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OCR for page 713
Breaking Waves in the Ocean and around Ships
M. P. TuTin ~ anc3 M. Lancirini 2
~ Ocean Engineering L`aboratory, UCSB, mpt@engineering.ucsb.edu
2 INSEAN, The Italian Ship Mode! Basin, Italy, maulan@waves.insean.it
ABSTRACT
A wide ranging summary of research on ocean and ship
breaking waves carried out within the Ocean Engineer-
ing Laboratory at the University of California, Santa
Barbara, beginning in 1985, is presented in a coherent
fashion. Emphasis is given to the role of naturally aris-
ing wave modulations in their crucial effect on wave
breaking, as well as on the essential role which wave
breaking plays in wave evolution. The OEL criterion
for the onset of wave deformation and breaking is dis-
cussed and its mathematical derivation is given; its ex-
perimental validation through surface orbital velocity
measurements is reviewed. The recent development of
a gridless numerical simulation method for the study
of post breaking wave flows is discussed, and exam-
ples of splashing flows are given, including bores, and
bow breaking waves produced by a Wigley hull. Com-
parison with laboratory and field observations are made
whenever possible.
1 FORWARD
In 1985 we began our studies of breaking waves at the
newly founded Ocean Engineering Laboratory within
the University of California at Santa Barbara. We
began with Raymond Cointe's analysis of the steady
breaker above a submerged hydrofoil, Cointe and Tulin
(1994), where good experimental data existed, Duncan
(1983), and for which we obtained some predictions of
breaking and breaker characteristics, based on a breaker
model consisting of an essentially stagnant eddy, Fig-
ure 1, riding on the forward face of the leading wave
in the wave train behind the hydrofoil and held in place
by shear stresses along the dividing streamline. This
model was quantified utilizing independent measure-
ments of the turbulent shear stresses, and it was shown
that the hydrostatic pressure acting on the dividing
streamline underneath the eddy creates a trailing wave
which largely cancels that trailing wave which would
exist in the absence of breaking. The 'wave' resistance
of the hydrofoil thus manifests itself in the momentum
flux of the residual trailing wave plus the breaker resis-
tance, i. e. the momentum flux in the breaker wake. The
total resistance for a fixed hydrofoil speed was shown
to have a minimum corresponding to a particular value
of the trailing wave steepness resulting in an onset con-
dition for breaking.
~ =* at
~ ." ~
~ . ~
~ I, ~ ~ · ~ ~ · ~
~ /~
a~ q~ (~a,)
M-~S now
(~) ~ =~:n
Figure 1: Physical model of a quasi-steady spilling
breaker.
In contrast to the quasi-steady hydrofoil
breaker, energetic waves in the ocean are transient by
their nature, and subsequently we have sought to gain
a fundamental understanding of the latter, and of their
effect on the nonlinear dynamics of ocean wave evolu-
tion. We early concluded, Li and Tulin (1992), that en-
ergetic breaking was closely associated with these non-
linear dynamics, especially wave modulation (groups)
caused by instability of the Benjamin-Feir type. As
time went on we utilized various methods, as necessary,
including analysis, exact numerical simulations, and a
variety of experiments in our small and large wind wave
tanks, Figure 2, including the use of radar.
In response to needs of the Navy, we also
OCR for page 714
,.,-------- HN-DRAULICS
~ ' - - - -
~ 5~7E~IAI~ER
~ 4--------- HONEN-COMB
turned our attention to those breaking ship waves pro-
duced by the bow and which contributed to the per-
vasive surface wakes seen behind ships. For these
studies we have relied heavily on nonlinear 2D + t nu-
merical simulations, and these have been recently ex-
tended to the post-breaking regime dominated by sur-
face turnover and splashing.
It now seems time to present a wide-ranging
summary of the various results which have been ob-
tained. This is done first through an extended Overview,
followed by two separate sections with more detail;
Section 3 is devoted to the mechanism of breaking, and
Section 4 to the numerical treatment of wave splash-
ing and to splashing bow waves on ships and their near
wake.
We are concerned here with an exposition of
our own work, and this should therefore not be con-
strued as another attempt to review the widespread lit-
erature. For that there exist several recent reviews,
Longuet-Higgins (1996) and Melville (19961. The first
of these gives particular emphasis to theoretical work
on the stability of steep waves, and the second to exper-
imental and field observations.
INLET \AiIND ~JNNEL
.
Figure 2: The large OEL Wind-Wave Tank.
TEST SECTION --------'
In, '\
_________.~_______________________________l ________' ~
BEAC.H
- = ~ =
. . An. ~ ~ .. . ~ ~ . '. . ~~ '$ . . . ~ ~ Ad. ~ ~ I. ~ ~ ~ .. ~t . . ~ . .~ . ~ ~ ~ ~~ . ~ ~
——————————————————————————————————————————————————————————————————————————t———————————————————————————————————————————————————————————————————
.~ ~ '.
':----- GOVE - NVIRES
A large number of talented individuals con-
nected with the OEL have taken part in the investi-
gations which led to the present results, and this pa-
per is fondly and gratefully dedicated to all of them,
who have taught me so much. Their names will be
found as authors of the OEL related papers in the Ref-
erences. During the last few years we have much ben-
efited from a close working collaboration with my co-
author, Maurizio Landrini, of INSEAN, Roma, who is
now in charge of our numerical simulations of breaking
and post breaking ship waves. He has written Section
4.
The investigations reported on here were all
supported by the Office of Naval Research, the inves-
tigations of ocean-like breakers by the Ocean Tech-
nology Division under a sequence of Program Officers
(Gene Silva, Steve Ramberg, and Tom Swean jr.), and
the investigations of ship breakers first by Jim Fein,
and then by the Ship Hydrodynamics Program under Ed
Rood. Their faithful support of the OEL from its first
beginnings until the present time has been absolutely
crucial to its development and its productivity, and we
cannot thank them enough.
OCR for page 715
2 OVERVIEW
2.1 1985
Our own view of both ocean and ship breakers has been
radically transformed since we began our studies of
breaking in 1985.
The description of ocean waves was then, as
it largely remains now, stochastic in nature, and the
prototype wave was considered synonymous with the
Stokes wave. Prevailing notions then related the break-
ing of energetic waves to the Stokes Limiting Wave, ei-
ther through the Longuet-Higgins (1969, 1977, 1985)
crest acceleration criterion, or through a wave steep-
ness criterion determined experimentally; the Phillips
energy spectrum, for ocean waves, which had enjoyed
wide popularity, was entirely based on the notion that
the saturated (breaking) wave at every scale shared the
same geometry, depending only on 9, as in the case of
Stokes waves.
Theorists had by then suggested that wave
modulations of nonlinear origin might be crucial for
ocean modeling, Yuen and Lake (19821; field observa-
tions of waves breaking at the peak of wave groups had
been made, and systematic and insightful laboratory ex-
periments on a large scale were being carried out. How-
ever, the significance of modulations for the breaking of
waves, whether due to natural instability (group forma-
tion) or to wave-wave interactions, had not been well
appreciated or understood. In fact, a small minority
of the publishing wave community even accented the
. , ~ · ..
existence of wave groups In the ocean, notable among
them Su (1986), and Goda (19761.
By 1985, a large body of literature existed on
nonlinear wave interactions, originating in the pioneer-
ing work of Phillips (1960), on the four wave interac-
tion mechanism, and in the form of the kinetic theory
of waves by Hasselman (1962) and Zakharov (1968),
and their new ideas were widely accepted within the
theoretical ocean community and even incorporated
into wave forecasting methods, Swamp Group (19851.
In these kinetic equations the effect of breaking and
wind pumping on the wave dynamics has been ignored.
The reason has been given that these real processes
are slower than wave interactions and therefore non-
competitive. In wave forecasting this neglect would be
fatal and therefore wave energy (or wave action) evolu-
tion is driven by wind pumping suitably reduced by en-
ergy dissipation due to breaking, all suitably parameter-
ized in terms of the wave energy. Wave momentum con-
siderations are neglected throughout, and this creates a
vacuum regarding the evolution of wave frequency, a
vacuum which is filled with certain largely empirical
results from wave interaction calculations. This unsat-
isfactory situation continues, in fact, until the present
time, although the effects of breaking on wave dynam-
ics, particularly frequency downshifting, has been elu-
cidated through experiments, Tulin and Waseda (1999),
and theory, Tulin ~ 19961.
~~ r
_, 1
In 1985, the situation regarding the divergent
bow waves from ships, and their breaking was under-
developed and unsatisfactory. The only coherent view
of these waves was as a part of the traditional Kelvin
wave pattern, represented by the divergent portion of
the ship's wave amplitude spectrum. The use of the
Neumann-Kelvin method for the prediction of wave re-
sistance continued this spectral representation of the
1 1
convergent waves in that it incorporated the Havelock
singularity based on linear considerations, as had the
Michell theory and its slender body representation. Dis-
quiet was injected into this situation by the experimen-
tal studies in Tokyo, Inui et al.,tl979), and Miyata,
(1980), which showed the appearance of a single almost
straight wave crest at the bow, whose angle of inclina-
tion depended on the ship speed. Neither of these ob-
served characteristics were understandable in terms of
conventional theory, and it remained for nonlinear sim-
ulations, based on the 2D + t approximations, much
later to reproduce and explain these results, Tulin and
Wu (19961.
The splashing, mixing, air entrainment, and
noise generation which occurred in the aftermath of
breaking of both energetic ocean waves and divergent
bow waves, and are of such great interest today, had
been seldom studied, with the exception of the hy-
draulic jump, or bore, about which a little was known.
Our state of ignorance was nowhere more critically re-
vealed and our curiosity prodded than by the stimulat-
ing reviews of Peregrine (1981), Melville (1996), the
experiments of Bonmarin (1989), and our own exper-
imental observations. Our own understanding of both
ocean and ship breaking waves has been achieved in
good part because of our high resolution simulation
methods, LONGTANK and 2D+t, both based on bound-
ary elements, and it was natural for us to extend simula-
tions into the regime of splashing. This has required the
use of a gridless Euler solver, based on a Smoothed Par-
OCR for page 716
Down shining of spectral pleat
1 ..—
1
1~
1 04 ---
a~
~$
Go
107 -
J -A Aims A
Bond V\Javex
{: · ~ j ~ ~ ~6.~:3m /
: ~ /: ~ 4~ ~ ~ I\\ /~:'\~Z7.0~'
~ ~ J ~ J: ~y I\~!~': 18.~ \'
I' side bands
~ I ~ ~ <~ ~ · ~ ~ ~ ~ S,.Om
---t-— = -- ~ ; . ~ _~
1~0 2*0 3.0 ~ 0
frequency (Hz)
:~:
Of
Figure 3: Wave energy density spectra at various fetches (9-36 m) from the wavemaker, measured with wave wires
in the large GEL Wind Wave Tank. The evolution of the instability until breaking and subsequent downshifting is
demonstrated.
0.~}5
0.00
O.C5
-O 1 0 20 30 4~0
X/;\
fiO 60 70 80
Figure 4: Modulating waveforms in space, created by a wavemaker at the far left, for a sequence of times. The growth
of the instability, wave group formation, and partial recurrence can be seen. LONGTANK simulation.
OCR for page 717
ticle Hydrodynamics Lagrangian technique and which
we call SPlasH.
2.2 Wave Modulation
The instability growth and modulation of a nearly
monochromatic wave has been demonstrated both in
the laboratory, Lake et al. (1977), Melville (1982), Su
and Green (1985) and Tulin and Waseda (1994), and in
numerical simulations, Wang et al. (1994), and the con-
nection with breaking has been conclusively demon-
strated.
The evolution of a wave system through mod-
ulation, breaking, and downshifting is seen through
the measured spectra taken in the OEL wave tank
(175'Lx14'Wx8'D), Figure 3. A time-distance simu-
lation of a similar wave system, using the OEL numer-
ical wave tank, LONGTANK, is shown in Figure 4.
In the Ocean Engineering Laboratory (OEL)
at UCSB we regularly produce waves breaking in self-
excited wave groups for the purpose of studying the
radar back scatter from a low grazing angle coherent
radar.
We have also used wave groups in the OEL
tank to study the loadings on floating bridge structures,
Welch et al. (1996), and to study the response of flexible
vertical cylindrical structures to excitation in deformed
and breaking waves, Welch et al. (19994.
Do such wave group modulations occur in the
ocean, and do they cause wave breaking? The evidence
is that they do. The earliest observations and connec-
tions between ocean wave groups and breaking include:
Donelan et al. (1972), Rye (1974), Goda (1976), Su et
al. (1982) and Holthuijsen and Herbers (19861. De-
spite these, there has been much controversy amongst
ocean engineers, where the concept of a stochastic sea
has proven so useful and has therefore become deeply
imbedded. Most of all, the visualization and measure-
ment of wave groups has not been clear-cut, and a clear
overall picture of how they exist in the ocean and under
which conditions has not yet been obtained.
Recently, however, low grazing angle (6°) co-
herent radars when viewing wind waves in the field,
have produced remarkable range-time R—t maps which
unmistakably show the existence of wave groups, Werle
(1995), Smith and Poulter (1993), Ivanov (19931. Fig-
ure 5 is an actual radar image together with an explana-
tory sketch. These maps show bursts of scattering along
straight lines in the R—t diagram and the analysis of
these shows that the slope of the lines conforms to the
law cg = dR/dt and the temporal spacing of the bursts
is 2T. A large number of these maps have been ob-
tained in sheltered waters, but they have been seen in
the open ocean (Ivanov) and more experiments are un-
derway.
. .ei ...
~ir.~;ji- <'?.;0 'i. 2~.0 .~Si~) Mali ,'j-;'0 ~ 9 I
in ~_~,,~, .
,~ ~
5~-j
.........
art....
`' AX ~ ~~ ; w
has >. I ~3>es K ¢
War ~~ ~ ~ jetty 6s3~ ~ god? z 7~
< ~ <9 I' .~i~i~~t)~~, ~'9~.31t
~ ji3kS;,$
Figure 5: Top: range-time radar return from a low graz-
ing angle, coherent radar mounted on a cliff overlook-
ing Lock Linnhe; the bursts are breaking events. Bot-
tom: a range-time schematic allowing a quantitative
analysis of the upper figure.
There is of course a theoretical basis for these
nonlinear wave groups. In the first place, it had been
shown long ago by Benjamin and Feir (1967) that
monochromatic waves are unstable to small sidebands,
although the theory does not allow for spatial variations
in the wave system, only temporal. The space-time evo-
lution of the resulting wave train was later studied, both
theoretically and experimentally by Lake et al. (19771;
they employed the weakly nonlinear model (the nonlin-
ear cubic Schrodinger equation, NLS) first introduced
by Benny and Newell (19671.
Later, exact nonlinear simulations showing
wave group modulations were first carried out by Dold
and Peregrine (1986), and subsequently a technique
was developed at the OEL for simulations of very long
wave trains, specifically for the purpose of understand-
ing breaking in wave groups. These allowed long time
exact inviscid studies of wave instability growth, wave
OCR for page 718
group formation, and wave breaking, see Wang et al. 2.3 Wave Deformation
(19941.
50 ~~ ~ ~ {\ ~~ \/ \ \~\
-50 .
j50
-50
~ ° RAAA~,: . . :: :: ~ :;; . ::A:: ': '. :: ::
-50 _
70
t (see)
Figure 6: Temporal wave records in the large GEL
Wind Wave Tank. Fetches from the wavemaker: 27 m
(upper); 18 m (middle); 5 m (lower). Strong grouping
and breaking (upper), and energy concentration under
the group peak.
What are the possible effects of wave group
modulations? Most important, wave energy becomes
concentrated under the peak of the envelope, and this
effect is strongest near breaking as shown in Figure 6,
a wave elevation time series taken in the GEL tank
of modulating wave groups near breaking. A sum-
mary of available information on this concentration ef-
fect of grouping is shown in Figure 7, where data on
the relation between Hmpo/2 and the wave steepness
before modulation, aOkO, are shown, based on labo-
ratory measurements, Su and Green (1985), and sim-
ulations, Wang et al. (19941. Under realistic condi-
tions (ka ~ 0 12), the wave height at the peak of the
envelope is seen to be almost double its initial value.
This alone, points up the potential importance of wave
groups for us. Connected with this fact, there is ample
evidence connecting breaking with wave groups, and it
is during breaking that heightened crests and steepened
wave faces occur. We discuss these subsequently.
(1.3 _
. ~ 0. ~ /,
Error :Bar ,~
MAX. Waive In ru;S ;~x~ eta 'l963~~i }
(}reallying in LIT, iierie5 ~
Breaking in LONOTANK, Series l
Breakj~2 ire t.~CTANK, Series 4-- .'
. .:t ._
__t . , . __ ~
0. ~ 9.~; (3.3
Figure 7: The wave height at breaking Hmk~o/2 vs. the
initial wave steepness aOkO, from tank experiments, Su
and Green (1985), and simulations by LONGTANK.
Surface wave kinematics approaching breaking and
during the breaking process change dramatically from
those we expect based on quasi-sinusoidal wave forms.
First of all, the almost circular motion of individual sur-
face particles becomes disrupted, resulting in a much
larger down-wave drift than Stokes, which varies as
(ak)2. This effect, in fact, occurs before breaking ac-
cording to LONGTANK simulations, see some examples
in Figure 8. Then, as the face progressively steepens
in time, see Figure 9, the motion of a large part of it
becomes directed toward the normal to the surface, and
the particles are accelerated down tank, see Figure 10.
0,65 I........
onto ~
~ o.~5 ~
onto ~
o.~6 ~
~ . ~
of
-~.5
Figure 8: Top: Particle trajectories, non-breaking mod-
ulating wave, LONGTANK. The dots are separated by
constant intervals in time. Note occasional large ex-
cursions, when group peaks have passed by. Bottom:
Horizontal, u, vs. vertical, v, velocity for a particle,
which eventually made a large excursion.
As a result of both increase in H and the rise of
the crest, the amplitude of the crest is much increased,
see Figure 9. Of course, this crest rise is very impor-
tant in the estimation of platform requirements. A crest
rise factor, AmaX/H' in the range 0 74 to 0 80 has been
found in LONGTANK simulations. This is larger than
the value of 0 6 suggested for engineering applications,
Stansberg (1991), but close to the value 0 77 quoted by
Kjeldsen (1990) for breaking waves in his tank experi-
ments.
These deforming waves found in LONGTANK
simulations have also been observed in tank measure-
OCR for page 719
omit
If
;~ m ~ ~ ~ ~ r ~ ~ · I i
0.05: ~
CASE 2. 5S
2342 22.4 2Z+6 Z£.8 33.D
Figure 9: Wave Deformation from Inception (left) until Jet Formation (right), LONGTANK.
meets at the OEL. A set of wave profiles measured with
a 14 wire linear array in the large UCSB wave tank is
shown in Figure 11 compared with a simulation; the
wave length is 2 3 m and the modulating waves were
measured at about 100' from the wavemaker.
0.05 _
A_
_
Q.04
._
0.03
4.3 1 4.33 4.33 4.34 4.35
x (m)
Figure 10: The velocity field within a breaking wave
with jet, LONGTANK.
~ ~ ~ .. _ ._ . ... ..... . ........ .
C., ~
C- _ ~ , ~
a. ~ , . _ a ..... ... i . _. ...... . . ... .
Figure 11: A comparison of breaking wave profiles
measured in the large OEL wind-wave tank (dots) with
LONGTANK simulations (lines).
The connection between asymmetric deforma-
tion (front face steepening) and the breaking of extreme
waves at sea was first made by Kjeldsen and Myrhaug
(1980), based on observations. They claimed that ex-
treme waves are accompanied by a shallow trough and
a steepened wave face; they defined a deformation pa-
rameter, see Figure 12, and a critical minimum value for
breaking. The LONGTANK simulation of wave breaking
in wave groups and the OEL measurements clearly con-
firm the K-M ocean observations, and partially confirm
the value of their deformation parameter see Wang et
al. (19941; in the light of the breaking criterion found in
LONGTANK, the deformation has to be viewed as a con-
sequence of the breaking process and not the cause of
it; and the K-M deformation parameter serves to quan-
tify the deformation just before jet formation, and not
as a criterion for breaking.
= ~ _
Figure 12: Extreme wave, deformation sketch af-
ter Kjeldsen and Myrhaug (19801. For breaking:
f0'/H]~/A'] > 3 5(K — M), and 3 48 - 5 29
(LONGTANK)
2.4 Inception and Mechanism of Breaking
We now have a clear understanding of the exceptional
role played by wave modulation in the initiation of
wave deformation and breaking, and furthermore we
possess a simple criterion for the initiation of this pro-
cess as the wave passes through the modulation peak.
In Section 3 of this paper we present theory which ex-
plains the mechanism associated with this criterion, and
also its experimental verification through surface parti-
cle tracking experiments.
As we have noted, wave modulation in the
ocean is a result of wave instability of the Benjamin-
Feir type, and this newly described mechanism leads to
breaking at group-averaged steepnesses as low as about
0 1. This explains those field observation, Holthuijsen
and Herbers (1986), Weissman et al. (1984), which
came to our attention at the beginning of our work, and
OCR for page 720
0.10
~ -
0.05
3~00
t ~ 1. ~ 74T
, << t- ~~194T
~ Ob ~ ~ ~ ~ ~ ~~ ;. ilk ~ =~; ~ ~ ~ ~;J , - , .i, ., ,,, ~ ,, ,,~ *; ~ ;,1 , ~~. ~~
of o.4 0~6 off t.o
x/A
O ~O . . .
Owe
e.o~
e.oz
Figure 13: Wave breaking in LONGTANK, hobo
formation of a plunging jet.
~ ~ 1.] - I.19 ~
OHS O^~O
;~iA
O
= 0 28. (a): waveforms in the initial stage of breaking. (b): the
OCR for page 721
-T ~ 1-
Z.Q _ ~
, Threshold line in~dicat~ng inception of bresk~rlg
j x o ~ hchteved: 1~l n,ax~mum ~ decreasir~g
L x + ~ Tending to break inevitably in ~ quarter of period I
_
.5
3
I_
Sc'
L~ o ~ .........
Q.ZQ
~-
LI L wave-w~e interacdon
I -- wave g~oups~ shallow water
wave gems, deep water
~ ~ ~ ~ ~ . a.
.... ..... . ~ ., . . *. . . . . .. . ... . .. ~ .. .. . . .. ~ . . ~ ... .. .
.. ....
~ 0 0
0 0
x~,+x ~ x~x~ ~ * x+ ,, x,,]
~ ~ ~~ x
0.25
arks
Figure 14: The ratio (crest particle velocity)/(wave group velocity), ~c/Cg' vs. aOkO The horizontal line demarcates
waves which did not proceed to breaking from those which did. The value of oc is determined just prior to wave
deformation and jet formation in the case of the breaking waves.
for which no explanation could be found in either lab-
oratory derived breaking measurements, Ochi and Tsai
(19831; Ramberg and Griffin (1987), or in prevailing
theory, Longuet-Higgins (1969, etch. These aforemen-
tioned field measurements of wave breaking at rela-
tively low values of wave energy, together with field
observations of wave breaking in wave groups, Donelan
et al. (1972), and the field observations and classifica-
tion of wave deformation associated with large waves,
Kjeldsen and Myrhaug (1980), all provided those clues
which we can easily read today. Indeed, the obser-
vation of breaking in wave groups has first led us in
the late 1980's to emphasize the phenomena of wave
instability and group formation. Analytical studies of
these processes, Tulin and Li (1992), convinced us that
the actual process of wave deformation and breaking
were probably inaccessible to treatment via the usual
weakly nonlinear approaches, perturbation expansion
theory for instance. This led us to the development of
the high resolution numerical wave tank, LONGTANK,
and to numerical experiments of wave group formation
leading to breaking, wave deformation and of the in-
evitable formation of the jet and of its growth and de-
scent in a ballistic trajectory, Figure 13. Most remark-
ably, a simple criterion for the initiation of deformation
was found from the LONGTANK studies:
"that upon passing through the peak of a modulation
group, when the orbital velocity at the wave crest, tic,
exceeds the wave group velocity dw/dk = cg, then the
wave crest and trough both rise, the frontface steepens,
the wave crest sharpens, and eventually a jet forms at
the crest, leadingfinally to splashing and a breakdown
of the wave."
This criterion was found to apply not only for
deep water waves where dc~/dk ~ c/2 where c is the
wave celerity, but for modulating waves in shallow wa-
ter too, where dc~/dk is significantly smaller, Figure 14.
In the comparisons shown there, the Stokes second-
order dispersion relation was used to calculate cg.
Theoretical understanding has been elusive,
but we have finally succeeded in demonstrating that the
usual almost sinusoidal motion of propagating waves
fails for modulating waves when in deep water the
aforementioned criterion, oc = cg, is reached.
Recently we have measured the orbital veloc-
ity of waves in the presence of mechanically generated
wave groups, and confirmed this criterion experimen-
tally as described in Section 3.
The Effects of Breaking on Wave Evolution
We know now that the application of wave theories to-
ward understanding of real ocean waves requires the in-
troduction of real effects like wind pumping and dissi-
OCR for page 722
pation. After all, the waves only exist because of the
wind, and the most systematic measurements of wave
growth (fetch limited growth) reveal a scaling of the
fetch laws with the wind speed. We know, too, that
for moderate to equilibrium wave systems the breaking
dissipation is of the same order as the wind pumping.
We have ourselves tried to understand and
quantify the role of real effects in the dynamic evolu-
tion of waves. Our first attempts, described in (A) be-
low involved the use of the modified complex evolution
equation (NLS), where we have introduced the break-
ing dissipation as a right-hand forcing term, as others
before us, Trulsen and Dysthe (19901; Hara and Mel
(19911. Unlike most previous work, however, we have
introduced a dissipation dependent upon (Ica)4; we had
discovered that only for this particular power law de-
pendency on steepness would the breaking dissipation
balance the wind input for waves in equilibrium, pro-
vided that Toba's law of individual waves is invoked.
Later we have confirmed that this law is consistent with
the observed fetch laws.
In the mid 90's we have realized the great
importance of momentum considerations in addition
to energy. In particular, we discovered that the dual
conservation laws for energy and momentum can be
combined to produce a law for wave speed evolution,
and can quantify downshifting, Tulin (19961. This is
briefly described in (B) below.
(A) The NLS allows the calculation of possible shapes
of the equilibrium modulational envelope; however,
without consideration of wind pumping and dissipation,
the envelope of groups which actually arise in nature
cannot be predicted. Therefore, Li and Tulin (1993),
used this model equation including real effects to pre-
dict envelope shapes in general. A major result of this
study was a prediction of the number of waves in a spa-
tial group as a function of the ratio of maximum and
minimum wave steepness, in the group. For values
of this ratio between about 0 3 and 0 08, the number
of waves varies between about 2 and 7. These results
are consistent with the observations of Holthuijsen and
Herbers (19861. Note that the number of waves in the
temporal group would be twice the quoted numbers.
Later, Li and Tulin (1994) thoroughly considered the
long time evolution of sideband systems with wind and
dissipation, using their modified NLS and dynamic sys-
tems analysis; they discussed the strange attractors and
possibilities of chaotic behavior.
(B) As breaking proceeds, water flows into the plunging
jet, carrying with it, both kinetic and potential energy
E and momentum q. Upon splashing, these are lost to
the organized wave motion and these losses must be ac-
counted for in the further evolution of the wave. This
requires conservation laws for both wave energy and
momentum, which may be constructed on the basis of
weak non-linearity. We have first done this on a heuris-
tic basis, Tulin (1996), and furthermore shown that a set
of evolution equations for E-c correspond to the orig-
inal, E-q pair. More recently we have derived these
same conservation laws from variational considerations
including a rigorous inclusion of breaking in the vari-
ational formulation; this was done by introducing, in
addition to the usual wave Lagrangian, a work func-
tion representing the effect of breaking and wind input
on the flow dynamics, and by a rigorous definition of
this function, Tulin and Li (19991. In addition, we have
shown that our set of conservation laws for energy and
wave speed can be combined into the complex Landau-
Ginzberg or NLS equation, giving the latter new mean-
ing, and extending it to include the effect of breaking in
a rigorous way. Here is the newly modified NLS, valid
for all (X,T):
AT + COAX + i(C0/4~)AXX + i~2~A~2A =
A [ ~ A ~ ~ - i4-y / ~ AN dX]
(1)
where (X,T? are the long time and space scales,
A = aeon, where ~ is the wave phase, e,..
and Db are the rate of the wind pumping and
breaking dissipation, respectively; e,,, ~ grad,
Db ~ 9 ~ A ~ 4 where the constants of proportionality fol-
low from the field observations; by = (~1) > 0.
A consequence of the law for the wave celerity
is that breaking leads to a continual increase in the wave
speed (frequency downshifting) at a rate controlled by
the breaking process. The way in which this works
was studied in our large wind-wave tank through labo-
ratory observations of wave group evolution, including
breaking effects, Tulin and Waseda (19991. A crucial
aspect is the cooperation of breaking dissipation and
momentum loss acting together with near-neighbor en-
ergy transfer in the discretized spectrum, this transfer
being due to detuned resonance acting over a limited
time (less than 50 wave periods). This is a very dif-
ferent point of view than is commonly incorporated in
large-scale wave prediction modeling, where the con-
servation of wave momentum is ignored.
OCR for page 723
2.6 On the Modeling of Splashing, and its Conse-
quences
We are indebted to the early, careful, experimental ob-
servations of Bonmarin (1989) for some knowledge of
the splashing process, which follows wave breaking.
These depended heavily on photographic visualization.
Recently we have undertaken detailed studies of the en-
tire splashing process and its mechanics, making use of
a high-resolution numerical simulation, SPlasH. This is
described in Section 4. The simulations reveal the pres-
ence of the partial forward ricochet of the plunging jet,
the collapse of the cavity under the jet, and a backward
counterjet created in reaction, and of the eventual for-
mation of a dipole structure with downward momentum
behind the splashing breaker. These all seem consistent
with Bonmarin's observations. We have systematically
studied the special case of the propagating bore in shal-
low water, as this has been much studied in the past, and
therefore offers an opportunity for validation of the sim-
ulation. We have found the simulated propagation of
the bore in excellent agreement with predictions based
on mass and momentum conservation. The simulation
method allows for arbitrary variations in the fluid den-
sity, and we intend to extend the present simulations to
include the presence of entrained air in the large scale
flow structures and to the eventual fate of this air.
2.7 Surface Tension Effects; Microbreakers
The local curvatures in the wave geometry are largest
at the crest of a deformed wave in its later prejet stage
and in the jet itself, whose thickness may only be a few
percent of the wave amplitude. We have carried out
LONGTANK simulations of modulating and breaking
waves, including the effect of surface tension, for waves
of varying length down to 25 cm. Scaling shows that
the effect of surface tension increases with reduction
in the length of the wave. For those waves, simulations
showed that surface tension does not effect the breaking
criterion and has insignificant effect on the wave defor-
mation except in reducing somewhat the largest curva-
tures at the crest of the wave prior to breaking. Surface
tension (or the length scale) does, however, have a pro-
nounced effect on the jetting process for waves shorter
than about 2 m in length. For 1 m waves the jet is con-
siderably rounded and weakened, which explains both
visual and radar observations of breaking waves in the
OEL large wind-wave tank, where much less energetic
splashing is observed for waves 1 m and shorter. For
waves between about 25 and 75 cm the jet does not ap-
pear at all, to be replaced by a forward facing bulge
growing out of the wave crest, Figure 15.
Figure 15: The effect of surface tension on the jet
development. Wave profiles were computed with
LONGTANK. Microbreakers have wavelength below lm
and surface tension reduces the jet to a bulge.
Nonjetting waves of this kind have been
called micro-breakers. Unlike energetic breakers where
the shape continually evolves in transient fashion, con-
tinually changing over a time interval of about one wave
period, the microbreaker can propagate for a consider-
able distance without significant change of shape. This
can be seen in radar tracks in range-time diagrams,
since the bulges on the microbreaker crests provide a
relatively strong specular radar return, Fuchs and Tulin
(2000), Figure 16. A striking feature of the micro-
breaker bulge is the appearance at the foot of the bulge
of a small region of very high curvature, see Figure 15,
It is well known that vorticity is generated at a free sur-
face with curvature, and there is ample evidence, Dun-
can (1994, 1996), Dommermuth and Mui (1996), that
a strong effect of this kind does originate at the foot
of the bulge, causing dissipation and probably resulting
eventually, for very short waves (sub-microbreakers),
in the separation of the bulge from the main wave flow
through a strong shear layer, forming a cap on top of the
wave, as noted by Ebuchi et al. (19871. This small scale
flow then bears a family relation to the steady hydrofoil
breaker, Figure 1, differing in its more or less symmet-
ric placement over the wave crest. Ebuchi and Toba's
waves were wind generated, and it may very well be
that the wind stress plays an important role in allowing
the propagation of this unusual wave.
The microbreaker itself is the consequence of
OCR for page 735
It can be seen, that a portion of the particles
originally in the first plunger are now riding on the
moving front of the propagating wave, while the oth-
ers are captured by the cavities and effectively mixed
with other portions of fluid. In following cycles, all the
tracked particles will be captured by the rotating struc-
tures and fresher particles will feed the bore front.
A more quantitative analysis is obtained by the
two bottom plots, where the components of the velocity
of the center of mass of the jet-particles
Vc=/ pvdA// pdA (18)
jet jet
are plotted in time. The free fall stage in the first part
of the jet-evolution can be clearly seen in the right plot,
and accompanied by an almost steady translation in the
horizontal direction (c; left plot). The instant of the
impact is evidenced by the sharp changes of Vc,~, Veal
The first drops down to less than 25% of its initial
value: this corresponds to the splitting of the jet into
two portions with backward and forward motion, bal-
ancing each other and thus reducing Vc,~. The vertical
component bounces up, changing in sign, because both
portions are moving upward, and then oscillates consis-
tently with the orbital motion of the particles entrapped
in the cavities. Eventually, Veal attains an almost zero
value, with small amplitude oscillations related to the
orbital motion inside the vertical structures rotating in
an uncorrelated way. After some oscillations with de-
creasing amplitude, due both to the orbital motion and
to the multiple splash-up, the horizontal component at-
tains the uniform velocity of the moving piston.
We have analyzed several cases, in the range
U/~ ~ ~ 0 5, 0 95 ]. For larger U. a more vigorous
formation of the backward plunging jet and MODE B
interaction has been observed, with initial formation of
stronger dipole-like structures, followed by a predom-
inance of MODE A interaction with vertical regions
more confined to the free surface. A slower motion
of the piston leads to less pronounced splash-up cy-
cles, producing weaker vertical structures of opposite
sign, mainly due to MODE A interaction. In the present
simulations, for U/~<0 6, dipole-like structures
are never observed. In all cases, MODE B appears con-
nected with the initial breaking events, when the most
energetic plungers are generated.
Plots in Figure 27 give an indication of this
transformation by comparing some global characteris-
tics of the plunging jet and those of the correspond-
ing splash-up, for two values of the piston velocity
U = A. Many parameters are involved and we sim-
.~
of
o...~. ::
o..:t
it:
1
~'31
to..
o.~:.[ .
.........................
~ x.-; ...
As...........
OT.~..~S if,
if, ~ ~ ~ ~ x ~ ~ ~ ~,,~. ~.~::1.
...... ,
x amp. .~ x 0 ::
· . ... 'I. :2.2: . :x.'.
. -.-- - - - - - ~
go. ~ ~ ~ .. A:.
Figure 27: The first plunging event and the corre-
sponding splash-up for U/~ = 0 95 (top) and
U/~ = 0 7 (bottom). The configuration of the
splash-up is reported when the upward-raised water
ply report some of them, just to show the complexity of
the underlying dynamics involved. All the quantities
are referred to a unit slice of fluid in the direction or-
thogonal to the plane of motion. The mass in the jet at
the touch down is 2 55 kg and 0 618 kg for the two
cases, respectively, and the corresponding cavity en-
trapped is 45 4 cm2 and 13 9 cm2. Despite the rather
large differences in jet masses, the masses involved in
the splash-ups are comparable: 4 51 kg and 3 45 kg.
This is probably due to the difference in the impacting
angle (21.1° vs. 18 6°), implying different depths of
penetration of the jet. It is remarkable that the mass of
OCR for page 736
the splash-up is greater than the original jet which gave
rise to it.
The entire history of the bore from initiation of
the motion involves the pile up of the water at the pis-
ton, the propagation of the wave, overturning and ini-
tial splashing followed by a series of splashes. Finally
the splashes ceases and a surfing eddy remains on the
face of the equilibrium bore, analogous to the breaker
above a hydrofoil, Figure 1; this was observed in our
simulations and in experiments reported by Peregrine
(19831. In this condition, the velocity of the breaking
front attains a constant value: i.e., in the frame of refer-
ence of the front, the velocities far upstream and suffi-
ciently downstream are constant.
For hydraulic jumps, by using mass and mo-
mentum conservation principles, we can relate the di-
mensionless velocity of the bore
Fo = (~) = 2 (ho) ( + ho) (19)
to the upstream, ho, and downstream, hi, water lev-
els. In Table 1, data from SPlasH simulations are col-
U/ N/4 h ~
0-95 0-212
0-9 0-21
0-S 0-19
0-7 0-~S
0-6 0-17
0-5 0-15
hi /ho Fo
2-12 1-82
2-1 1-8
1-9 1-66
1-8 1-59
1-7 1-52
1-5 1-37
Lath
1-80
1-78
1-64
1-57
1-50
1-36
SPIasH
1-81
1-78
1-63
1-57
1-47
1-38
Table 1: Comparison between the computed velocity,
~SPlasH, of the bore front and the theoretical value, Hth,
predicted by global mass and momentum conservation
principles, ci Equation (191. Velocities are expressed
in m/s.
lected and compared with the theoretical prediction.
The agreement is generally within 1%, and 2% at max-
imum, quite reasonable for such complex, non-steady
dissipating flows.
The result strongly suggests that inertial (Eu-
ler) effects may dominate the observable phenomena,
including strong mixing of surface water when passing
through the bore.
4.3 Breaking Bow Waves
Breaking ship waves have always captured the inter-
est of hydrodynamicists and naval architects because of
their role in contributing to the resistance of the hull.
More recent, however, is the interest of the Navy in the
very long narrow wakes behind ships which can be ob-
served remotely, and which originate abeam the ship
through extensive breaking of diverging bow and stern
waves.
The following analysis of this problem is lim-
ited to fine ships with a sharp stem, for which fun-
damental understanding can be gained by an approxi-
mate quasi-three dimensional model based on the idea
that longitudinal gradients of relevant flow quantities
are small compared with vertical and transverse gradi-
ents. The introduction of this seminal idea is not new,
and a historical recollection of slender-body theory for
ship hydrodynamics is given by Maruo (1989) and by
Fontaine and Tulin (19981.
More specifically relevant to our purposes,
nonlinear versions of this parabolized model for ship
flows have been presented by Faltinsen and Zhao
(1991), Maruo and Song (1994) and Tulin and Wu
(19961. In the last two, the calculations are carried
out in two dimensions, vertical and transverse, and suc-
cessively in time and the method has been consistently
named 2D+t.
Tulin and Wu (1994) presented a thorough
analysis of the genesis of diverging bow waves, and
a detailed parameter investigation is reported in Wu
(19971. In particular, the method exhibited the advan-
tage of high resolution, sufficient to capture breaking,
and even to trace the jet overturning.
An example of this ability is presented in Fig-
ure 28, where free surface profiles for successive time-
steps are superimposed, showing one of the key fea-
tures of the slender-hull bow wave system: the splash,
Tulin and Wu (19941. For this ship (a Wigley hull),
the free-surface flow is not much decelerated before
the stem, but upon reaching it, is deviated sharply up-
wards, rises on and eventually levels off and falls down.
An entire thin sheet is formed in this process and ap-
pears as a splash on either side of the hull. The re-
laxation of these splashes is the prime source of di-
vergent waves. In the present case, the large beam-
to-length ratio, B/L = 0 2, makes the radiated waves
large enough to break, following an evolution similar
in many ways to that described in Sections 2-3, with
crest-rising, front-steepening and jet formation.
Some genuine three-dimensional effects, such
as upstream influence and breaking before the bow, be-
come increasingly important as the ship becomes full
OCR for page 737
0.1
oboe
0.06
0.04
0.02
o
-0.02
~ /L
1
at
Fr=0.4
-0.04 ~ 0 1 0 2 0 3 X / L 0 4
0.1
0.08
0.06
0.04
0.02
o
-0.02
it_
1
Fr=0.5
-0.04 ~ 0 1 0 2 0 3 X / L 0 4
~ ~~-~ = #~ = #
F[oudc numDcr F[= /.
OCR for page 738
-0.4 ~ ~ .2—
0.3
0.4
0.05
O
-0.05
-0.1
Figure 29: Perspective view (from upstream) of the wave pattern generated by a Wigley hull (B/L = 0 1, D/L = 0 ~
Fr = 0 461. BEM computation (blue lines) starts from the bow, up to C/L = 0. Black solid lines are made by the.
uppermost layer of fluid particles from SPlasH computations, initialized by BEM.
and slow and are lost to this parabolized theory.
In the following we apply a version of SPlasH
to study the splashing of the forward wave generated by
a Wigley hull. The computation is initiated by a 2D+t
code based on BEM, and continues up to the detection
of jet overturning. Once this is accomplished, a sub-
domain, surrounding the breaking area, is defined and
filled in with particles. The initial velocity and pressure
of the particles is determined by BEM but, so far, we
have not fully coupled the two algorithms and, on the
outer SPH boundary, stationary conditions have been
imposed during all the following evolution. Therefore,
preliminary sensitivity tests have been conducted to de-
termine the vertical and horizontal dimensions of the
SPH domain to assure the invariance of the results dur-
ing the entire simulation time.
A perspective view of this compound simula-
tion is presented in Figure 29. The view point is lo-
cated upstream the bow, and y—z configurations for
increasing time are plotted in a 3D fashion by using
C/L = Ut/L as longitudinal coordinate, with the ori-
gin located midship. Ship cross sections (red lines)
are plotted only for those slices obtained by BEM, and
the corresponding free surfaces are represented by blue
lines. For this Froude number, Fr=0 46, we observe
a substantial splash at the bow, with height compara-
ble with the draft and maximum located around 25%
L from the stem line. The gravitational collapse of the
splash is accompanied by the radiation of a wave, which
steepens and forms a plunging jet, almost touching the
free surface for C/L = 0.
From this configuration on, the SPH computa-
tion has been started and the resulting free-surface pat-
tern is reported in the same figure by black lines. Only
the layer of particles at the free surface is plotted and
its horizontal extent corresponds to the actual computa-
tional domain used by SPlasH.
A more detailed analysis is given in Figure 30,
where an enlarged view of the wave pattern during the
splash (top) is presented together with some selected
particle distributions (bottom). Upon adopting a stern
viewpoint, we observe that the breaking crest is propa-
gating with an almost unchanged phase speed, though
the simulation time is rather short to be conclusive and
OCR for page 739
n-1
n
-0.05
_
I. -
~A'9
MA
,70.2
a
-
Figure 30: Breaking of the bow wave generated by a Wigley hull (B/L = 0 1, D/L = 0 1, Fr = 0 461. Details
from Figure 29. Only the uppermost layer of particles from SPlasH computations is plotted. Red colored sections are
reported in Figure 31.
.~1 i1
~ In-
,.1 .~'~
-.l,(M ~
i:
0_~._O
O.Ci4 ~
.':~.O'' ~
Ci-= ~
001[
t-.,~L
. . . . . . . . . . . . .
;.....i....i..........i....i....~.....i....i..........i....~..........i....i..........i....i..........i....i....~.....i....i....~.....i....i....~.....i....i....~.....i....i.....~....i....i....1....i...'
hi ~ 01.~.: 0.~4 0.~ Hi ~ 014 ~ 4~ ~ ~
Ci I
....
.
. i i i i ~ i i i ~ ~ i i ~ i i ~ i i ~ i '
O~3 Pi 3~ D 34 ~ Id, (1.:3S
.i ~ i l ~ i ~ ~ i ~ '
;-14 04 044
Figure 31: Particle distributions from Figure 30 showing splashing and formation of vertical regions.
OCR for page 740
the analysis hampered by the spreading of particles
around the crest emerging from the breaker.
More clear is the presence of a slower trace,
originating from the splash-up, and left on the free sur-
face behind the breaking crest. The analysis of parti-
cle distributions, colored according to the vertical dis-
tance from the highest point, reveals its origin. The im-
pacting jet, plot A, causes splash-up and cavity forma-
tion, plot B. A backward-facing jet is created, with a
MODE B interaction, plot C, and eventually resulting
in a dipole structure left behind the propagating crest,
plot D through E. This dipole is weaker with respect to
those observed for breaking bores with large U. Fur-
thermore, the overall downward-backward motion of
the fluid past the crest corresponds to the possible self-
convection of the dipole, which remains close to the
free surface. In any case, because of its vertical nature,
the dipole structure does not follow the breaking crest,
leaving its own signature on the free surface.
An analysis of the motion of the particles ini-
tially forming the plunging jet shows strong similarities
with that presented in Figure 26, for the breaking bore.
Also in this case, the jet flows in a backward stream,
captured in the nascent cavity, and another one feeding
the splash-up. It is worth stressing, in this case, that
the fluid particles emerging forward from the splash-up
in plots B-C, are then "surfing" the wave crest after the
splashing event, plots D-F, thus resembling the weak
eddy steadily moving with the hydrofoil breaking wave,
as discussed at the beginning of this paper.
We found a striking similarity between the
present results and the experimental observations by
Lamarre, reported in Melville (1996, Figure 21. Clearly,
on a longer time scale, the degassing stage will be pro-
foundly effected by the two-phase nature of the bubbly
flow created during the breaking.
For bow-flared ships and practical Froude
numbers, wave breaking takes place much closer to, or
at the bow. This has been shown in Tulin and Wu (1996,
Figure 13), where the first stage of the gravitational col-
lapse of the splash at the bow, with formation of a strong
plunger, has been computed for a frigate ship.
Although the application of the SPH technique
to study these flow conditions requires some improve-
ments (e.g. modeling of curved boundaries, and match-
ing with an outer solution for long-time simulations),
we can already get some basic insights by considering
the flow forced by an inclined piston. In particular, we
Fr = (:).4
0.2~
it
-0.1
0.2
0.1
n2
0.1
n
-n4 -n? -n2 -a 7 n n 7 n2 n? n4 n.S
Figure 32: Top: sketch of the model problem adopted
to study a breaking splash. Bottom plots: evolution of
the splash breaking near the piston. Jet particles are
marked in black to put in evidence the ricochet.
selected an angle of 66 8° by taking the slope at the wa-
terline of the DTMB Model 5415 (ci top plot in Fig-
ure 32) in the bow region. In our simulations we used
a constant horizontal velocity, U/~/~ = 1 0. For the
actual ship and Fr=0 41, the expansion velocity of bow
cross sections is even higher but drops down sharply. In
the bottom plots of Figure 32, we observe a quick pil-
ing up of the water against the piston, with a maximum
run-up of about 2ho, and the formation of a rather thick
jet, eventually collapsing down in the form of a plunger.
In contrast to previous results, the impacting fluid is not
creating a crater, even though a portion of the impacting
jet is still deflected inside the nascent cavity. The evo-
lution on a longer time scale is significantly effected by
finite depth, and is less relevant to the ship problem.
OCR for page 741
Much remains to be done on ship breakers but
the present results are very encouraging.
ACKNOWLEDGEMENTS
The ONR sponsors have been gratefully acknowledged
in the Forward. I would mention especially our con-
structive relationship with Dr. Ed Rood, who provided
the incentive for the present splashing studies.
Among the many OEL researchers who have
contributed to our understanding of breaking waves I
would specifically mention four of those who not only
received their PhD's at UCSB, but stayed on as Post
Graduate Researchers: Pei Wang (Mrs. Y.T. Yao), who
conceived of the special domain decomposition tech-
nique and implemented it for the high resolution simu-
lation of long wave trains and the breaking process- she
is the Mother of LONGTANK; Yi Tao Yao, who utilized
LONGTANK So brilliantly in studies of wave breaking,
who discovered the breaking criterion, simulated sur-
face tension effects, and who collaborated in a host of
studies of ocean wave behavior; J.J. Li, who made many
deep and scholarly mathematical studies of wave dy-
namical behavior in connection with ocean wave mod-
eling; Takuji Waseda, who developed much of our ex-
perimental systems for wind wave studies, pioneered in
the development of our experimental wave group capa-
bility, and carried out a series of profound experimen-
tal studies of wave breaking; Ming Wu, who continued
the 2D + t simulations of ship bow waves initiated by
Maruo and Song, and used them intensively and fruit-
fully in systematic studies of bow waves and of deck
wetness.
I am very grateful to my old colleague, Profes-
sor Hajimi Maruo, of Yokahama, for spending two pro-
ductive years at the OEL, during which time the 2D + t
simulations were begun. I must also thank Dr. Em-
manuel Fontaine, now of IFP in Paris, for his vital role
while a visiting researcher at the OEL in initiating our
work on gridless Euler simulations, and his early explo-
rations and use of the SPH method in splashing studies.
I thank my colleague and co-author Maurizio
Landrini for agreeing to the present collaboration which
brings him from Rome to the OEL for six months a
year, together with Andrea Colagrossi of Rome, who
has participated in many of the SPlasH computations.
The successful collaboration was made possible only
through the agreement of Admiral U. Grazioli, the Pres-
ident of INSEAN, and of Dr. U. Bulgarelli, the Scien-
tific Director, to whom we are grateful. Finally, many
thanks to the tireless staff of the OEL, who have helped
in the preparation of this paper, and especially Ms. San-
dra Jeppesen, and Mr. Daniel Matsiev.
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DISCUSSION
B Beck
University of Michig m, USA
Looking et your figures, it Up 915 that m the
2D+t computations, the diverging wave crest is c
straight line he figure using SPH in the cross
flow plane seems to indicate She breaking wave
front slows down so that the diverging wave
crest is parabolic Could you please comment on
this?
AUTHOR'S REPLY
In Figures 3 0 Ed 31, the growing separation of
the dipole vorticcl sh ucture Ed the cresting
breaker can he een h --. .nxal . tact re is
slow r Ed would appear to follow c roughly
parabolic hack, although its asymptotic motion
carmotyetbek ow fromfhelimited
calculations which w have done he crest of
the divergent wave is 19 ter Ed its shucture
spreads with time his conesponds physically
to the growth in extent of the broken water just
behind She crest Ed which would appear es c
foam scar in She oce m in front of this scar is She
small so fing breaker his marks the front of She
broken wave A line d awn th ough She center of
the surfing eddies back to the touchdow of the
original jet would appear to be recsorurbly
straight Agam, further calculations in time must
be mad to under t Ed She fate of the cm bng
eddy it would seem re tSornble to expect that it
must eventually disappear
Representative terms from entire chapter:
breaking waves
