Numerical Study on the Nonlinear Characteristics of Longitudinal Motions and Wave Loads for SWATH Ship in Regular Head Waves

DENG Lei,DONG Wen-cai,YAO Chao-bang

(Dept.of Naval Architecture Engineering,Naval University of Engineering,Wuhan 430033,China)

Numerical Study on the Nonlinear Characteristics of Longitudinal Motions and Wave Loads for SWATH Ship in Regular Head Waves

DENG Lei,DONG Wen-cai,YAO Chao-bang

(Dept.of Naval Architecture Engineering,Naval University of Engineering,Wuhan 430033,China)

Based on the viscous flow field solved by RANS equations and VOF method,the nonlinear characteristics of longitudinal motions and wave loads for Small Waterplane Area Twin Hulls (SWATH)in regular head waves are investigated.Comprehensive comparison between the numerical results and experimental data is carried out to validate the proposed approach.Further,the influence of the wave height on the 1st and 2nd harmonics of heave and pitch motions,vertical accelerations as well as wave loads are analyzed in details.The study indicates that there are nonlinear effects for the longitudinal motions and wave loads of SWATH ship in regular head waves,which is particularly significant at the resonance region.Additionally,the nonlinear characteristics are much stronger for the vertical acceleration as well as wave loads than for the heave and pitch motions.

SWATH;RANS;motion in waves;variation of wave height; nonlinear characteristics

0 Introduction

In the theoretical framework of ship seakeeping,the ship advancing in waves is normally assumed as a linear system[1],which is characterized by the additivity and homogeneity.However,the assumption above is only effective under the circumstance that the ship motion amplitude is relatively small.If the motion amplitude is increasing,the ship responses will tend to show the nonlinear characteristics,which are mainly reflected in two aspects：(1)The response transfer functions relative to the wave amplitude are not independent of the wave amplitude; (2)There are 2nd and higher harmonic components in ship responses.

The experiment of ship responses in different wave heights for a model of the S-175 containership advancing in regular head waves was carried out by O’Dea[2],which showed that the motion transfer functions decrease with the increasing wave amplitude as the 2nd and 3rdharmonics increase steeply.In addition,Adegeest[3]presented experimental data of vertical loads induced by regular waves on two Wigley hulls.Strong nonlinear effects were observed, which include 2nd harmonics as large as 85%of the 1st harmonic.Another model test of S-175 containership in different wave heights was explored by Fonseca[4],which indicated that the nonlinear behaviors were more significant for the vertical loads than for the motion responses.

The studies mentioned above are mainly focused on the monohulls.For some special ship forms such as the SWATH consisting of the lowerhulls,struts and cross structure,the nonlinear behaviors with the water entry of cross structure and lowerhull emergence seem to be apparent as the motion amplitude is increasing with the height of incident wave.However,the nonlinear characteristics of ship responses with the wave height are rarely discussed for this kind of ship.

For domestic and foreign researchers,model test method[5-6],potential theory method[7-8]and CFD method[9-11]are the main tools to investigate the SWATH ship motion in waves.The model test is the most efficient apparently but may take a long period and need relatively high cost.A revised potential theory method proposed by Lee[12-13]is usually adopted to dispose the effects of stabilizer fins and viscous issue for SWATH motion in waves,whose accuracy is influenced by the ship form,airfoil of stabilizer fins as well as the selection of viscous coefficient. The increasing development of computational techniques and numerical approach has made the CFD method possible to study on this issue.Comparing with model test and potential theory methods,CFD method is more efficient at a lower cost and can be effective to take into account for the viscous and all nonlinear effects such as the free surface broken,green water on deck and water entry of cross structure sufficiently.

By the CFD method based on RANS equations,present paper validates the availability for the SWATH ship motion in regular head waves and investigates the 1st and 2nd harmonics of heave and pitch motions,vertical accelerations as well as wave loads in different wave height conditions for SWATH ship.The nonlinear relationships between the ship responses and incident wave height are obtained.Further,the variation patterns and internal reasons of ship responses versus the wave height are explored.

1 Numerical wave tank

1.1 Governing equations and turbulence model

RANS(Reynolds-averaged Navier-Stokes)equations are the governing equations of viscous hydrokinematics and hydrodynamics,which are adopted as the basic equations to solve the viscous flow field in present study.The specific forms of RANS equations can be written as：

where ρ is the fluid density,μ is the fluid viscosity,p is the static pressure,fiis the unit massforce,and ui,ujare the velocity components respectively.In this study,the fluid is liquid water.The RNG k-ε turbulence model is selected to close the hydrodynamic problem.The free surface is handled with a VOF-type free surface capturing approach[14].

1.2 Wave generation and damping

According to the linear wave theory in infinite deep water,the wave generation model is implemented into the RANS code,whose regular waves are defined as

The velocities of flow field are determined as

where ζ is the wave elevation,a is the wave amplitude,k is the wave number,ω is the natural frequency of the wave and u,v and w are the longitudinal,horizontal and vertical velocity component of fluid particles,respectively.

A sponge layer is set at the end of tank to absorb the wave reflection,which is achieved by the resistance-term addition into the momentum source function.Of special note is that the sponge layer makes vertical velocity of fluid particles decay forcedly.

2 Geometry,conditions and numerical model validation

2.1 Geometry and conditions

The studied ship model is a SWATH,which is installed with the bilge keel and stabilizer fins.Fig.1 and Tab.1 show the geometry and principal particulars of this model.

The computational domain and boundary conditions are shown in Fig.2.Due to the symmetry of the ship hull,symmetry boundary condition is applied and only half of the ship hull is modeled in present simulation.6-DOF module is used to solve the heave and pitch motions of the SWATH ship in regular head waves[16].In the solution process of ship motions,two coordinate systems are adopted：(1)The earth-fixed coordinate system,whose origin coincides with the starboard keel midpoint of theship in static situation;(2)The ship-fixed coordinate system,whose origin is located at the center of gravity.The computational domain in the earth-fixed coordinate system is-0.5λ（+ 2.5LWL）＜x＜0.5LWL+λ,-LWL＜y＜yGand-LWL＜z＜D+0.5LWL,where λ is the incident wavelength and yGis the horizontal coordinate of ship center of gravity in earth-fixed coordinate system.

Fig.1 Three dimensional model

Tab.1 Principle particulars of model

Fig.2 Computational domain and boundary conditions

Fig.3 Mesh for numerical simulations

2.2 Validation of numerical model

The time history of incident wave height is illustrated in Fig.4,whose monitor point is set at 0.7m ahead of the ship model.The incident wave height is H=1/50LWLboth for λ/LWL=3 and λ/LWL=5.It is observed that the wave height curves are relatively steady and relative errors between average wave heights obtained by simulation and the theoretical wave heights are less than 5%.

Fig.4 Time history of the incident wave height(H=1/50LWL)

The ship responses here are focused on the heave,pitch,vertical acceleration,heave force and pitch moment,which are denoted as Za,θa,A,Fzand My,respectively.For the 1st and 2nd harmonic amplitudes,superscripts(1)and(2)are added to the symbols above.As an example,Za(1)and Za(2)represent the amplitude of 1st and 2nd harmonics of heave motion,respectively.

In order to make the comparison and analysis of numerical results convenient,the transfer functions are defined to represent the 1st harmonics of ship responses.For the heave motion,the transfer function is defined as the ratio for its 1st harmonic amplitude to the incident wave amplitude and the 2nd harmonic amplitude is also normalized by the incident wave amplitude.Specific definitions of other nondimensional ship responses are listed in Tab.2.Similarly,the 1st and 2nd harmonic amplitudes of ship responses are nondimensionalized by the same way,which is easy to compare the magnitudes of the harmonics.

Tab.2 Definitions of the nondimensional ship responses

Fig.5 provides the calculated transfer functions of motions at Fr=0.236 and H=1/50LWLfor the ship model advancing in head waves,which are compared with the experimental data. From the figure below,the motion transfer functions obtained from the experimental data and the calculated ones both have resonance peaks for the nondimensional wavelength ranging from 3 to 4.Additionally,the calculated results agree well with the experimental data,which indicates that the proposed numerical model in present study is reliable and validated.Therefore,further investigation of the nonlinear characteristics of SWATH ship longitudinal motions and wave loads in different wave heights will still be based on this model.

Fig.5 Comparisons of motion transfer function between calculated results and experimental data

3 Nonlinear characteristics of the longitudinal motions and wave loads

In order to explore the impact of wave height on nonlinear characteristics of longitudinal motions and wave loads for SWATH ship,the numerical simulation is mainly focused on the regular head wave conditions in this section.The SWATH model here is advancing with the speed Fr=0.236 in regular head waves,in which the wavelength is ranging from λ/LWL=1.5 to λ/LWL=7.5.Three kinds of wave heights including H=1/50LWL,1/30LWLand 1/15LWLare selected to evaluate the influence of the wave height on the ship responses.Furthermore,wave height H=1/10LWis added to capture the nonlinearities around the resonance region of the ship motions.

3.1 Longitudinal motion responses

Fig.6 illustrates the transfer functions of heave and pitch motions versus the nondimensional wavelength at several values of wave heights.Fig.7 shows the variation of motion transfer functions with the wave height for two wavelengths(λ/LWL=3 and λ/LWL=4)around the resonance region.

From the Fig.6 and Fig.7,the transfer functions of heave and pitch motions decrease with the increasing wave height around the resonance region(λ/LWL=3～4).Compared with the condition of H=1/50LWL,the maximum reduction of heave transfer function is about 26%for H= 1/10LWL.As for the pitch motion,the significant reduction is even as high as 40%between H=1/50LWLand H=1/10LWL.For the wavelength away from the resonance region,the transferfunctions are approximately independent of the wave height,which can be explained below：(1)The large amplitude motion is dominated around the resonance region for the SWATH ship; (2)As the wave height increases,the phenomenon of the water entry of bow and cross structure may lead to the steep increase of restoring force and moment,which would make the value of motion transfer function decrease.

Fig.6 Transfer functions of heave and pitch motions

Fig.7 Variation of motion transfer functions with the wave height

At λ/LWL=3,it can be also observed that the wave height exerts little influence on the transfer function of heave motion as the incident wave height is relatively small.However,the transfer function will be decreasing acutely if the wave height is up to a certain level(from H=1/ 15LWLto H=1/10LWL),which is related to the fact that the waterplane area increases significantly for the water entry of cross structure with the large amplitude motion of SWATH ship. In comparison to the heave motion,pitch motion is more sensitive to the wave height.From the figures above,the transfer function of pitch decreases continuously and sharply with the increasing wave height.

Fig.8 presents the ship motions and wave patterns at different wave heights for λ/LWL=3. Fig.9 illustrates the side profiles of ship motion and wave pattern between two lowerhulls. From the figures below,it is apparent that the motion amplitude is rather small and the nonlinear phenomenon is not significant at H=1/50LWL.For H=1/30LWL,the amplitude of ship motion is relatively large and the nonlinear phenomenon of the water entry of bow and lowerhull emergence starts to show up.For H=1/15LWL,the motion amplitude is continuing to increase and the nonlinear behaviors are significantly presented,in which the water entry of bow andlowerhull emergence are rather obvious.For H=1/10LWL,the motion amplitude reaches its highest level and the water entry of cross structure is even emerging.

Fig.8 Ship motions and wave patterns at different wave heights(λ/LWL=3)

Fig.9 Ship motions and wave patterns between two lowerhulls at different wave heights(λ/LWL=3)

Fig.10 illustrates the variation of the 2nd harmonics of heave and pitch motions versus the wave height for two wavelengths(λ/LWL=3 and λ/LWL=4)around the resonance region.

From the Fig.10,it is observed that the 2nd harmonics of heave and pitch motions are much smaller than the 1st harmonics and magnitude of the former is only 1%～3%of that of the latter for H=1/50LWLand H=1/30LWL.Additionally,the 2nd harmonics of ship motions are increasing along with the wave height and the increasing trend seems to accelerate with the increasing wave height.In comparison to the case of H=1/15LWL,the 2nd harmonic amplitude of heave motion is around 235%of that for H=1/10LWLand the 2nd harmonic amplitude of pitch motion is about 284%.At this moment,the magnitude of 2nd harmonics reaches approximately 10%of the 1st harmonics both for heave and pitch.The phenomenon above can be furtherexplained by the fact that large amplitude motion results in nonlinear behaviors of the water entry of bow and cross structure as well as the lowerhull emergence around the resonance region and the nonlinear behaviors become stronger with the increasing wave height,which causes the acute increase of the 2nd harmonics of motions.

Fig.10 Variation of the 2nd harmonic amplitudes of heave and pitch motions with the wave height

Fig.11 illustrates the transfer functions of the vertical acceleration at bow(19#),midship (the center of gravity)and stern(1#)versus the nondimensional wavelength for several values of wave heights.Fig.12 presents the variation of transfer functions of the vertical acceleration for two wavelengths(λ/LWL=3 and λ/LWL=4)around the resonance region.

Similarly,the transfer functions of the vertical acceleration decrease with the increasing wave height in proximity of resonance region(λ/LWL=3～4),whose reduction is up to 12%for bow(19#),26%for midship and 38%for stern(1#)at λ/LWL=3.However,the transfer functions of the vertical acceleration depend only slightly on the wave height for the wavelength range except the resonance.

Fig.11 Transfer functions of the vertical acceleration

Through the comparison between the transfer functions of the vertical acceleration in Fig. 11,it is not difficult to find that the decreasing of transfer function with the increasing wave height is the most rapid for stern,less for midship and the slowest for bow around the resonance region,which indicates that the impact of wave height on the 1st harmonics of the vertical acceleration is gradually apparent from bow to stern.From the Fig.11(b),the wave height is increasing from H=1/15LWLto 1/10LWLat λ/LWL=3 and the transfer function of the vertical acceleration at the center of gravity is decreasing rapidly,which is also existed in the heave motion response.

Fig.12 Variation of transfer functions of the vertical acceleration with the wave height

Noteworthily,the 1st harmonic of the vertical acceleration at the center of gravity is usually smaller than that at the bow and stern for traditional monohulls.However,the 1st harmonic of the vertical acceleration of SWATH ship at the center of gravity is greater than that at bow in present study.To explore the abnormal phenomenon,Fig.13 illustrates the time histories of the vertical displacement at ship bow(19#)and stern(1#)in the condition of H=1/30LWLand λ/LWL=3,where Zheaveis the vertical displacement caused by heave motion,Zpitchis that caused by pitch motion and Zt,the sum of Zheaveand Zpitch,is the total vertical displacement. There is no doubt that the vertical displacement caused by heave is the same for everywhere at the ship and the vertical displacement at the center of gravity is only contributed by heave motion.

It is clear that there is phase difference between the vertical displacements caused by heave and pitch motions both for bow and stern.Owing to the phase difference,the total vertical displacement at bow is smaller than the vertical displacement caused by heave motion, whereas the total vertical displacement at stern is greater than the vertical displacement caused by heave.As the acceleration is the 2nd derivation of the displacement,the variation pattern of the vertical acceleration is the same as that of the vertical displacement.

Fig.13 Vertical displacements at bow and stern in time history(H=1/30LWL,λ/LWL=3)

Fig.14 presents the variation of the 2nd harmonics of the vertical acceleration versus the wave height at the center of gravity for two wavelengths(λ/LWL=3 and λ/LWL=4),which is around the resonance region.The 2nd harmonics of the vertical acceleration are much smaller than the 1st harmonic and the magnitude of former is only 2%～4%of that of latter in the condition ofH=1/50LWLand H=1/30LWL.Additionally,the 2nd harmonic amplitude of the vertical acceleration at the center of gravity for λ/LWL=3 increases with the increasing wave height more steeply than that for λ/LWL=4.Due to the reason that the ship encounter frequency for λ/LWL=3 is closer to the resonance frequency than that for λ/LWL=4,the nonlinear characteristics of ship motion are so significant that the 2nd harmonics of the vertical acceleration increase steeply for λ/LWL=3.

3.2 Wave loads

Fig.15 illustrates the transfer functions of the heave force and pitch moment versus the nondimensional wavelength for several values of wave heights.The variations of the 1st and 2nd harmonics of wave loads versus the wave height for two wavelengths(λ/LWL=3 and λ/LWL=4)are shown in Fig.16 and Fig.17,respectively.

Fig.14 Variation of the 2nd harmonic amplitudes of the vertical acceleration with the wave height at the center of gravity

Fig.15 Transfer functions of wave loads

Fig.16 Variation of transfer functions of wave loads with the wave height

From the Fig.15 and Fig.16,the transfer functions of the heave force and pitch moment decrease with the increasing wave height for λ/LWL=3～4.However,the transfer functions of loads are approximately independent of the wave height for other wavelength,which is not around the resonance region.In contrast with the heave force,the pitch moment is more sensitive to the wave height.In Fig.17,the 2nd harmonics of loads are increasing along with the increasing wave height both for the heave force and pitch moment.

Fig.17 Variation of the 2nd harmonic amplitudes of wave loads with the wave height

The ratios between the 2nd and 1st harmonic amplitudes of ship responses at λ/LWL=3 are listed in Tab.3,in which ACGis the vertical acceleration at the center of gravity.From the data below,the ratios are increasing with the increasing wave height for all modes of ship response.In comparison to the heave and pitch motions,the ratios for the vertical acceleration, heave force and pitch moment at the center of gravity increase more rapidly versus the wave height,which are up to 25.69%,25.64%and 21.70%respectively at H=1/10LWL.Such phenomenon indicates the strong nonlinear characteristics of the vertical acceleration as well as the wave loads.

Tab.3 Ratios between the 2nd and 1st harmonic amplitudes of ship responses(λ/LWL=3)

4 Conclusions

The responses of SWATH ship in regular head waves are investigated with RANS code in this study.Nonlinear characteristics of ship longitudinal motions and wave loads in different incident wave heights are analyzed in details.Some findings are briefly concluded as follows：

(1)The transfer functions of longitudinal motions and wave loads are decreasing along with the increasing wave height in proximity of the resonance region(λ/LWL=3～4).However, the transfer functions above are approximately independent of wave height for the wavelength is not around the resonance region.

(2)Around the resonance region,the 1st harmonic of the pitch motion and vertical acceleration at bow and stern as well as the pitch moment is more sensitive to the wave height than that of the heave motion and vertical acceleration at the center of gravity as well as theheave force.The wave height exerts little impact on the latter three modes as the wave height is relatively small.However,the nonlinear behaviors of ship motion such as the water entry of cross structure are so significant that the transfer functions would decrease steeply for the latter three modes as the wave height continues increasing to a high level.

(3)Around the resonance region,the 2nd harmonics of longitudinal motions and wave loads increase with the increasing wave height.Of particular note is that such increasing speed of the 2nd harmonics accelerates with the increasing wave height.Additionally,the nonlinear characteristics are much stronger for the vertical acceleration as well as wave loads than for the heave and pitch motions.

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迎浪規(guī)則波中小水線面雙體船縱向運動及波浪載荷非線性特性數(shù)值分析

鄧磊，董文才，姚朝幫
（海軍工程大學艦船工程系，武漢430033）

基于RANS方程和VOF模型求解船體粘性興波流場，開展了小水線面雙體船（Small Waterplane Area Twin Hulls, SWATH）迎浪規(guī)則波中縱向運動及波浪載荷的非線性特性研究。通過數(shù)值計算結(jié)果與模型試驗結(jié)果的對比分析，驗證了文中方法的有效性；在此基礎上，較為系統(tǒng)地分析了SWATH船的垂蕩及縱搖運動響應、垂向加速度和波浪載荷的一階及二階量隨入射波高的變化規(guī)律，指出SWATH船的運動響應及載荷與波高存在非線性的關系，尤其體現(xiàn)在響應共振區(qū)附近；相比于船體垂蕩和縱搖運動，垂向加速度及波浪載荷的非線性特性更為顯著。

小水線面雙體船；RANS；波浪中運動；波高變化；非線性特性

U661.32

：A

國家自然科學基金資助項目（50879090），國家自然科學基金資助項目（51509256）

鄧磊（1990－），男，海軍工程大學艦船工程系碩士研究生，E-mail：hgdenglei@163.com；；董文才（1967-），男，海軍工程大學艦船工程系教授，博士生導師，E-mail：haigongdwc@163.com；姚朝幫（1987-），男，海軍工程大學艦船工程系講師，E-mail：hgycb2004111@163.com。

1007-7294（2017）03-0249-14

U661.32

：A

10.3969/j.issn.1007-7294.2017.03.001

Received date：2016-07-15

Foundation item：Supported by the National Natural Science Foundation of China(No.50879090); The National Natural Science Foundation of China(No.51509256)

Biography：DENG Lei(1990-),male,master student of Naval University of Engineering,E-mail：hgdenglei@163.com;

DONG Wen-cai(1967-),male,professor/tutor,E-mail：haigongdwc@163.com.