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SHIP STABILITY STUDY IN THE COASTAL REGION: NEW COASTAL WAVE MODEL COUPLED WITH A DYNAMIC STABILITY 983
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Ship Stability Study in the Coastal Region: New Coastal Wave
Model Coupled with a Dynamic Stability Model
R.-Q.Lin, W.Thomas (Naval Surface Warfare Center, Carderock Division, USA)
ABSTRACT
Dynamic Stability assessments of Navy ships can readily be performed for deep ocean regions. Recent emphasis on
joint operations in littoral regions has identified the need to include the effects of coastal wave dynamics in the stability
assessments because the waves are significantly different from deep ocean waves. In coastal regions, the major source
function, nonlinear wave-wave interactions significantly increase as water depth decreases, significantly reducing the
fetch limit, causing the wind generated waves to grow faster and steeper than in the deep ocean. Coast region wave
dynamics are further complicated by the very low frequency waves, such as, edge waves, bottom topographic waves, as
well as solitons etc. Wave-current interactions become very important in littoral regions, and it can generate coastal-
trapped waves and seamount trapped waves.
Waves in coastal regions can grow faster and steeper than in the deep ocean. This raises the possibility that dynamic
capsize is more likely in coastal waters in comparison to deep-water. In this paper we will conduct a comparative
assessment between deep ocean capsize predictions and coastal waters capsize predictions. Wave predictions using the
New Coastal Wave Model (Lin et al) will be coupled with the FREDYN dynamic stability model (De Kat et al) in the
capsize assessment of a frigate-type vessel. The New Coastal Wave Model won the competition in both accuracy and
efficiency in the international conference “Base Enhancement Wave Prediction” in 1998.
I. INTRODUCTION
One of the most critical factors for dynamic stability assessments of navy ships in the coastal region is the correct
estimation of the environment including wind, waves, currents, and storm surge events. Hull form geometry, load
configuration, heading and speed determine a ship's dynamic stability performance. In this study, we will use a naval
frigate to highlight differences in dynamic capsize behavior during the transition from the deep ocean to the coastal region
in the same wind environment. This naval frigate is displayed in Figure 1.
Figure 1. Isometric sketch of Naval Frigate.
A growth of the surface wave is based on the wind, wave breaking, wave-wave interactions, and wave-current
interactions. The last two mechanisms are especially important in shallow water. The nonlinear wave-wave interactions in
shallow water are order of magnitude greater than those in deep water (Lin and Perrie, 1999). Wave amplitudes can
substantially increased when the high frequency waves downshift to the lower frequency though the wave-wave
interactions. Therefore, the wind-waves grow faster and steeper in shallow water than those in deep water. In addition,
wave-current interactions increase with increases, where is current velocity and is group velocity. Current
velocity, usually increases with water depth decreases due to the continuity, but group velocity decreases with water
depth as Therefore, the wave-current interactions are far more important in shallow
water than those in deep water. Often the coastal trapped waves can be observed when the water is shallow and the water
depth varies rapidly (Lin and Huang, 1996). Therefore, the presence of larger and steeper waves in the littoral regions, can
expose a ship to a higher level of capsize risk in comparison to deep water. Furthermore, even in deep ocean, if the
bottom topography rapidly varies, and the free wave frequency is similar to the forced fre
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quency, resonant seamount trapped waves may also cause a ship to capsize.
In this study, we couple the New Coastal Wave Model (Lin and Huang, 1996a and b, Lin and Perrie, 1997; Lin and
Perrie, 1999) with time domain dynamic stability simulations to predict the implications of rapid changes in bottom
topography in two scenarios:
1) Gulf of St. Lawrence (east coast of Canada);
2) Fieberling Guyot in the eastern North Pacific Ocean (32°25' N, 127°47' W).
Unlike the State-of-Art Wave Model (WAM, SWAM) which uses parameterization to account for the wave-current
and wave-wave interactions, the New Coastal Wave Model is physics based on the first principle, which is suitable for
both deep ocean and shallow water. FREDYN is a time domain quasi-nonlinear dynamic stability model that can simulate
the capsize and broaching of intact and damage ships.
Dynamic stability assessments of surface ships usually rely on the standard equilibrium deep sea spectra, such as
JONSWAP, Pierson-Moskowitz, Neumann, Fisher and Roll, Darbyshire gravity wave spectra, etc. All these spectra
formulations are proportional to w−5, where w is intrinsic frequency. These formulations were based on the unit analysis
of Phillips (1958). Later Zakharov and Filonenko (1966) applied a conformal transformation to obtain the Kolmogorov
solution and demonstrated that the equilibrium of deep sea spectra are proportional to uw−4, where u is wind speed. These
were confirmed with observations (e.g. Toba, 1973; Donelan et al., 1985; as well as Phillips, 1985). However, variations
of the equilibrium range spectrum of wind generated waves in finite water are much complicated. The equilibrium energy
spectra are proportional to the range of w−4 to w−1. In this study, the ship stability analysis will be tested by the later
spectra formulations.
II. A COUPLED SURFACE WAVE MODEL AND SHIP DYNAMIC STABILITY MODEL
A. Surface Wave Model:
New Coastal wave model (Lin and Huang, 1996b) is based on the action conservation equation, which truly
calculates the wave-current interactions. The action conservation equation follows: is
(1)
where A is action density, λ is longitude, is latitude, θ wave propagated direction angle, ω is intrinsic frequency, t is
time, Sin, Sds, and Snl are wind input function, wave dispersion, and nonlinear wave-wave interactions, respectively.
The terms in left-hand side of the Equation (1) are based on the nonlinear kinematics (Lin and Huang, 1996b). This
is because the normalized wave steepness, is large even when the wave steepness (ak) is small for
Kh≪1, and where a is wave amplitude, K is wave number, and h is water depth. The characteristic propagation velocities
in Equation (1) are:
(2)
The terms in the right-hand side of the Equation (1) are source functions. Sin and Sds are based on Perrie and Lin
(1997) The formulation of Snl is based on Lin and Perrie (1997) and integration method of Snl is based on Lin and Perrie
(1999). The Snl is most accurate and efficient for both deep and shallow water (Jensen et al, 1998) for the existence
models. However, the formulation is still based on the weak-nonlinear theory (aK 0.3). Lin and Kuang
(2000) apply the Pseudo-spectrum method in order to obtain the Snl, which will be suitable for finite amplitude (strong
nonlinear wave-wave interactions). But it is still in a very primitive stage and not used in this study.
B. FREDYN Ship Dynamic Stability Model
FREDYN is a dynamic stability simulation program which has been developed by the Cooperative Research Navies
(CRNAV) Dynamic Stability working group. CRNAV members include the navies of Australia, Canada, the Netherlands,
United Kingdom, United States as well as the United States Coast Guard and Maritime Research Institute Netherlands
(MARIN). [deKat 98] A formal quality assurance effort has been included in the development of FREDYN ensuring that
it remains in compliance with NSWCCD ISO 9001 standards.
FREDYN models a vessel as a free running intact or damaged vessel under the control of an autopilot.
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SHIP STABILITY STUDY IN THE COASTAL REGION: NEW COASTAL WAVE MODEL COUPLED WITH A DYNAMIC STABILITY 985
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The hull form is modeled from the keel to the edge of the main deck and the motions are solved in the time domain for six
degrees of freedom. The model consists of a nonlinear strip theory approach, where linear and non-linear potential flow
forces are added to maneuvering and viscous drag forces. The architecture of FREDYN is based on the De Kat and
Pauling model and in essence, superimposes the relevant force contributions in the equations of motion as follows [deKat
and deKat]:
INERTIAL FORCE=FROUDE-KRYLOV FORCES
+WAVE DIFFRACTION FORCES
+WAVE RADIATION FORCES
+VISCOUS FORCES
+HULL RESISTANCE FORCES
+PROPELLER FORCES
+RUDDER AND SKEG FORCES
+WIND FORCES
+FORCES DUE TO INTERNAL LIQUID OR DAMAGE FLUID
The Froude-Krylov forces are evaluated up to the instantaneous free surface and include hydrostatic effects. Linear
theory is used in the time domain to estimate the diffraction and radiation forces where a correction is made to the
convolution integrals to account for large amplitude motions. Viscous effects comprise roll damping due to hull and bilge
keels, wave-induced drag due to orbital velocities, and calm water-maneuvering forces. Viscous drag due to cross flow
velocities is estimated empirically, using section-dependent drag coefficients derived from segmented model test results.
Propeller and rudder interaction is also modeled, including the effect of orbital velocities.
FREDYN has the capability of simulating the motion behavior of an intact or damaged monohull vessel in extreme
waves and wind. The ship motions, including capsizing, broaching, and surfriding events, can be displayed in real time
animation on the screen of a Pentium III computer.
III. CAPSIZE PREDICTIONS:
Capsize assessment methodologies often focus on dynamic capsize predictions in deepwater. The need to incorporate
hazard scenarios in capsize assessments has recently been identified by [Alman et al. 1999]. A logical extension to
capsize assessment is littoral regions. As previously stated, we will investigate two scenarios representing two coastal
cases:
1) Gulf of St Lawrence (east coast of Canada);
2) Guyot in the eastern North Pacific Ocean com pared with the deep water case.
A. Case I: Saint Lawrence Gulf
The first scenario involves the transit of a naval frigate in the Gulf of Saint Lawrence southeastward from the mouth
of the Saint Lawrence River Island to the Atlantic Ocean at a speed of 15 knots. In this limited-fetch region, 40-knot
winds have been blowing from the North east over the past 48 hours.
The bottom topography for the Gulf of Saint Lawrence is displayed in Figure 2.. In this figure, the yellow color is
land, and the contour lines in ocean are at 100 meters intervals. Near Îles de la Madeleine is a shallow water area, where
the 400-meter water depth from 400 meters rapidly decreases to less than 100 meters, from north of Îles de la Madeleine.
A semi diurnal tidal current (period is 12 hours) dominates this region. Figure 3 shows the observed current
distribution in the highest tidal current case (Bedford Oceanography Institute of Canada). The wind driving current (NE)
is greater than the tidal current (E-W, 0.2 meters), but the tidal current and its associated water depth variation still
significantly effect the wave height and wave periods. The tidal current contributions are about 30% of the total forcing
(Lin et al, 2000).
Figure 4 illustrates significant wave heights, simulated by New Coastal Wave Model (Lin et al, 1996a, b, 1997, and
1999). The maximum wave height is 11.8 m, which occurs along the north boundary of Îles de la Madeleine, and the
contour line is 2 meters. The wave period distribution (New Coastal Wave Model by Lin et al) is displayed in Figure 5.
The maximum wave period of 16.5 second appears along the north boundary of Îles de la Madeleine, and the contour line
is 2 second. However, the most probable period is about 12 second. For the same wind conditions, the wave height is
about 8.7 meters and the wave period is about 12 second in deep water. These phenomenon along the north boundary of
Îles de la Madeleine is called coastal trapped wave (Lin et al, 2000).
The fate of the imaginary naval frigate is outlined in the track in Figure 6. In the deepwater area, at Location “A”,
waves are coming predominantly from 30 degrees forward of the port beam with roll angles approaching 40 degrees.
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Heading control is challenging, with occasional broaching, yet the ship is able to handle the situation. (See Figure 7.) As
the ship enters the coastal trapped wave area, steep 11.8-meter waves are encountered from the port

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Figure 2. Bottom topography of St Lawrence River and Gulf in Canada. Contour line is 100 meters. Intended transit route of
ship included.
Figure 3. St Lawrence River and Gulf current distribution Figure 5. New coastal model wave period distribution for
in the highest current case. (Bedford Oceanography St. Lawrence River and Gulf. (Lin, 2000)
Institute of Canada)
Figure 4. New coastal wave model significant wave Figure 6. Track of naval frigate with capsize location.
height for St. Lawrence River and Gulf. (Lin, 2000)
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beam. The ship capsizes after the passage of a steep 12-meter wave. (See Figure 8.)
Figure 8. Capsize in region of trapped coastal waves.
Figure 7. Naval frigate motion prediction at Location “A”.
B. Case II: Pacific Ocean Seamount
In the eastern North Pacific Ocean stands a seamount, which extends above the 5000-meter depth ocean floor. (See
Figure 9.) The Fieberling Guyot is 4500 meters in height and 40 km in width with the top 500 meters below the surface.
To the typical navigator aboard ship, the nearly 500 meter clearance between the underside of a ship and the top of the
seamount would provide no basis for safety concerns, as the ship would not be exposed a grounding hazard. There would
be no hesitation in driving a ship right over the seamount if it was on the ship's transit route.
Haidvogal et al (1991) pointed out that due to the free wave frequency of Fieberling Guyot is equivalent to the
forced frequency, significant seamount trapped waves occur. The free wave frequency depends on the Burger Number,
where N is buoyancy force, ( ρ is density of the water), W is width of the seamount, h is the
height of the seamount, and fc is the Coriolis force. The forced frequency is tidal current in this case. The up welling and
down welling are more than 5 times greater than the far field.
In this scenario, a 20 m/s wind blows from the west for 48 hours continuously. The surface current distribution
surrounding Fieberling Guyot is displayed in the vector diagram to the left of the corresponding time series plot of current
on top of the seamount. (See Figure 10.) The tidal current is 0.2 m/s in far field, but the current is 1.05 meters on top of
the seamount. Figure 10 is predicted by the Coastal Current Model (Haidvogal et al, 1991). Figure 11 shows the
significant wave height distribution in this area (New Coastal Wave Model, Lin et al.), with contour lines of 0.5 meters.
The significant wave height in far field (deep water) is about 9.05 meters and on top of the Fieberling Guyot is about
11.48 meters. The far field energy density spectrum is displayed in Figure 12. The wave energy sharply increases on top
of the seamount, as displayed in Figure 13. The peak amplitude of the energy
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MODEL
Figure 9. Bottom topography of Fieberling Guyot.
Figure 10. Surface currents present at the top of the seamount.
SHIP STABILITY STUDY IN THE COASTAL REGION: NEW COASTAL WAVE MODEL COUPLED WITH A DYNAMIC STABILITY
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density spectrum on the top of the seamount is 1.7 times of those in the far field (New Coastal Wave Model by Lin et
al.).
In the FREDYN simulation, the frigate approaches the seamount from the northwest at a speed of 15 knots. (See
Figure 14.) The frigate experiences roll amplitudes has high as 45 degrees with several broaching events, producing yaw
amplitudes as high as 15 degrees. (See Figure 15.). As the frigate transits over the seamount, in stern quartering seas, the
presence of larger, (11.48 meter) and steeper waves induce a capsize due to loss of stability on the wave crest, as shown in
Figure 16.
IV. CONCLUSIONS:
Rapid changes in bottom contours can significantly influence wave fields, as illustrated in the two scenarios
described in this paper. Even when the water is fairly deep, changes in water depth, coupled with surface currents and
wind interactions can produce waves locally which can be substantially larger and steeper, increasing the likelihood of
capsize for some ships. Because in the coastal region the current significantly increases, the wave-current interactions
become very important. The coast trapped waves and seamount trapped waves often occur and can cause ships to capsize.
Furthermore, the nonlinear wave-wave interactions are order of magnitude increase with water depth decreases. A wave
growth needs a much less fetch in shallow water than those do in deep water.
V. ACKNOWLEDGEMENT
This work is supported by grants from the Office of Naval Research under ILIR program though the David Taylor
Model Basin, Naval Surface Warfare Center, Carderock Division.
Figure 11. Coast wave model height prediction at the
Figure 12. Deep water farfield wave spectra.
Fieberling Guyot.
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seamount.
Figure 13. Energy spectrum at the top of the seamount.
Figure 14. Track of naval frigate with capsize at the
seamount.
approaching seamount.
Figure 15. Motion predictions
for
naval
SHIP STABILITY STUDY IN THE COASTAL REGION: NEW COASTAL WAVE MODEL COUPLED WITH A DYNAMIC STABILITY
frigate
Figure 16. Capsize ofr naval frigate passing over the
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VI. REFERENCES
Alman, P.R., Minnick, P.V., Sheinberg, R.Thomas III, W.L., “Dynamic Capsize Vulnerability: Reducing the Hidden Operational Risk”, SNAME
Transactions, 1999, Vol. 107. In Print.
Donelan, M.A., Hamilton, J. and Hui, W.H., “Directional Spectra of Wind-generated Waves”. Phil. Trans. R. Soc. Lond. A. Vol. 315, 1985, pp. 509–
562.
De Kat, J.O., R.Brouwer, K., A.McTaggart, W.L. Thomas, “Intact Ship Survivability in Extreme Waves: New Criteria from a Research and Navy
Perspective”. Fifth International Conference on Stability of Ships and
Lin, R.-Q. and W.Perrie, “A New Coastal Wave Model, Part III. Nonlinear Wave-wave Interaction.” J. of Physical Oceano., Vol. 27, 1997, pp. 1813–
1826.
Lin, R.-Q. and W.Perrie, “Wave-wave Interactions in Finite Depth Water”. J. of Geophy. Res., Vol. 104, No C5, 1999, pp. 11193–11213.
Lin, R.-Q., W.Perrie, and B.Bash, “Sea State Forecasting in the St. Lawrence River and Gluf: Preliminary Operational Implementation.” Technical
Report of Bedford Oceanography Institute. 2000, In press.
Lin, R.-Q. and W.Kuang, “A Pesudo-spectrum Wave Model.” To be submitted J. of Physical Ocean.
Perrie, W. and R.-Q.Lin, “Relating Nonlinear Energy Cascades to Wind Input and Wave Breaking Dissipation”. Nonlinear Ocean Wave, Advances in
Fluid Mechanics. Computational Mechanics Publications. 1997, pp. 61–88.
Phillips, O.M., “The Equilibrium Range in the Spectrum of Wind-generated Waves.” J. of Fluid Mech., Vol. 4, 1958, pp. 426–434.
Phillips, O.M., “Spectral and Statistical Properties of the Equilibrium Range in Wind-Generated Gravity Waves.” J. of Fluid Mech., Vol. 156, 1985, pp.
505–531.
Toba, Y., “Local Balance in Air-Sea Boundary processes. III. On the Spectrum of Wind-Wave.” J. of Oceano. Soc. Japan. Vol. 29, 1973, 209–220.
Zakharov, V.E. and Filonenko, N.N., “The Energy Spectrum for Stochastic Oscillation of a Fluid's Surface”. Doklady Akad. Nauk, 170, 1968, pp.
1292–1295.
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DISCUSSION
J.de Kat
Maritime Research Institute
I would like to commend the authors for presenting a paper that for the first time couples a wind-wave-current model
to a large amplitude, time domain simulation tool for capsize assessment purposes.
As the authors point out correctly, wave conditions can become more onerous in shallow water or in the presence of
current. The way the data are presented e.g. in figures 11, 12 and 13 provides a comparison of the statistical wave
properties, including spectral densities. To assess the difference in probabilities of occurrence of critical waves between
the deep water and shallow water case, it would be of interest to compare the joint probability density functions of wave
height and period of the individual waves. From such plots it should become apparent to what extent waves with critical
length and steepness are more likely to occur in non-deep water conditions.
In the case of the seamount the increased surface (tidal) current velocity has a significant influence on wave
steepening effects. This effect should be highest when the current opposes the wind and wave direction; presumably for
this study the tidal current runs in east-west direction at that location, although it is not quite clear from figure 10. If the
current were to run at an oblique angle with respect to the incoming wave system, this would result in short-crested waves
above the seamount.
The program FREDYN does not account for wave-current interaction and uses the principle of linear superposition
of wave components in deep water conditions. Would the authors recommend further enhancement to the simulation
model to account for shallow water or certain nonlinear wave effects?
Lastly, the predicted capsizes shown are associated with one wave realization for each case; have the authors
performed simulations in different wave realization conditions for the same critical sea state, which also resulted in
capsize events?
AUTHOR'S REPLY
Thank you for your comments. The replies are following:
1) The traditional critical conditions for the non-deep water is tanh kh<1, where k is wave number, h is water
depth. However, in this study, we introduce a new concept of non-deep water conditions: the resonant
internal waves transfer substantial energy from the bottom to the sea surface and surface waves are strongly
affected by this bottom energy. The forcing depends on the bottom topography, buoyancy force (water
density varies with water depth), Coriolis force, as well as external forcing, such as tidal current. The
traditional non-deep water limit is significantly reduced under the new concept of the non-deep conditions.
For example, the Fieberling Guyot seamount trapped waves are affected by the bottom, which is 500 meters
below the sea surface.
2) In the case of the seamount, the increasing internal waves have a significant influence on wave steepening, as
well as the surface tidal current. Therefore, the effect should be highest when the tide is highest and the tidal
current is equal zero in far field, but on the top of the seamount, there are significant currents, which is due to
the resonant internal wave effects as showed in Figure 10a. The current on top of the seamount in Fig. 10a
almost propagated in the same direction of the swell, so we should not observed short-crested waves.
3) The current usually significant increasing with water depth decreasing. This is due to the continuity
principles, as well as nonlinear wave-current interactions. Sometimes, the wave-current may cause the
current significant increasing, for example, the seamount trapped wave case in this study, especially during
the storm surges. The huge current may direct effect the ship motion, so if we can consider the direct current
effects in Fredyn, the simulations may be more accurate.
4) Yes, we used different wave realization conditions for the same critical sea state, which also results in capsize
events.
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