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BASIC STUDIES OF WATER ON DECK 126
Basic Studies of Water on Deck
M.Greco1, O.M.Faltinsen 1, M.Landrini2
of Marine Hydrodynamics—NTNU, Trondheim—Norway. 2 INSEAN, The Italian Ship Model Basin, Roma
1Department
—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 water on deck is
carried out by using an approximate hydrodynamic model. In particular, the unsteady interaction between free surface and
ship is analyzed by solving the inviscid two-dimensional fully nonlinear problem numerically. Both water on deck
resembling dam breaking as well as due to plunging waves are investigated.
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 importance of hydroelasticity must then be
assessed. Moreover, the fluid motion onto the deck may affect roll stability of smaller ships 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 subsequent requirements of the
Norwegian Petroleum Directorate are given by Ersdal & Kvitrud (2000).
However, the numerous physical aspects determining the phenomenon make it difficult to clearly identify 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 characterizing a ship bow is far from being clarified. Sometimes, it is not even clear whether they
enhance or reduce 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) observed 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 introduced 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 interest. Maruo &
Song (1994) studied the shipping of the water for high speed vessels by using a Slender Body Theory. This may
also have relevance for slender ship bows at moderate forward speed. Buchner & Cozijn (1997) analyzed the bow deck
wetness for moored ships, assuming a two-dimensional problem in the longitudinal ship direction. They presented both
numerical 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
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BASIC STUDIES OF WATER ON DECK 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 studied 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 experimental results for the water height at the inflow boundary. In the simulation, the
ship motion is not taken into account. The author concluded that the flow interactions 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 fundamental aspects of deck wetness at the bow region of a FPSO unit in
head sea conditions. It implies that forward speed effects are not investigated. The problem is idealized by considering the
two-dimensional flow in the longitudinal plane of the ship and solving numerically the resulting fully nonlinear unsteady
problem.
After a preliminary validation, the effects of several 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 problem. 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 nonlinear phenomena. Ship motion, figure 1.B, can either enhance 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 conditions. 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 relevant, 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 longitudinal ship direction (Tulin
& Wu 1996). 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.
Figure 1: Some physical aspects involved: Wave-body interaction (A), body motions (B), three-dimensional effects (C).
The water motion is believed unaffected by the viscosity and a potential flow model is adopted. Actually, 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 relatively large spatial scales. Finally, the structural deformations 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 Ω is bounded by the free surface, ∂ ΩFS, the
instantaneous wetted portion of the ship, ∂ ΩBO, and a control surface ∂ Ω∞ in the
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BASIC STUDIES OF WATER ON DECK 128
far field. Deep water conditions are considered, with incident Stokes waves approaching the body from left to right.
Figure 2: Sketch of the numerical 2-D problem.
The problem is governed by the Laplace equation for the velocity potential
(1)
is a point in the fluid domain and t is the time. Along the free surface and the impermeable boundaries, the
where
following kinematic constraint applies
(2)
where is the displacement velocity of the surface, and is the unit normal vector, pointing out of the fluid
domain. The body motion is prescribed a priori. On the free surface, the dynamic condition requires constant pressure.
Upon choosing a Lagrangian description of the free surface ∂ ΩFS, the kinematic and dynamic boundary conditions,
respectively, read
(3)
Here pa is the atmospheric pressure, g is acceleration of gravity and ρ 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 • ΩFS and • ΩBO, is solved. In the second step, the free
surface conditions are stepped forward in time to update geometry and boundary data.
The kinetic problem is solved through the Green's second identity
(4)
where
(5)
θ is the inner angle at the point on the boundary, and
(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
The right vertical and the horizontal portions of the surface ∂ Ω∞ are chosen far enough from the body to give
negligible contributions. Along the vertical upstream barrier (left side in the sketch 2), both φ and ∂φ/∂n are specified by a
truncated Fourier representation of the Stokes wave in deep water for arbitrary steepness (Bryant 1983). The vertical extent
of the barrier is truncated at a suitable large depth, while the horizontal location is chosen far enough (order often
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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 boundary. 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 φ−φsto , where φsto is the velocity potential of the Stokes wave.
Similarly is done for the vertical component of the velocity in the kinematic condition. The Lagrangian drift of surface
points is eliminated through periodic regridding of the upstream region. Invariance of the results have been checked by
increasing the upstream length of the domain. Stokes wave conditions have been checked inside the domain Ω without
body.
For points on the boundary, where only φ (on the free surface) or ∂ φ/∂ n (on the body) are specified, the integral
representation (4) provides the relevant integral equations to evaluate the remaining boundary data.

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BASIC STUDIES OF WATER ON DECK 129
Once this is accomplished, φ and ∇ φ can be evaluated everywhere in Ω. The solution of the integral equations is obtained
by using a panel method with piecewise 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 surface. The tangential velocity ∂ φ/∂ τ is simply determined by finite difference operators, while the normal
velocity component is obtained from the integral equations. Higher order schemes (i.e. Landrini et al 1999) may lead to
numerical difficulties at the body-free surface 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 appropriate. The continuity of the potential is assumed at those points where the free surface meets a solid boundary.
Though no rigorous justification is available, this procedure gives convergence of the numerical results under grid
refinement (Dommermuth & Yue 1987). Occasionally, when the contact angle becomes too small, numerical problems still
may occur and the jet-like flow is partially cut (Zhao & Faltinsen 1993).
A standard Runge-Kutta fourth order algorithm is adopted to step forward in time the free surface evolution equations.
This requires the solution of four integral equations each physical time step. Though less demanding schemes are
conceivable, we preferred this for the simplicity in changing dynamically the time step. We found this crucial to keep under
control the accuracy 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
example, 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 experiments 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 resulting flow
field resembles the one after a dam breaking. Hence, we have further validated our model by studying this problem.
Numerical results agreed well with small time expansion analytical solutions and experimental 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 3). Once the freeboard is exceeded, the
fluid velocity relative to the the ship determines whether the deck will be wetted or the water will be deviated in the
opposite direction.
Figure 3: ‘Kutta' condition enforced at the edge of the deck.
Our numerical solution has been compared with the quasi-two-dimensional experiments by Cozijn (1995). A
wavemaker was used to generate regular waves, interacting 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
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BASIC STUDIES OF WATER ON DECK 130
by a video camera. Figure 4 shows the comparison between numerical (solid lines) and experimental (circles) free surfaces
profiles (for wave height H=0.128 m, frequency ω=5rad/s, experimental sequence coded 5:36:00–19 from the test No.
A03). We have chosen the free surface configuration just before the shipping 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. Nevertheless, 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 1979). This is supported by the observation that the
measured shape appears “rounded” and highly curved in proximity of the contact point. It would be relevant comparing
experiments with larger scale. Anyway, since the deviation between the two results is strongly localized, it is believed
unimportant in terms of effects on a superstructure 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 reaction point of view.
Figure 4: Water on deck of a rectangular structure due to incoming regular waves (H=0.128 m, ω=5 rad/s, initial freeboard
f=0.1 m). Snapshots of the free surface. Experimental data (circles) are from Cozijn (1995).
We now consider a more realistic set of parameters. 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 simulation. The draft of the ship is
D=17.52 m, while the relative 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 incident waves with
wavelength λ/L=0.75 and height H/λ=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 (1995). In the experimental set up the water level sensors were
located along the centerplane of the ship. In this case, in spite of the three-dimensionality, the comparison indicates
satisfactory numerical prediction of the propagation of the water front. It is observed that the scale of this experiment is
quite larger than the one used by Cozijn, which supports our conjecture about the role of surface tension in the previous
case. In more detail, in location A we observe a strong local effect associated with the first event of water on deck and a
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rather large overprediction 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

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BASIC STUDIES OF WATER ON DECK 131
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 resulting amount of water wetting the deck reduces, as well as the propagating flow
velocity.
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 (1995). twod is the time instant when the shipping starts.
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 Q0 associated with the incoming waves above the mean free surface level over a distance of a wavelength.
Figure 6: Sketch of the main geometrical parameters
considered.
Figure 7: Influence of ship length and stem overhang on
the bow deck wetness.
At first we consider cases where the body motion 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 overhang angle α and of the length-
to-draft ratio L/D of the ship, for H/λ=0.06 and λ/D=6.6. The results are presented in figure 7, where the relative amount of
shipped water is plotted versus f/H. As expected Q/Q0 is strongly influenced by the freeboard of the ship. The length L does
not affect directly the deck wetness severity, while it will through the length-to-wavelength ratio and then through the
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induced body motions, here not considered. As expected, a positive bow stem overhang reduces the relative amount of
ship-ped 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 (α=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/λ,

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BASIC STUDIES OF WATER ON DECK 132
and wavelength-to-draft ratio, λ/D, on Q/Q0 by keeping the body parameters fixed. As expected, as H/λ increases, a larger
amount of ship-ped water is computed, with an almost linear trend for small f/H. For larger f/H, a marked nonlinear
behavior of Q/Q0 is observed as the wave steepness increases. This is natural because 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 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.
Figure 8: Influence of nonlinearity of incoming waves on Figure 9: Influence of wavelength-to-draft ratio on the bow
the bow deck wetness. deck wetness.
Figure 10: History of the relative amount of shipped water Q/Q0. Cases A–E are described in table 1.
The previous analysis considered the first water on deck occurrence. We now consider longer evolutions to analyze the
history of shipping events. Figure 10 gives Q/Q0 as a function of the time, assuming as time origin the instant twod1 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/Q0 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 characterized by the interaction with irregular waves. However the results show that if two
succeeding waves with nearly the same height and wavelength cause deck wetness, the last one gives the most severe
condition.
Table 1: Synopsis of cases considered for studying the history of water shipping.
α λ/L H/λ
case
00
A 0.33 0.064
450
B 0.33 0.064
00
C 0.33 0.095
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00
D 0.05 0.095
00
E 0.67 0.095
In more detail, figure 10 shows that the worse deck wetnesses happen for the steeper conditions (cases C and E). The
corresponding Q/Q0 tends almost to the same value, confirming the steepness as the most important 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,

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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 a=45°), shows a certain effect of the stem overhang in reducing the severity of the
deck wetness.
Figure 12: Influence of body motion. a) restrained body
Figure 11: First four water on deck events for case A (cf. conditions, b) forced heave initially in phase with the
tab. 1). For each event, free surface configurations are water at the bow, c) and d) forced heave initially out-of-
plotted for the maximum freeboard exceedance (circles) phase with the water at the bow.
and for zero flux entering the deck (solid lines).
For the case A, figure 11 shows free surface profiles 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 constant Q/Q0 previously reported. We define, conventionally, the
beginning of the water shipping twod as the time just after the freeboard exceedance for which we have a positive inflow
onto the deck. Further tlast means the time when the shipping of water on deck ends. The flux of water onto the deck is then
zero. With these definitions, we observed that the time scale involved in a “water on deck cycle” does not change markedly
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and remains (40%) of the wave period T.
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 motions of the ship but simply prescribed a priori the motion. In particular,
since we are dealing with bow deck

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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 neglected. Since the stem overhang angle had a small influence, 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 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
(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 parameters 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 recent 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, impacts with superstructures are
likely to occur. This phenomenon 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
instability 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 1999).
Similar circumstances in complex combination with ship motions can cause these extreme events. So far, we have not tried
to model more complex incoming waves than regular steep waves. In spite of this, we have analyzed some extreme cases to
gain some insight, 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=1.0625). 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 λ/L=1.022 and H/λ=0.095 and focused our attention on the interaction of the body with the leading
wave which is characterized 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 t0 with an amplitude A and a
phase β, in the form
(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 β is selected to give a sudden vertical displacement of the ship at t=t0. Some of cases studied,
and discussed in the following, are summarized in table 2.
Table 2: Plunging wave analysis: summary of cases presented.
β
case f/H A/H t0 /T twod/T
a 0.6 0. – – 10.795
−900
b 0.6 0.5 10.626 10.841
−5 0
c 0.6 0.5 10.783 10.844
−100
c1 0.6 0.25 10.783 10.795
−20.5 0
c2 0.6 0.125 10.783 10.790
−110
d 0.6 0.5 10.783 10.783
e 0.5 0. – – 10.777
In figure 13, restrained body conditions are considered, and some free surface configurations are presented. The wave,
reaching the bow, is steep and unsymmetric 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
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BASIC STUDIES OF WATER ON DECK 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 opposite effect and is the main reason why the wave is not breaking before the water
impact on the deck house.
Figure 14: Plunging wave analysis: case b.
Figure 13: Plunging wave analysis: case a.
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 parameters). In the first case, the motion is excited with a phase such that the
instantaneous freeboard at t=t0 is higher than the wave elevation at the bow. By ‘instantaneous freeboard' we consider the
mean freeboard plus the change in vertical position due to heave. The upwards motion of the bow causes lower trough
ahead of the breaking wave and a bow impact occurs. Air entrapment 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 upwards motion of the bow, in facts, limits the increase of the vertical velocity of
the fluid after the impact.
Figure 15: Plunging wave analysis: case c. Figure 16: Plunging wave analysis: case d.
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 t0 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 velocity 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 amplitude A/H=0.25 and A/H=0.125 but maintaining
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BASIC STUDIES OF WATER ON DECK 136
the same instantaneous freeboard at t=t0 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 superstructure than for the bow or the
deck. With A/H=0.25, in particular, the faster rate of the water region to become shallower steepens the wave propagating
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 decreased (case c2, figure 18), the velocity of the wave front becomes larger relative to the plunging jet velocity.
This implies that the impact with the superstructure occurs from the deck. The plunging wave hits the water 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.
Figure 17: Plunging wave analysis: case c1. Figure 18: Plunging wave analysis: case c2.
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 eventually caused the
more common dam breaking type of event. This fact supports the conjecture that a plunging wave on a deck completely dry
is probably due to a “ready-to-break” wave rather than to bow interaction with incoming waves.
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Figure 19: Set up (top) and simulation of the impact with a vertical structure following a dam breaking
Impact with obstacles The fine details of the “external” flow field have a limited influence on the impact

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BASIC STUDIES OF WATER ON DECK 137
because of the small time- and space-scales involved. Therefore, here we study the impact occurring with superstructures
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 removed and the flow develops
along the horizontal deck, fig. 19.a, finally impacting against the vertical wall. The fluid is violenty deviated vertically
upwards, fig. 19.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 handled by the present method, though we believe they are not
relevant to compute the structural loads. As time increases, under the reaction of the gravity, the fluid acceleration
decreases and the upward velocity in the jet decreases until becomes negative. The motion of the water is reversed in a
waterfall, fig. 19.c, overturning in the form of a large wave plunging onto the deck, fig. 19.d. The numerical simulation is
eventually stopped due to numerical break-down.
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 infinite deck. The gravity plays a
minor role since the vertical acceleration of the fluid around the contact point is (5g). In particular, in figure 20, the free
surface close to the wall after the impact is shown in comparison 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 ∆τimp= 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
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=0.6m. 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.
Figure 20: Free surface and pressure distribution during the initial stage of the impact. Solid lines: present numerical
simulations; •: similarity solution from Zhang et al (1996); ∆τimp=τ−τimp . τimp=initial non-dimensional impact time.
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We now discuss in more detail the impact pressures. Right plots of figure 20 present the pressure distributions
corresponding to the free surfaces configurations on the left. According to the numerical results (solid lines), at each time
instant the maximum value

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BASIC STUDIES OF WATER ON DECK 138
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 almost equal to the atmospheric one, conventionally set to zero. The bullet symbols
represent the pressure distribution for the zero gravity impact of a fluid wedge with a flat wall. In particular, this has been
evaluated by solving numerically the boundary value problem for the velocity potential along the wall and taking the values
of both φ and its normal derivative on the free surface from the similarity solution by Zhang et al (1996). At the beginning,
the agreement between the two different results is good as for the corresponding free surfaces. For longer time, the pressure
distributions seem to diverge faster than the free surface elevations. As can be expected, in the “exact” computations, the
maximum pressure decreases while in the zero gravity case remains constant. In facts, as time and water level along the
obstacle increase, the −ρ∂φ /∂ t 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 important to introduce the generalized force ∫ pψnds, where ψn 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
hydroelastic response of the structure. On this ground, both the pressure distribution and its time evolution are important
for structural analysis and hydroelastic effects should be considered if the time duration of the loading over the analyzed
structural part is the same order or smaller than the highest natural period for the considered structural part (Faltinsen
1999).
In figure 22, the dashed lines give the time evolution of the pressure measured by Zhou et al (1999) by a gauge,
sketched at the top of the figure, with circular area of diameter 0.09 m centered at the location C along the wall (cf. top of
fig. 19). We have shifted the time origin consistently with that discussed before for comparing the water level evolution.
The dash-dotted 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 refinement and local regriding have been used to achieve invariance of
the solution and to rule out the dependence
Figure 21: Experimental (Zhou et al 1999, h =0.6 m) and
Figure 22: Top: position of the pressure gauge in
numerical water level hw (meters) at (x/h)A=3.721 and (x/
experiments by Zhou et al (1999, h=0.6 m). Bottom:
h)B=4.542 as a function of time. Positions A and B
experimental and numerical evolution of the pressure along
definited in top plot of figure 19.
the vertical wall (see top plot of figure 19).
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BASIC STUDIES OF WATER ON DECK 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 introduce significant differences in the measured data. This seems to be plausible also in the present case (cf. fig. 21).
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 particular, 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 surface close to the contact point. This is not visible in the dam
breaking free surface profiles in Dressler (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 perfectly 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 after the dam release giving rise to a very peculiar local flow. During the evolution, a hump becomes
apparent in the most advanced flow region. Experimental water levels in Zhou et al (1999) are given only for one test case
and our guess can not be confirmed by other comparisons. Anyway, according to the numerical simulation the
experimental pressure curve is rather close to the pressure evolution at the lower location of the transducer (solid lines),
indicated with the letter D in the sketch above figure 22.
As previously discussed, water on deck can result in impact of fluid with superstructures. As an example of a complex
interaction between the shipped water and the structures, we consider the first water on deck event for case e in table 2. The
ship motion is restrained and the freeboard relative to the wave height is f/H=0.5. The flow evolution along the deck is
presented in the top plot of figure 23. The motion of the fluid along the deck, resulting in a first impact against the deck
house, is accompanied by a plunging wave hitting the upper part of the vertical wall. The related pressure distributions
along the structure are estimated numerically by using the similarity solution as previously described
Figure 24: Case e in table 2. Impact due to the plunging
wave (solid lines) and to the wave front along the deck
( •) . Left: similarity solutions for free surface. Right:
pressure on the wall. Initial impact positions in (0, 0).
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.
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BASIC STUDIES OF WATER ON DECK 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 (0,0). It is observed that the maximum non-dimensional pressure due to the plunging impact is significantly
larger. The impact velocity itself is slightly higher than in the other situation 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 reduced to a
simple two-dimensional wave-body interaction problem. The related unsteady fully nonlinear free surface flow has been
solved numerically. Reasonably good agreement with experimental and analytical results enable us to use this simple
model to gain some fundamental insights concerning the water on deck occurrence, the flow field over the deck and the
impact with superstructures.
In particular, an analysis on the parameters dependence of deck wetness has been carried out, showing that
- For long wavelengths λ relative to the draft D, the wave steepness H/λ mainly determines water on deck
occurrence and severity. The relative amount of shipped water depends nonlinearly on H/λ.
- For small λ/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 pronounced 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 della 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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BASIC STUDIES OF WATER ON DECK 142
DISCUSSION
G.Chahine
Dynaflow, Inc., USA
To enforce the Kutta Condition at the corners (intersection horizontal plane and vertical plane) did you have to use
double nodes (nodes where there are 2 normals)?
AUTHOR'S REPLY
In the practical implementation we looked for a robust treatment of the flow at the corner at the initiation of and during
the shipping of water.
Before the shipping of water, the corner is not wetted and no special treatment is required. More relevant at this stage
is the implementation of a decision criterium of shipping of water. In particular, we allow the fluid to leave tangentially the
stem for a fraction of time step and evaluate the velocity component in the direction of the deck. In case of inward motion,
the free surface is cut at the corner and the shipping of water starts.
At this stage, the corner is fully wetted and a different treatment is adopted. We have not modeled the vortex shedding
that, in principle, should take place there. Further, in the discretization of the integral equation, at the corner we define the
unit normal vector as the vector along the bisector. This avoid the use of a double-node and, in a way, is equivalent to
smooth the discontinuity of the geometry.
[1] C.C.Mei, The Applied Dynamics of Ocean Surface Waves.
Singapore: World Scientific (1983), pp.740.
DISCUSSION
D.K.P.Yue
Massachusetts Institute of Technology, USA
For both long and short waves, the authors find that stronger wave reflection by the ship causes 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 dam-breaking theory. What causes this conflict?
AUTHOR'S REPLY
Consider two wave systems with the same steepness and different wavelengths, say L_a