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24thSymposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
A Finite Amplitude Steady Ship Motion Model
Ray-Qing Lin (David Taylor Mode! Basin, Naval Surface Warfare Center,
Carderock Division)
Weijai Kuang (University of Maryland Baltimore County)
ABSTRACT
We present preliminary results from our new ship
motion model that includes both strong and weak,
three-dimensional interactions between environmental
surface waves and ship bodies in arbitrary water depth.
The linear solutions of steady flow using the new
model agree well with those obtained using Green
function methods. When the Froude number is large,
the fully nonlinear solutions using the new model are
significantly different from linear solutions, even in the
calm water. The interactions between the ship and
incident gravity waves are completely different from
those in linear solutions even with small Froude
numbers and moderate amplitude surface waves, (for
example, Froude number =0.25, and significant wave
height of about 3 meters). The fully nonlinear solutions
show that interaction with incident waves strongly alter
the ship-generated waves, resulting in solutions that can
not be represented by any linear superposition of the
responses of the ship to regular waves.
INTRODUCTION
Modern ship motion prediction in irregular seas has
been studied for the half century since St. Denis and
Pierson (1953) first applied the principle of linear
superposition to the responses of ships to regular
waves.
However, linear ship wave solutions do not apply
to many real problems, such as high speed ships, and
large amplitude ship motions during storm events. In
these problems, nonlinear wave interactions become
very important, through alterations to linear wave
propagation. To understand better the effect of the
nonlinearity, several studies have been conducted on
nonlinear ship waves, such as Liu et al (1992), Lin et al.
(1990, 1994), and Xue (1997~. These studies have
significantly advanced our understanding on the issues.
But, as Beck and Reed (2000) pointed out, these studies
are still in the weakly nonlinear regime (the
nonlinearity parameter is less than 0.2 in all the
studies), and thus have not resolved strong nonlinear
ship-wave interactions.
It is very possible that nonlinear interactions
between surface waves and the ship are comparable to
or even stronger than the linear terms (e.g. Lin and
Segel, 19884. In other words, the nonlinearity parameter
can be on the order of one or greater, outside the
domain of applicability of the weakly nonlinear studies.
If interactions between the surface gravity waves and
between the surface waves and ship waves, which have
been neglected in the weakly nonlinear studies, are
included, nonlinearity becomes even more important
for seakeeping. Since nonlinear wave-wave
interactions increase significantly as water depth
decreases, they can be even more dominant in coastal
regions. Therefore, nonlinear wave-wave interactions
may significantly impact the possibility of ship capsize
(tin and Thomas, 2000~. These problems can not be
properly addressed in the previous models of ship
motions and ship-wave interactions.
This motivates us to develop a new ship motion
model capable of (1) resolving arbitrary nonlinear ship-
wave interactions, (2) investigating impact of surface-
wave-wave interactions on ship motions, and (3) being
flexible on water depth. To achieve these goals, one
must not only carefully examine the various
mechanisms that affect ship motions, but also develop
suitable numerical algorithms for computational
^~ . .
err~c~enc~es.
Studies of surface waves (Lin, 2000) demonstrate
that large amplitude wave-wave interactions, such as
those in winter storms and in hurricane events, transfer
the wave energy rapidly from low frequency to high
frequency waves (direct cascades), as well from high
frequency to low frequency waves (indirect cascades).
Thus, high frequency (i.e. small wavelength) waves
must be well resolved if these physical processes are to
be included in a model. Because of wave propagation,
these small-scale flow structures fill the whole spatial
domain of our interest. On the other hand, ship body-
wave interaction occurs around the ship boundary,
where the pressure imposed on the ship by the fluid
generates flow associated with ship motion, demanding
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Representative terms from entire chapter:
surface waves
fine numerical resolution in the region near the ship.
This imposes many numerical challenges in balancing
the effort on to resolve global, wave motion in the
entire domain and complex local flow near the ship
boundary.
Traditional numerical algorithms (e.g. finite
element method) and global numerical schemes (e.g.
spectral methods) are not very efficient in modeling the
complicated ship-wave interaction problems.
Therefore, we have developed a new, mixed-type
approach that is based on pseudo-spectral method, but
with added local analysis near the ship boundary, so
that both global and local flow features are efficiently
resolved.
In this paper, we shall focus on solving fully
nonlinear seakeeping problems. We refer the reader to
Lin et al (2002) for the details of our algorithm.
This paper is organized as follows: the
mathematical model is described in Section 2. The
benchmark results are given in Section 3. Our new
nonlinear ship motion results are shown in Section 4,
and conclusion are given in Section 5.
MATHEMATICAL MODEL
The basic equations of the flow in the reference
frame moving with the ship include the equation for a
potential, incompressible fluid:
ad ~ + Vh2¢ = 0, for -H < z
The nonlinear terms in the equations (2) and (3) are
solved on collocation points of Founer series (8), which
are then transformed back into spectral space via FFT.
The pressure on the ship body boundary ~ is first
calculated on the (irregular) finite element gods and
then converted on to (regular) collocation points.
Instead of periodic boundary conditions used in our
surface wave-wave interaction model (tin and Kuang,
2002), we introduce an open boundary condition for the
far field flow, permitting both incoming arid outgoing
waves across the boundaries (but the total flux is
conserved). For the details, we refer the reader to (tin
et al. 2002~. The wave numbers (km, kn) are determined
by the dimension of the domain in our modeling and the
truncation order (M, N).
KELVIN WAVES OF A SOURCE-SINK PAIR
One good example for benchmarking our model is
to solve the Kelvin waves generated by a source-sink
pair moving in calm water, because the linear solutions
of this problem have been well resolved with a steady
Green function (e.g. Yang, 20011. Furthermore, both
the new model and Green function method by Yang can
use the exact same and size and same truncation level
to obtain the solutions, which allows the comparison of
the results from the two models. For our
benchmarking, we consider a source-sink pair located at
x0 = (x0, ye, z0) = (+50, O. -2), with the strength q = +
0.1 and the Froude number F = 0.25. Other conditions
are the same as those defined in Yang et al, (20001. To
avoid the singularity of the pair, we follow Miloh and
Tyvand (1993) to define the mirror image (about z = 0)
of the pair at X0m = (150, O. 21. The corresponding
potential function is of the form
¢= qL ~ ..
4~ lie x—x0 ~ ~ x x0 ~ )
It is explicitly that ~ = 0 at z = 0, but at adz ~ o. We
also notice that the potential velocity is symmetric
about y = 0.
Because the solutions are symmetnc, we show only
half distributions of the solutions in Figure 1. The
linear solution of our model is displayed in the upper
half of the figure, while the linear solution by Green
function (Yang, 2001) is shown in the lower half. From
the figure we find that both solutions agree very well.
Our model is further benchmarked for the Wigley Hull
at constant speed in calm water as showed in Fig. 2.
The line represents the nonlinear solution from our
model and the points represent the measurement by
Tokyo University (Yang, 2000) at the Froude number
of 0.25. The Fig. 2 showed that the nonlinear wave
profile by our new model agrees well with the
experimental data. The linear wave profile agrees well
with the experimental data as well when Froude number
equal to 0.25, but the difference in bow waves between
the linear solutions and nonlinear solutions increases as
the Froude number increases. When the Froude number
is high, such as 0.408, the nonlinear solutions are much
more accurate than linear solutions. The detail is in Lin
et al. (20021.
v.o
0.4
ns
-0.3
-0.4
Linear solution by new
model
(a/ Green function
-0.5 0
Figure 1 shows that linear Kevin waves generated by a pair
moving submerge source and sink by Green function and
new model.
Figure 2 shows the wave profiles of Wigley Hull, where the
line represents the nonlinear solution by our model and the
points represent the experimental data by Tokyo University
(Yang et al, 2000).
NEW MODEL RESULTS ON SHIP-SURFACE
WAVE INTERACTION
Coupling our fully nonlinear ship motion model
and our fully nonlinear surface gravity wave model (tin
and Kuang, 2002), we are now capable of studying
nonlinear interactions between ship body and
environmental surface gravity waves. In our initial
studies, the environmental surface gravity waves are
obtained from our new wave model (tin and Kuang,
2002) or standard spectrum (Hasselmann and
Hasselmann, 19811. In Figure 3 we show the surface
wave density of the JONSWAP spectrum with
Hasselmann-Mistsuyasu directional spreading. The
significant wave height is 3 meters. The lines A, B. C,
D, and E in the figure are the energy distribution in the
directional angles of 0°, 15°, 30°, 45°, and 60° (0°
corresponding to +x direction). The ship body in our
study is assumed a simple ellipsoid, with the (non-
dimensional) boundary defined by
X2 + Y2 + Z2 = 1,
where a = 0.55, b = 0.05, c = 0.1. The nonlinear ship
wave profiles for various Froude numbers are shown in
Figure 4, (a) for Froude number = 0.25; and (b) for
:~ it: I: f:.
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Wave Profiles Affected by Surface Gravity Wave (Fr=0.25)
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: ~ : :~: ::: :~ ~ ~~ ~ : ~ : ~ ~~ ~ - ~ :: ~
War—
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::
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:~::~::::~::: :~: 1 :::: ::: : : ~~ ~~ if:
:: :::: ::::::::: I : ::: ::::: :: :: ::: ::
:~ ::::: ::: :~ if:::: :~: i::: ; ~ ~ ~ :~: ::: i: ~~ ~ ~ :~ : : ~~ i: ~ ~ ~~ ~ ~
::::: :: : ~ 1 ~ : ~~ ~.~,:~:~ : : ~::~ ~ : i:: : :,
Wave Profiles Affected by Surface Gravity Wave (Fr=0.3 16)
~~:~ Am::
:~: a ''' ~ t. ,~1~ ~ .2' Bad :~ - Cast A:" *is ~~ .
I: ~~ ~~ ~ t~i.~:Y. - [~ ~~
Figure 3 reference JONSWAP gravity wave spectrum with
Hasselmann-Mistsuyasu directional spreading (Hasselmann
and Hasselmann, 1981). The spectrum is in energy density-
frequency coordinate, where lines A, B. C, D, and E represent
the angles 0°, 15° 30° 45° 60°, with 0°, toward +x coordinate.
x/x! (xI-400m)
Figure 4 shows the wave profiles for a simple ellipse ship
body. The blue line represents the wave profile around the
ship body in calm water, the red line represents the wave
profile around the ship body impacted by the surface wave in
Figure 3; (a) Froude number=0.25; (b) Froude
number=0.3 16.
Froude number = 0.316. In the figure, the red lines are
the nonlinear ship wave profiles in calm water (i.e. no
interaction with external surface waves). The blue lines
are the ship wave profiles after the ship wave and
external surface wave have interacted, but with the
external surface wave subtracted to show the resulting
ship wave profile. Therefore, the difference between the
red and blue lines represents the external surface waves
true effect on the ship waves. From the figure we can
observe significant differences between two kinds of
wave profiles, indicating strong effects of surface
gravity waves on ship wave patterns (and on ship
motion). Furthermore, the surface wave effects increase
as the Froude number increases. Our new findings are
very different from those of previous model studies
(Noblesse et al., 1995, and 1997~.
To understand better how the environmental
surface waves affect ship motion, we eliminated the
short waves and assume that the surface gravity waves
are only swells with long wavelengths that are greater
than the ship length (lOOm). Since the ship waves are
generally short wavelength waves, by separating the
spatial scales of surface waves and of ship waves, we
can observe better the spatial and temporal changes of
the ship waves. Our numerical results are shown in
Figure 5 and Figure 6.
Figure 5 present the total free surface elevation A=
;+ ~ (where ~ is the elevation of environmental
surface waves, and ~ is the elevation of ship motion)
for (a) Fr= 0.25 and (b) Fr= 0.316. The wavelengths
of the surface waves are at least 225m, more than twice
of the ship body length. The results in Figure 6 are
similar, but for the shorter surface waves (of the
wavelength ~ 150m). In both figures we can observe
significant changes in ship waves under the influences
of environmental surface waves. Comparing the results
in Figures 5 and 6, we find that ship wave patterns are
more affected near the ship body in shorter
environmental surface waves in Figure 6. This is in
particular clear in the solutions shown in Figure 6b
because Froude number in b is greater than a. The
explanation is on the difference in time scales of two
kinds of waves: the wave frequency difference between
the ship waves and the environmental surface waves in
Figure 6 is smaller than that in Figure 5. When the ship
wave and surface wave both have similar wave length
and frequencies, the resonant phenomena occur, and the
surface wave impact on the ship wave certainly become
stronger.
The numerical results obtained so far from our new
ship motion model (see Figures 4, 5, and 6) are very
different from those obtained via linear superimposed
theory (e.g. Nobless et al., 1995, 19971. Our new model
results show that regardless the wavelength and wave
height, any surface gravity wave will significantly
modify the ship waves patterns. We also find that the
^..4 ~
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:~
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~ ~4
~ I. ~ ~:
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::::: :: : ::~:: ::~::
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Figure 5 shows that wave-wave interactions between the
surface waves (a swell) and ship body; (a) Froude
number=0.25; (b) Froude number=0.316.
degree of the surface wave impact depends on ship
speed (i.e. Froude number Fr) and the frequency
difference of the two kinds of waves. The faster the
ship (i.e. the greater Fr) or the smaller the frequency
differences, the stronger the impact of surface gravity
waves on ship waves. Of cause, we anticipate that
wave height, as well as water depth also affect the ship
wave patterns. We shall continue our studies on the
ship-wave interactions under various ship geometric
. . .
conditions.
Figure 6 is the same as Figure 5, except the surface waves
are shorter; (a) Froude number = 0.25; (b) Froude number =
0.316.
DISCUSSION
Coupling our fully nonlinear surface wave model
(tin and Kuang, 2002) and our new filly nonlinear ship
motion model, we are able to study the interactions
between the ship body and the surface gravity waves.
The linear results by the new model agree very well
with previous results by Green function (Yang et al.,
2000~. The fully nonlinear results from our new model
demonstrate that surface waves generate profound
impact the ship wave patterns. This impact can not be
described by simple linear superposition of the two
kinds of waves. Our numerical results also
demonstrated that the impact of the surface gravity
waves on ship waves increases with the ship velocity
(i.e. the Froude number in non-dimensional
description). The impact also depends on the frequency
differences of the two types of waves, thus directly
depends on the wavelengths of surface waves and ship
dimension. From our results, we find that when the two
length scales are comparable, the surface wave impacts
are the strongest. This is consistent with the resonant
wave-wave interaction properties.
Our nonlinear results agree well with the well-
known theory (Phillips, 1960, Lin and Perrie, 1997a
and b): the nonlinear interactions do not change the
total wave energy, but result in energy transfer from
high frequency domain to low frequency domain
(indirect cascades) and vice versa (direct cascades).
This is in particular significant in finite amplitude
wave-wave interactions, and in shallow water.
Therefore, ship wave patterns impacted by surface
gravity waves are usually significant.
Although our equations (1~-~5) are only valid in the
reference frame attached to a steadily moving ship, they
can be easily modified for an arbitrarily moving ship by
adding an acceleration term in (2~. Our numerical
algorithm does not need to be modified to
accommodate this more general case, provided that the
numerical domain in our model is sufficiently large. We
need to point out here that by combining the advantages
of local algorithms in the vicinity of the ship boundary,
and the advantage of spectral method for the rest of the
numerical domain, our model is computationally
efficient. In fact, our benchmarking test demonstrated
that the CPU time of our model is orders of magnitude
less than those of the previous models, including
i Super Green Functions . This computational efficiency
could allow us for real time simulation for naval
applications.
ACKNOWLEDGEMENTS
This work is supported by grants from the office of
Naval Research under ILIR program through the David
Taylor Model basin, Naval Surface Warfare Center,
Carderock division. W.K. is also supported by NSF
CSEDI program under Grant EAR0079998. We thank
Dr. Arthur Reed in Hydrodynamics Directorate,
Carderock Division and Dr. Chi Yang at George Mason
University. Their help is necessary for this work.
Finally we would like to thank Terry Applebee, the
Department Head of Seakeeping, in Hydromechanics
Directorate, Carderock Division of David Taylor Model
Basin, who helped us in many ways.
REFERENCES
Beck, R. F.. and Reed, A. M. "Modern Seakeeping
Computations for Ships". 23r~ Naval Hydrodynamics
Symposium. France. 2000.
Hasselmann, S. and Hasselmann, K., "A Symmetrical
method of Computing The Nonlinear Transfer in
Gravity Wave Spectrum". Hamburger
GeophYsikalische Einzelschriften., 1981, pp. 158.
Lin, C. C. and Segel, L., A., "Mathematics Applied to
Deterministic Problems in the Natural Sciences".
Classics in Applied Mathematics, SIAM, MacMillan,
New York. 1988. pp. 609.
Lin, R.-Q. and Perrie, W., " A new coastal wave
model, Part m. Nonlinear wave-wave interaction for
wave spectral evolution.". J. PhYs. Oceano~r. Vol. 27.
1997a. pp. 1813-1826.
Lin, R.-Q. and Perrie, W., "A new coastal wave model,
Part V. Five-wave interactions". J. Phys. Oceano~r.,
Vol 27, 1997b. pp. 2169-2186.
Lin, R.-Q. and Thomas, W., "Ship Stability in the
Coastal Region: New Coasatal Wave Model Coupled
with a Dynamic Stability Model",. 23r~ Naval
Hydrodynamics Symposium, held in France, Val-de-
Reuil. 2000.
Lin, R.-Q., "A new coastal wave model" Recent Res.
Developments in Phys. Ocean. 2000 pp.49-58.
, ,
Lin, R.-Q. and Kuang, W., "Nonlinear Wave-wave
Interactions of Finite Amplitude Gravity wave in a
Global Statistic Spectrum Wave Model in Finite Water
Depth", submitted to Journal of Fluid Mechanics. 2002.
Lin, R.-Q., Kuang, W., and Reed, A., "Finite
amplitude wave-wave interactions between the arbitrary
ship body and surface wave, Part I. Ship wave in calm
water". To be submitted to JFM. 2002.
Lin, W. M. and Yue, D. K. P., "Numerical Solution for
Large-Amplitude Ship Motions in Time-Domain",
Proceeding of Eighteenth Symposium on Naval
Hydrodynamics, The University of Michigan, Ann
Arbor, Michigan. 1990.
Lin, W. M., Meinhold, M. J. Salvesen, N., and Yue, D.
K. P., "Large-Amplitude Motions and Wave Loads for
Ship Design", Proceeding of Twentieth Symposium on
Naval Hydrodynamics University of California Santa
, ,
Barbara, California. 1994.
Liu, Yuming, Dommermuth, D. G., and Yue, D. K. P.,
" A High-Order Spectral Method for Nonlinear Wave-
Body Interactions". Journal of Fluid Mechanics 245.
1992. pp. 1 15-136.
Miloh, T. and Tyvand, P., "Nonlinear transients free-
surface flow and dip formation due to point sink". Phys.
Fluid A. 1993. pp. 1368-1375.
Noblesse, F., and Chen, X. B., "Decomposition of
free-surface effects into wave and near-field
components", Ship Technology Research vol. 42. 1995.
pp. 167-185.
Noblesse, F., Yang C., and Chen, X.-B., " Boundary-
Integral Representation of Linear Free-Surface
Potential Flows". Journal of Ship Research. 1997. pp.
10-6.
Phillips, O. M., "On the dynamics of unsteady gravity
waves of finite amplitude". J. Fluid Mech.. Vol. 9,.
1960. pp. 193-217.
St. Denis, M. and Pierson, W. J., "On the Motions of
Ships in Confused Seas", SNAME Transactions, Vol.
61. 1953.
Xue, Ming, '1hree-Dimensional Fully-Nonlinear
Simulations of Waves and Wave Body Interactions".
Ph.D. Thesis, Department of Ocean Engineering. MIT.
1997. pp. 408.
Yang, C., provided the linear solution by Green
function. 2001.
Yang, C., Noblesse, F., and L 0 inner, .R.,
"Verification of Fourier-Kochin Representation of
Waves". Ship tech. Res. 2001.
DISCUSSION
Choung Mook Lee
Pohang University of Science and Technology,
Korea
The author's contribution to nonlinear numerical
solution for ship-wave and sea wave interaction
problem is appreciated. I would like to point out
that the origin of the coordinate system should be
defined in more conventional way such as the
coordinate origin is set at either at the aft peak or
fore peak, or at the mid-ship. I think it could
have been better to show the effect of the
nonlinear ship and sea wave interactions by
choosing one-dimensional (or regular) head
waves for the sea waves rather than taking the
irregular waves with directional spreading. This
step would show how ship waves nonlinearly
interact with the linear sea wave. Then, show
the present results in which the results of the
nonlinear interaction of ship wave and nonlinear
sea waves are shown.
AUTHORS' REPLY
We presented a model for nonlinear ship-surface
wave interactions that used a pseudo-spectral
method with boundary finite elements. The
model was intended to study nonlinear
interactions between arbitrary ship bodies and
surface wave environments, but not nonlinear
effects on seakeeping. Lee requests that we
consider the nonlinear effects resulting from the
symmetric propagation of one-dimensional
environmental waves parallel to the motion of a
ship.
Nonlinear ship and environmental surface wave
interactions depend on the water depth and wave
numbers, amplitudes, and propagation directions.
Ship wave numbers depend on ship profiles and
speeds. Because nonlinear interactions vanish for
parallel ship and environmental wave
propagation directions (tin and Perriel), i.e.
when the incoming waves are one-dimensional
and move parallel to the ship, the significant
nonlinear effects can only be addressed by
considering two-dimensional incoming waves.
Our simulation results illustrate this point.
Figure 4a of Lin and Kuang2 presents a two-
dimensional environmental wave with a
significant wave height of lm. Figure 4b of and
Lin and Kuang2 presents a ship wave for a
simple ellipsoid for Fr=0.25 in calm water.
Figure 1 shows the resulting nonlinear wave-
wave interactions between the ship and the two-
dimensional environmental waves. Figure 7a of
Lin and Kuang2 presents a one-dimensional
environmental wave, again with a significant
wave height of lm. Whereas the 'nonlinear'
ship wave pattern for the one-dimensional
environmental wave in Figure 2 almost
resembles that of the original ship wave pattern,
the pattern for the two-dimensional wave in
Figure 1 differs significantly. Because nonlinear
effects vanish, one can simply linearly
superimpose the symmetric one-dimensional
incoming environmental and ship waves.
Figure 1
Figure 2
~~-~::::n:~
:~::: o
: ;0,3
REFERENCES:
DISCUSSION
Choung Mook Lee
Pohang University of Science and Technology,
Korea
The author's contribution to nonlinear numerical
solution for ship-wave and sea wave interaction
problem is appreciated. I would like to point out
that the origin of the coordinate system should be
defined in more conventional way such as the
coordinate origin is set at either at the aft peak or
fore peak, or at the mid-ship. I think it could
have been better to show the effect of the
nonlinear ship and sea wave interactions by
choosing one-dimensional (or regular) head
waves for the sea waves rather than taking the
irregular waves with directional spreading. This
step would show how ship waves nonlinearly
interact with the linear sea wave. Then, show
the present results in which the results of the
nonlinear interaction of ship wave and nonlinear
sea waves are shown.
AUTHORS' REPLY
We presented a model for nonlinear ship-surface
wave interactions that used a pseudo-spectral
method with boundary finite elements. The
model was intended to study nonlinear
interactions between arbitrary ship bodies and
surface wave environments, but not nonlinear
effects on seakeeping. Lee requests that we
consider the nonlinear effects resulting from the
symmetric propagation of one-dimensional
environmental waves parallel to the motion of a
ship.
Nonlinear ship and environmental surface wave
interactions depend on the water depth and wave
numbers, amplitudes, and propagation directions.
Ship wave numbers depend on ship profiles and
speeds. Because nonlinear interactions vanish for
parallel ship and environmental wave
propagation directions (tin and Perriel), i.e.
when the incoming waves are one-dimensional
and move parallel to the ship, the significant
nonlinear effects can only be addressed by
considering two-dimensional incoming waves.
Our simulation results illustrate this point.
Figure 4a of Lin and Kuang2 presents a two-
dimensional environmental wave with a
significant wave height of lm. Figure 4b of and
Lin and Kuang2 presents a ship wave for a
simple ellipsoid for Fr=0.25 in calm water.
Figure 1 shows the resulting nonlinear wave-
wave interactions between the ship and the two-
dimensional environmental waves. Figure 7a of
Lin and Kuang2 presents a one-dimensional
environmental wave, again with a significant
wave height of lm. Whereas the 'nonlinear'
ship wave pattern for the one-dimensional
environmental wave in Figure 2 almost
resembles that of the original ship wave pattern,
the pattern for the two-dimensional wave in
Figure 1 differs significantly. Because nonlinear
ettects vanish, one can simply linearly
superimpose the symmetric one-dimensional
. . .
Incoming environmental and ship waves.
: ~~ ::::
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ear and nonlinear computations and by varying
the incident wave amplitudes. The diffraction
phenomena will be independent of incident wave
amplitude (the magnitude of the diffracted waves
will be linear in incident wave amplitude).
AUTHORS' REPLY
V.
Figure 2
REFERENCES:
1. Lin, R. Q., and W. Perrie, "Wave-wave
interactions in finite water", J. Geophys. Res.
Vol. 104, No. C5, 1999, 11193-11213.
2. Lin, R. Q. and W. Kuang ,"A Finite
Amplitude Steady Ship Motion Model",
submitted to JMST, 2002
DISCUSSION
Arthur M. Reed
Naval Surface Warfare Center, Carderock, USA
The authors are to be congratulated on an in-
formative paper introducing a new
computational technique. It will be interesting to
see if the pseudo-spectral approach can be
successfully extended to a wide variety of
problems of interest to the naval hydrodynamics
community.
Regarding the modification of Kelvin waves by
ambient waves as shown in Figure 3. You state
that the modifications are due to nonlinear wave-
wave interactions, which may well be the cause
of the effects you observe. However, it seems
more likely that the cause of the changes in the
wave profiles along the ship while it is in in-
cident waves is wave diffraction the presence
of the ship diffracts the incident waves, such
effects become more significant as the wave
lengths become shorter. These are exactly the
same conditions/situations under which wave-
wave interactions become more significant.
I'm not sure how you would sort out the two
separate phenomena, perhaps by comparing lin-
Dr. Reed pointed out that the modification of
Kelvin waves by ambient waves as shown in
Figure 3 in the original paper should be
considered as a nonlinear effect, which includes
ambient wave effect on the ship wave and the
ship diffraction of the incident waves. He also
inquired how to obtain the nonlinear effect.
Indeed in the Figure 3, we should say that the
modification of Kelvin waves by ambient waves
is the total nonlinear effect. However, if the
ambient waves are swell, which is much longer
than the ship waves, then we can say the
modification is due to the ship wave losing its
energy to ambient wave, that is, as a long wave
interacts with a short wave, the long wave
absorbs the short wave and grows (tin and
lounge.
The following is an example to illustrate our
computed nonlinear effect. Environmental
waves are shown in Fig. 1. A ship wave in the
calm water is shown in Fig. 2. Fig. 3 shows that
linear super imposed the ship wave in Fig. 2 on
environmental waves in Fig. 1. Figure 4 show
the nonlinear interaction between the
environmental waves in Fig. 1 and ship wave in
Fig. 2. The difference between Figure 3 and 4 are
the nonlinear effect.
REFERENCE:
1. Lin, R. Q., and W. Kuang, "A Study of Long
Wave and Short Wave Interactions by Using A
New Spectrum Model", Proceeding of 6th
International Workshop on Wave Hindcasting
and Forecasting. In press.
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