Applied Ocean Research 51 (2015) 2 9 3 - 3 0 8
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Applied Ocean Research
E L S E V I E R
journal homepage: www.elsevier.conn/locate/apor
O C E A N
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R E S E A R C l
Development of a core mathematical model for arbitrary manoeuvres
of a shuttle tanker
S. Sutulo, C. Guedes Soares*
Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Portiigal
CrossMark
A R T I C L E I N F O A B S T R A C T
Article history: A 3 D 0 F s t i l l - w a t e r m a n o e u v r i n g m a t h e m a t i c a l m o d e l e m b r a c i n g a l l r e g i m e s o f m o t i o n o f a s i n g l e - s c r e w Available online 12 February 2015 s h u t t l e L N G c a r r i e r has b e e n d e v e l o p e d . B e s i d e s f o u r - q u a d r a n t m o d e l s f o r h u l l f o r c e s , m a i n p r o p e l l e r a n d t h e r u d d e r , t h e m o d e l d e s c r i b e s also t h e a c t i o n o f a l a t e r a l t h r u s t e r a n d o f a s s i s t i n g t u g b o a t s . E n v i r o n m e n -Keywords: t a l f a c t o r s i n c l u d e c o n s t a n t w i n d a n d c o n s t a n t c u r r e n t . R e s u l t s o f o f f - l i n e s i m u l a t i o n s p a r t l y c o m p a r e d Ship m a n o e u v r a b i l i t y v v i t h a v a i l a b l e t r i a l d a t a are g i v e n . Mathematical model © 2 0 1 5 E l s e v i e r L t d . A l l r i g h t s r e s e r v e d . Low-speed manoeuvring Simulation Lateral thrusters Tugs
W i n d and current action
1. I n t r o d u c t i o n H a n d l i n g t h e l i q u e f i e d n a t u r a l gas i n e v i t a b l y i n c l u d e s processes of b e r t h i n g a n d u n b e r t h i n g o f an LNG c a r r i e r ( s h u t t l e t a n k e r ) t o a n d f r o m a f l o a t i n g j e t t y (FLNG barge). I n g e n e r a l , s u c h m a n o e u v r e s are c o n s i d e r e d as v e r y c o m p l i c a t e d i n s e a m a n s h i p p r a c t i c e a n d r e l a -t i v e l y p r o n e -t o acciden-ts, especially -t o r a m m i n g . A l -t h o u g h i n -t h e m a j o r i t y o f cases s u c h accidents d o n o t r e s u l t i n disasters o r o t h e r grave consequences as t h e b e r t h i n g m a n o e u v r e s are p e r f o r m e d at l o w v e l o c i t i e s , t h e v e r y f a c t o f h a n d l i n g a v e r y d a n g e r o u s a n d easily i n f l a m m a b l e cargo sets a u g m e n t e d s t a n d a r d s f o r t h e m a n o e u v r i n g safety. The l a t t e r is s u p p o r t e d b y a p p r o p r i a t e m a n o e u v r i n g q u a l i -ties o f all i n v o l v e d c r a f t , r e a s o n a b l e s t e e r i n g t e c h n i q u e s a n d tactics, a n d also b y e x t e n s i v e t r a i n i n g o f h u m a n o p e r a t o r s w i t h t h e h e l p o f c o m p u t e r i z e d b r i d g e s i m u l a t o r s . Success o f t h i s t r a i n i n g d e p e n d s h e a v i l y o n t h e q u a l i t y o f i m p l e m e n t e d core m a t h e m a t i c a l m o d -els. A c o n s i d e r a b l e n u m b e r o f s u i t a b l e m a t h e m a t i c a l m o d e l s c o v e r -i n g t h e range o f " n o r m a l " or m o d e r a t e m a n o e u v r e s p e r f o r m e d w -i t h t h e m a i n c o n t r o l devices ( l i k e n o r m a l r u d d e r s ) a n d w i t h o u t r e v e r s -i n g t h e m a -i n p r o p u l s o r can be eas-ily f o u n d -i n t h e l -i t e r a t u r e [ 1 - 4 ] . The p a r a m e t e r s o f these m o d e l s are t h e n t o be d e t e r m i n e d f r o m CFD c o m p u t a t i o n s , c a p t i v e m o d e l tests o r b y i d e n t i f i c a t i o n f r o m f r e e r u n n i n g m o d e l s e x p e r i m e n t s or f u l l - s c a l e t r i a l s (e.g. [ 5 - 7 ] ) .
This is m u c h less t r u e f o r t h e s o c a l l e d l o w s p e e d m o d e l s a p p l i -cable i n f a c t t o t h e w h o l e r a n g e o f m o t i o n s i n c l u d i n g such h a r d
* Corresponding author.
E-mail address: c.guedes.soares@centec.tecnico.ulisboa.pt (C. Guedes Soares).
0141-1187/$ - see f r o n t m a t t e r ® 2015 Elsevier Ltd. A l l rights reserved. http://dx.doi.org/10.1016/j.apor.2015.01.008
m a n o e u v r e s as t h e crash stop, c r a b b i n g a n d r o t a t i o n o n t h e spot. Probably, o n e o f t h e f i r s t c o n t r i b u t i o n s b e i n g at t h e same t i m e v e r y i n f o r m a t i v e b e l o n g s t o [ 8 ] w h o s u p p o s e d t h a t a l l h y d r o d y n a m i c forces d e p e n d e d o n 4 d i m e n s i o n l e s s a n g u l a r p a r a m e t e r s a l l d e f i n e d i n 4 q u a d r a n t s . The h u l l f o r c e m o d e l w a s based o n a h e u r i s t i c d e c o m p o s i t i o n i n t o : ( 1 ) i d e a l f l u i d ( i n e r t i a l ) e f f e c t s , ( 2 ) h u l l l i f t i n g forces, ( 3 ) h u l l c r o s s f l o w e f f e c t s , a n d ( 4 ) h u l l resistance i n l o n g i t u -d i n a l m o t i o n . The p r o p o s e -d 4 - q u a -d r a n t p r o p e l l e r m o -d e l w a s base-d o n a p i e c e w i s e a p p r o x i m a t i o n o f t h e t h r u s t a n d t o r q u e c o e f f i c i e n t s , a n d , b e i n g v e r y c o n v e n i e n t , w a s used w i t h s m a l l m o d i f i c a t i o n s i n t h e p r e s e n t s t u d y a n d is d e s c r i b e d i n d e t a i l i n t h e m a i n p a r t o f t h e p a p e r . F i n a l l y , t h e r u d d e r ' s m o d e l i n t h e s l i p s t r e a m w a s b a s e d o n a c o m b i n a t i o n o f t a b u l a t e d r u d d e r characteristics, o n a m o r e o r less t y p i c a l s i m p l i f i e d s c h e m e f o r p r o p e l l e r race a c t i o n and, as c l a i m e d , o n several a d hoc rules, o m i t t e d , h o w e v e r , f r o m t h e p u b l i c a t i o n .
A n k u d i n o v et a l . [ 9 ] n o t i c e d t h a t q u a d r a t i c ( w i t h a b s o l u t e v a l u e o p e r a t i o n w h e n a p p r o p r i a t e ) d i m e n s i o n a l p o l y n o m i a l s are s t r u c t u r a l l y a p p l i c a b l e t o h a r d m a n o e u v r e s b u t the r e g r e s s i o n c o e f f i c i e n t s w e r e s u p p o s e d t o be e s t i m a t e d s e p a r a t e l y i n f o u r r e g i o n s : ( 1 ) m o d e r a t e d r i f t angles ahead, ( 2 ) large d r i f t angles ahead, ( 3 ) large d r i f t angles astern, a n d ( 4 ) m o d e r a t e d r i f t angles a s t e r n . N e c e s s i t y t o m a t c h f o u r separate regressions m a k e s t h i s a p p r o a c h s o m e w h a t c l u m s y . K o b a y a s h i a n d A s a i [ 1 0 ] i n t r o d u c e d t w o l i m i t i n g v a l u e s o f t h e F r o u d e n u m b e r . A b o v e t h e h i g h e r v a l u e a u s u a l cubic d i m e n s i o n -less m o d e l f o r m o d e r a t e m a n o e u v r i n g w a s a s s u m e d a p p l i c a b l e w h i l e b e l o w t h e l o w e r v a l u e a s p e c i a l l y d e v i s e d s e c o n d o r d e r q u a s i - p o l y n o m i a l d i m e n s i o n a l r e g r e s s i o n m o d e l was u s e d . I n t h e i n t e r m e d i a t e r e g i o n a l l h u l l forces a n d m o m e n t s w e r e o b t a i n e d as a l i n e a r b l e n d o f t h e h i g h - s p e e d a n d l o w - s p e e d m o d e l . The l o w - s p e e d
294 S. Sutulo, C. Guedes Soares/Applied Ocean Research 51 ( 2 0 I S j 293-308 m o d e l w a s used also i n a s t e r n m o t i o n b u t w i t h d i f f e r e n t r e g r e s s i o n c o e f f i c i e n t s . K h a t t a b [ 1 1 ] also a s s u m e d t h a t d i f f e r e n t p h y s i c a l p h e n o m e n a associated w i t h t h e l i f t (side f o r c e ) w i t h a n d w i t h o u t s e p a r a t i o n a n d w i t h t h e c r o s s - f l o w d r a g d o m i n a t e f o r d i f f e r e n t i n t e r v a l s o f t h e d r i f t angle. F u l l regressions f o r t h e s w a y f o r c e a n d t h e y a w m o m e n t are s u b d i v i d e d i n t o t h r e e s i n g l e - v a r i a b l e parts ( d e p e n d i n g o n t h e d r i f t angle a n d t w o d i m e n s i o n l e s s y a w r a t e p a r a m e t e r s ) v a l i d , h o w e v e r i n t h e w h o l e range o f m a n o e u v r e s . The regression c o e f f i c i e n t s w e r e chosen i n s u c h a w a y t h a t t h e regressions m a t c h a s y m p t o t i c a l l y s o m e e a r l i e r d e v i s e d p o l y n o m i a l m o d e l w h i l e c o e f f i c i e n t s c o r r e -s p o n d i n g , -say, t o c r a b b i n g w e r e e -s t i m a t e d w i t h t h e c r o -s -s - f l o w d r a g t h e o r y . It c a n be also n o t i c e d t h a t l a t e r p u b l i c a t i o n s a v o i d e d g i v i n g m o r e or less d e t a i l e d d e s c r i p t i o n s o f m a t h e m a t i c a l m o d e l s e s p e c i a l l y f o u r q u a d r a n t ones [ 1 2 ] . I n fact, p r a c t i c a l l y a l l s h i p h a n d l i n g s i m -u l a t o r s i m p l y core m o d e l s e x a c t l y o f t h e e x t e n d e d t y p e [ 1 3 ] , B -u t these m o d e l s are t y p i c a l l y c o n s i d e r e d as p r o p r i e t a r y and, as f a r as i t c o u l d be d e t e c t e d o r suspected, are m o r e o f t e n t h a n n o t based o n c e r t a i n t r i c k s a n d q u i c k "at h a n d " s o l u t i o n s m a y b e i n d e e d n o t d e s e r v i n g a d v e r t i s e m e n t . I n t h e p r e s e n t paper, a n a t t e m p t is u n d e r t a k e n t o d e v e l o p a c o m -p r e h e n s i v e u n i f i e d v e r s a t i l e m a t h e m a t i c a l m o d e l s u i t a b l e f o r all types o f m a n o e u v r e s i n s t i l l w a t e r i.e. n o w a v e e x c i t a t i o n is c o n -s i d e r e d . Special a t t e n t i o n w a -s p a i d t o the m o d e l b e i n g r e l a t i v e l y c o n s i s t e n t f r o m the v i e w p o i n t o f basic p r i n c i p l e s [ 1 4 ] . A l t h o u g h t h e r e s u l t i n g m o d e l is s i m i l a r i n m a n y respects t o o t h e r p u b l i s h e d m o d -els a n d is t h u s based o n s o m e synthesis, i t c o n t a i n s also a n u m b e r o f n o v e l t i e s a n d i m p r o v e m e n t s n o t e n c o u n t e r e d i n t h e l i t e r a t u r e . A l l o f t h e m are d e s c r i b e d i n d e t a i l a n d i t is b e l i e v e d t h a t these e l e m e n t s c o u l d b e c o m e u s e f u l i n o t h e r a p p l i c a t i o n s . The m o d e l w a s a p p l i e d t o a s h u t t l e LNG c a r r i e r Galea, f o r w h i c h s o m e t r i a l data w e r e a v a i l a b l e . M o s t l y , these f u l l - s c a l e r e s u l t s w e r e used f o r c o m p a r i s o n s w i t h t h e p r e d i c t e d b e h a v i o u r o f t h e s h i p . C o n -t r a r y -t o n o r m a l p r a c -t i c e f o l l o w e d b y d e v e l o p e r s o f s h i p h a n d l i n g s i m u l a t o r s [ 1 5 ] the m o d e l w a s n o t s u b j e c t to e x t e n s i v e a d j u s t m e n t s e x c e p t f o r l a t e r a l t h r u s t e r a c t i o n c o m m e n t e d i n t h e m a i n t e x t b o d y . C o m p u t e r i m p l e m e n t a t i o n o f t h e d e v e l o p e d m o d e l w a s p e r f o r m e d as series o f e x t e n s i o n s o f t h e o b j e c t - o r i e n t e d m u l t i - m o d e l code d e v e l o p e d e a r l i e r b y t h e a u t h o r s [ 1 6 ] . Finally, i t is w o r t h w h i l e t o n o t e t h a t t h e p o p u l a r t e r m " l o w -speed m a n o e u v r i n g " is s o m e w h a t m i s l e a d i n g a n d n o t v e r y exact. I n fact, e v e r y t i m e i t is used i t goes a b o u t the h a r d m a n o e u v r e s m e n t i o n e d above. I n d e e d , s u c h m a n o e u v r e s can o n l y be p e r f o r m e d at s l o w speed as o t h e r w i s e p o w e r r e q u i r e m e n t s w o u l d g r e a t l y exceed t h e a v a i l a b l e f r o m t h e m a i n e n g i n e . O n t h e o t h e r h a n d , n o r m a l m a n o e u v r e s reachable w i t h c o n s t a n t e n g i n e s e t t i n g s a n d w i t h t h e n o r m a l r u d d e r can be e x e c u t e d at any speed, a t least w i t h t h e t u r b i n e - d r i v e n vessel w h i c h , c o n t r a r y t o diesel engines, does n o t possess a s t a l l ( i d l e ) r o t a t i o n f r e q u e n c y . For m o d e r a t e a n d l o w values o f t h e F r o u d e n u m b e r , all reasonable m a n o e u -v r i n g c h a r a c t e r i s t i c s r e m a i n p r a c t i c a l l y i n d e p e n d e n t o f speed and, w h e n p r o p e r l y rescaled i n t i m e , the b e h a v i o u r o f t h e s h i p m u s t be i n d e p e n d e n t o f t h e a p p r o a c h speed. H o w e v e r , t h e r e is a g e n e r a l b e l i e f t h a t a n y s h i p is w o r s e c o n t r o l l a b l e i n l o w speed [ 1 7 ] a n d t h a t is w h y a u x i l i a r y s t e e r i n g devices are i n d i s p e n s a b l e . T h e r e are t w o possible causes o f t h i s p a r a d o x o r r a t h e r m i s u n d e r s t a n d i n g . First, a l t h o u g h t h e response t o c o n t r o l actions r e m a i n s a d e q u a t e , i t becomes r e a l l y s l o w e r i n t h e d i m e n s i o n a l t i m e w h i l e the h u m a n o p e r a t o r ' s o w n t i m e scale r e m a i n s t h e same a n d the s u b j e c t i v e per-c e p t i o n t e l l s a b o u t w o r s e per-c o n t r o l l a b i l i t y . The seper-cond per-c i r per-c u m s t a n per-c e is t h e i n c r e a s e d s e n s i t i v i t y t o e x t e r n a l factors, s u c h as w i n d a n d c u r r e n t , because r u d d e r s t e e r i n g forces are a p p r o x i m a t e l y p r o p o r -t i o n a l -t o -t h e square o f -t h e s h i p speed. I -t is, h o w e v e r , m e a n i n g l e s s -to discuss t h e loss o f c o n t r o l l a b i l i t y at l o w s p e e d i f t h e l e v e l o f e x t e r n a l i n f l u e n c e is n o t s p e c i f i e d .
Fig. 1 . Frames of reference.
2. G e n e r a l s t r u c t u r e of s h i p m a n o e u v r i n g m o d e l
2.1. Frames of reference and kinematic parameters
The used f r a m e s o f r e f e r e n c e are s h o w n i n Fig. 1 i n t h e h o r i z o n t a l p l a n e v i e w : t h e E a r t h - f i x e d f r a m e O f ??f w i t h t h e axis O f d i r e c t e d v e r t i c a l l y d o w n w a r d s is used t o describe t h e s h i p t r a j e c t o r y a n d to d e f i n e t h e steady w i n d c h a r a c t e r i z e d b y t h e w i n d speed Vw or by i t s m a g n i t u d e Vw = | V w | a n d b y t h e w i n d angle Xw. S i m i l a r l y , the s t e a d y u n i f o r m c u r r e n t v e l o c i t y is Vc also d e f i n e d b y t h e c o u p l e (Vc, Xc)- The o r i g i n 0 is p l a c e d o n t h e u n d i s t u r b e d f r e e surface a n d its l o c a t i o n i n t h e h o r i z o n t a l plane, as w e l l as t h e o r i e n t a t i o n o f t h e t w o h o r i z o n t a l axes c a n be a r b i t r a r y . As a r u l e , t h e y are c h o s e n f r o m con-v e n i e n c e c o n s i d e r a t i o n s . For instance, i f s o m e s t a n d a r d m a n o e u con-v r e as t h e t u r n i n g c i r c l e is c o n s i d e r e d , i t is n a t u r a l t o m a t c h t h e o r i g i n w i t h t h e ship's l o c a t i o n at t h e s t a r t o f t h e m a n o e u v r e i.e. w h e n the c o r r e s p o n d i n g o r d e r is g i v e n . I n t h e case o f b e r t h i n g / u n b e r t h i n g , t h e E a r t h - f i x e d f r a m e t y p i c a l l y m a t c h e s t h e m o o r e d p o s i t i o n o f the s h i p . The s h i p m o t i o n is d e s c r i b e d b y t h e g r o u n d v e l o c i t y v e c t o r Vc a n d b y t h e a n g u l a r v e l o c i t y v e c t o r S2. I n t r o d u c i n g t h e a i r s h i p veloc-i t y v e c t o r V ^ a n d t h e s h veloc-i p v e l o c veloc-i t y v e c t o r V ( m a g n veloc-i t u d e o f t h e latter is t h e ship's l o g speed) i t is easy t o e s t a b l i s h t h e f o l l o w i n g relations:
V c = V ^ + V w , V G = V - h V c . It f o l l o w s f r o m these r e l a t i o n s t h a t V A + V W = V + V C , V = V c - V c , (2) V A = V C - V W = V + V C - V W .
The w i n d a n d c u r r e n t v e l o c i t i e s can be also r e p r e s e n t e d t h r o u g h t h e i r p r o j e c t i o n s o n t h e E a r t h axes ( t h e v e r t i c a l c o m p o n e n t s are s u p p o s e d to be z e r o ) :
Vw,| = Vw cos Xw;
Vv,, = Vv
s i n Xw,Vcj = cos Xc; Vcq = Vc s i n Xc •
As usual, t h e b o d y axes Cxyz are i n t r o d u c e d w i t h t h e o r i g i n C l y i n g at a n y p o i n t i n t h e c e n t r e p l a n e a l t h o u g h u s u a l l y p r e f e r r e d is e i t h e r t h e c e n t r e o f mass or t h e p o i n t b e l o n g i n g also t o t h e m i d s h i p p l a n e a n d t o t h e e q u i l i b r i u m w a t e r p l a n e . The p o s i t i o n o f t h e body
S. Sutulo, C. Guedes Soares /Applied Ocean Research 51 (2015) 293-308 295
axes w i t h r e s p e c t t o t h e E a r t h f r a m e is d e s c r i b e d b y t t i e advance f c. t r a n s f e r ric, s u b m e r g e n c e f o h e a d i n g angle p i t c h a n g l e 6 a n d r o l l angle (p. P r o j e c t i o n s o f t h e v e l o c i t y V o n t o the b o d y axes x, y, z are d e n o t e d as u, v, w a n d c a l l e d v e l o c i t i e s o f surge, s w a y , a n d heave r e s p e c t i v e l y a n d t h o s e o f t h e a n g u l a r v e l o c i t y S2 are p , q, a n d r a n d called v e l o c i t i e s o f r o l l , p i t c h a n d y a w . T h e s u b s c r i p t s w , c, G, A can be a d d e d to a n y v e l o c i t y to r e f e r to t h e c o r r e s p o n d i n g c o m p o n e n t . I t f o l l o w s f r o m Eqs, ( 1 ) - ( 3 ) t h a t (5)
The f o r m u l a e above are v e r y i m p o r t a n t as t h e y d e t e r m i n e p e c u -l i a r i t i e s o f t h e s h i p m o t i o n i n p r e s e n c e o f w i n d a n d c u r r e n t . T h e w i n d a n d c u r r e n t v e l o c i t i e s , w h i c h are p r e s e n t t h e r e , are:
Uc = Vcf cos l/r + Va, s i n
Vc = - V ^ j s i n i/f + Vcq cos f ,
( 6 )
Uw = Vw^ cos i/f -i- Vw,, s i n
Vw = -Vwi s i n i/f + Vw,, cos i/f.
The m a i n state v a r i a b l e s o f t h e m a n o e u v r i n g s h i p w i l l be fc. ijc, ir, r, a n d i i , v a l t h o u g h each o f t h e l a t t e r t w o can be s u b s t i t u t e d w i t h a c o r r e s p o n d i n g g r o u n d or air v e l o c i t y . T h e f o l l o w i n g a u x i l i a r y v a r i a b l e s can o f t e n b e c o m e u s e f u l : • t h e d i m e n s i o n l e s s v e l o c i t i e s o f s w a y a n d y a w , V , I'l ' ^ = V ' '' = V' w h e r e V= | V | a n d L is t h e s h i p ' s r e f e r e n c e l e n g t h ; • t h e d r i f t angle ( w i t h r e s p e c t t o w a t e r )
{
- a s i n i / at u > 0, - T r s i g n y 4 - a s i n i / at u < 0; • t h e air d r i f t angle PA - a s i n i ; ^ at U/i > 0, -TT s i g n i»/! + a s i n at ii/i < 0,(7)
( 8 ) ( 9 )w h e r e v'^ = VA/VA a n d V^ = \V/\\is t h e ship's a i r s p e e d equal t o t h e r e l a t i v e w i n d speed as m e a s u r e d o n t h e s h i p ; • t h e course angle v ( w i t h r e s p e c t t o w a t e r ) ( 1 0 ) ( 1 1 ) • t h e course ( a n g l e ) x o v e r t h e g r o u n d sin
2£
atfc >
0, i]c > 0, k ; Tt + asin ^ a t fc < 0, 2jt + asin 'k a t fc > 0, i]c < 0. sc The u s u a l d i m e n s i o n l e s s v e l o c i t i e s i / a n d f are n o t c o n v e n i e n t f o r h a r d m a n o e u v r e s as t h e y t e n d t o i n f i n i t y as t h e s h i p speed t e n d s to zero. T h e d r i f t angle is used i n s t e a d o f t h e d i m e n s i o n l e s s v e l o c i t y o f s w a y a n d the f o l l o w i n g g e n e r a l i z e d d i m e n s i o n l e s s v e l o c i t y o f y a w can be i n t r o d u c e d :r " = ( 1 2 )
It is clear t h a t f e [ - 1 , 1 ] r e m a i n i n g a l w a y s finite.
The p l a n e r" (Fig. 2 ) can be used t o r e p r e s e n t d o m a i n s o f a r b i -t r a r y ( a l l i m a g i n a b l e ) m a n o e u v r e s ( -t h e large r e c -t a n g l e ) , m o d e r a -t e
•180 (
+ 1 . 0
+ 1 8 0 deg
Fig. 2 . Sketcli of manoeuvring domains.
m a n o e u v r e s ( t h e s m a l l e r h a t c h e d r e c t a n g l e ) , a n d w e a k m a n o e u -vres ( t h e s m a l l e s t filled r e c t a n g l e ; w e a k m a n o e u v r e s are t y p i c a l l y course k e e p i n g a n d s m a l l course changes). O f course, t h e b o u n d -aries s e p a r a t i n g t h e d o m a i n s f r o m each o t h e r are f u z z y a n d t r a c e d b a s i n g o n r a t h e r a r b i t r a r y e x p e r t e s t i m a t e s .
2.2, Equations o! motion
I n l o w - s p e e d m a n o e u v r i n g , e f f e c t s r e l a t e d t o t h e v e r t i c a l planes are n e g l i g i b l e a n d i t can be a s s u m e d t h a t t h e m a n o e u v r i n g m o t i o n is o n l y p e r f o r m e d i n t h e h o r i z o n t a l p l a n e . T h e k i n e m a t i c d i f f e r e n -t i a l e q u a -t i o n s l i n k i n g -t h e ship's p o s i -t i o n i n -the E a r -t h axes w i -t h -t h e v e l o c i r i e s p r o j e c t i o n s are t h e n f C = UQ COS ir-Vc s i n l/r, //c = I ' c s i n -I-i^G cos i / f , ( 1 3 ) iA = r. T h e d y n a m i c e q u a t i o n s o f m o t i o n o f t h e s h i p as a rigid b o d y are: (m + iJ.u)u - mvr - mxcr^ =XH +XP+XK+XA +XQ,
(m + iJ.iiW + imxc + inef + mur ^Yu + Yp + Ys + Yr + YA + Ya, ( 1 4 ) (mxc + Ai26)i' + (lu + fJ-eeV + mxgur = NH + Np + Nu + Nr + + NQ,
w h e r e m is t h e mass o f t h e s h i p ; /xy are t h e a d d e d masses, x c . y c are t h e c e n t r e - o f - m a s s c o o r d i n a t e s ; Izz is t h e m o m e n t o f i n e r t i a i n y a w ;
X, y, N are t h e forces a n d m o m e n t s o f surge, s w a y , a n d y a w r e s p e c
-t i v e l y a n d e a c h o f -these c o m p o n e n -t s is d i v i d e d i n -t o s u b c o m p o n e n -t s d e s c r i b e d b y t h e f o l l o w i n g s u b s c r i p t s : H—the h u l l h y d r o d y n a m i c forces, P—the p r o p e l l e r forces, R—the r u d d e r forces, T—the t h r u s t e r forces, A—the a e r o d y n a m i c forces, Q,—forces f r o m t h e tugs.
3. H y d r o d y n a m i c l i u l l f o r c e s T h e h y d r o d y n a m i c f o r c e s o n t h e h u l l are d e s c r i b e d u s i n g t h e m e t h o d p r o p o s e d b y S u t u l o [ 1 8 ] w h i c h , i n its p r e s e n t i m p l e m e n -t a -t i o n , r e p r e s e n -t s a c o n -t i n u a -t i o n o f -t h e s -t a n d a r d I n o u e m o d e l o n -t o the d o m a i n o f a r b i t r a r y m a n o e u v r e s . A c c o r d i n g t o t h i s m e t h o d , t h e h u l l s w a y f o r c e a n d y a w m o m e n t are r e p r e s e n t e d as: XH=X"^iV^ + Lh'^)LT, YH = Y"^{V^ +Lh-'^)LT, ( 1 5 ) NH = N"^{V^ + L^r^)L^T,
w h e r e X " , Y", Af" are t h e g e n e r a l i z e d f o r c e a n d m o m e n t c o e f f i c i e n t s . As t o t h e surge f o r c e , i t t u r n e d o u t t h a t t h e m o s t c o n v e n i e n t a p p r o a c h is t o t r a n s f o r m t h e surge f o r c e m o d e l s u g g e s t e d b y I n o u e et al. [ 1 9 ] . Besides m o v i n g f r o m t h e d i m e n s i o n a l t o t h e d i m e n s i o n -less f o r m , t h e s t r u c t u r e w a s c o r r e c t e d i n s u c h a w a y t h a t t h e s u r g e
296 S. Sutulo, C. Guedes Soares/ Applied Ocean Research SJ (2075; 293-308 f o r c e v a n i s h e s w h e n t h e s h i p is r o t a t i n g o n t h e s p o t i n d e p e n d e n t l y o f t h e d r i f t a n g l e ( s t r i c t l y s p e a k i n g , t h e r e w i l l be s o m e n o n z e r o s u r g e f o r c e f o r a s h i p w i t h o u t m i d s h i p s y m m e t r y b u t o b v i o u s l y i t w i l l n o t b e s i g n i f i c a n t a n d i n case o f absence o f e x p e r i m e n t a l d a t a s h o u l d be b e t t e r z e r o e d ) . T h e n , t h e g e n e r a l i z e d s u r g e f o r c e c o e f f i c i e n t is ( 1 6 ) w h e r e R r ( t i ) is t h e t o t a l resistance c u r v e as f u n c t i o n o f t h e surge v e l o c i t y a n d d e f i n e d f o r b o t h p o s i t i v e a n d n e g a t i v e a r g u m e n t , r^s is t h e a s t e r n c o r r e c t i o n c o e f f i c i e n t . Cm ^ 0.625 is t h e c o r r e c t i o n f a c t o r [ 1 9 ] . As t h e d r a g c u r v e i n a s t e r n m o t i o n is u s u a l l y u n a v a i l a b l e a n d n o t s u p p o r t e d w i t h s t a n d a r d r e s i s t a n c e p r e d i c t i o n m e t h o d s [ 2 0 ] , i t w a s a s s u m e d h e r e t o be o d d i.e. RT{-U) = -RJ{U). H o w e v e r , as u s u a l l y t h e h u l l is o p t i m i z e d f o r t h e a h e a d m o t i o n , t h e c o r r e c t i o n c o e f f i c i e n t r^is is a s s u m e d t o be 1.15 at u < 0 a n d 1.0 o t h e r w i s e . T h e d i m e n s i o n l e s s s w a y a n d y a w c o e f f i c i e n t s c a n be m o d e l l e d as f o l l o w s : CyO = Cyt
1
4^.'r; Cy2 = -lK{r)+Y^(-r)]+lY':;
Cy3 CyS CyS C„3 = CnS •• Cm •-Cn8 =[n(^) -
n ( - T ) ] ; cy4 =2^[y;(r)+y;(-r)];
2jr
1
[y;(t)-Y;(-r)-2y^]; CyS-
rl'rr- Cyl l y-6
(20)
c„o = N«; c„, =
--[NJ(T)
+ Ni(-T)]; c„2=-^TOr)+ N;(-r)]
- 2^[N;(r) + N;(-r)] - 2N2; c„4 = ^m^)" N^C-r)];
gL[N;,„+NJ(r) +
N ; ( - r ) - 2 N S ] ;c„6
= ^ W r ) - N;(-r)];(21)
12 , . - N , V + ( i - i ) [ W + N;(-r)l;
i 1 _ 1 _ V2^2" Ï 6
[ N ; ( T ) - N J ( - T ) ] ; c g : w h e r e r is t h e t r i m , p o s i t i v e b y t h e s t e r n a n d n o n d i m e n s i o n a -l i z e d b y t h e m e a n d r a u g h t ; Yg a n d are t h e g e n e r a -l i z e d s w a yY " { ^ , r " ) = Cyor" -I- Cy\ s i n ^ s i n OT" s i g n r " + C y 2 s i n , S c o s ^r" + Cy-i s i n 2 / J c o s -r" + C j , 4 C 0 S ; ö s i n O T " + Cys cos 2^6 s i n OT"
-(-CyeCosyS ('cos ^ r " - cos s i g n r " + Cy7(cos2^- cos4;ö)cos^r"sign;ö-|-Cy8sin3/3cos ^ r " ; ( 1 7 )
N"(y8, r " ) = C|,or" -(- Cni s i n 2/Ö cos ^ r " -1- c„2 s i n /3 cos ^r" + Cns cos 2,0 s i n nr" + Cn4 cos ,0 s i n Trr" -(- c„5(cos 2,0
- c o s 4 , 8 ) s i n : T r " -1- c „ 6 C 0 S fiicosfi- c o s 3 / 3 ) s i g n r " + c „ 7 s i n 2 / 3 ^cos ^ r " - cos ^ r " j + c„8 s i n / 3 ^cos ^ r " - cos ^ r " j
+ cn9 s i n 2 / ? ( c o s J r " - cos ^ r " ) s i g n r " . ( 1 8 ) _ The r e g r e s s i o n c o e f f i c i e n t s Cy^ni i n ( 1 7 ) a n d ( 1 8 ) w o u l d be b e s t d e t e r m i n e d a f t e r c a p t i v e m o d e l tests c a r r i e d o u t f o r each p a r t i c u -lar h u l l i n t h e w h o l e r a n g e o f t h e d r i f t a n g l e a n d d i m e n s i o n l e s s rate o f y a w . H o w e v e r , s u c h e x p e r i m e n t a l d a t a are o b t a i n e d r a r e l y a n d , as a bypass s o l u t i o n , t h e s e c o e f f i c i e n t s c a n be c h o s e n t o p r o v i d e a s y m p t o t i c e q u i v a l e n c e t o p o l y n o m i a l m o d e l s d e v i s e d f o r m o d -e r a t -e m a n o -e u v r -e s . Th-es-e p o l y n o m i a l m o d -e l s c a n b-e d -e v -e l o p -e d s p e c i f i c a l l y f o r t h e g i v e n h u l l or b o r r o w e d f r o m s o m e database m e t h o d , l i k e t h a t b y I n o u e et a l . [ 1 9 ] . These m o d e l s are n o r m a l l y w r i t t e n i n t e r m s o f p r i m e d v a r i a b l e s i / , r' b u t c a n be r e - w r i t t e n i n t e r m s o f f5 a n d r':
y"(/3, r ' ) = -Y- sin^ + Y^r' - V;,,^ s i n / ï | s i n ^ | - Y'^^, s i n ^ l r ' l + y ; | , , r ' i r ' | , N"{l}, r') = -N;, sinp + N f r ' + s i n ^ P r' - ( N ; „ - N[,) sinfi r'^ + JV;.,,|r'|r'|, ( 1 9 ) w h e r e t h e c o e f f i c i e n t s Y(„ NL.. are f u n c t i o n s o f t h e h u l l ' s p a r r|r| riculars a n d t r i m a n d are e s t i m a t e d a c c o r d i n g to I n o u e et a l . [ 2 1 ] . T h e n , t h e r e g r e s s i o n s ( 1 7 ) a n d ( 1 8 ) c a n be r e - w r i t t e n i n t e r m s o f t h e same v a r i a b l e s as ( 1 9 ) u s i n g t h e r e l a t i o n r" ^ r' - ^r'^ a n d t h e n a s y m p t o t i c a l l y m a t c h e d t o ( 1 9 ) , T h e i r s t r u c t u r e , w h i c h m a y s e e m at first s i g h t n o t q u i t e n a t u r a l a n d e v i d e n t , w a s c h o s e n t o m a k e t h i s m a t c h i n g p o s s i b l e . A s r e s u l t , t h e f o l l o w i n g r e l a t i o n s c a n be e s t a b l i s h e d : f o r c e a n d y a w m o m e n t c o e f f i c i e n t s d e t e r m i n e d f o r t h e h u l l r o t a t -i n g o n t h e spot, a n d Ygg -is t h e g e n e r a l -i z e d y a w m o m e n t c o e f f -i c -i e n t c o r r e s p o n d i n g t o t h e p u r e l a t e r a l m o t i o n i.e. t o t h e c r a b b i n g w i t h -o u t r -o t a t i -o n . If these d a t a are absent, w h i c h is t y p i c a l , t h e y can be e s t i m a t e d u s i n g the m e t h o d f r o m ( V o y t k u n s k y , 1 9 8 5 ) or esti-m a t e d w i t h t h e c r o s s - f l o w d r a g t e c h n i q u e . Because o f t h e absence o f d a t a r e l a t e d to t h e a s t e r n m o t i o n w i t h m o d e r a t e d r i f t angles, i t w a s a d d i t i o n a l l y a s s u m e d t h a t t h e h u l l is s y m m e t r i c w i t h respect t o t h e m i d s h i p p l a n e . A t t h e s a m e t i m e , t h e f a c t t h a t a t r i m b y the s t e r n w i l l act as a t r i m b y t h e b o w i n a s t e r n m o t i o n is a c c o u n t e d f o r . Finally, i t m u s t be n o t e d t h a t t h e o r i g i n a l m a t h e m a t i c a l m o d e l b y I n o u e et a l . [ 1 9 , 2 1 ] is 4 D 0 F i n v o l v i n g also t h e r o l l e q u a t i o n and d e p e n d e n c e o f t h e y a w m o m e n t o n t h e i n s t a n t a n e o u s r o l l angle. T h e back i n f l u e n c e o f t h e r o l l m a y b e c o m e s i g n i f i c a n t f o r r e l a t i v e l y f a s t vessels. The s h i p c o n s i d e r e d i n t h e p r e s e n t s t u d y is m a r g i n a l i n t h i s r e s p e c t a n d u s i n g a 4 D 0 F m o d e l c o u l d be d e s i r a b l e . H o w e v e r , as t h e i n v e s t i g a t i o n w a s h e r e m o r e c o n c e n t r a t e d o n t h e l o w - s p e e d m a n o e u v r i n g a n d c r e a t i o n o f u n i f o r m l y v a l i d r e g r e s s i o n s i n c l u d i n g r o l l e f f e c t s is p r o b l e m a t i c a n d c e r t a i n l y n o t s t r a i g h t f o r w a r d , the c h o i c e w a s m a d e i n f a v o u r o f a 3 D 0 F m a n o e u v r i n g m o d e l w h i c h finally p r o v e d q u i t e a d e q u a t e i n t h e p r e s e n t case. 4. P r o p e l l e r f o r c e s T h e p r o p e l l e r ' s l o n g i t u d i n a l f o r c e Xp = re. w h e r e Te is t h e effec-t i v e effec-t h r u s effec-t w h i c h is s u p p o s e d effec-t o be p o s i effec-t i v e w h e n effec-t h e p r o p e l l e r is
S. Sutiilo, C. Guedes Soares /Applied Ocean Research 5 Ï (2015) 293-308 297 r u n n i n g a h e a d a n d is n e g a t i v e w h e n w o r l < i n g astern. The e f f e c t i v e t h r u s t is o b t a i n e d f r o m t h e o p e n - w a t e r t h r u s t Tas re = T ( l - f p ) , ( 2 2 ) w h e r e tp is t h e t h r u s t d e d u c t i o n f r a c t i o n . For a n y g i v e n f i x e d - p i t c h p r o p e l l e r , the t h r u s t w i l l d e p e n d o n t h e c u r r e n t r o t a t i o n f r e q u e n c y ( r p s ) n , w h i c h is a s s u m e d p o s i t i v e i n ahead r o t a t i o n and o n t h e i n s t a n t a n e o u s p r o p e l l e r advance v e l o c -i t y UpA = u ( l - Wp), w h e r e Wp -is t h e p r o p e l l e r w a k e f r a c t -i o n . To s i m u l a t e a r b i t r a r y m a n o e u v r e s , t h e p r o p e l l e r o p e n - w a t e r charac-t e r i s charac-t i c s m u s charac-t be d e f i n e d i n f o u r q u a d r a n charac-t s i.e. f o r any signs o f n a n d
UpA- H o w e v e r , as s t u d i e s o f p r o p e l l e r h y d r o d y n a m i c s w e r e m a i n l y
d r i v e n b y i n t e r e s t s o f p r o p u l s i o n c a l c u l a t i o n s a n d p r o p e l l e r design, r e l a t i v e l y f u l l sets o f data c o r r e s p o n d i n g t o v a r i o u s p r o p e l l e r series a n d f o r f u l l range o f t h e n u m b e r o f blades, p i t c h a n d disc r a t i o , are o n l y a v a i l a b l e f o r t h e f i r s t q u a d r a n t i.e. w h e n Up^ > 0, n > 0 [ 2 2 ] . As t o 4 - q u a d r a n t c h a r a c t e r i s t i c s , s y s t e m a t i c data a n d a p p r o x i m a t i o n s [ 2 3 ] w e r e a p p a r e n t l y o b t a i n e d f o r o n l y the B4.70 p r o p e l l e r w h i l e o n l y selected p i t c h r a t i o s w e r e c o v e r e d f o r p r o p e l l e r s w i t h o t h e r n u m b e r o f blades a n d d i f f e r e n t v a l u e s o f t h e disc r a t i o . I n t h e p r e s e n t s t u d y , a n a p p r o x i m a t e a l t e r n a t i v e s o l u t i o n based o n r e s c a l i n g o f t h e 4 q u a d r a n t m o d e l f o r a u n i q u e base p r o -p e l l e r m o d e l is -p r o -p o s e d . T h e -p o s s i b i l i t y o f s u c h a -p -p r o a c h f o l l o w s f r o m t h e f a c t t h a t n e a r l y a l l p r o p e l l e r t h r u s t a n d t o r q u e c o e f f i -c i e n t -c u r v e s are a p p r o x i m a t e l y e q u i d i s t a n t i n t h e first q u a d r a n t a n d c a n be t r a n s f o r m e d each to o t h e r r e - s c a l i n g t h e axes. I n fact, t h i s o p e r a t i o n is desirable e v e n i f t h e a c t u a l p r o p e l l e r ' s data are a v a i l -able as a l l m e t h o d s f o r p r e d i c t i n g t h e s h i p resistance, t h r u s t a n d p r o p e l l e r - h u l l i n t e r a c t i o n are a p p r o x i m a t e a n d m u s t be a d j u s t e d to r e p r o d u c e the a c t u a l p r o p u l s i o n p o i n t . T h e a d j u s t m e n t p r o c e -d u r e -d e p e n -d s o n t h e a v a i l a b l e i n p u t -data a n -d i n a r a t h e r t y p i c a l case, w h e n t h e d e s i g n rps n^, t h e d e s i g n speed V^, t h e c o r r e s p o n d -i n g t o r q u e Qpd a n d t h e p r o p e l l e r d -i a m e t e r Dp are s p e c -i f -i e d , t h e a d j u s t m e n t is p e r f o r m e d as f o l l o w s :
1. T h e d e s i g n drag RjaiVci), t h e s t r a i g h t r u n w a k e f r a c t i o n Wpo a n d t h e t h r u s t d e d u c t i o n c o e f f i c i e n t tpo are e s t i m a t e d u s i n g s o m e a p p r o p r i a t e m e t h o d [ 2 0 ] . T h e d e s i g n a d v a n c e r a t i o is t h e n = Vdi^-WpoVinaDp). 2. T h e a d v a n c e c o r r e c t i o n f a c t o r kj is c o m p u t e d a s s u m i n g t h a t t h e p r o p e l l e r w o r k s i n t h e d e s i g n c o n d i t i o n s at m a x i m u m e f f i c i e n c y ( i n m o s t cases, t h e p r o p e l l e r s are d e s i g n e d m e a n i n g t h i s t a r g e t ) : kj = JoptlJd' w h e r e Jopt is t h e v a l u e o f t h e a d v a n c e r a t i o c o r r e s p o n d -i n g t o t h e r e f e r e n c e s c r e w p r o p e l l e r . 3. T h e t h r u s t a n d t o r q u e c o r r e c t i o n f a c t o r s kp a n d /<Q are d e f i n e d as f o l l o w s : /<r = Rpd Toptind,kjVa)il-tpo)'
Qopt(nd,/<;Vd)
( 2 3 )w h e r e Topt a n d Qopt are t h e values o f t h e t h r u s t a n d t o r q u e p r o -d u c e -d b y t h e p r o p e l l e r m o -d e l at t h e s h i f t e -d -d e s i g n p r o p u l s i o n p o i n t , w h i c h is o p t i m a l f o r t h e r e f e r e n c e p r o p e l l e r . A f t e r t h e a d j u s t m e n t is c o m p l e t e d , t h e e f f e c t i v e t h r u s t a n d t o r q u e are c o n t i n u o u s l y c o m p u t e d as T e = ( l - t p ) k T f > l d C r ( y B ) V | , (l = ka^AdDpCa(YB)Vi, ( 2 4 ) w h e r e f p is t h e c u r r e n t v a l u e o f t h e t r u s t d e d u c t i o n c o e f f i c i e n t ;
Ad = 7 r D ^ / 4 is the p r o p e l l e r disc area; YB is t h e e f f e c t i v e b l a d e
a d v a n c e angle, a n d VB is t h e e f f e c t i v e t o t a l b l a d e v e l o c i t y . A c c o r d i n g
t o O l t m a n n a n d Sharma [ 8 ] t h e g e n e r a l i z e d t h r u s t a n d t o r q u e c o e f f i c i e n t s are
q^ijCOSj/s cos/B + Cfi^sin YB sin ys
a t cos KB > Ü . 9 3 3 6
o t h e r w i s e
( 2 5 )
w h e r e t h e c o e f f i c i e n t s C° ^ C^^Q are o n l y d e f l n e d f o r t h e a 5-b l a d e d p r o p e l l e r w i t h t h e p i t c h r a t i o 0.745 a n d t h e e x p a n d e d area r a t i o 0.6; s i n y g ; kjUpA cos YB - Vg, a n d Vcp = O.lymDp, VB =
{kjUpAT
The w a k e f r a c t i o n c o e f f i c i e n t Wp a c c o r d i n g to I n o u e et a l . [ 1 9 ] c a n be a p p r o x i m a t e d b y
Wp = Wpoe ( 2 6 )
w h e r e /3p is the local g e o m e t r i c (i.e. w i t h o u t the h u l l ' s i n f l u e n c e ) s i d e w a s h angle. A l t h o u g h t h e a p p r o x i m a t i o n ( 2 6 ) w a s p r i m a r i l y d e v i s e d f o r o n l y m o d e r a t e m a n o e u v r e s , i t is e v i d e n t t h a t i t w i l l g i v e r e a s o n a b l e e s t i m a t e f o r a l l /3p e [ - TZ, K].
The r o t a t i n g p r o p e l l e r w i l l , i n g e n e r a l , p r o d u c e also t h e s w a y f o r c e Yp a n d t h e y a w m o m e n t Np. These forces are, h o w e v e r , i n s i g n i f l c a n t o n m o d e r a t e - s p e e d vessels e x c e p t f o r t h e case w h e n t h e p r o p e l l e r is w o r k i n g a s t e r n . I n t h i s case, t h e s o - c a l l e d H o v g a a r d f o r c e w i l l appear due to t h e i n f l u e n c e o f t h e t a n g e n t i a l i n d u c e d v e l o c i t i e s i n t h e s l i p s t r e a m o f a h e a v i l y l o a d e d p r o p e l l e r . T h e H o v -g a a r d f o r c e m u s t be a c c o u n t e d f o r w h e n t h e s h i p is s t o p p i n -g a n d b a c k i n g . D i r e c t i o n o f the H o v g a a r d f o r c e d e p e n d s o n t h e d i r e c t i o n o f r o t a t i o n o f t h e p r o p e l l e r . I n t h e d e s c r i b e d m o d e l t h e H o v g a a r d f o r c e is m o d e l l e d u s i n g a v e r y s i m p l e m e t h o d s u g g e s t e d i n [ 2 4 ] a n d d e f i n i n g t h i s f o r c e as a c o n s t a n t f r a c t i o n o f t h e t h r u s t : YP = ICHT, w h e r e t h e c o e f f i c i e n t d e p e n d s o n t h e c o n f i g u r a t i o n o f t h e a f t e r b o d y , v a r i e s m a i n l y f r o m 0.4 t o 0.8 b u t m u s t be a d j u s t e d a f t e r t r i a l s w h e n e v e r p o s s i b l e . T h e n , t h e y a w m o m e n t w i l l be Np = YpXpH, w h e r e t h e e f f e c t i v e abscissa XPH, a c c o r d i n g t o B r i x ' data varies f r o m 0.78xp to 0.94xp. O b v i o u s l y , t h i s m o d e l f o r t h e H o v g a a r d e f f e c t is s o m e w h a t s i m p l i s t i c as A n i s s i -m o v a a n d S o b o l e v [ 2 5 ] d e -m o n s t r a t e d t h a t d e p e n d e n c e o f t h e f o r c e a n d m o m e n t o n t h e a d v a n c e r a t i o is i n f a c t r a t h e r c o m p l i c a t e d . U n f o r t u n a t e l y , c o n t r a r y t o w h a t w a s c l a i m e d b y A m b r o s o v s l c y a n d K a t z [ 2 6 ] , a t t e m p t e d i m p l e m e n t a t i o n o f t h e f o r m u l a e p r o p o s e d i n t h e c i t e d p u b l i c a t i o n d i d n o t r e s u l t i n r e a s o n a b l e r e s p o n s e o f t h e s h i p . 5 . R u d d e r f o r c e s 5.J. General remarl<s D e s c r i p t i o n s o f p r a c t i c a l m a t h e m a t i c a l m o d e l s o f r u d d e r f o r c e s v a l i d i n m o d e r a t e m a n o e u v r i n g are a b u n d a n t i n t h e l i t e r a t u r e . A l l o f t h e m p r e s u m e ; t h a t t h e flow attacks t h e r u d d e r m o r e o r less f r o m t h e l e a d i n g edge a n d t h e a t t a c k angles d o n o t e x c e e d t h e s t a l l a n g l e . M o r e o v e r , a l t h o u g h t y p i c a l m o d e r n r u d d e r s h a v e a s m a l l aspect r a t i o XR p r a c t i c a l l y n e v e r e x c e e d i n g 2.0 w h i c h r e s u l t s i n a s e n s i t i v e n o n l i n e a r i t y o f t h e i r l i f t c u r v e , t h i s c i r c u m s t a n c e is o f t e n n e g l e c t e d a n d t h e r u d d e r ' s c h a r a c t e r i s t i c is l i n e a r i z e d [ 1 9 , 2 7 ] . I n f l u -ence o f t h e s l i p s t r e a m is p r i m a r i l y a c c o u n t e d f o r o n t h e basis o f t h e a c t u a t o r disc t h e o r y b u t m a n y e m p i r i c c o r r e c t i o n s a n d , o f t e n , i n c o n v e n i e n t s t r u c t u r e o f t h e f o r m u l a e f o r t h e p r o p e l l e r i n f l u e n c e o n t h e r u d d e r i n f l o w v e l o c i t y a n d s i d e w a s h angle m a k e d i f f i c u l t t h e i r g e n e r a l i z a t i o n t o a r b i t r a r y m a n o e u v r e s . T h e m o d e l p r o p o s e d i n t h e p r e s e n t p a p e r is f r e e o f s u c h p r o b l e m s a n d r e m a i n s v a l i d i n a n y s i t u a t i o n a l t h o u g h i t is n o t u n i f o r m l y a c c u r a t e . W h i l e i n m o d -e r a t -e m a n o -e u v r i n g i t c o u l d b-e v a l i d a t -e d a g a i n s t m u l t i p l -e m o r -e or less a c k n o w l e d g e d m o d e l s , t h e d i r e c t v a l i d a t i o n w a s t o o d i f f i c u l t i n s u c h s i t u a t i o n s as large a t t a c k angles o f t h e r u d d e r , e s p e c i a l l y
2 9 8 S. Sutulo, C. Guedes Soares /Applied Ocean Research 51 (2015) 293-308 w h e n t h e flow c o m e s f r o m t h e t r a i l i n g edge. A s s i m i l a t i o n , a n a l y s i s a n d g e n e r a l i z a t i o n o f v a r i o u s ideas and p a r t i a l m o d e l s s u g g e s t e d b y o t h e r specialists s e r v e d as i m p o r t a n t assets i n d e v e l o p i n g t h e m o d e l d e s c r i b e d b e l o w . H o w e v e r , special a t t e n t i o n w a s g i v e n to c o n t r i b u t i o n s b y S ö d i n g [ 2 8 , 2 9 ] , O l t m a n n a n d S h a r m a [8 ], a n d Kose [ 3 0 ] . 5.2. Basic assumptions A l l e n g i n e e r i n g m e t h o d s f o r e s t i m a t i n g c o n t r i b u t i o n o f t h e r u d -d e r i n m a n o e u v r i n g m o t i o n are base-d o n m o r e o r less n a t u r a l , t h o u g h n o t v e r y r i g o r o u s a s s u m p t i o n s . The f o l l o w i n g a s s u m p t i o n s w e r e t a k e n h e r e :
1 . The t o t a l r u d d e r area AP=ARO+ARP is c o m p o s e d o f t w o p a r t s :
ARO is t h e area o u t s i d e t h e s l i p s t r e a m w h i l e ARP is t h e area i n s i d e
t h e s l i p s t r e a m . If t h e r e is n o d i r e c t data a b o u t t h e s e areas, i t is u s u a l l y a s s u m e d t h a t (27) w h e r e hpp is t h e h e i g h t o f t h e r u d d e r ' s p a r t w a s h e d b y t h e s l i p -s t r e a m . 2. The s l i p s t r e a m j e t is s u p p o s e d t o be c i r c u l a r c y l i n d r i c a l f r o m t h e r u d d e r ' s l e a d i n g t o t r a i l i n g edge. 3 . V a l u e s o f t h e r u d d e r f o r c e c o e f f i c i e n t s ( l i f t c o e f f i c i e n t CRL, d r a g c o e f f i c i e n t Q D , n o r m a l f o r c e c o e f f i c i e n t CRJV, a n d t h e t a n g e n t i a l f o r c e c o e f f i c i e n t CRT) are t h e same f o r its p a r t s o u t s i d e a n d i n s i d e t h e s l i p s t r e a m . In a d d i t i o n , i t is a s s u m e d t h a t t h e t a n g e n t i a l f o r c e c a n a l w a y s be n e g l e c t e d .
4. The flow a r o u n d t h e r u d d e r is s u p p o s e d to be u n i f o r m a n d h o m o -g e n e o u s w i t h i n each o f the p a r t s o u t s i d e a n d i n s i d e t h e p r o p e l l e r race. T h e v e l o c i t y o f t h e r u d d e r w i t h respect to w a t e r is VR a n d VRP i n each area r e s p e c t i v e l y a n d t h e c o r r e s p o n d i n g a t t a c k angles are UR a n d URP. Each a t t a c k angle is c h a n g i n g w i t h i n t h e i n t e r v a l [ - ; r , 7 r l .
5.3. Representation of forces on tlie rudder
D e p e n d i n g o n w h e t h e r t h e f o r c e a c t i n g o n t h e r u d d e r is a n a l y z e d i n t h e i n f l o w v e l o c i t y axes o r i n t h e b l a d e f i x e d f r a m e , c o n s i d -e r -e d ar-e -e i t h -e r t h -e l i f t L a n d t h -e d r a g D or t h -e n o r m a l N a n d th-e t a n g e n t i a l T f o r c e c o m p o n e n t s . The l a t t e r a p p r o a c h has c e r t a i n a d v a n t a g e s . The p l o t i n Fig. 3 w a s r e c a l c u l a t e d f r o m t h e l i f t a n d d r a g data o b t a i n e d f o r a n o n - s y m m e t r i c 5% p r o f i l e i n c i r c u l a r w i n d t u n n e l tests [31 ] a n d c l e a r l y s h o w s t h a t the t a n g e n t i a l f o r c e c a n be p r a c t i c a l l y a l w a y s n e g l e c t e d . A s s u m i n g t h a t t h e r u d d e r n o r m a l f o r c e JV is p o s i t i v e w h e n d i r e c t e d t o t h e r u d d e r ' s p o r t s i d e , t h e r u d d e r - o r i g i n a t e d s u r g e f o r c e XR a n d s w a y f o r c e YR w i l l be: XR = -NsmSR, YR =-{1+aH)N cos SR, ( 2 8 ) w h e r e SR is t h e r u d d e r d e f l e c t i o n angle w h i c h is p o s i t i v e w h e n t h e r u d d e r is d e f l e c t e d t o the s t a r b o a r d , a n d an is t h e r u d d e r - h u l l i n t e r a c t i o n f r a c t i o n . T h e r u d d e r y a w m o m e n t is t h e n NR = YRXRH, (29) w h e r e XRH is t h e r u d d e r ' s h y d r o d y n a m i c abscissa a c c o u n t i n g f o r t h e s h i f t o f t h e s w a y f o r c e a p p l i c a t i o n p o i n t d u e to r e a l i z a t i o n o f its p a r t o n t h e h u l l . I n o u e e t a l . [ 1 9 ] r e c o m m e n d XRH = XR, w h e r e XR is 1 2 0 180 2 4 0 a, deg 3 0 0 3 6 0
Fig. 3. W i n d tunnel data for tangential and n o r m a l force coefficients CT, CN as func-t i o n of func-the afunc-tfunc-tack angle.
t h e actual abscissa o f t h e r u d d e r ' s stock, a n d a H = 0.633Cij - 0.153. A c c o r d i n g to S ö d i n g [ 2 8 ] XRH = XR 0 . 3 1 OH = C R / T - 1 - 0 . 4 6 ' 1 (30) l + ( 4 . 9 e R / r + 3 b R / r ) ^
w h e r e ep is the average g a p b e t w e e n t h e r u d d e r ' s l e a d i n g edge. A c c o r d i n g t o t h e a s s u m p t i o n s ( l ) - ( 4 ) f o r m u l a t e d above, the r u d d e r n o r m a l f o r c e c a n be r e p r e s e n t e d as N = C w ( a R ) i - ^ A R o + CdaRp)kd^ARp 2 (31) w h e r e < 1 is t h e j e t d e f l e c t i o n r e d u c t i o n f a c t o r s p e c i f i e d later. The n o r m a l f o r c e c o e f f i c i e n t d e p e n d e n c y Ct~i(a) m u s t be d e f i n e d i n d i f f e r e n t r e g i m e s : • T h e p r e - s t a l l r e g i m e w h e n t h e l e a d i n g edge is m o v i n g a h e a d w i t h respect t o t h e w a t e n I f t h e s t a l l angle i n ahead m o t i o n is as, t h e n a t |o?| <as t h e response w i l l be d e f i n e d b y [ 1 ] :
CNo(a) = CDR(a;)sina-+-CiR(a;)coso;. H e r e :
CLR{a) = iC°;^ +C^^\a\)a,
CDR(a) = CDRO + CDRI C f ^ ( a ) ,
w h e r e v a r i o u s c o e f f i c i e n t s are: C f l COS A A / A 2 / C O S 4 A 4 - 4 + 2a„ t h e r u d d e r l i f t g r a d i e n t . (32) (33) (34) (35) - t h e i n f i n i t e aspect r a t i o l i f t g r a d i e n t , a n d Ooo ^ 0.9 is t h e viscosity
c o r r e c t i o n f a c t o r , y l is t h e s w e e p angle close to z e r o i n m o s t cases. T h e n ,
S. Sutulo, C. Guedes Soares / Applied Ocean Research SJ (2015)293-308 299 a n d CD 0 . 1 + 0 . 7 / ) ' f o r f a i r e d t i p , 0 . 1 + 1 . 6 5 ' f o r s q u a r e d t i p ; ( 3 7 ) b' = btjbr is t h e r u d d e r t a p e r r a t i o ; fat is t h e t i p c h o r d , a n d br is the r o o t chord.CüRo is t h e p r o f i l e d r a g c o e f f i c i e n t ( 0 . 0 0 6 5 f o r the N A C A - 0 0 1 5 p r o f i l e ) , a n d
1
CDRI = ( 3 8 )
I n a s t e r n m o t i o n i.e. w h e n t h e r u d d e r is m o v i n g w i t h i t s t r a i l -i n g edge f o r w a r d , t h e f l o w w -i l l be n o t s t a l l e d w h e n \a\ >7T-asr, w h e r e asr<oc5 is t h e s t a l l a n g l e i n t h e b a c k w a r d m o t i o n o f t h e r u d -der b l a d e . T h e n , i t can be a s s u m e d t h a t i n t h i s d o m a i n t h e n o r m a l f o r c e c o e f f i c i e n t d e p e n d e n c y Ci^r{<x) can be r e p r e s e n t e d as CNr(Q') = C N o ( 7 r - | o ; | ) s i g n a . ( 3 9 ) I n t h e i n t e r m e d i a t e i n t e r v a l , i.e. w h e n | a ; | e ( a s . Jt-Usr), t h e n o r m a l f o r c e c o e f f i c i e n t c a n be s u p p o s e d t o change, i n t h e f i r s t a p p r o x i m a t i o n , l i n e a r l y : CN(a') = ( a + / ) | a | ) s i g n Q ' , w h e r e t h e l i n e a r a p p r o x i m a t i o n c o e f f i c i e n t s are: ü s + a s r - n a = CNo{oes)-asb.
5.4. Rudder inflow velocities and attack angles
The r u d d e r a t t a c k a n g l e is d e f i n e d as
Q'R = S R - ^ R - 5 R O ,
( 4 0 )
( 4 1 )
( 4 2 ) w h e r e 5^ is t h e r u d d e r d e f l e c t i o n angle, fin is the l o c a l d r i f t angle, a n d SRO is t h e n e u t r a l ( b a l a n c i n g ) d e f l e c t i o n angle. T h e a t t a c k angle m u s t , h o w e v e r , be n o r m a l i z e d t o f a l l i n t o t h e i n t e r v a l [ - n, K]. In t h e case w h e n 2 j r > |Q'R| > ; r :
aR:=aR-271 sign ap. ( 4 3 )
Let us c o n s i d e r t h e r u d d e r g e o m e t r i c (i.e. n o t a c c o u n t i n g f o r t h e h u l l a n d p r o p e l l e r i n f l u e n c e ) v e l o c i t i e s UR a n d VR. S i m i l a r v e l o c i t i e s f o r t h e p r o p e l l e r a h e a d o f t h e r u d d e r w i l l be up a n d Vp. A l i these v e l o c i t i e s g e n e r a t e the s p e e d m a g n i t u d e s VR = \ / U R +V'^ a n d VR = • y / ü j + l ^ , a n d the k i n e m a t i c s i d e w a s h angles fip a n d PP d e f i n e d i n s u c h a w a y t h a t UR = VRCOS,ÖR, fR = - V R s i n ^ R ; Up = VpcosPp, Vp = -VpslnPp. ( 4 4 ) The n e x t step is to l i n k t h e s t a n d - a l o n e - r u d d e r v e l o c i t i e s UR a n d VR t o t h e r u d d e r s a p p a r e n t v e l o c i t i e s b e h i n d t h e h u l l URA a n d VRA c o n n e c t e d s i m i l a r l y t o VRA a n d PRA. The l o n g i t u d i n a l v e l o c i t y is m o d i f i e d b y t h e r u d d e r w a k e f r a c -t i o n W R : " M = I ' K ( 1 - W R ) = U ( 1 - W R ) , w h e r e , s i m i l a r l y to t h e p r o p e l l e r , WR = WRoe'^'^R, ( 4 5 ) ( 4 6 ) The last e q u a t i o n is t h e e m p i r i c f o r m u l a i n t r o d u c e d b y I n o u e e t a l . [ 1 9 ] w i t h K l = 4 . 0 a n d WRO = 0.4 ( t h e l a t t e r v a l u e w a s r e c o m -m e n d e d b y Kose [ 3 0 ] ) a l t h o u g h b o t h p a r a -m e t e r s c a n be a d j u s t e d . As w i t h t h e p r o p e l l e r , t h e f o r m u l a ( 4 6 ) gives reasonable e s t i m a t e f o r any possible v a l u e o f PR. T h e h u l l ' s i n f l u e n c e o n t h e t r a n s v e r s e r u d d e r v e l o c i t y is f o r m u -l a t e d b y I n o u e et a-l. [ 19] d i r e c t i y i n t e r m s o f t h e s i d e w a s h ang-le b u t t h i s b e c o m e s i n c o n v e n i e n t w h e n a p p l i e d t o a r b i t r a r y m a n o e u v r e s . H o w e v e r , i t can be r e l a t i v e l y easily r e - f o r m u l a t e d i n t e r m s o f t h e r u d d e r t r a n s v e r s e v e l o c i t y as f o l l o w s : ( 4 7 ) VRA=KV{PR)VR, w h e r e t h e f u n c t i o n Kp{) is d e f i n e d as: min(/C2, 1<3\P\) a t \P\ < pu ; { a, + b,\p\ a t \p\^[Pt, P2I 1.0 at \P\>P2, ( 4 8 ) w h e r e K2 = 0.5, K3 = 0 . 4 5 a n d a„ = 0.5 - b.p^; bv = 0.5/(/J2 - P\ )• A s s u m i n g t h a t t h e h u l l ' s i n f l u e n c e v a n i s h e s at \P\ > f , i t is r e a -sonable t o set ,01 = 1.3 a n d ,82 = f .
T h e special s i d e w a s h angle ^ R i n Eq. ( 4 7 ) is r e s t o r e d i n t h e s t a n d a r d w a y f r o m UR a n d VR = v+kxRr w h e r e k is a d j u s t a b l e a n d I n o u e et a l . [ 1 9 ] r e c o m m e n d k = 2.0 w h i c h is i n a g r e e m e n t w i t h t h e data p r e s e n t e d b y Kose [ 3 0 ] . Necessity o f i n t r o d u c i n g t h i s special s i d e w a s h a n g l e is d u e to the f a c t t h a t t h e h u l l ' s i n f l u e n c e d e p e n d s o n t h e q u a s i - v e l o c i ties y a n d r i n a m o r e c o m p l i c a t e d w a y t h a n c o u l d have b e e n o b t a i n e d p o s t u l a t i n g its d e p e n d e n c e o n t h e g e o m e t r i c s i d e w a s h a n g l e o n l y . I n t r o d u c t i o n o f separate s t r a i g h t e n i n g f a c t o r s f o r t h e s w a y a n d y a w p a r t s o f the s i d e w a s h c o u l d b e a n a l t e r n a t i v e .
5.5. Influence of propeller slipstream
T h e a p p r o x i m a t e m a t h e m a t i c a l m o d e l f o r t h e p r o p e l l e r i n f l u -ence d e s c r i b e d b e l o w is based o n t h e classic a c t u a t o r disc t h e o r y . A c c o r d i n g t o t h i s t h e o r y , a n y p r o p e l l e r is r e p r e s e n t e d as a t h i n disc o f t h e area AQ p l a c e d i n t o t h e u n i f o r m flow w i t h t h e v e l o c i t y VA d i r e c t e d a l o n g the n o r m a l to the disc. The disc is s o m e h o w g e n e r -a t i n g -a u n i f o r m j e t w h o s e -axis is c o l l i n e -a r w i t h v^. I n t h e p r e s e n t a p p l i c a t i o n , i t c a n be a s s u m e d t h a t \VA\ = \UPA\ a n d t h e n , as e s t a b -l i s h e d i n t h e a c t u a t o r disc t h e o r y [ 3 2 ] , t h e a x i a -l i n d u c e d v e -l o c i t y at i n f i n i t y is Wo, = " P / I ( V ' 1 + C T / I - 1 ) w h e r e t h e l o a d i n g c o e f f i c i e n t is 2\T\ CTA = PUpA^O ( 4 9 ) ( 5 0 ) It f o l l o w s f r o m ( 4 9 ) t h a t t h e t o t a l j e t v e l o c i t y at i n f i n i t y b e h i n d t h e disc is
llco = UpA + Wfloo = UpA \ / 1 + C M ( 5 1 )
T h e l a t t e r f o r m u l a c a n be r e - w r i t t e n i n t h e f o l l o w i n g e q u i v a l e n t f o r m w h i c h c a n be u s e d also i n the b o l l a r d r e g i m e : llco = w h e r e •pA • aOoo 2 i r i ( 5 2 ) ( 5 3 ) " " " O - = pAo is t h e s q u a r e d i n f i n i t y a x i a l i n d u c e d v e l o c i t y i n b o l l a r d r e g i m e . T h e a x i a l i n d u c e d v e l o c i t y is v a r y i n g a l o n g t h e j e t t h e o r e t i c a l l y r e a c h i n g its u l t i m a t e v a l u e at i n f i n i t y b e h i n d t h e disc. Its v a l u e i n the disc is Wao = ( l / 2 ) w a « ) . The s e c t i o n a l area o f t h e j e t is v a r y i n g i n v e r s e l y t o s a t i s f y t h e c o n t i n u i t y e q u a t i o n . O f c o u r s e , i n t h e r e a l fluid t h e j e t a f t e r c e r t a i n r e g i o n o f c o n t r a c t i o n a n d a c c e l e r a t i o n w i l l s t a r t t o d i s s i p a t e i n v o l v i n g a d d i t i o n a l fluid b u t l o o s i n g its average v e l o c i t y . H o w e v e r , t h e p e r f e c t fluid m o d e l s t i l l w o r k s a d e q u a t e l y at all d i s t a n c e s w h e r e t h e r u d d e r can be l o c a t e d .
300 S. Sutulo, C. Guedes Soares / Applied Ocean Research 51 (2015)293-308 The l o n g i t u d i n a l v e l o c i t y o f the p a r t o f t h e r u d d e r i n s i d e t h e s l i p s t r e a m w i t h respect t o w a t e r is Upp = UpA
+
Wa ( 5 4 ) w h e r e t h e a x i a l i n d u c e d v e l o c i t y d e p e n d s o n t h e d i s t a n c e b e t w e e n t h e p r o p e l l e r a n d t h e r u d d e r a n d can be r e p r e s e n t e d as Wa(x) = T^Kkw{>!.)Wa ( 5 5 ) w h e r e K is t h e e m p i r i c c o r r e c t i o n f a c t o r i n t r o d u c e d b y Kose [ 3 0 ] w i t h t h e r e c o m m e n d e d v a l u e 0.68, a n d t h e d i s t a n c e f a c t o r 1 + K{T) s i g n T ( 5 6 ) w h e r e t h e r e l a t i v e s i g n e d d i s t a n c e f r o m t h e p r o p e l l e r to t h e r u d d e r 2{xp-xp)^ X a n d KiT) Dp '-signT 1 at x T > 0, 0.7 at x T < 0 ( 5 7 ) ( 5 8 ) A t t e n t i o n m u s t be p a i d t h a t the t h u s d e f i n e d f a c t o r kw is v a l i d f o r a n y s i g n o f t h e t h r u s t a l t h o u g h t h e p r o p u l s o r w o r k s a p p r o x i m a t e l y as a disc o f sinks f r o m t h e side t o w h i c h t h e t h r u s t is d i r e c t e d w h i l e i t p r o d u c e s a j e t i n the o p p o s i t e d i r e c t i o n . The c o e f f i c i e n t K is e m p i r i c . The t r a n s v e r s e c o m p o n e n t o f t h e r u d d e r - i n - t h e - s l i p s t r e a m v e l o c i t y is u s u a l l y a s s u m e d t o be t h e same as o u t s i d e t h e s l i p -s t r e a m i.e. vpp = vpA. T h i -s d e f i n i t e l y can be a p p l i e d o n t h e -s u c t i o n side o f t h e a c t u a t o r disc. H o w e v e r , e v e n w i t h s t r o n g s i d e w a s h , i n t h e v i c i n i t y o f t h e p r o p e l l e r disc the j e t k e e p s its d i r e c t i o n a l o n g t h e s h a f t axis as the t r a n s v e r s e m o m e n t u m is s t i l l n o t t r a n s m i t t e d t o i t . Because o f t h i s , at x T > 0 i t is m o r e r e a s o n a b l e t o a s s u m e t h a tVRP = ( x ^ / a + x^)vpA> w h e r e a is an e m p i r i c c o n s t a n t .
Due t o t h e j e t ' s c o n t r a c t i o n , t h e s l i p s t r e a m - w a s h e d p a r t o f t h e r u d d e r area ARP d e p e n d s o n the d i s t a n c e x. I n g e n e r a l , i t also d e p e n d s o n t h e r e l a t i o n b e t w e e n t h e r u d d e r h e i g h t a n d p o s i t i o n a n d o n h o w i t is p o s i t i o n e d w i t h respect t o t h e p r o p e l l e r axis. A t t h e s a m e t i m e , t h e e f f e c t i v e r a d i u s o f t h e j e t YRP can be e s t i m a t e d as rRp - " o / U j ! P • ( 5 9 ) As has a l r e a d y b e e n m e n t i o n e d , t h e d e f l e c t e d r u d d e r w i l l also d e f l e c t t h e j e t r e d u c i n g its o w n a t t a c k angle. A c c o r d i n g to S ö d i n g [ 2 8 ] t h e c o r r e s p o n d i n g f a c t o r ki f r o m Eq. ( 3 1 ) can be e s t i m a t e d as '<d = \URA/URP'{, w h e r e / = 2 [ 2 / ( 2 + d / b R ) ] ^ a n d d = ( V ^ / 2 ) r R p . 6. T h r u s t e r f o r c e s
M a n y vessels are n o w a d a y s e q u i p p e d w i t h side t u n n e l t h r u s t e r s t o i m p r o v e t h e i r l o w s p e e d c o n t r o l l a b i l i t y . These t h r u s t e r s p r a c t i -c a l l y a l w a y s w o r k i n t h e b o l l a r d r e g i m e w h i -c h m a k e s t h e t h r u s t p e r se a l m o s t i n d e p e n d e n t o f t h e ship's m o t i o n b u t t h e i n t e r a c t i o n w i t h t h e h u l l is r a t h e r c o m p l i c a t e d a n d a f f e c t s s i g n i f i c a n t l y t h e e f f e c t i v e s w a y f o r c e a n d y a w m o m e n t r e s u l t e d f r o m t h e t h r u s t e r s ' a c t i o n . The a c t u a l t h r u s t T p r o d u c e d b y a l a t e r a l j e t t h r u s t e r w i t h o u t a c c o u n t f o r t h e h u l l i n f l u e n c e can be r e p r e s e n t e d as r = kpTo, w h e r e fcj- s [ - 1 , 1 ] is t h e c o n t r o l p a r a m e t e r or t h e r e l a t i v e t h r u s t ( i t s v a l u e - 1 c o r r e s p o n d s to t h e m a x i m u m t h r u s t to t h e p o r t s i d e , a n d +1—to t h e s t a r b o a r d ) , a n d TQ is t h e m a x i m u m a b s o l u t e t h r u s t w h i c h is t h e m a i n c h a r a c t e r i s t i c o f t h e g i v e n t h r u s t e r a n d is r e l a t e d t o t h e t h r u s t e r d r i v e ' s p o w e r PQ as [ 2 4 ] :
w h e r e t h e c o n s t a n t CQ = 0.15 s / m a c c o u n t s also f o r the h u l l pressure r e - d i s t r i b u t i o n .
T h e t h u s d e t e r m i n e d t h r u s t T i s e f f e c t i v e o n a d e e p l y s u b m e r g e d t h r u s t e r at zero surge ( a d v a n c e ) speed o f t h e s h i p . In a m o r e g e n e r a l case, t h e s w a y f o r c e a n d t h e y a w m o m e n t f r o m a n y s i n g l e t h r u s t e r s h o u l d be c a l c u l a t e d as f o l l o w s :
Yp = k „ ( / i R ) / i y ( Ü ) T , N r = k h ( h R ) / < N ( i i ) r x T , ( 6 1 ) w h e r e k], is t h e s u b m e r g e n c e c o r r e c t i o n f a c t o r , hp^hlR is t h e r e l -a t i v e s u b m e r g e n c e o f t h e t h r u s t e r , h is t h e s u b m e r g e n c e o f its axis, R is t h e t h r u s t e r ' s r a d i u s , ky, /<N are t h e l o n g i t u d i n a l v e l o c i t y c o r r e c t i o n factors, 0 = \u\/Wj is t h e a b s o l u t e v a l u e o f t h e r e l a t i v e l o n g i t u d i n a l v e l o c i t y o f t h e ship, Wj is t h e e f f e c t i v e j e t v e l o c i t y , xp is t h e t h r u s t e r ' s abscissa.
D a t a o n t h e s u b m e r g e n c e c o r r e c t i o n f a c t o r a c c o u n t i n g f o r the loss o f t h e t h r u s t e r ' s e f f e c t i v e area a n d f o r t h e surface w a v e m a k i n g can be f o u n d i n [ 3 3 ] a n d i t can be a p p r o x i m a t e d as
k,, = m i n ( 1 . 0 , Qh + b^hR), ( 6 2 )
To = CQPO ( 6 0 )
w h e r e a/, = 1/3, Ö/, = 1 7 / 3 0 . T h i s f a c t o r does n o t a c c o u n t f o r aera-t i o n w h i c h can r e s u l aera-t i n m o r e d r a s aera-t i c f a l l o f aera-t h e aera-t h r u s aera-t b u aera-t w h i c h is u s u a l l y a v o i d e d i n n o r m a l o p e r a t i o n . As t o t h e v e l o c i t y c o r r e c t i o n f a c t o r s , t h e y also d e p e n d o n the t h r u s t e r ' s l o c a t i o n : near t h e s t e r n o r n e a r t h e b o w . E x p e r i m e n t a l d a t a o n ky f o r o n l y t h e b o w t h r u s t e r at a h e a d s h i p s p e e d b e l o n g -i n g t o C h -i s l e t t a n d B j 0 r h e d e n are p r e s e n t e d b y F a l t -i n s e n [ 3 3 ] . Br-ix [ 2 4 ] p r o v i d e s data f o r b o t h b o w a n d s t e r n t h r u s t e r s i n ahead and a s t e r n speed b u t as f u n c t i o n s o f t h e d i m e n s i o n a l v e l o c i t y u . The v a l u e o f t h e j e t v e l o c i t y Wj at those tests is n o t g i v e n e x p l i c i t l y b u t a r e m a r k a b o u t t h e p r o p e r t i e s o f t h e c o r r e c t i o n f a c t o r s m a d e pos-s i b l e a p p r o x i m a t e r e pos-s t o r a t i o n o f Wj. In g e n e r a l , t h e j e t v e l o c i t y can be f o u n d as ( 6 3 ) w h e r e AQ is t h e t h r u s t e r ' s s e c t i o n a l area. T h e n , B r i x ' d a t a w e r e a p p r o x i m a t e d w i t h s e c o n d or t h i r d -d e g r e e algebraic p o l y n o m i a l s a + bu + cü^ + -dü^ s e p a r a t e l y f o r e a c h case a n d s a f e l y c o n t i n u e d o u t s i d e t h e t e s t e d r e g i o n w i t h c o n -s t a n t value-s a l t h o u g h t y p i c a l l y e x p e r i m e n t a l data -s h o w ( f u r t h e r ) r e c u p e r a t i o n o f t h e t h r u s t . Values o f t h e a p p r o x i m a t i o n c o e f f i c i e n t s are g i v e n i n Tables 1 a n d 2. I n a d d i t i o n , t h e C h i s l e t t a n d B j 0 r h e d e n data c l e a r i y s h o w a small i n t e r v a l o f c o n s t a n t values e q u a l t o u n i t y f o r ü < 0 . 1 2 . T h i s is also j u s t i f i e d b y t h e process' p h y s i c s as a t s m a l l r e l a t i v e l o n g i t u d i -n a l speed -n o j e t a t t a c h m e -n t or p r e s s u r e r e - d i s t r i b u t i o -n ca-n occur. U n f o r t u n a t e l y , t h e B r i x data w e r e o b t a i n e d w i t h a t o o large step a n d c o u l d n o t s h o w t h i s e f f e c t . So, i t w a s d e c i d e d t o a d d t h e p o i n t ( 0 . 1 2 , 1.0) t o a l l B r i x data. The r e s u l t i n g a p p r o x i m a t e d responses t o g e t h e r w i t h t h e e x p e r i m e n t a l d a t a b o r r o w e d f r o m B r i x [ 2 4 ] and, i n o n e case, F a l t i n s e n [ 3 3 ] are s h o w n i n Figs. 4 a n d 5 w h e r e the a b s o l u t e v a l u e o f t h e r e l a t i v e l o n g i t u d i n a l v e l o c i t y is s h o w n o n the h o r i z o n t a l axes. 7. A e r o d y n a m i c f o r c e s T h e a e r o d y n a m i c f o r c e s can b e r e p r e s e n t e d as XA = CX{I3A,^C,I]C)^/^L, y , = C . ( / ^ „ f c , . c ) ^ A „ (64) NA = CdpA,k^ilc)^AdoA,
S. Sutulo, C. Guedes Soares/ Applied Ocean Research 51 (2015) 293-308 301 Table 1
A p p r o x i m a t i o n coefficients for ky(ü).
Thruster u a b c d Bow >0 1.236133984568 - 2 . 4 6 6 3 8 1 9 9 7 5 6 5 1.621751193502 0.0 <0 1.110973606193 - 1 . 2 6 1 2 0 0 4 6 7 6 8 1 0.9375221889657 0.0 Stern >0 1.319106928965 - 3 . 3 0 2 0 0 7 1 0 9 0 0 9 3.075565631253 0.0 <0 1.289253261375 - 2 . 5 4 0 6 3 4 4 1 3 5 0 9 1.610802566528 0.0 Table 2
A p p r o x i m a t i o n coefficients for /(«(fi).
Thruster 11 a b c d
Bow >0 1.334253478 - 3 . 5 3 4 6 4 0 5 3 7 6.647727158 - 3 . 6 6 9 9 8 6 7 2
<0 1.121478745 - 1 . 6 1 0 9 8 1 2 1 6 0.6918614180 0.0
Stern >0 1.172907048 - 1 . 9 6 5 4 1 0 9 7 8 2.054377528 0.0
<0 1.135322201 - 1 . 3 4 9 2 9 2 1 4 2 1.529502839 0.0
w h e r e CX,Y,N are t h e f o r c e / m o m e n t aeroidynamic c o e f f i c i e n t s , is t h e air d e n s i t y , Ai is t h e ship's l a t e r a l area, LQA is t h e ship's l e n g t h o v e r a l l .
The a e r o d y n a m i c c o e f f i c i e n t s are s h o w n here as d e p e n d e n t o n t h e ship's p o s i t i o n . This m a k e s sense w h e n a n o n - h o m o g e n o u s w i n d field, as f o r instance, b e h i n d s o m e obstacle, is c o n s i d e r e d . A t present, data o n a e r o d y n a m i c c h a r a c t e r i s t i c s o f the s h i p h u l l s i n n o n - h o m o g e n o u s w i n d are p r a c t i c a l l y absent a n d t h e n t h e c o e f f i c i e n t s w i l l o n l y d e p e n d o n t h e air d r i f t angle. I n d e p e n d e n c e o f t h e a e r o d y n a m i c f o r c e s o n t h e a n g u l a r v e l o c -i t y o f y a w -is c o m m o n l y a c k n o w l e d g e d b u t -i t -is n o t so e v -i d e n t a n d deserves e x p l a n a t i o n . A s s u m i n g , h o w e v e r , i n t h e case o f a h o m o g e -n o u s w i -n d , t h a t t h i s d e p e -n d e -n c e exists a -n d t a k i -n g i -n t o a c c o u -n t Eqs. ( 5 ) , ( 8 ) a n d ( 9 ) i t is p o s s i b l e e s t a b l i s h a c o m p l e t e l i s t o f a r g u -m e n t s f o r a n y a e r o d y n a -m i c c o e f f i c i e n t a n d to e x p a n d i t i n t o a m u l t i v a r i a t e Fourier series ( h e r e : w i t h r e s p e c t t o t h e zero p o i n t , u p t o t h e firstorder t e r m s a n d f o r o n l y s w a y f o r c e f o r s i m p l i c -i t y ) :
