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& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

is typinformip alway cowate

andmathemore

vering

of a ship in waves, several simplied mathematical models have

shipiable, the

encounter frequency and heading angle. Moreover, in their pre-

Contents lists available at SciVerse ScienceDirect

lse

Ocean Eng

Ocean Engineering 38 (2011) 19341945they developed a simulation-oriented mathematical model for aE-mail address: yhwankim@snu.ac.kr (Y. Kim).been developed. For instance, Hirano et al. (1980) estimated diction method, wave drift force was not calculated exactly. Inorder to consider memory effects in simple way, Sutulo andSoares (2008, 2009) adopted auxiliary state variables method,which is much simpler than convolution integral method. Also

0029-8018/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2011.09.023

n Corresponding author. Tel.: 82 2 880 1543; fax: 82 2 876 9226.waves with the model of an S-175 container ship.Meanwhile, in order to predict the maneuvering performance

accurate computation of the convolution integral is not simple,and more computational time is required due to time variantUeno et al. (2003) carried out turning, zig-zag, and stopping tests inregular waves with a VLCC model. Recently, Yasukawa (2006a,2006b) performed a turning test in both regular and irregular

order to take into account unsteady memory effects duringmaneuvering in waves. Although this method is more relthan other methods which do not include memory effecteffective manner.Despite the high cost, a free-running model test in waves is the

most reliable method to estimate wave effects on ship maneuver-ing. Several results of the free-running tests have been introduced:Hirano et al. (1980) performed a turning test in regular waves.

forces depending on encounter frequency in a time-domainsimulation.

In the above mentioned methods, the memory effect due toship motion was not considered. However, Bailey et al. (1997) andFossen (2005) adopted a linear convolution integral formula inwill increase, and propeller andTherefore, if the effect of wavesresponses can be included in themaneuvering, the estimation will berequires the integration of maneuship is steering in waves, wave-induced motion and drift forcerudder efciency will alter.

modeling has been implemented by using a strip method.Ottosson and Bystrom (1991) introduced a more simpliedShip maneuvering performancewater. This provides very valuable idesign stage. However, since the shmaneuvering performance in a seawcantly different from that in a calmcally predicted in calmation at the initial shipays sails in waves, thendition may be signi-r condition. While the

corresponding motionmatical model of shipreliable. Such demandand seakeeping in an

maneuvering performance by using 3-DOF equations of motionin calm water, computing only wave drift force. In his work, thewave-induced motion is neglected. McCreight (1986) developed anonlinear maneuvering model in waves, in which the hydrody-namic forces related to wave-induced motion such as waveexciting force, and the added mass and wave damping wereevaluated in a body-xed coordinate system. In his study, his

method. Added mass and damping coefcients were assumed asconstant values, which were obtained for mean encounter fre-quency during maneuvering motion. Later, Fang et al. (2005)developed a mathematical model to calculate the hydrodynamicNumerical analysis on ship maneuverin

Min-Guk Seo, Yonghwan Kim n

Department of Naval Architecture and Ocean Engineering, Seoul National University, 5

a r t i c l e i n f o

Article history:

Received 24 March 2011

Accepted 10 September 2011

Editor-in-Chief: A.I. IncecikAvailable online 13 October 2011

Keywords:

Ship maneuvering in waves

Seakeeping analysis

Mean drift force

Time-domain simulation

Rankine panel method

a b s t r a c t

This paper considers a num

waves and resultant ship

developed to solve the wa

coupled with a modular-ty

mean drift force, which ca

using a direct pressure i

second-order mean drift

presence of incident wav

coefcient 0.7 and the S-1

observed in this study inc

regular waves. The prese

trajectories.

journal homepage: www.ecoupled with ship motion in waves

wanak-Ro, Gwanak-Gu, Seoul 151-744, Republic of Korea

cal analysis of ship maneuvering performance in the presence of incident

otion responses. To this end, a time-domain ship motion program is

body interaction problem with the ship slip speed and rotation, and it is

4-DOF maneuvering problem. In this coupled problem, the second-order

lay an important role in the ship maneuvering trajectory, is estimated by

ration method. The developed method is validated by observing the

e, and planar trajectories in maneuvering tests with and without the

The comparisons are made for two ship models, Series 60 with block

containership, with existing experimental data. The maneuvering tests

e a zig-zag test in calm water, and turning tests in calm water and in

results show a fair agreement of overall tendency in maneuvering

vier.com/locate/oceaneng

ineering

maneuvering model (MMG model) is integrated with the

motion: wave-induced motion regarded as high-frequencymotion, and maneuvering motion regarded as low-frequencymotion. Maneuvering motion slowly varies, compared withwave-induced motion; therefore, the two equations of motionare treated separately.

In the case of the wave-induced motion, the adoption of thepotential theory is a typical approach. Under the assumption ofinviscid, incompressible ow with irrotational motion, velocitypotential f can be introduced, which satises the followingboundary value problem:

r2f 0 in fluid domain 4

@f@n

U!U n! @ d!

@tU n! on body surface 5

@

@tU!rfUr

zzx,y,t 0 on z zx,y,t 6

@

@tU!rfUr

fgz 1

2rfUrf on z zx,y,t 7

where U! u0yr0 i

!v0xr0 j!. z refers to wave elevation.

M.-G. Seo, Y. Kim / Ocean Engineering 38 (2011) 19341945 1935seakeeping model.The developed computer program is veried through the

comparison with published experiment data, e.g. the zig-zag testof Series 60 (CB0.7) in calm water, and the turning test of theS-175 containership in calm water and in waves. Computationalresults show good correspondence with the experimental data.

2. Boundary value problems

When the vessel is traveling at a non-constant speed in waves,the problem becomes more complicated than conventional sea-keeping problems. Although the forms of the boundary valueproblem are the same as the conventional problem, it shouldinclude the temporal and spatial variations due to the change ofheading speed and angle. Also, to solve this problem, the stronginuences of the nonlinear viscous component need to beconsidered.

2.1. Coordinate system

To solve the ship maneuvering problem in waves, the presentstudy adopts two coordinate systems. One is a space-xedcoordinate system X

! X,Y ,Z with the positive Z-axis pointingupwards. The other is a body-xed coordinate system x

! x,y,z,which translates with forward speed, u0, slip speed, v0, androtates with rotation, r0. Fig. 1 shows the coordinate system usedin the present study. In this gure, c0, b, and d denote the shipheading angle, drift angle and rudder angle, respectively. Inaddition, A, l, and w are the wave amplitude, wave length, andwave angle, respectively.

In the body-xed coordinate system, the wave-induced bodymotion is written as follows:

d! x!,t x!T t x

!Rt x! 1

where x!

T x1,x2,x3 and x!

R x4,x5,x6 are the displacementsslender ship maneuvering in waves by using hydrodynamic forcesdepending on encounter frequency. Recently, the second-orderwave effect was considered more accurately by Skejic andFaltinsen (2008). They computed the second-order mean driftforce using many different methods and compared each method.Also, they proposed a two-time scale model that separated low-frequency motion (maneuvering motion) and high-frequencymotion (seakeeping motion). Yasukawa (2006a, 2009) also con-sidered the second-order mean drift force to compute turningtrajectory in regular and irregular waves by using momentumconservation method. However, the effect of the ship speed wasnot included thoroughly. All of the previous studies adopted thetwo-dimensional strip method to consider the wave-induced shipmotion. Recently, Lin et al. (2006) and Yen et al. (2010) solved theship maneuvering problems in waves using a three-dimensionalpanel method. In their study, the nonlinear ship motion program,LAMP (Large Amplitude Motion Program), was extended. Also Yenet al. (2010) considered the second-order mean drift force usingdirect pressure integration method.

In the present study, the ship maneuvering problem in wavesis solved by using the time-domain ship motion program, WISH(computer program for nonlinear Wave Induced load and SHipmotion analysis), which is based on a B-spline Rankine panelmethod. To this end, two main areas are extended from theseakeeping program: extension to large lateral motions and thecoupling of seakeeping and maneuvering models. In thisapproach, the second-order mean drift force is calculated usinga direct pressure integrated method, and the modular-typeof translational and rotational motions. The relation of twocoordinate systems can be

X xcosc0tysinc0tX0tY ysinc0tycosc0tY0tZ z

9>=>; 2

where X0(t), Y0(t), and c0(t) indicate the x- and y-directionalposition and heading angle of the ship motion center at time t inthe global coordinate system. For instance, by using Eq. (2), thevelocity potential of regular incident wave of wave angle w can bewritten as follows:

fI gA

o ekz sinkX coswkY sinwot

gAo

ekz sin kxcoswc0tkysinwc0t

kX0tcosykY0tsinyotg 3

2.2. Seakeeping problem

Ship motion in waves can be decomposed into two types of

Fig. 1. Coordinate system.In this formulation, not only forward speed u0 but also slip speed

M.-G. Seo, Y. Kim / Ocean Engineering 38 (2011) 193419451936v0 and rotation r0 are considered. In addition, the variations ofthese values should be considered at each time step.

In this study, the NeumannKelvin linearization method isadopted for the linearization of the boundary value problem. Thetotal velocity potential f and wave elevation z are then decom-posed as follows:

f x!,t U!tU x!fI x!,tfd x!,t 8

z x!,t zI x!,tzd x!,t 9where U!U x! indicates the uniform ow potential. fI and zI arethe incident wave potential and elevation, respectively. In addi-tion, fd and zd are the disturbance potential and elevation,respectively. The linearized boundary conditions then take thefollowing forms:

@fd@n

X6j 1

@xj@t

njxjmj

@fI@n

on SB 10

@zd@t

U!Urzd @fd@z on z 0 11

@fd@t

U!Urfd gzd on z 0 12where m1,m2,m3 0 and m4,m5,m6 n! U

!. mi is the so

called m-term, which includes the effects of interaction betweenthe steady and unsteady ows.

It should be mentioned that a different linearization methodcan provide a different solution to the above boundary valueproblem. That is, two linearization methods are popular for theseakeeping problem: NeumannKelvin linearization and double-body linearization. Kim and Kim (2010b) showed that the solu-tions of the two formulations can differ. In this study, theNuemann-Kelvin linearization is adopted.

The ship motion can be obtained by solving an equation ofmotion as follows:

Mjkf xkg fFF:K:jgfFH:D:jgfFRes:jg 13where Mjk is the mass matrix of the ship, andfFF:K:jg, fFH:D:jg, fFRes:jg are the Froude-Krylov, hydrodynamic andrestoring forces, respectively.

2.3. Maneuvering problem

In the ship maneuvering problem, 4-DOF motions are consid-ered in the space-xed coordinate system. For this problem, a setof modular-type equations is used, which can be written asfollows:

m _u0v0r0 XHXPXRXWm _v0u0r0 YHYRYWIxx _p0 KHKRKWIzz _r0 NHNRNW 14where X, Y, K, N represent the surge, sway, roll and yaw direc-tional components, respectively, and the subscripts H, P, and Rdenote the hydrodynamic forces on the ship hull, propeller, andrudder, respectively. In addition, the subscript W denotes thesecond-order mean drift force which should be obtained from theseakeeping analysis. The hull force consists of linear and non-linear components due to motion, turning, resistance and so on.Some parts of the hull force are determined by the potentialtheory described above. The other parts of the hull force compo-nents can be obtained from empirical formulae or a model test.Particularly, the propeller and rudder forces can be obtained from

empirical formulae.3. Numerical methods

3.1. Seakeeping problem

The seakeeping problem is solved by using a time-domainRankine panel method. Particularly, the present study focuses onthe linear boundary value problem. To this end, the WISHprogram, which solves the seakeeping problem at a constantmoving speed, is extended to include the ships lateral androtational motions. The WISH program has been developed atSeoul National University under the support of several largeshipbuilding industries, and it adopts a Rankine panel methodbased on a B-spline basis function for physical variables (Kimet al., 2008; Kim and Kim, 2010a).

To solve the linear boundary value problem, Greens 2ndidentity is applied by discretizing the boundary surface. Theintegral equation is expressed as follows:

fdaSBfd@G

@ndSaSF

@fd@n

GdS aSB@fd@n

GdSaSFfd@G

@ndS 15

In the present study, velocity potential, wave elevation, andnormal ux along the uid boundary are approximated using theB-spline basis function.

fd x!,t Xj

fdjtBj x!

zd x!,t Xj

zdjtBj x!

@fd@n

x!,t Xj

@fd@n

j

tBj x! 16

where Bj x! denotes the B-spline basis function, andfdjt,zdjt, and @fd=@njt denote the coefcients of dis-turbed potential, wave elevation, and the normal ux at the j-thdiscretized panel, respectively. By substituting Eq. (16) into Eqs.(10) and (15), the normal ux of velocity potential on the freesurface and the velocity potential on the body surface can beobtained. The wave elevation and velocity potential on the freesurface can then be obtained by solving the differential equationsof Eqs. (11) and (12).

For the time integration of free surface boundary conditions, amixed explicitimplicit Euler scheme is applied. The kinematicfree surface boundary condition is solved explicitly to obtain thedisturbed wave elevation, while the dynamic free surface bound-ary condition is solved implicitly to predict the velocity potentialon the free surface in the next time step, i.e.

zn1d zndDt

PU!n

,znd ,fnd 17

fn1d fndDt

Q U!n

,zn1d ,fn1d 18

where P and Q are the forcing functions, which contain all otherterms in the free surface boundary condition. In addition, theequation of motion can be solved by applying a multi-step timeintegration method. The 4th-order prediction-correction methodis used in this study.

The radiation condition is satised by applying the concept ofan articial wave absorbing zone. An articial wave absorbingzone is distributed around the truncated boundary of the freesurface, and the kinematic free surface boundary condition ismodied to include an articial damping mechanism. In thisstudy, the following equation used by Nakos (1993) is applied:

@

@tU!Ur

zd

@fd@z

2nzdn2

gfd 19where n denotes damping strength.

3.2. Maneuvering problem

3.2.1. Hull force

The hull force for the 4-DOF motion dened in Eq. (14),F!

H XH ,YH ,KH ,NH, can be decomposed in generic form as

F!

H F!

pot: F!

lift F!

visc: 20where F

!pot: is the hydrodynamic force contributed purely by

potential ow. The remaining two terms are for viscous effects.F!

lift is the hull lift force, and F!

visc: is the additional viscousdamping force on the ship hull (Lin et al., 2006). These forces arethe typical components of the conventional maneuvering equa-tion. For instance, the following forms are dened by Yasukawa(2006a):

XH X _u _u0Y _vv0r0Xvvv20Xvrv0r0Xrrr20Ru0YH Y _v _v0X _uu0r0Yvv0Yrr0Yvvvv30Yvvrv20r0Yvrrv0r20Yrrrr30KH WGZN _fzHYHNH N _r _r0Nvv0Nrr0Nvvvv30Nvvrv20r0Nvrrv0r20Nrrrr30 21

These terms can be categorized into the three components inEq. (20). For example, X _u _u0, X _uu0r0, Y _v _v0, Y _vv0r0, and N _r _r0 arerelated to inertial force, and these terms can be part of F

!pot. And

Yvv0, Yrr0, Nvv0 and Nrr0 are related to wave damping and lifting

force, and these terms can be included in F!

pot: and F!

lift both. Inthe present study, F

!pot: is directly obtained by using the WISH

program. The hull lifting force, F!

lift, is modeled as the lift force on

Fig. 2. Data exchange process.

X Y

Z

Fig. 4. Panel model for S-175 containership.

C

XW

0 0.5 1 1.5 2 2.50

3

6

9

12Experiment ( = 180)Present ( = 180)Experiment ( = 90)Present ( = 90)

CY

W

0 0.5 1 1.5 2 2.50

3

6

9

12Experiment ( = 90)Present ( = 90)

CN

W

0 0.5 1 1.5 2 2.50

2

4

6Experiment ( = 90)Present ( = 90)

/L

/L

M.-G. Seo, Y. Kim / Ocean Engineering 38 (2011) 19341945 1937Fig. 3. Coupled time-marching.

Table 1Principal particulars of Series 60 (CB0.7) and S-175.

Series 60 (CB0.7) S-175

Hull particularsLength, Lpp 100.0 m 175.0 m

Breadth, B 14.28 m 25.4 m

Draft, d 5.7 m 9.5 m

Block coefcient, CB 0.7 0.572

Displacement 5863 ton 24,739 ton

Roll period, TF 20.08 s

Propeller particularsDiameter, DP 4 m 6.507 m

Pitch/diameter ratio, p 1.1 0.73

Rudder particularsArea, AR 8.54 m

2 32.46 m2

Aspect ratio, l 2.17 1.83/L

Fig. 5. Comparison of mean drift forces on S-175 containership: Fn0.15.(a) Surge-directional mean drift force. (b) Sway-directional mean drift force.(c) Yaw-directional mean drift moment.

an equivalent plate, i.e.

F!

lift 1

2grLTU2CL 22

where L, T denote the ship length, draft respectively, and CLdenotes lifting coefcient of a low-aspect-ratio plate. In thisapproximation, g is adopted to compliment the error due to thedifference between a plate and a real ship, and its magnitude iswithin the range of 1.02.0.

Nonlinear components such as Xvvv20, Xvrv0r0, Xrrr20, Yvvvv

30,

Yvvrv20r0, Yvrrv0r20, Yrrrr

30, Nvvvv

30, Nvvrv

20r0, Nvrrv0r

20, Nrrrr

30, and

N _f in Eq. (21) should be included in the additional viscousdamping force, F

!visc:. Moreover, the viscous component of Ru0,

i.e. viscosity resistance, should also be included in F!

visc:. In thepresent study, F

!visc: is calculated using the experiment data or

empirical formulae introduced by Yasukawa (2006a, 2006b),except for the viscous roll damping term, N _f. The viscous rolldamping moment is calculated by the WISH program whichapplies the equivalent linear damping coefcient of the followingform:

b 2bm m 1c

p23

where wp denotes the effective wake fraction which is dependenton the ship forward and slip speeds, and its form is as follows:

wp wp0exp8:0b2p 27

where wp0 refers to the wake fraction at straight-ahead condition,and bp (bl0pr00) represents the geometrical inow angle.

3.2.3. Rudder force

The hydrodynamic force acting on the ship by rudder actioncan be written as the following forms:

XR 1tRFN sind

t/(L/U)

,

0 (d

eg)

0 2 4 6 8-20

-10

0

10

20Experiment ()Present ()Experiment (0)Present (0)

(deg

/sec

)

0

0.3

0.6ExperimentPresent

M.-G. Seo, Y. Kim / Ocean Engineering 38 (2011) 19341945193844 44 44 44

where m44, m44(N) and c44 denote the roll moment of inertia, theadded roll moment of inertia at innite frequency, and therestoring coefcient, respectively. In fact, 2

m44m441c44

pis the critical damping coefcient, and the magnitude of viscousdamping coefcient can be adjusted using b.

3.2.2. Propeller force

In this computation, the propeller force is obtained by usingthe following equation:

XP 1tprn2D4PKT 24where tp, n, Dp and KT are the thrust-deduction fraction, propellerrevolution per second, propeller diameter, and thrust coefcient,respectively. Thrust coefcient, KT is expressed by using a second-order polynomial form of advance ratio, shown below:

KT Jp J0 J1Jp J2J2p 25

where J0, J1, J2 are constants, which can be obtained from an open-water test, and Jp denotes the advance ratio written as

Jp u01wp

nDp26

Y/L

X/L

-5 -4 -3 -2 -1 0 1

-1

0

1

2

3

4

5

ExperimentPresent

Fig. 6. Comparison of turning trajectories in calm water: S-175 containership,

port-side turning test, d351.t/(L/U)0 2 4 6 8

-0.6

Fig. 7. Comparison of 10111 zig-zag test in calm water: Series 60 (CB0.7),r 0

-0.3Fn0.20. (a) Rudder and ship heading angle. (b) Yaw velocity.

the rst-order pressure multiplied by the rst order displace-ment, and other terms are contributions by the inertia force andmoment coupled with the rst-order angular motions. It shouldbe noted that only a linear solution is needed for the mean driftforce computation; therefore there is no need to solve thecomplete second-order boundary value problem.

3.2.5. Time integration method

For the simulation of ship trajectory by using the maneuveringequations shown in Eq. (14), an explicit Euler method is applied inthis computation, i.e.

un10 un0Dt

1mmvn0rn0XnHXnPXnRXnW

vn10 vn0Dt

1mmun0rn0YnHYnRYnW

pn10 pn0Dt

1Ixx

KnHKnRKnW

rn10 rn0Dt

1IzzNnHNnRNnW 32

It should be mentioned that the body motion considered in themaneuvering problem is much slower than that of the seakeepingproblem. Since the ship speed considered in maneuvering is slow,the simple explicit Euler scheme shown in Eq. (32) is acceptable,provided the time segment is not very large. The ship position inthe space-xed coordinate system, i.e. X0, Y0, K0, and N0, can becalculated by one more time integration of Eq. (32).

3.3. Interaction between maneuvering and seakeeping problems

M.-G. Seo, Y. Kim / Ocean Engineering 38 (2011) 19341945 1939YR 1aHFN cosdKR zRYRNR xRaHxHFN cosd 28where d, and xR and zR denote the rudder angle, and the x- andz-directional center of normal force acting on the rudder, respec-tively. In addition, tR, aH, and xH represent the interaction of thehull and rudder which can be obtained by an experiment or anempirical formula. The rudder normal force FN can be written inthe following form:

FN 1

2rARU2R

6:13ll2:25 sinaR 29

here AR and l denote the area and aspect ratio of the rudder,respectively. UR and aR denote the speed and angle of effectiveinow into the rudder, respectively. The effective inow speedand angle is affected by the hull wake, the straightening effect ofthe hull, propeller, and ship motion. In order to estimate anaccurate rudder force, it is important to estimate these two valuesaccurately. To calculate these two quantities, semi-empiricalformulae are used. In this study, the experiment data andcoefcient values proposed by Yasukawa (2006a, 2006b) areapplied.

3.2.4. Drift force

In the presence of an incident wave and resultant body motion,a ship experiences planar drift motion during turning motion. Anaccurate prediction of such drift motion is one of the essentialelements in the prediction of motion trajectory. Two majoranalytical methods are used to calculate the second-order meandrift force. One is the far-eld method based on the momentum-conservation theory proposed by Maruo (1960), and the other isthe near-eld method by integrating pressure on the body sur-face. Recently, for the steady speed case, Joncquez (2008, 2009)and Kim and Kim (2010b) introduced a formulation to computeadded resistance on ships, particularly by using their time-domain Rankine panel methods.

In the present study, the near-eld method is adopted toestimate the mean drift force. By using Bernoullis equation andTaylors expansion, the second-order mean drift force can beformulated as follows:

F!

W rgZWL

1

2zx3z4yx5x2U n!dlr

ZZSB

1

2rfIfdUrfIfdU n!dS

rZZ

SB

d!Ur @fIfd

@tU!UrfIfd

U n!dSr

ZZSB

gzU n!2dS

rZZ

SB

@fIfd@t

U!UrfIfd

U n!1dSrZZ

SB

gx3x4yx5xU n!1dS

30where z, z represents total wave elevation and vertical positionrespectively and d

!is displacement of certain point due to the

ships motion x!T x!

R x!. n!1 and n!2 are the rst andsecond-order normal vectors and are expressed as following:

n!

1 x!

R n!

x!

T n! x!

R x! n!

8