Study on added resistance of a ship under parametric roll motion

E L S E V I E

C o n t e n t s lists available at S c i e n c e D i r e c t

Ocean Engineering

journal homepage: www.elsevier.com/locate/oceaneng

Study on added resistance of a ship under parametric roll motion

Jae-Hoon Lee, Yonghwan Kim

Department of Naval Architectiu-e and Ocean Engineering, Seoul National Univeisitij, Republic of Korea

CrossMark A R T I C L E I N F O Keywords: Parametric roll Nonlinear roll Added resistance Rankine panel method Ship stability

A B S T R A C T

Parametric roll of sliip is a rare event occurring in specific conditions, but it can cause the dynamic roll instability. Since this phenomenon is dependent on ship speed and wave frequency, the development of roll motion and the resultant added resistance which can cause speed change should be considered simultaneously when parametric roll occurs. This paper considers a numerical study on the added resistance of a modern containership under parametric roll. A time-domain Rankine panel method adopting a weakly nonlinear formulation is adopted to obtain ship motions and added resistances in waves. Seakeeping computations in regular head-sea conditions are performed wth and wlhout parametric roll, and the increment of added resistance is investigated mth regard to the components classified in the dkect pressure integration method. Furthermore, according to the decouphng phenomena between the components ofthe vertical motions and the roll motion, a correlation betiveen the parametric roll and the added resistance is derived. Lastly, a simple prediction method for the added resistance in irregular parametric roll motion is suggested based on the correlation, and its accuracy and efficiency are discussed by comparing the prediction with the results of the direct numerical computation.

1. I n t r o d u c t i o n P r e v e n t i o n of t h e p a r a m e t r i c r o l l p h e n o m e n a , w h i c h i s o n e of t h e d y n a m i c stability p r o b l e m s o f s h i p s , h a s b e e n a m a t t e r of c o n c e r n , b e c a u s e t h e r e s o n a n t r o l l m o t i o n i s excited r a p i d l y d u r i n g s e v e r a l e n c o u n t e r p e r i o d s o f w a v e s . A l t h o u g h there h a v e b e e n attempts to e x a m i n e a direct c o n t r o l o n t h e roll m o t i o n , s u c h as t h e a p p l i c a t i o n o f the bilge k e e l a n d t h e active fin s t a b i l i z e r ( L e v a d o u a n d van't V e e r , 2 0 1 1 ) , a n d t h e U - t a n k ( H o l d e n , 2 0 1 1 ) , t h e o p e r a t i o n a l g u i d a n c e f o r a s h i p crew's d e c i s i o n s u p p o r t s y s t e m c a n s e r v e a s a p r i o r c o u n t e r -m e a s u r e i n o r d e r to a v o i d v u l n e r a b l e e n v i r o n -m e n t a l a n d o p e r a t i o n a l c o n d i t i o n s to t h e p h e n o m e n a ( S o n g et a l . , 2 0 1 3 ) . T h e c o n d i t i o n s a r e d i r e c t l y r e l a t e d to t h e e n c o u n t e r w a v e f r e q u e n c y ; therefore, the f o r w a r d s p e e d a n d t h e h e a d i n g angle o f s h i p s s h o u l d b e c o n t r o l l e d f o r i t s p r e v e n t i o n . W h e n t h e excitation o f p a r a m e t r i c r o l l starts u n d e r t h e v u l n e r a b l e c o n d i t i o n s , t h e s p e e d o f s h i p is c h a n g e d s i m u l t a n e o u s l y d u e to t h e a d d e d r e s i s t a n c e o c c u r r e d i n t h e p h e n o m e n a . T h e r e f o r e , i n o r d e r t o t a c k l e v u l n e r a b i l i t i e s r e l e v a n t to t h e p a r a m e t r i c r o l l , t h e s p e e d v a r i a -tions s h o u l d b e a c c o u n t e d i n n u m e r i c a l s i m u l a t i o n s . T o this e n d , t h e a c c u r a t e p r e d i c t i o n of t h e a d d e d r e s i s t a n c e i n d u c e d b y t h e l a r g e -a m p l i t u d e r o l l m o t i o n is -also r e q u i r e d . I n other w o r d s , \vith r e g -a r d to the p a r a m e t r i c r o l l p h e n o m e n a , t h e a d d e d r e s i s t a n c e o f a s h i p i s c o n s i d e r e d i n t e r m s o f the d y n a m i c s t a b i l i t y i n w a v e s , not t h e efficiency

d u r i n g s h i p operations. H i s t o r i c a l l y , t h e a d d e d r e s i s t a n c e d u e to w a v e s h a s b e e n ^videly investigated b e c a u s e t h e a c t u a l p e r f o r m a n c e of s h i p s i n s e a w a y s i s d e t e r m i n e d b y t h e r e s i s t a n c e . S e v e r a l p r e v i o u s r e s e a r c h e r s c o n d u c t e d e x p e r i m e n t s f o r t h e a d d e d r e s i s t a n c e s o n t h e t y p i c a l s h i p m o d e l s s u c h as t h e s e r i e s 6 0 m o d e l s ( G e r r i t s m a a n d B e u k e l m a n , 1 9 7 2 ) , t h e S 1 7 5 c o n t a i n e r s h i p ( F u j i i a n d T a k a h a s h i , 1 9 7 5 ) , a n d t h e W i g l e y m o d e l s ( J o u r nee, 1 9 9 2 ) . I n t h e c a s e s o f a n a l y t i c a l a n d n u m e r i c a l a p p r o a c h e s , t w o m a j o r m e t h o d s h a v e b e e n i n t r o d u c e d to a n a l y z e t h e a d d e d r e s i s t a n c e p r o b l e m . F i r s t , a f a r - f i e l d m e t h o d , w h i c h w a s o r i g i n a l l y d e r i v e d b y M a r u o ( 1 9 6 0 ) a n d f u r t h e r r e f i n e d b y N e w m a n ( 1 9 6 7 ) , i s b a s e d o n a m o m e n t u m c o n s e r v a t i o n theory. T h i s m e t h o d i s s i m p l e a n d efficient t h a t does n o t i n v o l v e s o l v i n g a b o u n d a r y v a l u e p r o b l e m f o r t h e p r e s s u r e a c t i n g o n a b o d y . I n t h e m e t h o d , h o w e v e r , t h e r e a r e difficulties i n h a n d h n g a p r o p e r c o n t r o l s u r f a c e . A l t e r n a t i v e l y , a n e a r -field m e t h o d , w h i c h r e p r e s e n t s direct i n t e g r a t i o n o f t h e s e c o n d - o r d e r p r e s s u r e o n t h e b o d y s u r f a c e h a s also a p p l i e d to t h e c a l c u l a t i o n o f a d d e d r e s i s t a n c e . O n e o f t h e advantages o f t h e n e a r - f i e l d m e t h o d is t h e d e c o m p o s i t i o n o f a d d e d r e s i s t a n c e , w h i c h enables t h e p h y s i c a l o b -s e r v a t i o n b y a c o m p o n e n t a n a l y -s i -s a n d a n e x t e n -s i o n to c o n -s i d e r a t i o n -s for n o n l i n e a r i t i e s i n t h e p h e n o m e n o n . F a l t i n s e n et a l . ( 1 9 8 0 ) v a l i d a t e d t h e c o m p u t a t i o n r e s u l t s o b t a i n e d b y t h e n e a r - f i e l d m e t h o d , a n d f o r m u l a t e d a s i m p l i f i e d a s y m p t o t i c a p p r o a c h to e n h a n c e t h e r e s u l t s for s h o r t w a v e s . G r u e a n d B i b e r g ( 1 9 9 3 ) also adopted t h e m e t h o d a l o n g

* Corresponding author.

E-mail address: yhwankim@snu.ac.kr (Y. Kim). http://dx.doi.Org/10.1016/j.oceaneng.2017.08.015

Received 31 May 2017; Received in revised form 27 July 2017; Accepted 11 August 2017 0 0 2 9 - 8 0 1 8 / © 2017 Elsevier L t d All rights reseiTed

w i t h the f r e q u e n c y - d o m a i n w a v e G r e e n f u n c t i o n to evaluate the r e s i s t a n c e i n d u c e d b y waves o n a floating body a d v a n c i n g w i t h a s m a l l s p e e d . I t s h o u l d be noted t h a t most of t h e r e s e a r c h e s i n the early stages w e r e b a s e d on t h e l i n e a r potential theories; h e n c e , the effects of n o n l i n e a r free s u r f a c e flows a n d h u l l g e o m e t r i e s w e r e not i n c l u d e d .

R e c e n t l y , n o n l i n e a r a p p r o a c h e s u s i n g c o m p u t a t i o n a l fluid d y n a m i c s ( C F D ) s i m u l a t i o n h a v e b e e n a p p l i e d o w i n g to t h e d e v e l o p m e n t of c o m p u t i n g power. A s i m u l a t i o n m e t h o d c a l l e d t h e W I S D A M - X w a s u s e d b y O r i h a r a a n d IVIiyata ( 2 0 0 3 ) a n d O r i h a r a et a l . ( 2 0 0 8 ) to e x a m i n e the a d d e d r e s i s t a n c e for different b o w s h a p e s above the m e a n -w a t e r l e v e l . I n t h i s m e t h o d , the R e y n o l d s - a v e r a g e d N a v i e r - S t o k e s e q u a t i o n ( R A N S E ) w a s solved u s i n g a finite v o l u m e m e t h o d ( F V M ) i n a n o v e r l a p p i n g grid s y s t e m . F u r t h e r m o r e , Y a n g ( 2 0 1 5 ) developed a C a r t e s i a n - g r i d - b a s e d m e t h o d for s o l v i n g the E u l e r e q u a t i o n , a n d i n v e s t i g a t e d t h e effects of the n o n U n e a r b o w w a v e s o n the a d d e d r e s i s t a n c e s for different w a v e a m p l i t u d e s . I n the o p e r a t i o n o f a large m o d e r n s h i p , the m a j o r i s s u e s are not o n l y the efficiency of r e d u c i n g g r e e n h o u s e gas e m i s s i o n s , b u t also the p e r f o r m a n c e of s h i p s i n rough seas. T h e r e f o r e , the t r e n d of a d o p t i n g t h e n o n l i n e a r s i m u l a t i o n is expected to c o n t i n u e . H o w e v e r , t h e r e h a v e b e e n l i m i t a t i o n s i n the a p p l i c a t i o n of t h e C F D methods,, b e c a u s e of h i g h c o m p u t a t i o n a l costs a n d a strong d e p e n d e n c y o n the grid s y s t e m .

I n t h e c a s e of p a r a m e t r i c roll p h e n o m e n a , the t i m e - d o m a i n s i m u l a t i o n s f o r n o n l i n e a r s h i p m o t i o n s h a v e b e e n c o n d u c t e d . T h e m o s t i m p o r t a n t p a r a m e t e r i n t h e n u m e r i c a l s i m u l a t i o n is a n a c c u r a t e p r e d i c t i o n of the t i m e - v a r y i n g n o n l i n e a r r e s t o r i n g forces i n w a v e s . F o r a n efficient c o m p u t a t i o n , a w e a k l y n o n l i n e a r a p p r o a c h , w h i c h c o n s i d e r s the p a r t i a l n o n l i n e a i i t i e s of h u l l geometry, w a s u s e d i n m a n y p r e v i o u s r e s e a r c h efforts. F o r e x a m p l e , F r a n c e et a l . ( 2 0 0 3 ) a n d S h i n et a l . ( 2 0 0 4 ) appUed t h e R a n k i n e p a n e l m e t h o d ( R P M ) to evaluate not only t h e susceptibility criteria b u t also t h e a m p l i t u d e of roU m o t i o n f o r large c o n t a i n e r s h i p s . S p a n o s a n d P a p a n i k o l a o u ( 2 0 0 7 ) a n a l y z e d the p a r a -m e t r i c r o l l of a fishing v e s s e l i n r e g u l a r w a v e s u s i n g the i -m p u l s e r e s p o n s e f u n c t i o n ( I R F ) m e t h o d . K i m a n d K i m ( 2 0 1 1 b ) developed a m u l t i - l e v e l a n a l y s i s , w h i c h i n c l u d e level-by-level appUcations of a n a n a l y t i c a l f o r m u l a b a s e d o n the m e t a c e n t r i c height ( G M ) v a r i a t i o n , the I R F m e t h o d , a n d the R a n k i n e p a n e l m e t h o d to c o m p a r e the p r o p e r t i e s of e a c h m e t h o d .

F o r r e a l i s t i c s i m u l a t i o n s of p a r a m e t r i c r o l l m o t i o n s , t h e speed v a r i a t i o n due to r e s i s t a n c e c a u s e d b y severe r o l l m o t i o n s h o u l d be c o n s i d e r e d along >vith o c c u r r e n c e s a n d m a g n i t u d e s o f the p h e n o m e n a . T h e r e f o r e , t h e r e h a v e b e e n attempts to develop the s u r g e - r o l l c o u p l e d m o d e l to a c c o u n t for t h e i n t e r a c t i o n b e t w e e n t h e d e v e l o p m e n t o f p a r a m e t r i c roll a n d the s p e e d v a r i a t i o n ( V i d i c - P e m n o v i c a n d J e n s e n , 2 0 0 9 ) . A l s o , B r e u a n d F o s s e n ( 2 0 1 0 ) a p p l i e d the s p e e d a n d h e a d i n g c o n t r o l to t h e c o u p l e d m o d e l f o r m i t i g a t i o n o f t h e p h e n o m e n a . H o w e v e r , i n m o s t of the p r e v i o u s studies, a relatively s i m p l e m e t h o d w a s a p p l i e d to estimate the a d d e d r e s i s t a n c e , c o n s i d e r i n g only for the s u r g e a d d e d m a s s a n d F r o u d e - K r y l o v a n d r e s t o r i n g forces. O n the o t h e r h a n d , to p r e d i c t the a d d e d r e s i s t a n c e i n d u c e d b y p a r a m e t r i c r o l l m o r e accurately, L u et a l . ( 2 0 1 5 ) a d o p t e d a r o l l m o t i o n o b t a i n e d b y m o d e l e x p e r i m e n t s to a r e v i s e d f o r m u l a f r o m t h e l i n e a r f a r - f i e l d m e t h o d of M a r u o ( 1 9 6 0 ) . H o w e v e r , a c c o r d i n g to a Unear perspective, t h e m e t h o d d i d not c o n s i d e r t h e n o n l i n e a r effects of t h e large-a m p l i t u d e rofl m o t i o n . T h e r e f o r e , large-a n efficient n o n h n e large-a r large-a p p r o large-a c h is r e q u i r e d to a c c o u n t for the effects o n t h e a d d e d resistance.

I n t h e p r e s e n t study, t h e t i m e - d o m a i n 3 - D R a n k i n e p a n e l m e t h o d developed b y K i m et a l . ( 2 0 1 1 ) is a p p l i e d to p r e d i c t s h i p m o t i o n s a n d a d d e d r e s i s t a n c e of a c o n t a i n e r s h i p u n d e r the p a r a m e t r i c roll p h e n o m -e n a . T h -e n -e a r - f i -e l d m -e t h o d d -e r i v -e d b y J o n c q u e z ( 2 0 0 9 ) a n d K i m a n d K i m ( 2 0 1 1 a ) , w h i c h is the direct p r e s s u r e integration m e t h o d f o r the e v a l u a t i o n of a d d e d resistance, is m o d i f i e d b a s e d o n a w e a k l y n o n l i n e a r a p p r o a c h . I n o t h e r w o r d s , the h i g h e r o r d e r r e s t o r i n g a n d F r o u d e -K r y l o v forces at t h e a c t u a l w e t t e d s u r f a c e a r e i n c l u d e d to c o n s i d e r the n o n l i n e a r effects i n d u c e d b y p a r a m e t r i c roll m o t i o n . T h r o u g h the d e c o m p o s i t i o n a n a l y s i s a c c o r d i n g to c o m p o n e n t s of the n e a r - f i e l d

Fig. 1. Coordinate system and notations in Ranldne panel method.

m e t h o d , t h e i n c r e a s e d r e s i s t a n c e due to the p a r a m e t r i c roll i n a regular w a v e is investigated. A l s o , the c o r r e l a t i o n bet^veen the p a r a m e t r i c r o l l m o t i o n a n d the a d d e d r e s i s t a n c e is d e r i v e d b a s e d o n t h e d e c o u p h n g p h e n o m e n a b e t w e e n the c o m p o n e n t s of v e r t i c a l m o t i o n s Qieave a n d p i t c h m o t i o n s ) a n d r o l l m o t i o n , a n d t h e l i m i t a t i o n s o f t h e r e l a t i o n s h i p are also c o n f i r m e d b y c o n s i d e r i n g t h e w e a k l y n o n l i n e a r c o u p l i n g i n l a r g e - a m p l i t u d e s h i p m o t i o n s . U l t i m a t e l y , a s i m p l e p r e d i c t i o n m e t h o d f o r the a d d e d r e s i s t a n c e i n i r r e g u l a r p a r a m e t r i c r o l l is suggested b a s e d o n the c o r r e l a t i o n , a n d its a c c u r a c y a n d efficiency a r e d i s c u s s e d i n c o m p a r i s o n w i t h the r e s u l t of the d i r e c t t i m e - d o m a i n s i m u l a t i o n s .

2 . M a t h e m a t i c a l b a c k g r o u n d

2.1. Boundary value problem: Rankine panel method

S h i p m o t i o n s c a n b e defined i n a m e a n - b o d y fixed c o o r d i n a t e c o n s i d e r i n g a ship's a d v a n c e m e n t w i t h a f o r w a r d s p e e d , U i n w a v e s as s h o w n i n F i g . 1. H e r e , A, co, a n d /? a r e the w a v e a m p l i t u d e , f r e q u e n c y , a n d h e a d i n g angle, respectively. I n a d d i t i o n , the p r o b l e m d o m a i n s , S B a n d S F , denote the b o d y s u r f a c e a n d the free s u r f a c e , respectively. W h e n a s h i p is a s s u m e d to b e a rigid body, it h a s s i x degrees of f r e e d o m ( D O F ) w i t h t h e t r a n s l a t i o n , = (fi, & ) , a n d t h e r o t a t i o n , TR = iU< ?5' ió); the h n e a r d i s p l a c e m e n t i n d u c e d b y w a v e s c a n b e w r i t t e n as follows:

s'cx, 0 = & ( ' ) + & ( ' ) X ? . ( 1 )

U n d e r t h e a s s u m p t i o n o f a n i n c o m p r e s s i b l e a n d i n v i s c i d fluid w i t h i r r o t a t i o n a l m o t i o n , the velocity potential, (p satisfies t h e f o l l o w i n g b o u n d a r y v a l u e p r o b l e m : ^(p = 0 in fluid domain ( 2 ) — = U-li -I /! on Sfl dn dl dl

+ V 0

L z - fCv,)', f ) ] = o onz = f(-v,.v,

r)

(lip

Vip-Vip on z = f

(.V,

v, t)

<P(x,t) = 0(x) + 4>,(t,t) + ,p,i(y,t) ( 6 )

f ( ? , f ) = f,C?,

0 + (7)

w h e r e <I> is t h e b a s i s p o t e n t i a l w h i c h h a s the o r d e r of 0 ( 1 ) . ipi a n d a r e t h e velocity potential a n d elevation of i n c i d e n t w a v e , respectively. S i m f l a r l y , ipd a n d a r e those o f d i s t u r b e d w a v e , respectively. T h e o r d e r s of i n c i d e n t a n d d i s t u r b e d c o m p o n e n t s a r e 0 ( E ) . B y s u b s t i t u t i o n s o f E q s . ( 6 ) a n d ( 7 ) i n t o E q s . ( 3 ) - ( 5 ) , the h n e a r i z e d b o u n d a r y c o n d i -t i o n s c a n be d e r i v e d a c c o r d i n g -to -t h e double-body ( D B ) l i n e a r i z a -t i o n ,

s u c h that: dn ("1. " 2 . "3) («4, 1(5, »6)

(«(1, lUj,

lili)

(1114, Ills,

«l6)

(7?-v)(y - V * )

d-0

(u - y^yvQ = —Q + ^ + {u

d0_

+ (f/ - y0).v4>,

2.2. Equation of motion: weakly nonlinear approach

T h e e q u a t i o n o f m o t i o n c a n b e d e f i n e d f o r t h e w e a k l y n o n l i n e a r s h i p m o t i o n a s foUows:

[ M U f l = ( F F X I N O , , . + {fkes.lNon. + (fk.D.I + (Fviscoml ( 1 3 ) w h e r e [Af] i s t h e m a s s m a t r i x o f a s h i p . { f F . K . } N o n . a n d {FRes.}N„n. i n d i c a t e t h e n o n l i n e a r F r o u d e - K r y l o v a n d r e s t o r i n g f o r c e s , respectively. I n a d d i t i o n , { F H . D . } a n d {iviscous} denote t h e h y d r o d y n a m i c force due to w a v e s a n d t h e v i s c o u s d a m p i n g force, s u c h a s a r o l l d a m p i n g , respectively. I n t h e w e a k l y n o n l i n e a r a p p r o a c h , t h e h y d r o d y n a m i c f o r c e i s o b t a i n e d a c c o r d i n g to the l i n e a r b o u n d a r y c o n d i t i o n s , w h f l e t h e

n o n l i n e a r F r o u d e - K r y l o v a n d r e s t o r i n g forces a r e e v a l u a t e d at t h e actual wetted s u r f a c e s o f t h e s h i p i n t i m e - d o m a i n . I n other w o r d s , t h e p a r t i a l n o n l i n e a r i t y i n d u c e d b y t h e h u l l geometry is c o n s i d e r e d i n t h i s a p p r o a c h , w h i c h is also k n o w n a s a "blended m e t h o d " b e t w e e n h n e a r a n d n o n l i n e a r m e t h o d s ( J e n s e n et a l . , 2 0 0 0) . T h e r e f o r e , t h e n o n l i n e a r p r o b l e m , s u c h as t h e p a r a m e t r i c r o l l p h e n o m e n a , w h e r e t h e p r e d i c t i o n for t h e n o n l i n e a r r e s t o r i n g f o r c e i s i m p o r t a n t c a n b e t a c k l e d efficiently w i t h o u t a f u l l y n o n l i n e a r c o m p u t a t i o n . T h e n o n l i n e a r F r o u d e - K r y l o v forces a r e c o m p u t e d t h r o u g h t h e velocity p o t e n t i a l o f a n i n c i d e n t w a v e b e l o w the m e a n - w a t e r level, w h i l e t h e first-order p e r t u r b e d p o t e n t i a l w i t h respect to t h e w a v e elevation i s u s e d above t h e w a t e r level as follows

iiii(.\,y, z, t)

—e*'^ s i n ( i ( . v - f Ut)cos /i + ky sin fi - o)t) f o r < 0

• ün{k (.V -I- C//)cos fi -I- ky sin fi - oit) f o r 0 < z < f . ( 1 4 ) T h e n o n l i n e a r force c a n b e o b t a i n e d b y d i r e c t integration o f t h e h y d r o d y n a m i c p r e s s u r e i n d u c e d b y t h e i n c i d e n t w a v e s o n t h e exact wetted s u r f a c e as follows:

{ P F X I N ™ . = - P

f f

u-vé, + v0-v<j>i + ]-y<Pry<Pi

JJSR üt 2 ndS.

( 1 5 ) S i m i l a r l y , t h e n o n l m e a r r e s t o r i n g force is c a l c u l a t e d b y s u b t r a c t i n g the l i n e a r h y d r o s t a t i c force at t h e m e a n - b o d y p o s i t i o n f r o m t h e force at the a c t u a l b o d y p o s i t i o n a s follows: z)ndS pg f f {-z)ndS. ( 1 6 ) T h e v i s c o u s d a m p i n g force i n r o l l m o t i o n s h o u l d b e m o d e l e d , b e c a u s e the a m p l i t u d e of m o t i o n is s e n s i t i v e to t h e v i s c o s i t y o f t h e fluid. T h e v i s c o u s f o r c e i s g e n e r a l l y p r o p o r t i o n a l to q u a d r a t i c o f t h e r o l l a n g u l a r velocity, b u t t h i s s t u d y adopts a n e q u i v a l e n t h n e a r d a m p i n g a s suggested b y H i m e n o ( 1 9 8 1 ) . B e c a u s e t h e effects o f v a r i a t i o n o f r e s t o r i n g force a r e m o r e d o m i n a n t i n t h e o c c u r r e n c e a n d m a g n i t u d e of p a r a m e t r i c roll t h a n t h o s e o f d a m p i n g forces, t h e c o m p u t a t i o n s a r e c a r r i e d o u t b a s e d o n a given l i n e a r d a m p i n g f o r c e f o r a n e a s i e r n u m e r i c a l i m p l e m e n t a t i o n .

I n this study, to m o d e l t h e v i s c o u s roU d a m p i n g , a h n e a r d a m p i n g coefficient w h i c h i s e x p r e s s e d a s t h e r a t i o to t h e c r i t i c a l d a m p i n g f o r roll m o t i o n i s adopted.

: 2 ) ' ^ ( M 4 4 + /i44_„)C44 _{( 1 7 )}

w h e r e J444,„ a n d C44 i n d i c a t e t h e i n f i n i t e - f r e q u e n c y a d d e d m o m e n t o f i n e r t i a a n d t h e l i n e a r r e s t o r i n g coefficient i n roll m o t i o n , respectively. T h e v a l u e o f y, w h i c h r e p r e s e n t s t h e ratio o f d a m p i n g f o r c e to t h e critical d a m p i n g , c a n b e d e t e r m i n e d b y c o n s i d e r i n g t h e d e c r e a s i n g t e n d e n c y o f r o l l a m p h t u d e i n a f r e e - r o l l - d e c a y e x p e r i m e n t . H o w e v e r , the ratio i s v a r i e d d e p e n d i n g o n h u l l s h a p e , s h i p s p e e d , a n d a p p e n -dages. I n t h e p r e s e n t study, t h e v a l u e s i n t h e r a n g e o f 0 . 0 3 - 0 . 1 0 w h i c h are a p p r o p r i a t e f o r a t y p i c a l h u l l a r e u s e d to c o n s i d e r t h e effects o f different roll d a m p i n g forces.

2.3. Prediction of added resistance: weakly nonlinear near-field method

T h e a d d e d r e s i s t a n c e o f a s h i p i s p r e d i c t e d b y a d o p t i n g t h e n e a r -field m e t h o d , n a m e l y , d i r e c t i n t e g r a t i o n o f s e c o n d - o r d e r p r e s s u r e o n a b o d y s u r f a c e . T h e n e a r - f i e l d m e t h o d b a s e d o n t h e l i n e a r d i s p l a c e m e n t Oinear-motion-based m e t h o d ) w a s d e r i v e d b y J o n c q u e z ( 2 0 0 9 ) a n d

Kirn a n d K i m (201 l a ) for t h e time-domain 3 - D R a n k i n e p a n e l m e t h o d . T h e s e c o n d - o r d e r f o r c e c a n b e f o r m u l a t e d u s i n g t h e p e r t u r b a t i o n o f p h y s i c a l a n d g e o m e t r i c a l v a r i a b l e s s u c h a s l i n e a r d i s p l a c e m e n t , h y d r o -d y n a m i c p r e s s u r e , w a v e elevation, a n -d n o r m a l v e c t o r o f t h e s u r f a c e

Table 1

Classification of components in added resistance. Total added resistance

Formulation

F2 - (/) + (//) + i m + UV) + (V) + (V/) + iVI!) + {FF.K.)H.O.T. + (^^Res.lH.O.T,

Linear-motion-based method Weakly nonlinear method

Component (J) Component (11) Component (i/i) Component {IV) Component (V) Component (VT) Component {VII) Higher-order

Froude-Krylov force, {FF.KLIH.O.T. Higher-order restoring force, {FRes.ln.o.T.

- A ? 4 K - « 3 + f 4 y ^ f e . r ) ) ' - ^

[ y

} j

- P L s vi-iu - i v * ] - v * ({• - (Ö + - Ö - v ) ) - ^ 'IL

-'lli- y \ 2 ) ) sin a

-P Jfs„s--'"i'is

(U - v*)-v(,/i, + it,j) + g(£j -f Uy - fe.v)|-7?K;s

P fj'^^ (yV(,/l, -h ^,,)-V(#, -f ^ j ) trlS

- 0 [L ? - v - y - v * + -!-v*-v* -TTirfs -"a ( 2 j p /?! l - y - v * -1- - v * - v * -Tnrfs

irPsL, (f - (& + Uy - fe.v))--?—rft 2 •'"^ sin a ri sni rt - 2 / ' s ^ i ( ^ ' - ( & - ^ ^ 4 y - ^ 5 - v ) ) - • - " 4 , ( - ( ^ - | v * ) - V * j ( f - (Ö + f4V - i5-v)) -pJwL^-'^ - ^V*J-V* (C - (Ö -f {4y - fe-^))-;r^ ' -P //^^ - (t' - V*)-V#,, [TtidS

'PJISB^''^ - t / - V * 4--iv*-V* -Tii'K

- / ) ^ ^ H 7 - V | - Z / - V * 4- | V * - V * -ItdS

„ rr

l'*'

C^K.KjNün. + PJ (U - V0)-Witi, [ndS •ll (

w i t h r e s p e c t to the m e a n - b o d y p o s i t i o n . T h e r e f o r e , t h e force c o n t a i n s o n l y the q u a d r a t i c c o m p o n e n t s o f the h n e a r s o l u t i o n s . H o w e v e r , i n the p r e s e n t w e a k l y n o n l m e a r p r e d i c t i o n , t h e c o m p o n e n t s of t h e a d d e d r e s i s t a n c e i n d u c e d b y t h e F r o u d e - K r y l o v a n d h y d r o s t a t i c f o r c e s are r e p l a c e d b y the n o n l i n e a r forces at the a c t u a l s u r f a c e s of t h e b o d y i n w a v e s . I n other w o r d s , t h e h i g h e r - o r d e r f o r c e s a r e c o n s i d e r e d b y s u b t r a c t i n g the first-order f o r c e s f r o m t h o s e i n E q s . ( 1 5 ) a n d ( 1 6 ) . T h e a d d e d r e s i s t a n c e denotes the m e a n v a l u e o f the h i g h e r - o r d e r f o r c e i n the l o n g i t u d i n a l d i r e c t i o n of a s h i p . S i m i l a r c o n s i d e r a t i o n s for h i g h e r - o r d e r h o r i z o n t a l drifting effects on a s h i p i n a s s o c i a t i o n ivith t h e i n c i d e n t w a v e c a n b e f o u n d i n Z h a n g et a l . ( 2 0 0 9 ) .

F o r a p h y s i c a l o b s e r v a t i o n , the a d d e d r e s i s t a n c e i s d e c o m p o s e d into n i n e i n t e g r a l t e r m s as d e s c r i b e d i n T a b l e 1. H e r e , Til a n d 7?2, are t h e first o r d e r a n d t h e s e c o n d o r d e r n o r m a l vectors, respectively. 0 - ^ 6 & 0 - ^ 4 - f o fo 0 "2 2fofo 2fofo 0 • (fo2 + fo2) 2fofo 0 0 • ( f o '

+

,

I 1'

m

s o l v i n g a h i g h e r - o r d e r b o u n d a r y v a l u e p r o b l e m .

3 . A n a l y s i s r e s u l t s

3.1. Ship model

T h e p r e s e n t n u m e r i c a l s i m u l a t i o n s a r e c o n d u c t e d o n a 6 5 0 0 T E U c o n t a i n e r s h i p . T h e geometry of t h e s h i p h a s a large b o w flare angle a n d a n o v e r h a n g i n g t r a n s o m , w h i c h l e a d s to t h e large v a r i a t i o n i n w a t e r p l a n e a r e a a c c o r d i n g to the r e l a t i v e p o s i t i o n of t h e s h i p to the w a v e e l e v a t i o n a s s h o w n i n F i g . 3. I n the b o w section, t h e r e i s a s m a U v a r i a t i o n i n the w a t e r l i n e , w h i l e t h e s t e r n s e c t i o n s h o w s a l a r g e r c h a n g e b e c a u s e of t h e l o c a t i o n of w a v e a n d b o d y s h a p e . T h i s c h a n g e r e s u l t s i n t h e v a r i a t i o n of the t r a n s v e r s e stability, w h i c h i s r e p r e s e n t e d b y t h e v a l u e of t h e G M . T h e r e f o r e , it i s k n o w n t h a t t h i s type of a s h i p i s v u l n e r a b l e to the p a r a m e t r i c r o l l p h e n o m e n a . T h e p r i n c i p l e d i m e n s i o n s o f t h e s h i p m o d e l are s u m m a r i z e d i n T a b l e 2 . A U of t h e s i m u l a t i o n s a r e c a r r i e d out i n h e a d s e a (/? = 1 8 0 . 0 ° ) a n d at 5 k n o t s f o r w a r d - s p e e d c o n d i t i o n s . I n appUcations of t h e R a n k i n e p a n e l m e t h o d , the d i s c r e t i z e d p a n e l s

Fig. 3. Variation in water plane area and transverse stability: A/L = 0.005, X/L = 1.0.

Table 2

Principle dimensions of 6500 T E U containership.

Designation _{6500 T E U containership}

LBP (m) 286.30

Beam (m) 40.30

Draft (m) 13.13 (AP), 12.97 (FP)

GM (m) 1.38

Natural period of roll (s) 30.21

(a) linear panel

(b) nonlinear panel

Fig. 4. Examples of solution grids.

are d i s t r i b u t e d on t h e b o d y a n d the free s u r f a c e s . I n case o f the U n e a r b o d y p a n e l s , t h e grids a r e c l u s t e r e d n e a r the fore a n d aft body, w h e r e the f l o w p a t t e r n s a r e m o r e c o m p l i c a t e d t h a n those n e a r the m i d - s h i p . O n t h e other h a n d , t h e n u m b e r o f the l i n e a r free s u r f a c e p a n e l s v a r i e s a c c o r d i n g to the length o f i n c i d e n t w a v e s . T h e r a d i u s o f t h e t o t a l d o m a i n i s five t i m e s t h a t of t h e wavelength, w h i c h c o n s i s t s of t h r e e w a v e l e n g t h s f o r the p r o b l e m d o m a i n a n d two wavelengths for the artificial d a m p i n g zone. I n a d d i t i o n , the n o n U n e a r b o d y p a n e l s a r e m o d e l e d u p to a c e r t a i n height above t h e m e a n - w a t e r level to c o n s i d e r

Table 3

Grid systems of hnear panels in grid convergence test.

Grid number Number of body panels Number of free surface panels

Grid 1 750 1500 Grid 2 1500 1500 Grids 1500 2500 Grid 4 2500 2500 Grid 5 2500 4000 Gride 4000 4000

the n o n l i n e a r geometric effects at the exact wetted s u r f a c e s . F i g . 4 s h o w s a n e x a m p l e of the l i n e a r a n d n o n U n e a r s o l u t i o n grids. F o r t h e grid convergence test, s e v e r a l g r i d s y s t e m s of t h e l i n e a r p a n e l s are adopted as d e s c r i b e d i n T a b l e 3 .

3.2. Motion response and added resistance in regular luaves

F i r s t l y , the r e s p o n s e s o f tlie v e r t i c a l m o t i o n s w i t h o u t the p a r a m e t r i c roll are validated, b e c a u s e the motions are directly relevant to the a d d e d resistance, especially, the c o m p o n e n t s d e p e n d i n g o n t h e relative w a v e elevation a n d the r a d i a t i o n . A s t h e r e is n o e x p e r i m e n t a l data f o r the s h i p m o d e l , c o m p u t a t i o n r e s u l t s of t h e p r e s e n t R a n k i n e p a n e l m e t h o d are c o m p a r e d w i t h those o b t a i n e d b y the I m p u l s e r e s p o n s e f u n c t i o n ( I R F ) m e t h o d . T h e p r e s e n t I R F m e t h o d i s b a s e d o n the w o r k of F o n s e c a a n d S o a r e s ( 1 9 9 8 ) . I n t h i s m e t h o d , h y d r o d y n a m i c coeffi-cients a n d w a v e excitation forces a r e c o m p u t e d b y the t w o - d i m e n s i o n a l strip t h e o r y ( S a l v e s e n et al., 1 9 7 0) , a n d t h e n the fi-equency-domain solutions a r e c o n v e r t e d to the t i m e - d o m a i n solutions b y c o n v o l u t i o n integral. T h e n o n l i n e a r F r o u d e - K r y l o v a n d r e s t o r i n g forces are also c o n s i d e r e d a c c o r d i n g to the w e a k l y n o n l i n e a r a p p r o a c h . T h e details o f appUcations of the I R F m e t h o d for t h e s a m e s i m u l a t i o n c o n d i t i o n s c a n be f o u n d i n L e e a n d K i m ( 2 0 1 6 ) .

F r o m F i g . 5, it c a n be o b s e r v e d t h a t t h e r e a r e g o o d c o r r e l a t i o n s bet^veen the a m p l i t u d e s of the r e s p o n s e s c o m p u t e d b y t h e two different n u m e r i c a l methods. T h e p h a s e angles of t h e r e s p o n s e s t h a t affect the relative w a v e elevation a r e also s i m i l a r i n t h e t w o m e t h o d s . F u r t h e r m o r e , the w e a k l y n o n U n e a r m o t i o n s a r e c o m p a r e d to the m o t i o n i n the U n e a r a p p r o a c h , w h e r e l i n e a r r e s t o r i n g coefficients a n d F r o u d e - K r y l o v forces e v a l u a t e d at t h e m e a n - b o d y p o s i t i o n a r e adopted. F o r relatively s m a U w a v e a m p h t u d e s (A/L = 0 . 0 0 5 - 0 . 0 1 0 ) , t h e w e a k l y n o n l i n e a r a n d the l i n e a r m o t i o n s s h o w a g o o d a g r e e m e n t as w e a k r e s o n a n c e s of t h e v e r t i c a l m o t i o n s o c c u r f o r the s l o w f o r w a r d - s p e e d ( 5 k n o t s ) c o n d i t i o n . H o w e v e r , w h e n t h e w a v e l e n g t h i s s i m i l a r to t h e length o f t h e s h i p ( a ) ( L / g ) ' ^ ^ = 2 . 5 0 ) , d i s c r e p a n c i e s b e t w e e n the h e a v e motions o f the two a p p r o a c h e s c a n b e c o n f i r m e d due to t h e large v a r i a t i o n of the w a t e r p l a n e a r e a i n t h e w a v e ( F i g . 3 ) . O n t h e other h a n d , the v a r i a t i o n is less effective o n the p h a s e angels of s h i p m o t i o n s . I t i s w i d e l y k n o w n t h a t a s e c o n d - o r d e r v a l u e is m o r e s e n s i t i v e to t h e g r i d s y s t e m t h a n a U n e a r m o t i o n response. T o o b t a i n a n a c c u r a t e a n d a c o n v e r g e d solution, the g r i d convergence test for a d d e d r e s i s t a n c e o n the s h i p m o d e l is c a r r i e d out. F i g . 6 s h o w s the c o m p u t a t i o n r e s u l t s o f the l i n e a r - m o t i o n - b a s e d m e t h o d for the g r i d s y s t e m s , w h i c h are p r e s e n t i n T a b l e 3 . O v e r t h e g r i d s y s t e m , w h i c h is c o m p o s e d of 2 5 0 0 a n d 4 0 0 0 p a n e l s o n the m e a n b o d y s u r f a c e a n d f r e e s u r f a c e , r e s p e c -tively ( G r i d 5 ) , the convergent solutions c a n b e s e e n f o r t h e different wavelengths. I n addition, b e c a u s e t h e w a v e r a d i a t i o n is m o r e d o m i n a n t t h a n t h e w a v e diffraction for a l o n g w a v e l e n g t h , t h e a d d e d r e s i s t a n c e i s m o r e stable along w i t h t h e g r i d s y s t e m t h a n t h a t f o r a s h o r t wavelength. N e x t , the a d d e d r e s i s t a n c e s o b t a i n e d b y b o t h the U n e a r m o t i o n -b a s e d m e t h o d a n d t h e w e a k l y n o n l i n e a r m e t h o d a r e c o m p a r e d as s h o w n i n F i g . 7. T h e results o f the w e a k l y n o n U n e a r m e t h o d , w h i c h are not sensitive to the n o n U n e a r p a n e l s a r e slightiy different w i t h those of the U n e a r m e t h o d at t h e p e a k o f t r a n s f e r f u n c t i o n . T h e r e a r e two r e a s o n s for these d i s c r e p a n c i e s . F i r s t , the different h e a v e m o t i o n s o f

o,.| 0.: tS 180 S m 0 £ -180 •

IRF.LiiKii-IRF. Weakly nailine.ir, A/L=0.005 IRF. Weakly ixililinear, A.'L=0.010 RPM, LiiKar

r RPM, Weakly rcmlinear, At-O.OOS 3 RPM. Weakly nailiiiear,A4.-0.010

IRF.LiiKii-IRF. Weakly nailine.ir, A/L=0.005 IRF. Weakly ixililinear, A.'L=0.010 RPM, LiiKar

r RPM, Weakly rcmlinear, At-O.OOS 3 RPM. Weakly nailiiiear,A4.-0.010

IRF.LiiKii-IRF. Weakly nailine.ir, A/L=0.005 IRF. Weakly ixililinear, A.'L=0.010 RPM, LiiKar

r RPM, Weakly rcmlinear, At-O.OOS 3 RPM. Weakly nailiiiear,A4.-0.010

f

1

\

i

>—(i—

I

15

:

5

5 4 4.5

^ o.s

IRF. Weakly nonlinear. A/L=O.0OS IRF.Weaklynoiiliiiear, A/L=O.0:0 ^ RPM, Linear

r RPM. Weakly irailiiKar, A/I.-0.005

) RPM. Weakly nonlinear. A/L-0.010

- _i

IRF. Linear

IRF. Weakly nonlinear. A/L=O.0OS IRF.Weaklynoiiliiiear, A/L=O.0:0 ^ RPM, Linear

r RPM. Weakly irailiiKar, A/I.-0.005

) RPM. Weakly nonlinear. A/L-0.010

A

IRF. Linear

IRF. Weakly nonlinear. A/L=O.0OS IRF.Weaklynoiiliiiear, A/L=O.0:0 ^ RPM, Linear

r RPM. Weakly irailiiKar, A/I.-0.005

) RPM. Weakly nonlinear. A/L-0.010

A A -5 180

I

<

-(a) heave motion

(b) pitch motion

Fig. 5. Vertical motion responses.

A >A-0.30 T- - yJL'O.Vl - ?A-1.00 • >A=I.'IO A >A-0.30 T- - yJL'O.Vl - ?A-1.00 • >A=I.'IO

' - - -

_'

- -

'

, 1

\

Fig. 6. Grid convergence test for added resistance.

t h e l i n e a r a n d n o n h n e a r a p p r o a c h e s c h a n g e the c o m p o n e n t s d e p e n d i n g o n t h e relative w a v e elevation a n d the r a d i a t i o n . S e c o n d l y , i n the h n e a r m e t h o d , t h e n o n l i n e a r effects o w i n g to b o d y geometry a r e c o n s i d e r e d o n l y u s i n g t h e i n c l i n a t i o n angle, ( a ) , i n the w a t e r U n e i n t e g r a l t e r m , w h i l e the w e a k l y n o n l i n e a r m e t h o d u s e s t h e f o r c e s e v a l u a t e d at the exact w e t t e d s u r f a c e . T h e r e f o r e , the t e r m c a n b e different for the w a v e l e n g t h s w h e r e l a r g e r v a r i a t i o n s o f the w a t e r p l a n e a r e a o c c u r . E x c e p t t h e p e a k r e g i o n , t h e results of the two a p p r o a c h e s a r e s i m i l a r for s m a U w a v e a m p l i t u d e s , w h i c h indicate that the p r e s e n t w e a k l y n o n -l i n e a r m e t h o d p r o v i d e s a consistent p r e d i c t i o n c o m p a r e d to f h e U n e a r m e t h o d .

3.3. Added resistance in regular parametric roll motions

T h e s i m u l a t i o n of t h e p a r a m e t r i c roU f o r a r e g u l a r w a v e i s p e r f o r m e d b a s e d o n t h e p r e s e n t R a n k i n e p a n e l m e t h o d a l o n g w i t h

"ffl

"<

A Linear

T Weakly uolilm«ir.A'L-0.005 (4000 poncla)

O Weakly nonlinear. A/L=0.010

Wetik^' nonlinear. A/t=0 fXlS (aXlf) pineU) Weakly lwiainoar.A4,"0.005(6(XK;nuicU)

•

s

"O 0.5 I 1.5 2 2.5 VL

Fig. 7. Added resistance on ship: linear vs. weakly nonlmear methods.

t h e w e a k l y n o n U n e a r a p p r o a c h , a n d t h e r e s u l t s a r e c o m p a r e d w i t h those o f t h e I R F m e t h o d . I n b o t h t h e n u m e r i c a l m e t h o d s , a n i m p u l s i v e roU angle ( 3 . 0 - 1 0 . 0 ° ) is i m p o s e d at a n i n s t a n t to m o d e l t h e i n i t i a l d i s t u r b a n c e o f roU m o t i o n , w h i c h a r e i n d u c e d b y e n v i r o n m e n t a l conditions s u c h as gusts or c u r r e n t s as s h o w n i n F i g . 8. A f t e r the i n i t i a l c o n d i t i o n , the r e g u l a r p a r a m e t r i c roU is excited r e s o n a n t l y b y t h e v a r i a t i o n of t h e t r a n s v e r s e stability i n t h e w a v e . T h e r e a f t e r , t h e roll m o t i o n r e a c h e s a q u a s i - s t e a d y state as t h e d e v e l o p m e n t of m o t i o n is b o u n d e d b y b a l a n c i n g t h e d e c r e a s e d v a r i a t i o n o f t h e n o n U n e a r r e s t o r -i n g f o r c e a n d t h e -i n c r e a s e d roU d a m p -i n g f o r c e at a large h e e l angel. F i g . 9 s h o w s t h e quasisteadystate a m p l i t u d e s c o m p u t e d f o r c o n d i -tions of different r e g u l a r w a v e s a n d d a m p i n g f o r c e s . T h e w a v e c o n d i t i o n s a r e c h o s e n a c c o r d i n g to the v u l n e r a b l e c o n d i t i o n s to p a r a m e t r i c roU p h e n o m e n a ; the w a v e l e n g t h i s e q u i v a l e n t to the length of a s h i p , a n d t h e e n c o u n t e r w a v e frequency, cOe is a p p r o x i m a t e l y twice t h a t o f n a t u r a l roll frequency, a„ a n d the w a v e a m p l i t u d e is l a r g e r t h a n

t(g/L)"

Fig. 8. Time histories of regular parametric roU: A/L = 0.005, XjL = 1.35, y = 0.03.

a c e r t a i n t h r e s h o l d v a l u e . T h e two different n u m e r i c a l m e t h o d s give s i m i l a r overall t r e n d s for the o c c u r r e n c e s a n d t h e m a g n i t u d e s of t h e roll m o t i o n s . T h e slight d i s c r e p a n c i e s m a y r e s u l t f r o m the different m e t h o d s u s e d for the c a l c u l a t i o n o f the h y d i ' o d y n a m i c forces; t h e 3 - D b o u n d a r y v a l u e p r o b l e m i s s o l v e d to o b t a i n the f o r c e i n the R a n k i n e p a n e l m e t h o d , w h i l e t h e h y d r o d y n a m i c coefficients c a l c u l a t e d b y t h e 2¬ D s t r i p t h e o r y (S a l v e s e n et a l , 1 9 7 0 ) are a d o p t e d i n the I R F m e t h o d . T h e n a t u r a l roll f r e q u e n c y i n a w a v e v a r i e s w i t h the different a d d e d m o m e n t of inertia i n roll m o t i o n , w h i c h l e a d s to different p r e d i c t i o n s of the p a r a m e t r i c roU b e c a u s e o f the c h a n g e d v u l n e r a b l e c o n d i t i o n r e l a t e d w i t h t h e w a v e frequency.

I n t h e p r e s e n t study, to investigate the effects o f p a r a m e t r i c roU o n a d d e d r e s i s t a n c e , the n u m e r i c a l s i m u l a t i o n s a r e p e r f o r m e d w i t h a n d w i t h o u t p a r a m e t r i c r o l l for the s a m e w a v e c o n d i t i o n . F o r t h e case w i t h o u t p a r a m e t r i c roll, t h e i m p u l s i v e roll angle i s not i m p o s e d d u r i n g t h e s i m u l a t i o n , a n d t h e b o t h h n e a r a n d w e a k l y n o n l i n e a r a p p r o a c h e s a r e a d o p t e d for the c o m p u t a t i o n s . O n the other h a n d , i n the s i m u l a t i o n of p a r a m e t r i c roll, t h e i m p u l s i v e angle is i m p o s e d at a c e r t a i n i n s t a n t f o r t h e d e v e l o p m e n t o f r o l l m o t i o n as s h o w n i n F i g . 8. A l s o , o n l y the w e a k l y n o n l i n e a r a p p r o a c h w h i c h c a n c o n s i d e r the v a r i a t i o n o f t r a n s v e r s e stability i n w a v e s is a p p l i e d for t h e s u m i l a t i o n . I f t h e p a r a m e t r i c r o l l does not o c c u r f o r a w a v e c o n d i t i o n , the w e a k l y n o n l i n e a r a p p r o a c h p r o v i d e s the s a m e r e s u l t s regardless o f t h e i m p u l s i v e r o l l angle s i n c e t h e r o l l m o t i o n converges to zero f r o m the i n i t i a l c o n d i t i o n due to d a m p i n g f o r c e s .

F o r a v u l n e r a b l e c o n d i t i o n o f p a r a m e t r i c r o l l , t h e h i g h e r - o r d e r surge forces o n t h e s h i p c a l c u l a t e d u s i n g the l i n e a r - m o t i o n - b a s e d m e t h o d a n d the w e a k l y n o n h n e a r m e t h o d are c o m p a r e d as s h o w n i n F i g . 10. I n the h n e a r m e t h o d , t h e s i n u s o i d a l solution, w h i c h i s c o m p o s e d of the p r o d u c t s of the h n e a r p h y s i c a l a n d g e o m e t r i c a l q u a n t i t i e s of the e n c o u n t e r w a v e f r e q u e n c y , oscillates two t i m e s faster

(2<0e) t h a n the l i n e a r q u a n t i t i e s w h e n the p a r a m e t r i c roll i s n o t excited. O n the other h a n d , slight m o d u l a t i o n s exist i n the r e s u l t o f t h e w e a k l y n o n l i n e a r m e t h o d o w i n g to t h e h i g h e r - o r d e r (over s e c o n d - o r d e r ) effects i n the n o n l i n e a r r e s t o r i n g a n d F r o u d e - K r y l o v f o r c e s . W h e n t h e h e a v y p a r a m e t r i c roll m o t i o n o c c u r s , t h e b e h a v i o r o f t h e force i s c h a n g e d severely. T h e stronger n o n l i n e a r a p p e a r a n c e s c a n b e s e e n , a n d t h e c o m p o n e n t of, cOe, i s d o m i n a n t i n t h a t the roll m o t i o n oscillates w i t h t h e n a t u r a l roll f r e q u e n c y (a)„ = <»e/2).

F i g . 11 s h o w s the a d d e d r e s i s t a n c e , that i s , the t e m p o r a l m e a n v a l u e of h i g h e r - o r d e r s u r g e force. F o r the w a v e c o n d i t i o n s w h e r e p a r a m e t r i c r o l l does not o c c u r , o n l y t h e c o m p u t a t i o n r e s u l t s w i t h o u t t h e p h e n o m -e n a a r -e plott-ed. F o r th-e v u l n -e r a b l -e c o n d i t i o n s (_A/L = 0 . 0 0 5 , y = 0 . 0 3 , A / L = 1 . 2 5 - 1 . 6 5 ; A/L = 0 . 0 1 0 , y = 0.10, X/L = 1 . 0 5 - 1 . 7 5 a s s h o v m i n

F i g . 9 ) , the a d d e d r e s i s t a n c e s i n c r e a s e significantly due to p a r a m e t r i c r o l l m o t i o n s c o m p a r e d to t h o s e i n t h e a b s e n c e of t h e p h e n o m e n a . T h e effects o f t h e n o n h n e a r r e s t o r i n g a n d F r o u d e - K r l y o v forces a t the exact w e t t e d s u r f a c e s f o r large r o l l m o t i o n s a r e a c c o u n t e d i n the i n c r e m e n t s of a d d e d r e s i s t a n c e s b a s e d o n t h e w e a k l y n o n h n e a r a p p r o a c h . I t s h o u l d be noted that t h e i n c r e a s e d a d d e d r e s i s t a n c e is not d i r e c t l y relevant to t h e w a v e a m p l i t u d e , b e c a u s e t h e r e a l v a l u e s of the i n c r e m e n t s a r e s i m i l a r f o r different r e g u l a r w a v e c o n d i t i o n s . H o w e v e r , t h e steady-state a m p l i t u d e s of t h e roll m o t i o n s a r e s i m i l a r for the two c o n d i t i o n s due to t h e different r o l l d a m p i n g f o r c e s . T h e r e f o r e , the i n c r e a s e d r e s i s t a n c e i n d u c e d b y the p a r a m e t r i c r o l l s e e m s to b e closely r e l a t e d w i t h t h e m a g n i t u d e of t h e r e s u l t i n g roll m o t i o n . F o r a p h y s i c a l o b s e r v a t i o n , e a c h c o m p o n e n t of the a d d e d r e s i s t a n c e i n t h e w e a k l y n o n U n e a r m e t h o d d e s c r i b e d i n T a b l e 1 is i n v e s t i g a t e d as s h o w n i n F i g . 1 2 . T h e c o m p o n e n t s w i t h or w i t h o u t p a r a m e t r i c roU a r e c o m p a r e d to d e t e r m i n e the c o m p o n e n t that c o n t r i b u t e s to t h e i n -c r e a s e d r e s i s t a n -c e . W h e n the roU m o t i o n is not ex-cited, it -c a n be s e e n t h a t t h e w a t e r U n e i n t e g r a l t e r m , (ƒ), a n d the h i g h e r o r d e r F r o u d e -K r y l o v force, w h i c h d e p e n d on t h e relative w a v e elevation, a r e d o m i n a n t . I n a d d i t i o n , t h e r a d i a t i o n c o m p o n e n t s , s u c h a s (IU) a n d ( V ) as o b s e r v e d i n T a b l e 1, h a v e significant i n f l u e n c e s for r e l a t i v e l y long w a v e l e n g t h s ( A / L = 1 . 0 - 2 . 0 ) . T h e s e c o m p o n e n t s v a r y s i g n i f i c a n t l y o w i n g to the p a r a m e t r i c r o U , a n d stiU r e m a i n t h e m a i n t e r m s i n t h e a d d e d r e s i s t a n c e . T h e largest c o m p o n e n t i s t h e F r o u d e - K r y l o v f o r c e e v a l u a t e d at t h e a c t u a l w e t t e d s u r f a c e c o n s i d e r i n g h e a v y r o l l m o t i o n . I t s h o u l d b e n o t e d t h a t t h e t e r m d e p e n d i n g o n the k i n e t i c energy of t h e fluid, ( / V ) , b e c o m e s t h e s e c o n d c o n t r i b u t o r to the r e s i s t a n c e ; h o w e v e r , t h i s t e r m i s not s i g n i f i c a n t i f t h e p a r a m e t r i c r o l l does not o c c u r .

T h e p r e s e n t r e s u l t s of the i n c r e a s e d a d d e d r e s i s t a n c e c a n n o t b e v a l i d a t e d as t h e r e a r e no e x p e r i m e n t a l d a t a a n d other c o m p u t a t i o n

f

\

(a)^/L=0.005, y=0.03 (b)^/L=0.010, y=0.10

(a) without parametric roll

i\

\

(\

A

'\

A

n

\

V

1

1

\l

\

1

\

(b) with parametric roll

82 84 86 88

Fig. 10. Time liistories of Iiiglier-order surge force: A/L = 0.005, A/L = 1.35, y = 0.03.

8

m

'<

$ 6

;

-y- - wo' piirainctric roll (weakly nonlinear) 0 — w/parametric rol! (weakly nonlinear)

,

^ — - wo,''parnmelric mil (linear)

-y- - wo' piirainctric roll (weakly nonlinear) 0 — w/parametric rol! (weakly nonlinear)

, o - O O • _O O 03 l.l 1.2 1.3 14 1.5 1.6 1.7 1.1

A wo/ p.irainelric roll (linear) • wo/ paniinctric roll (weakly nonlinear)

Q — w/pamnwiric roll (weakly nonlinear) A wo/ p.irainelric roll (linear) • wo/ paniinctric roll (weakly nonlinear)

Q — w/pamnwiric roll (weakly nonlinear)

-• , o O O' O. O 'O. O O. 1 r2 1.4 1.6 l.i

(a)/i/I=0.005,y=0.03 (b)^/L=0.010, }'=0.10

Fig. 1 1 . Added resistance on ship in regular parametric roll.

r e s u l t s . I t is m e r e l y c o n f i r m e d t h a t t h e e x p e r i m e n t a l d a t a of L u et a l . ( 2 0 1 5 ) for a s i m i l a r s h i p m o d e l s h o w s s i m i l a r t e n d e n c i e s ivith t h e r e s u l t s of t h i s study; t h e i n c r e m e n t i n t h e r e s i s t a n c e is 3 or 4 t i m e s u n d e r p a r a m e t r i c roll m o t i o n s o f a p p r o x i m a t e l y 3 0 ° . A s the m a i n c o m p o n e n t s , s u c h as ( i ) , ( / ƒ / ) , ( T V ) , a n d ( V ) , a r e d e p e n d e n t o n t h e e l e v a t i o n or t h e velocity potential o f the d i s t u r b e d w a v e , strict c o n s i d e r a t i o n s for the d i s t u r b a n c e s h o u l d be c a r r i e d out to obtain a n e n h a n c e d s o l u t i o n . A s s h o w n i n F i g . 13, a s y m m e t r i c a n d significantly severe d i s t u r b e d w a v e s a r e generated n e a r the s h i p ; especially i n t h e

b o w r e g i o n b e c a u s e of the excitation o f t h e p a r a m e t r i c roll. I n t h e p r e s e n t study, the l i n e a r potential a n d t h e elevation a r e c a l c u l a t e d w i t h respect to t h e m e a n - b o d y position; h e n c e , t h e r e is a h m i t a t i o n to i n c l u d e t h e n o n l i n e a r i t i e s i n the d i s t u r b a n c e i n d u c e d b y t h e l a r g e a m p U t u d e m o t i o n . T h e r e f o r e , a n a p p l i c a t i o n of fully n o n U n e a r c o m p u -tation, f o r e x a m p l e , the C F D m e t h o d m a y b e r e q u i r e d f o r m o r e accurate p r e d i c t i o n of the t e r m s related w i t h the n o n l i n e a r d i s t u r b a n c e .

^ 3 CO < 2

(I) wo'paninielric roll

i F , ^ ! „ o j no/paranKUicroll { l ' ï t l H o T wo/parametric roll - A (I) vv/paramelric roll

- • """ f F j , I HOT paramelrii; roll

• O - - - IF, } „ wl paramelric roll

_ 3

"< 2

(III) wol parametric rol I (IV) wo/ parametric roll (V) wo/ parametric roll (III) w/parametric roll (W) w/panimeiric roll (V) w/ parametric roll

-I

' \ x 1 1.2

Fig. 1 2 . Components of added resistance: A/L = 0.010, y = 0.10.

(a) without parametric roil (b) with parametric roll

Fig. 13. Disturbed wave contours around sliip: A/L = 0.010, A/L = 1.35, y = 0.10.

3.4. Correlation betiueen parametric roll and added resistance

U n d e r t h e a s s u m p t i o n t h a t the h i g h e r - o r d e r q u a n t i t i e s i n the p r e s e n t w e a k l y n o n l i n e a r a p p r o a c h are neghgible, the a d d e d r e s i s t a n c e c a n b e r e g a r d e d as a m e a n v a l u e of q u a d r a t i c c o m p o n e n t s of h n e a r v a r i a b l e s . T l i e p r o d u c t s of the quantities of the v e i t i c a l m o t i o n s ( o s c i l l a t i n g w i t h w j a n d t h e roU m o t i o n ( o s c i l l a t i n g w i t h w,, = 0)^/2) are r e p r e s e n t e d b y the c o m p o n e n t s ^^nth f r e q u e n c i e s of coJ2 (differ-e n c (differ-e - f r (differ-e q u (differ-e n c y ) a n d 3coJ2 ( s u m - f r (differ-e q u (differ-e n c y ) ^vithout a m (differ-e a n v a l u (differ-e . E v i d e n t l y , t h e w a v e excitation f o r c e o n the h e a v e a n d p i t c h m o t i o n s m a y c o n t a i n the c o m p o n e n t of the n a t u r a l roll f r e q u e n c y d u e to a n o n h n e a r c o u p l i n g i n l a r g e r o l l m o t i o n . H o w e v e r , the r a d i a t i o n i s m o r e d o m i n a n t t h a n the d i f f r a c t i o n for longer w a v e l e n g t h s ; h e n c e t h e n o n l i n e a r effects a r e a l s o negligible. A s a result, t h e i n c r e a s e d r e s i s t a n c e (,RU,/PR - RWO/PR) i s c o n s i d e r e d to d e p e n d o n l y o n the r e s u l t i n g p a r a m e t r i c roll. I n t h i s study, t h e s e c o n d - o r d e r v a l u e i s a s s u m e d to b e p r o p o r t i o n a l to the s q u a r e of the a m p l i t u d e o f the r o l l angle a c c o r d i n g to t h e d e c o u p l i n g p h e n o m e n a b a s e d o n a U n e a r p e r s p e c t i v e . A s s h o w n i n F i g . 14, the c o r r e l a t i o n c a n b e v a l i d w h e n t h e a m p l i t u d e is s m a U e r t h a n 3 0 ° . A b o v e t h i s angle, t h e c o r r e l a t i o n i s c h a n g e d o w i n g to s t r o n g e r n o n U n e a r effects s u c h as the w e a k l y n o n h n e a r c o u p U n g o f t h e v e r t i c a l a n d roU m o t i o n s a n d the h i g h e r

-i

Fig. 14. Correladon between parametric roll and increased added resistance.

o r d e r (over s e c o n d - o r d e r ) r e s t o r i n g a n d F r o u d e - K r y l o v forces. F u r t h e r m o r e , t h e r e l a t i o n s h i p i s s h g h t l y different a c c o r d i n g to the m a g n i t u d e of the roU d a m p i n g force, w h i c h c a n affect the p e r i o d a n d a m p l i t u d e of the p a r a m e t r i c r o l l . I n t h e p r e s e n t s t u d y , the v i s c o u s d a m p i n g force i s m o d e l e d b y a U n e a r d a m p i n g . T h e r e f o r e , the effects of n o n U n e a r d a m p i n g force on the p a r a m e t r i c roU a n d t h e r e s u l t i n g a d d e d r e s i s t a n c e s h o u l d b e t h o r o u g h l y e x a m i n e d i n the f u t u r e . B y a d d i n g n o n l i n e a r velocity c o m p o n e n t s to t h e l i n e a r d a m p i n g , t h e p r e s e n t methodolog}' c a n b e a p p l i e d f o r t h e e x a m i n a t i o n , a n d ^vill also p r o v i d e a s o l u t i o n of i m p r o v e d a c c u r a c y .

3 . 5 . Weakly nonlinear coupling phenomena

I n the w e a k l y n o n l i n e a r a n a l y s i s , t h e n o n h n e a r c o u p h n g bet\veen the r e s t o r i n g forces o f v e r t i c a l m o t i o n s a n d roU m o t i o n is a c c o u n t e d , a n d the v a r i a t i o n s o f roU r e s t o r i n g m o m e n t s due to w a v e s a n d v e r t i c a l m o t i o n s l e a d to the p a r a m e t r i c r o l l p h e n o m e n a . W h e n t h e a m p l i t u d e of roll m o t i o n i s not l a r g e , the effects o f r o l l m o t i o n o n v e r t i c a l m o t i o n s c a n b e r e g a r d e d negligible. U n d e r h e a v y roll m o t i o n s , h o w e v e r , t h e r e e x i s t n o n l i n e a r c o u p U n g effects of t h e p a r a m e t r i c r o l l o n the v e r t i c a l m o t i o n s s i n c e the a c t u a l w e t t e d s u r f a c e v a r i e s s i g n i f i c a n t l y a c c o r d i n g to the r o U m o t i o n s , w h i c h affect the stiffiiess for the v e r t i c a l m o t i o n s . L u

et a l . ( 2 0 1 3 ) i n v e s t i g a t e d t h e effects b y c o n d u c t i n g a f r e e - r u n n i n g e x p e r i m e n t a n d a n u m e r i c a l s i m u l a t i o n b a s e d o n t h e 3 - D O F c o u p U n g m a t h e m a t i c a l m o d e l . F i g . 15 s h o w s t h e t i m e h i s t o r i e s o f m o t i o n s of s h i p u n d e r p a r a m e t r i c roU. T h e v e r t i c a l m o t i o n s a r e c h a n g e d severely a c c o r d i n g to o c c u r r e n c e o f t h e p h e n o m e n a w h e n the a m p l i t u d e of roll m o t i o n i s above 3 0 ° . E s p e c i a l l y , t h e h e a v e m o t i o n v a r i e s m o r e greatly c o m p a r e d to the p i t c h m o t i o n , a n d t h e s t r o n g n o n U n e a r b e h a v i o r s , s u c h as s u b h a r m o n i c c o m p o n e n t s , a r e c o n f i r m e d . A s s h o w n i n F i g . 16,

not o n l y the a m p l i t u d e s , b u t a l s o the m e a n v a l u e s of v e r t i c a l m o t i o n s are c h a n g e d d u e to t h e r e s t o r i n g c o u p U n g . T h e s e t e n d e n c i e s b e c o m e i n t e n s i f i e d as the a m p l i t u d e of t h e r o l l angle i n c r e a s e s .

T h e c h a n g e d v e r t i c a l m o t i o n s a g a i n affect t h e p a r a m e t r i c roU a n d the r e s u l t i n g a d d e d r e s i s t a n c e s . F i r s t , t h e n a t u r a l r o l l firequency a n d the v a r i a t i o n of t r a n s v e r s e s t a b i l i t y i n w a v e s are c h a n g e d due to the r e s u l t i n g m e a n v a l u e s (steady p o s t u r e of s h i p i n w a v e s ) a n d a m p l i -t u d e s , respec-tively. T h e s e v a r i a -t i o n s h a v e p a r a m e -t r i c r o l l l e s s divergen-t e v e n for a large w a v e a m p l i t u d e ( d e t u n i n g p h e n o m e n a ) s i n c e t h e s h i p gets out of t h e v u l n e r a b l e c o n d i t i o n s r e l a t e d w i t h t h e n a t u r a l roU f r e q u e n c y ((U„ = 0)^/2). S p a n o s a n d P a p a n i k o l a o u ( 2 0 0 9 ) c o n f i r m e d the d e c a y of p a r a m e t r i c r e s o n a n c e for h i g h e r w a v e s t e e p n e s s , a n d the