Advanced Series on Ocean Engineering  Volume 9
OFFSHORE STRUCTURE MODELING
Subrata Kumar Chakrabarti World Scientific
OFFSHORE STRUCTURE MODELING
ADVANCED SERIES ON OCEAN ENGINEERING Series EditorinChief
Philip L F Liu (Cornell University) Vol. 1 The Applied Dynamics of Ocean Surface Waves by Chiang C Mei (MIT) Vol. 2 Water Wave Mechanics for Engineers and Scientists by Robert G Dean (Univ. Florida) and Robert A Dalrymple (Univ. Delaware) Vol. 3 Mechanics of Coastal Sediment Transport by J: rgen Fredsee and Rolf Deigaard (Tech. Univ. Denmark) Vol. 4 Coastal Bottom Boundary Layers and Sediment Transport by Peter Nielsen (Univ. Queensland) Vol. 5 Numerical Modeling of Ocean Dynamics
by Zygmunt Kowalik (Univ. Alaska) and T S Murty (Inst. Ocean Science, BC) Vol. 6 Kalman Filter Method in the Analysis of Vibrations Due to Water Waves by Piotr Wilde and Andrzej Kozakiewicz (Inst. Hydroengineering, Polish Academy of Sciences) Vol. 7 Physical Models and Laboratory Techniques in Coastal Engineering by Steven A. Hughes (Coastal Engineering Research Center, USA) Vol. 8 Ocean Disposal of Wastewater by Ian R Wood (Univ. Canterbury), Robert G Bell (National Institute of Water & Atmospheric Research, New Zealand) and David L Wilkinson (Univ.
New South Wales) Vol. 9 Offshore Structure Modeling by Subrata K. Chakrabarti (Chicago Bridge & Iron Technical Services Co., USA)
Forthcoming titles: Water Waves Propagation Over Uneven Bottoms by Maarten W Dingemans (Delft Hydraulics) Tsunami Runup by Philip L F Liu (Cornell Univ.), Costas Synolakis (Univ. Southern California), Harry Yeh (Univ. Washington ) and Nobu Shuto (Tohoku Univ.) Beach Nourishment : Theory and Practice by Robert G Dean (Univ. Florida)
Design and Construction of Maritime Structures for Protection Against Waves by Miguel A Losada ( Univ. da Cantabria) and Nobuhisa Kobayashi (Univ. Delaware)
Advanced Series on Ocean Engineering  Volume 9
OFFSHORE STRUCTURE MODELING
SUBRATA KUMAR CHAIRABARTI Chicago Bridge & Iron Technical Services Co. Plainfield , Illinois
USA
World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 9128 USA office: Suite 1B , 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
OFFSHORE STRUCTURE MODELING Copyright 0 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form orbyanymeans, electronic ormechanical, including photocopying, recordingorany information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA.
ISBN 981 0215126 9810215134 (pbk)
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
ACKNOWLEDGEMENTS Since joining CBI twentyfive years ago I have been involved in model testing of offshore and marine structures. Many colleagues of mine have helped me understand tricks of model testing over the years. In particular, Erik Brogren provided guidance on model construction details and Alan Libby explained intricacies of instrumentation design. Many experts reviewed the chapters of this book. Dr. Devinder Sodhi and Prof. Tom Dawson reviewed Chapter 2. Prof. Dawson also checked sections of Chapter 7. Dr. E.R. Funke reviewed Chapter 4. Prof. Christian Aage took the time to review Chapters 4, 7 and 10 and provided valuable comments. Chapter 5 was reviewed by Prof. Bob Hudspeth, Dr. E. Mansard, and Dr. Andrew Cornett. Dr. Erling Huse commented on Chapter 6. Chapter 7 was read by Prof. Li and Dr. Ove Gudmestad. Prof. S. Bhattacharyya improved on Chapter 8. Chapter 9 was reviewed by Dr. O. Nwogu. Keith Melin again reviewed the entire book and provided many editorial and other comments which improved its quality. I am grateful to these individuals. I am, however, responsible for any shortcomings in this book.
CBI Technical Services provided the secretarial help which made this book possible. Many individuals helped in putting the manuscript together, of which the most noteworthy is Ms. Danielle Cantu who finalized the manuscript in its printed form by retyping and reformatting it many times. Finally, I acknowledge the patience of my wife Prakriti (Nature) for providing me the time at home over the last 3 years to complete the book.
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DEDICATION This third book by the author is dedicated with pride to the following people that had the greatest contribution in shaping his professional life.
° His professor and Ph. D. thesis adviser, Dr. William L. Wainwright who gave him the first lessons in writing a technical paper which became the first publication by the author. ° His first supervisor, Mr. William A . Tam, then Director of Marine Research, who had faith in the author's ability and gave him an opportunity to work on hydrodynamic related subjects. Dr. Basil W. Wilson, then consultant of Chicago Bridge and Iron Co. who taught the author many of the basics of ocean engineering and mooring systems. The author had the good fortune then in cooperating with Dr. Wilson on several research projects.
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PREFACE The offshore industry has matured over the years through innovation , initiatives and experience . The industry has advanced a long way to its present stage from its first installation of an exploration structure in coastal waters of the Gulf of Mexico in the 1940's. The early structures were fixed to the ocean floor and looked much like the electrical transmission towers common on land. Today the shape , size and type of offshore structures vary depending on the required applications . Offshore structures are abundant in all parts of the world . In addition to structures fixed at their base , moored floating structures and vertically tethered structures have been installed for exploration, production, storage and offshore processing of crude oil . While offshore exploration has been relatively trouble free , there have been a few catastrophic failures . Therefore, design of these structures for accident free operation under anticipated conditions is vital for the continued success and growth of the offshore industry . One of the means of verifying the design of a structure is the testing of scale model of the structure in a simulated ocean environment during its design phase. Model testing has been performed through the history of mankind . Systematic hydraulic scale model testing goes back to the nineteenth century. However , even today modeling is partly an art as well as a science . In modeling , certain laws of similarity are followed. Several text books are available that deal with these similarity laws. In many cases, these laws can not all be satisfied in a model test. In these cases, it is necessary to selectively distort some of these scaling laws to perform the model tests . Although this distortion is somewhat of a compromise , valid modeling results can still be expected. While the technical literature dealing with offshore structures discusses model testing, a comprehensive book in this area discussing design, construction, instrumentation , testing and analysis of physical model is lacking . It is desirable that a single text contain the theoretical and practical aspects of physical modeling. Such a book should be valuable to engineers dealing with the design , construction, installation and operation of offshore structures . This requirement inspired me to write this book on modeling . This book provides reasonably detailed coverage of the technology of model testing. As such , it has applications throughout the entire field of engineering , reaching far beyond its focus on offshore structures. It should be equally appealing to engineers and scientists involved with the design and construction of unique structures. The Introduction discusses the general need for model testing . A brief history of testing has been given in this section . Some of the general structural areas where model testing has been required have also been mentioned . Chapter 2 describes the modeling laws. It begins with the general requirements for similarity . A general discussion of the famous Buckingham pi theorem and an application of the pi theorem has been included. A few specific examples of modeling are discussed here including structural modeling, testing in uniform flows and modeling distortion . These are considered unique cases that
x Preface
may be applicable to an offshore structure . In hydraulic testing, however, Froude model law is discussed in detail in this section . A few textbooks that deal with the similarity laws and model testing have been discussed and referenced here. The methods of model construction are considered in Chapter 3. The physical requirements of the model necessary for scale testing are explained. Generally, the construction technique varies based on these requirements. For example, the fixed structures are generally used in the measurement of loads and stresses imposed by the environment . In this case, the dynamic properties of the structure are not a concern . This, however, is not true for compliant or floating structures. The compliant structure, in addition, must satisfy additional scaling laws. The static and dynamic properties that must be satisfied by these structures are discussed in detail . The techniques used in verifying these properties before testing can take place are illustrated. The testing of offshore structures requires specialized facilities. Many of the small facilities that exist at the universities and other educational institutions are used as teaching tools in discussing the needs and methods of model testing. However, many larger commercial facilities are in existence in various parts of the world. Many of these facilities are described in detail in Chapter 4 including their capabilities and limitations. This section will be useful to a design engineer in choosing a suitable facility for his particular test requirements. The important components of these testing facilities , such as the wave generators, the current generators , the towing carriages and beaches are described. The most important feature of these testing facilities is the wave generation capability. A few theories of wave generation and beach reflection are presented for those interested in designing wavemakers. These facilities are used in duplicating the ocean environment. The modeling of the ocean environment is the subject of Chapter 5. Simulation of various types of waves, such as random twodimensional and threedimensional waves, wave groups , and higher harmonic waves , are discussed in detail. The wind and current generation and cogeneration of current and waves are also explained in this section . Another important requirement of any model testing is the measurement of the responses of a structure model . This includes the inputs to structures from external sources and the corresponding outputs. In hydraulic testing, this measurement is further complicated by the presence of fluids. Various types of measuring instruments and measuring techniques are introduced in Chapter 6. Design methods of a few specialized instruments are described and methods of the waterproofing these instruments are discussed . The calibration procedure for special instruments is shown. Typical calibration curves of a few of these are also included. The important considerations in the recording of data output from these instruments are given here.
The actual modeling of various offshore structures is described in Chapters 79. Various areas covered in these chapters are outlined in the following:
Preface
FIXED STRUCTURES
OFFSHORE OPERATIONS
FLOATING STRUCTURES
Gravity Platform Storage Structures Piled Jackets Subsea Pipelines
Transport of Jackets Towing of Structures Launching of Structures Submergence of Structures
Buoys Single Point Moorings Moored Tankers Tension Leg Platforms
Pipelaying
Compliant Structures Semisubmersibles
xi
Chapter 7 deals with the fixed structures . The methods of installing load measuring devices on fixed structures and associated problems are described in detail. Various examples are included. Actual recorded data are given to illustrate the validity as well as inadequacies of the techniques . Offshore operations include special techniques in delivering the completed structure to the offshore sites. These require launching, transportation, submergence and installation of these structures . These various stages are generally model tested to insure a proper installation procedure and to identify potential unforeseen problems. This is the subject of Chapter 8. Chapter 9 covers the area of seakeeping tests. These tests include floating structures moored to the ocean floor by mooring lines, articulated columns, vertically tethered structures, e.g., tension leg platforms and compliant structures. Special care is needed to insure proper duplication of model response without introduction of additional effects through setup. Example cases are discussed and illustrated. In all these tests , data are recorded by the various instruments installed on the model. These data require special routines to reduce raw recorded data to a usable form for application in a design and analysis. Special care is taken to avoid spurious data entering into the test data. The data analysis techniques that may be used in reducing test results are described in Chapter 10. Examples are taken from specific tests to illustrate these techniques. It should be noted here that the dimensional units in describing quantities in the book are somewhat mixed. This reflects the slow transition in this country from the English to the metric system. Wherever possible dual units have been provided. In general metric system has been followed. However, with conversion factor noted, some examples have been left in the English units. It is hoped that readers will have appreciation for both systems in following the material in the book. A list of symbols has been included at the end of the book . Attempt has been made to maintain consistency throughout the book . A few variables have been used with dual meanings in two different parts of the book with little confusion . Some local variables have not been included in the list to limit its size . They have been defined locally where they appear. Subscripts and superscripts have been defined in the list. A list of all the abbreciations appearing in the book has also been included for convenience.
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Preface
As is evident from its contents, this book should be a valuable addition to the library of all offshore engineers and naval architects whether they are involved in the research, design, construction or offshore operations. All hydraulic and ocean engineering curriculums in universities offer a course in modeling . This book should be a very appropriate reference for such a course. It is written such that it may be used as a text for a junior or senior level course in a four year engineering curriculum . Since the book deals with many subjects in modeling that go beyond the specifics for offshore structures, it should be found useful by all engineers and scientists interested in structural or hydraulic testing. Finally, it should also be valuable as a reference to many model testing facilities as a complement to their expertise in the area.
TABLE OF CONTENTS
DEDICATION ...................................................................................................... V ACKNOWLEDGEMENTS .................................................................................. vii PREFACE ............................................................................................................ xi
1.0 INTRODUCTION ........................................................................................... 1 1.1 MODEL TYPES ...................................................................................... 1 1.2 BRIEF HISTORY OF MODEL TESTING ........................................... 2 1.3 PURPOSE OF MODEL TESTING ....................................................... 4 1.4 MODELING CRITERIA ....................................................................... 6 1.5 PLANNING A MODEL TEST ............................................................... 9 1.6 REFERENCES ........................................................................................ 11
2.0 MODELING LAWS ........................................................................................ 12 2.1 GENERAL DISCUSSIONS OF SCALING LAWS AND METHODS 12 2.2 BUCKINGHAM PI THEOREM ............................................................ 14 2.2.1 Dimensionality of Wave Motion ................................................... 17 2.3 NONDIMENSIONAL HYDRODYNAMIC FORCES .......................... 17 2.4 FROUDE 'S MODEL LAW .................................................................... 19 2.5 SCALING OF A FROUDE MODEL ..................................................... 21 2.5.1 Wave Mechanics Scaling ............................................................... 21 25.2 Current Drag Scaling .................................................................... 28 2.5.3 Wave Drag Scaling ........................................................................ 29 2.6 HYDROELASTIC STRUCTURAL SCALING .................................... 33 2.7 DISTORTED MODEL ........................................................................... 35 2.8 REFERENCES ........................................................................................ 37
3.0 MODEL CONSTRUCTION TECHNIQUES ................................................... 40 3.1 GENERAL REQUIREMENTS FOR MODELS ................................... 40 3.2 MODEL TYPES ...................................................................................... 41 3.3 ENVIRONMENTAL LOAD MODELS ................................................. 41 3.3.1 OTEC Platform Model ................................................................. 42 3.4 SEAKEEPING MODEL ......................................................................... 45 3.4.1 Tanker Model ................................................................................ 47 3.4.1.1 Wood Construction of Model .............................................. 47
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3.4.1.2 Fiberglass Construction of Model ........................................ 49 3.4.2 Submergence Model ...................................................................... 50 3.4.2.1 Construction Technique ...................................................... 51 3.4.2.2 Static and Dynamic Properties ............................................ 53 3.4.3 Tension Leg Platform Model ........................................................ 53 3.4.3.1 TLP Hull ............................................................................. 53 3.4.3.2 Tendons and Tendon Attachment Joints .............................. 55 3.4.3.3 TLP Model Deployment ...................................................... 56 3.4.4 Jacket Launching Models ............................................................. 56 3.4.4.1 Jacket Model ....................................................................... 56 3.4.4.2 Barge Model ....................................................................... 57 3.5 CONSTRUCTION OF A MOORING SYSTEM .................................. 57 3.5.1 Mooring Chains ............................................................................. 57 3.5.2 Mooring Hawsers .......................................................................... 59 3.6 MODEL CALIBRATION METHODS .................................................. 62 3.6.1 Platform Calibrations ................................................................... 63 3.6.1.1 Weight Estimate .................................................................. 63 3.6.1.2 Center of Gravity Estimate .................................................. 64 3.6.1.3 Estimate of Moments of Inertia ........................................... 66 3.6.1.4 Righting Moment Calibration .............................................. 66 3.6.2 Tendon Calibrations ..................................................................... 67 3.6.2.1 Dry Creep Characteristics .................................................... 69 3.6.2.2 Dry Static Stiffness ............................................................. 70 3.6.2.3 Dry Dynamic Stiffness and Damping .................................. 71 3.6.2.4 Hysteresis Effect Under Dry Dynamic Loading ................... 71 3.6.2.5 Wet InPlace Static Stiffness ............................................... 71 3.7 REFERENCES ........................................................................................ 74
4.0 MODEL TESTING FACILITY ....................................................................... 75 4.1 TYPE OF FACILITY ............................................................................. 75 4.2 WAVE GENERATORS .......................................................................... 75 4.3 MECHANICAL WA VEMAKER ........................................................... 77 4.3.1 Hinged Flapper Wave Theory ...................................................... 78 4.3.2 Wedge Theory ............................................................................... 82 4.4 PNEUMATIC WAVE GENERATOR ................................................... 86 4.5 DESIGN OF A DOUBLE FLAPPER WAVEMAKER ......................... 88 4.5.1 Wetback and Dryback Design ...................................................... 90 4.5.2 Hydraulic and Pneumatic Units ................................................... 90 4.5.3 Control System for TwoBoard Flapper ...................................... 90 4.5.4 Waveboard Sealing and Structural Support System ................... 91 4.6 A TYPICAL WAVE TANK ................................................................... 91
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4.6.1 Low Frequency Wavemaker ......................................................... 94 4.6.2 High Frequency Wavemaker ........................................................ 96 4.7 DESIGN OF MULTIDIRECTIONAL WAVE GENERATOR............ 97 4.7.1 Actuator and Control .................................................................... 100 4.8 A MULTIDIRECTIONAL TANK ........................................................ 101 4.9 CURRENT GENERATION ................................................................... 103 4.9.1 A Typical Current Generator ....................................................... 103 4.9.2 Local Current Generation ............................................................ 105 4.9.3 Shear Current Generation ............................................................ 106 4.10 WIND SIMULATION .......................................................................... 109 4.11 INSTRUMENTED TOWING STAFF ................................................. 110 4.12 PLANAR MOTION MECHANISM ..................................................... 111 4.12.1 Single Axis Oscillator ...................................................................112 4.13 LABORATORY WAVE ABSORBING BEACHES ............................ 113 4.13.1 Background on Artificial Beaches ............................................... 115 4.13.2 Progressive Wave Absorbers ....................................................... 117 4.13.3 Active Wave Absorbers ................................................................ 118 4.13.4 Corrected Wave Incidence ........................................................... 118 4.14 REFLECTION OF REGULAR WAVES ............................................. 120 4.14.1 Two Fixed Probes ......................................................................... 122 4.14.2 Three Fixed Probes ......................................................................123 4.15 REFLECTION OF IRREGULAR WAVES ......................................... 126 4.16 LIMITED TANK WIDTH ..................................................................... 128 4.17 TESTING FACILITIES IN THE WORLD .......................................... 130 4.17.1 Institute of Marine Dynamics Towing Tank , St. John's, Newfoundland, Canada ................................................................130 4.17.2 Offshore Model Basin, Escondindo , California .......................... 131 4.17.3 Offshore Technology Research Center, Texas A&M University, College Station, Texas ........................................................... 132 4.17.4 David Taylor Research Center, Bethesda, Maryland ................. 132 4.17.5 Maritime Research Institute , The Netherlands (MARIN) ......... 133 4.17.6 Danish Maritime Institute, Lyngby, Denmark ........................... 134 4.17.7 Danish Hydraulic Institute, Horsholm , Denmark ...................... 135 4.17.8 Norwegian Hydrodynamic Laboratory, Trondheim , Norway (MARINTEK) ............................................................................... 135 4.18 REFERENCES ....................................................................................... 135
5.0 MODELING OF ENVIRONMENT ................................................................. 139 5.1 WAVE GENERATION .......................................................................... 139 5.1.1 Harmonic Waves ........................................................................... 140
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5.1.2 NonHarmonic Waves ................................................................... 140 5.1.3 Imperfect Waves ........................................................................... 142 5.1.4 Shallow Water Waves ................................................................... 144 5.2 RANDOM WAVE SIMULATION ........................................................ 146 5.2.1 Random Phase Method ................................................................. 147 5.2.2 Random Complex Spectrum Method ........................................... 149 5.2.3 Random Coefficient Method......................................................... 150 5.3 WAVE PARAMETERS .......................................................................... 151 5.3.1 Wave Groups .................................................................................151 5.3.2 Wave Asymmetry .......................................................................... 152 5.3.3 Group Statistics ............................................................................. 152 5.4 HIGHER HARMONIC WAVES ........................................................... 155 5.5 GENERATION OF MULTIDIRECTIONAL WAVES ....................... 159 5.5.1 Procedure for Simulation of Sea State ......................................... 161 5.5.2 MultiDirectional Generation Theory .......................................... 162 5.6 GENERATION OF STEEP WAVES .................................................... 166 5.7 GAUSSIAN WAVE PACKET FOR TRANSIENT WAVES ................ 169 5.8 CURRENT GENERATION ................................................................... 173 5.8.1 Wave Current Interaction ............................................................ 176 5.8.1.1 Interaction Theory ............................................................... 176 5.8.1.2 Combined WaveCurrent Tests ........................................... 178 5.9 WIND GENERATION ........................................................................... 181 5.10 REFERENCES ....................................................................................... 184
6.0 INSTRUMENTATION AND SIGNAL CONTROL ........................................ 190 6.1 DATA ACQUISITION ...........................................................................190 6.1.1 Transducer .................................................................................... 190 6.1.2 Signal Conditioning....................................................................... 191 6.1.3 Data Recorder ...............................................................................192 6.2 BONDED STRAIN GAUGE .................................................................. 193 6.3 POTENTIOMETER ............................................................................... 194 6.4 DISPLACEMENT AND ROTATIONAL TRANSDUCERS ................ 194 6.5 VELOCITY TRANSDUCERS ............................................................... 196 6.6 ONE DIMENSIONAL CURRENT PROBE .......................................... 197 6.7 TWO DIMENSIONAL CURRENT PROBE ......................................... 198 6.7.1 Basic Properties ............................................................................. 198 6.7.2 Design Information ....................................................................... 200 6.7.3 Calibration and Testing ................................................................ 201 6.8 AN ALTERNATE CURRENT PROBE ................................................. 204 6.9 ACCELEROMETERS ........................................................................... 206
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6.10 PRESSURE TRANSDUCER......... 208 6.11 WAVE ELEVATION ..................... 209 6.12 FREEBODY MOTIONS ..................................................................... 212 6.13 FORCE MEASUREMENT ................................................................... 215 6.14 XY INSTRUMENTED SECTION ........................................................ 218 6.14.1 Experimental Setup ...................................................................... 220 6.15 DESIGN OF A LOAD CELL ................................................................ 221 6.15.1 Shear Force and Bending Moment .............................................. 222 6.15.2 Layout of Strain Gauges .............................................................. 223 6.15.3 Calibration of Cell ........................................................................ 224 6.16 TWOFORCE DYNAMOMETER ....................................................... 225 6.17 TOWING STAFF INSTRUMENTATION ........................................... 226 6.18 MECHANICAL OSCILLATION OF A FLOATING BODY ............. 227 6.19 DATA QUALITY ASSURANCE .......................................................... 229 6.20 REFERENCES ....................................................................................... 230
7.0 MODELING OF FIXED OFFSHORE STRUCTURES ................................... 232 7.1 DESIGN LOAD COMPUTATIONS ...................................................... 232 7.2 SMALLMEMBERED FIXED STRUCTURES ................................... 233 7.2.1 Morison Equations ........................................................................ 235 7.2.2 Description of a Test Setup ........................................................... 236 7.2.3 Pressure Profile Around a Cylinder ............................................. 240 7.2.4 Multiple Cylinder Tests ................................................................ 240 7.3 SEABED PIPELINE TESTING ............................................................. 241 7.3.1 Theoretical Background ............................................................... 242 7.3.2 Model Testing ................................................................................ 246 7.3.2.1 Pipeline Model .................................................................... 246 7.3.2.2 Wave Tests .......................................................................... 247 7.3.2.3 Force ServoControl Mechanism ......................................... 249 7.4 LARGE FIXED STRUCTURES ............................................................ 250 7.4.1 OTEC Platform ............................................................................. 251 7.4.2 Triangular Floating Barge ............................................................ 254 7.4.2.1 Description of Model .......................................................... 254 7.4.2.2 Test Setup ........................................................................... 255 7.4.2.3 Test Results ......................................................................... 257 7.4.3 Large Based Structures ................................................................ 257 7.4.4 OpenBottom Structures ............................................................... 263 7.4.5 Gravity Production Platform ........................................................ 265 7.4.5.1 Model Description ............................................................... 266 7.4.5.2 Test Setup ........................................................................... 266
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7.4.6 SecondOrder Loads on a Cylinder .............................................. 269 7.4.7 Scaling of Fixed Elastic Structures ............................................... 277 7.4.8 Elastic Storage Tank Model .......................................................... 282 7.4.8.1 Elastic Model Design .......................................................... 283 7.4.8.2 Elastic Model Tests ............................................................. 286 7.5 SCOUR AROUND STRUCTURES ....................................................... 288 7.5.1 Factors Influencing Scour ............................................................. 289 7.5.2 Scour Model Tests ......................................................................... 290 7.5.3 Scaling of SoilStructure Interaction ............................................ 292 7.5.3.1 Noncohesive Soil ............................................................... 292 7.5.3.2 Model Calculations ............................................................ 296 7.5.3.3 Cohesive Soil ..................................................................... 300 75.4 Scour Protection ............................................................................ 300 7.6 WIND TUNNEL TESTS ......................................................................... 300 7.7 REFERENCES ........................................................................................ 301
8.0 MODELING OF OFFSHORE OPERATIONS ................................................ 305 8.1 TYPES OF OFFSHORE OPERATIONS .............................................. 305 8.2 TOWING OF A BARGE ........................................................................ 306 8.2.1 Scaling Technique ......................................................................... 306 8.2.2 Barge Test Procedure .................................................................... 308 8.2.3 Data Analysis and Results ............................................................. 309 8.3 SUBMERSIBLE DRILLING RIG TOWING TESTS .......................... 312 8.3.1 Test Description ............................................................................ 313 8.3.2 Test Results .................................................................................... 314 8.4 TOWING OF A BUOYANT TOWER MODEL ................................... 315 8.4.1 Model Particulars .......................................................................... 317 8.4.2 Towing Tests .................................................................................. 317 8.43 Bending Moment Tests ................................................................. 319 8.4.4 Scaling to Prototype ...................................................................... 321 8.5 LAUNCHING OF OFFSHORE STRUCTURES .................................. 323 8.5.1 A Unique Launch .......................................................................... 324 8.5.1.1 Modeling of SoftVolume Cans ........................................... 324 8.5.1.2 Submersible Rig Model ....................................................... 329 8.5.1.3 Launching of Mat on Cans .................................................. 332 8.5.1.4 Seakeeping of Rig on Cans .................................................. 332 8.5.1.5 Deballasting of Cans ........................................................... 332 8.6 JACKET STRUCTURE INSTALLATION ........................................... 333 8.6.1 Scaling of Jacket Installation Parameters .................................... 333 8.6.2 Launching Test Procedure ............................................................ 335
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8.6.3 Side Launching of Structures ....................................................... 339 8.7 STAGED SUBMERGENCE OF A DRILLING RIG ............................ 339 8.7.1 Test Results .................................................................................... 341 8.8 DYNAMIC SUBMERGENCE OF A SUBSEA STORAGE TANK ..... 342 8.8.1 Model Testing ................................................................................ 343 8.9 OFFSHORE PIPE LAYING OPERATIONS ........................................ 345 8.9.1 Pipeline Similarity Laws ............................................................... 347 8.9.2 Partial Geometric Similarity ........................................................ 348 8.10 REFERENCES ....................................................................................... 352
9.0 SEAKEEPING TESTS .................................................................................... 354 9.1 FLOATING STRUCTURES .................................................................. 354 9.2 METHOD OF TESTING FLOATING STRUCTURES ....................... 355 9.3 SINGLE POINT MOORING SYSTEM ................................................ 358 9.3.1 Articulated Mooring Towers ........................................................ 361 9.4 TOWERTANKER IN IRREGULAR WAVES .................................... 369 9.5 TESTING OF A FLOATING VESSEL ................................................. 370 9.6 TENSION LEG PLATFORMS .............................................................. 372 9.6.1 Model Testing Program of a TLP ................................................ 374 9.6.2 Typical Measurements for a TLP ................................................. 375 9.6.3 Wave Frequency Response of a TLP ............................................ 375 9.6.4 Low and High Frequency Loads .................................................. 376 9.7 DRIFT FORCE TESTING OF A MOORED FLOATING VESSEL... 381 9.7.1 Test Setup ...................................................................................... 381 9.7.2 Hydrodynamic Coefficients at Low Frequencies ......................... 382 9.7.2.1 Free Oscillation Tests .......................................................... 382 9.7.2.2 Forced Oscillation Tests ...................................................... 382 9.8 DAMPING COEFFICIENTS OF A MOORED FLOATING VESSEL382 9.8.1 Tanker Model ................................................................................ 384 9.8.2 Semi submersible Model ............................................................... 385 9.8.3 Heave Damping of a TLP Model .................................................. 387 9.9 MODELING AIR CUSHION VEHICLES ............................................ 392 9.10 ELASTIC FLOATING VESSEL .......................................................... 394 9.11 MODELING OF A LOADING HOSE ................................................. 395 9.11.1 Hose Model ................................................................................... 395 9.11.2 Hose Model Testing ...................................................................... 399 9.12 MOTIONS IN DIRECTIONAL SEAS ................................................ 401 9.13 REFERENCES ...................................................................................... 402
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10.0 DATA ANALYSIS TECHNIQUES ............................................................... 406 10.1 STANDARD DATA ANALYSIS ........................................................... 406 10.2 REGULAR WAVE ANALYSIS ............................................................408 10.2.1 Standing Wave .............................................................................. 409 10.2.2 Reflected Wave .............................................................................. 410 10.2.3 Spurious Wave Data ...................................................................... 410 10.3 IRREGULAR WAVE ANALYSIS ........................................................ 411 10.3.1 Fourier Series Analysis ................................................................. 411 10.3.2 Wave Spectrum Analysis .............................................................. 412 10.3.3 Wave Group Analysis ................................................................... 414 10.3.4 Statistical Analysis ........................................................................ 416 10.4 ANALYSIS OF DIRECTIONAL WAVES ........................................... 417 10.5 FILTERING OF DATA ......................................................................... 419 10.6 RESPONSE ANALYSIS ........................................................................ 422 10.6.1 Frequency Domain Analysis ......................................................... 423 10.6.2 Linear System ................................................................................ 427 10.6.3 Theory of Cross Spectral Analysis ............................................... 429 10.6.4 Error Analysis ............................................................................... 432 10.6.5 Example Problem .......................................................................... 434 10.6.6 Nonlinear System .......................................................................... 438 10.7 ANALYSIS OF WAVE FORCE COEFFICIENTS ............................. 439 10.7.1 Fourier Averaging Method ........................................................... 441 10.7.2 Least Square Technique ................................................................ 443 10.8 FREE VIBRATION TESTS .................................................................. 445 10.8.1 Low Frequency Hydrodynamic Coefficients ............................... 445 10.8.1.1 Linear System ..................................................................... 445 10.8.1.2 Nonlinear System ................................................................ 449 10.8.2 Mechanical Oscillation .................................................................. 451 10.8.3 Random Decrement Technique .................................................... 456 10.9 REFERENCES ................................................................................. 457
LIST OF SYMBOLS ............................................................................................ 461 LIST OF ACRONYMS ......................................................................................... 464 AUTHOR INDEX ................................................................................................. 465 SUBJECT INDEX ................................................................................................. 468
OFFSHORE STRUCTURE MODELING
"1 contend that unless the reliability of small scale experiments is emphatically disproved, it is useless to spend vast sums of money on full scale trials which afterall may be misdirected unless the ground is thoroughly cleared beforehand by an exhaustive investigation on a small scale." William Froude, 1886
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CHAPTER 1 INTRODUCTION 1.1 MODEL TYPES Engineers constantly deal with models . Models are fundamental to communication. We learn from models . These models may be physical, mathematical, graphic or semantic. Mathematical models and methods are invaluable in guiding experimental work. The converse is also true . Regardless of the mode of model construction, each of these models represents a set of facts and has an intrinsic value . Each. allows us to confirm a new concept by demonstrating the suitability, workability, and constructability of the concept. Models are used as tools that verify a design. Models may be classified in two major categories : display models and engineering models . Display models are generally used to explain a concept or sell a product. An example of such a model of an exploration drilling structure is shown in Fig. 1 . 1. These models provide visual aid for engineering drawings and may also have moving parts. This latter category of display models is called a working model. It may be properly scaled and generally shows all of the major components. However, some of the details may be omitted for simplicity. The working schemes of these models are designed to demonstrate or highlight any special feature of the product. Engineering models, on the other hand, are used to collect data useful in the design of the product under consideration. These models may be divided into two major categories: constructability models and measurement models . The constructability models are built to scale to insure that the concept design is feasible for construction. Instrumentation is generally not used on these types of models . Often, a new process or a system may be verified with a model of this type . In this case, a portion of the system or an entire pilot plant is built and operated as a proof of the concept. For example, the habitability of a floating offshore hotel is the most important aspect of its operation which may be model tested at an early stage of development. Quite often, improvements to the original process are developed by this type of modeling. The results , however, are not directly scalable to the full size process or prototype, due to possible additional constraints.
2 Chapter 1 Introduction
FIGURE 1.1 DISPLAY MODEL OF A DRILLING RIG The measurement models, on the other hand, are mainly used to obtain engineering data from a scale model for direct use in the design or operation of the prototype . These model properties and their test environment follow certain similarity laws, often derived from a dimensional analysis. Important nondimensional quantities are taken into account in the modeling laws. 1.2 BRIEF HISTORY OF MODEL TESTING From the beginning of recorded history, models have been used not only to visualize various structures (e.g., pyramids) but also as a working plan from which prototype structures have been constructed . In particular, ship models have enjoyed a long and useful history dating back at least to the time of the Pharaohs (as witnessed by ship models in the uncovered tombs). Ship designers and shipwrights used essentially the same model techniques well into the 1600's. These models were , in essence, the precursor of analog computer, the basic precept being  as the model behaves, so will the prototype. Even without a strong analytical background , this method has been successfully used for centuries.
Section 1 . 2 Brief History of Model Testing 3
Stability models for ships have been (and in some cases still are) used to determine placement of cargo and ballast by the cargo officer on board ship. Working mechanical models came into use during the industrial revolution, and several fine examples have been preserved in the British museum and the Smithsonian. In these cases, the design of the prototype structure was scaled directly from the working model. Many of these models were tested for considerable lengths of time prior to building the prototype so that areas of insufficient structural strength could be incrementally strengthened until such time as the fatigue limits were satisfactory with respect to the expected life cycle of the prototype. The first major insight into the phenomena of modeling fluid mechanics was gained by Reynolds and Fronde, wherein they developed criteria for both the viscous and inertial effects respectively. They were followed by Lamb, Stokes, Boussinesq and others until the present state of hydrodynamics was reached. Although Osborne Reynolds ( 1842 1912) was a vigorous investigator of almost all physical phenomena, he was one of the first to elaborate upon scale effects in the areas of marine propulsions and hydrodynamic research. While exploring the apparently discordant results of experiments on the head loss during flow in parallel tubes by Poiseuille and Darcy , Reynolds argued that since viscous forces tend to produce stability while inertia forces tend to cause instability , the change of one type of motion to another could rely upon the ratio of these forces and would occur at some critical value of this ratio . The expression is now universally called the Reynolds number. William Fronde ( 18101879) proposed the idea of a bilge keel on the trial voyage of a ship to observe the rolling motion of the vessel. His first paper dealt with this rolling motion which he judged to have a greater need for study than the problems associated with hull form. In 1870 he began a series of experiments to study the resistance of ships using a towing tank . In 1876, the British Association set up a committee of which Fronde was a member to study the propulsion stability and seagoing quality of ships. These experiments led to his statement now known as Froude 's law of comparison: "In fact, we are thus brought to the scale of comparison which was just now enunciated, that the entire resistance of a ship and a similar model are as the cube of their respective dimensions if their velocities are as the square roots of their dimension".
4 Chapter 1 Introduction
1.3 PURPOSE OF MODEL TESTING Models of physical systems are, in essence, systems from which the behavior of the original physical system (i.e., prototype) may be predicted. The use of models is particularly advantageous when the analysis of the prototype is complicated or uncertain and when construction of the prototype would be uneconomical and risky without preliminary prediction of performance. Model testing offers great savings when compared to full scale tests. However, it can still become an expensive undertaking . The larger the model, the better the test data, and the easier it is to scale up to the prototype values; but the cost of the model test can also increase substantially . Therefore, the model test should be planned carefully to reduce the time and cost of testing , while maintaining reliability. While there are a number of peripheral reasons for model testing, the major justifications for model testing are as follows: • Investigate a problem or situation which can not be addressed analytically. • Obtain empirical coefficients required in analytical prediction equations. • Substantiate an analytical technique by predicting model behavior and direct correlation between the predicted behavior and the actual behavior. • Evaluate the effect of discarded higher order terms in a simplified analytical prediction model by correlating the discrepancy between the predicted model behavior and the actual model behavior. Thus, model testing is an experimental procedure which is most generally used where the analytical techniques fail to predict the expected behavior of the prototype either within the tolerances required or within the confidence level required for good design. Models may be, generally, classified in three groups. The first group consists of scale replicas of the prototypes, in which the behavior of the model is identical in small nature with that of the prototype, but responses differ in magnitude with the chosen scale factor. A second group is that of distorted models, in which a general resemblance exists between model and prototype; but some factors , such as distance (for example,
Section 1 . 3 Purpose of Model Testing 5
water depth in an open channel flow), are distorted . The distortion of a model requires the use of prediction factors in translating model behavior to prototype behavior. A third class of models is known as the analog . It consists of systems which are dissimilar in physical appearance to their prototypes but which are governed by the same class of characteristic equation, such as the Poisson or the Laplace equation. In such models, each element in the prototype has a corresponding item in the model. For example, a simple onedegreeoffreedom mechanical vibrating system may have, as an analog, an RCL electrical circuit where the inductance (L) of the model is proportional to the mass of the prototype, capacitance (C) to spring constant, resistance (R) to viscous damping , applied voltage to driving force, charge to displacement, current to velocity and so on. Sometimes, a mechanical simulator is applied to the model subjected to waves which simulate , for example, the wind load. Examples of mechanical simulators will be described in Chapter 4. In hydrodynamics and in free surface fluid flow problems (e.g., open oceans or rivers, etc .), the models are, generally, exact duplicates of the prototype [Kure (1981 )]. Occasionally, there is a need for some distortion in certain directions due to the limited size of a testing facility and choice of the scale factor. While this book is focused on physical modeling , there is another area of modeling that is gaining popularity and confidence quickly. This is the area of computer modeling which is essentially theoretical analysis of a problem using computational techniques . Numerical modeling, especially finite element modeling, is rapidly increasing in power and sophistication . At some point, engineers may be able to conduct fullscale modeling of a numerical prototype in a numerical ocean, with full accounting of inertial and viscous forces . Such numerical models will be created and tested with a possible savings of both time and money. Numerical sea state need not be limited by wave making machines, and conditions leading to distortions in model geometry and properties need not be relaxed . Numerical models will obviate the need for separate environmental load models and seakeeping models. Computers will become a more useful engineering tool as they continue to increase in speed and memory and decrease in cost . Numerical wind tunnels are gaining acceptance . There has been some discussion about the development of a numerical wave tank (Newman, MIT, during Weinblum lecture series in 1990 ). Digital computer models are replacing/supplanting some scale models, for example, in the analysis of airplanes and missiles . It is, however, too early to numerically model a complex offshore structure in a realistic sea state. Scale models constitute the most accurate and state of the art engineering tool available today. Given the physical nonlinearities and geometric complexities of many offshore structures, it is unlikely that virtual offshore modeling will become a routine procedure for years to come.
6 Chapter 1 Introduction
While computer models have been increasingly successful in simulating an everwidening range of engineering problems, it is nevertheless essential that advances in these models are validated and verified against experiment . Experimental measurements are themselves conditioned to the requirements of the computational models. Hence it is important that scientists working on experiments communicate with researchers developing computer codes as well as those carrying out measurements on prototypes. Numerical experiments will probably never replace wave tank experiments completely, because many physical uncertainties will still prevail in a numerical model. It is expected that physical and numerical experiments will complement each other and guide the development of an efficient structure design. 1.4 MODELING CRITERIA In building an engineering model, one tries to produce a facsimile of a particular product that will perform in a manner similar to the actual product being designed and constructed (called the prototype). The model may be full size, as is often the case in automotive and other industries where the product is relatively small and mass producible; or it may be small relative to the prototype as is often the case with very large structures of unique design as well as expensive design of limited products . In either case, the model must act in a manner physically similar to the prototype, and the similarity must be governed by rules that quantify the actions and allow them to be scaled up for use in predicting the action of the prototype . In some instances, the model may be larger than the prototype , e.g., in scaling small components or parts. The rules for quantifying and scaling model responses are called the laws of similitude . For the study of fluid phenomena, there are three basic laws . The first is geometric similitude or similarity of form. Under this law, the flow field and boundary geometry of the model and of the prototype must have the same shape. Consequently, the ratios of all model lengths to their corresponding prototype lengths are equal. The second basic law of similitude is kinematic similitude, or similarity of motion. According to this law, the ratios of corresponding velocities and accelerations must be the same between the model and the prototype. Thus, two different velocity components in the model must be scaled similarly with respect to the corresponding velocities of the prototype . Given geometric similitude , in order to maintain kinematic similitude, dynamic similitude or similarity of forces acting on the corresponding fluid must exist . Five forces that may affect the fluid structure interaction due to the flow field around the structure are the forces due to pressure,
Section 1.4 Modeling Criteria 7
gravity, viscosity, surface tension and elasticity. Thus, the ratios of these forces between the model and the prototype must be the same. For wave action, the surface tension is generally quite small and neglected. The elasticity is generally ignored for large offshore structures. The ratio of the inertia force to the viscous force is called the Reynolds number, while the ratio of the inertia force to the gravity force is the Froude number. While the Reynolds and Froude effects are generally present, the Froude number is considered the major scaling criterion in the water wave problems. Geometric similarity ensures the identity of functional relations among different parameters . Thus, it is a necessary condition for the existence of dynamic similarity. Distortion is common in hydraulic modeling (e.g., estuary) which invariably provides deviation from geometric similarity. For example, the distribution of velocity of a flow in a crosssection depends very strongly on the crosssection geometry. Hence, by distorting the model and thus the shape of the crosssection, the velocity distribution and its effects are inevitably distorted. Due to economic and practical reasons, in a conventional modeling technique, water is used as the model fluid . If the model operates with the prototype fluid (i.e. water) then the realization of dynamic similarity of a gravity dependent phenomenon in a small scale model is impossible in principle. It is well known that with water both Reynolds and Froude similitude cannot be achieved. The importance of any Reynolds number, however, decreases as its value increases . This phenomenon will be illustrated further in discussing specific modeling problems. Hence, when modeling hydrodynamic phenomenon, the usual practice is to build the model as large as possible and, therefore , minimize the importance of the viscosity parameter. Sometimes , turbulence is artificially induced in the flow ahead of the model to obtain the effect of a large Reynolds number. There are three major factors that influence the scale selection in waves: (1) model construction, (2) tank blockage, and (3) wave generation capability. For building models, generally a larger scale results in easier construction and material selection. In terms of tank blockage, the smaller the scale is, the better will be the quality of waves in the tank with less contamination due to reflection from the model. Finally, the wavemaker must be capable of producing the scaled wave heights and periods desired for the tests. Each of these criteria may lead to different "best scales". The wave generating properties in the tank, having fixed limitations, generally control the model scale.
8 Chapter 1 Introduction
The scale is chosen as a compromise between cost of the project and the technical requirements for similitude. It should be noted that the popularly held belief that the larger the scale model, the better it is, may not always be true. Several other considerations dictate the scale selection . Sometimes, more than one model is justified to study the different phenomena experienced by a structure. For example, the entire structure may be at one scale , whereas a small section of it may be studied using a larger scale model.
FIGURE 1.2 MOTIONS OF A RESIDENT TANKER AND MOORING TOWER (SCALE = 1:48) This is illustrated by an example of a buoyant tower. A permanently moored buoyant towertanker system is often used offshore for processing and storing crude oil. A 1:48 model of the tower and tanker moored with a yoke is shown in Fig. 1.2 being tested in a wave tank. This model allowed the study of the response of the
Section 1 .5 Planning A Model Test 9
overall system. The mooring system is often designed with a quick disconnect mechanism for emergency detachment . As a demonstration model for the design concept of the mechanical system, a 1 : 12 model of the quick disconnect assembly is shown in Fig. 1.3. The ship yoke and support system is mounted on a plywood barge. The operators could board the barge and manage the winch controls to simulate This larger model is used for training personnel and give qualitative docking . information on sea state limitations acceptable for such an operation.
FIGURE 1.3 MODEL OF RIGID YOKE ARM CONNECTING ARTICULATED TOWER AND TANKER UNDER TEST (SCALE = 1:12) While it is not possible to recommend an optimum scale factor for a structure without investigating all parameters of importance , a common scale factor used in water wave effects in a tank is 1:50. A range of scales for a wave tank is typically between 1:10 and 1:100. 1.5 PLANNING A MODEL TEST For a model test to be successful , proper planning is required . Care should be taken to consider all aspects of model testing with the ultimate goal in mind. It is often
10
Chapter 1 Introduction
Design of Model Test Scaling Laws
Model Parameters
1 Choice of Test Facility
Model Design & Drafting Model Construction [Model Calibration
Design of Instrumentation Calibration of Instrumentation Setup of Test in Wave Tank Debugging the System InPlace Calibration Test Runs I Analysis of Test Data
Documentation
FIGURE 1.4 EXECUTION OF A MODEL TEST
Section 1.6 References
11
a common mistake to expect too many results from one model test. Instead, the test plan should focus on a few most important aspects of modeling . This will help in choosing a proper scale factor and other parameters of the test. A general procedure for a model test is outlined in Fig. 1.4. One of the most important criteria for successful model testing is the evaluation of the scaling law. The scaling parameters for the results sought in a model test must be established first before a scale factor is chosen . Once the scale factor is established, the input parameters may be computed. This may help in deciding on the best testing facility from those available. Concurrently, the model sizes may be determined , and the design of the model may continue. In the design of the model, proper attention must be given to the attachment and effect of instrumentation on the model . The model, instrumentation and the wave tank must be properly calibrated before the model is placed in the tank. For example, the random wave to be used in testing should be generated in the absence of the model in the tank. The inplace calibration of the model is very important. Valuable information may be obtained from the pre and posttest calibration. A check of the calibration during the test runs insures accuracy of collected data. Also, testing should be properly documented for future reference. Any unusual observations must be noted by the test engineer in the lab notebook . These areas are explored in more details in the future chapters. 1.6 REFERENCES 1. Kure, K., "Model Tests With Ocean Structures", Applied Ocean Research, Vol. 3, No. 4, 1981, pp. 171176.
CHAPTER 2 MODELING LAWS 2.1 GENERAL DISCUSSIONS OF SCALING LAWS AND METHODS Most physical systems can be investigated through small scale models'whose behavior is related to that of the prototype in a prescribed manner [Soper ( 1967)]. The problem in scaling is to derive an appropriate scaling law that accurately describes this similarity . This requires a thorough understanding of the physical concepts involved in the system. One method of relating the model properties to the prototype properties is the parametric approach in which the Buckingham Pi Theorem is applied to all applicable variables to derive a group of meaningful dimensionless quantities. This method assumes that nothing is known about the governing equations of the system. If, however, these equations are known apriori, then the scaling law may be deduced from these equations by writing them in dimensionless form. There are two generally accepted methods by which scaling laws relating two physical systems are developed . The two physical systems in this context are the prototype and the model . The first method is based on the inspectional analysis of the mathematical description of the physical system under investigation. Fluid mechanics problems deduce the condition of similitude from the Navier Stokes equation . A method of similarity analysis is adopted here (see references " at the end of this chapter). The dynamics of the physical system are described by a system of differential equations. These equations are written in nond mensional terms . Since the simulated physical system duplicates the fullscale system, these nondimensional quantities in the Then, the equality of the differential equations must be equal for both . corresponding nondimensional parameters governs the scaling laws. This method assures similarity between the two systems but is dependent upon knowing explicitly the governing equations for both the prototype and the model. A second method is based upon the well known Buckingham Pi Theorem . In this approach, the important variables influencing the dynamics of the system are identified first. This is the most important step in the similarity analysis by this method. If a significant parameter is omitted , then the resulting scaling laws will be erroneous. On the other hand, if too many variables including those least significant are included, then the scaling laws become too complicated and, often, impossible to satisfy . It should be realized that a complete similitude cannot be obtained except at a onetoone scale.
Section 2. 1 General Discussions of Scaling Laws and Methods
13
Therefore, the parameters of least significance are neglected. The scale is chosen as a compromise between cost, complexity and technical requirements for similitude. Once the variables are identified, their physical dimensions are noted . Based on the Buckingham Pi Theorem (to be introduced shortly), an independent and convenient set of nondimensional parameters (pi terms) is constructed from these variables. The equality of the pi terms for the model and prototype systems yields the similitude requirements or scaling laws to be satisfied. The model and prototype structural systems are similar if the corresponding pi terms are equal. In order to arrive at the prototype values from the model test results, similitude relationships are used to formulate the system equations and prediction equations. Let us define the characteristic equation of a system as
n = $(nl, R2.... NO
(2.1)
where the function 0 is generally an unknown. The quantities nl,...mrn are the dimensionless group on which dimensionless it is considered dependent. The corresponding characteristic equation for the model is nm = Otn (It lm, n2m.... nnm)
(2.2)
where the subscript in stands for model. The two phenomena between the prototype and the model are identical if the functions 0 = 4 which provides the system equations tri = trim i = 1, 2,...n
(2.3)
In this case n=nm
(2.4)
This relationship provides the prototype values from the model test and is called the prediction equation of the system . Sometimes, due to the limitation in scaling, distortion modeling is used. Then
rim =eni
(2.5)
n = anm
(2.6)
so that
In this case, the relationship between e and a has to be established.
14
Chapter 2 Modeling Laws
2.2 BUCKINGHAM PI THEOREM The Buckingham Pi theorem may be stated as follows. Let the number of fundamental units, e.g., M,L ,T (mass, length, time) needed to express all variables included in a problem be R. Let the number of variables employed to describe a phenomenon be N. It can be shown that the equation giving the relation between the variables will contain NR dimensionless ratios which are independent . In other words, only NR dimensionless quantities are required to establish the functional relationship. The relationship among the variables is expressed by an exponential equation. Then the values of the exponents are solved for , assuming dimensional homogeneity. This is illustrated with the following example [Pao ( 1965)]. Consider a stationary sphere of diameter D immersed in an incompressible fluid flowing past the sphere in a steady flow. The flow around the fixed sphere will introduce a drag force , FD, whose magnitude will depend on several parameters: namely, the diameter of the sphere, D; the approaching velocity of flow, v; and the fluid properties, i.e., density, p, and viscosity, p.. Therefore , a functional relationship is expected between the drag force and these variables, FD = 4(D, v, p, µ) (2.7) An exponential form for this relation is FD = CD°vbp` t4 (2.8) where C is an arbitrary dimensionless constant. Converting this equation to their dimensional equivalent in an MLT system gives a
TL=L°(T)6(L)c(LT
(2.9)
J
Equating the exponents of each dimension for dimensional homogeneity, we have For M: c+d=1 For L: a+b3cd=1 For T: b  d = 2 This gives us 4 unknowns and 3 equations. unknown,
Writing the equation in terms of one
Section 2.2 Buckingham Pi Theorem
FD = C D2d v2d pld N.d
15
(2.10)
Rearranging terns, (2.11) Dp  lv p) Note that the term within parenthesis is the definition of Reynolds number. Then, the general form of the relationship becomes FD
=
CW(Re)
(2.12)
where FD is the nondimensional drag force on the sphere which is represented as a function of Reynolds number (Re). Note that Eq 2.7 involved 5 variables in an MLT system so that only (53 =) 2 nondimensional quantities are needed (Eq. 2.12) for a functional relationship. An experiment with a sphere in steady flow produces such a relationship. Thus, the dimensionless form of a property of the system can be uniquely determined by a functional relationship in terms of N3 nondimensional variables. Dynamic similarity means that all properties expressed as dimensionless forms are identical in model and prototype and the corresponding scale is equal to unity [Yalin (1982)]. This method is simple and attractive in that it does not depend on any mathematical relations which may not even be known for the particular system. It relies only on the parameters themselves. The best way to illustrate this is through an example. Example 1 In fluid mechanics, the common dimensional variables are obtained in three categories • Geometry of the structure boundary in the flow field, e.g., length, width, etc. • Fluid properties, e.g., density, viscosity, etc. • Properties of fluid motion, e.g., pressure, velocity, etc.
16
Chapter 2 Modeling Laws
Consider this general problem of a structure immersed in a fluid flow . The parameters to be included are written as an arbitrary function of a dimensionless constant it, (2.13)
It =^(P,v,l,,l2,p,µ,a,E,g) Relating it in two different unit systems , we obtain,
p,v,ti,1Z,PµB
v>l,>12,P,µ,a ,E,B) (2.14)
where p = ML'72p
fluid pressure
v = LT'v
fluid velocity
i, = Ll,
longitudinal dimension
1Z = L/2 transverse dimension p = ML'p ML 'T' t
fluid mass density fluid dynamic viscosity
(Y = MT2a
surface tension
E = ML;'T'2E
elasticity modulus
g = LT2g
acceleration due to gravity
The basic independent dimensional units are L, M and T and there are 9 independent variables. The number of independent dimensionless quantities is 9  3 = 6. The dimensionless quantities are chosen as P/Pv2 > l2 /l, ' µ/l, pv, a/p ljv2 , E/Pv2 and gl1 /v2
Note that these dimensionless quantities are not unique. For example, any two of these quantities may be combined to form a new one. However, the total number will remain at 6 in this example. Then, the general solution is given by it = $(!z /l, , P/Pv2, pvl,/µ , v2/g 2, , pv2!, /a, pv2/E)
(2.15)
It is noted that the quantities pv€ /µ, v2/g€ , pv2€ /a and pv2/E are known as Reynolds number, Froude number, Weber number and Cauchy number, respectively.
Section 2.3 Nondimensional Hydrodynamic Forces 17
2.2.1 Dimensionality of Wave Motion Let us consider the dimensional aspect of wave motion analysis. There are several parameters that are used in describing a twodimensional progressive wave. Some of these parameters are the wave height (H), wave period (T), water depth (d), wave length (L), wave frequency (w), wave number (k) and wave speed (c). Many of these parameters are interrelated . The independent quantities that are necessary and sufficient to characterize the wave motion are H, d, T and g [Sarpkaya and Isaacson (1981)]. All other quantities are related to these four independent variables in a manner prescribed by a particular wave theory. The twodimensional coordinate system (x, y) and time (t) are also needed for a complete description of a spatial and time dependent variable . Consider the horizontal water particle velocity: u = ^(H, k , w, g,x, y ,t)
(2.16)
Applying the pi theorem, there are 8 variables giving 6 dimensionless variables (in an L, T system) having the relationship u Hw =+(ky,kH,(02/gk, kx,wt)
(2.17)
For linear theory, the dependence on kH may be waived and the water particle velocity is given by, 2u __ coshky cos(kxwt) (2.18) Hw cosh kd
along with the dispersion relationship W2 = tanh kd gk
(2.19)
These two equations satisfy the functional relationship in Eq. 2.17. 2.3 NONDIMENSIONAL HYDRODYNAMIC FORCES The principal types of forces encountered in a hydrodynamic model test are: Gravity force: FG = Mg Inertia force: FI = Mu Viscous force: FV =µA (du/dy) Drag force: FD = 1/2 CDpAu2
18
Chapter 2 Modeling Laws
Pressure force: F = pA Elastic force: Fe = EA in which M = mass of the structure ; u, u = velocity and acceleration of fluid (or structure); y = vertical coordinate ; A = area; and p = pressure of fluid.
TABLE 2.1 COMMON DIMENSIONLESS NUMBERS IN FLUID FLOW PROBLEMS DIMENSIONELSS NUMBER
DEFINITION
REMARK
Froude Number, Fr Reynolds Number, Re Strouhal Number, St
v2/gD pvD/µ feD/v
KeuleganCarpenter Number, KC Ursell Number, Ur Cauchy Number,Cy
vT/D HL2/d3 v2/E
Inertia/Gravity Inertia/Viscous Vortex Shedding Frequency Period Parameter Depth Parameter Elastic Parameter
Hydrodynamic scaling laws are determined from the ratio of these forces. The dynamic similitude between the model and the prototype is achieved from the satisfaction of these scaling laws. Several ratios may be involved in the scaling . One of these may be more predominant than the others. In most cases, only one of these scaling laws is satisfied by the reduced scale model of the prototype structure. Therefore, it is important to understand the physical process experienced by the structure and to choose the most important scaling law which governs this process. From the above forces, the following ratios may be defined: Froude Number, Fr Inertia Force/Gravity Force, FI/FG Reynolds Number, Re Inertia Force/Viscous Force, FI/FV Iverson Modules, Iv Inertia Force/Drag Force, FI/FD Euler Number, Eu Inertia Force/Pressure Force, FI/Fp Cauchy Number, Cy Inertia Force/Elastic Force, Fl/Fe There are a few other dimensionless numbers one experiences in fluid flow. The common dimensionless numbers are listed in Table 2.1. Of these dimensionless scaling laws, the most common in the water wave problem is the Froude's law. While the Reynolds number plays an important role in
Section 2.4 Froude 's Model Law
19
many fluid flow problems, the Reynolds similitude does not practically exist in scale model technology. 2.4 FROUDE'S MODEL LAW The Fronde number considers the effect of gravity on the system in question. Thus, it contains the gravitational acceleration term. The Froude number is defined as the ratio of the inertia force to the gravitational force developed on an element of fluid in a medium. Let us consider an element of fluid as a block having dimensions dx, dy, and dz. The gravitational force on the block is given by: W = pg dx dy dz (2.20) The inertia force is given by the product of mass and acceleration, F, = p dx dy dz(du/ dt) (2.21) where u = fluid block velocity which may be defined as dy/dt. Then, the ratio of the inertia force to the gravitational force is obtained as F, udu W gdy
(2.22)
Dimensionally then the Froude number is given by F, u2 Fr = , W gi
(2.23)
Sometimes, the square root of the quantity on the righthand side is defined as the Fronde number, Fr. In the case of water flow with a free surface, the gravitational effect predominates. The effect of other factors, such as viscosity, surface tension, roughness, etc., is generally small and can be neglected. In this case, Froude's law is most applicable. The Froude number for the model and the prototype in waves can be expressed by Z 2
Fr= = gip giar
where the subscripts p and in stand for prototype and model, respectively.
(2.24)
20
Chapter 2 Modeling Laws
From geometric similarity,
(2.25)
lp = ,n where X is the scale factor for the model. Then
(2.26)
up = Jum
Similarly, force is given by F = Mg, where M is the displaced mass (= p13 ). Considering the same fluid density between the model and prototype, MP = X Mm and therefore , Fp =)OFm
(2.27)
Consider the example of a moored ship as a spring mass system. The mass of the ship between the prototype and the model is related by Eq. 2.27 . The (linear) spring constant K having the unit of force/length should be related by Kp =)?Km
(2.28)
using Froude's law. The natural period of the system is given by TN_2x(M)2
(2.29)
Then, the ratio of the natural periods between the model and the prototype is given by TNp = jTNm
(2.30)
In Froude scalings, the acceleration in the model equals the acceleration in the prototype. For example, the acceleration of water particles under waves is given by the relationship up=um
(2.31)
The advantages of the choice of Froude 's law are not only that it directly scales the most important criteria of the mechanism, but also that there is a large background
Section 2.5 Scaling of a Froude Model 21
of experimental procedures and data reduction techniques available from the years of Froude scale modeling. 2.5 SCALING OF A FROUDE MODEL A general assumption is made here that the model follows the Froude's law. The common variables found in the study of fluid mechanics are grouped under appropriate subheadings and are listed in Table 2.2. The units of these quantities are listed in the MLT (mass lengthtime) system. If the variable is dimensionless, the "units" column includes the entry "NONE". Using Froude's law and the scale as A„ the suitable multiplier to be used to obtain the prototype value from the model data is shown. Where applicable, appropriate assumptions or short definitions are included under "Remarks ". In the following sections, specific examples are considered from solid and fluid mechanics to show how the Froude models address the scaling criteria. From these examples, it should be clear that while Froude models do not scale all of the parameters, they satisfy the most important and predominant factor in scaling a system in wave mechanics, namely inertia. 2.5.1 Wave Mechanics Scaling As shown in the table of variables (Table 2.2), the wave height, wave length and water depth scale linearly (as 1/%.) in a Froude model. The time and wave period scale as 1/4k. The wave force and moment scale as l /7.3 and 1/?4, respectively. In the study of wave mechanics (especially in the wavestructure interaction problem), three nondimensional numbers are most important. They are (in the order of importance): Froude number, Reynolds number and Strouhal number. Another dimensionless number, the KeuleganCarpenter parameter, is also preferred in showing the dependence of the inertia and drag coefficient. In many problems with waves, inertia is the most predominant force in the system. That is why Froude scaling is used more extensively than any other in the model study. The wave force on a structure whose members are one order of magnitude smaller than the length of the wave depends on the • Reynolds number; and thus, in a Froude model, this force does not necessarily scale as X3. This will be illustrated in a subsequent section. The total wave force f per unit length dC on a small vertical cylindrical member of diameter D (Fig. 2.1) is obtained from inertia and drag effects added together [Morison, et al. (1950)]. It is written as
22 Chapter 2 Modeling Laws
TABLE 2.2 MODEL TO PROTOTYPE MULTIPLIER FOR THE VARIABLES COMMONLY USED IN MECHANICS UNDER FROUDE SCALING VARIABLE
UNIT
SCALE FACTOR
REMARKS
GEOMETRY Any characteristic dimension of the object
Length
L
Area
L2
X2
Surface area or projected area on a plane
Volume
L3
X3
For any portion of the object
Angle
None
1
e.g., between members or solid angle
Radius of Gyration
L
A.
Measured from a fixed point
Moment of Inertia Area
L4
A.4
Moment of Inertia Mass
ML2
A,5
Taken about a fixed point
Center of Gravity
L
A.
Measured from a reference point
Time
T
x,1/2
Same reference point (e.g., starting time) is considered as zero time
Acceleration
LT2
1
Rate of change of velocity
Velocity
LT 1
.1/2
Rate of change of displacement
KINEMATICS & DYNAMICS
Section 2.5 Scaling of a Froude Model
VARIABLE
TABLE 2.2 CONTD. UNIT SCALE FACTOR
REMARKS
Displacement
L
X
Position at rest is considered as zero
Angular Acceleration
T2
)L1
Rate of change of angular velocity
Angular Velocity
T1
XI/2
Rate of change of angular displacement
Angular Displacement
None
1
Zero degree is taken as reference
Spring Constant (Linear)
MT2
X2
Force per unit length of extension
Damping Coefficient
MT1
A5/2
Resistance (viscous) against oscillation
Damping Factor
None
1
Ratio of damping and critical damping coefficient
Natural Period
T
?.1/2
Period at which inertia force = restoring force
Momentum
MLT1
X7/2
Mass times linear velocity
Angular Momentum
ML2T1
x,9/2
Mass moment of inertia times angular velocity
Torque
ML2T2
X4
Tangential force times distance
Work
ML2T2
;,4
Force applied times distance moved
Power
ML2T3
A.7/2
Rate of work
23
24 Chapter 2 Modeling Laws
VARIABLE Impulse
TABLE 2.2 CONTD. REMARKS SCALE UNIT FACTOR X7/2 Constant force times its short MLT1 duration of time MLT2
),3
Action of one body on another to change or tend to change the state of motion of the body acted on
Stiffness
ML3T2
?,5
Modulus of elasticity times the moment of inertia, El
Stress
MLIT2
Moment
ML,2T2
X4
Applied force times its distance from a fixed point
Shear
MLT2
X3
Force per unit cross sectional area parallel to the force
Section Modulus
L3
X3
Area moment of inertia divided by the distance from the neutral axis to the extreme fiber
Kinetic Energy
ML2T2
X4
Capacity of a body for doing work due to its configuration
Pressure Energy
ML2T2
X4
Energy due to pressure head
Potential Energy
ML2T2
X4
Capacity of a body for doing work due to its configuration
Friction Loss
I ML2T2 I
X4
Loss of energy or work due to friction
Force, Thrust, Resistance
STATICS
Force on an element per unit area
HYDRAULICS
Section 2.5 Scaling of a Froude Model 25
VARIABLE
TABLE 2.2 CONTD. UNIT SCALE FACTOR
REMARKS
SCOUR Particle Diameter
L
X
For same prototype material
Free Settling Velocity
LT1
IFX
Final velocity of a freely falling particle in a medium
Sediment Number
None
1
Nondimensional no. based on velocity and particle size
Shield's Number
None
1
Nondimensional no. based on velocity and particle size
WAVE MECHANICS Wave Height
L
Consecutive crest to trough distance
Wave Period
T
Time between two successive crests passing a point
Wave Length
L
Celerity
LT1
Particle Velocity
LT1
IX
Rate of change of movement of a water particle
Particle Acceleration
LT2
1
Rate of change of velocity of a water particle
Particle Orbits
L
X
Path of a water particle (closed or open)
X
Distance between two successive crests at a given time Velocity of wave (crest, for example)
26 Chapter 2 Modeling Laws
VARIABLE
TABLE 2.2 CONTD. UNIT SCALE FACTOR
REMARKS
Wave Elevation
L
A.
Form of wave (distance from still waterline)
Wave Pressure
ML71T2
A.
Force exerted by a water particle per unit area
KeuleganCarpenter Parameter
None
1
Dependence of hydrodynamic coefficients on this parameter
Displacement (Volume)
L3
A,3
Volume of water moved by a submerged object (or part thereof)
Righting & Overturning Moment (Hard Volume)
ML2T2
X4
Moment about a fixed point of a displaced weight and dead weight, respectively
Natural Period
T
Metacenter
L
A.
Instantaneous center of rotation
Center of Buoyancy
L
A.
Distance of C.G . of displaced volume from a fixed point
Soft Volume
L3
A,3
Volume of trapped air in a member
Buoyancy Pickup per Unit Angle
L3
A,3
Increase in displaced volume per unit tilt angle
STABILITY
Period of free oscillation in still water due to an initial disturbance
Section 2. 5 Scaling of a Froude Model 27
VARIABLE
TABLE 2.2 CONTD UNIT SCALE FACTOR
REMARKS
PROPERTIES Density
ML3
1
Mass per unit volume
Modulus of Elasticity
ML'1T2
?
Ratio of tensile or compressive stress to strain
Modulus of Rigidity
ML1T2
.
Ratio of shearing stress to strain
FIGURE 2.1 DEFINITION SKETCH OF WAVE FORCE ON SMALL CYLINDER
f =PC. 4 DZ u+ 2 pCDDI ul
u
(2.32)
28 Chapter 2 Modeling Laws
The first term on the right hand side depends on the inertia force which is proportional to the water particle acceleration, ii. The second term is the drag force proportional to the square of the water particle velocity, u. All the quantities on the right hand side follow Froude scaling with force except for the two quantities, CM and CD. The hydrodynamic coefficients, CM (inertia coefficient), and CD (drag coefficient), are nondimensional. It has been found [Sarpkaya and Isaacson (1981)] that they are functions of the KeuleganCarpenter parameter, KC, (defined as uoT/D) and the Reynolds number, Re,(defined as u0D/v) where the subscript o refers to maximum value and v=p/p. Therefore, it is important to understand how these parameters scale from the model to the prototype. According to Froude's law, the velocity and wave period scale as the square root of the scale factor, while the linear dimensions scale linearly. Therefore, (KC) p = (KC) m
(2.33)
whereas 3
(Re)p = V (Re)m
(2.34)
Because the KeuleganCarpenter number follows Froude's law, dependence on KC ensures that the model values are applicable to prototype. However, if the quantities strongly depend on Reynolds number, direct scaling is not possible. Moreover, for a small scale model in a wave tank, the prototype Reynolds number can not even be approached due to low fluid velocity in model (Eq. 2.34). Therefore, as CM and CD are strong functions of Re, the results from the model test are not directly applicable to the design. Fluid flow past a small member of a structure creates a low pressure behind the member and causes vortices to shed from the surface of the member. This vortex formation behind the member is found to be a function of the Strouhal number. The eddyshedding frequency, fe, in the Strouhal number is dependent on Re. Therefore, the Strouhal numbers in a model and a prototype are different and do not follow Froude's law. 2.5.2 Current Drag Scaling The drag force per unit length exerted on a bluff cylindrical member by a uniform current flow is proportional to the square of the mean current velocity U. The force is given as
Section 2.5 Scaling of a Froude Model 29
fD 2PDCDUz
(2.35)
The drag force acts parallel to the component of current that is normal to the member axis (i.e., current acts on the projected area normal to flow). The drag coefficient is based on data from Hoerner (1965). Experiments have shown that the flow characteristics in the boundary layer are most likely to be laminar at Re < 105, whereas the boundary layer is turbulent for Re > 106 [Berkley ( 1968)]. Thus, most small model proportions and test conditions, scaled by Fronde number , will result in laminar flow conditions while fullscale conditions are evidently turbulent . Thus, in reality, two different scaling laws (e.g., Froude and Reynolds) apply simultaneously to the model. Since both scaling factors cannot be satisfied concurrently during model tests, it is convenient to employ the Froude scaling process, and allowances are made for the variation in Reynolds number. The dependence of the drag coefficient on the Reynolds number is quite strong in so far as the value of the Reynolds number tends to characterize the flow regime as laminar, transition or turbulent. Once the flow regime is turbulent , the drag coefficient is only weakly dependent on the Reynolds number. The turbulent flow may be verified by visual examination and pressure transducer data. It may be confirmed that the resistance is only weakly a function of the Reynolds number within the turbulent region, by testing two models with identical shapes but with different scales. By using a half size and full scale model , the Reynolds number will be doubled, and the dependence of the resistance and, therefore, the drag coefficient on the Reynolds number will be determined . When this relationship is confirmed , model data can be applied to prototype designs. A practical answer to this problem in model is to deliberately "trip" the laminar flow by some kind of roughness near the bow of the structure. In testing tanker models, various methods have been employed including struts placed upstream of the vessel, wires attached at a point just aft of the bow or sandstrips, studs or pins attached directly to the hull. It has been shown that studs appear to be the most effective method of stimulating turbulent flow at lower velocity and over a broader region of the wetted surface area. 2.5.3 Wave Drag Scaling For flow past a circular cylinder, the low pressure region present on the downstream side (the wake) accounts for a major part of the drag force . In steady flow, the drag coefficient as a function of Reynold's number is known for a smooth circular cylinder. However, the variation of CD against Reynolds number in waves has
30
Chapter 2 Modeling Laws
not been established for high Re. Moreover, the problem is complicated by the fact that the velocity is not constant, but rather changes in magnitude between 0 and a maximum value (depending on wave parameters) as well as in direction . Thus, how the Reynolds number should be defined is open to question . Usually, Re is given in terms of the horizontal water particle velocity for a fixed structure. For a moving structure, Re may be defined in terms of the maximum relative velocity between the structure and the fluid. The drag coefficient for steady flow has been observed to decrease with the increase in Reynolds number except for a small region at the start of the supercritical zone. Drag coefficient is not well defined at very high Reynolds number even for a steady flow . It is also a function of cylinder surface roughness, and other conditions. However, assuming that the roughness coefficient between the prototype and model is about the same, and leaving out the region between subcritical and supercritical zone , it may be observed that the CD in model is generally higher than that in the prototype . One way to circumvent this problem is to test the model at higher Reynolds number to try to reach the prototype Re so that a CD value for the prototype may be established. However, since Re scales as X312, it is generally not possible to increase the velocity , u, within the range of the wave generating tank to obtain prototype Re. This could result in a serious modeling problem in special cases.
10001
MODEL O
0.1 t I 0.01 0.1 1
O
w a} O
10 102 103 104 105 106 107 108 Re
FIGURE 2.2 STEADY DRAG COEFFICIENTS FOR SMOOTH CIRCULAR CYLINDER
Section 2.5 Scaling of a Froude Model
31
For structures whose dimensions compare to the wave length, wave forces are mainly inertial, and the drag forces are generally one order of magnitude smaller than the inertia forces . The velocitydependent drag force on a structural member is 90° out of phase with the acceleration dependent inertia force . The resultant maximum force (which is the sum of the inertia and drag force ) is only slightly higher than the inertia force alone . However, for a member which is small compared to wave length, there is considerable influence of drag on the resultant maximum force. Since the drag force on the model (scaled according to the Froude 's law) is greater than that on the prototype, the influence is expected to be higher in the model. This is illustrated by the following example [Chakrabarti ( 1989)]: Example 2 A section of a cylinder of diameter 14.6m (48 ft) and of length 0.3m (1 ft) is at the still water level in a 146m (480 ft) water depth. The cylinder is to withstand a wave of 15 sec period and 30.5m (100 ft) height. A model is built using a scale factor of X = 48. Calculate the phase shifts of the force in the model and prototype . Then, change the cylinder diameter to 1.5m (4.8 ft) and repeat the calculations above . Assume CM = 2.0 for both and obtain CD from the steadystate uniform flow curve (Fig. 2.2). Use Froude's law for modeling. Prototype Reynolds number, Rep =
V
(2.36)
where D = 14.6m (48 ft) and v = 0.15 x 105 m2/s (1.575 x 105 ft2/sec). The maximum water particle velocity is calculated using linear wave theory: uo = L =6.5in1 (21.2 ft/sec) (2.37) Then the prototype Rep 6 .46 x 107. The model has a diameter of 0.3m (1 ft) and a length of 6.3mm (1/4 in). Model water depth is 3m ( 10 ft), wave period is 2.16 sec and wave height 0.63m (2.08 ft). The kinematic viscosity of fresh water is 0.105 x 105 m2/s (1.126 x 105 ft2/sec). The water particle velocity amplitude, no = 0.94 m/s (3.06 ft/sec)
(2.38)
The model Reynolds number, Rem = 2 .72 x 105. From the uniform flow curve (Fig. 2.2), the drag coefficients,
32
Chapter 2 Modeling Laws
= 0.6
(2.39)
CDm =1.1
(2.40)
CDV
Maximum Inertia Force (Eq. 2.32):
Prototype :2 x 1.98 x (n / 4 x 482 ) x l x 8. 88 = 63.6kips Model: 2xL94x(n/4x12 ) xl/48x8.88 =0.58lbs Scale = )3
Maximum Drag Force (Eq. 2.32):
Prototype : 0.5x 0.6 x L98 x 48x Ix 21. 22 = 12.8kips Model: 0.5xl.1xL94xlxl/48x3.062 =0.2llbs Scale =)2.84 Note that the prototype drag is only 20 percent of the inertia force, while the model drag is nearly 36 percent of its inertia force. Maximum total force on the prototype = 63.6 kips, phase shift = 90°. Maximum total force on the model = 0.58 lbs., phase shift =90° ,
In the second case, the member diameter is 1/10th so that the prototype Reynolds number is 6.46 x 106 while the model Reynolds number is 2.72 x 104. From the steadystate curve, then, the prototype CDp = 0.6 and the model CDm = I.I. The calculated maximum forces on a 0.3m (1 ft) long prototype and a 6.3mm (1/4 in.) long models section are Inertia DW9 Prototype: 0.636 kips 1.28 kips Model: 0.0058 lbs 0.021 lbs Scale =
X3
X2.84
Thus, in this case, the prototype drag force is 201 percent of the inertia, and the model drag force is 362 percent of the inertia force. Maximum total force on the prototype = 1.28 kips, phase shift= 30°. Maximum total force on the model = 0.021 lbs, phase shift = 16°.
Section 2.6 Hydroelastic Structural Scaling 33
Thus, the two examples presented show the two extreme cases of inertia dominance and drag dominance depending only on the member size. It also demonstrates that the model drag force is relatively higher than the prototype drag force compared to the inertia forces (which scale as X3). Thus, the phase shift of the maximum total force is higher in the model than that in the prototype. 2.6 HYDROELASTIC STRUCTURAL SCALING In a hydrodynamic model study, it is a common practice to measure forces exerted on the model. Stresses are generally calculated from this measurement rather than measured directly. In order to measure the stresses, an elastic model is required. Thus, not only is the Fronde similitude maintained, but the Cauchy similarity between the model and the prototype is generally required. In a Fronde model, the quantities, e.g., crosssectional area, moment of inertia, section modulus, etc., will follow Fronde scaling. However, if the same prototype material is used to build the model, the stiffness of the model, EI, where I is the moment of inertia, will not scale the prototype stiffness. Thus, the model behavior will not correctly predict the prototype performance, as is illustrated by the following examples. Let us consider the fiber stress in a beam of a given section (Fig. 2.3). The moment of inertia of the section with respect to the neutral axis is I and the distance to the outer fiber from the neutral axis is represented by y. If F is the force acting on the section, then the shear stress is given by F/A where A is the crosssectional area. If MB is the bending moment on the section , then the bending stress is written as MBy/I. Therefore, the total stress on the extreme fiber of the section becomes: S= F +May
(2.41)
A I
Now, if the force and moment on the model are such that Fp=X3Fm and MBp=X4MBm, the stress will scale as So
=
AS,,,
(2.42)
Consider the section on a cantilever beam of length a with F acting at the free end; the maximum deflection at the free end is obtained from S _ Fes 3EI
(2.43)
34
Chapter 2 Modeling Laws
For the deflection to scale linearly, (Smax)p =' (Smax)m, we have F
I t/2
i
'12
W F,2
F,2
F/2 1 111111111111
SHEAR Fj2
MOMENT
MB= Fr/4
EI
DEFLECTION
SECTION A' N.A.
FIGURE 2.3 SCALING FOR SIMPLY SUPPORTED BEAM (EI)o =?5(EI),,,
But
(2.44)
Section 2.7 Distorted Model
35
(2.45)
I, = ),°I,„
so EP
=
U.
(2.46)
Thus, to scale the stiffness of the structure, a suitable material should be chosen of the model so that the Young's modulus scales linearly with the scale factor (e.g., for X =48, equivalent to prototype steel, the model Em should be 625,000 psi). Often, however, the same material as the prototype is used in model testing. In this case, the model is said to be distorted. Since Ep = Em, the stiffness for a Froude model will scale as (EI)p = %4(EI)m
(2.47)
Thus, for the deflection to scale linearly (Sp = XSm), the model should be subjected to a (distorted) force and moment given by the relationship to the prototype as F,
_
MB,
VF.
=)
(2.48)
MBm
(2.49)
In this case , the stress Sp=Sm
(2.50)
Since the model in this case is stiffer, the force and moment applied to obtain the scaled down deflection are larger so that the stress level is the same as the prototype. 2.7 DISTORTED MODEL Long models are defined as those that are large in one dimension compared to the other dimensions . This type of model may require distortion of scales in the short direction . In fluid mechanics , this distortion of the long model may be necessary to avoid the problem of viscous boundary layers at a rigid boundary or capillary effect. A model is distorted when it has more than one geometric scale . For example, the vertical scale (0) in the model is different from the horizontal scale (X). The ratio between a and a, is called the rate of distortion [LeMehaute (1976)]. In general, scale models used in the study of water waves are not distorted with the exception of a few special cases. Let us consider the wave velocity by linear theory which is given as
36
Chapter 2 Modeling Laws
(2.51)
c = 8T tanh kd 2n
For the wave velocity to scale properly (as /) according to Froude's law, the depth parameter (d/L) must be the same between the model and the prototype. Since the ratio of wave lengths Lp/Lm is given by the horizontal scale a ., it is clear that the ratio of the water depth (dp/dm) should also be X. Then the wave motion is in similitude. In the case of long waves , however, distortion is possible. Since, in long waves, tanh kd may be approximated by kd, the wave velocity is given by c
=
gd
(2.52)
In this case, the water depth in the model can be distorted by using a scale (3, different from the wave length scale X. Then, the velocity scale is given by
CP
dp
2 1 tit
(2.53)
C.
The wave period (or time) scales as a combination of the two scale factors T _( l ( Ll
Tp
 L.l dp
(2.54) YO
The distorted models are further discussed in Chapter 9. Let us consider another example of structural modeling . In this case, it is desired to model an elastic underwater oil storage tank made of rubber like material . In this case, the differential equations of motion have been derived by LeMehaute ( 1965) to describe the motion of two fluids separated by the membrane and subjected to surface waves. The similarity requirements deduced from these equations give
^gDP. of _% gDP.PO l (2.55) L Po P L Po J.
Section 2.8 References 37
and
[P R]
[p .R].
(2.56)
where pw,, po = densities of water and oil, respectively, D = depth of oil in the tank, R = radius of the bag, e = membrane thickness and E = modulus of elasticity of membrane. From the first relationship, Dp
=XDm
(2.57)
and w Pol _ Pw  Po
(2.58)
Po Jp L Po J,
This latter condition is satisfied by using lighter fluid for oil in the model, e.g., kerosene, alcohol or gasoline, to account for the difference between the sea water and fresh water. For the model of elasticity, we have (eE)IP X2(eE)T
(2.59)
Thus, the material and thickness of the model should be chosen such that their product satisfies the relationship in Eq. 2.56. For example, for a Firestone product 6.4 mm (1/4 in.) thick and at a scale of 1:30, a 0.4 mm (1/64 in.) thick rubber material will satisfy the scaling law. The choice of lighter fluid in the model has the added advantage here to model the Reynolds number associated with the motion inside the tank. Thus, 3
VP
=
X2
v,,,
(2.60)
where v = kinematic viscosity. For example, v for crude oil is 0.9 x 105m2/s (104 ft2 /sec); while for alcohol and gasoline , it is 0.15 x 105 m2/s (1.6 x 105ft2/sec) and 0.46 x 106m2/s (5 x 106ft2/sec), respectively. 2.8 REFERENCES 1. Berkley, W.B., "The Application of Model Testing to Offshore Mobile Platform Design", Society of Petroleum Engineers of AIME, Paper No. SPE 2149, Houston, Texas, 1968.
38 Chapter 2 Modeling Laws
2. Bridgeman, P.W., Dimensional Analysis, Revised Edition, Yale University Press, New Haven, Connecticut, 1965. 3. Buckingham, E., "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations ", Physics Review, Volume 4, 1914, pp. 345376. 4. Chakrabarti, S.K., "Modeling of Offshore Structures (Chapter), Application in Coastal Modeling, V.C Lakhan and A .S. Trenhaile (Editors), Elsevier Oceanography Series 49, 1989. 5. Hansen, A.G., Similarity Analysis of Boundary Value Problems in Enginccdag, PrenticeHall Inc., Englewood Cliffs, New Jersey, 1964. 6. Haszpra, 0., Modelling Hydroelastic Vibrations, Pitman Publishing, London, 1979. 7. Hoerner, S.F., FluidDynamic Drag, Published by the Author, Midland Park, New Jersey, 1965. 8. Langhaar, H.L., Dimensional Analysis and Theory of Models, John Wiley & Sons, New York, New York, 1951. 9. LeMehaute, B., "On Fronde Cauchy Similitude ", Proceedings on Specialty Conference on Coastal Engineering, Santa Barbara, California, ASCE, Oct., 1965. 10. LeMehaute, B., An Introduction to Hydrodynamics and Water Waves, Springer Verlag, New York, New York, 1976. 11. Morison, J.R., O'Brien, M.P., Johnson, J.W., and Shaaf, S.A., "The Forces Exerted by Surface Waves on Piles", Petroleum Transactions , AIME, Vol. 189, 1950, pp. 149157. 12. Murphy, G., Similitude in Engineering, Ronald Press Co., New York, New York, 1950. 13. Pao, R.H.F., Fluid Mechanics, John Wiley and Sons, Inc., New York, New York, 1965. 14. Sarpkaya, T. and Isaacson, M., Mechanics of Wave Forces on Offshore Structures, van Nostrand Reinhold Co., New York, New York, 1981.
Section 2. 8 References
39
16. Sedov, L., Similarity and Dimensional Methods in Mechanics, Academic Press, New York, New York, 1959. 17. Skoglund, VJ., Similitude Theory and Applications , International Textbook Co., Scranton, Pennsylvania, 1967. 18. Soper, W.G., "Scale Modeling ", International Science and Technology, Feb., 1967, pp. 6069. 19. Szucs, E., Similarity and Models, Elsevier, Amsterdam, The Netherlands, 1978. 20. Yalin, M.S. "Discussion on Hydraulic Modelling in Maritime ", Institute of Civil Engineers, Conference Proceedings, Thomas Telford Ltd., London, 1982, pp. 912.
CHAPTER 3 MODEL CONSTRUCTION TECHNIQUES
3.1 GENERAL REQUIREMENTS FOR MODELS The art of designing and building a working model that is both accurate in its properties and capable of withstanding the prolonged water environment is both difficult and time consuming . The models are usually equipped with a variety of instrumentation and proper provisions must be given for their attachment. Waterproofing of the instrumentation and their attachment to the model is critical to the success of the model test . Quite often, models must also possess surface finishes that are attractive for photographic requirements and visible in contrast under water. Models are made of a variety of materials, such as, wood , metal, Fiberglass, plastics, concrete or composite materials. Quite often , models are built from multiple materials for different model components. Selection of the most suitable material to build a particular model depends on several factors . One of the most important of these is the size of the model . The other is the modeling criterion. Most offshore structure models are built following Froude 's law, and geometric similitude is almost invariably maintained wherever possible. The size of the model is generally limited by the dimensions of the testing facility and its simulation capability of the environment. If standard parts needed for the model are available in the selected modeling material , then the computed scale factor may be changed somewhat to enable the use of these standard items , such as standard material thickness, pipes, angles, etc . The weight of the model is also an important consideration due to the limits in the available handling equipment . Of course, cost plays a major role in the choice of the scale factor. While the larger the model, the more expensive it may be to build , the small size of a model may sometimes make it expensive as well, as a result of the difficulty in fabricating and handling small, delicate components. Thus, cost should be considered in the decision of a model scale, X. The instrumentation should also be considered along with model size. For example, force scales as A,3 (Froude's law). Therefore, if loads on a structure are measured, the size of the model loads may be a controlling consideration in determining the limits of the model scale . If the model loads are too small or too large, they may be difficult to measure with acceptable accuracy.
Section 3.3 Environmental Load Models
41
3.2 MODEL TYPES The model for an offshore structure may be classified in the following categories depending on the purpose of testing: • Models for environmental load measurement; • Seakeeping models; • Elastic models; and • Models of attachments (e.g., mooring or anchoring system) Different techniques are used in designing and building these types of models . Examples are given here (as well as in Chapters 7, 8, and 9) to illustrate the model construction techniques in these areas. The scaling technique and building of an elastic model will be addressed in Chapter 9. 3.3 ENVIRONMENTAL LOAD MODELS The models used to determine environmental loads are held fixed in waves, wind and/or current. For the measurement of loads, the model is supported on load cells. In order that the weight of the model may be supported on load cells, the structure must be strong enough to represent a rigid body . Sometimes, additional internal support is needed to develop the structural strength required by the model . The outer dimensions of the model are maintained; however, the thickness of the wall or the internal geometry is not necessarily scaled . The model is ballasted in the tank so that the load on the load cells is within the range of measurement . Additional ballast may sometimes allow the tension load cells to register both tensile and compressive (i.e. oscillatory) loads without going slack. Lead (or steel) weight placed inside the model is a quite common method of ballasting a model used for load measurement . Sometimes, the model is flooded in place inside the wave tank . Ballasting with water , however, should be given careful consideration in order not to introduce any problem with internal sloshing. Communication of flow inside the model should always be avoided to insure the measurement of true external loads . Communications may introduce internal pressure which will alter the external loads registered. The interior surfaces of these models are primed where possible . The exterior is generally primed and topcoated with high visibility epoxy paint. Often contrasting radial and circumferential striping is applied to the model so as to provide improved visibility and photographic definition. For surface piercing structures, it also provides qualitative data on the wave run up at the model.
42 Chapter 3 Model Construction Techniques
The problems associated with the construction of an environmental load model are best described through an actual example of the construction of a model. The model chosen provides most of the common design techniques necessary for an environmental load model . A shelfmounted Ocean Thermal Energy Conversion (OTEC) platform model was tested in a wave tank to determine wave loads . The model construction involved several features described above. 3.3.1 OTEC Platform Model A large shelfmounted OTEC platform model was constructed for load determination tests. The platform concept included large power modules represented by submerged blocks which required verification for wave loads in the design calculation. As shown in Fig. 3.1, the OTEC platform frame consisted of 4 vertical columns, a deck, 2 horizontal pontoons and 2 horizontal braces . This configuration was identified as the baseline structure . The columns emanated from the circular pods (as shown in Fig. 3.1) and were connected at the top by a stiffened deck and at the bottom in pairs by 2 horizontal pontoons . The prototype and the corresponding model dimensions for the members of the baseline platform are shown in Table 3.1. The platform members by themselves did not provide sufficient rigidity. Therefore, additional members were introduced without affecting the load measurement. Two smaller tubular braces were introduced to connect the 4 pods. These members connected the pods on opposite faces where there are no pontoons. TABLE 3.1 BASIC OTEC PLATFORM MEMBER DIMENSIONS MODEL SCALE = 1:25.75
DESCRIPTION
DIMENSIONS QUANTITY PROTOTYPE MODEL
Vertical Columns Pods Pontoons Platform lft = 0.3 m; =diameter
4 4 2 1
300 x 320 52.54 x 40 300 x 117.5 220 x 220
1.16$ x 12.46 2.040 x 1.55 1.160 x 4.56 8.54 x 8.54
While these strength members should be avoided under water where they will experience wave loads, they could not be avoided in this case since rigidity was considered more important. More acceptable method to insure rigidity is to use internal members which are not directly exposed to waves . Both the horizontal
Section 3.3 Environmental Load Models 43
pontoons and braces were reinforced internally for strength and rigidity with rectangular box beam tubing . The box beam was welded to the pods and vertical columns which provided additional rigidity to prevent racking of the structure. The tubing was reinforced with "TEE" sections made from 12.7 mm ( 1/2 in.) steel plate. The resulting section is shown in Fig. 3.2. In addition, 9.5 mm (3/8 in.) diameter smooth rods were used as diagonal braces on all 4 vertical faces of the baseline model. These members provided the required rigidity in the platform without altering the global wave loads by a measurable amount.
FIGURE 3.1 LOAD MEASURING MODEL ON A GENERIC SHELF MOUNTED OTEC Since the platform deck was above the water surface and outside the reach of waves, it was designed to provide rigidity. The deck was framed out of 2 sizes of rectangular steel tubing . The tubing was 76.2 mm x 38.1 mm (3 x 11/2 in.) with a 6.3 mm (1/4 in.) wall thickness. A diagonal cross brace made of 63 .5 mm x 63.5 mm x 4.7
44 Chapter 8 Model Construction Techniques
mm (21 /2 x 21/2 x 3/16 in.) steel angle and 63.5mm x 4.7 mm (21/2 x 3/16 in.) flat bar was provided for rigidity. The deck was bolted directly to the top of the 4 columns. The columns . and pontoons were made of 356 mm ( 14 in.) outside diameter (12.2 m or 40 ft prototype) steel tubing with a wall thickness of 0.20 in. The circular pods were made from rolled 3.2 mm ( 1/8 in.) thick steel plate. Two horizontal plate stiffeners acting as the upper and lower pod surfaces were used to secure a column to a pod. These plates were cut from 4.7 mm (3/16 in.) plate. The pods and columns were cut to allow for the passage of the reinforcing box beam. Figure 3.2 shows a typical pod/column assembly. The OTEC model consisted of the welded steel frame over which a variety of submerged power modules (modeled as boxes) were mounted . All weld seams were ground smooth and the entire model was painted with a high visibility yellow paint. Selected reference points were marked in colored tape to provide easy identification of model components when viewing the test videotape.
FIGURE 3.2 LOAD CELL MOUNTING DETAIL
Section 3.4 Seakeeping Model
45
Note that a large environmental load model is typically supported on three load cells rather than four for ease in alignment . The frame was supported on three load cells, each capable of measuring forces in three orthogonal directions. One cell was mounted just outside of each pod on the west side of the northsouth vertical plane of symmetry (see Fig 3 .3). The remaining cell was located at the midpoint of the horizontal brace on the east side of the plane of symmetry . Each load cell was attached to the box beam frame which runs through the horizontal pontoons and braces . Figure 3.2 shows a cutaway of a typical load cell assembly. The model was fitted on the load cells in the dry, and 19.1 mm (3/4 in.) diameter holes were cut in the model 's box beam frame for passage of the 12.7 mm (1/2 in.) threaded rod used to attach the model to the load cells . The top of the vertical load cell was slightly recessed inside the horizontal cells. Three flat washers and a spherical washer were installed between the top of the vertical cell and the bottom of the box beam. The spherical washer was used to avoid any misalignment. In order to install the model with load cell assembly on the tank floor , a baseplate 3 in x 3 in x 19 . 1 mm (10 ft x 10 ft x 3/4 in.) thick was installed prior to the installation of the OTEC model . This plate may be seen in Fig . 3.2 on which the load cells were mounted. The weight and size of this plate provided enough friction to keep the plate and the model fixed in the tank in waves . This is a quite common method of model mounting. The thickness of the steel plate does not significantly affect the water depth . Installation can be accomplished in the dry, and in place calibration can be performed outside the tank . Then any problems with the load measurement or the set up may be corrected easily. However, care should be taken that the load cells are capable of withstanding the dry weight of the model. If the load cells are not capable of holding the model weight in the dry, installation must be made in the water (which may be made shallow for ease of handling). In this case the inplace calibration is carried out under water by loading the model in different directions with known weights and recording the load cell readings along the axis . The crossaxis sensitivity of the load cell (see Chapter 6) is also established at the same time. Adjustments may be necessary if the crosstalk is unacceptably high. The load cells are bolted to the baseplate. Additional lead weights may be placed on the corners of the baseplate if needed to hold the model in place. The baseplate may also be "stitch welded " to a steel section of the wave tank or bolted to the concrete floor to prevent it from sliding in waves. 3.4 SEAKEEPING MODEL Unlike fixed models for measuring environmental loads , the seakeeping models are allowed to respond with the environment . Therefore, in addition to the
46 Chapter 3 Model Construction Techniques
N W m
FIGURE 3.3 LOAD CELL LOCATIONS outside geometry, the dynamic properties of the structure must be modeled using Froude's law. Some of these properties include the displacement, center of gravity, moments of inertia and natural periods of oscillation. The scaling factors for these properties have been given in Table 2.2. Typical acceptable tolerances for these parameters are as follows: • Weight, length and center of gravity (G) within 3 percent; • Displacement, center of buoyancy, (B) and the distance GB within 3 percent; • Stiffness (i.e., EI factor) within 5 percent; and • Moments of inertia within 5 percent.
Section 3.4 Seakeeping Model 47
There are two types of floating vessels: transport vessels and moored vessels. The transport vessels include ships and barges which move from one location of the ocean to another carrying goods and services. The other type of structures includes articulated towers, tension leg platforms and moored tankers. They are connected to the ocean bottom by some mechanical means. Therefore, modeling these structures involve modeling not only the floating structure but also the mooring system. Here, construction of three seakeeping models and one launching model is exemplified. 3.4.1 Tanker Model A model scale is chosen first based on several factors already discussed. For the type of tanker, the model plans, namely, body, halfbreadth and profile drawings, and a table of offsets , are obtained or developed. Next, an estimation of the target weight of the finished model without ballast is made . This is done to ensure that the ballasting will achieve both the static and dynamic characteristics of the tanker . Tanker models are built with a variety of materials, e.g., wax, wood, fiberglass, plastic, etc. The choice of the material is based on economy , usage, applicability, and longevity of the model as well as model construction capabilities. It is preferable to build the model upside down on a perfectly flat worktable. This allows a close control of the model dimensions and facilitates dimensional check during construction. 3.4.1.1 Wood Construction of Model The general method used to produce the wood model is to prepare a laminated wood block that approximates the shape of the ship by cutting each laminate along waterplane lines. The final shape of the model is achieved by removing excess material by planing, spokeshaving and scraping . If a numerically controlled (NC) machine is available and suitable for the model size , it provides the final shape. Final form and fairness are verified with metal station and waterplane templates. From the lines drawing or table of offsets of the model, a halfbreadth set of thin sheet aluminum female station and half station templates are constructed. The templates are rough cut with a band saw and machined to a scribed line. Each template is marked for centerline and deckline and for benchmark (inspection) references. The tolerance on all templates generally is 0.1 percent or less . Several fullbreadth templates are also constructed for inspection of twist and flatness of the construction. All templates are made with a reference edge so that assembled alignment can be inspected and aligned by a long straight edge or machinists' optical transit.
48
Chapter 3 Model Construction Techniques
Since the basic manufacturing method with wood utilizes a cutout and laid up block, the direction of wood grain is an important consideration to minimize water contact and absorption through the end grain . Therefore, all wood is examined and the components are laid with the grain in one direction. Waterplane lines are then transferred to a selected plank and rough cut, 1 .6  3.2 mm ( 1/16 1/8 in.) oversized based on model size. Interior cuts of the waterplane plank are made in order to arrive at a full thickness of not less than 10 % of the ship beam with an additional thickness allowed at the bow. Successive layers of waterplane line laminates are then built up by applying glue to both bonding surfaces . Joints in adjacent laminates , if required, are alternated to opposite ends so that successive layers will not have joints in close approximation. Construction of the bottom of the ship is by lamination, with a minimum of 2 layers. No metallic fasteners are used in the fabrication of the block to facilitate machine use and finishing . Bar clamps with transverse strongbacks are used to ensure uniform clamping pressure. Reduction of the rough block is performed by hand planing, spokeshaving, scraping and sanding (Fig. 3.4). Alternatively, a numerically control machining device is used , if available. During the rough and final finishing, the hull is checked frequently with the previously prepared station templates. Final sanding is performed with successively finer grits until a smooth fair surface (such as 63 RMS ) is achieved.
FIGURE 3.4 WOOD CONSTRUCTION OF U.S.S. MIDWAY MODEL
Section 3.4 Seakeeping Model 49
Two coats of penetrating wood sealer are applied to the inside and outside of the model . While the outside is still wet with sealer, the hull is lightly machine sanded. This technique fills slight surface imperfections and any open grains to produce a higher grade surface finish . The final outside sealer coat is wiped to ensure uniform absorption and penetration . All holes or imperfections are filled with putty and painted with one coat of epoxy primer. Two coats of finish paint are applied by spraying the model. The completed model is inspected by positioning the station templates on the reference drawing and over the model; permissible tolerance is determined with a feeler gauge. 3.4.1.2 Fiberglass Construction of Model The fiberglass model may be constructed from a prepared polyurethane foam block that approximates the shape of the model. Fiberglass is applied to this form using a hand layup method to produce a hull 3 .2  6.3 mm (1/8  1/4 in.) thick, depending on model size. Final hull form is achieved by filling and sanding the surface to a smooth finish. The procedure for this construction method is described below. Male aluminum station and half station templates cut 3 .2  6.3 mm ( 1/8  1/4 in.) smaller than the finished dimension are prepared, aligned and stack drilled. Large diameter tubular aluminum spacers are cut to station lengths taking into consideration the template thickness. The male templates and station spacers are strung on threaded rod to form a skeleton frame of the model . Figure 3.5 shows the fabrication of a barge model. Aluminum split plates are attached to this frame which serve as full size station templates. Medium density polyurethane foam is sprayed between the station templates and allowed to rise over the templates . When the foam has cured, excess foam is removed until the edge of the male templates appear and a smooth and fair hull form is obtained. Several coats of fiberglass epoxy resin is applied to the foam form to produce a gel coat. Fiberglass cloth layers pretailored to the mold are then applied. Laminating resin is applied to the cloth by brush, roller or spray and worked into the cloth until no air pockets or resinstarved areas exist . The exterior of the hull is covered by not less than three layers of the fiberglass material and saturated with the gel coat. Final form of the model is achieved by hand and machine sanding until a fair and smooth model is obtained. All holes and imperfections are filled with a commercial epoxy fiberglass compound. Final form is determined by checking with female station and waterplane templates (Fig. 3.6).
50
Chapter 3 Model Construction Techniques
FIGURE 3.5 FIBERGLASS MODEL CONSTRUCTION OF HEAVY WORK BARGE If the model hull form is essentially a faired, single curved surface, the model may be constructed of thin wood or polyurethane strips and applied to small buttock frames representing the hull shape along its length. Fiberglassing would then proceed as presented in the previous paragraphs to build up the final thickness. The model is usually constructed and finished keel up on a centerline and station reference drawing. Therefore, the final smooth and fair form of the model is ensured . Also, the application of station templates always occurs at the correct interval and is perpendicular to the line of the keel . The use of accurate metal form templates permits easy, accurate , and reproducible inspection. The hull points are checked with surface height gauges and optical transit sighting. 3.4.2 Submergence Model The purpose of the submergence tests is to verify the stability of the structure during the submergence procedure . The model used for the submergence tests should not only be able to satisfy the dynamic properties of the structure but also be provided with ballast compartment to allow prescribed and systematic ballasting of the model. In this case, the outside dimensions of the model as well as the internal dimensions of the ballast tanks must be properly modeled . In the following, the
Section 3.4 Seakeeping Model
51
construction procedure adopted for the submergence model of a submersible exploratory drilling structure is described.
FIGURE 3.6 FIBERGLASS MODEL OF A 300 ,000 DWT TANKER; DIMENSIONAL CONTROL CHECK WITH FULL TEMPLATE 3.4.2.1 Construction Technique The general layout of the model is shown in Fig. 3 . 7. The model consists of a large foundation mat, three large columns holding the platform and several tubular structural members. The model's mat is constructed with watertight bulkheads to divide the mat into several separate chambers representing the prototype mat's ballast tanks , fuel tanks and drill water tanks. The mat is constructed of clear lexan sheet material except for the curved bow section which is machined from a block of clear plexiglas . The use of clear material allows the ballast level in all tanks to be observed during the testing. The geometry of the mat as well as the wall thickness are correctly scaled for similarity. The model mat is glued together at all joints except where the mat bottom attaches to the walls and bulkheads . This joint is screwed together and sealed with silicone caulking to allow for any required access to the tanks.
52 Chapter 3 Model Construction Techniques
The model caissons are constructed to properly model the outer dimensions of the prototype caissons. Internal details are not required except that any ballast tanks inside are provided . These tanks can be filled through a hole in the top of the caisson if required for testing. The tubular braces on the model are of plastic material and are sized to model the outer diameter of the prototype braces. 3 0*
705 ;,L EAGLE DRILL DECK
^ i T '^
FIGURE 3.7 LAYOUT OF A SUBMERSIBLE DRILLING MODEL (SCALE 1:48) The detailed superstructure on the model is not required for the submergence test. However, a simple platform is needed so that ballast weights may be attached to it for proper scaling of the platform properties. In order to submerge the model according to a prescribed procedure, water ballast must be added to the mat chambers in a specified sequence. An elaborate ballasting system including plastic tubings to each ballast compartment for water transfer and air venting may be added to accommodate an external ballasting sequence. To accomplish a simpler but controlled ballasting in the current model, a valve consisting of an inner and an outer sleeve is installed in each of the mat compartments. When the inner sleeve of the valve is oriented such that the inner and outer sleeve holes are not aligned, no water may enter the chamber. To insure water tightness, the inner sleeve is coated with high vacuum silicone grease so that it seals to the inner wall of the outer sleeve. To further insure a positive seal, a grease retainer is
Section 3..4 Seakeeping Model
53
glued to the bottom of the mat . To fill the ballast tank, the inner sleeve is rotated so that the inner and outer sleeve holes are aligned. With the valve in this position, water enters the chamber through the lower sleeve holes and the displaced air is vented through the upper holes. 3.4.2.2 Static and Dynamic Properties The model in this case may be constructed with the mat, caissons and superstructure all being reasonably represented . Therefore, the weight distribution of the model approaches the weight distribution of the prototype structure , and the model tends to approximate the dynamic characteristics of the prototype. Any variation may be adjusted with additional weights. After the model is constructed, its weight and C .G. location are determined. The longitudinal C.G. position may be found by balancing the model on a long 12 .7 mm (1/2 in.) diameter rod. The vertical C.G. location may be determined by hanging the model by the bow caisson so that it balances with the mat's base in a vertical plane. Lead weights are added to the model as necessary to model the prototype weight of the lightship (unloaded) plus the upper deck variables (pipe , machinery, etc.). The weights are positioned so that the C.G. of the lightship and upper variables is correctly modeled (Fig. 3.7). The verification of the dynamic properties of such models will be described in conjunction with the description of the following structure model. 3.4.3 Tension Leg Platform Model The complete Tension Leg Platform (TLP) consists of a floating steel (or concrete) platform attached to a bottom founded template. Each of the columns of the TLP platform is attached to the ocean floor by means of a tendon. For the purpose of seakeeping tests, the hull and the tendons are modeled without the bottom template. The tendon models are attached to the floor of the wave tank. Similarly, the deck structure need not be modeled, but structural support is provided for the integrity and rigidity of the model. The following describes the construction of a TLP model. 3.4.3.1 TLP Hull The structural components of the TLP platform can be divided into four basic categories: (1) the deck support , (2) the columns, (3) the pontoons and (4) the bracing. The chosen scale and various dimensions of a TLP in prototype and model scale are given in Table 3.2. The structural components of the platform model are fabricated
54 Chapter 8 Model Construction Techniques
from carbon steel. Figure 3.8 shows a photograph of the finished model prior to placement in the wave tank. The deck support consists entirely of 9.5 mm (3/8 in.) thick steel rectangular tubes. Four large rectangular tubes run from column to column forming the square perimeter of deck support Smaller rectangular tubes bisect the deck span in both the longitudinal and transverse directions. TABLE 3.2 PROTOTYPE AND MODEL TLP SCALE = 1:16 COMPONENT
PROTOTYPE
UNIT
MODEL
UNIT
Column Pontoon Vertical Braces Inclined Braces
40 28 8 6
ft ft ft ft
30 21 6 4.5
in. in. in. in.
61,440 8
kips sec
15,000 2
lbs sec
Displacement Heave Period
The columns were 1 in (39 in.) in diameter and 3.7 in ( 12 ft 11/2 in.) long made out of 4.7 mm (3/16 in.) thick steel plate. The bottom cap plate was also 4.7 mm (3/16 in.) thick. The columns were stiffened internally with four equally spaced ring stiffeners and longitudinal stiffening between the upper two ring stiffeners. In order to accommodate the tendons a 25.4 mm (1 in.) diameter schedule 80 pipe was run vertically through the center of each column extending 25.4 mm ( 1 in.) below the lower face of the cap plate. At the top of each column formed a "chair " consisting of two parallel beams spanning the columns to hold the tendons. The pontoons were 533 .4 mm (21 in.) diameter circular tubes formed from 4.7 mm (3/16 in.) thick steel plate. The braces were constructed entirely out of schedule 40 pipe. The outside diameter of the bracing ranged from 133.4 mm (51/4 in .) diameter to 190.5 mm (7 1/2 in.) diameter. All joints were welded completely around and ground smooth. To insure that the flotation pontoons were air tight, each pontoon was individually pressure tested. Subsequent to welding and pressure testing, the model was given one coat of primer and two coats of yellow paint
1
Section 3..4 Seakeeping Model
55
3.4.3.2 Tendons and Tendon Attachment Joints At a scale of 1:16, the typical water depth for a TLP is on the order of 30.5m (100 ft) in the model scale . Without modeling this depth it is not possible to model both axial and bending stiffness of the tendon. For the TLP dynamics, the axial stiffness is considered more important than bending stiffness. The tendon 's length must be distorted in the model due to limited tank depth available . The axial stiffness was modeled by a suitable plastic rod. The bending stiffness was higher in the model compared to scaled prototype value. The tendons consisted of a 15.9 mm (5/8 in.)
FIGURE 3.8 A TENSION LEG PLATFORM MODEL FLOATING IN WAVE TANK diameter steel rod endconnected to a 19 . 1 mm (3/4 in.) diameter acetal plastic rod. The end to end connection of the steel to plastic rod was provided by a 8900N (2000 lb) load cell for the measurement of axial load. The steel rod was threaded at the ends and was 4.07 m (13 ft 41/8 in.) long. It extended approximately 76.2 mm (3 in.) above the top of the tendon chair. The rod passed through the column (through the 25.4 mm or 1 in. diameter schedule 80 pipe ) and terminated at the submerged load cell approximately 158.8 mm (61/4 in.) below the bottom of the column. The lower end of the steel rod threaded directly into the top of the load cell. The upper end of the acetal plastic rod was capped with a 19.1 mm (3/4 in.) tube to 19.1 mm (3/4 in.) Swagelok end connector. The end connector threads into a 19.1 mm (3/4 in.) pipe cap that has a 12.7 mm (1/2 in.) fine thread stud welded to its outside face. The 12.7mm (1/2 in.) stud threads into the bottom of the load cell. The lower end of the plastic rod also has a 19.1 mm (3/4 in.) tube Swagelok end connector. This Swagelok threaded directly into the bottom plate located in the steel floor slab. Three nuts were located at the top end of the steel rod. Once the TLP was in the operational mode, the three nuts were used to vary the pretension acting on the tendons . By turning the lower nut against the tendon chair (while preventing the upper two nuts from rotating), the column could be pulled
56
Chapter 3 Model Construction Techniques
down against the tendon or eased up from the tendon , thus changing the static pretensile load on the tendon. 3.4.3.3 TLP Model Deployment Once the platform had been constructed, the wave tank was filled with water and the platform was placed in the tank. The platform was then floated to a convenient position, the tank drained and the model was allowed to settle on the wave tank floor. The dry model calibrations (described later ) were then performed. Subsequent to the dry calibrations, the tank was partially filled and the floating natural period tests and wet inclination tests were done . After these tests were complete, the steel and plastic tendons were cut to length and the wave tank was filled further. At an appropriate water level, the filling of the wave tank stopped, the steel rods were dropped through the columns from above and then divers installed the plastic tendons from below . With the tendons in place, the water level was raised further until the TLP reached the desired testing draft of 2.1 in (6 ft 101/2 in.). To remove the model, the tendon installation procedure was reversed. 3.4.4 Jacket Launching Models Jacket structures are generally transported to the installation site by barges. Model tests are required to simulate transportation, launch from the barge and upending of a barge mounted jacket. The transportation phase of the testing program is conducted to identify the dynamic characteristics of the jacket /barge combination as well as to determine the tie down forces and towing loads. The results are generally compared with numerical predictions of the jacket/barge behavior. 3.4.4.1 Jacket Model A Froude scale model of the jacket is constructed to the external dimensions of the prototype . All structural members, including skirt pile guides and conductor guide framing are modeled to the specified tolerances . Mud (foundation) mats, barge bumpers, etc., are usually modeled in simplified form. The model jacket structure is often of composite construction in that the main structural members (legs, horizontals and Xbracing ) may be constructed of thin walled aluminum tubing. All of the other structural members may be fabricated of plastic tubing . Because of the complexity of the model , a welding torch can not reach some of the smaller, more complicated tube intersections. These intersections may be held in place with epoxy glue. All ends of the plastic tubular members are plugged internally and tested for leaks before attachment to the main structure . The model is tested for leaks by coating all joints with a soap film and then applying air pressure to the
Section 3.5 Construction of Mooring System 57
aluminum tubing. This form of construction provides a rigid and strong model with the specified mass, center of gravity and buoyancy. Lifting eyes are provided for handling the model. The model is painted with high visibility paint and reference points marked with colored tape. The model is constructed to permit individual flooding of the volumes within the jacket legs during upending . Provisions are also made for the addition of lead shot, etc., internally in the event scaled weights of ballast water can not be provided. Alternatively, external lead weights are provided for attachment to the structure for simulation of added ballast during up ending tests. All of the model properties are determined, verified and documented. 3.4.4.2 Barge Model A barge model for jacket launching may be fabricated principally of marine plywood and sheet aluminum . Double curved surfaces may be formed from polyurethane foam. The entire outer hull is then covered with several layers of fiberglass cloth and resin. Internal hull construction is provided with suitable framing to provide adequate strength and rigidity for withstanding handling and testing. Water tight internal compartments are also provided to allow for adjustments in draft and trim. Sealed access openings are provided in the deck to allow for adjustment of draft, trim and mass properties, and for simulation of ballast conditions as specified. The barge model is provided with the launch ways, tilt beam assemblies and mooring points scaled from the prototype. A variable speed electric winch of suitable size is installed in the barge model . Friction brakes consisting of pivoting roller assemblies are provided as an adjustable friction device between the jacket skids and the barge tilt beams. Instrumentation is provided for the measurement of line pull , rocker arm loads and motions of the barge . The barge model is painted with a color contrasting with that of the jacket. Different draft lines are marked as required. The launch barge details have been further discussed in Chapter 8. 3.5 CONSTRUCTION OF MOORING SYSTEM Several types of mooring systems are used with floating structures. The most common of these are mooring chains and mooring hawsers. Examples are provided here for the construction of modeled mooring chains and hawsers. 3.5.1 Mooring Chains The mooring chains are modeled to prototype characteristics by dividing by the proper scale factor (Table 3.3). The chains used in the tests are selected on the
58
Chapter 3 Model Construction Techniques
basis of their dry weight as this is the characteristic which is considered most important and can be easily modeled. TABLE 3.3 MOORING CHAIN CHARACTERISTICS (SCALE 1:60) DESCRIPTION
PROTOTYPE
Number Pairs Angle Between Pairs Angle Between Chains in Pairs Length (Fore & Aft Chains) Length(Side Chains) Pretension Angle to Horizontal (with 100% Tanker) Linear Weight (in Water) Elasticity (for Full Length) Soil Friction Coefficient
12 6 60 deg
MODEL Actual Scaled Prototype 12 12 6 6 60 deg 60 deg
2 deg
2 deg
2 deg
2,600 ft 2,600 ft
43.33 ft 43.33 ft
43.33 ft 19.75 ft
60 deg
60 deg
60 deg
305 lbs/ft
0.085 lbs/ft
0.084 lbs/ft
166 kips/ft
3.84 lbs/m.
3.85 lbs/m.
1.0
1.0
0.42
The stiffness of the chain is determined by placing a length of the chain in a tension test machine and measuring its elongation at various loadings . Table 3.3 lists the prototype chain characteristics, theoretical model chain characteristics and the actual chain characteristics for a given mooring system . Due to limited tank width, the model chains in the transverse direction are shorter than the scaled prototype value. If the stiffness of the actual chain is greater than what is desired , springs are installed in the system to decrease the spring constant. The required spring stiffness is determined by the following equation: 1 K=K i 2
+K
(3.1)
Section 3.5 Construction of Mooring System
59
where K = desired spring constant of the system ; Kl = spring constant of the chain; and K2 = required spring constant in the added spring system . For example, if K = 980.1 N/m (67.15 lbs/ft) and Kl = 10,319 N /m (707 lbs/ft), this results in a K2 of 1080 N/m (74 lbs/ft). The length of these springs should be chosen so that the spring range is not exceeded during the entire loading sequence . For the above example, if each spring chosen has a constant of 388 .3 N/ n (26.6 lbs/ft) and six parallel sets of two springs in series are used , it results in a composite spring constant of 1164 .8 N/m (79. 8 lbs/ft) and provides the needed extension in the springs during testing. A static inplace calibration of the installed mooring system is performed by hanging a series of known weights on a string from the bow and stem of the ballasted tanker over a pulley and measuring the displacement of the tanker. An example data set is plotted in Fig. 3.9. 3.5.2 Mooring Hawsers Let us consider mooring hawsers based on a prototype 610 mm (24 in.) circumference nylon hawser with a breaking strength of 646 .5 metric tons ( 1425 kips). Load strain characteristics of the prototype hawser is defined by the equation: F = 0.125 [( 100)(e)]2
(3.2)
where F = load as a percent of breaking strength and s = percent strain . The actual load at any given strain , then, is the breaking strength times F . Table 3 .4 contains the prototype and desired model characteristics ( at a scale of 1:48) of the hawser for the example system. TABLE 3.4 NYLON HAWSER CHARACTERISTICS ITEM
PROTOTYPE
MODEL
Circumference 24 in. 1/4 in . Rubber Bands Breaking Strength 1425 kips 12.9 lbs Length 344.5 ft 7.2 ft Actual leng th of rubber band strand is about 15 in. Since the prototype hawser is nonlinear, the model hawser is also made nonlinear. One method of manufacturing the model hawser is to use rubber bands of unequal lengths banded together at the ends. For convenience in modeling , the rubber bands are usually chosen to have equal width . The load elongation curve of the prototype hawser is segmented into several parts, each of which may be represented by a mean
60
Chapter 3 Model Construction Techniques
LOAD RINGS
NYLON HAWSER
LENGTh O. ONAIN ON BOTTOM
ANCHOR RADIUS
8
6  CALCULATIONS TEST RESULTS 4
2
4
6
8 24
16
8
8
16
24
MODEL DISPLACEMENT. INCHES FIGURE 3.9 LOAD DISPLACEMENT CURVE MEASURED ON MODEL IN PLACE
Section 3.5 Construction of Mooring System
61
slope. The initial slope (stiffness) at the lowest strain is matched by a rubber band of a given width and length such that this length of a rubber band can stretch to the full elongation required of the model hawser. Then the width and length of the band represents the first segment of the loadelongation curve. For the same width , the second length of the rubber band is chosen in such a way that it provides no load (i.e., remains slack) over the first segment, but begins to stretch to pick up load over the second segment. Thus, there is a corresponding rubber band length for each segment of the stressstrain (i.e. loadelongation) curve. Additional lengths of band are added, which correspond to the strain for each loading point and the combined strain of the rubber bands match the slope of the particular segment . Sometimes, multiple bands or bands of unequal width are needed to match the segment load versus elongation characteristics of a particular mooring hawser. As the hawser is stretched, a different number of bands are loaded. Since the number of bands loaded at one time varies with the applied load and resultant strength , the response of the model hawser is nonlinear. The loadelongation characteristics of a prototype hawser is given in Table 3.5. The characteristics of the model hawser (scale 1:42) made of rubber bands are checked by applying a weight to one end and measuring the displacement. Figure 3.10 shows a comparison of theoretical and actual hawsers for the example problem.
TABLE 3.5 LOADELONGATION CHARACTERISTICS OF A HAWSER (SCALE = 1:42)
STRAIN
PROTOTYPE LOAD (kips)
MODEL LOAD (lbs)
.02 .04 .06 .08 .10 .125 .150 .175 .20
7.13 28.50 64.13 114.00 178.13 278.32 400.78 545.51 712.50
.096 .385 .866 1.539 2.404 3.757 5.410 7.363 9.619
62 Chapter 3 Model Construction Techniques
3.6 MODEL CALIBRATION METHODS Special calibration procedures are needed to determine the model properties of a floating platform and the corresponding mooring system . The platform calibrations are done to determine the following properties: • weight, • center of gravity, • mass moments of inertia, and • metacentric height. N
I.0
0 THEORETICAL O ACTUAL s
4 0
s M
2
4
6
8
E
12
MODEL LOAD. Ibs FIGURE 3.10 MODEL LOAD VS. STRAIN FOR A MOORING HAWSER ARRANGEMENT
Section 3. 6 Model Calibration Methods
63
The platform model is usually designed so that it is quite light compared to the required displacement. In order to model the specified weight, (lead and /or water) ballast is placed on the model at various positions . The positioning of the weight placement is dictated by the results of the center of gravity and moment of inertia tests. The term "unballasted model" refers to the platform without ballast , mooring components and instrumentation, and the term "ballasted model" refers to the model with ballast, mooring system and instrumentation. After the model construction is complete, the C.G. test and the mass moment of inertia test are performed on the unballasted model. Calculations are then made to determine the lead weight placement necessary to bring the unballasted model C .G. and moments of inertia to the required values. The calibration method for the properties of a tanker has been described through worked out examples by MunroSmith (1965). Bhattacharyya ( 1978) also presented several cases of model calibration. 3.6.1 Platform Calibrations The dry calibration of a dynamic model platform is performed after the model construction is complete, but before the ballast weights are placed . The amount and location of the ballast weights is determined based on the initial calibration. A final check calibration is usually performed after the placement of the ballast weights. It is always wise to perform a wet calibration of the platform before the starting of the test runs. 3.6.1.1 Weight Estimate Several evaluations may be made during various stages of model construction to estimate the weight of the unballasted platform model. These evaluations can be summarized as follows: a. Hand calculations of each structural component weight during design stage. b. Measurement of the component weights once they are available. c. Measurement of free floating draft of the completed model. d. Hanging model with a scale or a load cell inline with the cable and recording the model weight (based on the load cell output). The setup for this method is shown in Fig. 3.11. Each weight estimate resulting from the above four methods is adjusted for components not included in the original calculation. An example of the results of the four different weight evaluations for a TLP model is given in Table 3.6.
64 Chapter 3 Model Construction Techniques
3.6.1.2 Center of Gravity Estimate Small models may be balanced on a knifeedge along a particular model axis for the determination of the center of gravity along that axis. A round rod may also be used for this purpose instead of a knifeedge. This method works quite well for long models that are easy to handle. The model is placed transverse on its sides on the rod and moved until the two sides tend to balance and a small displacement on either direction of the rod provides a bias in that direction. Then the distance from the edge of the model to the center of the rod gives the location of the C.G. The setup for the center of gravity estimation for large models is shown in Fig. 3.11. This setup shows the model hanging from a universal joint such that it is free to swing in the roll and pitch directions. By lifting the bow of the model and simultaneously recording the lifting load (using a load cell) and the angle of inclination (using an inclinometer), the C .G. is calculated. The formula used to calculate the C.G. arises from the static equilibrium of moments as follows: sing = Fdl Wd,,B
(3.3)
TABLE 3.6 MODEL TLP PLATFORM WEIGHT ESTIMATES METHOD a b c d
PREDICTED WEIGHT (lbs 14845 15062 15209 15162
% DIFFERENCE FROM METHOD d 2.1 0.7 0.3 0.0
where 0 = model's angle of inclination, F = lifting force, dl = horizontal moment arm from lifting point to rotational point, W = model weight and do = distance from C.G. to rotational point (e.g., universal joint axis). Since all other quantities are known or measured, dcg may be computed from Eq. 3.3. Generally, several different inclination angles are applied to the model, and a line of best fit to these lifting load/angle points is obtained. The rotational point in this example of a TLP is 3537 mm (139.25 in.) above keel. Therefore, the best fit line to the dry inclination curve (for small angles) is expressed as
Section 3.6 Model Calibration Methods
8sine{W
d1 (139 .25KG) F
65
(3.4)
where 0 is in radians and the term in brackets is the slope of the best fit line . Knowing dl (in inches), W and the slope, one can solve for KG (in inches), the distance from the structure keel to the C.G.
FIGURE 3.11 PLATFORM CALIBRATION TEST SETUP C.G. AND MOMENTS OF INERTIA TESTS
66
Chapter 3 Model Construction Techniques
3.6.1.3 Estimate of Moments of Inertia The analysis of the motions of a floating body requires that the model be scaled properly for the rotational properties of the prototype . The moments of inertia (or the radii of gyration) of a floating model for the roll , pitch or yaw motion may be determined from a swing frame built from structural members (e.g., Ibeams). The weight, center of gravity and moment of inertia of the frame alone are known (i.e., measured) a priori. The model is mounted on the frame, which is placed on two knife edges. The vertical C.G. of the model is estimated by measuring the angle of tilt of the frame with an angle sensor by applying a known moment statically . By balancing the moments on the knife edge, the vertical C.G. is estimated. The model may be swung freely in roll, pitch or yaw. The periods of motion of the model are determined with a stopwatch or the recording of an angle sensor. The moments of inertia of the system for a particular degree of freedom may be estimated from the natural period. An alternative method uses the center of gravity test setup described in the previous section (Fig. 3.11). A sonic wave probe is mounted horizontally on a stand adjacent to the model. The model is given an initial rotational displacement and then allowed to swing freely about the point of rotation (e.g., the universal joint axis). By measuring the relative displacement of the swinging model with the probe, the natural period of displacement time history is determined from the average time period between each successive pair of crests. The mass moment of inertia of the model is then calculated based on the oscillation "swing" period of the model. The calculation of the moment of inertia is given by the following formula: ( Ig 2n )
z Wdcg g dCg
(3.5)
where Ig = mass moment of inertia about pitch axis through the C.G. (kg  m2), TN = measured natural period of swing (s), and W = model weight (N). 3.6.1.4 Righting Moment Calibration A wet inclination test is performed to measure the metacentric height (GM) of the platform in the pitch direction. The platform is floated freely and known weights are placed on the center of the bow deck support. The trim angle of the model as measured by an inclinometer is recorded. The applied heeling moment equals the
Section 8.6 Model Calibration Methods 67
amount of added weight times its moment arm ; thus the equilibrium of static moments requires the following: Fd1= W(GM)
sinO
(3.6)
where the right hand side is the righting moment term . For small angles, 9sin 0 =LW(GM)
JF
(3.7)
where 0 = trim angle in radians , dl = horizontal moment ann for applied weight, W = model weight, GM = metracentric height and F = applied weight. Several angles of trim are applied to the model. Then, the wet inclination curve is a line of best fit through these points. Similar to the dry inclination curve, the term in the bracket of the above equation represents the slope of the best fit line to the wet inclination curve. Knowing the slope, "dl" and "W", the GM may be calculated. An example of a submersible drilling rig is included here . The intent of the test run was to confirm the righting moment characteristic curves for the drilling rig. The test was performed on a 1 :48 scaled model. Two horizontal cables, one connecting the right side of the mat's bottom to an anchor located at one end of the tank and the other connecting the left side of the deck to a weight over a pulley on the opposite end of the tank were used to induce the righting moment on to the rig . The two points of load application are on the plane of the longitudinal center of gravity , located 743 mm (29.25 in.) from the bow of the model. Depending on the tilt angle (0), the moment arm (D) in inches varied: D = [29.25  (1.1 +L2)tan0]cos0 (3.8) where Ll = distance (in inches) between point of load application at the mat and the centerline of the model and L2 = distance (in inches) between point of load application at the deck and centerline of the model . The tilt angle was measured by an inclinometer located at the top of the deck. The data are tabulated on Table 3.7. The relationship of tilt angle to the horizontal distance GZ is plotted on Fig . 3.12. The model properties are adjusted to satisfy the theoretically computed curves for the prototype. 3.6.2 Tendon Calibrations Unlike fixed or floating models, the anchoring system requires different modeling and calibration procedures. Mooring chain and hawser models have
68
Chapter 3 Model Construction Techniques
already been described . Another example is the TLP tendon . Risers fall in the same category. Tendons are generally modeled using plastics. Plastic rods experience creep which should be stabilized before testing with tendon rods should commence. The tendon calibrations are done in wet and dry environments to determine: • The dry creep characteristics; • The dry static stiffness; • The dry dynamic stiffness and damping; • The effect of hysteresis under dry dynamic loading; and • The wet inplace static stiffness. 14'
12' L•3
10'
8' 1
TEST RESULTS CALCULATIONS 4'
2'
50
10°
15°
20°
25°
30°
TILT ANGLE° DEGREES FIGURE 3.12 RIGHTING MOMENT CURVE FOR UNDAMAGED SUBMERSIBLE
Section 3.6 Model Calibration Methods
69
During the dry tendon tests, data is collected to determine the creep, stiffness and damping properties of individual tendons which are loaded in the axial direction. A dry tendon test setup is presented in Fig . 3.13, showing a tendon hanging vertically from a roof beam with a load cart attached to the lower end of the tendon. The axial load in the tendon is varied by changing the amount of weight contained in the cart. TABLE 3.7 TILTING CHARACTERISTICS OF A SUBMERSIBLE MODEL (SCALE 1:48)
F
L1  18.5" L2 = 10.25"
D= [29.25  (Ll + L2) tan e]cos 0
F (lbs) 5 10 12 15 17
0 (deg) 2.15 7.21 9.57 12.67 15.08
Model D (in.) 28 .15 25.40 24.06 22.23 20.76
M (in. lbs) 140.76 254.08 288 .79 333.44 352.98
GZ (in .) 1.29 2.33 2.64 3.05 3.23
Prototype GZ (ft) 5.15 9.30 10.57 12.20 12.92
3.6.2.1 Dry Creep Characteristics Since the tendons are under tension for an extended period of time while the TLP is in the water, it is necessary to determine the rate of creep of the tendon material. Vendor supplied information indicates that stiffness of plastic rod is influenced by the
70 Chapter 3 Model Construction Techniques
extension of the material due to creep ; hence, it is necessary to perform stiffness tests with the effects of creep eliminated. Because of this, many of the dry tendon tests are done after the tendons have been loaded for several days. In a test of a tendon under load, approximately 7 days were required to achieve 95 percent of the material creep, although this may vary considerably depending on the material being used . For this particular test, DELRIN was used because of its desirable overall characteristics.
FIGURE 3.13 TENDON CALIBRATION TEST SETUP 3.6.2.2 Dry Static Stiffness Stiffness tests under static conditions are accomplished by initially loading the tendon with a given pretensile load and then incrementally increasing the load to a prescribed maximum. Upon reaching the maximum, the load is incrementally
Section 8.6 Model Calibration Methods 71
decreased to a prescribed minimum (less than the pretensile load), and then the load is increased back up to the pretensile load. These load changes are done in equal increments. By measuring the tendon length at each increment (using a measuring tape), the tendon stretch as a function of tendon axial load is determined. This procedure also identifies any hysteresis present in the material . The load range used for the static stiffness tests is chosen based on the tendon loads expected during operation. 3.6.2.3 Dry Dynamic Stiffness and Damping The dry dynamic stiffness and material damping tests may use the same setup as the dry static stiffness tests . Instead of incrementally loading the rod (as was done for the static stiffness test), a weight may be simply dropped on the (preloaded) weight cart which is attached to the bottom of the hanging tendon (Fig. 3.13). The resulting free oscillation of the axial load (e.g., strain) is recorded from a load cell or a strain gage. To conduct damping tests using load cells , the load cell is connected to the top of the hanging tendon so that the measurement axis of the cell is in line with the tendon axis . For dry material damping tests, a strain gauge may be mounted directly on the plastic rod. Both methods give a time history of the oscillating tendon load from which the stiffness and damping values are calculated. Strain gauges are preferred over load cells to avoid the inclusion of the stiffness of the load cell in the measurement. 3.6.2.4 Hysteresis Effect Under Dry Dynamic Loading The dry static stiffness tests show that the plastic material (DELRIN) does possess some hysteretic properties. Therefore, it is necessary to determine the significance of the hysteresis under dynamic loading conditions. Stiffness and damping tests which use both a strain gauge and a load cell together may be conducted to make this determination . The effect of hysteresis is such that as the tendon axial load decreases to zero , the corresponding strain decreases to some value greater than zero. By simultaneously recording the rod stress and strain (using the load cell and the strain gauge) during testing, the stress and strain during dynamic loading and unloading may be obtained. 3.6.2.5 Wet InPlace Static Stiffness The wet tendon tests are conducted on all four tendons simultaneously with the TLP in the wave tank , and they determine the total effective stiffness of the tethered TLP in the surge and heave directions . The setups required for the wet tests are shown in Figs . 3.14 and 3.15, respectively.
72 Chapter 3 Model Construction Techniques
If the plastic (DELRIN) tendons are used, the static stiffness of the four tendons should be estimated about seven days after the TLP had been deployed in the wave tank in order to account for material creep . The static stiffness of the four tendons acting together in the surge direction is determined by horizontally loading the structure using a pulley system (Fig. 3 . 14). Known horizontal loads are applied to the structure by placing known weights on the platen. The surge displacement of the structure may be measured by an optical system. r INPLACE TLP
NYLON S AT RING
PULLEY ATTACHED TO FLOOR OF INSTRUMENTATION BRIDGE
z_^_WEIGHT PLATEN
TENDONS (LOAD CELLS NOT SHOWN)
FIGURE 3.14 TENDON CALIBRATION TEST SETUP IN THE SURGE DIRECTION With the TLP in the operating mode, the axial force in the tendons is varied by changing the water level in the wave tank (e.g., changing buoyant force). The loads in the tendons are measured by the load cells . By taping rulers (0.25 mm or 0.01 in. resolution) to each column and using a surveyor 's level (Fig. 3 . 15), the change in elevation of each column may be measured . This change in elevation is a direct reflection of the tendon elongation. By incrementally varying the water level, the desired load range in the tendons can be achieved.
Section 3.6 Model Calibration Methods 73
SURVEYORS LEVEL NORTH WALL OF WAVE TANK SIGHTED ON SCALE
..SINPLACE TLP
SCALE
SCALEtt*
TENDONS F5(LOAD CELLS NOT SHOWN)
FIGURE 3.15 WET TENDON CALIBRATION TEST SETUP IN THE HEAVE DIRECTION
74 Chapter 3 Model Construction Techniques
3.7 REFERENCES 1. Bhattacharyya, R., Dynamics of Marine Vehicles, John Wiley and Sons, New York, New York, 1978. 2. MunroSmith, R., Notes and Examples in Naval Architecture, Edward Arnold (Publishers) Ltd., London, England, 1965.
CHAPTER 4 MODEL TESTING FACILITY 4.1 TYPE OF FACILITY The facility for model testing of an offshore structure generally consists of a wave generating basin , a towing tank and a current generating facility. It is advantageous to simulate the wave and current generation in a single basin so that their combined interaction with the structure model may be investigated . Sometimes, wind generating capability is also required. The wind generation in a wave basin is often accomplished using a series of blowers located just above the water surface near the model. Most of the wave tanks built preeighties are twodimensional , i.e., they are capable of generating waves that travel in one direction only. The long period waves in the ocean may exhibit unidirectional behavior . However, windgenerated ocean waves are generally multidirectional. In order to generate multidirectional waves, the wave basins must have widths comparable to their lengths. Many modern facilities are capable of producing multidirectional waves. This chapter will describe different types of wave generators, their design methods and the generation capabilities of many commercial wave testing facilities. The wind and current generating techniques will also be discussed. 4.2 WAVE GENERATORS In the natural sea state , the period range of the power spectrum of waves having appreciable energy contents varies from about 5 seconds to 25 seconds. If a largest recommended scale ratio for model testing in a wave tank is 1:100, then the shortest wave required in the tank is 0.5 second (2 Hz). Therefore, the wavemaker [Biesel (1954)] in the tank should be capable of producing waves of frequencies up to 2 Hz. There are many types of wave generating devices employed in model basins. They may be classified in two general categories : active and passive . The active generators consist of mechanical devices of various sorts displacing the water in direct contact with the generator. By controlling the movement of the device, the wave form is created. On the other hand, the passive wavemakers have no moving parts in contact with the
76 Chapter 4 Model Testing Facility
water. They use air pressure to generate oscillations of the water . Various types of active and passive generators are summarized in Fig. 4.1. All wavemakers in the figure are of the active type except for the types R and S. CONVERTIBLE
TO /
II ^I
OR 7'7'Tl ' 77'f
7'T199
O TO OA SIMPLE FLAP
PISTON ®
^^
l
PARTIAL FLAP . PAPTI AL PISTON 2 ACTUATORS
CARRIAGE AND FLAP 2 ACTUATORS
PARTIAL FLAP . PARTIAL PISTON PREDE FIRED . ADJUSTABLE RATIO
DOUBLE ARTICULATED FLAP 2 ACTUATORS
ll
PARTIAL FLAP . PARTIAL PISTON PREDEFINED MON ADJUSTABLE RATIO
DOUBLE ARTICULATED FLAP SINGLE ACTUATOR
IT
•
BOTTOM MOUNTED PISTON
VERTICALWEOGE
NOWITONTAL WEDGE
CARRIAGE {TRANSIENT WAVESI
ROLLING CYLINDER
DUCK
I\ +(Z r
1
T
TO ^ NMOED MITIAL
PNEUMATIC PLENUM
FLAP . PARTIAL PISTON . ADJUSTABLE RATIO
PNEUMATIC INEYRPIC )
T
F
a
v
z= fIWIGER
WAVE BOARD ON SMOULDER
DEPTH TRANSITION
NO DEPTH TRANSITION
FIGURE 4.1 VARIETY OF WAVE GENERATOR SCHEMATICS [Ploeg And Funke (1980)]
Section 4 .3 Mechanical Wavemaker 77
The wedge shaped wave generator is best suited for high frequency waves but has limited success in generating low frequency waves . On the other hand, the pneumatic wavemaker can generate low frequency waves quite well but is limited to high frequencies of about 1 Hz due to the quick response time required to form these waves. A wide frequency range (say , 0.5  10 seconds) can be best accomplished by a double articulated mechanical flapper . The top flapper generates waves between the frequencies of 1.0 Hz to 2.0 Hz, while the bottom flapper produces the lower frequency waves . Note that a random wave of broad banded spectrum requires a large frequency range. In this case, both flappers may be needed to produce this spectrum. A schematic of the double flapper arrangement is shown in Fig . 4.2. Typical dimensions for a water depth of 5.5 in (18 ft) are shown on the figure. Modes 1 and 2 are for high and low frequency waves respectively, while the combination of the two (Mode 3) is needed for a wide banded wave spectrum.
^
6.5'
tR111RN RAPPER HMM
FIGURE 4.2 DOUBLEHINGED WAVE GENERATOR 4.3 MECHANICAL WAVEMAKER There are two main classes of mechanical type wavemakers that are installed in facilities to generate waves [Patel and Ionnau (1980)]. One of them moves horizontally in the direction of wave propagation in a simple harmonic motion (to generate sinusoidal waves) and has the shape of a flat plate driven as a paddle or a piston . The other type moves vertically at the water surface in a simple harmonic motion and has the shape of a wedge [Bullock and Murton (1989)]. In the design of
78 Chapter 4 Model Testing Facility
these wavemakers [Gilbert, et al. (1971)], a boundary value formula is developed based on the type of wave generator and its motion. 4.3.1 Hinged Flapper Wave Theory Consider a flapper hinged at the bottom spanning the width of the wave tank [Galvin (1966), Hudspeth, et al. ( 1981 )]. For the purposes of generating the theory, the flapper is assumed to undergo small simple harmonic motion. The problem is treated as twodimensional . The water is assumed incompressible and inviscid and flow irrotational. Thus, a velocity potential , $ may be assumed to describe the flow. This wave generator theory mainly follows the work done by Biesel and Suquet ( 1953). Also see Hyun (1976). The velocity potential 4(x,y,t) for small amplitude waves generated by a simpleharmonic flap motion about x = 0 is governed by the Laplace equation, where (x,y) is the coordinate of a point in a twodimensional space and t is time:
a,+ aY2 +
=0, 05x 3d. Thus, at x > 3d, Ay = A sinh ky cos((ot  kx) (4.14) which we can rewrite for the still water surface as Ay = a cos(wt  kx) (4.15) where a is the wave amplitude = A sinh kd = H/2. With the aid of Bernoulli's equation, the pressure up to first order is given by
P=P+Pg(dy)
(4.16) =Pw+Pi
+Pg(dy)
where 'Y cos (wtk x ) Pw =Pga Cosh cosh kd
(4 . 17)
pi PgA. tan k. d cos k„ y exp[k„ x] sin wt
(4.18)
and
nsl
At x = 0, Eq. 4.17 becomes
Section 4 .3 Mechanical Wavemaker
81
(4.19)
Pw = pga coshky cosou = C, coswt cosh kd
and Eq. 4. 18 becomes (4.20)
Pi= IpgA, tan k„d cosk„ysinwt=C2sinwt
Note that pw and pi are 90° outofphase with respect to each other. Thus, the maximum value of Cl cos wt + C2 sin wt is C12 +C22 whereas its phase is tan 1 G C2 For a flaptype wave generator, the displacement of the wavemaker is written in terms of its stroke at the still water level, S, as
i;(y) = S(y8d), for &d 5y5d
(4.21)
t(y) = 0, for y < 8d
(4.22)
where D is the flap draft and Sd is the elevation of the hinge point (Sd=dD). Substituting the above equation into Eqs. 4.10 and 4.11 A = 2S
kD AM kd cosh kd +cosh kSd kD(sinh kd cosh kd + kd)
(4.23)
A2Sk„Dsink„d +cosk„dcosk„Sd " k„D(sink„dcosk„d+k„d) (4.24)
The relationship between wave height (H) and stroke (S) can be established by combining Eq. 4.23 and the expression for 'a' in Eq. 4.15.
S_ H
kD[sinh kd cosh kd + kd ] 4sinh kd[kD sinh kd coshkd+cosh
k&d]
(4.25)
82 Chapter 4 Model Testing Facility
Substituting this relationship into Eqs. 4.24 and 4.20, we can write an expression for pi as a function of H and T. For the case of random waves [Hudspeth, et al. (1978)], the wave profile can be represented by the sum of an infinite number of sine waves as (4.26)
r)(t)_ I Hnsin (wnt+en)
The wave height at a particular period corresponding to con over a frequency band Mw can be determined from Hn = 2 2S(w)Ow
(4.27)
where the energy density spectrum may be represented, for example, by the PiersonMoskowitz spectrum model, S(w) = 0.0081g2w exp[1.25(w / w
o)Y]
(4.28)
where wo = 40.16 1g / H, and Hs = significant wave height. The solution for the flapper motion is obtained by the linear superposition of the individual HT pair in Eq. 4.26 and relating it to Eq. 4.25. In addition to the forces needed to create the wave propagation, additional forces are required to counteract the static water pressure and to overcome the inertia of the flapper while in motion. 4.3.2 Wedge Theory Consider a plunger of arbitrary triangular section and of width, b, and depth, D, at the still water level (Fig. 4 .3). In developing the theory , the plunger is assumed to move harmonically at a frequency , w, in small amplitudes in the vertical direction. Since the wedge spans the width of the tank, the problem is treated as twodimensional. The wedge is assumed to travel a short distance so that linear theory may be applied. This allows the formulation of the problem [Wang (1974)] in terms of a velocity potential 0 that satisfies Laplace's equation (Eq. 4.1). The freesurface linearized boundary condition in this case is written as
Section 4 .3 Mechanical Wavemaker
83
FIGURE 4.3 COORDINATE SYSTEM FOR A PLUNGER z a
+!
^=0aty=0
(4.29)
The normal fluid velocity at the plunger surface is equal to the velocity component, vn, of the forced oscillation of the plunger in the same direction,
4an =v
(4.30)
In addition, the radiation condition dictates that at a large distance from the plunger, the waves are outgoing progressive waves. The solution is obtained through a conformal transformation from the (plunger) zplane and its image in the free surface to a reference ^plane of a unit circle (Fig. 4.4) in which z = (x,y) and (r,O) and y is measured positive upwards from the still water level. z
N
= ^ + E az^ i^
(4.31)
84
Chapter 4 Model Testing Facility
X
Z  PLANE
FIGURE 4.4 CONFORMAL TRANSFORMATION where z and t; are complex variables and the coefficients an are real numbers . In terms of coordinates (x, y) and (r, 0), we have X =rsin6 +^a,, sin(2n1)0 ao i r
(4.32)
Y = rcosO + IN (1) w 1 a2n1 cos(2nl)A 2m, a, I r
(4.33)
Note that the first term (only ao) in Eqs. 4.32 and 4.33 provides the fundamental shape of the plunger as a circular cylinder. If additionally al is considered nonzero (only ao and al), it yields a family of elliptic cylinders. The solution for ^ is constructed as a superposition of the source and multipole potentials at the origin in terms of these coefficients, having the form $=wSao[(A.s  Bic + ^1Pys^yn sin wt+( B$S + A^c+ Y1Q2m$2m)cosou (4.34)
]
in which the source potential e'"(xcosxy Zksinxy)dx ^s=ice " si n kx r u x +k
(4 . 35)
^c = Ice kr coskx
(4.36)
Section 4 .3 Mechanical Wavemaker 85
and the multipole potentials are expressed as cos2mO r2+
cos(2m  1)A N (2n1)a„_,co2m+2n1)01 I (4.37)
[ (2m_l)r' +(1) (2m+2n1) r2n'+Zw ,
The coefficients A, B, P2m and Q2m are determined from the normal velocity boundary condition at the body. All other boundary conditions including the radiation condition are automatically satisfied by ^. Note that the solution is linear with respect to the plunger amplitude, S. The function 4s is a two dimensional source , while $2m are multipole functions which represent only local disturbances in the vicinity of the (plunger) body and contribute no waves at infinity. 4s and Oc behave like outgoing progressive waves on the free surface and approach regular sinusoidal waves asymptotically as x increases. The above derivation is essentially taken from Wang (1974). After the velocity potential 0 is determined, the form of the free surface is obtained from the linearized condition 1
rl(x,t) =8 at at y = 0
(4.38)
At the point of interest away from the plunger, the waves will have the form tl(x,t)=kaoxS A2+B2 sin(wtkx+e)
(4.39)
B e=tan' A
(4.40)
where
The nondimensional amplitude of the wave, A is given as A=S=kba A2+B2/R
(4.41)
R=b=1+a2.,
(4.42)
where
ao R_,
Thus, A is dependent on the frequency parameter kb. Since potential 0 on the body is known, the hydrodynamic pressure from Bernoulli's equation and corresponding forces on the plunger may be determined.
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Chapter 4 Model Testing Facility
Wang (1974) presented many results on the amplitude ratio A for various plunger geometries. Typical plots of wave amplitude as a function of kb are shown in Fig. 4.5. As the ratio of depth to width decreases, the wave amplitude increases for the same stroke. Also, there is a general tendency of increase of amplitude with kb even though there is an optimum value at higher D/b values. The plot represents a plunger with a (= Aw/bD) having a value of 0.5, where Aw, is the crosssectional area of the submerged wedge (Fig. 4.3). 0.0.5 1.5 D/b  15
2.0 A in
3.0
0.5 3.5
2
3
4
kb
FIGURE 4.5 DESIGN CURVES FOR A WEDGESHAPED WAVEMAKER [Wang (1974)] 4.4 PNEUMATIC WAVE GENERATOR Unlike the flapper or the plunger , a pneumatic wave generator has no moving parts in direct contact with the water . The heart of a pneumatic wave generator is a blower with a low pressure head. The blower is connected to a partially immersed plenum chamber which is open at the bottom and situated at one end of the tank (Fig. 4.6). A flapper valve is placed at the bifurcation between the outside vent duct and the plenum chamber duct and connects alternately the discharge and the intake of the blower to and from the plenum chamber. This introduces a pressure differential in the
Section 4..4 Pneumatic Wave Generator 87
plenum chamber which alternately draws water up into the plenum chamber and then pushes it down . The plenum contains baffles to minimize transverse waves as well as to help distribute the air uniformly in the chamber. Usually, there are several openings to the plenum from the flapper valve for an even distribution of air . The cyclic motion of the water in the plenum chamber generates the waves in the tank. The position of the flapper valve is controlled by an electric or hydraulic servo system . The system accepts both a flapper position feedback signal from a transducer at the flapper as well as a reference signal , and operates an actuator to cause the flapper position to match the reference . The amplitude and frequency of the generated waves are directly related to the amplitude and frequency of the reference signal.
AIR INTAKE ,
AIR EXHAUST
FIGURE 4.6 PNEUMATIC AND PLUNGER WAVEMAKERS Because air is used to produce the wave form , the control mechanism for the flapper at the low air pressure as well as the air compressibility make the transfer function quite soft. A plot of the magnitude of the reference signal (or equivalently, the amplitude of the flapper motion) versus the generated wave in the tank is shown in Fig. 4.7. The direct drive wavemaker uses a similar signal to produce wave form (Fig. 4.8), but the relationship of inputoutput signal is different. In fact, the waves formed pneumatically grow slowly at the lower amplitudes and increase in height rapidly only as the signals approach maximum levels. Moreover, this phenomenon also limits the
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Chapter 4 Model Testing Facility
height of the high frequency waves . The pneumatic wave generators are not particularly suitable for high frequency waves (z 1 Hz ), since the flapper valve is usually incapable in handling the passage of high air volumes at these high frequencies.
Nu El
186
WAVE MAKER FEEDBACK
16 16 WAVE PROBE 1
m
m
F9
m
m
FIGURE 4.7 COMPARISON OF PNEUMATIC WAVEMAKER SIGNAL AND WAVE PROFILE AT MIDTANK 4.5 DESIGN OF A DOUBLE FLAPPER WAVEMAKER A double flapper wavemaker consists of two pivoted flappers, an actuation system, driven hydraulically, and a control system (Fig. 4.2). In accordance with the wave generation theory, the force and velocity requirements of the actuating devices of both flappers can be determined. The hydraulic demand depends on the work done by the actuator, which is obtained as the product of the acting force and its displacement or stroke. By superimposing the force requirements of the upper and lower flappers, the maximum output demands of. the hydraulic units are computed. Usually, the lower flapper has a much higher input requirement due to higher imposed force as well as a greater stroke length.
Section 4.5 Design of a Double Flapper Wavemaker 89
AA/) VP1 TMV^M VV
A AA Ah AA WAVE PROBE BS
h Hd 10h ^ VVV ^V^l r 4 5.9 7.5 11.1 WEDGE FEED BACK
12.5
15.8
17.5
29.8
22.5
25.8
FIGURE 4.8 COMPARISON OF DIRECT DRIVE WAVEMAKER SIGNAL AND WAVE PROFILE AT MIDTANK Although it is feasible to have water on both sides of the flapper, the side of a double flapper wavemaker away from the waves is often maintained dry. Both types of system exist among existing facilities . For example, the facility at the Stevens Institute of Technology at Hoboken, New Jersey, is wet on both sides . On the other hand, the MARINTEK facility at Trondheim , Norway, has a dry back side. Both systems have advantages and disadvantages . The wet wet system does not require the power to balance the static water pressure on the flapper. The possibility of leakage at the ends of the flapper is not a concern . The drawback is in the area of regular maintenance of the hydraulic units which are submerged for this system . Having water on both sides, it requires an additional force for wave generation since it must push water in both directions . Also, it demands a second wave absorber to dissipate the waves on the back side of the flapper. For the wave generator with water on one side, most of the force requirement is in counteracting the hydrostatic pressure . This force is a function of the water depth and is invariant of the type and size of waves generated. A separate system (from the
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Chapter . 4 Model Testing Facility
hydraulic driving system), such as a pneumatic or nitrogen gas system , can be used as a medium to generate the force required to counteract the static pressure. Then, the flow and power requirements of the hydraulic system can be greatly reduced. 4.5.1 Wetback and Dryback Design A dryback design is generally considered better in terms of the quality of wave generation and active absorption of the wave incident on the flapper. For a dryback design, a seal keeps the water from entering the back side of the flapper or wave board. A "rolling" seal is suitable to withstand many cycles of flexing and deformation from the motion of the wave board. It is more economical to operate than the wetback design even though additional mechanical or hydraulic equipment are needed to withstand the additional static load. It is, however , not very suitable for translational mode or multimode operation. 4.5.2 Hydraulic and Pneumatic Units An open or a closed loop hydraulic system may be chosen to drive the mechanical flappers. A closed loop system, however, is more accurate. A variable displacement pump is used to move the pumping element to the opposite position, reversing the direction of oil flow with the shaft being driven in the same direction. The shifting is regulated by a servo valve mounted on the pump (or on the actuator) with the hydraulic fluid supplied by a pressure compensated pump . Several pumps may be required to ensure a good response for the servo valve. The servo valves are controlled by a simple control signal. To assure equal displacement of the actuators, a set of flow control valves are provided which are controlled from the feedback of the actual movement of the actuator. The pressure of the hydraulic system is regulated by relief valves located inside the main housing. An appropriately sized pressure vessel may be used as an accumulator for the control of the pressure of a pneumatic or nitrogen system to overcome static head. With a higher pressure source (cylinders for feeding) on one end and a lower pressure source (cylinders for drawing) on the other, the pressure of the pneumatic system can be adjusted to accommodate various water depths. 4.5.3 Control System for a TwoBoard Flapper The flapper is driven by an electrically controlled hydraulic system. The control system consists of a command signal generator, a crossover network and dual servoamplifiers to supply current to the electrically controlled valves in the hydraulic flapper drive system.
Section 4.6 A Typical Wave Tank
91
The command signal generator outputs an analog signal representing the desired flapper position for each point in time during operation. The generator has the capability to provide both regular and random command signals . Signal generators are available commercially that are specifically designed for this purpose. A programmed microcomputer may also be used to generate the command signal. After the signal is output, it is fed to a crossover network where it is split into high and low frequency signals to drive the upper and lower flappers respectively. This task may be accomplished by a computer. The servoamplifiers in the system accept the outputs of the crossover as command signals as well as a position feedback signal from the flapper to produce a drive signal to the servovalves that is proportional to the difference between the command and feedback signals , thus reducing any error in the generating wave. 4.5.4 Waveboard Sealing and Structural Support System For a dryback system, a seal is required between the ends of the flappers and the side walls of the tank . Usually, a stainless steel wear plate is mounted on the concrete side wall. A plastic channel which is held in place by an inflatable air seal runs against the plate (Fig. 4. 9). The air bag is maintained at a given pressure provided by an air cylinder and a compressor system. For the joint between flappers, a urethane/butyl fabric can be applied to prevent leakage. Orings are used to seal both ends of the fabric. Several facilities, whether single or double flapper, use this type of system, including the U.S. Naval Academy at Annapolis, Maryland, and Oregon State University at Corvallis, Oregon. In a simple design , a rubber fabric folded between the flappers and the wall has been used with some success. To relieve the nitrogen counterbalance system, a structural system is used to support the actuators. It also provides structural constraint to the flappers in the case of the hydraulic system. The load on the flappers is transmitted through the structure onto the back wall and the floor of the wave tank (Fig. 4.2). 4.6 A TYPICAL WAVE TANK The wave tank at CBI's Marine Research Facility is 76.2m (250 ft) long, 10m (33 ft) wide and 5.5m (18 ft) deep. A 2.4m (8 ft) diameter pit is located in the center of the tank and extends an additional 4.1m (13.5 ft) below the tank floor. The layout of the wave tank showing the various components of the facility is shown in Fig. 4.10. The rail mounted traveling bridge is the test control center. The bridge supports the signal conditioning and data acquisition equipment and the controls for wave generator operation. A builtin hydraulic system provides power for transporting the bridge along the length of the tank. An electromechanical servo valve controls the hydraulic system and maintains the bridge at a constant speed. The maximum velocity of the bridge is 1.4m/s (4.5 ft/sec), which is suitable for lowspeed towing tests. A
92 Chapter 4 Model Testing Facility
second bridge on the same rails generates a speed of up to 3 .65 m/s (12 ft/sec) and may be used for higher speed towing of models. CONCRETE WALL
CONCRETE ANCHOR
STAINLESS STEEL WEAR PLATE
PLASTIC
INFLATABLE AIR SEAL STEEL CHANNEL
FLAPPER
FIGURE 4.9 SEALING ARRANGEMENT OF A DRY BACK FLAPPER SYSTEM PLAN VIEW Because the water depth in relation to the wave length has a significant influence on the wave kinematics, it is often necessary to vary the water depth in the wave basin based on the scale chosen and the prototype simulation depth desired. There are two basic approaches to achieve this . One is to have an adjustable basin floor (e.g., CBI, and MARINTEK). The other is to build a wave generator that can be adjusted vertically along the basin wall and then change the water level in the basin [e .g., National Research Council (NRC), Canada]. The customary method of changing the water depth in a tank is by draining the water from the tank. However, the pneumatic type of wave generator installed at a fixed elevation in the CBI tank, does not permit the water depth to be varied by lowering the water level. The elevation of moveable false floor sections in the tank (Fig. 4.10) can be adjusted to simulate ocean bottom conditions . The level of water intake at the pneumatic wavemaker and the high frequency wave generator is maintained for all floor
8.0 SECTION AA INSTRUMENT
15°
BRIDGE
PNEUMATIC \ADJUSTABLE CONCRETE WAVE MAKER PANEL FLOOR 0 TO 5.5 M., 0 TO 9
°4
STEEL PANEL WITH CURRENT 3X3 WINDOW GENERATORS
PLAN VIEW NOTE: ALL DIMENSIONS
FIGURE 4.10 LAYOUT OF CBI WAVE TANK
ARE IN METERS
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Chapter 4 Model Testing Facility
configurations. Therefore, the height of the waves generated at these wavemakers is expected to be about the same for all water depths. The waves are, of course , modified by the bottom conditions (arrangement of the floor sections ) as the waves travel down the length of the tank. The wave absorbing beach is graded stone in a rack covered with a plastic mesh. The rack is inclined at an angle of 15° to the still water surface and extends 3m (10 ft) below the surface . The wave energy absorbing efficiency of this beach is quite good having a reflection coefficient of 5 to 10 percent depending on the wave period. This beach in conjunction with the moveable floor slabs can be used to simulate near shore conditions . This is particularly useful for performing seakeeping tests of vessels in the surf zone. 4.6.1 Low Frequency Wavemaker This wave generator at this facility is a pneumatic type, consisting of an open bottom plenum chamber partially immersed in water and a blower with a low pressure head . Both the suction side and pressure side of the blower are connected to the plenum chamber by a flapper valve (Fig. 4.6). The system has maximum response capabilities between periods of 2.5 and 3.5 seconds, where maximum wave heights of 508 mm (20 in.) to 559 mm (22 in.) can be produced. The response capability of the wave generator declines at lower periods. At a period of 1 second, the maximum wave height that can be generated is approximately 51 mm (2 in.). Types of waves generated in commercial twodimensional tanks are similar, differing only in their ranges. Regular waves, wave groups and random waves can be generated in the tank at this facility . The signal generation for driving the wavemaker is also typical. The wave form reference signal is generated by a minicomputer. For regular waves, the software generates a sinusoidal signal of constant amplitude and period. The maximum response of the pneumatic wavemaker at various wave periods between 1 and 4 seconds is shown in Fig. 4.11. For random waves , the software decomposes the desired spectrum into a large number of frequency components. The wave reference signal is initially calculated and stored as a digital time series. To generate the analog reference signal required by the servo controller , the stored time series is output through a digital to analog converter under the control of a dedicated microcomputer. The generated wave is then analyzed . The resulting energy spectrum is plotted by the data acquisition system and compared with the desired theoretical spectrum. The reference signal can then be modified as required to insure a satisfactory correlation between the generated and desired spectra . The modified
Section 4. 6 A Typical Wave Tank 95
reference signal is stored as a time series and can be repeated on demand , thus insuring consistency between test conditions.
     DEEPWATER BREAKING WAVE LIMIT  •  PLUNGING WEDGE WAVEMAKER  PNEUMATIC WAVEMAKER N
10
N
OD 
OL 2.0 2.5 3.0 WAVE PERIOD (SEC)
3.5
4.0
FIGURE 4.11 HEIGHT OF REGULAR WAVES BY THE CBI PNEUMATIC WAVEMAKER Figure 4.12 is a comparison of a measured model tank spectrum and a theoretical wave spectrum; in this case, a JONSWAP type wave spectrum was used. As described above, a digital time series was executed to model the theoretical frequency distribution, which is plotted with the heavier line on Fig . 4.12. The recorded signal from a wave probe in the tank was analyzed in the frequency domain to produce the curve plotted with a lighter line on the same figure. The probabilistic estimates of random waves are made by assuming that the wave profile follows a stationary Gaussian random process and that the corresponding wave heights follow a Rayleigh distribution . Figure 4.13 shows the wave height distribution for the wave analyzed in Fig . 4.12. The solid line represents the cumulative probability
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Chapter 4 Model Testing Facility
of the normalized wave height (H/Hrms, Hrms = root mean square wave height) with the Rayleigh distribution . The wave tank data shown on the plot is grouped in small bands of wave height . The probability of occurrence is computed for each band. 4.6.2 High Frequency Wavemaker For generating the higher frequency waves than can be produced with the pneumatic generator, a plunging wedgetype wave generator is used (Fig. 4 . 14). This system consists of a wedge shaped plunger that spans the width of the tank and is vertically driven by two parallel hydraulic cylinders. This wave generator is controlled by the same dedicated microcomputer that controls the pneumatic generator.
t
9
 THEORETICAL Hs 9.22 (in)  MEASURED Hs 9.69 (in)
8
k4
0.4
0.8 1.2
1.6
2.0
FREIXIENCY, Ht
FIGURE 4.12 COMPARISON BETWEEN A JONSWAP TARGET SPECTRUM WITH A MEASURED WAVE SPECTRUM GENERATED BY PNEUMATIC WAVEMAKER
Section 4.7 Design of Multidirectional Wave Generator 97
An example of a comparison between the energy density spectra of a wave produced and the Bretschneider model is shown in Fig. 4.15. The model has a significant wave height of 66 .5 mm (2. 62 in.) and spans the high frequency segment of the wave generation capability.
0.00
0.40
0.80 1.20 1.60
2.00
2.40
WAVE HEIGHT / RMS HEIGHT
FIGURE 4.13 RAYLEIGH DISTRIBUTION OF A SINGLE WAVE RECORD GENERATED BY THE PNEUMATIC WAVEMAKER 4.7 DESIGN OF MULTIDIRECTIONAL WAVE GENERATOR A "serpentine" wavemaker is a machine in which the wave board moves in a snake like movement and produces progressive waves that propagate at oblique angles to the front of the wave board . The initial designs, over 25 years ago, produced monochromatic , but oblique longcrested waves . They had a common mechanical drive for all segments of the serpentine, but an individual phase adjustment for each segment [Miles, et al. (1986)]. With the advent of faster and cheaper controls and digital
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Chapter 4 Model Testing Facility
hardware, it has been possible to provide individual articulation of each segment of the wave board, thus allowing the simulation of multidirectional ocean waves. The most common method of wave generation is with the articulated wave boards. This concept of individually controlled generator segments was first implemented in 1978 [Salter (1981]. Today, there are numerous laboratories in the world that have facilities capable of reproducing multidirectional waves.
D300U0LPSI, 12" STROKE ENDED HYDRAULIC CYLINDER (TYP.)
FIGURE 4.14 WEDGE TYPE PLUNGER WAVEMAKER Table 4.1 summarizes many of the installations of multidirectional wave facilities. As shown, the majority use the flapper (i.e., rotational) mode. Others use the plunger or the piston. Some of the machines are of the "dryback design", while the remainder are of the "wetback design". As can be seen in the table, a few are particularly suited for offshore engineering application (as opposed to coastal zone simulations) because of their water depth and wave generation capabilities.
Section 4.7 Design of Multidirectional Wave Generator
99
1.0
0.8
0.6
0.4
0.2
0.0 0.4 0.6
0.8
1.0 1.2 1.4 1.6
1.8
2.0
FREQUENCY, HZ
FIGURE 4.15 COMPARISON BETWEEN BRETSCHNEIDER SPECTRUM WITH MEASURED WAVE SPECTRUM BY THE COMBINATION GENERATOR
In a large rectangular basin, wave generators may be located on two sides with absorbers on opposite faces . Generally, one set of generators are designed in segments to produce shortcrested (directional random ) waves, while the other is used to produce longcrested waves . This permits the generation of the interaction effect of short crested locally generated storm with the longcrested swells . However, more recent laboratories (e.g., NRCIMD at St. John's, Newfoundland) use segmented machines on two sides. The design of multidirectional wave generators requires the choice of the width of the wave board segment . The smaller the segment width, the larger is the angle of obliqueness and the better is the quality of oblique waves . However, a unit having more, smaller segments has a bigger cost associated with it. On the other hand, the smaller width segments require less driving power compared to larger segments requiring more actuator power and complicated support system. Therefore , a compromise is required between the angular spreading , the power limits of commercially available actuators and the overall cost of the system.
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TABLE 4.1 BASINS WITH SEGMENTED WAVE MACHINES [Miles, et al.(1986)] Name of Basin
Mode
Segment
Stroke /2 (t)
Action
Elec Elec Elec Elec Hydr Hydr Elec H dr
Depth (in)
of
Width
Height
(in)
Se ments
(in)
(in)
27x11 30x48
1 .2 2.0
80 80
0.3 0.31
0.7
15°
50x72 30x20 59x111 variable 30x50
0 10. 3.0 0.76 1.3 0.32.9
144 60 60 80 64
0.5 0.5 0.46 0.33 0.5
1.3 1.5 0.76 1 .28 2.0
16.5° 16.7° 0.15m 0.40m 0.40m
Size
U Edinburg HRS NEL MARIN EK DHi CERC DHL NRCC
No.
Basin
Flap Flap Flap Flap Flap Piston Variable Variable
Dry/ Wet
Dry Wet Dry Dry Dry Wet Wet Wet
Regular Wave Ht. (m)
.22 .50 .40 .50 .30 .70
In the design of wave boards, the water depth and the range of required frequencies dictate the choice of generating mode. The simplicity of design and proven performance of the hinged board generally make it the preferred choice over other designs such as the plungers [Miles, et al. (1986)]. The translational and rotational modes of a flat board offer simple and reliable calculations for wave height generation. The height of the wave board and its stroke height are determined from the maximum design wave height for the facility , assuming the translation mode of the wave board for wave generation. However, this translational mode is not appropriate for the shorter period wave generation , first because of the difficulty and inaccuracy in controlling the short stroke required for these waves. Secondly, there is a tendency of waves to develop in cross mode from the unused trapped energy at the wave board. If these modes have a vertical position control to vary the immersion depth, then they work well in all depths . KRISO in Korea has opted for this solution in their wave tank in which the water depth varies from 0.5 to 4m. However , these short period waves can be generated more efficiently in a rotational mode . The intermediate waves can then be generated by a combination of translational and rotational motion (Fig. 4. 1E). A dual actuator system is necessary for these operations. A more versatile wave machine covering shallow to deep water waves can be achieved , but at an additional expense. Sometimes, a mechanical linkage is used in conjunction with a single actuator to limit cost. In this case, the generating mode has to be mechanically changed from one linkage to the other. One wave board system installed at NRC, Ottawa, Canada, is shown in Fig. 4.16. 4.7.1 Actuator and Control The actuator for the wave board can be. either an electrical or a hydraulic device. The hydraulic actuator is considered to be a better choice because it is less expensive,
ii
Section 4.8 A Multidirectional Tank 101
more reliable and has a better frequency response. The electric actuator can be economical when the force requirement per unit board length is small. The Moog actuators are most commonly used in the laboratory. In designing an actuator, the natural frequency of the board linkage and actuator system should be investigated (which should be about 5 Hz or more). The servo control for the actuator can be analog, digital or a hybrid. Most twodimensional machines use analog control for the actuators. For a threedimensional machine, however, it becomes more complicated and expensive. Hybrid controllers consist of an analog control loop which is tuned digitally. They are more expensive, but, at the same time, considerably flexible. Digital servo controllers, on the other hand, use software to control the system and, hence, have the greatest flexibility. Standard microcomputer components may be used for this system. Software may also be implemented in the control loop to help in the active wave absorption.
FIGURE 4.16 DEEPWATER MULTIDIRECTIONAL WAVE BASIN AT NRC, OTTAWA, CANADA [Miles, et al. (1986)] Details of a multimode segmented wave generator design may be found in Miles, et al. (1986). 4.8 A MULTIDIRECTIONAL TANK The multidirectional wave basin at the Hydraulic Laboratory, National Research Council of Canada, Ottawa, is equipped with a segmented wave generator capable of producing multidirectional sea states in the test basin. The basin is 50 m wide, 30 m long and 3 m deep (Fig. 4.17). The basin has a central pit 5 m in diameter
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Chapter 4 Model Testing Facility
and 13 m deep . The segmented wave generator occupies one side of the basin while the wave energy absorbers made of perforated layers of metal sheeting (see Section 4.12) are placed along the other side of the basin. SEGMENTED WAVE GENERATOR
FIGURE 4.17 PLAN VIEW OF WAVE BASIN The wave generator consists of 60 segments or wave boards driven individually by a servo controlled hydraulic system . The individual wave boards are 2 .0 m high and 0.5 m wide , driven by Moog hydraulic actuators with a maximum stroke of ±0.1 m. The displacement of the actuator is mechanically amplified by a factor of 4 through a lever arm. The boards operate in the piston or translational mode (for shallow water waves), or flapper or rotational mode, or a combination of the two (for deep water waves). The machine is also vertically relocable to accommodate different water depths. The control system for the directional wave generator is a microprocessor based digital system with a four unit modular design . Each unit is responsible for the control of 16 segments and performs several real time tasks . The drive signals are initially synthesized on a VAX workstation . The multidirectional wave generation signals are usually based on linear theory and produce a sinusoidal (serpentine or snakelike) motion along the length of the wave generator . However, other methods such as side wall reflection techniques and nonlinear wave theory are also used.
Section 4. 9 Current Generation
103
4.9 CURRENT GENERATION The generation of a uniform (as opposed to local) current in the wave tank requires a closed loop for the circulation of water. The closed loop can be provided if the tank has a false floor. For any given system, the speed of current will depend on the water depth. It is also possible to connect the two ends of the tank with piping and a pump to draw water from one end (e.g., the beach end) and pump it back to the tank at the other end (e.g., the wavemaker end). This system is most practical for shallow water, since current is inversely proportional to water depth . Both systems can be designed to reverse the current flow, permitting the generation of current with or against the wave direction. Several wave tanks are equipped with current generation capabilities . Some of these facilities are CBI, and MARINTEK (using water jets under the false floor). The technique of generation is similar, but the mechanical systems are unique . An example of current generation and the type of current profile generated is illustrated in the following section. 4.9.1 A Typical Current Generator In the CBI wave tank, inline currents are generated by a 67 KW (90 HP) Lister diesel engine which drives two 762 mm (30 in.) propellers beneath the beach . Two sets of hydraulic lines are routed from the diesel to two underwater 44 KW (60 HP) reversible hydraulic motors. Each motor is mounted at one end of a convergingdiverging nozzle located underneath the beach as shown in Fig. 4. 10. Each nozzle can be independently controlled. Inside the nozzles are 762 nun (30 in.) diameter, three bladed propellers mounted on roller bearings. These propellers are of the high thrust type. The current speed can be controlled by either the engine speed or the pressure controls. Current can be directed to flow either with or against the direction of wave travel by reversing the rotation of the hydraulic motors. A series of flow straighteners are used to restrict the deviation in the mean current velocity to less than 10%. A single unit of flow straightener consists of corrugated plastic sheets bolted together and is 1.2 m (4 ft) wide, 1.2 m (4 ft) long and 1 m (31/4 ft) high. The units are interlocked to span the width of the tank and are located upstream of the model. Figures 4. 18a and 4.18b present the current profiles for flow in both directions. The mean value and the variation (± one standard deviation) have been plotted . Figure
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Chapter 4 Model Testing Facility
4.18a presents the vertical profile for currents inline with the direction of wave propagation for maximum and halfmaximum settings. Figure 4.18b presents a similar profile for current flowing in the opposite direction. Current in this direction is slightly more uniform than the in line current because the current passes through the beach prior to impinging on the model, functioning as an additional flow straightener. Figures 4.18c and 4.18d present the horizontal velocity distribution measured at 356 mm ( 14 in.) from the bottom in a 2 .lm (7 ft) water depth for current in both
OMEANVAWE ± 1 STANDARD DEVIATION
S
b A
s
rt t
TAL YEI NOIAMN
u NOROONDL VELZTY ( FT./sm)
TY ( FT/$EC)

d T
R
psTANCE
A
CROSS
IU
( )
FIGURE 4.18 VERTICAL AND HORIZONTAL CURRENT PROFILES
Section 4 . 9 Current Generation 105
directions. Both profiles are nearly uniform across the measured range. The distance across the tank is plotted as the abscissa and is referenced to the tank centerline. 4.9.2 Local Current Generation Local currents can be generated through the use of portable electrically driven propellers. These motors can be rigidly mounted to the wave tank rail or instrumentation bridge, and the motor speed , spacing between motors and their depth can be varied to obtain the desired current velocity and profile. If the generators are mounted alongside the tank, cross current may be generated transverse to the waves (Fig. 4.19). This is often an important design consideration for a moored tanker at a site where current exists transverse to the predominant wave direction . The current generation units are extendable from the surface with long shafts in order to produce vertical velocity profiles and can actually be used to simulate shear current . The spacing between motors and distance from the test section are adjusted during calibration prior to testing. The speed of the motors is varied to allow adjustment of the current velocity once the motors are in place . Any fluctuation in the generated current may be minimized by using flow straighteners.
FIGURE 4.19 CROSS CURRENT AND WIND GENERATION SETUP IN TANK
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Chapter 4 Model Testing Facility
Local and cross currents are sometimes produced by arranging pipes close to the model with flows from an external source . This method is generally inadequate for developing a uniform current flow field around the model. 4.9.3 Shear Current Generation The shear current is a common occurrence in the ocean. Generally, the surface current velocity is higher than the bottom current. The profile with depth may be approximately represented by a linear or bilinear shear current . Near the bottom, the boundary layer effect provides a parabolic profile reducing current at the bottom to zero. In deep water in particular, the shear current may have a significant influence on a component of an offshore structure. One example of such a component is a riser, or a group of risers.
10CURRENT FLAW WAVE TANK
FLOOR SLAB
FIGURE 4.20 SHEAR CURRENT GENERATION IN TANK,
Section .4.9 Current Generation 107
In order to generate shear current of a given profile, the flow at various elevations must be controlled to provide the desired profile in the tank. One method of generation may use a flow control mechanism at the forward end of the flow straighteners. This flow control can be achieved by installing butterfly valves at the inlet to each flow straightener (Fig. 4.20). Alternately, materials of varying porosity, e.g., foam or sponge, may be inserted in the tubes (Fig. 4.20, inset) to reduce the flow by a desired amount. Thus, by restricting the flow through the rows of tubes in the flow straightener by various degrees with depth, a positive or negative (linear) shear current may be generated. A bilinear shear current or simultaneous vertical and horizontal shear current profiles can also be created using this technique. 36
30
0 0.00 0.50 1.00 1 .50 2.00 2.50 3.00 3.50 VELOCITY (ft/sec) ■
@500 psi
O @2000 psi
❑
@1000 psi
A @2500 psi
♦
@1500 psi @3000 psi
FIGURE 4.21 POSITIVE SHEAR CURRENT PROFILES INTANK
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Chapter 4 Model Testing Facility
In a test at the CBI wave tank, flow straighteners consisting of smooth 102 mm (6 in.) diameter, 1.2 m (4 ft) long cylinders were placed spanning a 1.2 m (4 ft) deep, 3 m (10 ft) wide channel . Uniform fluid flow was created downstream of the flow straighteners. Shear currents were generated by providing increasing levels of resistance to flow at prescribed elevations . This resistance was achieved by placing a mesh screen in front of the flow straighteners covering the width and depth of the tank . This mesh consisted of polyethylene cloths with a diamond shaped mesh pattern. The cloths were commercially available in 914 mm (36 in.) wide rolls in various sizes with nominal openings of 3 mm ( 1/8 in.), 5 mm (3/16 in.), 6 mm (1/4 in.), 13 mm (1/2 in.), 19 mm (3/4 in.) and 25 mm ( 1 in.). 38
30
t
i
0.00 0.50 1.00 1 . 50 200 2.50 3.00 3.50 VELOCITY (8/8ic)
FIGURE 4.22 NEGATIVE SHEAR CURRENT PROFILE (see Fig. 4.21 for symbols)
Section 4.10 Wind Simulation 109
The cloths were cut into 152 mm (6 in.) widths, and various combinations of mesh openings were tried at each elevation . The added resistance provided by this cloth and the subsequent shear pattern developed proved to be highly nonlinear , and a trial and error approach was applied. The final mesh pattern with overlaying cloth layers is shown in Table 4.2. This mesh screen was then turned upside down with most resistance on top of the 914 mm (36 in.) depth to produce a negative shear. These shear currents are shown in Figs . 4.21 and 4. 22 for various hydraulic settings of the current generator (Section 4.8.1). TABLE 4.2 MESH CONFIGURATION FOR GENERATION OF SHEAR CURRENT ELEVATION (IN.) MESH OPENINGS (IN.) 3036 1 2430 3/4 1824 1/2 3/4 1218 1/4 612 3/16 1/4 06 1/8 1/8 1/8 1/8 BACKTOBACK LAYER NO. 1 2 3 4 4.10 WIND SIMULATION The wind load may be simulated with the help of blowers strategically placed in front of the model . They are conveniently placed on the face of a traveling bridge used to house the instrumentation and controls (Figure 4.19). In this case , the superstructure of the hull requires modeling . This allows variation of the steady wind load due to the motion of the model hull and may introduce a low frequency variation in load if a wind spectrum is simulated . Instead of modeling the wind speed , the steady wind load is sometimes modeled by adjusting the speed and location of the fans. It is also common to simulate the horizontal wind load by the use of an appropriate weight hung from a pulley. A stiff line (e.g., Kevlar or steel cable) is used to suspend the weight , thus avoiding any spring effect. The pulleys should be low friction type (e.g., aircraft pulleys, use of ball bearings). The line is attached at or near the deck of the structure . Special care should be taken to ensure that the system does not add any significant amount of damping in the floating system in surge and pitch . This may be verified by studying and comparing the free damped oscillation of the system with and without the wind load applied to the system. While the wind loads on the structure may be quite important for a particular design, e.g., the floating moored structure , useful modeling of these loads is limited by
110
Chapter 4 Model Testing Facility
the scaling problems associated with them. If the wind load alone is important , e.g., on the superstructure of an offshore platform, Reynolds number is more important and wind tunnel tests are performed on that portion of the structure model. In these cases, it is easier to achieve Reynolds number similarity. 4.11 INSTRUMENTED TOWING STAFF For performing towing resistance tests, the wave tank's carriage is generally fitted with an instrumented staff. The staff is arranged so that any combination of roll, pitch and heave motion of a floating model may be accomodated during a test. Instrumentation is installed to measure the loads on the model in the downtank , crosstank and vertical directions. Moments about any of these axes can also be measured . The staff is designed so that drag forces on the model can be counterbalanced by a weight on an adjustable arm. When the drag force is counterbalanced in this way, a more sensitive load cell can be used to measure the downtank force, and more accurate measurements of the effects of drag reducing modifications to a model can be made . A sketch of the towing staff is shown in Fig. 4.23.
TRANSDUCERS TO MEASURE HEANE AND SWAY FORCES, AND PITCH , ROLL AND YAW MOMENTS
FIGURE 4.23 INSTRUMENTED TOWING STAFF
Section 4.12 Planar Motion Mechanism
111
4.12 PLANAR MOTION MECHANISM A planar motion mechanism (PMM) is used for forced oscillations of moderately heavy models . There are several variations in the design of a PMM . Some are quite complicated. In a simple design, the mechanism may be driven by two parallel hydraulic cylinders (Fig. 4 .24) that can be controlled to produce pure translation, pure rotation or combined translation and rotation of the attached model . By varying its orientation in the setup , the translational axis of the mechanism can be aligned with the surge, sway, or heave axis of the model . Rotation of the model can take place about any axis that is normal to the mechanism 's translational axis . The maximum displacement and velocity requirements of either hydraulic cylinder are prescribed in their design.
POSITION POSITION REFERENCE I REFERENCE
I
2
SERVO CONTROLLER FEEDBACK 2:g
ROTATION
MODEL
TRANSLATION
FIGURE 4.24
PLANAR MOTION MECHANISM
112
Chapter 4 Model Testing Facility
The extension of each cylinder (measured by a position feedback transducer) is controlled by a servo system and is driven to match a computer generated position reference signal. If the planar motion mechanism is driven by servocontrolled hydraulic cylinders rather than cams or some other form of eccentric drive, the motions produced are not limited to regular sinusoidal motions, but can be random and may contain any number of frequency components in a prescribed range. Software is written to tailor the frequency distribution of the drive signal to match any desired spectrum model including white noise, if desired. The drive signal may be generated from a stored time series (similar to wave generation), and the motions can be duplicated for different test configurations. The planar motion mechanism can be used to determine the stability and control characteristics and the hydrodynamic added mass and damping coefficients in all six degrees of freedom for wave tank models such as surface ships or submerged structures. 4.12.1 Single Axis Oscillator A single axis oscillator is a simple PMM used to force models in a single direction in the wave tank (see Fig . 4.25). The oscillator is driven by a servo controlled hydraulic cylinder and is capable of generating velocities and displacements up to the design value of the mechanism. The control system of the oscillator is identical to that used in the planar motion mechanism. The position reference signal used to control the extension of the cylinder is generated by a microcomputer from a stored digital time series. The system can generate regular sinusoidal motions or random motions containing frequency components in its design range. Figures 4.25 and 4.26 illustrate two of the tests that have been performed using the single axis oscillator . Figure 4.25 shows a test of a resonant chamber being studied as a heave damping addition to a deepwater caisson structure. The chamber was driven vertically in still water to confirm that it was properly tuned so that the water inside the chamber oscillated 180° out of phase with the motion of the chamber. The load cell at the top of the chamber was used to measure the vertical load generated by the oscillating water column . A capacitance wave staff fixed to the cylinder was used to determine the free surface elevation of the water in the chamber. The second test, shown in Fig. 4.26, used the oscillator to force horizontal motion at the top of a model riser. The riser was one component of a deepwater mooring system. The riser was instrumented with strain gauges at three elevations to measure the bending stresses induced by the forced motion. Single period sinusoidal motions and random multifrequency motions were produced.
Section 4. 13 Laboratory Wave Absorbing Beaches 113
FIGURE 4.25 FORCED OSCILLATION OF A HEAVE DAMPING SYSTEM 4.13 LABORATORY WAVE ABSORBING BEACHES It is important to simulate open ocean conditions in a laboratory environment as closely as possible. One of the factors that contaminate the generated wave at the test site is the reflection of the incident waves from the end walls of the wave basin. The coastal and offshore structure models are placed near the center of the basin. Even when the basin is long, the model experiences reflection from the end wall during the course of a test run of reasonable duration . In order to dissipate the wave energy and minimize the problem of wave reflection , wave absorbers are generally installed at the
114 Chapter .4 Model Testing Facility
POSITION REFERENCE SERVO CONTROLLER
CONTROL
POSITION FEEDBACK CYLINDER EXTENSION TRANSDUCER TRANSLATION DIRECTION
SERVO VALVE
FIGURE 4.26 FORCED OSCILLATION TEST OF A DEEP WATER RISER end opposite to wave generation . The amount of wave reflection is quantified as a wave reflection coefficient (Cr) which is defined as the ratio of reflected wave height to the incident wave height (expressed as a percentage). For a solid vertical wall, the reflection coefficient is 100 percent, i.e., the magnitude of the reflected wave is equal to that of the wave incident upon it. For an efficient beach , the reflection coefficient should be consistently less than 10 percent and preferably less than 5 percent [Jamieson and
Section 4 . 13 Laboratory Wave Absorbing Beaches
115
Mansard ( 1987)] over the range of wave heights and periods that the basin is capable of producing. Since the wave absorber takes up valuable space from the wave basin, the design of an efficient wave absorber is always a challenge . If there is only limited space available for a wave absorber, efficient wave absorption may be very difficult to achieve. The most commonly used wave absorbers are beaches of constant slope which extend near the bottom of the basin. It is constructed of concrete, sand, gravel or stones. The slope of these beaches must be mild for efficient wave energy absorption. Typical slopes are from 1:6 to 1:10 [Ouellet and Datta (1986)]. A beach with variable slope may reduce the total required length of the beach. A parabolic slope is sometimes used along with surface roughness and porous materials . The position of the parabolic beach can be made adjustable with the water depth in order to maintain its efficiency. Geometry of a variety of existing beaches in testing facilities is shown in Fig. 4.27. Another concept for laboratory beaches is a progressive wave absorber. The concept of a progressive wave absorber [LeMehaute (1972)] consists of material whose porosity decreases towards the rear of the wave absorber. LeMehaute used aluminum shavings to achieve this. The shavings were more compacted (less porous) away from the wavemaker. To achieve the same efficiency as a beach with uniform porosity, the length of the beach required was reduced using variable porosity. The same principle was adopted by Jamieson and Mansard (1987) with perforated upright wave absorber. The upright wave absorber consisted of multiple rows of perforated vertical metal sheets . The porosity of the sheets decreased towards the rear of the absorber. This provided an efficient wave absorber over a relatively short length with about a 5 percent reflection coefficient. This absorber will be discussed in further detail in Section 4.12.2. In prototype situations, an upright caisson breakwater having a perforated wall in front of an impervious back wall [Jarlan (1961 )] is occasionally used to reduce the high reflections associated with a solid wall breakwater. 4.13.1 Background on Artificial Beaches The use of permeable material (e.g., crushed rocks, cast concrete products, wire mesh, etc.) in the sloping beaches is well known in both the field and the laboratory. If sufficient space is available, a long absorber with a low surface slope can result in very efficient wave reflections ( such as the twodimensional tank at the Oregon State University, Corvallis). However, limited space sometimes necessitates the use of an absorber with minimum length and steeper slopes. Laboratory tests have been
116 Chapter 4 Model Testing Facility
0 WATER LEVEL PERFORATED
15'
PANELS 3
FIGURE 4.27a BEACH GEOMETRIES performed to determine the optimum porosity, steepness and length of the absorbers [Beach Erosion Board (1949), Straub, et al. (1956)]. It was concluded that a material with a high porosity is desirable if minimum length is needed. The best absorption was obtained with a porosity of 60 to 80 percent.
Section 4. 13 Laboratory Wave Absorbing Beaches
117
WATC 2
PCZFO¢ATGO PANEL
A•OM
METAL GRID WATER LEVEL 0
8.4 m
I
BEACH INSTALLATION
8.84m
FIGURE 4.27b BEACH GEOMETRIES Even though mention was made earlier about perforated plates being used as a beach [Straub, et al. (1956)], Jarlan (1960) originally suggested the vertical porous wall breakwater, and presented experimental data on one. Jarlan ( 1961) examined the effects of mass transfer in waves through a perforated vertical wall. The theoretical development followed by Jarlan was based on acoustic theory. This theory demonstrated a simple relationship between the reflection coefficient and porosity. 4.13.2 Progressive Wave Absorbers Progressive wave absorbers are upright wave absorbers for use in laboratory wave tanks. They require limited space compared to conventional beaches. They are efficient in variable water depths and over a wide range of wave conditions. The wave
118
Chapter 4 Model Testing Facility
efficient in variable water depths and over a wide range of wave conditions. The wave absorber is constructed of multiple rows of vertical sheets of varying porosity and spacing between sheets. An open tubular framework supports the sheets and allows some variations in the overall geometry depending upon the test condition. They can also be used as the tank side absorbers (locally, if needed) to absorb reflected waves from large 3D structures [Jamieson, et al. (1989), Clauss, et al. (1992)]. They are equally applicable in multi directional basin. These absorbers have been successfully used in the NRC, Canada multidirectional test basin . They have been tested in laboratory waves at NRC. The reflection coefficients in both regular (Fig. 4 .28) and irregular waves have been limited between 3 and 7 percent over a wide frequency range. These absorbers have also been placed at the Texas A&M, College Station wave basin. 4.13.3 Active Wave Absorbers If a wave basin with a wave generator at one end of the tank and a wave absorber at the other end is considered an "open loop" system, then there is also a closedloop system available. In this case , a conventional active wave generator may be used both as a generator and an absorber of wave energy . A closedloop control system is a device that will generate only the wave train required, while at the same time absorbing the unwanted reflection components returning to the wave generator after reflection from the beach . This mode of operation requires a sophisticated control system and actuating mechanism . Thus, it is feasible to generate a train of progressive waves without rereflection at the generator of waves reflected by the structure being tested in the basin. 4.13.4 Corrected Wave Incidence A correction to the measured wave at a point in the tank due to beach reflection can be made by using a simple reflection theory developed by Ursell, et al. (1960). The reflection effects modify the generated wave in a tank of finite length in the following way. Even if a beach exists at the far end, the reflection coefficient is not zero. Successive reflections occur between the beach and the wave generator . There is a time lag in the reflections to take place , but each reflection becomes an order of magnitude smaller, and a steady state is eventually reached . The initially generated wave train, called the primary incident wave , is partially reflected from the opposite end to form a primary reflected wave component which is then reflected from the generator surface to form a secondary reflected wave, and so on . The primary reflected wave is most significant, and the higher order teens in subsequent reflections may usually be considered insignificant . This is particularly valid if the wave generator has active absorption capability.
Section 4. 13 Laboratory Wave Absorbing Beaches
h  1.8m T 1.746
L  4.63m
119
h/L  0.389
Wave Absorber No. 81W f  4.8m m  24 sheets !/L  1.037 s  20cm n  5,5,5,10,10,10, 15,15,15 ,20.20,20,25,25,25,30,30, 30,40,40,40,50,50,50% Location and porosity of perforated sheets Envelope of water surface motion 5 5 5X10 10 10 15,15'1.5 20 20 20 25 2 25,30 30 30 ♦ 0,40 40 50 50 50%
^_.
c
z
bz
'
=
SWL 96
1
i Antinode Node 0 (L/4)
1 Water
1
Antinode (L/2)
Node (3L/4)
L  4.63m
r
particle
t motion Antinode IL) 4
(a) Location of Perforated Sheets with Respect to a Pure Standing Wave. Cr  Hr/ Hi  1.0, in Front of an Impervious Vertical Wall (Linear Theory) 10 r a U
❑
❑ ❑
❑
(c) ❑
6 ev• F
5 U
I
10
T
b)^ (
I
I
0 0 •
All sheets in place
(d)
Ti.)
20
20
M15
15
610
610
d l
I
All sheets in place 5% sheets removed
25
25
T
F
5
5 0
I
I
1
I
I
1
I
I
0 L 0.0 6 Hi/L 0.06 Hi/L 0 0.02 0.04 0.04 LJ L L I L1 Lr_I 30 Hi(cm! I' 30 Hi(cm) 0 10 20 10 20 ❑❑ All sheets in place °° 5%.10% and 15 % sheets removed * 5%,10%,15% and 20% sheets •• 5% and 10% sheets removed removed °° 5%.10% and 15% sheets removed 0 L 0
0.02
FIGURE 4.28 PERFORATED SHEET LOCATION AND CORRESPONDING REFLECTION COEFFICIENT [Jamieson and Mansard (1988)]
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Chapter 4 Model Testing Facility
The primary incident wave and its partial reflection from the wave absorber will create a partial standing wave system . Ursell, et al. (1960) showed that under this condition the measured wave amplitude at a point x in the tank is given by a2 = a`2 {1+2e cose + 2e cos(2kx +e) + 2e2 cos2kx +2e2 }
(4.43)
where ai = primary incident wave amplitude , k = wave number, e and unknown parameters due to reflection . Considering e to be small, a = a.1l+e cos(2kx +e) + e cose +O(e2
)l
(4.44)
Thus, 'a' represents a spatially slowlyvarying wave amplitude with peaks and valleys. Moreover, the envelope is a sinusoid having amplitudes between amax and amin where amax =ai ( l+e+e cose )
(4.45)
amin = ai (1 e + e cose)
(4.46)
Writing Cr = reflection coefficient, it may be computed from C=a_a;, a,,,,, +a^;,
(4.47)
Substituting the values of amax and amin, C'
_ e l+ecose
(4.48
)
If the measured wave amplitude along the spatial coordinate x is recorded by a traversing wave gauge such that its maximum and minimum values are found, then the reflection coefficient, Cr may be computed . Moreover, at these locations, 2kx + c = 0 or It, respectively, so that a may be evaluated. Then from Eq . 4.48, c is determined and hence ai. 4.14 REFLECTION OF REGULAR WAVES The reflection of waves from the beach of a laboratory wave tank is an important parameter to consider in a model test. Moreover , the reflection parameter is often desired of a model being tested in a tank, particularly if the model is a twodimensional structure, e.g., a breakwater.
Section 4. 14 Reflection of Regular Waves
121
There are several methods available to compute the reflection coefficient. As stated earlier, the reflection coefficient is defined as the ratio of the amplitude of the reflected wave to the corresponding amplitude of the incident wave. Note that for a regular wave, the frequency of the reflected wave is the same as that of the incident wave. As shown earlier, the reflected wave creates a standing wave system in the tank having a node and an antinode . Therefore, the reflection coefficient may be computed from the wave heights measured by a single traversing wave probe. A second approach may use two fixed wave probes instead of a traveling wave probe. In this case, the two probes are located at distances of L/4 and 112 from the reflecting structure , where L is the wave length . This assumes that the waves are linear and the points of node and antinode as well as the wave length are known . However, in many instances, such as perforated or sloped structures (beaches), the phase of the reflected wave is not known apriori. In general, there are three unknown parameters due to the reflection of a regular wave: the incident wave height, the reflected wave height and the phase difference between the two. Assuming ai (= H/2) and ar to be the amplitudes of the incident and reflected waves (Cr = ar/ai) and a to be the phase difference , the free surface elevation, Tin, for a wave probe at a location given by xn is T1, = aicos(kx, ox)+ a, cos(kx, cut+e), n =1,2,... (4.49)
in which k and w are the wave number and wave frequency given by the linear wave theory for a given wave period, T, and a water depth, d. Assuming the location of the first probe to be xI and writing the distance between the nth probe and first probe to be Sn, we have X. = x, + S„
(4.50)
kx, = kx, +A,
(4.51)
or
where An = kSn. Then, the profile for the nth probe is given by % = at cos(kx, +A, (ut)+ a, cos(kx, +A„ +(x e) In complex notation,
(4.52)
122 Chapter 4 Model Testing Facility
ri„ _ Jai exp (flaco )+ a,exp[i(kx, e)]} exp(i(ot) (4.53) _ {aiexp[i(kx, +A„)] +a, exp[ i(kx, +A„  e)]}exp(i(ut) The recorded elevation at the nth probe may be written in teens of an amplitude, An, and a Phase, Sn, relative to the first wave probe record (i.e., 811' = 0) and is given as (4.54)
T6(') = A. cos(oxe , S.('))
where e I is the phase angle corresponding to the first wave probe and the superscript in refers. to the measured quantities. An equivalent expression may be written in the complex domain as in Eq. 4.53. 4.14.1 Two Fired Probes Using the expressions for the assumed and actual elevations in Eqs. 4.53 and 4.54 and using the complex equivalent form , the following set of equations may be written [Goda and Suzuki (1976), Isaacson (1991] aj exp(ikx„) +a, exp[i(kx„ e)] = A. exp[i(e , +S,)], n = 1,2
(4.54)
The incident and reflected wave amplitudes are derived from this set of equations.
ai =
a,
)r
(4 . 55)
[4+A;2AA 2 co4AZ 5 , )r
(4.56)
1 [4+A zz2A, A2co s(A Z +8 2lsin A2 1
= 2Isin A z
The phase difference between the two is written in terms of ? = 2kxl  e, A2 a?a2 cos?, = ' 2a1ar
(4.57)
Section 4. 14 Reflection of Regular Waves
123
From Eqs. 4.55 and 4. 56 the method fails for A2 = 0, it, 2n.... etc. which corresponds to a probe spacing of multiples of half wave lengths. Therefore, .2 should be outside the range of 0.4L < X2 < 0.6L [Goda and Suzuki (1976)]. 4.14.2 Three Fixed Probes The derivation of the incident and reflected wave is based on the waves being represented by the sine function (linear theory). Any nonlinearity in the wave will produce error in the measured amplitudes and phases for the two probe arrangement. This error may be reduced if additional probes are used so that a least square estimation is possible. If a three probe arrangement [Mansard and Funke ( 1980)] is used, then five measured quantities, namely, three wave amplitudes (An, n = 1,2,3) and two phase angles (82 and 83) may be used to derive the three unknowns. For convenience, let us write Eqs. 4.53 and 4.54 as follows: TI. = [bi exp(iA„)+b, exp(iA„)]exp(i(ot)
(4.58)
I1.() =B exp (iwt) n=1,2,3
(4.59)
where bi = a,exp(ikx,)
(4.60)
b, = a, exp[i(kx, e)] B. = A. exp[i(e,
(4.61) +S„)]
(4.62)
In order to minimize the error in the estimate in the least square sense, the error term is written as E2 = I [bi exp(iA„)+b, exp(iA„) _B.]2
The quantity E2 is minimized in the usual leastsquares sense giving following two equations in the unknown complex quantities bi and br. exp(iA„
)[b, exp(iA„) +b, exp(iA.) B„ =0
3
1 exp( iA„ 1
)[b; exp(iA„) +b, exp( iA„)  B. } = 0
(4.63) rise
to the
(4.64)
(4.65)
124
Chapter 4 Model Testing Facility
Once bi and br are known, ai, ar and X may be computed. ai = IXii ar = IXrI X = Arg(Xi)  Arg(Xr)
(4.66) (4.67) (4.68)
S2S3  3S4 X, = S 5
(4.69)
= S,S4  3S3 X, S
(4.70)
where
3
and 3
S, = Yexp(i2A„)
(4.71)
3
S2 = Yexp(i2A„) 1
(4.72)
A. exp[i(S„ +A„)]
(4.73)
S4=IA. exp[i(8„A.)]
(4.74)
S5=S1S29
(4.75)
S3 = 3
The method will fail when S5 = 0. For equal probe spacing A, this occurs when A = 0, it, 2n.... For unequal probe spacing such that µ = A2/A3, the condition S5 = 0 corresponds to sine A3 + sin2 (ILA3)+ Sin2(A3  PA3) = 0
(4.76)
which gives Sin A3= sin (AA3) =0
(4.77)
Section 4. 14 Reflection of Regular Waves
125
This occurs where A3 = 0, it, 2n, ... and A2 = 0, it, 2n.... Since in a three probe arrangement the number of measurements available is more than the number of unknowns , one may use fewer measurements to solve for these unknowns. Since only three unknowns are involved , the three wave heights at the three probes are sufficient [Isaacson ( 1991 )]. This eliminates the phase angles in the computation. In terms of earlier notation using Eq . 4.55, the measured wave amplitudes are related to the unknowns as A„2 = a? +a; +2aia , cos(? + 2A„ ),n =1, 2,3
(4.78)
where A. = 2xl  e and Al = 0 as before. Solving for ? from Eq. 4.78 cosX=fi
(4.79)
(2A„) f„ sink= f1cos sin(2A„ )
(4.80)
where
f=
2 
a; 2  aT 2
2a1a,
(4.81)
Eliminating A. from Eqs. 4.79 and 4.80 and after some algerbraic manipulation, a^ +a2 = A
(4.82)
2aia, = I'
(4.83)
where A
A, sin[2(A3A2)]A;sin(2A3)+ A;sin(2A2) sin[2(A3 A2)]+sin(2A2)sin(2A3)
(4.84)
2A 2 Az A vz 1 A,z+A32 ] +L sinA3 ] } (4.85) I 2 [ 1COSA3
126
Chapter 4 Model Testing Facility
The solution for ai and ar in terms of A and IF is
ai = 2( A+t+ A1')
(4.86)
a,2( A+I' AI')
(4.87)
Once ai and ar are known, the incident wave height, the reflection coefficient and the phase angle for the reflected wave may be computed. This method fails when the denominator in Eq. 4.84 becomes zero. Then, in terms of A3 sin(2µA3 ) sin(2A3 )+ sin(2A3 2µA3) = 0
(4.88)
which gives A3 = nn or nn/µ or nn/(1  µ) for an integer n [Isaacson (1991)]. For example, when µ = 0.4 or 0.6, the method fails when A3 = 0, it, 5n/3, 2n, 5n/2.... Moreover, in certain areas, the solution becomes imaginary. For example, when k is near unity and the height measurement is not accurate, the method may fail. As can be expected, the leastsquares method which uses the maximum number of measured variables in the computation is the most accurate. The recommended spacing for this method is µ = 0.45 or 0.65 [Isaacson (1991)]. 4.15 REFLECTION OF IRREGULAR WAVES Similar to regular waves, it is important to know the reflection characteristics of a random wave in a wave tank [Mansard and Funke (1987)]. A simplified approach is to obtain the reflection characteristics of the wave tank over the range of its frequency generation by studying regular waves. Waves generated by the wavemaker get reflected back and forth between the beach and the wavemaker. Therefore, the wave system when steady state is reached is a superposition of positive and negative wave trains of the same frequency. For a wave of amplitude a and frequency w, the wave profile of the multiwave reflection system [Goda and Suzuki (1976)] is given by the infinite series rl = a{cos(kx wt ) +C, cos(kx 2kl +wt)+CRC, cos(kx+2kl wt)+ CRC,Z cos(kx 4kl +wt)+CRZC,Z cos( kx+4kl (ot)+...}
(4.89)
Section 4 . 15 Reflection of Irregular Waves
127
where 1= length of the wave tank, and Cr and CR = reflection coefficients of the beach and wavemaker respectively . This reduces to a closed form as a
r1=
[12C,CR cos2kl
+C,2CR21
1/2 {co kxCote)
+C, cos(kx 2kl +wt+e)}
(4.90)
and e=tan
C,CRsin 2kl 1C,CR cos2kl
(4.91)
Thus, the two wave trains propagate in the opposite directions due to multireflection. In order to solve for the incident and reflected wave amplitudes , ai and ar respectively, wave probes are placed at a known distance , S, apart. The observed profiles of composite waves for a given frequency at these stations (xl and x2) are given by fl, =(r), +rl,)_, = A, coscut+B, sin wt
(4.92)
T12 =(rf, +rf,)X2 = A2 coscot +B2 sin cot
(4.93)
Al =a, cos4i, +a, cosh,
(4.94)
B1 = a; sin di,  a, sin cb,
(4.95)
A2 = a, cos(k8+4D) +a, cos(kS+g,)
(4.96)
B2 = a, sin (kS+i) a, cos(k8+0, )
(4.97)
(D, = kx, +e,
(4.98)
(D, = kx1 +Er
(4.99)
where
128
Chapter 4 Model Testing Facility
and ei and er are the phase angles associated with the incident and reflected waves. From the above equations, the amplitudes of the incident and reflected waves may be estimated. in , coskSB, sin kS)2 + (B2 +A, sin kSB, cos k8)2 (4.99) 1 kSl (AZ A a` 2lsin
in A 2  A, cos kS + B, sin kS)2 + ( B2  A, sinks  B,coskS)2 } (4.100) ar 2 1 6 1 {((A2
For the irregular waves, the principle of superposition is applied. The maximum number of components that can be analyzed is limited by the Nyquist frequency (i.e. onehalf the number of data sampling). Fast Fourier Transform technique is used to compute the quantities Al thru B2 for all component waves. The spectra of incident and reflected waves are obtained by smoothing the periodograms based on the estimates of ai and ar. An example of reflection coefficient due to a submerged, upright breakwater is taken from Goda and Suzuki.(1976). The crest of the breakwater was 60 mm below the mean water level (MWL) in a water depth of 410 mm. The wave gauges were located 4.8 and 5.0 in from the breakwater face. The significant wave height of the irregular wave was Hs = 82 mm with a period Ts = 1.44 sec. The duration of the wave record was 68.3 sec at a sampling rate of 1/15 sec. The results are shown in Fig. 4.29. The spectra for the incident and reflected waves show the divergence of spectral density near f = 0 and f = 1.97 Hz as may be expected. Therefore, the resolution is effective within a certain range of wave frequency, as demonstrated below. fmin : S/Lmax = 0.05
(4.101)
fmax : S/Linin = 0.45
(4.102)
where Lmin and Lmax are wave lengths corresponding to frnin and fmax. 4.16 LD41TED TANK WIDTH Wave tank model tests may exhibit large experimental scatter due to wall interaction effects. Therefore, in model testing, it is important to know the effect of the tank walls on the measurements. For a long narrow tank, the side walls may have an influence on the measured forces on a fixed structure placed on the center line of the tank. The interaction becomes significant if the structure transverse dimension becomes
Section 4. 16 Limited Tank Width
0.01 [1 J1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1
129
11111
0 0.5 1.0 1.5 2.0 2.5 Frequency f, Hz
FIGURE 4.29 REFLECTION FOR IRREGULAR WAVE ON A SUBMERGED, UPRIGHT BREAKWATER [coda and Suzuki (1976)] the same order of magnitude as the width of the tank . The common rule of thumb used in reducing this effect on the measurement of the inline force is to use a model such that the ratio of tank width to the model transverse dimension is at least 5 : 1. For example, for a vertical cylinder, the walls should be at least 2.5 diameters from the center lines of the cylinder. In towing tank hullresistance tests, the ship bow waves should not reflect from the tank walls back onto the model . This rule limits the possible model length for a given tank width. The interference effect of the tank wall for a ship model towed in a tank is given [Goodrich (1969)] in Fig. 4.30. According to this figure, for a given tank breadth to model length ratio (B/f ), any combination of the wave length/model length ratio and Fronde number that lies above a horizontal line through minimum 0 value will cause a
130
Chapter 4 Model Testing Facility
wall effect. The figure, however, does not give any indication of the degree of wall effect. In the experiments with floating vessels in a wave tank , the motion, in particular heave motion, produces radiated wave which will invariably radiate from the tank walls. In general, the radiated waves decrease in magnitude with distance from the model. This factor determines the optimum size of a model that may be tested in a given tank without significant adverse effect. Numerical computations have been carried out by Calisal and Sabuncu ( 1989) with floating vertical cylinders to determine the effect of the tank wall on the hydrodynamic coefficients . The numerical results show that the added mass coefficients exhibit peak values at resonant frequencies . These frequencies correspond to frequencies of waves with a wave length equal to the tank width and increase with the decrease in tank width . Several analytical and experimental data on a variety of submerged objects [Vasques and Williams (1992)], [Yeung and Sphaier (1989)] have been presented to quantify this effect. The forced oscillation test of a floating cylinder in still water will also experience similar interaction from the radiated waves reflected back from the tank walls. 4.17 TESTING FACILITIES IN THE WORLD There are numerous testing facilities that exist in various countries in the world. Many of these are used in the commercial and defense related testing of marine structures. A few of these are summarized here. Note that the larger facilities used for contract activities are mainly included. Many of the smaller research facilities that exist in universities have not been included here . One may refer to the International Towing Tank Conference (ITTC) manual for their description.
4.17.1 Institute of Marine Dynamics Towing Tank, St. John 's, Newfoundland, Canada Deep Water Wave Tank Tank Size: 200 in Long, 12 in Wide, 7 in Deep Carriage Speed : 10 m/s Waves: Regular and Irregular, l in Wavemaker: The wavemaker is of dualflap, dry back construction. The lower board operates at up to 0.1 Hz, while the upper board can operate to 1.8 Hz frequency . Each wave board is powered by a single actuator in the center of the
Section 4.17 Testing Facilities in the World
131
board . Two additional actuators on either side of the center provide hydrostatic support and compensation for the wave load . The sides are sealed by pneumatically pressurized sealing gaskets. Beach: The beach consists of corrugated plates bolted to a rigid framework. The surface has a large circular profile of 30 m radius and extends 20 m into the tank at the toe . Transverse wooden slats are placed near the water surface . A portion of the beach can be lowered for access to the dock area. 4.17.2 Offshore Model Basin, Escondido, California Tank Size: 90 m Long, 14.6 m Wide, 4.6 m Deep
2
I I I I I I I 1.0 Tank braadM B 2.0 3.0 L I Model length ` O I I I 0 0.1 0.2 0.3 L 0.4 0.5 016 0.7 0.06 0. 10 0.15 0.20 Fronde number , F, ed  V Length ratio 171
FIGURE 4.30 TANK WALL INTERFERENCE IN SIIIP RESISTANCE TESTS [Goodrich (1969)]
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Chapter 4 Model Testing Facility
Deep Section: Circular Pit 9 m Deep Carriage Speed: 6 m/s (20 ft/sec.) Transverse Rotating Subcarriage Waves: Regular and Irregular ; 0.74 m (29 in.) Wavemaker: The wavemaker is a servocontrolled , hydraulically driven, bottom pivoted, single flap wave board . Waves are generated by a prerecorded tape created on a minicomputer. Beach: Metal Shavings
4.17.3 Offshore Technology Research Center, Texas A&M, College Station, Texas Tank Size: 45.7 m (150 ft) Long x 30.5 m (100 ft) Wide x 5.8 m (19 ft) Deep Deep Section: 16.7 m (55 ft) Deep Pit With Adjustable Floor Wavemaker: Programmable, Hydraulically Driven Hinged Flap Waves: Regular and Irregular Seas Maximum Wave Height: 0.8m (34 in.) Frequency Range: 0.54.0 sec Beach: Progressive Expanded Metal Panels 4.17.4 David Taylor Research Center, Bethesda, Maryland Maneuvering and Seakeeping Facilities (MASK) Tank Size: 79.3 m (260 ft) Long x 73.2 m (240 ft) Wide x 6.1 m (20 ft) Deep Wavemakers: 8 Pneumatic Wavemakers on One Side 13 Pneumatic Wavemakers on adjacent side Waves: Regular and Irregular Waves MultiDirectional Waves Programmed on Magnetic Tape
Section .4.17 Testing Facilities in the World
Maximum Height 0.6 in (2 ft) Wave Length 0.9  12.2 in (3  40 ft) Beaches: Concrete Wave Absorbers With Fixed Bars Carriage Speed: 7.7 m/s (15 knots) Deep Water Basin Tank Size: 846 in (2,775 ft) Long x 15.5 in (51 ft) Wide x 6.7 in (22 ft) Deep Waves: Maximum Height 0.6 in (2 ft) Wave Length 1.5  12.2 in (5  40 ft) Carriage Speed: 10.2 m/s (20 knots) High Speed Basin Tank Size: 905 in (2,968 ft) Long x 6.4 in (21 ft) Wide x 3 in (10 ft) (for 1/3 a ), 4.9 in (16 ft) (for 2/3 P) Deep Waves: Maximum Height 0.6 in (2 ft) Wave Length 0.9  12.2 in (3  40 ft) Carriage Speed: 35.8  51.2 m/s (70  100 knots) 4.17.5 Maritime Research Institute, Netherlands (MARIN) Seakeeping Basin Tank Size: 100 in Long x 24.5 in Wide x 2.5 in Deep Deep Section: Pit Depth 6 in Waves: Regular and Irregular Waves 0.3 in significant; 0.73 sec Carriage Speed: 4.5 m/s Wave and Current Basin Tank Size: 60 in Long x 40 in Wide x 1.2 in Deep
133
134 Chapter .4 Model Testing Facility
Deep Section: Pit Depth 3 m Waves: Regular and Irregular Waves Carriage Speed: 3 m/s Current Speed: 0.1  0.6 m/s Deep Water Towing Tank Tank Size: 252 m Long x 10.5 m Wide x 5.5 m Deep Carriage Speed: 9 m/s High Speed Towing Tank Tank Size: 220 m Long x 4 m Wide x 4 m Deep Wavemaker: Hydraulic flaptype Waves: Regular and Irregular Wave Maximum height: 0.4 m sig. Period range: 0.3  5 sec. Carriage: Manned, motor driven; Unmanned, jetdriven Carriage Speed: 15 m/s and 30 m/s Beach : Lattice on circular arc plates 4.17.6 Danish Maritime Institute, Lyngby, Denmark Tank Size: 240 m Long x 12 m Wide x 5.5 m Deep Wavemaker: Numerically Controlled Hydraulic Double Flap Wavemaker Waves: Regular and Irregular Waves Maximum Height: 0.9 m Period Range: 0.5  7 sec Carriage Speed: 0  11 m/s(accuracy ± 0.2 percent)
Section 4.18 References
135
4.17.7 Danish Hydraulic Institute, Horsholm, Denmark [Rage and Sand (1984)] Tank Size: 30 m Long x 20 m Wide x 3 m Deep Deep Section: 12m Deep in Center Wavemaker: 60 Flap Type Hydraulically Driven Wavemakers on One Side Controlled by a Minicomputer Waves: Maximum Height  0.6 m Period Range  0.5  4 sec 4.17.8 Norwegian Hydrodynamic Laboratory, Trondheim, Norway (MARINTEK) The Ocean Basin [Eggestad (1981)] Tank Size: 80 m long x 50 m wide x 10 m deep Wavemaker: Hinged double flap, 144 individually controlled; Hydraulically driven hinged type Waves: Regular and Irregular, 0.9 m Current: Max speed 0.2m/s
4.18 REFERENCES 1. Aage, C. and Sand, S.E., "Design and Construction of the DM 3D Wave Basin ", Symposium on Description and Modelling of Directional Seas, Copenhagen, Denmark, 1984. 2. Beach Erosion Board , "Reflection of Solitary Waves", BEB Technical Report No. 11, 1949. 3. Biesel, F. and Suquet, F., "Les Appareils Generateurs de Houle en Laboratoire", La Houille Blanche , Nos. 2, 4 and 5, 1953 St. A. Falls, No. 6, 1953. 4. Biesel, F., et al., "Laboratory Wave Generating Apparatus", St. Anthony Falls Hydraulic Laboratory Report No. 39, Minneapolis , MN., March, 1954.
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Chapter 4 Model Testing Facility
5. Biesel, F., "Wave Machines ", Proceedings of First Conference on Ships and Waves, Stevens Institute of Technology, Hoboken, New Jersey, 1954. 6. Bullock, G.N. and Murton , G.J., "Performance of a Wedge Type Absorbing Wave Maker" , Journal of Waterway , Port, Coastal and Ocean Engineering, Vol. 115, No. 1, Jan., 1989, pp. 117. 7. Calisal, S.M. and Sabuncu, T., "A Study of a Heaving Vertical Cylinder in a Towing Tank", Journal of Ship Research, Vol. 33, No. 2, June 1989, pp. 107114. 8. Clauss, D., Riekert, T. and Chen, Y., "Improvements of Seakeeping Model Tests by Using the Wave Packet Technique and Side Wall Wave Absorber", Proceedings on Eleventh International Offshore Mechanics and Arctic Engineering Symposium, ASME, Calgary, Canada, 1992. 9. Eggestad , I., "NHL Ocean Laboratory: Engineering and Construction of Building and SubSystems ", Symposium on Hydrodynamics in Ocean Engineering, Norwegian Hydraulics Laboratory , Trondheim, Norway, 1981. 10. Galvin, C.J., "Heights of Waves Generated by a Flap Type Wave Generator", CERC Bulletin, Corps of Engineers , Department of Army, Vol. II, 1966, pp. 5459. 11. Gilbert, G., Thompson, D.M. and Brewer , A.J., "Design Curves for Regular and Random Wave Generators ", Journal of Hydraulic Research , ASCE, Vol. 9, No. 2, 1971. 12. Goda, Y. and Suzuki, Y., "Estimation of Incident and Reflected Waves in Random Wave Experiments ", Proceedings of Fifteenth Coastal Engineering Conference, Vol. 1, 1976, pp. 828845. 13. Goodrich, G.J., "Proposed Standards of Seakeeping Experiments in Head and Following Seas", Proceedings on Twelfth International Towing Tank Conference, 1969. 14. Hudspeth, R.T., Jones, D.F. and Nadi, J.H., "Analyses of Hinged Wavemakers for Random Waves", Proceedings of Tenth Coastal Engineering Conference, ASCE, 1978, pp. 372387. 15. Hudspeth, R.T., Leonard, J.W., and Chen, MC.,"Design Curves for Hinged Wavemakers: Experiments ", Journal of the Hydraulics Division , ASCE, Vol. 107, No. HY5,May., 1981 , pp. 553574.
Section 4 .18 References
137
16. Hyun, J.M., "Theory for Hinged Wavemakers of Finite Draft in Water of Constant Depth", Journal of Hydronautics, Vol. 10, No. 1, 1976, pp. 27. 17. International Towing Tank Conference Catalogue of Facilities, Sixteenth 1TTC Information Committee. Annapolis, MD, 1979. 18. Isaacson, M., "Measurement of Regular Wave Reflection", Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 117, No. 6, 1991, pp. 553569. 19. Jamieson, W.W. and Mansard, E.P.D., "An Efficient Upright Wave Absorber", Proceedings on Coastal Hydrodynamics, University of Delaware, ASCE, 1987, pp. 124139. 20. Jamieson, W.W., Mogridge, G.R. and Brabrook, M.G., "Side Absorbers for Laboratory Wave Tanks ", International Association for Hydraulic Research Proc. XXIII Congress , Ottawa, Canada, August 1989. 21. Jarlan, G.E., "Note on the Possible Use of a Perforated , Vertical Wall Breakwater", Proceedings of Seventh Conference on Coastal Engineering, Aug., 1960. 22. Jarlan, G.E., "A Perforated Vertical Wall Breakwater  An Examination of MassTransfer Effects in Gravitational Waves" , The Dock and Harbour Authority, Apr., 1961, pp. 394398. 23. LeMehaute, B., "Progressive Wave Absorbers", Research, IAHR, Vol. 10, No. 2, 1972, pp. 153169.
Journal of Hydraulic
24. Mansard, E.P.D. and Funke, E.R., "The Measurement of Incident and Reflected Spectra Using a Least Squares Method ", Proceedings of Seventh International Conference on Coastal Engineering, Sidney, Australia, ASCE, Vol. 1, 1980, pp. 154172. 25. Mansard, E.P.D. and Funke, E.R., "On the Reflection Analysis of Irregular Waves", National Research Council of Canada, Hydraulic Laboratory Technical Report TRHY017, 1987. 26. Miles, M.D., Launch, P.H. and Funke, E.R., "A Multimode Segmented Wave Generator for the NRC Hydraulics Laboratory ", Proceedings of TwentyFirst American Towing Tank Conference , Washington, D.C., 1986.
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27. Ouellet, Y. and Datta, I., "A Survey of Wave Absorbers ", Journal of Hydraulic Research, IAHR, Vol. 24, No. 4, 1986, pp. 265280. 28. Patel , M.H. and Ionnau, P.A., "Comparative Performance Study of Paddleand WedgeType Wave Generators ", Journal of Hydronautics , AIAA, Vol. 14, No. 1, Jan., 1980, pp. 59. 29. Ploeg, J. and Funke, E.R., "A Survey of Random Wave Generation Techniques", International Conference on Coastal Engineering , ASCE, 1980, pp. 135153. 30. Salter, S.H., "Absorbing WaveMakers and Wide Tanks", Conference on Directional Wave Spectrum Applications, Berkeley; California, ASCE, 1981. 31. Straub, L.G., Bowers, C.E. and Herbich, J.E., "Laboratory Tests of Permeable Wave Absorbers", Proceedings of Fifth Conference on Coastal Engineering, 1956, pp. 729742. 32. Ursell, F., Dean, R.G. and Yu, Y.S., "Forced Small Amplitude Water Waves: A Comparison of Theory and Experiment", Journal of Fluid Mechanics , Vol. 7, Part 1, 1960, pp. 3352. 33. Wang, S., "PlungerType Wavemakers: Theory and Experiment", Journal of Hydraulic Research, Vol. 12, No. 3, 1974, pp. 357388. 34. Vasquez, J.H. and Williams, A.N., "Hydrodynamic Loads on a ThreeDimensional Body in a Narrow Tank," Proceedings of Offshore Mechanics and Arctic Engineering Conference, ASME, Calgary, Canada, 1992, pp. 369376. 35. Yeung, R.W. and Sphaier, S.H., "Wave Interference Effects on a Truncated Cylinder in a Channel", Journal of Engineering Mathematics, Vol. 23, 1989.
CHAPTER 5 MODELING OF ENVIRONMENT
5.1 WAVE GENERATION In the testing of an offshore structure model , the environment experienced by the structure requires proper simulation in the laboratory . This environment generally consists of waves, wind, current and possibly earthquake . This last item is not covered in this book. The excitation due to earthquake loading is generally simulated by placing the model on a shaker table. For example, a shaker table may be placed at the bottom of the wave tank, and the instrumented structure may be attached to it . The shaker table may then simulate the earthquake motion (similar to the planar motion mechanism described in Section 4.12). The energy and frequency content of the earthquake may be simulated by one of the methods to be described for wave generation. There are many different methods that are employed in generating irregular waves [Kimura and Iwagaki ( 1976), Ploeg and Funke ( 1980), Johnson (1981 ), Mansard and Funke ( 1988)]. It should be recognized that the irregular wave may not have the statistical features possessed by the random wave but some control may have been exercised on either their frequency or time domain characteristics . The following are some of the common methods of irregular or random wave generation: • Harmonic Synthesis  These waves may be generated by a mechanical gear system, special purpose electronic device or by computer. It uses a range of discrete frequencies producing irregular wave form. •
Frequency Sweep, by varying the speed of the actuator drive motor.
• Pseudo Random Noise , which is generated by computer or an electronic device. •
True Random White Noise and Analog Shaping Filters [Loukakis ( 1968)].
• Synthesis by Fourier Transform Technique by on or off line computer.
140
Chapter 5 Modeling of Environment
• Reproduction of Prototype Wave Train in time domain. This may be accomplished by magnetic tape , whether analog or digital, and online computer [Gravesen, et al. (1974)]. •
Wind alone or in combination of one of the above methods.
A compensation technique is often used to account for the transfer function of the wave board or the servo . If a discrete time driving signal is used involving a digital to analog converter, a smoothing routine , such as analog lowpass filter or straight line interpolator, is recommended. The type of waves that are generated in a wave tank depends on the type of wave generator and the method of wave generation. There are several forms of waves generated in the laboratory. What waves are generated in a given test depends on the application and location of the offshore structure being tested. 5 .1.1 Harmonic Waves Waves are generated in the tank by scaling down the sea wave parameters by the scaling laws described in Chapter 2. Regular harmonic waves are very useful tools in understanding the physics of the fluidstructure interaction, even though such waves of significance are seldom found in nature . The generation of regular sinusoidal waves is quite straightforward once the height and period of the wave are scaled down. Then, knowing the transfer function of the given wave generator, the proper setting for the wave height may be established. 5.1.2 NonHarmonic Waves There are several techniques used in wave synthesizing through the Fourier transform. For a nonharmonic wave synthesizer, a large frequency band in the waves is generally simulated . The harmonic relationship between contributing frequency components is avoided . In one method, the frequencies are located at the center of the adjacent frequency bands having equal areas. Thus, the components within the individual bands have the same amplitude [Borgman (1969), Medina , et al. (1985), Told (1981)]. The Fourier amplitude coefficients are considered proportional to the square root of the desired spectral density. Phases are obtained from a selected random number generator [Goda (1970)]. A more realistic simulation of a Gaussian stochastic process uses a "white noise" complex spectrum . The time series is generated using statistically independent Gaussian random numbers. They are then "filtered" by the square root of the desired spectral density by crossmultiplication [Goda (1977)]. In a slightly different approach [Medina and Diez (1984)], random chisquared distributed numbers with two degrees of freedom are used. After crossmultiplication with the
Section 5.1 Wave Generation
141
desired spectral density, the square root provides the amplitudes which are paired with the uniformly distributed random phase . The complex spectrum in polar coordinates is inversely Fourier transformed to yield the desired time series. Alternatively, an iterative Fourier technique [Funke and Mansard (1979)] is used to achieve a specified distribution of energy in the frequency as well as the time domain. Sometimes, a white noise having "equal" amplitudes at all frequencies within a band is generated. Filtered white noise generators are random noise sources which never repeat. They may be used in a model test in obtaining a response transfer function from one single run. There are many energy spectrum models that have been proposed . Some of these are PiersonMoskowitz (PM), Bretschneider, JONSWAP and Ochi and Hubble [Chakrabarti ( 1993)]. While generation of waves in the tank to match a given energy density spectrum model is a common occurrence , it is sometimes desirable to duplicate the natural sea waves. In this case , a record of the short term wave activity at a specific site is available. This record is scaled down and reproduced point by point in the wave tank. The digital data is converted to the analog signal , rectified to account for the wave generator transfer function and then is used to drive the wave generator. Sometimes, an extreme wave height in a time series is achieved by experimenting with different random number seeds until the desired zero crossing wave height is found in the time series . The same technique is applicable to achieve a particular degree of wave grouping once the energy density is matched. No matter how perfect the simulation technique is, the physical model is handicapped by several factors causing departure from the theoretical model: • Imperfect knowledge of the dynamic transfer characteristics of the wave generator; • Reflection from side walls, beach and generator; • Distortion by mechanical device of the wavemaker in contact with water; • Wavewave energy transfer; • Resonance in oscillation from inline or lateral standing wave; and • Recirculation due to a net mass transport in the tank. For the generation of waves, a digital input signal is computed from the target spectrum. The digital signal uses a wave machine transfer function . The transfer function generally accounts for the relationship between the mechanical displacement to the water displacement, the hydraulic servo control system, and the dynamic effect of the lowpass filter. This compensation can be effectively implemented through a Fourier transform technique.
142
Chapter 5 Modeling of Environment
However, since this is a linear process, it does not account for the nonlinear effect in the wave generation . The discrepancy between the desired and output signal can, however, be compensated in an iterative method by generating and recording the wave a few times until a satisfactory recording of the wave at the test site is achieved. Unfortunately, this technique would not be applicable if any reflected wave is generated at the test site (for example , between a structure and the tank wall). Hence this iterative technique is generally adopted when calibrating the waves before placing the test structure in position. 5.1.3 Imperfect Waves An efficient wavemaker can reach a wave steepness of about 0.1 tanh kd which is about 70 percent of the theoretical limit . The excess energy fed into an inefficient wavemaker not used in the production of waves leads to standing , crossmode waves and wave breaking. The pneumatic wavemaker described in Chapter 4 is capable of producing waves of period, T > 1.0 sec. If one attempts to generate waves below 1 second with this wavemaker , large transverse waves are produced in the tank which propagate in the tank in crossmode. The period of this wave corresponds to the standing wave oscillation based on the width of the tank . A similar observation can also be found when generating shallow water waves by flapper motion of a conventional type of wave board or deep water waves by translatory motion. Another area of concern is spurious long waves created while simulating nonlinear waves by linear wave generation theory. This aspect will be discussed in further details in Section 5.4. In some studies , the nonlinear wave components have a large influence on the outcome, e.g., in the mooring tests of a vessel or in overtopping of structures. Nonlinear waves are known to contain, according to second order theory, high harmonics which are locked to the fundamental waves and therefore , move at the same celerity as the fundamental components. Traditional first order theory used in the laboratory reproduction of waves cannot take into account the locked nature of these harmonics and inadvertently produces undesirable free waves (called as spurious or parasitic waves) in model studies . Since the free waves move at their own celerities, they interact with the faster running locked harmonics along the length of the tank, thus resulting in profile changes such as the ones depicted in Fig . 5.1b. These undesired waves can however, be eliminated by a second order wave generation technique. With this technique, a stable second order wave moving down the tank without any appreciable profile change can be realized (see Fig. 5.1a). An example of the difference in the forces measured on a moored vessel model in a tank with and without this compensation is seen in Fig. 5.2. Forces are underestimated by about 15 percent without compensation of the undesired waves.
Section 5.1 Wave Generation 143
In the case of irregular waves, wave components of higher frequencies consisting of twice the fundamental (i.e. 2fi) as well as the sum of two fundamentals (i.e., fi + fj) are generated. These are also locked components and propagate at the same celerity as their fundamentals . Spurious, undesired free waves are also encountered in this case and their interactions with the locked waves during propagation result in substantial changes in the high frequency tails of their spectra (see for instance Fig. 5.3). It should, however, be pointed out that even if these free waves are eliminated by adopting a second order generation technique, variations in high frequency tails would still be encountered during propagation because of the following reason . The high frequency tail of a nonlinear wave spectrum is generally composed of two types of wave components: high frequency linear components which are integral parts of the spectrum and locked high harmonics produced by the interaction of components which belong to the low frequency part. Since linear and locked components travel with different propagation celerities , the shape of the high frequency tails of the spectrum can be expected to change [see Sand and Mansard (1986)]. litORDER GENERATION
2ndORDER GENERATION 0.20 0.10
0 • 4.4m 0.00 0.10 0.20
0.10 0.00 0.10 0.20 0.10 0 • 14.0.
0.00 0.10 0.20 0.10 0 • 19.5. 0.00 0.10 0.00
6.00
12.00. T  2.0.
0.00
6.00
12.00.
N • 0.14$,
FIGURE 5.1 REGULAR WAVE PROFILES DURING PROPAGATION (a) WITH AND (b) WITHOUT SECOND ORDER COMPENSATION [Mansard and Funke (1988)]
144
Chapter 5 Modeling of Environment
(a)
(fad)
.010
j.006 0.a 002
0.2
1
L
SURGE SWAY (m) (m)
I
I
I
j
J
HEAVE ROLL PITCH YAW (m) (fad )
( radl (fad)
WITH COMPENSATION WITHOUT COMPENSATION
IT) 30
(b) 25
STERN STERN STERN
BOW BOW BOW
BREAST SPRING SPRING BREAST (MOORING LINES)
FIGURE 5.2 EFFECT OF SECOND ORDER LONG WAVE COMPENSATION ON THE RESPONSE OF A MOORED VESSEL [Mansard and Funke (1988)] 5.1.4 Shallow Water Waves If the water depth is properly scaled in the tank , the form of the regular shallow water waves is automatically formed during its propagation down the tank. Very long period shallow water waves in the tank suffer from the soliton effect [Chakrabarti (1980, 1990)]. In this case the regular wave breaks down into several peaks called solitons, each traveling at a different speed . These waves reform as the front travels
Section 5.1 Wave Generation
145
down the tank. An example of soliton recorded in 1.53 m (5 ft) of water is shown in Figure 5.4. m2/H.zx103 10
(a) 8 6 x0 4
f 0 0
0.5
1.5Hz
1.0
10
(c) 8 6 x=20m
f 0.5
1.0
1.5Hz
(a) 8
6 x=29m 4
2 f 0 0
1.0
1.5Hz
FIGURE 5.3 SPECTRA OF FIRSTORDER WAVE GENERATION AT VARIOUS DISTANCES X FROM PADDLE WAVEMAKER [Sand and Mansard (1988)]
146
Chapter 5 Modeling of Environment
f^ Aj^ N fk_^ ^j ^j uk.), ^kj U^'j T=7.5 SEC 0
10
20
30
40
TIME  SECONDS FIGURE 5.4 REGULAR LONG PERIOD WAVES IN SHALLOW WATER IN TANK Kjeldsen and Myrhaug (1979) advanced the concepts of horizontal and vertical asymmetry parameters to characterize the nonlinear asymmetries found in natural waves. A technique was developed by Funke and Mansard (1982) to simulate these asymmetries during laboratory reproduction of waves . For this, two operations are needed. First, the wave crests are amplified and the trough is correspondingly attenuated . The duration of the trough is lengthened by a factor at the same time the crest duration is shortened by a comparable factor such that the wave period remains unchanged . The crest amplification factor is derived such that the areas under the crest and the trough are equal. The second distortion appears in the asymmetry of the crest (called the crest front steepness). Here, the distortion in the wave crest is achieved by shifting the peak of the crest forward by a factor . The amount of distortion is limited by the amount of modification it causes in the amplitude spectrum . These distortions are simulated in the wavemaker command signal. 5.2 RANDOM WAVE SIMULATION As we have already discussed, there exists a variety of techniques of reproducing random sea waves in the laboratory . Techniques of producing waves in the laboratory basins have evolved from the simple electro mechanical systems in the past to the presentday sophisticated hydraulic electric servo systems , controlled by online computers and capable of producing a large number of different time series of water
Section 5.2 Random Wave Simulation 147
surface elevations from a sea state defined solely by its spectrum or just by its significant wave height and peak period . Because of this possibility of variations due to different wave synthesis techniques (in addition to already existing differences in experimental setups), there is no assurance that testing the same model in different laboratories will give similar results . Therefore, an accurate definition of the required sea state has become very important. For example, matching just the significant wave height and period may not be sufficient to produce similar results . Groupiness in waves may be an important input parameter in defining a model test. The duration of random wave record should be sufficiently large for analysis; but, in any case, should not be less than 180 times the mean wave period Tm. Usually one starts with a given wave energy density spectrum for the simulation of sea waves in a laboratory. There are several different methods for wave generation based on this energy density. Two of the most common Fourier techniques [see Rice (1945)] are the Random Phase Method ( RPM) and Random Complex Spectrum Method (RCSM). The former is spectrally deterministic , while the latter is spectrally nondeterministic. The RPM method has been claimed [Tucker, et al. (1984] to be incorrect in representing nature and produces insufficient variability of wave parameters. For long non repetitive wave records (approaching infinity), the two methods are identical. Even for short records generated numerically as well as physically , the differences in some parameters are found to be small . This area will be discussed further in Section 5.3.3. 5.2.1 Random Phase Method In the socalled spectrally deterministic method , a smooth wave spectral density function is specified from which the adjacent frequency components are derived. These frequency components are not necessarily statistically independent . Therefore, in the strictest sense, the resultant wave trains are not part of a Gaussian stochastic process . The phases for each frequency component are obtained randomly; hence, it is called the random phase method . However, numerical simulation with a "random generator" may be restarted from the same initial condition (called seed) every time and is often cyclical over a finite number of entries into the generator . Such sources of random numbers are referred to as pseudo random number generators. By using different seeds, different sequences of random phases can be generated and thus different time domain characteristics of the sea state . This technique is often employed to achieve different degrees of grouping or different wave height statistics in the simulated waves. The sea surface is generally assumed to be Gaussian with a zero mean. Simulation of the sea surface usually consists of a finite number of Fourier components as a function of time:
148
Chapter 5 Modeling of Environment
I an N
71(t) =
co s(27rf
t +e,)
(5.1)
where fn and an are frequency and amplitude chosen from the wave energy spectrum and en is a random phase angle. This method always reproduces the wave energy density spectrum , S(f). The method, however, does not strictly model a random Gaussian surface except in the limit as N+°° [Tucker, et al. (1984)]. The smooth spectrum is subdivided in N equal frequency increments of width Af over the range of frequencies 0 and fm, where fm is the maximum generated frequency . Then, the spectrum density S(nAf) where n refers to the nth increment is converted into a Fourier amplitude spectrum a(nAf): an=a(nAf)= 2S(n4f)Af,n=1,2...N
(5.2)
The value of Af (in a Finite Fourier Transform simulation) is chosen such that
_ 1
(5.3)
TR
where TR is the length of the time series to be synthesized. The periodicity of the time series is also TR. S(f) is not generally provided at this sampling rate and , therefore, the resampling of S(f) is usually carried out. The phase spectrum e(nAf ) is created at each frequency, nAf, from a random number generator with a uniform probability distribution between n and +n. The target time series is derived by an inverse Fourier transform: r1(t) = Y a(nAf) co4 2xnAft + e(nAf )], 0 U
4
Re > 30,000
5
NR m  N, , p (order of magnitude)
6
NSM = N,
test. While a direct scaleup is often not possible, the model test will provide the general pattern of scour, erosion, etc., and provide an order of magnitude expected in the prototype when scaled up. In this section, computations are carried out to investigate how certain prototype materials may be scaled in a model test . A sample of sand material d50 = 0.1 mm is selected here. A scale factor for the model is taken as X = 36. The water particle velocity at the model base is 0.073 m/s (0.24 ft/sec) (for a prototype value of 0.44 m/s or 1.43 ft/sec). The model current velocity is 0.085 m/s (0.28 ft/sec) (for a prototype value of 0.51 m/s or 1.68 ft/sec). These are based on Froude's law of similitude. The model materials are chosen from sand and crushed walnut shells . The results are summarized in Tables 7.4 and 7.5. The smallest wholegrain sand particle has a mean diameter of 0.01 mm while the smallest crushed walnut was found to be about 0.2 mm. These materials are chosen for the comparison in the tables. Table 7. 4 summarizes the properties of model material of sand having a grain size of 0.01 mm used to model a prototype sand material of mean diameter of 0.1 mm. It is
298
Chapter 7 Modeling of Fixed Offshore Structures
assumed that fresh water will be used in the test . The table shows that the particle fall velocities and the grain size Reynolds number do not scale . The scaling of the particle fall velocity may not be very important as long as the shear velocity of the model material is larger than the model particle fall velocity. This latter criterion is satisfied for the selected model material . Similarly, the modeling of the entrainment function and sediment number is generally met. Assuming that Re > 30,000, the viscosity scale effect is small and the pore pressure will be adequately duplicated in the model . One might expect that the model test will produce results characteristic of the prototype behavior. TABLE 7.4 SUMMARY OF MATERIAL PROPERTIES FOR PROTOTYPE AND MODEL SAND FOR A SCALE OF 1:36 Quantity
Prototype
Model
Remarks
Fluid Kinematic Viscosity, V (ft2/sec)
1.59 E3
1.22 E3
Sea vs. Fresh Water
Fluid Density, p (slugs/ft3)
1.98
1.94
Sea vs. Fresh Water
Material Size, d50 (mm)
0.1
0.01
Smallest Whole Grain Sand
Specific Gravity, s
2.65
2.65
Particle Fall Velocity, uF (ft/sec)
0.0191
0.00025
Does Not Scale
Shear Velocity, u* (ft/sec)
0.0458
0.01
u* > uF
Turbulent Settling Velocity, uT, (ft/sec)
0.1305
0.0413
Same Order of Magnitude
Grain Size Reynolds Number, NR
0.923
0.026
Does Not Scale
Entrainment Function,
0.123
0.058
Roughly Equal
Sediment Number, N
10.96
5.82
Roughly Equal
The same prototype material of d50 = 0.1 mm is also modeled by crushed walnut shells which are lighter and have a specific gravity of 1.35. The results are
Section 7.5 Scour Around Structures
299
summarized in Table 7.5. This modeling is better since it scales all items quite well except the entrainment function. TABLE 7.5 SUMMARY OF MATERIAL PROPERTIES FOR PROTOTYPE AND MODEL WALNUT SHELL FOR A SCALE OF 1:36 Quantity
Prototype
Model
Remarks
Fluid Kinematic Viscosity, v (ft2/sec)
1.59 E3
1.22 E3
Sea vs. Fresh Water
Fluid Density, p (slugs/ft3)
1.98
1.94
Sea vs. Fresh Water
Material Size, d50 (mm)
0.1
0.2
Crushed Walnut Shells
Specific Gravity, s
2.65
1.35
Particle Fall Velocity, uF (ft/sec)
0.0191
0.021
Scales Well
Shear Velocity, u* (ft/sec)
0.0458
0.01
u* > UF
Turbulent Settling Velocity, uT, (ft/sec)
0.1305
0.085
Scales Reasonably Well
Grain Size Reynolds Number, NR
0.923
0.524
Scales Well
0.123
0.014
Does Not Scale Well
10.96
2.82
Same Order of Magnitude
Entrainment Function, Sediment Number, Ns
Even though the model sand scales several of the criteria set forth in Table 7.3, the size of the model sand of d50 = 0.01mm is much smaller than the lower limit of noncohesive sediment which is generally assumed to be about 0.08mm. Therefore, the model sand may not be suitable to model the noncohesive prototype sediment. On the other hand, the model walnut shell scales the majority of the criteria in Table 7.3 and as such is suitable for modeling the prototype sediment and duplicating prototype scour effects.
300
Chapter 7 Modeling of Fixed Offshore Structures
The above is a brief and simple explanation on scaling of scour in a model test. If one is interested in the details of sediment transport similitude requirements for movablebed models, reference is made to the works of Kamphuis ( 1982) and Yalin ( 1971). They have explained the various similitude requirements for this type of modeling and the consequence of not fulfilling some of the criteria. 7.5.3.3 Cohesive Soil In structure soil interactions, the cohesive soil resistance to the structure is significantly less than the resistance created by cohesionless soils . This conclusion is based on the assumption that the soil is undrained , and the additional loading is balanced by an increase of pore pressure . Structures exposed to wave loading oscillations may develop an almost undrained soil response. Modeling of cohesive soil in a scale model is a complex problem. While tests have been done at small scale to establish cohesive soil properties, structural modeling with cohesive soil is uncommon. There are significant time effects on the lateral soil resistance to structures supported by cohesive soils. The effects are caused by several factors, such as • Aging of the disturbed soil; • Pore pressure dissipation from the strained soil; and • Rate of external loading on the structure (due to waves). The question of time scale for scour is not answered very easily without close correlation with prototype observations , and few, if any, are available . For this reason small scale testing with cohesive soil is avoided. 7.5.4 Scour Protection When a structure is considered vulnerable to scour that may result in loss of stability, protective measures are taken to ensure that stability is maintained. Scour protection can be classified as either active or passive. Active scour protection is defined as any kind of protection that reduces the scour potential of the flow near the seabed (i.e., reduces the disturbing forces). Passive scour protection is any method that increases the ability of the foundation to resist scouring elements (i.e., increases the restoring forces). Many devices are available commercially that may be placed at the ocean floor to reduce the fluid flow near the structure base and lower the potential of scour. Model tests are often conducted to investigate the effect of a scour protecting device. Tests are conducted with and without the device to study the extent of scouring and its prevention.
Section 7.7 References
301
7.6 WIND TUNNEL TESTS The superstructure , such as the exposed deck of an offshore structure experiences wind loads. Often, these structures are tested in wind tunnels to determine such loads in model scale . This area of testing has been considered to be outside the scope of this book . The following briefly describes wind tunnel testing. The effect of wind on the exposed offshore platform is an important design consideration. When the wind flows over the platform, its flow pattern changes, introducing a force on the platform superstructure . The adverse wind condition also affects several operational efficiencies of the platform , such as the drilling operation, transport operation, etc. A realistic assessment of these problems is made [Littlebury (1981 )] by model studies in a boundary layer wind tunnel . The conventional aerodynamic wind tunnel generating laminar flow is generally unsuitable for offshore platform tests. Geometric similarity of airflow in the boundary layer over the sea is obtained in terms of several nondimensional parameters , such as wind speed profile, intensity of turbulence and length scale ratio of turbulence. A typical scale for a wind tunnel test of an offshore platform is between 1:100 and 1: 200. To generate a stable boundary layer within this range of scale , the boundary layer is started upstream of the flow development section so that a reasonably stable boundary layer may be achieved at the test section . For this purpose a barrier of spires is used shaped in such a way that the required amount of shear is produced . The tunnel floor is roughened to stabilize the turbulent shear flow. For a model scale of 1:100 to 1:200, it is virtually impossible to maintain the Reynolds similitude . However, for bluff bodies representing the offshore platform, the flow effect is independent of the Reynolds number so that the distortion in Reynolds scaling may not be important . The tunnel wind speed is often about 100 m/s. An electronic balance (dynamometer) is normally fitted in the tunnel to measure the forces and moments acting on the model structure. 7.7 REFERENCES 1. Abramson, H.N., "The Dynamic Behavior of Liquids in Moving Containers", Applied Mechanics Reviews, Vol. 16 , No.7, 1963, p. 501. 2. Allender, J.H., and Petrauskas, C., "Measured and Predicted Wave Plus Current Loading on a LaboratoryScale, Space Frame Structure," Proceeding on Nineteenth Annual Offshore Technology Conference , Houston, Texas, OTC 5371, 1987, pp. 143151.
302
Chapter 7 Modeling of Fixed Offshore Structures
3. Brogren, E.E., and Chakrabarti, S.K., "Wave Force Testing of Large Based Structures," Proceedings of Twentyfirst American Towing Tank Conference, Washington, D.C., August, 1986. 4. Carstens, M.R., "Similarity Laws for Localized Scour ", Hydraulics Division , ASCE, Vol. 92, No. HY3, May, 1966.
Journal of the
5. Chakrabarti, S.K., "Inline and Transverse Forces on a Tube Array in Tandem with Waves," Applied Ocean Research, Vol. 4, No. 1, 1982, pp. 2532. 6. Chakrabarti, S.K., "Recent Advances in HighFrequency Wave Forces on Fixed Structures," Journal of Energy Resources Technology, Trans.ASME, Vol. 107, Sep., 1985, pp. 315328. 7. Chakrabarti, S.K., "Wave Forces on Offshore Gravity Platforms," Journal of Waterway, Port, Coastal and Ocean Engineering, Vol.112, No.2, March 1986, pp. 269283. 8. Chakrabarti, S.K.,"Wave Forces on an OpenBottom Submersible Drilling Structure ," Applied Ocean Research, Vol.9, No.1, 1987, pp. 16. 9. Chakrabarti, S.K., "Correlation of Steady SecondOrder Force on a Fixed Vertical Cylinder," Applied Ocean Research, Vol.9, No. 4, 1987, pp. 234236. 10. Chakrabarti, S.K., Nonlinear Methods in Offshore Publishers, Netherlands, 1990.
Engineering, Elsevier
11.. Chakrabarti, S.K., Hydrodynamics of Offshore Structures, 2nd Edition, Computational Mechanics Publications, Southampton, U.K., 1993. 12. Chakrabarti, S.K., and Naftzger, R.A., "Wave Interaction with a Submerged OpenBottom Structure," Proceedings on Eighth Annual Offshore Technology Conference, Houston, Texas, OTC 2534, 1976, pp. 109123. 13. Chakrabarti, S.K. and Tam, W.A., "Interaction of Waves with Large Vertical Cylinder," Journal of Ship Research , Vol.19, March, 1975, pp. 2333. 14. Clauss, G.F. "Flat Foundations for Offshore Structures ," Marine Technology, Vol. 18, 1987, pp. 2331.
Section 7.7 References
303
15. Cotter, D.C., and Chakrabarti, S.K., "Wave Force Test on Vertical and Inclined Cylinders ," Journal of Waterway, Port, Coastal and 'Ocean Engineering, ASCE, Vol. 110, No.1, Feb., 1984, pp. 114. 16. Dawson , T.H., "Scaling of Fixed Offshore Structures ," Ocean Engineering, Vol. 3, 1976, pp. 421427. 17. Dean, R.G., "Stream Function Representation of Nonlinear Ocean Waves", Journal of Geophysical Research, Vol. 70, No. 18, 1965, pp.45614572. 18. Dean , R.G., "Methodology for Evaluating Suitability of Wave and Force Data for Determining Drag and Inertia Forces ," Proceedings on Behavior of Offshore Structures, BOSS '76, Vol. 2, Trondheim, Norway , 1976, pp. 4064. 19. Eatock Taylor, R., and Hung , S.M., "Mean Drift Forces on an Articulated Column Oscillating in a Wave Tank," Applied Ocean Research, Vol.7, No. 2, 1985, pp.66. 20. Hallermeier, R.J., "Terminal Settling Velocity of, Commonly Occurring Sand Grains," Sedimentology, Vol. 28, No. 6, 1981 , pp. 859865. 21. Hansen , D.W., Chakrabarti, S.K., and Brogren, E.E., "Special Techniques in Wave Tank Testing of Large Offshore Models," Presented at Marine Data Systems International Symposium , New Orleans, Lousiana., April 1986 , pp. 223331. 22. Herbich , J.B., Schiller, R.E., Watanabe, R.K. and Dunlap , W.A., Seafloor Scour, Marcel Dekkar, Inc., New York, N.Y., 1984. 23. Kamphuis , J.W., "Coastal Mobile Bed Modelling from a 1982 Perspective," C.E. Research Report No. 76, Queen's University , Kingston, Ontario, 1982. 24. Karal, K., "Lateral Stability of Submarine Pipelines," Proceedings on Ninth Annual Offshore Technology Conference, Houston, Texas , OTC 2967,1977. pp. 7178. 25. Littlebury, K.H., "Wind Tunnel Model Testing Techniques for Offshore Gas/Oil Production Platforms ", Proceedings on Thirteenth Annual Offshore Technology Conference , Houston, Texas, OTC 4125, 1981 , pp. 99103. 26. Lovera, F. and Kennedy , J.F., "FrictionFactors for FlatBed Floors in Sand Channels," Proceedings ASCE, Vol. 95, 1969, pp. 12271234.
304 Chapter 7 Modeling of Fixed Offshore Structures
27. Monkmeyer, P.L. "Wave Induced Seepage Forces on Embedded Structures", Proceeding of Civil Engineering in the Oceans IV, ASCE, San Francisco, Calif., 1979. 28. Sarpkaya, T., "InLine and Transverse Forces on Cylinders in Oscillatory Flow at High Reynolds Numbers ," Proceedings on Eighth Annual Offshore Technology Conference, Houston, Texas, OTC 2533, 1976, pp. 95108. 29. Sarpkaya, T. and Isaacson. M., Mechanics of Wave Forces on Offshore Structures, Van Nostrand Rheinhold, New York, New York, 1981. 30. Vasquez, J.H. and Williams, A.N., "Hydrodynamic Loads on a ThreeDimensional Body in a Narrow Tank ," Proceedings of Offshore Mechanics and Arctic Engineering Conference, ASME, Calgary, Canada, 1992, pp. 369376. 31. Yalin, M.S., Theory of Hydraulic Models, MacMillan Press, London, England, 1971. 32. Yeung, R.W. and Sphaier, S.H., "Wave Interference Effects on a Truncated Cylinder in a Channel", Journal of Engineering Mathematics, Vol. 23, 1989.
CHAPTER 8 MODELING OF OFFSHORE OPERATIONS
8.1 TYPES OF OFFSHORE OPERATIONS Offshore structures are built on shore or in sheltered coastal waters and transported to site by many different means. After the structures reach their destination , they are submerged and installed at site at the prescribed location . Thus, a completed offshore structure goes through three distinct stages before it is ready for operation at the site. These stages are transportation, launching and submergence. This chapter deals with these three operations for various offshore structure configurations. There are two common types of transportation operations for offshore structures. If an offshore structure is small in size or if the structure does not have a large buoyancy module, it is carried on a barge . One such structure is a jacket type drilling and production structure. These structures are built horizontally on launch pads near the water. They are pulled onto a barge which may be temporarily ballasted to be at proper level to accommodate the structure. The barge is then towed to the installation site before launching the structure at its final location . The launching of the jacket from the barge, i.e., sliding it down into water from the rails on the deck on which it sits and the critically controlled prescribed submergence onto the bottom in a vertical position are very tricky operations . They almost invariably require model testing before the structure is actually taken out from the launching pad. Even though this procedure for smaller jackets has become a routine procedure , for larger structures, model testing is a must, considering the investment and risk involved. The second type of transportation involves large volume structures which can be towed by themselves either vertically or horizontally at a shallow draft . Structures that are large near their bases , such as gravity production and storage structures, are generally built vertically in a graving dock. Before towout to the final destination of the structure, the graving dock is flooded and opened to the sea . Tug boats are used to tow the structure into the sea . Sometimes when the structure reaches deep water, it is ballasted down further for added stability during the towing operation. Some of the large concrete structures are actually built on water in deep water fjords. In this case, the structure is continually submerged as it is built up vertically . The buoyancy sections are pressure tested before final towing . The structure is deballasted to a predetermined
306
Chapter 8 Modeling of Offshore Operations
towing draft before towout . After final destination is reached these structures receive ballast water in a prescribed manner in their ballast chambers and vertically settle down to the sea floor. There are other large volume structures that are built horizontally in a graving dock. An example of this type of structure is a buoyant articulated tower. These structures are towed in a near horizontal position to their final destination by tug boats. If the tower carries a gravity base connected by a universal joint, the base is anchored to the tower by temporary tie rods (to prevent damage to the universal joint ). At the site, the tie rods are released, and the tower is swung down into position by flooding the ballast tank. The tower assumes different angles during swing down . These operations of towing and submergence for both vertical and horizontal modes are routinely model tested. The towing, launching and submergence model testing for different types of offshore structures are described in the following sections. Some of the modeling problems and scaling techniques of evaluating model data are discussed. 8.2 TOWING OF A BARGE In this section, the technique of towing a barge model and the scaling up of the data for evaluating the prototype behavior is described . Tests of tanker models are similar. A Froude model of a barge is towed in loaded as well as various ballast drafts at constant towing speeds . This test determines experimentally the towing resistance of the hull of the model. Accordingly, the load encountered by the towing mechanism is recorded. This load is scaled up to the prototype value. The resistance characteristics of the prototype barge is presented as a function of its speed. 8.2.1 Scaling Technique A ship in uniform horizontal motion , with its vertical plane of symmetry parallel to its velocity, experiences a hydrodynamic force (resistance) in a direction opposite to its velocity and lying in the plane of symmetry . A small lift force in the vertical direction is also experienced by the ship, which, however, is usually ignored . This resistance principally arises from two major categories: (i).one ofviscousorigin which essentially. is due to the shearing forces acting tangetially to the hull surface, and (ii) one of pressure origin which is due to . a component resulting from all pressure forces acting normally to the hull surface . Though both these categories interact with each other, they are usually considered separately . The hull being at the free surface, the variation of pressure forces over the hull generates a system of surface waves on water and the pressure distribution
Section 8.2 Towing of a Barge
307
itself is altered due to these waves. The wave system which the ship carries with it represents a drain of energy necessary to maintain it and therefore this is called wavemaking resistance. It turns out that wavemaking resistance is the predominant part of the second category. In scaling the model barge tow test results to the prototype, we assume that the total resistance Rt can be decomposed into Rt =R f +Rr
(8.1)
where Rr is frictional (viscous) resistance and Rr is residuary resistance whose dominant part is wavemaking resistance. Usually, the resistances are expressed in terms of coefficients which are normalized values of the resistances with respect to 0.5pAU2. Therefore, from Eq. 8.1 the corresponding coefficients are related by Ct
=C
f+Cr
(8.2)
It may be noted that only Froude's similarity is used and Reynolds number dependent part of drag is calculated using a friction formula. This is because if both Fr and Re are to be kept constant for both prototype and the model, for ship problems one needs a fluid whose kinematic viscosity is about 1/,,32 of that of water . For X= 25 this represents a factor of 1/125th, and no such fluid is known . The frictional resistance coefficients, C f are functions of Re, while the residual resistance coefficient Cr is principally a function of Fr. The geometrically similar model is towed at speeds corresponding to the prototype obeying Fronde's similarity. The residuary resistance coefficient, Cr, being a function of the Fronde number will have the same value between the model and the prototype: C =C rp mi
(8.3)
The friction coefficient Cf for both the model and the prototype is calculated using either ITTC formula [Muckle and Taylor (1987)]: C = 0.075 f (log,. Re2)2
(8.4)
308
Chapter 8 Modeling of Offshore Operations
or the Schoenherr line. For the prototype, an allowance of about 0.0004 is usually added to the above formula to improve model prototype correlation . The following steps then give the scaling procedure: • measure Ran at Um & compute Can by dividing by 2 pmAmUm2
• compute Cf n for Rem = Umt m / vm using Eq. 8.4
• C =C C^ lm an=C lp
• compute Cfp for Rep = Upt p/vp using Eq. 8.4 & augment it by +0.0004.
• Ctp = C fp +Crp
• Compute Rtp (i.e., multiply by 2 ppAPUp2) which is the total resistance of the prototype. The plot of Eq.8.4 is shown in Fig.8.1. It should be noted that t used in the formula is usually taken to be the length at waterline, which therefore can vary with the draft. The density and the viscosity difference of sea water and the water medium where towing tests are done are accounted for in the above. 8.2.2 Barge Test Procedure A typical towing test setup is shown in Fig. 8.2. The barge model is towed by the overhead bridge using a hollow rectangular towing staff. The lower end of the staff is attached near the middle of the model at the SWL by a pivot connection so that the barge model is free to pitch. The upper end is free to move up and down through roller guides to allow the barge to heave. By proper choice of the rollers and guides, the friction in the system is minimized . Thus, the staff allows two degrees of freedom. The towing load is measured by a load cell which increases the measured load by a factor of 6 through the placement of the load cell and moment anus (Fig.8.2). The load cell is calibrated in place in the dry by imposing loads at the bottom of the staff with known forces.
Section 8.2 Towing of a Barge
309
MODEL .005 .004 PROTOTYPE
cr .003 .002 Al
10
10' REYNOLDS 5 R
FIGURE 8.1 SKIN FRICTION DRAG COEFFICIENT The scaling procedure described in the previous section is illustrated with the results of a towing test (). = 48) performed in a towing tank. The speed ranged from 2.1 to 8.4 m/s (4.1 to 16.4 knots) in the prototype scale (0.311.22 m/s or 14 ft/sec model scale). These tests were repeated by towing the free model with a bridle arrangement so that the model had all six degrees of freedom. A sketch for this test setup is shown in Fig. 8.3. The bridle is taken over a pulley to a ring load cell with a spring located in between the load cell and the point of attachment to reduce shock loads. After the tests with the undisturbed flow were completed, a 19 mm (3/4 in.) strip of silver duct tape was attached just behind the bow thruster and another one about 0.6 in (2 ft) further behind in order to assist in the flow separation without introducing significant additional resistance . This is one of the commonly accepted methods of artificially stimulating turbulence in the flow in this type of tests. This method tends to equalize the distortion in the Reynolds number scaling discussed above. Studs are also used on the model surface for this purpose. 8.2.3 Data Analysis And Results The length, e at the waterline is used as the characteristic length in evaluating the Fronde and Reynolds numbers. The mass density, and the corresponding kinematic viscosity of fresh water are corrected to provide the sea water values. Table 8.1 presents the towing speed, resistance coefficients and resistances for the model and prototype respectively at a given draft. The table was prepared following the scaling procedure outlined earlier. At a towing speed of 0.84 m/s (2.75 ft/sec) corresponding to 5.8 m/s (11.28 knots) for the prototype or less, the Froude number is 0.15 or less, and
310
Chapter 8 Modeling of Offshore Operations
accordingly in these cases the wavemaking resistance should be small. For Froude numbers higher than 0.15, the total load will consist of viscous drag and pressure drag..
FIGURE 8.2 TOWING OF A MODEL WITH TOWING STAFF Plots of friction drag loads by the Schoenherr line as well as the measured loads versus model towing speeds (and Froude number) are shown in Fig. 8.4 for the loaded barge. Note that the experimental data are closer to the Schoenherr curve for Froude number less than 0.15 and drag is primarily frictional. Small bow waves were observed to form during these tests. The towing test data using both towing staff and a bridle arrangement are shown. The difference between the measured data and the calculated friction drag is the residual resistance (e.g., wavemaking drag). This residual resistance is multiplied by X3 where ) = 48 to scale up to the prototype value. Then the total towing load for the prototype is obtained by adding the residual
TABLE 8.1 MODEL AND PROTOTYPE RESISTANCE ON BARGE AT LOADED DRAFT Wetted Area, A. = 24.2 sq.ft Length, e m = 10.7 ft Mass Density of Water, pm =1.94; p, =1.98lbssec' /ft4 Kinematic Viscosity, v=1.174x 10 5ftZ /sec(@62°F)
TOWING SPEED Up
U.
MODEL RESISTANCE Rtm
Ctm
Cfm
ft/sec ft/sec lbs x 103 x 103 7.3 1.05 0. 14 5.40 4.73 9.0 1.30 0.20 5.03 4.51 10.4 1 .50 0.26 4.91 4.38 12.5 1 .80 0.37 4.86 4.22 14.2 2.05 0.48 4.86 4.11 15.9 2.30 0.58 4.66 4.02 17.7 2.55 0.73 4.78 3.93 19.3 2.78 0.85 4.68 3.86 21.1 3.05 0.98 4.48 3.80 22.9 3.30 1.20 4.70 3.74 24.9 3.60 1.45 4.76 3.68 3.85 26.7 1.64 4.71 3.63 28.4 4. 10 2.00 5.06 3.59 1FT=0.3M; 1LB=4.45N; °C=5(°F3 2)/9
PROTOTYPE RESISTANCE Ctm = C,P
x 103 0.67 0.52 0.53 0.64 0.75 0.64 0.85 0.82 0.68 0.96 1.08 1.08 1.47
Cfv
x 103 1.77 1.72 1.69 1.65 1.63 1.60 1.58 1.56 1 .55 1.53 1 .51 1.50 1 .49
Cf1 +0.4
C,P
R,p
x 103 2.17 2. 12 2.09 2.05 2.03 2.00 1.98 1 .96 1.95 1 .93 1.91 1.90 1.89
x 103 2.84 2.64 2.62 2.69 2.78 2.64 2.83 2.78 2.63 2.85 2.99 2.98 3.36
lbs 8,283 11,803 15,642 23,201 30,942 36,841 48,940 57,160 64,633 82,499 102,330 117,266 149,592
312
Chapter 8 Modeling of Offshore Operations
FIGURE 8.3 TOWING OF A MODEL WITH A TOWING BRIDLE resistance to the prototype friction drag (which uses Schoenherr friction coefficient plus 0.0004 for roughness allowance). Test runs with the bridle arrangement which allowed all six degrees of freedom for the barge model showed similar results without appreciable difference. The resistance with the turbulence simulator was generally found to be about 510 percent higher. 8.3 SUBMERSIBLE DRILLING RIG TOWING TESTS Unlike a selfpropelled barge or ship, large offshore structures for the most part require external power for transportation to the offshore site. Gravity submersible drilling rigs with large mats (Fig.3.7) are designed to be towed by tugs to the work site and be submerged to rest on the bottom by controlled flooding of ballast chambers in the mat and caissons. The following describes a test designed to study such a rig's towing characteristics under different environmental conditions . The test was done at a scale of 1:48. The model was constructed with the mat, caissons, and superstructure all reasonably representing the prototype dynamically as well as statically.
Section 8.3 Submersible Drilling Rig Towing Tests
313
x
2.75 MEASURED DATA
2.5 SYMBOL
TOWING METHOD
BON THRUSTER
TIRBULENCE SIMLLATOR
BRIDLE STAFF
OPEN
X
OPEN
NO NO
BRIDLE •
ORIOLE
OPEN PLUGGED
YES YES
1.5
2.0 2.5 TOWING SPEED, ft/sec
2.25 2.0 1.75
1.0 .75 .5 .25 _I
0 O
.5
1
I
1.0
I
I
I
I
I
.055 .068 .096
I
I
1
1
1
3.0 3.5 4.0 4.5 I
.123 .15 .177 .285 .218 FROIAE MNBER
FIGURE 8.4 FRICTION DRAG FORCE ON MODEL AT LOADED DRAFT 8.3.1 Test Description To test the rig's towing characteristics, the model is pulled along the tank with an overhead bridge. In this respect it should be noted that the large bridge moving on rails fails to model the characteristics of a tug boat. Thus, the interaction effect between the two floating bodies is not accounted for and the effect of the motion of the tug on the towing load is not considered in these tests . Nonetheless , such tests to represent towing by tugs are quite common, and determine the powering requirements. The model is ballasted to the desired towing draft by filling various ballast chambers. The nonlinear stiffness characteristics of the towing line is modeled. The towing line is usually attached to a bridle which, in turn, is attached to the model. In the case of the aforementioned rig model, the towing draft was set at 64 mm (2.5 in.)
314 Chapter 8 Modeling of Offshore Operations
which is 3m ( 10 ft) in the prototype. A nylon line of diameter 3 mm (1 /8 in.) was used with the bridle arrangement. The model is towed in still water, in waves and, sometimes, with wind as well. The survivability of the rig is also tested with the model held stationary in waves by the tow line . The test is designed to give an indication of the maximum sea state the rig can endure while in transit. The significant height of the sea state is increased until the rig's stability appears to be marginal. In some instances, the rig actually capsizes with increased wave height during testing . The critical towing line load is monitored during these tests . The damage stability of the rig is also examined during this type of test. To simulate damage, one critical tank in the model is flooded . The rig's behavior in waves in this damaged condition is examined. In towing tests, the speed is increased in increments up to a maximum design speed for the rig . The towing tests are repeated in waves. The waves are chosen to represent the expected sea states enroute during the period of towing . The rig model under discussion was designed to be towed at (prototype ) speeds between 0 .51 m/s and 2.56 m/s (1 and 5 kt) and a wave of significant height of 1.2 in (4 ft). Wind loads are generated using fans . To determine the wind loads actually seen by the model, the fans or blowers are aimed at the model while it is stationary in still water. The mean wind load is taken to be equal to the mean load recorded by the tow line load cell. The above model was towed at a speed of 2 m/s (4 kt) in a 30.7 m/s (60 kt) wind with a head sea and at 1 and 2 m/s (2 and 4 kt) in a 10.2 m/s (20 kt) wind with a 30° heading. 8.3.2 Test Results While the results of these tests are specific for the particular structure tested, they are described here to illustrate the type of results expected from these tests. The results are useful in determining the power requirement of the tug boat as well as the stability of the structure in the environment encountered during towing to the site. The adequacy of the towing line is also determined. The results of towing test in still water are presented in Fig 8 .5, which is a plot of the average tow line load versus the towing speed for the five runs that were made. The results shown have been scaled up from the model test using Froude scaling, and therefore represent prototype values . The scaling distortion due to viscous effects has not been included in this scaling.
Section 8.4 Towing of a Buoyant Tower Model
315
ED
N
DRAFT= 3m (10ft)
0 J
N
6
2
3
4
5
6
TOWING SPEED. KNOTS FIGURE 8.5 TOWING TESTS OF A RIG MODEL IN STILL WATER 8.4 TOWING OF A BUOYANT TOWER MODEL A large buoyant tower is transported from its construction site in floating condition with the help of a tug boat . Depending on the available draft and route of tow, the tower may be towed horizontally or vertically . The towing characteristics of a buoyant tower model is studied in a wave tank test . In one such test, the buoyant tower consisted of a gravity base connected to the tower by a universal joint . The tower itself had three sections separated by bulkheads : a large buoyant section at top, a middle section that is usually flooded during operation , and a lower ballast section. The ballast, usually concrete, is placed in an annular space in the ballast section before towing. The ballast section is flooded at site for swingdown and submergence. The tower has a fluid swivel at the top which allows the tanker to weathervane about the tower.
316
Chapter 8 Modeling of Offshore Operations
RING (TYP)
Y8' 00 X . 049" WALL
8" OD X . 050" WALL
"LOVEJOY" NRB4 UNIVERSAL JOINT
FIGURE 8.6 MODEL OF A BUOYANT ARTICULATED TOWER
Section 8.4 Towing of a Buoyant Tower Model
317
8.4.1 Model Particulars Models of this type are generally made of standard aluminum tubes and sheets. The outer dimensions are scaled following the chosen scale factor . The overall weight and location of the C.G. are adjusted to conform to the scaled down prototype values. The bulkheads are located at the scaled locations . The thickness of the materials used at various elevations and stiffening rings inside the tower are not usually scaled. The yoke and swivel connection is placed on the top of the model. The boat bumper is placed on the top shaft near the still water level (SWL). A drawing of the current model is shown in Fig. 8.6. The yoke (not shown) made a 48° angle to the tower centerline during the tow out operation. Ballast weights in the form of lead shots are placed at selected locations to match the scaleddown prototype weight, center of gravity in air (and the natural period in air as well) for the condition tested . It is advisable to avoid placing ballasts on the outside of the submerged portion of the model. In the present case ballast weights in the form of lead shots inside 19 mm dia x 0.3 m long (3/4 in. x 1 ft) PVC tubes were placed in the annular space in the ballast tank. A base was constructed from the same material. The base was attached to the bottom of the tower by means of a commercially available universal joint. For all the tests performed here, the ballast in the base was adjusted so that the base remained level under water and had a small net downward load. 8.4.2 Towing Tests The model test simulated a towing scenario where two identical tugs towed the structure in parallel by means of two identical towing lines. The stiffness of the towing lines was simulated in the model , with strands of rubber bands of unequal lengths which modeled their nonlinear behavior. Each scaled line had a total length of 427 m (1400 ft) including a 64 nun (21/2 in.) dia 305 m (1000 ft) long steel wire in series with two 279 mm (11 in.) circumference, 40 m (131 ft) long nylon ropes. The theoretical loadelongation curve for one such line and the actual calibration curve of the model towing line are shown in Fig. 8.7. In order to study the optimum tow orientation of the tower, several horizontal and vertical tow positions are tested. In the horizontal position, the tow line may be attached to the base end as well as the yoke end. In the vertical position, the tow line may be attached at different points on the tower. The overhead bridge is normally used as the carriage for towing. If a tug boat is employed for towing the structure, it allows the study of the snap loads on the line due to the relative motions of the boat and the tower.
318' Chapter 8 Modeling of Offshore Operations
The vertically floating tower was towed using a tug boat model to determine if the relative motion of the boat and the tower produced an increase in the towline load. The tug boat was remotely controlled . The simulated towline described before was connected to a bridle located 102 mm (4 in .) (4.9 in or 16 ft prototype) above the bottom of the buoyancy chamber of the vertically floating tower . The draft of the tower was 2210 mm or 87 in . ( 106 in or 348 ft prototype). The yoke was turned towards the towing direction so that the tower made an angle of 7.7 degrees in the equilibrium position with the line slack. A sketch of the test setup is shown in Fig. 8.8.
[I M' 2Y," DIA WIRE ROPE 1 31 . 5'  II" CIRC . NYLON ROPE  MODEL •
'•°" THEORETICAL
is
i7
3
4
5
6
7
B
9
is
LOAD. LBS
FIGURE 8.7 CALIBRATION CURVE FOR TOWING LINE Before starting the test , the speeds of the tug boat pulling the tower and the bridge were synchronized. As expected , the maxima appeared as sharp peaks indicating an impact type loading due to the variation in the motions between the tug boat and the tower . The magnitudes and duration of these snap loads are difficult to scale, however. The design of the tow line should accommodate these snap loads. In the design, it is advantageous to choose a towing speed such that the line does not become slack from the oscillating wave loads during towing in expected sea states.
Section 8.4 Towing of a Buoyant Tower Model
319
REMOTELY CONTROLLED IT.O®0 H.P. TUG BOAT
WAVE DIRECTION
a
FIGURE 8.8 TOWING TEST SETUP FOR A VERTICALLY FLOATING SPM 8.4.3 Bending Moment Tests During the horizontal tow in waves, the tower experiences a bending stress (due to hogging and sagging along its length) in the central shaft that may be severe from the design point of view. A model test may be designed to determine the bending
320
Chapter 8 Modeling of Offshore Operations
moment in the central shaft of the model. Two methods may be employed to determine this moment in a model test. The first one is to build a model that scales the stiffness of the prototype in bending . Desired locations on the tower may then be strain gauged directly. This is expensive and difficult to achieve (See Chapters 7&9). A second method is to segment the model at the desired point of measurement and a load cell is used to connect the two, which allows the measurement of the moment . One such test is described below. The model is segmented at the points where the bending moment is desired, and the sections are pinned and mounted on bearings (Fig. 8 .9) so that they are free to move in the vertical plane at these points. Load cells are attached offcenter between the separated sections to record the loads due to bending in the vertical plane. Thus, the recorded load times the distance of the load cell from the center of the shaft produces the bending moment. Two points on the tower were chosen  one 953 mm or 37.5 in. (45.7 in or 150 ft prototype) from the bottom U joint pin and the other 1270 mm or 50 in. (61 in or 200 ft prototype) from the pin. The center of the load cells was located at 84 mm (35/16 in.) from the center of the shaft.
FIGURE 8.9 ARRANGEMENT OF LOAD CELL FOR THE BENDING MOMENT TEST It is critical that the two shafts at the cut do not touch during the test runs. The effect of the friction in the bearings and hinges on the load cells is usually not known. However, in order to avoid any effect from this uncertainty, the load cells should be recalibrated in place for appropriate scale factors. The inplace calibrations for the tower showed a small amount of hysteresis and the best slopes (in the least square sense) were
Section 8.4 Towing of a Buoyant Tower Model
321
considered to be the calibration constants. Figure 8 . 10 shows the floating tower being tested in regular waves with the bending moment cells. 8.4.4 Scaling To Prototype In this type of tests , the model is built and the tests are performed on the assumption that the Froude law is applicable. It is known that in an inertia predominant system, the data between the model and prototype may be scaled following Froude 's law. This assumes that the forces due to waves are mostly inertial, and any nonlinear drag effect is negligible . In the present case , this latter assumption is not necessarily true . Since the drag coefficients in the prototype (in the turbulent region) are generally smaller than those in the model , Froude scaling is expected to predict higher loads , thus giving conservative results . This is, however, not necessarily true for the motion and other responses . In this type of tests, if the model test results are scaled up to the prototype values irrespective of the inertia or drag predominance, care should be exercised in interpreting the prototype results.
FIGURE 8.10 SHAFT BENDING MOMENT MEASURED ON SPM IN WAVES An example of how the towing loads should be scaled up if the drag coefficient (CD) is known as a functions of Reynolds number (Re) is shown here. In a flow normal to the axis of a circular cylinder , the drag coefficient as a function of Reynolds number is known (Fig. 2. 2). Note that this graph is not strictly applicable for the buoyant vertical tower since the tower was not of uniform diameter nor was it strictly vertical during towing . Therefore, the values obtained from the use of this graph are for illustrative purposes only.
322
Chapter 8 Modeling of Offshore Operations
The tubes making up the tower model and their frontal areas are tabulated below.
(in.)
Frontal Area (in.2)
0.49 ft/sec
0.73 ft/sec
8.0
184
2.6
3.0
5.2
6.3 4.5 5.0
46 170 14
2.0 1.5 1.6
3.0 2.2 2.4
4.1 2.9 3.2
Tube Dia
Reynolds Numbers (x10') at Speeds 0.93 ft/sec
The drag coefficient in this range of the Reynolds number is about 1.2. For the base, approximate drag coefficients are obtained from Hoerner ( 1965) as follows: CD
pyramid base edge
Frontal Area (in.2) 14.9 49.5
cylinders
39.3
0.5
Shape
1.0 0.5
These values are assumed not to vary with scale. In case of the prototype, the corresponding drag coefficients for the tower are shown in the following table.
Tube Dia. (ft) 32 25 18 20
Frontal Area (ft2) 2944 736 2720 224
: 2 kts S R CD x Oe 6 8.7 0.43 6.8 0.42 0.42 5.0 5.3 0.42
S : 3 kts Re CD x10'6 13.0 0.41 10.0 0.43 7.3 0.42 0.43 8.0
Speed : 4 kts Re CD x106 17.0 0.41 14.0 0.41 9.6 0.43 10.0 0.43
During towing, the tower was not vertical . The frontal area was adjusted due to the angle of inclination of the tower. The towing load was calculated from
F = . pUZCDAcose (8.5)
Section 8.5 Launching of Offshore Structures
323
where U = towing speed, A = frontal area, and 0 = angle into the direction of tow from vertical. The above formula was used to compute towing loads in the model scale. These values are compared with the actual measured model loads in the following table: Force in Pounds (model)
U
9
ft/sec
(deg)
Tower
Base
Total
Experimental
0.47 0.71 0.96
19.7 29.6 37.6
0.74 1.54 2.57
0.09 0.20 0.37
0.83 1.74 2.94
0.86 1.56 2.41
The calculated values compare favorably with the experimental data despite the approximation involved in the evaluation of CD. A similar table is then developed for the prototype: U (kts)
0 (deg)
Tower
1.93 2.91 3.94
19.7 29.6 37.6
30.3 68.1 105 .1
Force in Kips (prototype) Base Total Model Projected 10.0 22.1 40.9
40 90 146
95 173 267
The last column in the preceding table represents the scaled up model data using Froude scale and disregarding the Reynolds number effect . As noted earlier, these values are much higher than the calculated prototype values using appropriate drag coefficients . Thus, the model data should be scaled up not by Froude scaling but by proper care of the CD values. The tests showed that a horizontal tow should be used to avoid the potential risk of high tow line loads associated with the vertical tow. The tow line loads can be reduced by using a softer spring in the tow line. This permits limited tower motion and minimizes peak loads on the tow line. 8.5 LAUNCHING OF OFFSHORE STRUCTURES Most offshore structures are built on land close to water . Use of a graving dock is a popular facility for building and launching largevolumed structures offshore before towing to the site for installation . The Khazzan oil storage structures described earlier were built this way. Some of the large concrete structures in Norway are built in deep
324 Chapter 8 Modeling of Offshore Operations
water in fjords by ballasting the structure as the construction progressed. Then the structures are deballasted to a shallow draft before towing out of the fjords. 8.5.1 A Unique Launch Submersible drilling rigs with large bases were built by CBI at its Pascagoula Construction Yard in Mississippi. In the yard the launching dock had a much greater elevation than the existing water level adjacent to the dock. Therefore, a unique, patented launching procedure was developed. Openbottom buoyancy cans (up to 10.7m or 35 ft in diameter) were designed to support the structure on a cushion of compressed air during launch and load out in the sheltered bay off the Gulf of Mexico. Before the final towing of the rig under its own buoyancy in deep water, the system of buoyancy cans was disconnected from the rig by deairing and thereby submerging the cans in place. Since this launching procedure had not been used in the past, a test was conducted to investigate the launching sequence and the stability of the rig on the flotation cans. A simple model of the rig was built at a scale of 1:48. Nine buoyancy can models for supporting the rig were also built. The procedure outlined in the launching sequence was followed in the test in 15 discrete steps, and the pressures in each buoyancy can were monitored and adjusted as necessary. Air supply reservoirs were connected to each flotation can to accurately model the soft volume needed in the test with respect to the prototype can volume. (Note that soft volume refers to the air trapped in an openbottom enclosure as opposed to hard or buoyant volume in a closed container or vessel). The seakeeping characteristics of the rig were studied while supported by all nine cans in wind and waves. Finally, the cans were ballasted beneath the structure, allowing the rig model to submerge to its design towing draft. 8.5.1.1 Modeling of SoftVolume Cans Under normal testing conditions, the atmospheric pressure in the laboratory is close to the atmospheric pressure prevailing at the site. The hydrostatic head in a soft volume depends on the pressure difference between the inside and outside air pressure and, as such, poses no scaling problem. However, this head is directly related to the compressibility of air inside the soft volume which depends on the absolute pressure. Assuming that air follows Boyle's gas law, PV = constant
(8.6)
where P = absolute pressure (atmospheric + hydrostatic) of air in the soft volume, and V = trapped air volume. This law must be met in the model as well as the prototype. If the geometric similarity is maintained between the model and prototype, then the
Section 8. 5 Launching of Offshore Structures
325
volume, V can be modeled properly. However, the pressure, P, includes the atmospheric pressure which does not scale . The pressure in the model environment is higher than the scaled down value since the scale factor is greater than one . In fact, the distortion is greater as the scale factor becomes larger . There are two ways to circumvent this problem . In the model, the soft volume members may be placed under vacuum (of appropriately scaled negative pressure or below atmospheric pressure) which is difficult to accomplish. The other method is to distort the soft volume so that the gas law is satisfied in the model scale . This is achieved by connecting additional volume to the soft volume members . It is clear that as the pressure changes, the amount of this extra volume also changes . Thus, in a test, e.g., during deballasting, a variable reservoir volume is needed for each can. After the submersible rig has been launched from the dock, it is supported by freely floating soft volume cans. These support cans are modeled as shown in Fig. 8.11. Let us consider a prototype can of unit cross sectional area and small thickness initially floating in Position 1 with its freeboard equal to Hop . The internal pressure, Pip, corresponds to a differential head , Hip, between the water level inside and outside. If Po represents atmospheric pressure, then Plp=Po+H1p
(8.7)
and the air volume in the can is Vlp,
(8.8)
Vlp=Hop+Hip
AHP
PS
Ham=
Hw H8
PP VIP
PZPVn Q
OIP I
POSITION I
POSITION 2
FIGURE 8.11 MODELING AN OPENBOTTOM SUPPORT CAN Note that the volume has been normalized by the crosssectional area of the cans. if the floating can is displaced by an amount, AHp, to Position 2, its internal pressure will now be P2p.
326
Chapter 8 Modeling of Offshore Operations
P2p = Po + H2p (8.9) where H2p is the differential head between the water level inside and outside of the can. The new volume is V2p. V2p = Hop + AHp + H2p (8.10) Assuming the products PV to be constant between Positions 1 and 2, (Po + H1pXHop + Hip) = (Po + H2p)(Hop + AHp + H2p) (8.11) Solving this equation for the differential pressure head at Position 2, we get / 1/2 H2P 2 ((Po+Hop+AHp)2+4{Po(HI,_AHp)+HI,(H.,+HIp)})
2(Po+Hop+,&Hp)
(8.12)
When the can is modeled with a scale factor of A, the gas law gives us
(8.13)
where Vr is the volume of air that must be added to the can volume to compensate for the fact that Po will not be scaled in the model. Solving for this added volume, we get Po(^ 1)(HIp H2p AHp)
(8.14) Vr H2pH1P
Since the above relation assumes the density of the supporting liquid to be equal for the prototype and model and we are representing a prototype that floats in seawater by a model floating in fresh water, the equation for Vr must be modified to account for this difference.
Section 8.5 Launching of Offshore Structures 327
In the following, the subscript sw refers to seawater and the subscript fw refers to fresh water. If we consider the same prototype can floating in both seawater and fresh water with an identical mass of air in the can for both cases, then PpswVpsw = PpfwVpfw (8.15) For the can to be floating in equilibrium in both cases, the upward force on the can must be the same whether the can is floating in seawater or fresh water. If the upward forces are equal then the following must be true assuming that the buoyancy force from the submerged can wall is nearly the same for both cases: Ppsw = Ppfw
(8.16)
Vpsw = Vpfw
(8.17)
and
The can pressures are taken to be Ppsw = Po + Hlpsw 7sw
(8.18)
Ppfw = Po + Hlpfw 7fw
(8.19)
and
where 7 is the water density. Solving for the fresh water differential head we get ( 8 . 20)
Hlpsw
H1 Pfw =
Iw
which shows the fresh water differential head to be equal to its corresponding sea water head multiplied by the ratio of seawater density to fresh water density. If we multiply the differential pressure head terms in Eq . 8.14 by the ratio 7sw/Yfw, we get
Po(x 1)[ V = r
(H1P , H2psw) 
Jµ 7
AHP] I l 1 H2psw  HI, )
(8.21)
fiv
which gives the reservoir volume that should be added to the model can floating in fresh water in teams of the differential head of the prototype can floating in saltwater. The reservoir volume, Vr, allows us to correctly model the compressibility of the soft air volume.
328
Chapter 8 Modeling of Offshore Operations
Modeling the prototype seawater with fresh water also affects the model's freeboard. If we again consider the same prototype can floating in both seawater and fresh water, then Vpsw = Hopsw + Hlpsw
(8.22)
Vpfw = Hopfw + Hlpfw
(8.23)
and
Using Eq. 8.20 for Hlpfw and the equality in Eq. 8.17, and solving for Hopfw gives us
(8.24)
H0 fr = Hopsw + H1P , 1 Y Y SW JW
The fresh water freeboard of the model, Homfw, will, therefore, be H0Pfr Homfw x
(8.25)
or Hops, + HIP, (I  
(8.26)
Homfw =
Since the model should float at  H,,p_ Homfw x
(8.27)
to correspond to the prototype, the model must be raised in the water by an amount AHomfw where
Hops, + H1Ps 1AHom _ (Hs
or
7fiv
(8.28)
Section 8.5 Launching of Offshore Structures
Hlps,(1 l l Yfµ AHomfw  X
329
(8.29)
To raise the model to the correct elevation we can subtract the amount AHomfw from the reservoir volume . This will in effect move the volume AHomfw from the reservoir into the can , causing it to float at the correct freeboard . The compressibility of air is still correctly modeled because the can reservoir system volume remains constant. Our equation for Vr is now
Po(Xl)l s. (Hlpsw H2psw)AHp] yr
_
^
l'YSW Yfi, HtpM J_ + (8.30)
l ^ H2p Hlpsw / Yfw
which gives us the reservoir volume that will allow us to correctly model the compressibility of air and cause the model to float with the correct freeboard. The volume calculated by Eq. 8.30 will have dimensions of cubic units per square unit of model cross sectional area . An example of the extra volume needed for proper modeling is given as follows. Let us assume a scale factor of 1:48 for the model . Also, consider a prototype freeboard for a can to be 2 .6 m (8.5 ft) and the prototype pressure to be 1.78 m (5.85 ft) of seawater. Then, the calculation based on the above formulas requires an extra reservoir volume in the model (with fresh water) of 0.24 m3 (8.55 cu. ft). This is a substantially large volume compared to the model volume. This volume should be provided in the form of closed air reservoir on land which is connected to the openbottom can by flexible pipes. 8.5.1.2 Submersible Rig Model The section of a submersible drilling rig model near and under the water was modeled using a scale factor of 1:48 . The geometric similarity was maintained only where it was required for this test. The legs and super structure of the rig were not modeled. The top and bottom of the mat structure were fabricated from single pieces of plexiglas and secured to the side walls with adhesive and screws . A three legged adjustable platform was constructed . Weights were bolted to the platform and properly
330
Chapter 8 Modeling of Offshore Operations
positioned to place the center of gravity of the completed structure at the point shown in Fig. 8.12.
3.5
11.75" 49.5" P  PORT S  STARBOARD
17.625" TOP OF .25" thk 14.5" DIA PLATE
5  ID LB WEIGHTS
I
ID
I FIGURE 8.12 COMPLETE PLATFORM MODEL WITH BALLASTS
Section 8. 5 Launching of Offshore Structures
331
Since the top and bottom of the mat were transparent , positioning of the launch cans was accomplished by marking circles of various colors in the appropriate places TABLE 8.2 LOCATION OF ROD AND RIG ON LAUNCH PLATFORM Q1
I
I S
B
O
J
STEP
"0" OVERHANG (IN)
I
14.75
19.25
15.50
2
14.75
16.83
17.93
3
19.25
16.00
14.25
4
23.75
14.48
11.28
"S' CG FROM STERN (IN)
"6. CG FROM BULKHEAD (IN)
5
23.75
13.70
12.85
6
28.25
12.53
8.73
7
33.00
10.93
5.58
8
33.00
8.93
7.58 4.60
9
37.50
7.40
10
42.00
3.75
3.75
II
42.00
3.73
3.78
12
43.75
2.94
2.81
13
45.50
2.00
2.80
14
45.5
2.18
1.83
15
FLOATING ON LAUNCH CANS
and matching and centralizing larger colored circles placed on the top surface of the launch cans. Each launch can was handled individually and was connected to its reservoir placed outside the tank via tygon tubings. Each launch can pressure was measured by a 203 nun (8 in .) well type, Dwyer Manometer. The launch cans were scaled geometrically. The compensating scaled
332
Chapter 8 Modeling of Offshore Operations
reservoir volumes were provided with valved bulkhead fitting at the bottom of the container. Water was then added to the container via this fitting to provide a liquid piston to vary the volume of the reservoir for various stages of the launching procedure. The water level of the wave tank was adjusted to 54 mm or 21/8 in. below the top of the leveled launching platform . This corresponded to a prototype dock height of 2.6 m (8.5 ft) above the water level. 8.5.1.3 Launching of Mat on Cans During the launching, the mat was supported at the bow by the floating soft volume cans and at the stem by a 13 mm (0 .5 in.) diameter aluminum rod. The aluminum support rod at the stem was positioned to support the model at the centroid of the launch beam reaction. The launch procedure was simulated in fifteen discrete steps (Table 8.2). The structure ' s overhang past the bulkhead face, the launch beam reaction centroid location, and the pressure in each can were precalculated. For each launch step , the rig and the support rod were positioned at the desired locations and the mat was held level by hand as the cans were pressurized to the specified values . The mat was then released and the angle from horizontal was noted. The pressures in the cans were then adjusted as necessary for the cans to support the model parallel to the water and the new pressures were noted . On the final launch step the model was allowed to float free of the bulkhead, supported entirely by the nine soft volume cans. The submersible model was successfully launched several times following the procedure outlined through computation . The pressures in each can during the launching steps were noted . Observing the can pressures during the launch serves basically to check the statics calculations made earlier. Throughout the launching, the model appeared to be quite stable. 8.5.1.4 Seakeeping of Rig on Cans The seakeeping of the rig when supported by the cans was tested by floating the model in waves. The model was oriented so that the waves approached it from several different directions . The rig exhibited good seakeeping characteristics when exposed to wind and waves . The model experienced little motion in roll or pitch.
8.5.1.5 Deballasting of Cans Lowering the rig from its position on top of the cans to the point where it floats on its own requires that air be removed from the cans . The model was lowered into the water by venting air from several different combinations of cans . During each
Section 8.6 Jacket Structure Installation
333
deballasting test, the rig began to level itself as soon as the mat entered the water and was supported by its own buoyancy. To observe the behavior of the rig in a situation where air is lost from only one can while the rig is supported on the cans, the rig was floated level using the can pressures determined from the launching tests and then unbalanced by venting air from one can at a time . A stability problem was noted when the #5s can was lost. After the rig tilted approximately 46 degrees, the #2p and #3p cans were vented to bring the rig back to a nearly level position. A few tests were performed initially without the auxiliary volumes . The system appeared very stiff and a small change in pressure in the cans affected the behavior of the rig on cans visibly. After comparing the behavior of the model during the stability test with and without the auxiliary , volumes containing air, it was apparent that the extra  air volumes did influence the model behavior to a considerable extent. The operational tests performed on the launching of the submersible rig model was generally qualitative in nature . However, they were valuable in learning the limits of the operation and provided the operations crew a firsthand experience in the rigs behavior during the various stages of the launching procedure. 8.6 JACKET STRUCTURE INSTALLATION The installation of a steel jacket platform requires a three step procedure sequentially [Bhattacharyya ( 1984)]. These are loadout, launching and upending. The loadout is carried out on launch barge . In this case, the jacket is fastened to the launch barge forming a coupled jacketbarge system. In launching, the jacket slowly slides off the barge. Until this moment they act as a unit . During upending of the jacket, the dynamics of the jacket in swingdown is an important consideration. For this phase, a derrick barge is often used to install small jackets. 8.6.1 Scaling of Jacket Installation Parameters During loadout, the parameters that are of importance and need consideration in model testing are the ballasting , tension in the derrick winch, fender reactions, mooring line forces and rocker arm reactions [see Graff ( 1981) p. 113]. In launching, the surge of the barge is important. In addition, the rocker arm reactions, the trajectory entry velocity, and sinking of the jacket, trim and draft of the barge are also of interest. The forces during this operation include viscous, inertial and gravitational forces and static and dynamic pressure forces . Both Reynolds and Froude similitudes are not possible. The viscosity is considered to have a modest influence on the results of such a test using a Froude model. All other parameters , e.g. displacement, velocity, friction
334 Chapter 8 Modeling of Offshore Operations
factor, trim angle, hook elevation must follow Froude's law. Since the launch velocity is low, the flow is laminar, in which regime the Reynolds force cannot be ignored. However, jacket geometry is prone to induce turbulence . Additionally, turbulence flow may be artificially induced as is done in a ship model test. In upending, the crane hook load, hook elevation, jacket ballast and orientation, righting moment and seabed bearing load are important . In this case, the forces include buoyancy and jacket inertia. In launching, the modeling of the jacket trajectory is of fundamental importance including the attitude of the jacket along the trajectory. Geometric similitude is assumed in all cases . In loadout, the winch, fender and mooring line forces must be related to the jacket weight and the friction forces. Thus, a characteristic equation can be written as F F W f(µ'w 'w)
(8.31)
where Fm , Fw, Ff = mooring line, winch and fender forces, respectively, W = jacket weight and µ = coefficient of friction between the jacket and the barge launch rails. For the barge ballast and rocker arm hinge reaction , the characteristic equations are
W f (P)
(8.32)
and F,=f
(8.33)
where Fb, Fr = barge ballast and rocker arm reaction and x = jacket travel. While the model must be geometrically similar to the prototype for loadout and upending tests , certain less important functional fittings , e.g. bollard eyes, structural members, rocker arm hinges , deck openings, winch assembly, etc. may be designed only from the point of view of practical convenience . Similarly, in loadout, only the weight similarity of the jacket is necessary , while, in upending, the buoyancy distribution of the jacket is also necessary . If the model material and fluid density are different from the prototype, proper adjustment in the member dimensions (e.g. thickness of tubular members) is needed to model weight and buoyancy simultaneously. For ballasting the model, a different liquid density may have to be chosen for filling the tubular members such that the modeling of the tubular cross section is satisfied. In
Section 8.6 Jacket Structure Installation
335
order to maintain the weight, C.G. and moment of inertia of the jacket at each level, additional members in the form of flat plates , rods, lump weights, etc. may have to be added and some smaller members omitted. 8.6.2 Launching Test Procedure One of the most critical steps in the operation of an offshore steel jacket is the installation operation [Graff (1981)]. The installation operation consists of loadout, launching and upending in sequence. Due to the high risk involved, a physical simulation with a scale model is often recommended to verify the intended procedure. Model testing of the three step procedure is considered a necessary and powerful tool before the full scale structure is loaded out. A model test on launching helps address the following areas: • amount of launching force required to initiate and then maintain the motion of the structure, taking the draft and trim conditions of the barge into account. •
structural strength of the rocker arm and barge hull necessary during launching.
• difference in the trajectory between a singlehinged and a double hinged rocker arm. • required buoyancy and its distribution over the structure , including any added buoyancy. • values of hydrodynamic force coefficients (CD,CM) needed to analyze the structure trajectory. Launching structures from a barge at site is a common procedure offshore in the installation of a structure on the ocean bottom . The procedure involves pulling or pushing the structure longitudinally on skid rails using jacks and /or winches. In loadout and launching operations, the jacket and launch barge work as a system. In model testing, the jacket and the barge must be properly modeled. Jacket models are usually made from plastic or aluminum alloy . Standard tube sizes are chosen in the jacket model . Therefore, this criterion generally governs the actual model scale after preliminary scale selection based on the facility. This may lead to some peculiar scale factor . The members which cannot be modeled even by the smallest tube size available are often omitted or modeled incorrectly. The similarity in buoyancy is achieved closely , but not necessarily exactly. Certain additional members such as flat plates, and solid rods are included judiciously so that the level and face
336
Chapter 8 Modeling of Offshore Operations
configurations remain more or less unaltered. Additional lumped masses are attached to the jacket model to satisfy the overall requirements of weight , inertia and C.G. In order to satisfy the Cauchy scaling law, the model material should have a lower density than the prototype. When this is desired , perspex is often found to be the suitable material for both the jacket and the barge. For the loadout operation [Bhattacharyya, et al ( 1985)] a quay of suitable height is built on one side of the wave tank . The winch drive may be simulated by a variable speed electric motor fitted at forward of the barge model . The fender reaction and winch pull are monitored by load cells during the load out. The rocker arm hinge reaction is also recorded (Fig. 8 .13). The loadout operation is performed in discrete steps . For a given jacket , the tide range within which it can be loaded out can be found from the simulation results. This, in turn , gives the ballast rates needed for any given tide range to accomplish the loadout. During the launching operation, the history of the jacket trajectory is of importance. The entry velocity of the jacket is its velocity at the instant of launch and is of great significance [Bhattacharyya, et al. (1985)]. The estimation of the prototype launch velocity, however, is not easy because it depends on the coefficient of friction between the surface of the barge launch beams and the jacket launch trusses , variable trim angle as well as the travel. These parameters are not well defined quantitatively. Typical dynamic friction coefficient may be taken as 0.02 and an initial trim angle as 3 degrees. Since the full scale friction is often an unknown, it is customary to verify that the jacket launching is possible over a range of friction coefficients. This requires controlled variation of friction between the sliding surfaces. In one method [Rowe and Clifford (1981)] the launch slideways are replaced with free running rollers which may be rotated about a vertical axis thus pointing all the way from along to across the running axis of the roller. When the rollers line up, the slideway friction is negligible. When they are turned 90° on the barge, full material friction is encountered. Friction may be varied between these two values. A system of side rollers is used to guide the jacket. In an alternate method, Bhattacharyya, et al. (1985) employed small rollers fitted at suitable intervals over the length of the launch beam. To achieve a predetermined entry velocity, a pulley was fitted onto the deck in a reverse rigging arrangement. The jacket was pushed by an Lshaped hook so that it was disconnected the instant the rocker arm tilted. The speed of the electric motor was calibrated for specific entry velocity. A limit switch was fixed onto the side of one of the rocker arms so that it recorded the pulses caused by the contact with a few curved protrusions fitted at several points to the side of the launch truss along its length. Entry velocity during launch could be obtained from the relative positions of these pulse records.
Section 8. 6 Jacket Structure Installation
LAUNCH BEAM
ROCKER ARM
LAUNCH BEAM
ROCKER ARM
337
HINGE
LOAD CELL
77 77 WITH LOAD CELL
FIGURE 8.13 ROCKER ARM ARRANGEMENTS IN MODEL [Bhattacharyya, et al. (1985)] Upending is done in the following steps: the main legs of the jacket are ballasted; the derrickbarge crane is then used to lift the jacket until it is nearly vertical. At this point it is further lowered to the seafloor by controlled flooding of the jacket legs and auxiliary buoyancy tanks (if present). The procedure is followed in stages in a model test. Since the force system represented by the ballasts and crane hook load are similar, the prediction to the prototype values from the model test results is straightforward.
338
Chapter 8 Modeling of Offshore Operations
SINGLEBARGE METHODS
W/L
W/L
CONVENTIONAL LAUNCHING (SIDEWAYS FROM BARGE SIDE)
CONVENTIONAL LAUNCHING (TOP END FROM BARGE STERN)
TWINBARGE METHODS
(I) CONVENTIONAL LAUNCHING (FROM BARGE STERN)
W/L
II
(11) BOTTOM END LAUNCHING (FROM BARGE STERN)
(111) TOP END LAUNCHING (FROM BARGE STERN)
FIGURE 8.14 VARIOUS LAUNCH TEST SCHEMES [Seldta, et al. (1980)] In the model, caps are fitted to the top ends of the main legs and flexible hoses are connected to them for ballasting. Air vents are provided to the legs for air escape. Alternately, flood valves in the model may be operated remotely, although this is difficult to control. Slings are attached and connected to a single lifting hook representing the crane. The hook line is fitted with a load cell for the measurement of
Section 8.7 Staged Submergence. of a Drilling Rig
339
the hook load. The ballasting is done in incremental steps , the steps being smaller near large expected moments . Draft marks may be placed on four corners of the jacket to determine trim, heel and draft during upending. 8.6.3 Side Launching of Structures As the structures become large and heavy, longer and stronger launching barges with large winch capacity is required. Side launching is an alternative procedure that may reduce these excess requirements imposed on jacks or winches. Launching of structures from barges may be accomplished by many different methods. Model tests to compare five of these methods were carried out by Sekita, et al. (1980) on a template type jacket at a scale of 1:60. The conventional method consists of top end launching from the barge stern . Variations of this method include side launching from a single barge by heeling the barge over , side launch from twin barges astern , bottom and top end launching respectively from barge side . These are illustrated in Fig 8.14. The measurement system is schematically illustrated in Fig . 8.15. The jacket is moved on the barge by a direct torque motor whose driving power and speed are controlled electrically by voltage and mechanically by gears . Both singlehinged and doublehinged rocker arms and skid rails with and without nonfriction coating may be used in the tests. The side launching from a single barge seemed quite favorable from the model test results . In this case, the heeling characteristics of a barge may be advantageously harnessed to deploy the structure, thus reducing the required capacity of the launching equipment. Also, the vertical motion of the structure was small minimizing the risk of ramming the structure into the seabed . The twinbarge launching of the structure seemed to have special advantages as well , requiring smaller barges, more control of the structure and lower requirement of launch equipment capacity. 8.7 STAGED SUBMERGENCE OF A DRILLING RIG The upending or submergence test of an offshore structure model follows one of the following methods : dynamic and staged. For certain structures , staged upending provides the necessary information to verify the success of the planned procedure. It is generally easier to accomplish . In the case of the jacket structure model, a staged upending is generally preferred . For certain other structures, dynamic upending is preferred to investigate any instability during this procedure. The rate of flooding of the structure model is an important consideration in these types of tests. In either case, the
340
Chapter 8 Modeling of Offshore Operations
model is built as a dynamic model in which the weight, C.G., moment of inertia and GM are modeled accurately.
pM
MOTOR
^ LOAD CELL
%0%^ SPRING  STRAIN GAGE 0 ACCELEROMETER OO POTENTIOMETER
L BARGE SURGE
SURGING FORCE
JACKET ACCELERATIO
DYNAMIC AMPLIFIER
ROCKER ARM REACTION
STRESS IN THE HULL
R OCKER ARM R OTATION
JACK ET MOV EMENT
MOTOR RV RPM BALANCER
DIGITAL RECORDER
FIGURE 8.15 MEASUREMENT SYSTEM DURING JACKET LAUNCHING [Seldta, et al. (1980)] The technique for the introduction of ballast depends on whether the ballasting procedure models dynamic or staged upending scenarios . In the staged upending, the rate of ballast addition is not as important as the seakeeping characteristics at the successive ballast conditions . In this type of testing program, the model is held inplace and lump weights, such as, lead sheets are attached to the model where the ballast needs to be introduced . The model is then released , and the motions in that simulated environment are monitored . A practical example of a planned submergence of a drilling structure is illustrated here. The submersible drilling rig described earlier was tested for submergence in shallow water. In this case a complete , model with legs and superstructure was required. Moreover, the dynamic characteristics of the rig was also modeled. A staged submergence test was carried out by ballasting the rig in controlled steps (Fig. 8.16). The model was submerged in the tank following a controlled ballasting
Section 8.7 Staged Submergence of a Drilling Rig 341
procedure. The structure had 28 ballast tanks in the mat and required an 11 step ballasting procedure to bring it to the bottom from its loaded draft . During the testing, the lower variables (e.g., drill material, and fuel oil) were modeled by completely filling the forward pair of drill water tanks and the forward pair of fuel oil tanks. This was preferred rather than distributing the lower variable among all the drill water and fuel tanks in order to minimize the free surface effect of the liquid variables.
FIGURE 8.16 BALLAST TANKS OF SUBMERSIBLE FILLED IN PRESCRIBED SEQUENCE In modeling the prototype weight, the model weight was not directly scaled from the prototype value but was corrected to account for the fact that the model floats in fresh water rather than seawater. This was accomplished by multiplying the scaled prototype weight by the ratio of fresh water density to seawater density. This allowed the prototype draft to be correctly modeled. 8.7.1 Test Results The model was submerged successfully following the prescribed submergence procedure . The model did, however, exhibit some behavior characteristics worth noting. During submergence, the side to side list of the model was difficult to control. Unlike the prototype control of filling rates, there was no way to contror the filling rate of the model ballast tanks and the amount of ballast in each tank was estimated by visual observation. Therefore, most of the list may have been due to an imbalance of ballast in the model. Also contributing to the list was the ballast free surface effect. The effect on the model's list was greatest at the draft where the top of the mat was just below the
342
Chapter 8 Modeling of Offshore Operations
water surface . At this draft, the model would not float level but would list to one side or the other. With the ballast evenly distributed side to side, the model could be made to list to either port or starboard by tilting the model by hand. The fact that the model would list to either side but not remain level in the water at this draft illustrates the effect of the ballast shifting from one side of the tank to the other. Although the ballast tanks contain baffles that reduce sloshing in the prototype , the baffles have holes in them to allow the tanks to be filled using one water inlet and to be vented at one outlet. These holes allow water to flow through the baffles fast enough to produce instability due to free surface in the ballast compartments as the model lists slowly to one side or the other. When the model was tested with the full upper variable load in place, the maximum list observed was approximately 2 degrees to the side . The bow to stem trim of the model appeared not to be affected much by the ballast free surface, but depended mostly on the foreaft balance. 8.8 DYNAMIC SUBMERGENCE OF A SUBSEA STORAGE TANK The rate of ballast addition is important in a dynamic upending test . If this type of test is required, two valves are installed in the ballast chamber during model construction. One valve is placed at the bottom of the chamber for ballast water entry. The second valve is installed at the top of the chamber for air venting . The water level in the chamber is monitored with a wave probe . Alternatively, the drop in water level in an outside storage tank containing ballast water is monitored . The valve setting is modified until the desired flow rate is achieved. Three "Khazzan Dubai " subsea oil storage tanks were submerged and anchored to the floor of the Arabian Gulf. Chamberlin (1970) described the design, construction, and installation of these structures between the late sixties and early seventies. The submergence of this structure required a unique technique and the method was verified through scaled model testing before the actual operation . The structure is composed of an open inverted funnelshaped outside surface , a center bottle and a ringwall. Khazzan Dubai tanks were constructed on shore in a graving dock complete with a large topside platform . The resulting higher center of gravity increases the importance of an accurate prediction of structure 's stability during submergence. Figure 8.17 shows a schematic of the tank as it arrived at the site floating on a pressurized cushion of air [Bums and Holtze (1972)]. The center of gravity of the structure at this orientation is substantially above the center of buoyancy. However, the structure remains vertical because its outer ringwall which is ballasted with concrete raises the metacenter sufficiently to provide stability.
Section 8.8 Dynamic Submergence of a Subsea Storage Tank
343
As air is vented from the roof (Fig. 8.18a), the structure's draft increases until the ringwall is totally submerged. The metacenter at this point is suddenly lowered, and the structure slowly begins to tilt, raising the wall partially out of water to regain stability. As more air is vented, the tilt becomes large enough to allow air to escape from under the wall as shown in Fig 8.18b. The sudden release of air reduces the pressure supporting the structure, and the unbalanced weight of the structure causes a dynamic descent. This process continues with an increasing draft and angle of tilt until the structure is supported by the central pressure vessel, referred to as the "bottle". Angular moments carry the structure beyond the point of static equilibrium to the most severe draft and heeling angle, as seen in Fig. 8.18c. The structure starts to oscillate about the point of static equilibrium point until the motion is damped out.
FIGURE 8.17 500,000 BBL SUBMERGED STORAGE STRUCTURE When the structure comes to rest, as in Fig. 8.18d, water is pumped into the bottle to reduce the effective buoyancy until the center of gravity is below the center of buoyancy (Fig. 8.18e). At this time the structure returns to the vertical position and continued water pumping causes a controlled descent to the seafloor. 8.8.1 Model Testing A 1:48 scale model of the storage tank was constructed using Fronde similitude. The model materials were chosen so that each component would have approximately the correct density and center of gravity. The result was a model with accurate weight, center of gravity, and dimensional properties. The dimensional accuracy is required to insure that buoyancy forces are correct. The proper weight distribution yields a correct mass moment of inertia, and avoids the necessity of using large weights to adjust the center of gravity.
344 Chapter 8 Modeling of Offshore Operations
FIGURE 8.18 SUBMERGENCE SEQUENCE OF A STORAGE TANK The ringwall was formed from a rectangular aluminum bar, and holes were drilled to simulate pile sleeves . An extra set of holes was drilled and filled with foam to adjust the overall density to that of the prototype ringwall . A thin molded fiber glass roof was used to give the proper thickness . This roof was not sufficiently strong to support the ringwall ; therefore, wire ties were added as shown in Fig. 8.19.
Section 8. 9 Offshore Pipe Laying Operations
345
The platform and supports were made from PVC plastic, and the remainder of the structure from aluminum. After construction, the ballasted model was carefully weighed and leveled in the water. Removable weights in the shaft allowed for modeling both Khazzan Dubai 2 and 3, which have slightly different centers of gravity. The model test was performed in a wave tank (Fig. 8.20). Motion picture film was used to record the tests , and framebyframe analysis of the film yielded the time history plot in Fig . 8.21. The behavior of the prototype subsequently found during submergence of Khazzan 2 & 3 in the Persian Gulf confirmed the observed mode of submergence of the storage tank model.
FIGURE 8.19 INTERNALS OF STORAGE TANK MODEL 8.9 OFFSHORE PIPE LAYING OPERATIONS In offshore pipe laying operations, complete similarity cannot be achieved in a model scale . Distortion is necessary in proper scaling of the pipeline . The modeling should include not only the steady state condition but the dynamic response of the pipeline under prevailing environmental condition as well. In modeling offshore pipe
346
Chapter 8 Modeling of Offshore Operations
FIGURE 8.20 FINISHED STORAGE TANK MODEL BEING TESTED 4
 MODEL TEST  ANALYTICAL S
10 20 30 40 S0 SO PROTOTYPE TIME (SECONDS)
n
FIGURE 8.21 SCALEDUP MODEL TEST RESULT ON SUBMERGENCE ANGLE VERSUS TIME [Burns and Holtze (1972)] laying, there are several scaling laws to be considered and complete geometric similarity is not possible by fulfilling these laws. Therefore , compromise is made in this
Section 8.9 Offshore Pipe Laying Operations 347
selection. Some of the scaling laws are relaxed while others are satisfied by changing the model pipe geometry. 8.9.1. Pipeline Similarity Laws In a pipeline operation there are four major forces that should be considered in the scaling laws. They are the gravity forces , inertia forces, elastic forces and frictional forces. The ratio of these forces between the prototype and model may be conventionally expressed in teens of scale factors . Assuming two separate scale factors , one the linear dimensions, ? and one for time , ti, the ratio between the inertia forces on the prototype and the model [Clauss and Kruppa ( 1974)] is: __Pp"' ^r Pm tiz
(8.32)
where p is the mass density of the surrounding fluid, Fl is the inertia force and r denotes ratio between the prototype and model values. Similarly, the ratios for the gravity, elastic and frictional forces are FGr
=Y)o
Fe.=
(8.34)
lm Ep
z
EM
(8.35)
and F f, vPpp
(8.36)
h VmPm ti
respectively, where 7 and E are the specific gravity and Young 's modulus of pipe material. For perfect similarity, the ratio of these forces to the inertia force must be unity. Thus, for the frictional (viscous ) forces, (8.37)
which gives
(8.38)
348
Chapter 8 Modeling of Offshore Operations
This is equivalent to stating that the Reynolds number for both the model and prototype is the same. As illustrated earlier , in hydrodynamic model testing the viscous forces cannot be scaled adequately and are generally corrected for in scaling up the model test results. Considering the similarity of gravity and elastic forces , we have
, = (Pp /P.) X vz = (Pp / P.) 7i Yp/Y. Ep/Em
(8.39)
The first equality corresponds to the Fronde number while the second one refers to the Cauchy number. These identities can, therefore, be fulfilled simultaneously, if the model and prototype materials are not identical . In fact, the material must fulfill the following requirement from Eq.8.39: (Ep/Em)
(8.40)
Yp/Ym
This limits the choice of model scales . One can show [Clauss and Kruppa (1974)] that there is no suitable material which permits testing at model scales 105, X 5100. 8.9.2 Partial Geometric Similarity Full scale pipes are generally coated with a thick concrete jacket for added weight and stability. The coating, however, does not contribute to its section modulus for bending stress calculations. The elastic deflection on a pipe element is obtained from dZMe+ W F d2y0 a dx2 dj2 cos8
(8.41)
where MB and Fg are the moment and horizontal force on the element and W is the pipe weight per unit length . Nondimensionalizing with respect to the water depth d (so that z=x/dand y=y/d). Z ( !E) 2 1 1/2 Z =0 F„d d +Wd Zf l+I 2
J
For large elastic deflections, the beam theory suggests that
(8.42)
Section 8.9 Offshore Pipe Laying Operations
El d2y/dx2 MB =7 [
349
(8.43)
( d_ )2 ]3(2
where I is the cross sectional moment of inertia . If this expression is substituted in Eq.8.42 and the equation is normalized by dividing throughout by El/d, then the nondimensional coefficients of the second and third terms in Eq . 8.42 become
Full Scale
30 x
7s
D2p/Dlp = 0.95 Concrete coated
N
088
Steel Pipe D,Y P
in Seawater
07 /12071
150
09 11240 085 11270)
oat 1 7001
Dl m/Dm = 0.6
1
Mercury filled Perspex Pipe
5
0 0
in Freshwater
0.2 0.4
0.6D 0.8
1.0
Dim
FIGURE 8.22 SCALING PARAMETERS FOR MODEL PIPE DIAMETERS [Clauss & Kruppa (1974)]
350
Chapter 8 Modeling of Offshore Operations
3
C' EI
and
2
Cz =Fd EI
(8.44)
This requires that three quantities Wd, FH and EI/d2 have the same ratios between the model and prototype. These quantities are the gravity, external and elastic forces, respectively. The outer diameter of the model and prototype pipes are scaled according to X (= Dp/Dm). Assuming a model crosssection similar to Fig. 8.22 [Clauss and Kruppa (1974)] for the corresponding prototype pipe crosssection, the hydrodynamic inertia forces will scale properly. In this case, Eq.8.32 becomes
(8.45)
Fir = Pe
where
_ P Ip D lp z  D 2p z + Pc(1  Dtp z)+ Pp
Pp Pp
(8 . 46)
Pm 1DIM e+P.fm^ (Dtmz D2mz)+D2mz+P! Pm Pm
and 151 = Du D, 52 = D2/D, pp = mass density of steel, pm = mass density of model pipe and insert rod, pc = mass density of concrete coating, pf = mass density of surrounding fluid, pfi = mass density of liquid filling. This equation ensures similar mass per unit length of pipeline including an allowance for the hydrodynamic mass. The force ratio for the gravity force is similarly obtained Fcn = Y eX,3 where
(8.47)
Section 8.9 Offshore Pipe Laying Operations 351
P D lp2D 2p 2 + P c (1D lp 2) Y Y
P
e Ym
P
P P rfin D lm 2+P m(D lm2 _D 2m2)+D 2m 2m Pm
(8 . 48)
Similarly, the ratio for the elastic forces is derived as Fez = Eelellx
(8.49)
E Ee =p E.
(8.50)
where
and le
_ Dlp4D2p'
DIMa+D2Ma
8.51)
in which the effect of the concrete jacket has been neglected as not contributing to the prototype section modulus . Since all force ratios have to be identical , one obtains
lvx?. ¢  I _ L ?i  ( Pe Ye Eele
(8.52)
A  Ee le Ye
(8.53)
and
Obviously, for complete similarity, Pe=Ye=Ee=le
(8.54)
The model scale and time scale are affected by the material properties as well as the model pipe cross sectional geometry . For the horizontal tension forces
352
Chapter 8 Modeling of Offshore Operations
(8.55)
FHr =7e'
Other forces will be hydrodynamic in nature and may depend on Reynolds number, for example the transverse current force. An example of the scale factor for a model pipe is given here. The prototype pipe is considered to be of steel construction coated with concrete such that D2p/Dlp = 0.95. The specific gravity of the empty prototype pipe in air is given by
Ye = 75 D1p2_ 2p2 +2(1D1p2)
l
Pp
(8.56)
1
The model pipe is represented by a perspex pipe and an insert rod , the annular space being filled with mercury. In this case , D lm = 0. 6. For a range of D lp, the possible values of X as a function of D2m/Dlm are shown in Fig . 8.22. Thus, the model scale can be varied by changing the diameter of the insert rod. For a given pipeline specific gravity, 'ye, the corresponding values of time r and force Fjr are given in Fig. 8.23.
ru
D2m
1100000 60
Djp/Dp=0.88
F Ir C D = 0.95 1000000 .5 2N`1P
Dim
ae
600000 +40
06 FIr
600x00
Concrete coated Steel Pipe in Seawater
30
ac £00000 20
200000 10
50
100
150
250 A 300
0& 0
D1n/Dm=0.6 Mercury filled Perspex Pipe in Freshwater
FIGURE 8.23 SCALING PARAMETERS FOR TIME AND FORCE [Clauss & Kruppa (1974)]
Section 8. 10 References
353
8.10 REFERENCES 1. Bhattacharyya, S.K., "On the Application of Similitude to installation Operations of Offshore Steel Jackets," Applied Ocean Research, Vol.6, No.4, 1984, pp.221226. 2. Bhattacharyya, S.K., Idichandy, V.G., and Joglekar, N.R., "On Experimental Investigation of LoadOut, Launching and Upending of Offshore Steel Jackets", Applied Ocean Research, Vol.7, No.1, 1985, pp.2434. 3. Burns, G.E., and Holtze, G.C., "Dynamic Submergence Analysis of the Khazzan Dubai Subsea Oil Tank," Proceedings on the Fourth Annual Offshore Technology Conference, Houston, Texas, OTC 1667, May, 1972, pp. H467478 4. Chamberlin, R.S., "Khazzan Dubai 1: Design, Construction and Installation," Proceedings on the Second Annual Offshore Technology Conference, Houston, Texas, OTC 1192, May 1970, ppI440454. 5. Clauss, G., and Kruppa, C., "Model Testing Techniques in Offshore Pipelining," Proceedings on the Sixth Annual Offshore Technology Conference, Houston, Texas, OTC 1937, May, 1974, pp. 4754. 6. Graff, W.J., Introduction to Offshore Structures Design. Fabrication. Installation, Gulf Publishing Co., Houston, TX., 1981. 7. Hoerner, S.F., Fluid Dynamic Drag, Published by the Author, Midland Park, New Jersey, 1965. 8. Muckle, W. and Taylor, D.A., Muckle 's Naval Architecture, 2nd Edition, Butterworths & Co. Ltd ., London, England, 1987. 9. Rowe, S.J. and Clifford, W.R.H, "Model Testing of Launching and Installation of Steel Production Platforms," Offshore Structures: The Use of Physical Models in their Design, Construction Press, Lancaster, 1981. 10. Sekita, K., Sakai, M., and Kimura, T., "Model Tests on Various Launching Methods for Large Offshore Structures," Proceedings on the Twelfth Annual Offshore Technology Conference, Houston, Texas, OTC 3836, May, 1980, pp. 369378.
CHAPTER 9 SEAKEEPING TESTS 9.1 FLOATING STRUCTURES Seakeeping characteristics of a floating structure that is in motion under its own power or moored to either the seafloor or to another structure by some mechanical means determine its ability to survive the environment . The motion and component loads of the floating system are generally computed analytically and verified with model tests. While the seakeeping of rigid structures are commonly model tested, flexible structures are modeled as well . In general, the structure is allowed to undergo six independent degrees of motion  three transitional and three rotational. These motions are schematically shown in Fig . 9.1. Often the structure is restrained to have fewer degrees of freedom due to the type of mechanical connection used to fasten it to the seafloor or to another vessel or structure. HEAVE Y
SURGE
1E
(C.G.)A  * X % ani i
z SWAY FIGURE 9.1
DEFINITION OF SIX DEGREES OF MOTION OF A FLOATING BODY
Section 9.2 Method of Testing Floating Structures
355
Examples of such structures are tankers, barges, buoys, compliant towers, floating production systems and tension leg platforms. Compliant towers are typically designed for deep water applications and are allowed to flex in the waves. In designing compliant towers, the structural stiffness in sway is kept low such that its first natural period is outside the range of large wave energy that may excite the structure . Floating Production Systems (FPS) consist of a semisubmersible platform or a Mobile Offshore Drilling Unit (MODU) which is moored in deep water with mooring lines . The Tension Leg Platform (TLP) is a semisubmersible floater that is held on location by a vertical mooring system called tendons . Drilling and production facilities are supported on the floater. The TLP has gained widespread acceptance as the best choice for the development of deepwater oil fields. In order to design any of the above structures, the motions of the structure should be known in addition to the wave forces acting on it. The motion response of these structures is routinely obtained through model testing. Modeling is quite straightforward if the structure size is large such that the inertial force predominates. The scaling generally follows Froude's law even though special measures are sometimes taken to correct for the Reynolds scaling effects. For flexible members, Cauchy scaling is additionally superimposed. Model tests of a few of these structures and some of the problems associated with them are discussed in this chapter. 9.2 METHOD OF TESTING FLOATING STRUCTURES A floating platform model is generally placed near the middle of the test basin and anchored to the basin floor (or its sides or another structure model) by means of anchorline models. Of general interest are the motions of the platform and the loads experienced by the mooring lines when subjected to the model waves, wind and currents. The motion and associated loads occur at several frequencies from very low to very high, depending on the sea state, type of the platform and the mode of anchoring. The modeling of the natural period of motions of the platform and the corresponding damping is extremely important in the test set up. The system is often nonlinear due to nonlinear mooring line characteristics . One should take special care in modeling these lines within their range of interest. The offset of a moored floating structure and the mooring line loads are just two of several aspects determined by model tests. These quantities are highly dependent on the damping of the system. Therefore, modeling the damping experienced by a floating structure is an important consideration . One possible source of error in modeling damping is the mechanical friction introduced in the test setup. As discussed earlier (Chapter 5), a weight hanging over a pulley system has been a common method of introducing steady wind and current loads. The lines used for this system (or just to keep a model in position, for example, from any sway motion) may
356
Chapter 9 Seakeeping Tests
introduce additional unwanted friction in the system. The mechanical connection used for the motion measurement may also provide additional frictional force in the system. These sources of friction could be significant for the low frequency motion of a moored floating vessel which is very sensitive to the overall damping in the system . It will be shown through computations and experiments that in model testing it is safest and best to avoid mechanical friction whenever possible . If mechanical connections are unavoidable due to other constraints , then they should be thoroughly investigated for suitability and error before embarking on a test . The following analysis closely follows the work of Huse ( 1989). Let us consider a single degree of freedom system having hydrodynamic as well as friction type damping . The differential equation is written as Mz+boz /IA+biz +b&z+Kx = Focoso)t
(9.1)
where bo, bl, and b2, are the coefficients of zeroth (Coulomb friction type), linear and quadratic damping terms. The linear and quadratic terms appear from the hydrodynamic damping while a mechanical system in the set up is the source of zeroth damping. The energies associated with these forces are respectively: Eo = 4boxo
(9.2)
2 E, = ab.wxo
(9.3)
E2 = (8 / 3)b2w2x03
(9.4)
whereas the energy input into the system from the extinction force is
(9.5)
Ef =itFoxo
The quantity xo refers to the amplitude of the displacement x. The nonlinear damping term has been linearized in the above expression . By equating the input energy to the dissipated energy, the external force, Fo becomes:
F o+ 3b2w2xx = nI 4b0+7[bwx
)
(9.6)
Section 9.2 Method of Testing Floating Structures
357
Now consider two systems in a test setup where one setup introduces mechanical damping while the other does not. Writing the amplitudes of oscillation for the two systems as xof and xon respectively, and applying Eq. 9.6 for both xof and xon, we have xof =
[_U, ± (%
2 4uxu3 x (2u2)
(9.7)
where u,
=
ab,w
(9.8)
=3bxwx
u3 = 4bo  u,xon  uxxonx
(9.9)
(9.10)
This expression provides the error in the motion amplitude due to the additional friction
term. Let us consider a numerical example for a tanker model undergoing low frequency oscillation in still water. A typical model of a 100,000 dead weight ton (dwt) tanker at a scale of 1:65 has the following dimensions: displacement 0.49m3 length between perpendiculars 3.76m breadth 0.61m draft 0.26m wetted surface area 4.0m2 A surge oscillation is carried out under the following condition to determine viscous damping of the hull in still water: stiffness , K 22.61N/m surge amplitude without friction, xon O.lm linear damping coefficient, b1 0.0 quadratic damping coefficient, b2 38.4 Ns2/m2 The undamped natural period in surge is 30 seconds . A correction factor R is defined as the ratio between the surge amplitude measured without and with friction. The values of R for different values of bo are computed from Eq.9.7:
\^
INN
SINGLE ANCHOR LEG MOORING
CATENARY ANCHOR LEG MOORING
cINN, I RIGID ARM MOORING
4


\
EXPOSED LOCATION S.B.M. SPAR ARTICULATED SPAR BUOY
MOORING TOWER
FIGURE 9.2 EXAMPLES OF SINGLE POINT MOORING TERMINALS [Pinkster and Remery (1975)] w cn co
360
Chapter 9 Seakeeping Tests
In addition to the environmental conditions, the characteristics of the single point mooring terminal must be known . For example, for a CALM system, the following information is needed: • buoy: dimension, weight, center of gravity, bow hawser location, • anchor chains: number, length, weight per unit length, breaking strength, elastic limit and elongation , pretension and buoy connection points, • moored vessel : principal dimensions , displacement, draft, center of gravity, righting moments, radii of gyration, and natural period, and • bow hawser: length, weight per unit length, hawser connection point and load versus elongation curve. The model scale for a given basin is chosen based on water depth, wave generating capability and accuracy and magnitude of measurements, as usual . Froude's law is used throughout for modeling an SPM terminal . For testing in wind, waves and current, it is a normal practice to use a scale factor ranging between 50 and 85. This range of scale factor allows accurate measurement of the important variables. Models of SPM terminals are constructed of different types of materials: wood, metal, synthetic foam, plastic, etc . In practically all cases, components are made as rigid as possible since the tests are aimed at the determination of rigid body behavior and not the elastic effect of construction elements . The center of gravity of the model is determined in air and adjusted, if necessary , by an inclining test (Chapter 3). The mass moment of inertia is checked by a pendulum test . The model anchor chains are made to conform to correct scaled length and weight. The elastic stretch of the model chains is achieved by adding a coil spring at the end of the chain . Thus, the caternary characteristics and elastic properties are simultaneously scaled. The model of the moored vessel, such as a tanker , is commonly constructed of wood and fiberglass . In most cases, available stock models are used which may determine the model scale for the SPM test . Tankers are fitted with deck and superstructure. The geometry and location of the superstructure is important for the simulation of wind loads. The mass moment of inertia in yaw and pitch are adjusted in air by the pendulum method . The transverse stability is determined by inclining tests in calm water. Similarly, the roll natural period in water is modeled at the same time.
Section 9. 3 Single Point Mooring System
361
In some SPM model tests , underbuoy hoses or floating hoses are also modeled. These hoses consist of strings of large diameter (305610 mm or 1224 in.) flexible hoses. The elements of these strings are 0 .912 m (340 ft) long, bolted together. A hose element consists of a flexible middle section made of steel coils embedded in rubber and rigid end connection ending in a flange . Models of such hoses may be made in a similar manner. They consist of coil springs covered with latex. The extremities of each model element end in a rigid part with a flange, to which the next section may be connected. The measurements during SPM tests generally consist of (1) forces in anchor chains and bow hawsers, (2) motions of the buoy, (3) axial forces and bending moments in underbuoy and floating hoses , and (4) tanker motions. Before the wave tests commence, the static values of forces and motions are recorded. The SPM system is displaced from the equilibrium in an incremental manner and the overall stiffness of the system is established (static load  deflection curve). Also, the current loads and the combined wind and current loads are recorded. 9.3.1 Articulated Mooring Towers An articulated loading tower capable of mooring large oilexport tankers operate in moderate water depths (up to about 150 m or 500ft). The towers are maintained upright as a result of their buoyancy sections, and are considered large enough to be treated as rigid body system without appreciable deformation of their members. The motion of the tower and the tanker as well as the nonlinear restoring force characteristics of the hawser are needed for the design of this system. The tower must survive the largest storm ( e.g., 100 year) expected during its lifetime . The towertanker system must operate at a given sea state based on the location of operation. Model tests of these articulated towers require the measurement of the motions of the model in two perpendicular planes and the three component loads at the universal joint. An XYZ load cell can accomplish the necessary load measurement when mounted on the base and connected to the universal joint of the tower (Fig 9.3). The tower motion may be measured either by a direct contact method or can be measured indirectly. In the direct contact method , the displacement at the tower top or the rotation at the base of the tower may be measured . In the latter case , two RVDT's (section 6.4) may be placed at right angles to each other . In this case, the RVDT's must be waterproofed properly for underwater application. Displacement at the top of the tower may be measured in several different ways . Potentiometers (such as 10turn pots) are used for this purpose (with negators on the opposite side to maintain steady tension in the line). Two potentiometers at right angles to each other will measure the displacement at the top of the tower . The error due to the threedimensionality of the tower top motion may be minimized by making the horizontal lines to the
362
Chapter 9 Seakeeping Tests
potentiometers long at the equilibrium position so that the effect of the vertical displacement of the tower top is negligible.
FIGURE 9.3 AN ARTICULATED TOWER TEST SETUP Another direct contact measurement is a gimballed staff attached to the tower top with a universal joint (Fig 9. 3). RVDTs and potentiometers may be employed in the dry for measurements of two angles as well as displacements . A simple geometric relation (Fig. 9. 4) will provide the two angles of the tower oscillation. For example, the inline oscillation angle at the universal joint is obtained as follows:
Section 9.3 Single Point Mooring System
0(t)= sin `I sin{ s'+s, +s(t)}Is,s, S(t)e,e ,
363
(9.11)
in which the subscripts 0 and 1 refer to the tower equilibrium values, and values under static load respectively . The quantity 3(t) is measured from static position at the gimbal during waves.
FIGURE 9.4 GEOMETRY OF A GIMBALLED STAFF MEASUREMENT This method can minimize the effect of friction on the measurement . However, the mass of the staff will have some effect on the measured data . In particular, the vertical load is found to be affected by this rod . A measurement of the tower
364
Chapter 9 Seakeeping Tests
motion with the gimballed staff was made where the vertical pin load at the universal joint was also measured. In order to study the effect of the rod on the vertical load, tests were repeated with and without the rod for the motion measurement. The measured mean values of the transfer function for the vertical pin load in regular waves are shown in Fig. 9.5. It is clear that the gimbal transfers some of the vertical load in the load cell. The noncontact method of measurement most commonly used now employs LEDs on the tower with a system similar to SELSPOT (Section 6.12) In this case, the errors appear only by the measurement system, but not due to setup in the testing. 4
x NO GI MBAL
x
Nk
WITH GIMBAL
3
J
x x x
8 x
X e
o
e
e x
H 1.0
1.5
2.0
2.5
3.0 3.5
4.0
4.5
WAVE PERIOD, SEC. FIGURE 9.5 VERTICAL PIN LOAD WITH AND WITHOUT GIMBALLED STAFF
Section 9.3 Single Point Mooring System
365
In an analytical development , the oscillation of the tower about the pivot point in the plane of the wave propagation may be modeled as a forced damped springmass system with linear and nonlinear damping and solution obtained in a closed form [Chakrabarti and Cotter ( 1979)]. Data from a model test on an articulated tower are compared here with this simple solution. The model resembles the tower at the Statfjord " B " location of the North Sea operated by Mobil . The test consisted of the model tower placed in the wave tank on a universal joint. Regular wave runs were made with no steady load on the tower. The comparison of the measured maximum responses (normalized with respect to the wave height) is made with the theoretical results . The dynamic tower oscillation is shown in Fig . 9.6. The comparison is good in spite of any out of plane motion of the tower neglected in the analysis. The natural period of tower is outside the test range near 3.5 sec.
 THEORY 0 EXP. DATA N
108.9 '=
4" O.D. 91 0 y i SWL
M
N
UNIVERSAL JOINT O
0 .50
.75
1.00
1.25
1.50 1.75 2.00 2.25
2.50
2.75
3.00
WAVE PERIOD (SEC.)
FIGURE 9.6 COMPARISON OF INLINE OSCILLATIONS OF AN SPM The response of the tower oscillation due to irregular waves is shown in Fig. 9.7. The theoretical response spectrum is computed from the transfer function shown in Fig.
366
Chapter 9 Seakeeping Tests
9.6. The RAO is assumed to be linear so that the response spectrum may be simply obtained as the product of the square of the RAO and the wave energy density spectrum (Chapter 10). The second peak in the measured spectrum is at the natural frequency of the tower. When subjected to oscillatory waves, an articulated rigid tower is free to move in all planes about its pivot point near the ocean floor. Therefore, the transverse force due to separated flow introduces a motion of the tower normal to the inline motion. The magnitude of this motion depends on the magnitude of the transverse force, which in turn depends on the nature and number of eddies shed by the tower . An example of the transverse force on a submerged instrumented section of a fixed vertical tower versus the inline force as a function of time for about two cycles is given in Fig. 9.8.
0.2
0.4
0.6
0.6
1.0
WAVE FREQUENCYHZ
FIGURE 9.7 COMPARISON OF TOWER OSCILLATION IN IRREGULAR WAVES The oscillation of waves past a fixed tower may be equivalently represented by an The approach is to oscillate the tower oscillating tower without the wave motion . harmonically in an otherwise still fluid. While the effect of free surface of the wave cannot be reproduced in this method, a submerged tower section experiences similar motion. This method of testing is sometimes preferred over the test in waves because the tests can be more easily controlled, uncertainty of water particle kinematics can be avoided and higher values of Reynolds number can be achieved compared to a wave test. Moreover, these results may be directly applied to cases where added mass and drag
Section 9.3 Single Point Mooring System
367
FIXED TOWER
N 0
O
1 MECHANICAL TOWER
,,, V^< Z//, ^ "" ^^ I/ ^ 0 d
FREE TOWER
1.0
0.5 0 0.5
1.0
1.5
LOAD, LBS
FIGURE 9.8 LOADS ON AN INSTRUMENTED SECTION OF A BUOYANT ARTICULATED TOWER
368
Chapter 9 Seakeeping Tests
coefficients are required for moving members of an offshore structure, e.g., an articulated mooring tower, riser, or tendon of a TLP. In comparison to a fixed tower, the resultant force profile on the instrumented section of the tower harmonically forcedoscillated in still water shows (Fig. 9.8) that the transverse force is on the same order of magnitude as the inline force, but the tower experiences predominantly double frequency. The KC number in both cases is about eight while the Re number is on the order of 105. In a mechanical oscillation test, noise in the measured data is commonly found. Although the cylinder angle is generally a smooth sinusoidal function of time, the loads in the inline direction show high frequency noise. This noise occurs because a servocontrolled device tends to overcompensate for errors between the reference and feedback signals and therefore oscillate about the mean reference value. This 'hunting' has little effect on the cylinder' s angular position (the feedback to the drive system) but does cause cylinder acceleration that appears as spikes in the inline loads, especially where the reference signal changes most rapidly. A digital filter may be used after the data has been collected to remove any noise or unwanted data. The desired result of the mechanical oscillation test is the estimation of the added mass coefficient (CA), the drag coefficient (CD) and the lift coefficient (CL). These coefficients are formulated as functions of the KeuleganCarpenter number and the Reynolds number. Problems may be encountered in a test with a free tower in waves in which the purpose is to compute hydrodynamic coefficients from measured local loads on the submerged section of the tower. When the damping and the drag terms are relatively small, the tower begins to move in phase with the driving force of the waves. At this point the tower velocity is in phase with the water particle velocity (this tends to make the relative velocity term quite small) and the tower acceleration and water particle acceleration are in phase which causes the load to be the residual of the two dominant loads. Although both of these loads may be estimated to within 10% accuracy (which may easily be done as evidenced by the first two phases of the test), the residual load may be over 100% in error. This obviously makes the use of load data from the free tower in the evaluation of the hydrodynamic coefficients questionable. With the tower free to oscillate in the same waves as the fixed tower the corresponding twodimensional force is shown in Fig. 9.8. The KeuleganCarpenter number based on relative velocity between the tower and the wave is of the order of one. At this KC number, the transverse force on the tower is small and the transverse force has the same frequency as the inline force.
Section 9.4 Tower Tanker in Irregular Waves
369
WAVE
TOWER OSCILLATION
0 M
aA 1
MOORING LINE
LOAD M
20
30
40
50
TIME (SECONDS) FIGURE 9.9 MEASURED TOWER OSCILLATION AND MOORING LINE LOADS IN IRREGULAR WAVES FOR AN SPM 9.4 TOWERTANKER IN IRREGULAR WAVES Articulated towers may be used to moor a shuttle tanker with a single point mooring hawser. Besides the extent of oscillation of the tower, the load experienced by the hawser is an important design criterion. Model tests are often conducted to study this interaction phenomenon. An example of the resonance of the tower near its natural period is evident in Fig. 9.9 in which the towertanker system is subjected to a simulated irregular wave. The low frequency of surge motion of the tanker coexists with the high frequency tower motion. However, the subharmonic oscillating tower motion
370
Chapter 9 Seakeeping Tests
occurred only for a certain range (e.g., 3242 sec) of tanker position in surge as shown in Fig. 9.9. This arises from certain instability experienced by the nonlinear hawser e.g., in a catenary moored tanker (Fig. 9.10), which may become slack during the oscillation in waves if the steady loads are not large enough . The result of this nonlinearity has been found even in regular waves in model tests as an occasional increase in the hawser load. This is illustrated in Fig . 9.11. The model test results have been scaled up to the prototype values. The wave elevation, SPM and tanker motion have been presented in meter while the hawser load is in metric ton. Note that the wave is regular of a period of about 14 sec. The tanker motion shows the long period oscillation and high frequency wave motion superimposed on it . The hawser experiences slackness during motion and thus shows spikes. At an intermediate range of data, the tower motion becomes violent associated with a high load in the hawser. This instability was first discovered from model test results and may be explained mathematically as the Mathieu type instability generated from the Duffing's equation [Chakrabarti (1990)].
FIGURE 9.10 TEST SETUP FOR A CATENARY TANKER MOORING SYSTEM 9.5 TESTING OF A FLOATING VESSEL A floating structure is one designed to accommodate the motions caused by wave action, rather than to resist motion. The testing of a floating vessel differs significantly from testing of a fixed structure. On a fixed structure, the overall exciting load and,
Section 9.5 Testing of a Floating Vessel
371
possibly, local pressures are important . Sometimes, other wave effects such as run up may also be important. However, instrumenting, constructing, and modeling a floating structure are more complex. As previously described for articulated towers, the loads at critical points as well as the overall motions are needed in tests of a floating structure model. Unlike an articulated tower which has limited degrees of freedom, a rigid floating structure, in general, has all six degrees of freedom. A photograph showing the test setup and instrumentation of a floating storage vessel is shown in Fig. 9.12.
WAVE ELEVATION (M) 40 4SPM X MOTION (M) 0 10 20 HAWSER TENSION (T) 300 200 100 0 TANKER X MOTION (M) 10 20 30 40 700
900
800
1000
TIME
FIGURE 9.11 RESPONSES OF AN SPM IN REGULAR WAVES [Chantrel, et al. (1987)] The vessel in this case has two degrees of freedom of importance. The vessel is an open bottom buoyant cylindrical tank in which the draft and anchoring force are maintained automatically by the geometry of the buoyancy section. It has a single tensioned anchoring leg so that the tank appears as an inverted double pendulum . In this case, the displacement of the top of the storage tank in waves is an important design
372
Chapter 9 Seakeeping Tests
factor. This may be measured by a gimballed staff equipped with a potentiometer. An example of the comparison of this measured displacement with computed values for a two degrees of freedom motion analysis is shown in Fig. 9.13.
FIGURE 9.12 FLOATING STORAGE TANK MODEL IN WAVE TANK 9.6 TENSION LEG PLATFORMS A tension leg platform is a floating production platform which is anchored to the ocean floor by vertical mooring lines . The mooring lines are called tendons (or tethers), because they are held in high tension by the excess buoyancy of the floating platform. Thus, the inservice draft of the platform is considerably higher than (about twice) that of the floating hull. The stiff mooring system permits limited motions of the platform in heave, pitch and roll when subjected to waves. The geometry of the hull and the tendon placement are almost invariably symmetric so that the pitch and roll periods are about the same. The period range of appreciable ocean waves generally is 620 seconds . The typical natural periods in heave , pitch and surge of a TLP are such that practically no wave energy exists in the sea at these periods. Thus , virtually no amplification in these responses are expected from the linear wave exciting forces arising from a wave spectrum. However, the combination of multiple frequencies in a wave or the • higher components of a nonlinear wave may produce exciting forces near the natural frequencies in these motions both in the low frequency (e.g., surge period) and in the
Section 9.6 Tension Leg Platforms
2 Theory 0 Test data
I 5
T1=7.0 sec T2=25.0 sec
0. 5
(a) E 3.
T1=3.9 sec
°
T2 =23.5 sec
1.5 0
1.0
0.5
(b) 1.0
1.5
2.0 2.5 Wave period , in seconds
3.0
3.5
FIGURE 9.13 DISPLACEMENT OF TOP OF STORAGE TANK FOR TWO BALLAST CONDITIONS IN REGULAR WAVES (T1, T2  Natural Periods)
373
374
Chapter 9 Seakeeping Tests
high frequency (e.g., heave and pitch period) regions. The low frequency force is known as drift force while the high frequency force is commonly referred to as springing force. The springing force on a TLP arises from the heave force or the pitch moment. These exciting forces being of higher order are an order of magnitude lower compared to the firstorder exciting forces at the wave frequencies. However, the damping in the TLP system both for the high frequency vertical motion and low frequency horizontal motion is very small. Thus, the dynamic amplification near the natural period is relatively large. The damping measurements of floating structures are discussed in Section 9.8. Typically, the natural periods of a TLP in heave and pitch for the deep water applications (greater than 305 in or 1,000 ft) are between one and five seconds. On the other hand , the system is soft in the direction of surge because the restoring force in the vertical tendons in the surge direction is generally small. The natural period of the TLP in surge (or sway) is large, being of the order of 100 seconds or more. A TLP experiences two types of forces: a primary exciting force at the wave period and a secondary nonlinear exciting force having steady and oscillating components outside the range of typical wave periods. In particular, a high frequency springing force develops in the tendon in the pitch natural period. 9.6.1 Model Testing Program for a TLP The objectives of a TLP test program are to determine: • Overall dynamic behavior of the TLP in wave, wind and current environment • Global response characteristics, such as six degrees of motion, and tendon loads • Excitation and damping for the slow drift response of the TLP in random wave fields • High frequency response and damping of the TLP motion The hull construction (Chapter 3) consists of building the columns and pontoons made of steel or aluminum, generally rolled and welded. Access is provided in the hull for ballasting. It is preferable to ballast the model with dead weights anchored inside the hull rather than with water. Having a free water surface inside the hull can further complicate the dynamic behavior of the structure. The hull is checked by pressure testing to verify that it is watertight. The deck is not replicated in the model except that framing is needed for the rigidity of the model. A flat deck is provided for mounting the
Section 9.6 Tension Leg Platforms
375
desired instruments (e.g., optical light sources, accelerometers, tendon tensioning device). Moreover, the deck allows to study the underdeck wave effects. The properties of the hull are established by swinging the model in air . Simple calculations provide the location of ballasts to model the center of gravity and mass moments of inertij for various angular directions. The tendons are modeled to achieve the scaled weight and buoyancy characteristics. The scaled axial and bending stiffness of the tendon are desirable in the model tendon . However, in reality, both of them are difficult to achieve. For a TLP model, the axial stiffness is more important. The effective axial stiffness can be provided by an adjustable cantilever spring at the bottom of a tendon . The stiffness is established by loading tendon /spring combination in an axial direction . The bending stiffness may be achieved by the proper choice of the tendon material (see Section 3.4.3.2). The tendon loads are monitored by a threecomponent load cell mounted between the spring and the foundation. The foundation consists of a mounting plate or frame on which the tendon load cells are mounted. The plate is either welded or bolted to the tank floor. Sometimes, weights are placed on the plates sufficient to withstand the shear load or the vertical load at the base, simply to avoid attachment to the wave tank floor. 9.6.2 Typical Measurements for a TLP In a model test of a TLP, the following measurements are desired: • Six degrees of motion • Accelerations at extreme deck locations • Wave profile in front of the model • Wave profile in line with the model center for phases • Tendon loads • Wind and current loads, if simulated in tests • Photographic and video recording. These measurements allow the evaluation of the performance of the TLP design. 9.6.3 Wave Frequency Response of a TLP An example of the motions of a TLP model at various wave frequencies is shown in Fig 9.14. The TLP system for the test case was designed to have natural frequencies in the wave period range. An external electromagnetic damping system was introduced in each leg of the fourlegged platform in order to study the effect of additional damping in the system. This excess damping was expected to be
376
Chapter 9 Seakeeping Tests
introduced in the prototype by an equivalent system . In this case the damping factor is duplicated in the model . The external damping, as may be expected, reduced the amplification near the natural period.
1.5
 Co mpu t e d  
M e a sur ed x Without Dampi ng_ o With Damping
x xx
X
x
0 0 0
Ilk 0
1.0 2.( WAVE PERIOD, SEC.
FIGURE 9.14 RESPONSE OF A TLP MODEL EQUIPPED WITH A MECHANICAL DAMPER [Katayama, et al. (1982)] 9.6.4 Low and High Frequency Loads The second order inertial load on structures may be important , particularly when the natural period in a particular degree of motion is such that the structure is excited at the higher (or lower) harmonic frequency of this load. This phenomenon may be
Section 9.6 Tension Leg Platforms 377
investigated in the laboratory by measuring either the second order load or the nonlinear responses due to this load. Generally, the secondorder loads are small, but the corresponding responses are large . The difficulty in the direct measurement of the second order load on a structure was addressed in Chapter 7. This section describes the relative ease in measuring the corresponding responses. 6 4
2 0
2 4 6
r7 PHASE
PROBE  I
T_V_"
F_
OF WATER
400
400
Iu Iu °I VI Uf ' YY^I 0.0 2.S 5.0 7.S 10.0 TOTAL HORIZONTAL LOAD  POUNDS
12.5
15.0
17.5
20.0
FIGURE 9.15 MEASURED FORCES ON A FIXED CYLINDER DUE TO A WAVE GROUP An example of the forces measured on a fixed vertical cylinder due to a wave group is shown in Fig. 9.15. The wave profile corresponds to waves of frequencies 0.44 Hz and 0. 88 Hz. The wave energy density spectra show peaks at the above frequencies. The measured forces show the presence of higher frequency components. The force spectrum have additional peaks present corresponding to second harmonics of the above frequencies, and their sum frequency component at 1.32 Hz . These higher order loads are generally small, being on the order of 3 to 5 percent of the first order loads. The effect of these exciting forces on a moving structure due to dynamic amplification is illustrated by another example . The above cylinder was moored in a floating position with linear fore and aft springs. The springs are chosen such that the
378
Chapter 9 Seakeeping Tests
natural period of the system in surge was about 30 seconds . The springs were pretensioned to avoid slackness during testing . Regular wave groups having three frequency components were chosen . The frequencies were chosen such that the difference frequency between two adjacent frequencies was equal to the natural frequency of the system . Figure 9. 16 shows the wave and the corresponding mooring line loads. It is clearly seen that the response is amplified at the natural frequency (due to low damping) such that its amplitude is larger than the amplitude at the wave frequency.
e
. V I WPUMN 4
2
PHrSE t yE PRO
4 SER
xx ING LI
LOAD
POUNID
4
0 10 20 30 40 BOW MOORING LINE LOAD  POUNDS
70
FIGURE 9.16 MOORING LINE LOADS DUE TO A WAVE GROUP ON A MOORED FLOATING CYLINDER An example of the response due to high frequency springing force on a four column TLP is taken from DeBoom, et al. (1984). Comparison of the high frequency tendon force in regular waves is made in Fig . 9.17 with the corresponding measured data from a wave tank test. The phase relationship among the forces and wave profiles suggested that the springing force is a result of the pitching force and motion of the TLP.
Section 9.6 Tension Leg Platforms
F2 F3 Computed Measured • Fi & F2 Fi F4  0 F3 & F4
7.
5. •
2. E
•
c
0 w 0 a: 0
7.5
0 w H w ^ 5.0
8
n b
n
2.5
I I
I 9v ' ^ 1 _^
I
'
0 0.5
1.0 1.5
2.0
2.5
2CJ1(rad/sec)
FIGURE 9.17 NORMALIZED TENDON LOADS IN REGULAR WAVES ON A TLP [DeBoom, et al. (1984)]
379
380
Chapter 9 Seakeeping Tests
A typical TLP test setup is shown in Fig. 9.18. The setup includes a false floor in the wave tank which allows generation of current as well as deep water waves . A deeper circular pocket in the tank allows a larger model for deep water depth typical of a TLP without scale distortion . The tendons are shown to go through the false floor.
CURRENT & WAVES FLEXJOINT (TYP)
TENDON (4 TYP) SEALPLATE
CONCRETE SLAB
p SITEE E L CURRENT STEEL ry I I I SLAB AA o TANK 0 I I M iTENSION TIEBACKS BOTTOM (4 TYP )
FIGURE 9.18 A TYPICAL TLP TEST SETUP IN A WAVE TANK EQUIPPED WITH A DEEP SECTION The springing forces on a large scale (1:16) TLP model were measured in a test in a wave tank [Petrauskas and Liu (1987)] with a fourlegged TLP hull connected to the sea floor with four vertical tendons. The tendon lengths were distorted due to large scale, but axial stiffness was modeled . The springing forces arised from the resonant
Section 9.7 Drift Force Testing of a Moored Floating Vessel
381
pitch periods of about 3 seconds. Regular waves at twice the pitch period amplified the resonant springing force in the tendon. The amplification of the force at the tendon due to a random wave is clearly shown in Fig. 9.19. The tendon load at the wave frequencies is almost negligible compared to the resonant load at twice the wave frequencies. 9.7 DRIFT FORCE TESTING OF A MOORED FLOATING VESSEL The measurement of slow drift oscillation is demonstrated in a tanker test set up. In general, a large floating structure moored at location with a soft mooring system will experience long period oscillation when subjected to irregular waves. In practice, such systems are found in exposed location tanker mooring systems. The following illustrates the setup and measurement technique using a simple system.
__ wave spectrum tendon load spectrum ^I
IIlI I Iilll I IIIY1 J YLq` N
0.00
'
0.25
J 0.50
Frequency, Hz
FIGURE 9.19 WAVE SPECTRUM AND CORRESPONDING TENDON LOAD SPECTRUM ON A TLP [Petrauskas and Liu (1987)] 9.7.1 Test Setup The test setup is intended to produce results on the wave drift force in surge. Two identical spring sets are attached with aircraft cables on the fore and aft side of a tanker model, influencing its surge motion. Springs are chosen to be linear in the range of the anticipated loads. The points of attachment on the deck of the vessel restrict the pitch of the vessel to some extent. However, by making the lines long (of the order of 4.66.1 m or 1520 ft), this error is minimized.
382
Chapter 9 Seakeeping Tests
Loads in the mooring lines and the threedirectional motions (surge heave and pitch) of the floating vessel are generally measured during these tests . The springs are pretensioned so that they never go slack during the tests. 9.7.2 Hydrodynamic Coefficients at Low Frequencies It is important that the low frequency added mass and damping coefficients of a floating vessel be known in addition to the wave drift forces on them so that the mooring line loads may be accurately predicted. These coefficients are obtained in two ways: ( 1) the free oscillation of the moored vessel in surge is obtained by what is commonly known as the pluck test, or extinction test in still water ; (2) the transient oscillation of the vessel at a low frequency caused by a few initial regular waves is filtered to obtain the long period of decaying oscillation. Thus , the variation of the added mass and damping coefficients, not only with the vessel geometry and spring systems, but also with the wave height and frequency can be studied. 9.7.2.1 Free Oscillation Tests The model which is setup as a springmass system is oscillated in surge by displacing the model structure from equilibrium and then releasing it. The free oscillations of the model in surge are recorded by the load cells and LEDs. In still water the extinction test produces a lowfrequency decaying oscillation in the surge motion or in the mooring line load which may be described by a differential equation. The method of analysis for the added mass and damping is discussed in Chapter 10. 9.7.2.2 Forced Oscillation Tests In regular waves , the long period oscillating drift force component is absent. However, the initial waves introduce a transient lowfrequency model oscillation which decays at a rate depending on the damping in the system . Thus, the natural period of the system and its damping factor at different exciting wave frequencies can be determined from these runs . These quantities can be used to determine the added mass and damping coefficients for various wave frequencies. 9.8 DAMPING COEFFICIENTS OF A MOORED FLOATING VESSEL The extent of motion of the system at its natural frequency is controlled by the system damping . Therefore, knowledge of the damping of such a dynamic system is an important consideration in its design . The damping of a moored system arises from two natural sources, namely, material characteristics and hydrodynamic forces. The material damping of the mooring lines is small and is generally determined from
Section 9.8 Damping Coefficients of a Moored Floating Vessel
383
model tests of the component material. The hydrodynamic damping is considered to be the limiting factor in the low frequency motion of the moored vessel . In a lowdamped system at resonance , the motion amplitude is approximately proportional to the system damping. The hydrodynamic damping may be characterized as originating from four sources: 1. radiation wave damping proportional to structure velocity, 2. linear viscous damping proportional to structure velocity, 3. nonlinear viscous damping proportional to square of the structure velocity, and 4. wave drift damping
1.2 • Still Water  Waves w/o Current Waves w/Current
A
0.9
III
0.6
Ali
0.6
^ ^
V
J
0.9
1.2 0
10
20
30 40 50 60
70
80
90
Time (seconds)
FIGURE 9.20 DAMPED OSCILLATION IN STILL WATER, WAVES AND WAVES WITH CURRENT
100
384
Chapter 9 Seakeeping Tests
Model test results on various streamlined floating bodies e.g., tankers [Wichers (1988)] have shown that the nonlinear (drag type) viscous damping in slow drift oscillation in waves is negligible , and most of the damping is linear viscous. For other structures including vertical columns at the water surface , such as semisubmersible, the nonlinear viscous damping is important . For a catenery moored system, the drag forces on the mooring lines themselves and the friction between mooring lines and seabed also contribute to the slow drift damping [Huse and Matsumoto ( 1988 , 1989)]. Once nonlinear viscous damping has been eliminated from the total damping, the damping can be considered to be the linear superposition of damping sources. Thus, the damping coefficient, C, can be described by C=CR+CV+CW+CC
(9.11)
where CR = radiation damping , CV = viscous damping , CW = damping in waves (wave drift damping), and CC = damping in current. An example of the damped oscillation of a moored tanker in still water, in waves, and in waves plus current is shown in Fig. 9 .20. Note that for the same spring constant the natural frequencies of the moored vessel in still water are the same as those in regular waves and current . However, the damping coefficients in waves are, generally, higher than the coefficients in still water . When the current is additionally introduced, the amount of damping increased as evidenced by the faster decay. Figure 9 . 21 presents the natural logarithm of the amplitude versus cycle for the same test runs as presented in Fig . 9.20. Even in the presence of current, the damping is found to be linear with the tanker surge velocity (Fig.9.21). Nonlinear viscous damping contributions will show up as deviations from a constant slope . The lines from each scenario are clearly linear. Structures consisting of submerged cylindrical members, however, show evidence of nonlinear damping . A simplified method of handling the nonlinear term in the secondorder equation of motion [Faltinsen , et al. (1986)] is described in Chaper 10. 9.8.1 Tanker Model A 67,000 dwt tanker model , representing the ESSO Houston , was ballasted to 40 percent capacity and moored fore and aft . The fore and aft mooring lines were instrumented with ring type load gauges . The horizontal lines were long enough such that the effect of pitch and heave on the line loads was negligible. The loads were converted to the displacement of the tanker by dividing by the spring constant.
Section 9.8 Damping Coefficients of a Moored Floating Vessel
385
4. 5 
.0 
.5
a Still Water o Waves w/o Current + Waves w/Current
.0 0
2
4 6
8
10
12
Cycle
FIGURE 9.21 NATURAL LOG OF AMPLITUDE OF OSCILLATION VERSUS CYCLE Extinction tests without waves were performed to determine the still water damping of the tanker at its surge natural frequency. A series of different spring sets were used to vary the resonant frequency. In order to determine the surge damping of the tanker in waves, the tanker was displaced horizontally in head sea from its equilibrium position and held in place until waves generated by the wavemaker reached the tanker. Then the tanker was released and the decayed oscillation in waves were recorded. The high frequency response was filtered out and the low frequency oscillation was analyzed to determine the added mass coefficient and damping factor. An example of the surge calculated from the recorded mooring line load before and after digital filtering of the high (wave) frequency response is shown in Fig. 9.22. 9.8.2 Semisubmersible Model A semisubmersible model consisting of two pontoons of rectangular crosssection and six columns with (roughly) elliptical crosssection was similarly tested in still water, waves and current. Table 9.1 presents the value of the spring constant and the resulting system natural period for the various spring sets tested with the semi model. The surge added mass coefficient of the semisubmersible in still water is found to be insensitive to the natural period. The results are presented in Fig. 9.23 and compared to the added mass coefficients previously obtained for the tanker. The added mass coefficient of the semisubmersible is seen to be a factor of 2 to 3 times larger than for
386
Chapter 9 Seakeeping Tests
the tanker. This is mainly due to the difference in geometry between the streamlined tanker hull and the presence of semisubmersible columns. 2.0 1.5 ^
1.0
1.51
10
20
40
.
60
80
100
120
Time (seconds)
2.0 1.5 1.0
1.5' 2.0 10 20
40
60
80
100
120
Time (seconds)
FIGURE 9.22 DAMPED OSCILLATION OF MOORED TANKER IN WAVES The decaying oscillation curve was analyzed for the damping factor and drag coefficient using a simplified nonlinear theory (see Chapter 10). A typical XY relationship is presented in Fig . 9.24. With reference to this figure, the Y intercept is the damping factor, and the slope is a function of the nonlinear drag coefficient. The damping factor and the drag coefficient in still water for the semi submersible and the tanker are compared in Fig. 9.25. This figure demonstrates that the damping factors for both floating vessels are similar in still water once any nonlinear contributions are removed. The system damping of the semi submersible has a much stronger contribution from the nonlinear term , resulting in higher overall damping . As can be seen in Fig 9. 25, the nonlinear term in still water tends to be relatively independent of the
Section 9.8 Damping Coefficients of a Moored Floating Vessel
387
oscillation period . The damping factor for the semisubmersible in the presence of waves exhibited far greater damping than the tanker. TABLE 9.1 SPRING SETS FOR STILL WATER SURGE OSCILLATION TEST Spring Set
Spring Constant (lbs/ft)
Natural Period (sec)
A B C D E F G
1.05 1.32 1.75 2.63 3.76 5.26 8.77
23.27 20.75 18.01 14.69 12.31 10.40 8.03
0.16 T 0.14 +
I
■
I
semi
11
I
I
0.12
S 0.04
11 tanker
0.02
fl
I I I ❑I I I I 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 24.00 NATURAL PERIOD (sec)
FIGURE 9.23 COMPARISON OF ADDED MASS COEFFICIENT IN STILL WATER 9.8.3 Heave Damping of a TLP Model As with other floating vessels , the TLP system experiences damping from two natural sources, material properties and hydrodynamic effects . Sometimes, mechanical
Chapter 9 Seakeeping Tests
388
dampers [Katayama, et al. (1982)] or other active damper systems are introduced externally in order to reduce tendon loads. The material damping appears from the tendons and their attachments to the TLP as well as to the sea floor. The subsea template also provides some damping. The hydrodynamic damping appears in the form of the radiation damping as well as viscous damping as discussed previously. The rediation damping at the high frequency is generally quite small.
Co
O O
9j.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
XAXIS
FIGURE 9.24 FITTING OF DATA ON DAMPED OSCILLATION OF SEMI The firstorder loads on a TLP do not induce resonant motions in the system, particularly, in the tethers. However, the tethers may still experience resonance at the secondorder (sum frequency) components in heave and pitch. The amount of damping limits amplitude of tether loads due to springing. Due to inadequate theory in computing this damping, experiments are carried out to determine the damping coefficients in heave. Since the damping coefficient at the high frequency is extremely small , care must be taken in setting up the test so that accurate information may be obtained from the test.
Section 9.8 Damping Coefficients of a Moored Floating Vessel
389
In a test with a TLP column model [Huse (1990)], heave damping coefficients were obtained. The model represented a TLP column of 25m diameter and 37.5m draft at a scale of 1:20. The steel model had a sharp corner at the lower edge with a corner radius of less than 0.1mm. The model was suspended from the middle of two 6m long horizontal steel beams providing the springs of the oscillatory system in the vertical direction. The ends of the beam were welded to rigid structures to avoid additional damping introduced in the system from any fasteners. The spring constant of the suspension was high (1.414 x 106 N/m.).
0.014
■
D 0.012 A M P 0.01 N G F A C T 0 R
■
0 0
S
0.008
8
0.006
A
■
semi
#
9 ❑A tanker
0.004
❑ large displacement 0.002
0 medium displacement small displacement {
2.5 T
I 2 + I
I
I
Semi
1
0.5 +
❑
tanker ❑
0 ❑ I 6.00
8.00
10.00
ID 12 . 00
14.00
II 16. 00
18.00
I I I 20 .00
22.00 24.00
NATURAL PERIOD (sec)
FIGURE 9.25 DAMPING FACTORS AND DRAG COEFFICIENTS OF A TANKER VS. SEMI IN STILL WATER
390
Chapter 9 Seakeeping Tests
The model was ballasted with steel and water . The steel ballast inside was secured to the hull. The mass of the model was 3900 kg. The vertical and angular motions were measured by accelerometers. The tests were carried out as heave decay tests. The heave amplitudes ranged from 0.1 to 2 .0 mm. The angular motions were analyzed to ensure pure heave motions of the column . The frequency of vertical oscillation in water was 3.03 Hz. To start the test, the model was pulled upward from equilibrium by a crane and the steel wire connecting the model was cut. The subsequent heave oscillation was recorded. The initial displacement of the model was controlled by an electronic balance. Tests were also carried out in the dry to determine material damping of the setup so that the decay due to hydrodynamic damping alone could be established by subtraction. The hydrodynamic damping was found to be mainly linear, the quadratic damping being less than 10 percent . The linear damping coefficient was found to be bl = 122.6 N s/m or a normalized damping coefficient bl/M = 0.03.
LOAD CELL 7 77
FIGURE 9.26 DAMPING TEST SETUP OF A TLP COLUMN
Section 9. 8 Damping Coefficients of a Moored Floating Vessel
391
In another test with a floating vertical caisson [Chakrabarti and Hanna (1990, 1991 )], the added mass and damping coefficients in heave were evaluated. The model consisted of a single vertical cylinder scaled (1:20) from a vertical column of a typical TLP. A 0.76 in (21/2 ft) diameter model was selected for this test. The draft of the model (about 1.22 in or 4 ft) also represents the typical draft of a TLP. The model surface was painted to a smooth finish. The bottom edges were sharp. The ratio of the tank width to model radius was 26.4. A limiting ratio of 10 for the tank blockage effect was reported by Calisal and Sabuncu (1989). Tests were run in still water and in waves . The cylinder was held in place in the water by two vertical sets of springs ( Fig. 9.26). The springs were linear and pretensioned and were arranged so that they could be easily changed during the test. Load cells were attached to the ends of springs to record the loads (and hence, motion). The cylinder was displaced vertically from its equilibrium position and released, and the decaying oscillation was recorded. The test was repeated twice for nine sets of springs. In a first series of tests [Chakrabarti and Hanna ( 1990)], the motions of the caisson were recorded (to examine the effect of any pitching motion of the cylinder) with the help of two potentiometers placed at two diametrically opposite faces (Fig. 9.26). These potentiometers introduced substantial mechanical damping making the damping factor values erroneous. Subsequent tests were made without the potentiometers. O N 6
O SPRING SET IN AIR e SPR IN G SET IN WATER %K LINEAR DIFFRACTION
go
e e
ge
0 0.4
0 0.8
®
®
1.2 1.6 2.0
8 2.4
2.8
NATURAL PERIOD , SEC FIGURE 9.27 DAMPING OF A TLP LEG SPRING SET IN WATER VS. AIR
392
Chapter 9 Seakeeping Tests
In order to study hydrodynamic damping, it is important to extract the material damping in the spring system used in the tests . In this case , the spring sets were hung in air with an equivalent dead weight representing the pretension in the springs. The weight was displaced vertically an amount equivalent to the displacement of the caisson in water. The loads in the spring system were measured with a load cell at the fixed end of the setup . The results are shown in Fig . 9.27 for various spring sets. The material damping is generally quite small compared to the hydrodynamic damping, but not negligible. An interesting observation of the test results is that the heave damping for the vertical cylinder is linearly dependent on the cylinder velocity in low amplitude heave. For larger amplitude of oscillation, there is some evidence of nonlinearity. However, for a TLP caisson , the heave is expected to be small . The damping which is composed of radiation and linear viscous components is mostly of the order of 1 percent of critical . However, it tends to decrease with higher frequencies. At a frequency near 2 Hz, the damping is about a third of 1 percent . Several runs made with different (but small) initial displacements show the consistency of the data. At a higher frequency of 3Hz [Huse ( 1990)] the damping factor was even lower. 9.9 MODELING AIR CUSHION VEHICLES Froude scaling is used for conventional vessels in scaling the model test results on wave induced motions and loads to their corresponding prototype values . In this case, the model and waves follow the usual geometric scaling . Speed of the vessel is based on Froude similitude between the model and the prototype. For this model , all the scaling laws for various quantities are known (Chapter 2). However, this scaling law is found to be inapplicable to the air cushion vehicles (ACV) such as surface effect ships and hovercraft. The ACV is a seagoing vessel that has its weight supported primarily by an increased static air pressure (compared to the atmosphere ) within the confines of rigid sidewalls that penetrate the water surface. Because atmospheric pressure enters in the modeling of the prototype (see also Section 8.5), an interesting scaling problem arises in this case. The following discussion is essentially taken from Kaplan (1989). For a Froude model, accelerations scale as 1:1. However, for an ACV, model tests produced unconservative full scale values for the heave acceleration . This is mainly due to the effect of nonscaling of the ambient atmospheric pressure . The pressure affects the natural frequency and damping of the coupled pressure vertical motion mode. A simple linear analysis in head seas with adiabatic laws for an air cushion shows [Kaplan (1989)] a natural frequency for the coupled heave pressure mode of
Section 9. 9 Modeling Air Cushion Vehicles
coy = r'Y8(1+Pa'Po)12 hb
L
J
393
(9.13)
and the corresponding damping factor K Cy = 2coZK,
(9.14)
where
K,
Pa Abbe
(9.15)
y l+pa/po
and
Ks = 2 PaQo  Pal ap IoPo
(9.16)
The quantities in Eqs . 9.13  9. 16 are pa= density of air, Ab = cushion area, hb = height of cushion, 7 = ratio of specific heats for air, pa = atmospheric pressure, po = equilibrium cushion pressure (gauge), Qp = volume of air delivered by the fan at a pressure po, and (aQ/ap)p = slope of the fan characteristic curve at po. The details of the derivation of these relationships may be found in Kaplan ( 1989). The purpose of presenting the results here is to discuss the scaling laws that may be used for an ACV. The damping of the motion is found to depend primarily on the slope of the pQ fan curve which increases with a decrease in the slope. In a Froude model, the frequency scales as 11.. Thus, a model which is 1/25 in size in linear dimension (i.e. , = 25), will have a corresponding frequency 5 times that for a fullscale vehicle. For an ACV, the equilibrium cushion pressure as well as the cushion height in model scale varies as 1/X. Therefore, the natural frequency in model scale will vary approximately as X due to the large value of the ratio pa/po in Eq. 9.13 . Similarly, the damping factor is proportional nearly to ? rather than being the same as the prototype value in a Froude model . From Eq. 9.13 the natural frequency cuy decreases with the increasing size of the ACV. In this case a resonant effect may be expected for a full scale ACV. However, in a Froude model, this resonance may be missed since the model natural frequency (with same atmospheric pressure) will be too large and thus unconservative .
394 Chapter 9 Seakeeping Tests
One approach to avoid this problem in model testing is to carry out the model testing in a tank facility where the ambient atmospheric pressure in the facility itself may be properly scaled by depressuring the entire tank facility. Such a facility exists at MARIN, The Netherlands. The facility, however, does not have a wavemaker. A mechanical oscillator may be able to simulate the vertical motion. This then can illustrate the effects of ambient atmospheric pressure on the scaling relations. A more acceptable method of prediction of the ACV dynamic motion characteristics is to simultaneously develop a computer simulation model and a model testing program. The computer model is validated with the results from extensive series of model tests at smaller scales, by ensuring good correlation. The computer model is then applied directly to the full scale case using appropriate full scale parameter values to predict the design information on the dynamic characteristics of the ACV. 9.10 ELASTIC FLOATING VESSEL In designing a ship hull, the stresses or strains in the structure of the ship due to the distributed loads from waves must be known. Usually the stresses are computed theoretically from the computed or measured distributed loads (such as pressures on the hull). Measurements of the strain in the hull are also made in a model test. For the measurement of bending moment at a few crosssections, the model hull is distorted by segmenting it at these locations and backbone beams of proper bending stiffness are mounted across these sections which are fitted with strain gauges to measure strains due to waves. A more appropriate and complete modeling calls for an elastic ship model of suitable material [Lin, et al. (1991)] which follows the appropriate law of similitude. The material (such as foamed vinyl chloride) should be of stable characteristics and of small Young's modulus. The model ship, in this case, must satisfy all the geometric, hydrodynamic (i.e. Froude) and structural (i.e. Cauchy) conditions of similitude. Considering the longitudinal bending for the structural similitude, the relative displacement or strain in the longitudinal direction between the model and the full size ship must be the same Msy = constant EI
(9.17)
which gives the relationship for the bending stiffness (flexural rigidity) (EI)p =;.5(El)m
(9.18)
Section 9.11 Modeling of a Loading Hose
395
where MB is the vertical bending moment and y is the vertical distance to the center line of the hull . The natural frequency corresponding to the vertical bending mode of the dry hull satisfies the condition wm
=
'f%wP
(9.19)
9.11 MODELING OF A LOADING HOSE Another example of an interesting modeling problem is that of a very flexible object having both axial and bending stiffness of significance, such as a floating hose. A loading hose is generally used to transfer oil from an articulated loading tower (or buoy) to a shuttle tanker. The success of a single point mooring terminal greatly depends on the performance of the hose in waves and an effective life span of the hose string. One of the higher maintenance items that both floating buoys and articulated towers have in common is the loading hose string . A floating hose string permits the mooring of side loading tankers as well as bow loading tankers . The single point mooring dynamics coupled with the prevailing sea subject the hose string to an extremely complex system of motions and loads . This together with the flexibility of the hose string leads to problems of ( 1) chafing from contact with the tower, ship, and the hose itself, (2) kinking due to high local bending stresses , and (3) entanglement with the tower or hawser. Since all of these problems may put the mooring system out of service temporarily, a reduction of the severity of any of these problems will lead to more time online, longer hose service life, and lower annual costs. 9.11.1 Hose Model In modeling a hose system [Brady, et al. (1974)], the outside dimensions, mass and buoyancy of the hose are considered. In addition to scaling the geometry , the axial and bending stiffness are also modeled. When modeling underbuoy hoses, buoyancy tanks, beads and tie wires are also scaled with appropriate elasticity and net buoyancy. There are many techniques that may be used to model these parameters . However, satisfying all properties at the same time is not a simple task. Note that different parts of the hose section, such as floating hose, submerged hose, etc. have different properties . The validity of simple Froude scaling of the hose is subject to question, but considering that inertia plays a dominant role in the hose dynamics, it is a reasonable choice . One method of modeling described below [Young , et al. (1980)] illustrates some of these difficulties. The hose models were made of helical springs coated with an elastomer. The spring essentially provides the axial stiffness while the required bending stiffness is provided in combination . The elastomer also gave the hose model its outer geometry.
396
Chapter 9 Seakeeping Tests
Two types of hose modules in the hose strings were constructed from the same basic geometry floating hose and submarine hose . The inertial properties of the hose were modeled as closely as possible . In particular, the reserve buoyancy of floating hose sections and the submerged weight of submarine hose with flotation collars in place were modeled to obtain the scaled mass . Likewise, the outside diameter was modeled to obtain reasonable values for displaced volume and virtual mass. The theoretical values of these properties were obtained from the manufacturer's catalogue. To model the floating hose sections , it was necessary to increase the outside diameter of the existing models of submarine hose and to provide additional buoyancy. Washers and grommets were fabricated from 6. 4 mm (1 /4 in.) closed cell foam rubber. The outside diameter was cut to the scaled diameter while the inside diameter was such that the grommets would just slide over the submarine hose model. To prevent significant changes in the bending stiffness , the grommets were spaced 9.5 mm (3/8 in.) apart on center. However, insufficient buoyancy was obtained by the addition of grommets. Buoyant hose string Buoyancy modules Wrings (retainers) Ballast Spring covered with latex
End connector
Grommets Corks Submarine hose string
FIGURE 9.28 TYPICAL FLOATING AND SUBMARINE HOSE MODEL SECTION WITH THEIR COMPONENTS
Section 9.11 Modeling of a Loading Hose
397
To compensate for the additional buoyancy required , short cork segments were inserted in the model hose bore. By placing the segments inside , the correct buoyancy could be obtained without affecting the added mass of the model . In the few instances where additional weight was required to ballast a model , it was added by distributing the required weight of lead shot along a piece of monofilament line and threading it through the model hose bore . Figure 9 .28 shows typical completed floating and submarine segments with their components. TABLE 9.2 LOADING HOSE STRING COMPONENTS (Code Letter Identifies Components in Fig. 9.29) PROTOTYPE
MODEL
DESCRIPTION
SIZE
BUOYANCY FULL OF OIL (kg)
Full Length Reinforced Submarine Hose
12" IDx35'
417
15.6
0.33
3.8
b
Half Length Reinforced
12" IDx35'
377
15.2
0.32
3.4
c d e f g
12" IDx35' 12" IDx35' 12"IDx35' 10" IDx35' 10" IDx35'
391 732 604 266 1122
15.2 22.8 19.7 12.9 22.2
0.32 0.47 0.41 0.27 0.46
3.5 6.6 5.5 2.4 10.2
h i i k
Submarine Hose Floating Hose Reducing Floating Hose Submarine Hose Extra Buoyant Tanker Rail Hose with Auxiliary Equipment Bead Float for 10" End Bead Float for 10" Body Bead Float for 12" End Bead Float for 12" Body

81 59 113 81
30.1 26.4 34.8 30.1
0.63 0.55 0.73 0.63
0.7 0.5 1.0 0.7
1
Supporting Buoy with

570


5.2
12" IDx35'
1073
15.25
0.32
9.7
12" ID 24" IDx35'
24"IDx35'
4004 2746 4882 4146
27.5 27.3 39.3 35.2
0.57 0.57 0.82 0.73
0.7 16.4 11.3 20.0 17.0
20" IDx35'
2866
31.0
0.65
11.8

261 236
47.3 43.5
0.98 0.91
2.4 2.1
CODE LETTER a
in n o p q r s t u
Chain Heavy Coflexip Type Hose 45° Elbow Submarine Hose Submarine Hose Floating Hose Lightweight Reducing Floating Hose Lightweight Floating Hose Bead Float for 24" End Bead Float for 24" Body
24" IDx35' 24" IDx35'
APPROXIMATE OUTSIDE DIA. (in.)
OD (in.)
BUOYANCY (gru)
tin=25.4mm;1ft=0.3m
Table 9.2 lists all the prototype hose components used to assemble the hose configurations in this case along with their outside diameter and buoyancy. The eight hose configurations are shown in Fig . 9.29. A code letter is included in Table 9.2 to identify each hose component in the configuration.
398
Chapter 9 Seakeeping Tests
2
]
5
6
T
Bi8
20
2i
22
2J
2.
^\ 2J 26
OgKIWRATION N0.8
CONFIGURATION NO. T
FIGURE 9.29 HOSE STRING CONFIGURATIONS 18
Section 9. 11 Modeling of a Loading Hose
399
Starting at the hose to tower connection for configuration 1, the string consists of eight sections of 0.3 in (12 in .) diameter submarine hose with buoyancy collars attached to permit the hose to lie in a shallow U shape. The downstream end receives additional support from a buoy located at the junction of sections 5 and 6. Sections 9 through 20 are of 0.3 in (12 in.) dia. floating hose. Section 21 is a 0.3 in (12 in.) dia. to 0.25 in (10 in.) dia. reducing floating hose, and it is followed by four lengths of 0.25 in (10 in.) dia. submarine hose with flotation collars located to provide a very shallow submerged loop. The last section is an extra buoyant tanker rail hose with ancillary equipment attached . Configurations 2 through 8 are simple modifications of the basic hose string . The loading at the tower end of hose is a critical design consideration that should be measured in a model test. Load cells may be placed to measure bending and axial load at the hose to tower connection . Load cells are aligned such that bending is measured from a vertical plane radially away from the tower and at right angles to the radius, as shown on Fig. 9.30. 9.11.2 Hose Model Testing In an SPM model testing, attached with hose , visual observations and video recordings are made to study the behavior of particular hose strings. Such things as erratic motion, snaking, flexing etc . are observed during the testing . Snaking is a phenomenon where the hose takes on a sine wave shape transverse to the direction of wave travel and in the plane of the water surface. Configurations that reduce the hose loads at the tower also reduce the amplitude of the transverse waves. Thus, hose designs that reduce tower loads may reduce axial loads in the tower end of the hose string as well. Since the hose loop travels along its axis and since its flexing action is similar to a catenary, the downstream bending moments would be expected to be larger than the upstream moments . An obvious way to eliminate this moment is to put a swivel at the tower hose connection. This was done in Configuration 2. The result, as anticipated, reduced the bending moments measured in the load cell. This, however, allows the hose loop to come much closer to the tower when the tower makes a downstream excursion in waves. Configuration 4 is the basic string with the first two sections made with hose having the weight and the geometry of a 0 .3 in (12 in.) dia . coflexip hose (the stiffness was not modeled). The effect of this loop modification was to significantly reduce the dynamic loads in the tower to hose connection. Likewise, the snaking action of the hose was attenuated. In Configuration 5, a snubber line was connected to the two ends of the tower hose loop. Results show a significant reduction in loads at the tower connection. These
400
Chapter 9 Seakeeping Tests
FIGURE 9.30 LOAD CELL STRING AT TOWER CONNECTION reduced loads compare on an equal basis with Configuration 4. The general motion appeared to be somewhat better because snaking in this hose string was of smaller amplitude than the others tested. Configuration 6 is a hose string modeled after a monobuoy configuration. There is no submerged loop at the tower . The hose connects to the tower horizontally at the mean water line. This string showed the worst performance of the series . It was the. only one in which jerking of the hose at the tower could clearly be observed . The loads were the highest and snaking was as severe as any of the configurations tested. Configuration 7 was a combination of Configurations 4 and 5, whose purpose was to see if the two would work together to further reduce the dynamic loads at the tower, which was found to be the case. The submerged hose loops have been shown to dampen oscillatory motions for different hose sizes and at different hose string locations. Drag appears to play a significant role in the action of submerged hose loops and may well be an important influence on the entire hose string. This casts serious doubt on the ability of the Froude model to accurately represent a prototype hose structure.
Section 9. 12 Motions in Directional Seas
401
9.12 MOTIONS IN DIRECTIONAL SEAS The discussion thus far has been generally directed to motions of floating bodies in twodimensional waves . However, the method outlined earlier is equally applicable to threedimensional waves as well. The 3D waves are generally simulated in a small region of the test basin . Therefore it is important to plan the test setup carefully. The test and measurement system is otherwise similar to the 2 D wave tanks. 1.2
s I s 3 s •6 s•0
0.9
i
0.6
%
0.3
EK 0.3
0.4
0.5 0.6 0.7
0.8
0.9
FREQUENCY, Hz
FIGURE 9.31 MEASURED ROLL RESPONSE AMPLITUDE OPERATOR IN MULTIDIRECTIONAL WAVES [Nwogu (1989)] The 3D waves have a considerable influence on moored systems. Considering that the motion response is linear, the transfer function in a directional sea can be obtained in a similar fashion as in the case of a unidirectional sea. In this case, however, the additional dependency on the angle 0 is attached to the transfer function, H((' ,O). Then the motion response spectrum in a particular mode of motion j of
402
Chapter 9 Seakeeping Tests
a floating vessel may be obtained from this transfer function Hj and the directional sea S(O) as ( S.,j (w)= Jn n
H1 (w,0
2
S(w,0)dO (9.20)
The transfer function for a multi directional sea state represents directionally averaged values since they depend only on the wave frequency . The effect of wave directionality may be clearly found in the response amplitude operator in the frequency domain. An example for the roll response of a barge in several multidirectional seas (s=1,3,6,eo) is shown in Fig . 9.31 [Nwogu ( 1989)]. The roll is seen to increase in the multidirectional seas. The directionality of the waves is found to result in a decrease in the surge and pitch motion amplitudes and an overall increase in the response of sway, roll, and yaw motions. The effect on the heave of a floating moored system is mixed, but small . The low frequency motion, such as that found in surge is affected more by the directional sea. This has a direct influence on the mooring line loads . Tests on CALM systems should generally be performed in a 3 D tank for a more realistic assessment of the responses of the system. 9.13 REFERENCES 1. Brady, I., Williams, S., and Golby, P., "A Study of the Forces Acting on Hoses at Monobuoy Conditions", Proceedings on the Sixth Offshore Technology Conference, Houston, Texas, OTC 2136, 1974, pp. 10571060. 2. Calisal, S.M. and Sabuncu, T., "A Study of a Heaving Vertical Cylinder in a Towing Tank", Journal of Ship Research, Vol. 33, No. 2, June 1989, pp.107114. 3. Chakrabarti, S.K., and Hanna, S.Y., "Added Mass and Damping of a TLP Leg", Presented , at the Twentysecond Annual Offshore Technology Conference, Houston, Texas, OTC 6406, May 1990, pp. 559571. 4. Chakrabarti, S.K., and Hanna, S.Y., "High Frequency Hydrodynamic Damping of a TLP Leg", Proceedings of the Offshore Mechanics and Arctic Engineering Symposium, Stavanger, Norway, Vol. 1, Part A, June 1991, pp. 147152. 5. Chakrabarti, S.K., "Moored Floating Structures and Hydrodynamic Coefficients",, Proceedings on the Ocean Structural Dynamic Symposium, Corvallis, Oregon, Sept 1984, pp. 251266.
Section 9.13 References
403
6. Chakrabarti, S.K., "Wave Interaction on a Triangular Barge", Proceedings on the Fifth International Offshore Mechanics and Arctic Engineering Symposium, Tokyo, Japan, ASME, 1986. 7. Chakrabarti, S.K., and Cotter, D.C., "Analysis of a TowerTanker System", Proceedings on the Tenth Offshore Technology Conference, Houston, Texas, OTC 3202, 1978, pp. 13011310. 8. Chakrabarti, S.K. Nonlinear Methods in Offshore Engineering , Elsevier Publishing Co., Netherlands, 1990. 9. Chakrabarti, S.K., and Cotter, D.C., "Motion Analysis of an Articulated Tower", Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, Vol. 105, August 1979. 10. Chakrabarti, S.K., and Cotter, D.C., "Nonlinear Wave Interaction With a Moored Floating Cylinder", Proceedings on the Sixteenth Annual Offshore Technology Conference, Houston, Texas, OTC 4814, May 1984. 11. Chakrabarti, S.K. and Cotter, D.C., "Interaction of Waves with a Moored Semisubmersible", Proceedings on the Third International Offshore Mechanics and Arctic Engineering Symposium , ASME, New Orleans, Louisiana, Feb., 1984, pp. 119127. 12. Chantrel, J. and Marol, P., "Subhannonic Response of Articulated Loading Platform", Proceedings of the Sixth International Offshore Mechanics and Arctic Engineering Symposium, ASME, Houston, Texas, 1987, pp. 3543. 13. Cotter, D.C. and Chakrabarti, S.K.., "Effect of Current and Waves on the Damping Coefficient of a Moored Tanker", Proceedings on TwentyFirst Annual Offshore Technology Conference, Houston, Texas, OTC 6138, May 1989, pp. 149159. 14. DeBoom, W.C., Pinkster, J.A., and Tan, P.S.G., "Motion and Tether Force Prediction for a TLP," Journal of Waterway, Port, Coastal and Ocean Division, ASCE, Vol. 110, No. 4, November 1984, pp. 472486. 15. Faltinsen, O.M., Dahle, L.A. and Sortland, B., "Slowdrift Damping and Response of a Moored Ship in Irregular Waves," Proceedings on Fifth International Offshore Mechanics and Arctic Engineering Symposium, Tokyo, Japan, ASME, 1986.
404
Chapter 9 Seakeeping Tests
16. Goodrich, G.J., "Proposed Standards of Seakeeping Experiments in Head and Following Seas", Proceedings on Twelfth International Towing Tank Conference, 1969. 17. Huse, E. and Matsumoto , K. "Practical Estimation of Mooring Line Damping", Proceedings on Twentieth Offshore Technology Conference, Houston, Texas, OTC 5676, 1988, pp. 543552. 18. Huse, E. and Matsumoto, K., "Mooring Line Damping Due to First and SecondOrder Vessel Motion," Proceedings on TwentyFirst Annual Offshore Technology Conference , Houston, Texas, OTC 6137, May 1989, pp. 135148. 19. Huse, E., "Effect of Mechanical Friction in Model Test Setup," MARINTEK Project Report No. 511151.00.03, Trondheim, Norway, 1989. 20. Huse, E., "Resonant Heave Damping of Tension Leg Platforms", Proceedings on TwentySecond Offshore Technology Conference, Houston , Texas, OTC 6317, 1990, pp. 431436. 21. Kaplan, P., "Scaling Problems of Dynamic Motions in Waves from Model Tests of Surface Effect Ships and Air Cushion Vehicles," Preprint 89FE1, Joint ASCE/ASME Mechanics, Fluids Eng., and Biomechanics Conference , San Diego, California, July 1989, 9 pages. 22. Katayama, M., Unoki, K., and Miwa, E., "Response Analysis of Tension Leg Platform with Mechanical Damping System in Waves ," Proceedings on Behavior of Offshore Structures, MIT, Boston , Vol. 2, 1982, pp. 497522. 23. Lin, J., Qui, Q., Li, Q. and Wu, Y.," Experiment of an Elastic Ship Model and the Theoretical Predictions of its Hydroelastic Behavior ," Proceedings on Very Large Floating Structures ,Univ. of Honolulu, Hawaii, 1991, pp. 265276. 24. Nwogu, 0., "Analysis of Fixed and Floating Structures in Random Multidirectional Waves," Ph.D. Thesis, Univ. of British Columbia, Vancouver, B.C., Canada, 1989. 25. Petrauskas , C. and Liu, S.V., "Springing Force Response of a Tension Leg Platform", Proceedings on the Nineteenth Annual Offshore Technology Conference, Houston, Texas, OTC 5458, 1987, pp. 333341.
Section 9. 13 References
405
26. Pinkster, J.A. and Remery, G.F.M., "The Role of Model Tests in the Design of Single Point Mooring Terminals," Proceedings of Seventh Annual Offshore Technology Conference, Houston, Texas, OTC 2212, 1975, pp.679702. 27. Tan, S. G., and DeBoom, W.C., "The Wave Induced Motions of a Tension Leg Platform in Deep Water ", Proceedings of the Thirteenth Annual Offshore Technology Conference , Houston, Texas, OTC 4074, 1981. 28. Wichers, J.E.W., "A Simulation Model for a Single Point Moored Tanker", Maritime Research Institute, Wageningen, The Netherlands, Ph.D., Delft U. of Technology/MARIN Publication No. 797, 1988, 243 pages. 29. Young, R.A., Brogren, E.E. and Chakrabarti, S.K., "Behavior of Loading Hose Models in Laboratory Waves and Currents," Proceedings of the Twelfth Annual Offshore Technology Conference, Houston, Texas, OTC 3842, May, 1980.
CHAPTER 10 DATA ANALYSIS TECHNIQUES
10.1 STANDARD DATA ANALYSIS One of the most important aspects of a model testing program is the analysis of the data collected during test runs. Some model tests are only qualitative , requiring minimal instruments. These are quite helpful in evaluating a conceptual design or an offshore operation . However, the majority of the tests are designed to provide many useful data that are applied in verifying the design of the particular system or operational procedure . This chapter will deal with the most common data reduction procedures that are used in the analysis of the test results. There are two types of data reduction routines . Standard routines are commonly used for all test data collected in a wave tank environment. These are used in any model testing work. Many of these programs are commercially available today, although most testing facilities find it advantageous to develop their own softwares. The second type of routines are specialized software written for specific types of tests performed. There could be numerous programs developed for model test data. Those that have been chosen for discussion here relate to the model testing covered in the earlier chapters . It is recognized that descriptions of many special data reduction routines have been left out in this chapter . Included is only a representative sampling from the testing described earlier in Chapters 5, 7, 8 and 9. Since it is important to adequately quantify the waves generated in the tank , a major part of this chapter deals with both the 2D and 3D waves (Chapter 5). Data reduction on the wave forces on fixed models (Chapter 7) includes computation of hydrodynamic coefficients which is discussed here. Evaluation of towing loads have been adequately described in an earlier chapter (Chapter 8) and hence is omitted here. Computations of hydrodynamic damping and associated quantities of floating moored models (Chapter 9) are also addressed in this chapter. The recorded data contain three generic types of measurement errors: calibration errors, bias errors and random errors. Calibration errors result from the variation between the measuring device's actual input/output relationship and the calibration curve used with the device . This type of error will result in a deviation
Section 10. 1 Standard Data Analysis
407
(systematic or scatter) about the recorded values when they are expressed in engineering units . Bias errors evolve from data reduction procedures such as the windowing operations associated with the calculation of power spectral densities, and they appear with constant magnitude and direction from one analysis to the next. Random errors are the result of averaging operations that must be performed over a finite number of sample records. Time history records are the primary source of information from the wave tank testing. The time history data are first inspected for evidence of spurious signals. The following estimates of statistical and probabilistic parameters are generally obtained for further analysis:  mean value;.  root mean square;  standard deviation;  random error;  bias error;  calibration error or uncertainty;  probability and cumulative distribution functions plotted in normal and/or Rayleigh distribution format, as appropriate;  Kurtosis and skewness of record;  spectral (autospectral) density, including other spectral moments as appropriate for single peak spectra to estimate spectral bandwidth, peak frequency, etc. Some of the standard data reduction routines that are available at any model testing facility and their function are as follows: • Plotting package  ability to plot the recorded data in the time domain. • Summing routine  ability to add two or more channels in the time domain from one or more data files. This enables the development of force and moment traces from direct load cell readings, for example. • Maximum/Minimum routine  to compute the mean maximum and minimum amplitudes of a regular wave trace along with standard deviations of data from the mean values. • Phase routine  ability to compute the phase difference between two channels in which phase angles between corresponding peaks in two traces are found.
408
Chapter 10 Data Analysis Techniques
• Normalization  ability to generate tables of transfer functions by normalizing amplitude of measured response time history by the amplitude of regular wave time history . For irregular waves, the corresponding spectra are normalized. Some of the other data reduction routines that would be available at a model testing facility will be discussed subsequently in greater detail. 10.2 REGULAR WAVE ANALYSIS Regular waves generated in a tank are generally twodimensional in character and have a single frequency associated with them . While the input signal to the wavemaker in generating these waves is a sinusoid , the actual waves generated in the tank may not have a sinusoidal form due to the gravity , tank bottom and other effects. Moreover, the single frequency waves may produce responses that have multiple frequencies . This is particularly possible in responses for moving structures. An example of this may be found in Fig. 7 . 1 in which the measured forces have multiple Theoretical analysis justifies the frequencies due to a single frequency wave. existence of these multiple frequencies in the response due to a regular wave (Section 10.6.1). In a wave tank of limited size, the waves at the test site are often contaminated by the presence of standing waves and reflected waves. It is important that these effects are isolated from the measured data before further analysis is carried out.
NODE 
ANTINODE
L/2
DISTANCE, X FIGURE 10.1 STANDING WAVE PROFILE IN A TANK
Section 10.2 Regular Wave Analysis
409
10.2.1 Standing Wave A standing wave is encountered in a closed or partially open basin when the confined water is disturbed. A standing wave produces a vertical oscillation with time t at a given location (Fig. 10.1). The amplitude of the standing wave is expressed as tl= Hcoskxcoswt
(10.1)
where H is the height of the standing wave. According to this equation, a standing wave experiences the same amount of oscillation about the x axis from one cycle to the next. Also, where cos kx = 1, it experiences the maximum vertical oscillation which is called the antinode. The point, where cos kx = 0 (1/4 the wave length L away), experiences no vertical oscillation and is called the node. A standing wave in a basin may have one or more nodes and antinodes. The oscillation period (sloshing) in the basin is called the natural period. There are several modes of natural period in the basin depending on its length (or width for cross oscillation). For a rectangular basin, the nth mode of oscillation has a wave length given by L_ 2l n=1,2,... n
(10.2)
where I = length of basin. The sloshing period may be computed from the dispersion relationship: _ 4xl 2 T„ ng tanh kd
(10.3)
where d is the uniform water depth in the basin. For a given test basin, the longitudinal and transverse sloshing periods should be computed from the above relationship. Testing at these natural periods may then be avoided. Tests at these periods have indicated that the wave heights may vary considerably from point to point in the longitudinal direction. If the sloshing period is discerned in the test data, its effect may be filtered out through digital filtering.
410
Chapter 10 Data Analysis Techniques
10.2.2 Reflected Wave Another area of data contamination is the reflected wave from the walls of the wave tank. These waves have the same frequency as the incoming wave and, therefore, are indistinguishable from the incident wave . The methods of computing the wave reflection coefficient have been detailed in Chapter 4. These may be established for various wave frequencies in a given wave basin before any testing commences in the basin. Most test basins have this information available for their clients to review. Note that while it is important to know the amount of reflection in a basin, data measured in the test area include the reflected wave and its effect on responses. Sometimes it is desirable to reduce the effect of reflection . Methods to implement this have been discussed in Chapter 4. 10.2.3 Spurious Wave Data The data collected in a wave tank environment suffer from several other shortcomings [Mansard and Funke ( 1988)]. Some of the spurious data are removed from the recorded data before any analysis is performed. The preprocessing of data is essential to reduce the error in subsequent analysis. The first and foremost of preprocessing is the mean and trend removal. Most data have an offset from zero . The trends could be linear or secondorder. The linear trend appears as a drift in the mean value. The secondorder trend, while generally quite small, describes a parabolic drift of the mean value. These trends are removed from the wave data before temporal (zerocrossing) or spectral analysis is performed . However, a word of caution is warranted here . Some of the response data will produce a zero shift from the waves which is physical . Therefore, while the wave elevations may be assumed to have a zero mean for linear Gaussian waves , nonlinear waves or responses may have a nonzero mean which should not be removed before examining the physical phenomenon carefully. In the determination of the amplitudes or heights of a record (e.g., the wave height), the maximum recorded value of the sample between zero crossings is considered the crest (or minimum for troughs). This method works well for samples taken at a high sampling rate . However, if the sampling rate is low, it may introduce error in the height value. In this case, a parabolic fit to the three adjacent highest samples is made and the
Section 10.3 Irregular Wave Analysis
411
highest point is chosen as the crest value . The (wave) height, then, is obtained from the adjacent crest and trough thus chosen. 10.3 IRREGULAR WAVE ANALYSIS Offshore structures are often tested in irregular or random waves . The generation of these waves in the tank has already been discussed in Chapter 4. Some of the ambiguities in the irregular wave data are similar to the regular wave data discussed in the earlier section . The analysis of the random wave data can be made both in the time domain and in the frequency domain . Obviously, the analysis time is lengthy in the time domain, but it has the advantage of maintaining any nonlinearity present in the data. The spectrum analysis is generally linear in nature and readily provides a transfer function between the waves and responses . However, nonlinear frequency domain analysis methods are currently being developed. Some of the data analysis methods commonly used with the wave tank data analysis is described subsequently. 10.3.1 Fourier Series Analysis Fourier series analysis is a useful tool in analyzing data with multiple frequencies. It allows the determination of amplitudes and phases of different frequencies present in a time history of recorded data. Thus , a onetoone correspondence between the responses and wave at given frequencies is possible. If a small enough frequency increment is chosen , then it is possible to decipher all of the frequency components present in a system of recorded data. The derivation of amplitudes and phases is as follows. Let the time history of the signal recorded be given by f(t) and the length of the record be TR. Then, the Fourier representation of the signal f(t) is given by N
f(t)=ao+
2n at 2nttt a. cos +l, sin
TR T.
where N is the number of finite Fourier components of interest and ao, an , and bn are constants which are evaluated from the integrals
412
Chapter 10 Data Analysis Techniques
ao = T ITa R JJ00
(10.5)
f (t)dt
^T^ 2 2 f (t)cos a„=T R
Ltdt
b _ 2 IT. 2nnt dt u f(t )sin L TR
(10.6)
(10.7)
The coefficient ao is often expected to be near zero for irregular waves . However, as stated before, it may be nonzero for the recorded responses, e.g., loads. A numerical integration routine is applied to the integrals in Eqs. 10.5  10.7. Once the Fourier coefficients are known , the amplitude and phase of the harmonic components are computed from A=
a„Z+b„'
(10.8)
and e„=tan'b", n=1,...,N a„
(10.9)
It is quite common to find higher harmonics of wave and response records. The above analysis allows one to study the harmonic components of the wave and the corresponding response. 103.2 Wave Spectrum Analysis It is a common practice to represent a measured sea state in terms of its spectral density. Because of the short duration of the wave records, they suffer from spectral variability . One source of the variability is the algorithm used in the derivation of the spectral parameters , e.g., the peak frequency . Another variable is the choice of the upper and lower cutoff frequencies in order to avoid background noise. In order to overcome these and other difficulties and source of variability, Mansard and Funke (1988) advocated spectral fitting . In this case, a mathematical model , e.g., a JONSWAP
Section 10.3 Irregular Wave Analysis
413
spectrum, is fitted to the spectral estimate and parameters are derived from the best fit curve (e.g., by a least square technique). Once the mathematical description is known, all statistical parameters may be easily derived from this model without additional variability. A bias correction factor is often applied to the parameter estimate to obtain an enhanced estimate of the parameter. For example, it is customary to use cutoff frequencies in the spectral estimates to derive moments of the spectrum. However, the imposition of upper and lower cutoff frequencies will underestimate the zeroth moment, mo. The integration of a smooth spectral model between 0 and infinity will provide the theoretical value of mo. Then, a bias correction factor for mo may be defined as ^S(f )df CZ ° f S(f)df
(10.10)
where fl and f2 are the lower and upper cutoff frequencies. The significant wave height may be modified by Hs =CHs
(10.11)
For a JONSWAP spectrum (peakedness parameter, y = 1 to 12), if the cutoff frequencies are chosen as fl = 0.5fo, and f2 = 2.5fo, where fo is the peak frequency, then the correction factor is obtained [ Mansard and Funke (1988)] from c2(7) =1.0015+V[19.9178(y+2.6937) (10.12) The energy (or power) spectrum is computed using an FFT algorithm. The scale factor and sampling rate are chosen such that the number of data points is a power of 2. As an example, let us consider a prototype system which has a natural period of 90 seconds and we require 100 cycles of data for proper data reduction. Let the scale factor for the model test be chosen as 1:100 and a sampling rate of 9 Hz is considered sufficient. Then, 0 = 900 sec. Test duration in tank = 90 x 100 1.115
414 Chapter 10 Data Analysis Techniques
Number of data points = 900 x 9 = 8100 In this case the total number of data points is chosen as 8,192 being the nearest power of 2. In order to obtain a smooth (averaged) estimate, the test record is divided into several intervals, such as 8 intervals of 1024 points each for the above example. A spectrum estimate Sk for the kth interval is made . Then the average (smoothed) estimate is obtained from
1i Sk(w) S(w)= 8 k.1
(10.13)
The main characteristics of the spectrum is given in terms of the moments of the spectrum, ^2 S(w)w"dco Jcu
(10.14)
where mn is defined as the nth moment of the spectrum computed between the cutoff frequencies col and on. Then, the significant wave height is obtained from the formula [Longuet  Higgins (1957)] Hs = 4j
(10.15)
The mean and zerocrossing periods are defined as T.=2x TO nil
(10.16)
TZ =2a F ;IZI
(10.17)
and
10.3.3 Wave Group Analysis The motions of moored floating structures are very sensitive to low frequency secondorder forces. The square of the wave envelope is important when considering this low frequency behavior since the slowlyvarying secondorder response of the vessel
Section 10.3 Irregular Wave Analysis
415
is significantly affected by grouping patterns of waves in the time domain. Therefore, particular attention must be paid in the generation of random waves so that the generated wave grouping is accurate when compared to the theoretical grouping. The groupiness function is defined as G(µ)=8rS(w)S(w+µ)dw=16 f°°R2(u)cos(Wu)du 71 JO
(10.18)
4
 TARGET  MEASURED
3
0 0.0
0.5
1.0
1.5
2.0
FREQUENCY, Hz
FIGURE 10.2 MEASURED VERSUS COMPUTED GROUPINESS FUNCTION FOR A JONSWAP SPECTRUM where R(u) is the autocorrelation function . The second equality avoids cumulative numerical errors . An example of the measured versus computed JONSWAP spectrum groupiness function is shown in Fig. 10.2. The groupiness function is plotted versus frequency. The trend and correlation shown are typical.
416
Chapter 10 Data Analysis Techniques
103.4 Statistical Analysis For signals recorded during a test in irregular waves, a statistical analysis is performed on each measurable quantity and the following statistical parameters are sought: • Mean valueThis quantity is obtained as the mean of the data over the recorded length. x=
1N I xi N ;.,
(10.19)
where xi are the values of the response at each sample point and N is the total number of sample points in the length of the record. • Root mean square valueThe rms value of the response is calculated from the definition N
X_ =[..
2
X1 ] j., z ]
(10.20)
• Maximum positive valueThis is the maximum recorded value, Xmax, on the positive side of the zero axis. • Maximum negative valueThis is the minimum recorded value, Xmin, on the negative side of the zero axis. • Significant height valueThis significant height (2x)1/3 is defined (by Rayleigh distribution) as the average value of the highest onethird heights (i.e., peaktopeak) in a record. It is also obtained from Eq. 10.15. • Significant positive amplitudeThis quantity is the average of the highest onethird positive amplitudes in a record. • Frequency response functionsThe transfer function for the response under consideration may be computed from a single irregular test. This technique will be described in a subsequent section.
Section 10.4 Analysis of Directional Waves 417
• Peak frequencyThe frequency at which the energy spectrum value is maximum. Because of the erratic profile of the measured wave spectra , it is generally not sufficient to simply choose the frequency at which the spectrum peaks . A better alternative has been recommended by the IAHR ( 1986). In this method, a threshold of 80% of the maximum spectrum value is chosen and the two frequencies, low and high for the first and second threshold crossings are computed . The centroid of this portion between the two frequencies is taken as the peak frequency. The simple method of determining peak value shows a much larger variability (about 25% greater) than the centroid method [ Mansard and Funke ( 1988)]. The centroid is given by
J
02
0 =ml2
E"l
wS( w )dw
(10.21) S((O)dw
where col and cu2 are the frequencies corresponding to 80% (60% has also been proposed as a modification) threshold value. Peak frequency has also been defined as fwS"(w)dw wo =
(10.22) fo S"(w)dw
where values of n have been proposed as 8 [Read(1986 )] or 6 [ Mansard & Funke (1988)]. Because of the asymmetry in the spectrum shape, a bias may be introduced in the computation for which a correction factor has been proposed (Eq. 10. 10). For a JONSWAP spectrum with low 'y value, this factor is about 1.02. 10.4 ANALYSIS OF DIRECTIONAL WAVES For the analysis and description of multidirectional waves, the wave field requires simultaneous measurement of the water surface elevation at a number of
418
Chapter 10 Data Analysis Techniques
neighboring locations . Typically, four or more wave probes are arranged near the desired location for the wave measurement. Alternatively, orthogonal horizontal components of the water particle velocities along with the water surface elevation may also be measured . Of the methods that can be used in the estimation of the directional distribution of wave energy, the most common are the Direct Fourier Transform (DFT) method , parametric method, the Maximum Likelihood Method (MLM) and the Maximum Entrophy Method (MEM). Most of these use a cross spectral method in the computation of the wave energy as a function of wave frequency and direction of wave propagation. The direct transform method and the parametric method are the earlier methods and have been described by Chakrabarti (1990). The first was proposed by Barber (1963) while the second was applied by LonguetHiggins, et al (1963). These techniques are limited in the resolution of the directionality of waves. The MEM is considered more useful in discerning the details of a directional sea and has become the most commonly applied method for the estimation of multidirectional spectrum. The MEM is found to resolve directional seas better than the MLM. However, it does not converge for very narrow spreading functions (i.e., large values of s). The spectral derivations by these methods have been detailed in the thesis by Nwogu (1989) and the reader is referred to his work for details. These methods will be briefly discussed. For the analysis of the directional waves, data is usually available in a series of time history of water surface elevation (or slopes or velocities) at various locations of the open water. These time series are transformed to the frequency domain by crossspectral density calculation. The cross spectral calculation is similar to the autospectral estimation of wave energy except that it works with two different time histories. Initially, a covariance function of the profiles is obtained from
Ry(ti)=j
Ti(t)tlj(t+ti)dt
(10.23)
where TR is the record length and i and j refer to two separate probe readings. The coincident spectra and quadrature spectra are computed as the cosine and sine transforms of the covariance function
Section 10. 5 Filtering of Data
Cy (CO) = is R,1(t)coswtdt
a! ((0)
=IT• R+1 (t)sinwtdt
419
(10.24) (10.25)
The twodimensional ((o,6) energy spectrum is the Fourier transform of the co and quadspectra. Knowing the distance Dij among probes i and j and the corresponding angles to the reference axis j , the directional spectra may be written as
S(w,O) _ 1 [C+1(w)cos{kD^1 cos(tt1e)}+Qj(o)sin{kDt1 c,4( i18)}] (10.26)
The summation is taken over a pair of probes among the total number of probes used. This is a simple estimate of the directional spectral form by the direct transform method. Other more commonly used methods are much more complex mathematically as summarized in the thesis of Nwogu (1989). Standard routines are available at facilities capable of generating 3D waves . One important point to note is that these are estimates of true spectrum and are subject to errors . The more the number of recordings, the better is the estimate. Generally 4 to 5 wave probes suffice. They are arranged in several geometric patterns and research has been done to examine the optimum configuration [Chakrabarti and Snider (1972)]. Fewer time history recordings are needed if orthogonal water particle velocities are simultaneously measured along with a wave profile. Mathematical routines are available [Nwogu ( 1989)] to reduce this type of data. 10.5 FILTERING OF DATA Filtering permits isolation of components of waves at one frequency band from those at a different frequency band. Filters are of two types : analog and digital. Analog filters are electronic in nature and have been described in Chapter 6. Digital filtering may be accomplished in the time domain or in the frequency domain. A digital filter is described as a transfer function which when applied to the original time history function (or its Fourier transform) will provide the filtered data. Thus, mathematically
420
Chapter 10 Data Analysis Techniques
8(t)=f(t)" h(t)=r f(ti) h(tti)d'c
(10.27)
where f(t) is the input function such as the wave profile, h(t) is the weight function and g(t) is the resulting response function. The asterisk denotes convolution given by the integral on the following line . Alternatively, in the frequency domain Eq. 10.27 may be written as
G(w)= F((O)H(w)
(10.28)
where H( (o) is the transfer function corresponding to its time history representation. They are related to each other by a Fourier cosine transform h(t) = 9 r I H((O)I coqwt  4'((O)]dw
(10.29)
Digital filtering of the recorded data is performed for a number of different purposes: • removing noise from data • preparing data for spectral analysis • removing transient data • separating low and high frequency data • comparing inputoutput relationship between two channels There are three types of digital filters: low pass, high pass and band pass. Low pass filters eliminate high frequency components and may be used for smoothing data by removing short period wave components riding on the wave frequencies, for example. High pass filters may be used to eliminate low frequency transients such as the decaying natural period oscillation at the start of a test run. Band pass filters retain the middle frequency band of interest. There is also a band reject filter which rejects a frequency band from the middle of the spectrum. If a record f(t) consists partly of signal s(t) and partly of noise n(t), then f (t) = s(t)+n(t)
(10.30)
Section 10. 5 Filtering of Data
421
In the frequency domain this becomes: F(w) = S(w) + N(w)
(10.31)
An ideal filter to remove noise will have the form H(w) = S(w) S((O)+ N(W)
(10.32)
However, in practice numerical leakage occurs with digital filters and distortion in the amplitude of data is evident due to this leakage . As explained earlier, filtering may be accomplished in the time domain or in the frequency domain . In the time domain a convolution integral is performed with the function h(t) (Eq .10.27). There are many mathematical expressions of h(t). One description of the weight function is P
H(w) = W(0)+2E W(p)cos((OpAt)
(10.33)
P=1
where W(0) = 2At( w2  w,) / a :W (p) = [sin (wz pAt)  sin ((o, pAt)] lap
(10.34)
and where P is the number of weights . The weight functions are generally modified with a "window " to reduce spurious data in the filter . The larger the value of P the sharper is the cutoff frequency of the filter and the leakage in the filtered data is correspondingly less. This is illustrated in Fig. 10.3 in which a band pass filter has been constructed using P=30 and 50. The filter with P=50 appears to be sharper, but both filters deviate from the ideal band pass filter. It should also be kept in mind that this procedure loses original recorded data at its both ends through numerical integration (convolution). The phase information in the data may be restored with the knowledge of the number of weights and the time increment. For example, data lost in the beginning is equivalent to P(At)/2. Shifting data by this amount will restore the original phase. It is generally customary to filter all channels in a test run using the same filter parameters . This allows the phase relationship to be maintained among them.
422
Chapter 10 Data Analysis Techniques
In the frequency domain analysis, the time domain data are transformed into frequency domain using a Fast Fourier Transformation (FFT) routine. Then, the energy between the prescribed frequency limits (depending on high, low or band pass filter) is removed. The remaining modified frequency domain representation of the data is transformed back into the time domain by an inverse FFT routine. The entire process can be accomplished by one FFT routine with a positive (plus) and a negative (minus ) exponent. There are many FFT routines commercially available [e.g., Brigham (1974)].
 design digitol  30 digital  50
0.5
1.5
2
2.5
3
3.5
4
4.5
Wave Pedod. See
FIGURE 10.3 IDEAL AND DIGITAL BAND PASS FILTER 10.6 RESPONSE ANALYSIS In analyzing the responses of a model in waves, it is customary to obtain the transfer function between the response and the wave. The transfer function relates the response amplitude to the wave frequency and amplitude. In regular waves, it is straightforward to compute the transfer function. In random waves, there are several methods to compute the transfer function.
Section 10.6 Response Analysis
423
10.6.1 Frequency Domain Analysis The frequencydomain analysis is suitable for linear systems . They are also carried out for a weak nonlinear system. Sometimes , the responses are separated into linear and nonlinear signals and the transformation is carried out separately and then results added to obtain the total effect. An example (Fig. 10.4) is given here for such an analysis for a tensionleg platform model subjected to a steady current and random waves [Botelho, et al (1984)]. In this case, the total displacement of the TLP is composed of four parts : (1) steady offset due to current, (2) primary response due to the wave frequency, (3) steady drift offset, and (4) slowly varying response at the low frequency. A flow chart to analyze the extreme response due to these four components is shown in Fig. 10.5.
Start of Steady State Towing Response Due to Tow Only INo Waves)
FIGURE 10.4 COMPONENTS OF HYPOTHETICAL RESPONSE OF A TLP MODEL TO RANDOM WAVES AND CURRENT [Botelho, et al. (1984)]
424 Chapter 10 Data Analysis Techniques
Potential Drift Force RAO
T
L. ^
Regular Wave
V ' scous Drift Force RAO Current f
f
Steady Drift Response
V Offset Due to Current Only rms of Primary Response
Rayleigh Distribution
Expected Maximum Primary Response
Combine
Drift Force DAF ectrum
N Sp
f
L^f
Integral of Product
rms of Slowly Varying Drift Response Rayleigh Distribution Expected Maximum Slowly Varying Drift Response
Total Expected Extreme Response
FIGURE 10.5 FLOW CHART FOR A FREQUENCY DOMAIN PROCEDURE [Botelho, et al. (1984)] The open boxes in the frequency domain represent appropriate model tests to derive the required RAOs. It is also possible to obtain this information through computer programs or a combination of the two methods . The wave spectrum may similarly be a measured
Section 10.6 Response Analysis
425
wave spectrum or a theoretical model . The subsequent analysis is carried out mainly in the frequency domain. The first order and secondorder responses are treated individually and then combined to obtain the total extreme response . An appropriate distribution function should be selected, e.g. Rayleigh . Linear combination of extremes is simple, but may be conservative. The required input to this procedure are the wave spectrum, current profile and transfer functions for the surge displacement at the firstorder and secondorder respectively. The response due to current is obtained separately , disregarding interaction. The transfer functions are obtained from the tests in regular waves . In this case, the response amplitude at the wave frequency is normalized by the corresponding wave amplitude . This provides the firstorder Response Amplitude Operator: XaM ((0) = Xa (CO) (10.35)
where Xa is the firstorder response and a is the regular wave amplitude. The RAO is corrected to account for the Doppler frequency shift due to the current. The correction consists of shift in the RAO from the encounter frequency (frequency in the presence of current) to the wave board frequency (i.e., at zero current). The secondorder RAO is obtained from the regular wave steady drift force measured in the mooring system and the dynamic ampl ification factor (DAF). The steady drift force is calculated from the test data as the mean over a complete number of cycles . The steady drift force is normalized by the square of the wave amplitude and divided by the spring constant to give
X(z)(W)= F(W) Ka2
(10.36)
where F = steady drift force in regular waves and K = spring constant of the system. Botelho (1984) obtained the secondorder steady force separately from current and waves as shown in Fig. 10.5. The current load is simulated by towing the model in tank before and during waves. Note that while the two steady components are obtained
426
Chapter 10 Data Analysis Techniques
separately in one run , any interaction effect is included in the second component because of the method of testing used here. The spectrum of the primary response is computed from the product of the wave spectrum and the square of the firstorder RAO. The root mean square (rms) of the primary response is the square root of the area under the computed response spectrum. a(1) =
[fo
[ 1fa(1) (w)12 S(w)dw]
112
(10.37)
where a(1) is the rms of the primary response and S(w) is the wave spectrum. The steady drift force in random waves is computed from
FS = 2 F(w)S(w)d(o
(10.38)
The spectrum of the slowlyvarying displacement is computed by first computing the secondorder displacement RAO from
Xa(2)((O ) = R 2)((o)(DAF ((o))
(10.39)
where X(2) = F/K is the steady secondorder displacement. The slowlyvarying displacement spectrum is computed numerically from S,(w)=8f [Xa (2)( + S(o)S(w+µ)dµ (10.40)
Then the rms of the slowlyvarying drift, a(2) is given by
1/2
a(2) =[ f S,(w)dw] (10.41)
Section 10.6 Response Analysis 427
10.6.2 Linear System One of the goals of most model testing is to develop the Response Amplitude Operator. The RAO is the magnitude of the linear transfer function between the forcing function and the response function. A computer program determines this RAO given two measurements recorded as a function of time. A simple calculation of the RAO from the spectral density is given below. A more detailed theoretical derivation is discussed in the following section. 4
2
0.5
1.0
1 .5
2.0
FREQUENCY, Hz
FIGURE 10.6 RAO COMPUTED FROM FORCE AND WAVE SPECTRUM
428 Chapter 10 Data Analysis Techniques
The time domain records of the measured wave and structure responses are first converted to the frequency domain . Consider that S,,(w) [i.e .. S(w) rewritten for clarity here] is the autospectral energy density spectrum of the wave and Syy(w) is the corresponding energy density spectrum of a measured response . The transfer function is then computed from
LSxx( )1
H(w) _ Syy(w) z
(10.42)
w
An example of this method is illustrated in Fig. 10.6. The measured response and wave spectra are given by the top two curves from which the RAO (bottom curve) has been derived. This method may produce uncorrelated signal noise especially at the tail ends of the spectrum (removed in Fig . 10.6). Additionally, the information on the phase relationship between the response amplitude and wave amplitude is lost through this method. In order to avoid these problems, a cross spectral density analysis method may be employed . To apply this method , the cross spectral density function Sxy(w) between the wave and the response under consideration is computed. Then, 0w)+iQ( (O)=
(10.43) S"(CO)
where C and Q are the coincident and quadrature spectra as functions of frequency, w, and i is the imaginary quantity . The transfer function is computed as the amplitude H(w) = [C2 ((o) + Q2 ((o )] 2
(10.44)
and the corresponding phase angle is obtained from = tan'
c(w)
w) J l ^w
(10 . 45)
A coherence function that states the degree of correlation between the input and output signal may be calculated from the spectral estimate . The coherence function is obtained from
Section 10.6 Response Analysis
2(w) Sxy(w)
429
(10.46)
[Sxx(w)S,((O)
The correlation between the input and output signals is good as the value of 7 approaches unity, while a poor correlation is indicated as 7 approaches zero. 10.6.3 Theory of Cross Spectral Analysis As shown here, the transfer function under the cross spectral theory is a complex quantity and retains the amplitude and phase relationship between the forcing function and the response function. The following presentation documents the theoretical background, possible calculation errors and one of the example problems used to verify the computation method. Consider the equation of motion in time for any linear dynamic single degree of freedom system (SDOFS) my(t) + C$(t) + Ky (t) = x(t)
(10.47)
where y = response function and x = forcing function. The forcing function x(t) can be any deterministic or random function. Equation 10.47 can also be expressed by the following convolution (or Duhamel) integral [Boas (1966)]
y(t)=f x( t)h(t'0ddt
(10.48)
where the function "h" is known as the "weighting function" for the system. By taking Fourier transform of both sides of Eq. 10.48 and making an appropriate substitution, the following frequency domain relation results: Y(w) = H(w)X(w) (10.49)
430
Chapter 10 Data Analysis Techniques
where H((o) = Fourier transform of h(t), Y(w) = Fourier transform of y(t), X(w) = Fourier transform of x(t). Keep in mind that "H", "Y", and "X" are all complex quantities. Equation 10.49 is the frequency domain SDOFS equation of motion, and it directly results from applying the convolution theorem to Equation 10.48 [Brigham (1974), pg. 58]. The quantity H(w) is known as the transfer function for a linear system. It may be expressed in terms of the system properties . By assuming a complex forcing function of unit strength and solving the equation of motion for H (w) [Clough and Penzien (1975)], the system property description of H(w) (i.e. the exact transfer function) becomes: 1 (K(o2m)iwC H(w)= _ w2m+icoC+K ( K (02m)2 +(OL
=A(K wz m)iA(coC)
(10.50)
where A = [(Kw2m)2 +(WC)2]1. The magnitude of the exact transfer function (i.e. the RAO) is:
RAO =
IH(wA=
A2(K(0
2m)2
+A2(wC)2
(10.51)
Thus if the system properties (i.e. "C", "K", & "M") are known one can find the RAO from Eq. 10.51. If the system properties are not known , but the forcing time function and the response time function can be measured , the RAO may be calculated from the following spectral relations [Bendat and Piersol ( 1980)]:
5,'(f)=IH(f)12s.(.f)
(10.52)
s(j) =H(f)S (f)
(10.53)
from which equations 10.42 and 10.43 are derived. Note that w = 271f here. The forcing autospectrum is written as
Section 10. 6 Response Analysis 431
S,(f)=lim TE[X(f,T)X*(f,T)]
(10.54)
while the response autospectrum becomes Syy (f)= im T E[Y (f ,T)Y*(f ,T)] T
(10.55)
The crossspectrum between force and response function is given by Sxy( f)= lirn
TE[X(f,T)Y *(f,T)]
(10.56)
The variables "X" and "Y" are defined in Equation 10.49. The notation E [ ] denotes the average value of the quantity inside the brackets. The superscript asterisk implies the complex conjugate. By substituting Eqs. 10.54 and 10.56 into Eq. 10.53, Eq. 10.49 will result. The substitution of Eqs. 10.54 and 10.55 into Eq. 10.52 yields: IY(f )I2 =IH(f)121X(f )12 (10.57) Eq. 10.57 shows that only the moduli of the Fourier transforms are used in Eq. 10.52, hence all phase information is lost, whereas Eq. 10.53 retains the phase information. The ideal although unrealistic situation for applying Eqs. 10.52 and 10.53 is to have time records of infinite length from which to take the Fourier transforms. In reality, the time record length TR must be finite, and for this reason, all of the Fourier transforms are a function of TR as well as f [Bendat and Piersol (1980)]. The "real life" situation also dictates that the Fourier transforms shown in Eqs. 10.54  10.56 be the average transform which is obtained by averaging the transforms of many subrecords. These subrecords are obtained by dividing one long time record into a finite number of sections (or subrecords). If the long time record depicts a stationary and ergodic process, increased averaging will insure that the measured Fourier transform will approach the true transform (i.e. increased averaging means less random error, thus less Fourier Transform dependence on TR). The variables Sxx and Syy are the (average) autospectrums obtained from the average Fourier transforms, and "Sxy" is the (average) crossspectrum obtained from the average Fourier transforms.
432
Chapter 10 Data Analysis Techniques
The spectrums shown in Eqs. 10.52 and 10.53 are referred to as "double sided spectrums". This terminology arises from the fact that a Fourier transform of any time function (or record) will result in an even frequency function i.e., the frequency function is symmetric about zero [Brigham ( 1974), pg.132]. For many applications, the area under the frequency function is very important. By multiplying the positive frequency portion of the frequency function by two and zeroing out the negative frequency portion, the area is preserved and one is left with a "single sided spectrum". This manipulation is acceptable since the primary quantities of interest are the area and the frequency distribution, neither of which will be affected. The loss of the negative frequencies is inconsequential since they have no physical meaning . In equation form, the single sided spectrum (G) and the double sided spectrum (S) are related as follows: (10.58)
G = 2S, f >0
Most FFT programs calculate only the positive frequency portion of the Fourier transform. They do not calculate the negative portion. The resulting spectra are single sided spectra i.e., twice the positive frequency portion of a double sided spectrum. 10.6.4 Error Analysis Since discrete frequency increments are used for the Fourier transform instead of the ideal continuous case, negative bias errors will result. This means that all of the measured density spectra will be slightly smaller than the actual spectrum. This problem will always occur for "density" functions, such as those used here [Bendat and Piersol (1980), pg. 41]. The bias error normalized with respect to the quantity being measured associated with any density spectrum is as follows: 2
eb
3 (BW)
*100
(10.59)
where eb = % maximum normalized bias error , Mf = frequency increment, BW = spectrum halfpower bandwidth . The quantity "Fb" is the "maximum" normalized bias error. This maximum is measured by locating the largest peak in the spectrum and calculating the bias error associated with that peak (i.e. by dividing Mf by the bandwidth). Bias errors less than 2% are considered negligible.
Section 10. 6 Response Analysis
433
The inability to take infinite time records introduces random errors. The normalized random error formulas are shown in Table 10.1. The coherence function ('y) indicates the amount of nonlinearities, noise and spectral bias errors present in the measurements. A completely noisefree linear system (containing no bias errors in its spectral measurements) will have a coherence of one for all frequencies . A system whose output is completely unrelated to its input will have a coherence of zero for all frequencies. A coherence of 0.6 or above is considered acceptable . It can also be shown that the coherence represents the ratio of the crossspectral RAO to the autospectral RAO; thus for any system exhibiting a coherence less than one, the autospectral RAO will always be larger than the cross spectral RAO. TABLE 10.1 RANDOM ERRORS IN SINGLE INPUT/OUTPUT PROBLEMS QUANTITY
G,,(f), G YY (f)
GzY( f ^
Y (f)
RANDOM ERROR
QUANTITY DESCRIPTION
autospectrum ne
1 Y,y(f) ne
crossspectrum modulus
1  Y2. (f)
coherence function
y, (.f) 2ne
IHc(f)I
1_YZ^r ( f )^s
crossspectral RAO
Y,(f) 2ne IHa(f)I
1 Y2".(f)]z Y2^ (.f) 2ne
= the number of averages (i.e., subrecords or sections)
autospectral RAO
434
Chapter 10 Data Analysis Techniques
Assuming that the coherence will be some value less than one, the table shows that the autospectral RAO will always possess more random error than the crossspectral RAO. The RAO random error formulas also show that as long as the coherence is nonzero and the bias errors are not excessive, the RAO may be estimated to any degree of accuracy one desires simply by increasing the number of averages. Assuming that the time record being analyzed is of a given duration, increasing the number of averages means decreasing the subrecord length. Whenever the subrecord length is decreased, the frequency resolution becomes coarser (Of equals the reciprocal of the subrecord length ) and this implies an increase in the bias error. It is evident that there are conflicting requirements between bias error and random error considerations when working with a time limited set of data. If it is impossible to increase the duration of the run, one must find the appropriate number of averages that will give a reasonable random error and the appropriate subrecord length to give an acceptable bias error. If the duration of the run is not time limited, the user must first determine the proper subrecord length yielding an acceptable bias error and then multiply that subrecord length by the required number of averages (to reduce the random error) to obtain the necessary run duration. As previously mentioned, the subrecord length equals the reciprocal of the frequency resolution (Of). To determine the necessary subrecord length, one must estimate the halfpower bandwidth of the data being analyzed and then solve the bias error formula for M. 10.6.5 Example Problem If the forcing function is known as a function of time, Equation 10.47 may be numerically solved in time by using a piecewise linear acceleration scheme [Clough and Penzien (1975)]. In the example, it is assumed that a measured irregular wave time trace is the forcing function. The linear acceleration scheme was used to solve for the response, then the response time trace was created. The values used for the example are: K = 0.80 lb/ft, M = 0.075 slugs, C = 0.0583 slugs/sec, fN = .52 Hz The exact transfer function is as follows:
Section 10.6 Response Analysis
435
H((O) = A[0.80 w2 (0.075)]  Ai [w(0.0583)] (10.60)
A = 1 [0.80 w2 (.075)]2 +[w(.0583)]2
(10.61)
The wave was measured in the wave tank using a 5 hertz lowpass Butterworth filter. A spectrum of the measured wave form (Fig. 10.7) shows the peak at approximately 0.5 Hz. One will note that there is relatively little energy between 2 and 5 HZ. Since the Butterworth filter removes frequencies above 5 Hz, it is safe to say that absolutely no energy exists in the recorded wave data above 7 Hz. Any energy above 7 Hz that is present in the spectrum can only be due to numerical leakage [Brigham (1974), pg.140]. Figures 10.8 and 10.9 show the exact RAO and phase functions for this system. These exact functions are plotted as solid lines on the corresponding function estimate plots. Figure 10.10 present the response function energy density spectrum generated by Eq. 10.61.
W
J
a
0.2
0.4
0.6 0.8
1.0
1.2
WAVE FREOUENCY, Hz
FIGURE 10.7 MEASURED WAVE SPECTRUM For the case of Hanning smoothing , Figure 10.8 shows that the spectral analysis estimates of the RAO are very close to the true RAO function. Figure 10. 9 shows an
436
Chapter 10 Data Analysis Techniques
excellent estimation of the phase function below 5.5 Hz . Figure 10.11 shows that the coherence function has a value very close to unity at frequencies below 5.5 Hz. Above 5.5 Hz the coherence falls far below unity . The coherence function for any system can fall below the value of one due to any of (or combination of) the following reasons: 1. Extraneous noise is present in the measurements. 2. Resolution bias errors are present in the spectral estimates. 3. The system relating y(t) to x(t) is not linear. 4. The output y(t) is due to other inputs besides x(t).
 EXACT TRANSFER FUNCTION  AUTOSPECTRAL  CROSS SPECTRAL +
O
0.0
1.0
2.0
3.0 4.0
5.0
6.0
WAVE FREQUENCY. Hz
FIGURE 10.8 SYSTEM RAO Since the output data being analyzed is contrived data, causes 1,3, and 4 will not pose a problem. This leaves cause 2, resolution bias errors, as the only possible reason that could cause the coherence to drop. Large bias errors will occur in spectral estimates at frequencies where the frequency resolution is not fine enough to pick up all of the peaks and troughs of the spectrum . This can typically occur at frequencies where the spectrum is extremely peaked , or jagged. Although the magnitude of the calculated spectra is very small above 5.5 Hz, an expanded scale view of the spectra reveals a very jagged spectra above 5.5 Hz, hence large bias errors are very likely to be present.
Section 10.6 Response Analysis
437
S
 EXACT  CROSSSPECTRAL
2
4
6
8
It
12
WAVE FREOUENCY. Hz
FIGURE 10.9 SYSTEM PHASE ANGLE Estimates of the frequency response function gain IH(f)I and phase (e) are also analyzed in order to determine the existence of random and bias errors utilizing the following guidelines:  If 'y,,y (f) falls broadly over a frequency range where IH(f)I is relatively small, this might indicate measurement noise in output and/or contributions of other uncorrelated inputs.  If If y (f) falls broadly over a frequency range where IH(f)I is not near a minimum value and Gxx(f) is relatively small, then measurement noise at the input should be suspected.
438
Chapter 10 Data Analysis Techniques
 If IH(f)1 peaks sharply at system resonance frequencies and 7xy does not, then system nonlinearities might be suspected, as well as resolution bias errors in the spectral estimates. To distinguish between resolution problems and system nonlinearities, the spectral estimates may be repeated utilizing a different sampling window. 3
9
0.2
0.3
0.4
0.5 0.6
6.7
0.8
WAVE FREQUENCY. Hz
FIGURE 10.10 RESPONSE ENERGY DENSITY FUNCTION The above example discusses linear responses and methods of computing them for a given wave input. An example of motion response was shown but the method is similar for other linear responses. If a higher order response is expected from a wave input then special care should be taken to account for these higher order responses. The case of secondorder response is discussed below. 10.6.6 Nonlinear System When the higher order responses are expected, such as the low frequency response of a moored floating system, then the secondorder spectrum of the wave profile is also matched in the random wave generation. The secondorder spectrum is obtained as
S(2) (o)) =8J Sxx(ll)S.(o µ)dµ
(10.71)
Section 10.7 Analysis of Wave Force Coefficients
439
One of the significant problems associated with the determination of the second order transfer function is the length of the record required to obtain a stable estimate of the secondorder transfer function . Tests of this nature are typically half an hour long or longer [Pinkster and Wichers ( 1987)]. For statistical analysis the time history should be considerably longer (> 1 hr). Assuming H2((o) as the quadratic transfer function, the total response spectrum is computed from Syy(w)=I H1(w)I2S.(w)+85 I H2(µ)I2 IH2(WµA2S.(µ)Sxs(wµ)dli
( 10.72)
N
I
2
4
6
8
E
12
WAVE FREQUENCY, HZ
FIGURE 10.11 COHERENCE FUNCTION 10.7 ANALYSIS OF WAVE FORCE COEFFICIENTS Wave forces on small tubular members of a jacket type structure (as well as other cylindrical structural members) are determined experimentally . The method . of testing and some of the results from these tests have been given in Chapter 7.
440
Chapter 10 Data Analysis Techniques
Typically, the wave profile at the test cylinder and the wave force on a small section of the cylinder are recorded as functions of time. The velocity and acceleration time histories are computed from the wave profile using a particular wave theory (e.g., linear theory or stream function theory ). Sometimes, the particle velocities at the test section are directly measured in addition to the wave profile. The acceleration profile is obtained by numerical differentiation of the velocity profile. An example of these measurements is shown in Fig. 10.12. The wave represents a rather long period wave in shallow water. The horizontal and vertical components of the particle velocity are measured at a submerged point at the same location as the wave profile measurement. Note that the horizontal velocity profile is in phase and the vertical velocity profile is out of phase with the wave profile. The profiles contain multiple frequencies. The differentiation and integration of a single frequency and a double frequency profile are shown in Figs. 10.13 and 10.14. The latter is obtained by a Fourier series technique. VERTICRL VELOCITY (FT/SEC) of
HORIZONTAL
VELOCITY (FT/SEC)
of
0J
0 WAVE PROBE (INCHES OF WATER)
0 iD
1
2
3
TIME  SECONDS
FIGURE 10.12 MEASURED WAVE PROFILE AND WATER PARTICLE VELOCITIES IN TANK
Section 10.7 Analysis of Wave Force Coefficients
441
The purpose of measuring forces on structural members in waves is to compute the hydrodynamic coefficients, CM and CD, for different wave conditions. The equation for the forces of the type of Eq. 7.1 is applied in this analysis. Notice that for a vertical cylinder, the maximum inertia and drag forces are 90° out of phase of each other. However, calculating CM when drag is zero and CD when inertia is zero is generally not adequate . It is more customary to compute the average values of these coefficients over a wave cycle. Thus, it is assumed that the coefficients CM and CD are constant throughout a given wave cycle . There are two acceptable methods used in the computation of hydrodynamic coefficients. 10.7.1 Fourier Averaging Method The Fourier averaging analysis is applied to cases where the wave kinematics are sinusoidal. In cases where the forces are obtained through a simple harmonic motion (whether of the fluid or the cylinder) this method is applicable . When wave profiles are applied on a fixed cylinder and linear theory is used to derive the wave kinematics, the Fourier averaging technique may also be used. DIFFERENTIRTION OF CHANNEL I N
V ii 'J 'J 'J V J V INTEGRATION OF CHANNEL 1 m1
mJ
WAVE PROBE  INCHES OF WATER
8
12
16
20
TIME  SECONDS
FIGURE 10.13 DIFFERENTIATION AND INTEGRATION OF A WAVE PROFILE
442
Chapter 10 Data Analysis Techniques
In this method, the force profile is fitted to a Fourier series. The first two terms of the series are fitted to the inertia and drag components of the Morison equation. The coefficients of these terms provide average values of CM and CD. The higher order terms in the Fourier series are grouped together to form the error term or the remainder term, AR. The necessity for the term AR is associated with the fact that the point values of CM and CD in a wave cycle deviate from their average values over the cycle. In this respect, the method of analysis applied here is similar to the one adopted by Keulegan and Carpenter (1958) in analyzing data from a cylinder test. On the assumption that the kinematics are represented by harmonic function, the force time history on a unit length of a cylinder is represented by the Morison equation as follows: f (0) =CMA,uo sin O +CDADU2'jC0S0lcosO+AR(0) (10.73)
INTEGRATION
WWWWWWWWW
DIFFERENTIATION
1VAVAVAVVVVVAVN1AVAVAVAVA V A V A V A V A ORIGINAL DATA
8
12
16
20
TIME  SECONDS
FIGURE 10.14 DIFFERENTIATION AND INTEGRATION BY FOURIER TECHNIQUE
Section 10. 7 Analysis of Wave Force Coefficients
443
where 0=ox and the kinematics no and 60 are the particle velocity and acceleration measured at the center of the instrumented section . The coefficients CM and CD are evaluated over one wave cycle from
i f (0)sinOde
C. = ^u
,
(10.74)
o
and 3 2x
CD
= 2 Jo f(0) cosede 8A°ua
(10.75)
The remainder term is the difference between the measured force and the predicted force based on CM and CD and should be expected to be small. Also, by the nature of the computation , AR should contain higher harmonics of the force. An example of the comparison between the two forces and the resulting error is shown in Fig. 10. 15. The computation has been carried out on the measured forces on a 0.3m (1 ft) submerged section of a vertical cylinder . The remainder term is principally second harmonic, which is the next higher term for a Fourier series for f(0). 10.7.2 Least Square Technique The Fourier series method is not applicable to a case where the kinematics are nonlinear (i.e., nonsinusoidal). This will occur when the nonlinear wave profiles are involved (Fig. 10. 12). In this case , a least square method is applicable . This method minimizes the difference between the measured and predicted forces over one wave cycle in a least square sense . Thus, using AR as the difference between the two traces having N samples over a wave cycle , we write (AR)2 =^^fn CMT„I C°Tn°]2 n= 11
(10.76)
where TnI and TnD are quantities in the inertia and drag parts of the Morison equation (minus the coefficient) at every time increment in a cycle (n = 1, ...N). Note that average values of CM and CD over one wave cycle which minimize the left hand side of the equation are sought. Also, if At is the sampling rate , then T = NAt where T is the
444 Chapter 10 Data Analysis Techniques
AR is minimum for the chosen values of CM and CD, we
wave period. Since
I
MEASURED COMPUTED
.0
/ / /
Al \ 1) .0
/ /
1 1
41
\
l 1
/ I
) 1
/
U 0 .0(
11
I
1 (
I 1
\
^
I
CO
\
/
1
C .
1
/
 MEASURED COMPUTED
I`
.04 0
0.5
1.0
1.5
2.0
2.5
TIME (SECONDS)
FIGURE 10.15 MEASURED FORCE VERSUS COMPUTED FORCE ; RESIDUAL UNACCOUNTED FORCE set to zero the derivatives of AR with respect to CM and CD respectively. This provides two equations in the two unknowns CM and CD. Solving for CM and CD from these equations , the following expressions are derived (the summations of the products in these expressions run from 1 to N, i .e. use one cycle of wave data).
Section 10.8 Free Vibration Tests 445
T (10.77)
T^ITnD)2 L.I(T I )2 (TD)z _I(
XiT
D
( E T ) _Xf.T D 2
!
IT
1
T
D
(10.78)
CD X(T„ z)2X(TD)2 X(TITD)2
For a sinusoidal description of the wave kinematics, either of the two methods can be used and should provide similar results. 10.8 FREE VIBRATION TESTS Any test setup in the wave tank, whether for load tests or motion tests, may be treated as a spring mass system. Therefore, valuable information may be obtained from the free vibration of the system. For fixed structures, the vibration frequency determines if problem will be encountered from the dynamic amplification of the system (Section 7.5.7). For floating structures, information regarding the system natural period and damping may be determined from the vibration analysis. The data analysis is similar in both cases and is described below. 10.&1 Low Frequency Hydrodynamic Coefficients The magnitude of damping determines the extent of motions and corresponding mooring loads in a moored floating platform near its natural frequency (refer to Section 9.8). The free oscillation of the platform takes place at the natural frequency. In an experimental setup, this oscillation may be easily measured when the platform is disturbed from its equilibrium position . The platform returns to its equilibrium position and the duration of oscillation depends strictly on the damping of the system. 10.8.1.1 Linear System The lowfrequency hydrodynamic coefficients of the platform in still water can be determined from the recorded extinction curve. The equation of motion is described by a secondorder differential equation having a single degree of freedom:
446
Chapter 10 Data Analysis Techniques
(Mo+MJW+Cz+Kx = O
(10.79)
where x is the surge amplitude and dots represent first and second derivatives, and MO and K are the structure displacement (mass) and linear spring constant of the spring set, respectively. These quantities are measured directly . The quantities Ma and C are the added mass and linear damping coefficients , respectively. They are considered to be functions of the frequency of oscillation , cud. Note that in this case , cud is the damped natural frequency of the system . Values for Ma and C are determined in the following manner. By assuming a solution to Eq. 10 .79 of the form x=est and defining M = MO + Ma, the equation can be rewritten in the form I sZ+Ms+M e"=0
J
(10.80)
and thus s K sly t ( 2M 2M) M
(10.81)
The damping factor, t;, is defined as the ratio of the amount of damping C present in the system to that amount of damping , Cc, which will cause the part of the equation under the radical to go to zero. Therefore,
2M
M
WN
(10.82)
and C
(10.83)
2M  cuN
In case of light damping, the radical is imaginary and Eq. 10.81 can be written as
S,,2 =
[c
1  v ]wN
±4
(10.84)
Section 10. 8 Free Vibration Tests
447
The general solution to Eq. 10.79 is
(10.85)
x=xoexp(Cw,,t) sin[ 1Vw,,,t+e, in which x0 is the magnitude of oscillation at t=0 and a is its phase angle.
The solution of the equation of motion given by Eq . 10.85 represents harmonic oscillation values of subsequent amplitudes of oscillation in which the amplitudes decay exponentially. If two consecutive absolute values are given by IxkI and IxkII , then the logarithmic decrement is defined as
i
l I I
(10.86)
S = In xk I  ln xk which gives
(10.87)
The logarithmic decrement may be quite accurately related to the damping factor simply by
S
=
XC
(10.88)
For small values of ^ the error is small (for example, for 0. 1, the error is about 0.5 percent). The term x0exp(l (oNt) represents the curve that can be drawn through the succeeding peaks of the damped oscillation. Strictly speaking , the curve does not pass exactly through the peaks , but a small difference is usually neglected. If the natural logarithm of these peaks is taken, the quantity CWN represents the slope, m, of the line that can be drawn through the converted values . The frequency of the damped motion, wd, is also obtained from Eq . 10.85 , and thus we obtain two equations and two unknowns: m=Z(O'
(10.89)
448
Chapter 10 Data Analysis Techniques
Wd =WN
(10.90)
1SZ
The terms on the left hand side of Eqs. 10.89 and 10.90 are obtained by fitting exponential curves to the decayed oscillation data (Fig. 10.16). Once the values of (ON and ^ are known from the above equations, the added mass and damping coefficients are computed: Ma=MMO=
K
2 M0
(10.91)
N
and C=2Ml co,
(10.92)
Therefore, knowing the extinction curve for a moored floating structure, the damping of the system may be established by a simple analysis. This is illustrated by an example based on Fig. 10.16.
1.0
0.8 0.8
H n j  mci
0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 0
50
100
150
200
250
300
Time (seconds)
FIGURE 10.16 EXTINCTION TEST OF A TANKER IN SURGE
Section 10.8 Free Vibration Tests
449
The extinction curve represents the free oscillation of a moored tanker. The displacement of the tanker is MO = 372 .3 kg (25 .5 slugs) and the spring constant K = 102 N/m (7.0 lbs/ft). The least square analysis described above gives an added mass coefficient and a damping factor of 0.049 and 0.008 respectively . The natural period between the positive peaks no. 2 and 3 in Fig . 10.16 is measured as 12.3 sec. Then, Ma =17. lkg(1. 17slug )
(
10.93)
and Ca = M = 0
0.046
(10.94)
Also, the amplitudes of peaks 2 and 3 are 0 .235 and 0.22 in (0.77 and 0.72 ft) respectively. Therefore, 8= 2 (In 0 .235  In 0.22) = 0.03 (10.95) so that
0.03 _ 0.01 777 0. 03'
(10.96)
This example shows that two peak values in the extinction curve determine the unknowns .
In general, a least square fit of all data peaks in the extinction curve is
desirable.
10.8.1.2 Nonlinear System When nonlinear damping is present, the equation of motion for the damped free oscillation of a moored floating vessel, e.g., a semisubmersible in surge, is given by Mz +Cx
+b2lxlx + Kx=0
(10.97)
where M = total mass of the vessel in water, and b2 = nonlinear damping coefficient. Since this equation is nonlinear , it is difficult to solve in a closed form . Therefore, the following simplification is made. On the assumption that each half cycle of the decayed oscillation is reasonably sinusoidal , the nonlinear term is linearized by a
450
Chapter 10 Data Analysis Techniques
Fourier series expansion as 8
klx=
(10.98)
WNXkX
3
where WN = frequency of oscillation corresponding to the natural frequency of the system, and xk corresponds to the amplitude of the kth oscillation cycle. Upon substitution of Eq. 10.98 in Eq. 10.97, a linearized equation (with respect to time) is obtained Mx+Cs+ wNxkx+Kx = 0
(10.99)
Writing C' = C + 37 w NXk
(10.100)
Eq. 10.99 becomes the familiar form as in Eq. 10.79 whose solution may be written in the form similar to Eq. 10.85 with C replaced by ^', where C' is the damping factor including the linearized term, _ C 2MwN
( 10 . 101 )
Then, using Eqs. 10.100, 10.101 and 10.88 In xk_1 = 2n xk+1
[c^
2Mw
wN xk 1 =!k I
2
^ b2 1 6
M M 3TN
( 10 . 102 )
where TN is the natural period of oscillation [Chakrabarti and Cotter ( 1990)]. In terms of the traditional damping factor , ^, a more convenient nondimensional form may be written as
I in 1
C+ 4 Xk M
(10.103)
Assuming that the nonlinear damping term may be represented by the Morison equation drag term,
Section 10.8 Free Vibration Tests
451
(10.104)
b2=2pACD
where A = the projected area of the vessel in the direction of flow, Eq. 10.103 reduces to (10.105) 21tx Ink ( )Xk k+ I +3apMD
which is the equation of a straight line with the lefthand side representing the Yaxis and xk the X axis. Thus, knowing the peak values of the oscillation , the points (X,Y) from the measured data may be fitted to a straight line by the least square method (Fig. 9.24). Then, the quantities CD and ^ may be obtained from the slope and intercept of the fitted line. It should be noted that for sufficient accuracy in these estimates, a reasonable number of peaks are required . However, for a highly damped system, the amplitude reduces to a small value rather quickly and the estimates in these cases are rough. Because waves introduce further damping in the system , the resulting traces may be difficult to analyze by the above method. The added mass coefficient is computed from the measured natural period, TN, the spring constant, K, and the displaced mass, MO, using the formula KTz C. 41[2
Mo J l Mo
(10.106)
10.8.2 Mechanical Oscillation Consider the case of a floating structure model which is attached to a mechanical system similar to the ones described in Chapter 4. Consider also that the structure is forced to oscillate sinusoidally in a prescribed direction at a given amplitude and frequency, and the resulting forces are recorded . Assuming linear damping, the equation of motion due to sinusoidal oscillation has the form Mz+Cx+Kx = FO sin ot (10.107)
452
Chapter 10 Data Analysis Techniques
where x = prescribed oscillation , and F0 = exciting force amplitude . On the assumption that the structure oscillation is described by x = x0 sin((Ot  e), where e = phase angle by which the oscillation lags the force , the solution of Eq. 10.107 may be written after eliminating time, t, as (KM(0 2 )core+Cwsine= Fo xo
(10.108)
(KM(o2)sineCwcose=0
(10.109)
which reduces to
(KM(02)2+(Cw)2 []2 xo
(10.110)
Cw=(Kmw2)tan e
(10.111)
and
Equation 10 .110 may be equivalently written as: (wN2/w2 1)2
+(C/Mw)2 =(Fo/M(O&xo)2
(10.112)
Note that x0, and F0 are known from the test . From Eqs . 10.108 and 10. 109, K M(02 = F 0 cose xo
(10.113)
F Cw=sine
(10.114)
and
xo
The added mass coefficient and damping factor may be derived from the above expressions as C =MMo M 0
(10.115)
Section 10.8 Free Vibration Tests
453
CYLINDER EXTENSION  INCHES
TOP Z !ORD CELL u1  LBS
4.
6.
8.
10.
12.
TIME  SECONDS
FIGURE 10.17 FILTERED MEASURED DATA ON FORCED OSCILLATION TEST where M = K Fo cosE z w w z xo
(10.116)
r _C_ Fosinc C 2wxo MK
(10.117)
and
in which MO = displaced mass of the structure. The quantities F0, x0 and E are determined from the test data by a Fourier series analysis. Note that some noise may be present in the data from the vibration of the hydraulic cylinder. These are digitally filtered out by choosing only the first harmonic data through the use of the Fourier series analysis.
454
Chapter 10 Data Analysis Techniques
While similitude does not strictly exist between forced oscillation and response to waves, it is often a common practice to simulate wave tests with forced oscillation. A
a a O z o
m
01 0 CO0
0.8
1.6
2.4
3.2
4.0
4.8
OSCILLRTION PERIOD (SEC)
FIGURE 10.18 TRANSFER FUNCTION FOR FORCE vertical cylinder was tested mechanically in the heave mode. Both the oscillation and force traces were passed through a digital filter in order to maintain the phase and amplitude relationship . A sample trace of measured displacement and load after filtering is shown in Fig. 10. 17. The normalized force and the phase angle are plotted in Figs. 10.18 and 10.19 versus the oscillation frequency. The force is normalized by M0w2x0 (see Eq. 10.112). Once the quantities on the right hand side of Egs.10.116 and 10.117 are computed, the added mass coefficient and damping factor are known. These are plotted versus the oscillation frequency in Figs . 10.20 and 10.21. The area of frequency (0.4 to 0.5 Hz), where the phase angle is significantly different from 0 or 180 degrees, is expected to produce the most reliable results in this method. Near 0 and 180, the stiffness or inertia force predominates and the accuracy in the damping coefficient diminishes.
Section 10. 8 Free Vibration Teats
cbo
0.8 1.6 2.4 3.2 4.0 0.8 OSCILLATION PERIOD (SEC)
455
4.8
FIGURE 10.19 PHASE ANGLE BETWEEN MEASURED OSCILLATION AND FORCE
a)
0.8
1.6
2 .4
3.2
4.0
4.8
OSCILLATION PERIOD (SEC)
FIGURE 10.20 ADDED MASS COEFFICIENT FROM FORCED OSCILLATION TESTS
456
Chapter 10 Data Analysis Techniques
10.83 Random Decrement Technique Since amplitudes and frequencies in a random wave are variable, lowfrequency hydrodynamic coefficients are also expected to be time dependent. However, for engineering calculation purposes, we can assume the hydrodynamic coefficients to be time invariant and take on their average values. The average values will, however, be wave spectrum dependent. Co
U 07 Cr Ucb Z
N O
n O Q
^ O O k Q._...f/j.._.C..........._.._.Q ._O. O 0 0 O O ^ O k O O
t $
oO i
8
0 0.8 1.6 2.4 3.2 4.0 OSCILLRTION PERIOD (SEC)
4.8
FIGURE 10.21 DAMPING FACTOR FROM FORCED OSCILLATION TESTS Since the low frequency response is irregular, the standard technique of free oscillation may not be used to determine hydrodynamic coefficients. In this case, they may be determined by the random decrement technique [Yang, et al. (1985)]. The random decrement technique helps us to obtain curves of free extinction from the surge motion time history in irregular waves. Once the curve of free extinction is obtained, the hydrodynamic coefficients can be determined by the same method outlined in the case of regular waves. The basic concept of the random decrement method is based on the fact that the random motion response of the platform due to irregular waves may be viewed to consist of two parts: 1) the deterministic part (impulse and/or step function) and 2) the random part (assumed to have a stationary average). By averaging enough samples of the
Section 10.9 References 457
same random response, the random part of the response will be averaged out, leaving the deterministic part of the response alone. The deterministic part is the freedecay response from which the added mass and damping may be computed as before. To obtain this decayed oscillation, the oscillation time history is divided into an ensemble of segments of equal lengths. Each segment begins at the same chosen level and half the segments have positive initial slope while the other half have negative slope. These segments are then ensemble averaged, giving a signature starting ' at the chosen level. The initial slope is zero because an equal number of positive and negative slopes have been chosen in the ensemble. The constant level is arbitrary and can be chosen as a fraction of the zeromean rms value of the random time history. What remains on averaging is a decayed oscillation of the signature. The accuracy of the signature depends on the total record length and the number of averages performed. 10.9 REFERENCES 1. Arhan, M. and Ezraty, R., "Statistical Relations Between Successive Wave Heights," Oceanologica Acta, 1978, Vol. 1, No. 2, pp 151158. 2. Barber, N.F., "The Directional Resolving Power of an Array of Wave Detectors," Ocean Wave Spectra. PrenticeHall, Inc ., New Jersey, 1963, pp. 137150. 3. Bath, M., Spectral Analysis in Geophysics, Elsevier Scientific Publishing Co., Amsterdam, Holland, 1974. 4. Bendat, J.S., and Piersol, A.G., Engineering Applications of Correlation and Spectral Analysis, John Wiley and Sons, Inc., New York, New York,1980, pp.4142, 5455, 7881, 264283. 5. Bloomfield, P., Fourier Analysis of Time Series: An Introduction, John Wiley and Sons, New York, New York, 1976. 6. Boas, M.L., Mathematical Methods in the Physical Sciences, John Wiley and Sons, Inc., New York, New York, 1966, pp. 607611.
458
Chapter 10 Data Analysis Techniques
7. Botelho, D.L.R., Finnigan, T.D. and Petrauskas, C., "Model Test Evaluation of a FrequencyDomain Procedure for Extreme Surge Response Prediction of Tension Leg Platforms," Proceedings of Sixteenth Annual Offshore Technology Conference, Houston, Texas, OTC 4658, May 1984, pp. 105112. 8. Brigham, E.O., The Fast Fourier Transform, Prentice  Hall, Inc., New Jersey, 1974, pp.5861, 132146, 148169. 9. Chakrabarti, S.K., and Cotter, D.C., "Damping Coefficient of a Moored Semisubmersible in Waves and Current," Proceedings of the Offshore Mechanics and Arctic Engineering Symposium, Houston, Texas., Vol. 1, Part A, February, 1990, pp. 145152. 10. Chakrabarti, S.K., and Snider, R.H., "Design of Wave Staff Arrays for Directional Wave Energy Distribution," Underwater Technology , Vol. 5, October., 1972. 11. Clough, R.W., and Penzien, J., Dynamics of Structures , McGrawHill, New York, New York, 1975, pp. 8385,118128,482484. 12. Cotter, D.C., and Chakrabarti, S.K., "Effect of Current and Waves on the Damping Coefficient of a Moored Tanker," Proceedings on TwentyFirst Annual Offshore Technology Conference, Houston, Texas., OTC 6138, May 1989, pp. 149159. 13. Dean, R.G., "Methodology for Evaluating Suitability of Wave and Force Data for Determining Drag and Inertia Forces," Proceedings on Behavior of Offshore Structures, Vol. 2, Trondheim, Norway, 1976, pp. 4064. 14. IAHR, "List of Sea State Parameters," Joint Publication by the IAHR Section on Maritime Hydraulics and PIANC, Brussels, Belgium, 1986. 15. Keulegan, G.H., and Carpenter, L.H. "Forces on Cylinders and Plates in an Oscillating Fluid," Journal of the National Bureau of Standards, Vol. 60, No. 5, May, 1958, pp. 423440.
Section 10. 9 References
459
20. LonguetHiggins, M.S., "The Statistical Analysis of a Random Moving Surface, " Philosophical Trans. Royal Society, London, Vol. 249, Ser. A, 1957, pp 321387. 21. LonguetHiggins, M.S., Cartwright, D.E., and Smith, N.D., "Observations of the Directional Spectrum of Sea Waves Using the Motions of a Floating Body," Ocean Wave Spectra, PrenticeHall Inc., New Jersey, 1963, pp. 111132. 22. Mansard, E.P.D. and Funke, E.R., "On the Statistical Variability of Wave Parameters," National Research Council, Canada, Technical Report TRHY015,1986. 23. Mansard, E.P.D. and Funke, E.R., "On the Fitting of Parametric Models to Measured Wave Spectra," Proceedings of the Second International Symposium on Wave Research and Coastal Engineering, University of Hanover, Mass., 1988. 24. Nwogu, 0., "Analysis of Fixed and Floating Structures in Random Multidirectional Waves," Ph.D. Thesis, University of British Columbia, Vancouver, B.C., Canada, 1989. 25. Otnes, R.K., and Enochson, L., Digital Time Series Analysis, John Wiley and Sons, New York, New York, 1972. 26. Otnes, R.K., and Enochson L., Applied Time Series Analysis. Vol. 1 Basic Techniques, John Wiley and Sons, New York, New York, 1978. 27. Pinkster, J.A. and Wichers, J.E.W., "The Statistical Properties of LowFrequency Motions of Nonlinearly Moored Tankers ", Proceedings on the Nineteenth Offshore Technology Conference , Houston, Texas, OTC 5457, 1987, pp.317331. 28. Read, W.W., "Time Series Analysis of Wave Records and the Search of Wave Groups," Ph.D. Thesis, James Cook University of North Queensland, Australia, 1986.
460
Chapter 10 Data Analysis Techniques
29. Remery, G.F.M., "Model Testing for the Design of Offshore Structures," Proceedings of Symposium on Offshore Hydrodynamics , Publication No. 325, N.S.M.B ., Wageningen, 1971. 30. Yang, J.C.S., et al., "Determination of Fluid Damping Using Random Excitation," Journal of Energy Resources Technology, ASME, Vol. 107, June 1985, pp. 220225.
LIST OF SYMBOLS a cylinder radius a wave amplitude ai incident wave amplitude ar reflected wave amplitude A crosssectional or surface area bp, b1 , b2 damping coefficent B beam of model c celerity or wave speed C damping coefficent or Chezy coefficient CD drag coefficient Cf frictional coefficient CL lift coefficient CM inertia coefficient Cr residual coefficient or reflection coefficient at beach CR reflection coefficient at wavemaker Ct total resistance coefficient Cy Cauchy number d water depth d50 mean material grain size D structure diameter D(w,O) directional spreading function E modulus of elasticity Eu Euler number f force per unit length or frequency fD drag force for CD = 1 ff friction factor f1 inertia force for CM = 1 fL lift force F total force FD drag force Fe elastic force FG gravity force FH horizontal component of force F1 inertia force Fr Froude number Fv viscous force or vertical component of force g gravitational acceleration H wave height H0, H1 , H2 hydrostatic heads Hs significant wave height
462 List of Symbols
Iv Iverson number k wave number ka diffraction parameter K spring constant l , t structure dimensional length L wave length M mass of structure MB bending moment N number of frequency components or data points NR grain size Reynolds number NS sediment number P, p fluid pressure Pa, Po atmospheric pressure p (.) probability function r, 0 cylindrical polar coordinates R hydraulic radius Re Reynolds number Rf frictional resistance Rr residual resistance Rs soil resistance force Rt total resistance s specific gravity or wave spreading index S() spectral energy density t time T wave period TR length of time series Tz zerocrossing period u horizontal water particle velocity uF free fall velocity UT turbulent settling velocity u* shear velocity u horizontal water particle acceleration U current velocity UW wind velocity v fluid velocity V volume of structure w normal component of water particle velocity W (model) weight W(•) filter weight function x horizontal (longitudinal) coordinate y vertical coordinate (measured from seafloor or SWL as specified) z lateral coordinate
List of Symbols
0 vertical scale 7 specific weight or specific heat Af frequency increment At time increment water particle displacement in x or y direction Ax or Ay e wave component phase angle or pipe submergence ratio tl wave profile 0 direction of wave or polar coordinate scale factor friction coefficient or dynamic viscosity v kinematic viscosity 4 wavemaker displacement or entrainment function it dimensionless quantity (or = 3.1416) p mass density of water Pa mass density of air Ps mass density of steel a surface tension ay standard deviation of y ti shear stress or scale factor for time velocity potential or angle of friction CO circular wave frequency (= 2ttf) Pe encounter wave frequency in current
Superscripts A normalized quantity nondimensional quantities • first derivative with respect to time •• second derivative with respect to time * quantity in the presence of current
Subscripts e equivalent quantity in related to model o,0 refers to amplitude (e.g., uo, fo) or deep water (e.g., dO, LO) p related to prototype w related to wind
463
LIST OF ACRONYMS CBI Chicago Bridge & Iron Co. (Plainfield) CERC Coastal Engineering Research Center COV Coefficient of Variation DCDT Direct Current Displacement Transducer DHI Danish Hydraulic Institute ( Denmark) DHL Danish Hydraulic Laboratory (Denmark) DTRC David Taylor Research Center
HRS Hydraulic Research Station (U.K.) IMD Institute of Marine Dynamics (Newfoundland) ITTC International Towing Tank Conference KRISO Korean Research Institute of Ship (Korea) LED LightEmitting Diodes
LVDT Linear Variable Differential Transformer MARIN Maritime Research Institute, Netherlands MARINTEK Norwegian Wave Basin Facility, Trondheim MASK Maneuvering and Seakeeping Facilities (DTRC) NEL National Engineering Laboratory (U.K.) NIST National Institute of Standard Testing NRCC National Research Council of Canada (Ottawa) OTEC Ocean Thermal Energy Conversion RVDT Rotary Variable Differential Transformer SPM Single Point Mooring SWL Still Water Level TLP Tension Leg Platform
AUTHOR INDEX
Aage, C., 135 Aas, B., 177, 178, 185 AbelAziz, H.S., 205207, 230 Abramson, H.N., 283, 301 Abuelnaga, A., 205207, 230 Aguilar, J., 188 Allender, J.H., 239, 301 Anastasiou, K., 186 Anderson, C.H., 166, 184 Arhan, M., 457 Barber, N.F., 418, 457 Bath, M., 457 Battjes, J.A., 152, 184 Bazergui, A., 217,230 Beach Erosion Board Bendat, J.S., 430432, 457 Bergman, J., 170, 185 Berkley, W.B., 29, 37 Bhattacharyya, R., 63, 74 Bhattacharyya, S.K., 333, 336337, 353 Biesel, F., 75, 78, 135, 136 Bishop , J.P., 199, 230 Bloomfield, P., 457 Boas, M.L., 429, 457 Borgman, L.E., 140, 149, 184, 185 Bothelho, D.L.R., 423425, 458 Bowers, C.E., 138 Brabrook, M.G., 137 Brady, I., 395, 402 Brevik, I., 177, 178, 185 Brewer, A.J., 136 Bridgeman, P.W., 38 Brigham, E.O., 432, 435, 458, Brogren, E.E., 260, 301, 303, 405 Brown, D., 225226, 231 Buckingham, E., 38 Bullock, G .N., 77, 136
Burns, G.E., 342, 346, 353 Burrows, R., 186 Causal, S.M., 130, 136, 391,402 Carpenter, L.H., 442, 458 Carstens, M.R., 291, 295, 301
Carter, D.J.T., 189
Chakrabarti,S.K., 31, 38, 141, 144, 150152,172,185,223,232,233, 238,240,250,259,260,262264, 266, 277, 301303, 365, 370, 391, 402403, 405, 418419,450,458 Challenor, P.G., 189 Chamberlin, R.S., 342,353 Chang, P.A., 196, 211, 231 Chantrel, J., 371, 403 Cartwright, D.E., 459 Chen, D.T., 186 Chen, MC., 136, 186 Chen, Y., 136 Clauss,G., 118, 136, 170, 185, 268, 302, 34750, 352353 Clifford, W.R.H., 336, 353 Clough, R.W., 430, 434, 458 Cornett, A., 161, 163, 164, 166, 185 Cotter, D.C., 233, 238, 302, 365, 403, 450,458 Dahle, L.A., 403 Datta, I., 115, 138 Davis, M.C., 167, 185 Dawson, T.H., 277, 279, 280, 282, 302 Dean, R.G., 138, 234, 239, 302303, 458 DeBoom, W.C., 378379,403405
Diez, JJ., 140, 188 Dunlop, W.A., 303 Eatock Taylor, R., 277, 303 Eggestad, I., 135, 136 Elgar, S., 152154, 185 Enochson, L., 459 Eryzlu, N.E., 230 Ezraty, R. 457 Faltinsen, O.M., 403 Feldhausen, P.H.1185 Finnigan, T.D., 458 Forristall, G.Z., 163,185 Frederiksen , E., 186 Funke, E.R., 76, 123, 126, 137139, 141,143146, 186 188,410, 412,459
Gabriel, D., 186
466
Author Index
Galvin, C.J., 78, 136 Gilbert, G., 78, 136 Goda, Y., 122, 123, 126,128129, 136, 140, 151, 152, 186 Golby, P., 402 Goldstein, R.J., 190, 231 Goodrich, G.J., 129, 131, 136, 404 Graff, W.J., 333, 335, 353 Gravesen, H., 140, 186 Guza, R.T.. 185 Hanna, S.Y., 391, 402 Hansen, A.G., 38 Hansen. D.W., 257,303 Hardies, C.E., 197, 231 Haszpra, 0., 38 Hedges, T.S., 177178, 180, 182, 186 Herbich, J.B., 138 , 290, 303 Herbich, J.E. Hirayama, T., 189 Hoerner, S.F., 29, 38, 353 Holton, C., 188 Holtze, G.C., 342, 346, 353 Huang, N.E., 177, 186, 189 Hudspeth, R.T., 78, 82, 136, 149, 152,
186,188189 Hung, S.M., 277, 303 Huse, E., 356,384,389,392, 404 Hyun, J.M., 78, 137 Idichandy, V.G., 353 Ionnau, P.A., 77, 138 Isaacson, M., 17, 28, 38, 122, 125, 126, 137, 223, 232, 233, 303 Iwagaki, Y., 139, 187 Jamieson, W.W., 1145,118,119,137 Jarlan, G.E., 115, 117, 137
Jeffreys, E.R., 163, 186 Joglekar, N.R., 353 Johnson, B., 139, 166, 184, 187188 Jones, D.F., 136 Kaplan, P., 392, 393, 404 Karal, K., 244, 303 Katayama, M., 376, 388, 404 Kennedy, J.F., 294, 303 Keulegan, G.H., 442,458 Kimura, A., 139, 152, 187, 353 Kinsman, B., 169, 185, 187
Kirkegaard, J., 186 Kjeldsen, S .P., 146, 166, 187 Kowalski, T., 225226, 231 Kruppa, C., 347350, 352353 Kure , K., 5, 11 Langhaar, H.L., 38 Launch, P.H., 137 LeMehaute, B., 3536, 38, 115, 137 Leonard, J.W., 136 Li, Q., 404 Libby, A.R., 172, 185 Lin, J., 394,404 Littlebury, K.H., 301, 303 Liu, S.V., 380381,404 LonguetHiggins, M.S., 151, 167, 176, 187, 414, 418, 459 Loukakis, T.A., 139, 187 Lovera, F., 294, 303 Mansard, E.P.D., 114115,119,123, 126, 137, 139, 141, 143146, 155157, 159160, 186188, 459 Marol, P., 403 Mason, W.G., 186 Matsumoto , K., 384,404 McGuigan, S., 231 McNamee, B.P., 197, 231 Medina, J.R., 140, 152, 186, 188 Mei, C.C., 176, 188 Miles, M.D., 97, 100101, 137, 161, 163, 164, 166, 185 Miwa, E., 404 Mogridge, G.R., 137 Monkmeyer, P.L., 263, 296, 303 Morison, J.R., 21, 38 Myrhaug, D., 146, 166, 187 Muckle, W., 307, 353 MunroSmith, R., 63, 74 Murphy, G., 38 Murton, G.J., 77, 136 Naftzger, R.A., 263264, 302 Nath, J.H., 136 Nwogu, 0., 401402,404,418419,459 Ochi, M.K., 183 184, 188 Otnes, R.K., 459 Ouellet, Y., 115,138 Pao, R.H.F., 14, 38
Author Index 467
Patel, M.H., 77, 138 Penzien, J., 430, 434, 458 Petrauskas, C., 239, 301, 380381, 404, 458 Piersol, A.G., 430432,457
Pinkster, J.A., 163, 188, 358359, 403, 405, 439, 459 Ploeg, J., 76, 138 139, 188 Qui, Q., 404 Ratcliffe, T.J., 231 Read, W.W., 417,459 Remery, G.F.M., 358 359,404,460 Rice, J., 231 Rice, S.O., 147, 152, 188 Riekert, T., 136
Rowe, S.J., 336, 353 Sabuncu, T., 130, 136, 391, 402 Sakai, M., 353 Salsich, J.O., 168, 188 Salter, S.H., 98, 138 Sand, S.E., 38, 135 , 143, 155, 157, 159161, 188 Sarpkaya, T., 17, 28, 38, 223 , 232, 233, 239, 303 Saucet, J.P., 230 Schiller, R.E., 303 Sedov, L., 39 Sekita, K., 338340, 353 Seymour, R.J., 185 Sharp, B.B., 198, 231 Shin, Y.S., 183 184, 188 Skoglund, V.J., 39 Smith, N.D., 459 Snider, R.H., 185 , 419, 458 Soper, W.G., 12, 39 Sortland , B., 403 Sphaier, S.H., 130, 138, 304 Stevens, L.K., 231 Stewart, R.W., 176, 187 Straub , L.G., 116117, 138 Suquet, F., 78, 135 Suzuki, Y., 122, 123, 126, 128129, 136 Szucs, E., 39 Takekawa, M., 189 Takezawa, S., 167, 189 Tam, W.A., 259, 302
Tan, P.S.G., 403405 Taylor, D.A., 307, 353 Thompson, D.M., 136 Timoshenki, S.P., 222,231 Toki, N., 140, 189 Tuah, H., 149, 189 Tucker, M.J., 147, 148, 189 Tung , C.C., 177, 186, 189 Unoki, K., 404 Ursell, F., 120, 138 Vasquez, J.H., 130, 138, 303 Vledder, V., 152, 184 Wang, S., 82, 86, 138 Watanabe, R.K., 303 Wichers, J.E.W., 405,459 Williams, S., 402 Williams, A.N., 130, 138, 303 Wu, Y., 404 Yalin, M.S., 15, 39 Yang, J.C.S., 456,460 Yeung, R.W., 130, 138, 304 Young, R.A., 395,405
Yu, Y.S., 138 Zarnick, E.E., 167, 185
SUBJECT INDEX Accelerometers, 206208, 244 Added mass, 112, 130 , 237, 446, 448, 451, 452, 455 Air cushioned vehicle, 392394 Aliasing, 192 Amplifier, 191, 230 Analogtodigital, 190 Arctic structure, 260 Articulated tower, 89, 47 , 237, 306, 315323, 361370 Autocorrelation function, 415 Barges, 47, 49, 50, 5657 , 214, 254, 305312 Beaches, 9394, 103104, 112, 118, 126,131 135,141,169 Bending moment, 33, 206, 222, 224, 319321 Bias, 275276, 406407, 432 Boundary Value Problem, 7879, 159 Breaking Wave, 168, 177 Buoy, 227228 Calibration, 56, 59, 6263 , 6773, 201, 209, 215, 219, 223224, 229230, 255, 274 Catenary, 370 Cauchy number, 18, 33, 283 Celerity, 25, 36, 167 Coefficient of variation, 238 Coherence, 428, 433 , 436439 Computer, 5, 6 Critical damping, 446 Crosscorrelation, 418, 428429, 431 Crosstalk, 45, 201 , 219, 225, 229 Current, 28, 29, 75, 103105, 173181, 423424 Current probes, 162, 174, 196206, 229, 234 Cylinder, 27, 2931 , 112, 129, 130, 220, 233241 , 269, 272, 280, 377378
Damping coefficient, 112, 237, 446, 449 Damping, 23, 71, 172, 244, 274275, 356357, 382392, 446, 449450, 452,456 Decay function, 445451, 457 Distorted model, 4, 13, 3536, 345 Drag coefficient, 21, 2831, 199, 202203,232,309,321 Drag force, 14, 1718, 2829, 3133, 110, 198, 200, 227, 232, 240, 295, 322 Diffraction parameter, 233 Drift force, 381382, 425426, 481, Drilling platform, 260, 265, 266, 324, 329332 Dynamic similarity, 6, 15, 18 Dynamometer, 225226 Energy spectrum, 94, 96, 141, 147149, 162, 164, 173, 177178, 182, 413 Euler number, 18 Extinction curve (see Decay function) Filtering, 192, 211, 419422 Firstorder theory, 17, 143, 148, 275 Flexible model, 3335, 277288, 394400 Floating storage model , 370373 Flow straightener, 103, 105 Force transducers (see load cells) Fourier series, 237, 411412, 441443 Fourier transform, 128, 148150, 166 Frequency domain, 423426 Froude, 3, 1822, 2829, 31, 34, 36, 40, 46, 56, 129, 182, 257, 279, 283, 28995, 321 Geometric similarity, 6, 7, 20, 40, 348352 Hanning, 435
Subject Index
Heave motion, 110, 130, 212, 226, 228, 257 Hysteresis, 71, 320 Inertia coefficient, 21, 28, 203, 232 Irregular wave (see Random wave) Iverson modulus, 18 Jacket structure, 56, 218, 234, 305, 333339 JONSWAP, 95, 96, 141, 158, 160, 165, 180, 413, 415, 417 KeuleganCarpenter number, 18, 21, 26, 28, 233, 237, 246, 248 Laplace equation, 78, 82, 305 LED, 212214 Lift coefficient, 232, 236 Load cell, 41 , 4445, 55, 6365, 71, 215224, 228229, 255, 259, 266267, 271273
Logarithmic decrement , 447, 449 LVDT, 191, 194195, 199202, 204209, 215217, 225 Mechanical oscillator, 111114, 227229,249,250,368 Metacenter, 26, 62, 66, 67 Mooring line, 5763, 67 Morison equation, 27, 177, 203, 232, 235240 Natural frequency (see Natural period) Natural period, 20, 26, 56, 66, 214, 220221, 274275, 409 Nonlinear damping, 356, 449451 Nyquist frequency, 150 Onesided spectrum, 432 Oil storage tank, 257, 259, 263264, 282, 342346 Peak frequency, 417 Peakedness parameter, 413 PiersonMoskowitz (PM), 82, 141, 155, 158, 160, 180 Potentiomenter, 194, 227, 229
469
Pressure transducers, 208209, 229230 Probability distribution, 95, 150151, 153, 260, 265268, 289 Production platform, 256266 Random error, 238, 275276, 406, 433
Random wave, 77, 82, 92, 126, 139, 146151, 160, 169, 177178, 365366, 411417 Rayleigh distribution, 9597, 151 Reflection coefficient, 114128, 410 Response Amplitude Operator (RAO), (see Transfer function) Reynolds, 3, 15, 18, 19, 21, 2832, 37, 110, 182, 233, 237, 290299, 321 Riser, 112, 218, 240, 248 Roll, 66, 110, 212, 226, 401 Run length, 151, 152, 154 RVDT, 195 Scale factor, 20, 40, 46, 57 Scaling laws, 12, 18 Scour, 25, 246, 263, 288300 Secondorder theory, 143, 155157, 269, 425426, 438439 Semisubmersibles, 158, 385389 Shear force, 33, 222, 224, 295 Shortcrested sea, 99, 159166 Significant wave height, 82, 97, 128, 151, 154, 180, 314, 413, 416 Single point mooring, 158, 358371 Ship (see tanker) Sloshing (see Standing wave) Soft volume, 26, 324325 Spectral moment, 414 Spreading function, 161166 Stability, 50, 242, 266, 300, 314, 332333 Standard deviation Standing wave, 141, 142, 272, 282283, 408409
470
Subject Index
Steady force, 423424 Stiffness, 46, 244, 283285, 320, 356357 Strain gauge, 71, 112, 191, 193, 198, 204206, 213, 215217, 219, 223, 224, 320 Strouhal number, 18, 21, 28 Submergence, 5052, 339345 Tanker, 20, 29, 47, 48, 63 Tendons, 53, 55, 56, 59, 6773, 158, 269 Tension Leg Platform, 47, 53, 55, 64, 158, 269, 372, 374376, 379380, 423 Transfer function , 141, 149, 156, 158, 160, 161, 165, 167, 169, 172, 177, 252, 355, 416, 422425, 427431, 434436, 355, 402, Ursell number, 18 Velocity meter (see current probes) Wall effect, 128131 Waterproofing, 40, 204, 206, 209, 216, 219, 224, 240 Wave asymmetry, 152 Wave basin (see Wave Tank) Wave direction, 159, 160, 401402, 417419 Wave generation, 139, 162166 Wave group, 151154, 377, 414415 Wave heights, 17, 25 Wave length, 17, 25, 409 Wavemakers, 75, 78, 81, 8791, 94, 96101, 126, 130, 167, 168, 171, 173 Wave period, 17, 25 Wave probes, 121125, 162, 190, 209211, 229230 Wave profile, 82, 85, 88, 89, 148, ` 162, 170, 409 Wave reflection (see Reflection coefficient
Wave tank, 55, 71, 103, 105, 112, 128, 130135, 172, 213 White noise, 139, 141, 149 Whitstone bridge, 193, 209, 217, 222, 224 Wind, 75, 105, 109110, 1\40, 159, 161, 181184,314 Wind tunnel, 110, 300301 Zerocross period, 414
Advanced Series on Ocean Engineering  Vol. 8 OCEAN DISPOSAL OF WASTEWATER by I R Wood (Univ. Canterbury), R G Bell (National Institute of Water and Atmospheric Research, New Zealand) & D L Wilkinson (Univ. of New South Wales) The ocean is the ultimate sink for all liquid waste and has for many years been the recipient of both treated and untreated sewage waste. This book offers a comprehensive study on the subject of ocean disposal of these effluents. The early chapters cover the philosophy of outfall design, properties of sewage from developed towns and an overview of water quality regulations in New Zealand, Great Britain and the U.S. Alternative ways of satisfying these regulations are discussed. The book also provides information required to design outfall pipelines and diffusers. The methods of calculating the initial dilution and the investigations necessary to compute the further dispersion of the effluent are discussed. A brief discussion of the problems of salt water intrusion, of outfall construction and post construction monitoring is presented at the end of the book.
2127 he
ISBN 9810215126