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OCR for page 126
Basic Studies of Water on Deck
M. Wreck, O.M. Faltinseni, M. Lancirini2
Department of Marine Hydrodynamics - NTNU, Trondheim - Norway.
2INSEAN, The Italian Ship Mode! Basin, Roma - Italy.
ABSTRACT
Extreme wave-body interactions may cause shipping and
flowing of water on the main deck of ships (water on
deck). In this paper, the role taken by some of the main
geometric and kinematic parameters involved in the wa-
ter on deck is carried out by using an approximate hy-
drodynamic model. In particular, the unsteady interac-
tion between free surface and ship is analyzed by solv-
ing the inviscid two-dimensional fully nonlinear prob-
lem numerically. Both water on deck resembling dam
breaking as well as due to plunging waves are investi-
gated.
INTRODUCTION
In rough sea conditions, both moored vessels (such as a
Floating Production Storage and Offloading Unit, FPSO)
and ships in transit can suffer shipping of water on the
deck. When a sufficient amount of water comes onto the
deck, a flow with increasing velocity develops, possibly
hitting obstacles on its way. Water impacting against the
deck and superstructures may cause both high pressures
in confined regions and contribute to global ship loads.
Localized structural damages as well as excitation of
global response of the ship are expected. The impor-
tance of hydroelasticity must then be assessed. More-
over, the fluid motion onto the deck may affect roll sta-
bilitv of smaller shins and cause capsizing.
Different incidents occurred in the latest years to
FPSO units that motivated experimental investigations
and suggested some modifications of design rules. An
overview of the most important ones and of the subse-
quent requirements of the Norwegian Petroleum Direc-
torate are given by Ersdal & Kvitrud (20001.
However, the numerous physical aspects deter-
mining the phenomenon make it difficult to clearly iden-
tify the design parameters relevant for the occurrence
and severity of water on deck, and for its consequences
to the ship. The effect of geometric parameters charac-
terizing a ship bow is far from being clarified. Some-
times, it is not even clear whether they enhance or re-
duce the deck wetness. As an example, O'Dea & Walden
(1984, experiments in regular waves) reported that a
larger bow flare angle reduces the deck wetness, while
Lloyd et al (1985, experiments in irregular waves) ob-
served more frequent freeboard exceedances and deck
wetness for more heavily flared bows. On this ground,
fundamental investigations are necessary to improve this
lack of knowledge and to develop numerical tools of
practical use.
The conventional way of estimating water on deck
is to combine a probabilistic analysis (Ochi 1964) with a
linear hydrodynamic analysis. It implies that the above
water hull form is not included in the hydrodynamic
analysis. The important hydrodynamic variable is the
linear relative vertical motion between the ship and the
water. Often only the incident wave and not the local
wave accounting for the presence of the ship is used in
this context. An effective freeboard is sometimes in-
troduced for a ship with forward speed. This accounts
empirically for the steady wave profile and the sinkage
of the ship.
Details of the flow near the ship require a local
quantitative analysis of the specific conditions of inter-
est. Maruo & Song (1994) studied the shipping of the
water for high speed vessels by using a 2 --D Slender
Body Theory. This may also have relevance for slender
ship bows at moderate forward speed. Buchner & Co-
zijn (1997) analyzed the bow deck wetness for moored
ships, assuming a two-dimensional problem in the lon-
gitudinal ship direction. They presented both numeri-
cal simulations and experiments for a simple prototype
problem but no comparison between simulations and
measurements was presented.
In head sea conditions, the most severe water on
OCR for page 127
deck events are concentrated in the bow region. In these
cases, after the water exceeds the freeboard, Buchner
(1995) observed a marked similarity between the flow
of the shipped water and the one generated after the
breaking of a dam. Consistently, some authors stud-
ied the motion of the shipped water along the deck by
shallow water models. The reliability of this type of
approach is dependent on how the initial conditions as
well as the inflow boundary conditions are determined.
A sensitivity analysis in terms of the inflow velocities
has been carried out by Mizogushi (1989) by comparing
numerical results and experimental data for the S-175
container ship. The shallow water equations have been
solved for the three-dimensional problem, using exper-
imental results for the water height at the inflow bound-
ary. In the simulation, the ship motion is not taken into
account. The author concluded that the flow interac-
tions and the efflux occurring between the deck area and
the outer region represent important items in the water
on deck phenomena.
In this paper an attempt is made to study the fun-
damental aspects of deck wetness at the bow region of
a FPSO unit in head sea conditions. It implies that for-
ward speed effects are not investigated. The problem
is idealized by considering the two-dimensional flow in
the longitudinal plane of the ship and solving numeri-
cally the resulting fully nonlinear unsteady problem.
After a preliminary validation, the effects of sev-
eral physical parameters affecting type and severity of
water on deck are discussed. The flow along the deck
is also studied, with emphasis on pressures and impact
loads with superstructures.
ASSUMPTIONS AND MODELING
Many physical aspects determine the considered prob-
lem. Wave-ship interactions (cf. figure 1.A) modify
significantly the wave pattern with respect to that of the
incident waves. This is related to local effects and to
wave reflection, which in rough sea are highly nonlin-
ear phenomena. Ship motion, figure 1.B, can either en-
hance or prevent the deck wetness occurrence.
In this paper the heavy water on deck is analyzed
by fully retaining the nonlinearities associated with body
and free surface motions. The focus is on deck wetness
at the bow region for a FPSO unit in head sea condi-
A) local effects
....and wave reflection
B)
/ ~
me,
,..iy
.....
Figure 1: Some physical aspects involved: Wave-body
interaction (A), body motions (B), three-dimensional
effects (C).
lions. A blunt ship bow is assumed. Therefore, the
problem is simplified by considering the two-
dimensional flow in the longitudinal plane of the ship.
Clearly, figure 1.C, three-dimensional effects are rele-
vant, though less than for cases with forward speed. In
the latter, the steady wave pattern is characterized by an
increase of the water level due to a local disturbance and
bow waves generation which by themselves decrease
the effective freeboard, non uniformly along the longi-
tudinal ship direction (Turin & Wu 19961. Ship sinkage
due to forward speed may also matter. However, a two-
dimensional analysis can give important insights of the
phenomenon and useful information about the effects of
the parameters involved.
The water motion is believed unaffected by the
viscosity and a potential flow model is adopted. Actu-
ally, the edge of the deck could be a source of separation
and vortex shedding, which are not presently modeled.
Surface tension effects are neglected because of the rel-
atively large spatial scales. Finally, the structural de-
formations are not considered and the body is assumed
perfectly rigid.
In the following, we consider the prototype two-
dimensional problem sketched in figure 2. A frame of
reference fixed with respect to the fluid at great depth
is considered. The fluid domain Q is bounded by the
free surface, IFS, the instantaneous wetted portion
of the ship, ~QBO, and a control surface MOO in the
OCR for page 128
In the second step, the free surface conditions are stepped
Incident Wave
z ~ · forward in time to update geometry and boundary data.
. :j \ ~ The kinetic problem is solve 1 through the Green's
QBO ~ second identity
by
Q ~ C(\P) (I') = /0 (~0 - ~ ~ G)4SQ, (4)
.......................................................................................
where
Figure 2: Sketch of the numerical 2-D problem.
~ 27 Ace
C(P~) = ~ O
P ~ Q U be ,
far field. Deep water conditions are considered, with ~ ~ p ~ I
incident Stokes waves approaching the body from left
to right.
The problem is governed by the Laplace equa-
tion for the velocity potential ,o(P, t):
V2; = 0 ~t,~P~Q, (1)
where P is a point in the fluid domain and t iS the time.
Along the free surface and the impermeable boundaries,
the following kinematic constraint applies
0~ ~ At, VP ~ OFFS U ~QBO (2)
where VHQFB is the displacement velocity of the sur-
face, and r' is the unit normal vector, pointing out of
the fluid domain. The body motion is prescribed a pri-
ori. On the free surface, the dynamic condition requires
constant pressure.
Upon choosing a Lagrangian description of the
free surface OFFS, the kinematic and dynamic bound-
ary conditions, respectively, read
D! ~ V[,VP ~ OFFS (3)
~ D; ~ ~V,P~2 _ 9Z — pPa
Here Pa is the atmospheric pressure, 9 is acceleration
of gravity and p is the mass density of the fluid. The
first equation satisfies condition (2), while the second
follows from the Bernoulli equation.
As usual in the mixed Eulerian-Lagrangian method
for free surface flows (Longuet-Higgins and Cokelet 1976,
Faltinsen 1977), the resulting problem is split into two
sequential steps. In the first one, the kinetic problem for
the velocity potential, with mixed Dirichlet-Neumann
boundary conditions along OFFS and RIO, iS solved.
(5)
is the inner angle at the point P on the boundary, and
G(P,Q) = in(R) R = UP—A, (6)
is the fundamental solution of the Laplace equation in
two dimensions. The right-hand-side of (4) must be
interpreted as a principal value integral when P ~ an.
The right vertical and the horizontal portions of
the surface Alp are chosen far enough from the body
to give negligible contributions. Along the vertical up-
stream barrier (left side in the sketch 2), both ,o and
0,o/0n~ are specified by a truncated Fourier representa-
tion of the Stokes wave in deep water for arbitrary steep-
ness (Bryant 19831. The vertical extent of the barrier is
truncated at a suitable large depth, while the horizontal
location is chosen far enough (order of ten wavelengths)
so that within the time scale of the simulation (at most
the order of ten wave periods) disturbances due to wave
reflection are small in proximity of the inflow bound-
ary. Residuary (high frequency) effects are removed
by using a damping layer technique (Israeli and Orszag
1989) close to the barrier. In particular the dynamic
condition in (3) is modified by introducing a damping
term proportional to ,o—,°s~O, where ,°s~O is the velocity
potential of the Stokes wave. Similarly is done for the
vertical component of the velocity in the kinematic con-
dition. The Lagrangian drift of surface points is elim-
inated through periodic regrinding of the upstream re-
gion. Invariance of the results have been checked by
increasing the upstream length of the domain. Stokes
wave conditions have been checked inside the domain
Q without body.
For points on the boundary, where only ,o (on
the free surface) or 0,o/0n~ (on the body) are specified,
the integral representation (4) provides the relevant inte-
gral equations to evaluate the remaining boundary data.
OCR for page 129
Once this is accomplished, ,o and V,o can be evaluated
everywhere in Q. The solution of the integral equa-
tions is obtained by using a panel method with piece-
wise linear shape functions both for the geometry and
for the boundary data. The collocation points are taken
at the edges of each element, resulting in a continuous
distribution of the velocity potential along the free sur-
face. The tangential velocity 0,o/~ is simply deter-
mined by finite difference operators, while the normal
velocity component is obtained from the integral equa-
tions. Higher order schemes (i.e. Landrini et al 1999)
may lead to numerical difficulties at the body-free sur-
face intersection point and therefore are not adopted
here. Clearly, the use of a lower order method requires a
finer discretization in region with high curvature of the
boundary, or where the thickness of the fluid layer along
the body is small. This has been achieved dynamically
during the simulation by inserting new points where ap-
propriate. The continuity of the potential is assumed at
those points where the free surface meets a solid bound-
ary. Though no rigorous justification is available, this
procedure gives convergence of the numerical results
under grid refinement (Dommermuth & Yue 19871. Oc-
casionally, when the contact angle becomes too small,
numerical problems still may occur and the jet-like flow
is partially cut (Zhao & Faltinsen 19931.
A standard Runge-Kutta fourth order algorithm
is adopted to step forward in time the free surface evo-
lution equations. This requires the solution of four inte-
gral equations each physical time step. Though less de-
manding schemes are conceivable, we preferred this for
the simplicity in changing dynamically the time step.
We found this crucial to keep under control the accu-
racy of the solution during the development of jet flows,
impacts, and breaking waves.
PHYSICAL INVESTIGATIONS
Water on deck is a complex phenomenon and it is worth
to identify separate stages of evolution, even if they are
strongly connected with each other. In particular we
consider: i) the run-up of the water at the bow, ii) the
water shipping onto the deck, iii) the subsequent flow
developing along the deck, and finally iv) the impact
with ship equipments or deck house. Recently Greco
et al (2000) studied separately some of these aspects by
considering some suitable prototype problems both to
achieve a validation of the numerical method, and to
gain a first understanding of the phenomenon. For ex-
ample, the study of the interaction of a solitary wave
with a plane wall showed that weak nonlinear theories
underpredicts the maximum run-up with respect to ex-
periments and nonlinear solution, for wave amplitudes
large enough. This suggests a limited applicability of
simplified models to predict freeboard exceedance, at
least without recurring to some empirical corrections.
When the wave elevation exceeds the freeboard,
the water can flow over the deck and, very often, the re-
sulting flow field resembles the one after a dam break-
ing. Hence, we have further validated our model by
studying this problem. Numerical results agreed well
with small time expansion analytical solutions and ex-
perimental data.
In the following, we analyze more directly the
phenomenon of water on deck occurrence and the flow
in case of impact with superstructures.
Preliminary studies and validation As preliminary
studies and validation of water on deck occurrence we
consider the case of a fixed rectangular body under the
action of regular waves. This case is used to define the
proper treatment of the flow field when the freeboard
is exceeded. In particular, we adopted a 'Kutta' like
condition, enforcing the flow to leave tangentially the
bow when water reaches the instantaneous freeboard, f,
(see figure 31. Once the freeboard is exceeded, the fluid
'Kutta' condition
a_
Ad\ .............
Figure 3: 'Kutta' condition enforced at the edge of the
deck.
velocity relative to the the ship determines whether the
deck will be wetted or the water will be deviated in the
. . .
Opposlte dlrectlon.
Our numerical solution has been compared with
the quasi-two-dimensional experiments by Cozijn (19951.
A wavemaker was used to generate regular waves, in-
teracting with a rectangular bottom mounted structure,
placed at the opposite end of the tank. The freeboard f
was 0.1 m, and the water on deck events were recorded
OCR for page 130
by a video camera. Figure 4 shows the comparison
between numerical (solid lines) and experimental (cir-
cles) free surfaces profiles (for wave height H=0.128 m,
frequency Cal = 5 rad/s, experimental sequence coded
5:36:00-19 from the test No. A031. We have cho-
- H = 0.025 m - rum.
~ At = 0.04 s l o exp. I
''''''''''~_
Hi:
1 1 1 1 1 1 1 1
~_~
_,,~
- 8
_ 1 1 1 1 1 1 1 1< 1 1 1 1 1 1 1 1 1 1 1
Figure 4: Water on deck of a rectangular structure due
to incoming regular waves (H=0.128 m, Cal = 5rad/s,
initial freeboard f = 0.1 m). Snapshots of the free
surface. Experimental data (circles) are from Cozijn
(19951.
sen the free surface configuration just before the ship-
ping of water (see figure 4.1) and the following seven
wave profiles, with a time interval of dt = 0.04 s. The
global behavior of the free surface is well reproduced
by the numerical solution, confirming the efficiency of
the adopted model and the limited effect of the sharp
corner in the model (as we said we have not modeled
the vortex shedding from the edge of the deck). Later
in the evolution, the numerical solution predicts a fluid
front moving faster than in the experimental case. Nev-
ertheless, the water level along the deck is rather similar
for the two results. A possible reason of the differences
could be related to surface tension effects, which are
not presently modeled. In facts, the thickness of the
fluid layer is of order 0.01 m, and the high curvatures
call for a more complex description of the dynamics
of the contact point (cf. Dussan 19791. This is sup-
ported by the observation that the measured shape ap-
pears "rounded" and highly curved in proximity of the
contact point. It would be relevant comparing experi-
ments with larger scale. Anyway, since the deviation
between the two results is strongly localized, it is be-
lieved unimportant in terms of effects on a superstruc-
ture hit by the water along the deck. Differences in the
pressure over the structure are expected in a very small
time initially that is unimportant from the structural re-
action point of view.
We now consider a more realistic set of parame-
ters. In particular, we have chosen a FPSO unit in long
and steep head sea regular waves (Buchner 1995) to fix
the main parameters for our two-dimensional simula-
tion. The draft of the ship is D=17.52 m, while the rela-
tive length and freeboard are respectively L/D = 14.86
and f/D = 0.507. In the experiments, a superstructure
is located at a distance dS/D = 2.05 from the bow. For
simplicity, we have assumed a straight vertical bow and
restrained the body motions. The following analysis is
for the first water on deck occurrence caused by inci-
dent waves with wavelength A/L = 0.75 and height
H/A = 0.09. Top plot of figure 5 shows the free surface
just after the start of the water shipping and during the
later evolution, after the impact with the superstructure.
The four bottom plots report the time evolution of water
level at locations A-D (shown in the top plot) along the
deck and the numerical results are compared with the
(three-dimensional) experiments by Buchner (19951. In
the experimental set up the water level sensors were lo-
cated along the centerplane of the ship. In this case, in
spite of the three-dimensionality, the comparison indi-
cates satisfactory numerical prediction of the propaga-
tion of the water front. It is observed that the scale of
this experiment is quite larger than the one used by Coz-
ijn, which supports our conjecture about the role of sur-
face tension in the previous case. In more detail, in loca-
tion A we observe a strong local effect associated with
the first event of water on deck and a rather large over-
prediction of hw with respect to the experiments. We
recall that geometrical details of the bow and the actual
wave induced body motions hamper the possibility of a
finer comparison. The relative difference is smaller for
locations B-C. For the two most advanced (C, D), the
numerical results underpredict the experimental values.
This is somehow consistent with an increasing three-
dimensional behavior of the flow developing along the
deck which gives additional contributions to the water
OCR for page 131
length.
of/ ~ x/D=0.34
I_______ ~ x/D=0.88
~ x/D=1.21
1 1 1 1 1 1 1 1 1 1 1 >IX E=1.5'
10
hw
() _
.
10
hw
5
O
_A
-1
- 1
-tail I I I I I I I Or.
O t t wad
I- rum. sol. ~
B I 3D expert|
_C
/'
I,
- 1 1 1111 1 1 1 1 1 1 1
at:
_D
5 0
-
.
- /^ ``
— 1/ ~ ~
_ ~ `_
- 1 1 1 / 1 1 1 1 1 1 1
t twod 5
Figure 5: Water level hw (meters) as a function of time
(seconds) at locations A-D along the deck. Numerical
results (solid lines) and 3D experiments (dashed lines)
by Buchner (19951. twos is the time instant when the
shipping starts.
level through the lateral ship sides. When the number
of shipping events increases, the local wave steepness
in the bow region decreases and in this case the result-
ing amount of water wetting the deck reduces, as well
as the propagating flow velocity.
Effect of main geometric parameters A simplified
parametric analysis of the deck wetness can be made
taking the amount of shipped water, Q. as measure of
the water on deck severity. Systematic variations both
of body geometry and of incoming wave characteristics
have been considered. In particular, we have considered
the geometric parameters sketched in figure 6, with the
draft of the ship as reference length. Finally, the amount
of shipped water is made dimensionless by the amount
of water QO associated with the incoming waves above
the mean free surface level over a distance of a wave-
Incident Wave
... H. :~ ~
1 _1 1 _1
~ ~1 1— ~1
~ L
Figure 6: Sketch of the main geometrical parameters
considered.
At first we consider cases where the body mo-
tion is suppressed. We have considered four freeboard-
to-wave height ratios, f/H=0.05, 0.24, 0.36 and 0.55,
and we have analyzed the influence of the stem over-
hang angle or and of the length-to-draft ratio L/D of
the ship, for H/A = 0.06 and A/D = 6.6. The results
,' Ha/= 0.06
to.
o-
.
Am.
~ oo
Coo
45°
—~ ~ ~
1-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.2 04 :/H 06
Figure 7: Influence of ship length and stem overhang on
the bow deck wetness.
L/D
10
20
_ 20
are presented in figure 7, where the relative amount of
shipped water is plotted versus f/H. As expected Q/QO
is strongly influenced by the freeboard of the ship. The
length L does not affect directly the deck wetness sever-
ity, while it will through the length-to-wavelength ra-
tio and then through the induced body motions, here
not considered. As expected, a positive bow stem over-
hang reduces the relative amount of ship-pea water due
to a larger wave reflection by the ship. However, in the
present case, the deck wetness severity does not change
dramatically in the two considered cases (or = 0°, 45°~.
This is more evident for larger values of f/H, which are
the most interesting from a practical point of view.
Figures 8-9 show the influence of steepness, H/A,
OCR for page 132
and wavelength-to-draft ratio, A/D, on Q/QO by keep-
ing the body parameters fixed. As expected, as H/A in-
creases, a larger amount of ship-pea water is computed,
with an almost linear trend for small f/H. For larger
f/H, a marked nonlinear behavior of Q/QO is observed
as the wave steepness increases. This is natural be-
cause Q becomes stronger dependent on the wave crest
flow, which will have an increased nonlinear behavior
with increased wave steepness. In the second figure,
the effect of the wavelength-to-draft ratio is shown for
a constant wave steepness and zero stem overhang. The
0.3
Q
Go
0.2
0.1
o
0.2 04 :/H 06
Figure 8: Influence of nonlinearity of incoming waves
on the bow deck wetness.
~ = 0
H//
-a 0.032
-a 0.064
~ 0.095
deck wetness severity changes a lot from case to case,
though the nonlinearities associated with the incoming
waves are the same. In particular, the worst conditions
occur for large wavelength-to-draft ratios, for which a
weaker wave reflection is observed, and which are also
the more interesting from a practical point of view.
0.4 _
Q
Qo
0.3 _
0.2
0.1
o
=o°,H/~=0.095 N/D
* 1.0
~ 6.4
-a 10.0
-I I I I I I I I I I ~ ~ I ~ I l - ~~-l-l-ht1 1 ~ I I
0.2 04 :/H 06
Figure 9: Influence of wavelength-to-draft ratio on the
bow deck wetness.
0.4 _
Q
Qo -
0.2 _
~ ~ ~~ ,
, ~ I ~ ~
,
,
I'
~~ ,
,!
lo,
1
-a A
-6 B
-A C
-a D
-a E
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 25 (t—tWoDl)/T 5
Figure 10: History of the relative amount of shipped
water Q/QO. Cases A-E are described in table 1.
f/H = 0.24
1
The previous analysis considered the first water
on deck occurrence. We now consider longer evolu-
tions to analyze the history of shipping events. Figure
10 gives Q/QO as a function of the time, assuming as
time origin the instant tWodl of the first event. A fixed
f/H = 0.24 is considered for the cases summarized in
table 1. For all of them, we observe large changes of
Q/QO with respect to the first water on deck occurrence.
Interestingly, on a longer time-scale, it tends to reach a
more defined value with almost the same periodicity as
the incoming waves. Clearly, this result is not general
because more realistic sea-state conditions are charac-
terized by the interaction with irregular waves. How-
ever the results show that if two succeeding waves with
nearly the same height and wavelength cause deck wet-
ness, the last one gives the most severe condition.
case
A
B
C
D
E
0°
45°
0°
oo
oo
A/L
0.33
0.33
0.33
0.05
0.67
H/A l
0.064
0.064
0.095
0.095
0.095
Table 1: Synopsis of cases considered for studying the
history of water shipping.
In more detail, figure 10 shows that the worse
deck wetnesses happen for the steeper conditions (cases
C and E). The corresponding Q/QO tends almost to the
same value, confirming the steepness as the most impor-
tant wave parameter for long waves (in both cases the
wavelength is large with respect to the draft). In case D,
the steepness is the same but with shorter wavelength,
OCR for page 133
equal to the draft, and the shipped water is comparable
to that computed for a longer less steeper wave (case A).
Case B (same parameters as case A but with or = 45°),
shows a certain effect of the stem overhang in reducing
the severity of the deck wetness.
D
D
D
Q/QO= 0. 1 5
(tlast tWod)/T= 0. 42
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Q/QO= 0.37
- ( tlast twOd) /T = 0 . 42
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0oOOOOOOOo
:~ ~ —- = On
WoD 3
- Q/QO= 0.29
(tlast tWod)/T= A. 45
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
oOOOOOOOOo
~ Q/QO=0.30
( tlast twOd) /T = 0 . 4 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x/D
Figure 11: First four water on deck events for case A
(cf. tab. 11. For each event, free surface configurations
are plotted for the maximum freeboard exceedance (cir-
cles) and for zero flux entering the deck (solid lines).
For the case A, figure 11 shows free surface pro-
files for the first four water on deck events. In particular,
in each plot, two configurations are given: the one with
maximum freeboard exceedance (circles) and the one
with zero flux of water onto the deck (solid lines). In
the four cases, the wave pattern in front of the body is
not exactly the same because of the complex features
of the reflected wave field. In spite of this, the wave
forms in the very near field and on the deck attain a
more defined pattern, consistently with the almost con-
stant Q/QO previously reported. We define, convention-
ally, the beginning of the water shipping twos as the time
just after the freeboard exceedance for which we have
a positive inflow onto the deck. Further toast means the
time when the shipping of water on deck ends. The flux
of water onto the deck is then zero. With these defi-
nitions, we observed that the time scale involved in a
"water on deck cycle" does not change markedly and
remains (9 (40%) of the wave period T.
a) ~ ~ = l
D
D
D
nOo0000o
. .
,C,~, ~ ~
·. oooooi WoD 1
Q/QO= O. 1 6
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
b)
. ~
~ A/H=0.25
Q/QO= 0.06
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
c)
, ~
-I.............
~ A/H=0.25
Q/QO= 0. 30
d)
·—
· —
L.~e, ~
~ A/H=0.50
Q/QO= 0.5 1
. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x/D
Figure 12: Influence of body motion. a) restrained
body conditions, b) forced heave initially in phase with
the water at the bow, c) and d) forced heave initially
out-of-phase with the water at the bow.
Effect of body motion Body motions play a major
role in determining occurrence and severity of water on
deck. Here, we did not solve for the wave induced mo-
tions of the ship but simply prescribed a priori the mo-
tion. In particular, since we are dealing with bow deck
OCR for page 134
wetness in head sea, we have studied the effect of forced
heave motion. The effect of the pitch angle both on the
outer wave field and on the flow along the deck is ne-
glected. Since the stem overhang angle had a small in-
fluence, we could argue that the pitch angle should not
be important for shipping of water on the deck. But the
pitch angle may have a larger effect for the flow on the
deck. Top plot of figure 12 gives the first water on deck
occurrence for a studied situation (case E from table 1)
where we have considered a freeboard ratio f/H = 0.55.
In this case the body is constrained and Q ~ 16 % of
Q0. For the same parameters, plots b) through d) show
the flow when a forced heave motion is excited just at
the beginning of the water shipping, according to
(3 (t) = ~ A Hit—twos] sink 2T (t—twos], (7)
H being the Heaviside function. The arrows in the plots
indicate the direction of the heave motion. In particular,
for the last case the shipping starts with a heave initially
downwards. Heave is upwards in the end.
In plot 12.b the motion is initially in phase with
local wave motion. The amplitude-to-wave height ratio
A/H is 0.25. The phenomenon appears qualitatively less
severe. The amount of shipped water Q is only the 6%
of Q0. However this nice situation is unlikely to occur
in the case of a FPSO unit for the wave-body parame-
ters we have chosen. Conditions of out-of-phase body
motions are more reasonable and can make the water on
deck much more severe than in the restrained body case.
A heave amplitude A/H= 0.25, third plot, increases the
amount of shipped water with a factor 1.9 relative to
case a), while for an amplitude equal to the one half of
incoming waves, fourth plot, the factor becomes 3.2, to
reach 6.2 in the case not shown with A/H= 1.
Occurrence of waves plunging on the deck The flow
along the deck resembles the one after a dam breaking
in the most common type of green water event. More re-
cent experiments in irregular seas (MARINTEK 2000)
showed that water on deck can also occur in the form of
waves plunging directly on the deck. In this case, im-
pacts with superstructures are likely to occur. This phe-
nomenon appears like a 'single' event associated with
a very steep, almost breaking, incoming wave, usually
with smaller background waves. Actually, we cannot
classify this as 'freak wave' but it is known that instabil-
ity and modulation of wave groups in open sea can lead
to the formation of steep highly energetic waves. Their
interaction with structures is a known cause of highly
nonlinear force components (Chaplin et al 1997, Welch
et al 19991. Similar circumstances in complex combina-
tion with ship motions can cause these extreme events.
So far, we have not tried to model more complex in-
coming waves than regular steep waves. In spite of this,
we have analyzed some extreme cases to gain some in-
sight, with some emphasis on the effect of body motion.
The geometric parameters have been deduced from the
MARINTEK experiments (L/D=13.75, f/D=0.8,
dS/D=1.06251. Considering the limited role of the stem
overhang, we have approximated the bow with a straight
vertical wall. We have considered a wave train of long
steep (eventually) regular waves with A/L = 1.022 and
H/A = 0.095 and focused our attention on the interac-
tion of the body with the leading wave which is char-
acterized by higher steepness and strong tendency to
break. This makes our analysis more consistent with
the features observed in the experiments. Forced heave
motion is excited at a time instant to with an amplitude
A and a phase 4, in the form
(3 (t) = A Hit—to] singlet (t—to) + Hi, (8)
where T is the wave period. Wave generation starts at
t = 0, with the upstream section located 5 wavelengths
ahead of the bow and the phase angle ~ is selected to
give a sudden vertical displacement of the ship at t=to.
Some of cases studied, and discussed in the following,
are summarized in table 2.
case
a
b
c
cl
c2
d
e _
f/H
0.6
0.6
0.6
0.6
0.6
0.6
0.5
A/H
0.
0.5
0.5
0.25
0.125
0.5
O.
T
T
-900
- So
—100
-20.50
—110
_ _
Table 2: Plunging wave analysis
presented.
to/T
10.626
10.783
10.783
10.783
10.783
twod /T
10.795
10.841
10.844
10.795
10.790
10.783
10.777
summary of cases
In figure 13, restrained body conditions are con-
sidered, and some free surface configurations are pre-
sented. The wave, reaching the bow, is steep and un-
symmetric but its tendency to break is reduced during
the run-up along the bow. The water shipping starts
with already quite large horizontal velocities of the fluid
making the phenomenon less similar to the dam break-
OCR for page 135
ing problem. Though the shallower water conditions on
the deck would amplify the original tendency to wave
breaking, the fast motion of the wave front has oppo-
site effect and is the main reason why the wave is not
breaking before the water impact on the deck house.
2
1
D
o
-1
1
D
o.s
o
D
a)
-10 -8 x/D -6
Figure 13: Plunging wave analysis: case a.
- b)
1 1 1 1 1 1 1 1 1
-8 -7
1 1 1 1 1 1 1 1 1 1 1
x/D -6
Figure 14: Plunging wave analysis: case b.
The heave motion largely affects the phenomenon.
In the next figures 14-16, a heave with amplitude A/H
= 0.5 is considered (see table 2 for the other param-
eters). In the first case, the motion is excited with a
phase such that the instantaneous freeboard at t=to is
higher than the wave elevation at the bow. By 'instanta-
neous freeboard' we consider the mean freeboard plus
the change in vertical position due to heave. The up-
wards motion of the bow causes lower trough ahead of
the breaking wave and a bow impact occurs. Air en-
trapment and (probably) a complex two phase flow are
expected to occur. Upon neglecting these phenomena
and "stretching" our simulation further, we observe that
the shipping of water is not particularly severe. The up-
1.3 - C)
~ at. ~ 1- -- ,
_ . .....
. .....
. .....
'em:'"' 1 "' 1 1 1 1 1 1 1 1 1 1 1 1 1
-7 x/D -6
Figure 15: Plunging wave analysis: case c.
d)
.: ~< ~ ~ ,
_ , ~ ,
_ . . . ~ ,
''"' :""
_.....
'I .'1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-7 x/D -6
Figure 16: Plunging wave analysis: case d.
wards motion of the bow, in facts, limits the increase of
the vertical velocity of the fluid after the impact.
In figures 15 and 16 the heave motion is excited
later than in the previous case b. This is done in both
situations at the same instant but with different initial
phase. This means a different instantaneous freeboard,
in particular for the case d the wave elevation at the bow
is equal to the instantaneous freeboard. The larger to
eliminated the bow impact. However other interesting
phenomena occurred. In case c, we observe an initial
local breaking tendency of water along the deck. But
this is prevented by an increase of the horizontal veloc-
ity of the contact point between water and deck. The
subsequent flow is like the one after a dam breaking. In
case d, the amount of shipped water is larger and the
upwards motion of the ship results in a wave plunging
onto the deck. The three considered situations could
have quite different consequences on the ship.
If we modify case c by taking a heave ampli-
tude A/H = 0.25 and A/H = 0.125 but maintaining
OCR for page 136
1.5
D
1
o.s
1.5
D
1
o.s
-7 x/D -6
Figure 17: Plunging wave analysis: case cl.
I : : ~ ~
. .. it, ~
_ . . . . ,
_ ....
i , .. ..
_ . .
1 1 .'~ .I' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-7 x/D -6
Figure 18: Plunging wave analysis: case c2.
the same instantaneous freeboard at t = to we get the
results shown in figures 17 and 18, respectively. As
we can observe the water on deck is still quite serious
but the consequences are more dangerous for the su-
perstructure than for the bow or the deck. With A/H
= 0.25, in particular, the faster rate of the water re-
gion to become shallower steepens the wave propagat-
ing along the deck. A rather thin jet develops. The
jet evolves faster than the water-deck contact point and
eventually hits the superstructure. After the impact the
simulation was continued by a local matching with the
similarity solution by Zhang et al (1996) for an infinite
asymmetric fluid wedge hitting a wall (see sketch at the
top of figure 24). If the heave amplitude is further de-
creased (case c2, figure 18), the velocity of the wave
front becomes larger relative to the plunging jet veloc-
ity. This implies that the impact with the superstructure
occurs from the deck. The plunging wave hits the wa-
ter mass rising along the vertical wall after the impact.
This causes an air pocket to be formed. The relative
velocity between developing plunging jet and the wave
front depends on the rising rate of the deck. This has
an important influence on the possibility of a plunging
breaker hitting the superstructure.
The analysis here considered is not complete and
therefore no conclusive statement about the occurrence
of the plunging wave event can be given. However, in
the cases so far studied, the run-up along the bow even-
tually caused the more common dam breaking type of
event. This fact supports the conjecture that a plung-
ing wave on a deck completely dry is probably due to
a "ready-to-break" wave rather than to bow interaction
. . .
with Incoming waves.
~ -
hl
o
2r
h
O 1 1 1
o
EM |dam position| |wall|
~ ~ 1
- b) ;=2.6
1 1 1 1 1 1 1 1 1 1 1 —~—
~ c) :=5.6 <~
1 1 1 1 1 1 1 1 1 1 1
d) :=6.2
1~ 1
2 4 x/h
Figure 19: Set up (top) and simulation of the im-
pact with a vertical structure following a dam breaking
(7 = ty/~75)
Impact with obstacles The fine details of the "exter-
nal" flow field have a limited influence on the impact
OCR for page 137
because of the small time- and space-scales involved.
Therefore, here we study the impact occurring with su-
perstructures simplifying the initial conditions by using
the flow generated after a dam breaking.
The considered case is sketched in the top plot of
figure 19. A reservoir of water with height h and length
2h, closed by a dam, is placed at a distance 3.366 h from
a vertical obstacle. For t = 0, the dam is suddenly re-
moved and the flow develops along the horizontal deck,
fig. l9.a, finally impacting against the vertical wall. The
fluid is violenty deviated vertically upwards, fig. l9.b,
rising along the wall in the form of a thin jet. At this
stage, formation of spray and fragmentation of the free
surface may occur. These finer details cannot be han-
dled by the present method, though we believe they are
not relevant to compute the structural loads. As time in-
creases, under the reaction of the gravity, the fluid accel-
eration decreases and the upward velocity in the jet de-
creases until becomes negative. The motion of the wa-
ter is reversed in a waterfall, fig. l9.c, overturning in the
form of a large wave plunging onto the deck, fig. l9.d.
The numerical simulation is eventually stopped due to
numerical break-down.
0.2
0.!
- ~ z/h
_
- x/h
' '''- ___
O ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _
0.2 - ~ z/h
- x/h
o ~ 'a"'-' -_' t
xNNNNNNNNNxNNNNx
-I I I I I I I I I I I I I I I I I I I It
z/h
x/h ;
0.2
0.1
o
0.4
During the first stage of the impact against the
wall, the flow resembles the one due to a (half) wedge
of fluid hitting the structure, with the rest of the flow
field roughly unchanged with respect to the case of inn-
nite deck. The gravity plays a minor role since the ver-
tical acceleration of the fluid around the contact point
is (dreg). In particular, in figure 20, the free surface
close to the wall after the impact is shown in compari-
son with the zero gravity similarity solution by Zhang
et al (1996), for an infinite wedge of fluid hitting a flat
structure at 90°. The two solutions remain in qualitative
agreement even for a non-dimensional time /\Timp =
0.1338 after the impact.
The two plots in fig. 21 show the evolution in
time of the height of the water hw at the locations (x/h)A
3.721 and (x/h)B = 4.542 along the deck shown in
fig. 19. In agreement with the experiments (dashed lines),
the numerical simulation (solid lines) shows a first stage
characterized by a sudden rise of the water level hw,
followed by a slower growth. The simulation is then
stopped because of the surface breaking. Nevertheless,
in section B. a third stage with a steeper increase of hw
is captured. This is due to the water overturning which
gives an additional contribution to the latest part of the
_ I- numb solid z/h
i. simil. sol.l ~
- Aiimp=0.01 38 p/pg h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
- n u m so l . z/h
- Chimp= 0.0243 p/pg h
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
~|-num~ sol l z/hi
at. simil. sol.l ~
~ =0.0738 p/pg h
_ - ~
_~N
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
- ~ z/h ,
_ ~ :
- x/h t
0.2 ~
1
-num. sol.| z/hi
1- slmll. sol.l ~
AfEmre=O ~ 338 p/pq h
%
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 2
Figure 20: Free surface and pressure distribution during
the initial stage of the impact. Solid lines: present nu-
merical simulations; ·: similarity solution from Zhang
et al (19961; /\Timp = ~—Timp. Timp = initial
non-dimensional impact time.
evolution. Location A will be influenced later by this
phenomenon. The measured data by Zhou et al (1999)
are for an initial height of water h= 0.6 m. Due to
lack of sufficient details about the experiments, the time
when the experimental hw gets a non-zero value was set
equal to the numerical one. The two types of curves fit
quite well until breaking occurs.
We now discuss in more detail the impact pres-
sures. Right plots of figure 20 present the pressure dis-
tributions corresponding to the free surfaces configura-
tions on the left. According to the numerical results
(solid lines), at each time instant the maximum value
OCR for page 138
o.s
hw
o
o.s
hw
o
0 1
sum. sol. |
- I -- exp. data I
) ~1 1 1 1 1 1 1 1 1
sum. sol. |
- I -- exp. data 1,'
of,
I, `.
, . , ~ ~
(x/ h )A= 3. 72 1
. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
, ~ ~~ ( X/ h ~ s—4 642
1
2 t (S) 3
Figure 21: Experimental (Zhou et al 1999, h
= 0.6 m) and numerical water level hw (meters) at
(x/h)A = 3.721 and (x/h)B = 4.542 as a function of
time. Positions A and B definited in top plot of figure
19.
Of the pressure is located at the initial impact position
and gets the highest values just after the impact. In the
region of the thin jet along the wall, the pressure is al-
most equal to the atmospheric one, conventionally set
to zero. The bullet symbols represent the pressure dis- A/ h
tribution for the zero gravity impact of a fluid wedge pg
with a flat wall. In particular, this has been evaluated ° s
by solving numerically the boundary value problem for
the velocity potential along the wall and taking the val-
ues of both to and its normal derivative on the free sur-
face from the similarity solution by Zhang et al (19961.
At the beginning, the agreement between the two dif-
ferent results is good as for the corresponding free sur-
faces. For longer time, the pressure distributions seem
to diverge faster than the free surface elevations. As
can be expected, in the "exact" computations, the max-
imum pressure decreases while in the zero gravity case
remains constant. In facts, as time and water level along
the obstacle increase, the—p 0:o/0t contribution to the
pressure reduces.
In our simulation we have assumed a rigid wall
but the pressure distribution could be influenced also by
possible hydroelastic effects. In this case, it is impor-
tant to introduce the generalized force ~ p Ends, where
All is an eigenmode for the local structural vibration.
While the generalized forces related to the high initial
values are modest (z = 0 is a structural node), smaller
(but large enough) values of the pressure, distributed
on a larger portion of the wall, may excite a hydroe-
lastic response of the structure. On this ground, both
the pressure distribution and its time evolution are im-
portant for structural analysis and hydroelastic effects
should be considered if the time duration of the load-
ing over the analyzed structural part is the same order
or smaller than the highest natural period for the con-
sidered structural part (Faltinsen 19991.
In figure 22, the dashed lines give the time evo-
lution of the pressure measured by Zhou et al (1999) by
a gauge, sketched at the top of the figure, with circu-
lar area of diameter 0.09 m centered at the location C
along the wall (cf. top of fig. 191. We have shifted the
time origin consistently with that discussed before for
comparing the water level evolution. The dash-dotted
pressure ~~
area ~/ (a - ~ C: z/h=0.27
n 7/h=n 1
1
; -- sum., pos.C
~ sum., pos.D
iNlllIif~d __ experiments
. ~ ~
~1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
t (S)
Figure 22: Top: position of the pressure gauge in exper-
iments by Zhou et al (1999, h=0.6 m). Bottom: exper-
imental and numerical evolution of the pressure along
the vertical wall (see top plot of figure 191.
lines are the numerical results at that location. The two
curves get non-zero values almost at the same instant,
confirming the global agreement between the numerical
simulation and the experiment. A certain gap between
theory and experiments is apparent, though. Mesh re-
finement and local regriding have been used to achieve
invariance of the solution and to rule out the dependence
OCR for page 139
on the discretization parameters. On the other hand,
the complexity of the experiment makes it difficult to
identify the error sources because, according to authors
comments, it was difficult to achieve repeatability of the
results. It can be observed that, for the actual scales
of the experiment, even a deck not perfectly dry (for
example because of a previous experiment) can intro-
duce significant differences in the measured data. This
seems to be plausible also in the present case (cf. fig.
211. As said, though the experimental and numerical
evolutions are globally in good agreement, close to the
instant when the water level gets a non-zero value the
numerics underpredicts the measured data. In particu-
lar, the measured hw gets a local maximum not present
in the numerical results. These experimental features
can be converted from a temporal to a spatial point of
view. In particular they suggest a hump in the free sur-
face close to the contact point. This is not visible in the
dam breaking free surface profiles in Dressier (1954)
and could be due to the presence of a layer of water
before the dam breaks. Dam breaking experiments by
Stansby et al (1998) show that, if the deck is not per-
fectly dry due to leakage (in those experiments a film of
water with a thickness about 1-2 mm was downstream
to the dam) a horizontal bulge of fluid develops just af-
ter the dam release giving rise to a very peculiar local
flow. During the evolution, a hump becomes apparent
- e)
.. 1
7 x/D
i.3s~
-A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-6 s.g x/D
-6
1 1 1 1
-s.8
Figure 23: Case e in table 2. Top: flow onto the deck
and impacts with the deck house from the deck and due
to a plunging wave. Bottom: details of the plunging
wave impact.
in the most advanced flow region. Experimental wa-
ter levels in Zhou et al (1999) are given only for one \
test case and our guess can not be confirmed by other - 4,\
comparisons.Anyway,accordingtothenumericalsim- U/V —04 0.0 Ut ~t
ulatlon the experimental pressure curve IS rather close i | 46° 17° ~~\t
to the pressure evolution at the lower location of the ~ 16° 0° V \
transducer (solid lines), indicated with the letter D in
the sketch above figure22. s ~ ~z/vt ~ N z/Vt: l
As previously discussed, water on deck can re- - x/v ~ \ \ ~ p/p V2
suit in impact of fluid with superstructures. As an exam- ~ ! - - ~
pie of a complex interaction between the shipped water - . ~ ! /
and the structures, we consider the first water on deck ° ~ \ --em
event for case e in table 2. The ship motion is restrained ! - ~
and the freeboard relative to the wave heightisf/H=0.5. ~ /~~ |_pung waveimpact |
The flow evolution along the deck is presented in the / ~ - ~ . impact from deck
top plot of figure 23. The motion of the fluid along -s ~i ~ ~ ~ ~ ~ ~ i'~ ~ ~ ~ ~ i
the deck, resulting in a first impact against the deck
Figure 24: Case e In table 2. Impact due to the plunging
house, Is accompanied by a plunging wave hitting the wave (solid lines) and to the wave front along the deck
upper part of the vertical wall. The related pressure dis- Alp. Left: similarity solutions for free surface. Right:
tributions along the structure are estimated numerically pressure on the wall. Initial impact positions in (0, 04.
by using the similarity solution as previously described
OCR for page 140
(i.e. by studying the zero gravity impact due to infinite
fluid wedges) and are given in the right plot of figure 24.
The corresponding free surface profiles are shown in the
left plot, where the initial position of the impact for both
cases coincides with (0,01. It is observed that the maxi-
mum non-dimensional pressure due to the plunging im-
pact is significantly larger. The impact velocity itself is
slightly higher (Vl>/~ = 1.3) than in the other situa-
tion (Vl>/~ = 1.11. This circumstance, and the fact
that likely the plunging wave impact will occur far from
a structural node, make this event more critical than the
impact occurring at the deck level.
CONCLUSIONS
The phenomenon of bow deck wetness of a moored
ship in regular head waves has been idealized and re-
duced to a simple two-dimensional wave-body interac-
tion problem. The related unsteady fully nonlinear free
surface flow has been solved numerically. Reasonably
good agreement with experimental and analytical re-
sults enable us to use this simple model to gain some
fundamental insights concerning the water on deck oc-
currence, the flow field over the deck and the impact
with superstructures.
In particular, an analysis on the parameters de-
pendence of deck wetness has been carried out, show-
ing that
- For long wavelengths ~ relative to the draft D,
the wave steepness H/A mainly determines wa-
ter on deck occurrence and severity. The relative
amount of shipped water depends nonlinearly on
Hi/.
- For small A/L, where L is ship length, the bow
wave reflection reduces or prevents the shipping
of water, even for large wave steepnesses.
- The stem overhang reduces the relative a-mount
of shipped water, but its positive effect is less pro-
nounced with respect to that of the freeboard.
The occurrence of the less common "plunging
wave water on deck" has also been investigated. Wave-
body interaction by itself seems unable to cause a wave
to plunge directly onto a completely dry deck, and the
occurrence of this extreme and dangerous event appears
more related to the interaction with a steep wave already
prone to break. However, the influence of ship motion
to enhance or reduce the severity cannot be excluded.
It is fully realized that three-dimensional flow
coupled with the ship dynamics have to be introduced
in the future to predict quantitatively water on deck.
ACKNOWLEDGEMENTS
This research activity has taken place at the Strong Point
Centre on Hydroelasticity in Trondheim, supported by
NTNU and MARINTEK. The research has also been
supported by the Italian Ministero dei Trasporti e delta
Navigazione through INSEAN Research Program 2000-
02. The first author is a Ph.D. student at NTNU and is
also associated with INSEAN.
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OCR for page 142
Representative terms from entire chapter:
free surface
DISCUSSION
G Clrhme
Dynafl .., hoe, USA
To e force the Kutta Condition et She com rs
(intersection hori ontcl plume md vertical plume) did
you have to use double nodes (nodes where Here are
2 normals)?
AUTHOR'S REPLY
In the practical implementation w looked for c
robust treatment of the flow et the com r et the
initiation of md during the shipping of water
Before the shipping of water, the comer is not wetted
md no special treatment is reqmred More relevmt
et f is stage is the implementation of c decision
critermm of shipping of water in particular, we allow
the fluid to leave tmgenticlly the tem for c fraction
of time step md evaluate the velocity component m
the dimection of the deck in case of inward motion,
the free surface is cut et the corner Ed the shipping
of water starts
At f is tage, the corner is f fly w tted md c different
treatment is adopted We have not modeled the
vortex shedding that, in principle, should take pk e
there Fmther, m the discreti ction of the integral
equation, et the corner w define the unit normal
vector es the vector along the bisector This avoid the
use of c double-node md, in c way, is equivalent to
smooch the discontimmity of the geometry
[1] CC Mel, The Applied Dynamics of Ocem
Surface Waves
Singapme: World Scientific (1933), pp 740
DISCUSSION
D 1; P Yue
Massachusetts institute of Tech olo a, USA
For both long md short waves, She mthors find that
stronger wave reflection by She ship ceases less
shipping of water on deck In general, one would
expect that stronger wave reflection leads to larger
local wave height which produces more water on
deck according to the dambrecking theory Whet
c mses f is co flint?
AUTHOR'S REPLY
Consider two .. a ve ystems with the same steepness
mddiffe~entwavelengfhs,sayL c
