The relationship between these various forces is shown graphically in Figure 2-14. For a panel that remains fully submerged, the buoyancy force is constant and equals the weight of fluid displaced by the panel and some dynamometer hardware. In this case, isolation of the hydrodynamic force is relatively straightforward. Isolation of the hydrodynamic force component becomes more difficult when the submergence of the panel varies with time. For 43 Figure 2-14: Sketch of the fluid, hydrodynamic and buoyancy forces on an armour panel. this case, the magnitude of the time varying buoyancy force Fb (t) can be estimated from the instantaneous elevation of the waterline on the surface of the rubble-mound r)s (i) as Fb (t) = \Fb (t)\ = { Fb ,max F (Vs(t)-r,,,0\ 0 for r]s (t) > r]s<o + I sin a , for T)Sto < T}s (t) < T)3io + / sin a , for T)s (t) < T]sfi , (2.29) where Fbmax is the buoyancy force when fully submerged, 77S)o is the largest value of r)s(t) for which the panel remains entirely dry, and I sin a is the effective vertical distance between the lowest and highest parts of the panel. The time-varying hydrodynamic force is computed according to F(t) = F / ( t ) - F 6 ( t ) . (2.30) For the directions parallel and normal to the surface of the rubble-mound, Equation (2.30) 44 " 0.05 - -0.05 o > - - 0 . 1 w 0.15 -0.2 3000 Time (s) Figure 2-15: Separation of hydrodynamic and buoyancy forces. yields FP(t) = FfjX(t)-Fb(t)sma FN(t) = Ffty(t)-Fb(t)coBa (2.31) (2.32) where FftX(t) and Ffiy(t) are the x and y components of the total fluid force including buoyancy. The effectiveness of this procedure was evaluated in several experiments in which the water level in the basin was slowly raised or lowered so that the entire fluid force acting on the panels was due only to buoyancy. These experiments were also used to determine the most appropriate values of I and 77S)o for each panel. Figure 2-15 shows a typical result from one of these tests, during which the free surface slowly falls through the lower panel of the series 5 rubble-mound'. Measured time series of fluid force magnitude Ff (t) and waterline elevation r)s (t), and computed time series of buoyancy force magnitude Fb (t) and hydrodynamic force magnitude F (t) = yJFp(t) + F%(t) are shown. Equations (2.29), (2.31) and (2.32) have been applied to estimate 45 the buoyancy force from the waterline elevation and isolate the hydrodynamic portion of the fluid force. This procedure has produced an estimate of the hydrodynamic force close to zero as required. Deviations of F (t) from zero are due in part to the assumption of a linear relationship between water level and buoyancy that is not exact, particularly near the lower and upper edges of the panel. In spite of these deviations, the buoyancy and hydrodynamic components of the fluid force are generally well estimated over a wide range of water level. This procedure will be somewhat less effective in waves since the free surface will no longer remain horizontal. Because of this, the most accurate determination of slope-normal and slope-parallel hydrodynamic forces are obtained on armour panels located below the still waterline that remain fully submerged under wave attack. In this thesis, consideration of the slope-normal and slope-parallel hydrodynamic forces will be restricted to those panels located entirely below the still waterline that remain either fully or mostly submerged under wave attack. The horizontal component of the hydrodynamic force in-line with the direction of wave propagation is denoted by Fn(t) and can be written as FH(t) = FP(t) cos a - FN(t) sin a . (2.33) Substituting Equations (2.31) and (2.32) yields FH(t) = Ff>x(t) cosa - FftV(t) s ina (2.34) which indicates that FH (t) is independent of buoyancy and can be computed directly from the orthogonal components of the total fluid force. The horizontal hydrodynamic force will be used to investigate the spatial variation of hydrodynamic forcing over the surface of a rubble-mound, including locations above the still waterline that are only intermittently submerged. 46 2.3.10 Measurement of Damage The response of a rubble-mound breakwater to wave attack is usually expressed in terms of the degree of damage to the armour layer. Vidal et. al. (1991), following the work of Losada et. al. (1986), define the following four distinct degrees of damage that can be recognized by visual assessment. 1. Initiation of damage (ID): the outer armour layer displays holes larger than the average void space. 2. Iribarren's damage (IR): the failure area on the outer armour layer is sufficiently large to expose the units of the inner armour layer to direct wave attack and possible displacement. 3. Start of destruction (SD): a number of units of the inner armour layer are displaced so that holes are clearly visible. 4. Destruction (D): removal of material below the armour layer. Many definitions for a quantitative damage parameter have been proposed by various au-thors, including measures obtained from the number of armour stones displaced from a test section Nd, and the average eroded area Ae. The average eroded area is commonly obtained by differencing profiles of the armour surface before and after a duration of wave action. Broderick and Ahrens (1982) defined a damage parameter based on eroded area as SA = - (2.35) -^n50 In physical terms, SA represents the number of squares, with sides of length D „ 5 o , that fit into the average eroded area. The volume eroded from a test section of width I can be expressed in terms of the average eroded area as lAe, or in terms of the number of displaced armour stones as , where n is the porosity of the armour layer. These equivalent expressions of volume can be combined with Equation (2.35) to form a damage parameter expressed in terms of Nd 47 instead of Ae as S „ = & , (2.36) where I is the width of the test section and n is the porosity of the armour layer. SA and SN provide equivalent expressions of damage based respectively on the average eroded area and the number of displaced stones. Vidal and Mansard (1995) suggest approximate threshold values for a visual damage parameter Sy corresponding to each of the four distinct damage conditions identified by Vidal et. al. (1991) such that Sy — 1-0,2.5,4.0 and 9.0 correspond respectively to initiation of damage (ID), Iribarren's damage (IR), start of destruction (SD) and destruction ' (D). The breakwater section in channel 3 was used to monitor damage to the armour layer due to wave attack. This structure was identical to the rubble-mound in channel 2, except that the armour layer was comprised entirely of granite armour stones that were free to move in response to the wave-induced forcing. On this structure, the lower layer of armour stones were painted yellow so that they could be easily identified. Armour stones in the surface layer were not painted. Damage to the rock armour was recorded by three methods: colour photographs; counting the number of displaced armour stones; and profiling the surface of the armour layer before and after a duration of wave attack. Profiles of the armour layer were obtained by an electro-mechanical device developed at the N R C , consisting of a counter-balanced rotating arm mounted to a traversing carriage riding on a horizontal beam. Potentiometers are used to record the horizontal displacement of the carriage and the rotation of the arm as functions of time. For this study, the tip of the rotating arm was equipped with a 5 cm diameter plastic wheel. A n algorithm was developed to compute the wheel's elevation z as a function of the horizontal distance a; as it traversed the surface of the test section. Eight profiles z (x), taken at 8 cm intervals across the width of the test section were used to obtain an average profile zave (x) for the structure. Profiles were taken prior to wave attack, and after various durations of wave action. The vertical difference between initial 48 1 0.8 1 "•6 N 0.4 0.2 0 0.05 £ 0 N < -0.05 Initial profile Final profile 0.25 0.5 0.75 1.25 1.5 x (m) Figure 2-16: Profiles of the series 3 rubble-mound with SA = 7.2. and final average profiles is S I 1 1 1 1 1 1 1 11 • * *w , i i 1 1 1 1 1.75 &Zave (^) — Zave,i (-^) ^ave,f J (2.37) defined such that erosion is positive. The average eroded area Ae is obtained by integration of the positive values of Azave (x) between the toe x = Xt and crest x = xc of the breakwater test section: = / Azave (x) dx, Azave (x) > 0. JXt (2.38) The average accreted area Aa is given by integration of the absolute value of the negative values of Azave (x). The accreted area is often slightly less than Ae due to overall consolidation of the armour during wave attack. Typical examples of initial, final and difference profiles are shown in Figure 2-16 for a case during test series 3 with Ae = 0.0127 m 2 and SA = 7.2. Three methods are used to asses the damage to the armour layer: 49 • visual assessment in terms of Vidal's definitions and the parameter Sy', • measurement of the average eroded area AE and computation of SA', • counting the number of displaced stones Na and computing SN-In situations with little damage, random noise in the computed vertical distance Azave (x) contributes to a positive bias in the damage estimate SA- It is also difficult to differentiate damage (the removal of armour stones) from settlement of the rubble-mound when profiling is used. Fortunately, it is relatively easy to obtain an accurate count of the number of displaced stones, and therefore a reliable estimate of 5V, in these situations. Overall, SN was found to be the most accurate measure of damage for low damage levels, while SA is most appropriate for moderate to high levels of damage when counting individual stones is less precise, and the inaccuracies inherent in profiling become small compared to the total volume of erosion. A single damage parameter S, defined by S= < SN SN < 1 aSN + bSA 1<SN<3 (2-39) SA SN > 3 where a = 1 — (SN — 1) /2 and b = (SN — 1) /2 is used here to quantify the damage response of the armour layer. This method is similar to the one used by Vidal and Mansard (1995). 2.3.11 Wave Synthesis The test structures were exposed to a large number of regular and irregular wave conditions. Irregular waves were used because they provide an authentic simulation of waves in nature. Although they are not so commonly encountered in nature, regular waves were extensively used in order to simplify the study of the wave-induced forcing of the armour and the relationships between this forcing and the characteristics of the incident waves, the flows on the surface of the rubble-mound, and the damage response of the armour. 50 Regular waves were synthesized with periods between 1.0 and 3.0 s, and heights between 9 and 24 cm. The longer-period and higher regular waves were strongly non-linear, exhibiting non-symmetric profiles and significant energy content at higher harmonics of the fundamental frequency. Irregular waves were synthesized from a family of JONS W A P wave spectra Sv, defined by Sr, if) ccg-( 2 7 T ) 4 / 5 exp (2.40) where a = exp (2.41) 0.07 a = (2.42) for f < fp, 0.09 for f>fp. In the above, fp is the peak frequency at which S^ (/) is a maximum while a and 7 are parameters that control the scale and peakedness of the spectrum. For wave synthesis, the spectrum is specified using 7, fp and a spectral estimate of the significant wave height Hmo, defined by / roo (2.43) H m0 in which case, a is scaled to satisfy Equations (2.40) - (2.43). Irregular waves were synthesized from J O N S W A P spectra with 7 = 1 and 5, fp = 0.6, 0.5, 0.4 Hz, and Hm0 between 10 and 16 cm. Several of these spectra are shown in Figure 2-17. In all cases, ten minute long time series of water surface elevation 77 (t) were synthesized from S ,^ (/) by the random phase method as described by Funke and Mansard (1984). These time series were used to compute control signals for the wave machine. 51 0.01 0 0.5 1 1.5 2 f (Hz) Figure 2-17: Selected J O N S W A P wave spectra. 2 .3 .12 Test C o n d i t i o n s a n d P r o c e d u r e s R e g u l a r W a v e Tests Regular waves generate the same fluid kinematics and forces with each wave cycle. The damage response to this consistent loading is normally rather fast, and equilibrium conditions are often achieved in a short period of time. Hudson (1958) subjected each test structure to 30 minutes of wave attack. Ahrens (1975) used between 400 and 1900 wave cycles, depending on the wave period. In this study, regular wave tests were generally continued for a duration corresponding to 1000 wave cycles. In several cases, the test duration was extended to verify that 1000 cycles were sufficient to approach equilibrium damage conditions. In other cases, tests were stopped short to prevent complete failure of the rubble-mounds. Samples of transducer outputs were generally obtained over 100 wave cycles near the beginning of each test. The general procedure followed for each regular wave test is outlined below. 52 • Construction or re-construction of the test structure. • Profiles and photographs of the initial condition. • Offsets for transducers in still water. • Start-up of wave generation. • Sampling of transducer outputs. • Visual observation of wave interactions with the rubble-mound and the progression of damage. • Shut-down of wave generation. • Profiles, photographs and visual assessment of the final condition. After tests with minimal damage, only the surface layer of armour stones was re-built during the re-construction phase. In other cases with more significant damage, both layers of armour stones were re-built. After severe damage, the filter layer was also re-built. During test series 2, 4, 5 and 6, additional tests were performed without damage observations in order to obtain samples of transducer outputs in wave conditions below those required to initiate damage and above those required for failure of the armour. The range of regular wave conditions and the number of tests performed during each test series are summarized in Table 2.4. Transducer outputs were obtained in 153 regular wave tests, while damage assessments were made after 68 of these. Irregular Wave Tests Irregular waves cause a wide variety of fluid kinematics and forces on the surface of a rubble-mound. Under these conditions, longer test durations are generally required to approach the equilibrium damage level. A n infinite duration of stationary irregular wave attack will eventu-53 Test Series Series 2 Series 3 Series 4 Series 5 Series 6 Totals Regular waves Wave periods, T (s) 1, 1.5, 1.8, 2, 2.3, 2.5, 3 1.5, 2, 3 1.5, 2, 3 1.5, 2, 3 , 1.5, 2, 3 1 - 3 Range of wave height, H (cm) 1 0 - 1 6 9 - 1 5 9 - 2 3 9 - 2 2 9 - 2 1 9 - 2 3 No. of wave conditions 45 22 44 26 16 153 No. of tests with damage assessment 20 20 18 10 0 68 Irregular waves Wave periods, Tp (s) 1.7, 2.0, 2.5 1.7, 2.0, 2.5 1.7, 2.0, 2.5 1.7, 2.0, 2.5 2.0, 2.5 1 . 7 - 2 . 5 Range of wave height, H3 (cm) 9 - 1 5 9 - 1 3 1 0 - 1 5 1 0 - 1 6 1 3 - 1 4 9 - 1 6 No. of wave conditions 24 10 13 9 3 59 No. of tests with damage assessment 12 10 6 9 0 37 ally produce a true equilibrium level of damage response. Clearly, infinite test durations are not practical, and a finite test duration must be selected that allows significant damage to occur. Thompson and Shuttler (1976) tested structures protected by riprap for durations up to 5000 irregular wave cycles, calculated based on the mean zero-crossing wave period T m , with intermediate surveys of damage every 1000 waves. Van der Meer (1988) used test durations corresponding to 1000 and 3000 average wave periods. Based on empirical analysis of his own tests and those of Thompson and Shuttler, van der Meer suggests that the damage after Nw waves S(NW), can be given by 5 (Nw) = 1.3 (1 - exp [-0.0003.Nu,]) 5 (5000) (2.44) where 5(5000) is the damage after 5000 waves. In terms of 5(3000), the damage after 3000 irregular waves, this relation is equivalent to 5 (Nw) = 1.685 (1 - exp [-0.00037V^]) 5 (3000). (2.45) Van der Meer also proposed the simpler expression 5 (Nw) = 0.0Uy/NZS (5000) (2.46) valid for Nw < 8000, which in terms of 5 (3000) is equivalent to 5 (Nw) = 0.018\/A^5 (3000), for Nw < 8000. (2.47) In this study, irregular wave tests were generally continued until the test structures were exposed to 3000 wave cycles. This duration was adopted as being sufficient to capture the changes in damage response due changes in wave conditions and structure characteristics. In some cases, test durations were extended up to 6000 wave cycles, while in others, damage 55 assessments were made at intermediate intervals. Equations (2.45) and (2.47) provide a means to estimate the equilibrium damage level based on the damage observed after 3000 waves. Test procedures for tests with irregular waves were similar to those presented above for the regular wave tests. Samples of transducer outputs in irregular waves were obtained for 600 s near the beginning of each test. This duration equals the recursive period of the wave machine control signal, and is thus just sufficient to capture all unique events contained within the irregular wave test. On several occasions, samples were also obtained near the middle, and towards the end of an irregular wave test. Analysis of these samples indicates that the irregular waves remained stationary throughout the test duration. The range of irregular wave conditions and the number of tests for each test series are summarized in Table 2.4. Transducer outputs were obtained during 59 irregular wave tests, while damage assessments were made for 37 of these. 2.3.13 Wave Reflections When testing coastal structures in waves, it is desirable to minimize the build-up of reflected wave energy within the test facility. Waves reflected from test structures within the basin and from the boundaries of the test facility can propagate back to the wave board where they are re-reflected towards the test area. These reflected wave components add to the waves being generated by the wave machine, resulting in an increase in wave energy over time. The test layout was designed to minimize wave reflections, so that the wave energy incident to the rubble-mounds would remain stationary over time. Reflection analyses have been performed for several tests with regular and irregular waves to establish typical levels of reflected wave energy at several locations within the test basin. Wave reflections are determined from water surface elevation signals recorded by an array of five gauges following the iterative least squares technique described by Mansard and Funke (1980). Typical results from this analysis are presented in Figure 2-18 for the case of irregular waves with Hs = 14 cm and Tp = 2 s in test series 4. This figure shows the incident wave spectrum 56 O 30 24 18 12 C r at wave machine C r in Bide channel Cr at rubble-mound S. 0.25 0.5 0.75 1 1.25 f (Hz) 1.5 1.75 0.003 0.0024 0.0018 0.0012 0.0006 N Figure 2-18: Reflection analysis of an irregular wave test (test series 4, Hs = 14cm, Tp = 2s). Sr) (/) measured near the wave machine and three frequency dependent reflection coefficients defined by Cr (/) = 100 • f i n c i d e n t (2.48) ti reflected (/) where ^incident (/) a n d i?reflected(/) represent the frequency dependent wave heights of the incident and reflected wave components respectively. Reflection coefficients are shown for three different locations: (1) near the wave machine; (2) within side-channel 1; and (3) in front of the rubble-mound breakwater in channel 2. Over the frequency band containing significant wave energy, 0.4 < / < 1.25 H z (as indicated by £ , , ( / ) ) , reflections in front of the series 4 rubble-mound are approximately 20 %, while reflections in the side channel and near the wave machine are both less than 5 % . Analyses from other tests show similar results. These results indicate that wave reflections within the basin are generally well controlled by the porous gravel beaches installed between the test channels and along the rear boundary of the basin. 57 Chapter 3 Damage of Rock Armour This chapter focuses on the damage response of rock armour subject to wave attack. Section 3.1 contains a brief review of some of the more significant equations that have traditionally been applied to design armour layers for coastal structures. Consideration of these design equations indicates which parameters are most important, and the general manner in which these dominant parameters are related to the initiation and growth of damage. This review is provided to outline existing approaches to designing armour layers and to set a background for the developments and analyses presented in subsequent chapters which focus on the failure mechanisms, surface flows and forcing responsible for wave-induced damage. Experimental results on the initiation of damage in both regular and irregular waves are presented in Section 3.2. These results are compared to predictions from the two design methods most commonly used in current practice, namely the Hudson equation and van der Meer's design equations for static stability. The chapter concludes with an examination of the influences of wave height, wave period, structure slope, structure permeability and storm duration on the damage response of rock armour. The ability of a rubble-mound breakwater or revetment to survive wave attack depends, among other factors, on the performance of the armour used on the sea-ward face of the struc-ture. Removal of armour stones can quickly lead to erosion of underlying layers and failure of 58 the structure to the point that it can no longer perform its design service. The relationship between incident waves and the response of the armour depends on a very complex interaction of both deterministic and stochastic processes. Armour stability describes the ability of armour stones on the surface of a rubble-mound to resist displacement under wave attack. The stability of an armour stone depends on the relative magnitude of the forces acting to displace it (driving forces), and those mobilized to resist its displacement (resisting forces). Driving forces generally include contributions from components of the gravitational force as well as hydrodynamic forces resulting from the velocity and acceleration of fluid flow on the surface of the rubble-mound and within the armour layer. Resisting forces also generally include contact forces transmitted by adjacent stones in addition to contributions from components of the gravitational and hydrodynamic forces. Near the threshold of motion, driving forces and resisting forces are in approximate balance, such that only a small increment to the driving force is required to initiate displacement of the armour stone. Damage to an armour layer occurs when armour stones lack sufficient stability to remain stationary under wave attack. Damage is commonly quantified in terms of the number or volume of displaced armour stones. Different levels of damage may be acceptable for different types of design. Four different levels of damage, namely: initiation of damage (ID); Iribarren's damage (IR); start of destruction (SD); and destruction (D) are defined in Section 2.3.10. The initiation of damage condition is commonly adopted as a tolerable level of damage for operational design, while a more advanced damage level such as the initiation of destruction condition is adopted for ultimate design. 3.1 Design Equations Design equations are generally used for preliminary design of rubble-mound structures, includ-ing initial cost and performance comparisons of various alternative structure geometries and 59 armouring schemes. Many different design equations have been proposed and are commonly used throughout the world. Sixteen of these were reviewed and compared by the P I A N C (Per-manent International Association of Navigation Congresses) 2 n d Waves Commission (PIANC, 1976). They found large discrepancies between the various design formulae, and concluded that the stability formulae commonly used to design rock armour for coastal structures had "significant limitations". Design guides such as the "Shore Protection Manual" (CERC, 1984), prepared by the Coastal Engineering Research Center of the U.S. Army Corps of Engineers, and the "Report of P I A N C Working Group 12" (PIANC, 1992) strongly encourage the use of hydraulic model tests to evaluate the performance of the proposed design under site-specific conditions. Hydraulic model tests allow for optimization of the proposed design and provide a means to check on failure mechanisms that are not considered by design equations. The design equations considered herein attempt to predict the size of armour stones required to prevent excessive damage to conventional multi-layer rubble-mound structures under wave attack. Some of them can also be used to estimate the level of damage that will accrue from a specific duration of wave attack, or to estimate the most severe wave conditions that can be resisted by a particular combination of rubble-mound geometry and armour stone. 3.1.1 Governing Variables Various authors, including Raichlen (1974), Thompson and Shuttler (1976) and van der Meer (1988) have compiled lists of governing variables that influence the performance of rubble-mound armour. These lists include variables related to environmental conditions that are in general beyond the control of a designer, and variables related to the physical characteristics of the rubble-mound. A typical list of environmental variables includes: • the incident wave field r\ (x, t); • the duration of wave activity, characterized by the number of waves Nw; 60 • the angle of wave attack; • the water depth h(x, t) at the rubble-mound ; • the density of the water p; • the viscosity of the water v; • the acceleration due to gravity g. Three of these variables are functions of spatial position x and time t. In earlier work, the incident wave field is parameterized in terms of a design wave with height H. More recent works have adopted a spectral parameterization of the incident wave field. For example, van der Meer (1988) specified irregular waves in terms of significant wave height Hs, and average wave period Tm. Physical variables describe the geometry of a rubble-mound and characteristics of its con-stituent materials, including the size, shape and grading of armour and filter layers. A typical list might include: • the nominal diameter Dn5Q of armour stones; • the grading D^/D\^ of armour stones; • the density of armour stones pa; • the thickness of the armour layer ta; • the shape and surface roughness of armour stones; • the construction method; • the nominal diameter Dnso of filter stones; • the grading D^/D\s of filter stones; • the thickness of the filter layer; 61 • the permeability of the core; • the seaward slope angle a with respect to horizontal. The shape, roughness and gradation of armour stones and the construction method affect the porosity of the armour n, defined as the volume of voids divided by the total volume. These factors also affect the degree of interlocking between armour stones, which is commonly parameterized by the internal friction angle <p. Design equations for armour stability are semi-empirical relations that include a few dom-inant environmental and physical variables, such as fluid density p, armour stone density pa, armour stone size Dn5o, slope angle a , wave height H, and in some cases, wave period T. 3.1.2 Design Equation of Iribarren The work of Iribarren (1938) can be considered as the common starting point for many design equations in use today. Iribarren's equation can be written H cos a — s i n a AD^~o = JOT* ( 3 > 1 ) where A = is the relative density of the armour material. The median mass of armour stones M50 is related to the nominal diameter by M50 = paD^5Q. Iribarren's equation includes five of the most important fundamental variables; p, pa, Dnso, a and H. The coefficient K must be chosen to account for the influence of all other variables. For rock fill, a value of K = 0.015 was recommended. Although the level of damage is not explicitly considered, Iribarren's equation is intended to give the threshold for "no damage" to the armour. Solutions to Equation (3.1) with K — 0.015 are plotted in Figure 3-1 as a function of slope cot a . Iribarren's equation indicates that smaller wave heights are required to cause damage on steeper slopes. It also indicates cot a = 1 (a = 45°) as the limiting stable slope, which is in general agreement with natural repose angles commonly encountered for rock fill. 62 0 0 2 3 4 5 6 c o t ( a ) Figure 3-1: Solutions to Iribarren's design equation with K = 0.015. 3.1.3 Design Equation of Hudson Based on the work of Iribarren and results from small-scale physical model tests with regular waves, Hudson (1958) proposed a design equation that can be written Hudson's equation contains the same two environmental and three physical variables considered by Iribarren. The only difference between these two equations is the effect of the slope angle a. Hudson's equation includes the damage coefficient KQ to account for the influence of all variables that are not explicitly included; This equation is currently recommended in the "Shore Protection Manual" (CERC, 1984) (abbreviated hereafter as SPM-84). SPM-84 offers a wide selection of Kp depending primarily on: the type and shape of armour units; the thickness of the armour layer; the method of placement; whether the design waves are breaking or non-breaking; the angle of wave attack; the porosity of the underlayer material; and the tolerable H = ( K D c o t a ) 1 / 3 . (3.2) AD n 5 o 63 3 Q < 1.5 -o o c 2.5-1 -0.5 " K D = 2 0 0 2 3 4 5 6 cot(a) Figure 3-2: Solutions to Hudson's design equation for rock armour with KD — 2 and 4. damage level. For rough angular quarry stones placed randomly in two layers and exposed to non-breaking waves, KD = 4 is recommended for less than 5% damage. For waves that break onto the structure, KD = 2 is recommended. For smooth rounded stone, KD values of 2.4 and 1.2 are recommended for non-breaking and breaking waves respectively. Solutions to Hudson's design equation with KD = 2 and 4 are shown graphically in Figure 3-2. As with Iribarren's equation, enhanced stability is predicted on milder slopes; however, Hudson's equation fails to approach a reasonable asymptote on very steep slopes with cot a ~ 1. The values of KD recommended by SPM-84 suggest that waves that break on a rubble-mound are more damaging than those that do not break. In particular, waves with height H that break on a rubble-mound are predicted to cause the same damage as non-breaking waves that are 2 7 % higher. Unfortunately, SPM-84 provides little guidance as to when the breaking and non-breaking KD values should be used for design. One possible interpretation is that recommending different KD values for breaking and non-breaking waves represents an attempt to account for the effects of wave period on the nature of wave interactions with a 64 rubble-mound. Breaking waves can be interpreted as those waves that interact by plunging or collapsing onto the rubble-mound, while non-breaking waves may refer to those that interact by surging. Breaking is associated with steeper waves (shorter periods) while surging prevails for longer period waves that are less steep. A n alternative interpretation is that the breaking wave KD values are meant to apply to structures located in relatively shallow water and preceded by a steep foreshore, while non-breaking values are meant to apply to structures located in deep water, or structures preceded by a shallow foreshore which are less likely to encounter breaking waves. In SPM-84, damage is expressed in terms of the volume of armour units displaced as a percentage of the total volume of armour units on the sea-ward face of the rubble-mound from the midpoint of the crest down to a level equivalent to one wave height below the still water level. The relation between the SPM-84 definition of damage and the dimensionless damage parameter S commonly used elsewhere depends greatly on the geometry of the structure in question, and in particular on the volume of armour exposed to wave attack. For a uniformly sloped structure with a constant thickness of armour, the relation can be expressed as where D% is the percentage damage according to the SPM-84 definition, ta is the thickness of armour, wc is the crest width, e c is the elevation of the crest, h is the water depth and H is the design wave height. The ratio S/D% varies significantly for different rubble-mounds, and is between 0.6 and 1.25 for the structures considered herein. If a representative value of S/D% = 0.8 is selected, the SPM-84 criterion for less than 5 % damage is approximately equivalent to 0 < S < 4. This is a rather broad criterion for the initiation of damage compared to the criteria 5 = 1 adopted by Vidal and Mansard (1995), and S ~ 2 or 3 considered by van der Meer (1988). Although the degree of damage is not explicitly considered by the Hudson equation, the preceding analysis suggests that it corresponds to conditions close to the initiation (3.3) 100 • Dl 65 of damage threshold. Hudson's equation was originally developed for regular waves. However, extensions have been proposed so that it may be applied to design rubble-mound armour subject to the irregular wave attack that occurs in nature. SPM-84 recommends that Hi/W, denned as the average height of the highest 10 % of waves, be used in place of the regular wave height H, which gives ^ = (KD cot a) 1 / 3 (3.4) for irregular waves. Previous versions of the "Shore Protection Manual" had recommended that the significant wave height Hs be used to characterize irregular waves. For wave heights that satisfy the Rayleigh distribution, i?i/io = 1.27Hs. According to the Hudson equation, the required mass of armour stones increases with H3; therefore, the switch from Hs to -ffi/io amounts to a two-fold increase in the required mass of armour stones (1.273 = 2.05). A recent report from P I A N C Working Group 12 (PIANC, 1992) concludes that Equation (3.4) leads to unnecessarily conservative design when used with the KD values recommended in SPM-84. Medina and McDougal (1988) proposed a modification that accounts for the effects of irreg-ular waves and storm duration. They suggest that the most probable maximum wave height {Hrnax)mode likely to occur in Nw waves, given according to the Rayleigh distribution by {Hmax)rnode ~ ^s\l ~ > (3-5) be used in place of the regular wave height H. The substitution is made assuming that the standard Hudson equation is valid for Nw = 1000, and leads to an extended Hudson's equation given by which includes variables to account for the effects of irregular waves and storm duration. Ir-regular wave heights are generally well represented by the Rayleigh distribution in deep and 66 \ intermediate water depths; however, in shallow water, and particularly within the surf zone, wave heights can deviate significantly from the Rayleigh model. In such cases, Equation (3.6) must be applied with caution. 3.1.4 Design Equation of Losada and Gimenez-Curto Losada and Gimenez-Curto (1979) developed a predictive equation for rock armour stability based on analysis of experimental data in regular waves from three sources. Their equation can be written H = {A - & ) exp [B (£ - &)] r 1 / 3 (3.7) A £ > n 5 0 in which the effects of wave period and structure slope are represented through the surf similarity or Iribarren parameter, denned as t a n a gT2 / o ON where LQ = gT2/2ir is the deep-water wave length. A and B are coefficients that depend primarily on the type of armour and the structure slope, while £o = 2.65 tan a denotes a minimum value of £ limited by wave steepness. Battjes (1974) used the surf similarity parameter to indicate the type of wave breaking that prevails on a constant slope as plunging breakers 0.5 < £ < 2 , collapsing breakers £ ~ 3 , surging breakers £ > 4 . These three different breaker types are sketched in Figure 3-3. A more thorough discussion of these different breaker types, including their influence on surface flows and the wave-induced forcing of the armour layer is presented in Chapter 5. Solutions to Equation (3.7) for rubble-mounds with quarry-stone armour at slopes cot a = 1.5, 2 and 3 are shown in Figure 3-4. Equation (3.7) predicts minimum stability for £ c r = 67 plunging: \<2 Figure 3-3: Sketch of plunging, collapsing and surging breakers. 68 < o =4 0 1 2 3 4 5 6 7 ( = t a n a ( g T 2 / 2 T r H ) 1 / 2 Figure 3-4: Solutions to the design equation of Losada and Gimenez-Curto (1979). £o — 1/5 between 2 and 3.5, which corresponds roughly to the transitional regime between surging and plunging breakers. In the regime of surging breakers (£ > 4) stability increases on milder slopes. Figure 3-4 indicates that wave period has an opposite effect on stability in the plunging and surging wave regimes. Stability increases gradually with larger T for surging waves, but decreases rapidly with larger T for plunging breakers. Although this equation overestimates the influence of surf similarity on stability for plunging breakers, it represents a significant development because it was among the first to use the surf similarity parameter to account for the influence of wave period on the stability of rock armour. 3.1.5 Design Equation of Hedar Hedar (1986) presents design formulae for the stability of rock armour on rubble-mound break-waters that account for the effects of permeability. These formulae are based on a re-analysis of physical model tests with regular waves initially reported in Hedar (1953 and 1960). Hedar de-veloped separate equations for very permeable and very impermeable structures which indicate 69 Underlayer Uprush Downrush both 7 = a + ((j) - 48°) B = tan 4> + tan a y = a-((j>- 48°) B = tan ^ — tan a permeable A = 0.33 = 3-6 — exp [—4 tan 7] A = 1.0 /j (7) = 13.7 + exp [4 tan 7] impermeable A = 0.41 /i(7) = 3.3 - exp [ -4 tan 7] A = 1.6 = 16.5 + exp [4 tan 7] Table 3.1: Parameters for the stability design equation of Hedar (1986). enhanced stability on more permeable structures. Hedar also considered wave interactions with steep and mild slopes as two separate processes. Hedar reasoned that the limiting condition for stability on steeper slopes with cot a < 3 occurs during the downrush portion of the flow cycle, while uprush governs stability on mild slopes with cot a > 4. Uprush and downrush were considered equally important for slopes with cot a ~ 3.5. Hedar's formulae can be written as a single equation in terms of the nominal stone diameter required for stability as Ai50 _ ^ y / 3 f A(hb +0.7Hb) (tan 0 + 2) 6J \ B A / i (7) cos 4>+m a J (3.9) where Hb is the breaking wave height and hb is a corresponding water depth. The parameters A, B, 7, and /1 (7) depend on permeability and whether uprush or downrush prevails, and are summarized in Table 3.1. Graphical solutions to Hedar's design formula for permeable and impermeable rubble-mounds with (j) = 45° and hb = 1.28Hb are presented in Figure 3-5 as a function of slope cot a . Hedar's formula shows increasing stability on milder slopes for cot a < 3.5, but decreasing stability on milder slopes for cot a > 3.5. Within the range 1.5 < cot a < 3.5, the effect of slope is nearly identical to that predicted by the Hudson formula. As with Iribarren's formula, this equation predicts reasonable behaviour on very steep slopes, such that H/ADn5o —> 0 as a —* (j). 70 1 - I m p e r m e a b l e 0.5 - P e r m e a b l e 0 0 2 3 4 5 6 c o t ( o c ) Figure 3-5: Solutions to the design equation of Hedar (1986) for permeable and impermeable structures (c/> = 45°, hb = 1.28Hb). 3.1.6 Design Equation of van der Meer Van der Meer (1988) developed design formulae for the static stability of rock armour based on analysis of a large number of small and large scale tests with irregular waves. Separate equations were developed to predict stability in plunging and surging waves. They can be written where P is a special permeability factor and £ m = tan ay/gT~%j2ixWs is a surf similarity param-eter for irregular waves denned in terms of the average wave period Tm and the significant wave height Hs. Van der Meer's formulae also include the damage level S, and the number of waves Nw. Van der Meer considered the "initiation of damage" threshold to be given by S ~ 2 or 3, while "moderate damage" was associated with S between 5 and 12, depending on the structure 0.2 for surging waves, for plunging waves, (3.10) (3.11) 71 Figure 3-6: Values of the coefficient P recommended for various structures (from van der Meer (1988)). slope. The permeability factor P is an empirical coefficient (not related to the coefficient of per-meability k) intended to represent the relative overall permeability of the armour, filter and core layers of various different rubble-mound breakwaters. Recommended values of P vary between P = 0.1 for a fully impermeable structure, and P = 0.6 for an homogeneous structure comprised entirely of armour stones. Figure 3-6 is reproduced from van der Meer (1988) and shows the values of P recommended for various structures. This figure provides some guidance with respect to the selection of an appropriate value of P; however, van der Meer recommends that the ultimate selection rest on the judgement and experience of the designer. According to van der Meer's formulae, minimum stability occurs at the value of f m given 72 by (,m,cr = ( 6 . 2 P a 3 1 v / t a n ^ ) ^ , (3.12) which varies depending on the structure slope and permeability. For the rubble-mound break-waters considered herein, £ m ] C r ranges between 3 and 5. Larger values of £ m ) C r apply to the steeper, less permeable structures. Van der Meer associated minimum stability with the oc-currence of collapsing breakers. The effect of surf similarity on stability contained in van der Meer's design equations for irregular waves is similar to that predicted for regular waves by the design equation of Losada and Gimenez-Curto (1979). In addition to the five fundamental variables considered by Iribarren and Hudson (p, pa, DnbOy H, a) , van der Meer's equations include variables to account for the effects of wave period (T m ) , overall permeability (P), the duration of wave attack (Nw) and the level of damage (5). The equations were developed for irregular waves whose heights satisfy the Rayleigh distribution. Using different equations for plunging and surging waves reflects the notion that the process leading to the dislodgment of armour stones is different in these two wave breaking regimes. Van der Meer suggests that the difference is related to the relative importance of wave uprush for plunging waves, and the dominance of wave downrush in the surging wave regime. Stability is minimized when collapsing breakers prevail and both uprush and downrush flows contribute to the displacement of armour units. With damage level S as the dependent variable, van der Meer's design equations can be written as S = P - ° ' 9 r 2 ' 5 ( t a n a ) 2 5 ^ ( A ^ o ) - 5 ^ 3 7 5 (3.13) for plunging waves and / si \ -2.5P S= (J^j P a 6 5 T - ^ ( t a n a ) 2 - 5 - 5 P v ^ ( A D n 5 0 ) - 5 F r 2 ^ (3.14) for surging waves. These equations suggest that the type of wave breaking significantly affects 73 the influence of specific structural and environmental variables on the resulting damage. For example, Equation (3.13) indicates that increasing wave period leads to greater damage in plunging waves, but Equation (3.14) indicates that the opposite is true for surging waves. Both equations indicate greater damage on less permeable rubble-mounds (with smaller P). For surging waves, this holds true in spite of the positive exponent for the variable P. This permeability effect is similar to that indicated for regular waves by Hedar (1986). Van der Meer's design equations erroneously suggest that some damage will result under all wave conditions with Hs > 0. Clearly, waves with heights that are well below design conditions will not damage a rubble-mound. These equations do not accurately reflect the relation between damage and wave height in the limit as S (or Hs) approaches zero. Van der Meer's equations should therefore only be applied to wave conditions that are sufficiently severe to cause some damage. 3.2 Initiation of Damage The initiation of damage threshold refers to wave conditions that are just sufficient to initiate damage to an armour layer. Near this threshold, the driving and resisting forces acting on some armour stones are in approximate balance such that over time, a few of the least stable stones are displaced. Those stones that are poorly interlocked, and therefore cannot mobilize contact forces of sufficient magnitude to resist the hydrodynamic driving forces are selectively displaced from their initial locations. Initial damage typically develops just below the still water level, which suggests that driving forces are maximized in this zone. Stones are sometimes displaced a small amount into more stable positions; however, they are more commonly transported down the rubble-mound slope to new locations well below the still waterline. Initial damage typically accumulates towards an equilibrium level such that continued stationary wave attack will not lead to further damage. Once damage reaches this equilibrium, more severe wave conditions are required to advance the damage beyond this level. 74 In regular waves, each wave generates a nearly identical cycle of fluid flow on the surface and within the permeable sections of a rubble-mound. These regular cycles of surface and internal flows exert hydrodynamic forces on armour stones that are approximately consistent from wave to wave. (Small deviations can result from variations in the turbulent fluctuations released through wave breaking.) Tests with various regular waves near to the initiation of damage threshold suggest a fairly distinct transition between waves that are damaging and those that are not. This behaviour is consistent with the uniformity of hydrodynamic forcing due to regular waves. With irregular waves, each wave generates a different cycle of surface and internal flows on the rubble-mound. These flow cycles exert hydrodynamic forces on armour stones that vary considerably from wave to wave. For irregular wave conditions that are close to those required to initiate damage, only a small number of individual waves exert sufficient hydrodynamic forces to destabilize armour stones. Damage accumulates relatively slowly under these conditions. As slightly more energetic waves are used, a greater proportion of waves exert forces that are sufficient to destabilize armour stones. This results in an increase in both the rate of damage growth and the equilibrium damage level. Tests with various irregular waves near to the initiation of damage threshold suggest a more gradual transition between non-damaging and damaging wave conditions than for regular waves. This trend is consistent with the distributed character of the hydrodynamic forcing due to irregular waves. The progression of damage to an armour layer in response to wave attack is a random process that depends on the placement and interlocking of individual armour stones. As such, the progression of damage will vary between repeated trials of the same wave condition. Near the initiation of damage threshold, repeated experiments with identical waves were found to produce significantly different amounts of damage. This variability in damage response makes it difficult to precisely identify the initiation of damage threshold. In fact, the wave height required to initiate damage is not a fixed quantity, but varies in response to the specific arrangement and degree of interlocking of armour stones near the still waterline. Each repeated test produces 75 a different damage response because the arrangement of armour stones differs each time the armour is re-built. The damage response from each test provides a single measure of a random process. Near the initiation of damage threshold, the variability in damage responses was found to be quite large in proportion to the final damage level. This variability is apparent in the results that are presented below. Following Vidal and Mansard (1995), the criterion 5 = 1 is used to identify initiation of damage conditions. For the N R C experiments, 5 is computed by combining measures of Sjy and SA according to Equation (2.39). For the exploratory tests, 5 = SN, and is computed according to Equation (2.36). For the N R C tests, 5 = 1 corresponds to the displacement of 9 armour stones from the 65 cm wide test structure in channel 3. For the exploratory tests, 5 = 1 corresponds to the placement of 11 armour stones since a slightly wider test section was used. Figure 3-7 shows a photograph of the armour on the series 3 rubble-mound at a damage level of 5 = 1. 3.2.1 Regular Waves Results on the initial damage of rock armour in regular waves were obtained during test series 2, 3, 4 and 5 of the N R C experiments. Salient characteristics of structures used in these test series are summarized in Table 2.2. Test series 3 considered a basic rubble-mound breakwater consisting of an impermeable core, a filter layer and two layers of armour stones at a plane slope of cot a — 1.75 in 55 cm of water. For test series 2, the same structure was tested at 40 cm water depth. For test series 4, the filter and impermeable core were replaced by a permeable core. For test series 5, the slope was reduced to cot a = 3. These structures are representative of rubble-mound breakwaters commonly encountered in practice. Each structure was tested in a variety of regular waves. Wave periods of T = 1.5, 2.0 and 3.0 s were typically used, while wave heights were varied in approximately 1 cm increments around the initiation of damage threshold. Each structure was exposed to attack by Nw = 1000 regular waves for each combination of H and T. When damage occurred, the structure was 76 Figure 3-7: Photograph of the initiation of damage condition (S 77 re-built prior to the next test. At each wave period, the damage accumulated after 1000 regular waves was plotted versus incident wave height. A damage curve of the form was fitted to these results using a least squares error procedure to evaluate the coefficients a and b. This form of damage curve reflects the reafity that S increases rapidly with increasing wave height and also approaches zero for HS > 0. The initiation of damage wave height HID is obtained by setting S = 1 in Equation (3.15), which yields Data for the case of regular waves with T = 2 s on the series 4 rubble-mound are shown in Figure 3-8 together with the fitted damage curve. For this case, HID = 15.6 cm. In all cases, the veracity of the HID values obtained from this curve fitting method were verified using plots such as the one shown in Figure 3-8. According to this method of analysis, all of the test results at a single wave period are used to obtain a single value of Hi p. Initiation of damage wave heights for test series 2, 3 and 4 (all with cot a — 1.75) are summa-rized in Figure 3-9, which shows non-dimensional initiation of damage wave height HID/AD^Q plotted against surf similarity £ = tan ay/gT2/2nH. The non-dimensional HID for each struc-ture show some scatter, but no obvious trend with £. For these three rubble-mounds, Hudson's design formula (Equation (3.2)) with KD = 4 predicts a constant non-dimensional initiation of damage wave height of Hw/ADn5o = 1.91. This prediction is indicated in Figure 3-9 by a solid horizontal line. Although individual data points deviate from this line, the observed HID are on average in fairly close agreement to the predictions of Hudson's design equation. The degree of scatter evident here is quite moderate when compared to the variability of experimental results (3.15) (3.16) 78 ra 3 n50 Figure 3-8: Damage curve for regular waves (test series 4, T — 2 s). Figure 3-9: Regular wave height required to initiate damage (cot a = 1.75). 79 2.5 o 2 in a Q < 1.5 0.5 M H u d s o n , series 3 H u d s o n , series 5 • Series 3 H Series 5 0 1 2 3 4 5 6 7 f = t a n a ( g T 2 / 2 r r H ) 1 / 2 Figure 3-10: Effect of slope angle on the regular wave height required to initiate damage. in regular waves published elsewhere (i.e. Ahrens (1975) and Thomsen et. al. (1972)). Initiation of damage wave heights for the permeable series 4 rubble-mound are consistently larger than for the less permeable structure, which suggests that armour stability is marginally enhanced on more permeable structures. This result is consistent with the design equations of Hedar (1986) and of van der Meer (1988). The similarity of HID results from test series 2 and 3 suggest that water depth does not have a significant effect on the stability of armour stones over the range of depth considered here. Water depth can be expected to become important where it is sufficiently shallow to limit the heights of incident waves. Non-dimensional HID results from test series 3 (cot a = 1.75) and 5 (cot a = 3) are presented in Figure 3-10 as a function of £. Predictions from Hudson's design formula (Equation (3.2)) with KD = 4 are shown by horizontal lines. The results and prediction for the series 3 rubble-mound are identical to those shown previously in Figure 3-9. Figure 3-10 indicates that larger wave heights are required to initiate damage on the milder sloped, series 5 rubble-mound. This 80 is consistent with all of the design equations considered in Section 3.1. These results also suggest that HID for the series 5 rubble-mound varies with £ such that larger heights are required to initiate damage at smaller values of £. This implies that wave period has a considerable effect on armour stability for this milder slope. Hudson's design equation fails to predict this wave period effect; however, the observed trend in HID with £ is consistent with predictions from the design equations of Losada and Gimenez-Curto (1979), and of van der Meer (1988). In terms of the type of wave breaking that prevails at different values of £, the data from test series 5 shown in Figure 3-10 suggest that waves that interact with a rubble-mound as collapsing breakers (£ ~ 3) are more damaging than those that interact as plunging breakers (£ < 2). The data from test series 2, 3 and 4 shown in Figure 3-9 indicate that surging breakers (£ > 4) and collapsing breakers are similarly damaging. These trends are consistent with van der Meer's design formulae for relatively impermeable rubble-mounds (test series 2, 3 and 5). The factors responsible for these trends in damage response, including variations in surface flows and forcing of the armour, will be considered in Chapters 5 and 7. 3.2.2 Irregular Waves Results on the stability of rock armour in irregular waves were obtained in both the exploratory and N R C experiments. These experiments are described in Sections 2.2 and 2.3 respectively. In the N R C experiments, irregular waves with Tp = 2 s were most commonly used. Waves with Tp = 1.67 and 2.5 s were also applied to some structures. Several different significant wave heights were applied at each peak wave period. The damage after 3000 stationary irreg-ular waves was determined from profiles of the rubble-mound and by counting the number of displaced armour stones. The rubble-mound was re-built after each test of 3000 waves. The significant wave height required to initiate damage HSJD is determined by fitting a damage curve to the results with constant peak period and interpolating the value of HS corresponding to S = 1. This procedure is similar to that used to determine HID f ° r regular waves. Figure 3-11 shows a damage curve from test series 3 for irregular waves with Tp = 2.5 s. 81 2.5 w 1.5 0.5 1.6 1.7 1.9 2 2.1 H / A D a * r 2.2 2.3 2.4 n50 Figure 3-11: Damage curve for irregular waves (test series 3, Tp — 2.5s). The method of testing used in the exploratory tests differed from that of the N R C tests, and is similar to that used by Thomsen et. al. (1972). In these tests, irregular waves with the same peak period and progressively larger wave heights were applied without rebuilding the rubble-mound between different wave conditions. This test procedure simulates the evolution of wave climate during the growth of a storm. Each wave height was applied until equilibrium damage was attained. Equilibrium damage was considered to prevail when the rate of damage growth with time approached zero. In practice, the criterion that no incremental damage occurred during a 600 s duration of wave activity was used to identify this condition. This duration equals the recursive period of the wave machine control signal, and was thus sufficiently long to include all unique waves. Figure 3-12 shows the growth of damage with increasing test duration for one of these tests. Salient characteristics of the three rubble-mound breakwaters investigated in the exploratory tests are summarized in Table 2.1. Structure E l consists of an impermeable core, a thin filter layer and two layers of finely graded armour stones with Z?U50 = 4.4 cm at a slope of cot a — 1.5. 82 4 3 ra 2 1 1 1 • H 6 / A D n 5 0 = 1 20 • - A H s / A D n 5 0 = 1 57 • -• H e / A D n 5 0 = 1 83 • H B / A D n 5 0 = 2 17 • • • • A A A • 1 A A 1 1 0 1000 2000 3000 4000 5000 N w Figure 3-12: Growth of damage for an exploratory test (structure E2, Tp = 2 s). Structure E2 is a variant of E l with a milder slope of cot a = 3, while structure E3 is a second variant of E l in which the filter and impervious core are replaced by a permeable core. Structures E l and E3 are slightly steeper versions of the breakwaters used in N R C test series 2, 3 and 4; while structure E2 is similar to the structure used in N R C test series 5 and 6. Test structures E l , E2 and E3 were located in 40cm water depth on a foreshore with slope cot a = 20. Significant wave heights required to initiate damage on the more permeable test structures (E3 and series 4) are presented in Figure 3-13. Here, wave heights are presented in the non-dimensional form HsjD/ADn5o as a function of surf similarity £ m = tcmay/gT%l/2iTHs. Figure 3-13 also shows predictions of HSJD based on van der Meer's design equations (Equations (3.10) and (3.11)). For the E3 structure, these were applied using Nw = 1000, P = 0.4, 5 = 1 and cot a = 1.5. For the series 4 rubble-mound, Nw = 3000, P = 0.4, 5 = 1 and cot a = 1.75 were used. Figure 3-13 indicates that the significant wave heights required for initial damage observed in 83 2 3 4 5 £ = tana ( S T V 2 T T H ) 1 / 2 Figure 3-13: Significant wave heights required to initiate damage on permeable-core structures compared to predictions of van der Meer's design formulae with S = 1. these experiments are considerably greater than those predicted by the design equations of van der Meer. These results suggest that the predictive equations provide conservative estimates of the initiation of damage threshold for these permeable rubble-mounds. On average, the HSJD predicted by van der Meer's design equation are approximately 2 2 % less than the observed values. Part of this discrepancy may be related to uncertainty in the selection of the most appropriate value of the permeability factor P. It is also possible that the roughness, angularity, fine gradation and placement density of the armour stones used in these tests combine to give a more stable armour layer than was used in the experiments of van der Meer (1988). A third possibility is that the damage level S = 1 is below the range of S for which van der Meer's equations are valid. In the present study, mild damage has been precisely quantified by counting the number of displaced armour stones, whereas van der Meer relied solely on profiling the rubble-mound. The difference in measurement techniques used could contribute to the bias observed between the present results and predictions from van der Meer's design equations. As 84 f = t a n a (gT2/2*U )1/2 Figure 3-14: Significant wave heights required to initiate damage on permeable-core structures compared to predictions of van der Meer's design formulae with 5 = 2. mentioned previously, van der Meer's design equations erroneously predict some finite damage at very low wave heights, where in reality no damage would occur. These equations incorrectly represent the relation between 5 and HS for very small values of 5 . Figure 3-14 shows that the agreement between observed and predicted values of HSJD are significantly improved if the design equations of van der Meer are applied with 5 = 2 instead of 5 = 1 , while all other values remain unchanged. Wi th 5 = 2, the bias towards lower predicted values of HSJD is reduced to 1 0 % . The observed trend of HSJD with £ m is quite similar to that predicted by van der Meer's design equations, and suggests that wave period has some influence on the initiation of damage threshold in the plunging wave regime, and that minimum stability prevails for £ m ~ 4. Figure 3-15 shows the same HSJD results for the permeable E3 and series 4 rubble-mounds compared to predictions from Hudson's design equation, extended to account for irregular waves and storm duration as suggested by Medina and McDougal (1988) (Equation (3.6)). 85 — I 1 1 I 1 1 • • • • • Hudson, E3 Hudson, series 4 • E3 • Series 4 I I I I I I 0 1 2 3 4 5 . 6 7 £ = tana (gT 2 /2T T H ) 1 / 2 s m v & m ' a' Figure 3-15: Significant wave heights required to initiate damage on permeable-core structures compared to predictions from Hudson's design formula extended to account for irregular waves and storm duration. These estimates were obtained using KD — 4, with NW = 1000 for the E3 tests and NW = 3000 for test series 4. Agreement between the observed and predicted values of HSJD is quite good near to £ ~ 4, which suggests that Equation (3.6) gives a decent estimate of the minimum significant wave height required to initiate damage on these rubble-mounds. However, the extended Hudson formula fails to account for the effect of wave period on the stability of the armour. Initiation of damage significant wave heights for test structures with an impermeable core are compared in Figure 3-16 to estimates from van der Meer's design formulae with 5 = 2. For the E l and E2 structures, these formulae were applied with NW — 1000 and P = 0.2, while NW — 3000 and P = 0.2 were used for the series 2 and series 3 rubble-mounds. Between 1 and 4 results are available for each structure, and in several cases these results show significantly different values of HSJD for similar wave conditions. On average, the milder sloped E2 structure (cot a — 3) is more stable than the steeper rubble-mounds. The estimates of van der Meer's 86 2.5 I D c < 1.5 Q a l 0.5 2.5 C < 1.5 K 1 0.5 O O E l v a n der Meer, E l H E2 v a n der Meer, E2 • Series 2 v a n der Meer, series 2&3 • Series 3 i i i i i i 1 2 3 4 5 6 f = t a n a (gT 2 /2T T H ) 1 / 2 Figure 3-16: Significant wave heights required to initiate damage on impermeable-core struc-tures compared to predictions of van der Meer's design formulae with 5 = 2. design formulae tend to form a lower bound to the observed values. The overall trend towards greater stability at lesser £ is similar to that predicted by the design equations of van der Meer. In Figure 3-17, the same data are shown compared to estimates of the Hudson design equation as extended by Medina and McDougal (1988) to account for irregular waves and storm duration. These estimates were obtained from Equation (3.6) with KD = 4. Although individual data points show significant deviations from the predictions of the modified Hudson formula, in general they suggest that it can be used to estimate the minimum values of HSJD for these relatively impermeable breakwaters. 3.3 Influence of Wave Height Wave height is by far the dominant environmental variable affecting the stability of armour stones. According to linear wave theory, the average energy density of waves can be written in terms of their wave height as ^H2. In shallow water, the average forward energy flux is 87 2.5 2 -a < 1.5 0.5 cx IX) IX • o "~~0~ o E l H u d s o n , E l IX E2 H u d s o n , E2 • S e r i e s 2 " H u d s o n , s e r i e s 2 & 3 • S e r i e s 3 1 1 1 1 1 1 1 2 3 4 5 6 t a n a ( g T ^ / 2 T T H ) 1 / 2 Figure 3-17: Significant wave heights required to initiate damage on impermeable-core struc-tures compared to predictions of Hudson's design formula extended to account for irregular waves and storm duration. y/gh^-H2. Wave height can thus be considered to parameterize the energy density and energy flux of waves incident to a rubble-mound. As waves propagate onto a rubble-mound, the kinetic and potential wave energies are either reflected, dissipated or transmitted, depending on the characteristics of the waves and the structure. Energy dissipation occurs through wave breaking, bottom friction, and internal flows within permeable zones of the breakwater. The complex fluid flows that result exert forces on armour stones that can lead to damage of the armour layer. The strong influence of wave height on damage can be seen in the typical damage curves shown for regular and irregular waves in Figures 3-8 and 3-11 respectively. Design formulae for rubble-mound armour also predict a strong dependence on wave height. In particular, van der Meer's equations give S oc i f 3 7 5 for plunging waves and 5 oc H^+2bP for surging waves. Van der Meer's equations indicate that the influence of wave height is fundamentally different for plunging and surging waves, which suggests that different mechanisms are responsible for 88 D% 0-5 5-10 10-15 15-20 20-30 30-40 40-50 5 H/HD%=0 2 6 10 14 20 28 36 1.00 1.08 1.19 1.27 1.37 1.47 1.56 Table 3.2: Wave height multipliers for different damage levels (from SPM-84). generating damage in these two regimes. SPM-84 presents multipliers that can be applied to H for levels of tolerable damage greater than 5 %, including a value of 1.56 for the removal of 50 % of a rough quarry-stone armour layer. The multipliers H/HD%=Q are listed in Table 3.2, along with equivalent 5 values computed according to 5 = 0.8D%. These wave height multipliers can be used to develop an extended form of Hudson's design equation that explicitly includes the level of damage in terms of 5 , as = 0 . 9 5 0 1 5 {KD cot a) 1 / 3 . (3.17) A-L>n5o This extended Hudson design formula is equivalent to Equation (3.2) for 5 = 2. Wi th 5 as the dependent variable, this equation becomes 5 = 2.02 • Kj™ ( tana) 2 ' 2 2 ( ^ ) 6 ' 6 ? , (3-18) which suggests that 5 oc H667. This trend between damage and regular wave height is similar to that predicted by van der Meer's formulae for surging irregular waves on permeable rubble-mounds. The extended Hudson equation, given by either Equation (3.17) or (3.18), suffers the same deficiency mentioned previously for van der Meer's design formulae; namely that it indicates some finite damage for all values of H > 0, and cannot therefore be considered accurate for small 5 . 89 3.4 Influence of Wave Period For a rubble-mound with fixed slope angle subject to attack by waves with constant wave height, the wave period controls the type of wave breaking that occurs, which in turn affects the flow throughout the armour layer and the forces acting on, and stability of armour stones. Giinbak (1979) investigated the effect of wave period on the flow characteristics and stability of rubble-mound breakwaters, and concluded that minimum stability in regular waves occurs for £ ~ 3 where collapsing breakers prevail. He applied the term "breakwater resonance" to describe the surface flows associated with minimum stability. Giinback further observed that resonance conditions generate a strong eddy-like turn of the velocity field near the point where the downrush meets the incident wave crest, which produces large velocities and accelerations directed away from the surface of the rubble-mound. Other investigators, including Ahrens (1975), Losada and Gimenez-Curto (1979), Sawaragi et.al. (1983) and van der Meer (1988) have also identified collapsing breakers as the critical type of wave breaking for stability. Van der Meer's design equations indicate that the critical value of surf similarity for minimum stability in irregular waves is given by which depends on permeability and slope angle. These design equations also indicate that damage increases with increasing wave period according to S oc T^5 for plunging waves ( £ m < £m,cr), but decreases gradually with increasing wave period according to S oc T ~ 5 P for surging waves ( f m > 3.5 Influence of Slope Angle The angle of the sea-ward slope of a rubble-mound influences the stability of armour stones in two ways. (3.19) 90 Series 3 Series 5 Series 5 / Series 3 slope cot a /Up , average of observations • 7 $ P - , Hudson /*' D , van der Meer 1.75 1.77 3 2.18 1.71 1.22 1.20 1.31 Table 3.3: Influence of slope angle on initiation of damage wave height for regular waves. 1. It influences the type of wave breaking that occurs for a given wave condition, which in turn affects the fluid kinematics on and within the rubble-mound, and the resulting hydrodynamic forcing applied to the armour stones. 2. It influences the proportions of stone weight that act normal and tangential to the surface. Stability is reduced as greater proportions of stone weight act tangentially down-slope. In the limit as a approaches the friction angle <f>, armour stones become unstable without any additional hydrodynamic forcing. This factor acts to reduce armour stability on steeper slopes. Hudson's design formula, extended to include the damage level, gives S oc ( tana) 2 ' 2 2 (Equa-tion (3.18)), while van der Meer's formulae give S oc ( tana) 2 ' 5 for plunging waves (Equation (3.13)) and S oc ( t a n a ) 2 ' 5 _ 5 P for surging waves (Equation (3.14)). The Hudson and plunging-wave van der Meer equations predict a similar positive trend between damage and slope angle. However, the van der Meer equation for surging waves predicts a dependence on slope angle that is reduced by permeability to the extent that damage is independent of slope angle for permeable rubble-mounds with P = 0.5. Results from test series 3 (cot a = 1.75) and 5 (cot a = 3) can be compared to assess the effect of slope angle on the regular wave height required to initiate damage. Average values of non-dimensional HID for these tests are summarized in Table 3.3 along with predictions from the design equations of Hudson and van der Meer. On average, 22 % higher waves were required to initiate damage on the milder sloped series 5 rubble-mound. For these slopes, Hudson's design equation predicts a percentage increase in HID given by ^ / l ^ ) 1 / 3 — 1 • 100 = 2 0 % , while 91 E l E2 E2 / E l slope cot a 1.5 3 2 f!nID , average of observations 4 ^ 2 - Hudson 5?P,ID , van der Meer 1.51 2.10 1.39 1.26 1.50 Table 3.4: Influence of slope angle on initiation of damage wave height for irregular waves. van der Meer's equations predict an increase that varies with wave period, but averages to 31 % for the periods considered here. (To generate this estimate, van der Meer's equations were applied with 5 = 2, Nw = 1000, P = 0.2, cj = £ m and H = l.27HS.) The observed influence of slope angle on HID compares well to the estimate based on Hudson's design equation. The influence of slope angle on the significant wave height required to initiate damage in irregular waves can be assessed by comparing results for structures E l (cot a = 1.5) and E2 (cot a = 3). Results from this comparison are presented in Table 3.4. On average, 3 9 % larger waves were required to initiate damage on the milder sloped E2 rubble-mound. This increase lies between the 2 6 % estimated by Hudson's design equation and the 5 0 % predicted by the design equations of van der Meer. A l l results in regular and irregular waves indicate an increase in armour stability on milder slopes. 3.6 Influence of Core Permeability On rubble-mound breakwaters where the core is relatively impermeable, the flows generated by wave attack are concentrated on the surface and within the armour layer. On more homogeneous structures, the wave induced flows are able to penetrate further into the structure, and tend therefore to be less concentrated on the surface. Permeable rubble-mounds feature an increased capacity for energy dissipation which, improves the stability of the armour and reduces wave reflections. Results on the wave height required to initiate damage from test series 3 and 4 can be compared to assess the effect of core permeability on armour stability for regular waves. The 92 Series 3 Series 4 Series 4 / Series 3 core &j}D j average of observations 7 $ P ~ , Hudson Jilp , van der Meer impermeable 1.77 permeable 2.09 1.17 1.00 1.21 Table 3.5: Influence of permeability on initiation of damage wave height for regular waves. E l E3 . E3 / E l core ^p,ID , average of observations , Hudson A U „ 5 0 ' ?p.ID , van der Meer impermeable 1.51 permeable 1.76 1.16 1.00 1.19 Table 3.6: Influence of permeability on initiation of damage wave height for irregular waves. comparison is summarized in Table 3.5, which shows that an average increase in HID of 1 7 % was observed for the more permeable series 4 structure. This increase agrees well with the 21 % increase predicted by van der Meer's design formulae when applied with P = 0.2 for test series 3 and P = 0.4 for test series 4. Permeability effects are not considered by the Hudson equation. A similar comparison of the influence of core-permeability under irregular waves is summa-rized in Table 3.6. In this case, the average observed increase in HSJD of 16% for the more permeable E3 rubble-mound is again in good agreement with the 19 % increase predicted by the design equations of van der Meer. These results indicate that armour stones tend to be more stable on more permeable structures. 3.7 Influence of Storm Duration Among other factors, armour damage depends on both the intensity and duration of wave attack. Once the intensity of incident waves exceeds the threshold required to initiate damage, the damage that accrues depends on the duration of wave attack. This reflects the progressive nature of the erosion process, which can be described as follows. Typically, loose armour stones near the still waterline are gradually rocked and shifted by the more severe waves into less 93 stable positions and are eventually displaced and transported down the rubble-mound. This frees up space for neighbouring stones which in turn are shifted into less stable positions and eventually displaced. Damage generally progresses with time so long as the severity of waves is maintained above the initiation of damage threshold, and marginally stable armour stones remain exposed to wave action. As the number of marginally stable armour stones is reduced (through shifting and displacement), the rate of erosion tends to decrease until an equilibrium condition develops in which all remaining armour stones are sufficiently stable to resist erosion, regardless of the duration of further wave activity. Van der Meer's design formulae (Equations (3.13) and (3.14)) indicate that the duration of wave attack and the resulting damage are related according to 5 oc y/Nw. This relation can be used to generate an estimate of the damage after Nw waves 5 (Nw), in terms of the damage after 3000 waves 5(3000), as s { N w ) = 7m'5 ( 3 0 0 0 ) = ^ s ? ' 5 ( 3 0 0 0 ) • ( 3 - 2 0 ) A slightly different estimate of the relation between damage and storm duration for irregular waves can be obtained from the extended Hudson equations presented earlier in this chapter. Equations (3.17) and (3.6) can be combined to yield which includes the effects of both storm duration and damage level. This reduces to Equation (3.4) for 5 = 2 and Nw = 1000, if H1/10 = \.27Ha is assumed. Wi th 5 as the dependent variable, Equation (3.21) becomes s = h1^2'22 ( T A N A ) 2 ' 2 2 (lnNwf33 ( A ^ ) 6 W > ( 3 - 2 2 ) which predicts that 5 oc ( ln iV^) 3 - 3 3 . According to Equation (3.22), damage after Nw waves can 94 2.5 o o o CO TO 2 TO 1.5 0.5 A V n i i e x t e n d e d H u d s o n van der Meer yu. of o b s e r v a t i o n s /x + a of o b s e r v a t i o n s - a of o b s e r v a t i o n s A A A J _ 1000 2000 3000 4000 N 5000 6000 7000 Figure 3-18: Growth of damage with storm duration for irregular waves. be expressed in terms of damage after 3000 waves as s ^ ) = ( « ) 3 3 3 - S ( 3 0 0 0 , = 1 ; T - S ( 3 0 0 0 ) ' < 3- 2 3 ) Data on the growth of damage with increasing storm duration are available from 12 irregular wave tests from the exploratory and N R C test programs. Results from each test were normalized as S (Nw) /S (3000) with Nw = 750, 1500, 3000, and 6000. The mean value a and the mean ± one standard deviation a of normalized damage are plotted in Figure 3-18 together with predictions from Equations (3.20) and (3.23). These two equations predict a very similar relation between damage and storm duration that is in good agreement with the present results. The good agreement is not particularly surprising, since the results are presented in normalized form, and are therefore forced to pass through the points (0,0) and (3000,1). As a result of this normalization, cr = 0 at Nw = 3000. These results show that the rate of growth of damage is initially rapid, but continually decreases at extended storm durations. 95 Chapter 4 Failure Mechanisms for Armour Stones 4.1 Forces on Armour Stones Consider an armour stone at rest on the surface of a rubble-mound breakwater with slope cot a subject to gravitational, contact and fluid forces, as sketched in Figure 4 -1 . The forces acting are • F'b{t) — buoyancy force; • = weight of the stone in air; • F^w(t) = submerged weight = F'w -• F'(t) = hydrodynamic force due to wave attack; • F'N(t) = component of F'(t) normal to the surface; • F'P(t) = component of F'(i) parallel to the surface; • F'c(t) = forces resulting from contacts with neighbouring stones. 96 Figure 4-1: Definition sketch of the forces acting on an armour stone. Primes are used here to distinguish the forces acting on a single armour stone from those acting on an armour panel, while symbols written in bold face denote vectors whose direction may change with time. Except for the weight F^, all of these forces can vary with time. The buoyancy force Fb\t) remains constant while the stone is fully submerged, but is time-varying while the free surface passes over the stone, and equals zero while the stone is not submerged. For the period of interest during which the stone remains fully submerged, the magnitude of the buoyancy force is equal to the weight of water displaced by the stone, and can be written F'b = pgDn where Dn = (M/ pa)x^ is the nominal stone diameter, M is the stone mass, p is the density of water, pa is the stone density and g is the acceleration due to gravity. The buoyancy force always acts vertically upwards, opposite to the gravitational acceleration. The weight of the stone in air can be written F^ = pagD^. The submerged weight is given by the vector sum of the stone weight and maximum buoyancy force, which can be written F'sw = (pa — p) gD^ and always acts vertically downwards for realistic armour stones that are more dense than water. 97 Wave attack generates unsteady fluid flow around the armour stone. This flow causes the pressure distribution on the surface of the stone to deviate from that under static conditions. The time-varying hydrodynamic force F'(t) can be considered as the net force resulting from an integration of the pressure deviations due to fluid flow at each instant in time. In Figure 4-1, the hydrodynamic force is shown separated into two orthogonal components, parallel and normal to the surface of the rubble-mound. The magnitude of the slope-parallel component is denoted F'P(t) and is defined positive up-slope while the slope-normal component is defined positive away from the rubble-mound and is denoted by F'N(t). The magnitude of the hydrodynamic force is F'{t) = |F'(i)| = ^Fl2{t) + F'2{t). The hydrodynamic force is commonly assumed to consist of contributions that are propor-tional to the velocity and acceleration of the surrounding fluid. For example, Kamphuis (1966) modelled the fluid force acting on sediment particles on a horizontal bed under waves as a sum-mation of components due to skin friction, form drag, lift, added mass and pressure. Denoting the velocity parallel to the bed by u, the forces due to skin friction and form drag were taken to be proportional to u\u\, the lift force was taken to be proportional to u2, while the added mass and pressure force were assumed proportional to the acceleration ^ . Kobayashi and Otta (1987) used a similar approach to model the fluid force acting on rubble-mound armour. They considered force components due to drag oc u\u\, inertia oc and lift oc u2, where u is now the velocity parallel to, and just above the surface of the rubble-mound. This representation of hydrodynamic force is similar to the approach of Morison et. al. (1950) for wave forces on piles. According to this formulation, drag and inertia forces act parallel to the surface and thus contribute to F'p(t), while the lift force acts normal to it and contributes to FN(t). A n armour stone will remain stable so long as the fluid, gravitational and contact forces, and the moments generated by them, are in equilibrium. This stable condition will hold so long as sufficient contact forces can be mobilized to offset the fluid and gravitational forces acting on the stone. Clearly, in the case of fluid loading directed into the rubble-mound, very large contact forces will be mobilized to resist the external loads such that the stone wil l always 98 remain stable. However, for fluid loads directed away from the rubble-mound, such as sketched in Figure 4-1, resisting contact forces may be severely limited, in which case the stone may become unstable and be readily displaced. Armour stones on the surface of a rubble-mound feature a variety of geometries and contacts with neighbouring stones which influence their ability to mobilize resisting contact forces. The fluid loads on each armour stone will also vary due to factors such as: the shape, orientation and location of the stone; its proximity to neighbouring stones; and spatial variations in the external flow field. Considering that both the character of fluid loading and the ability of individual stones to mobilize resisting contact forces are variable, it is not surprising that damage to rubble-mound armour is a stochastic process that develops gradually over a range of wave heights. 4.2 Failure Mechanisms for Armour Stones In this section, five different modes of failure are considered for armour stones on the surface of a rubble-mound under wave attack. For each of these, the motion threshold for individual armour stones is developed by considering the balance of driving and resisting forces on a single stone. The analyses are restricted to the case of fully submerged armour stones. A failure index is defined for each failure mode in terms of the ratio of driving and resisting forces. 4.2.1 Lifting Failure Mode Lifting failure refers to the translation of an armour stone normal to the surface of the rubble-mound. In this failure mode, the normal component of the hydrodynamic force acts to de-stabilize the armour stone. The magnitude of the time-varying driving force can thus be written F'dr(t) — F'N(i). This driving force is resisted by the slope-normal component of submerged weight F{.wcos(a) and by any contact forces that can be mobilized opposite to F'N(t). The magnitude of the time-varying resisting force can be written F/.e(t) = F'sw cos(a) + F'c(t) where 99 F'c{t) denotes the contact force mobilized normal to the surface of the rubble-mound specifically to resist lifting failure. For a stable armour stone, force equilibrium implies that F'N(t) = F'swcos(a) + F'c{t). (4.1) To remain stable against lifting failure, the armour stone must be able to mobilize sufficient resisting contact force to maintain this equilibrium. The maximum value of resisting contact force that can be mobilized will depend on the geometry of the stone and its relation with its neighbours. The 'weakest' stones on a rubble-mound will not be able to mobilize any contact force against motion away from the surface of the structure, such that F'c(t) < 0. These weakest stones will be the first ones displaced through lifting failure. The time-varying lifting failure index that applies to these weakest armour stones is denoted R'L (t) and is defined by the ratio of the driving force to the maximum resisting force, which can be written R' _ Kyi*) _ F'N(t) (A ox R ^ --FJT- FL^cMoT) • ( 4 2 ) This failure index is defined such that R'L < 1 indicates a stable situation, while R'L > 1 indicates that the stone is unstable and could be displaced through translation normal to the surface of the rubble-mound. (Displacement will occur if sufficient impulse acts to overcome the stone's inertia and accelerate it into motion.) Minimum stability against lifting failure occurs at the instant when R'L(t) is a maximum. According to this failure index, the stability of armour stones depends only on their sub-merged weight, the normal component of hydrodynamic forcing and the slope of the rubble-mound. Unstable conditions (R'L > 1) are indicated whenever F'N(t)/F'sw > C O S ( O J ) while sta-bility (R'L < 1) prevails for all hydrodynamic forcing with F'N(t)/F'sw < cos(o;). Slightly greater hydrodynamic forcing is required to precipitate instability on more mildly sloping structures. In the absence of waves, F'N(t) = 0 and R'L(t) = 0. 100 4.2.2 Hudson Failure Mode The design equation for the stability of rock armour proposed by Hudson (1958) is considered in some detail in Section 3.1.3. It can be written as H = (KDcota)l/s . (4.3) Hudson arrived at this design equation by considering the balance between a fluid force and a resisting force at the motion threshold of an armour stone. Hudson formulated the fluid force as F' = CfpgD2H where Cf is an unknown force coefficient and D is a representative dimension of the armour stone. Friction between armour units was explicitly neglected and the resisting force was assumed to equal the submerged weight of the armour stone, which was written as F'sw — g(pa — p)kvD3, where kvD3 = D„ represents the stone's volume. At incipient instability, Hudson assumed these two forces to be in balance so that F' = F'sw, which implies that F' acts vertically upwards, opposite to the submerged weight. This force balance can be written Hudson used experiments in regular waves to conclude that the coefficients on the right hand side of Equation (4.4) could be replaced by ( i^^cota) 1 / 3 where KQ is a "damage coefficient" that is now considered to depend on many factors, including: the shape, roughness and packing of the armour stones; the number of layers of armour stones; the type of wave breaking; the permeability of the rubble-mound; the duration of wave loading; and the tolerable level of damage. A failure index can be defined using the same balance of forces used by Hudson to develop his design equation. Denoting the driving force by F'dr(t) = F'(t) = JF$(t) + F'p2(t), and the 101 resisting force by F'Te = F'sw, leads to a Hudson failure index defined by m)_^m+m. (4,, "re sw According to this failure index, the stability or failure of armour stones depends only on the magnitude of the fluid force and the submerged weight. The index is not influenced by the direction of the fluid force or the slope angle of the structure. In the absence of waves, F'P(t) = F'N(t) = 0 and R'H(t) = 0. 4.2.3 Sliding and Rolling Failure Modes Kobayashi and Otta (1987) considered three failure modes for rock armour, namely lifting, sliding and rolling. They concluded that the criteria for incipient motion in the rolling and sliding failure modes were identical, and that this criterion governed the stability of armour stones under realistic situations. Sliding failure refers to the translation of an armour stone, either upwards or downwards along the surface of the rubble-mound. Driving forces for this mode are the slope-parallel components of the fluid and gravitational forces acting on an armour stone, which can be written F'dr(t) = \F'P(t) — F ^ s i n a l , where the absolute value is taken to account for driving forces acting both up-slope and down-slope. In this mode, resisting contact forces are mobilized through friction against adjacent armour stones. The resisting frictional force is expressed in terms of the friction angle (j) and the slope-normal component of the fluid and gravitational forces, and can be written F^e(t) = tan <t>[F'sw cos a — F'N(i)]. The failure index for this sliding mode is given by the ratio of these two forces RI m - EiM - \FP(t)-F'sw sin a\ "sUdeW p U t ) t a n r w C O g a _ ^ ( t ) ] - (4.0) This index indicates the relative stability of armour stones against both up-slope and down-slope 102 Figure 4-2: Definition sketch of forces for rolling failure. sliding failure. Rolling failure refers to the down-slope rotation of an armour stone about a point of support, such as the point A in Figure 4-2. In this sketch, the hydrodynamic force is assumed to act at the stone's centre of gravity, denoted as point O, and the location of A is assumed such that an increase in the slope angle a to the natural angle of repose cp places point O directly above point A. Thus, in the absence of hydrodynamic forcing, the armour stone becomes unstable when a exceeds <$>. For a < <f>, the stability threshold is obtained when the moments taken about the point of support sum to zero. Denoting the distance between A and O by kDn, this condition can be written as By substituting sin (<f> — a) = sin <j> cos a — sin a cos <p and dividing through by kDn cos </>, this becomes kDn [F'N(t) sin cf> - F'P(t) cos 4> - F'sw sin (<f> - a)] = 0 . (4.7) F'N(t) tan 4> - Fp(t) — F'sw [tan 0 cos a — sin a] = 0 (4.8) 103 which is equivalent to F>wsma-FP(t) tan0[i^cosa-i^(t)] " ' ; Equation (4.9) defines the relationship between hydrodynamic and gravitational forces at the stability threshold of the armour stone sketched in Figure 4-2. The left side of Equation (4.9) can be treated as an index for the down-slope rolling failure of armour stones, which is defined as r>i m _ F'swsma-F'P(t) Stable conditions are predicted for R'rou < 1, while unstable conditions prevail for RroU > 1. The two indices developed here for sliding and rolling failure (defined by Equations (4.6) and (4.10) respectively) are equivalent whenever the slope-parallel component of the hydrodynamic force satisfies Fp(t) < . F ^ s i n a . Under this constraint, the numerators of Equations (4.6) and (4.10) are equal, i.e. \Fp(t) - F'sw sin a\ = F'sw sin a - F'P{t) for FP(t) < F'sw sin a . (4.11) This constraint includes all conditions in which the slope-parallel component of fluid force acts down-slope (F'P(t) < 0) and thus can be assumed to apply for all situations in which down-slope rolling might reasonably occur. This implies that the failure index for down-slope rolling failure can be combined with the index for up-slope and down-slope sliding failures. The resulting rolling-sliding failure index for an armour stone is defined by & m \F'P{t) - F^w sina\ R R { t } ~ tan t[F>w cos a-F>N(t)} ' The complex dependence of R'R on both the slope-normal and slope-parallel components of fluid force can be seen in Figure 4-3, which shows a contour plot of R'R in the force plane defined by F'P/F'SW and F'N/F'SW for a structure with cot a = 1.75 and (f> = 40°. This figure 104 -1.0 -0.5 0.0 0.5 1.0 F'/F' P sw Figure 4-3: Contour plot of the rolling-sliding failure index (cot a = 1.75, c/> = 40°). applies to the rubble-mounds investigated during series 2, 3 and 4 of the N R C experiments. Two wedge-shaped zones with R'R > 1, to the left and right of the line F'P/F'SW — s ina , indicate the potential for sliding failure in the down-slope and up-slope directions, respectively. The zone for down-slope rolling failure is identical to that for down-slope sliding. According to this failure mechanism, the magnitude and direction of the fluid force are both important factors affecting stability. For a given magnitude of fluid force, stability is minimized when the force acts down-slope and away from the rubble-mound (F'P < 0 and F'N > 0). When the fluid force acts in other directions, greater magnitudes are required to produce equally unstable conditions. The rolling-sliding failure index becomes negative for F'N/F'SW > cos a . Thus, F'N = F'swcosa represents an upper limit to the range of slope-normal fluid force for which Equation (4.12) remains valid. Figure 4-4 shows a similar contour plot of R'R for the milder sloped rubble-mound used in test series 5 and 6, for which cot a = 3 and <f> = 40°. On this milder sloped structure, more 105 0 F' /F' 1 P' 1 s Figure 4-4: Contour plot of the rolling-sliding failure index (cot a = 3, (j) = 40°). fluid force is required to precipitate down-slope failure while less force is required to initiate up-slope instability. Without waves, F'N(t) = F'P(t) = 0 and R'R(t) reduces to . , . t a n a RR(t) = without waves. H W tan0 (4.13) Thus, without waves, RR(t) = 0.68 and 0.4 for the rubble-mounds with cot a = 1.75 and cot a = 3 respectively. 4.2.4 Shields Failure Mode In this section, a failure index for armour stones on the surface of a rubble-mound is developed from consideration of the shear stress acting on the armour and the well known Shields curve. Shields (1936) presented a curve to describe the incipient motion of sediments on a flat horizontal bed subject to steady flow. Shields assumed the forces acting to move sediment grains to be 106 Figure 4-5: Definition sketch for Shields failure on a horizontal surface. proportional to the shear stress exerted by the fluid on the bed, and to the Reynolds number of the flow. Others (summarized by Nielsen (1992), Sleath (1984) and Yalin (1977)) have extended this work to permeable beds, rippled beds, beds on mild slopes, and beds under oscillatory flow. The condition obtained by Shields for steady flow can be derived by considering a particle on the surface of a horizontal bed subject to shear stress To(£), as sketched in Figure 4-5. For steady flow, the bed shear stress may be time-varying due to turbulent fluctuations. The fluid force is assumed to depend on the Reynolds number Re* = u*~Dn/v where u*~ — \JTQ/p is the time-averaged shear velocity, TQ is the time-averaged value of To(t), and v is the kinematic viscosity. Re* is considered to characterize the pattern of fluid flow around the particle. The time-averaged fluid force can be written while the submerged weight is given by F'sw = (pa - p) gD3. At the threshold of motion, the moments about a point of rotation (point A in Figure 4-5) are in balance, which provides F' = %Dn-MRe*) (4.14) (4.15) 107 where l\ and l2 are the respective moment arms of the fluid force and submerged weight. The point of action and direction of F' are assumed to depend on the pattern of fluid flow around the particle, thus h can be written l\ — Dnf2(Re*)- The moment arm for the submerged weight is not dependent on the flow field, thus I2 can be written I2 = k\Dn. Using these relations, the criterion for incipient motion in steady flow becomes Substituting T c for the critical value of ro corresponding to the initial motion of particles, this expression reduces to Shields used experimental data to define the relationship between r c /(p a — p)gDn and Re* for incipient motion. For rough turbulent flow, f(Re*) was found to approach a constant value approximately equal to 0.05. A similar development can be followed to derive the incipient motion criterion for an armour stone subject to steady flow on an inclined surface. The case with steady flow down a slope inclined by a with respect to horizontal is sketched in Figure 4-6. For consistency with sign conventions adopted earlier, shear stresses are negatively valued for this case. The magnitude of the fluid force can be written F' = \fo\D2fi(Re*), the submerged weight is F'sw = (pa — p)gD3, and the moment arm for the fluid force about the point of rotation is l\ = Dnf2(Reif). On the inclined surface, the moment arm for the submerged weight becomes a function of the slope angle a , and can be written 1% = Dnfs(a). By considering the components of F'sw normal and parallel to the surface, /3 (a) can be expanded to fz (a) = k\ cos a — k2 s ina . Taking moments about the point of rotation (point A in Figure 4-6), and substituting these expressions for F', 75£>2/i(ite,) • Dnf2(Re.) - [Pa - p)gDl • kxDn = 0 . (4.16) Tc ki = /(ite,) . (4.17) (Pa ~ p)gDn /i(i?e*) • f2{Re*) 108 Figure 4-6: Definition sketch for Shields failure on an incline. JFJIOJ k and l2 yields I TO I Dnfi(Re*) • Dnf2(Re*) - (pa - p)gD^ • Dn (fct cosa - A; 2sina) = 0 (4.18) which is equivalent to (Pa - p)gDn {cos a sin< fi{Re*) • f2(Re*) V h (4.19) Equation (4.19) applies to the specific case of negative shear stress acting down-slope. A more general form that applies to both negative shear acting down-slope and positive shear acting up-slope is = /(.Re.) fcosa + ^-^-sma] . (4.20) V T 0 | « i J |T0| (pa - p)gD, The constant fraction k2/k\ can be evaluated by considering conditions as a —> <j> with 7^ < 0 When the slope angle approaches the natural angle of repose for the particles, very little down-109 slope shear stress is required to initiate motion. This condition is equivalent to a—></> l im [cos a — fci sina] = 0, (4.21) which is solved for A i = cot 0. Thus, the Shields criterion for the incipient motion of particles on an inclined surface subject to steady flow can be written as where r c is the critical value of |ro| at which the bed particles begin to move. Equation (4.22) predicts a lower critical shear stress for the case of down-slope flow (TQ < 0) than for up-slope flow (TQ > 0). Equation (4.22) reduces to Equation (4.17) for the case of a horizontal bed where The application of this expression to armour stones on a rubble-mound under wave attack is somewhat uncertain. Komar and Miller (1974) were among the first to indicate that the value of /(.Re*) for initial motion on a horizontal bed under oscillatory flow could be obtained from Shields' curve for steady flow. Recent surveys by Sleath (1984) and Nielsen (1992) support this practice, provided that the maximum shear stress on the bed is used instead of TQ. Sleath (1984, page 256) writes, "In steady flow r c is the critical value of the shear stress ro on the bed at which the grains of sediment first begin to move but in oscillatory flow, it ought to be taken as the total horizontal force per unit area acting on the bed rather than just the shear stress." In unsteady flow, horizontal pressure gradients within the fluid make a significant contribution to the total force on the bed particles. For turbulent oscillatory flow, the contribution due to pressure gradients is particularly strong for larger bed particles in flows with smaller water particle orbits. The shear stress responsible for the motion of armour stones on a rubble-mound is taken to be the total slope-parallel hydrodynamic force acting on the surface layer of armour. For a (4.22) a = 0. 110 single armour stone, the time-varying shear stress acting on the armour can be represented by Mt) = ^fyY^F'pit) (4.23) where n is the porosity of the armour layer and £> 2/(l — n) represents the surface area occupied by the stone. To obtain an estimate of the critical shear stress for oscillatory flow on an incline, Equation (4.22) can be modified to rc(*) = f(Re*){pa ~ P)gDn [cosa + cot s inaj (4.24) where Re* = u*Dn/v, u» = y/To/p, TO is the amplitude of To(t), and Tc(t) is the critical value of fb corresponding to the initial motion of bed particles. The critical shear stress is time-varying for the case of oscillatory flow because different levels of shear amplitude are required to initiate motion during the periods of down-slope and up-slope flow. A time-varying Shields failure index for armour stones on a rubble-mound can be formulated by normalizing the instantaneous level of shear acting on the armour by the value of critical shear required to initiate motion as predicted by Equation (4.24). This normalization can be written ism _ bse>!— . ( 4 . 2 S ) T ° W f(Re*)(pa - p)gDn (cos a + r^ j r cot cp sin a) By substituting Equation (4.23) for To(i), and noting that (pa - p) gD3 = Fj.w, a Shields failure index can be written in terms of forces on a single armour stone as m ) = M i l . d-n)I^WI ( 4 . 2 6 ) T c [ t ) VF^ f cosa + | | ^ | c o t 0 s i n a j where the Shields parameter f(Re*) has been denoted by The value of ^ most appropriate for armour stones on a rubble-mound is not clear; however, ^ ~ 0.05 can be used as an initial 111 -1 -0 .75 -0 .5 -0 .25 0 P ' sw Figure 4-7: Variation of the Shields failure index (<fi — 40°, n = 0.4 and ^ = 0.05). estimate for rough turbulent unsteady flow. As with the other failure indices defined previously, R's(t) is defined such that values less than unity indicate stable conditions while values greater than unity suggest an unstable armour stone. According to R's(t), stability depends on the component of hydrodynamic force parallel to the surface of the rubble-mound as well as the slope angle, friction angle, porosity, and Shields parameter. Figure 4-7 shows the variation of Shields failure index, computed with 4> — 40°, n = 0.4, and \I> = 0.05, as a function of F'P/F'SW for two slopes, cot a = 1.75 and 3. These curves represent estimates of the relative instability of armour stones on the steep and milder sloped rubble-mounds investigated in the N R C experiments. For each structure, zones of up-slope and down-slope instability are demarcated by F'P = 0. For up-slope failure (Fp > 0), Equation (4.26) predicts a marginal increase in the fluid force required to de-stabilize armour stones on the steeper rubble-mound. For down-slope failure, the Shields index predicts that roughly twice the fluid force is required to de-stabilize armour stones on the milder sloped structure. According to this figure, the minimum magnitude of fluid force required to precipitate failure 112 in the Shields failure mode is significantly less than for the other failure modes considered previously. This suggests that ^ > 0.05 may be more appropriate for armour stones on a rubble-mound. Without waves, F'p(t) = 0 and R'g(t) = 0. 4.3 Relation of Forces on an Armour Panel to Forces on Indi-vidual Stones In the previous section, four different failure indices are developed from consideration of the forces acting on individual armour stones. Each of these indices describes the relative stability of an armour stone with respect to a particular mode of failure. Failure indices for the lifting, Hudson, rolling-sliding and Shields failure modes are defined by Equations (4.2), (4.5), (4.12) and (4.26) respectively. Each of these failure indices can be considered as a ratio of driving forces to resisting forces. Unstable conditions are associated with index values greater than unity, while stable conditions prevail for index values less than unity. The maximum index value is of particular interest since it indicates the minimum stability for the armour stone. Measurements of the fluid force acting on armour stones under wave attack have been obtained by means of a pair of armour panels. Each panel represents a rigid assembly of 50 individual armour stones. In this section, the relationship between fluid forces acting on an armour panel and those acting on individual armour stones is investigated. 4 . 3 . 1 Fo rces Each armour panel is a rigid assembly of 50 individual armour stones. Under wave attack, the fluid force acting on a panel will be equal to the vector summation of the fluid forces acting on each constituent armour stone. Denoting the time-varying fluid force on a panel by F(t) and the fluid force on a stone by F'(i), this relationship can be written (4.27) 113 where N is the number of individual armour stones in the panel. For the armour panels considered here, N — 50. At any instant, the fluid forces acting on the constituent armour stones will likely vary. Factors contributing to this spatial variation include: the shape of the stone; the orientation of the stone; the location of the stone; the proximity and geometry of adjacent stones; and local variations in the external flow field. These factors influence the fluid force by altering the pressure distribution on the perimeter of each stone. The magnitude of the hydrodynamic force acting on an armour panel is denoted by F(t) = \F(t)\ while the magnitude of the hydrodynamic force acting on an armour stone is written F'(t) = |F'(i)|. Since the forces on each constituent armour stone can act in different directions, these force magnitudes are related by F(t) = |F(t)| < £ |F'(i)| = f> ' ( i ) . (4.28) The equality in Equation (4.28) applies to the special case in which the hydrodynamic forces on all stones within an armour panel act in the same direction. The magnitude of the hydrodynamic force on an armour panel provides a lower bound estimate to the sum of the force magnitudes on each constituent armour stone. The slope-normal and slope-parallel components of the fluid force on an armour panel and the constituent armour stones are related according to N FN(t) = ^ ^ ( t ) , (4.29) FP(t) = f > P ( t ) . (4.30) The average components of fluid force on each of the constituent armour stones are given by Fkaveit) = ^ E ^ ( t ) = ^ ( t ) , (4.31) F'p,ave(t) = jjY,*p(t) = ±FP(t). (4.32) 114 F'Nave(t) and F'Pave(t) represent spatial averages of the slope-normal and slope-parallel fluid force components acting on individual armour stones within an armour panel. Similarly, the net submerged weight on a patch of N stones is equal to the sum of the submerged weights of the constituent stones, i.e. N (^t) = EF-(*)- (4-33) Two submerged weights are defined for an armour panel. The true submerged weight FSW:Panei represents the submerged weight of the aluminum model rocks while Fsw is defined to represent the submerged weight of an equivalent patch of granite armour stones. These two submerged weights are related by Fsw = Fsw^panei • (4.34) Ppanel P where ppanei = 2700 kg/m3 is the density of the aluminum used to fabricate the model rocks and pa = 2680 kg/m3 is the density of the granite armour. 4.3.2 Failure Indices Each of the failure indices developed in Section 4.2 can be considered as a ratio of driving forces to resisting forces for a particular mechanism or mode of armour stone displacement. Here, these indices will be represented by the generic form = f{§ (4.35) where R'(t) is a generic failure index for a single armour stone and F'r(t) and F^Jjt) are the driving and resisting forces, respectively. At any instant, different values of R' will apply to different armour stones. Variations in R! among individual stones may be due to variations in the driving forces, the resisting forces, or both of these. The value of the failure index R'(t), 115 spatially averaged over a patch of N armour stones can be written ^,w4E^)4E(tH)- ( « 6 ) R'ave(t) represents the average failure index value for a patch of N armour stones. At any instant, some stones will feature R! > R'ave, while R' < R'ave will hold for others. If the failure index captures the relative instability of armour stones in the mode of failure that governs their initial motion, the likelihood of stone displacement from this patch will be in some way proportional to the magnitude and character of R'ave(t). In particular, since the condition of minimum stability corresponds to the maximum index value, the likelihood of stone displacement can be expected to be proportional to the maximum value of R'ave(t). The armour panels provide measurements of the net fluid forces acting on a patch of armour stones. These net fluid forces can be used to construct net driving and resisting forces for the armour stones in the panel. The net driving force Fdr(t) represents the sum of driving forces on each constituent stone, and can be written as Fdr(t) = Y,F*r(t) (4-37) while the net resisting force Fre(t) represents the sum of resisting forces on the constituent stones, and can be written as N Fre(t) = Y,Fre(t)- (4-38) These net forces can be used to form a failure index for the armour panel according to m ~ " • ( 4 ' 3 9 ) R(t) will be referred to as the "panel index" for the generic failure mode. In general, R(t) and R'ave(t) are not identical; however, their relationship can be explored 116 by considering the driving and resisting forces on the constituent stones at each instant, F'dr and F'Te, as random variables. The generic failure index for individual stones R' can be written as a product of these random variables F' M re (4.40) A well known theorem from multivariate statistics states that the first-order estimate of the mean value of a product of random variables is equal to the product of the means. This can be expressed symbolically as E[R'} ~E[Fdr]-E F' \- re (4.41) where E [] denotes the mean or expected value of the bracketed quantity. Furthermore, if the random variables are statistically independent, then this estimate is exact, i.e. E[R!]=E[F'dr].E F' re for independent Fdr and F'Te When applied to Equation (4.36) at each instant in time, this theorem permits (4.42) Rave{t) = E[R'{t)]^E[F'dr(t)].E (4.43) which is equivalent to ENFdr(t) ENFre(t) = R(t) . (4.44) Thus, the panel index R(t) is a first-order estimate of the average index value for the con-stituent armour stones. If the driving and resisting forces on individual stones are statistically independent, then R'ave(t) = R(t) for independent F'dr(t) and F r ' e(i) . (4.45) 117 In this case, the panel failure index is identical to the average failure index for the individual constituent stones. In either case, the failure index obtained from force measurements on an armour panel R(t) will be proportional to R'ave(t), such that R(t) can be used as an indicator of the relative stability of the individual armour stones represented by the panel. The failure indices for the lifting, Hudson, rolling-sliding and Shields failure modes, devel-oped in Section 4.2 in terms of the forces acting on an individual armour stone, can be written in terms of the forces acting on a patch of armour stones as: Fsw(t)cos(a) R„(t) - \W)-sm*Fsw(t)\ . , , R ( ) tantlcosaFs^-FNit)}' V ' m ) = d - ) W ' ) i ( 4 . 4 9 ) tyFsw (t) [cos a + \Fp\t)\ cot 0 sin a J For the lifting, Hudson and Shields failure modes, the driving forces are functions of the fluid force while the resisting forces are functions of the submerged weight, which depends only on the stone size. Numerous factors contribute to the variation of fluid forces on individual armour stones, including: the stone shape and orientation; the stone location; the proximity and geometry of adjacent stones; and local fluctuations in the external flow field. Resisting forces are developed through gravitational forcing, while driving forces are developed through hydrodynamic forcing. Since different forcing mechanisms govern the driving and resisting forces, it is not unreasonable to assume some independence between them. This suggests that the panel indices for the lifting, Hudson and Shields failure modes will be similar to the average value of the failure indices for the individual constituent armour stones. For the rolling-sliding failure mode, the driving and resisting forces are both functions of the submerged weight and the fluid force. In this case, statistical independence between F'dr(t) and F^.e(t) is less likely, with the result that will only be an approximation to R'Rave(t). 118 4.4 Wave-Induced Forcing Required for Damage In Section 4.2, four failure indices are developed to quantify the relative stability of armour stones based on the ratio of driving forces to resisting forces at the incipient motion threshold of different hypothetical failure modes. These indices are denned such that stable conditions are indicated for index values less than unity, while potentially unstable conditions are predicted for index values greater than unity. In the case of the Shields failure mode, the index value associated with incipient motion could differ from unity due to uncertainty in the value of the Shields parameter that is most appropriate for armour stones on a rubble-mound. For all indices, the likelihood that damage will occur increases with increasing index value and is greatest when the index value is maximized. In this section, failure indices computed from forces measured on an armour panel in regular waves are compared to observations of the damage to the armour. These comparisons provide an indication of the index values, and thus the wave loads required for damage, and are also used to calibrate the armour panel force measurements in terms of damage. These comparisons show that the initiation of damage is well correlated to the level of shear stress acting on the armour, and suggest that a critical level of shear stress must be exceeded before damage can occur. 4.4.1 Ca lcu lat ion of Fai lure Indices Time series of the four failure indices defined by Equations (4.46) - (4.49) were computed from the forces measured on the instrumented armour panels. Analysis was restricted to test series 3, 4 and 5, in which both armour panels were located below the still waterline. Characteristics of the rubble-mounds for these tests are summarized in Table 2.2. Test series 3 and 4 considered impermeable and permeable versions of a relatively steep rubble-mound with cot a = 1.75, while test series 5 considered a milder sloped impermeable structure with cot a = 3. In these tests, the lower panel remained fully submerged during all wave conditions while the upper 119 panel became partially emerged during attack by the larger waves. Where required, the forces on the upper panel have been compensated for time-varying buoyancy effects as described in Section 2.3.9. Figure 4-8 shows short segments of FN (t), Fp (t), RL (t), RH (t), RR (t) and Rs (£) for the upper panel of the series 3 rubble-mound in waves with H = 13cm and T = 2s. Figures 4-9 and 4-10 show equivalent time series segments for the upper panel of the series 4 and series 5 structures in similar waves. The failure indices were computed using 0 = 40°, n = 0.4, and $ = 0.05. These figures indicate that the temporal variation of fluid forces measured on the armour panel are generally consistent during each cycle of wave attack. Comparison of Figures 4-8 and 4-9 suggests that the permeability of the rubble-mound core has only a small effect on the fluid forces acting on armour stones below the still waterline. As a result, the failure indices computed for these two structures are quite similar, but not identical. The influence of core permeability on the hydrodynamic forcing of the armour layer is considered further in Section 5.3.4. Comparison of Figures 4-8 and 4-10 suggests that structure slope has a more pronounced effect on the character and magnitude of fluid forces and failure indices. In particular, the maximum values of FN (t) are slightly larger, and prevail for a shorter duration on the milder sloped series 5 rubble-mound. Positive maxima of Fp (t) are also larger on this structure. The temporal variation and positive peak values of all the failure indices differ as a result of the differences in fluid forcing on the steep and mild slopes. Slope angle has the most influence on the rolling-sliding and Shields failure indices. In the case of the Shields failure index, Figure 4-10 shows that the maximum value of Rs (t) coincides with Fp (t) that are positive, which indicates minimal stability against up-slope failure for the milder sloped rubble-mound. In contrast, Figure 4-8 shows that Rs (£) predicts minimum stability for down-slope failure on the steeper series 3 rubble-mound. In these wave conditions, the Shields failure index predicts that down-slope failure is most likely for structures with cot a = 1.75, but that up-slope failure is most likely for milder sloped structures with cot a = 3. The influence of slope angle on the wave induced forcing of the armour layer is considered further in Chapter 5. 120 60 62 64 66 68 70 Time (s) Figure 4-8: Time series of forces and failure indices under regular waves (upper panel, test series 3, H = 13cm, T = 2s). 121 60 62 64 66 68 70 T i m e (s) Figure 4-9: Time series of forces and failure indices under regular waves (upper panel, test series 4, H = 13 cm, T = 2 s). 122 60 62 64 66 68 70 T i m e (s) Figure 4-10: Time series of forces and failure indices under regular waves (upper panel, test series 5, H — 14 cm, T = 2 s). 123 For each time series, the maximum values due to 100 consecutive waves were obtained and averaged to give a representative peak value. Peak values are denoted here by an accent 'circumflex'; i.e. Rs denotes the peak value of the Shields failure index, while peak values for the lifting, Hudson and rolling-sliding failure indices are denoted by RL, R-H and RR respectively. The peak index value is representative of the least stable condition during a cycle of wave attack. These peak index values provide four different non-dimensional measures of the severity of wave attack on the patch of armour stones represented by the armour panel. 4.4.2 Calculation of Damage Values The measurement and analysis of armour damage is described in more detail in Sections 2.3.10 and 3.2.1. In summary, damage describes the removal of armour stones from the surface of a rubble-mound, and is quantified by the parameter 5 , such that 5 = 0 represents no damage, 5 ~ 1 corresponds to the initiation of damage, 5 ~ 4 denotes the start of destruction and destruction ensues for 5 ~ 9. Quantities corresponding to initiation of damage conditions (5 = 1) are denoted by the subscript "ID" • Damage results in regular waves with different heights at a single wave period are used to construct damage curves that describe the relationship between damage and wave height. A typical damage curve is shown in Figure 3-8. These curves are used to interpolate and extrapolate observed damage results to other test conditions for which damage values were not obtained directly by experiment. In this way, damage values are obtained for every regular wave test condition. 4.4.3 Failure Indices and Damage Shields Failure Index Figure 4-11 shows peak values of the Shields failure index Rs for the upper panel of the series 4 rubble-mound plotted against the damage 5 resulting from 1000 cycles of regular wave attack. The data are grouped according to wave period T = 1.5, 2 or 3 s. Initiation of damage occurs 124 Figure 4-11: Relationship between damage and Rs (upper panel, test series 4). in waves that exert forces on the armour panel that result in Rs ^ 7. Waves that generate forces such that Rs < 7 are generally insufficient to produce damage. For waves that are sufficient to cause damage, these results suggest a strong correlation between damage and the peak value of the Shields failure index for the upper panel. This is consistent with the premise that more damage occurs in waves that exert greater hydrodynamic forces oh the armour stones. The relationship between damage and the peak value of the Shields failure index is virtually independent of wave period. Figure 4-12 is similar to Figure 4-11 except that Rs values for the lower panel are shown as a function of the same damage values. In this case, the peak value of the Shields failure index associated with the initiation of damage varies slightly with wave period, but can be taken as R-SJD — 3.5. The difference in Rs values shown in these two figures reflect the variation of hydrodynamic forcing with location on the surface of the rubble-mound. Larger peak index values are obtained on the upper armour panel, which suggests that greater fluid forces act on armour stones located closer to the still waterline. This is consistent with the fact that initial 125 6 5 <K 4 3 2 1 0 * A V So ° ° T = 1.5 s T = 2 s T = 3 s _L Figure 4-12: Relationship between damage and Rs (lower panel, test series 4). damage tends to occur just below the still waterline. Kobayashi et. al. (1990c) considered the variation of armour stone loading and stability with location using numerical simulations of wave interaction with a rubble-mound. They concluded that the critical location for initial damage is approximately 0.75H vertically below the still waterline. This is slightly below the elevation at which initial damage tends to occur. These results suggest that initial damage may be better correlated to failure indices computed for the upper panel. Figure 4-13 shows peak values of the Shields failure index Rs for the upper panel plotted against the damage S after 1000 regular waves for all tests on the series 3, 4 and 5 rubble-mounds. The data fall into two distinct groups, depending on the seaward slope angle of the structure. Rs values for the milder sloped series 5 rubble-mound (with cot a = 3) are roughly one-third those of the steeper, series 3 and 4 structures (with cot a = 1.75). R-SJD — 8 f ° r the steeper structures, while RS,ID — 2.7 can be taken for the milder slope. These results suggest that the Shields index (denned by Equation (4.49)) fails to properly account for the effect of slope angle on the relative stability of armour stones. 126 16 14 12 10 <K 8 6 4 2 0 • Series 3 • Series 4 M Series 5 • • • • • • • . • • • • • • • • • % u m M M M M M M M M 1 1 1 1 2 3 4 Figure 4-13: Relationship between damage and Rs (upper panel, test series 3, 4 and 5). On the steeper structures, the peak values of the Shields failure index are associated with down-slope failure of armour stones. This can be seen from the time series shown in Figures 4-8 and 4-9, where the maximum values of Rs (t) during each flow cycle coincide with Fp (t) that are negative. In contrast, the times series in Figure 4-10 indicate that for the milder sloping series 5 rubble-mound, the peak values of Rs (t) coincide with positive FP (£), which suggests up-slope failure. Observations of the initial motion of armour stones on this structure generally do not support displacements up-slope. This suggests that for the cot a = 3 rubble-mound, the Shields failure index predicts either excessively unstable conditions for up-slope failure or excessively stable conditions for down-slope failure. The Shields failure index appears to quantify reasonably well the relative stability of armour stones on a fixed slope in various wave conditions, but fails to quantify the relative stability of armour stones on different slopes. The results for test series 3 and 4 shown in Figure 4-13 suggest that for a fixed slope, the wave-induced forcing required to initiate damage can be reasonably well described by a single value of Rs over the range of wave period and core-permeability 127 considered in these experiments. These data also show a definite positive correlation between Rs and damage for S > 0. Waves that exert forces leading to greater Rs on the upper panel tend to produce more damage. This suggests that the peak value of the Shields failure index on the upper panel can be used as an indicator of the damaging effect of different waves. Moreover, since the Shields index is calculated from the shear stress or the slope-parallel component of the hydrodynamic forcing, this result suggests that slope-parallel forces exert a controlling influence on the damage response of the armour. The results in Figure 4-13 can be used to calibrate the wave loads measured on the upper armour panel in terms of their damaging effect on armour stones. For the cot a = 1.75 struc-tures, damaging waves are those that exert sufficient fluid force to give Rs > 8 on the upper armour panel. For the cot a = 3 structure Rs on the upper panel must exceed 2.7 for a wave to be considered damaging. The relationship between S and Rs on the upper panel of the series 3, 4 and 5 rubble-mounds can be represented by S ~ < a - (RS- RSJD) + 1 for RS > RSJD ~ V ° v J (4.50) 0 for Rs < RSJD - l/a where a = 2 and RSJD = 8 for the steeper structures with cot a = 1.75, and a — 6 and RSJD = 2.7 for the milder sloped series 5 rubble-mound with cot a = 3. Equation (4.50) represents a calibration of the fluid loads measured on the upper armour panel to the damage resulting from 1000 cycles of wave attack. Figure 4-14 shows that Equation (4.50) provides a fair representation of the relationship between the peak value of the Shields failure index and damage to the armour layer. Some of the scatter in this Figure can be attributed to uncertainties in the determination of damage response, particularly at low damage levels. As mentioned previously in Section 3.2, the level of damage resulting from a single experiment represents a single outcome of a stochastic process that depends on the construction of the armour layer. Repeated experiments with the same wave conditions will yield different levels 128 16 14 12 10 m <« a 6 4 2 0 • Series 3 • Series 4 M Series 5 • %-• H H - K - M M M M M - - — • " Figure 4-14: Calibration of the Rs to damage (upper panel, test series 3, 4 and 5). of damage response. It is possible to interpret Equation (4.50) as a type of transport relationship for armour stones. In this equation, S represents the damage or volume of erosion accumulated over a duration corresponding to 1000 wave cycles, and can thus be interpreted as a transport rate dS/dNw. With reference to Equation (4.26), the term Rs — RS,ID can be written as \%I lTc ~ \TQ\IDITC where \TQ\II) denotes the actual value |fo| corresponding to the initiation of damage and r c is a theoretical estimate of the critical shear stress. Using these alternative symbols, Equation (4.50) can be written as dS a ..^ . _ . ^ - = - • (M - |roU . (4.51) This equation suggests that the rate of growth of damage is proportional to the excess shear stress |fb| — |TO|/£, acting on the armour stones. It also indicates that for damage to occur, the amplitude of oscillatory shear stress fluctuations must exceed a threshold level given by |TO|/£J-129 The form of the relationship between transport rate and excess shear stress shown in Equa-tion (4.51) is similar to many of the equations that have been proposed to predict the bed-load transport of sediments under steady flow. Consider for example the well known bed-load trans-port equation for granular sediments proposed by Bagnold (1956). Yalin (1977) shows that Bagnold's equation can be written where qs is the sediment transport rate (in terms of weight per unit width of flow), TO is the shear stress acting on the bed, r c is the critical or threshold value of TQ and b ~ 4.25 for large particles in rough turbulent flow. According to Bagnold's equation, the rate of transport of sediments as bed-load is also proportional to the excess shear stress To — r c. The similarity of Equations (4.52) and (4.51) suggests that the initiation and growth of armour damage on a rubble-mound under wave attack has some commonality to the bed-load transport of sediments. Rolling-Sliding Failure Index The rolling-sliding failure index for an armour panel is defined by Equation (4.48) and depends on both the slope-parallel and slope-normal components of hydrodynamic force. Figure 4-15 shows peak values of the rolling-sliding index for the upper panel of the series 3, 4 and 5 structures plotted against the damage due to 1000 regular waves. The data again fall into two groups, depending on the seaward slope angle; however, there is also more variation with wave period than for the Shields index. The variation with slope angle suggests that the rolling-sliding index also fails to properly account for the effect of slope angle on the relative stability of armour stones. Figures 4-8 and 4-9 show that on the steeper rubble-mounds, RR (t) is maximized when large slope-parallel fluid forces acting down-slope (Fp (t) < 0) coincide with large slope-normal fluid forces directed away from the surface (Fjv (t) > 0). Forcing of this type was previously identified as the most critical for down-slope rolling-sliding failure. Figure (4.52) 130 4 — 3.5 -3 -2 . 5 -« <« 2 1.5 1 0.5 0 • S e r i e s 3 • S e r i e s 4 M S e r i e s 5 • 1 Figure 4-15: Relationship between damage and RR (upper panel, test series 3, 4 and 5). 4-10 shows that on the milder sloped rubble-mound, ##(£) is maximized with FN (t) > 0 and Fp (i) ~ 0, a condition which requires larger fluid forces to induce the same degree of instability. This difference in the character of fluid loading contributes to lesser values of RR for the milder sloped rubble-mound. For the steeper series 3 and 4 structures, larger values of RR are associated with initial damage in 1.5 s waves than at longer periods. However, on the milder sloped series 5 structure the trend is reversed, so that marginally smaller values of RR are associated with initial damage in 1.5 s waves. This variability with wave period may be related to the different types of wave breaking that occur on the two slopes. Alternatively, the variation with wave period may be an indication that the propensity for damage may depend on the magnitude of the de-stabilizing force and the duration over which it acts, and not just the force magnitude. This additional variability of RR with wave period makes it somewhat less useful as a general indicator of the relative damaging effect of various wave conditions. That RR is less well correlated to damage than the peak value of the Shields failure index 131 <K 0.6 I 1 i i i i i • S e r i e s 3 • S e r i e s 4 H S e r i e s 5 • 1 • • M • " * M 1 1 " 1 I ' M . " M • M M N i 1 1 1 1 1 0 1 2 3 4 5 6 7 S Figure 4-16: Relationship between damage and RH (upper panel, test series 3, 4 and 5). could be due in part to the fact that there is likely to be less statistical independence between the driving and resisting forces for this mode. This issue is considered in Section 4.3.2. As a consequence, the rolling-sliding failure index computed from force measurements on a panel may not provide an adequate measure of the index values for the individual constituent armour stones. This would weaken the correlation between S and RR. L i f t i n g and H u d s o n Fa i lure Indices Figures 4-16 and 4-17 show peak values of the Hudson and lifting failure indices for the upper panel of the series 3, 4 and 5 structures plotted versus armour damage after 1000 regular waves. The scatter evident in these figures suggest that these indices fail to capture the relationship between wave loading and armour damage. This conclusion is supported by the fact that these figures indicate initiation of damage index values scattered over a broad range well below unity; an improbable result considering that the formulation of the indices imply unstable conditions for index values greater than unity. It is evident that armour stones become unstable and begin 132 Figure 4-17: Relationship between damage and RL (upper panel, test series 3, 4 and 5). to fail by other mechanisms well before lifting or Hudson failures contribute to damage. Since the peak values of the Hudson and lifting indices are non-dimensional representations of the peak values of the hydrodynamic force magnitude F and the slope-normal force component F^ respectively, these results suggest that F and F^ do not exert a controlling influence on the damage response of the armour. E m p i r i c a l Fa i lu re Index The four failure indices derived from theoretical analysis of the incipient stability of an armour stone on a sloping rubble-mound fail to provide a unified indicator of damage that applies equally to all test conditions. In the following, an empirical failure index is developed to correct this deficiency. Results shown in Figure 4-13 suggest that the Shields failure index provides a measure of the relative stability of armour stones that is fairly well correlated to observed damage on both the impermeable series 3 and more permeable series 4 rubble-mounds (both with cot a = 1.75) 133 8 7 6 5 u <K 4 3 2 1 0 • Series 3 • Series 4 Series 5 t • MFT FT • * • • Figure 4-18: Relationship between damage and RE (upper panel, test series 3, 4 and 5). in a variety of wave conditions. For these structures, damage is fairly well correlated to the slope-parallel component of fluid force acting down-slope. However, the Shields failure index, defined by Equation (4.49), fails to properly account for the effect of slope angle. This deficiency can be overcome by using an empirical expression to account for the effect of structure slope in place of the expression derived in the development of the Shields failure index. Good agreement with the experimental results is obtained if the effect of slope angle is parameterized by the factor (cot a) 2 / 3 . This factor can be combined with the force balance for down-slope failure described by the Shields index to form an empirical failure index i?£;(£), defined by **(*) = - ( « * " ) ' * F s w { t ) (4.53) Figure 4-18 shows peak values of RE{t) for the upper panel of the series 3, 4 and 5 rubble-mounds as a function of damage after 1000 regular waves. This Figure suggests that RE provides a measure of the forces acting on armour stones that is well correlated to damage for all regular 134 wave conditions on the series 3, 4 and 5 rubble-mounds. This implies that the initiation and early growth of damage depends on the magnitude of the shear stress acting down-slope on the armour layer below the still waterline. 135 Chapter 5 Surface Flows and Wave Forces Waves propagating onto a rubble-mound breakwater exert hydrodynamic forces on armour stones that vary considerably with time and location on the surface of the structure. The magnitude and character of the hydrodynamic forces reflect the unsteady fluid flow on the surface and within the porous outer layers of the rubble-mound. These unsteady kinematics and the resulting hydrodynamic forces are influenced by fundamental properties of the incident waves, such as height and period, as well as properties of the structure such as slope angle and permeability. The stability of rock armour on rubble-mounds under wave attack is fundamentally de-pendent on the hydrodynamic forces acting on the armour stones. In spite of this, relatively little is known about the wave-induced forces acting on the armour, or the fluid flows that are responsible for them. Sigurdsson (1962) measured regular wave forces on a spherical armour unit placed at various locations on several different idealized breakwater sections. The idealized sections consisted of three layers of identical spheres placed with porosity n = 0.48 on plane slopes of cot a = 1.5 and 3. No attempt was made to separate the hydrodynamic force from the buoyancy force. Sigurdsson identified eight different types of flow that were responsible for different kinds of forcing. He concluded that the most important hydraulic forces occur below the still waterline 136 under the toe of an advancing breaker or when water is flowing out of the breakwater. He also found considerable impact forces directed up-slope at higher elevations. The lowest elevation of wave retreat was identified as an important factor controlling the distribution of hydraulic forces with depth. Jensen and Juhl (1988) and Juhl and Jensen (1989, 1990) measured wave forces on an idealized two-dimensional breakwater using two layers of horizontal pipes as armour. The pipes were arranged to give a porosity of n — 0.4. Forces due to regular and irregular waves were measured on nine pipes located on the surface of the idealized rubble-mound at different elevations above and below the still water level. They used a simple stability model to express the measured forces in terms of the pipe weight required to resist roll-up or roll-down. They found that the most dangerous forces were associated with roll-down below the still waterline, and that wave period had a significant influence on the forces. The most dangerous forces occurred in the transitional regime between plunging and surging breakers. Hettiarachchi and Holmes (1988) measured wave-induced hydraulic forces on model break-water sections armoured with a single homogeneous layer of hollow block armour units. Force measurements were obtained on a single armour unit installed near the still waterline. They found that the upward normal force, which acts to lift the armour unit away from the surface of the structure, occurred during the downrush phase of surface flow, and that the permeability of the core did not cause a significant change in either the normal or parallel forces. Sakakiyama and Kaj ima (1990) measured the total hydraulic force acting on a tetrapod (a type of concrete armour unit) located near the still waterline on a breakwater slope at four different model scales. They concluded that relatively larger forces act at smaller scales. T0rum (1994) measured regular wave loads on a single irregularly shaped model armour stone with a mass of 0.152 kg located 10.3 cm below the still waterline on the surface of a reshaped berm breakwater. Measurements of fluid kinematics above the instrumented stone were used to model the slope-parallel force component as a summation of drag and inertia forces calculated from slope-parallel kinematics. The slope-parallel force was found to be drag 137 dominated, such that the force peaks occurred in phase with the peaks of slope-parallel velocity. The slope-normal force was modelled using a similar drag and inertia formulation, augmented by an additional term to represent lift force. In this case, drag and inertia forces were calculated from slope-normal kinematics, while the lift force was computed from the slope-parallel velocity. The peaks of the slope-normal force led those of the slope-parallel force, and were significantly smaller in magnitude. T0rum concluded that the slope-normal force is not dominated by lift, and could not be adequately modelled by the assumed augmented formulation of the Morison equation. In this section, measurements of the hydrodynamic forces acting on the instrumented armour panels in various regular and irregular wave conditions are presented. The nature of the surface flows on a rubble-mound will also be considered, since an understanding of these flows is an essential ingredient in the interpretation of the wave-induced forcing. Analysis of these flows and forcing provides insight into the mechanisms responsible for the hydrodynamic forces acting on rock armour, and the influences on this forcing due to changes in wave height, wave period, breaker type, core permeability and slope angle. 5.1 Wave-Induced Surface Flows Rubble-mound breakwaters and coastal revetments are structures that are typically designed to block the propagation of wave energy. This is achieved through a combination of energy reflection and dissipation. More energy is generally reflected from steeper, less permeable rubble-mounds. Energy dissipation occurs primarily through the creation of turbulence in wave breaking, through friction against the surface of the structure, and through viscous losses developed through internal flows. 138 5.1.1 D e s c r i p t i o n As waves propagate onto a rubble-mound, they generate unsteady oscillatory flows over the surface and within the permeable zones of the structure. The character of these flows depends on features of the incident waves such as height and period, the type of wave breaking that occurs, and on characteristics of the structure such as its slope, permeability and composition. In regular waves, each wave produces essentially the same cycle of surface and internal flows. In irregular waves, the pattern of flow for each wave varies in response to the wave characteristics and the flow conditions on the structure at the beginning of the wave interaction. Waves propagating onto a rubble-mound undergo significant transformations into either plunging, collapsing or surging breakers. The type of breaking that prevails depends primarily on the slope of the structure, and on wave steepness.. Plunging breakers prevail for steeper waves on milder slopes, while surging breakers prevail for less-steep waves on steeper slopes. The influence of these different types of wave breaking on the character of the flows on a rubble-mound is considered in more detail in Section 5.1.4. Irrespective of the type of wave breaking that prevails, the water in the advancing wave crest generally has sufficient inertia to continue travelling up the slope as a surface flow. This up-slope flow, called uprush, causes the elevation of the waterline on the surface of the structure to rise rapidly. Some of this rising volume of water flows into the permeable outer layers of the structure. The amount of infiltration generally increases on more permeable structures. While uprush continues, the following wave trough arrives at the base of the slope. This causes down-slope flow over the lower portion of the rubble-mound. At this time, flow on the lower portion of the structure travels down-slope while the uprush flow continues to advance at higher elevations. These opposing flows quickly reduce the depth of water on the surface of the structure. Gravitational and frictional forces combine to decelerate the uprush flow. After reaching a maximum elevation, the direction of flow reverses and the water begins to recede down-slope. At this time, surface flows over most of the structure are directed down-139 slope as gravitational forces act to drain water from the surface and permeable outer zones of the structure. This down-slope flow is called downrush. On relatively impermeable structures downrush is concentrated near the surface, but on permeable structures it is generally more distributed throughout the porous outer layers. Seepage out from the porous zones of the rubble-mound generally occurs below the still waterline towards the end of downrush when the water level outside the structure recedes below the internal phreatic surface. This condition provides the hydraulic gradient required to drive seepage flows. Below the still water level, the change from down-slope to up-slope flow occurs under the front of an advancing wave crest. When this front is very steep, such as for plunging and collapsing breakers, the change in flow direction occurs rapidly and is associated with large accelerations directed up-slope and away from the surface of the rubble-mound. Surging waves generally induce weaker accelerations and more gradual reversals in the direction of flow. Runup and rundown refer to the maximum and minimum elevations respectively of the wa-terline on the surface of the structure with respect to the still water level. These characteristics of the surface flow depend primarily on wave height, but are also significantly influenced by wave period, surface roughness and the type of wave breaking that prevails. 5.1.2 Types of Wave Break ing On sloping rubble-mound structures such as revetments and breakwaters, incident waves tend to break as surging, collapsing or plunging breakers. Sketches of these different breaker types are shown in Figure 3-3. Characteristics of these different types of wave breaking have been described by many authors, including Galvin (1968), Battjes (1974), Giinbak (1979) and Basco Battjes (1974) linked the type of wave breaking on a particular slope to the surf similarity parameter (1985). (5.1) 140 which represents a ratio of the slope angle to the deep water wave steepness of regular waves. Battjes' classification system can be written as £ < 2 plunging breakers, £ ~ 3 collapsing breakers, (5.2) £ > 4 surging breakers. Plunging, collapsing and surging breakers are not discrete forms. In reality, a continuous range of breaker types occur for different values of £• Plunging breakers prevail over a wide range of £ less than 2, while surging breakers prevail over a wide range of £ greater than 4. Collapsing breakers represent a transitional form between these two extremes that generally occurs over a narrow range of £ around 3. Plunging breakers prevail for steep waves on mild slopes and are characterized by a large-scale, highly visible forward overturning of the wave crest. This overturing crest forms a falling jet that typically impacts the receding downrush flow from the previous wave. A n abrupt release of wave energy to turbulent flow results throughout the water column. In most cases, and particularly for small £, the impact of the falling jet is cushioned by a sheet of water remaining from the downrush of the preceding wave. Plunging breakers generally entrain more air than other breaker types. They also generate large fluid accelerations below the front of the steep, unstable, advancing wave crest. Surging breakers prevail for less steep waves on steeper slopes. The term 'surging breaker' is somewhat misleading since these waves do not really break in the same sense as do plunging or collapsing breakers. Surging breakers are semi-standing waves that generate turbulence through boundary shear against the structure. Surging breakers generally feature large runup accom-panied by strong velocities during uprush and downrush, but somewhat weaker accelerations under the front of the advancing wave crest. Infiltration and seepage flows through the surface of the rubble mound are most noticeable for surging waves. When the water level outside the structure exceeds the level of the internal phreatic surface, water flows into the permeable outer 141 layers. As the external water level recedes below the internal phreatic surface, water flows out of the permeable outer layers. Infiltration generally occurs above the mean water level during uprush while seepage is concentrated below the mean water level during downrush. Collapsing breakers are an intermediary or transitional form between plunging and surging breakers. They are characterized by a pronounced steepening of the wave crest, so that the front is nearly vertical, which then collapses without overturning or forming a well defined jet. The wave crest tends to collapse directly onto the structure and is generally less well cushioned by downrush from the previous wave. Thus, for collapsing waves, the intense velocities and accelerations associated with wave breaking are more likely to act directly on the armour layer. Particularly dangerous conditions for armour stability arise when the intense acceleration under a collapsing wave crest is combined with seepage flow out of the structure. Ahrens (1975) was among the first to identify collapsing breakers with minimum stabil-ity for armour stones. Since then, a substantial body of research, summarized by van der Meer (1988) and Bruun (1985) and supported by the results presented in Chapter 3, suggests that the stability of rubble-mound armour is minimized in the range of collapsing breakers. Losada and Gimenez-Curto (1979), and van der Meer (1988) further suggest that the value of £ corresponding to minimum stability also depends on slope angle such that minimum stability occurs for slightly larger values of £ on steeper slopes. 5.1.3 Effect of Breaker Type on Water l ine M o t i o n s Excursions of the waterline on the surface of a rubble-mound can be used as a quantitative description of the overall surface flow. Runup and rundown refer respectively to the maximum and minimum elevations of the waterline with respect to the still water level. For each regular wave test, a representative value of runup 7)s was determined by averaging the maximum values of r]s (t) over 100 consecutive waves. A representative value of rundown f]s was determined by a similar averaging of the minimum values of r]s (£). The total height of the waterline excursions is Hn„, which for regular waves can be obtained as H„s = fja — fjs. Runup, rundown and the 142 height of the waterline excursions can be expressed in dimensionless form as a fraction of the incident wave height H. Losada and Gimenez-Curto (1981) review a variety of experimental results on runup and rundown on rough permeable slopes in regular waves. They conclude that the results in each data set can be reasonably well represented by exponential functions of the surf similarity parameter £: | = a ( l - e x p [ 6 £ ] ) , (5.3) | = c(l-exp[de]) , (5.4) where a, b, c and d are coefficients that were determined by least squares. Although different coefficients were determined for each data set, typical values for quarry stone armour are a = 1.4, b = —0.6, c = —0.9 and d = —0.4. A prediction for the height of waterline motions can be determined from these curves as ^ = a (1 - exp [&£]) - c (1 - exp [<%]) . (5.5) Figure 5-1 shows HVs/H as a function of £ for the regular wave tests in series 3 (impermeable core, cot a = 1.75), 4 (permeable core, cot a = 1.75) and 5 (impermeable core, cot a = 3) together with the predictions of Equation (5.5). These results show a trend towards smaller waterline excursions for plunging breakers and larger waterline excursions for surging breakers that is in overall agreement with the predictive equation developed from Losada and Gimenez-Curto (1981). The variation of HVJH with £ is particularly strong for plunging breakers in the region with £ < 2.5. For a fixed slope, the height of waterline motions increases with increasing wave period. For a fixed wave period, waterline motions are generally greater on steeper rubble-mounds. Figure 5-1 indicates that the influences of wave period and slope angle on the waterline motions on a rubble-mound can be reasonably well represented by the surf 143 — i r ——a ~° ° O £ o A O S e r i e s 3 A S e r i e s 4 cxi S e r i e s 5 J : I 0 1 2 3 4 5 6 7 8 f = t a n a (glZ/2nH)1/Z Figure 5-1: Height of waterline motions in test series 3, 4 and 5. similarity parameter. 5.1.4 Effect of Breaker Type on Kinematics Flows on the surface of a rubble-mound exposed to wave attack are unsteady, spatially variable, and feature a small but significant two-dimensional structure related to infiltration and seepage through the surface. The spatial and temporal distribution of kinematics on a rubble-mound depends in a complex manner on the character of the incident waves, the type of wave breaking, and properties of the mound itself. Measurements of velocity near the surface of a breakwater under regular wave attack have been made by Mani et. al. (1994), T0rum and van Gent (1992) and Sawaragi et. al. (1983). Mani et. al. obtained measurements of the Lagrangian velocity on the surface of downrush flows by tracking the trajectory of a float dropped onto a structure near the time of maximum runup. They concluded that maximum down-slope velocities occur below the still waterline. Sawaragi et. al. obtained Lagrangian measurements of velocity by tracking the motion of neutrally buoy-2.5 2 -1.5 0.5 144 ant pieces of sponge over the surface of rubble-mounds with slope cot a = 2. Velocities were non-dimensionalized by the factor \fgH.. They concluded that the maximum slope-parallel ve-locities and accelerations varied with the surf similarity parameter, the ratio of wave height to water depth, and the roughness and permeability of the slope. Up-slope and down-slope velocities and down-slope accelerations were maximized for 2 < £ < 3, while up-slope accel-erations increased with decreasing £ for £ < 3. They found substantially larger kinematics on smooth impermeable slopes than on permeable rubble-mound slopes. They also concluded that the maximum up-slope and down-slope velocities generally occur below the still waterline. T0rum and van Gent used a Laser Doppler Velocimeter to obtain Eulerian measurements of slope-normal and slope-parallel velocity at several locations below the still waterline on a berm breakwater in regular waves. They recorded larger slope-parallel velocities during uprush than in downrush, and larger slope-normal velocities directed away from the surface. Even though regular waves were used, they found that the slope-normal velocities were quite irregular. Kinematics were measured during series 4 and 5 of the N R C experiments as described in Section 2.3.5. Slope-parallel and slope-normal velocities were measured by a bi-directional electromagnetic velocimeter located 4 cm above the surface of the rubble-mound and centered between the lower and upper armour panels. This location is 12 cm below the still water level and is shown as measurement point 1 in Figures 2-8 and 2-9. A second electromagnetic velocimeter was used to record slope-parallel velocity at measurement point 2, located above the geometric centre of the lower armour panel. Point 2 is 18 cm below the still water level on the series 4 rubble-mound, and 16 cm below the still water level on the milder sloped series 5 structure. In the following, velocity time series measured at point 1 under plunging, surging and collapsing breakers are presented to illustrate the effect of breaker type on the kinematics at a point below the still waterline. Figures 5-2, 5-3 and 5-4 show 10 s long segments of the waterline elevation r]s (t) and slope-parallel and slope-normal components of velocity u (t) and w (t) at measurement point 1, recorded under plunging (£ = 1.5), surging (£ = 5.3) and near-collapsing '145 Series 5, T = 1.5 s, H = 16.5 cm, £ = 1.5 0.15 ~ -0.15 64 66 Time (s) 70 Figure 5-2: Waterline motion and velocities under plunging breakers. Series 4, T = 3 s, H = 16.6 cm, £ = 5.3 0.15 -0.15 64 66 Time (s) 68 70 Figure 5-3: Waterline motion and velocities under surging breakers. 146 S e r i e s 5 , T = 3 s, H = 1 6 . 6 c m , | = 3 .1 TO 0.3 0.15 -0.15 1.5 \ 1 6, 0.5 >> 0 o o -0.5 ii -1 > -1.5 60 62 64 66 Time (s) 68 70 Figure 5-4: Waterline motion and velocities under near-collapsing breakers. (£ = 3.1) breakers respectively. The parallel velocity u(t) is defined positive up-slope while the normal velocity w(t) is defined positive away from the surface. The plunging and near-collapsing data were obtained on the series 5 rubble-mound (cot a = 3) in waves with T = 1.5 and 3 s, while the surging data is from the series 4 structure (cot a = 1.75) in waves with T = 3 s. A constant wave height of H = 16.5 cm was used for these tests. These figures are plotted to the same scale to facilitate comparison between them. Plunging breakers (Figure 5-2) generate significantly smaller waterline motions than either surging or collapsing breakers. This is consistent with the trend shown in Figure 5-1. The waterline motions in plunging waves are also more irregular than in either collapsing or surging breakers. This irregularity can be attributed to the turbulent nature of the plunging breaking process, and the relatively large amount of entrained air. Each wave crest breaks in a slightly different manner, which affects the uprush and downrush flows for each wave cycle. Positive parallel velocities prevail for a shorter period and reach a larger maximum than negative parallel velocities. This indicates a relatively short period of strong up-slope flow followed by a longer 147 period of weaker down-slope flow. Passage of the overturing crest front produces a large, steady, repeatable positive acceleration from the maximum down-slope velocity to the up-slope maximum. This strong acceleration begins while the waterline is still falling. Negative accelerations are also quite strong immediately following the maximum up-slope velocity, but they do not prevail for as long as the positive accelerations. Velocities normal to the slope are much smaller than the parallel velocities. The largest positive normal velocities coincide with the sudden reversal from down-slope to up-slope flow under the front of the plunging wave crest. Figure 5-3 shows that surging breakers produce a substantially different pattern of surface flow. Each surging wave crest conveys a relatively greater volume of water onto the rubble-mound, and since turbulent losses through wave breaking are diminished, this water is able to flow further up the slope, resulting in approximately twice as much runup as for plunging waves. Another characteristic of surging breakers is that the wave period is generally longer than the duration required for a full cycle of uprush and downrush. This can be seen in Figure 5-3, which shows that the waterline elevation remains constant at its minimum level for over a third of each flow cycle. This relatively stagnant period occurs between the end of each downrush flow and the start of the next uprush. Slope-parallel velocities are close to zero for a portion of this relatively stagnant period. Arrival of a new wave crest induces a fairly strong positive acceleration in the parallel flow, which quickly reaches its maximum up-slope velocity. Negative parallel accelerations are weaker, but prevail for a longer duration, and lead to a maximum down-slope parallel velocity that is slightly weaker than the up-slope maximum. Positive parallel velocities prevail for approximately one-third of the flow cycle and are slightly stronger than the negative parallel velocities acting down-slope. At the completion of downrush when the waterline has stopped falling, the parallel velocities become very weakly negative and remain thus until a new wave crest arrives and the flow cycle begins anew. Velocity normal to the slope is maximized towards the end of the downrush flow. This maximum is related to seepage flow from the permeable zones of the rubble-mound. Passage of a new wave crest also 148 produces positive normal velocities, but these are generally weaker than those associated with seepage flow towards the end of downrush. Figure 5-4 shows that near-collapsing breakers produce an intermediary cycle of flow between these two extremes that has some similarity to the flows under both plunging and surging breakers. The height of waterline motions in collapsing waves is similar to, but slightly less than the height under surging breakers; however, the temporal variation of the waterline is perhaps more similar to that for plunging breakers. The period of relatively stagnant flow seen under surging breakers is not present under these collapsing breakers. For collapsing breakers, the wave period is closely matched to the period required for a complete cycle of runup and rundown. Giinbak (1979) used the term "resonance" to describe conditions when "every run-down meets the new run-up from a breaking wave at the breaking point", and associated this condition with minimum stability for armour stones. These "resonant" conditions generally occur under collapsing breakers. Figure 5-4 shows that collapsing breakers produce stronger up-slope and down-slope flows parallel to the surface than plunging or surging breakers. In particular, negative parallel velocities are stronger and prevail for a longer duration. Arrival of the collapsing wave crest produces a strong, fairly continuous positive acceleration to the parallel flow that is similar to that under plunging breakers. The negative parallel acceleration is weaker but prevails for a longer duration. The normal velocity signal is relatively weak and irregular; however, the maximum normal velocity generally occurs under the front of the collapsing wave crest, and coincides with the transition between downrush and uprush. Figures 5-2, 5-3 and 5-4 illustrate that the motion of the waterline on the surface of a rubble-mound, and the kinematics measured at a point below the still water level, vary with the wave period and slope angle in a manner that can be explained in terms of the type of wave breaking that prevails. 149 5.1.5 Peak Slope-Parallel Velocities Positive and negative peak values of parallel velocity have been computed for all regular and irregular wave tests in which velocities were measured at points 1 and 2, as shown in Figures 2-8 and 2-9. For regular waves, the positive peak value, denoted by u, represents an average of the maximum positive velocities recorded during 100 consecutive flow cycles. Similarly, the negative peak value, denoted by u, represents an average of the strongest negative velocities recorded during 100 consecutive flow cycles. Irregular waves produce a wide range of velocity fluctuations that can only be characterized in a statistical sense. Characteristic values of the velocity due to irregular waves have been obtained using zero-upcrossing analysis of the velocity time series u(t). According to this procedure, the time-varying flow is separated into flow cycles based on consecutive zero-upcrossings. A single maximum and minimum value are obtained for each flow cycle. A l l minima are negative while all maxima are positive. These minima and maxima are analyzed in a statistical sense to obtain characteristic values for the unsteady velocity. The significant positive peak value, denoted by U 1 / 3 , represents the average value of the largest one-third of the maxima. The significant negative peak value, denoted by U 1 / 3 , represents a similar averaging of the largest (the strongest velocities) one-third of the minima. A more extreme characterization of the unsteady velocity is provided by the quantities •u1/2o and Ui/20, obtained by averaging the largest one-20 t h of the maxima and minima respectively. For breaking waves in shallow water, the velocity v of water in the wave crest is approx-imately equal to the wave celerity \fgh. At the point of breaking, the water depth can be estimated as a fraction of the wave height: h = kH where k depends primarily on slope angle and wave steepness, but is generally slightly less than unity for steep slopes. These equations can be combined to give a rough estimate of the velocity of water in the crest of a wave breaking on a rubble-mound as v~7kgH. (5.6) This approach was used by Hudson (1958) to develop an estimate of the wave force acting on 150 ° ° o o ~ • Series 5, T=1.5 S M Series 5, T=2 s • Series 5, T = 3 s _ J I 0 1 2 3 4 5 6 7 f = t a n a ( g T 2 / 2 7 r H ) 1 / 2 Figure 5-5: Peak slope-parallel uprush velocities at point 1 under regular waves in test series 4 and 5. armour stones. Setting k = 1, the factor \fg~H can be used to non-dimensionalize the peak values of velocity measured near the surface of a rubble-mound. This factor has been used previously by Sawaragi et. al. (1983). U p r u s h Ve loc i t y Peaks Positive non-dimensional peak velocities at measurement point 1 on the series 4 (cot a = 1.75, permeable core) and series 5 (cot a = 3, impermeable core) rubble-mounds are shown in Figure 5-5 as a function of the surf similarity parameter. For conditions with £ > 3 in which surging breakers prevail, u/\JgH is approximately constant. In this regime, the peak uprush velocity is roughly proportional to the velocity of the water in the advancing wave crest, and is not significantly influence by wave period. However, for smaller £ where plunging breakers prevail, these results show a strong dependence between normalized peak velocity and wave period. For plunging breakers, u/VgH increases strongly with increasing wave period. The stability design equations of van der Meer (1988) are considered in Section 3.1.6. They suggest that for plunging 151 1.1 • 1 S» 0.9 0.8 0.7 0.6 0.5 M H M M 0 Series 4, T=1.5 s 1X1 Series 4, T=2 s o Series 4, T=3 s 1.2 1.1 1 ™ 0.9 0.8 <3 0.7 0.6 0.5 0.4 M M M 0 Series 4, T=1.5 s • Series 5, T=1.5 s * Series 4, T=2 s M Series 5, T = 2 s ° Series 4, T=3 s • Series 5, T = 3 s o M ° o ^ OO O o o o 0 2 3 4 5 £ = t a n a ( g T 2 / 2 T T H ) 1 / 2 Figure 5-6: Peak slope-parallel uprush velocities at point 2 under regular waves in test series 4 and 5. waves, damage to armour stones increases with increasing wave period. Assuming that damage to armour stones is in some way proportional to the intensity of kinematics on the surface of a rubble-mound, this trend towards increased damage under longer period waves is consistent with the results on peak positive parallel velocities shown here. These results also suggest that for plunging waves, u/y/gH varies with H such that larger non-dimensional velocities occur under higher waves. This trend is consistent with the findings of Sawaragi et. al. (1983). The largest positive non-dimensional peak velocity measured at point 1 is u/^/gH — 1.06. This result agrees well with T0rum and van Gent (1992), who report a maximum velocity of l.Am/s in waves with H = 17.5 cm and T = 1.8 s, which gives ii/^JgH = 1.08. Figure 5-6 shows ujy/gH at measurement point 2 for the same regular waves on the series 4 and 5 rubble-mounds. The peak velocities at point 2 are consistently smaller than those at point 1, which indicates that stronger up-slope velocities prevail closer to the still water level. This result is consistent with the fact that damage to rubble-mound armour initiates 152 • U j / g , ser ies 4 • Ujyg, ser ies 5 A Uj/go, ser ies 4 A u i/20' s e i " i e s 5 1 1.5 2 2.5 3 3.5 4 £ = t a n a ( g T 2 / 2TTH ) 1 / 2 Figure 5-7: Influence of significant wave height on peak uprush velocities at point 1. near the still water level. These data also show a stronger dependence between u/y/gH and H for £ < 3, which suggests that the peak positive parallel velocity at point 2 is less dependent on the velocity of water in the wave crest at breaking. This is consistent with the fact that measurement point 2 is located at a lower elevation, further away from the wave crest. The variations in peak uprush velocity with wave height and wave period for irregular waves are similar to the trends observed in regular waves. In the plunging wave regime, the non-dimensional velocity increases with increasing wave height and increasing wave period. In the collapsing-surging regime, the influence of wave height is weaker, and the influence of wave period is negligible. The influence of significant wave height on peak uprush velocities can be seen in Figure 5-7, which shows &1/3/'y/g~H~s and UI/2Q/\JgHs at measurement point 1 for various wave conditions with constant peak period Tp = 2 s plotted against the irregular wave surf similarity £ m = tana\/gT£l/2irHs. It is evident that the influence of significant wave height on the non-dimensional peak uprush velocity is stronger on the milder sloped series 5 rubble-mound where plunging breakers prevail. Peak uprush velocities at the l / 2 0 t h level are 153 • Uj / g , ser ies 4 • u 1 / / 3 , ser ies 5 A u 1 / 2 0 , ser ies 4 A U l / 2 0 ' s e r i e s 5 1 1.5 2 2.5 3 3.5 4 £ = t a n a ( g T 2 / 2TTH ) 1 / 2 Figure 5-8: Influence of irregular wave period on peak uprush velocities at point 1. on average 4 2 % greater than at the l / 3 r d or significant level. This compares well with the theoretical estimate of 40 % that is obtained if the velocity peaks due to individual waves are assumed to satisfy a Rayleigh probability distribution. Probability distributions are considered further in Section 7.3. Figure 5-8 shows that the influence of wave period on the peak uprush flow velocities is confined to the plunging wave regime. In this figure, results from measurement point 1 are plotted for different irregular wave tests with virtually uniform significant wave height Hs = 14 cm. In most cases, differences in the mean wave period Tm reflect a change in the spectral peak period Tp; however, in several cases they result from a change in the shape of the wave spectrum. These results suggest that spectral shape has negligible effect on the peak uprush velocities, provided that the mean wave period is used to characterize the period of the sea state. 154 >3 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1 -1.2 0 O Series 4, T=1.5 s • Series 5, T=1.5 s -Series 4, T = 2 s M Series 5, T=2 s • 0 Series 4, T = 3 9 • Series 5, T = 3 s H M S o o o o 0 o o M M 2 3 4 5 f = t a n a ( g T 2 / 2 7 r H ) 1 / 2 Figure 5-9: Peak slope-parallel downrush velocities at point 1 under regular waves in test series 4 and 5. Downrush Velocity Peaks Negative non-dimensional peak velocities at measurement point 1 on the series 4 (cot a = 1.75, permeable core) and series 5 (cot a = 3, impermeable core) rubble-mounds in regular waves are shown in Figure 5-9 as a function of the surf similarity parameter. In all cases, these non-dimensional peak velocities vary with H such that higher waves generate stronger down-slope flows. This indicates that peak down-slope velocities at point 1 are not proportional to the velocity of water in the breaking wave crest. The dependence on wave height is particularly strong for 2 < £ < 4, corresponding to the transition between plunging and surging breakers. This wave height dependence is consistent with the results of Sawaragi et. al. (1983). Aside from the strong dependence on H, these results show that wave period has a significant influence on u/y/gH', such that longer waves generate stronger down-slope velocities. The velocity depen-dence is stronger for £ < 3 where plunging breakers prevail. Wave period has a similar effect on peak up-slope velocities over the same range of £. As mentioned previously, this increase in 155 >3 -0.2 — -0.3--0.4 --0.5 --0.6 --0.7 --0.8 --0.9 --1 --1.1 --1.21— 0 o , o . O O ° Series 4, T=1.5 s • Series 5, T=1.5 s * Series 4, T = 2 s H Series 5, T = 2 s o Series 4, T = 3 s • Series 5, T=3 s I I I I 3 4 5 6 £ = t a n a ( g T 2 / 2 7 T H ) 1 / 2 Figure 5-10: Peak slope-parallel downrush velocities at point 2 under regular waves in test series 4 and 5. the intensity of parallel velocities under longer period plunging breakers is consistent with the form of van der Meer's design equation for plunging waves, which predicts greater damage for longer period waves. Figure 5-9 shows that the strongest negative non-dimensional velocities were recorded under waves for which £ ~ 3, which suggests that collapsing breakers prevailed. Large collapsing breakers produce stronger down-slope flows below the still waterline than sim-ilarly sized plunging or surging breakers. This may explain why collapsing breakers are most damaging to the stability of armour stones. Figure 5-10 shows u/\/g~H at point 2 for the same regular waves. Peak down-slope velocities at this lower location are consistently significantly less than higher up the rubble-mound at point 1. This confirms that parallel velocities closer to the still waterline are more intense than those lower down the slope. This result is consistent with the fact that initial damage to the armour tends to occur just below the still waterline. At measurement point 2, u/\fg~H is influenced by wave height for £ < 4, and by wave period for £ < 2.5, in a manner that is similar to what has 156 -0 .3 -0 .4 - 0 . 5 C M \ -0 .6 K ° -0 .7 BJ) \ c \ -0 .8 -0 .9 -1 -1 .1 - 1 . 2 a U l / 3 ' ser ies 4 • U l / 3 ' ser ies 5 A U l/20' ser ies 4 A U l/20- ser ies 5 1.5 2 2.5 3 3.5 * = t a n a ( g T 2 / 2TTH ) 1 / 2 Figure 5-11: Influence of significant wave height on peak downrush velocities at point 1. been described for point 1. Figure 5-10 also suggests that stronger downrush velocities were measured on the milder sloped series 5 rubble-mound. This may be related to the fact that measurement point 2 is located at a higher elevation on the series 5 rubble-mound, or to the difference in core permeability for these two structures. Downrush flow velocities due to irregular waves are also significantly influenced by both wave height and wave period. The influence of significant wave height can be seen in Figure 5-11, which shows Uij^/y/gHl and W i ^ o / v ^ i ^ s at point 1 in several different wave conditions all with Tp — 2 s, plotted against the irregular wave surf similarity £ m . On both structures, sea states with larger significant wave heights generate stronger non-dimensional peak downrush velocities. In all cases, the downrush velocity peaks are slightly weaker than the corresponding uprush velocity peaks for the same wave condition. The influence of wave period on the peak downrush velocities for irregular waves can be seen in Figure 5-12. Here, downrush velocity peaks at point 1 are shown for several different wave conditions, all with Hs ~ 14 cm. There is a trend towards stronger downrush flows with longer 157 -0.3 -0.4 -0.5 CM -0.6 Til -0.7 ' — ' d \ -0.8 -0.9 -1 -1.1 -1.2 • U l / 3 ' ser ies 4 • U l / 3 - ser ies 5 A U l/20' ser ies 4 • U l/20' ser ies 5 1.5 2 2.5 3 3.5 | = t a n a ( g T i / 2TTH ) 1 / 2 Figure 5-12: Influence of irregular wave period on peak downrush velocities at point 1. period waves on both structures, but the trend is particularly strong on the milder sloped series 5 rubble-mound where plunging breakers prevail. On both structures, flow velocities during downrush are more strongly influenced by wave period than uprush flow velocities. This can be explained in physical terms by considering that downrush flows are more fully developed at longer wave periods. The volume of water is greater, the water flows down from a higher elevation, and the downrush extends further down the slope. iti/20 ' s o n average 6 2 % greater than U\/^. Since the theoretical ratio for the Rayleigh distribution is 4 0 % , this suggests that the peak downrush velocities for individual flow cycles are not Rayleigh distributed. 5.2 Vertical Distribution of Peak Horizontal Wave Forces The wave-induced forcing of armour stones on a rubble-mound is both temporally and spatially varying, and these variations are coupled to each other so that the temporal fluctuations vary 158 with elevation, or alternatively, the vertical distribution of forcing varies with time. This section is devoted to an analysis of the vertical distribution of the wave forces acting on the armour layer, as measured by the armour panels. This analysis shows that the horizontal force directed opposite to the direction of wave propagation (away from the rubble-mound) is maximized below the still waterline, and that the horizontal force acting in the direction of wave propagation (into the structure) is maximized above it. The armour panels are sensitive to both hydrodynamic and buoyancy forces. When installed at, or above the still waterline, the panels become intermittently submerged under wave attack. Under these conditions, the buoyancy force is time-varying, depending on the instantaneous position of the waterline and the concentration of air bubbles in the water surrounding the panel. The horizontal component of the fluid force is particularly attractive in these circumstances since it remains independent of the buoyancy force at all times. Because of this, the panel provides an accurate measure of the horizontal component of the hydrodynamic part of the fluid forcing, even when it is only intermittently submerged. The operations used to obtain the time-varying horizontal hydrodynamic force F}j{t) for an armour panel are described in Section 2.3.9. Fn(t) represents the horizontal component of the net hydrodynamic force acting on a fixed patch of armour stones and is defined to be positive in the direction of wave propagation. The positive and negative peak values of the horizontal force are denoted by FH and FJJ respectively, and represent the peak horizontal forces acting in-line with, and opposite to the direction of wave propagation. These peak values are obtained by averaging the local maxima and minima of Fn(t) over 100 consecutive regular wave flow cycles. The negative peak value will be more significant in terms of the stability of armour stones and the potential for damage to the armour layer. The elevation of an armour panel can be defined in terms of the elevation of the panel centroid zc with respect to the base of the rubble-mound. The panel centroid represents the geometric centre of the armour panel and is located below the surface of the armour layer by a distance equal to half the thickness of a single layer of armour stones. The difference between 159 Series 2 Series 3 Series 5 Series 6 Slope, cot a 1.75 1.75 3 3 /s ina (cm) 11.4 11.4 7.3 7.3 Water depth, h (cm) 40 55 55 40 Lower Panel Elevation of centroid zc (cm) 31.5 31.5 33.5 33.5 zc — h (cm) -8.5 -23.5 -21.5 -6.5 Upper Panel Elevation of centroid zc (cm) 43.5 43.5 41.5 41.5 zc — h (cm) 3.5 -11.5 -13.5 1.5 Table 5.1: Elevation parameters for test series 2, 3, 5 and 6. the panel elevation and the water depth is zc — h, such that zc — h>0 indicates a panel centered above the still waterline while zc — h < 0 indicates one located below it. It is important to note that the panels provide a measure of the net force on the surface layer of armour integrated over a vertical distance given by I sin a where a is the slope angle of the rubble-mound and I ~ 23 cm is the average dimension of the panel parallel to the surface of the structure in the direction of wave propagation. The vertical distance / s ina equals 11.4cm for the steeper slope with cot a = 1.75, and equals 7.3 cm for the milder slope with cot a — 3. Various elevation parameters for test series 2,3,5 and 6 are summarized in Table 5.1. In test series 2 and 3, the same steeply sloped impermeable core structure was tested at two different water depths (h = 40 and 55 cm respectively) without changing the location of the armour panels. In test series 2, the. still waterline lay between the lower and upper panels so that zc — h= —8.5 for the lower panel and zc — h — 3.5 for the upper panel. In test series 3, both panels were submerged. Test series 5 and 6 are related in a similar manner. In the following, measurements of peak horizontal forces on the upper and lower panels during test series 2 are combined with those from test series 3 to provide an indication of the vertical distribution of the wave-induced loads on a patch of armour stones on a rubble-mound with slope cot a = 1.75. Combining results from these two test series provides information at four elevations ranging between —23.5 cm < zc - h < 3.5 cm. Results from test series 160 5 are combined with those from test series 6 to provide the same information for a rubble-mound with slope cot a = 3 at four elevations over the range —21.5 cm < zc — h < 1.5 cm. It should be emphasized that this analysis provides only an approximate indication of the vertical distribution of wave-induced forcing. The veracity of this approach is affected by several factors. • Testing a structure at two different water levels is not the same as physically moving the panels to a new elevation and repeating the tests at the same water level. The change in water depth between test series 2 and 3, and 5 and 6 may have an influence on the wave-structure interaction process around the still waterline which will in turn affect the forcing of the armour in this region. Any such water depth effect will compound the differences in the forcing experienced by the armour panels due to their change in location relative to the still waterline. • The two armour panels represent a different random arrangement of 50 armour stones and will therefore experience slightly different forcing, even when located at the same position in identical waves. • The panels provide a measure of the net hydrodynamic force integrated over a patch of armour stones that extend over a finite vertical distance. For this analysis, the horizontal component of the net hydrodynamic force is assumed to act at the elevation of the panel centroid, which implies that the horizontal forcing is assumed to be uniformly distributed over the panel. In practice, the elevation of the point of action of this force will vary with time, particularly when the panel is only partially submerged. In all cases however, the vertical variation will be limited to the vertical span of the panel, equal to / sin a. In spite of these factors, the vertical force distributions resulting from this analysis provide some important insight into the spatial variation of the wave-induced forcing of the armour layer. 161 -5 N o > W -20 -10 •15 -251— -30 - 4 -A L _ ' / ' / / / / / / / / / • • l \ \ © / / / / <!> f I ; I / I / I / I / I ; (b A -20 -10 10 20 o H = 10 c m • H = 10 c m A H = 15 c m • H = 15 c m 30 P e a k H o r i z o n t a l F o r c e , F . F „ (N ) H H Figure 5-13: Influence of wave height on the vertical distribution of the peak horizontal forces (cota = 3, T = 3s). 5 . 2 . 1 I n f l u e n c e o f W a v e H e i g h t The most prominent effect of increased wave height is to increase both the positive (inward) and negative (outward) peak values of horizontal force at all elevations. This trend is common to both slopes at each of the regular wave periods T = 1.5, 2, and 3 s considered in this study. In some cases, wave height also has a significant influence on the shape of the force distribution. Consider for example the vertical distributions of peak horizontal force shown in Figure 5-13 for the case of 3 s waves on the milder sloped rubble-mound with cot a — 3. Vertical distributions of FH and FH are shown for two different wave conditions: H = 10 and 15 cm with T = 3 s. The surf similarity parameter for these conditions is £ = 3.9 and 3.2 for the 10 and 15 cm heights respectively, which indicates that semi-surging and near-collapsing breakers prevail. It should be stated that these distributions do not represent the force conditions at any instant in time, but rather indicate the peak forces that occur during the regular wave flow cycle. The peak forces at each elevation generally occur at different times. For both wave heights, the horizontal 162 £ 0 O XI "5 N ti O • F - l -4-> > •10 -15 W -20 -25 / \ \ \ I -30 -20 -10 0 0 A / / / / i i 6 A 10 20 P e a k H o r i z o n t a l F o r c e , F . F „ (N) H H o H = 10 c m • H = 10 c m A H = 15 c m • h- H = 15 c m 30 Figure 5-14: Influence of wave height on the vertical distribution of the peak horizontal forces (cot a = 1.75, T = 2 s). force away from the rubble-mound is maximized below the still waterline over the region with — 15 cm < zc — h < —5 cm, while the horizontal force into the structure is maximized above the still waterline where zc — h > 0. These trends hold over the complete range of wave conditions considered in this study. The greatest increase in forcing with wave height occurs away from the structure at zc — h — —13.5 cm, which corresponds to the upper panel during test series 5. Figure 5-14 shows a second example of the influence of wave height on the vertical distri-bution of the peak horizontal forces for the case of 1.5 s waves on the steeper structure with cot a = 1.75. The surf similarity parameter for these conditions is £ = 3.4 and 2.8 for the 10 and 15 cm heights respectively, which suggests that near-collapsing breakers prevail. The main dif-ference between the distributions shown in Figures 5-13 and 5-14 is the strength of the negative (outward) force peaks below the still waterline. On the steeper slope, the negative forcing is particularly intense in the region zc — h~ —12 cm, while the negative forcing is more vertically uniform on the milder slope. Aside from this difference, the vertical distributions of the negative 163 and positive peak horizontal forces shown in these two figures are quite similar. This similarity-is at first somewhat surprising, considering that both the wave period and the slope angle are different for these two cases. However, the similarity is less surprising when one considers that comparable types of wave breaking prevail in both cases. In both of these cases, the strongest horizontal forces away from the rubble-mound occur between —15 cm < zc — h < —10 cm, and correspond to the upper armour panel during test series 3 and 5 where h = 55 cm. These results suggest that forcing in this region is most critical to the stability of armour stones and the potential for damage to the armour layer. This conclusion is in good agreement with the numerical simulations of Kobayashi et. al. (1990c) which indicate that armour damage is most likely to occur near the elevation zc — h ~ —0.75H. Higher waves increase the hydrodynamic forcing of the armour over a wide range of elevation. In a general sense, higher waves have a pronounced effect on the magnitude of the hydrodynamic forcing, but have a weaker effect on the character of the vertical distribution of the forcing. Thus, changes in wave height have a similar effect on both the spatial and temporal variations of hydrodynamic forcing: - in both cases, changes in wave height have more influence on the magnitude than on the pattern or character of the forcing. The effect of wave height on the temporal variation of hydrodynamic forcing is considered in Section 5.3.1. 5.2.2 Influence of Wave Period Wave period has a complex influence that tends to alter the shape of the vertical distribution of hydrodynamic forcing. This complexity can be attributed to the relationship between wave period, slope angle and the type of wave breaking that occurs on a rubble-mound. Examples of the influence of wave period on the vertical distribution of the peak values of the horizontal hydrodynamic force are shown in Figures 5-15 and 5-16 for the steeper and milder sloped rubble-mounds respectively. In both figures, vertical distributions of FH and FH are shown for three regular wave conditions: H = 15 cm with T = 1.5, 2, and 3 s. These wave conditions result in collapsing and surging breakers on the steeper slope (2.8 < £ < 5.5), and 164 u A "5 " -10 o '•2 -15 CO. > CO W -20 -25 - NV \ \ \ \ \ \ \ \ \ \ \ \ \ w \ \\ \ w PA / / / ft* i I II i M I 11 I I I I I I \ I \ I I I I <!> A <i> i -30 -20 -10 0 10 20 30 40 O F H , T=1.5 s , £ = 2.8 • F H , T=1.5 s , £ = 2.8 A F H , T=2 a, £ = 3.7 A F H , T = 2 s , £ = 3.7 O F H , T = 3 s, £ = 5.5 • F H , T=3 s, £ = 5.5 50 P e a k H o r i z o n t a l F o r c e , F „ , F u (N) H H Figure 5-15: Influence of wave period on the vertical distribution of peak horizontal forces (cot a = 1.75, H = 15 cm). ? 0 o J3 "5 N G~ O +J > W -20 -10 -15 -251— -30 • A » I I I I II I II I U I U I II n / ii \\\ v. w PP A / /  / <m i \ \ i \ \ © A / / / / / ; i i i i i i i i i i i i <t> d A -20 -10 10 O F H , T = 1.5 s , £ = 2.8 • F H , T=1.5 9, £ = 2.8 A F H , T = 2 a, £ = 3.7 A F H , T = 2 s , £ = 3.7 O F H , T = 3 s , £ = 5.5 • F H , T=3 s , £ = 5.5 20 30 40 50 P e a k H o r i z o n t a l F o r c e , F „ , F u (N) H H Figure 5-16: Influence of wave period on the vertical distribution of peak horizontal forces (cot a = 3, H = 15 cm). 165 plunging and collapsing breakers on the milder slope (1.6 < f < 3.2). On the steeper slope (Figure 5-15), there is a trend towards reduced forcing with increasing period well below the still waterline, and an opposite trend towards increased forcing with increasing period above the still waterline. These trends affect both the positive (inward) and negative (outward) force peaks in a similar manner. The increase in forcing above the still waterline at longer wave periods is likely related to the greater energy contained in the uprush flows, and the increases in runup that occur under pure surging breakers. The reduction in forcing well below the still waterline is likely related to the less intense flow reversals that occur under the more stable crests of the longer period waves. At all three periods, the strongest force away from the rubble-mound occurs at zc — h — —11.5 cm, which corresponds to the upper panel on the series 3 rubble-mound. The strongest force away from the structure results from the 1.5 s waves that are nearest to collapsing. On the milder slope (Figure 5-16), the influence of wave period on the forcing of the armour is more uniform with depth. The peak inward forces at all elevations are greatest for the 2 s waves (for which £ = 2.2), and smallest for the 3 s waves (for which £ = 3.2). In the opposite direction (outwards), there is a significant trend towards stronger forcing at longer wave periods (except for the lowest elevation). This trend is most pronounced at the elevation zc — h = —13.5cm, which corresponds to the upper panel on the series 5 rubble-mound. Again, the strongest force away from the structure results from the waves that are nearest to collapsing. 5.3 Temporal Character of Wave Forces The forces presented here are the hydrodynamic component of the fluid force acting on an ar-mour panel comprised of 50 armour stones. These forces represent the net hydrodynamic forces acting on a section of armour consisting of 50 individual armour stones. These are obtained by resolving the armour panel dynamometers as described in Section 2.3.8, and isolating the hy-drodynamic forces from those due to buoyancy as described in Section 2.3.9. When considering 166 these hydrodynamic forces, it is useful to keep in mind that a single panel represents a patch of armour with a submerged weight of 62 N. The relationship between the forces measured on the armour panel and those acting on individual armour stones is considered in Section 4.3.1. The magnitude of the hydrodynamic force is denoted F(t), while the orthogonal force com-ponents normal and parallel to the slope are denoted F/v (t) and Fp (t) respectively. FN (t) is defined positive away from the rubble-mound, while FP (t) is positive up-slope. Maximum and minimum values are denoted by the subscripts " m a J c " and " m i n " respectively. For each of these force quantities, the positive peak value, denoted by the accent " A " , represents an average of the maximum values during 100 consecutive regular wave flow cycles. Similarly, the negative peak value, denoted by the accent " v " , represents an average of the minimum values during 100 consecutive regular wave flow cycles. The peak values of the hydrodynamic force parallel to the surface of the rubble-mound in the up-slope and down-slope directions are denoted by Fp and Fp respectively, while Fjv represents the peak value directed away from the structure, normal to the surface. Short portions of FN (t), FP (t), and F (t) measured on the upper panel of the series 3 rubble-mound in regular waves with H = 13.4 cm, T = 1.5 s are shown in Figure 5-17 together with the time series of waterline elevation r/s (£). A single 1.5 s long flow cycle is shown commencing with a zero-upcrossing of 77^  (£). This structure has an impermeable core with slope cot a = 1.75. The centre of the upper panel is located just below the point of minimum rundown. The surf similarity parameter for this test is £ = 2.9, which suggests that collapsing breakers may prevail. Periods of uprush and downrush are defined based on the motion of the waterline, such that uprush prevails for ^ > 0 while downrush prevails for ^ < 0. The normal and parallel forces in Figure 5-17 are both oscillatory, but they deviate signifi-cantly from the fundamental sinusoidal form. Fluctuations of the parallel force component lag behind those of the normal force, indicating a rotating force vector that sweeps out an irregular quasi-elliptical path during the flow cycle. The direction of rotation of the hydrodynamic force vector is identical to the orbits of water particles under progressive waves. 167 N P 60.2 60.4 60.6 60.8 61 61.2 61.4 61.6 Time (s) Figure 5-17: Waterline motion and hydrodynamic forces under regular waves with £ = 2.9 (test series 3, upper panel, H = 13.4 cm and T = 1.5 s) In Figure 5-17, the parallel force acts down-slope (Fp(t) < 0) for approximately 8 0 % of the flow cycle, reaching a minimum of FP>mm = —15 TV at time t = 61 s. At this time, the waterline is falling and surface flows around the armour panel are directed down-slope. These negative parallel forces are caused by downrush flow under the falling waterline. The zero-upcrossing of the parallel force is roughly coincident with the arrival of the advancing wave crest and the transition from downrush to uprush. Positive Fp(t) prevail for a shorter duration but reach a larger maximum value of Fpm3X = 19 iV. The positive parallel forces are caused by up-slope flows generated by the advancing wave crest. The normal force component shown in Figure 5-17 is also strongly asymmetric and features a short duration of small negative forces followed by a long duration of relatively large positive forces. The positive normal forces reach a maximum of -F^max = 37 N that is roughly coincident with the zero-upcrossing of the parallel force and the flow reversal under the toe of the advancing wave crest. These large positive normal forces result from a combination of seepage flows out 168 from the porous surface layers of the rubble-mound and strong accelerations under the toe of the breaking wave crest. The total hydrodynamic force on the armour panel reaches a maximum value of F m a x = 38 N that is dominated by the normal force component and occurs with Fp(t) > 0 and F^it) > 0, which indicates that the maximum hydrodynamic force on the armour panel acts away from the surface of the rubble-mound and slightly up-slope. Observation of the tests, combined with computer animation of the measured quantities and correlations between measured forces and measured kinematics indicate that the large positive normal force results primarily from a superposition of effects from two different processes. One of these is the strong fluid acceleration that occurs under the toe of a plunging or collapsing wave crest. The second process relates to seepage flow due to drainage from the porous zones of the rubble-mound. When the external waterline is lowered, water drains down and out of the structure through the porous zones. This water seeps out through the surface, contributing to significant slope-normal hydrodynamic forces. Both of these mechanisms contribute to the large positive normal force seen in Figure 5-17. Estimates of the magnitude of the seepage force that might act on the surface layer of armour can be obtained by analogy to the conditions within a saturated granular soil subject to a steady hydraulic gradient i. The seepage force per unit volume acting in the direction of flow is pgi. The seepage force acting on a group of iV particles, each occupying a volume of D3/ (1 — n), is given by Npg^^i. These forces can be expressed in terms of the bulk internal flow velocity v by using the Forchheimer equation (written in Section 2.3.2 as Equation (2.8)) to describe the dependence between v and i. In terms of the bulk internal flow velocity, the expression for the seepage force on a group of N particles is Npg (av + bv2), where a and b are the coefficients in the Forchheimer equation that characterize the flow resistance of the material. The critical hydraulic gradient ic is the value of i at which the seepage force is equal to the 169 submerged weight of the porous material. This condition can be written as _ N(pa-p)gD3n50 C " ^ ^ 5 0 / ( l - n ) - A ( 1 ( 5 " 7 ) If the direction of seepage flow is vertically upwards, opposite to the gravitational acceleration, the soil mass will loose all strength when i = ic and liquefaction will ensue. The critical value of the bulk internal flow velocity vc corresponding to i = ic is obtained from Equation (2.8) as -a + sjd2 + 4bic vc = ^ , (5.8) which for a <C b simplifies to vc ~ y/ic/b. For the armour panels used in this study (with N = 50, Dn5o = 0.042 m, n = 0.41), the seepage force (expressed in Newtons) evaluates to 61.6z. Taking A = 1.68 and n = 0.41, the critical value of hydraulic gradient for the armour is ic = 1. This implies that an armour panel will experience a force of 62 N (equal to its submerged weight) when subject to seepage flows driven by a critical hydraulic gradient. The critical hydraulic gradient can be assumed as an absolute upper bound to the gradients that occur within an armour layer under wave attack. This assumption implies that the seepage forces acting on an armour panel will always be less than 62 N. Seepage flows are most evident towards the end of downrush, when the external waterline falls well below the internal phreatic surface. At this time, the slope of the free surface within the armour layer may approach the slope of the rubble-mound. Assuming that quasi-stationary conditions prevail, the hydraulic gradient within the armour under these conditions can be estimated as i ~ sin a . Since the seepage force is directly proportional to the hydraulic gradi-ent, this suggests that the seepage force will increase on structures with steeper slopes. This approach yields an estimate of the seepage force on an armour panel that evaluates to 31N for the steeper structures with cot a = 1.75, and to 19 TV for the milder sloped structure with 170 cot a = 3. Figure 2-5 indicates that the flow resistance of the densely packed armour can be well characterized by the Forchheimer coefficients a = 0.1 and b — 29. Equation (5.8) yields vc = 0.18 m/s as the critical bulk velocity for flow within the armour layer. This bulk velocity corresponds to an average interstitial velocity o£vc/n = 0.44 m/s. The bulk flow and interstitial velocities will be lower for weaker hydraulic gradients. Recordings of slope-normal velocity obtained 4 cm above the surface of the armour layer (examples are shown in Figures 5-2 - 5-4) indicate peak velocities normal to the surface of the armour in the order of 0.4 m/s. These velocities are similar to the average interstitial velocities predicted within the armour layer under a critical hydraulic gradient. Clearly, the bulk internal flow velocity within the armour layer will be significantly less than the slope-normal velocity outside the structure. In addition, the hydraulic gradients and seepage flows within the armour layer of a rubble-mound under wave attack are unsteady and spatially varying. These aspects will affect-the seepage forces acting on a patch of armour stones located on the surface of the armour layer. In spite of these concerns, the theoretical estimates of seepage force obtained by assuming quasi-stationary conditions towards the end of downrush, combined with the measurements of slope-normal velocity, together suggest that seepage flows are an important forcing mechanism within the armour layer that should not be neglected. 5.3.1 Influence of Wave Height The general influence of wave height is to alter the magnitude, but not the character of the hydrodynamic forcing of the armour. This is particularly so if the type of wave breaking does not change rapidly with wave height. Increases in wave height tend to increase all aspects of the forcing, including the negative and positive peak values of the slope-normal and slope-parallel force components. The dramatic effect of wave height on the forcing of armour stones is clearly illustrated in 171 H = 9 . 2 c m , £ = 4.7 H = 1 2 . 8 c m , £ = 3.9 — - H = 1 8 . 7 c m , £ = 3.3 -0.3 -0.15 0 0.15 0.3 F / F p ' sw Figure 5-18: Influence of wave height on the hydrodynamic forcing of the armour (lower panel, series 4, T = 2 s). Figure 5-18, which shows hodographs of the hydrodynamic forcing on the lower armour panel during three tests with 2 s regular waves from series 4 in which only the wave height was varied. Figure 5-19 shows another example of the effect of wave height for the case of forcing on the upper panel of the series 5 structure in 1.5 s waves. In these figures, each curve represents the average path mapped out by the rotating force vector in the vertical plane aligned with the direction of wave propagation during 100 consecutive regular wave flow cycles. The force measurements for single flow cycles are generally quite repeatable (examples can be seen in Figures 4-8 - 4-10) and do not deviate significantly from the average. The average hodograph is thus very similar to the hodographs for individual flow cycles. The hodographs are shown in the force plane defined by Fp/Fsw and FN/FSW where the abscissa and ordinate represent non-dimensional forms of the slope-parallel and slope-normal forces respectively. This coordinate system is the same one used in Section 4.2.3 to plot contours of the rolling-sliding failure index. Indeed, the failure index contours shown in Figures 4-3 and 4-4 can be used as a guide to assess 172 — H = l l . l c m , £ = 1 . 9 — H=13.8cm, £ = 1 . 7 — -H=17.6cm, £=1.5 -0.2 -0.4 -0.2 0.2 0.4 F / F P ' sw Figure 5-19: Influence of wave height on the hydrodynamic forcing of the armour (upper panel, series 5, T = 1.5 s). the differences between force hodographs in terms of the potential effect on the stability of armour stones. Rolling-sliding failure index contours for the series 3 and 4 structures are shown in Figure 4-3, while index contours for the series 5 rubble-mound are shown in Figure 4-4. Figures 5-18 and 5-19 show typical examples of how increasing wave height affects the hydrodynamic forcing of a patch of armour stones located below the still waterline. The forces in Figure 5-18 were recorded under semi-surging breakers, while those in Figure 5-19 were obtained under plunging breakers. Measurements at different locations, on different structures, and at different wave periods all show a similar trend. For each wave height, the shape of the force hodograph is similar but the magnitude of the forcing is consistently greater. The forcing directed away from the surface of the rubble-mound increases more rapidly with wave height than the forcing directed into the structure. The rate of increase of the slope-parallel component of the forcing is similar in the up-slope and down-slope directions. 173 0.2 40 0.15 30 -0.1 -20 59.5 59.75 60 60.25 60.5 60.75 61 61.25 T i m e (s) Figure 5-20: Waterline motion and hydrodynamic forces under regular waves with £ = 3.9 (test series 3, upper panel, H = 13.3 cm and T = 2 s). 5.3.2 Influence of Wave Period Wave period has a dramatic and complex effect on the temporal character and magnitude of the hydrodynamic forcing of armour stones on a rubble-mound. The effect of wave period is coupled to the slope angle of the structure, since both of these factors influence the type of wave breaking that prevails. The general nature of this period effect is considered here with reference to force measurements from three different tests where only the wave period was varied. Time series of waterline elevation and the hydrodynamic forces recorded on the upper panel from two different series 3 tests with wave periods of 2 s and 3 s are shown in Figures 5-20 and 5-21 respectively. For the 2 s waves in Figure 5-20, H = 13.3 cm and £ = 3.9; while H = 13.1 cm and £ = 5.9 for the 3 s waves in Figure 5-21. The 3 s waves interact with the rubble-mound in a pure surging motion while the 2 s waves are in transition between collapsing and surging. The differences in the time-varying forces shown in Figures 5-17, 5-20 and 5-21 are due entirely to the effects of wave period, since virtually identical wave heights were used for each test. 174 60.4 60.8 61.2 61.6 62 62.4 62.8 Time (s) Figure 5-21: Waterline motion and hydrodynamic forces under regular waves with £ = 5.9 (test series 3, upper panel, H = 13.1 cm and T = Ss). With 2 s waves (Figure 5-20), the positive normal force exhibits two distinct peaks at times t = 60.5 and 61.1 s. These two peaks roughly coincide with the end of downrush and the start of uprush. The first peak is due primarily to seepage flows out from the porous zones of the rubble-mound, while the second peak is primarily related to the sudden flow reversal under the advancing wave crest. Normal forces due to seepage flows are maximized near the end of downrush and diminish gradually with time thereafter as the water stored in the porous zones of the rubble-mound is released. These strong normal forces coincide with negative parallel forces which combine to yield a resultant force vector directed down-slope and away from the surface of the rubble-mound. This type of forcing reduces the stability of armour stones. The second positive peak in the normal force near t = 61.1s is roughly coincident with the zero-upcrossing of the parallel force, and results from the large accelerations accompanying the sudden flow reversal that occurs under the leading edge of an advancing wave crest. In the case of forcing by 1.5 s waves (shown in Figure 5-17), the positive normal forces from these two 175 mechanisms overlap to form a single maximum that is larger than either of the two separate maxima observed in the 2 s waves. The temporal character of the parallel force due to the 2 s waves is quite similar to that observed for the 1.5 s waves, but the maximum and minimum values are both somewhat weaker. Time series of the forcing on the upper panel of the series 3 rubble-mound due to 3 s waves (shown in Figure 5-21) indicate a significantly different behaviour for the normal force component under surging breakers. In this case, positive normal forces reach a single maximum that coincides with the end of the downrush portion of the flow cycle. The second normal force peak observed near the start of uprush in the 2 s waves is no longer present. The surging breakers that prevail in this case induce a more gradual flow reversal and relatively weaker accelerations under the toe of the advancing wave crest, which result in significantly diminished normal forces towards the end of the flow cycle. Under surging breakers, the peak positive normal force on armour stones below the still waterline is due mainly to seepage flows which peak at the end of downrush. The dramatic effect of wave period on the forcing of armour stones is clearly illustrated in Figure 5-22, which shows hodographs of the hydrodynamic forcing on the upper armour panel during the same three tests from series 3 considered above in which only the wave period was varied. The hodographs in Figure 5-22 provide an alternative representation of the forces shown in Figures 5-17, 5-20 and 5-21. These figures clearly show that the magnitude and character of the hydrodynamic forces acting on a patch of armour stones located just below the still waterline depend strongly on wave period. Results from the lower armour panel also show a strong dependence on wave period, which suggests that the effect of wave period is not confined to a single location on the rubble-mound. For these conditions (H ~ 13 cm; cot a = 1.75; T — 1.5, 2 and 3s for which £ = 2.9, 3.9 and 5.9 respectively), wave period has a particularly strong influence on the positive peak values of the parallel force Fp and the normal force FN, such that longer period waves exert lower forces. Period has a weaker influence on the negative peak value of the slope-parallel force Fp, and almost no influence on the negative peak value of 176 0.6 — T = 3s, £ = 5.9 — T = 2s, £ = 3.9 T=1.5s, £ = 2.9 -0 .2 -0 .4 -0 .2 0 0.2 0.4 sw Figure 5-22: Influence of wave period on the hydrodynamic forcing of the armour (upper panel, series 3, H ~ 13 cm). forcing directed up-slope and away from the surface of the rubble-mound, but has less influence on the forcing acting down-slope and inwards. Some of this dependence on period can be explained in terms of the type of wave breaking that prevails. The three cases shown in Figures 5-17, and 5-20 - 5-22 correspond to near-collapsing, semi-surging and pure surging breakers. The peak value of the force magnitude F is greatest for collapsing breakers, and decreases with increasing wave period. The timing of the peak force with respect to the flow cycle on the surface of the rubble-mound also varies with wave period. In surging waves, the peak occurs near the end of downrush (when ^ < 0 with r]s < 0), while in collapsing breakers, the peak occurs near the start of uprush (when ^f- > 0 with rjs < 0). With collapsing breakers, the peak force acts away from, and nearly perpendicular to the surface of the rubble mound, and is caused by the strong flow reversal under the breaking wave crest. However, with surging waves the peak force results from draining flows and features the slope-normal force F^. Thus, wave period has a very strong influence on the hydrodynamic 177 a significant down-slope component. Several of the failure mechanisms considered in Chapter 4 suggest that the direction of the hydrodynamic force has a strong influence on the relative stability of armour stones. In general, hydrodynamic forces acting away from the surface of the rubble-mound tend to reduce the stability of armour stones, while forces acting down-slope are more dangerous than those acting in the up-slope direction. 5.3.3 Influence of Slope Angle Comparisons of the forces measured on the milder sloped series 5 rubble-mound (cot a = 3) to those recorded on the steeper series 3 structure (cot a = 1.75) under similar incident waves clearly indicate that slope angle has a strong influence on the nature of the hydrodynamic forces acting on armour stones below the still waterline. The influence of slope angle is coupled to the wave period, since both factors influence the type of wave breaking that prevails. Consideration of results for the upper and lower armour panels also indicates that the influence of slope angle on forcing varies with the elevation. Changes in slope angle alter both the slope-normal and slope-parallel components of hy-drodynamic forcing. A common aspect of this alteration is that the peak parallel force acting down-slope tends to be weaker on the milder sloped structure. This tendency generally holds at both the upper and lower panels for most of the wave conditions considered in this study. This reduction in down-slope forcing is consistent with the idea that downrush and seepage flows will be less intense on a milder slope and more intense on a steeper one. In the following, data from a single series 5 test are shown to illustrate the effects of slope angle on forcing for a specific wave condition. One flow cycle of ns (t) and Fp (£), F^ (t) and F (t) on the upper panel of the series 5 rubble-mound in regular waves with T — 2 s and H = 14 cm are shown in Figure 5-23. These waves are nearly identical to those considered for Figures 5-20 and 5-25. On the steeper structures, these waves interact as semi-surging breakers (£ = 3.9), but on the milder slope, they interact as semi-plunging breakers (£ = 2.2). These different types of wave breaking alter the temporal and spatial distribution of kinematics on the 178 59.7 60 60.3 60.6 60.9 61.2 61.5 T i m e (s) Figure 5-23: Waterline motion and hydrodynamic forces under regular waves with £ = 2.2 (test series 5, upper panel, H — 14cm and T = 2s). rubble-mound which in turn influence the character and magnitude of the hydrodynamic forces acting on the armour. The forces for the tests shown in Figures 5-20 and 5-23 are also shown together as hodographs in Figure 5-24. This figure shows that the positive peak values of the normal and parallel forces on the series 5 rubble-mound are both greater than those measured on the steeper series 3 structure in similar waves. In the case of the parallel force, Fp on the milder slope is 7 7 % greater than on the steeper one. This large increase in positive forces is caused by the strong positive accelerations and velocities which occur under the crests of plunging breakers. As discussed above, there is also a significant reduction in the peak parallel force acting down-slope on the milder slope. The wave-induced forces acting on armour stones on different slopes are more alike when similar types of wave breaking prevail on each slope. This can be seen by comparing the forces measured on the series 5 structure under semi-plunging breakers (£ = 2.2) shown in Figure 5-23 to those recorded on the series 3 structure under collapsing breakers (£ = 2.9) shown in 179 T = 2 s, H = 1 4 cm Series 3, cota = 1.75, £ = 3.9 Series 5, cota = 3, £ = 2.2 Figure 5-24: Influence of slope angle on the hydrodynamic forcing of the armour (upper panel, H ~ 14cm, T = 2s). 180 Figure 5-17. Although both the wave period and structure slope angle are different in these two tests, the nature of the wave-induced forcing on the armour stones is quite similar. This can be attributed to a similarity in the type of wave breaking that prevails on each slope. This observation reinforces the conclusion that the strong influences of slope angle and wave period on the wave-induced forces acting on armour stones can be at least partially explained in terms of the type of wave breaking that prevails. Since breaker type has been successfully parameterized in terms of the surf similarity parameter, this suggests that the wave-induced forces on armour stones could also be considered in terms of £. 5.3.4 Influence of Core Permeability The effects of permeability on the internal flows within a rubble mound under wave attack are described by Bruun (1985) and Hall (1987, 1989). On conventional structures with permeable armour and filter layers over an impermeable core, internal flows are confined to the permeable outer layers. On these structures, internal flow travels up and down through the armour layer, virtually parallel to the slope. In contrast, on structures with permeable cores, some of the flow introduced by each wave penetrates into, and is dissipated within the core. It seems reasonable, therefore, to expect a trend towards weaker slope-parallel surface flows and stronger slope-normal flows on structures with more permeable underlayers. The permeability of the core is typically only sufficient to admit a portion of the water volume incident with each wave, such that the armour layer continues to convey a significant portion of the internal flow. The structures investigated during test series 3 and 4 represent two versions of a rubble-mound which differ only in respect to the permeability of the materials below the armour layer. The permeability of the armour and core for these structures is considered in Section 2.3.2. A comparison of the hydrodynamic forces measured on the armour under similar wave conditions during these two test series indicate that increased permeability generally gives rise to greater forces normal to the surface and lesser forces parallel to the surface. This result is consistent with the idea that slope-parallel flows near the surface of the armour layer should be weaker 181 p 0.15 0.05 -0.05-59.25 59.5 59.75 60 60.25 60.5 T i m e (s) 60.75 61.25 4) O O 6-Figure 5-25: Waterline motion and hydrodynamic forces under regular waves with £ = 4 (test series 4, upper panel, H = 12.8 cm and T = 2 s). and slope-normal surface flows should be stronger on rubble-mounds with more permeable underlayers. Figure 5-25 shows the waterline elevation and hydrodynamic forces on the upper panel of the series 4 rubble-mound for one cycle of regular wave attack with T = 2 s and H = 12.8 cm, for which £ = 4.0. The influence of core permeability on the forces on a patch of armour stones can be discerned by comparing the measurements shown in this figure to those shown previously in Figure 5-20, since nearly identical wave conditions were used in each test. Hodographs of the hydrodynamic force on the upper and lower armour panels during these two tests are shown in Figures 5-26 and 5-27 respectively. The results for both panels indicate that increased core permeability gives rise to larger slope-normal forces and smaller slope-parallel forces. In particular, the normal forces directed into the rubble-mound during uprush are roughly twice as large on the series 4 rubble-mound with the permeable core. This increase in normal force is consistent with an increase in the infiltration of water into the structure across the surface of the armour layer. Positive normal 182 0.6 0.4 0.2 -0.2 -0.4 T = 2s, H = 1 3 c m , £ = 4 1 1 \ - < \ N \ \ // // /'/ A / / \ \\ \ \\ - \ 1 1 \ \\ \\ \ \ \ \ \ j\ / 1 / 1 / / / | \ \ / / V j / / / \ y \ s i 1 •0.2 0.2 P ' sw Series 3, impermeable core Series 4, permeable core 0.4 Figure 5-26: Influence of core permeability on the hydrodynamic forcing of the armour (upper panel, T = 2 s, H ~ 13cm, £ = 4). 183 0.3 0.15 -0 :15 •0.15 T = 2s, H = 1 3 c m , £ = 4 1 / \ / / \ \ / / |: \ / / ' \ / 1 ' \ '' / / ' ^ > / /l / 1 / 1 1 1 j J / \ v ^-^ \ \ \ ' \ / / / / i 0.15 F / F p ' s Series 3 , impermeable core Series 4 , permeable core 0.3 Fi gure 5-27: Influence of core permeability on the hydrodynamic forcing of the armour (lower panel, T = 2s, H-~ 13cm, £ = 4). 184 forces are also slightly increased on the more permeable rubble-mound. The bi-modal character of the positive normal force initially identified in Figure 5-20 still persists on the more permeable rubble-mound, but the two force peaks are slightly less distinct. Smaller parallel forces acting down-slope during downrush were recorded on the more per-meable structure at both elevations. This reduction in down-slope forcing has an important effect on the stability of the armour layer and its ability to resist damage under wave attack. Armour stones tend to be more stable on more permeable structures because of this reduction in slope-parallel forcing. This reduction suggests that the permeable core acts to reduce the downrush kinematics on the surface of the rubble-mound and within the armour layer. For the upper panel, the character and magnitude of the parallel forces acting up-slope are very similar, but for the lower panel, the up-slope forcing is also weaker. These results, considered together with similar results for other wave conditions suggest that when compared to the effects of wave period or wave height, core permeability has a relatively small influence on the overall character and magnitude of the hydrodynamic forces acting on armour stones below the still waterline. The improved performance of armour stones on more permeable structures appears to be related to a reduction in the slope-parallel hydrodynamic forcing acting down-slope. Although the influence of core permeability on forcing is small (relative to the effects of other factors such as wave height and wave period), it can have an important beneficial effect on the stability and performance of the armour layer. 5.4 Character of the Maximum Wave Force The largest hydrodynamic force acting on a patch of armour stones during regular wave attack can be completely characterized by its magnitude F, direction of action 6 [FJ , and phase $ (P1^. For this analysis, consecutive zero-upcrossings of the waterline on the surface of the rubble-mound are used to define the beginning and end of each flow cycle. The phase of the peak force <& (P) is defined between 0° and 360° such that $ (Pj = 0° indicates that the peak 185 force coincides with the initial zero-upcrossing of rjs (f), and $ yFj = 180° indicates that the peak force occurs midway between the two zero-upcrossings. The direction of action G (J?^J is also defined between 0° and 360° such that 0 [F^j = 0° indicates a force acting up-slope, parallel to the surface of the rubble-mound, and 6 [FJ = 90° indicates a force directed normal to, and away from the surface of the structure. The magnitude, direction and phase of the maximum wave-induced force acting on a patch of armour stones on a rubble-mound depend on many factors, including the character of the incident waves, the geometry of the structure, and the location of the armour stones. In the following, characteristics of the peak wave-induced force acting on a single patch of armour located below the still waterline on three different structures are presented. This analysis will focus on data from the upper panel during test series 3, 4 and 5 because this panel experiences the strongest outward horizontal forces, is located high enough on the rubble-mound to be within the zone in which initial damage generally occurs, yet remains sufficiently submerged to capture hydrodynamic forces throughout the complete flow cycle. This analysis provides some insight into the mechanisms responsible for the peak hydrodynamic force, and its relation to the type of wave breaking that prevails on the rubble-mound. 5.4.1 Magnitude of the Maximum Wave Force Hudson (1958) assumed the maximum wave-induced force acting on armour stones to be pro-portional to the height of incident waves at breaking. This assumption led him to the stability design formula shown as Equation (3.2), which has become widely used for the initial design of rubble-mound armour. Results from this study indicate that the magnitude of the maxi-mum hydrodynamic force acting on a patch or armour stones below the still waterline depends strongly on wave height, but is also significantly influenced by wave period and the slope of the rubble-mound. For waves with constant period on a fixed slope, the relation between F and H is approximately linear, which supports the assumption of Hudson (1958). However, these results also indicate that wave period has a significant influence on the relationship between F 186 • S e r i e s 3 , T = 1 .5 s M S e r i e s 3 , T = 2 s • S e r i e s 3 , T = 3 s o S e r i e s 4 , T = 1 .5 s S e r i e s 4 , T = 2 s O S e r i e s 4 , T = 3 s 0.25 Figure 5-28: Influence of core permeability, wave height and period on peak hydrodynamic force (upper panel, test series 3 and 4). and H, such that larger forces generally result from waves that break by collapsing. This find-ing supports the use of design equations that predict a variation in stability with wave period such that stability is minimized for collapsing breakers. The formulae proposed by Losada and Gimenez-Curto (1979) (Equation (3.7)) and van der Meer (1988) (Equations (3.10) and (3.11)) are of this type. The influence of wave period was not considered by Hudson. F recorded on the upper armour panel of the series 3 (impermeable core, cot a = 1.75) and series 4 (permeable core, cot a = 1.75) rubble-mounds in regular waves are shown in Figure 5-28 as a function of incident wave height H. The data are grouped by test series and wave period. The range of F on each structure at any given wave height illustrates the significant influence of wave period on the peak magnitude of hydrodynamic loads. These results indicate that variations in wave period can increase the peak hydrodynamic load by 50 % or more for the range of conditions considered in these experiments. Slightly larger peak hydrodynamic forces were consistently recorded on the more permeable series 4 structure than on the less permeable 187 • S e r i e s 3, T = 1 . 5 s M S e r i e s 3, T = 2 s • S e r i e s 3, T = 3 s o S e r i e s 5, T = 1.5 s * S e r i e s 5, T = 2 s o S e r i e s 5, T = 3 s 0 0.05 0.1 0.15 0.2 0.25 H ( m ) Figure 5-29: Influence of slope angle, wave height and period on peak hydrodynamic force (upper panel, test series 3 and 5). rubble-mound. This slight increase in F with greater core permeability prevails for all three wave periods, and is somewhat surprising, considering that damage to rubble-mound armour generally decreases on more permeable structures. The relationship between damage and the wave-induced forces on the upper armour panel are considered in Chapter 4, where it is shown that damage is more strongly correlated to the slope parallel component of the fluid force than to F. The effects of core permeability on the forcing of the armour are considered further in Section 5.3.4. The influence of slope angle on the peak wave-induced force can be seen in Figure 5-29, which shows F recorded on the upper panel of the steeper sloped series 3 (impermeable core, cot a = 1.75) and milder sloped series 5 (impermeable core, cot a = 3) rubble-mounds in regular waves. The influence of slope angle is compounded by wave period. F is reduced by approximately 4 0 % on the milder slope in 1.5 s waves, but in 2 s waves, F is roughly 2 0 % larger. Structure slope has negligible effect on F in waves with T = 3 s. Both wave period and 188 400 350 -300 55 250 33 <fc-II "3* 200 150 -100 50 0 0 o o c p o o 0 o ° o O o e gr- ooo'gr0 0 o n o o •e—Q-O O O o O 3 4 5 £ = t a n a ( g T 2 / 2 7 r H ) 1 / 2 Figure 5-30: Influence of surf similarity on the relation between peak hydrodynamic force and regular wave height (upper panel, test series 3, 4 and 5). structure slope influence the type of wave breaking that prevails on a rubble-mound, which in turn affects the fluid kinematics and the hydrodynamic forces acting on the armour stones. The combined influence of wave period and structure slope on F can be partially described in terms of the surf similarity parameter. Consider the simple model F = k(£).H (5.9) where the peak wave induced force F on a patch of armour stones is linearly related to the incident wave height H by the coefficient k (£) that depends on the surf similarity £. Values of k (f) for the upper armour panel in test series 3, 4 and 5 are shown in Figure 5-30. For £ < 2.5 where plunging breakers prevail, A; (£) can be well represented by k (£) = 100 • £, while for £ > 4 where surging breakers prevail, the data indicate that k (£) is constant and approximately equal to 200 N/m. These functions are shown in Figure 5-30 by solid lines. Over the critical range with 2.5 < £ < 4 in which collapsing breakers can occur, the relation between F and H is highly 189 ° F l / 3 . s e r i e s 3 * F l / 3 ' s e r i e s 4 D F l / 3 ' s e r i e s 5 0.05 0.075 0.1 0.125 0.15 0.175 0.2 H (m) Figure 5-31: Influence of significant wave height, core permeability and slope angle on F\/3 (Tp = 2s). varied and cannot be described as a function of £ alone. The largest peak forces are observed in waves for which £ ~ 3; however, the condition £ ~ 3 is not sufficient to ensure the occurrence of particularly large peak forces. For £ ~ 3, the largest peak forces occurred in 1.5 s waves while the smallest occurred in 3 s waves. This suggests that F/H oc 1/T for £ ~ 3. A special analysis algorithm was developed and applied to determine characteristic peak values of the hydrodynamic force magnitude for irregular waves. Standard zero-crossing analysis is clearly not adequate since F(t) is always greater than zero. Individual flow cycles were defined by consecutive zero-upcrossings of the waterline elevation r)s(t). A single local maximum of F(t) was identified for each flow cycle, and these maxima were analyzed in a statistical sense to determine characteristic values. The significant value of the force peaks, denoted by F1/3, represents the average value of the largest one-third of all local maxima. Figure 5-31 shows an example of the influences of wave height, core permeability and slope angle on the significant value of the peak hydrodynamic force for irregular wave conditions 190 with identical peak periods of Tp = 2 s. These forces were measured on the upper armour panel during test series 3,4 and 5. The wave conditions were synthesized from identically shaped spectra, and the average wave period for each sea state is Tm ~ 1.6 s. The data for each structure suggest an approximately linear relationship between significant wave height and significant peak force. Results for the series 3 (impermeable core, cot a = 1.75) and series 4 (permeable core, cot a = 1.75) structures indicate that the armour on the more permeable structure experiences slightly stronger hydrodynamic forces. This result is the same as for regular waves. The influence of core permeability on armour layer forcing is considered further in Section 5.3.4. For the wave conditions shown in Figure 5-31 where Tm ~ 1.6 s and Tp = 2 s, slope angle has a strong and fairly consistent effect on the significant peak force such that armour stones on the milder slope experience weaker forces. This slope effect is coupled to the wave period. The combined effect of wave period and slope angle on the irregular wave force peaks is quite similar to their effect in regular waves, provided that the average period (and not the peak period) is used to characterize the irregular waves. This can be seen by comparing the effect of slope angle shown for irregular waves in Figure 5-31, to that shown for regular waves in Figure 5-29. In regular waves, peak forces on the milder slope are 4 0 % less at T = 1.5 s, but 20 % greater at T = 2 s. The effect of slope angle on the significant peak forces in irregular waves with Tm ~ 1.6 s is very similar to its effect on peak forces in regular waves with T = 1.5 s. This result indicates that the average wave period provides a superior characterization of an irregular sea state in terms of the consistency between the effects of regular and irregular waves on the armour layer. Under irregular waves, the variation in the magnitude of the peak hydrodynamic force with wave height, core permeability, and slope angle is similar to that which occurs under regular waves. In both cases, the peak forces increase as an approximately linear function of wave height, but are also significantly influenced by the wave period and the slope angle. The influences of wave period and slope angle are coupled to each other and depend on the type 191 of wave breaking that prevails on the slope. Greater core permeability causes the peak forces to increase slightly. These conclusions apply equally well to peak forces at the significant level and more extreme levels, including Fi/20-Relation Between Maximum Wave Force and Waterline Motion In Section 5.1.3, flows on a rubble-mound were characterized in terms of the motion of the waterline on the surface of the structure. Excursions of the waterline were found to increase with increasing surf similarity, and the rate of increase was rapid for plunging breakers, and gradual for collapsing and surging breakers. Several different indirect measures of flow velocity on a rubble-mound slope can be con-structed from the motion of the waterline. If the vertical motion of the waterline is assumed to be sinusoidal, such that *(*) = ^ s i n ( ^ ? ) » (5-10) then the maximum vertical velocity of ns can be written as v = ^ (5.11) while the maximum velocity parallel to the surface can be written as va = — ^ - . 5.12 T s m a v and va represent estimates of the maximum vertical and slope-parallel velocity of the waterline expressed in terms of the height and period of waterline motions. While v and iia are not true flow velocities, they can be expected to be in some way proportional to flow velocities on the surface of the structure, and thus useful in the analysis of the wave-induced forcing of armour stones. The relationship between va and F at the upper panel of the series 3, 4 and 5 structures in 192 60 50 40 2 — 30 20 10 o*2 A A O i A O. A A A A ' A A A > A ^ ° XI XI XI A XI X l " * Series 3, cota = 1.75 Series 4, cota = 1.75 Series 5, cota = 3 J _ 0.25 0.5 0.75 1 £ (m/s) 1.25 1.5 1.75 Figure 5-32: Relationship between peak hydrodynamic force and va (upper panel, test series 3, 4 and 5). regular waves is shown in Figure 5-32. This data indicates that the relationship between F and ia depends on the slope of the rubble-mound, but for a constant slope is relatively independent of wave height, wave period and the permeability of the rubble-mound core. For a given slope angle, the peak wave-induced force acting on armour stones below the still waterline is roughly proportional to the velocity of waterline motions on the surface of the rubble-mound. Figure 5-33 shows that the same peak forces are more consistently related to v, the vertical component of waterline velocity. The data in this Figure show a definite positive correlation that can be well represented by an equation of the form F = a-vb . (5.13) Minimum error is obtained with a = 76 and b = 0.89, for which the coefficient of determination is r 2 = 0.86. This function is shown in Figure 5-33 as a solid curve. This result suggests that: 193 ( l 1 1 I I 1 I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 V (m/s) Figure 5-33: Relationship between peak hydrodynamic force and v (upper panel, test series 3, 4 and 5). • v can serve as an indicator of the peak hydrodynamic force acting on armour stones below the mean water level; • the observed relation between F and v is relatively insensitive to wave height, wave period, structure slope and core permeability for the conditions of these experiments; • the observed relation between F and v applies for plunging, collapsing and surging break-ers. 5.4.2 Phase of the Maximum Wave Force The phasing of the peak hydrodynamic force with respect to the cycle of flow on the surface of the rubble-mound can provide useful insight into the mechanism contributing to the peak force and its dependence on breaker type. The phase of the peak hydrodynamic force $ [P^j indicates the part of the flow cycle during which the hydrodynamic force is maximized. During test series 3, 4 and 5, both armour panels registered $ [P^j between 180° and 360° for all wave 194 360 m 300 <U U •S 270 240 210 180 I . I 1 1 1 1 1 - A A A o Ser ies 3 -- "J, " * X A 0 A<A A Ser ies 4 -\ « A o 1X1 « ^ Ser ies 5 o A A 1 1 1 1 1 A A A A ^ A 1 <A { a o A i 2 3 4 5 6 £ = t a n a ( g T 2 / 2 T T H ) 1 / 2 Figure 5-34: Influence of surf similarity on the phase of the peak hydrodynamic force (upper panel, test series 3, 4 and 5). conditions, which suggests that the largest hydrodynamic forces on armour stones below the still waterline occur during the second half of each flow cycle. As discussed in Section 5.3, force peaks near the end of downrush can be attributed to drainage flows down the surface of the slope combined with seepage flows out from the permeable zones of the rubble-mound. Force peaks near the beginning of uprush are associated with the sudden flow reversal under a steep advancing wave crest. Downrush and seepage flows tend to produce larger forces in surging waves, while sudden flow reversal tends to produce the largest forces in plunging breakers. In the following, the value of $ (^Fj is analyzed as an indicator of the mechanism responsible for the peak hydrodynamic force. For armour stones below the mean waterline, $ [FJ ~ 240° can serve as an approximate phase boundary between these two forcing regimes, such that downrush and seepage flows are responsible for the peak hydrodynamic force for 180° < $ [FJ < 240° while sudden flow reversal dominates F for 240° < $ ( F ) < 360°. Figure 5-34 shows $ (P^j computed for the 195 360 330 m 300 <D I-£ 270 240 210 180 A A 44 A * M A Upper panel A Lower panel 1 1 2 3 A * A * A A A A A A A _ J A A A I 4 5 £ = tana (gT2/2-nE)l/2 Figure 5-35: Phase of the peak hydrodynamic force on the upper and lower panels (test series 4). upper armour panel of the series 3, 4, and 5 rubble-mounds in regular waves as a function of surf similarity. This data indicates that the transition from (F) > 240° (flow reversal dominated) to $ (^FJ < 240° (downrush and seepage flow dominated) is rather abrupt, and occurs for £ between 4.5 and 5. This transition occurs in the regime of surging breakers. These results suggest that for £ > 5, the peak wave-induced force on armour stones below the still waterline is generated by downrush and seepage flow, while for £ < 4.5, the peak force is related to the strong accelerations that prevail under the front of each steep, advancing wave crest. The phase of the peak hydrodynamic force on the lower and upper panels of the series 4 test structure in regular waves is plotted in Figure 5-35 as a function of surf similarity. For £ < 4.5, the peak force on the lower panel precedes the peak force on the upper panel, which supports the assertion that the force peak results from a mechanism propagating onto the structure, such as the sudden flow reversal under a steep advancing wave crest. In contrast, for £ > 5, the peak force on the lower panel lags slightly behind that on the upper panel, which is consistent with 196 135 120 -S 105 a SB •S 90 <t>H ® 75 60 45 o° o Series 3 A Series 4 M Series 5 O o XI 0.01 0.02 0.03 0.04 0.05 H / L Q = 2 7 r H / g T 2 0.06 0.07 Figure 5-36: Influence of wave steepness on the direction of the peak hydrodynamic force (upper panel, test series 3, 4 and 5). a forcing mechanism progressing down the rubble-mound, such as downrush and seepage flow. 5.4.3 D i r e c t i o n of t h e M a x i m u m W a v e F o r c e Consideration of the failure mechanisms for armour stones presented in Chapter 4 suggests that their stability depends on the strength of the wave-induced forces and on the direction in which these forces act. For lifting failure, minimum stability occurs for wave-induced forcing acting normal to, and away from the surface of the rubble-mound. Minimum stability for the Shields failure mode occurs for forcing acting tangentially down-slope, while minimum stability for combined rolling-sliding failure occurs for forcing that acts obliquely down-slope and away from the surface. According to these failure modes, the most critical directions for wave-induced forcing lie between 90° and 180° with respect to the positive slope-parallel force. Figure 5-36 shows the direction of the peak wave-induced force 0 [fi^J on the upper armour panel of the series 3, 4 and 5 rubble-mounds in regular waves as a function of wave steepness 197 H/LQ = 2-nH/gT2. For steeper waves with H/L0 > 0.02, 9 [Pj is fairly constant near 80° for all test series, which indicates that the peak wave-induced force on the upper armour panel results from a large normal force and a relatively small parallel force acting up-slope, and that this condition is not dependent on slope angle or core permeability. For waves with lower steepness, 9 (P^ is consistently greater than 90°, indicating that the peak wave-induced force acts obliquely down-slope from normal to the surface of the rubble-mound. This condition is potentially more damaging to the stability of the armour. In all conditions 9 lies between 45 and 135°, which implies that the slope-normal force is always larger than the slope-parallel force at the time of maximum forcing. These results differ from those of T0rum (1994). T0rum reports measurements of the regular wave-induced forcing on a single armour stone located on the surface of a berm breakwater. His measurements indicate slope-normal forcing that is roughly half the magnitude of the slope-parallel component at the time of maximum forcing. These measurements are likely to be strongly influenced by: the geometry and orientation of the instrumented stone; its location on the rubble-mound with respect to neighbouring stones and the still waterline; and the force measurement system. Factors such as these are likely responsible for much of the difference between the results of T0rum (1994) and the present study. 198 Chapter 6 Wave-Induced Stresses The armour panels used in this study provide a steady, repeatable measure of the net fluid forces exerted on a patch of surface layer armour stones due to wave attack. In this chapter, the hydrodynamic forces acting on the armour panels are expressed in terms of stresses acting normal and parallel to the surface of the rubble-mound. Characteristic stress values recorded at two locations on three different structures in a variety of regular wave conditions are compared to assess the influences of wave height, wave period, structure slope and core permeability on the wave-induced stresses acting on rubble-mound armour. The orthogonal components of the hydrodynamic force acting on an armour panel parallel and normal to the surface of the rubble-mound slope are denoted by Fp(t) and respec-tively, where Fp(t) is defined positive up-slope and Fpj(t) is defined positive away from the structure. These hydrodynamic force components can be expressed as a shear stress r(£) and a normal stress o~(t), defined by TW = ^ , (6.1) " W = ^ , (6.2) where A is the surface area occupied by a panel. r(t) and a(t) represent the shear and normal stresses acting on a rectangular patch of armour stones due to wave-induced fluid flows. The 199 shear and normal stresses defined by Equations (6.1) and (6.2) include contributions from all hydrodynamic forcing mechanisms, including drag, inertia, seepage and lift forces. In this study, A — 0.15 m 2 for each panel. If the thickness of the surface layer of armour is assumed equal to the nominal stone diameter Dn5o, then the surface area can be expressed in terms of the nominal diameter, the number of constituent armour stones N, and the porosity of the armour layer n, as A = i V £ ) 2 5 0 / (1 — n). Alternatively, the surface layer of armour may be defined to have thickness Dn5o/ (1 — n) 1/ 3 , in which case the surface area can be written as A = NDn50/ (1 — n) 2/ 3 . Based on consideration of A, N, Dn5o and n values for each of the armour panels, the former (thinner) definition of surface layer thickness was found to be most appropriate. For a typical porosity of n — 0.4, the thicker definition for layer thickness leads to a 15 % reduction in surface area, which corresponds to a 15 % increase in shear and normal stress. Representative peak stresses are obtained by the same procedure applied to determine peak values for other force quantities. In regular waves, average maximum values, denoted by f and <7, are obtained by averaging the maximum stresses recorded during each of 100 consecutive flow cycles. Average minimum values, denoted by f and a, are obtained by a similar averaging of minimum stresses. A third quantity, referred to as the peak-to-peak value, is defined for each stress component to characterize the magnitude of stress fluctuations during a typical cycle of flow. The peak-to-peak values of the shear and normal stresses due to regular waves are defined by 1 N TPP = ^ E ( m a x [ T ( < ) ] - m i n [ r ( £ ) ] ) (6.3) 1 N APP = ^Et"1^^*)] -min[<7(t)]) (6.4) where max [ ] represents the local maximum value during a single flow cycle, min [ ] represents the local minimum value during the same flow cycle and the summation is taken over N = 100 consecutive flow cycles. For regular waves, Tpp — f — f and <TPP — a — a. 200 The stress fluctuations under irregular wave attack are quantified by characteristic values derived from zero-crossing analysis in a similar manner as for other force quantities. Thus, at the significant stress level, f i / 3 represents the average value of the largest one-third of all local shear stress maxima during individual flow cycles and represents the average value of the strongest one-third of all local minima. The significant peak-to-peak shear stress Tpp, 1/3 represents the average value of the largest one-third of all peak-to-peak shear stress values for individual flow cycles. Note that T p p i / 3 / f i / 3 — f i / 3 . Other characteristic values are similarly defined for the normal stress component and for more extreme stress levels, including the one-20 t h level. These peak stress quantities can be conveniently non-dimensionalized by the factors pgH or pgHs, which represent the pressure under a static column of water with height equal to the wave height H or Hs. 6.1 Temporal Variations of Velocity, Shear and Normal Stress In this section, the shear and normal stresses on a patch of armour stones below the still waterline due to regular waves are compared to the parallel component of velocity above the centre of the patch. Time series of the parallel velocity measured at point 2, and the shear and normal stress acting on the lower armour panel are used to illustrate the relationship between surface flows and the stresses on the armour layer. In this study, flow velocities were measured using a pair of bi-directional electromagnetic velocimeters located 4 cm above the upper surface of the instrumented armour panels at loca-tions 1 and 2 shown in Figures 2-8 and 2-9. These locations are believed to be either outside, or near the outer edge of the boundary layer. Nielsen (1992) gives the approximate displacement thickness 8 of a boundary layer under oscillatory flow as 6 ~ 0.5fwa (6.5) 201 where fw is the wave friction factor and a is the amplitude of water particle orbits just outside the boundary layer. The displacement thickness is but one of several dimensions commonly used to define the thickness of a boundary layer. It is generally much less than the thickness at which the velocity defects near the boundary are less than 1 % of the free stream velocity. Theory developed for conditions of smooth laminar oscillatory flow predicts that the maximum velocity at the elevation equal to the displacement thickness is approximately 2 5 % less than the maximum free stream velocity at the top of the boundary layer. Above the elevation 1.5<5, the maximum velocities in the boundary layer differ from the free stream velocity maxima by less than 10%. At the elevation 2(5, the boundary layer velocity equals the velocity in the free stream, while further away from the surface, the velocity in the boundary layer can exceed the free stream velocity by up to 8 %. For rough turbulent flow, Kamphuis (1975) indicates that the wave friction factor can be well estimated by where ks is the Nikuradse roughness and the factor a/ks is known as relative roughness and represents the amplitude of the oscillatory flow relative to the roughness of the surface. a/ks provides an indication of the relative importance of drag and inertia forces to the shear stress and can thus be considered as a form of Keulegan-Carpenter number. Inertia forces are more important for smaller a/ks. Wave friction factors are considered further in Section 6.4.1. Equa-tion (6.6) may be re-arranged to Kamphuis (1975) indicates that ks ~ 2Dgo ~ 2.5Dn5Q. Wi th ks = 2.5L>n5o = 10.5 cm, this (6.6) o = 0.295fc s/- 4/ 3 (6.7) whence the displacement thickness can be estimated by (6.8) 202 Series 5, T = 1.5 s, H = 16.5 c m , £ = 1.5 Cu m - 1 2 5 ' 1 1 1 1 60 62 64 66 68 70 Time (s) Figure 6-1: Shear and normal stresses on the lower armour panel and slope-parallel velocity at point 2 under plunging breakers (test series 5, H = 16.5cm, T — 1.5 s). equation gives 8 = 2.9 cm for /„, = 0.15, and 8 = 2.3 cm for fw — 0.3. These estimates of displacement thickness support the assertion that the points of velocity measurement are sufficiently far away from the surface so that the velocities are not significantly affected by boundary layer effects. Figures 6-1, 6-2 and 6-3 show short segments of the parallel velocity u (t) recorded at point 2 and the shear r (t) and normal a (t) stresses recorded on the lower armour panel in three different flow conditions, representative of plunging, surging and collapsing breakers respectively. The data in Figures 6-1 and 6-3 was obtained on the series 5 rubble-mound (impermeable core, cot a = 3) in waves with H ~ 16.5 cm and T = 1.5 and 3 s respectively, for which £ = 1.5 and 3.1. The data in Figure 6-2 was obtained on the steeper series 4 rubble-mound (permeable core, cot a = 1.75) in waves with H ~ 16.5 cm and T = 3 s for which £ = 5.3. In all flow conditions, the positive and negative peak shear stresses coincide with positive and negative accelerations in the surface flow. The strongest shear stresses tend to coincide with the strongest 203 Figure 6-2: Shear and normal stresses on the lower armour panel and slope-parallel velocity at point 2 under surging breakers (test series 4, H — 16.6cm, T = 3s). Figure 6-3: Shear and normal stresses on the lower armour panel and slope-parallel velocity at point 2 under near-collapsing breakers (test series 5, H = 16.6 cm, T = 3 s). 204 accelerations. Shear stresses on armour stones below the still waterline depend mainly on slope-parallel accelerations, and are not strongly influenced by slope-parallel velocities. This behaviour is consistent with the nature of shear stresses on rough impermeable beds under oscillatory flows at large Reynolds number and small relative roughness as described by Sleath (1984) and Nielsen (1992). Under plunging breakers (Figure 6-1) the shear stress features one positive peak per flow cycle that coincides with a strong positive acceleration in the surface flow. This strong acceleration occurs" under the front of the steep plunging wave crest. Under surging breakers (Figure 6-2), two positive shear stress peaks occur during each flow cycle, and both of these coincide with separate periods of positive fluid acceleration above the surface. Under plunging breakers, negative accelerations are initially strong, but gradually become weaker as the direction of surface flow reverses from up-slope to down-slope. This reduction in the strength of negative parallel fluid accelerations above the surface induces a similar reduction to the negative shear stress acting on the armour stones. Under surging and near-collapsing breakers, relatively constant negative accelerations prevail for a short portion of each flow cycle. During these periods, negative shear stresses are also fairly constant, even though the parallel velocity is changing rapidly. Stronger accelerations tend to exert larger shear stresses while weaker accelerations induce smaller stresses. These observations suggest that the shear stresses acting on armour stones below the still water level are dominated by pressure gradients that are proportional to parallel accelerations of the water on the surface of the structure. Normal stresses are in general inversely related to the flow velocity, such that negative a(t) prevail for up-slope flows with positive u(t), and positive a{t) prevail during down-slope flows with negative u(t). This relationship is particularly clear under plunging breakers as shown in Figure 6-1. This behaviour is consistent with the assertion that uprush flows drive infiltration into the porous armour layer and that seepage occurs during downrush. The relatively strong negative stresses measured under plunging breakers may be related to the cushioned impact of the overturning wave crest. Under surging breakers (Figure 6-2), negative stresses also prevail during uprush when u(t) > 0, but are generally weaker than those under plunging breakers. 205 1 1 : o U p p e r p a n e l • L o w e r p a n e l o O O • o o o •• • • • • I I 0 1 2 3 4 5 6 7 8 £ = t a n a ( g T 2 / 2 7 r H ) 1 / 2 Figure 6-4: Peak-to-peak shear stress from test series 3 (cota = 1.75, impermeable core). Positive normal stresses under surging breakers prevail from the start of downrush to just after the start of strong uprush flow. The fluctuations in <x (£) during this period are not obviously related to the velocity or acceleration of the external flow parallel to the surface of the rubble-mound. Consideration of the slope-normal component of velocity measured at point 1 indicates that the largest normal stresses are coincident with the strongest seepage flows. 6.2 Peak Shear Stress Data from test series 3, 4 and 5 are used below to investigate the effects of wave characteristics, core permeability and slope angle on the shear stresses acting on armour stones below the still waterline. The peak-tc-peak shear stress is considered since it captures the intensity of both the up-slope and down-slope forcing in a single measure. Figures 6-4 to 6-6 show non-dimensional peak-to-peak shear stresses measured on the upper and lower armour panels in regular waves during test series 3, 4 and 5 respectively. In each figure, TPp/pgH is plotted as a function of the surf similarity parameter £. Non-dimensional 206 0.2 % 0.15 Q. o. o. K 0.1 0.05 o° ° 0.25 0.2 \ a a. h 0.1 0.05 1 1 1 1 1 1 -A U p p e r p a n e l A Lower p a n e l -A A ^ A A A A A A A A A A A ^ A A A -A , A A A A A A . A A ^ OA 4 A ^ A A A * A A A *• A A A A A A A * ^ * A 4 A A J A A . A A A * A A A A A A 1 1 1 1 A A i i 0 1 2 3 4 5 6 7 ( £ = t a n a ( g T 2 / 2 7 r H ) 1 / 2 Figure 6-5: Peak-to-peak shear stress from test series 4 (cot a = 1.75, permeable core). bD 0.25 0.2 0.15 a. a. h o.i|-0.05 o U p p e r p a n e l • Lower p a n e l o o o o 2 3 4 5 6 £ = t a n a ( g T 2 / 2 T r H ) 1 / 2 Figure 6-6: Peak-tc-peak shear stress from test series 5 (cot a = 3, impermeable core). 207 peak-topeak shear stresses on the upper panel vary between 0.09 and 0.18, while those on the lower panel vary between 0.06 and 0.16. Overall, these large variations in Tpp/pgH indicate that wave characteristics and structure properties have a considerable influence on the shear stresses acting on armour stones below the still waterline. T0rum (1994) computes a shear stress of T = 216 F a for a specific regular wave with H = 0.2 m and T — 1.8 s. In non-dimensional terms, this is equivalent to r/pgH — 216/(9810 • 0.2) = 0.11, which is in general agreement with the magnitude of shear stresses reported here. In every test, larger peak-to-peak shear stresses were recorded on the upper armour panel. This indicates that larger shear stresses act on armour stones located closer to the still waterline. This is consistent with observations of armour damage under wave attack which indicate that initial damage generally occurs near the still waterline. This result is also consistent with the numerical simulations of Kobayashi et. al. (1990c) which suggest that armour damage is most likely to occur at an elevation approximately 0.75H below the still waterline, and that the likelihood of damage decreases below and above this elevation. Figure 6-4 shows non-dimensional peak-to-peak shear stresses on the series 3 rubble-mound with an impermeable core and slope co ta = 1.75. On this structure, wave period has a significant effect on Tpp, such that larger stresses result from shorter period waves (smaller values of £). Expressed in terms of the type of wave breaking that prevails for various £, this data suggests that collapsing breakers exert significantly greater shear stresses on armour stones below the still waterline than surging breakers do. Figure 6-5 shows that wave period has less effect on the peak-to-peak shear stresses acting on the more permeable series 4 rubble-mound (permeable core, cot a = 1.75). On this structure, shear stresses under collapsing breakers are only slightly larger than those exerted by surging breakers. Figures 6-4 and 6-5 indicate that the more permeable core of the series 4 structure modifies the flow on the slope in a manner that reduces the shear stresses acting on armour stones below the still waterline, particularly for short period waves that form collapsing breakers. This reduction in shear stress on the more permeable structure is consistent with observations of 208 oi— -0.02 -0 .04 -0 .06 -0.08 -0.1 -0.12 0 series 3 A series 4 . A ^ A O O D A O A A A ^ A A A A A A A A O A N A O A A O O A A A A A O O 3 4 5 £ = t a n a ( g T 2 / 2 T T H ) 1 / 2 Figure 6-7: Influence of core permeability on peak down-slope shear stresses (upper panel, test series 3 and 4). the damage to rubble-mound armour which indicate that armour stones have greater stability on more permeable slopes. As well as reducing the peak-to-peak shear stress, greater core permeability reduces the strength of the peak shear stress acting down-slope. The reduction persists over a wide range of wave conditions. This permeability effect can be seen in Figure 6-7, which shows values of f/pgH for the upper panel of the series 3 and 4 rubble-mounds plotted against surf similarity. Down-slope shear stresses are weaker on the more permeable series 4 structure for all values of £. Consideration of similar data for the lower panel indicates that this permeability effect extends to lower elevations as well. This reduction in down-slope shear is the principle factor contributing to the enhanced stability of armour stones on more permeable structures. Non-dimensional peak-to-peak shear stresses recorded on the milder sloped series 5 rubble-mound (impermeable core, cot a = 3) are shown in Figure 6-6. On this structure, the largest non-dimensional shear stresses result from waves with T = 2 s for which £ ~ 2.5, which suggests 209 that semi-plunging breakers prevail. Smaller shear stresses result from longer and shorter period waves with T — 3 and 1.5 s. This data indicates that for plunging breakers, Tpp/pgH increases with increasing wave period, such that larger stresses result from longer period waves. Thus, the influence of wave period on shear stresses below the waterline in plunging breakers is opposite to that observed for surging waves on the steeper series 3 rubble-mound. The design equations for rubble-mound armour proposed by van der Meer (1988) are con-sidered in Section 3.1.6. These equations indicate that damage increases with increasing wave period for plunging waves, but that damage decreases with increasing wave period for surging waves. The variation of damage with wave period proposed by van der Meer is identical to the variation in shear stress with wave period observed in these experiments. The design equation of Losada and Gimenez-Curto (1979) is considered in Section 3.1.4. They suggest that the critical value of surf similarity for minimum armour stability in regular waves varies with slope angle according to £ C T = 2.65 tan a - 4 (6-9) B where B depends on slope angle and the type of armour stone. Equation (6.9) indicates that minimum armour stability occurs for lower values of £ on milder slopes. By substitut-ing B = —0.66 (as recommended for quarry-stone with cot a = 2), estimates of the critical surf similarity for rubble-mounds with slopes cot a = 1.75 and 3 can be obtained as £ c r = 3.0 and 2.4 respectively. The measurements of normal and shear stresses presented here are consistent with this variation in minimum stability with slope angle. On the steeper series 3 rubble-mound, maximum shear and normal stresses were observed for £ ~ 3, but on the milder sloped series 5 structure, maximum stresses were recorded in waves for which £ ~ 2.5. The influence of core permeability and slope angle on the significant peak-to-peak shear stresses under irregular waves can be seen in Figure 6-8. Data are shown for the upper panel on the series 3 (impermeable core, cota = 1.75), series 4 (permeable core, cota = 1.75) and series 5 (impermeable core, cota = 3) rubble-mounds. Greater core permeability clearly reduces the 210 0.16 0.09 1.5 2 2.5 3 3.5 4 * = t a n a (ST2/2TIH ) 1 / 2 4.5 Figure 6-8: Influence of core permeability and slope angle on significant peak-tc-peak shear stresses (upper panel, test series 3, 4 and 5). intensity of the shear stresses acting on the surface layer of armour. The average reduction in significant peak-to-peak shear stress due to enhanced permeability is 12%. The stress peaks are weaker in both the up-slope (9% less on average) and down-slope (16% on average) directions. Results from the lower panel confirm that this permeability effect persists at lower elevations. A similar permeability effect is also evident at more extreme stress levels. The enhanced stability of armour stones on more permeable rubble-mounds can be attributed to the reduction in the shear stresses acting on the armour layer. The data presented in Figure 6-8 indicate that slightly weaker significant peak-to-peak shear stresses occur on the milder slope. For these wave conditions, semi-surging breakers prevail on the steeper slope while plunging breakers prevail on the milder slope. Because the effects of slope angle are coupled to the wave period, this result is particular to the two slope angles and the range of irregular wave periods considered in this study and cannot be generalized to other slope angles and wave conditions. Stresses measured in regular waves over a wider range 211 0.25 0.2 0.15 0.1 0.05 A Qo A A 0 C A O A A O A A . A AA A A A A O o O O 0 S e r i e s 3 A S e r i e s 4 o A A . A A A A A A A A O A o o o 3 4 5 £ = tana ( gT 2/27rH) 1 / 2 Figure 6-9: Influence of surf similarity and core permeability on peak normal stress (upper panel, test series 3 and 4). of wave period provide a better indication of the combined influences of slope angle and wave period (see Figures 6-4 and 6-6). Consideration of fi/3 and f x/3 for these tests indicates that the up-slope stress peaks are marginally greater on the milder slope. This increase in up-slope shear is offset by a greater reduction in down-slope shear. The shear stress fluctuations on the milder slope are considerably more asymmetric than those on the steeper slope. 6.3 Peak Normal Stress In the following, the influence of wave characteristics, core permeability and slope angle on the peak normal stress acting on armour stones below the still waterline is investigated using data from test series 3, 4 and 5. Non-dimensional maximum peak normal stresses a/pgH recorded on the upper panel of the series 3 (impermeable core, cota = 1.75) and 4 (permeable core, cota = 1.75) rubble-mounds in regular waves are plotted in Figure 6-9 against surf similarity. This data indicates that the permeability of the rubble-mound core has a minor 212 -1 1 o S e r i e s 3 o S e r i e s 5 o o o o o o _1 1 0 1 2 3 4 5 6 7 8 £ = t a n a (gl2/2nH)l/z Figure 6-10: Influence of surf similarity and slope angle on peak normal stress (upper panel, test series 3 and 5). effect on the peak normal stress acting on armour stones just below the still waterline, such that larger stress peaks tend to occur on the more permeable structure. This permeability effect is particularly evident when the results at each wave period are considered independently. Such analysis indicates that the effect of permeability is fairly consistent over the range of conditions considered in this study. Figure 6-9 also shows a trend towards larger a at smaller £, such that significantly greater normal stresses result from waves for which £ ~ 3. This suggests that collapsing breakers exert larger normal stresses on armour stones below the still waterline than surging breakers do, and are thus more likely to cause damage to the armour. The strong normal stresses under collapsing breakers result from a superposition of the forcing due to seepage outflows and the intense flow reversals under the collapsing wave crest. The influence of slope angle on the maximum peak normal stress acting on armour stones just below the still waterline can be seen in Figure 6-10, which shows values of o/pgH recorded on the upper panel of the series 3 (cot a = 1.75) and 5 (cot a = 3) rubble-mounds as a function Q. <b 0.2 0.15 0.1 ° O O o o o *0 0<s<> 213 T o U p p e r p a n e l • L o w e r p a n e l o o o o o o • -_1_ I 0 1 2 3 4 5 6 7 8 £ = t a n a (gT 2 /27TH) 1 / 2 Figure 6-11: Influence of surf similarity and elevation on peak normal stress (upper and lower panels, test series 3). of surf similarity. The influence of slope angle on a varies with wave period. For longer waves with T = 3 s, similar peak normal stresses occur on both slopes. Slightly larger peak normal stresses occur on the milder slope in 2 s waves, but much smaller peak normal stresses occur on the milder slope in short waves with T = 1.5 s. The period-dependent effect of slope angle on cr cannot be entirely explained by the type of wave breaking that prevails on the rubble-mound. &/pgH is maximized at lesser £ on the milder slope. This is identical to the trend observed for shear stresses, and further supports the assertion that the critical surf similarity corresponding to minimum stability varies with slope angle, and is lower for milder slopes. Figure 6-11 shows that the peak normal stresses acting on armour stones below the still waterline vary strongly with elevation, such that larger stresses consistently prevail closer to the still waterline. Results from test series 4 and 5 also support this conclusion. Figure 6-11 also suggests that the variation in a with elevation depends on the type of wave breaking that prevails on the slope and is greater for collapsing breakers than for surging breakers. The trend 0.2 X 0.15 <b 0.1 0.05 o o°9 °o° ° 214 S3 <b 0.2 0.175 ~ « 0.15 -0.125 0.1 0.075 0.05 o series 3 A A A series 4 A * *° ° O A * A o series 5 O A © o 1.5 2.5 3 3.5 4 £ = t a n a ( g T 2 / 2 T T H ) 1 / 2 4.5 Figure 6-12: Influence of core permeability and slope angle on significant peak normal stresses (upper panel, test series 3, 4 and 5). towards increasing a with decreasing £ that occurs at the upper panel does not persist at lower elevations. This suggests that the mechanisms responsible for the intense normal stresses under near-collapsing breakers have a localized effect that is concentrated near the still water level. Significant peak normal stresses on the series 3, 4 and 5 structures under irregular waves are compared in Figure 6-12. These stress peaks were obtained on the upper panel. They suggest that core permeability has little influence on the peak values of the normal stress acting away from the surface of the armour layer under irregular wave attack. The normal stress peaks at higher stress levels are similarly unaffected by the difference in core permeability. This behaviour is slightly different from the trend observed under regular wave attack. The reason for this difference is not clear, but it could be a consequence of the limited range of wave periods considered in the irregular wave tests. Figure 6-12 shows a strong dependence between cr-^ and slope angle, such that the normal stress peaks are significantly weaker on the milder slope. This slope effect is similar to that seen in Figure 6-10 for regular waves with T = 1.5 s. This 215 similarity is consistent with the fact that the mean wave period for most of the irregular wave conditions is Tm ~ 1.6 s. 6.4 Friction Factors Flows on the surface of a rubble mound under wave attack are in some aspects similar to wave driven flows on a horizontal seabed. While both flows are fundamentally oscillatory, flows on a rubble-mound contain proportionately more energy at higher frequencies and feature more asymmetric velocity fluctuations and larger accelerations. Flows on a rubble-mound also vary greatly with elevation. Near the base of the structure, flows are similar to those on a horizontal seabed at similar depth; however, approaching the still waterline, velocities and accelerations are significantly amplified. Above the point of minimum rundown, the surface of the rubble-mound is only intermittently submerged. In this zone, the depth of flow can vary greatly throughout a wave cycle. When the water level outside the structure exceeds the level of the internal phreatic surface, water flows into the permeable outer layers. As the external water level recedes below the internal phreatic surface, water flows out of the permeable layers. Infiltration flows generally occur above the mean water level during uprush while seepage flows are concentrated below the mean water level during downrush. 6.4.1 Wave Fr ic t ion Factor Shear stresses on an impermeable horizontal bed under oscillatory flow are commonly expressed in terms of a wave friction factor fw, defined by / » = | r (6-IO) Pamax where T m a x is the maximum shear stress at the bed, and u m a x is the maximum orbital velocity just outside the boundary layer. Riedel et. al. (1972) and Kamphuis (1975) report results on fw, obtained with a shear plate in an oscillating water tunnel, as functions of the orbital amplitude 216 Reynolds number Re = umaxa/u and the relative roughness a/ks, where v is the kinematic viscosity, a is the amplitude of water particle orbits just outside the boundary layer, and ks is the Nikuradse roughness, which can be written as ks ~ 2Dgo — 2.5D„50- Different flow regimes were delineated, corresponding to laminar, smooth turbulent and rough turbulent flow. For rough turbulent flow, the wave friction factor was found to be independent of Re and could be well represented by i ^ + l 0 E ( i ^ ) = -a35+i,0g(£) or by the simpler expression fw = - j for - < 100 . (6.12) Swart (1976) suggested the following empirical relation for wave friction factor fu 0.00251 exp 5 - 2 1 (it) for £> 1.57, (6.13) 0.3 for £ < 1.57 . that sets an upper bound to the value of fw for small a/ks. Kaj iura (1968) derived theoretical expressions for wave friction factors under smooth and rough turbulent flow using unidirectional turbulent boundary layer theory. Sleath (1984) indicates that for a/ks < 30, Kajiura's formula for rough turbulent flow can be written as a \ 2/3 . a fw = 0.35 ( - for — < 3 0 . (6.14) Predictions of fw according to Equations (6.12), (6.13) and (6.14) are compared graphically in Figure 6-13 over the range 0.5 < a/ks < 5. Kamphuis (1975) suggested the following criterion for the lower limit of rough turbulent flow: 217 Re = > 200-?- * H - for rough turbulent flow. (6.15) v ks\j fw Substituting Equation (6.12) for fw yields / a \ 1 3 7 5 a Re > 447 — for rough turbulent flow with — < 100. (6.16) \ fcs / ks For rough turbulent oscillatory flow, Nielsen (1992) and Sleath (1984) indicate that the shear stress acting on the bed at small values o£a/ks is almost totally due to pressure gradients across the roughness elements on the surface, and is therefore proportional to the acceleration of the fluid above the bed. This implies that the maximum shear stresses on the bed coincide with maximum accelerations in the flow above the bed. Numerical models of wave interaction with rubble-mound slopes that use a friction fac-tor / to characterize the energy dissipation due to flow over the surface of the slope are de-scribed by Thompson (1988), Allsop et. al. (1988), Kobayashi and Wurjanto (1992), Kobayashi 218 et. al. (1990a, 1990b), van Gent (1992), T0rum and van Gent (1992), and Wurjanto and Kobayashi (1993). In each of these models, the friction factor for a particular slope is assumed to be inde-pendent of the wave conditions that prevail, and thus independent of relative roughness. The value of friction factor used with each numerical model is typically selected on the basis of calibration to physical model tests. Thompson (1988) considered friction factors of 0.3 and 1.0 for rough slopes and 0.01 for smooth slopes. Allsop et.al . (1988) also used / = 0.01 for a smooth slope. Kobayashi and Wurjanto (1992) used / = 0.3 for simulations of the regular wave attack on a rough permeable slope, while Kobayashi et. al. (1990a, 1990b) used / = 0.05 and 0.1 in simulations of irregular waves on a rough impermeable slope. Van Gent (1992) used a friction coefficient of 0.2 to simulate the flow on and within various permeable and imperme-able structures, but did not report any calibrations to flows on real structures. T0rum and van Gent (1992) found fair agreement between velocities measured on a berm breakwater under regular waves and the predictions of van Gent's numerical model with / = 0.15. Wurjanto and Kobayashi (1993) used / = 0.05 and 0.1 to simulate irregular wave flows on a permeable slope. This brief survey indicates that a considerable range of friction factors have been used in numerical simulations of wave interaction with rubble-mound slopes. 6.4.2 Friction Factors for Rock Armour Water particle orbits on the surface of a rubble-mound can be estimated by integration of the Eulerian velocity signal. In particular, the water particle displacement parallel to the face of the structure at time t\ is given by rh sfa) = / u(t)dt . (6.17) Jo This integration is performed in the frequency domain using Fast Fourier Transforms. The orbital amplitude a for a single flow cycle can be taken as half of the peak-to-peak excursion of 219 s(t): a = i (max [s(t)] - min [s(t)}) . (6.18) where max[s(£)] and min[s(£)] are the local maximum and minimum values of s(t) during a single flow cycle. A more representative amplitude for a duration of wave attack can be obtained by averaging the amplitudes over a number of flow cycles: jf E a = £f E (m a x - m i n [*(*)]) (6-19) The representative orbital amplitude is equal to Spp/2 where sw represents the average value of the peak-to-peak excursions of the water particle orbits parallel to the surface. For regular waves, Spp is obtained from s (£) by averaging the peak-to-peak excursions over 100 consecutive flow cycles. For irregular waves, the average is taken over all flow cycles in the record. The quantity Spp/2 thus represents the average orbital amplitude for both regular and irregular oscillatory flow. Other characteristic values of orbital amplitude can be defined for irregular waves by averaging over a fraction of the total number of flow cycles. For example, the signifi-cant orbital amplitude 5^,1/3/2, can be obtained as the average of the largest one-third of all orbital amplitudes. However, using the average orbital amplitude (rather than the significant amplitude) to characterize an irregular wave condition is favored because of its consistency with the fact that the period dependence of irregular wave conditions can be well parameterized by the average wave period Tm. A peak-to-peak value of slope-parallel velocity upp is similarly defined by 1 N Upp = — E ( m a x _ m m MOD • (6.20) These peak-to-peak measures of displacement and velocity are used to define an orbital ampli-tude Reynolds number for flows on the surface of a rubble-mound as 220 0 1 03 10 6 r 10c 10 4 10-= 10' 10J 0.5 o OO Limit for rough turbulent flow o Re pp 1.5 2 a / 5 D 2.5 3.5 P P n50 Figure 6-14: Orbital amplitude Reynolds number at velocity measurement point 2. Repp — " • p p o p p lAppSpp (6.21) The relative roughness of the flow on a rubble-mound can be written as Spp/2 2.5Dn5o 5Dn5o _ Spp (6.22) The quantity sPp/5Dn5Cj is equivalent to a/ks, and therefore provides an indication of the relative importance of inertia and drag forces to the total shear stress. Figure 6-14 shows values of Repp in regular waves at the lower point of velocity measurement (point 2), in which the solid line indicates the threshold criterion for rough turbulent flow given by Equation (6.16). The data span the range of Spp/bDn^o (relative roughness) from 0.8 to 3.7 and indicate that rough turbulent flow prevails for all test conditions. Measurements of the shear stress on the lower armour panel and the kinematics at point 2 are used to construct friction factors for the armour layer. Three separate friction factors are 221 computed: one applies to the complete flow cycle; while the others apply for the uprush and downrush portions respectively. For regular waves, these three friction factors are defined by fpp = ^ ( rpp/2) _ 4T~PP £ Q r c o m p i e t e £ o w cycles, (6.23) p(upp/2y pupp 2f fu = -TZ for uprush, (6.24) pu1 —2f fd = —Tr for downrush. (6.25) puz For irregular waves, different friction factors apply for each exceedence level of the shear and velocity distributions. These can be written in a general form as 2 ( T p p ) i / n / 2 j 4r I / fpp l / n - — ^ — ' 2 = — T ' f ° r complete flow cycles, (6.26) P ( « B , , l / n / 2 ) / W p P ' 1 / n 2f\in fu,i/n = —— for uprush, (6.27) fd,i/n — — f ° r downrush. (6.28) PUl/n where the subscript denotes a quantity obtained as the average of the largest one -n t h of values for all individual flow cycles. Definitions for peak-to-peak, uprush and downrush friction factors at the significant level are obtained by setting n — 3 in Equations (6.26) - (6.28). The definitions for regular waves given by Equations (6.23) - (6.25) are equivalent to the specific case where n = 1 and all flow cycles are included in the averaging operation. The relationship between friction factors at one level and those at another will depend on the distributions of the peak values of shear stress and velocity that are involved. In particular, the ratio of fpPii/n to /Pp,i/3 can be written /pp,l/3 Tpp,l/3 \upp,\/n) This ratio can be evaluated for known distributions. If the peak-to-peak shear stresses and the 222 peak-to-peak velocities both follow the same distribution, then fpp,i/n = UPPA/3 fpp,l/3 upp,l/n (6.30) Moreover, if they satisfy the Rayleigh distribution, Sarpkaya and Isaacson (1981) indicate that the two friction factors will be related by fpp,l/n 1-416 /PP,I/3 vlriTn) + ^ {1 - erf [y/h^n)]} where erf [ ] is the error function defined by (6.31) erf [x] = 4= r exp (-t2) dt . (6.32) Jo v ' For n = 20, Equation (6.31) evaluates to /pp,l/20 /PP,1/3 1-4 (6.33) Friction factors for the complete flow cycle jw measured at the lower armour panel during regular wave attack of the series 4 and series 5 rubble-mounds are favorably compared in Figure 6-15 to estimates of fw computed from Equation (6.12) and plotted as a solid line. The measured friction factors vary between 0.13 and 0.59 over the range of relative roughness 0.9 < 5 £ ? P p s o < 3.7. Data from the two different rubble-mounds show good agreement with each other; however, there is evidence of a slightly stronger variation with relative roughness than predicted by Equation (6.12). The power function that best fits the data is which is plotted as a dashed line in Figure 6-15. Overall, these results suggest that the relation 223 0.8 0.7 fw Kamphuis (1975) 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 S / 5 D pp Figure 6-15: Comparison of measured peak-to-peak friction factors and predictions of fw from Kamphuis (1975). between friction factor and relative roughness on a rubble-mound under wave attack is quite similar to that for a rough, impermeable, horizontal bed under oscillatory flow. Moreover, these results suggest that wave friction factors might be used to estimate the shear stresses acting on rock armour under wave attack. Regular wave friction factors for the uprush and downrush portions of the rubble-mound flow cycle are presented in Figures 6-16 and 6-17. In these figures, as in Figure 6-15, the power curve that best fits the data is plotted as a dashed line while the solid line represents Kamphuis's predictions of fw computed from Equation (6.12). Most of the data for the uprush friction factor suggest that fu < fw, while the majority of data for the downrush friction factor suggest that fd> fw. In other words, the peak shear stress acting down-slope is generally larger than that predicted by Equation (6.12), while the peak shear stress acting up-slope is generally smaller. The difference between fu and is greater for the steeper, more permeable series 4 rubble-mound. These differences between fpp, fu, and fa may be caused by several factors. 224 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 f w K a m p h u i s (1975) - - 0 . 4 3 ( S p p / 5 D n 5 0 ) - 1 1 1 A Ser ies 4 -o©-P o 1.5 2.5 3.5 s / 5 D pp ' n50 Figure 6-16: Comparison of measured uprush friction factors to predictions of fw from K a m -phuis (1975). 0.8 0.7 0.6 " 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1.5 PP 2 / 5 D 2.5 3.5 n50 Figure 6-17: Comparison of measured downrush friction factors to predictions of fw from Kamphuis (1975). 225 0.81 1 1 r 0 0.5 1 1.5 2 2.5 3 pp ' n50 Figure 6-18: Comparison of measured significant peak-to-peak friction factors to predictions of fw from Kamphuis (1975). • The significant asymmetries of the flow on the surface of a rubble-mound, examples of which can be seen in Figures 6-1 - 6-3. The flow cycles on the surface of a rubble-mound are generally more asymmetric than those that prevail over horizontal beds under waves. • The permeability of the armour layer. Infiltration prevails during uprush while seepage flow prevails during downrush. These flows normal to the surface will certainly have an effect on the structure of the boundary layer, thereby altering the relationship between the peak velocity and the peak shear stress. • The time-varying depth of flow. Figure 6-18 shows that significant peak-to-peak friction factors /P P )i/3 for the armour on the series 4 and 5 rubble-mounds during irregular wave attack are also in good agreement with predictions of fw from Equation (6.12). These results suggest that Equation (6.12) can also be used to predict the significant level of shear stress on an armour layer under irregular wave 226 0.8 0.7 0.6 -0.5 o 2 0-4 cu 0.3 0.2 0.1 0 f w,l/3 w.1/20 A fpp,l/20 S e r i e S 4 ° f P P > i / 2 0 Series 5 0 0.5 PP 1.5 / 5 D 2.5 n50 Figure 6-19: Comparison of measured and predicted values of /ppj/20-attack, provided that the velocity fluctuations are known and that the relative roughness is defined in terms of the average orbital amplitude. A n estimate of the variation of friction factor with relative roughness at any level (specified in terms of n) can be obtained by combining Kamphuis' equation for fw (Equation (6.12)) with Equation (6.31) to yield fw,l/n 0 5 6 (ft) -3/4 v/uTH + {l - erf [y/tojn)] } (6.35) This model assumes that the peak-to-peak shear stresses and peak-to-peak velocities are both Rayleigh distributed. It predicts that the friction factor based on Tppl/2o a n d upp,1/20 w i H be 2 9 % less than fWti/$ for all values of relative roughness. Figure 6-19 shows that Equation (6.35) provides a reasonable estimate of fppj/20 for the armour of the series 4 and 5 rubble-mounds under irregular wave attack. Deviations between the measured values of / P P i i/2o a n d the predicted values of fWi\/2o can be partially attributed to the fact that the distributions of 227 peak-to-peak shear stress and peak-to-peak velocity are only approximately Rayleigh. Constant friction factors between 0.05 and 1.0 have been used in numerical models of wave interaction with rubble-mound slopes. These values are in general agreement with the results reported here. However, the present results confirm that the friction factor for rock armour on rubble-mound structures is a variable quantity that, for rough turbulent flow, depends on the relative roughness in a manner that is similar to that for oscillatory flow over a rough, imper-meable, horizontal bed. The friction factor for rock armour can be reasonably well predicted by Equation (6.12) for regular waves and Equation (6.35) for irregular waves. 228 Chapter 7 Irregular Wave Effects Irregular waves breaking on a rubble-mound breakwater create a sequence of irregular oscil-latory flow cycles on the surface and within the porous outer layers of the structure. As the wave crest meets the rubble-mound, water either surges or plunges up onto the surface of the structure depending on the type of wave breaking that prevails. While the following wave trough propagates onto the rubble-mound, this water has a chance to drain off and out of the structure. Depending on the permeability and thickness of the outer layers and core, this water either runs down outside the structure as a surface flow, percolates down through the porous outer layers, or propagates through the core. On permeable structures, downrush flow occurs through a combination of these three mechanisms, while surface flow tends to dominate on less permeable rubble-mounds. The character of these irregular, semi-oscillatory cycles of surface flow depend in part on the volume of water conveyed onto the rubble-mound, the height to which this water reaches, and the duration available for return flow. These aspects are con-trolled by characteristics of the incident wave such as the wave height and period, but also depend on the initial state of flow on the rubble-mound at the arrival of the wave crest. In this way the surface and internal flows (and through them, the wave-induced forces exerted on the armour) due to an individual irregular wave depend on the character of the incident wave as well as the character of the preceding wave. 229 In this chapter, the forcing of rock armour subject to irregular wave attack is primarily analysed as a sequence of forcing events caused by successive waves. The distributions of peak forces are presented and compared to the distributions of incident wave height and the height of waterline excursions on the surface of the structure. Extreme values for these quantities are also considered. Since waves in nature are irregular, the shape of these distributions strongly influences the performance of real structures. This analysis is extended to identify some common features of individual irregular waves that are responsible for exerting particularly large de-stabilizing forces on the armour layer. Waves with these features are likely to be the most damaging ones in an irregular wave train. 7.1 Irregular Waves in Design Formulae A n irregular wave climate contains a random sequence of waves with varying height, period and surface profile. When attacking a rubble-mound structure, these waves generate various flow patterns that exert a wide range of peak forces on the rock armour. Statistical quantities are commonly used to characterize such variable processes. For example, wave heights are commonly characterized by parameters of the form Hi/n which represents the average height of the highest one-nth of all waves in a record, with n typically equal to 1, 3, 10, 20 and 100. Hi/3 is by definition equal to the significant wave height Hs while Hi/i is identical to the average wave height Have. A similar approach can be used to define characteristic values for any random variable, including peak values of force quantities such as the Shields failure index Rs- Accordingly, Rsti/n denotes the average value of the largest one-nth of all Rs values due to an irregular wave train. Equations for the design of rock armour commonly make use of a single wave height pa-rameter to represent the effect of an irregular wave condition. In doing so, the wave heights incident to the toe of the structure are implicitly assumed to satisfy the Rayleigh distribution. This assumption is generally valid in deeper water, but does not necessarily hold in shallow 230 water where wave heights are depth-limited. The "Shore Protection Manual" (CERC , 1984) recommends that i?i/io be used in place of the regular wave height H to characterize the effect of irregular waves on rock armour. A recent report from Working Group 12 of the Permanent International Association of Navigation Congresses (PIANC, 1992) suggests that the use of HI/IQ in the Hudson formula may be overly conservative. For wave heights that are Rayleigh distributed, van der Meer (1988) presents equations in terms of significant wave height HS. However, for wave heights that are not Rayleigh distributed, he suggests that B~2%, equal to the height exceeded by 2% of all waves, be used as the characteristic wave height for design. In all of these approaches, the duration of wave attack is considered as a separate, independent parameter. V idal and Mansard (1995) suggest that the effect of irregular waves on a rubble-mound can be characterized by the parameter #150, equal to the average height of the largest 150 waves incident to the structure. This parameter depends on both the heights of the largest waves as well as the duration of wave attack, such that i f 150 increases with increasing duration. Clearly, damage to an armour layer due to irregular wave attack depends on those waves that exert forces of sufficient magnitude and character to de-stabilize armour stones. Damage represents an integration or accumulation of the effects of each de-stabilizing wave. Both the frequency of de-stabilizing waves, and their intensity, influence the damage resulting from a duration of irregular wave attack. Since wave height is the dominant wave characteristic influ-encing the hydrodynamic force on armour stones, the waves responsible for the most damaging conditions will likely be among the highest waves incident to a rubble-mound. In the literature, there is general agreement that damage to rock armour due to irregular waves is controlled by a few of the largest waves attacking the structure; however, a consensus has not yet been reached regarding the most appropriate way to parameterize their effect. 231 • previous 7V Figure 7-1: Definition sketch for zero-crossing analysis of waves. 7.2 Wave Characteristics and Force Quantities for Individual Irregular Waves Irregular waves can be analysed as a sequence of individual waves with different heights, periods, and surface profiles. Zero-crossing analysis of the water surface elevation n(t) is commonly used to define individual waves. For this study, zero-upcrossing analysis, in which waves are defined by pairing each crest to the following trough, was found to be the most useful method to identify individual waves. This is consistent with the findings of Kobayashi et.al. (1990c). Using this method, each wave starts with a crest and ends with a trough. Zero-upcrossing analysis of 77(f) from probe 10, located in the side-channel (channel 1) adjacent to the toe of the rubble-mound test sections (see Figure 2-3), is used to define the characteristics of incident waves. 7.2.1 Definition of Wave Parameters Figure 7^1 shows a definition sketch of fundamental time-domain wave parameters. Additional 232 wave parameters are defined as follows. • Horizontal asymmetry: fi = ac/H. • Vertical asymmetry: A = (Tc)/T. • Skewness: 01 ~ —r n l3/2 where fj denotes the mean value of n (t) between t\ and £5. Atiltness: ft_ j f w w - ^ ^ ( , 2 ) where r)(t) denotes the first time derivative of 77 (t). Percentage difference in wave height: AH = 100 • (H — i f previous)/H'. Percentage difference in wave period: A T = 100 • (T — Tp r e vious)/T'. Percentage difference in upcrossing and downcrossing height: A i f u _ d = 100 • (H — Hd)/H. Percentage difference in upcrossing and downcrossing period: ATu_d — 100 • (T — Td)/T. Average potential wave energy: Vw=% C772 (i) dt • (?-3) Total potential wave energy: PW = T -Vw = Y J** V2 (t) dt . (7.4) 233 7.2.2 Detection of Individual Cycles of Surface Flow Individual flow cycles on the rubble-mound are defined using zero-upcrossing analysis of the waterline elevation on the surface of the structure rjs(t). In the N R C experiments, waterline elevation was measured by a water level gauge installed parallel to, and just above the surface of the rubble-mound in the central test channel, as described in Section 2.3.4. In general, zero-crossing analysis of 77 (t) and 77^  (£) produced a different number of incident wave and surface flow cycles. Moreover, the surface flow cycles lagged the incident wave cycles by a duration corresponding approximately to the time required for waves to propagate over the horizontal distance h cot a between the toe of the structure and its intersection with the free surface. A n algorithm was developed to pair individual wave and surface flow cycles based on the time lag between them and their relative duration. Waves that could not be reliably matched to surface flow cycles were excluded from subsequent analyses. Typically, only small waves were excluded in this procedure. 7.2.3 Force Quantities for Individual Waves Time series of the wave forces acting on the two instrumented armour panels were computed as described previously for regular waves. Load cell reactions were resolved into slope-normal and slope-parallel force components as described in Section 2.3.8. Compensation for time-varying buoyancy was applied where required as described in Section 2.3.9. Failure indices for lifting R-L(t), Hudson R-H(t), rolling-sliding i?#(£) and Shields Rs(t) failure modes were computed according to Equations (4.46), (4.47), (4.48) and (4.49) respectively. Figure 7-2 shows short segments of rj(t), rjs (t), F (t) and Rs (t) from a test of the series 5 rubble-mound in irregular waves with Hs = 0.14 m and Tp — 2 s. Force quantities are presented for the upper armour panel, located below the still waterline as shown in Figure 2-9. Peak values of force quantities, including the failure indices, corresponding to each individ-ual surface flow cycle are determined by detecting local maxima during the appropriate time 234 0.2 450 451 452 453 454 455 456 457 458 459 460 T i m e (s) Figure 7-2: Time series of ??(£), r?s (£), F (t) and Rs (t) in irregular waves (upper panel, test series 5, Hs = 14 cm, Tp = 2 s) 235 interval. These peaks are then paired to individual waves by accounting for the time lag between incident waves and surface flow cycles. 7.3 Distributions and Extreme Values 7.3.1 Distribution of Wave Height The distribution of individual wave heights in an irregular wave record is commonly assumed, particularly in deep water, to follow the Rayleigh probability distribution. According to the Rayleigh distribution, the cumulative probability P (H) of the random variate H can be written P(H) = I ! -e x p [ - ? fey for H > 0, 0 otherwise. (7.5) The probability density p(H) of the Rayleigh distribution is given by dP{H) _ \ f 7 f c e x p [ - f ( i f ey for H > 0, 0 otherwise. (7.6) Incident irregular wave heights in test series 3, 4 and 5, (with h — 0.55 m) are generally well described by the Rayleigh distribution. A typical example of the agreement is presented in Figure 7-3, which shows the probability density histogram of incident wave heights for the wave record shown in Figure 7-2 (test series 5, Hs — 14 cm and Tp = 2 s) together with the theoretical Rayleigh density function. In Figure 7-3, individual wave heights are normalized by the average wave height Have. 7.3.2 Distribution of Waterline Height Zero-upcrossings of the waterline elevation on the surface of a rubble-mound are used to identify cycles of surface flow. In a general sense, the intensity of the surface flow during each cycle will be in some way proportional to the vertical height of waterline motions, Hna. Thus, HVs is 236 1.2 0.9 0.6 0.3 M e a s u r e d R a y l e i g h * -I 0.5 1.5 2 H / H 2.5 3.5 Figure 7-3: Comparison of measured wave heights to the Rayleigh distribution (test series 5, Hs = 14cm, Tp - 2 s). considered here as a general indicator of the intensity of surface flows on a rubble-mound. The height of waterline motion due to regular waves is considered in Section 5.1.3. The distribution of runup on rock slopes exposed to normally incident irregular waves was studied by van der Meer and Stam (1992). They analysed runup data from a large number of physical model tests and found that the majority of runup distributions were close to the Rayleigh distribution, but that a considerable number showed significant deviations. Regression analysis was used to fit the runup observations to a two-parameter form of the Weibull extreme value distribution. Their expression for fjStP, the elevation above the still waterline exceeded by p • 100 % of the runup events is fjs,P = b(-\np)1/a (7.7) where p is the probability of occurrence (a value between 0 and 1), b is the scale parameter and a is the shape parameter. The Rayleigh distribution is obtained for a = 2. The scale parameter 237 is given by b = OA(^Y1/\cata)-1<*H, (7-8) while the shape parameter is given by 3 ^ 3 / 4 a = < for U < ( 5 . 7 7 P a 3 v / t a n ^ ) ( p + 1 ° - 7 5 ) 0 . 5 2 P - ° - 3 ^ v ^ o t ^ for £ m > (5.77P 0 - 3 V / tar7 C : ) (7.9) In these expressions, P is the same notional permeability factor described in Figure 3-6. While originally developed to describe the probability distribution of runup events, these equations might also be used to describe the distribution of the height of waterline excursions HVs, particularly if runup and rundown heights are linearly related. Assuming that the Weibull distribution and the shape parameter proposed by van der Meer and Stam (1992) can also be used to describe the distribution of Hfla, the height exceeded by p • 100 % of the surface flow cycles can be written as HT,StP = c{-\npf'a (7.10) where c is a new scale parameter. The mean value of this Weibull distribution is fj, = cT 1 + (7.11) where T [ ] denotes the gamma function of the bracketed quantity, defined as r [x] = / tx~l exp ( -£) dt , x > 0 Jo (7.12) The distribution of HVs normalized by the average height of waterline excursions can thus be written *lri3,ave T 1 + -where the shape parameter a is obtained from Equation (7.9). 238 0 0 . 5 1 1.5 2 2 . 5 3 H / H , H77 / H77 ave s s.ave Figure 7-4: Comparison of measured wave heights and waterline heights to the Rayleigh distri-bution (test series 5, Hs = 14 cm, Tp = 2 s). In some cases, waterline heights from test series 3, 4 and 5 can be fairly well described by the Rayleigh distribution. Agreement between the measured and the Rayleigh distribution is particularly good for the milder sloped series 5 rubble-mound (cota = 3, impermeable core). Cumulative distributions of the wave height and waterline height from the same series 5 test considered previously (with Hs = 14 cm, Tp = 2 s) are compared in Figure 7-4 to the Rayleigh distribution and the Weibull distribution computed from Equation (7.13) using the shape parameter fitted by van der Meer and Stam (1992). The excellent agreement between the normalized distributions of wave height and waterline height shown in this figure suggests that is a linear function of H for this structure under these waves. The Weibull distribution adapted from van der Meer and Stam (1992) fits the observed waterline excursion heights slightly less well than the Rayleigh distribution for this case. The height of waterline oscillations on the steeper series 3 and 4 rubble-mounds (both with cota = 1.75) are generally less well described by the Rayleigh distribution. A n example 239 Figure 7-5: Comparison of measured wave heights and waterline heights to the Rayleigh distri-bution (test series 3, Hs = 13 cm, Tp = 2 s). of waterline heights that deviate significantly from the Rayleigh distribution and from the distribution of incident wave height are presented in Figure 7-5, which shows data for the case of irregular waves with Hs = 13 cm, Tp = 2 s incident to the series 3 rubble-mound ( cota = 1.75, impermeable core). In this case, waterline heights are distributed over a fairly narrow range compared to both theoretical distributions, and to the observed distribution of incident wave height. The Weibull distribution adapted from van der Meer and Stam (1992) suggests a distribution that is broader than Rayleigh, while the observations are more narrowly distributed. Several factors may contribute to the significant disagreement between these measurements and the adapted Weibull distribution. • Van der Meer and Stam (1992) warn that their analysis should not be applied to slopes steeper than cot a — 2. This condition is not satisfied for the structures tested in series 3 and 4 which have co ta = 1.75. 240 • The distributions considered here, relating to the height of the flows on the surface of a rubble-mound, depend on the distribution of the incident wave height and thus also on the water depth in front of the structure. The water depths used in the present investigation differ from those used in the experiments of van der Meer and Stam. • The shape parameter obtained by van der Meer and Stam from analysis of runup heights may not apply to the total height of waterline motions. • Because the water depth on the surface of a rubble-mound during runup events is often very shallow, the measurement of waterline motions, including runup heights, is very sen-sitive to the elevation of the gauge above the surface of the armour layer. The sensitivity of the gauge to splashing water and spray is also an issue. Differences in the behavior and location of the gauge used to measure waterline motions in these experiments and those of van der Meer and Stam could be a factor. 7.3.3 Distribution of Peak Hydrodynamic Force The relationship between the peak value of the hydrodynamic force acting on a patch of armour stones F, and the incident wave height H under regular wave attack is considered in Section 5.4.1. The relationships between F and H shown there in Figures 5-28 and 5-29 can be fairly well represented by an equation of the form F = aHb (7.14) where the coefficients a and b are both dependent on the slope angle a and the regular wave period T. This relation can be used to derive the expected distribution of hydrodynamic force peaks in irregular waves as follows. The probability density function of normalized hydrodynamic force peaks p ( ) can be 241 written as j>).r^(»)!smH (7.15) \FaveJ \HaveJ d[F/Fave) where PRay(H/Have) is the Rayleigh probability density function of normalized wave heights. In terms of normalized variables, Equation (7.14) becomes H 1 (7.16) from which d(H/Have) If F \ l , b - 1 d(F/Fave) b\Fave) Substitution into Equation (7.15) gives the probability density function for normalized hydro-dynamic force peaks as p(£)=^(£)K£)1A1' (7-i8) Equation (7.18) defines a family of distributions that depend on the coefficient b and will be collectively identified as the "modified-Rayleigh" distribution. For the special case of b = 1, Equation (7.18) reduces to p(j-)=PRay(-]f-) for 6 = 1 , (7.19) in which case the distributions of peak hydrodynamic force and wave height are both Rayleigh. For b > 1, peak forces will be more broadly distributed and extreme values of F will exceed those predicted by the Rayleigh distribution. In contrast, b < 1 leads to a narrower distribution of force peaks in which extreme values are less than those predicted by the Rayleigh model. Probability densities for three modified-Rayleigh distributions with b = 0.7,1.0 and 1.5 are shown in Figure 7-6. The modified-Rayleigh distribution with b = 1 is identical to the Rayleigh distribution. 242 1.2 m o d - R a y l e i g h , b= 1.0 m o d - R a y l e i g h , b= 1.5 m o d - R a y l e i g h , b = 0.7 •i 1 3.5 Figure 7-6: Modified-Rayleigh distributions with b = 0.7, 1.0 and 1.5. Hydrodynamic force peaks on the upper panel of the series 5 rubble-mound can be well de-scribed by a modified-Rayleigh distribution of the form given by Equation (7.18) with PRay [HH ) given by Equation (7.6) and b = 1.2. Figure 7-7 shows the cumulative distribution of normalized hydrodynamic force peaks for the same series 5 test considered previously (with Hs = 14 cm, Tp = 2 s), together with the theoretical Rayleigh and modified-Rayleigh distributions. This result is typical of those for other irregular wave tests on the series 5 structure, and suggests that for this structure, the peak hydrodynamic force acting on armour stones just below the still water level is related to the incident wave height by an expression of the form F = aH12 . For the steeper series 3 and 4 structures, the distribution of hydrodynamic force peaks is similar to the distribution of wave heights and can be reasonably well described by the Rayleigh distribution. As a typical example, Figure 7-8 shows the cumulative distributions of normalized peak hydrodynamic forces and wave heights compared to the Rayleigh model for the case of irregular waves with Hs = 13 cm, Tp = 2 s incident to the series 3 rubble-mound. This result is consistent with a linear relationship between force peaks and wave height of the form F = aH . 243 0 0.5 1 1.5 2 2.5 F / F ave Figure 7-7: Hydrodynamic force peaks compared to the Rayleigh and modified-Rayleigh distri-butions (upper panel, test series 5, Hs = 14 cm, Tp = 2s). 0 0.5 1 1.5 2 2.5 3 H / H , F / F ave ave Figure 7-8: Cumulative distributions of wave heights and force peaks compared to the Rayleigh distribution (upper panel, test series 3, Hs = 13 cm, Tp = 2s). 244 The small differences in the distributions of peak hydrodynamic force, wave height and waterline height on the milder sloped series 5 structure and the steeper series 3 and 4 rubble-mounds is likely related to the type of wave breaking that prevails on each slope. Surf similarities for the series 5 structure in irregular waves ranged between £ m = 1.6 and 2.2, suggesting a prevalence of plunging breakers. For the series 3 and 4 structures, £ m ranged between 2.8 and 4.2, suggesting more frequent occurrence of collapsing and semi-surging breakers. The prevailing type of wave breaking has a strong influence on the character of surface flows, and thus also has a significant effect on the hydrodynamic forces acting on the armour. 7.3.4 Distribution of Shields Failure Index Results in regular waves, presented in Section 4.4, suggest that the level of armour damage S after Nw = 1000 waves is proportional to the peak value of the Shields failure index Rs for the patch of armour stones just below the still waterline. The calibration between S and Rs is written in Equation (4.50). Under the assumption that Equation (4.50) can be applied to irregular waves, the armour damage resulting from a duration of irregular wave attack will depend on the distribution of Rs, and in particular the extreme values of Rs beyond the initiation of damage threshold RSJD-The distribution of Rs on armour stones located below the still waterline for all irregular wave tests on the series 3, 4 and 5 structures is found to generally follow the distribution of wave heights, and can be fairly well represented by the Rayleigh distribution. Cumulative distributions of normalized Rs and normalized wave heights are shown in Figure 7-9 for the series 5 test considered previously (with Hs = 14 cm and Tp = 2 s), and in Figure 7-10 for the series 3 test considered previously (with Hs — 13 cm and Tp = 2 s). These results are typical of those in other wave conditions and for the series 4 structure. The similarity between the distributions of H and Rs is consistent with a linear relationship of the form Rs = aH . 245 0 0.5 1 1.5 2 2.5 3 H / H , R\ / R\ ave S S.ave Figure 7-9: Wave heights and peak values of the Shields failure index compared to the Rayleigh distribution (test series 5, upper panel, Hs — 14cm, Tp = 2s). 0 0.5 1 1.5 2 2.5 3 H / H , R\ / R\ ave S S.ave Figure 7-10: Wave heights and peak values of the Shields failure index compared to the Rayleigh distribution (test series 3, Hs = 13 cm, T p = 2 s). 246 7.3.5 Extreme Values The performance of rock armour exposed to random wave attack, including the onset of damage and even the level of damage that ultimately occurs, depends critically on the few waves that exert the largest de-stabilizing forces. For this reason, the extreme statistics of the forcing, in -cluding the maximum force that occurs over a duration of random wave attack, are of particular interest. These extreme statistics are considered in this section. The distributions of incident wave height, the height of waterline motions, the peak hy-drodynamic force and the peak value of the Shields failure index (proportional to the shear stress) have been considered in previous sections of this chapter. In each case, the observed distributions have been compared to the theoretical Rayleigh distribution. Many of the ob-served distributions are similar to the Rayleigh distribution, but significant deviations from the Rayleigh model were also found in some cases. In spite of these deviations, the Rayleigh distribution serves as a useful basis from which to study extreme values. Sarpkaya and Isaacson (1981) show that characteristic values of the Rayleigh distributed random variable Y\jn are related to the average value Yave by where erf [ ] is the error function previously defined in Equation (6.32). This equation can be for any Rayleigh distributed random variable. Another extreme value that is often important is the maximum value that can be expected to occur over a certain duration of wave attack containing Nw waves. Estimates of this overall maximum can be determined from the distribution of the maxima that occur in numerous sam-ples, each containing Nw waves. Sarpkaya and Isaacson (1981) give formulae for the expected maximum value (mean) and the most probable maximum value (mode) that apply when the underlying quantity satisfies the Rayleigh distribution. The expected maximum value E [ym ax] (7.20) used to estimate extreme values, such as Yi/ioo (the average of the largest 1 % of all values), 247 can be expressed in terms of the average value Yave and the number of waves Nw as y/]nNw + 0.2886 for large JV, (7.21) For Nw > 50, this expression is accurate to within 3 %. The most probable maximum value, denoted here by M [Ymax], can be estimated from Over long durations (with very large Nw), the expected and most probable values of Ymax approach the same value. Over shorter durations, the actual maximum realized during any single trial can deviate substantially from these central measures. These formulae can be applied to predict the maximum value that is likely to occur over a specified duration of wave attack for any quantity whose distribution is close to the Rayleigh distribution. The accuracy of the prediction will improve as the underlying distribution approaches the Rayleigh model. The theoretical expressions for extreme values based on the Rayleigh distribution presented above can be used as a benchmark from which to study extreme values for quantities that have an important influence on the performance of rock armour under random wave attack. The nature of the distributions of four important quantities, namely the incident wave height, the height of waterline motions, the peak hydrodynamic force acting on a patch of armour stones, and the peak value of the Shields failure index (proportional to the shear stress) have been considered earlier in this chapter. Consideration will now be given to the extreme values for these four elements. Mean values and standard deviations of Hi/n/Have with n — 3, 10, 20 and 100 are sum-marized in Table 7.1 for all irregular wave tests on the series 3, 4 and 5 rubble-mounds. A number of different sea states with various significant wave heights, peak periods and spectral shapes were used in each test series. Ten different irregular wave conditions were used during ave 2 , ~ —pvlnA 7 ™ f ° r large Nw. (7.22) 248 H\/n/ Have Series 3 mean std. dev. Series 4 mean std. dev. Series 5 mean std. dev. Rayleigh 1/n 1/3 1.51 0.01 1.50 0.01 1.51 0.01 1.60 1/10 1.88 0.03 1.86 0.04 1.87 0.03 2.03 1/20 2.08 0.04 2.05 0.07 2.06 0.06 2.24 1/100 2.54 0.14 2.43 0.19 2.44 0.14 2.66 Table 7.1: Hi/n/HaVe for irregular waves on the series 3, 4 and 5 rubble-mounds. Hns,l/n/HnStave Series 3 mean std. dev. Series 4 mean std. dev. Series 5 mean std. dev. Rayleigh 1/n 1/3 1.41 0.06 1.43 0.03 1.50 0.02 1.60 1/10 1.65 0.14 1.75 0.08 1.89 0.04 2.03 1/20 1.78 0.18 1.96 0.14 2.08 0.06 2.24 1/100 2.00 0.24 2.33 0.19 2.38 0.07 2.66 Table 7.2: HVstl/n/HristaVe for irregular waves on the series 3, 4 and 5 rubble-mounds. test series 3, while thirteen and nine different sea states were employed during test series 4 and 5, respectively. Theoretical values based on the Rayleigh distribution, computed from Equation (7.20), are also tabulated. These values confirm that on average, the Rayleigh model provides a good description of the distribution of incident wave height. However, there is some evidence that the Rayleigh distribution slightly overpredicts the height of the largest waves. This trend is consistent with results from numerous previous studies (e.g. Sarpkaya and Isaacson, 1981). Similar summaries of characteristic extreme values for the height of waterline motions HVa,i/n/Hfj^ave, the peak hydrodynamic forces Fi/n/Fave, and the peak Shields failure index values Rs,i/n/Rs,ave are presented in Tables 7.2, 7.3 and 7.4, respectively. Maximum values for these four quantities are considered together in Table 7.5. In this last table, the maxima are normalized by the average value and compared to estimates of the expected and most probable normalized maxima computed from Equations (7.21) and (7.22). These estimates depend on the number of waves Nw in each record and the form of the Rayleigh distribution. The means and standard deviations for the maxima in Table 7.5 were computed from the same groups of tests with different sea states used to obtain the characteristic extreme values summarized in ' 249 F\/n/Fave Series 3 mean std. dev. Series 4 mean std. dev. Series 5 mean std. dev. Rayleigh 1/n 1/3 1.57 0.05 1.56 0.06 1.74 0.03 1.60 1/10 1.85 0.07 1.83 0.10 2.29 0.10 2.03 1/20 1.97 0.08 1.94 0.11 2.50 0.16 2.24 1/100 2.19 0.12 2.14 0.12 2.84 0.19 2.66 Table 7.3: Fi/n/Fave for irregular waves on the series 3, 4 and 5 rubble-mounds. Rs,l/n/Rs,ave Series 3 mean std. dev. Series 4 mean std. dev. Series 5 mean std. dev. Rayleigh 1/n 1/3 1.53 0.05 1.54 0.06 1.47 0.04 1.60 1/10 2.10 0.13 2.12 0.13 1.86 0.12 2.03 1/20 2.46 0.21 2.42 0.15 2.09 0.19 2.24 1/100 3.15 0.32 2.96 0.30 2.47 0.30 2.66 Table 7.4: Rs,i/n/Rs,ave for irregular waves on the series 3, 4 and 5 rubble-mounds. Series 3 mean std. dev. Series 4 mean std. dev. Series 5 mean std. dev. 377 37 376 29 370 30 M{Ymax/Yave] 2.75 0.02 2.75 0.02 2.74 0.02 F [Ymax/Yave\ 2.84 0.02 2.84 0.02 2.84 0.02 Hmax / Rave 2.76 0.22 2.58 0.27 2.59 0.18 Hr)s,max/H^save 2.12 0.28 2.47 0.20 2.47 0.11 Fm&x/ Flve 2.29 0.09 2.22 0.15 2.97 0.21 Rs,m&x/Rs,ave 3.47 0.40 3.14 0.36 2.53 0.30 Table 7.5: Normalized maximum values of H, HVs, F and Rs for irregular waves on the series 3, 4 and 5 rubble-mounds. 250 Tables 7.1 - 7.4. The statistics in Tables 7.2 and 7.5 indicate that the Rayleigh distribution tends to overpre-dict the height of the largest waterline motions on the surface of all three structures considered in this study, and that the overprediction is most severe for the steep impermeable rubble-mound examined in test series 3. These results are consistent with the measured distributions considered previously in Section 7.3.2. Tables 7.3 and 7.5 show that the distribution and extreme values of the peak hydrodynamic force acting on armour stones depend significantly on the structure slope. On the steeper structures with slope co ta = 1.75 (tested in series 3 and 4), peak forces are distributed more narrowly than the Rayleigh distribution, but the opposite holds for the more gently sloping structure with co ta = 3 considered in test series 5. In other words, the Rayleigh distribution tends to overpredict the largest forces that occur on the steeper slope, but slightly underpredicts the largest forces on the milder slope. The Rayleigh distribution can thus be said to provide a conservative estimate of extreme forces on steeply sloping structures, but a non-conservative estimate of extreme forces on milder slopes. These observations concerning extreme values are consistent with the shapes of the distributions of F considered previously in Section 7.3.3. Lastly, Tables 7.4 and 7.5 show that structure slope has an even stronger, but quite different influence on the distribution and extreme values of the Shields failure index (proportional to the shear stress on the armour). The peak values of Rg tend to be quite broadly distributed (compared to the Rayleigh model) on the two steeper slopes and more narrowly distributed on the milder slope. As a result, the Rayleigh model underpredicts the peak shear stresses on the steeper series 3 and 4 structures (with co ta = 1.75), and overpredicts the peak shear stresses on the milder sloped series 5 rubble-mound (with co ta = 3). This slope dependence is most noticeable for the more extreme values of Rs- In particular, during test series 3 the maximum value of the Shields failure index over 377 waves was on average 3.47 times the mean value. This is considerably more than the factor of 2.84 expected according to the Rayleigh distribution, or the factor of 2.53 observed for test series 5. These statistics demonstrate that structure slope 251 has a strong influence on the extreme values of the Shields failure index (shear stress) and that the extreme values can deviate significantly from the Rayleigh model. In summary, the distribution of the forces acting on armour stones under irregular wave attack, including the strength of the largest forces, has an important influence on the perfor-mance of the armour layer. In general, the Rayleigh distribution can be used to provide a rough first approximation to the shape of the distribution and to the largest forces acting on the ar-mour stones. However, different aspects of the forcing, such as the largest magnitude overall, or the peak slope-parallel component, can deviate significantly from the Rayleigh model. The deviations certainly depend on the structure slope and are likely related to the type of wave breaking that prevails on the rubble-mound. 7.4 Variability of Irregular Wave Forcing Previous sections in this chapter have focussed on the distributions of wave height, waterline height and of the peak values of wave-induced forces on rubble-mound armour under irregular wave attack. Analysis in this section focuses on the variability of the forces exerted on armour stones due to individual waves within an irregular wave train. Data for this analysis is drawn from the forces measured on the upper armour panel of the rubble-mound breakwaters studied in test series 3, 4, and 5. As described in Section 2.3, each of these structures was exposed to a variety of irregular wave conditions synthesized from different wave spectra. A single large data set containing close to 1000 different individual wave events was assembled for each test series by combining the waves from three different irregular wave records. In each case, the three chosen records featured a uniform significant wave height and peak periods Tp = 1.67, 2.0 and 2.5 s. This provided data sets with a wide variety of wave height - wave period combinations. Individual waves and corresponding surface flow cycles are defined using zero-upcrossing analysis as previously described in Section 7.2. Peak force quantities for each individual wave 252 0 0.05 0.1 0.15 0.2 0.25 H ( m ) Figure 7-11: Peak hydrodynamic force versus wave height for individiual irregular waves (upper panel, test series 3). are set equal to the local maxima of the appropriate time series during the duration of the corresponding surface flow cycle. Wave parameters are computed for each individual wave according to the definitions presented in Section 7.2.1. Results on the forcing of rubble-mound armour due to individual irregular waves clearly show that on average, hydrodynamic forcing increases with increasing wave height, but that wave height alone is insufficient to accurately predict the forcing due to an individual wave. In spite of this, it is still useful to quantify the variability of forces due to individual waves for potential application to stochastic analysis and probabilistic modelling of the performance of rubble-mound armour under irregular wave attack. Standard linear regression analysis is used below to investigate the dispersion or variability in the peak forces caused by incident waves with a given height. Figure 7-11 shows peak values of hydrodynamic force F on the upper panel of the series 3 rubble-mound plotted against wave height H for 858 different individual irregular waves. Linear 253 regression analysis of F on H assumes that the expected value of F given a particular value of H is given by the straight line E F\H = a + bH . (7.23) The method of least squares (e.g. Ang and Tang, 1975) is used to determine the regression coefficients a and b that minimize the overall error in F between the data points and the regression line. The least squares regression of F on H is shown as a solid line in Figure 7-11. The general trend towards increasing peak forces with increasing wave height is reasonably well represented by this linear model, however, individual data points show significant deviations from the regression line. The dispersion of peak forces about the regression line can be quantified in terms of the conditional standard deviation of F given H, denoted crF^H. Assuming a constant variance about the regression line, o~p\H can be written as AF\H = °-/Vl-r-2 (7.24) where ap is the standard deviation of F data and r is the correlation coefficient. The correlation coefficient provides a measure of the degree of linear interrelation between the two variables, and is defined by r = —^ -L i- 7.25 o-F • crH where p denotes a mean value and a represents a standard deviation. For the data in Fig-ure 7-11, the correlation coefficient is r = 0.797 and the conditional standard deviation of hydrodynamic force peaks is crF^H = 4.76 N. If the data points are normally distributed about the regression line, approximately 95% of the data will lie within ± 2 conditional standard deviations of the regression line. Thus, the value of F for a given value of H will lie between E F\H ± 2<Jp|^  with 95% confidence. The ratio between the 95% confidence half-bandwidth and the expected value E F\H 254 Nw Hs (m) r (N) F\H] (N) AF\H E 2(7 F\H E F\H 100 (%) Series 3 Series 4 Series 5 858 882 757 0.114 0.136 0.141 0.797 0.776 0.783 4.76 6.16 5.76 23.18 30.00 22.82 41 41 51 Table 7.6: Linear regression statistics between F and H for test series 3, 4 and 5. computed from the regression equation can be expressed in percentage terms as m 100% = f V , „ • 100% , E F\H a + bH (7.26) which decreases with increasing H. The variability of peak hydrodynamic forces due to individ-ual waves with height H can be characterized by Equation (7.26), evaluated at the significant wave height Hs. For the series 3 data in Figure 7-11, the 95 % confidence half-bandwidth for F is 2aF]H = 9.52 N, which at Hs = 0.114 equals 41 % of the regressed estimate E F\H = 23.18 N. Thus, the data in Figure 7-11 suggest that the peak hydrodynamic force due to an irregular wave with height H = Hs will lie between 23.18 N ± 41 %, 19 times out of 20. Statistics from linear regressions of F on H for individual irregular waves on the series 3, 4 and 5 structures are summarized in Table 7.6. These regressions feature a consistent degree of correlation; however, the 95 % confidence half-bandwidth for the milder sloped series 5 rubble-mound represents a larger (51%) proportion of the regressed estimate at Hs. These results clearly show that the peak hydrodynamic forces exerted on armour stones due to individual irregular waves are quite variable, and that the variability cannot be explained in terms of incident wave height alone. Values of F under regular waves, presented in Section 5.4.1, indicate that wave period and slope angle have significant influence on the peak magnitude of the hydrodynamic force acting on a patch of armour stones. On the milder sloped series 5 rubble-mound, F for a given regular 255 IE I I Ed I 30 20 10 -10 -20 # o oo o o oo Oj oof r 0 0.8 1.6 2.4 3.2 4 £ = t a n a (gT 2 / 2 T T H ) 1 / 2 4.8 Figure 7-12: Influence of surf similarity on the variation of wave forcing due to individual irregular waves (upper panel, test series 5). wave height is greatest for waves with period T = 2 s (for which £ ~ 2). On the steeper structures in test series 3 and 4, F increases with decreasing wave period such that the largest F values result from waves with T = 1.5 s (for which £ ~ 3). Extrapolated to irregular waves, these results suggest that peak hydrodynamic forces on the upper panel of the series 5 rubble-mound due to individual irregular waves could be maximized at wave periods for which £ ~ 2. Figure 7-12 shows the dispersion in F values, defined as the difference between the observed values and expected values E F\H computed from the linear regression equation, plotted against surf similarity £ for 757 different irregular waves incident to the series 5 rubble-mound. These data indicate that the largest positive deviations in F tend to result from waves for which £ ~ 2. However, it is important to note that only a small minority of waves with £ ~ 2 impart unusually large forcing. Thus, proportionately larger forcing on armour stones can, but does not necessarily result from waves with periods such that £ ~ 2. This result suggests that the significant variations in peak hydrodynamic force exerted by individual irregular waves cannot 256 0.2 0.1 2 fa -0.1 50 37.5 25 12.5 0 1 ^\ 7 I I 1 I •A 1 1 1 1 I \ f \ i \ / V / i A J7 \ .'/ \ •>/ N* . . / l l i "V V •/ V V \~ / \ / l l l l 1 1 1 1 1 ~J V, // 1 \ — •1 A 1 / V \ JJ \ i i V T^ i i i i i 356 357 358 359 360 361 362 Time (a) 363 364 365 366 Figure 7-13: Repeatability of irregular waves and wave-induced forcing (test series 5). be explained by the combined effects of wave height and wave period. 7.4.1 Repeatability of Measurements Some of the variation in the peak wave forces observed for individual irregular waves with similar height can be attributed to measurement inaccuracy and other experimental errors. The variability due to these sources can be partially assessed by considering the repeatability of measurements obtained in irregular waves. In order to assess the repeatability of the forces measured due to individual irregular waves, three repeated trials of a single irregular wave train on the series 5 rubble-mound have been analysed. Identical 8 s long segments of n (t) and F (t) measured during each of these trials are presented in Figure 7-13. Both n(t) and F (i) show good repeatability between repeated trials. The variation in n (t) is not large, however, it is enough to have a small effect on the characteristics computed for individual waves. Similarly, 257 small variations between these three F (t) signals suggest that the peak hydrodynamic force F due to a particular wave can vary in the order of 5 %. The good repeatability of the measured water surface elevations and hydrodynamic forces recorded on the armour panel lends confidence to the data on wave characteristics and forcing due to individual irregular waves. The segments of F (t) shown in Figure 7-13 contain the maximum hydrodynamic force recorded throughout the entire 10 minute duration of irregular wave action, which occurs at t ~ 362.2 s. This force peak is exerted by the moderately high upcrossing wave formed by the low crest at t ~ 259.5 s followed by the deep trough at t ~ 361.2 s. Both the preceding and following waves have greater height, but exert less force on the armour stones of the rubble-mound. This is a specific example of the general conclusion that the force exerted by an individual irregular wave cannot be accurately predicted on the basis of wave height alone. In this case, the wave that exerts the maximum force has a smaller height, a longer period and a deeper trough than its neighbours. Kobayashi et. al. (1990c), using numerical simulations of wave interaction with a rubble-mound, also identified a case in which the largest force exerted by a sequence of irregular waves was due to a moderately high wave with a particularly deep trough. 7 .5 Indicators of Damaging Waves In traditional design formulae for rubble-mound armour, the damaging effect of waves are typically quantified in terms of their height. The design equation proposed by Hudson (1958), written in Section 3.1.3 as Equation (3.2), is an example of a design equation that follows this approach. Recently, design equations have been developed that use more than one parameter to identify the damaging effect of various waves. As an example, the design equations proposed by van der Meer (1988), presented as Equations (3.10) and (3.11) in Section 3.1.6, use both the significant wave height Hs and the surf similarity £ m = ta.n a x/gT^/faHs) to parameterize the damaging effect of a spectrum of irregular waves. Surf similarity depends on Hs as well as 258 the average wave period Tm. In this section, two approaches are developed to identify the individual irregular waves that are most damaging to rubble-mound armour. In both approaches, the damaging effect of individual waves is quantified by the peak value of the Shields failure index on the armour stones located below the still waterline. In the first approach, single wave parameters other than wave height are explored. This analysis suggests that the potential energy of the incident wave is a better indicator of its contribution to damage than the wave height. In the second approach, the combined influence of wave height, wave period, and the eight additional parameters identified in Section 7.2.1, related to the wave shape and the character of the preceding wave, are explored. 7.5.1 Wave Potential Energy The peak value of the Shields failure index Rs is identified in Section 4.4.3 as a force quantity that is fairly well correlated to the extent of armour damage in regular waves. (The Shields failure index is proportional to the shear stress acting on the armour.) Greater damage to the armour layer was observed in waves that exerted larger peak values of the Shields failure index. This quantity provides a means to quantify the relative contribution of an individual irregular wave to the resulting armour damage. For a particular slope angle, waves that exert hydrodynamic forces leading to greater Rs are more likely to cause damage than those associ-ated with lesser Rs- In other words, the relative contribution of an individual wave to armour damage can be quantified by the value of Rs measured on an armour panel located below the still waterline. Figure 7-14 shows values of Rs on the upper panel of the series 5 rubble-mound (cot a = 3, impermeable core) for 840 different individual irregular waves plotted against wave height H. The correlation coefficient for these data is r = 0.777 and the least squares linear regression of Rs on H is drawn in the figure as a solid line. For this structure, the value of Rs corresponding to the initiation of damage in regular waves is RSJD — 2.7. The data show a definite trend towards increasing Rs values with increasing wave height that is fairly well represented by the 259 6.25 3.75 1.25 0.06 0.12 0.18 0.24 0.3 H ( m ) Figure 7-14: Peak value of the Shields failure index versus wave height for individual irregular waves (upper panel, test series 5). linear regression equation, particularly for the smaller waves. In some cases, individual Rs values induced by the largest waves greatly exceed the expected values E Rs\H computed from the regression equation. This data suggests that Rs values (and by association the relative contribution of an individual wave to armour damage) cannot be precisely determined from wave height alone. Figure 7-15 shows the same Rs data plotted against a non-dimensional form of the potential energy of each incident wave Pw (defined in Section 7.2.1). The linear correlation coefficient for these data increases to r = 0.841, and even the largest values of Rs are in better agreement with the linear least squares regression of Rs on Pw. This suggests that for the series 5 rubble-mound, the potential energy of an incident wave is superior to wave height as an indicator of the relative contribution of individual waves to armour damage. Linear correlation coefficients for similar regressions between Rs and selected wave parame-ters for the series 3, 4 and 5 rubble-mounds are summarized in Table 7.7. Correlation coefficients 260 6.25 0 8 16 24 32 40 P / P g ( A D ) 3 w ' ^ to v n50 7 Figure 7-15: Peak value of the Shields failure index versus potential wave energy for individual irregular waves (upper panel, test series 5). Series 3 Series 4 Series 5 Nw 922 936 840 H 0.811 0.825 0.777 p 0.804 0.809 0.841 vw 0.844 0.864 0.784 Table 7.7: Linear correlation coefficients between Rs and selected wave parameters (upper panel, test series 3, 4 and 5). 261 15 12 9 m <K 6 3 0 0 0.125 0.25 0.375 0.5 0.625 V / p g ( A D ) 2 w ' ^ & v n50 ' Figure 7-16: Peak value of the Shields failure index versus average potential wave energy for individual irregular waves (upper panel, test series 3). between Rs and H equal 0.811, 0.825 and 0.777 for the series, 3, 4 and 5 rubble-mounds, re-spectively. Correlation coefficients between Rs and Pw for these three structures are 0.804, 0.809 and 0.841, while correlation coefficients between Rs and the average potential energy Vw are 0.844, 0.864, and 0.784. These results suggest that the average potential energy of the incident wave Vw is superior to wave height as an indicator of Rs on the steeper series 3 and 4 rubble-mounds, while the total potential energy is superior to wave height on the milder-sloped series 5 structure. Plots of the relations between Rs and Vw for test series 3 and 4 are presented in Figures 7-16 and 7-17 respectively. The solid line in these figures shows the least squares regression equation computed for the data. The potential energy of an incident wave, either in total, or averaged over a wave length, provides a better indication than wave height of the peak value of the Shields failure index resulting from an individual irregular wave. The potential energy of incident waves can therefore also be expected to provide a better indication than wave height of the contribution to damage 262 15 12 9 ra <« 6 3 0 0 0.125 0.25 0.375 0.5 0.625 V / p g ( A D K n )2 w ' ^ to v n50 7 Figure 7-17: Peak value of the Shields failure index versus average potential wave energy for individual irregular waves (upper panel, test series 4). of individual irregular waves. 7.5.2 Wave Shape and the Sequencing of Successive Waves For a single irregular wave of given height and period, the process of wave - structure interaction whereby kinetic and potential wave energy is transformed into waterline motions and surface flow kinematics on a rubble-mound will likely be influenced by: • the shape of the incident wave (related to the temporal distribution of wave energy); and • conditions on the rubble-mound at the beginning of the wave interaction (related to the initial conditions for the wave - structure interaction). Many parameters have been proposed to quantify variations in the shape of individual irreg-ular waves from 77 (4). Four parameters are adopted here to quantify these variations, namely 263 horizontal asymmetry a., vertical asymmetry A, skewness f3\, and atiltness /?3. Definitions of these parameters are presented in Section 7.2.1 Conditions on a rubble-mound at the beginning of a wave interaction (initial conditions for that interaction) depend on the surface flows generated by the preceding wave, and thus depend on the character of the preceding wave and its own interaction with the structure. The time lag between successive waves and their relative wave height can influence the surface flow kinematics generated by the second wave and the forces exerted on the armour. Bruun and Gunbak (1978) and Gunbak (1979) used the term "breakwater resonance" to describe a particular state of sequencing when, "every run-down meets the new run-up from a breaking wave at the breaking point." According to Gunbak, in regular waves these conditions occur for £ ~ 3, and are associated with minimum stability for the rubble-mound armour. Sawaragi et. al. (1982) indicate that for regular waves on a smooth impermeable slope, "resonance" occurs for 2 < £ < 3, and is associated with particularly large surface flow velocities. Four parameters are adopted here to quantify the relation of an individual irregular wave to the preceding wave. These are the percentage difference in wave height AH, the percent-age difference in wave period AT, the percentage difference in upcrossing and downcrossing height AHu-a; and the percentage difference in upcrossing and downcrossing period A T u _ d . Definitions of these parameters are also given in Section 7.2.1. For this analysis, parameters for an individual irregular wave are denoted by the subscript ' V , as in height Hi and period T, etc., in order to differentiate them from the regular wave with height H and period T. Fai lure Surface U n d e r Regu la r Waves Peak values of the Shields failure index in regular waves are used to construct surfaces of Rg as a function of H and T for the upper armour panel in test series 3, 4 and 5. Each of these surfaces, denoted by Rs (H,T), are developed as follows. Cubic polynomials Rs (H) are fitted using least squares to describe the relationship between Rs and H at each of three wave periods, 264 T(s) d Figure 7-18: Failure surface Rs {H,T) under regular waves (upper panel, test series 5). T = 1.5, 2, and 3 s. The surface Rs (H,T) is defined by the family of quadratic polynomials Rs (T) that intersect the three cubics at successive values of H. The resulting surface is cubic in wave height and quadratic in wave period. Figure 7-18 shows a view of Rs (H,T) computed for the upper panel of the series 5 rubble-mound. This surface shows that over the considered range of wave height and period, Rs is an increasing function of wave height that also depends significantly on wave period. On the series 5 rubble-mound, the most de-stabilizing regular waves are those with large heights and long periods. The surface in Figure 7-18 can be used to estimate the value of Rs that would result from any combination of regular wave height and period. Since this surface has been interpolated from a limited number of experiments, and assumes a smooth variation, the true value of Rs for a particular combination of height and period may deviate somewhat from this estimate. The surface cannot be reliably extrapolated beyond the range of height and period 265 of the original data, and is therefore restricted to the range 1.5 < T < 3 s and 11 < H < 19 cm. R e m o v i n g the Influence of He ight and P e r i o d Each individual irregular wave with wave height Hi and wave period Ti produces a peak value of Shields failure index Rsti- The value of Rs,i produced by each individual wave will depend on its height and period, as well as additional factors including the shape of the wave and initial conditions for the wave interaction with the rubble-mound. This can be expressed symbolically as Rs,i = fi (Hh Tu Si) (7.27) where e{ includes the effects of all factors except for wave height and wave period. Using the calibration surface generated from results in regular waves to represent the variation of Rs,i with Hi and Ti, Equation (7.27) can be written Rs,i = RS(Hi,Ti)-f2(ei) . (7.28) where the effects of all other factors are represented by the term fi (ei). The quantity T, denned by r = . Rs'{— = f2 (a) (7.29) Rs(Hi,Ti) J 2 K t l K ' represents the peak value of the Shields failure index due to an individual irregular wave nor-malized by the peak index value of an equivalent regular wave with H = Hi and T — Ti. Variations in T among individual irregular waves will depend on the additional factors ei be-yond the wave height and period. T > 1 indicates individual waves that are more damaging than their regular wave counterparts, while T < 1 indicates less damaging individual waves. T = 1 indicates individual irregular waves that are equally damaging as their equivalent regular wave counterparts. 266 1.4 1.2 1 0.8 0.6 0.4 0.1 0.12 0.14 0.16 0.18 0.2 H ( m ) Figure 7-19: T versus H for individual irregular waves (upper panel, test series 5). Figures 7-19 and 7-20 show values of T computed for the set of individual waves incident to the series 5 rubble-mound plotted against wave height Hi and wave period Ti, respectively. The solid lines in these and subsequent figures show the linear least squares regression of T on the abscissa variable and are included to illustrate any linear trend that exists between the two variables. Values of F for individual waves show significant deviations from unity; however, these deviations are poorly correlated to both Hi and Ti, which suggests that the influences of wave height and period have been effectively removed. The remaining variation in T can be attributed to other factors influencing the wave interaction with the rubble-mound, such as the shape of each wave and the character the preceding wave. The 238 points shown in each of these figures represent all waves from the original set of 840 waves that fall within the range of the calibration surface shown in Figure 7-18, that is 3 < T, < 1.5 s and 11 < Hi < 19 cm. The mean value of r for these waves is 0.83, which suggests that on average, individual irregular waves exert 83 % of the peak Shields failure index of an equivalent regular wave. The standard error 267 1.4 1.2 0.8 0.6 0.4 * o 0 o 0 0 o / ° o . o 0 o • % ° o o 0 , o ° °P 1.5 1.8 2.1 2.4 T ( S ) r = - 0 . 1 0 2.7 Figure 7-20: T versus T for individual irregular waves (upper panel, test series 5). in r for this data is 0.18, which by Equation (7.26) yields a 95% confidence half-bandwidth of 43 % of the mean value. This result suggests that individual irregular waves exert de-stabilizing forces on rubble-mound armour that deviate significantly from the forces exerted by equivalent regular waves with the same height and period. Furthermore, it suggests that additional factors beyond upcrossing wave height and upcrossing wave period influence the de-stabilizing forces exerted by individual irregular waves. Charac te r of the M o s t D a m a g i n g Waves The most damaging waves to a rubble-mound are those that exert the greatest de-stabilizing forces on armour stones. De-stabilizing forces can be reasonably well quantified by the Shields failure index, such that larger peak index values are associated with greater damage. A long irregular wave train contains many individual waves with similar heights and periods. More detailed examination of these waves shows that although the wave heights and periods 268 are similar, the shape of the waves can differ significantly. Some waves feature unusually large crests while others feature deeper, wider troughs. Four wave parameters are defined in Section 7.2.1 to quantify differences in wave shape: vertical asymmetry /x; horizontal asymmetry A; atiltness /%; and skewness Similar waves can also differ in their relationship to adjacent waves. In some cases the pre-ceding wave will be smaller, in others it will be nearly the same, while in others it will be larger. The character of the preceding wave governs conditions on the rubble-mound at the beginning of each wave interaction. Regular waves represent the extreme case in which all neighbouring waves are identical. Four parameters to quantify the character of the preceding wave are defined in Section 7.2.1: height difference AH; period difference A T ; and the differences between the zero-upcrossing and zero-downcrossing definitions of wave height AHu_d and period AT„_<2. The parameter T, defined by Equation (7.29), represents the de-stabilizing force exerted by an individual irregular wave, compared to that of a regular wave with identical height and period. Variations in T away from unity suggest that other factors, unrelated to the wave height and period, have some influence on the de-stabilizing force exerted by an individual irregular wave. Figure 7-21 shows values of T for the surviving sub-set of 238 individual waves incident to the series 5 rubble-mound, plotted as a function of horizontal asymmetry. These data indicate a moderate trend towards greater T at lesser horizontal asymmetries which suggests that waves with deeper troughs are on average more damaging to armour stones below the still waterline. The linear correlation coefficient for this data is r = —0.484. Waves with particularly deep troughs are one of the critical forms identified by Kobayashi et. al. (1990). Figure 7-22 shows the same values of T plotted against AHu^d, the percentage difference between the zero-upcrossing and zero-downcrossing definitions of wave height. In this case, a moderate positive correlation exists between T and AHu-d, for which the linear correlation coefficient is r = 0.457. This result suggests that waves preceded by a shallow trough are more likely to exert stronger de-stabilizing forces than waves that are preceded by a deep trough. 269 Figure 7-22: Difference in upcrossing and downcrossing wave height as an indicator of damaging irregular waves (upper panel, test series 5). 270 M = a c / H Figure 7-23: Horizontal asymmetry as an indicator of damaging irregular waves (upper panel, test series 5, Hi > 15 cm). This is one of the dangerous wave conditions identified by Bruun (1985) as a "wave jump". These observations are strengthened if the set of individual waves is restricted to a narrow band of wave height. Figures 7-23 and 7-24 are similar to Figures 7-21 and 7-22, except that only values for the 67 largest individual irregular waves with Hi > 0.15 m are considered. For this restricted set of waves, the correlations between F and fi, and F and AHu-d are stronger such that r = —0.596 and r = 0.545, respectively. Correlation coefficients between F and selected wave parameters for the complete set of indi-vidual irregular waves incident to the series 3, 4 and 5 rubble-mounds are summarized in Table 7.8. A l l waves within the range of each calibration surface Rs (Hi, Ti) are considered in these regressions. Horizontal asymmetry p and the difference between upcrossing and downcrossing wave heights AHu_d are clearly less significant indicators of damaging waves for the steeper series 3 and series 4 rubble-mounds. The three most significant parameters for the series 3 structure are atiltness the difference in height AH and the difference in period A T , while 271 Figure 7-24: Difference in upcrossing and downcrossing wave height as an indicator of damaging irregular waves (upper panel, test series 5, Hi > 15 cm). Wave Parameter Series 3 Series 4 Series 5 Nw 151 361 238 linear correlation coefficient -0.080 -0.299 -0.484 A -0.007 0.110 0.179 /% -0.274 -0.243 -0.048 Pi -0.041 -0.221 -0.375 AH 0.244 0.138 0.208 AT 0.237 0.222 0.219 AHu_d 0.227 0.237 0.457 ATu-d 0.185 0.147 0.112 Table 7.8: Linear correlation coefficients between V and selected wave parameters for individual irregular waves (upper panel, test series 3, 4 and 5). 272 horizontal asymmetry /z, atiltness Ps, and the difference between upcrossing and downcross-ing wave height AHu-d are the most significant indicators of damaging waves for the series 4 rubble-mound. However, the weakness of the correlations between T and these parameters imply that none of the parameters investigated here can be used to reliably indicate damag-ing waves for these steeper structures. One possible explanation for the weak correlations on the series 3 and 4 structures is that the critical forcing events on these steeper slopes tend to result from drainage and seepage flows that peak towards the end of downrush, while on the milder slope, the critical forcing events tend to occur under the steep advancing wave crests and are generated by flows on the surface of the armour. The seepage flows that contribute to the critical forcing events on the steeper slope are apparently less sensitive to the shape and sequencing of the incident waves than are the surface flows that dominate on the milder slope. The drainage and seepage flows travel through the permeable zones of a rubble-mound, which can be considered to act as a filter that smooths and moderates the variations due to differences in wave shape and sequencing. For the milder sloped series 5 rubble-mound, wave skewness P\, together with horizontal asymmetry [i and the difference between upcrossing and downcrossing wave heights AHu_d can be used to identify the most damaging waves in an irregular wave train. 273 Chapter 8 Conclusions From the perspective of a coastal engineer faced with the immediate challenge of designing a rubble-mound structure, the most important conclusion of this study is that the results and analysis presented herein support the design equations for rock armour recently proposed by van der Meer. These equations include the effects of wave height, wave period, slope angle, permeability and storm duration in a manner that is generally consistent with the findings of this study. This endorsement is based on an assessment that considers results on the damage response combined with results on the wave-induced forcing of the armour layer. However, this endorsement must be supplemented by the recommendation that van der Meer's equation should not be applied for damage levels less than S = 2. In this study, physical experiments with hydraulic models of rubble-mound breakwaters have been used to investigate various aspects of the process whereby wave attack causes damage to rock armour. Different rubble-mound breakwaters, including structures with impermeable and permeable cores, and seaward slopes between cot a — 1.5 and 3, were tested in a variety of reg-ular and irregular wave conditions. The N R C experiments were designed to allow simultaneous measurement of incident waves, flows on the surface of the test structure, wave-induced forcing on a portion of the armour layer, and damage to the armour. The author is not aware of any previous studies in which these different quantities have been measured together. 274 A new form of instrumentation, called an armour panel, has been developed to measure the hydraulic forces acting on a portion of an armour layer. These force measurements were a key and unique aspect of the experiments. Two armour panels were fabricated, and these instruments provided robust and informative measurements of the fluid forces acting on realistic sections of rock armour with minimal distortion to the model rubble-mound. The armour panels provide a measure of the total hydraulic force acting on a section of the armour layer, but do not provide information on the forces acting on individual armour stones. For cases where the panels were installed below the still waterline, the hydrodynamic component of the fluid forces was obtained by removing the forcing due to buoyancy. The forces measured by an armour panel can be interpreted as a spatially averaged measure of the hydraulic forces acting on armour stones located on the surface of an armour layer. 8.1 Surface Flows Both a qualitative description and a quantitative analysis of the surface flows resulting from wave attack have been presented. Wave period and structure slope both have a significant effect on the character of surface flows that can be at least partially explained in terms of the type of wave breaking that prevails. Plunging breakers prevail for steeper waves on milder slopes, while surging breakers prevail for less steep waves on steeper slopes. Collapsing breakers are a transitional form of wave breaking that can occur between these two more common forms. Because the armour layer is permeable, infiltration and seepage flows through the surface of the structure exist even on structures with an impermeable core. Infiltration occurs at all elevations, but is strongest above the still waterline. Seepage flows are concentrated below the still waterline, and can generate important de-stabilizing forces on the armour. Vertical excursions of the waterline on the surface of a structure provide a quantitative measure of the overall surface flow. The observed variation in waterline excursions can be fairly well expressed as a function of the surf similarity parameter £. Measured waterline excursions 275 have been favorably compared to predictions derived from the work of Losada and Gimenez-Curto (1981). Measurements of the velocity of surface flows were obtained on two different structures at two locations, both below the still waterline. For most wave conditions, peak slope-parallel velocities during uprush and downrush were less than y/gH, however, peak uprush velocities slightly greater than yfgH were observed with high waves for £ ~ 2, while peak downrush velocities slightly greater than \JgH were observed with high waves for £ ~ 3. The strongest slope-parallel flows tend to occur under waves with 2 < £ < 4, corresponding to the transitional region between plunging and surging breakers where collapsing breakers can prevail. More intense uprush and downrush flows occur closer to the still waterline than at lower elevations, and the variation with depth is more prominent for downrush flows. Wave period has a strong influence on slope-parallel flows under plunging breakers such that longer period waves produce stronger flows. Wave period has little influence on the strength of slope-parallel flows under surging breakers. For surging breakers, the influence of wave height on the intensity of slope-parallel uprush and downrush flows is reasonably well described by the factor \fg~H-, but for plunging breakers, wave height has a stronger influence on the maximum slope-parallel velocities. 8.2 Wave-Induced Forcing The wave-induced forcing of the armour is unsteady and spatially varying. The most dangerous forces affecting the stability of armour stones occur below the still waterline. At this critical elevation, the largest slope-normal forces are generally greater than the largest slope-parallel forces. The temporal variations of the forcing depend strongly on the type of wave breaking that occurs on the slope. Under plunging breakers, the strongest forces result from the sudden flow reversal that occurs under the steep wave crest. Under surging breakers, the largest forces result from seepage flows that occur towards the end of the downrush phase of the surface flow cycle. Collapsing breakers are particularly damaging to the armour layer because these two 276 forcing mechanisms tend to occur simultaneously and reinforce each other. At lower elevations, the slope-normal forces are generally weaker, and the forcing is dominated by accelerations in the surface flow. The slope-parallel and slope-normal components of the hydrodynamic force acting on a section of armour stones have been expressed in terms of a shear stress r and a normal stress a. Friction factors have been developed which relate the peak values of the shear stress to the peak values of the slope-parallel flow velocity above the surface of the armour layer. The friction factor for rock armour on a rubble-mound structure under wave attack is a variable quantity that depends on relative roughness in a manner that is similar to that observed for oscillatory flow over a rough impermeable horizontal bed. The observed variation of friction factor with relative roughness under both regular and irregular waves can be fairly well described by the predictive equation for wave friction factor proposed by Kamphuis (1975). Increases in wave height tend to increase the hydrodynamic forcing of the armour at all elevations and at all times throughout a flow cycle. Thus, changes in wave height can be said to alter the magnitude, but not the character of the forces acting on the armour. Wave period and slope angle both have a strong influence on the character (the spatial and temporal variations) of the forcing. The effects of wave period and slope angle are coupled to each other through their influence on the type of wave breaking that occurs on the slope. Analysis of several different force quantities including the magnitude of the maximum hy-drodynamic force, the peak shear stresses, and the peak normal stress acting away from the structure, has shown that in each case, the wave-induced forcing is maximized within the range of surf similarity between 2 < £ < 4. This range corresponds to the transition between plunging and surging breakers where collapsing breakers can occur. For £ > 4 where surging breakers prevail, forcing tends to decrease with increasing wave period, but for £ < 2 where plung-ing breakers prevail, forcing tends to increase with increasing wave period. These results on the variation of armour forcing with changes in wave height, wave period and slope angle are generally consistent with the design equation proposed by van der Meer in 1988. 277 Changes in the permeability of the material below the armour layer also affect the character of the forcing on the armour. The slope-parallel forces are reduced, and the slope-normal force acting into the structure is increased on rubble-mounds with a more permeable core. These trends in forcing are the result of increases in the volume of infiltration, and the amount of energy dissipation through internal flow that occur on more permeable structures. The enhanced stability of armour stones on more permeable structures is due to the reduction in the shear stresses acting on the armour layer. 8.3 Wave-Induced Damage The damage of rock armour due to wave attack is a progressive stochastic process. This randomness makes it difficult to precisely identify the wave conditions required to initiate damage to an armour layer. Results on the wave conditions required to initiate damage in regular and irregular waves have been compared to predictions from the design equations of Hudson (1958) as implemented in the "Shore Protection Manual" (CERC, 1984), and of van der Meer (1988). The influences of wave height, wave period, storm duration, slope angle, and core permeability on the damage of rock armour have been considered. The effects of wave period, storm duration and permeability are not considered in the Hudson equation. The observed influences of these factors are consistent with the design equation proposed by van der Meer. Van der Meer's equations are recommended for preliminary design with irregular waves, but should be limited to damage levels greater than 5 = 2. Theoretical analysis of the forces acting on armour stones at their incipient motion threshold has been applied to develop failure indices that quantify the relative stability of armour stones in five possible failure modes. These failure indices have been used to interpret the forces measured on sections of the armour layer, and to study the relationship between the wave-induced forcing and the damage of armour stones under regular wave attack. Down-slope rolling and sliding of armour stones were the dominant failure modes (mechanisms of displacement) observed on the 278 structures considered in this study. Up-slope failures should be more prevalent on structures with more gradual slopes. The initiation and early growth of damage in regular waves is found to be closely linked to the peak shear stress acting in the down-slope direction on the critical section of armour below the still waterline. The hydrodynamic forcing must exceed a threshold level to initiate damage. 8.4 Irregular Wave Effects Damage due to irregular wave attack is controlled by those waves that exert forcing of sufficient magnitude and character to de-stabilize armour stones. Although the forcing exerted by an individual irregular wave is, on average, proportional to the wave height, the most damaging waves in an irregular wave train are often not the waves with the largest height. Zero-upcrossing analysis was found to be the most appropriate method to define individual irregular waves with respect to their effect on a rubble-mound structure. Results from regression analysis for many different individual irregular waves have been presented. These results show that the potential energy of each incident wave, calculated by integration of the wave profile in the time domain, provides a slightly more linear indicator of the damaging effect of individual irregular waves than does wave height. Probability distributions of force quantities, including peak values of the hydrodynamic force and peak values of the Shields failure index (proportional to the shear stress), have been pre-sented and compared to theoretical distributions and to observed distributions of wave height and the height of waterline motions. In these experiments, incident wave heights are generally well described by the Rayleigh distribution. In most cases, the distributions of peak forces observed for a section of armour below the still waterline can be approximated by the Rayleigh distribution, however significant deviations from the Rayleigh model sometimes occur. A mod-ified form of the Rayleigh distribution has been developed which provides a better description of the observed force distributions. Extreme statistics for four important quantities (H, HVs, F 279 and Rg) have been analysed and compared to theoretical estimates of extreme values based on the Rayleigh distribution. This analysis indicates that the slope of the rubble-mound structure has a significant influence on the distributions of armour forcing, and that the influence of core permeability is negligible. In general, the Rayleigh model provides only a rough approximation of the largest forces that act to destabilize armour stones under random wave attack. The structure slope is also shown to have an influence on the distribution of the height of waterline motions, such that a narrower range of motion tends to prevail on steeper slopes. The forces generated by the individual waves in an irregular wave train are considered and found to be highly varied. While much of this variability can be attributed to differences in the height and period of each wave, some of it is due to additional factors including differences in the shape of each wave and the sequencing of successive waves. A n analysis of the contributions due to these additional factors is presented which indicates that on milder slopes, the most dangerous irregular waves will likely feature a large upcrossing wave height with a deep trough and will likely follow a wave with a much shallower trough. This conclusion could not be generalized to steeper slopes. 8.5 Applications to Numerical Modelling Numerical models of wave interaction with rubble-mound structures are quickly advancing to the point where they may soon be used to assist in design. Data obtained in this study could be used to calibrate an existing numerical model, or alternatively, results from this study could be used to guide the development of a new numerical model that could include a prediction of the forcing and damage of the armour layer. • Results on the surface flow kinematics presented in Section 5.1 can be used to calibrate, or verify the performance of the modelling component that simulates surface flows. Similarly, results on the wave-induced forcing presented in Chapter 5, and the wave-induced stresses presented in Chapter 6 represent a valuable resource to those attempting to extend their 280 models of fluid flow to include predictions of the hydrodynamic forcing of the surface layer of armour. • The development of wave friction factors for rock armour presented in Section 6.4 provides a means to compute the peak shear stresses acting on an armour layer based on the roughness of the armour and the kinematics of the surface flow. The friction factor concept could be applied within a numerical model of surface flows to compute shear stresses acting on the armour. • The analysis of failure mechanisms for armour stones presented in Chapter 4 suggests that the initiation and early growth of damage to an armour layer depends primarily on the magnitude of the down-slope shear stress. This result could be applied within a numerical model to predict the initiation and development of damage based on the shear stresses computed over the surface of the structure. 8.6 Summary of Principal Conclusions A summary of the principal conclusions of this study is presented in the following. These conclusions respond to the specific objectives described in Section 1.2. • A new form of instrumentation, called an armour panel, has been developed which pro-vides excellent quality measurements of the hydraulic forces acting on a patch of rock armour. • The wave-induced forcing of rock armour is unsteady and spatially varying. The most dangerous forces affecting armour stability prevail below the still waterline where the strongest horizontal forces act opposite to the direction of wave propagation (away from the structure). Above the still waterline the strongest horizontal forces act into the core of the structure. 281 The prevailing type of wave breaking has a strong influence on the magnitude and tempo-ral character of the surface flows and the forcing of the armour. Slope-parallel velocities and de-stabilizing wave-induced forces tend to be maximized over the transitional regime between plunging and surging breakers where collapsing breakers can occur. Under plung-ing breakers, the strongest forces result from the sudden flow reversal that occurs under the steep wave crest. Under surging breakers, the strongest forces result from seepage flows that occur towards the end of the downrush phase of the surface flow cycle. Col -lapsing breakers are particularly damaging to the armour layer because these two forcing mechanisms tend to occur simultaneously and reinforce each other. Changes in wave height have a strong influence on the magnitude of the wave-induced forcing of armour stones, but have a lesser effect on the temporal character of the forcing. In contrast to this, changes in wave period or structure slope can have a much stronger influence on the temporal character of the forcing, particularly over the range of surf similarity 2 < £ < 4 corresponding to the transition between plunging and collapsing breakers. The strong influences of wave period and structure slope are coupled to each other through their combined influence on the type of wave breaking that prevails. The initiation and early growth of damage is closely linked to the peak down-slope shear stress acting on the critical section of armour below the still waterline. The enhanced stability of armour stones on more permeable structures is due to a reduc-tion in the shear stresses acting on the armour. Friction factors have been developed and successfully applied to relate the slope-parallel component of the forcing on the armour layer to the velocity of the surface flows. The friction factor for rock armour on a rubble-mound structure under wave attack is a variable quantity that depends on relative roughness in a manner that is similar to that observed for oscillatory flow over a rough, impermeable, horizontal seabed. 282 • The peak forces acting on rock armour due to irregular wave attack are highly varied. Most of the variability is due to differences in the height and period of each wave, but some further variation results from additional factors including differences in the shape of each wave and the sequencing of successive waves. • The variation in peak forces due to irregular waves can be approximated by the Rayleigh distribution, which can be applied to generate rough estimates of extreme values, including the maximum forcing that can be expected to occur over a duration of random wave attack. • Overall, the wave-induced hydrodynamic forces acting to dislodge armour stones on a breakwater vary with wave height, wave period, slope angle and core permeability in a manner that is generally consistent with predictions of damage obtained from the stability design equations of van der Meer (1988). This agreement suggests that these equations provide a reasonable representation of the process whereby wave attack induces fluid flows which exert the forces responsible for damage, and provides a rational basis for recommending these equations for preliminary design of armour layers. 8.7 Recommendations for Further Study This thesis has focussed on the wave-induced hydrodynamic forces acting on sections of rock armour located primarily below the still waterline on several different rubble-mound structures in a variety of regular and irregular wave conditions, and the relationships between these forces and characteristics of the incident waves, properties of the structure, flows on the surface of the structure, and damage of the armour layer. The investigations presented here could be extended in several areas. • The concept of wave friction factor was originally developed to describe the maximum shear stress on a rough, flat, horizontal, impermeable bed under simple harmonic flow. 283 In this study, friction factors were developed and used to relate the magnitude of the shear stress observed on a section of rock armour to the complex oscillatory flow on the inclined permeable surface of a rubble-mound structure under wave attack. The armour panels developed for this study could be used in a more systematic investigation of friction factors for rock armour, including the influences of relative roughness, bed permeability, flow asymmetry, and varying flow depth. Seepage flows have been identified as an important forcing mechanism for armour stones on a rubble-mound. There is a need for further investigation of this process, including simultaneous measurements of forces and pressure gradients within the armour layer. The present investigations are focussed on rubble-mound structures of conventional de-sign, yet many real structures include non-conventional design elements such as composite slopes or re-shaping berms. The approach used in this study could be applied to inves-tigate the wave-induced flows, forcing and damage response of the armour on various non-conventional rubble-mound structures, including berm breakwaters or reef breakwa-ters. In fact, the armour panels have already been used in a separate study of the forcing due to seepage and overtopping flows on rock-fill dams. The performance of rubble-mound structures under non-perpendicular or even multidirec-tional wave attack is an important issue that is not addressed in the present investigation. To properly investigate these effects, the armour panel dynamometers used in this study would need to be re-designed to provide measurements of the forcing