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HANDBOOK OF COASTAL AND OCEAN ENGINEERING

edited by

Young C Kim California State University, Los Angeles, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF COASTAL AND OCEAN ENGINEERING Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any 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., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-929-1 ISBN-10 981-281-929-0

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

YHwa - Hdbk of Coastal & Ocean.pmd

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Preface Although coastal and ocean engineering is a very ancient field with the construction of Port A-ur near the mouth of the Nile in 3,000 BC, significant advances in this field have been made in the last several decades. The rise of interest in this field can be seen from the number of attendees by academics and practitioners in international conferences. The first International Conference on Coastal Engineering was held in Long Beach, California in 1950 with less than 100 people. When the same conference was held in San Diego, California, in 2006, over 1000 delegates attended. In the last several decades, the world has seen significant coastal and ocean engineering projects, one of which is the Delta Project in the Netherlands. This project was designed to shorten and strengthen the total length of coast and dykes washed by the sea by closing off the sea arms in the Delta region. Other noteworthy coastal engineering projects include the Kansai Airport Project in Japan and, in recent years, the construction of mobile barriers at inlets to regulate tides in the Venice Lagoon known as the Venice Project. Interest in coastal and ocean engineering has arisen in recent years as humankind experiences coastal disasters that derive from coastal storm, hurricane and coastal flooding and seismic activities such as tsunamis, and the impacts of climate change which result in sea-level rise. The tsunami activity in Sumatra in December 2004 affected countries throughout the Indian Ocean and resulted in the loss of thousands of lives. Hurricane Katrina in New Orleans also claimed many lives with property damage exceeding $63 billion. Global warming and sea-level rise will affect shoreline retreats, inundate low coastal areas, damage coastal structures, and accelerate beach erosion. The need for better understanding of our coastal and ocean environment has risen considerably in recent years. This handbook contains a comprehensive compilation of topics that are the forefronts of many technical advances in ocean waves, coastal and ocean engineering. It represents the most comprehensive reference available on coastal and ocean engineering to date, and it also provides the most up-to-date technical advances and latest research findings on coastal and ocean engineering. More than 70 internationally recognized authorities in the field of coastal and ocean engineering contributed papers on their areas of expertise to this handbook. These international luminaries are from highly respected universities and renowned research and consulting organizations from all over the world.

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Preface

This handbook provides a comprehensive overview of shallow-water waves, water-level fluctuations, coastal and offshore structures, ports and harbors, coastal sediment processes, environmental problems, sustainable coastal development, coastal hazards, physical modeling, and coastal engineering practice and education. This book is an essential source of reference for professionals and researchers in the areas of coastal engineering, ocean engineering, oceanography, meteorology, and civil engineering, and as a text for graduate students in these fields. This handbook will be of immediate, practical use to coastal, ocean, civil, geotechnical, and structural engineers, and coastal planners and managers as well as marine biologists and oceanographers. It will also be an excellent source book for educational and teaching purposes, and would be a good reference book for any technical library. I would like to express my indebtedness to those who guided me and supported me as a mentor and a colleague throughout my professional life. They are: Professor Robert L. Wiegel, University of California, Berkeley Professor Joe W. Johnson, University of California Berkeley Professor Robert G. Dean, University of Florida Professor Fredric Raichlen, California Institute of Technology Professor Raymond C. Binder, University of Southern California Professor Shoshichiro Nagai, Osaka City University Dr Basil Wilson, Science Engineering Associates Dr Lars Skjelbreia, Science Engineering Associates Dr Bernard LeMehaute, University of Miami Professor Richard Silvester, University of Western Australia Mr Orville T. Magoon, Coastal Zone Foundation Professor Billy L. Edge, Texas A&M University Professor Michael E. McCormick, US Naval Academy Professor Yoshimi Goda, Yokohama National University and ECOH Corporation Professor Philip L.F. Liu, Cornell University Professor Forrest M. Holly, The University of Iowa Dr Etienne Mansard, National Research Council, Canada Professor J. Richard Weggel, Drexel University Mr Ronald M. Noble, Noble Consultants, Inc. I also wish to express my indebtedness to those who nurtured me from my early teen years and changed my course of life. They are: Dr Helen Miller Bailey, East Los Angeles College Mr H. Karl Bouvier, Jet Propulsion Laboratory I extend my gratitude to my wife, Janet, for her constant support, encouragement, patience, and understanding while I was undertaking this task and to my daughter, Susan Calix, for proofreading some of the materials.

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Finally, I wish to express my deep appreciation to Ms Kimberley Chua of World Scientific Publishing Company who gave me invaluable support and encouragement from the inception of this handbook to its realization. Young C. Kim Los Angeles, California January 2008

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Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section 1: Shallow-Water Waves 1. Wave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and T. L. Walton

1

2. Wavemaker Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. T. Hudspeth and R. B. Guenther

25

3. Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. B. Guenther and R. T. Hudspeth

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4. Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Goda

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5. Aeration and Bubbles in the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori, S. Kakuno and D. T. Cox

115

6. Freak Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori

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7. Short-Term Wave Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Kimura

151

Section 2: Water-Level Fluctuations 8. Generation and Prediction of Seiches in Rotterdam Harbor Basins . . . . . M. P. C. de Jong and J. A. Battjes

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9. Seiches and Harbor Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. B. Rabinovich

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10. Finite Difference Model for Practical Simulation of Distant Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. B. Yoon

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Section 3: Coastal Structures 11. Tsunami-Induced Forces on Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Nistor, D. Palermo, Y. Nouri, T. Murty and M. Saatcioglu

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12. Nonconventional Wave Damping Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci

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13. Wave Interaction with Breakwaters Including Perforated Walls . . . . . . . . K.-D. Suh

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14. Prediction of Overtopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. van der Meer, T. Pullen, W. Allsop, T. Bruce, H. Sch¨ uttrumpf and A. Kortenhaus

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15. Wave Run-Up and Wave Overtopping at Armored Rubble Slopes and Mounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Sch¨ uttrumpf, J. van der Meer, A. Kortenhaus, T. Bruce and L. Franco

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16. Wave Overtopping at Vertical and Steep Structures . . . . . . . . . . . . . . . . . . . T. Bruce, J. van der Meer, T. Pullen and W. Allsop

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17. Surf Parameters for the Design of Coastal Structures . . . . . . . . . . . . . . . . . . D. H. Yoo

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18. Development of Caisson Breakwater Design Based on Failure Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Takahashi

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19. Design of Alternative Revetments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Pilarczyk

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20. Remarks on Coastal Stabilization and Alternative Solutions . . . . . . . . . . . K. Pilarczyk

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21. Geotextile Sand Containers for Shore Protection. . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci and J. Recio

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22. Low Crested Breakwaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Lamberti and B. Zanuttigh

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23. Hydrodynamic Behavior of Net Cages in the Open Sea . . . . . . . . . . . . . . . . Y.-C Li

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Section 4: Offshore Structures 24. State of Offshore Structure Development and Design Challenges . . . . . . . S. Chakrabarti

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Section 5: Ports and Harbors 25. Computer Modeling for Harbor Planning and Design . . . . . . . . . . . . . . . . . . J.-J. Lee and X. Xing

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26. Prediction of Squat for Underkeel Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, M. Vantorre, K. Uliczka and P. Debaillon

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Section 6: Coastal Sediment Processes 27. Wave-Induced Resuspension of Fine Sediment . . . . . . . . . . . . . . . . . . . . . . . . . M. Jain and A. J. Metha

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28. Suspended Sand and Bedload Transport on Beaches . . . . . . . . . . . . . . . . . . . N. Kobayashi, A. Payo and B. D. Johnson

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29. Headland-Bay Beaches for Recreation and Shore Protection . . . . . . . . . . . J. R.-C. Hsu, M. M.-J. Yu, F.-C. Lee and R. Silvester

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30. Beach Nourishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and J. D. Rosati

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31. Engineering of Tidal Inlets and Morphologic Consequences . . . . . . . . . . . . N. C. Kraus

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Section 7: Environmental Problems 32. Water and Nutrients Flow in the Enclosed Bays . . . . . . . . . . . . . . . . . . . . . . . Y. Koibuchi and M. Isobe

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Section 8: Sustainable Coastal Development 33. Socioeconomic and Environmental Risk in Coastal and Ocean Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Losada, A. Baquerizo, M. Ortega-S´ anchez, J. M. Santiago and E. S´ anchez-Badorrey 34. Utilization of the Coastal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-H. Hwung

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Section 9: Coastal Hazards 35. Ocean Wave Climates: Trends and Variations Due to Earth’s Changing Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. D. Komar, J. C. Allan and P. Ruggiero

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36. Sea Level Rise: Major Implications to Coastal Engineering and Coastal Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Ewing

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37. Sea Level Rise and Coastal Erosion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. F. Stive, R. Ranasinghe and P. J. Cowell

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38. Coastal Flooding: Analysis and Assessment of Risk . . . . . . . . . . . . . . . . . . . . P. Prinos and P. Galiatsatou

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Section 10: Physical Modeling 39. Physical Modeling of Tsunami Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, H. Yeh and D. T. Cox

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40. Laboratory Simulation of Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. P. D. Mansard and M. D. Miles

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Section 11: Coastal Engineering Practice and Education 41. Perspective on Coastal Engineering Practice and Education . . . . . . . . . . . J. W. Kamphuis

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Contributors Jonathan C. Allan Coastal Field Office Oregon Department of Geology and Mineral Industries Newport, Oregon [email protected] William Allsop Technical Director HR Wallingford Wallingford, UK [email protected] Elena Sanchez Badorrey Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Asuncion Baquerizo Associate Professor CEAMA — Universidad de Granada Granada, Spain Jurjen A. Battjes Emeritus Professor Environmental Fluid Mechanics Section Delft University of Technology Delft, The Netherlands [email protected] Michael J. Briggs Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected]

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Tom Bruce School of Engineering and Electronics University of Edinburgh Edinburgh, UK [email protected] Subrata Chakrabarti Joint Professor, Civil and Mechanical Engineering University of Illinois at Chicago Chicago, Illinois [email protected] Peter J. Cowell Associate Professor School of Geosciences Institute of Marine Science University of Sydney Sydney, Australia Daniel T. Cox Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Robert G. Dean Graduate Research Professor of Coastal Engineering, Emeritus Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]fl.edu Pierre Debaillon Research Hydraulic Engineer Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF) Compiegne, France [email protected] Martijn P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology Presently at Delft Hydraulics Delft, The Netherlands

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Lesley Ewing Senior Coastal Engineer California Coastal Commission San Francisco, California [email protected] Leopoldo Franco Professor of Coastal Engineering Department of Civil Engineering University of Rome 3 Rome, Italy [email protected] Panagiota Galiatsatou Research Associate Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Yoshimi Goda Professor Emeritus Yokohama National University Adviser to ECHO Corporation Tokyo, Japan [email protected] Ronald B. Guenther Professor Emeritus Department of Mathematics Oregon State University Corvallis, Oregon [email protected] John Rong-Chung Hsu Professor Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Honorary Research Fellow School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Robert T. Hudspeth Professor and Director, Emeritus Coastal and Ocean Engineering Program Oregon State University Corvallis, Oregon [email protected] Hwung-Hweng Hwung Professor of Hydraulic and Ocean Engineering Director of Tainan Hydraulics Laboratory Department of Hydraulic and Ocean Engineering National Cheng Kung University Tainan, Taiwan [email protected] Masahiko Isobe Professor and Special Adviser to the President Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Mamta Jain Coastal Engineer Halcrow Inc. Tampa, Florida [email protected] Bradley D. Johnson Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi Shohachi Kakuno Professor and Vice President Department of Civil Engineering Osaka City University Osaka, Japan [email protected] J. William Kamphuis Professor of Civil Engineering, Emeritus Department of Civil Engineering Queen’s University Kingston, Ontario, Canada [email protected]

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Akira Kimura Professor Department of Social Systems Engineering Tottori University Tottori, Japan [email protected] Nobuhisa Kobayashi Professor and Director Center for Applied Coastal Research University of Delaware Newark, Delaware [email protected] Yukio Koibuchi Assistant Professor Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Paul D. Komar Professor of Oceanography College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, Oregon [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics Technical University of Braunschweig Braunschweig, Germany [email protected] Nicholas C. Kraus Senior Scientist Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected] Alberto Lamberti Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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Fang-Chun Lee Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Jiin-Jen Lee Professor of Civil and Environmental Engineering Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California [email protected] Yu-Cheng Li Professor School of Civil Engineering Dalian University of Technology Dalian, China [email protected] Miguel A. Losada Professor Research Group on Environmental Flux Dynamics CEAMA — Universidad de Granada Granada, Spain [email protected] Etienne P. D. Mansard Executive Director Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada [email protected] Ashish J. Mehta Professor of Coastal Engineering Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]fl.edu Michael D. Miles Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada

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Nobuhito Mori Associate Professor Disaster Prevention Research Institute Kyoto University Kyoto, Japan [email protected] Tad S. Murty Adjunct Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Ioan Nistor Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Younes Nouri Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada Miquel Ortega Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Hocine Oumeraci University Professor Leichtweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany [email protected] Dan Palermo Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected]

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Andres Payo Graduate School of Science and Technology University of Kumamoto Kumamoto, Japan Krystian W. Pilarczyk (Former) Manager, Research and Development Hydraulic Engineering Institute Rykswaterstaat Delft, The Netherlands HYDROpil Consultancy Zoetermeer, The Netherlands [email protected] Panayotis Prinos Professor of Hydraulic Engineering Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Tim Pullen Senior Engineer HR Wallingford Wallingford, UK [email protected] Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology Russian Academy of Sciences Moscow, Russia Department of Fisheries and Oceans Institute of Ocean Sciences Sidney, B.C., Canada [email protected]. Roshanka Ranasinghe Associate Professor UNESCO-IHE/Delft University of Technology Delft, The Netherlands [email protected] Juan Recio Leichweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany

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Julie D. Rosati Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Corps of Engineers Mobile, Alabama [email protected] Peter Ruggiero Assistant Professor Department of Geosciences Oregon State University Corvallis, Oregon [email protected] Murat Saatcioglu Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Juan M. Santiago Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Holger Schuttrumpf Professor and Director Institute of Hydraulic Engineering and Water Resources Management RWTH — Aachen University Aaachen, Germany [email protected] Richard Silvester Professor Emeritus School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Marcel J. F. Stive Professor and Director Delft Water Research Centre Department of Hydraulic Engineering Delft University of Technology Delft, The Netherlands [email protected] Kyung-Duck Suh Professor Department of Civil and Environmental Engineering Seoul National University Seoul, Korea [email protected] Shigeo Takahashi Executive Researcher and Director Tsunami Research Center Port and Airport Research Institute Yokosuka, Japan takahashi [email protected] Klemens Uliczka Research Hydraulic Engineer Federal Waterways Engineering and Research Institute (BAW) Hamburg, Germany [email protected] Jentsje van der Meer Principal Van der Meer Consulting Heerenveen, The Netherlands [email protected] Marc Vantorre Professor Division of Maritime Technology Ghent University Ghent, Belgium [email protected]

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Contributors

Todd L. Walton Director Beaches and Shore Resource Center Florida State University Tallahassee, Florida [email protected] Xiuying Xing Graduate Research Assistant Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California Harry Yeh Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Dong Hoon Yoo Professor Department of Civil Engineering Ajou University Suwon, Korea [email protected] Sung Bum Yoon Professor Department of Civil and Environmental Engineering Hanyang University Ansan, Korea [email protected] Melissa Meng-Jiuan Yu Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Barbara Zanuttigh Assistant Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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The Editor Young C. Kim, PhD, is currently a Professor of Civil Engineering, Emeritus at California State University, Los Angeles. Other academic positions held by him include a Visiting Scholar of Coastal Engineering at the University of California, Berkeley (1971); a NATO Senior Fellow in Science at the Delft University of Technology in the Netherlands (1975); and a Visiting Scientist at the Osaka City University for the National Science Foundations’ US–Japan Cooperative Science Program (1976). For more than a decade, he served as Chair of the Department of Civil Engineering and recently he was Associate Dean of the College of Engineering. For his dedicated teaching and outstanding professional activities, he was awarded the university-wide Outstanding Professor Award in 1994. Dr Kim was a consultant to the US Naval Civil Engineering Laboratory in Port Hueneme and became a resident consultant to the Science Engineering Associates where he investigated wave forces on the Howard-Doris platform structure, now being placed in Ninian Field, North Sea. Dr Kim is the past Chair of the Executive Committee of the Waterway, Port, Coastal and Ocean Division of the American Society of Civil Engineering (ASCE). Recently, he served as Chair of the Nominating Committee of the International Association of Hydraulic Engineering and Research (IAHR). Since 1998, he served on the International Board of Directors of the Pacific Congress on Marine Science and Technology (PACON). He currently serves as the President of PACON. Dr Kim has been involved in organizing 10 national and international conferences, has authored three books, and has published 52 technical papers in various engineering journals.

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Chapter 1

Wave Setup Robert G. Dean Department of Civil and Coastal Engineering University of Florida, Gainesville, FL, USA [email protected] Todd L. Walton Beaches and Shores Resource Center Florida State University, Tallahassee, FL, USA [email protected] Wave setup is the increase of water level within the surf zone due to the transfer of wave-related momentum to the water column during wave-breaking. Wave setup has been investigated theoretically and under laboratory and field conditions, and it includes both static and dynamic components. Engineering applications include a significant flooding component due to severe storms and oscillating water levels that can increase hazards to recreational beach goers and can contribute to undesirable oscillations of both constructed and natural systems including harbors and moored ships. This chapter provides a review of the knowledge regarding wave setup and presents preliminary recommendations for design. It will be shown that wave setup is not adequately understood quantitatively for engineering design purposes.

1.1. Introduction Wave setup was brought to the attention of coastal engineers and scientists in the 1960s (i.e., see Ref. 1, p. 245) after the initial theoretic developments of LonguetHiggins2 and Longuet-Higgins and Stewart3,4 along with limited field observations and laboratory studies supported the existence of wave setup, the magnitude of which was observed to be in the order of 10–20% of the incident wave height. It was noted in early field observations that water levels on the beach were higher than those recorded by a tide gauge at the end of a pier suggesting a wave setup physically forced by wind waves and swell. 1

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Fig. 1.1. Definition sketch. Energy and momentum are transferred from winds to waves in the generating area. The waves convey energy and momentum to the surf zone where the waves break. Upon breaking, the energy is dissipated and the momentum is transferred to the water column resulting in longshore and onshore forces exerted on the water column.

Wave setup is the additional water level that is due to the transfer of wave-related momentum to the water column during the wave-breaking process. As waves approach the shoreline, they convey both energy and momentum in the wave direction. Upon breaking, the wave energy is dissipated, as is evident from the turbulence generated; however, momentum is never dissipated but rather is transferred to the water column resulting in a slope of the water surface to balance the onshore component of the flux of momentum (see Fig. 1.1). If waves are irregular, in addition to a steady wave setup, the setup includes a dynamic component that oscillates with the wave group period and there may be a weak resonance within the nearshore amplifying this oscillating component. These have been termed infragravity waves and are more dominant for narrow banded spectra both in frequency and in directional spreading. The oscillatory component is denoted “dynamic wave setup” in this chapter. This chapter discusses the significance of wave setup to coastal engineering design, provides a review of the classical linear wave theory of wave setup, reviews results from laboratory and field studies, summarizes results and recommends preliminary design approaches for the static component. To provide a “look ahead,” we will see that the phenomenon of wave setup is not yet adequately understood for satisfactory engineering calculations and that the effects of profile slope are very

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significant. The interested reader is also referred to an earlier review article on wave setup by Holman.5 1.2. Engineering Significance of Wave Setup Wave setup (both static and dynamic components) is relevant to a number of engineering applications. The contributions of wave setup under extreme storm events can be substantial, adding several feet to the elevated water levels. The interaction of wave setup with vegetation differs from wind surge and thus it is important to differentiate the two components, for example in ascertaining the benefits of wetlands in reducing wave setup. Finally, the oscillating component of wave setup is relevant to beach safety in some locations and to many natural and constructed coastal systems that have the capability to resonate including harbors and moored ships. 1.3. Terminology and Related Considerations Standard terminology defines the water level in the absence of wave effects as “still water level,” whereas wave setup will cause a departure from the still water level and this water level including the effects of the waves is the “mean water level.” As implied, the mean water level is determined as the average of the fluctuating water level over a suitable time frame usually taken as a number of multiples of the short wave period, say the spectral peak. In considering wave setup, often the location of interest is that of the maximum wave setup at the shoreline. This raises the question of whether wave setup is defined at elevations above the maximum rundown, say on the beach face where the water is present over only a portion of the wave period. Since wave setup is defined as the mean water level, over what period should the water surface be averaged on the beach face which is “wetted” over only a portion of the wave period? If the time average is over only the portion of the period that water is present, in the upper limit, the maximum setup will be the maximum runup. For purposes here, wave setup will usually be defined only for conditions where water is present over a full wave period. When calculating wave runup on a structure such as a levee or revetment, the question arises whether it is appropriate to first calculate wave setup and then add the wave runup which is usually empirically based on model results. In the more recent empirical results (e.g., the TAW method, see Ref. 6), the runup is expressed as a proportion of the significant wave height at the base of the steeper slope (e.g., at a revetment or levee). The wave runup determined in the model on which the method was based generally included some wave setup (or setdown) seaward of the toe of the slope and included wave setup landward of the toe of the slope. Thus, in the application of interest, the most appropriate approach is to calculate and include wave setup at the toe of the slope; however, recognizing that the measured landward runup includes setup, no additional setup should be added explicitly landward of the toe of the slope.

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1.4. A Brief Review of Wave Setup Mechanics 1.4.1. Static wave setup for monochromatic waves Longuet-Higgins2 and Longuet-Higgins and Stewart3,4 were the first to formalize the notion of wave momentum flux and its relationship to wave setup. The momentum flux, Sij , is a second-order tensor given by 
1 , Sxx = E n(cos2 θ + 1) − 2 
1 2 (1.1) Syy = E n(sin θ + 1) − , 2 E sin 2θ, 2 where E is the wave energy density, n is the ratio of wave group velocity to wave celerity and θ is the angle between the wave direction and the x-axis. The term Sxy reads “the flux per unit width, in the x-direction, of the y-component of momentum,” etc. The steady-state equations of motion obtained by time averaging over the short wave period are, including the effects of wind stress and bottom friction:   1 ∂Sxx ∂Sxy ∂ (η wind + ηwave ) =− + − τsx + τbx ∂x ρg(h + η) ∂x ∂y Sxy = Syx =

and ∂ (η wind + η wave ) 1 =− ∂y ρg(h + η)





(1.2)

∂Syy ∂Syx + − τsy + τby . ∂y ∂x

In the above, ηwind is the surge component due to the wind stress, ηwave is the wave setup, ρ is the mass density of water, g is the gravitational constant, h is the local water depth, τsx and τbx are the surface and bottom shear stresses, respectively, and similarly for the y-direction. The coordinate direction, x is oriented shoreward and a right-handed coordinate system is considered. The most simple solution is for waves propagating directly shoreward (Sxy = 0) in which the surface and bottom stresses are considered negligible, and all variables are considered uniform in the y-direction. The resulting equation is   1 ∂Sxx ∂η wave =− . (1.3) ∂x ρg(h + η) ∂x To proceed, we need to determine a boundary condition for η wave ,a at the seaward end of the surf zone. Longuet-Higgins7 has shown that in the absence of energy dissipation, the following general relationship for η applies η=C−

1 2 (u − w2 )η=0 , 2g

(1.4)

a For purposes of convenience, hereafter the subscript on η wave will be omitted such that the wave setup is simply denoted as η.

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where u2 and w2 represent the time averages of the square of the first-order horizontal and vertical wave velocities evaluated at the mean water surface, respectively. Equation (1.4) is a type of a Bernoulli equation for unsteady flows which, when evaluated at the break point and considering no wave setup in deep water to evaluate the constant, C = 0, the setup is negative (setdown) and given by ηb = −

Hb2 kb , 8 sinh 2kb hb

(1.5)

where Hb is the breaking wave height and kb is the wave number at breaking. For shallow water conditions and depth limited breaking (Hb = κ(hb + η b )), Eq. (1.5) yields ηb = −

κHb . 16

(1.6)

As an example, for a κ value of 0.78, the wave setdown is approximately 5% of the breaking wave height. With the seaward boundary condition now established, for the case of shallow water wave-breaking and the consideration of depth limited breaking across the surf zone, the wave setup is η=−

3κ2 /8 κHb + (hb − h) . 16 (1 + (3κ2 /8))

(1.7)

It is noted that in the above equation, the bottom shear stress has been taken as zero and that a shoreward directed bottom shear stress on the water column as would occur due to undertow would increase the wave setup. As examples, the ratio of wave setup to breaking height at the still water line (h = 0) and at the location of maximum wave setup (η = − h) for a κ value of 0.78 are
 5κ η(h = 0, κ = 0.78) = = 0.198 (1.8) F0 |κ=0.78 ≡ Hb 16(1 + (3κ2 /8)) κ=0.78 and Fmax |κ=0.78

η(h = − η, κ = 0.78) ≡ = F0 Hb

   3κ2  5κ  1+ = = 0.244 . 8 κ=0.78 16 κ=0.78 (1.9)

It is seen that the wave setup is strongly dependent on the value of the breaking ratio κ which will be shown to decrease with decreasing beach slope. Figure 1.2 presents the ratios, F0 and Fmax versus κ. It is useful to relate κ in an approximate manner to beach slope. Although there is not a one-to-one correspondence, Fig. 1.3 is based on the Dally et al.8 wave-breaking model and provides an approximate correspondence between uniform profile slope and the associated κ value. It is evident that the Dally et al. model provides reasonable κ values for smaller beach slopes (say less than about 0.06), but the κ values are too large for steeper slopes.

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0.4

Fo Fmax

Fo and Fmax

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Kappa Fig. 1.2.

Values of F0 and Fmax versus wave-breaking index, κ (kappa).

2.0

Kappa

1.5

1.0

0.5

0.0 0.00

0.02

0.04

0.06

0.08

0.10

Profile Slope

Fig. 1.3. Relationship between profile slope and κ (kappa) value. Based on Dally et al.8 wavebreaking model.

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1.4.2. Effects of wave nonlinearity Wave nonlinearity depends on the following parameters: H/L0 and h/L0 . The nonlinearity is exhibited in the wave profile by peaked crests and flatter troughs and increases with wave height and shallow water. Somewhat surprisingly, the momentum fluxes in shallow water are less for nonlinear waves than for linear waves of the same height. This is primarily because the momentum fluxes are proportional to wave energy (Eq. (1.1)) and the wave energy is proportional to the root-mean square of the water surface displacement that is less for nonlinear waves with long troughs and peaked wave crests. Figure 1.4 presents the ratio of nonlinear to linear momentum fluxes as determined by Stream Function wave theory.9–11 The reason that the quantities for nonlinear waves are greater in deep water than for linear waves is that the nonlinear calculations extend up to the actual free surface whereas the linear quantities only extend up to the mean free surface. 1.4.3. Role of wave directionality

1.2

b =1

H/ H

b

/H H

b=

b =0

0.6

H/ H

.2 5

0. 50

=0 .7 5

0.8

.0

1.0

H/ H

Ratio of Nonlinear to Linear Momentum Flux

Equation (1.1) demonstrates that for a given wave height, the maximum shoreward flux of onshore momentum occurs for normally incident waves (θ = 0◦ ). Thus as expected for directional waves, the Sxx term is reduced. However, this reduction is relatively small as can be demonstrated by considering a breaking wave direction of 30◦ relative to a beach normal (this represents a reasonably large wave obliquity

0.4

0.2

0.0 10-3.000

2

3

4 5 6

10-2.000

2

3

4 5 6

10-1.000

2

3

4 5 5

100.000

2

3

4 5 6

101.000

h/Lo Fig. 1.4. Ratio of nonlinear to linear wave momentum flux, Sxx , for forty stream function wave combinations.12

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at breaking). The reduction in Sxx for a wave of given height and for shallow water conditions is 16.7%. 1.4.4. Effects of vegetation The effects of vegetation have been shown to result in a reduced setup and, in some cases, may cause a setdown.12 For linear waves, vegetation protruding through the water surface experiences a net drag force (quadratically related to velocity) on the vegetation in the direction of wave propagation and, of course, there must be an equal and opposing force exerted on the water column. This opposing force acting on the water column partially counteracts the force due to momentum transfer and thus reduces the wave setup (similar to an offshore directed wind stress). For linear waves and vegetation which is submerged during the entire wave passage, no net vegetation-related force exists on the water column and thus there is no effect on the wave setup. However, due to the character of nonlinear waves with higher and shorter shoreward velocities under the wave crests, even if the vegetation is fully submerged during the passage of the wave, a net drag force is induced on the vegetation in the wave propagation direction again resulting in a reduction in the wave setup and, for some cases, a wave setdown. 1.4.5. Dynamic wave setup It is noted that theoretical formulations of the dynamic wave setup must include the time dependent terms in the counterparts of Eq. (1.2). The dynamic wave setup or “surf beat” was first identified through field observations and measurements by Munk13 and Tucker.14 A number of theoretical treatments of dynamic wave setup based on various hypotheses have been developed with each focusing on a different mechanism. These include Symonds et al.15 (time-varying breakpoint), Symonds and Bowen16 (trapping of long waves by longshore bars), Schaffer and Svendsen17 (reinforcement of incoming and reflected long waves), etc. Kostense18 conducted laboratory experiments to investigate the dynamic setup component and found that the results were in qualitative agreement with the theory of Symonds et al.15 We can apply the results for monochromatic waves to investigate the approximate dynamic wave setup for a simple irregular wave case. Consider a bichromatic wave system with wave heights H1 and H2 (H1 > H2 ) and a small frequency difference between the two components. If the resulting wave group varies so slowly that static conditions occur within the surf zone, Eq. (1.7) applies and is written as η = F Hb ,

(1.10)

where Hb is the breaking wave height and F is a proportionality factor depending on whether the referenced setup is at the still water shoreline or the maximum wave setup (see Eqs. (1.8) and (1.9)). The maximum and minimum wave setup values are: η max =

F (H1 + H2 ) ,

ηmin =

F (H1 − H2 ) .

(1.11)

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Table 1.1. Static and dynamic wave setup characteristics for a biharmonic wave system. H2 /H1 0.2 0.4 0.6 0.8 1.0

η max /F H1

η avg /F H1

(η max − η avg )/η avg

1.2 1.4 1.6 1.8 2.0

1.01 1.04 1.09 1.16 1.27

0.19 0.35 0.47 0.55 0.57

It can be shown that the average wave setup depends on the ratio H2 /H1 as shown in Table 1.1. The fourth column presents the ratios of the maximum dynamic wave set amplitude to the average wave setup component. In the case with H2 = H1 , the dynamic wave setup displacement from the mean setup equals 57% of the average wave setup (Table 1.1, Column 4). In the above, we have examined the dynamic wave setup for the case of a simple bichromatic wave system in which the difference in frequencies of the two components was fairly small. For the case of a wave spectrum, the situation is much more complex with, for the case of a narrow spectrum, the group envelope varying according to the Rayleigh distribution. For the case of a wide spectrum, the dynamic component is reduced considerably.

1.5. Laboratory and Field Measurements of Wave Setup Having reviewed the theory of wave setup and its relationship to various factors, the two sources available for evaluation are laboratory and field data. 1.5.1. Laboratory experiments on wave setup Many laboratory investigations of static and dynamic wave setup have been conducted. The results of an early laboratory investigation with monochromatic waves by Bowen et al.19 are shown in Fig. 1.5. For this study, the ratio of maximum wave setdown and wave setup on the beach face to breaking wave height are − 0.035 and + 0.316, respectively, compared with − 0.049 and + 0.244 on the beach face for a κ value of 0.78. The effect of beach slope has been noted earlier and the relatively large beach slope of 0.082 in these experiments is undoubtedly a contributor to the large setup value. Later, laboratory investigations have included examination of irregular waves including measurements of water particle velocities and pressures which form the basis of the Sxx momentum flux component. Battjes20 conducted one of the earliest laboratory studies of wave setup due to irregular waves. Setup was measured through bottom mounted manometers and it was found that the wave setup was less than predicted. It was hypothesized that this difference was possibly due to air in the water column of the manometers. The entire setup was shifted landward relative to the theoretical and this delay was later attributed to a “roller” that is transported along with the wave crest region and

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Fig. 1.5.

Measured wave setup and setdown in the laboratory.19

conveys wave energy and momentum landward prior to transfer of the momentum to the water column and the associated wave setup.21 Later, Stive and Wind22 conducted a very detailed laboratory investigation in which they demonstrated the role of wave nonlinearity. In this study, the momentum flux components (velocities and pressures) were measured to the degree possible and it was found that the calculated wave setup based on nonlinear wave theories was in much better agreement with measured wave setup than calculations based on linear wave theories. In these comparisons, it was not necessary to introduce the roller concept. The two laboratory studies reviewed above have focused on static setup and it has been noted that irregular waves also produce dynamic wave setup. Hedges and Mase23 have presented an interesting reanalysis of earlier runup laboratory measurements by Mase24 in which irregular waves provided the forcing.b The planar slopes represented in the data were: 1:5, 1:10, 1:20, and 1:30. Figure 1.6 presents an example of the form in which the data were plotted where the horizontal axis is b Walton25

was the first to analyze the Mase data to extract the static wave setup.

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Fig. 1.6.

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Variation of nondimensional runup with Iribarren number.23

the Iribarren number, ξ0 defined as tan α ξ0 =  H/L

(1.12)

in which tan α is the profile slope. The interpretation of Fig. 1.6 is that for a zero slope (zero Iribarren number), there would be no short wave runup; therefore, the intercept represents the sum of the static and dynamic components of wave runup. Equations of the following form were fit to plots of the type of Fig. 1.6: Rchar Schar = + cchar ξ0 , H1/3 H1/3

(1.13)

where the subscript “char” refers to the percent associated with the variable; for example, the 2% runup is defined as R2% . It was found that both Schar and cchar were Rayleigh distributed with S1/3 and c1/3 equal to 0.27 and 1.04, respectively, where only the first term represents wave setup and is of interest here. The results for Schar can be interpreted in terms of the static and dynamic wave setup components. As an example, Smean = 0.17 and S2% = 0.37. Thus, the 2% value of the nondimensional dynamic setup defined here as ∆S2% is ηdyn,2% = (S2% − Smean ) = (0.37 − 0.17) = 0.20 . (1.14) ∆S2% = H1/3 Thus, the mean setup at the still waterline is 17% of the significant wave height measured at the toe of the slope and the 2% dynamic component at the still waterline is 20% of the significant wave height at the toe of the slope or slightly larger than the mean wave setup. These results are interesting and of reasonable magnitudes; however, there are two problems with recommending them for universal application. First, we know that the mean setup depends on the slope (through the κ dependency

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as discussed earlier), and the second is that the oscillating wave setup component should depend on the width of the input spectrum. Referring to Fig. 1.6, which is one of several similar plots presented in the Hedges and Mase paper, since each plot may include a mix of beach slopes, the slope dependency is not resolved in the Smean results which of course are derived from the y-intercept of these graphs. Secondly, these experiments were not designed to evaluate the effect of spectral width and the spectral characteristics included in the experiments are not known. However, it is of interest to identify the “representative” κ value and beach slope associated with a Smean = η avg /H1/3 of 0.17. Referring to Fig. 1.2, we see that the associated κ value is approximately 0.63 for F0 . Based on the Dally et al.8 breaking wave model, the associated beach slope from Fig. 1.3 is 1:29 compared to the beach slopes in the Mase experiments ranging from 1:30 to 1:5. 1.5.2. Field experiments on wave setup The paragraphs below describe several field experiments and observations of wave setup. An early study of wave setup comprised a pair of observations at an exposed coastal site (Narragansett Pier, RI) and a calmer water site (Newport, RI) where, at the latter, wave action was assumed not a factor and was found to show an approximate 3 foot water level difference during the peak of the 1938 hurricane storm surge.26 In a second early field experiment on wave setup at Fernandina Beach, Florida, Dorrestein27 placed transparent plastic tubes with lightweight floats to track the water surface on the beach in the zone of wave setup. To obtain the setup records, 16 mm movie film recorded the tracked surface of the floats. A float type tide gage on the end of a fishing pier provided offshore water level records. Through analysis of the tide gage records and the beach placed setup gages, Dorrestein27 evaluated the setup (with respect to the end of the pier) and compared observational results to existing setup theory. He found the measured setup in four of five experiments to be larger than the computed setup. One shortcoming of Dorrestein’s work is that the water level records were only 72 s in length and thus subject to considerable scatter and large standard deviation as later noted by Holman and Sallenger.28 Although rationale was provided by Dorrestein27 for possible differences between measured and computed setup in this early experiment, large discrepancies between measured and analytically or numerically computed setup still exist today. A North Sea field wave setup experiment was conducted on the Island of Sylt by Hansen.29 Utilizing a combination of ultrasonic wave gages and pressure sensor wave gages out to a distance of 1280 m from shore (10 m depth), Hansen29 found good correspondence of data to an empirical expression provided by: η = 0.3Hos = 0.42Horms .

(1.15)

Hansen also noted the maximum wave setup to be approximately 50% of the significant breaking wave height. It is not clear as to the methodology utilized to obtain η max in this field experiment.

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A wave setup field experiment was conducted as part of the Nearshore Sediment Transport Study at Torrey Pines Beach, San Diego, California by Guza and Thornton.30 The Torrey Pines Beach face was gently sloping (beach slope ≈ 0.02) and the beach material was a moderately sorted fine grain sediment (≈ 0.1 mm). A dual wire resistance runup meter was used for the recording and estimation of the wave setup. It should be noted that the measurements of the wave setup were considered to be the average runup determined from wires placed approximately 3 cm above the beach level rather than an actual water level at one location in these experiments. Offshore pressure sensors outside the surf zone at mean depths of 7 to 10.5 m were used for estimating wave height with recording lengths of 4096 s. Guza and Thornton30 note specific problems in the data set, which are typical of field measurements, i.e., the difficulty in obtaining a common datum for the offshore wave measurements and the beach wave setup measurements. Results of their measurement program suggest an empirical relationship as follows: η = 0.17Hos = 0.24Horms

(1.16)

with scatter that suggests η/Hos ranging approximately from 0.05 to 0.50 for individual experiments. Holman and Sallenger28 conducted a field experiment for measuring wave setup as well as other surf zone parameters at the U.S. Army Corps of Engineers field research pier in Duck, NC, USA. Data on water level at the shoreline were collected using longshore looking time lapse photography from Super-8 movie cameras mounted on the research pier scaffolding. The beach at the experiment site had a very steep foreshore (∼ 1 on 10) while the offshore profile slope is much milder (∼ 1 on 100). Beach material was bimodal in size with a median sand size of 0.25 mm and a coarse fraction of 0.75 mm. Results of the experiments showed considerable scatter and dependence on tide level. Regression lines were fit to the data (segmented by tide levels) with results as follows for high tide and mid-tide data: η = 0.35ξ0 + 0.14 (high tide) , (1.17) Hs η = 0.46ξ0 + 0.06 (mid-tide) . (1.18) Hs As most of the data fell in a range of ξ0 = 1 to 2, the maximum setup was noted to be of the same order as the significant wave height in many of the experiments, much higher than theoretically suggested values. Note that in terms of Horms (based on consideration of monochromatic theory results) the setup would be much higher than most other studies show or suggest. Although Holman and Sallenger28 conclude from their experiments that the setup is dependent on the Iribarren number, it is not entirely clear from their data, especially for higher waves (i.e., see Fig. 1.4, Ref. 28). An additional problem that must be considered when computing the Iribarren number for real beaches and irregular waves is how to define beach slope. It should be noted that video camera (visual) approaches estimate setup via the measurement of the water surface elevation on the beach (similar to the Guza and Thornton measurements) rather than an actual vertically fluctuating water level. The anomaly between dependence of

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setup on Iribarren number as noted by Holman and Sallenger28 is likely due to the aforementioned relationship between the wave-breaking coefficient, κ, and beach slope. Nielsen31 and Davis and Nielsen32 conducted a novel setup experiment on Dee Why Beach, in New South Wales, Australia using a set of manometer tubes as shown in Figs. 1.7 and 1.8 from Davis and Nielsen.32 The tubes were deployed throughout the beach face and surf zone. A total of 120 setup profiles were measured in 11 days. Wave heights Horms ranged from 0.6 to 2.6 m in height and significant wave periods (Ts ) ranged from 5.8 to 12.1 seconds. A shoreline setup of about 40% of Horms was found although Davis and Nielsen32 point out that there is reason to believe that the surf zone characteristics influence the relationship between wave height and setup magnitude, and also note a problem of defining beach slope via the Iribarren number. Nielsen31 and Davis and Nielsen32 also observe that a major portion of the setup occurred on the beach face as shown in Fig. 1.9. Nielsen31 points out that previous field investigations have typically measured the mean water level elevation on the beach as opposed to the average fluctuating mean water level in the vertical plane (i.e., the wave setup as usually defined), and that the two measurements are often different in part due to the beach permeability, which in turn is related to beach material size. The issue of extracting wave setup from runup and rundown on the beach is illustrated in Fig. 1.10. King et al.33 collected wave setup data at Woolacombe Beach in North Devon, U.K. which faces the North Atlantic Ocean. The beach face slope varied between 1 on 40 at high tide and 1 on 70 at mid-tide level with a tidal range of 3 m at neap and as much as 9 m at springs. Beach face material consisted of fine sand with

Fig. 1.7.

Manometer setup of Davis and Nielsen32 for measuring setup.

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Fig. 1.8.

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Schematic diagram of apparatus (from Ref. 32).

Fig. 1.9. Dimensionless setup versus total depth where much of the setup occurs on the beach face (from Ref. 32). In this figure, B and D are equal to η and h as used in this chapter, respectively.

90% in the 0.125 mm to 0.25 mm size range. Pressure transducers were utilized to collect wave and setup information at various stations across the beach and also in a longshore direction to assess the spatial variability of the mean setup. Both tripod mounted and buried pressure transducers were utilized. The buried pressure transducers were 50 to 80 cm below the beach surface and were protected by a porous cover. Instruments collected pressure data which were then transformed to water level data over 4096 second intervals. Data did not include sampling in very shallow water and the maximum wave setup was estimated by extrapolating the

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Fig. 1.10. Illustration of differences between mean water level (MWL) shoreline and mean water line on beach.

water surface from the most shoreward water stations. Wave setup estimated from the data showed the wave setup to be roughly: η = 0.10H os = 0.14H orms

(1.19)

with most of the values of η/H orms between 0.11 and 0.15. The authors do not speculate as to why such low values of setup (compared to analytical results) were found in this measurement program. Yanagishima and Katoh34 discuss field measurements of mean water level near the shoreline on the Pacific Coast of Japan as measured by an ultrasonic wave gage mounted on a pier where the mean depth of water was ∼ 0.4 m. The setup was determined via a multiple regression approach on 1305 sets of (20 minute records) data taking into account astronomical tide, wind setup, and atmospheric pressure head components of mean water level. Their data included 91 records in which the offshore wave height was above 3 m. Yanagishima and Katoh’s34 regression analysis suggested the following relationship:  0.2 Los η = 0.0520H os , (1.20) Hos which can be formulated in terms of Iribarren number for their beach slope (1 on 60) to the following: η = 0.27H os (ξ0 )0.4 = 0.38H orms (ξ0 )0.4 .

(1.21)

Yanagishima and Katoh34 noted reasonable agreement with the theory of Goda35 (to be discussed later). Even higher values of setup would be expected on the beach face in accord with theory and findings of other researchers. Greenwood and Osborne36 conducted field measurements on a Georgian Bay Beach, in Lake Huron, Ontario, Canada. Lake Huron has no measurable tide and the beach profile at the site had a slope of 0.015 with a steeper sloped (0.031 to 0.047) inshore bar. Setup was measured using surface piercing resistance wire wave staffs with the shoreward most gage being in approximately 0.4 m of water depth. Measured setup values were found as follows: η = 0.19H os = 0.27H orms .

(1.22)

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It is again noted that even higher values of setup would be expected on the beach face in accord with theory and experience of other researchers. Further work by Hanslow and Nielsen37,38 utilized the manometer tube deployment shown in Fig. 1.7 on three additional beaches (Seven Mile, Palm, and Brunswick) in New South Wales, Australia. With beach face slopes ranging from 0.03 to 0.16 and mean grain sizes of swash zone beach material ranging from 0.18 to 0.5 mm, shoreline beach setup was measured using 20 minute record averages. Using the data from these three beaches as well as earlier measurements at Dee Why Beach (see Refs. 31 and 32), linear least square relationships were fit to the data as follows: η = 0.27H os = 0.38H orms

with R = 0.65

(1.23)

  η = 0.040 Hos L0 = 0.048 Horms L0

with R = 0.77 ,

(1.24)

or

where a somewhat higher value of explained regression was noted using wave height and wave period. Data and regression lines for these two relationships are shown in Figs. 1.11 and 1.12. The improvement in fit due to inclusion of the deep water wavelength is not evident visually. A significant finding of these studies was that a major portion of the setup occurred on the beach face (see Fig. 1.9). Further measurements on wave setup at two river entrances is also discussed in Hanslow and Nielsen37 and Dunn et al.39 with the result that the wave setup at river entrances was found to be (somewhat surprisingly) negligible. Lentz and Raubenheimer40 report on a field experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 11 pressure sensor gages and 10 sonar altimeters extended across the surf zone from 2 to 8 m of water depth. Close agreement with Longuet-Higgins radiation stress theory for wave setup was noted

Fig. 1.11.

Empirical relationship between setup and wave height (from Ref. 38).

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Fig. 1.12.

Empirical relationship between setup and wave parameters (from Ref. 38).

although the lack of setup measurements in shallow water (< 2 m) did not allow conclusions regarding the maximum setup that might be expected on the beach. Raubenheimer et al.41 report on a second field experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 12 buried pressure sensor gages were employed across the surf zone from the shoreline to 5 m of water depth. Again good agreement with Longuet-Higgins and Stewart42 radiation stress theory was noted by integration of the cross-shore momentum equation to estimate the wave setup for water depths greater than 1 m but the theory was found to under-predict wave setup in shallow water (h < 1 m). The lack of setup measurements on the beach face did not allow conclusions regarding the maximum setup that might be expected on a beach although an empirical equation was provided to estimate wave setup at the SWL line as follows: η SWL −1/3 = 0.019 + 0.003βf , Hos

(1.25)

where βf is the average slope across the surf zone. Raubenheimer et al.41 suggest that theory under-predicts the setup by a factor of 2 for water depths less than 1 m. Stockdon et al.43 using video shoreline water level time series determined wave setup and wave runup results during 10 diverse field experiments (four from Duck, NC, USA; four from West Coast beaches in California/Oregon, USA; and two from Terschelling, The Netherlands). These wave setup results were analyzed to provide empirical parameterizations for wave setup under many natural beach conditions as follows:  η = 0.385βf H0 L0 , (1.26) which, assuming that tan βf ≈ βf can be expressed in terms of the Iribarren number as η = 0.385ξ0 H0

(1.27)

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and for extremely dissipative beaches η 0.043 = , H0 H0 /L0

(1.28)

where H0 is the effective deep water wave height, L0 is the deep water wave length associated with the peak spectral period and βf is the average slope over a depth range defined in terms of the standard deviation of the water surface displacement. It should again be noted that the video camera (visual) approach estimates setup via the mean of measurements of the water surface elevation on the beach rather than the mean of fluctuating water levels at one location. Results from nine of the field experiments presented here have been analyzed to determine the average ratio of wave setup at the still water line to significant wave height and its associated standard deviation. The ratios at the still water line were determined to be 0.191±0.100. Several caveats apply to these results. In cases where the beach slope and/or the deep water wave steepness was incorporated into the expression presented, these were taken as 0.01 and 0.04, respectively. Some of the published expressions were in terms of the breaking wave height and some in terms of the deep water wave height and no attempt was made to differentiate between breaking and deep water wave heights. The Holman and Sallenger results were not included in these results as they appeared to be anomalously high. Finally, the wave setup ratio at the intersection of the mean water line intersection with the beach profile would be greater than the average ratio (0.191) above. Also, although not examined in detail here, the dynamic wave setup which increases with energetic narrow spectra, would also contribute to the total wave setup. It is relevant to note that results from field measurements are often not consistent, possibly due to: (1) Limited measurement distances across the nearshore. (2) Use of many different approaches to measure/evaluate setup (i.e., videos, pressure sensors, runup gages, manometers, etc.). (3) Inherent difficulties in obtaining a consistent datum for nearshore measurements and offshore measurements. (4) A clear definition of setup on the beach face is lacking due to the nature of the permeable beach and the difficulty of sub-aerial setup measurements. 1.6. Published Guidance on Wave Setup for Engineering Applications Several sources of wave setup recommendations are available; two are reviewed here. The U.S. Army Corps of Engineers 1984 Shore Protection Manual (SPM) presents a graphical method to calculate wave setup at mid-depth of the surf zone. This method, developed for irregular waves, is presented in Fig. 1.13 in which the normalized setup has been multiplied by a factor of 2 to transfer approximately the results to the still water shoreline. The effect of beach slope and deep water wave steepness in Fig. 1.13 are evident.

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Fig. 1.13. Nondimensional wave setup versus deep water wave steepness and profile slope by the 1984 Shore Protection Manual recommendations as incorporated in Appendix D of FEMA44 Guidelines. Note that the normalized setup has been multiplied by a factor of 2 to transfer the setup from the mid-depth of the surf zone as it appears in SPM to the approximate still water level contour. Note: S in this figure is equal to η in this chapter.

Fig. 1.14. Nondimensional wave setup by Goda versus deep water wave steepness and relative water depth within the surf zone. Profile slope = 1:100.

Goda35 has presented guidance for static and dynamic wave setups due to irregular waves. The guidance for static setup and a profile slope of 1:100 is shown in Fig. 1.14. The effects of various deep water wave steepness values are illustrated in Fig. 1.14.

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Table 1.2. Comparison of nondimensional wave setup by SPM and Goda methods for irregular waves. η/H0 H0 /L0

SPM

Goda

0.005 0.01 0.02 0.04 0.08

0.154 0.135 0.120 0.103 0.097

0.122 0.102 0.083 0.065 0.049

Note: Values in SPM method have been multiplied by 2.0 to transfer from surf zone mid-depth to still water line.

Table 1.2 presents a comparison of ratios of nondimensional wave setup values at the still water shoreline as recommended by SPM and Goda. In examining the results in Table 1.2, recall that an additional wave setup occurs from the still water line to the location where the maximum setup intersects the beach profile.

1.7. Summary and Recommendations The reviews of theory, laboratory and field data, and published guidance for engineering applications presented here have identified static and dynamic wave setup components as contributing to the deviation from still water level in the surf zone and their relevance to engineering design. Examination of the static wave setup has reinforced the effect of beach slope on wave setup. The theory presented here does not account for the onshore bottom stress acting on the water column due to undertow. The available field measurement results exhibit a wide range of wave setup to wave height ratios. Some of this variability is undoubtedly due to the effect of profile slope, which is not accounted for explicitly in some of the analyses and part is due to the effect of wave-breaking in depths greater than the shallow water limit. Design methodology should account for the static and dynamic wave setup components. In determining the wave setup to include in design, the characteristics of the particular application of interest should be compared with those of the various field and laboratory experiments available including those referenced here. The dominant role of beach slope should be recognized. The preliminary results presented here of η/Hs = 0.191 ± 0.100 may serve as a useful guide for the static wave setup component. It is hoped that further research with improved instrumentation, modern surveying techniques, and more diverse field site studies will help to clarify both the static and dynamic wave setup components for future design applications.

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References 1. A. T. Ippen, Estuary and Coastline Hydrodynamics (McGraw-Hill Book Company, 1966), 744 pp. 2. M. S. Longuet-Higgins, Radiation stress and mass transport in gravity waves with application to “surf beats”, J. Fluid Mech. 13(4), 481–504 (1962). 3. M. S. Longuet-Higgins and R. W. Stewart, A note on wave set-up, J. Marine Res. 21, 4–10 (1963). 4. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves; a physical discussion, with applications, Deep-Sea Res. 11, 529–562 (1964). 5. R. A. Holman, Wave set-up, Handbook of Coastal and Ocean Engineering, Vol. 1, ed. J. Herbich (Gulf Publishing, 1990), Chapter 11, pp. 635–646. 6. van der Meer, TAW Technical Report: Wave Run-up and Wave Overtopping at Dikes, Technical Advisory Committee for Flood Defence, Delft (2003). 7. M. S. Longuet-Higgins, On the wave-induced difference in mean sea level between the two sides of a submerged breakwater, J. Marine Res. 25(2), 148–153 (1967). 8. W. R. Dally, R. G. Dean and R. A. Dalrymple, Wave height variation across beaches of arbitrary profile, J. Geophys. Res. 90(C6), 11917–11927 (1985). 9. R. G. Dean, Stream function representation of nonlinear ocean waves, J. Geophys Res. 79(30), 4489–4504 (1965). 10. R. G. Dean, Evaluation and Development of Water Wave Theories for Engineering Application; Volume I: Presentation of Research Results; Volume II: Tabulation of Dimensionless Stream Function Variables, Special Report No. 1, Published by U.S. Army Corps of Engineers, Coastal Engineering Research Center (1974). 11. R. A. Dalrymple, A finite amplitude wave on a linear shear current, J. Geophys. Res. 87(C1), 483–491 (1974). 12. R. G. Dean and C. J. Bender, Static wave setup with emphasis on damping effects by vegetation and bottom friction, Coast. Eng. 13, 149–156 (2006). 13. W. R. Munk, Surf beats, Trans. Am. Geophys. Union 30, 849–854 (1949). 14. M. J. Tucker, Surf beats: Sea waves of 1 to 5 minutes period, Proc. Roy. Soc. A 202, 565–573 (1950). 15. G. Symonds, D. A. Huntley and A. J. Bowen, Two-dimensional surf beat: Long wave generation by a time-varying breakpoint, J. Geophys. Res. 87(C1), 492–498 (1982). 16. G. Symonds and A. J. Bowen, Interaction of nearshore bars with wave groups, J. Geophys. Res. 89(C2), 1953–1959 (1984). 17. H. A. Schaffer and I. A. Svendsen, Surf beat generation on a mild-slope beach, Proc. ASCE Int. Conf. Coastal Engineering (1988), pp. 1058–1072. 18. J. K. Kostense, Measurements of surf beat and set-down beneath wave groups, Proc. ASCE Int. Conf. Coastal Engineering (1984), pp. 724–740. 19. A. J. Bowen, D. L. Inman and V. P. Simmons, Wave set-down and set-up, J. Geophys. Res. 73, 2569–2577 (1968). 20. J. A. Battjes, Set-Up Due to Irregular Waves, Report No. 72–2, Communications on Hydraulics, Delft University of Technology, Department of Civil Engineering (1972), 13 pp. (Paper also presented at the 13th Int. Conf. Coastal Engineering, Vancouver, B.C.). 21. I. A. Svendsen, Wave heights and set-up in a surf zone, Coast. Eng. 8, 302–329 (1984). 22. M. J. F. Stive and H. G. Wind, A Study of Radiation Stress and Set-Up in the Nearshore Zone, Publication No. 267, Waterlopkundig Laboratorium, Delft Hydraulics Laboratory (1982), 25 pp.

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23. T. S. Hedges and H. Mase, Modified Hunt’s equation incorporating wave setup, J. Waterway Port Coast. Ocean Eng. 130(3), 109–113 (2004). 24. H. Mase, Random wave runup height on gentle slopes, J. Waterway Port Coast. Ocean Div. 115(WW2), 649–661 (1989). 25. T. L. Walton, Interim guidance for prediction of wave run-up on beaches, Ocean Eng. 19(2), 199–207 (1999). 26. T. Saville, Experimental determination of wave set-up, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce Washington, D.C. (1961), pp. 242–252. 27. R. Dorrestein, Wave set-up on a beach, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce, Washington, D.C. (1961), pp. 230–241. 28. R. A. Holman and A. H. Sallenger, Setup and swash on a natural beach, J. Geophys. Res. 90(C1), 945–953 (1985). 29. U. A. Hansen, Wave setup and design water level, J. Waterway Port Coast. Ocean Div. 104(WW2), 227–240 (1978). 30. R. T. Guza and E. B. Thornton, Wave set-up on a natural beach, J. Geophys. Res. 96(C2), 4133–4137 (1981). 31. P. Nielsen, Wave setup: A field study, J. Geophys. Res. 93(C12), 15643–15652 (1988). 32. G. A. Davis and P. Nielsen, Field measurement of wave set-up, ASCE Int. Conf. Coastal Engineering, Malaga, Spain (1988), Chapter 38, pp. 539–552. 33. B. A. King, M. W. L. Blackley, A. P. Carr and P. J. Hardcastle, Observations of wave induced setup on a natural beach, J. Geophys. Res. 95(C12), 22289–22297 (1990). 34. S. Yanagishima and K. Katoh, Field observation on wave setup near the shoreline, Proc. 22nd Int. Conf. Coastal Engineering, Vol. 1, ASCE, New York, N.Y. (1990), Chapter 7, pp. 95–108. 35. Y. Goda, Random Seas and Design of Maritime Structures (World Scientific Publishing Co., 2000), 443 pp. 36. B. Greenwood and P. D. Osborne, Vertical and horizontal structure in cross-shore flows: An example of undertow and wave setup on a barred beach, Coast. Eng. 14, 543–580 (1990). 37. D. J. Hanslow and P. Nielsen, Wave setup on beaches and in river entrances, 23rd Int. Conf. Coastal Engineering, Venice, Italy (1992), pp. 240–252. 38. D. J. Hanslow and P. Nielsen, Shoreline set-up on natural beaches, J. Coast. Res. SI15, 1–10 (1993). 39. S. L. Dunn, P. Nielsen, P. A. Madsen and P. Evans, Wave setup in river entrances, Proc. 27th Int. Conf. Coastal Engineering, ASCE, New York, Sydney, Australia (2000), pp. 3432–3445. 40. S. Lentz and B. Raubenheimer, Field observations of wave setup, J. Geophys. Res. 104(C11), 25867–25875 (1999). 41. B. Raubenheimer, R. T. Guza and S. Elgar, Field observations of wave-driven setdown and setup, J. Geophys. Res. 106(C3), 4629–4638 (2001). 42. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves: A physical discussion with applications, Deep Sea Res. 11(4), 529–562 (1964). 43. H. F. Stockdon, R. A. Holman, P. A. Howd and A. H. Sallenger, Jr., Empirical parameterization of setup, swash, and runup, Coast. Eng. 53, 573–588 (2006). 44. FEMA, Guidelines and Specifications for Flood Hazard Mapping Partners, Appendix D: Guidance for Coastal Flooding Analysis and Mapping, Map Modernization Program, Washington, D.C. (2003). Also available at: www.fema.gov/fhm/ dl cgs.shtm.

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Chapter 2

Wavemaker Theories Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] The fundamental solutions to the wavemaker boundary value problem (WMBVP) are given for 2D channels, 3D basins, and circular basins. The solutions are given in algebraic equations that replace integral and differential calculus. The solutions are generic and apply to both full- and partial-draft piston and hinged wavemakers; to double-articulated wavemakers, and to directional wave basins. The WMBVP is solved by conformal mapping and by domain mapping. The loads on a wavemaker are connected to the radiation boundary value problem for semiimmersed bodies and demonstrate the connection of these loads to the added mass and radiation damping coefficients required to compute the dynamic response of large Lagrangian solid bodies.

2.1. Introduction Wavemaker theories play several important roles in coastal and ocean engineering. The most important role is the application to laboratory wavemakers for both wavemaker designs and wave experiments. A second role for wavemaker theories is to compute a scalar radiated wave potential to compute the wave-induced loads on large solid bodies applying potential wave theory. The displacements and rotations of a semi-immersed six degrees-of-freedom large Lagrangian solid body are related to the displacements and rotations of wavemakers. The boundary between a planar wavemaker and an ideal fluid requires special care because the fluid motion

25

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is an Eulerian field with time and space as the independent variables, and the planar wavemaker is a Lagrangian solid body with time and the wavemaker as the independent variables. Consequently, the kinematic boundary condition will be different from the free surface boundary that separates two Eulerian fluid fields of air and water. The boundary between the fluid and wavemaker separates an Eulerian field (the fluid) from a Lagrangian body (the wavemaker), and the wavemaker kinematic boundary condition (WMKBC) must convert the Lagrangian wavemaker motion to a Eulerian field motion in order that the independent variables for both dependent motion variables are equivalent. This may be accomplished by multiplying the Lagrangian motion of the wavemaker by the unit normal to the boundary. Because the unit normal is a function of space and the Lagrangian wavemaker motion is a function of time, the product will produce a motion that is a function of both space and time that are the independent variables of the Eulerian fluid field. Although this fact is not central to the WMBVP, it is an important connection between the WMBVP and the radiation potential boundary value problem for semi-immersed large Lagrangian solid bodies.1 The formulae for computing the two fundamental fluid unknowns for an incompressible fluid of the velocity q(x, z, t) and the pressure p(x, z, t) from a scalar velocity potential Φ(x, z, t) are given first. The classical linear WMBVP for dimensionless 2D planar wavemaker is reviewed for two types of double-articulated planar wavemakers. The sway X1 (t) displacement of a full-draft piston wavemaker and the roll Θ5 (t) rotation of a hinged wavemaker are connected directly to the sway displacement and the roll rotation of a semi-immersed large Lagrangian solid body. In this review, integral calculus formulae for computing the integrals that are required to compute the coefficients of the eigenseries for the fluid motion, to compute the loads on the wavemaker and the average power required to generate the propagating waves are replaced by generic algebraic formulae. For example, an integral equation that is required to compute the nth eigenseries coefficient Cn for the nth eigenfunction Ψn (Kn , z/h) from a wavemaker shape function χ(z/h) may be computed symbolically and expressed by a dimensionless algebraic formula In (α, β, b, d, Kn ), that is given by
Cn = h

0

−1

χ(z/h)d(z/h)Ψn (z/h)d(z/h) = In (α, β, b, d, Kn ).

(2.1)

The coefficient in (2.1) may then be computed very efficiently by substitution into algebraic formulae in all subsequent applications. Next a dimensionless theory for both amplitude-modulated (AM) and phase-modulated (PM) circular wavemakers is reviewed. Then, a dimensionless theory for double-actuated wavemakers is reviewed. Following that, a dimensionless directional wavemaker theory for large wave basins based on a WKBJ approximation1 is reviewed. Next, a theory for sloshing waves due to transverse motions of a segmented wavemaker in a narrow wave channel is reviewed. Then, 2D planar wavemakers are mapped to a unit circle by conformal mapping and to a fixed rectangular domain by domain mapping; and both the linear and nonlinear wavemaker solutions are computed numerically.

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2.2. Planar Wavemaker in a 2D Channel Two generic planar wavemaker configurations are shown in Figs. 2.1(a) and 2.1(b). The fluid motion may be obtained from the negative gradient of a dimensional scalar velocity potential Φ(x, z, t) according to q(x, z, t) = u(x, z, t)ex + w(x, z, t)ez = −∇2 Φ(x, z, t), where the 2D gradient operator in (2.2a) is given by ∇2 (•) =

∂(•) ∂(•) ex + ez . ∂x ∂z

Fig. 2.1(a).

Definition sketch for a Type I planar wavemaker.

Fig. 2.1(b).

Definition sketch for a Type II planar wavemaker.

(2.2a)

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The total pressure field P (x, z, t) may be computed from the unsteady Bernoulli equation according to 

 ∂Φ(x, z, t) 1 2 P (x, z, t) = p(x, z, t) + pS (z) = ρ − |∇Φ(x, z, t)| + Q(t) − ρgz, ∂t 2 (2.2b) where Q(t) = the Bernoulli constant; and the free surface elevation η(x, t) for zero atmospheric pressure according to 1 η(x, t) = g



 ∂Φ(x, η, t) 1 2 − |∇Φ(x, η, t)| + Q(t) ; ∂t 2

x ≥ ξ(η, t);

z = η(x, t). (2.2c)

The scalar velocity potential must be a solution to the Laplace equation ∇22 Φ = 0;

x ≥ ξ(z, t);

−h ≤ z ≤ η(x, t),

(2.3a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

x ≥ ξ(−h, t);

z = −h.

(2.3b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):   ∂ 2Φ 1 ∂ ∂Φ dQ − − ∇Φ · ∇ |∇Φ|2 + = 0; + g ∂t2 ∂z ∂t 2 dt

x ≥ ξ(η, t);

z = η(x, t). (2.3c)

Kinematic WaveMaker Boundary Condition (KWMBC): A Stokes material surface for planar wavemaker is W (x, z, t) = x − ξ(z, t), and the Stokes material derivative gives the KWMBC from ∂Φ ∂ξ ∂Φ ∂ξ DW = + − = 0; Dt ∂x ∂t ∂z ∂z

x = ξ(z, t);

−h ≤ z ≤ η(t).

(2.3d)

Kinematic Radiation Boundary Condition (KRBC): A KRBC is required as x → +∞ for uniqueness to insure that propagating waves are only right progressing or that evanescent eigenmodes are bounded. For a temporal dependence proportional to exp ±iωt, the KRBC may be expressed by  lim

x→+∞

 ∂ ± iKn Φ(x, z, t) = 0. ∂x

(2.3e)

A velocity potential ϕ(x, z) may be defined by the real part of Φ(x, z, t) = Re{ϕ(x, z) exp −i(ωt + ν)},

(2.4)

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where Re{•} means the real part of {•}; and ν = arbitrary phase angle. The linearized WMBVP for kh = O(1) is1 ∇22 ϕ(x, z) = 0;

0 ≤ x < +∞;

∂ϕ(x, z) = 0; ∂z

0 ≤ x + ∞;

∂ϕ(x, z) − k0 ϕ(x, z) = 0; ∂z  lim

x→+∞

η(x, t) = Re

z = −h,

(2.5b)

z = 0,

 ∂ − iKn ϕ(x, z) = 0, ∂x x = 0;

 −iω ϕ(x, 0) exp −i(ωt + ν) ; g

p(x, z, t) = ρ

(2.5a)

0 ≤ x < +∞;

∂ϕ(x, z) ∂ξ(z, t) exp −i(ωt + ν) = − ; ∂x ∂t 

−h ≤ z ≤ 0,

∂Φ(x, z, t) ; ∂t

(2.5d)

−h ≤ z ≤ 0, x ≥ 0;

0 ≤ x < +∞;

(2.5c)

z = 0,

z = 0,

(2.5e)

(2.5f)

(2.5g)

where k0 = ω 2 /g. Because the boundary conditions defined by (2.5b)–(2.5e) are prescribed on boundaries with constant values of the independent variables x and z, a solution by the method of separation of (independent) variables may be computed.1 The instantaneous wavemaker displacement ξ(z, t) from its mean position x = 0 is assumed to be strictly periodic in time with period T = 2π/ω, and may be expressed by     S χ(z/h) exp −i(ωt + ν) ξ(z/h, t) = Re i (∆/h)   S = χ(z/h) sin(ωt + ν). (2.6) (∆/h) The specified shape function χ(z/h) for the Type I wavemaker shown in Fig. 2.1(a) is valid for either a double-articulated piston or hinged wavemaker of variable draft and is given by the following dimensionless equation for a straight line2 : χ(z/h) = [α(z/h) + β][U (z/h + 1 − d/h) − U (z/h + b/h)],

(2.7a)

where α, β = dimensionless constants; U (•) = the Heaviside step function with two boundary conditions given by [S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h + ∆/h) = S,

(2.7b)

[S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h) = Sb ,

(2.7c)

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that may be solved simultaneously for the dimensionless coefficients α, β to obtain α = (1 − Sb /S);

β = ∆/h + α(1 − d/h − ∆b /h − ∆/h).

(2.7d,e)

The coefficients α and β for the specified shape function χ(z/h) in (2.7a) may be obtained for the Type II wavemaker shown in Fig. 2.1(b) by substituting ˆ S = S¯ + S;

¯ Sb = S;

∆=h−b−d

into the following boundary conditions2 in (2.7b) and (2.7c):   ˆ (S¯ + S) ˆ χ(z/h = −b/h) = S¯ + S, 1 − b/h − d/h 

 ˆ (S¯ + S) ¯ χ(z/h = −1 + d/h) = S, 1 − b/h − d/h

(2.8a)

(2.8b)

that may be solved simultaneously for the constant coefficients α, β to obtain  ¯ d b S Sˆ ; β =1− − . (2.8c,d) α= ¯ ˆ ¯ ˆ h S+S S+S h 2.2.1. Eigenfunction solution to the WMBVP Because all of the boundary conditions defined by (2.5b)–(2.5e) are now prescribed for constant values of the independent variables (x, z) and the dimensionless parameter kh = O(1), a solution by separation of independent variables1 is suggested according to ϕ(x, z) = X(x) • Z(z).

(2.9)

The eigenseries solution may be written compactly as1,3,4

Φ(x, z, t; Kn ) = Cn cosh Kn (z + h) exp +i(Knx − ωt + ν),

(2.10a)

n=1

where Kn = k for n = 1 and Kn = +iκn for n ≥ 2 provided that ko h − kh tanh kh = ko h + κn h tan κn h = 0;

n > 2.

(2.10b)

The eigenseries (2.10a) may be separated into a propagating Φp (x, z, t; k) and evanescent eigenmodes Φe (x, z, t; κn) or “local ” wave components3 according to Φ(x, z, t; Kn ) = Φp (x, z, t; k) + Φe (x, z, t; κn ) 

Cn cos κn (z + h) exp +i(Kn x − ωt + ν). = C1 cosh k(z + h) + n=2

(2.10c)

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The wave number k = 2π/λ where λ = wavelength. Because the numerical value of kh must be computed from an eigenvalue problem in the vertical z coordinate, equivalence of the eigenvalue k to the wave number 2π/λ requires a pseudo-horizontal boundary condition of periodicity given by k = 2π/λ and ϕ(x + λ, z) = ϕ(x, z). It is computationally efficient to normalize the eigenseries in (2.10a) according to cosh Kn h(1 + z/h) ; n = 1, 2, . . . , Nn where the nondimensional normalizing constant Nn is  2kh + sinh 2kh  
0 ; n = 1,  4kh Nn2 = cosh2 Kn h(1 + z/h)d(z/h) =  2κn h + sin 2κn h −1   ; n ≥ 2. 4κn h Ψn (Kn , z/h) =

(2.11a)

(2.11b) (2.11c)

The eigenseries in (2.10a) may be written as an orthonormal eigenseries by Φ(x, z, t; Kn ) =

∞

Cn Ψn(Kn , z/h) exp i(Knx − ωt − ν),

(2.12)

n=1

where the orthonormal eigenfunction Ψn (•,•) is dimensionless. 2.2.2. Evaluation of Cn by WM vertical displacement χ(z/h) The following dimensionless coefficient computed from (2.5e) will replace integral calculus with algebraic substitution for the coefficients Cn in the eigenseries (2.12):
−b/h [α(z/h) + β]Ψn (Kn , z/h)d(z/h) In (α, β, b, d, Kn ) = −1+d/h

    d b     sinh K K h 1 − d − K b sinh K h 1 −   n n n  n h h  α =   (Kn h)2 Nn  b     − cosh Kn h 1 − + cosh Kn d   h    b β (2.13) + sinh Kn h) 1 − − sinh Kn d (Kn h)Nn h that is dimensionless when α and β are given by (2.7d) and (2.7e) or (2.8c) and (2.8d). The coefficients Cn may be computed algebraically by (2.13) from the KWMBC (2.5e) to obtain Sωh In (α, β, b, d, Kn ), Kn ∆ and the orthonormal eigenseries (2.12) is given by Cn = i

(2.14)

Φ(x, z, t; Kn ) =

∞

iSωh In (α, β, b, d, Kn )Ψn (Kn , z/h) exp i(Kn x − ωt − ν). Kn ∆ n=1

(2.15)

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2.2.3. Decay distance of evanescent eigenmodes n ≥ 2 Numerical solutions and experimental measurements of ocean and coastal designs require that the KRBC (2.5d) be applied far enough away so that only the propagating eigenmode for n = 1 in (2.12) is measurable. The evanescent eigenseries in (2.12) for n ≥ 2 will decay spatially at least as fast as the smallest evanescent eigenvalue κ2 . This eigenvalue must be κ2 h > (n − 3/2)π = π/2. If the smallest value for κ2 h > π/2, then κ2 > π/2h and ϕ(x, z) ∝ exp −(πx/2h). For the values of the evanescent eigenseries to be less than 1% of their values at the wavemaker, ϕ(x, z) ∝ exp −(πxd /2h) = 0.01 and πxd /(2h) = 4.6 ≈ 3π/2, and the minimum decay distance is xd ≥ 3h. 2.2.4. Transfer function for wave amplitude from wavemaker stroke The average rate of work or power done by a wavemaker of width B is1
˙ τ = Pτ = B W

τ +1




0

h

p(x, z, τ )u(x, z, τ )d(z/h)dτ,

τ

(2.16a)

−1

where the temporal averaging operator is defined by
•τ =

τ +1

(•)dτ,

(2.16b)

τ

and  ˙ τ = Pτ = W

ρω 3 S 2 Bh4 ∆2 2kh

 I12 (α, β, b, d, k),

(2.16c)

so that all of the average power from a wavemaker is transferred to only the propagating eigenmode. The average energy flux in a linear wave is given by1  ˙ τ = E

ρgBA2 2

CG ,

(2.16d)

where the group velocity CG is given by1 CG =

  2kh C 1+ . 2 sinh 2kh

(2.16e)

Equating (2.16c) to (2.16d) gives the following transfer function for a planar wavemaker:  A ko h = Ψ1 (k, 0)I1 (α, β, b, d, k). (2.16f) S kh

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2.2.5. Hydrodynamic pressure loads (added mass and radiation damping) The wave loads on a planar wavemaker may be estimated by integrating the total pressure over the wetted surface of the wavemaker, i.e., 

 F n = dS, (2.17a,b) P M r×n 0

S

where the outward pointing unit normal n points from the wavemaker into the fluid, and the pseudo-unit normal n for the rotational modes is given by n = r × n = (z + h − d)nx ey = ny ey .

(2.17c)

Force. For the Type I piston wavemaker of total width B, the horizontal component of the pressure force on the fluid side only may be computed from the real part of 
−b/h

Cn Ψn (Kn , z/h)d(z/h) exp −i(ωt + ν) F1 (t) = Re iρωBh n=1

−1+d/h

= −F1 cos(ωt + ν − α1 ),

(2.18a)

where the static component of the pressure force on the fluid side only is Fs = −

ρgBh2 [1 − 2(d/h) + (d/h)2 − (b/h)2 ]. 2

(2.18b)

The hydrodynamic component of F1 (t) may be separated linearly into a propagating and an evanescent component that are related to the piston wavemaker translational velocity and acceleration, respectively, from the real part of F1 (t) = −Re{[λ11 (Sω) + µ11 (−iSω 2 )] exp −i(ωt + ν)} = −µ11 (−Sω 2 sin(ωt + ν)) − λ11 (Sω cos(ωt + ν))

(2.18c)

¨ 1 (t) − λ11 X˙ 1 (t)}, = Re{−µ11X

(2.18d)

where the added mass coefficient µ11 may be computed from the evanescent eigenmodes only, and the radiation damping coefficient λ11 may be computed from the propagating eigenmode only. The average power may be computed from λ11 (Sω)2 . −F1 X˙ 1 t = 2

(2.19a)

Equating (2.19a) to (2.16d) yields  λ11 =

A1 S1

2

ρBh CG , ko h

(2.19b)

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that relates the radiation damping coefficient to the square of the ratio of the radiated wave amplitude to the amplitude of the wavemaker displacement. Moment. For the Type I wavemaker of width B, the dynamic pressure moment on one side only of the wavemaker may be computed from the real part of  
 −b/h 

d z 2 Cn M5 (t) = Re iρωBh 1+ − h h −1+d/h n=1   z  × Ψn (Kn , z/h)d exp −(ωt + ν) h = −M5 cos(ωt + ν − α5 ),

(2.20a)

and the static component of the pressure moment on the fluid side only is       2   3  2  3 2 d d b ρgBh3 b d b −3 . 1− 1− +2 +3 − Ms = 6 h h h h h h (2.20b) The pressure moment M5 (t) in (2.20a) may be separated linearly into a propagating and an evanescent component that are related to the rotational velocity and acceleration from the real part of1   Sω 2 M5 (t) = −Re µ55 −i ∆(1 + ∆b /∆)    Sω + λ55 −i exp −i(ωt + ν) , ∆(1 + ∆b /∆) ¨ 5 (t) − λ55 Θ ˙ 5 (t), M5 (t) = −µ55 Θ

(2.21a)

where µ55 = ρBh4

I 2 (α, β, b, d, κn ) n , κn h n=2

λ55 = ρωBh4

I12 (α, β, b, d, k) . kh

(2.21b)

(2.21c)

2.3. Circular Wavemaker Havelock5 applied Fourier integrals to develop a theory for surface gravity waves forced by circular wavemakers in water of both infinite and finite depth. The fluid motion may be obtained from the negative gradient of a scalar velocity potential Φ(r, θ, z, t) according to q(r, θ, z, t) = −∇Φ(r, θ, z, t),

(2.22a)

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where the 3D gradient operator ∇(•) in polar coordinates is  ∇(•) =

∂ er + ∂r

  1 ∂ ∂ eθ + e3 (•). r ∂θ ∂z

(2.22b)

The total pressure field P (r, θ, z, t) may be computed from the unsteady Bernoulli equation in polar coordinates according to P (r, θ, z, t) = p(r, θ, z, t) + pS (z)   ∂Φ(r, θ, z, t) 1 − |∇Φ(r, θ, z, t)|2 + Q(t) − ρgz, =ρ ∂t 2

(2.22c)

where Q(t) = the Bernoulli constant, and the free surface elevation η(r, θ, t) for zero atmospheric pressure according to 1 η(r, θ, t) = g



Q(t) ∂Φ(r, θ, η, t) 1 − |∇Φ(r, θ, η, t)|2 + ∂t 2 ρ

r ≥ b + ξ(θ, η, t);

 ;

z = η(r, θ, t).

(2.22d)

The scalar velocity potential Φ(r, θ, z, t) must be a solution to the continuity equation ∇2 Φ =

1 ∂ r ∂r

 ∂Φ 1 ∂2Φ ∂2Φ r + 2 + = 0, ∂r r ∂θ2 ∂z 2

r ≥ b + ξ(θ, z, t);

0 ≤ θ ≤ 2π;

−h ≤ z ≤ η(r, θ, t),

(2.23a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

r ≥ b + ξ(θ, −h, t);

0 ≤ θ ≤ 2π;

z = −h.

(2.23b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):   ∂Φ dQ ∂ 1 ∂ 2Φ + g − − ∇Φ · ∇ |∇Φ|2 + = 0; ∂t2 ∂z ∂t 2 dt r ≥ b + ξ(θ, η, t);

0 ≤ θ ≤ 2π;

z = η(r, θ, t).

(2.23c)

Kinematic WaveMaker Boundary Condition (KWMBC): ∂Φ ∂ξ 1 ∂Φ ∂ξ ∂Φ ∂ξ + − − = 0; ∂r ∂t r2 ∂θ ∂θ ∂z ∂z

r = ξ(θ, z, t);

−h ≤ z ≤ η(b, θ, t).

(2.23d)

Two types of circular cylindrical wavemaker displacements ξ(θ, z, t) may be analyzed, viz., amplitude-modulated (AM) and phase-modulated (PM) wavemakers.

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The distinction between these two types is in the azimuthal θ dependency of the wavemaker displacement ξ(θ, z, t) from its mean position r = b, given by   cos mθ sin(ωt + ν) mS (2.23e) χ(z/h) . ξ(θ, z, t) = (∆/h) sin(ωt + ν + mθ) (2.23f) Kinematic Radiation Boundary Condition (KRBC):   ∂ ± iKn Φ(r, θ, z, t) = 0; r → ∞. √ lim |Kn r|→+∞ ∂r

(2.23g)

Finally, physically realizable solutions to (2.23a) must be periodic in θ; i.e., Φ(r, θ, z, t) = Φ(r, θ + 2π, z, t).

(2.23h)

1

The dimensional WMBVP may be scaled and linearized by expanding the variables in perturbation series with a dimensionless perturbation parameter ε = kA. A scalar radiated velocity potential ϕ(r, θ, z) may be defined by the real part of Φ(r, θ, z, t) = Re{ϕ(r, θ, z) exp −i(ωt + ν)}.

(2.24)

A linearized WMBVP may be obtained by setting the dimensionless parameter kA = ε = 0 and by requiring that kh = O(1). This linearized WMBVP is ∇2 ϕ(r, θ, z) = 0;

b ≤ r < +∞; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

∂ϕ(r, θ, z) = 0; ∂z

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = −h,

∂ϕ(r, θ, z) − ko ϕ(r, θ, z) = 0; ∂z  √

lim

|Kn r|→+∞

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = 0,  ∂ − iKn ϕ(r, θ, z) = 0, ∂r

∂ϕ(r, θ, z) ∂ξ(θ, z, t) exp −i(ωt + ν) = − ; ∂r ∂t  η(r, θ, t) = Re

(2.25b)

(2.25c)

(2.25d)

r = b; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

 −iωϕ(r, θ, z) exp −i(ωt + ν) ; g

(2.25e) b ≤ r < ∞; 0 ≤ θ ≤ 2π; z = 0,

P (r, θ, z, t) = {p(r, θ, z, t)} + ps (z) = Re{−iωρϕ(r, θ, z) exp −i(ωt + ν)} − ρgz, ϕ(r, θ, z) = ϕ(r + λ, θ, z);

(2.25a)

ϕ(r, θ, z) = ϕ(r, θ + 2π, z).

(2.25f) (2.25g) (2.25h,i)

The specified wavemaker shape function χ(z/h) is valid for either a doublearticulated piston or hinged circular AM or PM wavemaker of variable draft that

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37

Definition sketch for circular wavemaker.

is shown in Fig. 2.2 is identical to (2.7) for a 2D planar wavemaker with the dimension b replaced with a and the stroke S replaced with the azimuthal stroke m S. The solution to the WMBVP (2.25) may be compactly expressed by the following orthonormal eigenseries: m ϕ(r, θ, z)

=

∞

(1) Cmn Ψn (Kn , z/h)Hm (Kn r)MA(P ) (mθ),

(2.26a,b)

n=1

where the azimuthal mode function is  cos mθ ; MA(P ) (mθ) = exp −imθ

m ≥ 0 and integer,

(2.26c,d)

and where (2.26a) represents an AM wavemaker; (2.26b) represents a PM wavemaker; Ψn (Kn , z/h) = the orthonormal eigenseries defined in (2.11); (1) Hm (Kn r) = the Hankle function of the first kind. When K1 = k and Kn = iκn for n ≥ 2 and integer, (1) Hm (iκn r) = Jm (iκn r) + iYm (iκn r) =

2 −(m+1) i Km (κn r), π

(2.26e)

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where Km (•) = the Modified Bessel (or Kelvin) function of the second kind of order m. The coefficients Cmn may be computed by expanding the KWMBC in an eigenseries following the procedure in (2.14) and obtaining Cmn = −

m Sj hω

In (α, β, a, d, Kn )

(1) Kn ∆ Ln (Hm (Kn b))

Ln (Zm (ζn )) =

;

n ≥ 1 and integer,

(2.26f)

dZm (ζn ) 1 = {Zm−1 (ζn ) − Zm+1 (ζn )}; dζn 2

(1) (ζn ). Zm (ζn ) = Jm (ζn ), Ym (ζn ), Km (ζn ), Hm

(2.26g)

The solution to (2.25) is given by the real part of the following eigenseries expansion: m Φj (r, θ, z, t)

[m Sj hω]

=

m Φpj (r, θ, z, t)

+ m Φej (r, θ, z, t) [m Sj hω]

  (1)   I1 (α, β, a, d, k) Ψ1 (k, z/h)Hm (kr)    (1)  k∆   L1 (Hm (kb))   = −Re  ∞ 

In (α, β, a, d, κn ) Ψn (κn , z/h)Km (κn r)       +  κn ∆ Ln (Km (κn b)) n=2       −i(ωt+ν) . (2.27a,b) × MA(P ) (mθ)e      Because the asymptotic behavior of the evanescent eigenseries Km (κn r) depends on the mode m(1) , it is not possible to specify a minimum distance from the wavemaker equilibrium boundary at r = b where the evanescent eigenvalues are less than 1% of their value at the circular wavemaker boundary. The wave field must be computed far away from the wavemaker, and it is understood that far away must be computed uniquely for each radial mode m for either an AM or PM circular wavemaker. The evaluation of the power, forces, and moments, and added mass and radiation damping coefficients for both AM and PM circular wavemakers are given by Hudspeth.1 2.4. Directional Wavemakers Directional wavemakers are vertically segmented wavemakers that undulate sinuously and, consequently, are also called snake wavemakers. Segmented directional wavemakers may be driven either in the middle of each vertical segment or at the joint between vertical segments. Because of these two methods of wave generation, parasitic waves are formed along the wavemaker due to either the discontinuity of

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the wavemaker surface (middle segment driven) or of the derivative of the wavemaker surface (joint driven). A dimensional scalar spatial velocity potential ϕ(x, y, z) may be defined by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.28a)

The dimensional linear fluid dynamic pressure field p(x, y, z, t) and 3D fluid velocity vector field may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) , ∂t

q(x, y, z, t) = −∇3 Φ(x, y, z, t), ∇3 (•) =

∂(•) ∂(•) ∂(•) ex + ey + ez . ∂x ∂y ∂z

(2.28b) (2.28c) (2.28d)

The dimensional WMBVP for directional waves is given by ∇23 ϕ(x, y, z) = 0;

x ≥ 0;

B ≤ y ≤ +B;

−h(x, y) ≤ z ≤ 0,

(2.29a)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.29b)

∂ϕ = 0; x ≥ 0; y = ±B; −h(x, y) ≤ z ≤ 0, ∂y  x = 0, ∂Φ(x, y, z, t) ∂ξ(y, z, t)  =− ; −B ≤ y ≤ +B,  ∂x ∂t −h(0, y) ≤ z ≤ 0,  lim

x→+∞

 ∂ − iKn ϕ(x, y, z) = 0, ∂x

∂ϕ(x, y, z) = −∇2 ϕ(x, y, z) · ∇2 h(x, y); z = −h(x, y), ∂z   2 ∂ ∂2 (•), ∇22 (•) = , ∂x2 ∂y 2  ˆ z) U(y, ξ(y, z, t) = Re i [∆U (y, a)][∆U (z, b, d)] exp −i(ωt + ν) , ω  ˆ (y, z) = U

Sω ∆/ho

(2.29c)

(2.29d)

(2.29e)

(2.29f) (2.29g)

(2.29h)

Γ(y)χ(z/ho ),

∆U (y, a) = U (y + a− ) − U (y − a+ ), ∆U (z, b, d) = U (z + h − d) − U (z + b),

(2.29i)

(2.29j)

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Fig. 2.3.

Definition sketch for rectangular directional wave basin.

where ko = ω 2 /g and where a± denotes the (possibly nonsymmetric) transverse ends of the directional wavemaker in the transverse y-direction in Fig. 2.3. The solution to the WMBVP in (2.29) is given by the following set of orthonormal eigenfunctions: ϕ(x, y, z) = i

g

ζn (x, y)Υn (Kn , z/h), ω n=1

  Ψ1 (k, z/h) ; Ψ1 (k, 0) Υn (Kn , z/h) =  Ψ1 (κn , z/h); ko h = Kn h tanh Kn h = 0;

(2.30a)

n = 1,

(2.30b)

n ≥ 2,

(2.30c)

n = 1, 2, 3, . . . ,

(2.30d)

where K1 = k and Kn = +iκn for n ≥ 2 and Ψn (•) is defined in (2.11). The orthonormal eigenfunctions (2.30b) and (2.30c) are applicable strictly only for constant depth wave basins; however, they may be applied to slowly varying depth wave basins if (2.30b) and (2.30c) are considered to be evaluated only locally over relatively small horizontal length scales (e.g., several wavelengths λ), where the depth may be considered to be locally equal to a constant by a Taylor series expansion of the depth.1 Substituting (2.30a) into (2.29a) yields the following 2D Helmholtz equation: ∇22 ζ(x, y) + Kn2 ζ(x, y) = 0;

x ≥ 0;

−B ≤ y ≤ +B.

(2.31)

Alternatively, for wave basins with mildly sloping bottoms, the mild slope equation may be applied according to1 ∇2 • (CCG ∇2 ζ(x, y)) + Kn2 CCG ζ(x, y) = 0,

(2.32)

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where the wave group velocity CG is given by (2.16e). If the product CCG is a constant, (2.32) reduces to the 2D Helmholtz equation (2.31). Applying the WKBJ approximation1 for the x-dependent solution in the method of separation of variables to (2.32) yields the following solution1 :   g Φ(x, y, z, t) = Re i ζ(x, y)Υn (Kn , z/h) exp −i(ωt + ν) ω    M   g

[C(x)CG (x)]x=0     Ξm (µm , y/B) Amn i  ω m=0 n=1 C(x)CG (x) = Re 
x        × Υn(Kn , z/h) exp i Qmn dξ exp − i(ωt + ν)  (2.33a)

Ξm (µm , y/B) =

cos µm B(y/B − 1) ; Mm 

Qmn =

Mm = 1 + δm0 ;

Kn2 − µ2m ;

m > 0;

µm =

mπ 2B

K1 = k > µ2m .

(2.33b–d)

(2.33e)

The coefficients Am may be computed from (2.29d) by expanding the wavemaker shape function in orthonormal eigenfunctions1 and are given by k

Amn

S Qmn (∆/hx0 )

= −Ψn (Kn , 0)In (α, β, b, d, k)
×

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B).

(2.34)

2.4.1. Full-draft piston wavemaker The prescribed transverse y-component of the snake displacement of a full-draft (b = d = 0) piston (α = 0 and β = 1) wavemaker may be expressed as Γj (qj , y/B) =

+∞

c˜j exp i[qj B(y/B + νy )],

(2.35a)

j=−∞

where the coefficients c˜j may be computed from the integral in (2.34) by
cmj = =

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B)

Ra+ ,a− + iIa+ ,a− . mB(qj2 − µ2m )(1 + δm0 )

(2.35b)

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If the full-draft piston snake wavemaker spans the entire width of the wave basin so that a± = ±B, then (2.35b) reduces to the integral in (2.34) and  4qj B cmj =

sin[qj B(νj − 1)] + (−1)m sin[qj B(νj + 1)] −i{cos[qj B(νj − 1)] + (−1)m cos[qj B(νj + 1)]} ((qj B − (mπ)2 )2 )(1 + δm0 )

.

(2.35c)

Values for cmj for (possibly nonsymmetric) values for a± are given by Hudspeth.1

2.5. Sloshing Waves in a 2D Wave Channel A long rectangular wave channel with a horizontal flat bottom, two rigid vertical sidewalls, and a wavemaker may generate either 2D, long-crested progressive waves or two types of transverse waves, viz., (1) sloshing waves that are excited directly by transverse motion of the wavemaker or (2) cross waves that are excited parametrically by the progressive waves at a sub-harmonic of the wavemaker frequency. The WMBVP for 3D sloshing waves is identical to (2.5) for planar 2D wavemakers except for the KWMBC at x = 0 and an additional kinematic boundary condition on the sidewalls of the 2D wave channel at y = ±B in Fig. 2.4. The kinematic and dynamic wave fields may be computed from a dimensional 3D scalar

Fig. 2.4.

Definition sketch for a sloshing wave channel.

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velocity potential Φ(x, y, z, t). The fluid velocity q(x, y, z, t) may be computed from the negative directional derivative of a scalar velocity potential by q(x, y, z, t) = −∇Φ(x, y, z, t). (2.36a) The dimensional fluid dynamic pressure field p(x, y, z, t) may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) . ∂t

(2.36b)

A spatial velocity potential ϕ(x, y, z) may be defined by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.36c)

The WMBVP for sloshing waves is given by the following: ∇2 ϕ(x, y, z) = 0;

x ≥ 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0,

(2.37a)

∂ϕ(x, y, −h) = 0; x ≥ 0; −B ≤ y ≤ +B; z = −h, ∂z

(2.37b)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.37c)

∂Φ(x, y, z, t) ∂ξ(y, z, t) =− ; x = 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0, ∂x ∂t ∂ϕ = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y  ∂ lim − iKn ϕ(x, y, z) = 0, x→+∞ ∂x     x ≥ 0, −iω −B ≤ y ≤ +B, Φ(x, y, 0, t) ; η(x, y, t) = Re  g z = 0.

(2.37d)

(2.37e)



A solution to (2.37) is given by the following eigenfunction expansions1 : 

˜ Φ(x, y, z, t) = Re ψn (x, y)Ψn (Kn , z/h) exp −i(ωt + ν) ,

(2.37f)

(2.37g)

(2.38a)

n=1

ϕ(x, y, z) = ψ˜n (x, y)Ψn (Kn , z/h), 

η(x, y, t) = Re ζn (x, y) exp −i(ωt + ν) ,

(2.38b) (2.38c)

n=1

ω ζn (x, y) = −i ψ˜n (x, y)Ψn (Kn , 0), g

(2.38d)

g ζn (x, y) , ψ˜n (x, y) = i ω Ψn (Kn , 0)

(2.38e)

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where ζn (x, y) is sometimes referred to as a displacement potential. The scalar potential (2.38a) may be expressed from (2.38d) and (2.38e) as 

g Ψn (Kn , z/h) Φ(x, y, z, t) = Re exp −i(ωt + ν) . (2.39) i ζn (x, y) ω Ψn (Kn , 0) n=1 The WMBVP may be expressed in terms of a displacement potential ζn (x, y) by   x ≥ 0, 2 2 ∂ ζn (x, y) ∂ ζn (x, y) 2 −B ≤ y ≤ +B, (2.40a) + + K ζ (x, y) = 0; n n  ∂x2 ∂y 2 −h ≤ z ≤ 0.   x = 0,

∂ζn (x, y) Ψn (Kn , z/h) ω (2.40b) = i U (y, z) B ≤ y ≤ +B,  ∂x Ψn (Kn , 0) g n=1 −h ≤ z ≤ 0.   ∂ (2.40c) lim − iKn ζn (x, y) = 0. x→+∞ ∂x ∂ζn (x, y) = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y

(2.40d)

where (2.40a) is the 2D Helmholtz equation.9,10 Because the boundary conditions are prescribed on boundaries that are constant values of (y,z), a solution to the WMBVP (2.40) may be computed by the method of separation of variables and is given by1 Φ(x, y, z, t)                   g = Re i  ω                

 M

 Ψ1 (k, z/h)   Cm1 Υm (y/B)  exp iPm1 x   Ψ1 (k, 0)  m=0    

 Ψ1 (k, z/h)   exp + Cm1 Υm (y/B)    Ψ (k, 0)                 

                 

             1 m=M+1 exp −i(ωt + ν) ,   − Ξm1 x          

  Ψn(κn , z/h)    exp + Cmn Υm (y/B)       Ψ (κ , 0) n n  m=0 n=2     − Qmn x (2.41a)

where Υm (µm , y/B) =

cos µm B(y/B − 1) ; Mm µm =

mπ , 2B

2 Mm = 1 + δm0 ;

(2.41b)

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Pm1 =

K12 − µ2m = Ξm1 =



k 2 − µ2m ;

 µ2m − k 2 ;

Q2mn = µ2m − Kn2 ; n ≥ 1: K1 = k > µm : Qm1 = i

k < µm

45

k > µm ; m ≤ M,

k < µm ; m ≥ 0;

(2.41d)

n ≥ 1,

(2.41e) m ≤ M,

n = 1: k = µm : Qm1 = 0  : Qm1 = µ2m − k 2 = Ξm1 > 0; m > M, 

(2.41c)

m > M,

 k 2 − µ2m = iPm ;

n ≥ 2 : Kn = iκn : Qmn =

FA

µ2m + κ2n > 0,

(2.41f)

(2.41g) (2.41h)

where M is the maximum integer value for m in order for µm < k. Substituting (2.41a) into (2.40b) yields the following solutions for Cmn 1 :

  ω Ψ1 (Kn , 0) +1  y  0  z   z  z y d d U y, Ψ1 K n , Υ m µm ; Cm1 = g Pm1 B −1 h h h B −1

Cm1 = −i

Cmn = −i

ω Ψ1 (Kn , 0) g Ξm1 ω Ψn (Kn , 0) g Qmn




+1

d

y B

−1




+1

d

y

−1

B




m≤M (2.42a) 0   z   z  z y U y, Ψ1 K n , Υm µm , ; d h h h B −1




m>M +1 (2.42b) z   z  0   z y U y, Ψn K n , Υ m µm , ; d h h h B −1 m ≥ 0, n ≥ 2.

(2.42c)

The first three transverse eigenmodes are illustrated in Fig. 2.5. 2.6. Conformal Mappings Conformal and domain mappings are applications of complex variables to solve 2D boundary value problems. Conformal mapping is an angle preserving transformation that will compute exact nonlinear solutions for surface gravity waves of constant

Fig. 2.5.

First three transverse eigenmodes in a 2D wave channel.

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form that may be treated as a steady flow following a Galilean transformation from a fixed inertial coordinate system to a noninertial moving coordinate system. Domain mapping is a transformation of the wavemaker geometry into a fixed computational domain where a solution may be computed efficiently. 2.6.1. Conformal mapping1 Conformal mapping of the WMBVP provides a global solution that accurately accounts for the singular behavior at all irregular points. The irregular points in the physical wavemaker domain are transformed into both weak and strong singular kernels in a Fredholm integral equation. The two irregular points on the WMBVP boundary are located at (1) the intersection between the free-surface and the wavemaker boundary and (2) the intersection between the horizontal bottom and the wavemaker boundary. These two irregular points exhibit integrable weakly singular kernels. The far-field radiation boundary exhibits a strongly singular kernel and significantly affects the solution. The irregular frequencies3,4 are included in the global solution by the Fredholm alternative. A theory for the planar WMBVP computes a global solution by a conformal mapping of the physical wavemaker boundary to a unit disk that includes the motion of an inviscid fluid near irregular points that are illustrated in Fig. 2.6. A numerical solution to Laplace’s equation in a transformed unit disk may be computed from a Fredholm integral equation. The WMBVP defined by (2.5) is transformed into complex-valued analytical functions where the complex-valued coordinates are defined as z = x + iy. The coordinates for the semi-infinite wave channel in Fig. 2.1(a) must be transformed to complexvalued coordinates z. The fluid velocity q(x, y, t) and dynamic pressure p(x, y, t) may be computed from a scalar velocity potential Φ(x, y, t) according to q(x, y, t) = −∇Φ(x, y, t);

p(x, y, t) = ρ

∂Φ(x, y, t) . ∂t

(2.43a,b)

The WMBVP and Type I wavemaker shape function are given by (2.5)–(2.7). There are both Irregular (I) and Regular (R) points at the intersections between the Smooth (S) and Critical (C) boundaries B1 and B2 in the WMBVP as illustrated in Fig. 2.6 where these two boundary intersection points are identified as P1 and P2 . The classification of the boundary points P1 and P2 in Fig. 2.6 depends on (1) the boundary conditions ϕi (Pj ) and (2) the continuity of the boundaries Bm and their derivatives where i, j, and m = 1 or 2. A conformal mapping of the semi-infinite wave channel strip in the physical plane will yield a Fredholm integral equation,6,7 where these critical points may be transformed to singular points that are integrable over a smooth continuous mapped boundary. 2.6.1.1. Conformal mapping to the unit disk 1 Two conformal mappings are: (i) the physical z-plane to a semi-circle in the Z-plane shown in Fig. 2.7; and (ii) a semi-circle in the upper Z-plane to a unit disk in the

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Fig. 2.6. Combinations of Irregular (I) and Regular (R) boundary points P1 and P2 between Smooth (S) and Critical (C) boundaries B1 and B2 intersections in the WMBVP.8

Fig. 2.7. Mapping of the semi-infinite strip in the lower half x–y-plane in the physical z-plane to the upper half X–Y -plane in the Z-plane.8

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Fig. 2.8.

Mapping of the upper half-plane in the Z-plane to the unit disk in the Q-plane.

Q-plane shown in Fig. 2.8. The Schwarz–Christoffel transformation dz C1 √ =√ dZ Z +1 Z −1

(2.44a)

may be integrated to obtain z = x + iy  h = − Log[−Z − Z 2 − 1], π

(2.44b)

where the Log[•] denotes the principal value of Log[•] and the argument of the Log[•] is −π ≤ arg < π. Inverting (2.44b) yields Z = X + iY = − cosh(πz/h),

(2.44c)

where ! πy " cos , h h ! πx " ! πy " Y = − sinh sin . h h

X = − cosh

! πx "

(2.44d) (2.44e)

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The radiation boundary in the z-plane transforms to a semi-circle in the Z-plane by     1 2πy 2πx cos + cosh , (2.44f) R2 = X 2 + Y 2 = 2 h h  πx   πy  Y tan θ = = tanh tan . (2.44g) X h h Details of the transformation of the WMBVP are given by Hudspeth.1 2.6.1.2. Mapping Z-plane to a unit disk The upper-half-plane of the Z-plane may be mapped into a unit disk shown in Fig. 2.8 by the following bilinear transformation:   i−Z Q = ξ + iζ = exp(iθ0 ) , (2.45a) i+Z that may be inverted to obtain  1−Q , Z = X + iY = i 1+Q 

(2.45b)

and the mapping function coordinates are ξ=

1 − X2 − Y 2 , X 2 + (Y + 1)2

ζ=

2X , X 2 + (Y = 1)2

(2.45c,d)

that may be transformed into the cylindrical coordinates for the unit disk in Fig. 2.8 by   2X (X 2 + Y 2 − 1)2 − 4X 2 ζ . = arctan r2 = ξ 2 +ζ 2 = ;  = arctan (X 2 + (Y + 1)2 )2 ξ 1 − X2 − Y 2 (2.45e,f) Details of the transformation are given by Hudspeth.1 A numerical solution to the transformed WMBVP may be computed from the following Fredholm integral equation1 :   
+π  ˆ r , ) ∂G(r, r ˆ , , ) ˆ ∂ Φ(ˆ ˆ 2π ˆ ˆ r , ) ˆ + G(r, rˆ, , ) ˆ rˆd, ˆ π Φ(r, ) = − −π Φ(ˆ ∂ˆ r ∂ˆ r (2.46a) where G(r, rˆ, , ) ˆ = − ln[ρ(r, rˆ, , )]; ˆ

ρ2 (r, rˆ, , ) ˆ = r2 − 2rˆ r cos( ˆ − ) + rˆ2 . (2.46b,c)

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Fig. 2.9. Nodal points on the unit disk in the Q-plane and the corresponding nodal points on the wavemaker in the physical z-plane.8

Numerical solutions to (2.46a) may be computed by discretizing the unit disk boundary as shown in Fig. 2.9. The numerical details regarding the evaluation (2.46a) at the two weakly singular irregular points at B and D in the physical z-plane in Fig. 2.7 and the strongly singular point at ±π that is the vertical radiation boundary A–E at +∞ in the physical z-plane in Fig. 2.7 are tedious.8 Global numerical solutions may be computed for both the linear and the nonlinear WMBVPs. 2.6.1.3. Conformal mapping to the unit disk 2 The wavemaker geometry shown in Fig. 2.10 is mapped to the unit disk by two transformations. The WMBVP is given by (2.5) and the WM shape function is χ(y/h) = [α(y/h) + β][U (y/h + 1 − b0 /h) − U (y/h + a0 /h)].

(2.47)

In order to transform the wavemaker geometry to a Jacobian elliptic function, it must be rotated and translated as shown in Fig. 2.11. The 90◦ rotation to the z  -plane is given by z  = x + iy  = iz = −y + ix.

(2.48a)

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Fig. 2.10.

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51

WMBVP11 with the six critical boundary points at a–a0 –b–b0 –c–d.

Fig. 2.11. Rotation and translation of the physical wavemaker rectangular strip in the z-plane to the w-plane.11

The horizontal shift to the left in the z  -plane is given by z  = x + iy  = w − h/2 = −y − h/2 + ix.

(2.48b)

In order to map the WM geometry in the z-plane to the semi-circle in the Z-plane shown in Fig. 2.12 as a Jacobian elliptic function, the rotated and translated strip in the z  -plane must have the dimensions of −K ≤ ξ ≤ +K and 0 ≤ ζ ≤ K  , where K = h/2 and K  = 3h = 6K. This requires a coordinate amplification given by 2K  (x + iy  ) h  h 2K −y − + ix . = h 2

w=

(2.48c)

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Fig. 2.12.

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Mapping of the wavemaker semi-circle in the Z-plane to the unit disk in the Q-plane.11

The Schwarz–Christoffel transformation from the w-plane to the Z-plane is dw = 

ˆ CkdZ (a − Z)(b − Z)(c − Z)(d − Z)

.

(2.48d)

The following change of variables: ˆ dZ = adZˆ : κ = a/c; Cˆ = c, Z = aZ; modifies (2.48d) to the following Jacobian elliptic function:
Z dZˆ = sn−1 [Z, κ], w= 1 [(1 − Zˆ 2 )(1 − κ2 Zˆ 2 )] 2 0

(2.48e)

where sn[Z, κ] = the Jacobian elliptic function of modulus κ or sine amplitude function.9 Define kˆ = sn−1 [1, κ],

(2.48f)

and the mapping of the rectangle {x1 , x2 ; y1 , y2 } = {0, 3h; 0, h} is given by Z = X + iY "   # $ % & ! ˆ  sn − 2K  y + h2 , κ dn 2Kx h h ,k " ! =  1 − dn2 #− 2K $y + h % , κ& sn2 2Kx , kˆ  h 2 h ! " ! "  # $ % & # $ % & h 2K h 2Kx ˆ 2Kx ˆ   cn − 2K y + , κ dn − y + , κ , sn , k cn , k h 2 h 2 h h " ! , +i # 2K $ % &   h 2Kx ˆ 2 2 1 − dn − h y + 2 , κ sn h ,k (2.48g) where sn[•, •] in the copolar half-period trio in (2.48g) is defined in (2.48e), and cn[•, •] and dn[•, •] are defined by cn 2 [•, •] = 1−sn 2 [•, •], dn 2 [•, •] = 1−κ2 sn2 [•, •]. The mapping from the semi-circle in the Z-plane to the unit disk in the Q-plane

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Fig. 2.13.

53

Transformed boundary conditions mapped to arcs on the perimeter of the unit disk.11

is shown in Fig. 2.12; and the mapping to the unit disk in the Q-plane is shown in Fig. 2.13. The mapping of the Z-plane to the Q-plane is given by Q=

ˆ )2 + 2iZ(1 − α ˆ )2 i−Z −α ˆ (i + Z) (1 − α ˆ )2 − Z 2 (1 + α , = i+Z −α ˆ (i − Z) (1 − α ˆ 2 ) + Z 2 (1 + α) ˆ 2

(2.49a)

where −1 < α ˆ < +1. Changing variables to circular cylindrical coordinates by (1 − α ˆ 2 ) − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ )2 , (1 − α ˆ )2 − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ2 )   2X(1 − α ˆ2) , θ(X, Y ) = arctan (1 − α) ˆ 2 − (X 2 + Y 2 )(1 + α) ˆ 2

R2 (X, Y ) =

(2.49b) (2.49c)

the unit disk may be transformed into functions of the copolar trio of Jacobian elliptic functions. The transformed WMBVP in circular cylindrical coordinates is given by Hudspeth.1 A general solution to the transformed WMBVP may be written as10 ϕ(R, θ) =

N

n=0

R

n



 a ˆn ˆ cos nθ + bn sin nθ , 1 + δn0

(2.50)

where δij is the Kronecker delta function. Substituting (2.50) into the generic boundary conditions on each of the six arcs on the perimeter of the unit disk illustrated in Fig. 2.13, multiplying each of these six boundary conditions by a member of the set of the orthogonal series in (2.50), integrating over the interval of orthogonality −π ≤ θ ≤ +π yields the following matrix equation for each of the coefficients a ˆn and ˆbn AB = H.

(2.51)

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Fig. 2.14.

Physical fluid domain.12

2.7. Domain Mapping Domain mapping of the WMBVP12 follows the theory by Joseph.13 The physical fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a fixed computational fluid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank–Nicholson (C–N) method.14,15 At each iteration of the C–N method, the potential field is computed by the conjugate gradient method.15 The wavemaker motion Ξ(y/h, t) is assumed to be periodic with period T = 2π/ω, and the WMBVP with the surface tension Tˆ is given by  0 ≤ y ≤ Γ(x, t), 2 ∇ Φ(x, y, t) = ∆Φ(x, y, t) = 0; (2.52a) Ξ(y/h, t) ≤ x ≤ L. Tˆ ∂ 2 Γ(x, t) ∂Φ(x, y, t) 1 ρ ∂x2 + |∇Φ(x, y, t)|2 −  + gΓ(x, t) = 0.  2 3/2 ∂t 2 ∂Γ(x, t) 1+ ∂x

(2.52b)

∂Φ(x, y, t) ∂Γ(x, t) ∂Φ(x, y, t) ∂Γ(x, t) − + = 0; ∂y ∂x ∂x ∂t Ξ(Γ(x, t), t) ≤ x ≤ L; ∂Φ(x, y, t) = 0; ∂y ∂Φ(x, y, t) = 0; ∂x

y = Γ(x, t).

Ξ(Γ(x, t), t) ≤ x ≤ L;

x = L;

(2.52c) y = 0.

0 ≤ y ≤ Γ(L, t).

∂Φ(x, y, t) ∂Ξ(y/h, t) ∂Ξ(y/h, t) ∂Φ(x, y, t) =− + ; ∂x ∂t ∂y ∂y



x = Ξ(y/h, t), 0 ≤ y ≤ Γ(0, t).

(2.52d)

(2.52e)

(2.52f)

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The initial conditions for t = 0 are Γ(x, 0) = H(x); ∂Γ(x, t) = 0;

(2.52g)

 Ξ(x, 0) ≤ x ≤ L.

Φ(x, Γ(x, 0), 0) = 0;

(2.52h) (2.52i)

The physical fluid domain shown in Fig. 2.14 may be mapped into a dimensionless fixed rectangle of dimensions 0 ≤ ξ ≤ 1 by 0 ≤ ζ ≤ 1 by the transforms ξ=

x ; L

ζ=

y ; Γ(x, t)

τ = ωt;

Γ(x, t) , h

(2.53a–d)

P (x, y, t) , ρA2 ω 2

(2.53e,f)

γ(ξ, τ ) =

and dimensionless variables by q(ξ, ζ, τ ) = −

ϕ(ξ, ζ, τ ) =

∇Φ(x, y, t) ; Aω

Φ(x, y, t) ; Ahω

w(ζ, τ ) =

p(ξ, ζ, τ ) =

Ξ(y/h, t) ; S

ˆ= T

˜ T . ρALhω 2

(2.53g–i)

Because ζ is a function of both x and y in (2.53b), transforming partial derivatives with respect to x must be done with some care.12 Details of these lengthy transformations and the transformed WMBVP in the fixed mapped domain are given by Hudspeth.1 References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. R. T. Hudspeth, J. M. Grassa, J. R. Medina and J. Lozano, J. Hydraulic Res. 32, 387 (1994). 3. F. John, Commun. Pure Appl. Math. 2, 13 (1949). 4. F. John, Commun. Pure Appl. Math. 3, 45 (1950). 5. T. H. Havelock, Phil. Mag. 8, 569 (1929). 6. P. R. Garabedian, Partial Differential Equations (Wiley, Inc., New York, 1964). 7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). 8. Y. Tanaka, Irregular points in wavemaker boundary value problem, PhD thesis, Oregon State University (1988). 9. G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Theory and Technique (McGraw-Hill Book Co. Inc., New York, 1966). 10. R. B. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations (Dover Publications, Inc., New York, 1996). 11. P. J. Averbeck, The boundary value problem for the rectangular wavemaker, MS thesis, Oregon State University (1993). 12. S. J. DeSilva, R. B. Guenther and R. T. Hudspeth, Appl. Ocean Res. 18, 293 (1996).

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13. D. D. Joseph, Arch. Rational Mech. Anal. 51, 295 (1973). 14. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied Numerical Methods (John Wiley and Sons, New York, 1965). 15. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (SpringerVerlag, 1984).

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Chapter 3

Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] The Wiggins–Holmes extension of the generalized Melnikov method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to nonautonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in order to demonstrate that cross waves are chaotic. The GMM is a global perturbation analysis about a manifold of fixed points that are connected by separatrices for higher dimensional nonlinear dynamical systems. The Luke Lagrangian, density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates that are the time-dependent components of three velocity potentials that represent three standing waves. The Hamiltonian for these cross waves is homomorphic to the Hamiltonian for a parametrically excited pendulum that is an example of a Floquet oscillator that may be approximated by the Mathieu equation. Neutral stability curves measured from wave tank data motivated the inclusion of dissipation in the Luke Lagrangian density function for cross waves. An integral containing a generalized dissipation function that is proportional to the Stokes material derivative of the free surface is added to the Luke Lagrangian integral so that dissipation is correctly incorporated into the dynamic free surface boundary condition. The generalized momenta are computed from the Lagrangian; and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains nonautonomous components and is transformed by three canonical transformations in order to obtain a suspended system for the application of the Kolmogorov–Arnold–Moser (KAM) averaging

57

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58

operation and the GMM. The system of nonlinear nonautonomous evolution equations determined from Hamilton’s equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic orbits are computed from the unperturbed autonomous system. The nondissipative perturbed Hamiltonian system with surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic orbit; and, consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents. A chaos diagram of the largest Liapunov exponent demonstrates regions in the Floquet forcing parameter space of possible chaotic motions; and the range of values of the Floquet parametric forcing parameter ε and of the wavemaker dissipation parameter α in the α−ε space where chaotic motions may exist.1,3

3.1. Introduction The cross waves shown in Fig. 3.1 are excited parametrically by progressive wavemaker waves at a sub-harmonic of the wavemaker frequency.1 Parametrically excited standing cross waves that oscillate in a direction transverse to the wavemaker forcing with crests perpendicular to the wavemaker are analyzed by the generalized Melnikov method (GMM) and by the Generalized Herglotz Algorithm (GHA) extended to nonautonomous systems. The cross wave wavelengths and wave modes

Fig. 3.1.

Mode 2 cross wave in a 2D wave channel.1–3

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possible are determined by the channel width L
c = 2(channel width)/n, where n = the mode number and is equal to the number of half-wavelengths across the channel. Floquet instabilities occur when the cross wave wavelengths also satisfy the frequency dispersion equation for surface gravity waves. Energy is transferred from the progressive wavemaker waves to the parametrically forced cross waves through the spatial mean motion of the free surface and their growth is due to the rate of working of the transverse stresses of the progressive wavemaker waves. The cross wave instability is shown to be homomorphic to the Floquet oscillator instability2 by constructing the neutral Floquet stability diagram shown in Fig. 3.2. The simultaneous generation of a primary resonance (ωp : ωc = 2 : 1) and a secondary resonance (ωp : ωc = 1 : 1) may be observed in the cross wave data. The criteria for horseshoe maps and for homoclinic/heteroclinic orbits are selected to test whether or not cross waves are a chaotic dynamical system. Specifically, the GMM provides local criterion for the transverse intersection of stable and unstable manifolds of the perturbed system and for the resulting chaotic motion near the unperturbed (undamped and unforced) homoclinic/heteroclinic orbits. In order to apply the GMM to a suspended dynamical system that will survive the KAM averaging operation, the nonautonomous Hamiltonian is transformed by three separate canonical transformations by applying the GHA for nonautonomous transformations. Three canonical transformations are applied in order to (1) eliminate cross product terms by a rotation of axes; (2) to eliminate two degrees of freedom, and (3) to suspend the nonautonomous terms in the wavemaker forcing component of the Hamiltonian by applying the Hamilton–Jacobian transformation. The Liapunov characteristic exponents represent an alternative criterion for diagnosing the chaotic behavior of a dynamical system by measuring the mean rate of exponential divergence of nearby trajectories, and is computed numerically for the perturbed dissipative Hamiltonian when the GMM fails to predict chaos.

Fig. 3.2. Neutral Floquet stability diagram for mode 2 cross waves (o = mode 2 cross waves and
= no cross waves).1,2

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Fig. 3.3.

Definition sketch of fluid domain.

3.2. Hamilton’s Principle for Cross Waves The fluid is assumed to be incompressible and inviscid and the flow to be irrotational. The dimensional fluid particle velocities u
and the dimensional total pressure in the fluid P
are computed from ¯
ϕ
; u
= − ∇

P
1 ¯

2 = −g
z
+ ϕ
t
− |∇ ϕ| ; ρ
2

(3.1a,b)

where ϕ
= a velocity potential; ρ
= the fluid mass density; and g
= the gravitational acceleration. The fluid domain is the 3D rectangular wave channel shown in Fig. 3.3. 3.2.1. Variational principle The Lagrangian L
with kinematic surface tension T
is given by L
=




V
(t
)


(ζ
− 1)
1 ¯

2 dSη
; |∇ ϕ | − ϕ
t
+ g
z
dV
+ T
2 ζ
S



(3.2a)

η

where η
= the free surface profile and where dSη

= ζ
dx
dy
;

¯
ϕ
|2 ]1/2 . ζ
= [1 + |∇

(3.2b,c)

Generalized Hamilton’s principle with dissipation.4 The first variational of (3.2) is
δ

t
2 t
1

L
dt
+




t
2

t
1

F




Dη
Dt




dt
= 0;

(3.3a)

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where the generalized dissipation function per unit mass density F
(Dη
/Dt
) is F






Dη
Dt






= −α

g
κ



Sη



Dη
δη

dSη
; Dt
ζ


(3.3b)

where α = a dimensionless damping parameter and κ
= the cross wave wavenumber. The dimensional boundary-value-problem may be obtained by requiring that the variation of φ
and η
vanish at the arbitrary values t
1 and t
2 , and is given by δL
+

d dt









 1

2 |∇ ϕ | −ϕ
t
+g
η
−T
∇
(∇
η
/ζ
) δη
dx
dy


−b
χη 2 z =η


2



ˆ s
) − (∇ ϕ )δϕ dV + (ϕn
+ Us
· n V
(t
) S








ηn (3.4) × δϕ dSs
+ T
δη /ζ dSη
,


δϕ
dV
=

(t )

b


Sη



where n
denotes a normal derivative. The Lagrangian in (3.2) reduces to1  1

ˆ s
ϕ
dS
ϕ ∇ ϕ dV + ϕ
+ Us
· n 2 n V
S





b


1 b + g
η
2 dx
dy
+ T
(ζ
− 1)dx
dy
. 2 −b
χη
−b
χ



1 L =− 2








2










(3.5)

η

The velocity potential ϕ
is the field variable, and the free surface η
may be expressed in terms of ϕ
by the free surface boundary conditions and the contact line conditions at the vertical sidewalls.1,3 The velocity potential ϕ
and the free surface displacement η
are assumed to be linear combinations of the progressive wavemaker wave and the parametrically excited cross wave given by the following: ϕ
(x
, y
, z
, t
) = ϕ
p (x
, z
, t
) + ϕ
c (y
, z
, t
),











η (x , y , t ) =

ηp
(x
, t
)

+

ηc
(y
, t
).

(3.6a) (3.6b)

The dimensional variables may be scaled by the following scales1,5: x
=

x ; k


y
=



h=κh; ωp
2 = g
k
τλ ; τ12 = 1 + τ ;

y ; κ


z
=

z ; κ






t ; g
κ


(3.7a–g)





ξ=k;

b=κb; τ = T
κ
2 /g
;

τλ = 1 + (τ /λ4r ); λ2r = κ
/k
;

t
= √

ω
c = g
κ
τ12 ; 2

(3.7h–m)

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 ϕ
p

=

ϕp a
p

ηc


=

g
; k


a
c ηc ;

 ϕ
c

=




χ =

ϕc a
c

a
w χ;

g
; κ


ηp
= a
p ηp ;

La
2 g
L =
c
. kκ

(3.7n–s)




Scaling by (3.7) yields the following dimensionless scaling parameters1,3: ε = κ
a
c ;

β = ωc
/ωp
;

γ = k
a
w ;

Γ = a
p /a
c ;

(3.8a–d)

and the following dimensionless field variables: ϕ(x, y, z, t) = ϕc (y, z, t) + Γλr ϕp (x, z, t); η(x, y, t) = ηc (y, t) + Γηp (x, t).

(3.9a,b)

The independent variation of ϕ
and η
vanish at arbitrary temporal values t
1 and t
2 in Hamilton’s principle (3.3) yielding the following scaled boundary value problem: 1 ϕxx + ϕyy + ϕzz = 0; γχ ≤ x ≤ ξ, y ≤ |b|, −h ≤ z ≤ εη, (3.10a) λ4r   1 z = εη, γχ ≤ x ≤ ξ, y ≤ |b|, ϕ η + ϕ η ϕz = −τ1 ηt + ε x x y y ; λ4r (3.10b)     ε 1 2 1 ϕ + ϕ2y + ϕ2z − τ1 ϕt + η = τ ηxx + ηyy − αϕz ; 2 λ4r x λ4r  γχ ≤ x ≤ ξ, (3.10c) y ≤ |b|, z = εη  ˆ = 0; ∇ϕ · n

y = |b|,

−h ≤ z ≤ εη,

γχ ≤ x ≤ ξ

z = −h,

y ≤ |b|,

γχ ≤ x ≤ ξ,

γ ϕx = − λ4r τ1 χt + γλ4r ϕz χz ; ε

x = γχ,

y ≤ |b|,

−h ≤ z ≤ εη,

(3.10d,e)

(3.10f)

plus an appropriate radiation condition at x = ξ. The free surface curvature requires a dynamical constraint that is given either by the contact line condition ηn

= 0 or by the edge constraint boundary condition η
= 0.1,3 The wavemaker perturbation forcing parameter γ is smaller than the Floquet parametric forcing parameter ε because experiments demonstrate that the standing cross wave amplitude becomes larger than the wavemaker forcing amplitude as t → ∞. The parameter ordering is 0 < γ 2 < εγ < ε2 < γ < ε < 1 or

0<

γ2 γ < γ < < 1. ε ε

(3.11a,b)

The higher order terms that will be neglected are O(ε2 ), O(εγ), and O(γ 2 ); but terms O(γ/ε) will be retained. Neglecting these higher order terms, the nondimensional Lagrangian integrals may be approximated by the following linear

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combination1,3 : Lχ = −

γτ1 2ε
b

+ −b







b −b

0

−h

[(ϕc + Γλr ϕp )χt ]x=0 dzdy

[εχt (ηc + Γηp )(ϕc + Γλr ϕp )]x,z=0 dy,


b
0 Γ [ϕpx (ϕc + Γλr ϕp )]x=ξ dzdy Lξ = − 3 2λr −b −h
b + [εϕpx (ηc + Γηp )(ϕc + Γλr ϕp )]x=ξ,z=0 dy, Lη =

1 2




−b b

−b




ξ

(3.12a)

(3.12b)

{τ1 (ηc + Γηp )t [(ϕc + Γλr ϕp )

0

+ ε(ηc + Γηp )(ϕc + Γλr ϕp )r ] + (ηc + Γηp )2 }z=0 dxdy
γ b − {χ[τ1 (ϕc + Γλr ϕp )(ηc + Γηp )t + (ηc + Γηp )2 ]}x,z=0 dy, (3.12c) 2 −b  


b   2 τ b ξ Γ2 2 Γ 2 2 2 Lτ = η + ηcy dxdy − η + ηcy dy. γχ 2 −b 0 λ4r px λ4r px −b x,z=0 (3.12d) The wavemaker forcing χ is specified to be t χ = f (z) sin β



f (z) = 1 + f (z) = 1,

z , for a full draft hinge, h for a full draft piston;

(3.13a,b)

and the deepwater cross wave and wavemaker wave are approximated by ϕc (y, z; t) = q(t) cos(y − b) exp[z], ϕp (x, z; t) = [Q1 (t) cos x +

Q2 (t) sin x] exp[z/λ2r ],

(3.14a) (3.14b)

where the variables q(t), Q1 (t), and Q2 (t) are the generalized coordinates. The freesurface displacement η is a solution to the linearized inhomogeneous boundary value problem given by L η η = [Lpη ηp − fp ] + [Lcη ηc − fc ] ≡ 0; Lpη ηp − fp = 0; ηpx = 0;

x = 0,

(3.15a)

Lcη ηc − fc = 0;

(3.15b,c)

ηcy = 0;

(3.15d,e)

y = |b|,

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where (3.15d and 3.15e) are the contact line conditions and where Lpη ηp =

τ ηp − ηp , λ4r xx

fp = αλr ϕpz − τ1 λr ϕpt ;

Lcη ηc = τ ηcyy − ηc ,

fc = αϕcz − τ1 ϕct ;

z = 0,

z = 0.

(3.15f,g) (3.15h,i)

Substituting (3.14a) and (3.15h,i) into (3.15c) yields ηc =

1 (−qα + τ1 q) ˙ cos(y − b). τ12

Substituting (3.14b) and (3.15f,g) into (3.15b) yields   −Q2 α + λ2r τ1 Q˙ 2 −Q1 α + λ2r τ1 Q˙ 1 cos x + ηp = λr τλ λr τλ  

√ λ2p τ . × sin x + 2 exp − √ x λp τ

(3.16)

(3.17)

The Lagrangian may be decomposed into the following components: L = L0 + Lε + Lγ + O(ε2 , εγ, γ 2 );

(3.18a)

where   bΓ2  2    (t) + Q22 (t)) + Q1 (t)Q2 (t)] [ξ(Q −   1   2         2 2 bτ Γ 1 4 2 2 ˙ ˙ ˙ ˙ L0 = ; + [ξλ τ ( Q (t) + Q (t)) + 4τ Q (t) Q (t)] 1 2 r λ 1 2   2λ2r τλ2             bξ 2 2   − (q (t) − q˙ (t)) 2 εbΓτ1 τ εbΓ Lε = − 3 q 2 Q˙ 2 − q qQ ˙ 2; 2λr τλ 2τ1 λ3r

(3.18b)

(3.18c)

   √ t q q˙ Γλr τ1 Q1 Γ2 τ12 2 ˙ ˙ Q1 (λr Q1 + τ Q2 ) + f1 + Lγ = −γb cos β εβ βτλ 2β    √ t Γ2 τ 2 −q˙2 + q2 Γ2 Q21 − 2 12 (λ4r Q˙ 21 + λ2r τ Q˙ 1 Q˙ 2 + τ Q˙ 22 ) + , + γb sin β λr τλ 2 (3.18d) where f1 is an integral of (3.13a or 3.13b), L0 represents the free oscillations, Lε represents the Floquet parametric forcing of the cross wave by the progressive wave, and Lγ represents the completely nonautonomous perturbation. The analogy to the nonautonomous Hamiltonian for a Floquet oscillator follows.

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3.2.2. Legendre transform The Legendre transform of the Lagrangian yields the following Hamiltonian: ˙ t), H(q, p, t) = pq˙ + P1 Q˙ 1 + P2 Q˙ 2 − L(q, q,

(3.19)

where p = (p, P1 , P2 ) are the conjugate momenta corresponding to the generalized coordinates q = (q, Q1 , Q2 ). The conjugate momenta are computed from pi =

∂(L0 + Lε ) ; ∂ q˙i

p1 = p = bξ q˙ −

i = 1, 2, 3, εbΓ qQ2 ; 2τ1 λ3r

  2τ ξ Q˙ 1 + 4 Q˙ 2 ; λr τλ   2 2 2 εbτ1 Γτ 2 bτ1 Γ λr 2τ ˙ ˙ ξ Q2 + 4 Q1 . p3 = P2 = − 3 q + 2λr τλ τλ λr τλ p2 = P1 =

bτ12 Γ2 λ2r τλ

(3.20a) (3.20b) (3.20c) (3.20d)

Inverting (3.20b–3.20d) yields q˙1 = Q˙ 1 =

p εΓqQ2 ; + bξ 2τ1 ξλ3r

λ4r τλ P1 −

Q˙ 2 =

2τ ξ P2 2 bτ1 Γ2 λ6r ξ

λ4r τλ P2 −

−

2τ ξ P1 2 2 bξτ1 Γ λ6r

(3.21a)

ετ 2 q 2 ; Γλ9r ξ 2 τ1 τλ

(3.21b)

ετ q 2 . 2Γξλ5r τ1

(3.21c)

+

Substituting (3.18), (3.20), and (3.21) into (3.19) yields the following Hamiltonian: H(q, p, t) = H0 (q, p) + Hε (q, p) + Hγ (q, p, t) + O(ε2 , εγ, γ 2 ),

(3.22a)

where the free oscillation component is H0 =

bΓ2 1 p2 (ξ(Q21 + Q22 ) + Q1 Q2 ) + bξq 2 + 2 2 2bξ +

2τ P1 P2 (P12 + P22 ) − 2 2 6 2; 2bβ 2 Γ2 ξ bτ1 Γ λr ξ

(3.22b)

the Floquet parametric forcing component is Hε =

εΓ ετ ετ 2 2 qpQ + q P − q 2 P1 ; 2 2 2τ1 ξλ3r 2τ1 Γξλ5r τ1 Γξ 2 λ9r τλ

(3.22c)

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and the nonautonomous perturbed component from the wavemaker forcing is   √   Q1 (λ2r P1 + τ P2 ) bΓλr τ1 Q1     +    f1      εβ βξλ2r   t       γ cos   √     2 β   2τ Q1 (λr P2 + τ P1 ) pq − + . Hγ = 2 6 βξ λr τλ 2βξ            √      (λ2r P1 + τ P2 )2 1 2 p2 t   2 2  −γ sin  + bq − bΓ Q1 − 2 2 2 6 β bτ1 Γ ξ λr 2 2bξ 2

(3.22d)

Although the nonautonomous Hamiltonian in (3.22d) is homomorphic to a nonautonomous Floquet Hamiltonian, a subtle but important distinction exists. The nonautonomous component in a Floquet Hamiltonian is due to parametric forcing; while, in contrast, the nonautonomous component in (3.22d) is due to external wavemaker boundary forcing that generates the progressive wave that then parametrically forces the cross waves. This sequence of external forcing to parametric forcing distinguishes cross wave parametric forcing from the simple pendulum Floquet parametric forcing. Damping forces. The scaled Hamilton’s principle (3.3) is given by1,3
δ

t2

t1


L dt =

t2

t1

3 


Di δqi dt = −

i=1

t2




t1

b −b




ξ

(αϕz δη)dxdydt,

(3.23a)

0

where D = (D1 , D2 , D3 ) is a set of generalized components of the damping force corresponding to the set of generalized coordinates q = (q, Q1 , Q2 ) and where

δη =

3   ∂η i=1

∂qi

δqi +

 ∂η δ q˙i . ∂ q˙i

(3.23b)

Integrating by parts, the last term1,3 reduces D to the canonical variables q, p   ∂ ∂η ∂η + Di = α −ϕz dxdy; i = 1, 2, 3, ϕz ∂qi ∂t ∂ q˙i −b 0      2  αεbΓ α α bξ p+ qQ2 + q, D1 = τ1 2τ12 λ3r τ12         2 α 2ατ αεbΓτ 2 α bξΓ2 2 D2 = P1 − P2 − q + Q1 , λ2r τ1 ξλ6r τ1 τλ λ9r ξτλ λ2r τλ    2  2     α αεbΓτ 2α bτ Γ2 α bξΓ2 2 P2 + q + Q1 + Q2 . D3 = τ1 λ2r 2λ5r τλ λ6r τλ2 λ2r τλ


b




ξ



(3.24a) (3.24b) (3.24c) (3.24d)

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3.3. Hamilton’s Dynamic Equations and Canonical Transformations Hamilton’s equations of motion are given by q˙i =

∂H ; ∂pi

p˙ i = −

∂H − Di ; ∂qi

i = 1, 2, . . . , N.

(3.25a,b)

If the canonical transformation Q = Q(q, p, t), P = P(q, p, t) is computed from a generating function F (u, U, t) computed from the GHA-Type I (Appendix A), then the Hamiltonian K for the new set of variables is given by K(Q, P, t) = H[q(Q, P, t), p(Q, P, t), t] +

∂F (u, U, t) ; ∂t

(3.26a)

and the transformed Hamilton’s equations of motion are N

∂K  ˜ ∂qj Q˙ i = + ; Dj ∂Pi j=1 ∂Pi

i = 1, 2, . . . , N,

N  ∂K ˜ j ∂qj ; P˙i = − − D ∂Qi j=1 ∂Qi

i = 1, 2, . . . , N,

˜ i (Q, P, t) = Di [q(Q, P, t), p(Q, P, t)]; D

i = 1, 2, . . . , N.

(3.26b)

(3.26c) (3.26d)

3.3.1. Three canonical transformations and analyses by the GMM In order to apply the GMM to the dynamical system in (3.26), canonical transformations are required first to simplify terms in H0 ; second to simplify terms in Hε ; and third to suspend nonautonomous terms in Hγ . The GHA will be applied to two of the following three canonical transformations, because the rotation of axes transformation is well known so that the GHA will not be applied for that transformation even though it is still applicable. (1) Rotation of axes. The cross product terms in (3.22b) may be eliminated by q = q˜,

p = p˜,

(3.27a,b)

˜ 1 cos θ + Q ˜ 2 sin θ, Q1 = Q

P1 = P˜1 cos θ + P˜2 sin θ,

(3.27c,d)

˜ 2 cos θ − Q ˜ 1 sin θ, Q2 = Q

P2 = P˜2 cos θ − P˜1 sin θ,

(3.27e,f)

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that satisfy the Poisson bracket requirements in Appendix A. The rotation angle θ = π/4 eliminates the cross product terms, and the transformed Hamiltonian is given in component form by ˜ε + H ˜ γ (t) + O(ε2 , εγ, γ 2 ), ˜ =H ˜0 + H H→H where

   2   bξ 2 p˜2 P˜12 bΓ ξ ˜ 2     q˜ + + Q1 +    2 2   2 2bξ 2 2bβ Γ ξ            

    2 2 ˜2 P ξ bΓ 2 ˜ ˜ , + Q2 + H0 =   2 2bβ 2 Γ2 ξ          

    2 2 2 ˜ ˜   τ ( P − P ) bΓ   1 2 2 2  ˜  ˜ + (Q2 − Q1 ) +    2 4 bτ1 Γ2 λ6r ξ 2 

(3.28a)

(3.28b)

4 ˜2 − Q ˜ 1 ) − ετ√(2τ + ξλr τλ ) q˜2 P˜1 ˜ ε = √ εΓ q˜p˜(Q H 2 2τ1 ξλ3r 2 2τ1 Γξ 2 λ9r τλ

+

ετ (−2τ + ξλ4r τλ ) 2 ˜ √ q˜ P2 , 2 2τ1 Γξ 2 λ9r τλ

(3.28c)

   ˜1 + Q ˜ 2) bΓλr τ1 (Q     √ f   1      2εβ               √ √     2 2  ˜ 2 )((λr − τ )P˜1 + (λr + τ )P˜2 )  ˜1 + Q t ( Q       γ cos     +   2     β 2βξλ   r           √ √   2 2 ˜ ˜ ˜ ˜ ˜ τ (Q1 + Q2 )((λr + τ )P2 + (−λr + τ )P1 ) p˜q˜ . Hγ = − +   2 λ6 τ   βξ 2βξ   r λ         √ √     2 2 2 2 ˜ ˜   ((λ − τ ) P + (λ + τ ) P ) bΓ   1 2 r r 2   ˜ ˜    + Q ) −  ( Q 1 2   2 2 2 6   2 t   2bτ1 Γ ξ λr       −γ sin     2   β 1 p ˜   2   + b˜ q − 2 2bξ 2 (3.28d) The first three energy square brackets [•] in (3.28b) are the action variables in the next canonical transformation. (2) Action/Angle Transformation. The three new canonical variables are given by pˆ =

bξ 2 p˜2 q˜ + , 2 2bξ

bΓ2 ξ ˜ 2 P˜12 Pˆ1 = , Q1 + 2 2bβ 2 Γ2 ξ

bΓ2 ξ ˜ 2 P˜22 , Q2 + Pˆ2 = 2 2bβ 2 Γ2 ξ

(3.29a–c)

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that satisfy the Poisson brackets. The Herglotz auxiliary functions Xi are the ratios p; X1 = q˜/˜

˜ 1 /P˜1 ; X2 = Q

˜ 2 /P˜2 . X3 = Q

(3.30a–c)

The angle variables are qˆi =

3 

q˜j

j=1

∂F ∂ p˜j + ; ∂ pˆi ∂ pˆi

i = 1, 2, 3,

(3.31a)

where qˆ = tan−1 (bξX1 );

ˆ 1 = β tan−1 (bβξΓ2 X2 ); Q

ˆ 2 = β tan−1 (bβξΓ2 X3 ). Q (3.31b–d)

The canonical transformation to action/angle variables is given by the following:  q˜ =

 ˜1 = Q

 ˜2 = Q

2Pˆ1 sin bξΓ2

2Pˆ2 sin bξΓ2





ˆ1 Q β

ˆ2 Q β

2ˆ p sin qˆ, bξ

p˜ =





2bξ pˆ cos qˆ,



 ,

P˜1 = Γβ

,

P˜2 = Γβ



2bξ Pˆ1 cos



 2bξ Pˆ2 cos

(3.32a,b)

ˆ1 Q β

ˆ2 Q β

,

(3.32c,d)

,

(3.32e,f)



that satisfy the Poisson brackets.1,3 The transformed Hamiltonian is ˜ ⇒H ˆ =H ˆ0 + H ˆε + H ˆ γ (t) + O(ε2 , εγ, γ 2 ), H     ˆ1 2β 2 τ Q 2β 2 τ 2 ˆ P1 cos − λ6r ξτ12 β λ6r ξτ12      ˆ2 ˆ1 ˆ2 ˆ1 ˆ2 Q Q Q P P 2 2 2 × Pˆ2 cos − sin + sin , β 2ξ β 2ξ β

(3.33a)



ˆ 0 = [ˆ p + Pˆ1 + Pˆ2 ] + H

(3.33b)

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    

ˆ2 ˆ2   Q Q   ˆ   P2 cos 2ˆ − cos 2ˆ q+ q−     β β   εˆ p ˆ Hε =   

   4 bξ 3 λ3r τ1    ˆ1 ˆ1   Q Q   ˆ   − cos 2ˆ q+ − P1 cos 2ˆ q−   β β εβτ pˆ −  2 bξ 5 λ9r τ1 τλ     

ˆ1 ˆ1   Q 1 Q   4   (2τ + ξλr τλ ) Pˆ1 cos − cos 2ˆ q+       β 2 β          

       ˆ2  ˆ1 Q 1 Q 4 ˆ , × q− + (2τ − ξλr τλ ) P2 cos − cos 2ˆ  2 β β                   ˆ ˆ   Q 1 Q 1   2 2   cos 2ˆ q + − cos 2ˆ q − −     2 β 2 β (3.33c)    √     ˆ1 ˆ2 bλr f1 τ1  Q Q     √ + Pˆ2 sin Pˆ1 sin       β β βε ξ                    √      2 4   ˆ   + τ )(−2τ + ξλ τ ) (λ Q   λ 1 r r   ˆ     P + cos 1     6 2     λ ξ τ β λ   t r     γ cos           β   √      2 4   ˆ + τ )(−2τ + ξλ τ )  (λ  Q λ 2   r r   ˆ   P + cos   2   6 ξ2τ  λ β    λ  r                  pˆ sin(2ˆ q)   + ˆ . Hγ = 2βξ    

        ˆ   √ −β 2   2   ˆ1 cos Q1 P (λ − τ )     r   6τ 2   λ β     r 1               2        ˆ √   Q   2   2   ˆ  γ  + τ ) cos + (λ P   2 t   r     β   − sin       ξ β       

  2         ˆ ˆ     Q Q 1 2   ˆ1 sin ˆ2 sin     + + P P        β β            2 2 + pˆ(sin qˆ − cos qˆ) (3.33d)

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The nonautonomous perturbed component in (3.33d) will not survive the KAM averaging theorem, and some of the nonautonomous terms may be suspended by applying the following Hamilton–Jacobian nonautonomous canonical transformation. (3) Hamilton–Jacobian transformation. The following nonautonomous transformations: qˆ =

t + q + Q1 , 2β

ˆ 1 = t + 2βQ1 , Q

ˆ 2 = 3t + 2β(q + Q1 + Q2 ) Q

(3.34a–c)

satisfy the Poisson brackets. The Herglotz auxiliary functions Xi = Xi (p, q) satisfy the nonzero Jacobian condition given by ∂(ˆ q, X) 4β 2 = 0. = ∂(p, q) qQ1 Q2

(3.35)

Solving (3.35) for qi = qi (p, X), substituting qi (p, X) into (3.34), and inverting yields p=

X1 ˆ 1 ); P1 = −X2 (t − Q ˆ 1 ); (2β qˆ − Q 2β 2β −X3 ˆ 2 ). P2 = (2t + 2β qˆ − Q 2β

(3.36a–c)

The generating function F (•, •, •) for the nonautonomous canonical transformation may be computed from the following indefinite integral: F (ˆ q, p(ˆ q, X, t), t)  3
Xi   ∂P1 ∂P2 ∂p
=− q
+ Q1 ∂X
+ Q2 ∂X
dXi ∂X i i i i=1 =

−1 ˆ 1 )2 + X2 (t − Q ˆ 1 )2 + X3 (2t + 2β qˆ − Q ˆ 2 )2 ]. [X1 (2β qˆ − Q 4β 2

(3.37)

Computing the three remaining conjugate momenta from pˆi = −

3  j=1

qj

∂F ∂pj − ; ∂ qˆi ∂ qˆi

i = 1, 2, 3

(3.38a)

1 Pˆ2 = P2 . 2β

(3.38b–d)

gives pˆ = p − P2 ,

1 Pˆ1 = (P1 − p), 2β

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The transformations in (3.34) and (3.38) satisfy the Poisson brackets, and the first square bracket term [•] in (3.33b) may be transformed to [ˆ p + Pˆ1 + Pˆ2 ] =

  1 P1 1− (p − P2 ) + . 2β 2β

(3.39)

The primary Floquet resonance condition is, approximately, ωc
− ωp
≈ O(ε) so that a detuning parameter Ω may be defined by     ωp
1 1− = 1 − = εΩ; 2ωc
2β

(3.40)

and the primary parametric Floquet resonance is ωp
: ωc
= 2:1. A generating function is now given by   1 ˆ 1 ˆ 1 ) + 1 P2 (2t + 2β qˆ − Q ˆ 2 ); F (ˆ q, p, t) = p −ˆ q+ P1 (t − Q Q1 + 2β 2β 2β

(3.41)

and the transformed Hamiltonian H(p, q, t) with autonomous and nonautonomous terms by ˆ − ∂F ⇒ H = H0 + H0 (t) + Hε + Hε (t) H ∂t + Hγ + Hγ (t) + O(ε2 , εγ, γ 2 ),

(3.42a)

where     βτ 1 βτ 1 1 (P P2 , + + − − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12 √ ε(p − P2 ) P1 − p Hε = εΩ(p − P2 ) +  [(2βτ 2 − ξλ4r τλ (λ2r − βτ )) cos(2q)], 4 2bβξ 5 λ9r τ1 τλ  √   γ bf1 λr τ1  2β − 1  P1 − p sin(2Q1 ) + γ Hγ = 4βξ 2ε 2ξβ 3 H0 =

× (P2 − p) sin 2(q + Q1 ),

(3.42b) (3.42c)

(3.42d)

and where the nonautonomous terms are given by Fadel.5 (4) Transformation to original variable (qorig , porig ) ⇒ (q, p). In order to compute the generalized damping forces in (3.24), the original canonical variables (qorig , porig ) must be expressed as functions of the final transformed canonical

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variables (q, p) by

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  2 t , p − P2 sin q + Q1 + bξ 2β     t , porig = 2bξ p − P2 cos q + Q1 + 2β     1 t Q1orig = √ P1 − p sin 2Q1 + β Γ 2bβξ    3t + P2 sin 2(q + Q1 + Q2 ) + , β     bβξ  t P1orig = Γ P1 − p cos 2Q1 + 2 β    3t , + P2 cos 2(q + Q1 + Q2 ) + β     1 t √ Q2orig = − P1 − p sin 2Q1 + β Γ 2bβξ    3t + P2 sin 2(q + Q1 + Q2 ) + , β      bβξ t − P1 − p cos 2Q1 + P2orig = Γ 2 β    3t + P2 cos 2(q + Q1 + Q2 ) + . β qorig =

(3.43a) (3.43b)

(3.43c)

(3.43d)

(3.43e)

(3.43f)

3.3.2. Transformed damping forces The transformed damping forces are1,3    √ t 2bβξα      p − P2 cos q + Q1 +     τ1 2β         √       2    t 2bξα     + p − P sin q + Q +   2 1 2   τ 2β   1           εα t , D1 = √ + p − P sin q + Q + 2 1  2β  2λ3r ξ βτ12                  3t     P sin 2(q + Q + Q ) +   2 1 2     β         ×              t     − P1 − p sin 2Q1 + β

(3.44a)

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    √ t bβαΓ(2τ + λ4r ξτλ )      √ P − p cos 2Q + 1 1   6 2ξ   β τ τ λ   1 λ r       √         4    3t bβαΓ(−2τ + λ ξτ ) λ   r   √ + P cos 2(q + Q + Q ) +   2 1 2   6 2ξ β   τ τ λ 1 λ r             2 2Γεατ t − 9 2 2 (p − P2 ) sin 2(q + Q1 ) + , D2 =   λr ξ τλ 2β         √     2    Γ bξ α t     √   + P − p sin 2Q + 1 1   2τ   β λ 2β   r λ       √     2      Γ bξ α 3t     + √ P sin 2(q + Q + Q ) + 2 1 2 2 β λr τλ 2β (3.44b)        t     − P − p cos 2Q + 1 1  √      β   αΓ bβξ      √          2τ      2λ   3t r 1     + P cos 2(q + Q + Q ) +   2 1 2   β                 εαΓτ t   + ) sin 2(q + Q ) + (p − P 2 1 5 D3 = , λ ξτ 2β r λ          √       t α2 Γ b(2τ − λ4r ξτλ )       √  P − p sin 2Q + + 1 1   2 6τ   β λ 2βξ   r λ          √     2 4    3t Γ b(2τ + λ ξτ ) α   λ r     √ P sin 2(q + Q + Q ) + + 2 1 2  2 6 β  λr τλ 2βξ (3.44c) that are completely autonomous and will survive the KAM averaging operation. Averaging H(q, p, t) over the dimensionless cross wave period 2π yields

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 )     βτ 1 βτ 1 1 (P P2 + + − = − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12

 √ (p − P2 ) P1 − p 2 4 2 + ε Ω(p − P2 ) +  [(2βτ − ξλr τλ (λr − βτ )) cos(2q)] 4 2bβξ 5 λ9r τ1 τλ 

 √   2β − 1 bf1 λr τ1   (P2 − p) sin 2(q + Q1 ) . P1 − p sin(2Q1 ) + +γ 4βξ 2ε 2ξβ 3 (3.44d)

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The averaged system of autonomous equations is1,3   2P1 + P2 − 3p     √ −a cos[2q] + εΩ + εa 1 3     P − p   1         sin[2Q1 ] 2 − γc2 sin[2(q + Q1 )] + α d1 , − γc1 √ q˙ = ε P1 − p               d (P − p) + d (P − p)   2 1 3 2     √ sin[2q] − αε P1 − p      4εα3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )]   p˙ = ,  α     + αd4 P2 − p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1    4γ     − P − p cos[2Q ] − 2γc (P − p) cos[2(q + Q )] c 1 1 2 2 1   ε 1 ˙ , P1 = 2  (4λr − 1)(P2 − p) − P1       +α − αεd P − p(p − P ) cos[2q] 5 1 2 2λ2r τ1   τ − λ4r ξτλ P˙ 2 = αP2 , λ6r ξτ1 τλ   (p − P2 ) sin[2Q1 ]     √ √ cos[2q] + γc + εa a 3 1   1 P1 − p ε P1 − p ˙ , Q1 =   P2 − p     sin[2q] + α2 d6 + αεd3 √ P1 − p     a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )]  , Q˙ 2 =    + α2 d7 + αεd2 P1 − p sin[2q]

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(3.45a)

(3.45b)

(3.45c)

(3.45d)

(3.45e)

(3.45f)

where ai , ci , and di are tabulated in Appendix B. 3.4. Application of the GMM and Largest Liapunov Characteristic Exponent The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped α = 0 and unperturbed γ = 0) system. The unperturbed vector field may be computed by setting the perturbation (wavemaker forcing) parameter γ = 0 and the dissipation parameter α = 0 in (3.45), and are q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos(2q), P1 − p

(3.46a)

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p˙ = 4εa3



P1 − p(p − P2 ) sin(2q),

P˙1 = P˙2 = 0,

(3.46b) (3.46c,d)

(p − P2 ) Q˙ 1 = a1 + εa3 √ cos(2q), P1 − p  Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos(2q),

(3.46e) (3.46f)

where the coefficients a1 , a2 , and a3 are defined in Appendix B. The unperturbed vector field α = γ = 0 in (3.46) is a three degrees of freedom Hamiltonian system with (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . An important consequence of the Hamilton–Jacobian canonical transformation is that the unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) is independent of Q1 and Q2 . Hyperbolic saddle points. For every (P1 , P2 ) ∈ R2 , the (q − p) components of the unperturbed vector field α = γ = 0 in (3.46) possess a hyperbolic saddle point that varies smoothly with P1 and P2 and is given by p0 (P2 ) = P2 ;

2q0 = nπ;

n = 0, 1, . . . ,

  a1 − εΩ 1 √ , cos−1 2 2εa3 P1 − P2  2 a1 − εΩ P1 ≥ P2 + . 2εa3

q0 (P1 , P2 ) =

(3.47a,b)

(3.47c)

(3.47d)

The fixed point (q0 , p0 ) is a hyperbolic saddle point if the determinant satisfies # # # ∂ q˙ ∂ q˙ # # # # ∂q ∂p # # # = (a1 − εΩ)2 − 4(P1 − P2 )ε2 a23 < 0. (3.48) # ∂ p˙ ∂ p˙ # # # # ∂q ∂p # (q=q0 ,p=p0 ) The symmetry properties of Hamiltonian systems imply that the stable and the unstable manifolds of the hyperbolic saddle point (q0 , p0 ) have equal dimensions in the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The unperturbed system α = γ = 0 has a 4D (R2 × T2 ) normally hyperbolic invariant manifold given by the union of the hyperbolic saddle points (q0 , p0 ) according to M = {(q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P1 , P2 ); p = p0 (P2 )}. (3.49) The normally hyperbolic invariant manifold M has 5D (R1 ×R2 ×T2 ) stable manifold W s (M) and unstable manifold W u (M) that coincide along the 5D heteroclinic manifold H = W s (M) ∩ W u (M) − M, where W s (M) and W u (M) are the set of

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initial conditions that approach the hyperbolic saddle points as t → ±∞ under the action of the unperturbed flow. Dynamics on M. The unperturbed vector field α = γ = 0 restricted to the normally hyperbolic invariant manifold M may be determined from1,3 P˙1 = 0;

P˙2 = 0;

Q˙ 1 = a1 ;

Q˙ 2 = a2 − a1 .

(3.50a–d)

The unperturbed vector field α = γ = 0 restricted to M has the form of two degrees of freedom, 4D phase space (T1 × R1 × T2 ), completely integrable Hamiltonian system with the Hamiltonian given by the level energy surfaces H(P1 , P2 ) = H(q = q0 , p = p0 ; γ = 0) = a1 P1 + (a2 − a1 )P2 = E,

(3.51)

where E is a constant energy that allows the phase space motion to be reduced from four dimensions (R2 × T2 ) to three dimensions. Because P1 = P2 = 0, the constant energy surfaces may be reduced to a 1D surface in the 3D constant energy space. On this 1D surface, the angular motion is parameterized by the two frequencies σ1 = a1 , σ2 = a2 − a1 .

(3.52a,b)

The angular components of the motion on the normally hyperbolic invariant manifold M are given by Q1 (t) = a1 t + Q1 (0);

Q2 (t) = (a2 − a1 )t + Q2 (0).

(3.53a,b)

The normally hyperbolic invariant manifold M is a two-parameter (P1 , P2 ) family of 2D tori shown in Fig. 3.4. A 2D torus on M may be defined by  $ (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P¯1 , P¯2 ); ¯ ¯ Υ(P1 , P2 ) = , p = p0 (P¯2 ); P1 = P¯1 ; P2 = P¯2 (3.54) where each 2D torus is invariant. The level energy surface is a family of concentric tori as shown in Fig. 3.5. Because the frequencies of motion σ1 and σ2 in (3.52) are independent of P1 or P2 , they do not change from one concentric torus to another concentric torus. The motion on the surface of the invariant torus Υ ⊂ M is quasiperiodic. When the frequency ratio σ1 /σ2 = a1 /(a2 − a1 ) is an irrational number, the motion on the surface of the 2D invariant nonresonant torus may no longer be periodic; i.e., trajectories wind densely on the surface of the torus and never close on themselves. This 2D torus has a 3D (R1 × T1 ) stable manifold W S and unstable manifold W U that coincide along the 3D (R1 × T1 ) heteroclinic manifold H defined for fixed values of P1 and P2 , as shown in Fig. 3.4. On a constant level energy surface, neither the nonresonant invariant tori nor the stable and unstable manifolds may be isolated. In addition, the 2D nonresonant invariant torus has a 2D center manifold corresponding to nonexponentially expanding or contracting directions tangent to the normally hyperbolic invariant manifold M.

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Fig. 3.4.

3D unperturbed heteroclinic manifold.1

Fig. 3.5. Motion of a phase space point for an integrable Hamiltonian system with two degrees of freedom. (a) Invariant tori in a 3D constant energy space E; and (b) the flow on a 2D torus.1,3

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Heteroclinic orbits. For fixed values of (P1∗ , P2∗ ) ∈ R2 , the (q, p) components of the unperturbed vector field α = γ = 0 in (3.46) possess a 1D heteroclinic orbit connecting the hyperbolic saddle points (q0 , p0 ). The heteroclinic orbit lies on each of the level energy surfaces defined by (3.51) and are solutions to

H(γ = 0) − (a1 P1∗ + (a2 − a1 )P2∗ ) = 0.

(3.55)

Values for q on the heteroclinic orbit are given by & %  1 q = cos−1 (a1 − εΩ)/2εa3 P1∗ − p . 2

(3.56)

In the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 ×R1 ×R2 ×T2 , the heteroclinic manifold H may be determined by substituting   2   a1 − εΩ a1 − εΩ √ √ , sin[2q] = 1 − , (3.57a,b) cos[2q] = 2εa3 P1 − p 2εa3 P1 − p into the unperturbed vector field α = γ = 0 in (3.46), and integrating to obtain1,3 √ A ph (t) = P2 + sech2 [ At] + p(0), (3.58a) B 

√ √ 1 A −1 (3.58b) − qh (t) = tan tanh( At) + q(0), 2 a1 − εΩ P1h (t) = P1 (0), P2h (t) = P2 (0),

(3.58c,d)

Q1h (t) = a1 t − qh (t) + Q1 (0) + q(0) 

√ √ 1 A −1 tanh( At) + Q1 (0), = a1 t − tan − 2 a1 − εΩ Q2h (t) = (a2 − a1 )t + Q2 (0), A = [−(a1 − εΩ)2 + (P1 − P2 )B] > 0,

(3.58e) (3.58f)

B = 4ε2 a23 ,

(3.58g,h)

π q = qn (0) = (2n + 1) ; n = 0, 1, 2, . . . . (3.58i,j) 2 The trajectories of the unperturbed system α = γ = 0 along the 5D phase space (R1 × R2 × T2 ) heteroclinic manifold H may be expressed as p(0) = 0,

Ψ(P1 ,P2 ) = {qh (t), ph (t), P1 (0), P2 (0), Q1h (t), Q2h (t)}.

(3.59)

The 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T × R × R × T is a direct product between a 4D space with coordinates (q, p, P1 , P2 ) and a 2D torus with angular 1

1

2

2

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Fig. 3.6.

Unperturbed 4D phase space.

coordinates (Q1 , Q2 ). Because P1 and P2 are constants, the motion in the 6D phase space is reduced to the four dimensions (q, p, Q1 , Q2 ) shown in Fig. 3.6. Small perturbations are anticipated to break up the geometric structure of the unperturbed system α = γ = 0 and to separate the manifolds. The behavior of the perturbed systems γ > 0 and α > 0 near the unperturbed heteroclinic manifold H is required in order to apply the GMM. The distance between the stable and unstable manifolds of any surviving invariant set in the perturbed system must be computed at a point P on the unperturbed heteroclinic manifold H. Two perturbed systems are evaluated: (1) γ > 0, α = 0 and (2) γ > 0 and α > 0. 3.4.1. Geometric structure for γ > 0 and α = 0 The perturbed vector field for nonzero wavemaker forcing γ > 0 and no dissipation α = 0 is given by q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos[2q] P1 − p

sin[2Q1 ] − γc1 √ − γc2 sin[2(q + Q1 )], ε P1 − p

    4εa3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )] ,  p˙ =  +αd4 P2 − α p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1  4γ P˙1 = − c1 P1 − p cos[2Q1 ] − 2γc2 (P2 − p) cos[2(q + Q1 )], ε

(3.60a)

(3.60b)

(3.60c)

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P˙2 = 0, (p − P2 ) sin[2Q1 ] Q˙ 1 = a1 + εa3 √ cos[2q] + γc1 √ , P1 − p ε P1 − p  Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )],

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(3.60d) (3.60e) (3.60f)

where (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The perturbed vector field is computed from 5D level energy surfaces defined by

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 ).

(3.61)

Because the perturbation is Hamiltonian, the 3D level energy surfaces are preserved. In the 4D normally hyperbolic invariant manifold of the unperturbed space, the locally stable and unstable manifolds and the flow describe the geometric structure of the perturbed phase space given by the perturbed normally hyperbolic locally invariant manifold, the locally stable and unstable manifolds, and the persistence of the 2D nonresonant invariant tori Υγ (P1 , P2 ). Persistence of M. The perturbed system γ > 0 and α = 0 possesses a 4D normally hyperbolic locally invariant manifold Mγ given by   (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :     Mγ = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ); .     p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.62)

On Mγ there are locally stable and unstable manifolds that are of equal dimensions and are close to the unperturbed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold Mγ intersects each of the 5D level energy surfaces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem.1,3 The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. KAM Theorem.1,3 The KAM theorem determines whether the recurrent motions occur on the perturbed normally hyperbolic locally invariant manifold Mγ and whether any of the two parameter families of 2D nonresonant invariant tori survive the Hamiltonian perturbation. The unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) satisfies the following nondegeneracy (or nonresonance) condition: # 2 # ∂ H # # ∂P 2 # 1 # 2 # ∂ H # # ∂P ∂P 2

1

# # # # (a1 − εΩ)2 # =− < 0. # 4(P1 − P2 )2 ∂ 2 H ## ∂P22 #(q=q0 ,p=p0 ;γ=0)

∂ 2 H ∂P1 ∂P2

(3.63)

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Most of the 2D nonresonant invariant tori Υ(P1 , P2 )) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold Mγ and are KAM tori. In the phase space of the perturbed system γ > 0 and α = 0, there are invariant tori that are densely filled with winding trajectories that are conditionally periodic with two independent frequencies σ1 and σ2 . The resulting conditionally-periodic motions of the perturbed system are smooth functions of the perturbation γ. A generalization of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds.1,3 In order to determine if chaos exists, two measurements are required in order to determine whether or not W s (Υγ ) and W u (Υγ ) intersect transversely. Melnikov integral. The distance between W s (Υγ ) and W u (Υγ ) at any point P ∈ H may be computed from


βn

∂H γ (P1 ,P2 ) (Ψ )dt, ∂Q1

(3.64a)

2c1  P1 − p sin(2Q1 ) + c2 (P2 − p) sin[2(q + Q1 )], ε

(3.64b)

M(Q1 (0)) = −

lim

αn ,βn →∞

−αn

where Hγ =

where Hγ = γH γ in (3.61), and where1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))]     4c1 
βn   P1 − ph (cos(2qh − 2a1 t)) −  dt  ε lim   αn ,βn →∞ −αn − 2c2 (P2 − ph ) cos(2a1 t) + sin[2(Q1 (0) + q(0))]     4c1 
βn   P1 − ph (sin(2qh − 2a1 t)) −  dt ,  ε lim   αn ,βn →∞ −αn + 2c2 (P2 − ph ) sin(2a1 t) (3.64c) that after retaining only even integrands reduces to1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))][I1 + I2 + I3 ],  $
−2c1 (a1 − εΩ) βn I1 = lim cos(2a1 t)dt , n→∞ ε 2 a3 −αn ' (
√ 2c2 A ∞ cos(2a1 t)sech2 ( At)dt , I2 = B −∞  $ √
∞ √ −2c1 A I3 = sin(2a1 t)tanh( At)dt . ε 2 a3 −∞

(3.65a) (3.65b)

(3.65c)

(3.65d)

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The Melnikov integral M(Q1 (0)) = 0 when ¯ 1n (0) = (2n + 1) Q1 (0) = Q

π − q(0); 4

n = 0, 1, 2, . . . ,

¯ 1n (0)) ∂M(Q ¯ 1n (0) + q(0))][I1 + I2 + I3 ] = 0, = −2 sin[2(Q ∂Q1 (0)

(3.66a)

n = 0, 1, 2, . . . . (3.66b)

Consequently, the stable W s (Υγ (P1 , P2 )) and unstable W u (Υγ (P1 , P2 )) manifolds of the KAM tori Υγ (P1 , P2 ) intersect transversely yielding Smale horseshoes1,3 on the appropriate 5D level energy surfaces. This implies multiple transverse intersections and the corresponding existence of chaotic dynamics in the perturbed system γ > 0 and α = 0. 3.4.2. Geometric structure for γ > 0 and α > 0 The perturbed dissipative system α > 0 and γ > 0 possesses a 4D normally hyperbolic locally invariant manifold Mγα that is given by

Mαγ

  (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :       = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ), .       p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.67)

The manifold Mγα has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds; and if these manifolds intersect transversely, then the Smale–Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Υγα (P1 , P2 ) may be located on Mγα by averaging the perturbed dissipative vector field γ > 0 and α > 0 restricted to Mγα over the angular variables Q1 and Q2 . The averaged equations have a unique stable hyperbolic fixed point (P1 , P2 ) = (0, 0) with two negative eigenvalues provided that the determinant1,3 # # # ∂ P˙1  ∂ P˙ 1  # # # # ∂P ∂P2 ## ν 2 γ 2 (λ4r ξτλ − τ ) 1 # > 0. (3.68) # #= # ∂ P˙  ∂ P˙  # 2λ8r ξτ12 τλ 2 2 # # # # ∂P1 ∂P2 The Melnikov method fails to predict chaos for this dissipative system when α > 0.1,3 Liapunov characteristic exponents (LCE). Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain toward an invariant surface or an attractor of lower dimensionality than the original space.

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For the 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 , there are six real exponents that may be ordered as µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 ≥ µ6 , where µ1 is the largest Liapunov characteristic exponent (LCE). The LCE is calculated from the first variation of the perturbed dissipative vector field γ > 0 and α > 0 according to  ∂ q˙  ∂q   ∂ p˙     δ q˙  ∂q     δ p˙   ˙    ∂ P1  ˙     δ P1   ∂q    δ P˙  =  ∂ P˙2  2      δ Q˙   ∂q  1    ∂ Q˙ 1  δ Q˙ 2  ∂q    ∂ Q˙ 2 ∂q

∂ q˙ ∂p

∂ q˙ ∂P1

∂ q˙ ∂P2

∂ q˙ ∂Q1

∂ p˙ ∂p

∂ p˙ ∂P1

∂ p˙ ∂P2

∂ p˙ ∂Q1

∂ P˙ 1 ∂p

∂ P˙1 ∂P1

∂ P˙ 1 ∂P2

∂ P˙1 ∂Q1

∂ P˙ 2 ∂p

∂ P˙2 ∂P1

∂ P˙ 2 ∂P2

∂ P˙2 ∂Q1

∂ Q˙ 1 ∂p

∂ Q˙ 1 ∂P1

∂ Q˙ 1 ∂P2

∂ Q˙ 1 ∂Q1

∂ Q˙ 2 ∂p

∂ Q˙ 2 ∂P1

∂ Q˙ 2 ∂P2

∂ Q˙ 2 ∂Q1

∂ q˙  ∂Q2   ∂ p˙    ∂Q2    ∂ P˙ 1   ∂Q2    ∂ P˙ 2   ∂Q2    ˙ ∂ Q1   ∂Q2    ˙ ∂Q 



δq



   δp       δP1     δP  .  2    δQ   1 δQ2

(3.69)

2

∂Q2

For 20 different values of the dimensionless damping parameter and of the dimensionless Floquet parametric forcing parameter, the largest LCEs were calculated and a chaos diagram for the positive values of the largest Liapunov exponents identified the parameter space in which chaotic motions exist.1,3

Appendix A The extension of the Herglotz algorithm to nonautonomous dynamical systems (GHA) significantly reduces the effort required to suspend the nonautonomous Hamiltonian component in (3.33d).

Generalized Herglotz algorithm (GHA) The Herglotz algorithm for autonomous dynamical systems4 may be generalized by: (1) including time t; and (2) defining a generating function with a nonzero determinant of second derivatives. The GHA transforms a set of 2N variables (u, v) to a set of 2N new variables (U, V) by choosing N new variables Ui and then computing the remaining N new variables Vi uniquely from the chosen Ui so that the transformation (u, v) → (U, V) is canonical as shown by satisfying the Poisson bracket conditions.

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GHA:Type I Equate N new variables to N old variables Ui = Ui (u, v, t) such that the Poisson brackets satisfy [[Ui , Uj ]]uv = 0 and the determinant # ∂U # 1 # # ∂v1 # # # · # # # # ∂UN # ∂v1

·

·

·

∂U1 ∂vN

·

∂Ui ∂vj

·

·

·

·

·

∂UN ∂vN

# # # # # # # = 0. # # # # #

(3.A.1)

Equate N Herglotz auxiliary functions Xi (u, v) to the absolute value of the ratios of the old variables in either of the following forms:  |ui /vi | Xi = Xi (u, v) = ; i = 1, 2, . . . , N, (3.A.2a,b) |vi /ui | such that the Jacobian of the new variables U and the Herglotz auxiliary functions X is nonzero ∂(U, X)/∂(u, v) = 0. Solve (3.A.2) for vi = vi (u, X); substitute vi (u, X) into Ui (u, v, t) and invert to obtain ui = ui (U, X, t). Compute the generating function F (u, U, t) from dF (u(U, X, t), U, t) =

N 

(vi dui − Vi dUi ),

(3.A.3)

i=1

 N   ∂F ∂F dXi + dUi ∂Xi ∂Ui i=1   N N N    ∂uj ∂uj  vj dXi + vj dUi − Vi dUi  . = ∂X ∂U i i i=1 j=1 j=1

(3.A.4)

Equate the coefficients of like differentials and compute: (1) the generating function F (u, U, t) =

N
 i=1

N Xi  j=1

vj

∂uj dXi
+ C(U, t), ∂Xi


(3.A.5)

and (2) the new N variables from Vi (u, v, t) =

N  j=1

vj

∂F ∂uj − ; ∂Ui ∂Ui

i = 1, 2, . . . , N.

(3.A.6)

In order to compute the transformed Hamiltonian in terms of the new variables (U, V), the inverse canonical transformation (u(U, V, t), v(U, V, t)) must be computed,

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and the new Hamiltonian K(Q, P, t) is given by K(Q, P, t) = H(q(Q, P, t), p(Q, P, t), t) +

∂F (u, U, t) . ∂t

(3.A.7)

The GHA may also be applied to equate N old variables ui = ui (U, V, t) to obtain N new variables Ui = Ui (u, v, t).1,3 Appendix B

τ1 =

√ 1+τ

κτ = κ(1 + τ )

τλ = 1 + (τ /λ4r )

    h  2  λ 1 − exp − ; for a full draft piston,   
0  r λ2r z f1 = f (z) exp 2 ds =     λr h −h  λ4r h  ; for a full draft hinge − 1 + exp − 2  h λ2r λr a1 = −

1 βτ + −8βξ 2ξλ6r τ12

c1 =

√ bf λ τ 1 r 1 4 2ξβ 3

d1 =

a2 = −

1 βτ 1 + − β 8βξ 2ξλ6r τ12

c2 =

2β − 1 4βξ

1 d2 =  8 2bβξ 3 λ3r τ12

2βτ 2 − ξλ4r τλ (λ2r − βτ )  8 2bβξ 5 λ9r τ1 τλ

d3 =

τ (2τ + λ4r ξτλr )  8 2bβξ 5 λ9r τλ2

d4 =

a3 =

τ τ 2 − λ2r τλ d5 =  1 2 2bβξ 3 λ5r τλ τ12

−τ + λ4r ξτλ d6 = 4βλ6r ξτλ2

1 4



τ − λ4r ξτλr 2 2 + τ1 βλ6r ξτλ2



τ + λ4r ξτλ (λ2r − 1) λ6r ξτ1 τλ

1 d7 = 4



−2 τ + λ4r ξτλ + τ12 βλ6r ξτλ2



References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. C. M. Bowline, R. T. Hudspeth and R. B. Guenther, Applicable Anal. 72, 287 (1999). 3. R. T. Hudspeth, R. B. Guenther and S. Fadel, Acta Mechanica 175, 139 (2005). 4. R. B. Guenther, H. Schwerdtfeger and G. Herglotz, Vorlesungen u ¨ber die Mechanik der Kontinua (1985). 5. S. Fadel, Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension, PhD dissertation, Oregon State University, USA (1998).

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Chapter 4

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone Yoshimi Goda ECOH CORPORATION, 2-6-4 Kita-Ueno Taito-ku, Tokyo 110-0014, Japan [email protected] A review is made on the statistical features of breaking waves in the nearshore waters. Inherent variability of the breaker index for regular waves is examined with the revised Goda’s formula. The incipient breaking height of significant wave is about 30% lower than that of regular waves. Nonlinearity of random waves is strongest at the outer edge of surf zone, but it is destroyed by wave-breaking process inside the surf zone. The wave height distribution is the narrowest in the middle of the surf zone, but it returns to the Rayleigh near the shoreline. Large differences among various wave models are noted for prediction of wave heights in the surf zone.

4.1. Introduction Breaking waves exert strong actions on maritime and coastal structures, while wave dissipation through breaking plays a major role in the generation of nearshore currents. Without good understanding of wave-breaking process, we cannot pursue any study for coastal engineering works. Nevertheless, wave breaking is an elusive phenomenon. Not many people spend enough time to observe wonderful pictures of wave deformation by breaking and regeneration after breaking. One needs some kind of a pier at a beach to have a good look of waves that break and rush toward the shore. Otherwise, one should work for some hours in a laboratory to measure waves at various locations along a wave flume. Our knowledge on wave breaking mostly comes from various literature based on previous research works. Quite a number of people use the formulas, diagrams, and other information listed there without examining the credibility of the information. For example, many people regard the breaker index, or the ratio of wave height to water depth at breaking, as a deterministic value without paying consideration to the fact that the breaker height exhibits large fluctuations even for a given wave 87

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condition and that the breaker index has been obtained by drawing a mean curve among scattered data. Nowadays, many researchers are developing various numerical models of random wave transformations. They have to adopt some kind of wave-breaking criteria and energy dissipation mechanism so that the model can reproduce wave deformation in the nearshore waters. However, modelers seem to pick up whatever is readily available without deliberation on the physical features of the wave-breaking process and the appropriateness of the breaking model. The present chapter is a revised version of the author’s paper1 presented at the 4th International Conference on Asian and Pacific Conference 2007, in Nanjin, China. It aims at providing coastal engineers and researchers with the most advanced knowledge on the statistics of wave breaking in the nearshore waters so that they can make a correct approach to the problems related to wave-breaking processes.

4.2. Physical Definitions of Wave Breaking Waves are defined as breaking when the crest starts to contain foams or when water particles jump out from the wave crest. It is a disruption of smooth water surface, and breaking waves are often classified into three types: spilling, plunging, and collapsing breakers. In the sea, there is no pure spilling breaker, because it is always accompanied with a small portion of plunging water. For example, we can observe from a window of an airplane such a scene of wave breaking in deep water that a crest of large wave makes plunging, leaves a patch of white foam behind, and moves forward with blue color. The following wave grows in height and breaks. It is a manifestation of wave energy being transported by the group velocity, which is one-half the phase velocity in deep water. Theoretically, three criteria can be cited. The first is the condition that the horizontal velocity of the water particle at the wave crest becomes equal to or greater than the phase speed of wave profile. The second is the upward vertical acceleration of the water particle at the wave crest to be equal to or to exceed the gravitational acceleration. The third is the vertical gradient of the total pressure at the wave crest to be zero or negative. The first criterion has been employed by mathematicians to find out the limiting waves of permanent form on a horizontal bed. Conventional perturbation techniques are ineffective to derive the limiting waves, and the approach specific to the limiting form needs to be employed. These limiting waves are characterized with the angular crest having the angle of 120◦ . Yamada and Shiotani2 have produced the most reliable computation results so far, which are summarized by Goda3 as reproduced in Table 4.1. The symbol H denotes the wave height, h is the water depth, L is the wavelength, L0 is the deepwater wavelength, C is the wave celerity, and ηc is the crest elevation. The subscripts “b” and “A” refer to the quantities at breaking and those of the small amplitude waves (Airy’s wave), respectively.

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Table 4.1. Characteristics of breaking waves of permanent type (after Goda3 based on Yamada and Shiotani2 ). Hb /L0

hb /LA a

hb /Lb

Cb /CA a

Hb /Lb

Hb /hb

ηc /Hb

infinity 0.935 0.471 0.286 0.1856 0.1117 0.0763 0.0474 0.0284 0.01669 0.01095 0.00575 0.00239 0.001144 0.000437 0

infinity 0.935 0.474 0.300 0.216 0.1510 0.1198 0.0915 0.0694 0.0525 0.0422 0.0306 0.01953 0.01351 0.00833 0

infinity 0.7686 0.4011 0.2597 0.1885 0.1331 0.1050 0.07915 0.05909 0.04398 0.03499 0.2483 0.01570 0.01075 0.00660 0

1.193 1.189 1.181 1.154 1.143 1.134 1.141 1.156 1.174 1.193 1.207 1.231 1.244 1.257 1.263 1.285b

0.1412 0.1409 0.1386 0.1277 0.1115 0.08997 0.07410 0.05771 0.04430 0.03371 0.02720 0.01962 0.01260 0.00871 0.00538 0

0 0.1791 0.3456 0.4919 0.5912 0.6683 0.7059 0.7293 0.7496 0.7666 0.7774 0.7904 0.8028 0.8099 0.8160 0.8261b

0.6706 0.6765 0.6908 0.7165 0.7619 0.7939 0.8392 0.8766 0.9061 0.9242 0.9453 0.9649 0.9757 0.9849 1.0000b

Notes: a LA and CA denote the wavelength and celerity of small amplitude waves. b These values are those of solitary wave computed by Yamada et al.4

The limiting height of solitary wave is (H/h)b = 0.8261 by Yamada et al.,4 instead of the often-quoted value of (H/h)b = 0.78 by McCowan.5 The limiting wave steepness of (H/L)b = 0.142 by Miche6 is calculated for waves of nonlinear waves, and it becomes Hb /(L0 )A = 0.1684 when the linear deepwater wavelength (L0 )A is used. The second criterion is for the breaking of standing waves, but no theoretical computation of limiting standing waves has been made. The third criterion has been proposed by Nadaoka et al.,7 for defining wave breaking in numerical time-domain computation. Zero vertical gradient of total pressure implies no presence of water below the wave crest, i.e., wave breaking.

4.3. Parameters Governing Breaker Index Because wave breaking attracts attention of many researchers, there have been proposed a number of formulas to describe the ratio of wave height to water depth. Kaminsky and Kraus8 called this ratio as the breaker height-to-depth index, but the present chapter employs the term of breaker index for the ratio of wave height to water depth at breaking for the sake of simplicity. Kamphuis9 has compared 11 formulas for testing of goodness-of-fit with his 225 sets of hydraulic model tests. Rattanapitikon and Shibayama10 have collected 574 data points from 24 papers/reports and calculated the root-mean-square errors of 24 breaking index formulas. Both authors have proposed their own formulas by modifying some of the previous ones.

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Rattanapitikon et al.11 have further added 121 data points in large-scale flume tests and proposed revised formulas. Similar to the approach by Rattanapitikon et al.,11 the breaker index formulas can be categorized into the following four functional forms: Hb /hb = f1 (0) = constant ,

(4.1)

Hb /hb = f2 (hb /L0 or hb /Lb ) ,

(4.2)

Hb /hb = f3 (s) ,

(4.3)

Hb /hb = f4 (s, hb /L0 or hb /Lb ) ,

(4.4)

where s denotes the bed slope. The wavelength L0 is the deepwater wavelength given by L0 = g/(2π)T 2 , where T is the wave period. Because the relative water depth hb /L0 is easily converted to hb /Lb through the dispersion relationship, the two relative depths hb /L0 and hb /Lb are interchangeable. The formula of Hb /hb = 0.78 is a typical example of Eq. (4.1). There are some other formulas using the parameter of deepwater wave steepness H0 /L0 . They can be useful for predicting the breaker height of regular waves. Nevertheless, they cannot be applied for breaking of random waves, because individual zero-crossing waves in a train of random waves are unrelated to individual waves in deepwaters. Thus, there is little room for the parameter H0 /L0 to function in the breaker index for random waves. Performance of a breaker index formula can be judged by the magnitude of the bias of the predicted breaker height from the observed heights. It should also be examined with either the root-mean-square error of predicted breaker heights or the correlation coefficient between prediction and observation. The root-mean-square error analysis by Rattanapitikon and Shibayama10 is not conclusive in differentiating the merits of four functional forms, but they recommend a certain modification of the slope effect in the function f4 (s, hb /L0 ), apparently indicating their preference of this functional form. Kamphuis9 calculated the correlation coefficients between 11 formulas and his laboratory data. By assigning the best-fitting value to the proportionality coefficient of each formula, he obtained the determination coefficient R2 = 0.69 to f1 (0), R2 = 0.67 to f2 (hb /L0 ), R2 = 0.84 to f3 (s), and R2 = 0.88 to f4 (s, hb /L0 ). His result clearly suggests the necessity to include both the parameters of bed slope and relative water depth in the breaker index formula.

4.4. Breaker Index for Regular Waves and Its Scatter 4.4.1. Scatter of regular breaking waves In 1970, Goda3 presented a diagram of breaker index curves of regular waves for four bed slopes, based on the laboratory data from eight sources, which included his own large-scale tests with Hb = 0.43 to 0.93 m; Rattanapitikon et al.11 did not

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analyze this data set. Later, Goda12,13 approximated the breaker index curves with the following empirical formula:
  Hb hb  A 4/3 1 − exp −1.5π 1 + 15s : A = 0.17 . (4.5) = hb hb /L0 L0 Equation (4.5) was derived by a graphical curve-fitting technique without direct comparison with the original laboratory data. Rattanapitikon and Shibayama10 have recommended a modification of the slope effect term of (1 + 15s4/3 ) into (1.033 + 4.71s − 10.46s2 ). Upon reexamination of the original laboratory data, the author has also recognized a necessity of modifying the slope effect term. The revised formula is as follows:
  hb  Hb A 1 − exp −1.5π 1 + 11 s4/3 : A = 0.17 . (4.6) = hb hb /L0 L0 Muttray and Oumeraci14 found the best-fitting coefficient of Eq. (4.5) being 0.167 instead of 0.170 for the slope of 1/30. When Eq. (4.6) is applied to their data, the coefficient would have the value of 0.173. Comparison of the laboratory data of breaker index with Eq. (4.6) is shown in Fig. 4.1 for five groups of bed slopes, i.e., 1/7 to 1/12, 1/20, 1/30, 1/50, and 1/200 to 1/200. (See Goda3 for description of the laboratory data employed here.) Recent data sets by Li et al.15 for s = 1/30 and s = 1/50, Li et al.16 for s = 1/200, and Lara et al.17 for s = 1/20 are also added. It is clear in Fig. 4.1 that the value of the breaker index increases as the bed slope becomes steep. Thus, it is absolutely necessary to incorporate the slope effect into the breaker index formula. Because the experimental data are scattered around the index curves, the upper and lower bound curves with the range of 87%–115% of the value by Eq. (4.6) are drawn in Fig. 4.1. A quantitative evaluation of the degree of the scatter is made by means of the relative error of the breaker index, i.e., E = (1 − γmeas/γest ), where γmeas is the measured value of Hb /hb and γest is the predicted value by the breaker index formula of Eq. (4.6). The mean and the standard deviation of the relative error are calculated for each group of the bed slope. The mean Emean indicates a bias of the breaker index and the standard deviation of E represents the degree of scatter of the breaker index. Because E is defined as the relative error, the standard deviation σ(E) is equivalent to the coefficient of variation (CoV). A positive bias indicates a tendency of overestimate, while a negative bias shows an underestimate. Table 4.2 lists the bias and CoV of the breaker index of Eq. (4.6) for the data of various bed slopes. The slope data of 1/9 and 1/12 are excluded from the analysis because of their small sample sizes. The bias varies from −2.9% to +6.2% depending on the bed slope, but the formula of Eq. (4.6) can be regarded as yielding reasonable estimates of the breaker heights. The scatter of the data as expressed by CoV is about 5% to 7% for the bed slope of 1/200 to 1/50; it increases as the slope becomes steep, and it takes the value of 14% for the bed slope of 1/10. Such scatter of data represents an inherent stochastic nature of wave-breaking phenomenon. It resides in the data set itself, being independent of the breaker index formula being applied.

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2.0

Breaker Index, γ = Hb /hb

Slope = 1/9 to 1/12

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Kishi-Iohara 1/9 Iversen 1/10 Goda et al. 1/10 Galvin 1/10 Bowen et al. 1/12 Index curve 1/10 Upper 115% Lower 87%

0.01

0.1

Relative water depth, hb /L0

(a) Breaker index data for the slope of 1/9−1/12. 1.5 Breaker Index, γ = Hb /hb

Slope = 1/20 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Iversen 1/20 Galvin 1/20 Toyoshima 1/20 Lara 1/20 Index curve 1/20 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(b) Breaker index data for the slope of 1/20.

Breaker Index, γ = Hb /hb

1.2

Slope = 1/30

1.0 0.9 0.8 0.7 0.6 0.5 0.4

0.3 0.001

Iversen 1/30 Mitsuyasu 1/30 Toyoshima 1/30 Li Y.C. et al. 1/30 Index curve 1/30 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(c) Breaker index data for the slope of 1/30. Fig. 4.1.

Comparison of breaker index formula Eq. (4.6) with laboratory data of regular waves.

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Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7

Slope = 1/50

0.6

Iversen 1/50 Mitsuyasu 1/50 Li Y.C. et al. 1/50 Index curve 1/50 Upper 115% Lower 87%

0.5 0.4

0.3 0.001

0.01 Relative water depth, hb /L0

0.1

(d) Breaker index data for the slope of 1/50.

Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7 0.6

Slope = 1/100 to 1/200

0.5 0.4

0.3 0.001

Goda 1/100 Li Y.C. et al. 1/200 Index curve 1/100 Index curve 1/200 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(e) Breaker index data for the slope of 1/100–1/200. Fig. 4.1.

Table 4.2.

(Continued )

Bias and CoV of the breaker index formula of Eq. (4.6).

Bed slope

No. of data

Bias = Emean (%)

CoV = σ(E) (%)

1/10 1/20 1/30 1/50 1/100 1/200

29 47 73 28 32 19

−0.5 +3.9 +1.4 −2.9 +6.2 +3.5

14.0 11.3 8.6 4.8 5.5 7.4

Sources Iversen, Goda et al. Iversen, Galvin, Toyoshima et al., Lara et al. Iversen, Mitsuyasu, Toyoshima et al., Li et al. Iversen, Mitsuyasu, Li et al. Goda Li et al.

Even under a well-controlled laboratory test, the breaking point fluctuates over some distance and the breaker height varies from wave to wave. Smith and Kraus18 reported on their regular wave tests that “despite care in conducting the tests and use of the average value of the given quantity (i.e., over 10 waves), wide scatter appeared in some quantities and must be considered inherent to the breaking process of realistic waves.” One cause of the data scatter is the presence of small-amplitude, long-period oscillations of water level in a laboratory flume, but the breaking process itself is triggered by many small factors beyond the control of experimenters. We should regard the wave-breaking phenomenon as stochastic one and accept a certain range of natural fluctuation. As listed in Table 4.2, the coefficient of

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variation is large for steep slope and it becomes small for gentle slope. In any research work involving wave-breaking phenomena, due consideration should be given to such stochastic nature of breaker heights. 4.4.2. Scatter of random breaking wave heights The first effort to identify the breaker heights of individual waves among trains of irregular waves was undertaken by Kimura and Seyama19 and Seyama and Kimura,20 who analyzed about 1000 individual breaking waves for each slope of 1/10, 1/20, 1/30, and 1/50 with the aid of a video camera and wave gauges. The breaker index of individual waves varied over a wide range, which was equivalent to CoV of 18% to 23% (the present author’s visual inspection of scatter diagrams). In order to reduce the scatter of data, they proposed to employ an artificial water depth below the mid-level between the wave crest and trough of individual breaking waves and succeeded in reducing CoV to the values between 8% and 11%. Because Eq. (4.5) yielded the breaker index larger than most of the observed value, they proposed its modified version. When the revised breaker index formula of Eq. (4.6) is employed, however, the center line of the scatted data appears to be at the level of 85% (s = 1/10) to 95% (for other slopes) of the predicted value. Black and Rosenberg21 made observation of individual breaking waves on a natural beach with a depth of 1.0–1.5 m at Apollo Bay in southern Australia. The median value of the breaker index was about 84% and 87% of those by Eqs. (4.5) and (4.6), respectively. Another observation was made by Kriebel22 in a large wave flume with a bed slope of 1/50 for waves with a significant height of 0.46 m and peak period of 2.9 s. When he applied Eq. (4.5) to individual breaker heights, he found that the value of the proportionality coefficient A fitted to the data varied from about 0.09 to 0.18 (against the original value of A = 0.17) with the mean of 0.142 (84% of Eq. (4.5)) and the standard deviation of 0.017 (equivalent to CoV of 12%). Li et al.23 also reported the result of their measurements of random breaking waves on the slope of 1/50 and 1/200, recommending the coefficient value of A = 0.150 with the standard deviation of 0.031. It should be recalled that the random wave breaking model by Goda24 in 1975 had already incorporated the variability of breaker heights by assigning a variable probability of individual wave breaking. The probability was assumed to increase linearly from 0 to 1 in the range of the wave height from 71% to 106% of the height predicted by Eq. (4.5), corresponding to A = 0.12 to 0.18. Therefore, it is expected that the median value of individual breaker heights would be smaller than those predicted by Eq. (4.5) or (4.6). 4.5. Breaker Index for Random Waves 4.5.1. Incipient breaking index of significant wave Equations (4.5) and (4.6) are examples of the breaker index for regular waves. There are some people who try to apply such breaker index formulas to coastal

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waves or random waves, but such application does not yield correct answers. The breaker index for regular waves may be utilized for the highest wave in an irregular wave train, but it cannot be applied for the significant wave, the root-mean-square wave, or any other characteristic wave. When random waves approach the shore, breaking of individual waves occurs gradually with large waves first at the far distance, medium waves next at some distance from the shore, and small waves near the shoreline. The variation of significant wave height from the offshore toward the shore is so gradual that we cannot employ the concept of wave-breaking line, which is so obvious in the case of regular waves. Against such difficulty of defining the breaking point of significant wave, Kamphuis9 introduced the definition of “incipient wave breaking.” He measured cross-shore variations of significant wave height beyond and across the surf zone, drew a curve of wave shoaling trend in the outside of the surf zone and a curve of wave height decay within the surf zone, and called the condition at the cross-point of the two curves as the incipient wave breaking. By using the data of the significant wave height at incipient breaking, he calibrated 11 breaker index formulas and determined the best-fitting proportionality coefficient. For the formula of Eq. (4.5), he obtained the proportionality coefficient of A = 0.12 for significant wave height. Li et al.23 have also presented a data of the breaker index of (H1/3 /h)b on the slope of 1/200, which is fitted to Eq. (4.5) with a modified constant value of 0.12 for the initial stage of breaking. Their breaking condition was some observation of large individual breaking waves in an irregular wave train. Goda24 has prepared a set of diagrams depicting variations of significant wave height across the surf zone (reproduced as Figs. 3.29 to 3.32 in Ref. 25). The boundary lines of 2% decay in these diagrams approximately correspond to the breaker index with A = 0.11, and the water depth (h1/3 )peak at which the significant wave takes a peak value within the surf zone (Fig. 3.34 in Ref. 25) corresponds to A = 0.11–0.13. Therefore, the incipient breaker index of the significant wave can be expressed with the following formula: 
   H1/3, b 0.12 (hb )incipient  4/3 = 1 − exp −1.5π 1 + 11 s . (4.7) hb hb /L0 L0 incipient Thus, the incipient breaker index of significant wave is about 30% lower than that of regular waves. The incipient breaking of significant wave corresponds to the condition that the high waves of upper several percent among individual waves have begun to break. 4.5.2. Laboratory data of breaker index of random waves After incipient breaking, the percentage of wave breaking increases as waves proceed across the surf zone. The ratio of the significant wave height to the water depth gradually increases toward the shoreline. Ting26,27 made detailed laboratory investigations of random wave deformations on a uniform slope of 1/35, using frequency spectra of broad- and narrow-band with the peak enhancement factor of 3.3 and 100, respectively. Waves had the significant height of Hs = 0.15 m and the spectral

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Breaker indices, H1/3 /h & Hrms /h

1.0

0.6 0.4 0.3 0.2

0.10

Fig. 4.2.

Slope = 1/35

0.8

Broad spec. H1/3 /h Narrow spec. H1/3 /h Breaker envlp. H1/3 /h Broad spec. Hrms /h Narrow spec. Hrms /h Breaker envlp. Hrms /h

0.01

0.1 0.03 Relative depth, h/L0

0.3

Breaker indices for H1/3 and Hrms on s = 1/35 with the data by Ting.26,27

peak period of Tp = 2.0 s. He recorded wave profiles at an offshore station with the depth of 0.457 m and at six stations on the slope with the depth of 0.27–0.0625 m. Waves at the six stations on the slope had the percentage of breaking ranging from 5% to 94% (the case of broad-band spectrum). Wave records were analyzed by the zero-downcrossing method, and calculated results of characteristic wave heights and periods are presented in tabular forms. From these results, the ratios of H1/3 and Hrms to the local depth (inclusive of mean water level change) are calculated and plotted against the relative water depth h/L0 , as shown in Fig. 4.2. The curves denoted as breaker envelopes are calculated by Eq. (4.6) for s = 1/35 with the proportionality coefficient of A = 0.145 (85% of regular waves) for the significant height H1/3 and to A = 0.111 (65%) for the root-mean-square height Hrms . Because the percentage of breaking waves is high in these data, an A value higher than that for incipient breaking fits to the data. The breaker index data for Hrms by Tick is higher than the value proposed by Sallenger and Holman,28 who gave an expression of Hrms /h = 3.2s + 0.32 without inclusion of the relative depth (h/L0 ) term. They converted the orbital velocity spectra to the surface wave spectra with the transfer function based on the linear theory, and estimated the energy-based Hrms , which must have been smaller than the statistical Hrms value based on direct measurement data of surface profiles. 4.5.3. Description of field data employed for analysis In the field wave observation at a fixed station, it is not feasible to judge whether individual waves are at the stage of breaking or not, unless simultaneous measurements with video cameras are taken. However, we may find out an upper limit of significant wave height for a given water depth by taking an envelope of many data at different relative water depths. For this type of analysis, stationary wave

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Summary of stationary coastal wave data employed in the present analysis.

Observation station Rumoi Port Yamase-Domari Port Tomakomai Port

Kanazawa Port Caldera Port, Costa Rica Sakata Port

97

Type of wave gauge

Water depth (m)

Sampling interval ∆t (s)

Significant height H1/3 (m)

Significant period T1/3 (s)

No. of data

Step-resistance ”

−11.5 −12.7

0.5 0.5

2.2–7.1 1.9–6.2

5.9–11.7 7.7–15.6

44 9

” ” Ultrasonic ” ”

−10.8 −13.8 −20 −20 −18

1.0 0.5 0.5 1.0 0.5

2.9–5.8 2.6–2.8 2.4–2.5 1.0–6.8 1.5–3.6

7.7–10.9 6.7 – 7.5 6.9–7.4 1.0–6.8 14.2–18.4

9 2 2 13 50

Pressure

−14.5 −10.5

0.5 0.5

1.7–9.7 1.7–6.1

6.3–13.4 6.5–15.0

123 123

(Source: Goda and Nagai,29 Goda24,30 ).

observation data analyzed by Goda and Nagai29 and the data of long-traveled swell recorded with an ultrasonic wave sensor reported by Goda30 were utilized. Table 4.3 lists the characteristics of these field data. Waves were recorded by means of either step-resistance gauges or ultrasonic wave sensors so that they were reliable registration of surface wave profiles. The data at Tomakomai and Kanazawa as well as Caldera were measured with ultrasonic wave sensors. They are not analyzed for breaker limits but for wave nonlinearity effects to be described in Sec. 4.6. Table 4.3 also lists the wave records at Sakata Port measured by means of pressure gauges, which were utilized by Goda24 for calibration of his random wavebreaking model. Although there remains a problem of pressure conversion to surface profiles, the conversion error would have been small because of the relatively shallow water depth at Sakata stations (10.5 and 14.5 m). They were included in the present analysis to increase the size of database. Other sources of nearshore waves are the photogrametric measurement data by Hotta and Mizuguchi31,32 as well as by Ebersole and Hughes.33 Hotta and Mizuguchi mobilized 11 motion-picture cameras set on top of a coastal observation pier at Ajigaura Beach, Ibaragi, Japan. They took film pictures of instantaneous water surfaces simultaneously at some 60 surveyor’s poles erected in the nearshore waters on a line perpendicular to the shoreline stretched over a distance of about 120 m. Films of surface wave records were taken every 0.2 s for an effective duration of 760 s. The beach profile in September 1978 had a trough at the distance of 25 m from the shoreline and the slope of about 1/60 offshore of the trough. The beach profile in December 1978 was somewhat uniform without any bar or trough, and the slope was about 1/70. The water depth inclusive of tides at the poles varied from 0.1 to 2.7 m. Photogrametric measurements of nearshore waves were also executed by Ebersole and Hughes33 during the DUCK85 campaign in Duck, North Carolina, USA with the cooperation of Dr. Hotta who brought twelve cameras with him and took charge of filming. They referred to this technique as “the photopole method”; this terminology is employed in the present chapter. Over the distance of 64 m,

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Summary of photopole measurements.

Date

No. of data

h (m)

H0 (m)

T1/3 (s)

H0 /L0

1978/09/05 1978/12/13–14 1985/09/04–05 1986/09/11–19

54 175 99 140

0.6–2.7 0.1–1.8 0.5–2.4 0.4–3.7

0.7 0.5–0.7 0.3–0.5 0.5–1.1

8.4 7.2–7.9 10.3–11.1 5.4–11.5

0.0064 0.0059–0.0081 0.0017–0.0028 0.0027–0.0254

12 poles were erected in the initial depth ranging from 0.4 to 1.9 m. Measurements were taken for nine runs during the fourth and fifth days of September 1985. With the variation of the tide level, the actual water depth varied from 0.5 to 2.4 m. The effective duration of wave recording was about 650 s, judging from the number of waves and average periods listed by Ebersole and Hughes.33 The beach profile was nearly flat for about 25 m from the shoreline with the depth of about 0.5 m below the mean sea level, and it had the slope of about 1/30 beyond that. Another series of photopole measurements were carried out during the SUPERDUCK campaign in 1986. Dr Hughes kindly supplied the author with the data files of measured wave statistics. The number of poles was increased to 20 and the water depth inclusive of tides varied from 0.4 to 3.7 m. The beach profile during SUPERDUCK is not known, but it would have been nearly the same as DUCK85 because of the same season. All the photopole measurement data were analyzed by the zero-downcrossing method, and various statistical wave characteristics were calculated. Table 4.4 lists the summary of the photopole wave measurement conditions. The significant wave period T1/3 has been converted from the pole-averaged values of either the mean period Tmean or the spectral period Tp by assumption of T1/3 = 1.05Tmean or T1/3 = 0.95Tp, which would be appropriate for swell of very low steepness. The offshore wave height H0 was converted from the significant height H1/3 measured at the most offshore pole using the shoaling coefficient; no refraction effect was taken into consideration as no information of wave direction was available. All the waves were swell of very low steepness ranging from 0.0017 to 0.0081, except one case of SUPERDUCK with H0 /L0 = 0.0254. 4.5.4. Field data of breaker index for energy-based significant waves Coastal surface wave data, pressure-converted wave data, and three sets of photopole data are plotted together in Fig. 4.3 in the form of Hm0 /h versus h/L0 . The energybased significant wave height Hm0 defined by Eq. (4.8) is employed here instead of the zero-crossing height H1/3 , because the latter is greatly enhanced over Hm0 by strong effects of wave nonlinearity and it is not representative of breaking-dissipated wave energy level; this aspect is discussed in Sec. 4.6. √ (4.8) Hm0 = 4.0ηrms = 4.0 m0 , where m0 denotes the zeroth moment of frequency spectrum.

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Step-resistance ( h=11-14m) Presssure gage (h=10-14m) Photopole Ajigaura (h=0.1-2.7 m) Photopole DUCK85 (h=0.5-2.4m) Photopole SUPERDUCK (h=0.5-3.2 m) Breaker envelope (80% limit)

Wave height ratio, Hm0 /h

0.8 0.6 0.4

0.2

0.1 0.001

Fig. 4.3.

0.01 0.1 Relative depth, h/L0

1

Breaker index for Hm0 based on the field wave data.

Coastal wave data, on the other hand, were not much affected by wave nonlinearity effects, and the zero-crossing significant height H1/3 was almost the same as Hm0 . Many data points in Fig. 4.3 belong to nonbreaking condition, but what interests us is the upper envelope which provides an estimate of the upper limit of breaking wave height. The curve of dash-dot line in Fig. 4.3 has been calculated by Eq. (4.6) for the slope of s = 0.0143 (1/70) with the coefficient being reduced to A = 0.136 (80%). Similar with the laboratory data shown in Fig. 4.2, the wave height ratio Hm0 /h is higher than the incipient breaker index of significant wave expressed by Eq. (4.7). It is because the breaker index increases inside the surf zone as the percentage of breaking waves increase. It is seen that the energy-based significant wave height Hm0 on gentle slopes does not exceed 0.7 times the local water depth except for the low-steepness swell in very shallow water. For the range of h/L0 > 0.03, the upper limit of significant wave height is about 0.6 times the local water depth. Some data points above the dash-dot curve are those of DUCK85 and SUPERDUCK, which were conducted on the beach steeper than the beach in Ajigaura.

4.6. Evolution of Wave Nonlinearity Across Surf Zone 4.6.1. Variations of skewness and kurtosis across the surf zone Ocean waves are characterized with almost linear property, as evidenced by the Gaussian distribution of surface elevation. Wave linearity is the basis of spectral

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representation and analysis. Deviation from the linearity is measured with the values of the skewness and kurtosis of the surface elevation with reference to the mean water level during a wave record. The skewness is zero when a distribution is symmetric with respect to the mean, and takes a positive value when a distribution is asymmetric with a long tail toward the right side (large value). The kurtosis takes a value of 3.0 for the Gaussian distribution. When the mode of distribution has a sharp peak and the distribution has long tails in both the left and right sides, the value of kurtosis becomes much larger than 3.0. The degree of positive skewness and the deviation of kurtosis from 3.0 are the measure of the strength of wave nonlinearity. The skewness of ocean waves is less than 0.5, and the kurtosis is below 4.0 for most cases, and thus the nonlinearity of waves in deepwater is weak. The variations of the skewness and kurtosis of field waves are examined with coastal surface waves listed in Table 4.3 (excluding the pressure sensor data of Sakata Port), and the photopole data in Ajigaura, DUCK85, and SUPERDUCK listed in Table 4.4. The data of skewness and kurtosis of September 5th in Ajigaura were kindly provided by Dr Hotta. The data of the photopole measurements of SUPERDUCK provided by Dr Hughes had the data of skewness only. As waves approach the shore, wave nonlinearity is enhanced and both the skewness and kurtosis increase significantly. Figure 4.4(a) exhibits the increase of the skewness with the nonlinearity parameter Π1/3 , which was introduced by Goda34 with the following definition: Π1/3 =

H1/3 2πh coth3 , LA LA

(4.9)

where LA denotes the wavelength calculated by small amplitude wave theory or Airy’s theory. The data are grouped by the range of the offshore wave steepness H0 /L0 : the first group for 0.001 < H0 /L0 < 0.0029, the second group for 0.003 < H0 /L0 < 0.0049, the third group for 0.0050 < H0 /L0 < 0.0099, the fourth group for 0.010 < H0 /L0 < 0.029, and the fifth group for 0.030 < H0 /L0 < 0.049 (Legends are shown with abbreviated figures). The data shown in the left diagram of Fig. 4.4 are those outside the surf zone. Because the boundary of surf zone is difficult to be set for random waves, an arbitrary boundary of h/H0 = 2.5 is employed here to separate the wave data outside and inside the surf zone. As shown in the left diagram, the skewness outside the surf zone shows a clear correlation with the wave nonlinearity parameter. The skewness begins from the value of zero at Π1/3 = 0, increases almost linearly with Π1/3 , and attains the value of 2.0 around Π1/3 = 4. The dashed line represents a semitheoretical relationship, which is based on the analysis of finite amplitude regular wave profiles by Goda34 with the consideration of the probability of individual wave heights according to the Rayleigh distribution. Variation of the skewness inside the surf zone (h/H0 < 2.5) is shown in Fig. 4.4(b). The ordinate is the ratio of offshore wave height to water depth, H0 /h, which increases rapidly as waves approach the shore. There is a clear trend of skewness decreasing toward β1 = 0 with the increase of the height-to-depth ratio

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2.5 0.001 0.5–1. For a cover layer lying on a geotextile on sand or clay, without a granular filter, the leakage length cannot be determined because the size of bf and k cannot be calculated. The physical description of the flow is different for this type of structure. For these structures there is no such a theory as for the blocks on a granular filter. However, it has been experimentally proved that Eq. (19.1) or (19.17) are also valid for these structures. It can be concluded that the theory has led to a simple stability formula [Eq. (19.17)] and a subdivision into four types of (block) revetment structures: • • • •

(a1) (a2) (a3) (a4)

cover cover cover cover

layer layer layer layer

on on on on

granular filter possibly including geotextile, low stability; granular filter possibly including geotextile, normal stability; geotextile on sand; clay or on geotextile on clay.

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Lower and upper value for F .

Type

Description

a1.1

Pitched irregular natural stones on granular filter Loose closed blocks/basalt on granular filter, low stability Loose blocks on granular filter, normal stability Loose blocks on geotextile on compacted sand/clay Linked/interlocked blocks on geotextile on good clay or on fine granular filter

a1.2 a2 a3 a4

Low F

High F

Usual F

2.0

3.0

2.5

3.0

5.0

3.5

3.5

6.0

4.5

4.0

7.0

5.0

5.0

8.0

6.0

The coefficient, F , is quantified for each structure type by way of fitting Eq. (19.1c) or (19.17) to the large collection of results of model studies from all over the world. Usually, only large-scale studies are used because both the waves and the wave induced flow in the filter should be well represented in the model. In the classification of structures according to the value of (k
D/kf bf ), the upper limit of (k
D/kf bf ) is 10 times the lower limit. Therefore, the upper limit of F of each structure type (besides a1.1) is assumed 100.33 = 2.14 times the lower limit, since F = f (k
D/kf bf )0.33 . A second curve is drawn with this value of F . In Table 19.3, all available tests are summarized and for each type of structure a lower and upper boundary for the value of F is given (see also an example in Fig. 19.11). The lower boundary gives with [Eq. (19.1c)] a stability curve below which stability is guaranteed. Between the upper and lower boundaries, the stability is uncertain. It depends on various unpredictable influences whether the structure will be stable or not. The upper boundary gives a curve above which instability is (almost) certain. The results for structure type a3 (blocks on geotextile on sand) may only be applied if the wave load is small [Hs < 1 or 1.5 m (max.)], or to structures with a subsoil of coarse sand (D50 > 0.3 mm) and a gentle slope (tan α < 0.25), because geotechnical failure is assumed to be the dominant failure mechanism (instead of uplift of blocks). A good compaction of sand is essential to avoid sliding or even liquefaction. For loads higher than H = 1.2 m, a well-graded layer of stone on a geotextile is recommended (e.g., layer 0.3–0.5 m for 1.2 m < H < 2.5 m). The results for structure type a4 can be applied on the condition that clay of high quality and with a smooth surface is used. A geotextile is recommended to prevent erosion during (long duration) wave loading. The general design criteria for geotextiles on cohesive soils are given by Pilarczyk.27 In the case of loose blocks, an individual block can be lifted out of the revetment with a force exceeding its own weight and friction. It is not possible with the cover layers with linked or interlocking blocks. Examples of the second type are: block mattresses, ship-lap blocks, and cable mats. However, in this case, high forces will be exerted on the connections between the blocks and/or geotextile. In the case of blocks connected to geotextiles (i.e., by pins), the stability should be treated as

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Fig. 19.11. Example of stability function for type a1.2 (loose closed blocks on granular filter); low stability. Table 19.4. Recommended values for the revetment parameter F for blockmats (the lower values refer to blocks connected to geotextile while the higher ones refer to cabled blocks). Type of revetment

F (−)

Linked blocks on geotextile on compacted sand or good clay For uncertain conditions/mediocre clay use Linked blocks on a granular filter:

5–6 4–5

— Favorable construction — Normal construction — Unfavorable construction

5–6 4–5 3–4

for loose blocks in order to avoid the mechanical abrasion of geotextiles by moving blocks. The lower boundary of stability of cabled mats can be increased by a factor of 1.25 (or 1.5, if additionally washed-in with granular material) in comparison with loose blocks. Such an increase of stability is only allowable when special measures are taken with respect to the proper connection between the mats. The upper boundary of stability (F = 8) remains the same for all systems. Application of this higher stability requires optimization of design (including application of geometrically open but stable filters and geotextiles).6,9 To be able to apply the design method for placed stone revetments under wave load to other semi-permeable systems, the following items may be adapted: the revetment parameter F , the (representative) strength parameters ∆ and D, the

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design wave height Hs , and the (representative) leakage length Λ. Table 19.4 gives an overview of usable values for the revetment stability parameter F in the blackbox model for linked blocks (block mats). The terms “favorable,” “normal,” and “unfavorable” refer to the composition of the granular filter and the permeability ratio of the top layer and the filter layer.9 In the case of fine granular filter and relatively permeable top layer, the total composition can be defined as “favorable.” In a case of very coarse granular layer and less permeable top layer, the composition can be defined as “unfavorable.” In a case of blocks connected to a geotextile and concrete-filled mattresses on a filter layer, the construction can be usually defined as between “unfavorable” and “normal,” and the stability factor F = 3.0–3.5 (max. 4.0) can be applied. For blockmats and permeable mattresses on sand F = 5 (max. 6.0) can be applied. The higher values can also be used in cases that the extreme design loading is not very frequent or when the system is (repeatedly) washed in by coarse material providing additional interlocking. This wide range of recommended values for F only gives a first indication of a suitable choice. Furthermore, it is essential to check the geotechnical stability with the design diagrams (see Fig. 19.5 and for a full set of diagrams see Refs. 26 and 27). 19.4. Stability Criteria for Concrete-Filled Mattresses 19.4.1. Concrete mattresses Characteristics of concrete mattresses are the two geotextiles with concrete or cement between them. The geotextiles can be connected to each other in many patterns, which results in a variety of mattress systems, each having its own appearance and properties. Some examples are given in Fig. 19.12.

Fig. 19.12.

Examples of concrete-filled mattresses.

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Fig. 19.13.

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Principles of permeability of Filter Point mattress.

The permeability of the mattress is one of the factors that determine the stability. It is found that the permeability given by the suppliers is often the permeability of the geotextile, or of the so-called Filter Points (Fig. 19.13). In both cases, the permeability of the whole mattress is much smaller. A high permeability of the mattress ensures that any possible pressure buildup under the mattress can flow away, as a result of which the uplift pressures across the mattress remain smaller. In general, with a subsoil of clay and silty sand, the permeability of the mattress will be higher than the permeability of the subsoil. Therefore, the water under the mattress can usually be discharged without excessive lifting pressures on the mattress. The permeability of the mattress will be lower than the permeability of the subsoil or sublayers if a granular filter is applied, or with sand or clay subsoil having an irregular surface (gullies/cavities between the soil and the mattress). This will result in excessive lifting pressures on the mattress during wave attack. 19.4.2. Design rules with regard to wave load The failure mechanism of the concrete mattress is probably as follows: • First, cavities will form under the mattress as a result of uneven subsidence of the subsoil. The mattress is rigid and spans the cavities.

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• With large spans, wave impacts may cause the concrete to crack and the spans to collapse. This results in a mattress consisting of concrete slabs which are coupled by means of the geotextile. • With sufficiently high waves, an upward pressure difference over the mattress will occur during wave run-down, which lifts the mattress (Fig. 19.1). • The pumping action of these movements will cause the subsoil to migrate, as a result of which an S-profile will form and the revetment will collapse completely. It is assumed that local settlement of the subsoil will lead to free spans of the concrete mattress. Then, the wave impact can cause the breaking of these spans, if the ratio of Hs /D is too large for a certain span length. A calculation method is derived on the basis of an empirical formula for the maximum wave impact pressure and the theory of simply supported beams. The collapsing of small spans (less than 1 or 2 m) is not acceptable, since these will lead to too many cracks. The empirical formula for the wave impact is16 : Fimpact = 7.2Hs2 tan α, ρg

(19.18)

where Fimpact = impact force per m revetment (N). Calculation has resulted in an average distance between cracks of only 10–20 cm for a 10-cm thick mattress and wave height of 2 m. This means that at such a ratio of Hs /D the wave impacts will chop the mattress to pieces. For a mattress of 15 cm thick and a wave height of 1.5 m, the crack distance will be in the order of 1 m. Apart from the cracks due to wave impacts, the mattress should also withstand the uplift pressures due to wave attack. These uplift pressures are calculated in the same way as for block revetments. For this damage mechanism the leakage length is important. In most cases, the damage mechanism by uplift pressures is more important than the damage mechanism by impact. The calculated representative values of the leakage length for various mattresses are presented in Table 19.5. The results of calculated stability for various values of leakage length (permeability) are presented in Fig. 19.14.

Table 19.5.

Estimated leakage length for concrete mattresses.

Mattress Leakage length Λ(m) Standard — FP FPM Slab Articulated (Crib) a Good

On sanda

On sandb

On filter

1.5 1.0 3.0 0.5

3.9 3.9 9.0 1.0

2.3 2.0 4.7 0.5

contact of mattress with sublayer (no gullies/cavities underneath). assumption: poor compaction of subsoil and presence of cavities under the mattress. b Pessimistic

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Fig. 19.14. Calculation results for concrete mattresses (Hs /∆D < 4 because of acceptable crack distance due to impacts on spans).

Taking into consideration the above failure mechanisms, the following design (stability) formula has been derived for the mattresses [similar to Eq. (19.1c)]:

Hs F Hs = 2/3 with = 4, (19.19) ∆D ∆D max ξop where D ∆ ρs F

= = = =

mass per m2 /ρs (which can be called Deffective or Daverage ), relative volumetric mass of the mattress (−) = (ρs − ρw )/ρw , volumetric mass of concrete (kg/m3 ), stability factor (see below).

For an exact determination of the leakage length, one is referred to the analytical model.9,16 However, besides the mattresses of a type as, for example, the tube mat (Crib) with relative large permeable areas, the other types are not very sensitive to the exact value of the leakage length. It can be recommended to use the following values of F in design calculations: F = 2.5 or (≤ 3) — for low-permeable mattresses on (fine) granular filter, F = 3.5 or (≤ 4) — for low-permeable mattress on compacted sand, F = 4.0 or (≤ 5) — for permeable mattress on sand or fine filter (Df 15 < 2 mm). The higher values can be applied for temporary applications or when the soil is more resistant to erosion (i.e., clay), and the mattresses are properly anchored.

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19.5. Stability of Gabions and Stone-Mattresses 19.5.1. Introduction Gabions are made of rectangular baskets of wire mesh, which are filled with stones. The idea of the protection system is to hold the rather small stones together with the wire mesh. Waves and currents would have easily washed away the small stones, but the wire mesh prevents this. A typical length of gabions is 3–4 m, a width of 1–3 m, and a thickness of 0.3–1 m. The gabions with small thickness (less then 0.5 m) and large length and width are usually called Reno-mattresses or stone-mattresses. An important problem of this protection system is the durability. Frequent wave or current attack can lead to a failure of the wire mesh because of the continuously moving grains along the wires, finally cutting through. Another problem is the corrosion of the mesh. Therefore, meshes with plastic coating or corrosion resistant steel are used. On the other hand, the system is less suitable where waves and currents frequently lead to grain motion. 19.5.2. Hydraulic loading and damage mechanisms Wave attack on gabions will lead to a complex flow over the gabions and through the gabions. During wave run-up the resulting forces by the waves will be directed opposite to the gravity forces. Therefore, the run-up is less hazardous than the wave run-down. Wave run-down, as it was already mentioned in Sec. 19.2, will lead to the following mechanisms: The downward flowing water will exert a drag force on top of the gabions and the decreasing phreatic level will coincide with a downward flow gradient in the gabions. During maximum wave run-down, there will be an incoming wave that a moment later will cause a wave impact. Just before the impact, there is a “wall” of water giving a high pressure under the point of maximum run-down. Above the run-down point the surface of the gabions is almost dry and therefore there is a low pressure on the gabions. The interaction of high pressure and low pressure is shown in Fig. 19.1. A simple equilibrium of forces leads to the conclusion that the section from the run-down point to the phreatic line in the filter will slide down: • if there is insufficient support from gabions below this section, • if the downward forces exceed the friction forces: (roughly) f < 2 tan α with: f = friction of gabion on subsoil and α = slope angle. From this criterion, we see that a steep slope will easily lead to the exceeding of the friction forces, and furthermore, a steep slope is shorter than a gentle slope and will give less support to the section that tends to slide down. Hydrodynamic forces, such as wave attack and current, can lead to various damage mechanisms. The damage mechanisms fall into three categories: (1) Instability of the gabions (a) The gabions can slide downwards, compressing the down slope mattresses. (b) The gabions can slide downwards, leading to upward buckling of the down slope mattresses.

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(c) All gabions can slide downwards. (d) Individual gabions can be lifted out due to uplift pressures. (2) Instability of the subsoil (a) A local slip circle can occur, resulting in an S-profile. (b) The subsoil can wash away through the gabions. (3) Durability problems (a) Moving stones can cut through the mesh. (b) Corrosion of the mesh. (c) Rupture of the mesh by mechanical forces (vandalism, stranding of ship, etc.).

19.5.3. Stability of gabions under wave attack An analytical approach of the development of the uplift pressure in the gabions can be obtained by applying the formulae for the uplift pressure under an ordinary pitched block revetment, with leakage length: Λ = 0.77D. With this relation the stability relations according to the analytical model are also applicable to gabions. Substitution of values, which are reasonable for gabions, in the stability relations provides stability function which indeed match the line through the measured points.16 After complicated calculations the uplift pressure in the gabions can be derived.16 The uplift pressure is dependent on the steepness and height of the pressure front on the gabions (which is dependent on the wave height, period, and slope angle), the thickness of the gabions, and the level of the phreatic line in the gabions. It is not dependent on the permeability of the gabions, if the permeability is larger than the subsoil. The equilibrium of uplift forces and gravity forces leads to the following (approximate) design formula: Hs −2/3 = F · ξop ∆D

with 6 < F < 9 for slope of 1:3 (tan α = 0.33)

(19.20a)

or, using Pilarczyk’s equation (19.2) with b = 2/3 and F = 9 as an upper-limit (Fig. 19.15):


Hs ∆D

 = cr

F cos α 9 cos α = 2/3 , b ξop ξop

(19.20b)

where Hs ∆ D F ξop Tp

= = = = = =

significant wave height of incoming waves at the toe of the structure (m), relative density of the gabions (usually: ∆ = ∆t ≈ 1), thickness of the gabion (m), stability factor, √ breaker parameter = tan α/ (Hs /(1.56Tp2) wave period at the peak of the spectrum (s).

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Fig. 19.15.

Summary of test results (Ashe1 ) and (Brown3 ) and upper-limit design curve.

For practical applications, F ≤ 7 is recommended in (19.20b) or F = 6 and b = 0.5 in Eq. (19.2) or (19.20b). It is not expected that instability will occur at once if the uplift pressure exceeds the gravity forces. On the other hand, the above result turns out to be in good agreement with the experimental results. The experimental verification of stability of gabions is rather limited. Small scale model tests have been performed by Ashe1 and Brown3 ; see Fig. 19.15.

19.5.4. Motion of filling material It is important to know if the filling material will start to move during frequent environmental conditions, because it can lead to rupture of the wire mesh. Furthermore, the integrity of the system will be affected if large quantities of filling material are moved. During wave attack the motion of the filling material usually only occurs if ξop < 3 (plunging waves). Based on the van der Meer’s formula for the stability of loose rock5,7,8 and the assumption that the filling of the gabion will be more stable than loose rock, the following criterion is derived (van der Meer formula with permeability factor: 0.1 < P < 0.2; number of waves: N < 5000; and damage level: 3 < S < 6): Hs F = , ∆f D f ξop

(19.21)

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where: ∆f = relative density of the stones in the gabions (usually: ∆f ≈ 1.65), Df = diameter of stones in the gabion (m), and F = stability factor, 2 < F < 5. In all situations the stone size must be larger than the size of the wire mesh in the basket; this defines the minimum size.

19.6. Scour and Toe Protection Toe protection consists of the armoring of the beach or bottom surface in front of a structure which prevents it from scouring and undercutting by waves and currents. Factors that affect the severity of toe scour include wave breaking (when near the toe), wave run-up and backwash, wave reflection, and grain size distribution of the beach or bottom materials. Toe stability is essential because failure of the toe will generally lead to failure throughout the entire structure. Toe scour is a complex process. Specific (generally valid) guidance for scour prediction and toe design based on either prototype or model results have not been developed as yet, but some general (indicative) guidelines for designing toe protection are given in Refs. 4, 5, 22, and 31. The maximum scour force occurs where wave downrush on the structure face extends to the toe and/or the wave breaks near the toe (i.e., shallow water structure). These conditions may take place when the water depth at the toe is less than twice the height of the maximum expected unbroken wave that can exist at that water depth. The width of the apron for shallow water structures with a high reflection coefficient, which is generally true for slopes steeper than about 1 on 3, can be planned based on the structure slope and the expected scour depth. The maximum depth of a scour trough due to wave action below the natural bed is about equal to the maximum expected unbroken wave at the site. To protect the stability of the face, the toe soil must be kept in place beneath a surface defined by an extension of the face surface into the bottom to the maximum depth of scour. This can be accomplished by burying the toe, when construction conditions permit, thereby extending the face into an excavated trench the depth of the expected scour. Where an apron must be placed on the existing bottom, or can only be partially buried, its width should not be less than twice the wave height. If the reflection coefficient is low (slopes milder than 1 on 3), and/or the water depth is more than twice the wave height, much of the wave force will be dissipated on the structure face and a smaller apron width may be adequate, but it must be at least equal to the wave height (minimum requirement). Since scour aprons generally are placed on very flat slopes, quarrystone of the size (diameter) equal to 1/2 of the primary cover layer probably will be sufficient unless the apron is exposed above the water surface during wave action. Quarrystone of primary cover layer size may be extended over the toe apron if the stone will be exposed in the troughs of waves, especially breaking waves. The minimum thickness of cover layer over the toe apron should be two quarrystones. Quarrystone is the most favorable material for toe protection because of its flexibility. If a geotextile is used as a secondary layer it should be folded back at the end, and then buried in cover stone and sand to form a Dutch toe. It is recommended to provide an additional flexible

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Fig. 19.16.

Alternative toe protections.

edge (at least 1 m) consisting of loose material which may easily follow the scour at the toe. Some alternative designs of toe protection are shown in Fig. 19.16. The size of toe protection against waves can be roughly estimated by using the common formulae on slope protection and schematizing the toe by mild slopes (i.e., 1 on 10) or by using formulae developed for breakwaters.

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Toe protection against currents may require smaller protective stone, but wider aprons. The necessary design data can be estimated from site hydrography and/or model studies. Special attention must be given to sections of the structure where scour is intensified; i.e., to the head, the areas of a section change in alignment, the channel sides of jetties, and the downdrift sides of groynes. Where waves and reasonable currents (>1 m/s) occur together, it is recommended to increase the cover size at least by a factor of 1.3. Note that the conservatism of the apron design (width and size of cover units) depends on the accuracy of the methods used to predict the waves and current action and to predict the maximum depth of scour. For specific projects, a detailed study of scour of the natural bottom and at nearby similar existing structures should be conducted at a planned site, and/or model studies should be considered before determining a final design. In all cases, experience and sound engineering judgment play an important role in applying these design rules.

19.7. Protection Against Overtopping If a structure (revetment) is overtopped, even by minor splash, the stability can be affected. Overtopping can: (a) erode the area above or behind the revetment, negating the structure’s purpose; (b) remove soil supporting the top of the revetment, leading to the unraveling of the structure from the top down; and (c) increase the volume of water in the soil beneath the structure, contributing to drainage problems. The effects of overtopping can be limited by choosing a higher crest level or by armoring the bank above or behind the revetment with a splash apron. For a small amount of overtopping, a grassmat on clay can be adequate. The splash apron can be a filter blanket covered by a bedding layer and, if necessary, by riprap, concrete units, or asphalt. No definite method for designing against overtopping is known due to the lack of the proper method on estimating the hydraulic loading. Pilarczyk (1990) proposed the following indicative way of design of the thickness of protection of the splash area (Fig. 19.17): Hs 1.5 cos αi  , = Rc ∆Dn ΦT ξ 2b 1 − R n

(19.22)

where Hs = significant wave height, ξ = breaker index; ξ = tan α(Hs /Lo )−0.5 , α = slope angle, αi = angle of crest or inner slope, L0 = wavelength, b = coefficient equal to 0.5 for smooth slopes and 0.25 for riprap, Rc = crest height above still water level, Rn = wave run-up on virtual slope with the same geometry (see Fig. 19.17),

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Fig. 19.17.

Definition of splash area.

D = thickness of protective unit (D = Dn for rock), and ΦT = total stability factor equal to 1.0 for rock, 0.5 for placed blocks, and 0.4 for block mats. The length of protection in the splash area, which is related to the energy decay, depends on the permeability of the splash area. However, it can be roughly assumed as equal to: Ls =

ψ  T g(Rn − Rc ) ≥ Lmin , 5

(19.23)

with a practical minimum (Lmin ) equal at least to the total thickness of the revetment (including sublayers) as used on the slope. ψ is an engineering-judgment factor related to the local conditions (importance of structure), ψ ≥ 1, usually 1 < ψ < 2. Stability of rockfill protection of the crest and rear slope of an overtopped or overflowed dam or dike can also be approached with the Knauss formula.17 The advantage of this approach is that the overtopping discharge, q, can be used directly as an input parameter for calculation. Knauss analyzed steep shute flow hydraulics (highly aerated/turbulent) for the assessment of stone stability in overflow rockfill dams (impervious barrages with a rockfill spillway arrangement). This kind of flow seems to be rather similar to that during high overtopping. His (simplified) stability relationship can be re-written to the following form: √ q = 0.625 g(∆Dn )1.5 (1.9 + 0.8Φp − 3 sin αi ),

(19.24)

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where q = maximum admissible discharge (m3 /s/m), g = gravitational acceleration (9.81 m/s2 ), Dn = equivalent stone diameter, Dn = (M50 /ρs )1/3 , ∆ = relative density; ∆ = (ρs − ρw )/ρw , αi = inner slope angle, and Φp = stone arrangement packing factor, ranging from 0.6 for natural dumped rockfill to 1.1 for optimal manually placed rock; it seems to be reasonable to assume Φp = 1.25 for placed blocks. Note: When using the Knauss formula, the calculated critical (admissible) discharge should be identified with a momentary overtopping discharge per overtopping fraction of a characteristic wave, i.e., volume of water per characteristic wave divided by overtopping time per wave, roughly (0.3–0.4)T (T = wave period), and not with the time-averaged discharge (q).

19.8. Joints and Transitions Despite a well-designed protective system, the construction is only as strong as the weakest section. Therefore, special care is required when designing transitions. In general, slope protection of dike or seawall consists of a number of structural parts such as: toe protection, main protection in the area of heavy wave and current attack, upper slope protection (very often grass mat), berm for run-up reduction, or as maintenance road. Different materials and different execution principles are usually applied for these specific parts. Very often a new slope protection has to be connected to an already existing protective construction which involves another protective system. To obtain a homogeneous strong protection, all parts of protective structures have to be taken under consideration. Experience shows that erosion or damage often starts at joints and transitions. Therefore, important aspects of revetment constructions, which require special attention are the joints and the transitions; joints onto the same material and onto other revetment materials, and transitions onto other structures or revetment parts. A general design guideline is that transitions should be avoided as much as possible, especially in the area with maximum wave attack. If they are inevitable, the discontinuities introduced should be minimized. This holds for differences in elastic and plastic behavior and in the permeability or the sand tightness. Proper design and execution are essential in order to obtain satisfactory joints and transitions. When these guidelines are not followed, the joints or transitions may influence loads in terms of forces due to differences in stiffness or settlement, migration of subsoil from one part to another (erosion), or strong pressure gradients due to a concentrated groundwater flow. However, it is difficult to formulate more detailed principles and/or solutions for joints and transitions. The best way is to combine the lessons from practice with some physical understanding of systems involved. Examples to illustrate the problem of transitions are given in Fig. 19.18.

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Fig. 19.18.

Transitions in revetments.

As a general principle, one can state that the transition should be of strength equal to or greater than the adjoining systems. Very often it needs reinforcement in one of the following ways: (a) increase the thickness of the cover layer at the transition, (b) grout riprap or block cover layers with bitumen, and (c) use concrete edge strips or boards to prevent damage progressing along the structure. Top edge and flank protection are needed to limit the vulnerability of the revetment to erosion continuing around its ends. Extension of the revetment beyond the point of active erosion should be considered but is often not feasible. Care should therefore be taken that the discontinuity between the protected and unprotected areas is as small as possible (use a transition roughness) so as to prevent undermining. In some cases, open cell blocks or open block mats (eventually vegetated) can be used as transition (i.e., from hard protection into grass mat). The flank protection between the protected and unprotected areas usually needs a thickened

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Fig. 19.19. Construction aspects of revetments; examples of toe protection and placing of block mats (mattresses), and some methods of anchoring.

or grouted cover layer, or a concrete edge strip with some flexible transition, i.e., riprap. 19.9. General Construction (Execution) Aspects Revetments are constructed in a number of phases, for example: — construction of the bank/dike body, — placement of toe structure,

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— placement of revetment sublayers (clay and/or filter layers), — laying the blocks or mattress, — anchoring the mattress and, possibly, applying the joint filler. A well-compacted slope is important in order to produce a smooth surface and thus ensure that there is a good connection between the mattress and the subsurface. When laying mattresses on banks, it is strongly recommended that they are laid on undisturbed ground and that areas excavated too deeply are carefully refilled. Before using a geotextile, the slope must be carefully inspected for any projections which could puncture the material. When laying a mattress on a geotextile, care must be taken to ensure that extra pressures are not applied and that the geotextile is not pushed out of place. Geotextile sheets must be overlapped and/or stitched together with an overlap of at least 0.5–1.0 m to prevent subsoil being washed out. This is particularly important if the mattress is laid directly on sand or clay. Block mattresses are laid using a crane and a balancing beam. The mattress must be in the correct position before it is uncoupled because it is difficult to pick up again and also time-consuming. Provided that part of the mattress can be laid above the water line, it can generally be laid very precisely and joints between adjacent mattresses can be limited to 1–2 cm. Laying a mattress completely under water is much more difficult. The spacing between the blocks of adjacent mattresses, nonetheless, should never be more than 3 cm. Once in place, mattresses should be joined so that the edges cannot be lifted/turned up under the action of waves. Loose corners are particularly vulnerable. In addition, the top and bottom edges of the revetment should be anchored, as shown in Fig. 19.19. In such a case, a toe structure is not needed to stop mattresses sliding. More information on execution aspects of revetments can be found in Refs. 4, 5, 7, 8, 9, and 31

19.10. Conclusions The newly derived design methods and stability criteria will be of help in preparing the preliminary alternative designs with various revetment systems. However, there are still many uncertainties in these design methods. Therefore, experimental verification and further improvement of design methods are necessary. Also, more practical experience at various loading conditions is still needed.

References 1. G. W. T. Ashe, Beach erosion study, gabion shore protection, Hydraulics Laboratory, Ottawa, Canada (1975). 2. BAW, Code of practice: Use of geotextile filters on waterway, Bundesanstalt f¨ ur Wasserbau, Karlsruhe, Germany (1993). 3. C. Brown, Some factors affecting the use of Maccaferi gabions, Water Research Lab. Australia, Report 156 (1979). 4. CEM, Coastal Engineering Manual (US Army Corps of Engineers, Vicksburg, 2006).

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5. CIRIA-CUR-CETMEF, The Rock Manual (CIRIA, London, 2007). 6. CUR, Filters in Hydraulic Engineering (Civil Engineering Research and Codes (CUR), Gouda, the Netherlands, 1993) (in Dutch). 7. CUR/CIRIA, Manual on Use of Rock in Coastal Engineering, CUR/CIRIA Report 154 Gouda, the Netherlands (1991). 8. CUR/RWS, Manual on Use of Rock in Hydraulic Engineering, CUR Report 169, Gouda, the Netherlands (1995). 9. CUR/TAW, Design Manual for Pitched Slope Protection, CUR Report 155 (A.A. Balkema, Rotterdam, 1995). 10. DVWK, Guidelines for water management no. 306: Application of geotextile in hydraulic engineering, German Association for Water Resources and Land Improvement (DVWK), Bonn, Germany (1993). 11. Flexible Armoured Revetments, Proceedings of the International Conference, Thomas Telford Ltd., London (1984). 12. FHWA, Geosynthetics Design and Construction Guidelines, Federal Highway Administration, FHWA-HI-95-038, Washington, DC (1995). 13. G. Heerten, Dimensioning the filtration properties of geotextiles considering long-term conditions, Proc. 2nd Int. Conf. on Geotextiles, Las Vegas (1982). 14. G. Heerten, Geotextiles in coastal engineering, 25 years experience, Geotex. Geomemb. 1(2), 119–141 (1984). 15. R. D. Holtz, B. R. Christopher and R. R. Berg, Geosynthetic Engineering (BiTech Publishers Ltb., Richmond, Canada, 1997). 16. M. K. Breteler, K. W. Pilarczyk and T. Stoutjesdijk, Design of alternative revetments, Proc. 26th Int. Conf. on Coastal Engineering, Copenhagen (1998); http://www. wldelft.nl/rnd/publ/search.html (insert for author: Breteler or Pilarczyk). 17. J. Knauss, Computation of maximum discharge at overflow rock-fill dams, 13th Congress des Grand Barrages (ICOLD), New Delhi, Q50, R.9 (1979). 18. S. T. Maynord, Corps riprap design guidance for channel protection, in River, Coastal and Shoreline Protection, eds. R. Thorne Colin et al. (John Wiley & Sons, Chichester, UK, 1995). 19. C. R. Neill, Stability of coarse bed material in open channel flow, Edmonton (also, IAHR Congress, Fort Collins) (1967). 20. PIANC, Guidelines for the design and construction of flexible revetments incorporating geotextiles for inland waterways, Report WG 4, PTC I, Supplement to Bulletin No. 57, Brussels, Belgium (1987). 21. PIANC Bulletin, Special issue on propeller jet action, erosion and stability criteria near the harbour quays, Pianc Bulletin no. 58, Brussels, Belgium (1987). 22. PIANC, Guidelines for the design and construction of flexible revetments incorporating geotextiles in marine environment, Report WG 21, PTC II, Supplement to Bulletin No. 78/79, Brussels, Belgium (1992). 23. PIANC, Guidelines for the design of armoured slopes under piled quay walls, Supplement to Bulletin No. 96, Brussels, Belgium (1997). 24. K. W. Pilarczyk (ed.), Coastal Protection (A.A. Balkema, Rotterdam, 1990), www.enwinfo.nl (select English, downloads). 25. K. W. Pilarczyk, Simplified unification of stability formulae for revetments under current and wave attack, in River, Coastal and Shoreline Protection, eds. R. Thorne Colin et al. (John Wiley & Sons, Chichester, UK, 1995). 26. K. W. Pilarczyk (ed.), Dikes and Revetments (A.A Balkema, Rotterdam, 1998). 27. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, 2000), www.balkema.nl.

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28. K. Pilarczyk, International perspectives on coastal structures uses, in Advances in Coastal Structure Design, eds. K. Mohan Ram et al. (ASCE, 2003). 29. EAU, Recommendations of the committee for waterfront structures, German Society for Harbour Engineering (Ernst & Sohn, Berlin, 2000). 30. RWS, The Closure of Tidal Basins, Rijkswaterstaat (The Delft University Press, Delft, 1987). 31. SPM, Shore Protection Manual, U.S. Army Corps of Engineers, Vicksburg (1984). 32. G. Van Santvoort (ed.), Geotextiles and Geomembranes in Civil Engineering, revised edition (A.A. Balkema, Rotterdam, 1994).

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Appendix A. Comparative Stability of Revetments (Pilarczyk, 1990, Coastal Protection) Limits φ(rock) = 2.25 ctgα ≥ 2

Criterion Hs cos α cos α = ψu · φ b = ψu 2.25 b ∆m D ξp ξp

0.5 ≤ b ≤ 1.0 Sublayer

Riprap (2 layers) Riprap (tolerable damage)

Granular Granular

Pitched Stone

1.00 1.33 1.50

Poor quality (irregular-)stone Good quality (regular-)stone Natural basalt

Granular Granular Granular

Loose closed blocks; Hs < 1.5 m Loose (closed-)blocks Blocks connected to geotextile Loose closed blocks Cabled blocks/Open blocks (>10%) Grouted (cabled-) blocks/Interlocked blocks adequately designed Surface grouting (30% of voids) Pattern grouting (60% of voids)

Geotextile on sand Granular Granular Geotextile on clay Granular Granular

Blocks/Blockmats

Grout

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Gabions Fabric Containers Grass

2.00 2.50 2 + 3.0 2 + 2.5 1.00 1.50 2.00

Open stone asphalt; Up ≤ 6 m/s Open stone asphalt; Hs < 4 m

Geotexile on clay Sand asphalt

Gabion/mattress as a unit, Hs < 1.5 Stone-fill in a basket, dmin = 1.8 Dn Pm  1 less permeable mattress Pm ≈ 1 (Pm = ratio permeab. top/sublayer Pm
2 Permeable mattress of special design Grass-mat on poor clay; Up < 2 m/s Grass-mat on proper clay; Up < 3 m/s/s

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Chapter 20

Remarks on Coastal Stabilization and Alternative Solutions Krystian Pilarczyk (Former) Hydraulic Engineering Institute Public Works Department, Delft, The Netherlands HYDROpil Consultancy, 23 Nesciohove 2726BJ Zoetermeer, The Netherlands [email protected] Erosion control and coastal stabilization are common problems in coastal engineering. A brief overview of some available alternative systems for shore stabilization and beach erosion control is presented. Special attention is paid to artificial reefs and geosystems. Geosystems (geotubes, geocontainers, etc.) have gained popularity in recent years because of their simplicity in placement, cost effectiveness, and environmental aspects. However, all these systems have some advantages and disadvantages, which have to be recognized before application.

20.1. Introduction Some coastal environments may be regarded as rather stable (rock and reef coasts) while others are more vulnerable (sand and mud coasts, soft cliffs). In this last case, coastal users and managers all over the world are frequently faced with serious erosion of their sandy coasts. Possible causes of erosion include natural processes (i.e., action of waves, tides, currents, sea level rise, etc.) and sediment deficit due to human impact (i.e., river regulations, sand mining, and coastal engineering works). Countermeasures for beach erosion control function depend on local conditions of shore and beach, coastal climate, and sediment transport. Continuous maintenance and improvement of the coastlines, together with monitoring and studies of coastal processes have yielded considerable experience on various coastal protection measures all over the world. This contribution presents an overview of the various available methods for shore stabilization and beach erosion control, with special emphasis on the novel/alternative systems in various design implementations. More detailed information on other coastal protection systems and measures applied nowadays throughout the world can be found in extensive list of references.15,16,51,52,54,80 521

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20.2. General Approach 20.2.1. Types and functions of coastal structures In general, a coastal structure is planned as a practical measure to solve an identified problem. Starting with identification of the problem (e.g., shoreline erosion), a number of stages can be distinguished in the design process for a structure: definition of functions, determination of boundary conditions, creating alternatives, geometrical design, and the final choice of functional solution. After the choice of functional solution has been made the structural design starts including creating structural alternatives (i.e., using different materials and various execution methods). The final choice will be made after verification of various structural solutions in respect to the functional, environmental, and economic criteria. Coastal management, in its broadest sense, must take into account all factors, which bearing on the future of the coastal zone. Politics, policy making, planning, economy, and a multitude of economic, and noneconomic users (e.g., natural reserve areas), environment protection and, by and large, sustainable development, all play significant roles and provide both motivation and background for coastal management. The factors of coastal management may well entail many scientific and engineering disciplines other than coastal engineering and at sites and locations far removed from the costal zone. Some of these factors interact with one another; others are incompatible. The extent to which this applies in any particular region, area, or specific site needs careful evaluation and compromise solutions. This is one of the major roles of coastal management. The primary objectives of a typical coastal management study are to formulate long-term engineering planning, including financial strategies for the future usage, development, and conservation of the coastal zone. In this process, priorities should be defined both for new works and essential maintenance, with estimates for contingency items to cover emergency situations, which inevitably occur (see Fig. 20.1). The key element in any coastal management study is thorough understanding of coastal processes by which is meant the interaction between the hydraulic environment of winds, waves, tides, surges, and currents with the geological conditions in the coastal zone. To be effective, this may require a very broad view to be taken on a regional basis in the first instance. A regional cell could then be subdivided into smaller cells once the basic coastal processes had been established, and so on, with decreasing cell sizes until the cell in question becomes the specific one of the project itself. It is only in this way that the impact of new works in the coastal zone can be evaluated satisfactorily or long-term planning undertaken. The basic tools of the coastal engineering are still fairly limited and comprise cross-shore structures (such as groins, jetties spurs, etc.) shore-parallel structures (offshore breakwaters, seawalls, revetments generally close to shoreline), and dikes, headland structures, and artificial beach nourishment (see Fig. 20.2). • Groins generate considerable changes in wave and circulation patterns but their basic function — to slow down the rate of littoral drift — is sometimes overlooked. In the absence of beach nourishment, groins can redistribute the existing supply and, in a continuous littoral system, may be expected to create a deficiency at

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Analysis of coastal strategies.

the downdrift end where the uncontrolled drift rate is re-established. Without an adequate supply of beach material, groins are of limited value. In addition to controlling the rate of drift, groins are also used extensively to control the distribution of material along a frontage and to limit the temporary effects of drift reversal. There are unfortunately many examples where either bad design or failure to provide for the downdrift consequences has resulted in an adverse effect

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Fig. 20.2.

Schematic presentation of various shore protection measures.

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• •

• • •

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on the coastline. In other instances, failure to maintain groin systems might be more worse than having no groins at all. Offshore breakwaters are usually provided either to reduce wave energy at shoreline structures or to modify the wave climate and enhance sediment transport patterns so as to improve beach levels and create desirable beach features, such as salients.17,22 Offshore breakwaters can be shore-connected or detached, submerged or emerging, alongshore or oblique. Perched beach is a system consisting of a submerged breakwater (“sill” or “reef”), usually located not far away from shoreline, and artificial beach nourishment providing sand to the area extending between sub-aerial beach and the sill crest. Seawall (wording sometimes used interchangeably with bulkhead) is either a retaining wall intended to hold or prevent sliding of the soil behind it or a massive structure whose primary purpose is to protect the backshore from heavy wave action. Sometimes one speaks of “beach wall” or “shore wall.”15,16,52,53 Revetment is placed on a slope to shelter the adjacent uplands from erosion. Wave reflection, a serious disadvantage of vertical-wall bulkheads (seawalls), does not accelerate toe erosion as strongly at revetments as it does at seawalls.51 Dikes are generally intended as means of flood prevention. The crest of a dike is elevated high enough to counteract or confine overtopping in rare storm surge events.53,55,56 Beach nourishment or fill (or recharge), consists of importation of granular material to beach from an external source.17,53 It is not new, and has been used in some countries for decades, but is now being applied to an increasing extent and in a greater variety of ways. The resulting beach provides some protection to the area behind it and also serves as a valuable recreational resource. The beach fill functions as an eroding buffer zone, and its useful life will depend on how quickly it erodes. One must be prepared to periodically renourish (add more fill) if erosion continues. Headland control has been devised by analogy to the Nature’s efforts to keep in equilibrium a certain crenulate shape of erosion bays sculptured for thousands or so years. The crenulate shaped bays can be kept in equilibrium by use of a system of headlands. The headland system is claimed to be in feedback with coast and to combine the advantages of groins and detached breakwaters (shore-parallel or oblique).64

All forms of shore protection (i.e., groins, breakwaters, seawalls, revetments, bulkheads, beach-fill, etc.) have certain advantages and disadvantages. A shoreparallel (or oblique) breakwater, placed near the shoreline or offshore and designed either to intercept a portion of moving sediment or to protect a placed beach-fill, has the potential to perform close to the ideal for many types of coastal environments. A number of innovative solutions are given by Silvester and Hsu.64 Recent experience with design of beach stabilization structures is reviewed by Bodge.12 Various low-cost, environment-friendly, emergency and temporary measures, and their combinations provide alternatives to the principal measures. These systems are often appropriate for application only in sheltered waters. Inherent in the concept of environmental friendliness and low cost is the assumption on the equal importance

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of physics, engineering, ecology, and economy. The approach in this contribution is to provide a background for some of these alternatives. 20.2.2. Functional requirements and design It is common knowledge that the realm of coastal processes and the interactions of offshore breakwaters and beaches can be broadly classified as far-field and nearfield phenomena. The scales are somehow arbitrary but can be roughly identified as those greater and smaller, respectively, than characteristic dimensions of a structure or coastal feature. The design procedures for coastal structures should embody functional/ geometrical design and structural design reflecting, respectively, the far-field and near-field requirements imposed on structures. This corresponds to the division of design procedures into two basic groups concentrating on the overall layout and configuration of a structure as a whole, and its interaction with the coastal environment to produce desirable sedimentation patterns and coastal management effects stability and reliability of the structure and its components, hence dimensioning of structural constituents, associated with possible unavoidable and undesirable hazards due to the loadings exerted by the coastal environment. In other words, the first group involves design parameters producing the best environmental effectiveness of a structure in “ideal” conditions, i.e., upon negligence of possible “harmful by-effects,” such as different modes of failures and instabilities, both overall and internal. The second group is concerned about these “by-effects” and provides the tools, which secure the integrity and proper operation of the structure and its components. As in many other engineering activities, the design of coastal structures should encompass the following considerations and stages (see also Fig. 20.3): (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

specification of the structure’s function(s); description of the physical environment (boundary conditions); selection of envisaged construction technologies; inclusion of design of the structure’s operation and maintenance; conceptual design; preliminary design and selection of alternatives; geometrical dimensioning basing on far-field considerations; detailed design basing on near-field factors, including structural design; inclusion of possible construction constraints affecting the design; inclusion of some design flexibility allowing for redesign basing on monitoring of the operation and effectiveness of the structure after construction.

Functional requirements and design outline are depicted in Fig. 20.3. It is seen that the design in various stages is verified through the use of simulation models at different levels of complexity. Boundary conditions (bottom) constitute input to both design considerations and the models employed, while the functional requirements (top) ensure evaluation of the suitability of the design and provide design objectives at the same time.

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Design process.

The starting point in the design process consists of the identification of the beach erosion problem, followed by the selection of the type of protection measure; the final design can incorporate the risk analysis. Attention is drawn to the proper choice of the shore protection measure. The selection is usually affected by the cost. For example, beach nourishment can be cost-effective for low sediment deficit but might be comparable with offshore breakwaters for higher erosion rates. Aside from the cost, many other aspects must also be taken into account upon selection of the shore protection measure. Not shown in the drawing are legal restrictions, regional constraints and priorities, construction, operation and maintenance aspects, etc. Coastal structures are constructed to protect life and property against storm surges, to combat erosion and/or to create (often artificial) beaches for recreational purposes, and to preserve the natural environment. However, the absolute safety of an area or structure is nearly impossible to achieve. Therefore, it is much better to speak about the probability of failure (or safety) of a certain protection system. To implement this concept, all possible causes and outcomes of failure have to be analyzed. This concept is actually being developed for breakwaters52 and the dike and dune design, mostly in the Netherlands (see, www.enwinfo.nl).18,53,55 The “fault tree” is a handy tool for this aim. In the fault tree all possible modes of failure of elements, which can eventually lead to the failure of a structure section and to inundation are included. They can also badly affect the behavior of the structure, even if the latter is properly designed on the whole. Although all categories of

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events which may cause the inundation of a land or damage of structure are equally important for the overall safety, the engineer’s responsibility is mainly limited to the technical and structural aspects. In the case of coastal structures the following major events can be distinguished: • • • • • •

overflow or overtopping of the structure (i.e., instability of the superstructure); erosion or instability of slopes; instability of inner sections leading to progressive failure; scour and instability of toe-protection; instability of the foundation and internal erosion (i.e., piping); instability of the whole structure.

20.3. Alternative Systems for Coastal Protections Various coastal structures, as already discussed, can be applied to solve, or at least, to reduce erosion problems. They can provide direct protection (seawalls, dikes, revetments) or indirect protection (groins and offshore breakwaters of various designs), thus reducing the hydraulic load on the coast (Fig. 20.4). Rock and concrete are usually the construction materials. However, there is a growing interest both in developed and in developing countries in low cost or novel methods of shoreline protection particularly as the capital cost of defense works and their maintenance continues to rise. The shortage of natural rock in certain geographical regions can also be a reason for looking to other materials and systems. Despite this interest there is little published and documented information about the performance of low cost or patented structures especially at more exposed wave climate. Novel systems as geosystems (geotubes, geocontainers, geocurtains) and some other (often patented) systems (Reef Balls, Aquareef, prefabricated units, beach drainage, etc.) have gained popularity in recent years because of (often but not always) their simplicity in placement and constructability, cost effectiveness, and their minimum impact on the environment. These new systems were applied successfully in number of countries and they deserve to be applied on a larger scale. Because of the lower price and easier execution, these systems can be a good alternative for traditional coastal protection/ structures. The main obstacle in their application is, however, the lack of proper design criteria. An overview is given on application and performance of some existing novel systems and reference is made to the actual design criteria. Additional information on these systems can be in references and on websites. 20.3.1. Low-crested structures Low-crested and submerged structures (LCS) as detached breakwaters and artificial reefs are becoming very common coastal protection measures (used alone or in combination with artificial sand nourishment).5,9,17,22,57 As an example, a number of systems and typical applications of shore-control structures is shown in Figs. 20.4–20.7.

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Fig. 20.4.

Fig. 20.5.

Examples of shore-control and low-crested structures.

Definitions and objectives of low-crested/reef structures.

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Fig. 20.6.

Fig. 20.7.

Example of Aquareef.33

Example of Reef Balls units.31

The purpose of LCS structures or reefs is to reduce the hydraulic loading to a required level allowing for a dynamic equilibrium of the shoreline. To obtain this goal, they are designed to allow the transmission of a certain amount of wave energy over the structure in terms of overtopping and transmission through the porous structure (emerged breakwaters) or wave breaking and energy dissipation on shallow crest (submerged structures). Due to aesthetical requirements low freeboards are usually preferred (freeboard around SWL or below). However, in tidal environment and frequent storm surges they become less effective when design as a narrow-crested structures. That is also the reason that broad-crested submerged breakwaters (also called, artificial reefs) became popular, especially in Japan (Figs. 20.5, 20.6 and 20.10).5,33,40,41,48,60−62 They can also be an important measure in combating the effect of sea-level rise; the water level will gradually rise but the wave heights and thus, coastal erosion, run-up and overtopping can be reasonable reduced.

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However, broad-crested structures are much more expensive and their use should be supported by a proper cost-benefit studies. On the other hand, the development in alternative materials and systems, for example, the use of sand-filled geotubes as a core of such structures, can reduce effectively the cost.24,45,56 The upgrading of (integrated/multidisciplinary) design criteria for LCS structures recently took place in the scope of European project DELOS22 ; (see also: www.delos.unibo.it). The relatively new innovative coastal solution is to use artificial reef structures called “Reef Balls” as submerged breakwaters, providing both wave attenuation for shoreline erosion abatement, and artificial reef structures for habitat enhancement.7 An example of this technology using patented Reef BallTM is shown in Fig. 20.7. Reef Balls are mound-shaped concrete artificial reef modules that mimic natural coral heads. The modules have holes of many different sizes in them to provide habitat for many types of marine life. They are engineered to be simple to make and deploy and are unique in that they can be floated to their drop site behind any boat by utilizing an internal, inflatable bladder. Worldwide a large number of projects have already been executed by using this system. The first applications were based purely on experience from previous smaller projects. Since recently, more well-documented design criteria are available. Stability criteria for these units were determined based on analytical and experimental studies. For high energetic wave sites the units can be hydraulically anchored with cables to the sea bed. Wave transmission was studied in Canada.4 Technical design aspects are treated by Harris.31 20.3.2. Prefabricated systems There exist a number of other novel and/or low-cost materials and methods for shore protection (gabions and stone mattresses, open stone asphalt, used tire pile breakwaters, sheet pile structures, standing concrete pipes filled with granular materials, concrete Z-wall (zigzag) as breakwater, geotextiles curtains (screens), natural and mechanical drainage of beaches, and various floating breakwaters, etc.)54,56 Most of them are extensively evaluated and documented. However, more recently, a new family of prefabricated concrete elements as SURGEBREAKER offshore reef system, BEACHSAVER reef, WAVEblock, T-sill elements, and others have been developed and applied.54 The details on these systems can be found in references and on the websites. However, because of very narrow crest these prefabricated breakwaters are only efficient during mild wave conditions and their effects usually disappear during storm conditions, and because of scour and/or settlement, even losing their stability. The recent evaluation on performance of prefabricated, narrowcrested breakwaters can be found on the US Army website and publications.66 Some of these breakwaters are applied for comparison with other systems in recent US National Shoreline Erosion Control Development and Demonstration Program (Section 227), and more reliable information on the effectiveness of these systems can be expected within a few years (http://chl.erdc.usace.army.mil/CHL.aspx?p = s&a=PROGRAMS;3). The website provides details on sites/systems, and also documentation, if available.

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Fig. 20.8.

Principle of distorted ripple mat and application.50

20.3.3. Some other systems 20.3.3.1. Distorted ripple mat A new concept for creating shore accretion is actually developed and applied in Japan. A distorted (precast concrete blocks) ripple mat (DRIM) laid in the surf zone induces a landward bottom current providing accretion of a shore; see Fig. 20.8.50 The strong asymmetry of (artificial) ripple profile generates current near the bottom to one direction and thus sediment movement, whose concentration is high near the bottom, can be controlled with only very little environmental impact. The hydraulic condition on which the distorted ripple mat can control the sediment transport most effectively is studied experimentally and numerically and its capability to retain beach sand is tested through laboratory experiments and field installation. The definite onshore sediment movement by the control of DRIM is expected if the relative wave height H/h is less than 0.5, where H is the wave height and h is the water depth. The optimum condition for the efficient performance of DRIM is that d0 /λ > 1.7, where d0 is the orbital diameter of water particle and λ is the pitch length of DRIM, and this condition coincides with the condition in which natural sand ripples grow steadily. 20.3.3.2. Beach drainage (dewatering) systems Beach watertable drainage is thought to enhance sand deposition on wave uprush while diminishing erosion on wave backwash (Fig. 20.9). The net result is an increase in subaerial beach volume in the area of the drain. The larger prototype drainage by pumping installations used in Denmark and Florida suggest that beach aggradation

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Fig. 20.9.

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Principles of beach drainage.

may be artificially induced by beach watertable drainage. The state of the art of this technique is presented by Vesterby.77 An extensive evaluation of drainage systems is provided by Curtis and Davis18 and actual information can be found on the website: http://shoregro.com/. It is concluded that the drainage system has, in general, a positive effect on diminishing the beach erosion; however, its effectiveness is still difficult to control. The idea to achieve lowering of the water table without pumps by enhancing the beach’s own drainage capacity or hydraulic conductivity through the use of strip drains has been applied in Australia and in Japan.39

20.4. Some Remarks on Wave Transmission and Coastal Response 20.4.1. Wave transmission For shoreline control the final morphological response will result from the timeaveraged (i.e., annual average) transmissivity of the applied systems. However, to simulate this in the designing process, for example, in numerical simulation, it is necessary to know the variation in the transmission coefficient for various submergence conditions. Usually, when there is need for reduction in wave attack on structures and properties the wave reduction during extreme conditions (storm surges) is of interest (reduction of wave pressure, run-up and/or overtopping). In both cases the effectiveness of the measures taken will depend on their capability to reduce the waves; the submergence ratio and crest width are important factors (see Fig. 20.10).14,19,21,22,26,57,63,75,76,81 The transmission coefficient, Kt , defined as the ratio of the wave height directly shoreward of the breakwater to the height directly seaward of the breakwater, has the range 0 < K < 1, for which a value of 0 implies no transmission (high, impermeable), and a value of 1 implies complete transmission (no breakwater). Factors that control wave transmission include crest height and width, structure slope, core and armor material (permeability and roughness), tidal and design level, wave height, and period. As wave transmission increases, diffraction effects decrease, thus decreasing the size of sand accumulation by the transmitted waves and weakening the diffractioncurrent moving sediment into the shadow zone.30 It is obvious that the design rules

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Fig. 20.10.

Prototype measurements for Yugawara reef, Japan.3,49

for submerged structures should include a transmission coefficient as an essential governing parameter. Example of the transmission over the submerged structures (Aquareef) is shown in Fig. 20.11. More detailed descriptions of the functional and technical design of these reefs can be found in Hirose et al.33 The construction of detached breakwaters and, especially, artificial reefs (= submerged breakwaters with broad crest) is very popular and advanced in Japan. Their application had already started in the 1970s, supported by extensive model studies. The design techniques were gradually improved by using the results of a large number of prototype measurements and by monitoring completed projects (Fig. 20.10). Examples of prototype measurements in Japan and the Japanese design procedure can be found in Refs. 25, 27, 49, 68, and 82. 20.4.2. Layout and morphological response Most commonly an offshore obstruction, such as a reef or island, will cause the shoreline in its lee to protrude in a smooth fashion, forming a salient or a tombolo. This occurs because the reef reduces the wave height in its lee and thereby reduces the capacity of the waves to transport sand. Consequently, sediment moved by longshore currents and waves builds up in the lee of the reef.9–11,35,36,46 The level of protection is governed by the size and offshore position of the reef, so the size

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Fig. 20.11. General transmission characteristic for Aquareef33 (Ht is the transmitted wave height recorded on the landward side, H1/3 and L1/3 are the significant wave height and wavelength at the toe of the rubble mound, B is the crown width, and R is the submergence of the crown).

of the salient or tombolo varies in accordance with reef dimensions. Of course, one can expect this kind of morphological change only if the sediment is available (from natural sources or as sand nourishment). The examples of simple geometrical empirical criteria for the layout and shoreline response of the detached, exposed (emerged) breakwaters can be found in Refs. 20, 32, and 53. To include the effect of submergence (transmission) Pilarczyk57 proposes, at least as a first approximation, adding the factor (1− Kt) to the existing rules. Then the rules for low-crested breakwaters can be modified to (for example): • Tombolo: Ls /X > (1.0–1.5)/(1 − Kt ) or X/Ls < (2/3 to 1)(1 − Kt ),

or

X/(1 − Kt ) < (2/3 to 1)Ls . • Salient: Ls /X < 1/(1 − Kt ) or X/Ls > (1 − Kt ) or X/(1 − Kt ) > Ls . For salients where there are multiple breakwaters: GX/L2s > 0.5(1 − Kt ), where Ls is the length of a breakwater and X is the distance to the shore, G is the gap width, and the transmission coefficient Kt is defined for annual wave conditions. The gap width is usually L ≤ G ≤ 0.8Ls , where L is the wavelength at the structure defined as: L = T (gh)0.5 ; T = wave period, h = local depth at the breakwater. As first approximation, Kt = 0 for emerged breakwaters and Kt = 0.5 for submerged breakwaters can be assumed for average annual effects. These criteria can only be used as preliminary design criteria for distinguishing shoreline response to a single, transmissive detached breakwater. However, the range of verification data is too small to permit the validity of this approach to be assessed for submerged breakwaters.

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In general, one may conclude that these simple geometrical design rules, both for emerged as well as submerged breakwaters, are of limited value for design calculations because they do not include the effect of the rate of sediment transport, which can be very different for a specific coast. It is supported by the studies on effect of offshore breakwaters along UK coast.6,67 On the other side, it can be stated that numerical models (i.e., Genesis, Delft 2D-3D, Mike 21, etc.) can already be treated as useful design tools for the simulation of morphological shore response to the presence of offshore structures. Examples can be found in Refs. 13, 17, 28–30, 42, and 70. As mentioned above, while considerable research has been done on shoreline response to exposed offshore breakwaters, very little qualitative work has been done on the effect of submerged offshore reefs, particularly outside the laboratory. Thus, within the Artificial Reefs Program9 (www.asrltd.co.nz), Andrews2 examined aerial photographs seeking cases of shoreline adjustment to offshore reefs and islands. All relevant shoreline features in New Zealand and eastern Australia were scanned and digitized, providing 123 different cases. A range of other statistics, particularly reef and island geometry, was also obtained. Some of these results are repeated below. To examine the effects of wave transmission on limiting parameters, data for reefs and islands were considered separately. The data indicated that tombolo formation behind islands occurs with Ls /X ratio of 0.65, and higher and salients form when Ls /X is less than 1.0. Therefore, for islands the Ls /X ratios determining the division between salients and tombolos are similar to those from previously presented breakwater research. Similarly, data resulting from offshore reefs indicate that tombolo formation occurs at Ls /X ratios of 0.6 and higher, and salients most commonly form when Ls /X is less than 2. The data suggests that variation in wave transmission (from zero for offshore islands through to variable transmission for offshore reefs) allows a broader range of tombolo and salient limiting parameters. Thus, a reef that allows a large proportion of wave energy to pass over the obstacle can be (or must be) positioned closer to the shoreline than an emergent feature. Thus, from natural reefs and islands the following general limiting parameters were identified: • Islands: Tombolos form when Ls /X > 0.65; Salients form when Ls /X < 1.0; • Reefs: Tombolos form when Ls /X > 0.60; Salients form when Ls /X < 2.0; • Nondepositional conditions for both shoreline formations occur at Ls /X < ≈0.1. The choice of the layout of submerged breakwaters can also be affected by the current patterns around the breakwaters. The Japanese Manual (1988) provides (indicative) information on various current patterns for submerged reefs.82 However, for real applications it is recommended to simulate the specific situation by numerical or hydraulic modeling.

20.5. Geosystems in Coastal Applications 20.5.1. General overview Geotextile systems utilize a high-strength synthetic fabric as a form for casting large units by filling with sand or mortar. Within these geotextile systems a distinction

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Examples of application of geosystems: (a) Geotube as a breakwater, (b) BEROSIN

can be made between: bags, mattresses, tubes, containers, and inclined curtains.56 All of which can be filled with sand or mortar. Some examples are shown in Fig. 20.12. Bags are suitable for slope protection and retaining walls or toe protection but the main application is the construction of groynes, perched beaches, and offshore breakwaters. The tubes and containers are mainly applicable for construction of groynes, perched beaches, and (offshore) breakwaters, and as bunds for reclamation works (Fig. 20.13). Geotubes can form an individual structure in accordance with some functional requirements for the project but they can also be used complementary to the artificial beach nourishment to increase its lifetime. Especially for creating the perched beaches, the sand tubes can be an ideal, low-cost solution for constructing the submerged sill.34,45,56 Geotubes and geocontainers can also be used to store and isolate

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Example of reef structure composed with geotubes24 ; http://geotecassociates.com/.

contaminated materials from harbor dredging, and/or to use these units as bunds for reclamation works. An interesting application for shore erosion control is the geocurtain known under the name BEROSINTM,71 [Fig. 20.12(b)]. The BEROSIN curtain is a flexible structure made of various woven geotextiles, which after placing by divers near the shore and anchoring to the bed catches the sand transported by currents and waves providing accretion on a shore and preventing the erosion. The horizontal curtain (sheet) can be easily spread by a small workboat and two divers. The upper (shoreside) edge, equipped with some depth-compensated floaters, should be properly anchored at the projected line. The sea-side edge is kept in position by the workboat. By ballasting some of the outside pockets at the lower edge with sand or other materials and with the help of divers, the lower edge is sinking to the required position. The proper choice of permeability of geotextile creates the proper conditions for sedimentation of suspended sediment in front/or under the curtain and at the same time allowing the water to flow out without creating too high forces on the curtain and thus, on the anchors. In case of Pilot project at the coast of Vlieland (NL), some of the horizontal curtains placed in the intertidal zone have provided a growth of a beach/foreshore of 0.5–1.0 m within a week while others within a few weeks.56 These geocurtains can also be applied for the construction of submerged sills and reefs. In the past, the design of geotextile systems for various coastal applications was based mostly on vague experience rather than on the general valid calculation methods. However, the increased demand in recent years for reliable design methods for protective structures have led to the application of new materials and systems (including geotextile systems) and to research concerning the design of these systems. Contrary to research on rock and concrete units, there has been no systematic research on the design and stability of geotextile systems. However, past and recent research in the Netherlands, USA, Germany, and in some other countries on a number of selected geotextile products has provided some useful results which can be of use in preparing a set of preliminary design guidelines for the geotextile systems under current and wave attack.31,56,58 The recent, large scale tests, with large geobags, can be found on the website: http://sun1.rrzn.unihannover.de/fzk/e5/projects/dune prot 0.html.

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When installing geosystems, one should see to it that this does not take place on a rough foundation. Sharp elements may easily damage the casing of the element. Geosystems must usually not be filled completely. With a fill ratio of approximately 75% an optimum stability of the elements is reached. A sound soil protection is necessary if gravel (sand) sausages are used in circumstances where they are under attack of flow or waves. Stability criteria are discussed by Pilarczyk.56 The main (large) fill-containing geosystems (geobags, geotubes, and geocontainers filled with sand or mortar) and their design aspects are briefly discussed below. For more detailed information, the reader will be guided to the relevant manuals and publications (see references and websites). 20.5.2. Geobags Geobags can be filled with sand or gravel (or cement, perhaps). The bags may have different shapes and sizes, varying from the well-known sandbags for emergency dikes to large flat shapes or elongated “sausages” (see Fig. 20.14). The most common use for sandbags in hydraulic engineering is for temporary structures. Uses for sandor cement-filled bags are, among other things: • • • •

repair works; revetments of relatively gentle slopes and toe constructions; temporary or permanent groynes and offshore breakwaters; temporary dikes surrounding dredged material containment areas.

Fig. 20.14.

Application of geobags (sand or cement-filled).

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Because this material is easy to use and cheap, it is extremely suitable for temporary structures. A training groyne is a good example. The working of a groyne is difficult to predict in advance. That is why it is a good procedure to make such a construction using a relatively cheap product first, to see how one thing and another works out, and subsequently either make improvements or, after some time, a permanent structure. However, it is recommended not to use sand-filled geobags for a wave height above 1 m and/or a flow velocity above 2 m/s. The geosystems filled with sand cannot be used at these more extreme conditions because the sand in the systems is no longer internally stable. Sandbags can be placed as follows: (1) As a blanket: One or two layers of bags placed directly on the slope. An “interlocking” problem arises if the bags are filled completely. The bags are then too round. A solution is not to fill the bags completely, so that the sides flatten out somewhat, as a result of which the contact area becomes larger. (2) As a stack: Bags stacked up in the shape of a pyramid. The bags lie halfoverlapping with (usually) the long side parallel to the shoreline. The stacking of geotubes increases their stability due to interlocking with the neighboring bags. More recently, prototype experience and large scale tests with large geobags, are described by Restall,58 and more detailed information can be found on the website: http://sun1.rrzn.uni-hannover.de/fzk/e5/projects/dune prot 0.html. 20.5.3. Tube system Geotube is a sand/dredged material filled geotextile tube made of permeable but soil-tight geotextile.56 The desired diameter and length are project specific and only limited by installation possibilities and site conditions. The tube is delivered to the site rolled up on a steel pipe. Inlets and outlets are regularly spaced along the length of the tube. The tube is filled with dredged material pumped as a water–soil mixture (commonly a slurry of 1 on 4) using a suction dredge delivery line (Fig. 20.15). The choice of geotextile depends on characteristics of fill material. The tube will achieve

Fig. 20.15.

Filling procedure of geotube.

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its desired shape when filled up to about 70–80% (or a height equal to about a half of the flat width of the tube); a higher filling grade is possible but it diminishes the friction resistance between the tubes. Tube can be filled on land (e.g., as dikes for land reclamation, bunds, toe protection, or groyns) or in water (e.g., offshore breakwaters, sills of perched beaches, dikes for artificial islands, etc.). The tube is rolled out along the intended alignment with inlets/outlets centered on top. When a tube is to be placed in water, the effects of buoyancy on the tube geotextile prior to filling as well as on the dredged material’s settling characteristics must be considered. In order to maximize inlet/outlet spacing, an outlet distant from the inlet may be used to enhance the discharge of dredged slurry and thereby encourage and regulate the flow of fill material through the tube so that sufficient fill will flow to distant points. Commonly, the filter geotextile on both sides of a tube (against scour) and flat main tube are fully deployed by floating and holding them in position prior to beginning the filling operation. The geotextile filter is often furnished with small tubes at the edges which when filled with sand holds the filter apron in place. The required length of apron is usually two times the local wave height.

20.5.3.1. Shape and mechanical strength of geotubes For the selection of the strength of the geotextile and calculation of a required number of tubes for a given height of structure, knowledge of the real shape of the tube after filling and placing is necessary. The change of the cross-section of the tube depends on the static head of the (sand) slurry. Depending on this static head, the laying method, and the behavior of the fill-material inside the tube, it is possible that the cross-sectional shape of the filled tube will vary from a very flat hump to a nearly fully circular cross-section. More recently, Silvester64 and Leshchinsky43,44 prepared some analytical or numerical solutions and graphs allowing the determination of the shape of sand- or mortar-filled tubes based on some experiments with water. The Leshchinsky’s method combines all the previous developments and can be treated as a design tool. The design of the shape of the geotube is an iterative process. To obtain a proper stability of the geotube and to fulfill the functional requirements (i.e., required reduction of incoming waves/proper transmission coefficient, the width, and the height of the tube must be calculated). If the obtained shape of geotube does not fulfill these requirements, a new (larger) size of a geotube must be taken into account or a double-line of tubes can be used.

20.5.4. Container system Geocontainer is a mechanically filled geotextile and “box” or “pillow” shaped unit made of a soil-tight geotextile.56 The containers are partially prefabricated by sawing mill widths of the appropriate length together and at the ends to form an elongated “box.” The “box” is then closed in the field, after filling, using a sewing machine and specially designed seams (Fig. 20.16). Barge placement of

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Fig. 20.16.

Example of split barge and filling and placing of geocontainer.

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the site-fabricated containers is accomplished using a specially configured bargemounted crane or by bottom dump hoppers scows, or split barges. The containers are filled and fabricated on the barge and placed when securely moored in the desired position. Positioning of barge for consistent placement — a critical element of constructing “stacked” underwater structures — is accomplished with the assistance of modern surveying technology. The maximum volume of applied geocontainers was up to now 1000 m3 . The advantages of these large barge-placed geocontainers include: • Containers can be filled with locally available soil, which may be available from simultaneous dredging activities. • Containers can be relatively accurately placed regardless of weather conditions, current velocities, tides, or water depths. • Contained material is not subject to erosion after placing. • Containers can provide a relatively quick system buildup. • Containers are very cost competitive (for larger works). 20.5.4.1. General design considerations When applying geobags, geotubes, and geocontainers, the major design considerations/problems are related to the integrity of the units during filling, release and placement impact (impact resistance, seam strength, burst, abrasion, durability), the accuracy of placement on the bottom (especially at large depths), and the stability under current and wave attack. The geotextile fabric used to construct the tubes is designed to: • • • • •

contain sufficient permeability to relieve excess water pressure, retained the fill-material, resist the pressures of filling and the active loads without seams or fabric rapture, resist erosive forces during filling operations, resist puncture, tearing, and ultraviolet light.

The following design aspects are particularly of importance for the design of containers: (a) change of shape of units in function of perimeter of unit, fill-grade, and opening of split barge, (b) fall-velocity/equilibrium velocity, velocity at bottom impact, (c) description of dumping process and impact forces, (d) stresses in geotextile during impact and reshaping, (e) resulting structural and executional requirements, and (f) hydraulic stability of structure. Some of these aspects are briefly discussed hereafter.

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Fig. 20.17.

Development of forces during dumping of geocontainers.

20.5.4.2. Dumping process of containers and practical uncertainties A summary of various forces during the dumping and placement process is given in Fig. 20.17.56 (1) The required perimeter of geotextile sheet must be sufficient enough to release geocontainer through the given split width b0 for a required cross-sectional area of material in the bin of barge Af (or filling-ratio of fill-material in respect to the max. theoretical cross-section). The derivation of the required minimum length of perimeter of geotextile sheet is given by Pilarczyk.56 After opening of the split of a barge the geocontainer is pulled out by the weight of soil but at the same time the friction forces along the bin side are retarding this process. Due to these forces the tension in geotextile is developing at lower part and both sides of the geocontainer. The upper part is free of tension till the moment of complete releasing of geocontainer. The question is how far we are able to model a friction and the release process of geocontainer. (2) Geocontainer will always contain a certain amount of air in the pores of soil and between the soil and the top of (surplus) geotextile providing additional buoyancy during sinking. The amount and location of air pockets depends on soil consistency (dry, saturated) and uniformity of dumping. The air pockets will exert certain forces on geotextile and will influence the way of sinking. The question is how to model in a proper way the influence of soil consistency and air content on shape and stresses in geotubes/geocontainers. (3) The forces due to the impact with the bottom will be influenced by a number of factors: • consistency of soil inside the geocontainer (dry, semi-dry, saturated, cohesion, etc.) and its physical characteristics (i.e., internal friction);

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• amount of air; • permeability/air tightness of geotextile; • strength characteristics of geotextile (elasticity/elongation versus stresses, etc.); • fall-velocity (influenced by consistency of soil; saturated soil diminish amount of air but increases fall speed); • shape and catching surface of geocontainer at impact including effect of not horizontal sinking (i.e., catching of bottom with one end); • type of bottom (sand, clay, soft soil, rock, soil covered with rockfill mattress, etc.) and/or type of sublayer (i.e., layer of previous placed containers). During the impact the cross-sectional shape of geocontainer will be undergoing a continuous reshaping; from cone shape, first probably into a transitional cylindrical shape, and through a certain relaxation, into a semi-oval shape or flat triangular/rectangular shape dictated by soil type, perimeter, and elongation characteristics of geotextile. The question is how far we are able to model this impact phenomena and resulting forces/stresses in geotextile. The impact forces with the bottom are a function of the fall velocity (dump velocity) of a geocontainer. (4) In final situation the geocontainers will perform as a core material of various protective structures or as independent structure exposed to loading by currents and waves, and other loadings (ice, debris, ship collision, vandalism, etc.). In most cases the geocontainers will be filled by fine (loosely packed) soils. The question is how these structures will behave in practice under various types of external and internal loadings. Practical note: The prototype experience indicates that geocontainers with volume up to 200 m3 and dumped in water depth exceeding 10 m have been frequently damaged (collapse of seams) using geotextile with tensile strength lower than 75 kN/m, while nearly no damage was observed when using the geotextile with tensile strength equal to or more than 150 kN/m. This information can be of use for the first selection of geocontainers for a specific project. The placing accuracy for depths larger than 10 m is still a problem.8 20.5.5. Durability of geotextiles Durability of geotextiles is a frequently asked question especially concerning the applications where a long life-span is required. Geotextile is a relatively new product. The first applications are from 1960s. Recently, some 30-year-old geotextiles used as a filter in revetment structures in the Netherlands have been evaluated. In general, these geotextiles were still in a good condition. The technology of geotextiles is improved to such an extent that the durability tests under laboratory condition indicate the life-time of geotextiles at least of 100 years (when not exposed to UV radiation). There is no problem with durability of the geosystems when they are submerged or covered by armor layers. However, in case of exposed geosystems the UV radiation and vandalism are the factors, which must be considered during the design.

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All synthetics are vulnerable to UV. The speed of UV degradation, resulting in the loss of strength, depends on the polymer used and type of additives. Polyesters (PET) are by nature more light stable than, for example, polyamide (PA) and polypropylene (PP). The long-term prototype tests in the Netherlands indicated that the geosynthetics under water and in the intertidal zone show very little degradation in strength in comparison with geosynthetics placed on land; in the inter-tidal zone the geosynthetics are covered very soon by algae which provide very good UV protection. To avoid the problem with light degradation the fabrics must be properly selected (i.e., polyester) and UV stabilized. As the period in which the fabric is exposed is short (in terms of months), no serious problems are to be expected. In case of more or less permanent applications under exposed conditions the fabric must be protected against direct sunlight. There are a number of methods of surface protection for geosystems. To provide additional UV and abrasion protection to the exposed sections of tubes, a coating of elastomeric polyurethane is often used. This coating, however, has a tendency to peel after about a number of months and therefore, has to be re-applied. The permanent surface protection by riprap or block mats is a rather expensive solution and it will normally be applied only when it is dictated by necessity due to a high wave loading or danger of vandalism or other mechanical damage, i.e., boating, anchoring, etc. In other cases it will be probably a cheaper solution to apply a temporary protection of geotextile tubes by an additional layer of a strong geotextile provided with special UV-protection layer. This extra protection can be realized by adding the highly UV-stabilized nonwoven fleece needled onto the main fabric. The function of this felt layer is also to trap the sediment particles and algae, which give again extra UV protection.

20.6. Remarks on Stability Aspects Structural design aspects of low-crested structures are relatively well described in a number of publications.1,15,16,22,23,37,51,52,65,68,69,74,78,79,82 Some useful information on the design of breakwaters on reefs in shallow water can be found in Jensen et al.38 Usually for submerged structures, the stability at the water level close to the crest level will be most critical. Assuming depth limited conditions (Hs = 0.5 h, where h = local depth), the (rule of thumb) stability criterion becomes: Hs /∆Dn50 = 2 or, Dn50 = Hs /3, or Dn50 = h/6, where Dn50 = (M50 /ρs )1/3 ; Dn50 = nominal stone diameter and M50 and ρs = average mass and density of stone. The upgraded stability formulae for LCS structures, including head effect and scour, can be found in Delos report22 (www.delos.unibo.it). It should be noted that some of useful calculation programs (including formula by van der Meer)72,73 are incorporated in a simple expert system CRESS, which is accessible in the public domain (http://ikm.nl/rwscress/ and http://www.ihe.nl/ we/dicea). Useful information on functional design and the preliminary structural design of low crested-structures, including cost effectiveness, can be found in CUR report.17 Alternative solutions, using geotubes (or geotubes as a core of breakwaters), are treated by Pilarczyk.56 An example of this application can be found in Refs. 21, 24, and 57.

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20.7. Conclusions The author does not intend to provide the new design rules for alternative coastal structures. However, it is hoped that this information will be of some aid to designers looking for new sources, which are considering these kinds of structure and improving their designs. Offshore breakwaters and reefs can be permanently submerged, permanently exposed or inter-tidal. In each case, the depth of the structure, its size, and its position relative to the shoreline determine the coastal protection level provided by the structure. To reduce the cost some alternative solutions using geosystems can be considered. The actual understanding of the functional design of these structures needs further improvement but may be just adequate for these structures to be considered as serious alternatives for coastal protection. Continued research, especially on submerged breakwaters, should further explore improved techniques predict shore response and methods to optimize breakwater design. A good step (unfortunately, limited) in this direction was made in a collective research project in the Netherlands.17 Research and practical design in this field is also the focus of the “Artificial Reefs Program” in New Zealand (www.asrltd.co.nz), the International Society for Reef Studies (ISRS) (www.artificialreefs.org), the European Project DELOS (Environmental Design of Low Crested Coastal Defence Structures; http://www.delos.unibo.it), and recent US National Shoreline Erosion Control Development and Demonstration Program (Section 227). Some useful documents can also be found on the website: http://www.citg.tudelft.nl/live/pagina.jsp?id=4de0d195-5207-4e6784bb-455c5403ae47&lang=en; www.hydraulicengineering.tudelft.nl. Also, the past and recent research in the Netherlands, USA, Germany, and in some other countries on a number of selected geotextile products (geosystems) has provided some valuable results, which can be of use in preparing a set of preliminary design guidelines for the geotextile systems under current and wave attack. The following conclusions can be drawn on application of geosystems based on the actual developments and experience. • Geosystems offer the advantages of simplicity in placement and constructability, cost effectiveness, and minimal impact on the environment. • When applying this technology the manufacturer’s specifications should be followed. The installation needs an experienced contractor. • When applying geotubes and geocontainers the major design considerations/problems are related to the integrity of the units during release and impact (impact resistance, seam strength, burst, abrasion, durability, etc.), the accuracy of placement on the bottom (especially at large depths), and the stability. • The geotextile systems can be a good and mostly cheaper alternative for more traditional materials/systems. These new systems deserve to be applied on a larger scale. However, there are still many uncertainties in the existing design methods. Therefore, further improvement of design methods and more practical experience under various loading conditions are still needed. • The state of the art of the actual knowledge on the geosystems in hydraulic and coastal engineering can be found in Refs. 8, 34, 45, 56, and 58.

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These new efforts will bring future designers closer to more efficient application and design of these promising coastal solutions. The more intensive monitoring of the existing structures will also help in the verification of new design rules. The intention of this literature search is to uncover the technical information on these systems and make them available for the potential users. It will help to make a proper choice for specific problems/projects and it will stimulate the further developments in this field. International cooperation in this field should be further stimulated.

References 1. J. Ahrens, Characteristics of Reef Breakwaters, USAE CERC TR 87-17, Vicksburg (1987). 2. C. J. Andrews, Sandy shoreline response to submerged and emerged breakwaters, reefs or islands, unpublished thesis, University of Waikato, New Zealand (1997) (www.asrltd.co.nz). 3. T. Aono and E. C. Cruz, Fundamental characteristics of wave transformation around artificial reefs, 25th Coastal Eng., Orlando, USA (1996). 4. H. D. Armono and K. R. Hall, Wave transmission on submerged breakwaters made of hollow hemispherical shape artificial reefs, Canadian Coastal Conf. (2003). 5. T. Asakawa and N. Hamaguchi, Recent developments on shore protection in Japan, Coastal Structures and Breakwaters’91, London (1991). 6. P. Axe, S. Illic and A. Chadwick, Evaluation of beach modelling techniques behind detached breakwaters, ASCE, Proc. 25th ICCE, Orlando, USA (1996). 7. T. Barber, What are Reef Balls, Southwest Florida Fishing News (2000). 8. A. Bezuijen et al., Placing accuracy and stability of geocontainers, 3rd EuroGeo, Munich, Germany (2004), http://www.wldelft.nl/rnd/publ/search.html (insert for Author: Bezuijen). 9. K. Black and S. Mead, Submerged structures for coastal protection, ASR, Marine and Freshwater Consultants, New Zealand (1999) (www.asrltd.co.nz). 10. K. Black and C. J. Andrews, Sandy shoreline response to offshore obstacles, Part I: Salient and tombolo geometry and shape, J. Coastal Res. Special Issue on Surfing (2001). 11. K. Black, Artificial surfing reefs for erosion control and amenity: Theory and application, J. Coastal Res. 1–7 (2001). 12. K. R. Bodge, Beach fill stabilization with tuned structures; experience in the Southeastern USA and Caribbean, Coastlines, Structures and Breakwaters’98, London (1998). 13. K. J. Bos, J. A. Roelvink and M. W. Dingemans, Modelling the impact of detached breakwaters on the coast, 25th ICCE, Orlando, USA (1996). 14. M. Buccino and M. Calabrese, Conceptual approach for prediction of wave transmission at low-crested breakwaters, J. Waterways, Port, Coastal Ocean Eng. (May/June) (2007). 15. CEM, Coastal Engineering Manual, US Army Corps of Engineers, Vicksburg (2006). 16. CIRIA-CUR-CETMEF, The Rock Manual, CIRIA, London (2007) (also CUR/ CIRIA, Manual on use of rock in coastal engineering, CUR/CIRIA report 154, Gouda, the Netherlands, 1991). 17. CUR, Beach nourishments and shore parallel structures, R97-2, Centre for Civil Engineering Research and Codes (CUR), P.O.Box 420, Gouda, the Netherlands (1997).

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18. W. R. Curtis and J. E. Davis, Field evaluation/demonstration of a multisegmented dewatering system for accreting beach sand in a high-wave-energy environment, WES/CPAR-CHL-98-1 (1998). 19. K. d’Angremond, J. W. van der Meer and R. J. de Jong, Wave transmission at lowcrested structures, 25th Int. Conf. on Coastal Eng., Orlando, Florida (1996). 20. W. R. Dally and J. Pope, Detached breakwaters for shore protection, Technical report CERC-86-1, U.S. Army Engineer WES, Vicksburg, MS (1986). 21. Delft Hydraulics, AmWaj Island development, Bahrain; physical modelling of submerged breakwaters, Report H4087 (2002). 22. DELOS, Environmental design of low crested coastal defence structures; D 59 DESIGN GUIDELINES, EU 5th Framework Programme 1998–2002, Pitagora Editrice Bologna (2005) (www.delos.unibo.it). 23. Ch. Fleming and B. Hamer, Successful implementation of an offshore reef scheme, 27th Coastal Engineering, Sydney (2000). 24. J. Fowler, T. Stephens, M. Santiago and P. De Bruin, Amwaj Islands constructed with geotubes, Bahrain, CEDA Conf., Denver, USA (2002), http://geotecassociates. com/. 25. H. Funakoshi, T. Siozawa, A. Tadokoro and S. Tsuda, Drifting characteristics of littoral sand around submerged breakwater, Hydro-Port’94, Yokosuka, Japan (1994). 26. Y. Goda, Wave damping characteristics of longitudinal reef system, Advances in Coastal Structures and Breakwaters’95, London (1995); M. D. Groenewoud, J. van de Graaff, E. W. M. Claessen and S. C. van der Biezen (eds.), Effect of submerged breakwater on profile development, 25th ICCE, Orlando, USA (1996). 27. T. Hamaguchi, T. Uda, Ch. Inoue and A. Igarashi, Field experiment on wavedissipating effect of artificial reefs on the Niigata Coast, Coastal Engineering in Japan, JSCE 34(1) (1991). 28. H. Hanson and N. C. Kraus, GENESIS: Generalised model for simulating shoreline change. Report 1: Technical Reference, Tech. Rep. CERC-89-19, US Army Eng., WES (1989). 29. H. Hanson and N. C. Kraus, Shoreline response to a single transmissive detached breakwater, Proc. 22nd Coastal Eng. Conf., ASCE, The Hague (1990). 30. H. Hanson and N. C. Kraus, Numerical simulation of shoreline change at Lorain, Ohio, J. Waterways, Port, Coastal Ocean Eng. 117(1) (1991). 31. L. E. Harris, www.artificialreefs.org/ScientificReports/research.htm; Submerged Reef Structures for Habitat Enhancement and Shoreline Erosion Abatement; FIT Wave Tank and Stability Analysis of Reef Balls, http://www.advancedcoastaltechnology.com/science/DrHarrisWavereduction.htm. 32. M. M. Harris and J. B. Herbich, Effects of breakwater spacing on sand entrapment, J. Hydraulic Res. 24(5) (1986). 33. N. Hirose, A. Watanuki and M. Saito, New type units for artificial reef development of eco-friendly artificial reefs and the effectiveness thereof, PIANC Congress, Sydney (2002) [see also 28th ICCE, Cardiff (2002)]. 34. G. Heerten, Geotextiles in coastal engineering, 25 years experience, Geotex. Geomem. 1(2) (1982). 35. J. R. C. Hsu and R. Silvester, Accretion behind single offshore breakwater, J. Waterway Port Coastal Ocean Eng. 116, 362–381 (1990). 36. J. A. Jimenez and A. Sanchez-Arcilla, Preliminary analysis of shoreline evolution in the leeward of low-crested breakwaters using one-line models, EVK3-2000-0041, EU DELOS workshop, Barcelona, 17–19 January 2002.

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37. K. Itoh, T. Toue and D. Katsui, Numerical simulation of submerged breakwater deformation by DEM and VOF, Advanced Design of Maritime Structures in the 21st Century, Yokosuka, Japan (2001). 38. Th. Jensen, P. Sloth and V. Jacobsen, Wave dynamics and revetment design on a natural reef, 26th Coastal Engineering, Copenhagen (1998). 39. K. Katoh, S. Yanagishima, S. Nakamura and M. Fukuta, Stabilization of beach in integrated shore protection system, Hydro-Port’94, Yokosuka, Japan (1994). 40. T. Kono and S. Tsukayama, Wave transformation on reef and some consideration on its application to field, Coastal Eng. Jpn. 23 (1980). 41. Y. Kuriyama, K. Katoh and Y. Ozaki, Stability of beaches protected with detached breakwaters, Hydro-Port’94, Yokosuka, Japan (1994). 42. M. Larson, N. C. Kraus and H. Hanson, Analytical solutions of the one-line model of shoreline change near coastal structures, J. Waterway Port Coastal Ocean Eng., ASCE, 123(4) (1997). 43. D. Leshchinsky and O. Leshchinsky, Geosynthetic confined pressurized slurry (GeoCops): Supplement notes for Version 1.0, May (1995) (Nicolon/US Corps). 44. D. Leshchinsky, O. Leschinsky, H. I. Ling and P. A. Gilbert, Geosynthetic tubes for confining pressurized slurry: Some design aspects, J. Geotech. Eng., ASCE, 122(8) (1996). 45. C. R. Lawson, Geotextile containment: International perspectives, Proc. Seventeenth GRI Conf., Geosynthetic Institute, Philadelphia, USA, December (2003), pp. 198–221. 46. D. Ming and Y.-M. Chiew, Shoreline changes behind detached breakwater, J. Waterway Port Coastal Ocean Eng. 126(2) (2000). 47. A. Nakayama, N. Horikosi and H. Kobayashi, The planning and design of multipurpose artificial barrier reefs, Coastal Zone’93, Coastline of Japan II, New Orleans (1993). 48. A. Nakayama, M. Yamamoto, J. Yamamoto and A. Moriguchi, Development of waterintake works with submerged mound (WWSM), Hydro Port’94, Yokosuka, Japan (1994). 49. S. Ohnaka and T. Yoshizwa, Field observation on wave dissipation and reflection by an artificial reef with varying crown width; Hydro-Port’94, Yokosuka, Japan (1994). 50. N. Ono, J. Irie and H. Yamaguchi, Preserving system of nourished beach by distorted ripple mat, Coastal Engineering, Lisbon (2004), http://www.civil.kyushu-u.ac.jp/ engan/drim.html. 51. PIANC, Guidelines for the design and construction of flexible revetments incorporating geotextiles in marine environment, Brussels, Belgium (1992). 52. PIANC, Analysis of rubble mound breakwaters, PIANC, P.T.C. II/WG. 12, Suppl. To Bulletin 78/79, Brussels (1992). 53. K. W. Pilarczyk (ed.), Coastal Protection (A.A. Balkema, Rotterdam, 1990). 54. K. W. Pilarczyk and R. B. Zeidler, Offshore Breakwaters and Shore Evolution Control (A.A. Balkema, Rotterdam, 1996), www.balkema.nl. 55. K. W. Pilarczyk, Dikes and Revetments (A.A. Balkema, Rotterdam, 1998). 56. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, 2000), www.balkema.nl (select Pilarczyk). 57. K. W. Pilarczyk, Design of low-crested (submerged) structures: An overview, 6th COPEDEC, Sri Lanka (2003), www.enwinfo.nl (select English, downloads). 58. S. Restall et al., Australian and German experiences with geotextile containers for coastal protection, 3rd EuroGeo, Munich, Germany (2004). 59. A. Sanchez-Arcilla, F. Rivero, X. Gironella, D. Verges and M. Tome, Vertical circulation induced by a submerged breakwater, 26th Coastal Engineering, Copenhagen (1998).

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60. T. Sawaragi, I. Deguchi and S. K. Park, Reduction of wave overtopping rate by the use of artificial reef, Proc. 21st Int. Conf. Coastal Eng., ASCE (1988). 61. T. Sawaragi, Detached breakwaters; short course on design and reliability of coastal structures, Venice, 1–3 October 1992. 62. T. Sawaragi, Coastal Engineering-Waves, Beaches, Wave–Structure Interactions (Elsevier, 1995). 63. S. R. Seabrook and K. R. Hall, Wave transmission at submerged rubble mound breakwaters, 26th Int. Conf. Coastal Eng., Copenhagen (1998). 64. R. Silvester and J. R. Hsu, Coastal Stabilization, (Prentice Hall Inc., Englewood Cliffs, 1993), http://www.worldscibooks.com/engineering/3475.html. 65. D. Smit et al., Submerged-crest breakwater design, Advances in Coastal Structures and Breakwaters’95, London (1995). 66. D. K. Stauble and J. R. Tabar, The use of submerged narrow-crested breakwaters for shoreline erosion control, J. Coastal Res. 19(3), 352 (2003), http://chl.erdc.usace. army.mil/CHL.aspx?p=m&a=MEDIA. 67. F. Thomalla and Ch. Vincent, Designing offshore breakwaters using empirical relationships: a case study from Norfolk, UK, J. Coastal Res. 20(4) (2004). 68. T. Uda, Function and design methods of artificial reef (in Japanese), Ministry of Construction, Japan (1998) (see also, Coastal Zone’93). 69. US Army Corps, Engineering design guidance for detached breakwaters as shoreline stabilization structures, WES, Technical Report CERC–93–19, December 1993. 70. S. C. Van der Biezen, J. A. Roelvink, J. Van de Graaff, J. Schaap and L. Torrini, 2DH morphological modelling of submerged breakwaters, 26th ICCE, Copenhagen (1998). 71. van der Hidde, BEROSIN, Bureau van der Hidde, Harlingen, P.O.B. 299, the Netherlands (1995). 72. J. W. van der Meer, Stability of breakwater armour layers, Coastal Eng. 11 (1987). 73. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Doctoral thesis, Delft University of Technology (1988) (Also: Delft Hydraulics Communication No. 396). 74. J. W. van der Meer, Low-crested and reef breakwaters, Delft Hydraulics Rep. H 986 (1990). 75. J. W. van der Meer, Data on wave transmission due to overtopping, Delft Hydraulics (1990). 76. J. W. van der Meer and K. d’Angremond, Wave transmission at low-crested structures, Coastal Structures and Breakwaters (Thomas Telford, London, UK, 1991), pp. 25–40. 77. H. Vesterby, Beach drainage — state of the art — Seminar on Shoreline Management Techniques, Wallingford, 18 April 1996. 78. C. Vidal, M. A. Losada, R. Medina, E. P. D. Mansard and G. Gomez-Pina, A universal analysis for the stability of both low-crested and submerged breakwaters, 23rd Coastal Engineering, Venice (1992). 79. C. Vidal, I. J. Losada and F. L. Martin, Stability of near-bed rubble-mound structures, 26th Coastal Engineering, Copenhagen (1998). 80. N. Von Lieberman and S. Mai, Analysis of an optimal foreland design, 27th Coastal Engineering 2000, Sydney (2000). 81. T. Wamsley and J. Ahrens, Computation of wave transmission coefficients at detached breakwaters for shoreline response modelling, Coastal Structures’03, Portland, USA (2003). 82. K. Yoshioka, T. Kawakami, S. Tanaka, M. Koarai and T. Uda, Design manual for artificial reefs, in Coastlines of Japan II, Coastal Zone’93 (ASCE, 1993).

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Chapter 21

Geotextile Sand Containers for Shore Protection Hocine Oumeraci∗ and Juan Recio Leichtweiss-Institute for Hydraulic Engineering and Water Resources, Technical University Braunschweig, Beethovenstr. 51a, 38106 Braunschweig, Germany ∗ [email protected] This chapter aims at providing a brief overview on geotextile sand containers (GSCs) applied in coastal engineering for shore protection. First, the engineering properties required for the geotextile used for sand containers as well as the durability and the lifetime prediction issue are discussed. Second, some example applications are provided to illustrate the versatility of GSCs as an appropriate soft shore prediction alternative to conventional hard coastal structures made of rock and concrete units. However, the major part of the chapter is aimed to address the hydraulic stability of the containers constituting a shore protection structure subject to wave attack. For this purpose, simple formulae are first proposed for the stability of the slope and crest containers. The processes which may affect the hydraulic stability are then discussed to highlight the necessity of developing more process-based stability formulae. New stability formulae are finally proposed which can also account explicitly for the effect of deformation of the containers. Finally, a discussion is provided on the comparative analysis of the stability of the slope and crest containers with and without consideration of the deformation effect.

21.1. Introduction In view of the increasing storminess associated with climate change and its effect on coastal flood and erosion, more versatile materials and solutions are required for the design of new, cost-effective shore protection structures as well as for the reinforcement of existing threatened coastal barriers and structures, including dune reinforcement and scour protection. In search for low-cost, soft, and reversible solutions, the concept of geotextile sand containers (GSCs) as “soft rock” for coastal defense structures was introduced for the first time in the 1950s in the form of “sand bags” for the construction of a dike to close the inlet “Pluimpot” in the Netherlands.53 553

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At the same time some test groins were constructed by using “sand bags” at the German North Sea coast.10,57 Later, GSCs have mainly been used as temporary shore protection measures due to the difficulties associated with the assessment of the long-term performance. Meanwhile, significant advances have been made with respect to the following issues: • improvement of the long-term performance of geotextiles (additives and stabilizers against UV-radiation, coating against abrasion, etc.), • assessment of the durability and lifetime prediction (accelerated testing, standards, etc.),3,14,46 • survey of GSC-built structures and analysis of past experience with respect to the degradation mechanisms, and • understanding of the mechanisms of failure, including hydraulic instability under severe wave action.32,34 These advances, together with numerous advantages of GSCs as “soft rock,” have contributed to extend the use of GSCs to permanent coastal defenses, including a wide range of types of structures such as seawalls, revetments, groins, artificial reefs, offshore breakwaters, perched beaches, dune reinforcement, core of rubble mound structures, scour protection, etc. (Fig. 21.1). Generally, in coastal engineering any containment of sand encapsulated in geotextile to build flexible and erosion-resistant gravity structures is called “geotextile sand container.” A variety of size and shapes of GSCs have been used, including “geotubes,” “geocontainers,” and “geobags.” The latter have a volume of about 0.05–5 m3 and are generally filled offsite. Geocontainers are much larger (sausage shape up to 700 m3 ) and generally filled in split bottom barges. Geotubes have a diameter up to 5.5 m and are filled directly at the location where they are built (see Fig. 21.2 and Table 21.2). In most cases, it is advantageous to use smaller volume containers because: (i) they are more versatile and can be adapted to build any type of structures; (ii) they can better fulfill any requirements with respect to structure slope and geometry (better tolerance); (iii) maintenance and remedial work are much easier in case of vandalism or failure by wave action; (iv) less tensile strength is required and less change of shape will be experienced, thus resulting in longer lifetime; (v) higher density of the sand fill can be achieved; (vi) less risk of liquefaction of the sand fill is expected and thus less GSCdeformations; (vii) simpler mobilization of the required equipment; and (viii) the smaller the containers, the larger the self-healing capacity of the structure. However, larger containers may be required, for instance, in the case of higher waves forces and for temporary structures. A brief illustration of the container sizes used in coastal engineering is given in Fig. 21.2.

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Fig. 21.1.

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Coastal engineering applications of geotextile sand containers.

In the following, some basic information on geotextiles, including a discussion on the durability issue are first provided. Second, the versatility of the use of GSCs as a soft shore protection alternative to conventional hard structures made of rock and concrete units is illustrated by some example applications. The major part of the chapter will, however, focus on the hydraulic stability of the GSCs under wave action. For this purpose, simple formulae for the stability of slope and crest GSCs will first be proposed which do not take explicitly into account the effect of GSC-deformation and friction between containers. A detailed description of the processes, which may lead to failure under wave action, is then provided to illustrate the necessity of developing more process-based stability formulae. Finally, the new detailed stability formulae and the simple formulae are comparatively analyzed to stress the effect of GSC-deformation on the hydraulic stability.

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Range of nonwoven geotextile sand container sizes applied in coastal engineering.

21.2. Geotextile Properties and Durability of Geotextile Sand Containers 21.2.1. Required properties of the geotextile for sand containers Geotextiles and geomembranes, including their related products, are called geosynthetics;19,50 i.e., fabrics that are specially manufactured for civil and environmental engineering applications. While geomembranes are impermeable to water, geotextiles are permeable. The most widely used polymer for geotextiles is polypropylene (PP > 90%), followed by polyester (PET ≈ 5%) and polyethylene (≈ 2%). Based on the manufacturing process two major categories of geotextiles may be distinguished: nonwoven (≈ 60%) and woven geotextiles (≈ 40%). Nonwoven geotextiles are composed of directionally or randomly oriented fibers which are mechanically (needle punching), chemically, or thermally bonded into a loose web. Woven geotextiles are obtained by interlacing two or more sets of yarns (one or several fibers), using conventional weaving processes with a weaving loom. The yarns can be mono-filament, slit film, fibrillated or multi-filament. Besides these two main groups there are also knitted and stitched geotextiles. The main engineering properties of nonwoven and woven geotextiles are comparatively summarized in Table 21.1.

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Geotextile Sand Containers for Shore Protection Table 21.1.

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Properties of nonwoven and woven geotextiles (After Lawson and Kemplin, 1995)∗ .

Types of geosynthetics Nonwovens Heat-bonded Needle-punched Resin-bonded Wovens Mono-filament Multi-filament Flat tape Knitted Weft Warp Stitch-bonded

Tensile strength (kN/m)

Extension at max. load (%)

Apparent opening size (mm)

Water flow rate (volume permeabiltiy) (liters/m2 /s)

Mass per unit area (g/m2 )

3–25 7–90 5–30

20–60 30–80 25–50

0.02–0.35 0.03–0.20 0.01–0.25

10–200 30–300 20–100

60–350 100–3000 130–800

20–80 40–1200 8–90

20–35 10–30 15–25

0.07–4.0 0.05–0.90 0.10–0.30

80–2000 20–80 5–25

150–300 250–1500 90–250

2–5 20–800 30–1000

300–600 12–30 10–30

0.20–2.0 0.40–1.5 0.07–0.50

60–2000 80–300 50–100

150–300 250–1000 250–1000

∗ C. R. Lawson and G. T. Kempton, Geosynthetics and Their Use in Reinforced Soils (Terram Ltd., UIC, 1995).

As shown in Table 21.1, nonwoven and woven fabrics have significantly different properties which can be exploited to produce the best solution for each specific need, including composite fabrics to combine the advantages of both types. The lower tensile strength of nonwoven geotextiles as compared to woven geotextiles might represent a drawback, if the containers have to accommodate very large stresses during installation without failure. Generally, however, the stresses on the containers are much lower in service than during installation. The higher capacity for elongation of nonwoven geotextiles can to some extent compensate the disadvantage of lower tensile strength in the sense that it allows to accommodate large strains without failure. This is particularly important when the container is required to reshape during installation and in service (adaptation to scour development and settlement, self-healing effect). Project experiences have shown that the final dimensions of sand-filled containers made of nonwoven geotextile are comparable to those made of woven geotextile. Based on tests with both materials — woven and nonwoven — the final height of hydraulically filled geotextile containers (tubes) was approved to be 80% of the theoretical diameter.17 The hydraulic permeability of geotextile used for sand containers is particularly important when subject to cyclic wetting and drying (e.g., in tidal regime). Water should be drained from the sand container fast enough to avoid excess pressure build up and to ensure overall stability. Therefore, it is generally required that the geotextile should have a much higher permeability than the sand fill without losing the finer fractions. Alternatively, the geotextile can be selected to fulfill the commonly used filter criteria. Nonwoven geotextiles have a higher permeability and a higher fine retention capability than their woven counterparts, but it should be stressed that the permeability is a function of the fabric thickness, and thus depends on its compressibility under normal stresses. The abrasion resistance is particularly important in the surf environment where coarse angular sand, shell fragments, or coral debris are present. The larger thickness of nonwoven geotextiles and the ability of their structure to retain sand material

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make them more resistant. For testing abrasion resistance of different geotextile products, a special test setup has been introduced by the Federal Waterways Engineering and Research Institute (BAW, Germany) already about 30 years ago.47 For geotextiles being used in the surf environment abrasion tests, standard test setup (as described in BAW Guidelines for Testing Geotextiles for Navigable Waterways4,47) are recommended. The puncture resistance is important in the case of vandalism, drift ice, drift wood, or dropped rock material during construction. The catalogue for geotextile testing in waterway engineering established by the German Institute for Waterway Engineering (BAW) also includes a puncture test setup which allows a 1:1 test according to the actual stone drop energy expected for the specific job conditions.23,47 Due to their high elongation capacity and to the retained sand within the fabric structure, nonwoven geotextiles are more susceptible to limit damage from puncture, including vandalism. A higher friction angle between sand containers is desirable to enhance the hydraulic stability against wave and current actions. Due to their structure, nonwoven geotextiles provide a higher friction (see also Sec. 21.4.25). The resistance against UV-radiation and oxidation still represents one of the most critical issues which can limit the service time of exposed GSCs. The resins, fibers, and yarns of the used woven and nonwoven fabrics need special stabilization packages to avoid deterioration already during production and to guarantee longterm stability for more than 50 years. As they are reasonable cost factors, the special stabilization packages need to be described in the specifications or material certificates with, e.g., confidential information to the certification institute. In general, increasing the thickness of fibers and yarns and increasing weight of the fabric improve UV resistance and long-term stability.49 Dug-up operations from woven and nonwoven geotextiles carried out in the seventies have already shown up to 25 years long-term stability with no indication of further concern to the upcoming years. This is also linked to the fact that the design parameters for surviving handling and installation are giving much reserve for the container simply lying on the sea bed after successful installation. Well-designed and well-installed woven and nonwoven geotextiles for coastal engineering applications have indeed shown that they can be resistant over a long term.9 Despite the significant progress in the use of UV-stabilizers, coating or/and armoring of the exposed geotextile containers still remain the sole alternative to achieve a satisfactory lifetime without damage. Simultaneously, the coating/armoring will also protect the geotextile against abrasion, vandalism, drift ice, and drift wood. The resistance against chemical and microbiological attack of polymers (e.g., polypropylene) is very high, so that no significant loss of strength is expected during the design lifetime. The ability of geotextiles to enhance marine growth and to attract/support diverse invertebrate communities also becomes an important issue, if maximizing the biodiversity of the recruiting communities is relevant for the choice of the type of geotextile. First, results of investigations comparing woven and nonwoven geotextiles in Australia have shown that the latter are more favorable in this respect.5 The marine growth may represent an enhancement of the resistance against

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UV-radiation and abrasion. However, it is still not clear whether it may cause serious changes or detrimental effects on the mechanical properties of the geotextile over lifetime. 21.2.2. Durability and lifetime prediction Unlike conventional construction materials such as rock and concrete, synthetic geotextiles are, even after 50 years experience, for most coastal engineers relatively new products. Therefore, their degradation and long-term performance are still not well understood. Instead of trying to answer the most frequently asked question: “How long will a geotextile structure last?”, it is more practical from the engineering point of view to ask, how long must a geotextile structure last. The expected lifetime when designing “permanent” shore protection structures is typically 20–100 years. Modern geotextiles are designed to be resistant to degradation from UV-radiation, chemical/biological attack, abrasion, and hydraulic loading. Generally, a lifetime in the order of 20–25 years can be expected if damages during construction and through vandalism are avoided. However, not all applications in coastal engineering require such a level of lifetime, for instance, in the case of temporary protection measures. Although lifetimes up to 100 years have been suggested based on accelerated testing and extrapolation, the following question still remains unanswered: “How to predict/achieve a 100 years or more lifetime for geotextile structures applied for shore protection?” Since much is known about the degradation mechanisms and degradation rates of polymer materials, and how these can be reduced or prevented (Fig. 21.3), it seems reasonable to use this knowledge:

Fig. 21.3.

Degradation mechanisms and reduction procedures.

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• to develop further “index” tests similar to those proposed in Annex B of the European Standards EN 13249–13257 and EN 13265 which are believed to ensure a minimum durability of 25 years and • as a basis for planning and interpreting site monitoring (see ISO 13437), including the development of further procedures and techniques to reduce the degradation rates. Ideally, a degradation curve over the design lifetime for each relevant property of the geotextiles such as tensile strength, specific mass in g/m2 , elongation capacity, and hydraulic permeability should be determined, together with the associated acceptable degradation limits at which the geotextile cannot perform its primary function (Fig. 21.4). The most authoritative evidence for long-term durability is generally obtained from monitoring of the degradation under real service conditions. In fact, the empirical evidence of long-term durability from retrieved (nonexposed) geotextiles has shown that the reduction of tensile strength and other important engineering properties strongly depends on the prevalent service conditions and therefore significantly differs from one site to another. Samples of PP (polypropylene) and PET (polyester) nonwoven geotextiles retrieved from 25 sites in France lost up to 30% of their tensile strength after 10–15 years service time as filters, separators, and drains, while no chemical/biological effects were identified.51 A further interesting case was reported by Lefaive,21 showing that the reduction in tensile form strength after 17 years of the same PET straps embedded in a

Fig. 21.4.

Degradation curves (principle sketch).

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concrete facing wall and in the backfill was completely different: 2% in the backfill with pH = 8.5 and up to 40% at the transition between the wall and the backfill where pH values of 13–14 and temperature up to 30◦ C prevailed. This case well illustrates the contribution of alkaline surface attack (25%), internal hydrolysis (5–10%), and mechanical damage to degradation of PET fibers or yarns. The effect of UV-radiation is illustrated by a case reported by Troots et al.,52 where woven PET samples were retrieved after 13 years from an earth embankment: while along the section of the slope covered by vegetation and bitumen to provide protection against UV-attack no significant changes of the geotextile properties occurred; a reduction up to 50% of the tensile strength was identified in the nonprotected part. These and further numerous examples from the literature show that the results/ data from retrieved (essentially nonexposed) fabrics, although very valuable, have serious limitations when intended to be used for the prediction of long-term durability and lifetime of permanent structures. Moreover, the obtained data are often incomplete and relate to conditions that are generally far from those for which the prediction/assessment is being made (Fig. 21.5). In order to make the best use of field data some brief recommendation for future site monitoring are given in Plate 21.1. These limitations and the urgent necessity for both users and manufacturers to predict lifetime of geotextiles have led to the development of accelerated tests which also have serious limitations (Fig. 21.5).

Fig. 21.5. Limitations of present approach to predict/assess lifetime of permanent geotextile structures for shore protection.

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Plate 21.1.

Fig. 21.6.

Recommendations for future site monitoring.

Principle and limitations of accelerated tests.

The principle of accelerated testing is briefly summarized in Fig. 21.6, showing that (i) generally only one dominant degradation mechanism can be considered, thus ignoring the interaction with other mechanisms and (ii) the approach cannot be applied to all degradation mechanisms.

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Moreover, if two degradation mechanisms occur sequentially (e.g., UV-radiation followed by mechanical degradation), then the two mechanisms are analyzed separately and the predicted lifetimes simply added. Further difficulties arise when extrapolating the short lifetime obtained from accelerated tests at increased load frequency/intensity and increased temperature to predict longer lifetime at service conditions. Power laws are often used for extrapolation. For more details, refer to Greenwood and Friday.7 Ideally, the results of both site monitoring and accelerated tests in combination are expected to provide the best basis for the prediction of long-term durability and lifetime. However, a consistent methodology to combine both approaches still needs to be developed (Fig. 21.5). In summary, it can be stated that geotextile applications, although using previous weaker versions of polymer material, performed relatively well over many decades when not exposed to UV-radiation.56 Most failures observed were rather due to faulty design, incorrect choice of material, and poor quality of installation. The knowledge available on the degradation mechanisms, although still limited, allows to predict rationally lifetime up to about 25 years. Since rational prediction cannot foresee problems for which there is no empirical evidence or scientific basis, the primary goal of future research toward the assessment of durability and lifetime is to improve the understanding of all relevant degradation mechanisms — separately as well as in combination — by making use of both site monitoring and laboratory testing. Future significant improvement of UV-resistance and aesthetical aspects as well as the improvement of the long-term performance of GSCs against large wave loads represent further R&D challenges toward avoiding to cover or armor geotextile structures.

21.3. Example Applications of Geotextile Sand Containers for Shore Protection The types of GSCs used in coastal engineering as temporary or permanent structures are generally referred to in the literature as geotubes, geocontainers, or geobags (see Table 21.2 and Sec. 21.1). For permanent coastal structures, small volume containers offer more advantages and are therefore often preferred (see Sec. 21.1). In particular, they are more versatile in application and are used for different class of structures, including dune reinforcement, seawalls and revetments, detached breakwaters and artificial reefs, groins, etc. (see Fig. 21.5). Comprehensive largescale model investigations on GSCs used for the scour protection of monopile foundation of offshore wind turbines have also been conducted by Gr¨ une et al.8 The results are described in more detail in a final report by Oumeraci et al.29 Reviews of example applications related to geotextiles in general but, also including geocontainers, can be found in Heerten,11,12 van Santvoort et al.,53 and Pilarczyk.31 Comprehensive reviews on the applications of geosynthetics in hydraulic engineering and for the protection of land fill (including coastal areas) which may also provide valuable information and inspiration for the application

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564 Table 21.2. Type 1. Geotubes

2. Geocontainers

3. Geobags

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Types of geotextile containments used in coastal engineering.

Volume (m3 ) Sand fill Generally >700 m3

On site

Generally 100–700 m3

Split bottom barge Offsite

0.05–5 m3

Shape

Applications

• • • Cylindrical/pillow • (D < 5 m) • Cylindrical (D = 1–55 m)

Pillow, box, mattress

Groins Containment dikes Nonpermanent structures Reef structures (surf zone) Defense structures against tsunami As soft rock units to build any type of coastal structures. Also for scour protection and dune reinforcement

in coastal engineering are given by Heibaum et al.15 and Kavazanjian et al.,18 respectively. More specific reviews related to geocontainers are provided by Fowler and Trainer,6 Lenze et al.,22 Lawson,20 Saathoff et al.,48 and Jackson et al.16 Rather than trying to duplicate the examples from the aforementioned reviews and to provide a further comprehensive review, it is attempted in the following to select only few examples from three classes of structures and applications: (a) long-shore barriers in the form of seawalls, revetments and dune reinforcement, (b) cross-shore barriers in the form of sea groins, and (c) a new possible application of GSCs as a core of rubble mound structures. The latter type of application may also become particularly important when armoring the GSC-structure is required due to too severe wave attack, abrasion, UV-radiation, and vandalism. 21.3.1. Revetments, seawalls, and reinforcement of beach–dune system Most of the applications of geotextile containment in coastal engineering belong to this type of shore protection; i.e., the containment is built directly along the shoreline to prevent erosion and to stabilize a beach–dune system during storm surge (Fig. 21.7). For this purpose, different types of containments have been applied, very often as a last defense line in combination with beach nourishment. An impressive example of the performance of such a last defense line behind a beach nourishment is the wrapped sand containment needle-punched composite geotextile (woven PP slit film and nonwoven PET) to reinforce a dune on the island of Sylt (North Sea, Germany) is shown in Fig. 21.8. The stability of this stepped barrier was successfully tested in the Large Wave Flume (GWK) of Hannover. Since 1990, it survived several storm surges with water levels of about 2.5 m above normal and wave heights up to 5 m. Only the sand cover was removed, confirming that the nickname “Bulletproof Vest” commonly given to this type of construction is appropriate. More details on the design and construction of this shore protection are given by Nickels and Heerten26 and Lenze et al.22 Similar geotextile sand containments have also been used successfully at many other sites. Some of them are well documented by ACT.1

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Fig. 21.7. Seawall made of geotextile sand containers (principle sketches): (a) plan view and (b) cross-section.

Fig. 21.8. Geotextile containment for dune reinforcement, Sylt/Germany (extended and modified from Ref. 26).

More flexible and much simpler in both engineering design and installation, but equally efficient are smaller volume containers (Fig. 21.9). Moreover, smaller containers have many advantages over larger containers and tubes (see Sec. 21.1 and Ref. 2). A comparative analysis of containers with V = 0.75 m3 and containers with a 30 times larger volume for dune reinforcement which confronted the pros and cons of both methods from the client and contractor view point clearly resulted in the selection of the smaller containers.2

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(a)

(b)

(c) Fig. 21.9. Beach and dune reinforcement with geotextile sand containers. (a) Beach reinforcement in Australia, (b) Stockton Beach Revetment,48 and (c) Dune Reinforcement in Wangerooge, North Sea/Germany.54

Using geotextile containment for a long-shore barrier in the form of seawalls, revetment or beach-dune reinforcement has several advantages over hard barrier such as rock structures (see Sec. 21.1), but there are also some drawbacks. The most important drawback is that the sand cover has to be fully or partially rebuilt after each important storm surge because a natural recovery is not always possible. Due to their lower permeability and larger slope steepness, GSC-structures have generally higher reflection coefficients and higher wave overtopping rates than rubble mound structures (Fig. 21.10).

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(a)

(b) Fig. 21.10. Wave reflection and overtopping performance of GSC-seawalls.28 (a) Overtopping performance and (b) reflection performance.

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Fig. 21.11.

Down drift erosion induced by seawalls.

Moreover, it should be kept in mind that a long-shore GSC-barrier also induces a down drift erosion similar to those induced by hard structures (Fig. 21.11), including erosion of the foreshore. 21.3.2. Sea groins Geotextile sand containers, including geotubes, have often been used for emerged and submerged groins (Fig. 21.12). Generally, the containers are uncovered and directly exposed to wave impact, abrasion, and UV-radiation. Therefore, the fabric should be heavily treated for UV-stability and should consist of an inner layer for strength and an outer layer for robustness, durability, and abrasion resistance (geocomposite). Sometimes, the

Fig. 21.12.

Sea groins (principle sketches).

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fabric is made in a color that blends naturally with the beach environment.24 Heavy-duty UV stabilized nonwoven needle-punched geotextiles with high tenacity polyester thread in all seams (min. 80% of basic fabric strength) have also been successfully used.44 Despite large displacement that occurred during storms, the groin continued to provide protection and even withstood abrasion and UV attack over 10 years.48 To reduce the costs in the case of larger groins and larger projects, the groin core can be made of smaller container with lower requirements while strengthening the outer protective layer.24 Further example applications are provided by Saathoff et al.48 and Restall et al.44,45 21.3.3. Core of rubble mound structures There are several reasons which might lead the engineer and other decision-makers in practice to use sand instead of conventional quarry run for the core of rubble mound breakwaters and structures, including among others: (i) Nonavailability of rock material in sufficient quantities and at affordable costs. (ii) Avoiding sediment infiltration through rubble mound structures which may result in the shoaling of navigation channels and harbor basins, and thus in higher maintenance dredging costs. (iii) Reduction of wave transmission through the structure which might particularly be crucial in the case of long waves. On the other hand, the use of sand as a quasi-impervious core instead of quarry stone would result in an increase of (i) wave setup and runup at the structure, (ii) wave overtopping, (iii) wave reflection, which might be detrimental to the stability of the structure, to the operation on and behind the breakwater (due to excessive overtopping) as well as to navigation and seabed stability. Moreover, serious difficulties arise in practice when trying to design and construct the filter to protect the sand core against wash out by wave action. Applying geometrically closed filter criteria would result in very complex, multiple, and relative thin filter layers which will not only be very costly and very difficult to build in larger water depths, but also might certainly fail due to the almost unpredictable very complex loading conditions of the sand core under cyclic pulsations by waves and entrained air at the interface with the last filter layer. Such failures have indeed been observed under both laboratory and field conditions in the past. Laboratory evidence has also shown that introducing the so-called “geometrically open filter” criteria to design a “hydraulic sand-tight filter” may reduce the number of filter layers. However, the main practical difficulties mentioned above will remain, including those associated with the long-term stability of the sand-core due to the high complexity of the loading and its uncontrollability during the entire storm duration and over the life cycle of the structure.

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Geotextile filters might present themselves as an alternative to the very complex, costly, and uncertain filter made of multiple layers of granular material. However, geotextile mats are not only difficult to install under waves and currents, but also may introduce a shear surface which might be detrimental to the stability of the armor layer. Therefore, geotextile filters need careful installation by experienced contractors with special equipment, and the design should address the shear plane between the geotextile filter and the layers below and above. Also, “sandmats” as composite products of nonwoven geotextiles and sand needle-punched together may be a suitable solution.13,14 A more feasible alternative is to use a core made of GSCs. This will not only allow to overcome the aforementioned core stability problems, but also to provide (i) a better erosion stability of the core and (ii) an increased stability against seismic loads as compared to a core simply made of loose sand. However, many of the drawbacks mentioned above remain with respect to wave setup, runup, overtopping, reflection, and armor stability in comparison to a conventional core. Therefore, an extensive research program has been initiated at LeichtweissInstitute to study both hydraulic performance and armor stability, including the processes involved and the development of prediction formulae for the design of a class of rubble mound structures with a core made of geotextile sand containers (Fig. 21.13). The first phase of this research program which is concerned with hydraulic model tests to study in the twin-wave flumes of LWI the hydraulic performance and the armor stability of a rubble mound breakwater made of GSCs as compared to its conventional counterpart with a core made of quarry stones, has been completed

Fig. 21.13.

Class of geocore structures in comparison to conventional rubble mound structures.

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and the results are published by Oumeraci et al.30 Prediction formulae for wave reflection, wave runup and overtopping, wave transmission, and armor stability have been determined for the nonconventional breakwater in comparison with the conventional rubble mound breakwater.30 21.4. Hydraulic Stability of Geotextile Sand Containers Depending on the prevailing wave loads and degradation mechanisms, GSCs may experience different types of failure modes: (i) hydraulic failure modes, (ii) geotechnical failure modes, and (iii) failure modes related to the geotextile itself (Fig. 21.14). In the following, only the first type which is related to hydraulic stability under wave loads will be addressed. First, simple stability formulae without explicit account of the effect of deformation will be proposed separately for the containers on the slope and the containers on the crest of the structure (called hereafter “slope containers” and “crest containers”). Second, the necessity of a better understanding of the processes responsible for the deformation of the sand containers under wave loads as well as their effect on the hydraulic stability is illustrated by some selected results from recent research. Finally, more detailed stability formulae are proposed which can also explicitly account for the deformation effects, including a comparison with the simple stability formulae.

Fig. 21.14.

Failure modes for geotextile sand containers.

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21.4.1. Simple stability formulae Due to the different wave loads and boundary conditions which prevails on the slope and on the crest of a coastal structure, a different stability behavior and thus different stability formulae are expected for the containers on the slope and those on the crest. The following results are extracted from the research report27 of two comprehensive laboratory studies: small-scale model tests performed in the wave flume of Leichtweiss-Institute, using 1 liter sand containers subject to random waves up to 20 cm height and large-scale model tests in the large wave flume of the Joint Coastal Research Centre (FZK) of both Universities of Hannover and Braunschweig, using 150 liter sand containers subject to random waves up to 1.6 m height.27,28 21.4.1.1. Stability of slope containers The sand containers on the slope which are located around the still water level are repeatedly moved up and down by the waves rushing up and down the slope, leading to an incremental seaward displacement of the containers. This dislodgement/pull out effect is illustrated by Fig. 21.15 as observed in the wave flume and in the field. Based on the HUDSON-formula for the hydraulic stability of rock armor units (nondeformable) and similarly to WOUTERS,56 a stability number Ns is formulated and postulated to be a function of the surf similarity parameter ξ0 , which includes both the slope steepness tan α as well as the significant wave height Hs and the wavelength Lop (Plate 21.2): Ns =


Cw =√ . ξ0 −1 ·D Hs

ρE ρW



(21.1)

 With the surf similarity parameter ξ0 = tan α/ Hs /Lop expressed in terms of the deepwater length Lop = gTp2 /2π (Tp = peak period of wave spectrum) the following stability formula is obtained in terms of the characteristic size D of the container: 3/4

D=

Hs

1/2

· Tp · (tan α)1/2 .  2π 1/4
ρE Cw · d ρW − 1

(21.2)

Defining the characteristic size D as D = lc sin α according to Wouters56 and according to the principle sketch in Plate 21.2, Eq. (21.2) can be reformulated in terms of the length lc of the slope containers as: 3/4

lc = Cw ·




2π g

Hs · 1/4


 Tp

ρE ρW

.  sin 2α −1 2

(21.3)

The empirical parameter Cw was determined by stability tests in a large and a small wave flume to Cw = 2.75. In Fig. 21.16 only the results of the large-scale model tests are plotted to illustrate that the threshold curve between stable and unstable containers is obtained for Cw = 2.75.

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(a)

(b)

(c)

Fig. 21.15. Hydraulic failure modes of slope containers. (a) Wave uprush and downrush on slope containers, (b) pull-out effect in the FZK-large wave flume, and (c) pull-out effect in a dune reinforcement (Courtesy of Heerten).

For Cw = 2.75 and g = 9.81 m/s2 , Eq. (21.3) reduces to:  Tp   − 1 · sin(2α)

3/4

lc =

1.74 ·




with Hs = Tp = α= ρE =

significant wave height (m), peak period of waves (s), slope angle of structure (◦ ), bulk density of GSC (kg/m3 ),

Hs ρE ρW

·

(21.4)

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Plate 21.2.

Fig. 21.16.

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Stability of slope containers based on Hudson formula.

Stability of slope containers from large-scale model tests (modified from Ref. 28).

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density of water (kg/m3 ), (1 − n) · ρs + ρW · n (with ρE ≈ 1800 kg/m3 for sand), porosity of fill material (–), density of grain material (kg/m3 ).

The stability formula in Eq. (21.4) expressed in terms of the required container length lc is plotted in Fig. 21.17 for a structure slope angle α = 45◦ in order to illustrate the sensitivity to the significant wave height Hs and to the peak period Tp . To compare the results with those of the Hudson-formula for rock armor, the volume of the tested slope containers (V = lc ·0.46lc ·0.14lc = 0.065lc3) is considered. For Hs = 1.5 m and Tp = 4 s, about the same required weight for slope containers with ρE = 18,000 kg/m3 is obtained as for a rock armor with ρE = 2700 kg/m3 and Kd = 2.0. Using a 10 times higher Kd -value (Kd = 20) and applying Hudsonformula in this specific case for slope containers would indeed provide the same result as the proposed formula. However, such an approach is not applicable as the stability of slope container is very sensitive to the wave period and the Kd value is expected to be a function of both wave height and wave period, due to the deformation effect caused by wave action. In fact, if for the same wave height Hs = 1.5 m the wave period is increased from Tp = 4 s to Tp = 6 s the required length lc will increase by more than 20% (Fig. 21.17). Moreover, due to the effect of deformation of the slope containers on the long-term stability, it is not advisable to use unprotected slope containers for design wave heights of Hs > 1.5–2.0 m.

Fig. 21.17.

Required length of slope containers for hydraulic stability.

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Fig. 21.18.

Hydraulic failure modes of crest containers.

21.4.1.2. Stability of crest containers The sand containers on the crest of the structure may fail due to two possible mechanisms (Fig. 21.18): (i) uplifting during the wave uprush process and shoreward displacement by the wave overtopping flow, (ii) dislodgement and pull-out effect similar to the mechanism observed for the slope containers. Due to the boundary conditions of the crest containers which are more critical than those of the slope containers (no overburden from upper layers), it is expected that the stability of the crest containers will be more critical than that of the slope containers if the crest level of the structure is not high enough. The relative freeboard Rc /Hs , therefore, represents the most important influencing parameter. In fact, it was difficult to identify a noticeable effect of the surf similarity parameter

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Fig. 21.19.

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Stability of crest containers form large-scale model tests.28

ξ0 on the stability number Ns as clearly observed for slope containers (Fig. 21.19). However, plotting at the top left of Fig. 21.19, the stability number Ns against the relative freeboard Rc /Hs shows that the stability number of the crest containers increases with increasing relative freeboard Rc /Hs according to the following linear relationship (Fig. 21.19): Ns =


Rc  = 0.79 + 0.09 , Hs −1 ·D Hs

ρE ρW

(21.5)

(see Plate 21.2) with D = lc sin α substituted in Eq. (21.5), a stability formula for crest containers is obtained in terms of the required container length lc : lc =


ρE ρW

−1




Hs

 . Rc 0.79 + 0.09 H sin α s

(21.6)

In a similar way as in Fig. 21.17 for slope containers, the required container length lc in Eq. (21.6) is plotted against the design wave height Hs for different relative freeboard Rc /Hs = 0 − 2.0 (Fig. 21.20). Considering exemplarily a typical value Rc /Hs = 1.2 and Hs = 1.5 m, a container length lc = 3.15 m and a container weight W = 36.6 kN are obtained which are much larger than those obtained for the slope containers in Fig. 21.17.

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Fig. 21.20.

Required length of crest containers for the hydraulic stability.

21.4.2. Processes affecting the hydraulic stability 21.4.2.1. Position of the problem and need to improve process understanding As observed in the experiments and shown in Fig. 21.21, the dislodgment and pull out of the slope containers by wave action, including the sliding and overturning of crest containers are strongly affected by the deformation of the sand containers. Simple stability formulae like those proposed in Sec. 21.4.1 cannot explicitly account for the deformation and other mechanisms affecting the hydraulic stability. An improved understanding of the processes and mechanisms is needed in order to (i) possibly avoid failure (engineering judgment), (ii) develop more process-based stability formulae (see Sec. 21.4.3). For this purpose it is necessary to address the following aspects: (i) hydraulic permeability of GSC-structures and its effect on the stability, (ii) wave loads and identification of the most critical loading case and location of the containers, (iii) internal movement of sand fill and its effect on the stability, (iv) effect of the friction angle between geotextile containers on the stability, and (v) effect of the container deformations on the stability. The content of this section and next section represents a brief summary of selected key results which have essentially been obtained from comprehensive laboratory studies, including several types of experiments in combination with numerical

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Fig. 21.21.

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Observed failure mechanisms under wave loads.

studies using a CFD-code (RANS-VOF model COBRAS developed at Cornell University) partially coupled with a CSD-code (FEM-DEM code “UDEC” developed by Itasca). These studies were performed in the framework of the PhD-research work by Recio37 and the results are described in more detail in several research reports33,35,36,38–40 and in a PhD thesis.37 21.4.2.2. Hydraulic permeability of GSC-structures Surprisingly, no information on the assessment of the hydraulic permeability of structures made of GSCs could be found in the published literature, although permeability represents an important parameter for both hydraulic stability and hydraulic performance (wave transmission, reflection runup, and overtopping). Moreover, reliable permeability values of GSC-structures are also needed for numerical simulations. Comprehensive laboratory experiments supplemented by numerical simulations using the COBRAS-code were performed for the first time to investigate the hydraulic permeability of several types of GSC-structures for different modes of placement and sizes of the containers, and based on the achieved results then to develop a conceptual model for the assessment of the hydraulic permeability.37,38,41 The key results may be summarized as follows: (i) The permeability of a GSC-structure is essentially governed by the size of the gaps between the containers. The flow through the sand fill can therefore be neglected in the computation. (ii) The hydraulic permeability of a GSC-structure is generally more than 10 times higher and 10 times lower than the permeability of sand (k ≈ 10−3 m/s) and gravel (k = 10−1 m/s), respectively (see Fig. 21.22).

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Fig. 21.22. Hydraulic geocontainers.38,41

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permeability

coefficients

for

different

mode

of

placement

of

(iii) The mode of placement of GSCs generally affects the permeability of the GSC-structure (Fig. 21.22). However, randomly placed GSCs and longitudinally placed GSCs have approximately the same permeability coefficient: k = 2.4 cm/s and k = 2.3 cm/s, respectively. To achieve a lower permeability an “interlaid” placement (gaps of the longitudinally placed containers are blocked by transversally placed containers) is required, allowing to reduce the permeability up to k = 1 cm/s and even to k = 0.5 cm/s. (iv) A conceptual model and a systematic procedure to predict the hydraulic permeability is proposed by Recio and Oumeraci.38,41 Validation by experimental data show that uncertainties less than 30% would be expected when applied to prototype containers. 21.4.2.3. Wave-induced loads and critical location of slope containers Based on the pressure recorded by load sensors around an instrumented container placed at different locations along the slope of the GSC-structure subject to both breaking and nonbreaking waves, the horizontal and vertical components of the total wave force on the instrumented containers is obtained for each time during a wave cycle. A typical result of the measurement is exemplarily provided in Fig. 21.23

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Fig. 21.23.

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Wave-induced force: Identification of critical container location.38– 40,42

to illustrate that the highest horizontal force in seaward direction and the highest vertical force in upward direction occur on the container located just below still water level. Using pressure measurements within the gaps between slope containers for both breaking wave impacts and nonbreaking waves, it is found that the former are less critical for the hydraulic stability than the nonbreaking waves rushing up and down the slope and causing more damage. In fact, the impact pressure induced by breaking waves is of much shorter duration. Moreover, it is strongly damped when propagating inside the gap (Fig. 21.24). 21.4.2.4. Movement and redistribution of sand inside the containers induced by wave action The analysis of the video records of the movement of colored sand inside a transparent permeable container built in the slope of the GSC-structure (Fig. 21.25) and subject to wave action has shown that (Fig. 21.26): (i) Noticeable sand movements occur only for larger waves that are capable to substantially move the front part of the container up and downward during the wave runup and rundown process [Fig. 21.26(b)]. After about 30 wave cycles the internal sand movement decreases significantly due to the accumulated sand at the seaward front of the container [Fig. 21.26(c)]. (ii) Due to the sand fill redistribution at the seaward front of the container, the latter deforms, thus offering a larger impact surface area to the mobilizing wave forces and reducing the contact area with neighboring containers [Fig. 21.26(c)].

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Fig. 21.24.

Fig. 21.25.

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Impact pressure propagation inside a gap between geotextile sand containers.42

Permeable transparent container for the investigation of internal movement of sand.38

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Internal movement and redistribution of sand fill in the transparent container.42

With the increased mobilizing forces and the decreased resisting forces, an incremental lateral seaward displacement of the deformed container occurs (pull-out effect) which causes again the start of the internal sand movement in a similar way as during the first wave cycles [Fig. 21.26(a)]. These results have considerable practical implications as the internal sand movement is responsible for the deformations of the container which affect • the hydraulic stability by reducing the contact area between GSCs and by increasing the drag and lift forces due to the increased exposed areas, • the crest level of a GSC-structure. Even in the laboratory it is observed that the height of sand containers is reduced by 4% after placement under water and by further 6% due to wave action. As a result, a total reduction of the height of the GSC-structure of about 10% was observed.38,41 Since these effects are strongly dependent on the adopted sand fill ratio, future research and design guidance should be directed toward the definition of an optimal sand fill ratio by accounting for the deformation properties of the geotextile and

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Table 21.3.

Results of friction tests in a shear box25 (30 cm × 30 cm).25 Measured peak friction stress τ (kN/m2 ) at normal stress σ (kN/m2 ) σ = 50 kN/m2

σ = 100 kN/m2

Nonwoven versus nonwoven geotextiles

27 29

48 47

Woven versus woven geotextiles

16 14

29 26

Interface

by balancing the advantages and drawbacks of high sand fill ratio. Moreover, this issue should also be explicitly addressed in future standards and guidelines due to the considerable effect of the sand fill ratio on stability and long-term performance. 21.4.2.5. Friction between geotextile sand containers Friction angles between woven geotextiles vary from about 12◦ (e.g., MirafiGT500) and 20◦ (e.g., Geolon PP120S), while for nonwoven geotextiles values of 20–26◦ (mechanical bound) or even 20–30◦ (thermal bound) are more common. Generally, the friction behavior between geotextiles increases with the surface roughness of the geotextiles. The higher the transmittable shear stress τ in relation to the induced normal σ stress in an interface (τ /σ), the higher is the friction behavior. Table 21.3 shows shear box test results for different interfaces at normal stresses of 50 kN/m2 and 100 kN/m2 .25 The results of numerical simulations using the partially coupled CFD and CSD codes (COBRAS and UDEC) previously validated by laboratory data, as shown in Fig. 21.27, highlight the significant effect of friction between the GSCs on the hydraulic stability. The effect of the friction angle on the stability is particularly important within the range of the practically relevant values (15–30◦), implying that the friction between the containers should be explicitly considered in future stability formulae. 21.4.2.6. Effect of container deformations on the stability It was shown in Sec. 21.4.2.4 that the sand fill is redistributed, resulting in deformations of the containers (Fig. 21.26). These deformations affect the stability of the containers in the following manner: (i) Reduction of the stability against sliding caused (a) by the increase of the drag and uplift forces as a result of the increased exposed areas AS and AT (Fig. 21.28) as well as (b) by the decrease of the resisting force as a result of the decreased friction area between the containers (Figs. 21.28 and 21.29). (ii) Reduction of the stability against overturning caused by the increase of the mobilizing drag and uplift forces as mentioned above, but also by the seaward

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Fig. 21.27.

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Effect of friction between GSCs on hydraulic stabilization.42

Fig. 21.28. Increase of exposed areas of drag and uplift forces and decrease of friction areas due to container deformation.

shift of the center of gravity of the deformed container leading to a reduction of the resisting moment (Fig. 21.30). A closer experimental and numerical examination of the variation of the “effective” contact areas between containers during wave action (i.e., those areas

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Fig. 21.29.

Fig. 21.30.

Effect of container deformation on sliding stability.38,42

Effect of container deformation on overturning stability.42

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Reduction of effective contact areas between containers during wave action.37,42

which contribute to the resistance against dislodgment by friction) has shown that the “effective” contact areas (i) decrease due to the upward movement of the front part of the containers (Fig. 21.31). (ii) increase with increasing slope angle of the GSC-structure. 21.4.3. Process-based stability formulae The insight in the processes affecting the hydraulic stability as described in Sec. 21.4.2 has clearly highlighted the necessity of an explicit account of these processes in future stability formulae, at least those processes which mostly affect the stability such as the effect of deformations of the container and the friction between the containers. In order to examine more closely the effect of deformation on the stability, new process-based stability formulae are proposed: first, without any account of

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Fig. 21.32.

Sliding stability formulae without deformation effect.

the deformation effect, and then correction factors are introduced to account for the deformation effect. Finally, a comparative analysis between the results of the stability formulae with and without deformation effect is provided for both slope and crest containers. It should be underlined that the new formulae proposed below were derived analytically for the geometry of the containers commonly used in coastal structures, i.e., with a container length lc which is twice the container width (lc /2) and five times the container height (lc /5) resulting in the following relationships for the volume of the container V and the application area of the drag and uplift forces AS and AT , respectively [Fig. 21.32(a)]. V = 0.1lc3 , AS = 0.1lc2 ,

(21.7)

AT = 0.5lc2 . These relationships provide the geometrical parameters that govern the resisting forces (weight) and the mobilizing forces (drag, inertia, and uplift forces), thus allowing to express the stability formulae in terms of the container length lc (see also Sec. 21.4.1). If, however, other container geometries, and thus other relationships which differ from those in Eq. (21.7) are adopted, the stability formulae can be modified accordingly. Further indications on how to proceed with such modifications and on the limitations of the proposed stability formulae will also be given below.

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21.4.3.1. Stability formulae without deformation effect (a) Stability against sliding: A sand container is stable as long as the resisting force FR = µ(FGSC − FL ) generated by the resulting normal force (FGSC − FL ) due to friction remains larger than the drag force FD and the inertia force FM (Fig. 21.32): µ (FGSC − FL ) ≥ (FD + FM ) .

(21.8)

And with the relative density parameter ∆ = (ρE /ρW ) − 1 (see Plate 21.2): 

µ ∆gV − 0.5CL AT · u

2



∂u 2 . ≥ 0.5CD AS u + CM V ∂t

(21.9)

Given the considered container geometry, relationships similar to those in Eq. (21.7) which provide V , AT , and AS as a function of the container length lc can be obtained and substituted in Eq. (21.9) which is then solved to obtain either the required length lc or the required mass WGSC of the container. Using, for instance, the geometry described by Eq. (21.7), the stability formulae are given in Fig. 21.32 in terms of the required length lc or mass WGSC of the container. (b) Stability against overturning: A sand container is stable as long as the stabilizing moment induced by the weight of the container under buoyancy FGSC remains larger than the mobilizing moment induced by the drag, inertia, and uplift forces FC , FM , and FL (Fig. 21.33): FGSC · rs ≥ FD · rh + FM · rh + Fc · rs

(21.10)

(ρE − ρw )gV · rs ≥ 0.5ρw CD u2 As · rh + ρw CM

∂u V · rh + 0.5ρW CL u2 AT · rs . ∂t

(21.11)

Given the considered container geometry, relationships similar to those in Eq. (21.7) which provide V , As , and AT , but also the lever arms rs and rh as a function of the container length lc can be obtained and substituted in Eq. (21.11). The latter is then solved to obtain either the required length lc or the required mass WGSC of the container. Using, for instance, the geometry described by Eq. (21.7), the overturning stability formulae are obtained in terms of the required length lc or mass WGSC of the container (Fig. 21.33). More details on the force coefficients CD , CM , and CL as well as on further input parameters required in the stability formulae summarized in Figs. 21.32 and 21.33 will be given in Sec. 21.4.3.2. 21.4.3.2. Stability formulae including deformation effect The effect of the container deformations on the stability is explicitly accounted for by introducing analytically derived deformation factors into the formulae for the drag force, lift force, inertia force, and resisting forces for both hydraulic failure

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590

Fig. 21.33.

Overturning stability formulae without deformation effect.

modes37,43 : sliding and overturning. The deformation factors are obtained as correction factors, describing the changes of (Plate 21.3): The resisting force FR against sliding. The correction factor KSR is obtained as the ratio of the effective weight contribution to the frictional force FR with and without deformation. The resisting moment against overturning. The correction factor KOR is obtained as the ratio of lever arm rs of the container weight FGSC under buoyancy with and without deformation effect. The mobilizing drag, lift, and inertia forces contributing to sliding. The correction factors of the drag force FD and lift force FL (KSCD and KSCL ) are obtained as the ratios of the areas AS and AT with and without deformation effect. The correction factor KSCM is assumed to be 1.0 since the container volume V remains constant. The mobilizing moment induced by the drag, lift, and inertia forces. The correction factor for the moments induced by the drag and lift forces (KOCD and KOCL ) area is obtained as the ratio of describing the changes of both surface areas (AS and AT ) and lever arms (rm and rs ) of the drag and lift forces FD and FL . The correction factor KOCM is obtained as the ratio of lever arms rsn of the inertia force with and without deformation effect. The values of the correction factors suggested in Table 21.4 are derived on the basis of a number of simplifying assumptions (see Ref. 37 for more details).

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Plate 21.3.

Stability formulae including the effect of deformation.

Table 21.4.

Deformation factors and force coefficients.

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Defining of parameters and typical values to be used in the stability formulae.

Among the most important assumptions the following are noteworthy: sand fill ratio of 80% and a slope angle of the GSC-structure of 45◦ . Indications to account for other slope angles and recommendations on further research to overcome most of the simplifying assumptions are given by Recio.37 Moreover, the force coefficients CD , CL , and CM are also given as a function of the Reynolds number for different locations and boundary conditions which may represent different practical applications (scour protection on the sea bed, artificial reef, slope containers, and crest container of a surface piercing structure such as revetments, seawalls, groins, etc.). The proposed values of CD , CM , and CL have been determined on the basis of systematic laboratory experiments.35 In Plate 21.4 the parameters used in the stability formulae described in Figs. 21.32–21.35 are defined and typical values are also given, including some remarks on the limitations of the suggested values. 21.4.3.3. Comparative analysis of stability formulae with and without including deformation effect In order to illustrate the effect of deformation the results of the new more processbased stability formulae with and without consideration of the deformations of the container as proposed in Secs. 21.4.3.2 and 21.4.3.1 are compared in Figs. 21.35 and 21.34 for a sloping revetment with an angle of 45◦ , a water depth d = 4 m at the structure, and a peak period Tp = 6 s of the waves. Moreover, the simple stability formulae for slope containers (Fig. 21.34) and crest containers (Fig. 21.35) proposed

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Fig. 21.34.

Stability formulae for slope containers: Comparative analysis.

Fig. 21.35.

Stability formulae for crest containers: Comparative analysis.

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in Sec. 21.4.1 are also considered to illustrate the difference with the results from the more process-based formulae. Depending on the range of the design wave height Hs , the following remarks may be drawn from the comparison of the formulae for the slope containers (Fig. 21.34): (i) For smaller design waves (Hs ≤ 1.5 m): the simple stability formulae proposed in Sec. 21.4.2 are too conservative and the deformation effect on the stability obtained from the comparison of the new formulae proposed on Sec. 21.4.3 are relatively small. (ii) For larger design waves (Hs ≥ 2.5 m): the simple stability formulae become more unsafe with increasing wave height. The effect of deformation also increases with the increase of the design wave height. Depending on the range of the design wave height Hs the following remarks may be drawn from the comparison of the stability formulae for the crest containers (Fig. 21.35): (i) For smaller design waves (Hs ≤ 1.5 m): the simple stability formula proposed in Sec. 21.4.2 is slightly conservative and becomes unsafe as soon as the significant wave height exceed 1.5 m. It can therefore be used instead of the more processbased formula only for Hs ≤ 1.5 m. However, the effect of deformation is higher than in the case of slope containers. (ii) For larger design waves (Hs ≥ 2.5 m): the simple stability formula by Oumeraci et al. (2003) becomes more unsafe with increasing wave height Hs . The effect of deformation on the stability also increases with increasing wave height. Comparing Figs. 21.34 and 21.35 also confirms that for commonly used relative freeboards Rc /Hs in the order of 1.2, much larger containers are required for the crest than for the slope of the structure. Moreover, it also shows that the effect of deformation on the stability dramatically increases with increasing design wave height and is much more pronounced for the crest containers than for the slope containers.

21.5. Concluding Remarks After about 50 years of successful experience of geotextile applications in coastal engineering, applications for shore protection are well established. Most failures which have yet been experienced are rather due to bad design, bad choice of material, and/or bad installation. Geotextile sand containers (GSCs) represent nowadays a soft and low cost alternative to conventional hard structures made of rock and concrete. Moreover, GSCmade structures are environmentally more appropriate and more easily reversible as they need essentially sand as construction material which is generally available at any coastal site. As “soft rock” GSCs can be manufactured at any size and used to build any type of shore protection structure, including scour protection, dune reinforcement, and repair of undermined structures.

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However, several problems still need to be solved in order to make use of the full potential of GSCs, particularly including (i) the long-term durability and lifetime prediction and (ii) the hydraulic stability under severe wave action. The facts, limitations and research needs related to the durability and lifetime issue may be summarized as follows: (i) Although field evidence (mostly nonexposed geotextile) is available over about 50 years, useful information extracted from nonretrieved samples is often very incomplete or very limited as the results can hardly be transferred to other sites, to present geotextile products and to time durations and conditions beyond the experienced service time/conditions. (ii) Results of accelerated tests — even in combination with those from site monitoring and retrieved samples — are still very limited when trying to predict lifetime of more than about 25 years. In fact, a systematic methodology to combine both laboratory and field monitoring approaches for this purpose is still missing. (iii) Recommendations for future research in the mid-term and long-term should focus on two directions: (a) improvement of the understanding of the degradation mechanisms, including physical, biological, and chemical processes, both isolated and in combination, and (b) development of a consistent framework for the assessment of long-term durability and lifetime (up to 100 years and more) based on the results of the above and including site monitoring, laboratory testing, and theoretical/numerical modeling. (iv) Meanwhile, in order to contribute to solve the present problems in practice, the following two recommendations might be helpful. (a) Apply engineering judgment based on the present knowledge of degradation mechanisms rather than relying on “extrapolation approaches” to predict lifetime and (b) apply where feasible and necessary well-established measures to enhance long-term performance, including, for instance, appropriate stabilizers and additives (e.g., against UV-radiation), more robust geotextiles (e.g., multi-layer), and geotextile coating (e.g., against abrasion and vandalism), sand covering (e.g., against UV-radiation and vandalism and to enhance aesthetical aspect), rock covering (e.g., against ice loads, debris, very high waves, UV, and vandalism) and setup of a consistent maintenance plan. Moreover, the collaboration with experienced companies with regard to geotextile production, manufacturing of sand containers, filling, handling, and placing is also strongly recommended. Regarding the hydraulic stability under severe wave action, the present state of knowledge, the limitations, and the needs for future research may be summarized as follows: (i) The large experience and stability formulae available for rock and concrete armor units cannot simply be transferred to GSCs, essentially due to the deformation of the GSCs under very severe wave attack. (ii) The deformations of the GSCs are essentially induced by the internal movement of the sand fill of the containers.

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(iii) The effect of the deformations of the GSCs on the hydraulic stability rapidly increases with the severity of wave attack, depending on the size and sand fill ratio of the containers as well as on the degree of exceedance of the wave loads required for the inception of the internal movement of the sand fill. (iv) Besides, the effect on the hydraulic stability, the internal sand movement in submerged geotextile containers may lead to a substantial reduction of the height of GSC-structures (up to about 10%) when subject to severe wave attack. (v) The deformation of GSCs affects the hydraulic stability, essentially due to two mechanisms: (a) reduction of the contact areas between GSCs caused by the uplift of the containers by wave action, thus decreasing the stabilizing forces/moments and (b) increase of the surface areas exposed to drag and lift forces which represent the main destabilizing forces/moments. Therefore, the deformation effect should be accounted for explicitly in the stability formulae. (vi) Friction between GSCs affects the hydraulic stability much more than commonly assumed in past and present design practice. Therefore, and due to the possible changes in service life, friction should be incorporated explicitly in stability formulae. (vii) The hydraulic permeability of structures made of GSCs is not only important for the prediction of the hydraulic performance (e.g., wave transmission, runup, overtopping, etc.), but also slightly affects the hydraulic stability. However, no clear correlation could be found between stability and permeability for the range of practical permeability coefficients of GSC-structures in the order of k = 1–3 cm/s. The permeability of a GSC-structure is essentially determined by the gaps between the containers, so that the flow through the sand fill itself can be neglected. Therefore, the hydraulic permeability essentially depends on the mode of placement of the containers. For randomly placed containers and longitudinally (in wave direction) placed containers, the permeability coefficient is in the order of k = 2.5 cm/s. (viii) The effect of breaking wave impact on sliding and overturning stability of slope containers has been found much less than expected, due to the potential of the GSCs to effectively damp impact pressure propagation inside the gaps. More efficient to destabilize the slope containers are the uprush and downrush of the longer nonbreaking waves and partially breaking waves. (ix) The proposed simple stability formulae derived in Sec. 21.4.1 on the basis of the Hudson-formula take additionally into account the effect of the wave period for the slope containers and that of the relative freeboard (Rc /Hs ) for the crest containers. In both cases, the effect of container deformations is taken into account implicitly through the empirical parameters determined from laboratory experiments. These formulae are conservative for waves up to about Hs = 1.5 m and can thus be used for design wave heights not larger than about 2 m. For higher waves (Hs > 2 m) the effect of deformation on the stability becomes more important and must therefore be considered more explicitly in order to ensure long-term stability performance.

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(x) The more detailed stability formulae proposed in Sec. 21.4.3 do not only take into account the most relevant processes, but also allow us to better quantify the effect of deformation on the required size of the container as a function of the incident wave height for both slope containers (Fig. 21.34) and crest containers (Fig. 21.35). In fact, these new process-based formulae clearly highlight the effect of container deformation which dramatically increases with the design wave height and is much more pronounced for the stability of the crest containers than for that of the slope containers. Moreover, they also show that even for the commonly used relative freeboard of about Rc /Hs ≈ 1.2 much larger containers are required for the crest than for the slope of a GSCstructure to ensure hydraulic stability. (xi) Although significant advancement has been achieved in the understanding of the processes and mechanisms responsible for the hydraulic failure of GSCs, more systematic research is further needed to investigate and better control the friction between GSCs, the effect of the sand fill ratio, the effect of the slope steepness of the GSC-structure, the internal movement of the sand fill, the container deformations and their more explicit consideration in both stability formulae and numerical simulations. A fully coupled CFD and CSD model system well-validated by experimental data will be needed as a necessary tool in combination with new laboratory experiments to achieve these goals.

Acknowledgments The financial support by StAUN Rostock for the experimental investigations described in Sec. 21.4.1 and by NAUE GmbH & Co. KG for those described in Sec. 21.4.3.2 are gratefully acknowledged. Sections 21.4.2 and 21.4.3 are essentially the part of the PhD-Thesis of the second author which has been supported by the Deutsche Akademische Austausch Dienst (DAAD). The first author would like to thank Prof. Heerten, Mr. Pilarczyk, and Mrs. Werth for their valuable comments and for reading the manuscript. Thank is also due to Mrs. Werth for providing the first author with valuable information and references on the properties and durability of geotextiles. References 1. ACT, Innovative Technology for Coastal Erosion Control — Subsurface Dune Stabilization, Advanced Coastal Technologies, LLC (2006), http://bcs.dep.state.fl.us/ innovative/report/appedic c/06 Subsurface Dune Protect.pdf. 2. J. Buckley and W. Hornsey, Woorin beach protection–chasing the tide sand fill tubes versus sand filled containers, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 761–764. 3. CEN, Guide to Durability of Geotextiles and Geotextile Related Products (European Normalization Committee (CEN/TC189/WGS/N210/Paris), 1998). 4. EAG-CON, German recommendations on geotextile containers in hydraulic engineering, WG-UG5 “Geotextile Containers of AK 5.1” Geosynthetics, Geotechnics and Hydraulic Engineering (German Geotechnic. Society (DGGT), 2008).

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5. R. A. Edward and S. D. A. Smith, Optimising biodiversity of macro invertebrates on artificial reefs, Poster presentation AMSA National Conf., Brisbane, Australia (2003). 6. J. Fowler and E. Trainer, Overview of geocontainers projects in the United States, Proc. Western, Dredging Conf., USA (1998). 7. H. H. Greenwood and A. Friday, How to predict hundred year lifetimes for geosynthetics, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 1539–1542. 8. J. Gr¨ une, U. Sparboom, R. Schmidt-Koppenhagen, Z. Wang and H. Oumeraci, Stability test of geotextile sand containers for monopile scour protection, ASCE Proc. Int. Conf. Coastal Eng., San Diego, USA (2006), pp. 5093–5105. 9. G. Heerten, Long-term experience with the use of synthetic filter fabrics in coastal engineering, Proc. 17th Int. Coastal Engineering Conf., Vol. 3, Sydney, Australia (1980), pp. 2174–2193. 10. G. Heerten and F. F. Zitscher, 25 Jahre Geotextilien im K¨ ustenschutz — Ein Erfahrungsbericht. 1. Nationales Symposium Geotextilien im Erd-und Grundbau, Mainz, Germany, Forschungsgesellschaft f¨ ur Straßen- und Verkehrswesen (Hrsg.), K¨ oln, Germany (1984), pp. 7–15 (in German). 11. G. Heerten, Geotextiles in coastal engineering — 25 years experience, Geotextiles and Geomembranes, Vol. 1 (Elsevier, 1984), pp. 119–141. 12. G. Heerten, A. Jackson, S. J. Restall and F. Saathoff, New development with mega sand containers of non-woven needle-punched geotextiles for the construction of coastal structures, ASCE Int. Conf. Coastal. Eng., Sydney, Australia (2000). 13. G. Heerten, The challenge for the use of geosynthetic construction materials in environmental, coastal and offshore engineering applications, Offshore Arabia 2006, Conf. and Exhibition, Dubai, United Arab Emirates (2006). 14. M. Heibaum, Geosynthetic containers — A new field of application with nearly no limits, Proc. 7th Int. Conf. Geosynthetics, Nice, France (2002). 15. M. Heibaum, A. Fourie, H. Girard, G. B. Karunararne, J. Lafleur and E. M. Palmeira, Hydraulic applications of geotextiles, Proc. Int. Conf. Geosynthetics, Yokohama, Japan, Millpress, Rotterdam (2006), pp. 79–120. 16. L. A. Jackson, R. Tomlinson, I. Turner, B. Corbett, M. D’Agatha and J. McGrath, Narrow artificial reef — Results of 4 years of monitoring and modifications, Proc. 4th Int. Surfing Reef Symposium, Manhattan Beach, California, USA (2005). 17. Th. Jansen, Erprobung und Verwendung von sandgef¨ ullten Kunststoffschl¨ auchen aus Gewebe und Vlies beim Seedeichbau in der Leybucht, Ostfriesland, 1. Kongress “Kunststoffe in der Geotechnik” K-Geo, deutsche Gesellschaft f¨ ur Erd- und Grundbau e.V., Hamburg (1988) (in German). 18. E. Kavazanjian, Jr., N. Dixion, T. Katsumi, A. Kortegast, P. Legg and H. Zanziger Geosynthetic barriers for environmental protection of landfill, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 121–152. 19. R. M. Koerner, Designing with Geosynthetics, 5th edn. (Pearson Prentice Hall, Ltd., London, 2005), 796 pp. 20. C. R. Lawson, Geotextile containment for hydraulic and environmental engineering, Proc. Geosynthetics Int. Conf. (Mitchpress, Rotterdam, 2006), pp. 1–48. 21. E. Lefaive, Durability of geotextiles: The French experience, Geotextiles and Geomembranes 7, 553–558 (1988). 22. B. Lenze, G. Heerten, F. Saathoff and K. Stelljes, Geotextile sand containers — Successful solutions against beach erosion at sandy coasts and scour problems under hydrodynamic loads, Proc. EUROCOAST, Littoral 2, Porto, Portugal (2002), pp. 375–381.

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23. MAG, Code of Practice — Use of Geotextile Filters on Waterways (Federal Waterways Engineering and Research Institute (BAW), Karlsruhe, 1993), www.baw.de. 24. A. McClarty, J. Cross, L. Gilbert and O. M. James, Design and construction of coastal erosion protection groyne using geocontainers, Langebaan, South Africa. Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006). 25. NAUE, Scherversuche im Labor der Firma NAUE f¨ ur Soil Filters Australia, NAUE Fasertechnik (2002). 26. H. Nickels and G. Heerten, Objektschutz Haus Kliffende, HANSA, Band 137(3), 72–75 (2000) (in German). 27. H. Oumeraci, M. Bleck, M. Hinz and S. K¨ ubler, Large-scale model test for hydraulic stability of geotextile sand containers under wave attack, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 878 (2002) 62 and Annexes. 28. H. Oumeraci, M. Hinz, M. Bleck and A. Kortenhaus, Sand-filled geotextile containers for shore protection, Proc. COPEDEC, Colombo, Sri Lanka (2003). 29. H. Oumeraci, J. Gr¨ une, H. Sparboom, R. Schmidt-Koppenhagen and Z. Wang, Investigations on scour and scour protection for monopile foundation of wind offshore turbines, Forschungszentrum K¨ uste (FZK), Res. Report (2007) 79 and Annexes (in German). 30. H. Oumeraci, A. Kortenhaus and K. Werth, Hydraulic performance and armour stability of rubble mound breakwaters with core made of geotextile sand containers — Comparison with conventional breakwaters, Proc. Int. Conf. Coastal Structures, Venice, ASCE (2007). 31. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, The Netherlands, 2000). 32. J. Recio and H. Oumeraci, Effect of the deformation on the hydraulic stability of revetments made of geotextile sand containers, Proc. Int. Symp. “Tsunami Reconstruction with Geosynthetics”, Bangkok, Thailand (2005), pp. 53–68. 33. J. Recio and H. Oumeraci, A numerical study on the hydraulic processes associated with the instability of GSC-structures using a VOF-RANS model, LeichtweissInstitute for Hydraulic Engineering and Water Resources, LWI Report No. 941 (2006). 34. J. Recio and H. Oumeraci, Processes affecting the stability of revetments made with geotextile sand containers, Proc. Int. Conf. Coastal Engineering, ICCE 2006, San Diego, USA (2006). 35. J. Recio and H. Oumeraci, Geotextile sand containers for coastal structures, hydraulic stability formulae and tests for drag, inertia and lift coefficients, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 936 (2006). 36. J. Recio and H. Oumeraci, Preliminary experiments and numerical simulations of solitary wave acting on a submerged-filter-reef, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 945 (2006). 37. J. Recio, Hydraulic stability of geotextile sand containers for coastal structures — Effect of deformations and stability formulae, PhD-Thesis, Leichtweiss-Institute for Hydraulic Eng., Technical University Braunschweig (2007). 38. J. Recio and H. Oumeraci, Permeability of GSC-structures, model tests and analyses, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 943 (2007). 39. J. Recio and H. Oumeraci, “Coupled” numerical simulations on the stability of coastal structures made of geotextile sand containers, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 942 (2007).

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40. J. Recio and H. Oumeraci, Effect of the deformations on hydraulic stability of coastal structures made of geotextile sand containers, Geotex. Geomemb. J. 25, 278–292 (2007). 41. J. Recio and H. Oumeraci, Hydraulic permeability of GSC-structures: Laboratory results and conceptual model, Geotex. Geomemb. 26, 473–487 (2008). 42. J. Recio and H. Oumeraci, Processes affecting the hydraulic stability of GSC structures: Experimental and numerical investigations, Coastal Eng. 56, 632–658 (2009). 43. J. Recio and H. Oumeraci, New stability formulae for coastal structures made of geotextile sand containers, Geotex. Geomemb. 56, 260–284 (2009). 44. S. J. Restall, L. A. Jackson, G. Heerten and W. P. Hornsey, Case studies showing the growth and development of geotextile sand containers — An Australian perspective, Geotex. Geomem. 20(5), 321–342 (2002). 45. S. J. Restall, W. P. Hornsey, H. Oumeraci, M. Hinz, F. Saathoff and K. Werth, Australian and German experiences with geotextile containers for coastal protection, Proc. Euro-Geo 3, Munich, Germany (2005), pp. 141–146. 46. A. L. Rollin, Long term performance of geotextiles, 57th Canadian Geotechnical Conf., Session 4D (2004), pp. 15–20. 47. RPG, Guidelines for Testing Geotextiles for Navigable Waterways (Federal Waterways Engineering and Research Institute (BAW), Karlsruhe, 1994), www.baw.de. 48. F. Saathoff, H. Oumeraci and S. Restall, Australian and German experiences on the use of geotextile containers, Geotex. Geomem 25, 251–263 (2007). 49. H. F. Schroeder, H. Bahr, P. Herrmann, G. Kneip, E. Lorenz and I. Schmuecking, Durability of polyfine geosynthetics under elevated oxygen pressure in aqueous liquids, Proc. Second European Geosynthetics Conf. EUROGEO, Bologna, Italy (2000). 50. S. K. Shukla and J. H. Yin, Fundamentals of Geosynthetics Engineering (Routledge, 2006). 51. M. Solton, B. Leclerq, J. L. Paute and D. Fayoux, Some answer’s components on durability problems of geotextiles, Proc. 2nd Conf. Geotechnics 2, 553–558 (1982). 52. G. H. Troost, G. Den Hoedt, P. Risseeuw, W. Voskamp and H. M. Schmidt, Durability of a 13 year old embankment reinforced with polyester woven fabric, 5th Int. Conf. Geotextiles, Geomembranes and Related Products, Singapore, IGS (1994) pp. 1185–1190. 53. G. van Santvoort (ed.), Geotextiles and Geomembranes in Civil Engineering (A.A. Balkema, Rotterdam, The Netherlands, 1994). 54. J. Vohlken, H. Lind and J. Witte, Dune reinforcement with geotextile sand containers, HANSA 4, 60–62 (2003) (in German). 55. J. D. M. Wisse, C. J. M. Broos and W. H. Boels, Evaluation of the life expectancy of polypropylene geotextiles used in bottom protection structures around the Ooster Schelde storm surge barrier — A case study, Proc. 4th Conf. Geosynthetics (1990), pp. 697–702. 56. J. Wouters, Slope Revetment — Stability of Geosystems, Delft Hydraulics Report H. 1939 (1998) Annex 7 (in Dutch). 57. F. F. Zitscher, Kunststoffe f¨ ur den Wasserbau, Bauingenieur-Praxis, Heft 125, Verlag von Wilhelm Ernst & Sohn, Berlin, M¨ unchen, D¨ usseldorf (1971).

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Chapter 22

Low Crested Breakwaters Alberto Lamberti∗ and Barbara Zanuttigh Department of Civil Engineering (DISTART), University of Bologna Viale Risorgimento, 2; 40136 Bologna, Italy ∗ [email protected] The chapter describes typical features of low crested breakwaters, their hydraulic stability, effects on waves, induced circulation, erosion, and problems related to construction and maintenance, providing basic tools for design.

22.1. Introduction As low crested breakwaters (LCBs) we mean breakwaters that are frequently overtopped or weakly submerged. They are in any case breakwaters, i.e., the height over the bed of these structures is sufficient to cause systematic wave breaking on the crest (submergence less than one significant wave height), but they are also low, i.e., not so high to make overtopping rare (freeboard less than one significant wave height). In this sense they are distinguished from artificial reefs (deployed mainly to enhance fisheries) or bed protections on one hand, that are deeply submerged and do not cause breaking, and by usual harbor breakwater on the other hand that are high enough not to cause significant overtopping. LCBs mimic in some manner the functioning of coral reefs, that however have usually a much wider crest and a narrower range of submergence/emergence due to their generation mechanism. As a consequence they are sometimes named also “reefs.” LCBs are usually built with natural rock for the defence of beaches (Fig. 22.1), even if some examples of LCBs protecting the main breakwater from highest waves or protecting an external harbor from excessive agitation and currents (Fig. 22.2) can also be found. Beach defence breakwaters are usually built in a few meter water depth within the surf zone at design conditions. The freeboard (positive or negative) is usually of the same order of magnitude as the tidal range, i.e., in contrast with most harbor breakwaters, freeboard variability in lifetime is substantial. 601

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Fig. 22.1. Detached LCBs protecting the beach of Skagen (DK, 57◦ 44 07 N 10◦ 37 50 E; coordinates are given to let the reader obtaining a plan view from Google-Earth).

Fig. 22.2. Jetties protecting Ijmuiden entrance (NL, 52◦ 27 55 N 4◦ 32 18 E) during a storm surge event. From http://www.vandermeerconsulting.nl/.

As it will be shown later, the height of the structure in this case is not greater than a few armor stones, posing serious problems in building the traditional filter structure separating the armor from the sandy bed. LCBs are usually located in the surf zone where design wave height is depth limited and, therefore, lasts for long periods. It is therefore unsafe to accept any,

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even small, permanent damaging rate in design conditions, since the damage will cumulate in a few years at a significant level. For all the reasons sketched above, LCBs cannot be treated as a small scale version of rubble mound harbor breakwaters, even if they share some evident common features. Due to their wide use in beach defence, however, they deserve some special attention. This contribution is dedicated to highlight differences between high and low breakwaters. We summarize the experience accumulated in the last two decades on the subject, making use of several results focused within DELOS research project, that we had the opportunity to coordinate; we acknowledge at the same time the contribution of project partners, evident to everyone will read one at least of the two major deliverables of the project: the special issue of Coastal Engineering8,24,28,29,35,46,57,68 and the Design Guidelines.5 The sections are arranged as follows: Section 22.2 — typical features, Section 22.3 — hydraulic stability, Section 22.4 — wave reflection and transmission, Section 22.5 — overtopping, Section 22.6 — induced circulation, Section 22.7 — scour and erosion, Section 22.8 — construction and maintenance problems.

22.2. Typical Layout and Sections LCBs were used since long time as detached emergent breakwaters for the defence of beaches. The low crest was motivated by economy reason in conditions where some agitation in the protected area could be accepted and sometimes required in order to avoid silt sedimentation. The problem of visual impact was not posed at that time. Later during the last two decades of the 20th century, in Europe particularly along the Mediterranean coasts subject to intensive tourism exploitation, the aim of reducing visual impact of defence works promoted the use of submerged breakwaters. Figure 22.3 shows one of the most deeply submerged breakwater constructed in Pellestrina (near Venice, IT) and completed in 1997.29 It is a 9 km long defence work, where a 4.5 × 106 m3 capital nourishment of 0.2 mm sand was protected by a LCB system. The offshore part of the LCB is a submerged breakwater defending the beach against offshore sand losses. The groins limit the long-shore displacement of sand and are characterized by an emerged part in the aerial beach zone and a submerged continuation to reach the submerged longitudinal breakwater. Average water depth at structure is 4.5 m. Design submergence is −1.5 m a.m.s.l. Spring tidal range near Venice is 1.0 m, but high tide combined with storm surge caused by Scirocco wind may reach 1.5 m a.m.s.l. Incident significant wave height with return period 10 years is 3.70 m, therefore, extreme waves at the structure are depth limited. Since the construction of Venice outlet jetties (end of the 19th century), the beach, deprived of any natural sand nourishment, suffered a slow but continuous erosion. The beach, after the construction of the defence system, showed to be rather stable with low southerly directed littoral drift.

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Fig. 22.3. Pellestrina (IT, 45◦ 15 57 N 12◦ 18 10 E) offshore submerged breakwater and groins. Courtesy of Consorzio Venezia Nuova.

Fig. 22.4.

Lønstrup defense section (DK, 57◦ 28 37 N 9◦ 47 44 E). From Burcharth et al.6

Figure 22.4 shows the section of a Danish beach protection example at Lønstrup in northern Jutland.29 Mean tidal range at the site is 0.3 m; exceptional high tide may reach 1.5 m; wave are usually depth limited. Longshore net transport is great, approx. 600 × 103 m3 /year. Beach sand size is 0.2 mm. A 1.1 km long segmented breakwater was constructed to protect the cliff and the village from an ongoing 1.5 m/year erosion. To maintain the shoreline position, a 20–30 × 103 m3 /year nourishment was required even after the breakwater construction.

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Fig. 22.5. Sea Palling defence section (UK, 52◦ 47 32 N 1◦ 36 11 E). Adapted from Halcrow;21 see also Sea Palling site at http://argus-data.wldelft.nl/index.html, reefs located north of the video station.

Another example of a typical emerged breakwater for coastal defence is represented by detached breakwater built near Sea Palling in North Norfolk, UK, Thomalla and Vincent.48 The defence system was motivated by severe flooding in 1953 (causing seven casualties in the area) and consisted of a continuous seawall protecting dune foot, requiring for its stability an adequate beach at the toe. In order to reduce wave intensity on the wall and required nourishment volumes in the most critical part of the defence, nine breakwater (or reefs) were constructed since 1993. Figure 22.5 shows the section of the first built reefs 5–8.21 The site is characterized by 3.0 m spring tidal range (from −1.3 to +1.7 m over datum -ODN-); extreme level reach 3.7–4.0 m ODN with return period 50–100 years. Offshore significant wave height reaches 7.9 m with return period 100 years, therefore, extreme waves are depth limited at the structure. Longshore tidal currents are relevant with a peak south-easterly flood of 0.8 ms−1 and a peak spring ebb to the northwest of 0.6 ms−1 . The net longshore sediment drift has been estimated to be 100–150×103 m3 per year in a southerly direction. The sandy beach shows to be highly volatile, with temporary spatial average erosion under severe storms reaching up to 70 m3 /m. Finally, an example of low crested harbor breakwater is worth to be presented. The outer breakwater prolongation in Leix˜ oes harbor (near Porto, PT) is selected, detailed, described in Vera-Cruz and Reis de Carvalho.58 The two long jetties enclosing the 100 ha outer harbor were built in 1884–1892. The outer harbor was not designed to provide load/unload capacities, but the limited extension of the initial dock area forced ships to stay anchored or moored in it. A storm in January 1917 caused severe damages to several of these ships. The construction of an outer vertical wall breakwater was therefore initiated in 1934, but during the first winter the last 120 m of the 400 m long built breakwater were destroyed. Based on economy considerations, a submerged breakwater was proposed. The design was verified with a hydraulic model carried out in Lausanne (CH), showing that the structure safety and harbor tranquility resulted satisfactory. The main environmental conditions are: water depth at roundhead −15.0 m, tidal excursion 0.0–4.0 m, significant wave height 9.5 m, wave period 13–19 s. The construction of the submerged extension was carried out in years 1937–1942 and its behavior was satisfactory, requiring only minor maintenance works. In 1966, in order to provide berthing facility to 100,000 t oil-carriers, the construction of berth A along the outer breakwater and of a wave wall with crest elevation 15 m a.s.l. was decided. The designed wall, apparently tested with regular waves, was protected by a steep (3:4) 40 t tetrapod slope resting on the crest of the 90 t cube submerged

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breakwater. In 1972, two years after the construction ended, a replacement of more than one-third of the placed tetrapods was necessary, and a similar replacement was required in 1974. In 1979, the last 90 m of the wave wall were destroyed. Breakage of tetrapods was observed in prototype that, together with the insufficient characterization of design waves (regular rather than irregular waves with equal significant wave height), were recognized as the main causes of the failure. The construction of some protection works of the roundhead was decided and, after comparison of several alternative solutions in a hydraulic model, the construction of a submerged breakwater with section similar to the previous one but shifted 52 m offshore was selected, aiming to reduce through breaking the wave height incident on the wave wall. The construction of the selected design (see Fig. 22.6) ended in 1982; the crest was constructed 1 m higher than shown in the drawing anticipating a 2 m settlement. The behavior of the submerged breakwater was evaluated satisfactory in 1993 either regarding its stability under wave action or regarding the protective effect on tetrapod stability and wave wall overtopping. The prototype verification appears significant since during the 11 year monitoring period a 9.0 m peak significant wave height event occurred (January 1985), reaching almost design intensity. PERFIL 6 - 6 0

L = 65,0 m - Solução 1 L = 40,0 m - Solução 2 L = 52,0 m - Solução 3 (ADOPTADA)

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Fig. 22.6. Terminal section and layout of the Leix˜ oes harbor breakwater (PT, 41◦ 10 22 N 8◦ 42 30 W). From Vera-Cruz and Reis De Carvalho.58

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22.3. Armor Stability in Shallow Water and Significant Tide Excursion A low breakwater compared to a high one is less exposed to breaker impact and to wave downrush, since the breaker falls on the crest often covered with water and the impact finds vent shorewards in the absence of a wave wall. This section is dedicated to describe this effect and to provide proper tools for armor design. Systematic laboratory experiments regarding rock armor stability in low crested breakwaters were carried out by several authors. Givler and Sørensen20 provided results of 2D experiments with regular waves, van der Meer51 and Burger7 with irregular waves. Vidal et al.59,60 Kramer and Burcharth,27 and Kramer et al.28 performed experiments with long crested and short crested irregular waves in a wave basin observing also damages at roundheads. A synthetic presentation of these results with a critical analysis is presented in Burcharth et al.6 on which this section is essentially based. Experiments are described in detail in the referred papers. All experiments are carried out on a fixed bed, where no settlement or erosion can take place, and therefore they do not provide information on these failure modes. Results refer to where the damage takes place and how severe the damage is. Assuming equal armor stone size at trunk and roundhead, all authors agree on where first and how damage occurs, see Fig. 22.7. • For submerged or zero freeboard breakwaters (Rc ≤ 0), hydraulic damage occurs first at trunk crest and results in a shorewards displacement of stones; roundhead in not more critical than the trunk and damage is distributed all around the head. • For slightly emerged breakwaters (0 < Rc < Dn50 ), the crest remains the most critical part of the trunk with some stones thrown shorewards and other dragged Seaward and middle head Leeward head

Stability number Hs/∆Dn50

5

4

3

X D

Trunk seaward slope Trunk crest

º

Trunk leeward slope

2

1

Least stable section, Eq. (22.1) 0 -3

-2

-1

0

1

Normalized freeboard Rc/Dn50 Fig. 22.7.

Partial and global start of damage curve, from Burcharth et al.5

2

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down the offshore slope; leeward roundhead becomes the first damaged part of a breakwater. • For emerged LCBs (Dn50 < Rc ), the seaward slope is the most critical part of the trunk with most stones dragged down the slope and the leeward roundhead is the first damaged part of a breakwater. • The crest width, built with stones of the same size as the armor slope, has no evident effect on hydraulic stability. • In all the mentioned tests, slope was 2:3 or 1:2 and no significant effect of the slope was observed. Burger7 showed that gradation and stone shape within reasonable limits (D85 /D15 < 2.5, L/D < 3) have not an evident effect on stability. Based on all the available information Burcharth et al.6 proposed the following formula representing the initiation of damage in some part of the breakwater
2 Hs Rc Rc = 0.06 − 0.23 + 1.36 (22.1) ∆Dn50 Dn50 Dn50 for relative crest freeboard less than 1.9. Above this value the stability number remains constant and equals to 1.14, the minimum of Eq. (22.1). This value is significantly lower than the one that can be derived from Hudson formula (∼ =1.5). This is due partially to the fact that no damage is assumed in this formula against a moderate damage (5%) accounted for in Hudson formula and partially to the deliberate use of Hs as the characteristic wave height (Hudson formula is used sometimes with H1/10 and some other with Hmax ). Most LCBs are placed in the littoral zone and in design conditions are subjected to depth breaking waves. For many of these the mean water level varies in a relatively wide range, regularly due to tide and occasionally due to storm surge. If waves are depth limited and the breaking limit is represented as a limit ratio γ between significant wave height and the local water depth, among the incident wave height Hi , the water depth at structure toe hs , the structure height hc , and structure freeboard Rc the relation holds Hi = γ · hs = γ · (hc − Rc ),

(22.2)

where the breaker index γ depends on seabed slope, wave steepness, and the characteristic wave height in use (significant, extreme . . . ). Figure 22.8 shows a typical situation at failure boundary. The shape of the two domains (failure domain and possible sea conditions) is such that stability is assured if breaking limit does not cross the start of damage curve or is tangent to it as a limit. This leads to the hydraulic stability condition under variable water level and depth limited waves characterized by the significant wave height: γs /∆ Dn50 ≥ . hc 1.36 − (γs /∆ − 0.23)2 /(4 · 0.06)

(22.3)

If the tangency point does not lie within the possible sea level range or breaking conditions are unlikely, the condition contains an implicit safety margin.

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Fig. 22.8.

609

Failure representation under variable water level, from Burcharth et al.6

Table 22.1. Stability condition and critical degree of submergence for different foreshore slope and breaker index. Wave solicitation Foreshore slope 1/∞ 1/200 1/100 1/40 1/20 — —

Tangency point

Hsi /hs

Rc /hc

Dn50 /hc

Hsi /∆Dn50

0.40 0.45 0.50 0.55 0.60 0.65 0.70

−0.02 −0.08 −0.16 −0.25 −0.36 −0.48 −0.64

0.18 0.21 0.23 0.26 0.29 0.33 0.37

1.39 1.46 1.5 1.6 1.7 1.8 1.9

Results for the minimum stone size according to Eq. (22.3) for relative stone density ∆ = 1.6 and different values for significant breaker index γs are reported in Table 22.1. The breaker index values are evaluated according to van der Meer52 with sop = 0.03 and hs /Lop = 0.05. For a gentle foreshore slope the following simple and cautious rule of thumb is found Dn50 ≥ 0.3 · hc .

(22.4)

The ratio Dn50 /hc imposes serious restrictions to the structure design. A double armor layer with Dn50 /hc > 0.4 require that at least part of the filter is placed below the natural bed level; this could be beneficial in case a bed erosion might occur. A value Dn50 /hc > 0.3 implies that there is no sufficient space for a core and the two layer armor stones rest directly on a coarse filter. Only when Dn50 /hc < 0.2 there is space for a conventional core and filter layer.

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It should be noted that Eq. (22.1) is based on model tests with sea-side slope 1:2 (some 1:1.5). For milder slopes, smaller armor stones can presumably be stable. According to Hudson formula, for a slope 1:5, an increase of the stability number for emerged breakwaters by a factor 1.36 is expected; a smaller increase in stability is argued in the case of submerged crest because the crest (including edges) is the most damaged part in this case. Also, some laboratory test cases support this conclusion (Ostia IT, Ferrante et al.18 ; Sea Palling UK, Leasowe Bay UK, Barber and Davies1 ). Offshore and inshore the armor slopes, a toe berm is normally constructed with the function of supporting armor and preventing the propagation of damage resulting from scour or settlement. Regarding berm stability no specific analysis was carried out on low breakwaters. Design rules suggested by van der Meer et al.56 were evaluated as reasonably safe for the offshore berm of low breakwaters in Burcharth et al.6 For a sufficiently wide crest, the breaker will fall on the crest and the inshore toe may be designed with the same material as the offshore one. For narrow and submerged breakwaters, the breaker caused by the offshore slope can fall beyond the crest causing a severe solicitation of the inshore toe berm; care should be paid in this case. It is worth noting that the above formulae are derived from fixed bed models, and do not represent any effect of bed mobility and/or erosion, whereas in most prototype conditions the fine sand seabed is subjected to intense mobilization and transport. Indeed, the observed damage of prototype rubble mound (as described by Lamberti et al.29 ) is often the consequence of geotechnical or morphodynamic instabilities. First of all, when proper bed protection is not used, dumping of stones onto bed cause their rapid sinking already during construction followed by a slower settlement. Moreover, breaking waves and related strong currents produce local scour (Sumer et al.46 ) which affects toe stability either directly (stone sliding in scour hole) or indirectly (increase of water depth and breaker height). To avoid sinking of the rubble stone material into a sandy bed it is necessary to separate the two materials with an appropriate filter: widely dispersed small stone layers, geotextiles, or mattresses. If sinking of stones stops within construction period, the rubble mound volume used for construction must be oversized and that is all; if sinking last longer, the structure must be designed and initially built higher (see, e.g., the Leixoes example). The overheight will correspond to the anticipated settlement, and depends on seabed characteristics, construction method, and structure height. For instance, a large rubble mound built on muddy seabed by means of floating equipment demands a large overheight, whereas a low mound structure placed on a coarse sandy seabed by landbased equipment running on already placed material demands much less overheight (but great overvolume) as settlement will be almost completed during construction. Burcharth et al.6 describes in detail some settlement cases. Statistics from a much wider set of cases from Japan is presented in Uda.49 Depending on the nature of the bed material the statistics in Table 22.2 was observed. The fraction of settling breakwater increases from 10% on rock (probably degradation of armor units), up to 43% on gravel, 63% on sand and 100% on sylt. The settlement height is highly variable in the range 0–2.80 m, with average and modal value around 1.00–1.20 m.

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Table 22.2. Settlement distribution of detached breakwater as function of bed material. Settlement Bed material

Exist

Nil

Unknown

Total

Rock Gravel Sand Silt

10%(19) 43%(139) 64%(678) 100%(1)

73%(141) 56%(183) 33%(347)

17%(34) 1%(5) 3%(42)

100%(194) 100%(327) 100%(1067) 100%(1)

Total

53%(837)

42%(671)

5%(81)

100%(1589)

Results are also given for scattering (displacement, hydraulic damage) of armor units. Globally scattering was observed in 14.2% of the 1,552 cases, but the frequency varies significantly between breakwaters showing evident settlement, for which the scattering frequency was 17.9%, and those for which no settlement was observed, for which scattering frequency was 9.4%. The frequency difference is highly significant, proving that scattering and settlement are related. In our experience hydraulic damage, i.e., the removal of stones from the breakwater crest, is able to justify only a minor part of crest settlement. On the other side, bed erosion around the structure and stone sinking preferably along the structure perimeter are the cause of block dilation on the slopes and therefore of reduced interlocking and instability. The mutual influence of settlement and block scattering is therefore bilateral and can explain the evident correlation observed.

22.4. Effect on Waves: Transmission and Reflection The main effect of a LCB compared to a high one is that a significant fraction of incident energy can pass over its crest and generate waves behind the structure. The increase of wave transmission leads, on one hand, to lower beach protection from erosion or to less calm areas for swimming purposes inshore the LCBs. On the other hand, it guarantees a high degree of water recirculation and thus a water quality improvement within the protected cell. Transmission, as well as reflection, can be characterized by a global intensity factor, normally the transmission and reflection coefficients, and by a description of energy distribution within the energy containing frequency range. The EU-projects DELOS (www.delos.unibo.it) and CLASH (www.clash-eu.org) collected and generated wide datasets from tests on all kind of structures, where the transmission and reflection coefficients were available in almost all cases. An extensive and homogeneous database on wave transmission (van der Meer et al.57 ) and on wave reflection (Zanuttigh and van der Meer64 ) were prepared, whose analyses produced for LCBs the major outcomes synthesized below. For rubble mound LCBs, to be able to estimate the transmission coefficient Kt also in presence of submerged structures with very wide crests, the following two formulae are given for design conditions (Briganti et al.3 ; van der Meer et al.57 ).

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Equation (22.5a), which is the formula by d’Angremond et al.,12 is used for Bc /Hsi < 8; Eq. (22.5b) for Bc /Hsi > 12: Rc Kt = −0.40 + 0.64 Hsi




Bc Hsi

−0.31

(1 − e−0.50ξop )

(22.5a)

with imposed lower and upper limit values Ktmin = 0.075, Ktmax = 0.8 and Kt = −0.35

Rc + 0.51 Hsi




Bc Hsi

−0.65

(1 − e−0.41ξop )

(22.5b)

with Ktmin = 0.05, Ktmax = 0.93 − 0.006(Bc /Hsi ). A linear interpolation between the two is performed in the range 8 < Bc / Hsi < 12. In Eqs. (22.5) Rc is the structure freeboard (negative for submerged structures), Bc is the crest width, and ξop is the surf similarity parameter based on structure offshore slope and peak wave period. Equations (22.5) were applied in the range −5 ≤ Rc /Hsi ≤ 5. Not all LCBs are of the rubble mound type. Sometimes smooth and impermeable structures exist, for example, LCBs covered with asphalt or armored with a block revetment. Often the slope angles of these structures are gentler (1:3 or 1:4) than for rubble mound structures, mainly for construction reasons. Wave transmission over smooth LCBs is completely different from rubble mound structures. First of all, the wave transmission is larger for the same crest height, simply because there is less energy dissipation by friction and structure permeability. Furthermore, the crest width has less or even no influence on transmission, as also on the crest there is lower energy dissipation. Only for very wide (submerged) structures there could be some influence on the crest width, but this is not a case that will often be present in reality as asphalt and block revetments are mainly constructed in the dry and not under water. The presence of tide makes it possible to construct these structures above water. For smooth LCBs the prediction formula by van der Meer et al.57 is   Rc −0.5ξop + 0.75(1 − e ) (22.6) Kt = −0.3 Hsi with imposed lower and upper limit values Ktmin = 0.075, Ktmax = 0.8, and limitations: 1 < ξop < 3, 1 < Bc /Hsi < 4. Transmitted spectra are often different from incident spectra. Waves breaking over a LCB may generate two or more transmitted crests on the lee side per incident crest. The effect is that more energy is present at higher frequencies than in the incident spectrum. Usually the peak period is quite close to the incident one, but the mean period may decrease considerably. If the reduction of wave energy is mainly led by the dissipations due to the flow through the armor layer, however, higher frequencies may be cut. A first analysis on this topic can be found in van der Meer et al.,53 who developed a simple model where in average the 60% of the transmitted energy is present in the range f < 1.5fp and the other 40% of the energy is evenly distributed between 1.5fp and 3.5fp. This scheme was analyzed little further by

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Briganti et al.3 and by van der Meer et al.,57 concluding that the redistribution is applicable only to submerged rubble mound structures. Particular care to the representation of the shape of the transmitted wave spectra was paid by Lamberti et al.30 The transmitted wave is reconstructed from the incident wave as the sum of the overtopped and filtered components. Incident waves transform seawards the structure into overtopping events and then regenerate, leeward the structure, due to perturbations caused by impulsive overtopping volumes. The overtopping and filtration discharges are added and converted into displacements of an ideal wave maker placed leeward the structure. A sample of the reconstructed transmitted wave spectrum with comparison of an experimental one is given in Fig. 22.9. The approach describes fairly the spectrum modification. Related to spectral change is the effect on obliquity described below. Similarly, for traditional structures several formulae exist that predict the reflection coefficient Kr as function of the Iribarren–Battjes surf similarity parameter ξ and are empirically determined for different types of armor units, of tested conditions and layouts. The only formula that can be applied, with varying coefficients (a, b) for different kinds of structures is the one by Seelig and Ahrens42 Kr =

a1 · ξ 2 . ξ 2 + b1

(22.7)

The coefficients (a1 , b1 ) are, respectively, equal to (0.6, 6.6) for rubble mound structures, and to (1.0, 6.2) for smooth slopes.

Fig. 22.9. Measured transmitted spectra (solid line with +), transmitted spectra components due to filtration (dot line) and to overtopping (dash line), in case of abundant overtopping. From Lamberti et al.30

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A more recent formula by Zanuttigh and van der Meer64 exists that • can be applied to all kinds of structures or revetments in design conditions, provided only that the roughness factor γf is known, • represents physical bounds, • was validated against a wide database. This formula reads: Kr = tanh(a · ξob ),

(22.8)

where a = 0.167 · [1 − exp(−3.2 · γf )]; b = 1.49 · (γf − 0.38)2 + 0.86 and ξo is the Iribarren–Battjes parameter calculated with the mean (−1, 0) spectral period at the structure toe. Based on CLASH results, the parameter ξo is used in this formula since it allows to represent also bi-modal spectra or shallow water with a flat spectrum where a peak period is not well defined. For the most typical cases, the coefficients in Eq. (22.8) are: γf = 0.40, a = 0.12, b = 0.87 for rock slopes; γf = 0.55, a = 0.14, b = 0.90 for rock slopes with impermeable core; γf = 1.0, a = 0.16, b = 1.43 for smooth slopes. Based on the DELOS database, van der Meer et al.,57 proved that as expected low crest breakwaters have smaller reflection than high ones, due to the fact that more energy passes over the structure. Moreover, the key parameter to describe the reduction of Kr is the relative crest height Rc /Hsi . Based on these observations, van der Meer et al.57 provided a first rough reduction factor for Kr as function of Rc /Hsi to be generally applied to formulae for well-emerged structures. The dependence of Kr on Rc /Hsi can be appreciated for permeable structures in Fig. 22.10, where the measured value of Kr is a dimensionalized by the value 1.20

1.00

K rm/K rc

0.80

0.60

UPC UCA UPD UFI Seabrook

0.40

0.20

0.00 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Rc/Hsi Fig. 22.10. Measured to computed values based on Eq. (22.10) of the reflection coefficient with varying degree of overtopping.

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obtained from Eq. (22.6). Similarly to van der Meer et al.,57 Zanuttigh and van der Meer64 proposed an extension of their formula to rubble mound LCBs by including a simple linear regression on Rc /Hsi . Reflection is not influenced if the relative crest height is > 0.5 and decreases to 0 when submergence is twice the wave height. The formulae (22.5), (22.6), and (22.7) are given for perpendicular waves only. Based on a limited set of data, van der Meer et al.57 and Wang et al.61 analyzed the effects of oblique wave attack on wave transmission. With increasing incident wave angle βi , Kt tends to decrease and the obliquity of the transmitted wave βt is a little smaller than the incident βi . The influence of wave obliquity on transmission is more evident for smooth structures, where it results proportional to cos2/3 β, and hardly observable for rubble LCBs. The deviation of transmitted waves (obliquity reduced to about 80%) can be interpreted as refraction of the average wave due to spectral change and reduced nonlinearity of transmitted waves. Explanation of effects is provided in Wang et al.63 Analysis of the effect due to wave obliquity on Kr can be found in Wang et al.,62 who derived for smooth LCBs a Kr reduction proportional to cos2/3 β.

22.5. Wave Overtopping Overtopping refers to the process of water passing over the structure crest. If the structure is not continuously submerged the process can be naturally interpreted and made up of several events strictly related to the passage of wave crests and characterized by a certain mass and momentum (or volume and velocity). The consequences of overtopping are dependent on specific statistics of the process related to the considered effect: for instance, the extreme momentum of one overtopping crest may be responsible for damaging exposed elements, whereas the mean volume per unit time (or mean overtopping discharge) is responsible for flooding. Nevertheless, the last is the standard parameter for characterizing overtopping intensity, probably because it is the easiest to measure. The total elapsed time and the total volume of water passing over the structure are simply the sum of elapsed time and passed volume within each wave. This trivial observation is the basis of the relation among mean overtopping discharge qot , overtopping probability Pot , mean overtopping volume ·Vot , and mean period of well formed waves Tm : qot

 Volumes Vot  Not · Vot  = waves = Pot , = Periods N · T Tm w m waves

(22.9)

where the overtopping probability Pot is the ratio between the number of overtopping waves Not and the number of waves Nw . Since every wave whose runup exceeds the structure crest (and only these waves) cause overtopping, the overtopping probability at the offshore edge of the crest is equal to the probability that a single wave runup is higher than the crest. For irregular waves, van der Meer and Stam54 suggested for runup at rubble mound structures a Weibull probability distribution, with parameters k1 and k2 related to

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incident waves and slope characteristics, from which the overtopping probability can be derived: Prob(Ru ≤ z) = 1 − exp(−(z/k1 )k2 ) ⇔ Pot = exp(−(Rc /k1 )k2 ),

(22.10)

where k1 = 0.4Hsi s−0.25 cot gα−0.2 , som is the mean wave steepness and α the om −0.75 structure offshore for plunging waves (ξm < 2.5) and k2 = √ slope; k2 = 3.0ξm −0.3 P 0.52·P ξm cot α for surging waves (ξm > 2.5), ξm is the Iribarren number based on mean wave period and P is the notional structure permeability. Later, van der Meer and Janssen55 provided a simpler expression for the runup distribution at dikes: the shape parameter is constant k2 = 2, i.e., the distribution is Rayleighian, and the location parameter is proportional to incident wave height k1 = 0.81 · γh · γf · γβ · min(γb · ξop , 2) · Hsi . The Rayleigh distribution fits in any case rather well the distribution of runup. In Eq. (22.9) Pot is the factor controlling the order of magnitude of mean overtopping discharge and is the expression of the main effect of relative crest elevation Rc /Hsi , as it can easily vary by orders of magnitude as a consequence of modest variations of structure crest elevation, see the CLASH database (Steendam et al.44 ; wwwclash-eu.org). The data obtained in wave basin by Zanuttigh and Lamberti67 for perpendicular and oblique layouts of LCBs show that the pattern and values of Pot are in good agreement with predictions from Eq. (22.10) and that the overtopping volumes can be well approximated by a Weibull distribution, as in van der Meer and Janssen.55 The mean value of such distribution Vot  shows a more or less parabolic relation with the runup Ruot , that is the median potential runup of waves causing overtopping (Fig. 22.11). Ruot is derived from Eq. (22.10) as the value giving exceeding probability just half the overtopping probability. As suggested by Pilarczyk,38 when overtopping is rare, Vot  is almost proportional to (Ruot − Rc )2 ; this implies a fixed shape of the overtopping crests. The scaled volumes, however, increase significantly as soon as Pot exceeds 0.4–0.5. The location parameter k1 of the volume distribution (typical overtopping volume) can be obtained from the mean value as k1 = Vot /Γ(1 + k2−1 ). The shape parameter k2 of the distribution increases with increasing Pot (Fig. 22.12). Its values range from something below 1 up to 3; the lowest values refer to the case of “rare” overtopping and the highest values to very frequent overtopping. If due attention is paid to difference in overtopping frequency and parameter uncertainties, the lowest figures are not substantially different from the value 3/4 suggested by van der Meer and Janssen.55 In conclusion, for LCBs that are overtopped by most waves, not just by the highest (Pot > 0.4 or Rc < k1 ), the crest level is so low in the approximate Rayleigh distribution of runup that the shape of all overtopping characteristics distributions become more symmetric and less variable (the shape parameter in a Weibull distribution becomes greater) and the overtopping events become longer. The combined model of a Weibull (or Rayleigh) distribution of runup, providing the overtopping probability, and a shape model for overtopping crests, as the one suggested by Pilarczyk,38 represents properly the process for high and low freeboards if the shape parameters are assumed variable with relative freeboard or overtopping probability.

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8 F=0, Layout 1, Narrow berm F=0, Layout 1, Wide berm F=3, Layout 1, Narrow berm F=3, Layout 1, Wide berm F=0, Layout 2, Narrow berm F=3, Layout 2, Narrow berm F=0, Layout 2, Wide berm F=3, Layout 2, Wide berm

7

/(Ruot-Rc)2

6 5

F

Ru

4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pot Fig. 22.11. Mean overtopping volumes Vot  versus overtopping probability Pot , emergent and eres et al.8 zero-freeboard structures. In the plot, F = Rc . From Cac´

3.0

F=0, Layout 1, Narrow berm

F=0, Layout 1, Wide berm 2.5

F=3, Layout 1, Narrow berm

F=3, Layout 1, Wide berm Rayleigh 2.0

F=0, Layout 2, Narrow berm

F=3, Layout 2, Narrow berm

k2

F=0, Layout 2, Wide berm F=3, Layout 2, Wide berm 1.5

Exponential 1.0

0.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pot Fig. 22.12. Shape parameter k2 of the Weibull volume distribution versus overtopping probability Pot , emergent and zero-freeboard structures. From Cac´eres et al.8

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The extrapolation below the aforementioned limits of formulae adapted to high structures has two contrasting effects: the intrinsic limitation Pot < 1 is not recognized and the factor is overestimated; the increasing duration of overtopping events is not recognized and the mean volume is underestimated. The total result on the discharge is weak and may be confused with formulae uncertainty. No specific formula exists so far to estimate the overtopping discharge at LCBs. The widely adopted formula for surging waves by van der Meer 
qot Rc  , (22.11) = 0.2 · exp −2.6 3 Hsi γf γβ gHsi where the symbol γ denotes the runup reduction factors accounting for slope roughness (γf ) and wave obliquity (γβ ), whose values and expressions can be found in TAW,47 tends to overestimate the overtopping discharge for LCBs, even with the indeed small correction by Kofoed and Burcharth.26 Equation (22.11) can be thus used only when overtopping is rare, i.e., well-emerged LCBs or weak wave attack with respect to the crest freeboard. 22.6. Induced Setup and Planimetric Circulation Unlike high barriers, the flow rate over LCBs is abundant and related to the rear piling-up. Overtopped water accumulates behind the structure, establishing a higher mean water level, or piling-up, which forces return flows along different paths: water may return offshore through gaps, originating the so-called rip currents, and, if the crest is submerged, also over the structure itself. In this latter case, the flux over the crest during the wave cycle is alternately directed inshore and offshore, driven by waves and piling-up. Since LCBs are typically made of permeable rubble mound, return flow occurs also through the structure, due to the unbalance between the steady hydraulic gradient induced by piling-up and the breaking wave thrust. Gaps are normally shorter but proportionate to breakwater length (∼ = 1/3), but breakwaters, even when permeable, are far less permeable than the open gap, therefore, when the breakwater is emerged the overtopping discharge returns to the sea almost only through the gaps. The flux through gaps qG globally equals the net flux across the structure qIS , which is the algebraic sum of the inshore directed flux over the structure crest qD and the return flow across the structure qU , see Fig. 22.13. The return flow through gaps qG is frequently named also recirculation flow, since it is usually simulated in a wave flume by a recirculating system. The process can be schematized in analogy to a pump system (Lamberti et al.31 ; Zanuttigh et al.66 ). The head losses associated to return flow through gaps are represented by a characteristic curve and the relation between piling-up and net mass flux across the structure is similarly described by a barrier pumping curve. The system operational point at equilibrium is the intersection between the two curves and is the solution of the equation. The pumping curve was experimentally investigated in wave flumes equipped with a recirculation system and it was found to be approximately linear for emerged conditions (Ruol et al.40,41 ; Cappietti et al.10 ). Piling-up reaches its maximum p0 in

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qIS

zO

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qD

zI

qU qG

p ≡ zI − zo qG ⋅ LG LB

Fig. 22.13. At the left-hand side scheme of planimetric circulation; at the right-hand side, scheme of the model approach. From Lamberti et al.31

absence of recirculation and decreases to zero when qIS is recirculated without any flow resistance. For given wave conditions, p in confined conditions is quite greater than for a structure with gaps: indeed, the overall return flow resistance decreases with increasing the gap to structure length ratio LG /LB , and consequently the piling-up required to drive all the return flows is smaller (Martinelli et al.36 ). Waves and currents around LCBs can be estimated by means of physical or numerical modeling. Laboratory experiments are usually dedicated to the analysis of a specific layout, are time consuming and expensive, and may be affected by scale effects. It is thus common practice for design purposes to adopt numerical models for simulating waves and current fields. When the structure crest is fully submerged, standard wave averaged models (as) can be used with the assumption of negligible filtration through the structure. If the breakwater is deeply submerged, its permeability has secondary effects when compared to the return flow over breakwater crest. For this case (reef barrier) induced circulation and setup can be evaluated with a 2DH wave-current model. Depending on the relative depth (depth to wavelength ratio) the user is addressed to the following models: • Nonlinear Shallow Water Equations numerical integrator (Brocchini et al.4 ; Hu et al.23 ). It is a rather simple and robust phase resolving model that describes both nondispersive waves and currents at the same time, used since long time and expressly adapted to represent discontinuous bore propagation over an existing body of water or a dry beach. Its use is recommended if everywhere in the computation domain h/L < 1/20 or, that is equivalent, if h/gT 2 < 1/120 and the computation domain is not so wide to make the small dispersive effects relevant during propagation. • Boussinesq equations numerical integrator (Karambas and Koutitas25 ). It is a phase resolving model representing both wave and currents at the same time,

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recommended for greater relative water depth and/or wider space. The Boussinesq model is only a little more time consuming than the previous one and it is potentially less robust, but it is specifically designed to represent wave dispersion and it can be used with confidence almost up to the deepwater limit if it uses higher order approximations. • Phase averaged models (as MIKE 21 or LIMCIR, and many others, see Johnson et al.24 ). These models represent wave statistics (spectra) and current separately, based essentially on mild slope approximation. They are to be used when the relative water depth variation in a wavelength is small and the computational domain is much wider than the wavelength. When they are applied on a breakwater, which is invariably rather steep, they tend to overestimate wave height decay rate due to breaking, wave setup, and induced currents. For both submerged and emerged LCBs, accurate predictions with high computational efforts can be obtained from the 2DV RANS-VOF code COBRAS by Losada et al.,35 which allows to represent the structure porosity. Some analytical models are also available under simplifying assumptions: both Calabrese et al.9 and Bellotti2 provided the fluxes over an impermeable structure, the first including friction effects in a 2D context and the second one excluding friction effects in 3D conditions. A rapid and reasonable assessment of the 3D fluxes around and within LCBs can be derived from Zanuttigh et al.66 for all crest freeboards and wave attacks.

22.7. Scour and Erosion Wave breaking and currents generated by the presence of the LCBs induce in turn morphological changes in their vicinity within a timescale of the order of a storm, i.e., within a time frame of hours to days. A detailed description of seabed dynamics and timescales, together with a classification of morphological changes, can be found in De Vriend and Ribberink.13 In describing morphological effects of waves and currents, it is usual to distinguish between near-field scour and far-field erosion. First of all, scour occurs in the immediate vicinity of the structure (less than 1/4 the wavelength), and it is caused by quite different mechanisms from those responsible for far-field erosion. These mechanisms include wave streaming, horse-shoe vortices, local turbulence, and possible liquefaction due to vertical pressure gradients. Furthermore, different timescales are associated with the development of scour and far-field erosion. Similar considerations apply to the shoreline response induced by LCBs, i.e., the development of salient or tombolos, especially with regards to the timescale involved, which is larger than that associated with far-field erosion and typically ranges from months to years. Far-field erosion in the vicinity of LCBs is intimately linked to the onshore flow over the crest of the structures, which returns offshore mainly through gaps, and to the deflection and acceleration of longshore and return currents at the roundheads. Currents flowing offshore through gaps produce irregularly shaped erosion

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Fig. 22.14.

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Field surveys performed in 2004 in Lido di Dante. From Zyserman et al.68

areas resembling a tongue of fire. Similarly, currents bending in offshore direction at roundheads produce crescent-shaped erosion patterns. At some distance from the gaps, these offshore-directed currents bend back toward the structure and merge with the onshore-directed flow over the crest of the LCBs. An example of these sediment transport patterns can be seen in Fig. 22.14, which consists of a bottom survey carried out in Lido di Dante (Ravenna, Italy). The coastal defence in this site is composed of three groynes, two detached zero freeboard LCBs with a protected gap in between, and two low-crested connectors from the northern and southern groynes to the LCBs (Zyserman et al.68 ; Zanuttigh65 ). Deep crescent-shaped erosion areas at about 70 m from the two barrier roundheads can be recognized, which have a maximum depth of 1.5 m, a length of 150 m, and a width of 50 m. A flame-shaped erosion hole, 1.0 m deep, 120 m long, 50 m wide, is present seaward of the central gap, due to the rip current concentration and intensity. Deposition occurs seaward the detached LCBs, due to wave breaking at the offshore LCB slope and to return currents from the roundheads. Far-field processes can be best investigated through the analysis of detailed bathymetric surveys coupled with concurrent monitoring of hydrodynamic

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conditions at the site and information about bed sediment characteristics. Besides economical and technical difficulties, prototype observations are difficult to generalize due to the particular conditions of each site (Lamberti et al.29 ). Physical modeling is another valid alternative approach, and can also provide data for calibration of numerical models. Few physical model tests of LCS have been carried out in wave basins (Kramer et al.28 ) and most of them adopted a fixed bed and dealt only with hydrodynamics around impermeable submerged structures (Haller et al.22 ; Drønen et al.17 ). So far only two sets of movable bed experiments were performed in a wave basin with the aim of analyzing morphodynamic changes induced by LCBs. Both cases included detached rubble-mound breakwaters separated by gaps of different length, under regular (Van der Biezen et al.50 ) and irregular (Martinelli et al.36 ) waves. However, restrictions imposed by the limited extension and boundary conditions of laboratory facilities, and scale effects associated with movable bed models must be kept in mind when selecting this method to investigate far-field erosion. Wave and flow fields in the vicinity of coastal structures and the associated morphological response are thus usually assessed on the basis of coastal area numerical models, see, e.g., De Vriend et al.15 and De Vriend14 for a description. Several references can be found in the literature regarding applications of two-dimensional, depth-integrated (2DH), or quasi three-dimensional (Q3D) models to study sites or simplified cases. A nonexhaustive list of cases includes: large-scale and local effects of groynes (De Vriend and Ribberink13 ), rip channels and bars evolution (Damgaard et al.11 ; Ranasinghe et al.39 , where numerical results are compared with video imaging analysis), bed changes in presence of breakwaters (Leont’yev32 ; Nicholson et al.37 , which also includes a comparison of different model performances), intermediate beach sedimentation inside and close to harbors (Lesser et al.33 ), morphological response induced by LCBs (Zyserman et al.68 ; Martinelli et al.36 ; Zanuttigh65 ). Main advantages of numerical models are the lack of scale effects and the limited amount of work involved in modifying the model setup, which in turn allows testing a wide range of layouts at limited cost. Near-field erosion close to LCBs is related to two processes: steady streaming, due to the superposition of incident and reflected waves at the offshore structure slope, and plunging breakers inshore the structure. Steady streaming occurs both at the structure trunk and roundheads. The plunging breaker is basically the key element of the scour process inshore the roundhead. After its formation, the plunger travels some distance along the crest width of the breakwater, eventually descends toward the bottom, impinges on the bed, and mobilizes the sand leading to a scour hole. Whilst far-field erosion can be generally predicted based on numerical or physical modeling results, near-field erosion has been extensively analyzed by means of laboratory tests deriving some analytical expressions for the estimate of its depth and extension. Figures 22.15 and 22.16 show the scour depth and extension, respectively, due to steady streaming and plunging breakers. The plotted data are obtained under regular and irregular waves by Sumer et al.,46 Fredsøe and Sumer,19 Lillycrop and Hughes34 and Dixen.16 In all these works α = 1:1.5, with the exception of Lillycrop and Hughes34 where α = 1:2.

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Fig. 22.15. Depth S (to the left) and width Ly (to the right) of the scour hole induced by steady streaming. In the plots, F = Rc and H = Hsi . From Sumer et al.46

Figures 22.15 and 22.16 are based on a dimensional analysis, which allows to represent the characteristics of the steady streaming [Eq. (22.12)] and of the plungingbreaker [Eq. (22.13)] induced scour by the following nondimensional equations  h L Rc , , , Re, ϑ, α, L dn50 h
√  S Tw gH Rc = fp , ϑ, α, , Hsi h h

S =f Hsi




(22.12) (22.13)

where S is the scour depth, H is the significant incident wave height, h is the bottom depth in front of the structure, L is the incident wavelength, ϑ is the Shields’

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Fig. 22.16. Depth S (to the left) and width Ly (to the right) of the scour hole induced by plunging breakers. In the plots, F = Rc and H = Hsi . From Sumer et al.46

parameter, α is the offshore structure slope, dn50 is the nominal sediment diameter, √ Tw gH/h is the plunger parameter (Fredsøe and Sumer19 ) and represents the ratio between the amount of water in the plunging breaker entering in the main body of water and the distance to be penetrated, Rc is the crest height with respect to mean sea level (positive if the structure is emerged), Re = aUm /ν (being a the amplitude of the orbital motion of water particles, Um the maximum value of the orbital velocity at the bed, Tw is the wave period, ν the kinematic viscosity).

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Field surveys highlight appreciable erosions along the trunk of LCBs, offshore and inshore, and severe erosions at LCB roundheads and gaps. The erosion induced at the offshore slope appears less pronounced and dangerous than in the case of traditional emerged structures. This can be explained because of the lower reflection and of sediment transport toward the structure induced by wave overtopping, with consequent deposition offshore the LCB and flattening or canceling erosion holes. Erosion may induce displacements of the structure toe and sliding of the armor layer into the scour holes. The prolongation of such conditions in time can bring to the structure destabilization. It is therefore necessary to protect the LCBs with a sufficiently wide toe berm. Toe protection layer may be constructed in the form of a protection apron. The apron must be designed so that it will remain intact under wave and current forces, and it should be “flexible” enough to conform to an initially uneven seabed. With this countermeasure, scour can be minimized, but not entirely avoided. Some scour will occur at the edge of the protection layer, and consequently, armor stones will slump down into the scour hole. This latter process, however, will lead to the formation of a protective slope, a desirable effect for “fixing” the scour. The determination of the width of the protection layer is an important design concern. The width should be sufficiently large to ensure that some portion of the protection apron remain intact, providing adequate protection for the stability of the breakwater. On the basis of the works by Sumer and Fredsøe45 and Sumer et al.,46 it is recommended to estimate from the following empirical equation the width of the protection apron at the trunk section W (from the offshore structure edge) L (22.14) − αhs , 4 where hs is the structure slope. The validity range of Eq. (22.14) is h/L = 0.1 ÷ 0.2. On the inshore side, the same W can be cautiously assumed, in order to prevent also damages induced by wave overtopping. At the roundhead, the toe protection width can be calculated as
 Rc Rc Rc W = We , if > +0.9; W = 0.29 + 0.74 We , if < +0.9, Hsi Hsi Hsi (22.15) where We is the value of W for emerged breakwaters, which is evaluated following Fredsøe and Sumer19 as W =

We = A · Um · Tw ,

(22.16)

where A is 1.5 for complete scour protection and 1.1 for a scour protection which allows a scour depth of 1% of the breakwater width at the toe. The above equation (22.15) is based on experiments where the breakwater slope was 1:1.5. Therefore, for slopes milder than 1:1.5, the width necessary for protection might be reduced (and for steeper slopes increased). Furthermore, Eq. (22.15) is for scour protection against the local scour caused by the combined effect of steady streaming and stirring up of sediment by waves. Due considerations must be given to global scour caused by the far-field flow circulations around the breakwater. Finally, the recommended width is for protection at the offshore side of the head.

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Experiments show that berm width designed according to Eq. (22.16) is also able to protect the sand bed against the breaker-induced scour at the inshore side of the head. However, scour might occur in the protection layer itself due to wave plunging. Therefore, over-sizing of the protection material is recommended at the inshore side compared to the usual rule provided, for instance, by formula (22.17), van der Meer et al.,56 0.15 Hsi /(∆Dn50 ) = (0.24hb /Dn50 + 1.6) · Nod ,

(22.17)

where hb is the minimum water depth over the berm and Nod is the number of stones displaced from a one stone wide strip of the berm and for standard toe size (about five stones wide and two to three stones high at trunk) Nod = 0.5, 2, 4, respectively, for no damage, acceptable, and severe damage. The volume of the toe berm shall be such that its material is sufficient to protect the scour/erosion hole from further erosion without destabilizing the armor layer slope, i.e., its width should be around three times the erosion depth and its thickness at least four times its maximum stone size (SPM43 ; Burcharth et al.6 ). In this way, slide berm stones can form although dispersed a stable and continuous slope covering the sand bed. The erosion induced at gaps can both cause serious problem of structure stability and act as sink for sediments inside the protected area, making them first slide into the hole and then favoring their exit from the gap transported by rip currents. It is therefore necessary to adequately protect the gaps with a stable and flexible plateau that may follow bottom movements, usually consisting of the same material at the barrier toe. The objective must be to shift erosion from the structure at such a distance not to compromise structure stability. Gap protection shall be extended more in offshore than in inshore direction, although it is not realistic an offshore protection to the limit of the eroded area. The amount of material must exceed the strictly necessary quantity in order to fill the holes that invariably form at the protection boundaries. Maintenance works for restoring toe protection before structure damage occur should be planned. 22.8. Structure Construction and Maintenance As already mentioned, LCBs can be constructed by means of either floating or landbased equipment. The selection of the construction method depends on environmental conditions, like the water depth, tidal range, and wave climate. The selection has relevant effects on rapidity and accuracy of construction, as well as risk for the contractor. Land-based equipment (dumpers, front loaders, dozers, cranes including backhoes) is preferred when materials arrive to the construction yard by road and the breakwater is either placed in really shallow water and/or close to the shore, or constructed in a site where tidal range is large enough to make the site dry in each cycle for a few hours. Floating equipment (barges and crane on them) are preferred in frequently calm waters more than 3–4 m deep, and when materials are transported to the yard on barges.

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Land-based construction is more accurate and can be carried out usually under sea conditions that do not allow construction with the aid of barges. Moreover, the risk of damaging concrete units is lower. When both type are possible land-based is normally preferred. In many sites, however, depending on local conditions, different combinations of land and floating equipment are actually used. Where the tidal range is large, one can rather safely assume that for half of the daylight period water level is lower than the structure basement level (higher than m.s.l.); during the summer period this allows 8 hrs working period per day, that is normally considered quite sufficient to carry out safely a work on the upper beach, where land-based equipment are normally used. In beaches exposed to frequent wave activity, so that a barge cannot operate safely for long periods, land equipment are used also in the lower part of the intertidal beach and even below. LCBs are constructed by dumping rocks from lorries moving on the mound crest and placing armor with land-based cranes. If the structure is to be submerged, the mound is constructed emergent and advancing from land, with equipment moving on it; the emergent crest is lowered in a second phase, when the crane is retreating and crest material is dumped at the sides of the mound. In some other case the access causeway is temporary, and is removed at the end of works. Barge-mounted equipment are normally used in calm waters. Barges are rather insensitive to the short period waves (2–4 s) that are generated by local winds blowing in bays or among islands. The direct dumping from barges with the assistance of a floating crane for armor placement is the most common construction method in this environment. Material shuttle barges and crane barge for placement are usually separate, allowing crane barge to remain on station. Self unloading split, bottom-door, tilting, or side unloading barges are usually used to dump core material. For bedding layers, scour protection and berms flat deck barges will a bulldozer for discharge can also be used. For the placement of filter layers only side unloading or flat deck barges can offer a good placement precision. Thin layers (0.50 m) can be placed only with barges operating with a high precision positioning system in one or multiple passage. LCBs pose a challenging separation problem between bed, foundation mound, and armor, requiring a good placement precision. For operations the following site conditions shall be considered: current, wind, waves, tidal level variation, maneuvring space, water depth, and visibility, including seasonal variation. Positioning of barges is obtained by a roundabout anchoring system (usually six anchors) or by a dynamic and computerized positioning system only for large structures. Generally, in sheltered water (no severe currents and waves), a horizontal accuracy of 1 m can be achieved. In exposed conditions the accuracy will be less and will decrease with increasing water depth. Down-time caused by waves and wind is often determined by the influence on the positioning accuracy and on crane loading, rather than by operational limitation of the barge. Critical are the barge movements; short wind waves have less impact than long swell. Generally, wind waves should not exceed 1.0–1.5 m height, whereas a 0.5 m high swell might be critical for positioning and accurate dumping. More

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severe are the limits for placing heavy armor units with cranes: cranes are normally not designed to sustain lateral forces caused by swinging loads due to barge motions, and heavy concrete units must be laid on bed with a minimal velocity. For these reasons maximum allowable tilts during placement should not exceed a few degrees. Often it is not possible to build a perfect filter separating the mound breakwater from the bed, particularly under water. Some movement of sand in the mound pores under heavy waves can therefore be anticipated in these cases. Moreover, if the bed is composed of fine sand some liquefaction or partial liquefaction of the upper layers are possible in storm conditions. The sand body sustaining the breakwater weight should not be affected by, not even partial, liquefaction. For all the mentioned reasons wide toe berms are useful: they reduce filtration velocity below the breakwater body, they rest on and let sand at rest; if bed erosion occurs, material sliding from the berm protects the bed from further erosion. The state of berms is therefore an indicator of structure safety that should be monitored. If scour holes of the order of twice the stone diameter are shown by bathymetric surveys, or the berm width is reduced below 2–4 stones diameter, toe berm stability may be compromised and toe protection should be reinforced and widened. Since the functionality of a LCB is very sensitive to crest elevation, the crest settlement should be also monitored. As soon as the crest is lowered on the average more than half the armor stone size or anywhere more than one stone size, the armor layer should be recharged. The design should not foresee a too frequent maintenance (below 10 years) since this will probably induce risk for structure stability and cause disturbance to the surrounding environment due to repeated working activity. Nevertheless, a limited settlement of 3–5 cm/year is present also in properly designed and constructed breakwaters, as those described in Section 22.2. A settlement rate greater than 10–15 cm/year should be considered instead worth of reassessing the stability of the breakwater and of analyzing a reinforcement feasibility. Acknowledgments Most of the research on LCBs synthesized and discussed in this study has been carried out within the framework of the DELOS project (EVK3-CT-2000-00041), through which the support of EC is gratefully acknowledged. References 1. P. C. Barber and C. D. Davies, Offshore breakwaters — Leasowe Bay, Proc. Institution of Civil Eng. (1985), pp. 85–109. 2. G. Bellotti, A simplified model of rip currents systems around discontinuous submerged barriers, Coastal Eng. 51(4), 323–335 (2004). 3. R. Briganti, J. W. van der Meer, M. Buccino and M. Calabrese, Wave transmission behind low-crested structures, Proc. Coastal Structures 2003, ASCE (2003), pp. 580– 592 (2003).

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4. M. Brocchini, A. Mancinelli, L. Soldini and R. Bernetti, Structure-generated macrovortices and their evolution in very shallow depths, Proc. 28th Int. Conf. Coastal Eng. ASCE (2002), pp. 772–783. 5. H. F. Burcharth, S. Hawkins, B. Zanuttigh and A. Lamberti (eds.), Environmental Design Guidelines for Low Crested Coastal Structures (Elsevier, UK, 2007), 448 pp. 6. H. F. Burcharth, M. Kramer, A. Lamberti and B. Zanuttigh, Structural stability of detaced low crested breakwaters, Coastal Eng. 53(4), 381–394 (2006). 7. G. Burger, Stability of low-crested breakwaters, Delft Hydraulics Rep. H1878/H2415 (1995). 8. I. Cac´eres, A. S. Arcilla, B. Zanuttigh, A. Lamberti and L. Franco, Wave overtapping and inducing currents at emergent low crested structures, Coastal Eng. 52(10–11), 931–947 (2005). 9. M. Calabrese, D. Vicinanza and M. Buccino, Verification and recalibration of an engineering method for predicting 2D wave setup behind submerged breakwaters, Proc. Int. Coastal Symposium ’05, Hofn, Iceland (2005). 10. L. Cappietti, E. Clementi, P. Aminti and A. Lamberti, Piling-up and filtration at low crested breakwaters of different permeability, Proc. 30th Int. Conf. Coastal Eng., S. Diego, USA, Vol. 5 (2006), pp. 4957–4969. 11. J. Damgaard, N. Dodd, L. Hall and T. Chesher, Morphodynamic modelling of rip channel growth, Coastal Eng. 45, 199–221 (2002). 12. K. d’Angremond, J. W. van der Meer and R. J. de Jong, Wave transmission at low crested structures, Proc. 25th Int. Conf. Coastal Eng., ASCE (1996), pp. 3305–3318. 13. H. J. De Vriend and J. S. Ribberink, Mathematical modeling of meso-tidal barrier island coasts, Part II: process-based simulation models, Adv. Coastal and Ocean Eng., ed. P. L.-F. Liu, Vol. 2 (World Scientific, 1996), pp. 151–197. 14. H. J. De Vriend, Mathematical modelling of meso-tidal barrier island coasts, Part I: empirical and semi-empirical models, Adv. Coastal and Ocean Eng., ed. P. L.-F. Liu, Vol. 2 (World Scientific, 1996), pp. 115–149. 15. H. J. De Vriend, J. Zyserman, J. Nicholson, J. A. Roelvink, P. P´echon and H. N. Southgate, Medium-term 2DH coastal area modelling, Coastal Eng. 21(1–3), 193–224 (1993). 16. M. Dixen, Scour around the roundhead of a submerged rubble mound breakwater, Master’s thesis, Tech. Univ. Denmark (2003). 17. N. Drønen, H. Karunarathna, J. Fredsøe, M. Sumer and R. Deigaard, An experimental study of rip channel flow, Coastal Eng. 45, 223–238 (2002). 18. A. Ferrante, L. Franco and S. Boer, Modelling and monitoring of a perched beach at Lido Di Ostia (Rome), Proc. Int. Conf. Coastal Eng. 1992, Vol. 3 (1992), pp. 3305– 3318. 19. J. Fredsøe and B. M. Sumer, Scour at the round head of a rubble-mound breakwater, Coastal Eng. 29, 231–262 (1997). 20. L. D. Givler and R. M. Sørensen, An investigation of the stability of submerged homogeneous rubble mound structures under wave attack, H. R. IMBT Hydraulics Report IHL-110-86. Lehigh Univ. (1986). 21. Halcrow, Anglian coastal management atlas, Anglian water and the sea defence management study for the Anglian region, Supplementary studies report, Anglian water (1988). 22. M. Haller, R. A. Dalrymple and I. A. Svendsen, Experimental study of nearshore dynamics on a barred beach with rip channels, J. Geophysical Res. 107(C6), 1–21 (2002).

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23. K. Hu, C. G. Mingham and D. M. Causon, Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations, Coastal Eng. 41, 433–465 (2000). 24. H. K. Johnson, Th. Karambas, J. Avgeris, B. Zanuttigh, D. Gonzalez and I. Caceres, Modelling of wave and currents around submerged breakwaters, Coastal Eng. 52(10–11), 949–969 (2005). 25. Th. V. Karambas and C. Koutitas, Surf and swash zone morphology evolution induced by nonlinear waves, J. Wat., Port, Coastal and Ocean Eng. 128(3), 102–113 (2002). 26. J. P. Kofoed and H. F. Burcharth, Estimation of overtopping rates on slopes in wave power deviced and other low crested structures, Proc. 28th Int. Conf. Coastal Eng., Cardiff, UK (2002), pp. 2191–2202. 27. M. Kramer and H. F. Burcharth, Stability of low-crested breakwaters in shallow water short crested waves, Proc. Coastal Structures ’03, Portland, USA (2003), pp. 139–149. 28. M. Kramer, B. Zanuttigh, J. W. van der Meer, C. Vidal and X. Gironella, 2D and 3D experiments on low-crested structures, Coastal Eng. 52(10–11), 867–885 (2005). 29. A. Lamberti, R. Archetti, M. Kramer, D. Paphitis, C. Mosso and M. Di Risio, European experience of low crested structures for coastal management, Coastal Eng. 52(10–11), 841–866 (2005). 30. A. Lamberti, B. Zanuttigh and L. Martinelli, Overtopping and wave transmission: An interpretation of spectral change at low-crested rubble mound structures, Proc. 30th Int. Conf. Coastal Eng., Vol. 5 (2006), pp. 4628–4640. 31. A. Lamberti, L. Martinelli and B. Zanuttigh, Piling-up and rip-currents induced by low-crested structures in laboratory and prototype, Proc. Coastal Structures 2007 (2007). 32. I. O. Leont’yev, Modelling of morphological changes due to coastal structures, Coastal Eng. 38, 143–166 (1999). 33. G. R. Lesser, J. A. Roelvink, J. A. T. M. van Kester and G. S. Stelling, Development and validation of a three-dimensional morphological model, Coastal Eng. 51, 883–915 (2004). 34. W. J. Lillycrop and S. A. Hughes, Scour hole problems experienced by the Corps of Engineers; data presentation and summary, Miscellaneous papers, CERC-93-2, US Army Engineer Waterways Experiment Station, Vicksburg, MS (1993). 35. I. J. Losada, J. L. Lara, E. D. Christensen and N. Garcia, Modelling of velocity and turbulence fields around and within low-crested rubble-mound breakwaters, Coastal Eng. 52(10–11), 887–913 (2005). 36. L. Martinelli, B. Zanuttigh and A. Lamberti, Hydrodynamic and morphodynamic response of isolated and multiple low crested structures: Experiments and simulations, Coastal Eng. 53(4), 363–379 (2006). 37. J. Nicholson, I. Broker, J. A. Roelvink, D. Price, J. M. Tanguy and L. Moreno, Intercomparison of coastal area morphodynamic models, Coastal Eng. 31, 97–123 (1997). 38. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A. A. Balkema, Rotterdam, The Netherlands, 2000). 39. R. Ranasinghe, R. Symondsa, K. Black and R. Holman, Morphodynamics of intermediate beaches: A video imaging and numerical modelling study, Coastal Eng. 51, 629–655 (2004). 40. P. Ruol and A. Faedo, Physical model study on low-crested structures under breaking wave conditions, Proc. Int. MEDCOAST Workshop on Beaches of the Mediterranean and the Black Sea, Kusadasi, Turkey (2002), pp. 83–96.

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41. P. Ruol, A. Faedo and A. Paris, Physical model study of water piling-up behind low crested structures, Proc. 29th Int. Conf. Coastal Eng. (2004), pp. 4165–4177. 42. W. N. Seelig and J. P. Ahrens, Estimation of wave reflection and energy dissipation coefficients for beaches, revetments and breakwaters, CERC technical paper 81-1, Fort Belvoir, U.S.A.C.E., Vicksburg, MS (1981). 43. SPM, Shore protection manual, 4th edn., US Army Corps of Engrs (Coastal Engng. Res. Center, US Govt Printing Office, Washington DC, 1984). 44. G. J. Steendam, J. W. van der Meer, H. Verhaeghe, P. Besley, L. Franco and M. Van Gent, The international database on wave overtopping, Proc. 29th Int. Conf. Coastal Eng., Lisbon (2004), pp. 4301–4313. 45. B. M. Sumer and J. Fredsøe, Experimental study of 2D scour and its protection at a rubble-mound breakwater, Coastal Eng. 40, 59–87 (2000). 46. M. Sumer, J. Fredsøe, A. Lamberti, B. Zanuttigh, M. Dixen, K. Gislason and A. Di Penta, Local scour and erosion around low-crested structures, Coastal Eng. 52(10–11), 995–1025 (2005). 47. TAW, Wave run-up and wave overtopping at dikes, J. W. van der Meer (author), Tech. Report of the Tech. Advisory Committee on Water Defences in the Netherlands (2002). 48. F. Thomalla and C. E. Vincent, Designing offshore breakwaters using empirical relationships: A case study from Norfolk, UK, J. Coastal Res. 20(4), 1224–1230 (2004). 49. T. Uda, Statistical analysis of detached breakwaters in Japan, Proc. 21st Int. Conf. Coastal Eng. (1988), pp. 2028–2042. 50. S. C. Van der Biezen, J. A. Roelvink, J. Van de Graaff, J. Schaap and L. Torrini, 2DH morphological modelling of submerged breakwaters, Proc. 26th Int. Conf. Coast. Eng., Copenhagen, DK (1998), pp. 2028–2041. 51. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Delft Hydraulics Commun. 396 (1988), 214 pp. 52. J. W. van der Meer, Extreme shallow water wave conditions, Report H198. Delft Hydraulics Laboratory, The Netherlands (1990). 53. J. W. van der Meer, H. J. Regeling and J. P. de Waal, Wave transmission: Spectral changes and its effects on run up and overtopping, Proc 27th Int. Conf. Coastal Eng. (2000), pp. 2156–2168. 54. J. W. van der Meer and C. J. M. Stam, Wave runup on smooth and rock slopes of coastal structures, J. Waterway, Port, Coastal and Ocean Eng. 118(5), 534–550 (1992). 55. J. W. van der Meer and J. P. F. M. Janssen, Wave run-up and wave overtopping at dikes, Wave Forces on Inclined and Vertical Structures, eds. N. Kobayashi and Z. Demirbilek (1995), Ch. 1, pp. 1–27. 56. J. W. van der Meer, K. D. Angremond and E. Gerding, Toe structure stability of rubble mound breakwaters, Proc. Advances in Coastal Structures and Breakwaters Conf., Institution of Civil Engineers (Thomas Telford Publishing, London, UK, 1995), pp. 308–321. 57. J. W. van der Meer, R. Briganti, B. Zanuttigh and B. Wang, Wave transmission and reflection at low crested structures: Design formulae, oblique wave attack and spectral change, Coastal Eng. 52(10–11), 915–929 (2005). 58. D. Vera-Cruz and J. Reis de Carvalho, Macico submerso de pre-rebentacao das ondas como meio de proteccao de obras maritimas: O caso do quebra-mar de Leixoes, LNEC Memoria no 796, Lisboa (1993), p. 36.

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59. C. Vidal, M. A. Losada, R. Medina, E. P. D. Mansard and G. Gomes-Pina, An universal analysis for the stability of both low-crested and submerged breakwaters, Proc. 23rd Int. Conf. Coastal Eng., Venice (1992), pp. 1679–1697. 60. C. Vidal, M. A. Losada and E. P. D. Mansard, Stability of low-crested rubble mound breakwater heads, J. Wat., Port, Coastal, and Ocean Eng. 121(2), 114–122 (1995). 61. B. Wang, A. K. Otta and A. J. Chadwick, Analysis of oblique wave transmission at smooth and rubble mound structures, ICE Proc. Coastlines, Structures and Breakwaters, London, UK (2005). 62. B. Wang, J. W. van der Meer, A. K. Otta, A. J. Chadwick and J. HorrilloCaraballo, Reflection of obliquely incident waves at low-crested structures, Proc. Coastal Dynamics ’05 (2005). 63. B. Wang, A. K. Otta and A. J. Chadwick, Transmission of obliquely incident waves at low-crested breakwaters: Theoretical interpretation of experimental observations, Coastal Eng. 54(4), 333–344 (2007). 64. B. Zanuttigh and J. W. van der Meer, Wave reflection from coastal structures, Proc. 30th Int. Conf. Coastal Eng. (World Scientific Publising Co., 2006), Vol. 5, pp. 4337–4349. 65. B. Zanuttigh, Numerical modeling of the morphological response induced by lowcrested structures in Lido di Dante, Italy, Coastal Eng. 54(1), 31–47 (2007). 66. B. Zanuttigh, L. Martinelli and A. Lamberti, Wave overtopping and piling-up at permeable low-crested structures, Coastal Eng. 55(6), 484–498 (2008). 67. B. Zanuttigh and A. Lamberti, Experimental analysis and numerical simulations of waves and current flows around low-crested coastal defence structures, J. Wat., Port, Coastal and Ocean Eng. 132(1), 10–27 (2006). 68. J. Zyserman, H. K. Johnson, B. Zanuttigh and L. Martinelli, Analysis of far-field erosion induced by low-crested rubble-mound structures, Coastal Eng. 52(10–11), 977–994 (2005).

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Chapter 23

Hydrodynamic Behavior of Net Cages in the Open Sea Yu-Cheng Li School of Civil Engineering Dalian University of Technology, Dalian, China [email protected] Knowledge of the hydrodynamic behavior of net cage under the action of waves and currents is the basis of the design and management of net cages in the open sea. Techniques used to investigate the net cage have typically included the use of scaled physical and numerical models, and, where possible, field measurements. In this chapter, information on the hydrodynamic behavior of net cages in the open sea is focused on gravity cages. The main methods used for research into hydrodynamic behavior are introduced: physical tests and numerical computation.

23.1. Introduction Exposed net cages in the open sea are subject to wave and current action. Thus, knowledge of their hydrodynamic behavior under the action of waves and currents is important for the design and management of net cages in the open sea. There are many types of net cages, among which gravity cages are the most popular. Thus, in this chapter, information on the hydrodynamic behavior of net cages in the open sea is focused on gravity cages. The main methods used for research into hydrodynamic behavior are introduced: physical tests and numerical computation.

23.2. Experimental Methods 23.2.1. Modeling criteria for fishing nets in experiments Deepwater sea cage engineering has developed rapidly, but research into hydrodynamic characteristics is relatively weak. Model experiments are very useful methods for such studies, but the main difficulty is in obtaining reasonable simulation criteria for model tests of fishing nets. For such model tests, the geometric scale λ is usually 633

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greater than 20:1. If the model net is designed strictly according to geometric similarity principles, two difficult problems arise. First, the yarn used for the model net will be very thin, which is difficult to manufacture. Second, because the model yarn is so thin, the Reynolds number (Re) for the model net will change significantly relative to the prototype net, leading to a large difference in hydrodynamic behavior between the model and the prototype nets. Thus, special simulation methods are needed. Tauti’s criteria1,2 were developed in the 1930s during research into fishing net problems and are used only for conditions where a current is present. 23.2.1.1. Tauti’s simulation criteria1,2 Tauti’s simulation criteria involve two geometric scales for nets, λ and λ
, which are the global model scale and the model mesh scale, respectively: λ=

Lp Lm

and λ
=

dp ap = , dm am

(23.1)

where d is the filament diameter, a is half the mesh size, and subscripts p and m denote the model and prototype sizes, respectively. Based on geometric similarity and dynamic similarity, the following expressions can be obtained according to Fig. 23.1: rp Sp Tp Lp Fp Wp Sp = = = , Wm Sm rm Sm Tm L m Fm

(23.2)

where W and r are the gravity and the hydrodynamic force per unit area on the net, respectively, and T is the force per unit length on the net edge. When the net material is the same for the model and the prototype, the equation can be replaced by: λ2 λ
= λ2

Vp2 Tp Fp =λ = . 2 Vm Tm Fm

(23.3)

In trawl experiments, Tauti’s simulation criteria are suitable and used widely. However, in model tests of net cages, two limitations arise for practical experiments. First, in model tests the model mesh scale cannot be too large, so a greater current velocity is required. Second, Tauti’s simulation criteria cannot be applied to model tests under wave conditions because Eq. (23.3) is not satisfied if geometric similarity LT LT S rS WS rS WS Fig. 23.1.

Calculation model of forces acting on a microsegment of a fishing net.

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is applied in the presence of waves. Thus, for model tests of net cages in wave and current conditions, the following gravity simulation criteria are recommended. 23.2.1.2. Extended gravity simulation criteria3,4 The main principle for extended gravity simulation is as follows: (1) Two geometric scales for the nets are still used as in Tauti’s simulation criteria: λ is the global scale and λ
is the scale for the yarn diameter and net mesh size. (2) Since the porosity ratio for the model net is the same as for the prototype, external forces acting on the net will follow the gravity simulation criteria by a scale of λ3 . (3) The model net weight simulated by a scale of λ3 should be modified as follows:
 
1 1 πd21 4 − × 10 (23.4) ∆W = · (ρ − ρw ) · q · S, · λ
λ 4a1 µ1 µ2 where ρ is the density of the net material, ρw is the water density, q is the packing fraction of the filament, S is the area of the net, and µ1 , µ2 are the hanging ratios (Fig. 23.2) as defined in Eqs. (23.5) and (23.6). In the cross-wise direction of the net, the hanging ratio is defined as: µ1 = a/2L

(23.5)

and in the longitudinal direction as µ2 = b/2L.

(23.6)

Based on gravity simulation criteria, the following equation can be written: λ3 = λ2

Vp2 Tp Fp =λ = . Vm2 Tm Fm

(23.7)

23.2.1.3. Validation of extended gravity simulation criteria3,5 To examine the validity of extended gravity simulation criteria,3,5 special model tests are carried out according to Tauti’s simulation1,2 and extended gravity simulation criteria.3,5 In the model tests, λ = 20 and λ
= 2. In the model arrangement shown in Fig. 23.3, the net made of PE has 68×28 diamond meshes with knots. The a

b

L

Fig. 23.2.

Definition of mesh properties.

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0.4m

transducer

lower bar sinker

1m Fig. 23.3.

16

Sketch of the net model.

Gravity Tauti

Force (KN)

12 8 4 0 0

20

40

60

80

100

120

140

Velocity (cm/s) Fig. 23.4.

Comparison of force values between the gravity similarity and Tauti similarity criteria.

yarn diameter is 1.18 mm and the half-mesh opening is 10 mm. The sinker system consists of a lower bar and sinkers. The prototype mass of the weight system is 240 kg. To record the net configuration, two bulbs were fixed at the middle and bottom of the net. As shown in Figs. 23.4 and 23.5, both the hydrodynamic force and deflection of the model net are in agreement under the two similarity criteria, but the weakness in Tauti’s simulation criteria does not exist in the extended gravity simulation criteria. The √ latter can be applied in both current and wave conditions, and since Vm = Vp / λ, the requirement to create a current in the test facility is lower than for Tauti’s simulation criteria. 23.2.2. Scanning method for sea cage motion6 When analyzing cage motion responses in model tests, the data obtained for tracing points include motion trajectory, displacement, inclination angle, velocity, and acceleration. Data processing involves: (1) image acquisition; (2) scanning of tracing points; (3) coordinate transformation; and (4) data analysis, with (2) and (3) requiring the greatest focus. Detailed information on scanning methods for sea cage motion can be found in the literature.6

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Projected area(cm 2 )

800 600 400 200

Gravity Tauti

0 0

20

40

60

80

100

120

140

Velocity (cm/s) Fig. 23.5. criteria.

Comparison of projected area between the gravity similarity and Tauti similarity

23.3. Numerical Methods This section introduces simulation methods for the main parts of a gravity cage, including the float collar and fishing net, then the whole gravity cage can be simulated by connecting the main parts of cage.7–15 23.3.1. Model of the float collar14,15 23.3.1.1. Forces on the float collar In general, the float collar system of a gravity cage is at the water surface and double floating pipes bear the wave-induced loads. For simplicity, the float collar system is simplified to a double-column pipe system, as shown in Fig. 23.6. When calculating wave-induced forces on the float collar, the collar can be divided into many mini-segments. The forces acting on the whole collar can be

Fig. 23.6. Sketch of the float system for a gravity cage. (a) Simplified float system and (b) section of double pipes.

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u v

b

n

c

a d

u

ο Fig. 23.7.

Schematic diagram of a mini-segment.

obtained by summing the forces acting on each mini-segment. Figure 23.7 is a sketch of a mini-segment of float collar with a local coordinate system n, τ , v defined for each mini-segment. As to coordinate system, n and τ are in the normal and tangential direction of the mini-segment, respectively. Thus, v is normal to the mini-plane (Fig. 23.7). According to Brebbia and Walker,16 the n component of wave-induced forces on a mini-segment in local coordinates can be obtained as follows: Fn =

1 CDn ρAn · |(un − Un )| · (un − Un ) + ρV0 an + Cmτ ρV0 (an − U˙ n ), 2

(23.8)

where CDn and Cmn represent the drag and inertial coefficients of the n component, respectively; An is the effective projected area in the direction of the n component; and an and U˙ n represent the acceleration vectors for water particles and minisegments of the n components, respectively. Other parameters are as described above. The same expression can be applied to other wave-induced forces (Fτ , Fν ) of v components. Figure 23.8 is a sketch of a simplified pipe model for calculation. v n

e

urfac

es Wav

η

n

γ

v

Fig. 23.8.

Sketch of a simplified pipe model.

O

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The projected areas in different directions are as follows:  An = dn · li    1 Aτ = · r · φi · li  π   Av = dv · li ,

(23.9)

where li is the length of the mini-segment. Aτ is related to the arc area of the minisegment (r · φi · li ) in the water. φi is the corresponding central angle of the projected chord-length, which is calculated from cos(φi /2) = 1 − dn /r. For the normal projected area Av , when dn ≥ r, let dv = 2r. In addition to wave-induced forces, the float collar is also subjected to gravity, buoyancy, and mooring line forces. The gravity acting on the mini-segment can be written as: Gi = G/N,

(23.10)

where G is the total gravity acting on the floating system and N is the number of mini-segments. The buoyancy acting on a mini-segment can be calculated as: Ffi = ρg · Vi .

(23.11)

The relationship between mooring line forces and elongation can be obtained directly by experimental measurement. 23.3.1.2. Motion equation for the float collar The three-dimensional motions of the float collar include surge–sway–heave translation and roll–pitch–yaw rotation. In this section, two coordinate systems are adopted, the fixed coordinate system Oxyz and the body coordinate system G123, as shown in Fig. 23.9. The body coordinate system G123 is rigidly attached to the float collar and the coordinate axes 1, 2, 3 are principal axes with origin at the center of mass G. Initially, axes x, y, and z are parallel to axes 1, 2, and 3.

3 p

r G z

2

rp

rG 1 o

y

x Fig. 23.9.

Schematic diagram of the moving coordinate system for a rigid body.

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Six degrees of freedom are required to describe the motion of a float collar in general spatial motion, resulting in six equations of motion. According to Newton’s second law, under fixed coordinates, the three translational equations of motion are given by: x ¨G =

n 1  Fxi , mG

y¨G =

i=1

n 1  Fyi , mG

z¨G =

i=1

n 1  Fzi , mG

(23.12)

i=1

where Fxi , Fyi , Fzi are the components of the external force vector Fi (i = 1, n) along the fixed coordinate axes xyz, n is the number of external forces, and mG is the mass of the rigid body. Axes 1, 2, 3 are principal axes with origin at the center of mass G, and thus Euler’s equations of motion for a rigid body17 can be applied. Under the body coordinate system, the three rotational equations of motion are given by: I1 ω˙ 1 + (I3 − I2 )ω2 ω3 =

n 

M1i ,

I2 ω˙ 2 + (I1 − I3 )ω3 ω1 =

i=1

I3 ω˙ 3 + (I2 − I1 )ω1 ω2 =

n 

n  i=1

M2i , (23.13)

M3i ,

i=1

where subscripts 1, 2, 3 represent the body coordinate axes 1, 2, 3. ω1 , ω2 , and ω3 are the components of the angular velocity vector ω along the principle axes, M1i , M2i , M3i are the components of the moment vector Mi (i = 1, n) along the principle axes, n is the number of moment vectors, and I1 , I2 , I3 are the principal moments of inertia. Although six equations of motion have been set up, it is necessary to know the transformation relationship between the fixed coordinates and body coordinates before solving the equations. If Bryant angles φ1 , φ2 , φ3 (Ref. 18) are obtained, the transformation matrix [R] between the fixed coordinates and body coordinates can be expressed as:   cos φ2 cos φ3 cos φ1 sin φ3 sin φ1 sin φ3   + sin φ1 sin φ2 cos φ3 − cos φ1 sin φ2 cos φ3     sin φ1 cos φ3 [R] =  − cos φ2 sin φ3 cos φ1 cos φ3 ,    − sin φ1 sin φ2 sin φ3 + cos φ1 sin φ2 sin φ3  sin φ2 − sin φ1 cos φ2 cos φ1 cos φ2 (23.14) where [R]−1 = [R]T . The relationship between the fixed coordinates and body coordinates is given by:     x 1  2  = [R]  y  . (23.15) z 3

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The kinematic differential equations for the Bryant angles are given by: ω1 = φ˙ 1 cos φ3 cos φ2 + φ˙ 2 sin φ3 ,

ω2 = −φ˙ 1 sin φ3 cos φ2 + φ˙ 2 cos φ3 ,

ω3 = φ˙ 1 sin φ2 + φ˙ 3 .

(23.16)

The motion of the float collar can be described by three translational displacements of the centroid and three Bryant angles, which can be obtained by solving the simultaneous ordinary differential Eqs. (23.12), (23.13), and (23.16). The equations can be solved using the Runge–Kutta–Verner fifth-order and sixth-order method. During calculation, transformation between the fixed coordinates and body coordinates is carried out by matrix [R]. It should be noted that the forces and motions of the collar are dependent on the net through the mutual mass points attached to both the net and the collar. 23.3.1.3. Hydrodynamic coefficients for the float collar in waves Numerical research5,14 has shown that the tangential coefficient Cτ (0.4–0.8) for a pipe in waves is much greater than that of a circular cylinder fully submerged under the water surface. The effects of surface tension may account for this difference, since the pipe is floating on the water surface. Results for the normal coefficient Cn (0.6–1.0) are within the range reported by other researchers.19,20 Hydrodynamic coefficients calculated here for a float system under wave action will be favorable for the design of and further research into the behavior of net cages in the open sea. It is interesting to note that in wave-only conditions the float collar exerts approximately 90% of the total load on the gravity cage, whereas in current-only conditions the float collar accounts for 4

Submerged

I II III

1.0 0.84 1.96

1.0 1.12 1.92

1.0 1.11 1.9

1.0 1.43 1.4

1.0 1.63 1.97

1.0 1.16 1.5

1.0 0.64 1.41

1.0 0.63 1.4

1.0 0.81 1.51

1.0 1.57 1.57

1.0 1.03 2.27

1.0 3∼4 >4

a For

wave + current, the wave height was H = 35 cm and the current velocity was u = 17.29 cm/s.

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General comparison of the cage models under floating and submerged conditions (ratios).

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Hydrodynamic Behavior of Net Cages in the Open Sea

Table 23.2.

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for the same cage diameter and height. In other words, the advantages of the sea station cage can be attributed to largely losing the original aquaculture volume. Even for the same original volume as the gravity cage, management of sea station and quasi-sea station cages is inconvenient. From the viewpoint of cage motion, the sea station and quasi-sea station cages exhibit better stability, which is considered beneficial for fish in the cages. However, it should be noted that even if cages can endure the action of tidal currents and waves, there is no guarantee that fish in such cages will survive.

23.5. Calculation of Velocity Reduction Behind a Plane Fishing Net25 Part of the fluid will flow through adjacent areas of the fishing net owing to its blockage effect. However, it is difficult to determine the flux passing through adjacent areas because of the complexity of the interaction between the fluid and the fishing net. In general, the fluid velocity in adjacent areas decreases as the distance between the fluid and the fishing net increases. The effective adjacent area is related to the fabric property of the fishing net. Here, the area of the fishing net is discretized into two parts: the solid projected area of the twines and knots, and the area of the holes. The latter is treated the same as the effective adjacent area, which means that the average velocity in holes is assumed to be the same as that in the effective adjacent area. Therefore, the effective adjacent area is a key factor for proper evaluation of the velocity reduction behind a fishing net. Figure 23.15 shows the definition of the flowing areas. It is assumed that the cross-sectional area of the fishing net is A1 with solidity Sn and the effective adjacent area is A2 . Here, the variation in area introduced by changes in water level in different sections is neglected. The solid projected area of the fishing net is Sn A1 and the area of holes is (1−Sn )A1 . Therefore, considering the unit: cm

100

50

Area of net (A1, u 1)

70

50

39.6

200

Hypothetic effective adjacent area (A2,u 2)

Fig. 23.15.

Definition of the flowing areas.

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effective adjacent area, the total area affected by fluid passing through the fishing net is A = (1 − Sn )A1 + A2 .

(23.27)

According to mass conservation, we have: (A1 + A2 ) · u0 = [(1 − Sn )A1 + A2 ] · u2 ,

(23.28)

where u0 is the velocity of the far-field approach fluid, which is assumed to be undisturbed by the fishing net; u2 is the velocity in the adjacent area, which is the same as that in the holes of the fishing net. A coefficient k is introduced here to denote the ratio between the adjacent area to the solid projected area. It can be written as: k = A2 /(Sn · A1 ).

(23.29)

Substituting Eq. (23.29) into Eq. (23.28), we have: u2 =

1 + k · Sn · u0 , 1 − (1 − k) · Sn

(23.30)

As described above, the velocity of the fluid passing through the holes will decrease due to amplification of the sectional area behind the fishing net. According to mass conservation within the section of fishing net (excluding the adjacent area), we have: u1 = (1 − Sn ) · u2 =

(1 + k · Sn ) · (1 − Sn ) · u0 , 1 − (1 − k)Sn

(23.31)

i.e., u1 (1 + k · Sn ) · (1 − Sn ) = . u0 1 − (1 − k)Sn

(23.32)

Studies on the hydrodynamics of pile groups or double-circle cylinders reveal that the influence of adjacent piles can be neglected when the gap between them is four-fold greater than the pile diameter, i.e., the hydrodynamics of each pile can be treated separately. Therefore, the range influenced by one pile is three-fold the pile diameter. For the fishing net, we assume that the effective area influenced is in proportion to the solid projected area of the net. Here, the ratio k is supposed to be the value of Eq. (23.32), which can be written as: u1 (1 + 3Sn ) · (1 − Sn ) = . u0 1 + 2Sn

(23.33)

According to Eq. (23.32), if Sn = 0, i.e., there is no fishing net in the flow field, we have u1 = u0 . If Sn = 1, the fishing net is equivalent to a solid plate and the velocity behind the fishing net is equal to zero. Results under these two cases are reasonable. However, when Sn = 0 and k = 0, the fishing net exists in the whole section, which is similar to a trash rack. It is known that under this case, the velocity behind it will increase to some extent. However, the result calculated with

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Eq. (23.32) is u1 = u0 . Therefore, calibration should be made to Eq. (23.32). It is assumed that ∆u is the calibration velocity, which is the average velocity difference between the section behind the fishing net and that at in the far field in front of the fishing net when k = 0. ∆u can be determined from the equations for energy conservation and mass conservation as follows: H0 +

u20 u
2 = H
+ + hj , 2g 2g

H0 · u0 = H
· u
,

(23.34)

(23.35)

where H0 is the water level in the far-field section in front of the fishing net; H
is the water level behind the fishing net; u
is the flow velocity behind the fishing net; and hj is the local energy loss in this section of fishing net, which can be calculated according to the trash rack analogy. The general form of the local energy loss is as follows:
4/3 b u2 u2 (23.36) hj = ξ · 0 = β · · sin α · 0 , 2g t 2g where ξ denotes the local head loss coefficient and β is the shape factor of the trash rack. For the fishing net, β is equal to 1.79. b and t refer to the grid width and the gap between grids, respectively. Thus, b/t is the ratio between the solid projected area and the gap area. For the fishing net, we have b/t = Sn /(1 − Sn ). α in Eq. (23.36) refers to the angle between the cross plane of the trash rack and the horizontal plane, which is 90◦ here. Substituting the value of each parameter into Eq. (23.36) yields: 4/3 2
Sn u (23.37) hj = 1.79 × · 0. 1 − Sn 2g By combining Eqs. (23.34), (23.35), and (23.37), the flow velocity (u
) behind the fishing net can be determined. Then the value of ∆u is obtained according to: ∆u = u
− u0 .

(23.38)

The calibration velocity ∆u is then distributed according to the flowing area. Equation (23.32) is then written as: (1 + k · Sn ) · (1 − Sn ) 1 − Sn ∆u u1 = + · . u0 1 − (1 − k)Sn 1 − (1 − k)Sn u0

(23.39)

As shown in Eq. (23.39), the velocity reduction is related to the calibration velocity ∆u and the solid projected area of the fishing net. For a given fishing net, its fabric property is determined. The velocity reduction behind the fishing net can then be calculated using Eq. (23.39). A model experiment of velocity reduction behind a plane net was carried out in a flume tank (60 m long, 2 m wide, and 1.8 m high) equipped with a currentproducing system at one end.

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50

39.6

C1

A2

70

651

20

A1

Flow direction

FA

C2

A3

C3

A4

C4 Plane net Measuring Point

Fig. 23.16.

Measuring point

Model setting and measurement points during experiments.

Table 23.3. Comparison of the velocity reduction between calculated results and experimental data. Measured velocity and velocity reduction

Calculated result (1 − u
/u0 )

Position A u0 (cm/s)

Position C u
(cm/s)

1 − u
/u0

Eq. (23.39)

15.9 31.8 44.21 60.4

13.85 28.22 38.92 52.25

12.89% 11.26% 11.97% 13.49%

12.32% 12.11% 11.84% 11.31%

In Eq. (23.39), u0 is the far-field velocity, so velocities at position A in front of the plane net were selected for calculation. Velocities at position C were selected for validation of the calculation model proposed. The model setting and measurement points during experiments are shown in Fig. 23.16. As shown in Table 23.3, an average velocity reduction of 12.4% was observed behind the plane net. Results calculated according to Eq. (23.39) agree well with the experimental data. The calculated results are also consistent with experimental results presented by Fredriksson,26,27 in which an approximate 10% reduction in velocity was found for a sea-station cage. 23.6. Hydrodynamic Characteristics of a Single Gravity Cage Model tests were carried out to investigate the hydrodynamic behavior of a gravity cage of a given structure size. The geometric scale λ and model mesh scale λ
were 20:1 and 2:1, respectively. The gravity simulation criteria described in Sec. 23.2.1.2 were applied in designing the model net cage. Under current-only, wave-only, and wave + current conditions, the mooring line force and motion of the net cage were measured when the net cage was floating or submerged under water. In model tests,

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Table 23.4. Design of experimental conditions (H and T denote the wave height and period, respectively).

Wave H (cm) T (s)

1

2

3

4

5

6

7

8

9

I

J

K

20 1.2

20 1.4

20 1.6

25 1.4

25 1.6

25 1.8

30 1.4

30 1.6

30 1.8

35 1.6

35 1.8

35 2

Velocity (cm/s)

8

12

16

Transducer

z

0.43m

0.796m

20

Tracing point

Bottom collar Sinker

h=1.0m

No.

Sinking ring

x

2.92m

(a)

Transducer

z

Bottom collar Sinking ring

x

h=1.0m

H/3

0.796m

Sinker

2.92m

(b) Fig. 23.17. Sketch of the setup for the gravity cage model. (a) Floating condition and (b) submergence condition.

submergence depths of U1 and U2 , corresponding to h/3 and h/2 (h is the water depth), respectively, were used. The experimental conditions are shown in Table 23.4 and a sketch of the experimental model is shown in Fig. 23.17. The float collar with a diameter of 0.796 m was made of HDPE material. The cage net was made of PE with a mass density of 953 kg/m3 . The netting was knotless, with a mesh size of 20 mm and a yarn thickness of 1.18 mm. Mounted as diamond meshes, the net then formed an open vertical cylinder with a diameter of 0.796 m and a height of 0.43 m. The full-scale diameter of the net cylinder is assumed to be 16 m. The weight system of the gravity cage comprises a bottom collar, sinkers, and a sinking ring. The bottom collar is of steel, with a mass of 25 g in water, which is attached to the bottom of the net cylinder by a thin line with a length of 5 cm. There are 10 spherical sinkers, each with a mass of 3.75 g in water and a diameter of 3 cm. When the net cage is floating, the mass of the sinking ring is zero; when submerged, the mass is 248.2 g. Outer pipe perfusion was applied to submerge the

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cage, and the sinking ring was used to control the submergence depth of the cage. The water mass that was perfused into the pipe was 334.1 g. Using the data measured, empirical formulae for the mooring line force and motion were obtained by the least square method.

23.6.1. Mooring line force 23.6.1.1. Current only Under floating conditions, the empirical formula for the force of a single mooring line on the net cage is V2 F = 0.149 (correlation coefficient R = 1.000) . ρgD 2 B gh

(23.40)

At submergence depths of U 1 and U 2, the corresponding empirical formulae are F V2 = 0.237 (correlation coefficient R = 1.000) 2 ρgD B gh

(23.41)

V2 F = 0.217 (correlation coefficient R = 0.990) , ρgD 2 B gh

(23.42)

where F is the mooring line force (N ), V is the current velocity (m/s), ρ is the water density (kg/m3 ), D is the diameter of the float collar (m), B is the depth of the net cage (m), h is the water depth (m), and g is acceleration due to gravity (kg m/s2 ). 23.6.1.2. Wave only Under floating conditions, the empirical formula for the maximum force of a single mooring line of net cage is (correlation coefficient R = 0.979):
2
 H H F −3 −3 + 1.73 × 10 · = 1.83 × 10 · . (23.43) ρgD 2 B h h At submergence depth U 1, the corresponding empirical formula is (correlation coefficient R = 0.982):
2
 H H F −3 −3 = 3.33 × 10 · , (23.44) − 0.28 × 10 · ρgD 2 B h h where H is the wave height and other symbols are as previously described.

23.6.2. Motion of the net cage 23.6.2.1. Current only As shown in Fig. 23.18, the inclination of the floating collar system is not significant under current conditions when the cage is floating. Even for a prototype current velocity of 0.86 m/s, the inclination is only 1.17◦ . When the cage is submerged, the inclination of floating collar is significant. It can be concluded that current has a destabilizing effect when the net cage is submerged.

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20 Floating 15 Inclination/Deg

Submergence 10

5

0 0

0.2

0.4

0.6

0.8

1

Velocity/ms-1 Fig. 23.18.

Inclination angle for the cage under current conditions.

23.6.2.2. Wave only When a net cage is subject to wave action, the amplitude of the horizontal motion of tracing points is greater on the lee side than on the wave side of the floating collar. The empirical formula for the maximum amplitude of the horizontal motion of tracing points on the lee side of the floating collar is (correlation coefficient R = 0.986): ξF = 0.614 × D




H D

2

+ 4.56 × 10−3 ×




gT 2 h




− 0.175 ×

Uh h

 ,

(23.45)

where ξF denotes the maximum amplitude of the horizontal motion of tracing points on the lee side of the floating collar and Uh is the submergence depth. In contrast, the amplitude of the vertical motion of tracing points is greater on the wave side than on the lee side of the floating collar under wave action. The empirical formula for the maximum amplitude of the vertical motion of tracing points on the wave side of the floating collar is (correlation coefficient R = 0.991): ηF = 0.321 × D




H D



+ 3.05 × 10−3 ×




gT 2 h




− 0.217 ×

Uh h

 ,

(23.46)

where ηF denotes the maximum amplitude of the vertical motion of tracing points on the wave side of the floating collar.

23.7. Effects of Structural Arrangement on the Hydrodynamic Behavior of a Gravity Cage9,28,29 A change in the structural arrangement of a gravity cage may have effects on its hydrodynamic behavior. Structural arrangements include the weight system, structure size ratio, mesh type, etc. In this section, researches9,28,29 including model

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test and numerical simulation are introduced, which will give references of the design of net cage. 23.7.1. Effects of the weight system For gravity cage, the holding of available volume for fishing net may mainly depend on the function of weight system. Adding mass of weight system may be a good way to reduce the loss of netting volume, but increase in mass will lead to the tension increase in fishing net, and then the manufacture cost and risk of netting breakage will be higher. In practical application, bottom collar has been proposed and offer promise in maintaining net shape. In this section, the effects of weight system on hydrodynamic behaviors of gravity cage are introduced, according to our recent research.28,29 In our model tests, two types of weight system were applied: (1) sinker, and (2) bottom-collar sinker. Details of the weight systems are shown in Table 23.5. Other settings for the net cage are the same as in Sec. 23.6. A sketch of the model setup is shown in Fig. 23.19.

Weight mode Bottom collar (g) Sinker (g) Total mass (g) Prototype mass (kg)

Weight system models.

A1

A2

A3

B2

B3

0 38.8 38.8 310.4

37.5 38.8 76.3 610.4

100 38.8 138.8 1110.4

— 76.6 76.3 610.4

— 138.8 138.8 1110.4

0.43m

0.796m

Transducer

z

B4 — 176.3 176.3 1410.4

Tracing point

H=1.0m

Table 23.5.

Bottom collar Sinker

x

2.92m

(a) Tracing point

Bottom of net Sinker

x

H=1.0m

Transducer

z

0.43m

0.796m

2.92m

(b) Fig. 23.19. Sketch of the model setup. (a) Weight system comprising a bottom-collar sinker and (b) weight system comprising sinkers.

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60

60

B2

A1

B3

40

Force /(kN)

A2 Force/(kN)

FA

A3

20

0

40

B4

20

0 0.36

0.54

0.72

Velocity /m s

0.89

1.07

0.36

-1

0.54

(a) Fig. 23.20. mode B.

0.72

Velocity /ms

0.89

1.07

-1

(b)

Resultant forces in the two upstream mooring lines. (a) Weight mode A and (b) weight

1.0

Ratio

0.8

0.6 B2/A2

0.4

B3/A3 0.2 0.72

0.89 Velocity /cms

Fig. 23.21.

1.07 -1

Comparison of the resultant mooring-line forces for two different weight systems.

23.7.1.1. Mooring line force Values for the resultant forces in upstream mooring lines are shown in Fig. 23.20. When a net cage is subject to current action, the mooring line forces increase with increasing mass of the weight system for both the sinker system and bottom collarsinker system. For the same mass, the mooring line force is greater for the bottomcollar sinker system compared to the sinker system, as shown in Fig. 23.21. When a net cage is subject to wave action, the mass of weight system has little effect on the mooring line force, as shown in Fig. 23.22. For the same mass, the mooring line force is smaller for the bottom-collar sinker system than for the sinker system, as shown in Fig. 23.23. 23.7.1.2. Deformation Net deformation is an important consideration. A sea cage is a 3D structure and thus deformation of a fishing net is also a three-dimensional problem under current

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Force /(kN)

Force /(kN)

60

657

40

20

A1

40

20

B2 B3 B4

A2 A3 0

0 7.2 4m

8.0

8.0

8.9

5m

6m

7m

7.2

T/s H/m

8.0

4m

(a) Fig. 23.22. mode B.

8.0

5m

6m

8.9 7m

T/s H/m

(b)

Resultant forces in the two wave-side mooring lines. (a) Weight mode A and (b) weight

1.5

Ratio

1.2

0.9

B2/A2 B3/A3

0.6

Fig. 23.23.

7.2

8.0

4m

5m

8.0 6m

8.9 7m

T/s H/m

Comparison of the resultant mooring-line forces for two different weight systems.

and/or wave conditions. At present, there is no effective method for measuring the net deformation of sea cages. During the experiment, six diodes were arranged in two columns on the fishing net, as shown in Fig. 23.19. Images were gathered with a CCD camera set outside the glass wall of the wave-current flume. The rate of volume loss was obtained by comparing the dynamic areas enclosed by the six tracing points on the fishing net with that under static conditions. Under current conditions, nets with either the sinker system or bottom collarsinker system exhibit a decrease in the rate of area loss with increasing mass of the weight system, as shown in Fig. 23.24. For the same mass, the bottom-collar sinker system can improve net deformation compared to the sinker system, as shown in Fig. 23.25. When a net cage is subject to wave action, the mass of the weight system has little effect on the maximum rate of area loss, as shown in Fig. 23.26. With the same mass of weight system, the maximum rate of area loss is lower for the bottom-collar sinker system than for the sinker system, as shown in Fig. 23.27.

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90%

90%

A1 A2

60%

Rate of area loss

Rate of area loss

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A3

30%

0%

B2 B3

60%

B4

30%

0% 0.36

0.54

0.72 Velocity/ms

0.89

1.07

0.36

-1

0.54

0.72

(a) Fig. 23.24.

0.89

Velocity/ms

1.07

-1

(b)

Influence of weight size on net deformation. (a) Weight mode A and (b) weight mode B.

Rate of area loss

90%

B2 A2 B3 A3

60%

30%

0% 0.36

0.54

0.72 Velocity/m·s

40%

A1 A2

30%

A3 20% 10%

40% 30% 20% B2 10%

B3 B4

0%

0% 7.2 4m

8.0

8.0

8.9

5m

6m

7m

(a) Fig. 23.26. mode B.

1.07

Influence of weight style on net deformation.

Maximum Rate of area loss

Maximum Rate of area loss

Fig. 23.25.

0.89 -1

T/s H/m

7.2 4m

8.0

8.0

8.9

5m

6m

7m

T/s H/m

(b)

Influence of weight system on net deformation. (a) Weight mode A and (b) weight

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3

Ratio

2

1

B2/A2 B3/A3

0 7.2 4m Fig. 23.27.

8.0

8.0

8.9

5m

6m

7m

T/s H/m

Influence of weight style on net deformation.

23.7.2. Effects of RDH and mesh type on net deformation in current8 In this section, the numerical model described in Sec. 23.3 is used to analyze the effects of RDH (RDH = D/H, where D and H are the diameter and height of the gravity cage) and mesh type on net deformation in a current. 23.7.2.1. Model description In the numerical model, the net cage is designed at a model scale of 1:7.1, as shown in Fig. 23.28. The net cage model comprises a hoop, a net, and a number of weights attached to the bottom of the net. The top of the net is mounted on the hoop, which is kept in a fixed position during each test. The weights are suspended around the bottom of the net to stretch it and maintain its shape under the influence of a current. The hoop is made of stainless steel and has an overall diameter of 1.435 m and a rod diameter of 0.025 m. The hoop itself is designed to have no deformation. The full-scale diameter of the net cylinder is assumed to be 10 m. The net is made of nylon with a mass density of 1,130 kg/m3 . The total number of meshes in the circumferential direction is 252. The netting is knotless, with a Top of net cage fixed

Net

Current

Sinker

Fig. 23.28.

Sketch of the gravity cage model with a bottom-collar sinker-weight system.

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Properties of the net cages used for numerical simulation. No. of meshes

Cage no.

RDH

Mesh style

Diameter (m)

Height (m)

Circumference

Height

1 2 3 4 5 6

2.0 2.0 1.43 1.43 1.0 1.0

Square Diamond Square Diamond Square Diamond

1.435 1.435 1.435 1.435 1.435 1.435

0.717 0.717 1.0 1.0 1.435 1.435

248 210 248 210 248 210

40 25 56 35 80 50

Table 23.7.

Weight-mode setting.

Mass (g)

Bottom collar

One sinker (g)

Scaled weight system (g)

WA WB

0 931.2

87.3 29.1

1396.8 1396.8

mesh size of 32 mm and a yarn thickness of 1.8 mm. The netting itself is not scaled, and ordinary full-scale netting is used. Mounted as square meshes, the solidity ratio (S) of the netting is 0.225. The net then forms an open vertical cylinder with a diameter of 1.435 m. Keeping the cage diameter constant, three RDH values are applied: 2.0, 1.43, and 1.0. In each RDH mode, two types of knotless net are used and simulated, a square mesh and a diamond mesh. The yarn diameter and mesh size are set to 1.8 and 32 mm, respectively, for both net types. The properties of the net cages are presented in Table 23.6. The bottom of the net cage is not modeled. In this section, two types of weight system are applied: (1) a sinker (WA), and (2) a bottom-collar sinker (WB), as shown in Table 23.7. The gravity cage system is subjected to five different current velocity cases (Usteady = 0.11, 0.19, 0.26, 0.34, and 0.37 m/s) corresponding to the full-scale cases (Usteady = 0.3, 0.5, 0.7, 0.9, and 1.0 m/s) according to the gravity simulation criteria. After entering the net cage, the velocity of a fluid particle will decrease slightly. 23.7.3. Effect of RDH on net deformation Under different current velocities, the volume-holding coefficient (Cvh ) of the net cylinder can be calculated as: Cvh =

Vc , Vc0

(23.47)

where Vc is the volume of the net cylinder exposed to the current and Vc0 is the initial volume of the net cylinder not exposed to the current.

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23.7.3.1. Fishing net with square mesh For both weight systems the effect of RDH on net deformation is significant, as shown in Fig. 23.29. An increase in RDH helps to decrease net deformation. According to the simulation results, the bottom-collar sinker system holds the volume of the gravity cage better than the sinker system. As shown in Fig. 23.30, at low current velocity the increase in volume-holding coefficient is greater at lower RDH, but when the velocity increases to approximately 0.3 m/s, the opposite trend is apparent.

Volume holding coefficient [-]

Volume holding coefficient [-]

1.0 0.9 0.8 0.7 0.6

RDH=2.0 RDH=1.43 RDH=1.0

0.5 0.4

1.0 0.9 0.8 0.7 0.6

RDH=2.0 RDH=1.43 RDH=1.0

0.5 0.4

0.3 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.10

0.15

0.20

0.25

0.30

Velocity [m/s]

Velocity [m/s]

(a)

(b)

0.35

0.40

Fig. 23.29. Volume-holding coefficient under (a) the sinker system and (b) the bottom-collar sinker system at different RDH as a function of current velocity.

Increase in the volume holding coefficient[-]

0.09

RDH=2.0 RDH=1.43 RDH=1.0

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.10

0.15

0.20

0.25

0.30

0.35

0.40

Velocity [m/s] Fig. 23.30. Increase in the volume-holding coefficient with the bottom-collar sinker system for different RDH relative to the sinker system.

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Fig. 23.31.

Shapes of net cylinders with diamond and square meshes.

(a) Fig. 23.32.

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(b)

Initial net images (a) with a sinker system and (b) a bottom-collar sinker system.

23.7.3.2. Fishing net with diamond mesh For the same RDH, a net cylinder with a diamond mesh has the same volume as a net with a square mesh when not exposed to the current and not attached to the weight system, as shown in Fig. 23.31. A net cylinder with a diamond mesh exhibits significant initial deformation when attached to a weight system but not exposed to the current. Figure 23.32 shows the initial net deformation for the sinker system and the bottom-collar sinker system. With a decrease in RDH, the initial deformation increases. When RDH decreases to 1.0, the initial volume reduction is approximately 50% for weight system WA and 20% for weight system WB, so an RDH of 1.0 is not recommended for use in practice (simulated in this section). For a net cylinder with a diamond mesh, the volume-holding coefficient (Cvh ) is calculated on the basis of the simulation results, but here the symbol Vc0 in Eq. (23.47) is defined as the volume of the net cylinder that is not exposed to the current and not attached to the weight system. It can be concluded from the numerical results that the effect of RDH on net deformation is important with either type of weight system (Fig. 23.33). An increase in RDH is helpful for decreasing net deformation. In comparison with the bottomcollar sinker system, the effect of RDH on net deformation is more significant for the cage with the sinker system.

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Volume holding coefficient [-]

1.00

Volume holding coefficient [-]

0.80 0.75 0.70 0.65 0.60 0.55

RDH=2.0 RDH=1.43

0.50 0.45

0.95 0.90 0.85 0.80 0.75 0.70 0.65

RDH=2.0 RDH=1.43

0.60 0.55 0.50 0.45

0.40 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.10

0.15

Velocity [m/s]

0.20

0.25

0.30

0.35

0.40

Velocity [m/s]

(a)

(b)

Fig. 23.33. Volume-holding coefficient under (a) the sinker system and (b) the bottom-collar sinker system at different RDH as a function of current velocity.

Increase in the volume holding coefficient [-]

0.26 0.24

RDH=2.0 RDH=1.43

0.22 0.20 0.18 0.16 0.14 0.10

0.15

0.20

0.25

0.30

0.35

0.40

Velocity [m/s] Fig. 23.34. Increase in the volume-holding coefficient with the bottom-collar system at different RDH relative to the sinker system.

As shown in Fig. 23.34, the bottom-collar sinker system exhibits lower deformation than the sinker system with a lower RDH. For a gravity cage with a diamond mesh, the structure of the bottom collar is critical when the RDH is low.

23.7.4. Effect of mesh type on net deformation Diamond and square are two mesh types that are commonly used in fishing nets. To investigate the effect of mesh type on netting shape, net deformations of the two mesh types are compared in this section. Figures 23.35 and 23.36 show comparison

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Fig. 23.35.

Comparison image of net deformation with the sinker system (V = 0.37 m/s).

Fig. 23.36. Comparison image of net deformation with the bottom-collar sinker system (V = 0.37 m/s).

0.95

Volume holding coefficient [-]

Volume holding coefficient [-]

1.00

0.90 0.85 0.80 0.75 0.70

Square mesh Diamond mesh

0.65 0.60 0.55 0.10

0.15

0.20

0.25

0.30

0.35

0.40

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30

Square mesh Diamond mesh

0.10

0.15

0.20

0.25

Velocity [m/s]

Velocity [m/s]

(a)

(b)

0.30

0.35

0.4

Fig. 23.37. Comparison of the volume-holding coefficient with the sinker-weight system. (a) RDH = 2.0 and (b) RDH = 1.43.

images of net deformation for the sinker and bottom-collar sinker systems with RDH = 2.0, in which the dashed lines denote net deformation for the square mesh. More quantitative comparisons are shown in Figs. 23.37 and 23.38. According to Fig. 23.37, when the sinker system (WA) is applied, net deformation is greater for a diamond mesh than for a square mesh, but the discrepancy decreases with increasing current velocity. This discrepancy may be induced by the initial net deformation. Even if the net is not exposed to the current, net deformation for the diamond mesh is significant, whereas deformation for the square mesh is small. According to Fig. 23.38, when the bottom-collar sinker system (WB) is applied, net deformation for the diamond mesh is less than that for the square mesh with increasing current velocity. This is the reason why rotation of the bottom of the net is reversed for the two different mesh types, and with increasing current velocity

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Volume holding coefficient [-]

Volume holding coefficient [-]

1.00 0.95 0.90 0.85 0.80 0.75 0.70

Square mesh Diamond mesh

0.65 0.60 0.55 0.10

0.15

0.20

0.25

0.30

0.35

0.40

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30

665

Square mesh Diamond mesh

0.10

0.15

0.20

0.25

Velocity [m/s]

Velocity [m/s]

(a)

(b)

0.30

0.35

0.4

Fig. 23.38. Comparison of the volume-holding coefficient with the bottom-collar sinker system. (a) RDH = 2.0 and (b) RDH = 1.43.

the orientation of the bottom of the diamond-mesh net is more helpful in reducing net deformation.

References 1. M. Tauti, A relation between experiments on model and on full scale of fishing net, Bull. Jpn. Soc. Scientific Fisheries 3(4), 171–177 (1934). 2. M. Tauti, The force acting on the plane net in motion through the water, Bull. Jpn. Soc. Scientific Fisheries 3(1), 1–4 (1934). 3. Y. C. Li, F. K. Gui, H. H. Zhang and C. T. Guan, Simulation criterica of fishing net in aquaculture sea cage experiments, J. Fishery Sciences of China 12(2), 179–187 (2005) (in Chinese). 4. F. K. Gui, Y. C. Li and H. H. Zhang, The proportional criteria for model testing of force acting on fishing cage net, China Offshore Platform 17(5), 22–25 (2002) (in Chinese). 5. F. K. Gui, Hydrodynamic behaviors of deep-water gravity cage, Doctoral dissertation, Dalian University of Technology (2006) (in Chinese). 6. F. K. Gui, Y. C. Li, G. H. Dong and C. T. Guan, Application of CCD image scanning to sea-cage motion response analysis, Aquacultural Eng. 35, 179–190 (2006). 7. Y. C. Li, Y. P. Zhao, F. K. Gui and B. Teng, Numerical simulation of the hydrodynamic behavior of submerged plane nets in current, Ocean Eng. 33(17–18), 2352–2368 (2006). 8. Y. P. Zhao, Y. C. Li, G. H. Dong, F. K. Gui and B. Teng, Numerical simulation of the effects of structures ratio and mesh style on the 3D net deformation of gravity cage in current, Aquacultural Eng. 36(3), 285–301 (2007). 9. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, Numerical and experimental study of submerged flexible plane nets in waves, Aquacultural Eng. 38, 16–25 (2008). 10. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, Numerical simulation of the hydrodynamic behaviour of gravity cage in waves, China Ocean Eng. 21(2), 225–238 (2007).

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11. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, A numerical study on hydrodynamic properties of gravity cage in combined wave-current flow, Ocean Eng. 34, 2350–2363 (2007). 12. Y. P. Zhao, Numerical investigation on hydrodynamic behavior of deep-water gravity cage, Doctoral dissertation, Dalian University of Technology (2007) (in Chinese). 13. Y. P. Zhao, Y. C. Li, G. H. Dong and F. K. Gui, Wave theory selection in the simulation of gravity cage, Proc. 17th (2007) Int. Offshore and Polar Engineering Conf., Lisbon, Portugal, 1–6 July 2007, pp. 2222–2228. 14. Y. C. Li, F. K. Gui and B. Teng, Hydrodynamic behavior of a straight floating pipe under wave conditons, Ocean Eng. 34, 552–559 (2006). 15. Y. N. Zheng, G. H. Dong, F. K. Gui and Y. C. Li, Movement response of floating circle collars of gravity cages subjected to waves, Eng. Mech. 23(Sup I), 222–228 (2006) (in Chinese). 16. C. A. Brebbia and S. Walker, Dynamic Analysis of Offshore Stuctures (NewnesButterworths, 1979), pp. 109–143. 17. R. B. Bhatt and R. V. Dukkipati, Advanced Dynamics (Alpha Science International Ltd., UK, 2001), pp. 213–219. 18. J. Wittenburg, Dynamics of Systems of Rigid Bodied (B.G. Teubner, Stuttgart, 1977). 19. Y. C. Li and B. Teng, Wave Action on the Maritime Structures, 2nd edn. (The Ocean Press, Beijing, 2002), pp. 250–265 (in Chinese). 20. E. H. Hou and Q. L. Gao, Theory and Design of Fishing Gear (The Ocean Press, Beijing, 1998), pp. 41–47 (in Chinese). 21. A. Fredheim and O. M. Faltinsen, Hydroelastic anslysis of a fishing net in steady inflow conditions, in 3rd Int. Conf. Hydroelasticity in Marine Technology, Oxford, Great Britain, University of Oxford (2003). 22. B. W. Wilson, Elastic characteristics of moorings, ASCE J. Waterways and Harbors Division 93(WW4), 27–56 (1967). 23. Y. I. Choo and M. J. Casarella, Hydrodynamic resistance of towed cables, J. Hydronautics 5(4), 126–131 (1971). 24. Y. C. Li, F. K. Gui and F. Song, Comparison on the mooring line force and cage movement characteristics of gravity and sea staion cages, in Proc. 15th Int. Offshore and Polar Engineering Conf., Seoul, Korea (2005), pp. 187–193. 25. F. K. Gui, Y. C. Li, Y. P. Zhao and G. H. Dong, A model for the calculation of velocity reduction behind a fishing net, China Ocean Eng. 20(4), 615–622 (2006). 26. D. W. Fredriksson, M. R. Swift, J. D. Irish, I. Tsukrov and B. Celikkol, Fish cage and mooring system dynamics using physical numerical models with field measurements, Aquaculture Eng. 27(2), 217–270 (2003). 27. D. Fredriksson, Open Ocean Fish Cage and Mooring System Dynamics (UMI, USA, 2001). 28. Y. C. Li, Y. C. Mao and F. K. Gui, The influence of sinker forms and weight on mooring line force of graviy sea-cage, China Offshore Platform 1(1), 6–16 (2006) (in Chinese). 29. Y. C. Li, Y. P. Zhao, F. K. Gui and B. Teng, Numerical simulation of the influences of sinker weight on the deformation and load of net of gravity sea cage in uniform flow, Acta Oceanologica Sinica 25(3), 125–137 (2006).

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Chapter 24

State of Offshore Structure Development and Design Challenges Subrata Chakrabarti Civil and Mechanical Engineering Department University of Illinois at Chicago, 842 West Taylor Chicago, IL 60607, USA [email protected] This chapter will review and highlight the research being carried out today to meet the challenges in the design, and operation of offshore structures. The subject matter, while general in nature, will focus on one of the most unique areas in the offshore structure design, namely, the fluid-induced responses of offshore structures and the associated structural design consequences. Due to the rapid growth in the offshore field, particularly in deepwaters, this area is seeing a phenomenological advancement. The chapter will begin with an overview of the historical development of fixed and floating structures. It will state the design status for these systems. The fixed structure design is more mature today, even though many aspects of it still remain empirical. For floating structures the design procedure is still advancing and more research is ongoing in this area. These will be highlighted. The treatment of the individual components of the floating structure, namely, the floater, the mooring system, and the riser system including their interactive coupling effects with fluid will be discussed. The state-of-the-art in the treatment of the individual components of the floating structure, namely, the floater, the mooring system, and the riser system will be briefly described. The design methods for these offshore components will be included. The basic differences between the coupled and uncoupled systems and the complexity of the later method will be discussed. The chapter will conclude with a discussion of the present-day deepwater design challenges that remain and the research that is needed to meet these challenges.

24.1. Introduction of Offshore Structures Offshore structures are located isolated in waters of the ocean with no continuous access to dry land. Their design life-span ranges from a few years to as many as 25 years. In most cases, they may be required to stay in position in all weather 667

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conditions. Offshore structures are defined by either their function or their configuration. The functions of an offshore structure may be one of the following: • Exploratory Drilling Structures: A Mobile Offshore Drilling Unit [MODU] configuration is largely determined by the variable deck payload and transit speed requirements. • Production Structures: A production unit can have several functions, e.g., processing, drilling, workover, accommodation, oil storage, and riser support. • Storage Structures: These structures are used in storing the produced crude oil temporarily at the offshore site before its transportation to the shore for processing. Sometimes a structure may be used for multiple functions. The configuration of offshore structures may be classified by whether the structure is a fixed structure, either piled or gravity, a compliant or articulated structure, or a floating structure. The requirements of a floating structure are that it be moored in place and that the facility remains under the environment within a specified distance from a desired location achieved either by mooring lines, or by dynamic positioning system using thrusters, or a combination of the two. First, a short description of these structures and their applications will be discussed. Then the current state-of-the-art in the general hydrodynamic analysis associated with the design of these structures, and the future problems that need to be considered in these areas will be addressed. 24.2. Fixed Structures Fixed offshore structures composed of small tubular members are mainly used for the production of oil and gas. These structures may be composed entirely of

Fig. 24.1.

Historical development of fixed jacket structures (Courtesy Shell Oil Co.).

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small battered members which are piled in place. These are called jacket structures. The evolution of fixed jacket structures with water depth is shown in Fig. 24.1. Note that, within a short span of 40 years, the fixed structure extended from a water depth of 6 m to over 400 m. Exploratory drilling operations may be performed with fixed structures, called jack-ups, which are mobile. The jack-ups are usually buoyant during transit and are towed from station to station. Once they reach their destination at the drilling site, the small-membered legs of the jack-up are set on the ocean bottom and the deck is jacked up above the water level on these legs for the drilling operation. The gravity-type fixed structures, on the other hand, are mostly composed of large steel shell-type members, which may also be used for storage in addition to production of oil and gas. There have also been many concrete gravity structures in existence, most of which are located in the North Sea. 24.2.1. Design of fixed jacket structures The first step in the design of fixed structures is the computation of the forces on its members due to the extreme environment present at the site. Forces on the submerged members of the structure arise from currents, and waves. The fatigue life of the members is also of concern in the design of these structures. In fact, several structures have failed due to the fatigue failure of critical members in the structure. It is not uncommon to repair or replace the underwater members of a fixed jacket structure during its lifetime. Today’s design of fixed offshore structures with small tubular members is still based on empirical formulae. The forces on the member are the inline (direction of current and wave) and lift or transverse force. For example, for a streamline shape of an airfoil (a) the flow remains attached to the body giving rise to an inline mean drag force, but very little flow separation and lift (or transverse) force [Fig. 24.2(a)]. On the other hand, a bluff body, e.g., a circular cylinder, generates a large separation of flow in current behind the cylinder forming vortices, which remain attached to the body [Fig. 24.2(b)] at low Reynolds number (Re) and separate from it and move away with the flow at higher Re. These vortices are generally alternating in nature, at least at moderate Re. Because of this flow separation

U

(a)

(b)

Fig. 24.2. Flow effects on body shapes. (a) Steady flow past airfoil and (b) steady flow past bluff body.

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Fig. 24.3.

Lift and drag coefficients for Re = 200.3

and the presence of vortices, the pressure behind the body in steady flow is low compared to the forward of the body fluctuating asymmetrically. These forces are difficult to compute numerically specially at high Re and research in this area is continuing. The flow past the body, as shown in Fig. 24.2(b) for a cylinder in steady flow, produces an asymmetric (fore-aft) pressure distribution around the cylinder, which traverses the flow axis at a given (Strouhal) frequency at a low or intermediate Re. This pressure distribution, in turn, generates an oscillating lift (or transverse) force with near-zero mean, in addition to an inline mean drag force, and an oscillating inline force of smaller magnitude over the steady force. Since the vortex shedding behind the cylinder is alternating in nature, the transverse force will be periodic and its frequency (at low Re) will depend on the frequency of vortex shedding. This is demonstrated in Fig. 24.3 computed in a 2D numerical simulation. The result presented applies to a Reynolds number of 200. At higher Reynolds numbers, this shedding of vortices is more random and complex causing a more irregular large transverse force on the body, which, at the present time, causes significant computational difficulty. 24.2.1.1. Empirical formulae As shown in Fig. 24.3, the mean inline load is large and the transverse force has a zero mean. The fluctuating current load in the inline direction is generally ignored as being small in a fixed-structure design. In the empirical form, the mean inline current load per unit cylinder length, known as the drag force, is given by f=

1 ρCD AU 2 2

(24.1)

and the transverse current load per unit cylinder length, sometimes called the lift force, is written as fL =

1 ρCL AU 2 , 2

(24.2)

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where ρ = fluid density; A = projected cross-sectional area; U = steady flow velocity; D = cylinder diameter; CD = drag coefficient; and CL = lift coefficient. The Reynolds number: Re = UD/ν, and ν = kinematic viscosity of fluid. The values of the coefficients needed to compute these forces have been established through model tests. The drag coefficients for a smooth cylinder have been found to be functions of Re only as shown in Fig. 24.4. If the member surface is considered rough, then the roughness parameter of the member is considered in the choice of the coefficients (generally resulting in an increase in the values of the coefficient depending on the extent of surface roughness). The fluctuating nature of the lift force makes the values of CL in Eq. (24.2) a function of time and it is a common practice to represent CL by its maximum (or rms) value. Experimental data on CL versus Re have shown considerable scatter in the value of CL . Figure 24.5 shows their upper and lower range. Figures 24.4 and 24.5 (with appropriate roughness correction) are normally used for computing the forces on a small member of an offshore structure. Since waves are oscillatory, the inline wave force on a cylinder will also be oscillatory and will depend on both the water particle velocity and the water particle

Fig. 24.4.

Fig. 24.5.

Drag coefficient for a smooth circular cylinder in steady flow.

Lift coefficient for a smooth circular cylinder in steady flow.

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acceleration. No analytical or numerical solution exists for the forces which (per unit length) are computed from the well-accepted empirical formula, commonly known as the Morison equation,29 given as f = CM AI u˙ + CD AD |u|u,

(24.3)

where AI = ρ(π/4)D2 ; AD = 0.5ρD and u = wave particle velocity; u˙ = wave particle acceleration; CM = wave inertia coefficient; and CD = wave drag coefficient. Using linear wave theory the horizontal particle velocity and acceleration are obtained from gkH 2ω gkH u˙ = 2 u=

cosh ks cos(kx − ωt) cosh kd cosh ks sin(kx − ωt), cosh kd

(24.4) (24.5)

where g = gravitational acceleration; k = wave number; ω = wave frequency; H = wave height; s = elevation from the ocean floor to the mean water level (s = 0 to d); and d = water depth. Total force on a structure member requires integration over its length. These formulae using appropriate hydrodynamic coefficients (see Sarpkaya32 for experimental values) may be applied in the design of the members of fixed structure. The API, DNV, and other similar design guidelines provide suitable data to use for cylindrical structures. For example, the API RP 2A Section C3 Recommended values are given below: Coefficient values

Smooth cylinders

Rough cylinders

CM CD

1.05 0.65

1.2 1.6

When current is present in the direction of wave (negative sign indicates opposing current), the Morison equation is modified in the presence of current in terms of a relative velocity model: f = CM AI u˙ + CD AD |u ± U |(u ± U ).

(24.6)

In this case the values of the hydrodynamic coefficients will be different than those for the wave alone. Similarly for inclined cylinders, e.g., those found in the cylindrical bracings of a jacket structure, the modified Morison equation considers force normal to the cylindrical member based on the normal component of velocity and acceleration of the water particle at the point. 24.2.1.2. Blockage factor in steady flow Often the structural members (e.g., vertical production riser bundle) appear in close proximity in an array or a group as shown in Fig. 24.6. In these cases the flow is

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S i D

Fig. 24.6.

Group of cylinders in a matrix array.

blocked by the forward cylinders. API Guidelines2 give a composite blockage factor for a dense cylinder group in steady current as follows:
CBF = 1 +

N

i=1 (CD A)i 4A

−1 (24.7)

in which CBF = composite current blockage factor, N = no. of cylinders (risers) in the bundle, A = individual cylinder projected area, and A = overall structure area normal to the flow. 24.2.2. Design of large fixed structures Unlike small-membered offshore structures, wave forces on large structures may be computed by an elegant numerical method on the assumption that the flow past the structure remains essentially potential and the irrotationality assumption for the flow is valid. 24.2.2.1. Linear diffraction/radiation forces The general analytical approach based on linear theory (commonly called the linear diffraction/radiation theory) includes the diffraction and radiation effect from the submerged portion of a structure due to linear progressive waves. Several numerical methods may be used in solving the problem mathematically e.g., panel method, fluid finite element method, hybrid region method, etc. The most common method is the boundary element panel method. It makes use of Green’s mathematical function and Green’s theorem. The spatial part of the total wave potential is written as φ = φo + φs +

6 

φRn

(24.8)

n=1

in which φo = incident potential, φs = scattered potential, and φRn = radiated potential due to forced oscillation of unit amplitude in the nth mode. The spatial

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Fig. 24.7.

Illustrative sketch of the linear force calculation on a vertical cylinder.

incident potential at a point (x, y, z) is obtained from gH cosh ks exp(ikx) (24.9) 2ω cosh kd in which i = imaginary quantity. In the numerical computation, the submerged surface of the structure is discretized into small flat panels (Fig. 24.7). The scattered part of the velocity potential in the fluid due to the continuous source distribution on the structure surface is given as  1 φs (x, y, z) = σ(a, b, c)G(x, y, z; a, b, c)ds (24.10) 4π S φo (x, y, z) = i

in which σ represents the source strength, (a, b, c) represents the source point on the surface of the structure and (x, y, z) the field point in the fluid, ds is the flat area of the panel on the submerged surface. The function G represents the near-field Green’s function given in a series or an integral form. The source strength function σ, is computed from  ∂G (x, y, z; a, b, c)ds = −4πun (x, y, z), σs (a, b, c) (24.11) 2πσs (x, y, z) − ∂n S where un = known normal fluid velocity at (x, y, z) due to the incident wave. Equation (24.11) is solved numerically by assuming the field point to be on the structure surface and setting it up in a matrix form in terms of the centers of the panels. An N × N complex matrix is formed to describe Eq. (24.11), where N is the total number of panels. The solution for the source strengths σ at the center of each panel is obtained by the inversion of the complex matrix. For a large value of N , this computation is time consuming. For the radiated potential the right-hand side of Eq. (24.11) is replaced by the normal displacement of the body at one of six degrees of freedom. Since the only difference is the right-hand side, this equation may be solved at the same time once the inverted matrix is obtained. The radiated potential provides the added mass and damping coefficients of the structure in six degrees of freedom. Once the diffraction/radiation potentials are known at the center of each panel, the external forces on the submerged body due to the total diffraction and radiation potential, are obtained respectively from the integrals  FkD = −iρω (φo + φs )nk ds (24.12) S

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and  FkR = −iρω

S

φRk nk ds,

(24.13)

where k = 1, 2, . . . , 6 and S = submerged surface area up to SWL (Fig. 24.7). Thus, the exciting wave forces (and moments) by the linear wave theory are obtained from Eq. (24.12), while the hydrodynamic added mass and radiation damping are derived from Eq. (24.13) (see, Chakrabarti8 for details). 24.2.2.2. Time domain fully nonlinear forces Numerous articles are available on nonlinear wave-body interaction with offshore structures.5,23,34 Many of these not only considers the nonlinear forces on the floating structure, but the response of the structure as well. The consistent nonlinear numerical solutions are quite elaborate and extremely time-consuming. The fully nonlinear wave-structure interaction boundary-value problem may be solved by the mixed Eulerian–Lagragian (MEL) method4,23 without any analytical approximations. This method of solution requires prohibitively large computational efforts and is not yet practical for routine industry use. Moreover, several technical issues are yet to be satisfactorily resolved before this approach can be successfully applied for complex 3D offshore structures.33 To address the need of the industry, several timedomain solution methods have been proposed which minimize this excessive use of computational efforts, while accounting for the so-called essential nonlinearities by some approximate means.11,14,33 In most cases, the hydrodynamic interaction due to radiation and diffraction effects is linearized. This allows the use of the usual 2D or 3D linear diffraction/radiation theory. In this chapter, a 2D nonlinear wave-structure interaction problem is formulated using a potential-based fully nonlinear Numerical Wave Tank (NWT). Figure 24.8 shows the definition sketch for a vertical cylinder assumed frozen at an instant in wave. The theory is based (closely following the work of Kim and Koo24 ) on modedecomposition method, and MEL material-node time marching scheme,28 and uses the boundary element method (BEM). As in the case of linear diffraction/radiation theory, an ideal fluid is assumed so that the fluid velocity can be described by the gradient of velocity potential

Fig. 24.8.

Illustrative sketch of the fully nonlinear force calculation on a vertical cylinder.

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Φ. A Cartesian coordinate system is chosen such that z is positive upwards corresponding to the still water level. Then the governing equation of the velocity potential is given by ∇2 Φ = 0.

(24.14)

The boundary conditions consist of • Fully nonlinear dynamic free surface condition, satisfied on the exact free surface (thick line in Fig. 24.8): ∂Φ 1 = −gη − − |∇φ|2 . ∂t 2

(24.15)

• Fully nonlinear kinematic free surface condition, satisfied on the exact free surface: ∂η ∂Φ = − ∇φ · ∇η. ∂t ∂z

(24.16)

• Body boundary condition on the structure: ∂Φ = 0. ∂n

(24.17)

• Input boundary condition: At the inflow boundary, a theoretical particle velocity profile is fed along the vertical input boundary. The exact velocity profile of a truly nonlinear wave under the given condition is not known a priori. Therefore, the best theoretical wave profile is input along the input boundary. Since the fully nonlinear free-surface condition is applied in the computational domain, the input wave immediately takes the feature of fully nonlinear waves. Any unnecessary spurious waves inside the domain is accounted for and corrected. An integral equation in terms of two-dimensional Green function satisfying Laplace equation is adopted. To update the fully nonlinear kinematic and dynamic free-surface conditions at each time, Runge–Kutta fourth-order time-integration scheme7 and the MEL approach are adopted. Lagrangian approach for which the free-surface nodes move with water particle motion is used. At each time step, (i) the Laplace equation is solved in the Eulerian frame, and (ii) the moving boundary points and values are updated in Lagrangian manner. To avoid nonphysical sawtooth instability on the free surface in time marching, smoothing, and regriding schemes are used. In the case of fully Lagrangian approach, the free-surface nodes need to be updated and rearranged at every time step. The regriding scheme prevents the free-surface nodes from crossing or piling up on the free surface, and thus makes the integration scheme more stable. Near the end of the computational domain, an artificial damping zone is applied on the free surface so that the wave energy is gradually dissipated in the direction of wave propagation. The profile and magnitude of the artificial damping is designed to minimize possible wave reflection at the entrance of the damping zone, while maximizing wave energy dissipation. The most physically plausible open boundary condition is Sommerfeld/Orlanski outgoing wave condition.30 The Orlanski radiation condition, for example, was used

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by Boo et al.5 for the simulation of nonlinear regular and irregular waves and by Isaacson and Cheung21 for wave–current–body interaction problems. There exist other open-boundary conditions, such as absorbing beaches by artificial damping on the free surface6,19,27 or matching with linear time-domain solutions at far field.17 It is well known that a properly designed artificial damping on the free surface does not have to be far from the body and can damp out most of the wave energy, if its length is greater than two wave lengths. Therefore, it is more effective than the matching technique and ideal to damp out relatively short waves.24 When the simulation starts, a ramp-function7 at the input boundary may be applied. The ramp function prevents the impulse-like behavior at the start and consequently reduces the corresponding unnecessary transient waves, which usually wastes computational time to die out. As a result, the simulation is more stable and soon reaches the steady state. Accurate calculation of the time derivative of velocity potential is very crucial in obtaining correct pressure and force on the body surface at each time step. There are several ways to obtain this velocity potential. Backward difference is the simplest way using the potential values of previous time steps. In case of a stationary structure, more accurate finite-difference formulae7 can also be used. The wave force on the body surface is calculated by integrating Bernoulli’s pressure over the instantaneous wetted surface from the nonlinear wave. While the above development is shown for a wave force computation, the method can be easily extended to include moving structures as well.24

24.3. Floating Offshore Structures Floating offshore system (a variety of which is illustrated in Fig. 24.9) consists of three principal structural components: • Floating hull: facilitating the space for the operation of the production work, and storage for supplies,

TLP

Sem i Fig. 24.9.

SPA R Floating offshore structures.

FPSO

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• Mooring system: providing a connection between the structure and the seafloor for the purposes of securing the structure (generally called station-keeping) against environmental loads, and • Riser system: achieving drilling operation or product transport. The station-keeping may also be achieved by dynamic positioning system solely using thrusters, or in combination with mooring lines. The mooring lines and risers provide restoring forces to the floater. 24.3.1. Frequency domain approach The motion responses of most of the large floating structures (Fig. 24.9) may be obtained in the frequency domain, which is considered satisfactory for design purposes. The floating offshore structure is almost always taken as a rigid body in its response calculations. In the simplest analysis, the attachments to the body from the moorings and risers are treated as linear or nonlinear springs externally attached to the body. The computation of wave forces on the floating structure is carried out the same way as a large fixed structure shown in the earlier section. The environmental forces are determined at the equilibrium position of the body for a linear analysis (Fig. 24.7). Moreover, the free motions of the body generate the linear hydrodynamic added mass and damping effect. In particular, the radiated potential from the body produces a 6 × 6 force matrix. When nondimensionalized by the oscillation frequency (or frequency squared) and structure displacements, the radiated forces give rise to a 6 × 6 added mass coefficient and a 6 × 6 damping coefficient matrix. The motions are obtained by solving a coupled set of equations of motion. In a linear analysis, it is even possible to introduce a nonlinear damping in an approximate manner. If a Morison type nonlinear damping is included in a linearized form, the equations of motion become mk x ¨k +

 6   8 Mlk x Dlk |x˙ l0 |x˙ l + Clk xl = fk ; ¨l + Nlk x˙ i + 3π

k = 1, 2, . . . , 6,

l=1

(24.18) where mk = mass or moment of inertia in the kth mode, xk = displacement in the kth direction, dots are time-derivatives, and subscript 0 denotes amplitude, while the variables Mlk , Nlk , Dlk , Clk , = added mass, linear and nonlinear damping and restoring force coefficients, respectively due to l degree of freedom in the kth direction. The factor 8/(3π) in the nonlinear Morison damping term arises from the linearization. The restoring force includes the stiffness arising from the structure as well as the mooring lines. The right-hand side, fk , represents the six forces (moments) on the floater by the linear diffraction theory. The stiffness due to risers are generally ignored in this analysis, but can be easily accommodated in Eq. (24.18). The stiffness is assumed linear (or linearized) for a frequency-domain solution. The solutions for 6 DOF motion xm are obtained by the inversion of the 6 × 6 matrix on the left-hand side by assuming the motions to be harmonic.

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admass+1% damping

Heave, ft/ft

admass + 15%damp

6

no plate+1%damp

4 2 0 10

11

12

13

14 15 16 Period, sec

17

18

19

20

Fig. 24.10. Example heave response of a Truss Pontoon Semisubmersible including additional heave plate added mass and damping.

Note that Eq. (24.18) is still nonlinear in the motion amplitude due to the third term. A simple iterative technique is chosen to solve the above linearized equation. In the first step, a linear harmonic solution for xl0 = 0 (l = 1, . . . , 6) on the lefthand side is obtained. This first iteration value is introduced on the left-hand side of solution and the process repeated. Generally, between two and three iterations produce convergence in the results. An example of the computed heave motion from Eq. (24.18) of a floater (pictured on the left of Fig. 24.10) is given in Fig. 24.10. Different responses refer to with and without additional added mass and nonlinear damping generated from the heave plates at the bottom of the floater. 24.3.2. Time domain approach While the above simple linear (or linearized) solution is a useful design tool for a variety of floating offshore structures, it is limited by the linear restoring force, linear damping, and linear waves. Some of the nonlinear aspects of the motion analysis are well-established, including steady drift force, and second-order low frequency (slow drift) and high frequency (TLP tendon) loads. The second-order steady drift force is derived directly from the first-order potential. Therefore, the steady drift force is computed by the pressure-area method within the linear diffraction/radiation program. The low or high frequency force calculations are much more complex in terms of a quadratic transfer function, which is extremely time-consuming. In order to reduce the computation effort for routine application in a design, simplified assumptions are often applied using fewer frequency pairs around the resonance frequency. This type of approximation is a common design practice. In view of several nonlinearities in the offshore system, a frequency domain solution is not always sufficiently accurate and time domain solution is sought in these cases. In the time domain the analysis still assumes the floater to be a rigid

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body, and the solution is generated by a forward integration scheme. The analysis can easily accommodate the nonlinearities due to the excitation forces (e.g., wind, current, and waves), as well as damping and restoring forces typically encountered in a moored floating system. The waves can be single frequency or composed of multiple frequency components following a given wave energy spectral model (e.g., PM or JONSWAP). For the series representation of long crested waves for a given wave energy spectrum, the wave profile η may be decomposed into N components as: η(x, t) =

N 

an cos[kn x − ωn t + εn ],

(24.19)

n=1

where an represents the wave amplitude having frequency ωn and a randomly chosen phase angle εn . The corresponding linear and nonlinear wave excitation force, and structure velocity dependent radiation force for each of these wave components are computed from the linear diffraction/radiation theory at a given time step. At each time step, a set of second-order differential equations (similar to Eq. (24.18), but retaining all the nonlinearities present in the system) is solved to obtain the accelerations of the system. For a single degree of freedom system, the current value of acceleration is computed from the equation of motion for the total random force at time t, F (t): x ¨c = [F (t) − cx˙ p − kxp ]/m,

(24.20)

where the subscripts c and p stand for the current and previous values, respectively. The force time history F (t) may be composed of nonlinear wind and current forces including wind spectrum, the linear diffraction forces, second-order steady and oscillating forces, and Morison and lift forces. The solution is initiated with prescribed values for the displacements and velocities, and these values are calculated for the next time step from the derived accelerations by the forward integration scheme. e.g., finite-difference: xc + x ¨p ) ∗ dt x˙ c = x˙ p + 0.5 ∗ (¨

(24.21)

xc = xp + 0.5 ∗ (x˙ c + x˙ p ) ∗ dt.

(24.22)

The extension of this approach to include multiple floating structures, e.g., an FPSO and a shuttle tanker, interconnected by mooring lines is straightforward. For example, an example of a time history of responses for a single point moored tanker-tower system7 is shown in Fig. 24.11. The biggest difficulty in these computations of the vessel response is an accurate determination of the hydrodynamic damping, specially in slow drift oscillation of soft-moored vessels or high frequency load on the stiff vertically moored vessels, e.g., TLP. For example, the typical percent damping factor of a TLP in heave is found from model test to be about 0.05 for both round and square vertical columns, while the same for a horizontal pontoon are 0.176 and 0.278.20 The evidence in the literature of appropriate and accurate values of damping in real structures is generally rare.

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20 10

Wave profile

0 400 -10

500

600

700

WAVE -20

2

Tower Motion

1

0 400

500

600

700

OMEGA

-1

20 15

Tanker Surge

10 SURGE

5 400

500

600

700

4.E+04 LINE

Line Tension

2.E+04

0.E+00 400

Fig. 24.11.

500

600

700

Time history of motions and load due to a random wave on an SPM.

24.4. Mooring System During the earlier floating structure response analysis, the mooring lines are succinctly assumed to be merely a nonlinear spring and the effects of the environment on the lines themselves are ignored having small overall effect on the floater response. In the design of these lines themselves for a floating system, however, the environmental forces may be an important criterion. In the process of design of an offshore system, one first selects their layout, geometry, and the initial mechanical and structural properties. Different types of mooring and anchoring system for floating offshore structures of today are shown in Fig. 24.12. The lines (typically 8 to 12) are arranged in a symmetric arrangement except to make room for risers if needed. There are principally two types of mooring system in use today — catenary mooring and taut mooring system. The catenary lines often consists of chain-wire-chain combination. The taut lines could be steel wires or synthetic polyester. It is customary to use drag anchors for the catenary lines. For the taut polyester lines, a vertically loaded plate or suction anchor may be suitable.

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Fig. 24.12.

Schematic layout of mooring and anchoring system.

For the taut mooring system the restoring force comes primarily from the stretch in the line. The restoring force for the conventional catenary system results principally from the submerged weight and changes in the catenary shape.

24.4.1. Design of mooring lines The design of a mooring line is performed in several ways depending on the sophistication desired in the analysis. Quite often the design is static in which the loadelongation characteristics for a single line and a mooring spread are established from the horizontal excursion of the line, ignoring fluid forces from the environment on the line itself. For a composite catenary line, the customary catenary equation is used. For the taut lines, the stiffness under loads appears from the stretch in the lines from their elastic behavior. The static mooring design analysis is accomplished by the following steps. The load-elongation characteristics for each line are first computed, given the line end-point coordinates, line length, and submerged weight. The forces for all lines in the mooring spread are then summed based on their orientation, which yield the resultant horizontal and vertical restoring forces as a function of the displacement of the vessel. The overall mean offset of the vessel caused by the steady loads from wind, current, and wave drift is then estimated. Based on this steady load, the tension in the most loaded line is computed by displacing the vessel through this offset. If line length is insufficient, calculations are repeated with increased length. The safety factor for the most heavily loaded line is computed based on the breaking strength of the line and compared with the allowable (generally 2, Ref. 1). If it is too low, the design parameters, e.g., line pre-tension, material specifications, their end co-ordinates or number of lines, are adjusted and the calculations are repeated. The safety margin is checked again allowing one (generally the most loaded) broken line. In a quasi-static design, the line dynamics are still ignored, but the dynamic loads on the floater are included in the analysis. Thus, in addition to vessel offset from the mean wind, current, and wave drift forces on the vessel, the maximum excursion

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from the wave-induced vessel motion at wave and slow-drift frequency are computed. Then the maximum line tensions resulting from the total vessel offset for each possible environmental direction are determined. The line tensions are compared with the minimum breaking load given the safety factor. DNV16 currently suggests separate safety factors for the mean and dynamic loads to avoid excessive conservatism, since mean loads are much higher and more reliably predictable. As before, the maximum peak line loads with 1 line broken are recalculated. If the proposed mooring specification fails the safety factor test, then a new specification is tried. The dynamic design of the mooring lines includes additional loads on the mooring lines themselves. In this case the analysis is much more involved and a numerical method is needed. The dynamic mooring system design is a two-step procedure as follows. First, the motions of the floater independent of the line dynamics are calculated to determine the top-end oscillation of lines. Loads on the floater are the steady current, and steady and fluctuating wind, wave and wave drift. For the floating structure, mooring system is an external nonlinear stiffness term (step 1, Fig. 24.13) as shown earlier. Once the floater dynamics are known, the motions of the mooring line attachment (fairlead) points are determined. For the mooring lines, motions of the floating structure are included at the attachment point as an externally defined oscillation (step 2). To determine the mooring line dynamics, a time domain analysis is required here. External effects on the mooring lines include the following: topend motion, current load, wave load, seabed lateral friction, and soil spring and damping effect on the portion of the line on bottom. A lumped mass, finite element or finite difference scheme may be used to model the lines. The line is decomposed into a number of straight elements (bars) with linear shape function. The distributed mass plus hydrodynamic added mass is lumped at the element nodes. The hydrodynamic damping is included for the relative motion between the line and the fluid. Damping levels vary significantly depending on water depth, line makeup, offsets, and top-end excitation. Quite often a modified Morison equation is used to represent the environmental effect. At each time step, a standard set of matrix equation is developed composed of the inertia, damping, and stiffness matrix.

Fig. 24.13.

Schematic of two-step uncoupled analysis method.31

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It is often important to check the fatigue life of the mooring line. A mooring line fatigue analysis may be based on long-term cycles of the dynamic tension on the line due to time varying loads. In fatigue, the factor of safety (FOS) is recommended1 to be between 3 and 10 depending on the lines being inspectable or not. Because of the low level of experience of lighter polyester lines in deepwater applicable, the FOS is taken to be as high as 60.

24.5. Riser System As noted earlier, the riser is an important component of an offshore structure. For a floating structure there are a variety of risers in use today, which are schematically identified in Fig. 24.14. The vertical risers are still the most common and always used for the drilling operation. These risers for drilling and production operation are pretensioned (Top Tension Risers, TTR). The production risers may have a catenary shape (Steel Catenary Riser, SCR) or may be flexible as well. Flexible risers may have additional buoyancy elements giving it an S-shape. 24.5.1. Dynamics of risers In order to implement more advanced design of risers, the current research and development on riser technology follows the following paths: numerical analysis, laboratory testing, and in situ testing. The numerical simulation includes coupled structural finite element and computational fluid dynamics (FE/CFD) analysis. In addition, several commercially available large CFD programs are currently updated for offshore applications. Limitations of this analysis today are large computational time, limitations in flow solver, including convergence and high Reynolds number. They also need systematic validity with reliable benchmark tests. The small-scale

Flexible w/buoyancy

Free standing

Fig. 24.14.

Float Tensioned

SCR

Types of risers in use today.

TTR

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testing is generally done at the University laboratory water channels, and wave tanks. They are limited by small scales, modeling problems, and measurements. Their main purpose is to increase the understanding of the flow problems past risers and generating empirical coefficients. Field measurements often include insitu testing in full scale and face the constraints of cost, practicality, environmental limitation, and client confidentiality. In spite of these ongoing developments, today almost all riser design analyses are performed by empirical methods. Static analysis is generally based on steady current loads (uniform or shear) and provides the riser deflected shape, stresses at various points on the riser, top and bottom angles, and its structural mode shape under load. Dynamic analysis considers both the inline and transverse unsteady loads. In addition to the riser shapes and stresses, it provides the vortex induced vibration, and fatigue life of the riser. 24.5.1.1. Mathematical model In this section, a basic mathematical model for the riser analysis is developed. It should, however, be recognized that there are many approaches in the dynamic analysis of risers and their further details are outside the scope of this chapter. Therefore, the development below should be taken as an exercise to introduce the basic parameters and governing equations for the riser dynamics. 24.5.1.1.1. Governing equations Consider a vertical top-tensioned riser exposed to ocean waves and currents, as illustrated in Fig. 24.15. The waves and current are shown as collinear. The definition of the co-ordinate system is shown in the figure. Considering only the current loads, the motion of the risers is governed by the following equations. For inline static analysis, the right-hand side is the current force:    d2 d2 x d 1 dx EI(y) − F + m(y)¨ x + c(y)x˙ = ρCD (y)D(y)U 2 (y). (y) e 2 2 dy dy dy dy 2 (24.23) For transverse riser analysis, the right-hand side is the transverse (or lift) force:    d2 z d 1 dz d2 EI(y) 2 − Fe (y) + m(y)¨ z + c(y)z˙ = ρCL (y)D(y)U 2 (y), 2 dy dy dy dy 2 (24.24) where x = direction of current, y = vertical direction, z = transverse direction, m = total mass per unit length of riser section, c = damping coefficient of riser section, ρ = mass density of water, Fe (y) = effective tension due to axial tension and pressure force, U (y) = current velocity as a function of the vertical coordinate y, CD (y) = drag coefficient for the riser, and CL (y) = maximum lift coefficient for the riser. Note that the mass m includes the hydrodynamic added mass. It is assumed that the riser bottom end is connected to a frictionless ball joint. The upper end is constrained to the floater through the top tensioner, but free to move with the floater both in the horizontal and vertical directions.

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Fig. 24.15.

Schematic of vertical top-tension riser in current.

For a mathematical solution of the system governed by the above equations, suitable initial conditions are also needed. For example, the initial conditions may be given by the static solutions with zero initial velocities.9 The standard modal superposition method may be used to solve the problem. For the dynamic riser solution, the right-hand sides of Eqs. (24.23) and (24.24) are replaced by the forces due to wave. In this case the modified Morison equation may be used for the inline direction [Eq. (24.23)] and the appropriate form of the lift force is used in the transverse direction [Eq. (24.24)]. The static and dynamic loads may also be combined in a single analysis, even though it is rarely found in the literature. There are several empirical codes to analyze the VIV problems based on 2D measurements of cylinder models most of which analyze cross-flow response only. A few of these application tools are SHEAR7 (MIT), VIVA (MIT), VIVANA (MARINTEK/ NTNU), VICoMo (NTNU), and ABAVIV (Technip). Some of them also perform the in-line static deflection using amplified CD values from the crossflow oscillation. 24.5.2. CFD numerical model While the empirical method of riser design is the current accepted method, considerable efforts are being spent in the more pleasant numerical approach. In this case,

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a CFD analysis incorporating the fluid and the structure is desirable. Due to high Reynolds number the problem is quite involved and time-consuming. However, the recent results look promising.12 The numerical analysis applies the turbulent incompressible Navier–Stokes equations in describing the conservation of mass and momentum in three-dimensions. Among the numerical methods, most of the CFD codes are 2D on parallel planes. They include finite element, finite difference, discrete vortex, vortex-in-cell, and vortex tracking method. A few examples are FSI-Navsim (Norsk Hydro), USP, DeepFlow (IFP), VIVIC (Imperial College), TACC, Orcina VT, and Orcina WO of OrcaFlex. There are also a few large 3D Commercial Codes being adopted for offshore applications, e.g., Fluent and Acusolve. A fundamental problem in the simulation of long marine risers is the large scale of the computational fluid dynamics problem itself.12 The risers may have lengths of thousands of meters so that using three-dimensional CFD solutions seems impossible. A practical approach to riser VIV predictions that has been proposed is to combine a series of two-dimensional fluid flow solutions along the riser axis. These “strips” are pieced together with a structural model of the riser to obtain a prediction of the fluid–structure response. This method reduces the large 3D CFD problem to a large number of small 2D problems. However, this “strip” method has some serious shortcomings. In particular, the flow around bluff bodies is inherently three-dimensional so that the 2D strip solutions can only be expected to give approximate answers. In addition, the riser may be at a steep angle of attack to the flow as is often the case with steel catenary risers (SCRs) creating strong axial flow components. Also, VIV suppression devices like helical strakes that have a very strong three-dimensional wake cannot be modeled. Finally, the strip method requires that some kind of interpolation method be used to estimate the forces between the strips. There is no general rule available to make such interpolations. In addition to the sheer scale of the CFD problem, several other difficult problems remain. Although the structural response of the riser usually does not involve nonlinear material response, it usually includes nonlinear geometric effects, such as large displacements and rotations. The presence of these effects means that a nonlinear structural model must eventually be incorporated in the solution of riser problems. In addition, large motions of the generated meshes must be accommodated in the fluid flow solution. Finally, it cannot be expected that the flow environment remains constant over the entire length of the riser. Both flow speed and direction may change. Any general solution must be able to treat these effects. A simple linear structural model for the solutions is described here.12 The riser axis is oriented in the z-direction and the eigenmodes are assumed sinusoidal. Then the eigenvectors have the form:

nπy  , (24.25) ζin = sin L where the ζin is the eigenvector associated with the nth mode and the ith index indicates the x or z directions, L is the riser length, y is the distance along the riser axis, and n is the mode number. It should be noted that the risers modeled here are

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tension dominated in that the bending stiffness of the riser is negligible compared to the stiffness due to tension. Also, the use of a sine shape in Eq. (24.25) implies that the riser tension is constant along the riser length. With this approach, the motion of the riser is assumed to be a linear summation of the vibration modes. The response is found by solving the equation: yin } + [cni ]{y˙ in } + [kin ]{yin } = fin , [mni ]{¨

(24.26)

where yin are the modal amplitudes, and mni mni kin are the associated mass, linear damping and stiffness for each mode. The forces fin are computed at each time step using the corresponding eigenvector. The displacement of the riser for the particular mode is then computed before moving to the next time step. Modeling a full scale deepwater riser was possible with this analysis [Eq. (24.26)] using a medium size computational cluster. Satisfactory experimental validation was obtained from the solution within the limitations, uncertainties, and assumptions of the available data and CFD model.12 However, some refining of data is still needed for a successful correlation. This is illustrated by the following case. A test was performed (Fig. 24.16) by towing a long riser model in water and measuring the detailed inline and cross-flow response of the riser model.10 The test setup and input data were supplied to the available software, both research type and commercial codes, for a blind correlation check. The results from this analysis are given in Fig. 24.16. It shows the measured inline and cross-flow response compared with results from these codes. The names of the codes used in this blind test are shown on the left (see the original paper for their identification). It is clear that the correlation is quite varied with different degrees of success without a consistent trends in them. After this test, the results were made available to the users of the codes for further verification. The second correlation turned out much better than the first. Thus this type of analysis tool appears to have the capability to predict the results reasonably well, but needs further experience and validation. In order that such validation can be achieved more precisely, more accurate benchmark results should be available. Only then it is possible to make use of such tools in the design of deepwater risers with confidence. Because of the numerous uncertainties in the riser analysis, the factor of safety in the fatigue life in a riser design is considered large. The guidelines, e.g., API, suggest values of FS = 3, for inspectable riser applications, and FS = 10–20, for un-inspectable applications with increased uncertainty (e.g., VIV).

24.6. Uncoupled Analysis The uncoupled analysis is still the traditional design practice for floating production systems. As already explained, in the uncoupled formulation the numerical analysis tool is based on the hydrodynamic behavior of the floater, not influenced by the nonlinear dynamic behavior of the lines. Two distinct stages of design may be identified in an uncoupled methodology:

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-1

1

Cabin

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USP Water surface in the flume Riser:28mm diameter, 13.12m long The bottom of the riser is connected to a stiff frame (not shown) mounted on the carriage.

Incident velocity profile at the riser; maximum velocity 1m/s

Deep Flow

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Orcina VT

Orcina WO

VIVA

VIVANA

VICoMo

SHEAR7

ABAVIV In-line displacement

Fig. 24.16.

Cross-flow displacement

Mode shapes of riser response in steady current.

(1) The first stage is the design of the floating moored system, employing programs based on an uncoupled model, to analyze the hydrodynamic model of the floating hull and determine its motions. The contribution of the risers is ignored in this analysis, and the behavior of the mooring lines is approximated by scalar coefficients inserted in the 6-DOF equations of motion of the hull, to represent the contribution of the lines (in terms of mass, damping, stiffness, and current loading). In this methodology, the stiffness of the risers is generally not taken into account.

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(2) The second stage consists of the design of the risers. The floater motions that result from the first stage (expressed either as RAOs or time series) are applied to the top of each individual riser, and the structural response of each riser (subjected also to the wave and current loads) is assessed employing a finiteelement based program. In the analyses of the first stage, the values for the coefficients that approximate the behavior of the lines should be estimated by the engineer, determined via simplified analytical models, or calibrated using results from experimental tests. In this analysis procedure, generally few or no integration between the moored system and the risers take place. However, some level of integration between mooring lines and risers can be achieved even when only uncoupled analysis is used.35 Such refinements consist of employing enhanced procedures for the determination of the scalar coefficients introduced in the vessel equations of motion to represent the behavior of the mooring lines. For example, as a refinement to this two-step procedure, part of the current loads on the mooring lines may be transferred to the floater, as an excitation load at the fairlead point on the floater. Traditionally, “basic” design methodologies have considered only the hull damping, ignoring the contribution not only of the risers but also of the mooring lines to the damping of the system. More refined methodologies consider this contribution by, for instance, introducing a scalar coefficient calibrated from the results of an experimental decay test. The contribution of the mooring lines to the total mass of the system may be roughly approximated by introducing a scalar mass coefficient.

24.7. Coupled Analysis The effects of the appendages, namely mooring lines and risers, become increasingly important as the structure moves into deeper waters. Thus their effects in the total system may not be ignored in these cases. In a coupled system analysis, the mooring lines and risers are included as an integral part in the numerical model simultaneously with the floating structure. It has recently been recognized13,18,22,23,26,33,34 that the most accurate design methodology for floating offshore systems should employ analysis programs based on coupled formulations. Ideally such programs should incorporate, in a single code and data structure, a hydrodynamic model for the representation of the vessel, coupled to a 3D finite-element model for the representation of the hydrodynamic and nonlinear structural dynamic behavior of the mooring lines and risers. The characteristics and advantages of this so-called “fullycoupled” methodology are described by Correa et al.13 One characteristic of this methodology is its excessive computer costs; therefore, “hybrid” methodologies and analysis procedures are presented in order to circumvent this problem and to gradually advance toward a fully coupled and integrated design methodology. In a complete coupled analysis (Fig. 24.17) the 6 DOF motions of the floating vessel is solved at a given time step. The loads and motions of the top of each of the riser/mooring line are determined. A finite element (or lumped mass) method for

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Fig. 24.17.

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Schematic for a coupled system analysis.31

riser and mooring line is applied. For example, to compute the dynamic position of the entire mooring line at each time step in the simulation, a lumped mass method is used; see, for example, Derksen and Wichers.15 In this lumped mass method, the mass of the mooring line is lumped to a finite number of nodes that are connected by linear springs, corresponding to the axial stiffness of the line. Bending stiffness is not taken into account. The current (and wave) loads on the mooring line elements are computed. In the simulation, the mooring forces in the equation of motion are kept constant over a time step (typically taken as 0.1 seconds). During this interval, the equation of motion is integrated with a Runga–Kutta method and the position and velocity of the buoy at the new time step is obtained. The new position of the buoy is then used in the lumped mass method to compute the position of and the tension in the mooring lines at this new time step, after which the procedure is repeated until the simulation reaches its desired convergence. The new values of mooring line top loads and motions are imposed to the vessel and the 6 DOF of the vessel is solved again. Iterations continue until a convergence is reached and proceeded to the next time step.

24.8. Conclusions and Outlook While deepwater development is continuing at a steady pace and considerable progress has been made in the design of offshore structures, several challenges in the analysis of deepwater floating and fixed structures remain. For the fixed structure of small members the empirical Morison equation has been well-established and the values of the hydrodynamic coefficients have been prescribed in various design codes. While it is interesting to explore a more complete solution that does not depend on empiricism, it is unlikely that considerable effort will be made in this area as the deepwater developments do not call for fixed jacket-type structures. For large structures the theory is well-established and several commercially available codes are routinely used in the design of such structures. One area where further development is sought is in the application of shallow water LNG facilities. In this case the high wave overtopping on the deck may be a serious design consideration

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for which more complete reliable and practical CFD or NWT type solutions are desirable. The design of large floating structures has traditionally been carried out using the uncoupled analysis in which the dynamics of the floater are considered along with the mooring lines and risers treated as external stiffness terms. Once the floater dynamics are known, they are applied to the individual mooring line or riser in the design of these components. Today’s design of risers and mooring line is generally empirical in nature. Progress is being made in the development of more complete and sophisticated numerical analysis for their design. This area needs further development for improved efficiency and accuracy with less dependence on empiricism. They need careful validation before their practical application. Moreover, ultradeep water and relatively smaller size of floaters necessitated the interaction of the appendages and the environmental effects on themselves to be incorporated in a coupled analysis with the floater. This is mostly in the development stage, even though some limited design verification is currently being carried out by the offshore industry with this type of tools.

References 1. American Petroleum Institute, Recommended practice for design and analysis of station-keeping systems for floating structures, API-RP2SK, Washington, DC (1996). 2. American Petroleum Institute, Recommended practice for planning, designing and constructing fixed offshore platforms, API-RP2A, 21st edn, Washington, DC (2000). 3. P. W. Bearman, J. M. R. Graham, X. W. Lin and J. R. Meneghini, Numerical simulation of flow-induced vibration of a circular cylinder in uniform and oscillatory flow, Flow-Induced Vibration, ed. P. W. Bearman (Rotterdan, Balkema, 1995), pp. 231–240. 4. R. F. Beck, Time-domain computation of floating bodies, Appl. Ocean Res. 16, 267–282 (1994). 5. S. Y. Boo, C. H. Kim and M. H. Kim, A numerical wave tank for nonlinear irregular waves by 3D high-order boundary element method, Int. J. Offshore and Polar Eng. 4, 265–272 (1994). 6. M. S. Celebi, M. H. Kim and R. F. Beck, Fully nonlinear 3-D numerical wave tank simulation, J. Ship Res. 42(1), 33–45 (1998). 7. S. K. Chakrabarti, Numerical simulation of multiple floating structures with nonlinear constraints, J. Offshore Mechanics and Arctic Eng. ASME (May 2002). 8. S. K. Chakrabarti, Hydrodynamics of Offshore Structures (Computational Mechanics Publication, Southampton, UK, 1987). 9. S. K. Chakrabarti, Nonlinear Methods in Offshore Engineering (Elsevier Publishers, UK, 1990). 10. J. Chaplin, Vortex- and wake-induced vibrations of deep water risers, Fourth Int. Conf. Fluid Structure Interaction, Ashurst, Southampton, UK (2007). 11. A. S. Chitrapu and R. C. Erketin, Time-domain simulation of large amplitude response of floating platforms, Ocean Eng. 22(4), 367–385 (1995). 12. Y. Constantinides, O. H. Oakley Jr. and S. Holmes, CFD high L/D riser modeling study, Proc. 26th Int. Conf. Offshore Mechanics and Arctic Engineering, San Diego, CA, OMAE2007-29151, June 2007.

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13. F. N. Correa, S. F. Senra, B. P. Jacob, I. Q. Masetti and M. M. Mourelle, Towards the integration of analysis and design of mooring systems and risers, parts I and II: Studies on a dicas system, Proc. 26th Int. Conf. Offshore Mechanics and Arctic Engineering (2002). 14. J. O. De Kat and R. J. Pauling, The simulation of ship motions and capsizing in severe seas, Trans. of the Society of Naval Arch. and Marine Eng. 97, 139–168 (1989). 15. A. Derksen and J. E. W. Wichers, A discrete element method on a chain turret tanker exposed to survival conditions, Boss Conf. (1992). 16. Det Norske Veritas, Offshore standard position mooring, DNV-OS-E301, June 2001. 17. D. G. Dommermuth and D. K. P. Yue, Numerical simulations of nonlinear axisymmetric flows with a free surface, J. Fluid Mech. 178, 195–219 (1987). 18. J. M. Heurtier, P. Le. Buhan, E. Fontaine, C. Le Cunff, F. Biolley and C. Berhault, Coupled dynamic response of moored FPSO with risers, ISOPE, June 2001. 19. S. Y. Hong and M. H. Kim, Nonlinear wave forces on a stationary vertical cylinder by HOBEM-NWT, Proc. 10th Int. Offshore and Polar Engineering Conf., ISOPE Seattle, WA, Vol. 3, (2000), pp. 214–220. 20. E. Huse, Resonant heave damping of tension leg platforms, Proc. 22nd Offshore Technology Conf. Paper 6317 (1990), pp. 431–436. 21. M. Isaacson and K. F. Cheung, Time-domain solution for wave-current interactions with a two-dimensional body, Appl. Ocean Res. 15, 39–52 (1993). 22. C. H. Kim, A. Clement and K. Tanizawa, Recent research and development of numerical wave tanks-a review, Int. J. Offshore and Polar Eng. 9, 241–256 (1999). 23. C. H. Kim, Recent progress in numerical wave tank research: A review, 4th Int. Conf. of the Society of Offshore and Polar Eng., Osaka, Japan (1995), 9 pp. 24. M. H. Kim and W. C. Koo, 2D fully nonlinear numerical wave tanks, Numerical Modeling in Fluid-Structure Interaction, ed. S. K. Chakrabarti (WIT Press, Great Britain, 2005), Chap. 2. 25. M. H. Kim, 2D fully nonlinear numerical wave tanks, Numerical Modeling in FluidStructure Interaction, ed. S. K. Chakrabarti (WIT Press, UK, 2005), Chap. 2. 26. M. H. Kim, E. G. Ward and R. Haring, Comparison of numerical models for the capability of hull/mooring/riser coupled dynamic analysis for spars and TLPs in deep and ultra-deep waters, ISOPE, June 2001. 27. W. C. Koo and M. H. Kim, Fully nonlinear waves and their kinematics: NWT simulation VS experiment, Proc. 4th Int. Symp. Ocean Wave Measurement and Analysis, WAVES 2001, ASCE 2, 1092–1101 (2001). 28. M. Longuet-Higgins and E. D. Cokelet, The deformation of steep surface waves on water: I. A numerical method of computation, Proc. Royal Soc. London A 350, 1–26 (1976). 29. J. R. Morison, M. P. O’Brien, J. W. Johnson and A. S. Schaaf, The force exerted by surface waves on piles, Petroleum Transactions, American Institute of Mining and Metal Eng. 4, 11–22 (1950). 30. J. E. Orlanski, A simple boundary condition for unbounded hyperbolic flows, J. Comput. Phy. 21, 251–269 (1976). 31. H. Ormberg and K. Larsen, Coupled analysis of floater motion and mooring dynamics for a turret-moored ship, Appl. Ocean Res. 20(1–2), 55–67 (1998). 32. T. Sarpkaya, In-line and transverse forces on cylinder in oscillating flow at high reynolds number, Proc. Offshore Technology Conf., Houston, Texas, OTC 2533, (1976), pp. 95–108.

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33. D. Sen, Time-domain simulation of motions of large structures in nonlinear waves, Proc. 21st Int. Conf. Offshore Mechanics and Arctic Eng., OMAE2002–28033, Oslo, Norway (2002). 34. A. Tavassoli and M. H. Kim, Interactions of fully nonlinear waves with submerged bodies by a 2D viscous NWT, Proc. 11th Int. Offshore and Polar Eng. Conf., Vol. 3, ISOPE, Stavanger, Norway (2001), pp. 348–354. 35. S. F. Senra et al., Towards the integration of analysis and design of mooring systems and risers, Part I: Studies on a semisubmersible platform, Proc. Offshore Mechanics and Arctic Engineering Conf., OMAE2002-28046 (2002).

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Computer Modeling for Harbor Planning and Design Jiin-Jen Lee Sonny Astani Department of Civil and Environmental Engineering University of Southern California 3620 S. Vermont Avenue, Kaprielian Hall 200 Los Angeles, CA 90089-2531, USA [email protected] Xiuying Xing Sonny Astani Department of Civil and Environmental Engineering University of Southern California 3620 S. Vermont Avenue, Kaprielian Hall 229 Los Angeles, CA 90089-2531, USA [email protected] Harbors are built to provide a sheltered environment for the mooring of ships and vessels. For some wave periods the semi-enclosed harbor basin acts as a resonator to amplify the wave motions in the harbor due to the combined effects of wave diffractions, refractions, and multiple reflections from the boundaries. This undesirable wave motion could induce significant ship motions, damage ships and dock facilities, and delay loading and unloading activities if the resonant wave periods are close to that of the ship mooring system. Harbor planners and engineers need to model the wave induced oscillations as new harbor layouts are contemplated. This chapter presents a computer model to be used for predicting the response characteristics of arbitrary shape harbors with variable depth. The model incorporates the effects of wave reflection, refraction, diffraction, and dissipation losses due to boundary absorption, bottom friction, and energy losses due to the flow separation at the entrances. The model is applied to four real harbors and the model results have been shown to agree surprisingly well with the field data obtained from tsunami-genic events as well as hurricane induced wave motions. The computer model is shown to be an effective engineering tool for harbor

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planning and design to derive ways of eliminating or altering the harbor response so that the harbor may indeed provide a sheltered environment for moored ships and vessels.

25.1. Introduction Harbors are built to provide a sheltered environment for the mooring of ships and vessels. In order to accomplish this goal, marine structures such as breakwaters and jetties are constructed (either detached or undetached to the shore) to reduce the wave energy incident from the open sea. The effectiveness of the breakwaters and jetties in reducing the incident wave energy must be ascertained in the planning and design of harbors. Normally, the breakwaters and jetties are effective in reducing the incident wave amplitude for waves with shorter wave periods (in the order of 16 s or less). As the wave period increases, the effectiveness of the breakwaters or jetties in reducing the incident wave amplitude progressively decreases. For waves of shorter wave period the effect of wave reflection from the harbor boundaries is quite small, thus only wave refraction due to changing water depth and wave diffraction around the breakwaters and jetties need to be considered. When the wave period increases (thus the wavelength increases), the combined effect of wave diffraction, wave refraction, and wave reflection from the harbor boundaries is very significant. In fact, it is possible that for certain semi-enclosed harbor the combined effect of wave diffraction, refraction, and multiple reflections from the boundaries can cause significant increase in the wave amplitude compared with the incident wave amplitude. This is commonly referred to as “harbor resonance” due to long waves.36,37 Resonances or oscillations due to long period waves in bays and harbors have often been observed. For example, in response to the daily flooding and ebbing tide, the Bay of Fundy (at the border of eastern Canada and northeast of the United States) has produced the largest tidal range in the world (approximately 50 ft). At the Crescent City harbor region in northern California, larger than usual water surface elevations have been observed in response to a tsunami-genic event whether it is distant tsunami or local tsunami. The records at the tide gauge station located in Crescent City harbor will be discussed later in this chapter. The resonance in a harbor could result in large fluctuation of water level in certain areas and produce strong currents throughout the harbor. In addition, it may also induce large ship motions for moored ships, especially if the fundamental resonant period of the harbor is close to that of the ship mooring system. Such oscillations could last as long as several days, delaying the cargo loading and unloading activities, breaking the mooring lines, and even damaging the moored boats and dock facilities. As a result, some ships or boats may have to move out to the open sea to avoid those large oscillations within the harbor, resulting in significant economic loss.

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It is important for harbor planners and engineers to have the answers to important questions such as: (a) (b) (c) (d)

What are the important wave periods that the harbor would respond to? Are these wave periods close to the resonant periods of the ship mooring system? What amplification of wave and current can be expected? How will the harbor respond to different forcing functions such as tsunamis or hurricanes (called typhoons in the Pacific region) as opposed to long period swells? (e) Will the response characteristics change when the harbor layout is changed or modified?

In this chapter, we have attempted to show how one can use the computer model to arrive at the answers to those questions listed above. Even though no one can completely eliminate the need for physical models, the use of computer models can be shown to provide fast and reliable answers to the questions in such a way that the physical models, if absolutely needed, can be performed more efficiently and effectively to save time and money.

25.2. Comments on Computer Model Techniques and Previous Studies 25.2.1. General comments on computer modeling techniques As discussed in Ref. 37, there were three numerical methods to be employed for the computer models: (1) finite difference method; (2) boundary element method; (3) finite element method. The model could either be run in time domain with the results expressed as a function of time or in frequency domain with the results expressed as a function of wave frequency. When the time sequence of the dependant parameters (such as wave amplitude and water particle velocity field) is important, time domain simulations will be appropriate. For harbor planning and design purpose, usually the exact incident wave form is not known or not yet occurred, frequency domain computations would appear to be more appropriate, so that one can explore all possible scenarios in the model more effectively. We will first discuss the pros and cons of the three numerical methods used in the computer models. We must realize at the outset that no one particular computer modeling technique is superior in all applications. Each has its own pros and cons. Selection of a modeling technique will depend on the experience and background of the individuals, or the scope and objective of the modeling efforts. 25.2.1.1. Finite difference method Finite difference method discretizes the function on rectangular grids with equal or varying spacing. The function on a node is related to the values on the neighboring nodes which define the grid system. Spatial or time derivatives are approximated by the difference of the values on the neighboring notes or the successive time steps. The

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functions on each node of the whole system are calculated by solving a series of difference equations with certain boundary conditions. The finite difference method is not restricted to any governing equations and the approximation is straightforward. But the calculation might not always be stable for time dependent problems and the curved boundary can only be approximated by rectangular grids. The earliest use of the method was conducted by Leendertse.25 The two-dimensional depth-averaged governing equations containing nonlinear terms were solved to study the long wave propagation mechanism. Other examples can be found in Refs. 8 and 40. 25.2.1.2. Boundary element method Boundary element method formulates the solution inside a harbor domain by an integral equation along the boundary, and the boundary integral equation is converted into a matrix equation with approximation from each of the boundary elements. The method was first used for harbor resonance study by Lee18 and Hwang and Tuck13 independently. Physically, this method can be considered as distributing sources, sinks, and doublets with proper strength along the boundary to satisfy the governing equation and the boundary conditions. The unique virtue of this method is that it can reduce the domain of calculation by one dimension. A three-dimensional problem can be formulated to solve equations on the boundary surface and a two-dimensional problem can be reduced to integration along the line boundary. The limitation of this method is that it can only be used for wellknown differential equations such as the Laplace’s equation and the Helmholtz equation with known fundamental solutions. Examples of computer models based on boundary element method can be found in Lee,19 Lee and Raichlen,23 Grilli et al.,10 and Lee et al.20 25.2.1.3. Finite element method Finite element method divides the simulation domain into finite polygons or polyhedrons, called elements. A shape function is needed to approximate the solution inside each element. The boundary value problem is presented by a variational principle with a functional that contains a bilinear form for the unknown coefficients of the shape function. The functional is then minimized to obtain a series of linear algebraic equations about the unknown coefficients for the shape function, from which the sought unknowns at the nodal points can be obtained. If the functional is not attainable, weak form formulation or Galerkin method can be used. This makes the method applicable to a wide range of physical and engineering problems since almost any governing partial differential equation can be approximated by the Galerkin procedure. Another advantage of the method is that the boundary of the simulation domain can be more precisely approximated and finer grids can be easily applied for special local regions. One of the first applications of this method in this field was conducted by Chen and Mei.7 Mild slope equation was solved and an eigenfunction expansion was used at a common boundary in the exterior region of the harbor. It is also called hybrid finite element method. Examples of finite element model can be found in Refs. 9, 12, 21, 22, 26, and 30.

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25.2.2. Governing model equations Several different partial differential equations are available for implementation in the computer models with varying degree of success. The classic depth averaged shallow water equation originally used by Leendertse25 has been used by many to solve for the water surface elevation and velocity field. The classic Laplace’s equation or Helmholtz equation was used by many to solve for the velocity potential.10,19,20 Higher order Boussinesq type of equations for velocity potential have also been successfully applied to harbor modeling.26,44 An effective and convenient model equation which has been found to be well suited for combining the effects of wave refraction and diffraction in the coastal region is the two-dimensional elliptic mild slope wave equation first derived by Berkhoff.2 The mild slope equation can be written in the form as ∇ · ccg ∇φ + k 2 ccg φ = 0, where φ is the horizontal variation in velocity potential Φ,
 cosh k(z + h) Φ = R φ(x, y) exp(−iωt) cosh kh

(25.1)

(25.2)

in which ω and k are wave frequency and wave number. c = ω/k is the wave celerity, and cg = ∂ω/∂k is group velocity. The theory is restricted to irrotational and inviscid linear harmonic waves, and a slowly varying bathymetry condition. The loss of energy due to friction or breaking is not taken into account. It was found that the results of regular mild slope equation tend to overestimate the amplification factor of the harbor resonance because of ignoring the separation and dissipation losses.34 The effects of bottom friction and boundary absorption on wave scattering were examined by Chen.6 The regular mild slope equation (25.1) was modified by including a parameter λ: ∂ ∂φ λccg + k2 ccg φ = 0. ∂xj ∂xj

(25.3)

Similar approaches were developed by Lejeune et al.24 and Yu.43 The regular mild slope equation was modified through a parameter to account for the bottom friction. Shorling and breaking effects were included in many works such as by Balas and Inan1 and Massel.28 The nonlinear effects of higher order were investigated by Mei.29 Various extended approaches (so-called extended mild slope equation) were generated to improve the performance of the regular mild slope equation for abrupt and undulating topography. Terms of bottom curvature and slope were included into the regular mild slope equation by groups of researchers.4,5,31 Although different approaches were used to derive the equations, they all obtained equivalent formulae as ∇ · ccg ∇φ + (k 2 ccg + f1 ∇2 hg + f2 (∇h)2 gk)φ = 0

(25.4)

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in which f1 and f2 are functions of kh, and h is the water depth. The effects of curvature and slope terms were examined by Chandrasekera and Cheung5 and Lee and Yoon.17 Although the mild slope equation was obtained with the assumption of “mild slope,” the work by Booji3 showed that the regular mild slope equation is applicable for bottom slope as large as 1/3. Since the mild slope equation can be conveniently implemented in a finite element model, we will apply it to real harbors in this presentation. The basics of the model will be shown in the following section.

25.3. A Computer Model Using the Mild Slope Equation A hybrid finite element model using the mild slope equation will be briefly described herein. The model solves the mild slope equation over arbitrary shape harbor of variable water depth. It incorporated the effects of wave reflection, refraction, diffraction, and dissipation losses due to boundary absorption, bottom friction, and energy losses due to the flow separation at the entrances. An earlier version of the model has been applied successfully for the modeling of harbor responses of Los Angeles and Long Beach harbors by Lee et al.22 It has also been applied to model a large coastal region for assessing the potential coastal impacts of tsunamis surrounding the island of Taiwan.21 25.3.1. Governing equation The governing equation is the mild slope equation first derived by Berkhoff2 : ∇ · (CCg ∇φ) +

Cg ω 2 φ = 0. C

(25.5)

It can also be written in the form of (25.1). In which φ = φ(x, y) is the horizontal variation of the velocity potential Φ as shown in Eq. (25.2), C = ω/k is the wave celerity, and the group velocity is,   C 2kh C Cg = (1 + G) = 1+ (25.6) 2 2 sinh 2kh in which G = 2kh/sinh 2kh. 25.3.2. Boundary conditions The modeling area is divided into two parts, the finite inner area (region A) and the infinite outer area (region R), as shown in Fig. 25.1. The inner area includes the harbor and a connected semi-circular area. The half ring-shaped outer area has a radius of infinity. Solutions of both regions are matched at the connecting boundary ∂A. The water depth is variable in the inner area but assumed to be constant in the outer area. The energy dissipation is considered in the inner area but is negligible in the outer area.

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Fig. 25.1.

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Simulation domain of the hybrid FEM model.

25.3.2.1. Far-field boundary In the simulation domain, wave field includes incident waves, reflected waves, and scattered waves. φR = φI + φS ,

(25.7)

where φI , φR , and φS are incident wave potential (including incident wave and reflected wave), outer region wave potential, and scattered wave potential, respectively. For the far-field boundary, the Sommerfeld radiation condition is applied.   √ ∂ − ik φS = 0. (25.8) lim r r→∞ ∂r The scattered wave potential function satisfying the Helmholtz equation and the radiation condition can be expressed as φS =

∞ 

Hn (kr)(αn cos nθ + βn sin nθ),

(25.9)

n=0

where αn and βn are unknown coefficients and Hn is the Hankel function. The wave potential in the inner region is then obtained by solving the mild slope equation and by matching the solutions with outer region at ∂A. 25.3.2.2. Partially absorbing boundary The solid boundary such as a vertical wall or a natural beach and the energy releasing boundary such as a river outlet can be treated as that energy is partially

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absorbed and partially reflected. The energy flux out through the boundary is formulated using a second-order scheme as iα ∂ 2 φ ∂φ , = −iαkφ − ∂n 2k ∂s2

(25.10)

where α is the absorption coefficient with a range of 0 ≤ α ≤ 1. It should be mentioned that for partially absorbing boundaries, the changing of absorption coefficient α may represent different boundary conditions. When α = 0, it represents fully reflecting condition, i.e., ∂φ/∂n = 0. When α = 1, it represents fully absorbing condition. The relation between absorption coefficient and reflection coefficient R for an incident wave angel θi is R=−

(α cos θ i )2 − 2α cos θi + α2 . (α cos θ i )2 + 2α cos θi + α2

(25.11)

25.3.2.3. Wave transmission through breakwater Wave energy is considered as partially transmitted and partially absorbed when a wave passes through a porous breakwater. The transmitted wave potential φT is assumed to be proportional to the incoming wave potential φi . ∂φT ∂φi = KT = ikKT φi , ∂n ∂n

(25.12)

where KT is the transmission coefficient through the breakwater. 25.3.2.4. Entrance loss Quadratic law for head loss has been widely used for energy dissipation at harbor entrances and bottom friction.11,14,32,39,42 The quadratic entrance head loss at the harbor entrance is also applied in this model as follows: ∆H = fe

U2 U = fe |U0 | = Ke U 2g 2g

(25.13)

in which, Ke =

fe |U0 |, 2g

(25.14)

where fe represents the dimensionless entrance loss coefficient. |U0 | is the averaged velocity at the harbor entrance computed considering no entrance loss and U is the new entrance velocity to be computed considering the entrance loss in the model. Thus the relationship of the complex velocity potentials at the entrance can be written as φ1 = φ2 + ∆φ = φ2 +

g U fe |U | . iω 2g

(25.15)

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25.3.2.5. Bottom friction The energy dissipation due to bottom friction is described as an instantaneous energy flux throughout the bottom: Ef = τb Ub ,

(25.16)

where τb is the instantaneous complex shear stress at the bottom, which can be formulated by the water particle velocity near the bed, τb =

1 ρKb |Ub |Ub , 2

(25.17)

where Kb is a dimensionless friction coefficient. The particle velocity near the bed is    ∂φ  1 = ∇φ Ub = exp(−2iωt). (25.18) ∂s b cosh kh By introducing a bottom friction coefficient, fω = 1/2gKb |Ub |, the energy flux through bottom becomes  2 1 ρ Ef = fω (∇φ)2 exp(−2iωt). (25.19) g cosh kh The bottom friction coefficient can be obtained based on the bottom roughness study by Jonsson and Carlsen.16 The resulting formula is 0.75 0.25 αb fω = 0.2gωCbf

fω = 0.15gωαb

for 1.6 < for

αb < 100 Cbf

αb < 1.6, Cbf

(25.20)

where αb = a/sinh kh, Cbf is the Nikuradse roughness height, a is wave amplitude. 25.3.3. FEM scheme The variational principle method and shape function are used to derive the FEM matrix. With the shape functions, the velocity potential and its variation can be transformed as φ = N i φi

(25.21)

∇φ = ∇N i φi = B i φi

(25.22)

φS = NSi φiS

(25.23)

∂φS ∂NSi i = φ = PSi φiS . ∂r ∂r S

(25.24)

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The matrix equations can be written as [K][Ψ] + [Q] = 0, where [Ψ] is the unknown matrix, and 

[M ] [M2 ]T

K=

[M2 ] [M1 ]

(25.25)  (25.26)

in which,   Cg ω 2 T T N N dxdy − iωCg αN T N ds CCg B B − M = C A ∂B  2 1 i 2 T fω + iωCg Kt N N ds − B T Bdxdy cosh kh ∂T A ω

 g − iωCCg (1 − Kj )N T B − Ke B T B ds iω ∂E M1 =

CCg PST NS ds

∂A

M2 = − Q1 = − Q2 =

∂A

CCg N T

∂A

∂A

CCg N T PS ds ∂φI ds ∂nA

CCg PST φI ds.

(25.27)

(25.28) (25.29) (25.30) (25.31)

The quadratic Lagrange elements are used for the region shoreward of the imaginary common boundary ∂A. For the open sea region an eigenfunction expansion is used to represent its solution. Solutions for the two regions are matched at the common boundary ∂A. A substructure technique is used in the model for which the whole calculation domain is divided in several small domains. The matrix equations are generated in each subdomain. The solutions in each subdomain are solved separately based on their own boundary values. The solution for the entire domain is obtained by matching the results at the imaginary boundaries between those subdomains. This technique largely decreases the matrix size, thus effectively accelerating the calculation. 25.4. Application to Real Harbors As mentioned before, many harbors or bays have encountered large oscillations due to long waves. For some harbors, the problem was mainly induced by the

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earthquake generated tsunamis, such as the Crescent City harbor. Some of them were caused by the typhoons and winter storms, such as the Hualien harbor in Taiwan and the Pohang harbor in Korea. The oscillation modeling in the four basins were examined by the numerical model discussed in the previous section. The results will be presented one by one, and then conclusions will be drawn at the end of this section.

25.4.1. Crescent City harbor, California Crescent City harbor, located in northern California, is one of the oldest harbors in California. The surrounding area is well known for its tsunami vulnerability. Due to its location and topography along the Pacific coast (Fig. 25.2), many have described it as a “sitting duck” for tsunami waves originated from the Pacific Ocean. It was severely damaged by tsunamis in the past such as the one generated by Alaskan earthquake in 1964, in which 11 people were killed and the property loss was estimated to be tens of million US dollars.27,35 Most recently, another heavy loss

Fig. 25.2.

Location of Crescent City.

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Fig. 25.3. Simulation domain for Crescent City harbor (left) and locations of special interest as A, B, C, D, E, and the tide gauge (right).

in the harbor was caused by the tsunami produced by the Kuril Islands earthquake on November 15, 2006. The model region for the computer model is shown in Fig. 25.3 (left). Only the major grid blocks are shown, the model contains 9,709 finite elements and 39,688 nodes with eight incident wave directions (indicated as 1 to 8). The Crescent City harbor is shown in Fig. 25.3 (right) with five locations of special interest as A, B, C, D, E, and the tide gage noted. Figure 25.4 presents the computer simulation response curves with the two distinct resonant periods at the tide gauge station of Crescent City harbor under different scenarios of wave directions. The ordinate is the amplification factor defined as the wave height at the tide gauge station divided by the incident wave height. The abscissa is the dimensionless wave number kl (where k is the wave number, 2π divided by wavelength, and l is the characteristic length of the harbor which is the length from the outer harbor entrance to the facing coastal line about 4,363 feet in the present model). It clearly shows that 22.0 min and 10.3 min resonant periods existed at the tide gauge station. Figure 25.5 shows the response curves at locations A–E and tide gauge station with incident wave coming from direction 2 (oriented toward north). It provides a clear indication that the waves are amplified as the inner harbor region is approached for both the 22.0 min and 10.3 min resonant wave periods. Thus, it appears that the more inner the location is, the more vulnerable for tsunami hazards. It explained the damage occurred in the small inner harbor on November 15, 2006 during which strong currents were invoked by the large oscillations. This large oscillation pushed the boats against the berth facilities and colliding with other neighboring boats. An examination of the tide record associated with several different tsunami events indicates that water levels at the recording station at Crescent City harbor have been amplified from waves originated from near-field as well as that from

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Fig. 25.4.

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Response curves at tide gauge location for different incoming wave directions.

Fig. 25.5. Response curves at locations noted by A–E and tide gauge station with incident wave coming from direction 2.

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Fig. 25.6. Historical records at the Crescent City harbor for events on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) August 15, 2007.

far-field. Figures 25.6(a)–25.6(d) show tide gauge records of water surface elevations during the near-field earthquake generated in offshore of northern California (June 15, 2005, Magnitude 7.2), and the three far-field earthquakes. Two generated in Kuril Islands, Japan (November 15, 2006 and January 13, 2007, with Magnitude 8.3 and 8.1), and one generated in offshore of Peru (August 15, 2007, Magnitude 7.9). The spectral analysis of the records shown in Fig. 25.6 has been performed, and the spectral density distributions are correspondingly shown in Fig. 25.7. It can be seen that the dominant waves in those events, which have the highest energy density, are all around 21 to 22 min. The incoming wave directions in the four events are roughly from southwest for the northern California earthquake, from west for the two Kuril Islands earthquakes, and from south for the Peru earthquake. The simulated response curves for these incoming wave directions superimposed with the observed resonant waves corresponding to those events are plotted in Fig. 25.8. The results indicate that the actual resonant periods are very close to those computed, especially the fundamental mode at 22 min. Water depth affects the value of wavelength for a certain wave period. Since the actual water depth may be varied due to the tide condition, the wave period may change for a certain wavelength in different events. The slight variation of

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Fig. 25.7. Corresponding spectral density of the historical records at the Crescent City harbor on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) August 15, 2007.

the resonant wave period is reasonable. The results presented clearly indicate the effect of local topography in amplifying the incident tsunami waves whether they are near-field or far-field tsunamis.

25.4.2. San Pedro bay, California San Pedro bay in southern California is one of the most important economic zones in the United States. It includes the ports of Los Angeles and Long Beach (LA/LB port), which are one of the largest and busiest ports in the world. The simulation for San Pedro bay covers a very large area including the LA/LB harbor to capture the basin resonance characteristics. An air photo of the San Pedro bay is shown in Fig. 25.9 with the simulated region indicated in Fig. 25.9(a). The radius of the semicircle of the simulation domain is about 12 miles. The model grids contain 13,263 elements and 56,134 nodes. The location of tide gauge station LA Berth60 is also shown. The computed response curves at the tide gauge LA Berth60 for different incoming wave directions are plotted in Fig. 25.10. It can be seen that the most significant resonant wave period is about 60.0 min. Generally, the amplification factor increases for incident waves coming from south. Since waves can go inside the harbor

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Fig. 25.8. Simulated response curves with field observed dominant waves superimposed at Crescent City harbor for events on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) August 15, 2007.

Fig. 25.9.

Air photos of San Pedro bay with simulation domain imposed (a) and LA/LB port (b).

directly otherwise the reflection and diffraction due to the coast and breakwaters will weaken the wave field. Similar to the modeling for Crescent City harbor, the simulation results were compared with the tide gauge (LA Berth60) records. The ports of Los Angeles and Long Beach have experienced modifications and expansion over the years. The harbor layout is now different from that which existed several decades ago. Thus

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Response curve at LA Berth60 for different incoming wave directions.

we focused on the recorded data for recent tsunamis generated by: the earthquake in offshore of northern California on June 15, 2005; two major earthquakes near the Kuril Islands, Japan on November 15, 2006 and January 13, 2007; the Solomon Islands earthquake on April 1, 2007, with a magnitude of 8.1. Both local and distant tsunamis generated by those earthquakes reached San Pedro bay, causing water level oscillations. The tide records at station LA Berth60 were analyzed to obtain the periods of the dominant modes of oscillation at that location during those events. The tide gauge records and the corresponding spectral density distributions at LA Berth60 are shown in Figs. 25.11 and 25.12. One or two dominant waves occurred during those events, all of them are around 60 min. The recorded data on April 1, 2007 was not used in the modeling examination for Crescent City harbor because the tide record was not available during that period for Crescent City harbor. For further comparison, the observed dominant waves are superimposed on the response curves simulated with the corresponding incoming wave direction for each event. This is shown in Fig. 25.13. Roughly, the incident wave came from the northwest direction for the northern California offshore earthquake, from west for the Kuril Islands earthquakes, and from southwest for the Solomon Islands earthquake. Apparently, all the observed dominant waves have all been found to correspond to the first mode of the response curves, indicating that the computer model catches the resonance characteristics of the basin. It is interesting to note that the four tide records for Crescent City harbor shown in Figs. 25.6 and 25.7 had the dominant wave period at 22 min which was predicted by the computed response curves. However, the same tide records at San Pedro bay LA/LB harbor presented in Figs. 25.11 and 25.12 showed that the dominate

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Fig. 25.11. Historical records at the LA Berth60 for events on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) April 1, 2007.

Fig. 25.12. Corresponding spectral density of the historical records at LA Berth60 on (a) June 15, 2005, (b) November 15, 2006, (c) January 13, 2007, and (d) April 1, 2007.

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Fig. 25.13. Simulated response curves with field observed dominant waves superimposed at LA Berth60 for events on (a) June 15, 2005, (b) November 16, 2006, (c) January 13–14, 2007, and (d) April 1–2, 2007.

resonant modes are at 60 min. This again implies that the local topography and the plan form strongly affect the resonant response. Thus there is a need for performing the computer modeling whenever modifications of harbor layouts are planned. 25.4.3. Hualien harbor, Taiwan Hualien harbor is located in the eastern coast of Taiwan and facing the Pacific Ocean (shown in Fig. 25.14). It has a long history of harbor resonance problem induced by typhoons during the typhoon seasons. Excessive water surface oscillations in the harbor induce large ship motion, delay cargo loading or unloading activities, and damage ships and marine structures. In the last 20 years, on several occasions, harbor resonances have resulted in mooring lines being snapped, ships and dock facilities being severely damaged. To avoid damages as typhoons approached, the ships have been ordered to move out of the harbor! The simulation region is shown in Fig. 25.15 (left). The radius of the outside semi-circle of the domain is about 4 km. To provide clarity only the major grid blocks are shown. The present grid system contains 72,955 nodes and 17,823 elements. The bathymetry was obtained from the field survey data conducted by HMTC and other organization commissioned for this region.

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Fig. 25.14.

Location of Hualien harbor, Taiwan.

The model for the present condition is simulated and compared with the field data measured by HMTC during typhoon Tim in July 1994.38 The model results and field data at station #22 and #8 are presented in Fig. 25.16. It shows the amplification factor as a function of the incident wave period. It is seen that the comparison is quite good with respect to resonant periods, resonant bandwidth, and peak amplification factors. The results clearly indicate that there exists a broadband of resonant response for wave periods between 100 and 160 s and the computer model results appear to have captured the resonant modes correctly. 25.4.4. Pohang harbor, Korea Pohang new harbor, situated in the Yongil bay in the southeast of Korea, is the largest industrial harbor in Korea and is one of the largest industrial harbors in the world. The harbor handles cargos of steel company POSCO and other industrial complex in the region. The pier structure and the loading and unloading facilities are capable of handling 36 ships concurrently which handle 47 million tons yearly including 250,000DWT size ship. Figure 25.17 presents an air photo of the harbor region with the model grid layout superimposed (left). A close-up picture of the Pohang new harbor indicated by the rectangle in the left panel is also attached on the right. The finite element grid used in the computer model contains 5,950 elements and 24,853 nodes.

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Fig. 25.15. Air photos of Hualien harbor, Taiwan with simulation domain imposed (left) and stations #8 and #22 indicated (right).

Fig. 25.16. (right).

Simulated response curves and the observed field data at stations #22 (left) and #8

Fig. 25.17. Layout of simulation domain with mesh and incoming wave directions superimposed (left) and the close up photo of the Pohang new harbor (right).

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Due to its location and the coastline orientation, Pohang new harbor has been found to provide fairly good protection against typhoons coming from the south. However, seiche motion due to long period waves have occurred frequently which produced undesirable wave and ship oscillations in the harbor, especially during the season with waves coming from the northeast direction. Field measurement program has shown the existence of the long period wave oscillation within the harbor.15 Figure 25.18(a) presents the simulated response curve at Station T1 in Pohang new harbor at various wave frequencies for the existing harbor configuration. It covers the wave periods from 100 min to 60 s. The ordinate is the amplification factor defined as the wave height at Station T1 divided by the incident wave height. The abscissa is the wave frequency with unit at 10 cycles per minute (the right-hand limit corresponds to 60 s wave). Several resonant modes are clearly seen from Fig. 25.18. Figure 25.18(b) presents the spectral density curve for Station T1 covering the same frequency range based on the data obtained in the field measurement program.33 The first four resonant periods indicated by the four vertical lines in Fig. 25.18(a) are the resonant periods obtained from the field data shown in Fig. 25.18(b), i.e., 4,800 s, 1,650 s, 490 s, and 260 s. It is seen that the present computer results compare very well with the field data, especially for the first few resonant modes. It is also noted

Fig. 25.18.

Results of (a) numerical simulation and (b) field data at Station T1.

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Fig. 25.19.

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Results of (a) numerical simulation and (b) field data at Station T2.

that several resonant modes also exist for wave coming from the north direction for wave period between 60 and 120 s. Similar results for Station T2 are shown in Figs. 25.19(a) and 25.19(b) for computer results and field measurements. Again, the computer model appears to capture the major resonant response of the harbor region very well. 25.4.5. Summary The good agreements between the simulated results and the field observations in these harbor resonance studies prove that the numerical model is a very useful tool to investigate the resonance mechanism for long waves. The differences of the amplification factor caused by various incident wave directions are the results of wave reflection from the coast or outside jetties, as well as wave diffraction due to the natural obstacles or manmade breakwaters, both of which weaken the wave field inside the harbors or bays. Tide gauge data in three earthquake events (northern California earthquake on June 15, 2005, Kuril Islands earthquakes on November 15, 2006 and January 13,

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2007) was used in the result examinations for both San Pedro bay and Crescent City harbor. Although the incoming waves in the same events may be similar before they reached the west coast of the USA, the dominant waves in LA/LB harbor (about 1 h) are totally unrelated with those in Crescent City harbor (about 22 min). This provides proofs that harbor resonance is locally induced due to the harbor layout and the local geographical configuration.

25.5. Application for Harbor Improvements As shown in the previous section, resonant response in a bay or harbor is mainly influenced by the local geometrical layout and the bathymetry and less dependent on the origin of the incident waves. The question now is what we can do to alter the resonant response characteristic to minimize the negative effect of long period oscillations. An example of harbor resonance improvement is the modification of Pier J, which is located in Long Beach harbor in San Pedro bay. Figure 25.20 shows the air photo of Pier J and its surrounding area. As reported in Lee et al.,22 the original Pier J (left photo in Fig. 25.20) was modified to include the outside breakwater (right photo in Fig. 25.20). Thus the planforms of the two layouts are quite different. The response curves obtained from the computer model before and after the construction of the breakwater at the chosen locations #81 and #82 (as noted in Fig. 25.20) are plotted in Fig. 25.21. As can be seen from the results, the amplification factors for wave periods between 40 and 140 s have been greatly reduced due to the introduction of the breakwater. The major resonant mode at 130 s has been shifted to 170 s or longer period in the hope that the resonant periods associated with moored ships are avoided. Another example for harbor modification is the Pohang harbor in Korea. The results of Pohang harbor discussed in the previous section focus on the harbor layout condition about two decades ago. The field data used for the examination was obtained in 1987. Later on another breakwater was constructed inside the harbor and the entrance area was also modified. The old and new layouts are illustrated in Fig. 25.22. The computer model generated response curves before and after the modification at station T1 and T9 shown in Fig. 25.22 are plotted in Fig. 25.23. It can be seen

Fig. 25.20.

Air photos of Pier J with the surrounding area included.

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Fig. 25.21. Response curves before and after the construction of the breakwater at location #81 (left) and #82 (right), by Lee et al.22

Fig. 25.22.

Layouts of Pohang harbor with the old one on the left and new one on the right.

Fig. 25.23. Response curves before and after the modification at Station T1 (left) and T9 (right) in Pohang harbor.

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that the amplification factor is largely diminished for the wave period ranging from 100 to 200 s, a region in which severe ship motions were found to occur. Thus, the modification of harbor layout will be helpful in reducing the seiche problem in Pohang harbor. Different types of ships may be oscillated by different wave periods. Generally, the “troublesome” waves for ships have periods ranging from 100 to 300 s. The ship motion study, such as the modeling done by Tzong et al.41 will help determine the resonant waves for ships inside a particular harbor.

25.6. Concluding Remark As shown in this presentation, wave induced oscillation in complicated harbors can be obtained conveniently using the computer model. The good agreements, which are shown, between the prototype measurement and the computer generated results further reinforce the validity of the computer modeling technique. As mentioned earlier, once the “troublesome wave periods” are determined either by the observation or by computer simulation, they can be diminished or shifted by modifying the harbor configuration to avoid potential troubles. To investigate such problem in a harbor, using physical models will be very costly and time consuming. However, modifications in a numerical simulation can be done easily especially when the original grid is already generated. The computer model can serve as a very powerful and cost-effective engineering tool for harbor planning and design to provide a sheltered environment for moored ships and vessels. Even though we may not be able to completely eliminate the need of physical model for all applications, computer model can at least help plan the physical model in a more cost-effective way to save time and money.

References 1. L. Balas and A. I. Inan, A numerical model of wave propagation on mild slopes, J. Coastal Res. SI36, 16–21 (2002). 2. J. C. W. Berkhoff, Computation of combined refraction-diffraction, Proc. 13th Coastal Eng. Conf., ASCE, New York, NY (1972), pp. 471–490. 3. N. Booji, A note on the accuracy of the mild slope equation, Coastal Eng. 7, 191–203 (1983). 4. P. G. Chamberlain and D. Porter, The modified mild slope equation, J. Fluid Mech. 291, 393–407 (1995). 5. C. N. Chandrasekera and K. F. Cheung, Extended linear refraction-diffraction model, J. Waterway Port Coastal Ocean Eng. 123, 280–286 (1997). 6. H. S. Chen, Effects of bottom friction and boundary absorption on water wave scattering, Appl. Ocean Res. 8(2), 99–104 (1986). 7. H. S. Chen and C. C. Mei, Oscillations and waves forces in an offshore harbor, Report No. 190, Parsons Laboratory, MIT (1974). 8. W. J. Chiang and J. J. Lee, Simulation of large scale circulation in harbors, J. Waterway, Port, Coastal and Ocean Div. 108, WWI (1982).

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9. M. Ganaba, L. C. Wellford and J. J. Lee, Finite element methods for boundary layer modeling with application to dissipative harbor resonance problem, Chapter 15 of Finite Elements in Fluid, Vol. 5 (John Wiley & Sons, 1984), pp. 325–346. 10. S. T. Grilli, J. Skourup and I. A. Svendsen, An efficient boundary element method for nonlinear water waves, Eng. Analysis with Boundary Elements 6(2), 97–107 (1989). 11. K. Horikawa and H. Nishimura, On the function of tsunami breakwaters, Coastal Eng. J. 13, 103–112 (1970). 12. J. R. Houston, Interaction of tsunamis with the Hawaiian Islands calculated by a finite element numerical model, J. Physical Oceanography 8(4), 93–102 (1978). 13. L. S. Hwang and E. O. Tuck, On the oscillations of harbors of arbitrary shape, J. Fluid Mech. 42, 447–464 (1970). 14. Y. Ito, Head loss at tsunami-breakwater opening, Proc. 12th ASCE Conf. Coastal Eng. (1970), pp. 2123–2131. 15. W. M. Jeong, S. B. Oh, J. W. Chae and S. I. Kim, Analysis of the wave induced downtime at Pohang new harbor, J. Korean Soc. Coastal and Ocean Engineers 9(1), 24–34 (1997). 16. I. G. Jonsson and N. A. Carlsen, Experimental and theoretical investigations in an oscillatory turbulent boundary layer, J. Hydraul. Res. 14, 45–60 (1976). 17. C. Lee and S. B. Yoon, Effect of higher-order bottom variation terms on the refraction of water waves in the extended mild slope equation, Ocean Eng. 31, 865–882 (2004). 18. J. J. Lee, Wave induced oscillations in harbors of arbitrary shape, Report No. KH-R20, W. M. Keck Laboratory, Caltech (1969), p. 266. 19. J. J. Lee, Wave-induced oscillations in harbors of arbitrary geometry, J. Fluid Mech. 45(2), 375–394 (1971). 20. J. J. Lee, C. Chang and F. Zhuang, Interactions of transient nonlinear waves with coastal structures, 23rd Conf. Coastal Eng. (1992). 21. J. J. Lee and C. P. Lai, Assessing impacts of tsunamis on Taiwan’s and China’s southeast coastlines, Proc. ICCE 2006, San Diego (2006). 22. J. J. Lee, C. P. Lai and Y. Li, Application of computer modeling for harbor resonance studies of Long Beach and Los Angeles harbor basins, Proc. 26th Int. Conf. Coastal Eng. (1998), pp. 1196–1209. 23. J. J. Lee and F. Raichlen, Oscillations in harbors with connected basins, J. Waterways, Ports, Coastal and Ocean Eng. Div. 98(WW3), 311–332 (1972). 24. A. Lejeune, M. Lejeune and M. Sahloul, Wave plan computation method in study of the Calvi Bay erosion in Corsica, France, Int. J. Numer. Meth. Eng. 27, 71–85 (1989). 25. J. J. Leendertse, Aspects of a computational model for long-period wave propagation, Memo KM-5294-PR RAND Corp., Santa Monica, CA (1967). 26. T. G. Lepelletier and F. Raichlen, Harbor oscillations induced by non-linear transient long waves, J. Waterway, Port, Coastal and Ocean Eng. 113(4), 381–400 (1987). 27. O. T. Magoon, Structural damage by tsunamis, Coastal Engineering, Santa Barbara Specialty Conf., ASCE (1965), pp. 35–68. 28. S. R. Massel, Ocean Surface Waves: Their Physics and Prediction (World Scientific Publication, Singapore, 1995). 29. C. C. Mei, Mild-slope approximation for long waves generated by short waves, J. Eng. Math. 35, 43–57 (1999). 30. C. C. Mei, M. Stiassnie and D. K. Yue, Theory and Applications of Ocean Surface Waves (World Scientific Publishing Company, 2005).

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31. S. R. Massel, Expanded refraction-diffraction equation for surface waves, Coastal Eng. 19, 97–126 (1993). 32. J. W. Miles and Y. K. Lee, Helmholtz resonance of harbors, J. Fluid Mech. 67(3), 445–464 (1975). 33. Ministry of Construction, The Report of Wave Measurement and Numerical Experiment in Pohang New Harbor (1987), pp. 108–116. 34. M. Okihiro, R. T. Guza and R. J. Seymour, Excitation of seiche observed in a small harbor, J. Geophys. Res. 98(C10), 18201–18211 (1993). 35. D. M. Powers, The Raging Sea: The Powerful Account of the Worst Tsunami in U.S. History (Kensington Publishing Corporation, 2005). 36. F. Raichlen, Long period oscillations in basins of arbitrary shapes, Chapter 7, Coastal Engineering, Santa Barbara Speciality Conf. (1965), pp. 115–145. 37. F. Raichlen and J. J. Lee, Oscillation of bays, harbors, and lakes, Chapter 13, Handbook of Coastal and Ocean Engineering, ed. J. B. Herbich (Gulf Publishing Company, 1992). 38. C. H. Su and T. K. Tsay, Numerical simulation on harbor oscillations in Haw-Lien harbor, Report NO. MOTC-IOT-IHMT-NB9001-1, Institute of Harbor and Marine Technology Institute of Transportation, Tai-Chung, Taiwan (2002). 39. G. M. Terra, W. J. Berg and L. R. M. Maas, Experimental verification of Lorentz’ linearization procedure for quadratic friction, Fluid Dynamics Res. 36, 175–188 (2005). 40. V. V. Titov and C. E. Synolakis, Numerical modeling of tidal wave runup, J. Waterways, Port, Coastal and Ocean Eng. 124(4), 157–171 (1998). 41. T. J. Tzong, C. P. Lai, J. J. Lee and F. Zhuang, Ship motion modeling in Los Angeles and Long Beach harbors, Proc. ICCE 2006, San Diego (2006). ¨ Unl¨ ¨ uata and C. C. Mei, Effects of entrance loss on harbor oscillations, J. Waterways 42. U. Harbors and Coastal Eng. Div. 101(WW2), 161–179 (1975). 43. X. Yu, Finite analytic method for mild-slope wave equation, J. Eng. Mech. 122(2), 109–115 (1996). 44. J. A. Zelt and F. Raichlen, A Lagrangian model for wave-induced harbour oscillations, J. Fluid Mech. 213, 203–225 (1990).

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Chapter 26

Prediction of Squat for Underkeel Clearance Michael J. Briggs Coastal and Hydraulics Laboratory US Army Engineer Research and Development Center 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA [email protected] Marc Vantorre Ghent University, IR04, Division of Maritime Technology Technologiepark Zwijnaarde 904, B 9052 Gent, Belgium [email protected] Klemens Uliczka Federal Waterways Engineering and Research Institute Hamburg Office, Wedeler Landstrasse 157 D-22559 Hamburg, Germany [email protected] Pierre Debaillon Centre d’Etudes Techniques Maritimes Et Fluviales 2 bd Gambetta, BP60039, 60321 Compiegne, France [email protected] This chapter presents a summary of ship squat and its effect on vessel underkeel clearance. An overview of squat research and its importance in safe and efficient design of entrance channels is presented. Representative PIANC empirical formulas for predicting squat in canals and in restricted and open channels are discussed and illustrated with examples. Most of these formulas are based on hard bottoms and single ships. Ongoing research on passing and overtaking ships in confined channels, and offset distances and drift angles is presented. The effect of fluid bottoms or mud is described. Numerical modeling of squat is an area of future research and some comparisons are presented and discussed.

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26.1. Introduction When a ship travels through shallow water it undergoes changes in its vertical position due to hydrodynamic forces from the flow of water and wave-induced motions of heave, pitch, and roll. The focus of this chapter is on the former mechanism of ship squat. Squat is the reduction in underkeel clearance (UKC) between a vessel at-rest and underway due to the increased flow of water past the moving body. The forward motion of the ship pushes water ahead of it that must return around the sides and under the keel. This water motion induces a relative velocity between the ship and the surrounding water that causes a water-level depression in which the ship sinks. The effect of shallow water and channel banks only exacerbates these conditions. The velocity field produces a hydrodynamic pressure change along the ship similar to the Bernoulli effect in that kinetic and potential energy must be in balance.1 This phenomenon produces a downward vertical force (sinkage, positive downward) and a moment about the transverse axis (trim, positive bow up) that can result in different values of squat at the bow and stern (Fig. 26.1). This combination of sinkage and change in trim is called ship squat. Most of the time squat at the bow, Sb , represents the maximum value, especially for full-form ships, such as supertankers. In very narrow channels or canals and for high-speed (fine-form) ships, such as passenger liners and containerships, the maximum squat can occur at the stern Ss . The initial trim of the ship also influences the location of the maximum squat. The ship will always experience maximum squat in the same direction as the static trim.2 If trimmed by the bow (stern), maximum squat will occur at the bow (stern). A ship trimmed by the bow or stern when static will remain that way and will not level out when underway to offset the sinkage at the bow or stern due to squat. So why do we care about ship squat? For one thing, ship squat has always existed, but was less of a concern with smaller vessels and with relatively deeper channels. The new supertankers and supercontainerships have smaller static UKC and higher service speeds. Secondly, the goal of all ports is to provide safe and efficient navigation for waterborne commerce. Since operation and maintenance costs continue to escalate and can easily exceed $3M per vertical meter, it is imperative to minimize required channel depths and associated dredging costs. Finally, even though we have a pretty good handle on squat predictions, accidents continue to occur. Barrass3 noted that there have been 12 major incidents between 1987 and 2004. In 2007, this number of ship incidents had increased to as many as 82 that are partially attributable to ship squat.4 The luxury passenger liner QEII grounded

Fig. 26.1.

Schematic of ship squat at bow and stern.

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off Massachusetts in 1992 with a repair cost of $13M and another $50M for lost passenger bookings. In the early 1990s, the Maritime Commission (MarCom) of the Permanent International Association of Navigation Congresses (PIANC) formed a working group (WG30) to provide information and recommendations on the design of approach channels.5 In the past 10 years since the WG30 report, research in squat predictions was a dynamic area in naval architecture with new experiments to study the effects of fluid bottoms and passing and overtaking vessels, especially with the increasing size of the shipping fleet. Time domain Reynolds Average Navier–Stokes Equation (RANSE) numerical models are being developed to predict squat, but these models are still being validated. In 2005, the PIANC MarCom formed a new working group Horizontal and Vertical Dimensions of Fairways (WG49) to update the WG30 report on design of deep draft navigation channels.6 A summary of ship squat is presented in this chapter. In the second section, factors governing squat including ship characteristics, channel configurations, and combined factors are discussed. Some empirical formulas from the PIANC WG30 report are presented and compared in the third section. The fourth section presents some recent research on the effect of squat on passing and overtaking ships in confined channels by the Federal Waterways Engineering and Research Institute (BAW) in Hamburg, Germany, and the Flanders Hydraulic Research (FHR) Laboratory in Antwerp, Belgium. It also includes numerical modeling by Delft University of Technology and laboratory modeling by FHR on the effect of ship offset and drift on squat. The fifth section summarizes the recent studies at FHR on the effect of fluid bottoms (i.e., mud) on squat. The development of numerical models to predict ship squat is an ongoing research area. The current status of this development at Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF), France, is discussed in the sixth section. Finally, a summary and conclusions of ship squat issues is presented in the last section.

26.2. Factors Governing Squat Prediction of ship squat depends on ship characteristics and channel configurations. These factors are often combined to create new normalized parameters to describe the squat phenomenon. 26.2.1. Ship characteristics The main ship parameters include ship draft, T , hull shape as represented by the block coefficient, CB , and ship speed, VS (m/s) or VK (knots). Other ship parameters include the length between forward and aft perpendiculars Lpp and the beam, B. The CB is a measure of the “fineness” of the vessel’s shape relative to an equivalent rectangular volume with the same dimensions. The range of values of CB is typically between 0.45 for high-speed vessels and 0.85 for slow, full-size tankers and bulk carriers. The most important ship parameter is its speed VS . This is the relative speed of the ship in water, so fluvial and tidal currents must be included. In general,

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squat varies as the square of the speed. Therefore, doubling the speed quadruples the squat and vice versa. There are two calculated ship parameters that are based on the basic ship dimensions. The ship’s displacement volume ∇ (m3 ) is defined as ∇ = CB Lpp BT.

(26.1)

The CB can be determined from the ∇ if the other ship dimensions are known. The underwater midship cross-sectional area AS is generally defined as AS = 0.98BT.

(26.2)

The “0.98” constant accounts for reduction in area due to the keel radius.7 Some researchers ignore this and use a constant of “1.00” since the error is small relative to other uncertainties in the squat calculations. Finally, the bulbous bow and stern-transom are two other characteristics of a ship that affect squat. Many of the early squat measurements were made before bulbous bows were in use. Newer designs of bulbous bows, although mainly to reduce drag and increase fuel efficiency, also have an effect on squat. The newer “sterntransoms” on some ships are “blockier” (i.e., wider and less streamlined) than earlier ship designs and affect squat as they become more fully submerged with increases in draft.8 26.2.2. Channel configurations The main channel considerations are proximity of the channel sides and bottom, as represented by the channel depth h and cross-sectional configuration. If the ship is not in relatively shallow water with a small UKC, squat is usually negligible. Ratios of water depth to ship draft h/T greater than 1.5–2.0 (i.e., relatively deepwater) are usually considered safe from the influences of squat. The main types of “idealized” channel configuration are (a) open or unrestricted (U), (b) confined or restricted (R), and (c) canal (C). Figure 26.2 is a schematic of these three types of entrance channels for ocean-going or deep draft ships. Unrestricted channels are in relatively larger open bodies of water and usually toward the offshore end of entrance channels. Analytically and numerically, they are easier to describe and were some of the first types studied. Sections of rivers may even

Fig. 26.2. canal.

Schematic of three channel types: unrestricted or open, restricted or confined, and

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be classified as unrestricted channels if they are wide enough. The second type of channel is the restricted channel with an underwater trench that is typical of dredged channels. The restricted channel is a cross between the canal and unrestricted channel type. The trench acts as a canal by containing and influencing the flow around the ship, and the water column above the hT allows the flow to act as if the ship is in an unrestricted channel. The last type of channel is the canal. These channels are representative of channels in rivers with emergent banks. The sides are idealized as one slope when in reality they may have compound slopes with revetment to protect against ship waves and erosion. The canal may or may not be exposed to tidal fluctuations. For instance, the Panama and Suez Canals have a constant water depth. Many channels can be characterized by two or three of these channel types as the different segments or reaches of the channel have different cross-sections. Finally, many real-world channels look like combinations of these three types as one side may look like an open unrestricted channel and the other side like a canal or restricted channel with side walls. Most of the PIANC empirical formulas are based on ships in the center of symmetrical channels, so the user has to use “engineering judgment” when selecting the most appropriate formulas. New data are being collected for some of these more realistic channel shapes, so future formulas may account for these differences in channel shapes. Other important parameters necessary to describe restricted channels and canals are the channel width at the bottom of the channel W , trench height hT from the bottom of the channel to the top of the trench, and inverse bank slope n (i.e., run/rise = 1/ tan θ). The value of n, although not necessarily an integer, typically has a value such as 1, 2, or 3 representing side slopes of 1:1, 1:2, and 1:3, respectively. How does one define the width of an unrestricted or an open channel since there are no banks or sides? In 2004, Barrass had defined an effective width Weff for the unrestricted channel as the artificial side boundary on both sides of a moving ship where the ship will experience changes in performance and resistance that affect squat, propeller RPMs, and speed.3 His width of influence FB is defined for h/T values from 1.10 to 1.40 as 
7.04 B. (26.3) FB = Weff = CB0.85 Mean values of FB are of the order of 8B to 8.3B for supertankers (CB range from 0.81 to 0.87), 9B to 9.5B for general cargo ships (CB range from 0.68 to 0.80), and 10B to 11.5B for containerships (CB range from 0.57 to 0.71). The calculated cross-sectional area AC is the wetted cross-section of the canal or the equivalent wetted area of the restricted channel by projecting the slope to the water surface. It is given by AC = W h + nh2 .

(26.4)

For an unrestricted channel, use Barrass’s effective width Weff for channel width W and set n = 0 in the equation for AC .

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26.2.3. Combined ship and channel factors Several dimensionless parameters are required in the PIANC squat prediction formulas that are ratios of both ship and channel parameters. They include the depth Froude number Fnh and the blockage factor S. The most important dimensionless parameter is Fnh , which is a measure of the ship’s resistance to motion in shallow water. Most ships have insufficient power to overcome Fnh values greater than 0.6 for tankers and 0.7 for containerships. Most of the empirical equations require that Fnh be less than 0.7. For all cases, the value of Fnh should satisfy Fnh < 1, an effective speed barrier and the defining level for the subcritical speed range. The Fnh is defined as Vs Fnh = √ gh

(26.5)

with gravitational acceleration g (m/s2 ). The blockage factor S is the fraction of the cross-sectional area of the waterway AC that is occupied by the ship’s underwater midships cross-section AS defined as S=

AS . AC

(26.6)

Typical S values can vary from 0.03 to 0.25 or larger for restricted channels and canals, and to 0.10 or less for unrestricted channels.3,9 Higher values may occur, for example, the canal from Terneuzen (The Netherlands) to Ghent (Belgium) is operated with a blockage factor S = 0.275, and higher values will be evaluated in the near future.10 The value of S is a factor in the calculation of the ship’s critical speed in canals and restricted channels (see Sec. 26.3.3.3 and Appendix 26.A). 26.3. PIANC Squat Formulas 26.3.1. Background In 1997 the PIANC WG30 report included 11 empirical formulas and one graphical method from nine different authors for the prediction of ship squat.5 They were based on physical model experiments and field measurements for different ships, channels, and loading characteristics. The formulas included the pioneering work of Tuck11 , Tuck and Taylor,12 and Beck et al.,13 and the early research by Hooft,14 Dand,15 Eryuzlu and Hausser,16 R¨omisch,17 and Millward.18,19 The PIANC recommends that channels be designed in two stages. The first is the “Concept” Design where a “quick” or “ballpark” answer is desired. The WG30 report recommended the International Commission for the Reception of Large Ships (ICORELS) formula20 in this phase. The second stage is the “Detailed” Design phase where more accurate and thorough predictions and comparisons are required. The WG30 recommended the formulas by ICORELS, Huuska,7 Barrass,21,22 and Eryuzlu et al.23 in this second stage. All of these formulas give predictions of bow squat Sb , but only the R¨ omisch formula gives predictions for stern squat Ss for all channel types. The Barrass

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formula gives Ss for unrestricted channels, and for canals and restricted channels depending on the value of CB . Each formula has certain constraints that it should satisfy before being applied, usually based on the ship and channel conditions under which it was developed. Caution should be exercised if these empirical formulas are used for conditions outside those for which they were developed. In 2005 the PIANC MarCom formed WG49,6 which is in the process of reviewing and revising these formulas for an updated report on channel design (expected to be completed in 2010). There have been some new formulations since the WG30 report that are being evaluated. Barrass has continued to develop and refine his formulas and now has predictions for both Sb and Ss . Ankudinov et al.24 proposed the Maritime Simulation and Ship Maneuverability (MARSIM) 2000 formula for maximum squat based on a midpoint sinkage and vessel trim in shallow water. It is one of the most thorough and the most complicated formulas for predicting ship squat. The St. Lawrence Seaway (SLS) Trial and Very Large Crude Carriers (VLCC) formulas are based on the prototype measurements in the SLS by Stocks et al.25 Briggs26 developed a FORTRAN program to calculate squat using most of these formulas. It is not possible to include all the formulas in this chapter. We have selected a representative sample of formulas that can be used for both phases of design. Some are the “old tried and true” formulas and some are based on new research. The Concept Design phase is by definition the simplest, of course this does not necessarily mean that these formulas are any less accurate than some of the more complicated formulas. In the Detailed Design phase, it is usually a good practice to evaluate the squat with several of the formulas and calculate some statistics such as average and range of values. In some cases, the maximum squat values might be used in design for the case of dangerous cargo and/or hard channel bottoms. The user should always be mindful for the original constraints. Some of these constraints are very restrictive (especially for the newer vessels coming on line) as they are based on the limited set of conditions tested in physical models by the individual researchers. This does not mean that the particular formula would not be applicable if the constraints are exceeded by a reasonable amount. Therefore, the user should exercise Engineering Judgment when deciding the applicability of those predictions. Table 26.1 summarizes the applicable channel configurations and Table 26.1.

Channel configurations and parameter constraints for PIANC squat formulas. Configuration Code ID

Constraint Code ID

Formulas

U

R

C

CB

Barrass27

Y

Y

Y

0.5–0.85

1.1–1.4

Eryuzlu et al.23 Huuska7 ICORELS20 Yoshimura28 R¨ omisch17

Y Y Y Y Y

Y Y

Y

≥ 0.8 2.4–2.9 0.6–≥0.8 2.19–3.5

1.1–2.5 1.1–2.0

Y Y

Y Y

0.55–0.8

B/T

2.5–5.5 2.6

h/T

≥1.2 1.19–2.25

Notes: 1. Huuska/Guliev originally for Fnh ≤ 0.7.

hT /h

L/B

6.7–6.8 0.22–0.81 5.5–8.5 3.7–6.0 8.7

L/h

L/T

16.1–20.2

22.9

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parameter constraints according to the individual testing conditions for the formulas in this chapter. 26.3.2. Concept design 26.3.2.1. ICORELS The ICORELS formula20 for bow squat Sb is one of the original formulas from the PIANC WG30 report.5 It was developed for unrestricted or open channels only, so it should be used with caution if applied for restricted and canal channels. It is similar to Hooft’s14 and Huuska’s7 equations and is defined as S b = CS

2 ∇ Fnh  2 L2pp 1 − Fnh

(26.7)

where CS = 2.4 and the other factors have been previously defined. The Finnish Maritime Administration (FMA) uses this formula with different values of CS depending on the ship’s CB .29,30    1.7 CB < 0.70 CS = 2.0 0.70 ≤ CB < 0.80 . (26.8)   2.4 C ≥ 0.80 B The BAW, however, recommends a value of CS = 2.0 for the larger containerships of today which may have a CB < 0.70. Their research is based on many measurements along the restricted channel (side slope n varies from 15 to 40), 100-km long, River Elbe.31 The wider stern-transom ships (see Sec. 4.3) require CS = 3 because of the increased bow squat. The FHR has found CS ≥ 2.0 for modern containerships. They typically travel at much higher speeds than the ICORELS formula was originally developed, even in shallow and restricted waters. The Fnh are higher and in this speed range the effect of blockage S on the critical ship speed is considerable. For example, a very small S = 0.01 results in an important decrease in critical speed.10 26.3.2.2. Barrass The Barrass4,27 formula is one of the simplest and “user friendly” and can be applied for all channel configurations. Based on his earlier work in 1979,21 1981,22 and 2004,3 the maximum squat SMax at the bow or stern is determined by the value of ship’s CB and Vk as SMax =

KCB Vk2 . 100

(26.9)

According to Barrass,2 the value of CB determines whether SMax is at the bow Sb or stern SS (requires even keel when static). He notes that full-form ships with CB > 0.7 tend to squat by the bow and fine-form ships with CB < 0.7 tend to squat by the stern. The CB = 0.7 is an “even keel” situation with squat the same

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at both bow and stern. Of course, for channel design, one is mainly interested in the maximum squat and not necessarily whether it is at the bow or stern. This formula is based on a regression analysis of more than 600 laboratory and prototype measurements. Stocks et al.25 found that the Barrass formulas gave the best results for New and Traditional Lakers in the Lake St. Francis area (unrestricted channel) of the SLS. The BAW feels that the Barrass restricted formula is conservative for their restricted channel applications in the Elbe River. The coefficient K 4 is defined in terms of blockage factor S as K = 5.74S 0.76.

(26.10)

A value of S = 0.10 is equivalent to a “wide” river (unrestricted or open water conditions). The value of K = 1 and the denominator in the equation for SMax remains 100. If S < 0.10, the value of K should be set to 1. For restricted channels, a value of the order of S = 0.25 gives a value of K = 2, and the denominator becomes 50. Thus, the effect of K is to modify the denominator constant between values of 50 to 100. Constraints on these equations are 1.10 ≤ h/T ≤ 1.40 and 0.10 ≤ S ≤ 0.25. This equation can accommodate a medium width river with a value of S between the limits of S above. For ships in unrestricted channels that are at even keel when in a static condition (i.e., moored), one can estimate the squat at the other end of the ship (either bow or stern) based on SMax . Thus, if CB indicates the ship will squat by the bow, then this formula will give the squat at the stern, and vice versa:  Sb CB ≤ 0.7 2 [1 − 40(0.7 − CB ) ]SMax = . (26.11) SS CB > 0.7 26.3.2.3. Yoshimura The Overseas Coastal Area Development Institute of Japan32 and Ohtsu et al.33 proposed the following formula for Sb as part of their new Design Standard for Fairways in Japan. This formula was originally developed by Yoshimura28 for open or unrestricted channels typical of Japan. The range of parameters for which this formula is applicable is shown in Table 26.1. In 2007, Ohtsu34 proposed a small change to the ship velocity term Vs (last factor in the equation is now Ve ) to include S to improve its predictions in restricted channels and canals: Ve =

  Vs

Unrestricted



Restricted, canal

Vs (1 − S)

.

(26.12)

Their Sb predictions generally fall near the average for most of the other PIANC bow squat predictions, regardless of ship type:





3 2 CB 1 CB 1 Ve + 15 0.7 + 1.5 . (26.13) Sb = h/T Lpp /B h/T Lpp /B g

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26.3.3. Detailed design 26.3.3.1. Eryuzlu One of the more recent series of physical model tests and field measurements was conducted by Eryuzlu et al.23 for cargo ships and bulk carriers with bulbous bows in unrestricted and restricted channels. Their tests used self-propelled models with bulbous bows. Many of the early PIANC formulas did not have ships with bulbous bows. The range of ship parameters was somewhat limited with CB ≥ 0.8, B/T from 2.4 to 2.9, and Lpp /B from 6.7 to 6.8. The Eryuzlu formula should not be used for containerships unless they meet this CB criteria. They conducted some supplemental physical model tests with an hT /h = 0.5 and n = 2 to investigate the effect of channel width in restricted channels. The Canadian Coast Guard35 is using the Eryuzlu et al.23 formula exclusively. Stocks et al.25 recommended the Eryuzlu formula for the chemical tankers in the Lake St. Louis section (unrestricted channel) of the SLS. The Eryuzlu formula for Sb is defined as 2.289 −2.972

Vs h h2 √ Sb = 0.298 Kb . (26.14) T T gT Note that the Ship Froude number rather than Fnh is used in their equation since the ship draft T is used in the denominator instead of the channel depth h. The Kb is a correction factor for channel width W relative to ship’s B given by  W 3.1   < 9.61  W/B B . (26.15) Kb =  W  1 ≥ 9.61 B One should use the second value of Kb = 1 for unrestricted channels regardless of effective width Weff since the channel has no boundary effects on the flow and pressures on the ship. 26.3.3.2. Huuska/Guliev The next empirical formula in the Detailed Design phase is by Huuska.7 This Finnish professor extended Hooft’s work for unrestricted channels to include restricted channels and canals by adding a correction factor for channel width Ks that Guliev36 had developed. The Spanish ROM 3.1-99 (Recommendations for Designing Maritime Configuration of Ports, Approach Channels, and Floatation Areas37 ) and the FMA recommend the Huuska/Guliev formula for all three channel configurations. In general, this formula should not be used for Fnh > 0.7. The FMA29 also includes some additional constraints for lower and upper limits as follows (Table 26.1): • • • •

CB 0.60 to 0.80 B/T 2.19 to 3.50 Lpp /B 5.50 to 8.50 hT /h 0.22 to 0.81

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Fig. 26.3.

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733

Huuska/Guliev K1 versus S.

The Huuska/Guliev formula is defined as Sb = CS

2 ∇ Fnh  Ks . 2 L2pp 1 − Fnh

(26.16)

The squat constant CS = 2.40 is typically used as an average value in this formula. The value for Ks for restricted channels and canals is determined from

Ks =

7.45s1 + 0.76 s1 > 0.03 s1 ≤ 0.03

1.0

(26.17)

with a corrected blockage factor s1 defined as s1 =

S . K1

(26.18)

The correction factor K1 is given by Huuska’s plot of K1 versus S for different trench height ratios hT /h shown in Fig. 26.3. One should use a value of hT = 0 for unrestricted channels and hT = h for canals. Appendix 26.A contains a set of least square fit coefficients for Fig. 26.3 if one wants to program these curves.26 26.3.3.3. R¨ omisch R¨omisch17 developed formulas for both bow and stern squat from physical model experiments for all three channel configurations. His empirical formulas are some of

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734

the most difficult to use, but seem to give good predictions for bow Sb and stern squat Ss given by Sb = CV CF K∆T T Ss = CV K∆T T

(26.19)

where CV is a correction factor for ship speed, CF is a correction factor for ship shape, and K∆T is a correction factor for squat at ship critical speed. The value for CF is equal to 1.0 for the stern squat. The values for these coefficients are defined as

CV = 8

V Vcr

2

V − 0.5 Vcr



4 + 0.0625

2 10CB CF = Lpp /B  K∆T = 0.155 h/T .

(26.20)



(26.21) (26.22)

The ship critical or Schijf-limiting speed Vcr is the speed that ships cannot exceed due to the balance between the continuity equation and Bernoulli’s law.9,38,39 For economic reasons, maximum ship speeds are typically only 80% of Vcr . The Vcr (m/s) varies as a function of the channel configuration given by    CKU Vcr = Cm KC   CmT KR

Unrestricted Canal Restricted

.

(26.23)

The three-wave celerity parameters C, Cm , and CmT (m/s) are defined as C=

 gh;

Cm =

 ghm ;

CmT =

 ghmT .

(26.24)

The mean water depth hm (m) is a standard hydraulic parameter that is used for canals and restricted channels. It is defined as hm =

AC WTop

(26.25)

where WTop (m) is the projected channel width at the top of the channel equal to WTop = W + 2nh.

(26.26)

The relevant water depth hmT (m) is for restricted channels and is defined as hmT = h −

hT (h − hm ). h

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R¨ omisch’s KC versus 1/S.

Table 26.2. 1/S

1

6

10

20

30

∞

KC

0.0

0.52

0.62

0.73

0.78

1.0

R¨omisch’s correction factors KU , KC , and KR for unrestricted, canal, and restricted channels, respectively, are defined as


0.125 h Lpp T B 1.5


Arc sin(1 − S) KC = 2 sin 3

KU = 0.58

KR = KU (1 − hT /h) + KC (hT /h).

(26.27) (26.28) (26.29)

Note that the KR for the restricted channel is a function of both KU and KC . Table 26.2 lists R¨omisch’s limited dataset for KC as a function of 1/S (i.e., AC /AS ). Appendix 26.A contains more detailed descriptions of KC and some additional equations for defining it relative to Schijf’s limiting speed and his limiting Froude number FHL . 26.3.4. Example problems Three example problems are presented in this section to illustrate the different formulas for several channel and ship types. All are for bow squat Sb . Comparisons of the different formulas with the measured laboratory values are shown for each example in Figs. 26.4–26.6, respectively. Appendix 26.B contains worked examples for at least one Concept and one Detailed Design application for each example problem. 26.3.4.1. Example 1: BAW Post-Panamax containership in unrestricted channel The first example is for a Post-Panamax containership traveling at Vk = 13.3 kt (Vs = 6.84 m/s) in an unrestricted channel. This speed matches laboratory data ugge and Uliczka.40,41 This vessel is similar (Sb = 0.70 m) obtained at BAW by Fl¨ to the last generation Emma Maersk containership (launched in August 2006), but with a larger CB . The larger CB is not realistic for the newer containerships (most have CB < 0.7), but was tested by BAW by “lengthening” an existing model during design experiments. A comparable CB is of the order of 0.62 for a ship of this size. The dimensions of the ship and channel are listed in Table 26.3. Figure 26.4 shows comparisons among the Barrass, Eryuzlu, Huuska, ICORELS, R¨omisch, and Yoshimura formulas and the measured BAW laboratory values. The numerical values are from a numerical model described in Sec. 26.6. In general, the best formulas are the Yoshimura (Concept) and Eryuzlu (Detail) as they are slightly

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736

0.0

Bow Squat for BAW Hansa Container Ship - Unrestricted R N

R R N

-0.5

R R

R

N

R N

N

Sb , m

R

Barrass Eryuzlu Huuska ICORELS Romisch Yoshimura Numerical BAW

N

-1.0

R N R N

-1.5 Example

R N

Unrestricted Channel Bottom

-2.0 9

11

13

15

17

19

Vk, knots Fig. 26.4. Comparison of BAW’s experimental measurements, empirical formulas, and numerical model of bow squat for a Post-Panamax containership in an unrestricted channel (open water).

0.0

Bow Squat for FHR Tanker G, Condition C - Canal

R N

N

-0.5

Barrass Huuska Romisch Yoshimura Numerical FHR

R N

Sb, m

R R

Example R

N R

-1.0 R N

R R

Canal Bottom

-1.5

R

7

8

9

10

11

12

Vk, knots Fig. 26.5. Comparison of FHR’s experimental measurements, empirical formulas, and numerical model of bow squat for a Tanker “G”, in Condition C in a canal with vertical sides.

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0.0

FA

737

Bow Squat for Tothil Canadian Laker - Canal R N

R R R N

R R

N

-0.5

Sb , m

R N

R

-1.0

N

R

Barrass Huuska Romisch Yoshimura Numerical Tothil

N

Example

R

Canal Bottom

-1.5 4

5

6

7

8

Vk, knots Fig. 26.6. Comparison of Tothil’s experimental measurements, empirical formulas, and numerical model of bow squat for a Canadian Laker in a canal. Table 26.3. BAW’s Post-Panamax containership in unrestricted channel. Lpp (m)

B (m)

T (m)

CB

h (m)

400

50

17

0.84

19

conservative (i.e., larger than measured). The R¨omisch is slightly smaller than the measured values, but follows the trend very well. Appendix 26.B contains worked examples for the Concept Design formulas of Yoshimura and ICORELS and the Detail Design formulas of Eryuzlu and R¨omisch. 26.3.4.2. Example 2: FHR “G” Tanker in a canal with vertical side, Condition C The second example is for the “G” Tanker, Condition C in a canal with vertical sides (similar to a restricted channel) from FHR and Ghent University.42 The 1:50 scale laboratory experiments were performed in a 7.0-m-wide (350-m prototype) towing tank. The measured Sb = 1.18 m for the ship sailing at Vk = 10 kt (Vs = 5.14 m/s). The ship and channel characteristics are listed in Table 26.4. Figure 26.5 shows comparisons among the Barrass, Huuska, R¨ omisch, and Yoshimura formulas and the measured FHR laboratory values for the canal with vertical sides. The numerical values are from a numerical model that is described in Sec. 26.6. In general, the best formulas are the Yoshimura (Concept) and R¨omisch (Detail) as they are nearly exact or slightly conservative for the smaller ship speeds

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738 Table 26.4.

FHR “G” Tanker in restricted channel, Condition C.

Lpp (m)

B (m)

T (m)

CB

h (m)

hT (m)

W (m)

WTop (m)

n (deg)

180

33

13

0.85

14.5

14.5

350

350

0.0

(i.e., larger than measured). Appendix 26.B contains worked examples for the Yoshimura, Barrass (Concept), and Huuska (Detail). The R¨omisch is not included in the worked examples for this case as it has already been demonstrated. The Barrass is a little small, especially for higher ship speeds. The Huuska formula is conservative for all ship speeds. 26.3.4.3. Example 3: Tothil’s Canadian Laker in a canal The third example is for a Canadian Laker in a canal with sloping sides (typical canal). These data are from Tothil’s 1:48 scale model experiments.43 The measured Sb = 0.93 m for the ship traveling at 6.98 kt (Vs = 3.59 m/s). Ship and channel features are listed in Table 26.5. Figure 26.6 shows comparisons among the Barrass, Huuska, R¨ omisch, and Yoshimura formulas and the measured Tothil laboratory values for the canal case. The numerical values are from a numerical model that is described in Sec. 26.6. In general, the best formulas are the Barrass (Concept), Huuska (Detail), and R¨omisch (Detail). The Barrass is a good match for ship speeds less than 6.54 kt, but does not follow the measured values for increasing speeds. The Huuska is on the low side, but matches reasonably well until Vk exceeds 6.54 kt. The R¨omisch is on the low side, but follows the measured trend of the data for all speeds. The Barrass and R¨omisch formulas are included in worked examples in Appendix 26.B. 26.4. Recent Investigations of Ship Squat So far we have discussed the PIANC empirical formulas for predicting ship squat. These are based on “idealized” conditions with single vessels that are sailing along the centerline of symmetrical channels. Unfortunately, real-world channels and ship transits are seldom this simple. This section discusses some recent research in laboratory and field measurements of ship head-on passing encounters and overtaking maneuvers in two-way traffic, stern-transom effects, abrupt sills, and offset and drift angle effects for ships sailing off the centerline with drift angles. When two ships pass or overtake each other, the water flow and corresponding squat is affected as a function of the other ship’s size, speed, and direction of travel, and the channels configuration. Dand44 was one of the first to study this Table 26.5.

Tothil’s Canadian Laker in a canal.

Lpp (m)

B (m)

T (m)

CB

h (m)

W (m)

WTop (m)

n

215.6

22.9

7.77

0.86

9.33

72.3

105.9

1.8 (29 deg)

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phenomenon. He found increases in bow squat of 50–100% during passing and overtaking encounters. During the past 10 years, the BAW has conducted many field and laboratory studies to investigate ship–waterway interactions, especially head-on passing encounters and overtaking maneuvers of ships in restricted channels within German federal waterways. Preliminary studies of the dynamic response of large containerships in laboratory models have shown tendencies of reduced squat.40,41,45 These results were confirmed by additional model tests in restricted and unrestricted channels and field measurements along the Elbe River.31 The FHR (in cooperation with the Ghent University) has conducted laboratory experiments to study passing and overtaking in their automated towing tank as part of a larger study to improve their ship simulator for traffic in Flemish waterways.46 Finally, the Delft University of Technology47 had conducted some numerical modeling of the effects of ship offset and drift angles on ship squat. Thus, this section presents a summary of recent laboratory, field, and numerical investigations of ship squat in real-world situations including head-on passing encounters, overtaking maneuvers, wider stern-transoms, and ships with offset and drift angles. 26.4.1. Head-on passing ship encounters 26.4.1.1. BAW laboratory experiments Laboratory experiments were conducted at the BAW-DH shallow water basin to study squat as a function of ship size, hull form, draft, speed, direction of travel, and channel water level. This facility has approximate dimensions of 100-m length, 35-m width, and 0.7-m maximum water depth. Geometric and dynamic conditions were accurately scaled according to dimensional analysis at a scale of 1:40. A section of the River Elbe (i.e., restricted or confined channel) with a width of 1.0 km and length of 1.5 km was modeled. The cross-section had a channel depth h = 18.5 m, channel width of 265 m, and river width of 850 m. The results of a Panamax (PM) containership (PM32) and a Post-Panamax (PPM) bulk carrier (MG58) during head-on passing were investigated (Table 26.6). Note that the MG58 is the larger vessel. The two ships passed each other at a passing distance of 156 m (between course lines). The range of ship speeds for the two ships was approximately 7–14 kt for the PM32 and 7–12 kt for the MG58. A laser measuring system was installed on the self-propelled, cable-guided model ships to record their vertical behavior. Measurements were recorded over a distance of approximately 90 m, including acceleration and braking phases of each run. The velocity-independent precision of the laser system was ∆S