Table Of ContentECAFERP
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 off-
shore 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 engi-
neering, 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
xi
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
REHSILBUP NOTE
stnempoleveD ni 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 OT THIS VOLUME
.rD Khyruddin .A Ansari, Dept. of Mechanical Engineering, School of Engineering, Gon-
zaga University, Spokane, WA 99258
.rD .S .K Chakrabarti, 191 .E Weller Drive, Plainfield, IL 60544
.rD 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
.rD Masahiko Isobe, Department of Civil Engineering, University of Tokyo, Hongo 7-3-1,
Bunkyo-ku, Tokyo 113, Japan
.rD .M .H Kim, Associate Professor of Ocean Engineering, Civil Engineering Depart-
ment, Texas A&M University, College Station, TX 77843-3136
.rD Vijay .G Panchang, Department of Civil Engineering, University of Maine, Orono,
ME 04469-5711
.rD 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 EHT 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 engi-
neering 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 engi-
neer 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 sev-
eral 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.
Xlll ~176
CHAPTER 1
THE GREEN-NAGHDI THEORY
FO FLUID SHEETS ROF
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, 51
Equations for Shallow Water, 81
Restricted Theory and Constitutive Relations, 20
SUBSET THEORIES, 32
Level I Theory: Unsteady Flow and Uneven Bathymetry, 32
Level I Theory: Steady Flow and Constant Bathymetry, 52
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, 53
Integration, 53
Numerical Model, 73
2 Offshore gnireenignE
Wave-Maker, 40
Boundary Conditions, 14
laitapS and Time Integration, 42
EXAMPLES, 34
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, 15
Introduction
This chapter presents a mathematical theory for simulating wave transforma-
tion in shallow waters. The theory is intended for coastal engineering applica-
tions 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 characteris-
tics of wave trains in the shallow water regime. The GN approach, which is funda-
mentally different from the perturbation method based on developments in classi-
cal 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 verti-
cal 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 perturba-
tion approach in that the free surface and bottom boundary conditions are met
exactly, whereas the field equation is implicitly approximated. The result is a theo-
ry 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 pro-
vide 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 topog-
raphy 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 esti-
mation of surface wave parameters for coastal design studies. The theory is par-
ticularly suited for violent collision of waves with natural and man-made struc-
tures, 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 bound-
ary 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 deriva-
tion 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 develop-
ment of fluid sheet theory in an Eulerian frame 31 made this theory much easi-
er to apply to fluid flow problems. The reader is referred to the pair of papers by
Green and Naghdi 14, 51 for a definitive and highly mathematical exposition
4 Offshore Engineering
of the theory. In recent years, we have made significant advances in the adapta-
tion 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 Theory
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 equa-
tions 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 dis-
cuss 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 simpli-
fication 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
The Green-Naghdi Theory of Fluid Sheets for Shallow-Water Waves 5
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 approxima-
tion 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 alge-
braic 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 approxi-
mation 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 spec-
trum, 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 simplifica-
tion 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 decom-
posed 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 quanti-
ties 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 conver-
gence 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
