Developments in Offshore Engineering; Wave Phenomena and Offshore Topics

DEVELOPMENTS IN

OFFSHORE ENGINEERING

DEVELOPMENTS IN

OFFSHORE ENGINEERING

Gulf Publishing Company Houston, Texas

DEVELOPMENTS IN

OFFSHORE ENGINEERING Wave Phenomena and Offshore Topics

John B. Herbich, Editor in collaboration w i t h - -

Khyruddin A. Ansari Subrata K. Chakrabarti

Zeki Demirbilek John D. Fenton Masahiko Isobe

M. H. Kim Vijay G. Panchang Robert E. Randall

Michael S. Triantafyllou William C. Webster

Bingyi Xu

To my wife, Margaret Pauline, and children,

Ann, Barbara, Gregory, and Patricia, who have been sources of encouragement

and inspiration during the preparation of this book

Developments in Offshore Engineering

Copyright © 1999 by Gulf Publishing Company, Houston, Texas. All rights reserved. This book, or parts thereof, may not be reproduced in any form without express written permission of the publisher.

Gulf Publishing Company Book Division

P.O. Box 2608 73 Houston, Texas 77252-2608

109 8 7 6 5 4 3 2 1

Library of Congress Cataloging-in-Publication Data

Developments in offshore engineering : wave phenomena and offshore topics / John B. Herbich, editor in collaboration with Khyruddin A. Ansari... [et al.].

p. cm. Includes bibliographical references and index. ISBN 0-88415-380-0 (alk. paper) 1.Ocean engineering. 2.Wave mechanics. I. Herbich, John B.

II. Ansari, Khyruddin A.

TC 1650.D487 1998

620'.4162--dc21

Printed in the United States of America. Printed on Acid-Free Paper (~).

98-35547 CIP

Cover photos by Mieko Mahi. Cover design by Johnny Guzman.

iv

CONTENTS

Preface, ix

Publisher's Note, xi

Contributors, xii

About the Editor, xiii

Wave Phenomena

1 The Green-Naghdi Theory of Fluid Sheets for Shallow-Water Waves ZEKI DEMIRBILEK AND WILLIAM C. WEBSTER

Introduction, 2 Theoretical Basis, 3 Mathematical Formulation, 8 Subset Theories, 23 Solution Scheme, 35 Examples, 43 Conclusion, 50 References, 51

2 The Cnoidal Theory of Water Waves JOHN D. FENTON

Introduction, 55 Background, 57 Cnoidal Theory, 59 Presentation of Theoretical Results, 66 Practical Application of Cnoidal Theory, 73 Practical Tools and Hints for Application, 80 A Numerical Cnoidal Theory, 86

55

Accuracy of Methods, 90 Notation, 96 References, 98

3 Equation for Numerical Modeling of Wave Transformation in Shallow Water

MASAHIKO ISOBE

Introduction, 102 Basic Equations and Boundary Conditions, 103 Mild-Slope Equation, 109 Time-Dependent Mild-Slope Equations, 121 Parabolic Equation, 125 Boussinesq Equations, 134 Nonlinear Shallow-Water Equations, 138 Nonlinear Mild-Slope Equations, 141 Validity Ranges of Wave Equations, 150 Summary, 153 Notation, 155 References, 157

. . . . . . 101

4 Wave Prediction Models for Coastal Engineering Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 163 VIJAY G. PANCHANG, BINGYI XU, AND ZEKI DEMIRBILEK

Introduction, 163 Energy Balance Models, 164 Mass and Momentum Conservation Models, 172 Conclusion, 186 References, 189

Offshore Topics

5 Mooring Dynamics of Offshore Vessels . . . . . . . . . . . . . . . 195

KHYRUDDIN A. ANSARI

Introduction, 196 Mooring System Selection, 197 Generation of Mooring System Restoring Forces, 197 Computation of Environmental Forces, 224 Vessel Dynamic Analysis, 233

Effect of Vessel Hydrodynamic Mass on Vessel and Mooring Line Dynamic Responses, 240

Conclusion, 244 Notation, 248 References, 251

6 Cable Dynamics for Offshore Applicat ions . . . . . . . . . . . 256

MICHAEL S. TRIANTAFYLLOU

Introduction, 256 Cable Mechanics, 258 Motions of Moored Structures, 270 Experimental Testing of Cables and Mooring Systems, 274 Guidelines for Analyzing Multi-Leg Systems, 276 Overall System Damping Induced by the Mooring Lines, 276 Application to Multi-Leg System Analysis, 277 Summary, 282 Appendix 1: Derivation of the Equations of Cable Motion, 283 Appendix 2: Methodology for Calculating Vortex-Induced Motions, 288 Appendix 3: Methodology for Calculating the Interaction Between

Wave-Induced and Slowly Varying Motions, 291 References, 292

7 Model ing Laws in Ocean Engineering . . . . . . . . . . . . . . . . 295

SUB~TA K. CHAKr~BARTI

Introduction, 296 Modeling Laws, 297 Model Testing Facility, 312 Model Environment, 315 Examples of Modeling, 321 Notation, 332 References, 333

8 Hydrodynamics of Offshore Structures . . . . . . . . . . . . . . . 336

M. H. KfM

Introduction, 336 Wave Loads on Slender Bodies, 337 Wave Loads on Large Structures, 349 Motion Analysis, 371 References, 377

. ~

VII

9 U n d e r w a t e r A c o u s t i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8 2

ROBERT E. RANDALL

Introduction, 383 Sonar Equations, 389 Properties of Transducer Arrays, 394 Underwater Sound Projector, 403 Underwater Sound Propagation, 408 Transmission Loss Models, 426 Ambient Noise, 430 Scattering and Reverberation Level, 434 Target Strength, 439 Radiated Noise Levels, 440 Self Noise, 443 Detection Threshold, 447 Underwater Acoustic Applications, 449 References, 468

10 D i v i n g and U n d e r w a t e r Li fe S u p p o r t . . . . . . . . . . . . . . . 4 7 2

ROBERT E. RANDALL

Introduction, 473 Diving Physiology, 491 Gas Laws, 496 Operating Characteristics and Gas Supply Calculations for Diver Breathing Equipment, 502 Thermodynamics for Diving Systems, 514 Pressure Vessel Charging and Discharging Process, 519 Diving Gas Mixtures, 521 Control of Underwater Chamber Environment, 522 Mixing of Breathing Gases, 531 Carbon Dioxide Absorption in Diving Operations, 547 One-Dimensional Compressible Flow in Pipes, 548 Heat Transfer, 552 References, 559

A u t h o r I n d e x . . . . . . 561

S u b j e c t I n d e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 4

o o ~

Vl l l

PREFACE

There has been a need to update the Handbook of Coastal and Ocean Engi- neering series (published in 1990-1992), and this new book provides the latest state-of-the-art information, as well as research results, in this challenging field.

Recent increases in offshore industrial activity, such as oil/gas exploration and production, make this a timely collection of pertinent offshore engineering information. The book, drawing from experts and top researchers from around the world, presents current developments in a variety of ways that impact offshore and ocean engineering. The book also provides valuable insights into key aspects of several important offshore engineering subjects.

Deeper, tougher, faster, safer, and better are the demands placed on all aspects of today's offshore and ocean engineering efforts. The book assists those profes- sionals who must answer such demands to conceive, design, and implement the ways and means to succeed in a hostile marine environment.

The book covers wave phenomena and offshore topics. The first part of the book covers the Green-Naghdi and cnoidal wave theories, numerical modeling of wave transformation and nearshore wave prediction models. The second part of the book discusses the mooring dynamics of offshore vessels and cable dynamics for offshore applications, followed by modeling laws in ocean engineering, dynamics of offshore structures, and underwater acoustics.

This book represents the efforts of eleven experts from around the globe. In addition, it reflects the opinions of many engineers and scientists who provided assistance in developing this book. All chapters were peer-reviewed, corrected and finally reviewed by the editor. This effort took many months, evenings, and weekends. We hope that all mistakes were found and corrected. My deepest gratitude is extended to all the contributors to the book. Great appreciation is also extended to the reviewers: Dr. Jun Zhang, Dr. C. H. Kim, and Dr. Jack Y. K. Lou, Ocean Engineering Program, Civil Engineering Department, Texas A&M University, College Station, Texas; Professor Cengiz Ertekin, Ocean Engineering Department, University of Hawaii, Honolulu, Hawaii; Dr. C. C. Mei, Department of Civil and Environmental Engineering, Massachusetts Insti- tute of Technology, Cambridge, Massachusetts; Dr. Zeki Demirbilek, Coastal Engineering Research Center, U.S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Mississippi; Dr. Robert A. Dalrymple, Center for Applied Coastal Research, University of Delaware, Newark, Delaware; Dr. P. L. F. Liu, DeFrees Hydraulics Laboratory, Cornell University, Ithaca, New

ix

York; Professor Michael Triantafyllou, Ocean Engineering, Massachusetts Insti- tute of Technology, Cambridge, Massachusetts; Khyruddin A. Ansari, Depart- ment of Mechanical Engineering, Gonzaga University, Spokane, Washington; Dr. D. L. Kriebel, Millersville, Maryland; Dr. Steven Hughes, Coastal and Hydraulic Laboratory, U.S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Mississippi; Dr. Subrata K. Chakrabarti, Offshore Structure Analysis, Plainfield, Illinois; Dr. Aubrey L. Anderson, Department of Oceanog- raphy, Texas A&M University, College Station, Texas; Dr. Jan P. Holland, Applied Research Laboratory, Pennsylvania State University, State College, Pennsylvania.

The manuscripts were assembled for publication by Ms. Joyce Hyden, to whom I am most grateful. Without her expert help, this book would have taken much longer to produce.

I also wish to thank the many publishers and individuals who have kindly granted permission to reprint copyrighted materials.

John B. Herbich, Ph.D., P.E. W. H. Bauer Professor Emeritus Civil and Ocean Engineering Texas A&M University College Station, Texas and Vice-President Consulting & Research Services, Inc. Wailuku, Hawaii, and Bryan, Texas

PUBLISHER NOTE

Developments in Offshore Engineering is a collective effort involving many technical specialists. It brings together a wealth of information from world-wide sources to help scientists, engineers, and technicians solve current and long- range problems.

Great care has been taken in the compilation and production of this volume, but it should be made clear that no warranties, express or implied, are given in connection with the accuracy or completeness of this publication, and no respon- sibility can be taken for any claims that may arise.

The statements and opinions expressed herein are those of the individual authors and are not necessarily those of the editor or the publisher. Furthermore, citation of trade names and other proprietary marks does not constitute an endorsement or approval of the use of such commercial products or services, or of the companies that provide them.

xi

CONTRIBUTORS TO THIS VOLUME

Dr. Khyruddin A. Ansari, Dept. of Mechanical Engineering, School of Engineering, Gon- zaga University, Spokane, WA 99258

Dr. S. K. Chakrabarti, 191 E. Weller Drive, Plainfield, IL 60544

Dr. Zeki Demirbilek, Coastal and Hydraulics Laboratory, USAE Waterways Experiment Station, 3909 Halls Ferry Road, Vicksburg, MS 39180

Dr. John Fenton, Department of Civil and Environmental Engineering, University of Melbourne, Parkville, Victoria 3153, Australia

Dr. Masahiko Isobe, Department of Civil Engineering, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan

Dr. M. H. Kim, Associate Professor of Ocean Engineering, Civil Engineering Depart- ment, Texas A&M University, College Station, TX 77843-3136

Dr. Vijay G. Panchang, Department of Civil Engineering, University of Maine, Orono, ME 04469-5711

Dr. Robert E. Randall, Professor of Ocean Engineering, Civil Engineering Department, Texas A&M University, College Station, TX 77843-3136

Dr. Michael Triantafyllou, Professor of Ocean Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 5-323, Cambridge, MA 02139-4307

Dr. William C. Webster, University of California-Berkeley, 308 McLaughlin Hall, Berkeley, CA 94720

Dr. Bingyi Xu, Department of Civil Engineering, University of Maine, Orono, ME 04469-5711

~

Xll

ABOUT THE EDITOR

John B. Herbich, Ph.D., P.E., is the W.H. Bauer Professor Emeritus, Civil and Ocean Engineering, at Texas A&M University, College Station, Texas. He is a Fellow and Life Member of the American Society of Civil Engineers and many other engineering societies. Dr. Herbich received his B.Sc. degree in civil engineering from the University of Edinburgh, Scotland; an M.S.C.E. in hydrome- chanics from the University of Minnesota; and a Ph.D. in civil engineering from Pennsylvania State University.

Prior to joining Texas A&M University, Dr. Herbich was on the faculty of Lehigh University, Bethlehem, Pennsylvania (1957-1967) and a research engineer at the University of Delft, The Netherlands (1949-1950). He has served as project manager of a United Nations Development Program in Poona, India (1972-73); a visiting professor at the U.S. Army Corps of Engineers Waterways Experiment Station in Vicksburg, Mississippi (1987-88); and as a consultant for many U.S. and international governments and industries; and has served on several committees of the National Research Council. He is the recipient of the "International Coastal Engineering Award," American Society of Civil Engineers (1993) and the recipient of the "1995 Dredger of the Year Award," Western Dredging Association. Dr. Herbich is a registered professional engineer in Texas.

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This Page Intentionally Left Blank

C H A P T E R 1

THE GREEN-NAGHDI THEORY OF FLUID SHEETS FOR

SHALLOW-WATER WAVES

Zeki Demirbilek

US Army Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, Mississippi, USA

William C. Webster

University of California at Berkeley Berkeley, California, USA

CONTENTS

INTRODUCTION, 2

THEORETICAL BASIS, 3

Overview of Theory, 4

Approach, 7

MATHEMATICAL FORMULATION, 8

Governing Equations with General Weight Functions, 8

Discussion of the Generalized GN Theory, 15

Equations for Shallow Water, 18

Restricted Theory and Constitutive Relations, 20

SUBSET THEORIES, 23

Level I Theory: Unsteady Flow and Uneven Bathymetry, 23

Level I Theory: Steady Flow and Constant Bathymetry, 25

Level II Theory: Unsteady Flow and Constant Bathymetry, 27

Level II Theory: Steady Flow and Constant Bathymetry, 30

Level II Theory: Unsteady Flow and Uneven Bathymetry---Generalized Formulation, 32

SOLUTION SCHEME, 35

Integration, 35 Numerical Model, 37

2 Offshore Engineering

Wave-Maker, 40 Boundary Conditions, 41 Spatial and Time Integration, 42

EXAMPLES, 43

Example 1: Wave-Structure Interaction--Reflection of Waves from a Structure, 43 Example 2: Steep Nonlinear WavesmShoaling Waves on a Planar Beach, 47

CONCLUSION, 50

REFERENCES, 51

Introduction

This chapter presents a mathematical theory for simulating wave transformation in shallow waters. The theory is intended for coastal engineering applications involving propagation of time-dependent, nonlinear waves where existing theories may either be inapplicable or simple analytic/numerical solutions may be inappropriate. The theory detailed here is in essence a new-generation water wave theory for shallow to moderate water depths where seabed may be rapidly varying. The new theory is the generalized or unrestricted Green-Naghdi (GN) Level II theory, derived here specifically for water waves.

The GN theory employs the conservation principles and incorporates some of the most important mathematical features of the water wave equations. These include non-approximating the governing Euler's field equations and imposing proper boundary conditions necessary for capturing the bulk physical characteristics of wave trains in the shallow water regime. The GN approach, which is funda- mentally different from the perturbation method based on developments in classical wave theory begun by Stokes and Boussinesq in the last century, can only do this by introducing some simplification of the velocity variation in the vertical direction across the fluid layers or sheets. In contrast to the Stokes and Boussinesq theories, the equations of motion in the GN theory are obtained by enforcing exact kinematic and dynamic boundary conditions on the free surface and on the bottom, conservation of mass and of the 0-th and 1-st moments of momentum in the vertical direction. These conditions yield eleven coupled partial differential equations that can be reduced to three complicated governing equations by elimination of many of the variables. In summary, the GN theory is different from the perturbation approach in that the free surface and bottom boundary conditions are met exactly, whereas the field equation is implicitly approximated. The result is a theory that can predict the shape and behavior of a wave up to almost the breaking limit. The GN theory breaks down when the particle velocity at the crest equals the wave speed, the criterion for breaking in the exact theory.

By developing the unrestricted Green-Naghdi theory of fluid sheets, we provide a new wave theory consisting of a coupled, nonlinear set of partial differen-

The Green-Naghdi Theory of Fluid Sheets for Shallow-Water Waves 3

tial equations and integrate these in time and space to simulate either regular or irregular waves. The theory has been implemented in a numerical model that has been shown to reproduce with engineering accuracy the evolution of a wave of permanent form, from small amplitudes up to the breaking limit. The presented theory is a nonlinear numerical wave tank in which the specified seabed topography profile can be arbitrary and very irregular and numerical wave gauges can be positioned at will inside the computational domain to obtain snapshots and profiles of wave elevation and wave kinematics and dynamics. The types of coastal engineering studies that the GN theory can be applied to are many and include problems of both military and civil interest. The theory is purposely made to be versatile to permit decision makers, designers, and analysts to assess aspects of waves and wave-structure interaction problems arising in practice. One can evaluate, for instance, the effect of submerged obstacles on train of waves approaching a beach or landing zone during military operations, or the reflection of waves from sea walls or wave loads on spillway hydraulic gates, and the time history of bottom-mounted pressure gauge measurements for estimation of surface wave parameters for coastal design studies. The theory is particularly suited for violent collision of waves with natural and man-made structures, and their impact on shore-preventive hydraulic systems.

Theoretical Basis

This chapter presents a comprehensive description of a relatively new theory for modeling coastal waves. It also provides a general discussion and critique of various approaches for simplifying complex hydrodynamics of the wave boundary value problem, a derivation of the general Green-Naghdi (GN) theory of fluid sheets, the equations of motion of two-dimensional (2-D) shallow-water waves for GN Level I and II theories, and a concise description of the numerical methods used for integration of GN governing equations. Our intent was to assemble a document that a coastal engineer can use to understand this relatively new theory and its application to complex shallow-water wave problems. The entire theoretical formulation presented in this chapter is new, greatly improving upon several previously published reports, papers or dissertations. The derivation of the theory is in the form of a tutorial in which all of the intermediate steps are included, since no textbook or article is available in this level of detail.

During the two decades since its introduction [ 16], the theory of fluid sheets has been applied to a variety of fluid flow problems. These include studies of waves in shallow and deep water [14, 15, 27], the flow beneath planing boats [22], the waves created by a moving pressure disturbance [8-10], solitons [11], and wave reflection by obstacles [20], to name a few. In particular, the development of fluid sheet theory in an Eulerian frame [ 13] made this theory much easier to apply to fluid flow problems. The reader is referred to the pair of papers by Green and Naghdi [14, 15] for a definitive and highly mathematical exposition


of the theory. In recent years, we have made significant advances in the adaptation of GN theory to water waves [4-7], and developed computational models for wave simulations and wave-structure interaction problems in coastal and hydraulic engineering. Because our formulation is different from that of Green- Naghdi, we provide here a summary of our developments for completeness.

Overview of GN T h e o r y

Alternative approaches, with a variety of approximations and assumptions, exist for predicting wave motion in coastal waters. The classical equations of motion for fluid flow in three dimensions are a continuum model that embodies many assumptions. For ordinary fluids, such as water, the Navier-Stokes equations are universally recognized as a good model for the resulting flows. Howev- er, these equations are not "exact" equations but are an idealization similar in spirit to the idealization of space by Euclidean geometry. Even for simple free- surface problems, these equations and their simpler inviscid counterparts, the Euler equations, are difficult to solve. One popular approach has been to system- atically simplify the three-dimensional equations and their boundary conditions through a formal perturbation analysis until the resulting system can be solved. The theories of water waves developed by Stokes, Boussinesq, and others follow this type of development.

The GN fluid sheet theory offers an alternative in the form of a new model, that of a 2-D continuum of unsteady three-dimensional (3-D) flows. Although the examples cited here involve inviscid fluids, the development of GN theory is not at all limited to such fluids. The following discussion is aimed at exploring the difference between these two very different paths to simplification of the analysis of fluid flow problems. In either case, it is anticipated that the solutions obtained are approximate ones, because there really is no substitute for solving the 3-D equations exactly. Both approaches are called approximations, although it is clear that the meaning is not the same for each.

Before introducing details on the nature of GN theory, it is useful to first discuss the notion of approximation in general. An approximation approach for analyzing a given problem is usually chosen based on its ability to predict the phenomena that one is interested in and on its ease of use. The selection of an approximation scheme can be viewed as a type of non-zero-sum game where one attempts to make assumptions that will have a greater impact on the simplification of the analysis than on the accuracy of the prediction of the phenomena of interest.

Two observations from this discussion are significant. First, the choice of the approximation scheme depends on the specific answers for which one is looking (i.e., the choice depends on the context of the problem rather then its generic type). Second, the means of analysis change in time; that is, computations that 20 years ago would have required the world's largest computers can now be


accomplished faster and for a minimal cost on a personal computer. It is proper to think in terms of approximation schemes "appropriate for the current time." Because the evolution of a new computer generation appears to take only a few years, it seems natural that we will see a corresponding evolution in approximation schemes that will take advantage of these new resources. It is a thesis of our research that GN theory and, in particular, higher-level GN theory is appropriate for our time. Two developments lead to this conclusion: the emergence of low- cost, high-speed computation, and the emergence of sophisticated symbolic manipulation software that allows one to accurately perform calculus and algebraic manipulations on rather large systems of equations, such as those resulting from the GN formulations as we shall see later in this chapter.

Approximation schemes can be separated into different categories. Perturba- tion methods, both ordinary and singular, introduce some mathematical approximation to reduce the complexity of the model to the point where it can be solved. One advantage of these methods is that one obtains governing equations for the flow and from these, both specific solutions can be obtained and general- izations of the behavior of the flow can be made. On the other end of the spectrum, the original problem can be solved by purely numerical techniques. Finite difference, finite element, and panel methods are such schemes. These methods are comparable to physical experiments in that each computation yields another result corresponding to a single realization of the flow. Generalization about the behavior of the flow requires induction from many of these specific solutions. GN fluid sheet theory lies in the middle of this spectrum. It achieves simplification by reducing the dimensionality from three dimensions to two. This theory yields governing equations for the flow, which are solved numerically in a more efficient manner than those from the three-dimensional model.

Perturbation analyses introduces reference scales appropriate for the particular problem at hand. These scales are used to nondimensionalize the variables and to identify a nondimensional perturbation parameter (or parameters), which can be considered small (or large). For time invariant problems, the flow is decomposed into a sequence of flows of presumably decreasing importance, each of which is a correction to the sum of the previously computed flows. The assumed sequence is inserted into the field equations and boundary conditions and the perturbation parameters are used to segregate these into a corresponding sequence of perturbation problems. Typically, each of these problems is linear in the unknowns at its level, although it may involve higher-order terms of quantities determined already in previous (lower-order) solutions.

An implicit assumption is made that this sequence is convergent, but this is almost never proven. In some flow problems, such as two-dimensional water waves in both shallow and deep water, there is the evidence of a slow convergence only for constant depth problems [25]. For steady periodic waves, Fenton [12] demonstrated that no gain in accuracy would result by including terms beyond the fifth order and in this sense perturbation method was asymptotic


rather than convergent. In problems such as the flow about thin airfoils, the lack of convergence is well known and the procedure is not unique. The perturbation methods, which date back to Stokes and other early researchers, are often called "rational methods" because the assumptions are clear and testable, and the details can be embodied into a mathematical process through which theoretically one can obtain solutions to whatever level of accuracy one chooses, if the perturbation sequence converges. A particular advantage of the perturbation approach is that, because perturbation parameters are used to size quantities, the ingredients of this parameter give one insight into the types of problems for which the approximation is appropriate. In perturbation approach, the mathematical and physical orders of errors are not the same, although there is the wrong but quite common perception that they are the same or very close. The perturbation approach does not yield quantitative measures of the accuracy to be expected for a particular problem. This information can only be obtained from an analysis of higher order problems or from comparison with experiments.

Unsteady fluid problems are rather different. It is usually not feasible to consider the flow as a sum of linear perturbation problems that one can solve sequentially until sufficient accuracy has been obtained, unless some additional limiting assumption such as periodic motion is introduced. Generally, one must solve a single set of governing equations in time. Perturbation methods have also been used, for instance, by Wu [31 ] for the formulation of approximations appropriate for time-domain wave problems. For these problems, the introduction of scales permits grouping of terms of like size for a particular problem. One can obtain a variety of different sets of governing equations depending on the order of terms retained. That is, one can obtain a sequence of sets of governing equations for a given problem, each of which contains all of the terms of the previous sets plus those due to the retention of the next order of smaller terms. Presumably, this sequence of increasing complexity will produce solutions of increasing accuracy. Generally, all of these sets of governing equations will be nonlinear with the exception perhaps of the first. However, because we are throwing away parts of the exact problem, we can expect that some quantities, such as mass and momentum, may not exactly be conserved (although, if a consistent analysis has been performed, the errors should be of a size comparable to the first neglected order).

The limitations resulting from such an analysis can be subtle. Many perturbation schemes consider the fluid velocity to be a small perturbation to a reference velocity. The typical result is that any order of the perturbation theory is not Galilean invariant, because the terms that are needed to make it so are spread among several orders. One such example is the Korteweg de Vries equations for nonlinear, shallow-water flow.

In problems where viscosity is not important (or can be ignored) and the flow is initially quiescent, the field equation becomes Laplace's equation, which is linear. For these flows much of the focus of perturbation analysis is therefore on


the boundary conditions. Both the kinematic boundary condition on material surfaces, as well as the dynamic boundary condition on the free surface, are nonlinear. Expansions of these nonlinear conditions and grouping of terms by orders of the perturbation parameter(s) lead to a basis for selecting or discarding terms. For these flows, then, the perturbation method solves a problem where the exact field equation is satisfied, but where the boundary conditions are satisfied only approximately. In some sense, the uniqueness of these flow situations stems solely from the imposed conditions at the boundaries and it is worrisome to con- centrate the approximations there.

The GN theory is, however, quite the opposite in nature. The boundary conditions are met exactly, but the field equations are approximated. In this approach the dependence of the kinematic structure of the solutions along one coordinate direction is prescribed. This direction is the vertical (or x 3) direction for the theories discussed in this chapter. In many problems, such as shallow water problems, the depth of the fluid in the x 3 direction may be quite small; in others the fluid domain can be infinite in this direction. In either case the resulting theories are called fluid sheet theories. The assumed variation of the fluid velocity across x 3 will be expressed here as a finite sum of products. The first term in each product is a coefficient that depends on the remaining two horizontal coordinates (x 1 and x 2) and time, and the second term is a function of x 3 alone.

Approach

The present formulation starts with the governing equations for the generalized GN theory consisting of an exact statement of the conservation of mass, an approximate statement of the conservation of momentum, and exact statements for the various boundary conditions. In our formulation, x 3 no longer appears in the governing equations and wave kinematics are only functions of x l, x 2, and time. No scales are introduced and no terms are thrown out. Development of GN theory takes place in two steps: postulation of a set of governing equations, and verification that these equations satisfy certain physical requirements. The process presented below is one which is unique to this research, and although it closely follows earlier efforts in the subject area at the University of California, Berkeley, it is substantially different from that used by A. E. Green of Oxford University and P. M. Naghdi of University of California at Berkeley.

The first step of our formulation is a procedure for identifying a candidate fluid sheet model. In fact, there is no preferred method for this identification and at this point the model could just as well have been induced from the results of model tests. The same variational approach of Kantorovich and Krylov [17], used previously by Shields and Webster [27], is also used in the following section to derive the candidate model from standard 3-D equations. Although this commonality leads to some similarities between our derivations and those of Shields [26] and Shields and Webster [27, 28], and Webster and Kim [29], this


presentation in its totality is more comprehensive and complete than all earlier works. The approach we use here is a variation of the method of weighted resid- uals, and is therefore similar in nature to the procedure used in the development of finite elements. In this procedure, the dimensionality of the system of partial differential equations is reduced, rather than the system being replaced by the system of algebraic equations, as it would be in a Galerkin procedure. The derivation and fundamentals of the GN theory are provided in the following section. Later, we present a variety of GN evolution theories for steady and unsteady flows both for constant and variable bathymetry; a special solution scheme for GN equations; two example applications that illustrate the practical usage of GN theory are presented; two example applications that illustrate the practical usage of GN theory are presented; and the relationship of GN theory to other more common wave theories. Pertinent literature is cited in the References and Bibliography sections at the end of the chapter.

Mathematical Formulation

Governing Equations with General Weight Functions

Let x = x i (i = 1,2,3) be a system of fixed Cartesian coordinates in Euclidian space with base vectors e i, where e 3 is oriented vertically upwards. For convenience, x 3 is denoted by ~ in the subsequent development because this dimension plays a much different role than the other two dimensions. In the following, standard Cartesian tensor notation is used, with the summation convention implied for repeated indices. In many instances, however, the summation will be stated explicitly for clarity. Latin indices are used for quantities having three spatial components and take on values of 1, 2, 3; Greek indices take on the values of 1 and 2 only. A comma in the subscript denotes differentiation by the following variable or that corresponding to the subsequent index.

The fluid velocity vector at a point x and time t is given by v = v (x, t) = vie i with scalar components v 1, v 2, and v 3. The fluid is assumed to be bounded by two smooth and non-intersecting material surfaces. The material surfaces are given by ~ = ct(x l, x 2, t) and ~ = 13(x l, x 2, t), 13 > ix, respectively. For the reason previously stated the vertical component of the fluid kinematics is hereafter explicitly written while the horizontal components are identified with indicial notation for the sole purpose of keeping the equations as compact as possible. Because o~ and 13 both are material surfaces, on these surfaces the kinematic ("no leak") boundary condition is

D (~ - a) = 0

Dt

D (~ - 13) = 0 (la) Dt


Using indicial notation, we write Equation 1 a in scalar form as

[1~3 -- ~ , t -- V T 0[, T ]1;=~ "- 0

[1"3 -- ~,t -- V T [~,Y ]1;=[~. -- 0 (lb)

where the index Y takes on values of 1 or 2, denoting the horizontal components and derivatives with respect to these coordinates. These equations are the kinematic boundary conditions indicating that the motion of particles on the surfaces is identical to the motion of the material surfaces.

This discussion will be concerned only with an incompressible and inviscid fluid, and in this case the mass density of the fluid p is constant, the stress vector, t = - P i ei, and the body force is given by -pge 3. In a more general theory, t would have the form t. = x.. n., where "r.. is the stress tensor, and the body force i lj ] lj would be a general vector pf. Unit outward normal vectors on the top and the bottom surfaces are denoted by fi and fi, respectively. The three-dimensional equations for a general continuum (i.e., Euler equations) resulting from the conservation laws of mass and momentum are

Vi, i = 0

PV,t + p(V i V), i = -- P,iei - pge 3 (2)

The fundamental kinematic assumption that the velocity field can be approximated is introduced as

K v (x l, x 2, ~, t) = ~ W n (X 1, X 2, t) ~n (~)

n=0

where

(3)

W n (X 1, x 2, t ) = W i (X 1 , x 2, t ) e i

The implications of Equation 3 are important to the understanding of GN theory. In essence, we have assumed that the vertical variation of the velocity field may be represented by an arbitrary "shape" function ~'n(~) that depends upon ~ only. The coefficients W n are unknown time-dependent vectors to be determined as a part of the solution. The vectors W n correspond with the "directors" in the original work of Green and Naghdi [14, 15]. For each choice of K, a complete, closed set of equations is developed that is independent from those for a different value of K. Thus, the kinematic models form a hierarchy depending on K and increasing in complexity with K. Because this hierarchy is different from a perturbation expansion, we adopt here a terminology that describes the complexity of the theory, and henceforth, refer to a particular member of this hierarchy as the "K th level approximation."


The kinematic boundary conditions in Equation 1 may be rewritten using Equation 3 as

K K

E W2 ~n ( ~ ) = ~,t + E W ~ ~n (~)~,7 n=0 n=0 K K

n=0 n=0 (4)

The continuity equation, Equation 2a, likewise becomes

K K

EWm~,7 ~m + E W 3 ~,m,~ = 0 (5) m--0 m=0

It turns out that we need 4K+2 scalar equations in addition to the kinematic boundary conditions. We choose K+I scalar equations derived from conservation of mass and K+l vector equations derived from conservation of momentum (corresponding to 3K+l scalar equations for three-dimensional problems) to provide closure for the fluid sheet theory. It is convenient at this point to restrict the weighting functions to those that possess the following property

K

~gm,~ -- ar r r=0

(K < m) (6)

where the coefficients a m are constants. The function set {~m} is therefore a finite closed set under differentiation. Inserting Equation 6 into Equation 5, the continuity equation can be expressed as

m 0

(7)

The terms in braces are not a function of ~ and therefore, Equation 7 may be written as

m = 0 W~, + W 3 a r m=0

(8)

for r = 1 ..... K. Equation 8 is therefore an exact statement of conservation of mass for all flows which are subject to the kinematic approximation, Equation 3. Note that if the derivative of the r th weighting function with respect to ~ is not


expressible in terms of the previous orders of the weighting function set, more than K+I conditions will result from this procedure and the Krylov-Kantorovich method described below could be used to determine approximate equations for the conservation of mass.

There are many function sets that satisfy Equation 6 (for instance, exponential functions where m = K). Polynomial functions also satisfy Equation 6; these have been used in some earlier studies for shallow-water fluid sheet theory. Sim- ilarly, the sets {sinh(a~), cosh(a~)} and {sin(a~), cos(a~)}, a = a 0, a 1 . . . . . a n, also satisfy Equation 6. It is this property of the generalized GN theory that makes it unique because in essence we are not limiting the shape functions to any particular water depth. For example, consider waves in deep water. We know from experience that fluid velocities rapidly decay with depth. Therefore, a hyperbolic shape function set should be used in the derivation of GN theory for deep waters. On the other hand, for waves in shallow water we expect to model their velocity profile reasonably well with a polynomial set of base functions. Note that Boussinesq and other perturbations wave theories also use a polynomial set (at most quadratic) for shallow-water waves, but these theories do not allow the type of flexibility that GN theory offers for choosing the vertical distribution of fluid kinematics. Due to these constraints associated with the fluid kinematics, one can only model weak dispersion and weak nonlinearities with the Boussi- nesq and other wave theories.

If we were to substitute Equation 3 into momentum equations and require that the resulting equation be satisfied everywhere in the fluid domain (as we did for continuity), we would obtain more equations than K + 1 vector equations we need. This overspecification results from the presence of quadratic terms in Equation 2b. To circumvent this difficulty, we employ a weak formulation due to Kantorovich where the "shape functions" ~m are used as weighting functions to develop K + 1 approximate equations, which express the conservation of momentum in some integral sense. Multiplying Equation 2b by each ~'n(~) and integrating it through the vertical direction gives

[(p 1,'), t d-(pV i lP),i] ~ n ( ~ ) d ; = [ - P , i e i - p g e a ] k n (~)d~ (9a)

for n = 1 ..... K. Using the product rule of differentation, and noticing that ~'n is not a function of t, and that ~/ranges from 1 to 2, Equation 9a can be expressed as

[(P V),t = (P Vi l~),i] ~n d~ = [p V ~'n ],t d~

I2 d- (I 3 V v 1~), v ~n d~ + (P V3 1~),3 ~n d~ (9b)


The right side of Equation 9b may be rewritten as

-.io .17 .t7 13 [P ~' ~ ],t d~ + [p Vy ~' ~'n ],y d~ - p Vy F X n Y d~ n

4" [p V 3 F ~'n],3 d ~ - p v 3 F Xn, 3 (9c)

B e c a u s e ~n is only a function of ~, we h a v e ~n,y = 0 in Equation 9c. Using this fact and recognizing that the fourth integral is an exact integral, one obtains

f2 f2 f [(P I"),t + (P Vi l"),i] ~'n d~ = [p 1~' ~n ],t d~ + [p V T 11,' ~n ],T d~

s; + p V 3 V ~n = ~ - p V 3 v ~L n d~ tX

(10)

where ~,n'= 3~Ln/O ~, the prime is used to denote the differentiation with respect to ~. Similarly, the fight side of Equation 9a becomes (with application of Leib-

nitz' rule)

S • [ - P,iei - P g e3] ~n d~

.l~ s~ s~ = - ev P,'r ~'n d~ - e 3 P,3 ~n d~ - p g e a ~n d~

.r; s; s; = - % [P ;Zn ],'t d~ - e3 P X + e3 p :Lnd~ - p g e3 X,. d~ n IX

,Y

s; s: _ P e3 ~L n = 13 + e3 p Xnd ~ - p g e 3 ~L n d~ IX

(11)

Combining Equations 10 and 11 gives

s;' s: s;o [p V ~n],td~ + [p V T v ~n],7 d~ - V 3 v Knd~ + p v 3 v ~n = O~

= (-Pn,~, + f~n (~) P,T -- P~n (0~) (I,,T) e~,

e3 (12a)

for n = 0 ..... K. The lowercase variables, p and ~, are the pressures on the top and bottom surfaces, respectively, and Pn and Pn' are the n th integrated pressures and are defined as


Pn = p X n d~ Pn = p X n d~ (12b)

The expression for the velocity field in Equation 3 is now inserted into the left side of Equation 12:

f a P ~ W m , t ~m ~,n d~+I{ p W7 ~ r X W m ~,m ~ n d~ m=o r=O m=O ,?

f~ ~ ~ , ~ ~ ~=13 (13a) - p Wm~, m wa~,r~n d~+ P(WmWa~'m~'r~'n) ~=0~

m=O r=O m=O r=O

= P Wm,t ~m ~n d~ + P (Wm Wr ~),V ~r ~'m ~'n d~- m=0 m=0 r=0 m=0

d~ + P WmWr 3 (~m ~r ~'n) [3 Wm W3 ~m ~r ~n r=O m=O r=0 -- ~

K K K

= X PWm, t Ymn + E X P (WmWrV),~, Ymm m=0 m=0 r=0

- X X PWmW3 Yn + X X PWmW3 (~m ~r ~'n) m=0 r=0 m=0 r=0 = (X

where

Y m n = m ~'n d~, Y mm = m ~r ~'n d~, y n = m ~'r ~'n d ;

(13b)

(13c)

(13d)

Thus, the equations for fluid sheets (Equation 12) for an inviscid fluid can be written as

X p Wm, t Ymn + p (W m W~),~, Y mm- P Wm w3 yn m=O r=O r=O

+ X X p wm wr3 (~m ~r ~n) - - ( -Pn + [l~n ([~) ~,7 - P~n =01;

m=O r=O

(t~) ct,v ) e~ + (Pn - P g Yno - ~}~n ([3) + P~n (a))e3 (14)


for n = 0 ..... K. Equation 14 will be reduced further, but first we perform some algebraic manipulations for later use. In the continuity equation (Equation 5), we first change the dummy index m to r and then multiply it by ~m and ~n and sum it over m for m = 0 ..... K. The resulting equation is then integrated through the vertical direction. We find

K K ~ K K

~~ Y___a Y___a w~'~t ~r ~m ~n d~+ ~a Y___a Y___aW3~; ~'m ~'n d~=O (15, m=O r=O m=O r=O

After interchanging the order of summation and integration and using Equation 13c, we have

K K K K ~WrV,~ Ymrn + ~ ~ W r 3 Y ~ = 0

m=0 r=0 m=0 r=0

(16)

We now consider the left side (LS) of Equation 14 and use the chain rule for differentiation to expand the second term as

LS(14) = ~La P Win, t Ymn + p Win, 7 W~ Y mrn + 19 W m W~r Ymrn m=O r=O r=O

_ ~ pW m Wr 3 yn + P W m W 3 (~m ~r ~n) ( 1 7 ) =0~

r=0 m=0 r=0

After use of Equation 17 to replace the third term in Equation 18, the left side of Equation 14 becomes

---- ~ , -- Wr Ymn P Wm t Y mn + P Wm,~ Wr ~ Y mrn p Wm 3 r m=O r=O r=O

- PWmWr 3 yn r ,+ p W m W 3 (~ ~r~n) m =0~ r=0 m=0 r=0

= ~ (pWm,I Ymn + ~ PWmr Y m m - ~ P W m W3 [Y r mn +yn} m=0 r=0 r=0

-t- ~ y____a p Wm W3 (/~m /~r /~n) (18) m-O r=O


The value of [yr n + y~] can be determined by using integration by parts:

[Y~nn + Yn] = I~X'm ~n X'r d~ + I~X'm ~'r ~n d~ = I~~m (~n X'r)' d~

"-(~'m ~r ~n)]~ =~a --I~ ~n ~r Emd~

--(~m ~r ~n)]~-'~_ m ~ Yrn (19)

Upon inserting this identity into Equation 18 to replace the left side of Equation 14, we find

E D Wm, t Ymn + P Wm,7 Wr Y Ymrn + p W m W: ym (20) m=0 r=0 r=0 =(-Pn + P~n(~) ~,7- P~'n (0~) ~,7)e7+ (Pn -P g Yn0 -P~n (~)+ P~n (O~))e3

for n = 0 ..... K. Equation 20 is the conservation of momentum vector equation.

Discussion of the Generalized GN Theory

We have completed the derivation of the generalized GN equations for an inviscid, incompressible fluid. The GN governing equations for an inviscid flow include two kinematic boundary conditions stated in Equation 4, K conservation of mass equations defined in Equation 8, and K approximate conservation of momentum (vector) equations stated in Equation 14 (or 20). The present derivation has reduced the dimensionality of 3-D equations to a set of two dimensional equations in x I, x 2, and t. As such, these equations are reminiscent of the equations for a membrane, but unlike a membrane, the "fluid sheet" has a much greater kinematic complexity. For instance, a membrane has only one kinematic variable, the location of the membrane for a given x 1 and x 2. The fluid sheet has vectors W n, one of which may be identified as the "location" of the sheet, but the others of which are clearly kinematical ingredients that have no counterparts to a membrane.

In their original works, GN regard a set of governing equations as a postulated set of equations motivated by the 3-D equations. The equations are to be validated by comparison with the general theory or, if required, modified to reflect the physical principles embodied in the general fluid sheet model. It is fortunate that an ideal, incompressible fluid has such a simple constitutive relation, that is, its internal stresses are only pressures and these do not depend on the rate of strain


of the fluid. Only then does the variational procedure we have used in the previous derivation yield a set of governing equations that fits the mold provided by the GN fluid sheet model. One of the distinct advantages of fluid sheet theory is that it always results in approximate governing equations for unsteady 3-D flows. The specialization of these equations to either two dimensions or to steady flows presents no difficulty. However, many of the specialized numerical techniques used in fluid mechanics are developed with appeal to specialized methods that depend on the flow being steady or 2-D (or both), and are therefore limited in their applications.

Several observations can be made about the results we have obtained so far. It is useful at this point to provide a summary of the governing equations for the generalized GN theory. The pertinent equations include the following: Velocity profile given by Equation 3 is

K v (x I , x 2, ~, t )= ~ W n (x l, x 2, t) ~n (~)

n=O

where

W n(x 1,x 2 , t ) = W ~ ( x 1,x 2,t) e i (20a)

Kinematic boundary conditions expressed in Equation 4 are

K K

2 W ~ ~n(ix)=ix,t + 2 W ~ ~n(ix) ix,y n=o n=O K K

2 W ~ ~n(B)=~,t + 2 W ~ ~n(~)~, T n=0 n=0

(20b)

Conservation of mass is stated in Equation 8 as

~, + w 3 a m

m=O =o (2Oc)

Conservation of momentum as defined in Equation 20 is

2 PWm, t Ymn A- P Wrn, y Wrv Ymrn + p W m Wr 3 ym m=0 r=0 r=0

= (-Pn + f~ ~n (I ~) l~,y -- P ~n (IX) IX,7 ) e7

+ (Pn - P g Yn0 - P ~n (l ]) + P ~n (IX)) e 3 (20d)


where the indices r and n in the equations take values from 0 to K. The momentum equations are deceptively simple, but note that there are two levels of implied summation from index repetition. Actual evaluation of these equations is sufficiently tedious that it is impractical to carry out any but the first two levels without the use of some symbolic manipulation software packages to perform the calculus and the algebra. The resulting conservation of momentum equations are 3K scalar equations for 3-D flows and the variables include 3K unknown components of W n, K integrated pressures Pn' and two conditions on the bounding surfaces. On the top surface either [3 or ~ is unknown, depending on the problem. Similarly, on the bottom surfaca either tx or p is unknown. Thus, we have 4K + 2 unknowns and the same number of equations, and the system is closed.

The GN equations depend only on x 1, x 2, and t and have no explicit dependence on the variable ~. After the initial assumption of the form of the velocity distribution in the z direction was made, no terms were thrown out. The governing equations, like the conservation laws from which they were derived, are Galilean invariant. Because no scale was introduced, there is no explicit flow situation for which this theory is most applicable. The governing equations derived this way are, to be sure, an approximation. However, the limits of this approximation are implicit and must be determined by numerical or physical experiment. Because both the conservation of momentum laws and the boundary conditions are nonlinear, the resulting governing equations are nonlinear even for the lowest level theory.

In contrast, Boussinesq and Stokes wave theories are restricted by the assumptions regarding the magnitudes of the perturbation parameters that are inherent to the problem. For water waves, the perturbation parameters are the ratios of wave height to water depth and depth to wavelength and the variables of the problem are expanded in series containing these scales and then substituted into the governing equations and boundary conditions. Subsequently, based on the assumptions regarding the scaling, higher-order terms in the perturbation expansion are usually neglected. This is a systematic error-introducing process specific to all perturbation approaches.

Because the GN governing equations are approximate, they do not exactly satisfy Kelvin's theorem and the flow computed from these equations may not remain irrotational. Recall that irrotationality is not a property of an inviscid fluid, but rather a consequence of an assumption that the fluid is initially quiescent and is acted upon by conservative forces. Shields and Webster [28] have shown that the K th level shallow-water fluid sheet theory satisfies conservation of circulation in an average sense across the fluid domain and the flow remains approximately irrotational in an initial value problem when the initial state is quiescent. However, the treatment of steady flow (time invariant) problems does require some additional specifications of the average circulation (or of the vorticity distribution). Using a development that is an analog of the development of the 3-D equations (for instance, the Navier-Stokes equations), GN theory pro-


vides a continuum model of 2-D sheets with kinematic complexity. This development specifies the general form to be expected with arbitrary complexity. In this development, the kinematic ingredients are called "directors" and the sheet is a "directed fluid sheet" or "Cosserat" surface.

Later in this chapter and also in some previous work [25], the so-called "restricted" theory is used, corresponding to a restricted director as implied by GN. To obtain K + 1 equations, Shields had simply set W~ = 0 to satisfy the continuity equation; this was necessary because of the transformation he has introduced. As has been shown by Demirbilek and Webster [4], the meaning of the constraint is unclear with Shields' choice for the restricted director, and it appears to exclude the solution that may be possible otherwise. In the original GN formulations, the restricted theory meant the first level of the direct theory with constrained director. We extend the original concept here to the K th level theory by calling it a restricted theory if the K th 2-D velocity components are constrained, corresponding to constraint of the K th component of the directors. This generalized constraint for the restricted theory has a simple meaning, that is, the ~, functions (polynomials in this work) must be so chosen that the last term of W K is restricted and has no components in the x 1 or x 2 directions. This requirement is further discussed later.

Equations for Shallow Water

Nonlinear wave equations for shallow water are very important because linear wave assumption is no longer valid. Traditionally, the two-dimensional Boussi- nesq equations are considered appropriate to simulate the wave effects including shoaling, refraction, diffraction, and reflection. However, it is well known that Boussinesq equations [ 1, 21, 24] account for the effects of nonlinearity and dispersion to the leading order. The earlier forms of the Boussinesq equations were limited to roughly depth to wavelength ratio of 1/10. Recently, efforts were made by several investigators to derive alternative forms of Boussinesq equations that extend this limit to about 0.2 [2, 18, 19, 23, 30]. Of these improved models, the one by Nwogu [23] is the most promising because the equations have been obtained through a consistent derivation from the continuity and Euler equations. Nonetheless, all Boussinesq models are based on a vertical profile of horizontal velocity that is at most quadratic in z, and this representation will certainly not be adequate in the intermediate and deep-water limits. Further- more, the increased dispersive characteristics of Boussinesq models should not be construed as a parallel claim to increased nonlinear capabilities of these equations. In fact, all Boussinesq models are limited to the lowest order effects of nonlinearity and as such these models should not be expected to describe the low-frequency surf beat and high-frequency enhancements of the wave crests during shoaling of coastal waves. Boussinesq models should also be expected to fail when the local bed slope exceeds the relative water depth ratio, d ~ . It seems


natural that a better alternative for modeling waves in the shallow water is the GN theory because none of these constraints were used in the derivation of this theory we have presented here.

In this section, a special case of the GN theory for shallow water is provided by further reducing the generalized GN equations of the previous section. Fur- ther simplification of these equations is possible by an appropriate choice of the polynomial weighting functions. The equations for shallow-water were previously given by Green, Laws, and Naghdi [16] and Green and Naghdi [13-15], and Shields and Webster [27]. In this chapter these equations are re-derived based on the general derivation we have presented here and the present work yields equations that are identical to those given by Green and Naghdi [13, 14]; that is, these equations can algebraically be transformed into those of Green and Naghdi. Constraints of the restricted theory that existed in the earlier work of Shields and Webster have been removed in our generalized GN theory that com- prises a set of equations for an arbitrary level of this theory for water waves. The complete set of equations for the first two level theories for shallow-water waves, termed hereafter as Level I and Level II subset theories, will be presented later in this chapter.

If one chooses a polynomial weighting function set given as ~n(~) = ~n, it is then possible to further reduce the general equations we have derived so far. This choice also allows a direct comparison of GN and Boussinesq theories. In this case the inertia coefficients Ymn' Ymrn' etc. Can be expressed using a single function:

~f~n 1 ([~n+l txn+l) H = d~ = - (21) n n + l

from which one finds

Ymn = H(m+n) Ymrn = H(m+r+n)

ym = mH(m+r+n_l ) (22)

The velocity field is given by

K K V , y ' - Z WnT ~ n, V3 = Z w3~n

n=0 n=0 (23)

The equations for kinematic boundary conditions, conservation of mass and momentum can now be reduced using Equation 23. The kinematic boundary conditions become

K K

n--0 n=0 (24)


K K Z w3~n :~ , t + Z W~ [~n~,y n=0 n=0

(25)

The continuity equation becomes

K K Z Wn~,7 ;n + Z W3 n r =0 n=0 n=0

(26)

We now separate the K th term in the first summation and change the index n to n + 1 and obtain

K-1 WKV,v ;K + ~ {WnV, v + (n + 1) Wn3+, } ;n = 0

n=0 (27)

If Equation 27 is to hold everywhere, each coefficient of ~n must be set to zero, that is

WK~,~, = 0 (28)

WnV,~, + (n + 1) Wn3+l = 0 for n = 0, 1 . . . . . K - 1 (29)

Finally, the conditions for conservation of momentum become

Z D Wm, t H(m+n) + P H(m+r+n) Win,7 Wr y + D Wm w3 n(m+r+n-1) m=0 r=0 r=0

= (-Pn + f~ ~n (13) 13,7 -- P ~'n (~) 0~, 7 ) e~, + (Pn - P g YnO - f~ ~n ([~) + P ~n ((g))e3 (30)

for n = 0 ..... K. Equations 23 through 30 are equivalent to those previously given by Green and Naghdi [13] and Shields and Webster [27]. Equation 28 is the equation that is related to the so-called "restricted theory." Provisions of the restricted theory are discussed next.

Restricted Theory and Constitutive Relations

It is necessary to understand the restricted theory in terms of the original GN theoretical formulation. When GN introduced their theory, they restricted the last component of the director so that it remains vertical at all times. Specific constitutive equations are required for the 3-D response functions (see for example, the terms on the right side of Equation 20), which for a general fluid can be


considerably more complicated. The constitutive equations represent the material properties of the fluid and its particular geometry. In addition, the inertia coefficients, Ymn and the relationship of the velocity fields v to the director velocities W m need to be specified. GN chose the response functions so that the pressure was the only component that determines the mechanical power and the constraint responses were found such that the corresponding mechanical power is zero. They also assigned force vectors to obtain proper responses. This procedure is central to GN's approach and is what makes this theory self-consistent in its internal structure. Notably in the theory of plates and shells, GN theory shows self-consistency in its ability to satisfy exactly both dynamic and kinematic boundary conditions. Many competing approaches used to form the 3-D equations ended up having inconsistencies, and more specifically, both boundary conditions were not satisfied at the same time. If one starts with the right kinematic conditions, one ends up with the wrong dynamic conditions, and vice versa. The approach of GN can model a general fluid by specifying its constitutive equation without a conflict.

We have stated earlier that Shields [26] set WVK= 0, corresponding to a restricted director, to satisfy the continuity equation. This process provided K + 1 conditions from the continuity equation, but Shields was unable to offer a clear meaning of the constraint. The particular constraint chosen by Shields appears to exclude the solution that may be possible otherwise. Therefore, we further examine here the condition with no constraint by remembering that originally the restricted theory meant the first level of the direct theory with a constrained director. We extend this concept now to the K th level theory. To extrapolate GN's original definition to our generalized GN theory, we shall call it a "restricted theory" if the K th components of the two-dimensional velocity profiles are constrained, corresponding to constraint of the K th component of the directors. We note that the constraint defined in this fashion has a simple meaning. For example, for a 2-D flow, the most general solution of Equation 28 is

WlK = constant (31)

Consider now the 2-D, steady periodic waves. To determine the wave celerity for steady periodic waves, an additional assumption is needed to ensure that solutions are unique. Traditionally this is accomplished by adopting either of the two definitions of irrotational wave speed introduced originally by Stokes. For instance, Cokelet [3] defined the circulation per unit length, C as

C = ~ u dx (32)

where ~ is a wavelength and u is a horizontal component of the velocity. In the work of Cokelet [3], Equation 32 is satisfied by the choice of reference frame that travels with the wave speed c. Because the flow is assumed irrotational, Equation


32 holds at every vertical location in the fluid. According to Stokes' theorem, the vertical gradient of the averaged horizontal velocity is zero if the flow is irrotational. The wave speed defined by Cokelet is then that according to Stokes' first definition, which is based on the prior assumption of an irrotational flow.

Stokes' first definition has been used by most researchers. Because in the direct theory no assumption of an irrotational flow was made a priori , it would be more natural for us to adopt Stokes' second definition of the wave speed. However, to be consistent with previous work, Stokes' first definition is used here, and accordingly, additional irrotationality requirements are needed. The added irrotationality conditions can be treated analogously in the direct theory. Whether or not GN equations can be used to model irrotational flow requires additional research, and we put forward some thoughts on this subject in the following paragraphs.

Equation 32 will be used here for the condition of irrotationality. Recall that each weighting function in our generalized GN theory represents different vertical dependence. Because there is no vertical gradient of the averaged horizontal velocity for an irrotational flow, we may obtain K + 1 conditions for the requirement of an irrotational flow if Equation 32 is to be satisfied at any vertical location. These conditions are

~(W 1 + c) dx = 0 (33)

~o ~W~ dx = 0 for j = 1, 2 . . . . . K (34)

where ~, is a wavelength and c is the speed of a moving frame, which is the same as the wave speed. Equation 33 is the definition of wave celerity and is analo- gous to the first definition of Stokes, whereas Equation 34 is an expression for global irrotational requirement which may be used as a measure of the vorticity. The solution is not an irrotational solution in the integral sense if Equation 34 is not met. Conversely, if Equation 34 is satisfied, the definition of wave speed then becomes independent of the vertical location in the flow field, a condition that is therefore equivalent to the first definition of Stokes.

Both Equations 31 and 34 must be satisfied for shallow-water waves. For Equation 34, one of the options and clearly the simplest one is to set W 1 - 0, which is precisely the condition one needs for the "restricted theory." This proves that the restricted GN theory with velocity components having a polynomial vertical distribution is a viable choice for shallow-water waves. Because W~ = 0 corresponds to the statement of the restricted theory, we can use this restriction as an implicit assumption of the irrotationality of the flow. If the "restricted theory" were not deduced from the generalized GN theory, this assumption may be a necessary condition of irrotationality of flow, but it may not be a sufficient condition.


For example, Shields [26] stated that Level I theory does not admit shear flow solutions. This is not true in general, but it is true if one uses the restricted theory. It is possible to model waves with shear flow (or current) with the unrestricted Level I theory. Higher level theories are, of course, capable of modeling shear flow solution with a restricted theory. The premise that W ~ - 0 is a necessary condition of the irrotationality could also be explained from physical arguments. Consider for example, 2-D solitary waves for which WIK vanishes far upstream and downstream. Consequently, WlK is identically zero everywhere. This implicit condition has been used in some earlier studies [8, 11 ].

Based on both mathematical and physical arguments, we conclude here that the restricted theory can be used for modeling shallow-water problems whose fluid field is irrotational. When modeling rotational flow in any water depth, either a higher level theory or the unrestricted first level theory should be used depending on the accuracy of the solution desired and other conditions, if necessary. The restricted theory will henceforth be used for shallow-water problems unless otherwise stated.

Subset Theories

Level I Evolution Theory: Unsteady Flow and Uneven Bathymetry

This section presents two-dimensional equations for unsteady free surface flows over an even bottom and a basic overview of the treatment of these equations. The notation will be changed from tensor notation to a component notation in which x = x 1 is the flow direction and usage of ~ is retained to represent the vertical direction for convenience. The coordinate system is taken so that the bottom is expressed by r t) = 0 with p, the unknown pressure on the bottom. The upper surface 13(x, t) is a free-surface on which ~needs to be specified. With K = 1, Level I theory equations for this specific case are derived next.

The velocity profile is given by

u = u 0 + u 1 ~ (35)

w = w 0 + w, ~ (36)

where u and w are the horizontal and vertical components, respectively. The kinematic boundary conditions are

w 0 = 0 (37)

Wl ~ "-[~t + U0 [~x (38)


The continuity equation yields the following two conditions as

U 1 "-0 (39)

U0x + W 1 • 0 (40)

Equation 39 implies that the restricted theory is used. The conservation of momentum conditions yield the following equation as

Uot + 13 u0 Uox = ~ ~x Pox (41) P P

1 132 1 132 P Plx ( 4 2 ) Uot+ UoUox =lS xp p

1132 1132 P 1 132 w l t + u o + w 2 = - - + - - - g 13 ( 4 3 ) ~- ~- w,~ ~- P P

1133 1133 Po 1 13: 1 ~3Wl t + U0 + W 2 = + ~ - - - - g (44) W,x p p 2 Equations 41 and 42 are statements of conservation for the horizontal component of momentum equations for n = 0 and 1, respectively. Equations 43 and 44 are similar statements for vertical component of the momentum equation.

Because there are many unknown variables in this theory, it is convenient for computational purposes to reduce the system of equations to one with fewer unknowns and equations. The reduction is done by expressing vertical components of the velocity in terms of horizontal components of the velocity. We do this using the continuity equation and the kinematic boundary conditions. This reduction can be done for any level of the theory. Moreover, we note that some terms are decoupled from other variables. For instance P1 (PI~ in the K th level theory) is decoupled and occurs only in Equation 42. We discard this variable and its corresponding equation altogether because there is no particular interest in this unknown. We also note that the parameter p occurs only in Equation 43, which we use to express ~ and remove this variable from our equations, realiz- ing that if one wishes, it is always possible to compute p from Equation 43. Likewise P0 (P0, P1 ..... P,k-1 for the K th level theory) may be eliminated. As a result of this reduction process, the original set of eight equations reduces to a system with two equations and two unknown variables, 13 and u 0. For Level II theory, it is possible to reduce the system to three equations and three unknowns. For steady flow problems, one of the components of the horizontal velocity may be further eliminated.

The reduction process is important for further simplification of GN equationsl To illustrate how the actual reduction process works, consider now Equations 35


through 44. We express the vertical components of velocity in terms of the horizontal components of the velocity as

w o =0, w 1 = -U0x (45)

From Equations 43 and 45, an expression for pressure on the bottom surface is obtained as

m

P -2"1 ~2 1 ~2 1 132 2 P -- = Uox t - -~ u o U0x x + -~ Uox + -- + g ~ (46)

P

The unknown P1 occurs only in Equation 42 and because there is no interest in P1, we omit Equation 42 henceforth. The reduced set of differential equations with three unknowns is

[~t =--[U0 [~]x (47)

[~ U0t W [~ U 0 U0x = p [~x Pox (48) p

1 ~3 1133 1 [33 ~ + P0 1 --3 U0xt _ "3 u0 Uoxx + 3 U2x = _ f~ P P _ "2 g 132 (49)

After eliminating the variable Po between Equations 48 and 49, we find two equations with two unknowns, 13 and u 0 as

[~t = -- [U0 ~]x (50a)

Ouo (Ouo ~U0 ~ O2U0 ~2 ~3U0 ._ _ 3g - 3u 0 - 3[3 3 - - -~ - - 3[3 ~xx ~ /)x2/)t ~xx ~ ~ x k./)x )

~ ~2U0 0U0 ~2U0 ~2 O3U0 + 3[3 u o /)x C)X 2 1~2 I- U 0 2 T x 3 (50b)

Note that the Level I GN theory for an unsteady flow over an arbitrary bathymetry gives rise to some third-order, nonlinear evolution equations.

Level I Theory: Steady Flow and Constant Bathymetry

Steady flow equations may be obtained by setting the time derivative equal to zero in the above equations. We start with Equations 47 through 49 and three unknowns ([3, u 0, and P1), and attempt to further reduce these for the most general case of time-independent flows. The pressure on the upper surface is given by

f~=Pa - q (51)


where

TI3xx (52) q = ~](1 + 132x) 3

in which T represents the constant surface tension on the surface and Pa is the atmospheric pressure. Assuming an incompressible fluid, we may for convenience set Pa = 0 without loss of generality. Two of the three governing equations can be integrated. The integration of the conservation of mass statement given by Equation 47 (or Equation 50) with respect to x yields

u 0 ~ = Q (53)

where Q is the constant of integration. To determine a meaning of this constant, we consider the mass flux per unit span given as

u de = u o de = u o [3 = Q (54)

Thus, the constant Q represents the mass flux per unit span. Integration of Equa- tions 48 and 49 yields

Pox T 13x 13xx

1133u2 x 1133 Po 1 132 T[313xx - -~ u0 u0 xx . . . . g + (56) p 2 p ~/1+132x) 3

The pressure on the bottom is determined by

- 1 132 TI3 p = 1132u0u0 + U2x+ xx +gfl (57) P - -~ xx -~ p~J(l+132x)3

Equation 55 may be integrated with respect to x as

T Po / P = - Q u0 + + S (58)

P l+13x

where S is a constant of integration related to the momentum flux per unit span S* given by

f0 ~ 2 S * = [p/p + u ] d ~ = P 0 / p + Q u 0 (59)

from which the relation between S and S* is now obvious; S may be interpreted as the momentum flux per unit span less the momentum defect due to the effect of surface tension. Using Equations 54 and 58, we obtain from Equation 56 a single governing equation inclusive of [3 as


Q2 1 [32 T T IBxx 1 QZl3xx 1Q2 [32+ + _ g _ - 3 [3 13 2 p~/1 + ~2 x p~/(1 + [~2 x )3 + S = 0 (60)

Equation 60 may be integrated once more, first multiplying it by 13x/[~ 2 and then integrating. We multiply the resulting equation by [32 and obtain

1 Q2 2 ~3 ~2 [3 x + g + 2 R + 2S13+ 2T13

p41 + [3 2 _ Q 2 = 0 (61)

where R is another constant of integration. Ertekin [8] was the first who has studied the solutions of these equations for different values of Q, R, and S both for regular and solitary waves.

Level II Theory: Unsteady Flow and Constant Bathymetry

The Level II evolution equations developed by Demirbilek and Webster [4] include the set of equations listed next. Velocity profiles:

u = u0 + Ul ; + u2 ;2 (62)

w : w 0 + w , ; + w : ; 2 (63)

Kinematic boundary conditions:

w0=0

Wl [~+W2 ~2 = ~ t + u 0 [~x + u l ~ x

(64)

(65)

Continuity equation:

U 2 -" 0 (restricted theory)

UOx + W 1 = 0

U lx + 2 w 2 = 0

Conservation of momentum: n = 0, x-component:

(66)

(67)

(68)

1 ~2 +_~1 IB 3 1 ~Uot +~" Ult Ul Ulx +-~1 B2

1 ~2 1 ~3 + ~ U 1 W 1 + U 1 W 2 --IBx

e -3 o

lf)2 UO U lx +'~" U 1 UOx + [~ U 0 UOx

Pox

P (69)


n = 1, x-component:

1 13 2 1 13 3 1 i~ 4 1 13 3 1 13 2 Uot +3" Ult +'~" Ul Ulx +~" Ul UOx + ~ UO UOx

1~4 1~3 P Plx + - - Ul W2 + Ul Wl = ~ x

4 3 p p (70)

n = 2, x-component:

1 [33 1 ~4 1 [35 1 ~4 1 133 U0t + ' 4 " Ult + ~ Ul Ulx +~- Ul UOx + '~ U0 U0x

1 [3 5 1 ~4 = 1]2 f) P2x + - U 1 W 2 + U W 1 I~x 5 "4" 1 p p

n = 0, ~-component:

1 ~3 1 ~4 1 ~3 1 ~3 1 i] 2 wit + + u 1 + u 0 + u w "2 "3 W2t "4" W2x "3 W2x "3 i ix

1 1 ~4 ~3 1 132 + - - U 0 WIX + W 2 + W i W + Wl 2 = - - f ~ + - - - - g ~ 2 2" 2 ~" P

(71)

(72)

n = 1, ~-component:

1 133 1 ~4 1 ~5 1 ~4 1 ~4 1 5 Wit "F'~- W2t +-~ U 1 W2x -F-~- U 0 W2x +-~- U 1 Wlx + ~ ~3U 0 W 1

2 135 3 ~4 1 Po 1 ~2 +-- w 2+ w , w 2+ I ] 3 w ~ = - l ] p + - - - - g 5 p 2

(73)

n = 2, ~-component:

1 ~5 1 ~6 1 [35 1 ~5 1[~4 1 ~4 Wl t -I- W "F U --F U 0 "1- U 1 W -F U 0 W1X -4 "5 2t g l W2x 5 W2x ~ lx 4-

1 ~6 3 135 1 [~4 ~2 P P1 1 ~3 + - w E+ w l w + w 2 = - - - + 2 - - - - g (74) 3 5 2 "4 19 19 3

In this case, we have a total of eleven equations. This system of equations can be reduced by eliminating the variables w 0, w 1 and w 2, representing vertical components of the velocity. From Equations 66 through 68, we find

W 0 = 0 , W 1 " " - U0x, W 2 - - - l Ulx (75)

Using Equation 75, we write five governing equations in terms of 13, u o, Ul, PO and P1 as

~t = - Uo 13 + -~ Ul (76) x


1 132 1 133 I~Uot + ~ - Ult + g U 1U 1132 + UoU +13UoU =~13x Pox lx "2" lx Ox p p (77)

1~4 1 ~2 1 ~3 + u 1 u ~" not +'~ " nit "~"

= 13 ~x f~/p- Pax/P

1~3 + U o U lx 3 lx

1~2 + -~- U 0 UO x

(78)

1154 1 [35 1134 1 135 1 ~,3U0xt-g ~ xt ~ xx "~" Ulxx - U l U l - Uo + U x,2

3 ~4 1 ~4 1 [~3 1 ~3 +-- UOx U 1 Ul -- U 0 U 0 + U20 8 U o x x xx x

~ P Po 1 ~2 = - - - + ~ - - - g P P 2

(79)

1134 1 [35 1 1~6 1 [55 1 ~6 Ul2x 4 U~ - "i-o Ulxt - T2 Ul Ulxx - i-o u~ Ulxx + i 2

3 135 1 135 1 ~4 1 [~4 2 + - - Uox Ulx- U0xx-~ ~ ox 10 5 Ul U0 U0xx + U

= _ ~ 2 P + 2 P 1 1 ~ 3 (80)

At this point, one needs to search for some secondary variables that can be eliminated without loss of generality of the governing equations. For example, the variables P0 and P1 can be eliminated and surface tension may be set to zero, and thus ~ is zero in this case. With these simplifications, we obtain a set of three equations expressed in terms of variables u 0, u 1, and [5. These are

C 1 132/ ]~t =-- UO ~+'2"Ul X

(81)

()Uo ()Ul ()~ ()2Uo ~2 ()~ ()2Ul b 40 [~2 ()3Uo - 120 - ~ - 60 ~ . - ~ + 12015 ~xx /)x/)t + 60 ~xx /)x/)t ~)x 2/)t

~ ~U 0 ~ OU02 + 15 ~3 O3Ul = 120 g + 120 u o + 120 [3 ~- 20 13 (3 u o /)x2/)t ~x -~x ~x t)x ~Ul ~2 ~ ~I10 ~11 ~3 ~ OU12

+ ~ U l ) - ~ x +180 Ox Ox Ox t-60 Ox Ox 120~(u o

0[~ O2U 0 [~2 ~U0 02U0 ~3 0Ul 02U0 60 ~2 + ~ U 1 ) ~XX t)X 2 b 40 /)x ()x2 + 15 /)x ()x2 (Uo

(Equation 82 continued)


~ ()2Ul 1~3 t)Uo ~2Ul ~4 ~Ul ~2Ul -I- ~ U 1 ) ~X ~ + 30 t- 12 ~X2 ~X ~X2 ~X ~X2

~3U0 ~3 O3Ul + 3 ~ u 1) /gX, 3, - 3 (5u o + 4 ~ u 1) ~ x 3

10 ~2 (4 u o

(82)

t)U 0 t)U 1 ~ t)2U 0 - 60---~-- - 40 ~---~- + 60 [5 ~xx OxOt

+ 30 ~2 ~ ~L'2ul /)x ~x/)t

~- 15 132 /93u~ Ox2Ot

+ 5 [3 (8 u o + 313u 1) ~13 OUo ~[~ OUo2 + 6 [33 33Ul = 60 g + 60 u o Ot 6013

/)x 23t ~x - ~ x ~xx ~gx

~u~ fi E ~fl ~u0 ~u fl3 ~13 ~ul ~13 ~2u0 + 90 1 + 30 2 60 [3 (u o + 13u 1) Ox ~x ax /gx ~ ax ~ ~gx 2

+ 15132 ~Uo ~92Uo 133 ~9Ul /32Uo /3x /9x2 I-6 /gx ~9x2

+ 12133 /gu~ /32u----L1 + 5 [34 /3ul /92ul ~x ~x2 ~x ~x2

~3U 1 -133(6 u 0 +513 u l) ~x 3

~[5 ~2u 1 30 ~2 (u0 + fl u~) ~x ~x 2

O3U 0 3 [32 (5 u o + 4 [3 u 1 ) ~ x 3

(83)

See Demirbilek and Webster [4, 6] for further details.

Level II Theory: S teady F low and C o n s t a n t B a t h y m e t r y

The governing equations for Level II GN theory for two-dimensional steady flow with surface tension are provided here. In this case, we retain the surface tension in the equations and specify the pressure on the upper surface from Equations 51 and 52. The time derivative in the governing equations (Equations 76-80) may be set to zero. By integrating Equation 76 we find

u o [3 + -~ u 1 = Q (84)

where as before, Q may be interpreted as the mass flux per unit span. Note that an additional term appears in Equation 84 in comparison to Equation 53; this arises from the difference between the second level theory and the first level theory. Integrating the x-component of the momentum equation (Equation 77) and using Equation 83 yields


1 133 _ Q2 1 T + s (85) P0 / P =i2- u12 ~ + p41 +l~x 2

where S may be interpreted as the momentum flux per unit span less the momentum defect due to the effect of surface tension.

The variable p may be determined from Equation 72 if other variables were known. We note that as there is no interest in the variable Pz in this research, we omit the corresponding equation. Using the remaining equations, we eliminate u 0 and P0 to obtain three equations in three unknowns, 13, u 1, and P1 as

Xl3Px 13xx

p4il +

Plx t 1 [34 1 ]3 2 1 2

I "

+ ..... 0 u2+ Q Ul+ Q - p T2

t.

+ 1-~ Ul +2-4Q Ulxx

1 134 1 132 1 [33 1 Qz 13x + - U 1 Ulx + Q + 13 x u 2 _ __ 12 ~ Ulx 8" 2"

OX/( 1 + ]32)3" 13xx

(86)

1 134 7 ]32 1 133 2 1 1 135u~ x+ ]3 xu + ]3 xu + ]3 xul 2+ QI3Ulx]32 240 ~ lUlx "~ lx ~ g

1 132 1 1 Q2 13x 2 + T 1 133u2+ +Q S = 0 3 13 px/1 + 132x 12 2 g ~ -

(87)

3 135 2 1 [33 | Q2 N Ul +?Q Ul+- TI~ 2 } 1 136 p4(1 + 132 )3 [3xx + 2 - ~ Ul

1 ]34 1 + ~ Q Ulxx

40 240 [36 2 1 135 1 133 __ ~ Ulx + ~ I~x U 1 Ulx + "~" Q 13x Ulx

1 [34 2 1 [~2 2 1 Q2 2 Pl 1 133 + - - u12 13x + Q Ull3 x - 13x - 2 - - + - g = 0 16 ~ 4 p 3

(88)

Note that P1 may also be eliminated between Equations 87 and 88, but for steady flow equations there is no particular advantage in doing so. Shields and Webster [27] present solutions to these equations for both solitary waves and large- amplitude regular waves.


Level I I T h e o r y : U n s t e a d y F l o w a n d U n e v e n B a t h y m e t r y - - T h e G e n e r a l i z e d F o r m u l a t i o n

It has been shown by Demirbilek and Webster [4] that the governing equations for an unsteady flow over an uneven bed are rather complex, but these can be reduced to three equations and three unknowns, u 0, u l, and [3. Demirbilek and Webster derived these the governing equations for this case using an algebraic manipulation program called Mathematica TM. Because these equations will form the foundation for future research, a presentation of them in usual mathematical notation is made here. Note that these equations have a new time-independent variable, o~ that represents the vertical coordinate of the bottom. The equations and their solutions involve derivatives ~. We shall assume that these derivatives through the third derivative in x are bounded. The set of applicable equations for an uneven seabed that exclude the surface tension effects are listed here for completeness. These equations formed the basis of a numerical code in the implementation of a new generation wave theory for coastal engineering applications [5]. These are Governing equation 1:

~ ~)Uo ba /)13 - 2 ~ = - 2 ( a - 13) " ~ x - 2 ~ (u o + {~Ul) + 2 -~x (u~ + ~ ul)

/)u 1 - (a - 13) (a + 13) gx (89)

Uneven bottom governing equation 2:

Ou 0 O2(X Ou 0 O(X ~ Ou 0 - 120 (a - 13) - - ~ + 6 - ~ - ~ (tx - 13) 2 - - ~ - 120 -~x (ct - 13)/)--~ /)--~-

()(X ~2U 0 ~ ()2U 0 ~3U 0 + 120 -~X (Ct --]3) 2 /)t/)x 120 (tX -- 13) 2 ~---~ ~)t/)-----~ + 40 (Ct -- 13) 3 ~)t~)x 2

~13~ 2 ~U 1 ~213~ ~U 1 + 60 ~ (a -13)2 --~ - + 60 a - ~ (a - 13) 2 - ~ - 60 (a - 13) (a + [3) ~u~t

~0[ ~ OU 1 ~0[ O2U 1 - 120 o~ ~ (o~ - 13) ~xx ' ~ + 120 a -~x (a - 13) 2 ~)tOx

~1~ ~2Ul ~3ul - 6 0 (a - 13),. (a + 13) Tx ~ t ~ x + 5 (a - 13) 3 (5 a + 3 13) - ~ x - 2

~13 a13 ~u 2 ~Uo ~2Uo = 120 (or - 13) ~ g - 120 ( a -13)2 ~ x - ~ x + 40 (0~ -[3) 3 ~x bx 2

~u o i)a + 120 (a - 13) ~ (u o + (~Ul) + 120 -~x ( a - 13) U 1 (U 0 + Of, U l )



()(~ ()~ t)U 0 ()0{: ()2U 0 - 120 ~ ( a - 13) ~ x - ~ x ( - u ~ + ul ( a - 2[$)) - 120 -~x (t~ - 13)2 ~xx ~"

(U 0 + 13 U 1) + 120 (a - 13)2 ~ O2Uo 00~2 ()~ /)x /)x 2 ( u ~ ( - c t + ~ ) - ~ - x Ul

(u o + 13 ul) + 120 ~ (ct - [3) -~x (u~ + ct Ul) (u o + 13 u 1 )

( ~)2a ~)uO +60~)a ~)2 ()3a / - 60/)x-- T /)---~- -~X - - - ~ U l + 20 ~ - ~ (u o + a U 1 ) ((Z -- 13)2 (3Uo

O3Uo (4u o + (a + 3 13) u 1 ) - 60 (ct - 13)2 (o~ + 3 13) + (a + 2 13) Ul) - 10 (a - 13)3 ~x 3

t)~ ()Uo t)Ul F 5 (a -- 13) 3 (5 a + 3 13)/)2u~ /)ul /)a ~913 /)x /)x /)x ~'x -~ /)x + 120 ~ (a - 13) /)x

()Ul ( ~0~2 O2a~ ((2 a - [3)u o + ct 13 Ul) --~- x - 60 ~ x + 60 o~ /)x2 ) (or - [3) 2 (3 u o

/)u 1 + (ct + 2 13) Ul ) -~x + 20 (a - [3) (3 (a + 13) u o + (40~ 2 + O~

/)u 1 + ~2) u~) ~ - 60 (a - 13)2 (~ + 13) 13 a~x13 au~ 2~x. + 10 (a - 13) 3

t)U 0 ()2U 1 ()[~ t)Eu 1 (a + 313) ax ~ + 60 (a - [~)2 (a + 13) Tx (Uo + 13 u~) 7 x 2

Oct - 40 ~ (ct - 13)2 ((4 tx - [3) u o + o~ ( a + 2 13) u 1 )/)2ul~x 2

OU 1 ()2U 1 + (a - 13) 3 (7o~ 2 + 21 ot 13 + 12 132) ~x" ~xx 2"

03Ul - ( a - ~ ) 3 ( 5 ( 5 a + 3 1 3 ) u 0 + ( 7 a 2 + 2 1 a l 3 + 12132)Ul) ~xx 3 (90)

Uneven bottom governing equation 3:

()U 0 ~20~ t)U 0 - 60 (ct - 13) ( a + [3)---~-- + 20 ~ (tx - [3)2 (2or + 13)

t)Ot, ()~ t)Zu 0 ()(/, t)Zu 0 - 6 0 - ~ - x ( c t - 1 3 ) ( a +13)'/)x /)t 1-40~-x ( t ~ - ~ ) 2 ( 2 a +13) OtOx

O~ ()2U 0 Oauo - 60 (t~ - 13)2 (0~ + 13) ~XX OtOx ~ 5 (ix - 13)3 (50~ + 313)

~t~ 2 OU 1 ~21~ OU 1 + 20 - ~ x ( a - 13)2 (2o~ + [3) - ~ - + 20 ~ ~ (~ - 13)2 (2~ + [3)



Oul Oa O~ Oul - 4 0 ( a - ~ ) ( a 2 + a ~ + ~2) - - ~ - 6O a -~x (a - ~ ) (a + ~ ) -~xx -fit"

~(~ ~2U 1 + 4 0 a ~ x ( a - [3)2 ( 2 a + ~) OtOx - 30 (a - [3)2 (0~ + [3)2 ()~ (92Ul

Ox OtOx t)3U 1

+ 2 (a - 13) 3 (8 ~2 + 9 a [3 + 3 [32)/)-~x ~

2 013 015 OUo = 60(a - ~l) (a + [3) ~ g - 60 (a - [3) 2 (a + ~l)~x X Ox

()U 0 (92U 0 ()U 0 + 5 (a - [3) 3 ( 5a + 3[3) Ox Ox 2 ~ 60 (a - [3) ( a + [3) ~ (u o + a u 1 )

Oa + 60 -~x ( a - [3) ( a + [3) u I (u o + a u 1 )

( ~2a Ou~ Oa~2a 03a I - 15~x2 ~xx + lS"~x ~--~Ul + 5 ~ x 3 (u~ + a U l ) "

Oa O~ OUo (a - [3) 2 (4 (2a + [3)u o + 3(ct + ~)2 Ul ) _ 60 -~x (a - [3) (a + [3) ~x Ox

013 02Uo ( - u o + ( a - 2[3) Ul) + 60 (a - [3) 2 ( a + 13) ~xx /)x 2 (u~ + 13ul)

~0~ 2 ~ ~20~ ~ + 60 - ~ x (~ + ~) (~ - [~) ~ x Ul (Uo + 13 Ul) + 60 ~ - U (~ - [~) (~ + ~) ~xx

bO~ OEUo (U o + aUl) (U o + ~Ul) -- 5 -~X (a -- ~)2 OX 2 (8 (2a + [3) U o + (a 2 + 14a~

O3U 0 + 9 1 3 2 ) u l ) - ( a - [ 3 ) 3 0 x 3 (5(5a+313)u o + ( 7 a 2 + 2 1 a [ 3 + 1 2 1 3 2 ) u l )

O~ OUo OUl - 30 (a - [3) 2 ( a + 13) ( a + 313)~x x Ox Ox + 2 ( a - [3) 3 ( 8a 2 + 9a[3 + 3[32)

()2U 0 ()Ul 15 ~)Ot2 Ox 2 Ox ~ (a - [3) 2 (4 (2a + [3) u o + 3(a + [3) 2 u 1 )/)ul~x

~U 1 ()~ (~ -- [3) /)2a (a - [3) 2 (4 (2a + ~) u o + 3 (a + [3)2 u, ) ~ + 60 ~ x - 15aO- ~ -

~[~ ~U 1 (a + 13) ~ x ( (2a - 13) Uo + ,x 13 Ul) - ~ x + 5 (a - 13) (8 (a ~ + a 13 + ~ ) Uo

Oul 0[30Ul 2 + 3 ( a + ~ ) ( 3 a 2 +~2)Ul)-~-x - 3 0 ( a - ~ ) 2 ( a + [ 3 ) 2 ~ x Ox

()Uo ~2Ul 15 Oa + (a - [3) 3 (7 a 2 + 21 ct ~ + 12132) ~xx Ox 2 ~ ( a - ~1)2((7a 2



/92ul /913 + 2ct 13 - 132)Uo + 2ct (a + 13)2u~) ~9x2 + 30 (a - 13) 2 (a + 13) 2 7 x

/gul 0aul (a - 13) 3 (2 (8a 2 02u~ + 5 (a - ~)3 (a + [~)3 0x 0x 2 (Uo + ~ul) ~x 2

~3U 1 + 9a 13 + 3132) u 0 + 5 (a + 13) 3 Ul) ~x y (91)

S o l u t i o n S c h e m e

I n t e g r a t i o n

The shallow water GN wave evolution equations are quite complex from an algebraic standpoint although these can be integrated with little difficulty. In this section we consider the integration of GN equations. Our Level II GN wave evolution equations form a system of three coupled, partial differential equations that are first-order in time and third-order in space. These equations are subject to some boundary conditions at both ends of the domain (two-point boundary conditions). The key to an efficient scheme for their solution lies in the fact that the highest-order mixed derivatives are only first-order in time and second-order in space. We develop here a special solution scheme for these equations. A similar scheme has been used for integration of the evolution equations for deep-water waves [29]. Our solution scheme is not limited to three governing equations; in fact, any number is allowable so long as there are sufficient boundary conditions.

Ertekin [8] was the first who devised the basis of this elegant scheme for solving similar equations and boundary conditions for Level I GN theory. His analysis was simplified because there were only two governing equations, and these were not coupled in the time derivatives of variables. In fact, the equations solved by Ertekin are identical to our Equations 50 a and b. He integrated the first of these equations directly and the second equation, implicit in the spatial coordinate, was expressed as a tri-diagonal system of linear equations and solved using the Thomas algorithm. The Level II equations (i.e., Equations 81-83), are considerably more complex than those of Level I, and in addition, Equations 82 and 83 have the disadvantage of being coupled in the time derivatives of unknown variables. We will show that with some modifications, it is still possible to use essentially the same efficient solution scheme for Level II equations. Our modified solution procedure, a variation of the Thomas algorithm, results from a hybridiza- tion of Ertekin's method [8] and the scheme of Newman for coupled ordinary differential equations. In the interest of future work with the GN theory, this algorithm is described here in general terms. In particular, consideration will not be limited to the three equations at hand and instead we demonstrate how this solution scheme can be used for a more general system of equations.


Consider a system of K coupled, quasi-linear partial differential equations in the K-dependent variables. The variables are expressed here as a K-dimensioned vector, ~(x, t), whose equations have the special form as

A ~ + B ~ , x +C~,xx : g (92)

where the dot over ~ signifies a derivative with respect to time, and A, B, and C are K x K matrices, g is a K-dimensioned vector. A, B, C, and g are perhaps functions of x and ~ and its spatial derivatives, although this dependence will not be shown here in the interest of simplicity. It is assumed that the problem is posed as a two-point boundary-value problem in x and an initial-value problem in t. The domain of x over which a solution to the equations is desired may be assumed to have a uniform grid of x's, spaced a distance Ax apart. The i-th point on the grid will be denoted by x i = i Ax, for i = 1,ns. Time is also assumed to be discretized with intervals At, with tj = j At. The value of the solution vector ~(xi, tj) will be denoted by ~(ij), and similar superscripts will be used for the other vectors and matrices. The spatial derivatives will be approximated by central differences as

~,,~,j) ____ (~( '+ ' , j ) - ~ ( i - ' , j ) ) / 2 Ax

~ (i,j) =(~(i+l , j)-2~(i , j )+ ~(i-l,j)) /Ax2 (93) ,XX

With these approximations, Equation 92 can be written as

,~(i,j) ~ ( i - l , j ) + B( i , j )~( i , j )+ D( i , j )~ ( i+ l , j )= g(i,j) (94)

where

/~k(i'J ) -- / c(i 'J) AX 2 B(i'J) / 2Ax

c(i'J) / fi(i,j) _ _ 2 + A (i'j)

Ax 2

c(i,J) B(i,J)) b (i'j) -- + ~ (95)

Ax 2 2Ax

Suppose that the solution ~(0,j) = ~o) is known as the result of a boundary condition at this point. Then the solution at i = 1 can be readily found from Equation 93 as

__ ~(1) + ~.]( 1, j) ~(2, j) (96)


where

~0(1)- [n(1, j)]-1 [g(1, j)-/~k(1, j)~0(0)], and

~,l , j , = __ [B,I,j,]-1 D,l,j, (97)

This process can be repeated for the entire domain and we find

_ (i) + ~_~(i,j) ~(i+i,j) ~(i.j, ~0 ~0(i, _[~(i,j ,]-l[g(i,j ,_/~k(i,j , ~(i-1,]

~.~(i,j) ____ [fi(i,j)]-1 D(i,j) (98)

At the boundary point i = ns it is assumed that ~(ns+l,j) is known and given as a boundary condition at this point. The value of ~(nsd) may be found as

~ (ns.j) _ [B(ns, j, +/~k(ns, j, ~_~(ns-l,j,]-1 {g(nS, j, _/~k,ns, j, ~(ns-1, _ D(i,j, ~(ns+l,} (99)

With the value of ~(nsd) thus determined, all other values of ~(id) can be found by back substitution as

~(i - l,j)----~0 (i-l,j) + ~_~(i-l,j) ~(i,j) (100)

These values of ~(i,j) are then used to estimate the values of ~(i,j+l) (i.e., at the next time step). These are

~(i , j+l)- ~(i,j)+ ~(i, j)At (101)

Note that this estimate is only first-order accurate and could be unsatisfactory. However, we can obtain an estimate for ~(i,j+l) by using these new values and reap- plying the procedure at t = (j + 1) At. A new estimate for ~(i,j+l) can be formed as

~(i,j+l) _~(i,j) + (i,j) +~(i,j+l) Y (102)

This new estimate is second-order accurate in both At and Ax. Additional information on this solution algorithm is available [5, 8, 29]. Upon some modifications, the Thomas algorithm can be made applicable to 3-D GN equations.

Numerical Model

We have provided in a separate report [5] a detailed description of a numerical model developed on the basis of our shallow-water Level II GN theory presented here. This numerical model was developed for coastal engineering pro-


jects for military and civil works and is far superior in performance and in programming style to the programs used by Shields [26] and Shields and Webster [27]. This model is generally applicable to 2-D solutions of all sorts of water wave problems. A brief description of this model is provided here because it forms the foundation for future developments in this field of research.

The general modular structure of our GN model is depicted in a flow chart [5]. The part of the program unique to Level II GN theory is a subroutine that corresponds to Equations 89-91. Other principal subroutines include a coding of the Thomas algorithm, a standard linear equation inversion routine using Gauss- Jordan elimination, a digital smoothing filter to remove the spurious tipples near the wave maker, a routine that determines the various spatial derivatives of the bottom topography, and several small routines to perform standard vector and matrix operations to make the program stand alone. The model has been checked for accuracy, and applied to several practical problems [5].

For numerical simulations, the seabed bottom profile can be arbitrary; wave gauges may be positioned at will anywhere in the computational domain. Differ- ent kinds of simulations may be performed for problems of both military and civil interest. One can study, for instance, the effect of a submerged sandbar on a train of waves approaching a beach (of critical importance during military land- ings), the reflection of waves from a sea wall and forces on a sluice gate (important for the design of these civil works), or obtaining the surface time history from a wave pressure gauge under a train of near-breaking waves (of interest to scientific researchers). The robustness and correctness of the code was extensively tested for fluid flow problems with known analytical solutions. The model has been shown to reproduce with engineering accuracy the evolution of a wave of permanent form from small amplitudes up to almost-breaking limit [5].

Figure 1 shows the computational domain and discretization of the seabed topography. The bathymetry for typical problems may involve a spatial discretization with thousands of points. To increase the user-friendliness of the model, an option is provided in the model that allows for a simplification of the bathymetry; the user may describe the bottom profile as a polygonal shape with a series of straight lines connecting a sequence of arbitrarily spaced nodes. Either the simple representation or the actual bathymetry data define the bottom fluid sheet, ct(x) at each of the individual finite difference nodes. If the value of x for a given node falls outside of the range of bottom data, o~(x) is set to the closest input bottom data point. Linear interpolation between the two closest points is used if the value of x falls within the specified bottom data. Because the polygon does not have even second derivatives at the comers and because the theory involves derivatives up to and including the third derivative of t~(x), the final set of interpolated values of o~(x) are smoothed using a numerical filter.

In the computer code, the Level II GN equations are numerically integrated and results are stored into several files as the solution becomes available. Different numerical output options are available in the model; these may be tailored to user needs in a given application. An important feature of the model is the "re-start"


computational domain ,, ,~

|ls(x.t) " | t

o~(x) Z

\ / [xbottom (1), abottom(1)] [xbottom (nbottom), abottom (nbottom)

Figure 1. A typical flow for simulation by GN model: (a) computational domain, (b) discretization of the seabed topography.

capability that allows computations to be continued with the previously saved output. This feature is particularly useful when one is using a microcomputer to simulate wave evolution nearshore over several miles. In such cases, long simulations may have to be carded out in several steps using the restart feature of the numerical model. As designed, our computational model is in essence a "numerical wave-tank" that simulates the evolution of a train of 2-D regular waves and irregular sea states moving from deep waters to shallowers depths over waters of arbitrary bottom topography. The model simulations are based on the velocity profiles of u(x,z,t) = Uo(X,t) + Ul(X,t) z and v(x,z,t) = Vo(X,t) + vl(x,t) z + v2(x,t) z2; the solution of Equations 89-91 is obtained in three dynamic variables, 13(x,t), U o(X,t), and u l(x,t). In addition, the model also computes three dependent variables of practical interest, p, P0, and Pl, which are the pressure in the flow field, the integrated pressure from the ocean bottom to the crest of waves, and the first moment of the total pressure taken about the undisturbed free surface level.

The model assumes a wave time history at the left side boundary of the domain for waves moving from left to right. On this boundary, wave input includes the local wave height history, ~(t) as well as the corresponding history


of the other unknown variables, u0(t) and Ul(t). As a first-order approximation, time histories of the unknown variables may be specified from the solution to the steady flow equations (linearized for small wave amplitude), including for steep wave input to the model. For waves proceeding from left to right with a celerity c, the linear solution gives

I] (t) = ~0 cos [kx - tot] = [30 cos [k (x - c t)]

u0 = [30 12g [20 + 7(kh) 2 ] cos[tot] c [240 + 104 (kh) 2 + 3 (kh) 4 ]

ul =~0 120g(kh)2 c h [240 + 104 (kh) 2 + 3 (kh) 4 ]

I 24 g h [(kh) 2 + 10 ]

c = i240 + i04 (kh) 2 + 3 (kh) 4 ]

cos[tot]

(103)

where 130 and h are the wave amplitude and water depth, respectively. The linear small-amplitude waves are not exact boundary conditions for finite amplitude waves, and consequently there are often some small oscillations near the wave maker as the solution "finds itself." This problem is similar to the flow near a flap-type wave maker in a physical wave tank because the boundary condition at the flap is only roughly like a free water-wave. The numerical ripples at the boundary can cause problems if allowed to propagate. In our model, these are thus filtered out near the wave-maker; free waves elsewhere are not filtered. In computer implementation, the wave-maker input time histories are placed sequentially in the first three elements of the variable arrays so that spatial derivatives can be formed. An Orlansky open boundary condition is placed at the other end of the domain for a beach; a full reflection condition is also available for a wall-type boundary at the down-wave end of the domain.

In the wave-tank simulation, four major numerical tasks are performed by the model: generating waves at the ocean end of the tank, enforcing an appropriate boundary condition at the shore end of the tank, finding the solution of equations of a two-point boundary value problem in space at a given instant in time, and integrating these equations in time. Each task requires special techniques as described in detail by Demirbilek and Webster [5] and because these tasks are crucial, a brief discussion of each task follows.

Wave-Maker

The water in the numerical wave tank is quiescent at the beginning of a simulation and there are no waves. This is the only known exact solution for nonlinear waves passing over an uneven bottom. The computational domain is discretized into equal spatial steps of length dx for ns discrete points. For regular waves, dx is selected such that at the water depth corresponding to the origin of the computational domain each wavelength is about 100dx; the shortest wave is


at least 60dx for irregular waves. At each point in the computational domain, spatial derivatives up to the third order of the independent variables and the bottom profile ct(x) are computed and this requires adding two additional points to both the left and fight of the computational domain. The points are indexed from i = 1 to ns +4, where the computational domain corresponds to the points from i = 3 (at x = 0) to i = ns + 2 (at x = L). The points i = ns + 1 and i = ns + 2, respectively, correspond to the location of the actual wave-maker boundary condition, and the open boundary and reflection boundary condition (when applicable).

Waves are generated on the left-side boundary of the domain, propagating to the fight. The time step, dt, is chosen according to c dt = dx, where c is the regular wave celerity at the wave maker or the celerity of the wave of median frequency for an irregular wave train. For regular waves, time histories of each of the three independent variables 13, u 0 and u I are constructed by estimating the wavelength of a regular wave of the specified height and period in the water depth at the wave- maker (Equation 103). Time histories for irregular waves are constructed by inter- polating a series of tables to determine the first five Fourier coefficients of 13, u o, and u 1 for equivalent regular waves of this height and length.

These data are used to construct the required time histories of the independent variables at t = 0, dt, 2 dt ... nt dt, where nt dt = tmax is the length of the simulation. Because the fluid field is initially quiescent, the wave time history is "ramped," so that the wave-maker imposes only a little disturbance at first and gradually builds to the full disturbance level. The ramping is done by multiplying the time history from t = 0 to t = T (the period of the regular wave or the period of the median frequency wave of a spectra) by a bell-shaped function given by sin2[ r~/(2 T)]. This function and its slope attain zero values at t = 0, and 1 and 0 when t = T, forming rather continuous and smooth time histories even at t = T. This wave time history is identical with the desired spatial history for regular waves and is nearly identical for the case of irregular waves, especially if the spectrum is relatively narrow-banded. For waves of permanent form moving in the positive x-direction, 13(x,t) = 13(x - c t) and with the choice of dx = c dt, we then have [3[(x o + dx) - c t o] = 13[x o - c (t o - dt)]. The imposed time history can therefore be used to define [3(x,t), Uo(X,t) and Ul(X,t) at i = 1, 2 and 3, corresponding to the left-side edge of the computational domain and the two points outside of the domain. These points provide a boundary value for the GN equations at this end of the computational domain. The values of the three independent variables at i = 1, 2, and 3 are updated at each time step for simulating a wave-maker on the left-hand edge of the computational domain.

Boundary Conditions

In the numerical wave-tank, we can simulate two types of boundary conditions on the fight side of the domain: a reflection boundary condition and an open boundary condition. If waves are perfectly reflected from the fight-side boundary, we require that the boundary at i = ns + 2 act as a fluid "mirror." For


instance, we set 13 Ins+3 = 13 [ns+l for the points outside of the fluid domain on the right hand side, and likewise 13 [ns-~ = 13 Ins. We also set similar conditions on the

horizontal velocit~r u o Ins+3 = -u o ~s+l' Uo Ins+4 = -ul Ins' and U 1 Ins+3 = - u 1 Ins+l, and u 1 Ins+4 =-Uo Ins. Consequently, at the reflection boundary the wave amplitude is symmetric and the fluid velocity is anti-symmetric, resulting in zero fluid velocity at the reflected boundary. The reflection boundary is exact if the boundary is vertical and applies equally to linear and nonlinear waves.

The treatment of an open boundary for simulation of a situation, where waves must continue to propagate unfettered beyond the computational domain, is tricky and difficult. No exact equivalent for this boundary condition is known for nonlinear water waves. In other simulations, a variety of different methods have been used. We adopt for our purposes here the Schr6dinger condition, where the wave celerity is assumed known and is constant at the fight-side boundary. In this case, one can write simple expressions for the values of the three independent variables at the fight-side edge of the computational domain and the two points just outside of the domain. Tests have indicated that this condition simulates an open boundary condition for a considerable interval of time after the wave train impinges on this boundary, but for nonlinear waves, it eventually breaks down and leads to instabilities. There are two reasons for this to occur. First, instabilities result when there is the lack of conservation of mass in the computational domain. Second, instabilities result from anomalous wave-breaking with no rule of thumb that predicts when the wave breakdown might occur. To circumvent this difficulty in forming the open boundary condition for this model, the nonlinear celerity of waves of the same period and height as the incoming waves is computed for the water depth where the condition is enforced. Tests using this condition have been conducted and the results appear to be satisfactory for quite a long time after wave trains impinge on the boundary. The model boundary conditions have been successfully tested for a number of practical problems [5].

Spatial and Time Integration

For spatial integration of the GN equations, we use the Thomas algorithm as outlined earlier in this chapter. This algorithm involves several vector and matrix operations and the evaluation of coefficients corresponding to the discrete form of the partial differential equations. The solution obtained by this algorithm provides the first time derivatives of the three independent variables at each point in the computational domain. A second-order accurate modified Euler integration is then used to determine the value of the variables at the next time step. Tests have indicated that the flow near the wave-maker and near the right-hand boundary (if an open boundary is used) suffer from an alternate point instability, and if not corrected, the values of the three independent variables tend to oscillate high and low at neighboring points on the finite difference grid. A modest three-point averaging is used for smoothing only the first and last few points in the computational domain, which is sufficient to eliminate this cosmet-


ic blemish. The integration continues until either the maximum specified simulation time is reached or until a wave breaks; the latter is exhibited by a failure of the matrix inversion in the Thomas algorithm.

To test the overall applicability of the GN Level II theory for shallow-water waves, we have re-examined several examples from earlier studies which have used GN equations [8, 26]. These exercises were useful for checking the coding and fine-tuning the model. Upon confirming that the new model is able to reproduce the past works, we then moved on to test our model for a number of more complicated problems of interest to coastal engineers. Detail of the worked examples are described in a separate report [5] and in the subsequent publications; see References. In the interest of space, only two examples will be briefly described here.

Examples

Example 1" Wave-Structure Interaction--Reflection of Waves from a Structure

Example 1 is a test for evaluating the general applicability of our model for wave-structure interaction applications. The schematic in Figure 2 shows the geometry of a structure. With the vertical scale exaggerated, Figure 2 may represent water downstream of an hydraulic structure (say, tip of a jetty or a sluice gate) or a wall for flood control purposes. The coordinate x is measured from the left end of the domain. In the first 1,000 ft of the domain, the water depth is 30 ft downstream of the structure. This is followed by a smooth beach section of 200 ft in front of the structure. Water depth at the structure is 10 ft. This particular simulation examines 4-ft-high waves of 4-sec period progressing from deep

z t 1200'

10'

30'

X v

Figure 2. Configuration for Example 1" numerical simulation of waves with a structure. Distance x is measured from left boundary; structure is located at x = 1,200 ft. Waves start at the left boundary and move to the right.


water up to the structure and the subsequent wave reflection from the structure. The questions of interest may include how high will waves be at the structure, what forces will they exert on the structure, and where will the center of pressure of these forces be? We will see shortly that these and other practical questions can be investigated using this numerical wave-tank.

We select the computational domain to be 1,200 ft long (Figure 2) that provides us to observe the evolution of a large number of waves in the numerical tank and allows for a long period of simulated time to transpire before reflected waves from the structure begin to impact the wave-maker. Snapshots from our simulation are provided in Figures 3-7 which show that waves in the deep part of the tank are relatively short (about 80 ft long) and appear very much like the typical deep-water waves, which are sinusoidal. Waves near the structure at water depth of 10 ft look like the typical shallow-water waves that have sharper crests and long, flatter troughs. Some features of wave evolution are captured in Figures 3, 4, and 5 as snapshots of the water surface elevation at 10-sec simulation time interval. Before the wave train reaches the structure, the leading wave is noticeably higher than those that follow it. This is a direct result of the dispersive character of the finite depth waves. Figures 4 and 5 show that when the first large incident wave impacts the structure, it produces even larger waves over 10 ft in height. The primary large incident wave is then reflected and all subsequent waves in the train result in waves with an average of 8 ft in height. The envelope

6.0

4.0

2.0

0,0

-2.0

-4.0

I ! ,.,o=~ . = I I I

.................................. i ................................................................................................... i .................................................................

i

6,0

cO. 4.0

2.0

0.0

-2.0

-4.0 0 2OO 40O 6OO 8OO 1000 1200

x

Figure 3. Approaching wave profiles at t = 70, 80, 90 and 100 sec representing early development of waves in Example 1.

The Green-Naghdi Theory of Fluid Sheets for Shal low-Water Waves 4 5

' A k A r~ A-A~wb~,~--/~-/t i i i i .

6.o t l eo 1 ! ! A ]_ / 4 .0 . , ,

. . . . . . ~ ~ A A // l i ~LEL_/O ,.o j f l / \ i/\ / \ /N / \ / \ / \ / \ / I / ~ I~ 1T- /U_~ o.o

.... ~ v v v ~ v v v x J \J ~ J / v _ k ~ __V_ ~

. . . . . . , , Fia v~mvvvv! i iv';Ulill} JVxz.v~ -

6 .0 I I I r I ',,, 1NIOr ] l i / % i i

,.o ..... ~ l i "i ...... i . . . . . . . . . . .

0 2 o 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 lc

Figure 4. Wave profiles at t = 150, 160, 170, and 180 sec showing reflection of waves from the structure.

8 . 0 g a u g e 1 I / s . O ........................................................................................................................... l ............................................................. L ...........................................................

4 . 0 ........................................................................................................................... i .............................................................. f ........................................................... t

............................................................ ~ ............................................................ J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~. .................................................... 2 . 0

0 . 0 ' ' '

2 . 0 .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i .............................................................. i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 . 0 l I i

I . . . . g a u g e 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

......................................................................................................................................................................................... ~ ...........................................................

] 8 0 i i

6 0

oO . . . . . . . . . i ii i 0 . 0

- 2 . 0

- 4 . 0 1 I : 0 4 5 9 0 1 3 5 1 8 0 time

Figure 5. Wave elevations at three gauge locations, x = 800, 100, and 1,200 ft for the entire simulation time.

4 6 Offshore Engineering

,--, 2 0 0 0 . 0 r i I / gauge 1 atx-800' t .~'~ I-F iii i~ ~ ~ i ~ ~ ~ ~ i ~ ~ ~ ~ i ~ ~ ] ~ : , . . . . . . iiiiiiiiiiiiiiiiii 1 8 0 0 . 0

............................ ............................. i .............................................................. ! i 1 6 0 0 . 0 ............................................................ i .............................................................. i .............................................................. ~ ...........................................................

: i i

............................................................ i .............................................................. i .............................................................. i ............................................................ 1 4 0 0 . 0 I I 1

2 0 0 0 . 0 g a u g e 2 at x = 1 0 0 0 '

1 8 0 0 . 0

~o00 ~ iii ii •••••••••••••••••i•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••[••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••i••••••••••••••••••••••••••••••••••••••••••••••••••••••••••; 1 4 0 0 . 0 I I 1

-'-" 9 0 0 . 0 ! c~

oo.o . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 0 0 . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

300.0 I I 0 45 90 135 time 180

Figure 6. Dynamic pressures predicted at 2 ft off bottom at three gauge locations, x = 800, 100, and 1,200 ft.

7 5 0 0 . 0

....,,.

5 0 0 0 . 0 o

2 5 0 0 . 0

""i .............. i .............. ! ............... i m

2

0

-2

o.o ................................................... i ................................................................................................................................................................ 4 = u)

e~

0

-25oo.o - 6 r

o.. N

- 5 0 0 0 . 0 I I 8

1 0 0 1 0 1 2 0 1 3 0 1 4 0 150 160 170 180

time

Figure 7. Total force (integrated pressure) and center of pressure t ime history at x = 1,200 ft.


of these waves oscillates over a period of about five waves. The snapshot at t = 180 shows that the reflected waves are beginning to impinge on the wave-maker and an examination of the wave closest to the structure indicates the beginning of an instability resulting from this impingement. Further continuation of the simulation past this point is not meaningful. Interestingly, we should point out that a similar situation would also be encountered in a physical wave tank of limited size.

The time history of wave amplitude at three gauge locations positioned at x = 800, 1,000, and 1,200 ft, the last being at the structure, is shown in Figure 5. The standing wave pattern in parts of Figure 5 is a consequence of wave reflection from the structure and comparison of three gauges shows in particular that the second gauge is very nearly at a node. The time history from a pressure gauge placed 2 ft above the bottom is shown in Figure 6. This history resembles that of the wave elevation time history except that some frequency doubling can be seen once standing waves are formed. Figure 7 shows history of the total force per foot of the structure width and the location of pressure center (Po and pl/po, respectively in our equations). These results show that wave reflection from a structure placed a short distance immediately following a sloping beach can produce large standing waves near the structure, which give rise to some large oscillatory forces on the structure. The hydrodynamic effect of a sloping seabed near the toe of a breakwater, jetty, or seawall may have also been treated as a substantial berm placed in front of these structures. Wave simulations for sufficiently long time periods may be necessary for discerning slope effects for design and estimating wave forces on structures; it may not be possible to accurately estimate these effects from results of short simulations of limited duration.

Example 2: Steep Nonlinear Waves---Shoaling Waves on Planar Beach

This example is test for validation of the model predictions with the data from a laboratory experiment. In this case, numerical simulations of our GN model are compared to the measurements from a laboratory study for steep, near-breaking, nonlinear waves. As waves approach a beach, they steepen, become unsym- metrical and eventually break. Like all other inviscid theories, the GN Level II theory cannot predict the evolution of waves beyond the breaking point; our model can predict the evolution almost up to breaking point. Shoaling waves were also studied in a thesis by Shields [26] in an attempt to replicate the experiments conducted at the Technical University of Denmark (Hansen and Svendsen 1979). A small sketch of the test situation is shown at the top in Figure 8. Regu- lar waves are created in a tank 25 m long with a beach which begins 13 m downstream of the wave-maker. The beach has a slope of 1:34.26. The water depth at the wave-maker is 0.360 m. For purposes of our numerical modeling, the bottom of beach is not allowed to pierce the water surface and is kept at a water depth of

4 8 Offshore Engineering

0.040 m at a distance of 0.02396 m from end of the beach. The numerical simulation is driven with a train of regular waves generated at the wave-maker that has a height of 40 mm and a period of 3.33 sec.

Five wave profiles at times from 17.2 sec to 18.4 sec are displayed in Figures 8 and 9. We terminated our simulation when the first broken wave occurred approximately at 18.5 sec. Wave profiles near the beach at 0.1 sec intervals are captured in Figure 9 as solid lines. The model output data (shown as small squares in Figure 9) were fitted with spline curves to yield smooth wave shapes. Results of this simulation show the sharpening and steepening of the wave and the development of a "shoulder wave" just behind the main crest. For this partic-

L . 360jmm 2 5 m I

,-1~" ~ 13m "J-I \ .lop. - : ~.26

=:'o: F olo. ~"i?iii-_~;2;_;2-_-I.i ............. ;222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- 0 . 021 ~ l ......

1 . 1 7 . 5 s e c ! ............................ ~ ......................... ~ ............................................................... ~ . . . . . . . . . . . . . . . . i = z

. . . . . . . . . . . . . . . . . . . . . . . . . . . .~ . . . . . . . . . . . . . . . . . - - 1 - . . . . . . . . . . . . . . . . . . . . . . . . T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~i . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ............................ ! . . . . . .

0.10 1" | t . . . . . . | '

i i

~176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - A . . . . . . . . . . .

o.o,~ .......................................... ! ............................................. i ............................................ i ............................................. i ....................... -it .............. 1

m E

0 .02 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t = 18.1 sec I , ~ ~ i i i i i t

0 . 1 0 , / I I 0 . 0 8 t - - ........ ! ...=....!...l~....4 .....se...c ......... c ............................................. -~ ............................................. ! .............................................. ! ............................... * ........ q

~ 1 7 6 t-" .......................................... i ............................................ 4 ............................................. i ............................................. i ............................. II' ....... "-I 0 . 0 4 I_ .......................................... j ............................................. i ............................................. i ............................................. i ....................................

o.oo 0.02 ~ i i ) i i i i ! ~ i ) i i i i ) ~ i i ! i ! i ) ~ ~ ................................ i ................................... i ................................... i .................................... i .....................................

- 0 . 02 0 5 10 15 2 0 2 5

X

Figure 8. Wave profile snapshots at t = 17.2, 17.5, 17.8, 18.1, and 18.4 sec. These snapshots correspond to waves in tank just before the first wave breaks in Example 2.


0.1 O0

~ I 1 0.080 ........................................................................................................................................................... : ...................................................................................... I

0.060

0 . 0 4 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.020

. . . . . . . . t . . . . oooo

-0.020 .................................................. T ...................................................... ~ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

j f i I

- 0 . 0 4 0 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ...................................................... "., ...................................................... t ...................................................... I ..............................................

.............................................. iiiii i ...................................................... i ...................................................... ............................................ i .................................................... -0.1 O0 1 l [

21.5 22 22.5 23 23.5 24 X

Figure 9. Shoaling wave profiles at 0.1 sec apart as the first wave approaches the beach (1:34.26 slope)in Example 2.

ular wave train, the comparison of model curves to the laboratory traces was so good that it was impossible to differentiate one from the other. Comparison for two other wave conditions was not as good; the maximum differences of 14 and 20% were observed. Overall, the present results are similar in form to those obtained by Shields (1986), but there are some differences. Our simulation shows waves break farther up the beach from the point reported by Shields. In addition, wave profiles from the present simulation match better, both in amplitude and form, to the measurements of Hansen and Svendsen (1979).

In this simulation, the first wave in the train was purposely attenuated (or "ramped") as discussed earlier. We found that the ramping was necessary for most practical applications and that computations should not be continued when the first full wave breaks in the simulation. We also noticed that in this simulation although the subsequent waves in the wave train had the required height of 40 mm, the first wave was not high enough. As a result, the lower wave height caused the wave to break farther up the beach from the point reported in the measurements by Hansen and Svendsen (1979). It is possible to get around this problem by starting the simulation with an already developed train of regular waves and letting waves progress on shore. This is the approach Shields [26] has used in his thesis. This way his first wave was at the full height and he was able to compare directly with the experimental results. We chose not to tinker with the input waves in this simulation. Overall, this simulation showed that our Level II GN model was performing correctly and that it was able to reproduce


the measurements for near-breaking shoaling waves on a planar beach. Further validation of this model with field and laboratory data will be necessary for identifying its strengths and weaknesses.

Conclusion

In this chapter, we have presented details related to the development and some of the philosophy behind the GN theory of fluid sheets. The fundamental principles of the Level I and Level II GN theories have been described in detail here for completeness because GN theory is relatively new to most coastal engineers. Due to complexities of the GN theory and high level of mathematics involved, derivation of both the Level I and Level II theories are presented here in a systematic manner. The derivation presented here was first made manually, and subsequently it was verified with the powerful symbolic manipulator Mathe- matica TM. A modular Fortran program has been developed for shallow-water Level II theory based on the equations outlined in this chapter. The 2-D numerical model developed has been applied to solutions of many engineering applications. The numerical model allows for a general description of the sea bed profile and has two types of boundary conditions, open or reflective, for representing the down-wave beach end of the domain.

It will be necessary in the future to develop the Level III GN theory for coastal applications. The Level II theory described here can be extended to 3-D flows. The 3-D GN equations will be considerably more complex although these are not beyond the human capabilities at the present time due to the availability of extremely powerful mathematical and symbolic manipulators. The 3-D GN theory is particularly suited for coastal engineering applications because this theory can represent the combined effects of amplitude nonlinearities, frequency dispersion, refraction, shoaling, reflection, and diffraction of waves propagating over arbitrary seabed topography. The GN theory does this without introducing a scale parameter and consequently the problems of wave breaking and re-formation may realistically be represented by a 3-D time-dependent GN theory. The solution algorithm for 3-D dimensional flows will have to be redeveloped to accommodate the new domain.

The only drawback of the GN theory is perhaps the complexity of its mathematical formulation. There are other wave theories that are not as complex as GN theory. Boussinesq theory is one that is widely used for wave modeling in shallow water. It is appropriate to note that Boussinesq-type equations can be used to describe only weakly nonlinear and weakly dispersive wave propagation in shallow water. For propagation of short waves in intermediate depths, Boussi- nesq equations fail to describe wave evolution correctly and quite often these equations become unstable, numerically generating spurious short waves that cause erroneous results; these instabilities also occur in the shallow-water region. This behavior of the Boussinesq equations requires one to use different


equations in different water depths and somehow match them, which is a formi- dable task. This is not surprising because the Boussinesq equations are based on the assumption that the weak nonlinearity represented by the ratio of wave amplitude to water depth, a/d is in the same order of magnitude as the frequency dispersion represented by the square of the ratio of water depth to wavelength, (d/L) 2. It is this assumption that imposes a major limitation on Boussinesq-type equations, forcing these to strictly be applicable for relatively shallow depths. Generally, standard Boussinesq equations require water depth to be less than about one-fifth of the equivalent deep-water wavelength.

On the other hand, higher-order and weakly nonlinear Boussinesq models may offer an alternative to the assumption- and scale-free GN models, provided that two major obstacles related to Boussinesq models can be overcome. First, dispersive characteristics of the Boussinesq equations must be improved significantly in order to extend the range of applicability of these equations at least to the moderate water depth limit. This improvement is necessary for Boussinesq models to be used up to 30 m depths. Second, it is imperative that some computationally efficient Boussinesq models be constructed for practical problems because these usually require wave simulation over large regions for several days. The modified Boussinesq models with improved linear dispersion properties further subject to the parabolic approximation assumption (i.e., waves must propagate in a dominant direction) appear presently to be the only Boussinesq- type equations that can meet both requirements (Chen and Liu 1995). The parabolic approximation that converts an elliptic equation into a parabolic equation, reduces the computational efforts dramatically and eliminates the need of specifying the down-wave boundary conditions, which usually are unknown a priori

for most coastal applications. These gains are attainable at the expense of reduced accuracy of the Boussinesq models.

In closing, we have provided a bibliography in addition to references. These items are not directly mentioned in the text, but we found them quite useful in the course of our research and therefore decided to list them here.

References

1. Boussinesq, J., 1877. Essai sur la theorie des eaux courantes. Mem Pres. Acad. Sci. Paris Ser. 2 23, 1-680.

2. Chen, Y. and Liu, P. L.-F., 1995. Modified Boussinesq Equations and Associated Parabolic Models for Water Wave Propagation. J. Fluid Mech. 288, pp. 351-381.

3. Cokelet, E. D., 1977. "Steep Gravity Waves in Water of Arbitrary Uniform Depth." Phil. Trans. Roy. Soc. London A 286, pp. 183-230.

4. Demirbilek, Z. and Webster, W. C. 1992a. "Application of the Green-Naghdi The- ory of Fluid Sheets to Shallow-Water Waves. Report 1. Model formulation. US Army Wat. Exp. Sta., Coastal Engng. Res. Cntr. Tech Rep. No. CERC-92-11, Vicksburg, MS, 45 p.


5 . 1992b. User's Manual and Examples for GNWAVE. US Army Wat. Exp. Sta., Coastal Engr. Res. Cnjtr. Tech Rep. No. CERC-92-13, Vicksburg, MS, 55 p.

6 . 1993a. "Evolution of Time-Dependent Nonlinear Shallow-Water Waves." Advances in Hydro-Science & Engineering (ed. S. Y. Yang) I, pp. 1536-1544, Washington DC.

7 . 1993b. "Nonlinear Theories of Steady and Unsteady Shallow-Water Waves. "Invited presentation to the MEET'N'93 Conf., Univ. of Virginia, Char- lottesville, VA, June 6-9, 1993.

8. Ertekin, R. C., 1984. "Soliton Generation by Moving Disturbances in Shallow Water." Ph.D. thesis. Univ. of California, Berkeley.

9. Ertekin, R. C., Webster, W. C., and Wehausen, J. V., 1984. "Ship-Generated Soli- tons." Proc. 15th Syrup. Naval Hydrodyn., Hamburg, pp. 347-361.

1 0 . 1986a. "Waves Caused by a Moving Disturbance in a Shallow Channel of Finite Width. J. Fluid Mech. 169, pp. 275-292.

11. Ertekin, R. C. and Wehausen, J. V., 1986b. "Some Soliton Calculations." Proc. 16th Symp. Naval Hydrodyn., Berkeley, pp. 167-184.

12. Fenton, J. D., 1979. "A High-Order Cnoidal Wave Theory." J. Fluid Mech. 94, pp. 129-161.

13. Green, A. E., and Naghdi, P. M., 1984. "A Direct Theory of Viscous Fluid Flow in Channels." Archive for Rational Mechanics and Analysis 86, pp. 39-63.

1 4 . 1986. "A Nonlinear Theory of Water Waves for Finite and Infinite Depths. Philos. Trans. Roy. Soc. London A 320, pp. 37-70.

15.1987. "Further Developments in a Nonlinear Theory of Water Waves for Finite and Infinite Depths." Philos. Trans. Roy. Soc. London A 324, pp. 47-72.

16. Green, A. E., Laws, N., and Naghdi, P. M., 1974. "On the Theory of Water Waves." Proc. Roy. Soc. London A 338, pp. 43-55.

17. Kantorovich, L. V. and Krylov, V. I., 1958. Approximate Methods of Higher Analysis. P. Noordhoff Ltd., Groningen, The Netherlands.

18. Madsen, P. A., et al., 1991. "A New Form of the Boussinesq Equations with improved Linear Dispersion Characteristics." Coastal Engng 15, pp. 371-388.

19. Madsen, P. A. and Sorensen, O. R., 1992. "A New Form of the Boussinesq Equa- tions with Improved Linear Dispersion Characteristics. Part 2: A Slowly-Varying Bathymetry." Coastal Engng 18, pp. 183-204.

20. Marshall, J. S. and Naghdi, P. M., 1990. "Wave Reflection and Transmission by Steps and Rectangular Obstacles in Channels of Finite Depth." Theoretical and Computational Fluid Dynamics 1, pp. 287-301.

21. Mei, C. C. and L6Mehaut6, B., 1966. "Note on the Equations of Long Waves Over an Uneven Bottom." J. Geophy. Res. 71, pp. 393-400.

22. Naghdi, P. M. and Rubin, M. B., 1981. "On the Transition to Planing of a Boat." J. Fluid Mech. 103, pp. 345-374.

23. Nwogu, O., 1993. "Alternative Form of Boussinesq Equations for Nearshore Wave Propagation." J. Waterway, Port, Coastal Ocean Engng, ASCE 119, pp. 618-638.

24. Peregrine, D. H., 1967. "Long Waves on a Beach." J. Fluid Mech. 27, pp. 815-827.

25. Schwartz, L. W., 1974. "Computer Extension and Analytic Continuation of Stokes' Expansion for Gravity Waves." J. Fluid Mech. 62, pp. 553-578.


26. Shields, J. J., 1986. "A Direct Theory for Waves Approaching a Beach." Ph.D. thesis, Univ. of Calif. Berkeley, 137 p.

27. Shields, J. J. and Webster, W. C., 1988. "On Direct Methods in Water-Wave Theo- ry." J. Fluid Mech.197, pp. 171-199.

2 8 . 1989. "Conservation of Mechanical Energy and Circulation in the Theory of Inviscid Fluid Sheets." Journal of Engng. Math. 23, pp. 1-15.

29. Webster, W. C. C. and Kim, D.-Y., 1990. "The Dispersion of Large-Amplitude Gravity Waves in Deep Water." Proc. of the 18th Symp. on Naval Hydro., pp. 134-146.

30. Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R., 1995. "A Fully Nonlinear Boussinesq Model for Surface Waves. Part 1: Highly Nonlinear Unsteady Waves." J. Fluid Mech. 294, pp. 71-92.

31. Wu, T. Y., 1981. "Long Waves in Ocean and Coastal Waters." J. Eng. Mech. ASCE 107, pp. 501-522.

Bibliography

Benjamin, T. B., 1967. "Instability of Periodic Wave Trains in Nonlinear Dispersive Sys- tems." Proc. Roy. Soc. A. 299, pp. 59--67.

Benjamin, T. B. and Feir, J. E., 1967. "The Disintegration of Wave Trains on Deep Water. Part 1." J. Fluid Mech. 27, pp. 417-430.

Fenton, J. D. and Mills, D. A., 1976. "Shoaling Waves: Numerical Simulation of Exact Equations." Lecture Notes in Physics, IUTAM Symp., pp. 94-101.

Hansen, J. B. and Svendsen, Ib A., 1979. "Regular Waves in Shoaling Water: Experimen- tal Data." Series Paper No. 21, Inst. of Hydrodynamics and Hydraulic Engr., Tech. Univ. of Denmark.

Kaup, D. J. and Newell, A. C., 1978, "Solitons as Particles, Oscillators, and in Slowly Changing Media: A Singular Perturbation Theory." Proc. Roy. Soc. London A 361, pp. 413--446.

Lake, B. M., Yuen, H. C., Rungaldier, H., and Ferguson, W. E., 1977. "Nonlinear Deep- Water Waves: Theory and Experiment. Part 2: Evolution of a Continuous Train." J. Fluid Mech. 83, pp. 49-74.

Lamb, H., 1932. Hydrodynamics. 6th ed., Cambridge University Press, Cambridge, (1945, Dover, New York).

Longuet-Higgins, M. S. and Cokelet, E. D., 1976. "The Deformation of Steep Surface Waves on Water. I: A Numerical Method of Computation." Proc. Roy. Soc. London A 350, pp. 1-26.

Madsen, O. S. and Mei, C. C., 1969. "The Transformation of a Solitary Wave Over an Uneven Bottom." J. Fluid Mech. 39, pp. 781-791.

Mei, C. C., 1983. Applied Dynamics of Ocean Surface Waves. Wiley, New York. Miles, J. W., 1979. "On the Korteweg-de Vries Equation for a Gradually Varying Chan-

nel." J. Fluid Mech. 91, pp. 181-190. 1980. "Solitary Waves." Ann. Reviews of Fluid Mech. 12, pp. 11-43.

Munk, W. H., 1949. "The Solitary Wave and its Application to Surf Problems." Ann. N.Y. Acad. of Sci. 51, pp. 376---424.

54 Offshore Engineedng

Newman, J., 1968. "Numerical Solution of Coupled, Ordinary Differential Equations." lndust, and Engr. Chem. Fundamentals 7, pp. 514-517.

Peregrine, D. H., 1983. "Breaking Waves on Beaches." Ann. Rev. of Fluid Mech. 15, pp. 149-178.

Rienecker, M. M. and Fenton, J. D., 1981. "A Fourier Approximation Method for Steady Water Waves." J. Fluid Mech. 104, pp. 119-137.

Stiassnie, M. and Peregrine, D. H., 1980. "Shoaling of Finite-Amplitude Surface Waves on Water of Slowly Varying Depth." J. Fluid Mech. 97, pp. 783-805.

Svendsen, Ib. A. and Hansen, J. B., 1978. "On the Deformation of Periodic Long Waves Over a Gently Sloping Bottom." J. Fluid Mech. 87, pp. 433-448.

Vinje, T. and Brevig, P., 1981. "Numerical Simulation of Breaking Waves." Adv. Water Resour. 4, pp. 77-82.

C H A P T E R 2

THE CNOIDAL THEORY OF WATER WAVES

John D. Fenton

Department of Civil and Environmental Engineering The University of Melbourne Parkville, Victoria, Australia

CONTENTS

INTRODUCTION, 55

BACKGROUND, 57

CNOIDAL THEORY, 59

PRESENTATION OF THEORETICAL RESULTS, 66

PRACTICAL APPLICATION OF CNOIDAL THEORY, 73

PRACTICAL TOOLS AND HINTS FOR APPLICATION, 80

A NUMERICAL CNOIDAL THEORY, 86

ACCURACY OF THE METHODS, 90

NOTATION, 96

REFERENCES, 98

Introduction

Throughout coastal and ocean engineering the convenient model of a steadily- progressing periodic wave train is used to give fluid velocities, pressures, and surface elevations caused by waves, even in situations where the wave is being slowly modified by effects of viscosity, current, topography, and wind, or where the wave propagates past a structure with little effect on the wave itself. In these situations the waves do seem to show a surprising coherence of form, and they can be modeled by assuming that they are propagating steadily without change, giving rise to the so-called steady wave problem, which can be uniquely specified and solved in terms of three physical length scales only: water depth, wavelength, and wave height. In practice, the knowledge of the detailed flow struc-

55


ture under the wave is so important that it is usually considered necessary to solve accurately this otherwise idealized model.

The main theories and methods for the steady wave problem that have been used are Stokes theory, an explicit theory based on an assumption that the waves are not very steep and is best suited to waves in deeper water; cnoidal theory, an explicit theory for waves in shallower water; and Fourier approximation methods, which are capable of high accuracy but which solve the problem numerically and require computationally-expensive matrix techniques. A review and comparison of the methods is given in [42] and [ 16]. For relatively simple solution methods that are explicit in nature, Stokes and cnoidal theories have important and complementary roles to play, and indeed it has relatively recently been shown that they are more accurate than has been realized [ 16].

This chapter describes cnoidal theory and its application to practical problems. It has probably not been applied as often as it might. One reason is the unfamiliarity of the Jacobian elliptic functions and integrals and perceived difficulties in dealing with them. One possibility might be too that in the long wave limit for which the cnoidal theory is meant to apply, almost all expressions for elliptic functions and integrals given in standard texts are very slowly convergent (for example, those in [ 1 ]). Both of these factors need not be a disincentive; relatively recently some remarkable formulas have been given [ 18] that are simple, short, and converge most quickly in the limit corresponding to cnoidal waves. These will be presented below.

It may be, however, that I have inadvertently provided further reasons for not preferring cnoidal theory. In [13] I presented a fifth-order cnoidal theory that is both apparently complicated, requiring many coefficients as unattractive floating point numbers, and also gave poor results for fluid velocities under high waves. A later work [ 16], however, showed that instead of fluid velocities being expressed as expansions in wave height, if the original spirit of cnoidal theory were retained and they be written as series in shallowness, then the results are considerably more accurate. Also, that work showed that, in the spirit of Iwagaki [23], the series can be considerably shortened and simplified by a good approximation.

There have been many presentations of cnoidal theory, most with essentially the same level of approximation, and with relatively little to distinguish the essential common nature of the different approaches. However, there has been such a plethora of different notations, expansion parameters, definitions of wave speed, and so on, that the practitioner could be excused for thinking that the whole field was very complicated. The aim of this chapter is to review developments in cnoidal theory and to present the most modem theory for practical use, together with a number of practical aids to implementation. My hope is that this surprisingly simple and accurate theory becomes accessible to practitioners and regains its rightful place in the study of long waves.

Initially, the chapter presents a history of cnoidal theory, and describes and reviews various contributions. It then outlines the theory that can be used to gen-

The Cnoidal Theory of Water Waves 57

erate high-order solutions and presents theoretical results from that theory. This chapter contains the first full presentation of those results in terms of rational numbers, as previous versions have used some floating point numbers. It presents two forms of the theory: The first is a full third-order theory; the second is a fifth-order theory in which a coherent approximation is introduced that, it is suggested, is accurate for most uses of cnoidal theory. It is suggested that this be termed the "Iwagaki Approximation." Next it presents a detailed procedure for the application of the cnoidal theory, allowing for cases where wavelength or period is specified. The chapter then introduces some new simplifications and presents several practical tools and hints for the application of the theory, including a numerical check on the coefficients used, a simple test to check that the series are correct as programmed by the user, some simple approximations for the elliptic functions and integrals used, and techniques for convergence enhancement of the series. The chapter also presents a numerical cnoidal theory, which is a numerical method based on cnoidal theory. Finally, it examines the accuracies of the methods and suggests appropriate limits.

Background

There have been many books and articles written on the propagation of surface gravity waves. The simplest theory is conventional long-wave theory, which assumes that pressure at every point is equal to the hydrostatic head at that point, and which gives the result that any finite amplitude disturbance must steepen until the assumptions of the theory break down. In 1845 John Scott Rus- sell presented unsettling evidence that this is not the case when he published his observations on the "great solitary wave of translation" generated by a canal barge and seeming to travel some distance without modification. This was derid- ed by Airy ("We are not disposed to recognize this wave as deserving the epi- thets 'great' or 'primary' . . . , " [37]) who believed that it was nothing new and could be explained by long-wave theory. This is one aspect of the long-wave paradox, later resolved by Ursell [45], who showed the importance of a parameter that incorporates the height and length of disturbance and the water depth in determining the behavior of waves. The value of the parameter determines whether they are true long waves and show the steepening behavior, or whether they are "not-so-long" waves where pressure and velocity variations over the depth are more complicated, as is their behavior. The cnoidal theory fits into this latter category.

In 1871 Boussinesq [4] and in 1876 Rayleigh [37] introduced an expansion based on the waves being long relative to the water depth. They showed that a steady wave of translation with finite amplitude could be obtained without making the linearizing assumption, and that the waves were inherently nonlinear in nature. The solutions they obtained assumed that the water far from the wave was undisturbed, so that the solution was a solitary wave, theoretically of infinite


length. Cnoidal theory obtained its name in 1895 when Korteweg and de Vries [27] obtained their eponymous equation for the propagation of waves over a fiat bed, using a similar approximation to Boussinesq and Rayleigh. However, they obtained periodic solutions they termed "cnoidal" because the surface elevation is proportional to the square of the Jacobian elliptic function cn(). The cnoidal solution shows the familiar long, fiat troughs and narrow crests of real waves in shallow water. In the limit of infinite wavelength, it describes a solitary wave.

Since Korteweg and de Vries there have been several presentations of cnoidal theory. Keulegan and Patterson [26], Keller [25], and Benjamin and Lighthill [3] have presented first-order theories. The latter is particularly interesting, in that it relates the wave dimensions to the volume flux, energy and momentum of a flow in a rectangular channel (or per unit width over a flat bed) and showed that waves of cnoidal form could approximate an undular hydraulic jump. Wiegel [46, 47] gave a detailed presentation of first-order theory with a view to practical application, including details of mathematical approximations to the elliptic integrals. Laitone [28, 29] presented a second-order cnoidal theory in a formal manner, which provided a number of results, recasting the series in terms of the wave height/depth. However, the second-order results are surprisingly inaccurate for high waves (see, for example, [30]). The next approximation was obtained by Chappelear [7], as one of a remarkable sequence of papers on nonlinear waves. He obtained the third-order solution, and expressed the results as series in a parameter directly proportional to shallowness: (depth/wavelength) 2.

Iwagaki published his "Hyperbolic theory" in 1967, with an English version appearing a year later [23]. This was an interesting development, for it was an attempt to make the computation of the elliptic functions and integrals simpler by replacing all of them by their limiting behaviors in the limit of solitary waves, except for quantities related to wavelength. In this case, the cn function becomes the hyperbolic secant function sech, and other elliptic functions become other hyperbolic functions, giving rise to the name he proposed. This approach raises several interesting points, and it is further discussed below.

Tsuchiya and Yasuda in 1974, with an English version in 1985 [44], obtained a third-order solution with the introduction of another definition of wave celerity based on assumptions concerning the Bernoulli constant. In 1977 Nishimura et al. [33] devised procedures for generating high-order theories for Stokes and cnoidal theories, making extensive use of recurrence relations. The authors concentrated on questions of the convergence of the series. They computed a 24th- order solution, however, few detailed formulae for application were given.

Fenton [ 13] produced a method in 1979 for the computer generation of high- order cnoidal solutions for periodic waves. It had been observed that second- order solutions for fluid velocity were quite inaccurate [30], and it was desired to produce more accurate results, as well as try to make the method more readily available for practical application. As with Laitone's effort, results were expressed in terms of the relative wave height. The paper also raised some interesting points: How it is rather simpler to use the trough depth as the depth scale


in presenting results, and that the effective expansion parameter is not simply the wave height but is actually the wave height divided by the elliptic parameter m. For the expansion parameter to be small and for the series results to be valid, the short wave limit is excluded. In this way the cnoidal theory breaks down in deep water (short waves) in a manner complementary to that in which Stokes theory breaks down in shallow water (long waves) [14]. A solution was presented to fifth order in wave height, with a large number of numerical coefficients in floating point form. For moderate waves, results were good when compared with experiment, but for higher waves the velocity profile showed exaggerated oscillations and it was found that ninth-order results were worse. These results were unexpectedly poor.

Isobe et al. [22] continued the work of Nishimura et al. [33] and presented a unified view of Stokes and cnoidal theories. They proposed a generalized double series expansion in terms of Ursell parameter and shallowness, the square of the ratio of water depth to wavelength. They also proposed a boundary between areas of application of Stokes and cnoidal theory of U = 25, where U is the Ursell number, H~2/d 3.

In a review article in 1990, Fenton [16] considered cnoidal theory as well as Stokes theory and Fourier approximation methods such as the "stream function method." The approach to cnoidal theory in [ 13] was reexamined and some useful advances made. It was found that if the series for velocity were expressed in terms of the shallowness rather than relative wave height, as done by Chappelear [7], then results were very much better, and justified the use of cnoidal theory even for high waves. This would fit in with the fundamental approximation of the cnoidal theory being an expansion in shallowness. That review article also incorporated the fact that the wave theory does not determine the wave speed, and that neither Stokes' first nor second definitions of velocity are necessarily correct. In general one must incorporate the effects of current, as had been done using graphical means in [24] and [20], and analytically for numerical wave theories in [38] and for high-order Stokes theory in [14].

I now present the theory and results. The theory is essentially that described in [13] but with the advances made in [16] incorporated plus some more contributions introduced in this chapter. A number of suggestions for practical use are made, and then the performance of the theory is compared with other methods. One of those is a new numerical version of cnoidal theory. In general, the theory as presented here is found to be surprisingly robust and accurate over a wide range of waves.

Cnoidal Theory

The Physical Problem

Consider the wave as shown in Figure 1, with a stationary frame of reference (x,y), x in the direction of propagation of the waves and y vertically upwards


c F

T H

T h

r

d

~c

/ v

Figure 1. Wave train, showing important dimensions and coordinates.

with the origin on the flat bed. The waves travel in the x direction at speed c relative to this frame. This is usually the frame of interest for engineering and geophysical applications. Consider also a frame of reference (X,Y) moving with the waves at velocity c, such that x = X + c t , where t is time, and y = Y. The fluid velocity in the (x,y) frame is (u,v), and that in the (X,Y) frame is (U,V). The velocities are related by u = U + c and v = V.

In the (X,Y) frame all fluid motion is steady, and consists of a flow in the negative X direction, roughly of the magnitude of the wave speed, underneath the stationary wave profile. The mean horizontal fluid velocity in this frame, for a constant value of Y over one wavelength X is denoted by -U. It is negative because the apparent flow is in t h e - X direction. The velocities in this frame are usually not important, but they are used to obtain the solution rather more simply.

Equations of Motion in a Frame Moving with the Wave

We proceed to develop higher-order solutions for the problem where waves progress steadily without any change of form. If you are more interested in results than the details of the theory, you may proceed straight to the next section "Presentation of Theoretical Results."

It is easier to consider the equations of motion in the (X,Y) frame moving with the wave such that all motion in this frame is steady. If it is assumed that the water is incompressible and the flow two-dimensional, a stream function v(X,Y) exists such that the velocity components (U,V) are given by


U = ~ and V =-/)--~ (1)

and if the flow is irrotational, ~ satisfies Laplace 's equation throughout the flow:

/)X----- T - ~ = 0 (2)

The boundary conditions are that the bottom Y = 0 is a streamline on which y is constant:

v (x, o) = 0 (3)

and that the free surface Y = TI(X) is also a streamline:

V (X, 1"1 (X)) = - Q (4)

where Q is the volume flux underneath the wave train per unit span. The negative sign is because the flow relative to the wave is in the negative X direction, such that relative to the water the waves will propagate in the positive x direction. The remaining boundary condition comes from Bernoulli 's equation:

1 (U 2 -I- V 2 ) + gy + p R (5) 2 p

where g = gravitational acceleration p = pressure p = density R = Bernoulli constant for the flow, the energy per unit mass

If Equation 5 is evaluated on the free surface Y = rl(X), on which pressure p = 0, we obtain

1 ((/)N/'~ 2 + ~ , ~ - ) + g r l = R (6)

Y=~

We assume a Taylor expansion for ~ about the bed of the form:

d df y3 d3f = - sin Y �9 f ( X ) = - Y + (7)

dX dX 3! dX 3 "'"

as in [13], where df/dX is the horizontal velocity on the bed. We have introduced the infinite differential operator sin Y d/dX as a convenient way of repre-


senting the infinite Taylor series, which has significance only as its power series expansion

d y3 d 3 y5 d 5 sin y ~ = d y +

dx dx 3! dx 3 5! dx 5 "'"

It can be shown that the velocity components anywhere in the fluid are

U=/)xlt= . . . . . /) sinY d f ( X ) = - c o s Y d - f ' (X) /)Y /)Y dX dX

V = - ~ / ) ~ - - /) sinY d . f ( X ) = s i n Y d . f ' ( X ) (8) /)X ~)X dX dX

Further differentiation shows that the assumption of Equation 7 satisfies the field equation 2 and the bottom boundary condition 3. The kinematic surface boundary condition 4 becomes

d sin rl .-:7- f (X)= Q (9)

clx

This equation is a nonlinear ordinary differential equation of infinite order for the local fluid depth 11 and f' (X), the local fluid velocity on the bed, in terms of the horizontal coordinate X.

The remaining equation is the dynamic free surface condition, Equation 6. Substituting Equation 8 evaluated on the free surface we obtain

1 ( ( d d f )2 ( d d f ) 2) - cos 11 ~ . + sin 1"1 ~ . + gq = R (10) 2 dX dX dX dX

One of the squares of the infinite order operators can be eliminated by differentiating Equation 9:

d df dr I d df dQ �9 ~ + cos 1" I . . . . . 0 s i n r l d x dX dX dX dX dX

as Q is constant along the channel, to give

~ . ~ + grl= R 1 + ~ . ~ ) cos 1"1 dX dX (11)

Equations 9 and 11 are two nonlinear ordinary differential equations in the unknowns rl(X) and f'(X), the horizontal velocity on the bed. They are of infinite order and must be approximated in some way. It is possible that they could be solved as differential equations, but that would require an infinite number of boundary conditions. This and subsequent sections use two methods, one using


power series solution methods, the traditional way; and another using a numerical spectral approach based on assuming series of known functions.

S e r i e s S o l u t i o n

The equations have the trivial solution of uniform flow with constant depth: I"1 = h and f'(X) = U. We proceed to a series expansion about the state of a uniform critical flow. We will assume that all variation in X is relatively slow and can be expressed in terms of a scaled dimensionless variable txX/h, where tx is a small quantity that expresses the relative slowness of variation in the X direction, and h is the minimum or trough depth of fluid. At this stage, while solving the equations, it is more convenient to write them in terms of dimensionless variables. Let the scaled horizontal variable be 0 = ctX/h.

Writing 1"1, = rl/h and f, = f/Q, Equation 9 can be written

1 d - - sin rl,t~ �9 f, - 1 = 0 (12)

The dynamic boundary condition (Equation 11) can be written in terms of these quantities as

-- cos r l ,a �9 f2(0) + g,rl, = R, (13) 2 k, dOJ

where g , - g h a / Q 2 is a dimensionless gravity number (actually the inverse square of the Froude number)

R, -" Rh2/Q 2 is a dimensionless Bernoulli constant

The form of Equations 12 and 13 suggests that we u s e (Z 2 as the expansion parameter. We write the series expansions

N 11, = 1 + E o~2jYj (0) (14)

j=l

N

f~ = 1 + ~ (x2JFj (0)

j=l (15)

N

g, = 1 + E 0~2jgJ j=l

(16)

3 N R, = -~ + E o~2Jrj

j=l (17)


where N is the order of solution required. Now, these are substituted into Equa- tions 12 and 13. Grouping all the terms in tz ~ o~ 2, &,..., and requiring that the coefficient equation of each be satisfied identically, a hierarchy of equations is obtained that can be solved sequentially. At o~ ~ the equations are satisfied identically; at o~ 2 we obtain

F1 + Y1 = 0

F1 + Y1 + gl - rl = 0

with solution Y1 = -F1 and gl = rl- At the next order ct 4 we obtain

1 FI,,= F2 +V2 +F Vl- 0

1 1 F2 +Y2 + g2 - r2 - ~- FI"+ ~ -F2 +glY1 = 0

Z Z

and by subtracting one from the other, and using information from the previous order, we obtain

1FI,,_ 3 -~ -~ F2 + r2 - g2 + glF1 = 0 (18)

This is a differential equation of second order, which is nonlinear because of the F21 term. The usual way of integrating such an equation (for example [4 1 ], #3.3.3) is to write the F~' term as d(F~2/2)/dF 1, integrate the equation with respect to F 1, from which the solution for F 1 in terms of cn2(01m), a Jacobian elliptic function (see, for example, [1 ], #16), can be obtained. This is a rather complicated procedure. Here we prefer a rather simpler approach to solve the nonlinear differential equation by presuming knowledge of the nature of the solution. We write

F 1 = A 1 cn 2 (01m) (19)

where A 1 is independent of 0, and m is the parameter of the elliptic function. Using the properties as set out in [ 1], (#16.9 and #16.16), reproduced as Equa- tions 40 and 4 1 below, it can be shown that

d 2 d0 2 cn2(01m) = 2 - 2 m +(8 m - 4) cn2(0lm)- 6 m cn4(0im) (20)

Substituting into Equation 1 8 and collecting coefficients of powers of cn2(01m) we obtain


4 4 8 m 1 = - ~ m, ga = -~ (1 - 2m), r 2 -- g2 = ~ m (1 - m)

J J y

such that the first-order cnoidal solution is

11 ,=1+tX24 - - m c n 2 (O[m) 3

f,~=l (x 2 4 ' - -- m cn 2 (0]m) 3

g, = l + a 1 4 ~ ( 1 - 2m)

3 ~ 2 8 R, = - - + - ( 1 - m)

2 9

(21)

These solutions should have been shown with order symbols O(tx 4) after them, showing that the neglected terms are at least of the order of t~ 4. As this is obvious anyway, we choose not to do that here or elsewhere in this work, where the order of neglected terms is almost everywhere obvious.

The procedure described here can be repeated at all orders of t~ 2, and at each higher order a differential equation is obtained that is linear in the unknowns, and with increasingly complicated and lengthy terms involving the already- known lower orders of solution. The procedure has been described in some detail in [13]. At each order j, the solution for Fj and Yj involves polynomials in cnZ(0l m) of degree j. With increasing complexity, the operations quickly become too lengthy for hand calculation and it is necessary to use computer software. In 1979 I [13] used floating point arithmetic and a conventional lan- guage (FORTRAN); however, now it is much easier to use modem software that can perform mathematical manipulations. (I used the symbolic manipulation software MAPLE to prepare this chapter.)

After the operations have been completed, the solutions are power series in tx 2, up to the order desired, for rl,, f',, g, and R,. It is convenient to obtain the series for Q/~/gh 3 by taking the power series of g,-1/z, the series for R/gh by taking the power series of R,/g,, and the series for ~ / g h 3 by taking the power series of ~ , x Q/~gh 3, where ~ , is evaluated from

1 d ~ , = - - sin rl,Ct �9 f, (22)

Expressions for velocity components follow by differentiation. So far, all series have been in terms of ct 2. It is simpler to express the series in

terms of ~5, where

8 = 4t~2/3 (23)


as suggested by the results of Equation 21. Physical solutions could be presented in terms of these power series, and they do reflect the nature of the theory that the essense of the approximation is that the waves be long (ct small). However, most presentations have converted to expansions in terms of e = H/h, the ratio of wave height to trough depth. This can be done by expressing a series for e in terms of ~5 or o~ 2 by evaluating 11. - 1 with cn = 1. The series can then be revert- ed to express ~i or o~ 2 in terms of e, which can then be substituted.

The parameter m is determined by the geometry of the wave. As the function cn(0lm) has a real period of 4K(m), where K is the complete elliptic integral of the first kind [ 1 ], it is easily shown that cn2(0lm) has a period of 2K(m), and as the wave has a wavelength of ~, the elementary geometric relation holds:

o~-- = 2K (m) (24) h

The mean depth d is known in physical problems, but it has not entered the calculations yet. The ratio d/h can be obtained from the series for 11. = rl/h, by replacing each cn 2j by Ij, where Ij is the mean value of cn 2j (01m):

Ij = cn 2j (0lm) (25)

then, ([19], #5.13): I 0 = 1, I 1 = (-1 + m + e(m))/m, where e(m) = E(m)/K(m), and E(m) is the complete elliptic integral of the second kind, and the other values may be computed from

Ij+2 = 2j + 2 2 - Ij+ 1 + - 1 Ij, for all j (26) 2j + 3 ~, 2j + 3

This allows all quantities to be calculated with d as the non-dimensionalising depth scale. Similarly the mean fluid speed: U/x/-~, which is related to wave speed, can be obtained from the series for horizontal velocity U / ~ , by replacing each cn 2j by Ij.

Presentation of Theoretical Results

This section presents two sets of results. For the first time a complete solution is given in terms of rational numbers, whereas in [16] at least some floating- point numbers were used. First, a full solution is presented to third-order, which is a reasonable limit for space reasons. Next, a fifth-order solution is presented, but in which the approximation is made of setting the parameter m to 1 wherever it explicitly occurs in the coefficients of the series expansions. This makes feasible the presentation of the theory to two higher orders. Here we present the solutions; the application and use of these theoretical results are described in the subsequent section.


Features of Solutions

Although the underlying method relies on an expansion in shallowness, it is often convenient to present results in terms of expansions in wave height. It was found [13] that the best parameter for this was the wave height relative to the trough depth, H/h, which we denote by e. If the mean depth had been used, to give a series in H/d, many more terms would be involved, because, as Equation 35 shows, the expression for h/d involves a double polynomial in m and e = E/K of degree n at order n, such that, for example, Equation 27 for rl/h is a triple series in e/m, m, and cn 2, but the corresponding expression for rl/d would be a quadruple series in terms of e as well. It is, of course, a simple matter to evaluate q/h from the results given and then to obtain rl/d by multiplying by h/d.

The expression of the series as power series in e/m rather than e was suggested in [ 13], when it was observed that as m could be less than 1 it was better to moni- tor the magnitude of rdm than to have a power series in e with coefficients that are polynomials in l/m, which could become large without it being obvious.

I have experimented with presenting all the series in terms of o~ 2, which relates much more closely to the theory being based on an expansion in shallowness; however, for all the quantities of cnoidal theory but one, series in e/m gave more rapid convergence and better accuracy. The only exception is a very notable one, however, and that is the series for the velocity components. In [ 13] I presented results for fluid velocity that fluctuated wildly and were not accurate for high waves, and this has given cnoidal theory something of a bad name. However, in [16] the series were expressed in terms of tx 2 (actually ~5 = 4ct2/3). Much better results were obtained, and were found to be surprisingly accurate even for high waves, and that approach has been retained here.

In the presentation of results, the order of neglected terms such as O((E/m) 6) has not been shown, as it is obvious throughout.

A. Third-Order Solution

Here the full solution to third order is presented. This will be more applicable to shorter and not-so-high waves, where the parameter m might be less than, say, 0.96.

The symbol cn is used to denote cn(o~X/hlm) = cn(o~(x - ct)/hlm). Table 1 (page 81) provides a check, to indicate whether a typographical or transcription error might have been made: If the series expression is evaluated with all mathematical symbols on the fight set to 1 (a meaningless operation in the context of the theory), then the result should be the number shown in Table 1.


Surface Elevation

rl (Em/ + ( E__~.~2 ( _ 3 m2 cne + 3 me cn4 ) (27) -~-=1+ mcn2 \ m ) k, 4 4 )

3 61m2 + 111 m3~cn 2 + (61 m2 53 m3~cn4 + cn 6 + -8-0 80 J \80 20 J 80

Coefficient (x

34~~m ( ( _ 2 ) ( 1 7 / (_~)2 ( 1 11 111 m2)) ~ = 1+ - - - m + ~ - ~ m + - - 4 8 32 32 128

Horizontal Fluid Velocity in the Frame of the Wave

(28)

u_ (12 - - 1 +8

19 79 - ~ + ~ m - ~

40 40 +82

4

- - - m + m c n 2 3

79m2+cn2( 3 2) 2 - - m + 3 m - m cn 4 40 2

m + - m + cn 2 m - 3m 2 + - cn 4 4 4

+83

55 3,471 7,113 m2 2,371 m3 + cn 2 ~ m + ~ m - ~ 112 1,120 1,120 560 40

/ (-i-- ~ / 6m3 339m3 +cn 4 27m2 27m3 + _ cn 6 + 40 - - i f 5

9 27m2 9m3 ( 9 27m2 27m3 / m - ~ + -- + cn 2 - - -m + -

( 75m3/ 15m3cn6 75 m2 + __ _ cn4 --if- 4 2

3 _ (8 51 m2 51 m3 ) - - - m + 9 m 2 3m3 + cn 2 m + - - ..!.. ( y )4 16 16 8 16 16

/ )4'm3 45 m2 45 m3 + ~ cn6 +cn4 ~- --8"- 16

\ (29)

The leading term -1 should not cause concem, for if the wave is considered to be travelling in the positive x direction, then relative to the wave the fluid is flowing under the wave in the negative x direction with velocities of the order of the wave speed.


Vertical Fluid Velocity. This can be obtained from Equation 29 by using the mass conservation equation/)U//)X + OV/OY - 0, and the result from Equation 58 that d(cn(01m))/d0 =-sn(01m) dn(01m), with the result that each term in Equation 29 containing (Y/h) i cn j (txX/hl m ), for j > 0, is replaced by (t sn() dn()

( J ) i - ~ x (Y/h) i+l cnJ- (). Hence, if we write Equation 29 a s l

U ~~ii ~ ( h / 2 J k~ 0 - - 1 + cn2k() Oijk i=l j=0 =

where each coefficient Oij k is a polynomial of degree i in the parameter m, the vertical velocity component follows:

V ~ ~ (y)2j+l ~ 2j k ~ i j k+ 1 /-2-ff = 20: cn( ) sn() dn( ) 8 i cn 2(k-l) ( ) ~ / ~ l l i=l j=O k=l Discharge

l + ( ~ / / ~/gh 3

(.~Em)3 ( 11 + - ~ - t - ~ 140

1 / (~/~(9 7 imp) + m + ~ - ~ m - ~ 2 40 20 40

69 11 m2 3 mS ) m - l - ~ + 1120 224 140 )

(30)

(31)

Bernoulli Constant

. ~ (~ / ( 1 gh 2 2

/~/3( lO7 + - - ~ - t - ~

560

)(~/2(7 7 1 . . . . . m

+ m + 20 20 40

25 224

13 m2 13 m3 ) m + ~ - k - ~ 1,120 280

-- ~ m 2 /

(32)

Mean Fluid Speed in Frame of Wave

- /~ / / - ; / (~ )~ / U 13 ~ =1+ - e +

120

361 1,899 2,689 m2 13 + m - ~ + 2,100 5,600 16, 800 280

Wavelength in Terms of l-I/d

~ = 4 K ( m ) ( 3 H')-l/2('m---dJ 1+ - ~ ( H ) / 5 5 4 8

(m.~d)Z( 1 5 1 5 21m2 (1 + + ~ m - ~ + 32 32 128 8

(_~ 1 ); (~)3 1 ~lm2 + + ~ m e + m - 60 40 12

( 7 103 131m2/ / m+ e (33) m3+ 75 300 600

-- - - m - - w 3~ e 2

16 (34)


Trough Depth in Terms of H/d

( m ~ ) (m~d)2( 1 1 (1 1 ) ) (m~d/3 h = l + (1 m e)+ + m + m e + d 2 2 2 4

~ r n + ~ + - ~ + ~ m - ~ e+ - - - m e ~. 200 400 400 200 200 25 2

(35)

Fifth-Order Solution with Iwagaki Approximation

For waves that are not low and/or short, the values of the parameter m used in practice are very close to unity indeed. This suggests the simplification that, in all the formulas, wherever m appears as a coefficient, it be replaced by m = 1, which results in much shorter formulas. In honor of the originator of this approach [23], we suggest that this be termed the "Iwagaki approximation." Here, this is implemented (but in a manner different from Iwagaki's original suggestion) that wherever m appears as the argument of an elliptic integral or functionmsuch as the elliptic functions cn(0lm), sn(01m) and dn(0[m), and the elliptic integrals K(m), E(m) and their ratio e(m)-- the approximation is not made, as the quantities can be evaluated by methods that do not need to make the approximation.

This theory will be applicable for longer waves, where m > 0.96. Iwagaki [23] observed that in many applications of cnoidal theory m can be set to 1 with no loss of practical accuracy. He presented results to second order and termed the resulting waves "hyperbolic waves," because the Jacobian elliptic functions approach hyperbolic functions in that limit. In [16] theoretical results to fifth order were presented with this approximation, and it was shown that it was accurate for longer and higher waves. I, however, prefer not to use the term "hyperbolic waves" because in this work I present several useful approximations to the elliptic functions that have a wider range of validity than merely replacing cn() by the hyperbolic function sech(). The version of the theory that I present is a simple modification of the full theory: That wherever m appears explicitly as a coefficient, not as an argument of an elliptic integral or function, it is replaced by 1, but is retained in all elliptic integrals and functions.

The use of the Iwagaki approximation for typical values of m in the cnoidal theory is rather more accurate than the conventional approximations on which the theory is based, namely the neglect of higher powers of the wave height or the shallowness. For example, m = 0.9997 for a wave of height 40% of the depth and a length 15 times the depth; in this case the error introduced by neglecting the difference between m 6 and 16 (0.002) in first-order terms is less than the neglect of sixth-order terms not included in the theory (0.46 = 0.004).


All the results presented here agree with those presented in [ 16] (where some coefficients were presented as floating point numbers), except for two typographical errors in that work: In the equivalent of Equation 43 the term 3H/d should have appeared with a negative exponent, and in the equivalent of 44 a third-order coefficient (-e/25) was shown with the sign reversed [35].

Surface Elevation

r l _ l + e c n E + e 2 ( 3 3 / e 3 ( 5 151 101 ) . . . . cn 2 + -- cn 4 + cn 2 c n 4 + ~ cn 6 h 4 4 80 80

+ e 4 ( 8,209 2 11,641 4 112,393 17,367 ) - ~ c n + ~ - - - - c n - ~ c n 6 + ~ c n 8 6, 000 3, 000 24, 000 8, 000

e5 (364,671 2,920,931 2,001,361 17,906,339 cn 2 _ cn 4 + cn 6 _ cn 8 + 196, 000 392, 000 156, 800 1,568, 000

+ 1,331,817 cnlO / 313, 600

(36)

Coefficient r

5e 71 E 2 100, 627 E 3 16, 259, 737 e4 + ~ - +

J 8 128 179, 200 28, 672, 000 (37)

Horizontal Fluid Velocity in the Frame of the Wave

l1 9/) ~fgh - 1 + 8 - - - + cn 2 + - ~ + -- cn 2 - cn 4 + - - - cn 2 + - - c n 4 2 40 2 2 4

55 71 2 27 6 ( Y ) 2 ( 9 75 15 ) + ~ c n - ~ c n 4 + -- cn 6 + - - - cn 2 + ~ c n 4 - - - c n 6

112 40 10 5 4 8 2

cn 2 _ ~ c n 4 + m c n 6 16 16

-.I-6 4

r 11,813 53,327 13,109 1,763 197 _ ~ + ~ c n 2 _ ~ c n 4 + ~ c n 6 cn 8 22,400 42,000 3,000 375 125

( Y ) 2 ( 213 3,231 729 189 ) + cn 2 + ~ c n 4 - ~ c n 6 + ~ c n 8

80 160 20 10

( h i 4 ( 916 327 915 315 / + m cn 2 cn 4 + ~ c n 6 cn 8 32 32 16

( h ) 6 ( 3 189 63 189 / + - ~ c n 2 + ~ c n 4 - ~ c n 6 + ~ c n 8 80 160 16 64 j



+ 5 5

r 57,159 144,821 . . . . cn 2 _

98,560 156,800 294,000 298,481 13,438 lo

- ~ c n 8 + ~ c n 36,750 6,125

53,327 1,628,189 , cn 2 +

8,ooo 6,ooo _ 5'31____~9cnlO

125

213 cn 2 640

1,701 lO + ~ c n

20

9 + cn 2 +

160

(5 8( 9 cn 2

+ 4,480

1,131,733 757,991 cn 4 + ~ cn 6

73,500

192,481 cn 4 _ ~ cn 6 +

2,000

13,563 68,643 5,481 _ ~ c n 4 + ~ cn 6

267

64

640 32

987 cn 4 _ ~ cn 6 +

32

459 4 567 - ~ c n + ~ cn 6

1,792 256

11,187 cn

100 8

cn s

7,875 cn 8 567 cnlO 128 16 )

1,215 729 cnl0 _ ~ c n s + ~ 256 256 ) (38)

Vert ica l F lu id Velocity. In the same way as above, each term in Equation 38

containing (Y/h) i cnJ(), for j > 0, is replaced by a sn() dn() i - ~

(Y / h) i+l cn j-1 (). Hence, the expression for V ~ f ~ can be written ~ ( y ) 2 j + l ~

V k (i) ijk /-2-ff : 2 a cn() sn() dn() ~i cn2(k-l) () 2j + 1 i=l j=O k=l

(39)

where the coeff ic ients Oij k can be ext rac ted f rom Equa t ion 38, or f rom Table III

o f [16].

D i s c h a r g e

Q e 3132 3133 309 134 12,237 135 - 1 ~ + I

~/gh3 2 20 56 5 ,600 616, 000 (40)

B e r n o u l l i C o n s t a n t

R 3 e e 2 3 e 3 3 E 4 2, 427 e 5

gh 2 2 40 140 175 154 ,000 (41)


Mean Fluid Speed in Frame of Wave

~--U ( h ) ( l _ e ) + ( h ) 2 ( 3 5 ) ( H ) 3 ( 3 19 ) - ~ + ~ e + e ~/gh =1+ 20 12 56 600

(hi4( 309 3,719 ) (H)5 (12 ,237 __ 997,699 e) + 5~6--O0 ~" 2-i;()()-0 e + 616100--0 8, 820, 000

(42)

Wavelength in Terms of H/d

( H "~-1/2 ~ = 4K (m) ~3--~-) d

/ H ) / 5 23 / / d / 2 ( 21 1 3e2 ) 1 + - - e + + - - e + - 128 16 8

(H)3(20,127 409 7e2 1 e3 ) + ~ - ~ e + ~ + - - 179,200 6,400 64 16

( H ) 4 ( 1,575,087 + 1,086,367 e 2'679 e2 + 28, 672,000 1,792,000 25,600

13 e3 3 e4 ) +~ + v 128 128

Trough Depth in Terms of H/d

ff n ) fin) ff / (d/4( H 2 e 3 1 1 e2 573 h = 1 + - - (-e) + + - ~ e +-- + 7 d 4 25 4 2,000

) ( H ) 5 ( l e4 ) +-1 e3 + _ 302,159 e+ 1,779 e2 _~123 e3 + _ 4 1,470, 000 2, 000 400 4

(43)

57 e2 400

(44)

Practical Application of Cnoidal Theory

This section outlines the procedure for applying the previous results. First, it addresses the problem of obtaining solutions in a frame through which the waves move. We have not yet encountered this problem for the high-order cnoidal theory, as all operations were performed in a frame (X,Y) moving with the wave such that all fluid motion was steady.

T h e Firs t Step: S o l v i n g for P a r a m e t e r m

In practical problems, usually the water depth d and the wave height H are known, and either the wave length ;~ or period x is known. The problem is initially to solve for the parameter m. We now consider the different ways to do that whether the wavelength or period is known.


Wavelength Known. Either of the transcendental equations, Equation 34 for the full third-order solution or Equation 43 for the fifth-order Iwagaki approximation, can be used to solve for the parameter m. In the latter case one would of course check that the value of m so obtained was sufficiently close to unity that the Iwagaki approximation was justified. Both equations contain K(m) and e = e(m) = E(m)/K(m), for which convenient expressions are given below. The variation of K(m) with m is very rapid in the limit as m ~ 1, as it contains a singularity in that limit; hence, gradient methods such as the secant method for this might break down. i prefer to use the bisection method, for which reference can be made to any introductory book on numerical methods, [8] for example. This requires bracketing the solution, for which I use the range m = 0.5 to m = 1 - 10 -12, if 14-digit arithmetic is being used.

As an aside, here we develop an approximation for rn in terms of the Ursell number, which gives some insight into the nature of m. Consider Equation 24:

o~-- = 2K (m) h

If we introduce the first-order approximation from equation 28:

I ~ = ,

mh

and as the lowest-order result from Equation 35 is h/d = 1, we can write the lowest-order approximation to Equation 24 as

3 H )~ 2K (m) 4 t o d d

It is noteworthy that this can be written in terms of the Ursell parameter U = (H/d)/(d/~)2 = H)~2/d 3, giving

U = ~fmK (m) (45)

1 However, the limiting behavior of K in the limit as m ~ 1 is K(m) -- -~ log

( 16 ) ( [1] ' # 17 ~ 3 ~ which shOWs strOng singulal~ behaViOr in that limit' 1 -- m

and we can replace ~ - by 1 to give

1 16 a/-mK (m) -- ~- log ~ l - m (46)


Substituting this into Equation 45 and solving gives an explicit first-order approximation for m in the limit m ~ 1"

m -- 1 - 16 e-3~-3-U/4 (47)

This has some theoretical as well as practical interest, in that we have shown that the parameter m is related to the Ursell number, and as such it might be interpreted as a measure of the relative importance of nonlinearity to dispersion, which is a common interpretation of the Ursell number. Hedges [21] suggests that the boundary between the application of Stokes and cnoidal theory is U = 40, in which case, Equation 47 gives m -- 0.933. This is an indication that, roughly speaking, m is always greater than 0.93 when cnoidal theory is used within its recommended limits.

The Effects of Current on Wave Period and Fluid Velocities

A steadily-progressing wave train is uniquely defined by three physical dimensions: the mean depth d, the wave crest-to-trough height H, and wavelength ~,, such that it can be expressed in terms of two dimensionless quantities, usually H/d and L/d for shallow waves. In many situations the wave period is known, rather than the wavelength. In most marine situations waves travel on a finite current, and the wave speed and hence the measured wave period depends on the current, because waves travel faster with the current than against it. Most presentations of steady wave theory have used either one of two particular definitions of wave speed, such that (1) the wave moves such that the mean fluid speed at any point is the mean current observed, or, (2) that the depth-integrated mean fluid speed is the mean current observed. These are known as Stokes' first and second definitions of wave speed, respectively. However, in general, the speed depends on the current, which cannot be predicted by theory, as it is determined by other topographic or oceanographic factors. What the theories do predict, however, is the speed of the waves relative to the current, and this is the quantity U previously introduced.

The existence of a current has two main implications for the application of a steady wave theory. First, the apparent period measured by an observer depends on the actual wave speed and hence on the current such that the apparent period is Doppler-shifted. This means that without explicit allowance for the current, if the period is known instead of the wavelength, it is not possible to solve the problem uniquely. This will have a relatively small effect, of the order of the ratio of fluid speed to wave speed. The second main effect of current is more important if fluid velocities are to be calculated, and this is the additive effect it has on the horizontal fluid velocities, which will be of the order of the current relative to wave-induced fluid velocities. To determine these velocities it is necessary to know the current. If the current is not known, then the problem is


underspecified, and the error in fluid velocities thus computed will be of the order of the currents possible.

In the stationary frame of reference the time-mean horizontal fluid velocity at any point is denoted by ~l, the mean current that a stationary meter would measure. It can be shown that if the fluid flow is irrotational, on which the above theory has been based, that this is constant throughout the fluid. Relating the velocities in the two co-ordinate systems gives

Ul ---- C -- U (48)

If there is no current,__~ 1 = 0, and hence c = U, so that in this special case the wave speed is equal to U, previously introduced as the mean fluid speed in the frame of the wave. This is Stokes' first approximation to the wave speed, usually incorrectly referred to as his "first definition of wave speed," and is relative to a frame in which the current is zero. Most wave theories present an expression

m

for U, obtained from its definition as a mean fluid speed, and it is often referred to, incorrectly, as the "wave speed."

A second type of mean fluid speed is the depth-integrated mean speed of the fluid under the waves in the frame in which motion is steady. If Q is the volume flow rate per unit span underneath the waves in the (X,Y) frame, the depth-averaged mean fluid velocity i s -Q/d , where d is the mean depth. In the physical (x, y) frame, the depth-averaged mean fluid velocity, the "mass-transport velocity," is u2, given by

U2 -- C -- Q/d ( 4 9 )

If there is no mass transport, U2 = 0, then Stokes' second approximation to the wave speed is obtained: c = Q/d. This would be the case in a closed wave tank in a laboratory.

In general, neither of Stokes' first or second approximations is the actual wave speed, and in fact the waves can travel at any speed. Usually, the overall physical problem will impose a certain value of current on the wave field, thus determining the wave speed.

Wave Period Known, and Current at a Point Known. In many applications, instead of knowing the wavelength, one knows the wave period and current, in which case formulas based on Equations 48 or 49 can be used. In this case it is simpler to present separate expansions for the quantities which appear in the equations.

Equation 48 can be shown to give

H I - k - U - ~ - - 0


where x = wave period, as c = L/x by definition We can substitute this and rearrange the equation to give

Ul U ( h ) 1/2 )~/d + - o (50)

In the case that water depth d, wave height H, gravitational acceleration g, period x, and mean Eulerian current U l are known, the quantities ~ 1/xggd and x ~ can be calculated. The dimensionless trough depth h/d and dimensionless wavelength L/d are known as functions of the known wave height H/d and the as-yet- unknown m, as given by Equations 34 or 43 and 35 or 44. The quantity U ~ is given by Equation 33 or 42, which can be calculated also in terms of m and the known physical dimensions from

H H/d . . . . (51)

h h/d

With these quantities substituted, Equation 50 is now an equation in the single unknown m, and methods such as bisection can be applied to obtain a solution. Equation 50 is simpler than Equation 20 in [16], where I did not realize the series for the wavelength itself could be used so simply.

Wave Period Known and Mean Current Over the Depth Known. In the other case, where the depth-integrated mean current u2 is known, the equation to solve for m is

u- 2 Q (d /3 /2 )~/d I 0 (52)

where the procedure is the same as before, but the dimensionless discharge Q/~/gh 3, known as a function of e and m from Equations 31 and 40, appears instead of the mean fluid speed 0 / ~ . This is also a simpler formulation than my equation 25 in [16].

Wave Period Known, Current Not Known. In this case, the problem is not uniquely defined, and an assumption must be made for the current, and one of the above two approaches adopted. It must be recognized that any horizontal fluid velocities calculated have an error of the magnitude of the real current relative to the assumed current.

An Alternative Approach. Poulin and Jonsson [35] have expressed products of two series in Equations 50 and 52 as single power series. Thus, they provided

78 Offshore Engineedng

a power series for U / ~ and one for Q/~gd 3 in terms of the known H/d. Hence, in Equation 50, if one were to work to the full fifth-order accuracy of the current theory (the series was presented to fourth order only in [35]), then the series for U / ~ contains 21 terms, compared with the procedure adopted here, of evaluating the product of two series, that for U/~'gh with a total of 11 terms in the series and that for h/d with 12 terms, a total of 23 terms. (Similarly they expressed a term in h/d/cx as a single power series, which has now been super- seded by the realization that the expression is simply related to the wavelength).

Equivalently considering Equation 52, the series for Q/x/gd 3 in [35] (which is actually wrong at third and fourth orders as presented therein), would contain 21 terms at fifth order, compared with 6 terms for Q/x/gh 3 plus 12 terms for h/d, a total of 18 terms using the present approach.

Having formulated Equations 50 and 52, I deliberately chose the sequential evaluation of series (not "simultaneous" or "coupled" as stated in [35]), rather than combining the series, because it seemed that the necessity of providing more series expansions as part of the theory was not justified.

Application of the Theory

Having solved for m iteratively, the cnoidal theory can now be applied. Trough depth h: Equation 35 or 44 can be used to calculate h/d. This will

probably already have been calculated as part of the converged solution process for m.

Wavelength g: This follows easily from Equation 34 or 43, and also will probably already have been calculated.

Dimensionless wave height E = H/h: Equation 51. Coefficient cx: Equation 28 or 37. This is used as an argument of the elliptic

functions in all quantifies which vary with position and is used to calculate ~5. Shallowness parameter 5: It is more accurate to present results for fluid

velocity in terms of o~ rather than e = H/h, and it is more convenient to present the results in terms of 5 , rather than in terms of r [ 16], where

~5 = 4 ot 2 - (53) 3

Mean fluid speed in frame moving with wave U: Equation 33 or 42 is used to calculate U / ~ .

Discharge Q: Equation 31 or 40 is used to calculate Q/a/gh 3. Wave speed c: Follows from Equation 48 if the current at a point is known:

m

c = fil + U, or from Equation 49 if the depth-integrated mean current is known:

c = u2 + Q/d. Surface elevation: For a particular point and time (x,t) the elliptic function

cn(cz (x - c0 /h lm) can be computed using the approximation in Table 2 and Equation 27 or 36 used.


Fluid velocity components (u,v): Fluid velocities in the physical (x,y) frame are given by

u(x, y, t) = c + U(x - ct, y) (54)

where U(X, Y) is given by Equations 29-30 or 38-39. These equations can be written

u(x, y, t) c u 1+ (5 i

i=l j=O k=O cn 2k (o~(x - ct) / him) r k (55)

where the coefficients Oijk can be extracted from Equation 29, where each is a polynomial of degree i in the parameter m, or in the Iwagaki approximation where they are rational numbers, from Equation 38, or from Table III of [16]. The vertical velocity components follow, using the mass conservation equation, differentiating with respect to x and integrating with respect to y to give:

v(x, y, t) ~ ~ ( y ) 2 j + l = 2ix cn() sn() dn() ~5 i

i=l j=O i k

�9 Zcn2(k-1) (~(X -- ct)/hlm) 2j + 1 Oijk (56) k=l

It will be seen below that this theory predicts velocities accurately over a wide range of wave conditions.

Derivatives of fluid velocity: In some applications it is necessary to know the spatial and time derivatives of the velocity. These follow from differentiation of Equations 55 and 56 and the use of elementary properties of elliptic functions, and application of the mass conservation and irrotationality equations:

Ou = 20t cn() sn()dn()z.... ~i ...~

Ox i=l j=0 i

�9 Zcn2r - ct) / him) kr k (56a)

k=l ~ ~ i-1 I y I 2 j 1 i

/)u = 2 ~i Z Z cn2k(ct(x - ct)/him) j(I)ijk (56b) 3y

i=l j=l k=O

/)u /)u /)v /)v (56c) ------ C , - ' - - C - - ~gt /)x 3t 3x /)v ~)u /)v ~)u (56d)

3x ~gy /)y ~)x


Bernoulli constant R: Equation 32 or 41 is used to calculate R/gh. Fluid pressure p: By applying Bernoulli 's theorem in the frame in which

motion is steady, Equation 5 can be used to give an expression for the fluid pressure at a point:

1 v2 p(x, y, t) = R - gy - -~ [(u - c) 2 + ]

P

Practical Tools and Hints for Application

Here we provide some methods and results which may make the application of cnoidal theory somewhat simpler.

Numerical Check for Coefficients

In the presentation of high-order series results it is very easy to make errors. To check this, Table 1 provides a list of numbers, one for each of the Equations 27-44, which have been obtained by evaluating each of the expressions with all mathematical symbols set to 1. This is a meaningless operation physically, and the fact that the numbers from the full third-order theory and fifth order theory disagree does not imply that something is wrong. If a user checks his/her own calculations and does not obtain the values shown here, an error has been made by someone, and checking should be carried out. As a possible extra check, it should be mentioned that the fifth-order Iwagaki approximation as presented in [16] is believed to be correct as printed but for two sign errors: the exponent - ~ of 3H/d in Equation 19 (cf. Equation 43) and in the coefficient -e/25 in the third-order term of Equation 21 (cf. Equation 44).

Numerical Richardson Test for the Series Results

There is one simple method that can test whether or not a series solution to a problem is correct, and if not, at which order of accuracy it is wrong. It gives a simple answer as to whether all the series used in the computation are correct, but it does not reveal where any errors might be. The method, in the context of this work might prove helpful to a practitioner having written a program based on the theoretical results above who might want to check the accuracy of the series as programmed.

The method, proposed in [14], is based on Richardson's extrapolation to the limit. It can be used almost anywhere, but a simple test for some of the most important quantities previously presented would be to calculate the pressure at an arbitrary point on the free surface, where the method would test whether or not all of the expressions were correct: the elevation of the surface from Equa- tion 27 or 36, the coefficient ~ from Equation 28 and 37, the velocity compo-


Table 1 Values of Expressions Evaluated with All Symbols Set to 1

Quantity

h

u

Q

~h 3

R gh

U

d

Third-Order Full Solution

Equation

27

28

29/30

31

32

33

34

35

Check Value

Fifth-Order lwagaki Approximation

Equation Check Value

119 r-- 2----~/3 = 0.80513

33 = 0.05893

560

393. = 1.40357 280

547

280 = 1.95357

138 175

=0.78857

431 r-- - ~ / 3 = 1.55524

23 0.46 50

36

37

38/39

40

41

42

43

44

26,815,417 ~ = 0.80995 57,344, 000

4,572,863

2, 464, 000 =-1.85587

842,847 616,000

= 1.36826

295,783 154, 000

= 1.92067

158,576,387 194, 040, 000

=0.81724

6,826,061 ~ = 2.74904 4, 300, 800

2,176,261

1,470, 000 = 1.48045

nents f rom Equat ions 29 or 38 and 30 or 39, the mean fluid speed f rom Equat ion

33 or 42 and the Bernoul l i constant f rom Equat ion 32 or 41!

If one has a series approximat ion to a quanti ty that should be zero, such as

pressure, or testing the series by evaluat ing an identi ty such as H/h - (H/d)/(h/d),

then evaluat ing it will not give zero in general , but a finite error, such as non-

zero pressure at a point on the surface. W e denote this error by A, and suppose it

to be a funct ion only of an expans ion paramete r e (probably H/d in the present

context), for all other quanti t ies given numer ica l values. For example , we might

take a wave of length Md = 20, and calculate the pressure on the surface at (x -

ct)/h = 0.5. Now, if we assume that the error is propor t ional to the nth power of

e, then we can write, where e is the expans ion quanti ty, whe ther e, 8, or H/d,

A = ae n


where a is independent of e. Now, if we evaluate the error numerically for two different values of e" r 1 and E 2, to give A 1 and A 2, then we can eliminate a such that

A1 = E1

x; and this can be solved to give

log (A1/A2) n = (57)

log (el/e2)

thus giving a numerical estimate of the error. The error of this expression can be shown to be O(e), so to provide convincing evidence of the order of the theory, it is necessary to use a small value of e. In practice it is very reassuring to obtain a value of n = 5.98, for example, providing strong evidence that all series that have gone into the calculation are correct to fifth order.

F o r m u l a s a n d M e t h o d s for Elliptic Integrals and Functions

Elementary Properties of Elliptic Functions and Integrals. For elementary properties, see [1, 19, and 43], for example. Those sources contain several approximations, but the expressions given usually do not have the same remarkable accuracy as those given in [ 18] for the limit required for cnoidal theory of waves, m ---> 1. To further investigate the theory, see [11], which contains a refreshing different approach to the subject and inspired the following work, which was originally obtained in [18].

Approximations to Functions and Integrals. One perceived practical problem with the application of cnoidal theory has been that the theory makes use of Jacobian elliptic functions and integrals, which are seen as being difficult to calculate. This has some justification, as conventional formulas such as in [1] are very poorly convergent, if convergent at all, in the limit m ---> 1, precisely the limit in which cnoidal theory is most appropriate. However, alternative formulas can be obtained that are most accurate and remarkably quickly convergent in the limit of m ---> 1. This has been done in [18], which provides several useful express ions for both ell iptic funct ions and integrals . The formulas are dramat ica l ly convergent , even for values of m not in the m --> 1 limit. Convenient approximations to these formulas can be obtained and are given here in Table 2. For values of m likely to be encountered using cnoidal wave theory the formulas are probably accurate to machine accuracy. It is remarkable that even for m = ~, the simple approximations given in Table 2 are accurate to five significant figures. For the case m ,~ ~A, when cnoidal theory becomes less valid,


conventional approximations could be used, for which reference can be made to [ 18] or to standard references.

However, cnoidal theory should probably be avoided in this case.

Derivatives

cn(0[m) = - sn(0lm) dn(0lm), b0

sn(0[m) = cn(0lm) dn(0lm), b0

dn(0lm) = - m sn(0[m) cn(0[m) (58)

Table 2 Approximations for Elliptic Functions and Integrals in the Case Most

Appropriate for Cnoidai Theory, m > V2

Elliptic Integrals

Complete elliptic integral of the first kind K(m)

2 2(I + m I/4) K(m)= mi/4)2 log (1 + 1 - m 1/4

Complementary elliptic integral of the first kind K ' (m)

K' (m) - 2~

(1 + ml /4 ) 2

Complete elliptic integral of the second kind E(m)

E(m) = K(m) e(m), where

2m l , / e(m) -- + ~ + 2 - - - + 3 2 K K ' K -'7 24 (1 - q2 )2

where ql (m) is the complementary nome ql = e-~K/K'

Jacobian Elliptic Functions

sn(zlm) -- m "1/4 sinh w - q 2 sinh 3W

cosh w + q2 cosh 3w

f 1 l'' 1 ml 1 - 2q 1 cosh 2w cn(zlm) = 7 \ ~ ) cosh w + q~ cosh 3w

( / 1 m 1 1 + 2 q l cosh 2w

dn(z[m) = 7 t ,~ l ) cosh w + ql 2 cosh 3w

in which w = ~ z / 2 K '


Relations Between Squares

dn 2 (0lm)= 1 - m (1 - cn 2 (0lm)) sn 2 (01m) = 1 - cn 2 (01 m) (59)

Fourier Series for cn 2 Although not necessary for the application of the previous theory, there is an

apparently little-known Fourier series for the cn 2 function that might prove useful in certain applications. It is presented here, partly because in some fundamental references ([5], #911.01, [19], #8.146, 26) an incorrect expression (an odd function) is given. The correct expression is given in ([34], #2.23) as a Fourier series for dn2() which can be used to convert to a Fourier Series for cn20, which can be written

cn 2 (0lm) = 1 + ~ + ~., m mK 2 sinh ( j~K'/K cos (60) j=l

For typical shallow water waves, m ---> 1, and K ---> 0% such that the series would be slowly convergent, as would be expected for a wave that is so non-sinusoidal as a long wave with its long trough and short crest. The series could be recast to give a complementary rapidly-convergent formula that would involve a series of hyperbolic functions.

Convergence E n h a n c e m e n t of Series

The previous series are presented to third order for the full theory and to fifth order for the Iwagaki approximation. There are several techniques available for obtaining more accurate results by taking the series results and attempting to extract more information from the series than is apparently there.

The Shanks Transform. One simple way of doing this is to use Shanks transforms, which are delightfully introduced in [40], and used in the context of water wave theory to enhance the convergence of series in [12]. They take the first few terms of a series and attempt to mimic the behavior of the series as if it had an infinite number of terms. The method takes three successive terms in a sequence (such as the first, second, and third-order solutions for a wave property), and extrapolates the behavior of the sequence to infinity, mimicking, one hopes, the behavior of the series if there were an infinite number of terms. There is little theoretical justification for the procedure, but it can work surprisingly well. It is easily implemented: If the last three terms in a sequence of n.terms are

Sn- 2' Sn- 1' and S n, an estimate of the value of Soo is given by


Zoo S n (Sn - Sn-1)2 = - (61) (S n - s n _ 1)-(Sn_ 1 -Sn_ 2)

This is not the form usually presented, but it is that which is most suitable for computations, when in the possible case that the sums have nearly converged and both numerator and denominator of the second term on the right go to zero, the result is less liable to round-off error. The transform does indeed possess some remarkable properties. For example, it gives the exact sum to. infinity for geometric series, which can be verified by substituting S n = ]~j=0 r J, then Equa- tion 61 gives 1/(1 - r), the exact result for the sum to infinity.

The transform is simply applied and can be used in many areas of numerical computations. It gives surprisingly good results, but its theoretical justification is limited and sometimes it does not work well.

Pad6 A p p r o x i m a n t s . A form of approximation of the series that has more justification is the Pad6 approximation, where a rational function of the expansion variable is chosen to match the series expansion as much as possible, [2]. It was introduced to water wave theory by Schwartz [39]. The calculations for Pad6 approximants are not as trivial as for Shanks transforms; however, the properties are usually more powerful. The [i, j] Pad6 approximation is defined to be the rational function p(r162 where p(r is a polynomial of degree < m and q(r is a polynomial of degree < n, such that the series expansion of p(r162 has maximal initial agreement with the series expansion of the function. In normal cases, the series expansion agrees through the term of degree m + n , and it is this way that the coefficients in the two polynomials are computed. An example is (1 + x/2)/(1 - x/2) as the [1, 1] approximation to e x, which for small values of x is more accurate than the equivalent series with quadratic terms 1 + x + x2/2. Another example is where the function 1 + x + x 2 has as its [ 1, 1 ] approximant the function 1/(1 - x), and this, too, indicates that the first three terms of the series look like a geometric series.

Use of C o n v e r g e n c e Acce lerat ion P r o c e d u r e s in Cnoida l Theory . Tests of Shanks transforms and Pad6 approximants in applying the cnoidal theory described in this work indicate the Pad6 approximation is more powerful. How- ever, a limitation became quickly obvious, when at the first step in application, solving Equation 43 for the wavelength, approximating the quartic in H/d in the large brackets by a [2, 2] Pad6 approximant, with a quadratic in numerator and denominator, the latter passed through zero for an intermediate value of H/d, such that in the vicinity of that point very wildly varying results were obtained. This was sufficiently dangerous that generally [2, 2] or [3, 2] Pad6 approximants could not be recommended for the approximation of fifth-order cnoidal theory. Examining Pad6 approximants with a linear function in the denominator, I found


that, given an n-term series, the [ n - 1, 1 ] Pad6 approximant is, in fact, exactly equal to the Shanks transform of the last three sums in the series, as given in the previous equations. As the Shanks transform is more simply implemented, I will refer to the series convergence acceleration using this method by that name.

In practice, obtaining solutions for given values of wavelength and wave height, the use of the Shanks transforms everywhere gave better results than just using the raw series in the case of global wave quantities such as ct, Q, etc., which are independent of position; and it is recommended that for both third- order theory and the Iwagaki approximation that Shanks transforms be used to improve the accuracy of all series computations for those quantities. They are, of course, trivially implemented, given say, three numbers for the third, fourth, and fifth solutions.

For the surface elevation and the fluid velocity components, however, because they are functions of position, then depending on that position the series could show rather irregular behavior, and it was found that the Shanks transform results could also be irregular. As in [ 16], it is recommended that for quantities that are functions of position no attempts be made to improve the accuracy by numerical transforming of the results; but that for all other quantities, characteristic of the wave as a whole, the Shanks transform be applied to all numerical evaluations of series. This procedure was adopted for all the results shown further below.

A Numerical Cnoidal Theory

Introduction

The accuracies of various theories were examined by comparison with experimental results and with results from high-order numerical methods [ 16]. It was found that fifth-order Stokes and cnoidal theories were of acceptable engineering accuracy almost everywhere within the range of validity of each. For long waves that are very high however, even the high-order cnoidal theory presented above becomes inaccurate. In such cases the most accurate method is a numerical method. The usual method, suggested by the basic form of the Stokes solution, is to use a Fourier series that is capable of accurately approximating any periodic quantity, provided the coefficients in that series can be found. A reasonable procedure, then, instead of assuming perturbation expansions for the coefficients in the series as is done in Stokes theory, is to calculate the coefficients numerically by solving the full nonlinear equations. This approach began with Chappelear [6], and has been often, but inappropriately, known as "stream function theory" [10]. Further developments include those of [38]. A comparison of the various methods has been given in [42], which concluded that there was little to choose between them. A more recent development has been the simpler method and computer program given in [ 15].


This Fourier approach breaks down in the limit of very long waves, when the spectrum of coefficients becomes broad-banded and many terms have to be taken, as the Fourier approximation has to approximate both the short rapidly- varying crest region and the long trough where very little changes. More of a problem is that it is difficult to get the method to converge to the solution desired [9].

A new approach [17] describes a numerical cnoidal theory, which is to cnoidal theory what the various Fourier approximation methods are to Stokes theory. It solves the problem numerically by assuming series of cnoidal-type functions, but rather than solving them by analytical power series methods as previously mentioned, the coefficients in the equations are found numerically and there is no essential mathematical approximation introduced. The method is described here briefly.

T h e o r y

A spectral approach is used, in which all functions of x are approximated by polynomials of degree N in terms of the square of the Jacobian elliptic function cn2(01m) for the surface elevation and bottom velocity of the form suggested by conventional cnoidal theory:

N

rl, = 1 + E Yj cn2J(01m) j=l

(62)

N

f: = F o + ~ F j cn2J(0lm)

j=l

(63)

where the Yj and F. are numerical coefficients for a particular wave. Note that the N here is not t~e order of approximation but the number of terms in the series. Conventional cnoidal theory expresses the coefficients as expansions in terms of the parameter o~, which is related to the shallowness (depth/wavelength) 2 (Equations 14 and 15), and produces a hierarchy of equations and solutions based on series expansions in terms of t~, which is required to be small. In this work there is no attempt to solve the equations by making expansions in terms of physical quantities. The surface velocity components are then given by

ush d u, s . . . . cos ~XTI, �9 f, q

vsh d V,s = - - = sin txrl, �9 f" (64) q


On substituting these into the nonlinear surface boundary conditions (Equations 12 and 13) we have two nonlinear algebraic equations valid for all values of 0. The equations include the following unknowns" a, m, g,, R,, plus a total of N values of the Yj for i = 1...N, and N + 1 values of the Fj for i = 0...N, making a total of 2N + 5 unknowns. For the boundary points at which both boundary conditions are to be satisfied we choose M + 1 points equally spaced in the vertical between crest and trough such that:

cn 2 (0i[m) = 1 - i/M, for i = 0.. .M (65)

where i = 0 corresponds to the crest and i = M to the trough. This has the effect of clustering points near the wave crest, where variation is more rapid and the conditions at each point will be relatively different from each other. If we had spaced uniformly in the horizontal, in the long trough where conditions vary little, the equations obtained would be similar to each other and the system would be poorly conditioned. We now have a total of 2M + 2 equations, but, so far, none of the overall wave parameters has been introduced. It is known that the steady wave problem is uniquely defined by two dimensionless quantities: the wavelength 3Jd and the wave height H/d. In many practical problems the wave period is known, but [17] considered only those where the dimensionless wavelength 3Jd is known. It can be shown that Md is related to ct using the expression (24) which we term the "wavelength equation":

X d a - - -- - 2K(m) = 0 (66)

d h

where K(m) is the complete elliptic integral of the first kind, and where the equation has introduced another unknown d/h, the ratio of mean to trough depth.

The equation for this ratio is obtained by taking the mean of Equation 62 over one wavelength or half a wavelength from crest to trough:

N

d = 1+ E YJ cn2J(0lm) h

j=l

(67)

The mean values of the powers of the cn function over a wavelength can be computed from the recurrence relations (Equation 26) for the Ij such that Equa- tion 67 can be written

N d I + E Y j I j - - - = 0

h j=l

(68)

thereby providing one more equation, the mean depth equation.


Finally, another equation which can be used is that for the wave height:

H 110 riM ' i f = h h (69)

which, on substitution of Equation 62 at x = x 0 = 0 where cn(0[ m) = 1 and, because cn(~xMI m) = 0 from Equation 65, gives

H d N - 0 (70

d h j=l

the wave height equation. We write the system of equations as

e(z) = {e i (z), i = 1. . .2M + 5} = 0 (71)

where e i is the equation with reference number i, the 2M + 2 equations previously described plus Equations 66, 68, and 70, and where the variables that are used are the 2N + 5 unknowns previously described plus d/h:

z = {zj,j = 1. . .2N + 6} (72)

Whereas the parameter m has been used in cnoidal theory, it has the unpleasant property that it has a singularity in the limit as m ~ 1, which corresponds to the long wave limit; and as we will be using gradient methods to solve the nonlinear equations, this might make solution more difficult. It is more convenient to use the ratio of the complete elliptic integrals as the actual unknown, which we choose to be the first:

K(m) (73) zl = K(1 - m)

The solution of the system of nonlinear equations follows that in [15], using Newton's method in a number of dimensions, where it is simpler to obtain the derivatives by numerical differentiation.

As the number of equations and variables can never be the same (2M + 5 can never equal 2N + 6 for integer M and N), we must solve this equation as a generalized inverse problem. Fortunately, this can be done very conveniently by the singular value decomposition method (for example [36], #2.6), so that if there are more equations than unknowns, M > N, the method obtains the least squares solution to the overdetermined system of equations. In practice this was found to give a certain rugged robustness to the method, despite the equations being rather poorly conditioned.

The set of functions {cn2J(0lm), j = 0...N} used to describe spatial variation in the horizontal do not form an orthogonal set, and they all tend to look like one another, the result of which, although apparently an esoteric mathematical property, has the important effect that the system of equations is not particularly


well-conditioned, and numerical solutions show certain irregularities and a relatively slow convergence with the number of terms taken in the series. It was difficult to obtain solutions for N > 10. The Fourier methods, however, using the robustly orthogonal trigonometric functions, do not seem to have these problems. Fortunately, however, in the case of the numerical cnoidal theory, good results could be obtained with few terms.

For initial conditions in the iteration process, it was obvious to choose the fifth-order Iwagaki theory presented in Equations 36-44. The first step is to compute an approximate value of m and hence z 1 using the analytical expression for wavelength in terms of m from Equation 43, combined with the bisection method of finding the root of a single transcendental equation. After that the rest of the fifth-order expressions previously presented can be used.

Accuracy of the Methods

This section examines the applicability of the full third-order cnoidal theory, the fifth-order Iwagaki approximation, and the numerical cnoidal theory by considering several high waves and showing results for the surface profile and, possibly more importantly, for the velocity profile under the crest.

The Region of Possible Waves and the Validity of Theories

The range over which periodic solutions for waves can occur is given in Fig- ure 2, which shows limits to the existence of waves determined by computational studies. The highest waves possible, H = H m, are shown by the thick line, which is the approximation to the results of Williams [48], presented as Equa- tion 50 in [ 16]:

Hm o141o6 -- + 0.0095721 + 0.0077829

d

1+ 0.0788340 ~' /~-] z [~-/3 -- + 0.0317567 + 0.0093407 d

(74)

Nelson [31, 32], has shown from many experiments in laboratories and the field, that the maximum wave height achievable in practice is actually only Hm/d = 0.55. Further evidence for this conclusion is provided by the results of Le Mrhaut6 et al. [30], whose maximum wave height tested was H/d = 0.548, described as "just below breaking." It seems that there may be enough instabilities at work that real waves propagating over a flat bed cannot approach the theoretical limit given by Equation 74. This is fundamental for the application of the present theories. If indeed the highest waves do have a height to depth ratio of only 0.55, it seems that both fifth-order Stokes theory and fifth-order cnoidal theory are capable of giving accurate results over all possible waves [ 16].


0.8

0.6 Wave height/

depth H/d

0.4

0.2

EquatiOnWilliams(74) - - o Solitary wave

- Cases here (Table 3) * ~ -

- / \ Nelson H/d = 0.55 -

1 10 100 Wavelength/depth (.X /d)

Figure 2. The region of possible steady waves, showing the theoretical highest waves (Williams), the highest long waves in the field (Nelson) with cases reported on here and Hedges' proposed demarcation line between regions of applicability of Stokes and cnoidal theories.

Reference 16 proposes a formula for the boundary between the use of Stokes theory and cnoidal theory. It has been pointed out by Hedges [21 ], however, that a simpler criterion, and one agreeing more with the numerical evidence, is that cnoidal theory should be applied for

m? u = - ~ > 40 (75)

while for U < 40, for shorter waves, Stokes theory should be used. This line is plotted on Figure 2, and it shows an interesting and important property for small waves, that cnoidal theory should not be used below a certain wave height, even for very long waves! This was explained in [ 13], where it was shown that in the small amplitude limit, the waves tended to become sinusoidal and the parameter m became small, such that the effective expansion parameter e/m became large, even if e itself was not, and the series showed poor convergence.

Comparison of Theories and Numerical Methods

Now we examine the accuracy of the various theories over the range of possible waves, considering H/d = 0.55 and increasing the wavelength from 8 to 64, doubling each time. One with a height of 0.7, close to the theoretical maximum, will be considered.


Table 3 Wave Trains and Results

H/d X/d U m (3rd order) m (5th order)

0.55 8 35.2 0.9168 0.8964 0.55 16 141 0.9983 0.9980 0.7 32 717 1 - 0.14 x 10 -6 1 - 0.24 x 10-6 0.55 64 2,250 1 - 0.75 x 10 -13 1 - 0.11 x 10 -12

These cases are summarized in Table 3, which shows the wave dimensions, the Ursell number, and the value of m obtained by solving Equation 43.

Figure 3 shows the solution for the surface profiles obtained for a high wave of intermediate length, when conventional cnoidal theory has been considered not valid, and which falls outside Hedges' recommended boundary for cnoidal theory of U > 40, as can be seen on Figure 2. Four curves are plotted, results from the full third-order method, the fifth-order Iwagaki approximation, the numerical cnoidal theory previously described, and from the Fourier approximation method, which should be highly accurate in this relatively short wave limit. It can be seen that most results are almost indistinguishable at the scale of plotting, but that in this case of a relatively short wave, with m = 0.9, the Iwagaki approximation is

3rd order 5th, Iwagaki : .

Numerical -b- Fourier •

Figure 3. Surface profiles for Hid = 0.55, L/d = 8.


not so accurate, as expected. Whereas conventional cnoidal theory should not be particularly accurate in this shorter wave limit, as it depends on the waves being long for its accuracy, there is nothing in the numerical cnoidal method that necessarily limits its accuracy to long waves. In fact, for the initial conditions for the numerical method only cnoidal theory was used, and it was not accurate enough for waves shorter than this example. If Stokes theory could be modified to provide the initial conditions, there is no reason why the numerical cnoidal method could not be used for considerably shorter waves.

Figure 4 shows the velocity profiles under the crest for the same wave. It is clear that the numerical cnoidal method and the Fourier method agree closely, and possibly strangely, that the Iwagaki approximation is accurate, even for this wave with m --- 0.9. The third-order theory predicts the mean fluid speed under the wave poorly, but predicts the velocity variation in the vertical very well, so that the curve is displaced relative to the accurate results.

0.8

1.4

1.2

0.6

0.4

0.2

i I

u/J~

3rd order 5th, lwagaki

Numerical -b- Fourier -X--

y/d

0 I I i

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 4. Velocity profiles under crest for the same wave as the previous figure; U / ~ plotted.

Figure 5 shows the results for a longer wave, of X/d = 16. In this case, m = 0.998, and it is expected that the Iwagaki approximation would be accurate. It can be seen that even the third-order theory predicts the surface very accurately. For all subsequent cases studied, even for the higher wave with H/d = 0.7, the results for surface elevation were better even than this, and no more results for surface elevation will be presented here.


All theories

Figure 5. Surface profiles for H/d = 0.55, L/d = 16.

Figure 6 shows the velocity profiles under the crest. It is clear that the fifth-

order Iwagaki theory is highly accurate for practical purposes, but that the third-

order theory has a constant shift as before.

1.6

1.4

1.2

y/d 0.8

0.6

0.4

0.2

0 0.6

I I I I I

3rd order 5th, Iwagaki ~ -

Numerical -t--- Fourier -)~

// _

I I I I

0.1 0.2 0.3 0.4 0.5 e/,/~

Figure 6. Velocity profiles under crest for Hid = 0.55, Z/d = 16.


y/d

1.6

1.2

0.8

0.4

5th, Iwagaki -~-- Numerical, N = 5 - 9

Fourier -X---

0 I I I

0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 v/,/~

Figure 7. Velocity profiles under crest for high and long wave: H/d = 0.7, ~ d = 32.

Figure 7 shows the behavior of the numerical cnoidal method for very high and long waves, for a wave of length k/d = 32 and a height of H/d = 0.7, close to the maximum theoretical height of Hm/d = 0.737, calculated from Equation 74. There is evidence that no long wave in shallow water can exist at this height, and that a maximum of H/d = 0.55 is more likely [32]. This wave is sufficiently long that the Fourier method is beginning to be tested considerably, yet it is capable of giving results, provided sufficient numbers of Fourier terms are taken and sufficient steps in wave height are taken. The present numerical cnoidal theory is also capable of high accuracy, as demonstrated by the close agreement between the two very different theories. It used much smaller computing resources, typically using 9-10 spectral terms with the solution of systems of 25 equations compared with the Fourier method with some 25 spectral terms and some 70 equations. However, there are some irregularities in the solution, and the results for different values of N do not agree to within plotting accuracy. Although the method shows difficulty with convergence, it does yield results of engineering accuracy. It is still remarkable, however, that such a demanding problem can be solved with so few "spectral terms."

Figure 8 shows the velocity profile under the crest for a very long wave, L/d = 64, with Nelson's maximum height H/d = 0.55. The Fourier method took several steps to converge, so it was considered not worthwhile. The numerical cnoidal method performed quite well, although there was some variation between the solutions for N = 5 to N = 9. What is noteworthy, however, is that the fifth-order Iwagaki theory gave a good engineering accuracy solution to this problem.


1.6

1.2

0.8

0.4

I I

5th, lwagaki Numer ica l , N - 5 - 9

0 I I

0 0.1 0.2 0.7

I I I I

I I I

0.3 0.4 0.5 0.6 v/,/~

Figure 8. Velocity profile under crest for Hid = 0.55, L/d = 64.

Conclusions from Computational Results

The numerical cnoidal method has been shown to be accurate for waves longer than some eight times the water depth. It can treat very long waves rather more easily than Fourier methods can. As the theoretical highest waves are approached, however, the accuracy decreases to an approximate engineering accuracy. However, there is strong evidence that these waves cannot be achieved in practice. Throughout, however, for waves with an Ursell number greater than 40, and apparently even for high waves, the fifth-order Iwagaki theory presented in this work gave satisfactory engineering solutions to the problems studied.

Notation

a

c

cn(01m) D(e)

d dn(01m)

E(m) e(m)

e i

e(z)

constant in numerical test of order of accuracy wave speed elliptic function Polynomial in denominator of Pad6 approximant mean water depth elliptic function elliptic integral of the second kind = E(m)/K(m), ratio of elliptic integrals equation i in numerical cnoidal theory vector of errors in equations for numerical cnoidal theory


F i

Fij f'(X)

fP,

g g,

gj H

H m h

i(j) i

J K(m)

K'(m) k

M m

m 1 N

n

P Q

ql R

R,

S n

sn(0[m) t

U U U u

m

u 1

uz

U ~

U, s

V v

v ,

coefficients in expansion for f', coefficient in series for F i velocity on bed dimensionless velocity on bed gravitational acceleration = gh3/Q 2, dimensionless number, inverse of square of Froude number coefficients in expansion for g* wave height (crest to trough) maximum wave height possible for a given wavelength water depth under wave trough mean value of cn2J(0[m) integer used in sums etc. integer used in sums etc. elliptic integral of the first kind = K(1 - m), complementary elliptic integral integer used in sums etc. number of computational points in numerical cnoidal theory parameter of elliptic functions and integrals = 1 - m, complementary parameter number of terms in series or polynomial in numerator of Pad6 approximant order of errors or degree of polynomial or number of terms in series pressure volume flux per unit span perpendicular to flow = exp(-K/K'), complementary nome of elliptic functions Bernoulli constant (energy per unit mass) = RhZ/Q 2, dimensionless energy per unit mass sum to n terms of series elliptic function time velocity component in X co-ordinate mean value of fluid speed over a line of constant elevation = H ~2/d3, Ursell number velocity component in x direction of frame fixed to bed current at a point: mean value of u, averaged over time at a fixed point depth-averaged current: mean value of u over depth, averaged over time dimensionless velocity value of u, on surface velocity component in Y co-ordinate velocity component in y co-ordinate dimensionless velocity


V,S

W

X X,

X

Y Yj Y,

Y Z

zj Z

value of v. on surface = xz/2K', dummy variable = x - c t , horizontal co-ordinate in frame moving with wave crest =X/h horizontal co-ordinate in frame fixed to bed vertical co-ordinate in frame moving with wave crest coefficients in expansion for 11. = Y/h = Y, vertical co-ordinate in frame fixed to bed dummy argument used in elliptic function formulas variable j in numerical cnoidal theory, z 1 = K(m)/K(1 - m) = { zj, j = 1...2M + 6 }, vector of variables

Greek symbols

E

rl, 0

P I;

~j l lg

lg,

coefficient of X/h in elliptic functions and expression of shallowness error in any equation

4 a2 quantity used in series for velocity components 3

general symbol for expansion quantity of series: e, ~5 or H/d = H/h, dimensionless wave height water depth = rl/h, dimensionless water depth argument of elliptic functions, often aX/h in this work wavelength fluid density wave period velocity coefficients in cnoidal theory stream function = q /x ~/gh3/Q, dimensionless stream function

Mathematical symbols

o 0 [i,j]

order symbol: "neglected terms are at least of the order of" Pad6 approximant with ith and jth degree polynomials in numerator and denominator

R e f e r e n c e s

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19. Gradshteyn, I. S. and Ryzhik, I. M., 1965. Table of Integrals, Series, and Products. Academic, fourth edition.

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34. Oberhettinger, F., 1973. Fourier Expansions. Academic, New York & London. 35. Poulin, S. and Jonsson, I. G., 1994. "A Simplified High-Order Cnoidal Theory," in

Proc. Int. Symp. on Waves--Physical and Numerical Modeling, Vancouver, vol. 1, pp. 406--416.

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Steady Water Waves." J. Fluid Mech., 104, pp. 119-137. 39. Schwartz, L. W., 1974. "Computer Extension and Analytical Continuation of Stokes'

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Stokes, Cnoidal, and Fourier Wave Theories." A.S.C.E.J. Waterway Port Coastal and Ocean Engng., 113, pp. 565-587.

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44. Tsuchiya, Y. and Yasuda, T., 1985. "Cnoidal Waves in Shallow Water and Their Mass Transport," in Debnath, L. (ed.), Advances in Nonlinear Waves, pp. 57-76. Pitman.

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Trans Roy. Soc. London A, 302, pp. 139-188.

C H A P T E R 3

EQUATIONS FOR NUMERICAL MODELING OF WAVE TRANSFORMATION IN

SHALLOW WATER

Masahiko Isobe Department of Civil Engineering

University of Tokyo Bunkyo-ku, Tokyo, Japan

CONTENTS

INTRODUCTION, 102

BASIC EQUATIONS AND BOUNDARY CONDITIONS, 103

Basic Equations and Boundary Conditions for Waves on a Fixed Bed, 103

Basic Equations and Boundary Conditions for Waves on a Permeable Bed, 107

MILD-SLOPE EQUATION, 109

Derivation of Mild-Slope Equation, 109

Alternative Forms of Mild-Slope Equation, 112

Physical Interpretation of Mild-Slope Equation, 115

Mild-Slope Equation on a Slowly Varying Current, 116

Mild-Slope Equation with Energy Dissipation, 117

Mild-Slope Equation on a Permeable Bed, 118

TIME-DEPENDENT MILD-SLOPE EQUATIONS, 121

Derivation of Time-Dependent Mild-Slope Equations, 121

Alternative Forms of Time-Dependent Mild-Slope Equations, 122

Time-Dependent Mild-Slope Equations for Random Waves, 123

PARABOLIC EQUATION, 125

Derivation of Parabolic Equation, 126

Alternative Forms of Parabolic Equation, 127

Parabolic Equation for Large Wave Angle, 128

Parabolic Equation in Non-Cartesian Coordinates, 129

101


Weakly-Nonlinear Parabolic Equation, 131 Extension of Parabolic Equation Model, 133

BOUSSINESQ EQUATIONS, 134

Derivation of Boussinesq Equations, 134 Modified Boussinesq Equations, 135 Boussinesq Equations with Breaking Dissipation, 136 Boussinesq Equations for Waves on a Permeable Bed, 137

NONLINEAR SHALLOW-WATER EQUATIONS, 138

Derivation of Nonlinear Shallow-Water Equations, 138 Nonlinear Shallow-Water Equations on a Permeable Bed, 139

NONLINEAR MILD-SLOPE EQUATIONS, 141

Derivation of Nonlinear Mild-Slope Equations, 142 Relationship with Other Wave Equations, 144

VALIDITY RANGES OF WAVE EQUATIONS, 150

SUMMARY, 153

NOTATION, 155

REFERENCES, 157

Introduction

Waves transform in shallow water due to shoaling, refraction, diffraction, reflection, transmission, bottom friction, breaking, etc. Predicting wave transformation is indispensable in coastal and ocean engineering practices because wave action causes various important phenomena such as forces on structures and sediment transport. Analytical solutions, numerical models, and physical models can be used for the prediction. Among them, numerical modeling has achieved remarkable progress owing to development of wave theory and computer technology. A comprehensive review has been given in [56] for numerical models based on the mild-slope equation and its parabolic approximation, and the Boussinesq equations. This chapter presents various numerical model equations and their extended forms for predicting wave transformation in shallow water.

The mild-slope equation (MSE) includes the combined effect of refraction and diffraction of linear waves. Predictive models based on this equation have been used for a wide variety of engineering problems. Time-dependent forms of the MSE have been developed for improving numerical efficiency and treatment of boundary conditions and for extension to random waves. Parabolic approximations of the MSE have been developed to increase its computational efficiency remarkably.

Equations for Numerical Modeling of Wave Transformation 103

The Boussinesq equations are model equations for weakly nonlinear waves in shallow water. These equations have been solved both by direct numerical calculation and Fourier transformation. The equations have been modified to extend their applicable range to deeper water.

The nonlinear shallow-water equations are fully nonlinear wave equations in very shallow water in which hydrostatic pressure distribution is assumed. Because these equations do not require any empirical formula for energy dissipation due to wave breaking, they have specifically been used to predict the wave transformation in surf and swash zones.

Fully nonlinear and fully dispersive wave equations have been recently derived. These equations can reproduce even strongly nonlinear transformation. In addition, all the above model equations can be derived as special cases of the most generalized equations described in this chapter.

The following sections of this chapter derive simple versions of the basic model equations to clarify the concepts behind various theories. Then, the chapter presents extended versions especially for the analyses of wave-current interaction, wave dissipation, and waves on a permeable bed. The last section describes the applicable ranges of the model equations.

Basic Equations and Boundary Conditions

This section summarizes three-dimensional basic equations and boundary conditions for wave motions in a water body and permeable layer. The wave equations derived in the subsequent sections have a common characteristic in the sense that the basic equations are integrated in the vertical direction to yield horizontally two-dimensional (2D) equations. This reduction of the dimension simplifies the theory and makes it much easier to calculate wave transformation numerically.

Basic Equations and Boundary Conditions for Waves on a Fixed Bed

Basic Equations in Terms of Particle Velocity. In describing water waves, the viscosity and compressibility of water can usually be neglected. Then, the basic equations are the continuity equation for an incompressible fluid and the Euler equations of motion [68]. To integrate the equations in the vertical direction, symbols are defined to distinguish the vertical direction from the horizontal directions:

x 3 = (x, z) = (x, y, z) (1)

u3 = (u, w) = (u, v, w) (2)


V 3 -- V, - 0 x ' O y ' / ) z (3)

where x, y = horizontal coordinates z = vertical coordinate

u, v, w = components of water particle velocity in the x, y, and z directions, respectively

Vectors with subscript 3 denote three-dimensional quantities, whereas those without subscript are for horizontally 2D quantities.

With the above notations, the continuity equation, and the momentum equations in the horizontal and vertical directions (in 3D) are written as follows:

~W Vu + ~z = 0 (4)

~ u ~ u 1 + ( u V ) u + w . . . . V p (5)

~t ~z p

/)w /)w 1 0p + ( u V ) w + w = - - ~ (6 )

~t ~ - g p /)z

where p = pressure p = density of water g = gravitational acceleration.

B a s i c E q u a t i o n s in T e r m s o f V e l o c i t y P o t e n t i a l . Because the motion of an inviscid fluid starting from rest remains irrotational, the wave motion can be regarded as irrotational and thus the velocity potential ~3 exists:

U 3 = V3(~3 (7)

The continuity equation (4) is rewritten as Equation 8 and the momentum equations (5 and 6) can be integrated to yield the Bernoulli equation (9):

~2~3 - ' 0 (8) V2~3 = V2@3 + ~Z 2

~9~3 1 )2 P + (V3~) 3 + gz + -- = 0 (9)

/)t -2 p

The Laplace equation (8) and Bernoulli equation (9) are simultaneous partial differential equations in terms of ~3 and p, and equivalent to the continuity and momentum equations (4 to 6). Because the pressure p usually does not appear in


the boundary conditions, the Laplace equation is first solved to obtain ~3 for a given set of boundary conditions, and then the Bernoulli equation is used to determine p. This means that any velocity field expressed by a velocity potential can be generated by a certain pressure distribution and without any shear stress.

Boundary Conditions. The boundary conditions for water waves on a fixed bed consist of the dynamic and kinematic free surface boundary conditions (10 and 11), and the kinematic bottom boundary condition (12):

[~(~3 1 2 ] p = - p --~---+-~(u 2 +W ) + g ~ = 0 ( z = ~ ) (10)

w = ~ + (uV)~j (z = ~) (11)

w + ( u V ) h = 0 ( z = - h ) (12)

where ~ = water surface elevation h = still water depth

The dynamic boundary condition implies the constant pressure on the surface. The kinematic boundary conditions require that any water particle on a boundary remains on that boundary. One may obtain the equations for the latter condition by taking the total derivatives (D/Dt = ~//)t + (uV) + (w~/~z)) of z = and z + h - 0.

Non-dimensionalization. It is often important to know the order of magnitude of each term in the basic equations and boundary conditions. For this purpose, the dimensional quantities are non-dimensionalized as follows:

X = LlX*, z = hlZ*,

- - w =

~ = hieS*, h = hlh*,

[i = h l /L 1

t = ( L 1 / g ~ ~ ) t *

p = pghlEP*

(13)

where L1, h 1 = representative length scales in the horizontal and vertical directions

e, ~5 - orders of the non-dimensional wave amplitude and relative water depth, respectively


Then, Equations 4 to 6, 8, 10 to 12 are rewritten as

~)w* V ' u * + ~ = 0

/)z* (14)

[ + e ( u * V * ) u * + w* = V* * /)t* /)z*J P

(15)

+ e (u* V*) w* + w* /)w* ~t* /)z* = - 1 -/)z----- ~

(16)

1 t92~; V2(~ 4" ~2 ~)Z,2 = 0 (17)

E ()(~ + - - ( U .2 + ~2W'2) + Z* + p*=O (z* = e~*) /)t* 2

(18)

[ 1 P*= Ot~; e ; , - ~ - F + ~ ( u * 2 + 5 2 w * 2 ) + = 0 (z*=e~*) (19)

w, ~;* = + E [(u, V'K*] ( z , = e ; , ) (a0) ~t*

w* + (u* V*)h* = 0 (z* = - h*) (21)

Energy C o n s e r v a t i o n Equat ion . Because the mechanical energy is conserved for a flow of an inviscid fluid, the time rate of change of total energy in a certain volume V fixed in space is equal to the sum of the net energy inflow into it and the work done by pressure through its surface S:

] 0 ] 0"-t" (V3~) 3 + pgz dV = - -~ (V3~) 3 -/- pgz + p (u3) n dS (22)

where the subscript n denotes the component of a vector in the direction normal to the surface.

Because V is fixed in space and z is independent of t, the left side becomes

) L.S. = p ~- ~3(~3 (V3(~3) dV (23)

By rewriting the left side using the Bernoulli equation (9) and then converting the surface integral to a volume integral using the Gauss theorem, the right side becomes


~s ~r R.S. = D "-'~" (V3(~3)n as

(~'3 V3,3 ) dV = PfvV3 k ~9t (24)

Then the following energy conservation equation is obtained:

;v c3~3 = (25) V2~3dV 0 -57

To derive this equation, the momentum equations have been used through use of the Bernoulli equation, but not the continuity equation. This implies that, even if the continuity equation may not be satisfied, the total mechanical energy is conserved only by satisfying Equation 25. This is the case for the MSE.

Basic Equations and Boundary Conditions for Waves on a Permeable Bed

For waves on a permeable bed as depicted in Figure 1, the motion is to be analyzed both in the water column and in the permeable layer. For the water column, the basic equation and the free surface boundary conditions are the same as for the waves on a fixed bed. The basic equations and the boundary conditions on the interface and bottom are described in the following section [55].

Basic Equations in Terms of Particle Velocity. First, by denoting the three components of the seepage velocity as

Up3 "-(Up, Wp)"-(Up, Vp, Wp) (26)

h u

Figure 1. Definition sketch for waves on a permeable bed.


the continuity equation is expressed as follows:

~)Wp Yap + - ~ Z = 0 (27)

When an obstacle is fixed in an accelerating fluid, it exerts a force to the fluid. This can be expressed by introducing the virtual mass coefficient C m. Then, by adding the mass of water in the permeable layer, the apparent total mass C r per unit volume is obtained as:

C r = Ep + (1 - Ep)C M (28)

where Ep -- porosity

By using C r, the momentum equations can be derived as

VOUp ~Up] 1 Epl.) Cr L/)t + (UpV)Up + Wp ()Z J - -p Vpp - -~p Up

E2pCf [Up3[U p (29)

[~Wp ~Wpl = 1 ~pp Ep_~ E2pCf Cr L ~)t + (upV)wp + Wp ~Z J -g p ~z Kp Wp ~ p lu.I Wp (30)

where the last two terms on the fight side of these equations represent the linear and nonlinear resistance forces, respectively

and K = intrinsic permeability cPf= turbulent friction coefficient

= kinematic viscosity

The values of K_ and Cf are given in [87]. Investigation of relative magnitude of each term has s~own that the nonlinear resistance force cannot be neglected in usual situations, so that equivalent linear resistance should be substituted even for small amplitude waves [22].

Basic Equations in Terms of Velocity Potential. For a small amplitude wave motion in a permeable layer, nonlinear terms can be neglected and the seepage velocity can be expressed by a velocity potential:

Up3 = V3(~p 3 (31)

and the continuity and momentum equations are rewritten as

~2(~p3 --0 (32) V2~p3 = V2(~p3 + ~z 2


~)p3 Pp Cr Ot + gz + ~ + fpG~p 3 --0 (33)

P

where fp = coefficient for the linearized resistance force

Boundary Conditions. The boundary conditions at the interface are the continuity of the pressure and the flow rate:

p = pp (z = - h) (34)

w + (uV) h = Ep [Wp w (UpV) h] (z = - h) (35)

An impermeable bed is assumed at the bottom of the permeable layer. This yields the boundary condition:

Wp + (UpV) h t = 0 (z = - h t ) (36)

where h t = h + hp = total water depth.

Mild-Slope Equation

Two assumptions that enable a simple mathematical formulation of wave transformation on a sloping bottom are small amplitude and mild-slope assumptions. These lead to the mild-slope equation (MSE), which was first derived by Berkhoff [4] and then expressed in various ways (for example, [3, 33, 61, 86]). The MSE includes effects of both refraction and diffraction, and thus has widely been used in engineering practices. In the following, the MSE is derived from the energy conservation equation, which may give a clearer physical meaning. The procedure is somewhat similar to [3].

Derivat ion of Mi ld-S lope Equat ion

The basic equation and boundary conditions for small amplitude waves on a fixed sloping bottom are obtained from Equations 8, 10 to 12 as

V2~3 + 32~3 = 0 (37) ~9z 2

= __1 ~3~3 (z = 0) (38) g Ot


3~___L3 = 1 ~21~3 (Z = O) (39) OZ g Ot 2

+ (V~3) (Vh) = 0 (z = - h) (40) 3z

where Equations 10 and 11 are linearized and combined to yield Equations 38 and 39.

Upon the assumption of a mildly sloping bottom, the second term in Equation 40 can be neglected in the zero-order of the slope (i.e., for a horizontal bottom). The velocity potential can be expressed as follows:

~3 = Z ( z ) , ( x , t) (41)

where Z (z) o~ cosh k(h + z) (42)

Equation 39 requires the following dispersion relation:

0.2 = gk tanh kh (43)

and Equation 38 gives the water surface elevation in terms of the velocity potential at the still water level.

This solution agrees with the small amplitude wave theory on a horizontal bottom. The vertical distribution function Z satisfies

32Z - k 2 Z = 0 ( 4 4 )

~gz 2

()Z 0 .2 = ~ Z (z = 0) (45)

3z g

3Z = 0 (z = - h ) (46)

3z

In the first order of the bottom slope (i.e., in the equations with terms proportional to the bottom slope), only the vertically integrated energy conservation equation is considered instead of satisfying the continuity and momentum equations at each elevation. The energy conservation equation is obtained from Equation 25 for a water column that has a projection area of dx x dy and a height from the bottom to the surface:

/ V21~3 q- ~Z 2 (47)


where the velocity potential in the form of Equation 41 is considered and the upper limit of the integral is changed from the water surface to the still water level upon the small amplitude assumption. The vertical distribution function Z for a horizontal bed is regarded to give a good approximation even to that for a sloping bottom, but its derivatives may include a substantial error because of its sensitivity to the distribution function. Therefore, in the following derivation, Equation 41 is used for integration in the vertical direction but not in differentiat- ed forms.

By using the following identities:

ZV2~3 -- V(Z2V{~) + Z(V2Z)O (48)

Z2V~----- ZV(~3 - - Z(VZ)~ (49)

the first term in Equation 47 is rewritten as

0 V2~3dz _ V 2V@dz -- (Vh) (Z2Vt~)l_h + (V2Z) @dz

= V 2V,dz - (Vh) (ZV,3 - Z (VZ),)[_ h + (vEZ) ,dz (50)

The second term is reexpressed by using integration by parts twice and substituting Equations 39, 40, and 44 to 46 as

f._q O2Z ( ~ 3 OZ /1~ ( ~ ) 3 ~Z /! qZ 02(~3 dz = (~3 dz + Z (~3 -- Z I~3 0Z 2 h ~ ()Z ()Z t)Z ()Z -h

~ ~ k 2Z2t~dz- Z ~ 0 2 r 1 6 2 +Z(Vt~ 3)(Vh)l_h = h g Ot 2 g

(51)

Then, substitution of Equations 50 and 51 into Equation 47 yields

E( Oz) i (SOz) V 2dz Vt~ +k 2 2dz t~- g

= - (~:?V2Zdz) t~ - (Vh) (VZ) Zl_h t~

Z 02 ~2~ g /)t 2

(52)

where the terms on the fight side are of the second order in the bottom slope and therefore will be neglected hereafter. These terms have recently been included in a modified version of the modified slope equation MSE [8].

If the proportionality constant in Equation 42 is taken to give


cosh k(h + z) z = (53)

cosh kh

Equation 52 becomes

V(CCgVr + (k2CCg - 0.2)~ _ ~ = 0 (54)

which is the time-dependent form of the MSE [86]. Then, Equation 38 gives the water surface elevation:

1 /9r . . . . (55 )

g ~t

On assuming a sinusoidal oscillation:

~(x, t)= ~(x)e -i~ (56)

(x, t) = ~ (x)e -i~t (57)

the MSE can be obtained from Equation 54:

V(CCgVt~) + k 2 CCg~) = 0 (58)

From Equation 55, the complex amplitude of the water surface elevation is expressed as

~= i__~ ~) (59) g

This relationship is independent of the water depth for a given wave frequency. In spite of the assumption that results in the vertical distribution of the veloci-

ty potential as given by Equation 41, the MSE can be applied up to a ~ slope [6], which covers most practical situations (Figure 2). Recently, a finite series expression of the velocity potential has been adopted to improve the accuracy of the MSE [67].

Alternative Forms of Mild-Slope Equation

Alternative forms of the MSE can be derived by taking different proportionality constant in Equation 41. A useful example can be obtained by taking

Equations for Numerical Model ing of Wave Transformat ion 113

4- O.2 -

0.1

i q . g t j

.~ O.O& -

L

II.02

0.01 0.1

§

4.

I I I I i ,

0.2 0./. 1 2 /. Ws

2 1 0~ ~2 0.1 t~loJ

Figure 2. Reflection coeff ic ient as a funct ion of bot tom inclination. Lengths are normalized by deepwater wave number. Curve: refract ion-di f f ract ion model. Crosses: three- dimensional model [5].

(I)3 = Z' (z)(~' (x, t) (60)

Z 1 cosh k(h + z)

Z'-- ~fCCg "- 5/CCg cosh kh (61)

Then, the time-dependent form of the MSE and the expression of the water surface elevation become

V2t~'+(k 2 0'2 )(~ ' 1 ~)2t~ ' Cfg CCg t)t 2 - 0 (62)

1 ~ ' (63) ; = g~/CCg bt


and the MSE and the complex amplitude of the water surface elevation become

V2@ ' + k2@'= 0 (64)

gafCi2g (65)

where

r (x, t) = t~' (x)e -izt (66)

The Helmholtz equation (64) was obtained in [78] by transforming the dependent variable in Equation 58.

An interesting feature appears in the expression of the energy flux in terms of ~'. The energy flux due to waves is obtained from the work done by the dynamic pressure to the leading (second) order of the wave amplitude:

F = dUdZ (67)

where Pd = the dynamic pressure induced by waves

Using the linearized Bernoulli equation and the definition of the velocity potential,

[ z' Pd = Re -p /)t .] pt3 Re [i~' e -iot ] (68)

u = Re [V~3] = Z' Re [V~' e -i~ ] (69)

Substitution of Equations 68 and 69 into Equation 67 yields

F = po Re [i~' e -iot ] Re [V@' e -izt ] (70)

which shows that the proportionality constant in the expression of energy flux is pt~ and thus independent of the water depth. This implies that if ~' and its normal derivative are continuous at the boundary, the energy flux is also continuous. The boundary condition used in [20] can also be derived from this result.

Because the MSE is an elliptic equation, its numerical solution requires fairly long computational time, and thus some efficient numerical solution techniques have been investigated [51-54, 63, 75, 76, 96]. A technique that solves the wave ray and amplitude separately as in the case of refraction problem has also been developed [ 16, 98].


Physical Interpretation of Mild-Slope Equation

To give an idea of refraction and diffraction effects included in the MSE, the complex amplitude of the velocity potential is expressed in terms of the amplitude a and phase angle 0 [5, 16]:

~ - a e i0 (71)

On substituting this expression into the mild-slope equation (58), the following equations are obtained from the real and imaginary parts, respectively:

V(CCgVa) (V0) 2 -- k 2 + (72)

CCga

V (a 2 C C g V 0 ) = 0 (73)

By invoking V0 = k, the term in the parentheses in Equation 73 becomes t~a2Cg, which is essentially the energy flux. Therefore, Equation 73 represents the energy conservation. Equation 72 is essentially the same as the eikonal equation to determine the wave direction in refraction problems, but it includes an extra term on the right side. If, for example, there is a local maximum of a in a constant water depth, the second term in Equation 72 takes a negative value, which makes the magnitude IV01 of the real wave number smaller than that determined from the dispersion relation and thus is equivalent to a larger water depth at the location. This results in increase of the distance between two wave rays and decrease of the energy density and thus wave amplitude. This phenomenon is interpreted as the dispersion of wave energy due to diffraction.

For an alternative form of the MSE, substitution of the following to Equation 64:

~' = a' e i0' (74)

results in

V2a ' ( V 0 ' ) 2 -- k 2 + ~ (75)

a p

V (a'2V0 ') = 0 (76)

This equation implies that the transport velocity of a p2 is equal to the wave number vector V0'. Thus, if only the wave number vector but not necessarily the group velocity is calculated accurately, the distribution of a' can accurately be predicted.


Mild-Slope Equation on a Slowly Varying Current

A mild-slope equation in the presence of a slowly varying spatial current has been derived [33]. The waves and current motions are separated as

u = U + u w, w = W + w w (77)

where capital letters and suffix ties, respectively, and

denote the current- and wave-induced veloci- w

~1~3 (78) Ilw = Vl~3' Ww -" OZ

From the continuity equation and bottom boundary condition for current,

W = - z (VU) (79)

The assumptions on the magnitudes of the velocities are O (U) = 1 and O (Uw) = O (w w) = O (W) = e. Then, the basic equation and boundary conditions are written as

V2~3 + ()2~3 = 0 (80) ~Z 2 '

+ (UV)t~3 d- g~ = 0 (z = 0) (81) Ot

0,3 O~ = - - + ( u v K + ~ ( v u ) (z = o) (82)

bz bt

0,3 ~z

+ (V@3) (Vh) = 0 (z = - h)

Equations 81 and 82 are rewritten as

(83)

= _ 1 D~___ Z (z = 0) (84) g Dt

= + ~ (vu) (z = 0) (85) /)z Dt

D where ~ = m + (UV) (86)

Dt 3t


With these definitions, one finds the following mild-slope equation for waves on a slowly varying current [33]:

V(CCgVr + (k2CCg De D2r - 0 (87) - o 2)~ - (VU) Dt Dt 2 -

where the intrinsic frequency o satisfies the dispersion relation (43) and obtained from the local angular frequency to as

= to - k U ( 8 8 )

For a monochromatic and uni-directional wave field, the previous equation can be solved [49]. However, if a local wave field consists of more than one component due to more than one wave path from offshore region, the intrinsic frequency cannot be determined uniquely. This causes a problem in solving Equation 87. Various versions of Equation 87 for wave-current interaction have been reviewed and examined in [50].

Mild-Slope Equation with Energy Dissipation

An energy dissipation term may be introduced into the MSE as follows [ 13]:

V(CCgV$) + (k2CCg + io fD)$ = 0 (89)

By substituting Equation 71 into this equation, the following equation can be obtained from the imaginary part:

V(a2CCg V0) = - a2 fD (90)

from which fD is understood as an energy dissipation coefficient. Various formulas of fD are given for a porous bottom, viscous mud bottom, laminar bottom boundary layer, densely packed surface film, and others in [13].

An empirical formula is often used for wave breaking [25]. The breaking point is first determined in [93]:

Y =Yb (91)

where y denotes the ratio between the water particle velocity u e at the still water level and wave celerity C:

U c ), = --~ (92)

The value of y at the breaking point is given as

Tb = 0.53 - 0.3 exp[-3~/h/L o ]+ 5 tan 3/2 ~ exp [-45~/h/L o - 1) 2 ] (93)


where tan 13 = bottom slope L ~ = deep-water wavelength

Then, fD is given so that it yields constant wave height to water depth ratio 7s on the uniformly sloping beach and vanishes when the ratio becomes smaller than ~'r because of increase in water depth or others:

fD= '~ y s 2 7~ tanl3 (94)

~/s = 0.4(0.57 + 0.53 tan 13) (95)

~/r =0.135 (96)

Mild-Slope Equation on a Permeable Bed

The MSE on a permeable bed was derived in a manner similar to the MSE on a fixed bed [79]. The basic equation in the permeable layer is the Laplace equation:

~2(~p3 -- 0 (97) V2@p3 + ~Z 2 ....

The boundary conditions at the interface are Equations 34 and 35, which can be rewritten in terms of the velocity potential as

~ p 3 ~ 3 = Cr + fpl3@p 3 (Z = - h) (98) ~---~ ~t "

~3 I~P 3 ] Oz + (Vh) (V{~3) = Ep 3z + (Vh) (V~p 3) (z = - h) (99)

These interface boundary conditions do not include the continuity of the tangential velocity, which requires the introduction of boundary layer. However, the energy dissipation in the boundary layer is usually small compared to that in the permeable layer, so that it can be neglected [83].

The boundary condition (36) at the bottom of the permeable layer is also rewritten as

~r + (Vht) ) = 0 (Z = -- h t) (100) ~z (V~)p3

By using the separation of variable technique, the analytical solution is first derived for a horizontal bottom and interface:

Equations for Numedcal Modeling of Wave Transformation 119

(~3 = F (z)0(x, t) (101)

~p3 = Fp (z) ~ (x, t) (102)

where F(z)= E p sinh (khp)exp [k(h + z ) ] - ~Sp cosh [k(h + z)] ep sinh (khp) exp (kh) -/Sp cosh (kh)

(103)

cosh [k(h t + z)] (104) Fp(Z) = ep sinh (khp) exp (kh) - ~p cosh (kh)

~ip = ep sinh (khp) - (C r - ifp) cosh(khp) (105)

0 .2 - - gk I~p exp (kh) sinh (khp) - (Sp sinh (kh) ep exp (kh) s i n h ( k h p ) - ~p cosh (kh)

(106)

Next, by substituting Equations 101 and 102 into the following integral equation:

f qhF 2 -h V3t~3dz + ~_ s ht

(107)

the following MSE on a permeable bed is obtained:

V (otV~) + k 2 tx~ = 0 (108)

where o~ = tx 1 + Ep(C r - ifp)O~ 2

~1 = [32h { 2~~ [1 - exp(-2kh)] - 2~-~~ [1 -

1 [ sinh(2khp)] ~2 = - ~ 2 hp 1+ 2khp

~l = [ep exp(kh) sinh (khp) -/Sp cos (kh)] -1 ~ 2 -" ep exp(kh) sinh (khp) - (~Sp / 2) exp kh [~3 =-- (~p / 2) exp(-kh)

exp (2kh)]- 2~2~3 }

(109)

Figure 3 compares distribution of the root mean square (rms) water surface fluctuation calculated by using Equation 109 and measured data from a laboratory wave flume study. Instead of the wave height, the rms value, which represents the wave energy, is used for comparison because transmitted waves are not sinu-


o , :

D m "~ "": . . . . . . ; "" "' �9 .'..'7.' :.'" ": ; " ; . X-:"

' ,tl.l.,NrtB.'~.'% X ~

5 . 8

3 . 0

o-~ Z . 5 E U

2 . 0

1 . 0

0 . 8

O . O 0

�9 L ' �9 i .... �9 i '�9 " ' i " � 9 �9

I I i - 4 . 6 6 p m

,.. C A L C U L A T 1 ON T - - 1 . 8 2 el

�9 EXI)I~Is I M E N T D O -- O . 6 0 ,:m

Doo " 8 7 . 6 0 c m

B - - s O 0 c m

@

X,~

. , A , �9

t 2 3 4 6 6 7 a 9 x o

x (m)

A

E 2 0

0 9 1 0

! ! i - t . 4 7 r - CALCULAT I ON

T - - 1 . 8 1 s �9 E X P E R I M E N T

D s - - 8 . O 0 c m

Doo "- 3 9 . O 0 c m

_ B -- 2 3 8 O O c m

X, B X,h

, , , I . . . . . . I , , I . 0 1 2 s 4 8 6 7 8

X (m)

Figure 3. Distribution of root mean square values of water surface fluctuation due to a submerged permeable breakwater [79].


soidal due to nonlinear effect. A good agreement implies that both transmission and reflection coefficients are accurately calculated.

Time-Dependent Mild-Slope Equations

Time-dependent forms of the MSE have been proposed for improvement of numerical calculation [10, 52, 63, 71, 92] or for application to random wave analysis [26]. The following presents time-dependent mild-slope equations for monochromatic waves and then introduces those for random waves.

Derivation of Time-Dependent Mild-Slope Equations

For a sinusoidal oscillation expressed by Equation 56, the mild-slope equation (58) can alternatively be written in a hyperbolic form as

a2~ V(CCgVtD- n 0-~ = 0 (110)

cg 1( / where n - = - 1 + ~ (111)

C 2 sinh 2kh

A hyperbolic equation can be split into two simultaneous first-order partial differential equations. By considering physical meanings, the following two quantities that correspond to the water surface elevation and flow rate per unit width are introduced:

1 ~)~ _ _ . . . .

g at C 2

Q = V~ g

(112)

Then, by substituting the previous definitions into Equation 110, one can obtain an equation equivalent to the continuity equation. One can also obtain an equation similar to the momentum equation by cross differentiation of Equation(s) 112. These are written as

~9r 1 ~ + - V ( n Q ) = 0 /)t n

a Q + c2vr = o at

(113)


These two equations constitute a set of simultaneous partial differential equations in terms of ~ and Q, and equivalent to the MSE. These are called time- dependent mild-slope equations.

The definitions of the two dependent variables are the same as Nishimura et al. [71], but Copeland [10] used a different definition for Q, which leads to a physical meaning different from the flow rate by the factor of n = Cg/C.

Alternative Forms of Time-Dependent Mild-Slope Equations

Different forms of the MSE and different definitions of two dependent variables yield slightly different time-dependent mild-slope equations.

By substituting the following definition of the vertical distribution function:

~)3 "- z u (Z) ~)u (X, t) (114)

Z" Z 1 cosh k(h + z) (115) = ~ = ~ cosh kh

into Equation 52, the MSE becomes

V(C2V(~ it) -- ~)t2, = 0 (116)

Then, by defining

1 /)~"

~= g~-ff Ot

C 2 R= Vr (117)

another set of time-dependent mild-slope equations is obtained:

~ + VR =0 Ot

OR + cEv(~/n~)- ~ = 0

3t ~ v q l l p ~

(118)

In [29, 90], because the definition of the water surface elevation is different by the factor of x/if, a correction factor becomes necessary after solving the equations.

In Watanabe and Maruyama [92], the vertical distribution function is defined as

(~3 -- z u ' (Z)~ u' (X, t) (119)


Z 1 cosh k(h + z)

n n cosh kh

and then the MSE has the following form:

2 V(~") 1 ~)2(~H, V - -

n n ()t 2

(120)

=0 (121)

By defining

1 ~ "

gn /)t

C 2 Q = V~" (122)

gn

the corresponding time-dependent mild-slope equations have the following form:

- - + V Q = 0 Ot OQ c 2 - - + V(n~) = 0 Ot n

(123)

When breaking transformation is analyzed, an energy dissipation term is added to the second equation:

o~Q C 2 - - + V(n~) + foQ = 0 (124) /)t n

The numerical model based on the first equation of Equation(s) 123, and Equa- tion 124 has been tested for wave transformation due to refraction, diffraction, and breaking to predict wave-induced nearshore current and sediment transport and resulting bottom topography change. Figure 4 shows the comparison between calculated and measured breaking lines and wave height distributions.

Time-Dependent Mild-Slope Equat ions for R a n d o m Waves

A sinusoidal oscillation is assumed in the time-dependent mild-slope equations previously derived. Another time-dependent mild-slope equation is Equa- tion 54, which has been applied to the propagation of wave groups [41 ]. Because the coefficients in the equation are evaluated at a certain frequency and their

1 2 4 O f f s h o r e E n g i n e e r i n g

4.0-=

3.0-! E

g 2.0- O m

0.t 0.0

Incident waves

Breaking point [ Computed ( ~o/C' = 0.35)

0 Measured (Om ~ y~ 4m) �9 Measured (4m <.yr 8m)

1 . ' 1.o 2.0

{

3.0 Distance onshore x (m)

. i v �9 j /

4.0

off 5.0 ". . . . . . . . . , , �9 , w ! ,

z 4.0 ~ " ~ y=4. Y0 �9 �9 "~ . �9 �9 e

" 0.0 ~ " r * ' ~ ' i r 0.0 1.0 2.0 3.0 4.0

flu 5.0 , , , , , " , " , " '

~ 2.0 1.o �9 " ' " ' ~ " t >-o0I ,'v v a I ~ - - I i , l ; �9

~: 0.0 1.0 2.0 3.0 4.0

4.0 y=2m .. 3.01 _ _ . .~. . . . . . . . . " " " " ,

"~ 1.0 0.0 �9 i , L ~ . ~ , ,

0.4) 1.0 2.0 3.0 4.0 Distance onshore x (m)

C o m p u t e d

0 M e a s u r e d (Ore ~ ; u ~ 4 m )

A M e a s u r e d ( 4 m < t r~ 8 m )

F i g u r e 4 . L o c a t i o n o f b r e a k e r l i n e a n d c r o s s - s h o r e d i s t r i b u t i o n o f w a v e h e i g h t [ 9 2 ] .

changes due to the deviation of the frequency are correct to the first order, the model is valid only for a small range of frequency.

Time-dependent mild-slope equations, which are applicable for wider wave spectra, are derived as follows [26]. By considering that random waves consist of an infinite number of component waves with different frequencies, the sinusoidal oscillation with a representative angular frequency ~ is removed from ~':


t~' = ~' (X, t)e -iSt (125)

Hence, for an arbitrary angular frequency o = i5 + (Ao), ~' is written as

~' (x, t) = ~(x)e -i(A~ (126)

and thus

/ ) r - i ( A o ) r - - ( A ( ~ ) 2 ( 1 2 7 ) ~)t t)t 2 -

In an alternative form (64) of the MSE, k 2 is a function of the frequency and approximated by a Pad6 approximant:

k2 = b 0 + b l (AO)+b 2(AO) 2

1 - a 1 ( A o ) (128)

where the coefficients are determined according to the spectrum range of the irregular waves. Substitution of this approximation into Equation 64 with use of Equations 127 yields

V2~) ' - ial V2 + bor + ib 1 - ~ - b 2 0t 2 = 0 (129)

All the coefficients in this equation are independent of the frequency. There- fore, it is used to calculate ~' composed of infinite number of waves with different frequencies, and thus enables direct calculation of random wave transformation. It is noted that, as seen from Equation 76, the transport velocity of a '2 in Equation 76 is the same as the wave number vector. Thus, as long as the Pad6 approximant to k 2 is accurate, the shoaling coefficient for a '2 and then the water surface fluctuation by Equation 63 can accurately be predicted.

Figure 5 shows a comparison of water surface fluctuation of shoaling random waves.

Parabolic Equation

Numerical calculation of the MSE requires a fairly long computational time due to the elliptic characteristic of the equation. Since Radder [78], various parabolic approximations have been proposed to save computer resources. This section presents alternative forms of the parabolic approximation.


AAA A A I ~, P , , , , , A An ,A,. ,A~ ~ , A AAII (~n~)~ V V~V . . . . ~ ~ v 1 2 ~ v v v

-s t(s)

�9 111eas.

"I ",,. A / A - ] (cm)O[ O ' :j ' ''' ,5 i,/': ~'~; V" V ' V~'10 V V V: V'~ "~v2~5

t(s) Figure 5. Time history of water surface fluctuation of random waves in shoaling water; top figure: measured incident wave history, bottom figure: calculated and measured histories at 4m shoreward [26].

Derivation of Parabolic Equation

Parabolic equation can be obtained by eliminating the second derivative in one (x) direction in the elliptic type MSE. This is based on the assumption that the change of amplitude in the direction (wave ray direction) is small compared to that in the direction perpendicular to it (wave front direction).

An alternative form (64) of the MSE can be rewritten as

@2r @2r c3x-- T + ~ - ~ + k2@'= 0 (130)

On considering waves that propagate approximately in the x-direction, the phase change in the direction can almost be removed by

(~'= ~' e iJkdxf (131)

in which the change of ~ in the x-direction is assumed to be small. Then, in the expanded form of the first term on the left side of Equation 130:

O2C)X 2~, _ k ( ~)2 I]/' ~)X2 "~XO~/' oxOk Xl/, / eil kdx - + 2ik + i 7 - - kEy ' (132)

the first term is neglected in the parabolic approximation. Thus Equation 130 is approximated by a parabolic equation:


2ik -~x + ~)y2 + i ~x = 0 (133)

which, for constant k, reduces to

2ik ~9~' + = 0 (134) t)X ~)y2

This equation is the simplest form of parabolic equation.

Alternative Forms of Parabolic Equation

In the previous derivation, it is not always convenient nor possible to use k. Thus, the modified wave number K, which is usually obtained from a constant depth or a uniformly sloping bottom, is used to reduce the spatial variation of the wave phase:

1~'= tIa'e i~Kdx (135)

Then, on substituting the following relationship:

V, = qj,ei(~ Kdx- ~ kdx) (136)

into Equation 133, the following parabolic equation can be obtained:

t)tI~' t)2~IJ' [ ~)k ] 2ik /)--~-+ ~)y2 + i~x + 2 k ( k - K ) ~g'=0 (137)

Another slightly different parabolic equation often appearing in literature is obtained by substituting Equation 135 into Equation 130 and neglecting /92W'/3x 2 [91 ]:

2iK -~x + ~)y2 + i ~ + (k 2 ) = 0 (138)

A parabolic equation derived from the original mild-slope equation (58) is

~)~ 1 (9 ( ~ / [ i ~)(kCCg) 1 2 i k ~ - ~ CC + , + 2 k ( k - K ) ~ = 0 CCg Oy g CCg ~x

(139)


where t~ = tlJe iIK dx (140)

The difference in the basic parabolic equations (133, 137-139) previously described is also found in extended equations for weakly nonlinear waves or non-Cartesian coordinates.

Parabolic Equation for Large Wave Angle

When the wave angle relative to the x-axis is large, the assumption to derive a parabolic equation should be modified. One choice is to consider the large angle of propagation [34].

For progressive waves expressed by

i(kxx+kyy) t~' = ae (141)

the x-component k x of the wave number vector should satisfy

(142)

which is approximated by a Pad6 approximant:

a0+a kx - (143)

l + b 1

where a 0 = 0.998213736 a 1 = --0.854229482 b 1 = -0.383283081

These values assure the error to Equation 142 is within 0.2%. The corresponding parabolic equation is obtained as

O31lt' 2 i k / ) ~ ' + 2k 2 V' /)2~, 2ibl (144)

~x (a 0 - 1) + 2 (b 1 - a l) /) y2 k /)xOy 2

Figure 6 shows the validity of the previous equation for obliquely incident waves with an angle of 45 ~ .

Another method has been proposed for wide angle on the basis of Fourier transformation [12, 14, 89].


/ / ~ - - - ~ \ . \ i i ! 1 / / . _ , \\\\\

l / / [ \1 / ~/ \\~\, \ , // I

Figure 6. Amplitude contours as calculated by large-angle parabolic equation model for incident wave angles of 0 ~ (solid line) and 45 ~ (dash line) [34].

Parabolic Equation in Non-Cartesian Coordinates

Another way to consider the variation of wave direction is to use non-Carte- sian coordinates, which almost trace the wave propagation directions.

Parabolic Equation in Orthogonal Curvilinear Coordinates. Orthogonal coordinates that trace refraction patterns on a uniformly sloping bottom are used in [6 I, 91 ]. A more general expression is given in the following [24]. As shown in Figure 7, the coordinates ~* and rl* are defined so that rl* = constant and ~* = constant, respectively, coincide with the directions of wave rays and fronts of refracted waves due to a distribution of wave number K. Because the units of the coordinates are taken as phase in radians, the distances d~ and drl corresponding to d~* and dYl* are

d~ = h~d~*, dq = hndyl * (145)


~ hnd~? ~

hcdE.*

(wave front)

+

+ (t~ ~ + d~',r/* + d r / ' )

(wave ray)

Figure 7. Orthogonal curvilinear coordinates.

where the scale (conversion) factors h~ and h n are

hg = hn = 1/K

Then, substitution of the equation:

r = ~iJei~ Kh~d~*

into the MSE:

1 /) CCg h~h n /)~ * h~ ~)~* h~h

t) CCg + k2CCgr = 0 rl 0rl* hrl 0rl*

(146)

(147)

(148)

and neglecting the second derivative with respect to ~* yields

1 3~ 1 1 3 ( h~ 3W 2 ik ~ - ~ ~ ~ CC ~ ~

h~ ~ * CCgh~ h n ~ * ( g h n 3xl* ) [ i 1 ~ ]

+ (kCCghn) + 2k(k - K) ~ = 0 CCghn h~, ~9~*

(149)

This equation is modified for large angle and applied to wave refraction, diffraction, and breaking of random waves [25]. Figure 8 shows an example of application to a field in which the spatial distribution of the significant wave height and mean direction are compared between calculation and measurement.


Ore< Hi/3= im I

7. 10 6

. . 4 5

�9 0 100 500 lO00m

Figure 8. Comparison of the spatial distribution of the significant wave height (length of arrows) and mean direction (direction of arrows) between calculation by parabolic model (dash lines) and measurement (solid lines) [27].

Parabolic Equation in Non-orthogonal Coordinates. Non-orthogonal coordinates were introduced in [36, 57]. As seen from an example below, some extra terms are introduced due to non-orthogonality of the coordinates ~ and 11.

2iK---~-+ 2iK + ~ - - ~ J " ~ - + "~x

+ (k 2 - K 2 ) tIJ'

an 2 +2ax

(150)

Figure 9 compares wave height distributions calculated by using orthogonal and non-orthogonal coordinates with measurement.

Weakly-Nonl inear Parabol ic Equat ion

A parabolic equation for diffraction of weakly nonlinear waves of Stokes type was derived for constant water depth [99]. For variable water depth, [38] and [58] derived the following nonlinear equation of Schr6dinger type:

bY 2ik ~ + ~

bx

_ k 2 C Cg

/ i ] 1 0 C C g -I- , + 2k (k - K)

CCg Oy CCg Ox

D ~ = 0 (151)


o s % i .... i - o , I I I I I I I I ! !

a , I B , , ,I I

| | I -

' I ! i

\ - ] . . . . . . . . I i

4 . 1 . 8 - ~ B

e L_ o

;; ~'tl--'-9.0 i, . ~

I! O. 6 - - ' ~ : ~ 3 . 6 ~

I: I I~o.=4

" ~ " i ' . r . ~, . . - - / , " , ' ' r . - / / r , - r

~ l l l 1 1 1 5 h!-,0.4 """

i

3

�9 P

4 . 9 . 0

I

!

1 . 5

1 . o

am

-r. 3:

0 . 5

0 . 0

I I i I I i i [

I . / ' ~ ' ~ . \ . [

I ! I . . . . 1 I ! ! ! -1.s -1.o -o.s o.o o.s ~.o ~.s

y(m)

Figure 9. Wave height distribution along section B-B'; circle: measured, solid and dash- dot lines: non-orthogonal coordinates used in [36] and [57], respectively, dash line: orthogonal coordinates [24].

cosh 4kh + 8 - 2 tanh2 kh (Y k where D = e 2 E~ = ~ IqJl (15 2)

8 sinh 4 kh ' g

Figure 10 compares the results calculated by the linear and weakly nonlinear parabolic equations with laboratory data, which indicate the importance of wave nonlinearity.


. . . . . .

~-o.o ~ I ~, i'", .0 s'%% o �9 �9

1 z l m )

,o i TM'..d(Uc~,\ _ - ' ~ ~ , �9 . , .o , , , , ' ' ' ' ' ' ' ; ,: , ;

L-_ 0 ' 5 I0 15 ZO

y l m )

0 I 2 3 4 5 6 7 8 9 I0 II ( X - 10 .5 ) ( . . )

Figure 10. Comparison among wave height distributions measured (circles) and calculated by linear equation (dash line) and by nonlinear theory (solid line) around a shoal [39].

Because Stokes wave theory, as mentioned later, is valid only when the Ursell parameter is smaller than 25, these nonlinear equations could give even worse results than linear equations in very shallow water. A modified nonlinear term is proposed in [40]. This agrees with the previous theory in deep water and gives correction of wave celerity in terms of relative wave height in shallow water. Accord- ing to this modification, the coefficient D in Equation 151 is substituted by D':

tanh (kh + t'2 e) D ' = (1 + f~D) - 1 (153)

tanh kh

fl = tanh5 kh, f2 = [kh/sinh kh] 4 (154)

where D = theoretical coefficient given by Equation 152

Extens ion of Parabo l i c E q u a t i o n M o d e l

A parabolic equation with energy dissipation term has been proposed and applied [13]. Wave reflection that is neglected in basic parabolic equations is considered in [34, 59].

Applications of various parabolic equations to field are found in [ 15, 85].


Boussinesq Equations

Boussinesq equations are weakly nonlinear shallow water equations that include terms up to the orders of e 2 (relative wave height squared) and 152 (water depth to wavelength ratio squared). The Boussinesq equations for variable water depth were derived by Peregrine [77]. As the coefficients are independent of frequency, the equations can also be used for random wave transformation. More- over, inclusion of the second-order terms in relative wave height allows to calculate wave-induced nearshore current simultaneously with the wave field.

Derivation of Boussinesq Equations

Boussinesq equations can be derived by an iterative procedure. In reference to the non-dimensionalized basic equations and boundary conditions 14 to 21, zero-order terms are first considered to yield zero-order relations and then the Boussinesq equations are derived by considering the terms up to the order of e and 152. In the following derivation, dimensional equations (4 to 6 and 10 to 12) are used by considering the order of each term in the non-dimensional forms.

As seen from Equation 16, zero-order terms in Equation 6 are the two terms on the right side, which can be integrated with the boundary condition 10 to yield the hydrostatic pressure distribution:

p = pg(~ - z) (155)

Then, from Equation 15, the horizontal components of the water particle velocity u are independent of z in the leading order, which enables the integration of Equation 14 with the boundary condition (12) to yield the linear distribution of the vertical component of the velocity:

w = - z(Vu) - (Vhu) (156)

In the next iteration, the first term on the left side of Equation 6 should be included because, as seen from Equation 16, it is of order of 152. Substitution of Equation 156 into Equation 6 and integration gives pressure distribution modified by the vertical acceleration term:

I 1 p = p g (~ - z) + - ~ ~- (Vu) + z (hu) (157)

In the horizontal momentum equation 5, the third term on the left side is of orders higher than e 2 because u is independent of z to the leading order. Then, by substituting Equation 157 into Equation 5 and taking the depth average, the following momentum equation is obtained:


t)fi h 2 ~9 h ~9 + (fi V) fi + gV~ . . . . V(V fi) + V [V(hfi)] (158)

~t 6 0t -2 ~-

where fi is the horizontal components of the depth-average velocity. The depth- integrated continuity equation is derived by integrating Equation 4 from the bottom to the surface:

"/..2 + V[(h + ~)fi] = 0 (159) ~t

The Boussinesq equations have been verified for wave transformation in shallow water [1, 17, 19, 65, 80]. A numerical calculation method based on the Fourier transformation has been developed [37, 60].

Modified Boussinesq Equations

Because of the assumption, the previous Boussinesq equations can be applied only in shallow water. To maintain the error in the wave celerity within 5%, the water depth to deep-water wavelength ratio h ~ o must be smaller than 0.22. Modified Boussinesq equations have been developed by applying the method [94] to improve the accuracy of the wave celerity [64, 66]. Keeping the same order of theoretical accuracy, the equations have higher numerical accuracy in deep water.

By using the continuity equation (159), the momentum equation (158) can be written as

~}t [(h + ~)fi] + V{ (h + ~) [fi, fi]} + g (h + ~)V~ . . . .

h 2 ~} + ~ - - V [V(hu)]

2 /)t

h 3 ~)

6 Ot V(Vu)

(160)

where the operation [, ] denotes the matrix generated from the two vectors as [a, b] = ab t (b t denotes the transposed vector of b). The previous equation is expressed in terms of the flow rate Q per unit width:

+ $

h~} + -- [V, Q] (Vh)

h 2 ~9

3 ~t

h V (VQ) + o: (Vh) ~- (VQ)

(161)


The equation of leading order terms in the previous equation:

~Q + ghV~ = 0 (162)

/)t

is taken its divergence and then gradient and multiplied by h 2 to yield

h 2 t) ~- V(VQ) + gh 2 V(hV2~) + gh 2 ([V, V~]Vh) = 0 (163)

which has the same order of accuracy as Equation 161. Then, by adding B times the previous equation to the momentum equation (161), the following modified Boussinesq equation can be obtained:

J - g ~ ~ + g ( h + ; ) V ~ = B + h2 0 , -~- V(VQ) + Bgh2V(hV2~)

h O h b + Bgh 2 ([V, V;]Vh) + ~ Vh ~ (VQ) + ~ ~- [V, Q] (Vh) (164)

Equation 164 with B = 1/15 can be used for h ~ o < 0.5 within a 5% error in the wave celerity [66].

Modified Boussinesq equations are derived by taking the dependent variable as the horizontal velocity at an arbitrary elevation [73]. The elevation za optimum for the wave celerity within the range 0 < h ~ o < 0.5 was found to be -0.39h, which closely agrees with -0.40h obtained from Equation 164 with B = 1/15. The equations have been applied to random wave interaction [74]. Parabolic models have been proposed by applying Fourier transform to the modified Boussinesq equations [9].

Boussinesq Equations with Breaking Dissipation

A model was proposed to incorporate the effect of wave breaking in the Boussinesq equation [84]. The model introduces excess momentum flux R b due to breaking waves:

~5 r R b = [(C - fi), ( C - fi)] (165)

1 -- ~ r / ( h + ~)

w h e r e ~r = thickness of the surface roller determined in a heuristic geometrical way

C = the wave celerity vector

Figure 11 shows the water surface elevation, cross-shore distribution of the wave height, and wave-induced current calculated by this model [82].


15 16 17 18 19 20 21 22 23 24 25

0.02 m,,m/s

0.1- 1 ~,o.oa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "" "7 r"'~ ~r~ "~ 1

0.o,- 1

i 3= o.o2- , \ ] 0.0-

2 4 6 8 10 12 14 16 18 20 22 24 26 28

0.1-

.~o.oa . . . . . . . .

0"02-. - ' [

0.0- 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Diitonce (m)

Figure 11. The water surface elevation and wave-induced current calculated by Boussi- nesq equations incorporated with breaking model and comparison between calculated and measured cross-shore distribution of the wave height [82].

Other breaking models that are incorporated with the Boussinesq equations are proposed in [30, 31 ].

B o u s s i n e s q E q u a t i o n s for W a v e s on a P e r m e a b l e B e d

Equations for weakly nonlinear shallow water waves on a permeable bed can also be derived upon assumptions similar to those in Boussinesq equations. By following the derivation in [ 11 ], the following Boussinesq equations on a permeable bed are obtained:

- - + V[(h + ~)fi] + V[EphpUp] - 0 (166) /)t

3~ + (f ir) fi + gV~ . . . .

Ot h ~)

+ ~ -~-V[V(EphpUp )]

h 2 c)

6 Ot

h b V (Vfi) + -~ ~ V[V(hfi)]

(167)


[~p ] 1 cq Cr LOt + (UpV)gp + gV~ = ~ ~-~ V[V(h2g)] + ~-~ V[h(V(Ephpgp))]

-- C r ~- + -~p V(VUp) + -~- V(h - hp) (V~p)

hP2 V(Vhtgp)+(Vh)(Vhtup) - - ~ p uP ~ p (168)

Nonlinear Shallow-Water Equations

When the wavelength is extremely long compared to the water depth, the basic equations and boundary conditions are significantly simplified and nonlinear shallow-water equations are obtained [88]. Thus, vertical acceleration can be neglected and hydrostatic pressure distribution results. As can be understood by using the characteristics method, the equations do not allow waves of permanent form on a horizontal bed, which is due to the neglect of vertical acceleration. Thus, the nonlinear shallow-water equations should be used when wave transformation due to other effects such as sloping bottom and energy dissipation are predominant [7]. One advantage of using the equations is that wave breaking can be modeled as a discontinuity of the solution and therefore the model does not need any empirical relationship or constant [18, 32]. This gives a primary reason why these equations are used to analyze breaking wave transformation including run-up [23, 81 ].

D e r i v a t i o n o f N o n l i n e a r S h a l l o w - W a t e r E q u a t i o n s

Assuming that the water depth to wavelength ratio is extremely small, the momentum equation (16) in the vertical direction or, in dimensional form, (6) with the boundary condition (10) gives a hydrostatic pressure distribution:

p = pg (~ - z) (169)

Then the momentum equation (5) in the horizontal direction becomes

~u + (uV) u - - gV~ (170)

Ot

The continuity equation can be obtained by integrating Equation 4 from the bottom to the surface:


~_2 + V[(h + ~)u] = 0 (171) Ot

Adding a bottom friction term to the momentum equation, Kobayashi and his co- workers [43, 45-48, 95] studied extensively the wave transformation in the surf zones including run-up, set-up and reflection under various conditions of smooth and rough bottoms, gentle and steep slopes, and regular and irregular waves. Although discrepancy is found between calculation and measurement in wave shoaling and breaking inception, wave transformation in very shallow water has been accurately reproduced. Swash oscillation due to obliquely incident waves has also been analyzed by the nonlinear shallow-water equations [2, 42].

Nonl inear S h a l l o w - W a t e r Equat ions on a P e r m e a b l e Bed

Based on the long wave assumption, recently model equations have been derived for nonlinear shallow-water waves on a permeable bed [44]. In the following, the equations are presented for a three-dimensional case by using the present notations.

The continuity equations for the water and permeable layers are obtained by integrating vertically Equations 4 and 27, respectively.

t)~ ~- V[(h + ~) U] = - qb (172) ~)t

V[Ephpup] = qb (173)

where qb = flow rate per unit area from the water layer to the permeable layer The momentum equations for both layers can be obtained from Equations 5 and 29 upon the long wave assumption:

/) 1 tO " [ ( h + ~)u] + V{(h + ~)[u, u]} = - g(h + ~)V~- =L fb lulu-- qb Ub (174)

EpV t) [Eph ] + V {8php[Up } Ephp Up pUp , Up] = - gephpV~ - Ot

I V

2Cf Ep luplup + qbU b (175)

where the second term on the right side of Equation 174 represents the bottom friction with the friction coefficient fb" The last terms on the right sides of Equa-


tions 174 and 175 represent the momentum exchange between the two layers,

and u b = u for qb > 0 and u b = up for qb < 0. Figure 12 shows a good agreement between calculated and measured water-

line oscillations. The terms of momentum exchange consider the effect of boundary layer in

some sense. However, it is also interpreted that the boundary of the two control volumes for the water and permeable layers changes depending upon the sign of qb: at the lower limit of the boundary layer during qb > 0 and at the upper limit during qb < 0.

2

'i Z r o

- ' 1 -

. . . . . . . . . M e a s u r e d C o m p u t e d

"-2- 150 155 160 165 170

2

1 - i i o $ I i �9 i

e Z r o-

'~1- . . . . . . . . . M e a s u r e d C o m p u t e d

Zr

- 2 - 150 155 160 165 170

21 Run P3

-1 i ......... Measured Computed

" - - 2 | / ' " i - I " ' I "

150 155 160 165 170

Figure 12. Waterline oscillations due to random waves on a sloping beach [95].


The continuity equations for the two layers are combined to yield the total mass conservation equation:

~ t + V[(h + ~) u] + V [EphpUp ] - 0 (176)

The momentum equations (174 and 175) are rewritten with use of the continuity equations (172 and 173) as

an lulu- qb --~ + (uV)u + gV~ = - 2(h + ~) h -t- ~ (ub -- U) (177)

~Up EpV E2pCf qb -k (UpV)Up + gV~ = - Ephp ~ U p ]uplup -b (u b - Up) (178)

~t Kp ~ p Ephp

As compared to Equation 166, the continuity equation is the same as in the case of the Boussinesq type equations. If C r is taken to be unity as often assumed, the differences in the momentum equations are that Equations 167 and 168 include the effect of vertical acceleration, whereas Equations 177 and 178 include the bottom friction and momentum exchange. However, if the two control volumes are taken constantly outside of the boundary layer, the last terms in the momentum equations should vanish because u b = u for Equation 177 and u b = Up for Equation 178 at all phases of wave motion.

Nonlinear Mild-Slope Equations

Fully-nonlinear and fully-dispersive wave equations were first derived in [69, 70]. Other formulations are also found in [26, 72]. Equations are derived by expanding the dependent variable such as the velocity, velocity potential, or pressure into a series in terms of a set of vertical distribution functions, substituting the expression into the basic equations and then integrating them in the vertical direction. Because no assumption is made on the nonlinearity and dispersivity, the resultant equations can be used even for the analysis of strongly nonlinear wave transformation. Required accuracy can be achieved by increasing the number of terms in the series: usually by two or three terms.

The following section introduces derivation by using Lagrangian. Then, it is demonstrated that the model equations including the MSE, Boussinesq equations, and nonlinear shallow-water equations are derived as special cases [26].


Derivation of Nonlinear Mild-Slope Equations

A Lagrangian for water waves is written as [62]

ftt2ffA ~ i~t~3 l f , , _ [ ~t )2 1 (t)~z3)2 } s ,~]= _ + (Vr + ~ + g z , d z d g d t 1

(179)

For infinitesimal changes in ~3 and ~, the change in Lagrangian is expressed after integration by parts:

I 02(I)3 ) 80 dz dA dt ~L/~-'--fttl2IIAI_;h V2r 0Z 2

--I 1 f ~ l L c3t + + 2 ( ~ z 3) +g~ 8~z_;

+ ~ + (V~) (V~)3) - 8* + (Vh)(V~)3) + ~ dA dt z=~ z=-h

0r ; t2 1 dz dA + I~2 J~C I_~h '~-n (~ dz ds dt + I IA I_h [(~(I)]tl (180)

To terminate the Lagrangian with respect to ~3 and ~, each term in Equation 180 must vanish, which yields all the basic equation and boundary conditions (8, and 10 to 12) for water waves.

To derive two-dimensional model wave equations, the velocity potential (I)3 in three dimensions is expressed as a series in terms of a set of vertical distribution functions Zct which should be given a priori:

N (~3(X, Z, t) = Z Za (z; h(x)) f(x(x, t) - Zafa

0t=l (181)

Substitution of this expression into Equation 179 and analytical integration in the vertical direction yields:

s ~] = I]2 1 IAZ (f~, ~)dA dt I

(182)

oqf~ ar g - h2 all3 1 AN (Vf~) (Vfl3) where Z f~,--~-, r = ~ (r ) + ZI3 - ~ +

I I D~cfvfl3 (Vh) 2 + -- B#f~,f~) + C~f~ (Vf[~)(Vh) + ~- 2

(183)


Aixl3 = h ZixZBdz' Bix~ ----- h /)z /)z dz, Cix~ = h - - ~ Zi~ dz,

ozoo. oz,o. dz, :Za = f h ix dz (184)

To terminate the Lagrangian with respect to fix and ~, the following Euler equations are required:

I [ ] = ~ ,3(3fa/3t ) + V 3(Vfix) (185)

(186)

On substituting the definition (182) of ~, the following nonlinear simultaneous partial differential equations can be obtained:

a~ Z~ --~ + V (Aixl3 V f13) - B~I~ f~ + V (C~f~V h) - C~I~(V f~) (V h)

- D~f~(Vh) 2 = 0 087)

o3fl3 1 1 ,9Z~ ~9Z~ g~ + Z~ --~ + -~ Z~ Z~ (Vf~') (Vf[3) + 2 /)z /gz f'r f~

c)Z~ 1 ,gZ~ ~9Z~ + - ~ Z~ fr (Vfl3) (Vh)q 2 ~)h ~)h f~, fl3 (Vh)2 = 0 088)

where Z; a : Zaiz=; /)Z;a -- OZa[ (189) ' ~z ~z z=;

Upon the mild-slope assumption, terms of second order in the bottom slope are neglected to yield the following nonlinear mild-slope equation:

a; Z;a ~ + V(Aal3Vfl3) - Bal3 f13 + (CI3a - Cixl3)(Vfl3)(Vh)

~z~ + - ~ Z~ f, (V~)(V h)= 0 (190)


Of 13 1 g; + + Cvf )r +

az~ + Oh Z~ f~, (V fB) (V h) = 0

1 ~Z~ r 2 Oz Oz f~' f~

(191)

The unknowns in these equations are ~ and fa (~ = 1 to N). Once the equations are solved numerically, the velocity potential is determined by Equation 181. Although the mild-slope assumption has not been used to derive Equations 187 and 188, simplification to Equations 190 and 191 may usually be consistent with the selection of the vertical distribution functions Za because they are usually taken from the theory for waves on a horizontal bed.

Figure 13 compares calculated and measured water surface fluctuation at the shoreward edge of and behind a submerged breakwater. With only three terms, calculation gives a good agreement with measurement. Figure 14 compares calculated and measured water surface fluctuation and bottom velocity on a sloping bottom. In spite of strong nonlinearity of waves just prior to breaking, the agreement is good for both surface fluctuation and velocity.

Relationship with Other Wave Equations

Mild Slope Equation. Linearized forms of the nonlinear mild-slope equations ( 190 and 191) are

i9~ o z~ .-~-+ V(A~I 3 7 f13)- B~ f13 + (CI3a - C~13) (Vfl3) (Vh) = 0 (192)

/)f~ =0 (193) g~ + Z~ ~ t

When only one component is taken in the series and the vertical distribution function is taken from the small amplitude wave theory:

~3 (X, Z, t )= Z 1 fl (x, t) (194)

cosh k(h + z) Z 1 = (195)

cosh kh


2.Ore O.Tml.5mO.7m 2.1m waves_ P1 _----- I T

15cm

50cm , . ~ / / ~ . ~ �9

i i ~ l i I _ -J

P5 I

_Case 4 (T = 2.01 s, Ho = 5.0 era)

P3

-I

0 1 t/T 2

Figure 13. Water surface fluctuation at the shoreward edge of and behind a submerged breakwater. N indicates the number of terms taken in the series expansion of the velocity potential [26].

1 4 6 O f f s h o r e E n g i n e e r i n g

,, . . ~ , , ,,, ,

4 0 . 1 c m %

L .L .! . _,

0.4m l.Om 9.03m

C a s e 1 - 2 ( fp = 0 .50 Hz , H i t/3 = 5 .4 c m )

i n x = - 4 0 c m . . . . . . . m e a s ; N = 2; . . . . . . . N = 3 U / �9 I l l I 1 I t

k- -1 ~, n L . _ ..... . . . . . . .- . . . . . : ' . -_ . . . . ," �9 . - ~ . , -"--_ . ' . , q

- 1 0 1 i 1 i z t i i /

0 5 10 15 2 0

l = z 2 o , , , ,

- 1 0 / . . . . ! J ~ t ~ , ~ / 0 5 10 15 2 0

5 0 z = 720 c m T r r r ,

u.b 0

5 10 t ( s ) 1 5 2 0

Figure 14. Water surface fluctuation ~ and bottom velocity U b of random waves on shoaling water [26].

E q u a t i o n s 192 and 193 b e c o m e

~ + V V f 1 + - ( k 2 C C g ) fl = 0 a t g

(196)

af~ g~ + - ~ = 0 (197)


which yields a time-dependent form of the MSE:

V (CCg V fl ) -k- (k 2 CCg -- (y2) fl a2f, Ot 2

- ~ = 0 (198)

If more than one component is taken for which the vertical distribution functions are

cosh k a (h + z) Z a = (199)

cosh kah

2 _ gka tanh kah (200) (Ya

then Equations 192 and 193 become

0r t~ V 2 f13 o - - + g~ - a ~ f ~ = 0 Ot

(201)

a f ~ = o g~ + 3t

13=1

(202)

where

2 2 2 2 2 10"a - ~ (a ~ 13) 1 kaO'13 - k130'a

A ~ - k~ - k~ B ~ = k~ - k~

2 (a ]3) o 2 a ( 1 - n a) (a 13) ca na = =

(203)

From Equations 201 and 202, ~ can be eliminated to yield

g 13=1 0t2 o V 2 o + Aa~ f~ - Ba~f ~ = 0 (204)

Then, by~ assuming progressive waves with the angular frequency 8 and wave number k"

fa = aa ei(~-&) (205)


Equation 204 becomes

Bal3 al 3 f~2 o _ o = AaBal3

13=1 13=1

(206)

To have a nontrivial solution, k 2 is determined as an eigenvalue for a given 0. It can easily be proved that 1< = ka for 8 = Ga, and therefore the dispersion relation is exactly satisfied at the frequencies Ga (o~ = 1 to N). This suggests that the dispersion relation is accurately satisfied even if the frequency is not equal to either of the selected frequencies. Therefore transformation of random waves with a wide spectrum can accurately be calculated by Equations 201 and 202.

Nonlinear Shal low-Water Equations. These equations are obtained by taking one component with

Z 1 = 1 (207)

Then,

All = h + ~, Bll = 0, Cll = 0 (208)

and the nonlinear mild-slope equations (190 and 191) become

"/_2 + V[(h + ~) V t'1 ] = 0 (209) bt

c3f 1 1 )2 g~ + .-~- + ~ (V fl = 0 (210)

By rewriting these equations in terms of u:

u = V ~ 3 =Vf~ (211)

the nonlinear shallow-water equations are obtained:

+ V [(h + r~) u] = 0 (212) ~)t

~u ~ + (uV)u + gV~ = 0 (213) ~t


Boussinesq Equations. For two components with the following vertical distribution functions:

(h + z) 2 Z 1 = 1, Z 2 = ~ (214) h 2

the nonlinear mild-slope equations become

~9~ I (h + ~)3 ] ~+Ot V ( h + ~ ) V f 1+ -3~ ~ Vf 2

2(h + ~)~ - h 3 f2 ( V ~ ) ( V h ) = 0

(h + ~)2 (h - 2~) (V f2) (Vh) 3h 3

(215)

(h+~)2 ~)~ [(h+~) 3 (h+~)5 ] 4(h+~) 3 h 2 8t + V Vfl + V f2 -- 4 f2 3h 2 5-h~ 3h

(h + ~)2 (h - 2~) 2 (h 4- ~ ) 3 ~

- 3h 3 (V fl ) (V h) - h 5 f2 (V~) (Vh) = 0 (216)

,{ )2 f 1 2 (h + ~) g ~ + ~ + h 2 Ot +-2 Vfl + h ' 5 ~ V f 2 +-2 h 2

2 ( h + ~ ) ~ { ( h + ~ ) 2 ) h 3 - f2 Vf~+ h------T~Vf2 (Vh)=O

2

(217)

By invoking the orders of magnitude as

V h --, 0 (q~-), ~ - fl ~' 0 (E), f2 - 0 (e 2) (218)

Equations 215 to 217 are simplified as

8t ~ V (h +~) V fl +-~ Vf2 + ~ (Vf2)(Vh) = 0 (219)

~ + V Vfl - f 2 - (Vfl)(Vh)=O ~t ~ -3

(220)


t~fl /)f2 1 )2 g~ +---~- + - ~ + ~ (V f, = 0

Then, because

(h + z ) 2 2z(h + z) u=Vt~3 = V f l + ~ V f 2 - 3 f2 v h

h E h

(221)

(222)

1 1 ~=vf, +gvf: +gf~Vh

Equations 219 to 221 are combined to yield the Boussinesq equation:

a~ + V [(h + ~) fi] = 0

/)t

~ + ( f i V ) f i + g V ~ . . . . /)t

h 2 ~)

6 /)t

h ~ V (Vfi) + ~- ~ V [V (hfi)]

(223)

(224)

(225)

Validity Ranges of Wave Equations

In the MSE, the wave steepness HA, (H = wave height, L = wavelength) is assumed to be small, but the relative water depth h/L is arbitrary. On the other hand, in the Boussinesq equations, the relative wave height H/h and relative water depth squared (h/L) 2 are assumed to be small quantities with the same order of magnitude, i.e., the Ursell parameter U r = H LE/h 3 is of order of unity. These two sets of assumptions, respectively, lead to the Stokes and cnoidal wave theories for waves that propagate on a horizontal bottom without deformation.

There are only two independent non-dimensional parameters for waves of permanent form: H/L and h/L, or any combination of products of these two parameters. Perturbation expansion might be done with respect to two independent parameters instead of one parameter as in the Stokes and cnoidal wave theories. A regular perturbation solution can be found if the Ursell parameter U r = n L 2 ] h 3 = E 1

and relative water depth squared (h~) 2 - E 2 are chosen as the two parameters, i.e., in the velocity potential expanded into double power series [28]:

o o o o

m=l n=l

(226)

~)mn can be solved to any order of e 1 and E 2.

E q u a t i o n s f o r N u m e r i c a l M o d e l i n g of Wave Transformation 151

0 (~2 - ( h /L ) 2) S t o k e s w a v e s

) - - - - Q - - - - i

) - - - - 0 - - - - 4

) - - - - 0 - - - ~

) - - - - 0 - - - .

) - - - - 0 - _ _.~

) - - - ~

) - - - - ~

) - - - - 4

) - - - - ~

) - - - - ~

) - - - - 1

) - - - . . 1

) - - - - 4 ) . . . . .

) _ - - . ~ ) . . . . .

) - - - - 4 ) . . . . .

f

A . . . . . . . II

I I . . . . . , 4 ) 11

I I

" c n o i d a l ~ ,, w a v e s

l !

I I i I I

1 2 3 4 5 0 (~1 ~HL2/h3)

Figure 15. I l l u s t r a t i o n o f t h e s o l u t i o n in a d o u b l e power series.

The final solution can be illustrated in Figure 15. If the summation with respect to n in Equation 226, i.e., summation in the vertical direction in the figure, is taken first, the resultant single power series becomes equivalent to the Stokes wave solution. The order of the solution agrees with the number of solution lines included and the terms corresponding to the points on the lines are included in the solution. On the contrary, if summation is taken first with respect to m, i.e., in the horizontal direction, the resultant single power series becomes equivalent to the cnoidal wave solution. Each theory reflects its own process in taking the double summation. Then, it is understood that, even though other various wave theories can be established by first taking summation in inclined directions, finite-order solutions contain only finite number of terms in the double power series and therefore inferior to both the Stokes and cnoidal wave theories. For example, the theory for O[H/L] = O[(h/L) 4] corresponds to taking the first summation in the direction of 45 ~ and the first-order solution includes only one point in the figure. Thus, the Stokes and cnoidal wave theories are the only two useful theories for waves of permanent type.

As can also be understood from the previous discussion, the Stokes wave theory is valid in relatively deep water, whereas the cnoidal wave theory is valid in relatively shallow water. Figure 16 compares wave profiles calculated by a 5th-order Stokes wave theory, 3rd-order cnoidal wave theory, 5th-order stream function wave theory, and small amplitude wave theory in various water depths. In an intermediate water depth, all the three finite amplitude wave theories predict almost the same wave profile. However, in shallow water, the Stokes wave theory gives an unrealistic wave profile because the given wave parameters are out of the validity


~ 0

-1

S-5 h/Lo =0.641 SFM-S~ tt/ Lo =0.096 SFM-5- -~ . H/L, =0.128 5 - 5 ~ H/Lo =0.038

- - 0 ,

I I I - 1 i 1 I -0.5 0 0 .5 - 0 . 5 0

X/L X/L (a) h=100rn. T=10s. H=20m (b} h=lSm. T=lOs. H=6m

0.5

:z: ~- 0

-1 -0.5

h~ Lo = 0.032 ,.'",, / ~ ,'H/Lo =0.013

- s ~,/ ',,~.-.~..,:,' . / , h = 0 . 4

/ / \c-3 ~', /~"ss ' "~ I~o.St

I , I I 0 0.5

X/L

(c) h=5m. T=10s,H=2m

Figure 16. Comparison of wave profiles calculated by various wave theories (S-1- small amplitude wave, S-5: 5th-order Stokes wave, C-3: 3rd-order cnoidal wave, SFM: 5th- order stream function wave theory).

range of the Stokes wave theory. For the cnoidal wave theory, although the profile agrees fairy well with other theories even in deep water, other quantifies such as the velocity on the bottom cannot be predicted reasonably.

To assure the validity of the perturbation theories, the series solution should be convergent, which implies that higher order terms should be smaller than lower order terms. By taking the ratio between the second- and first-order terms for various quantities and using the approximations in deep and shallow water ( h ~ >> 1 and << 1), the following results can be obtained:

2nd order

1st order Stokes

~H/L (h/L >> 1) t Ur (h/L << 1) (227)

2nd order f h/L (h/L >>1) (228) o c

[H/h (h/L << 1) 1st order cnoidal


From these results, the validity range of the Stokes wave theory is limited by the wave steepness in deep water and by the Ursell parameter in shallow water, whereas that of cnoidal wave theory by the relative water depth in deep water and relative wave height in shallow water. This can be investigated further in a quantitative manner by calculating the errors generated by finite-order solutions in the two nonlinear surface boundary conditions. Figure 17 shows constant error lines for the dynamic surface boundary condition as well as breaking criteria obtained theoretically [97] and empirically [21]. Based on the error calculation, a diagram for the selection of a wave theory is proposed as Figure 18.

The validity ranges of the wave equations described in this chapter can roughly be understood from the previous discussion. Especially, the weakly-nonlinear MSE of Stokes type can be used only for small Ursell parameters, and the original Boussinesq equation cannot be used for large relative water depth. Although modified versions of these equations may give reasonable results for particular wave properties such as the wave celerity, modification consistent for all properties may not be possible. For more strict calculations, fully nonlinear mild-slope equations might be most promising.

Summary

This chapter deals with several types of equations for numerical modeling of wave transformation in shallow water. The mild-slope assumption, weakly nonlinear shallow-water wave assumption, and long-wave assumption lead to the MSE, Boussinesq equations, and nonlinear shallow-water equations, respectively.

0.1

0.01 0.001 0.01 0.1 1

h/Lo

Figure 17. Constant (1% of gH) rms error lines for dynamic surface boundary condition (S: Stokes wave, C: cnoidal wave, SFM: stream function theory).


0.1

tan#= 1/10.1/20, 1/30, 0-1150 God-, (1970) ~'"--v--,-TL..-.2L._~./.H/h=O.B3 Y. am, .ada and -'-" . ~ ",-""?7"--.....:..~'r"~.-~ .~ t l::inlotan~

i i H/h=0.4

C-3

2 / ~ S-5

0 . 0 1 t v t , ~ I , , , , , l , , , , , 1 , , 1 , , , , , , H 0 .001 0.01 0.1 I

h/Lo

Figure 18. A diagram for selection of a wave theory.

The fundamental form of the MSE is derived for small amplitude monochromatic waves, but it has been extended for wave transformation on a slowly varying current. The Boussinesq equations and nonlinear shallow-water equations do not necessarily assume oscillatory motions and thus can be used for predicting wave-current interaction even by their fundamental forms.

The energy dissipation, especially due to wave breaking, has high priority in extending the model equations derived on the basis of the momentum equations for an inviscid fluid. Terms representing the effect of breaking have been introduced into the MSE and Boussinesq equations. The nonlinear shallow-water equations have the advantage that the wave breaking is mathematically treated as the discontinuity of the solution and the energy dissipation is automatically considered.

Equations for waves on a permeable bed are also presented for the three models. They can also be used for predicting wave transformation over submerged breakwaters.

To increase computational efficiency, parabolic approximations have been proposed to the MSE of an elliptic type. Varieties of parabolic equations were presented to clarify the difference among them.

Time-dependent mild-slope equations have been proposed to improve numerical efficiency and treatment of boundary conditions, or to deal with random wave transformation. They were also described in this chapter.

As described in the previous section, each set of model wave equations has its own validity range. Reliable prediction cannot be performed outside of the


range. Nonlinear mild-slope equations are valid as long as the series expression for the dependent variable in terms of vertical distribution functions gives a good approximation. Strongly nonlinear and strongly dispersive wave transformation can be predicted accurately by the equations.

Because the progress in the numerical modeling of wave transformation in shallow water has been remarkable, reliability of predictive models has greatly improved. Numerical model equations will also be used for developing techniques to control not only wave height but also wave period and direction, which cannot be predicted by linear models.

Notation

a

C Cg

CM Cf C r F

fa(x, t) fo

g H h

h 1 hp

h t K,

k k s

L Lo L1 P

Pa

qb

amplitude wave celerity group velocity mass coefficient turbulent resistance coefficient in permeable layer apparent mass per unit volume of permeable layer energy flux per unit width coefficient for Z a in Equation 182 enery dissipation coefficient bottom friction coefficient linearized resistance coefficient in permeable layer gravitational acceleration wave height still water depth representative vertical length scale depth of permeable layer = h + hp, total depth intrinsic permeability wave number wave number vector Lagrangian defined by Equation 179 wavelength deepwater wavelength representative horizontal length scale pressure in water layer dynamic pressure pressure in permeable layer flow rate per unit width flow rate per unit horizontal projection area from water layer to permeable layer


R b t

U, V, W

u 3 u

u

u b

U c

Up, Vp, Wp

Up3

U w W

W w x, y, z

X

X 3 Z

Z" Z"

Z PtP

Za

7

~s Vr

13

V

~ , 1"1 ~

P

excess momentum flux due to wave breaking time components of water particle velocity in water layer in x, y, and z directions; u, v = horizontal components; w = vertical component = (u, v, w) = (u, v) depth-average horizontal velocity in water layer water particle velocity transported between water and permeable layer water particle velocity at still water level in wave propagation direction components of seepage velocity in permeable layer in x, y, and z directions; Up, Vp = horizontal components, Wp = vertical component --= (Up, Vp, Wp)

= (Up, Vp) depth-average horizontal velocity in permeable layer horizontal two-components of velocity induced by currents horizontal two-components of velocity induced by waves vertical component of velocity induced by currents vertical component of velocity induced by waves Cartesian coordinates; x, y = horizontal coordinates, z = vertical coordinate = (x, y) = (x, y, z) vertical distribution function defined by Equation 53 vertical distribution function defined by Equation 61 vertical distribution function defined by Equation 115 vertical distribution function defined by Equation 120 a set of vertical distribution function = U c/C, ratio of water particle velocity at still water level to wave celerity value of 7 at breaking point value of Y on uniform slope value of Y at recovery point = hl/L 1, relative water depth = a/h l, relative wave amplitude porosity water surface elevation kinematic viscosity general coordinates; either orthogonal or non-orthogonal coordinates non-dimensional general coordinates water density


Ij

G

% r

0'

v o.)

V3

intrinsic angular frequency representative angular frequency = (~ - D, deviation from representative angular frequency velocity potential in water layer velocity potential at still water level; defined by Equations 41 and 53 amplitude of velocity potential 4) at still water level velocity potential defined by Equations 60 and 61 amplitude of (l)' defined by Equation 125 defined by Equation 140 defined by Equation 135 defined by Equation 131 apparent angular frequency; = (~ in absence of current

Ox Oy Oz

= ,~yy

D 0 = - - + ( u v )

Dt Ot

(vector)3 three-dimensional vector (vector) horizontally two-dimensional vector

( ) quantity in permeable layer ( j~ quantity at water surface ( )0 quantity at still water level ( )* nondimensional quantity

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71. Nishimura, H., Maruyama, K., and Hiraguchi, H., 1983. "Wave Analysis by Direct Numerical Integration," Proc. 30th Japanese Conf. on Coastal Engrg., JSCE, pp. 123-127.

72. Nochino, M., 1994. "Fully-Nonlinear Coupled Vibration Equations for Irregular Water Waves and Their Basic Characteristics," Proc. Coastal Engrg., JSCE, vol. 41, pp. 16-20 (in Japanese).

73. Nwogu, O., 1993. "Alternative Form of Boussinesq Equation for Nearshore Wave Propagation," J. Waterway, Port, Coastal, and Ocean Engrg., ASCE, vol. 119, pp. 618-638.

74. Nwogu, O., 1994. "Nonlinear Evolution of Directional Wave Spectra in Shallow Water," Proc. 24th Int. Conf. on Coastal Engrg., ASCE, pp. 467-481.

75. Panchang, V. G., Cushman-Roisin, B., and Pearce, B. R., 1988. "Combined Refraction-Diffraction of Short-Waves in Large Coastal Regions," Coastal Engrg., vol. 12, pp. 133-156.

76. Panchang, V. G., Wei, G., Pearce, B. R., and Briggs, M. J., 1990. "Numerical Sim- ulation of Irregular Wave Propagation Over Shoal," J. Waterway, Port, Coastal and Ocean Engrg., ASCE, vol. 116, pp. 324-340.

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78. Radder, A. C., 1979. "On the Parabolic Equation Method for Water-Wave Popaga- tion," J. Fluid Mech., vol. 95, pp. 159-176.

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C H A P T E R 4

WAVE PREDICTION MODELS FOR COASTAL ENGINEERING

APPLICATIONS

Vijay G. Panchang and Bingyi Xu

Department of Civil & Environmental Engineering University of Maine, Orono, Maine, USA

Zeki Demirbilek

US Army Engineer Waterways Experiment Station Coastal Engineering Research Center

Vicksburg, Mississippi, USA

CONTENTS

INTRODUCTION, 163

ENERGY BALANCE MODELS, 164

MASS AND MOMENTUM CONSERVATION MODELS, 172

Mild-Slope Equation Models, 172 Approximate Models, 174

Complete Elliptic Models, 179

REFERENCES, 189

Introduction

This chapter provides a review of some state-of-the-art nearshore wave models that can be used for practical prediction in engineering studies. It describes modeling philosophy, strengths, and limitations of models based on the steady state energy equation and the steady and unsteady state mass and momentum equations.

Coastal engineering projects such as those dealing with wave agitation in harbors, beach protection, maintenance of navigational channels, studies of shoreline evolution, etc. typically require a detailed knowledge of the wave field in the project area. In general, however, the desired wave information is not readily available at the project site. Available information may consist of buoy and

163


satellite measurements, usually at some considerable distance from the project area. On the basis of these measurements, the engineer is required to estimate the wave conditions at various locations of interest to enable subsequent project calculations such as sediment transport rates, wave forces on structures, etc. Under these circumstances, computer simulation using numerical models constitutes an eminently cost-effective, efficient, and rational approach to generating the desired estimates.

Mathematical models of wave propagation attempt to simulate the various mechanisms that induce the transformation of waves. In coastal areas, the dominant mechanisms consist of shoaling, refraction, and diffraction due to bathymetric variations, diffraction by structures, reflections from coastlines, structures, and bathymetric variations, dissipation due to wave breaking, friction, and percolation, wind-induced growth, wave-current interactions, and wave-wave interactions. Over the years, researchers have developed many elementary models to describe these complex mechanisms individually (e.g., refraction models, shoaling models, diffraction models, etc. as described, for example, in SPM (1984)). However, except in very simple cases, the engineer has no prior knowledge of which of these mechanisms (or combination of mechanisms) is dominant for a particular project. Also, wave conditions in nature are usually the result of the complex combination of various physical processes. It is therefore more prudent to use a model that can incorporate as many of these mechanisms as possible. On the other hand, modeling all the relevant mechanisms simultaneously in a rigorous manner is beyond the realm of practical engineering; indeed, at present, it is not possible even in a research environment, because some of the physical processes defy complete description and others require exorbitant computational resources.

This chapter reviews some wave models that strive to provide an acceptable mix of simplification (e.g., two-dimensional vs. three-dimensional; steady-state vs. unsteady-state; linear vs. nonlinear; frequency domain vs. time domain; etc.), sophistication (in the inclusion of several mechanisms), and convenience of application from the standpoint of practical engineering. Most of the models described here or their variations are presently used for coastal projects. The purpose of this chapter is to briefly describe the overall philosophy, methodology, advantages, difficulties, and some applications of each modeling strategy.

The following sections describe two kinds of wave models commonly used for coastal engineering applications: models based on the conservation of energy and models based on the conservation of mass and momentum. Both types of models can, by some rigorous or approximate method, incorporate many of the mechanisms mentioned earlier. However, energy balance models cannot incorporate the effects of diffraction and reflections caused by bathymetric features and structures. Mass balance models, on the other hand, cannot describe the effects of wind-induced growth and the resulting wave-wave interaction. Pearce and Panchang [73] and Vogel et al. [90] describe some research in this context. As a result, these two categories of models are appropriate for different kinds of

Wave Prediction Models for Coastal Engineering Applications 165

applications (although, in practice, they are often used interchangeably). Con- cluding remarks address some general difficulties that a practicing engineer is likely to encounter while using numerical wave models.

Energy Balance Models

Models based on the conservation of energy are generally used for predicting the deep ocean wave climate. Perhaps the most sophisticated and well-known model of this type is a model called "WAM3G," which is a result of much international cooperation [43, 91 ]. Models of this type are based on a solution of the time-dependent energy-balance equation for the complete range of discrete frequency and direction components that comprise the wave energy spectrum. This equation, also known as the "radiative transfer" equation, may be written as

~)E(x, y, t; f, 0)

~t + V �9 [ Cg(x, y; f) E (x, y, t; f, 0)] = Sw+ Sn+ Sd+ Sf+ S p (1)

The variable of interest in this equation is E (x, y, t; f, 0), which represents the spectral energy for frequency f and direction 0 at location (x, y) and time t. Esti- mating this variable allows the calculation of the significant wave height, peak frequency, and other parameters. The two terms on the left represent, respectively, the temporal rate of change of the spectral energy and the propagation of this energy at (frequency and depth dependent) group velocity C g. On the fight, S w represents the input of energy from the wind, S n represents the redistribution of wave energy between different wave components that arises from nonlinear interactions, S d represents dissipation due to breaking, Sf represents losses due to bottom friction, and S p represents losses due to percolation. Many different parametric forms have been suggested for these source/sink terms Si; even these simplified forms are complicated and difficult to compute. We refer the reader to WAMDI [91], Young [96], and Komen et al. [43] for additional details.

Energy balance models are further classified as either spectral or parametric, depending on the method used to solve Equation 1. Spectral energy models usually first discretize the wave frequency and direction spectra into many frequency and direction bands, and solve the energy-balance equation for each component. For instance, if 10 frequency bands and 10 direction bands are used, the 100 unknowns comprising the spectrum at each grid point in the time and space domains must be obtained (e.g., the WAM3G model [91]). Parametric models reduce the degrees of freedom (conventionally) in the following way: directionality is integrated out of the energy-balance equation by assuming a cosm(0) distribution, and the frequency spectrum is parameterized in terms of a small number of parameters (e.g., 5 for the JONSWAP spectrum). The energy equation is rewritten with this small number of parameters at each grid point as the unknown variables (e.g., [26]). Both types of models are generally solved in a time-dependent manner on a finite-difference grid.


While WAM and other models based on the radiative transfer equation are quite sophisticated, they are not well-suited to the estimation of nearshore wave conditions for coastal engineering projects. The strength of these models is their ability to model the effects of time-varying wind conditions on waves (e.g., large storms). Obviously, these models require wind velocities at every grid point as a function of time. Such wind-fields can be obtained from atmospheric models, which are usually run at large meteorological centers such as the Fleet Numerical Oceanographic Center and the European Centre for Medium Range Weather Forecasting; these runs involve length scales far greater than those pertaining to coastal engineering projects. Typically, therefore, the wave models also are run on the scale of large ocean bodies. The simulation of time-varying wind and wave fields usually necessitates a model resolution that, from a coastal engineering standpoint, is relatively coarse; model grid sizes are typically of the order of several kilometers. For example, the finest resolution used by the US Naval Oceanographic Office for this type of operational modeling is 0.25 ~ lati- tude x 0.25 ~ longitude. These features thus clearly preclude routine application of these types of models for coastal engineering projects, which require a more localized and detailed description of the wave field; their value is limited to the provision of deep ocean wave conditions, which serve as offshore boundary conditions for nearshore wave prediction models that can be used for coastal engineering applications.

Models based on the energy equation can, however, be used for coastal engineering applications if some simplifying assumptions are made. The engineer is often interested in estimating wave conditions at a nearshore site that result from strong steady winds blowing in a largely onshore direction. The domain under consideration may be of the order of, say, 20 km x 20 km. An example of such conditions, dealing with the design of a disposal facility for dredged material in Saginaw Bay (Michigan), is shown in Figures 1 and 2 [5, 23]. During a storm in May 1981, the winds held steady most of the time, at about 12 rn/s at a direction of 40 ~ . This direction is along the axis of the bay and (fortunately) along the alignment of three wave gauges shown in Figure 1. Measurements from these gauges (Figure 3) also displayed relatively steady state behavior for about a day, with wave heights declining from about 6 ft at gauge A to about 3 ft at gauge B (due to the various bathymetric effects) and then increasing to about 4.5 ft at gauge C (due to wind effects). Under such conditions, modeling via the energy equation can be simplified considerably by restricting its application only to "forward propagating" spectral components (namely, those in a window of +__90 ~ about the main wind direction) and by discarding the unsteady state term. The governing equation is then a time-independent first-order partial differential equation that can be solved by finite-differences.

The following describes some recent developments in "stationary" energy balance models. In these models, because the propagation of wave energy is modeled rather than the actual wave motion, fairly coarse resolution (typically 250 m) can be used. (In mass balance models, on the other hand, the grid sizes

Wave Prediction M

odels for Coastal Engineering A

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must be a small fraction of the wavelength.) However, the resolution must be fine enough to represent the bathymetry adequately. Basic input to these models consists of water depths at each grid point and wave conditions along the upwave (offshore) model boundary.

The model "HISWA" is perhaps the most widely used stationary energy balance model. It was developed by Holthuijsen et al. [29] at the Technical Univer- sity of Delft, within the overall framework described above. However, the model equations are based on the conservation of wave action (A = E/o) rather than energy (E), which enables the incorporation of wave-current interaction (~ is the angular wave frequency). In the absence of currents, the conservation of wave action reduces to the conservation of spectral energy. Also, additional simplifications are made in that all discrete frequency and directional components are not modeled. The philosophy used in the development of HISWA is that the spectral approach is computationally too demanding for nearshore regions where a high spatial resolution is required (in addition to the frequency-direction dis- cretizations), while the conventional parametric approach (as previously described) involves too drastic a reduction in the information-bearing components (directionality becomes important due to bathymetric and possibly boundary effects). HISWA therefore retains the directionality, by solving the steady-


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state energy equation for several direction bands, say 10, like the spectral models. Rather than solve for (either the parameters or the discrete components of) the frequency spectrum, the energy balance equation is then solved (for each directional band) for two frequency-integrated measures that are usually of engineering interest, viz. the total energy and the mean frequency. This somewhat unconventional modeling strategy is in between that of the usual spectral and parametric wave models in its degrees of freedom: there are 20 unknowns at each grid point. A model based on this strategy has also been constructed at the Danish Hydraulics Institute.

The model attempts to simulate wave refraction, shoaling, wave-current interaction, wind growth, bottom dissipation, and wave breaking. Of these, the last three mechanisms are included in a parameterized fashion, and the associated


parameters can help in tuning the model if necessary. If wave-current interaction is to be modeled, a steady-state current velocity (calculated previously, perhaps with a tide model) must be provided at each grid point as input. Model input also consists of the sizes of the forward propagating window and the directional interval. For practical reasons, this must be specified to be somewhat less than _+90 ~ about the main axis. Although it is desirable to model as much of the directional energy distribution as possible, specifying a large value (approaching _+90 ~ ) increases the computational requirements dramatically. This occurs for two reasons: (1) the number of discrete directional components for which the model equations are solved increases; and (2) as a result of the model stability criteria, the size of the window has an inverse effect on the spatial resolution. Typical values for the window are _+60 ~ , although higher values have been used by Bondzie and Panchang [5].

Engineering investigations with HISWA appear widespread, e.g., in the San Ciprian Harbor, Spain [4], the Rhine Estuary off the Norwegian Coast [28], the Texel tidal inlet and Haringvliet Estuary [90], the German Wadden Sea [ 18], the Columbia River entrance [89], Saginaw Bay [5]. Several other investigations with this model (e.g., in the coastal regions of Holland, North America, Italy, and Australia) were presented at the HISWA Modeling Conference in Renesse (Holland) in December 1991. In general, the model has produced fairly admirable and useful results. At times, tuning and/or sensitivity studies pertaining to the various physical processes were necessary (e.g., [5]), although generally favorable comparisons with data have been reported. A sample result from Bondzie and Panchang [5] for Saginaw Bay is shown in Figure 4. The reduction in wave height and period observed at gauge B is reproduced in this case; however, the subsequent growth in wave height at gauge C is not. This growth, though, is mainly due to a regeneration process, i.e., growth of high-frequency components when substantial energy in lower frequencies is already present (as input). The formation of a bi-peaked spectrum and/or interfrequency energy exchange are not modeled in HISWA; these mechanisms occurring at this site thus violate the basic model parameterizations. A similar situation is reported by Holthuijsen and Booij [27] in the Haringvliet area.

Bondzie and Panchang [5] also undertook a rigorous study of model performance for cases with complex bathymetry. Bathymetric configurations (such as shoals) known to cause significant diffraction effects were selected. As noted earlier, diffraction usually cannot be incorporated into the energy-balance based models. As a result these models can encounter difficulties in regions where refractive ray tracing analyses predict unrealistically high accumulation of wave energy. However, complex bathymetries that induce refraction-diffraction abound in several coastal regions (e.g., Lie and Torum [49] describe the shoals


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off the Norwegian coast), and if wave generation by wind is to be included, the engineer's options are limited to these energy-balance models. At other times, he has no a priori knowledge of the extent to which diffraction may be important. Traditionally, spectral energy models have dealt with this problem by plac-


ing an artificial upper bound on the energy of each spectral component (e.g., the British Meteorological Office model [24]; the Australian Defence Force Acade- my Model [96]). Unlike these models, however, HISWA does not propagate discrete frequency components. Rather, the propagation of frequency-integrated measures is modeled, and no artificial cap on the energy is used. By comparing model results with data and other mathematical model solutions for wave propagation over shoals, Bondzie and Panchang [5] inferred that the degree of spreading affects the performance of HISWA considerably. Many realistic sea states with crossed seas (short waves with broad and narrow input spectra) were handled quite well by HISWA. As the input spreading decreased (unidirectional spectra or swell-like conditions), large errors were seen.

A more recent development in this type of modeling attempts to retain all frequency components (rather than just two frequency-integrated measures as in HISWA). This model, called SWAN, is under development by Holthuijsen and Booij at Technical University Delft. Holthuijsen et al. [28] briefly described an application of this model to the Haringvliet estuary. A comparable model, called STWAVE, has been developed by Resio [77-80]; at present, this model is being evaluated and upgraded by the US Army Corps of Engineers.

Mass and Momentum Conservation Models

Mild-Slope Equation Models

Models based on the conservation of mass and momentum are most widely used for predicting the transformation of waves in shallow water, i.e., the effects of varying coastal bathymetry and geometry and of structures like breakwaters, etc. For small amplitude water waves, the motion of the fluid may be assumed to be incompressible, inviscid, and irrotational. The equation of motion then reduces to the three-dimensional Laplace equation for the velocity potential. Although developing a three-dimensional numerical model from the Laplace equation is straightforward, it has little use in coastal engineering practice (as a result of its exorbitant memory requirements which limit its application to very small domains). For monochromatic waves over slowly varying bathymetry, the Laplace equation can be vertically integrated leading to the following "mild- slope equation" [2, 3]:

V �9 (CCg Vt~) + k2CCgO = 0 (2)

where t~ is the two-dimensional wave potential that is related to the three-dimensional velocity potential as follows:

t~v (x, y, z, t) cosh k (h + z) = e-i~ y) (3) cosh kh


The water surface elevation can be written as

~(x, y, t )= q( x, y)e -i~ = i_..~ t~(x, y)e -i~ (4)

where k = wave number G = angular frequency

C, C g = wave and group velocities, respectively

The derivation of the mild-slope equation (2) may be found in Berkhoff [2], Smith and Sprinks [83], and Isobe (see Chapter 3 of this handbook). It may be noted that Equation 2 may be transformed into the Helmholtz equation:

V2(I ) + K2tI) = 0 (5)

which is often encountered in acoustics. In Equation 5, O(x, y) is a modified potential and K(x, y) is a modified wavenumber (as described, for example, in Radder [76], Panchang et al. [69]).

Equation 2 simulates wave refraction, diffraction, and reflection in domains of arbitrary shape. However, various other mechanisms also influence the behavior of waves in coastal areas, and several extensions of the mild-slope equation have been developed to include these additional mechanisms, although in an approximate way. First, the effects of frictional dissipation [9, 15, 52] and wave breaking [ 14, 17] may be included as follows:

V �9 (CCgV~) + (CCgk 2 + iow + iCgoqt)~ = 0 (6)

where w = bottom friction factor ), = wave breaking parameter

Second, Kirby and Dalrymple [40] have shown that the use of an amplitude- dependent dispersion relation instead of the usual linear dispersion relation can lead to improved solutions of Equation 2. Third, the effect of currents on waves may also be included in the mild-slope equation [34, 45] through the incorporation of additional terms and a Doppler shift for the wave frequency in the dispersion relation. While useful in shallow areas where tidal or other currents may modify the wave heights, this approach requires a specific wave direction, which may not be readily defined in areas with complex wave behavior. Further research is therefore needed in this area.

Finally, the derivation of Equation 2 assumes that Vh/kh = O(~t) << 1, which implies that it is valid only for "mild" slopes. Recently, Massel [59, 60], Cham- berlain and Porter [8], and Porter and Staziker [75] have derived new forms of the vertically integrated wave equation that are applicable for "steep slopes." These new extensions involve the addition to Equation 2 of terms describing the


bathymetric variations to a higher order. Because these additional terms are relatively simple to calculate, models based on Equation 2 or 6 may easily be extended to provide a wider range of applicability.

Equation 2 is an inseparable elliptic partial differential equation that requires the simultaneous solution of the discretized equations over the entire model domain. Although models based on this approach were developed during 1970s and '80s [ 10, 11, 30, 31], their application to many routine wave propagation studies extending over many wave lengths is difficult, owing to excessive computing demands. Consider, for example, the case of wave propagation in Homer Spit (Alaska), studied by Ebersole et al. [22] using an approximate method, on a computational domain of roughly 100 • 80 deep-water wavelengths. If the complete elliptic boundary-value problem is to be solved for such a case with 10 points per wavelength, one is confronted with the task of solving 800,000 simultaneous equations in complex variables. An elliptic model that uses a direct solver based on Gaussian elimination would require the storage of a coefficient matrix containing approximately 6.4 x 1012 complex elements. Even when possible, the solution of such large systems by this method may require hundreds of CPU hours. It is clear that the direct method may be conveniently used only on rather small domains. These difficulties spawned the development of several approximate or simplified models in the eighties. These models greatly enhance the ability to handle large domains, but by compromising some of the physics contained in the original equation. In general, they can simulate wave propagation in one dominant direction with weak lateral scattering. Some of the simplified models are described next. However, when an engineering problem precludes the use of these approximate models, one must resort to a complete elliptic model. To overcome the associated problems, extremely fast and efficient solvers have been devised and new representations of the open boundary conditions, leading to improved model performance, have been developed in the past few years. This greatly enhances ability of the elliptic Equation 2 to handle large complicated domains with reasonable computer resources. These developments are described later in this chapter.

Approximate Models

The Parabolic Model. One class of approximate mild-slope equation models is based on a "parabolic approximation" of Equation 2. By splitting the wave field into forward and backward propagating components with major directions _+ x, Equation 5 can be recast in the form of two coupled equations. If the reflected component is neglected, a parabolic equation results (e.g., Radder [76])"

~O (i K 1 ~K i ~2 / - + ~ ~ ~ ~x 2K ~gx 2K ~9y2

(7)


where x = the "principal" wave propagation direction

Because the splitting technique is not unique, it is possible to obtain other forms as well (e.g., Kirby [35, 36]). Because the parabolic equation can be solved as an initial value problem, solutions can be obtained over very large computational domains simply by "marching" one row at a time.

The computational convenience offered by the parabolic model comes, of course, at a cost. In Figure 5, the wave vector relationships associated with various wave equations are compared. These relations are obtained by substituting in these equations a wave-like solution �9 ~ exp(imx+iny) where m and n are components of the wave vector k in the x and y directions, respectively. It can be seen that the parabolic equation matches the analytical curve only in a small window about the +x axis. Because the approximation completely fails in the -x direction, the model is not applicable to cases where submerged or surface- piercing structures (seawalls, islands, etc.) or bathymetric slopes create significant reflections propagating in t h e - x direction. Also, this method is inapplicable when the wave energy is scattered at a wide angle to the principal direction.

Because they are easy to code, a large number of parabolic models have been developed, e.g. including REFDIF [40, 42], CREDIZ [19], etc. They are usually based on finite-differences, and some include other mechanisms, such as nonlinear effects [39, 40], wave-current interaction [34], wave energy dissipation [15], etc. To allow for a locally varying principal wave direction in the wave field, new

(+x)

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Parabolic models

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forms of the parabolic approximation were also derived by using orthogonal curvilinear and non-orthogonal coordinates [32, 37, 41, 88]. Simulation of spectral transformation can be conveniently made by linear super-position of the discretized spectral components. For example, Panchang et al. [67] used a parabolic model to successfully simulate a series of laboratory experiments of narrow and broad directional wave spectra passing an elliptic shoal.

While using the parabolic approximation in a domain with arbitrarily varying bathymetry, special attention should be paid to the two lateral boundary conditions. Except for a few special cases (such as in some laboratory experiments where two lateral vertical walls would guarantee a fully reflecting boundary condition), the lateral boundary conditions are unknown. The use of approximate boundary conditions may sometimes significantly affect the results in the computational domain, especially when oblique incident wave trains are considered. One may be able to improve the boundary approximation by using parabolic model based on orthogonal curvilinear and non-orthogonal coordinates. In general, it is better to expand the computational area well beyond the area of interest. A detailed discussion of lateral boundary conditions for Cartesian parabolic wave models can be found in Dalrymple and Martin [15a], who also devised "perfect boundary conditions," which are very effective if the bathymetry outside the two lateral boundaries can be approximated by a constant depth.

Despite the inherent approximations, the strength and usefulness of parabolic models for dealing with large domain problems are often incomparable. Parabolic models have been used in several coastal applications. Vogel et al. [90] and Liu and Tsay [52] report a satisfactory match between their model results and field data in their studies off the European coast and off the North Carolina coast, respectively. Figure 6 shows an example of a REFDIF simulation performed at the US Army Engineer Waterways Experiment Station as part of beach erosion study for the Revere Beach area in Massachusetts [85]. Model results were generally found to be consistent with observed geomorphological patterns.

The Model RCPWAVE. By introducing the wave irrotationality equation explicitly and separating the mild-slope equation (1) into real and imaginary parts, Ebersole [21] obtained three coupled equations. One of these coupled equations is still elliptic. However, when reflection is neglected, it is possible to march the solution into the domain using specified incident wave conditions. Because the solution involves the simultaneous solution of three equations on a given row, iteration is required on each row, and pure refraction is used as a first guess in the iterations.

Like most Cartesian parabolic models, the RCPWAVE model introduces errors when the incident angle of wave approach varies considerably from the x- direction. Further, it has been shown that the form of the equations used in this model tends to produce an excessively smoothed wavefield [38]. Finally, for some grid sizes and wave frequencies, the solution does not converge. A stability analysis of the finite-difference equations is necessary to determine the opti-


_ . . . . ~ : i : . . ' - ~ - - . . .

................. : : ~ 0 , : . . - ~ _ . . .... ...... �9 .- .....

(::) .~...

" " . . . . . . . . . . . . . ............... . ....... "i : - - - - ~ ~ . o -~ -.. ~ ~ . ~ . ~ ~.--,,. ~!! ..... . . . . . - ~ - ............. ~ . ~ ~.. o ~ - - ~ - , ~ , ',, 0 " - 7 ..... " " " �9 -- ~ ~ - ~ ~ . . . . . . . . . "- . . . . . 9 - " . 0 b ~ . ,

i -2 " " . . . . . 0 9 ..... - . . . " . . . . . . ' .-

o . / ~ - r ~ �9 ~ : ~ .'" . .:- . . . �9

o .................. ........"

0 "''''''""" .... i . "

0 10 20 30 40 50 60 70 80 9 0 100 110 120 130 140 150 160 X GRID CELL NUlVIBER

Figure 6. Modeled wave heights (normalized) near Revere Beach (Massachusetts). Period = 15 s, waves incident from left. Cell size = 200 ft in each direction.

mum aspect ratio for the grid. However, the nature of the equations precludes a conventional stability analysis. Extensive numerical tests suggest that the grids have a 2:1 aspect ratio (alongshore to cross-shore dimensions). This requirement is akin to the EVP model discussed below.

Despite these shortcomings, the RCPWAVE model has been shown to be a versatile engineering tool. The US Army Corps of Engineers frequently uses this to make wave calculations in coastal regions, e.g., in Homer Spit, Alaska [22] and off the North Carolina coast [21, 22]. As an illustration, we show in Figure 7 the wave heights predicted by the RCPWAVE model along the 14-ft depth contour near Grand Isle, Louisiana. These simulated wave conditions, pertaining to three different incident wave periods, were then used as input to a shoreline evolution model [25]. This indicates the usefulness of this model to the engineer investigating various scenarios.


1.4

.~ 1.2

E

~. 0.8

0.6

Nearshore Wave Height Transformation Coefficients

[, . . . . . . o , , 1 H = 1.0 m, = 7.3 deg, T = 5 . 6 s ~ - - H = 1.0 m, = 7.3 deg, T = 7.3 s -4--" H = 1 . 0 m , e = ' / . 3 d e g , T = 3 . 5 s ~

_ I

Pr

d Isle Model Reach t

i

, ) /

l i i i I t I I

50 100 150 200 250 300 350 400 Alongshore Coordinate (cell spacing = 100 ft)

Figure 7. Modeled wave heights (normalized) along the 14-ft contour near Grand Isle (Louisiana).

The EVP Model. The Error Vector Propagation (EVP) model [66] contains an appealing algorithm to solve elliptic type problems. It is different from the previous two approximate methods in that it can include the effects of reflection. The procedure consists of making an initial guess along the first row, and marching the solution down the domain to the other end.

Because the assumed solutions on the first row are only guesses, the computed solution at the other end will not generally satisfy the boundary condition there. The degree of mismatch is therefore computed at this boundary on each grid point. The mismatch vector is related to the vector of errors along initial row through a correlation matrix. This matrix can be determined and may then be inverted easily to give the initial row of errors [66]. The initial guesses are then corrected, and another marching sweep of the grid produces the desired solutions.

Although this procedure averts the need for storing large matrices as described earlier (the correlation matrix is quite small), marching the solution from one boundary is inherently unstable for most elliptic problems. For the wave equation (4), though, the march becomes stable for Ax < 2/K and Ay > 2/K. It is shown [66] that this latter requirement leads to an approximation of the analytical wave vector relationship by the eyeball-shaped curve shown in Figure 5. The approximation is valid if the wave scattering is at small angles from the x-direction, i.e.,


the model requires paraxiality. However, it can be used to simulate two-way wave propagation, unlike the parabolic models generally used (Figure 5).

The EVP model has been tested against data obtained from several hydraulic model studies [66]. Although excellent results were obtained, the model has not found its way into engineering practice, partly because of the subsequent development of more sophisticated techniques of solving elliptic problems, described in the following section.

Complete Elliptic Models

For constant depths, the Helmholtz equation (4) can be solved quite easily by the boundary element method. For arbitrary bathymetries, however, Equation 1 must be solved by either the finite-difference or the finite-element method. For problems with simple geometries, the finite-difference method is attractive because it is much easier to construct a numerical grid and to code than finite- element models. One such model has been developed by Panchang et al. [69] and has been successfully applied to several test cases. Li and Anastasiou [48] have also developed a model based on the multigrid technique.

In general, though, the topography of a coastal region is very complex. Domains such as harbors involve breakwaters, other structures, and have arbitrary shapes that cannot be properly represented by a finite difference grid. Also, the grid resolution should ideally be dependent on the wavelength. For a uniform finite-difference grid, the resolution would be dictated by the shallowest areas, leading to extremely fine grids all over the domain. This results in an exorbitantly large set of linear equations, creating computational difficulties. The finite-element method can overcome these difficulties, because it can better represent the actual shape of the domain and allows flexibility in the construction of elements.

In the 1980s, finite-element models based on the complete elliptic equation (1) were developed by Tsay and Liu [86] and Chen and Houston [ 10]. The model by Chen and Houston [10], called HARBD, has been used by the Army Corps of Engineers in several harbor development projects (e.g., Lillycrop et al. [50]) and by Okihiro et al. [65] for a study of infragravity waves in Barbers Point Harbor (Hawaii). However, these models solve the discretized linear system of equations by the direct method (i.e., Gaussian elimination), which makes excessive demands on computer resources, as stated earlier. Hence these models can be applied only to very small domains or to very long waves. (An earlier version of HARBD was used by Houston [30] for a very successful simulation to tsunami propagation near the Hawaiian islands.) In the past two decades, considerable effort has been devoted to enhancing the capabilities of finite-element elliptic water wave models [ 10, 30, 31, 45, 66, 72, 86, 87, 95]. Perhaps the most important development is the ability to handle extremely large domains, thanks to the use of iterative solution procedures that avert the necessity of storing large matrices [47, 69]. The remainder of this section will focus on these more recent developments.


Unlike direct methods (i.e., Gaussian elimination), indirect (iterative) methods do not require the storage of the coefficient matrix [A] for a linear system of equations [A] { t~ } = { f} that results from the discretized mild-slope equation, and hence can be used on large domains. For convergence of the iterative procedures, [A] must usually be strictly diagonally-dominant, or it must be symmetric and positive-definite. However, matrix [A] that is obtained from a finite-difference or finite element discretization of the mild-slope equation does not satisfy these properties. Conventional iterative methods (e.g., Jacobi, Gauss-Seidel, SOR, etc.) therefore do not guarantee convergence for the problem. The remedy used by Panchang et al. [69] relies on the following Gauss transformation:

[A*] [A] {~} = [A*] {f} (8)

where [A*] is the complex conjugate transpose of [A]. The system matrix [A*][A] is now symmetric and positive definite, allowing the use of the iterative techniques. The conjugate gradient (CG) algorithm is a particularly attractive choice, because it has been shown to converge several orders of magnitude faster than many other schemes. Convergence by this method is guaranteed and can be accelerated by preconditioning techniques [69]. Therefore, solutions to very large problems by the CG method can be obtained with modest computer resources. Li [47] has recently suggested modifications to these techniques that further enhance convergence.

As in most wave models, the open boundary can be problematic in elliptic wave models. The traditional approach, used by Chen [9] and Tsay and Liu [86], is based on a Bessel-Fourier series representation for the scattered waves that satisfies the Sommerfeld radiation condition [62]. However, this approach is subject to the following limitations: (a) the exterior region must have a constant depth, (b) the exterior coastlines must be collinear, and (c) the exterior coastlines must be fully reflecting. These limitations are often not met in nature, and Xu et al. [95] have demonstrated the spurious effects generated as a result. These are illustrated in Figure 8a. (The semicircle in Figure 8 is used to separate the computational domain from the exterior region.) In particular, chaotic phase pattern stems from full reflections from the exterior that diffract into and contaminate the results in the computational area. Xu et al. [95] have designed new techniques, based on matching a radial parabolic approximation for the scattered waves on the semicircle with the mild-slope equation in the interior. The results obtained with this approach, shown in Figure 8b, display a much more orderly and acceptable phase pattern.

This method has been successfully applied to various wave propagation situations in the context of finite-difference [69, 94] and finite-element [95] models. A finite-element version called CGWAVE [72] has recently found widespread use in many coastal and harbor engineering projects. As mentioned earlier, the grid sizes employed in such models should be much smaller than both the local wavelength and the scale of local bathymetric variations. Finer resolution is also


1. O0

0.80

O. 60

O. 40

0.20

0.00

-0 .20

-0 .40

-0, 60

-0 .80

-1. O0

Figure 8a. Model simulation in Toothacher Bay via traditional treatment of open boundary conditions. 20-second waves incident from top. Figure shows cosine of phase angle [95].

desirable at places where the change of amplitude in space is rapid (e.g., near "caustics"). Experience suggests that, in general, ten or more points per wavelength are preferable and six should be regarded as minimum. The creation of such a grid-network and the proper specification of boundary conditions has generally been regarded as a very difficult task, and this has precluded more widespread application of finite-element wave models.

In recent years, a few sophisticated grid generation packages have been developed for coastal applications. One of these is "FASTTABS," a 2D hydrodynamic modeling package developed by the US Army Corps of Engineers. FAST- TABS includes an efficient finite-element grid-generator developed by the Engineering Computer Graphics Laboratory at Brigham Young University. However, this grid-generator (and others like it) was designed for other types of applications, mainly hydrodynamic modeling of flows in rivers and estuaries. It is not directly suited to the needs of an elliptic coastal wave model. For the wave model, wavelength-dependent grids must be constructed and, as described later,


l .O0

0.80

O. 60

0.40

0,20

O. O0

-0 .20

-0 .40

-0, 60

-0 .80

- l . O0

Figure 8b. Model simulation in Toothacher Bay via improved treatment of open boundary conditions. 20-second waves incident from top. Figure shows cosine of phase angle [95].

the open boundary needs special attention and the construction of a semicircle. Further, reflection coefficients, which may vary from one part of the coastal boundary to another, are also to be specified as input data for the model. These features are generally not available in grid-generators for flow models. CGWAVE therefore contains additional programs that interface the wave calculations with the FASTTABS for pre- and post-processing.

CGWAVE has been verified against several test-cases, which include harbor resonance, scattering by islands, transformation by complex bathymetries, nonlinear amplitude dispersion [69, 94, 95] and for the cases of breaking, dissipation, and reflection of long waves on a planar beach [9, 46]. A simple example describing the use of the model for positioning breakwaters is described in Pan- chang et al. [69]. Some of these applications involve tens of thousands of nodes. Figure 9 shows a sample application of this model to Kahului Harbor where about 23,000 elements were used to provide resolution of approximately 12


0.60

90 1.20

0.60

0.60

1.5

~-0.90

0.

0.]0

~ .60 0.60 O.3O

/ .~0~ 0-. ~.~0 ~ _ ~ ~.

m 0.60~

1 . 0 0

0 .70

0 . 5 0

0 . 3 0

0 . 1 0

- 0 . 1 0

- 0 . 3 0

- 0 . 5 0

- 0 . 7 0

- 1 . 0 0

�9 ~ ! ~ ' i,

Figure 9. Modeled wave conditions in Kahului Harbor. Waves of height 5 m and period 30 s incident from top. Coastal boundary reflectivity 0.4. (top) wave amplitudes; (bottom) cosine of phase angle.


points per wavelength. Figure 10 shows the simulated wave conditions when one of the breakwaters is extended by about 130 m. As a result of the smaller harbor entrance, lower wave heights are seen to occur in many areas of the harbor. Because the differences for the two layouts vary with the incident wave condition, the use of such models for harbor development is evident.

Kostense et al. [45] describe application of a comparable model called PHAROS in the area around the Venice lagoon; however, the numerical techniques associated with this model are not described in detail. See also Mattioli [61 ].

Time Dependent Models

The weakly-dispersive linear shallow water wave theory assumes two parameters are small:

A p - kh << 1 and e - - - << 1 (9)

h

where A = wave amplitude

0.60

0.60

,60 ~ / -'~ 0.3o

~.3o o.~o" ~:~o~ , ,

Figure 10. Modeled wave amplitudes in Kahului Harbor with extended breakwater.


In some coastal problems, these assumptions may not be valid due to a decrease in water depth in nearshore areas and an increase in wave amplitude caused by shoaling and refraction. For weakly nonlinear and moderately long waves in shallow water (O(e) = O(~t 2) < 1), Peregrine [74] has derived the following Boussinesq equations for waves over variable depth:

+ V . [(h + ~) fi] = 0 (10) o~t

~fi h h 2 + ft. Vfi + gV~ = V [V. (hil t)] - 7 [V. fit] (11)

where fi = depth-averaged velocity in the horizontal plane

Traditionally, these two-dimensional equations have been considered appropriate for simulating nearshore wave propagation effects such as shoaling, refraction, diffraction, and reflection. However, it is well known that the Boussinesq equations account for the effects of nonlinearity and dispersion only to leading order [63, 74] and they are therefore limited to depth to wavelength ratios of about 1/10. Recently, efforts to extend this limit to about 0.2 have been made [12, 57, 58, 64, 93]. While these involve improving the linear dispersion characteristics, this should not be construed as improving the nonlinear capabilities of these equations. Also, these advanced Boussinesq models are based on a vertical profile of horizontal velocity that is at most quadratic in z, and this representation will certainly not be adequate at the intermediate and deep water limits. Thus, in view of the weak nonlinearity and dispersivity, Boussinesq models should not be expected to describe the low-frequency surfbeat and high-frequency enhancements of the wave crests during shoaling of coastal waves. Boussinesq models should also be expected to fail when the local bed slope exceeds the relative water depth ratio, d/L. The Green-Naghdi theory may be a better choice than the Boussinesq theory for modeling waves in the shallow water because the former is not subject to these constraints. A comprehensive description of the Green-Nagh- di theory for coastal waves is presented in Chapter 1 of this book [20].

Numerical Boussinesq-type wave models are generally based on advancing a finite-difference solution in time. This is often extremely time-consuming if one is dealing with short waves in a large coastal domain. In addition to spatial discretization, temporal discretization is also needed for the integration of these equations. In computing the solutions at a certain time level, only the information from the previous time level is required. Therefore, the time-dependent models require modest computer storage that increases roughly linearly with the number of grid points. However, integration over an extremely large number of wave periods with small timesteps (usually governed by an appropriate stability condition) is required for model spin-up. Additionally, numerically generated short waves will produce erroneous results and could cause instabilities [12].


Although a few two-dimensional Boussinesq models have been developed for coastal applications [1], in a recent simulation of a hydraulic model study involving waves passing over complex bathymetry, Wei and Kirby [92] found that the results of a mild-slope equation model matched data somewhat better than their Boussinesq model did. They pointed to the need for additional research in this area. At present, Boussinesq models are generally used in small regions or for obtaining a detailed picture of one-dimensional wave propagation problems. Chen and Liu [12] have developed a parabolic model based on the modified Boussinesq equations of Nwogu [64] to relax some of the computational limitations of these models. (See also Liu et al. [55].)

Somewhat comparable but linear time-dependent equations may be obtained from the mild-slope equation also. Equation 2 is an elliptic stationary equation with time-harmonicity implicit therein. By introducing the time variable explicitly, it may be transformed into three coupled time-dependent hyperbolic equations:

Oh C /)U C /)V i i = 0 (12)

/)t Cg ~x Cg ~)y

3U Oh ~ + C C g ~ = 0 (13) Ot /)x

OV Oh ~ + C C g = 0 (14) /)t Oy

where h(x,y,t) = r exp(-io't) U(x,y,t) and V(x,y,t)= functions of vertically integrated water velocities

in the x- and y- directions

These equations are integrated in time under the sustained influence of the boundary conditions, until a time-harmonic steady state is reached. This represents the solution to the original mild-slope equation (2). Several models based on the above hyperbolic equations have been developed [13, 56, 68, 81]. But the treatment of open boundaries for the time-dependent equations is still rather difficult, and an area of active research. Different models have handled the boundary problem in different ways, e.g., Madsen and Larsen [56] have used "sponge layers."

Conclusion

This review, although not exhaustive (see also [53, 54]), may give the impression that the field of coastal engineering is littered with complex numerical wave models involving complicated model equations and numerics, rendering the task of model selection, application, and interpretation of results somewhat difficult. This impression is indeed true if wave models are compared to hydrodynamic


flow models, which are perhaps applied to coastal systems more frequently. Although several hydrodynamic models exist, they all have, by and large, the same objective: the simulation of tidal and wind-driven circulation in coastal areas of the order of a few kilometers (e.g., [71 ]). As a result, model equations are largely similar (usually the shallow water equations); differences in the models are mainly confined to numerical techniques (explicit versus implicit schemes; finite differences versus finite elements, etc.) Of course, individual cases do require special attention, e.g., density effects may be important if there is fiver input, etc. In general, the longer time scales associated with the mechanisms modeled (typically, 12-hour tidal fluctuations), the relatively small size of the model domains (in terms of the tidal wavelengths), and the frequent availability of water level data at coastal stations enable the performance of reliable simulations with reasonable effort, using various friction coefficients as tuning parameters.

Modeling of surface gravity waves, on the other hand, presents the coastal engineer with a far more complex job. The short time scales associated with wave motion involve more complex physics and mathematics and the large model domains (in terms of wavelengths) necessarily involve more complex numerics. The mechanisms being simulated are not only more chaotic, but also more diverse, e.g., wave generation occurring over a distance of several kilometers, transformation of waves over bathymetric variations with length scales that vary from a few meters (e.g., shoals) to several hundred meters (e.g., navigation channels), the effects of breakwaters with widely varying orientation and length. It is this complexity and diversity that has resulted in the development of a wide range of wave models.

In spite of this diversity in modeling techniques, it must be noted that the models are all usually validated only for idealized laboratory conditions (although many involve fairly complex features) or for other mathematical (closed form) solutions. In general, it is difficult to convincingly demonstrate that the models accurately reproduce the wave patterns seen in nature. As this review demonstrates, the models involve several assumptions and simplifications in the physics and the numerics. Further, adequate prototype data are almost never available to validate the wave models. In fact, even model forcing functions such as input wave conditions, wind data, etc. are rarely available for the model domain, and even if data were available, they would always contain, to varying (but unknown) degrees, the effects of mechanisms not being modeled. (As noted earlier, a unified model to simulate all mechanisms is not available for engineering purposes.)

This description of the state-of-the-art is not intended to imply that coastal engineering wave models are of little use. On the contrary, the development of such models is indispensable, because the engineer often has no other cost-effective alternative tool at his disposal. (Laboratory modeling as an alternative is often not possible, and in any case, not devoid of difficulties.) Rather, the description is intended to caution the engineer that wave model results cannot be.


considered as the ultimate solution, and that models must be used with due attention to their limitations and the results interpreted carefully. The fact that a wave model has been satisfactorily tested against laboratory data and/or other mathematical solutions enhances our confidence in the model. After all, if they do not work for the controlled test-cases, they are likely to be even less reliable in the natural environment. To that extent, and also for directing future research, such testing is invaluable. However, for most cases, making only a few model runs is rarely sufficient. Model results are usually sensitive to various features unique to the particular model, and nearly all model results are sensitive to grid resolution and the location and the treatment of the open boundary, as shown in Figure 8. In addition, the modeler must be aware that even seemingly simple problems present complications in wave modeling. For example, the impedance boundary condition used on solid boundaries (e.g., coastlines and structures) is often of the form:

/)tI) = i 1 - K r t~ (15) ~gn 1 + K r

where K r = reflection coefficient

While this is correct for normally incident waves, the reflection coefficient for oblique incidence must be a function of the wave angle [33] as well as the wave period. However, for most problems, the wave angle is not known a priori . In fact, in areas with complex wave reflections from several boundaries, a definition for a unique wave angle does not even exist. The reflection coefficients, usually obtained from laboratory experiments, may not be directly applicable. Such features exacerbate the difficulties of model-data comparison.

This review has shown that developments in the past two decades have placed a wide array of coastal engineering wave models at the disposal of the project engineer. However, none of these models can be used as a "black box." It is imperative that the user understand the complexities of the physics and the numerics associated with a particular model to obtain results that are meaningful to his goals. The Saginaw Bay simulations described here and in Bondzie and Panchang [5] highlight this point. In that study, a perfectly satisfactory reason was found for the model's inability to reproduce the observed wave growth between gauges B and C. Had no data been available, the modeler would have been prudent to obtain more conservative estimates by adjusting the wind growth coefficients. With due care, it is almost always similarly possible to obtain reliable and useful results for project calculations and the recent development of numerical wave models has allowed engineers to successfully tackle a variety of practical cases dealing with sediment transport studies [84], harbor design [6, 7, 51], forces on structures, etc. It is, however, appropriate to conclude that additional research in wave model development is vital to reducing the burden on the practicing engineer.


Acknowledgments . Partial support of this work was received from the Maine Sea Grant College Program (National Oceanic & Atmospheric Administration, contracts NA36RG0110 and NA56RG0159), and from the University of Maine Center for Maine Studies. The Coastal Research Program of the US Army Corps of Engineers also provided support for this work (contract DAC 39-95-M-4459). Permission was granted by the Chief of Engineers to publish this chapter.

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C H A P T E R 5

MOORING DYNAMICS OF OFFSHORE VESSELS

Khyruddin Akbar Ansari

School of Engineering Gonzaga University

Spokane, Washington, USA

CONTENTS

INTRODUCTION, 196

MOORING SYSTEM SELECTION, 197

Selecting Mooting Line Components, 197

Classes of Mooting Systems, 200

GENERATION OF MOORING SYSTEM RESTORING FORCES, 200

Quasi-Static Cable Analysis, 202

Dynamic Cable Analysis, 210

COMPUTATION OF ENVIRONMENTAL FORCES, 224

Forces Due to Wind, 225

Forces Due to Current, 227

Forces Due to Waves, 228

VESSEL DYNAMIC ANALYSIS, 233

Vessel Equations of Motion, 233

Solution Technique, 235

Numerical Results and Observations, 236

EFFECT OF VESSEL HYDRODYNAMIC MASS ON VESSEL AND MOORING LINE DYNAMIC RESPONSES, 240

Added Mass Coefficients, 241

Numerical Results and Observations, 241

CONCLUSION, 244

ACKNOWLEDGMENTS, 248

NOTATION, 248

REFERENCES, 251

195


Introduction

With the increase in the world's need for energy, offshore oil suddenly became quite noticeable as a potential source. The exploration, production, and transportation of such an energy source often requires the use of offshore vessels. If a floating ocean structure is involved, it would require some form of a mooring system to maintain its position and control excursions within imposed operational constraints. The present-day need for offshore operations involving the use of moored vessels stationed in relatively deep water and subjected to hostile environmental conditions has made moored vessel dynamics an important technical area.

The performance of any mooting system is a function of the size and type of vessel moored, the environmental forces acting, water depth, and soil conditions of the seabed. Relative to present-day offshore activity, newer vessels have become more complex and so have their mooring systems. Consequently, mooring system design is no longer a simple task, because the various components of a mooting system must meet all the prescribed requirements pertinent to operating life, maximum permissible excursions, and the site environment. Under trying sea conditions, a proper choice of anchors, clump weights, chains, and cables becomes vital for keeping a vessel on site and for mooting system survival.

In designing a mooting system, one must first define the structure or vessel and select a mooring system, then apply suitable mathematical models and analysis techniques to check out its adequacy and station-keeping capability. The designer(s) must then determine a possible range of vessel excursions and cable tensions, along with possible failure modes, such as the breaking of a cable or the dragging of an anchor. Also, cost requirements must be reviewed relative to available budget and then, finally, after an acceptable compromise is achieved, the design must be finalized.

The purpose of this chapter is to discuss the basic concepts and tools needed to analyze and realistically predict the dynamic behavior of an offshore vessel subjected to environmental forces such as those from winds, waves and ocean currents and moored in open waters with a conventional anchoring system, which is typically a multi-leg combination of anchors, clumped weights, chains, and cables. For an analyst to assess the dynamic response of a moored vessel realistically, he or she must be able to (1) represent the entire system by a proper mathematical model, (2) obtain quantitative estimates of all the significant environmental loads acting on it, and (3) determine and then include the stiffness characteristics of the anchoring system in the dynamic analysis of the vessel. Once the total excitation due to environmental loadings and mooting line restoring forces is obtained as a function of time, the vessel equations of motion can be formulated and numerically solved to yield useful response information that can then be used to advantage in designing a mooting system.

Mooring Dynamics of Offshore Vessels 197

The behavior of a mooring line resembles that of a nonlinear spring with tension-displacement characteristics that depend on its length, weight, elastic properties, anchor holding capacities, and water depth. A general description of the various components in a mooring system is presented along with analysis tools to determine and include its stiffness characteristics in the dynamic analysis of the vessel. Numerical results are generated for a typical mooring line used on a production barge.

The external forces causing excitation of an offshore vessel exposed to an open ocean environment are mainly due to winds, waves, and currents. Although these forces, their origins, and their occurrence are not completely known, there is an abundance of statistical data available, based on which predictions can be made and the necessary forces calculated from empirical relations, charts, tables, etc. Equations for the computation of environmental loadings are presented along with a procedure for generating a wave force time history from a random sea spectrum for use in the analysis. Numerical results for a typical situation are then generated.

As the chapter progresses, it develops the mathematical model and the associated equations of motion of a moored offshore vessel subjected to time-varying forces. It then generates vessel motion and line tension time histories and compares them with and without the inclusion of the effect of the dynamics of the mooting cables.

Because the inclusion of added mass is generally an important consideration in vessel dynamic analysis, this chapter presents information regarding the calculation of added mass coefficients and examines the effect of the hydrodynamic mass terms on the station-keeping response of the vessel from a mooring system design standpoint.

Finally, it presents conclusions with respect to doing an offshore vessel dynamic analysis and designing an anchoring system based on the studies and results presented.

Mooring System Selection

Selecting Mooring Line Components

Mooring line components that include metallic and nonmetallic ropes, chains, links, and connecting hardware come in all types, materials and sizes, and, consequently, their choice, which is a function of the application, life expectancy, and the restraints involved, can be painful and cumbersome. Water depth is a demanding requirement for mooring, and, often, tradeoffs must be made between cost, ease of operation, weight, etc. The types of components that can be used are dictated by environmental conditions and operational factors. For instance, in areas where biological attacks are probable, the use of fiber rope is not feasible. Design criteria for mooring line components are also based on fac-


tors such as vessel size, environmental loads, operational constraints, available line hardware, consequences of failure, and cost considerations.

Wire Ropes. Ropes made of metallic wire are used extensively as mooring lines. These ropes have excellent strength-to-size ratio but poor strength-to- weight ratio. They are easy to handle and their cost is relatively low. In many instances, they may be used to resist fish bites. However, they are susceptible to corrosion, fatigue, and kinks. Often, metallic ropes are covered by a waterproof jacket of hard plastic, such as polyurethane or polyethylene, thus providing protection against corrosion and abrasion. Most wire ropes are made of carbon steel, but stainless steels and other alloys are also becoming popular in ocean applications because of their higher breaking stress and corrosion resistance characteristics. Because a higher strength-to-weight ratio is a desirable characteristic for a mooring line, ropes made of higher strength steels are generally preferred. Other factors like size and weight being equal, the better rope for mooring line use would be the stronger rope. Metallic ropes offer a definite advantage in that they have little ductility and thus elongation is small and occurs only at high tension. The main disadvantages associated with metallic ropes are their weight and short life expectancy. Furthermore, too many mooring legs may be required with metallic wire rope in deep-water applications.

Synthetic Fiber Ropes. Ropes constructed of nylon, dacron, kevlar, polypropylene, polyethylene, etc., are often used as mooting line components. These ropes do not corrode or deteriorate appreciably in sea water. Their strength to immersed weight ratio is excellent and they are easy to handle. How- ever, they are susceptible to fish bites and, consequently, the use of small size fiber ropes in ocean depths where fish attacks are likely to occur has often resulted in mooting losses. At high stresses, plastic flow of the fibers can occur resulting in premature failure. Because of the low allowable load per leg, an excessive number of mooring legs may be required with fiber ropes. Also, stretch is excessive, being of the order of 20%, and this can be intolerable in a tight mooting system. However, this feature could be used to advantage in rough weather conditions.

Chains. Often used for mooring in shallow water, with increase in water depth chains become less feasible because of weight, cost, and the high loads they impose on the vessel. As a result, mooting systems for deep-water applications often use lighter components. At times, where necessary, chain lengths are inserted in deep-sea mooting lines to provide higher strength and abrasion resistance. Furthermore, because of its weight, a length of chain attached to an anchor will reduce the vertical pull on the anchor. The biggest advantage with chains is that the larger catenary allows more lateral vessel excursion. Other advantages are their long life span and high strength.


Hollow Cylindrical Links (HCL). In deep water, solid chains will break under their own weight and wire ropes cannot be obtained in large enough sizes to handle the forces involved. Fiber rope is probably alright, although its strength may only be marginal. Because of these drawbacks associated with chains and ropes, the use of hollow cylindrical links has been suggested. The hollow cylindrical link (HCL) is a metallic tension member with the strength to weight-in-water ratio improved by addition of buoyancy such as floats. This is an advanced concept developed to demonstrate the potential of meeting most extreme drag requirements. It is made up of pinned links with each link incorpo- rating an integral buoyancy chamber and does not use winches. However, it is more expensive than nylon or chain, but nonetheless holds promise of a long life span of 10 to 40 yr.

Anchors. Proper selection of anchors is vital for vessel station-keeping and mooring system survival. The function of an anchor is to resist both horizontal and vertical components of line tension. To this end, an anchor must be designed for a good combination of deadweight and lateral pull resistance. In other words, the anchor must be large enough to provide the necessary holding power but at the same time not so large and heavy that the handling of it should become a problem. The performance of an anchor is a function of its type, mass, and soil properties at the seabed. Anchors are broadly classified in three categoriesD deadweight anchors or simply clumped weights, embedment anchors, and piles.

Deadweight Anchors. Cast iron clumps, concrete castings, old railroad car wheels, bundles of anchor chain and almost anything that is compact can be classified as a deadweight anchor. This anchor is simple, inexpensive, and offers good resistance to vertical pull. However, it is limited to applications where the horizontal pull on it is small.

Embedment Anchors. These are designed to dig into the seabed as they are pulled until they are seated firmly in the ocean bottom, thus offering a large resistance to lateral pull. Examples are the Navy Stockless, Stato, Danforth, Lightweight, Boss, and Mushroom anchors, which perform well on sandy, muddy, and clayey bottoms. The main disadvantages of this type of anchor are the necessity for dragging it on the ocean bed until it digs and holds and its tendency to pull out when a vertical force is applied.

Piles. Piles are vertical members of square or circular cross section that resist lateral as well as vertical pull. They can be constructed with steel, reinforced concrete or prestressed concrete. While the passive soil pressure provides the lateral resistance, resistance to vertical pull is provided by the friction between the soil and the pile. The main advantage with piles is their high holding power in sandy soils. High installation costs and difficulty of handling are the major drawbacks.


Classes of Mooring Systems

There are essentially two classes of flexible mooring systems: single-leg moored system, and multileg moored system.

Single-Leg Moored System. This is basically a simple mooring system in which only one mooring line is used to constrain the floating structure. The principal advantage of this system is that it is simple because of fewer lines, anchors, etc. However, there is one major drawback: If the mooring line is lost for some reason, the vessel is at the mercy of the environmental forces.

Multileg Moored System. When two or more mooring lines are used to constrain the floating structure, the mooring system is called a multileg moored system. The main advantage of such a system is the reliability stemming from the minimized vessel motions. However, there are certain disadvantages with this system:

1. It is difficult to pretension each of the mooring lines correctly. 2. If an anchor is lost during rough weather, it is unlikely that it can be reset at

that time, and the probability of damage is high because of possible entan- glement with other lines at the surface.

3. Alignment of the vessel with the weather is not easy and several mooring line winches together with a manned crew to operate these winches will be required on the barge for constant adjustment of the line tensions.

Generation of Mooring System Restoring Forces

In a series of papers, Childers [ 1-4] discusses mooting system considerations from a practical standpoint and outlines the advantages of multicomponent cable systems over single component lines. Niedzwecki and Casarella [5] developed an algorithm for solving the dimensionless form of the catenary equation for mooting lines, which are made up of chain or rope or a combination of the two. However, clump weights and additional anchors are not treated in their work and line extensibility is not accounted for either. Further, they ignore the effect of line dynamics and concern themselves strictly with the equilibrium configuration of the cable.

Schellin et al. [6] use a quasi-static approach and evaluate probability functions of the magnitudes of line tensions and vessel motions.

Nath and Felix [7] consider a single point mooting system with a uniform cable and predict mooring line motion and tensions resulting from oscillating wave forces. Their numerical model, however, is not good for all ranges of water depths and wave conditions.


Wilson and Garbaccio [8] also consider a uniform cable to determine the dynamic tensions in a mooring line. One difficulty is that their method, though quite elegant for uniform cables, tends to be rather long and time-consuming for any given case. Also, because the formulation assumes a uniform cable, multicomponent lines with clump weights cannot be handled.

Dominguez and Smith [9] present a method that extends the static analysis of a non-elastic cable to a dynamic capability using a discrete parameter model composed of linear springs. This method, however, is only applicable when cable displacements are small and it does not consider vessel motions at the upper end of the cable.

To predict velocities at which an anchor can impact the ocean floor during an anchor-last deployment procedure, Nath and Thresher [10] and Thresher and Nath [11] employ a lumped mass representation in which the mooring line is modeled as a group of discrete masses interconnected by springs. They conclude that with an appropriate integration scheme, the lumped mass model should fur- nish a useful tool for mooring line dynamic analysis.

Leonard and Nath [12] examine basic foundations, similarities, and fundamental differences between the finite element and lumped parameter methods for analysis of oceanic cables under hydrodynamic loads, presenting the continuum approach as a ground truth prior to discussing various aspects of the approximate method derivations. It is their conclusion that so long as attention is paid to defining the boundary conditions and the degree of discretization, which may be different for the two methods, both methods can be equally accurate.

Skop and O'Hara [13] have developed the method of imaginary reactions to determine the static cable configuration using rapidly converging iterations for cables with concentrated loads. However, this method requires a "good" initial guess. For a "bad" initial guess, the method does not converge very rapidly. A faster numerical procedure for finding the complete geometry and static end forces for a plane cable element has been developed by Peyrot and Goulois [ 14]. This method uses the static catenary equations and requires the specification of the anchor and the fairleader positions.

Griffin and Rosenthal [ 15] discuss uses of slack cables in marine and offshore applications, including deep-water moorings and the guy lines of towers and platforms. Their examples use typical methods developed for predicting the dynamics of slack cables with and without arrays of attached discrete masses.

Nakamura et al. [ 16] present a time-domain modified approximate method for calculating the dynamic tension of a mooring chain, which includes the effect of hydrodynamic drag. Calculated results are compared with experimental ones and agreement seems to be good.

Tuah and Leonard [17] discuss a finite-element model for predicting the dynamic viscoelastic response of a synthetic cable. The model employed, which is a three-parameter linear viscoelastic model, is quite suitable for dynamic analysis of complexly connected and curved systems of cables in the ocean.


Quasi-Static Cable Analysis

This mooring system is a conventional network of multicomponent lines, each of which is either a single cable connecting to a bottom anchor or a multi-component combination of anchors, clump weights, chains and cables as shown in Figure 1. The behavior of each mooring line is best described as that of a nonlinear spring with tension-displacement characteristics that depend upon its length, weight, elastic properties, and water depth.

If the station-keeping response of a moored offshore vessel is deemed to be outside the exciting frequency range of the mooring system, the mooring lines would only respond statically to the inplane motions of the vessel, and therefore, the static catenary equations can be used. Under this assumption, Ansari [18] presents an analysis to determine the tension-displacement characteristics of a slack mooring line made up of anchors, clump weights, chains, and cables, and shows how the effect of cable behavior can be included in the dynamic analysis of a moored offshore vessel. In deriving the various configuration equations, use is made of the catenary relationships pertaining to a static mooring system configuration and, where necessary, continuity of slope and displacement is incorporated with due regard for force equilibrium and anchor holding power considerations. The maximum limiting tension is determined from the several breaking strengths and anchor capacities associated with the various cable configurations that can occur.

HORIZONTAL PR , ,OJECTION xt.

SEA LEVEL" - '~ 4 , . , r " ]

FNRLEADER

HORIZONTAL TENSION

~ENT ( 5 ) LENGTH - SL3 4 UNIT WEIGHT - w 3

CLUMP WEIGHT OR ANCHOR WITH HOU~NG FACTO~ F=:~

SEGMENT ( 2 ) LENGTH �9 SL23 UNIT WEIGHT �9 w2

SEGMENT ( I ) LENGTH �9 S L. UNIT WEIGH~'Ze w I

ANCH()R WITH HOLDING FACTOR F!

BOTTOM

Figure 1. Typical mult icomponent mooring line.


Catena ry Equat ions . For a uniform cable segment hanging freely under its own weight w per unit length as shown in Figure 2, the governing differential equation is

dEz w ds = - - ~ (1)

dx 2 H dx

where H = horizontal component of the cable tension ds = an infinitesimal element of the cable

w H ~

_. Xr

f v

Figure 2. Uniform cable hanging freely under its own weight.

Upon integration and inclusion of boundary conditions, the following relationships for the horizontal projection X c and the catenary height Z c can be derived:

"E wsc+ X~ = ~ sinh -1 - sinh -1

w H (wx v) ( v/l Z c = - - cosh + sinh -1 - cosh sinh -1

w H

(2)

where Sc = cable length V/H = the slope at the bottom of the catenary

C o n f i g u r a t i o n E q u a t i o n s . Several configurat ions of the mul t icomponent mooting line can occur as shown in Figures 3-7. Using Equation(s) 2, the governing system equations can be written in general as


ALL OF SEGMENTS I AND 2 AND PART OF SEGMENT :3 ON BOTTOM WITH BOTH ANCHORS HOLDING.

SLI2 L~ ~L~

I -I- ~L . . . . . 2~~ X=n

F " -

J ! i 3 '

N

A

3 - - ,, .._L_ X c~4

XL ,, r

Figure 3. Multicomponent line in configuration 1.

ALL OF SEGMENT I AND PART OF SEGMENT 2 ON BOTTOM WITH BOTH ANCHORS HOLDING.

SLI2 _ ~ S~ L

XL


N

SEGMENT I ONLY ON BOTTOM WITH BOTH ANCHORS HOLDING.

XL



PART OF SEGMENT I ONLY ON BOTTOM WITH ANCHOR I HOLDING.

D2R

XL


N

NO CABLE ON BOTTOM WITH ANCHOR I HOLDING.

SL

~ )0r ~ X=23


N

xc,2- c, Esinh-l/sc'2cl + tan0'/- sinh-'ttan0 )l E { xc t 212 = C 1 c o s h 12 + sinh-1 ( tan ]31) - c o s h {sinh -l(tan ]31)

C1

V 1 = Htanl]l, V2L = V1 + wiSe , V2R = WE + V2L, I]2L = tan -1 V2L ,2 n

~2R = tan-1 V2-~RH ' Xc23 = C2 Isinh-1 (Sc23c 2 +V2R) - s i n h - l H V2R]H

(equation continued on next page)


223 = C2 Ic~ (XC23c 2 + sinh-1 V2R)--c~ ( s i n h - l H V2R31H

V3 =V2R +W25c23, Xc34 =C3 Isinh-1 (5C34c3 + ~--)-3/-sinh-1 --~]

234--C3[c~ Xc34C3 +sinh-1--~)-c~ V 4 --V 3 +W35c34,2=212 +223 +234 ,X L =X c12 +Xc23+ xc34+Xc~

(3)

with the following applicable to specific configurations as given. Configuration 1 (Figure 3)"

S c,z -5c23 =B1 =I~2L "-1~2R =Vl "-V2L =V2R =V3 "-0

5c23 = 122(C3 +Z],Xcon =SKI2 +Sc23 +5L34-5c34 (4)

Configuration 2 (Figure 4):

Sc12 = BI =I~2L =I~2R =vl =V2L =V2R =0 8c23 = SLl 2 + SL23 - Xcon, 8c34 = SL34 (5)


5c,2 = ~1 =Vl =V2L =~2L =0,5c23 =S Ll3, 5C3 4 = SL3 4 , Xco n = SL,E, V2R = Htan ~2R (6)


[~1 =Wl = 0 , S C - X c o n, , = S 12 = SLI2 5c23 -" 5L23 5c34 L34 (7)


Sc12 = SL12, Scz 3 "- SL23,5c34 "- 5L34, Xco n = 0 (8)

In the foregoing equations, C k is the same as H/w K (K = 1, 2, 3) and V K denotes the vertical tension component at the point K on the mooring line. The subscripts 2R and 2L refer to points on the line just to the right or left, respectively,


of anchor 2, and the quantity Xco n refers to the length of line lying on the ocean bottom, which, in the analysis, is considered fiat.

Cable Stretch. The effect of cable stretch is accounted for by including effective elastic moduli and stress areas for the different cable segments, the unit weights of which will now vary with their lengths. The stretch of any cable segment (i), whose initial length under no load is SLi j, would then approximately be

(Tavg)ij SLij

(EoAi j) , ( j = i + l )

where (Tavg)i j = average tension in that segment Eij = effective elastic modulus Aij = its effective stress area

Anchor Model. In the case of the multicomponent cable using "gravity"-type anchors such as the "Navy Stockless," the anchor holding power is a function of the configuration the line is in. It is:

W 1F 1 + W2F2 W1F 1 + (W 2 - Htan~2R)F2

W1F l (W 1 - Htan[~l)F 1

in Configurations 1 and 2 in Configuration 3 in Configuration 4 in Configuration 5

where W 1, W 2 = anchor weights F1, F 2 = holding factors

H = horizontal tension component in the line [~2R, ~1 = the bottom angles at the anchors

The anchor or anchors involved will start dragging toward the barge fairleader when the horizontal component of cable tension exceeds the total holding power of the anchoring system. In Configuration 3, the anchor that is closer to the vessel will lift off the ocean bottom and thus cease to hold when H becomes equal to or greater than the horizontal tension given by the relationship

W 2 = Htan~2 R (9)

which marks the transition from Configuration 3 to Configuration 4. The holding factors F 1 and F 2, which are functions of many parameters such as anchor mass, anchor type, and soil properties at the seabed, must be estimated as realistically as possible, because mooring system behavior relies heavily on the magnitudes assigned to them.


Solution Technique. Because a direct solution cannot be obtained in the case of a multicomponent system, iteration must be resorted to with some known quantity, such as the water depth, employed as the iteration parameter. The solution proceeds as follows:

1. Determine the maximum horizontal tensions for the various mooring line configurations, and thence the limiting condition Hma x used to generate the X L versus H table, by considering the several breaking strengths, anchor capacities, and line tensions associated with these configurations.

2. Evaluate the static cable configuration pertinent to each H considered in the range of horizontal tension values assigned and thus generate a table that gives the horizontal projection X L as a function of the horizontal tension H.

3. Convert the information generated in step 2 to a numerical table of cable tensions interpolated for discrete displacements between the barge fairleader and the farthermost anchor. This table is used for looking up or inter- polating a cable tension for a known vessel displacement at any integration step in the vessel dynamic analysis.

Example Problem. As an application of the analysis presented in the previous paragraphs, the dynamic response of a typical mooring line used on a production barge moored in water 50 ft (15.2 m) deep is examined. The line analyzed, which is shown in Figure 8, is a 500 ft ( 152.40 m) length of 2~-in. (54-mm) chain with a l0 kip (44.4 kN) clump weight positioned 150 ft ( 45.7 m) from an anchor pile. For purposes of modeling, the anchor pile is treated as a 100-kip (444-kN) anchor and the chain forward of the clump weight is broken up into two segments.

ANCHOR PILE

I00 KIP ANCHO

LEADER

WATERLINE i - - - " ~ " ~ - - CHAIN OCEAN BOT TOM"--~

CLUMP WEIGHT

Figure 8. Typical mooring line example. Conversion factor: 1 ft = 0.3048 m; 1 Ib-force = 4.44N.

Figure 9 shows the characteristic shape of the tension-displacement curve generated for the mult icomponent line. As the line tension is gradually increased, the line length on bottom starts decreasing and the mooring line configuration steadily changes as shown in Figure 9 until the limiting condition is reached. In this example, the limiting tension, Hma x, is the one at which the anchor starts dragging in Configuration 5. The hump between points C and D on


X

(n Z bJ I-

0

fL :E 0 0 .J F- Z 0 N a: 0 x

LIMITING TENSION HMAX AT WHICH ANCHOR STARTS DRAGGING

LINE IN CA:~FIG.5

NO LINE ON OCEAN BOTTOM AT THIS P O I N T ~

CLUMP WEIGHT LIFTS OFF OCEAN BOTTOM ~ D LINE IN CONFIG. 4 I

J

CLUMP WEIGHT LIFTS OFF OCEAN BOTTOM . . . . ~ - LINE IN CONFIG. 2

_ . ,LINE IN CONFI~I t A ' I , , 440 455 47O 485 500

HORIZONTAL PROJECTION (XL.), FEET

500

-400

300

200

-IO0

Figure 9. Tension-displacement characteristics g e n e r a t e d for the multicomponent mooring cable by quasi-static cable analysis. Conversion factors: 1 ft = 0.3048 m; 1 Ib- force = 4.4N.

the curve is due to the presence of the clump weight, which lifts off the ocean bottom at the horizontal tension level denoted by point C.

For this type of shallow water mooring system, the horizontal tension induced in the mooring line for any vessel displacement is very sensitive to the pretension level in the line. That is, a 10-kip (44.4-kN) horizontal pretension would generate a horizontal tension of 30 kips (133.44 kN) for a 5-ft (1.5-m) displacement while a 25-kip (11.2-kN) horizontal pretension would generate a horizontal tension of 320 kips (1,423.4 kN) for the same displacement. Thus, higher pretensions would generate higher line tensions for the same vessel displacement and consequently higher maximum values also. On the contrary, if the pretensions are too low, the restoring forces generated in the mooting lines would not be large enough to limit the vessel displacements within the required operational constraints. As a result, while the maximum values of tension would remain low, the vessel displacements themselves could get quite large.


Dynamic Cable Analysis

The foregoing "quasi-static" cable analysis clearly addresses the dynamics of the anchoring system in a static manner, whereby a static equilibrium state is assumed at each time step. This assumption is held valid because the response of the moored vessel is normally outside the frequency range of the mooring system. However, this kind of analysis ignores the effect of line dynamics which, in some situations, as Ansari and Khan [ 19] have shown, may prove to be a significant element in the dynamic analysis of a moored offshore vessel. In an effort to predict mooring system behavior in a way that is realistically feasible as well as useable, Khan and Ansari [20] model each mooring line as a multi-segment, discrete dynamic system. The equations of motion are formulated and then numerically solved to develop tension-displacement characteristics. This information can then be used in providing nonlinear restoring forces in the dynamic analysis of the moored offshore structure, as shown by Ansari and Khan [ 19].

The mooring system considered is a network of multicomponent lines each of which is a combination of clumped weights, chains, and cables. The mathematical model of each mooring line is a multidegree of freedom system obtained from breaking up the line into a series of finite partitions or segments whose masses are lumped at appropriate nodes as shown in Figure 10. The total number of such nodes used should be large enough to model the basic motions of the mooring array. However, this is a function of the accuracy desired. Each seg-

~ F a l r l e a d e r

~7 Water l ine . ~n/' I Vessel ~ . _ / , ~ 1 mn

/

(xi,Y i~ ~~_ei+l

(x t - 1 ,Yt-

/ /

(x2 , y 2 ) / /

Anchor (xx ,y l,) j~ z~Ql.Jm2 . . . .

(xo ,y o ) ~ , , ~ / Ox I

,

Figure 10. Mathematical model of an n-segment cable.


ment of the line between two lumped masses or nodes is treated as a massless, inextensible cylindrical link.

For applications using chains and metal ropes, the assumption of inextensibili- ty is justifiable. For example, for a 2~-in. (54-mm) chain-cable type multicomponent line with a modulus of elasticity of 30 x 106 psi (20.65 x 106 N/cm), the strain is less than 10 -3 in./in., and thus, the total elongation for a 500-ft (152.4-m) mooring line is only 3 in. (76.2 mm) for an average tension of 100 kips (444 kN).

A global frame of coordinates is selected with the anchor at (x o, Yo)- Coordi- nates x i, Yi, and O i are chosen as shown in Figure 10 to describe the motion. The O i are measured from the horizontal axis and are positive counterclockwise. To make the formulation more general, anchor motions are included in the analysis. For sufficient anchor holding capacity, however, a stationary anchor at the origin can be considered. The primary interest being one of stationkeeping, vessel motion is restricted to the horizontal plane, which imposes a constraint on the motion of the surface end of the cable at the vessel fairleader. Because of seawater movement, the mooring line would be subjected to drag as well as damping. The added mass effect from acceleration of the fluid around a link can be included in the form of a fractional mass added to each lumped mass.

The modified Lagrange's equations for cable motion permitting the use of holonomic constraints [21, 22] are

- + = {Qk } + Xeaek e = l

, (k = 1,2,3, . . . , N) (10)

where T, V = kinetic and potential energies of the system, respectively Pl, P2 ..... PN = a set of coordinates used to define the system motion

N = total number of such coordinates 2L e = Lagrange multipliers Qk = nonconservative generalized forces acting in each coordi-

nate direction Pk aek = coefficients defined by

af, aek = 8Pk

/?= 1,2 . . . . . cand k = 1,2 . . . . . N (11)

in which the fe are a total of c constraint equations.

Three-Segment Model. Before extension to an n-segment line, the three segment model of Figure 11 will first be analyzed. The kinetic and potential energies in this case are


(Vessel moves only In h o r i z o n t a l p l a n e )

T / "*

Y, +

/ ~ ml

. . . . ~ , - I - t - ' - - 0 ~0 ~ 1 " + " (xo .yo) J

. . . . . . . . . . . ~>

[ , D l s p l a c e m e n t , H " x + ~ 9.j cos Oj

. . . . . . . . J ' !

Figure 11. Mathematical model of three-segment cable.

1 1 ( x 2 + ~ , 2 ) T = 2 - M A ( 2~ku+~,u/+3] 2"ml

1 1 (k~ + y22)+ m3x32 + - m 2 2

V = MAgy 0 + m~gy 1 + m2gy 2 + m3gh

(12)

where M A is the anchor mass, m i (i = 1, 2, 3) is the mass lumped at the ith node, x o and Yo are the anchor coordinates in the fixed frame of reference and h is the height of the floating vessel above the ocean bed. The constraints imposed on the motion of the mooring line yield the fol lowing six constraint equations:

fl - ( x l - x0 - el cos O 1 ) = 0

f2 - (Yl - Y0 - gl sin O 1) = 0

f3 - - ( X 2 - - X 1 - - ~ 2 COS O 2 ) = 0

f4 - (Y2 - Yl - g2 sin 0 2 ) = 0

f5 - ( x 3 - x 2 - ~3 c o s 0 3 ) = 0

f6 - ( h - Y2 - f3 sin 0 3 ) = 0

Here, the ten coordinates employed are

Pl = Xo, P2 = Y0, P3 = xl

P4 = Yl, P5 = x2, P6 "- Y2

P7 = x3, P8 = 01, P9 = 02 Pl0 = 03

(13)

(14)


Application of Equation 10 leads to

MAX = Qxo - -~1

MAY + M g g = Qyo - ~2

mlxl = Qxl + ~1 - ~,3

mlYl + mlg = Qyl + ~,2 - ~4 (15)

m2x2 = Qx2 + ~,3 - ~,5 m2Y2 + m2g = Qy2 + ~4 - ~6

m3x3 = Qx3 + ~,5, ~2 = ~,ltan01 ~ 4 ---- ~,3tan02, ~6 ---- ~,5 tan 03

Elimination of the Lagrange multipliers ~'1, ~'2 . . . . . ~'6 and coordinates x 1, Y l, x2, Y2, and x 3 through algebraic manipulation and simplification eventually gives five differential equations in the five unknown variables x o, Yo, 01, 02, and 03 in matrix form as follows

[A] + [ . ] {F,} + + (16)

where [A], [B] = 5 x 5 matrices whose elements are functions of the instantaneous line configuration (0 l, 02, 03)

{ F } s = force column matrices { q} = representation of the system unknown variables x o, Yo, 01, 02,

and 03 , shown as follows: [A] =

(M A + m 1

+ m 2 + m3) 0 - ( m I + m 2 + m 3) sin 0 l

0 [M A cos01 + - ( m I + m 2 + m 3)

(m 1 + m 2 + m 3 ) sin 2 01

sin 01 ]

- [ m l s i n 0 1 m l c o s 0 1 [ m l cos(01 - 01)

sin(01 - 0 2 ) +(m E + m 3) +mE

COS 02

sin (01 - 02 ) / sin 01

A o

cos 02

sin(01 - 0 2 ) + m 3

q COS 03

sin 01A

- [ m l s i n 0 2 + m 3 m 2 c o s 0 2 [ m 2 c o s ( 0 2 - 0 1 )

sin (02 - 03) sin(02 - 0 3 ) . + m 3 COS 03 q COS 03

�9 sin 0

- ( m 2 + m 3 ) s i n 0 2 - m 3 sin03

- ( m 2 + m 3 ) s i n 0 1 sin - m 3 sin01 sin03

02

sin(01 - 02 ) sin(01 - 02 ) (m 2 + m 3) m3

cos 02 cos 02

�9 sin 02 �9 sin 03

I m2 cos (02 - -02)

sin (02 - 03 ) + m 3

03 7 COS

�9 sin 02 / ._l

sin (02 - 03 ) m3

cos 03

�9 sin 03

(17)

0 m 3 m 3 cos 01 m 3 cos 02 m 3 cos 03


[B] =

0 0 - [ (m 1 +m 2 +m3)cosO1] (m 2 +m2)cosO !

0 0 - [ (m 1 +m E +m3)sin01 cos01] (m E + m3)sinO1 cosO 2

O O I m l s i n ( O l - 0 2 ' I (m2+m3)sin(Ol-O2)cOsO 2

"3 + (mE + m3 ) sin(01 - 02) 1[ cos 0

COS 02 d

0 0 Im2 sin (02 -- O1) Im2 sin(02 - 02)

sin (02 - 03 ) / sin(O2 - 03 ) + m 3 cos 01 + m 3

cos 03 -J cos 03 .__0 0 - m 3 sin 01 - m 3 sin 0 2

cos 02]

m 3 cos 03

m 3 sin 01 cos 03

t sin(O1 - 0 2 ) cos 0 m 3 cos 02

�9 cos 03

sin (02 - 03 ) m3

COS 03

�9 COS03

-- m 3 sin 03

(18)

X o

Yo q = gl 01

g2 02 g3 03

(19)

--MAOcos 01 {FI} = 1 - m l g cos O1 [-m, oC~

(Qxo + Qxl + Qx2 + Qx3) Qyo cos 01 + (Qxl + Qx2 + Qx3) sin 01

{F2} = (Qyo c o s 01 - Qxl s in 01)

(Qy2 c o s 0 2 - Qx2 sin 0 2) 0

0 0

sin (0 2 - 01) {F3}= (Ox2+Ox3)"

cosO 2 sin (0 3 - 0 2)

Q x 3 " cos 0 3

(20)

n-Segment Model. To make the formulation more general, models with different number of segments have been studied. The equations obtained have shown a general pattern enabling extension to an n-segment mathematical model


as follows. For the n-segment line of Figure 10, the kinetic and potential energy expressions can be seen to be

1 . 2 . 2 T = - M A (~:o 2 + ~'o 2) + + 2 mjxj mjyj

j = l j = l [ n_X ] V = MaY o + m n h + mjyj g

j = l

(21)

in which (x o, Yo) are anchor coordinates. Prob lem constraints are represented with the following 2n equations:

f 2 i - 1 - - [Xi -- ( X i - 1 + ~i COS0i ) ] : 0 , for i = 1 , 2 , - - . , n

f2i - [ Y i - - ( Y i - 1 + g i s i n 0 i ) ] = 0 , for i = l, 2, " ", n - 1 (22)

fEn -- [h - (Yn-1 + gn sin 0 n)] = 0

For this model, the total number of coordinates represented in Lagrange ' s modified equations is

N = n + c + 1 (23)

where n = the number of line segments c = the total number of constraints, that is, 2n

It can be shown that the motion of the anchor is given by

M A + m i i 0 - m i ej s in0j0j - m i ej0] cos0j = Qxo + Qxi

i=l i=l j=l i=l j=l "=

[ x I x n i

M A + tan 01 m~ 9o - tan 01 m~ gj sin 0j0j (24)

i=l i=l j=l

"2 - tanO 1 m i gjOj cosOj = Qyo + tanO1 Qxi - M g g

i=l j=l i=l

In addition, "n" other equations for the n segments can be written down in matrix form as follows:

o~ = 1,2,. . . . . n

1~ = 1,2,. . . . . n (25)


where a , 13 = indices employed to denote rows and columns, respectively

Aal 3, Ba13 and F a are given by

n Aal 3 = m a cos (0 a - 013). �9 + (sin(O a - Oa+l)sin 013 / cosOa+l) E m i

i=5 Bal 3 = masin (0 a - 013). �9 + sin (0 a - 0a+ 1) cos Oa+ 1

sin0 �9 sin cos x mi}

- { m a cos Oa}~0 - { m a g c o s Oa} + {Qya cosOa - Q x a sinOa}

- { s i n ( 0 a - 0a+ l ) / c~ s i=a+l

a = 1, 2 , . . . , n - 1 ;13 = 1, 2 , . . . , n

(26)

An13 = m n cos 013,

F n = -mn~ 0

Bnl 3 = - m n sin 0~,

(13= 1,2 . . . . . n) (27)

In the above, the multiplying factor �9 and the index 5 are defined as

1, a___13 ~= O, a<13

a + l , a>_13 8= O, a<13

(28)

Generalized Forces on Cable. The virtual power due to generalized forces applied through infinitesimal virtual velocities compatible with system constraints is

P - E Qi ~ 19i (29) i=l

where Qi = the generalized force associated with the generalized coordinate Pi

Equating this, for the n-segment model, with the power resulting from exciting forces applied through infinitesimal virtual velocities compatible with system constraints, yields the generalized forces as


Qxo = F,,~, Qyo =FV,

Qxi = F~, Qyi = FV, Q0i = 0 ,

(i = 1, 2 , . . . , n)

(3o)

where Fi I-I, F v = horizontal and vertical components of the nonconservative force acting at a lumped node in the horizontal (x) and vertical (y) directions, respectively.

Subscript A refers to the anchor, while subscripts i (1, 2 . . . . n) provide a representation of the nodes.

Computation of External Forces Acting on Nodes. In general, the forces acting on a mooring line segment can be categorized as (1) constant and/or time- dependent wave and current forces; (2) wind forces on the exposed part of the line; (3) for a steady flow, forces from Morison's formulation. Although the formulation permits the application of any forcing functions at cable nodes, wind and wave forces will not be included for the sake of simplicity, because the cable, for all practical purposes is almost entirely submerged in the water. For steady flow below the surface, Morison's formula [23] is appropriate for computation of forces on a submerged cable. The force exerted by a fluid on an accelerating body is composed of two components, one depending upon friction effects and the other upon displaced fluid inertia. For cylindrical cable segments in a steady flow this is [7, 23]

e l pwfDu lul§ pwfm -if7 (3~)

where f = force per unit length of the cylindrical cable segment Pw = mass density of seawater

u = segment velocity D = diameter

C o = drag coefficient C m = the added mass coefficient

The second term in Equation 31 is due to the added mass effect, which is included in the inertia force computation. Consequently, only the first term in Equation 31, representing the effect of friction, need be considered in the computation of the exciting forces. The virtual mass to be lumped at a node would then be

m' = (m + CmPw D2rc / 4) g (32)


where m = mass per unit length of the cable segment = segment length

C m = added mass coefficient

The drag forces in tangential and normal displacement directions of the pth segment of the mooring line can be written as (see Figure 12)

= - Iv#Iv - - Iv Ivv

(33)

where the dimensional coefficients Cp v and Cp N can be expressed in terms of the non-dimensional coefficients C T and C n as follows:

1 C T = -~- CTPw~Dp~p,

1 C y = -~ CNPwDp/~p �9

(34)

where ~/prp, ~r = average tangential and normal velocities of the pth segment Dp, (p = diameter and length of the pth segment

C T, C N = tangential and normal drag coefficients for cylindrical links

Appropriate values of these coefficients are recommended by Casarella and Par- sons [24] and Berteaux [25]. Pertinent information can also be found in [23] and [26]. Considering the velocities along and normal to the segment as the averages of the corresponding velocities at the nodes on the opposite ends of the segment, it can be shown that

V

T P

Op

Figure 12. Average velocities and drag forces along and perpendicular to the pth link.


~N = ~/0 COS 01 -- X0 sin 01 + �89 el{~ 1

V1 v = x0 cos 01 - 3'0 sin 01 p-1

vN = )/0 COS 0p -- X0 sin 0p + E ~J{~J COS (0p -- 0j) + 1 ep{~p j=l p-1

vT = 3'0 sin 0p + Xo cos 0p + E ~J{~J sin (0p - 0j) j=l

(p = 2, 3,--., n)

(35)

The normal and tangential drag force components can then be found from Equa- tion(s) 33. Typically, the tangential drag for mooring lines is small and may be neglected in comparison with other forces, if desired. The generalized forces Qxp and Qyp acting at a node p can be considered as the average of those at the connecting links, that is,

1 (Fp + Fp+ ) Qxp = ~ 1 ,

1 (FY +FY+ ) Qyp = ~- 1 ,

(p = 1, 2 , . . . , n - l ) 1

Qxn =-~ Fx ,

1 Qyn =-~ Fy,

(36)

where superscripts x and y denote force components in x and y directions, respectively.

In a situation where anchor motion is possible, the effect of added mass has to be incorporated in the anchor inertia and the drag forces on the anchor must be included too, using

1 AV2 F A = -~ PwAs C (37)

where A s = anchor projected area CD A = anchor drag coefficient Pw = mass density of seawater V = anchor speed

Drag coefficients for several body shapes and various values of Renolds number are listed in [23].


Numerical Solution. The set of mooting line equations of motion derived for the n-segment model can be written in the form

[M] {~i} = {F} (38)

where [M] = pseudo-mass matrix of functions of the qs {F} = a pseudo-force column matrix made up of functions of the qs and

the qs

For sufficient anchor holding capacity, the motion of the anchor can be neglected and the anchor can be restrained at the origin. In this case, the anchor motion given by Equation(s) 24 need not be included. A minor modification to Equation 25 obtained by dropping the ~0 and Y0 terms of Equation(s) 26 will yield a complete set of n equations of motion with only 0s as the variables. These equations are coupled, nonlinear differential equations that can be solved numerically.

A "starting" static equilibrium configuration [13, 14, 18] can be assumed to provide the initial conditions for the problem. If the values of 0s and 0s are known at any instant, the substitution in [M] and {F} will lead to a set of n simultaneous algebraic equations with 0s as unknowns. Elements of [M] and {F} are first computed from the known values of 0s and 0s at the first step and then the simultaneous equations are solved bv Gauss's method of elimination to find the fis, which leads to the solution for the subsequent time step. Essentially, we have a set of n coupled nonlinear differential equations to be solved at each time step. The solution to these equations may be obtained by Runge-Kutta calculations. A subroutine subprogram that performs Runge-Kutta calculations by Gill's method [27] is used to solve these second-order differential equations. This is a self-starting method, which yields a step-by-step-solution up to the given time limit, once the initial 0 and 0 values are furnished. The time step can be monitored during the computations, if desired.

Mooring Line Tensions. An analysis of forces at the surface end of the cable yields

T H = Fn H - m n ~ n

T v = F v - m n g (39)

where F H, F v = horizontal and vertical components, respectively, of external forces acting at the nth mass

T H, T v = horizontal and vertical components of the tension in the nth segment

~(n = acceleration


Xn = XO ~j (Oj sin Oj + Oj cos Oj)

j=l

(40)

Similarly, the tension components in adjacent segments i and (i + 1) are seen to bear the following relationships (see Figure 13):

TH = TiH I + FH -- mi x0 gj(0j sin0j + 0j cos 0j)

j=l

"2 T V = T iV1 -t- Fi V - m i g + Yo + ej(i}j cosej - ej sin ej)

j=l

(41)

where F H, F v = horizontal and vertical components of external forces acting at the ith mass m i

Starting with the nth segment, thus, tensions in all segments of the cable can be found using Equations 39-41. The values of x 0, Y0, Xo, Yo, 0i, t~i, and (~i are known at any time, and, in turn, tensions at that time can be computed, yielding the entire tension time history from the differential equation solution.

t Y

l g

/ ~ ' v

t s Imp1+ 1

TI+ I / - J

',,4.' 1 J =i-1 / m i g

m:l. ~":t

�9 --==--=lb- X

Figure 13. Horizontal and vertical equilibrium of mass, m i.


Mooring Line Dynamic Tension-Displacement Characteristics. Relations may now be obtained giving time-displacement, time-tension, and time-horizontal tension profiles. With the help of these, a horizontal tension vs. horizontal cable projection curve can be generated providing the tension-displacement characteristics of the mooting lines for specified configurations or displacement ranges. Using curve-fitting techniques, polynomial forms for these curves can then be obtained.

The "starting" or "initial" configuration for a given displacement range must, however, be specified, which can be furnished through the static catenary equations [ 13, 14, 18]. Effects of elastic stretch, thermal elongation, etc. can also be easily included in the generation of the "initial" static cable configurations. The dynamic solution assumes, however, that the cable remains inextensible as it goes from one configuration to the next.

As a moored offshore structure subjected to environmental forces moves from its initial position, some of the mooting lines can become taut while others can slacken. The technique employed can handle both taut and slack cables as well as single-component and multicomponent combinations with or without clump weights.

Computer Program Development. Using the analysis technique presented in this chapter, a digital computer program has been developed for generating the dynamic tension-displacement characteristics of a mooring line. This program performs a dynamic time domain analysis of mooring line motions. A flow diagram illustrating the general procedure on which the development of the program is based is shown in Figure 14.

Example Problem. To be able to compare "dynamic cable" responses with the results of the "quasi-static" analysis the same mooring line of the example of the previous sub-section is reexamined using the dynamic model equations. For purposes of modeling, the line is broken up into ten segments, with the clump weight put in as an additional mass at the appropriate node as shown in Figure 15.

Values of Cm, CN' and C T used for the cable were 0.5, 1.4, and 0.0112, respectively. For low Reynolds numbers, the drag coefficient values employed seem reasonable. The mooring line is analyzed for five "starting" configurations, which are obtained from the static catenary equations, and are shown in Figure 15. Tension-displacement profiles are then developed for specified line configurations as the initial conditions. These are shown in Figure 16. The jagged appearance of the curves at the upper end of the horizontal projection range is probably due to the way in which the line has been discretized and the initial conditions prescribed for the cable dynamics solution. The hump at the upper


START

, I , -

< inp uc

Se t initial c o n f i g u r a t i o n o f moor ins l l n e u s i n 8 one of the

methods e=ployin s s ta t ic catensry e ~ u a c l o n s , w i t h segment masses

lumped a t d i s c r e t e nodes

. . . . . . _

Manipulate equation (16) in the form:

CM]{~')- {r) _

Calc. elements of matrix [HI and v e c t o r {F] which a r e f u n c t i o n s o f p's and p's, usinS equation (18).

! , ~ ,, ,

So lve s e t o f n - s l m u l t a n e o u s a l g e b r a i c e q u a t i o n s in m a t r i x form

by Gauss - Jo rdon method, to y i e l d [p'}

t a l c . p and ~ u s i n g B u n g e - K u t t a c a l c u l a t i o n by G i l l ' s method

" " I ' " - - -

Csl r v e s s e l p o s i t i o n , h o r l z o n t a l and ver t ica l compt, of tension in each segment , u s i n s e q n s , (30)-(32)

. . . . . . . . _

. ' j

YES

. . . . ~J / ,

Cenerate horizontal component of tension vs. horizontal

displacement of vessel data (tens ton-disp lacemen c

c h a r a c t e r l s r o f i n d i v i d u a l a o o r i n 8 l i n e s )

l Develop equation of polynoLtal of tension-diap lacement

characteristics using curve- f i t t t n s t e c h n i q u e s

Figure 14. General scheme for developing mooring line dynamic tension-displacement characteristics.

2 2 4 Offshore Engineering

[ . Con,i or,,,on, 1 -

~ A "~ O _ _ ~ A 5 _ . . . . . -r" O 1()0 v 200 300 400 500

Distance in ft. e~ ~ ' . 5 0 . . . . . . . . . . . e~ J= O~ "~ 0 - t-

O I oo 200 300 400 500 Distance in ft.

e,d ~- 50 r " O) "~ 0 "1-

o

A

. . . . Conf igurat io

1 o0 200 300 400 500 Distance in ft.

50 . . . . . . . . . . . . . . - r- o~

::l: 0 1 O0 200 300 400 500 Distance in ft.

= 5 0 . . . . . . . . i

"-~e 0 I -!- O 1 O0 200 300 400 500

Distance in ft.

Figure 15. L ine s ta r t ing configurations obtained from static catenary equations. C o n v e r - s ion factor: 1 f t = 0 .3048 m.

right comer of the profile generated with configuration 4 as the initial condition, for instance, is because the line that is initially in configuration 4 (with the clump weight just off the ocean bottom) immediately goes into configuration 3 (clump weight on bottom) upon the start of the dynamic solution. A similar effect is seen in the quasi-static analysis (Figure 9) when the clump weight suddenly lifts off the ocean bed giving a "jerk" to the mooring cable.

Computation of Environmental Forces

Before the response of an offshore vessel can be realistically computed, an analyst must have quantitative estimates of all the significant environmental forces acting on it. This section presents the necessary equations for the computation of loadings due to wind and current on an offshore vessel, and illustrates, by an example, a practicable procedure for generating a wave force time history from a given random sea spectrum. Once the total excitation is obtained as a function of time, vessel equations of motion can be formulated and numerically

Mooring Dynamics of Offshore Vessels 22.5

=- 180 z" _o z

'" 140 I---

100 o o ~ o

60 o

20 ~.- 490

/ "J "" As Initial Condition ./Confl - - Cable Dynamics 5

Included ] Conf |

- - - Cable Dynamics Conf3" / 4* /~ - - Ignored Conf 2" /

Conf 1"

I I I 491 492 493 494 495" 496' 497

HORIZONTAL PROJECTION, FT

Figure 16. Tension-displacement profiles generated for the multicomponent mooring cable by dynamic cable analysis. Conversion factors: 1 ft = 0.3048 m; 1 Ib-force = 4.44N.

solved using standard dynamical analysis methods to generate useful response information.

The external forces causing excitation of offshore vessels exposed to an open ocean environment are characterized by winds, waves, and currents. These forces, their origins, and their occurrence, are not yet completely known. A lot of statistical data have been collected, on the basis of which predictions are made, and the resulting forces are calculated from empirical relations, charts, and tables. Wind and current forces are usually described in terms of two components: a drag force in the direction of flow, and a lift force perpendicular to the flow direction. Wave forces are computed using the principles and procedures of spectral analysis.

Forces Due to Wind

Because of the stochastic nature of wind, its properties vary with time and location. In common meteorological practice, wind velocity is predicted as an average over a certain interval of time, varying from 1 to 60 min or more [28]. Although offshore structures are excited by fluctuations in velocity around the mean value, these aerodynamic forces, in general, are found to be small in comparison with hydrodynamic forces. Under these conditions, a steady value of wind velocity, both in magnitude and direction, is considered by most researchers [6, 28, 29]. Therefore, excitation due to wind is generally taken in


the form of steady forces and moment. The two-fold role of wind may be studied as the forces due to a stream of air on the exposed portion of the structure, and/or exerted by the wind on the water surface, causing disturbances of the still level and leading to wind-generated waves and currents.

Remery and Van Oortmerssen [28] and Ewing [30] have presented wind-load data for tanker-shaped bodies, ships, and offshore structures. For ship-shaped bodies, these forces may conveniently be described as steady longitudinal and transverse components, acting on the midship section, and a steady yawing moment. These are

(1) 2 X w = . PaVwCxw ((X) AT

Yw = " PaVwCyw((Z) AL

M w = PaV~Cmw (a) ALL

(42)

where X w = steady longitudinal wind force Y w = steady transverse wind force M w = steady yaw moment

Pa = air density V w = wind velocity

ct = wind direction A x = exposed transverse area A L = exposed lateral area

L = length of vessel

The coefficients Cxw, Cyw, and Cmw, which depend on the angle of incidence of the wind, can be expanded in a Fourier series as a function of the angle of incidence:

o o

Cxw = a~ + E anC~ net n=l

C yw = E b n sin no~

n=l

Cmw = E bn sin n a n=l

(43)

where a 0, a I ..... an, and b 1, b 2 ..... b n are Fourier coefficients given in Reference [28] for a fifth-order representation of wind data, which is sufficiently accurate for preliminary design purposes.


Forces Due to Current

The complex nature of the occurrence of current is understood to be a result of several independent phenomena. These may be (1) the ocean circulation system, resulting in a steady current; (2) cyclic changes in lunar and solar gravities causing tidal current; (3) wind and differences in sea-water density.

Tidal currents are of major importance while working in areas of restricted water depth and can, occasionally, attain values up to 10 kt; and a 2- or 3-kt tidal current speed is very common. The magnitude of tidal currents is predicted by considering many (at least 60) parameters. In the case of offshore vessels, currents at the surface are the governing ones.

Because the direction of various components will be different, total measured current is split into two or more components, e.g., a tidal component and a non- tidal component. Because the variation in current velocity and direction is very slow, its effect is often considered as steady for the sake of analysis.

Current data for the design of ships and offshore structures are presented by Ewing [30]. Current forces and moment may be assumed to be made up of two parts: (1) a viscous part, due to friction between the structure and the water which is negligible for blunt bodies, and due to pressure drag; and (2) a potential part, with a component due to circulation around the object, and another component due to the free water surface (wave resistance) which is mostly small.

The forces and moment due to current on a ship-shaped structure can be writ-

ten as [28]

Xc wV: xc,o,

Yc-- (1.) pwW2fyc ((z) mLs

Mc ._ (1.)pwW2fmc((X) ALS t

(44)

where X c = steady longitudinal current force Y c = steady transverse current force M c = steady yaw current moment Pw = water density V c = current velocity ot = current incidence angle

ATs = submerged transverse area ALS = submerged lateral area


Here, Cxc, Cyc, and Cmc are coefficients depending on the current direction. As in the case of wind-load coefficients, these coefficients are expanded in a Fourier series as a function of the angle of current incidence:

o o o o

C Y c = E b n sin nc~ and Cmc -- E b n sin no~

n=l n=l

(45)

For mooring problems, the longitudinal current force is not too important and may consequently be neglected. However, for head sea and quartering sea conditions, its magnitude may be estimated using [28]

0-075 ] [cos c~ I Xc = ( ln~ n--?)2 " �89 2 coscx (46)

where Rn=Reyn~ /VcC~176 ) v L

v = kinematic viscosity of water S = wetted surface

For deep water, average values of coefficients for a fifth-order Fourier representation are given in [28]. For shallow water, the current force and moment coefficients are multiplied by a factor that is a function of the water depth/draft ratio [28].

Forces Due to Waves

There are many sea spectrum formulas available in the literature with significant differences among them [31-34]. These differences may be due to terminology, notation, and parameters used for the ordinates of the spectral curves. For the purpose of selecting a suitable formula to be used in analysis, these differences should be studied thoroughly so that any possible error resulting from confusion may be eliminated. Cuong et al. [35] have described a method that uses the Inverse Fast Fourier Transform to obtain digital random time histories with specified stochastic properties. Such random records, which are generated by summing a finite number of sine waves, have proven to be very useful and often necessary in projects involving computer simulation or model towing tank experiments.

The spectral density SF(O) of the wave force can be found by first computing the autocorrelation function RF(X), which can be expressed as [36, 37]

4 2 2 R r (I:)= r~uKclG[Ru(a:) / C~u] + KiR a (1:) (47)


in which

G(r) = 4r 2 + 2)sin -1 ( r )+ 6r (1 - r) ~ (48)

and K d and K i are constants determined from the drag and inertia coefficients C d and C m, respectively. The drag and inertia constants are given as

0wCc

Ki = PwAcCm (48a)

where L c, A c = vessel characteristic length and area, respectively

2 is the variance of the wave velocity u, and may be obtained The quantity Ou from S u (co) using

~2 ~0S u = u(C~ at~ (49)

and R u (x) and R a (1;) are the autocorrelation functions of velocity and acceleration. A useful approximation to G(r) is provided by the series [36]

G ( r ) = - - l [ 8 r + 4 / 3 r 3 + l / 1 5 r S + . . . ] (50)

In principle, the spectral density SF(C0) of the wave force can be derived by taking the Fourier transform of RF(X). However, because of the presence of higher powers of Ru(Z) and the drag component of the wave force spectrum, which involves a series of self-convolutions of the velocity spectrum, this becomes a complex process. Taking only the linearized form of Equation 50 as a reasonable approximation, Equation 47 yields

RF(I:) = 8 K2O2uR u (I:) + Ki2Ra (1:) (51) /1;

The force density SF(C0) can now be found by taking the Fourier transform of Equation 51, which yields

SF(C0) = __8 K~O2uS u (CO) + K~S. (CO) (52)

where Su(CO ), S a((O ) = spectral densities of the velocity and the acceleration, respectively


Linear theory gives the horizontal velocity u and the acceleration a of a component wave train in complex form as follows [37]"

u ( t )= H u (00) rl (t) and a (t) = H a (00) T I (t) (53)

where k = wave number (= 2re/Z)

Hu(00), Ha(00) = complex receptances given by

H u (00) = 00cosh [k(z + d)] / sinh (kd)

H a (00) = - i00 2cosh[k(z + d)] / sinh(kd) (54)

and

rl (t) = (H s / 2) e -it~ (55)

gives the wave elevation, with H s representing the significant wave height. Here, the parameter z is zero at the vessel center of mass, and d is the water depth. The corresponding velocity and acceleration spectra are related to the spectral density Sn(00) of the wave elevation by

Su (o,)- II-Iu (56)

S a (00) - - [H a (00)12 Srl (00)

Equations 55 and 56 yield, after simplification,

Su (00) = 00 2 cosh2[k(z + d)] / sinh 2 (kd) �9 Sn(00)

Sa (00) __ 004 cosh2 [k(z + d)] / sinh 2 (kd). Sn (00) (57)

On substitution of these values of Su(00) and Sa(00 ) in Equation 52, the spectral density of the wave force can be obtained as

2 2 SF(00 ) "- Srl (00) {002[c0sh2 (k (z + d) ) / sinh 2 (kd) ] [(8 / rt) KaO u + Ki 2 ] } (58)

Equation 58, obtained for the spectral density of the wave force, is related to the sectional force on the structure as computed in Morison's equation. However, to obtain the spectral density SF~(00) of the total force on structure, this force has to be integrated over the characteristic length of the vessel, which, in this case, is the draft.

The wave force spectrum, which is a mathematical representation of the total force exerted by a random seaway on an offshore structure or vessel, is built up,


theoretically, by an infinite number of sine-wave components superimposed in a random manner [33, 34, 38, 39, 40]. In practice, however, this spectrum can be approximated by the superposition of a large number of sine waves rather than by an infinite number, each component having its own frequency, amplitude, and phase lag. Once the individual components are generated, the total force F(t) can be computed using

where

F (t) = s F j cos [ ~jt + e(~j ) ] (59)

j=l

F~ = amplitude of a component wave = 42SFv(CO j) ~coj

~j = frequency of the jth component &oj = width of the jth frequency segment

e(~j) = random phase lag of each component in relation to any arbitrary reference, is uniformly distributed between 0 and 2rr [32, 33, 34, 39, 40, 41].

It is to be noted that while the effect of regular waves is a second-order drift force that gives rise to an almost static shift of the position of the moored vessel, irregular waves, on the other hand, are composed of first-order, high-frequency oscillatory forces of large magnitude and varying periods (6-12 sec) superimposed upon slowly-varying, low-frequency (20-200 sec) drift forces of smaller magnitude, which arise due to second-order hydrodynamic effects and which are proportional to the square of the wave height. From a standpoint of vessel station-keeping, the large first-order wave forces, because of their high-frequency characteristics, are in a region of low system response and consequently have little or no effect on vessel motions. The second-order low-frequency drift forces, however, despite their lower magnitudes, are in a region of maximum system response and consequently give rise to large drifting motions or oscillations of the vessel in the horizontal plane. In this region then, most of the applied load must be absorbed by the mooring lines. Owing to its importance in determining shut-down conditions, "slow-drift" response, which is due to the soft mooring stiffness and large vessel mass and inertia, and which is characterized by the mean values of surge, sway, and yaw motions, is a significant consideration in mooring analysis. Also, because the mooring system must be able to withstand the second-order drift force and still be flexible enough to allow movement or oscillation about the mean vessel position, proper prediction of the forces due to waves is important in evaluating vessel station-keeping.

For the class of problems intended to be solved with the analysis presented in this work, the system response is generally between 20 and 200 sec, which is outside the range of normal waves (6-12 sec), reinforcing the significance of


long-term drift forces as opposed to short-term forces from each individual wave. Methods of determining the drift force in irregular waves have been discussed by Hsu and Blenkarn [42], Remery and Hermans [43], Remery and Van Oortmerssen [28], and Pinkster [44].

As an illustration, the time history of wave forces due to a 6-ft (1.82-m) random beam sea with a characteristic wave period of 5.5 sec acting on a barge 210 ft (64 m) x 60 ft (18.3 m) x 14 ft (4.3 m) with a draft of 7.5 ft (2.3 m) in water 50 ft (15.3 m) deep is computed using a computer program based on the technique presented in this chapter. The wave spectrum employed is the modified Pierson-Moskowitz spectrum. The wave force spectrum and the corresponding wave force time history generated are shown in Figures 17 and 18. The total number of intervals used for discretization of the force spectrum was 26.

0

0

o

z Q

W o ~ o , ~ -

-

) . 3 0 2 - 5 4 4 . 7 8 7 , 0 2 9 . FREOUENCY, RAD/SEC.

Z5

Figure 17. Wave force spect rum for the modi f ied P ierson-Moskowi tz wave spec t rum (T s - 5.55 sec; H s - 6 ft). Conversion factors: 1 Ib = 4.44N, 1 ft - 0.3048 m.


_, _ _ | _ J - :

�9 �9

0

~ 8

,

8

I I I ! I I 1 I I I I �9 1 aO.O0 23.92 47.84 71.76 95.68 119.60 143.52 J

T I M E I N SECONDS

Figure 18. Sine-wave representation of wave force. Conversion factor: 1 Ib = 4.44N.

Vessel Dynamic Analysis

Vessel Equations of Motion

Most advanced dynamic analysis procedures [45, 46, 47] attempt to derive the vessel equations of motion in six degrees-of-freedom: surge, sway, heave, roll, pitch, and yaw (see Figure 19). These six coupled equations of motion are then solved as simultaneous equations, which is a complex process requiring large computer time. However, as discussed in [18] and [48] and under the assumption that the dynamic "station-keeping" response can adequately be described by a three degree-of-freedom mathematical model representing the inplane motions of surge, sway, and yaw, the equations of motion of a moored vessel with lateral symmetry can be written as

Surge: ( M + mxx) ~ + Bxx/C + Qx (x, y, (x) = F x (t) Sway: ( M + myy) y + Byyy -I- Qy (X, y, (X) = Fy (t)

Yaw: ( I + maa) 6~ + Baa6~ + Q~ (x, y, (x) = M z (t)

(60)

where x, y, (x = surge, sway, and yaw displacements (x is positive forward, y is positive port, and (x is positive bow to port)

/c, ;r & = surge, sway, and yaw velocities


Z

(6) yaw I (3) heave

(2) sway

tch

X

( I ) surge

Figure 19. Definition sketch of the six components of motion of a floating structure.

~(, ~,, a = surge, sway, and yaw accelerations M = vessel mass

I = vessel mass moment of inertia about yaw axis through the barge center of mass

mxx, myy, maa = frequency-independent time-average added masses Bxx, Byy, Baa = first-order damping coefficients of vessel resistance to

velocity Qx, Qy, Qa = nonlinear restoring forces and moment induced by the

mooring system F x, Fy, M z = external nonlinear time-varying forces and moment

from waves, wind, current, etc., acting on vessel.

In these equations, all cross-coupled added mass terms have been dropped, because the mean response of the vessel is assumed to be independent of the coupling effects of the fluid mass [18]. Cross-coupling damping terms are also excluded because these are negligible for zero vessel speed for a vessel having lateral symmetry [49]. The nonlinear restoring forces Qx, Qy, and Q~, due to the dynamic behavior of the mooring system are functions of the vessel position at any time and, for a multi-line mooring system with N lines, can be expressed as:

N N

Qx = - E TH cos~j,Qy = - E T j H sin~j j = l j = l

N

Qa : - E TJ n (xF sin 13; - yF COS ]3~)

j=l

(61)


where 13j = angle that the jth mooring line makes with the x-axis

TH= 13~= horizontal component of the tension in that mooring line angle that the jth mooring line makes with the vessel longitudinal axis in the horizontal plane

(x F, yF, Z F) = initial coordinates of the jth fairleader referred to a system of axes at the initial position of the vessel center of mass

As the vessel center of mass moves away from the origin, the new coordinates of the jth fairleader with respect to a system of axes at this new position can be found through the coordinate transformation equations. Note that ~j is the same as [~j at the initial vessel position when the vessel longitudinal axis coincides with the x-axis. Computations of ~j and 13~ can be performed using coordinate transformation and given vessel geometry [50].

It is important to note that in classical ship motion theory, equations of motion are typically formulated in the frequency domain by employing hydrodynamic coefficients that are dependent on the frequency of motion. Cummins [51 ] and Van Oortmerssen [52] have discussed a conversion to a time-domain form by which a convolution term is added to the vessel equations of motion. This convolution integral involves retardation functions that are representative of the memory effect of the past motion of the ship and arise because of an assumed impulsive nature of the forcing function. A review of Van Oortmerssen's work [52] suggests that even the most important retardation functions, which are those for uncoupled motions, drop down from a high value to a drastically low value quite fast and start oscillating about a zero level. Because the very short period of time in which this drop occurs can be considered as a period of transience and should not have much of an effect on the steady-state behavior of the system, the distant past motion of the vessel reflected by the inclusion of the convolution term is seemingly negligible. Because the purpose of the present analysis is to evaluate the "long-term" response of the system (in the order of tens of seconds), Equation(s) 60, though not precisely correct, can be regarded as acceptable and reasonable approximations, in which constant added mass and damping coefficients, as computed for different frequencies and then averaged, are used. The values of these constant coefficients, so long as they are reasonable, should not have much of an effect on average system response values.

Solution Technique

The equations of motion of a moored structure with lateral symmetry are found to be second order, nonlinear differential equations which are implicitly coupled due to the nonlinear restoring forces and moment induced by the mooring system. For a three-degree-of-freedom mathematical model, there are three


equations representing the inplane motions of surge, sway, and yaw. Initially, an equilibrium configuration of the moored system is assumed to provide the initial conditions for Runge-Kutta calculations. At any time step then, the horizontal projection of each mooting line is computed. Associated cable tensions can be determined from the tension-displacement profiles along with the required forcing functions Qx, Qy, Q~ for particular values of 13's, and 13" s, which can be calculated for the time step in question.

Other external time-varying forces and moment from waves, wind and current, etc., Fx(t), Fy(t), and Mz(t), in surge, sway, and yaw directions are computed for a particular sea state, and the vessel equations of motion solved by Gill's method [27] using Runge-Kutta calculations.

This is a self-starting procedure that gives a solution at every time step up to the upper limit of the time provided by the user, once initial conditions are furnished. The accelerations computed at any time step provide the solution for the subsequent time step. This is numerically valid so long as the time step used is reasOnably small. Thus, the values of x, y, ct, k, y, &, ~(, ~,, and ~ are all known for a discrete time domain with small time increments which can be monitored during computation. The moored system can be studied for any representative position at a given time, and time histories of surge, sway, and yaw motions of the vessel together with those of line tensions can easily be obtained.

Numerical Results and Observations

Practical applications of this analysis include pipelaying, derrick and trench- ing barges, and other production platforms requiring multileg mooring systems. This chapter illustrates analysis methods by examining the dynamic response of a typical floating production vessel moored by a multileg system of six multicomponent cables in 6-ft (1.82-m) random beam seas, as shown in Figure 20. The vessel attached to the floating end of each cable is 210 ft (64 m) x 60 ft (18.3 m) x 14 ft (4.3 m) with a 7.5-ft (2.3-m) draft in water 50 ft (15.2 m) deep. Figure 8 shows a typical mooring line employed. Average vessel added mass coefficients for surge, sway and yaw motions are estimated using the formulas suggested in the literature for ship-shaped structures [53-56]. Environmental loadings from winds, current, and waves are calculated using the formulas given in the preceding section of this chapter.

Results generated with the effect of cable dynamics ignored (quasi-static analysis) are presented in Figure 21, which shows the time history of the sway motion of the vessel and the tension in line 2 obtained with an initial line pretension of 20 kips (89 kN). The surge and yaw motions were observed to be small and are therefore not plotted. The results indicate that the mooting system functions mostly in configurations 1 to 3 with the clump weight sitting on the ocean bottom. At maximum vessel sway, the clump weight has been lifted off the ocean bottom, but the line tension is only 30% of its limiting value and the


Anchor (5)

Anchor (4|

~ . .

Anchor 131

Anchor (6)

Anchor (1)

Anchor (2)

t Beam Sea

Figure 20. Configuration of moored system used in example problem.

I),o!

I.O,

I- w 3.0. tl.

~ o ~ z

_1 ~_ -6 .o .

- 9 . 0

"SWAY MOTION

a a a A

' ~o ,oo L=O0 TIME, SECONI~

CONFIG.4

u)

CONFIO. 3

o ol z

CONFIG.Z

~----MAXIMUM LINE TENSION

TENSION IN LINE 2 Iso

il ,100

CONFIG. I ~ - ~ " "

TIME, SECONDS

Figure 21. Sway motion and line tension time histories generated with quas i -s ta t ic cable analysis. Conversion factors: 1 ft = 0.3048 m; 1 Ib-force = 4.44N.


anchor still holds. Thus, the multicomponent mooring system has been used in a satisfactory manner. While surge and yaw motions were quite small in comparison, the average sway was found to be 2.1 ft (0.64 m) with an average line tension of 28 kips (124 kN).

Figure 22 presents results generated with the effect of cable dynamics included in the analysis. The analysis was carried out with a time step of 0.075 sec for cable dynamics and 0.1 sec for vessel dynamics. It can be seen that convergence towards the steady dynamic oscillation is quite rapid. The differences between the amplitudes from the first to the last cycle shown are rather small for the tension in line 2 and for sway motion, which are the most significant behavior parameters to be studied in the case of an offshore vessel exposed to a random beam sea environment. Some small fluctuations in the surge and yaw response amplitudes are noted but, because these motions are small, these variations can be disregarded. In view of the damping force provided by the water, there is, therefore, reason to believe that steady-state conditions have been achieved. With the cable dynamics included, the analysis yields an average sway of 1.8 ft (0.55 m) with an average line tension of 59 kips (262 kN) as shown in Figure 22.

Surge and yaw motions are still quite small, which is reasonable for the system in the environment considered. At maximum vessel sway, the system functions in configuration 4 in which the clump weight is lifted off the ocean bottom at a line tension of 98 kips (435 kN). However, for the most part, the system operates satisfactorily in configurations 1 to 3 with the clump weight sitting on bottom. The results show that while the maximum tension in line 2 obtained in this case is less than that obtained with the quasi-static analysis, the average tension with the cable dynamics included is 111% higher. The inclusion of the effect of cable dynamics has resulted in somewhat higher maximum tensions in lines 4, 5, and 6 but lower maximum tensions in lines 1, 2, and 3. Average values of tensions in all lines are much higher than those of the quasi-static analysis. A comparison of tension-displacement characteristics for line 2 with and without the cable dynamics effect (Figure 16) indicated that, although, in the displacement range of cable configurations 4 and 5, the line tension does not go up with the inclusion of cable dynamics; it does shift to higher values in configurations 1 to 3, which are the configurations in which the mooting system mostly functions. Because the average tension in line 2, with cable dynamics is, in general, higher, the average sway is reduced by 14%. Details are given in Table 1. These observations suggest that, to predict mooring system performance and to evaluate moored vessel response in a realistic and meaningful way, the effect of cable dynamics must be included in the dynamic analysis of the vessel.

From the lateral motion responses shown in Figures 21 and 22, it is interesting to note that the response periods appear to be between 35-40 sec, which correspond to exciting forces in the 0.16-0.18 rad/sec frequency range, while, in fact,


0.16 C 6

E 0.00 G u

w

i~ -0.16

Surge Mot,on . . . . . . .

A A,A A

-0 .32 / . . . t l f . . . . . j . t ._. t j r ,

6.00

2.00 E=

-2.00

c~ -600 0.20 iYaw Motion

o

c 0.10 E

m

~.=_ 0 . 0 0 " i I I I �9 o -0. I0 . . . . . . . . .

T I ension In' L ine2- . . . . . . . Maximum Line Tension

.~ 87

c o t _ ~ A I ~ v e r a g e Linon_ ~ 3 71 Tens, I--

,,= 55 | U

39 I I , !. ! ~ l.. 20.00 38.67- 57.35 76.03 94.70 113.38 132.05 150.73

T i m e i n S e c o n d s

Figure 22. Vessel motion and line tension time histories generated with dynamic cable analysis. Conversion factors: 1 ft = 0.3048 m; 1 Ib-force = 4.44N.


Table 1 Comparison of Dynamic Cable Analysis Results

With Those of Quasi-Static Analysis

Variables Maximum Minimum Average

Units* A B A B A B

Standard Deviation A B

Barge Motions

Surge Sway Yaw

Feet 0.4 0.2 -0.5 -0.3 0.0 -0.0 0.2 0.1 Feet 7.1 7.6 -4.5 -5.5 2.1 1.8 2.0 2.4

Degrees 0.4 0.2 --0.2 -0.1 0.1 0.1 0.1 0.1

Cable Tensions

Line #1 Kips 94 74 16 35 31 49 11 6 Line #2 Kips 126 98 11 40 28 59 13 10 Line #3 Kips 98 70 16 35 31 49 12 6 Line #4 Kips 42 63 13 34 20 43 4 5 Line #5 Kips 41 77 8 38 14 47 4 6 Line #6 Kips 40 65 13 34 20 43 4 5

*Conversion Factors: 1 ft = 0.3048 M; 1 kip = 4.44 kN A Cable Dynamics Ignored B Cable Dynamics Included

these low frequencies do not even appear in the wave force spectrum shown in Figure 17. This observation reinforces the importance of the effect of the second-order interaction between the waves and the structure in the form of a drift force that gives rise to the large, low-frequency, drifting motions of the moored vessel, as depicted in Figures 21 and 22.

Effect of Vessel Hydrodynamic Mass on Vessel and Mooring Line Dynamic Responses

Because the inclusion of added mass is generally an important consideration in the dynamic analysis of an offshore vessel, the effect of the hydrodynamic mass terms on the station-keeping response of an offshore construction vessel is studied in this section from a standpoint of mooring system design.


Added Mass Coefficients

While the ideal way would be to include added mass coefficients as functions of the frequency using two-dimensional [57-60] or preferably three-dimensional [61-67] computational procedures, there are ways by which they can be reduced to their frequency-independent forms [51, 52]. However, one could also generate them using the available formulas in the literature for ship-shaped structures such as the following [53-56, 68]: Surge added mass coefficients:

Cx~B2D Cmx = 1 + ~ [53] (62)

8M

Sway added mass coefficients:

B~ Cmy : 1 + ~ [53]

16D 2pD2jL

Cmy - 1 + ~ [54] (63) Mn

2D Cmy - 1 + ~ [55]

B

Yaw added mass coefficient:

1 CmyML 2 C m t x - - 1 + - [50, 56] (64)

3 I

where C x = inertia coefficient plotted in Figure 23 as a function of the barge beam-to-length ratio

B = barge beam D = draft L = barge length p = mass density of sea water

M = vessel mass I = yaw mass moment of inertia of the vessel J = coeff ic ient whose value is not known accura te ly but can be

obtained approximately from Figure 24

Numerical Results and Observations

To investigate the effect of the added-mass terms in the vessel equations of motion, the dynamic station-keeping response of a typical construction vessel


2.4

X 0

2.2 \ l-- Z W (.J e.O b_ b.. Ld E3 1.8 \ 0

<[ 1.6 F- rY LJ Z ~.~ 1.4

1.2

1.0

~ T H E D I

EXPERIMENT

~Y

->-\

0.1 0.2 0.5 l,O ~.0 5.0 10.0

BARGE B E A M - LENGTH RATIO

Figure 23. Dependence of surge added mass on barge dimensions.

2.4

2.2

2.0

I - z LU 1.8 O m LL LL LU 1.6 O

1.4 /

. / /

/ f -

1.2

1.0 3 5 7 9 11 13 15 17 19

L/2D RATIO

Figure 24. Dependence of coefficient J on barge parameters.

Mooring Dynamics of Offshore Vessels 2 4 3

moored by a multi-leg anchoring system in 3.5-m beam and quartering sea conditions was studied using the mathematical model depicted by Equation(s) 60. The vessel employed is 200 m long x 40 m beam x 15 m deep, with a draft of 10 m, in water 300 m deep, and the dynamic effect of the mooring cables is ignored in the analysis. The plots sketched in Figures 25-28 show that the average system response in both beam and quartering sea conditions is, in general, not affected by the inclusion of the added mass terms.

On the other hand, the peak excursions and maximum line tensions seem to vary, in general, with any changes in the added mass coefficients. For the range of added mass coefficients used in this study, the maximum percentage variations in these responses are given in Table 2. From the results generated, it can be concluded that the added mass terms in the equations of motion have little or no influence on the average station-keeping response of a construction vessel moored with a conventional multileg anchoring system in an open ocean environment. On the contrary, the peak excursions and the maximum line tensions

I .... BEAM-SEAS " QUARTERING SEAS 60- M A X I M U M E X C U R S I O N S

( / } CMYAV=I.O rv, 5 0 - 1.5 W 2.0

W ~ - - ~ - ~ - - - ~ - ~ "

v 40 . .. ~ ~ ~ I I \ z E] I - - - I

c,q 30 "

(.3 X 20 ILl "

I , I LD cv Z) tO - CMYAV = 1.0 -2 .0 O0 QUARTERING SEAS _ \ " " ~

IAVG- BEAM SEAS o L _~. I . . . . . I

1.0 L5 2.0

CMX, CMY

Figure 25. Surge excursion as a function of added mass in beam and quartering seas.


GO

(/) cy' 50 - W I-- W Z v 40 -

Z O I - - - I

c,q 3 0 " n,,

o X W 20 -

>- <~

C/j

- ' B E A M S E A S " - - Q U A R T E R I N G S E A S I "

MAXIMUM EXCURSIONJ C M Y A V = i I ~ ~

~VG-QUARTERING SEAS

CMYAV 1.0-2.0

AV~-- B E A M_ .,SEAS . . . .

1.0 1.5 2.0

CMX, CMY

Figure 26. Sway excursion as a function of added mass in beam and quartering seas.

generated are, in general, affected by changes in the added mass coefficients. It is therefore recommended that, in situations where information regarding peak excursions is deemed crucial in the design process, but sufficiently accurate values of the added mass coefficients are not available, their inclusion should be properly considered by overdesigning the mooring system to the degree needed. However, where the average station-keeping response is the controlling design element, it would seem adequate to compute average frequency-independent added mass coefficients from available formulas and include them in a time- domain vessel dynamic analysis.

Conclusion

This chapter presents a general discussion on selecting mooring systems and mooring line components, along with analysis techniques to include the effect of

Mooring Dynamics of Offshore Vessels 2 4 5

20 t BEAM SEAS QUARTERING SEAS

W 15 -- W iv

L~ W

v

10- - Z n I - - I

O

) s - <[ >--

MAXIMUM EXCURgInNS - CMYAV =I.0 ~ ,

,.. /

. / ,

0 ' 1.0 1.5 2.0

CMX, CMY

Figure 27. Yaw motion as a funct ion of added mass in beam and quartering seas.

quasi-static as well as dynamic cable behavior in the dynamic analysis of a moored offshore vessel. If the station-keeping response of the vessel is deemed to be outside the exciting frequency range of the mooring system, a "quasi-static" cable analysis using the static catenary equations can be resorted to. Howev- er, if the effect of line inertia is not to be ignored, a "dynamic" cable mathematical model can be employed, which is a multi-degree-of-freedom system generated by breaking up the line into a series of finite, massless, and inextensible segments with masses lumped at appropriate nodes.

Application of the modified Lagrange's equations permitting holonomic constraints results in a matrix equation for the n-segment cable representing a set of nonlinear, coupled, differential equations that can be solved by Runge-Kutta's method to yield cable displacement and tension time histories for the various configurations occurring. Initial conditions needed for the initiation of the dynamic solution can be provided through the static catenary equations [ 13, 14, 18].

Although the dynamic solution is restricted to inextensible cables, effects such as elastic stretch and thermal elongation can be easily incorporated in the generation of the initial conditions. Once the dynamic solution is computed, the


2500

20OO

(A Z p.

bJ Z v 1500

Z r ' l

(A Z bJ P" 1000

bJ _J <r (.J

500 - -

- - - - - - - - BEAM SEAS QUARTERING SEAS

MAXIMUM TENSIONS C.MYAV = 1.0

L5 " - " " " - - - 2.0

AVG-BEAH SEAS . . . . . . . . . .

CHYAV - 1.0-2.0 /

0 |

1.0 1.5 2.0

CHX, CHY

Figure 28. Cable tension as a function of added mass in beam and quartering seas.

stiffness characteristics of the mooring line can be generated from the tension and displacement time histories. With the analytical procedure discussed, cable segments having different sizes and material properties can be included, any possible anchor motion can be allowed, and both taut and slack cables as well as single component and multi-component combinations with or without clump weights can be handled with ease. The dynamic cable analysis presented should serve as a useful tool in developing realistic mooring line restoring forces for use in the mooring analysis of offshore vessels anchored with multi-point, multicomponent cable systems.

The section entitled "Computation of Environmental Forces" discusses estimation of realistic environmental loadings on an offshore vessel in an open ocean environment. It presents the equations for computing wind and current forces and illustrates a procedure for generating a wave force time history from a given random sea spectrum. Once the time history of the total excitation force is


Table 2 Maximum Percentage Variations in System

Response for Range of Added Mass Coefficients Used

Maximum Percentage Variation

Surge Sway Yaw Line Sea Direction Excursion Excursion Excursion Excursion

Beam sea 50 91 289 56 (4.5-17 ~ )

Quartering sea 32 14 133 15 (6-14 ~ )

computed, the vessel differential equations of motion can be assembled and solved using standard dynamical analysis methods.

Most advanced dynamic analysis procedures attempt to derive the equations of motion of the moored offshore vessel using six degrees of freedom, namely, surge, sway, heave, roll, pitch, and yaw. However, because the solution of these six coupled, nonlinear, differential equations is a complex, time-consuming procedure, the dynamic "station-keeping" vessel response is often described by a three-degree-of-freedom mathematical model representing the inplane motions only. For a moored vessel with lateral symmetry, this model is held adequate and valid [ 18, 48, 49]. To start with, an equilibrium configuration of the moored system is assumed to provide the initial conditions for the dynamic solution. At any time step, then, the horizontal projection of each mooring line is computed with associated cable tensions determined from the tension-displacement profiles. Other time varying environmental forces from waves, winds, etc. can also be computed for a specified sea state and the vessel equations of motion solved at each time step. Thus, the moored system can be studied for any representative position at a given time, and vessel motion as well as line tension responses can be easily determined and evaluated. For the practical example analyzed, results have shown that to predict mooring system performance and to evaluate moored vessel response in a realistic way, the effect of line dynamics must be included in the dynamic analysis of the vessel. The dynamic cable analysis presented suggests a viable means of handling the problem.

While the ideal way of including the effect of added mass is to put in added mass coefficients as functions of the frequency in a frequency-domain vessel dynamic analysis, there are ways by which a time-domain dynamic analysis can be done using frequency-independent time-averaged values [51, 52], and this chapter presented pertinent information on this. To investigate the effect of vessel-added mass, the dynamic response of a typical construction vessel moored in open waters by a multileg anchoring system has been studied. The results gener-


ated have shown that vessel-added mass has little or no influence on the average station-keeping response of the vessel. Peak excursions and maximum line tensions, however, are affected by changes in the added mass coefficients. It is therefore recommended that, in situations where the mean station-keeping response is of significance, it would be sufficient to include, in a time-domain dynamic analysis, time-averaged added mass coefficients as computed from formulas in the literature [53-56]. However, where peak response is deemed crucial, especially in situations where the added mass coefficients available may not be very accurate, their influence should be properly accounted for by overdesigning the mooring system to the degree needed.

It is difficult to pinpoint a specific mooring system for an application unless the physical features of the vessel, the environmental conditions, and the operational constraints are known explicitly. However, by proper selection of the line parameters with the help of the techniques of analysis given in this chapter, the characteristics of a mooting system can be tailored to any specified requirements.

Acknowledgments

I am grateful to Dr. John B. Herbich, editor of this text, for inviting me to make this contribution. Thanks are also due to Sandy Hank and Sara Waldroup for their accurate typing of this manuscript.

Notation

A

Ac A L

ALS As AT

ATS a (t)

a 0, a 1, a2 ..... a n Aa~

aek

Bxx, Byy, Baa

b 1, b 2 . . . . . b 1, b n C

C D

C T, C N

effective stress area of cable segment characteristic area of vessel (beam x length) exposed lateral area of vessel submerged lateral area of vessel projected surface area of the anchor exposed transverse area of vessel submerged transverse area of vessel water acceleration Fourier coefficients for representation of wind loads an n x n pseudo-mass matrix a coefficient in the modified Lagrange ' s equation [ -

t)fe//)pk] first-order damping coefficients of vessel resistance to velocity Fourier coefficients for representation of current loads cable parameter ( = H/w) drag coefficient for cylindrical cable links tangential and normal drag coefficients for cylindrical links tangential and normal drag coefficients for line segment

Mooring Dynamics of Offshore Vessels :249

Cm Cx

Cxw, C w, Cmw Cxc' ~yc' Cmc

Cmx, Cmy, Cmot

D d E

F n

FH, F v

F x, Fy, M z

{F1}, {F2}, {F 3} F 1, F2

fl f g

H h

Hs Hu(O~), Ha(O)

I L

Lc

M [M] M A Mc Mw

m m p m i

anchor drag coefficient inertia coefficient for cylindrical cable links inertia coefficient in computation of vessel surge added mass wind-load coefficients depending on wind direction current-load coefficients depending on current direction frequency-independent time-average vessel added mass coefficients diameter of cylindrical link water depth effective elastic modulus of cable segment amplitude of total wave force for jth wave component horizontal component of nonconservative force acting at a cable node horizontal and vertical components of nonconservative forces acting at anchor horizontal and vertical components of nonconservative forces acting at ith mass of cable dynamic model tangential and normal components of drag forces acting on the pth link of cable dynamic model external time-varying environmental forces and moment on vessel pseudo force column matrices in cable dynamic model anchor holding factors "c" number of constraint equations in Lagrange's formulation force per unit length of cylinder in Morison's formula acceleration due to gravity horizontal component of line tension height of floating vessel center of mass significant wave height complex wave velocity and acceleration receptances vessel mass moment of inertia about yaw axis length of vessel characteristic length of vessel which, in this case, is the draft length of ith cable segment vessel mass pseudo-mass matrix for cable dynamics anchor mass steady yawing moment due to current load steady yawing moment due to wind load mass per unit length of cable segment virtual mass lumped at node in cable dynamic model cable segment mass lumped at ith node


mxx, myy, m(x a N n

Pk

5P

Qi

Qx,Qy,Qa

Q• Qyi

{q}

Rn

RF('I;) Ru(X), Ra(X)

S

SL Sc

S~(co) s

Su(0)), Sa(0))

SF(00) SFT(CO)

T Tavg

Till, T v

T s u

u(t) V

V c

Vw

w

W 1, W 2

vessel added masses and inertia number of mooring lines used for constraining the vessel total number of line segments in cable dynamic model kth co-ordinate (k = 1, 2 . . . . . N) used in modif ied Lagrange' s equation virtual power nonconservative generalized force associated with the generalized co-ordinate Pi nonlinear restoring forces and moment induced by mooring system x and y components of generalized forces acting at ith mass of cable dynamic model representation of system unknown variables in cable dynam-

ic model V c cos (x

vL Reynolds number

autocorrelation function of spectral density of wave force autocorrelation functions of velocity and acceleration spectral densities wetted surface of vessel cable segment length suspended catenary length spectral density of wave elevation length along cable spectral densities of wave velocity and acceleration spectral density of sectional wave force spectral density of total wave force cable segment tension; system kinetic energy average segment tension horizontal and vertical components of tension in ith cable segment significant wave period velocity of flow relative to cable segment wave velocity vertical component of segment tension; system potential energy current velocity wind velocity average tangential and normal velocities of pth segment of cable submerged weight per unit length of cable segment anchor weights


X c

Xcon XL

x0, Y0

Xi' Yi x,y,a

X c, Xw Yc, Yw

Z Z

Pa'Pw 0 i v

Tl(t) (~j)

horizontal projection of suspended cable segment line length on bottom total horizontal line projection co-ordinates of anchor with reference to a fixed frame of coordinates x, y co-ordinates of ith lumped mass in cable dynamic model surge, sway, and yaw displacements of vessel steady longitudinal current and wind forces steady transverse current and wind forces catenary height coordinate in vertical direction which is zero at vessel center of mass angle at bottom of catenary, bottom angle at anchor instantaneous angle that jth mooting line makes with x-axis instantaneous angle that jth mooring line makes with vessel longitudinal axis in horizontal plane Lagrange multipliers air and sea water densities angular displacement of ith cable segment kinematic viscosity of water wave elevation random phase lag for j th wave component distr ibuted between 0 and 2r: circular frequency of jth wave component width of jth frequency segment in wave force spectrum

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49. Zarnick, E. E., and Casarella, M. J., 1972. "The Dynamics of a Ship Moored by a Multi-Legged Cable System in Waves." Reprints of the Eighth Annual Conference and Exposition of the Marine Technology Society, Washington, D.C., pp. 67-96.

50. Khan, N. U., 1983. "The Effect of Cable Dynamics on the Motions of Moored Off- shore Structures." M. S. Thesis, University of Petroleum & Minerals, Dhahran, Saudi Arabia, Nov.

51. Cummins, W. E., 1662. "The Impulse Response Function and Ship Motions." David Taylor Model Basin Report No. 1661, Oct.

52. Van Oortmerssen, G., 1976. "The Motions of a Moored Ship in Waves." Publica- tion No. 510, Netherlands Ship Model Basin, Wageningen, The Netherlands.

53. Wilson, B. W., 1960. "Mooring of Ships Exposed to Waves." Texas A and M Uni- versity, Technical Report No. 204-2, Nov.

54. Lewis, F. M., 1967. "Hull Vibration of Ships." Principles of Naval Architecture, New York: SNAME, pp. 718-751.

55. Costa, F. V., 1965. "Analytic Study of the Problem of Berthing--Analytical Treat- ment of Problems of Berthing and Mooting." Proc. of the Nato Advanced Study Institute, ASCE Publication, pp. 157-173.

56. Salveson, N., Tuck, E. O., and Faltinson, O., 1970. "Ship Motions and Sea Loads." SNAME Transactions, vol. 78, pp. 250-287.

57. Frank, Werner, 1967. "Oscillation of Cylinders in or Below the Free Surface of Deep Fluids." Report No. 2375, Naval Ship Research and Development Center, Washington, D.C., Oct.

58. Vugts, J. H., 1968. "The Hydrodynamic Coefficients for Swaying, Heaving and Rolling Cylinders in a Free Surface," International Shipbuilding Progress, vol. 15, pp. 251-276.

59. Kim, C. H., 1969. "Hydrodynamic Forces and Moments for Heaving, Swaying and Rolling Cylinders on Water of Finite Depth," J. of Ship Research, vol. 13, pp. 137-154.

60. Takaki, M., 1977. "On the Hydrodynamic Forces and Moments Acting on the Two-Dimensional Bodies Oscillating in Shallow Water." Rep. Res. Inst. Appl. Mech., Kyushu Univ., vol. 25, pp. 1--64.

61. Kim, W. D., 1966. "On a Freely Floating Ship in Waves." J. of Ship Research, vol. 10, pp. 182-200.

62. Garrison, C. J., 1974. "Hydrodynamics of Large Objects in the Sea, Part 1 - Hydrodynamic Analysis." J. ofHydronautics, vol. 8, pp. 5-12.

63. Garrison, C. J., 1975. "Hydrodynamics of Large Objects in the Sea, Part 2 m Motions of Free-Floating Bodies." J. ofHydronautics, vol. 9, pp. 58-63.

64. Garrison, C. J., 1974, "Dynamic Response of Floating Bodies." Proc. Offshore Tech. Conf., Houston, Paper No. OTC 2067, vol. II, pp. 365-377.

65. Loken, A. E. and Olsen, O. A., 1976. "Diffraction Theory and Statistical Methods to Predict Wave Induced Motions and Loads for Large Structures," Proc. Offshore Tech. Conf., Houston, Paper No. OTC 2502, vol. 1, pp. 792-820.


66. Faltinsen, O. M. and Michelsen, F. C., 1974. "Motions of Large Structures in Waves at Zero Froude Number." Proc. Int. Symp. On the Dynamics of Marine Vehicles and Structures in Waves, Univ. College, London, pp. 91-106.

67. Garrison, C. J., 1978. "Hydrodynamic Loading of Large Offshore Structures. Three-dimensional Source Distribution Methods," in Numerical Methods in Off- shore Engineering, Zienkiewicz, O. C., Lewis, R. W. and Stagg, K. G. (eds.), Wiley, Chichester, England, pp. 97-140.

68. Ansari, K. A., 1989. "Effect of Vessel Hydrodynamic Mass on the Station-Keeping Response of a Moored Offshore Vessel." Int. J. of Energy Research, vol. 13, pp. 573-579.

C H A P T E R 6

CABLE DYNAMICS FOR OFFSHORE APPLICATIONS

M. S. Triantafyllou

Department of Ocean Engineering Massachusetts Institute of Technology

Cambridge, Massachusetts, USA

CONTENTS

INTRODUCTION, 257

CABLE MECHANICS, 258

Modeling Cable Dynamics, 258

Linear Dynamics of Cables, 259

Nonlinear Dynamics of Cables, 264

Low-Tension and Snapping Cables, 266

Synthetic Mooring Lines, 268

MOTIONS OF MOORED STRUCTURES, 270

Amplitude and Time-Scale Expansion, 270

Slowly Varying Large Amplitude Motions, 272

Fast Varying, Wave-Induced Motions, 273

Vortex-Induced Motions, 273

Design Application, 274

EXPERIMENTAL TESTING OF CABLES AND MOORING SYSTEMS, 274

GUIDELINES FOR ANALYZING MULTI-LEG SYSTEMS, 276

OVERALL SYSTEM DAMPING INDUCED BY THE MOORING LINES, 276

APPLICATION TO MULTI-LEG SYSTEM ANALYSIS, 277

Storm Conditions, 278

Estimate Damping Coefficient, 279

Comparison With Rules, 282

SUMMARY, 282

Acknowledgments, 283

256

Cable Dynamics for Offshore Applications 257

APPENDIX 1: DERIVATION OF THE EQUATIONS OF CABLE MOTION, 283

APPENDIX 2: METHODOLOGY FOR CALCULATING VORTEX-INDUCED MOTIONS, 288

APPENDIX 3: METHODOLOGY FOR CALCULATING THE INTERACTION BETWEEN WAVE-INDUCED AND SLOWLY VARYING MOTIONS, 291

REFERENCES, 292

Introduction

Multi-leg mooring systems are used to anchor floating offshore structures. The mooring lines provide a holding force to prevent the structure from drifting away from its average position. At the same time the fast, wave-induced motions of the structure cause substantial dynamic forces on the mooring lines, which exhibit complex dynamic behavior, while vortex-induced vibrations amplify the hydrodynamic loads. This chapter provides an overview of the basic mechanisms of cable response and cable-fluid interaction as they pertain to marine mooring cables.

Mooring systems consisting of several lines are routinely used to position floating drilling or production systems, and their use is constantly extended to greater depths. Mooring systems require a careful dynamic analysis to ensure their reliability. This has been recognized by regulatory bodies (such as the American Petroleum Institute, Det Norske Veritas, and others), which currently require a dynamic analysis as an essential step before installation of a floating structure.

Some of the crucial parameters of a mooring system, such as the overall damping coefficient, are difficult to estimate with the state of the art. This chapter provides ways to perform the dynamic analysis of a mooting system, including novel methods to estimate reliably all the critical parameters of the system.

A typical mooting system has several mooting lines, each consisting of wire, chain, synthetic rope, or a combination of these components. The excitation consists of three separate types: slowly varying, large amplitude motions; fast varying, wave-induced motions; and very high frequency, vortex-induced vibrations. The interactions among the three types of excitation render a complete analysis very difficult, hence simplifications are introduced to make a solution computationally tractable.

The following sections outline first the dynamic properties of cables with special emphasis on the response to each of the three separate excitations. They address separately the behavior of synthetic mooring lines, whose dynamic behavior remains largely unknown. Next, the sections provide a methodology for decomposing the overall problem into three interacting components, making the solution tractable. Appendixes outline: (1) a recently developed methodolo-


gy to account for the vortex-induced vibrations of cables, which are known to cause significant drag amplification, hence affecting the response to the other two types of motion as well; and (2) The interaction effects caused by combination of wave-induced motions and slowly varying motions, and especially their effect on the drag coefficient.

Finally, we outline a methodology for evaluating the slowly varying drift motions, and, in particular, in obtaining accurate estimates for the damping coefficient of the moored structure, which is a critical quantity in designing the multi-leg mooring system. This leads to a theory for assessing the dynamic performance of multi-leg moored systems in a realistic sea environment and a design procedure, fully accounting for the detailed mechanics of the individual lines of the system.

Cable Mechanics

A cable is not a rigid structure, but constitutes a mechanism, whose shape changes substantially as a result of the applied load. The fact that the shape, as well as the tension of the cable, depend in a nonlinear manner on the external load is one of the basic difficulties in analyzing cable systems.

Modeling Cable Dynamics

The dynamics of cables of interest to offshore applications have wavelengths much larger than the diameter of the line, hence the cable can be modeled as a slender structure. The bending stiffness of wires and synthetic ropes is, under normal operating conditions, very small compared to the tension stiffness; and in chains, of course, it is totally absent. Bending stiffness can be very important for cables under very low tension and for all wires and ropes locally, especially near the ends where boundary layers form [ 13, 14, 17].

Hydrostatic and fluid dynamic forces are important and must be modeled. The effect of the hydrostatic force has been incorporated through the concept of the effective tension. This has been covered already in great detail in the literature [6, 25]. The unsteady fluid force is typically decomposed into an excitation force, an acceleration-dependent part (added mass force) and a velocity-dependent part (drag force). In the presence of a current, the added mass formulation becomes quite complex, particularly if the cable has a substantial axial steady velocity component [2, 23, 25]. The drag force is often modeled through the so- called separation principle, i.e., the decomposition of the force along the tangential direction as well as a second direction contained in the normal plane and parallel to the component of the relative velocity on the normal plane.

When a cable moves with large amplitude motion, or in the presence of an ocean current, vortices form causing substantial unsteady forces, which lead to vortex-induced vibrations (VIV). As a result of VIV, the drag coefficient may be


amplified by a factor often in excess of 3. The mechanism of VIV formation is a self-limiting process; hence, vibrations never exceed amplitudes of one to two diameters; still, their effect on cable loading is profound.

The derivation of the equations of motion is given in Appendix 1. The details of the derivation are very important, because they point to the source of certain problems that have made past cable simulation difficult. For example, when the tension is small, cables and chains lose their principal stiffness mechanism, and the governing equations become "ill-posed" [40, 41 ]. Hence, an explosive instability dominates the response and no convergent numerical solution can be found. By including a small bending stiffness term we correct the problem of "ill-posedness," hence bending terms must be included even when studying chain dynamics just to fix this problem. The real physical mechanism at work in chains, which prevents the onset of an explosive instability, is the finite length of the individual links making up the chain, which for very small wavelengths possess appreciable moment of inertia. Because only macroscopic properties are of interest, it is irrelevant which corrective means to employ. In the case of the chain, for example, it would be a waste of computational effort to model the dynamics of each individual link separately.

An example is shown in Figure 1" a chain hanging under its own weight in air, with the lower end free, is driven harmonically from the top in transverse resonant motions [12]. Due to the low tension at the free end, the chain undergoes very large amplitude motions resulting in the collapse of part of the chain against itself. Simulation is impossible unless bending stiffness is included, due to the explosive instability of the governing equations. As shown in Figure 1, however, a solution is obtained in the presence of a small bending term, and this solution does not depend on the value of the bending stiffness. The two results shown differ by a factor of 10 in the value of the bending stiffness, but the resulting response is nearly the same.

Linear Dynamics of Cables

The small amplitude motions of lightly damped cables are characterized by natural frequencies and modes, which are influenced substantially by the curvature. Irvine & Caughey [16] explored in depth the response of shallow-sag cables stretching quasi-statically, and demonstrated the presence of mode cross- over phenomena. Such mode cross-over occurs when two in-plane frequencies corresponding to different modes of a horizontal catenary have the same value for certain parametric values. The basic parameter controlling the formation of cross-over is E:

EAZ = T ( )2 (1)


0.220

0.153

x(t)

0.087

Er. I .OE(.6)

El'-1.0E(-7)

t-2.75 s t,,3.18 $ t,-3.40 s

0.020 . . . . . -0.200 -0.100 0.(100 0.100 0.200

Ylt)

Figure 1. Resonant response of a chain to top excitation at three successive time intervals for two values of bending stiffness El (1 x 10 -6 and 1 x 10-7).

where E = Young 's modulus A = cross-sectional area T = static tension at mid-point w = weight per unit length L = length of the cable

If f~ denotes the non-dimensional frequency,

CO

CO 1

where

(1) 1 = - ~

m - mass per unit length

(2)

(3)

then there is frequency coalescence (two frequencies corresponding to different modes having the same value) when L/rt = 2n, where n = 1, 2, 3 ....


In the case of inclined catenaries two frequencies come very close together but do not coincide (avoided crossings) and the original mode shapes mix to produce hybrid modes [34]. The controlling parameter is a similar parameter, )~*:

(~*)2 EA wL 2 ~2 = ~ ( - ~ - ; - ) c o s

T* (4)

where ~) = inclination angle of the cable chord T* = static tension at the point where the local static angle is equal to the

inclination angle

Similar phenomena of frequency coalescence and avoided crossings exist for catenaries traveling between two fixed points, as well as other mechanical systems [3, 35, 39]. Figure 2 shows the avoided crossings region of an inclined catenary, the progressive formation of hybrid modes and the large amplification in the dynamic tension observed.

Triantafyllou & Triantafyllou [39] showed that frequency coalescence (mode crossover) and avoided crossing phenomena can be described, and their properties explained, within the same mathematical context by allowing the parameters of the system to become complex through analytic continuation. The resulting complex frequency coalescence for certain parametric values is associated with a branch point in the parameter space, which causes large sensitivity to parametric changes and formation of hybrid modes. Hybrid (mixed-shape) modes form because the characteristic modal matrix of the analytically continued system is not diagonalizable and, instead, contains Jordan form blocks, which admit generalized eigenvectors. A consequence of the existence of a branch point in the parameter space is that when the parameters of the real system are allowed to vary, a crossing of the branch cut in the parametric plane causes mode-switch- ing, as observed in all systems exhibiting avoided crossings studied so far.

Triantafyllou & Yue [45] studied the forced response of a synthetic cable, which exhibits hysteretic damping. They find that the apparent damping coefficient of the transverse vibrations of a highly extensible hysteretic cable increases by an order of magnitude for those parametric values that result in having avoided crossings phenomena between two (damped) natural frequencies corresponding to a transverse and an elastic mode, respectively. This is important for applications, since the elastic frequencies of synthetic ropes are close to the wave frequency range. Figure 3 shows the maximum transverse response q of a synthetic cable forming a horizontal catenary, due to harmonic, resonant transverse excitation at one end. The principal parameters are: the non-dimensional curvature at the mid-point, or; A = EAfro; and to = D./~t; where EA is the elastic stiffness, T O is the static tension at the mid-point, f~ is the excitation frequency, and ~"~t is the first transverse natural frequency. The parameters were chosen such that two natural modes, one corresponding to a transverse and one to an elastic

(text continued on page 264)

262 O

ffshore Engineering

10 - \ I

, + l -

10-I - i Figure 2. Formation of hybrid modes: (a) Mode transition for various

lo-? /' values of Ah ((1) 0.1, (2) 1.7, (3) 1.9, (4) 2.1, (5) 2.3, (6) 2.5, 2.7, (8)

- - f i r s t m. 8.0); (b) First two natural frequencies as function of k and for various ,/'

- - - Second met values of the inination angle I$, and E = w w ; (c) dynamic tension for the first two modes of unit amplitude as function of )L.

1 I I L I I I f 10 100 low

2. 2

1 .m

1.2s

I .a7

' 0.M

0.M

0.43

0.21

0.w

i

A = EA/To

- llne 1 line 2 lhna 3 lina 4

---- -.-.-. . . . . . . . . . . .

I

0.00 0.02 0.04 0.06 0.08

Figure 3. (a) Peak, resonant transverse response amplitude of a horizontal catenary as function of A = EAn, when the first elastic mode interacts with the second transverse mode; (b) Amplitude of transverse response (s) and axial response (p) as function of the damping coefficient p. (Mode 1 : line 1 q, line 2 5p; Mode 2 line 3 q, line 4 5p.)


(text continued from page 261)

natural mode, have frequencies very close to each other (A --- 4, co = 2, cz = 0.2): The transverse response falls by an order of magnitude due to the formation of hybrid modes [45].

Figure 3 also shows the unusual behavior of the forced cable response near the avoided crossings region as function of the damping coefficient 13: The transverse response first falls and then increases as damping increases.

Nonlinear Dynamics of Cables

It is important to view cables as mechanisms, rather than simple structural elements, because they substantially change their shape when subjected to external loads. As a result, one faces first a nonlinearity called geometric, because the internal forces depend on the shape, which is not known before obtaining the solution. Also, cables are far stiffer along their axis rather than in the transverse direction, because a motion in the axial direction causes the cable to stretch considerably, employing its large elastic stiffness; whereas a motion in the transverse direction causes second order stretching only [16, 17]. This results in a stiff system of equations, which cause numerical problems; also, it is the basic source of the avoided crossings and frequency coalescence phenomena mentioned above.

Under dynamic conditions, drag along the cable becomes a dominant transverse load preventing the cable from moving laterally, hence forcing it to stretch significantly [36]. The higher the frequency, the more pronounced this effect is, because drag varies as the square of frequency, until ultimately the cable responds almost entirely through stretching. Hence, it is possible that small, higher frequency motions (such as wave-induced motions), can cause large tensions, comparable to those caused by large-amplitude, low-frequency motions (such as the slow drift motions). In this respect dynamic effects must be an intrinsic part of cable analysis and design when wave effects are of importance.

Figure 4 shows the dynamic stiffness (tension per unit imposed amplitude) for a wire in water forced to move harmonically at its upper end in the axial direction as function of the frequency of excitation. The response is typical of cables forming a catenary shape in water: The stiffness is nearly equal to the catenary stiffness at very low frequencies, and increases rapidly to reach a value equal to the elastic stiffness (EA/L), at a frequency o~ = 0.7 r/s. Beyond this frequency, the response is influenced by the first elastic ~equency, and the dynamic tension at the top end of the cable first becomes zero, and then increases to very large values at the elastic resonance, which is at about co = 3.9 r/s. Up to the frequency of maximum stiffness, COp, the dynamic tension is nearly constant along the cable; for higher frequencies there is considerable variation along the cable length, eventually leading to the formation of elastic modes. The results shown in Figure 4 are for a typical offshore line in 3,000 ft of water depth, consisting of


350000. ' r /A (N/m)

300000.

250000.

200000.

1 50000.

100000.

50000.

0.00 0.60 1.20 1.80 2.40 3.00 3.60 4.20 ,,., (,',=,~/,)

Figure 4. Dynamic stiffness (dynamic tension T divided by the amplitude of motion A) for a catenary in water forced to move sinusoidally in the axial direction at its upper end as a function of the frequency (0.

4,500 ft of 5-in. chain and 7,000 ft of 3.75-in. wire. The static tension at the top is 2.93 x 106 N, the chain lying on the floor has length 3,000 ft with an equivalent spring constant of 9.93 x 106 N/m and damping constant 1.44 • 107 N s/m. The lack of resonance peaks for the transverse natural frequencies and the large levels of dynamic tension demonstrate the significant effects of the drag force, which render results for in-air response of cables meaningless for applications in water [21, 31, 36].

Cable-bottom interaction is another possible source of dynamic amplification. In the absence of violent dynamic interaction, one must at the very least include the portion of the cable lying on the floor in the cable modeling; otherwise, substantial overestimation of the dynamic tension may result. This is due to the fact that part of the cable lying on the floor is capable of stretching, depending on the frictional support provided by the soil. As shown experimentally by Papazoglou et al. [26], once the portion of the cable that participates in stretching is known, it is a simple matter of introducing an equivalent spring and damper to account for the effect of the cable on the bottom. Determining the portion lying on the


floor, however, requires accurate knowledge of the soil parameters. In the case of violent dynamic interaction, as, for example, in the case of a sinker (a clump weight attached to the cable near the touch down point), full dynamic modeling is essential. In three-dimensional cable response one must consider lateral (out- of-plane) friction between cable and bottom, as well as in-plane motion. These items require and deserve further investigation.

A wave frequency analysis can be performed either in the frequency or in the time domain. Presently, a well established theory exists for including the nonlinear drag force in a frequency domain program through equivalent linearization [21, 31, 38, API rules 1991 ]. Other nonlinearities can also be included, while the Gaussian assumption can be eliminated [24]. Including nonlinearities such as cable-bottom interaction, snapping response, and material nonlinearity can improve the accuracy of response; the drag nonlinearity, however, must always be included, otherwise the results are certainly in error.

Vortex-induced vibrations and the interaction of slowly varying and wave- induced motions amplify the drag coefficient. Complex calculations are required to calculate the drag coefficient amplification, while the theory is still under development. Appendixes 2 and 3 provide an outline of an existing methodology, which has been implemented in a numerical simulation package of programs [42].

Low-Tension and Snapping Cables

Due to the dynamic tension build-up, there is a distinct possibility that the total tension will become negative for part of the time, resulting in loss of tension and then sudden recovery, accompanied by impulsive motions (snapping response). Depending on the conditions of the problem, snapping of underwater cables can cause softening or hardening of the tension response, with substantial dynamic effects [24, 26]. The principal parameters determining the nature of snapping are the free-falling velocity of cable in water and the value of the elastic stiffness. The free-falling velocity controls the amount of sag that will build up when the cable is free to fall. The larger the sag the smoother the resulting tension is, because the cable can absorb large amounts of energy by expending them through the fluid drag.

Milgram et al. [24] and Papazoglou et al. [26] describe a "clipping" cable model, where the total tension is set equal to zero when the compatibility relations provide negative tension, reverting to a freely falling model, until tension is restored. Simulation is in very good agreement with experiment as shown in Figure 5. The experiments were conducted on a horizontal cable with length L = 67.06 m, diameter d = 9.53 mm, Young's modulus E = 4.80 x 101~ N/m 2, mass per unit length m = 0.388 kg/m. A spring was inserted in line with the wire to ensure proper similitude with a full scale wire, for reasons explained in the section on testing cables experimentally. A new generation of codes allows simulation of snapping cables without clipping models [ 12, 32].


.ol t.oo 5 0

2 0 0

= 3 0 "-"

i ,o '~176 (J

o'~o I=

- ~ o

- 1 0 0

- 3 0

- 5 0 - 2 0 0 0 5 10 15 20 25 3 0

Time (s)

Figure 5. Dynamic tension (Ib) as function of time for a towing cable. Solid line: theory; broken line: experiment. Amplitude of excitation equal to 11.3 diameters, frequency 0.63 rad/s.

Occasionally, a line may go totally slack; then we classify its behavior as that of a low-tension cable, resulting possibly in a snapping response identified above. There are cases, however, when a cable has very low static tension by design. For example, freely moving tethered underwater vehicles cannot sustain large forces in their tethers, because their mobility would be seriously restricted. In some applications, in fact, the tether is simply a fiber optic line that must not carry any appreciable tension. Such tethers are classified generically as low tension cables and require special attention and treatment: Low-tension cables respond in an intrinsically nonlinear manner, because tension constitutes the principal restoring mechanism; when tension is small, the nature of cable response varies rapidly with the motions. Even mildly low tension causes changes in the behavior of a cable. Cables under zero tension are impossible to simulate in the absence of bending stiffness. In the limit of zero tension, the dynamics exhibit peculiar properties in the steady-state [39], or impulsive response in the transient state [ 12, 40, 41 ]. For practical applications, such behavior is to be expected from moorings in severe storms, particularly in the least loaded lines.

Figure 6 shows the tension at three different points of a cable used to tether a remotely operated vehicle as it maneuvers in the water. Such tethers are long and under very low static tension, so as not to cause unbalanced forces on the small vehicles. It is seen that dynamic tension builds up and goes to zero quickly. The impulsive nature of the dynamic tension is an intrinsic feature of low- tension cables [12]. The tether shown in Figure 6 has length L = 100 m, diameter d = 1.14 cm, Young's modulus E = 200 GPa, and is freely floating. The vehicle has mass M = 360 kg.


300.

20O.

TENS.

0.

0. t75. 350. 525. T I M E

. L S - 0 m . . - - - - . S - 5 0 m . . . . . . S - t 0 0 m

Figure 6. Tension as function of time at three locations along a cable tethering a maneuvering underwater vehicle.

Synthetic Mooring Lines

For certain applications synthetic cables may have significant advantages over metallic cables, because they are considerably lighter, very flexible, and can absorb imposed dynamic motions through extension without causing excessive dynamic tension [4]. Their material and mechanical properties, however, are not as well known as for traditional ropes, leading to over-conservative designs that strip them of their advantages. In marine applications, for example, synthetic ropes subject to dynamic loads are often designed with factors of safety in excess of ten. Also, the large potential energy stored in extensible cables can cause significant damage to humans and property if the line ruptures under high tension.

Extensible synthetic cables are characterized by a significantly smaller value of Young's modulus, compared with metallic cables, and hence large extensibility under normal operating conditions. Whereas the maximum strain in a metallic cable under breaking tension is of the order of 2%, some synthetic cables reach or exceed a maximum strain of 25%. Also, the stress-strain relation of synthetic cables is often nonlinear and hysteretic, whereas wire ropes have a linear stress-strain relation, their damping is small and it is caused primarily by strand frictional interslippage.


In wire ropes the first elastic frequency is far removed from the first transverse natural frequency in air. Typically, the first elastic frequency corresponds to the 30th transverse frequency, and for most offshore applications elastic modes may be safely ignored for low-frequency and wave-induced motions. VIV motions are caused by transverse, self-limiting excitation and are very rarely associated with elastic waves, except when one of the ends is free. On the contrary, synthetic ropes have elastic frequencies corresponding to low-order transverse modes (first elastic mode corresponding to the fifth transverse mode, for example). As a result, elastic modes are often within the wave frequency range, while due to the high extensibility of the line, VIV motions, although primarily excited by transverse force, can couple through structural and fluid nonlinearities with elastic modes. This coupling is particularly important for certain parametric combinations that cause hybrid (mixed-shape) modes as outlined in the section on linear dynamics of cables.

Impulsive and transient motions of synthetic cables are governed by three characteristic speeds: The speed of elastic waves, c e, governed by the effective Young's modulus; the speed of transverse waves, c t, governed by the tension in the line; and the contraction speed of the cable, c c < Cee o, when tension decreases suddenly, where e o is the strain in the line prior to the change in tension. In the case of total loss in tension, the speed of contraction is equal to its maximum value and causes the catastrophic effects noted in rupturing lines. The equations become ill-posed, however, because the speed of transverse waves vanishes unless a bending stiffness is included. The bending stiffness must be chosen judi- ciously: it must be large enough to be numerically significant to ensure a well- posed problem; and small enough so as not to influence the solution numerically.

Because of their small weight in water, synthetic lines form shallow catenaries in the absence of a current force. As a result, when the moored structure is forced to move the lines are often susceptible to impulsive loading, as they go from slack to taut condition very quickly. A wire or chain, by contrast, builds a sufficiently large catenary so that when loaded it can absorb a large amount of energy simply by reducing the catenary (the drag force helps to absorb significant amounts of energy).

The stress-strain relation in synthetic lines is both nonlinear and hysteretic. When loading is imposed very slowly, the nonlinear form of the stress-strain form is clearly obtained and is often of the stiffening type. For fast-varying loading, hysteretic damping is observed, hence energy is dissipated per cycle of axial loading. Although the stress-strain relation varies with the history of loading, after a few cycles the variation is small and constant parameters can be used [ 1 ]. Because the stress strain relation is of the stiffening type, sudden loading may result in the formation of a shock wave traveling along the cable. Reflection of the shock can cause very significant tension amplification, leading possibly to rupture of the line [32]. Figure 7 shows the formation of shock waves in four horizontal catenaries in air. The static curvature is gradually increased from the


first to the fourth catenary by reducing the initial tension. One end is forced to move away from the other end, causing a shock wave that travels down the cable and upon reflection becomes larger. When the line is taut (Figure 7a), the reflected wave has maximum dynamic tension and rupture may occur. The least tensioned catenary (7d) shows significant effects from the curvature and the maximum dynamic tension is obtained at some intermediate point along the cable. In all four cases, however, a clear shock wave forms [32]. The synthetic cable used has length L = 600 m, diameter d = 0.05 m, and density p = 1,140 kg/m 3. The static tension T s and sag-to-span ratio, respectively, are (a) 20,000 N, 0.08; (b) 10,000 N, 0.16; (c) 5,000 N, 0.3; (d) 2,000 N, 0.66.

Motions of Moored Structures

The equations of motion of a moored structure involve forcing from several sources of excitation containing a wide range of frequencies. As a result, the solution of the overall problem is intractable and simplifications must be intro-. duced.

Amplitude and Time-Scale Expansion

To make cable analysis tractable, one must first recognize the three distinct time scales involved in the dynamics of mooring systems [33]:

1. The low-frequency motions caused by the current, the low-frequency wind forces, and the low-frequency (second order) wave forces. Typical periods are of the order of 100 s or larger; motions of the order of 5% of the water depth, larger (as a percentage of depth) for shallow water moorings.

2. The wave-frequency motions imposed by the motions of the moored structure. The direct action of the waves on the mooring lines is typically small compared to the energy transmitted by the structure. Typical periods are in the range of 6 to 20 s, motions of the order of 10 ft.

3. The vortex-induced motions, which are typically smaller than two cable diameters, with periods of the order of 1 sec, or smaller. These motions, although infinitesimal compared to the other two, cause drag coefficient amplification from a nominal value of 1.2 (for Reynolds numbers below 3 • 105) up to a value of 3.0 or even higher.

To these time scales one must add the elastic frequencies of the cable, which are usually sufficiently removed from the low- and wave-frequency ranges; they may contribute however to the vortex-induced motions, and especially to problems involving impulsive loading (such as deployment of heavy objects, or tight moorings near docks).

Cable D

ynamics for O

ffshore Applications

0. -0.10 0.30 0.70 1.1 0

Axial DistanceILength Axial Distance/Length

Axial DistanceILengtb Axial Distance/Length

Figure 7. Shock formation in impulsively loaded synthetic cable. Tension as function of the length at successive time steps. Static tension: ?

(a) 20,000 N; (b) 10,000 N; (c) 5,000 N; (d) 2,000 N.


Numerical analysis of cable structures, the development of fast computers, and ongoing developments in vortex-induced oscillations, have created the proper background for studying cable dynamics in their generality.

In the current state-of-the-art the following terms can be included for simulating cable dynamics [42]:

1. Geometric and material nonlinearity, particularly important at low and moderate frequencies.

2. Fluid drag, important at almost all frequencies. Vortex-induced vibrations affect the fluid drag in a complicated manner. Similarly, wave-frequency motions interact with slowly varyi'ng forces to alter the equivalent drag coefficient. Both interaction mechanisms must be included for properly simulating cable mechanics.

3. Bending stiffness for wires, particularly important for snapping conditions. 4. Elastic stretching, important at low and moderate frequencies. 5. Elastic waves for high-frequency excitation, and for transients involving

impact. Nonlinear stress-strain relation with hysteretic effects is essential in modeling synthetic lines.

6. Cable-bottom interaction, particularly important for low and moderate frequencies.

7. Attached submerged and floating buoys and masses, including the capability for interaction with the water surface and the bottom.

Significant simplification is achieved when specialized forms of the equations are developed, properly derived for the amplitude and frequency range under consideration. This is studied in the next sections.

Slowly Varying Large Amplitude Motions

Moored offshore structures have a low resonant frequency, because the effective stiffness of the mooring system is relatively small compared to the large mass of the structure. The resulting natural periods are often larger than 100 s. The effective damping of the structure is typically 10% to 40% of the critical, hence large amplitude resonant oscillations can be excited as a result of the slowly varying wave and wind forces. Hence, a dynamic calculation of low-frequency motions may give substantially larger tensions than a quasi-static one. In fact, the principal parameter in controlling these slowly varying tensions is the damping of the structure, a major part of which often comes from the drag of the mooring lines. Such a dynamic analysis must combine the inertia of the structure and the stiffness of the line, hence simplified models of the line may be employed.

Because the time scales of these motions are separated by at least an order of magnitude from each other, one may employ separate models for each time


scale. This scheme, however, must necessarily be iterative, because, as a rule, higher frequency dynamics depend parametrically on lower frequency dynamics, while lower frequency dynamics depend dynamically on higher frequency dynamics. For example, as the static tension varies slowly, the amplitudes of both the wave frequency and vortex-induced motions vary accordingly, because the principal restoring force is the static tension; the change, however, is very slow compared to the time scales of the motions considered, hence the effect of the low-frequency motion is only parametric. On the other hand, the high- and wave-frequency motions alter the damping of the low-frequency motion, at time scales comparable to the low-frequency motion, causing dynamic effects.

To predict the proper drag coefficient to be used in the low-frequency analysis, one must perform both a wave-frequency analysis and a vortex-induced motion analysis. Because the high-frequency analyses depend parametrically on the outcome of the low-frequency analysis, and the low-frequency analysis damping depends on the high-frequency motions, an iterative approach is necessarily employed.

Fast Varying, Wave-Induced Motions

The wave-induced motions of a floating structure are normally influenced very little by the mooring lines [37], hence each mooring line can be simulated assuming an imposed upper end motion, as calculated by the hydrodynamic analysis of the structure.

The principal nonlinearity to be included is the drag force, which makes the dynamics of submerged cables totally different from the dynamics of cables in air in the wave frequency regime, as explained in the section on nonlinear cable dynamics.

Vortex-Induced Motions

Vortex-induced vibrations (VIV) are of small amplitude (at most equal to two diameters) and high frequency (higher than typically 2 Hz). The viscous forces never reach the high levels of wave-induced motions, while they are affected by flow instability effects resulting in the formation of a staggered array of vortices [ 18].

As a result, the dynamic model most suitable for VIV is a high-frequency model, i.e., that of a string with variable tension in a flow with variable features. Most structural nonlinearities are quite weak at such small amplitudes of vibration. Also, with the exception of very low tension cables, bending effects are restricted to the ends of the cable only [ 17]. The resulting simplification in cable equations is of great help, because the fluid mechanics of VIV are very complex.


Design Application

The previously described capability is instrumental for design. The time scale separation is best suited for such purposes. One must recall that for ship dynamics there are different models for low-frequency motions (maneuvering model, to include viscous separation, and lift-induced forces) than for high-frequency motions (which must include the waves and free-surface effects); hence the similar separation of cable models is not only convenient, but necessary within the state of the art. Also, one must recall that the principal purpose of such a capability is to evaluate the low-frequency motions and total tensions given the wave and wind statistics and the form of the sheared current; the higher frequency models are required for properly accounting for low-frequency damping. At the same time these calculations may be used to obtain a first estimate of the overall tensions to be encountered, which affect the design of the mooring system.

Experimental Testing of Cables and Mooring Systems

When testing cables in water it is essential to violate the similarity in some of the scaling parameters, for practical reasons. The question then arises which non-dimensional parameters can be safely ignored, and what is the maximum allowable scaling ratio. These questions were addressed in detail in Triantafyl- lou [37] and Papazoglou et al. [26]. One of the principal conclusions is that the proper scaling of the elastic stiffness is essential for proper testing of the dynamics of cables, despite the fact that it is routinely violated because it is very difficult to satisfy.

As outlined in Papazoglou et al. [26], when a single line is tested dynamically in a tank in the presence of regular waves and current, there are nine parameters to consider: (1) the cable length to diameter ratio, (2) the cable length to water depth ratio, (3) the cable weight to top static tension ratio, (4) the Reynolds number in the nominal current based on the cable diameter, (5) the cable density to water density, (6) the ratio of the elastic stiffness to the catenary stiffness, (7) the ratio of the amplitude of wave-induced motions imposed at the top to the cable diameter, (8) the first non-dimensional transverse natural frequency of the cable, and (9) the ratio of the imposed wave-induced amplitude in the x- versus the amplitude in the z- direction (as well as the relative phase between the imposed motions).

Once the Reynolds number is above a threshold value, strict similitude is not necessary, provided the full scale tests are subcritical (typically below a value of 300,000), while the slenderness ratio (first parameter) is unimportant once it exceeds a value of typically 100. The second and third parameters must be kept the same for geometric similarity, while the sixth is equally important for dynamic similarity. The remaining parameters are rather easy to satisfy, with the possible exception of the eighth parameter, which strongly affects the development of vortex-induced oscillations. At this point, however, it seems dubious


that one would try to simultaneously test in model scale for slowly varying and wave-induced motions, as well as vortex-induced oscillations.

For deep-water applications one is often forced to violate similitude in the diameter and hence the static tension to satisfy the third parameter. For example, a cable in 3,000 ft depth tested in a tank 24 ft deep has a scaling ratio of more than 100. This would require a diameter of 5 in. to be scaled down to 1/20 in., which is not expedient for Reynolds number scaling, and one normally uses a larger diameter model cable. The significance of this violation is not overwhelm- ing and can be handled safely, provided the elastic stiffness is properly modeled.

In the section on the nonlinear response of cables we outlined the increasing importance of the elastic stiffness on the nonlinear cable response in water. Whereas the elastic stiffness has very limited importance for static, or quasi-static response, it has great importance for dynamic response, because the cable is resisted by fluid drag to move laterally, and hence responds by stretching. With- out properly scaling the elastic stiffness, therefore, the dynamic experimental results may be off by an order of magnitude [37]. The remaining question is whether, and how such similarity can be achieved.

For the model to have a properly scaled elastic stiffness, the Young's modulus of the material must scale down linearly with the scaling ratio. In most cases this is impossible: Either such material does not exist, or its density and other properties cause additional problems to scaling the remaining parameters. There is a solution to this problem, through the use of an inserted elastic spring in the line, provided caution is used to avoid certain resonance conditions.

Indeed, when a soft spring is inserted in series with the line, the equivalent elastic stiffness can be scaled appropriately [26]. The dynamics of most cables employed in moorings do not involve elastic waves and the stretching is slowly varying along the cable length; this is sufficient to justify the use of the spring, provided the model-scale cable does not resonate due to the newly formed natural frequency between the soft spring and the mass of the cable.

In the case of multiple-segment cables, and especially when submerged buoys are involved, the insertion of several springs, one per separate cable segment, becomes essential. Indeed, by inserting a spring at several places along the cable, elasticity is properly scaled, while the springs are stiff enough to prevent spurious axial resonances between the springs and the mass of the cable or the buoys.

Cable-bottom interaction is another possible source of error in modeling. In the absence of violent dynamic interaction, one must at the very least model the portion of the cable lying on the floor, again through inserting a properly selected spring, which models and scales down the equivalent elasticity of the cable lying on the bottom, otherwise substantial overestimation of the dynamic tension may result. This is due to the fact that part of the cable lying on the floor is capable of stretching, depending on the frictional support provided by the soil. As shown experimentally by Papazoglou et al. [26], once the portion of the cable that partic-


ipates in stretching is known, it is a simple matter of introducing an equivalent spring and damper to account for the effect of the cable on the bottom.

Guidelines for Analyzing Multi-Leg Systems

The American Petroleum Institute (API), Det Norske Veritas (DNV), and other organizations have developed recommended practices for the design of mooring systems. A common feature of the most recent rules is that they require considering two dynamic mechanisms:

1. The resonant large amplitude motions of the vessel at low frequencies, due to the natural frequency of the vessel mass (and added mass), and the effective mooting stiffness.

2. The dynamic tension amplification due to small amplitude high frequency wave-induced motions.

The principal oustanding issue is the evaluation of the damping coefficient of the structure, especially the components due to second-order wave forces (wave drift damping) and due to viscous forces on the mooring lines. For an analysis of the other components see Triantafyllou et al. [44].

Overall System Damping Induced by Mooring Lines

There are two drag coefficient amplification mechanisms to consider, as previously outlined above: amplification due to wave-frequency, low-frequency oscillations; and amplification due to VIV.

A computer code [42] has been implemented to model the amplification due to wave-frequency, low-frequency oscillations as follows: First, a drag coefficient is assumed and a frequency-domain code is used, as developed in Tri- antafyllou et al. [36], to predict the standard deviation of the cable motion due to upper end wave-induced motions. Also, a simulation is made of the slowly varying motion (accounting properly for the local shear current velocity) assuming resonant motion at the natural frequency of the structure. A number of time instances is considered within a period of oscillation, and at each time instance the experimental data base is used at each point of the cable to determine the amplification factor. Using the new drag coefficient, the calculations are repeated until convergence is achieved.

A second computer code has been implemented to model the amplification due to VIV as follows: First a simulation is made of the slowly varying motion (accounting properly for the local shear current velocity) assuming resonant motion at the natural frequency of the structure. A number of time instances is


considered within a period of oscillation, and at each time instance a calculation of the VIV response and drag amplification is made. The VIV calculation is made by modeling the cable as an infinite-length string as previously outlined. Because the velocity varies along the length, the excitation has variable frequency equal to the local Strouhal frequency (as determined by the normal fluid velocity).

As shown in Triantafyllou et al. [43] for a typical mooting line (which is described in the section on nonlinear cable dynamics), in 3,000 ft of water depth, the average drag coefficient due to VIV increases from the nominal value of 1.16 to a value of 1.35. The amplification factor due to interaction of the wave- induced motion and the slowly varying motions is equal to 2.11. This results in an equivalent drag coefficient for the slowly varying, resonant motions of c D = 2.85. The resulting damping coefficient, B, for a motion at resonant frequency ( t . 0 n - - 0.031 rad/s, i.e., period T n " - 203 s) and amplitude 4.6 m is B = 17,400 N s/m. If the nominal value of c D = 1.16 were used, the damping coefficient would have been underestimated by almost 60%.

Calculations were made in Triantafyllou et al. [43] for the Jack Bates semi- submersible, a modem fourth-generation semi-submersible, with the following dimensions (in ft) (L x B x D) = 370 x 255 x 140; operating draft 75 ft; and operating displacement = 103,000 kips [43]. The rig uses a conventional 8-point spread mooring system consisting of: 3~6 in. • 2,000 ft RQ3 chain, and 33A in. x 7,000 ft six-strand wire. Calculations showed that the mooting damping is of the order of 15% of critical damping, and hence as significant as the vessel hydrodynamic damping; the mooting damping amplifies due to WFmLF motion interactions, and mooring damping is mainly from the most loaded windward.

Application to Multi-Leg System Analysis

To demonstrate the principal points made herein with an application to a complex system, we will evaluate the forces and motions of a moored semi-submersible. The semi-submersible has length 86 m, width 70 m, and draft 17 m at operating displacement of 22,000 tons. An eight leg multi-leg mooring system is employed in water depth of 2,000 ft (609.6 m). The lines are assumed to be sym- metrically arranged and identical, consisting of a first chain segment (starting from the anchor), and a second wire segment. Table 1 provides the details of the parameters of each line.

The horizontal component of the pretension, the angle of the mooring line direction in the horizontal plane with respect to the x-axis, and the x and y location of the fairlead attachment points for the mooting lines are given in Table 2. The vertical distance is 4 m below the service waterline. The x-axis coincides with the north-west direction.


Segment Number I Table 1

Length segment = 800 m Mass/unit length = 106.7 kg/m Added mass/unit length = 7.6 kg/m Weight/unit length = 1,044 N/m Diameter = 0.137 m EA = 412,000 kN Breaking tension = 4,600 kN

Segment 2

Length segment = 1,450 m Mass/unit length = 27.1 kg/m Added mass/unit length = 4.04 kg/m Weight/unit length = 214.5 N/m Diameter = 0.08 m EA = 491,000 kN Breaking tension = 4,600 kN

Table 2

Horiz. Pretension Angle X Y (kN) (deg) (m) (m)

750 22.5 48.00 34.00 700 67.5 43.00 34.00 200 112.5 -43.00 34.00 100 157.5 -48.00 34.00 100 202.5 -48.00 -34.00 200 247.5 -43.00 -34.00 700 292.5 43.00 -34.00 750 337.5 48.00 -34.00

Storm Condit ions

The survival storm is specified as a co-linear combinat ion of waves described by a J O N S W A P spectrum with significant wave height 10 m, peak period 12 s and peakedness factor 3.3; wind with velocity 25 rn/s described by the Harris spectrum; and current with surface velocity 1 m/s and velocity 0.6 m/s at a depth of 150 m, varying linearly in between. The steady wave drift force is evaluated


at 270 kN, the average wind force at 2,000 kN, and the current force at 600 kN. The direction of the waves, current, and wind is from north-west.

Estimate Damping Coefficient

First, we find the position of the multi-leg system under the influence of the static forces. The force is acting in the x direction with magnitude F x = 2,870 kN, and the moment is negligible. The resulting motion is 20.37 m in the x- direction, providing a stiffness of K x = 79,520 N/m.

When a drag force acts on a structure:

F D =0"5pCD Ap uIuI (5)

where C D = drag coefficient Ap = projected area U = velocity

Then, if U consists of a steady part U o and an unsteady part u, the damping coefficient for the unsteady velocity can be approximated by

B D -- pc D mplU[= 2 F D / U (6)

The moored semi-submersible has a slow drift natural frequency in the x-direction which is found, on the basis of the stiffness of the mooring system K m = 79,520 N/m and the mass augmented by the added mass M = 2.7 x 107 kg, as equal to ( t} n = 0.05427 (period T = 115.8 s).

The damping ratio, ~, is found as:

~= BD (7) 2Mr n

As a result we find the current drag damping ratio as, ~c = 0.27 (27% of critical); and the wind damping as ~w = 0.055 (5.5% of critical).

The wave drift damping requires hydrodynamic calculations and the use of the sea spectrum. For the given submersible and the specified storm we find the wave drift damping ratio as ~s = 0.082 (8.2% of critical).

The damping coefficient induced by the mooting system, Bm, requires the use of cable programs in an iterative manner, so we describe its evaluation in detail:

Mooring System Drag. The following data must be provided first: (a) the slowly varying force spectrum; (b) the wind force spectrum. The first item can be evaluated hydrodynamically either in its completeness or using Newman's approximation. The wind forces can be found using a standard drag coefficient and the wind spectrum after care has been taken to limit the bandwidth of the


wind forces based on the proper dimension of the structure (in this case the length of the semi-submersible).

First, we assume a value for the damping induced by the mooting lines, which is expected to be anywhere from 10% up to 80% of the total damping force. A first estimate can be ~m = 0.l 0 ( 1 0 % of critical). Then we find the slow drift resonant oscillations induced by the slowly varying wave forces and the slowly varying wind (gusting) forces. The resonant oscillations can be found by time domain simulation and frequency domain calculations.

Using frequency domain calculations, we find that the standard deviation of the slow drift oscillation is G m ----- 3 .8 m, while the standard deviation of the velocity is Gv = 0.18 m/s.

Now the vortex-induced vibration codes must be run with the statistics of the slow drift oscillation and the incoming current for each individual line. It is easier to assume that the structure is performing an equivalent harmonic oscillation at the natural frequency o~ n = 0.0543 r/s and with amplitude a equal to the significant amplitude, i.e., a --- 2(y m, although a more complete evaluation may be required in the end. The computer codes perform an evaluation of the vortex- induced oscillations of the wire section of the line (chain does not undergo any vortex-induced oscillations) and drag amplification of each line at several times within a period of oscillation. As a result, an average drag coefficient for each line is provided (which has been amplified as a result of the vortex-induced vibrations of the wire). Table 3 summarizes the drag amplification found for each line due to the vortex induced oscillations.

Next, we evaluate the dynamic response in waves of each mooting line as a result of the motions imposed by the semi-submersible at the upper end of the cable. The Response Amplitude Operators (RAO) are needed for the semi-sub-

Table 3 Average Drag Coefficient for Each Line Due to Vortex-Induced Oscillations

Line Number Drag Coefficient

1 1.35 2 1.28 3 1.20 4 1.2 5 1.3 6 1.2 7 1.28 8 1.35


mersible as well as the spectrum of the storm. As a result of the dynamic analysis of each line in the storm, the following are found for each line: The statistics of the transverse velocity of the cable along the cable as function of the length; based on these statistics and using the method outlined before we find the drag coefficient amplification at each point along the cable. An average is found typically for each line, as shown in Table 4.

Table 4 Drag Coefficient Amplification for Each Line Due to

Slow Motion-Wave Motion Interaction

Line Number Drag Amplification

1 2.41 2 2.08 3 2.61 4 2.44 5 2.44 6 2.61 7 2.08 8 2.41

Next, the final evaluation of the drag coefficient of each line for the slow-drift oscillations is made as follows" The drag coefficient is equal to the product of the drag coefficient given in Table 3 times the amplification factor found in Table 4. The final result is given in Table 5.

Table 5 Overall Drag Coefficient for Each Line

Line Number Drag Amplification

1 3.25 2 2.66 3 3.13 4 2.88 5 2.88 6 3.13 7 2.66 8 3.25


Next, the mooring line dynamic program is used to evaluate the reaction force at the fairlead due to imposed slow drift oscillations using the drag coefficient found in Table 5. The component of the force in phase with the velocity provides the mooring line damping coefficient. Based on these calculations we find that the damping coefficient is B d = 1.07 x 106 Ns/m, resulting in damping ratio ~d = 0.365 (36.5% of critical). Hence, the mooring line damping constitutes 47.1% of the total damping of the system.

It must be noted that if the drag amplification is not considered, i.e., if the nominal value of c D = 1.2 is used, then the damping coefficient due to mooring lines is found equal to B d = 0.407 x 106 Ns/m, resulting in damping ratio ~d = 0.137 (13.7% of critical), i.e., it is underestimated by 62%. The overall system damping is underestimated by about 30%. This points to the fact that mooring line damping is a very important quantity that must be evaluated with sufficient accuracy. In this example, the error resulting from underestimating the mooting line exceeds the total value of the wave drift damping, which requires elaborate hydrodynamic calculations.

The process must be repeated, because a new estimate of the damping coefficient of the mooting system provides a new estimate of the maximum resonant response of the moored semi-submersible. Usually the second iteration has already produced an acceptable solution, so the iterative process is not unduly laborious.

On the basis of the calculations above, the slow drift motion is repeated and the final value obtained for the standard deviation is 3.48 m.

Comparison with Rules

We proceed to finally calculate the tension in the most loaded line, for a storm with duration 5 hours and in accordance with the API rules. The highest tension occurs in line 1 with average tension 1,695,000 N, maximum slow drift dynamic tension (3.18 times standard deviation) 576,100 N, and significant wave-induced dynamic tension (2 times standard deviation) 198,900 N. The total tension is equal to 2,470,000 N, i.e., 53.7% of the breaking tension, which is Tbr = 4,600,000 N, hence satisfying the API rules.

Summary

Mooring lines are used to anchor large structures, i.e., prevent them from drifting away from their average position and from performing large-amplitude, low-frequency resonant oscillations. At the same time, the mooring lines are forced to move at the top with the wave-induced motions of the structure, and are subject to flow-induced vibrations. Cables have complex mechanics of their own, which interact with fluid forces to create several important mechanisms for dynamic tension amplification and generation of damping forces. Recent progress in developing codes for the simulation of cable structures, conceptual


developments in the proper modeling of low-tension cables, and the development of experimentally-based models for fluid forces, allows a comprehensive analysis of mooring line dynamics, suitable for design.

Because of the superposition of excitation forces of many time scales, the problem becomes tractable for design applications only by separating the equations through an amplitude and time-scale expansion. Three separate equations describe: (a) slowly-varying large amplitude motions, which are very important for studying the resonant motions of a moored structure; (b) wave-induced motions of moderate amplitude, which are important to evaluate the peak dynamic tensions due to the seaway, as well as the drag coefficient amplification caused by nonlinear superposition of slow and wave-induced motions; and (c) very fast, small-amplitude vortex-induced motions, which affect the drag coefficient. All three sets of equations are coupled parametrically, hence analysis is still computationally intensive. Reliable results can now be obtained, however, which can assist the practicing engineer in choosing optimal configuration.

Acknowledgments

The author wishes to acknowledge support from the Office of Naval Research (Ocean Engineering Division) under grant numbers N00014-89-J-3061 and N00014-95-1-0106. Some of the vortex-induced vibration experimental results were obtained in a Joint Industry Project with Noble & Denton, supported by Amoco Production Co., Arco Oil and Gas Co., B P Exploration Inc., E1 Dorado Engineers Inc., Exxon Production Research Co., NCEL, and Reading and Bates Drilling Co.

Appendix 1: Derivation of the Equations of Cable Motion

We derive the cable equations by considering the line to have very small diameter compared to the wavelength of dynamic motion, but we include bending terms. The configuration of the centerline describes the motion of a slender structure. Let R (s, t) denote the position vector at time t from the origin to a material point of the cable, identified by the Lagrangian coordinate s, which is equal to the unstretched length of the line from some fixed origin. The material point is taken to be at the centerline of the rope. Let V denote the velocity vector:

_. OR v= Ot (8)

Let m 1 denote the mass per unit (stretched) length, T the tension acting in the tangential direction-~, i.e.,


-. ~R 1 ~R t . . . . . (9)

~S 1 1 + e c)s

where s 1 denotes the stretched length, i.e. ds 1 = ds (1 + e), where e is the strain. If G 1 denotes the vector of distributed force per unit stretched length, and F the vector shear force, then the equations of motion are:

/) [mldSl~] = ~ OF dS+~ldSl (10) tT ds] + Os

Using conservation of mass:

mldS 1 =mds (11)

where m denotes the mass per unit unstretched length, we arrive at the final form of the equations of motion:

~r~ ~9 ~F m-~- = ~s [Tt] +--~s + 81 (1 +e) (12)

Simple continuity assumption for the vector R (s, t) provides the compatibility relations. Hence, from the condition"

~9 /)R b 8R ~ ~ = - - ~ (13) ~)s ~)t 8t 8s

we derive on the basis of the definitions (Equations 8 and 9) that:

f = ~- [(1 +e) t] (14)

The compatibility relations can be the cause of the "ill-posedness" of the cable equations in the absence of bending stiffness, because when tension becomes negative there is no physical mechanism to keep the cable configuration smooth, hence waves of infinitely small wavelength will be generated to represent the discontinuous slope of a buckling, perfectly flexible string.

Finally, a stress-strain relation is required, which is usually of the form:

T = f ( e , 6 ) (15)

where f(.) denotes the function or functional relating the tension to the strain e. We explicitly put down the time derivative of the strain, 6, because it is useful in representing a hysteretic material. Synthetic cables may also require an equation of the form:

fl (T, T)= f2 (e, e) (16)


where t denotes the time derivative of the tension T, and fl, f2 are functions determined by fitting experimental data. Bitting [1] provides some empirically derived models of this form.

The bending terms can be very complicated, but for the present development we adopt only a simple beam model, neglecting shear deformation and rotary inertia effects. Under such conditions and for linear stress-strain relation as far as the bending moment is concerned, the vector moment M = (M t, M n, Mb), where M t is the torsional moment and (Mn, Mb) the components of the bending moments, is given as"

1VI = 1~ ~ (17)

and the shear force in the absence of distributed external moments is connected to . . . . r

the internal moment vector M, simply as: . . . , .

-- aM t x F + =0 (18)

as

where f~ is the Darboux vector [ 10] and the matrix E is equal to [22]"

_ (Oltt o o / E = [ ~ EInn EInb

EIbn Elbb

(19)

where G is the shear modulus. For example, for a circular-section cable of radius r and density Pc, Ibn = Inb = 0, while:

2 = - - r (20) Itt Pc 2

Xr4 (21) Inn = Ibb = I = Pc 4

It should be noted that whenever a derivative of a vector f = (ft, fn' fb) is taken, projected in the local tangential-normal binormal system, the following rule must be used:

a ~=(aft afn afb as a~-' as'as )+~x~ (22)

a ~=(aft afn afb - at ~ ' at ' at )+~xf (23)


Three Euler angles are needed to define the cable configuration at each point: two angles, @ about the y-axis and 0 about the new x-axis are needed to specify the orientation of the tangential vector t (Figure 8); the third, ~, about the x-axis is needed to define the material torsion [22]. Then the Darboux vector ~ = (f~l, ~'~2' ~"~3 ) and the angular velocity i5 = (o)1, (02, 0)3) can be found as:

co 1 = - sin (0)-~- (24)

~0 co 2 =---~- (25)

~, co s = cos (0)-~- (26)

O, f21 = - s i n ( 0 ) ~ s (27)

~0 f~2 =-~-s (28)

x Xx=X2=t

t *

�9 Z 2 ---- b

y -

s z

Figure 8. Reference systems and Euler angles.


f~3 = cos (0) -ffS- s (29)

The governing equations can be projected on any convenient reference system. I prefer the local tangential, normal, binormal directions (denoted in the following as t-n-b system), because of certain advantages it offers, most notably the decomposition into transverse and longitudinal waves, which is natural to this system; and the relative ease of determining the fluid drag force. Some disadvantages exist, because it is a non-inertial system, and hence a fixed Cartesian system may have advantages for certain applications. In the t-n-b system the equations become:

3u 3T m(-47- + o~2w- ~3v) = - z - + G t (1 + e ) + f~2F b - f~3Fn

Ot Os (30)

Ov OFn m (--~- +0~3u-OllW) = T ~ 3 +G n ( l+e)+---~s + ~ 3 F b tan(0) (31)

3w OFb m(--~- + O~lV-O~2u)=- T f~ 2 + G b (1 +e)+---~s - f ~ 3 F n tan(0) (32)

where v = (u, v, w), F = (F n, Fb), G 1 = (G t, G n, Gb). The compatibility relations are:

3u 3e ~ 4- ~"~2W -- ~'-~3V -" (33) Os

~ v ~ - t - f~3u- f~lW = o~ 3 (1 +e) (34) 3t

~ w + f~ lv - f~2u = - 0~2 (1 +e) (35)

3t

while for a circular section cable:

3f~ 2 F b = E I 3s - E I f~2 tan (0) (36)

3f~3 F n = - E I - ~ s + E I f13f~2 tan (0) (37)

When the cross-section is circular, no torsional motion results from bending, unless external torsional moments are applied.


Appendix 2: Methodology for Calculating Vortex-Induced Motions

The relative velocity between the cable and the fluid is almost never uniform: Often the oncoming flow is sheared; also, a curved cable is subject to a variable normal velocity component even when the flow is uniform, because of the variation of the inclination angle. Full-scale data [9, 38] indicated that one of the principal effects of shear in the normal component of the flow is a beating motion in cables. A theoretical explanation has been provided in the cited references, and a numerical model is explained in detail in Triantafyllou et al. [43, 44] and Triantafyllou and Grosenbaugh [47].

As a result of this simplifying fact, sectional forces measured on a rigid cylinder undergoing beating oscillations in a stream of steady velocity U (the steady velocity representing the low-frequency relative velocity), can provide the force information needed to perform numerical calculations. Some representative cases of force measurement in the laboratory have been presented [7], while a very dense grid of parametric changes has been conducted at the MIT Testing Tank Facility within a Joint Industry Project [43, 44]. The parameters changed were:

1. Shape of the cylinder (smooth cylinder, wire rope, chain, riser shape) 2. Maximum amplitude of vibration 3. Frequency of oscillation 4. Beating frequency of oscillation

The data base has been incorporated in a numerical scheme as follows: The cable is presumed to be modeled adequately by a string equation with variable properties (mass, tension), placed transversely to a current with nominal velocity V. The transverse response of the string is denoted by y(t, s):

~2y ~Y ~ I ~Y] m - ~ + b - - ~ - = ~ s s T(s)-~- s +f(s , t ) (38)

where s = Lagrangian coordinate along the string T(s) = static tension

m = mass augmented by the added mass per-unit-length b s = structural damping per-unit-length

It is further assumed that the characteristic wavelength of the string oscillations is much smaller than the length of the string, hence the response is effectively that of an infinitely long string, f(s, t) is the remaining fluid force (i.e., in addition to the added mass force), which can be decomposed into the following form [43, 44]:


f s,t, Iv s.t, 1 (r (s, t) Le~ -bhV(S't)

where ~(s, t) is the slowly varying envelope of the velocity v(s, t):

v (s, t) = ~-~ y (s, t)

The damping coefficient, b h, and Leo are given approximately as:

tovd'

(39)

(40)

(41)

(42)

The values of ~,, C o, and of the function H(to v) are determined from experimental data, where tov is the local Strouhal frequency (to = 2rtStU/d with St --- 0.17). The basic point of this decomposition is that the total fluid force can be decomposed first into an acceleration-dependent part (added mass force) and a velocity dependent part; and, more importantly, that the velocity-dependent part can be further decomposed into a term that varies in time in phase with the velocity but is independent of its amplitude, and a second term which is a linear function of the velocity [43, 44, 47]. The linearity of hydrodynamic damping in VIV was explicitly stated for the first time in Ramberg et al. [29].

Note that in all equations above the velocity U is a function of s, hence Equa- tions 41 and 42 must be evaluated separately at each point of the cable.

Full correlation is presumed within half wavelength, following the experimental measurements and observations of Ramberg and Griffin [28] and Gharib [5]. The drag force is determined from the experimental data base after the local beating frequency and maximum and minimum amplitude of vibration have been calculated numerically.

There is a difficulty in obtaining the solution, because one must make sure that the excitation is indeed properly correlated with the velocity. This is straightforward in time-domain simulations, because one must calculate the envelope of the velocity at each time step, before using Equation 39. Often, however, frequency domain techniques are employed, resulting in considerable savings in computational expense; then, an additional requirement must be imposed, to ensure that the excitation is properly correlated, viz.

I lim l f o 12 1 2Ilim lf0~ 1 T ~ T f (S, t) V (S, t) dt =-~ Lo T ~ T v (S, t) v (s, t) dt (43)


where

Lo = / l p d U 2 ) C o (44)

Figure 9 shows a comparison between the predicted ampli tude of vibration and measu red data f rom Griff in [8]. Very good corre la t ion be tween theory and experiment is established. Further comparisons are shown in Table 6.

10.0

6.0 >..~" 4.0

2.0 oJ E m 1.0

.~ 0.6

c~ 0.4 0

0.:2

t~ 0.1

MARINE STRUCTURES

.~- MARINE CABLES _l

LIMITING DISPLACEMENT

(~ Present calculations I ~

I I .... I I I. I I , I ~ i . I 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 5.0 10.0

Reduced Damping, ~slli

Figure 9. Maximum double amplitude of vortex-induced vibration versus reduced damping [8, 43, 44].

Table 6 Test Cases for VIV Response

Case Ref. A/d Cd No Description Exper. Theory Exper. Theory

1 [50] RUN 1A - - 0.80 2.47 2.21 2 [50] RUN 2A - - 0.79 2.24 2.05 3 [9] transient-- 0.68 1.95 2.08 4 [9] RUN 4+adverse current ~ 0.72 2.15 1.95 5 [9] RUN 4+favour. current ~ 0.70 - - 1.91

6 [48] St Croix exper. .25-.5 0.40 1.50 1.61 7 [48] Castine exper. 0.5-1. 0.65 - - 1.98 8 [48] Lawrence exper. 0.2-0.5 0.2-0.4 ~ 1.30


Appendix 3: Methodology for Calculating the Interaction Between Wave-Induced and Slowly Varying Motions

The nonl inear form of the drag force provides for a potent ia l ly strong amplification of the drag coefficient. Indeed, a s imple use of the Mor i son equat ion for a cyl inder undergoing harmonic veloci ty oscil lat ions of f requency co = 2n/T and ampli tude u o in-line with a current of veloci ty U, provides the ampl i f ica t ion

factor, (x:

Ix ( x : ~ (45)

T U 2

I, : / 0 T (U + u <t)>lu + u (t) Idt (46)

u ( t ) = u o cos ((ot) (47)

which is plotted in Figure 10. The theoretical predict ion for the amplif icat ion is not valid when U < u o, because the cyl inder re-enters its wake, hence the drag coefficient is not constant. This is evident f rom Figure 10. One should note the large ampl i f i ca t ion factors ach ieved expe r imen ta l ly . Ex tens ions for r a n d o m waves can easily be achieved using Figure 10.

o �9 -. 4

o

u 3 ~

~ 2

<-- theoretical curve

x *

x *

o

~ 9

x

x - 0.50 knots 0 - U . 7 2 k n o t s

* - 1.00 knots + - i.50 knots

+ ..... :~.

[ / . 'br~

0 0 0.5 ! 1.5 2 2.5 3 3.5 4

Figure 10. Drag coefficient amplification factor as function of Uo/U w, where U o is the low frequency velocity and U w the amplitude of wave induced velocity. Solid line theory; x, o, *, + denote experimental results at various Reynolds numbers.


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5. Gharib, M., 1991. "Ordered and Chaotic Vortex Streets Behind Vibrating Circular Cylinders," Marine Industry Collegium Symposium, April 23-24, MIT, Cam- bridge, Massachusetts.

6. Goodman, T. R. and Breslin, J. P., 1976. "Statics and Dynamics of Anchoring Cables in Waves," J. Hydronautics, 10, pp. 113-120.

7. Gopalkrishnan, R., Grosenbaugh, M. A. and Triantafyllou, M. S., 1992. "Ampli- tude Modulated Cylinders in Constant Flow: Fundamental Experiments to Predict Response in Shear Flow," Proc. Third Int. Symp. on Flow-Induced Vibrations and Noise, ASME, Anaheim, California.

8. Griffin, O. E., 1981. "OTEC Cold-Water Pipe Design for Problems Caused by Vortex-Excited Oscillations," Ocean Engng., 8, pp. 129-209.

9. Grosenbaugh, M. A., 1991. "The Effect of Unsteady Motion on the Drag Forces and Flow-Induced Vibrations of a Long Vertical Tow Cable," Int. J. Offshore and Polar Engng., 1, pp. 18-26.

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11. Hover, F. S., Grosenbaugh, M. A. and Triantafyllou, M. S., 1994. "Calculation of Dynamic Motions and Tensions in Towed Underwater Cables," IEEE Journal of Oceanic Engineering, 19 (3), pp. 449-457.

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15. Huse, E. and Matsumoto, K., 1989. "Mooting Line Damping Due to First and Sec- ond Order Vessel Motion," OTC # 5676, Houston, Texas.

16. Irvine, H. M. and Caughey, T. K., 1974. "The Linear Theory of Free Vibrations of a Suspended Cable," Proc. of the Royal Society, A341, pp. 299-315.

17. Irvine, H. M., 1993. "Local Bending Stresses in Cables," Int. J. of Offshore and Polar Engineering, 3 (3), pp. 172-175.

18. King, R., 1977. "A Review of Vortex Shedding Research and Its Applications." Ocean Engng. 4, pp. 141-171.


19. Koterayama, W., 1989. "Viscous Damping Forces for Slow Drift Oscillation of the Floating Structure Acting on the Hull and the Mooring Lines," Proc. Eighth Intern. OMAE, The Hague, The Netherlands.

20. Kuttler, J. R. and Sigillito, V. G., 1981. "On Curve Veering," J. of Sound and Vibration, 75, pp. 585-588.

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22. Landau, L. D. and Lifshitz, E. M., 1959. Theory of Elasticity, Pergamon Press, New York.

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24. Milgram, J. H., Triantafyllou, M. S., Frimm, F. and Anagnostou, G., 1988. "Sea- keeping and Extreme Tensions in Offshore Towing," Trans. of the Society of Naval Architects and Marine Engineers, 96, pp. 35-72.

25. Paidoussis, M. P., 1987. "Flow-Induced Instabilities of Cylindrical Structures," Applied Mechanics Reviews, ASME, 40, pp. 163-175.

26. Papazoglou, V., Mavrakos, S., Triantafyllou, M. S., 1990. "Scaling and Model Testing of Cables in Water," J. of Sound and Vibration, 140, pp. 103-115.

27. Perkins, N. C. and Mote, C. D., 1986. "Comments on Curve Veering in Eigenvalue Problems," J. of Sound and Vibration, 106, pp. 451-463.

28. Ramberg, S. E. and Griffin, O. M., 1976. "Velocity Correlation and Vortex Spac- ing in the Wake of a Vibrating Cable," J. Fluids Engng., 98, pp. 10-18.

29. Ramberg, S. E., Griffin, O. M. and Skop, R. A., 1975. "Some Resonant Properties of Marine Cables with Application to the Prediction of Vortex-Induced Structural Vibrations," Ocean Engineering Mechanics, ASME Winter Annual Meeting, Houston, Texas, pp. 29-42.

30. Simpson, A., 1972. "On the Oscillatory Motions of Translating Cables," J. of Sound and Vibration, 20, pp. 177-189.

31. Tein, Y. S. D., Chantrel, J. M., Marol, P., and Huang, K., 1987. "Rational Dynamic Analysis for Turret Mooring Systems," OTC # 5527, Houston, Texas.

32. Tjavaras, T., 1995. "Dynamics of Highly Extensible Cables," Ph.D. thesis, Massa- chusetts Institute of Technology, Cambridge, MA.

33. Triantafyllou, M. S., 1982. "A Consistent Hydrodynamic Theory for Moored and Positioned Vessels," J. Ship Res., 26, pp. 97-105.

34. Triantafyllou, M. S., 1984. "The Dynamics of Taut Inclined Cables," Quart. J. of Mechanics and Applied Mathematics, 37, pp. 421-440.

35. Triantafyllou, M. S., 1985. "The Dynamics of Translating Cables," J. of Sound and Vibration, 103 (2), pp. 171-182.

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39. Triantafyllou, M. S. and Triantafyllou, G. S., 1991. "Frequency Coalescence and Mode Localization Phenomena: A Geometric Theory," J. of Sound and Vibration, 150, pp. 485-500.

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C H A P T E R 7

MODELING LAWS IN OCEAN ENGINEERING

Subrata K. Chakrabarti

Offshore Structure Analysis, Inc. Plainfield, Illinois, USA

CONTENTS

INTRODUCTION, 296

Benefits of Model Testing, 296

Choice of Scale, 297

MODELING LAWS, 297

Geometrically Similar Structures, 298

Geometrically Similar Motions, 298

Geometrically Similar Systems, 299

Scaling Laws, 302

Froude Model, 303

Reynolds Model, 304

Cauchy Model, 305

Distorted Model, 307

Secondary Scale Effects, 310

MODEL TESTING FACILITIES, 312

Coastal Modeling, 312

Offshore Modeling, 314

MODEL ENVIRONMENT, 315

Modeling of Waves, 316

Modeling of Wind, 319

Modeling of Current, 320

EXAMPLES OF MODELING, 321

Types of Measurements, 321

Coastal Structures, 323

Fixed Offshore Structures, 326

295


Floating Offshore Structures, 328 Hydroelasticity, 330

NOTATION, 332

REFERENCES, 333

Introduction

Physical models are a close representation of reality in which a prototype system is duplicated as closely as possible in a (generally) smaller scale. The purpose of the model is to approximate and anticipate the prototype behavior through certain prescribed modeling laws. There are many modeling approaches that are followed in the study of the natural system. The most important of these are physical models and mathematical models. Physical models may be scaled hardware models or computer analog models. This chapter is limited to the case of scaled physical models and the scaling laws associated with the physical models used in ocean engineering.

For hundreds of years, models have been used as a working plan from which prototype structures have been designed, modified, and constructed. Ship models have long enjoyed the usefulness of modeling for various purposes, such as in determining the placement of their cargo and ballast. Working mechanical models came into use during the industrial revolution. Systematic hydraulic scale model testing goes back to the nineteenth century. Coastal modeling has been helpful in the development of many harbors, breakwaters, mooring piers, etc.

Benefits of Model Testing

One of the principal benefits of model testing is that valuable information is provided, which can be used to predict the potential success of the prototype at relatively little cost. The physical model provides qualitative insight into a physical phenomenon that is not fully understood. The use of models is particularly advantageous when the analysis of the prototype structure is very complicated. In other situations, models are often used to verify simplified assumptions that are involved (or inherent) in most analytical solutions, including higher-order effects. An example of this is the discovery of slow drift oscillation of a moored floating tanker through model testing before a theory describing the second- order oscillating drift force and the associated motion was derived. Model test results are also employed in deriving empirical coefficients that may be directly used in a design of the prototype.

Modeling Laws in Ocean Engineering :297

Choice of Scale

The choice of scale for a model test often is limited by the experimental facilities available. However, within this constraint, optimum scale should be determined by comparing the economics of the scale model with that of the experiment. It should be kept in mind that too small a scale may result in scale effects and error. Too large a scale is often very expensive and may introduce problems of physically handling the model.

The primary purpose of wave tank study is to obtain reliable data by minimizing scale effects and measurement error. The common range of scale for studies such as breakwater stability are 1:50 to 1:20 in two-dimensional test tanks, and 1:150 to 1:80 for three-dimensional basins. The desired range of scale for offshore structures in a two-dimensional wave tank is between 1:100 and 1:10.

When Reynolds effect (such as presence of drag force) is important, a large scale is recommended to minimize the problem of scale effects. However, inertial wall effects must also be considered and avoided in this case. For a circular cylinder, the diameter should not exceed 1/5th the width of the tank. When three-dimensional structures are tested in a wave tank, then transverse resonance may exist in the tank from its sidewalls. This may be remedied by using lateral wave absorbers.

Modeling Laws

Modeling laws relate the behavior of a prototype to that of a scaled model in a prescribed manner. The problem in scaling is to derive an appropriate scaling law that accurately describes this similarity. A parametric approach of relating the model properties to the prototype properties is used when little is known about the governing equations of the system. In this case, the Buckingham Pi Theorem is applied to all applicable variables to derive a group of meaningful dimensionless quantities. If, however, the governing equations are known a priori then the scaling laws are derived directly from those equations in nondimensional form.

Dimensional analysis is useful, particularly when the phenomena are so complex as to preclude differential analysis. It provides the rationale for all model experiments. Typically in dimensional analysis one lists the various quantities thought to be significant in a certain natural phenomenon and seeks a functional relationship between dimensionless parameters composed of these quantities [ 1 ]. The physical variables change in magnitude over the domain under investigation. However, arguments appear in a functional relation with these quantities, which have the nature of coefficients, and do not change in numerical magnitude when the physical system alone changes. These are called dimensional constants. In dimensional analysis, the use of physical constants and the controlling equations in which the constants are employed must go hand in hand. The con-


trolling equations are those physical laws that express proportionalities between the various concepts employed in analyzing the problem at hand. The dimensional constants are treated as secondary variables.

In the Buckingham Pi Theorem, the most important variables governing the system are identified first. The parameters of least significance are omitted. The number of variables is limited to a manageable quantity. The dimensions of these variables are used to construct an independent and convenient set of dimensionless parameters (pi terms). In an M, L, T (mass length, time) system, if the number of dimensional variables is N, then there would be N-3 nondimensional parameters that are independent. The scaling law similitude is satisfied and the model and prototype structural systems are similar if the corresponding pi terms are equal.

Geometrically Similar Structures

Geometrically similar structures may have different dimensions, but must have the same shape. Assuming two structures are geometrically similar, a constant scale ratio exists between them

g P = a (1)

~m

where/p, {m = any two homologous dimensions of the two structures

Geometrically Similar Motion

Two geometrically similar structures will undergo geometrically similar motion when their homologous points will follow similar paths in proportional times [2]. Therefore, their times will have a scale ratio b.

tp ~ = b (2) tm

Then, the velocity ratio will be obtained as

Up l p t m a

u m lmt p b (3)

Also, the acceleration ratio can be determined as

Llp lpt 2 a

-- 1 m t p -----~m 2 -- b''~" (4 ) fi--~--

The ratios in Equations 1-4 may be all different. However, for the above similarities only two of them can be independent.

Modeling Laws in Ocean Engineering 299

D y n a m i c a l l y S i m i l a r S y s t e m s

Consider the masses mp and m m as the homologous elements of two similar structures in similar motions. The masses have a constant ratio c"

3 mp = pplp =c

m m pml3m (5)

so that

P p _ C

Pm a3 (6)

where p = density All corresponding impressed forces must then be in a constant ratio and similar direction:

4 2 Fp _ mpfip _ pplpt m ca - �9 - 2 b 2 F m mmU m Om 14 tp

(7)

which is constant, because Pp/Pm' ljp/lm' and tp/t m are constant. Therefore, geometrically similar structures in similar motions having similar mass systems are similarly forced. When any three of these ratios are fixed, the fourth one is automatically determined.

The laws of dynamically similar fluid motions are derived by three different methods: (1) Newtonian or integral method, (2) differential method and (3) dimensional method [2]. In each case the physical quantities governing the flow are based on experience. In the Newtonian method, the ratio of acceleration forces is equated to several forces of corresponding impressed forces, thus obtaining specific conditions for dynamical similarity. In the differential method, the differential equations for the two fluid motions assumed to be dynamically similar are written. Then the ratios of corresponding terms are equated to obtain the scaling laws.

To illustrate, let us consider an example of an open channel flow problem [3]. For a uniform steady velocity U through a uniform cross-section A of a channel of width w, the continuity equation is written as

~z ~(UA) w ~ + ,3x -o (8)

where z = elevation of the free surface The momentum equation is expressed as


t)U t)U t)z kU 2 --+ U = -- + ~ (9) 0t ~ x - g ~)x R

where R = hydraulic radius k = a constant

The relationship between the prototype and model variables involved in these two equations in terms of the respective scale factor is written as

Vp = ~v Vm (10)

where v = variable under consideration ~, = corresponding scale factor

p, m = prototype and model, respectively

Note that ~'A = ~'x~'z �9 Then the previous equations become

~,x~z ~Z m ~Lt Wm "~m + ~Lu ~Lz

O(Um Am) ~Xm

=0 (11)

and

~x ~U ~ U m ~z ~Zm +~Lk~Lx km U2 ~Lt ~LU Oqtm +Um ~Xm - g ~J ~Xm ~R Rm

(12)

For similarity to follow, Equation 11 gives

~,x (13) ~,u =-~-t

which follows from the kinematic similarity. It follows from Equation 12 that

~,z =1 (14)

and

~'k ~'x =1 (15)

The first of these (Equation 14) is shown to be a consequence of Froude's law. Therefore, for similarity between the model and prototype in this open channel flow, the identities in Equations 13-15 must be satisfied.


In the dimensional method, the unknown forces, F on two homologous fluid elements are equated to the sum of all the terms from the flow-governing quantities arranged in power products each having the dimension of F. Then taking the ratio of F forces, the scaling laws are derived. This is illustrated by the following example.

Consider that the force F on a structure in a fluid stream depends solely on density p, diameter D, gravity g, velocity u, dynamic viscosity ~t and Young's modulus due to elasticity E. Thus, there are 6 independent variables in a mass (M), length (L), and time (T) system. The number of possible independent products having the dimension of F will be 6 - 3 = 3. To form these, any triad pXDYuZ of the six independent quantities is chosen and multiplied in succession by the remaining variables g, l.t, and E. Thus,

P1 =paDbuCg; P2 =pdDeufkt; P3 =P gDhuiE (16)

Equating the dimensions of each product to [F] = [ML/T2], the first one yields

~5- Lu @ = T 2 (16a)

On equating the indices of L, M, and T successively, we have:

- 3 a + b + c + l = l ; a = l : - c - 2 = - 2 (17)

Thus, a = 1, b = 3, c = 0, and P1 = pD3g �9 Similarly, P2 = Du~t and P3 = DEE" Then, a general resistance equation for the specified dynamical conditions may be written as

(gD ~ E / (18) F=(pOEu 2)f u2 , Du ' pu 2

where f = a functional relationship among the nondimensional quantities within the parentheses

~t = px), a9 being kinematic viscosity

The dimensional method seems to be a straightforward algebraic operation requiring less knowledge about the physical system than the other two. In reality, however, all three methods require adequate judgment about the physical quantities involved in the dynamical system and their relative importance. Knowledge and understanding of the physical phenomena are necessary to con- sciously neglect those that are of secondary importance and to determine important scale effects.


The derived working formula from any of these methods contains dimensionless coefficients that are determined from model tests and applied to the prototype system operating under similar dynamical conditions.

Scaling Laws

Hydrodynamic scaling laws are derived from the ratio of forces commonly encountered in a hydrodynamic model test. These are listed in Table 1.

Table 1 Common Dimensionless Quantities in Ocean Engineering

Symbol Dimensionless Number Force Ratio Definition

F Froude Number Inertia/Gravity u2/gD R Reynolds Number Inertia/Viscous uD/~ E Euler Number Inertia/Pressure P/pu 2 C Cauchy Number Inertia/Elastic puE/F e K Keulegan-Carpenter Number Drag/Inertia uT/D S Strouhal Number feD/u U Ursell Number HLE/d 3

In Table 1, u = fluid velocity amplitude fe = vortex (eddy) shedding frequency H = wave height L = wavelength

The last two quantities are not derived from the force ratios. Strouhal number is the nondimensional vortex shedding frequency, while Ursell number is considered the nondimensional depth parameter.

The typical current or wave-structure interaction problem involves Froude Number, Reynolds number, and Keulegan-Carpenter number. For vibrating structures in fluid medium, the Strouhal number is important. The Cauchy number plays an important role for an elastic structure. In wave theories, the Ursell number determines the nonlinearity in waves and chooses applicable wave theories.

The frequency of vortex shedding, fe, from a stationary circular cylinder of diameter D in a fluid stream of velocity u has been shown to be a linear function of R = uD/a9 over a wide range of Reynolds number R. Thus, it is interesting to seek the relationship between the Strouhal number S = feD/U and R [4]. It is generally accepted that S -- 0.2 in the range 2.5 x 10 2 < R < 2.5 x 10 5.

Beyond this range, S increases up to about 0.3 and then, with further increase in R, the regular periodic behavior of u in the wake behind the cylinder disap- pears. The general behavior of S versus R is shown in Figure 1. Some variation


0.5

0.4

S 0.3

0.2

0.1 . ~ - -

j v

2 3 4 5 6 7 log R

Figure 1. Relationship between Strouhal number and Reynolds number from Popov [4].

in this trend has been observed in experiments by several investigators particularly outside the constant range of S.

The most common of the dimensionless scaling laws presented in Table 1 in the water wave problem is Froude's law. The Reynolds number is also equally important in many cases. However, Reynolds similarity is quite difficult, if not impossible, to achieve in a small-scale model. Simultaneous satisfaction of F and R is even more difficult. The Froude law is the accepted method of modeling in hydrodynamics.

Froude Model

The Froude number has a dimension corresponding to the ratio of u2/gD as previously shown. Defining F as

2 U

F = ~ (19) gD

the Froude model must satisfy the relationship:

U 2 2 p Um

gDp gD m (2.0)


Assuming a scale factor of ~, and geometric similarity, we have

Dp = ~D m (21)

Then, from Equation 20:

= ~ - u (22) Up m

Other variable quantities are derived from dimensional analysis as follows:

Mass mp = ~3 mm (23)

Force Fp = ~3 Fm (24)

Fluid Acceleration tip = tl m (25)

Time tp ---- ~f-~m (26)

Reynolds Model

If a Reynolds model is built, it will require that the Reynolds number between the prototype and the model be the same. Assuming that the same fluid is used in the model system, this means that

upDp = u m D m (27)

If a scale factor of ~, is used in the model, then this equality is satisfied if

U m - '~Up (28)

In other words, the model fluid velocity must be ~, times the prototype fluid velocity. In general, this is difficult to achieve, especially if a small-scale experiment is planned. In automobile drag tests, because a large scale model (~, = 1-2) is often used, this scaling is easier to achieve. In this case:

Rp = ~3/2R m (29)

Therefore, the larger the scale factor, the larger is the distortion in the Reynolds scaling. In fact, it is possible that the model flow will be laminar while the prototype flow falls in the turbulent region. Experiments have shown that the flow characteristics in the boundary layer are most likely to be laminar at R < 105, whereas the boundary layer is turbulent for R > 106. In this case two different scaling laws apply, namely, both Froude and Reynolds, which cannot be satisfied simultaneously unless different fluids are used, which may not be practical.


It is most convenient to employ Froude scaling and to account for the Reynolds disparity by other means discussed in the following sections.

One method of achieving a proper Reynolds number effect at the boundary layer is to deliberately trip the laminar flow in the model by introducing roughness on the surface of the model. This works because once the flow regime is turbulent, the drag effect is only weakly dependent on the Reynolds number. In testing tanker models, often external means, such as studs, pins, or sandstrips attached near the bow, are used to induce turbulence. Turbulent flow can also be tripped ahead of the model by introducing a mesh barrier.

Sometimes, fluid of lower viscosity than water is used to increase the value of Reynolds number in the model. For equality of both Froude and Reynolds number, a fluid whose kinematic viscosity is about 1 / ~ of that of water should be used. When ~, is large, this is difficult to achieve. In ship or barge towing tests, corrections are made in the friction factor based on the respective Reynolds number before the data on model towing resistance are scaled up to the prototype value. If this difference is ignored in scaling, the (scaled up) prototype data will generally be conservative.

Cauchy Model

Hydroelasticity deals with the problems of fluid flow past a submerged structure in which the fluid dynamic forces depend on both the inertial and elastic forces on the structure. It is well known that for long slender structures, the stiffness of the structure is important in measuring the response of the structure model in waves.

It is often desired to test structures to determine stresses generated in its members due to external forces, for example, from waves. In this case the elasticity of the prototype should be maintained in the model. Therefore, in addition to the Froude similitude, the Cauchy similitude is desired.

The Cauchy similitude requires that stiffness such as in bending of a model must be related to that of the prototype by the relation:

(EI)p = ~5 (EI) m (30)

where I = moment of inertia This provides the deflection in the model, which is 1/~, times the deflection in the prototype; also, stress must be similarly related, such that, t~p = ~,t~ m, (Froude's law). For example, for a cantilever beam the maximum deflection is given by ~max- PP/(3EI) where P is the load at the end of the cantilever of length 1. Equation 30 satisfies Froude's law for this relationship.

Because the section moment of inertia satisfies

Ip -~4 Im (,31)


we have

Ep = ~ E m (32)

Thus, the Young' s modulus of the model material should be 1/~, times that of the prototype. Assuming steel for the prototype material (E = 2.07 x 108 kPa or 30 x 106 psi) and ~, = 36, the model E m should be 5.7 x 10 I~ (83,300 psi).

Let us consider modeling a membrane-type oil storage tank anchored at the bottom of the ocean. The modeling involves two liquids of different densities separated by a flexible rubber-like material, which models the skin of the storage tank [5 ].

If D is the average oil depth in the storage tank and p and Po are the densities of water and oil, similitude shows that the following identity holds:

[ o00o I E o00o] Po p 19~ m

(33)

Considering the relationship, D p = ~D m, we have

00o I 00o I Po p 19~ m

(34)

This condition may be fulfilled if the scale model tank is filled with kerosene, alcohol, or gasoline to correct the density difference between sea and fresh water. The elasticity of the tank requires that

eE I eE I P~ 'p m Po R

(35)

where e = membrane thickness E = elasticity R = radius of the storage tank (Rp = ~Rm)

Then,

eEl =~2 e~o ] 9o p m

(36)

Assuming Po being (practically) the same between the prototype and the model

(eE)p = ~2 (eE) m (37)


Assuming a scale of 1:30, a 6.4-mm (�88 thick composite rubber product may have an elastic similitude with a 0.4-mm (Y64-in.), thin rubber-like material. In this case, the Reynolds similitude is also achieved for the motion within the tank by the relation:

~gp -- ~3/2 ~)m (38)

For example, the kinematic viscosity of oil is 0.93 x 10 -5 m2/s (10 -4 ft2/s), for alcohol it is 0.15 x 10 -5 m2/s (1.6 • 10 -5 ft2/s) and for gasoline 0.46 x 10 -7 ma/s (0.5 x 10 -6 ft2/s).

Distorted Model

The choice of scale factor depends on several items, such as accuracy of model construction, measurement accuracy, and ability to collect accurate data. Small models are not necessarily the most economical. For very small models the cost of operation may actually be high because of the difficulty of achieving reliable measurements. The smallest scale to which a Froude model may be built is determined by the influence of the fluid viscosity and surface tension. For example, test results from small-scale spillway models are commonly scaled up to the prototype value using Froude scaling, disregarding any effect from viscosity or surface tension. The lowest useful model scale is often determined on the basis of past experience.

Maxwell and Weggel [6] showed that the minimum energy flux for a uniform flow per unit weight flux over a unit width in a wide rectangular channel (such as river flow) is given by

F 2 =1 - ~ 1 (39) W

where F = Froude number defined as q /V~3 W = Weber number defined as 3~2/o

q = fluid discharge per unit width y = depth of flow 3' = specific weight of fluid o = coefficient of surface tension

The Weber number in general is extremely large. In this case, the Froude model is not a problem as F approaches one. However, it is possible that the depth of flow in model scale is so shallow that the Weber number will have some influence in Equation 39. In this case a distorted model may be necessary, especially if space is restricted.

Riser Modeling. Let us consider the modeling of a floating platform connected to the ocean floor in deep water by a riser system [7]. Because of the depth


limitation in laboratory testing facilities, scaling of the riser in deep water often requires distortion in the overall length of the riser. The motion of the platform under wave action will obey the Froude similitude provided the mass and volume of the platform are distributed accordingly such that mp = ~g3m m. Because of the scale effect due to distortion, the motion of the deeper section of the riser will not be in similitude. On the other hand, the interest is higher at the intersection of the riser and the platform where the forces exerted on the riser will be the highest. The moment at the top of the riser (where it is connected to the platform) is given by

M = Ta/T-E-] 8 + M H (40)

where T = tension in riser EI = riser stiffness

8 = angle of riser M H = moment due to hydrodynamic force

Inertial forces due to the fluid will be in proper similitude as long as the external riser diameter is scaled according to geometric scale. By necessity, the cross- section of the riser will probably be modeled by a solid rod. The mass of the riser is modeled such that the density of the model equals the average density of the prototype riser, including the steel annulus and the contained water:

Pm Am =po Aw + PR AR (41)

where p = density A = cross-sectional area

W, R = water and riser, respectively

This will ensure proper inertia of the riser. The drag force will generally not be in proper similitude due to the Reynolds

effect. However, this effect is small compared to the inertia force near the water surface. The Cauchy similitude is ensured if the following relationships are satisfied:

Mp =~4 Mm (42)

~)p =~Jm (43)

so that

(TEI)p = ~8 (TEI)m (44)

To determine the relationship for the elasticity between the model and the prototype, the expressions of T and I for the model and the prototype are examined.


The moment of inertia I for the model is expressed as

I m - - ~R 4 4

where R m = model riser radius, while that of the prototype is

Ip ~ XR3pARp

where R = outer radius of riser p ARp = thickness of the annulus

Similarly, the tensions are expressed as

Tm = ~ R 2 E Alm m lm

and

Alp Tp = 2nRpARpEp

lp

A1 where m is the relative elongation. Note that for similitude,

1

Substituting the above relationships:

Ep = ~, Rp E m

Z~ m Alp

1 m lp

(45)

(46)

(47)

(48)

(49)

Note that if we assume that T_ = ~,3Tm, then the modulus of elasticity between the model and prototype are re~ated by

R Ep = ~, P E m (50)

2ARp

The first condition (Equation 49) is more appropriate for the reaction of the platform upon the riser. It also approximates well the motion of the riser in similitude. In this case, the quantities such as the inertia forces, bending forces, and tension are in approximate similitude from the platform to the bottom.

For a scale of ~, = 200, a wire of polytetra-flurethylene with a relative density of 2.2 and E = 4 x 105 kPa (0.58 x 105 psi) is appropriate for the similitude requirements. Even at this scale the depth in the wave tank may have to be distorted for very deep water (see Figure 2). If the length of the riser is distorted so that

lp = ~,lm~ (51)


".~.'."::'-";~:.: :?'~;~:~'"::!';~" ~~'~":~" ;-~ i: ~ ~ " : - - - ~ :_: ~~ ~!~ ';.'"~!?"-;;~;;"~:-:~!i.:@:~~@:i~:~ '

)~ WELL

: ' , ; ,~: , ,TCIRCULAR HOLE IN THE " ]'~:':- -" I' : ~ ' " : " BOTTOM OF THE TANK

WAVE TANK.

i!~ii~!!!!~:i~:!~i~i~i~i~i~ii~iiiii!ii!ii~!~!~i~i~i~!!!~!i~i~!~i~!!~!ii~i~.i.: ;i!;?i ..i:~:i: .... : : :~: ~i:i ~..~'~..i!~.i~ii~i:~:~i~!~.~i!i~i!!i~i~i~i~i~!!!?~!ii~i~i~?~.::!ii!ii!~ii~ I:

Figure 2. Distorted riser model taken from LeM6haut6 [7].

then considering Alp = ~A1 m, the modulus of elasticity is related by

E p - ' ~ R-------Lp Em ARp 2 ~ f 2

where ~5 is the distortion factor.

(52)

S e c o n d a r y Scale Effects

In all cases of scale model studies, one should ensure that capillary effects and viscous damping are negligible [7, 8]. Generally, viscous damping is not a significant concern for most wave tanks having water depths much larger than 10 cm.

Capillary Effects. The capillary effect influences waves by introducing a wave damping effect. This in turn distorts the wave celerity (or wave length). To a first order of approximation (Airy theory), the wave celerity is given by


A 2 n ) 2~d C 2 = / g L ' + ~ tanh L--- 7-

~2~ p L ' (53)

where c - L'/T and A = surface tension (74 dynes/cm for air-water interface at 20~ If L is the wave length for A = 0, then

where AL is the scale effect on wavelength due to capillary effect. For small 2nd/L',

AT. 1.55 • 10 -3 -- ( 5 5 )

L dT 2

where d is in cm. If 2red is large L

AL T 8

L (0.105) 4 (56)

AT. AL LeM6haut6 [5] has plotted versus T and d. The value of .....

L L 0.01 when T > 0.5s and d > 20 mm (0.79 in.).

is smaller than

Density Effects. In a wave tank almost invariably fresh water is used to represent the sea water found in a prototype application. This creates a small difference in the density, which is about 3%. This difference reflects a similar change in the measured forces which should be corrected.

Let us consider the example of a rockfill breakwater. Considering stability, the minimum weight of rocks or concrete blocks may be approximately given the Hudson Formula [5] by

W = Pb H3

(Pb--Ps ] 3 Ps (57)

where PD = density of block Ps = density of sea water

For similitude, the model densities must be related to the prototype densities by the relation:


Pb Pm

Pb-Ps Pm - P

P~ P

(58)

where Pm = model block density Assuming that the block densities between the model and prototype are the same (Pm = Pb)' the ratio of the prototype block weight to the model block weight is obtained as

W m Wp = (1.10 to 1.15) A, 3 (59)

Thus, the error resulting from the use of the fresh water is about 10-15%.

Model Testing Facility

The model testing facility [9] for both coastal and offshore model testing should consist of the following capabilities: model building, instrumentation, simulation of environment, and the software to record and analyze data. The physical facility should consist of a basin with the capability of generating waves, wind, and current. An efficient wave absorption system is also essential in a basin [ 10]. The simultaneous generation of waves and current allows study of their combined interaction with the model. The wind effect is simulated on the superstructure of the model (the portion above the water) and is often accomplished using a series of blowers located just above the water surface near the model.

The earlier wave tanks built prior to 1980 only produce waves that travel in one direction. These are suitable for reproducing long-period ocean waves that are unidirectional. Wind-generated multidirectional waves require facilities that can generate multidirectional waves. These facilities have widths comparable to their lengths. Many modem facilities have this capability [ 11 ].

Coastal Modeling

For coastal structure modeling shallow basins are required. In most cases, multidirectional wave capability is desirable. The models, such as breakwaters and jetties, require a large area. Because large scale factors are generally used to model the coastal structures, only small waves are required. The waves generated undergo considerable refraction and diffraction from the structure model.

One such facility is located at the U.S. Army Corps of Engineers Waterways Experiment Station (WES) in Vicksburg, Mississippi. The directional spectral wave generator (DSWG) of WES Coastal and Hydraulic Laboratory is capable


of producing natural sea states and is particularly suitable for coastal model studies [12, 13]. It is the only facility of its kind and size in the United States that produces waves in a coastal environment. The DSWG basin is 29.3m (96 ft) long and 34.7 m (114 ft) wide. The water depth in the basin is variable up to a maximum of 0.6 m (2 ft). For the wave absorbing system, portable metal frames at a 37 ~ slope are installed along the basin perimeter (Figure 3). Wave absorption and energy dissipation are provided by two layers of 50.8-mm (2-in) horse- hair sandwiched between two layers of expanded metal. The DSWG consists of 60 portable paddles in 4 modules, each 0.76 m (2.5 ft) high and 0.46 m (1.5 ft) wide. It is a wet-back design [22] with no bottom or end seals which permits water to reside behind the paddles. The paddles operate in a piston (translational) mode for a maximum stroke of _+ 152.4 mm (6 in). Directional waves are generated using the snake principle. The wave height ranges from 25.4-45.2 mm (1 to 7 in.) while the wave period varies from 0.75 s-4.0 s.

Another multi-directional wave basin performing coastal modeling is the Hydraulic Laboratory at the National Research Council of Canada, Ottawa [ 14]. The basin, which is 50 m (164 ft) long, 30 m (98.4 ft) wide, and 3 m (9.8 ft) deep

Figure 3. Waterways Experiment Station 3-D shallow water wave basin.


Figure 4. National Research Council, Canada 3-D wave basin.

(Figure 4), is equipped with a segmented wave generator capable of producing multi-directional seas. The segmented wave generator occupies one end of the basin and perforated layers of metal sheeting acting as wave absorbers are located at the other end as well as along the sides (Figure 5). The perforations and spacing of the metal sheeting may be changed to minimize the wave reflections.

The wave generator consists of 60 segments or wave boards driven individually by a servo-controlled hydraulic system. The individual wave boards are 2.0 m (6.6 ft) high and 0.5 m (1.6 ft) wide, driven by Moog hydraulic actuators with a maximum stroke of _ 0.1 m (0.33 ft). The displacement of the actuator is mechanically amplified by a factor of 4 through a lever arm. The boards operate in the piston or translational mode (for shallow-water waves) or flapper or rotational mode, or a combination of the two (for deep-water waves). The machine is also vertically movable to accommodate different water depths.

Offshore Modeling

For offshore structure modeling, a two-dimensional wave basin with a mechanical wavemaker is often used. There are two main classes of mechanical type wavemakers. One of them moves horizontally in the direction of wave propagation and has the shape of a flat plate driven as a flapper or a piston. The other type moves vertically at the water surface and has the shape of a wedge. In deeper water, a double flapper is often used. A double flapper wavemaker consists of two pivoted flappers, an actuation system driven hydraulically and a

Model ing Laws in Ocean Engineering 3 1 5

i i i j

SEGI~NTED WAVE GENERATOR I I I I, i" I I i, i I ' l I I' I I I i I I .... i": i I I"':I I I f I '7

t REMOVABLE I 51DE WALL l

N ~' / 3e . k ~ _

I I J l .

f GATE

Figure 5. Plan v iew of NRC wave basin showing wave absorbers.

control system (Figure 6). For a flapper type wavemaker, the backside (outside the basin) may be wet or dry. Both have advantages and disadvantages, which are considered in the design of such a system. The dry-back system appears to be more popular.

The wave basin sometimes has a false bottom, which is adjusted to obtain the required scaled water depth. In this way a facility may be made suitable for both deep- and shallow-water testing. The MARINTEK facility at Trondheim, Nor- way, has a 10 m (32.8 ft) deep basin, an adjustable false bottom, and multi- directional wave generation capability.

Another multi-directional wave basin is located at Texas A&M University, College Station. The Offshore Technology Research Center (OTRC) facility at Texas A&M University is suitable for testing of offshore structures. The dimensions of the facility are 45.7 m (150 ft) long, 30.5 m (100 ft) wide and 5.8 m (19 ft) deep. It has a deep pit in the middle which has an overall depth of 16.8 m (55 ft). The basin is equipped with a hydraulically-driven hinged flapper. The beach has a design similar to the NRC facility consisting of the progressive expanded metal panels. The period range of the waves generated in the basin is 0.5 to 4.0 s and the maximum wave height is about 0.86 m (2.83 ft).

Model Environment

In model testing, the environment experienced by the structure should be properly simulated in the laboratory. Two of the major environmental parameters required in coastal and offshore testing are waves and wind.


i 2.~, i r--"

TOP

HII~

I I .

CONCRETE PEOESIAL

HOOEI NOII2 N00E3

J

Figure 6. Double flapper wave generating system (1 ft = 0.3 m).

M o d e l i n g o f W a v e s

Modeling of regular waves is straightforward. The regular waves are given in terms of a wave height and a wave period. These quantities are properly reduced to the model scale by the selected scale factor. The waves of the given height are generated by the harmonic oscillation of the wavemaker at the required amplitude.

There are several methods available to generate irregular or random waves. Waves are often generated in the model basin to simulate one of many energy spectrum models proposed to represent sea waves. For the generation of waves, a digital input signal is computed from the target spectrum, considering the transfer function for the wavemaker. The transfer function generally accounts for the relationship between the mechanical displacement of the wavemaker to the water displacement, and the hydraulic servo control system.

Unid irec t iona l Waves . Two of the most common methods of wave generation [15] in the basin are the Random Phase Method and the Random Coeffi- cient Method. The former is spectrally deterministic, while the latter is non- deterministic.

The sea surface is generally assumed to be Gaussian with a zero mean. Simu- lation of the sea surface [16, 17] usually consists of a finite number of Fourier components as a function of time.

N

rl(t) = X an c~ (2x fnt + En) (60) n = l


where rl(t) is the generated surface profile having the energy density of a specified spectral model. The quantities a n and fn are the amplitudes and frequencies of the wave components and are obtained as follows.

The spectral model is subdivided into N equal frequency increments as shown in Figure 7 having width Af over the range of frequencies between the lower and upper end of the frequency spectrum, fl and f2" The Fourier amplitude a (nAf) is obtained from the spectrum density S (nAf) as

a n = a(nAf) = ~/2S(nAf)Af, n = 1,2 ..... N. (61)

The frequency fn is chosen as the center frequency of the nth band width in the spectral model. The corresponding phase E n is created from a random number generator with a uniform probability distribution between - ~ and +r~. The quantity fn is sometimes chosen arbitrarily within the nth band width to provide further randomness.

The duration of the random wave time history record should be sufficiently large for data analysis, but should have a minimum of at least 200 cycles for statistical stability in data analysis. Because of the method of generation of the

IF)

o X I'~1.

U

0 O - re

* - ' 0 XE)

j= 0

~o

,j 0 . 0 5

' I ~ ' I 6 o.3 o.6 o:9 , r o d / s e e

H s = 20 FT

' i ' i i o.,o f, o.~5 o'.2o o:2s f , HZ

'i'.z ~'.5

Figure 7. Decomposition of energy spectral density into time domain simulation (1 ft = 0.3 m).

N/2

time series, when inverted by Fourier transform it yields a close match to the target spectrum. An example of a Pierson-Moskowitz spectral model is shown in Figure 8.

The above method does not produce random amplitudes of the wave components. In the Random Coefficient Method, the sea surface profile is obtained from

n = l

rl(t) = ~ ( b n cos 2/1;fnt + c n sin 27ffnt) (62)

1 . 0

where the coefficients b n and c n are considered to be independent random variables that have Gaussian distribution with a common variance

2 S(fn)A (3" n =

The coefficients b n and c n have a joint probability density function [ 18].

1 -(b2n +Cn) P(bn,cn) = 2no/" exp 2o2

(63)

(64)

0 . 8

e~

>:

z

r162

r

~ o 6


0 . 4

0 . 2 - -

,e/•,X

r

0 . 0 I 0.4 0 . 6 0 . 8 1 . 0 1 2 1 . 4 1 6 1 . 8 2 . 0

FREQUENCY, HZ

Figure 8. Pierson-Moskowitz spectral model from a wave tank simulation time history (1 in = 2.54 cm).


Writing b n = a n cos E n and c n = a n sin e n, it can be shown that a n has a Rayleigh distribution.

Multi-directional Waves. In the case of a directional sea, the directional spreading function will be specified in addition to the specified energy density spectrum [ 1 5, 1 9, 20]. Then, generalizing Equation 60 we have

N

r l (x ,y , t )= E an cos[kn (XCOS0 n + y s i n 0 n ) - 0 3 n t+an] n=l

(65)

in which the amplitude a n now includes the spreading function:

a n = ~ / 2 5 ( 0 3 n ) D ( 0 3 n , 0 n ) A 0 3 A 0 (66)

where 03n = 2~fn and the directional spectrum is obtained as

S (03, 0) = S (03) D (03, 0) (67)

The common form of D(03, 0) is given as

D(03,0)=C (s)cos 2s ( 0 - 0 o) (68)

where 0 o is the principal direction of wave and s is the spreading index. The coefficient C(s) is defined as

F ( s + l ) C(s)= F (s + 1 / 2)-~-~ ' s= 1,2 .... (69)

where F is the gamma function. The most common value in use is s = 2. A plot of D(03, 0) for different values of s is given in Figure 9.

Modeling of Wind

The wind effect on a coastal or offshore structure, if it is deemed important, is generated with blowers positioned strategically in front of the model. In this case, the model superstructure must be accurately modeled. The wind loads on the structure may be particularly important in the design of such structures as a floating moored structure. However, it should be emphasized that these loads in the model system are limited by the associated scaling problems. Wind loads are functions of Reynolds number and R is (an order of magnitude) smaller in the model compared to the prototype R. Therefore, it is possible that the prototype wind effect falls in the turbulent region while the corresponding model wind effect is in a laminar region. In this case, the model test results may be considered conservative. While wind velocity is often taken as a steady value, the wind

320 Of fshore Eng ineer ing

0

1.0

O. 75

0 . 5

O. 25

I

!

!

/ 11 ,=~

# # ~

S

I

2

- 5

I 0

- 180 - 9 0 0 9 0

t

t

t

t !

�9 i

1 8 0

0 d e 9 .

F i g u r e g. S p r e a d i n g funct ion for d i rect iona l sea .

spectrum may be important in some applications. The frequency range of a wind spectrum is quite broad-banded, often covering a range from 0.005 to 1 Hz [ 18].

M o d e l i n g of Current

The modeling of current in a laboratory test with or without waves is an important consideration in coastal and offshore modeling problems. In current modeling in a facility, the uniformity and distribution of current should be carefully investigated. The generation of current is simplified if a closed loop is placed in the facility. It is often achieved by pumping water into and out of the two ends of the tank by a piping system. If a false bottom exists in the facility, underwater pumps can circulate the water in a loop above and below the false floor. Counter-current is generated by reversing flow. This allows wave-current interaction study on the model. If an installed current generation is not available,


local currents are often generated by placing portable current generators in the basin. These may take the form of series of hoses with outside water source or portable electric outboard motors. Uniformity of flow is achieved by proper control of the velocity. Flow straighteners, such as a tube bundle, may be accommo- dated in the basin to stabilize the flow.

The uniformity and distribution with depth of the current profile is established at the test site by a series of current probe placed vertically. The temporal variation of current should be limited to 10% or less for a steady current test. It is advisable to establish the current profile in the facility before the model is placed in the tank. The current velocity required for the test will depend on the scale factor and is generally scaled with Froude scaling. If current is an important consideration in the testing, the scale factor should be chosen such that the available current can simulate the desired environment. If current is inadequate or unavailable, it is sometimes simulated by attaching the model to a towing car- riage and towing the model at steady speeds down the tank with or without waves. While towing does not duplicate the current effect exactly, it is generally considered acceptable.

Examples of Modeling

This section provides examples of modeling of various types of ocean structures. While the examples relate to specific structures, the methods employed may be applied to a large variety of ocean structures. The first section describes the common instruments used in measuring responses in these model tests.

Types of Measurements

In model testing, the simulated environment and the responses of the model structure to that environment are measured. Usually, the test environment is intended to scale a specified ocean environment. To verify that the sea state has been properly modeled in the laboratory test, measurements are made with the wave height gauge (e.g., resistance or capacitance wave probe) and current meters. These instruments are commercially available. The instruments are placed near the model, often across in line with the model to determine phase relationship between the model response and the environment experienced by the model.

Structure responses of interest might include loads and stability due to the presence of the environment on a bottom mounted structure, motions of a floating or moored structure and stresses on individual members or components of a structure. The interaction of waves with a structure may also be of importance in a design. For example, wave reflection or the runup of waves on the face of a structure can be an important consideration in the design of a breakwater or an offshore platform. Various specialty instruments [21, 22] are used in these mea-

3~ . Offshore Engineering

surements. These instruments are often specially designed to meet the requirements of the model. For example, load cells are designed to fit the model and its mounting system in the tank in the range of expected loads.

Instruments are electrically connected to an automatic data acquisition system (DAS) so that the transducer signal may be automatically recorded. A simple schematic of a data acquisition system is shown in Figure 10. The typical transducer signal is such that its output is given in microvolts. It is first amplified by a gain factor to a 0-5-volt or 0-10-volt limit. The signal is then conditioned, which may include analog filtering of noise and other unwanted signals and converted from the analog to digital form through an A/D converter. The digital signal is then stored into a computer memory for later viewing and analysis. Today these operations may be accomplished efficiently on a desktop personal computer.

Transducers receive a physical input from the test setup during testing such as displacement, acceleration, force, etc. for the model subjected to a model environment and produces an equivalent electrical output. The transducer is designed so that this transformation from the measured response to volts is in the linear range for the level of response expected, which allows a single scale factor for conversion of the output to the required engineering unit. A few common means of measuring an input signal include a bonded strain gauge, a linear variable differential transformer (LVDT), and a capacitance probe. These components are placed in a transducer stock, which is designed to measure an expected response in a model test.

For example, the strain gauge is glued strategically on a tension/compression member of a load cell designed for the desired load range. The load cell is attached between the model and the mounting system. As the model is subjected to waves, the load imposed by the wave on the model is recorded by the load cell. Before this placement, these instruments are placed on a specially designed calibration stand and calibrated over the range of expected values. For example, the load cell is fixed on the calibration stand and known weights are hung in the direction of measurement from the load cell in increments and the associated voltages are recorded. In case of a capacitance wave probe, the calibration is achieved by placing it submerged at the water surface and moving it up and down in water. The linearity of the instrument is verified and a scale factor in

RESPONSE TRANSDUCER AMPLIFIER

SIGNAL CONDITIONER

A~ CONVERTER

COMPUTER DATA BUS

Figure 10. Schematic of a data acquisition system.


terms of the response unit per volt is generated. This factor is used to multiply the voltage output during the testing in the DAS.

Coastal Structures

In many cases the modeling of coastal and offshore structures is quite similar. Many of the scale effects discussed earlier are equally applicable to both types of structures. In this section a scaling technique of a typical coastal structure is discussed. The structure considered is a rockfill breakwater. Most of the material here is taken from LeM6haut6 [8].

Let us consider a rough permeable structure such as a rockfill or rubble mound breakwater. Areas of interest for a breakwater are wave runup and overtopping as well as stability of armor units. There are two aspects of a breakwater of interest--one is the reflection of waves from it, while the other is the wave transmission through i t--which are discussed here. For a reflection coefficient, the size of the rocks and the void coefficient (permeability) of the first top layers are important. On the other hand, the composition of the core of the breakwater is important for the transmission coefficient.

For reflection coefficient, the size of the rocks of the first layers are scaled. If, in addition, the flow is fully turbulent in the model, then Froude similitude applies. The scaled rocks must also have the same shape and the same void coefficients. However, if a very small scale model is used, such that the flow is laminar, the reflection in the model will be larger and the transmission will be smaller than the scaled values, and the scaling to the prototype will not be valid.

For the similitude of energy dissipation on the structure,

Dp =~,KD m (70)

where D = rock diameter K = coefficient smaller than unity

This requires that the model rock diameter is increased compared to the scaled (by factor ~,) value.

The similitude of energy transmission requires calculations of flow through the breakwater. This is achieved if the void coefficient is invariant and the coefficient K is computed properly. But K is found to be a function of Reynolds number for the flow through the breakwater.

The similitude condition requires that the ratio of wave height to the average width of the core be the same,

p m

B = average width line (71)


According to the law of flow through porous medium:

H = C (R) u2

where R = Reynolds number ~ = void coefficient

(72)

Following Darcy law, for small Reynolds number, it reduces to uF (e) where

( l - e ) 2 1 F(~) = ~5 ~5

The quantity C is the dimensionless constant. Therefore,

o p m

Assuming Froude similitude,

p m

If it is assumed that ep = E m (same rock size distribution), then Fp = F m and

(73)

(74)

(75)

p m

This equality is satisfied if Equation 70 is satisfied. Reynolds number in this case is related by

R p = ~3/2 K R m (77)

From this, it can be determined that (Cp = KCm)

C ( R p ) = K C ( R p / ~ 3/2 K) (78)

Once C (R) is known, then K is known as a function of the Reynolds number for the prototype. It is convenient to replace the Reynolds number by a function of H/B, D p, and e. Multiplying by 2gD(D/v) 2 and eliminating R, Equation 78

expresses K as a function of ~, and --H D3Es. This is shown as a nomograph in B

Figure 11. The value of C(R) is obtained from experimental data.

The value of H is taken as the amplitude at the model in front of the breakwater and B is the width of the core of the breakwater. For fully turbulent flow, K =

Mode l ing Laws in Ocean Engineer ing 325

I000

I 0 0

~ . . . . . I

lo Z ~ " . . . .

. . . . . _ .

I0 -6 tO-l, 10-4 i0-]I IO-Z H

B

iO-a I0 o IOJ i01

Figure 11. Nomograph for the similitude of permeability taken from LeM6hau t6 [7].

1. For laminar flow in the prototype system, Froude similitude is valid as long as K = k-1/4.

For the similitude of wave reflection by the adjustment of roughness,

C(Rp) Up = KF m C(Rp/~3/2K) (79)

where F is proportional to the head loss coefficient. From experiments, it has been determined that

(D~ ]/3 (D~ ]/3 Fm k-d-jp = Fp k'-d'Jm

where d is a significant length, e.g., water depth. Because

(80)

dp - " ~ d m (81)

K is obtained as a function of ~, and the Reynolds number from

C(Rp) K a/3 C (Rp/~3/2K) (82)

For large values of R, C m = Cp and K = 1. Otherwise, for similitude K must be less than unity. Because C is not well known in this case, a nomograph is difficult to obtain.

If the breakwater is allowed to overtop, then the combination of overtopping and permeability makes it very difficult to determine the scale effect in a model test of this kind. The large variation of the permeability for a small variation of the void coefficient makes the similitude difficult in this case.

The effect of underpressure acting on the top layers can be reproduced in similitude only if the core material is increased in size following the rule of


similitude of permeability. Top layers are associated with wave breaking and corresponding turbulence. Here, the value of the coefficient K is near unity. However, for the core K is smaller than unity. Therefore, the model results may be considered conservative.

Fixed O f f s h o r e S t r u c t u r e s

Fixed offshore structures are often treated as rigid. However, for long slender structures the deflection of the top (near attachment to the deck) may be important in which case slenderness should be considered in modeling.

Vertical Piles. For coastal and offshore structures whose diameter is small compared to wavelength, the wave forces are computed by the Morison equation [23], which is written as the horizontal force on a unit length of vertical pile of diameter D as

f = p C m A fi + 0.5 p CD Dlulu (83)

where A =/I;D2/4 p = mass density of fluid u = particle velocity fi = acceleration due to waves at a point where force f is computed

C M, C D = dimensionless constants

Unfortunately, in the usual sense of dimensional analysis, these constants are generally not the same in the model and prototype. They are functions of at least two nondimensional parameters. One of them is the Reynolds number defined in terms of the velocity amplitude in waves (R = uD/~). The other quantity is sometimes considered as the Iverson's number given by

flD I v = u2 (84)

However, for linear waves, fi= r in which case"

2r~D 2rt I v . . . . (85)

uT K

where K = Keulegan-Carpenter number. This last nondimensional number is satisfied in a Froude model. Because both K and R cannot be satisfied in a model, for all practical purposes, wave forces on small piles cannot be studied accurately on small scale models [24] unless the ratio for the inertia force to drag force (Iverson number) is large. This is true for large piles where the ratio of diameter D to length L exceeds 0.5 (D/L > 0.5). In this latter case, small scale models will

reproduce the prototype effect very well. On the other hand, for small values of D/L, the larger the scale model, the better the results, so that the Reynolds number created in the model places it near the turbulent region, away from the laminar flow.

Jacket Structures. The use of scaling laws in the analysis and design of an offshore structure is common [25]. Here a simple scaling law is developed for fixed offshore structures assuming a rigid deck and an elastic structure (Figure 12) having a linear structural response [26]. The scaling laws are derived based on dimensional analysis.

From physical considerations, the displacement of the deck of the structure U is related to the following dimensional variables"

U = U(m s, m, 9, g, H, co, E, L) (86)

where m, m s = mass of the deck and support structure, respectively E = elastic modulus of the structural members L = structure length

(87)

The dimensional analysis will give

U = f ( H r pco2HL3 mco 2 msco2 /

L L ' g EL 2 EL EL

RIOID DECK


i11111111111111 l /1/1/f/il I lll1711-lllflll I I I

U II

Figure 12. Modeling of an elastic jacket structure with a rigid deck from Dawson [26].


Thus, the nondimensional displacement on the left side will remain invariant under change of scale as long as all quantities on the right side are equally invariant. The first term on the right side represents a geometric ratio in which the wave height varies with L (both varying as ~,). The second term is the ratio of wave acceleration to the gravitational acceleration. The scaling is satisfied by Froude modeling if

tOp -- ~ r n (88)

The third represents the ratio of wave to elastic force. This ratio is satisfied only if

Ep = ~E m (89)

The last two terms represent the ratio of both the deck-inertia and the structure- inertia forces, respectively, to the elastic forces. These ratios are satisfied if

mp = ~3mm;msp = ~,3msm (90)

The deck-inertia ratio can be easily satisfied by a suitable determination of deck mass. However, it cannot be met, in general, for the structure-inertia ratio because its mass is proportional to the structure density Ps and the characteristic volume L 3 of its members. Suitable choice of the model material is needed to satisfy Equation 90.

Floating Offshore Structures

In addition to the geometric similarity, a floating offshore structure model must satisfy the dynamic properties of the prototype [27]. These include the location of the center of gravity, mass moments of inertia or radii of gyration. An example of a tanker model attached to a buoyant tower is shown in Figure 13. For a mooring system [28], the geometry and weight density of the cable and chain must also be modeled.

In many floating structure tests, the Reynolds effect is important in addition to the Froude scaling effect. Resistance testing is one area of model testing where the Reynolds effect cannot be ignored. The Reynolds effect is corrected in this case in the following manner (see Chakrabarti [9] for details). The total towing resistance in model scale is established from test results. This measured resistance consists of frictional resistance and wave-making resistance. The model frictional resistance is computed using available frictional drag coefficient versus the Reynolds number curve (see Chakrabarti [22] for details) corresponding to the model Reynolds number. This value is subtracted from the measured total resistance. The balance represents the wave-making resistance that follows the Froude law of similitude. The resistance value is scaled up to the prototype value (usually by the factor ~3 for forces). The frictional resistance for the proto-


Figure 13. Modeling of tower-tanker system in a wave tank.

type is likewise computed from the Schoerner curve corresponding to the prototype Reynolds number. The total resistance is obtained by the addition of these two quantities. Similar corrections are necessary for a semisubmersible in transit (or in station-keeping in current). In this case, additional corrections are necessary for the submerged columns and pontoons. These members are generally circular in cross-section, for which steady drag coefficients are known (as functions of R).

Prediction of the prototype translational and rotational motions of a floating structure based upon model test results are expected to be quite reliable. These motions are primarily influenced by the dynamic characteristics of the platform in waves that satisfactorily follow Froude similitude. The only area where such similitude is questioned is near resonance if nonlinear damping is present in the system.


H y d r o e l a s t i c i t y

In traditional model testing, the model is considered a rigid structure and the model deformation and associated interaction with waves is ignored. While this approach is generally acceptable, for structures in deep water or for mooting systems in conjunction with a floating structure such simplification is not acceptable. This coupling of external load with structure response is termed hydroelasticity.

In these cases, an elastic model that properly considers the scale effects is designed and constructed. Sometimes it is not possible to build an elastic model due to lack of suitable material, scaling problem, etc. Often in these cases, a segmented model can be built, where individual segments are properly modeled and the segments are then hinged together with rigid intermediate sections. The elasticity (such as in bending) is imposed at the hinges. The number and stiffness of these hinges are chosen to provide the scaled mode shapes of the model at its scaled natural frequencies.

Cable Modeling. Modeling very long cables in a laboratory facility and maintaining experimentally realistic diameters, requires combining the use of proper elastic materials, the role of the drag coefficient in conjunction with buoyant devices, and increased kinematic viscosity of the test fluid [29]. If the two-dimensional differential equations of motion of a cable in a flowing fluid are properly nondimensionalized, the coefficients of the equations become the basis for determining the nondimensional parameters for dynamic similarity requirement. Following this procedure, the five nondimensional parameters determine the scaling law for a cable system [29]:

p T CDL E u 2 , ~ , , ~ (91)

Pc mgL D pcgL 'gL

where Pc = density of cable material T = cable tension m = cable mass per unit length L = stretched cable length D = cable diameter E = cable modulus of elasticity

The first term is the density ratio, the second term is the nondimensional cable tension, the third term is the hydrodynamic force, the fourth term gives the Cauchy similarity, while the last term is the Froude number. The Reynolds number based on cable diameter is involved indirectly with the drag coefficient, C D.

The requirements for scaling a large cable structure in a laboratory are governed by its length (Lp = ~Lm). Once the length scale is chosen, the flow velocity


is determined from the Froude number. In addition, the density ratio is fixed, which determines the modulus of elasticity for the model cable (Ep = ~Em).

Let us consider a prototype anchored cable where

Assume

L = 1,220 m D = 18.5 mm Pc = 2.94 g/cm 3 E = 3.07 x 104 MPa u = 0.257 m/s (0.5 knot) v = 1.3 x 10 -6 m2/s

C o = 1.5 = 1,000.

The model parameters are derived from the previous nondimensional quantities assuming their equality between the model and the prototype. The results are summarized in Table 2 for three different fluid viscosities assuming the same free stream velocity. Materials, such as plasticized polyvinylchloride (PVC), can be used in the model cable to provide the required modulus of elasticity. The proper density for the material may be achieved by impregnation of powdered lead. The diameter of the model is determined from the appropriate nondimensional quantity by making proper adjustment of the drag coefficient based on the Reynolds number.

Model 1 gives the required size assuming that the fluid properties between the model and prototype are the same. It can be seen that both the diameter and cable tension at its upper end are extremely small and cannot be used in a model test. Changing the scale to ~, = 500 does not improve this situation much. On the other hand, if the viscosity is increased by a factor of 10, the diameter and tension are significantly increased. This improvement may be achieved by either adding a polymer or using a different fluid, such as glycerin, in the model. Increasing the viscosity in the model by a factor of 100 (model 3) produces acceptable diameter and tension.

Table 2 Prototype and Model Parameters for a Riser in a

Free Stream Velocity of 0.5 Knot [29]

L (m)

U v E D (m/s) (m2/s) (MPa) (mm) R

T C D (mN)

Prototype 1,220 0.257 Model 1 1.22 0.0081 Model 2 1.22 0.0081 Model 3 1.22 0.0081

1.3 x 10 -6 3.07 X 10 4 18.08 1.3 x 10-6 30.7 0.154 13 x 10-6 30.7 0.375

130 x 10-6 30.7 1.009

3,571 1 .50 11.1 x 10 6

0.971 12.8 0.81 0.236 31.1 4.78 0.064 83.7 34.6


Notation

A cross-sectional area a wave amplitude

B average width of porous breakwater core C Cauchy number

C D drag coefficient C M inertia coefficient

D diameter D(.) directional spectrum function

d water depth E Euler Number E Young's modulus e membrane thickness F force

F e elastic force F Froude number f force per unit length f wave frequency

fe eddy-shedding frequency g gravitational acceleration H wave height I Iverson number I sectional moment of inertia

K Keulegan-Carpenter Number L wavelength

[L] unit of length e characteristic length

M w hydrodynamic moment M moment

[M] unit of mass m mass

m s mass of structure p pressure

p (.) probability density R radius of object or hydraulic radius R Reynolds Number S energy spectral density S Strouhal Number s wave spreading function

T cable tension or wave period [T] unit of length

t time


U displacement U Ursell Number u flow velocity fi flow acceleration

W Weber Number W weight

x horizontal coordinate y vertical coordinate

Af frequency increment AR thickness of annulus

~5 distortion factor e phase angle F gamma function

specific weight of fluid 11 wave profile

angle 0 wave direction ~, scale factor tx dynamic viscosity n 3.1416 p mass density of water

Pb density of armor block Pc density of cable material Po mass density of oil Ps mass density of sea water

variance a9 kinematic viscosity of water o~ 2nf

subscripts n wave component

m model p prototype

References

1. Kroon, R. P., 1971. "Dimensions," J. of Franklin Institute, vol. 292, no. 1, July, pp. 45-55.

2. Zahm, A. F., 1928. "Theories of Flow Similitude," Report of National Advisory Committee for Aeronautics, Report #287, Washington, D.C., May, pp. 187-194.

3. Doodson, A. T., 1949. "Tide Models," Dock and Harbour Authority, vol. XXIX, no. 339, Jan., p. 223.


4. Popov, S. G., 1968. "Dependence Between the Strouhal and Reynolds Numbers in Two-Dimensional Flow Past a Circular Cylinder," NASA Technical Translation, NASA "IT F-11,763, NASA, Wash., D.C., July, 7 pages.

5. LeM6haut6, B., 1965. "On Froude Cauchy Similitude," Proc. on Specialty Confer- ence on Coastal Engineering, Santa Barbara, CA, ASCE, Oct.

6. Maxwell, W. H. C. and Weggel, J. R., 1969. "Surface Tension in Froude Models," J. of Hydraulics Division, ASCE, vol. 95, no. HY2, March, pp. 677-701.

7. LeM6haut6, B., 1976. "Similitude in Coastal Engineering," J. of the Waterways Harbors and Coastal Engineering Division, ASCE, vol. 102, no. WW3, Aug., pp. 317-335.

8. LeM6haut6, B., 1967. "Scale Effects Due to Waves on Short Structures," Tetra Tech Report No. TC 114, Pasadena, Calif., Mar., 48 pages.

9. International Towing Tank Conference Catalogue of Facilities, 1979. Sixteenth ITTC Information Committee, Annapolis, MD.

10. Ouellet, Y. and Datta, I., 1986. "A Survey of Wave Absorbers," J. of Hydraulic Research, IAHR, vol. 24, no. 4, pp. 265-280.

11. Johnson, B., 1981. "A State-of-the-Art Review of Irregular Wave Generation and Analysis," Proc. of Sixteenth International Towing Tank Conference, Leningrad, USSR, U.S. Naval Academy Report No. EW-9-82, Sept.

12. Briggs, M. J. and Hampton, M. L., 1987. "Directional Spectral Wave Generator Basin Response to Monochromatic Waves," Technical Report CERC-87-6, Coastal Engineering Research Center, Department of the Army, Vicksburg, MS.

13. Briggs, M. J., Borgman, L. E., and Outlaw, D. G., 1987. "Generation and Analysis of Directional Spectral Waves in a Laboratory Basin," Proc. of Offshore Technolo- gy Conference, OTC #5416, Houston, Texas.

14. Miles, M. D., Laurich, P. H., and Funke, E. R., 1986. "A Multimode Segmented Wave Generator for the NRC Hydraulics Laboratory," Proc. of Twenty-first Ameri- can Towing Tank Conference, Washington, D.C.

15. Kimua A. and Iwagaki, Y., 1976. "Random Wave Simulation in a Laboratory Wave Tank," Proc. of Fifteenth Coastal Engineering Conference, Honolulu, Hawaii, July, pp. 368-387.

16. Borgman, L. E., 1969. "Ocean Wave Simulation for Engineering Design," J. of the Waterways and Harbors Div., ASCE, vol. 95, no. WW'4, Nov., pp. 557-583.

17. Ploeg, J. and Funke, E. R., 1980. "A Survey of Random Wave Generation Tech- niques," Proc. of Seventeenth International Conference on Coastal Engineering, Sydney, Australia, pp. 135-153.

18. Chakrabarti, S. K., 1990. Nonlinear Methods in Offshore Engineering, Elsevier Publishers, Netherlands.

19. Comett, A. and Miles, M. D., 1990. "Simulation of Hurricane Seas in a Multidirec- tional Wave Basin," Int. Conf. on Offshore Mechanics and Arctic Engineering, Houston, Texas, Feb., pp. 17-25.

20. Gravesen, H., Frederiksen, E., and Kirkegaard, J., 1974. "Model Tests with Direct- ly Reproduced Nature Wave Trains," Proc. of Fourteenth Coastal Engineering Conference, Copenhagen, Denmark, June, pp. 372-385.

21. Goldstein, R. J., 1983. Fluid Mechanics Measurements, Hemisphere Publishing Corporation, Washington, D.C.


22. Chakrabarti, S. K., 1995. Offshore Structure Modeling, World Scientific, Singa- pore, 1995.

23. Morison, J. R., O'Brien, M. P., Johnson, J. W., and Schaaf, A. S., 1950. "The Force Exerted by Surface Waves on Piles," Petroleum Transactions, American Institute of Mining and Metal Engineering, vol. 4, pp. 11-22.

24. Chakrabarti, S. K., 1989. "Modeling of Offshore Structures" (Chapter 3, Applica- tion in Coastal Modeling, V. C. Lakhan and A. S. Trenhaile (eds.), Elsevier Oceanography Series 49.

25. Bhattacharya, S. K., 1984. "On the Application of Similitude to Installation Opera- tions of Offshore Steel Jackets," Applied Ocean Research, vol. 6, no. 4, pp. 221-226.

26. Dawson, T. H., 1976. "Scaling of Fixed Offshore Structures," Ocean Engineering, vol. 3, pp. 421-427.

27. Goodrich, G. J., 1969. "Proposed Standards of Seakeeping Experiments in Head and Following Seas," Proc. of Twelfth International Towing Tank Conference.

28. Pinkster, J. A. and Remery, G. F. M., 1975. "The Role of Model Tests in the Design of Single Point Mooting Terminals," Proc. of Seventh Annual Offshore Technology Conference, OTC #2212, Houston, Texas, pp. 679-702.

29. Parnell, L. A. and Hicks, J. C., 1976. "Scale Modeling of Large Elastic Undersea Cable Structures," ASME Winter Annual Meeting, New York, 76-WA/OCE-8, Dec., 8 pages.

30. Hansen, D. W., Chakrabarti, S. K., and Brogren, E. E., 1986. "Special Techniques in Wave Tank Testing of Large Offshore Models," presented at Marine Data Sys- tems International Symposium, New Orleans, Louisiana, April, pp. 223-231.

C H A P T E R 8

HYDRODYNAMICS OF OFFSHORE STRUCTURES

M. H. Kim

Ocean Engineering Program Civil Engineering Department

Texas A&M University College Station, Texas, USA

CONTENTS

INTRODUCTION, 336

WAVE LOADS ON SLENDER BODIES, 337

WAVE LOADS ON LARGE STRUCTURES, 349

Fully Nonlinear Wave-Body Interaction Problem, 350 Perturbation Approach for Weakly Nonlinear Free-Surface Problems, 351

MOTION ANALYSIS, 371

REFERENCES, 377

Introduction

As the world's energy need increases, more deepwater oil fields are being explored and developed. For example, Shell's Auger tension-leg platform (TLP) set a world record in water depth in 1994 and currently produces oil and gas at 870-m water depth. The challenge to produce oil in deeper water depth continues today. As an example, a new TLP and subsea production system are planned at 1,200-m and 1,500-m water depths in the Gulf of Mexico.

As water depth increases, the safety, structural integrity, mooring, and maintenance of a system become more and more difficult and challenging. Besides, the system has to be desirably functional and cost-effective. Therefore, many conventional platform concepts may not be appropriate for the ultra-deepwater development. In view of this, several novel compliant platforms have been proposed and their feasibility is being actively studied. One of them is the spar platform, a large deep-draft hollow vertical cylinder, which was successfully installed in the Gulf of Mexico in 1996. For the reliable design of those deepwa-

3,36

Hydrodynamics of Offshore Structures 337

ter compliant platforms, more accurate hydrodynamic loading and response predictions including nonlinear effects are required.

Most compliant platforms are designed so that their natural frequencies are far below or above typical wave frequencies to minimize wave-induced motions. However, the system is still subjected to the second-order (and higher-order) sum- and difference-frequency wave loads causing resonant high-frequency (springing) or low-frequency (slowly-varying) responses. In many cases, the slowly-varying horizontal responses of moored platforms can be much greater than wave-frequency motions mainly due to the small damping at such low frequencies, which implies that nonlinear dynamic analyses have to be used for the reliable design of those structures.

This chapter summarizes linear and nonlinear wave-body interaction theories and motion analyses, and describes the most important up-to-date information pertaining to nonlinear wave force and motion analyses and computations for slender and large-volume structures. However, it is almost impossible to include all the pertinent subjects in this single chapter; therefore, more emphasis was put on the recent development of nonlinear slender-body or diffraction/radiation theories, which are rarely covered in existing textbooks. However, for the self-containment, relevant fundamental subjects are also included. For more basic and broader topics of hydrodynamics of offshore structures, readers are directed to [7, 19, 55, 68].

Wave Loads on Slender Bodies

When the size of a body is small compared to predominant wavelengths, say diameter to wavelength ratio D/L < 0.2, it can be assumed that the incident wave field is not significantly deformed by the presence of the body. In this case, wave loads can be calculated from the Morison equation [51 ], where the total force is simply given by the linear combination of drag and inertia forces. To calculate such forces, only the kinematics of incident waves are required. Although there exists no rigorous theoretical basis for the Morison's formula, it has been widely used and has produced reasonable results for numerous ocean engineering problems in the absence of equally simple and effective alternative approximate methods.

According to the Morison equation, wave forces on slender members consist of inertia and drag forces:

C I pV dv 1 =

dt 2 (1)

where Cp C o = inertia and drag coefficients, respectively p, S, ~' = fluid density, frontal area, and displaced volume of a structure

= (u, v, w), wave particle velocities d/dt = total derivative


The inertia and drag coefficients C I and C D are functions of Keulegan-Carpenter number (KC), Reynolds number (R), and surface roughness. For low KC numbers, potential theory can be used, and then C I = 1 + C M, where C u represents added-mass coefficient (added mass normalized by displaced mass). A vast library of experimental data on C I and C o for a variety of geometries is available from numerous laboratory and field tests [7, 68], which allows a designer to choose appropriate values. However, only limited information is available for the cases of small KC number and large R, which is of practical importance. More research is

Iner t ia C o e f f i c i e n t

3.0

2.5

2 . 0 -

o- 1.5

1 . 0 -

0.5

0.0 ' I ' i ' I ' I ' I '

0.0 10.0 20.0 30.0 40.0 50.0 60.0

KC

D r a g C o e f f i c i e n t

2.5

2.0

~-- =.5

1.0

0.5

0 . 0 ' , ' , ' I ' , ' , " ' i

0.0 10.0 20.0 30.0 40.0 50.0 60.0

KC

Figure 1. Inertia and drag coefficients vs. KC for a smooth circular cylinder in waves (from Chakrabarti, 1980). Shaded region represents the scatter of data.


needed in this area. Typical C x and C o values obtained for a smooth circular cylinder from the wave-tank test of Chakrabarti [8] are shown in Figure(s) 1. The range of Reynolds number for these figures was 20,0(g)-30,000 and the stream function wave theory and least square method were used to determine the force coefficients. On the other hand, the values of C I and C o for a circular cylinder recommended by API are 1.5-2.0 and 0.6-1.0, respectively.

In the following analysis, a Cartesian coordinate system with the origin on the mean free surface and z-axis positive upward is used. If Airy's linear wave theory is used for 9, we can write

dv /)~(1) gA cosh k(z + h) sin (kx- to t ) (2) ~(1) _. r o l l ) , d-~ = ~9---~-' a n d O~ 1) = (1) cosh kh

where g = gravitational acceleration A = wave amplitude h = water depth

The wave circular frequency and the wavenumber k satisfy the linear dispersion relation; co 2 = kg tanh kh. Then, the x-component of Equation 1 has the form:

F x = - F I sin O + F o cos O Icos el (3)

where O = kx - o t Considering the phase difference between inertia and drag forces, the maximum force by a regular wave can be obtained as follows:

F 2 Fma x = F D + (If F1 <1)or Fma x = F I (If FI > 1) (4)

4FD 2Fo 2FD

The relative importance of the inertia and drag forces can be judged by the magnitude of KC number, which is defined as

where VT/D

V = magnitude of wave particle velocity T = wave period D = characteristic length of a body (diameter in case of circular cylin-

ders)

The KC number can be interpreted as the ratio of the drag force to the inertia force. In other words, if the KC number is small (say less than 5), inertia forces are dominant over drag forces, whereas if KC number is large (say greater than 40) drag forces are more important. If the horizontal particle velocity of Airy's linear wave theory on the surface of deep water is used, V = Ao) (A = wave amplitude and co = wave circular frequency) and T = 2fifo, and therefore KC = 2rtA/D, which implies that drag forces become more important for higher waves.


The wave kinematics used in Equation 1 are usually calculated from an adequate wave theory in view of the fact that acceleration fields are difficult to measure. To obtain more accurate force results, the inertia and drag forces must be integrated up to the instantaneous free surface. It is well known that the wave kinematics above the mean sea level (MSL) can be greatly overestimated when linear wave theory is used for large-amplitude waves. To remedy this problem, numerous simple extrapolation or stretching techniques have been suggested. For instance, a linear extrapolation using the slope at MSL can give better prediction than the direct use of linear wave theory. As can be seen in Figure 2, this method gives reasonable results for regular waves when compared with measurement but may greatly overestimate the horizontal particle velocity of bichromatic or irregular waves. For irregular waves, better comparison with experiments can be made by using the Wheeler's stretching technique [71 ], where the vertical coordinate is modified so that the velocity at the crest is the same as that at the trough. For instance, the Wheeler's stretching technique uses the following expression for the horizontal wave particle velocity u:

cosh k(z + h) ( h ) gHk h +

u = cosO (5) 2to cosh kh

where rl(x, y, t) represents free-surface profile. Zhang et al. [74, 75] extensively investigated wave kinematics above MSL using the so-called "hybrid method," which includes rigorous nonlinear theory for long and short wave interaction. They showed that this method better predicts wave kinematics for a variety of sea conditions (see Figure 2), including broad-band sea spectra, than the other stretching techniques or wave theories, including Stokes high-order wave theory.

When slender members are not perpendicular to the wave direction, the wave particle velocity and acceleration are decomposed into normal and tangential components, and only the normal components are used for the Morison equation. According to Chakrabarti et al.'s [9] experiments, the C I values are found to be relatively unaffected by the angle of inclination, while the C D values seem to increase slightly with the angle.

When bodies move with ambient flows, the Morison's formula can be rewritten using relative velocities and accelerations:

F=CIpVdV-CMPV ~+ CDPS(~ ~)~-- (6> d--t 2 -

where ~, ~ = body velocity and acceleration vectors The second term (added mass term) can be combined into the mass inertia term to be the total inertia by virtual mass. Using the relative velocity, the drag force contributes to both excitation and damping. When currents exist, the current velocity can be vectorially superposed to the wave particle velocity. In this case, the wave frequency co should be changed to the encounter frequency

O' A

(a) Vewx~W pmme ~w~ c~ ~J'~176 o/ SWL

/ . i " J ~

/ , /

10-

o- ~ .to-

rn

.,1)

(b) Velodly Prcd~ under Trough

o /

~ / " " /

/ " /~

/,/~

...'._ / / - - - w h i r r s ~ r

�9 ,.,,.,......

m

1o 2o 30 4o 8o

Figure 2a. Horizontal velocity profiles for a regular wave (T = 1.0 s, H - 9.7 cm) under (a) crest and (b) trough.

Figure 2b. Horizontal velocity profiles for a dual component wave (T --- 0.6 and 1.8 s, H = 2.7 and 7.8 cm) under (a) crest and (b) trough.

o

~ -10

ul

/s" /~ o,,"

o~;O"" /.... ~

.,o / 7 "~

~

lO

SWL

- - -whee~ ~ r m ~ ----- urw,~ F.~mW~n - --Vertk~a F_xtra~a~n

- ! ; 0 ,o

Horizontal Velocity (crn/s)


0~ e = o~ + kU cos where U = current velocity

I] = wave heading

similar to the Doppler frequency shift. The use of the drag force model may not be fully justified when the frequency

of body oscillation is quite different from the frequency of ambient fluid motions. In view of this, alternative expressions other than Equation 6 have also been proposed. For example, the independent flow drag-force model that treats body and fluid velocities separately was suggested by Moe and Verley [49].

The drag force is nonlinear, and thus prohibits linear spectral analyses in the frequency domain, which are an order-of-magnitude more efficient in many applications. Therefore, it is quite useful to linearize the drag force. For a regular wave, we can use the equal-energy-dissipation-per-cycle principle and obtain the following equivalent linear drag coefficient:

- 81+ I V l+r I = c~ Vr and c e = - ~ r (7)

where V r = V - - ~. The same result can also be obtained by using an alternative least square method. Figure 3 compares the pitch response of an ALP (see Fig- ure 4) by using the equivalent linear drag coefficient with that obtained from the direct time-domain integration with the nonlinear drag. It is seen that c e produces quite reasonable results for two different wave amplitudes. From the same figure, it is clear that viscous drag forces have to be included when potential damping (or wave damping) is very small.

When random waves and uniform currents coexist, the following formulas can be used for the statistical linearization of drag forces:

~rl~r[ = Co + C~ Vr

where ~r = ~ + U - ~ (8)

2 ] t 2 Co =OVr[ (1- )(2Z(y) - 1) + 2yz (y)] (9)

c e = 2OVr [ y ( 2 Z ( y ) - 1) + 2z(y)] (10)

where

- x 2 / 2 e

]( = U / ( Y V r , Z(X) = ~ , and

x

Z(x) = ~ z (t)dt (11)


0 . ~ _ I I �9 I i l ! i I I I I I I I I I_ II �9

II

0.15 - I . ,

is1\ i - I 1 \ ~ 0.10 - W

Z O

B

, / /

o.o o. t o.= o s 0.4 0 5 o l WAVE FltEQU]~NCY ( r e d / s )

Figure 3. Pitch response amplitudes of an ALP as functions of wave frequency: ( ) A = lm; ( - - - - - - - - - ) A = 3m; ( - - - - - ) Potential only. Marks are obtained from a time- domain computation including quadratic drag forces.

When there exists no current (U = 7 = 0), c e is reduced to the well known formu-

la; c e = Ov r ~/-8/rr,. On the other hand, the drag force can also be expressed as a sum of various har- monics using Fourier series, i.e.,

COS OICOS O I = ~ a n cos nO

n = 0

(12)

where the Fourier coefficients can be obtained f rom the integral

= -- cos O Icos O cos n O d O an ~;

After integration, it can be shown that a n = 0 for n = 0,2,4, ..., and

8 8 8 = = ~ , a s = - ~ , etc.

al - ~ ' a3 15re 105re

(13)

(14)

F rom Equat ion 14, it is seen that the nonlinear drag force produces higher harmonic forces at 3 o~ and 5 co etc., which are small compared to wave- f requency forces but may cause high-frequency resonant structural responses, such as the ringing of a TLP [17].


47.4m , ~ . , -

I ~ uoyaaey tank

_ . _ , . . = , . . . , .

75.2m

It 7.7m I

Figure 4. Articulated Loading Platform.

If KC number is small and the natural frequencies of a dynamic system are lower than predominant incident wave frequencies, such as articulated loading platforms (ALP) and spar platforms, it may be important to include the higher- order terms of the inertia force. Noting that the total derivative of Equation 6 consists of temporal and convective accelerations, the second-order horizontal inertia force on a slender body [35], for instance, can be written as

~U(2) C F~x 2) = (p V + m a) /gt + (p V + m a) U (1) ~u(1) + ~)(1) ~u(1) Ox Oy

W(1) ~U(1) / ~W (1) ma~(x 2) Oz ) + ma (u(1)- ~(xl)) 3"~"-- + (15)

in which m a is the sectional added mass in the x direction. By including more higher-order terms, a variety of higher-harmonic forces can be generated in a similar manner. The first contribution in the right side of Equation 15 is due to the second-order temporal acceleration ~)u(2)//)t, the second term results from the second-order convective acceleration, and the third term was called the axial


divergence term by Rainey [65]. Rainey derived Equation 15 based on the total energy and momentum of a system including both the structure and fluid. In Rainey [65], there exist additional contributions from the ends of a structure called free-surface and immersed-end point loads. The free-surface point load is third order, while point loads at the other end can be second-order and is given by -m a (u (1) -6x(1))(w (1) - 6z (1)) evaluated at the immersed end. The fourth term is due to the second-order body acceleration and can be combined into the body inertia term. Kim & Chen [35] used Equation 15 to compute the moment quadratic transfer function (QTF; the second-order excitation caused by dual waves of unit amplitude) for an ALP (see Figure 4) and compared the results with the corresponding results from the second-order diffraction theory [33]. The second- order diffraction theories are explained in the next section. Figures 5a and b show this comparison. It is seen that the nonlinear slender-body formulas produce reasonable QTF results compared to much more theoretically complicated and computationally intensive second-order diffraction method.

When time-domain approach is used, the nonlinear sectional forces must be integrated over the instantaneous wetted body surface. In this case, the effects of body motions and free-surface fluctuation can straightforwardly be included. However, if frequency-domain approach is preferred, the additional contributions due to the free-surface fluctuation and body motions have to be explicitly derived. For example, the integration of the first-order sectional inertia force (pV + ma)(~u(1)/~)t) and added mass force ma~(l) over the fluctuating part of the wetted surface produces the following second-order contribution, in which the integrand is assumed to be constant over the small region:

~(1) ~)u(1)

{(pS(z)+ m a (z)) t)t -ma~(xl)}dz=(PSw + mawlTl(1) ~)u(1)/)t

o

- ma ~ ' rl~l'lz=0 (16)

where S w and maw are the sectional area and added mass of waterplane. On the other hand, an additional sectional force caused by rotational body motions is given by

dFx = (P V + ma) gAk0(l/( k(z + h) c~ k(z + h) + sinh k(z + h) ) c o s h kh (17)

where 0 (1) is the inclination angle of a body in the plane of in-line forces. For illustration, the second-order slender body equations were used to calcu-

late the difference-frequency moment QTF spectrum in unidirectional irregular waves for the ALP shown in Figure 4 [35]. The results are compared with more



7

o

o

E-

Figure 5a. Amplitudes of the pitch-moment QTF, Fp, from the second-order diffraction theory (top) and slender-body approximation (bottom).


a.

t

~

< >

> X !

~e~ uer~C ~ 1 grad~5

I.Z

irad/s)

Figure 5b. Amplitudes of the pitch-moment QTF, F q, from the second-order diffraction theory (top) and slender-body approximation (bottom).



accurate second-order diffraction computation as shown in Figure 6. It is seen that reasonable slender-body results can be obtained in the low frequency range where the natural frequency of the ALP is located. A similar comparative study was also conducted for a spar platform and the results are summarized in Ran et al. [66, 67], Mekha et al. [48], and Cao and Zhang [4]. In these papers, respective numerical predictions are compared with the same experimental results. Because only the second-order incident wave field is required to derive the second-order force on a slender body, it is straightforward to derive similar second- order slender-body formulas for multi-directional waves [35].

When Morison equation is used for a moving slender body whose natural frequencies are small, the radiation damping (or wave damping or potential damping) due to the generation of outgoing waves is relatively unimportant compared to viscous damping. However, its contribution is expected to grow with frequency or body size. If necessary, the radiation damping can be calculated from the following Haskind relation [22] that relates radiation damping b to wave exciting forces F:

2/~

b i i = k 8~pgCgA 2 I IFi ([3)12d~ 0

(18)

~

6 of"" ( 9 "

1

Z , i (~ ~x~

N' o:i.,'

••slender body theory

/ i diffraction theory / ' ,, -/ , �9 .... , ...... , '>

~.e ~ 0.0 0.2 0.4 0.6 0.8

f r e q u e n c y ~ (rad./see.)

i

Figure 6. Difference-frequency second-order moment spectra for an ALP by second- order diffraction and slender-body theory. As an input spectrum, one-parameter Pier- son-Moskowitz spectrum with H s = 5.5 m was used.


where C g is group velocity and 13 is wave heading. The exciting force F(13) can be calculated from the Morison's inertia-force term. Figure 7 shows the comparison between the radiation damping calculated from Equation 18 and that calculated from the exact radiation theory.

Finally, the lift forces occurring in the direction perpendicular to the drag force due to an asymmetrically separated flow behind a blunt body can be calculated from the same formula as the drag force with the drag coefficient C o replaced by the lift coefficient C L. For uniform steady flows, the frequency f of this lateral force can be predicted using Strouhal number; S r = Df/V. In case of circular cylinder, the empirical formula S r = 0.23/Ct) is valid for a wide range of Reynolds numbers excluding the transitional regime. For smooth circular cylinders in unsteady oscillatory flows, the lift coefficient approaches a value of about 0.25 for relatively large values of Re and KC, i.e., Re > 105 and KC > 40. The predominant lift force frequency is a multiple of the wave (or oscillation) frequency and the multiplier increases with KC. Therefore, the lift force can also cause higher-harmonic responses of a TLP. At higher values of Re (>5 • 104), the lift-force frequency is also dependent on Re. More detailed information on C L and f can be found in [7].

Wave Loads on Large Structures

When the size of a body is not small compared to predominant wavelengths, say D/L > 0.2, the incident wave field is significantly deformed by the presence

t 0 . , f

.q...

EOoT. E.

,o u

J ~ / / t r . ..,o Z '~ y o.f

o.f,~

0.2 0.4 0.6 0.8 1.0 1.2 1.4

f requency ~ ( r ad . / sec . ) -o

Figure 7. First-order overtuming moment ( ) and radiation damping ( - - - - ) from Morison and Haskind equations. For comparison, marked lines are obtained from diffraction computation.


of the body. In this case, Morison equation cannot produce accurate results and the diffraction theory considering the effects of wave scattering around the body has to be used. In this case, KC number is usually very small and inertia forces dominate drag forces. Therefore, the use of potential-based diffraction theory can be justified in calculating wave loads.

Fully Nonlinear Wave-Body Interaction Problem

Assuming ideal fluid and neglecting surface-tension effects, the fluid motion can be described by the velocity potential ~(x, y, z, t), which satisfies the following governing equation and boundary conditions:

V 2 t I ) -- 0 inside the fluid (19)

Ot

0~ 1 Oa, u _ ~ + V~ V ~ + g q o O x 2

DFSC on the free surface (20)

0rl U + V ~ . Vrl = 0z KFSC on the free surface (21) 0t

0~ -- 0n = VB" fi on the body (22)

On = 0 on the impermeable bottom (23)

where U = forward speed of a body in the positive x direction (or equivalently, uniform current traveling in the negative x direction)

~)B = body velocity fi = (n x, ny, nz) = outward normal vector z = rl(x, y, t), the instantaneous free surface profile

In addition to the above boundary conditions, an appropriate open boundary condition has to be imposed on the far field boundary surrounding the computational domain. The most physically plausible open boundary condition is Som- merfeld/Orlanski outgoing wave condition [61 ], which is given by

+ C (t) = 0 on the far field (24) 0t

This radiation condition, for example, was used by Boo et al. [3] for the simulation of nonlinear regular and irregular waves in a numerical wave tank and by


Isaacson & Cheung [23] for wave-current-body interaction problems. There exist other open-boundary conditions, such as absorbing beaches by artificial damping on the free surface [2] or matching with linear time-domain solutions at the far field [ 14]. It is well known that the artificial damping on the free surface can damp out most wave energy if the length of damping zone is about one wave length, and therefore, it is more effective for short waves. When the matching technique is used, the matching boundary has to be located far away from a body to minimize the difference between the inner nonlinear solution and outer linear solution, which consequently increases the size of the computational domain. Clement [ 13] recently developed a hybrid method that uses both the artificial damping on the free surface effective for short waves and the piston absorber on the radiation boundary effective for long waves. The performance of a particular open-boundary closure can depend upon the type of boundary value problem and numerical methods to be used. Kim [25] and Clement [13] give more detailed discussions including pros and cons of various open-boundary conditions.

To solve the fully nonlinear boundary value problem at each time step, a BEM with mixed Eulerian and Lagrangian (MEL) scheme can be used, i.e., the linear field equation is solved in an Eulerian frame by a Rankine-source-based integral equation; then the fully nonlinear boundary conditions are used to track individual Lagrangian points on the free surface. The dynamic and kinematic free surface conditions (DFSC and KFSC) and body-boundary condition must be satisfied on the time-varying free surface and moving body surface, which requires rediscretizing the computational domain at each time step. In other words, when BEM is used, the influence coefficient must be set up and the resulting full matrix should be inverted at each time step. As a result, the requi- site CPU time is substantial even with very efficient algorithms. In particular, the reduction of CPU time in solving the matrix equation becomes more important as the matrix size increases.

For a large matrix, the well designed iterative method is known to be much more efficient than conventional Gauss elimination methods [73]. The stability related to the time integration of the free surface is another numerical problem that has to be carefully examined. For instance, the saw-tooth instability on the free surface can be artificially removed by using an appropriate smoothing technique [44] or regriding technique [14]. Using the MEL method, two- or three- dimensional overturning breaking waves can also be simulated [44, 73].

Perturbation Approach for Weakly Nonlinear Free-Surface Problems

In view of the computational burden in solving the fully-nonlinear formulation (Equations 19-24), alternative perturbation approaches have been used assuming that wave and body motions are not excessive but weakly nonlinear. For exam-


ple, the velocity potential is expressed as a perturbation series and Taylor expansion is used for the free-surface and body-boundary conditions with respect to their respective mean positions. Then, the computational domain can be fixed for all time steps, which greatly reduces the overall computational time. By introducing two smallness parameters e - kA and ~5- U/~gD and collecting terms of respective orders, the boundary value problem can be rearranged as follows:

Double-Body Flow Problem at O(~5). The leading order potential at 0(8) represents the steady disturbance due to the forward movement of a body (or the presence of a body in uniform currents). Assuming small forward speed and neglecting ship-wave-like wave field of O(~52), the steady disturbance potential D �9 satisfies the Laplace equation, no flux condition at the sea floor, and the ensu- ing boundary conditions:

3 0 = 0 on the mean free surface SF(Z ---- 0) (25)

3z

B 3 0 --

= U. fi on the mean body surface S B (26) On

�9 -~ 0 at far field S c (27)

Due to the wall-like boundary condition on the free surface, the solution can be regarded as fluid motion by the forward movement of a double body (a body plus its image) in an unbounded fluid.

Linear Problem at O(e). The boundary value problem for the first-order unsteady wave potential �9 <1) can be obtained from Equations 19-24 by collecting terms up to O(e~5). Then, the potential satisfies Laplace equation in the fluid domain, no flux condition on the bottom and body surface, outgoing wave condition at S c, and the following kinematic and dynamic free-surface conditions on z = 0 :

~(I)(1) ()1~(1) t - U ~ - - ~ ~ I ~ ( 1 ) ~ ~I~(1) 4- ~ ~)1~(1) 4- I] (1) ~ 2 ~

3z 3t 3x 3x 3x 3y 3y ~ x 2 (28)

3(I )(1) 0(I )(1) 3 ~ 3(I )(1) 0 ~ 3(I )(1) 4- gl] (1)- U ~ = on S F

3t 3x 3x 3x 3y 3y (29)

When U = 0, we obtain the well-known classical linearized kinematic and dynamic free-surface conditions, i.e,. 30(1)/~z = Orl(1)/3t and grl ~1) - -()tI)(1)fi)t which can be combined into the following single condition:


()2(I)(I) ()(I)(I) Ot ------T- + g 0-----~ = 0 on SF

On the other hand, the body boundary condition for tl) (I) is given by

~)(i)(1) ~)1~(I) . = .

On at n + [(W V)(~(i) _ ((~(i) V) W] fi on S B

where Vr V(t~- Ux) a=_~+t~x~ ~ = (~x' Ey, ~z)' the translational body displacements ~ = (~x, ~y, ~z), the rotational body displacements

(30)

(31)

The symbol ~ represents the position vector of the body surface. When U = 0, Equation 31 is reduced to the zero-speed linearized body boundary condition:

oq(i:)(1)/o-}n = (~9(~(I)/0t). fi on S B (32)

The classical zero-speed linearized wave-body interaction problem has extensively been studied by numerous researchers. In the context of linear wave-body interaction theory, the velocity potential can be decomposed into incident, diffraction, and radiation potentials, i.e., ~ = OI + ~D + OR" The diffraction potential represents the scattering of incident waves around a fixed body, while radiation potential describes outgoing waves generated by the motion of a body in otherwise calm water. The sum of three velocity potentials then fully represents the interaction of incident waves with a floating body. For given incident waves, the body-boundary conditions of the diffraction and radiation potentials can be rewritten as

O ~ ) / O n = - O ~ l ) / O n and O~)/On=(O~(1)/Ot)-fi onS B (33)

By integrating the linear hydrodynamic pressure p(1) = _ p(/)O(1)//)t ) over the mean body position~ first-order wave exciting forces can be obtained as follows"

~(1) __ _ p ~ f ~((I)~ 1)+ (I~)(D 1))

Ot SB

ridS (34)

Similarly, by integrating the dynamic pressure caused by the normalized radiation potential ~i of unit body velocity in mode i (t)tpi/0n = n i on SB), the added mass and radiation damping (or wave damping or potential damping) of mode i and direction k can be obtained from the following integral:


gg mai k + ibik = - p J J tPinkdS

SB

(35)

In the context of linear wave-body interaction theory, the added mass and radiation damping have to satisfy the following identity called Kramers-Kronig relation:

o o if m i i (oo) = m i i (tO) + - - Ri i (t)sin cotdt CO

0

(36)

where Rii(t ) is the retardation function and it is the Fourier-cosine-transform pair of the radiation damping bii(0)):

o o 2f Rii ( t ) = - bii (r.o)coscotdo

0

(37)

Besides Haskind and Kramers-Kronig relations, other hydrodynamic relations between diffraction and radiation problems also exist as detailed in Mei [47]. These hydrodynamic relations can be used to check the accuracy of indepen- dently developed diffraction and radiation codes.

When linear theory is used, analytic diffraction and radiation solutions are available for several simple geometries. For example, the first-order velocity potential and horizontal wave force for a bottom-mounted vertical circular cylinder of radius a are given by the following explicit solutions [46]:

igA cosh k(z + h) ~(1) (r, O,z)= ~ 13n i n [Jn (kr)

co cosh kh n=0

1 Jn (ka)

- ~ H n (kr)] cos nO Hn(ka)

(38)

FxO) _ 4pgA tanh kh - k 2 H ~ ( k a )

(39)

where O(x, y, z, t) = Re{ ~(x, y, z)e -i~ } F(t) = Re { Fe -ic~ }

Jn' Hn = first-kind Bessel and Hankel functions of order n 130= 1 13n = 2 for (n > 1)

The first-kind Hankel function H n = Jn + iYn' where Yn is the second-kind Bessel function of order n. To approximately estimate wave loads on a deep


truncated vertical cylinder, the potential (Equation 38) can be integrated from the cylinder bottom to the waterline. This kind of approximation was used to calculate wave loads on the four columns of a TLP and the results were compared favorably with BEM solutions [26, 33]. Using the exact solution (Equa- tion 39), the frequency-dependent inertia coefficient for a bottom-mounted vertical circular cylinder can be derived as follows:

~: YI' (ka) CI = 4 (ka)21H~ (ka)12

(40)

This frequency-dependent inertia coefficient can be used with the Morison equation (Equation 1) to indirectly account for wave diffraction effects. For a small cylinder, i.e., ka << 1, the limiting value C I = 2 can be obtained from the small- argument asymptotics of Bessel and Hankel functions. Figure 8 plots the frequency-dependent inertia coefficient as function of ka. It decreases monotonical- ly with ka and the result agrees well with Chakrabarti & Tam's [5] experiments.

On the other hand, the surge radiation potential of a bottom-mounted vertical circular cylinder has the following expression [29, 30]:

_

U

m

I I I I 1 I I

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 8. Inertia coefficient for a vertical circular cylinder by theory (circle) and by experiment (square) from Chakrabarti & Tam (1975).


(r,0,z) = (4s inhkhcoshk(z + h) H 1 (kr) {Pl k(2kh + sinh 2kh) H~ (ka)

m~14sin~mhcOs~m(z+h) K l ( ~ m r ) / + , cos 0 lC m (2~Cmh +sin 2~: m h ) K 1 (~c m a)

(41)

where K 1 = second-kind modified Bessel function of order 1 ~m = eigenvalues of the equation; -o~2/g = ~ tanrda

The second term results from the evanescent modes which are related to local standing waves around a body. For a given accuracy, more evanescent modes need to be included as water depth increases. The series eventually becomes an integral for infinitely deep water.

The MacCamy & Fuchs' diffraction solution (Equation 38) and the radiation solution (Equation 41) for a single bottom-mounted vertical cylinder can be extended to multiple cylinders as detailed in Linton & Evans [39] and Kim [26, 35]. Explicit solutions are available both in diffraction and radiation problems. These analytic solutions are computationally very efficient and particularly useful when the number of columns is large. For illustration, Figure 9 shows a snapshot of diffracted wave field around the ISSC TLP calculated from the analytic solutions. The particulars of the ISSC TLP are summarized in Eatock-Taylor & Jefferys [15]. When multiple columns exist, wave elevation at some locations can be greatly amplified due to a strongly reinforcing interaction between the diffracted waves produced by each column. Therefore, for multi-column structures, this kind of dynamic swell-up should be carefully checked to provide sufficient air-gap (deck clearance). The interaction of multiple arbitrary three- dimensional bodies can be studied by a more general interaction theory.

To solve the diffraction/radiation of arbitrary three-dimensional bodies, discretization-based numerical methods have to be used. If Green theorem is applied for d~ (~) and Green function G, the following Fredholm second-kind inte-

" D gral equation for g)~) can be obtained:

4 r t c ~ ) + ~ ~ ~) ~G ~) ~1) d S = - ~ G 3 n dS

SB SB

(42)

where c = solid angle G = linear free-surface Green function [56, 70] satisfying the same gov-

erning equation and boundary conditions as d) (1) except the body- " D

boundary condition. If Rankine sources (or simple sources) are used as G, the integral domain has to cover the entire boundary of a computational domain. Thus, the simple-source-based BEM is not computationally efficient for linear free-surface problems.


# 2

# l

Figure 9. Free surface elevation TI/A around ISSC TLP for wave number ka = 0.8.

To solve the integral equation, the body surface can be discretized by many quadrilateral panels on which potential values are constants (constant panel method, CPM). Then, the integral equation can be reduced to a linear simultaneous equation for unknown velocity potentials on each panel and the resulting algebraic equation can be solved by an appropriate matrix solver. For example, Figure 10 shows a typical discretization of the surface of the ISSC TLP by 4048 constant panels [37]. Alternatively, the body surface can be discretized by high- order curved elements and the variation of the potential over the element can be described by high-order shape functions. This kind of approach is called higher- order boundary element method (HOBEM). In general, HOBEM requires more human effort in programming but it usually converges faster and is computationally more efficient than CPM. Using curved boundary elements, the possible leak of the body surface can also be eliminated. For instance, Liu et al. [40] used 152 9-node quadratic elements (see Figure 11) and quadratic shape functions to produce almost equivalent results of Korsmeyer et al. [37]. In particular, HOBEM is more robust than CPM in calculating waterline integrals or the first- and second-order spatial derivatives of the velocity potential on the body surface [43]. Figure 12 shows the comparison of the numerical and analytic first and second spatial derivatives of the velocity potential on the surface of a translating sphere with unit velocity in an unbounded fluid. The numerical solution was obtained by HOBEM.

There was a comparative study to examine the reliability and accuracy of various discretization-based computer programs developed by numerous institu- tions worldwide, in which first-order hydrodynamic loadings on the ISSC TLP were compared. The result is summarized in Eatock-Taylor & Jefferys [ 15]. The


Figure 10. ISSC TLP by 4048 constant panels.

!

m-7--,

X

Figure 11. ISSC TLP by 38 quadratic elements per quadrant.

computed results were widely scattered about the mean values and the comparative study typically illustrated that the correctness, accuracy, and convergence of panel programs have to be carefully checked before intensive usage. A similar comparative study was later conducted for the mean drift forces and slowly varying motions of a moored vessel and a deep draft floater, which showed even greater standard deviations. The result is reported in Faltinsen [20].

When the size of a floating body is very large, such as offshore airports, the elastic deformation can be significant and one must consider the hydrodynamic

Hydrodynamics of Offshore Structures 3 5 9

. 6 -

0 . 4 - -

~ 0 . 2 -

�9 ~ o . o

~ -0.2

-0.4 -

- 0 . 6 ~

~ . . n ..-.~" ~--i~.. .~

..... f ,i I I 1 I I I '

1030 1032 1034 1036 1038 1040 1042 1044 1046 1048

(a) Node number

1.5

1 .0 -

-~ o.o

-o.s N

-1.0

-1.5 . . . . . . . I i - i I I I 3 I

1030 1032 1034 1036 1038 1040 1042 1044 1046 1048

(b) Node number

Figure 12. First- and second-order derivatives of velocity potential on the body surface. Lines are analytic solutions and circles, squares, and triangles are numerical x, y, and z derivatives, respectively.

interaction by these additional elastic modes [41 ]. In many cases, the wet elastic deformation (wet mode) may not be known in advance, and the hydrodynamic interaction and the equations of rigid and elastic motions have to be solved simultaneously [21]. Otherwise, the radiation problem needs to be generalized including pre-estimated elastic modes [58].

When forward velocity (or current speed) is small but not zero, a more complicated boundary value problem including Equations 28, 29, and 31 must be solved. This small forward speed assumption means that only ring wave systems are generated and short ship-wave systems are disregarded. If current speed is small, the effects of flow separation are unimportant and the use of potential theory can be justified. As an indication of the applicability of the potential theory, it may be assumed that flow separation effects need not be considered when the


relevant KC number is less than about 3 [23]. Zhao et al. [76] experimentally proved that flow separation around a hemisphere does not occur when KC < 1 and current speed is less than the maximum wave induced velocity.

Recently, several numerical methods for the wave-current-body interaction problem have been developed mostly in the frequency domain. For example, Zhao et al. [76] used a matching technique in which numerical inner solutions satisfying Equations 28 and 29 are matched with simpler asymptotic outer solutions at the far-field matching boundary, while Nossen et al. [59] or Wu and Eatock-Taylor [72] used an integral equation method with the forward speed Green function satisfying the far-field free-surface condition. Analytic solutions for bottom-mounted vertical cylinders were also derived by Emmerhoff and Sclavounos [18]. Alternatively, the integral equation can be solved in the time domain using an appropriate time-integration scheme and the open boundary condition suitable for time-domain approach, such as Orlanski condition [23, 27].

As can be seen in Equations 28 and 31, the free-surface and body-surface conditions contain second-order spatial derivatives, which cannot be robustly computed when constant panel methods are directly used [38]. If a body surface is smooth, the integrals containing the second spatial derivatives may be transformed by Stokes' theorem to more complicated forms containing only the first- order spatial derivatives [59]. However, the use of this modified formula may not be fully justified for multiply-connected bodies having sharp comers, such as TLPs. Otherwise, higher-order boundary element methods need to be used. It is already mentioned that the HOBEM is more robust than conventional constant panel methods in calculating second-order spatial derivatives [43].

In Figure 13, the interaction of radiated waves generated by a heaving hemisphere with uniform currents is calculated from a time-domain higher-order boundary element method [27]. The top figure (o~ ~ = 0.83) shows a perfectly axi-symmetric radiated wave field when currents are absent. The bottom figure (F n = U/~/-~ = 0.1) shows that the symmetry of radiated waves is distorted due to the presence of the current. As can be seen in the figure, when wave and current travel in the same direction, the wave amplitudes decrease and wavelengths become longer. On the contrary, when they move in the opposite direction, wave amplitudes increase and wavelengths are shortened. Figure 14 shows the change of wave-frequency forces, weather-side wave run-up (rl/A), and second-order mean drift forces on a bottom-mounted vertical cylinder (radius = water depth) due to coplanar or adverse steady uniform currents. It is seen that wave run-up and drift forces are more sensitive to current speed than wave-frequency forces. The wave run-up can be increasedby more than 40% due to moderate currents, and therefore, this effect needs to be considered in determining minimum air gap. The drift force on a vertical cylinder increases in coplanar currents and decreases in adverse currents for all wave frequencies considered. The drift force can also be increased by about 50% in moderate current conditions, as shown in this

[ I . .


(a)

(b)

Figure 13. Three dimensional and contour plots of wave elevation for a half immersed sphere undergoing heave motion: (a) no current, (b) with current.

example. The change of mean drift forces with current speed near U = 0 can be used to estimate the wave drift damping in the following manner:

~(2) ~(2) 1 ~)~(2) [ +0(U2) = u=0 + U 3U u=0

(43)

The coefficient 3F(2~/OU proportional to the small forward velocity U is called wave drift damping. Faltinsen [19], for example, compared the time series of the free decay test of a moored TLP in calm water with that in waves. The decaying rate of slowly-varying surge motions in waves was found to be greater than that in calm water. The decaying rate had a tendency to be proportional to wave amplitude squared. The increment of damping in waves was attributed to the

3 6 2 Of fshore Eng ineer ing

3 . 5 '

<l:: 3 . 0 - (N

X

u . 2 . 5 -

. 0 . . . . . . .

0.2

~ . ~ 1 7 6

. . ~ . : : Fn=0.0 . . ~ - ' ,~'~_ - - - Fn = +0.1

..... t .... I I I " ' I I t

0.4 0.6 0.8 1.0 1.2 1.4 1,6

(a) ka

2 . 5 ] . . . . .

- - Fn = 0.0 | ........ Fn = + 0 . 1 . . . . . . . -

2 . 0 1 - - - - - F n = - 0 . 1 1 1 1 s "

2

1.0

0.5 0.4 0.6 0.8 1.0 1.2 1.4 1.6

(b) ka

1.8

2.0

o) 1.5 o

. ~

"0 1.0

t: 0.5

. F n = 0.0 ......... Fn = +0.04 - - - Fn = -0.04 . . . . Fn = +0.1

I ~ ---- - Fn = -0.1

. . . . . . . ' " . . . . . . . . . . . . . . . .

. . . . . ~ " ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . , . . , . . . . ~ . . . _ . . . . . . .

~ i ""

I . . _ , . _ �9 - �9 - , r , - �9

0.4 0,6 0.8 1.0 1.2 1.4 1.6

(c) ka F igure 14. (a) Wave exc i t i ng force, (b) run-up, (c) mean dr i f t f o rce for a b o t t o m - m o u n t e d ver t ica l cy l i nder Fx/pgaA2.

H y d r o d y n a m i c s of Of fshore S t ruc tu res 3 6 3

wave drift damping. The phenomenon is similar to the added resistance of a ship in waves. Clark et al. [12] suggested a very simple formula to calculate wave drift damping from only the zero-forward-speed mean-drift-force results. In their numerical examples with vertical cylinders, it is shown that the simplified formula, although it does not seem to be universally applicable, can produce the equivalent results of Emmerhoff & Sclavounos [ 18].

Figure 15 shows the wave drift damping of the ISSC TLP as function of dimensionless wavenumber ka. It is interesting that wave drift damping of multiple cylinders fluctuates a lot and can be negative at some frequencies due to interactions between columns. This negative wave drift damping, which contributes to reduce the total fluid damping at particular wave periods, was also reported in other literature (e.g., [59]).

Second-Order Problem at O(82). The second-order wave-body interaction problem can be formulated by applying perturbation and collecting terms of 0(82) from Equations 19-23 with U = 0. Then, the kinematic and dynamic free-surface conditions can be combined into the following single free-surface condition"

4 0

%

3 0 -

2 0 -

1 0 -

- 1 0 -

- 2 0 -

/ i

f \ . ~

~\ - \ . \. / jl \ \ , ' i i

'\. J

- 3 0 ' l I I = I 0 . 0 0 .2 0 .4 0 .6 0 .8 1 .0

/,:a

1.2

F igure 15. Surge w a v e drift d a m p i n g as funct ion of d i m e n s i o n l e s s w a v e n u m b e r ka for w a v e h e a d i n g s 13 = 0 ~ ( ) , 13 = 45 ~ ( - - - - - - - ) , and 13 = 90 ~ ( - - - ) .


( () t) )(~(2)= 1 ()(I)(1) t) (()2(I)(1) ~-7 +g~zz g at c)z ~)t 2 + g

~O(1) ) ~ )2 ~Z - - ~ " (viI)(1) (44)

The second-order potential can be decomposed into incident, diffraction, and radiation potentials like the linear wave-body interaction theory, i.e. ~(2) = O(i 2) -F (I) (D2) --..(2). +~R The second-order incident wave potential (Stokes second-order wave) can be obtained when O(~) = ~(i ~) in the fight side of Equation 44. For monochromatic waves, it is given by

3coA 2 cosh 2k(z + h) s in(2kx- 2rot) (I3{2) "- 8 sinh 4 kh (45)

It is well known that the Stokes high-order waves can be used when the Ursell number L2H/h 3 is less than about 26. Therefore, in shallow water, it is valid only for very small amplitude waves.

On the other hand, the second-order diffraction potential ~ ) satisfies Laplace equation, no-flux condition at the bottom, body-surface and far-field conditions, and the free-surface condition (Equation 44) excluding the products of incident waves in the fight side, which have already been used for the second-order incident wave potential. The body boundary condition for the second-order diffraction potential has the form:

+ ~ ' - fi-( r) -- fi.(l~ (1). V) V(I 3(1) 3n 3n 3t

+ l , -

(46)

The matrix H (2) is second order and consists of the quadratic products of the first-order rotational motions:

/~ 2y + ~2 0 0 z

y O x+Oz o

f~x f~z - 2 ~"~y ~'~z ~"~2x + ~"~y

As pointed out by Ogilvie [60], the expression of the H matrix depends on the sequence of rotations. The order roll-pitch-yaw is used here. Finally, the far- field condition of the second-order diffraction potential is given by [50]:


r 2) - EL(0) [cosh ~]2k 2 (1 +cos0) (z + h)]e ikr(l+c~ - 4 ;

EF(0) eik2 r + ~ coshk 2(z+h) (48)

where (I) (2) = Re[~)(2)e -2i~ k 2 = wavenumber associated with 2oo

The first term on the right side represents the asymptotic form of the locked- wave potential related to the inhomogeneous free-surface forcing (or free-surface pressure) in Equation 44, while the second term represents the free-wave potential (outgoing waves) that satisfies the homogeneous free-surface condition. Along the line of 0 = n (weather side), the locked-wave potential forms a standing wave and its depth attenuation is extremely slow, which seems to be related to the well known phenomenon in geophysics called "microseism" [45]. Kim & Yue [30] computed the second-harmonic pressure field around a vertical cylinder and actually showed that the second-order pressure attenuates slowly near the weather side. Newman [57] subsequently carried out the asymptotic

r at large depths and explicitly showed that O~D2) decays alge- analysis of �9 D braically with depth with the rate of O(1/z):

z for z >> 1 (49) #~)or ~/r2+z 2 (~ r2+z2+x)

Because of the slow depth attenuation of the second-order potential compared to the exponential decay of the first-order potential, the second-order pressure may be greater at large depths than the first-order pressure, which implies that second-order wave loads can be important to deep-draft structures, such as tension- leg platform [28]. For illustration, the second-harmonic pressure at various angular positions of a bottom mounted vertical cylinder is calculated using the complete second-order diffraction theory [30] and the result is shown in Figure 16. We can see that the weather-side pressure attenuates slowly as predicted by the asymptotic analysis of Newman [57]. Kim & Yue [32] report that the second-order difference-frequency pressure also attenuates slowly with depths, which may cause appreciable slowly-varying excitations to deep-draft structures.

(2) can be Upon using Green theorem, the second-order diffraction potential ~O obtained from:

0G § dS + I I QG+ dS (50) 4 n c , ~ ) + ~ , ~ ) - - ~ n dS = ~ a + B - c3n g

SB SB SF


4.0 I I - - __~ I . . . . . . . . l . . . . . . . . . ]

T

A ~ N ..%

3.0

2.0

1.0

X ~

\ \

0.2 0.4 0.6 0.8 1.0 -z/h

Figure 16. Modulus of the double-frequency hydrodynamic pressure due to the second- order potential on the side of a vertical cylinder (a = h,co%/g = 1.2) at different angular posi t ions" O = 0 ( - - - ) ; O = 1/~ ( _ �9 _ ) ; O = 1/=~ ( - - - - - ) ; O = %re ( ); and 0 = ~ ( ).

where c represents the normalized solid angle and G § is the free-surface Green function evaluated at 2o). The symbols B and Q represent the fight side of the free-surface and body-surface conditions of Equations 44 and 46. Compared with Equation 42, this integral equation has an additional integral over the entire free surface which makes the second-order problem much more complicated and challenging than the first-order problem. The integral is highly oscillatory and its integrand decays slowly with radial distance; therefore, any direct application of numerical quadratures is to be computationally very costly. An efficient algorithm using an analytic free-surface integration with the far-field asymptotics of the integrand is introduced in Kim and Yue [30]. A simpler approach including only the leading asymptotic terms of Bessel and Hankel functions was also introduced in Eatock-Taylor and Hung [16] and Abul-Azm and Williams [ 1 ]. In the previous integral equation, both B and Q contain second-order spatial derivatives of the first-order potential, and thus HOBEM may be more efficient and robust than CPM [43]. When CPM is used, the order of derivatives needs to be reduced by one by using Stokes theorem [30].

After solving for ~(o 2), the second-order wave exciting force can be obtained as follows


~(2) =_p ~ / 9 ((I)~ 2)/)t+ (I)(D2)) ridS+ T pg ~ rl~')2 l~ldl + fi(1)x (F(1)+ ~;(1))

S B WL

--p V (I)(1)" V (I)(1) +~(1)"'~-t- V (I)(1) ridS

SB _ r t ~ ( 1 ) t ~ ( 1 ) ~ l( pgAw t~-x "'z '~f + ~2~ 1) ~ l ) y f ) (51)

where Tlr = TI (1) -- -z=(1)_ yf~(x l) + xf~l) lq =fi/41 -n2 z

A w = waterplane area W E = waterline

(xf, xvf) = center of floatation of waterplane F(1) = _pgAw(E~l)+ yf.Q(1)_ x # ~ l ) )

H

When there is no body motion, Equation 51 can be simplified to

~(2) = _ p ~ / ) (O~2)0t+O~))ridS- p f ~ ( 1 V O(

SB SB Pg ~ (1)2-

+ T .~ 1] Ndl

WL

1). V (i)(1) ] dS

(52)

Molin [50] suggested an alternative formula to compute F (2) without explicitly solving for O~), i.e.,

~ , ~ ) n k d S : f ~ t p : B - o n d S + g ~ Qtpk+dS SB SB SF

(53)

where tp[ is an assisting linear radiation potential evaluated at 2(o. Equation 53 can be used when only the integrated quantities, such as forces, are needed.

Using Equation 53 and the first-order analytic solutions, Equations 38 and 41, the second-order force on a bottom-mounted vertical cylinder can be obtained in a semi-analytic manner [30]. First, the second-order mean drift force can be calculated from the following explicit equation:

11;2 (ka) 3 1 + sinh2-kh

oo [1- n (n + 1) / (ka) 2 ]2

E ' rJ '2 (ka)+yn21 (ka)] n=0 [Jn2 (ka)+ Yn 2 (ka)]t n+l (54)


On the other hand, the second-harmonic force due to the quadratic products of first-order variables is given by

F (2) = 2 ipg aA2 ~ ( -1) n

qx n(ka)2 ~_ H n ( k ~ ' n § )

3 - 2kh n(n + 1) 2kh ~ + ( 1 + ~ sinh 2 kh (ka) 2 sinh 2kh

)) (55)

The double-frequency Froude-Krilov type force due to second-order incident waves is

FIZZ ) = -3ir~pgaA2j1 (2ka) 2 sinh2 kh

(56)

Another second-order double-frequency force component results from the body- surface inhomogeneity, i.e., the first term on the right side of Equation 53:

(Z)_ggaAZ3~ikZJ ~ (2ka)tanhkh ( 4sinhkzh H o H 1 (kza)

x - sinh 4 kh 2k2h + sinh 2 k2h k 2 H~ (k2a)

4 sin KZm h I-I m K 1 (KZm a) / + , (57) ) m=l 2~C2mh +sin 2K:2m h 1(2m K] (K:2m a)

1 / s inh(2k+k2)h s inh(2k-k2)h = + where H 0 -~ 2k + k 2 2 k - k 2 _l(sin(2k+~2m)h s in (2k -~ 2 )h) __ + m

rim 2 2k + K~2m 2 k - K~2m

and

(58)

The last contribution comes from the flee-surface integral, i.e., the second term of Equation 53:

Fp(2) ffklkrq x = Pg A28~i l (kr)(Pld(kr) (59)

where

Z / P P Q1 (kr) = ( -1) n x Z n (kr)Zn+ 1 (kr) - Jn (kr)Jn+l (kr)

n=0

+ [Z n (kr)Zn+ 1 (kr) - Jn (kr)Jn+l (kr)] [ n(n(kr) - - - X - + 1) + 3 tanh 22 kh - 1 (60)

Hydrodynamics of Offshore Structures 36g

p

Jn(ka) and Zn(kr)= J n ( k r ) - ~ H n ( k r ) (61)

I Hn(ka)

The surge radiation potential in Equation 59 is given by Equation 41 evaluated at frequency 2o) and z = 0 = 0. Semi-analytic solutions for multiple bottom- mounted cylinders can also be obtained in a similar manner [36].

Figure 17 shows the comparison of the first- and second-order diffraction computation with measured horizontal forces on a bottom-mounted vertical circular cylinder. As can be seen in the figure, the second-order solution typically gives small correction to the first-order solution and the second-order result correlates better with Chakrabarti's [6] experimental results. The second-order correction of the shallow cylinder is appreciably increased for small ka mainly due

o

=;

o

=.E r 4

o

- ' t ' " w - ~ "

.

1 . . . . . . l _ _ . �9 . , . _ , i . .

o.8 t . l 1.4 t.7 2

Figure 17. Comparison of the first-order ( ~ �9 -----) and the total (first- plus second-order) (-------) maximum horizontal forces on the bot tom-mounted uniform vertical cylinders (radius a, h = 1.1 6a and 4a) as a function of the incident wavenumber ka. For the h = 1.1 6a case, the results are compared to the experimental measurements (,) of Chakrabart i (1978) for measured wave amplitudes in the range of A/h = ~ - ~0 as indicated by the shaded area. For h = 4a, wave amplitude of A/h = V40 is used in the computations.


to the increased effects of second-order incident waves. For the deeper cylinder, the second-order correction generally increases with ka (or a/L). For some geometries, however, the second-order correction can be significant. For instance, Figure 18 shows the comparison of measured horizontal forces on a conical gravity platform with the numerical predictions by the first- and second- order diffraction theory. In this case, it is seen that the second-order force can be as large as the first-order force and only the second-order results correlate well with Jamieson et al.'s [24] experimental results.

" " 0

K

o

0 N

0 i O " "~ 0

1 "1' 1 1

L ~ ~ L ~ . - - - - . J . - - - ~ - .

4.0 8 .0 t 2 . 0 t 6 . 0 2 0 . 0

tt=e (see)

Figure 18. Compar ison of the compu ted f i rs t -order ( D _ D ) and total (first- p lus second-order) (--- * ---) hor izontal forces on a conical gravity plat form (h = 135 ft, a = 55 ft, toe angle = 45 ~ wi th measured data (----) (Jamieson et al., 1985) for a regular incident wave (H = 11.5 ft, T = 7.2 sec)

To calculate the second-order wave loads in random waves, it is necessary to obtain the sum- and difference-frequency quadratic transfer functions (QTFs; second-order forces due to bichromatic incident waves of unit amplitudes) occurring at CO i d- (Oj and O) i -- O)j, respectively. The details can be found in [31 ]. The second-order sum-frequency wave excitation can cause the springing responses which have to be taken into consideration in the fatigue analysis of a taut-moored system, such as a TLP [28]. The importance of the second-order difference-frequency wave loads to the large-amplitude slowly varying responses of a moored platform has already been mentioned in the preceding section.

Finally, second-order wave forces tend to be more sensitive to the directional spreading of incident waves than first-order wave forces. For instance, Figure 19 compares the difference-frequency wave force by the bichromatic waves propagating in the same direction with that by the same bichromatic waves propagating in the opposite direction [33]. It is observed that the latter can be several times greater than the former in the practically important wave frequency region. This example implies that the assumption of wave uni-directionality may not be conservative so far as the second-order wave loading is concerned.


2.5

2.0

:# ~ ,,~ ~.5

a'P 1.0 r~

0.5

0.0

"X

r"

I 01 1 1 1 1 16 0.4 .6 0 8 1.0 1.2 1.4 1.

t ."rn 8

1.8

1.5

m 1 " . z . .

:Z ~ .< ~1 0.9

0.3

0.0

A,

\

, 1 I 1 I I I I 0.4 0.6 0.8 1.0 1.2 1.4 1.6

V m a

Figure 19. Amplitudes of difference-frequency surge forces on a bottom-mounted vertical cylinder (h/a = 2) as functions of dimensionless wavenumber, Vma -- (COl2 + o~2)a/2g, for 13j = 13 t = 0 (top) and 13i = 0&13, = ~ (bottom): (o)12 - (0~a/g = 0 (.), 0.25 (+), 0.5 (• and 0.75 (A).

Motion Analysis

In the preceding sections, how to calculate wave forces, added mass, and radiation damping on large or slender structures was explained. In the context of linear theory, the motion of a floating body can then be obtained from


([m] + [m,aj ])~ (') + ([bij ] + [Bij ])~ (') + ([cij ] + [Kij ])~ (') = F(') (62)

where, ~ = (F. x, .'~,y, ~'z' ~"~x' ~"~y' ~')z )T' [m], [mai.], [bij], and [cij], are mass, added mass, radiation damping, and hydrostatic coef~cient matrices, respectively. The symbols [Bij] and [Kij] are linearized viscous damping and linearized mooring- line stiffness, respectively. The mass matrix is given by

rm 0 0 0 mZg -myg"

0 m 0 -mZg 0 mXg 0 0 m myg -mxg 0

0 -mZg myg Ixx Ixy Ixz mZg 0 -mXg Iy x Iyy Iy z

- m y g mXg 0 Izx Izy Izz

(63)

where m = body mass (Xg, yg, Zg) = coordinate of the center of gravity

I = mass moment of inertia

As was seen in Equation 7, the linearized viscous damping depends on the unknown motion amplitudes; therefore, an iterative procedure has to be used when viscous damping is not negligible. For moored structures, a major part of B may come from the viscous damping of mooting lines (mooting-line damping), as was pointed out by Webster [69]. The hydrostatic restoring coefficients are only geometry-dependent and can be calculated from the following formulas:

C33"-Pg~snz dS, C34----Pg~sYnzdS,

Can = Pg~s YZnzdS + Vpgzb - mgZg, B

c55 = Pg~s X2nzdS + VPgzb - mgZg, B

c56 = - pgVy b + mgyg

c35 = - pg~sxnzdS,

c45 -- - pg~sXynzdS,

C46 "- -- pgVx b + mgxg,

(64)

where cij = cji for all i,j in the above except for C46, C56 and for all other values of indices cij = 0. The symbols V and (x b, Yb, Zb) represent the displaced volume and the coordinate of the center of buoyancy of a body.

When the natural frequencies of a system are much smaller (or larger) than predominant wave frequencies, the slowly varying (or high-frequency) responses excited by the second-order difference-frequency (or sum-frequency) forces can be large, and thus it is important to include second-order difference-frequency (or sum-frequency) wave loads for the motion analysis of such platforms. When vis-


cous damping, mooring system, and motions are strongly nonlinear, the nonlinear equation of motion has to be integrated in the time domain. In this case, the nonlinear hydrodynamic loading at the instantaneous body position and the equation of motion of a nonlinear system have to be solved simultaneously, which requires tremendous computing time. For a more efficient time-domain simulation, the following pseudo nonlinear time-domain equation can be used [66, 67]:

([m] + [maij (~176 ~ + Rij ( t - x) ~ ('c) dx + ([cij ] + [Kij (~)1) ~ = F (1)

+ + + gwo (65)

where FVD = nonlinear viscous drag force proportional to relative velocity squared and integrated over the instantaneous wetted body surface

Fwo = wave drift damping

The convolution integral is used to account for the frequency-dependent radiation damping. The solution of Equation 65 in the time domain with low-frequency drift forces included is critically dependent on a good estimate of the damping force acting on the vessel at low frequencies, where radiation damping is usually very small. The level of damping force governs the maximum slow-drift motions and mooting line tensions, and therefore, it has to be reasonably estimated or verified through model experiments. Using Equation 64, Ran et al. [65] numerically simulated in the time domain the motions of a large floating spar platform (diameter = 40.5 m, draft = 198.2 m) in regular, bichromatic, and random waves and compared the results with experiments. Figure 20 shows the predicted and measured pitch responses in a regular wave. Figure 21 shows a similar comparison for a bichromatic wave. In both cases predicted motions are in good agreement with measured motions. In Fig- ure 20, it is seen both in theory and experiment that wave frequency motions ride on larger slowly varying motions.

In random waves, the first- and second-order wave loads obtained either from slender-body theory or diffraction theory, can be expressed as follows using the two-term Volterra series expansion:

N E _ _ ,. -iwit F(1)(0=Re ~iqe

i=l

(66)

N N E E * - -i(wi-wj)t F (2) ( t)= Re {AiAjfij (wi,w j)e i=l j=l

-i(w i +w j )t + (W i, W j) e } +AiAj f i j (67)


0.01o-

O.OtO

0.005

0,000

- .005

- ,010

- .015 0 100 200 300 400 500 600 700 800 900

0.015

Iq V

J~

r

0.010

0.005

0.000

- .005

- .010

- .015 0 1 O0 200 300 400 500 600 700 800 900

Time (s)

Figure 20. Spar pitch responses in regular waves (H = 3 m, T = 14 s) from experiment (top) and computation (bottom).

0.06-

oo' ioo J I~ , i , , , i O . O 0

0 200 400 600 800 1000 1200 i 400

0.06-

0.04-

~'~ 0.02

0.00

- .02

- . 0 4

- . 0 6

o 200 ~ o ~ % - ~ o 8;0 lo'oo 12'oo ,4oo

Time (s)

Figure 21. Spar pitch responses in bichromatic waves (H = 6 m, T = 12 s and H = 6 m, T = 14.2 s) from experiment (top) and computation (bottom).

Hydrodynamics of Offshore Structures 3 7 5

where A = complex wave amplitude N = number of wave components

fi, ~ = linear and quadratic force transfer functions, i.e., respective wave forces due to unit-amplitude monochromatic and bichromatic incident waves

When the natural frequencies of a moored platform are very small, it is reasonable to approximate fij by its diagonal component fit which can be more easily calculated [29, 54]. The wave drift damping in random waves can also be simulated in the same manner as Equation 67. Figure 22 shows measured and numerically predicted pitch response spectra of a spar platform in the uni-directional irregular waves characterized by a JONSWAP spectrum with significant wave height H s = 13.1 m, peak period Tp = 14 s, and overshoot parameter 7 = 2. It is clearly seen that the slowly varying pitch responses are much greater than wave frequency motions and the use of Equation 65 can reasonably reproduce such results. As another example, Figure 23 illustrates the relative importance of various TLP offset contributions as function of sea states [52]. It shows that second- order mean and slowly varying offsets are much greater than first-order horizontal motions for a variety of sea states. The detailed spectral and statistical analyses including second-order wave loads are summarized, for example, in Kim and Yue [29, 31, 32].

Finally, for a moored deep-water platform, hull-mooting coupling effects and mooting-line damping may be significant. In such a case, hull motion should be solved considering the dynamic effects of mooting lines. In other words, the hull

j I ,~ o . o o e

O.OO5-

u 0 . 0 0 4 - O3

0 . 0 0 3 -

0

0 . 0 0 2 -

=0 0 . 0 0 1 -

O3

o.ooo 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Frequency (rad/s)

Figure 22. Measured (----) and numerically simulated ( - - - - - ) pitch response spectra.

376 Offshore Engineer ing

~ ~:7:~; .-.:~ . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i:i::;ili!!iii! i:i:: ::: _ _ . . ~ 1 st Order Wave

ili~!!i!ii!iill '

B 2nd Order Wave IEI 1st Order Wave I I Dyn Wind B Mean Offset

t0 11 5 13 15 16.5 18 17 H 8 10 12 14 16 18 20 ~p

Figure 23. Relat ive impor tance of the various offset con t r ibu t ions as func t ion of sea states (Natvig et al., 1992).

and mooring line responses should be solved simultaneously as an integrated system. These kinds of dynamic effects of tethers or mooring lines have not been considered in the conventional uncoupled analyses, where hull responses are solved treating mooring lines as massless springs, and then the mooring dynamics are subsequently solved by imposing prescribed top motions. The accuracy of this kind of uncoupled analysis has not been thoroughly assessed due to lack of bench-mark hull-mooting coupled analysis results. The examples of hull-mooting coupled dynamic analysis are given in Pauling & Webster [63], Kim et al. [26], and Ran and Kim [66, 67]. The coupled dynamic analysis programs are particularly useful for the prediction of the transient burst-type responses, called "tinging" [52] of a taut-moored system, such as TLPs, in the random waves representing survival conditions.

It should also be remarked that a dynamic system may also experience nonlinear and chaotic behaviors due to system or mooring line nonlinearities [ 11 ]. For example, an articulated loading platform (e.g., Figure 4) behaves like an inverted pendulum and its restoring force for pitch is given by (Bz b - WZg) sin0, where B = buoyancy force, W = weight, and z b and Zg are distances from the pivoting point to the center of buoyancy and center of gravity, respectively. If pitch angles are not small, the first two terms of the Taylor series of sin0 should


be used, i.e., sin0 = 0 - 03/3 !, which leads to a Duffing equation including cubic nonlinearities. This Duffing equation can be solved analytically by several standard analysis techniques, such as multiple-scale method. As a result of the cubic nonlinearity, the amplitudes and frequencies of the system responses are modified compared to the linear solutions. In addition, the system can be unstable under certain conditions and may experience typical nonlinear behaviors, such as jump phenomena and sub- or high-harmonic resonances. In many cases, the mooring line restoring forces for large displacements can also be modeled by cubic nonlinearities. A similar approach can also be applied to study large- amplitude roll motions of a ship. This kind of nonlinear analysis is usually limited to simple mathematically-idealized dynamic systems and may not be directly applicable to complicated real systems.

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C H A P T E R 9

UNDERWATER ACOUSTICS

Robert E. Randall



CONTENTS

INTRODUCTION, 383

Early History, 383

Underwater Acoustic System Categories, 385

Underwater Sound Fundamentals, 387

Decibel Scales, 388

SONAR EQUATIONS, 389

Active Sonar Equation, 390

Passive Sonar Equation, 391

Transient Form of the Sonar Equations, 392

PROPERTIES OF TRANSDUCER ARRAYS, 394

Array Gain, 395

Receiving Directivity Index, 397

Transducer Responses, 398

Beam Pattern, 399

Special Arrays, 401

UNDERWATER SOUND PROJECTOR, 403

Explosives as Sound Projectors, 405

UNDERWATER SOUND PROPAGATION, 408

Spreading Laws, 409

Cylindrical Spreading, 409

Multiple Constant-Gradient Layers, 422

TRANSMISSION LOSS MODELS, 426

Transmission Loss for Ray Diagrams, 426

Sea Surface and Bottom Loss, 427

382

Underwater Acoustics 383

Transmission Loss Model for Mixed Layer Sound Channel, 428

Deep Sound Channel Transmission Loss Model, 428 Arctic Propagation, 429

Transmission Loss Models for Shallow Water, 429

AMBIENT NOISE, 430

SCA'I~ERING AND REVERBERATION LEVEL, 434

TARGET STRENGTH, 439

RADIATED NOISE LEVELS, 440

SELF NOISE LEVELS, 443

Sources of Self Noise, 443

Flow Noise, 446

DETECTION THRESHOLD, 447

UNDERWATER ACOUSTIC APPLICATIONS, 449

Seismic Exploration, 449

Analysis of Seismic Reflection Data, 450

Acoustic Position Reference System for Offshore Dynamic Positioning, 452 Short Baseline System, 455

Acoustic Depth Sounders, 457

Side-Scan Sonar, 459

Subbottom Profiling, 465

Acoustic Positioning and Navigation, 466

Acoustic Doppler Measurements, 467

REFERENCES, 468

Introduction

Early History

In the 15th century, Leonardo da Vinci stated that if a ship is stopped and a person places one end of a long tube in the water and places the other end to the ear, then the person can hear other ships at a great distance. Placing a second tube similarly to the other ear provided the opportunity to estimate the direction to the ship. The first mathematical treatment of sound theory was completed by Newton in 1687, and it related the propagation of sound in fluids to physical properties of density and elasticity. Bernoulli, Euler, LaGrange, d'Alembert, and Fourier all contributed to the theory of sound during the 18th and 19th centuries. In 1827, Colladon and Sturm measured the speed of sound in water using a light flash coupled with the sounding of an underwater bell to obtain 4707 ft/s at 8~ Rayleigh


published his famous "Theory of Sound" in 1877, and it is republished as Strutt (1945). The magnetostrictive and piezoelectric effects, discovered respectively by Joule in 1840 and Curie in 1880, are used to produce transducers that generate and receive underwater sound. Fessenden developed the first high powered underwater sound source in 1912 and later developed the first commercial application for an underwater acoustic device in which a foghorn and underwater bell were used to determine distance from shore. He also designed a moving coil transducer, called the Fessenden oscillator, for echo ranging.

During World War I (1914-18), a system of underwater echo-ranging was developed under the acronym ASDIC (Allied Submarine Devices Investigation Committee). The principle of echo-ranging was that a pulse of sound was transmitted into the water, and any reflection (echo) from a submarine was received by a hydrophone. The received signal was heard on headphones, and the time delay between transmission and reception was used as a measure of the range of the submarine. If sound transmission could be made directional, then target directions could be determined. In the United States, Hayes pioneered the field of passive sonar arrays at the New London Experiment Station, currently the New London Division of the Naval Underwater Warfare Center (NUWC), in New London, Connecticut. In the United States, the term ASDIC was replaced by SONAR, which is an acronym for Sound Navigation and Ranging. Langevin used the piezoelectric effect in underwater sound equipment to detect submarine echoes at distances as great as 1500 m.

In 1925, the Submarine Signal Company coined the word fathometer, a trade- mark of the Raytheon Company, that was an acoustic device used by ships in the US and Great Britain for depth sounding. Adequate sonar systems were developed and produced in US in 1935, and US ships were equipped with underwater listening and echo ranging equipment. In 1937, Spilhaus invented the bathythermograph that measures the temperature versus depth of water. This device was installed on all submarines to measure the temperature profile of the ocean to assist in the determination of characteristics of sound propagation and sonar detection. Surface vessels were equipped in 1938 with echo ranging equipment and operators searched in bearing with headphones and a loudspeaker. Submarines were equipped with line hydrophone arrays with headphones for listening.

After WWlI, underwater sound applications greatly expanded. Signal processing developments have been greatly advanced and target noise has been greatly reduced. Some underwater acoustic applications since then are:

�9 Fishing aids (locating commercial fish). �9 Ocean engineering/oceanography (telemetry of data, acoustic Doppler cur-

rent meter, acoustic release mechanism, vertical echo sounders, dynamic positioning systems).

�9 Geophysical research (oil exploration).


�9 Underwater communications (surface ships, submarines, divers, remotely operated vehicles).

�9 Navigation (depth sounders, beacons, transponders, acoustic speedometers, upward-looking depth sounders for navigating under ice).

�9 Underwater search and hydrographic surveying (side scan sonar, subbottom profilers, depth sounders).

�9 Coastal processes and Dredging (sediment thickness and characteristics, acoustic flow meters).

�9 Acoustic tracking ranges (surface ships, submarines, torpedoes).

Sonar systems that occur in nature are found in porpoises, whales, and bats for navigation. Porpoises and bats also use their natural sonar capability for search, detection, and localization of food sources. A valuable source of reference texts have been developed [ 1, 5, 7, 8, 13, 16, 17, 19, 24, 29, 37, 38, 40--43, 50].

Underwater Acoustic System Categories

Underwater acoustic systems can be divided into four categories such as active sonar systems, seismic systems, underwater communications and navigation systems, and passive systems. Examples of these systems are

1. Active Sonar Systems

�9 Active echo ranging sonar is used by ships, submarines, helicopter, fixed underwater installations, and sonobuoys to locate submarine targets. These sonars use a short pulse of sound that is transmitted into the water by a sound projector (transducer). For reception, the same transducer or a second transducer is used as a hydrophone to receive the returning sound signal (echo). Fixed transducer arrays such as line, conformal, cylindrical, and spherical are used as well as towed line arrays.

�9 Torpedoes use moderately high frequencies to echo range on targets and then steer on reflected signals.

�9 Depth sounders send short pulses downward and time the bottom return. �9 Side-scan sonars are used for mapping ocean terrain at fight angles to a

ship' s track. �9 Subbottom profilers are used for showing features of the ocean subbottom

directly beneath the ship's track. Frequently, the transducers for side-scan and subbottom sonars are contained together in a hydrodynamical ly designed tow body to collect seafloor and subbottom information simultaneously.

�9 Fish finding aids are forward or side looking active sonars for spotting fish schools. These sonars may also use multiple beams.

�9 Diver hand held sonars are for diver location of underwater objects.


�9 Position marking beacons transmit sound signals repeatedly. �9 Position marking transponders transmit sound only when interrogated. �9 Acoustic flow meters and wave height sensors are used to measure flow

rate and wave height, respectively. Acoustic Doppler current profilers (ADCP) measure ocean currents.

�9 Fluid levels in tanks are frequently measured using active sonar systems. �9 Sonobuoy is a floating buoy that is equipped to send and receive acoustic

signals. It is used as the link between an aircraft and underwater explosive source when used to track submarines.

2. Seismic Systems

�9 Seismic profilers are used to explore the ocean subbottom, or sub-floor. The acoustic pulses used are basically unidirectional pressure pulses that are generated by explosive charges, underwater arcs (sparkers), compressed air release (air guns), and electromagnetic devices (thumpers). These seismic devices produce results that show the geological features of the ocean floor.

3. Underwater Communications and Telemetry Systems and Navigation

�9 Underwater telephone is a device used to communicate between a surface ship and a submarine or between two submarines.

�9 Diver communications use a full face mask that allows the diver to speak normally underwater and a throat microphone to acquire speech signals. A transducer is used to transmit the signal. The same transducer is used to receive, and the signal is passed to the diver via an ear piece.

�9 Telemetry systems transmit data from a submerged instrument to the surface. �9 Doppler navigation uses a pair of transducers pointing obliquely down-

ward to obtain speed over the bottom from the Doppler shift of the bottom returns. A pulsed Doppler system uses only a single transducer.

�9 Acoustic tracking ranges are used for tracking submarines, surface ships, and torpedoes.

4. Passive Systems

�9 Passive sonars use a hydrophone array that detects acoustic radiation from another vessel or object (e.g., line, conformal, cylindrical, and spherical hydrophone arrays and towed arrays used by submarines).

�9 Acoustic mines explode when the received acoustic radiation reaches a certain value.

�9 Torpedoes home on acoustic radiation of a submarine or ship.


Underwater Sound Fundamentals

Sound is the small amplitude periodic variation in pressure, particle displacement, and particle velocity in an elastic medium. Sound is produced by mechanical vibration and the energy from the vibrating source is normally transmitted as a longitudinal wave (to and fro motion). Sound waves are longitudinal waves, because the molecules transmitting the wave move back and forth in the direction of propagation of the wave, producing alternate regions of compression and rarefaction. The main reason acoustic waves are used in underwater detection and communications is that electromagnetic waves do not propagate for long distances in water except at very long wavelengths. At these long wavelengths, the waves are not useful for most underwater search purposes, but long wavelength electromagnetic waves are used in communication systems.

Transmission of sound waves is very complicated, so plane waves of sound are studied, and these are the simplest type of wave motion propagated through a fluid medium. For a plane wave, the acoustic pressures, particle displacement, and density changes have common phases and amplitudes at all points on any given plane perpendicular to the direction of wave propagation. Plane waves are easily produced in a rigid pipe with a vibrating piston. In a homogeneous medium, plane wave characteristics are attained at large distances from their source.

The term particle of the medium is understood to mean a volume element large enough to contain millions of molecules so that it may be considered a continuous fluid, yet small enough so that the acoustic variables of pressure, density, and velocity can be considered as constant throughout the volume element. For the case of a plane wave of sound, the acoustic pressure (p) is related to the particle velocity (u) by

p=pcu (1)

where p - pressure p = density c = propagation velocity of the plane wave

pc = called the specific acoustic resistance u = particle velocity

The specific acoustic resistance for seawater is 1.5 x 105 g/cm2s and for air is 42 g/cm2s. This equation is also known as Ohm's Law for acoustics.

The energy involved in propagating acoustic waves through a fluid medium is kinetic energy due to the particle motion and potential energy resulting from the stresses in the elastic medium. For a plane wave, the acoustic intensity (I) of a sound wave is the average rate of flow of energy through a unit area normal to the direction of wave propagation. The average intensity is


~2 I = ~ ( 2 )

pc where P2 = time average of the instantaneous acoustic pressure squared

If the units of pressure (p) are gtPa (1 BPa = 10 -5 dynes/cm2), density (p) are gm/cm 3, wave propagation velocity (c) are cm/s, then the units of intensity are ergs/cmZ-s. Intensity is commonly expressed in units of W/cm 2, which is also power per unit area, where one watt is equal to 107 ergs/s, and then the expression for intensity is

p2 I = ~ x 1 0 -7 ( 3 )

pc

For transient signals, the energy flux density (E) in W-s/cm 2 is a useful term, and it is the integral of the instantaneous intensity.

f ~ ~o p--~-2 E = dt = pc dt (4)

Spectrum level refers to the level of a sound wave in a frequency band 1 Hz wide, and the band level refers to a level in a frequency band greater than 1 Hz wide. The units of frequency are hertz (Hz), which is defined as 1 cycle per second (Hz = s-l).

D e c i b e l S c a l e s

In practice it is common to describe sound intensities and pressures using logarithmic scales known as sound levels. Two reasons for using logarithmic scales are the very wide range of sound pressures and intensities encountered in the ocean and the fact that the human ear subjectively judges the relative loudness of two sounds by the ratio of their intensities. The most generally used logarithmic scale for describing sound levels is the decibel scale. The intensity level (N) of a sound of intensity I 1 is defined by

N = 10 log ~I 1 (5) I2

where 12 is the reference intensity. Also used is the sound pressure level that is defined as

N = 20 log Pl (6) P2


where P2 is the reference pressure. The reference level must be known to ensure proper interpretation of the decibel (dB) value. Previous reference pressures are 1 dyne/cm 2 and 0.0002 dyne/cm 2, and the latter reference pressure (0.0002 dyne/cm 2) is the threshold of hearing for humans. The current reference pressure is (1 ktPa) where 1 ktPa is equal to 10 -5 dyne/cm 2. Also, 1 Pascal is equal to 1 psi multiplied by 6895. It is often necessary to convert from one reference (P2) to another (P3)- An expression for converting a sound pressure level referenced to an acoustic reference pressure (P2) to a level with a new reference pressure (P3) is

Np3 = Np2 + 20 log P___L2 (7) P3

For example, it is desired to express 125 dB relative to 0.0002 dyne/cm 2 (P2) in dB relative to 1 dyne/cm 2 (P3)" Using Equation 7, the new sound level is determined as

= 1 2 5 + 20 log ~0"0002 =125 - 74 = 51 dB rel dyne/cm e (8) Np3 1

If the 125 dB is expressed relative to the current reference pressure of 1 ktPa, the result is 151 dB re 1 ~tPa.

The level of a sound wave is the number of decibels by which its intensity, or energy flux density, differs from the intensity of the reference sound wave. For clarity the level should be written as "N dB re 1 ktPa" that means the sound level in decibels relative to the intensity of a plane wave of acoustic pressure equal to 1 micropascal.

Sonar Equations

The sonar equations are a means for determining the effects of the medium (ocean), the target, and the sonar equipment. These equations are one of the design and prediction tools available to the engineer for underwater sound applications. The practical uses of the sonar equations are (1) prediction of sonar equipment performance and (2) sonar design.

The total acoustic field at a receiver is defined as a desired portion (signal) and undesired portion (background). The background is the noise or reverberation (back scattering of output signal). The objective of the design engineer is to find a means for increasing the overall response of the sonar system to the signal and decreasing the response to the background. A sonar system is just accom- plishing its purpose when the signal level exceeds the background level by an amount sufficient enough for the observer to distinguish the signal from the noise. The sonar parameters are defined in relation to the equipment, medium, and target as tabulated in Table 1.


Table 1 Classification of Sonar Parameters

Equipment related parameters

Medium related parameters

Target related parameters

Projector source level Noise level Receiving directivity index Detection threshold Transmission loss Reverberation level Ambient noise level Target strength Target source level

(SL) (NL) (DI) (DT) (TL) (RL) (NL) (TS) (SL)

Active Sonar Equation (Noise Limited)

To describe the meaning of the sonar parameters, consider the example of an active sonar (Figure 1). The sound source produces a level of SL dB at a reference distance (e.g. 1 yd or 1 m), and it acts as the receiver as well. The radiated sound signal is reduced by the transmission loss when it reaches the target. The level at the target is then SL - TL. As a result of reflection and scattering by the target whose target strength is TS, the reflected or backscattered level is SL - TL + TS at a distance of 1 yd from the acoustic center of the target in the direction back toward the source. In traveling back, the level is again attenuated due to transmission loss and consequently, the level on return to the receiver is SL - 2TL + TS. If the ambient background noise (NL) is isotropic, then the background noise level is reduced by the directivity index (DI), and the relative noise level is NL - DI. When the noise level is not isotropic, the directivity index is replaced by the array gain (AG), which is discussed later. Therefore, the signal to noise ratio is SL - 2TL + TS - (NL - DI). Generally, detection is the purpose of the sonar, so some means of determining when the target is present is needed. A detection threshold (DT) is established such that when the signal to noise ratio is above the DT then a decision is made either by a person or the electronics that a target has been detected. The active sonar equation, Equation 9, is now written as an equality in terms of the detection threshold as

SL - 2TL + TS = NL - DI + DT (9)

This equation applies to a monostatic sonar that means the source and receiver are coincident and the target return is back toward the source. A bistatic sonar has the source and receiver separated or in different locations, and therefore the transmission loss (TL) values from source to target and target to receiver are not always the same.

Detection Threshold (DT)

Directivity Index (DI) or

Array Gain (AG)

Noise Level (NL)

Electronics

Headphones

Source Level (SL) at 1 yd (TS)

One-way Transmission Loss (TL)


Figure 1. Definition of sonar parameters for active sonar equation.

Active Sonar Equation (Reverberation Limited)

A modification of the active sonar equation is necessary when the background is reverberation instead of noise. In this case DI is not correct as defined for noise. For reverberation background, the term in Equation 9 (NL - DI) is replaced by an equivalent reverberation level RL. An increase in system performance can not be achieved by raising the source level further once the system reaches its reverberation limit. This is in contrast to the noise limited case where increases in source level produce corresponding increases in performance. The reverberation limited active sonar equation then is

SL - 2TL + TS = RL + DT (10)

Passive Sonar Equation

In the case of a passive system, the target itself produces the signal by which it is detected. Therefore, the source level parameter refers to the level of the radiated signal of the target at the distance of 1 yd, or 1 m. Also, the target strength (TS) parameter is no longer meaningful. Again, the directivity index is replaced by the array gain (AG) when the noise level is not isotropic and if AG is known. The transmission loss only occurs from the target to the receiver. Thus, the passive sonar equation is


SL - TL = N L - DI + DT (11)

Table 2 contains definitions of sonar parameters. The performance figure (SL - ( N L - DI) is the difference between the source level and the noise level measured at the receiver. The Figure of Merit (SL - (NL - DI + DT)) is the maximum allowable transmission loss in passive sonars, or the maximum allowable two-way loss for active sonars when TS is 0 dB. When the detection threshold (DT) is zero, then the sonar equations are a statement of the equality between the desired part of the acoustic field (signal---echo or noise from target) and the undesired part (background of noise or reverberation). This equality holds at only one range in most cases. Echo and reverberation decrease with range, and noise is usually a constant.

Transient Form of the Sonar Equations

Previous sonar equations were written in terms of the average acoustic power per unit area of the sound emitted by the source or received from the target. Average implies a time interval. The time interval causes uncertain results whenever short transient signals are used or whenever severe distortion is present due to scattering from the target. A more general approach is to write equa-

Table 2 Summary of Sonar Parameter Definitions [43]

Parameter Name Symbol Reference Definition

source intensity Source Level SL 1 yd from source 10 log on acoustic axis reference intensity *

Transmission Loss TL 1 yd from source and at target

signal intensity at lyd

10 log signal intensity at target or receiver

echo intensity at 1 yd from target Target Strength TS 1 yd from acoustic 10 log center of target incident intensity

noise intensity Noise Level NL At transducer 10 log location reference intensity *

Receiving DI At transducer t0 log Directivity Index terminals

noise power generated by equivalent nondirectional transducer

noise power generated by actual transducer

Reverberation At transducer 10 log

Level RL terminals reverberation power at transducer terminals

power generated by signal of reference intensity

At transducer Detection Threshold DT terminals 10 log

signal power to just make decision

noise power at transducer terminals

*Reference intensity is that of a plane wave with rms pressure of 1 ~tPa.


tions in terms of energy flux density (E) that is the acoustic energy per unit area of wave front.

o o if E = pc p2(t)dt (12)

o

The intensity (I) is the mean square pressure of the wave divided by the specific acoustic resistance and averaged over a time interval (T).

I = 1 i p2(t____~) dt (13) T .I pc

0

For long-pulse sonars, T is the duration of the emitted pulse and is nearly equal to echo duration. For short-pulse sonars, the duration of the echo is vastly different from the emitted pulse, and it can be shown that the intensity form of the sonar equations can be used if the source level is defined as

SL = 10 log E - 10 log x e (14)

where x e is the duration of echo of source at 1 yd and measured in energy flux density units of a 1 l.tPa plane wave over an interval of 1 s. For pulsed sonars emitting a flat topped pulse of constant source level (SL') over a time interval x o, and because energy density is the product of average intensity times duration, then the energy flux density (E) may be expressed as

10 log E = SL' + 101ogx e (15)

By combining Equations 14 and 15, an equation for the source level of pulsed sonars is

SL = SL' + 10 log a:o/a: e (16)

For long-pulsed sonars, x o is equal to x e and SL and SL' are identical. For short- pulsed sonars, x e is greater than x o. For active short-pulse sonars (Figure 2), echo duration x e is a parameter in its own fight. The echo duration (Xe) consists of three components that include duration of emitted pulse at source (x0), additional duration caused by two way propagation (Xm), additional duration caused by the target (xt). Thus, the echo duration is

"1; e = I; 0 -t- "gm + "l;t (17)

Typical examples of the three components of an echo duration are tabulated in Table 3.

The sonar equations have several limitations. These include the requirement to correct the source level for short pulse sonars. Correlation sonars must account


short pulse near source

1 . , . .

_ ~ ~ m ~ ~ l ~ Bear target

~o + 1; m

i , ~ _ , J ,, , ,

, . . ,

echo

Figure 2. Characteristics of echo duration for short pulse sonars.

Table 3 Examples of the Components of Echo Duration [43]

Duration of the emitted pulse at short ranges (xo)

Duration produced by multipaths ('lim)

Duration produced by submarine target (xt)

Explosives: 0.1 ms SONAR: 100 ms Deep Water: 1 ms Shallow Water: 100 ms Beam Aspect: 10 ms Bow-stern : 100 ms

for correlation loss. Additional limitations are related to the medium being inhomogeneous, having irregular boundaries, and the fact that one boundary is in motion. In addition, some of the sonar parameters fluctuate randomly with time. There are also unknown changes in equipment and platform conditions. There- fore, the sonar equations yield a time averaged result of a stochastic problem.

Properties of Transducer Arrays

Underwater sound equipment provides the means to detect the existence of an underwater sound wave. This equipment consists of a hydrophone array that converts acoustic energy into electrical energy and this energy is then fed into a signal processing unit to display it. Transducers are devices that convert sound and electrical energy into each other. Hydrophones are transducers that convert sound into electrical energy. A projector is a transducer that converts electrical


energy into sound energy. The conversion of sound into electrical energy and vice versa is accomplished with the use of materials that have certain properties.

Piezoelectric materials such as quartz, ammonium dihydrogen phosphate (ADP), and Rochelle salt acquire a charge between the crystal surfaces when placed under pressure, and conversely they acquire a stress when a voltage is placed across the surfaces. For example, the electrical potential across a piezoelectric material may be varied periodically at the frequency of a desired sound signal, and as a result, the material vibrates at the desired frequency. Elec- trostrictive materials have the same effect as piezoelectric materials. However, these materials are ceramics that have been properly polarized. Examples are barium titanate and lead zirconate. Magnetostrictive material changes dimensions when it is subjected to a magnetic field, and conversely its magnetic field is changed when it is stressed.

The design of transducers is an art and is a special technology in its own right. Some research and measurement activities use single elements of piezoelectric or magnetostrictive material as hydrophones. In most other cases, hydrophone arrays are used that consist of a number of elements spaced in a particular way. Arrays have the advantages of being more sensitive, possessing directional properties, and having a greater signal to noise ratio than "small" single elements or elements the size of those used in the array. A large single element the size of the array has the same beamwidth characteristics as the array, but it must be mechanically steered.

Array Gain

One of the important advantages of using a hydrophone array is the improvement of signal to noise ratio. The improvement of signal to noise ratio is measured by the array gain (AG)

(S / N)array (18) AG = 10 log (S / N)element

One approach to evaluating AG is to consider the directional patterns of the signal [S(0,~)] andjnoise [N(0,~)] fields along with the beam pattern of the array as expressed in the following equation

f S(O, r b (0, r dn / J" N(O, r b (0, r

AG = 10 log 4g t' /'4~ (19) df~

4x 4n

Each integral in the numerator is the directional pattern of signal or noise weighted or multiplied by the beam pattern and integrated over all solid angle (t2).


The second approach considers the coherence of signal and noise across the dimensions of the array. Coherence is the degree of similarity of either the signal waveform or the noise wave form between any two elements of the array. The coherence is measured by the cross correlation coefficient (p) of the outputs of the different elements of the array.

Z Z ( P s ) i j

i j (20) AG = 10 l~ Z Z ( P n ) i j

i j

For amplitude shading

ZZaiaj (Ps) i j i j

A G = 1 0 l o g Z Z a i a j ( P n ) i j

i j

(21)

where Ps and Pn are the cross correlation coefficients of signal and noise respectively. The term a i is the root mean square (rms) voltage produced by the ith element due to signal or noise. Expressions for Ps and Pn are tabulated in Table 4.

When both the signal and noise are completely coherent across an array, there is no array gain (AG = 0). If the signal and noise are completely incoherent across an array, then the array is unable to distinguish between signal and noise. For a perfectly coherent signal in incoherent noise, then

(Pn)ij = 0 where i ~ j and (Pn)ij = 1 when i = j

and

(22)

AG = 10 log n (23)

where n is the number of elements. For a perfectly coherent signal in only partially coherent noise, then

(Pn)ij = p i ~ j and (Pn)ij =1 i = j (24)

and

n AG = 10 log (25)

l + ( n - 1 ) p

The gain of an array depends on the statistics of the desired and undesired por- tions of the sound field in which the array operates.


Table 4 Expressions for Cross Correlation Coefficients [43]

Signal, Ps Isotropic Noise, Pn

Single frequency, cos (tyt w sin(cod) no time delay \ c )

cod c

Single frequency, cos t.0(x w + x e) time delay sin( )

cod c

COS CO "C e

Note: co = 2x (frequency) x w = signal travel time between array

elements = (d/c)cos 0

d = array element separation

x e = electrical (steering delay) 0 = the angle to the line joining the

two elements c = sound velocity

Receiving Directivity Index

When a signal is a unidirectional plane wave (perfectly coherent) and when the noise is isotropic (same in all directions, N(0,t~) = 1) then the array gain reduces to the quantity called directivity index.

5 df2 47c

AG = DI = 4/ t - 10 log 2n n/2 (26)

I b ( O ' * ) d " f ~ b (O,t~)cosOdO df~ 4 x

0 - ~ / 2

If the beam pattern has rotational symmetry and is nondirectional in the plane in which t~ is determined, then

4 ~ DI = 10 log x / 2 (27)

2r~ ~ b(O) cos 0 dO

- x / 2

For simple arrays (line and circular plane) the directivity index can be computed using expressions tabulated in Table 5.

The directivity index (DI) is restricted to the special case of a perfectly coherent signal in isotropic noise, but these conditions seldom occur in the real ocean.


Table 5 Mathematical Expressions for Directivity Index of Simple Transducers [43]

Transducer Type Beam Pattern

Function [b(O)] Directivity Index (DI)

Continuous line with length L, where L >>

Piston with diameter D in infinite baffle where D >> ~,.

sin ( - ~ ) sin012 10 log ( ~ )

( -~1 sin0

sinO

Line array with "n" elements equally spaced a distance "d" apart.

Two element line array spaced "d" apart sin sin 0]

2 sin ( -~) sinO

10 log

10 log

l+- - n

p=l

,n ~ / 2p~d

1+ sin( /

2~d

Signals transmitted in the ocean are perfectly coherent only at short ranges, and at long ranges signals are received from various directions and along different paths. Noise is anisotropic in the real ocean. Thus, the directivity index is only useful for approximate calculations and array gain (AG) should replace DI in sonar calculations when the characteristics of signal and noise are known.

Transducer Responses

Normally a hydrophone linearly transforms sound energy into electricity and the proportionality factor for this transformation is called the response. It relates the generated voltage to the acoustic pressure of the sound field. The receiving


response is the number of decibels relative to 1 volt produced by an acoustic pressure of 1 l.tPa and is expressed as N dB re 1V/ll.tPa. For a response o f -80 dB re 1V/ll.tPa, the voltage generated is determined by

V - 8 0 = 20 l o g -

1

logv = - 4 . 0

v = 0.0001 or 10 -4 volts (28)

Thus, the hydrophone generates an rms voltage of 10 -4 volts when placed in a plane wave sound field having an rms pressure of 1 l.tPa. Transmitting current response is the number of decibels relative to 1 micropascal measured at 1 meter (1 yard), produced by 1 amp into the terminals of the projector and is expressed as N dB re 1 ~Pa/A.

For a transmitting response of 100 dB re 1 ktPa/A at 1 m (add 0.78 dB for 1 yd), the acoustic pressure generated at 1 m is determined as follows

20 log p = 100 1

log p = 5

p = 105 kt Pa at lm (29)

Beam Pattern

The response of a transducer array varies with direction relative to the array. This very desirable property of array directionality permits the determination of the direction of arrival of a signal and also facilitates resolution of closely adjacent signals. In addition, directionality reduces noise relative to the signal arriv- ing from other directions. The response of an array varies with direction in a manner specified by the beam pattern of the array. Expressions for simple arrays are tabulated in the previous Table 5.

The beam pattern is

b(O, (~)- V 2 (0, ~) (30)

and response V(0,t~) is normalized such that V(0,0) is equal to one. For hydrophones, b(0,~) is the mean square voltage produced by an array of unit response when sound of unit pressure is incident on it in the direction (0,~). For projectors, b(0,~) is the mean square pressure produced at unit distance when unit current is fed into the projector. The direction of (0 = 0, ~ = 0) is arbitrary, but is usually taken as the direction of maximum response (acoustic axis). Direc- tion for which b(0 = 0, t~ = 0) is called the acoustic axis of the array. Decibels are normally used in describing responses and beam patterns. Arrays can be


steered mechanical ly and electrically. Electrical steering is accomplished by inserting appropriate phasing or time delay networks in each element 's circuitry. Examples of beam patterns are illustrated in Figure 3, and an example calculation is illustrated in Table 6. The significance of the -3 and - 6 dB beamwidths (Figure 4) is that they are the �89 and �88 power points for the system, respectively.

Acoustic Axis

Line Transducer

_ Acoustic Axis

Circular Plane Array

Figure 3. Three dimensional beam patterns for line and circular plane arrays [43].

Table 6 I l lus trat ive Ca lcu la t ion of the B e a m P a t t e r n for a Line A r r a y

Given:

Find:

n = 5, equally spaced line array d = 0.5 ft, c = 5000 ft/s, f = 5 kHz b(O)

Solution" X c 5000 f 5000

lft

b(O) = sin I ( - ~ 3 sin 01 2

n sin [(_~d_) sin ~ =

2 sin/' ~ 1 . _{ sin !7.854 sin ~ l 2

5 sin [(~ 10_~.5) sin 01 5sin (1.571 sin 0)J

For 0 = 0, the value of b(0) in the limit is equal to 1, and therefore 10 log b(0) is 0 dB. A tabular listing of 0 and 10 log b(0) is shown below. The angle 0 must be converted to radians (multiply by re/180) to use in above equation to get results below.

0 (deg)

0 1 5

10

10log b(0) 0 10log b(0) 0 10log b(0) (dB) (deg) (dB) (deg) (dB)

0 15 -6.883 35 -12.05 -0.02613 20 15.30 40 -13.03 -0.6621 25 -24.83 45 -16.55 -2.775 30 13.98 50 -24.96

0 (deg)

60 70 80 90

10 log b(O) (dB)

-19.88 -14.95 -14.04 -13.98

dB


width at -3 dB

acoustic axis

"~--"--elements , , , , _ , . . . .

Figure 4. Definition of the beam width at -3 dB for a line array.

Special Arrays

When an array consists of individual elements that are directional, then the product theorem is used to determine the array beam pattern. This theorem states that the beam pattern of an array of identical equally spaced direct ional hydrophones is the product of the pattern of each hydrophone alone, and the beam pattern of an identical array of nondirectional hydrophones as described in Equation 31.

b n (0, (I)) = V 2 (0, r = b (0, (I)) bnond (0) (31)

Mills Cross is an array of two line transducers in which the outputs are multiplied, or correlated together. When the two line arrays have "n" elements the beam pattern of a Mills Cross array with 2n elements is the same as a rectangular array with n 2 elements. The Mills Cross beam pattern is the same as for a rectangular array on the major axes only, and the beams are much wider than that for a rectangular array everywhere else. Its advantages are that it is light in weight and economic to build. The disadvantages are lower array gain and lower sensitivity.


Shaded Area

m m

3 dIB do~ total l ~ v

o degre

s ~ 7 e 9 =~ zo 30 40 5o 6o7oeo ,oo I J ~ A m y ke= l~ in w a v e ~

Figure 5. Beam width for a line array with elements spaced L/2 apart for various steering angles to broadside [12].

When the beam pattern of an array of particular geometry is controlled or changed, it is called shading (Figure 6). For amplitude shading, the responses of individual array elements are adjusted to provide the most desirable pattern. The arrays are usually adjusted so that the maximum response is at the center and least response at the ends. Superdirectivity results in very narrow beams and the elements are spaced less than �88 wavelength apart with signs, or polarities, of adjacent elements reversed. Phase shading varies the spacing of array elements, but has not been commonly used. In adaptive beam forming, a null can be placed in the beam pattern to cancel out an undesired signal (noise) in a certain direction.

Previous arrays have been linear and additive. The output is linearly proportional to acoustic pressure, and the outputs are simply added together. In the case of multiplicative or correlative arrays, the outputs are multiplied together. These types of arrays find applications in conditions of high signal-to-noise ratio and when narrow beams or high resolution are needed and when a reduction in size or number of elements over a linear array is necessary.


N a m e Unshaded Two element, end weighted Two element, center weighted Binomial Dolph-Chebyshev

Shad ingFo~ula 1,1,1,1,1,1 1,0,0,0,0,1 0,0,1,1,0,0 0.1,0.5,1,1,0.5,0.1 0.3,0.69,1,1,0.69,0.3

Line Tvoe

. , .

%<.:..~.:

-I0

Ii lira;! ,,-' ',,_ ii,,. !f II l t i , , \ \\1 -4Oo m lO 3O 4O SO r~l ro lio

0 (degrees)

Figure 6. Beam pattems for different shading [43].

gO

Underwater Sound Projector

Active sonars use a projector to generate acoustic energy. The projector normally consists of an array of individual elements that produce a directional beam in a desired direction. Source level (SL) designates the amount of sound radiated by a projector. Transmitting directivity index is the difference between the level of the sound generated by the projector and the level that would be produced by a nondirectional projector radiating the same total amount of acoustic power (Figure 7). Transmitting directivity index is defined as

DI T = 10 log ld (32) Inond

The relation between source level and radiated acoustic power is determined by assuming a nondirectional projector is in a homogeneous, absorption-free medium. At a large distance r, the intensity of the sound emitted by the projector will be I r. For r very large, the plane wave assumption is valid and

f . . . . . . . - . / nondirectional projector

\ \

\ \

/ / /

/ /

/ I

I t !

. . . . . . . . . . J.. I I

\ \ /

/


' Inond N / / , . _

F directional projector

acoustic axis

I d

Figure 7. Comparison of directional and nondirectional sound sources.

2 ir = P_._r_r • 10-7 ( W / c m 2 )

pc

where Pr = rms pressure in dynes/cm 2 9 = density in gm/cm 3 c = velocity of sound in cm/s

(33)

For p = 1 gm/cm 3, c = 1.5 • 105 cm/s, and converting to yards

I r -- 5 . 5 8 X 10 -9 p2 (W/yd z) ( 3 4 )

In the case of a nondirectional projector, this intensity corresponds to a radiated power output of

Pr = 4rt r2Ir = 70 • 10 -9 p2 r 2 (W) (35)

At a distance of 1 yd, the power is

p = 70 • 10 -9 p2 (36)

where Pl = rms pressure at 1 yd in dynes/cm 2 Converting to decibels and recalling that SL is 10 log p~ referenced to 1 ~Pa, then

10 log P = - 171.5 + SL (37)

If the projector is directional, then

SL = 171.5 + 10 log P + DI T (38)

Underwater Acoust ics 4 0 5

which is illustrated in Figure 8. The quantity P is the total acoustic power radiated. In the case of an electroacoustic projector, P is less than the electric power (Pe) input to the projector, and the difference is related to the projector efficiency (1] = Pe/P). The source level is then expressed as

SL = 171.5 + 10 log Pe + 10 log r I + DI T (39)

Typical ranges of shipboard sonar parameters are an acoustic radiated power (P) of 300-50,000 W, a transmitting directivity index (DIT) of 10-30 dB, a source level (SL) of 210-240 dB, and an efficiency (1"1) 20-70%. Limitations of sonar power are due to cavitation and interaction of elements. Cavitation bubbles form on the face of the projector when the power is increased to a certain value. It is also possible to have cavitation several feet, or about one meter, away from the face of the transducer where the beam starts to form. The cavitation threshold may be raised by increasing frequency, decreasing pulse length, or increasing depth. Interaction between sonar elements can also reduce the sonar's power when one element absorbs power from the other.

Explosives as Sound Projectors

Explosive charges of material ranging from a few grains to a few pounds in weight are commonly used as underwater sound sources. When explosive material

230

220

210

- - 1 9 0

180

r,~ 170

J /

f

J J / / j

I / / j

/ I . , i /

=P

/ �9 " j

f /

/ J J

1 1 I I I 1 ] l I I I I I I I 1111

2 5 t0 2O 5O1O0 Acoustic Power Output (W)

/ j r

�9

1 6 0 ' ' ** ' * j =*'* 0.t 0.2 Q5 I t,(XX) t 0 ,000

Figure 8. Source level as a function of acoustic power for different transmitt ing directivity index [43].


is detonated, a pressure wave initiated inside the material propagates into the surrounding medium (Figure 9). The pressure signature of a detonating explosion consists of shock wave and small bubble pulses at short ranges. For long range, the signature is complicated by refraction and multipath-propagation effects in the sea.

The shock wave has a pressure time relationship

t

P = P0 e to (40)

where p = instantaneous pressure at t Po = peak pressure occurring at time t = 0 t o = time constant of the exponential pulse (time to decay to 0.368 Po)"

P0 2.16 • 104 / wl/3 / 1"13 = psi r

(41)

t o 58wl/3( wl/31 --0.22 r

(42)

where w = charge weight (lb) r = range (ft)

pressure

sea surface �9 i ! i i i ii

d e p t h !

, ,f....~.. ,.

gas bubble 6 ( ! \~. .~

h o c k w a v e

[ I 1st 2nd ] r d ~ , p u ~ pulse pulse ~.

, t . . . . . . t . . . . . . . . . J J i ~ ; 0

i ' i / ""~, 0 ". /

" / bubble migrat ion as it , . . . . . . . . . . . .

e x p a n d s and col lapses

�9 ~'= t i m e

F i g u r e 9. Shock wave and bubble migration for explosive source.


Values for Po and t o are plotted in Figure 10 and for a 1 lb TNT charge. Bubble pulses are the series of positive pressure pulses emitted by the pulsat-

ing gas globe at the instant of minimum volume. The amplitude of the pulse decreases progressively as the energy is dissipated. The time interval T(s) between the shock wave and the first bubble pulse is

Kw 1/3 T = (43)

(d + 33) 5/6

where K is a proportionality constant, w is the charge weight (lb), and d is the depth of detonation below sea level (ft). The constant (K) depends on type of explosive (e.g. K is 4.36 for TNT). Advantages of explosive sound sources include being very mobile, easily launched, detonated at any depth, nondirectional, and a short high power broad band pulse. The disadvantages are that explosive sound sources are not very repeatable, the processing of the received signal is difficult because of its short duration, and reverberation can occur. Explosive sources are used for research, antisubmarine warfare (ASW), and seismic profiling.

10 e

e | :

t 0 6 _

] -

�9 1 0 ~

=')

1 I I J I I I I I , I I l = = = = = = I ==~= 2 3456e,0 20 5o ,00 200 soo

i t t~e, r (yd)

Z 4 0

e ! 1

22o

! 200

Figure 10. Peak pressure and for I Ib charge of TNT [3].


' ~ 1,000

1 I00

1

74

l 1 1 I I l l 1 | I 1 l I 1 1 I I I I 1 1 1 1 I

~0 tO0 ~,000

Range, r (yd)

Figure 11. Time constant for I Ib explosive charge of TNT [3].

Underwater Sound Propagation

The flow of acoustic energy from a source to a receiver is described in terms of its intensity at a reference distance such as 1 yd, or 1 m, from the source and the reduction in intensity between this point and the receiver. The transmission (propagation) loss is the reduction in intensity between the reference point and receiver. Transmission (propagation) loss is affected by spreading and attenuation. S.preading is the result of acoustic energy becoming diluted as it spreads over a larger area, and thus intensity is reduced. Near the source, spreading is spherical and the intensity reduction is proportional to the inverse square of the distance. At larger distances the spreading is affected by refraction (bending of rays along paths that the waves travel). Attenuation is the loss of energy from the sound wave as a result of absorption and scattering. Absorption is caused by the conversion of acoustic energy into heat (frictional effects). Scattering is the process whereby objects in the medium cause some of the energy to be deflected in various directions.

Absorption is dependent on the acoustic frequency and is very severe at high frequencies. Included in the scattering losses is the loss of energy resulting from reflection at the bottom and surface of the ocean. This loss is due in part to the scattering of sound in other directions and to transmission of energy to the adjoining medium. Scattering is a function of sea state, but surface losses are usually small (1 dB). Bottom losses can be severe (20 dB) and are strongly


affected by the nature of the bottom. Hard bottoms are good reflectors and soft bottoms are poor reflectors.

Spreading Laws

Spherical Spreading. For spherical spreading, consider a source located in a homogeneous, unbounded, and lossless medium as shown in Figure 12. Then, the power (P) generated by the source is radiated equally in all directions so as to be distributed over the entire surface of a sphere surrounding the source.

Because power equals intensity times area, then

P = 4~;q 2 11 = 4~;r~ 12 = .... (44)

If r 2 is expressed in yards and r 1 is taken as 1 yd, then Equation 44 reduces to the inverse square law

I 2 = 1 (45)

or the transmission loss is expressed as

TL = 10 log I-L1 = 10 log r 2 (46) 12

which is called the spherical spreading law. This law shows that the intensity decreases as the square of the range, and the transmission loss (TL) increases as the square of the range.

Cylindrical Spreading. Spreading that occurs in a medium between two parallel planes and at a certain range from a source is called cylindrical spreading. In this case, the power (P) is radiated over a cylindrical surface and is expressed as

P = 2r~qHI 1 = 27~r2HI 2 = ... (47)

r

[ ~ r . ) . -~ ~ "-I~ power (p)

~ ~ - - 4 g q2

unbounded medium

Figure 12. Schematic of spherical spreading.


[ ~ ~ ' ~ : ~ medium between IH ~ 2xrlH twO parallel planes

Figure 13. Schematic of cylindrical spreading.

If r 2 is set to 1 yd, then

P

2r~qH TL = 10 log I-L = 10 log = 10 log r 2 (48)

12 P

2r~reH

which is called the cylindrical spreading law. Cylindrical spreading occurs at moderate and long ranges whenever sound is trapped by a sound channel.

No spreading occurs for propagation in a lossless tube or pipe of constant cross section. The area over which the power is radiated is constant, and therefore the intensity and TL are independent of range. For time stretching (hyperspherical spreading), the signal from a pulsed source is spread out in time due to multipath propagation as the pulse propagates through the ocean, especially in areas such as the deep ocean sound channel. A summary of these spreading laws is tabulated in Table 7.

Absorption of Sound in the Sea

The loss of sound intensity in the sea due to absorption is the result of the conversion of acoustic energy into heat (frictional) as sound propagates through the medium. The loss of intensity for a plane wave propagating in the horizontal (x) direction may be written as

Table 7 Summary of Spreading Laws [43]

Intensity Varies Transmission Type as Range (r) Loss (dB) Propagation

No spreading r ~ 0 Cylindrical r -1 10 log r Spherical r -2 20 log r Hyperspherical r -3 30 log r

Tube Between parallel planes Unbounded Unbounded with time stretching


dI = - nIdx (49)

where n is a proportionality constant and the negative sign indicates dI is a decrease in intensity. Solving the differential equation by separation of variables yields

dI = - ndx (50)

I

Integrating yields

l n ( ~ 2 ) = - n(r2 - r l) (51)

which can be reduced to

10 log 12 = 10 log I 1 - 10n(r 2 - r 1) l og e (52)

Now, let ~ be (10 n log e), and then

10log/i ): rl) where o~ is the logarithmic absorption coefficient expressed in dB/kiloyard

The absorption of sound is due to viscosity of pure water and the presence of dissolved salts in the water whose effect is dominant in seawater when the frequency is below 100 kHz. Francois and Garrison [15] presented a relationship for evaluating absorption that considers the sum of the effects of boric acid, magnesium sulfate and pure water. The expression for sound absorption (a) is

B1Dlfl f2 B2D2f2 f2 0~ = f2 +f2 + f2 + f2 + B3D3 f2 (54)

where f is frequency in kHz, fl and f2 are relaxation frequencies in kHz, and ct is the absorption coefficient in dB/km (multiply by 1.0936 to get dB/kyd). The first term of Equation 54 represents the effect of boric acid in seawater, and the equations for evaluating the coefficients B 1, D1, and fl are

8.86 x 10 (0"78pH- 5) B 1 = C D 1 =1

(S ~0.5 (4 1245 / fl = 2 . 8 \ - ~ j 10 - ~ J

c = 1412 + 3.21T + 1.19S + 0 .0167d (55)


where c is sound speed in m/s, T is temperature in ~ S is salinity in parts per thousand (0/00), and d is depth in m. The second term in Equation 54 accounts for the effects of magnesium sulfate, MgSO 4, in seawater and the coefficients B 2, D 2, and f2 are determined from

21.44S B 2 = ~ ( 1 + 0.025T)

c D 2 = 1-1.37 x 10-4 d + 6.2 x 10 -9 d 2

(8-1990 / 8.17x10 273~-TJ

f2 = 1 + 0.0018(S- 35)

(56)

The third term represents the contribution of pure water to absorption, and the coefficients B 3 and D 3 are evaluated using

D 3 = 1 - 3.83 • 10 -5 d + 4.9 x 10 -1~ d 2

forT < 20~

B 3 = 4.937 x 10 -4 - 2 . 5 9 x 10 -5 T+9.11 x 10 -7 T 2 -1 .50 x 10 -8 T 3

for T > 20~

B 3 = 3.964 • 10 -4 - 1.146 x 10 -5 T + 1.45 x 10 -7 T 2 - 6.5 x 10 -l~ T 3 (57)

Figure 14 developed by Francois and Garrison [15] shows the variation of the absorption coefficient (ix) as a function of frequency from 0.1 to 1000 kHz at zero depth (surface) for a salinity of 35 o/oo and pH of 8.0. The accuracy of the predicted absorption coefficients is estimated by the developers as _+5% for the ranges of 0.4 to 1000 kHz, -1.8 to 30~ and 30 to 35 o/oo.

Spherical Spreading and Absorption

Propagation measurements made in the ocean indicate that spherical spreading together with absorption yields a reasonable approximation to measured data for a wide variety of conditions. Therefore, transmission loss may be expressed by

TL = 20 logr + t~(r x 10 -3) (58)

where r is the range in yards and ct is the absorption coefficient in dB/kyd. The procedures and results for evaluating the absorption coefficient and transmission loss using Equation 58 are tabulated in Table 8 using the following example. Consider an active sound source operating at a frequency of 50 kHz and located at a depth of 4,000 ft where the temperature is 8~ pH is 8, and salinity is 30 o/oo. It is desired to detect a target at a range of at least 4,000 yd. If the major causes of transmission loss are spherical spreading and absorption, predict the magnitude of the two way transmission loss.


I 0 0 0

I00

I0

Temperature ('C)

Seawater / / / ////// s=ss oIoo/ / II/Pure w ~ r

o.o,~ o / ~ / , o / / / e q ~ = o ,, , (mrr=~)

oDo( 0.1 I I 0 I 0 0 I O 0 0

Freqae~y (kHz)

Figure 14. Absorption coefficient and correction for depth over useful sonar frequency range for salinity of 35 o/oo and pH of 8 [15].

Speed of S o u n d in the Sea

The speed of sound in water has been determined both theoretically and experimentally. One equation is that developed by Leroy [23]:

c = 1492.9 + 3 ( T - 1 0 ) - 6 x 10 -3 (T-10)2 - 4 x 10 -2 ( T - 18) 2

+ 1 . 2 ( S - 3 5 ) - 10 -2 (T - 1 8 ) ( S - 35) + d /61 (59)

Another more recent sound speed equation is that developed by Mackensie [26]:

c=1448.96 + 4 .591T- 5.304 • 10 -2 T 2 + 2.374 x 10-4T 3 + 1.340 (S - 3 5 )

+ 1.630 • 10-2d + 1.675 x 10 -7 d 2

- 1.025 x 10-2T ( S - 3 5 ) - 7.39 • 10-13Td 3 (60)

(continued on page 415)


Table 8 Illustrative Example for Evaluating Transmission Loss in the Ocean

Given: f = 50 ld-Iz r = 4, 000 yd pH = 8

d = 4 , 0 0 0 f i T = 8 ~ S = 3 0 o / o o

Find: Two way transmission loss (2 x TL)

Soln: TL = 20 log r + tx r x 10 -3

Use Equation 12 to evaluate coefficients for first term of Equation 11

c = 1412 + 3.21 (8~ + 1.19 (300 / oo) + 0.0167 (4000 ft / 3.28 ft / m) = 1493.7 m / s

8.86 x 10 (~ (pH)- 5) 8.86 x 10 (0.78(8)-5) B 1 = = = 0.1031 dB / (km - kHz)

c 1493.7

D 1 =1

1245 ) ( 30 /0 . 5 1245 ) 014- 1014 fl = 2.8 k , - ~ j 1 =2.8k,-~) = 0.962 kHz

Using Equation 13 to obtain coefficients in term 2 of Equation 11

B 2 = 2 1 . 4 4 S ( 1 + 0 . 0 2 5 T ) 21.44 3 0 o / o o c 1493.7 m ! s

(1 + 0.025 (8~ = 0.517 dB / ( k m - kHz)

02 :l_l.37x,0 d+6.2x,0_9d2_1_1.37x10 / ( )2 + 6.2 x 10 -9 = 0.5512 3.28 ft / m 3.28 ft / m

1990 ] 8.17 x 10 8- 27---3-~+T)

f2 = 1 + 0.0018 (S - 35)

8 - 1990 ) 8 . 1 7 x 1 0 273+8~

= = 68.28 kHz 1 + 0.0018 (30 - 35)

Using Equation 14 to obtain coefficients for term 3 of Equation 11

03 1 383x,05d+49x,oo 2:l 383x,05(4 ft) ,olo(40 ft/2 + 4.9 x = 0.8546 3.28 ft / m 3.28 ft / m

Because temperature (T) is less than 20~

B 3 = 4.937 x 10 -4 - 2.59 x 10-ST + 9.11 x 10-7T 2 - 1.50 x 10-8T 3

B 3 = 4.937 x 1 0 - 4 - 2.59 x 10-5 (8~ + 9.11 x 10-7 (8~ 2 - 1.50 x 10-8 (8~ 3 = 0.000337dB / (km - kHz 2)

B1Dlfl f2 B2D2f2 f2 o ~ = ~ + ~ + B 3 D 3 f2

f2 + f2 f2 + f2

(0 .1031)(1) (0 .962)(50) 2 (0 .517) (0 .551) (68 .28) (50) 2 tx = + + (0.000337) (0.8546) (50) 2

(50) 2 + (0.962) 2 (50) 2 + (68.28) 2

tx = 7.61dB / km, or ct = 7.61dB / km (1.0936 km / kyd) = 8.3dB / kyd

Evaluate transmission loss (TL) using spherical spreading and absorption (Equation 15)

TL = 20 log r + a (rx 10 -3 ) = 20 log (4, 000 yd) + 8.3 dB / kyd (4 kyd) = 72.0 + 33.2 = 105 dB

Therefore the two way transmission loss = 2(105) = 210 dB


where c = sound speed (m/s) T = temperature (~ at the depth S = salinity (ppt) d = depth (m)

The range of validity for the Mackensie [26] equation is �9 0~ < T _< 30~ 30 o/oo < S < 40 o/oo, and 0 m < d < 8,000 m. The expressions are good for practical work and shows that sound speed increases with temperature, salinity, and depth.

Sound Speed Structure in the Ocean

The sound speed profile, variation of sound speed with depth, is illustrated in Figure 15. The surface layer is where the sound speed is subject to daily and local changes in heating, cooling, and wind action. Seasonal thermocline is the negative thermal or velocity gradient that varies with season. In the summer and fall, the near surface waters are warm and seasonal thermocline is well defined, but in the winter and spring, it tends to merge and be indistinguishable from the surface layer. The main thermocline is affected only slightly by seasonal changes, and it is here that the major decrease in temperature occurs. The deep isothermal layers of the deep Atlantic, Pacific, and Indian ocean waters the temperatures are about 32.5 to 35~ and the sound velocity increases are due mostly to the effect of depth only. Characteristic sound speed profiles for deep ocean locations are shown in Figure 16. Temperature is an important factor affecting sound speed and example monthly and daily temperature variations are illustrated in Figure 17.

Instruments Used to Measure Sound Speed. A bathythermograph measures temperature as a function of depth as it is lowered into the water. A velocimeter measures sound speed in terms of sound travel time over a fixed path. The principle of the velocimeter is that a projector sends an initial pulse and a receiver accepts the signal and triggers a second pulse by the projector. This results in continuous repetition, and the repetition frequency of the pulses is a measure of the sound speed. Such a sound velocimeter is also called a "sing around velocimeter."

A third instrument (Figure 18) is the expendable bathythermograph (XBT) that measures temperature versus depth without having to retrieve the sensing unit. The operational principle is that a thermistor probe sinks at a known constant rate. The probe is connected to electronic equipment on ship by a fine wire that is paid out from spools on the launch vessel and probe. The thermistor bead changes its resistance with temperature, and the trace obtained is resistance (temperature) versus time (depth). The bathythermograph trace is converted to sound speed without considering the effects of salinity, and its trace is comparable with that of the velocimeter that does account for changes in salinity.


147g 4850 m ft

915 3000

DEPTH

1829 6000

2744 9000

SOUND SPEED 1494 1509 1524 m/= 4900 4950 5000 ft/s ..=

Surfo~e - ~ ~ ~ ~ _ _ Loyer ~ ~ S| Thermocllne

_

Figure 15. Typical sound speed profile in the deep ocean.

1 2 4 3

012 i 1 ] 4 i '" i" i i

! l

4,850 4,890 4,930 4,970 5,010 5,050 Speed (rUs)

Figure 16. Example sound speed profiles for various deep ocean locations. 1. Antarct ic Ocean (60~ 2. North Pacific (45 to 55~ 3. Southern oceans (45 to 55~ 4. Pacific and South Atlantic (40~ and Indian Ocean under influence of Red Sea outf low, and North Atlantic under influence of Mediterranean Sea outf low (US Navy 1970).


0.9 ~ 1.2 ~ 1.3 ~ 3.0 ~ 53" 4.5" 4.0" 2.8 ~

0615 0800 t000 1200 1407 t600 1800 0610 ( = ) L o r ~ - e ( h r )

o

. . . , ~ ~ ~ �9 ~ ~h ~. ~-8 I,,- U) h . r-. h- (I) 0D r h.. I~-. . ,

�9 0 �9 0 �9 �9 0 0 0 0 �9 �9 �9 0 �9

~~'~176 I I F I Ir I I r I I r r IF" r i W l i l I1 2~176 I ! I I I I I ! Ir r w i t P ' l l I1 ~ r I f i ! a r l [ f l !1 4 0 0

t3 t7 26 9 t3 26 11 28 18 31 17 12 3 30 t7 27 25 t9 ' - . v - ' ~ ' ' v "-----e ~ ~

M o t / ~ ' i l M o y J u n e J u l y A u g S e p t O c t N o v D e c J o n Feb

s Momh Figure 17. Time (a) and monthly (b) variability of temperature in ocean waters near Bermuda [43].

ship d isp lay of ! t t e m p e r a t u r e vs t ime ( d e p t h ) ~ 1 [ ~ ~ wire

thermis tor

wire spool

Figure 18. Schematic of expendable bathythermograph.


Snell's Law

Figure 19 shows a plane wave that is traveling downward in the first medium at a ray angle 01 with the boundary. The wave front at the instant when the ray AB reaches the boundary is shown by BB' which is perpendicular to AB and A'B'. As the wave crosses the boundary, the speed of propagation is suddenly changed from c 1 to c 2. Therefore, while ray A'B'C'D' is traversing the distance B'C', the ray ABCD traverses a different distance BC. When ray A 'B 'C 'D ' reaches the point C', the wave front lies along the line CC'. To locate the point C, swing an arc of radius BC about the point B and then draw a tangent line from C' to the arc. The time to travel from B'C' and BC is a constant, and the magnitude of BC is C2At. The tangent line determines the direction 0 2 in which the ray BCD travels.

BC B'C'

C 2 C 1

BC = BC' c o s 0 2

B ' C ' = BC' cos02 (61)

Therefore

BC' cos 0 2 BC' COS 01

C 2 C 1

COSO 2 COSO 1

C 2 C 1 (62)

A A' c = c 1

B' @1

~ . . . . . . . . . . . . , , O'

" D ~ D '

C - - C

c 2 > c 1

Figure 19. Schematic illustrating Snell's Law.


For many layers

C 1 C 2 C 3 = ~ = ~ = . . . = const = c v (63)

COSO 1 COSO 2 COSO 3

Approximating the ocean with layers in which the sound speed is constant and letting the number of layers approach infinity and the thickness of the layer go to zero, then limitations of Snell's Law are that it is valid only when the speed of sound is a one dimensional space function. The constant c v applies only to a particular ray and is the speed of propagation at the depth at which the ray is horizontal.

The critical angle (0c) is defined in Figure 20 and Equation 64.

cos 0 c = c---L (c 1 < c 2) (64) C2

When a ray in the slower medium is incident upon the boundary at an angle 01 > 0 c with the horizontal, the ray enters the faster medium and is bent toward the horizontal. At the critical angel 0 c the refracted ray travels along the interface (02 = 0). When the incident ray is more nearly horizontal than the critical ray (01 < 0 c) the ray does not enter the faster medium but is totally reflected. When a ray is incident from the faster medium, there is no critical angle. Refraction occurs for all angles of incidence, and the angle between the refracted ray and the horizontal will never be less than 0 c.

The Ray Solution of the Wave Equation

In the real ocean, the speed of sound varies spatially and the solution of the basic differential equation (wave equation) is generally not possible. Therefore, approximate methods are necessary and one of these is the method of rays. In three dimensions, the wave equation is

~)2p-F ~)2p ~)2p 1 ~)2p (65) C)X 2 ~ ' ~ + /)Z'--T = C-- T at--g-

incident ray slow medium (el) incident ray x ~ ~

.. ~ , reflected ray ncldent ray. . ,., "~ \ ,_,.

. . . . . . . . . ~ ......... "':-->_.~_~..-,-',-"'" _ ~ layer interface

refracted ray

Figure 20. Critical angle for sound rays. fast medium (c2)


Families of rays are obtained as a solution to the simpler equation called the Eikonal Equation (Equation 66), which is a solution to the wave equation in special cases and under certain conditions is a good approximate solution.

( ow fow (ow (66)

where W (called the Eikonal)=

W (x, y , z ) = c--s176 (o~x + 13y + yz) (67) C

The direction cosines of the ray are defined as o~, 13, and 7 with respect to the x, y, and z coordinate axes, respectively. The speed of sound in water is c, and c o is an arbitrary reference sound speed. The criterion for the validity of Equation 66 is that the change in the velocity gradient over a wavelength is small compared to c/;k o. Mathematically

XoAg ~ < < 1 (68)

where Ag = the change in velocity gradient over the distance of one wavelength. The Eikonal equation is applied to typical ocean conditions where c is mainly

a function of depth (y), and the problem is two dimensional. In this case, x is the horizontal distance from the source, y is vertical distance from the source, s is distance from the source along the ray path, t is the time along the ray path, 0 is the angle of the ray relative to the horizontal and measured positively upward (Figure 21), and g is the velocity gradient (dc/dy). The objective is to determine

water surface x (horizontal distance) , . ,

Y

(depth)

wave front

/

Y

Figure 21. Schematic of coordinate system for ray.


the characteristics of the ray path by finding relationships between the variables x, y, s, t, and 0. The resulting solution to the Eikonal equation is four differential equations that are expressed as

dx Cv Cv = ~ cos 0 dO; dy = - - - sin 0 dO g g

d s = c--2-v dO; d t = dO g g cos 0

(69)

The slope of the ray is dy/dx = -tan 0. If g is assumed constant, then Equation(s) 69 can be integrated and the results are

C v C v x = ~ sin 0 + const ; y = ~ cos 0 + const

g g

c v 1 s = ~ 0 + const (0 is in radians) ; t = l n ~

g 2g

1 + sin 0

1 - sin 0 + const (70)

where c v is called the vertex velocity of the ray and is defined as c v = Co/COS 0 o. Sound rays are arcs of very large circles. For a positive velocity gradient the

origin of the ray arc is in the sky and in the case of a negative velocity gradient the origin is well below the bot tom of the ocean. The speed of sound at the source depth is Co, and the angle of inclination of a ray at the source is called the initial angle, 0 o. Considering a family of rays leaving the source in any one vertical plane, then each individual ray of the family is identified by its own particular value of 0 o. If the source is at depth Yo and the sound speed at Yo is c o, then the constants in Equation(s) 70 are evaluated from the initial conditions of (0 = 0o; x = 0; y = Yo; s = 0; and t = 0). The result is

C O Co x = ~ ( s i n 0 - s i n 0 0 ) ; Y-Yo = ~ ( c ~

g cos 00 g cos 00

c o 1 F l + s i n 0 l + s i n 0 0 s = ~ ( 0 - 0 0 ) " t = L l n ~ - l n ~

g cos 00 ' ~ 1 - sin 0 1 - sin 00 J (71)

Practical computations of ray paths in a constant-gradient medium are usually made by treating y as the independent variable and then (1) selecting a value for y, (2) solving the y equation for cos 0, (3) determining 0 and sin 0, and finally (4) computing x, s, and t. There is an ambiguity in algebraic sign in the determination of 0 and sin 0 because both plus and minus values of 0 give the same value for cos 0 (Figure 22). The physical significance of this ambiguity is demonstrated by the fact that a horizontal line (y = constant) intersects the ray circle at two points.

422. Offshore Engineering

y = const

F boundary between layers

~ •

Figure 22. Ambiguity of sign in ray equations.

Multiple Constant Sound Speed Gradient Layers

Constant-gradient layers have rays that are arcs of circles and curve upward when the gradient is positive and downward when it is negative. Consider the simple case (Figure 23) of two constant-gradient layers having different gradients. Any ray such as SA that leaves the source with a positive, upward angle continues to rise upward. A ray whose initial angle is slightly negative (i.e., below the horizontal) descends to a vertex and then rises again due to the upward curvature of the ray path. A ray such as SD has a large negative initial angle and crosses the boundary between layers with a slope greater than zero at E and penetrates into the second layer where it curves downward. Somewhere in between SD and SB there is a special ray whose vertex occurs at a point of tangency T on the boundary. This ray is the limiting case between SB and SD and is called the limiting ray. Beyond the vertex T, it splits into two branches, the upper branch TC and the lower branch TC'. Between TC and TC' is a region, according to ray theory, where no sound rays enter, and this region is called the shadow zone. Shad- ow zones occur in the vicinity of a region of maximum sound speed. Shadow zones do not truly exist in the strict sense of the real ocean because sharp discontinuous changes in velocity gradients do not occur in nature and ray theory breaks down whenever the velocity gradient changes rapidly with depth. How- ever, very pronounced shadow zones where the sound intensity is very small are

~.~ th

sound velocity (c) A B

source (S) shadow e

v

Figure 23. Example of ray paths for two constant-gradient layers.


actually observed in the ocean. The effects described above result in a continuous transition into the shadow zone rather than a mathematically abrupt one, but the spreading loss inside the zone is so high that detection is not likely. A similar pattern exists when the source is located in the lower layer and the limiting ray leaves the source with an upward angle.

The ray SPQR (Figure 24) leaves S horizontally, crosses the boundary at P, and is refracted upward in the lower layer. It vertexes at Q and T and the vertex at T is the same depth as the source. The ray oscillates up and down but remains within the boundaries of the two layers, and the area between these boundaries is called a sound channel. The ray SP'Q'R' is the limiting ray at boundary A, and it crosses the sound channel axis at Q' and vertexes at R' at depth C. Since the ray is horizontal at both A and C, it is evident from Snell's law that the speed of sound is the same at both depths. Thus, the limiting ray from S is confined to the channel A to C, and this is true for a source located at any depth within the channel. The ray that leaves at a steeper angle than SP'Q'R' will leave the boundary. Any ray such as SP"Q"R that leaves at a correspondingly steep downward angle vertexes at a depth Q" below C and upon returning upward leaves the channel at T". If the ray is sufficiently steep it may cross the lower boundary and leave the channel.

A region in which the sound speed passes through a minimum gives rise to a sound channel. The depth of this minimum is the axis of the sound channel. To determine the thickness of the sound channel, locate the smaller maximum sound speed above or below the channel axis. Draw a vertical line to where it intersects other gradient. The intersections are the limits of the sound channel.

Vertex. A sound ray vertexes when the ray angle 0 is zero. The location of a ray vertex is determined by setting 0 to zero and solving for values of x, y, s, and

sound velocity (c)

A v P' T"

. . . . . . "T

(s) depth (y)

Figure 24. Illustration of sound rays in a three constant gradient layer ocean medium. Point B is sound speed minimum and points A and D are sound speed maximums. Rays leaving S all curve downward.


t using Equations 71. As an example, the x location (x v) is determined for 0 = 0 and a constant gradient (g l) as

x v = - c o sin 00 / (gl cos 0 o) (72)

Limiting Ray. The limiting ray from a source in layer 1 vertexes at Y12 and the vertex speed c v is C12. TO locate the limiting ray, its initial angle (0 L) must be determined. This is found from Snell 's Law (Equation 63) using (c = c12, 0 - 0,

0 o = 0 L) as follows

C _ C O . C12 - C O ; t h e n C O S 0 L -- c o (73) cos0 cos0 o 1 C O S 0 L C12

The values of x, s, and t are determined by setting 00 = 0 L and 0 = 0 in Equation 71.

Ray Path in Othe r Layers . If the initial angle is large enough, the ray crosses into the layer below. The value of x at the boundary between layers 1 and 2 is

C 0 X12 = ~ (sin 012 -- sin 00) (74)

gl cos 00

To determine x at any depth in the second layer we return to the original equation. The vertex velocity (c v) is constant in all layers. The initial conditions in the second layer are x = x12 when 0 = 012. Therefore the increment of x in second layer is

CO X- X12 -" ~ (sin 0 - sin 012) (75)

g2 cos 0 0

At the boundary between layers 2 and 3, the above equation becomes

C0 X23 -- X12 = ~ ( s i n 023 - sin 012) (76)

g3 cos 0 o

The increment in the third layer is

X - X23 = CO (sin 0 - s i n 023 ) (77) g3 cos 0 o

and so on. The cumulative value is the sum of all the increments. For example, the total horizontal distance (xt) for the three layers is

X 1 = X 1 2 + ( X 1 2 - - X12 ) + ( X - X 2 3 ) (78)

The corresponding values of s and t are obtained in a similar manner. An example of a ray tracing problem is illustrated in Table 9.


Table 9 Example Ray Tracing for Constant Gradient Ocean M e d i u m

A sound source is located at 1500 m and the sound speed profile for that location is shown below. Determine the x and y coordinates for the vertex point of a ray which leaves the source with an initial angle of zero degrees.

o

1 ooo

2o0o depth C%~o

4OO0

50012

sou.d ~ (m/s) 14.80 1490 1500 1510 horizontal distance (x) --r ' l ' t = i ~ "

C o C v - -

cos 0 o

gl = 0 ; g 2 =

1 ,490 m / s

cos(O) = 1 ,490 m / s;

1 ,480 - 1, 500 m / s

2 , 0 0 0 - 1 , 0 0 0 m = - 0 . 0 2 s - l " g 3 =

1,510 - 1 ,480 m / s

5 , ~ - 2 , ~ m = 0.01 s -z

Angle of ray crossing interface 23

C o Y23 -- Yo = ' ( C O S 0 2 3 - c O S O o )

g 2 c O S O o

1490 2 , 0 0 0 - 1 , 5 0 0 = ( C O S 0 2 3 -- 1)

( - 0 . 0 2 )

500 (--0.02)

1, 490 + 1 = COS 023

0 2 3 ---- 6.64 ~ = - 6 .64 ~ (nega t ive s ign is inse r ted b e c a u s e a n g l e is m e a s u r e d d o w n w a r d )

Co (cOSOv -- C0S023) Yv --Y23 = g3 COSOo

1 ,490 m / s Yv - 2 , 0 0 0 = ~ (cos(O) - c o s ( - 6 . 6 4 ) )

0.01 S -1

yv = 3 , 0 0 0 m

Horizontal distance

c o x 2 3 =

Y2 cOSOo ( s i n 0 2 3 - s in 0 o ) =

1 ,490 m / s _ 0 . 0 2 s _ 1 ( s i n ( - - 6 . 6 4 ) - s i n ( O ) ) = 8 , 6 1 4 m

X v -- X23 ---- C o

g3 c ~ (s inO v - s in 0 2 3 ) ; x v - 8 , 6 1 4 m =

x v = 8 , 6 1 4 + 17 ,229 = 25 ,843 m

1, 4 9 0 m / s

0.01 s -1 (sin(O) - s i n ( - 6 . 6 4 ) ) ;

(table continued on next page)


Table 9 (Continued) Example Ray Tracing for Constant Gradient Ocean Medium

Distance along path

S23 = ~ c o

g2 cos 0 o

1,490m/s ( 6.64 ) (023 - 0 o ) = ~0-~-s- i- - 1--~--0 =8,634m

co '490m's( / s v -s23 c o s 0 - ~ ( 0 v - 0 2 3 ) ; s v - 8 6 3 4 m = ~ 0 - - 0.01 s -1

Time to travel along path

1 I l+sin023 l+s in0o] 1 ( 1 + s i n ( - 6 . 6 4 ) 1 + 0 t23 = ~ In - In = In - In = 5.8 s 2g 2 1- sin023 1- sinO o 2(-0.02 s -1 ) 1 - sin(-6.64) 1 - 0

[ 1 1 [ 1+0 1 + sin(-6.64) ] 1 lnl+sinOv Inl+sin023 ;t v 5.8s+ -l In - In tv -t23 = 2g 3 l - s i n e v 1-sin023 2(O.Ols ) 1 - 0 l-sin(-6.64)

=5.8+11.6= 17.4s

T r a n s m i s s i o n L o s s M o d e l s

Transmission Loss from Ray Diagrams

The intensity (I1) at 1 yd from a source located at O l and the intensity (12) at

location 0 2 are

AP AP I 1 = ~ ; 12 = ~ (79)

A m I A m 2

The transmission loss is then expressed as

TL = 10 l o g - - I1 = AA2 (80) I 2 Am 1

Figure 25. Schematic for transmission loss model using ray separation [43].


Because AA 1 = 2rt cos01A0 and A A 2 = 2r~rpAL = 2~rAh c o s 02, then

TL = 10 log rAh c~ (81) A0cos01

By Snell's Law, c~ = c-L and then (81a) cos01 c 1

TL = 10 log rAhc2 (82) A0c I

where r is the range (yd), Ah is the vertical separation (yd) of rays at P, A0 (radians) is the vertical angle of separation of rays at location 01, c~ is the speed of sound at 01, and c 2 is the speed of sound at 0 2. The assumptions are: (1) no crossing of acoustic energy between rays, (2) no scattering, and (3) no diffraction.

Sea Surface and Bottom Loss

The sea surface is both a reflector and a scatterer of sound. When the sea surface is rough, the reflection loss is not zero. For example, the surface loss is approximately 3 dB for 1-ft waves at a frequency of 25 kHz. At 30 kHz, the surface loss is 3 dB for 0.2 to 0.8 ft wave heights and at lower wave heights, the loss is less. The Rayleigh parameter (R)

R = kH sin 0 (83)

is used as a criteria for evaluating the roughness of sea surface where k is 2n/~,, H is the rms wave height, and 0 is the grazing angle. When R << l, the surface is primarily a reflector and when R >> 1, the surface is rough and acts as a scatterer.

The sea bottom is also a reflector and scatterer of sound. Its effects are more complicated because of its diverse and multilayered composition. The reasons are that the bottom is more variable in its acoustic properties (hard rock to soft mud) and the bottom is often layered so that sediment density and sound speed change with depth. Therefore, reflection loss of the seabed is not as easily predicted as for the surface. Complete or total reflection occurs at grazing angles less than the critical angle. For angles greater than the angle of intromission, all sound rays are totally transmitted into the bottom. Measured bottom losses for a 10 ~ grazing angle at 24 kHz are 16 dB for a mud bottom, 10 dB with mud-sand bottom, 6 dB for sand-mud, 4 dB for sand and stony bottom. All bottom materials tend to absorb sound. Sound attenuation (tx) in sediments is a function of frequency (~ = k f n) dB/m, where f(kHz), n = 1, k = 0.5 for porosity 35 to 60% (additional values for k can be located in Hampton [16]).


Transmission Loss Model for Mixed Layer Sound Channel

In many ocean areas of the world, the temperature profile regularly shows the presence of an isothermal layer just beneath the sea surface. The layer is maintained by turbulent wind mixing of the near surface water and is called the mixed layer sound channel or surface duct. In this layer of depth (H), the sound speed increases with increasing depth. A model for evaluating transmission loss (TL) in this layer is

TL = 10 log r 0 + 101ogr + (~ + ~L)r • 10 -3 (84)

where r = the range in yards = absorption coefficient in dB/kyd

r o and ff"L are defined below. If the layer sound speed gradient is constant, the radius of curvature (R = Co/g) is much greater than the layer depth (R >> H), and 0 o is small (sin 0 o << 1), then the skip distance of the limiting ray (x), the maximum angle of limiting ray (0o), the angle (0) of limiting ray at the source depth, and the range (ro) are determined from the following equations

x = ~/8RH = skip distance of limiting ray

00 = ~/2H / R = maximum angle of limiting ray, rad

0 = -~[ 2 ( H - ci) -- angle of limiting ray at source depth R

r~ = H - d 8 H - d

where H = thickness (ft) of the mixed layer d = depth (ft) of the source

Leakage of sound from layer is caused by scattering of sound out of the layer and transverse diffusion, and the leakage coefficient o~ is

~L = 2S~ f (86)

where H is the layer depth, S is the sea state, and f is frequency (kHz).

Deep Sound Channel Transmission Loss Model

The deep sound channel (sometimes called SOFAR channel). The axis of this channel (minimum velocity) varies from 1220 m (4000 ft) in mid-latitudes to


Baa~ (=1) 20 30 40 5O 60 70 80

I:

!

| v o v v v

Figure 26. Ray diagram for deep ocean sound channel for source near axis [43].

near the surface in polar regions. Long ranges can be obtained from a source of moderate acoustic power output located near the axis of this channel. The transmission loss model for a deep sound channel is

TL = 10 logr 0 + 10 logr + txr x 10 -3 (87)

where r o and r are ranges in yd and o~ is the absorption coefficient in dB/kyd.

Arctic Propagation Loss

Sound transmission reaches long ranges due to repeated reflection from under ice surface. Low propagation loss in the Arctic is caused by the positive gradient of the sound speed profile that results in continuous upward refraction of all sound rays. The rays then interact with ice-ocean interface and essentially results in a surface duct. The propagation loss is low because there is no interaction with the ocean bottom.

Transmission Loss Models for Shallow Water

Range of sound transmission in shallow water is effected by repeated reflections from surface and bottom. Marsh and Schulkin [27] developed semi-empirical expressions for evaluating the transmission loss that are based on measurements in the frequency range of 0.1 to 10 kHz. A parameter (F) is defined as

F = (d + H) (88)

where F = kyd d = depth of water, ft H = layer depth, ft


The equation used depends on the relationship between the range and the parameter F as follows For r < F

TL = 20 log (r x 10 -3) + cz(r x 10 -3) + 60 - k L (89)

For F < r < 8F

(r x 10 -3) _ 1/ TL = 15 log (r x 10 -3) + cz(r x 10 -3) + ~t F

+ 5 log F + 6 0 - k L (90)

For r > 8F

(r x 10 -3) _ 1/ TL = 10 log (r x 10 -3 ) -4- ct(r x 10 -3 ) 4- t3~ t F

+ 10 log F + 64.5 - k L (91)

where r is the range in yd, F is in kyd, and ot is the absorption coefficient in dB/kyd. Table 10 provides data for the shallow water attenuation coefficient (cz t) and the near-field anomaly k L.

Ambient Noise

Ambient noise is the composite noise from all sources at any specific point in the ocean except for desired signals and the noise inherent in the measuring equipment and platform. The ambient noise level (NL) is expressed as

NL = intensity of ambient background (92)

intensity of plane wave having rms pressure of 1 l-tPa

Common sources of ambient noise in deep water (Figure 27) are tides, seismic, turbulence, ship traffic, wind waves, and thermal noise. Pressure fluctuations resulting from tides and hydrostatic effects of waves cause very low frequency noise, but it is not too important at frequencies of interest in underwater sound. Tidal currents can cause flow induced noise. The intensity levels of ambient noise in the deep ocean vary with frequency and this variation with frequency is called the spectrum level of ambient noise that is expressed in dB re 1/.tPa as illustrated in Figure 27.

Constant seismic activity results in low frequency noise (<1 Hz). Turbulence in the ocean induces motion of the transducer, causing self noise, and pressure

I'-,

..1

0 lm

E~ I=

~J

l,,t

iJl

r~

ee~ I r~

rf~

r~

o

c,,,I ~

v'~ ~

o ~t-

oo ("~1 t-.-- ~c:)

~ c~

~

o~

~

~ ~

~ ~

.

ce~ t,,..

c,~1 o t---. t'--.

oo cr~

ir~

,.~ c,~

~t- oo

o ~

~ ~

~.


Q.

QQ

Q,) a,

E

Qa Q.

1 2 0

llO. -

90-

e0-

70

60-

5O

40-

30-

~l T i d e T u r b u l e n c e

I !

Wind Thermal I !-/"' ' " ! 1 I I S h i p p i n g

i I I I ~ t I I I -

a I I i \-8 to -101 i I \dS/octave I I I \ J I I

I I I ! \ i t

i ', I I I ~ - s ,o -6 I I I I ~,~s/~t,~ I , 20 soo ~ sozx~ , 6 /

1 I l 1 t l

1 tO tOO 1.000 10,0OO SOO~X~

Frequency (Hz)

Figure 27. Ambient noise spectrum level in the ocean [43].

changes associated with turbulence may be radiated. Ship traffic is the principal source of noise in the frequency range of 10-500 Hz. Surface waves create ambient noise between 500 Hz-25 kHz that correlates well with sea state or wind force. Causes include breaking white caps, flow noise (wind blowing over rough sea surface) and cavitation (collapse of air bubbles). Rough sea surface is a dominant noise source at 1-30 kHz. Thermal noise is the result of molecular agitation in the sea and is important typically at frequencies greater than 75 kHz, but it can be a limit when frequencies are above 40 kHz in calm seas. A summary of deep water ambient noise is shown in Figure 28. Data indicate the deep water noise is not isotropic, but is directional in the 10 to 500 Hz range where shipping noise dominates.

Intermittent sources of ambient noise do not persist over periods of hours or days. Any biological sounds from whales, porpoises, dolphins, shellfish cause noise in the 10-500 Hz frequency range. Noise from snapping shrimp occurs in the 500 Hz--20 kHz range. Heavy rain can result in 30 dB increase between 5-10 kHz. Steady rain can cause a 10 dB increase around 20 kHz. Seismic explosions from seismic surveys are another noise source. If shipping and biological noise are absent and wind is the primary contributor, then shallow and deep water noise levels are nearly the same. In general, shallow water is a noisy and highly variable environment for most underwater acoustic operations. Aver- age ambient noise level is higher in winter than in summer due to better sound transmission conditions. Shallow water ambient noise levels undergo wide vari-


I • I zo JL - I Z o =L

I 1 0 1 1 0

,oo \ ~L---, . . . . ,oo " ".""PP.'mJ

r 70 . ~ 70

~" - - ' . .L -~e . , , I s o

E , o , 4 0

I 1 J i l l I 1 1111 I 1 J i l l I I f i l l I I 1111 I i 1 2 S tO ~ SO tO0 ~ X ) SO01.000 ~0.000 tOO~XX) 5 0 0 . 0 0 0

wRq~mcy (Hz) Figure 28. Average deep water ambient noise spectra.

ations with respect to time and place and are a mixture of shipping and industrial, wind noise, and biological noise.

Examples of noise spectra for New York harbor in the daytime (AA), upper Long Island Sound (BB), and an average of many WWII measurements (CC) are shown in Figure 3a. Also shown are average subsonic measurements. Data are for the range of 1-20 kHz. Additional sources beyond those mentioned in deep water are: industrial activity, marine life, tidal currents. In coastal waters, wind speed appears to be the main cause of noise. Studies show the noise level is dependent on wind speed at frequencies between 10-3,000 Hz as shown in Figure 29.

For an infinite layer of uniform water with a plane surface along which the sources of noise are uniformly distributed, the ambient noise level (NL) should be independent of depth for frequencies <10 kHz. At frequencies >10 kHz absorption should cause NL to decrease with increasing depth. More realistic results must consider refraction and multipath propagation. At a location north of St Croix, Virgin Islands and offshore San Diego, California, the noise level in the 50 to 3,000 Hz range was found to generally decrease slowly with depth (1 dB/1,000 ft) for depths to 14,000 ft [42-46]. For a non-continuous ice formation, ambient NL of 5 to 10 dB higher than those measured at the same sea state in ice-free waters. With continuous ice coverage, very low noise levels occur for rising temperatures. Impulsive popping noises increase the levels for falling temperatures. The conclusion is that NL is highly variable and depends on ice conditions, wind speed, and snow cover.


I I I I I I i l i | I 111111 I 1 I l l l l l I I I I i i i i ~ . . ,,,

2.0 ~ .,,

,,o ~'/~ ,oo " r >~ ,o " < ~ ~ ~ . . . .

.~ ,o ~~..~ �9 o ~

4 0 l~l �9 I llllll I I IIIIII I II IIIIII I I IIIIII I I IIIIII

g~ ~ sO sO0 I.s tOs t ~ r~

Frequency 01~)

t lO

, , ~ 8 0

I 1 i ' 1 1 1 1 1 i i 1 I i 1 I i I I I i i i i i i ! i i ! l l l l l

- - Knudsen et at (1948)

-- Were (1962)

Wi.d Sp~.d Omot,)_

~ - - - 22 - 2 r -

, , ~'-- 7- I0-

~11 l 1 1 1 1 1 1 1 1 i i i 1 1 1 1 1 I I I l l l l J 1 1 1 1 1 1 1 1

t 0 t 0 0 ~ . 0 0 0 ~0.000 ~O(XO00

Frequency (Hz)

Figure 29. Typical noise levels in bays and harbors (a) and coastal waters (b) [43].

Scattering and Reverberation Level

Reverberation is the sum of the acoustic energy scattered back to the receiving array by inhomogeneities in the ocean medium, such as dust particles, schools of fish, sea mounts, and marine organisms. Reverberation can severely limit sonar system performance. Types of reverberation include volume, surface, and bottom reverberation. Volume reverberation occurs in the volume of the sea and is caused by marine life, inanimate matter, and the inhomogeneous structure of the sea itself. Surface reverberation is caused by scattering from waves and bubbles while bottom backscattering from the boundary and inhomogeneities in and on the sea bottom produces bottom reverberation.

A scattering strength parameter (Ss,v) is defined as

Ss, v = 10 log Isca----~t (93) Iinc

where Iscat is the intensity of sound scattered by an area of 1 yd 2 or volume of 1 yd 3 and Iinc is the intensity of the incident plane wave (Figure 30).

U n d e r w a t e r A c o u s t i c s 4 3 5

Equivalent plane wave reverberation level (RL) in sonar equations is the level of an ax ia l ly inc iden t p lane wave that wou ld p roduce the same hydrophone output level as the reverberation does. The propagation loss is assumed to be due to spherical spreading only. Other assumptions include: random, homogeneous distribution of scatterers, density of scatterers is large, pulse duration is short, and there is an absence of multiple scattering. The volume reverberation is evaluated by

RL v = S L - 40 log r + Sv + 10 logV

V = c 1 7 _ r 2 2 I1/ (94)

where c is velocity of sound, x is pulse duration, r is range, and ~ is the composite-transmit receive pattern beam width or equivalent two-way beam width. SL is source level, and S v is scattering strength.

Surface reverberation is reverberation produced by scatterers distributed over a nearly plane surface. The expression for equivalent plane wave level of surface reverberation is

RL s = SL - 40 log r + S s + 10 log A C'17

A = - - @ r (95) 2

where A is the area of the surface of scattering strength (Ss) lying within the ideal beam width (~) which produces the same reverberation as that actually observed. The ideal beam width (~) is tabulated in Table 11.

! inc

~1 , . . ~ _ / ' . , 4- . . . . . . . . ~.~

/ ..... , ,.t"," a l p \ / -.>1..- / ........ �9 . . . . . . .~ ",...

/ / / / " ~'x../. P . - / " " "~" . 1 " ' , r / / " ,A/ \ . , o ~ ",, / / / m , ~ . "'. \ / , ; , ~ ; ,,, ,, \

', ~ _ "I" 3 / .." \ \Unit Volume = I yd ..... /

! inc

/

~ : " , ......... �9 ........ "~.

" / yd \ \

Unit Area = 1 yd 2

Volume

I lscat Surface S = 10 log scat

S v = 10 log s I inc I lnc

F i g u r e 30. S c h e m a t i c o f v o l u m e and b o t t o m reverberation.


Table 11 Values of Equivalent Two-Way Beam Widths W and ~ in Log Units

Array 1O log ~ dB re I steradian 10 log �9 dB re I radian

Integral expression j,2n i,~/2

10 log | | b(O, tp) b'(O, tp) x cosO dO d~0 dO t/-lt/2 ~0 TM

10 log b(O, O)b'(O, (p)d( p

Circular plane array, ( ~ _ ~ / in an infinite baffle 20 log + 7.7 or 10 log ~" 2ha

of radius a > 2~, 20 log y - 31.6 10 log y - 12.8

+ 6.9 or

Rectangular array in an X2 10 log ~' infinite baffle, side a 10 log 4~ab + 7.4 or 2ha + 9.2 or

horizontal, b vertical, 10 log ya Y b - 31.6 10 log y a - 12.6 with a, b >> ~,

X X Horizontal line of 10 log + 9.2 or 10 log + 9.2 or

2nL 2nL length L > ~,

10 log y - 12.8 10 log y - 12.8

Nondirectional 10 log 4n = 11.0 10 log 2n = 8.0

(point) transducer

Note: y is the half angle, in degrees, between the two directions of the two-way beam pattern in which the response is 6 dB down from the axial response. That is, y is the angle from the axis of the two-way beam pattern such that b(y)b'(y) = 0.25. For the rectangular array, Ya and Yb are the corresponding angles in the planes parallel to the sides a and b.

2 0 0

600

1,000

1,400

1,800

2 , 2 0 0

Z,600 3,000

I ! u !

10 kH~

_ ~ / Night

" / / / ~ Day

t I , ,I I ,

- ~ - s o - ~ -7o

S, (~B)

. . . . . If I �9

60 25 kBz

../.- - 180

- 4 2 0 ~

~ o ~

78O

A l = l I

-.~o -so -eo -to S, (d]B)

Figure 31. Mean profiles of S v at two frequencies for six locations between Hawaii and California [2].


The scatterers that cause the volume reverberation are commonly biological in nature (marine life in the sea). The deep scattering layer (DSL) is a complex aggregate of different biological organisms and its scattering strength varies with frequency, location, season, and time of day. The DSL has a diurnal migration in depth, and it is at a greater depth by day than at night. A rapid depth change occurs at sunrise and sunset. The depth migration is over several hundreds of feet, and the DSL appears to adjust its depth to maintain a constant intensity of light illumination. Typical characteristics of DSL are that it rises at sunset and descends at sunrise, is expected at depths between 600-3,000 ft by day at mid-latitudes and shallower at night, has a volume scattering strength -70 to -80 dB near 24 kHz, and S v is frequency dependent between 1.6 and 12 kHz. Layers of scatterers occur in shallow water as well, and they are located directly under the ice in the Arctic.

The sea surface is frequently rough and has entrapped air bubbles just beneath surface which makes it a good scatterer. The sea surface scattering strength as a function of angle, frequency, and roughness (measured by wind speed) are illustrated in Figure 32. The sea floor is an effective reflector and scatterer of sound, and the variations of selected scattering strengths are shown in Figure 33, and the under ice reverberation data are shown in Figure 34. An example reverberation problem is tabulated in Table 12.

,~II:Xtl

~~176 ~ ~ 3 ; L - ~ s . l s e; x64.15

�9 ...:> ' , ~ , m~ ,z ~s x2.1'3 Z.t6 f ='__

-ZO x , . , 2 ~ ~ ' ~/-=-t'-t6 "5

-, ,__~,o~- ~ _ / ' 7 ~,o - - " "~U5 o,~,~ o4 Io

oZ.~O ~ Z Z " / , , ~ 4 . , K ~ / . ,,~.,~t5 x 1'/13/~)

A 7119155

:3,10 ^ T/21/~t -,o ~.~,~, ~, -~,," ~.~ / j

z.s ~ ^~o., ~ No. of pings averaged II II J" I | I t , I ,, | I ~ _ J

-500 SO 20 30 40 50 SO ~ 80 gO

Grazing Angle ((leg)

Figure 32. Variation of sea-surface scattering strength at 60 kHz with angle at different wind speeds off Key West, Florida [44].

E 0 0 L_

o o

~ ~

~ .

T ,

T Ill

lap) q~C~ms

8 "r-

! i

! �9

o u_

r~

-

o ~

,~ ~

T ,

T ,

(llP) q

~u~ts

~u.ua~:)S~l:)~8 a:).uapu~l

IR ;' 0

.0

f,,}

- IR .~

-N

~L

438


Table 12 Example of Reverberat ion Predict ion

Given: Echo ranging sonar, f = 50 kHz, SL = 220 dB, pulse duration 1 ms. Line transducer, 1 ft long, 100 ft above mud bottom, k - 0.1 at 50 kHz, Grazing angle 0=9 .5 ~

Find: RL for diagonal range to bottom of 200 yd.

Soln: c = k f = 0.1 (50000) = 5000 ft/s From Table 1

0.1 10 log �9 = 10 log "~n + 9.2 = 10 log 2n(1) ~ + 9 . 2

10 log �9 = -8 .8 = 0.13 rad or 7.5 degrees

A = nc'l: ~ = 5000 ft / s (0.001 s)

2 2(3 f t /yd) (0.13) (200 yd) = 22 yd 2

From Figure 33, Ss = -41 dB RL = SL - 40 log r + Ss + 10 log A = 220 - 40 log 200 - 41 + 10

log 22 = 2 2 0 - 9 2 - 41 + 13 RL = 100 dB re 1 ~tPa

Target Strength

The target strength (TS) refers to the echo returned by an underwater or sur-

face target, and is defined as

intensity of sound returned by the target

TS = 10 log at a distance 1 yd from its acoustic center _ I r

incident intensity from the source I i (96)

Assume the sphere in Figure 35 has a radius "a" and is an isotropic lossless reflector (echo distributed equally in all directions). The sphere is insonified by

a plane wave of sound intensity I i. The power intercepted by the sphere is

P = ~:a2Ii (97)

Using the isotropic assumption, the intensity of the reflected wave at a distance r

in yards from the acoustic center (center of sphere) is


: /

/

/ i

\

\ \

/ f "~'~,

lit' ., \

/ f - ............. --.,, \ \ \

" 1 i

..... --- J / / x,\ /

. . /

Figure 35. Incident and reflected waves for a sphere.

na2Ii a 2 I = - I ~

r 4 71; r 2 - i 4r 2 (98)

For r = 1 yd

a 2

I r = Ii 4

a 2 TS = 10 log =

4 (99)

Thus, the arbitrary reference often causes the target strength (TS) to be positive for targets. This should not be interpreted as meaning more sound is coming back from target than is incident upon it. It is a consequence of the reference distance. Theoretical target strengths of a number of geometric shapes and forms are presented in Table 13. These expressions are reasonable approximations for complex targets and are found to provide useful results when no measured data are available. Complex targets may be broken into elemental parts and by replacing each part by one of the various simple forms. The variation of target strength for a submarine at various aspect angles is illustrated in Figure 36 and nominal target strength values for different targets are tabulated in Table 14.

Radiated Noise Levels

Ships, submarines, and torpedoes are all sources of radiated noise. The machinery in these vessels generate vibrations that appear as underwater sound at a distant hydrophone after transmission through the hull and through the sea.

F o r m

Finite P l a t e u

any shape

Table 13 Target Strength for Simple Forms [43]

Target strength = 10 log t

Rectangular

Plate

Circular Plate

Ellipsoid

Average over-

all aspects--

Circular disk

Conical tip

Any convex

surface

Sphere- - la rge .

Spheremsmal l

Cy l inde r - -

Infinitely long

Thick

Cy l inde r - -

Infinitely long

Thin

Cy l inde r - -

Finite

Plate--Inf ini te

(plane surface)

2 a )

a 2

8

( 8 - ~ ] 2 t a n 4

qo 1 - cos2 qs

al a2

a 2

4

V 2 61.7 m ~4

a r

2

911;4 a 4 ~ r ~2

aL2/2~,

aL2/2~,(sin~/~) 2 COS20

Symbols

A = area of plate

L 1 = greatest linear

dimension of plate

L 2 = smallest linear

dimension of plate

a,b = side of rectangle

13 = ka sin0

a = radius of plate

13 = 2ka sin 0

a,b,c = semimajor axes

of ellipsoid

a = radius of disk

tp = half angle of cone

ala 2 = principal radii

of curvature

r = range

k = 2/7t X

a = radius of sphere

V = volume of sphere

X = wavelength

a = radius of cylinder


L = length of cylinder



13 = kL sin 0

Incidence Direction

Normal to plate

At angle 0 to normal

in plane containing

side a

At angle to 0 normal

Parallel to axis of a

Average over all

directions

At angle 0 with axis

of cone

Normal to surface

Any

Any

Normal to axis of

cylinder

Normal to axis of

cylinder

Normal to axis of

cylinder at angle 0

with normal

Normal to plane

Conditions

r > I-'~

k L 2 >> 1

a 2

r > m

k b > > l

a > b

a 2

r > ~

ka >> 1

ka, kb, kc

1

r >> a, b, c,

ka >> 1

(2a) 2 r > ~

0 < t p

kal ,ka 2 :,, 1 r > a

k a : , , 1

r > a

ka << 1

k r > > l

k a ~ 1 r > a

k a , , : 1

k a : ~ 1 r > LE/X

Note: All dimensions are in yards.


0 !

2 7 0 ~

1 8 0 ~

>

I

\

9 0 ~

Figure 36. Target strength variation for a submarine at different aspects [43].

Table 14 Nominal Values of Target Strength [43]

Target Aspect TS (dB)

Submarines Beam +25 Bow-stem + 10 Intermediate + 15

Surface ships Beam +25 (highly uncertain) Off-beam +15 (highly uncertain)

Mines Beam +10 Off-beam +10 to -25

Torpedoes Bow -20 Fish of length L (inches) Dorsal view 19 log L -54 (approx.) Unsuited swimmers Any -15 Seamounts Any +30 to +60

Passive sonar systems distinguish between radiated noise and a background of self and ambient noise. Noise spectra are of two basic types that are called broadband (continuous spectrum) and tonal noise (line component).

The radiated noise from vessels is usually measured by having the vessel move by a stationary distant hydrophone array system. The Atlantic Undersea Test and Evaluation Center (AUTEC) is instrumented for the measurement of


radiated noise from submarines. The measured noise levels are normally reduced to the 1 yd reference distance, and spherical spreading is normally used for this reduction.

Sources of radiated noise are machinery, propeller, and hydrodynamic. Machinery noise is vibration that is coupled to the sea by the hull. Noise origi- nates inside the vessel from rotating parts, reciprocating parts, cavitation and turbulence, and mechanical friction. Propeller noise is the result of cavitation at the propeller tip and on the blade surface, and it has a continuous spectrum. There is a critical speed at which the cavitation noise suddenly begins. For WWII subs, the critical speed was 3-5 kts at 60 ft (periscope depth), and the critical speed increased with increasing depth.

Damaged propellers make more noise than undamaged propellers. Turns and accelerations result in more noise. Singing propellers result when the propeller blades are excited by the flow into vibrational resonance. Propeller noise is amplitude modulated or has a beat that increases with rotation speed and can be used to estimate speed. Hydrodynamic noise is caused by irregular and fluctuating flow of fluid past the moving vessel. Flow noise is the result of flow of a viscous fluid over immersed bodies. Flow noise is a normal characteristic.

Tabular and graphical radiated noise level data for surface ships and submarines are contained in Table 15 and Figure 37 and Figure 38. These data are from the World War II era and in many cases are not representative of today's vessels.

Self Noise Levels

Sources of Self Noise

Self noise differs from radiated noise since the receiving hydrophone is located on the platform making the noise. Sources of self noise are the vessel's machinery (reduce by sound isolation), propellers (cavitation), and hydrodynam-

Table 15 Average Spectrum Levels for a 1 Hz Band for Several

Types of Surface Ships (dB re 1 ~tPa at 1 yd)

Frequency, Freighter, Passenger, Battleship, Cruiser, Destroyer, Hz 10 kt 15 kt 20 kt 20 kt 20 kt

100 152 162 176 169 163 300 142 152 166 159 153

1000 131 141 155 148 142 3000 121 131 145 138 132 5000 117 127 141 134 128

10000 111 121 135 128 122 25000 103 113 127 120 114

4 4 4 Of fshore Engineer ing

~ ' " ~ I ' ' ' ' " ' ' " " ' " " ' " ' " " ' ' ' " ' " " " " ~ s N l ~ I ~ " ~ ~ , ! ' l" Bottlesl~ps / A I E~.o,.,, I "

t4 I_ ,=r=4.9 / /m . =r=52 " ! 0 I y

I (orrie~ I / / / / , I i . ~ ~ . / I Freighte, i / ' s ~ ~ ~ ~ ' ~ I I - - ( N o ooo~ ~.o,),,,/--,~--~

,,~1 i ~"~Escortl I .'~.'~.~' 1 / / ~ ~ ; , I ,oo,, / | Potrol croft ~ ~ ( x ~ , ~

,,oI / ',~ _ __ c~'~r, 3 m ~ . ~ _ ~ , ~ ~ _ o ~ , , ~ ] . / / '

/ I / / / / [ , 7 ~ ~m~,e"8"1 i

tOO I / / / potrot boot I "1:~ ,,,--,,, _ l ~ ~ ~ ' ~ Y ' l ..2.~ j --/--,.~~

Tug (oceon- going-] . . . . ~.:' 4.:3 -.. 9o' I I / ,/'#' I so~>ge) , I

I / 7 - , , - .~.s I I | i I i l l i l t i t I l I | [ / ' [ I i I i I l | I l l l l m i l i l i l i l i l l | I I I | l i l l I l l i l

6 7 8 SO t5 20 30 3 4 5 6 7 8 t0 t5 20 30 5 6 7 8 10 15 20

F igure 37. Average rad ia ted spec t rum levels (s tandard dev ia t ion of o) for sur face sh ips [11].

160 r " - i f II w i l l ] 1 ! 1 I I I 1 - I " I I

"~ tso ,

1 4 0 -~ ~ ~ ~.....~Periscope Depth 'Speed r 130 . . . .

~ - - - - ~ ~ , ~ 110,, ~ ~ _ ~ ' - 6

r162 10o . . . . . . - ~

~ ,4o ~ . ~ . ~ . ~ Surface . . . .

I . , I 0 CJI

IOO[---:-l [_],llll l__ ~ 6 1 , mOO t . O 0 0 JO.O00

Frequency (1~)

Figure 38. Radiated noise level of several d i f ferent submar ines [22].


ics (water flow past hydrophone and vessel support structures). Figure 39 indicates that machinery is the predominant cause of self noise at low frequency and that the propeller and hydrodynamic noise are the main cause of self noise at high frequency. The self noise associated with machinery is relatively independent of the vessel speed, but the propeller and hydrodynamic noise are strongly affected by speed. Example data for self-noise on a destroyer and submarine are illustrated in Figure 40.

Speed

i

Propeller" Mac j Hydrodynamic

Ambient i

Low " High Frequency

k v

Figure 39. S p e e d a n d frequency relationship for self noise.

5O

| , , |

1 #

/ /

/ , r

/ / ~ / I J

/ / ' Propeller and h y d r ~ l ) ~ i c n ~i~

. . . . . . . i 0 S IO IS 20 2S

Speed (knm) (a)

(

\\

- . . . ,

i i l l l l ] I i l l l i six) r soo t~oo z.ooos~o

--i i

(b)

/

Figure 40. Self noise level for a destroyer (a) and submarine (b) [43].

/ /

/

. / . 2 �9 6


Flow Noise

Flow noise is a form of hydrodynamic noise and is the result of turbulent pressure fluctuations over the face of the hydrophone. Pressure fluctuations are the result of the turbulent boundary layer about the hydrophone. Surfaces should be free of roughness which extends through the laminar boundary layer and affects the turbulent flow. Otherwise, the surface is considered smooth. The coherence of turbulent pressures is determined by experimental correlation coefficients (p) have been evaluated as

PL (S) = e-~ longitudinal separation

PT (S) = e -Sisl transverse separation (100)

where S = Strouhal number (S = fd/u c) f = frequency d = separation distance

u e = convection velocity

The convection velocity represents the velocity at which turbulent eddies trans- late past the hydrophone.

A pressure transducer of finite size discriminates against flow noise to an extent determined by PL and PT" A discrimination factor 13 is defined as

R p

13 = - - (101) R

where R' is the mean squared voltage output of the array in the flow noise and R is the mean squared voltage output of a very small transducer in the same flow noise and having the same sensitivity as the array. Thus, the discrimination factor is a measure of the reduction of flow noise experienced by an array, and it is expressed in Equation 102 and illustrated in Figure 41 for rectangular and circular arrays for elements having a uniform response function. The parameter T should be much greater than one. Variable response functions were investigated by Randall [36] and were found to reduce the response of the array to flow noise in some cases.

0.659 RectangularArray: 13 = ~r where ), =

0.207 CircularArray: I] = T2 where T =

2 n f L

uc (102) 2 n f r

U c


I 0

.| o-, i

ODOI

Underwater Acoustics I 1 1 " 1 1 1 1 1 ' 1 1 �9 I I 1 1 1 ' ' " 1 I l l l l ~

I ' J O

\ kX .... o

- i i 1110

I I 1 1 1 1 1 1 l l I l I I I I I l

t.O ~O tOO

2 x f ~ 2 x f r Strouhal Number

Uc U c

Figure 41. Discrimination of rectangular and circular arrays against flow noise [10, 49].

Flow noise reduction is accomplished by making the hydrophone larger, moving the hydrophone forward, removing hydrophone from turbulent boundary layer, and ejecting polymer fluids. Domes reduce self noise by minimizing turbulent flow, delaying onset of cavitation, and transferring the flow noise source away from the transducer. The domes must be acoustically transparent, produce no large side lobes, be streamlined, and kept free of marine fouling.

Detection Threshold

Detection threshold (DT) is the ratio of the signal power (S) in the receiver bandwidth to the noise power (N) in a 1 Hz bandwidth measured at receiver terminals required for detection with some assigned confidence.

S DT = 10 l o g - (103)

N

Detection probability is for a correct decision that a signal is present, and false- alarm probability is for incorrect decisions when a signal is present. The threshold concept sets a threshold so that the decision "target present" means the threshold was exceeded.


I

C~

O.C

OCO01 0 .~ 0.2 ! 5 20 40 GO 80 ~ ~ ~ . 9 ~J.~Jg

Probability of False Alarm p(FA) %

Figure 42. Detection index (d) as a function of the probabilities of detection p(D) and false alarm p(FA) [43].

For a known signal in Gaussian noise, the detection index is

2E d = ~ (104)

No

where E = total input signal energy in the receiver band N O = noise power in a 1 Hz band

The detection threshold for a known signal power (S) and over a duration time (t) is

S d = = 10 log (105) DT 10 log ~-

No A t

For an unknown signal in Gaussian noise and band width (w), the signal to noise ratio is

s sw

N O N (106)


and the detection threshold is

S dw DT = 10 log . . . . 5 log ~ (107)

N o t

As a rule of thumb, the detection threshold for reverberation

dw DT R=51og , (108)

t

However, reverberation bandwidth w' is generally larger than w (receiver bandwidth). Therefore

dw DT R = - 10 log w' + 5 log ~ (109)

t

Underwater Acoustic Applications

Seismic Exploration

The purpose of seismic exploration is to search for ocean subbottom structural features that might have oil and mineral deposits. One of the necessary conditions for the formation of oil and gas fields are that the rock below the seabed has supported simple life. The adjacent rock stratum must be permeable to allow migration of hydrocarbon molecules in an upward direction. There must be an impermeable barrier to capture the upward migrating hydrocarbons, and the bar- tier must be strong enough to remain impermeable for millions of years. There must be a suitable structure or space below the impermeable barrier consisting of permeable rock to allow hydrocarbon accumulation. The combination of an impermeable barrier and permeable rock must have occurred before or during the migration of hydrocarbons. Therefore, the age of the structure is as important as the size and isolation of that structure.

The various types of subbottom structure where oil and gas are likely to be trapped include anticlines, salt domes, faults, pinchouts, and limestone reefs. An anticline is a common type of trap that occurs below the seabed, and if the anticline has an impermeable layer overlaying a permeable layer, then it may also serve as a hydrocarbon reservoir. Salt domes are formed when a mass of salt flows upward and results in a mushroom type structure beneath the sea bed. The salt structure is impermeable to hydrocarbons, and consequently, petroleum reservoirs may form around the sides of the dome when permeable layers inter- sect the salt dome. A fault that occurs in the seabed may result in an impermeable layer overlaying a permeable layer, and it is called a fault trap. Pinchouts occur when a reservoir bed gradually thins and eventually pinches out. Lime-


stone reefs are often covered by deposition of impermeable material and the reef material is usually porous and acts as a trap for petroleum.

The seismic survey principle uses an acoustic source that is activated at or near the water surface at a known time. Selected acoustic and energy sources for conducting marine seismic profiling are tabulated in Table 16. Acoustic waves radiate downward through the ocean waters and bottom sediments. When there is a major discontinuity between one type of rock and another, part of the signal is reflected back to the surface. Then, by measuring the time taken for this signal to reach and return from each stratum, an estimate of the depth of the stratum below the surface can be determined by assuming a sound speed in the layers. Hydrophones are placed in a straight line (i.e. streamer) at specific distances to record characteristics of the signals and times of their arrivals. Seismic records are stored and played back for interpretation, and grid lines are followed by the vessels conducting the seismic survey.

Analysis of Seismic Reflection Data. Seismic reflection data are recorded on a reflection record that is a record of the voltage output from each hydrophone as a function of time. Each line on the record represents a hydrophone in the array of hydrophones being towed behind the seismic vessel. The distance from the seismic source near the vessel to the individual hydrophones is known and constant. Also the time of initiation of the source signal (e.g., air gun, sparker, boomer) is known and recorded on the record. The reflection record has a series of timing lines or marks across the record at selected time intervals (e.g., 0.01 to 0.005 s). The seismic signal travels through the water and into the seafloor where a portion of the signal is reflected off the first subfloor interface and is received by the hydrophones in the seismic array. The reflected signal is received at different times for each hydrophone and recorded on the seismic

Table 16 Marine Seismic Reflection Profiling Methods and Equipment

Acoustic System Energy Source

Explosive Sources

Air Gun Aquapulse Sparkers

Boomer Vaporchoc

Dynamite Nitrocarbonitrate High pressure air escapes from chamber and oscillates like a bubble. Detonation of propane and oxygen Sound waves are generated by sudden discharge of current in the water between electrodes Disk moves against water High pressure stream injected into the water.


record. Reflections from all the layers detected are identified and marked on the reflection record. The seismic record is analyzed to determine the time that the reflected signal from each layer is received by each hydrophone. These data are then used to determine the depth to the layer. Using similar information for each initiation of seismic signal and the exact geographic location of the seismic vessel, the structure of the ocean subfloor is mapped. These results are subsequently analyzed to locate subfloor structure that is likely to have oil and gas. Explorato- ry drilling and coring is then used to confirm the presence of gas and oil.

Consider the simple case of a level reflecting bottom surface and uniform layer velocity as illustrated in Figure 43. The location (S) represents the seismic source and the location D represents a receiving hydrophone. The horizontal distance between the source and the hydrophone is denoted by "X" and the distance to the reflecting interface or layer depth is denoted by "Z." The straight line distance the signal travels from the source to the reflecting surface is determined by the product of the average sound speed (velocity) of the layer material and one half of the time it takes the signal to reach the hydrophone as determined from the seismic record.

Using the results of the well known Pythagorean theorem, an expression for the relationship between X, T , V and Z is obtained as

= + Z 2 (110)

and after rearranging, the expression is

S D L x J

reflecting surface i R

Figure 43. Simple seismic reflection where S is location of seismic source, D is location of receiving hydrophone, V is average sound velocity in the layer, and T is travel time of reflected ray.


T 2 X 2 4Z 2 = -~T + ~ V2 ( I I I )

This equation is a l inear express ion with a slope (1/V 2) and an in tercept (4Zz/V2). The seismic record analysis provides data for the time it takes the reflected signal to reach each hydrophone whose distance from the source is known and constant. Linear curve fits to the seismic record data of T 2 versus X 2 for example data are tabulated in Table 17.

Table 17 Linear Curve Fits to Example Seismic Record Data

Layer T 2 = f(X 2) Slope Intercept

1 T 2 = 6.7 x 10 -9 X 2 + 0.27054 6.7 X 10 -9 0.27054 2 T 2 = 5.74 x 10 -9 X 2 + 0.53445 5.74 • 10 -9 0.53445 3 T 2 = 4.98 x 10 -9 X 2 + 0.72866 4.98 X 10 -9 0.72866 4 T2= 5.12 x 10 -9 X 2 + 3.12385 5.12 • 10 -9 3.12385

The slope and intercept values are used to evaluate the average sound speed (velocity) (V) in the layer and the depth (Z) to the layer reflecting surface. Example results are tabulated in Table 18 showing the depth to the layer and the average sound speed in the layer. This method of analysis is very simplified, and the analysis of actual seismic records are much more compl ica ted . More advanced texts such as Coffeen [9] should be consulted for more details and explanations.

Acoustic Position Reference System for Offshore Dynamic Positioning

An offshore dynamic positioning system is used on offshore drilling vessels to maintain position of the vessel over wellheads located on the sea bottom. This system is composed of several basic elements as illustrated in Figure 44.

The purpose of the sensors is to gather information with sufficient speed and accuracy for the controller to calculate the thruster commands so that the vessel performs the desired task. Information required is vessel position, vessel heading, wind speed, and direction. The offshore position reference must include not only navigation but also an accurate and repeatable local position reference. The navigation position is the location on earth 's surface, and the local position needed for thruster control is the location relative to a point of interest on the


Table 18 Example Problem for Data from Seismic Record

Example Calculations for Slope and Average Sound Speed (Velocity) in Layer I

slope = T 2 / X 2 1 S 2 = ~ - = 6.7 • 10 -9 ~ f t 2

V2 _. 1 = 1.4925 x 108 ft 2 / sec 2 6.7 X 10 -9

V - 12,217 f t / s

4 Z 2 ft 2 intercept = - - ~ = 0.27054 - - - " T

0.27054 V 2 Z 2 =

4 Z = 3,177 ft

0.27054 ~ - 12,217

= 10.095 x 106 ft 2

Summary of Example Results

Layer Average Velocity Depth

1 12,217 3,177 2 13,200 4,825 3 14,175 6,050 4 13,975 12,350

ENvIRONME~rT Wind

Waves Position & Current Heading '

C ~ Thrust dThrus'ter' d--Vessel . . . . ~- -~ ,,. C~176 ] Comm=~l.l~ System - I~ Dynamics I

" ] HeadingSemors ~q . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 44. Schematic of an offshore dynamic positioning system.


seafloor. One of the commonly used local position sensors is an acoustic position reference system (Figure 45). The local acoustic systems are restricted to a relatively small coverage area. Types of acoustic position reference systems are the short and long baseline systems that use time of arrival, phase comparison, pinger, and transponder acoustic systems. Typically, the short baseline system uses phase comparisons and the long baseline system uses time of arrival. All systems depend on propagation of an acoustic signal from one point to another through the ocean medium. Therefore, the propagation characteristics of acoustic energy in water affect the performance of the system.

Acoustic systems operate by projecting acoustic energy into the ocean medium. In simple systems the acoustic energy travels only from the subsea beacon to receivers on the vessel. In more complicated acoustic systems the acoustic energy is transmitted from the vessel to a subsea transponder, then the transponder transmits an acoustic pulse back to the vessel. As previously mentioned, the transmitted acoustic signal is affected by the medium through which it must travel. The acoustic signal experiences transmission loss that can be estimated using the spherical spreading plus absorption equation,

TL = 20 log r + o~ r (112)

Ambient and self-noise are also problems that affect the system performance and range of operation.

Supporting Equipment

Vessel Mounted Acoustic Transducers

Propagation Path

. . . . . . . . . . . . . . . . . . . . . . . .

Subsea Acoustic Transducer

L. . . . . . . . . . . . . . . . . . . . . . . J

/ . . . . . . . . . . . . . " - ' - , ' - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,~I Signal Position FllP' Interface to Processor ~ Computer . . . .

DP System

: Position |

', Display i

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J

Figure 45. Schematic of basic acoustic position reference system.


Short B a s e l i n e S y s t e m . A geometrical pattern of transmitters or receivers is located either on the vessel or on the sea bottom. The short baseline system has the array located on vessel, and the long baseline system has the array on sea bottom. The basic surface mounted array geometry (short baseline system) is illustrated in Figure 46. The geometrical arrangement of the array elements and the subsea transducer are shown in Figure 46a. The geometry resultant range (Ro) and its projection on the XA-Z A and Y A-ZA planes is illustrated in Figure 46b, which also shows the angles 0vx and 0vy. These angles are used to determine the horizontal distances between the subsea transducer and the center of the plane containing the vessel mounted array. The vessel uses this information to maintain position relative to the subsea transducer using its dynamic positioning system. If the subsea transducer is located on a subsea wellhead, then the

YA

2d, 3 r - ~ - - - - - ~ 2 Horizoatai Phme of Elemems (1, 2, ;3 & 4)

2 ~ ~ ~ ~ Attacl~d to Botlom of Vemd

g

ZA T ~ r (X~. YA, ZA)

(a) Geometry for E ~ Array meat Sudace (Short Budime)

Y.

"7 ZA

, t X.

Tr~mlucer (XA, YA, ZA)

Co) Geometry for X u d Y Coordimates of Sulmea Tr, mdm~r Relative to Center of Eiemeut Array

Figure 46. Surface mounted array geometry [31].


dynamic positioning system can be used to keep the drilling vessel within tolera- ble limits of the wellhead.

The array elements are assumed to be in the same plane and arranged in a rectangular pattern with sides parallel to the acoustic system coordinate frame (X A, Y A, ZA)" The ranges (R l, R 2, R 3, and R 4) can be measured by the acoustic system. The resultant range R o represents the distance from the center of the vessel mounted array to the subsea transducer. The coordinates of the subsea transducer (reference point) are computed, as derived in Morgan [31 ], using

xA ER3-R2J . E J ' I R3 R2 ] - R4 - R 1 R1

YA =I R22dy-RllIR1 +R2]=[ R 3 2 2dy-N41IR3 +R412

- Oxen] (113)

The ranges for a short baseline system are required to compute the coordinates of the subsea transducer, and they can be de termined by knowing the t ime required for the acoustical signal to travel from the subsea transducer (time of departure) to the vessel mounted array elements. A technique for determining position without knowing the time of departure involves forming differences in the ranges such as

2XA dx R 3 -R 2 = ~ Z A X A R3 -R 2

tan0vx . . . . ZA 2dx

-- sin0vx (114)

Similarly, in the direction Y, the equations are

2Y A dy R 2 - R 1 = ~

ZA YA R2 - R 1

tan0vy = ~ _- Z A 2dy --- sin0vy (115)

To measure the difference in the two ranges (R2-R~), time of departure is not necessary

R 3 - R E = c 3 (t3 - to) - c2(tg - to)

R 3 - R 2 = c(t 3 - t2) if c 2 = c 3 (116)


This is valid even if the speed of sound in the medium is not constant along the ray path. It is necessary for the sound speed (c) for the two paths to be known and equal. The position of the vessel can then be computed from 0vx and 0vy

Ovx = sin -1 c(t3 - t 1) 2dx

Ovy ---- sin -1 c(t2 - t 1) 2dy

(117)

For small angles and angular ship motions, corrections for vessel motions are important and the reader is referred to Morgan [31 ] for these procedures.

Acoustic Depth Sounders

A typical echo sounder is shown in Figure 47, and it has an electric motor that drives a rotating stylus. At the point where the stylus passes over the zero of the range-scale, a device causes the closing of the transmission contacts, allowing a current to pass through the stylus and recording chart to the earthing plate (plat- en). A mark is made on the recording chart. The stylus travels across the chart range-scale until the echo pulse generates a current that is applied to the stylus, and another mark is made. The stylus next reaches the transmission point once more and the cycle is repeated. At the same time, the motor/gearbox drive moves the chart paper in a plane at fight-angles to that of the stylus movement, and the resulting succession of marks made by the stylus constitutes a time- depth graph.

The stylus speed is adjusted in calibration to be the same speed as that of the acoustic pulse traveling to and from the seafloor. The chart speed is simply a geared-down proportion of the motor speed and is not the same as the vessel's speed. The "time-depth" graph thus does not represent a vertical-horizontal distance profile and the resulting record is not a true profile of the seabed. The chart continues to move as long as the instrument is operating, regardless of whether the vessel is moving or stopped.

Figure 47 shows that one transmission mark and one depth mark provide but little information. The depth mark could be confused with other spots that inevitably appear on the chart due to pitting of electrical contacts and small midwater targets. Also, the attenuation of the transmitted acoustic power is such that, after the traveling to and from the sea bottom, the echo pulse may not be strong enough for the receiver circuitry to discriminate its arrival from the background noise. However, the succession of stylus marks usually produces a rec- ognizable profile of the bottom. The recorded depth measurements can be related to the position of the vessel using the event marker. Other marks may be made using mechanical cam or slip ring devices, such as time marks, depth scale graduations, and motor speed calibration lines.


Motor and Gearbox Transmission Switch

j ~ Psper Trace Movement I ! ' ' _ ~ : - - ~ _r--~. ~ i ~j

P;',I!, Depth Scale ~ ,J 11I i~ e : | : '. s ! , �9 Stylus Travel " ' ~i =' -J 1 F ~ rS~u' Belt Dflve

I~:" i' / t l + S , ~ - Trolley Bar

I | . i+. . /k ~ J ~ .:1 L. f

IlI~'~RDEll ,

Hull Plate Tram=Bit Receive Trmtsducer Tnmsducer

"t T Transmit Pulse Echo Pulse

Figure 47. Typical acoustic depth sounder [18].

The width of the recording paper is usually designed to represent a number of meters, or feet, of depth (e.g., 30, 50, or 100 m). With a width representing 0-100 m, an echo from a depth of over 100 m is usually not recorded. Also, a depth sounder operating at 30 kHz typically penetrates silty sediments and shows the seafloor and the consolidated matter beneath for several meters. How- ever, a 200 kHz system typically records reflections from the silt alone and often indicates a depth difference of a meter or more than that determined with the 30 kHz system. For this reason, depth sounders are commonly designed to operate at two frequencies simultaneously, especially when siltation and fluid mud are serious navigational problems such as found in the approaches to Europort and US ports along the Gulf of Mexico.


Side Scan Sonar

A side scan sonar is a line array that looks sideways or perpendicular to the survey vessel trace. It operates on two channels with one for each side of the vessel's track. It uses a very narrow beam in the horizontal plane to get high resolution along a strip of seafloor and a broad beam in the vertical plane. The sonar is usually housed in a towed body and images are constructed from successive scans to form a composite picture on a moving strip chart. A sketch of a typical side scan sonar is shown in Figure 48.

Surface Tow Vessel

Submerged Tow Bodl

Port Transducer

Tow Cable

.z-- Starboard Transducer

Swath Width \ Recorder Output of Bethymetry

Figure 48. Top view of a typical side scan sonar system.

A side scan sonar measures and displays ranges to targets from the tow body. The transducer produces the sound pulse and receives the echo. A graphic display shows the echoes and transmitted pulse, but in some cases the transmitted pulse is suppressed. The typical sequence of marks on the graphic display are the transmitted pulse, surface echo, bottom echo, and successive echoes from the seafloor at greater distances from the sonar (slant ranges). The horizontal distance can be calculated based on geometry. Acoustic shadows occur behind objects, and they are shown as white areas on the graphic display. The side scan sonar geometry is illustrated in the Figure 49. The depth of the side scan towbody below the water surface is designated by the numeral 1, and the height of the towbody above the seafloor is defined as numeral 2. Numeral 3 is the slant range to the target and numeral 4 represents the length of the acoustic shadow.


surface vessel . . . . . . . . . . . . .

ide scan Lowbody

3

sea floor I ' ' ' " ' " " ' ' . . . . . . . . . . . . . ' . . . . . . . - I

I L s h a d o w zone

Figure 49. Typical side scan sonar geometry.

Across-track range resolution is the ability to distinguish between two distant objects, and the theoretical minimum separation is one half the pulse thickness (spatial pulse length). Examples of minimum separation are 0.75 cm for 500 kHz sonar and 7.5 cm for 50 kHz. Across track range resolution improves with distance from the tow fish, and a side scan sonar can image targets as small as 1 cm in diameter. The along-track transverse resolution distinguishes between two distinct targets on the seafloor separated in the direction of tow. If the two objects are spread less than the spread of the beam, then the objects are merged on the graphic display. At closer ranges the beam is narrower and the two objects can be resolved. Resolution is also dependent on tow speed and the interval between pulses, but the beamwidth is the most critical factor. Higher frequency sonars have shorter pulse lengths. The narrower beams give better resolution, but the range will be decreased. Lower frequency side scan sonars have greater range but less resolution. Beamwidth can be narrow, horizontal plane (approximately 1 o), or broad, vertical plane (approximately 40~ Example beam patterns for a side scan sonar are shown in the Figure 50.

Absorption, spreading, and scattering tend to weaken a signal returning to the sonar. It is desirable for the sonar data of a given bottom type to look the same at 150-m range as it does at the 50-m range. Therefore, amplification of the returning signal is needed to overcome losses in particular operating areas, and this amplification is called time varied gain (TVG). Self noise and ambient noise can also reach the side scan sonar and interfere with the desired signals. Some detec- tive work is often necessary to determine what the noise source is and how to


~easity (dB)

Ltin Lobe

Back Front

tO"

Typical Vertical Beam Pattern

Typical Horizontal Beam Pattern

Figure 50. Beam pattems for common side scan sonar [20].

eliminate or deal with it. Sometimes you can just turn off a piece of equipment and the noise source is eliminated. Noise can enter the system acoustically or electrically. Acoustical noise enters the system from the water through the transducer as does the signal. Electrical noise is the result of the power supply, cable faults, grounding problems, and components failures.

Examples of interpretation of side scan sonar records are best illustrated in some selected records such as those shown in the following figures. Surface return echoes are illustrated in Figure 51. Multiple reflection echoes from a single target do not occur very often but do occur under certain circumstances. Examples of multiple reflections are shown in the Figure 52. Targets that pro- trude above the bottom block sound rays from reaching the bottom depending on the height of the target and sonar above the bottom. As a consequence a light area (shadow) appears on the sonar record as illustrated in Figure 53. A target projecting above the seafloor typically produces a dark mark followed by a light mark on the sonar record. A depression typically produces a light mark or area preceding the dark return. Examples of depressions are drag marks, trenches for pipelines or cables, and scours.

A target on the seafloor typically shows a shadow immediately following the dark mark of the target. A target above the bottom but below the sonar transducer shows the dark mark of the target followed by the seafloor and then a delayed shadow. A target in the water column that is above the sonar transducer typically shows the dark mark but no shadow follows. An example of a sonar record of the target in the water column is shown in Figure 54.

As a result of the side scan sonar geometry the ranges to the target are actually slant ranges. Thus, the distances on the sonar record to the target are not horizontal. Also, the differences in slant ranges to the leading and trailing edges of the target are always less than the real extent of the target. The closer the sonar



water surface . . . . . . . . . . . . . . . . . L

towbody(sonar)

water surface

towbody(sonar)

SONAR RECORD

surface surface return -] ~-return

SONAR RECORD

surface surface return -1 [ " r e t u r n

bottom output bottom bottom ---J Lrl L_ bottom pulse output

pulse

Figure 51. Schematic of surface retums for different depths of towbody.

su r face

soaar

pa ths

1-1

1-3-2 2-3-1 2-3-3-2

pipe

Actual Record

. . . ,~ '~. .~.~ . ..

� 9 �9 . ' ~ & �9 " .

Figure 52. Examples of multiple reflections and possible multipaths for side scan sonar.


water surface

S O W

~ONAR RECORD

output pulse depression

Figure 53. Example of depressions and projections in side scan records.

water surface

SONAR RECORD Ae~d Reem'd

output

I! m Figure 54. Example side scan record of targets.



is to the target, the more the distortion. Corrections can be made using horizontal offset calculations and target height calculation can also be made as shown in the Figure 55 and Equations 118 and 119. The horizontal offset (r t) to the base of the target is determined by

1

r t : ( s ~ - h s 2 ) ~ (118)

where h s = height of the tow body above the bottom s t = slant range to the target as measured from the side scan record.

The target height is evaluated by

ss(hs) h t = ~ (119)

S t + S s

where s s = length of the acoustic shadow as measured from the side scan sonar record

As an example, a survey vessel has conducted a side scan sonar survey at a speed of 3 m/s. The recorder line density was 50 lines/cm and the 40 m range was selected. Assuming the sound velocity was 1,500 m/s, determine the hori-

S \

surface vessel . . . . . . . . . . .

sea surface

~ _ _ _ ~ e scan towbody

, S t

h s

' , . . . . . . _ . . . . . . . . . . . . . , [

EL rt -1_1 t - L_ shadow zone . . w I

Figure 55. Horizontal offset and target height calculations for side scan sonar record.

0.25 cm


Across-track 2 4 6 8 10 12 14 16 18 20 cm

i I - I , I ! , t i' " i i.lil

i ~ [ . . . . 1 ! J t j ! surface bottom

Figure 56. Side scan sonar record example.

zontal d is tance (rt) to the center of buoy, its he ight (ht) above the b o t t o m and

d iame te r f r o m the part ial record shown in F igure 56.

The resul ts for the d iameter , r t and h t are tabula ted in Tab le 19. The d i ame te r

o f the buoy is 2 m and the hor izonta l d is tance to the cen te r o f the b u o y is 15.9 m

normal to t rack o f the survey vessel . The he igh t o f the cen te r of the b u o y above

the bo t tom is 2.1 m.

Table 19 Calculations from Example Side Scan Sonar Record

Across-track scale = range/paper width = 40m/20 cm = 2 m/cm Along-track scale = vessel speed • line density • range/750 = (3 m)(50 m)(40 m)/750

crn/s = 8 rn/cm Towbody depth = number of divisions (1 division = 1 cm) across track • across-track

scale = 1 cm (2 m/cm) = 2 m Depth of water = 4 cm (2 m/cm) = 8 m Height of towbody above the bottom (hs) = 8 m - 2 m = 6 m Horizontal distance to center of buoy (rt)

Equation 118 yields: r t = afst 2 - hs 2 = ~ [(8.5 cm)(2 m/cm)] 2 - [ 6 m] 2 = 15.9 m

Height to center of the buoy

hs(s s) 6m(4.5 cm • 2m/cm) 54 Equation 119 yields: h t . . . . . 2.1 m

s t + s s 8.5 cm • 2m/cm + 4.5 cm • 2 m/cm 26

Diameter of Buoy Diameter = number of divisions along track (1 division = 0.25 cm) • along track scale

(8 rn/cm) Diameter = 0.25 cm • 8 m/cm = 2 m

Subbottom Profiler

The opera t ion o f subbo t tom prof i lers is typ ica l ly in the low k H z range (e.g.

10 -50 kHz) . The use o f these l ower acoust ic f r equenc ies r educes the potent ia l


resolution but allows penetration into marine sediments, permitting geotechnical inspection of the seafloor, as well as the possible location of buried objects such as wreck artifacts, cables, or pipelines. Figure 57 shows a subbottom profiler output and demonstrates that brine jets with a salinity of approximately 230 o/oo (parts per thousand) are detected as the jet rises above the seafloor from nozzles in a buffed brine pipeline [51 ].

45

50

55

60

65

70

75

Depth (ft)

1~ ~ =1,~ ~.~ 11- ' . . . . . . . ~ . _~ , . , ~ . . . . . . ~1,._-.,:. . . . . . . . , , . , - t ..- ,-- I - ,, - - t . �9 ~ ".

I il ,~:

"'i: '-" ,,:c-a..:'.,:-,'~-':.U .... ~-I ,=..:-..:: ,-~::::. z,,:..:..:~.~.-: t i : : : , .~ i , :p i : ..... 1.11 .,;. : : t : ' . " " 1 ~ . . . . . . . " . . . . . . ' , t ~ ; . - ' : - ' - ~.'..~: . . . . . . ~ . ' 1 " . ' . ' - " ~ : : . . . . . . . ] l l ~ ,~ - " ,~11C~ I . ~ ' . ' - . ' : . . . . . . .

- : - ' : . . . . : = . . . . . . . . . . . . . ~' . . . . . . ~ .... : . . . . . . . . . . . . . . . . . �9 . . . . 10 | ~: . ' -~ . . . . . . . . . . �9 ~: .... " �9 '1i.~" I ~'-',: ....... :::~, ':::- . .: , ." :'-'-":~"-" ~:',:::":":i'" "," :':1 I ~ :-.-..-~::::.-,'. : : , : : .~~I.,:;-.

!i t

�9 " - , . ; ~ - . . . - . . . . . . . . . . . . ..: . . . . . ~ : ~ ' . . , i - . - : . - . ~ : : . . . . ~ ~ - - : - . ; . . . . : - : - . : r , - �9 . . . . . . :.. . . . . . ; ..... ..... - - , , - , , ..... . .

22

Depth (m)

Figure 57. Output from a subbottom profiler measuring brine jets issuing from bottom diffuser buried in seafloor.

Acoustic Positioning and Navigation

Transponders are deployed in pairs or in larger networks and can interact with each other to allow determination of their separation by acoustic pulse time-of- flight measurement. The simplest transponders are acoustic beacons that remain in a passive, listen-only mode until awakened by an interrogation pulse, usually transmitted by a searching recovery vessel. They then respond to further interrogation pulses allowing, to some degree, the surface vessel to position itself vertically above the beacon, at which time the transponding round-trip delay is at a minimum.

The method is extended in underwater navigation systems, by installing at least three slave transponder units at particular geographic locations. The master transponder is able to activate coded responses from each of the bottom mounted beacons (slaves) so that, as Figure 58 illustrates, slant ranges between the master and each of the slaves can be measured and, by a process of triangulation, location within the survey area is determined. There are long-baseline and short- baseline systems similar to that described for dynamic positioning systems. The


Figure 58. The circular acoustic navigation principle [8].

former terminology is for seafloor slaves and the latter is from ship- or platform- mounted equipment. The method illustrated in Figure 58 is spherical navigation. The mobile transponder is considered to be at the unique point of intersection of three hemispheres entered on each of the three slaves and of radii equal to each of the respective slant ranges. The intersection of only two such hemispheres produces a vertical semicircular locus of possible positions, rather than a uniquely determined location. Circular, or transponding navigation has the disadvantage that the mobile transponder must have an active transmitter.

Acoustic Doppler Measurements

The Doppler effect is the frequency shift that occurs when either an observer moves with respect to the transport medium, or the medium itself moves. The effect is used in several types of sonar measurement equipment among which is the Doppler current meter. This device, illustrated in Figure 59, is used to measure water velocity at a point, or movement of an object through the water. The operating principle is that of a high-frequency continuous wave sonar with spatially separated transmit and receive transducers. The horizontal flow component along the sound axis between heads A and B and heads C and D of the meter generates Doppler shifts of magnitude

AfAB = fo(V/C)COS0 and mfcd = fo(V/c)sin0 (120)

From these equations, speed v and the direction angle 0 are easily obtained. A similar principle is used in Doppler logs that allow the forward speed and

sideways drift of a submersible to be determined. The calculations involved are similar, except that an additional angular dependence is introduced by a downward depression angle of the beams. The Doppler shift information is contained


C

A

Figure 59. Doppler measurement of water velocity or vehicle velocity using underwater acoustics [8].

in backscattered, rather than reflected, sound. A range-gated Doppler log has been successfully operated for ocean remote current sensing. For this application, the concept is to inspect the Doppler shift on signals backscattered in the consec- utive range cells of a pulsed high-frequency sonar. The instrument takes advantage of the signal processing power available using modern microelectronics.

The Doppler principle has also been used with considerable success in monitoring seafloor geotechnical properties. In this case, a constant frequency 12- kHz transmitter is placed in the tail of a free-fall, torpedo-shaped penetrator. As the penetrator descends, it accelerates, with a corresponding Doppler shift, until it reaches a terminal velocity which, for a two-ton penetrator, can exceed 100 miles per hour (50 ms-~). At this speed the received signal at the surface is approximately 11.6 kHz. On impact the penetrator decelerates, and the Doppler shift decreases to zero. This allows the deceleration profile to be measured and the depth of penetration to be calculated. As a result, the sediment strength can be estimated remotely, without coring.

Water currents in the ocean and in the laboratory are now commonly measured using acoustic Doppler techniques. An example of an acoustic Doppler current meter is shown in Figure 60. A Doppler current meter has the capability of measuring currents at different depths. Its principle is based on the concept that the acoustic signal is reflected off particles in the water, and the speed of the particle causes a Doppler shift in the return signal. The resulting frequency shift is directly related to the water current in which the particle is traveling. This principle is also used in the laboratory to obtain three components of velocity, and an example of such an instrument is illustrated in Figure 61.


Figure 60. Typical acoustic Doppler current meter (courtesy of RD Instruments).

Transmit Receive Transducer Transducer

�9 d

b 30 ~ t d 4V

�9 �9

S~n~pling ~" VOlume

Figure 61. Acoustic Doppler velocimeter for laboratory (courtesy of Sontek).

References

1. Albers, V. M., 1965. Underwater Acoustics Handbook II, Penn State University Press, University Park, PA.

2. Anderson, V. C., 1967. "Frequency Dependence of Reverberation in the Ocean," J. Acoustical Society of America, 41:1467.

3. Arons, A. B., Yennie, D. R., and Cotter, T. P., 1949. "Long Range Shock Propaga- tion in Underwater Explosion Phenomena II," U.S. Navy Dept. Bur. Ord. NAVORD Rep. 478.

4. Brown, J. R., 1964. "Reverberation under Arctic Ice," J. Acoustical Society of America, 36:601.

5. Burdic, W. S., 1991. Underwater Acoustic System Analysis, 2nd Edition. Prentice Hall, Englewood Cliffs, N. J.

6. Burstein, A. W. and Keane, J. J., 1964. "Backscattering of Explosive Sound from Ocean Bottoms," J. Acoustical Society of America, 36:1596.


7. Clay, C. S. and Medwin, H., 1977. Acoustical Oceanography: Principles and Applications, John Wiley & Sons, N.Y.

8. Coates, R. F. W., 1989. Underwater Acoustic Systems, John Wiley & Sons, N.Y. 9. Coffeen, J. A., 1986. Seismic Exploration Fundamentals, 2nd Edition, PennWell

Publishing Co., Tulsa, OK. 10. Corcos, G. M., 1963. "Resolution of Pressure in Turbulence," J. Acoustical Society

of America, 35:192. 11. Dow, M. T., Emling, J. W. and Knudsen, V. O., 1945. "Survey of Underwater

Sound No. 4: Sounds from Surface Ships," National Defense Research Committee, Div. 6, Sec. 6.1, NDRC-2124.

12. Elliott, R. S., 1963. "Beamwidth and Directivity of Large Scanning Arrays," Microwave Journal, Vol. 6, p 53.

13. Etter, P., 1996. Underwater Acoustic Modeling; Principles, Techniques, and Appli- cations, E & FN Spon, N.Y.

14. Fisher, F. H. and Simmons, V. P., 1977. "Sound Absorption in Sea Water," J. Acoustical Society of America, 62:558.

15. Francois, R. E. and Garrison, G. R., 1982. "Sound Absorption Based on Ocean Measurements. Part II: Boric Acid Contribution and Equation for Total Absorp- tion," J. Acoustical Society of America, 72(6), December.

16. Hampton, L. L., (ed.), 1974. Physics of Sound in Marine Sediments, Plenum Press, New York.

17. Horton, J. W., 1959. Fundamentals of Sonar, 2nd Edition, US Naval Institute. 18. Ingham, A. E., 1984. Hydrography for the Surveyor and Engineer, 2nd ed., John

Wiley & Sons, N.Y. 19. Kinsler, L. E., Frey, A. R., Coppens, A. B. and Sanders, J. V., 1982. Fundamentals

of Acoustics, John Wiley & Sons, N.Y. 20. Klein & Associates, 1985. Side Scan Sonar Record Interpretation, Klein & Associ-

ates. 21. Knudsen, V. O., Alford, R.S. and Emling, J.W., 1948. "Underwater Ambient

Noise," J. Marine Research, 7:410. 22. Knudsen, V. O., Alford, R.S. and Emling, J.W., 1943. "Survey of Underwater

Sound No. 2: Sounds from Submarines," National Defense Res. Comm., Div. 6, Sec. 6.1-NDRC- 1306.

23. Leroy, C. C., 1969. "Development of Simple Equations for Accurate and More Realistic Calculation of the Speed of Sound in Seawater," J. Acoustical Society of America, 46:216.

24. Loeser, H. T., 1992. Sonar Engineering Handbook, Peninsula Publishing, Los Altos.

25. Mackensie, K. V., 1961. "Bottom Reverberation for 530 and 1030 cps Sound in Deep Water," J. Acoustical Society of America, 33:1498.

26. Mackensie, K. V., 1981. "Nine-term Equation for Sound Speed in the Oceans," J. Acoustical Society of America, 70:807.

27. Marsh, H. W. and Schulkin, M., 1962. "Shallow Water Transmission," J. Acousti- cal Society of America, 34:863.

28. McKinney, C. M. and Anderson, C. D., 1964. "Measurements of Backscattering of Sound from the Ocean Bottom," J. Acoustical Society of America, 36:158.


29. Milne, A. R., 1964. "Underwater Backscattering Strengths of Arctic Pack Ice," J. Acoustical Society of America, 36:1551.

30. Milne, P. H., 1980. Underwater Engineering Surveys, Gulf Publishing, Houston, TX.

31. Morgan, M., 1980. Dynamic Positioning of Offshore Vessels, Pennwell, Tulsa, OK. 32. National Defense Research Committee (NDRC), 1969. Physics of Sound in the

Sea, US Government Printing Office, Washington, D. C. 33. Officer, C. B., 1958. Introduction to the Theory of Sound Transmission with Appli-

cation to the Ocean, McGraw-Hill, New York, N. Y. 34. Patterson, R. B., 1963. "Back-Scatter of Sound from a Rough Boundary," J.

Acoustical Society of America, 35:2010. 35. Piggott, C. L., 1965. "Ambient Sea Noise at Low Frequencies in Shallow Water of

the Scotian Shelf," J. Acoustical Society America, 36:2152. 36. Randall, R. E., 1973. "Flow Noise Response of a Transducer with Radial Varying

Sensitivity," J. Acoustical Society of America. 37. Ross, D., 1987. Mechanics of Underwater Noise, Peninsula Publishing, Los Altos,

CA. 38. Ross, D., 1976. Mechanics of Underwater Sound, Pergamon Press, New York, N. Y. 39. Strutt, J. W. (Lord Rayleigh), 1945. Theory of Sound, Vols. I and II, New York:

Dover Publications, Inc. 40. Tolstoy, I. and Clay, C. S., 1966. Ocean Acoustics, McGraw-Hill, New York, N. Y. 41. Tucker, D. G. and Gazey, B. K., 1966. Applied Underwater Acoustics, Pergamon

Press, London. 42. Urick, R. J., 1984. Ambient Noise in the Sea, Peninsula Publishing, Los Altos, CA. 43. Urick, R. J., 1983. Principles of Underwater Sound, McGraw-Hill, New York, 3rd

Edition. 44. Urick, R. J. and Hoover, R. M., 1956. "Backscattering of Sound from the Sea Sur-

face: Its Measurement, Causes, and Application to the Prediction of Reverberation Levels," J. Acoustical Society of America, 28:1038.

45. Urick, R. J. and Saling, D. S., 1962. "Backscattering of Explosive Sound from the Deep-Sea Bed," J. Acoustical Society of America, 34:1721.

46. Urick, R. J., 1954. "Backscattering of Sound from a Harbor Bottom," J. Acoustical Society of America, 26:231.

47. US Navy, 1965. "An Interim Report on the Sound Velocity Distribution in the North Atlantic Ocean," U.S. Navy Oceanographic Office, Tech. Rept. 171.

48. Wenz, G. M., 1962. "Acoustic Ambient Noise in the Ocean: Spectra and Sources," J. Acoustical Society of America, 34:1936.

49. White, F. M., 1964. "A Unified Theory of Turbulent Wall Pressure Fluctuations," U.S. Navy Underwater Sound Laboratory Report 629.

50. Wilson, O. B., 1988. An Introduction to the Theory and Design of Underwater Transducers, Peninsula Publishing, Los Altos, Calif.

51. McLellan, T. N. and Randall, R. E., 1986. "Measurement of Brine Jet Height and Dilution," J. Waterway, Port, Coastal and Ocean Engineering, Vol 112, No. 2, p 200.

C H A P T E R 1 0

DIVING AND UNDERWATER LIFE SUPPORT

Robert E. Randall



CONTENTS

INTRODUCTION, 473

Diver Breathing Equipment, 474

Submarines, 476

Atmospheric Diving Systems, 482

Underwater Habitats and Hyperbaric Chambers, 483

Energy Systems for Diving Applications, 490

DIVING PHYSIOLOGY, 491

GAS LAWS, 496

Pressure and Temperature Relationships, 496

Equation of State, 497

Van der Waals Equation of State, 499 Beattie-Bridgeman Equation of State, 501

Virial Form of the Equation of State, 502

Law of Corresponding States, 502

OPERATING CHARACTERISTICS AND GAS SUPPLY CALCULATIONS FOR DIVER BREATHING EQUIPMENT, 502

SCUBA Demand-Regulator Apparatus, 503

Closed-Circuit Rebreathing Apparatus, 505

Semi-closed-circuit Breathing Apparatus, 506

Procedure for Evaluating Semi-closed Breathing Apparatus

Liter-flow Rate, 509

Open-Circuit Diving Rigs Breathing Gas Requirements, 511

472

Diving and Underwater Life Support 473

Relative Gas Flow Rates for Different Underwater Breathing

Apparatus, 512 Ventilation of Large Chambers, 513

THERMODYNAMICS FOR DIVING SYSTEMS, 514

First Law of Thermodynamics for General Open and Closed Systems, 514

Properties of the Perfect Gas, 517

Perfect Gas Processes, 518

PRESSURE VESSEL CHARGING AND DISCHARGING PROCESS, 519

Adiabatic Charging, 520 Non-Adiabatic Charging and Discharging, 520

DIVING GAS MIXTURES, 521

CONTROL OF UNDERWATER CHAMBER ENVIRONMENT, 522

Psychrometric Charts for Diving Applications, 523

Calculations Using Psychrometric Charts, 525

MIXING OF BREATHING GASES, 531

Mixing by Partial Pressures, 532

CARBON DIOXIDE ABSORPTION IN DIVING OPERATIONS, 547

Temperature Considerations, 548

ONE-DIMENSIONAL COMPRESSIBLE FLOW IN PIPES, 548

Isothermal Gas Flow with Friction, 550

Adiabatic Gas Flow with Friction, 551

HEAT TRANSFER, 552

Modes of Heat Flow, 552 Basic Laws of Heat Transfer, 553

Steady One-Dimensional Heat Conduction, 553

REFERENCES, 559

Introduction

Diving and underwater life support systems include submarines, underwater habitats, hyperbaric chambers, one-atmosphere diving suits, diving bells, and diver breathing equipment such as self-contained breathing apparatus (scuba), semi-closed breathing apparatus, closed-circuit breathing apparatus, and surface supplied breathing equipment. Diving systems allow divers, submariners, and aquanauts to live and work underwater and to view and record underwater phe-


nomena through the use of underwater video cameras and acoustic devices. Pressure vessels are used to house divers, store breathing gases, and enclose electronic and photographic equipment. To sustain life under the sea, the breathing mixture is often altered because of effects of high pressure requiring gas mixtures such as helium and oxygen, hydrogen and oxygen, and helium, nitrogen, and oxygen. In addition, the temperature, carbon dioxide, trace contaminant gases, and moisture content of the environment inside the habitat, submarine, and diving bell must be maintained within an acceptable range.

Diver Breathing Equipment

The breathing equipment used by working and recreational divers must supply the necessary breathing gas containing the proper amount of oxygen. Five general types of breathing equipment are the demand regulator or self-contained underwater breathing apparatus (scuba); semi-closed breathing apparatus; surface supplied deep-sea diving outfit, with or without carbon dioxide absorption; and closed circuit breathing equipment.

The self-contained underwater breathing apparatus (scuba) is the primary equipment used by recreational divers as shown in Figure 1. It consists of a pres-

Figure 1. Photograph of self-contained breathing apparatusmscuba (courtesy US Divers).


sure tank, a first stage regulator that reduces the tank air pressure to about 1033.7 kPa (150 psi) over ambient pressure, and a second stage demand regulator that delivers air to the diver at ambient pressure. The regulator is part of the diver's mouthpiece that supplies breathing gas when the diver inhales and is closed when the diver exhales. The breathing gas is stored in a compressed gas cylinder carried by the diver. The rate of breathing gas usage depends upon the exertion effort of the diver and water depth. Dive duration with this type of equipment can vary from approximately 12 min when divers are under heavy exertion in deep water (approximately 39.6 m or 130 ft) to several hours when diver exertion is light in shallow water.

Semi-closed breathing equipment, as illustrated in Figure 2, allows very efficient use of breathing gas. The breathing gas has oxygen supplied at a partial pressure just under the toxic limit of 1.2 ata (atmospheres absolute), and it is rebreathed after being passed through a carbon dioxide absorber until the oxygen partial pressure is reduced to 0.16 ata on exhalation. Thus, a large percentage of the oxygen is used as compared to about 20% for the scuba and surface supplied systems. The semi-closed breathing apparatus is the most economical

Figure 2. Photograph of semi-closed breathing apparatus (courtesy of US Navy).


breathing gas supply when used at moderate depths. It can be supplied by compressed gas cylinders carried by the divers and through an umbilical from a compressed gas supply located in a diver lockout chamber, diving bell, or surface vessel.

Closed-circuit breathing rigs have been developed such that none of the breathing gas is vented to the water. The diluent gas, typically helium, is added to fill the breathing bags and adjusted as required by the diving depth. Oxygen is added to the breathing mixture at the rate of consumption, and thus all the oxygen is used by the diver. When the diver exhales the breathing mixture, the exhaled gas is passed through a carbon dioxide removal device to cleanse the gas of all carbon dioxide, and the remaining diluent gas and unused oxygen is recirculated to the inhale bag and combined with oxygen supplied to replace the used oxygen. This system requires the automatic sensing of oxygen and carbon dioxide partial pressure. Closed-circuit systems are very advantageous for deep diving operations because no diluent gas is lost. Closed-circuit systems are not widely used by recreational divers, being limited mainly to deep diving military and scientific applications.

Surface supplied or open-circuit breathing equipment, Figure 3, is used at moderate depths with air supplied from air compressors or a bank of compressed air cylinders from a surface support vessel. The breathing gas is supplied to the helmet at a flow rate that is sufficient enough to dilute the carbon dioxide exhaled by the diver. This is common practice for commercial diving in shallow waters less than 30.5 m (100 ft) to 39.6 m (130 ft) of sea water. In some applications the diving helmet is equipped similar to a semi-closed breathing apparatus and the breathing gas is recirculated after passing through a carbon dioxide absorber. This is essential when the diving operation is at depths that require the use of helium, because it is uneconomical to exhale the helium to the surrounding water.

Submarines

Military. The United States and other nations use large submarines as part of their military defenses. These submarines remain unseen and undetected below the water surface and are capable of launching torpedoes, mines, and missiles to attack enemy ships, submarines and other inland targets [6, 8, 14, 22-24].

Military submarines use a unique classification that indicates the propulsion and weapon system used. An SS designation indicates the submarine is powered by diesel engines. These engines are used to propel the submarine on the surface and to charge electrical storage batteries that are used to provide the energy to propel the submarine when it is submerged. Diesel submarines must return to the surface, or use a snorkel to run the diesel engines that charge the batteries. Typically, diesel submarines operate submerged during the day and on the surface at night to charge their batteries. The snorkel is a large tube that is raised to the surface, and this tube provides an air supply for the diesel engines that


Figure 3. Photograph of surface supplied diving apparatus ( courtesy of US Divers).

allows the submarine to charge batteries while submerged at periscope depth (approximately 18.3 m or 60 ft). Extended submerged operations below periscope depth can extend for many days depending on the submarine's speed, use of auxiliary electrical equipment and the breathing atmosphere.

Submarines that are powered by nuclear energy are designated as SSN. The nuclear submarine has almost unlimited power and needs no access to surface air. Therefore, it is considered a true submarine because it can remain submerged indefinitely. These nuclear submarines can dive to depths in excess of approximately 457 m (1,500 ft) and travel over 111,180 km (60,000 nautical miles) before exhausting initial nuclear fuel supplies [8]. If these submarines are


armed with cruise missiles, they are designated SSGN, and with ballistic missiles they are designated SSBN.

Manned Research and Work Submersibles

The development of research and work submersibles began in the 1950s, and a selected summary of manned submersibles is tabulated in Table 1. A more detailed summary is found in Allmendinger [1] and Busby [2,3]. In 1952 the manned submersible Trieste was designed and built in Italy and was purchased by the US Navy. It made a historic dive in 1960 to 10,915 m (35,800 ft) in the Marianas Trench near Guam in the Pacific Ocean. Also, J. Y. Cousteau of France used the French-built shallow water diving saucer Denice and made hundreds of dives worldwide. The US Navy modified the Trieste and used Trieste H to investigate the tragic loss of a military submarine Thresher in 1963.

Alvin (Figure 4) is one of the most well-known and well-used US submersibles. Built by General Mills and Litton industries, it has made over 2,100 dives [1] during its first 25 years of operation and has been operated by the Woods Hole Oceanographic Institute with support from the US Navy. The original pressure hull was replaced with a titanium hull giving her a depth capability of 4,000 m (13,120 ft). In the late 1980s, Alvin was used in conjunction with the remotely operated vehicle (ROV) Argo-Jason to investigate the Titanic which rests on the ocean floor in the deep waters of the North Atlantic.

Figure 4. The manned submersible Alvin [1].

Diving and U

nderwater Life Support

479

0 o

m

o~

m

0 r~

0 0

Dm

r Z

-F ~

c~

.,,<~ .

..

.

~ o~.'~.~~.r-,~'o~~.~'~.~.~v~

~.

c~ ~


Perry Oceanographic and HYCO have constructed many manned submersibles for the offshore industry to complete work tasks related to the recovery of oil and gas from the Gulf of Mexico, North Sea and other offshore locations around the world. The PC-15 shown in Figure 5 is an example of this type of submarine.

Manned submarines that are operated by the Harbor Branch Oceanographic Institute and used for research are the Johnson-Sea-Link I and II. Originally, these were diver lockout submarines with an acrylic pilot sphere and an aluminum lock-out chamber. These submarines have been in operation for nearly 20 years, and they were used to assist in the space shuttle Challenger disaster search in 1985. The lockout chamber is now used primarily as an observation chamber.

Remotely Operated Vehicles. Unmanned underwater vehicles known as remotely operated vehicles (ROV) have been developed in the last 60 years. Their development has been strongly dependent on the development of reliable water-

Figure 5. Perry PC 15 submarine (courtesy of Perry Oceanographics).


proof electronic components, microprocessors, small sonars, mechanical arms, and underwater closed-circuit television. In 1975 the number and type of ROVs increased rapidly and their growth and use continues to grow as offshore development expands into deep water. There are essentially six types of remotely operated vehicles that include tethered, free-swimming, towed, bottom reliant, structurally reliant, untethered autonomous, and hybrid vehicles [11]. Listings of remotely operated vehicles and their technical details are available from various sources [3].

Recreational and Tourist Submarines . Tourist locations such as the Caribbean have experienced some development of submarines to take passen- gers for underwater tours. Popular theme parks in the USA have had submarine excursion tides for many years and one of the popular tides is the Jules Verne 20,000 Leagues Under the Sea ride at the Disney World theme park in Orlando, Florida. These tourist submarines provide characteristic viewing windows along the port and starboard sides for observation of the underwater environment by the submarine occupants. Two-person recreational submarines have also been developed for tourist locations with good underwater visibility.

Human Powered Submarines. In 1989 the first human powered submarine race was sponsored by the Perry Foundation and Florida Atlantic University, and it took place at Riviera Beach, Florida. This competition was organized to provide the opportunity for students and others interested in the advancement of underwater technology to participate in the design, construction, and racing of submarines that used only human power for propulsion and steering. No electrical energy was permitted, and the submarines were free-flooding, which eliminated the need for a pressure hull. The divers inside the submarine used a standard self-contained breathing apparatus (scuba).

As of this writing, there have been three International Submarine Races. The first race included 19 submarines built and raced by university students and the small submersible industry. The first overall performance winner was the US Naval Academy's submarine Squid. The 2rid International Submarine Race was conducted at the same location in 1991, and Florida Atlantic University's FAU Boat won the 100-m race with a speed of 4.7 knots. It also won the 800-m underwater oval race with two submarines simultaneously on the underwater race track. The 3rd ISR was staged off Ft. Lauderdale, Florida. Nearly fifty submarines participated over a two-week period with Tennessee Tech University's Tech Torpedo H winning the overall performance award. In 1994 the West Coast Submarine Invitational was held in the Offshore Model Basin in Escondi- do, California, and Florida Atlantic University's FA U Boat won the 100-m race with a speed of 10.94 kph (5.9 kts), which set a world speed record. At the 1996 World Submarine Invitational held in the Offshore Model Basin in Escondido, Califomia, the new speed record for the two-man submarines was set at 6.3 kts by the submarine Omer from Ecole de Technologie Superiere in Canada. Ocean


Engineering students at Texas A&M University have built three human powered submarines (Aggie Ray, Argo, and SubMaroon; see Figure 6) and participated in the races in 1991, 1993, and 1996.

Atmospheric Diving Systems

One atmosphere diving systems were developed to overcome the difficulties associated with decompression sickness and high-pressure nervous syndrome in water depths as deep as 550 m (1,800 ft). The diver in these systems remains at one atmosphere pressure and can work for relatively long periods without hav-

Figure 6. Texas A&M human powered submarines Aggie Ray (top) and SubMaroon (bottom).


ing to spend many hours or days in decompression. The JIM suit (Figure 7) was initially developed in England and has undergone design improvements to attain greater depth and flexibility capabilities. Greater bottom times, security, and protection from cold are additional capabilities provided by the one atmosphere diving systems. In 1976 the system was used for a dive in 275.8 m (905 ft) of sea water in the Canadian Arctic through a 4.9 m (16 ft) diameter hole in the ice, and the diver worked for a 6-hr period. Using saturation diving for the same operation would have required more than eight days of decompression. Newer JIM systems have been constructed with magnesium alloy and carbon steel fiber. The record dive for the JIM system is 548 m (1,780 ft) in the Gulf of Mex- ico. An adaptation of this one atmosphere diving system that allows a diver to operate at midwater locations is called the WASP (Figure 7) where the legs have been removed and movement in the water is accomplished by small propulsors similar to those found on small remotely operated vehicles (ROV). Advances in manipulator technology are expected to further improve the performance of divers working in these one atmosphere systems.

Underwater Habitats and Hyperbaric Chambers

The use of underwater habitats began in the early 1960s and peaked in the late 1960s. Their use diminished during the 1970s and in the 1990s there were very few in operation. NOAA's National Undersea Research Program (NURP) has operated the habitats Aquarius and La Chalupa in the Caribbean and off the Florida Keys. An excellent description of nearly sixty different habitats is presented by Miller and Koblick [15].

The early habitats were designed to determine the engineering feasibility and to evaluate man' s ability to survive undersea living. After the success of the initial habitats such as Man in the Sea, Conshelf, and SeaLab, the habitat construction increased, and they were used for observation stations, seafloor laboratories for scientists, and operational bases for working divers. Underwater habitats are designed to allow diver scientists, engineers, and technicians easy access to the ocean environment, which enables them to make observations, conduct experiments, and test new equipment and systems. Because the habitats are normally open to the ambient pressure at the habitat depth, the aquanauts, or divers, are saturated at the depth with the breathing gas mixture designed for the habitat or breathing apparatus used for the habitat excursions. Consequently, decompression is required at the completion of the habitat stay unless the depth is less than 10 m (33 ft) of sea water.

Habitats are often pressure vessels that can withstand external and intemal pressures during installation and decompression. The shape of the habitat is usually a cylinder or sphere because of their ability to best resist the pressure with minimum thickness of material and providing the most volume for space. Com- binations of spheres, cylinders, and hemispheres are often used to increase the space available for the divers.

Figure 7. JIM (top) and WASP (bottom) one atmospheric diving systems (NOAA 1991).


When designing habitats, various technical, logistical, and habitability considerations are investigated. Some of these considerations include comfort, simplicity, and functionality. Desired features for an underwater habitat were determined in a University of New Hampshire study and are tabulated in Table 2. The desirable features include an overall size of 2.4 m (8 ft) (diameter, height, width) and 11.6 m (38 ft) in length with separate wet room and living room. The interior needs to have temperature and humidity control, and the divers should have communications between themselves and the habitat. An external survival shelter and on-bottom and surface decompression capability are desirable.

More than 72 underwater habitats have been constructed around the world since 1962. These habitats range from simple shelters to sophisticated large facilities used for extended stays on the ocean bottom. Some of the more well- used habitats include Tektite, Hydrolab, and La Chalupa in the United States; Chernomor in Russia; and Helgoland in Germany. Several universities also ven- tured into the design and operation of underwater habitats that included Edalhab

Table 2 Underwater Habitat Desired Design Features [17]

General Separate Wet Room Living Room

Size: 8 ft (D, H, W) by 38 ft (L) 2.4 m by 11.6 m

Hemispheric windows Temperature and humidity

control Separate double chambers

Bottom and surface decompression capability

Suitable entry height off the bottom

Submersible decompression chamber for emergency escape

External survival shelter External lights at trunk

and viewports External breathing gas cylin-

der storage and charging Habitat to diver and diver

to diver communications Adjustable support legs, mobility Extemal or protected internal

chemical hood

Large entry trunk Bunks

Wet suit rack Hot shower

Hookah and built-in breathing system

Scuba charging

Wet lab workbench

Freezer for samples

Clothes dryer Diving equipment

storage Rebreathers

Microwave oven Food freezer and refrigerator

Water heater

Toilet

Individual desk and storage

Laboratory workbench

Trash compactor Library

Tapes, TV, radio, CD player

Emergency breathing system

Computer terminal


at the University of New Hampshire, Portalab at the University of Rhode Island, and Suny-Lab at the State University of New York.

The underwater habitat Hydrolab is shown in Figure 8 and is briefly described herein because of its simplicity, representative design, relatively low operating cost and its significant use over a nearly 20-year period. It was decommissioned by NOAA in 1985 and can be observed at the Smithsonian Museum of Natural History in Washington, DC. The habitat consisted of a 2.4-m (8-ft) diameter and 3.7-m (16-ft)-long steel cylinder with a hemispherical viewing port at one end. It was supported by four short legs providing a 3 ft clearance above the concrete support base. The habitat could be towed for short distances and was submerged by venting ballast tanks on the side of the main structure and several tanks in its base structure. Hydrolab was designed for depths 30.5 m (100 ft) or less and has been used at 13.7 m (45 ft) and 18.3 m (60 ft) with air as the breathing gas. An entrance hatch was located at the bottom and near one end of the cylinder, and it also was used as a lock when there was a pressure difference between the inside and ambient water pressure.

The habitat was furnished with three bunk beds, dehumidifier, air conditioner, folding chairs, sink, electric hot plate, and table surface. The close quarters are illustrated in Figure 9. A shower hose was located in the entrance trunk. A 7-m (23-ft)-long surface support vessel was moored to the top of the habitat. This boat-shaped vessel supplied electrical power, high- and low-pressure air and water, and communications through an umbilical. The habitat was used for many years by NOAA's Manned Undersea Research Program and based in St. Croix, Virgin Islands, and Freeport, Bahamas.

The habitat, Tektite, was a four-person habitat consisting of two vertical cylinders connected by a horizontal cylinder and mounted on a rectangular support base. The General Electric Company sponsored the construction and operation of Tektite. The vertical cylinders were divided into two compartments containing an equipment room, wet room, living quarters, and control room (dry scientific laboratory). The living quarters contained four bunk beds, small galley, storage, and entertainment equipment. Environmental control, food freezer, and toilet facilities were contained in the equipment room. Umbilicals from the shore provided air, water, electrical and communications to the habitat. Several hemispherical windows provided scientists a view of the surrounding area. A personnel transfer cap- sule (PTC) was lowered to the bottom for decompression, and the divers entered the PTC and were transported to the surface where they entered a deck decompression chamber. Tektite I and II missions were conducted in 1969-70 in the Virgin Islands and were the world's longest open sea saturation operation (59 days).

The habitat, Aquarius, was constructed by NOAA for their Manned Undersea Research Program in the late 1980s and has been used at various sites throughout the Caribbean Sea (Figure 10). Aquarius was designed to operate at depths of 36.6 m (120 ft) or less and has bunk beds to accommodate six scientists.



ii:Aii:~@:iiiii!ii!~ :.ii iiiii~i i ~i iii!iii

..... . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . ~ i ~ .......... . . . . . . . . . ~ ~ir

Figure 8. Underwater habitat, Hydrolab (top), and its support buoy (bottom) [17] (NOAA 1991).


Figure 9. Interior view of Hydrolab [15].


Figure 10. NOAA underwater habitat, Aquarius [17].


The use of underwater habitats has also been considered as subsea bases for offshore oil wellheads. The concept is to bring divers down by submersible to the underwater habitat, which is kept at one atmosphere. The submersible con- nects to the hatch on the habitat, and the divers are transferred to the habitat to conduct their maintenance work. A similar scenario is used to rescue personnel from large submarines.

The use of recreational habitats for facilities such as underwater hotels and restaurants has received some interest. Currently, there is a small habitat in the Florida Keys that can accommodate divers for about one week. It is located in approximately 9.1 m (30 ft) of water off Key Largo. Ocean engineering students at Texas A&M University have conducted preliminary designs for an underwater hotel and restaurant, and their "Underwater Wotel" received some interest but was not constructed. Several conferences have been devoted to the concepts of underwater cities. These areas for underwater facilities may expand in the years ahead to accommodate expanding populations in land-starved areas and to provide recreational facilities.

Hyperbaric chambers are facilities that are normally used to treat divers for decompression sickness, conduct controlled diving experiments, and provide a


means for divers to decompress after saturation diving. These chambers are found at medical hospitals, diving surface support vessels, US Navy submarine rescue vessels, and at operational diving locations where decompression is a common necessity. Most hyperbaric chambers are constructed with a double lock capability that allows medical doctors and technicians to enter the chamber to treat and observe patients and then leave, or to rotate personnel.

Energy Systems for Diving Applications

The power sources used for underwater applications include diesel generators, transmission lines from local commercial power utilities, closed-cycle diesel systems, electrical storage batteries, fuel cells, and small nuclear plants. Under- water habitats have commonly used diesel powered generators located on board a surface support ship, nearby buoy, or at a shore station with emergency facilities available. In 1969 the habitat Hydrolab used a fuel cell called Powercel 8 and built by Pratt and Whitney, for its total electrical power [ 15]. This fuel cell provided power for interior and exterior lighting, CO2 scrubbers, instruments, data recording, communication equipment, compressor, and a motor generator. It was located just outside one of the viewing ports so that the divers could mon- itor the generator status. After 1970 Hydrolab received its power from a small 7- m (23-ft) surface support vessel that was moored above the habitat and supplied air, water, electricity, and communications. A 12.3-kW (16.5-hp) diesel engine powering a 7.5-kW generator was located in the buoy along with a low- and high-pressure compressor. Storage facilities on the buoy included 946.3 1 (250 gal) of fresh water and 946.3 1 (250 gal) of diesel fuel, which provided approximately a one-month supply. The German habitat, Helgoland, used a similar buoy support system. Electrical power requirements for underwater habitats typically range from 10 to 100 kW [26].

For submersibles, batteries, fuel cells, and closed-cycle engines are being used for propulsion and energy requirements. Batteries used include the lead acid, nickel cadmium, and silver zinc type, and their energy densities are tabulated in Table 3. Lead acid batteries are heavy and significantly increase the habitat weight. These batteries also produce oxygen and hydrogen gases, as well as chlo- rine gas if exposed to sea water, that must be carefully vented during the charging

Table 3 Energy densities for submersible batteries [1].

Theoretical Actual Battery Type Watt-Hours/lb Watt-Hours/lb Watt-Hours/in 3

Lead acid 80-115 10-20 0.8-1.2 Nickel cadmium 105 12-19 0.7-1.25 Silver zinc 205 40-60 2.4-3.8


process. The silver zinc batteries have been shown to be very attractive for submersibles because of their high energy density. Fuel cells have been tried with some success, but they are expensive and have some safety concerns. Closed- cycle engines have also been tried, but the combustion by-products are a danger.

Battery energy sources are placed either externally or internally. The external storage requires the battery container be pressure compensated. Oil compensated batteries have been used for many submersibles, but they require maintenance and are less reliable than the dry storage containers. However, most submersibles have used a properly designed oil compensated battery for electrical energy. The internal container requires a pressure vessel, and the greatest danger is associated with handling the gases that occur during the charging process. Dry cells or rechargeable batteries are commonly used for emergency energy sources for communication equipment, carbon dioxide scrubbers, and emergency lighting.

Diving Physiology

Basic knowledge of the human circulatory and respiratory systems is necessary for the analysis and design of underwater life support systems. Some references that discuss diving physiology are Miles and Mackay [13], NOAA [17], Schilling [20], Schilling et al. [21], and USN [28]. In the human circulatory system (Figure 11), the heart's fight ventricle pumps blood to the lungs via the pulmonary artery and then from the lungs through the pulmonary vein to the left auricle. Blood is deficient in oxygen in the pulmonary artery but rich in carbon dioxide. The lungs are where the gas transfer of oxygen to the blood and carbon dioxide from the blood occurs. The blood returning from the lungs is rich in oxygen, and the heart's left ventricle pumps this oxygen rich blood to the upper and lower body. The blood that is depleted of oxygen returns to the fight atrium through veins and then is pumped to the lungs by the fight ventricle completing the circuit. The distribution of blood to body organs is accomplished by the con- tinual branching of arteries that become capillaries. Carbon dioxide and other substances are exchanged between the blood and body tissues through the thin walls of the capillaries. The blood from the capillaries flows into veins and is returned to the heart. The carbon dioxide is then transported from the heart to the lungs where it diffuses across membranes into the lung alveoli and is eventually exhaled.

For the respiratory system (Figure 12), breathing is the result of rhythmic changes in the volume of the chest wall cavity. These changes are the result of the muscular action of the diaphragm, pelvic muscles, and chest wall. This muscular action is controlled by the central nervous system, which is responding to changes in blood oxygen and carbon dioxide levels. The normal respiratory rate at rest is 12 to 16 breaths per minute. The actual volume change of air involved


Lun~J Pulmon~y Pulmon~'y Ao~tl ~'t cry vc=ns.

Tflcus valve

Lu~j r

v,dvc

Veins

y c~i l l~cs

Figure 11. Schematic of human circulatory system [20].

in a given breath is called the tidal volume which is illustrated in Figure 13. The respiratory minute volume is defined as the respiratory rate (breaths per minute) times the tidal volume (volume per breath). Large, average, and small lung vital capacity volumes are 6.0, 4.5, and 3.0 liters (0.210, 0.158, and 0.105 ft3), respectively.

During respiration, air is drawn into the lung alveoli where the exchange of oxygen and carbon dioxide occurs. The partial pressure of oxygen is less in the body tissues than in the blood because these tissues are continuously using oxygen. As a consequence of the partial pressure difference, oxygen diffuses from the blood to the tissues. Also, the partial pressure of carbon dioxide is greater in the tissues because it is being produced there. Therefore, carbon dioxide diffuses from the tissue to the blood. Changes in the gas exchange rate is accomplished


ConcMe

Sep~urn

Hairs -- - "'; ,.--

Herd Palete - ~ ~ . ,

Tongue /

Epiglottis (Cover ~. of Windpipe)

/ Larynx (Voice Box)

Trachea (Windpipe) Al~mll Bronchlil A~terf �9

Bron,

Pulmonary Venule

Sphenoldal Sinus

J Adenoid (Naso-Phar~geal Tons.)

Soft Palate

_--- Tonsil

(Food Tube)

Right Ltmg

Bronchus o

Pulmonary Vein

Pulmonary Artery

Cu~ Edge of Pleu~ (Haue)

Cut Edge e4 Oiephr4qlm

Pulmonary

Figure 12. Schematic of human respiratory system [17].

Volume

Maximum lunq extension

Inspiratory~ ~ 60 % vital c_a.p..acjty . . . . . . .

Vital Tidal Volume Capacity

40 % vital capacity . . . . . .

Expiratory Reserve_~.~

Residual Volume I 1

Time

Figure 13. Volume relationship for a normal breath.

Total Lung Volume


by increasing the chest movement, increasing heart action, and increasing partial pressure differences.

Respiratory problems can occur in diving operations, and hypercapnia (carbon dioxide excess) is one of the respiratory problems that occurs when the body is producing carbon dioxide faster than it is being eliminated. It can be caused by an excess of carbon dioxide in the breathing medium, inability to remove carbon dioxide from the breathing medium, or inadequate removal of carbon dioxide from the tissues or blood. A relationship for the physiological effects of carbon dioxide concentration and exposure period is shown in Figure 14. As shown on the fight side (40 days) of this figure, long term exposures to carbon dioxide should be limited to a partial pressure less than 0.005 ata (Zone A). Hypoxia (oxygen shortage) can occur when the oxygen partial pressure drops below 0.1 ata (atmospheres absolute), and then consciousness is usually lost. If oxygen is completely cut off for 3 to 5 minutes, irreparable damage to the brain is experienced.

12.0 0.12 0.12

,~ 10.0

~ 8 . 0

" ~ ~ ~

~ ~ 4 . o

0.I

_ Zone IV Dizziness , s tupor 0.08 ~.

Icl ~ ~ ~ 0.06 ~

Zone III D i s t r a c t i n g d i s c o m f o r t ~ I 0.04 "~

2.0 ~ . . . Z o n e II Minor perceptive chanl ~ 0.02 ~ B

' I " [ Zone I No effect 0.0 ' 1 . . . . . . 0 A - ~ t

0 20 40 60 80 Exposure Time (minutes)

40 Days

0. I0

0.08

0.06

0.04

0.02

0.005 0

Figure 14. Physiological effects of carbon dioxide concentration and exposure period.

Oxygen poisoning is another respiratory problem that occurs when there is an excess of oxygen, typically greater than 1.2 ata partial pressure. Oxygen poisoning is a time dependent event and the limits of oxygen partial pressure and concentration are illustrated in Figure 15.

The effects of water depth or increased pressure can also cause problems for divers. The most common problem is ear squeeze, but pressure effects can also cause problems in the sinuses and teeth. Gas embolism is another problem that can occur if the pressure is reduced too rapidly or the ascent from depth is too rapid. In this case, dissolved gas comes out of solution, forming bubbles in the tissues and blood. These bubbles may block the blood circulation to the lungs, brain, heart, and spinal cord, and death can result. Recompression is the only


1oo.o 8 ~ 611

~. 4c

~ 2~

�9 ~ ]0.0

I " 4 0 >

,,Q

. lw

g

2

1.0 0.~ 0.~ 0.4

0.2

0.1

10

--~~------___. ----~ Oxygen ~ , ~, ' " are RanQ~''"~ j JH ....... ~ fatal

First Sy Helpless

20 30 40 60 80 200 400 600800 2000 5000 I O0 1000 10000

Depth (ft)

Figure 15. Percentage of oxygen in breathing mixture as a function of depth and oxygen partial pressure.

treatment, and it requires the diver to be placed in a hyperbaric chamber. Subse- quently, the chamber pressure is usually, but not always, increased to the original depth of the diver and then a slow decompression procedure is followed to prevent the reoccurrence of the embolism.

Two other important consequences of water depth or increased pressure are decompression sickness (bends) and inert gas narcosis (rapture of the deep). The common inert gases (nitrogen and helium) are physiologically inert under normal pressure, but nitrogen has distinct anesthetic effects when its partial pressure is sufficiently high. Nitrogen narcosis usually begins between the water depths of 30.5 m (100 ft) and 45.7 m (150 ft). Decompression sickness occurs when the elimination of gases by the blood flowing through the lungs is slower than the rate of reduction of the external pressure. The amount of super saturated inert gas in the tissues can cause the inert gas to be released in the form of bubbles that cause rashes, block circulation, and distort body tissues. Again, the only treatment is recompression using decompression chamber treatment facilities.

An important consideration for the design of underwater life support systems is the amount of breathing gas required and oxygen consumed by the diver. Fig- ure 17 shows the average required respiratory minute volume and oxygen consumption for different working conditions. The amount of carbon dioxide produced is determined from the respiratory quotient (RQ), which is the ratio of the carbon dioxide produced divided by the oxygen consumed, and the respiratory quotient can vary from 0.7 to 1.1. The typical respiratory quotient is 0.9, but it is


100

90

~ E 80

~ 70

-~ 6O

~ 50

e 30 k,

~

~, 20

~ 10

�9 , 0

0 4

J / Severe Work / , / t " /

I .,- JtF Light Work Swim- 1 knot

_ X ~ r 0.85 knot (Average Speed)

Sw m-O. k .ot Slow)

P t ~ Sitting Quietly

t j . . . . . .

0.5 1 1.5 2 2.5 3 3.5 Oxygen Consumption (slm)

3.5

3 T

2.5 u

o 2 ~

1.5~

o

0.5~

Figure 16. Relation of respiratory minute volume in liters per minute (I/min) or cubic feet per minute (cfm) and oxygen consumption in standard liters per minute (slm) to type and level of exertion.

often selected as 1.0 for convenience. However, Guyton [7] indicates that for a normal diet the average respiratory quotient value is 0.82.

The units used to describe the respiratory minute volume are liters per minute and cubic feet per minute and both refer to the volumetric flowrate at the diver's depth, or pressure. Standard volumetric flow rate conditions such as standard cubic feet per minute (scfm) and standard liters per minute (slm) refer to conditions at the surface or standard atmospheric and temperature conditions. Standard atmospheric pressure is commonly assumed as 10.1 m (33 ft) of sea water, 101.33 kPa (14.7 psia), 1 ata, or 760 mm of Hg (mercury). Standard temperature is normally taken as 15~ (59~ However, the US Navy Diving Gas Manual [27] defines standard conditions as 70~ at 1 ata for gas flows in standard cubic feet per minute (scfm) and as 32~ at 1 ata for gas flows in standard liters per minute (slm). Flowrates in cfm or liters/min refer to diver gas flow requirements at depth.

Gas Laws

Pressure and T e m p e r a t u r e R e l a t i o n s h i p s

When pressure is measured relative to a perfect vacuum, it is called absolute pressure, and when it is measured relative to atmospheric conditions, it is called


,i standard atmospheric pressure

101.33 kPa 14.7 psia

10 m sea water 33 ft sea water

g a u g e - - - ~

pressure t

---~] lotalospheri c

I; bsolute ressure

vacu_..um (gauge)

absolute zero pressure (perfect vacuum)

Figure 17. Measured pressure relationships.

gauge pressure. The pressure relationships are illustrated in Figure 17. The standard atmospheric pressure is the average pressure found at sea level and is given as 10.1 m (33 ft) of sea water, 101.33 kPa absolute (14.7 psia), and 760 mm of Hg. Partial pressure is frequently used in diving and life support calculations. The partial pressure is the pressure a component of gas would exert if all the other gases were removed and the component gas occupied the volume alone. Dalton's Law of Partial Pressure states that the sum of the partial pressures of each component gas equals the total pressure of the gas mixture.

Temperature is measured as Centigrade (°C) or Fahrenheit (°F), and in absolute terms it is Kelvin (°K) or Rankine (°R), respectively. In equation form, the temperature may be expressed as

°K = °C + 273 or °R = °F + 460 (1)

The absolute temperature must be used in many thermodynamic relationships (e.g. equation of state) used in diver life support calculations. The expressions for converting degrees Centigrade (Celsius) to Fahrenheit and vice versa are

OF=9 -5 (o C) + 32 and °C = _5 9 (°F - 32) (2)

Equation of State

The equation of state (perfect gas law) expresses the relationship between pressure, density, and temperature for a gas and is

p = p R T (3)


where p = absolute pressure p = density R = gas constant T = absolute temperature

The gas constant is defined as

R = R u M

where R u = universal gas constant (1,544 ft-lb/mole-~ M = molecular weight

(4)

The gas constant and molecular weight for several gases are tabulated in Table 4. Because the density of a gas is the mass (m) per volume (V) of the gas, a

commonly used expression for the equation of state is

pV = mRT (5)

Other useful forms of the equation of state are

pv = RT (6)

R pv = u T (7)

M

pV = nRuT (8)

where v is the specific volume of the gas and n is the number of moles (n = m/M). Special cases of the perfect gas law are

Constant Temperature (Boyles Law) PlV1 = p2V2 (9)

Table 4 Molecular Weight and Gas Constant for Typical

Breathing Gas Components

Gas Molecular Weight (M)

Ibm Gas Constant (R)

ft-lb/lbm-~

Air Carbon dioxide

Helium Hydrogen Nitrogen Oxygen

29 44

4 2

28 32

53.35 35.11 386.2

766.53 55.16 48.29

Diving and Underwater Life Support 4 9 9

Constant Pressure (Charles Law) V] = V2, ' (10) T~ T2

General Gas Law plV1 = P2V2 (11) T1 "I'2

where the subscripts represent equilibrium state points for the gas. In some cases the perfect gas law is not satisfactory, and a real gas law that considers compressibility effects must be used. Two forms of the real gas law are

p v = Z R T or p V = ZmRT (12)

where Z is the compressibility factor that is a function of temperature and pressure. For a real gas, the compressibility factor (Z) is a function to temperature (T) and pressure (p), and for the special case of a perfect gas the compressibility factor is unity (Z = 1). These factors are normally plotted in the form of Z versus p for various temperatures as shown for helium in Figure 18 and air in Figure 19.

Van der Waals Equation of State

An equation for determining the state of a gas is the Van der Waals equation of state

5.50

5.00

4.50

4 .00

t~ 3.50

r 3.00

~ �9 ~ 2 . 5 0 . m

~, 2.00 0

ro 1.50

1.00

0.50

o.,

! If .......... 1 .50 , PV ~ ~ 5DO p , , = t Z -- - - 4~0 p=;= . . . . . �9 T

1.00- = = = = m ~ - ~ i - ~ i i i i i ,,so, , . , . 14. ; ps=e

t SO ,,i=

0.50 I 300 350 400 450 5 0 0 . 550 6'~ _I

r ....

, , h , . o ~ - , , , . . �9 1.so0. psi~ I t .. . . ~ j - h . " e ~ _ " - . d ~ . . . . . 9nO psia ~ ~ ! : - : ~ ""

-so,,~o, 6oo~,,o,, ! I I ', i ! I i j - 4 0 0 -350 -300 -250 -200 -150 -I00 -50 0 50 I00 150 200 25( 300

Temperature (~

Figure 18. Compressibi l i ty factor for helium [10].


i.S 2,000

Pressure (psia) 3,000 4,000 6,000 8,000

1.4

1.3

1.2

~z~ 1.0

�9 ~-. o.~

~ O.g

0.7'

0.6

0.5

0.4 100 200 300 400

Pressure (am)

Figure 19. Compressibility factors for air [10].

600

p __ R u T a

v - b v 2

w h e r e v = speci f ic v o l u m e (f t3/mole)

a, b = van der W a a l cons t an t s l i s ted in T a b l e 5

R u = un ive r sa l gas cons t an t

(~3)


Table 5 Approximate Values for the van der Waals Constants [9].

Gas a b

ata- ft 6 psia- ft 6

mole 2 mole 2

ft 3

mole

Air 344 5,052 0.587 Carbon dioxide 926 13,600 0.686

Helium 8.57 126 0.372 Hydrogen 62.8 922 0.427 Nitrogen 346 5,082 0.618 Oxygen 350 5,140 0.510

Water vapor 1,400 20,580 0.488

B e a t t i e - B r i d g e m a n Equat ion of State

Another equation of state used is that due to Beattie-Bridgeman

R u T ( l - e ) A p = v2 (V + B) V2 (14)

( a ) c where A = A o 1 - ; B = B o 1 - ; a n d e =

vT 3

and Ao, a, B o, b, and c are constants that are tabulated in Table 6 for various gases.

Table 6 Beattie-Bridgeman Constants [9].

Gas A o a B o b c(10 -4)

a ta - ft 6 psia- ft 6 ft 3 ft 3 ft 3 (ft 3 -- R 3)

mole 2 mole 2 mole mole mole mole

Air 334 4,910 0.309 0.738 -0.0176 406 CO 2 1,285 18,890 1.14 1.69 1.16 6,170 H2 50.7 745 -0.081 0.336 -0.698 4.7 He 5.6 82 0.958 0.224 0.0 0.37 N 2 345 5,070 0.419 0.808 -0.111 393

N20 1,285 18,890 1.14 1.68 0.116 6,170 0z 383 5,620 0.411 0.741 0.0674 449


Virial Form of the Equation of State

For great accuracy over large pressure ranges, the virial form of the equation of state is often used.

B C D pv = 1 + -- + + --3- (15) RT v ~ v

where B, C, D = virial coefficients Tabular results using this equation are found in USN [27] for breathing gases used in diving.

Law of Corresponding States

The law of corresponding states says that all gases have the same p-v-T behavior at the same reduced conditions. The behavior of all gases is found by using ratios of actual temperature and pressure to that of the critical temperature (Tc) and pressure (Pc), which are tabulated in Table 7. The reduced temperature (TR) and pressure (PR) are

T r = T / T c and Pr = P/Pc (16)

Figure 20 is a plot of the compressibility factor (Z) as a function of reduced temperature (Tr) and pressure (Pr)"

Operating Characteristics and Gas Supply Calculations for Diver Breathing Equipment

All breathing apparatus used in diving must, in some way, satisfy both respiratory-volume requirements and oxygen-supply requirements at all depths. Vari- ous types of apparatus having different characteristics have been developed,

Table 7 Approximate Critical Constants for Several Gases [9]

Critical Temperature (T c) Critical Pressure (Pc) Gas ~ ~ ata psia

Air 132.4 238.8 37.2 547.0 Carbon dioxide 304.1 547.8 72.9 1,071.0

Helium 5.2 10.0 2.3 33.8 Hydrogen 33.2 60.5 12.8 188.0 Nitrogen 126.0 227.2 33.5 492.5 Oxygen 154.3 278.1 49.7 730.9

Water vapor 647.3 1,165.4 218.2 3,206.2


t _

tL

.Q

i r E O o

1.16 1.12 1.08 1.04 1.00 0.96 0.92 0.88 0.84 0.80 0.76 0.72 0.68 0.64 0.60 0.56 0.52 0.48 0.44 0.4P1 0.36 0.32 0.28 0.24 0,20

D

Reduced Pressure Pr = ~c

Reduced Temperature T r = T~

P, Pc, T, and T c (absolute units)

Z= 1 for ideal gas

0 1 2 3 4 5 6 7 8 Reduced Pressure (Pr)

Figure 20. Compressibility factor for law of corresponding states [10].

each having applications of greatest suitability. The general characteristics of each type of system, and means of calculating or estimating the composition and flow rates of breathing gas required for a diving mission are discussed below.

Scuba Demand-Regulator Apparatus

Open-circuit scuba equipment uses a demand regulator with which gas is supplied with each inhalation in the exact quantity needed, thus conserving stored compressed gas during exhalation. However, all exhaled gas is vented to the surrounding water, so that the volume rate of usage is equal to the respiratory minute volume. The actual rate of stored gas usage and duration of the stored gas supply depends upon the rate of effort and depth of usage. Volume-flow rates required for breathing vary between 7 and 701/min, or 15 and 150 ft3/hr, at


the pressure corresponding to the water depth, depending upon level of effort. Figure 21 shows the endurance variations for demand-regulator scuba system using a single aluminum 90 scf tank. The endurance of other demand scuba systems depends primarily upon their tank capacity and pressure, and similar curves could be drawn.

The actual volume in standard cubic feet (scf) of air in a pressure vessel (scuba tank) at any pressure is given by

NVrPg W a = ~ (17)

Pt

where Va = actual volume of air available in the tank, scf N = number of tanks

Vr = tank rated capacity, scf pg = gauge pressure in the tank, psi Pt = rated tank gauge pressure, psi

The effect of temperature is calculated by

V t = ~ (18) %

where V t = volume of air adjusted for temperature V a - available air from previous Equation 17 T 2 = water absolute temperature T1 = absolute temperature of air under standard conditions (530 ~

The duration of air supply in minutes for a pressure vessel (scuba tank) leaving a reserve pressure is evaluated by [25].

120

| 0

=] 2o

1 ' \ ~ X L.--- No-decomp re,ui~ limit

= . . . . I I ! ~ . - . .

\ RMV ' 3 0 \ i ~ s '

0 10 20 30 40 50 60 70 80 90

Endurance (mlautes)

3000 psi 725 in 3 tank

Respiratory Minute Volume (RMV) 18 LPM Swimming 0.5 knot (slow) 30 LPM Swimming 0.85 knot (avg. speed) 40 LPM Swimming 1.0 knot 60 LPM Swimming 1.2 knot

Figure 21. Endurance of single aluminum 90 scf scuba flask (from 3,000 psi to a reserve of 500 psi) [27].


S NVr(Pg - Pro)(33)T2

B(D + 33)PtT 1 (19)

where S = duration in minutes Pm = minimum reserve pressure, psi B = breathing rate, ft3/min D = depth of water in feet

Closed-Circuit Rebreathing Apparatus

It is feasible to breathe pure oxygen in a closed system if the exhaled oxygen and carbon dioxide are passed over an absorbent that removes the carbon dioxide and if oxygen is added at the rate of consumption. Such an apparatus has been developed and used within the depth limitations imposed by oxygen toxicity. It is suitable for shallow depths, and it leaves no trail of bubbles. It is also very efficient in the use of oxygen, because all of the oxygen is actually used. Because of the efficient oxygen usage and the elimination of the diluent gas, sufficient oxygen can be stored for rather long diving periods, and the diving time is independent of depth. Diving periods are limited by the quantity of stored oxygen, level of exercise, rapid development of oxygen toxicity, and the amount of C02 absorbent provided, and these should be approximately balanced. Because of the serious danger of oxygen toxicity, this unit has very limited application.

Several experimental closed-circuit rigs have used diluent gas in breathing bags to permit extension of closed-circuit rigs to greater depths. The only diluent gas needed is that required to fill the breathing bags and to adjust their volume with changes in depth. Pure oxygen is added to the breathing mixture at the rate of consumption, so that all oxygen is used, and the duration of oxygen supply is independent of depth. These systems rely upon an automatic control system to sense and maintain oxygen partial pressure within tolerances of 0.2 to 1.2 ata. These systems should be advantageous for deep operations because they con- sume virtually no diluent gas and only minimum quantities of oxygen. All constant oxygen diving units using any inert gas diluent require special decompression tables.

Oxygen consumption rates have been measured in a series of test dives with the U.S. Navy Mark X closed-circuit apparatus, with values ranging from 0.4 to 1.1 slm under different diving conditions. Although higher consumption rates can be maintained for short periods, the measured average rate of 1.1 slm is considered to be near the maximum for extended time periods. Consequently, dive plans can be based upon an average oxygen consumption of 66 standard liters per diver hour, which is equivalent to 3.4 scf or 0.28 lb oxygen per hour.


Semi-closed-circuit Breathing Apparatus

The semi-closed-circuit breathing apparatus (Figure 22) is used at deep depths and is very efficient in use of the breathing gas. Theoretically, oxygen is supplied at a partial pressure of 1.2 ata, just below the toxic limit, and rebreathed through a carbon dioxide absorber until the oxygen partial pressure is reduced to 0.16 ata on exhalation. Up to 86% of the oxygen supplied is used as compared to only 20% for an open-circuit system. In addition, the high partial pressure of oxygen in the gas mixture minimizes the amount of mixed gas used for the required oxygen consumption. This permits flows of about 3.5 to 6 liters/min at the diver's depth. The determination of the gas flow into a semi-closed breathing apparatus requires an analysis similar to that used for the habitat or hard hat. However, partial oxygen volumes are used instead of partial carbon dioxide volumes.

In a typical helium-oxygen semi-closed-circuit apparatus suitable for saturated diving, breathing gas is supplied continuously at a flow rate (referred to as the "liter-flow" rate) sufficient to provide for oxygen usage of about 3 slm. The partial pressure of oxygen in the supplied mixture is 1.2 ata. The incoming gas enters an inhalation bag where it mixes with and is diluted by partially depleted gas. After inhalation, it is exhaled into the exhalation bag and then circulated through a CO2 absorber back to the inhalation bag. Surplus exhaled gas is vented from the exhalation bag at a rate that is a little less than the volume of the

r--I umbilical

emergency bottle gas

thpiece

emergency supply helium/oxygen

F regulator

valve exhale inhale c~ bag . ~ _ : . ~ bag

valve -~ ~ I c e

C02 I removal

I ~ 0 2 sensor

orifice ~~- - selector

valve

by-pass valve

Figure 22. Schematic of a semi-closed breathing apparatus.


incoming gas as a result of the CO2 being absorbed. The level of oxygen partial pressure in the inhalation bag varies with the rate of oxygen usage, from about 1.0 ata when at rest to 0.21 ata at maximum exertion, and the partial pressure of oxygen in vented gas varies from 0.96 to 0.16 ata. With constant mass flow of breathing gas, the most efficient use of the supply occurs only at maximum exertion, and at other conditions, the surplus oxygen is vented.

The quantity of oxygen that must be supplied is independent of depth if the range of oxygen partial pressures is maintained between 1.2 to 0.21 ata in the inhalation bag at all depths. This is accomplished by the selection of the oxygen concentration and injection flow rate. However, to avoid oxygen toxicity, the oxygen percentage in the make-up gas must decrease as depth increases, and thus the quantity of helium supplied with the oxygen increases with depth. Therefore, the total consumption of breathing gas increases in proportion to total pressure or depth.

The semi-closed-circuit apparatus is very economical in the use of breathing gas if operated over a moderate depth range. For a diving mission, the oxygen percentage is limited by the maximum depth planned, and the liter-flow rate is established to provide sufficient oxygen for exertion at the minimum depth. As the maximum and minimum depths become farther apart, the difference in oxygen partial pressure between breathing gas and vented gas is reduced, which reduces the efficiency of oxygen use and increases the required liter-flow rates.

Figure 23 is a chart for selecting the oxygen percentage and liter flow rate for a semi-closed-circuit breathing apparatus. It illustrates the relation of oxygen content of the breathing gas, partial pressure of oxygen in the inhalation bag, and liter flow to the diving depth. The oxygen percentage in the liter flow is found at the intersection of the vertical line of maximum depth and the sloping curve of 1.2 ata oxygen partial pressure. Then, the liter flow rate is found to the left at the same oxygen percentage, on the vertical line representing the minimum diving depth. When used in this way the conditions selected will provide 1.2 ata oxygen partial pressure in the inhalation bag when resting and consuming 0.5 slm oxygen at the greatest depth, and sufficient flow for severe exertion with consumption of 3 slm oxygen at the least depth. It should be noted that the oxygen partial pressure lines indicate higher oxygen percentages than in Figure 23 because they represent remaining oxygen partial pressure in the inhalation bag after some oxygen has been used.

With helium-oxygen scuba, used at depths to 61 m (200 ft), it is possible to use oxygen partial pressures in the range of 1.3 to 2.0 ata for short periods to extend mission duration or excursion depths because oxygen partial pressures in the inhalation bag approach these values only while at rest, when tolerance to high oxygen concentrations is maximum. In saturation-diving, mixed-gas, umbilical- supplied excursion systems it is common to use mixtures providing 1.2 ata oxygen partial pressure. The habitat partial pressure is maintained in the range of 0.3

o,n ~

o o

0

�9 ~Ol~l -~:~!"I u! u~

x 0

tu;):)J;)d

~

==_og z

..........

o

f, "~

~

u~ z

~ o

T;

-;

z,

, :E

z i

,., 4

~n a.

"

-- ~

-'1 .

u~ ::3

Q.r.- (D

::3

(D

.,.., ~

, �

9 ~

X O

.

0 _ox:

.'- ~

L_

0 e-

~"

0

C

~

_o~

"~.c_ if)

e-

~._~

._~ e-


to 0.32 ata. For nitrogen-oxygen scuba equipment, it is common to use oxygen partial pressure corresponding to those for surface supplied diving rigs.

The semi-closed-circuit breathing apparatus is currently the most versatile breathing apparatus available. Umbilical-supplied units are suitable for use at any depth to 305 m (1,000 ft), as demonstrated in experimental test-chamber dives, Sealab programs, and other saturation-diving missions. Accordingly, the rates of consumption of oxygen and of helium can vary with depth, and with range of depth required for specified missions.

Procedure for Evaluating Semi-closed Breathing Apparatus Liter-flow Rate

The procedure for calculating oxygen concentration in breathing gas and the liter-flow rate for semi-closed-circuit apparatus is

1. Select the maximum diving depth, minimum depth, and maximum oxygen usage rate.

2. Determine percentage of oxygen in the breathing gas that corresponds to 1.2 ata at the maximum diving depth.

3. Determine percentage of oxygen in exhalation bags that corresponds to 0.166 ata at the minimum diving depth or at the surface.

4. Calculate liter-flow rate for breathing gas using above oxygen percentages.

The diving depths are usually defined by the diving operation objectives. The maximum oxygen demand could be estimated on the basis of rate of effort planned, using Figure 16 as a guide. However, it is recommended that an oxygen usage of 3.0 slm be provided in order to permit extreme exertion. The value of 1.2 ata is low enough for extended periods, but higher partial pressures can be used for shorter periods within time limits shown in Figure 15. The partial pressure of oxygen in exhalation bags should be taken as 0.166 ata at the minimum diving depth. However, for dives from the surface, it is usual to provide enough oxygen for surface swimming. The liter flow can be calculated using

U L = (20)

( 1-01 ) 1 -u O1 -- 02 - - ' 3

where L = liter flow (slm) U = oxygen usage (normally 3.0 slm)

O1 = percentage (decimal) of oxygen in liter flow (1.2 ata at greatest depth) 02 = percentage (decimal) of oxygen at 0.166 ata at depth of interest 03 = percentage (decimal) of oxygen inhaled from the inhalation bag

(0.208 ata at least depth).


The slm of oxygen available in the breathing gas at the surface, or any other depth, is calculated from

U = L [ O I - 0 2 ( 1 - O 1 ) ] I 0 3 (21)

where U = oxygen the diver can use, slm

Example 1. A semi-closed-circuit breathing apparatus is to be used for diving work at a depth of 250 ft under saturated diving conditions. No changes in depth are anticipated and no decompression is required, as the diver will use a diving bell and a surface decompression chamber between dives. It is desired to evaluate the optimum oxygen percentage in the breathing mixture and the liter flow required for heavy exertion

1. Find pressure at 250 ft and calculate as follows:

Pata Depth 250 = ~ + 1 = ~ + 1 = 8.58ata (22) 33 33

2. Find oxygen percentages for oxygen partial pressures of 1.2, 0.166, and 0.208 ata at depth of 250 ft. The values are taken from the curves of Figure 15 or calculated as:

O l = l .O........~IDD =

Pata

0 2 = ppO___~ 2 = Pata

0 3 = ppO_._....~ 3 = Pata

1.2 = 0.1399 (13.99%)

8.58

0.166

8.58 = 0.0193 (1.93%)

0.208 = 0.0242 (2.42%)

8.58

(23)

3. Calculate liter-flow rate

U 3 L ~

~ ~ 3

= ~ = 24.4 slm 0.1229

0 .1399- 0"0193/11 -0.0242-01399 /

(24)

4. Find oxygen flow in breathing gas, slm

O s l m - " L x 01 = 24.4 x (0.1399)= 3.41 slm (25)


Example 2. Under the same conditions as Example 1, a supply of gas containing 10% oxygen is already mixed and available. Evaluate the liter-flow rate for this mixture.

O1 is taken as 0.10 (10%) to match the available gas. O+ is 0.0193 (1.93%) from Example 1. 03 is 0.0242 (2.42%) from Example 1.

U 3 L -

O1 -- O2/1 -- O 1 / 1 0 3 0" 10 - 0"0193 / 1 - 0 " 1 0 / 1 - 0.0242

3 = = 36.5 slm (26)

0.0822

Thus, with a 10% oxygen mixture, the liter-flow is 36.5 slm.

Open-Circuit Diving Rigs Breathing Gas Requirements

Simple open-circuit surface-supplied diving rigs are widely used for light activity and moderate depths using air as the breathing gas. The breathing air is supplied to the diving rigs from low to medium pressure air compressors or from a bank of compressed air cylinders. Factors to be considered in the use of surface supplied diving rigs include ventilating-air flow rate, pressure requirements, air compressor design, and environmental conditions. The breathing air is supplied to a helmet or mask within which the diver breathes normally. The helmet acts as a ventilated dead space within which breathing air is mixed with carbon dioxide exhaled by the diver. The air flowrate must be great enough to dilute the exhaled carbon dioxide to nontoxic levels for rebreathing, in the range of 0.01 to 0.02 ata partial pressure. The ventilation rate required is defined by

+~1= PataOslm ( R Q ) M 26.3(Ka _ K1Pata) (27)

where = volume rate of air required, scfm Osl m = oxygen requirements, slm RQ = respiratory quotient (volume of COe produced/volume of O2 con-

sumed) 26.3 = conversion factor for converting slm to scfm and accounts for the

different temperature standards (1 scfm = 26.3 slm) K 2 = desired partial pressure COa in inhaled air, ata K 1 = partial pressure CO2 in the air entering the compressor, ata

Pata = pressure at depth, expressed in ata as (D+33)/33 M = the mixing effectiveness factor (%CO2 in inhaled air)/(%CO2 in

vented air)


The following values can be used in almost all operations: 0sl m = 2.6 slm (enough for heavy to severe exertion); RQ = 0.9 (the highest value likely to occur); K 2 = 0.02 ata (Zone II, Figure 14) and F = 1.0 if actual values are not known. Substituting these values into Equation 27 yields

V = 0"0893Pata (28) 0 . 0 2 - K 1 P a t a

If the breathing air contains no carbon dioxide this equation reduces to

"Q = 4.5Pat a (29)

The mixing effectiveness factor (M) is a factor that corrects for the mixing of exhaled carbon dioxide with incoming breathing air inside the face mask/helmet. The ventilation rate and the internal design of the diving helmet or mask have the most influence on the mixing effectiveness factor. It is the ratio of the percent CO2 in inhaled air divided by the percent CO2 in the vented air and varies as

M = I

M < I

M > I

Completely mixed and the vented mixture contains the same percentage of carbon dioxide as the inhaled mixture. Imperfect mixing and flow is directed such that the inhaled mixture contains less CO2 than the vented mixture. Imperfect mixing but poor flow causes the inhaled mixture to contain more CO2 than the vented mixture.

The mixing effectiveness factor can be determined by measuring the carbon dioxide concentration in the exhaust and at the mouth during inhalation only while the diver performs hard work at various depths.

Relative Gas Flow Rates for Different Underwater Breathing Apparatus

Table 8 compares the relative breathing gas-flow rates needed by four basic types of underwater breathing equipment (demand-regulator scuba, a semi- closed-circuit system, a surface-supplied deep-sea diving outfit, and surface-supplied deep-sea diving outfit with carbon dioxide absorption provisions) under identical conditions. For these comparisons, the breathing gas is air containing 21% oxygen. Data are shown for three levels of oxygen consumption: 3.0 slm, corresponding to heavy exertion, 2.0 slm, corresponding to moderate exertion, and 1 slm, corresponding to light exertion or rest. From the table it can be seen that for heavy exertion, the gas flow required for the semi-closed-circuit system decreases markedly with increasing depth, and that air flow required for the other systems increases markedly with depth. Thus, the semi-closed-circuit system appears especially attractive for deep-diving missions, the scuba for shallow


dives, and the deep-sea rigs for all heavy-duty applications where surface supply of air is advantageous. It should be noted that, at shallow depths, 0.5 cfm of breathing gas does not supply enough oxygen for exertion in the deep-sea diving outfit with carbon dioxide absorption. Flow rate then becomes equal to that for the semi-closed circuit apparatus.

For all of the conditions listed in Table 8, the breathing gas required to supply the semi-closed-circuit system is less than for the demand-regulator system. However, this is not necessarily true of all missions. For example, the demand- regulator system operates on actual demand for breathing gas, which may be low if little exertion is required, while the semi-closed-circuit system requires continuous flow sufficient for the highest exertion level.

Venti lat ion of Large C h a m b e r s

An underwater habitat is a large chamber, and it must be ventilated at a rate that is sufficient to avoid excess carbon dioxide concentrations. If the atmosphere is assumed to be flushed continually with pure (no carbon dioxide) incoming air, a rather simple analysis can be used, and the result is

I - {' air t riaRT vt

Pco2 = �9 1 - e (30) Vair

Table 8 Comparison of Air-Flow Rates for Different Breathing Systems [27]

Air Supplied to Breathing Apparatus, slm

Depth Pressure 02 Demand Semiclosed Scuba Deep Sea Deep Sea ft m ata slm Rig 1 C02 abs 2

150 45.7 5.54 3.0 16.3 388 707 78.6 50 15.2 2.51 3.0 19.7 176 320 35.6 30 9.1 1.91 3.0 22.77 134 244 27.2 20 6.1 1.60 3.0 25.6 112 204 25.63 10 3.0 1.30 3.0 32.9 91 166 32.93 0 0 1.00 3.0 60.0 70 127 60.03

30 9.1 1.91 2.0 15.2 89 244 27.2 10 3.0 1.3 2.0 22.0 61 166 22.03 0 0 1.0 2.0 40.0 47 127 40.03

10 3.0 1.3 1.0 11.0 30 166 18.5 0 0 1.0 1.0 20.0 23 127 20.03

t4.5 cfm air as measured at working depth 20.5 cfm air as measured at working depth 3Flow rate needed to provide needed oxygen.


where Pco 2 m

R T

Vair t

Vt

= partial pressure of carbon dioxide at time t = mass flow rate of carbon dioxide = gas constant for carbon dioxide = absolute temperature = volumetric flow rate of air at depth = time = the total volume of chamber

If the time, t, is very long, then the steady state value of the carbon dioxide partial pressure is

hURT Pco2 = ---7----- (31)

Vair

A very long time is considered to be 3 to 5 time constants (t = 3V t/Vair o r t =

3Wt/~/air)-

Thermodynamics for Diving Systems

First Law of Thermodynamics for General Open and Closed Systems

A general open sys tem (Figure 24) is cons idered for the t he rmodynamic analysis of diving systems. An amount of heat (SQ) is transferred to the system, and the system does an amount of work (~SW). A mass of gas (rain) enters the system, and a mass (mout) leaves the system. The change in the total energy of the system (dE s ) is given by the first law of thermodynamics as

~w

i= ,, i,

. . . . . . . . . . I �9 l i~ ",n [ " : . . . . . . . . "'out system boundary

~Q

Figure 24. Schematic of open system for diving gas thermodynamics.


dE s = 8Q - (SW + Pout dVout - Pin d V i n ) + E i n - Eout (32)

where E = energy of mass entering and leaving the system pdV = flow work term representing the change in system energy due to

mass crossing the boundary

The energy term, E, represents three forms of energy" kinetic energy (KE = mV2/2), potential energy (PE = mgz), and internal energy (U). The total energy term, E, is written as

mV 2 E = ~ + mgz + U (33)

2

and on a per unit mass is given as

V 2 e = ~ + g z + u

2 (34)

Substituting in the first law of thermodynamics (Equation 32) yields

mV dE s = 8Q - 8W + PindVin - PoutdVout +

2

( mV2 ) - - - - -~ + mgz + U

out

+ mgz + U/i n

(35)

For a continuous process, Equation 35 is often used as a quasi-steady state rate equation that is obtained by differentiating with respect to time that yields

des v2 /} E V2 1 ~ - - - m u + ~ + g z = 0 - W + l i l in u + p v + ~ + g z dt dt 2 2

s in

E 1 - m o u t u + p v + ~ + g z 2

out

(36)

The continuity equation (Conservation of Mass) says that

d m s = l'i'lin - Iiqou t

dt (37)

where dm/dt = mass flow rate (rn) dQ/dt = rate of heat transfer Q dW/dt = rate of doing work ~r dV/dt = rate of change of volume (rhv)


The sign convention for work (W) done by system is positive (+), work done on system is negative (-) , heat (Q) added to system is positive (+), and heat removed from system is negative (-).

For steady state, the mass flow rate into the system is equal to the mass flow rate out of the system (lilin -- Iilout). The rate of heat transfer and rate of work do not vary with time, and the state of the fluid in the system at any point does not vary with time. Then, the first law (Equation 36) reduces to

E v2 ] wE v2 1 Q + u + p v + + g z + u + p v + ~ + g z (38) rh 2 in ria 2 out

If q is defined as Q/rh, w is ~r/rh, and the thermodynamic property enthalpy (h = u + pv) is used, then the result yields

[ ) 1 "-~ + gz2 + h2 + Cl = + gZl + hi + ~ in out

(39)

For many cases the change in elevation (z) is small and the work term (fv) is zero, and the result is

[ ] v2 ] - - ~ + h 2 +Cl = ~ + h 1

in out

Using the definition of stagnation enthalpy (ho), the first law reduces to

(40)

hoin + r = hoou t (41)

If flow is adiabatic (41= 0, then

I V 2 ] [ ~ 2 1 (42) - - ~ + h 2 = ~ + h 1 or hoi n = hoout in out

or the stagnation enthalpy is constant. The first law of thermodynamics for a closed system is similar to Equation 32

except that all terms dealing with energy transfer across the boundary are zero. The first law of thermodynamics for a closed system is

dE s = 8Q - 8W (43)

or

d(mses) = 8Q - 8W (44)

Because the mass of the system (ms) is constant, the expressions may be written as


8Q 8 w

m s m s = 8q - 8w (45)

and integrating between state points (1 and 2) yields

v~ - v? u2 - ul + + g (z2 - zl) = ql-2 - wl-2 (46)

2

When the velocity changes and elevation differences are negligible, the first law

per unit mass for a closed system is

u2 - Ul = ql-2 - Wl-2 (47)

Properties of the Perfect Gas

The internal energy of a system is the total molecular energy of the gas molecules in the system, and for a perfect gas, internal energy is commonly assumed to be a function of temperature only. Enthalpy is the sum of the internal energy (U) and the flow work (pV), and it is also assumed to be a function of tempera-

ture only. Thus,

u = f(T) and h = f(T) (48)

Therefore, the change in internal energy may be expressed as

du = CvdT

u 2 - u 1 = c v (T 2 - T l) (49)

where cv = specific heat for a constant volume process Similarly, the change in enthalpy is written as

dh = CpdT o r h 2 - h 1 = Cp ( Y 2 - T l)

R andcp- c v = R ; k = c p / c v ' c v = ~ ; c =

k - 1 P

kR

k - 1

where Cp = specific heat for a constant pressure process R = individual gas constant k = ratio of specific heats

(50)

The normal values given for Cp and Cv are accurate for moderate temperature and low pressure only.

There are two general expressions for the change in entropy of a perfect gas that are obtained from the first law of thermodynamics. The change in entropy is

expressed as


Tds = du + Pdv Tds = dh - vdP (51)

Using the definitions of internal energy, enthalpy, and the equation of state, the change in entropy (assuming constant specific heats) may also be expressed as

dT dv dT dP R ~ (52) ds=cv T + R ~ v or ds=cp T p

Perfect Gas Processes

Constant Volume Process. If only volume change work is considered, then the work term in the first law of thermodynamics for a closed system (Equations 46 and 47) is zero, and the heat transferred is

RdT dq =CvdT = ~ (53)

k - 1

Constant Pressure Process. When the pressure is constant, then the reversible work done is given by

2

W I _ 2 = ~ pdv = p ( v 2 - v 1 )

1

(54)

Isothermal Process. For a constant temperature or isothermal process, the term "pv" in the equation of state equals a constant, and therefore the work term in the first law may be expressed as

2 2

w,_2- pdv- 1 1

d v const ~ = PlVl ln(Pl/P2 ) (55)

V

Also, constant temperature means that the internal energy (u) and the enthalpy (h) are zero. The first law indicates that the heat transfer for this process is equal to the work done, and therefore the heat transfer is

ql-2 = RT In (pl/p2) (56)

Reversible--Adiabatic Process. For a reversible and adiabatic (isentropic) process and for a closed system, the pressure-volume relationship may be expressed as

pv k = constant (57)


where k = ratio of specific heats (Cp/Cv). Using the perfect gas equation between two equilibrium state points, expressions relating the temperature, pressure, and volume are obtained as

k-1 1

T2 P.2 k 72 T1 k-1 -~(= and ~ ( = (58)

and the heat transferred between states 1 and 2 (ql-2) is zero because the process is adiabatic.

Polytropic Process. A polytropic process is a general irreversible process in which there is heat transfer. For this process the properties are defined by

pv n = const (59)

where n is the polytropic exponent that can vary from 0 to oo. Like the isentropic process, expressions for the temperature, pressure and volume between two equilibrium state points are

n-I

-~( = -~( and -~1 ~ V 2 ) (60)

The relationship pl vk = p2 vk is true for a reversible, adiabatic (isentropic) process and is useful for predicting the state of a gas mixture after blowdown of a gas cylinder or hyperbaric chamber from a higher to a lower pressure.

Pressure Vessel Charging and Discharging Process

A pressure vessel (Figure 25) that is being charged or discharged is commonly analyzed using the assumption of no heat transfer (adiabatic). This assump-

Q

' ( ~ m , P , T - " " ~ m m . - - - - - ~ = in ~ - out pressure

ho. vessel h ~ ir l

Figure 25. Typical pressure vessel.


tion is valid only when the charging or discharging process is rapid. The charging and discharging process usually results in temperature changes and consequently undergoes a heat transfer process also. Analyses can be accomplished that include the heat transfer. The first law of thermodynamics is the basic principle used in the analysis of pressure vessels, and starting with Equation 36, it may be written as

d(mu) = 8Q + hoi n dmin - hoout dmou t (61)

where the stagnation enthalpy (hoin) is defined as hoi n = h + Vi~/2.

Adiabatic Charging

Adiabatic charging can be assumed only for a process occurring in a short time interval, which means the heat transfer will be insignificant. Considering partially filled pressure vessels, Equation 61 reduces to

d(mu) = hoi n dmin (62)

For a perfect gas hoi n = cpToi n and u = cvT, and then

kT~ -- 1

T-L = kT~ T1 (63) T1 TI m___2

m l

where T o is the stagnation temperature and

T2= k/pL ) (64) T

Non-Adiabatic Charging and Discharging

When the heat exchange cannot be neglected because the charging process occurs over significant time interval, the first law and standard heat transfer equations are used. The equations are linear and can be solved only if the mass flow rate is constant. This type of discharging is analyzed using the first law, heat transfer equations, and properties of ideal gas. The resulting differential equations must be solved numerically. If the mass flow rate is constant, then analytical solutions are possible.


Isothermal charging is valid when receivers have large thermal capacities, high heat transfer coefficients and low flow rates. For ideal gases and when T1 equals T2, then

P_._L2 = m_._L2 (65)

Pl m l

and for real gases

P_.L = Z2m~ 2 (66) p~ Z~m~

Adiabatic discharging occurs when a blowdown from a receiver is sufficiently rapid, then the expansion process can be considered reversible adiabatic (or isentropic). Several relationships can be derived from the first law and properties of ideal gases. These equations for constant volume receivers are

k-1 T2 m2 = V l = P2 k

-~l = \ ml ) ~,.v2 .]

P2 ( m 2 / k = ( V l ]k = ( T 2 / k / k - 1

-P-]-~ = t,. m, ) \ v 2 j ~-~J 1/k-I

mE Vl P2 T2

m l v2 (67)

Diving Gas Mixtures

At the temperatures and pressures observed in diving, all respired gases can be approximated as a mixture of perfect gases, it is necessary to find the average or total values of molecular weight (M), number of moles (n), and gas constant (R) for such mixtures so that the gas laws can be used. The total number of moles (nt) for "i "components of gas is

n t = n l + n 2 + n 3 + . . . + n i (68)

The mole fraction (x) and the sum of the individual mole fractions are defined as

Xl " - n l / n t =

X 2 = n 2 / n t =

n 1

n 1 + n 2 + . . . . n i

n2

n 1 + n 2 + . . . n i

(equation 69 continued on next page)


ni X i = ni/n t =

n I + n 2 + . . . n i and X 1 + X 2 + X 3 + . . . X i = 1 . 0 (69)

The average molecular weight (Mavg) and average gas constant (Ravg) are defined as

M a v g = X l M 1 + x 2 M 2 + " " x i M i and Ravg -- R u / M a v g (70)

Dalton's Law of partial pressure states that the partial pressure is the pressure a component of gas would exert if all the other gases were removed and the component gas occupied the volume alone. The gas laws are applied to each component gas in the mixture separately. The fraction of a gas by partial pressure is equal to its mole fraction. Many life-support computations depend upon the partial pressure of gases.

pl/Pt = n l / n t = x 1 (71)

The partial volume of a gas is the volume a component gas would occupy if it were at the temperature (T) and total pressure (Pt) of the mixture.

vt = v~ + v : +.. . vi (72)

Partial volume and partial pressure should never be used in the same law equation. Then

PtVl =nlRuT (73)

Because Pt and T are the same for the component gas and the entire mixture, then

V1 nl Pl . . . . . . X 1 (74) Vt nt Pt

Therefore, the partial volume fraction, partial pressure fraction, and mole fraction are the same for perfect gases. The physical properties of ideal gas mixtures in terms of the properties of the constituents of the mixture are tabulated in Table 9.

Control of Underwater Chamber Environment

The environment inside underwater habitats, diver lock-out chambers, submersible pilot chambers, personal transfer capsules, diving bells, and other underwater enclosures must be controlled. In addition to the requirement for the proper breathing gas mixture and supply rate, the environment must be maintained within reasonable limits of temperature and humidity. The increased pres-


Table 9 Summary of Physical Properties of Ideal Gas Mixtures

as a Function of Mixture Component Properties

Temperature Tt = T l = T2 . . . . T i

Mass m t = m I + m 2 + . . . m i Number of moles n t = n~ + n2 + �9 �9 �9 ni Mole fractions x I = nl/n t, x 2 = ne/n t xi = ni/nt Average molecular weight Mave = x 1 M 1 + x 2 M 2 + . . . x i M i

Pressure Pt = Pl + P2 + . . . Pi Volume Vt= Vl + V2 + . . . + Vi

Internal energy U t = U 1 -t- U 2 + . . . + U i or ut = m~u~ + mzu e + ... + m i u i

m t

Enthalpy h t = m~h 1 + m2h 2 +. . . + m i h i

m t

Specific h e a t s Cvt or ave -- m l C v I + m 2 C v 2 + . . + m i C v i

�9 c Pt or ave - m t

m l c p l -t- r r l2Cp2 + . . + m i C p i

m t

Gas constants R a v e = m~R 1 + m2R 2 + . . . miR i

m t

sures and various gas mixtures require the use of special psychrometric charts to evaluate the environmental conditions. These psychrometric charts are available

in the US Navy Diving Gas Manual [27]. Heating and cooling systems are needed to maintain the tempera ture and

humidity at desirable conditions. In normal air, these conditions are typically 23.9~ (75~ and 50% relative humidity, but if the water depth requires the use of hel ium/oxygen mixtures, then the desirable temperature condition is more like 29.4~ (85~ Heat transfer analysis of the pressure vessel system is necessary to determine the steady state and transient conditions inside the chamber.

Psychrometric Charts for Diving Applications

Psychrometric charts are commonly used to determine the effects of heating and cooling moist air at atmospheric pressures. These charts relate enthalpy changes to wet- and dry-bulb temperatures, relative humidity, dew point, and specific humidity. Definitions of terms used in psychrometric calculations are defined in Table 10. Standard psychrometric charts are available for a single gas mixture (air) and at a single pressure (14.7 psia), but they are not suitable for use with a range of gas compositions and pressures. Accordingly, special psychrometric charts suitable for use with diving-gas mixtures to depths of 1,300 feet

were developed by the US Navy [27]. Charts suitable for gases of differing molecular weight and density, required

that the mole be used as the unit of gas quantity. A mole is a quantity having a


Table 10 Definitions of Psychrometric Terms and Symbols

Term Symbol Definition

Humidity ratio W Ratio of the mass of water vapor contained in a given sample of moist gas to the mass of dry gas with which the water vapor is associated, pounds of water vapor per pound of dry gas.

Relative humidity Ratio of the mole fraction of water vapor in moist gas to the mole fraction of water vapor in saturated moist gas at the same temperature and pressure. It is normally expressed as a percentage.

Mole fraction Any given constituent in a mixture of gaseous substances is used herein as the number of moles of that constituent present in the mixture divided by the total number of moles of dry gas contained in the mixture. It is numerically equal to the volume fraction. Xw represents the mole fraction for water vapor, and Xws represents that for water vapor at saturation.

Dry-bulb temperature tdb The Fahrenheit temperature of moist gas at rest with respect to the temperature-measuring element.

Thermodynamic wet-bulb temperature twb

Temperature at which water (liquid or solid), by evaporating into moist gas at given dry-bulb temperature, tab, and humidity ratio, W, can bring the gas to saturation adiabatically at the same temperature twb while the pressure p is maintained constant.

Specific heat Cpm Specific heat at constant pressure, expressed as Btu/mole of dry gas

Additional nomenclature employed in the use of the diving-gas psychrometric charts

h

hm M

Q rh

P

Specific enthalpy, Btu/lb Molar enthalpy, Btu/mole Pound molecular weight, lb/mole Volumetric flow rate, ft3/min Heat flow rate, Btu/min Mass flow rate, lb/min Density, lb/ft 3

General subscripts used d w

m

Dry gas mixture Water vapor Mole as unit of weight


weight, in pounds, equal to the molecular weight of the dry gas mixture. A useful property of this unit is that the volume of a mole of gas is the same for any gas of any molecular weight, at any specified temperature and pressure. The molar volume and partial pressure of water vapor provide a means of determining the quantity of water per mole of gas. The psychrometric chart, Figure 26, is for use with gas mixtures at pressures from 14.7 psia to 200 psia. Figure 27 covers the pressure range from 100 psia to 600 psia. Corrections for moisture enthalpy as a function of temperature are obtained from Figure 28 and applied to the enthalpy data from Figure 26 and 27. The slope of constant total enthalpy lines for different gas mixtures are shown in the upper left area of Figure 26 and 27. The molar specific heats of air, nitrogen, and oxygen are approximately equal at 6.95 Btu/lb-mole and the molar specific heat of helium is 4.96 Btu/lb-mole. Lines for gas mixtures containing different percentages of helium are plotted between these limits.

To determine a constant enthalpy line, a line is drawn parallel to the appropriate line in the upper left area of the chart, using a parallel rule or drafting triangles. This line starts at the intersection of the wet-bulb temperature and the proper pressure curve and extends to the right until it reaches the desired value of dry-bulb temperature. The mole fraction of water vapor in the mixture at the dry-bulb temperature can be read from the right scale. The relative humidity can then be calculated as the ratio of this value to the value corresponding to the intersection of the dry-bulb temperature with the proper pressure curve. Alterna- tively, if dry-bulb temperature and relative humidity are known, then the wet- bulb temperature can be found. First, the moisture mole fraction is determined for a saturated gas at the intersection of the dry-bulb temperature and proper pressure curve. This value is multiplied by the relative humidity, and the result is plotted on the dry-bulb temperature line. Then, a constant-wet-bulb temperature line is drawn from this point to the left until it intersects with the proper pressure line. The wet-bulb temperature is read below this intersection on the temperature scale. Enthalpy is given for the moisture content alone, rather than for the moist gas mixture, and is expressed in Btu/mole of dry gas. Most of the enthalpy of the water vapor is in the latent heat of vaporization. Consequently, the water-vapor enthalpy is almost directly proportional to the amount of water vapor present. For this reason, both enthalpy and moisture concentration appear on the vertical scale of Figures 26 and 27 as a temperature correction.

Calculations Using Psychrometric Charts

Evaluate the moisture removal rate, energy requirement, and coil temperature to cool, dehumidify, and reheat a 10% 02 and 90% He gas mixture at 200 psia from 85~ dry-bulb temperature and 90% relative humidity to a temperature of 80~ and 50% relative humidity. The gas mixture flowrate through the environmental control system is 120 cfm. Also, find the wet-bulb temperatures for this gas mixture at 85~ dry-bulb temperature and at 90% and 50% relative humidities.


l - ==

o

N

mmm

3 0 4 0

i i i i i i i Slope of Constant

!

Wet Bulb Temperature Lines ~ . . 1 I 1 t J

~ 111 / |

/ / r

/

r J l l l l l l | / l r A l l F 4 l / I l l l l / l l | l A / / | l i b t

m m ~ _ ~ . - . ~ / /

7 e~ / ' , , Y , / ,~/ / / / / / / / " ~" / / ~ ' /

el 140C IDB

"3 _~ 1200 0 B

0 .04 "~ eO0

u r

- / ~ 200

0.00 0 so so 7o eo 90 ,oo ,,o ,zo ,30 Dry Bulb Temperature (~

Figure 26. Psychrometr ic chart for gas mixtures from 14.7 to 200 psia [27].

So lu t ion .

Moisture Removal Rate. Using Figure 27, the saturation line for 200 psia shows that the moisture mole fraction at 85~ and 100% relative humidity is 0.003 molew/mOled. Because relative humidity, r is

<~ = Xw/Xws (75)

The moisture mole fraction at ~ = 90% is

x w =0.90 (0.003) = 0.0026 molew/mole d (76)

and at = 50% is

x w =0.50 (0.003) = 0.0015 molew/mole d (77)

The 90 and 50% relative humidity points are shown on Figure 27 as Points A and B. From the density tables for the 10% 02 and 90% He mixture, the density values given for 420 ft depth (201.4 psia) are

Pd = 0.2349 Ibm/ft 3 at 80 ~ and Pd = 0.2307 lbm/ft 3 at 90 ~ (78)


8 ,-z

.,.,.. 3E 7 r,~

~,o r t~ 6 o

�9 .1- _..~ 5 �9 -~ ~

_o I 1.

.40 Percent Helium -~0 I inMixture -~6 [

\ \

-

'\\ . / 2 ( .'~

j " ~ . J

30 40 50

/ / / / /

/ / ,/ / / /

~/ ,/ ~ , .4/ / / / / /, / / /

, ' o #, / 2 ' / . / / o~i , / / /

o/ , /" / " A // //./,

/ , i - / - ~, / C /

/ / 2 . . . ~ > - > . ~ I J .. . . I "

60 70 80 90 100 110 120 Dry Bulb lemperature (~

0.008 160 O3 t13

0.007 ~" 14o'~

0.006 120 ~

F--- 0.005 ~ lOO,,,, ~ ,-.-

0.004 " ~ 80 O E

0.003 ~ 80 ~ N o

0.oo2 4o ._~

L -

I . U 0.001 ~ 20

=E

0.000 0 130

Figure 27. Psychrometric chart for gas mixtures from 100 to 600 psia [27].

and the molecular weight of the dry gas mixture is given as 6.803. The density at 850F is found by interpolation and the result is

o =0-23281bm/ft 3 P85 F (79)

The corresponding moist-gas-mixture enthalpies are computed by the following expression

h = [(Cpm) dtdb + hwm]/M d (80)

The value of the specific heat of the dry gas is computed as

(Cp) d = 6.95 (x02) + 4.965 (XHe) = 6.95(0.1) + 4.965(0.9)

= 5 .164Btu /mole F (81) d -

Using this value in the expression for enthalpy, the following values are obtained:

h A = [(5.164) (85) + 54] / 6.803 = 72.46 Btu / lb d (82)


IIIIIIII Add enthalpy correction /'Ill/Ill ,o va,ue o~,a,ne~,r0m / I I Fig. 26 or Fig. 27 III/I J' l / / IIIIII

t / I / J / I / / / / ' / / / /

/ ' / / ' l i ,'l l j , / /7 , / / I / I

, ~ / / / / / / / , / , t , ~ , , y j / I , , ~ / ~ / ' / / / / I

30 40 50 60 70 80 90 100 110 Dry bulb Temperature (~

0.08

�9 I l l / \ \ \ \ i

L _

I / \ \ ' , ~ ~ \ o \ 0.03 l\ \ \

i 1

I\ \, \oo2_~ I~ \ I \ ~

L -

0.01 o i

J 0.00 120 130

Figure 28. Correction for enthalpy of moisture content [27].

h c = [(5.164) (64.5) + 30] / 6.803 = 53.37 B t u / l b d (83)

Q = m d (hAhc) = 120(0.2328) ( 7 2 . 4 6 - 53.37) = 27.94 (19.09) = 533.3 B t u / m i n = 31,998 Btu / hr (84)

This is equivalent to approximately 2.7 tons of refrigeration where 1 ton of

refrigeration is equal to 200 Btu/min. In controlling the climate of a diving habitat, it may be necessary to reheat the

gas leaving the cooling coil (point C on Figure 27). If it is desirable to have the gas from the environmental control system returning at 80~ the reheat energy

required can be computed as

Ah64.5_80 = (Cp) d A tab ] M a = 5.164 (80 - 64.5) / 6.803 = 11.77 Btu / lb d (85)

Q = m d Ah = 27.94 (11.77) = 328.7 Btu / min = 19, 722 Btu / hr (86)


The moisture removal rate (MRR) is calculated as follows

W 1 = humidity ratio at 90% ~ and W E = humidity ratio at 50% ~)

Mw = 0.0026 ( 18 ) lbw W1 = Xwl M----T 6.803 = 0.00688~1ba (87)

M w = o . o 0 1 5 ( 18 ) lbw W2 = Xw2 M---d- 61803 = 0.00397~1ba (88)

( MRR = (W1 - W2) rh d = (0.00688 - 0.00397) [120

\

0.0811b m =117.11b m min d a y

mlnft3/(0"2328 lbm/-~ =

(89)

The cooling coil temperature is determined from the psychrometric chart as 64.5~ that is the intersection of the horizontal line from the point corresponding to 80~ and 50% relative humidity to the 100% saturation line for a pressure of 200 psia.

On Figure 27, the slope of the constant wet-bulb temperature line should be that corresponding to a specific heat of 5.164 Btu/mole a. Drawing lines of this slope from Points A and B to the 200 psia saturation line gives Points D and E. At saturation, wet-bulb and dry-bulb temperatures are the same; therefore, the following results are obtained:

twb = 84.5 ~ F when tab = 85~ F and ~ = 90% (90)

twb = 81. I~ when t db = 85~ and ~ = 50% (91)

One of the applications of a psychrometric chart is in systems where the mixture of two gas vapor streams occur. The underwater habitat is an example of humid air leaving the lungs and mixing with the gas vapor flow from the dehumidifier and heater. A mass balance on the water vapor may be expressed as

mv,1 +mv, z = (my,1 + my,2)

Because

(92)

W = ~ mv (93)

o r

m gas

Wlmgas, + W mgas W3 (mgas 1 + mgas (94)


Now the unknown mixed humidity ratio W 3 can be computed if mgas,1 and mgas,2 are known. A second parameter can be determined from an energy balance as follows

Hlmgas, l + H2mgas,2 = H 3 (mgas, l + mgas,2 ) (95)

Because lines of constant enthalpy (H) and humidity ratio (W) cross each other, they fix a specified location on the chart. In many cases, this third point lies on or very close to the straight line connecting points 1 and 2. Air leaving lungs (point 1) is located at (T L, Wl) and air leaving the dehumidifier and heater (point 2) is found at (T D, HE). The mixture condition (point 3) is located at the intersection of W 3 and H a lines. Equations 94 and 95 can be used to solve for either W 3 or H 3, and the corresponding W 3 or H a can be found on the line between points 1 and 2 on the chart.

Five basic engineering processes that can be accomplished with standard environmental control equipment are simple heating and cooling, dehumidification by cooling, adiabatic humidification, and chemical dehumidification. These processes are illustrated in Figure 30. Whatever the process, the heating and cooling loads in Btu/min can be computed by reading the total H at each state, taking the difference (H 1 - HE), and multiplying by the gas flow rate in lb/min as illustrated in

Cooling or Heating Load = Ii'lgas (H 1 - H E ) (96)

J t H !

100% relotive

T ~ TD TL Figure 29. Schematic of mixing process on psychrometric chart.


50% relative 100% relative humidity--~_ ~humidity

by cool,ng .... -,~ de'~mid,ificption T "~

Figure 30. Schematic of engineering processes on psychrometric charts.

Mixing of Breathing Gases

Procedures for mixing breathing gases include mixing by volume, weight, calibrated flowmeter, and partial pressure. When the breathing gas is not air, then mixtures of helium and oxygen are commonly used. When mixing by volume, the desired gases at ambient pressure are mixed by adding suitable volumes of each gas to a constant pressure gas vessel, and the volume of each gas is measured. The mixture is then analyzed, and the gas composition is adjusted as necessary. Finally, the gas mixture is compressed into high pressure storage vessels. Mixing by weight requires that the weight of gas mixture needed to fill the cylinder to the desired pressure be calculated. The percentage by weight of each constituent gas is then calculated, and the weight of each gas to be added is determined. The gas storage vessel is then placed on a very accurate scale, and the required weights of gases are added. Another mixing method is to mix the gases using calibrated flow meters. A very useful procedure is to mix the gas by partial pressure. In this case, each constituent gas is sequentially added to the storage vessel. The proportion by volume is in direct proportion to the partial pressure of each gas in the mixture. There are two approaches to mixing by par-


tial pressure, and these are the ideal or real gas methods. The ideal gas method uses the perfect gas law, and the real gas method uses the real gas law or uses tables of properties that consider the effects of compressibility.

Mixing by Partial Pressures

When two or more gases are mixed at constant temperature, the partial pressure of each gas in the mixture is proportional to the percentage by volume of that gas in the mixture, and the sum of the partial pressures of all gases must add to the total pressure. Consequently, it is possible to prepare a gas mixture in a high-pressure storage vessel by sequentially adding several gases. As each gas is added, the storage vessel pressure increases, and the pressure, as a percentage of the final pressure, is proportional to the percentage by volume of the gas. Isothermal conditions are assumed during the mixing process, and the initial and final temperatures are identical. If mixing is accomplished quickly, then the adiabatic compression in the receiving vessel causes a temperature increase that can result in a mixing error. This effect is commonly minimized by slowly adding the gas to the storage vessel such that the gas temperature can reach equilibrium with vessel metal temperatures or by waiting for temperatures to return to ambient and then adding additional gas to the vessel until the proper pressure is reached. A very accurate pressure gauge having many scale divisions and of suitable scale range is required for mixing gases by partial pressure.

When the receiving vessel is filled by the partial-pressure method it contains the proper amounts of oxygen and helium, but these gases may not be well mixed. At the high pressures used for gas storage, molecular diffusion is very slow, and gas drawn from the tank immediately after mixing can be significantly different in composition than the average for the entire receiver. Consequently, a storage period of many hours is recommended before use or mechanical agitation is recommended to improve the rate of mixing.

If small oxygen percentages are used for a helium-oxygen mixture, then the best accuracy is attained by adding oxygen first to the vessel using a low pressure gauge to measure its partial pressure and then adding helium to reach the final mixture pressure. When the oxygen percentage is high, or when helium is already in the vessel, oxygen is usually added after the helium, and it is necessary to know the partial pressure of helium and the final mixture pressure for the desired mixture.

When adding oxygen first, the density of oxygen at its partial pressure in the mixture is found by solving

32aPm (97) Po2 28a + 4


where p = density of oxygen at its partial pressure in the mixture Pm = density of the gas mixture at the final pressure

a = decimal percentage of oxygen in the mixture

The constants 32, 4, and 28 are the molecular weight of oxygen, helium, and the difference in molecular weight of oxygen and helium, respectively. The partial pressure of oxygen corresponding to this density is located in the oxygen density table (Table 11) by locating the computed density in the proper temperature column and finding the corresponding tabulated pressure. Oxygen is added to the vessel until this pressure is reached, and then helium is added until the final mixture pressure is reached. This procedure must be accomplished at a nearly constant gas temperature. Therefore, the receiving vessel is filled slowly, allowed to cool until the pressure is stabilized, and is then charged again to the desired final partial pressure.

If helium is to be added to the mixture first, the helium density at its partial pressure in the final mixture is determined using

4bPm (98) P He 32 -- 28b

where Pile = density of helium b = decimal percentage of helium in the mixture.

The other terms are the same as in the oxygen equation. Helium partial pressure is taken from the helium density table (Table 12). This real-gas mixture method cannot be used where a third gas such as nitrogen is present, as the nitrogen would affect the compressibility of the mixture and cause error.

As an example, consider refilling a partially filled tank with the same mixture. The tank contains a mixture of 10% oxygen and 90% helium at a pressure of 900 psia. It is to be refilled with the same mixture to a pressure of 3,000 psia using pure oxygen and helium. The temperature is assumed to be a constant 80~ What is the tank pressure after the addition of oxygen? The results of this example are tabulated in Table 14 and show that the calculated partial pressure of oxygen to be added is 184.6 psi using the real-gas method and 210 psi using the perfect-gas method. The 02 percentage of the gas mixture added is 10% using the real-gas method and 11.4% using the perfect-gas method. If the oxygen content of the initial mixture is different from that of the final mixture, then the same steps are followed, but al and a2 are different values and Pol and Po2 are found in different tables.



Table 11 Oxygen Density Data from Virial Equation of State [27]

Depth

ft m

0.0 79.3

191.7 303.8 416.1 528.4 640.8 752.8 865.2 977.5

1089.7 1314.4 1538.8 1763.5 1987.9 2212.7 2437.1 2661.5 2886.2 3110.6 3335.3 3559.7 3784.4 4008.8 4233.6 4458.0 4682.7 4907.1 5131.8 5356.2 5580.6 5805.4 6029.8

0.0 24.2 58.4 92.6

126.8 161.1 195.3 229.5 263.8 298.0 332.2 400.7 469.1 537.6 606.0 674.5 742.9 811.3 879.8 948.3

1016.8 1085.2 1153.7 1222.1 1290.6 1359.0 1427.5 1495.9 1564.4 1632.8 1701.2 1769.8 1838.2

Pressure

psia ata

14.7 5O

100 150 200 250 300 350 400 450 500 600 700 800 900

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700

1 3.40 6.81

10.21 13.61 17.01 20.42 23.81 27.22 30.62 34.02 40.83 47.63 54.44 61.24 68.05 74.85 81.65 88.46 95.26

102.07 108.87 115.68 122.48 129.29 136.09 142.9 149.7 156.51 163.31 170.11 176.92 183.72

30

0.08958 0.30555 0.61319 0.92293 1.2348 1.5486 1.8646 2.1826 2.5025 2.8244 3.1482 3.8015 4.4619 5.129 5.8023 6.4812 7.1651 7.8532 8.5447 9.2388 9.9346

10.631 11.328 12.023 12.716 13.405 14.091 14.771 15.446 16.114 16.775 17.427 18.071

40

0.08778 0.29934 0.60055 0.90365 1.2086 1.5154 1.8239 2.1343 2.4465 2.7602 3.0758 3.7117 4.3537 5.0015 5.6546 6.3124 6.9743 7.6396 8.3076 8.9777 9.6489

10.3206 10.9917 11.6615 12.3291 12.994 13.654 14.31 14.961 15.606 16.243 16.874 17.497

50

0.08605 0.29339 0.58845 0.88519 1.1836 1.4836 1.7852 2.0884 2.3931 2.6994 3.0071 3.6267 4.2517 4.8815 5.5159 6.1542 6.7959 7.4404 8.087 8.7352 9.3841

10.0331 10.6814 11.3283 11.973 12.615 13.253 13.887 14.515 15.139 15.756 16.366 16.969


Temperature (~

60 70 80 90 100 110 120

0.08439 0.08279 0.08125 0.07977 0.07834 0.07696 0.07563 0.28767 0.28216 0.27687 0.27178 0.26687 0.26214 0.25758 0.57682 0.56566 0.55493 0.54461 0.53467 0.52510 0.51586 0.86747 0.85048 0.83415 0.81846 0.80338 0.78886 0.77486 1.1596 1.1366 1.1145 1.0933 1.0729 1.0534 1.0345 1.4531 1.4240 1.3935 1.3692 1.3434 1.3187 1.2949 1.7481 1.7126 1.6786 1.6460 1.6147 1.5847 1.5558 2.0444 2.0023 1.9622 1.9237 1.8868 1.8514 1.8173 2.3421 2.2934 2.2468 2.2023 2.1597 2.1188 2.0794 2.6411 2.5855 2.5325 2.4817 2.4332 2.3868 2.36t0 2.9413 2.8787 2.819 2.762 2.7075 2.6554 2.6052 3.5455 3.4683 3.3949 3.3249 3.258 3.1942 3.1329 4.1542 4.0618 3.974 3.8905 3.8109 3.735 3.6622 4.7672 4.6589 4.5562 4.4586 4.3659 4.2775 4.1928 5.3838 5.2591 5.1409 5.0289 4.9225 4.8214 4.7245 6.0038 5.8619 5.7278 5.6009 5.4805 5.3663 5.2569 6.6265 6.467 6.3165 6.1742 6.0396 5.9119 5.7897 7.2514 7.0737 6.9064 6.7485 6.5992 6.4578 6.3226 7.878 7.6817 7.4972 7.3232 7.159 7.0037 6.8553 8.5056 8.2904 8.0882 7.898 7.7187 7.5493 7.3875 9.1336 8.8991 8.6791 8.4723 8.2777 8.094 7.9187 9.7613 9.5073 9.2693 9.0459 8.8357 8.6376 8.4487

10.3882 10.1145 9.8583 9.618 9.3923 9.1798 8.977 11.0135 10.72 10.4455 10.1884 9.947 9.72 9.5035 11.6367 11.3233 11.0306 10.7565 10.4996 10.258 10.028 12.257 11.924 11.613 11.322 11.049 10.793 10.549 12.874 12.521 12.192 11.884 11.596 11.326 11.068 13.486 13.114 12.767 12.443 12.14 11.855 11.584 14.095 13.703 13.339 12.998 12.679 12.381 12.096 14.697 14.287 13.905 13.548 13.215 12.902 12.604 15.295 14.866 14.467 14.094 13.746 13.42 13.109 15.886 15.439 15.023 14.635 14.272 13.933 13.609 16.47 16.006 15.574 15.171 14.794 14.441 14.104



Table 11 (Cont inued)

Depth

ft m

6254.5 6478.9 6703.6 6928.0 7152.8 7377.2 7601.9 7826.3 8050.7 8275.4 8499.8 8724.5 8948.9 9173.7 9398.1 9622.8 9847.2 10071.9 10296.3 10520.7 10745.5 10969.9 11194.6

1906.7 1975.1 2043.6 2112.0 2180.5 2248.9 2317.4 2385.8 2454.2 2522.7 2591.2 2659.7 2728.1 2796.6 2865.0 2933.5 3001.9 3070.4 3138.8 3207.2 3275.7 3344.1 3412.7

Pressure

psia ata

2800 190.53 2900 197.33 3000 204.14 3100 210.94 3200 217.75 3300 224.55 3400 231.36 3500 238.16 3600 244.96 3700 251.77 3800 258.57 3900 265.38 4000 272.18 4100 278.99 4200 285.79 4300 292.6 4400 299.4 4500 306.21 4600 313.01 4700 319.81 4800 326.62 4900 333.42 5000 340.23

30

18.705 19.33 19.945 20.55 21.145 21.728 22.302 22.864 23.416 23.958 24.489 25.01 25.52 26.021 26.512 26.993 27.465 27.927 28.381 28.826 29.263 29.691 30.112

40

18.111 18.717 19.313 19.9 20.478 21.047 21.605 22.154 22.693 23.222 23.741 24.251 24.752 25.243 25.725 26.197 26.661 27.117 27.564 28.003 28.433 28.856 29.271

50

17.565 18.152 18.732 19.303 19.865 20.418 20.963 21.498 22.024 22.542 23.05 23.549 24.04 24.922 24.995 25.459 25.916 26.364 26.804 27.236 27.661 28.078 28.488


Temperature (~

60 70 80 90 100 110 120

17.047 17.618 18.18 18.735 19.282 19.821 20.351 20.873 21.387 21.892 22.389 22.878 23.358 23.831 24.295 24.751 25.199 25.64 26.073 26.498 26.917 27.328 27.732

16.567 17.12 17.667 18.207 18.739 19.264 19.781 20.29 20.792 21.286 21.772 22.25 22.721 23.184 23.639 24.087 24.527 24.96 25.386 26.805 26.217 26.622 27.02

16.119 16.657 17.189 17.715 18.233 18.745 19.249 19.746 20.236 20.719 21.195 21.663 22.124 22.577 23.024 23.463 23.896 24.322 24.741 25.193 25.558 25.957 26.35

15.701 16.225 16.743 17.255 17.76 18.259 18.752 19.237 19.716 20.188 20.653 21.112 21.564 22.008 22.447 22.878 23.303 23.721 24.133 24.539 24.938 25.331 25.718

15.31 15.821 16.326 16.825 17.318 17.806 18.287 18.761 19.23 19.692 20.147 20.597 21.039 21.476 21.906 22.329 22.747 23.158 23.563 23.962 24.356 24.743 25.125

14.944 15.442 15.935 16.423 16.905 17.381 17.851 18.315 18.774 19.226 19.672 20.113 20.547 20.975 21.397 21.814 22.224 22.628 23.027 23.42 23.807 24.189 24.565

14.595 15.082 15.563 16.039 16.509 16.975 17.435 17.889 18.338 18.78 19.218 19.649 20.075 20.495 20.91 21 318 21 722 22 119 22 511 22 898 23.279 23.655 24.025


Table 12 Helium Density Data from Virial Equation of State [27]

Depth

ft m

0.0 79.3

191.7 303.8 416.1 528.4 640.8 752.8 865.2 977.5

1089.7 1314.4 1538.8 1763.5 1987.9 2212.7 2437.1 2661.5 2886.2 3110.6 3335.3 3559.7 3784.4 4008.8 4233.6 4458.0 4682.7 4907.1 5131.8 5356.2 5580.6

0.0 24.2 58.4 92.6

126.8 161.1 195.3 229.5 263.8 298.0 332.2 400.7 469.1 537.6 606.0 674.5 742.9 811.3 879.8 948.3 016.8 085.2 153.7 222.1 290.6 359.0 427.5 495.9 564.4 632.8 701.2

Pressure

psia ata

14.7 50

100 150 200 250 300 350 400 450 500 600 700 800 900

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

1 3.40 6.81

10.21 13.61 17.01 20.42 23.81 27.22 30.62 34.02 40.83 47.63 54.44 61.24 68.05 74.85 81.65 88.46 95.26

102.07 108.87 115.68 122.48 129.29 136.09 142.90 149.70 156.51 163.31 170.11

30

0.01119 0.03802 0.07590 0.11365 0.15126 0.18874 0.22609 0.26329 0.30038 0.33733 0.37416 0.44741 0.52017 0.59242 0.66418 0.73545 0.80624 0.87656 0.94641 1.01579 1.08472 1.1532 1.2212 1.2888 1.356 1.4227 1.489 1.5549 1.6204 1.6855 1.7501

40

0.01097 0.03726 0.07439 0.11139 0.14826 0.18500 0.22161 0.25810 0.29446 0.33069 0.36681 0.43866 0.51003 0.58092 0.65133 0.72128 0.79076 0.85979 0.92837 0.9965 1.0642 1.13145 1.1983 1.2647 1.3307 1.3963 1.4614 1.5262 1.5906 1.6545 1.7181

50

0.01075 0.03653 0.07294 0.10922 0.14537 0.18140 0.21731 0.25310 0.28877 0.32431 0.35975 0.43025 0.50028 0.56986 0.63897 0.70764 0.77587 0.84365 0.911 0.97793 1.04443 1.11051 1.1762 1.2414 1.3063 1.3708 1.4348 1.4985 1.5618 1.6247 1.6873


Temperature (~

60 70 80 90 100 110 120

0.01054 0.01034 0.01015 0.00997 0.00979 0.00962 0.00945 0.03583 0.03515 0.03451 0.03388 0.03327 0.03269 0.03213 0.07153 0.07019 0.06889 0.06765 0.06644 0.06528 0.06377 0.10712 0.10512 0.10318 0.10131 0.09951 0.09777 0.09609 0.14260 0.13992 0.13735 0.13487 0.13247 0.13016 0.12793 0.17794 0.17462 0.17141 0.16831 0.16534 0.16246 0.15968 0.21527 0.20919 0.20536 0.20166 0.19809 0.19465 0.19133 0.24829 0.24384 0.23921 0.23491 0.23076 0.22675 0.22289 0.28329 0.27802 0.27285 0.26805 0.26333 0.25876 0.25436 0.31818 0.31227 0.30657 0.30109 0.29579 0.29067 0.28573 0.35295 0.34641 0.3401 0.33402 0.32816 0.32249 0.31702 0.42215 0.41435 0.40684 0.3959 0.3926 0.38584 0.37932 0.4909 0.48186 0.47315 0.46476 0.45665 0.44882 0.44126 0.55921 0.54895 0.53906 0.52953 0.52032 0.51144 0.50285 0.62708 0.61562 0.60457 0.59391 0.58362 0.57368 0.56408 0.69451 0.68186 0.66967 0.6579 0.64654 0.63557 0.62497 0.76152 0.7477 0.73437 0.72151 0.7091 0.6971 0.68551 0.82811 0.81313 0.79869 0.78475 0.77129 0.75829 0.74572 0.89428 0.87816 0.86261 0.84761 0.83312 0.81912 0.80559 0.96003 0.94279 0.92616 0.9101 0.8946 0.87961 0.86512 1.02538 1.00703 0.98933 0.97223 0.95572 0.93976 0.92433 1.09033 1.07089 1.05211 1.034 1.01649 0.99958 0.98322 1.1549 1.1344 1.1145 1.0954 1.0769 1.0591 1.0418 1.219 1.1974 1.1766 1.1565 1.137 1.1182 1.1 1.2828 1.2602 1.2383 1.2172 1.1968 1.177 1.158 1.3462 1.3225 1.2996 1.2775 1.2562 1.2356 1.2156 1.4092 1.3845 1.3606 1.3376 1.3153 1.2937 1.2729 1.4719 1.4461 1.4213 1.3973 1.3741 1.3516 1.3299 1.5341 1.5074 1.4816 1.4566 1.4325 1.4092 1.3866 1.596 1.5683 1.5415 1.5156 1.4906 1.4664 1.443 1.6576 1.6288 1.6011 1.5743 1.5484 1.5234 1.4991



Table 12 (Continued)

Depth

ft m

5805.4 6029.8 6254.5 6478.9 6703.6 6928.0 7152.8 7377.2 7601.9 7826.3 8050.7 8275.4 8499.8 8724.5 8948.9 9173.7 9398.1 9622.8 9847.2 10071.9 10461.3 10520.7 10745.5 10969.9 11194.6

1769.8 1838.2 1906.7 1975.1 2043.6 2112.0 2180.5 2248.9 2317.4 2385.8 2454.2 2522.7 2591.2 2659.7 2728.1 2796.6 2865.0 2933.5 3001.9 3070.4 3189.1 3207.2 3275.7 3344.1 3412.7

Pressure

psia ata

2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000

176.92 183.72 190.53 197.33 204.14 210.94 217.75 224.55 231.36 238.16 244.96 251.77 258.57 265.38 272.18 278.99 285.79 292.60 299.40 306.21 318.01 319.81 326.62 333.4 340.2

30

1.8144 1.8783 1.9417 2.0048 2.0676 2.1299 2.1918 2.2535 2.3147 2.3756 2.4361 2.4963 2.5561 2.6156 2.6747 2.7835 2.792 2.8501 2.908 2.9654 3.0226 3.0795 3.136 3.1923 3.2482

40

1.7813 1.8441 1.9066 1.9687 2.0304 2.0917 2.1527 2.2133 2.2736 2.3335 2.3931 2.4524 2.5113 2.5699 2.6282 2.6861 2.7437 2.801 2.858 2.9146 2.971 3.027 3.0828 3.1382 3.1934

50

1.7495 1.8113 1.8727 1.9338 1.9945 2.0549 2.115 2.1746 2.234 2.293 2.3517 2.41 2.4681 2.5258 2.5832 2.6403 2.697 2.7535 2.8096 2.8655 2.9211 2.9763 3.0813 3.086 3.1404


Temperature (~

60 70 80 90 100 110 120

1.7187 1.689 1.6604 1.6327 1.6059 1.58 1.5549 1.7796 1.7489 1.7193 1.6908 1.6631 1.6364 1.6105 1.84 1.8085 1.778 1.7485 1.72 1.6924 1.6657 1.9001 1.8677 1.8363 1.8059 1.7766 1.7482 1.7207 1.9599 1.9265 1.8943 1.8631 1.8329 1.8037 1.7754 2.0194 1.9851 1.9519 1.9199 1.8889 1.8588 1.8298 2.0785 2.0433 2.0093 1.9764 1.9445 1.9137 1.8839 2.1373 2.1012 2.0663 2.0326 2.0000 1.9684 1.9377 2.1957 2.1588 2.123 2.0885 2.055 2.0227 1.9913 2.2588 2.216 2.1795 2.1441 2.1099 2.0768 2.0447 2.3117 2.273 2.2356 2.1994 2.1644 2.1805 2.0977 2.3692 2.3296 2.2914 2.2545 2.2187 2.1841 2.1505 2.4263 2.386 2.347 2.3092 2.2727 2.2373 2.201 2.4832 2.442 2.4022 2.3637 2.3264 2.2903 2.2553 2.5398 2.4978 2.4572 2.4179 2.3799 2.343 2.3073 2.596 2.5532 2.5119 2.4718 2.433 2.3955 2.3591 2.652 2.6084 2.5662 2.5255 2.486 2.4477 2.4106 2.7076 2.6633 2.6203 2.5788 2.5386 2.4596 2.4619 2.763 2.7179 2.6704 2.6319 2.591 2.5613 2.5129 2.8181 2.7722 2.7277 2.6848 2.6431 2.6028 2.5617 2.8728 2.8262 2.781 2.7973 2.695 2.654 2.6104 2.9273 2.8799 2.8341 2.7896 2.7466 2.7049 2.6645 2.9815 2.9334 2.8868 2.8417 2.798 2.7556 2.7146 3.0355 2.9866 2.9391 2.8935 2.8491 2.8061 2.7644 3.0891 3.0395 2.9915 2.945 2.9 2.8563 2.814


Table 13 Density Data for Helium/Oxygen Gas Mixture

(90% He/10% 02, Mavg = 6.803) from Virial Equation of State [27]

Depth

0.0 79.3

191.7 303.8 416.1 528.4 640.8 752.8 865.2 977.5

1089.7 1314.4 1538.8 1763.5 1987.9 2212.7 2437.1 2661.5 2886.2 3110.6 3335.3 3559.7 3784.4 4008.8 4233.6 4458.0 4682.7 4907.1 5131.8

m

0.0 24.2 58.4 92.6

126.8 161.1 195.3 229.5 263.8 298.0 332.2 400.7 469.1 537.6 606.0 674.5 742.9 811.3 879.8 948.3

1016.8 1085.2 1153.7 1222.1 1290.6 1359.0 1427.5 1495.9 1564.4

Pressure

psia ata

14.7 50

100 150 200 250 300 350 400 450 500 600 700 800 900

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300

1 3.40 6.81

10.21 13.61 17.01 20.42 23.81 27.22 30.62 34.02 40.83 47.63 54.44 61.24 68.05 74.85 81.65 88.46 95.26

102.07 108.87 115.68 122.48 129.29 136.09 142.90 149.70 156.51

30 40 50

0.01901 0.01863 0.01827 0.06461 0.06331 0.06208 0.12898 0.12640 0.12393 0.19310 0.18926 0.18557 0.25700 0.25189 0.24698 0.32066 0.31429 0.30818 0.38409 0.37648 0.36917 0.44729 0.43843 0.42992 0.51025 0.50018 0.49049 0.57299 0.56168 0.55082 0.63551 0.62298 0.61096 0.75985 0.74493 0.73059 0.88329 0.86601 0.84939 1.00585 0.98622 0.96736 1.12753 1.10561 1.08455 1.2484 1.2242 1.2009 1.3683 1.3419 1.3165 1.4875 1.4588 1.4313 1.6058 1.5750 1.5453 1.7233 1.6903 1.6586 1.8399 1.8048 1.7711 1.9558 1.9186 1.8829 2.0708 2.0316 1.9939 2.1852 2.1439 2.1041 2.2987 2.2555 2.2137 2.4114 2.3661 2.3226 2.5235 2.4762 2.4307 2.6348 2.5856 2.5383 2.7452 2.6942 2.6450


Temperature (~

60 70 80 90 100 110 120

0.01792 0.01758 0.01725 0.01694 0.01664 0.01635 0.01606 0.06089 0.05989 0.05863 0.05756 0.05654 0.05555 0.05456 0.12155 0.11926 0.11706 0.11494 0.11289 0.11091 0.10901 0.18201 0.17859 0.17529 0.17212 0.16906 0.16610 0.16325 0.24225 0.23771 0.23333 0.22912 0.22505 0.22112 0.21733 0.30230 0.29663 0.29118 0.28593 0.28085 0.27596 0.27123 0.36212 0.35535 0.34883 0.34254 0.33647 0.33062 0.32496 0.42175 0.41387 0.40628 0.39897 0.39191 0.38511 0.37853 0.48117 0.47220 0.46355 0.45522 0.44719 0.43943 0.43194 0.54037 0.53032 0.52063 0.51128 0.50228 0.49357 0.48517 0.59939 0.58824 0.57752 0.56717 0.55719 0.54755 0.53825 0.71679 0.70351 0.69072 0.67838 0.66648 0.65500 0.6439 0.83341 0.81801 0.80318 0.78889 0.77509 0.76177 0.7489 0.94923 0.93174 0.91491 0.89866 0.88301 0.86786 0.85326 1.06425 1.04474 1.02590 1.00775 0.99024 0.97331 0.95698 1.1785 1.1570 1.1362 1.1161 1.0968 1.0781 1.0601 1.2920 1.2684 1.2458 1.2238 1.2027 1.1823 1.1625 1.4048 1.3792 1.3546 1.3309 1.3079 1.2858 1.2644 1.5168 1.4893 1.4628 1.4372 1.4125 1.3887 1.3657 1.6281 1.5986 1.5703 1.5429 1.5165 1.4910 1.4663 1.7385 1.7072 1.6770 1.6479 1.6198 1.5926 1.5663 1.8484 1.8152 1.7832 1.7523 1.7225 1.6937 1.6658 1.9575 1.9224 1.8887 1.8560 1.8245 1.7941 1.7647 2.0659 2.0291 1.9934 1.9591 1.9260 1.8940 1.8630 2.1735 2.1349 2.0976 2.0616 2.0269 1.9933 1.9607 2.2807 2.2401 2.2011 2.1635 2.1270 2.0919 2.0579 2.3870 2.3448 2.3040 2.2646 2.2267 2.1900 2.1545 2.4926 2.4486 2.4063 2.3653 2.3258 2.2875 2.2506 2.5977 2.5520 2.5080 2.4653 2.4242 2.3845 2.3461



Table 13 (Continued)

Depth

ft In

5356.2 5580.6 5805.4 6029.8 6254.5 6478.9 6703.6 6928.0 7152.8 7377.2 7601.9 7826.3 8050.7 8275.4 8499.8 8724.5 8948.9 9173.7 9398.1 9622.8 9847.2 10071.9 10296.3 10520.7 10745.5 10969.9 11194.6

Pressure

1632.8 1701.2 1769.8 1838.2 1906.7 1975.1 2043.6 2112.0 2180.5 2248.9 2317.4 2385.8 2454.2 2522.7 2591.2 2659.7 2728.1 2796.6 2865.0 2933.5 3001.9 3070.4 3138.8 3207.2 3275.7 3344.1 3412.7

psia

2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400

ata

163.31 170.11 176.92 183.72 190.53 197.33 204.14 210.94 217.75 224.55 231.36

30 40 50

2.8550 2.8021 2.7511 2.9642 2.9093 3.8566 3.0725 3.0158 2.9613 3.1802 3.1218 3.0655 3.2872 3.2269 3.1689 3.3934 3.3315 3.2717 3.4990 3.4353 3.3740 3.6040 3.5386 3.4755 3.7082 3.6412 3.5765 3.8120 3.7432 3.6768 3.9150 3.8445 3.7767 4.0174 3.9453 3.8758 4.1192 4.0453 3.9745 4.2203 4.1450 4.0724 4.3208 4.2439 4.1698 4.4206 4.3423 4.2667 4.5199 4.4400 4.3628 4.6187 4.5372 4.4586 4.7167 4.6339 4.5540 4.8148 4.7300 4.6486 4.9113 4.8255 4.7426 5.0078 4.9204 4.8361 5.1037 5.0148 4.9293 5.1989 5.1089 5.0219 5.2937 5.2072 5.1138 5.3878 5.2951 5.2054 5.4817 5.3873 5.2964

3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000

238.16 244.96 251.77 258.57 265.38 272.18 278.99 285.79 292.60 299.40 306.21 313.01 319.81 326.62 333.42 340.23


Temperature (~

60 70 80 90 100 110 120

2.7019 2.6547 2.6090 2.5648 2.5221 2.4810 2.4410 2.8057 2.7567 2.7094 2.6636 2.6194 2.5768 2.5355 2.9088 2.8581 2.8092 2.7620 2.7163 2.6721 2.6295 3.0113 2.9589 2.9084 2.8596 2.8125 2.7668 2.7228 3.1131 3.0591 3.0070 2.9567 2.9082 2.8612 2.8157 3.2143 3.1587 3.1052 3.0534 3.0033 2.9550 2.9080 3.3148 3.2577 3.2026 3.1493 3.0979 3.0481 2.9999 3.4148 3.3562 3.2995 3.2448 3.1919 3.1408 3.0913 3.5141 3.4540 3.3959 3.3398 3.2854 3.2330 3.1821 3.6130 3.5513 3.4918 3.4342 3.3785 3.3247 3.2725 3.7111 3.6479 3.5871 3.5280 3.4710 3.4159 3.3623 3.8089 3.7441 3.6818 3.6213 3.5630 3.5065 3.4519 3.9058 3.8398 3.7760 3.7143 3.6544 3.5967 3.5408 4.0023 3.9348 3.8697 3.8066 3.7455 3.6864 3.6292 4.0983 4.0293 3.9628 3.8983 3.8360 3.7757 3.7172 4.1937 4.1233 4.0554 3.9895 3.9259 3.8644 3.8048 4.2886 4.2169 4.1476 4.0804 4.0155 3.9528 3.8918 4.3828 4.3098 4.2391 4.1706 4.1047 4.0405 3.9785 4.4768 4.4021 4.3303 4.2605 4.1931 4.1279 4.0548 4.5700 4.4941 4.4208 4.3499 4.2813 4.2149 4.1506 4.6628 4.5856 4.5110 4.4388 4.3690 4.3014 4.2357 4.7549 4.6764 4.6006 4.5273 4.4562 4.3874 4.3206 4.8467 4.7669 4.6898 4.6152 4.5429 4.4730 4.4053 4.9380 4.8568 4.7785 4.7027 4.6292 4.5581 4.4893 5.0287 4.9463 4.8667 4.7896 4.7150 4.6427 4.5729 5.1190 5.0352 4.9544 4.8761 4.8005 4.7272 4.6560 5.2087 5.1239 5.0418 4.9624 4.8854 4.8110 4.7389


Table 14 Example of Mixing Gases by Partial Pressure

Refilling a partially-filled tank with the same mixture. Tank contains a mixture of 10% oxygen and 90% helium at 900 psia. It is to be refilled with the same mixture to 3000 psia using pure oxygen and helium. Assume a constant temperature of 80~ Determine the tank pressure after addition of new oxygen.

Terms Example

t = mixing temperature

Pl = initial mixture pressure, psia

P2 = final mixture pressure, psia

Pml = initial mixture density

Pm2 = final mixture density PPol = oxygen partial press, initial mixture

PP02 = oxygen partial press, final mixture Pol = oxygen density, initial mixture

P02 = oxygen density, final mixture al = decimal fraction of oxygen in initial mixture

a2 = decimal fraction of oxygen in final mixture

Procedure---Real-Gas Method

80 ~

900 3,000

0.10

0.10

(1) Find Pml in Table 14 at 900 psia, 80~

(2) Find Pm2 in Table 14 at 3000 psia, 80~

1.0259 3.2026

32aPm I 32(0.10)(1.0259) _ 3.2829 (3) Find Po~, Pol - 28a + 4 28(0.10) + 4 6.8

= 0.48278

(4) Find PPol in Table 12 at Pol = 0.48278, t = 80~ (interpolate) 87.05 psia

32aPm2 32(0.10)(3.2026) 10.32 m

(5) Find Po2, Po2 = 28a + 4 28(0.10) + 4 - 6.8 = 1.5181

(6) Find PPo2 in 12 at P02 = 1.5181, t = 80~ (interpolate) (7) Find change in oxygen partial pressure,

Apo = ppo 2 - ppol (8) Find pressure after adding new oxygen, Pml + Apo

Change to gauge pressure (9) Find pressure after adding new helium

Change to gauge pressure

271.6 psia

271.6 - 87.05 = 184.6 psia

900 + 184.6 = 1084.6 psia

1 , 0 8 4 . 6 - 14.7 = 1,069.8 psig

3,000 psia

3,000 - 14.7 = 2985.3 psig

Procedure--Perfect-Gas Method

(1) App o = (P2 - pl)a = (Appo = change in oxygen partial pressure)

(2) Pl + Appo =

Change to gauge pressure

( 3 , 0 0 0 - 900)(0.10) = 210 psia

900 + 210 = 1110 psia = pressure after adding new oxygen

1 , 1 1 0 - 14.7 = 1095.3 psig



Carbon Dioxide Absorption in Diving Operations

Rebreathing of expired air or breathing gas is common in diving operations, including decompression chambers, underwater habitats, personnel transfer chambers, closed-circuit and semi-closed-circuit breathing apparatus, and deep- sea diving rigs In each application, it is necessary to eliminate virtually all of the exhaled carbon dioxide before rebreathing the gas to avoid toxic effects. This is accomplished with a carbon dioxide removal system (scrubber) consisting of a canister filled with a chemical absorbent through which the exhaled gas is passed. The design of the scrubber is such that all of the carbon dioxide that enters is removed, and the gas leaving is essentially free of carbon dioxide. It has been determined that about half of the theoretical absorption capacity of common chemical absorbents can be use in diving applications. However, at cold temperatures the use can drop to as low as 20%.

The amount of chemical absorbent required is determined by the quantity of carbon dioxide to be absorbed and the absorbent efficiency. The weight of carbon dioxide is determined by the number of divers, carbon dioxide production rate per diver, and the operating time. Absorbent efficiency depends on canister design and operating conditions. Highly efficient absorbers require good distribution of air flow through the absorbent bed, humidity above 70%, and no bed moisture condensation.

The choice of an absorbent is influenced by the conditions under which it is used. Baralyme is frequently used in underwater breathing equipment because it is not caustic if wetted. However, lithium hydroxide or Sodasorb can be used for hyperbaric chambers where flooding is an unlikely problem. The amount of carbon dioxide generated by a diver depends upon oxygen consumption and respiratory quotient (RQ), which is the ratio of carbon dioxide produced to oxygen consumed. At moderate depths the maximum rate of oxygen consumption is about 3 slm, but at very great depths this decreases to 2 slm because of increased breathing resistance. The respiratory quotient varies from 0.7 to 1.1, and the average value is 0.9. The maximum rate of carbon dioxide production for 3 slm oxygen consumption and 0.9 respiratory quotient is 2.7 slm, which is equivalent to 0.71 lb/hr, and this value is recommended for breathing rig scrubber design.

Measurements with closed-circuit and mixed-gas scuba have indicated that oxygen consumption rates over extended periods are below 1.5 slm, which is about half the maximum rate of 3.0 slm. Thus, the use of a carbon dioxide production rate of 0.71 lb/hr/diver provides a safety margin under usual working conditions. Table 15 summarizes the characteristics of three commercial carbon dioxide absorbents, Baralyme, lithium hydroxide, and Sodasorb. Baralyme is used in underwater breathing rigs and also for hyperbaric chambers. Sodasorb, a lower cost material, is usually not used in underwater breathing rigs because it


Table 15 Characteristics of Three Carbon Dioxide Absorbents [27].

Absorbent

Characteristic Baralyme Lithium Hydroxide Sodasorb

Absorbent density, lb/ft 3 Theoretical C O 2 absorption, lb CO2/lb Theoretical water generated, lb/lb CO2 Theoretical heat of absorption, Btu/lb CO2 Useful CO2 absorption, lb CO2]lb

(based on 50% efficiency) Absorbent weight, lb/diver hr (0.71 lb CO 2) Absorbent volume, ft3/diver hour Relative cost

65.4 28.0 55.4 0.39 0.92 0.49 0.41 0.41 0.41 6702 8752 6701

0.195 0.46 0.245 3.65 1.55 2.90

0.0558 0.0552 0.0533 2 8 1

I Based on generating gaseous H20. 2Based on calcium hydroxide reaction only.

forms a highly caustic solution with water. However, it has been used in scrubbers for hyperbaric chambers and in the US Navy's closed circuit underwater breathing apparatus. Lithium hydroxide is the lightest of the absorbents, but it requires the same canister volume as the others, is much more expensive, and forms a caustic solution with water.

Temperature Considerat ions

The rate at which carbon dioxide is absorbed in chemical absorbents is influenced by temperature, and is considerably lower at 40~ than at 70~ In some scrubbers that have been designed for adequate performance at 70~ the absorbing capacity at 40~ may be as little as 1,4 that at 70~ This effect is strongly dependent upon the canister design and the rate of carbon dioxide absorption. It is most evident in absorbers working at peak flow rates and least evident in oversized scrubbers and those used intermittently. It appears highly desirable to provide external insulation and heating of CO2 scrubbers for use in cold environments as a means of minimizing size and assuring that the design absorbent capacity can be obtained. This is also advisable as a means of avoiding moisture condensation. Another possible alternative for cold water environments is to design for about three times the absorbent capacity needed at 70~

One-Dimensional Compressible Flow in Pipes

In many diving applications, the diver breathing gas is transported through pipes and flexible hoses. The analysis of gas flows through these pipes common-


ly uses one dimensional compressible flow in pipes. The one-dimensional equations for compressible flow in pipes is obtained using the equation of state, energy, momentum, and continuity equations [5]. The control volume shown in Fig- ure 31 is used and the linear momentum equation may be expressed as

-Adp - % Pdx = GAdV (99)

where Xo = wall shear stress P = wetted perimeter A = pipe cross-sectional area V = average velocity G = mass flow rate per unit area

For constant mass flow rate per unit area (G) and wall shear stress (Xo), the momentum equation can be integrated over the pipe length (L) and the result is

(Pl - P2)A - Xo P L = G A (V 2 - V1) (100)

where subscripts 1 and 2 indicate the pipe inlet and outlet, respectively. The wall shear stress, %, may be expressed using the Darcy friction coefficient (f) as

( ~ ~ (101)

Combining these two equations and the continuity equation yields the differential form of the momentum equation

dp + pVdV + f 13rV 2 dx = 0 (102) 2 D

where D = inside pipe diameter The Mach number (M) and speed of sound (c) for a perfect gas are

M = V V V c 4kRT ~/k(p/p) c - 4kRT for a perfect gas (103)

where k = ratio of specific heats R = gas constant T = absolute temperature

Substituting these definitions into the momentum equation yields

dp kM2 dx kM2 dV ~ - ~ - ~ f ~ + �9 = 0 p 2 D V

(104)


I V P

T P

pipe wall f , ~ I( '- r o . = i r ,

"~-- % I _ r - dx

I II II I V+dV

" - 1

p+dp

T+dT

p+dp I

Figure 31. Schematic of the control volume for pipe flow.

The friction factor (f) is determined from the Moody diagram (Figure 32) and can be used for subsonic flow 0 < M < 1. The limiting length for continuous flow occurs when M equals 1 for adiabatic flow and when M equals 1/x/-k for isothermal flow.

Isothermal Gas Flow with Friction

The Reynolds number (Re) for gas flow in a pipe is

Re VDp = GD (105) ~t B

where B = dynamic viscosity of the gas. Using the equation of state, continuity, and Mach number for isothermal flow in the momentum equation yields

1 2pdp 2 dp dx - - + f - - =0 (106)

kM~ p~ p D

Integrating over the length of the pipe, L = ( x 2 - x 1), yields

- P ' pO = kM~ 2 In--P2 + f (106a)

o r

p2 _p~

pl 21n P__L + f

PlPl P2 (107)

When the pipe or tube is long, then the term (fL/D) is much greater than the term [2 In (Pl/P2)] where the subscript (1) indicates the pipe inlet and (2) is the pipe outlet. Therefore, the Equation 107 reduces to


0.06

0 O7

0.06

0.05

OO4

..q 0.03

._o

._o

0.O2

0.0!

I0 10 4

r

I 2 3 4 5 7 9 2 3 4 5 7 9 2 3 4 5 7 9

10 s

Reynolds Number

2 3 4 5 7 9 10 6

0.0.5

0.04 i !0.03

I0 7

).02

_ _ 0.015

0.01

0.008 ~ 0.006 ~

0.004 ~ u

._~ 0.002 ~

0.11111 0.0(~ 0.0006 0.0004

0.0002

0.0001

0.00005

Figure 32. Moody diagram, friction factor as a function of Reynolds number and relative roughness [16].

P ~ - P ] : k M 2 f L P ~ - P 2 2 = ~ G 2 1 L I f p2 ~- or p2 PlPl

(108)

There is a maximum length for which the particular isothermal flow will proceed continuously, and this is the length of pipe (x) at which the Mach number (M) is equal to 1/n/k (M = 1/~/k). If the pipe exceeds this limiting length, either a shock occurs or a back pressure adjustment modifies the inlet pressure flow conditions.

Adiabat ic Gas F low with Frict ion

An approximate solution for adiabatic flow (no heat transfer or the pipes are insulated) with friction may be obtained if the pressure density relationship (pp-k = const) is assumed. The momentum equation yields

1/k k p~

k + l Pl

O2pl ,k / )O2p ,k + In P2 + ~ ~ = 0 (109) (p(2k l) ,k_p~k+l)/k) p~ k pl 2 2D

The friction factor (f) varies along the pipe, but the incompressible flow friction factors can be used. The limiting pipe length for continuous adiabatic flow is reached at the distance (x) where the Mach number is equal to one (M = 1).


Heat Transfer

Heat transfer is the process by which heat energy is transported from one region to another as a result of a temperature difference between the two regions. All heat transfer processes involve the transfer and conversion of energy, and they obey the first and second laws of thermodynamics. The rate of heat transfer for a specific temperature difference is used to estimate the cost, feasibility, and size of different environmental control equipment (e.g., heaters, air conditioners, dehumidifiers, scrubbers, and humidifiers). Applications of heat transfer laws are needed for diving applications such as habitats, breathing gas supply systems, submersibles, diver lock-out chambers, diving bells, submarines, and diver protective clothing.

Modes of Heat Flow

Conduction is the heat transfer process by which heat flows from a region of higher temperature to a region of lower temperature as a result of direct physical contact. The resulting conditions are temperature equilibrium and continuous heat flow from the hot region to the cold region.

Convection is the heat transfer process where heat is transported by the combined action of heat conduction, energy storage, and mixing motion. The process includes heat flow by conduction from a surface at a higher temperature to the surrounding fluid. The energy increases the temperature and internal energy of fluid particles, and these fluid particles move to a region of lower temperature where they mix and transfer energy to other fluid particles. The types of convective heat transfer are called free and forced convection. In free convection, mixing is the result of the density difference caused by temperature gradients. In forced convection, mixing is induced by external equipment (e.g., pump or blower).

Radiation heat transfer is the process by which heat flows from a high temperature body to a low temperature body when these bodies are separated in space, even when a vacuum exists between them. All bodies emit radiant heat continuously, and the intensity of emission depends on surface temperature and the nature of the surface. Radiation heat transfer becomes more important as the body temperature increases.

The different types of heat transfer processes include steady, unsteady and periodic or quasi-steady state heat transfer. Steady-state means the rate of heat flow does not vary with time. Unsteady implies the rate of heat flow varies with time, and it is also known as transient heat flow. Periodic or quasi-steady state is the cyclic variation of heat flow.


Basic Laws of Heat Transfer

Conduction. The basic Fourier law of heat conduction is

Qk = - kA dT / dx (110)

where Qk = rate of conduction heat transfer (Btu~r) k = thermal conductivity (Btu/(hr ft ~ A = cross-sectional area measured perpendicular to direction of heat flow

dT/dx = temperature gradient

For steady state, the heat transfer due to conduction is

Ak AT Qk = - -~ (Thot - Tcold )= (L/Ak) (111)

The term L/Ak is called the thermal resistance (R k) and Ak/L is thermal conductance (Kk), where L is the distance over which the heat transfer occurs (e.g., wall thickness). Ranges of values of the thermal conductivity (k) are tabulated in Table 16 and illustrated as a function of temperature in Figure 33.

Steady One-Dimensional Heat Conduction. In many diving applications, the heat transfer due to conduction can be evaluated using relatively simple equations for the geometrical configurations such as a plane wall, hollow cylinder, and a spherical shell. These expressions are

Ak Plane wall: Qk = --7-- (TH - Tc) (112)

L

Table 16 Approximate Values for Thermal Conductivity [29]

Material Btu/hr ft ~ Watt/m ~

Gases at atmospheric pressure 0.004-0.10 0.0069--0.17 Insulating materials 0.02-0.12 0.034-0.21 Nonmetallic liquids 0.05-0.40 0.086-0.69 Nonmetallic solids (brick, stone, cement) 0.02-1.5 0.034-2.6 Liquid metals 5.0-45 8.6-76 Alloys 8.0-70 14-120 Pure metals 30-240 52-4 10


Thermal Conductivity BTU/hr ft ~

6O

i Copper

i I Aluminum -I

Zinc

Low.carbon Steel

-Z2 Lead

" ' / , ~ - R r �9 ~tCP[

Silico~ Cz=rbide

i,ll Magnesite Brick

" ~ Silica Brick _ .__ - - - - - - 1

0.8 ~ ~ ~ Missouri Firebrick 0.6 ,,--- ,1,,--

f 0.4 f ~ - -

Water

i ,,... 0.2 Hydroqen G a s ~ ~

0 . 1 , -

0.08 p,-;- 0 . 0 6

Oiotomoceous ~ o ~ _.,.

0 . 0 2 Me,n,~ ~ I I rTO'~ o.ol .. ~ / . . . . . . . l

0.008 - - (gaseous) 0.006 I I

200 600 ! 000 1400 1800 2200

Temperature ~

2600

Figure 33. Variation of thermal conductivity of solids, liquids, and gases with temperature [29]

Ti - T~ Q k Ti - T~ where R k In (r o / r i ) ( 1 1 3 ) Hollow Cylinder: Qk = In (r o / r i ) ' = R- - - - "~- ' = 2gk'--"-"-~

2~kL

S p h e r i c a l S h e l l : Q k = Ti - T~ Q k Ti - T~ w h e r e R k = r~ - ri r o / - - - ~ i ' = R--------~' 4 g r i r o k ( 1 1 4 )

4 ~ r i r o k


xi-% Composite walls: Qk = (115)

~ R n 1

where subscripts i, o = inside and outside subscripts H, C = hot and cold

L = the wall thickness of a plane wall, hol low cylinder, or sphere

n = number of wall layers of different materials

Convect ion. The heat transfer due to convection is

Qc = hcAAT (116)

where Qc = rate of heat transfer, Btu/hr A = heat transfer surface area

AT = temperature difference between surface temperature (T s) and fluid temperature (Too) away from surface

h c = average convection coefficient, Btu/hr ft 2 ~ Kc = average thermal conductance (K c = hcA) R c = average thermal resistance (R c = 1/hcA)

The convection coefficient (hc) is actually a difficult term to evaluate. It is a funct ion of the type of convect ion (free or forced) and is a funct ion of the Reynolds number, Nusselt number, and Prandtl number as defined in the following equations

Reynolds Number (R e) = VDp GD

= ~ (117) kt kt

Nusselt Number (Nu) = hoD

(118)

Prandtl Number (Pr) = pc p (119) k

For liquids flowing over a single cylinder, the heat transfer coefficient [ 12] may be determined from

hcD = r/0.35 + 0.56 Re5/' P r ~ (120) k L J

and for spheres in a gas the heat transfer coefficient may be determined from


hcD = 0.37 Re~ (121)

k

Radiat ion . For a perfect radiator (black body), the radiation heat transfer rate (Qr) in Btu/hr is

Qr = ~ n l T4 (122)

where ~ = Stefan-Boltzman constant whose value is 0.1714 • 10 -8 Btu/hr ft 2 OR4

A 1 = body surface area T = absolute temperature

For black body radiation between a body with a surface temperature of T1 and a surrounding temperature T 2, the heat transfer is expressed as

Qr = oA1 ( T4 - T4) (123)

and for imperfect radiators (gray bodies), the expression is

Qr = ~Alel (T4 - T 4) (124)

where e~ is called the emissivity of the gray body, which is the ratio of the emissivity of a gray body to that of a black body at the same temperature. Now considering the effect of geometry and remittances, the radiation heat transfer is

Qr = o'A,FI-2 ( T 4 - T 4 ) (125)

where F~_ 2 is the modulus that accounts for remittances and geometry.

Combined Heat Transfer. Typically, heat is transferred in a series of steps and by parallel modes. For example, one step would be conduction, second step would be convection, and then radiation. The combined heat transfer (Q) between the outside temperature (To) and the inside temperature (Ti) is expressed as

Q = T~ - - T i - - m T t ~ (126)

R 1 + R 2 + R 3 R1 + R 2 + R 3

It is sometimes convenient to combine the individual resistances into one quantity called an overall heat transfer coefficient (U) such that the product of U and the surface area (A) through which the heat is transferred is

1 UA = (127)

R 1 + R 2 + R 3


The overall heat transfer coefficient may be based on any chosen area, and therefore it must be stated to avoid confusion. The evaluation of the thermal resistances is often the most difficult part of the heat transfer problem.

Example of Underwater Heat Transfer Problem

A personnel transfer chamber (Figure 34) is spherical in shape and is made of aluminum with a 3-in wall thickness and has an inside diameter of 12 ft. Assume the average inside convection heat transfer coefficient (hi) is 2.5 Btu/hr ft2 ~ and the average outside coefficient (h o) is 125 Btu/hr ft2 ~ the thermal conductivity of the aluminum wall (kal) is 130 Btu/hr ft ~ the thermal conductivity of the insulation (k ins ) is 0.02 Btu/hr ft ~ The temperature difference between the outside and inside is assumed to be 30~ What is the effect of the thickness of insulation on the outside?

The resistances are calculated using Equation(s) 128 that follow and then substituting the results in Equation 126 to determine the heat transfer through the walls of the chamber with varying amounts of insulation thickness.

1 1 R 1 = ~ = - 8.84 x 10-4

h i A 1 2 . 5 4 1 1 ; ( 6 ) 2 -

to1 _ r i (6.25 - 6 ) = = 4.08 x 10-6 R2 = 41rirol k 4 I(6)(6.25)(130)

ro 2 _ rol to2 - 6 . 2 5 to2 - 6 . 2 5

R3 = 4~rol ro2 k - 416.25 ro2 (.02) 1.571 ro2

1 1 6.37 x 10-4 R 4 . . . .

hoA o 125 41r22 r22

AT Q= R 1 + R 2 + R 3 + R 4

30

8.84 x 10-4 + 4.08 x 10 -6 + r~ - 6.25 + 6.37 x 10 -4 1.571 ro2 ro22

(128)

The results of solving the equations shown in Equation 128 are tabulated in the spreadsheet shown in Table 17. The effect of insulation thickness on the amount of heat transferred through the chamber walls is illustrated in Figure 35. Without insulation the heat transfer is 33,164 Btu/hr and with only one inch of insulation the heat loss is reduced to 3,244 Btu/hr, which is approximately 10% of the no-insulation heat loss. Continued increase of insulation thickness does not result in similar heat loss reduction, so the 1 in. of insulation is considered a good value for the insulation thickness.


lation

F 0 2

Figure 34. Schematic of personnel transfer chamber for example problem.

Table 17 Summary of Results for Heat Transfer Calculations for

Personnel Transfer Chamber

hi (Btu/hr ft 2 F) ho (Btu/hr ft 2 F) kal (Btu/hr ft F) kins (Btu/hr ft F)

2.5 Pi 3.14156 125 rol 6.25 130 ri (ft) 6

0.02 A T (F) 30 ro2(ft) R1 R2 R3 R4 t (in) Q (Btu/hr)

6.25 6.333 6.417

6.5 6.583 6.667 6.75 7.25

0.000884 0.000884 0.000884 0.000884 0.000884 0.000884 0.000884 0.000884

4.08E-06 0 1.63E-05 0 33164 4.08E-06 0.00834 1.59E-05 1 3244 4.08E-06 0.01657 1.55E-05 2 1717 4.08E-06 0.02449 1.51E-05 3 1181 4.08E-06 0.03220 1.47E-05 4 906 4.08E-06 0.03982 1.43E-05 5 736 4.08E-06 0.04716 1.4E-05 6 624 4.08E-06 0.08781 1.21E-05 12 338

35000

30000

25000

20000 ,,..= 15000 c

m 10000

m 5000

0


. . . . . v

0 2 4 6 8 10 12

Insulation Thickness (in)

Figure 35. Example problem tabular and graphical heat transfer results.

References

1. Allmendinger, E. E., (ed.), 1990. Submersible Vehicle Systems Design, Jersey City: Society of Naval Architects and Marine Engineers.

2. Busby, R. F.; 1976. Manned Submersibles, Office of the Oceanographer of the Navy, Washington D.C.: Government Printing Office.

3. Busby Associates Inc., 1979. Remotely Operated Vehicles, Washington D.C.: Gov- ernment Printing Office.

4. Chuse, R. and Eber, S. M., 1984. Pressure Vessels: The ASME Code Simplified, sixth edition. New York: McGraw-Hill.

5. Daily, J. W. and Harleman, D. R. F., 1966. Fluid Dynamics. Reading: Addison- Wesley.

6. Friedman, N., 1994. U.S. Submarines Since 1945: An Illustrated Design History. Annapolis: Naval Institute Press.

7. Guyton, A. C, 1991. Textbook of Medical Physiology, 8th edition, Philadelphia: Saunders.

8. Hervey, J. B., 1994. Submarines, Vol. 7, Brassey's Sea Power: Naval Vessels, Weapons Systems, and Technology Series. London: Brassey's (UK).

9. Jones, J. B. and Hawkins, G. A., 1960. Engineering Thermodynamics. New York: John Wiley & Sons, Inc.


10. Kunkle, J. S., Wilson, S. D. and Cota, R. A. (ed.), 1970. Compressed Gas Hand- book. NASA SP-3045. Washington, D.C.: National Aeronautic and Space Admin- istration.

11. Marine Technology Society, 1984. Operational Guidelines for ROVs. Washington, D.C.: Marine Technology Society.

12. McAdams, W. H., 1954. Heat Transmission. New York: McGraw-Hill. 13. Miles, S. and Mackay, D. E., 1976. Underwater Medicine, fourth edition, Philadel-

phia: Lippincott Co. 14. Miller, D. 1991. Submarines of the World. New York: Orion Books. 15. Miller, J. W. and Koblick, I. G., 1995. Living and Working in the Sea, second edi-

tion, Flagstaff: Best Publishing Company. 16. Moody, L. F., 1944. "Friction Factors for Pipe Flow," ASME Trans., Vol. 66, pp

671--684. 17. National Oceanic and Atmospheric Administration (NOAA), 1991. NOAA Diving

Manual. Washington, D.C.: Government Printing Office. 18. Nuckols, M. L., Tucker, W. C. and Sarich, A. J., 1996. Life Support Systems

Design: Diving and Hyperbaric Applications. Needham Heights: Simon and Schuster Custom Publishing.

19. Randall, R. E., 1997. Elements of Ocean Engineering, Jersey City: Society of Naval Architects and Marine Engineers.

20. Schilling, C. W., 1965. The Human Machine. Annapolis: US Naval Institute. 21. Schilling, C. W., Werts, M. F, and Schandelmeier, N. R. (ed.) 1976. The Underwa-

ter Handbook: A Guide to Physiology and Performance for the Engineer. New York: Plenum Press.

22. Sharp, F., (ed.), 1994. Janes Fighting Ships. London: Butler and Tanner Limited. 23. Sweeney, J. B., 1970. A Pictorial History of Oceanographic Submersibles. New

York: Crown Publishers, Inc. 24. Terzibaschitsch, S., 1991. Submarines of the US Navy. New York: Arms and

Armour Press. 25. Tucker, W., 1980. Underwater Diving Calculations. Centreville: Cornell Maritime

Press. 26. University of New Hampshire (UNH), 1972. "The Impact of the Requirements of

the United States Scientific Diving Community on the Systems' Design, Operation and Management of Underwater Manned Platforms," Tech. Rep. No. 111, Manned Undersea Science and Technology Office, National Oceanic and Atmospheric Administration, Rockville, MD.

27. US Navy (USN), 1971. US Navy Diving Gas Manual. Washington, D.C.: Govern- ment Printing Office.

28. US Navy (USN), 1988. US Navy Diving Manual. Vol. 1, NAVSEA 0994-LP-001- 0910, Revision 2, Washington, D.C.: Government Printing Office.

29. Krieth, F., 1973. Principles of Heat Transfer, 3rd edition, New York: Intext Educa- tional Publishers.

AUTHOR INDEX

Allmendinger, E. E., 478 Anastasiou, K., 179 Ansari, K. A., 202, 210 Benjamin, T. B., 58 Berkhoff, J. C. W., 109 Bitting, K. R., 285 Blenkarn, K. A., 232 Bondzie, C., 170, 172, 188 Boo, S.Y., 350 Booij, N., 170 Boussinesq, J., 57, 58 Brown, J. R., 438 Burstein, A. W., 438 Busby, R. F., 478 Cao, P., 348 Caughey, T. K., 259 Chakrabarti, S. K., 328,338, 340,

355,369 Chamberlain, P. G., 173 Chappelear, J. E., 58, 59, 86 Chen, H. S., 179, 180, 186 Chen, W., 345 Chen, Y., 51 Clark, P. J., 363 Clement, A., 351 Coffeen, J. A., 452 Cokelet, E. D., 21 Copeland, G. J. M., 122 Cummins, W. E., 235 Dalrymple, R. A., 173, 176 Demirbilek, Z., 18, 27, 30, 32, 40 DeVries, J., 57 Dominguez, R. F., 201 Eatock-Taylor, R., 356, 360 Ebersole, B. A., 174, 176 Emmerhoff, O. J., 360, 363 Ertekin, R. C., 27, 35

Evans, D. V., 356 Ewing, J. A., 226, 227 Faltinsen, O. M., 358, 361 Felix, M. P., 200 Fenton, J. D., 5, 58, 59 Francois, R. E., 411, 412 Garbaccio, D. H., 201 Garrison, G. R., 411, 412 Gharib, M., 289 Green, A. E., 1, 2, 3, 7, 9, 19, 20 Griffin, O. E., 289, 290 Griffin, O. M., 201 Grosenbaugh, M. A., 288 Guyton, A. C., 496 Hampton, L. L., 427 Hansen, J. B., 49 Hedges, T. S., 15, 91, 92 Herbich, J. B., 248 Hermans, A. J., 232 Holthuisen, L. H., 166, 170, 172 Houston, J. R., 179 Hsu, F. H., 232 Hung, S. M., 366 Irvine, H. M., 259 Isaacson, M., 351 Isobe, M., 59, 173 Iwagaki, Y., 56, 58, 70, 79, 80, 81, 86,

92, 93, 94, 96 Jamieson, W. W., 370 Jefferys, E. R., 356 Jonsson, I. G., 77 Kantorovich, L. V., 7, 11 Keane, J. J., 438 Keller, J. B., 58 Khan, N. O., 210 Kim, C. H., 351,356, 376 Kim, M. H., 345,356, 365,366, 375

561


Kirby, J. T., 173, 175, 186 Kobayashi, N., 139 Koblick, I. G., 483 Komen, G. J., 165 Korsmeyer, F. T., 357 Korteweg, D. J., 57 Kostense, J. K., 184 Krylov, V. I., 7, 11 Laitone, E. V., 58 Larsen, 186 Laws, N., 19 LrMehautr, B., 90, 311,323 Leonard, J. W., 201 Leroy, C. C., 413 Li, B., 179, 180 Lie, V., 170 Lighthill, M. J., 58 Lillycrop, L. S., 179 Linton, C. M., 356 Liu, P. L.-F., 51, 176, 179, 180, 186 Liu, Y. H., 357 Mackay, D. E., 491 Mackensie, K. V., 413, 416, 438 Madsen, P. A., 186 Marsh, H. W., 429 Maruyama, K., 122 Massel, S. R., 173 Maxwell, W. H. C., 307 McKinney, C. M., 438 Mei, C. C., 354 Mekha, B. B., 348 Miles, S., 491 Milgram, J. H., 266 Miller, J. W., 483 Milne, A. R., 438 Morgan, M., 456, 457 Naghdi, P. M., 1, 3, 9, 19, 20 Nakamura, M., 201 Nath, J. H., 200, 201 Natvig, B. J., 376 Nelson, R. C., 95 Newman, J. N., 365 Nishimura, H., 58, 59, 122

Nossen, J., 360 Nwogu, O., 18, 186 O'Hara, G. J., 201 Ogilvie, T. F., 364 Okihiro, M., 179 Orlanski, J. E., 360 Panchang, V. G., 164, 170, 172, 173,

176, 179, 180, 188 Papazoglou, V., 265, 266, 274, 275 Patterson, G. W., 58 Patterson, R. B., 438 Pauling, J. R., 376 Peregrine, D. H., 134 Pinkster, J. A., 232 Porter, D., 173 Poulin, S., 77 Radder, A. C., 125, 173, 174 Rainey, R. C. T., 345 Ramberg, S. E., 289 Ran, Z., 348, 373, 376 Randall, R. E., 446 Rayleigh, L., 57, 58 Remery, G. F. M., 232 Resio, D. T., 172 Rosenthal, F., 201 Russell, J. S., 57 Saling, D. S., 438 Schellin, T. E., 200 Schilling, C. W., 491 Schulkin, M., 429 Schwartz, L. N., 85 Sclavounos, P. D., 360, 363 Shanks, D., 84, 86 Shields, J. J., 7, 19, 20, 21, 23, 31, 38,

47, 49 Skop, P. A., 201 Smith, C. E., 201 Smith, R., 173 Sprinks, T., 173 Staziker, D. J., 173 Svendsen, I. A., 49 Tam, W. A., 355 Thresher, R. W., 201

Index 563

Torum, A., 170 Triantafyllou, G. S., 261 Triantafyllou, M. S., 261,274, 276,

277, 288 Tsay, T.-K., 176, 179, 180 Tsuchiya, Y., 58 Tuah, H., 201 Urick, R. J., 438 Van Oortmerssen, G., 226, 232, 235 Verley, R. L. P., 342 Vogel, I. A., 164, 176 Watanabe, A., 122 Webster, W. C., 7, 18, 19, 20, 27, 30,

31, 32, 38, 40, 372, 376

Weggel, J. R., 307 Wei, G., 186 Williams, J. M., 90 Wilson, B.W., 201 Wu, G. X., 360 Wu, T. Y., 6 Xu, B., 180 Yasuda, T., 58 Young, I. R., 165 Yue, D. K. P., 261,365,366, 375 Zhang, J., 340, 348 Zhao, R., 360

SUBJECT INDEX

A

Absorption coefficient, 411

Acoustic applications, 384

communication, 385 depth sounders, 457 Doppler

measurements, 467

dredging, 385 fishing, 384 geophysical research,

384 navigation, 385 position reference

system, 452 positioning, 452, 466

scattering level, 439 seismic reflection

data, 450 short-baseline system,

455 side-scan sonar, 459 subbottom profiling,

465 underwater, 385, 386,

449 hydrographic, 385 search, 385

system active sonar, 385 passive, 386 seismic, 386

Added mass, 340, 354 Adiabatic

charging, 520 American Petroleum

Institute (API), 276

Analysis motion, 371

Analysis of multi-leg systems, 276

application, 277 guidelines, 276

Anchor, 199 deadweight, 199 drag coefficient, 219 embedment, 199 holding factor, 207 model, 207 projected area, 219

Application, 73, 78 Cnoidal theory, 73

wave length known, 74

wave period known, 75,76

effect of current, 75, 76, 77

fluid velocities, 75, 79 Approximants

Pad6, 85 Array

gain, 395 line, 400 special, 400

Atmospheric diving systems

JIM, 483,484 WASP, 483,484

B

Bathymetry, 23 constant, 25, 27, 30 uneven, 23, 32

Bathythermograph expendable, 417

Beam pattern, 399, 400 Boundary conditions, 2, 41,

103, 105 dynamic, 2 kinematic, 2, 10, 16 waves on a fixed bed,

103 waves on a permeable

bed, 107, 137 Boundary element method,

356 Boundary-value

complete elliptic, 174 conditions, 176 convergence, 180

Breathing closed circuit, 476 critical constants, 502 equipment, 474, 475 example, 510 gas, 495, 531 open circuit, 476, 511 rates, 512 Scuba, 474 semi-closed circuit,

506, 509 surface supplied, 476

C

Cable, 201,202 analysis, 202

dynamic, 210 quasi-static, 202

dynamics, 238 forces on, 216 non-elastic, 201 stretch, 207 tension, 246

564

Index 565

Cable mechanics bottom interaction, 265,

275 comparison with rules,

282 dynamic, 258,272

equations, 283 linear, 258 non-linear, 264 tension, 267,268

low tension, 266 modeling, 258 snapping cables, 266 structures, 272

numerical analysis, 272

synthetic lines, 268 maximum strain,

268 testing, 274

Carbon dioxide, 502 absorbents, 548

Carbon dioxide absorption Baralyme, 547,548 Lithium hydroxide, 547,

548 Sodasorb, 547, 548

Catenary, 203 equation, 203

Cavitation, 405 Chain, 198 Coefficient

absorption, 411 added mass, 218 cross correlation, 397 drag, 217, 218,219, 337,

338, 342 inertia, 337, 338, 342,

355 Coherence, 396 Compressible flow in pipes

adiabatic, 551 isothermal, 550

Computation, 224 environmental forces,

224 Computer program, 222

Conservation of mass, 16 momentum, 11, 16, 24

Convective acceleration, 344

D

Daltous Law, 522 Damping

coefficient, 279 estimate, 279 induced by mooring

lines, 276 reduced, 290

Density, 229 force, 229

Depth sounders, 457 Detection

threshold, 447 Det Norske Veritas (DNV),

276 Diffraction,

first order, 353,356 second order, 363,365

Diffraction equation, 377 Directivity

receiving index, 397 transmitting index, 403

Diving apparatus, 477 applications, 490, 523 atmospheric, 484 physiology, 491 systems, 482

Doppler acoustic, 467

Doppler frequency shift, 342

flow problem, 352 double body, 352

Doppler shift, 173 Drift forces, 361,362, 363 Duffing equation, 377 Dynamic positioning

acoustic position reference, 452

short baseline, 455

E

Eikonal equation, 420 Elasticity

modulus, 331 Elliptic

function, 82 integrals, 82

Energy systems batteries, 490 fuel cells, 490

Equation Bernoulli, 107 Boussinesq, 101, 134,

150, 153, 154 modified, 135 with breaking

dissipation, 136 continuity, 121 dissipation, 117

elliptic, 114 energy conservation,

106, 110 energy, 166 Euler

motion, 106 for shallow water, 18 Helmholtz, 114, 173 hyperbolic, 121, 186 Laplace, 172 mild slope, 172, 173 of motion, 2, 3, 172 parabolic, 175 partial differential, 174

Equation of state, 497 Beattie-Bridgeman, 501 Van der Waals, 499, 501 Virial form, 502

Equations frame moving with

wave, 60 motion, 60

Examples, 43, 47 Excursion, 243

surge, 243 sway, 244 yaw, 245


Explosives bubble pulse, 406, 407 charge, 405

FASTTABS, 181, 182 Flow

adiabatic, 551 compressible, 548 isothermal, 550 rate, 508 steady, 25, 30 unsteady, 23, 27, 32

Fluid incompressible, 172 inviscid, 172

Fluid sheets, 3 model, 7

Force density, 229 Forces

current, 217, 227 drag, 339, 342 drift, 232 environmental, 224 external, 217 inertia, 339 maximum horizontal,

369 on-cable, 216 restoring, 200 surge, 371 wind, 225

Fourier coefficient, 343

G

Gas constant, 522, 523 helium, 502 hydrogen, 502 law, 496, 499 mixing, 532 mixtures, 521

nitrogen, 502 Hollow cylinder link, 199 oxygen, 502, 505 Homer spit, 174

percent, 495 Hydrostatic restoring perfect, 517, 518 coefficient, 372 requirements, 512 Hyperbaric chamber, 483,

Gas laws 489 Baettie-Bridgeman, 501 ventilation, 513 Boyles Law, 498 Charles Law, 499 Equation of State, 497 I General Gas Law, 499 Law of Corresponding Index

States, 502 receiving directivity, Van der Waals, 499 397, 398 Virial, 502 Integration

Generalized GN theory, 15, spacial, 42 19 time, 42

Grazing Intensity, 338, 393 angle, 438

Green function, 356, 365 Green theorem, 356 K Grid network, 181

H

Habitat desired design, 485 underwater, 483,487-

489, 522 WOTEL, 489

Haringvliet estuary, 170 Haskind relation, 348 Heat transfer, 522, 553

combined, 556 convection, 555 example, 557, 558 flow, 552 laws, 553

combined, 556 conduction, 553 convection, 555 radiation, 556

modes of heat flow, 552 High-order boundary

element method, 357

Kahului, 182-184

Law closed system, 514

geometrically similar motions, 298 structures, 298 system, 299

modeling, 297 differential, 299 dimensional, 299

Newtonian, 299 open system, 514

Snell, 427 spreading, 409, 410

cylindrical, 409 spherical, 412

thermodynamics, 514 Layers

constant gradient, 425

Index 567

Lift force, 349 Limiting

ray, 424 Line tension, 237

time histories, 237, 239 Linearized drag, 342 Load

drag, 264, 266 non-linear, 266

transverse, 264

M

Mathematical formulation equations for shallow

water, 8 Mid-slope, 101, 102, 112,

115, 116, 122 energy dissipation, 101 non-linear, 141

Mixing breathing gases partial pressure, 532

Model Cauchy, 302, 305 distorted, 307 effects

density, 311 environment, 315 Froude, 302, 303, 324,

325 mixed layer, 428 Pierson-Moskowitz, 318 Reynolds, 302, 304, 324 scale, 297

secondary effects, 310 spectral 318 transmission loss, 426

Modeling current, 320, 331 facilities, 312, 313, 315 measurement, 321,322 piles, 326 structures

coastal, 312, 323

floating, 328 offshore, 307

waves, 316 multi-directional, 319 wind, 319

Models Australian Defence

Force Academy, 172

British Meteorological Office, 172

complete elliptic, 179 CGWAVE, 180, 182 diffraction, 164 elliptic, 174

coastal wave, 181 energy balance, 164 EVP, 178 HARBD, 179 HISWA, 166, 168, 170,

172 hydraulic, 186 mild-slope equation, 172 parabolic, 174-176 parameter, 165 PHAROS, 184 RCPWAVE, 176, 177 refraction, 164 shoaling, 164 STWAVE, 172 SWAN, 172 time-dependent, 184 WAMDI, 165 WAM3G, 165

Moisture removal, 526, 529 Mole fraction, 524 Molecular weight, 521 Mooring, 195

classes, 200 dynamics line dynamic tension-

displacement, 222 line dynamic response,

240

Moored system, 197, 200, 274

chain, 257 drag, 279

coefficient, 280, 281, 291

lines, 276 multi-leg, 200, 257 single-leg, 200 synthetic rope, 257 wire, 257

Morison equation, 377 Motion, 237

surge, 243 surge equation, 243, 244 sway, 237, 239 yaw, 239, 245

Motion of moored structures

amplitude, 270, 272 cable, 257, 258

vortex-induced vibration, 258

design, 274 fast-varying, 273

wave-induced, 273 semi-submersible, 277 slowly-varying, 257,

258, 291 large amplitude, 272

storm condition, 278 time-scale expansion,

270 vortex-induced, 257,

273, 288 wave-induced, 257, 258,

291

N

Navigation acoustic, 466

Noise ambient, 430


flow, 446 hydrodynamic, 443 radiated levels, 440 self-noise levels, 443

Non-adiabatic charging, 520 oxygen consumption,

496 pressure vessel, 519 psychrometric charts,

525,527, 530 rebreathing apparatus,

505 re mote 1 y-operated

vehicles, 480 respiratory problems, 494

Number Cauchy, 302, 305 Euler, 302 Froude, 302, 303, 307,

325,326, 360 Keulegan-Carpenter,

302-326, 339, 349 Reynolds, 302, 304, 324,

339, 349 Strouhal, 302 Ursell, 302 wave, 339 Weber, 307

Numerical model, 37 Numerical solution, 220

results, 236

O

Orlanski condition, 350

P

Parabolic, 101, 102, 112, 115, 116, 122, 125, 126, 127

approximation, 102 equation model, 133

for large wave angle, 128 in non-Cartesian

coordinates, 129 non-linear, 102, 138 non-orthogonal

coordinates, 131 orthogonal curvilinear

coordinates, 129 permeable bed, 101,107,

118 shallow water, 102 slowly-varying current,

101,116 time-dependent, 101,

123, 124 wave

validity ranges, 150 weakly-nonlinear, 102,

131 Perfect gas

properties, 517 Perfect gas process

constant pressure, 518 constant volume, 518 isothermal, 518 polytropic, 519 reversible-adiabatic, 518

Perturbation, 2, 11 method, 5, 6 parameter, 7 second-order problem,

363 wave-current-body

interaction, 352 weak nonlinear

problems, 351 Physiology

circulatory, 491 gas embolism, 494 hypoxia, 494 oxygen consumption, 496 oxygen poisoning, 494 respiratory, 491

Piles, 199 dynamics, 202

Pitch moment, 346, 347 Platform

articulated, 344 spar, 375 tension leg, 358

Pressure vessel adiabatic charging, 520 isothermal charging, 521

Projector explosive, 405 sound, 403 source level, 390

Propagation arctic, 429

Psychrometric chart, 523, 526, 527

terms, 524

Q

Quadratic transfer function, 373

R

Radiation, 354 Ray

incident, 419 reflected, 419 refracted, 419

Ray solution, 419 limiting, 424 path, 424 tracing, 425

Relative humidity, 524 Remotely operated

vehicles, 480 Retardation function, 354 Reverberation level, 434 Rhine estuary, 170 Ropes, 198

synthetic fiber, 198 wire, 198

Run-up, 360

Index 569

S

Saginaw Bay, 166 San Ciprian Harbor, 170 Scale

decibel, 388 Scattering

deep scattering layer, 437

strength, 434 Scuba, 474, 481,503 Sea

bottom, 427 surface, 427

Seismic exploration, 449 reflection data, 450

Side-scan sonar beam patterns, 461 geometry, 460 horizontal offset, 464 records, 462, 463 target height, 464

Slowly-varying motions, 354

Snell law, 418 Solution

comparisons, 91-96 fifth-order, 70 Iwagaki approximation,

70, 90 Newton, 89 series, 63, 67 third-order, 67

Solution scheme, 35 Sonar equations, 389

active, 385,390 noise-limited, 390 passive, 391 reverberation limited,

391 side-scan, 459 transient form, 392

Sound absorption, 410

explosives, 405,407 measuring, 416 projector, 403 propagation, 408 speed, 413,415,416 underwater, 387,403

Sound speed profile deep isothermal layer,

415,416 main thermocline, 415,

416 seasonal thermocline,

415,416 Spar pitch, 374 Spectra, 348 Spreading laws

cylindrical, 409 spherical, 409

Subbottom profiling, 465 Submarine, 476, 480, 481

Alvin, 478 human-operated, 481,

482 human powered, 481 military, 476 research, 478 tourist, 481

Submersible, 478,490 batteries, 490

T

Target strength, 439, 442

Temperature dry bulb, 524 wet bulb, 524

Testing coastal, 312 experiments, 274, 275 facilities

MARINTEK, 315 National Research

Council of Canada, 313

Texas A&M University, 315

Waterways Experiment Station, 313

model, 296 benefits, 296

Texel inlet, 170 Theoretical basis, 3

formulation, 3 Theory

Cnoidal, 59, 83 numerical, 86

hyperbolic, 58 Thermal conductivity, 554 Thermodynamics of diving

systems, first law, 514-516

Transducer arrays, 394 electrostrictive, 395 magnetostrictive, 395 piezoelectric, 395 response, 398

Transform Shanks, 84

Transmission loss

from ray diagrams, 426

sea surface and bottom, 427

model deep sound channel,

428 mixed layer sound

channel, 428 shallow water, 429

Transponders, 466

U

Underwater habitats Aquarius, 484 Chernomor, 485 Helgoland, 485


Hydrolab, 485, 486 La Chalupa, 485 Tektite, 485

Ursell parameter, 150, 153

V

Velocity particle, 107 potential, 104, 108, 110,

111,359 relative, 340 under crest, 341 under trough, 341

Ventilation large chambers, 513

Vertex ray, 423

Vessel, 240 analysis, 233,240 equation of motion, 240

dynamic analysis, 233

hydrodynamic mass, 240 Volterra series, 373

W

Wave amplitude, 339 Cnoidal, 150, 152

random, 123 condition, 166 damping, 354 deep-water, 11 diffraction, 173, 185 drift damping, 361 equation

vertical integration, 173

load on large structures, 349 on slender bodies, 337

non-linear, 47 period, 339

reflection, 173, 185 coefficient, 182, 188

refraction, 173, 185 shallow-water, 3 shoaling, 47, 185 Stokes, 150, 152 theories, 57

Airy, 57 Cnoidal, 57-59 long waves, 58, 59 Stokes, 59 stream function, 59

Wave-maker, 40 Wave spectrum,

230, 232 Pierson-Moskowitz,

232 Wave-structure interaction

reflection of waves, 2 Wet-bulb temperature, 529

Documents

Developments in Offshore Engineering; Wave Phenomena and Offshore Topics