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OCR for page 397
The Dispersion of Larg - Amplitude
Gravity Waves in Deep Water
W. Webster (University of California, Berkeley, USA)
D.-Y. Kim (Wageningen, The Netherlands)
Abstract: The Green-Naghdi (GN) theory of
fluid sheets is used to analyze large amplitude,
deep water waves in the time domain. Level III
theory is used to simulate a train of steep regular
waves and a random wave record corresponding
to steep seas measured during hurricane Cam-
ille. An analysis of the simulated random wave
record shows that the linear dispersion assumed
for referring a random wave train from one point
in space to another does not result in conservative
estimates of two important quantities used in de-
sign: the crest elevation and particle velocity un-
der the crest.
1. Introduction
The focus of this paper is on the behavior of
large amplitude water waves in deep water, with
a particular emphasis on the implications of this
behavior for the engineering analysis of the mo-
tions of and loads acting on ships and offshore
platforms. For the problems of greatest concern
here the waves are of a length scale comparable
to the horizontal dimensions of the ship or plat-
form. For such waves it is common (and reason-
able) to neglect both surface tension and viscosity,
and we shall do so here. During the last few
decades significant advances have taken place in
the understanding of these water waves in both
deep and shallow water, and there is a very large
literature on this research. We will not attempt
an exhaustive review this research here since
our interest is a fairly narrow one.
Much of the research into deep water
waves and their applications to design can be di-
vided into two principal and almost mutually ex-
clusive thrusts: the description of the kinematics
and the stability of regular, two-dimensional
waves of large amplitude (up to and including
breaking), and the description and measurement
of random wave systems using analyses which
rely on superposition and linearity.
This split has its counterpart in the design
ounce. A typical problem in design is to deter-
mine the adequacy of a structure under consid-
eration to withstand the forces imposed by the
largest waves it will encounter in its lifetime (the
so-called survival" problem). This problem can
be thought of as composed of two parts: a descrip-
tion of the wave situations (in the absence of the
structure) which would lead to the survival con-
ditions, and estimation of loads and motions
which result from the interaction of these waves
with the structure. The focus here is on the first
part, the description of the wave system,
although it is recognized that the second part is
probably the more difficult of the two.
This problem is at once nonlinear and
random, since the waves which lead to the sur-
vival conditions are likely to be breaking, or
nearly breaking, local storm waves. This design
problem causes a dilemma for the engineer,
since he often must choose between an analysis of
his structure based on the impingement of a sin-
gle, regular, large-amplitude wave (the design
wave approach) or an analysis based on the
impingement of a random wave system of super-
posed linear wave components (the spectral ap-
proach).
We note that the use of the spectral ap-
proach for the estimation of the motions in more
moderate seas where linear superposition is
probably not a bad assumption (the so-called
"operational" problem) has become almost uni-
versally accepted, since the use of linear, ran-
dom-wave analysis does have several advantages.
Its use brings with it the powerful theoretical
bases of time series analysis and stochastic pro-
cess theory. These provide a rational framework
for the estimation of the reliability and operability
of the structure. Further, since the wave compo-
nents in the spectral decomposition are linear,
one can treat with almost equal ease both the fre
Department of Naval Architecture & Offshore Engineering, University of California, Berkeley, CA 94707
Currently: Department of Naval Architecture, Seoul National University, Kwanak-Gu, Seoul, Korea
397
OCR for page 398
quency domain and time domain problems.
Both approaches to the more severe sur-
vival problem have advantages and disadvan-
tages. The loads used in the design wave ap-
proach reflect the sharpening of the crests and
the flattening of the troughs due to nonlinear ef-
fects, and these effects, in particular, often have
significant consequences on the wave loads on
offshore platforms and on the shipping of green
water on the deck of surface ships. The use of a
design wave does yield a deterministic load sys-
tem which is relatively easy to incorporate into a
design analysis. For this purpose, it is common
to use fifth or higher-order Stokes wave approxi-
mations or, more recently, the results of stream
function expansions (Dean, 1974, Chaplin, 1980~.
The methods used to determine these nonlinear
waves make use of approaches which can neither
be extended to three dimensions nor be general-
ized to arbitrary time-domain calculations in
which a representation of steep, random wave
systems can be made. Kinematic descriptions of
regular deep water waves (assuming one can ig-
nore viscosity and surface tension) are known to
great accuracy (Schwartz, 1974; Fenton, 1988~. It
is not difficult to formulate a second-order or
higher-order perturbation approximations for
non-linear waves in the time domain, but they
have been little exploited, if at all, in the design
process.
The random and three-dimensional char-
acter (short-crestedness) of a measured real
storm wave system is captured by the usual spec-
tral analysis approach. Time series analysis al-
lows identification of the spectral composition of
the wave surface elevation at the point of mea-
surement, and allows identification of some of
the directional character of the seaway if many
such points of measurement are made close by
concurrently. When the spectral representation
of the water surface is known, the prediction of
the pressures and velocities at and under the free
surface at the reference location is usually made
by associating the Fourier components of the
wave surface with linear (Airy) wave compo-
nents. This superposition is only valid if the orig-
inal wave system is of a height and character
which is consistent with linearization of the free
surface boundary condition. Such an assump-
tion becomes ever more questionable as the waves
become steeper and approach breaking. There
are a number of approaches whereby the inter-
pretation of the spectral decomposition is modi-
fied to improve the prediction of the pressure and
velocity fields corresponding to the free surface
description. We shall discuss one of these due to
Wheeler (1969) in a subsequent section of this pa-
per.
The prediction of the pressures and veloci-
ties at locations remote from the reference loca-
tion requires, in addition, an estimate of the dis-
persion of the waves. If one supposes the super-
position of Airy waves, then each component
travels at a different speed which is uniquely re-
lated to its own frequency. Thus, the phasing of
these components at the remote location is differ-
ent from that at the reference location. However,
it is known from the study of nonlinear regular
waves that steeper waves of the same length
travel faster than their less steep counterparts.
One can therefore anticipate that there will be a
nonlinear interaction between the component
waves which will affect their wave speed. For in-
stance, consider the case where one analyzes the
motions of a large ship in head seas and pre-
scribes the wave time history at one point on the
ship, say amidships. In order to perform this
calculation, it is necessary to predict the wave
environment over the whole length of the ship at
each instant in time. Since the length of typical
large ships is in the order of 400' to 1000' or more,
small differences in the estimated dispersion of
shorter waves may cause significant discrepan-
cies between the wave time history at the bow and
at the stern. Further, since the discrepancy at
the bow is of the opposite sense from that at the
stern (relative to a reference point amidships),
these discrepancies may become especially im-
portant for pitch or yaw motions which reflect the
difference in forces bow and stern.
Although linear ship motions analysis can
be considered state-of-the-art, nonlinear motions
analysis is not. In particular, much of the thrust
in recent years in nonlinear ship motions has re-
volved about the slow drift problem where second-
order forces and waves are taken into account.
These endeavors are extremely complex and the
prospect of accomplishing in the near future an
analysis correct to, say, the third order is not
bright. What is troublesome with this state of af-
fairs is the fact that the second-order wave prob-
lem predicts the same wave celerity as the linear
problem and has many of the dispersion charac-
teristics of Airy theory. The third-order solution
is the lowest order perturbation theory which pre-
dicts an increase in celerity of regular waves
with steepness similar to that observed in nature
and interactions between waves which lead to
"phase-locking".
In conclusion, it is fair to say that neither
design approach to the survival loading (design
wave or spectral decomposition) is wholly satis-
factory. It is the purpose of this paper to explore
the substantial gap which exists between these
two design approaches by presenting a different
model for the behavior of large-amplitude deep
water waves in the time domain. It is of particu
398
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tar interest to use this model to investigate the
dispersion of a random wave system from one lo-
cation to another so that some insight into the
omissions of current linear and second-order
theories can be obtained. The foundation for this
development is the Green-Naghdi theory of fluid
sheets (hereafter referred to simply as GN theory)
and, in particular, the extension of this theory to
deep water waves (Green & Naghdi, 1986 & 1987~.
This study could also have been performed using
other nonlinear formulations, but GN theory was
chosen since it is particularly efficient computa-
tionally.
Following the introduction of the GN gov-
erning equations for level III theory below, the
remainder of the paper will consist of two parts:
(a) validation of the theory using known results of
steep regular waves, and (b) use of the time-do-
main solution of these equations to simulate a
steep, random seaway.
2. GN level III theory of deep water waves
GN theory is a model for three-dimen-
sional fluid flow which, since it involves one
fewer independent space variable than three-di-
mensional space, is called a fluid sheet model.
The basis of this model is rather different from
traditional models derived from potential theory
using perturbation methods or from the special-
ized methods often introduced to compute with
high accuracy the characteristics of regular, two-
dimensional water waves. When viscosity and
surface tension are ignored and the fluid flow is
assumed to be irrotational, the field equation
(Laplaces's equation) is linear. The only nonlin-
earities are found in the boundary conditions on
the free surface. The treatment of the field equa-
tion and nonlinear boundary conditions by per-
turbation methods and GN theory are the an-
tithesis of one another.
In the perturbation method, the field equa-
tion is retained exactly and the boundary condi-
tions are approximated; in GN theory the field
equation is approximated and the full boundary
conditions are retained. However, it is not our
purpose here to give a detailed discussion of the
consequences of these different approaches. The
reader is referred to Green & Naghdi (1986, 1987)
for a precise exposition of GN approach to water
waves, and to Webster & Shields, (1990) for an
overview and commentary on the method.
Since perturbation parameters or scales
are not used in its development, the limits of ap-
plicability of GN theory are implicit and must be
determined by physical or numerical experi-
ment. For the problem of steep water waves we
choose GN level III theory, as defined in Webster
& Shields (1990~. Although this theory is com
plex, this level theory was necessary for the
treatment of even a narrow-banded spectrum.
We introduce a coordinate system Oxyz,
with the Oz axis oriented vertically up and the
Oxy plane horizontal and corresponding to the
undisturbed free surface. In the GN theory used
here, the vertical dependence (i.e., the depen-
dence on z) of the kinematics of the fluid flow is
restricted. That is, we introduce a set of func-
tions (n(Z) which will serve as a basis for the ver-
tical dependence. These functions play the same
role that "shape functions" play in finite element
analysis. We assume that the fluid velocity,
v~x,y,z;t) = (u,v,w) can be approximated with
three of these basis functions (for level III). Thus,
3
v~x,y,z;t) = I, vn~x~y;t) kn(Z)
n=1
~1
where vn = (un,vn,wn) are vector coefficients as-
sociated with the function An. Following Green
and Naghdi (1986), we select basis functions
given by
kn~z) = zonk) eaz n = 1 2 3 (2
where a is a constant, the choice of which will
discussed below. The exponential factor, \~ = eaz
was selected since it has the same form as the z
dependence found in the Airy wave solution. The
other terms in the basis can be regarded as sys-
tematic variations of the Airy wave velocity pat-
tern.
The kinematic assumption (I) is inserted
into the equations for conservation of mass, con-
servation of momentum (Euler's equations), and
the kinematic boundary condition on the free sur-
face, z = ,B(x,y;t). It is possible to satisfy all of
these equations identically except for conserva-
tion of momentum, which is satisfied only ap-
proximately. Euler's equations are multiplied by
\1' \2, \3 and integrated with respect to z. The
result is a set of three vector equations which re-
flect conservation of momentum in a weighted
average sense. These together with exact state-
meets of conservation of mass and the kinematic
boundary conditions are the evolution equations
for this model of the flow. The final evolution
equations can be expressed in rather compact
general form (equations 3.4, 3.8 & 3.11, respec-
tively, in Webster & Shields (1990~) but these equa-
tions will not be repeated here. The determina-
tion of the evolution equations in terms of deriva-
tives of the primary variables requires a prodi-
gious amount of algebraic manipulation. This
manipulation is, however, not difficult if one uses
any of the new symbolic processors now avail-
able. (A program called Mathematically was
used for this manipulation).
399
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The final set of four evolution equations for
unsteady two-dimensional flow is presented in
Appendix A. The components of the vertical
components of the vn (w~,w2,w3) have been elim-
inated, as have the so-called "integrated pres-
sures" pi, P2 and pa. The pressure on the free
surface, 13, is taken to be zero. The remaining
four variables are the free surface elevation,
p~x,t), and the three horizontal components of the
vn: ul~x,t), u2(x,t) and u3(x,t). These evolution
equations will be used for all of the nonlinear
computations in this paper.
3. Large-amplitude waves of permanent form
In this paper we use the GN level III the-
ory for time domain calculations of the dispersion
of random waves. The authors know of no high
accuracy calculations of these wave systems to
compare with. Since this theory is a new one, it
seems prudent to first compare the characteris-
tics of large amplitude regular waves predicted
by this theory with known very accurate results
for these waves.
To determine waves of permanent form,
the transformation 8/at ~ uO 3/0x is applied to the
evolution equations in Appendix A (these equa-
tions are Galilean invariant). The equations now
only depend on the primary variables ,B, us, u2
and U3 and their derivatives with respect to x.
The problem of determining a wave of permanent
form of a given length and elevation at the crest
can be posed as a two-point boundary value prob-
lem over a domain equal to the half-length of the
wave, with a symmetry condition imposed at both
ends of the domain and an elevation condition at
the crest imposed at one end of the domain. In
addition, global conditions stipulating that the
flow is irrotational on the average and the mean
water depth is zero need to be applied. A proce-
dure based on Thomas' method described by
Ertekin (1984) and generalized by Shields (1986)
was used to find these solutions. For the compar-
isons below, the x domain was discretized into
200 equally-spaced intervals (201 nodes) for one
wave length. Central difference formulas were
used throughout.
Several characteristics of these waves are
obvious candidates for comparison. These in-
clude: wave celerity, wave profile and velocity
profile. The parameters which are of importance
here are wave steepness, wave length and the
constant, a, which appears in the basis functions
(2~. As mentioned in the previous section, this
constant governs the exponential decay of the ve-
locities in depth. A value of a equal to the wave
number, k = 2~/\, where ~ is the wave length,
produces the same decay with z as predicted by
Airy wave theory and this choice yields the best
comparison with finite regular waves. We intro-
duce the notion of "bandwidth", the ability of GN
theory to predict waves of wave numbers which
are different from a. We anticipate that there
will be a range of wave numbers kit ~ a ~ k2 for
which the theory will produce satisfactory re-
sults. Accurate, high-order stream function re-
sults computed by Sobey (1989) are used for the
comparisons below.
a. Wave celerity.
The celerity, or phase velocity of the wave,
is the speed, uO, of the coordinate system neces-
sary to yield a time invariant wave form. Figure
1 shows the ratio of the celerity of infinitesimal
waves predicted by various levels of GN theory to
the celerity of Airy waves. It is seen that the
bandwidth of GN level I for a relative celerity er-
ror of, say, 2% is very narrow, that of GN level II
is broader and that of GN level III is broader still.
Since the focus of this paper is a random wave
train, it seemed appropriate to choose the theory
with the broadest bandwidth and therefore GN
level III was selected primarily on this basis.
1 1
>` 1
=
~ o.s
~ 0.8
._
a)
a) 0.7
c'
is
0.6
- /. . ~ levell
;,! . level 11
0 level 111
3
Figure 1. The ratio of celerity of infinitesimal
waves predicted by various levels of GN theory to
that predicted by Airy wave theory.
It is well known that the celerity of a regu-
lar wave depends on its steepness. Figure 2
shows the results of GN level III theory for waves
of various steepness for the special situation
where k _ a. The error between the GN results
(the line) and the stream function results (the
black squares) is much smaller than 1% and
cannot be detected on this figure. For values of k
different from a, Figure 3 shows the error in
celerity as a function of steepness. It is seen that
for values of 2.25 > a/k > 0.5, the celerity error is
within 1% for all values of steepness less than
0.12 (breaking waves correspond to a steepness of
400
OCR for page 401
about 0.14). That is, the celerity error is less than height). For wave of small wave height the eleva
1% for waves of one-half of the length of the wave lion varies almost sinusoidally in x. As the wave
for which k - a to waves which are well over steepens, the crest becomes sharper and the
twice the length of the wave for which k _ a). For trough flattens. Figure 5 shows the profiles for
waves of lower steepness the bandwidth Is some- waves of steepness 0.106 predicted for various
what larger, but the bandwidth is, of course, al- values of a/k. It is seen that even for this very
ways smaller than that for infinitesimal waves. steep wave, the variation in profiles is very little
for the range of am between 0.5 and 2.0.
1.10 ' ' ! ! ! 0
1.08 ~ · So fey ~Zz 0 4~ ~
106 F'''''''''''.'''''''''''''.''''''''''''''-.''''''''''''''.''''''''~'''''''-''''''''''''] ~ 0.2 ~:
~ ~ / ~ ~ ~O C
0 0.04 0.08 0.12
steepness, h/\ ~04 ) 0;785 1.57 2.36 3 14
Figure 2. The variation of celerity with steepness 2~ x/\
for GN level III theory for am = l.O (curve) and Figure 4. Wave profiles predicted by GN level III
for numerically accurate results (black squares). theory for regular waves of various crest heights
for a/lr = 1.
j ~ 2.25 c 0.5
B ~ W_~ Hi= ~L D IS /
0 0.02 0.04 0.06 0.08 0.1 0.12 -0.30 0.785 1.571 2.356 3.142
steepness, h/\ 2~ x/\
Figure 3. Error in prediction of the celerity of Figure 5. Profile of waves of steepness 0.106 pre
regular waves by GN level III theory as a dieted by GN level III theory for various values of
function of steepness for elk ~ 1. a/k.
b. Wave profile. c. Particle velocity
Figure 4 shows the wave profiles computed From the point of view of design of many
for waves of various elevations at the crest for the offshore platforms, the horizontal particle veloc
case k- a. These profile shapes deviate less than ity under the crest of the wave is probably the
one line width from high accuracy profiles (the most important. It is this characteristic of the
deviation is much less than 1% of the wave flow which causes the most significant loads on
40~
OCR for page 402
fixed (jacketed) platforms. Once again, the hori-
zontal velocity finder the crest of even very steep
waves is predicted with much less than a 1% er-
ror if k _ a. For k ~ a deviations occur. Figure 6
shows the variation with z of the horizontal veloc-
ity under the crest for a wave of steepness- 0.106.
The velocity is non-dimensionalized with the Airy
celerity. Curves for various values of a/k are
shown, where the value for a/k = 1 is coincident
with numerically accurate results. The water
surface at the crest of this wave is at a non-di-
mensional value of fizz/\ = 0.403, and the undis-
turbed water level corresponds to a value of z = 0.
Also shown on this figure is the horizontal parti-
cle velocity prediction from Airy wave theory.
0.5 ~ ~ ~ 1 ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~
-'I ° a''''''''''''''''''' ''''''.? ''''''''''''''''''''''~
a/k
- - -0.49
- -0.64
1 .0
- - - - 1 . 5 6
----- 1.96
- ----Airy
0 0.2 0.4 0.6 0.8
horizontal particle velocity / ~
Figure 6. Variation of horizontal particle velocity
under the crest of a wave of steepness 0.106 as a
function of non-dimensional altitude.
It is seen that Airy wave theory uniformly
over-predicts the horizontal particle velocity un-
der the crest for all z. In general, the prediction
of the GN level III theory is good, except very
near the crest. If 0.64 ~ a/k < 1.56 the error is ev-
erywhere less than 5% of the numerically exact
result. For a/k = 1.96 the error is 25% at the wa-
ter surface but becomes less than 5% for values of
2~z/l < 0.25; for elk = 0.49 the error is 14% at the
water surface but does not drop to less than 5%
unless fizz/\ < 0.
Separate investigations were also made at
different values of steepness and at different loca-
tions along the wave. At the crest at a steepness
of 12%, the error for 0.64 < a/k < 1.56 increased
slightly to 6%; that for a/k = 0.49 and 1.96 in-
creased to 20% and 35%, respectively. Beneath
the trough much smaller particle velocity errors
were observed for all values of a/k discussed here
and, thus, it appears that the crest is the most
critical location.
d. Summary of regular wave comparisons.
The bandwidth for particle velocity error is
much narrower than for either wave celerity er-
ror or for wave profile error. Let us denote the
wave length for which a/k = 1 by DO. The above
comparisons indicate that as long as 2/3 TO < ~ <
3/2 TO, we can anticipate errors of less than 1%
for celerity or wave profile and less than 6% for
horizontal particle velocities for a steepness up to
12%. These limitations imply that the GN level
III theory may be a good model for a steep, nar-
row-banded seaway.
4. Time domain results
Two different wave situations were investi-
gated using the time domain version of the
Green-Naghdi level III equations in Appendix A.
The evolution equations are second-order in time
and third order in space. At each instant the
time derivatives (on the left-hand side of each
equation in Appendix A) can be found as a solu-
tion to a two-point boundary value problem. Since
this is an initial-value problem starting from an
initially quiescent condition, the global conditions
for irrotationality or for mean water level used for
determining waves of permanent form need not
be applied here.
The difference formulation for the two-
point boundary value problem is the same
Thomas' algorithm used for the waves of perma-
nent form. Integration in time is performed us-
ing a modified Euler method. Both integrations
in space and time are second-order accurate and
variations in both time and space steps were
made to assure that convergence was adequate
(less than 1% error).
One of the particular advantages of GN
theory in general is that it yields differential
equations in the horizontal coordinates. Since
the computational effort required to solve the two-
point boundary value problem grows linearly
with the computational domain, the overall time
integration retains this property. In this sense,
the GN theory allows one to compute larger spa-
tial domains than, say, boundary element meth-
ods were the effort typically grows with at least
the square of the size of the number of nodes.
The left-hand boundary for both problems
below was considered to be a "wave-maker"
where values of ~B(xw,t), ul~xw~t)' u2(Xw~t) and
u3(xw,t) were prescribed (xw is the x location of
the wavemaker). In general, the values of the
three u's are not known a priori for the nonlinear
wave system. We used the values obtained from a
402
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linear solution of the GN level III equations to re-
late these quantities to p. The local disturbance
caused by the not quite correct values of the three
u's appeared to die out quickly (as it does with a
real wave-maker in a wave tank). In order to
avoid any initial disturbances, a cosine-squared
ramp was provided at the start of the wave-
maker. The ramp was applied only to the first
full cycle of the wave maker.
A simple Sommerfeld boundary condition
was imposed on the right-hand boundary for both
examples and this condition was based on the as-
sumption that all waves have a celerity equal to:
that of infinitesimal waves of length lO. In prac-
tice, little reflection was observed, but in both ex-
amples the right-hand boundary was taken far
enough away to minimize any possible adverse
consequences from reflections.
a. The generation of regular waves.
In order to assure that the time domain in-
tegration scheme and the wave-maker were per-
forming correctly, a set of regular waves was
generated. Although the internal calculations
were all performed non-dimensionally, the re-
sults are reported dimensionally, corresponding
to typical ocean wave scales. The computational
domain consisted of 1300 space steps of 12.28' and
the time steps were 0.2 sec. The waves had a
wave length of 809' and final height of 70'
(corresponding to steepness of 8.65%~. Figure 7
shows a snapshot of the wave elevation profile 160
seconds after the start-up of the wave-maker.
Since the wave packet had not progressed past
6000', the remainder of the computational do-
main is not shown. This figure shows that the
eldest two waves (the rightmost two waves) are
somewhat distorted, and that the train of notice-
ably steep, but regular waves follows.
Figure 8 shows the wave elevation time
history as seen by an observer 644' from the wave
maker. This observer sees almost twice as many
waves as seen in the surface elevation view be-
cause the group velocity is much less than the
celerity (a manifestation of dispersion). A very
short time after the time of figure 7, the leading
wave of this packet became quite steep and
reached a breaking condition. Local snapshots of
this process at 2 second intervals are shown in
Figure 9. The solution algorithm breaks down
when the wave reaches breaking conditions and
the computation can not continue.
It was not clear, at first, whether the
breaking wave at the front of the group was real
or simply an artifact of either the start-up of the
wave maker or of the GN theory. Longuet-
Higgins (1974) demonstrated, both theoretically
and experimentally, that the leading wave in a
packet of generated waves will steepen. Further,
his experiments showed that the leading wave
can break if the generated waves were steep
enough to begin with. Unfortunately, his linear
a)
a, -20
ct
~ 40
60
-
r
-
a
._
~ 20
Cl) 0-0
a)
~ -20
ct
-40 A,
P ~.i ~', ~
0 1000 2000 3000
4000 5000 6000
distance from wave maker Oft.
Figure 7. Wave surface elevation at time = 160 seconds after initiation of wavemaker
(regular waves of length = 809' and height = 70')
~.V V V ~V V V, V V V ~
1
120 160
Figure 8. Wave elevation time history at a point 644' away from wavemaker.
403
time (see)
OCR for page 404
100.
c
o
40
0' 20
a)
~0
V
-20
Figure 9. Details of breaking wave at leading edge of wave packet.
analysis was unable to predict such an occur-
rence. In order to investigate this further, we at-
tempted to duplicate the simulated wave situation
in the Ship Model Towing Tank at U.C. Berkeley.
To our surprise, we found that we were unable to
make a train waves at this steepness without the
wave in the front of the group breaking after
about 12-14 wave maker cycles (that is with 6 or 7
waves in the tank). The breaking occurred even
when the wave maker was turned on very
smoothly over three wave cycles.
b. Steep random waves.
In order to investigate the dispersion of
random waves, a numerical experiment was
conducted. An existing measurement of steep
waves, recorded during Hurricane Camille in
the Gulf of Mexico, August 16-17, 1969 was used
as a foundation for this experiment. The particu-
lar record covered 512 seconds in real time with
two measurements per second. Although this
record was taken in a water depth of 325 ft (which
corresponds to a depth at which shallow water ef-
fects are just beginning to be perceived), it was
felt that this sample was a good representative of
the survival conditions one might encounter.
The waves were simulated in exactly same
fashion as the regular waves in a. above were.
However, it was desired to generate a wave sys-
tem like that from Hurricane Camille at a given
reference point removed some distance from the
wave-maker. The coordinate system was chosen
so that this reference point was x = 0. A finite,
untruncated Fourier transform of the record was
determined,
2 nmax
p(t) = A, an sin At + bn cos Ant, (3
n=0
where
An
In =-
at nmaX
_ . ~.......... ....................................
~; ,
3500 4000 4500 5000 5500 6000
distance from wave maker (ft.)
fit is the time interval between
data points,
nma~ is the number of data points
in the record.
Using linear dispersion (Airy theory), the
record (3) was referred to a new location xw, as
suming that the waves are two-dimensional and
progressing in the positive x direction, yielding
2 nmax
p(t) = A, an sin ¢(t) + bn cos ¢(t), (4
n=0
where the phase ¢(t) = can (it (cog w)
For the numerical experiment described
below, xw was taken to be -644' (i.e. 644' up
weather from the point x = 0'). This new record
was used to drive the wave maker. Three wave
probes were "mounted" in the computational do
main, at x = 0', x = 400' and x = 800'. The dis
tance between the probe at the reference point at x
= 0' and that at x = 400' is comparable to the
length of a typical offshore platform; the distance
between the reference point at x = 0' and that at x
= 800' is comparable to the length of a typical
large ship. The constant a was selected to corre
spond to waves for which TO = 809' (i.e. a = 2~/809)
and the computation was run for the same 512
seconds of the original record with temporal
steps of 0.2 sec. and 1300 spatial steps of 12.28'.
Figure 10 shows a comparison of the wave
elevation measured at the probe at x = 0 and the
original Hurricane Camille record. In general
the two traces compare very well except near t =
130 and t = 450. The computed wave elevation
history is smoother than the original record pre
sumably because the bandwidth of the GN level
III theory is limited. We do note however, that
most of the waves do lie within the wave length
range of 500' to 1200' corresponding to the range
404
OCR for page 405
60
-
-
o
g
a)
a)
3 -2C
4G
20
4r
- ~Camille time history
''''''~''''-'''''-IT'----'-------------'--''''''-------~--'''''---''----------'''--''''------------------'----- ------------------------------------- -:- ill -
v GN level 111 simulation ~
, , . .
0100 200 300 400
time (see)
500
Figure 10. Comparison of GN level III simulation at x = 0 with recorded Camille time history
2/3 NO ~ ~ < 3/2 TO for which the GN level III the-
ory yields uniformly excellent results. The useful
part of the wave elevation records at the three
probes cover a somewhat smaller time interval
than the original Camille record because of the
time it takes for waves to progress from the
wavemaker to the probes.
In
fir
~ 600
.~
- 400
Cal
Q
o, 200
o probe at origin
O probe at400'
probe at 800'
Camille
Al ~ ~ ~ i . .~,
O0.2 0.4 0.6 0.8 1 1.2
frequency transect
Figure 11. Spectrum of the original Camille
record and that measured at the three wave
probes.
Finite Fourier transforms of these records
were also made and spectra formed. Figure 11
shows the spectrum of the original record, as
well as the spectra of the time histories recorded
at the three probes. These spectra have been
smoothed using a seven-point moving average.
60
40
20
0
-20
-40
All four spectra are very nearly the same except
the original Camille spectrum has a peak at a
wave frequency of about 0.45 ra/sec which does
not occur in the simulated record. The three
spectra from the wave probes can be considered
identical.
Several additional simulations were per-
formed using the same input record but with in-
put to the wavemaker multiplied by a factor. The
simulation with a factor of 1.2 (i.e., the input was
20% larger) produced waves which were almost
breaking. Larger factors produced waves which
did break and in these cases it was not possible to
complete the simulation.
The spectra of the time histories of the
three wave probes were all about equal for the
20% larger simulation and all were almost ex-
actly 44% larger than the corresponding spectra
for the original simulation, as one would antici-
pate. The actual wave profiles, although quite
similar in form, were measurably more "peaked"
near the highest waves. In the discussion below
we will use both the simulation using the orig~-
nal wavemaker input (labelled 100% Camille in-
put) and that resulting from the 20% larger input
(labelled 120% Camille input).
The time histories of the wave elevation at
x = 0' are now parts of a consistent description in
time and space of nonlinear wave systems, and
these descriptions afford an opportunity for
assessing of the effects of nonlinear dispersion.
Let us suppose that the time history recorded at
the numerical wave probe at x = 0 is a realistic
405
OCR for page 406
40
O 20
._
' O
a)
-20
-40
Figure 12. Comparison of waves at x = 400' predicted by GN level III and
by linear dispersion from x = 0' (100% Camille input).
record of a possible realization of a storm wave
system (its closeness to the measured Hurricane
Camille record lends credibility to this supposi-
tion). This time history will be taken as a refer-
ence time history. The time histories of wave ele-
vation at the probes at x = 400' and 800' recorded
in the simulations are part of a nonlinear wave
system, but can also be estimated from the refer-
ence time history at x = 0' using finite Fourier
transforms and linear dispersion (as was done in
(3) and (4) above). Such a process is spectrum-
preserving and therefore these estimated time
histories at the other two probes will have exactly
the same spectrum as the simulated time history
at x = 0'. However, the spectra of the simulated
time histories at the x = 0', 400' and 800' are all
sensibly the same (see Figure 10~. Thus, the two
sets of time histories: the nonlinear GN level III
simulation, and that derived by linear dispersion
will have essentially the same spectra at each
probe. In other words, each represents a differ-
ent realization of the same spectrum.
The comparison between the time histories
at the two alternate probe locations is essentially
a comparison between linear dispersion and non-
linear dispersion. Figure 12. shows a compari-
son of the time histories at x = 400' for the origi-
nal wavemaker input (100% Camille input). It is
clear that the character of both wave systems is
more similar than the comparison between the
simulation and the original Camille record, but
upon close examination one finds the trace from
nonlinear dispersion shows sharper peaks and
flatter troughs than that from the linear disper-
s~on.
Ll ' I ' I ' ' ' ' ! ' ' ' ' ! ' ' ' ' ! ' ' ' ' ! ' ' ' ' ! ' ' ' ' I ' ' ' '
GN level 111 simulation .-
II11,,,,1,,,,1,,,,,,,,,,,,,,1,,,,1,...
100 150 200 250 300 350 400 450 500
time (see)
40
20 O
.
0 '
a)
-20 ~
3
-40
The relation between linear and nonlinear
dispersion can be perhaps more clearly seen in
Figure 13. The top three graphs in this figure
show the results of matching the elevations of the
individual crests and troughs from the record
produced by the nonlinear dispersion simulation
and that from the linear dispersion. The values
resulting from each are plotted along a different
axis.
For the probe at x = 0' there is a perfect cor-
relation between the crest and trough elevations
derived from the time history and those derived
from the finite Fourier transform, since the
transform was determined using the whole wave
time history at this point and no terms were
thrown away.
The two time histories at the probes at x =
400' and 800' were not identical and, in some
cases, were not geometrically similar. Thus,
identifying the corresponding crests and troughs
was sometimes ambiguous and scatter occurred.
For these two probes, the points show a signifi-
cant deviation from the 45° line which would
indicate perfect correlation. It is seen that, in
comparison with the nonlinear dispersion simu-
lation, the linear dispersion results show smaller
crest heights and larger trough depths. This dis-
crepancy is worse for the probe at x = 800' than
for that at x = 400'.
What is important in design of many ships
and platforms is the combination of the maxi-
mum wave elevation at the crest and the horizon-
tal particle velocity at the crest. For the superpo-
sition of waves given by (3), Airy theory predicts
406
OCR for page 407
o - ~ - : - ~ ~ l l ~ ~ ~ ~ ~ ~ l
Hi: ~ ~ to
Juntas JalaaU.M - U°!SJadS!P J~aU!1 5-
uolsJads!p J~9U!1
of:- off- M~-
; I I > ~ ~ ~ O ,_
A o ~
g N ~0) S
uo!s~ads!p~eau!l 6u!qo~a'~s~alaa4M-Uo!s~ads!p~eau!l ;z x
Kit o ~ 9 r
~, ~, ~0 T ~--.- --- ----- ~--------~0 Cock
01 O O O O ~ O O O O O O
6ulq~a3~S Jala6UM
uo!~!sodwo~ap ~a!'n°d 91!U!d
407
OCR for page 408
the horizontal particle velocity, u~x,z,t), in the
waves to be
u~x,z,t)
2 nmax ((I)n ~
Ad, ~ e g (an sin cut + bn cos ont)
n=0
Referring again to Figure 6, it is clear that Airy
theory yields particle velocities which are much
too high. It is typical in many offshore applica-
tions to use an approximation called Wheeler
stretching" (Wheeler, 19691. In this approach,
the exponential decay factor in (5) is modified,
yielding
5(x,z,t)
2 nmax fin Z it)
Ad, ~ e g (an sin ant + bn cos Ate,
n=0
~6
where z' = z - p~x,t). That is, the value of z used
in the exponential decay measures the relative
distance below the free surface rather than the
absolute distance below the undisturbed free sur-
face level.
The bottom graphs in Figure 13 are corre-
lation plots showing particle velocities both under
the crest and under the troughs. It relates the
corresponding peaks in the horizontal particle ve-
locity at a distance 10' below the free surface at x
= 0' computed by using the GN level III model,
(1), and that determined using the finite Fourier
sum (3) and Wheeler stretching (6~. The sum in
(6) is unrealistically dominated by the large
number of high-frequency components in the fi-
nite Fourier decomposition when the exponential
decay factor is unity, as it is when z _ ~ (i.e., z' =
0~. The effect of these high frequency components
is unimportant for very small values of z' ~ O and
thus a value of z' = -10' was selected. The correla-
tion between the prediction of horizontal particle
velocity at x = 0' from GN level III theory and
Wheeler stretching is very good under the crest
(positive velocities) and only slightly less good
under the troughs (negative velocities). A signif-
icant deviation occurs only for the very highest
waves and the deviation remains no more than
about 5%.
Although we do not show the results here,
finite Fourier decompositions and estimates of
the particle velocities using Wheeler stretching
were performed for the GN level III time histo-
ries recorded at the other two wave probes. This
information was used to develop correlation dia-
grams similar to the bottom graph in Figure 13.
These correlations were almost identical in
character to the left-hand bottom graph in Figure
13. As a result, we conclude that if the time his-
tory of the wave elevation is known at the point of
interest, Wheeler stretching is a very good esti-
mator of the peak velocities beneath the crest of a
wave and a good estimator for the velocities be-
neath the trough of the wave.
Let us now investigate the situation when
the wave elevation history is not known at the
point of interest and must be determined by lin-
ear dispersion. The middle and right graphs on
the bottom of Figures 13 are correlation diagrams
resulting from comparing the horizontal particle
velocity at the probes at x = 400' and 800'. The GN
level III prediction is based on (1) using the val-
ues of us, us, Us and ,B determined at these loca-
tions by the nonlinear simulation; the spectral
method prediction is based on the Fourier de-
composition (3), linear dispersion (4) and Wheel-
er stretching (6~. It is obvious that the compari-
son at the probe at x = 400' is significantly poorer
than that at x = 0', and that at x = 800' is poorer
still. In particular, GN theory predicted particle
velocities under many of the crests in excess of 30
fps, whereas the linear dispersion result did not
predict any velocities this large.
When viewed as a whole, the graphs in
Figure 13 show that the relationship between hor-
izontal particle velocity discrepancy and the wave
crest and trough discrepancy is nearly constant.
That is, when the wave crest and trough predic-
tions are good, the horizontal particle velocity
predictions are good; when the wave crest and
trough predictions are poor, the particle velocities
are corresponding poor. It appears therefore that
linear dispersion is the weak link in the predic-
tion process rather than the Wheeler stretching.
Perhaps a more instructive view of the dif-
ference between linear dispersion can be gleaned
from a comparison of the wave elevation profiles.
Figure 14a shows snapshots of the wave elevation
computed using linear dispersion for x = -400' to
x = 2000' at 2 second intervals from t = 442 to t =
462 seconds. Figure 14b shows the same set of
snapshots for the simulated waves using GN
level III. Both sets correspond to the 120%
Camille input to the wavemaker and by construc-
tion, both sets of records have exactly the same
time histories at x = 0'. The linear dispersion
record shows many small wiggles which are the
result of the high frequency terms in the 2100
terms of the finite Fourier sum. In general these
effects are localized near x = 0.
A cursory glance shows that the two sets of
snaphsots are similar, but a significant differ-
ence occurs between t = 454 and t = 460. The GN
level III simulation predicts a large, nearly
breaking wave crest which persists for about 6
408
OCR for page 409
seconds and has a maximum elevation of 65'
above the undisturbed free surface. This wave
was the highest crest obtained by the simulation.
The corresponding wave for the linear dispersion
case is never more than 50' high and lacks the
coherence of the simulated wave. The difference
appears to be that the Airy wave components are
not "phase-locked" and can not remain together
for any length of time.
5. Conclusions
A nonlinear fluid sheet model for
predicting the dispersion of random wave sys-
tems in the time domain was introduced and
compared with known results for steep waves of
permanent form. It was found that the particu-
lar model used here, Green-Naghdi level III,
compared extremely well with the results for
wave celerity and wave profile for a wide range of
steepness and over a fairly broad bandwidth of
wave lengths. The comparison with wave parti-
cle velocities was good over a fairly narrow, but
useful, bandwidth of wave lengths.
The GN level III model was used to predict
the generation of regular waves and it was found
(and confirmed by laboratory experiments that
the leading edge of a packet of relatively steep
waves always appears to break before very many
waves are created.
This nonlinear model was also used to
model a real steep wave record, that measured
during Humcane Camille in 1969. The purpose
of this study was to investigate the effects of non-
linear dispersion. These results can be summa-
rized as follows: linear dispersion leads to under-
pred iction of both the wave elevation aru! the wave
particle velocities at a point remote from a loca-
tion where the wave elevation history is known.
This under-prediction may represent a signifi-
cant lack of conservatism in the use of the spec-
tral method for design to withstand survival con-
ditions.
The time histories at all probes either
recorded from the nonlinear simulation or from
linear dispersion from the probe at x = 0' all had
spectra which were sensibly the same. That is,
all were acceptable realizations of the same spec-
trum. Yet those time histories of either wave ele-
vation or particle velocity resulting from linear
dispersion did not compare well with those which
resulted from nonlinear dispersion. Thus, we
can further conclude that not all realizations of a
spectrum correspond to realistic wave systems, if
the waves high enough to lead to significant non-
linear effects.
Acknowledgement
This research was in part sponsored by
the Office of Naval Research, United States Navy,
under contract N00014-88-K-0002 with the Univer-
sity of California, Berkeley.
References
Chaplin, J. R. (19801. Developments of stream
function wave theory. Coastal Engineer-
ing, Vol. 3, pp. 179-205.
Dean, R. G. (1974), Evaluation and development
of water wave theories for engineering ap-
plication. U. S. Army Coastal Engineering
Research Center, Report SR- 1 (two vol-
umes).
Ertekin, R. C. (1984~. Soliton generation by mov-
ing disturbances in shallow water. Ph.D.
Thesis, Univ. of Calif. Berkeley. v + 352 pp.
Fenton, J. D. (19881. The numerical solution of
steady water wave problems. Comput.
Geosci., Vol 14, pp. 357-368.
Green, A. E. and Naghdi, P. M. (1986~. A nonlin-
ear theory of water waves for finite and in-
finite depths. Philos. Trans. Roy. Soc.
London Ser. A, Vol. 320, pp. 37-70.
Green, A. E. and Naghdi, P. M. (1987~. Further
developments in a nonlinear theory of wa-
ter waves for finite and infinite depths.
Philos. Trans. Roy. Soc. London Ser. A,
Vol. 324, pp. 47-72.
Longuet-Higgins, M. S. (1974), Breaking waves -
in deep or shallow water. 10th Symposium
on Naval Hydrodynamics, MIT, pp. 597-
605.
Schwartz, L. W. (1974) Computer extension and
analytic continuation of Stokes' expansion
for gravity waves. J. Fluid Mech. Vol. 62,
pp. 553-578.
Shields, J. J. (19861. A direct theory for waves ap-
proaching a beach. Ph.D. Thesis, Univ. of
Calif. Berkeley. v + 137 p.
Sobey, R. J. (1989~. Variations on Fourier wave
theory. International Journal for Numer-
ical Methods in Fluid Mechanics, Vol. 9,
pp.1453-1467.
Webster, W. C. & Shields, J. J. (19901. Appli-
cations of high-level, Green-Naghdi theory
to fluid flow problems. IUTAM Sympos-
ium on Marine Dynamics, Brunel Univer-
sity, London (in press).
Wheeler, J. D. (19691. Method for calculating
forces produced by irregular waves. Off-
shore Technology Conference, Houston,
Texas, OTC 1006.
409
OCR for page 410
50
o
-50
50
o
-50
50
o
-50
50
o
-50
50
o
-50
-400 0 400 800 1200
1600 2000
x (it)
Figure 14a. Wave profiles predicted by superposition of Airy waves and linear dispersion.
410
442
444
446
t = 448
t =450
t =452
t =454
t =456
t=458
t =460
t =462
/
OCR for page 411
so
o
- 5 0
50
o
- 5 0
- 5 0
50
o
- 5 0
50
o
- 5 0
-400 0 400 800 1200
Figure 14b. Wave profiles predicted by GN Level III theory.
411
1 600 2000
x (ft)
442
444
t = 446
t =448
t = 450
t = 452
t -454
t = 456
t=458
t =460
t =462
OCR for page 412
o
E::]
c
rat
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r a I
1 Cq ~ |
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rot ~+ ~ ~ | ~
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N ~_' ~N`=
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412
OCR for page 413
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OCR for page 414
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414
OCR for page 415
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415
OCR for page 416
DISCUSSION
Krish Thiagarajan
University of Michigan, USA
The authors have prescribed a certain velocity distribution at the
wave-maker position in their numerical wavetank. They also claim
that disturbances caused due to this specified distribution are localized
and die out quickly with horizontal distance. This claim may not be
entirely true. Satisfying the no-flow condition on the wave-maker
surface had been a known problem. Existing second order wave
generator theories (Ref[1] reviews some of them.) indicate the
existence of a second order free wave of frequency twice the
fundamental wave frequency generated by the wave maker. This free
wave is parasitic in nature as it travels along with the wave of
interest, i.e., it does not die out. My own experiments have
confirmed this (Reftl]). The above discussion, while pertaining to
a physical tank, may also be applicable to a numerical wave tank.
Ref.[1]: Thiagarajan, K. "An Experimental Study on Higher Order
Waves and Hydrodynamic Loading on Vertical Surface Piercing
Cylinders," M.Eng. Thesis, Faculty of Engineering, Memorial
University of Newfoundland, St. John's, Canada, 1989
i .
416
Representative terms from entire chapter:
water waves
