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Hydrodynamics and risk analysis of iceberg impacts with offshore structures 1989
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Title | Hydrodynamics and risk analysis of iceberg impacts with offshore structures |
Creator |
McTaggart, Kevin Andrew |
Publisher | University of British Columbia |
Date Created | 2011-01-20T18:28:03Z |
Date Issued | 2011-01-20T18:28:03Z |
Date | 1989 |
Description | The evaluation of design iceberg impact loads for offshore structures and the influence of hydrodynamic effects on impact loads are examined. Important hydrodynamic effects include iceberg added mass, wave-induced oscillatory iceberg motions, and the influence of a large structure on the surrounding flow field and subsequent velocities of approaching icebergs. The significance of these phenomena has been investigated using a two-body numerical diffraction model and through a series of experiments modelling the drift of various sized icebergs driven by waves and currents approaching a large offshore structure. Relevant findings from the hydrodynamic studies have been incorporated into two probabilistic models which can be used to determine design iceberg collision events with a structure based on either iceberg kinetic energy upon impact or global sliding force acting on the structure. Load exceedence probabilities from the kinetic energy and sliding force models are evaluated using the second-order reliability method. Output from the probabilistic models can be used to determine design collision parameters and to assess whether more sophisticated modelling of various impact processes is required. The influence of the structure on velocities of approaching icebergs is shown to be significant when the structure horizontal dimension is greater than twice the iceberg dimension. As expected, wave-induced oscillatory motions dominate the collision velocity for smaller icebergs but have a negligible effect on velocity for larger icebergs. |
Subject |
Icebergs Offshore Structures Ice Mechanics -- Mathematical Models Sea Ice -- Mathematical Models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2011-01-20T18:28:03Z |
DOI | 10.14288/1.0050512 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/30733 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0050512/source |
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H Y D R O D Y N A M I C S A N D R I S K A N A L Y S I S O F I C E B E R G I M P A C T S W I T H O F F S H O R E S T R U C T U R E S By Kevin Andrew McTaggart B. Sc. (Civil Engineering) Queen's University at Kingston M . E. Sc. (Civil Engineering) University of Western Ontario A T H E S I S S U B M I T T E D I N 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 O F 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 S T U D I E S C I V I L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A November 1989 Â© Kevin Andrew McTaggart, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Civil Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract The evaluation of design iceberg impact loads for offshore structures and the influ- ence of hydrodynamic effects on impact loads are examined. Important hydrodynamic effects include iceberg added mass, wave-induced oscillatory iceberg motions, and the influence of a large structure on the surrounding flow field and subsequent velocities of approaching icebergs. The significance of these phenomena has been investigated using a two-body numerical diffraction model and through a series of experiments modelling the drift of various sized icebergs driven by waves and currents approaching a large offshore structure. Relevant findings from the hydrodynamic studies have been incorporated into two probabilistic models which can be used to determine design iceberg collision events with a structure based on either iceberg kinetic energy upon impact or global sliding force acting on the structure. Load exceedence probabilities from the kinetic energy and sliding force models are evaluated using the second-order reliability method. Output from the probabilistic models can be used to determine design collision parameters and to assess whether more sophisticated modelling of various impact processes is required. The influence of the structure on velocities of approaching icebergs is shown to be signifi- cant when the structure horizontal dimension is greater than twice the iceberg dimension. As expected, wave-induced oscillatory motions dominate the collision velocity for smaller icebergs but have a negligible effect on velocity for larger icebergs. 11 Table of Contents Abstract ii List of Tables x List of Figures xii Nomenclature xvii Acknowledgements xxv 1 Introduction 1 1.1 Significance of Icebergs in Offshore Operations 1 1.1.1 Icebergs at Hibernia 3 1.1.2 Iceberg Management Methods 3 1.2 Iceberg Impact Forces 4 1.2.1 Impact Force as a Function of Iceberg Kinetic Energy 5 1.3 Hydrodynamic Considerations for Iceberg Collisions 6 1.3.1 Oscillatory Iceberg Motions 8 1.3.2 Modification of Iceberg Motions by a Fixed Structure 9 1.3.3 Variation of Added Mass with Proximity to Structure 9 1.4 Risk Analysis of Iceberg Collisions 10 1.4.1 Selection of a-Suitable Design Event 10 1.4.2 Methods of Ranking Collision Severity 12 iii 1.4.3 Methods of Evaluating Exceedence Probabilities 13 1.5 Scope of the Present Investigation 14 1.5.1 Verification of Numerical Ice Mass Motion Model 14 1.5.2 Numerical Study of Iceberg Collision Hydrodynamics 15 1.5.3 Physical Modelling of Iceberg Drift Motions Near a Structure . . 15 1.5.4 Kinetic Energy Reliability Model 16 1.5.5 Impact Force Reliability Model 16 2 Numerical Iceberg Motion Model 17 2.1 General Framework of Model 17 2.2 Oscillatory Iceberg Motions 18 2.2.1 Theoretical Basis of Diffraction Solution 18 2.2.2 Definition of Co-ordinate Systems and Modes of Motion 20 2.2.3 Re-formulation of Velocity Potential 22 2.2.4 Solution of Diffraction and Radiation Potentials 24 2.2.5 Solution of Equations of Motion 28 2.3 Mean Drift Motions 33 2.3.1 Evaluation of Zero Frequency Added Mass 35 2.3.2 Water Drag Forces 36 2.3.3 Current Field Near a Large Structure of Arbitrary Shape 38 2.3.4 Current Field Near a Large Cylindrical Structure 40 2.3.5 Wave Drift Forces 41 2.4 Verification of Numerical Model 45 2.4.1 Wave-Induced Motions 45 2.4.2 Wave Drift Forces 50 2.4.3 Zero Frequency Added Masses 50 iv 2.4.4 Boundary Element Computation of Current Near Structure . . . . 53 3 Hydrodynamic Aspects of Iceberg Collisions 56 3.1 Modelling of Iceberg Geometry 56 3.2 Effective Iceberg Added Mass 58 3.2.1 Transient Motions of a Floating Body 58 3.2.2 Effective Added Mass for Iceberg Drift Motions 60 3.2.3 Effective Added Mass for Iceberg Collisions 61 3.2.4 Effective Added Mass Example Problem 67 3.3 Iceberg Hydrodynamic Coefficients in Open Water . 75 3.3.1 Wave Drift Force Coefficients and Surge RAO's 75 3.3.2 Zero Frequency Added Mass Coefficients 83 3.3.3 Infinite Frequency Added Mass Coefficients 84 3.3.4 Water Drag Coefficients 87 3.4 Hydrodynamic Interactions between Icebergs and a Large Structure . . . 89 3.4.1 Configuration for Modelling Hydrodynamic Interactions 90 3.4.2 Variation of Zero Frequency Added Mass 91 3.4.3 Variation of Infinite Frequency Added Mass 92 3.4.4 Influence of Structure on Current Driven Impact Velocities . . . . 92 3.4.5 Variation of Wave Drift Forces 94 3.4.6 Variation of Oscillatory Surge Motions 101 3.4.7 Implications of Hydrodynamic Interactions for Iceberg Collisions . 102 4 Physical Modelling of Iceberg Drift Near a Large Structure 103 4.1 Similitude Requirements 104 4.2 Description of Experiments 105 4.2.1 Towing Tank 105 v 4.2.2 Structure Model. . 107 4.2.3 Iceberg Models . . 107 4.2.4 Model Iceberg Drag Coefficients 109 4.2.5 Environmental Conditions for Drift Studies 110 4.2.6 Experimental Procedure 112 4.3 Digitizing of Iceberg Trajectories 114 4.3.1 Video Camera Configuration 115 4.3.2 Reduction of Video Co-ordinates 115 4.4 Micro-Computer Digitizing System 117 4.4.1 Description of Hardware 119 4.4.2 Description of Software 120 4.5 Error Analysis of Digitized Co-ordinates â€¢ â€¢ â€¢ â€¢ 121 4.5.1 Mouse Operation Error 122 4.5.2 Error Due to Variation of z Co-ordinate 124 4.5.3 Summary of Digitizing Errors . . 126 4.6 Variability of Environmental Conditions 126 4.7 Analysis of Observed Trajectories . 127 4.7.1 Reduction of Iceberg Co-ordinates Relative to Structure . . . . . 128 4.7.2 Plotting of Iceberg Trajectories 128 4.7.3 Estimation of Impact Velocity and Eccentricity 130 4.7.4 Error Estimates for Velocities and Eccentricities 135 4.7.5 Presentation of Results 136 4.8 Numerical Modelling of Wave Tank Experiments 137 4.8.1 Simplified Numerical Drift Model 152 4.8.2 Comparison of Numerical and Physical Models 157 4.8.3 Comparison of Open Water Velocities for Wave Driven Motions . 160 vi 4.8.4 Discrepancies between Physical and Numerical Models 162 5 Kinetic Energy Design Collision Event 164 5.1 Design Criteria for Iceberg Collisions 165 5.2 Description of Second-Order Reliability Method 166 5.2.1 Introduction of Failure Function 167 5.2.2 Solution in Standard Normal Space 168 5.2.3 Failure Probability for Linearized Failure Surface 168 5.2.4 Failure Probability for Curved Failure Surface 170 5.2.5 SORM Solution for Dependent, Non-Normal Variables 173 5.3 Formulation of Failure Function for Iceberg Kinetic Energy 175 5.3.1 Idealization of Iceberg Geometry 177 5.3.2 Iceberg Added Mass 178 5.3.3 Iceberg Velocity 178 5.3.4 Summary of Failure Function 181 5.4 Application to Selection of a Design Iceberg Collision 181 5.4.1 Exceedence Probabilities for a Single Collision 182 5.4.2 Extension to Multiple Collisions 183 5.5 Example Design Problem for Large Iceberg Population 185 5.5.1 Design Criterion 185 5.5.2 Statistical Distributions for Input Variables 186 5.5.3 Correlated Input Variables 188 5.5.4 Computation of Exceedence Probabilities 188 5.5.5 Parameters for Design Collision Event 189 5.5.6 Exceedence Probabilities without Wave-Induced Motions 190 5.6 Example Design Problem for Small Iceberg Population 191 vii 6 Impact Sliding Force Design Event 194 6.1 Iceberg Loading of Offshore Structures , 195 6.1.1 Ice Forces for Head-on Impact 195 6.1.2 Ice Forces for Eccentric Impact 197 6.1.3 Selection of Effective Ice Pressure 198 6.2 Failure Function for Iceberg Impact Force Reliability Model 202 6.3 Impact Velocity Phase of Reliability Model 204 6.3.1 Equations of Drift Motion 205 6.3.2 Initial Conditions for Simulated Drift Trajectories 205 6.3.3 Hydrodynamic Coefficients for Drift Motions 207 6.3.4 Variation of Water Velocity 207 6.3.5 Iceberg Impact Velocity 208 6.4 Impact Force Model 208 6.4.1 Development of Impact Forces 209 6.4.2 Collision Geometry 209 6.4.3 Pressure Calculation Method 211 6.4.4 Iceberg Equations of Motion 212 6.4.5 Time-Stepping Procedure 213 6.5 Example Problem for Large Iceberg Population 215 6.5.1 Design Criterion and Input Distributions 215 6.5.2 Design Event for Constant Ice Pressure 217 6.5.3 Design Event for Varying Ice Pressure with Area 219 6.6 Example Problem for Small Iceberg Population 220 6.6.1 Design Collision for Constant Ice Pressure 221 6.6.2 Design Collision for Varying Ice Pressure 222 V l l l 7 Conclusions 224 7.1 Effective Added Mass 224 7.2 Oscillatory Surge Motions 225 7.3 Influence of Structure on Iceberg Hydrodynamics 226 7.4 Kinetic Energy Reliability Model 228 7.5 Impact Force Reliability Model 228 7.6 Recommendations for Future Research 229 Bibliography 231 ix List of Tables 2.1 Hydrodynamic Properties of Barge 46 3.1 Input Parameters for Comparison of Ramp and Time Stepping Models . 71 4.1 Specifications for BC Research Towing Tank â€¢ . 107 4.2 Dimensions of Model Icebergs 109 4.3 Environmental Conditions for Model Tests I l l 4.4 Number of Tests for Each Model and EnvironmentalCondition 113 4.5 Occurrence of Impacts for Experiments 130 4.6 Errors between Numerical and Physical Models for Collision Eccentricity 159 4.7 Errors between Numerical and Physical Models for Velocity Ratio . . . . 159 4.8 Comparison of Open Water Drift Velocities for First Wave Condition . . 161 4.9 Comparison of Open Water Drift Velocities for Second Wave Condition . 161 5.1 Input Distributions for Example Design Problem 186 5.2 Distribution Moments for All Icebergs and Impacting Icebergs 187 5.3 Results for Large Iceberg Design Collision 190 5.4 Results for Small Iceberg Design Collision 193 6.1 Input Distributions for Large Iceberg Example Design Problem 216 6.2 Results for Large Iceberg Collision, Constant Ice Pressure 218 6.3 Results for Large Iceberg Collision, Pressure-Area Variation 220 6.4 Results for Small Iceberg Collision, Constant Ice Pressure 222 x Results for Small Iceberg Collision, Pressure-Area Variation xi List of Figures 1.1 Major Iceberg Drift Paths 2 1.2 Schematic of an Iceberg Collision with a Fixed Structure 7 1.3 Boundary Element Grid for Three-Dimensional Solution, Elevation View 11 1.4 Boundary Element Grid for Two-Dimensional Solution, Plan View . . . . 11 2.1 Co-ordinate Systems for Iceberg Motion Model 21 2.2 Control Surface for Far-field Wave Drift Force Computation 43 2.3 Barge for Verifying Diffraction Computations 46 2.4 Barge Surge Motions in Head Waves 47 2.5 Barge Heave Motions in Head Waves . . . 47 2.6 Barge Pitch Motions in Head Waves 48 2.7 Barge Sway Motions in Beam Waves 48 2.8 Barge Heave Motions in Beam Waves 49 2.9 Barge Roll Motions in Beam Waves 49 2.10 Barge Wave Drift Forces in Head Waves 51 2.11 Barge Wave Drift Forces in Beam Waves 51 2.12 Zero Frequency Added Masses for Vertical Circular Cylinders 52 2.13 Zero Frequency Added Masses Approaching Structure 54 2.14 Current Velocity in Vicinity of Structure 55 3.1 Frequency Dependent Added Mass and Damping for Example Iceberg . . 68 3.2 History Term for Example Iceberg 69 xii 3.3 Effective Added Mass for Example Iceberg, Ramp Forcing Function . . . 70 3.4 Collision Duration for Ramp and Time Stepping Models . 73 3.5 Effective Added Mass for Ramp and Time Stepping Models 73 3.6 Impact Force for Ramp and Time Stepping Models 74 3.7 Ice Penetration for Ramp and Time Stepping Models 74 3.8 Computed Surge RAO's, Aspect Ratio = 0.1 . 78 3.9 Computed Surge RAO's, Aspect Ratio = 0.2 79 3.10 Computed Surge RAO's, Aspect Ratio = 0.4 79 3.11 Computed Surge RAO's, Aspect Ratio = 0.6 80 3.12 Computed Surge RAO's, Aspect Ratio = 0.8 80 3.13 RAO's from Numerical Model and High Frequency Approximation 83 3.14 Computed Wave Drift Coefficients, Aspect Ratio = 0.1 84 3.15 Computed Wave Drift Coefficients, Aspect Ratio = 0.2 85 3.16 Computed Wave Drift Coefficients, Aspect Ratio = 0.4 85 3.17 Computed Wave Drift Coefficients, Aspect Ratio = 0.6 86 3.18 Computed Wave Drift Coefficients, Aspect Ratio = 0.8 86 3.19 Computed Zero Frequency Added Masses 87 3.20 Computed Infinite Frequency Added Masses 88 3.21 Variation of Drag Coefficient with Aspect Ratio 89 3.22 Zero Frequency Added Mass near Structure 92 3.23 Infinite Frequency Added Mass near Structure 93 3.24 Relative Impact Velocities for Current Driven Head-on Collisions 94 3.25 Variation of Wave Drift Coefficients with Proximity to Structure 95 3.26 Influence of Wave Period on Wave Drift Forces near Structure 96 3.27 Open Water Wave Drift Force Coefficient 97 3.28 Wave Drift Force Variation Amplitude Coefficient 97 xiii 3.29 Wave Drift Force Variation Decay Coefficient 98 3.30 Wave Drift Force Variation Phase Coefficient 98 3.31 Influence of Aspect Ratio on Wave Drift Forces near Structure 99 3.32 Influence of Lateral Offset on Wave Drift Forces near Structure 100 3.33 Variation of Oscillatory Surge Motions near Structure 101 4.1 Elevation View of BC Research Towing Tank 108 4.2 Elevation View of Structure and Iceberg Models 108 4.3 Drag Coefficients for 400 mm Diameter Iceberg Model 110 4.4 Overhead View of 400 mm Iceberg Approaching Structure 114 4.5 Elevation View of Video Image Planes 117 4.6 xâ€”y and x'-y' Co-ordinate Systems and Relevant Dimensions 118 4.7 Schematic of Video Digitizing System 119 4.8 Elevation View of Reference Distance and Corresponding Image Distance 124 4.9 Observed Iceberg Trajectory and Displacement vs. Time Graphs 129 4.10 Observed Trajectory for Wave Condition 1, Model 1 131 4.11 Far-field and Near-field Eccentricities of Approach for an Iceberg 133 4.12 Iceberg Eccentricity and Velocity, Currents, Model 1 â€¢â€¢â€¢ â€¢ 4.13 Iceberg Eccentricity and Velocity, Currents, Model 2 139 4.14 Iceberg Eccentricity and Velocity, Currents, Model 3 140 4.15 Iceberg Eccentricity and Velocity, Currents, Model 4 141 4.16 Iceberg Eccentricity and Velocity, Wave Condition 1, Model 3 142 4.17 Iceberg Eccentricity and Velocity, Wave Condition 1, Model 4 143 4.18 Iceberg Eccentricity and Velocity, Wave Condition 2, Model 3 144 4.19 Iceberg Eccentricity and Velocity, Wave Condition 2, Model 4 145 4.20 Iceberg Eccentricity and Velocity, Wave-Current Condition 1, Model 1 . . 146 xiv 4.21 Iceberg Eccentricity and Velocity, Wave-Current Condition 1, Model 2 . . 147 4.22 Iceberg Eccentricity and Velocity, Wave-Current Condition 1, Model 3 . . 148 4.23 Iceberg Eccentricity and Velocity, Wave-Current Condition 2, Model 1 . . 149 4.24 Iceberg Eccentricity and Velocity, Wave-Current Condition 2, Model 2 . . 150 4.25 Iceberg Eccentricity and Velocity, Wave-Current Condition 2, Model 3 . . 151 4.26 Wave Drift Forces near Structure, No Current, 0.7 s Wave Period . . . . 153 4.27 Wave Drift Forces near Structure, No Current, 1.0 s Wave Period . . . . 153 4.28 Wave Drift Forces near Structure, Current, 0.7 s Wave Period 154 4.29 Wave Drift Forces near Structure, Current, 1.0 s Wave Period 154 4.30 Open Water Wave Drift Force Coefficients 155 4.31 Amplitude Coefficient for Variation of Wave Drift Forces 155 4.32 Decay Coefficient for Variation of Wave Drift Forces 156 4.33 Phase Coefficient for Variation of Wave Drift Forces 156 5.1 Standard Space for 2 Random Variables 169 5.2 Influence of Curved Failure Surface 171 5.3 Flow Chart for Second-Order Reliability Method 176 5.4 Kinetic Energy Exceedence Probabilities for a Single Collision 189 5.5 Comparison of SORM and Analytical Model Results 191 5.6 Kinetic Energy Exceedence Probabilities for Small Iceberg Example . . . 192 6.1 Ice Indentation for Korzhavin's Experiments 200 6.2 Pressure vs. Area Measurements for Ice Strength 202 6.3 Initial Conditions for Iceberg Drift Trajectory toward Structure 206 6.4 Eccentric Iceberg Collision Schematic, Plan View . . . 210 6.5 Large Iceberg Force Exceedence Probabilities, Constant Ice Pressure . . . 217 6.6 Large Iceberg Force Exceedence Probabilities, Pressure-Area Variation . 219 xv 6.7. Small Iceberg Force Exceedence Probabilities, Constant Ice Pressure . . . 221 6.8 Small Iceberg Force Exceedence Probabilities, Pressure-Area Variation . 223 xvi Nomenclat ur e a = acceleration (m/s2) or cylinder radius (m) A(u>) = Fourier transform of acceleration [A] = matrix for solution of wave source strengths [Ac] = matrix for solution of current source strengths Acw wave drift variation amplitude coefficient AF, BF, CF - coefficients for approximating iceberg trajectory in far-field AN,BN, CN â€” coefficients for approximating iceberg trajectory in near-field An = normal contact area between ice and structure (m2) At tangential contact area between ice and structure (m2) Aw water cross-sectional area of iceberg (m2) Axup = iceberg waterplane area (m2) [B] = matrix for solution of wave source potentials [Bc] â€” matrix for solution of current source potentials Bcw â€” wave drift variation decay coefficient (m - 1) [c] = hydrodynamic stiffness matrix â€” wave drift variation phase coefficient (m) cd = water drag coefficient Cj = added inertia coefficient CM added mass coefficient CW = wave drift force coefficient c ^wo = wave drift force coefficient in open water xvii d â€” water depth (m) [D'\ = diagonal matrix of effective standard deviations Di - iceberg diameter (m) Ds â€” structure diameter (m) E{x*) expected value of xr ec = eccentricity of iceberg impact (m) < = dimensionless impact eccentricity eF - approach eccentricity in far-field (m) = approach eccentricity in near-field (m) to - initial eccentricity of approach (m) < = dimensionless approach eccentricity F = iceberg impact force (N) {F>} vector of unit exciting forces Fa = air drag force (N) Fc - Coriolis force (N) Fc = maximum iceberg impact force (N) Fcr ice crushing force (N) Fd = water drag force (N) fe = forcing function (s) = Fourier transform of fe {Fe} = excitation force vector Fj = friction force (N) {/fc} = source strength vector for potential <j>k (dimesionless or m) FL = impact force load (N) FR â€” structure impact force resistance (N) xvm Ft = tangential force acting on iceberg (N) Fw = wave drift force (N) fx(X) = PDF for variable X FX(X) = CDF for variable X â€”â€¢ â€”Â¥ fx"(X) = joint probability density function of X Fx, Fy â€” x â€” y force components acting on iceberg (N) G = Green's function (m _ 1) or failure function G* = modified Green's function (m - 1 ) G(x',y', z') = co-ordinate system with origin at iceberg centre of gravity â€”â€¢ G(X) = failure function {G'(Y*)} = vector of first -order partial derivatives of G [G"(Y*)] = matrix of second-order partial derivatives of G h = iceberg draft (m) H â€” wave height (m) {H} = memory effect vector [H] = transformation matrix ht = total vertical dimension of iceberg (m) Hrms â€” root-mean-square wave height (m) Ha = significant wave height (m) I = indentation factor [7] = identity matrix Iij = iceberg moment or product of inertia (kg-m2) Iw = iceberg rotational inertia (kg-m2) Jo = Bessel function of first kind, order 0 k = wavenumber (m - 1 ) or contact factor xix = modified Bessel function of second kind, order 0 kp = ice force constant (N/m1/2) ks = video image scale factor (/pixel) KE - kinetic energy of impacting iceberg (Joules) L = distance from camera to x' â€” y' plane (m) L - history term (kg/s, kg-m/s, or, kg-m2/s ) Lt - distance from camera to x â€” y â€” z origin (m) M iceberg mass (kg) m = indentor shape factor M = iceberg mass matrix {M>} â€” vector of effective means Mv - virtual iceberg mass (kg) n = unit normal vector nc - number of collisions Ni - number of facets on iceberg surface = unit normal velocity for mode k â€” number of iceberg collisions during structure lifetime Nrv â€” number of input random variables Ns â€” number of facets on structure surface = directional cosines of normal vector pointing outward from body 0(x,y,z) = co-ordinate system with origin at centre of structure at the waterline Per â€” ice crushing pressure (N/m2) Pr = reference ice pressure (N/m2) Rij = element of original correlation matrix Roij element of modified correlation matrix Rp = structure force resistance (N) RKE â€” structure kinetic energy resistance (Joules) rti = moment arm for frictional torque acting on iceberg (m) sn = iceberg normal displacement relative to structure (m) snc = maximum normal displacement relative to structure (m) st â€” iceberg tangential displacement relative to structure (m) stc = maximum tangential displacement relative to structure (m) T = wave period (s) [T] = transformation matrix tc = collision duration (s) TC = wave period in presence of a current (s) TL = operating life for structure (years) TP = peak wave period TT = rotational torsion acting on iceberg (N-m) u,v = iceberg velocity components (m/s) ut = tangential velocity of iceberg face relative to structure face (m/s) Uw,vw = water velocity components (m/s) Vc = iceberg impact velocity (m/s) Vd = iceberg drift velocity (m/s) Vf = iceberg translational velocity after collision (m/s) VFXIVFV = far-field velocity components (m/s) Vn = velocity component in direction of vector component n (m/s) Vn = iceberg velocity normal to structure (m/s) VNX,VNV = near-field velocity components (m/s) Vr = iceberg velocity relative to current (m/s) xxi Va = iceberg significant oscillatory surge velocity (m/s) Vw = water velocity (m/s) Vwo = incident current velocity (m/s) {Y} = transformed vector of independent, standard normal variables {Y*} = design point in standard normal YQ = Bessel function of second kind, order 0 x = location of point in fluid domain {X} â€” vector of random variables {X*} = design point x"i y" = video image co-ordinates (pixels) xF,yF â€” iceberg co-ordinates in far-field (m) g,yg,Zg = location of iceberg centre of gravity relative to structure (m) %N,VN = iceberg co-ordinates in near-field (m) {Z} = random vector in transformed space z'b = z' co-ordinate of iceberg centre of buoyancy (m) ct = incident wave direction (Â°) {a} = vector of sensitivity coefficients j3 = reliability index or incident current direction (Â°) [r0] = transformation matrix accounting for correlation of variables A = iceberg penetration distance during collision (m) A c = maximum penetration distance (m) 8ij = correlation correction factor 8k = phase angle for mode k (Â°) AS = facet area (m2) = change in iceberg velocity in response to forcing function (s2/kg) xxii A y , = distance between reference points for evaluating ks (m) A y ; = image distance between reference points (pixels) e" = error in mouse digitizing position (pixels) < image error in scale reference distance (pixels) â€” error in reduced x co-ordinate (m) ty = error in reduced y co-ordinate (m) = error in z co-ordinate (m) G = amplitude of oscillatory motion for mode k (m or rad) a - response amplitude operator for mode k (m/m or /m) = vertical displacement of water surface (m) e = angle of camera orientation from vertical (Â°) Oc = contact point angle between iceberg and structure (rad) Or = iceberg rotational displacement (rad) A wavelength (m) [A] radiation damping matrix Ac = mean rate of iceberg collisions (/year) A L â€” length scale A, = time scale [A,] = viscous damping matrix = added mass matrix Pf = ice friction coefficient Pl = r'th moment of random variable x V - kinematic viscosity of water (m 2/s) i location of source on iceberg or structure surface [p] = correlation coefficient matrix xxiii p - water density (kg/m 3) Pi = ice density (kg/m3) a standard deviation o-c = uni-axial compressive strength of ice (N/m2) ax = standard error of reduced x co-ordiante (m) standard error of reduced y co-ordiante (m) drift velocity modification factor = oscillatory velocity modification factor $ = wave field velocity potential (m2/s) <t> complex velocity potential (m2) = water current velocity potential (m2/s) diffracted current potential (m2/s) *Â« incident current potential (m2/s) <f>k potential due to unit motions in mode k (m/m or /m) <f>(x) probability density function for standard normal distribution = cumulative distribution function for standard normal distribution â€” incident wave potential (m) </>7 = diffracted wave potential (m) u = angular wave frequency (rad/s) = iceberg rotatial velocity after collision (rad/s) uc = angular wave frequency in current (rad/s) Up peak angular wave frequency (rad/s) rotational velocity of iceberg (rad/s) xxiv Acknowledgement s I would like to express my appreciation to several people for their help during the course of my research. I am most grateful to the members of my advisory committee for their constructive comments. The co-operation and assistance of the staff of the BC Research Ocean Engineering Centre during the experimental program is greatly appreci- ated. I am also thankful to my various graduate student colleagues, particularly to those who helped with the experimental work at BC Research. The assistance of Professor Greg Lawrence with development of the micro-computer digitizing system is gratefully â€¢ acknowledged. Most importantly, I would like to thank my research advisor Professor Michael Isaacson for his enthusiastic supervision and careful guidance throughout the duration of my work. Financial support for this study was provided by the Natural Sciences and Engineer- ing Research Council of Canada. xxv Chapter 1 Introduction The quest for new petroleum reserves has led to exploration and production in off- shore regions. As the search for new reserves continues, increasingly hostile environments are being encountered. Structural designers are being faced with the combined challenges of high loads and limited previous experience in such regions. Among the various envi- ronmental loads encountered, ice loads originating from landfast or drifting ice are among the most severe in many polar and sub-polar regions. The prediction of ice loads is very difficult due to the complexities of ice mass dynamics, collision mechanics, and mate- rial properties. The inherent uncertainties regarding ice loading necessitates the use of probabilistic methods for the development of rational structural design criteria. 1.1 Significance of Icebergs in Offshore Operations Icebergs present one of the most severe threats to offshore installations in sub-polar regions, particularly off the coast of Newfoundland. The majority of icebergs are calved from glaciers on the coast of Greenland and are left to drift freely under the influence of winds, currents, and waves. Major, iceberg drift patterns around Greenland and the eastern coast of Canada are illustrated in Figure 1.1. 1 Chapter 1. Introduction 2 Figure 1.1: Major Iceberg Drift Paths (from Mobil [80]) Chapter 1. Introduction 3 1.1.1 Icebergs at Hibernia Among the areas most susceptible to icebergs is the Hibernia oilfield, whose location illustrated in Figure 1.1 is among major drift tracks. Annual records of the number of icebergs reaching 48Â° N have been compiled by the International Ice Patrol (IIP) and indicate that an average of 379 icebergs reached 48Â° N annually between 1900 and 1979 [62]. The average iceberg mass in the Grand Banks region is estimated to be approximately 0.6 million tonnes [80]. 1.1.2 Iceberg Management Methods When operating in iceberg infested waters, an appropriate iceberg management sys- tem is essential. The following three options are available for reducing the risk of iceberg collision damage: 1. Iceberg towing, 2. Iceberg avoidance, 3. Iceberg destruction, 4. Utilization of a structure capable of withstanding iceberg impacts. These four options will now be discussed in greater detail. Iceberg towing can be accomplished using one or more tugs [10]. One of the most difficult aspects of iceberg towing is the selection of an appropriate direction for applying a towing force. An incorrectly applied towing force can increase the risk of an iceberg collision. Adverse weather phenomena such as fog and heavy seas greatly reduce the feasibility of towing operations. Because of its limitations, iceberg towing should not be the sole component of an iceberg management system. Chapter 1. Introduction 4 Iceberg avoidance action in the form of site evacuation is often taken when an iceberg is approaching a semi-submersible drilling unit. A lead time of several hours is usually required for a semi-submersible to terminate drilling operations and to evacuate a site. Due to the unpredictability of iceberg motions, evacuation measures are often taken even when the probability of collision is relatively small. Iceberg destruction can be accomplished using explosives placed on an iceberg. Due to the inherent instability of many icebergs, the placement of explosives on an iceberg can pose great danger to a landing crew. Like iceberg towing, iceberg destruction oper- ations can be interrupted by adverse weather conditions. Another drawback of iceberg destruction is the possibility of of a large iceberg being broken into a small number of large segments, thus resulting in no effective reduction of the risk of iceberg impact. The remaining alternative is to build a structure capable of withstanding all iceberg impacts during its lifetime. The capability of withstanding iceberg collisions would prob- ably not be economically justifiable for a semi-submersible unit; however, such capability would be very desirable for a fixed production structure. The construction of an iceberg resistant gravity based production structure for operation off the coast of Newfoundland has been proposed by Mobil Oil Company [80]. 1.2 Iceberg Impact Forces When evaluating iceberg collisions, the structural designer is primarily interested in the maximum impact force experienced by the structure. A representative iceberg collision is illustrated in Figure 1.2. Several methods are available for calculating iceberg impact loads [18], and are discussed in greater detail in Chapter 6. Each method is based upon the following equation: F = p c r (A) A n ( A ) (1.1) Chapter 1. Introduction 5 where: An = normal contact area (m2) F = impact force (N) pcr â€” ice crushing pressure (N/m2) A = penetration distance (m) The crushing pressure pcr is a function of many variables, including ice compressive strength and the geometry of the collision. Determination of the normal contact area An is relatively simple given the geometry of the structure and ice mass. 1.2.1 Impact Force as a Function of Iceberg Kinetic Energy Various methods are available for calculating the maximum impact load experienced during a collision, as discussed by Croasdale [19]. For icebergs impacting a large structure, the limit momentum approach can be used to equate the kinetic energy of the impacting iceberg with the energy dissipated during the collision and to predict the maximum impact load. For a direct iceberg impact, the following relationship will exist: \ (1 + Cm) M Vc2 + [Ac Fe(A) d A = fAc F(A) d A (1.2) I Jo Jo where: Cm â€” iceberg added mass coefficient Fe = external environmental force acting on iceberg (N) F = impact force (N) M = iceberg mass (kg) Vc = impact velocity (m/s) A c = maximum penetration distance (m) Chapter 1. Introduction 6 The external force Fe acting on the ice mass will include components from ocean currents, wind, waves, and Coriolis force. Hydrodynamic added mass is excluded from Fe, and is incorporated in the term Cm. Because of the small magnitude of external forces relative to impact forces, the external force term is usually neglected in equation 1.2. If the iceberg were to roll from a metastable equilibrium to a stable equilibrium during a collision, then additional kinetic energy would be available to contribute to impact forces acting on the structure. For eccentric iceberg collisions, the amount of kinetic energy dissipated during impact can be significantly reduced. 1.3 Hydrodynamic Considerations for Iceberg Collisions Of the variables given in Equation 1.2, iceberg added mass and impact velocity will be primarily influenced by hydrodynamic effects. Although hydrodynamic effects have been largely neglected in previous studies, they can play a significant role in iceberg impacts [17,46,47,60]. The following three effects can influence the severity of iceberg collisions: 1. Wave-induced oscillatory iceberg motions, 2. Modification of iceberg motions by the presence of a large structure, 3. Variation of iceberg added mass with proximity to structure. These phenomenon will now be discussed in greater detail. Chapter 1. Introduction 7 Plan View Elevation View i t h hi Iceberg Structure / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Figure 1.2: Schematic of an Iceberg Collision with a Fixed Structure Chapter 1. Introduction 8 1.3.1 Oscillatory Iceberg Motions Velocity input values for iceberg collision models are usually obtained from hourly observations of iceberg positions (e.g. [6]), thus giving mean drift velocities. Ideally, instantaneous iceberg velocities should be used in iceberg collision models. Although full-scale instantaneous velocity observations are very difficult to obtain, oscillatory mo- tions can be computed relatively easily using numerical diffraction models. The problem of estimating maximum iceberg velocity is greatly simplified if mean and oscillatory ve- locities are assumed to be collinear. This assumption is quite reasonable considering the degree of interaction between winds, waves, and currents. The horizontal iceberg velocity will fluctuate during a given wave cycle, and will have a maximum value as follows: where: V = maximum velocity during wave cycle (m/s) Vd = drift velocity (m/s) Va = amplitude of oscillatory surge velocity (m/s) For an iceberg in regular waves, the surge velocity amplitude is easily expressed as: where: H â€”â€¢ wave height (m) ([ = surge response amplitude operator u â€” angular wave frequency (s _ 1) The significance of wave-induced iceberg motions has been illustrated by Lever and Sen [60] and Salvalaggio and Rojansky [99]. Lever and Sen report that the kinetic energy V = Yd + Vs (1.3) (1.4) Chapter 1. Introduction 9 of a 10,000-100,000 tonne iceberg can be increased by a factor of 3-6 when the wave- induced oscillatory velocity is added to the mean drift velocity. 1.3.2 Modification of Iceberg Motions by the Presence of a Fixed Structure Due to the large dimensions of a gravity based structure, the incident flow field will be modified significantly by its presence. It is likely that the zero normal flow boundary condition at the structure will reduce the velocity of impacting icebergs. Several inves- tigations have examined iceberg motions in the vicinity of a large structure. Hay and Company [35], Isaacson [43], and Isaacson and Dello Stritto [48] used a numerical poten- tial flow model to predict wave and current induced iceberg motions. References [35] and [48] indicate that negative wave drift forces near a large structure can prevent smaller icebergs from impacting. NORDCO [90] examined wave-induced iceberg motions in the presence of a gravity based structure. Both numerical and physical models were used to examine motions for icebergs of similar size to the structure. It was found that ice- berg motions were not significantly affected by the presence of the structure. A series of experiments was conducted by Salvalaggio and Rojansky [99] to examine wave-induced motions in the vicinity of a structure. They reported that no impact occurred if the iceberg length was less than approximately half the structure length, regardless of the incident wavelength. 1.3.3 Variation of Added Mass with Proximity to Structure Iceberg added mass is among the various factors contributing to the severity of collisions with an offshore structure. Application of equation 1.2 requires knowledge of iceberg added mass at the instant of impact. Investigations by Isaacson and Cheung Chapter 1. Introduction 10 [46,47], have examined the variation of iceberg added mass with proximity to a fixed structure. Their results indicate that iceberg added mass can increase significantly as the iceberg approaches the structure. Iceberg added mass is usually obtained using a form of the boundary element method. Isaacson [43] used a three-dimensional diffraction computation to obtain iceberg added mass in the vicinity of a structure. A typical three-dimensional boundary element discretization is illustrated in Figure 1.3. This method was restricted by numerical diffi- culties as the separation distance between the iceberg and structure became less than a typical facet dimension. An alternative two-dimensional boundary element method solution has been devel- oped by Isaacson and Cheung [46,47]. A two-dimensional discretization such as that shown in Figure 1.4 allows for the use of much smaller facet dimensions. Extension of two-dimensional results to three dimensions is done using correction factors from Yeung [119]. 1.4 Risk Analysis of Iceberg Collisions . Due to the complex nature of the offshore environment, probabilistic methods should be utilized for the development of structural design criteria. Output from a risk analysis should ideally give the probability of structural failure over a given period of time (e.g. 1 year or the life of the structure). 1.4.1 Selection of a Suitable Design Event When designing a structure to withstand iceberg collisions, it is necessary to select a suitable design event that the structure should be able to withstand [24]. The design event should have an associated impact severity level with a low probability of exceedence Chapter 1. Introduction 11 Iceberg Structure / / / / / / / / / / / / / / / / / / / / / / / / / / / v ]> / r; v v'/ /, < > ? ? Figure 1.3: Boundary Element Grid for Three-Dimensional Solution, Elevation View Figure 1.4: Boundary Element Grid for Two-Dimensional Solution, Plan View Chapter 1. Introduction 12 and should be described by the most likely combination of parameter values for the given severity level. Typical collision parameters will include the following: 1. Iceberg dimensions, 2. Impact velocity, 3. Eccentricity of impact. 1.4.2 Methods of Ranking Collision Severity The parameters of a design event will not only depend on the specified exceedence probability, but also on the method of ranking collision severity. For iceberg collisions â€¢with a large structure, the following criteria can be used for ranking collision severity: 1. Kinetic energy of impacting iceberg, 2. Kinetic energy dissipated during collision, 3. Maximum load experienced during collision. Iceberg kinetic energy is the simplest criterion for ranking impact severity. Although due consideration must be given to hydrodynamic effects, impact load computations are not required when determining iceberg kinetic energy. Ranking of collision severity accord- ing to dissipated kinetic energy is a more sophisticated method which can account for the influence of eccentric impacts. Both hydrodynamics and collision mechanics must be evaluated when determining kinetic energy dissipated during a collision. Ranking of collision severity according to maximum impact load requires the greatest degree of com- putational effort. Ideally a sophisticated model which incorporates both hydrodynamics Chapter 1, Introduction 13 and collision mechanics should be utilized. The primary advantage of load ranking ac- cording to impact force is that impact force is usually the quantity of principal interest to the structural designer. 1.4.3 Methods of Evaluating Exceedence Probabilities When evaluating the probability of exceedence for a given collision severity level, one of the following three approaches can be used: 1. Analytical methods, 2. Monte Carlo simulation, 3. First or second order reliability methods (FORM or S0RM) Application of analytical methods is mainly limited to relatively simple systems involving few random variables. Probability density functions are described by uni-variate distributions for uncorrelated variables or by multi-variate distributions for correlated variables. With the advent of computers, Monte Carlo methods have proven useful for the analysis of complex systems. Monte Carlo simulation utilizes random number generators to produce variables having specified statistical distributions, and is discussed extensively in references by Ang and Tang [2] and Naylor et al. [84]. A major disadvantage of Monte Carlo simulation is the large amount of computational effort required, particularly when evaluating small probabilities of exceedence. The first and second order reliability methods (FORM and SORM) are based on work originally proposed by Hasofer and Lind [34]. Detailed descriptions of reliability methods are given in reference books by Ang and Tang [2], Madsen et al. [67], and Thoft- Christensen and Baker [110]. Of key importance in the application of reliability methods Chapter 1. Introduction 14 is the concept of a failure function which is defined such that a value less than 0 indicates structural failure. Thus, the failure function can appear as follows: where: G = failure function L = structural load R = structural resistance X â€” vector of random variables Using Equation 1.5, the probability of failure for a given structure can be calculated using an efficient computer algorithm presented by Rackwitz and Fiessler [97]. If the exceedence probability for a given load level is required, the structure resistance R can be treated as a deterministic quantity representing the given load level. 1.5 Scope of the Present Investigation The present investigation examines the various hydrodynamic aspects of iceberg collisions with a large structure and their incorporation into probabilistic models for the evaluation of design iceberg collision events. The various phases of investigation are as follows. 1 .5 .1 Verification of Numerical Ice Mass Motion Model A numerical model developed by Isaacson [43] has been used for modelling ice mass motions in the vicinity of a fixed structure. The model utilizes potential flow theory to compute oscillatory and mean drift motions. The objective of the initial phase of G{X) = R - L (1.5) Chapter 1. Introduction 15 investigation was to verify the reliability of the computational model through comparison with published results from other sources. 1.5.2 Numerical Study of Iceberg Collision Hydrodynamics The ice mass motion model has been used to examine hydrodynamic effects for various collision scenarios. Areas of interest include the following: 1. Prediction of iceberg oscillatory velocities in open water, 2. Variation of mean and oscillatory iceberg velocities with proximity to structure, 3. Effective added masses for drift motions and collisions, 4. Variation of iceberg added mass with proximity to structure. A gravity based structure with a diameter of 100 m situated in a water depth of 100 m has been used as a representative large structure for computations. 1.5.3 Physical Modelling of Iceberg Drift Motions Near a Large Structure A review of published experimental work revealed that little physical modelling data were available for validating the numerical iceberg drift model. Consequently, an experimental investigation has been conducted to examine iceberg drift motions near a large structure. Drift motions driven by both currents and waves have been modelled in a series of towing tank tests. The influence of the structure on iceberg velocities and approach eccentricities has been examined and compared with results of the numerical drift model. Chapter 1. Introduction 16 1.5.4 Kinetic Energy Reliability Model A probabilistic model for determining an iceberg collision design event based on iceberg kinetic energy has been developed. The use of kinetic energy as a ranking criterion allows for the evaluation of a design event without computing impact forces. The model has been formulated such that statistical distributions for input variables can be obtained from available environmental data. A key aspect of the model is its inclusion of wave- induced oscillatory motions, which are evaluated using response amplitude operators obtained from diffraction computations. Output from the model can be used to assess the importance of iceberg geometry and wave-induced motions on iceberg kinetic energy. 1.5.5 Impact Force Reliability Model A second reliability model, based on iceberg impact force, has been formulated. It is likely that such an impact force reliability model would be used in the actual design of an offshore structure. The influence of the structure on drift motions of approaching icebergs is considered along with the variation of iceberg approach eccentricities. The relative importance of hydrodynamic effects compared to other phenomena, such as the variability of ice strength, can-be examined using the impact force reliability model. Chapter 2 Numerical Iceberg Motion Model A major objective of the present investigation was to examine the importance of hydrodynamic effects for iceberg collisions. The numerical model ICEM0T developed by Isaacson [43,48] has been used to model iceberg motions in the vicinity of a large structure, and to predict relevant collision parameters. This chapter describes the main features and computational methods of ICEM0T. 2.1 General Framework of Model Like previous numerical drift models [6,40], the present model utilizes a time-stepping procedure to compute iceberg drift motions driven by waves and currents. Due to the variation of currents and waves with proximity to the structure, wave drift forces and water drag forces can be re-evaluated at specified time intervals. Computations of oscil- latory motion amplitudes can be also made at specified time intervals. Small amplitude motions are assumed when computing oscillatory motions; thus, the variation in iceberg proximity to the structure during a single wave period is neglected. 17 Chapter 2. Numerical Iceberg Motion Model 18 2.2 Oscillatory Iceberg Motions The problem of predicting oscillatory iceberg motions is highly dependent on the size of the iceberg relative to the incident wavelength. A series of experiments by Lever et al. [59] indicated that an iceberg follows the motion of a fluid particle when the iceberg diameter to wavelength ratio Di/X is less than 1/13. Under such conditions, oscillatory motions can be determined merely' from a knowledge of the incident wave field. For larger icebergs (i.e. D./A > 1/13), the wave field is modified by the presence of the iceberg to such a degree that the diffracted wave field must be computed for accurate prediction of oscillatory motions. The diffraction problem is further complicated when a large structure is present in the wave field. The structure can have a significant effect on iceberg motions if its size is large relative to the iceberg. It is necessary to consider the presence of both the iceberg and the structure when solving for the diffracted wave field, which will also vary with the location of the iceberg relative to the structure. Due to the inherently large size of icebergs, diffraction effects can be very significant. It is likely that a petroleum production structure built in iceberg-infested waters would also be large enough to cause significant diffraction. Consequently, the numerical model utilizes a two-body diffraction solution to determine oscillatory motion amplitudes of an iceberg near a large structure. 2.2.1 Theoretical Basis of Diffraction Solution If a body is large enough that significant wave diffraction occurs, flow separation effects will be relatively unimportant [103], thus allowing the application of potential flow theory for predicting oscillatory motions. The diffraction problem is further simplified by utilizing linear wave theory, which can be considered valid when the incident wave height is small relative to incident wavelength and water depth. Chapter 2. Numerical Iceberg Motion Model 19 The velocity potential solution for the wave field must satisfy the continuity equation throughout the fluid domain: V 2 $ = 0 (2.1) where: $ = wave field velocity potential (m2/s) The potential flow linear diffraction solution must also also satisfy the following boundary conditions: | ? = % at 2 = 0 (2.2) oz dt â€” + rig = 0 at 2 = 0 (2.3) dz dn dn = 0 at z = â€”d (2.4) = Vn on Si (2.5) = 0 on Ss (2.6) where: d â€” water depth (m) g = gravitational acceleration (m/s2) n = unit normal vector pointing outward from body (m) S{ = iceberg surface Ss = structure surface Vn â€” velocity component in direction of n (m/s) z = elevation relative to still water level (m) r\ â€” vertical displacement of water surface (m) Chapter 2. Numerical Iceberg Motion Model 20 In addition to the above boundary conditions, the diffraction solution must satisfy a radiation condition which requires that the diffracted wave from a body must radiate outward from the body. 2.2.2 Definition of Co-ordinate Systems and Modes of Motion Before proceeding further with the formulation of the velocity potential solution, the co-ordinate systems (shown in Figure 2.1) and iceberg modes of motion will be defined. The G(x',y', z') co-ordinate system has its origin at the centre of gravity of the iceberg; thus, the origin of this system changes with time due to iceberg drift. The 0(x, y, z) system has its origin fixed at the waterline at the centre of the structure. Relationships between the two co-ordinate systems are as follows: x = xg + x' (2.7) V = yg + V (2-8) z = zg + z' (2.9) where: x,y,z = co-ordinates of point in 0(x,y, z) system (m) x',y',z' â€” co-ordinates of point in G(x',y',z') system (m) xg,yg,zg â€” co-ordinates of iceberg centre of gravity in 0(x, y, z) system (m) The respective axes of the two co-ordinate systems may be taken as parallel, as shown in Figure 2.1, since yaw motions are not considered. The six modes of oscillatory iceberg motion are taken about the origin of the G(x',y', z') system and are as follows: 1. Surge - translational motions along x' axis, Chapter 2. Numerical Iceberg Motion Model 21 Z y x Iceberg Z y Structure / / / / / / / / Figure 2.1: Co-ordinate Systems for Iceberg Motion Model Chapter 2. Numerical Iceberg Motion Model 22 2. Sway - translational motions along y' axis, 3. Heave - translational motions along z' axis, 4. Roll - rotational motions about x' axis, 5. Pitch - rotational motions about y' axis, 6. Yaw - rotational motions about z' axis. 2.2.3 Re-formulation of Velocity Potential The solution of the diffracted potential is greatly simplified if a complex velocity potential defined as follows is adopted: The complex potential consists of an incident potential </>0, a diffracted potential <f>7, and potentials (j>\ â€” 4>Q due to motions in each of the six component modes. Using unit forms of the above potentials, the complex potential of equation 2.10 can now be expressed as follows: $(x,y,z,t) = Re{<f>(x,y,z) exp(-iut)} (2.10) where: $ = real, time-varying velocity potential (m2/s) <f> = complex velocity potential (m2/s) (2.H) where: H = incident wave height (m) <f)0 = incident wave potential (m) Chapter 2. Numerical Iceberg Motion Model 23 (f>7 = diffracted wave potential (m) <f>k = radiation potential for motions in mode k (m or m2) Ck = complex response amplitude operator for mode k (dimensionless or m - 1 ) u> â€” angular frequency of incident waves (s - 1) The zero flow boundary condition on the iceberg and structure surfaces establishes the following condition for the unknown diffraction potential: -râ€” = â€” â€” on Si and Ss (2.12) 5n an The following equation is used for describing the incident wave potential: g coshfHz + d)) â€¢ \i <po - Tm\ exp !fc i cosa + ysina) (2.13) or cosh[A;aJ where: k = wavenumber (m _ 1) a = incident wave direction (Â°) The radiation potentials <f>i â€” <f>e must satisfy the zero flow condition at the structure: On = 0 on Ss for k = 1 - 6 (2.14) In addition, the normal velocity condition on the iceberg surface must be satisfied for each mode of oscillation as follows: â€” â€”i IU on Si for k = 1 â€” 6 (2.15) an where: nk = unit normal velocity for mode k Chapter 2. Numerical Iceberg Motion Model 24 The unit normal velocities are related to the iceberg geometry as follows: ni = n x (2.16) n 2 = n v (2.17) n 3 = n, (2.18) n 4 - y'nz -z'ny (2.19) n 5 = z n x â€” X Xiz (2.20) n 6 - x n y - V n x (2.21) where: n x , n y , n 2 = directional cosines of normal vector pointing outward from the iceberg The diffraction and radiation potentials must also satisfy the radiation condition mentioned previously. For the diffracted and radiated waves to propagate outward from the body, the following radiation equations must be satisfied at large distances from the body: y/r - ik<j))j - 0 as r - â€¢ oo for j = 1 - 7 (2.22) where: r = distance from body in horizontal plane (m) 2.2.4 Solution of Diffraction and Radiation Potentials The unknown potentials <f>i â€” fa are solved using a source distribution method as described by Faltinsen and Michelsen [28], Garrison [31], and Hogben and Standing [36]. Using a series of sources located on the iceberg and structure surfaces, the potential <Ì‚ Chapter 2. Numerical Iceberg Motion Model 25 at a point in the fluid domain can be represented by: (2.23) where: fk = source strength for potential <f>k (dimensionless or m) G â€” Green's function (m - 1) x = location of point in fluid domain Â£ = location of source on iceberg or structure surface Form of Green's Function The Green's function is of a form that satisfies the conditions of equations 2.1 - 2.4 and 2.22 regardless of the values of the source strengths fk(0- The s o u r c e strengths are then solved using the surface boundary conditions of equations 2.12, 2.14, and 2.15. Alternative integral and series forms of the Green's function [28,31,36] are as follows: + 4 Â£ A\ \ A c o s ^ 2 + *)] â„¢MC + d)) Koi^n) (2.24) Hj d + d â€” u G(x,y,z;(,r],() = 1_ J_ R + B! G(x,y,z;Z,rj,() =â€¢ 2 7T (l/ 2 - P) cosh[fc(z + d)] cosh[A:(C + ci)][y0(fcri) - iJo(kri)} k2d - v2d + v d/x (2.25) Chapter 2. Numerical Iceberg Motion Model 26 where: J 0 = Bessel function of first kind, order 0 KQ = modified Bessel function of second kind, order 0 Y0 â€” Bessel function of second kind, order 0 Terms fij in equation 2.24 represent the successive real roots of the following equation: Hj tan(/Xj d) + v = 0 (2.26) Additional terms of equations 2.24 and 2.25 are defined as follows: LJ2 v â€¢ = â€” (2.27) 9- R = yj(x - 0 2 + (y - V)2 + (*' - C)2 (2.28) R' = y/(x - 0 2 + (y " V)2 + (z + 2d + 02 (2-29) ri = y/(x - 02 + (V ~ n)2 (2.30) The symbol P.V. in equation 2.25 denotes that the principal value of the integral is taken. Numerical evaluation of the Green's function between points x and Â£ is done using the series formula for k r x > 0.1, while the integral form is used for k r a < 0.1. Discretized Solution of Source Strengths and Potentials To enable the solution of the source strengths fk(Â£)i th e iceberg and structure sur- faces are divided into discrete facets as shown in Figure 1.3 with point sources located at the centre of each facet. The normal derivative of the potential <j>k at each facet can be expressed as a function of the source strengths as follows: dt* 1 . 1 / , ddj dn \ /*.- + T- [â€¢ /*, ^ * = 1 - 7 (2.31) 2 4 7r JSi+Ss 3 on Chapter 2. Numerical Iceberg Motion Model 27 where: fk = source strength at facet j for potential k Gij = Green's function between centres of facets i and j (j)k. = potential at centre of facet i Using equation 2.31 and the required surface boundary conditions of equations 2.12, 2.14, and 2.15, seven matrix equations are formed for solution of the source strengths <f>kj'. [ A ] { f k } = {it} <* = 1 - 7 ' (2-32) The matrix [A] is independent of the potential <f>k being evaluated, with elements as follows: Aij = (2-33) 4 7r an Aij = - l- for i = j (2.34) where: ASj â€” area of facet j (m2) Once the source strength vectors have been solved, the potentials (f>k can be evaluated at the facet centres using a discretized form of equation 2.23: {<M = [B] {fk} (2.35) Elements of matrix [B] are: Bn = GH f o r iztJ (2-36) 4 7T B* = J- f ^ dS + 7 ^ ^ - ; 6) fori' = i (2-37) 4 7T JASi ti 4 7T where: G*(xi\^j) = modified Green's function (m - 1 ) Chapter 2. Numerical Iceberg Motion Model 28 Equation 2.37 is formulated so as to eliminate the singular 1/R term of equation 2.25. The integral in equation 2.37 is evaluated for quadrilateral facets using expressions from Hogben and Standing [36], while the modified Green's function G* is obtained from equation 2.25 as follows: G*(Â£ t;Â£) = G(x-;Â£) - ~ (2.38) 2.2.5 Solution of Equations of Motion Once the diffracted wave field and its associated unit potentials <f>x â€” </>r have been solved, the oscillatory iceberg motions can be evaluated as follows: (-u? ([rn] + [/*]) ([A] + [K]) + [c]) {('} = {F1} (2.39) where: [c] = hydrodynamic stiffness matrix {F'} = vector of unit exciting forces [m} = iceberg mass matrix [X] = radiation damping matrix [K} = viscous damping matrix {('} = vector of response amplitude operators Computation of the above terms will now be described in detail. Chapter 2. Numerical Iceberg Motion Model Mass Matrix The iceberg mass matrix [m] appears as follows: M 0 0 0 0 0 0 M 0 0 0 0 0 0 M 0 0 0 0 0 0 hi -/l2 -/ is 0 0 0 -hi- hi hz 0 0 0 ~hz ~hz hz (2.40) where: lij = iceberg moment or product of inertia (kg-m2) Equations for the moments and products of inertia are: hi = jv Plx'2dV (2.41) h2 = / pi x y dV Jv (2.42) hz = f Ptx'z'dV Jv (2.43) 122 â€” [ P,y'2dV Jv (2.44) hz â€” Jv P i y z dV (2.45) hz â€” [ Plz'2dV Jv (2.46) where: Pi = iceberg density (kg/m3) Chapter 2. Numerical Iceberg Motion Model 30 Added Mass Matrix The added mass matrix [u] is evaluated using the radiation potentials for each mode of motion. Terms of the added mass matrix are based on the following integral equation: lijk = - p Re | jT fa UJ dS } (2.47) where: n, = unit normal velocity for mode j (from equations 2.16 - 2.21) <f>k = unit radiation potential for mode k (m) p = water density (kg/m3) The discretized form of equation 2.47 is: ' fiik = -p Re | K nJ. ASi | (2-48) where: Ni = number of facets on iceberg surface = unit normal velocity for mode j at facet i (f)^ = unit radiation potential for mode k at facet i Radiation Damping Matrix The radiation damping matrix [A] is solved in a manner very similar to that used for evaluating the added mass matrix. Integral and discretized forms for the radiation damping matrix are: \ j k = - p c u l m j ^ fanjds} (2.49) Chapter 2. Numerical Iceberg Motion Model 31 Viscous Damping Matrix In addition to the radiation matrix, an empirical viscous damping matrix [Xv] is introduced. Viscous damping can be significant for heave, pitch, and roll motions [43,103]; thus, the viscous damping matrix is assumed to be of the form: [Al = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \v A33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2.51) For the present problem, viscous damping coefficients are determined from empirical viscous damping ratios Xv>. The following relationship is used for calculating viscous damping coefficients from damping ratios: A l = 2 A< yJCii (m30 + Hj) (2.52) where: Cjj = hydrostatic stiffness matrix term nijj - - mass matrix term fijj = added mass matrix term Chapter 2. Numerical Iceberg Motion Model 32 Hydrostatic Stiffness Matrix Like the mass matrix, the hydrostatic stiffness matrix is dependent only upon the iceberg geometry and appears as follows: [c] = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C33 C34 C35 0 0 0 c34 C44 0 0 0 0 C35 0 C55 0 0 0 0 0 0 0 Equations for the matrix are: c33 â€” C34 - C35 = c 4 4 = C55 = P 9 Awp - 99 I y' dA dA - P9 I x' - P9 y'2 dA + mg z'h JAWp - pg x12 dA - f m g z'b JAWV where: iceberg waterplane area (m2) (2.53) (2.54) (2.55) (2.56) (2.57) (2.58) z'b â€” z' co-ordinate of iceberg centre of buoyancy (m) Exciting Forces The exciting forces acting on the iceberg are caused by the incident and diffracted waves. Integral and discretized forms of the exciting forces for unit amplitude incident Chapter 2. Numerical Iceberg Motion Model 33 waves are: F'k = -pu2 j {fa + <f>7)nk dS (2.59) JSi N, K = -pu2 J2(^o, + h.)nktASt (2.60) t=i where: Ff. â€” force or moment for mode k in unit amplitude waves (N/m or N) Final Computation of Response Amplitude Operators Once the hydrodynamic coefficients and exciting forces are known, the complex response amplitude operators (RAO's) are computed by solving equation 2.39. Motion amplitudes and phase angles are subsequently computed, with phase angles defined as follows: 8k = t a n - ^ - ^ (2.61) where: 8k = phase angle for mode k (Â°) The physical meaning of the phase angle is dependent on the form of incident wave potential used. For the incident potential of equation 2.13, the phase angle represents a lag relative to an incident wave crest crossing the origin of the 0(x,y,z) co-ordinate system. 2.3 Mean Drift Motions The drift motions of an iceberg are caused by forces due to ocean currents, waves, winds and the Coriolis effect. When modelling iceberg motions in the vicinity of an Chapter 2. Numerical Iceberg Motion Model 34 offshore structure, Coriolis and wind drag forces are typically small relative to water drag and wave drift forces [44]. The numerical model utilizes the following equations for evaluating iceberg drift motions: 1 (M + / z n ) l y ~ (M + fi22) where: (Fdx + FWx) (2.62) ax,ay = iceberg acceleration components (m/s2) Fdx,Fdy = water drag force components (N) FWx,FWy = wave drift force components (N) i P22 = zero frequency added mass components (kg) The iceberg trajectory is then predicted using an Adams-Bashford time stepping method: (2.64) (2.65) (2.66) (2.67) where: u,v = iceberg velocity components (m/s) xg,yg = iceberg location components (m) For a typical simulation, the time stepping procedure begins when an iceberg enters the vicinity of a structure and continues until a collision occurs or the iceberg drifts freely past the structure u(t + At) = u(t) + At 2 [3 ax(t) â€” ax(t -At)} v(t + At) = v(t) + At 2 [3ay(t) - ay(t -At)] xg(t + At) = xg(t) + At 2 [3u(t) - u(t --At)} yg{t + At) = y9(t) + At 2 [3 v(t) - v(t --At)} Chapter 2. Numerical Iceberg Motion Model 35 2.3.1 Evaluation of Zero Frequency Added Mass Computation of an iceberg drift trajectory requires that the zero frequency added mass components and fi22 he known. The zero frequency added mass varies with the proximity of the iceberg relative to the structure; thus, f t n and / x 2 2 must be re-evaluated as the iceberg location changes. The procedure for computing zero frequency added mass is essentially the same as that used for evaluating wave frequency added mass. The principal difference in the numerical procedure is that the following zero frequency form of the Green's function given by Garrison [31] can be used for evaluating zero frequency radiation potentials: G(x,y,z;Â£,r),() = (2.68) Terms in the above equation are evaluated as follows: R = V2 + \z - c)T/2 (2.69) Ri = [r2 + (z + o 2] 1 / 2 (2.70) Rin = [r2 + (z - 2nd - C)2]1 / 2 (2.71) R^n = [r2 + (z + 2nd + C)T / 2 (2.72) RAU = [r2 + (z + 2nd - C) 2] 1 / 2 (2.73) = [r2 + (z - 2nd + C) 2] 1 / 2 (2.74) r = [ ( * - 0 2 + (y - v)2?/2 (2.75) Successive terms of equation 2.68 are computed until convergence is obtained. In practice, the evaluation of zero frequency added mass requires significant computational effort; thus, added mass components are usually re-evaluated only once every few time steps. Chapter 2. Numerical Iceberg Motion Model 36 2.3.2 Water Drag Forces Evaluation of the water drag forces acting on the iceberg would ideally consist of integration of the normal pressures and shear (viscous) stresses acting on the iceberg surface; however, due to the complex nature of the flow around the iceberg, drag force components are instead evaluated using an empirical drag force coefficient as follows: Fdl = \p{uw - u) \ VW - V \ CdAw (2.76) Fdy = l-P (vw - v) | Vw - V | Cd Aw (2.77) where: Aw = iceberg underwater cross-sectional area (m2) Cd â€” iceberg water drag coefficient â€”â€¢ V = iceberg velocity vector (m/s) Vw = water velocity vector (m/s) uw,vw = water velocity components (m/s) The empirical water drag coefficient Cd is highly dependent on iceberg geometry. For typical full-scale situations, Reynolds numbers are sufficiently high that drag coefficients can be considered independent of Reynolds number. The evaluation of water drag forces requires that the water velocity incident to the iceberg be known. If the structure is small relative to the iceberg, the free-stream current velocity is used for calculating water drag forces. Alternatively, if the structure is large relative to the approaching iceberg, then the modification of the flow field by the structure must be considered when evaluating water velocity. The water velocity Vw used in equations 2.76 and 2.77 is based on the water velocity surrounding the structure and is evaluated at the location of the iceberg centre of gravity. The modification of the flow field by the iceberg is implicitly considered in the iceberg drag coefficient for the above Chapter 2. Numerical Iceberg Motion Model 37 two cases. Among the forces not considered in equations when evaluating the iceberg drift motions in ICEMOT are the convective forces arising from the spatial variation of the water velocities in the vicinity of the structure. If the flow in the vicinity of the structure is considered to be ideal, non-uniform, steady, and having no vertical component, then the total hydrodynamic force acting on the iceberg can determined from Isaacson [45] to be as follows: Fx = (M + M i i ) K - Â« ) ^ f + (M + p22)(vw~v)^ - (2.78) Fy = {M + p22){vw-v)-^ + ( M +/zn) K , - u ) - Â»22 â€” (2.79) To examine the implications of convective forces on the trajectory of an iceberg ap- proaching a large structure, an iceberg moving with an incident current in the x direc- tion and with a small eccentricity y relative to the structure (see Figure 2.1) can be considered. Considering first the x component of force, as the iceberg approaches the structure duw/dx is less than zero and (uw â€” u) is greater than zero (neglecting wave drift force effects) so that the convective force component given as the first term on the right-hand side of equation 2.78 will be negative and cause the iceberg to decelerate in the x direction. Considering now the y component of force, for small non-zero values of y, both dvw/dy and (vw â€” v) are greater than zero so that the convective force compo- nent given as the first term of the right-hand side of equation 2.79 will be positive and will cause the iceberg to veer away from the structure. Significant coupling is likely to occur between the forces and motions in the x and y directions. The implications of the above convective forces on iceberg motions could be examined in greater detail by their incorporation into the numerical time-stepping model. Chapter 2. Numerical Iceberg Motion Model 38 2.3.3 Computation of Modified Current Field in the Vicinity of a Large Structure of Arbitrary Shape The current field in the vicinity of a large arbitrarily shaped structure is determined using a source distribution method similar to that used for computing the diffracted wave field. It is postulated that flow separation effects are significant only on the downstream side of the structure. Consequently, potential flow theory is used for modelling the flow field upstream of the structure during the approach of an iceberg. The current field must satisfy Laplace's equation throughout the fluid domain: V 2 $ c = 0 (2.80) where: $ c = water current velocity potential (m2/s) The zero flow boundary conditions must be met at the seabed, the free surface, and at the structure surface: 3$ c dz = Oa,tz = -d (2.81) = 0 at z = 0 (2.82) ^ = 0 on Ss (2.83) The current potential is divided into incident and diffracted components as follows: $c = $ c , + $ c d (2.84) where: $ C d = diffracted current potential (m2/s) $C|. = incident current potential (m2/s) Chapter 2. Numerical Iceberg Motion Model 39 A uniform incident current is assumed and has the following form: $o = Vwo (x cos/3 + y sin B) (2.85) where: Vw0 = incident current velocity (m/s) B â€” incident current direction (Â°) The diffracted current is represented by a series of point sources on the structure surface such that the integral form of the diffracted potential is: = J- I fc(OG(x;()dS (2.86) 4 7T JS* where: fc = source strength (m/s) The Green's function of equation 2.68 is used to represent the diffracted current potential because it satisfies all governing equations except for the boundary condition at the structure surface (equation 2.83). The diffracted current source strengths are determined by solving the following set of simultaneous equations: K H / c } = { - ^ } (2-87) where: [Ac] = matrix for solution of diffracted current source strengths {/c} = vector of source strengths on structure surface (m/s) {d$Ci I dn} = vector of incident current velocities normal to structure surface (m/s) Elements of matrix [Ac] are as follows: A 9 DC ACtJ = - \ forz = ; (2.89) Chapter 2. Numerical Iceberg Motion Model 40 It should be stressed that the diffracted current source strengths only need to be evaluated once and can subsequently be used to evaluate the current velocity at discrete points along the iceberg trajectory. Required equations for computing current velocity components are: N, A C fist uw(x9) = Vwocos(3 + - L â€” ( x - , Â£ ) / (2.90) â€¢=1 4 7T OX vâ€ž(*a) = Vwosm/3 + ^ _^_(f;,^.)/C j (2.91) where: Na â€” number of facets on structure surface x9 = location of iceberg centre of gravity (m) Equations 2.90 and 2.91 are re-evaluated as the iceberg location xg changes. 2.3.4 Computation of Modified Current Field in the Vicinity of a Large Cylindrical Structure Evaluation of the current field is greatly simplified if the structure is a vertical circular cylinder extending from the seabed to the free surface. The following analytical solution for two-dimensional flow around a circular cylinder [109] is used: 2 $c(r,6) = Vwo (r + - ) cos(6 - /?) (2.92) r r = yjx] + y\ (2.93) 0 = tan"1 ( Ì‚ 1 (2.94) where: a = structure radius (m) Chapter 2. Numerical Iceberg Motion Model 41 r â€” horizontal distance from centre of structure to centre of iceberg (m) 6 = angular co-ordinate between iceberg and structure (Â°) The above solution is well suited to practical application since offshore structures which are large enough to significantly modify current fields typically have shapes approximating vertical circular cylinders (e.g. [80]). An alternative procedure for computing the current field around a vertical structure of arbitrary plan shape has been presented by Isaacson and Cheung [46]. The wave drift force on an object is the mean wave force averaged over one wave period. The importance of wave drift forces on iceberg motions has been discussed by Hsiung and Aboul-Azm [40] and Isaacson [44]. The computation of wave drift forces has been discussed in detail by several authors (e.g. Faltinsen and Michelsen [28], Pinkster [96], Standing et al. [108]). If the assumption of linear wave theory is used, then the mean wave drift forces will be due primarily to the following sources: 1. Mean pressure increase due to relative water elevation on iceberg waterline, 2. Mean pressure decrease due to water velocity squared on iceberg surface. Using the first-order real velocity potential, the mean drift forces and moments can be expressed as follows: 2.3.5 Wave Drift Forces k = 1-6 (2.95) Chapter 2. Numerical Iceberg Motion Model 42 where: FWk = wave drift force or moment for mode k (N or N-m) wli â€” iceberg waterline Thus, the wave drift forces can be computed using the diffraction and radiation potentials which are determined for the computation of oscillatory motions. Evaluation of equation 2.95 can be done using either a near-field or far-field method. The near-field method involves the direct solution of equation 2.95 at the body surface as discussed by Pinkster [96]. Alternatively, the far-field method utilizes a vertical cylin- drical control surface with infinite radius r centred at the body, as shown in Figure 2.2. Wave drift forces result in the mean change in momentum of the wave field in the control volume. Development of the far-field method has been largely based on initial work by Newman [88]. The far-field method is limited to the evaluation of forces and moments in the horizontal plane (i.e. surge, sway, and yaw); however; only horizontal forces are required for the prediction of iceberg drift motions. The far-field method is used in the present model because of its greater computational efficiency and ease of programming. Application of the Far-field Method The procedure for computation of wave drift forces using the far-field method is given by Faltinsen and Michelsen [28]. For the evaluation of drift forces acting on an iceberg in the vicinity of a structure, the procedure must be modified for the presence of the second body, as discussed by Loken [65]. The horizontal drift force components are evaluated using the following equations from [28]: Chapter 2. Numerical Iceberg Motion Model 43 Figure 2.2: Control Surface for Far-field Wave Drift Force Computation Chapter 2. Numerical Iceberg Motion Model 44 F = puH //2_7T 4 sinh(iW) V V k - smh(2 Jfc of) + â€” p k ~2~ - smh(2 Ar d) + â€” r2-K Jo 2 F(a) cos cos 0 d0 7(a) + 7T cos a (2.96) p LO H 4 sinh(A:o?) ' ( T 1 . , /~ , Â« fc <Ì‚ - sinh(2 fc of) + â€” /) A; ~2~ - sinh(2 kd) + ^ 4 2 /â€¢27T / F 2(0) si Jo 2 F(a) cos s n 0 d0 1 \ N 7(a) + - j sin a F(0) exp [i 7(e)] = 2 7T (i/2 - P) exp 3TT â€” z Q(0 F(0) HUJ 8TT /o(f) + E(-Â«/*(flcO = tan -1 Im{F(0) exp[Â»7(0)]} Re{F(0) exp[Â»7(0)]} Re{F(fl) exp[7(fl)]} cos[7(0)] where: F = force contribution term (m 2 , 5/s) Q = diffracted and radiation source strength term (m/s) a = incident wave direction (rad) 7 = phase term (rad) v = u2/g (m _ 1 ) 0 = angular co-ordinate for location on control volume (rad) (2.97) k2 d â€” v2 d + ir k x / Q(t,V,Q cosh[fc(C -I- d)] exp[-ikÂ£ cos0 i k rj sin 0] dS (2.98) (2.99) (2.100) (2.101) Chapter 2. Numerical Iceberg Motion Model 45 The computational method utilizing the above equations proceeds in the following se- quence: 1. Q(0 is evaluated at each facet on the iceberg surface using equation 2.99, 2. Using equations 2.98, 2.100, and 2.101, F{6) and j(9) are evaluated at 90 discrete values of 6 between 0 and 2ir, 3. FWx and Fw are computed using numerical integration to evaluate equations 2.96 and 2.97. 2.4 Verification of Numerical Model Due to the complexity of the iceberg motion model, a series of numerical tests was conducted to verify the validity of the model. Output from the model was compared with analytical, numerical, and physical modelling results found in the literature. Areas of the model which were subjected to verification included the following: 1. Wave-induced oscillatory motions, 2. Wave drift forces, 3. Zero frequency added masses, 4. Boundary element evaluation of current field in vicinity of structure. 2.4.1 Wave-Induced Motions For verification of oscillatory motion computations, it was decided to examine the response of a barge which had been the subject of previous investigations [92,95,108]. Relevant parameters of the barge are given in Table 2.1 and Figure 2.3. Chapter 2. Numerical Iceberg Motion Model 46 Plan View 150 m 50 m Elevation View 5 l ^ T ^ l O m Figure 2.3: Barge for Verifying Diffraction Computations Table 2.1: Hydrodynamic Properties of Barge (from [92]) Length 150 m Breadth 50 m Draft 10 m Displaced volume 73750 m 3 za 0 m Transverse gyradius 20 m Longitudinal gyradius 39 m Vertical gyradius 39 m Natural heave period 10.4 s Natural pitch period 12.1 s Natural roll period 9.4 s Chapter 2. Numerical Iceberg Motion Model 47 1 .2- 1.0 â€¢C 0.8 - O < ^ 0.6- V I 04 0.2- 0.0 ICEMOT Ostergaard numerical A Pinkster experimental Wave Period T (s) Figure 2.4: Barge Surge Motions in Head Waves O as > l . O - i 0.8- 0.8 0.4 0.2 0.0 ICEMOT Ostergaard numerical A Pinkster experimental Wave Period T (s) Figure 2.5: Barge Heave Motions in Head Waves Chapter 2. Numerical Iceberg Motion Model 48 4 . 0 - 1 3 .0 - o < CC 2 . 0 H J C o 1.0- 0 . 0 - Ostergaard numerical A Pinkster experimental Wave Period T (s) - i â€” is 2 0 Figure 2.6: Barge Pitch Motions in Head Waves 2 . 0 - 1 1 . 5 - O < O J 1 0 - >> (0 in 0.5-1 o.o- ICEM0T Ostergaard numerical A Pinkster experimental A 10 â€” i â€” 15 Wave Period T (s) Figure 2.7: Barge Sway Motions in Beam Waves â€” i 2 0 Chapter 2. Numerical Iceberg Motion Model 49 1.5-. Jv_o 1.0- O < CC V > Â£ 0.5- 0.0- 47 ICEMOT Ostergaard numerical A Pinkster experimental i 5 1 0 1 5 Wave Period T (s) Figure 2.8: Barge Heave Motions in Beam Waves i 2 0 30-i 20-1 Kr O < o cn 10 IGEMOT Ostergaard numerical A Pinkster experimental Figure 2.9: Barge Roll Motions in Beam Waves Chapter 2. Numerical Iceberg Motion Model 50 Response computations were made for head waves and beam waves with periods ranging from 5 s to 20 s. The geometry of the barge was modelled using 142 facets. Viscous damping coefficients were set equal to zero for all cases. Figures 2.4 - 2.9 show comparisons with numerical results from Ostergaard and Schellin [92] and experiments by Pinkster [95]. The agreement among results supports the validity of the program ICE- MOT. Differences between numerical and physical response values are most pronounced near the natural heave and roll periods of the barge. These discrepancies are due mainly to the absence of viscous damping in the numerical models. 2 . 4 . 2 Wave Drift Forces Wave drift forces acting on the barge described above were also computed for verification. Results were originally compared with those published by Pinkster [95]; however, poor agreement was obtained. An explanation of the poor agreement between results was given by Standing et al. [108] who suggested that Pinkster's application of the near-field method caused significant errors arising from the evaluation of the waterline integral term of equation 2.95. Results from ICEMOT were subsequently compared with far-field computations by Standing et al. [108]. Figures 2.10 and 2.11 indicate relatively good agreement between results when one considers that wave drift forces are a second- order effect. 2 . 4 . 3 Zero Frequency Added Masses Zero frequency added mass computations were verified using values published by Yeung [119]. Using an analytical solution for velocity potential, Yeung computed added mass and damping coefficients for vertical circular cylinders. Added masses were eval- uated for cylinders having diameter D equal to water depth d, with the draft to water Chapter 2. Numerical Iceberg Motion Model 51 O.lO-i Cj 0.08- o Â£ 0.06-1 Â£ 0.04 > Â£ 0.02 0.00- ICEMOT o Standing Far- f i e ld i 10 â€”iâ€” 15 Wave Period T (s) Figure 2.10: Barge Wave Drift Forces in Head Waves i 20 ICEMOT 3 o Q > co o Standing Far- f ie ld Wave Period T (s) Figure 2.11: Barge Wave Drift Forces in Beam Waves Chapter 2. Numerical Iceberg Motion Model 52 depth ratio h/d varying between 0 and 1. The surface of each cylinder was modelled using between 100 and 200 facets, depending on its geometry. The results presented in Figure 2.12 indicate excellent agreement between the analytical and numerical solu- tions. It appears that the number of facets used for each cylinder was sufficient to obtain accurate results using the boundary element method. Relative draft h/d Figure 2.12: Zero Frequency Added Mass Coefficients for Vertical Circular Cylinders, D = d For the interaction case of an iceberg approaching a structure, the presence of the structure must be considered when evaluating iceberg added mass. Isaacson and Cheung [46] have indicated that facet sizes must be less than the distance between the iceberg and the structure for accurate added mass predictions using the boundary element method. To assess the validity of ICEMOT for the interaction case, iceberg added masses for a two-dimensional example were computed and compared with results by Yamamoto [118]. The iceberg and structure were modelled as surface-piercing vertical circular cylinders Chapter 2. Numerical Iceberg Motion Model 53 extending from the seabed, thus creating a two-dimensional flow field. Iceberg and struc- tural diameters were taken as 100 m, with a corresponding water depth of 10 m for the numerical computations. Added mass computations were made for several facet grids, with each grid having less than the 200 facet limit imposed by the program ICEMOT. Optimal results were obtained using a structure mesh of 32 x 2 facets (circumferential x vertical) and an iceberg mesh of 32 x 4 facets. It should be noted that inaccuracies in the boundary element method typically arise from flow sources at the centre of facets being too close together. The optimal grid described above would partly alleviate this problem due to staggering of iceberg and structure facet centres in the vertical direction. Computed added masses using the optimal grid are presented in Figure 2.13. For separation distances greater than half a facet dimension, computed added masses are within 4% of those given by Yamamoto [118]. The boundary element values rapidly diverge from actual values for smaller separation distances. Greater accuracy in computed added masses for small separation distances could be achieved by using smaller facets on the portions of the iceberg and structure surfaces near the points of closest approach. Alternatively finer meshes for the entire iceberg and structure surfaces could be used to obtain accurate values for a smaller separation distance s; however, the number of required facets for a three-dimensional boundary element solution would increase as 1/s2 with the corresponding computer memory and number of computations increasing as 1/s4. 2 . 4 .4 Boundary Element Computation of Current Near Structure The validity of the boundary element method for computing the current near a struc- ture was tested using a vertical circular cylinder extending to the seabed. The analytical Chapter 2. Numerical Iceberg Motion Model 54 1.5-1 - 1 5 -1.4 -1.3 -1.2 -1.1 -1.0 Iceberg Location xa/D* Figure 2.13: Zero Frequency Added Masses Approaching Structure solution for flow around a cylinder (equation 2.92) provided a convenient means for ver- ification. The diameter of the cylinder was set equal to the water depth. The cylinder surface was modelled using 144 facets; thus, a typical facet dimension was approximately l/7th of the cylinder diameter. The comparison of numerical and analytical current velocities presented in Fig- ure 2.14 indicates that the boundary element method gives excellent results even within one facet diameter.of the cylinder. Application of the boundary element for computing current fields is recommended when large structures of arbitrary shape are encountered. Chapter 2. Numerical Iceberg Motion Model 5 5 Location x/D* Figure 2.14: Current Velocity in Vicinity of Structure Chapter 3 Hydrodynamic Aspects of Iceberg Collisions As stated previously, a major objective of the present investigation is to examine the significance of hydrodynamic effects on iceberg collisions with a large offshore structure. Using the program ICEMOT, various aspects of collision hydrodynamics have been ex- amined, with particular attention given to supplementing existing knowledge regarding iceberg hydrodynamics. 3.1 Modelling of Iceberg Geometry The present study uses vertical circular cylinders for modelling iceberg geometry. Cylinders have been used to model icebergs by several previous investigators, including Arunachalam et al. [5], Hay and Company [35], Isaacson and Cheung [46], Kobayashi and Frankenstien [52], McTaggart and Davenport [76], and Salvalaggio and Rojansky [99]. Consequently, much data is available in the literature regarding the hydrodynamics of cylinders. Although the modelling of icebergs as random shapes would be more realistic, the modelling of random shapes would introduce several problems. The selection of represen- tative shapes would be difficult and highly subjective. In addition, the complex geometry of a random iceberg would require a large number of facets to accurately describe its ge- ometry. Further problems would be introduced by hydrodynamic stability requirements, 56 Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 57 most notably the vertical alignment of centres of gravity and buoyancy. Among regular shapes, the circular cylinder is convenient for hydrodynamic modelling because rotational motions in the horizontal plane do not require evaluation. When selecting a geometry for modelling icebergs, consideration should also be given to the mechanics of collisions. Results from Bass et al. [7] indicate that impact forces will be more severe if the impacting face of the iceberg is vertical; thus, a vertical circular cylinder will likely give a conservative impact force for a given impact scenario. It is assumed that l/7th of the volume of the modelled icebergs is above water, with hydrodynamic stiffness and inertial properties evaluated accordingly. A useful upper limit to iceberg aspect ratio h/Di is determined by hydrostatic stability requirements. For a vertical circular with l/7th of its volume above water, hydrostatic instability occurs when its aspect ratio exceeds approximately 0.86. The present investigation considers icebergs with aspect ratios to a maximum value of 0.8. Although the influence of non-cylindrical shapes on iceberg hydrodynamics has not been considered in the present investigation, reference can be made to studies by Bass and Sen [8] and Lever et al. [58,61]. Bass ad Sen used a numerical diffraction model to evaluate added mass and damping coefficients for various random iceberg shapes. Their results indicate that the greatest limitation imposed by a cylindrical iceberg geometry is that the variation of hydrodynamic coefficients with length/width ratio cannot be modelled. The influence of shape on wave-induced oscillatory motions was examined experimentally by Lever et al. for iceberg models having cubical, spherical, cylindrical, and trapezoidal geometries. For icebergs having non-vertical sides, the heave response was shown to be significantly different from icebergs having vertical sides due to the non- linearity of restoring forces in heave. Due to the absence of resonance effects for surge motions, the influence of iceberg shape on surge motions was much less than for heave motions. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 58 3.2 Effective Iceberg Added Mass Iceberg added mass is among the parameters that must be known when evaluating iceberg drift motions or for estimating the maximum energy that can be dissipated during a collision with an offshore structure. Due to the variation of added mass with frequency, the selection of a suitable value must be considered carefully. The numerical model of Isaacson [43] uses zero frequency added mass for the evaluation of iceberg drift motions, which are considered to be of low frequency relative to wave-induced oscillatory motions. Examination of various iceberg collision models [7,14,18,25,50] indicates that added mass is considered when evaluating iceberg kinetic energy and impact forces. Although not explicitly stated, it appears that zero frequency added mass is used in each of the above collision models. Detailed studies of ship added masses during collision have been presented by Motora et al. [82] and Petersen and Pedersen [94]. Effective added masses are developed based on the kinetic energy and momentum absorbed during rapid deceleration of ships in collisions. For collisions of short duration, the effective added mass approaches the infinite frequency value. Conversely, the effective added mass approaches the zero frequency value for collisions of long duration. 3.2.1 Transient Motions of a Floating Body Before considering effective added mass in greater detail, the basic principles of transient floating body motions will be presented. More detailed discussions of floating body transient motions are given by Mei [77] and Wehausen [115]. Recall that the oscillatory motion of a floating body are given by equation 2.39. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 59 Alternatively, transient motions of a floating body can be expressed as: [M + /*(Â«>)] {5(Â«)> + {H(t)} + [c] {*(*)} = {F.(0> (3-1) where: {Fe} = excitation force vector {H} = memory effect vector The memory effect vector {H} is dependent on the floating body motions prior to time t. Components of {H} can be evaluated as follows: 6 r* Hi = J2 Lai* ~ T) Xj(r) dr i = 1 - 6 (3.2) j=i J-Â°Â° where: Lij â€” history term (kg/s, kg-m/s, or kg-m2/s ) The excitation forces of {Fe} can be due to a variety of sources such as incident and diffracted waves, water drag forces, and forces experienced during a collision with another body. Solution of the convolution integral of equation 3.2 requires that the history term L be known for a given body. From Wehausen [115], L can be evaluated by taking the cosine transform of the frequency dependent added mass or the sine transform of the frequency dependent damping as follows: 2 fÂ°Â° â€¢Ì‚ tj(*) = ~ iPiji^) ~ cosutdt (3.3) = 2 / " * ) ^ ( 3 . 4 ) 7T JO LO Thus, if the frequency dependent added mass or damping of a floating body is known, the history term L can be easily evaluated. The transient motions of a floating body can then be rigourously evaluated using equation 3.1. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 60 3.2.2 Effective Added Mass for Iceberg Drift Motions Although equation 3.1 provides a correct method for evaluating transient iceberg motions, its form is inconvenient for predicting iceberg drift motions. Recalling that the drift motions of an iceberg are not subjected to hydrostatic stiffness effects, equation 3.1 can be re-written for iceberg drift motions in one direction as follows: [Af + /i(oo)] x(t) + f L(t - T)x(r)dT = Fe(t) (3.5) Jâ€”oo Evaluation of the convolution integral of equation 3.5 is simplified by considering the following properties of L: L(t) = 0 for t < 0 (3.6) L(t) Â» 0 for t>tL (3.7) where: tl = decay time for L Equation 3.7 suggests that L(t) can be considered equal to zero after a decay time for a given body. The convolution integral of equation 3.5 can then be re-written as follows: f L(t - T) x(r)dT = f L(T) x(t - T) d T (3.8) w fL L(T) x(t - r) d r (3.9) Jo The right hand side of equation 3.9 can be easily evaluated numerically. To simplify the evaluation of iceberg drift motions, it is convenient to develop an effective added mass coefficient Cm such that equation 3.5 can be approximated as follows: M (1 + Cm) x(t) = Fe{t) (3.10) A solution for effective added mass can be readily obtained if it is assumed that the excitation force Fe and associated acceleration x vary slowly relative to the history decay Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 61 time t^. This assumption is likely to be quite reasonable for a freely drifting iceberg, thus allowing equation 3.5 to be re-written as: Solution of the above integral is made using the following relation from Wehausen [115]: Thus, it is appropriate to use zero frequency added mass for the evaluation of iceberg drift motions due to slowly varying excitation forces. 3.2.3 Effective Added Mass for Iceberg Collisions When evaluating iceberg collisions with a structure, an estimate of added mass is required to assess the resulting collision forces. Effective added masses for head-on collisions can be based on either, of the following momentum and energy relationships for an iceberg crushing against a rigid structure: (3.11) (3.12) Taking co = 0 in equation 3.12, equation 3.11 can then be re-written as: [M + MO)] x = Fe (3.13) (3.14) (3.15) where: F = collision force between iceberg and structure (N) tc = collision duration (s) Vc = initial iceberg velocity (m/s) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 62 A = distance travelled by iceberg during collision (m) A c = maximum A for collision (m) Evaluation of effective added mass based on each of the above equations requires that the force integral on the right hand side be solved. A time-stepping procedure must typically be used, with transient iceberg motions during the collision ideally based on equation 3.5. The evaluation of effective added mass can be somewhat simplified if the variation of impact force with time is assumed to be of a known form. Maximum Impact Force The maximum impact force to occur during a head-on iceberg collision can be esti- mated using the kinetic energy balance of equation 3.15. The normal contact area for a cylindrical iceberg impacting a cylindrical structure can be approximated as follows (see Chapter 6): v 1/2 An (A) = 2ht Di A (3.16) 1 + Di/DsJ It is assumed in the above equation that the penetration distance A is small relative to the iceberg and structure diameters Di and Ds respectively. Substituting equations 1.1 and 3.16 into the kinetic balance of equation 3.15, the maximum iceberg displacement and impact force are as follows: 2/3 (3.17) 3 (1 + Cm) M Vc2 $ Per ht ( 1 + g . / D , ) 1/2 Fr 3 P l h\ (1 + Cm) Di M V} 1/3 (3.18) 1 + Di/D. Equation 3.18 is particularly useful for examining the relative importance of each collision parameter on impact severity. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 63 The assumption of a vertical circular cylinder for iceberg geometry will typically overestimate the contact area An for a given penetration distance A . For an iceberg of irregular geometry impacting an offshore structure, the reduction in contact area for a given penetration distance will lead to a smaller maximum impact force Fc and a larger maximum penetration distance A c . Collision Duration and Added Mass for Ramp Impact Force The duration of an iceberg collision can be estimated using the momentum balance of equation 3.14. No analytical solution for collision duration is readily available from the momentum balance; however, collision duration can be easily estimated if the variation of impact force with time is assumed to be of a known form. When considering how impact force pcr An varies with time, it should be noted from equation 3.16 that the impact force during a collision may be expressed in the form: F = kF A1/2 (3.19) where: kp = force constant (N/m 1/ 2) Thus, the iceberg velocity dA/di during the collision can be related to the force as follows: Given that the collision force F is equal to zero at the beginning of a collision and that the initial impact velocity dA/di is non-zero, dF/dt must approach infinity at the beginning of a collision. Although a step forcing function will satisfy the above requirements, results from numerical impact models by Bass et al. [7] and Duthinh and Marsden [25] indicate Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 64 F(t) Â± j? (3.21) that it is more reasonable to adopt a ramp impact force function as follows: 0 t < 0 and t > tc 0 < t <tc Note that the above forcing function does not satisfy the requirement that dF/dt ap- proach infinity at the beginning of a collision. Adopting the ramp force assumption, the momentum balance of equation 1.2 can then be expressed as: 1 (1 + Cm) MVC = ^ t\ Fc (3.22) Substituting equation 3.18 into equation 3.22, the collision duration for a ramp impact force is: t. (i + cm)Â±Â£- 4 pcr 2/3 (1 + Di/D,) Vc 1/3 (3.23) As stated previously, the assumption of a vertical circular cylinder iceberg geometry will likely lead to overestimates of the maximum impact force F c ; thus, the actual collision duration will likely be greater than the estimate given by equation 3.23. A technique for predicting effective collision added masses has been developed by Motora et al. [82], who assume that the variation of collision force with time is of a known form. If the hydrodynamic inertial and damping forces acting on the iceberg are taken as being linear, then a Fourier transform technique can be applied. The assumption of linear damping forces is valid provided that the damping forces arise from radiation damping rather than viscous damping. Although viscous damping can have a significant effect on heave, pitch, and roll motions, its influence is usually small for surge motions. For application of the Fourier transform technique, the response of an iceberg to the following ramp forcing function can be considered: 0 for t < 0 fe(t) = t for t > 0 (3.24) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 65 where: fe = forcing function (s) The external forcing function fe could be representative of the force applied to an iceberg by a structure during a collision. To obtain the Fourier transform of the iceberg accel- eration, the Fourier transform of the external forcing function fe can be multiplied by the acceleration admittance function of the iceberg oscillatory motions (equation 2.39) as follows: A{u) = -. > f a > Fe(w) (3.25) i w Mv(u) + X(u) where: A(u>) = Fourier transform of iceberg acceleration Fe{oj) = Fourier transform of external force fe(t) MV(LO) = virtual iceberg mass at frequency to (kg) Taking the inverse Fourier transform of equation 3.25 and then integrating with respect to time, Motora et al. [82] have shown that the change in velocity due to the ramp forcing function of equation 3.24 is: \t2 2 /â€¢Â«> A(w) coswi -t 2 fÂ° A " W " MM) -'Jo â€ž(0) 7T Jo M2{U)UJ2 + X2(co) CO2 2 [Â°Â° \(u) i M2(LO)UJ2 + X2(LO) U2 -d to 2 fÂ°Â° Alw) 1 , . . 7T Jo 2 )0J2 2(bj) J2 V ' where: Av = change in iceberg velocity in response to forcing function (s2/kg) Taking a momentum balance over the duration tc of the ramp force, the effective added mass can be obtained by: Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 66 M (1 + Cm) = & M g f (3.27) " A ^ i ( 3 , 2 8 ) Substituting equation 3.26 into 3.27, the effective added mass based on a momentum balance for a ramp force can be determined by: M (1 + Cm) = 1 4 1Â°Â° \(u) cosut MJti) ~ T T t f Jo M v 2 ( w ) w 2 + A2(w) CJ2 W 4 A 0 0 A(w) 1 I - 1 + T* 2 Jo M 2(u>W + A 2 M w2 w (3.29) Thus, the effective added mass for a given iceberg is merely a function of collision duration tc which can be estimated using equation 3.23. The assumption of a ramp impact force greatly simplifies the required computation for predicting effective added mass. To evaluate the influence of the shape of the assumed forcing function on added mass estimates, one can consider collision forces arising from step and ramp forcing functions, between which the actual forcing function will likely lie. The collision duration predicted for a step forcing function will be one half of the value predicted for a ramp forcing function; thus, the actual collision duration will be no less than half the value predicted by equation 3.23 provided that all parameters of equation 3.23 are known. With regard to the influence of forcing function on effective added mass, Motora et al. have shown that the momentum-based effective added mass for a step force can be up to 30% greater than the effective added mass for a ramp force of equivalent duration; however, differences between equivalent added masses for step and ramp forcing functions are typically much smaller for most collision durations. In summary, the influence of the assumed forcing function shape will likely have only a small effect on predicted equivalent added masses. It is likely that differences between an assumed ramp forcing function and the actual forcing function for an iceberg collision will result in only minor differences between Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 67 effective added mass estimates. Collision Duration and Effective Added Mass by Time-Stepping Assumptions regarding the variation of impact force with time can be avoided if a numerical time-stepping procedure is used to determine collision duration and effective added mass. The motions of an iceberg during a collision can be rigourously evaluated using the equation of transient motion for a floating body (equation 3.1). For a head-on impact with a structure, the equation of motion for an iceberg can be written as: Am - F { t ) ~ ! - ~ L { t ~ T ) A ( r ) d r (3 30) ( } " M[l + Cm(oo)] ( 3 " 3 0 ) Equation 3.30 can be evaluated numerically to determine iceberg displacement and as- sociated impact force at any time during a collision. Numerical integration of impact force F with respect to time t and distance A can be used to obtain estimates for effec- tive added mass based on momentum (equation 3.14) and kinetic energy (equation 3.15) respectively. 3.2.4 Effective Added Mass Example Problem Effective added masses and relevant parameters have been evaluated for an example cylindrical iceberg having a diameter D{ of 100 m, a draft h of 50 m, and situated in a water depth d of 100 m. Frequency dependent added mass and damping for surge motions were computed using the program ICEMOT, and were also obtained from Yeung [119]. Due to facet size limitations, added mass and damping were available for a maximum frequency LO of 1.25 rad/s. For higher frequencies it was assumed that added mass and damping coefficients reached their infinite frequency values of 0.321 and 0 respectively at to = 3.0 rad/s, with intermediate values in the range 1.25 rad/s < to < 3.0 rad/s Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 68 estimated using linear interpolation. Note that the upper frequency limit of 3.0 rad/s was selected such that the following relationship from Wehausen [115] would be satisfied: roo / (n(u) - n(oo)) dw = 0 (3.31) Jo Frequency dependent added mass and damping for the example iceberg are presented in Figure 3.1. CJ Di/g 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Frequency co (s_1) Figure 3.1: Frequency Dependent Added Mass and Damping for 100 m x 50 m Iceberg The history term obtained from the sine transform of the frequency dependent damp- ing (equation 3.4) is presented in Figure 3.2, and is shown to have a decay time tj, of approximately 20 s for the example iceberg. Estimates of tL for other icebergs can be developed using dimensional analysis. If the assumptions of inviscid flow and a vertical circular cylinder are maintained, ti is a function of the following variables: tL = f (D{ , h , d , g , p , (3.32) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 69 In terms of dimensionless variables, equation 3.32 can be re-written as: f(h/Di , h/d, Pi/p) (3.33) The computed values for the example problem suggest that t^^jg/Di will be of the order of 10 for an iceberg having an aspect ratio hjDi of 0.5 and a relative draft h/d of 0.5. t y/g/Di 0 . 5 - 0 . 4 - > â€¢ 0 . 3 - r-4 0 . 2 - 0 . 1 - 0 . 0 - u o -fcj - 0 . 1 - - 0 . 2 - Figure 3.2: History Term for 100 m x 50 m Iceberg Effective Added Masses for Ramp Forcing Function Effective added masses for a ramp forcing function obtained from equation 3.29 are shown in Figure 3.3. For collisions of short duration the added mass is approximately equal to the infinite frequency value, while the zero frequency added mass is approached for collisions of long duration. These results are in agreement with those presented by Motora et al. [82] and Petersen and Pedersen [94]. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 70 In considering the general applicability of computed added masses, it is useful to note that the added mass of a vertical circular cylinder subjected to a ramp forcing function can be expressed as follows: Cm = f ( A , h , d , g , tc , p , Pi) (3.34) In terms of dimensionless variables: Cm = f (ji/Di , h/d , tc sJg~JDi , pifp^j (3.35) The results of Figure 3.3 indicate that infinite frequency added mass can be used in iceberg collision design when tcyJg/D{ < 0.5, while zero frequency added mass should be used when tcyJg/Di > 3. For design purposes the zero frequency added mass provides a convenient and somewhat conservative value for effective added mass. tc Vg/Dt 0 1 2 3 4 5 6 7 Collision duration tc (s) Figure 3.3: Effective Added Mass for 100 m x 50 m Iceberg, Ramp Forcing Function Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 71 Table 3.1: Input Parameters for Comparison of Ramp and Time Stepping Models Water depth, d 100 m Structure diameter, Ds 100 m Iceberg diameter, /J, 100 m Iceberg draft, h 50 m Iceberg height, ht 57 m Impact velocity, Vc 1.0 m/s Ice pressure, pcr 0.02 - 5.0 MPa Comparison of Ramp and Numerical Impact Force Models To assess the validity of assuming a ramp impact force function, collision parameters have been evaluated using both the ramp impact force assumption and a numerical time stepping procedure based on equation 3.30. Input parameters for the example problem are given in Table 3.1. Note that computations were made for input ice pressures varying between 0.02 MPa and 5 MPa. Estimates of effective added masses and collision durations for the ramp force func- tion were made using successive iterations of equation 3.23 and effective added masses presented in Figure 3.3. Due to the relatively minor influence of effective added mass on collision duration, convergence was always achieved in two iterations. After determining effective added mass for a given collision, the maximum ice indentation A c and maximum impact force Fc were evaluated using equations 3.17 and 3.18 respectively. Iceberg collisions were also evaluated using a numerical time-stepping procedure based on equation 3.30, with a time interval of 0.01 times the collision duration estimated from equation 3.23. The convolution integral of equation 3.30 was evaluated numerically at each time step. Iceberg velocities and displacements during collisions were obtained by Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 72 numerical integration of equation 3.30. At the end of each collision, the effective added mass was determined by the momentum balance of equation 3.14. Results from the ramp force and numerical time-stepping models are shown in Fig- ures 3.4 - 3.7. Collision durations presented in Figure 3.4 indicate that the ramp force model over-estimates collision duration by approximately 50%. This result corresponds with numerical impact force models by Bass et al. [7] and Duthinh and Marsden [25], which suggest that the ramp force model will under-estimate forces in the early stages of a collision, thus leading to over-estimates of collision duration. Effective added masses presented in Figure 3.5 are in good agreement. According to Motora et al. [82], ramp force added masses for collisions of short duration should approach the infinite frequency added mass. The results presented for the ramp function approach a value of 0.29 rather than the infinite frequency value of 0.32. Errors in the estimated frequency dependent added mass and damping coefficients (Figure 3.1) are the likely source of this discrepancy. Penetration distances and impact forces for the two models are virtually identical. The excellent agreement between the two models is primarily due to the relatively minor influence of effective added mass on penetration distance A c (equation 3.17) and impact force Fc (equation 3.18). The above results indicate that the assumption of a ramp input force function can be used to estimate effective added mass. An estimate of collision duration can be made using equation 3.23. The appropriate added mass for collision design (i.e. infinite frequency or zero frequency) can then be selected based on the dimensionless parameter tcJg/Di and effective added masses given in Figure 3.3. er 3. Hydrodynamic Aspects of Iceberg Collisions 100q C/) .2 10 it-> cO u p Q C o Vi â€¢ fâ€”t "o u 1- 0.1 Ramp Force Model Time Stepping Model 1 1 1â€”râ€”r T â€¢ T T j 1 0.01 0.1 ~I 1 â€” V I T [ Ice Pressure pcr (MPa) Figure 3.4: Collision Duration for Ramp and Time Stepping Models 10 0 .8 -1 Ramp Force Model Time Stepping Model 2 3 Ice Pressure pcr (MPa) Figure 3.5: Effective Added Mass for Ramp and Time Stepping Models Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 74 2 0 0 0 Ramp Force Model Time Stepping Model 1 2 3 Ice Pressure per (MPa) I 5 Figure 3.6: Impact Force for Ramp and Time Stepping Models 1 0 o â€¢ r-t c <u X) CJ I â€” I 0) o 0 . 1 0 . 0 1 Ramp Force Model Time Stepping Model ~i 1 1â€”iâ€”r~ *~iâ€”iâ€”râ€”iâ€”r" T | - i 1 râ€”r- 0 . 1 1 Ice Pressure per (MPa) 1 0 Figure 3.7: Ice Penetration for Ramp and Time Stepping Models Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 75 3.3 Iceberg Hydrodynamic Coefficients in Open Water Information regarding iceberg hydrodynamics in open water (i.e. away from other bodies) was compiled using program ICEMOT and data in the literature. Open water hy- drodynamic coefficients could be used to estimate impact velocities with small structures, and to make preliminary estimates of impact velocities with large structures. Parameters of primary interest included the following: 1. Surge response amplitude operators, 2. Wave drift force coefficients, 3. Zero frequency added mass coefficients, 4. Infinite frequency added mass coefficients, 5. Water drag coefficients. The above parameters were compiled for a wide range of input conditions. 3.3.1 Diffraction Computations of Wave Drift Force Coefficients and Surge RAO's Wave drift force coefficients and surge RAO's are necessary for predicting iceberg drift and oscillatory velocities. The surge RAO ([ is the amplitude of oscillatory surge motions in unit amplitude waves, while the wave drift force acting on an iceberg can be expressed in terms of a wave drift coefficient as follows: (3.36) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 76 where: Cw = wave drift force coefficient Although previous investigators have computed wave drift force coefficients and surge RAO's for specific cases (e.g. Arunachalam et al. [5], Hsiung and Aboul-Azm [40], Isaac- son [43], Lever and Sen [59], NORDCO [90]), a comprehensive set of coefficients was lacking. The compilation of hydrodynamic coefficients is made manageable if the assumptions of cylindrical iceberg geometry, linear wave theory, and inviscid flow are adopted. Linear wave theory can be considered valid for wave heights small relative to water depth and incident wavelength, while the assumption of inviscid flow will be valid if the iceberg diameter is large relative to the distance traversed by water particles during a wave period. Wave drift coefficients and surge RAO's can subsequently be considered as functions of 3 dimensionless parameters: CÂ«, = f ( h/Di , h/d , ) (3.37) Ci = f ( h/Di , h/d , ^ ) (3.38) Parameter Ranges Adopted for Computations The three independent parameters of equations 3.37 and 3.38 were selected such that each parameter would have a limited range of validity or interest. Due to hydrodynamic stability requirements, a maximum iceberg aspect ratio h/Di of 0.8 was adopted. The dimensionless draft h/d (draft/depth) has a maximum possible value of 1. Boundary element computations introduce a further requirement that the gap between the seabed and iceberg bottom should be at least one facet diameter; thus, h/d was limited to a maximum value of approximately 0.9. The range of interest for the frequency parameter Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 77 u>2Di/2g was based on limiting values for Cw and ([. For u2Di/2g < 0.5, the incident wavelength becomes large relative to iceberg diameter. Experimental results by Lever et al. [59] indicate that the iceberg moves as a water particle in this low frequency range, with the corresponding surge RAO from linear wave theory as follows: C l = U n h f l T ) ( 3 - 3 9 ) Numerical results from the program ICEMOT and other sources (e.g. Hsiung and Aboul- Azm [40], Kobayashi and Frankenstien [52]) indicate that wave drift force coefficients approach 0 in this lower frequency limit. As the frequency parameter co2Di/2g becomes greater than 3 the incident wavelength becomes small relative to the iceberg diameter, requiring a large number of boundary element facets to maintain facet diameters less than A/8. Oscillatory motions become relatively small and wave drift forces approach the following high frequency limit given by Maruo [74] for waves propagating in the x direction: (3.40) where: â€”b,bâ€” points on waterline where fiwi = 0Â° and 180Â° respectively fSw[ = angle between waterline plane tangent and wave direction (Â°) For a vertical circular cylinder, the high frequency limit for Cw is 1/6. The frequency parameter range adopted for the diffraction computations was 0.5 < uj2Di/2g < 3.14, with the upper frequency limit being imposed by restrictions on facet size and the lower frequency limit governed by the small degree of wave diffraction for smaller icebergs. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 78 Discussion of Computed Surge Response Amplitude Operators Computed surge RAO's are presented in Figures 3.8 - 3.12. Due to the absence of resonance effects for surge motions, the presented RAO's exhibit a relatively smooth variation with the frequency parameter w2Di/2g. It appears that the numerical surge RAO's model actual conditions reasonably well, with high frequency and low frequency values agreeing with expected trends. 0-0 I i â€¢ â€¢ i | â€¢ i â€¢ i | i â€¢ â€¢ . | i â€¢ â€¢ i | â€¢ i i i | â€¢ i â€¢ - | ' ' i ' | 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 u zDi/2g Figure 3.8: Computed Surge RAO's, Aspect Ratio h/Di = 0.1 High Frequency Surge Response Amplitude Operators A major limitation of the boundary element method for evaluation of iceberg surge response is the requirement that facet sizes be less than approximately one eighth the incident wavelength. For icebergs which are large relative to incident wavelength, a pro- hibitively large number of facets is required to model the iceberg surface. An alternative Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 79 2.0 1.5H o < K 1.0- s_ W 0.5- 0.0 0.0 0.5 1.0 i 1 1 1 1 i 1 1.5 2.0 u zDi/2g 2.5 3.0 3.5 Figure 3.9: Computed Surge RAO's, Aspect Ratio h/Di = 0.2 2.0-, 1.5H O < & 1.0- <u fao s- W 0.5- 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 3.10: Computed Surge RAO's, Aspect Ratio = h/Di = 0.4 Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 80 O < K <30 6 0 S i 3 CO 2.0-1 1.5- l.OH 0.5H 0.0 I 1 1 1 1 i 1 1 1 1 i 1 1 1 1 I 1 1 1 1 I 1 1 1 1 i 1 ' I 0.0 0.5 1.0 1 5 2.0 2.5 3.0 3.5 uDi/2g Figure 3.11: Computed Surge RAO's, Aspect Ratio h/D{ = 0.6 2.0-i o < oi l.o on !_, 3 W 0 . 5 H 0.0 h/d â€¢ = 0.89 o = 0.8 A = 0.4 + = 0.0 _ , â€¢ , â€¢ | â€¢, , i . | , , i , i | , , . . i 0.0 0.5 1 0 1.5 2.0 2.5 3.0 3.5 uDi/Zg Figure 3.12: Computed Surge RAO's, Aspect Ratio h/Dx = 0.8 Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 81 solution has been developed for estimating iceberg surge response at high frequencies (i.e. w2Di/2g > 3 ). It is conservatively assumed that iceberg added mass and damping approach their infinite frequency values of Cm(oo) and zero respectively for u2Di/2g > 3 . The surge response of an iceberg can subsequently be written as follows: Using the solution of MacCamy and Fuchs (see [103]), the amplitude of the wave exci- tation force acting on a surface piercing circular cylinder extending to the seabed is as follows: where: a = cylinder radius (m) J[ = derivative of Bessel function of the first kind of order 1 Y{ â€” derivative of Bessel function of the second kind of order 1 For large values of ka, the wave field and associated excitation force acting on the cylinder will be dependent only on the cylinder geometry near the waterline; thus, equation 3.42 can be used to predict the excitation force on a body with a circular waterline at high frequencies. Evaluation of equation 3.42 is simplified if the following high frequency approximation is used: u? M [1 + Cm(oo)] (i = Fe (3.41) (3.42) (3.43) Substituting equation 3.43 into 3.42, the excitation force Fe is as follows: (3.44) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 82 The iceberg surge RAO for high frequencies can now be evaluated as: 7T (ka)^(h/Di)[l + C m ( o o ) ] (3.45) If the deep water dispersion relation valid for k d > n is adopted, equation 3.45 can be re-written as: To verify the high frequency surge approximation, results from equation 3.46 and the numerical diffraction model were compared for various iceberg aspect ratios h/Di and are presented in Figure 3.13. A frequency parameter value io2Di/2g = 3.14 was adopted because both the numerical model and high frequency approximation were postulated to be valid at this frequency. Diffraction computation results for deep water (i.e. d â€” oo) were used for comparison; however, the computed RAO's of Figures 3.8 - 3.12 indicate no variation of with depth for u>2Di/2g = 3.14. Figure 3.13 indicates excellent agreement between RAO's given by the numerical model and high frequency approximation for h/Di > 0-3. The finite draft of the iceberg appears to significantly influence the excitation force for h/Di < 0-3 and u>2Di/2g <= 3.14, thus leading to conservative estimates of oscillatory surge motions. It is suggested that equation 3.45 be used in conjunction with Figures 5.31 - 3.12 for estimating iceberg surge response when uj2Di/2g < 4. Discussion of Wave Drift Force Coefficients Compared with computed surge RAO's, the wave drift force coefficients presented in Figures 3.14 - 3.15 exhibit a more erratic variation with the frequency parameter to2Di/2g. It should be noted that large wave drift forces occur at the natural heave frequency of each iceberg due to the influence of heave motions on wave drift forces, as discussed by Faltinsen and Michelsen [28]. In reality, viscous damping will attenuate heave motions (3.46) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 83 0.5-1 O 0.4- â€¢ = Diffract ion Model o <: OS 0.3- o = A p p r o x i m a t i o n 3 in o.o oo 0.2 0.4 Aspect Ratio h/Di 0.6 0.8 Figure 3.13: Comparison of RAO's for Numerical Model and High Frequency Approxi- mation and associated wave drift forces. Aside from heave resonance effects, the high and low frequency behaviour of the drift force computations appears as expected. The computed high frequency values are particularly good when compared with the asymptotic limit of 0.167 for a vertical circular cylinder. 3.3.2 Z e r o F r e q u e n c y A d d e d M a s s C o e f f i c i e n t s As indicated previously, the zero frequency added mass of an iceberg can be used for predicting the drift motions of an iceberg and for estimating kinetic energy dissipated during collisions of long duration. For a vertical circular cylinder, its zero frequency added mass coefficient is a function of two dimensionless parameters: Computed zero frequency added masses are presented in Figure 3.19. The smooth Cm = f(h/D,, hid) (3.47) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 84 0.6 * i 0.5 c 5 0.4- U - l O 0.3- u Q > h/d â€¢ = 0.50 o = 0.25 A = 0.125 + = 0.05 x = 0.0 i ' 1 1 â€” , â€” r 1.0 1.5 20 co2Di / 2g Figure 3.14: Computed Wave Drift Coefficients, Aspect Ratio h/Di = 0.1 nature of the curves in Figure 3.19 indicate that the computed added masses can be used as a convenient, reliable source of Cm values for design purposes. 3 . 3 . 3 Infinite Frequency Added Mass Coefficients For iceberg collisions of short duration, the infinite frequency added mass can be used to estimate dissipated kinetic energy. Infinite frequency added masses can also be used to estimate iceberg surge response at high frequencies using equation 3.46. The boundary element solution presented in Chapter 2 can be used to determine the added mass of a floating body for oscillations of infinite frequency. The following Green's function has been developed by Garrison and Berklite [32] for the infinite frequency potential: Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 CO*Di /Zg Figure 3.15: Computed Wave Drift Coefficients, Aspect Ratio h/Di = 0.2 O.6-1 3 uzDi / 2g Figure 3.16: Computed Wave Drift Coefficients, Aspect Ratio h/Di = 0.4 Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 86 9 4 J c s- Q <D > 0.6- 0.5 .a 0.4-1 HO O 0.3- 0.2- 0.1 0.0 0.0 0.5 i 1.0 1.5 2.0 u2Di / 2g h/d â€¢ = 0.86 o = 0.75 A = 0.6 + = 0.3 x = 0.0 2.5 3.0 3.5 Figure 3.17: Computed Wave Drift Coefficients. Aspect Ratio h/Di = 0.6 3 Co CD > (0 0.6- 0.5- 3 0.4-1 U - l V Q 0.3 â€¢ - 0.2 Q 0.1 0.0 h/d â€¢ = 0.89 o = 0.8 A = 0.4 + = 0.0 0.0 0.5 i 1 1.0 1.5 2 2.0 cfDi / 2g 2.5 -as 3.0 3.5 Figure 3.18: Computed Wave Drift Coefficients, Aspect Ratio h/Di = O.S Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 87 with terms in the above equation being the same as those given for equation 2.68. The infinite frequency Green's function of equation 3.48 was added to the program ICEMOT to enable the computation of infinite frequency added masses. Computed infinite fre- quency added masses for vertical circular cylinders are shown in Figure 3.20. Unlike the zero frequency values, infinite frequency added masses are virtually independent of relative draft h/d. The maximum value of the infinite frequency added mass coefficient is approximately 0.5, compared to a maximum of 1.0 for zero frequency. 3.3.4 Water Drag Coefficients As mentioned previously, an empirical water drag coefficient is required for comput- ing iceberg drift motions in the program ICEMOT. Icebergs typically have sufficiently Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 88 6 Co w CO S CU < l.O-i 0.8- 0.6- 0.4- 0.2- 0.0 â€¢ = 0.1 o = 0.2 h/Di A = 0.4 + = 0.6 x = 0.8 -e- -B- -e- -B- 0.0 0.2 0.4 0.6 Relative Draft h/d 0.8 1.0 Figure 3.20: Computed Infinite Frequency Added Masses low Froude numbers that wave-making resistance is negligible [117]. Reynolds numbers for drifting icebergs are typically super-critical; therefore, drag coefficients can also be considered independent of Re. The drag coefficient of a cylindrical iceberg thus can be considered as a function of two dimensionless parameters: Cd = f(h/D{, h/d) (3.49) For h/d â€”> 1, the flow around the iceberg will be essentially two-dimensional and the associated drag coefficient will be approximately 1.2 [9]. If the water is deep (h/d â€”+ oo), Ca will depend on the iceberg aspect ratio h/Di as illustrated in Figure 3.21. For aspect ratios representative of icebergs (i.e. h/D( < 0.85), Cj is approximately constant at 0.70 provided that the relative iceberg draft h/d does not approach unity. Consequently, a. Cd value of 0.70 has been used for icebergs in the present investigation. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 89 l.O-i â€¢a 0.8- 0.6- <4â€”I CD O O 0.4- cd 0.2- 0.0 0.0 2.0 4.0 6.0 Aspect Ratio h/Di. 8.0 10.0 Figure 3.21: Variation of Drag Coefficient with Aspect Ratio h/D, (from Blevins [9]) 3.4 Hydrodynamic Interactions between Icebergs and a Large Offshore Structure For iceberg impacts against a large structure, the presence of the structure can significantly modify the hydrodynamics of the iceberg. Added mass and velocity will both be affected as an iceberg approaches a large structure. Physical modelling studies by Lever et al. [59] and Salvalaggio and Rojansky [99] have shown that hydrodynamic effects can prevent an iceberg from impacting a large structure. Numerical studies by Isaacson [43] and Isaacson and Dello Stritto [48] have confirmed this phenomenon. A comprehensive numerical investigation was conducted to examine iceberg hydro- dynamics in the vicinity of a large structure. Main areas of concern include the following: 1. Variation of added mass with proximity to structure, Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 90 2. Influence of structure on current field and associated iceberg motions, 3. Variation of iceberg wave drift forces with proximity to structure, 4. Variation of oscillatory surge motions with proximity to structure. 3.4.1 Configuration for Modelling Hydrodynamic Interactions When evaluating the hydrodynamics of an iceberg in the vicinity of a large structure, the number of relevant variables is significantly greater than for the open water case. As an example, the oscillatory response of a cylindrical iceberg in the vicinity of a surface- piercing cylindrical structure extending from the seabed is considered. The assumptions of linear wave theory and inviscid flow adopted for equation 3.38 are maintained. Addi- tional parameters are required for describing the relative size of the structure and location of the iceberg relative to the structure such that the number of dimensionless variables is increased from three for the open water case to six for an iceberg near a structure: where: Da â€” structure diameter (m) xg,y3 = location of iceberg relative to structure (m) The large number of variables for the interaction problem makes it difficult to present general results such as those given for open water motions. Instead, a particular struc- ture configuration was selected for modelling hydrodynamic interactions. The modelled structure was a surface-piercing circular cylinder extending from the seabed, with a di- ameter Ds of 100 m situated in a water depth d of 100 m. Similar configurations have been studied by Isaacson [43], NORDCO [90], and Salvalaggio and Rojansky [99], thus (3.50) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 91 providing a basis for comparing results. Although various values for iceberg diameter Di were used for modelling hydrodynamic interactions with the structure, an aspect ratio h/Di of 0.5 was maintained for all icebergs. 3.4.2 Variation of Zero Frequency Added Mass Computations were made to examine the variation of iceberg zero frequency added mass for head-on approaches toward the example structure. The structure surface was modelled using 64 facets; thus, a typical facet dimension was approximately 20 m. Ice- bergs having diameters of 10 m, 20 m, 50 m and 100 m were modelled, with 81 facets for each iceberg. Due to limitations of the boundary element method, reliable computations could be made for a minimum separation distance of one half of a facet dimension, or approximately 10 m. Computed added masses presented in Figure 3.22 indicate that zero frequency added mass increases as an iceberg approaches a structure. This phenomenon corresponds with the results of Yamamoto [118] for two-dimensional flow around two circular cylinders. When considering the influence added mass variation on iceberg collision severity, consideration should be given to the kinetic energy of the fluid-solid system. For an iceberg drifting toward a structure in an otherwise still fluid, the total kinetic energy of the iceberg and and fluid system will be as given by Sarpkaya and Isaacson [103]: KE = ^ (1 + Cm) M V2 (3.51) For a conservative system where energy is constant, any increase in iceberg added mass must be accompanied by a corresponding decrease in iceberg velocity; thus, the kinetic energy of an iceberg upon impact will be equal to its energy far away from the structure. It should be noted that iceberg kinetic energy could change with proximity to the structure if external forces from currents and waves were acting on the iceberg. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 92 1.0 0 8 e W 0.6# cd S - T3 < 0.4 0.2- 0.0- -e- -fl- Ice D i a m e t e r â€¢ = 100 m o = 50 m A = 20 m + = 10 m -200 -150 -100 -50 Iceberg Loca t i on x9 (m) Figure 3.22: Variation of Zero Frequency Added Mass with Proximity to Structure 3.4.3 Variation of Infinite Frequency Added Mass Infinite frequency added masses in the vicinity of the structure were also evaluated. The results presented in Figure 3.23 show that infinite frequency added mass is not significantly affected by the presence of the structure; however, it is possible that the influence of the structure could be more pronounced for smaller separation distances. 3.4.4 Influence of Structure on Current Driven Impact Velocities The presence of a structure in a current field will affect water velocities and associated water drag forces near the structure. A series of numerical tests was conducted for current driven collisions with the example structure. The flow field around the structure was computed using the analytical solution of equation 2.92. Head-on iceberg approaches Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 93 Ice D i a m e t e r â€¢ = 100 m o= 50 m A = 20 m m sâ€”â€”aâ€” ihifl a a Aâ€”Â»- + = 10 m U . U -1 1 1 1 â€¢ 1 â€¢ â€¢ . â€¢ 1 . 1 . r - , , , , , , -200 -150 -100 -50 0 Iceberg L o c a t i o n x9 (m) Figure 3.23: Variation of Infinite Frequency Added Mass with Proximity to Structure were simulated using a time-stepping procedure commencing at a location where the modification of the flow field by the structure was negligible (i.e. 5 Da from the structure face). Collision trajectories were simulated for icebergs having diameters D, of 5 m, 10 m, 20 m, 50 m, and 100 m. Due to the minimal computational effort required for generating trajectories, four different aspect ratios were used for each iceberg diameter. It was assumed that each iceberg had a water drag coefficient d of 0.70. Zero frequency added masses were taken from Figure 3.19, and were assumed to be independent of iceberg proximity to the structure. Figure 3.24 illustrates the ratio of impact velocity Vc to free-stream current velocity Vwo for the simulated trajectories. For a 10% or greater reduction in impact velocity, the relative iceberg size Di/D, must be less than 0.3. The influence of aspect ratio h/Di appears to be very small and is due solely to the variation of added mass coefficients. Co w w CO -a -a < l.O-i 0.8 0.6- 0.4 0.2- Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 94 Rel. Ice Diameter Di/Ds Figure 3.24: Relative Impact Velocities for Current Driven Head-on Collisions 3.4.5 Variation of Wave Drift Forces Previous investigations (e.g. Kokkinowrachos et al. [53], Loken [65], Van Oort- merssen, [112]) have indicated that the wave drift forces acting on a body can be signifi- cantly modified by the presence of a second body. The significance of this phenomenon on iceberg drift motions has been illustrated by Isaacson [44]. Using the program ICEMOT, wave drift forces were computed for icebergs at various locations in front of the example structure. The iceberg dimensions and facet grids were identical to those used for studying the variation of added masses with proximity to the structure. Computed wave drift coefficients for a wave period T of 12 s are presented in Figure 3.25. It appears that wave drift forces for the larger icebergs are only moderately affected by the presence of the structure during a head-on impact. Wave drift coefficients for the 50 m and 100 m icebergs fluctuate near their open water values, while forces acting on the 10 m and 20 m icebergs exhibit a pronounced variation with location. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 95 The development of negative drift forces suggests that the 10 m and 20 m icebergs will not impact the structure if no current is present. This result is in agreement with experiments by Salvalaggio and Rojansky [99], who observed that wave driven icebergs would not impact a structure when the relative iceberg size Di/Da was less than 0.5. Their experiments also indicated that the occurrence of iceberg impacts was independent of the incident wavelength. c a; a> o a > 0.3 0 .2 0.0 - 0 . 1 - 3 0 0 - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 Iceberg Location x, - 5 0 Figure 3.25: Variation of Wave Drift Coefficients with Proximity to Structure To examine the influence of wave period, additional computations for the 20 m iceberg were made for wave periods of 10 s and 14 s, as shown in Figure 3.26. For each wave period the wave drift force exhibits a cyclic variation with distance from the structure, with a cycle length equal to half a wavelength. In addition to the cyclic variation, the wave drift forces acting on the 10 m and 20 ni iceberg decay with distance from the structure. Diffraction computations far away from the structure confirmed this observation. The following equation is proposed for modelling wave drift forces acting Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 96 on an iceberg in front of a large structure: Cw = Cwo + ACw exp[BCuxg] cos[2k(xg - CCw)\ (3.52) where: Acw = amplitude term Bcw = decay term (m _ 1) Ccw = phase term (m) Cwo = open water wave drift coefficient 0.20-1 0.15 3 c cu .9 0.05 I- 0.00 Q cc - 0 . 0 5 -0 .10 - â€¢ Wave period T - â€¢ = 10 s 0 = 12 s A = 14 s /â€¢erf \ 7Â°- \ H Â©A / * \ ^ Y A - 3 0 0 - 2 5 0 â€¢ - 2 0 0 - 1 5 0 i i i i - 1 0 0 â€¢ â€¢ â€¢ - 5 0 0 Iceberg Location x, Figure 3.26: Influence of Wave Period on Wave Drift Forces near Structure Coefficients of equation 3.52 have been evaluated for icebergs with aspect ratios of 0.5 approaching the example structure. Wave drift coefficients for icebergs approaching the structure were computed using the program ICEMOT. Coefficients of equation 3.52 were then determined using a least squares fit to the computed wave drift forces and are presented in Figures 3.27 - 3.30. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 97 3 U Q > cd (1) -i-> C 0) a o 0.25-1 0.20- 0.15- 0.10 0.05- 0.00 10 20 30 40 Iceberg Diameter D% (m) 50 Figure 3.27: Open Water Wave Drift Force Coefficient Cu 0.25-1 0.20- 3 c E 0.15- o u <D 0.10- T3 a | 0.05H 6 < 0.00- 10 â€”r- 20 30 â€”râ€” 40 - 1 50 Iceberg Diameter Dx (m) Figure 3.28: Wave Drift Force Variation Amplitude Coefficient Acu Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 98 O 1 O 3 5.0 4.0- 3.0- 2.0- 1.0- 0.0- , - - 0~ G - ' Wave Period T â€¢ = 10 s o = 12 s A = 14 s â€”r~ 10 20 30 Iceberg Diameter Di (m) - 1 â€” 40 Figure 3.29: Wave Drift Force Variation Decay Coefficient Bcu 3 d 200 150 .S2' 100- O U <D W (0 ji Cu 50 Wave Period T â€¢ = 10 s o = 12 s A = 14 s A A- â€¢ A- -G â€¢A- .â€¢A I 50 â€¢ A â€”Tâ€” 10 I 20 I 30 40 I 50 Iceberg Diameter Di (m) Figure 3.30: Wave Drift Force Variation Phase Coefficient Cc u Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 99 Wave drift coefficients presented thus far have been for icebergs having aspect ratios h/Di of 0.5. Because wave drift forces are primarily due to effects near the waterline of an iceberg, it is suggested that the results presented above will not vary greatly with aspect ratio. Wave drift forces near a structure for 20 m diameter icebergs with aspect ratios of 0.2, 0.5, and 0.8 have been computed for a wave period of 12 s. The results presented in Figure 3.31 indicate that wave drift forces do not vary significantly with iceberg aspect ratio. 3 CO c 'o O o Q > O.lO-i 0.05 0.00 -0.05- -0.10 -300 -250 -200 -150 -100 Iceberg Location x9 (m) Figure 3.31: Influence of Aspect Ratio h/Di on Wave Drift Forces near Structure Wave drift forces near a structure will also vary with the iceberg lateral offset relative to the direction of propagation of incident waves. Drift forces for a 20 m diameter iceberg have been evaluated for offset distances of 25 m and 50 m. It is evident in Figure 3.32 that drift forces in the direction of wave propagation do not vary significantly with the lateral position of the iceberg. In addition to longitudinal wave drift forces, lateral drift forces were encountered for the offset distances of 25 m and 50 m; however, the lateral Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 100 drift forces were negligible in comparison to the longitudinal forces. 0 10-1 Offset Distance 3 â€¢ = 0 m c 0.05- o = 25 m CD > CO * A â€¢ â€¢ â€¢ A = 50 m .A 0.10 T â€” i â€” i â€” . â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” r â€¢â€¢â€”I -300 -250 -200 -150 -100 Iceberg Location xg (m) -50 o Figure 3.32: Influence of Lateral Offset on Wave Drift Forces near Structure To examine the influence of wave drift forces on iceberg drift velocity, it is useful to consider the equilibrium velocity of an iceberg in collinear waves and currents. Taking the acceleration from wave drift and water drag forces to be zero (equation 2.62), the equilibrium velocity for a cylindrical iceberg in collinear currents and regular waves is as follows: Equation 3.53 can be used to assess whether negative wave drift forces will have a sig- nificant effect on the velocity of icebergs approaching a large structure. (3.53) Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 101 0 . 2 - 0.0-1â€”iâ€”iâ€”iâ€”â€¢â€”râ€”-â€”'â€”iâ€”'â€”iâ€”'â€”â€¢â€”1â€”1â€”iâ€”â€¢ â€¢ 1 â€” 1 â€” i â€” ' â€” â€¢ â€” â€¢ â€” 1 1 â€” i â€” â€” ' ' i - 3 0 0 - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 - 5 0 0 Iceberg Location z , Figure 3.33: Variation of Oscillatory Surge Motions near Structure 3.4.6 Variation of Oscillatory Surge Motions The diffraction computations used for obtaining wave drift forces also provide os- cillatory motions near the structure. Figure 3.33 shows the variation of surge response near the structure for a wave period of 12 s. The degree of variation of surge RAO's with proximity to the structure decreases with increasing iceberg size. Experimental results by NORDCO [90] were useful for confirming the numerical results for the large icebergs. Their investigation examined wave-induced motions of icebergs in the vicinity of a large cylindrical structure. The relative size of icebergs to the structure, D{/Da, was approximately 0.7 for all tests. It was concluded that the oscillatory motions were not significantly affected by the presence of the structure. Chapter 3. Hydrodynamic Aspects of Iceberg Collisions 102 3.4.7 Implications of Hydrodynamic Interaction Effects for Modelling Iceberg Collisions A comprehensive series of numerical tests has been conducted to investigate hydro- dynamic interaction between icebergs and a large structure. The numerical results have been confirmed by available experimental data. Zero and infinite frequency added masses exhibited little variation with proximity to the structure. It is suggested that open water added mass can be used for estimating added mass upon impact. For current driven iceberg motions, the modification of the flow field by the structure will have a significant effect on impact velocities only if the relative iceberg diameter Di/Ds is less than 0.3. The prediction of current driven iceberg trajectories near a cylindrical structure requires little computational effort and can prove to be very useful. Wave drift forces showed significant variation near the structure when Di/Ds was less than 0.5. The development of negative wave drift forces for Di/Ds values of 0.1 and 0.2 indicates that impacts will not occur if drift motions are solely wave driven. The wave drift forces acting on smaller icebergs exhibit a cyclic variation with distance from the structure. An observed cycle length of a half wavelength should be considered when evaluating the number of drift force computations required for accurate modelling of drift trajectories. Due to the large degree of computational effort required for predicting wave drift forces, it is suggested that equation 3.52 be adopted for estimating wave drift forces on small icebergs approaching a large structure. Oscillatory surge response for smaller icebergs showed fluctuations of approximately 10% as the icebergs approached the structure. For D(/D3 > 0.5, the presence of the structure caused little variation in surge response and surge motions could be considered constant. Chapter 4 Physical Modelling of Iceberg Drift Near a Large Structure The numerical model described previously can be used for predicting iceberg drift motions in waves and currents near a large structure. To validate the numerical iceberg drift model, suitable physical modelling data were required. NORDCO [90] conducted a series of experiments of wave driven iceberg motions near a large structure, with icebergs of comparable size to the structure. Salvalaggio and Rojansky [99] examined wave driven drift motions for icebergs of various sizes relative to a large structure. Neither of the above investigations considered the influence of currents on iceberg motions near the structure. To supplement existing physical modelling data, a series of experiments was con- ducted to examine iceberg drift motions near a large structure. Iceberg motions driven by both waves and currents were examined. Iceberg size relative to the structure, Di/Ds, ranged between 0.1 and 0.8. The influence of the structure on iceberg drift motions was of principal interest. Specifically, the variation of iceberg velocity and trajectory eccen- tricity approaching the structure were evaluated and compared to the predictions of the numerical model. 103 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 104 4.1 Similitude Requirements Similitude requirements for physical modelling of offshore structures are well es- tablished, and have been discussed in various textbooks (Chakrabarti [16], Newman [89], Sarpkaya and Isaacson [103]). In addition to geometric similarity requirements, kinematic and dynamic similarity requirements must also be met or approximated. The modelling of a cylindrical iceberg approaching a surface piercing cylindrical structure extending to the seabed is considered in greater detail here. For an iceberg approaching a structure in collinear waves and currents, the iceberg impact velocity is a function of the following variables: Vc = / (Di , Ds , h , d , e0 , p , pi , Vwo , g , H , uc , v) (4.1) where: e0 = initial approach eccentricity (m) v = kinematic water viscosity (m2/s) LOC = angular wave frequency in current (rad/s) In terms of dimensionless variables, equation 4.1 can be re-written as: Vc _ , (Dj_ ]}_ h e0 ft y/g~D~ ~ J \ D s ' D i ' d ' 1/2(A + D.) ' p ' Vwo to2cd ul II Vwo Dj V9 Di g ' g v For practical reasons, water must be used in the model. With the exception of Reynolds number Vwo Di / v, all of the above dimensionless variables can be modelled correctly if geometric similarity is maintained and the following relationship is obeyed: Xt = \ 1 ' 2 (4.3) Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 105 where: Xr, = length scale Xt = time scale Reynolds numbers for small-scale models are typically much smaller than those for prototypes; thus, viscous effects are accentuated in physical models. Scaling problems associated with Reynolds number effects can be alleviated if sufficiently high Re values (e.g. Re > 10, 000) can be achieved during experiments. In practice, Reynolds number effects are minimized by using the largest possible model scale for a given situation. 4.2 Description of Experiments The experimental investigation was conducted in the towing tank at the BC Re- search Ocean Engineering Centre. The drift motions of four model icebergs near a large offshore structure were examined. Both waves and currents were modelled in the physical investigation. 4.2.1 Towing Tank Dimensions of the BC Research towing tank are given in Table 4.1, with an elevation view of the tank given in Figure 4.1. The towing tank is intended primarily for model testing of ships, and is equipped with an instrumented towing carriage for measuring drag forces on models. The velocity of the towing carriage is controlled manually from a station located at one end of the towing tank equipped with a manual control potentiometer and a digital read-out of the towing velocity. A Digital PDP 11/23 mini-computer system situated on the towing carriage is available for recording carriage velocity and forces acting on towed models. A series of horizontal planks located along the sides of the tank Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 106 at water level is used to reduce reflections of waves produced by towed models. In addition to towing capabilities, the BC Research towing tank has a computer controlled wave generator which is capable of producing regular sinusoidal and random uni-directional waves. The wave generator is of the flapper type, and is thus most ap- propriate for generating deep water waves (i.e. X/d < 0.5). Water elevations in the tank can be measured using resistance-type wave probes which can be connected to a micro-computer and a chart recorder. An IBM Personal Computer situated on the side of the towing tank completes a closed-loop system between the wave generator and wave probes, enabling constant monitoring and re-adjustment of the sea state produced at a given test section. A sloped beach consisting of perforated sheet metal has been installed at the downstream end of the tank to minimize wave reflection. The beach design was taken from Miles and Murdey [78], who reported reflection coefficients of less than 5% for wave periods less than 2 s and wave heights under 100 mm. Due to the possibility of undesirable phenomenon occurring when generating waves in the towing tank, care must be taken when preparing for an experiment. Sarpkaya and Isaacson [103] warn that cross-waves can develop when incident wavelengths are smaller than the tank width. During iceberg drift tests conducted by NORDCO [90] significant interference was observed between waves diffracted by a model structure and the side walls of the towing tank. This interference was largely due to the significant blockage (25%) of the towing tank width by a model structure. To allow for generated waves to reach a freely travelling form, the test section for an experiment should be located a sufficient distance away from the wave generator. The present set of iceberg drift experiments was conducted at the mid-length section of the towing tank. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 107 4.2.2 Structure Model The structure model was intended to be representative of that used in Chapter 3 for numerical modelling of iceberg motions near a large structure. Recall that the structure used for numerical modelling had a diameter Da of 100 m and traversed the entire water depth of 100 m. Due to the limited towing tank width, the model structure was given a diameter of 0.5 m, thus introducing a model length scale A^ of 1/200 and a tank width blockage of 14%. The influence of the tank walls on currents and waves near the structure appeared to be negligible during experiments. The resulting structure was rolled out of 0.8 mm thick sheet aluminum. The draft of the model structure was 1.6 m, which was considered sufficient to produce the desired flow field in the upper 0.5 m of water. Elevation views of the structure and iceberg models are given in Figure 4.2. Table 4.1: Specifications for BC Research Towing Tank Length 67 m Width 3.7 m Depth 2.4 m Maximum carriage speed 4.52 m/s Maximum wave height 0.25 m 4.2.3 Iceberg Models Four model icebergs were constructed, with dimensions as given in Table 4.2. Each model was constructed to have an aspect ratio h/Di of approximately 0.5. The two smaller icebergs were made from aluminum disks, with the centres machined to obtained Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 108 0 10 m â€¢ â€¢ Horizonta l scale Vert ical exageration - 5x Wave Generator Beach 7d Docking area Figure 4.1: Elevation View of BC Research Towing Tank Model 1 Model 2 Model 3 Model 4 Structure -O- 0 1 m _ J Scale / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Figure 4.2: Elevation View of Structure and Iceberg Models Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 109 the desired model draft. The two larger models consisted of shells made of sheet alu- minum. Ballast weights were added to both models to obtain the desired drafts. Table 4.2: Dimensions of Model Iceberg: s Model 1 Model 2 Model 3 Model 4 Diameter, D{ (mm) 51 100 253 400 Draft, h (mm) 28 51 132 197 Total height, ht (mm) 50 100 253 445 Centre of gravity, zg (mm) -10 -19 -81 -131 Mass, M (kg) .058 .423 6.65 24.8 The inertial properties of the model aluminum icebergs were somewhat different than those for prototype icebergs, which are of uniform density. The ballast for each model iceberg could have been adjusted to simulate the centre of gravity and radii of gyration for a prototype iceberg; however, this was not considered necessary because the objective of the experiments was simply to validate the numerical model. Measured model inertial properties were subsequently given as input to the numerical model. 4.2.4 Model Iceberg Drag Coefficients It was mentioned in Chapter 3 that a circular cylinder with an aspect ratio h/Di < 2 travelling through water at low Froude numbers (Fr < 0.15) will have a drag coefficient value of approximately 0.70. Although it was postulated that this Cd value would be valid for the model icebergs, there was some concern regarding the influence of low Reynolds numbers and associated viscous drag forces on model iceberg drag coefficients. A series of towing tests conducted on the 400 mm diameter model indicated that a drag coefficient of 0.7 was valid for the model icebergs. Drag coefficients from the towing tests are shown in Figure 4.3. It should be noted that the iceberg model was allowed to rotate freely Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 110 in pitch during the towing tests. The increased drag with Froude number is likely due to greater pitch angles at higher towing velocities, with a maximum pitch angle of 11Â° experienced for the highest towing velocity. 1.0' Co 0.8- C O CJ> 0.6 0.4- C5C CD - 0.2 0.0 0.00 Reynolds Number Re 50000 100000 Â£3 IE u | â€¢ E I o â€¢ 0.05 0.10 0.15 Froude Number Fr 0.20 Figure 4.3: Drag Coefficients for 400 mm Diameter Iceberg Model 4.2.5 Environmental Conditions for Drift Studies The motions of icebergs were examined for conditions involving currents only, waves only, and combined waves and currents. The five different environmental conditions are given in Table 4.3. Tests with currents are denoted by C, and tests including waves are denoted by W. To simulate currents around a fixed structure, the model structure was moved through the water using the towing carriage. Iceberg trajectory statistics reported by Pearson and Ro [93] indicated that a suitable prototype current velocity would be 0.5 m/s, Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 111 Table 4.3: Environmental Conditions for Model Tests c W l W2 WC1 WC2 Incident current, Vw0 (m/s) 0.1 0.0 0.0 0.1 0.1 Wave height, H (mm) 0 21 61 19 59 Wave period, T* (s) 0.0 0.7 1.0 0.7 1.0 "relative to structure corresponding to a model velocity of 0.04 m/s through Froude number scaling. The tow- ing carriage was capable of maintaining steady speed for a minimum velocity of 0.10 m/s. Consequently, a towing velocity of 0.1 m/s was used, corresponding to a relatively large prototype current of 1.4 m/s. To examine the influence of Reynolds number effects on the experiments it is useful to consider the Reynolds number VW0Dsjv for the structure in the current. For a prototype structure diameter of 100 m and a current velocity of 0.5 m/s, the Reynolds number is approximately 50 x 106. In comparison, the Reynolds number for the model structure is approximately 50,000. Corresponding Reynolds numbers for the model icebergs would be significantly smaller (e.g. 0.01-0.1 times smaller) due to the smaller iceberg sizes, and more importantly, due to the small relative velocities between the icebergs and the water. Detailed discussions on Reynolds number effects for circular cylinders have been given by Sarpkaya and Isaacson [103] and Simiu and Scanlan [105]. For the model structure and icebergs in the present study, Reynolds number effects cause the model drag coefficients to be somewhat larger than prototype drag coefficients. In addition, the wakes behind the models would be significantly different from the wakes behind the prototypes. At the relatively low Reynolds numbers corresponding to the experiments, periodic shedding of vortices alternating from each side of the model may arise and induce signif- icant periodic lift forces perpendicular to the direction of the relative velocity between Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 112 the water and iceberg. This vortex shedding typically occurs for Re values between 30 and 5000. In contrast, the turbulent wake behind the prototype icebergs and structure would induce relatively small lift forces. For tests involving waves, regular waves with nominal heights of 20 mm and 60 mm (4m and 12 m prototype) were generated. Wave periods of 0.7 s and 1.0 s were based on the following equation obtained by LeBlond et al. [57] relating wave periods to wave heights for observed ocean wave records from several sites off the Canadian east coast: Tp = 13.88 y/H. I g (4.4) For tests with simultaneous waves and currents, the wave periods of 0.7 s and 1.0 s were maintained relative to the moving structure. Corresponding wave periods of 0.74 s and 1.06 s were generated by the fixed generator, and were calculated using the following equation from Sarpkaya and Isaacson [103]: LOC = LO + Vwo k (4.5) where: Lo = angular wave frequency in reference frame moving with current (rad/s) LOC = angular wave frequency in fixed frame (rad/s) For the present series of experiments in which the structure was towed to simulate a current, LO is the frequency of the wave generator motion and LOC is the frequency of waves encountered by the structure. 4.2.6 Experimental Procedure Iceberg model drift motions were observed for trajectories commencing in front of the structure. For tests involving only wave driven motions, each model was launched from Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 113 a platform extended across the towing tank. The platform was located approximately 3 m in front of the structure. For tests requiring movement of the towing carriage and structure to simulate cur- rents, the icebergs were launched from a cantilever extending partially across the tank width. The cantilever was placed such that the structure and towing carriage could pass by freely. The starting point of the structure for each test was approximately 8 m from the launch point. The structure would then be accelerated to a steady velocity of 0.1 m/s, with the iceberg being launched when the structure was approximately 4 m away. In the planning stages of the experimental program, it was considered desirable to observe icebergs with both head-on and eccentric approaches. During the initial series of tests it was found that head-on approaches were difficult to obtain. It was ultimately decided to launch each model from a position centred with the structure. Despite the ini- tial centred iceberg position, significant lateral iceberg motions were observed for almost all tests. A summary of the number of tests conducted for each model and environmental condition is given in Table 4.4. Table 4.4: Number of Tests Conducted for Each Model and Environmental Condition C W l W2 WC1 WC2 50 mm model 8 6 9 12 15 100 mm model 8 6 7 10 15 250 mm model 20 6 9 11 15 400 mm model 6 6 8 0 0 Iceberg motions were monitored using an overhead video camera, with details of the camera configuration given in the next section. Figure 4.4 shows the 400 mm iceberg approaching the structure as viewed from the camera position. An electronic timer attached to the video camera displayed the time on the video tape to a resolution of Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 114 0.01 s. The motions of each iceberg were recorded until the iceberg drifted outside the camera field of view or until the iceberg drifted behind the structure. Figure 4.4: Overhead View of 400 mm Iceberg Approaching Structure 4.3 Digitizing of Iceberg Trajectories The videotape recorded during the experiments provided a permanent record of ob- served drift trajectories. A micro-computer system was developed for digitizing the drift trajectories from the videotape. Although the model iceberg motions consisted of both drift and oscillatory components, only the drift motions were of interest. The restriction of drift motions to the horizontal plane allowed for trajectory co-ordinates to be obtained using observations from a single camera. A time interval between digitized co-ordinates of the order of one second was considered adequate for analysing drift trajectories. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 115 4.3.1 Video Camera Configuration The video camera was placed on a balcony above the towing tank at an elevation of 5.2 m above water level and at a horizontal distance of 2.7 m from the centreline of the towing tank. This location and the settings of the camera correspond to a field of view at the waterline plane which was approximately 4 m x 3 m. All relevant dimensions were measured such that co-ordinates in the waterline plane could be evaluated from the video images as indicated below. 4.3.2 Reduction of Video Co-ordinates A suitable method was required for reducing observed co-ordinates from video images to co-ordinates in the waterline plane. The following three co-ordinate systems illustrated in Figures 4.5 and 4.6 were adopted: x-y-z: x-y plane coinciding with waterline plane. Origin coinciding with video image centreline (i.e. camera is in y-z plane), x'-y': Plane parallel to video image, passing through a given observation point. Origin aligned with x-y-z system origin, x"-y": Video image plane. Origin at the centre of the video image. Dimensions given in terms of image pixel units. Note that both the x'-y' and x"-y" planes are normal to the centreline of the video camera field of view and that the location of the x'-y' plane is dependent on the location of a given observation point. Scaling from the video image to the x'-y' system for a point can be easily carried out as follows: x' = ksLx" (4.6) Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 116 y' = KLy" (4.7) where: ks â€” video image scale factor (/pixel) L = distance from camera to x' â€” y' plane (m) x', y' = co-ordinates in x' â€” y' plane (m) x", y" = co-ordinates in x" â€” y" plane (pixels) The scale factor ks is dependent on the video camera system and zoom lens setting. The distance L from the camera to the x'-y' plane can be expressed as: L â€” Lt + y sin 9 â€” z cos 9 (4.8) where as indicated in Figure 4.6: Lx = distance from camera to x â€” y â€” z origin (m) 9 = angle of camera orientation from vertical (Â°) Equations 4.6 and 4.7 can then be re-written in terms of the x â€” y co-ordinate system: x â€” ks (Lt + y sin# â€” z cos 6) x" (4.9) y cosO + z s'm9 = ks (Lt + y sm9 â€” z cos9) y" (4-10) Actual reduction of video image co-ordinates can be done by re-writing equation 4.10 in closed form as: ks (Lt - z cos9) y" - z sm9 . y = Q 1, n â€¢ n (4-U) cos 9 â€” ks y" sin 9 Thus, the iceberg drift co-ordinates x and y can be determined provided that the other terms of equations 4.9 and 4.11 are known, including the z co-ordinate of the reference point on the iceberg used for digitizing. The scaling factor ks can be determined using two Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 117 observation points having known y, y", and z values. For the present set of experiments, a survey rod was laid across the top of the towing tank (0.3 m above water level) to provide reference points for determining the scale factor kâ€ž. C a m e r a Waterline p lane xâ€”y Figure 4.5: Elevation View of Video Image Planes 4.4 Micro-Computer Digitizing System The analysis of observed iceberg trajectories required that iceberg locations be dig- itized from the recorded videotape. A schematic of the system developed for digitizing iceberg locations is shown in Figure 4.7. Using the framegrabber, a single frame from the videotape can be displayed on the monitor of the micro-computer. A cursor is superim- posed on the monitor image, and can be moved anywhere on the image using a mouse. An iceberg location can be recorded by moving the cursor to a reference point on the iceberg and clicking the mouse. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure Elevation View 118 Camera Observation pointy Plan View Camera + y > + Observation point Figure 4.6: xâ€”y and x'-y' Co-ordinate Systems and Relevant Dimensions Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 119 Monitor (Optional) Video Cassette Recorder Framegrabber Micro-Computer Figure 4.7: Schematic of Video Digitizing System 4.4.1 Description of Hardware The system hardware shown in Figure 4.7 consists of standard off-the-shelf com- ponents. The video cassette recorder (VCR) uses the VHS tape format. The micro- computer is an Apple Macintosh II, which has a Motorola 68020 processor, Motorola 68881 co-processor, and 5 megabytes of random access memory (RAM). A Data Translation Limited QuickCapture framegrabber board was installed in an expansion slot of the Macintosh II. The framegrabber converts the analogue signal from the VCR to a digital pixel map of a single frame which is stored in RAM resident on the framegrabber board. The resolution of the QuickCapture pixel map is 640 pixels horizontally x 480 pixels vertically, corresponding to the resolution of the Macintosh monitor. The intensity of each pixel is stored as a grey level varying between 0 and 255. The on-board RAM of the QuickCapture hardware is 512 k, which is sufficient for storing a single video frame. A product review of QuickCapture has been given by West and Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 120 Newton [116]. 4.4.2 Description of Software Included with QuickCapture is software for various tasks, including displaying video images on the Macintosh II screen and storing images on disk. The software is intended primarily for graphic arts applications. Complementary software products for graphics work is available from other vendors. A review of available commercial software indicated that no products were suitable for the present digitizing application. Consequently, it was decided to develop a program for digitizing iceberg motion co-ordinates. The initial step in the development of the digitizing software was the selection of a suitable programming language. The acquisition and display of video images with QuickCapture is carried out using memory-mapped input and output (see Lawrence and Mauch [56]). Of the various common programming languages, C is among the most suitable for operation of memory-mapped systems. Graphical display of images on the Macintosh monitor was also a consideration when selecting a programming language. Graphics programming with the Macintosh II is most readily accomplished using the Macintosh Toolbox (see Apple [3]), a set of subroutines residing in read-only memory (ROM). Although documentation examples for the Toolbox are written in PASCAL, translation to C is very straightforward. Consequently, the C programming language was selected for writing the digitizing software. The resulting digitizing program ICEDIGITIZE was written to obtain iceberg co- ordinates for a given series of tests. The program consists of two distinct phases. In the first part of the program, input regarding the camera configuration and environmental conditions of a test series is read. The program user is asked to click the mouse on two reference points on a captured frame so that the scale factor ks for equations 4.9 and Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 121 4.11 can be determined. A lateral reference point on the structure is taken to determine the location of the structure relative to the centreline of the screen image. The second part of the program obtains iceberg and structure co-ordinates during the drift experiments. Beginning with the first frame of a test, the user aligns the mouse with the centre point of the iceberg at the waterline plane. The iceberg location is stored in memory, and the framegrabber automatically displays the next frame. The VCR is run continuously during digitizing; thus, the time interval between frames can be determined using the internal clock of the computer. The time interval between captured frames is typically 3 s. Digitizing of an iceberg location is indicated by a single mouse click. For some frames the location of the structure is also required. A double mouse click will store the structure location, and will delay the frame advance until a single click indicating an iceberg location is detected. Once all points for an iceberg trajectory have been entered, the program user can request that the digitized co-ordinates be written to an output file. -The user is asked to input the time displayed on the last video frame, thus allowing the display time for each iceberg point to be stored. One drawback of the digitizing system was the 3 s time interval between digitized points. A finer time interval was desired in many cases, particularly when the structure was being towed to simulate a current. More frequent observations for a given trajectory were obtained by rewinding the videotape and repeating the digitizing procedure several times. 4.5 Error Analysis of Digitized Co-ordinates Digitized iceberg co-ordinates were prone to various errors, of which the two primary sources were: Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 122 1. Variation of the mouse location from the actual point to be digitized, 2. Variation of the z co-ordinate of an observation point from its assumed constant An analysis was conducted to determine the magnitudes of errors from both of the above sources. 4.5.1 Mouse Operation Error Co-ordinates digitized using a mouse are subject to errors by the human operator. The magnitude is dependent mainly on the resolution of the image from which points are digitized. A trial of the digitizing program was conducted to examine the variability of digitized co-ordinates. The location of a stationary iceberg model was digitized 14 times. The digitized x" and y" co-ordinates had standard deviations oxn and a yÂ» of 0.7 pixel units. The magnitude of the mouse operation error for reduced co-ordinates can be deter- mined through equations 4.9 and 4.11. It should be noted that mouse operation error is not only caused by the error in digitizing a single point, but is also caused by errors in digitizing two reference points used to determine the scaling factor ks. The error analysis is simplified by approximating y" and z as zero in equation 4.11. The y co-ordinate for a digitized point can be estimated in terms of the distance between two reference points, denoted A y s , in the x-y plane, and Ay" in the image plane, as shown in Figure 4.8. The y co-ordinate is then: value. Ay, II (4.12) y = Ay? y where: Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 123 A y s = distance between reference points for evaluating ks (m) Ay" = image distance between reference points (pixels) If the reference image distance Ay" and the digitized image co-ordinate y" have errors of t" and e" respectively, the reduced y co-ordinate will have the following value: (y + <y) = ( A y f + e,, } (y" + O (4-!3) where: ey = error in reduced y co-ordinate (m) t" = error in digitized y" co-ordinate (pixels) e" = image error for reference distance (pixels) Applying a first-order binomial expansion for the denominator and discarding terms of order e2, the error due to mouse operation is: ey = e" ^ - es' (4.14) Ay;' 3 Ay'l K ' Note that each of the above error terms can have a positive or negative value. The magnitude of the mouse error term e" can be taken to be approximately 0.7 pixels, which corresponds to the standard deviation oy, obtained from digitizing a point several times. The magnitude of the error term e" for the image distance between two reference points can be estimated by assuming t" to have a normal distribution, thus giving <?&yn equal t O 0~y". When digitizing reference points for evaluating the scale factor the reference distance A y s was 1.0 m, with a corresponding image distance A y " of approximately 300 pixels. The standard error of reduced y co-ordinates due to mouse operation can thus be approximated from equation 4.14 as: <Ty = 2.3 mm + 0.003 \y\ (4.15) Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 124 Camera Video image plane x"â€”y t y Reference point 1 ^_ Waterline plane x-y Figure 4.8: Elevation View of Reference Distance Aj / 5 and Image Distance A y " Only points with \y\ < 0.5 m were of interest for the observed model trajectories; thus, the maximum standard error oy due to mouse operation is approximately 4 mm. A similar analysis could be conducted to assess errors in reduced x co-ordinates. Due to the inclusion of y in the equation for reducing x co-ordinates (4.9), the error analysis would be significantly more complex. However, it is expected that the standard error of x reduced co-ordinates due to mouse operation error is approximately the same as for y co-ordinates. 4.5.2 Error Due to Variation of z Co-ordinate When evaluating iceberg co-ordinates using equations 4.9 and 4.11, it was assumed that the observation point for each iceberg remained at the still waterline z = 0. Wave- induced motions caused the reference points on the icebergs to vary significantly from the assumed elevation of zero. Deviations for smaller icebergs likely approached the wave Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 125 amplitude, which was equal to 30 mm for the largest waves. The influence of varying z values on the calculated values of the reduced co-ordinates can be estimated using equations 4.9 and 4.11. The sensitivity of y co-ordinates to z fluctuations is: _ ks y" cos 6 + sinfl t y ~ ~tz cos9 - ks sinOy" [ } where: e2 = variation in z from assumed value (m) For the camera configuration used in the experiments, equation 4.16 exhibits little varia- tion with y". The error in y co-ordinates for the experimental configuration can thus be reduced to: ey = -0.5 ez (4.17) The corresponding standard error due to elevation variations is 0.5 oz, which will have a maximum value of approximately 15 mm for the smaller models in the 61 mm waves. The reduced x co-ordinates obtained from equation 4.9 are much less sensitive to variations in z values: Â£x = ks [ty s'mO â€” e2cos0] x" (4.18) Substituting known values into the above equation, the x error reduces to: tx = -0.00057/pixel x" ez (4.19) If x" and es have their maximum magnitudes of 320 pixels and 30 mm respectively, then the standard error of co-ordinates a; due to variation of z can be estimated to be 5 mm. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 126 4 . 5 . 3 Summary of Digitizing Errors The above analysis indicates that the standard error for digitized co-ordinates is approximately 10 mm. The largest co-ordinate errors will occur for y values of smaller models in large waves, with the standard error ay approaching 20 mm. Additional errors are introduced by the recorded time values of digitized points, which have a resolution of 0.1 s. For an iceberg model travelling at 0.1 m/s, the displacement error due to an average time error of 0.025 s will be 2.5 mm; thus, the error due to time resolution will contribute slightly to the error of reduced co-ordinates. 4.6 Variability of Environmental Conditions The variability of simulated currents and waves in the towing tank must also be considered when evaluating results from the physical modelling tests. Fluctuations in wave paddle motions and wave reflections from the ends and sides of tank were possible sources of irregularities for the generated wave conditions. Because the towing carriage was controlled manually using a potentiometer and digital readout of carriage velocity, deviations in simulated current velocities also existed. Wave records for each set of wave conditions were obtained from a resistance-type wave probe located in the test section that was connected to an amplifier and chart recorder. Due to the regular, sinusoidal nature of the waves generated for the experiments, measurement of wave heights from the chart record was straightforward. Chart records obtained for each of the four wave conditions indicated that the standard deviation of wave height for each record was between 2 mm and 3 mm, and appeared to be independent of wave height. The towing carriage velocity for each test was estimated by taking a least-squares Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 127 linear fit to observed structure co-ordinates. The estimated towing carriage velocities for all tests with a simulated current had a mean value of 0.110 m/s (compared with the target value of 0.1 m/s) and a standard deviation of 0.018 m/s. Although it was assumed that the towing velocity was constant for each test, the standard deviation of 0.018 m/s among tests can likely be considered as an upper bound to the variability of current velocity during a given test, with the actual variation of current velocity for a given test being closer to 0.01 m/s. To estimate the influence of variable current velocity and wave height on iceberg drift velocity, it is useful to consider equation 3.53 which gives the equilibrium velocity of an iceberg drifting in collinear waves and currents. If it is assumed that the iceberg responds instantaneously to changes in wave height, then the wave-induced iceberg drift velocity will have the same coefficient of variation as the incident wave height (approximately 15% for the 20 mm waves, 5% for the 60 mm waves). In reality, the iceberg drift velocity does not respond instantaneously to changes in wave height but requires time to accelerate to a new equilibrium velocity; thus, the variability of iceberg drift velocity due to variation of incident wave height is likely to be no greater than 5%. Because currents were simulated by towing the structure, the variability of the effective current-induced iceberg drift velocity during a given test is equal to the variability of the towing carriage velocity (approximately 0.01 m/s for a mean towing velocity of 0.11 m/s). In summary, the variability of environmental conditions likely resulted in a drift velocity fluctuation of the order of 10% during each test. 4.7 Analysis of Observed Trajectories Once digitizing of the iceberg co-ordinates from the videotape was completed, the observed trajectories were analysed to determine the influence of the structure on drift Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 128 motions of approaching icebergs. 4.7.1 Reduction of Iceberg Co-ordinates Relative to Structure The iceberg co-ordinates obtained from the videotape were subsequently reduced to those relative to the structure. Reduction of iceberg co-ordinates was particularly im- portant for tests involving movement of the structure to simulate a current. Due to the nature of the carriage control system, the structure velocity deviated from the desired value of 0.10 m/s. The observed structure trajectory for each test was approximated with a least squares linear fit; thus, it was assumed that the structure velocity was con- stant for a given test. Iceberg co-ordinates relative to the structure were then evaluated accordingly. 4.7.2 Plotting of Iceberg Trajectories The iceberg trajectory and displacement versus time relative to the structure were plotted for each test, as shown in Figure 4.9. The iceberg models were drawn to scale on the trajectory plots at the initial observation point, final observation point, and ob- servation point nearest the structure. A plot of the iceberg nearest the structure was very useful for indicating the occurrence of collisions and for indicating possible errors in reduced co-ordinates (e.g. overlap between the structure and iceberg). Qualitative observations of the plotted iceberg trajectories were useful for indicating whether collisions occurred during a given series of tests. Results from qualitative ob- servations are presented in Table 4.5 and indicate that the likelihood of impact appears small for Di/Ds < 0.2 when only waves are present. For the smaller wave condition, the two smaller models veered away from the structure long before impact could occur. A representative trajectory is shown in Figure 4.10. It appeared that the two smaller Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 129 0.6 0.4 H -0.4 H â€” 0.6 - j â€” i â€” i â€” i â€” i â€” | â€” i â€” i â€” i â€” i â€” | â€” i â€” i â€” i â€” ' â€” | â€” i â€” ' â€” 1 â€” 1 â€” i â€” 1 â€” 1 â€” 1 â€” 1 â€” I -2.0 -1.5 -1.0 -0.5 0.0 0.5 X (m) 0.5-, â€”2.0 - | â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” i â€” \ â€” i â€” i â€” i â€” i â€” i â€” i â€” ' â€” 1 â€” 1 â€” i 10 15 20 25 30 35 Time (s) Figure 4.9: Observed Iceberg Trajectory and Displacement vs. Time Graphs Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 130 Table 4.5: Occurrence of Impacts for Experiments Di/Da = 0.1 0.2 0.5 0.8 Currents Yes Yes Yes Yes Waves 1 (H = 21 mm) No No Yes Yes Waves 2 (H = 61 mm) Maybe Maybe Yes Yes Wave-currents 1 (H = 19 mm) Yes Yes Yes Yes Wave-currents 2 (H â€” 59 mm) Yes Yes Yes Yes models could make glancing impacts under the large wave condition. The qualitative observations are in general agreement with the numerical iceberg drift model and with the experimental observations of Salvalaggio and Rojansky [99]. Salvalaggio and Rojansky observed that wave driven icebergs near a structure would not impact when D{/D3 was less than 0.5. The numerical model results of Figure 3.25 indicate that negative wave drift forces can develop in front of a structure when D^/Ds is less than 0.4; thus, the experimental results observed for wave driven iceberg drift are as expected. It was also shown in Chapter 3 that the influence of the structure on current driven motions is negligible for most icebergs. The experimental observations confirm this result. For experiments combining waves and currents, the currents appeared to have a dominant effect on iceberg motions. 4.7.3 Estimation of Impact Velocity and Eccentricity The influence of the structure on observed model iceberg drift motions was of pri- mary interest. Specifically, the variation of iceberg velocity and approach eccentricity (see Figure 4.11) with proximity to the structure were to be considered. The approach eccentricity of an iceberg at a given point is defined as the closest distance between the Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 131 y component <s.u-| ' 1 ' 1 â€¢ 1 ' 1 ' 1 ' 1 ' I 1 I 0 20 40 60 80 100 120 140 160 Time (s) Figure 4.10: Observed Trajectory for Wave Condition 1, Di/Da = 0.1 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 132 iceberg and structure centres if the iceberg velocity (speed and direction) remains con- stant; thus, an approach eccentricity of zero is representative of a head-on approach. It should be noted that approach eccentricity varies with time as the iceberg location and velocity vary. A method was required for estimating observed iceberg drift velocities in the far-field (i.e. far away from the structure) and near-field (i.e. near the structure). Various meth- ods were considered for estimating observed drift velocities. Although finite difference methods could have been used for estimating observed drift velocities, it was felt that oscillatory motions in waves and errors in observed positions would lead to inaccurate estimates of drift velocities. It was decided that a least squares technique would provide the best estimate of model drift velocities. Far-field and near-field segments of observed trajectories were approximated as follows: xF(t) = AFX t2 + BFxt + CFX (4.20) yF(t) = AFY t2 + BFyt + CFY (4.21) ' xN(t) = ANX t2 + BNxt + CNX (4.22) yN(t) = ANY t2 + BNyt + CNY (4.23) where: AF,BF,CF = approximating coefficients for far-field motions AN,BN,CN â€” approximating coefficients for near-field motions XFIVF â€” iceberg co-ordinates in far-field (m) XNIVN = iceberg co-ordinates in near-field (m) Equation coefficients for the far-field were evaluated using a least squares fit to the first four points of an observed trajectory. To reduce the degree of numerical error, it was also stipulated that the observation points representing the far-field motions must traverse a er 4. Physical Modelling of Iceberg Drift Near a Large Structure 133 Far-field Eccentricity + + Structure + = Observed iceberg co-ordinate Near-field Eccentricity Structure Figure 4.11: Far-field and Near-field Eccentricities of Approach for an Iceberg Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 134 distance of at least 0.5 m; thus, more than four observation points were used in some cases. The quadratic approximation to the near-field trajectory was based on the iceberg observation point closest to the structure taken together with previous three observation points. A minimum traversing distance of 0.3 m was specified for the near-field obser- vation points. In cases where contact with the structure appeared possible, steps were taken to ensure that all near-field observation points were taken prior to impact. Iceberg velocity components in the far-field and near-field, corresponding to the time of the last observation point for each of the respective fields, were estimated by differ- entiating equations 4.20 - 4.23. The eccentricity of approach, illustrated in Figure 4.11, was calculated for each field as follows: eF CN VF - xF vFx \VN - Xn y Â£ Gfe) +1 (4.24) (4.25) where: eF = far-field eccentricity of approach (m) ejv = near-field eccentricity of approach (m) VFx,VFy = far-field velocity components (m/s) VJvi, V/vy = near-field velocity components (m/s) The corresponding dimensionless eccentricity of approach was given as: ' = 1/2 ( A + D.) ( 4 ' 2 6 ) A dimensionless eccentricity of 0 is representative of a head-on approach, while a value greater than 1 is indicative of no contact occurring between the iceberg and structure if the iceberg direction remains constant. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 135 4.7.4 Error Estimates for Velocities and Eccentricities The estimated iceberg velocities and eccentricities were subjected to errors caused by the digitized iceberg co-ordinates and the variability of the environmental conditions. It was shown previously that each digitized co-ordinate had. a standard error of ap- proximately 10 mm, and that the variable environmental conditions led to x velocity fluctuations of approximately 10%. Recalling that far-field and near-field velocity estimates were obtained from data points over distances of at least 0.5 m and 0.3 m respectively, a conservative estimate of the error in estimated velocities can be made by assuming that velocities were evaluated using a finite difference scheme as follows: V = X2 " X l (4.27) t2 ~ h where: ti = time of observation point at beginning of field (s) t2 = time of observation point at end of field (s) Xi = co-ordinate of observation point at beginning of field (s) x2 = co-ordinate of observation point at end of field (s) The corresponding error in the estimated velocity due to the error of digitized co-ordinates is as follows: ev = Â£ ! 2 " 'Xi (4.28) t2 â€” H where: ty â€” error in velocity estimate (m/s) Dividing equation 4.28 by equation 4.27, the error in the estimated velocity can be Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 136 expressed as a fraction of the velocity: V x2 â€” X\ â€” e (4.29) Taking the standard error for a digitized co-ordinate to be 10 mm, the numerator in the above equation has an expected magnitude of 14 mm, while the denominator is of the order of 0.3 m. Thus, the error in estimated velocities due to digitized co-ordinate errors is approximately 5%. The total error in estimated velocities caused by the variability of both digitized co-ordinates and environmental conditions is approximately 15%. Estimates of errors in eccentricities obtained from equations 4.24 and 4.25 can be made most easily by considering the limiting cases where Vx/Vy approaches zero and infinity: Thus, the estimated error in the approach eccentricity e due to the digitized co-ordinate error is approximately 10 mm. From equation 4.26, the corresponding error in the esti- mated dimensionless eccentricity e' is approximately 0.04. 4.7.5 Presentation of Results The far-field velocity and eccentricity were used to approximate open water values V0 and e0 respectively. Likewise, the near-field velocity and eccentricity were used to approximate "collision" values Vc and ec corresponding to the instant of contact or min- imum separation distance. Thus the experimental results for the far-field are expressed in terms of open water velocity V0 and corresponding dimensionless eccentricity e'0, while near-field results are expressed in terms of collision velocity Vc and corresponding di- mensionless eccentricity e'c. Graphs showing the variation of approach eccentricities and ev\ forV x /Vâ€ž-Â»0 (4.30) - \ex\ for Vx/Vy â€”> oo (4-31) Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 137 velocities are given in Figures 4.12 - 4.25. The graphs also show corresponding numerical predictions, which are described in detail in the following section. Results are shown for trajectories having initial approaches which would lead to contact with the structure (i.e. e'F < 1). If the iceberg model remained stationary in front of the structure due to negative wave drift forces, the estimated near-field velocity V}v cannot be equated to the collision velocity Vc; thus, results are not presented for tests with no currents and D, < 0.2. Due to problems with the towing carriage, no wave-current tests were conducted with the 400 mm diameter iceberg model. Results for the current driven motions (Figures 4.12 - 4.15) indicate that the influ- ence of the structure on approach eccentricity is significant for the smaller models. The deflection of the 100 mm model away from the structure is particularly striking (Fig- ure 4.13). Velocities of the current driven icebergs appear to decrease somewhat as they approach the structure. Figures 4.16 - 4.19 show significant scatter for the two larger icebergs when driven by waves only; however, it appears that the motions are relatively unaffected by the presence of the structure. The combined wave-current results are very similar to those shown for currents only. It appears that the iceberg motions for the wave-current tests were dominated by the currents. 4.8 Numerical Modelling of Wave Tank Experiments A series of numerical simulations was conducted to compare observed experimental results with the numerical iceberg drift model. Drift trajectories for icebergs approaching the structure were computed for the five different conditions and six initial dimensionless eccentricities varying between 0 and 1. Results for the numerical drift model are plotted with the experimental results in Figures 4.12 - 4.25. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 138 14 i â€¢ 1.2- u C 0.8H <u o o 0) 0.6 c o 0.4 - O O 0.2- â€¢ â€¢ Physical model Numerical model oo o.o 0.2 0.4 0.6 Initial eccentricity eâ€ž' 0.8 1.0 1.4 0> > 1.2- 1.0- .2 0.8H -l-> cd u >> 0.6-| 0.4 0.2- â€¢ Physical model Numerical model o.o- i â€” 0.2 0.0 0 4 0 6 Initial eccentricity e0' 0.8 1.0 Figure 4.12: Variation of Iceberg Eccentricity and Velocity, Currents, Di/Ds â€” 0.1 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 139 Initial eccentricity e<>' 1.4-1 1.2- 1 0 - â€¢ X 0.8 -<-> (0 s-, >, 0 6 O Â° 0.4 CD > 0.2-1 0 0 0.0 â€¢ â€¢ Physical model Numerical model 0.2 0 4 0 6 Initial eccentricity eÂ«' 0.8 10 Figure 4.13: Variation of Iceberg Eccentricity and Velocity, Currents, D{/Ds = 0.2 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 140 â€¢ Initial eccentricity e<>' 14 -i o (0 > 1.2- 1.0- 0.8- >, 0.6 O 0.4 0.2- 0.0- â€¢ 0.0 â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ Physical model Numerical model 0.2 04 0 6 Initial eccentricity e 0 ' 0 8 i o Figure 4.14: Variation of Iceberg Eccentricity and Velocity, Currents, D{jDs = 0.5 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 141 1.4 - i 1.2- r; 1 0 - C 0.8 o o <U 0.6 0.4 - O U 0 . 2 - 0.0-0.0 â€¢ â€¢ â€¢ Physical model Numerical model 0.2 0.4 0.6 Initial eccentricity e<>' 0.8 1.0 1.4 .X 0.8 0 . 2 - 0.0 - . â€¢ 1 1 1 u â€¢ Physical model Numerical model â€¢ i ' i ' 0.0 0.2 0.4 0 6 Initial eccentricity Go' 0.8 1.0 Figure 4.15: Variation of Iceberg Eccentricity and Velocity, Currents, Di/Ds = 0.8 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 142 1.4-1 O ->-> o o 1.0 0.0- â€¢ â€¢ â€¢ â€¢ â€¢ Physical model ' 1 ' 1 Numerical model i ' i â€¢ i 0.0 0.2 0.4 0.6 Initial eccentricity e0' 0.8 1.0 Figure 4.16: Variation of Iceberg Eccentricity and Velocity, Wave Condition 1, Di/Ds = 0.5 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 143 Figure 4.17: Variation of Iceberg Eccentricity and Velocity, Wave Condition 1, Di/Da = 0.8 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 144 1.4 -| 1.2- \ 1.0- u .2 0.8- -t-> CC s->, 0.6Ì‚ -4-> 'o Â° 0.4 > â€¢ 0.2 â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ â€¢ Physical model Numerical model o.o- 0.0 0.2 0 4 0 6 0.8 1 0 Initial eccentricity e0' Figure 4.18: Variation of Iceberg Eccentricity and Velocity, Wave Condition 2, D{/Ds = 0.5 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 145 1.4 1.2H ft) "c 0.8H u u 0) 0 .6H c o "55 0.4 O U 0.2- 0.0- â€¢ â€¢ 0.0 â€¢ â€¢ â€¢ â€¢ Physical model Numerical model 0.2 0.4 0 6 Initial eccentricity e0' 0.8 1.0 l - 4 - i CO o > 0.2- 0.0 â€¢ â€¢ - â€¢ Physical model - Numerical model â€¢ 0.0 i 0.2 i 1 i ' i ' i 0 4 0 6 0.8 1 Initial eccentricity e0' 1.0 Figure 4.19: Variation of Iceberg Eccentricity and Velocity, Wave Condition 2, D{/Da 0.8 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 146 Initial eccentricity e<>' Figure 4.20: Variation of Iceberg Eccentricity and Velocity, Wave-Current Condition 1, Di/D. = 0.1 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 147 Figure 4.21: Variation of Iceberg Eccentricity and Velocity, Wave-Current Condition 1, Di/D. = 0.2 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 148 1.4-1 1.2- \ l.oH â€¢2 0 .8H cd u > , 0.6- -Â»-> â€¢ p-H o 0.4 H 0 . 2 - â€¢ Physical model Numerical model o.o- o.o 0.2 0.4 0.6 Initial eccentricity e<>' 0.8 1.0 Figure 4.22: Variation of Iceberg Eccentricity and Velocity, Wave-Current Condition 1, DJD, = 0.5 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 149 1.4-1 1.2- â€¢ 1.0- 0.8- D â€¢ D â€¢ >, 0.6- -2 0.4 H 0.2- â€¢ Physical model Numerical model o.o- o.o 0.2 0.4 0.6 0.8 Initial eccentricity e<,' 1.0 Figure 4.23: Variation of Iceberg Eccentricity and Velocity, Wave-Current Condition 2, Di/D. = 0.1 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 150 Figure 4.24: Variation of Iceberg Eccentricity and Velocity, Wave-Current Condition 2, DilD. = 0.2 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 151 Figure 4.25: Variation of Iceberg Eccentricity and Velocity, Wave-Current Condition 2, DJD. = 0.5 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 152 4.8.1 Simplified Numerical Drift Model The simulation of iceberg drift trajectories near the structure could have been most readily accomplished using the program ICEMOT described in Chapter 2. Unfortunately, the number of diffraction computations required to determine wave drift forces near the structure would have made running the model prohibitively expensive. A simplified drift model was developed for simulating the observed experiments. The main purpose for developing the simplified numerical drift model was to reduce the computational effort required for estimating wave drift forces. Based on the results of Chapter 3, it was assumed that lateral wave drift forces (i.e. perpendicular to the incident wave direction) could be considered negligible. The equations of drift motion could then be written as: 1 - (1 + Cm) M ax = -p Cd Aw \VW - V\ (uw - u) + X-pgCwD%H2 (4.32) {I + Cm) May = l-p Cd Aw \VW - V\ (vw - v) (4.33) Wave drift forces for equation 4.32 were determined using the simplified wave drift force equation 3.52. Diffraction computations for each iceberg model and wave condition were made to determine wave drift forces at various locations in front of the structure. Computed wave drift forces are presented in Figures 4.26 - 4.29. Coefficients for the approximate wave drift force equation 3.52 were determined using a least squares fit to values from the diffraction computations, and are shown in Figures 4.30 - 4.33. Zero frequency added mass coefficients for equations 4.32 and 4.33 were taken from Figure 3.19. It was assumed that added mass was independent of proximity to the structure. Water velocities in the vicinity of the structure were evaluated using the analytical solution for flow around a two dimensional circular cylinder (equation 2.92). Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 0.20 Q > IS: 0.15- 3 O c cu O 0.10 O 0.05- 0.00 -0.05- -0.10 -2.5 Ice Diameter â€¢ = 50 m m o = 100 m m A = 250 m m + = 400 m m -2.0 -1.5 - 1 0 -0.5 Iceberg Location x, (m) Figure 4.26: Wave Drift Forces in Front of Structure, No Current, T = 0.7 s 0.20 -2.0 Iceberg Location xt (m) Ice Diameter â€¢ = 50 m m o = 100 m m A = 250 m m + = 400 m m -0.5 o.o Figure 4.27: Wave Drift Forces in Front of Structure, No Current, T = 1.0 s Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 0.20-1 Ice Diameter â€¢ = 50 m m o = 100 m m A = 250 m m + = 400 m m -2 .5 - 2 . 0 -1.5 -1 .0 - 0 . 5 Iceberg Location xt (m) o.o Figure 4.28: Wave Drift Forces in Front of Structure, Current, Tc == 0.7 s > c V o o O.6-1 0.5 0.4- 0.3- 0.2- 0.1 Ice Diameter â€¢ = 50 m m 0 = 100 m m A = 250 m m + = 400 m m o.o Iceberg Location xt (m) Figure 4.29: Wave Drift Forces in Front of Structure, Current, Tc = 1.0 s Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure a CJ C CD O CD O O Q CD > O.6-1 0.5- 0.4- 0.3- 0.2- 0.1- 0.0 0.0 T' c Vwo â€¢ = 0.7 s 0.0 m/s o = 1.0 s 0.0 m/s A = 0.7 s 0.1 m/s + = 1.0 s 1.0 m/s 0.1 0.2 Iceberg Diameter Di (m) Figure 4.30: Open Water Wave Drift Force Coefficients 9 u c CD O CD O U CD â€¢o 3 0.20-1 0.15- 0.10- Â£ o.OS- 'S, 6 < 0.00- 0.0 Tc Vu>o â€¢ = 0.7 s 0.0 m/s o = 1.0 s 0.0 m/s A = 0.7 s 0.1 m/s + = 1.0s 1.0 m/s 0.1 0.2 0.3 Iceberg Diameter Di (m) 0.4 Figure 4.31: Amplitude Coefficient AQW for Variation of Wave Drift Forces er 4. Physical Modelling of Iceberg Drift Near a Large Structure 03 2 Â«<- 0.4- CD o o X 0.2- CC o-cu Q l.O-i Tc â€¢ = 0.7 s 0.0 m/s 0.8- o = 1.0 s 0.0 m/s A = 0.7 s 0.1 m/s 0.6- . + = 1.0s 1.0 m/s 0.0- 0.0 - 1 â€” 0.2 0.1 0.3 Iceberg Diameter Di (m) 0.4 Figure 4.32: Decay Coefficient Bcw for Variation of Wave Drift Forces a to -l-> c CD o CD O U CD W CC .13 2.0 1.5- a I.O 0.5 0.0- 0.0 Tc Vu,o â€¢ = 0.7s 0.0 m/s o = 1.0 s 0.0 m/s A = 0.7 s 0.1 m/s + = 1.0 s 1.0 m/s H 1 G e 6 6 0.1 0.2 0.3 Iceberg Diameter Di (m) 3 0.4 Figure 4.33: Phase Coefficient Ccw for Variation of Wave Drift Forces Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 157 A water drag coefficient value of 0.7 determined by towing tests described earlier was used for all models. Each iceberg trajectory was evaluated commencing at an xg value of -3DS, and an initial yg value corresponding to the initial dimensionless eccentricity e'0. The initial velocity for each iceberg was taken as its open water equilibrium velocity V0, taken from equation 3.53 to be as follows: v " v - + \b\a ( 4 - 3 4 ) A time step interval of 0.1 s was used to evaluate each trajectory until one of the following occurred: 1. The iceberg impacted the structure, 2. The x velocity component became negative, 3. The iceberg drifted freely past the structure. The collision velocity Vc and impact eccentricity ec were then evaluated at the final time step. 4.8.2 Comparison of Numerical and Physical Models Output from the numerical model is plotted with the experimental data in Figures 4.12 - 4.25. The experimental data show significant scatter, with experimental trends generally following the predictions of the numerical model. In many cases the numerical model appears to be conservative, under-estimating the collision eccentricity e'c and over- estimating the velocity ratio Vc/V0. To provide a quantitative basis for comparing agreement between numerical and physical models, the standard error between values of collision eccentricity and velocity Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 158 ratio can be evaluated as follows: Nob, Â£ (eN - <? (4-35) Wc/Vo = \ -TJâ€” Â£ (VN/VF - VJV0)2 (4.36) \ 1* obs iâ€”i where: Nobs = number of observation for physical model with e'F < 1 ae>c â€” standard error for predicted collision eccentricity o~vc/v0 â€” standard error for predicted velocity ratio Standard errors for collision eccentricities and velocity ratios are presented in Tables 4.6 and 4.7. Observed trajectories for current driven motions (Figures 4.12 - 4.15) indicate signif- icant increases of approach eccentricities as the icebergs approach the structure. Collision eccentricity predictions by the numerical model tend to be conservative when compared with the experimental results. Observed and numerically modelled iceberg velocities ap- pear to be relatively unaffected by the structure. The 50 mm model exhibits high velocity ratios for e'c > 1 because of the high water velocities around the sides of the structure. Agreement between the physical and numerical models is reasonably good for the first wave condition with T = 0.7 s and H = 21 mm (Figures 4.16 and 4.17); however, the numerical model severely under-estimates impact velocities of the 250 mm iceberg for the second wave condition with T = 1.0 s and H = 61 mm (Figure 4.22). Examination of Figure 4.29 indicates that the numerical model predicts that a negative drift force will act on the 250 mm iceberg when it is immediately in front of the structure for a wave period of 1.0 s. The physical model indicates that the velocity of the 250 mm iceberg will be reduced in front of the structure, but not to the same degree as predicted by the numerical model. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 159 Table 4.6: Standard Errors between Numerical and Physical Models for Collision Eccen- tricity e'c Di/Ds = 0.1 0.2 0.5 0.8 Currents Waves 1 (H = 21 mm) Waves 2 (if = 61 mm) Wave-currents 1 (H =19 mm) Wave-currents 2 (H â€” 59 mm) 0.48 0.43 0.79 0.46 0.65 0.64 0.32 0.58 0.84 0.26 0.47 0.22 0.22 0.69 Table 4.7: Standard Errors between Numerical and Physical Models for Velocity Ratio ve/v0 DijDs = 0.1 0.2 0.5 0.8 Currents Waves 1 (H = 21 mm) Waves 2 (H = 61 mm) Wave-currents 1 (H = 19 mm) Wave-currents 2 (H = 59 mm) 0.23 0.20 0.86 0.11 0.30 0.88 0.14 0.27 0.48 0.33 0.67 0.09 0.45 0.60 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 160 Figures 4.20 - 4.22 indicate that the physical and numerical models are in relatively good agreement for the first wave-current condition (Tc â€” 0.7 s, H = 19 mm). For the second wave-current condition (Tc = 1.0 s, H = 59 mm), the large waves have a more dominant influence on iceberg motions (Figures 4.23 - 4.25). Collision velocities predicted by the numerical model are reduced to nearly zero because of negative wave drift forces in front of the structure, as shown in the diffraction computations of Figure 4.29. 4.8.3 Comparison of Open Water Velocities for Wave Driven Motions The results presented above indicate that significant discrepancies exist between the physical and numerical models when predicting velocities for wave driven drift motions. To examine the agreement between the numerical and physical models for the case of no structure, open water drift velocities for wave driven motions have been evaluated using equation 4.34 and compared with observed far-field velocities from the experiments. Although the observed far-field velocities were somewhat affected by the presence of the structure, they provide a useful source for comparison with predicted open water velocities. The open water drift velocities presented in Tables 4.8 and 4.9 show a high degree of variation of observed drift velocities among tests. The coefficient of variation for the drift velocity of a given model and wave condition is typically greater than 30%, and cannot be explained by digitizing or other errors. Consistent drift velocities were observed only for the iceberg with Di/Ds = 0.5. This was also the only iceberg for which good agreement was obtained between the numerical and physical models. Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 161 Table 4.8: Comparison of Open Water Drift Velocities for First Wave Condition, T = 0.7 s, H = 21 mm Di/D. = 0.1 0.2 0.5 0.8 Number of Tests 6 2 5 5 Mean Observed Vj (m/s) .008 .012 .049 .027 Standard Deviation of Vj (m/s) .004 .010 .009 .012 Wave Drift Coefficient Cwo .00295 .00736 .06242 .14682 Predicted V0 (m/s) .026 .030 .054 .068 Table 4.9: Comparison of Open Water Drift Velocities for First Wave Condition, T = 1.0 s, H = 61 mm Dt/D3 = 0.1 0.2 0.5 0.8 Number of Tests 9 8 9 8 Mean Observed Vj (m/s) .012 .024 .055 .044 Standard Deviation of V/ (m/s) .004 .005 .005 .017 Wave Drift Coefficient Cwo -.00015 -.00026 .00514 .06447 Predicted V0 (m/s) -0.017 -0.016 .045 .131 Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 162 4.8.4 Sources of Discrepancies between Physical and Numerical Models For iceberg motions driven by currents, it is likely that convective forces had a significant influence on the eccentricity of the icebergs as they approached the structure. As indicated in Chapter 2, convective forces will cause smaller icebergs to travel as water particles, thus causing the icebergs to drift around the side of the structure rather than to impact the structure. Neglecting convective forces in the numerical model was the most probable cause of the under-estimates of impact eccentricities for current driven motions. Discrepancies between the physical and numerical models are greatest when wave drift forces acting on an iceberg vary significantly with proximity to the structure. One possible explanation can be developed by examining the computed wave drift forces in Figures 4.26 - 4.29. The wave drift force acting on an iceberg at a given location is highly dependent on the wave frequency u>c and the incident current velocity Vwo. The frequency of waves relative to the iceberg will vary with the drift velocity of the iceberg; thus, wave drift forces are likely to be significantly influenced by iceberg drift motions. The influence of drift motions on wave drift forces is not included in the numerical diffraction model. The evaluation of wave drift forces using the far-field method by the program ICE- MOT could also be a source of significant errors. A major assumption of the far-field method is that the rotational orientation of a floating body about the x' and y' axes (i.e. pitch and roll) will vary negligibly during a given wave cycle; however, Pinkster [96] has noted that the oscillatory pitch and roll motions of a body in low frequency waves can be sufficiently large to affect wave drift force significantly. This effect would be most pronounced for the smaller icebergs of the present investigation whose wave-induced os- cillatory motions are relatively large, thus resulting in significant variation of the iceberg Chapter 4. Physical Modelling of Iceberg Drift Near a Large Structure 163 rotational orientation and corresponding geometry during a given wave cycle. Chapter 5 Selection of a Design Collision Event Based on Iceberg Kinetic Energy The development of environmental loading criteria is a key aspect in the design of offshore structures, with the selection of design loads most commonly done using prob- abilistic methods (see Vanmarke and Angelides [113]). Design load levels are associated with long return periods (e.g. 100 years) and/or acceptably low exceedence probabilities during the operating life of a structure (e.g. 10%). Probabilistic methods for evaluation of design iceberg collisions have been developed by Maes et al. [69] and Salvalaggio and Rojansky [99]. Collision velocities for both of these models are based primarily on iceberg drift velocities. To account for wave-induced motions of smaller icebergs, Salvalaggio and Rojansky applied a correlation coefficient of -0.4 between iceberg diameter and velocity. The incorporation of wave-induced motions into iceberg velocity statistics has been examined by Lever and Sen [60]. Using wave climate data and the surge response transfer function for a given iceberg, the oscillatory velocity probability density function (PDF) for the iceberg can be evaluated. Although this technique can be used to obtain the velocity PDF for a given iceberg, it appears to have limited application for the evaluation of design loads. A new approach for the selection of a design iceberg collision event is discussed in the present chapter. The load level associated with the design event is based on an 164 Chapter 5. Kinetic Energy Design Collision Event 165 acceptably low exceedence probability. A key aspect of the proposed model is its inclusion of wave-induced iceberg motions, which are computed using linear wave diffraction theory. Results from the model can be used to examine the importance of wave-induced motions for a design collision event. 5.1 Design Criteria for Iceberg Collisions The nature of a design collision will depend on the following two factors: 1. Exceedence probability associated with the design event, 2. Criterion used for ranking collision severity. The specified exceedence probability can be based on various factors, such as the con- sequences of structural failure. The magnitude of the design load will increase as the specified acceptable exceedence probability decreases. Guidelines for acceptable excee- dence probabilities are given in various design codes such as the Canadian Standards Association (CSA) Code for the Design, Construction, and Installation of Fixed Offshore Structures [15]. For design against iceberg collisions, the CSA code recommends that the design iceberg load should have an annual exceedence probability between 10 - 4 and 10~2, depending on the structure safety class and location. The design event will also depend on the method used for ranking collision severity. Iceberg kinetic energy and impact force are two alternative criteria which can be used for ranking collision severity. Evaluation of impact force requires that structure properties be known; thus, determination of a design event based on impact force is most suited to the final stages of design. Iceberg kinetic energy is a useful ranking criterion for preliminary design, and can be expressed as follows: KE = 1(1 + Cm)MVc2 (5.1) Chapter 5. Kinetic Energy Design Collision Event 166 where: Cm = added mass coefficient KE = iceberg kinetic energy (Joules) M = iceberg mass (kg) Vc = iceberg impact velocity (m/s) Hydrodynamic interactions between a structure and approaching icebergs are neglected when evaluating equation 5.1 in the present design model. Studies by Isaacson and Dello Stritto [48] and Salvalaggio and Rojansky [99] indicate that the presence of a large structure will reduce impact velocities of smaller icebergs; thus, neglecting hydrodynamic interactions will likely yield conservative results. 5.2 Description of Second-Order Reliability Method The second-order reliability method (SORM) was selected for evaluating kinetic energy exceedence probabilities because of its high degree of efficiency for analysis of complex systems. A general description of the method is given by Madsen et al. [67]. Application of the second-order reliability method was carried out using the computer program RELAN developed by Foschi [30]. In addition to being very efficient, the second- order reliability method gives output parameters which are very useful in engineering design. In the present context, the SORM model gives the most probable values of input random variables (e.g. iceberg mass and drift velocity) for a given kinetic energy level. Sensitivity coefficients for input random variables are useful indicators of the relative importance of each variable in determining iceberg kinetic energy. Chapter 5. Kinetic Energy Design Collision Event 167 5.2.1 Introduction of Failure Function The second-order reliability method is most commonly used to estimate the proba- bility of structural failure, which is expressed in terms of a failure function G(X). The relationship between the probability of structural failure and the failure function is as follows: X = vector of input random variables The probability of failure is estimated using an approximate solution to the integral of equation 5.3. The following guidelines should be followed when formulating the failure function G(X): 1. Statistical distributions for each of the input random variables should be known or easily obtainable, 2. Input random variables should ideally be independent, or have low correlation coefficients, 3. Combinations of input random variable values which defy prescribed physical con- straints should not be possible. With regard to condition 2 above, it should be noted that correlation coefficients be- tween input random variables can be given as input to the reliability model. However, P (failure) (5.2) (5.3) where: fx{X) = joint probability density function of X â€”* G(X) = failure function Chapter 5. Kinetic Energy Design Collision Event 168 correlation coefficients are only representative of the degree of linear dependence be- tween variables and cannot adequately describe non-linear relationships between random variables. 5.2.2 Solution in Standard Normal Space Due to computational limitations, the integral of equation 5.3 is not suited to direct evaluation but must instead be solved approximately. The following transformation is used so that the failure function can be expressed in terms of a vector of independent, standard normal variables: {Y} = [T] {X} (5.4) where: [T] = transformation matrix {Y} = transformed vector of independent, standard normal variables An example of the resulting standard space for two input variables is shown in Figure 5.1. The failure probability of equation 5.3 can be expressed in terms of the new standard space: P(failure) = I 4>(YX) <f>{Y2) <f>(Y3) ... 4>{YNR9) dY (5.5) JG(Y<O) where: Nrv = number of input random variables 4>(Yi) = probability density function for standard normal distribution 5.2.3 Failure Probability for Linearized Failure Surface Evaluation of the integral of equation 5.5 is greatly simplified if the failure surface G(Y) = 0 is approximated by a first-order Taylor series expansion about a design point Chapter 5. Kinetic Energy Design Collision Event 169 (0 s_ CO > "3 -pes ign point Y 6 o w c CO Fa i lu re r e g i o n G < 0 0 Safe r e g i o n 67 > 0 ~i 1 1 1 T r a n s f o r m e d v a r i a b l e Y i Figure 5.1: Standard Space for 2 Random Variables {y*} located on the failure plane. The selected linearization point {Y*} is the most probable occurrence of {Y} given G (Y) = 0. In terms of the standard space of Figure 5.1, the design point {Y*} is the point on the failure plane which is closest to the origin of the standard space. Solution of the design point is solved using an iterative procedure given by Madsen et al. [67]: 1 {<?(?â€¢)}. (5.6) w here: |G'(y*) j = vector of first-order partial derivatives of G The origin of the standard space is commonly taken as a starting point for the above equation, with convergence to the design point {Y'} typically obtained within a few iterations. Chapter 5. Kinetic Energy Design Collision Event 170 Once the design point {Y*} has been found, the reliability index (3 and associated failure probability can be determined. The reliability index can be expressed in terms of the effective mean and standard deviation of the failure function at the design point: 0 = ^ (5.7) {Y>} where: P = reliability index \IQ â€” effective mean value of failure function O~G = effective standard deviation of failure function Since the variables Yi are independent and standard normal, /? is easily found from {K*}: /?2 = J2Y;2 (5.8) The design point values also yield sensitivity coefficients which indicate the relative im- portance of the random variables. The sensitivity coefficients are evaluated by: K ) = \ {Yl (5-9) The sensitivity coefficients are useful indicators of which variables are dominant in causing structural failure. 5.2.4 Failure Probability for Curved Failure Surface The linearization of the failure surface G(Y) = 0 can introduce significant error in the evaluation of the failure integral of equation 5.5. Figure 5.2 illustrates two cases where failure probabilities are over-estimated and under-estimated respectively. An improved estimate of failure probability can be obtained based on the following second-order expansion about the design point {V*}: -Â» 1 x G(Y) = G(Y*) + {G'{Y*)} {Y - Y*} Chapter 5. Kinetic Energy Design Collision Event 171 P(failure) Overes t ima ted P(failure) U n d e r e s t i m a t e d Variable Yi Var iab le Yi Figure 5.2: Influence of Curved Failure Surface + \{Y ~ Y1T [G"(Y')} {Y - Y'} (5.10) where: J(j"(F*)j = matrix of second-order partial derivatives of G Utilization of the curved failure surface is simplified by making an orthogonal transfor- mation from Y to a new vector space Z containing a* as a unit normal vector. The orthogonal transformation is made using matrix [H], which contains a* as its n'th row: {Z} = [H]-1 {Y} (5.11) where: [H] = transformation matrix {Z} = vector in transformed space The failure surface in the transformed co-ordinate system is approximated to second-order Chapter 5. Kinetic Energy Design Collision Event 172 by: Zn = 0 + {Z'f [C] {Z'} (5.12) Terms in the above equation are: {Z'} = ( Z l 5 Z2, ZNrv.x) (5.13) C Â« = l f â€ž . â€ž - â€ž l l ( W [ G " ( y ) ] W ) , . - i,J = l , 2 , . . . , J V â€ž - l (5.14) Various methods are available for solving the failure probability associated with the second-order failure surface of equation 5.12. The following solution by Tvedt [111] is used by the computer program RELAN: F(G < 0) = A l + A 2 + A 3 (5.15) Ar = $(-/?) det ([7] + 2/? [C]) _ 1 / 2 (5.16) A 2 = (/3 $(-/*) - m)x (det ([7] +. 2/3 [CI) " 1 / 2 - det ([7] + 2 (0 + 1) [C]) - 1 / 2 ) (5.17) = (/? + 1) (0 *(-/*) - #/9))x (det([7] + 2/?[C]) - 1 / 2 - Re { det ([/] + 2 (/? + i) [C] ) - 1 ' 2 }) (5.18) where: [7] = identity matrix </>(/?) = PDF of standard normal distribution $(/?) = CDF of standard normal distribution Chapter 5. Kinetic Energy Design Collision Event 173 Results from equation 5.15 are typically very close to those obtained assuming a flat failure surface. 5.2.5 S O R M Solution for Dependent, Non-Normal Variables The SORM solution presented thus far is limited to the case of independent, stan- dard normal variables. For practical applications, a method is required for dealing with input random variables that are dependent and have various distribution types. A solu- tion can be obtained by transforming the input random variables {X} to the standard space set {Y} as given by Der Kiureghian and Liu [21]: {Y} = [To] [DT1 {X - M'} (5.19) where: \D'\ = diagonal matrix of effective standard deviations for {X} {M1} = vector of effective means for {X} [To] = transformation matrix accounting for correlation of {X} variables Evaluation of the above terms is now described in greater detail. Solution of Correlation Transformation Matrix Input to the reliability program includes a correlation matrix [R] for the random variables {X}. Solution of the transformation matrix [r 0] is based on the input corre- lation matrix. The input matrix [R] is modified to account for input variables having non-normal distributions to obtain a modified correlation matrix [Ro] as follows: Ron = RijStJ i,j = l,n (5.20) where: Chapter 5. Kinetic Energy Design Collision Event 174 Rij = element of original correlation matrix Roij = element of modified correlation matrix 8{j â€” correlation correction factor The correction factors <5,j are dependent on the input distributions for variables i and j. Semi-empirical <5,j values have been obtained by Der Kiureghian and Liu [21] for commonly used input distributions, and in most cases results in corrections of less than 5%. Once the modified correlation matrix [R0] is obtained, a Cholesky decomposition is performed to obtain a lower triangular matrix [Lo] defined by: [Ro] = [L0] [L0]T (5.21) The transformation matrix [r 0] is then taken as the inverse of [Lo]: [To] = [Lo]'1 (5.22) Evaluation of Effective Means and Standard Deviations The transformation to standard normal space requires that the effective means and standard deviations of the input variables be evaluated. Elements of [D1] and {M'} can be expressed as: Dl, = 1 = 1 , ^ (5.23) JX,{Xi) Ml = x{ - D'u [FXi(xi)] 1 = 1 , ^ (5.24) where: fXi(xi) = PDF for variable X{ Fxiixi) = CDF for variable X{ Chapter 5. Kinetic Energy Design Collision Event 175 Values for D'i{ and M[ are dependent on the the point {X} used for the transforma- tion. Equations 5.23 and 5.24 are evaluated for each iteration of the Rackwitz-Fiessler algorithm (equation 5.6) until convergence to the design point {X*} is reached. Flow Chart for SORM Algorithm A flow chart of the SORM algorithm is given in Figure 5.3. As stated previously, the correlation transformation matrix [r0] is only evaluated once. Successive iterations are made to converge on the design point {X*}, with the transformation from {X*} to {Y*} being re-evaluated during each iteration. Once convergence is obtained, the final probability of G < 0 is estimated using a second-order expansion of the failure surface about the design point. 5.3 Formulation of Failure Function for Iceberg Kinetic Energy Application of the second-order reliability method requires the formulation of an appropriate failure function G (X^j. The failure function for predicting kinetic energy exceedence probabilities has the following form: G(x) = RKE - KE(x) (5.25) where: KE â€” iceberg kinetic energy RKE â€” input kinetic energy level The SORM algorithm can be run for various RKE values to obtain the portion of the cumulative distribution function (CDF) relevant to structural design. Chapter 5. Kinetic Energy Design Collision Event 176 Read distributions for input variables. Read correlation matrix [R], Compute correlation transformation matrix [r 0]. Read initial guess for {X*}. Evaluate [T] = [To^D^iX - M'}. Evaluate {Y*} = [T]{X*}. I Compute new {Y*} and {X*} = [^{Y*} . I Convergence? No Yes Write FORM output. Compute SORM probability of failure. Write SORM output. Figure 5.3: Flow Chart for Second-Order Reliability Method Chapter 5. Kinetic Energy Design Collision Event 177 The selection of input random variables for the failure function is of critical impor- tance. For the present reliability model, the kinetic energy of an iceberg at a given site is expressed as a function of the following four variables: 1. Iceberg mass, M , 2. Iceberg aspect ratio, h/D{, 3. Iceberg drift velocity, Vd, 4. Significant wave height, Hs. The above random variables have been selected primarily because of the ease with which statistical data can be obtained for each variable. Iceberg dimensions and drift veloci- ties are recorded by petroleum operators working in iceberg-prone regions, while wave climate data is available for most offshore areas. The four variables selected for iceberg kinetic energy are likely to have low correlation coefficients, and are thus well suited to application of the second-order reliability method. 5.3.1 Idealization of Iceberg Geometry Due to the infinite number of possible iceberg shapes, it is necessary to select an idealized iceberg geometry. A vertical circular cylinder having 1/7'th of its mass above water is used in the present model. The geometrical properties of a cylindrical iceberg can be described by its mass M and aspect ratio h/Di (draft/diameter). The aspect ratio of a cylindrical iceberg will have an upper limit of 0.86 imposed by hydrostatic stability requirements. Chapter 5. Kinetic Energy Design Collision Event 178 5.3.2 Iceberg Added Mass Evaluation of iceberg kinetic energy requires that iceberg added mass be known. The added mass at the zero-frequency limit is used for the present application, and is considered to provide a conservative estimate of the iceberg kinetic energy that could be dissipated during an impact (see Chapter 3). The zero-frequency added mass coefficient for a cylinder can be expressed as a function of two dimensionless parameters: Since water depth d is constant for a given site, Cm can be expressed as a function of the iceberg geometry variables: When evaluating the kinetic energy failure function, equation 5.27 is treated as a deter- ministic relationship. Added mass coefficients are obtained by interpolation of tabulated values for equation 5.26 presented in Figure 3.19. 5.3.3 Iceberg Velocity The accurate evaluation of iceberg kinetic energy requires that both drift and oscil- latory velocity components be considered. If drift and oscillatory motions are assumed to be collinear, then the following expression from Lever and Sen [60] can be adopted: Cm = f(h/d, h/D^ (5.26) Cm = f(M, h/Di) (5.27) Ve = Vd + Vs (5.28) where: Vc = total collision velocity (m/s) Vd = drift velocity (m/s) V, = significant surge velocity (m/s) Chapter 5. Kinetic Energy Design Collision Event 179 Due to the influence of wave drift forces on iceberg drift motions [40,44], the assumption of collinear drift and oscillatory components is quite reasonable. Statistical data for iceberg drift velocity can be obtained from observed iceberg trajectories; thus, drift velocity is a suitable input variable for the probabilistic model. When evaluating the influence of oscillatory motions on impact velocity, consid- eration must be given to the variation of iceberg velocity during a wave cycle and to the variability of wave heights in a random wave field. When accounting for this, the significant wave height provides a reasonable nominal value of the height to be used in calculating oscillatory velocity amplitude since this represents the 86-th percentile of wave height. In summary, the significant oscillatory velocity amplitude Vs based on the significant wave height is used for predicting the oscillatory velocity on impact. Oscillatory surge velocity cannot be used as a model input parameter because of problems associated with obtaining oscillatory velocity measurements. An alternative method must be used for evaluating wave-induced motions. The following expression is adopted for evaluating the significant surge velocity: Vs = \HauPC[ (5.29) where: Hs = significant wave height (m) ([ = surge response amplitude operator iop = peak angular frequency (rad/s) The availability of wave climate data makes significant wave height a suitable input random variable for the probabilistic model. The oscillatory velocity can be subsequently evaluated using deterministic relationships. The peak wave period Tp is strongly dependent on the incident wave height. Lever and Sen [60] have adopted the following expression obtained by LeBlond et al. [57] using Chapter 5. Kinetic Energy Design Collision Event 180 a least squares fit to a variety of ocean wave records from the Canadian East Coast: Tp = 13.88 y/Hj~g (5.30) Similar expressions can be developed for other sites (e.g. see Thoft-Christensen and Baker [110]). Equation 5.30 indicates that the effect of increasing wave height on Vs will be offset by decreasing u>p. The surge RAO ([ ha equation 5.29 is evaluated using linear wave theory, which can be considered valid when wave height is small relative to incident wavelength and water depth. Using the idealized cylindrical iceberg geometry, the surge RAO can be expressed as a function of 3 dimensionless parameters: CJ = / (u2Di/2g , h/Di , h/d) (5.31) If water depth d is constant for a given site and to is considered to be a function of H3, equation 5.31 can be re-written in terms of the model input random variables: ([ = f(M,h/DiyH.) (5.32) Deterministic evaluation of surge RAO's is done using tabulated values from Figures 3.8 - 3.12 which were obtained from diffraction computations for 0.5 < u2Di/2g < 3.14. If an RAO is required for io2Di/2g < 0.5, it is conservatively assumed that the iceberg moves as a water particle at the water surface (equation 3.39). In the high frequency range u>2Di/2g > 3, the surge RAO is determined using the high frequency approximation of equation 3.46. When calculating iceberg impact velocity using the kinetic energy model, the influ- ence of the structure on iceberg motions is not considered. It was shown in Chapter 3 that the influence of a structure on iceberg drift velocity can be significant for wave driven drift motions when the iceberg diameter is less than half the structure diameter. Chapter 5. Kinetic Energy Design Collision Event 181 This phenomenon suggests that the significant wave height Hs could have a negative influence on the drift velocity Vd upon impact. It is suggested that the influence of the structure on impact velocity be neglected when performing the initial evaluation of a design event. If the design iceberg diameter is then determined to be less than half the structure diameter and the drift velocity is driven primarily by waves, detailed numerical and physical studies of the hydrodynamic interactions between the iceberg and structure would be warranted. 5.3.4 Summary of Failure Function The failure function for iceberg kinetic energy has been formulated in terms of four random variables as follows: G (M , h/Di, Vd , H,) = RKE - KE (M , h/Di, Vd , H.) (5.33) The failure function is used to estimate the exceedence probability of the kinetic energy level RKE f Â° r a n iceberg. Iceberg added mass and oscillatory velocity are evaluated deterministically using the four random variables as input parameters for computation. 5.4 Application to Selection of a Design Iceberg Collision The above method can be used to determine an iceberg collision event for structural design. The kinetic energy of the design collision is based on an acceptably low exceedence probability during the life of the structure. Because the failure function is formulated to compute exceedence probabilities for a single iceberg, results must be modified to account for the variable number of collisions which can occur during the structure lifetime. The general design process will now be described in detail. Chapter 5. Kinetic Energy Design Collision Event 182 5.4.1 Exceedence Probabilities for a Single Collision The failure function outlined in the previous section is well suited to estimating the kinetic energy CDF for a single collision, denoted FKE {KE \ nc = 1). For design purposes, the upper portion of this CDF is of primary interest (e.g. FXE(KE) > 0.99 ). Input distributions for ice mass and velocity must be modified to account for the greater likelihood of larger and faster icebergs impacting the structure, as proposed by Dunwoody [22]. Through application of Bayes' theorem [101], diameter and velocity PDF's for icebergs impacting a finite size structure are obtained by: / D , ( A K = 1) = D i + DÂ° fDt(Dt) (5.34) fvd(Vd\nc = l) = â€”fvd(Vd) (5.35) Pvd where: (Di) = diameter PDF for all icebergs (m _ 1) /D, (Di | nc = 1) = diameter PDF for impacting icebergs (m - 1) fvd (Vd) = drift velocity PDF for all icebergs (s/m) fvd (Vd | nc â€” 1) = drift velocity PDF for impacting icebergs (s/m) nc = number of collisions The mass PDF for impacting icebergs can be evaluated using equation 5.34. If a charac- teristic aspect ratio Hh/D{ is assumed, then the modified mass PDF is: /W(M|NC = 1) = XM^) + DS MM) (5' 36) 7 = ( â€” ^ â€” ( 5 . 3 7 ) where: Chapter 5. Kinetic Energy Design Collision Event 183 E(xT) = mean value of xr IM (M) - mass PDF for all icebergs (kg - 1) JM (M | nc = 1) = mass PDF for impacting icebergs (kg - 1) p â€” water density (kg/m3) Application of equation 5.36 requires that the mean value of M 1 / / 3 for all icebergs be evaluated. Various methods such as numerical integration are available for utilizing equations 5.35 and 5.36. Perhaps the most convenient method is to select parent mass and velocity distributions which have analytical solutions for the r'th moments: / oo fx{x) xT dx (5.38) â€¢oo Distributions which have analytical solutions for moments include the lognormal and gamma distributions (see Bury [12]). Solutions to equation 5.38 can be combined with equations 5.35 and 5.36 to obtain moments (e.g. mean, standard deviation, skewness coefficient) of the distributions for impacting icebergs. Once distributions have been selected for input variables, the SORM program can be run for various RKE values to obtain the upper portion of the kinetic energy CDF for a single iceberg collision. 5.4.2 Extension to Multiple Collisions Due to the possibility of multiple iceberg collisions during the life of a structure, the kinetic energy CDF obtained for a single collision must be modified accordingly (see Maes and Jordaan [70]). If the number of iceberg collisions occurring during the life of the structure is a fixed quantity NL, then the CDF of the extreme load can be determined Chapter 5. Kinetic Energy Design Collision Event 184 as: FKE (KE I n c = NL) = [FKE (KE \ nc - 1)}Nl (5.39) where: FKE (KE | nc â€” 1) = kinetic energy CDF for single collision FKE (KE \ nc = NL) = extreme value kinetic energy CDF for NL collisions NL = fixed number of collisions during lifetime In reality, the number of collisions to occur during the life of the structure is a random quantity and the following expression must be used to account for a variable number of collisions: oo FKE (KE) = Â£ pnL (nL) [FKE (KE \ nc = 1 ) P (5.40) nL=0 where: ni, = variable number of collisions during lifetime of structure PnL (uL) â€” probability mass function for number of collisions Evaluation of equation 5.40 is greatly simplified if the occurrence of iceberg collisions is modelled as a Poisson process. The probability mass function for appears as follows: Pn L(n L) = exp(-finL) (5.41) Substitution of equation 5.41 into equation 5.40 yields: CO FKE (KE) = Â£ exp(-/x n J ^ [FKE(KE\nc = l ) p (5.42) = exp { -nnL [1 - FKE(KE I nc = 1)] } (5.43) The simple form of this equation lends itself to easy application. If an acceptable ex- ceedence probability for the structure life is specified, then an associated exceedence probability for a single collision can be evaluated. The kinetic energy failure function for a single collision can then be used to determine the lifetime design load. Chapter 5. Kinetic Energy Design Collision Event 185 5.5 Example Design Problem for Large Iceberg Population The method described above has been used to determine a design event for a hy- pothetical structure. The operating environment for the example structure is similar to that for the Hibernia area off the coast of Newfoundland, with icebergs in the area being relatively large in size. A structure diameter of 100 m was chosen to operate in a water depth of 100 m. 5.5.1 Design Criterion The kinetic energy of the design event was specified to have a 10% exceedence probability during the life of the structure. For application of the kinetic energy failure function it was necessary to compute an associated exceedence probability for a single collision. The mean number of collisions expected during the structure lifetime can be ex- pressed as follows: where: TL = operating life for structure (years) Ac = mean rate of iceberg collisions (/year) A mean value of 20 collisions during the structure lifetime was used for the present problem. The associated exceedence probability of 0.0052 for a single collision was sub- sequently evaluated using equation 5.43. (5.44) Chapter 5. Kinetic Energy Design Collision Event 186 5.5.2 Statistical Distributions for Input Variables Statistical distributions for input variables were estimated from available data for the Hibernia oilfield and surrounding areas. A summary of input distributions is given in Table 5.1. Lognormal distributions have been used in the present example for modelling iceberg mass and drift velocity largely for the sake of computational convenience. In an actual design situation, care must be taken to select the statistical distribution model (e.g. lognormal or gamma) which best represents the variable under consideration. For each variable, particular care must be given to modelling the distribution tail (e.g. the upper or lower decile) that will have the greatest influence on the design load. Table 5.1: Input Distributions for Example Design Problem Variable Distribution Parameters M (106 tonnes) Lognormal [i = 0.50, a = 1.74 h/Di Uniform 0.1 - 0.8 Vd (m/s) Lognormal fi = 0.34, a = 0.29 Hs (m) Lognormal p â€” 2.44, a = 1.22 Statistical moments for iceberg masses at Hibernia were estimated from a graph presented by NORDCO [91]; thus, they can only be considered as rough estimates of actual values. Due to the greater likelihood of larger icebergs impacting the structure, the iceberg mass distribution had to be modified according to equation 5.36. Evaluation of statistical moments for impacting icebergs was done using the moment equation for a lognormally distributed variable (from Bury [12]): E(x r) â€¢= exp(r/i l n x + \ r 2al^ (5.45) where: Chapter 5. Kinetic Energy Design Collision Event 187 Table 5.2: Distribution Moments for All Icebergs and Impacting Icebergs Variable All Icebergs Impacting Icebergs M (106 tonnes) fi = 0.33, o = 1.05 p = 0.50, a = 1.74 Vd (m/s) y. = 0.20, a = 0.17 \i = 0.34, a = 0.29 E (xT) â€” expected value of xr Combining equation 5.45 with equation 5.36, the mean and standard deviation for im- pacting icebergs were evaluated as follows: fJ-M\nc=l E ( M 2 I n c = l ) Â°~M\nc=\ A characteristic aspect ratio of 0.45 was used for evaluating 7. Statistical moments for the general iceberg population and impacting icebergs are given in Table 5.2. Note that the mean and standard deviation of mass for impacting icebergs are much greater than values for the parent population. It was assumed that the mass of impacting icebergs was lognormally distributed; however, use of alternative statistical models could be examined if higher quality input data were available. Moments for iceberg drift velocity were taken from Pearson and Ro [93], who anal- ysed 9920 hourly drift trajectories for the Labrador Sea. It was assumed that drift = ' , E ( A + D. E ( M 4 / 3 ) + TE(M%) + D. M ( 5 ' 4 6 ) = 7 E(MW) + D S E (M?/3) + 1 E ( A + D . E ( M 9 ) ( 5 - 4 ? ) = ( Â£ ( M 2 | n c = l) - /4 |n c = 1 ) 1 / 2 (5-48) Chapter 5. Kinetic Energy Design Collision Event 188 velocity was lognormally distributed. Statistical moments for impacting icebergs were evaluated using equations 5.35 and 5.45 and are given in Table 5.2. Iceberg aspect ratio h/Di was assumed to be uniformly distributed between 0.1 and 0.8. The upper limit on h/Di was chosen to insure hydrostatic stability. The significant wave height parameters for Hibernia were taken from Neu [86]. Results from Neu's analysis indicate that a lognormal distribution provides an exceDent fit to observed wave data. 5.5.3 Correlated Input Variables Input to the SORM program can include correlation coefficients for random variables. Studies of observed iceberg dimensions by Brooks [11] and El-Tahan and Davis [26] indicate that iceberg aspect ratio decreases with increasing waterline length. This trend was modelled using a correlation coefficient of -0.1 between mass M and aspect ratio h/Di. Utilization of a negative correlation coefficient was necessary from a computational viewpoint to avoid iceberg drafts larger than the input water depth. The influence of wave drift forces on iceberg drift motions will likely cause a positive correlation to exist between drift velocity Vd and significant wave height Hs. This possible correlation was neglected for the present problem due to the lack of supporting field data. 5.5.4 Computation of Exceedence Probabilities The SORM program was used to estimate exceedence probabilities for various kinetic energy levels. Energy levels with exceedence probabilities near the design value of 0.0052 for a single collision were of primary interest. Results for the SORM computations presented in Figure 5.4 indicate that the design event for the specified design criterion has a kinetic energy of 2.1.x 109 Joules. Chapter 5. Kinetic Energy Design Collision Event 189 O.Ol-i cu o c cu â€¢o cu cu o X Cd Du O.OOlH â€” i â€” â€” r- 1 1 1 0 1 2 3 4 5 Iceberg Kinetic Energy (109J) Figure 5.4: Kinetic Energy Exceedence Probabilities for a Single Collision 5.5.5 Parameters for Design Collision Event Output parameters for the design collision event are given in Table 5.3. The design values for the random variables represent the most likely values for the given kinetic energy level while the magnitudes of the sensitivity coefficients indicate the relative im- portance of the variables. Other collision parameters based on the values of the input random variables for the design event are listed at the bottom of Table 5.3. The high sensitivity coefficients for M and Vd indicate that these variables are dom- inant for the design event. Particular emphasis should be given to the quality of the input data for these two variables. The low Ha sensitivity coefficient suggests that wave- induced motions have a minor influence on kinetic energy, with the significant oscillatory velocity being less than 1% of the drift velocity. Chapter 5. Kinetic Energy Design Collision Event 190 Table 5.3: Results for Large Iceberg Design Collision Variable Design Value Sensitivity M (106 tonnes) h/Di Vd (m/s) H. (m) 3.08 0.90 2.20 0.30 0.75 0.01 0.09 0.65 Other parameters associated with above design values: Di = 235 m, h = 69 m, Cm - 0.67, Tp = 6.6 s, u>lDi/2g = 10.9, CJ = 0.006, V, = 0.006 m/s 5.5.6 Exceedence Probabilities without Wave-Induced Motions Due to the small influence of Hs on the design event, it is suggested that wave- induced motions can be neglected when evaluating the upper portion of the kinetic energy CDF for the present example. Iceberg kinetic energy can be modelled using a lognormal distribution if the following assumptions are made: 1. Added mass is independent of iceberg geometry, 2. Wave-induced motions are negligible, 3. Iceberg mass and drift velocity are lognormally distributed. The resulting lognormal model for iceberg kinetic energy can provide a useful comparison with SORM results. If the design event added mass of 0.67 is treated as a constant, iceberg kinetic energy will have a mean value of 1.49 x 109 Joules and a standard deviation of 16.0 x 109 Joules. Figure 5.5 shows excellent agreement of exceedence probabilities evaluated using the Chapter 5. Kinetic Energy Design Collision Event 191 SORM and analytical models. It appears that wave-induced motions have a negligible influence on iceberg kinetic energy at the upper tail of the CDF. O.OOlH 1 1 1 1 1 0 1 2 3 4 5 Iceberg Kinetic Energy (109J) Figure 5.5: Comparison of SORM and Analytical Model Results 5.6 Example Design Problem for Small Iceberg Population The relative importance of wave-induced iceberg motions will increase as the size of impacting icebergs becomes smaller. This trend can be examined by varying the size of impacting icebergs for the example problem given in the previous section. If the parent iceberg population has masses which are 1/1000'th of those for the first example, then the mass distribution for impacting icebergs will have a mean value of 0.36 x 106 kg and a standard deviation of 1.19 x 106 kg. Note that the smaller iceberg population could be encountered at a structure site farther away from the original source of icebergs, or Chapter 5. Kinetic Energy Design Collision Event 192 could be caused by iceberg management methods such as towing of larger icebergs or site evacuation by a mobile drilling unit when large icebergs come within a prescribed distance. If the remaining input parameters remain the same as for the previous example, then the kinetic energy exceedence probabilities are as given in Figure 5.6. Using the acceptable exceedence probability of 0.0052 for a single collision, the kinetic energy of the design event is 5.5 x 106 J. Reliability model parameters for the small iceberg example given in Table 5.4 indicate that wave-induced motions dominate the impact velocity for the design collision. Due to the small size of the design iceberg relative to the structure (Di/Dâ€ž = 0.2), it is suggested that the influence of negative wave drift forces acting on the iceberg could significantly reduce the impact velocity predicted by the kinetic energy reliability model, and that a refined model accounting for this influence should be developed. O H O C <D TJ <D 0> 0.01-O X 0.0011 1 ' 1 1 1 1 0 2 4 6 8 10 Iceberg Kinetic Energy (106J) Figure 5.6: Kinetic Energy Exceedence Probabilities for Small Iceberg Example Chapter 5. Kinetic Energy Design Collision Event 193 Table 5.4: Results for Small Iceberg Design Collision Variable Design Value Sensitivity M (103 tonnes) 3.03 0.85 h/Di 0.45 -0.02 Vd (m/s) 0.47 0.31 H, (m) 3.76 0.43 Other parameters associated with above design values: Di = 20 m, h = 9 m, Cm = 0.55, Tp = 8.6 s, ulDi/2g = 0.55, ([ = 0.77, V, = 1.06 m/s Chapter 6 Selection of a Design Event Based on Global Impact Sliding Force In the previous chapter a probabilistic method was described for selection of a de- sign collision event based on iceberg kinetic energy. Although kinetic energy is a useful ranking criterion for preliminary design, the ultimate design criterion should be based on the expected mode of structural failure. The present chapter discusses the selection of an iceberg collision design event based on global impact sliding force, which is likely to determine whether structural failure will occur. Required input parameters include structure dimensions; thus, the model is most suited to the latter stages of design. A time-stepping procedure which considers hydrodynamic interaction is used to model the trajectory of an iceberg approaching a structure. Impact forces are subse- quently evaluated once a collision commences. The maximum horizontal force developed during the impact phase is used as the collision severity ranking criterion. The primary objective of the impact force reliability model is to illustrate the im- portance of hydrodynamic effects in collision risk analysis. Structural dynamic effects are neglected, and relatively simple methods are used for evaluating ice forces. Despite these shortcomings, the model is a useful starting point for more sophisticated examination of ice-structure interactions. 194 Chapter 6. Impact Sliding Force Design Event 195 6.1 Iceberg Loading of Offshore Structures The evaluation of ice loads on offshore structures has been the subject of extensive research in recent years, as described in reviews by presented by Cammaert and Mug- geridge [13], Nessim et al. [85], and Sanderson [100]. The prediction of ice forces on offshore structures requires an understanding of the possible failure modes for ice. In general, the following three failure modes are dominant: 1. Bending, 2. Buckling, 3. Crushing. The bending mode is likely to occur when a thin sheet of ice is forced upon the face of a sloping structure, while buckling can occur if a thin ice sheet encounters a vertical sided structure. If neither bending or buckling occurs, then the impact energy will be dissipated by crushing. Due to the relatively high aspect ratio of icebergs, failure will almost always be in the form of crushing. 6.1.1 Ice Forces for Head-on Impact Head-on impact is often used as a worst case scenario for predicting forces from iceberg collisions (e.g. Cammaert and Tsinker [14], Cox [18] Johnson and Nevel [50]). A head-on impact of an iceberg with an offshore structure is shown in Figure 1.2. The force acting on the structure during head-on impact can be expressed as follows: FCr = PcrAn(A) (6.1) where: Chapter 6. Impact Sliding Force Design Event 196 An = normal contact area between ice and structure (m2) Fcr = ice crushing force (N) pcr = effective ice crushing pressure (N/m 2) A = penetration distance of structure into ice (m) The normal contact area An is merely a function of the collision geometry. The effective crushing pressure pcr will depend on several factors which are discussed later in this Further insight can be gained by examining the relationship between iceberg kinetic energy and impact force. If environmental forces other than those influencing iceberg added mass are considered negligible relative t6 impact force, then the following rela- tionship can be used: where: Vc = initial iceberg velocity (m/s) A c = maximum penetration distance (m) Thus, impact will progress until the entire kinetic energy of the iceberg is dissipated. The added mass in equation 6.2 is an effective added mass based on a kinetic energy balance. It should be noted that the following momentum relationship is also valid for head-on chapter. (6.2) impact: (6.3) where: tc = duration of impact (s) Chapter 6. Impact Sliding Force Design Event 197 Unlike the effective added mass of equation 6.2, the effective added mass of 6.3 is based on momentum and will be slightly different than the effective added mass based on kinetic energy. 6.1.2 Ice Forces for Eccentric Impact Evaluation of collision forces is significantly complicated if eccentric impacts are considered. Time-stepping procedures for computing eccentric impact forces have been developed by Bass et al. [7], Duthinh and Marsden [25], and Salvalaggio and Rojansky [99]. Frictional forces which act tangentially on a structure will arise during eccentric impacts. The tangential force will apply torque to the structure and will also contribute to the sliding force. The kinetic energy balance equation for eccentric impacts must include energy dissipated through friction. In addition, the iceberg will possess some remaining kinetic energy after impact. The revised kinetic energy equation is as follows: where: C/ = coefficient of added inertia Ft = tangential force acting on iceberg (N) L = iceberg rotational inertia (N-m2) Sn = normal displacement of iceberg relative to structure (m) Snc â€” total normal displacement (m) St = tangential displacement of iceberg relative to structure (m) stc = total tangential displacement (m) vf = final translational velocity after collision (m/s) + \ (1 + Cm) M Vj + \ (1 +'d) Iio) (6.4) Chapter 6. Impact Sliding Force Design Event 198 u>j = final rotational velocity after collision (rad/s) The last term of the above equation is due to the rotational motion of the iceberg induced by the tangential force Ft. The tangential force component Ft is typically expressed in terms of a friction coefficient fij: Ft = fijFn (6.5) It should be noted that the existence of normal and tangential forces are conditional upon the iceberg motions. For the normal force to exist, the iceberg must be moving with a normal velocity into the structure. The existence of a tangential force component is dependent upon a corresponding normal force (equation 6.5). The tangential force acting on the structure will act in the direction of the tangential velocity of the iceberg relative to the structure. 6.1.3 Selection of Effective Ice Pressure The evaluation of ice loads requires that the effective ice pressure pCT be estimated during impact. Extensive reviews of methods for predicting ice pressures have been compiled by Nessim et al. [85] and Sanderson [100]. The most commonly used methods for estimating effective ice pressure are: 1. Constant pressure, 2. Korzhavin's equation, 3. Pressure vs. area. These three methods will be discussed in greater detail. Chapter 6. Impact Sliding Force Design Event 199 Constant Ice Pressure The simplest method for computing ice loads is to assume a constant ice pressure. The assumption of constant ice pressure lends itself to the development of analytical solutions for ice forces, such as those developed by Cammaert and Tsinker [14] and Johnson and Nevel [50]. Selection of a suitable effective ice pressure is difficult due to the wide range of published results. Bass et al. [7] cited laboratory tests giving iceberg crushing strengths ranging from 10 MPa to 30 MPa. Laboratory test results surveyed be Nevel [87] indicated crushing strengths in the range 6-10 MPa. Field measurements of the strength of iceberg ice conducted by Johnson and Benoit [49] indicated highly varying ice pressures. It should be emphasized that the assumption of a constant ice pressure does not account for dependence of ice strength on factors such as strain rate, impact geometry, confinement effects, and size of contact area. Korzhavin's Equation An empirical equation accounting for geometric effects on ice pressure has been devel- oped by Korzhavin [54]. The prediction of ice impact loads using Korzhavin's equation has been adopted by several investigators, including Cox [18] and Croteau [20]. Ko- rzhavin's results are based on laboratory measurements of forces acting on a long narrow indenter impacting a sheet of ice, as shown in Figure 6.1. Korzhavin's equation for effective ice pressure appears as follows: pCT â€” I m k oc (6-6) where: I = indentation factor k = contact factor Chapter 6. Impact Sliding Force Design Event 200 Figure 6.1: Ice Indentation for Korzhavin's Experiments m = indenter shape factor crc = uni-axial compressive strength of ice (N/m 2) The indentation factor I arises from confinement effects, which increase the effective ice pressure. Croasdale [19] reports I values ranging from 1.0 to 4.0, and generally increasing with the ratio of ice height to structure diameter (ht/Da using the notation of the present thesis). The indenter shape factor m exhibits little variation, ranging from 0.92 for a circular indenter to 1.0 for a flat indenter. Non-simultaneous failure of ice is modelled using the contact factor k, which generally varies between 0.4 and 0.7 (Nessim et al. [85]). The uni-axial compressive strength of ice oc is dependent upon several factors, including ice composition, temperature, and strain rate. Croasdale [19] has presented crc values ranging from 0.5 MPa to 5.5 MPa for strain rates of 10~6 s _ 1 - 10~2 s - 1 . Chapter 6. Impact Sliding Force Design Event 201 When selecting a method for predicting iceberg impact loads, it should be remem- bered that Korzhavin's equation has two major limitations. Its experimental basis is limited to the case of a narrow vertical structure impacted by a wide, thin ice sheet (see Figure 6.1). The second limitation is imposed by the small scale of the laboratory tests which are the basis of the equation. Recent full-scale measurements [102] indicate that ice pressures can exhibit significant scale effects. Pressure-Area Methods Ideally, design ice pressures should be based on full-scale field measurements. Al- though several full-scale ice force measurement programs have been conducted, results have only recently been released to the public domain. A summary of full-scale and lab- oratory pressure measurements has been compiled by Sanderson [102] (see Figure 6.2). It is evident that effective pressure decreases significantly with increasing cross-sectional area. Sanderson has proposed the following relationship for large areas (i.e. A > 0.1m2): Various explanations have been proposed for the dependence of ice pressure on scale. One possibility is the larger size of flaws which lead to lower ice strength at larger scales. The occurrence of non-simultaneous failure at larger scales is another hypothesis. While the above two explanations appear quite plausible, a comprehensive understanding of ice pressure scale effects is still lacking. The pressure-area curve of Figure 6.2 offers a useful empirical method for selecting design pressures; however, one should be aware of the limitations of the compiled results. The majority of large-scale ice force measurements to date have been for thin ice sheets with low aspect ratios (ht/Di). For icebergs impacting a large structures, aspect ratios are likely to be significantly greater, thus leading to three-dimensional confinement effects. It Per oc (6.7) Chapter 6. Impact Sliding Force Design Event 202 1000 0. ' UJ DC V) CO Ul rr o. 10 Z 1 g I-Â£ z ui ot o z 0.01 8 8 FRESHWATER ICE o Laboratory (S-2) & lake ice o River bridge piers FIRST YEAR SEA ICE â€¢ Laboratory & in situ tests â€¢ Islands & structures MULTI YEAR SEA ICE o Hans Island -y-Meso-scale models 0.001 10- s 10-' 10" 3 10 ' 2 10"' 10Â° 10' 10 J CONTACT AREA m ! 10 3 104 10 5 10 6 10' Figure 6.2: Pressure vs. Area Measurements for Ice Strength (from Sanderson [102]) is postulated that large-scale iceberg collision pressures could be somewhat greater than those shown in Figure 6.2; however, supporting field data is lacking. 6.2 Failure Function for Iceberg Impact Force Reliability Model A reliability model has been developed for determining a design iceberg collision based on global impact force. The model utilizes the second-order reliability method (SORM) to evaluate exceedence probabilities for iceberg impact forces. A description of the SORM has been given in the previous chapter. The failure function for the impact force model is: G (X) = FR - FL (X) (6.8) Chapter 6. Impact Sliding Force Design Event 203 where: FL = impact force load (N) FR â€” structure impact force resistance (N) G â€” failure function (N) X = vector of input random variables (N) The SORM program can be run for various FR values to obtain the desired portion of the impact force cumulative distribution function (CDF). The random variables used as input for the impact force failure function are as follows: 1. Iceberg mass, M (kg), 2. Iceberg aspect ratio, h/Di, 3. Iceberg drift velocity in open water, Vdo (m/s), 4. Significant wave height, Hs (m), 5. Initial dimensionless eccentricity of iceberg approach, e'0, 6. Reference ice crushing pressure, pr (N/m 2), 7. Ice friction coefficient, fif. Deterministic parameters given as input to the model are: 1. Water depth, d (m), 2. Structure diameter, Ds (m). Chapter 6. Impact Sliding Force Design Event 204 Evaluation of the failure function proceeds in two distinct phases. In the first phase, the trajectory of the iceberg is modelled beginning at a point far away from the structure. The trajectory evaluation phase is used to obtain the iceberg velocity and location upon impact. During the second phase, the maximum global impact force is evaluated to determine the value of the failure function. Output from the model is given as a force exceedence probability given a potential impact. A potential impact occurs when an iceberg has an initial velocity and approach eccentricity which would cause an impact if the iceberg velocity (speed and direction) remained constant. Due to the influence of the structure on wave and current fields, a potential impact will not always lead to an actual impact; thus, there is a finite probability of the failure function G (X) being equal to zero. 6.3 Impact Velocity Phase of Reliability Model The evaluation of impact velocity required the development of an iceberg trajec- tory model. Although a. general model for icebergs approaching a structure has been developed by Isaacson [43], this model requires significant computational effort due to repeated diffraction calculations. A simplified trajectory model was developed for use in the reliability program. An important aspect of the trajectory model is the assumption of collinear waves and currents which is likely to be quite reasonable if one considers the influence of wind on both waves and currents. Input drift velocities for the model are based on observed iceberg velocities; thus, the assumption of collinear waves and currents will not influence estimates of open water drift velocities. Chapter 6. Impact Sliding Force Design Event 205 6.3.1 Equations of Drift Motion The motions of an iceberg approaching a structure are modelled using a numerical time-stepping procedure. The direction of propagation for incident waves and the incident current direction are both along the x axis. The following equations are used to evaluate the x and y iceberg acceleration components: M ( l + C m ) ax - ^pCdAw Vw - V (uw - u) + 1-P9CWD, H2rms (6.9) M(l + Cm) ay = l-PCdAw \VW - V\ (vw - v) (6.10) Variations in water velocities can be modelled as an iceberg approaches a structure. Iceberg velocities are evaluated using an Adams-Bashford method: u(t + At) = u(t) + ^ [3 ax{t) - ax(t - At)} (6.11) v(t + At) = v(t) + ^ [3 ay(t) - ay(t - At)] (6.12) Iceberg displacements are computed using the following second-order Taylor series ex- pansion: xg(t + At) = xa(t) + At u(t) + ^ A * 2 ax(t) (6.13) yg(t + At) = yg{t) + At v(t) + l- At2 ay(t) (6.14) 6.3.2 Initial Conditions for Simulated Drift Trajectories The drift trajectory for a potential impact is modelled beginning at a point far away from the structure, as shown in Figure 6.3. The dimensionless initial eccentricity e'0 will vary between 0 and 1, with the corresponding dimensional eccentricity e0 being evaluated Chapter 6. Impact Sliding Force Design Event 206 as follows: e0 = e'0 l- (Di + D.) (6.15) The initial x3 value for trajectory simulations is â€”5 Da. The flow field is relatively unaffected by the structure at this location, with the x component of water velocity being equal to 99% of the incident velocity. Iceberg Figure 6.3: Initial Conditions for Iceberg Drift Trajectory toward Structure Due to the limited availability of ocean current data, input water velocities are based on iceberg drift velocity Vd and significant wave height Ha. The assumption of collinear waves and currents allows one to estimate the incident current velocity. Using a force balance between wave drift and water drag forces, the incident current velocity is: Vwo = Vd0 - HTma (6.16) Chapter 6. Impact Sliding Force Design Event 207 6.3.3 Hydrodynamic Coefficients for Drift Motions The evaluation of iceberg drift motions using equations 6.9 and 6.10 requires the following hydrodynamic coefficients: 1. Zero frequency added mass coefficient C m , 2. Water drag coefficient Cd, 3. Wave drift force coefficient Cw. Zero frequency added mass and water drag coefficients are assumed to be indepen- dent of iceberg proximity to the structure. Zero frequency added masses are obtained from tabulated values, as discussed in Chapter 5. A water drag coefficient of 0.7 is used for the cylindrical icebergs in the present model. It was shown in Chapter 3 that wave drift forces on smaller icebergs (Di/Ds < 0.5) can vary significantly near a structure. The numerical ice motion model suggests that neg- ative wave drift forces can prevent smaller icebergs from impacting a structure; however, the experimental results of Chapter 4 suggest that the numerical model underestimates impact velocities for small icebergs. For the impact reliability model it is conservatively assumed that wave drift coefficients do not vary with proximity to the structure. It is suggested that the influence of negative wave drift forces be investigated once a design event has been selected using the reliability model. 6.3.4 Variation of Water Velocity The current field near the structure is modelled using the analytical solution for flow around a circular cylinder (equation 2.92). Figure 3.24 illustrates that the modification of the flow field by the structure will have a negligible effect on impact velocities for large icebergs; thus, the modification of the flow field is neglected for D{ > Da. Chapter 6. Impact Sliding Force Design Event 208 6 . 3 . 5 Iceberg Impact Velocity The drift trajectory time-stepping procedure begins at an initial location, as shown in Figure 6.3, and continues until one of the following occurs: 1. xg becomes greater than 0 (the iceberg drifts freely past structure), 2. The iceberg collides with the structure. The impact force for a potential collision is zero if the iceberg drifts freely past the structure. If an impact occurs, the significant oscillatory velocity is added to the drift velocity to obtain the total impact velocity components: VCx = Vdx + V3 (6.17) VCy = Vdy (6.18) It is assumed that oscillatory motions are unaffected by the presence of the structure; thus, the method described in Chapter 5 for estimating oscillatory motions in open water is also used in the'interaction model. Diffraction computations results presented in Chapter 3 confirmed that the presence of the structure had a relatively minor effect on oscillatory motions. 6.4 Impact Force Model The second phase of the impact force reliability model involves the computation of iceberg forces acting on the structure. Impact forces are evaluated using a numerical time- stepping procedure similar to models developed by Bass et al. [7], Duthinh and Marsden [25], and Salvalaggio and Rojansky [99]. Due to the possibility of eccentric impacts, both radial and tangential forces can develop on the structure. Environmental forces from Chapter 6. Impact Sliding Force Design Event 209 waves and currents are assumed to be negligible during impact. Due to the modelling of iceberg geometry as a vertical circular cylinder, the contact area An and associated forces for a given penetration distance A will likely be greater than contact areas and forces for actual iceberg geometries. Thus, the idealized iceberg geometry will lead to conservative predictions of design impact forces. The inclusion of non-cylindrical iceberg geometries would be a justified improvement to the model if sufficient observations of iceberg geometries were available for a given site. Bass et al. [7] and Salvalaggio and Rojansky [99] have shown that impact forces can be significantly reduced if the contact face of an iceberg has a non-vertical profile. 6.4.1 Development of Impact Forces The present model considers impact forces due to ice crushing and friction. Iceberg geometry and velocity components associated with collision forces are shown in Figure 6.4. The magnitude of the crushing force is given by equation 6.1. For a cylindrical iceberg impacting a cylindrical structure, the crushing force Fcr is directed toward the centre of the structure. The existence of crushing forces requires that the iceberg have a positive normal velocity Vn toward the centre of the structure, as shown in Figure 6.4. The frictional force Fj is evaluated using equation 6.5. For the existence of a frictional force, it is necessary for the iceberg face to have a tangential velocity component Vt relative to the structure. The resulting frictional force Fj will act in a direction opposite to that of Vt. 6.4.2 Collision Geometry The evaluation of ice force requires that the normal contact area An be known. For the present case of an axisymmetric iceberg impacting an axisymmetric structure, Chapter 6. Impact Sliding Force Design Event 210 Figure 6.4: Eccentric Iceberg Collision Schematic, Plan View the impact geometry is dependent merely on the iceberg location in the x â€” y plane. Figure 6.4 illustrates the geometry of an iceberg collision, with the origin of the x â€” y plane located at the structure centre. The penetration distance of the structure into the iceberg is as follows: A = rt + r s - Jx* + yj (6.19) Solving for the normal contact area An is easily accomplished if the penetration distance A is broken into components A; and A , , as shown in Figure 6.4. The contact area can then be evaluated using either A,- or A , : An = 2 ht y/2 r s A,- - A? (6.20) = 2 ht y/2 n A, - A 2 (6.21) Chapter 6. Impact Sliding Force Design Event 211 Given that the areas given by equations 6.20 and 6.21 must be equal and that A = A,- -f A31 it is possible to solve for A3 given A: r A - 1 A 2 ^ - (6.22) Thus, the contact area An can be determined by successive evaluation of equations 6.19, 6.22, and 6.21. Once An is known, the crushing and frictional forces are evaluated using equations 6.1 and 6.5 respectively. The frictional force Fj will cause a rotational torque between the structure and iceberg. The effective moment arm for the torque acting on the iceberg is approximated as: rti = ri ~ \ A i ~ A * ( 6- 23) . where: rti = moment arm for frictional torque acting on iceberg (m) 6.4.3 Pressure Calculation Method The impact force reliability model allows the user to select one of the following two methods for evaluating effective ice pressure: 1. Constant pressure method, 2. P-Area method. Both methods are based on a reference ice pressure pr, which is an input random variable to the reliability model. For the constant pressure method, the effective crushing pressure pcr is equal to pr for the duration of a collision. Chapter 6. Impact Sliding Force Design Event 212 Effective ice pressure computed using the P-Area method is evaluated as follows: pr for An < 0.1m2 pr (An I 0.1m2)"0-4 for An > 0.1 m 2 The above equation is an approximation of the P-Area curve developed by Sanderson [102]. Equation 6.24 is evaluated at each time step of a collision. 6.4.4 Iceberg Equations of Motion Evaluation of iceberg motions during a collision is based on the normal and tangential forces acting on the iceberg during impact. It is useful to introduce the contact point angle 9C between the iceberg and the structure (see Figure 6.4). The tangential velocity of the iceberg relative to the structure can then be expressed as: ut = Lor rt{ + ug sin 9C â€” vg cos 0C (6.25) where: 9C â€” contact point angle between iceberg and structure (rad) u)T = rotational velocity of iceberg (rad/s) The sign convention for ut is such that a positive value will represent counter-clockwise rotation relative to the structure. The friction force Ff acting on the iceberg will act in a direction opposite to ut. Considering Fj to be positive if it induces counter-clockwise rotation of the iceberg, the friction force can be written as: Fj = -~pCTAnyij (6.26) \ut\ Once the crushing force Fcr and frictional force Ff are determined, the associated translational force components and torsion acting on the iceberg can be evaluated: Fx = Fcr cos#c + Ff sin0c (6.27) 6.24 Chapter 6. Impact Sliding Force Design Event 213 Fy = FCT sin0c - Fs cosdc (6.28) Tr = F}rti (6.29) where: Fx, Fy = x â€” y force components acting on iceberg (N) Tr â€” rotational torsion acting on iceberg (N-m) The corresponding equations of motion for the iceberg are: d ug Fx dt (1 -f Cn) M dvg _ F\, dt (1 + Cm) M dur Tr (6.30) (6.31) (6,32) dt h where: Izz = iceberg rotational inertia about z axis (N-m2) It should be noted that the added rotational inertia of a cylindrical iceberg is zero. As stated previously, the iceberg must have a positive normal velocity relative to the structure for crushing forces to develop. The normal velocity can be expressed as: Vn â€” â€”ug cos0c â€” Vg smOc (6.33) The development of frictional forces requires the existence of corresponding normal forces. In addition, the relative tangential velocity ut must be non-zero for frictional forces to exist. 6.4.5 Time-Stepping Procedure Iceberg motions during the impact phase are determined using a numerical time- stepping procedure. The impact phase commences when the iceberg first comes into Chapter 6. Impact Sliding Force Design Event 214 contact with the structure, and continues until the normal velocity Vn becomes negative. Translational and rotational accelerations are determined using equations 6.30 - 6.32, with associated velocities and displacements computed as follows: u 9 t+l V9 t + l + 2 At A* v Â« + T At r t+l 'd dt ( BUg V dt 'dVg , d t , ( f ^ I ( dt t-iJ t'-u t-u A 1 A 2 (9Ug\ Xgi+1 = x9, + Ai U g i + - At y-gjj ygi+i = y3, + &tvgi +\&t2 (^jA 1 A o (dujT + 1 = eri + Atu>ri +- At2 1 - ^ (6.34) (6.35) (6.36) (6.37) (6.38) (6.39) where: 6r = iceberg rotational displacement (rad) Of critical importance in the impact phase model is the selection of an appropriate time step size. Trial simulations indicated that 40 time steps were sufficient for obtaining accurate force predictions and energy balances. An initial estimate for the collision duration tc is made using the estimated collision duration for a head-on collision (equation 3.23). The impact phase model is then run using a time interval At of i c/100. If the numerical model requires less than 40 time steps, the model is re-run .using a finer time interval. Conversely, a larger time interval is selected if the initial simulation is not finished after 500 time steps. Special attention must be given to the evaluation of frictional forces in the time- stepping model. The friction force Fj will typically induce rotation of the iceberg, causing Chapter 6. Impact Sliding Force Design Event 215 its tangential velocity relative to the structure to be reduced to zero. Disregard for this phenomenon when running the time-stepping model can cause the tangential velocity ut to fluctuate about zero (Duthinh and Marsden, [25]). This problem was alleviated by restricting the maximum possible frictional force during any given time step: l*>l < (6-40) The impact force reliability model is based on the maximum sliding force experienced during a collision. The magnitude of the sliding force is giving by: The sliding force is evaluated at each time step and compared to the previous maximum value that has occurred during a collision. 6.5 Example Problem for Large Iceberg Population The impact force reliability model has been used to determine design iceberg colli- sions for an offshore structure with a diameter of 100 m located in water of 100 m depth. Design events have been evaluated for the case of constant ice pressure, and for varying ice pressure with contact area as given by equation 6.24. 6.5.1 Design Criterion and Input Distributions The design criterion for the impact force examples is the same as for the kinetic energy examples. The design load level will have an exceedence probability of 0.1 during the life of the structure. Given that 20 collisions are expected during the structure life, the acceptable exceedence probability for a single collision is 0.0052, as given by equation 5.43. (6.41) Chapter 6. Impact Sliding Force Design Event 216 Input distributions for the large iceberg example are given in Table 6.1 and are considered to be representative of conditions at the Hibernia oilfield off the coast of Newfoundland. Distributions for iceberg mass, aspect ratio, drift velocity, and significant wave height are the same as those for the large iceberg example of Chapter 5. The correlation coefficient for iceberg mass and aspect ratio is again assigned a value of -0.1. Table 6.1: Input Distributions for Large Iceberg Example Design Problem Variable Distribution Parameters M (106 tonnes) Lognormal p = 0.50, a --- 1.74 h/Di Uniform Range 0.1 - 0.8 Vdo (m/s) Lognormal p = 0.34, a = 0.29 Ha (m) Lognormal p = 2.44, a = 1.22 e'0 Uniform Range 0.0 - 1.0 pr (MPa) Lognormal p = 5.0, a = 5.0 Lognormal p = 0.08, a --- 0.04 Initial approach eccentricity e'0 is uniformly distributed between 0 and 1 for potential impacts. A compilation of measured ice pressures by Sanderson [102] suggests that the reference ice pressure pr should have a mean value of 5 MPa. Significant scatter of measured ice pressures indicates that the coefficient of variation for pr is of the order of unity. Laboratory studies by Forland and Tatinclaux [29] indicate that the ice friction coefficient p,j should have a mean value of approximately 0.08 and a standard deviation of approximately 0.04. Because ice pressure and friction coefficient must both be positive, they are both modelled with lognormal distributions. Chapter 6. Impact Sliding Force Design Event 217 6.5.2 Design Event for Constant Ice Pressure The reliability model has been run for various impact force levels to determine the sliding force exceedence probabilities for constant ice pressure presented in Figure 6.5. For the acceptable exceedence probability of 0.0052 for a single collision, the corresponding impact force is 6.2 x 109 N with design values and sensitivity coefficients as given in Table 6.2. 0.001 Design point Sliding Force (109N) Figure 6.5: Sliding Force Exceedence Probabilities for Large Iceberg Example, Constant Ice Pressure Due to the large size of the design iceberg, its oscillatory velocity and the influence of the structure on the iceberg drift velocity are both negligible. The sensitivity coeffi- cient for approach eccentricity e'0 has a value of -0.10, suggesting that the influence of eccentricity is relatively small. To examine the influence of eccentricity on impact force, the design collision was re-evaluated for a head-on impact keeping all other parameters the same. It was found that the impact eccentricity for the design event resulted in a 5% Chapter 6. Impact Sliding Force Design Event 218 Table 6.2: Results for Large Iceberg Design Collision, Constant Ice Pressure Variable Design Value Sensitivity M (106 tonnes) 2.86 0.74 h/D{ 0.41 0.27 Vdo (m/s) 0.56 0.40 Hs (m) 2.20 0.008 e'0 0.40 -0.10 pr (MPa) 9.41 0.46 uf 0.072 0.002 Other parameters associated with above design values: Di = 205 m, h = 84 m, Cm = 0.81, Tp = 6.6 s, u;2Di/2g = 9.54, ([ = 0.005, V, = 0.006 m/s, tc = 0.57 s force reduction. The very small sensitivity coefficient for the friction factor Uf suggests that the friction factor has a negligible influence on the design event, and that uj could be treated as a constant. Note that uj would likely become very important if the design event were based on the torsion acting on the structure. The small collision duration indicates that the infinite frequency added mass should be used for the effective iceberg added mass during impact. The infinite frequency added mass coefficient for the design iceberg is 0.31, compared with 0.81 for the zero frequency value. Despite the great difference in the added mass values, the impact force computed using the infinite frequency added mass is only 10% less than the force obtained using the zero frequency added mass. The sensitivity coefficient for aspect ratio h/Di is much greater for the impact force reliability model than for the kinetic energy model. Although aspect ratio does not Chapter 6. Impact Sliding Force Design Event 219 significantly affect iceberg kinetic energy, it has a major influence on impact force. 6.5.3 Design Event for Varying Ice Pressure with Area When the variation of ice pressure with area (equation 6.24) is incorporated into the reliability model, impact sliding forces become significantly lower as shown in Figure 6.6. The impact force of the design event is reduced to 0.43 x 109 N , only 7% of the design load for constant ice pressure. Design values and sensitivity coefficients are given in Table 6.3. Sliding Force (109N) Figure 6.6: Force Exceedence Probabilities for Large Iceberg Example, P-Area Variation The sensitivity coefficient for reference pressure pr is significantly greater when the P-Area force method is used. Accordingly, sensitivity coefficients for other variables such as iceberg mass and drift velocity are somewhat smaller. The collision duration of 3.8 s for the design collision is approximately 7 times greater than the corresponding duration for constant ice pressure. Due to the relatively long duration, effective added mass during impact is approximated reasonably well by the zero frequency added mass. Chapter 6. Impact Sliding Force Design Event 220 Table 6.3: Results for Large Iceberg Design Collision, P-Area Variation Variable Design Value Sensitivity M (106 tonnes) 1.73 0.62 h/Di 0.41 0.21 Vdo (m/s) 0.50 0.35 Ht (m) 2.21 0.012 e'0 0.41 -0.09 pT (MPa) 14.6 0.67 Ht 0.072 0.003 Other parameters associated with above design values: Di = 174 m, h = 71 m, Cm = 0.70, Tp = 6.6 s, LolDi/2g = 8.03, ([ = 0.008, Vs = 0.009 m/s, tc = 3.8 s 6.6 Example Problem for Small Iceberg Population To investigate the influence of iceberg size on collision properties, the impact force reliability model has been re-run for an iceberg population with masses that are 1/1000'th of those in the large iceberg example. A similar example problem was investigated using the kinetic energy reliability model. Input distributions are the same as those given in Table 6.1, with the exception of the mass distribution for impacting icebergs. The mass distribution for the smaller iceberg problem has a mean value of 360 tonnes and a standard deviation of 1190 tonnes. The specified exceedence probability for a single collision is 0.0052, as was used for the large iceberg example. Chapter 6. Impact Sliding Force Design Event 221 6.6.1 Design Collision for Constant Ice Pressure Sliding force exceedence probabilities for constant ice pressure are shown in Fig- ure 6.7. Parameters for the design load of 135 x 106 N are given in Table 6.4. 0.001 Design point 50 100 150 Sliding Force (106N) 200 Figure 6.7: Force Exceedence Probabilities for Small Iceberg Example, Constant Ice Pressure Due to the small size of the design iceberg relative to incident wavelength, oscillatory motions are significant. The oscillatory velocity of the iceberg is 2.6 times its drift velocity. The small size of the iceberg relative to the structure suggests that negative wave drift forces could prevent the design iceberg from impacting the structure; however, the variation of wave drift forces with proximity to the structure has been neglected in the reliability model, and thus a detailed investigation into the influence of variable wave drift forces would be worthwhile for the design event. As was observed for the large iceberg examples, the influence of collision eccentricity and frictional forces appears to be very small. The short collision duration again suggests Chapter 6. Impact Sliding Force Design Event 222 Table 6.4: Results for Small Iceberg Design Collision, Constant Ice Pressure Variable Design Value Sensitivity M (tonnes) 2500 0.74 h/Di 0.36 0.28 Vd0 (m/s) 0.31 0.10 H. (m) 2.80 0.22 e'0 0.38 -0.07 pT (MPa) 10.9 0.56 uf 0.072 0.002 Other parameters associated with above design values: Di = 20.6 m,h = 7.3 m, Cm = 0.49, Tp = 7.4 s, u>lDi/2g = 0.75, ([ = 0.68, Vs = 0.81 m/s, tc = 0.044 s that infinite frequency added mass should be used for evaluating impact forces. If the infinite frequency added mass of 0.24 is used, the resulting impact force for the design collision is reduced by 8%. 6.6.2 Design Collision for Varying Ice Pressure If the P-Area method is used for predicting ice pressure, impact forces are signifi- cantly reduced as shown in Figure 6.8. The design load level is 32 x 106 N , approximately 1 /4 of the load for constant ice pressure, with design collision parameters as given in Ta- ble 6.5. The sensitivity coefficients indicate that reference ice pressure becomes more important when the P-Area method is used, as was observed for the large iceberg exam- ple. As expected, the reduction of design load due to P-Area variation is not as great for the small iceberg example. Chapter 6. Impact Sliding Force Design Event 223 O h S l i d i n g Force (10 aN) Figure 6.8: Force Exceedence Probabilities for Small Iceberg Example, P-Area Variation Table 6.5: Results for Small Iceberg Design Collision, P-Area Variation Variable Design Value Sensitivity M (tonnes) 1350 0.69 h/Di 0.42 0.12 Vd0 (m/s) 0.29 0.04 Hi (m) 2.56 0.12 e'0 0.39 -0.10 Pr (MPa) 16.6 0.69 PJ 0.072 0.003 Other parameters associated with above design values: Di = 15.8 m, h = 6.7 m, Cm = 0.53, Tp = 7.1 s, = 0.63, C{ = 0.73, V, = 0.82 m/s, tc = 0.088 s Chapter 7 Conclusions Various factors influencing the risk of iceberg collisions with a fixed offshore structure have been examined. Special emphasis has been given to the evaluation of iceberg added mass and impact velocity, which are determined by hydrodynamic effects. Findings from hydrodynamic studies have been incorporated into probabilistic models for evaluating design events based on iceberg kinetic energy and impact force. The probabilistic mod- els can be used to examine the importance of hydrodynamic effects in the design of a particular structure. 7.1 Effective Added Mass Due to the transient nature of iceberg motions during drifting and collisions with structures, iceberg motions are dependent on hydrodynamic memory effects. It is useful to adopt an effective added mass for evaluating iceberg drift motions or iceberg kinetic energy. For a drifting iceberg subjected to forces which vary slowly with time, zero frequency added mass can be adopted for evaluating accelerations. Effective added mass is also useful for estimating the kinetic energy or momentum that can be dissipated during an iceberg collision with an offshore structure. For collisions of short duration (i.e. tcJg/Di < 0.5), the infinite frequency added mass can be used to approximate effective 224 Chapter 7. Conclusions 225 added mass. The effective added mass for long duration collisions (i.e. tcyg/Di > 3) will approach the zero frequency value. If a vertical circular cylinder is selected to model iceberg geometry, the aspect ratio h/Di is restricted to a maximum value of 0.86 due to hydrostatic stability requirements. The zero frequency added mass for a circular cylinder increases with relative draft h/d, and approaches a limiting value of 1.0. Alternatively, the infinite frequency added mass for a circular cylinder increases with aspect ratio h/Di, and approaches a limiting value of approximately 0.45 when h/Di approaches 0.86. Thus, the effective added mass for an impacting iceberg can vary considerably depending on collision duration. Despite the large difference between zero and infinite frequency added mass values, computations presented in Chapter 6 indicate that the maximum impact force will vary by only 5-10% depending on whether the zero frequency or infinite frequency added is used for force computations. It is recommended that zero frequency added mass can be used in iceberg collision design for all impact scenarios, with the resulting impact force levels having a reasonable degree of conservatism. 7.2 Oscillatory Surge Motions When evaluating iceberg impact velocities, both oscillatory and mean drift velocity components should be considered. Oscillatory motions can be particularly important for smaller icebergs. Due to the general availability of wave climate data, it is recommended that oscil- latory velocities be estimated using response amplitude operators (RAO's). For the low frequency range u>2 A / 2 g < 0.5, it is reasonable to assume that an iceberg will oscillate like a water particle, with its corresponding surge RAO given by equation 3.39. The MacCamy-Fuchs solution for the exciting force on a circular cylinder provides a useful Chapter 7. Conclusions 226 closed-form solution (equation 3.46) for high frequency surge motions (LO2 D{/2 g > 3). For the intermediate frequency range, computed RAO's from a numerical diffraction model such as those presented in Chapter 3 should be used for determining oscillatory motions. Results from the kinetic energy and sliding force reliability models indicated that oscillatory motions had a negligible influence on design impact velocities when the frequency parameter LO 2 -D,/2 g was greater than three for the design event; however, it should be remembered that the relative importance of oscillatory motions is dependent on the magnitude of drift velocities. 7.3 Influence of Structure on Iceberg Hydrodynamics The velocity and added mass of an iceberg can be significantly modified as it ap- proaches a large structure. Hydrodynamic interactions appear to be important when the iceberg diameter is less than half the structure diameter. The analytical solution for potential flow around a circular cylinder can be used for computing current driven iceberg motions near a large structure. Reasonably good agreement was observed between physical and numerical models for current driven mo- tions near a large structure. It is suggested that the inclusion of convective forces in the numerical drift model could lead to improved estimates of impact eccentricities for current driven motions. The modification of the flow field by the structure can reduce iceberg impact velocities by 10% or more when the relative iceberg size Di/Ds is less than 0.3. Experimental observations indicated that approach eccentricities can be significantly increased as icebergs come toward a structure when Di/Ds is less than 0.3. Surge response amplitude operators have been computed for icebergs in the vicinity Chapter 7. Conclusions 227 of a large structure. The variation in surge response with proximity to the structure appears to be relatively small. The greatest influence that a structure will have on motions of approaching icebergs is through the modification of wave drift forces. The numerical diffraction model indi- cates that wave drift forces vary significantly when the iceberg diameter is less than half the structure diameter. Wave drift forces vary cyclically with distance in front of the structure, with a cycle length equal to half the incident wavelength. Due to the large degree of computational effort required for determining wave drift force coefficients, it is suggested that equation 3.52 be used for estimating wave drift forces while evaluating drift trajectories in front of a structure. A comparison of physical and numerical model results revealed large discrepancies in iceberg impact velocities when wave drift forces varied significantly with proximity to a structure. Impact velocities predicted by the numerical model were significantly lower than velocities for the physical model. When calculating wave drift forces using the far-field method, various second-order effects such as the variation of iceberg rotational orientation due to oscillatory motions were ignored, thus leading to possible errors. Note that the wave drift force coefficient of a small iceberg in front of a structure is highly dependent on wave frequency; thus, the variation of effective wave frequency due to iceberg drift motions will likely have a significant influence on wave drift forces near a structure. In consideration of the above findings, it is suggested that hydrodynamic interac- tions between a structure and approaching icebergs can be neglected when designing for iceberg collisions in most cases. Although hydrodynamic interactions can significantly reduce drift velocities of smaller icebergs (Di/Ds < 0.5) in waves near a large structure, additional numerical and experimental studies need to be conducted before conclusive recommendations can be made regarding the degree of velocity reductions that will occur. Chapter 7. Conclusions 228 7.4 Kinetic Energy Reliability Model A probabilistic model has been presented for evaluating a design collision event based on iceberg kinetic energy. The model has been formulated such that input statistical distributions can be obtained from commonly available environmental data. Iceberg added mass and oscillatory motions are evaluated through deterministic relationships. The two example problems indicate that iceberg kinetic energy is relatively insen- sitive to aspect ratio h/Di\ thus, it would seem reasonable to infer that iceberg kinetic energy will exhibit little variation with iceberg geometry for a given mass. The exam- ple problem for a population of large icebergs indicated that wave-induced oscillatory motions had a negligible influence on iceberg kinetic energy for a design collision event; however, the wave-induced oscillatory component dominated the design impact velocity for a population of smaller icebergs. 7.5 Impact Force Reliability Model If failure of an offshore structure is most likely to be caused by the global load experienced during an iceberg collision, then the design collision should be evaluated using an impact force reliability model such as the one described in Chapter 6. The variation of current velocities near the structure is considered in the model; however, the variation of wave drift forces is neglected. Sensitivity coefficients for example problems indicated that iceberg approach eccen- tricity has a relatively small effect on design loads; thus, the assumption of head-on collisions would not be overly conservative. The extremely low sensitivity of the ice friction coefficient suggests that uj can be treated as a constant. It was previously shown that iceberg aspect ratio h/Di has a minor influence on Chapter 7. Conclusions 229 iceberg kinetic energy; however, sensitivity coefficients for the impact force examples show a strong dependence on aspect ratio. This result is due to the influence of contact geometry on impact forces. For the vertical sided cylindrical icebergs used in the impact force model, contact areas and associated forces are likely to be greater than values for actual iceberg geometries; thus, the present model can be considered to be conservative. The inclusion of random iceberg geometries would be a significant improvement to the model if sufficient field data concerning actual iceberg shapes were available. The inclusion of varying wave drift forces with iceberg proximity to a structure is an additional improvement to the impact force model that should be considered more closely. For a structure located in an area of relatively small icebergs (D{/Ds < 0.5), the influence of the structure on iceberg wave drift forces could significantly reduce the impact velocity for a design iceberg collision event. As shown in the example problems, the variation of ice pressure with contact area can have drastic effect on design forces. For a structure situated in a population of large icebergs, the design impact force for P â€” Area variation was shown to be less than one-tenth the corresponding design force based on constant ice pressure. As expected, the importance of P â€” Area effects increases with the size of impacting icebergs. There remains great uncertainty regarding ice pressures for large contact areas. Sources of this uncertainty include limited force measurements for large contact areas, and the lack of a comprehensive theory explaining the variation of ice pressure with contact area. 7.6 Recommendations for Future Research There remain many areas of theoretical and practical interest where further research is warranted. The evaluation of iceberg added mass near a structure using the bound- ary element method is limited by the number of facets that can be modelled. Reliable Chapter 7. Conclusions 230 values can only be obtained when the separation distance between the iceberg and struc- ture is greater than half a facet length. Isaacson and Cheung [46] have developed a two-dimensional model for predicting zero frequency added mass of an iceberg near a structure. Since the effective added mass of an impacting iceberg is likely to be closer to the infinite frequency added mass, it is suggested that a solution be found for infinite frequency added mass near a structure. Multi-directional wave effects have been neglected in the present study, resulting in over-predictions of wave drift forces and oscillatory motions. Multi-directional wave effects could significantly reduce impact velocities, particularly for smaller icebergs. For current driven iceberg motions near a large structure, further work should be done to examine the influence of convective forces on iceberg motions. Further experi- mental and numerical work should particularly address the influence of convective forces on iceberg approach eccentricities. Of the various hydrodynamic effects influencing iceberg collisions, the greatest uncer- tainty lies with the variation of wave drift forces on small icebergs near a large structure. A greater understanding of second-order effects influencing wave driven drift motions is required. Due to the sensitivity of wave drift forces to wave frequency, the influence of random waves on wave drift forces also needs to be addressed. 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