Wave-Induced Motions on aLaterally Drifting ShipBy Hironori Yasukawa, Faizul Amri Adnan and Kohei Nishi

ABSTRACTStrip theory equations presented by Adnan and Yasukawa (2007) are solved numerically in order to calculatehydrodynamic forces and wave-induced ship motions taking into account lateral drift. Calculated motions forvarious drift angles are compared with experiments for the container ship S-175. The results show that thepresented method captures the effect of drift on the wave-induced ship motions, and that this effect is notnegligible.

1 Introduction

Waves and wind continually decline ship from her course, and the helmsman may find itnecessary to take corrective actions. For head waves and wind, the lateral drift may not appear. Inreal seaways, however, lateral drift occurs due to lateral wind and wave forces acting on ships. Tanaka(2003) measured the hull drift angle of a VLCC (very large crude carrier) and a Pure Car Carrier(PCC) in real seaways using DGPS (differential global positioning system). The averaged values ofthe drift angle were 2.0 and 5.3◦ for the VLCC and PCC, respectively. It seems therefore appropriateto investigate the influence of lateral drift on seakeeping performance and to develop a predictionmethod for ship design.

Adnan and Yasukawa (2008) presented a method for the prediction of ship motions for alaterally drifting ship in regular waves. The special feature of the method is to take the lateral hulldrift effect into account in the framework of strip theory. An expansion technique in asymptoticseries as presented by Huijsmans and Hermans (1985), Huijsmans and Hermans (1989) was employedin the power value of the lateral speed parameter τ :

τ = V0ωe0/g (1)

V0 is the given lateral velocity due to hull drift, ωe0 is the encounter frequency and g is theacceleration of gravity. The ship motions ξj (j = 1 , 2 , .., 6) are described by complex numbers as

ξj = ξ(1)j + τξ

(τ)j (2)

ξ(1)j is the ship motion for zero drift angle and ξ(τ)j is the change of ship motion due to the hull drift

effect. Subscript j = 1 , 2 , ..., 6 refers to surge, sway, heave, roll, pitch and yaw, respectively. Surgemotion is not considered here. All quantities (ship motions, velocity potential, hydrodynamic forces,etc.) are expressed in the form of eq. (2). By neglecting the O(τ 2 ) term, two sets of motion equations

6

are derived: one set for ships without lateral drift, which is completely the same as the equations inthe New Strip Method, Watanabe et al. (1994), and the other set for additional motions induced by thelateral drift. By solving the second set of equations using the solutions of the first set of equations,ship motions for a ship moving laterally in waves are obtained.

When considering a laterally drifting ship in waves, the lift effect of ship hull cannot beneglected. Bertram and Thiart (2008) calculated ship motions in oblique waves taking into accountthe lift effect. However, the lift effect is not considered here since the Kutta condition determiningthe lift effect cannot be handled in the framework of the strip method.

Here a numerical method for solving the basic equations is presented and ship motion charac-teristics of a laterally drifting ship in regular waves are discussed. Strip method requires calculatingboth hydrostatic and hydrodynamic forces on 2D transverse cross sections of the ship hull. A 2Dcomplex function is introduced for the solution of the 2D boundary value problems for velocitypotentials with respect to additional motions induced by the lateral drift. Hydrodynamic forcesand wave-induced motions are computed for a container ship S-175 with various drift angles. Thecalculated ship motions are compared with the experimental data provided by Yasukawa and Adnan(2006).

2 Motion equations for a laterally drifting ship in waves

2.1 Coordinate systems

A ship advances in waves with average ship speed U∗ and hull drift angle β0 (Fig. 1) assumedconstant. Lateral drift occurs due to the effect of wind and wave forces. A small drift angle β0 isassumed to be given and the forward U0 and lateral V0 speed components defined in the ship fixedcoordinate system are respectively given by

U0 ' U∗, V0 ' −U∗β0 (3)

Incident Wavey

z

x�Ô

X

Z Y

O o

�Õ0

bU*

Fig. 1: Coordinate systems.

We consider two coordinate systems: O -XYZ fixed in space and o -xyz moving in steadytranslation at speed U∗ with heading angle ψ0 along the mean position of oscillating ship, Fig. 1.The o -xyz coordinate system does not rotate together with the ship around the x and y axes. The Zand z axes point vertical downward, and X -Y and x -y planes coincide with the still water surface.The heading angle ψ0 is defined as the angle between the X and x axes. The x -axis coincides withthe positive direction of the ship’s forward speed U0 . Origin o is located at LCB . The z -axis passesthrough the ship’s centre of gravity.

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The angle of incident wave χ is defined with respect to the negative direction of the X -axis.χ = 0 , 90 and 180◦ mean head, beam and following waves, respectively, when ψ0 = 0 . Theencounter wave frequency ωe0 of the ship is defined as

ωe0 = ω + νU0 cos(χ− ψ0 ) (4)

ν is the wave number and ω the circular frequency of the incident wave.

2.2 Motion equations in frequency domain

Motion equations in the time domain were presented by Adnan and Yasukawa (2008). In orderto calculate wave-induced motions, the motion equations in the time domain are usually transformedinto the frequency domain. The motion equations of heave and pitch motions in the frequency domainare represented as follows: (

H (1)33 −mω2

e0

)ξ(1)3 + H (1)

35 ξ(1)5 = haF (1)

3 (5)

H (1)53 ξ

(1)3 +

(H (1)

55 − Iyyω2e0

)ξ(1)5 = haF (1)

5 (6)

(H (1)

33 −mω2e0

)ξ(τ)3 + H (1)

35 ξ(τ)5 = haF (τ)

3 −6∑

j=2

H (τ)3j ξ

(1)j (7)

H (1)53 ξ

(τ)3 +

(H (1)

55 − Iyyω2e0

)ξ(τ)5 = haF (τ)

5 −6∑

j=2

H (τ)5j ξ

(1)5 (8)

The motion equations of sway, roll and yaw are:(H (1)

22 −mω2e0

)ξ(1)2 + H (1)

24 ξ(1)4 + H (1)

26 ξ(1)6 = haF (1)

2 (9)

H (1)42 ξ

(1)2 +

(H (1)

44 − Ixxω2e0

)ξ(1)4 + H (1)

46 ξ(1)6 = haF (1)

4 (10)

H (1)62 ξ

(1)2 + H (1)

64 ξ(1)4 +

(H (1)

66 − Izzω2e0

)ξ(1)6 = haF (1)

6 (11)

(H (1)

22 −mω2e0

)ξ(τ)2 + H (1)

24 ξ(τ)4 + H (1)

26 ξ(τ)6 = haF (τ)

2 −6∑

j=2

H (τ)2j ξ

(1)j (12)

H (1)42 ξ

(τ)2 +

(H (1)

44 − Ixxω2e0

)ξ(τ)4 + H (1)

46 ξ(τ)6 = haF (τ)

4 −6∑

j=2

H (τ)4j ξ

(1)j (13)

H (1)62 ξ

(τ)2 + H (1)

64 ξ(τ)4 +

(H (1)

66 − Izzω2e0

)ξ(τ)6 = haF (τ)

6 −6∑

j=2

H (τ)6j ξ

(1)j (14)

whereH (k)`j = −A (k)

`j ω2e0 + iωe0 B (k)

`j + C (k)`j (15)

m denotes the ship’s mass, Ixx , Iyy and Izz denote the moments of inertia with respect to x , y , z -axes,

respectively, ha denotes the amplitude of the incident wave, and A (k)`j , B (k)

`j , C (k)`j and F (k)

j denote theadded mass, damping, hydrostatic restoring force coefficients, and wave exciting force, respectively.In the notations, superscript (1) means the term independent from the lateral speed and superscript(τ) denotes linear terms with respect to τ (drift effect). Equations (5), (6), (9), (10) and (11) are theship motion equations for zero drift angle; equations (7), (8), (12), (13) and (14) are the equations formotion changes due to hull drift effect ξ(τ)j . A (k)

ij , B (k)ij and F (k)

j can be calculated by the integrationof 2D hydrodynamic forces over transverse cross sections of the ship hull. In order to obtain the 2Dhydrodynamic forces, 2D boundary value problems with respect to the velocity potentials are solved.

8

3 2D boundary value problems

3.1 Boundary conditions

The boundary conditions for the velocity potentials due to additional motions induced bylateral drift ϕ(τ)

j (j = 1 , 2 , 3 , S , where j = 1 , 2 , 3 mean radiation potentials for sway, heave and rollfor the flow in the transverse cross section, and j = S means the scattering potential) are definedfollowing Adnan and Yasukawa (2008). ϕ(τ)

j satisfy the 2D Laplace equation with respect to y and z .The boundary condition on the hull is:

∂ϕ(τ)j

∂N= 0 on SH (16)

SH denotes the hull surface, N the outward normal direction. The linearised free-surface condition iswritten as:

Kϕ(τ)j +

∂ϕ(τ)j

∂z= −2

(i∂

∂y+ ν0

)ϕ(1)j on z = 0 (17)

K = ω2e0/g and ν0 = ν sin(χ− ψ0 ). ν0 becomes zero for χ = 0 and 180◦ when ψ0 = 0◦. ϕ(1)

j isthe velocity potential for 2D floating body problems with zero speed and is assumed to be known. Inaddition, the radiation condition has to be satisfied.

3.2 Integral equation to be solved

In Green’s theorem, the velocity potential ϕ(τ)j (j = 1 , 2 , 3 , S) is expressed as

CPϕ(τ)j (P) =

∫SH+SF

[∂ϕ

(τ)j (Q)

∂NQ− ϕ(τ)

j (Q)∂

∂NQ

]G(P ; Q) dSQ (18)

P and Q mean field point (y , z) and singular point (η, ζ), respectively. SF denotes the undisturbedfree surface at z = 0 . When P is in the fluid domain, CP = 1 and when the P is situated on theboundary, CP = 0 .5 . G(P ; Q) is the 2D water wave Green function corresponding to a pulsatingsource, Kashiwagi et al. (2003):

G(y , z; η, ζ) =1

2π

[lg

RR1− 2FC (y − η, z + ζ)

](19)

withFC (y − η, z + ζ) = <

[e−Z E1 (−Z)

]− iπe(−Z) (20)

RR1

}=√

(y − η)2 + (z ∓ ζ)2 (21)

and

Z = K(z + ζ) + iK |y − η|, E1 (z) =

∞∫z

e−n

ndn

< denotes the real part of a complex potential. G(P ; Q) satisfies the following relation:

KG(P ; Q1 ) +∂G(P ; Q1 )

∂ζ= 0 (22)

Q1 denotes the singular point (η, 0) situated on the undisturbed free surface. When P is on the hullsurface, eq. (18 ) is rewritten as:

ϕ(τ)j (P)

2=

∫SH

[∂ϕ

(τ)j (Q)

∂NQ− ϕ(τ)

j (Q)∂

∂NQ

]G(P ; Q) dSQ − 2

∫SF

[ν0ϕ

(1)j + i

∂ϕ(1)j

∂η

]G(P ; Q1 ) dη

(23)

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The second integral term on the r.h.s. of eq. (23) is expressed as an integral over the infinite undisturbedfree surface. To avoid an infinite integral, the integral over the free surface is replaced by an integralover the submerged hull surface. Following Nossen et al. (1991) and Emmerhoff and Sclavounos (1991),

G∗(P ; Q) = 2

(i∂

∂η− ν0

)∂G(P ; Q)

∂K(24)

G∗(P ; Q) satisfies the Laplace equation. Since G∗(P ; Q) has no singularity at P = Q , the followingrelation holds:∫SH

[∂ϕ

(1)j (Q)

∂NQ− ϕ(1)

j (Q)∂

∂NQ

]G∗(P ; Q) dSQ +

∫SF

[∂ϕ

(1)j (Q)

∂z− ϕ(1)

j (Q)∂

∂z

]G∗(P ; Q) dSQ = 0

(25)Then eq. (23) is rewritten as

12ϕ(τ)j (P) =

∫SH

[∂ϕ

(τ)j (Q)

∂NQ− ϕ(τ)

j (Q)∂

∂NQ

]G(P ; Q) dSQ

−∫SH

[∂ϕ

(1)j (Q)

∂NQ− ϕ(1)

j (Q)∂

∂NQ

]G∗(P ; Q) dSQ (26)

This equation is composed of integrals only over the hull surface. In order to solve eq. (26), acalculation with respect to G∗ is needed which is shown in Appendix A. G∗ given by eq. (24) ismathematically singular in the far field but behaves well on the hull surface as pointed out by Yoshidaet al. (2003).

3.3 Solution of 2D problem

In order to verify the method described in 3.2, radiation forces (described by added massand damping coefficients) on a 2D half-immersed circular cylinder with slow drift were calculated.A close-fit method (constant panel method) was used for the calculations with approximately 50elements on the body surface. The method proposed by Haraguchi and Ohmatsu (1983) was appliedin order to remove the effect of “irregular frequencies”.

Fig. 2 shows added mass and damping coefficients for three different drifting velocities(τ0 = −0 .1 , 0 , 0 .1 ) vs. non-dimensional oscillating frequency ω′ (= ω

√b/g , where b denotes

the radius of the cylinder). Here, τ0 represents the effect of lateral drift velocity. It is defined asτ0 = V0ω/g , where V0 is drifting velocity. A 2D

11 , A 2D22 are the added mass coefficients, and B 2D

11 , B 2D22

are the damping coefficients for sway and heave motions, respectively. They are non-dimensionalisedas A 2D

ij = A 2Dij /ρ(2b)2 and B 2D

ij = B 2Dij /ρω(2b)2 .

The added mass and damping coefficients for τ0 = 0 agree well with the results in Kashiwagiet al. (2003). The coefficients for non-zero drifting velocity were compared with Prins and Hermans(1994). Fig. 3 shows these coefficients plotted vs. Froude number Fn based on radius of the cylinder forω′ = 1 .125 . The change of the coefficients with increase of Fn agrees well with Prins and Hermans(1994). The remaining discrepancies are mainly due to the higher order effect of the drifting velocityin the free-surface condition used by Prins and Hermans (1994).

4 Studied ship and calculation conditions

The S-175 container ship was selected for the calculations, Table 1 and Fig. 4. Model tests werecarried out in order to obtain validation data in Yasukawa and Adnan (2006). In 1, L denotes theship length between perpendiculars, B the breadth, d the draft, ∆ the displacement and xG the LCB

10

0 0.5 1 1.5 20.0

0.1

0.2

0.3

0.4

0.5

0.6A11−

ω’

τ0 = 0.0τ0 = −0.1

τ0 = 0.1

2D

0 0.5 1 1.5 20.0

0.5

1.0

1.5A22−

ω’

τ0 = 0.0τ0 = −0.1

2D

τ0 = 0.1

0 0.5 1 1.5 20.0

0.1

0.2

0.3

0.4

0.5B11−

ω’

τ0 = 0.0τ0 = −0.1

2D

τ0 = 0.1

0 0.5 1 1.5 20.0

0.2

0.4

0.6

0.8

1.0

1.2B22−

ω’

τ0 = 0.0τ0 = −0.1

2D

τ0 = 0.1

Fig. 2: Added mass and damping coefficients for a 2D half-immersed circular cylinder.

−0.05 0 0.05 0.1 0.150

0.1

0.2

0.3

0.4

Fn

A11 & B11

ω’=1.125Prins & Hermans(1994)Present

2D 2D− −

B11

A11

−0.05 0 0.05 0.1 0.150

0.1

0.2

0.3

0.4

Fn

A22 & B22

ω’=1.125

2D 2D− −

B22

A22

Fig. 3: Added mass and damping coefficients for a 2D half-immersed circular cylinder versus Froudenumber based on the radius (ω′ = 1 .125 ).

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Table 1: Principal dimensions of S-175 container ship.

Model ShipL 3.500 m 175.00 mB 0.508 m 25.40 md 0.190 m 9.50 m∆ 193.57 kg 24801 txG -0.051 m -2.55 mKM 0.210 m 10.50 mGM 0.025 m 1.25 mTφ 2.43 s 17.2 s

kyy/L 0.272 0.272

Fig. 4: Body plan of S-175 container ship.

X

Y

U

b

U

V

*

U *

0

0

0

c = 00

Incoming Head Waves

c = 90

Incoming Beam Waves

0

Fig. 5: Plane motions of laterally drifting ship in waves (ψ0 = 0◦).

12

coordinate from midships. GM , roll period Tφ and non-dimensional radius of gyration for pitch kyy/Lwere taken from the model test.

In the calculations, the selected ship speed U∗ corresponds to Fn = 0 .15 based on L . Ship’sheading ψ0 is assumed to be zero; two wave directions were selected (head and beam waves, χ = 0and 90◦, respectively). For each direction, hull drift angle was varied as β0 = 5 , 0 ,−5 ,−10◦. Fig. 5shows plane motions of the laterally drifting ship in head and beam waves.

5 Added mass, damping and wave exciting forces of a laterallydrifting ship

First, consider added mass Aij , damping Bij and wave exciting forces Fj acting on a laterallydrifting ship in waves, defined as:

Aij = A (1)ij + τA (τ)

ij , Bij = B (1)ij + τB (τ)

ij , Fj = F (1)j + τF (τ)

j

They are non-dimensionalised as

Aii =Aii

ρ∇for i ≤ 3 , Aii =

Aii

ρ∇Lfor i ≥ 4

Bii =Bii

ρ∇√

2g/Lfor i ≤ 3 , Bii =

Bii

ρ∇L√

2g/Lfor i ≥ 4

Fj =Fj

ρg∇for i ≤ 3 , Fj =

Fj

ρgL∇for i ≥ 4

where∇ denotes the volume displacement of the ship. Only the diagonal terms of added mass anddamping matrices are dealt with here.

5.1 Head waves (χ = 0 ◦)

Fig. 6 shows added mass and damping coefficients and Fig. 7 shows the non-dimensionalamplitude of the wave exciting forces in head waves for various drift angles β0 . Added mass anddamping without hull drift angle are shown as functions of wave encounter frequency. The addedmass and damping are presented as functions of the encounter frequency and wave direction.

0 5 10 15 200

0.20.40.60.8

11.21.41.61.8

2

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A22 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.5

1

1.5

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A33 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A55 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.05

0.1

0.15

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A66 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B22 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.5

1

1.5

2

2.5

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B33 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B55 χ=0o, Fn=0.15

CAL.

−

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B66 χ=0o, Fn=0.15

CAL.

−

Fig. 6: Added mass and damping coefficients for various hull drift angles in head waves (χ = 0◦).

The change of added masses A (τ)33 , A (τ)

55 , damping B (τ)33 , B (τ)

55 and wave exciting forces F (τ)3 ,

F (τ)5 due to hull drift becomes zero with changing heave and pitch for χ = 0 and 180◦. This is

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0 0.5 1 1.5 20

0.10.20.30.40.50.60.70.80.9

1

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F2|

/ρg

χ=0o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 202468

101214161820

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F3|

/ρg

χ=0o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 20

0.0020.0040.0060.008

0.010.0120.0140.0160.018

0.02

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F4|

/ρgL

χ=0o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F5|

/ρgL

χ=0o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F6|

/ρgL

χ=0o, Fn=0.15

CAL.

∆

Fig. 7: Amplitudes of wave exciting forces for various hull drift angles in head waves (χ = 0◦).

mathematically proven in Appendix B. In the calculation results, therefore, the added mass, dampingand wave exciting forces have the same values even if the drift angle is different. Some smallinfluence of the drift angle on the added mass A22 , A66 and damping B22 , B66 with respect to swayand yaw is observed. Even in the head waves, wave exciting forces with respect to sway, roll and yawF2 , F4 , F6 are induced, proportional to absolute value of the drift angle. This implies that sway, rolland yaw motions are induced when the ship moves with lateral drift even in head waves. Amplitudesof the exciting forces are the same for β0 between −5 and 5◦.

5.2 Beam waves (χ = 90 ◦)

Fig. 8 shows added mass and damping coefficients and Fig. 9 shows the non-dimensionalamplitude of the wave exciting forces in beam waves for various drift angles. The influence of thedrift angle on the added mass and damping coefficients for χ = 0◦ is more remarkable than forχ = 90◦. The hydrodynamic force components for lateral motions A22 , A66 , B22 , B66 , F2 and F4 arelarger than those for longitudinal motion A33 , A55 , B33 , B55 , F3 and F5 . B33 and B55 show similartrends versus drift angle β0 : their values increase with β0 in the range where non-dimensional wavenumber νL is larger than 10. B22 and B66 decrease with β0 for larger β0 . Drift angle influenceslongitudinal and lateral motions in a different way. Similarly, the tendency of the exciting forcechanges with the wave number and with the change from longitudinal to lateral motion.

6 Wave-induced motions of a laterally drifting ship: comparisonwith experiments

6.1 Ship motions with zero drift angle

Fig. 10 compares amplitudes of wave-induced motions for zero drift angle. Sway and heaveamplitudes are non-dimensionalised with respect to the incident wave amplitude ha ; roll, pitch andyaw amplitudes are non-dimensionalised using the incident wave slope haν . The pitch amplitudeis calculated accurately. The heave amplitude is also calculated accurately except for λ/L = 1 .0in head waves (χ = 0◦). This inaccuracy might be due to the poor prediction of the coupling ofhydrodynamic force terms with respect to heave and pitch as pointed out by Kashiwagi et al. (2000).

14

0 5 10 15 200

0.20.40.60.8

11.21.41.61.8

2

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A22 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.5

1

1.5

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A33 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A55 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.05

0.1

0.15

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

A66 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B22 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.5

1

1.5

2

2.5

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B33 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B55 χ=90o, Fn=0.15

CAL.

−

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

β0=0o

β0=−5o

β0=−10o

β0=5o

νL

B66 χ=90o, Fn=0.15

CAL.

−

Fig. 8: Added mass and damping coefficients for various hull drift angles in beam waves (χ = 90◦).

0 0.5 1 1.5 202468

101214161820

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F2|

/ρg

χ=90o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 202468

101214161820

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F3|

/ρg

χ=90o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F4|

/ρgL

χ=90o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F5|

/ρgL

χ=90o, Fn=0.15

CAL.

∆

0 0.5 1 1.5 20

0.20.40.60.8

11.21.41.61.8

2

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|F6|

/ρgL

χ=90o, Fn=0.15

CAL.

∆

Fig. 9: Amplitudes of wave exciting forces for various hull drift angles in beam waves (χ = 90◦).

From an industry perspective, the present method is sufficiently accurate. Calculated results for sway,roll and yaw amplitudes are smaller than the measured ones. However, the method captures well thetendency of the lateral motions vs. λ/L .

6.2 Ship motions with drift angle in head waves

Fig. 11 compares heave and pitch amplitudes in regular head waves for the drift angles 0 , −5and −10◦. As described in section 5.1, the changes of added mass, damping and wave exciting forcesdue to hull drift with respect to heave and pitch are zero in head waves. Therefore, the change ofheave and pitch due to hull drift should be zero. The calculated amplitudes for β0 = −5 and −10◦

are almost the same as those for zero drift angle, as expected. In the results of the experiments, theinfluence of the hull drift on the amplitudes of heave and pitch is negligibly small. Thus, the tendencyis the same in the calculations and the experiments.

Fig. 12 compares sway, roll and yaw amplitudes in regular head waves. Even though theheading angle is kept coinciding with the course, sway, roll and yaw motions are induced. Motion

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0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

CalExp

λ/L

|ξ2|

/ha

Fn=0.15, β0=0o

SWAY

χ=90o

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

CalExp

λ/L

|ξ3|

/ha

Fn=0.15, β0=0o

HEAVE

χ=0o

χ=90o

0 0.5 1 1.5 20

1

2

3

CalExp

λ/L

|ξ4|

/(h a

ν)

Fn=0.15, β0=0o

ROLL

χ=90o

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

CalExp

λ/L

|ξ5|

/(h a

ν)

Fn=0.15, β0=0o

PITCHχ=0o

χ=90o

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

CalExp

λ/L

|ξ6|

/(h a

ν)

Fn=0.15, β0=0o

YAW

χ=90o

Fig. 10: Comparison of ship motion amplitudes in following waves (β0 = 0◦).

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ3|

/ha

χ=0o, Fn=0.15

CAL.HEAVE

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ5|

/(h a

ν)χ=0o, Fn=0.15

CAL.PITCH

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ3|

/ha

χ=0o, Fn=0.15

EXP.HEAVE

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ5|

/(h a

ν)

χ=0o, Fn=0.15

EXP.PITCH

Fig. 11: Comparison of heave and pitch amplitudes in head waves for various drift angles (χ = 0◦).

16

amplitudes are approximately proportional to the absolute value of the drift angle. The calculationpredicts well fact observed in the experiments that sway and yaw amplitudes tend to be largest forλ/L = 1 .0 . However, the calculated sway amplitude is only half of the measured one. The calculatedroll and yaw amplitudes are much smaller than the measured ones. We attribute this poor accuracyto insufficient wave exciting roll moment acting on the laterally drifting ship in head waves.

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ2|

/ha

χ=0o, Fn=0.15

CAL.SWAY

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ4|

/(h a

ν)

χ=0o, Fn=0.15

CAL.ROLL

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

β0=0o

β0=−5o

β0=−10o

λ/L

|ξ6|

/(h a

ν)

χ=0o, Fn=0.15

CAL.YAW

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2β0=0o

β0=−5o

β0=−10o

λ/L

|ξ2|

/ha

χ=0o, Fn=0.15

EXP.SWAY

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6β0=0o

β0=−5o

β0=−10o

λ/L

|ξ4|

/(h a

ν)

χ=0o, Fn=0.15

EXP. ROLL

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2β0=0o

β0=−5o

β0=−10o

λ/L

|ξ6|

/(h a

ν)

χ=0o, Fn=0.15

EXP.YAW

Fig. 12: Comparison of sway, roll and yaw amplitudes for various drift angles inhead waves (χ = 0◦).

6.3 Ship motions with drift angle in beam waves

Figs. 13 and 14 compare the amplitudes of ship motions in regular beam waves for drift angles5 , 0 , −5 and −10◦. In general, the effect of the drift angle on the ship motions in beam waves ismore remarkable than in head waves. Fig. 15 compares ship motion amplitudes in regular beamwaves vs. drift angle β for λ/L = 0 .7 and 1 .2 . Pitch amplitude is relatively small and the tendencyvs. drift angle differs from the experiments. The calculated heave amplitude decreases with theincrease of the hull drift angle, which agrees with the experiment. The calculated amplitudes ofsway, roll and yaw increase with the increasing drift angle, which also agrees with the experiments.Thus, the present method captures the effect of hull drift angle on the ship motions in waves with theexception of pitch motions.

7 Conclusion

Strip-theory equations presented by Adnan and Yasukawa (2008) are solved numerically inorder to calculate hydrodynamic forces and wave-induced motions of a container ship S-175 forvarious drift angles. The calculated ship motions were compared with the experimental resultsprovided by Yasukawa and Adnan (2006). The present method can roughly capture the effect of hulldrift on the ship motions in waves. The following insights were obtained:

– The effect of hull drift on the wave-induced motions is not negligible.– Heave and pitch are not influenced by the hull drift in head and following waves.– In addition to heave and pitch, lateral motions (sway, roll and yaw) are also excited in head

waves for a laterally drifting ship. The magnitudes of the lateral motions are proportional tothe drift angle.

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0 0.5 1 1.5 20

0.20.40.60.8

11.21.41.61.8

2

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ3|

/ha

χ=90o, Fn=0.15

CAL.HEAVE

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ5|

/(h a

ν)

χ=90o, Fn=0.15

CAL.PITCH

0 0.5 1 1.5 20

0.20.40.60.8

11.21.41.61.8

2

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ3|

/ha

χ=90o, Fn=0.15

EXP.HEAVE

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ5|

/(h a

ν)

χ=90o, Fn=0.15

EXP.PITCH

Fig. 13: Comparison of heave and pitch amplitudes for various drift angles in beam waves (χ = 90◦).

– The amplitudes of sway, roll and yaw increase while the amplitude of heave decreases withthe increasing drift angle in beam waves.

Appendix A: Calculation method of G ∗In order to solve the integral equation (26), G∗(P ; Q) and ∂G∗(P ; Q)/∂NQ values need to be

calculated. G∗(P ; Q) can be expressed by substituting eq. (19) into eq. (24) as follows:

G∗(P ; Q) = −2iπ

∂2 FC

∂η∂K+

2ν0

π

∂FC

∂K(27)

Similarly, ∂G∗/∂NQ can be expressed as

∂G∗

∂NQ= N1 (Q)

∂G∗

∂η+ N2 (Q)

∂G∗

∂ζ(28)

Substituting eq. (27) into eq. (28), ∂G∗(P ; Q)/∂NQ is obtained as

∂G∗(P ; Q)

∂NQ= −2i

π

{N1 (Q)

∂3 FC

∂η2∂K+ N2 (Q)

∂3 FC

∂η∂ζ∂K

}+

2ν0

π

{N1 (Q)

∂2 FC

∂η∂K+ N2 (Q)

∂2 FC

∂ζ∂K

}(29)

In order to calculate G∗(P ; Q) and ∂G∗(P ; Q)/∂NQ , the derivatives of FC with respect to K , η and ζneed to be evaluated. Only the final results are given here:

18

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ2|

/ha

χ=90o, Fn=0.15

CAL.SWAY

0 0.5 1 1.5 20

1

2

3

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L|ξ

4|/(

h aν)

χ=90o, Fn=0.15

CAL. ROLL

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ6|

/(h a

ν)

χ=90o, Fn=0.15

CAL.YAW

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ2|

/ha

χ=90o, Fn=0.15

EXP.SWAY

0 0.5 1 1.5 20

1

2

3

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ4|

/(h a

ν)χ=90o, Fn=0.15

EXP. ROLL

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

β0=0o

β0=−5o

β0=−10o

β0=5o

λ/L

|ξ6|

/(h a

ν)

χ=90o, Fn=0.15

EXP.YAW

Fig. 14: Comparison of sway, roll and yaw amplitudes for various drift angles in beam waves (χ = 90◦).

−10 −5 0 50

0.2

0.4

0.6

0.8

1

1.2

CalExp

β(deg)

|ξ2|

/ha

Fn=0.15, χ=90o

SWAY

λ/L=1.2λ/L=0.7

−10 −5 0 50

0.20.40.60.8

11.21.41.61.8

2

CalExp

β(deg)

|ξ3|

/ha

Fn=0.15, χ=90o

HEAVE

λ/L=1.2

λ/L=0.7

−10 −5 0 50

0.5

1

1.5

2

CalExp

β(deg)

|ξ4|

/(h a

ν)

Fn=0.15, χ=90o

ROLL

λ/L=1.2

λ/L=0.7

−10 −5 0 50

0.1

0.2

0.3

0.4

0.5

CalExp

β(deg)

|ξ5|

/(h a

ν)

Fn=0.15, χ=90o

PITCH

λ/L=1.2λ/L=0.7

−10 −5 0 50

0.1

0.2

0.3

0.4

0.5

CalExp

β(deg)

|ξ6|

/(h a

ν)

Fn=0.15, χ=90o

YAW

λ/L=1.2λ/L=0.7

Fig. 15: Comparison of ship motion amplitudes in beam waves vs. drift angle (λ/L = 0 .7 and 1.2).

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∂FC (y − η, z + ζ)

∂K= −(z + ζ)

{<[e−Z E1 (−Z)

]− πie−Z

}+|y − η|

{=[e−Z E1 (−Z)

]− πe−Z

}− 1

K

(30)

∂2 FC (y − η, z + ζ)

∂η∂K= K(y − η)

{<[e−Z E1 (−Z)

]− πie−Z

}−sgn(y − η) {1 − K(z + ζ)}

{=[e−Z E1 (−Z)

]− πe−Z

} (31)

∂2 FC (y − η, z + ζ)

∂ζ∂K= −{1 − K(z − ζ)}

{<[e−Z E1 (−Z)

]− πie−Z

}−K |y − η|

{=[e−Z E1 (−Z)

]− πe−Z

}+ 1

(32)

∂3 FC (y − η, z + ζ)

∂η2∂K= −K {2 + K(z − ζ)}

{<[e−Z E1 (−Z)

]− πie−Z

}−K 2 |y − η|

{=[e−Z E1 (−Z)

]− πe−Z

}− K<

[1Z

]+ K

(33)

∂3 FC (y − η, z + ζ)

∂η∂ζ∂K= −K 2 (y − η)

{<[e−Z E1 (−Z)

]− πie−Z

}+sgn(y − η)K=

[1Z

]+ sgn(y − η)K {2 − K(z + ζ)}

{=[e−Z E1 (−Z)

]− πe−Z

} (34)

Appendix B: Heave radiation forces of O(τ) in head wavesConsider the boundary value problem for the heave radiation potential ϕ(τ)

2 with lateral driftin 2D hull section. Then ψ0 = 0 and χ = 0 , so that the boundary conditions described in 3.1,equations (16) and (17) become

∂ϕ(τ)2

∂N= 0 on SH (35)

Kϕ(τ)2 +

∂ϕ(τ)2

∂z= −2i

∂ϕ(1)2

∂yon z = 0 (36)

ϕ(1)2 denotes the 2D radiation potential for heave motions without lateral drift. The r.h.s. of eq. (36) rep-

resents the free-surface disturbance. The magnitude of this disturbance is proportional to ∂ϕ(1)2 /∂y .

The disturbance due to heave motion is symmetrical because the hull section shape is assumedsymmetrical, thus ϕ(1)

2 must be an odd function with respect to y . Therefore, its derivative withrespect to y is an even function. The resulting disturbance of the free-surface is then an even function,thus ϕ(τ)

2 must be an even function with respect to y .

Solving the boundary problem as mentioned above for ϕ(τ)2 , added mass M (τ)

22 and damping

N (τ)22 for 2D section shape can be calculated using the following equations, Adnan and Yasukawa

(2008):

M (τ)22 − i

N (τ)22

ωe0= −ρ

∫SH

(ϕ(τ)2 + i

gω2

e0

∂ϕ(1)2

∂y

)N2 ds (37)

The integral term on the r.h.s. of eq. (37) becomes zero because ∂ϕ(1)2 /∂y and ϕ(τ)

2 are even functions

with respect to y for a symmetrical hull section, then M (τ)22 and N (τ)

22 are zero in head waves.Here, the section at the aft end of the ship is assumed zero. If we know added mass and

damping for 2D section shape, the change of added mass and damping with respect to heave motiondue to lateral drift, A (τ)

33 and B (τ)33 , are written as follows, Adnan and Yasukawa (2008):

A (τ)33 =

∫L

[M (τ)

22 +gνω2

e0sin(χ− ψ0 )M (1)

22

]dx (38)

20

B (τ)33 =

∫L

[N (τ)

22 +gνω2

e0sin(χ− ψ0 )N (1)

22

]dx (39)

M (1)22 and N (1)

22 are added mass and damping with respect to heave motion for 2D section shape

without lateral drift. Then, A (τ)33 and B (τ)

33 become zero because χ = ψ0 = 0 , and M (τ)22 and N (τ)

22 arezero. Thus the change of added mass and damping acting on the ship with respect to heave motiondue to lateral drift must be zero.

A similar argument applies to the heave exciting force and hydrodynamic forces with respectto pitch for χ = 180◦. Thus changes of added mass, damping and exciting forces with respect toheave and pitch due to lateral drift are zero in head and following waves (χ = 0 and 180◦).

ReferencesAdnan, F. A. and Yasukawa, H. (2008). A strip method for a laterally drifting ship in waves. J. Marine Science and Technology, 12(3):139–

149Bertram, V. and Thiart, G. (2008). A Kutta condition for ship seakeeping computations with a Rankine panel method. Ship Technology

Research, 45:54–63Emmerhoff, O. J. and Sclavounos, P. D. (1991). The slow-drift motion of arrays of vertical cylinders. J. Fluid Mech., 242:31–50Haraguchi, T. and Ohmatsu, S. (1983). On an improved solution of the oscillation problem on non-wall sided floating bodies and a new

method for eliminating the irregular frequencies (Japanese). Trans. West-Japan Society of Naval Architects, Japan, 66:31–50Huijsmans, R. H. M. and Hermans, A. J. (1985). A fast algorithm for computation of 3-D ship motions at moderate forward speed. In

4th Int. Conf. Num. Ship Hydrodynamics, Washington, 19–26Huijsmans, R. H. M. and Hermans, A. J. (1989). The effect of the steady perturbation potential on the motions of a ship sailing in

random seas. In 5th Int. Conf. Num. Ship Hydrodynamics, Hiroshima, 375–389Kashiwagi, M., Kawasoe, K. and Inada, M. (2000). A study on ship motion and added resistance in waves (Japanese). J. Kansai Society

of Naval Architects, Japan, 234:85–94Kashiwagi, M., Takagi, K., Yoshida, T., Murai, M. and Higo, Y. (2003). Hydrodynamics of floating bodies (Japanese). Seizandou-Shoten

Publ., ISBN 4-425-71321-4Nossen, J., Grue, J. and Palm, E. (1991). Wave forces on three-dimensional floating bodies with small forward speed. J. Fluid Mech.,

227:135–160Prins, H. J. and Hermans, A. J. (1994). Time domain calculations of drift forces on floating two-dimensional object in current and waves.

J. Ship Research, 38:97–103Tanaka, Y. (2003). Economical speed and life cycle value of ship (Japanese). In 4th JTTC Symp. Ship Performance at Sea, Soc. Naval

Arch. Japan, 10–19Watanabe, I., Toki, N. and Ito, A. (1994). Strip method (Japanese). In 11th Marine Dynamics Symp. Application of Ship Motion Theory

to Design, Soc. Naval Arch. Japan, 167–218Yasukawa, H. and Adnan, F. A. (2006). Experimental study on wave-induced motions and steady drift forces of an obliquely moving ship

(Japanese). J. Japan Soc. Naval Arch. & Ocean Eng., 3:133–138Yoshida, M., Bao, W. and Kinoshita, T. (2003). A method to remove secularity in the solution of velocity potential – a calculation of

wave-drift added mass (Japanese). J. Soc. Naval Arch. Japan, 194:129–136

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