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SHIP MOTIONS AND LOADS IN LARGE WAVES 98
Ship motions and loads in large waves
Ryuji Miyake, Tsuguki Kinoshita, Hiroshi Kagemoto (The University of Tokyo, Japan)
Tingyao Zhu (Research Institute, Nippon Kaiji Kyokai, Japan)
ABSTRACT
Comprehensive experiments regarding ship motions and loads are conducted using a large container-ship model
advancing in a regular wave train. The emphasis is on the nonlinear characteristics of motions and loads in large waves.
Computations are also carried out and their results are compared with the measured ones. A CFD (computational fluid
dynamics) technique is used for the computation, in which the equations of motion of water particles and those of a ship are
solved simultaneously under the exact boundary conditions within the context of an inviscid flow. It is found that the
theoretical computation can account for the nonlinear natures of the measured motions and loads fairly well, while distinct
discrepancies exist from the measured results in some cases even in a moderate sea state.
1 INTRODUCTION
Facing the current keen international competitions, shipyard companies need a rational design basis that enables them
to construct cost effective ships. For that purpose, the expected maximum loads on a ship in her lifetime need be predicted
with good accuracy at the design stage of the ship. Recently several numerical tools that can predict the loads on a ship
advancing in large waves have been appearing (e.g. King, Beck & Magee 1988, Lin & Yue 1990, Huang & Sclavounos
1998). Although some comparisons of ship motions and loads predicted by such numerical tools with experimental ones
can be found sporadically, no systematic experimental studies on ship motions or loads in large waves have ever been
conducted, or at least, been published. Under these circumstances,
Table 2.1 The particulars of the model
Length (L) (m) 5.000
Breadth (B) (m) 0.617
Draft (m) 0.214
Displacement weight (N) 4000.4
Froude Number 0.171
Fig. 2.1 The container-ship model
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SHIP MOTIONS AND LOADS IN LARGE WAVES 99
ship motions and loads in large waves are experimentally investigated in a quite systematic manner using a large
container-ship model of 5m length. Six degrees of motions, vertical as well as horizontal bending moment and hydro
dynamic pressures acting on the ship hull are measured as the ship is freely advancing without any external restrictions in a
regular wave train. The height of the waves is varied to 5cm, 10cm, 15cm, which may correspond to about 2.6m, 5.2m, 7.8m
respectively in a real sea. The advancing direction of the ship relative to waves is varied from 180deg. (head waves) to
0deg. (following waves) with 30deg. interval while the wavelength(λ) is changed from λ/L=0.2 to λ/L=1.5.
The nonlinear characteristics that appear in motions, bending moments and hydrodynamic pressures are discussed in
detail while using the experimental results. Numerical computations based on a newly developed CFD (computational fluid
dynamics) code (Kinoshita, Kagemoto & Fujino 1999), which solves the Euler's equations of motion of water particles
together with the equations of motion of the corresponding ship under exact nonlinear boundary conditions, are also carried
out in order to examine to what extent the observed nonlinearities can be accounted for by the sophisticated numerical
method.
2 EXPERIMENT
The experiment was conducted in a water tank (length:50m, width:30m, water depth:2.14m) at the University of
Tokyo. The container-ship model used for the experiment is shown in Fig. 2.1 and its principal particulars are specified in
Table 2.1. Six degrees of motions, vertical as well as horizontal bending moment and hydrodynamic pressures acting on the
ship hull at 15 locations were measured as the ship was freely advancing without any external restrictions in a regular wave
train. Fig. 2.1 also shows the locations of the 15 pressure gauges (P1~P15) attached to the ship hull for the measurement of
hydrodynamic pressures. The fore part and the aft part of the model is connected at its longitudinal center via a force
transducer so that the vertical and the horizontal moment acting at the midship section can be measured. The six degrees of
motions were measured by optical-fiber gyros. The height of the waves was varied to 5cm, 10cm, 15cm, which may
correspond to about 2.6m, 5.2m, 7.8m respectively in a real sea. The advancing direction of the ship relative to waves was
varied from 180deg. (head waves) to 0deg. (following waves) with 30deg. interval while the wavelength(λ) was changed
from λ/L=0.2 to λ/L=1.5 (λ/L=0.2,0.3,0.4,0.5,0.625, 0.75,0.875,1.00,1.25,1.50). The ship speed U was kept to be
in all the experiments. (The ship speed may be lower than that of a conventional container ship, but
this was the maximum speed that could be achieved in the tank used.) The height of the waves and the ship speed were
controlled in a quite strict manner in such a way that if the desired
Fig. 3.1(1) The experimental results on heave motion amplitude(ZA)
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SHIP MOTIONS AND LOADS IN LARGE WAVES 100
waveheight and/or the desired ship speed was not attained, the experiment was carried out again till the prescribed values
of both the waveheight and the ship speed were attained within allowable differences (4% for the ship speed and 10%~5%
for the 5cm~15cm waveheight).
Fig. 3.1(2) The experimental results on roll motion Fig. 3.1(3) The experimental results on pitch motion
amplitude(ΦA) amplitude(θA)
3 RESULTS OF THE EXPERIMENT
3.1 Motion
Fig. 3.1(1),(2),(3) show some of the results on the motions of the model ship as a function of λ/L (wave length/ship
length). The vertical axes of all the figures represent the nondimensionalized motion amplitude of the Fourier component
that has the same frequency as the encountering frequency with the incident waves. The legends ‘5cm', ‘10cm', ‘15cm'
indicate that the corresponding results were obtained in waves of 5cm, 10cm, 15cm height respectively. Since all the
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quantities shown in the figures are normalized by the amplitude of the incident wave(ζA) or the maximum slope of the
incident wave(kζA), the results for ‘5cm', ‘10cm', ‘15cm' should coincide with each other if the phenomena were of linear
ones. However, the so-called wave-height effect is observed in some cases, particularly in heave motions in χ=120deg.,
90deg. and in roll motions in χ=90deg., 30deg. in such a way that

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SHIP MOTIONS AND LOADS IN LARGE WAVES 101
the response per unit incident-wave amplitude becomes smaller as the waveheight increases. The significant wave-height
effect observed in the roll motion in χ=30deg. in the long-wave range may be attributed to the nonlinear viscous-damping
characteristics that are known to be manifested in large roll motions. On the other hand, the amplitudes of the other motions
per unit incident-wave amplitude including those that are not shown in Fig. 3.1 vary little in all the three waveheights as
typically observed in the pitch motions (Fig. 3.1(c)).
3.2 Pressure
Fig. 3.2, Fig. 3.3, Fig. 3.4, Fig. 3.5 show the results on the pressure amplitudes measured at P1, P2, P5, P14 (see
Fig. 2.1) respectively. Like in Fig. 3.1, the vertical axes of the figures represent the nondimensionalized pressure amplitude
of the Fourier component that has the same frequency as the encountering frequency with the incident waves. ρ,g in the
vertical axes represent the fluid density and the gravitational acceleration respectively. As can be seen in Fig. 2.1, P1, P2,
P5, P14 are located along the same horizontal level (2cm below the calm water surface) on the weatherside of the hull. At
P1, the pressure amplitudes normalized by the incident-wave amplitude(ζA) differ significantly among those obtained in the
three waveheights, in which the pressure amplitudes per unit incident-wave amplitude become smaller as the waveheight
becomes larger. These nonlinearities with respect to waveheight are also observed in the pressures measured at P2, P5,
P14, especially in the short-wave range.
As for the pressures at other locations, Fig. 3.6(1) ~Fig. 3.6(4) show the pressures measured in χ=120deg. at P7, P10,
P12, P13, which are located around the cross section close to the midship (see Fig. 2.1). Again, although the magnitude is
not large, the nonlinearities with respect to waveheight are clearly observed, especially in the short-wave range. (The reason
why the results in =120deg. are shown in Fig. 3.6 is that it is the case in which the nonlinear characteristics are
manifested.)
Fig. 3.7(1),(2) show example time histories of the pressures measured at P1 and P10 respectively. As shown in
Fig. 3.7(1), the time history of the pressure at P1, which is located at the bow just below the calm water surface(see
Fig. 2.1), is truncated at a certain pressure level, which may correspond to the atmospheric pressure. This is a well-known
characteristic of the pressure on a ship hull near the calm water surface, which may come out of the water into the
atmosphere as a large displacement is induced in waves. This should be the main cause of the nonlinear wave-height effects
observed in the frequency characteristics of the pressure at P1 that were shown in Fig. 3.2(1), because the results shown in
the figure are those obtained by a Fourier decomposition while taking no consideration on the possible truncation of the
pressure time history. In order to confirm this speculation, an attempt was made in which the pressure amplitudes were
reanalyzed by a Fourier decomposition after sinusoidal time histories were reproduced from the truncated ones. The typical
results obtained in this manner are shown in Fig. 3.8(1), Fig. 3.8(2). In the figures, the results obtained by a Fourier
decomposition without accounting for the pressure truncation at the atmospheric level are shown in Fig. 3.8(1) while those
obtained with the consideration on the pressure truncation are shown in Fig. 3.8(2). As evident in the figures, the once-
seemed nonlinearities with respect to the waveheight turn out to be accounted for in most part if the truncation of pressure
time histories is taken into account. On the other hand, as is shown in Fig. 3.7(2), although the time history of the pressure
at P10, which is located well below the calm water surface, is never truncated because the corresponding portion is always
below the water surface, an appreciable nonlinearity still exists as we observed in Fig. 3.6(2).
Overall, although only a limited number of results are shown here due to the limitation of space, the nonlinearities with
respect to waveheight are particularly manifested in quartering waves (χ=120deg., 30deg.) in all locations of the pressure
gauges.
3.3 Be nding moment
Fig. 3.9, Fig. 3.10 show the results on the vertical bending moment and the horizontal bending moment respectively
acting at the midship section. The vertical axes of the figures represent the nondimensionalized amplitude of the
corresponding moment. The amplitude shown in the figures is that of the Fourier component which has the same frequency
as the encountering
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at P1
SHIP MOTIONS AND LOADS IN LARGE WAVES
Fig. 3.2 The experimental results on pressure amplitude(PA)
at P2
Fig. 3.3 The experimental results on pressure amplitude(PA)
102

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at P5
SHIP MOTIONS AND LOADS IN LARGE WAVES
Fig. 3.4 The experimental results on pressure amplitude(PA)
at P14
Fig. 3.5 The experimental results on pressure amplitude(PA)
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SHIP MOTIONS AND LOADS IN LARGE WAVES
Fig. 3.7 The example time histories of measured pressures
amplitude
Fig. 3.6 The experimental results on pressure amplitudes(PA) at P7, P10, P12, P13 in χ=120deg.
Fig. 3.8 The effect of the pressure truncation on a Fourier
104

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SHIP MOTIONS AND LOADS IN LARGE WAVES
bending-moment amplitudes(MZ) at the midship
Fig. 3.10 The experimental results on the horizontal
The nonlinearities
Fig. 3.9 The experimental results on the vertical bending-moment amplitudes(MY) at the midship
Fig. 3.11 The asymmetricity of the vertical bending
moment (χ=120deg.) frequency with the incident waves.
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SHIP MOTIONS AND LOADS IN LARGE WAVES 106
with respect to waveheight are observed again in the short-wave range(λ/L=0.2~0.8) in quartering head waves
(χ=120deg.) for both the vertical moment and the horizontal moment as the moment per unit waveheight becomes smaller
with the increase of the waveheight. They are also observed in the horizontal moment in the long-wave range in quartering
following waves (χ=30deg). The nonlinearities in χ=120deg. may be due to the nonlinearities of the pressures because, as
already shown, the nonlinearities of the pressures are commonly observed in χ=120deg. in the corresponding short-wave
range. On the other hand, the main cause of the nonlinearities observed in the horizontal moment in χ=30deg. may be the
strong nonlinear features of the roll motion as we observed in Fig. 3.1(2). Other than these, although they are not shown
here, both the vertical and the horizontal moments show fairly linear characteristics. It is known that the vertical moment on a
container ship possesses a crest-trough asymmetricity and this is reconfirmed in Fig. 3.11, in which the sagging and the
hogging bending moment at the midship measured from the moment level acting on the ship advancing in a calm water are
separately shown. (The vertical axis is not nondimensionalized but represents the moment itself.) It is evident that the
sagging moment is larger than the hogging moment and this tendency is enhanced significantly as the waveheight becomes
larger.
4 COMPUTATION
4.1 Basic idea of the computation
As for the theoretical estimation of ship motions and loads in large waves, quite a few works are now being
conducted, although the number of works that have been applied successfully to the computation of motions and loads of a
practical ship advancing in large waves is still limited. LAMP code developed by Lin, Yue et al. (1990) or SWAN code
developed by Sclavounos et al. (1998) account for the exact body boundary condition under the incident-wave surface
whereas the free surface condition is linearized about the incident wave surface based on a so-called weak-scatter
hypothesis. Comparisons of these calculations with experimental results on motions and loads in large waves can also be
found in some literatures. LAMP code, especially, has been extensively tested in comparing the results on such quantities
as motions, loads, waves and pressures (Lin & Yue 1994, Lin, Shin et al. 1997). Most of these current methods including
LAMP, SWAN are velocity-potential-based panel methods. Here, on the other hand, we apply a CFD (computational fluid
dynamics) for the calculation of ship motions and loads by directly solving simultaneously the equations of motion of water
particles and those of the corresponding ship together with the continuity equation of the fluid while discretizing the fluid
domain into hexahedral grids. Since most of the nonlinearities that are observed in the motions and loads in large waves can
be attributed to non-viscous forces, Euler's equations are used in the present computations as the equations of motion of
water particles. The present method, as described above, discretizes the fluid domain and solves fluid velocities and
pressures by a finite-difference scheme while satisfying the boundary conditions on the free surface as well as on the body
surface exactly within the context of an inviscid flow. Since, when viscous effects are neglected, a panel method based on a
velocity potential theory can be used, the volume-discretized CFD computation may not seem a good strategy. However,
the exact treatment of moving boundaries such as a free surface or a body surface in some cases turns out to be much easier
than the surface-discretized panel method. This is particularly true when strong nonlinearities associated with the motions
in large waves are involved.
Although the details of the present CFD calculation method can be found in Kinoshita et al. (1999), a brief explanation
is reproduced here to give a rough idea of the method.
4.2 Computation proce dure
The equations of motion of water particles (Euler's equation) are described with respect to a space-fixed coordinate
system (x, y, z) whereas those of a floating body are described with respect to a body-fixed coordinate system (X, Y, Z).
These two sets of motion equations and the continuity equation of the fluid are solved under the exact boundary conditions
on a free surface as well as on a body surface by a finite-difference scheme. The temporal derivatives are
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SHIP MOTIONS AND LOADS IN LARGE WAVES 107
evaluated by a first-order Eulerian scheme. As for the spatial derivatives, a third-order upwind-difference scheme is used
for the convection terms while a second-order central-difference scheme is used for the other terms. After obtaining the X,
Y, Z velocities of the body and the angular velocities of the body around the X, Y, Z axes, the locations and the attitudes of
the body with respect to the space-fixed coordinate system are updated at each time step.
Fig. 4.1 The hybrid grid system used for the CFD computation (one-half of the vertical cross section of the computational
domain)
4.3 Grid system
In a CFD computation, which uses a volume-discretized grid system, the choice of the grid system is one of the
crucial points for the successful computation. In the present calculation, a hybrid grid system is used in which a body-fixed
boundary-fitted grid system is used in the vicinity of a body whereas a space-fixed grid system is used away from the body
as shown in Fig. 4.1. The two grid systems are overlapped partially at the perimeter of each grid system. Although the
body-fixed grid system is fitted to the body boundary, it is not fitted to the free surface as shown in the figure, but, instead,
the free surface is displaced through the grids in such a way that the mass continuity of the fluid is satisfied in the grids
located along the free surface.
Fig. 4.2 The comparisons of the calculated and measured
results on motions in head waves
Fig. 4.3 The comparisons of the calculated and measured
results on pressures in head waves
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SHIP MOTIONS AND LOADS IN LARGE WAVES
Fig. 4.3 The comparisons of the calculated and measured results on pressures in head waves
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results on pressures in head waves
SHIP MOTIONS AND LOADS IN LARGE WAVES
Fig. 4.3 The comparisons of the calculated and measured
the waveheight (χ=180deg.)
Fig. 4.4 The nonlinearities of the pressures with respect to
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SHIP MOTIONS AND LOADS IN LARGE WAVES 110
Fig. 4.5 The comparison of time histories of measured and calculated pressures in head waves
4.4 Results and their comparison with experiments
4.4.1 Fre que ncy response characteristics
Fig. 4.2, Fig. 4.3 compare the results of the CFD computation on the motions and on the pressures respectively in head
waves of 10cm height with the corresponding experimental ones. Since the experimental results are the Fourier component
that has the same frequency as the encountering frequency of the incident waves, the calculation results were also Fourier-
decomposed so that the component that has the encountering frequency can be compared. The pitch motion is predicted
well by the present computation, whereas the calculation results on the heave motion are a little smaller than the
corresponding experimental ones. As for the pressures, the large pressures acting at the bow section P1 is predicted quite
accurately by the present calculation. On the other hand, pressures on P7, P8, P9, P10, P11, P12, P13, which are located
along the cross section close to the midship (see Fig. 2.1), are overestimated by a significant amount in the long-wave
range. It is characteristic that this tendency persists for all the pressures measured on the section. Although the detailed
cause of this discrepancy is the subject of a future study, this may have to do somehow with the underprediction of the
heave motion observed in Fig. 4.2(1). As we go further downstream along the ship hull, good agreements between the
present calculation and the experiment are recovered as we observe in the comparisons at P14, P15.
4.4.2 Nonlinearities with respect to waveheight
Fig. 4.4 shows to what extent the present CFD computation can reproduce the nonlinear characteristics with respect to
waveheight that are observed in the experimental results on the pressures. The significant nonlinear characteristics of the
pressure at the bow (P1) are accounted for fairly well by the present computation. The rather subtle difference of the
pressures per unit incident-wave amplitude in the three different waveheights observed at P4, P14 are also reproduced by
the present computation quite well. On the other hand, at P7, which is located on the cross section close to the midship, the
calculation results differ form the experimental ones in two respects, that is, (1) the magnitude of the calculation results is
significantly larger than that of the experimental ones in all the three waveheights, which was already noted in 4.4.1. (2) the
calculation results show distinct wave-height effect while the experimental evidence is that little wave-height effect exits
among the experimental results obtained in the three waveheights
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4.4.3 Time histories
As example comparisons on time histories, Fig. 4.5(1)~Fig. 4.5(3) compare the calculated and measured time histories
on pressures measured at P2, P10 and P12 respectively. The truncation of the pressure observed in Fig. 4.5(1) at the
atmospheric pressure level can be accounted for quite well by the present CFD computation. (The zero level in the vertical
axis does not necessarily represents the atmospheric pressure.) At P10, which is located deep enough to always remain
under the water surface, the

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SHIP MOTIONS AND LOADS IN LARGE WAVES 111
time history of the pressure shows a fairly sinusoidal nature and it is well reproduced by the present computation. As for the
pressure at P12 shown in Fig. 4.5(3), the trough of the computed time history is deeper than the measured one although the
distorted nature from a sinusoidal curve is somehow reproduced by the computation.
5 CONCLUSIONS
A comprehensive experiment on ship motions and loads in large waves was conducted using a large container-ship
model. Some of the results were compared with those obtained by an inviscid CFD computation. The followings may be
concluded from these studies.
(1) Nonlinearities with respect to waveheight are observed in such a way that the motion and load responses per unit
incident-wave amplitude acting on the ship become smaller as the waveheight increases. These nonlinear characteristics are
especially manifested in quartering waves.
(2) The time history of the pressure measured at the location close to the water surface is sometimes truncated at the
atmospheric pressure level as the corresponding point is exposed to the air due to the large motion of the ship. This may be
the main cause of the nonlinear characteristics of the pressure with respect to waveheight that are observed in the
experimental results. However, the pressures at the locations well below the water surface still show nonlinear
characteristics even though the corresponding points always stay under the water surface.
(3) The asymmetricity of the vertical bending moment acting at the midship is clearly observed in such a way that the
sagging moment is larger than the hogging moment. This tendency is significantly enhanced as the waveheight becomes
larger.
(4) The motions and loads are predicted fairly well by the presented CFD computation. The notable exceptions are the
pressures on the cross section close to the midship in head waves, where (a) the magnitude of the calculated pressures is
significantly larger than that of the measured ones, (b) the calculated pressures show distinct nonlinear characteristics with
respect to waveheight whereas the measured ones show fairly linear nature.
(5) Overall, the nonlinear characteristics that are observed in the measured pressures in large waves can be accounted
for fairly well by the presented CFD computation. On the other hand, there exist some distinct discrepancies in the
calculated motions and pressures from the measured ones even in moderate waveheight, which may indicate that the
presented CFD computation procedure still needs further improvements.
ACKNOWLEDGEMENT:
The authors would like to acknowledge Drs. Iwao Watanabe, Shigesuke Ishida, Katsuji Tanizawa of Ship Research
Institute, Ministry of Transport, Japan and Atsushi Kumano of Nippon Kaiji Kyokai for their support and valuable
comments in conducting the experiments.
REFERENCES
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Hydrodynamics, The Hague, The Netherlands, 1988.
W-M.Lin and D.K.P.Yue: Numerical solutions for large-amplitude ship motions in time domain, Proc. 18th Symposium on Naval Hydrodynamics, Ann
Arbor, Michigan, 1990.
Y.Huang a nd P.D.Sclavounos: Nonlinear ship motions, Journal of Ship Research, Vol. 42, No. 2, 120–130, 1998.
T.Kinoshita, H.Kage moto a nd M.Fujino: A CFD application to wave-induced floating-body dynamics, Proc. 7th Intl. Conf. on Numerical Ship
Hydrodynamics, Nantes, France, 1999.
W-M Lin, D.K.P.Yue: Large-amplitude motions and wave loads for ship design, Proc. 20th Symposium on Naval Hydrodynamics, Santa Barbara,
California, 1994.
W-M Lin, Y-S Shin, J-S Chung, S.Zhang a nd N.Salvesen: Nonlinear predictions of ship motions and wave loads for structural analysis, Proc. 16th Intl.
Conf. on Offshore Mechanics and Arctic Engineering, Vol. 1-A, Yokohama, Japan, 1997.
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