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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
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 <r1, (1) where ~ is the potential velocity and the subscript h indicate the horizontal differentiation. The momentum balance in the fluid domain is described by the dynamic equation, ~'P =_ gzP_(V0) vship(V¢) (V¢~+vV ¢, at z=0, (2) where Vship is the ship velocity, g is the gravitational acceleration, p is the pressure field, p is the fluid density and v is the fluid kinematic viscosity. It should be pointed out that viscous dissipation disappears for the potential incompressible flow (11. However, we keep this term in (2) to model the wave-breaking mechanism. The relation between fluid flow and wave elevation ~ is described by the kinematic equation at aZ ~ h0) Vship Vh~ · V¢, at z = ~ (3) On the ship body boundary ~ and at the bottom of the ocean z= -H. flow is impenetrable, i.e. (4) and At + (VhH) (Vim + vship ~ = 0, at z = -H. (5) where n is the unit normal of the ship boundary. When the bottom is flat, (5) can be simplified to ¢=0, atz=-H, az (6) We should point out here that the equations apply only to constant ship speed. In more general cases, an artificial forcing aVship/at must be added to (21. In the far field boundary rfar-f~eld' we use open boundary condition: 71 =~0, at x on rfar-field (7) In our modeling, the potential velocity ~ and wave elevation r1 are expanded in Fourier series: p~x,y,z,t)] ~11(X, y, z,t)] >~ [(/)ntn (t) cosh kmn (H + z): ei(kmX+kn I) + c c n`,n=0 71n~n (t) ~ (8) where k,nn = (km2+kn21~/2, M and N are truncation order, and c.c. stands for the complex conjugate parts. The ship body in our model is described by local grids that are generated by either a finite difference or a finite element methods. In our numerical simulation presented in this paper, we use the finite element ship body grids.
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 xx0 ~ ~ 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:. :::: - ::: :::: :: :: :~:~ :~: . ~:~ All' : :: : :,,~ 30::: J :: :::: 5 :: :~, Wave Profiles Affected by Surface Gravity Wave (Fr=0.25) :~ :.~'~:~.~ ' ~~ :: ail: ~l,~2Nt:t1,4~spt:~:~."R,= i: i: ~~ :~ ~~ :~ ~~ ~ ~~ ~ ~ : ~ : :~: ::: :~ ~ ~~ ~ : ~ : ~ ~~ ~ - ~ :: ~ War i', ~~ ~~ ~~-~:::~ ~~ ~~ :: ~~ ~~ ~~ ail:: i: ,, ~ ~~ ~ ~ ~~ ~~ ~~:~ ~:~ i: ~ : :: : :::::: :::::: :::: :::: : ~ ::: L::: :::::::::: : ~~ : i:: ::::: : : ~ - i: :~:~::~ ~ ~ ~ : ~~ ~~ i: : : i: :~ : : :::: 1 : :: :::: : i:: i: i:: ~ i: .: 1~ : : : : :: : :: ::::::: : ~ :: T: : :~:::: :: : i: :::::::::: ~ :~: 1 ::~::::: i:: : : ::::: :::: :: ~ :~ ::: ::: i: i: ~ ::: ,4: : :~:: :: :: :~: : ~ ~ if:: ::: :: :~::~::::~::: :~: 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 ~ : ~~.:~ ~ ::::: :: ::: :. ~ ::: : ~~ : ~ : :: : ~ ~ ~ : :~ :: :. :::: ::~ it: ~ :: ~ . ~ : ~~ : ~ :: ~ ~4 ~ I. ~ ~: : :~ :::::: ::::: : :~: ~~ :: :~ :: :: ~ ~ .,.1 :~: : oft ::::: :: : ::~:: ::~:: I: ~ ~:~:::: ~ :~ :: ~~ ~ ... ~ ~ ~ . . . ~ ~ ~~.~.~ . ~ .. . , . ... .^ . : if,. :::~::: : :: : .~s. : . ~ 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. : ~~ :::: I, ~ ::: ~ :~: In :~: Paula :::: ~ 4~ I: : ~ :~$ .. : :.o,4 I: :: as =- ~ ~ ~ S :: ~ Figure 1 at . x: :: ~
F of not 02 D.1 n 4.1 And And 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.
Qua of ~2 0.1 C atl.1 -02 -0.3 -0.5 D`5 0,4 03 ~2 O.t ~1 ~.3 Hi' .5 -0.5 MIX Figure ~ °51 OAT 03 Neal O. ~ ~ Figure 4.
