numerical analysis of added resistances of a large

Journal of Advanced Research in Ocean Engineering 3(2) (2017) 083-101 https://doi.org/10.5574/JAROE.2017.3.2.083

Numerical Analysis of Added Resistances of a Large Container Ship in Waves

Jae-Hoon Lee 1, Beom-Soo Kim 1, and Yonghwan Kim 1 * 1 Department of Naval Architecture and Ocean Engineering, Seoul National University, Korea

(Manuscript Received March 28 2017; Revised April 15, 2017; Accepted June 1, 2017)

Abstract In this study, the added resistances of the large container ship in head and oblique seas are evaluated using a

time-domain Rankine panel method. The mean forces and moments are computed by the near-field method, namely, the integration of the second-order pressure directly on the ship surface. Furthermore, a weakly nonlinear approach in which the nonlinear restoring and Froude-Krylov forces on the exact wetted surface of a ship are in-cluded in order to examine the effects of amplitudes of waves on ship motions and added resistances. The compu-tation results for various advance speeds and heading angles are validated by comparing with the experimental data, and the validation shows reasonable consistency. Nevertheless, there exist discrepancies between the numer-ical and experimental results, especially for a shorter wave length, a higher advance speed, and stern quartering seas. Therefore, the accuracies of the linear and weakly nonlinear methods in the evaluation of the mean drift forces and moments are also discussed considering the characteristics of the hull such as the small incline angle of the non-wall-sided stern and the fine geometry around the high-nose bulbous bow.

Keywords: Large containership, Added resistance in waves, Rankine panel method, Weakly nonlinear approach

1. Introduction

To meet the regulation of the International Maritime Organization (IMO) regarding the restriction of

greenhouse gas emissions from ships, the shipbuilding industry has been working towards building fuel-

efficient ships. In the regulation, various types of ships should meet a minimum energy efficiency require-

ment defined by the Energy Efficiency Design Index (EEDI). Therefore, the prediction of the added re-

sistance on a ship induced by waves is an important issue for the realistic evaluation of the required propul-

sion power in a seaway, as the magnitude of the resistance is approximately 15–30 % of calm-water re-

sistance. On the other hand, the reduced engine power to achieve the EEDI requirement may decrease ma-

neuverability in rough seas. The estimation of the sufficient power and controllability of steering device in

waves is directly related to the mean drift forces and moments acting on ships for various heading angles.

In other words, the second-order mean forces should be accurately considered for both efficiency and safe-

ty of ship navigation.

The mean drift forces and moments have been widely examined by analytical and numerical approaches.

Maruo (1960) originally derived the far-field method using the asymptotic expression of the velocity poten-

*Corresponding author. Tel.: +82-2-880-1543, Fax.: +82-2-876-9226,

E-mail address: [email protected]

Copyright © KSOE 2017.

Journal of

Advanced Research in Ocean Engineering

84 Jae-Hoon Lee, Beom-Soo Kim, and Yonghwan Kim Journal of Advanced Research in Ocean Engineering 3(2) (2017) 83-101

tial at the far-field and the momentum conservation theory. Newman (1967) extended this approach to the

evaluation of not only the second-order drift horizontal force but also the yaw moment. Combined with the

seakeeping analysis method, such as the strip-based STF method (Salvesen et al., 1978), the unified theory

(Kashiwagi, 1992), and the Rankine panel method (Liu et al., 2011; Seo et al., 2013), the momentum-

conservation method has been constantly developed and extensively used for the computation of added

resistance of a ship. The formulation of the far-field method is relatively simple, and the solution can be

obtained without solving a boundary value problem for the pressure acting on a floating body. However, as

the approach is based on the linear potential theory, there is a limitation in extending it to a nonlinear pre-

diction, and this is one of the major drawbacks.

On the other hand, the near-field method, which allows direct integration of the second-order quadratic

pressure, has been also applied for the added resistance problem. The formulation of this approach is ob-

tained by the perturbation of physical variables with respect to the mean-body position, which enables the

decomposition of added resistances for a physical observation. Pinkster (1979) derived the components of

the mean and low-frequency wave drifting forces on floating structures, and Faltinsen et al. (1980) comput-

ed the increased resistance in a seaway by using the near-field method, and modified the results for short

wave lengths by adopting a simplified asymptotic method. Thereafter, this method has been also developed

with the three-dimensional (3-D) panel method. For example, a frequency-domain wave Green’s function

method is adopted by Grue and Biberg (1993) to evaluate the wave-induced drift force and moment on a

body advancing with a small speed, and Jonquez (2009) and Kim and Kim (2011) formulated the near-field

method for the time-domain Rankine panel method. Nevertheless, it was reported that the panel method

approach yields underestimated results, especially for a fine hull in short wave lengths (Seo, et al., 2014).

Therefore, there have been efforts to enhance the method for prediction of added resistances by adopting

the linearization with respect to the steady wave elevation (Bunnik, 1999; Hermans, 2005).

Recently, applications of computational fluid dynamics (CFD) have been carried out to investigate the

nonlinear phenomena in the added resistance problem (Guo et al., 2012; Sadat-Hosseini et al., 2013; Ley et

al., 2014). To obtain a high-performance hull form in a seaway, the bow shape above the mean-water level

should be considered including the nonlinear wave diffraction effects. Orihara et al. (2008) used the Reyn-

olds-averaged Navier–Stokes equation (RANSE) solver called WISDAM-X for the examination of the

added resistance for different bow shapes. Furthermore, the evaluation of added resistance for a high wave

amplitude is relevant to a ship’s operation in adverse conditions. The Euler equation solver based on Carte-

sian-grid-based method was applied to calculate the effects of wave magnitude in a regular wave by Yang

(2015). However, there have been concerns regarding the applications of the CFD method due to its huge

computational cost and a strong dependency on the grid system.

This study considers the wave-induced added resistances, the drift sway forces, and the yaw moments of a

modern large containership. The target ship is a test model of the benchmark study of an EU-funded re-

search project called SHOPERA (Sprenger et al., 2015). The time-domain 3-D Rankine panel method de-

veloped by Kim et al. (2011) along with the near-field method of Kim and Kim (2011) are used to compute

the second-order mean forces at various advance speeds and heading angles of the ship. The simulation

results are validated with the experimental data obtained in the benchmark study. In the test conditions of

the measurement, the wave amplitude is relatively high for reflecting the environmental conditions of

heavy weather. Therefore, not only the linear-motion-based near-field method, but also the weakly nonline-

ar approach, which considers the higher-order restoring and Froude-Krylov forces at the actual wetted sur-

face of the body, are applied, and the effects of wave amplitude are investigated. Furthermore, an analysis

on the reason for discrepancies between the numerical and experimental results, especially for the specific

wave conditions, such as a higher speed, a short wave length, and a stern quartering sea, is also conducted.

From the analysis, the accuracy and the limitation of the present linear and weakly nonlinear methods for

the mean drift force problem are discussed by considering specific geometries of the ship model such as the

small incline angle of the non-wall-sided stern and the fine geometry around the high-nose bulbous bow.

Jae-Hoon Lee, Beom-Soo Kim, and Yonghwan Kim 85 Journal of Advanced Research in Ocean Engineering 0(0) (2017) 83-101

Fig. 1. Coordinate system and notations.

2. Mathematical Backgrounds

2.1. Boundary value problem

The coordinate system of a ship advancing with a forward speed, ( ),0,0U U=ur

, can be defined as shown

in Fig. 1. Here, β, A, and ω are the heading angle, the wave amplitude, and the frequency, respectively. To

define the boundary of the domain, SB and SF denote the body surface and the free surface, respectively. If

the ship is assumed to be rigid, the linear motion of the ship induced by the wave can be written as follows:

( ) ( ) ( ), T Rx t t t xd x x= + ´r r rr r

(1)

where the translation vector, ( )1 2 3, ,Tx x x x=r

, and the rotation vector, ( )4 5 6, ,

Rx x x x=r

, represent the six

degrees of freedom (DOF).

The linear potential theory is applied to the ship motion analysis. When assuming a fluid to be incom-

pressible and inviscid and the flow to be irrotational, a velocity potential (ϕ) can be introduced. For the

linearization of boundary conditions, the velocity potential and the wave elevation (ζ) are decomposed as

follows:

( ) ( ) ( ) ( ), , ,I dx t x x t x tf f f= F + +r r r r

(2)

( ) ( ) ( ), , ,I dx t x t x tz z z= +r r r

(3)

where Ф is the basis potential with the order of O(1), which is the double-body flow. ϕI and ζI are the velocity potential and the elevation of the incident wave, respectively. In addition, ϕd and ζd are the velocity potential and the elevation of the disturbed wave, respectively. Both incident and disturbed components have the order of O(ε). By using the decomposed variables, the linearized boundary value problem for ship motion can be derived as follows:

2 0 in fluid domainfÑ = (4)


( )

( )

( ) ( )( )( ) ( ) ( )( )

6

1

1 2 3

4 5 6

1 2 3

4 5 6

on

, ,

, ,

, ,

, ,

jd Ij j j B

j

n m Sn t n

n n n n

n n n x n

m m m n U

m m m n x U

xf fx

=

¶æ ö¶ ¶= + -ç ÷

¶ ¶ ¶è ø

=

= ´

= ×Ñ -ÑF

= ×Ñ ´ -ÑF

år

r r

rr

rr r

(5)

( ) ( )2

2 on 0d d

d I d IU zt z z

z fz z z z

¶ ¶ F ¶- -ÑF ×Ñ = + + -ÑF×Ñ =

¶ ¶ ¶

r (6)

( ) 1 on 0

2d

d d IU g U zt

ff z f

¶ é ù- -ÑF ×Ñ = - + ×ÑF - ÑF ×ÑF -ÑF ×Ñ =ê ú¶ ë û

r r(7)

where BS is the surface of the mean body, and the m-terms (mj) represent the effects of interaction be-

tween the steady and unsteady solutions in the problem of an advancing ship.

2.2. Equation of Motion

The equation of motion can be defined for the ship motion as follows:

[ ]{ } { } { } { } { }F.K. H.D. Res. viscousM F F F Fd = + + +&& (8)

where [M] is the mass matrix of ship. {FF.K.}, {FH.D.}, {FRes.}, and {FViscous} correspond to the Froude-

Krylov, hydrodynamic, restoring, and viscous damping forces, respectively.

In the linear approach, the Froude-Krylov force is obtained by integrating the linearized pressure of the incident wave potential on the mean-body surfaces of ship ( BS ), as follows:

{ } ( ). Lin. for =1,2,...,6.B

F K j I jSF U n dS j

tr f

ì ¶ üæ ö= - - -ÑF ×Ñí ýç ÷

¶è øî þòò

ur (9)

In addition, the hydrodynamic force is obtained by integrating the linearized pressure induced by the dis-

turbed wave potential and the basis potential, such that:

{ } ( ). . Lin.

1 for =1,2,...,6.

2BH D j d jS

F U U n dS jt

r fì ¶ üæ ö

= - - -ÑF ×Ñ - ×ÑF+ ÑF×ÑFí ýç ÷¶è øî þ

òòur ur

(10)

In the traditional linear equation of motion, ship motions are assumed to be small; therefore, a constant

restoring coefficient is adopted ({ } [ ]{ }Res. Lin. jjF C x=- ). In this study, for the roll motion for which the

viscous effects should be considered, the equivalent linear damping force ({ }viscous 4Lin.4 rollF b x= - & ) is adopt-

ed for easier numerical implementations. The linear damping coefficient, broll depends on the hull shape,

ship speed, and incident waves. The value can be determined from the free-roll-decay experiment (0.03–

0.10 for a typical hull).


On the other hand, in the weakly nonlinear approach, only the restoring and Froude-Krylov forces are

evaluated at the actual wetted surface of the ship, while the hydrodynamic force is evaluated at the mean

body, which is identical with the linear approach. This method is also known as a “blended method” be-

tween linear and nonlinear methods (Jensen et al., 2000) in which the partial nonlinearity induced by the

hull geometry is considered. In order to evaluate the nonlinear Froude-Krylov force, the incident wave po-

tential below the mean-water level takes its own linear value, while the incident wave potential above the

mean-water level takes the first-order perturbed value with respect to a wave elevation as follows

( )( )( )

( )( )

sin cos sin for 0

, , , .

sin cos sin for 0

kz

I

gAe k x Ut ky t z

x y z tgA

k x Ut ky t z

b b wwf

b b w zw

ì+ + - £ïï

= íï + + - < £ïî

(11)

By integrating the pressure induced by the incident waves on the exact wetted surface, the nonlinear

Froude-Krylov force can be obtained as follows:

{ }F.K. Non.

1 for =1,2,...,6.

2B

Ij I I I I jS

F U n dS jt

fr f f f f

¶ì ü= - - ×Ñ +ÑF ×Ñ + Ñ ×Ñí ý

¶î þòò

r (12)

Similarly, the nonlinear restoring force is calculated by subtracting the linear hydrostatic force at the

mean-body position from the force at the actual body position; it can be observed as follows:

{ } ( ) ( )Res. Non. for =1,2,...,6.

B Bj jj S S

F g z n dS g z n dS jr r= - - -òò òò (13)

2.3. Numerical method

In order to solve the linearized boundary value problem, an integral equation is derived by applying the

Green’s second identity:

.B F B F

d dd d dS S S S

G GdS GdS GdS dS

n n n n

f ff f f

¶ ¶ ¶ ¶+ - = -

¶ ¶ ¶ ¶ò ò ò ò (14)

In the Rankine panel method, the boundaries, i.e., the body and free surfaces are discretized into quadri-

lateral panels, and 3-D Rankine sources (G=1/r) are distributed on the panels to derive an algebraic equa-

tion for the unknown variables such as ϕd on SB and ∂ϕd/∂n on SF. In this study, furthermore, the variables at

the boundaries are evaluated by using the B-spline basis function; it can be written as follows:

( )

( )

( )

( ) ( )

( )

( ) ( )

( )9

1

,

,

,

d jd

d d

j j

dd j

tx t

x t t B xn n

x t t

ff

f f

z z

=

é ùé ù ê úê ú ê ú¶ ¶æ öê ú = ê úç ÷ê ú¶ ¶è øê úê ú ê úë û ë û

å

r

r r

r

(15)

where ( )B xr

indicates a B-spline basis function, and the subscript j are the coefficients of the variables at

the j-th discretized boundary panel, respectively. After solving the algebraic equation, ζd on SF, and ϕd on SF

at the next time step is obtained by the time integration kinematic and the dynamic free surface boundary


conditions of Eqs. (6-7), respectively. As a time-marching scheme, a mixed explicit–implicit scheme is

implemented as follows:

( )

( )

1

11 1

,

,

n nn nd d

d d

n nn nd d

d d

Pt

Pt

z zz f

f fz f

+

++ +

-=

D

-=

D

(16)

where P and Q denote the forcing functions, which contains all other terms in the free surface boundary

conditions. Also, an artificial damping zone at the truncated boundary of the free surface is applied to satis-

fy the radiation condition by attenuating outgoing waves. The kinematic free surface boundary condition is

modified according to the artificial wave-absorbing mechanism:

2

2 on 0dd d dU z

t z g

f nz nz f

¶¶æ ö- ×Ñ = - + =ç ÷

¶ ¶è ø

r (17)

where ν is the damping strength increasing gradually from the inner boundary to the outer boundary of the

zone. The details of the numerical implementations of this 3-D Rankine panel method in time-domain can

be found in Kim et al. (2011).

2.4. Prediction of the Second-Order Mean Drift Forces and Moments: Near-Field Method

In the near-field method, the second-order forces and moments are evaluated by direct integration of sec-

ond-order pressure on a body surface. To obtain the second-order pressure, the perturbations of physical

and geometrical variables, such as the ship motions, the hydrodynamic and hydrostatic pressures, the wave

elevation, and the normal vector of the surface, with respect to the mean-body position are used. In the

linear approach, only the quadratic terms of the linear values are considered, and the mean wetted surface is

employed as a problem domain. From the perturbation, the components of second-order force can be for-

mulated as in Table 1. In the equation, WL represents the waterline of the mean body. In addition, α in the

waterline integration indicates an angle between the mean waterline and the surface of the hull at the y-z

plane. By introducing the incline angle for a non-wall-sided hull, the force induced by the change of wetted

surfaces in waves can be calculated more precisely, which indicates a correction for the vertical hull slope.

The normal vector of each order derived by considering the linear translational and rotational motions of

the ship can be expressed as follows:

( )

( )

( )

( )

( )

( )

(0)

3

(1)

3

(2)

for 1,2,3

for 4,5,6

for 1,2,3

for 4,5,6

for 1,2,3

j

j

j

Rj

j

T Rj

j

j

T

n jn

x n j

n j

nn x n j

Hn j

nH x n

x

x x

x x

-

-

ì =ï

= í´ =ï

î

ì ´ =ï

= íé ù´ + ´ ´ =ï ë ûî

=

=´ + ´

r

r r

r r

r r r r r

r

r r r( )

( )( )

( )

2 25 6

2 24 5 4 6

2 234 6 5 6 4 5

0 0

1 where 2 0

2for 4,5,62 2

Rj

Hn j

x x

x x x x

x x x x x x-

é ù- +ì ê úï ê ú= - +í

ê úé ù´ =ïë û ê úî - +ë û

r r

(18)


Table 1. Classification of components in second-order force

Second-order

force ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } { }2 F.K. Res.H.O.T. H.O.T.

F I II III IV V VI VII F F= + + + + + + + +r

Formulation Linear approach Weakly nonlinear approach

(I)

( )

( )

( )

(0)2

3 4 5

(1)

3 4 5

(0)

3 4 5

1( )

2 sin

1( )

2 sin

1( )

2 sin

j

WL

j

WL

j

WL

ng y x dl

nU y x dl

nU y x dl

r z x x xa

r z x x xa

r d z x x xa

- + -

é ùæ ö- - - ÑF ×ÑF - + -ç ÷ê ú

è øë û

é ùæ ö- ×Ñ - - ÑF ×ÑF - + -ç ÷ê ú

è øë û

ò

ò

ò

uur

ur uur

( )

( )

( )

( )

(0)2

3 4 5

(0)2

3 4 5

(1)

3 4 5

(0)

3 4 5

1( )

2 sin

1( )

2 sin

1( )

2 sin

1( )

2 sin

j

WL

j

IWL

j

WL

j

WL

ng y x dl

ng y x dl

nU y x dl

nU y x dl

r z x x xa

r z x x xa

r z x x xa

r d z x x xa

- + -

- - + -

é ùæ ö- - - ÑF ×ÑF - + -ç ÷ê ú

è øë û

é ùæ ö- ×Ñ - - ÑF ×ÑF - + -ç ÷ê ú

è øë û

ò

ò

ò

ò

uur

ur uur

(II) (2)

Bj

Sgzn dSr- òò

-

(III) ( ) (1)

3 4 5

( )( )

( )B

I dI d

jS

Un dSt

g y x

f ff f

r

x x x

¶ +é ù- -ÑF ×Ñ +ê ú- ¶

ê ú+ + -ê úë û

òò

uur ( ) (1)

B

dd jS

U n dSt

fr f

¶é ù- - -ÑF ×Ñê ú¶ë ûòò

uur

(IV) (0)1

( ) ( )2B

I d I d jS

n dSr f f f f- Ñ + ×Ñ +òò (0)1

2BI d d d jS

n dSr f f f fæ ö

- Ñ ×Ñ + Ñ ×Ñç ÷è ø

òò

(V) ( ) (0)( )( )

B

I dI d jS

U n dSt

f fr d f f

¶ +é ù- ×Ñ - -ÑF ×Ñ +ê ú¶ë ûòòur uur

( ) (0)

B

dd jS

U n dSt

fr d f

¶é ù- ×Ñ - -ÑF ×Ñê ú¶ë ûòòur uur

(VI)

(1)

(2)

1

2

1

2

B

B

jS

jS

U n dS

U n dS

r d

r

æ öé ùæ ö- ×Ñ - - ÑF ×ÑFç ÷ç ÷ê ú

è øë ûè ø

é ùæ ö- - - ÑF ×ÑFç ÷ê ú

è øë û

òò

òò

ur uur

uur

(1)

(2)

1

2

1

2

B

B

jS

jS

U n dS

U n dS

r d

r

æ öé ùæ ö- ×Ñ - - ÑF ×ÑFç ÷ç ÷ê ú

è øë ûè ø

é ùæ ö- - - ÑF ×ÑFç ÷ê ú

è øë û

òò

òò

ur uur

uur

(VII) (0)1

2BjS

H x gz U n dSræ öé ùæ ö

- ×Ñ - - ÑF ×ÑFç ÷ç ÷ê úè øë ûè ø

òòr uur (0)1

2BjS

H x U n dSræ öé ùæ ö

- ×Ñ - - ÑF ×ÑFç ÷ç ÷ê úè øë ûè ø

òòr uur

{FF.K.}H.O.T. - { } ( )F.K. Non.B

II jj S

F U n dSt

fr f

¶æ ö+ - -ÑF ×Ñç ÷

¶è øòò

r

{FRes.}H.O.T. - { } ( )Res. 3 4 5Non.B B

j jj S SF gzn dS g y x n dSr r x x x+ + + -òò òò

The temporal mean values of the second-order forces indicate the drift forces due to waves. The details of

this formula can be found in the studies by Joncquez (2009) and Kim and Kim (2011).

Like the weakly nonlinear approach for the equation of motion, the near-field method can also be extend-

ed to the weakly nonlinear formulation by modifying the components related to the incident wave and the

hydrostatic pressure. In other words, the components of the restoring and Froude-Krylov forces are re-

placed by the higher-order forces evaluated at the exact wetted surfaces. The modified components of the

weakly nonlinear approach are shown in Table 1. The higher-order restoring and Froude-Krylov forces

({FF.K.}H.O.T. and {FRes.}H.O.T., respectively) are calculated by subtracting the first-order forces from the non-

linear forces. It should be noted that the coupling terms between the variables of disturbed wave and inci-

dent wave remain in the formulation. Zhang et al. (2009) carried out similar computations for higher-order

horizontal drifting effects on a ship considering the incident wave. For a typical ship model, the waterline-

integral term, (I), is known to be the main contributor to the second-order force. Although the vertical hull

slope correction using the incline angle is adopted in the linear approach, the force induced by relative

wave elevation can be obtained more accurately by the present weakly nonlinear approach, especially for a

non-wall-sided ship in large-amplitude motions.

3. Computation Results

3.1. Ship Model

The Duisburg Test Case (DTC) container vessel, which is a post-Panamax 14,000 twenty-foot equivalent

unit (TEU) container vessel, is chosen as the target ship. The main dimensions of the ship for real scale are

summarized in Table 2. This ship model has a relatively fine hull (low CB), and the geometry of the ship is


Table 1. Main dimensions of the DTC container ship.

Designation DTC container ship

LBP, L (m) 355.0

Beam, B (m) 51.0

Draft, d (m) 14.5

Block coefficient, CB 0.661

GM (m) 5.1

characterized by a large bow flare angle and an overhanging transom. These non-wall-sided shapes lead to

the large variation of the water-plane area according to the changes of the wave elevation at the fore and

the aft body. Therefore, the calculation of the waterline-integral term, (I) in Table 1 is significantly influ-

enced by the vertical hull slope correction factor (sinα), which is very small near the transom stern as

shown in Fig. 2.

In the present Rankine panel method, the hull and free surfaces are discretized respectively. On the hull,

the bow and stern regions are discretized finer than the mid-ship region, because the flows are more com-

plicated in these regions. On the free surface, the O-type grid system is applied to consider various heading

angles, and the panels are more clustered near the hull to obtain accurate disturbed wave patterns generated

by the ship. The total domain size of the free surface is five times that of the wave length, which contains

two wave lengths for the artificial damping zone located at the truncated boundary. In the weakly nonlinear

approach, the hull geometry above the mean-water level should be considered to include the exact wetted

surface; hence, the nonlinear hull panels are modeled up to a certain height. Fig. 3 shows an example of the

linear and nonlinear solution panels.

Fig. 2. 2-D strips and vertical hull slope correction factor (sinα) for ship model.

(a) Free surface panels (b) Nonlinear hull panels

Fig. 3. Examples of solution grids.


Table 2. Tests conditions of DTC container ship.

Froude number,

Fn

0.000 (zero speed),

0.052 (U=6 knots),

0.139 (U=16 knots)

Heading angle,

β (degree)

180.0 (head sea),

120.0 (bow quartering sea),

60.0 (stern quartering sea)

Wave amplitude,

A/L 0.01-0.02

Fig. 4. Linear panels and waterline around bow of ship.

Also, the ship has a high-nose bulbous bow to attenuate wave resistance, especially for the high-speed op-

eration. Owing this fine hull geometry, the waterline at the bow region has a sharp corner. These character-

istics of geometry require close considerations in the application of the 3-D Rankine panel method. Other-

wise, unstable numerical solutions for the velocity potential and the wave elevation may be obtained at the

tip of the bulb. This local phenomenon can significantly affect the prediction of the second-order force,

while there are little effects on the global behaviors such as ship motions. The Fig. 4 shows the example of

linear panels for the body and free surfaces, and the waterline around the bow of the ship.

3.2. Test Conditions

For the ship model, various experiments have been carried out in the benchmark study of EU-funded

SHOPERA project. The measurement data for the wave-induced added resistances, drift sway forces, and

yaw moments have been obtained by the captive model test (model scale of 63.65) in the towing tank and

the ocean basin at MARINTEK (Sprenger et al., 2015). It should be noted that the wave amplitude of each

condition in the experiment is relatively high as the aim of the benchmark study is to evaluate the accuracy

and the reliability of the current numerical methods for the prediction of the added resistance in adverse

conditions. In this study, therefore, the two conditions for wave amplitudes (A/L=0.01-0.02) are adopted for

the weakly nonlinear computations, which covers the range of wave amplitudes in the experiment of the

benchmark study. In addition, to access the sufficient propulsion power and steering device to maintain

maneuverability, the test conditions are assigned for various forward speeds and heading angles. All of test

conditions are summarized in Table 3.

3.3. In head sea conditions

3.3.1. Motion response

It is important to calculate the ship motion responses precisely because the motion is directly related to the

radiation component of the added resistance. At zero speed, the amplitudes of the vertical motions (heave

and pitch motions) obtained by the Rankine panel method is validated with the frequency-domain solutions

of the commercial software “WADAM” as shown in Fig. 5. Furthermore, the weakly nonlinear motions are

also compared with the linear motions. For short wave lengths (λ/L<0.5), the heave and pitch motions are

relatively small, and the results of linear and weakly nonlinear approaches show good correspondences. In


waves of larger wave lengths, on the other hand, the weakly nonlinear approach provides different motions

compared to those of the linear analysis according to the different wave amplitudes. Especially, when the

wave length is similar to the length of the ship (λ/L≈1.0), the water-plane area varies significantly accord-

ing to the wave elevation, which leads to the different restoring forces and motion responses. As the ad-

vance speed increases, this phenomenon becomes intensified as the wave length of the resonance in the

vertical motions gets closer to the length of the ship.

3.3.2. Added resistance

To access the accuracy and reliability of the present near-field method, computation results for added re-

sistances in head sea conditions represented in Table 3 are validated with the experimental data shown in

Fig. 6. At zero speed, the present linear approach of Table 1 gives similar results with the far-field solution

of WADAM for overall wave lengths, and good agreements between the numerical and experimental re-

sults can be observed when the vertical hull slope correction is adopted. However, if the ship is regarded

wall-sided (sinα=1), the discrepancies with the measurement data are confirmed especially for large wave

lengths.

(a) Fn=0.000 (zero speed)

(b) Fn=0.052 (U=6 knots) (c) Fn=0.139 (U=16 knots)

Fig. 5. Motion responses in head sea conditions.


In the linear near-field method, the total added resistance can be decomposed into the component of inci-

dent wave and disturbed wave. The contributions of incident wave and restoring force on the resistance are

defined as follows:

( )

( )

( )

( )

(0)2 (2)

3 4 5

(1)3 4 5

(0) (0)

(0)

1( )

2 sin

( )

1

2

B

B

B B

B

j

I I jWL S

II jS

II I j I jS S

jS

nF g y x dl gzn dS

U g y x n dSt

n dS U n dSt

H x gz n dS

r z x x x ra

fr f x x x

fr f f r d f

r

= - + - -

¶æ ö- - -ÑF ×Ñ + + -ç ÷

¶è ø

¶æ ö é ù- Ñ ×Ñ - ×Ñ - -ÑF ×Ñç ÷ ê ú¶è ø ë û

é ù- ×Ñë û

ò òò

òò

òò òò

òò

ur

ur ur

r

(19)

As shown in Fig. 7, the incident-wave component with or without the vertical hull slope correction are

significantly different. Especially, the component is converged to the zero value for short wave lengths

when the correction is adopted, which is physically appropriate since the ship motions are negligible for

short waves, so the contributions of incident wave calculated by the integration for the closed contour (the

mean-body surface of ship) should be zero. Furthermore, when the component is compared with the higher-

order restoring and Froude-Krylov forces of the weakly nonlinear approach, it is confirmed that the linear

and weakly nonlinear approaches are consistent when the correction is applied. The higher-order forces

computed by the surface integration at the exact wetted surface of ship considering incident wave elevation

can be regarded more accurate than the incident-wave component in the linear approach. On the other hand,

the disturbed-wave components (Fd; the rest of added resistance except the incident-wave components)

which is calculated by the same formulation in the two approaches are also quite different according to the

adoption of the correction. This phenomenon indicates that the waterline-integral term (I) induced by the

disturbed wave significantly varies depending on the considerations for the nonlinearities of non-wall-sided

geometry. Therefore, the inaccurate predictions for added resistances obtained by the linear and weakly

nonlinear approach without the correction can be explained by this inconsistency.

As the forward speed increases, the similar trends of prediction for added resistance can be seen; the line-

ar prediction corresponds to the measurement data, and the results are consistent with those of the weakly

nonlinear approach with the vertical hull slope correction. However, when the wave length is similar with

the length of ship (λ/L≈1.0), the weakly nonlinear results decreases compared to those of linear approach

due to the different component of incident wave while the disturbed-wave components are quite similar

according to the wave amplitude as seen in Fig. 7. These nonlinear effects with respect to the wave ampli-

tude result from the different vertical motions (Fig. 5). Therefore, the effects can be considered negligible

in short waves where the ship motions are relatively small. In other words, by using the weakly nonlinear

approach, only the nonlinearities of the radiation component which is related with the ship motions can be

accounted.



(b) Fn=0.052 (U=6 knots)

(c) Fn=0.139 (U=16 knots)

Fig. 6. Added resistances in head sea conditions: with (left) or without (right) vertical hull slope correction.



(b) Fn=0.052 (U=6 knots)

(c) Fn=0.139 (U=16 knots)

Fig. 7. Decomposition of added resistances in head sea conditions: incident-wave (left) and disturbed-wave (right) component.



(b) Fn=0.052 (U=6 knots)

(c) Fn=0.139 (U=16 knots)

Fig. 8. Wave contours around the ship in head sea conditions: λ/L=0.200

For a short wave, the hydrodynamic nonlinear effects of disturbed waves are intensified while the inci-

dent-wave component is negligibly small, which results in inaccurate computations of the added resistances.

Especially, the discrepancies in comparison with the measurement data occur in short waves at the higher

speed of ship (Fn=0.139) as shown in Fig. 6. On the other hand, the asymptotic formula of the National

Maritime Research Institute (NMRI; Kuroda et al., 2008) provides slightly better accurate prediction. This

formula was developed as a modification for the method by Fujii and Takahashi (1975), which is the semi-

empirical method for added resistance in short wave conditions by adopting some complement coefficients

to the drift force formula of a fixed vertical cylinder. When the wave length is small compared to the draft

of the ship, the wave-induced motion can be considered negligible, and the contribution of the shaded re-

gion (stern of ship) on the added resistance can be also neglected owing to the fully diffracted waves at the

bow region. Therefore, it can be inferred that the discrepancies at the high speed in short wave lengths re-

sult from the inaccurate disturbed waves at the bow and stern of the ship.

To investigate the phenomenon concretely, the wave contours near the ship for different ship speeds are

compared as shown in Fig. 8. At the high speed, stern waves are incurred in a short wave, which lead to the

negative force according to the waterline-integral term (I) in Table 1. When the vertical hull slope correc-

tion is not adopted (sinα=1.0), this force is attenuated, so the total added resistance obtained by the linear

approach gets closer to the experimental data. In other words, it can be found that the prediction of the add-

ed resistance changes according to the vertical hull slope correction at the overhanging transom stern of the

ship advancing at a high speed. On the other hand, there are no noticeable discrepancies between the results

with and without the correction factor at the zero and low speeds as the stern waves do not occur. The

overestimated stern waves at the high speed result from the limitation of the present Rankine panel method

to capture the steady flow and the short waves in the vicinity of the non-wall-sided stern. By considering

the geometrical characteristics of the ship model, therefore, the enhancements for the prediction of the dis-

turbed waves are required to improve the accuracy of the present method.


(a) Heave motion

(b) Roll motion

(c) Pitch motion

Fig. 9. Motion responses in oblique sea conditions: Fn=0.000 (zero speed), bow quartering sea (β=120.0 degree, left) and stern

quartering sea (β=60.0 degree, right).

3.4. In Oblique Sea Conditions

3.4.1. Motion Response

In oblique seas, all the 6-DOF motions are incurred due to waves. For the test conditions of oblique seas

in Table 3, the ship motions are computed by the present Rankine panel method. As shown in Fig. 9, it is

seen that the motion responses obtained by the linear and weakly nonlinear approaches are similar. Howev-

er, when the amplitude of the wave is quite large, there are slight discrepancies between the two results.

Especially for the heave motions, the discrepancies are intensified because the motions in oblique seas have

significant amplitudes even for a short wave length (λ/L≈0.5). On the other hand, the roll motions increase

continuously as the wave length increases. Due to these large-amplitude motions, the differences between

the linear and weakly nonlinear results are also confirmed for long wave lengths. As a result, the viscous

damping force in the roll motion becomes the important factor in the prediction of the responses of roll

motion.

3.4.2. Drift force and Moment

The computation results for the drift forces and the moments in the oblique sea conditions are validated

with the experimental data in the SHOPERA benchmark study. The vertical hull slope correction using the

incline angle, which is validated for head sea conditions, is also adopted for the prediction. As shown in Fig.

10, the drift surge forces obtained by the present method show a good agreement with the measurement

values and the results of WADAM for different heading angels. However, the discrepancies between the


results of the linear and the weakly nonlinear methods are confirmed in the peak region due to the different

motion responses of the two methods in Fig. 9. These discrepancies are intensified in stern quartering seas

because not only the heave motion, but also the pitch motions are different in the two methods. Also, in the

stern quartering seas, the waves proceed to the stern of ship where the wetted surfaces are varied signifi-

cantly according to the change of draft, which results in the different predictions of added resistances of the

linear and weakly nonlinear approaches. In the cases of the drift sway forces and the yaw moments, quite

accurate computation results can also be obtained in comparison with the measurement data. In contrast

with the drift surge force, the weakly nonlinear effects due to the wave amplitude become attenuated. How-

ever, some discrepancies between the numerical and experimental results for the drift yaw moments are

shown in short and stern quartering waves. This phenomenon is related to the unstable computations, which

are induced by the fine geometry around the high-nose bulbous bow.

(a) Surge force

(b) Sway force

(c) Yaw moment

Fig. 10. Drift forces and moments in oblique sea conditions: Fn=0.000 (zero speed), bow quartering sea (β=120.0 degree, left)

and stern quartering sea (β=60.0 degree, right).

Next, the drift forces and the moment for the low speed (Fn=0.052) are considered as shown in Fig. 11.

The computed drift surge and the sway forces also show good correlations with the measurement values.

Also, similar results are obtained by the linear and the weakly nonlinear approaches as the waves around

the bow of ship where the incline angle is relatively large become important for a ship with the forward

speed. However, the discrepancies between the drift yaw moments of the numerical and the experimental

results are confirmed in the short waves. To examine this phenomenon concretely, the wave elevation and


the hydrodynamic pressure distribution are investigated around the bow of the ship as shown in Fig. 12. For

a ship advancing forward, the fine geometry around the high-nose bulb can be regarded as a thin wall,

meaning that the bow waves have the characteristics of the corner flow in the potential theory without the

flow separation triggered by the viscous effects. Furthermore, in the present numerical approach, the Ran-

kine sources distributed at the bulbous bow and the free surface are in close proximity. Therefore, the wave

elevation and the pressure increase significantly at the tip of the bulb compared to those in the both bow

and stern quartering seas. This local phenomenon is intensified in the stern quartering waves as the waves

rotate more severely around the corner. These overestimated wave elevation and pressure make the predic-

tion of the yaw moment unstable and inaccurate, while there are only a few noticeable effects on the surge

force due to the thin bow geometry (small normal component in longitudinal direction). In conclusion, the

additional considerations in the present Rankine panel method are required for the accurate prediction of

the local-flow-induced yaw moment near the fine bow.

4. Conclusions

The drift forces and moments on the large container vessel at different advance speeds and heading angles

are investigated by applying the time-domain 3-D Rankine panel method. The computation results are vali-

dated with the experimental data measured in the previous benchmark study. By considering the geomet-

rical characteristics of the ship model, the following conclusions can be obtained:

(a) Surge force

(b) Sway force

(c) Yaw moment

Fig. 11. Drift forces and moments in oblique sea conditions: Fn=0.052 (U=6 knots), bow quartering sea (β=120.0 degree, left)

and stern quartering sea (β=60.0 degree, right).


(a) Bow quartering sea (β=120.0 degree) (b) Stern quartering sea (β=60.0 degree)

Fig. 12. Wave elevation and hydrodynamic pressure distribution around the bow of ship: Fn=0.052 (U=6 knots), λ/L=0.200.

� In the linear approach, the vertical hull slope correction using the incline angle should be adopt-

ed to include nonlinearities of the non-wall-sided geometry, which provides the consistent solu-

tion for incident-wave induced added resistance with the nonlinear restoring and Froude-Krylov

forces in the weakly nonlinear approach.

� Motion responses in head seas obtained by the linear and weakly nonlinear approaches show the

discrepancies when the wave length is similar with the length of ship due to the different restor-

ing forces. For the advancing ship, this phenomenon is intensified, which leads to different pre-

diction of the radiation component of the added resistance.

� The added resistances in head seas obtained by the present method and the experiment are in

good agreement for the overall test conditions except for the cases of very short wave lengths at

the high advance speed. The underestimated prediction may have resulted from the diffracted

waves inaccurately calculated at the non-wall-sided stern, which gives rise to the negative force

on the ship (pushing forward).

� In oblique seas, the discrepancies between the numerical and experimental results on the drift

yaw moments are confirmed for short waves due to the unstable solutions around the tip of high-

nose bulb. This phenomenon becomes intensified for the stern quartering seas due to the more

overestimated pressure and the wave elevation induced by the corner-flow behaviors in wave

propagation around the fine hull.

Acknowledgement

This study was funded by the Ministry of Trade, Industry and Energy (MOTIE), Korea, through the project

“Technology Development to Improve Added Resistance and Ship Operational Efficiency for Hull Form

Design” (Project No.10062881), and the Lloyd’s Resister Foundation (LRF)-Funded Research Center at

Seoul National University. Their support is acknowledged.

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