V o l . 6 4 15 M a y 2013 I S S N 0029-8018
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Speed loss of a vessel sailing in oblique waves
Zhenju Chuang* Sverre Steen
Nomegian University of Science and Teclmology, Department of Marine Tecltnotogy, NO-7491 Trondheim, Norway
A R T I C L E I N F O
Article history:
Received 13 December 2012 Accepted 24 February 2013 Available online 25 March 2013
Keywords:
Speed loss Oblique waves Added wave resistance
A B S T R A C T
In o r d e r to d e m o n s t r a t e the effect of oblique w a v e s on o c e a n - g o i n g v e s s e l b e h a v i o r in realistic sea states, this p a p e r a d d r e s s e s an effective and useful tool to p r e d i c t t h e ship's m o t i o n and p r o p u l s i o n s y s t e m b e h a v i o r w i t h s u f f i c i e n t a c c u r a c y , c o n s i d e r i n g the w a v e c o n d i t i o n s on the o p e r a t i n g route. S e a k e e p i n g m o d e l t e s t w a s c a r r i e d out w i t h a freely r u n n i n g m o d e l in o b l i q u e w a v e s i n the o c e a n b a s i n at the M a r i n e T e c h n o l o g y Centre, T r o n d h e i m , N o r w a y . A t i m e d o m a i n n u m e r i c a l s i m u l a t i o n s t u d y w a s p e r f o r m e d , a n d t h e r e s u l t s of the s i m u l a t i o n are c o m p a r e d w i t h the m o d e l test data.
D u e to the l e n g t h l i m i t a t i o n of the ocean basin, c o n v e r g e d speed in w a v e s c a n n o t be a c h i e v e d in all runs. A c o r r e c t i o n m e t h o d is proposed in this p a p e r to d e t e r m i n e the converged s p e e d f r o m n o n -c o n v e r g e d r u n s . T h e -c o r r e -c t i o n m e t h o d is b a s e d on the -c o n d i t i o n that the -converged s p e e d i n w a v e s is d e p e n d e n t on the b a l a n c e b e t w e e n resistance a n d p r o p u l s i o n force.
© 2 0 1 3 E l s e v i e r Ltd. A l l rights r e s e r v e d .
1. Introduction
if The s h i p b e h a v i o r i n a c t u a l w^eather c o n d i t i o n is c u r r e n t l y one o f t h e m a j o r c o n c e r n s f o r designers a n d ship o w n e r s as w e l l as f o r s h i p o f f i c e r s ( P r p i c - O r s i c aniJ Faltinsen, 2 0 1 2 ) . Speed loss o f a n o c e a n - g o i n g vessel can be c a t e g o r i z e d as v o l u n t a r y o r i n v o l u n t a r y ( F a l t i n s e n e t al., 1 9 8 0 ) . The f o r m e r is t o a v o i d s l a m m i n g , p r o p e l l e r r a c i n g a n d excessive s h i p m o t i o n . The l a t t e r is d u e t o a d d e d resistance f r o m w a v e s , w i n d , a n d c u r r e n t , as w e l l as r e d u c t i o n o f p r o p u l s i v e e f f i c i e n c y caused b y waves a n d increased resistance. This p a p e r f o c u s e s o n t h e i n v o l u n t a r y speed r e d u c t i o n . A r e l i a b l e speed loss p r e d i c t i o n r e q u i r e s i n t e g r a t e d k n o w l e d g e a b o u t r e s i s -tance, p r o p u l s i o n , s h i p m a c h i n e r y , seakeeping, s t e e r i n g a n d a u t o m a t i c c o n t r o l . I n o r d e r t o get a b e t t e r u n d e r s t a n d i n g a n d i n s i g h t i n t o t h e n a t u r e o f t h e ship speed d r o p process, t h e research r e p o r t e d i n t h i s p a p e r is based o n m o d e l tests a n d n u m e r i c a l s i m u l a t i o n s .
I n r e c e n t years, t h e r e has b e e n some research o n speed loss, f u e l c o n s u m p t i o n a n d GHG emissions. Prpic-Orsic a n d F a l t i n s e n ( 2 0 1 2 ) d e v e l o p e d a n u m e r i c a l m o d e l to p r e d i c t t h e speed loss o f a vessel i n i r r e g u l a r sea a n d i m p l e m e n t e d t h e v e n t i l a t i n g p r o p e l l e r m o d e l b y S m o g e l i ( 2 0 0 6 ) . T h e i r w o r k also c o v e r e d t h e e s t i m a t i o n o f CO2 e m i s s i o n f r o m a c o n t a i n e r s h i p o n the N o r t h e r n N o r t h A t l a n t i c r o u t e , w h i c h w a s based o n t h e m e a n speed d r o p , c o n s t a n t e n g i n e p o w e r a n d f u e l c o n s u m p t i o n . B u t t h e i r m o d e l lacks v e r i f i c a t i o n . C h u a n g a n d Steen ( 2 0 1 2 ) used m o d e l tests a n d a t i m e d o m a i n n u m e r i c a l m o d a l to s t u d y t h e speed loss d u e t o
*Tel.: + 4 7 96807451; fax; + 4 7 73595528.
E-mail address: chuang.zhenju@ntnu.no (Z. Chuang).
0029-8018/$-see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi,org/10.1016/j.oceaneng.2013.02.018 zig-zag m o t i o n i n w a v e s . T h e y p o i n t e d o u t t h a t t h e reasons f o r speed r e d u c t i o n f o r a s h i p d o i n g z i g - z a g m a n e u v e r i n g i n w a v e s are d u e t o y a w i n g , loss o f t h r u s t d u e t o s t e e r i n g a n d a d d e d w a v e resistance. H o w e v e r , m o s t o f t h i s w o r k w a s f o c u s e d o n h e a d sea c o n d i t i o n s .
Based o n the l i t e r a t u r e s t u d y , research o n the speed loss o f a vessel s a i l i n g i n o b l i q u e w a v e s is l i m i t e d . Carrica et a l . ( 2 0 0 8 ) a p p l i e d URANS analysis t o s t u d y t h e speed v a r i a t i o n d u r i n g b r o a c h i n g e v e n t o f a s h i p i n i r r e g u l a r q u a r t e r i n g seas. B u t t h e y i m p l e m e n t e d a s i g n i f i c a n t l y s i m p l i f i e d p r o p e l l e r m o d e l . A m o r e advanced p r o p e l l e r m o d e l is n e e d e d t o achieve m o r e a c c u r a t e results. H o w e v e r , several researchers have m a d e g r e a t c o n t r i b u -t i o n s o n h o w -t o p r e d i c -t a d d e d w a v e resis-tance i n o b l i q u e w a v e s . F u j i i a n d Takahashi ( 1 9 7 5 ) p r o v i d e d a s e m i - e m p i r i c a l f o r m u l a to calculate t h e w a v e resistance im r e g u l a r o b l i q u e w a v e s . The f o r m u l a gives a g o o d c o n s i d e r a t i o n o f a d d e d w a v e resistance c o m p o n e n t s b o t h d u e t o s h i p m o t i o n s a n d due to b o w r e f l e c t i o n , w h i c h is based o n t h e c o n t r i b u t i o n f r o m M a r u o ( 1 9 6 3 ) a n d H a v e l o c k ( 1 9 4 0 ) . 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 ) e v a l u a t e d t h e added resistance i n l o n g i t u d i n a l w a v e s b y e q u a t i n g t h e w o r k o f added resistance t o t h e e n e r g y c o n t a i n e d i n the w a v e s r a d i a t e d a w a y f r o m t h e ship. T h e i r w o r k has l a t e r been tested b y J o u r n e e ( 1 9 7 6 ) i n f o l l o w i n g w a v e s . His r e s u l t s s h o w s t h a t t h e m e t h o d o f G e r r i t s m a a n d B e u k e l m a n gives a c o n s i d e r a b l e n e g a t i v e a d d e d resistance o b t a i n e d i n d i f f e r e n t w a v e a n d speed ranges w h i c h gives a significant d e v i a t i o n f r o m experiments. Therefore, G e r r i t s m a and Beukelman m e t h o d needs m o d i f i c a t i o n i n o r d e r t o be a p p l i e d o n other t h a n head sea.
The p u r p o s e o f t h i s p a p e r is t o p u t e m p h a s i s o n t h e speed loss o f an o c e a n - g o i n g vessel i n h e a d sea, b o w sea, b e a m sea a n d q u a r t e r i n g sea c o n d i d o n s . C a l m w a t e r resistance, a z i m u t h
Z, Chuang, S. Steen / Ocean Engineering 64 (2013) 88-99 89
p r o p u l s i o n system, ship m a c h i n e i y , seakeeping, s t e e r i n g a n d a u t o m a t i c c o n t r o l are a l l i n c l u d e d b o t h i n m o d e l t e s t a n d t i m e d o m a i n n u m e r i c a l s i m u l a t i o n s . D u e t o t h e l i m i t a t i o n o f t h e e x p e r i m e n t a l e n v i r o n m e n t , c o n v e r g e d speed i n w a v e s c a n n o t be achieved i n a l l r u n s . A c o r r e c t i o n m e t h o d is also p r o p o s e d i n t h i s paper t o f i n d c o n v e r g e d speed f r o m n o n - c o n v e r g e d m o d e l tests.
2. Model test
2 . 1 . Model test set up
M o d e l tests t o o b t a i n speed loss i n o b l i q u e w a v e s have been c a r r i e d o u t i n t h e Ocean basin l a b o r a t o r y a t t h e M a r i n e T e c h n o l -ogy Center i n T r o n d h e i m , N o r w a y . The ocean b a s i n l a b o r a t o r y has a n e f f e c t i v e l e n g t h o f 65 m , w i d t h o f 50 m a n d d u r i n g t h e tests t h e m o v a b l e b o t t o m w a s set t o a d e p t h o f 4 m . A d o u b l e - f l a p w a v e m a k e r ( B M 2 ) a l o n g t h e 5 0 - m e t e r side is g e n e r a t i n g l o n g crested waves. A l o n g t h e 6 5 - m e ï è r side, t h e r e is a m u l t i - f l a p w a v e m a k e r ( B M 3 ) c o n s i s t i n g o f 1 4 4 i n d i v i d u a l l y c o n t r o l l e d f l a p s . W a v e a b s o r p t i o n beaches are i n s t a l l e d o n t h e t w o o p p o s i t e sides t o reduce t h e p r o b l e m s o f w a v e r e f l e c t i o n . A carriage is a v a i l a b l e t o f o l l o w m o d e l s w i t h f o r w a r d speed. The carriage carrie? p o w e r a n d signal cables t o t h e m o d e l , a n d c a n be used t o assist t h e m o d e l d u r i n g a c c e l e r a t i o n a n d d e c e l e r a t i o n phases. The m o d e l a p p l i e d d u r i n g t h e tests w a s a 1:16.57 scale m o d e l o f a n 8000DVVT t a n k e r d e v e l o p e d b y Rolls-Royce M a r i n e , Ship T e c h n o l o g y — M e r c h a n t . The m o d e l is b u i l t b y MARINTEK. T h e ship m o d e l has a w i d e t r a n s o m a n d c o n v e n t i o n a l b o w w i t h b u l b . The m a i n d i m e n s i o n s o f the ship m o d e l a n d b o d y p l a n are l i s t e d i n Table 1 a n d Fig. 1. The m o d e l is s e l f - p r o p e l l e d w i t h t w o m o d e l s o f A Z P 1 2 0 a z i m u t h t h r u s t e r p r o p e l l e r s . The m a i n p a r t i c u l a r s o f t h e p r o p e l l e r m o d e l s are l i s t e d i n Table 2 . I n o r d e r t o c o m p e n s a t e f o r t h e r e l a t i v e l y h i g h e r f r i c t i o n a l resistance o f t h e m o d e l , t o w r o p e f o r c e is a p p l i e d b y a n a i r f a n m o u n t e d o n t h e m o d e l . T h e t o w r o p e f o r c e is o b t a i n e d b y c o n t r o l l i n g t h e speed o f t h e f a n . The f a n was m o u n t e d o n a l o a d cell so t h a t t h e t h r u s t o f t h e f a n c o u l d be m e a s u r e d . H o w e v e r , d u e t o t h r u s t loss o f t h e f a n , t h e e f f e c t i v e f a n f o r c e is o n l y 0.67 t i m e s t h e f a n f o r c e m e a s u r e d d i r e c t l y . This has b e e n v e r i f i e d b y
Table 1
Principal dimensions of model hull.
UNIT Model toA Im] 7.142 Lpp | m | 6.832 D | m l 0.905 B | m l 1.147 T Im] 0.435 -8 1 I I I I I I I 1 I I I I I I I 1 I I I I I I I I I I I I I I , I I I I I I I I 1 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
Fig. 1. Body plan of the model (below-water part only).
Table 2
Principal dimensions of model propellers.
UNIT Model
Propeller diameter D [m] 0.199
Pitch ratio a t P / D = 0 . 7 [ . . . . ] 1.2
Blade area ratio 1 •••1 0.435
Number of blades [ . . . . ] 4
Table 3
Calibrated waves in model scale.
Wave No, Wave maker Wave type H(m) T(s) -'JLpp
1 BM2 REG 0,121 1.351 0,42 2 BM2 REG 0.121 1.842 0,78 3 BM2 REG 0.121 2.088 1 4 * BM2 REG 0.121 2.383 1,3 5 BM2 REG 0.061 2.383 1,3 6 BM2 REG 0,242 2,383 1,3 7 BM2 REG 0,121 2,579 1,52 8 BM2 REG 0,121 3,071 2,15 9 BM3 REG 0,121 1,351 0,42 10 BM3 REG 0,121 1,842 0,78 11 BM3 REG 0,121 2,088 1 12 BM3 REG 0.121 2.383 1,3 13 BM3 REG 0.121 2,579 1,52 Table 4
Test conditions for all the runs.
' Vm (m/s) Direction (deg) Wave no.
1.769 30 1,2,3,4,5,6,7,8 60 1,2,3,4,5,7,8 90 9,10,11,13 150 1,2,3,4,7,8 1.191 30 23,4,6,7 60 2,3,4,7 150 2,4 a d d i t i o n a l tests w h i c h c o m p a r e t h e f o r c e m e a s u r e d b y t h e l o a d c e l l a n d t h e n e t f o r c e a c c e l e r a t i n g t h e m o d e l . A c o n t r o l s y s t e m k e p t t h e p r o p u l s i o n p o w e r c o n s t a n t d u r i n g t h e tests, e q u a l t o t h e p o w e r r e q u i r e d t o r e a c h d e s i g n speed i n c a l m w a t e r . So t h e speed r e d u c t i o n s r e c o r d e d are o n l y d u e t o a d d e d w a v e resistance, s t e e r i n g , a n d r e d u c t i o n o f p r o p u l s i v e e f f i c i e n c y . A n a u t o p i l o t h e a d i n g c o n t r o l l e r is used o n t h e vessel i n o r d e r t o keep t h e d e s i g n e d h e a d i n g d u r i n g t h e tests. T h e m o d e l is f r e e a n d self-p r o self-p e l l e d d u r i n g tests. The o n l y c o n n e c t i o n b e t w e e n t h e m o d e l a n d carriage are cables t o relay t h e m e a s u r e m e n t s a n d p r o v i d e p o w e r t o t h e p r o p u l s i o n m o t o r s . The cables are h a n d l e d so t h e forces f r o m t h e cables o n t h e m o d e l are s m a l l . A n o p t i c a l t r a c k i n g s y s t e m is used t o t h e i n s t a n t a n e o u s p o s i t i o n o f t h e m o d e l . V e l o c i t i e s are d e r i v e d f r o m t h e m e a s u r e d p o s i t i o n s .
2.2. Test conditions and procedures
The a i m o f t h e tests is t o m e a s u r e t h e speed loss i n o b l i q u e w a v e s . Since a d d e d resistance is p r o p o r t i o n a l t o t t i e s q u a r e o f t h e w a v e a m p l i t u d e , t h e q u a l i t y o f t h e w a v e s are r e a l l y i m p o r t a n t f o r t h e results. A l l t h e w a v e s are c a r e f u l l y c a l i b r a t e d p r i o r t o t h e t e s t i n g , w i t h o u t t h e s h i p m o d e l i n t h e ocean b a s i n . T h i r t e e n w a v e s c o m b i n e d w i t h f o u r headings a n d t w o v e s s e l speeds are t e s t e d . The w a v e s used are l i s t e d i n Table 3 a n d t h e c o m b i n a t i o n o f speed, h e a d i n g a n d w a v e c o n d i t i o n s are s p e c i f i e d i n T a b l e 4 .
H e a d i n g angle is d e f i n e d as the angle b e t w e e n w a v e p r o p a g a -t i o n d i r e c -t i o n s a n d -t h e x - a x i s w h i c h is i n a b o d y fixed c o o r d i n a -t e s y s t e m p o i n t i n g f r o m f o r e p e r p e n d i c u l a r t o w a r d s a f t p e r p e n d i -cular. The scenarios o f t h e h e a d i n g angles i n t h e ocean basin are s h o w n i n Fig. 2. F o u r angles are tested, t h e y are 3 0 deg, 60 deg, 90 deg a n d 150 deg.
The m o t o r p o w e r is a d j u s t e d so t h a t a speed o f V,„ = 1.769 m/s ( F n = 0 . 2 2 ) is reached o n a s t r a i g h t l i n e course i n c a l m w a t e r . A l l s u b s e q u e n t tests m a r k e d w i t h this speed i n T a b l e 4 are p e r f o r m e d a t t h i s p o w e r s e t t i n g . The same p r o c e d u r e is used f o r t h e o t h e r speed, w h i c h is \/m = 1.191 m/s (Fn = 0.15). A c o n t r o l l e r is c o n t r o l l i n g t h e f a n used t o produce t h e t o w rope f o r c e , so t h a t t h e f a n gives t h e s p e c i f i e d f o r c e a c c o r d i n g t o t h e speed o b t a i n e d i n t h e a c t u a l c o n d i t i o n . H o w e v e r , d u e to a t h r u s t d e d u c t i o n e f f e c t
o f t h e f a n t h a t w a s discovered a f t e r t h e tests, the e f f e c t i v e t o w rope f o r c e a p p l i e d is o n l y 67% o f t h e c o r r e c t v a l u e . A h e a d i n g c o n t r o l l e r is used i n o r d e r t o keep t h e m o d e l r u n n i n g at t h e designed h e a d i n g i n t h e ocean basin. O u t p u t s are c o m m a n d e d r u d d e r angles t o t h e t w i n a z i m u t h p r o p u l s o r s .
The test p r o c e d u r e is as f o l l o w s : First, t h e m o d e l is t o w e d t o t h e c o r n e r o f t h e basin a n d set u p w i t h t h e c o r r e c t h e a d i n g d i r e c t i o n as s h o w n i n Fig. 2. T h e n , t h e p r o p e l l e r s are s t a r t e d i n i t i a l l y w i t h a h i g h p o w e r level t o accelerate t h e m o d e l . W h e n a p r e - d e f i n e d speed is reached, t h e p o w e r is r e d u c e d a u t o m a t i c a l l y t o t h e c o r r e c t level. W h e n t h e m o d e l is g e t t i n g close t o t h e e n d o f t h e basin, t h e p r o p e l l e r s are s t o p p e d a n d t h e m o d e l is s t o p p e d b y t i g h t e n i n g a rope f r o m t h e carriage, c o n n e c t e d t o the s t e r n o f t h e m o d e l . The v a l u e o f t h e p r e - d e f i n e d speed t h r e s h o l d is i m p o r t a n t
a
BM2
BM2
CO CDBM2
BM2
CO CDY \!/ \!/
Wave direction
Fig. 2. Scenarios of ttie model test setting for eacti vessel-wave direction relationship, (a) head sea (30 deg), (b) head sea (60 deg), (c) beam sea (90 deg) and (d) quartering sea (150 deg).
Z. Chuang, S. Steen / Ocean Engineering 64 (2013) 88-99 91
f o r t h e a b i h t y t o reach a c o n v e r g e d speed. Some t r i a l a n d e r r o r w a s used t o f i n d t h e r i g h t t h r e s h o l d speed f o r d i f f e r e n t t e s t c o n d i t i o n s . S t i l l , c o n v e r g e d speed w a s n o t r e a c h e d i n a l l cases. The reason f o r t h e r e l a t i v e l y s l o w speed convergence is t h e l o w r e s i s t a n c e / w e i g h t - r a t i o o f s u c h a d i s p l a c e m e n t s h i p m o d e l , so t h a t t h e r e s i d u a l t h r u s t - r e s i s t a n c e i m b a l a n c e r e s u l t s i n r e l a t i v e l y l o w a c c e l e r a t i o n / d e c e l e r a t i o n .
D u r i n g t h e tests t h e f o l l o w i n g variables are r e c o r d e d : vessel m o t i o n s i n six degrees o f f r e e d o m , vessel y a w a c c e l e r a t i o n ( i n t e r m s o f a rate g y r o ) , p r o p e l l e r t h r u s t , RPS, t o r q u e , u n i t t h r u s t , r e v o l u t i o n s o f t h e f a n , f a n f o r c e , a n d w a v e e l e v a t i o n i n t h e b a s i n a n d r e l a t i v e w a v e e l e v a t i o n a t t h e t h r u s t e r p o s i t i o n . S a m p l i n g f r e q u e n c y f o r m o d e l p o s i t i o n s is 25 H z a n d a l l t h e o t h e r m e a s u r e -m e n t s are s a -m p l e d at 2 0 0 H z . A rate l i -m i t e r w a s set e q u a l t o 3 0 deg/s t o l i m i t t h e m a x i m u m rate o f a z i m u t h angle response.
A f t e r a l l t h e w a v e s w e r e finished w i t h t h e first h e a d i n g , t h e m o d e l p o s i t i o n f o r t h e n e x t h e a d i n g w a s r e a r r a n g e d as s h o w n i n Fig. l ( b ) - ( d ) . W h e n a l l t h e w o r k w a s finished w i t h t h e first m o d e l speed, a l l t h e p r o c e d u r e s w e r e r e p e a t e d f o r t h e n e x t c a l m w a t e r vessel speed V m = 1 . 1 9 1 m / s .
2.3. Analysis of model test data
W h e n c o n v e r g e d speed is a c h i e v e d i n t h e m o d e l test, t h e t i m e series o f t h e r e c o r d e d speed is o s c i l l a t i n g a r o u n d a c o n s t a n t m e a n value. I n t h i s case, t h e m e a s u r e d vessel speed can be t a k e n d i r e c t l y b y a v e r a g i n g t h e d a t a w i t h i n t h e c o r r e s p o n d i n g t i m e p e r i o d . W h i l e , f o r n o n - c o n v e r g e d r u n s , t h e vessel speed k e p t decreasing o r i n c r e a s i n g w i t h i n t h e t i m e w i n d o w . Table 5 s h o w s t h e a c c e l e r a t i o n o f t h e m o d e l o f t h e t w o above cases. The a c c e l e r a t i o n h e r e is t h e average a c c e l e r a t i o n w h i c h w a s f o u n d by d e r i v a t i o n o f t h e l o w - p a s s filtered v e l o c i t y , so t h a t t h e w a v e e n c o u n t e r f r e q u e n c y v a r i a t i o n s are e x c l u d e d .
I n order t o decide i n a systematic and o r d e r i y m a n n e r i f t h e model's speed is converged or not, w e established a c r i t e r i o n based on the acceleration: I f a < 0.001 m/s^, t h e n the speed is regarded as converged speed; I f a > 0.001 m/s^, t h e n the speed is treated as n o n -converged speed. A p p l y i n g this c r i t e r i o n , there are seven o u t o f 3 6 runs w h e r e t h e speed is n o t s u f f i c i e n t i y converged, w i t h increasing or decreasing t r e n d t i l l the end o f t h e tests. I n Table 5, a l l t h e seven cases w i t h non-converged speeds and several cases w i t h s u f f i c i e n t l y converged speed are listed. Fig. 3 shows t w o examples o f the seven cases w i t h non-converged speeds.
As s h o w n i n Fig. 3, t h e m o d e l speed is s t i l l c h a n g i n g t h r o u g h -o u t t h e e n t i r e test. F-or t h i s case, t h e a t t a i n a b l e speed i n t h i s w a v e
Table 5
Overview of runs witii non-converged speed, compared with a few samples with converged speed. H r Heading Vo a (m/s^) (m x a/T,h,„„+,„„,„pe) x 100% (m) (s) (deg) (m/s) Converged 0.121 2.383 30 1.769 0.0008 2.1871 0.121 3.071 30 1.769 0.0000 0,0000 0.121 2.383 60 1.769 - 0 . 0 0 0 7 - 1 . 9 2 0 6 0.121 2.579 60 1.769 0.0008 2.2588 0.121 2.088 90 1.769 - 0 . 0 0 0 8 - 2 . 1 3 0 3 Non-converged 0.121 1.842 30 1.769 - 0 . 0 0 1 6 - 4 . 3 9 1 8 0.121 1.351 60 1.769 0.0013 3.6941 0.121 1.842 60 1.769 -0.001 - 2 . 6 7 6 7 0.121 2.088 60 1.769 0.0025 6.5907 0.061 2.383 60 1.769 0,0048 11.7753 0.121 1.351 90 1.769 - 0 . 0 0 3 0 - 8 . 6 9 4 5 0.121 2.579 90 1.769 - 0 , 0 0 3 6 - 1 0 . 3 0 c o n d i t i o n c a n n o t be t a k e n d i r e c t l y f r o m t h e m e a s u r e m e n t . So t h e m o d e l t e s t results have t o be c o r r e c t e d b e f o r e t h e y c a n be c o m p a r e d w i t h t h e n u m e r i c a l c a l c u l a r i o n r e s u l t s . A m e t h o d o l o g y f o r c o r r e c t i o n is o u t l i n e d i n t h e f o l l o w i n g . C o n v e r g e d speed is based o n t h e b a l a n c e b e t w e e n n e t t h r u s t force, t o w r o p e f o r c e a n d t h e s u m o f t h e c a l m w a t e r resistance and a d d e d w a v e resistance. Eq. ( 1 ) s h o w s h o w t h e a c c e l e r a t i o n o f t h e m o d e l is r e l a t e d t o t h e f o r c e balance i n t h e d i r e c t i o n o f t r a v e l . I n t h i s f o r c e balance, w e k n o w t h e c a l m w a t e r resistance Realm f r o m c a l m w a t e r m o d e l tests p e r f o r m e d e a r l i e r w i t h s o m e m o d i f i c a -tions w h i c h w i l l be s p e c i f i e d c l e a r l y i n S e c t i o n 2 . 1 . a n d Rcw is c a l c u l a t e d u s i n g t h e average v e l o c i t y o f t h e e f f e c t i v e t i m e w i n -d o w . T h e average values use-d-to p r e s e n t t h e spee-d loss a n -d o t h e r results are c o m p u t e d f r o m a n e f f e c t i v e t i m e w i n d o w o f b e t w e e n 2 0 t o 3 0 s f o r d i f f e r e n t c o n d i t i o n s . T h e e f f e c t i v e time w i n d o w s are chosen as close as t o t h e e n d o f t h e t e s t as possible, b u t b e f o r e t h e p r o p e l l e r s w e r e s t o p p e d a n d t h e m o d e l w a s t o w e d t o stop, so t h a t t h e speed w i t h the s m a l l e s t a c c e l e r a t i o n is c a p t u r e d . The t h r u s t Tthrust is m e a s u r e d d u r i n g t h e tests, as is also t h e t o w r o p e f o r c e ^towrope_effective — 0.67Ttowrope_measure.
m X a = Realm -l-'^OH'^rthrust X (1 -0-rtowrope_effective (1)
It is t h e n necessary t o d e c i d e t h é t h r u s t d e d u c t i o n f a c t o r o f t h e p r o p u l s i o n s y s t e m i n w a v e s . I f w e k n o w t h e a d d e d resistance d u e to w a v e s . Raw w e can easily d e t e r m i n e 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 i n w a v e s ( 1 - f ) . I d e a l l y , Rcaiu,+Raw s h o u l d be d e t e r m i n e d f r o m t o w i n g t h e m o d e l i n w a v e s a n d m e a s u r i n g t h e resistance. I n lack o f such measurements w e have c o m p u t e d Raw by the m e t h o d o f Loukakis and Sclavounos ( 1 9 7 8 ) o u t l i n e d i n Section 2.2.2.
N a k a m u r a a n d N a i t o ( 1 9 7 7 ) , Y a m a z a k i et a l . ( 1 9 7 8 ) a n d B h a t t a c h a r y y a ( 1 9 7 8 ) a r r i v e d at t h e c o n s i s t e n t c o n c l u s i o n t h a t t h e t h r u s t d e d u c t i o n f a c t o r i n w a v e s is s l i g h t l y l o w e r t h a n i n c a l m w a t e r . T h i s s m a l l d e v i a t i o n is c h a n g i n g w i t h d i f f e r e n t w a v e c o n d i t i o n s a n d F r o u d e n u m b e r . W i t h o u r data, w e can e i t h e r use t h e c o m p u t e d a d d e d resistance a n d t h e n find 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 f r o m t h e m e a s u r e m e n t s , o r w e c a n m a k e a n a s s u m p t i o n a b o u t 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 a n d find t h e a d d e d resistance f r o m t h e m e a s u r e m e n t s . I f n o t h i n g is k n o w n a b o u t 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 i n w a v e s , t h e best a s s u m p t i o n is t o use t h e c a l m w a t e r v a l u e . Since t h e r e s u l t s are q u i t e s e n s i t i v e to t h e c h a n g e o f p r o p u l s i o n factors, w e a p p l y b o t h m e t h o d s a n d c o m p a r e t h e r e s u l t s . Fig. 4 ( a ) s h o w s t h e t h r u s t d e d u c t i o n i n r e g u l a r w a v e s o f d i f f e r e n t w a v e l e n g t h s i n h e a d i n g d i r e c t i o n o f 3 0 , 6 0 a n d 90 deg. T h e d i f f e r e n c e s b e t w e e n t h r u s t d e d u c t i o n i n c a l m w a t e r a n d i n w a v e s are e v i d e n t i n t h e r a n g e o f L/). f r o m 0.4 t o 1.3. I n v e r y l o n g a n d v e r y s h o r t w a v e s , t h r u s t d e d u c t i o n i n w a v e s is f a i r l y close t o t h e one i n c a l m w a t e r . T h e r e a s o n f o r t h i s is p r o b a b l y t h a t i n t h e i n t e r m e d i a t e w a v e l e n g t h s , t h e s h i p e x p e r i -ence large m o t i o n s a n d a d d e d resistance, so t h a t flow c o n d i t i o n s i n t h e p r o p e l l e r area b e c o m e q u i t e d i f f e r e n t f r o m t h e c a l m w a t e r c o n d i t i o n . T h e c h a n g i n g o f w a k e f r a c t i o n i n d i f f e r e n t w a v e c o n d i t i o n s is s h o w n i n Fig. 4 ( b ) . T h i s is s i m i l a r t o results b y N a k a m u r a a n d N a i t o ( 1 9 7 7 ) , b u t t h e v a r i a t i o n i n the t h r u s t d e d u c t i o n a n d w a k e f r a c t i o n seen i n Fig. 4 is l a r g e r t h a n w h a t is s h o w n b y N a k a m u r a a n d N a i t o ( 1 9 7 7 ) . T h e r e a s o n f o r t h a t is p r o b a b l y t h e i n a c c u r a c y o f t h e c a l c u l a t e d a d d e d w a v e resistance, w h i c h is c l e a r i y a s h o r t - c o m i n g o f o u r m e t h o d . A n o t h e r source o f d i f f e r e n c e is o f course t h a t t h e h u l l f o r m s a n d t y p e o f p r o p u l s i o n s y s t e m d i f f e r s q u i t e s t r o n g l y b e t w e e n o u r case a n d N a k a m u r a a n d N a i t o ( 1 9 7 7 ) . N a k a m u r a a n d N a i t o s t u d i e d a s i n g l e s c r e w c o n -t a i n e r vessel w i -t h a c o n v e n -t i o n a l p r o p e l l e r , w h i l e w e s -t u d y a t a n k e r w i t h t w i n a z i m u t h p r o p u l s o r s . A l s o t h r u s t d e d u c t i o n v a r i a t i o n d u e t o s t e e r i n g w i l l p r o b a b l y c o n t r i b u t e t o t h e large d i f f e r e n c e .
I > E • Hm=0.061(m) Tm=2.383(s) Dir=60 deq 67 68 69 70 71 72 73 74 75 76 77 78 79 i time (s) t 81 82 83 I
Fig. 3. Two examples of not converged speeds from model tests with initial speed 1.769 m/s. (a) Beam sea, H=0.121 m, 1=1.315 and (b) wave direction 60 deg, H=0,061 m, r = 2 . 3 8 3 s . 0.6 • 0,5 • 0.4 • 0,3 4 0.2 • 0,1 • 0.0 • 0,1 -—m— _calm —<a— :_wave_ _Dir30
_DirêO t_wave. Dir90 t_wave. Dir90 i 0,4 0.6 0,8 1,0 1.2 1,4 1.6 1,8 2,0 2.2 2.4 2,6 . UX 0,4 0.3 0.2 - I S 0.1 0.0 -0.1 -0.2 —•~\v_ca!m —9— vv_vvave. Dir30 -A— w_wave. .Dir60 w_wave_ Dir90
0,2 0,4 0,6 0,8 1,0 1,2 1.4 1.6 1.8 2.0 2.2 2.4 2.(
Fig. 4. Thrust deduction and wake factorin waves, (a) Thrust deduction t and (b) wal<e fraction w.
The c o n v e r g e d speed is f o u n d b y r e q u i r i n g t h e a c c e l e r a t i o n t e r m i n Eq. ( 1 ) t o be zero. W e t h e n a r r i v e a t t h e f o l l o w i n g e q u a t i o n :
Rca\tn{V)-hRawiV)~-T,UrusdV) X (1 ~t)-rtowrope_effecdve(\^) = 0 (2)
A l l terms are speed dependent, except the t h r u s t d e d u c t i o n , w h i c h w e assume t o be constant. Rcatm(V) is k n o w n f r o m m o d i f i e d resistance tests. RawiV) is c o m p u t e d a c c o r d i n g to the m e t h o d o f Loukakis and Sclavounos (1978). Ttowrope.eifectiveC^) 's also a k n o w n f u n c t i o n o f the speed? w h e n e x p e r i m e n t a l u n c e r t a i n t y is neglected. In order to obtain the speed dependent l o n g i t u d i n a l t h r u s t force TthrastC^O. the f o l l o w i n g m e t h o d is used. Since the p o w e r P is c o n t r o l l e d t o be constant d u r i n g the w h o l e test, w e can assume t h a t t h e p o w e r is also constant i n the ship heading d i r e c t i o n w i t h i n this small speed v a r i a t i o n range, w h i c h means: Tthnistl^i) x x (/Q, =
TihmstiV) X V X w h e r e V, is the measured non-converged speed at
the end o f the test; f/pis propulsive efficiency, w h i c h is assumed t o be constant w i t h i n the small range o f Speed change. For larger velocity variations, w i t h s i g n i f i c a n t change o f p r o p u l s i o n point, the change o f propulsive efficiency m u s t be included. So the speed dependent l o n g i t u d i n a l t h r u s t force i n Eq. (2) can be w r i t t e n as TthmsdV) = Ta^nsstiV•i)y<V^/V, W h e r e r , | , „ 5 t ( l / , ) is k n o w n f r o m the m e a -surements.
The results o f this c o r r e c t i o n m e t h o d are presented i n Section 3.
3. Theoretical calculations
Ship m o t i o n s w e r e calculated i n ShipX Vessel Response (Fathi a n d H o f f , 2 0 0 8 ) by t h e s i m p l e b u t p o w e r f u l S a l v e s e n T u c k F a l t i n -sen (STF) s t r i p theory (Salve-sen et al., 1970). W a v e effects w e r e
considered by u p d a t i n g the first order a n d second order w a v e e x c i t a t i o n forces at each t i m e step and the r e t a r d a t i o n f u n c t i o n was used to consider the m e m o r y effect. Linear and n o n l i n e a r m a n e u v e r i n g h y d r o d y n a m i c forces w e r e o b t a i n e d t h r o u g h ShipX m a n e u v e r i n g P l u g - I n (Ringen, 2009). T h r u s t forces w e r e a p p l i e d b y i m p l e m e n t i n g a f o u r - q u a d r a n t propeller m o d e l . W h e n all t h e m o d u l e s w e r e ready, seakeeping c o m b i n e d w i t h m a n e u v e r i n g equations w e r e solved at each t i m e step i n the Vessel s i m u l a t o r V e s i m (Ringen a n d Fathi, 2008). Some o f the t h e o i y b e h i n d these calculations is g i v e n i n the f o l l o w i n g sections.
The n u m e r i c a l s i m u l a t i o n is p e r f o r m e d f o r f u l l scale. A l l t h e results are t h e n scaled t o m o d e l scale f o r c o m p a r i s o n s .
3.J. Calm water resistance
Calm w a t e r resistance is obtained f r o m the resistance test i n t h e t o w i n g t a n k i n IVIARINTEK ( A l t e r s k j a r , 2010). For f u l l scale speed range 0 - 6 k n , c a l m w a t e r resistance is o b t a i n e d based o n a s s u m i n g t h a t t h e w a v e m a k i n g resistance is negligible, so t h a t t h e residual resistance c o e f f i c i e n t CR is e f f e c t i v e l y zero. For 1 0 - 1 8 k n o t s , the value o f the residual resistance c o e f f i c i e n t is f o u n d f r o m c a l m w a t e r t o w i n g tests. The resistance scaling m e t h o d i n use here f o l l o w s the standard m e t h o d i n use at MARINTEK, w h i c h is a v a r i a t i o n o f t h e ITTC'78 m e t h o d . The m e t h o d used to find the t o t a l resistance f r o m m o d e l tests, v a l i d f o r the speed range 1 0 - 1 8 knots is o u t i i n e d first.
The r e l a t i o n b e t w e e n m o d e l t o t a l resistance Rjm a n d t h e m o d e l t o t a l resistance c o e f f i c i e n t C^,,, is:
Z. Chuang. S. Steen / Ocean Engineering 64 (2013) 88-99 93 w h e r e p,n is m o d e l t e s t w a t e r d e n s i t y , is t h e area o f s h i p m o d e l w e t t e d surface. T h e t o t a l resistance c o e f f i c i e n t can be c o m p u t e d u s i n g a f r i c t i o n l i n e a n d f o r m f a c t o r : Cm = CR + Cfm X (1 + k) + CgDm + CAppm (4) Cfm = ( 0 . 0 7 5 / ( l o g , o i ? N - 2 ) 2 ) is t h e f r i c t i o n resistance c o e f f i c i e n t i n m o d e l scale, w h e r e is Reynolds n u m b e r ; k is f o r m f a c t o r w h i c h is c a l c u l a t e d u s i n g t h e e m p i r i c a l t h e e q u a t i o n /(: = 0.6 (p + 145 ^^.s ^ j j j ^ ^ ^ {CB/LWL)\/B X (TA + TF). w h e r e CB is b l o c k c o e f f i c i e n t , LWL is l e n g t h o f t h e w a t e r l i n e , TA is d r a u g h t at a f t p e r p e n d i c u l a r , Tp is d r a u g h t at f o r w a r d p e r p e n d i c u l a r ; CBD = 0.29 y^iSB/S/Cp) i s ' t h e t r a n s o m s t e r n resistance c o e f f i -c i e n t , w h e r e SB is t r a n s o m s t e r n area a n d S is w e t t e d h u l l s u r f a -c e i n f r o n t o f t h e t r a n s o m ;
CAppm is m o d e l scale a p p e n d a g e resistance c o e f f i c i e n t , C A p p m= 0 i n t h i s case s t u d y ; H o w f u l l scale s h i p resistance is d e t e r m i n e d f r o m m o d e l t e s t r e s u l t s is b r i e f l y o u t l i n e d b e l o w . T h e r e s i d u a l resistance is F r o u d e scaled, w h i c h m e a n s t h a t t h e r e s i d u a l resistance c o e f f i c i e n t is e q u a l i n m o d e l a n d f u l l scale: CRHI = CRS
(5)
T h e f u l l scale t o t a l resistance c o e f f i c i e n t can be w r i t t e n as: (6) CTSI = CRS + (CF5 + zl CF)(1 + k ) + CBDS + CAPPS + CA A C f = ( 1 1 0 ( H X V s ) ° - ^ ' - 4 0 3 ) X C^s is a r o u g h n e s s c o r r e c t i o n . H = 1 5 0 ( 1 0 " ^ m m ) is used as a s t a n d a r d v a l u e . Cfy is t h e c o r r e l a t i o n a l l o w a n c e w h i c h a c c o u n t s f o r s y s t e m a t i c e r r o r s i n t h e s c a l i n g m e t h o d a n d m o d e l t e s t set-up.
For t h e l o w speed r a n g e 0 - 6 k n o t s , t h e r e s i d u a l resistance c o e f f i c i e n t is zero, so t h a t Eq. ( 6 ) is s i m p l i f i e d t o
CTS2 = (CFS + ACr)(\+k)+CBDS + CApps + CA (7)
Since t h e o r i g i n a l c a l m w a t e r tests i n t o w i n g t a n k h a p p e n e d i n 2 0 1 0 , t h e m o d e l h a d b i l g e keels a d d e d f o r t h e t e s t i n t h i s paper, a n d a larger s u p e r s t r u c t u r e w h i c h leads t o increased air resis-tance. Also, e x p e r i e n c e is t h a t t h e m o d e l resistance g e n e r a l l y increases w i t h t i m e , m a i n l y because o f surface finish d e t e r i o r a -t i o n . R e s u l -t i n g c a l m w a -t e r resis-tance c u r v e w a s c o r r e c -t e d f o r -t h e runs i n t h e ocean basin, a c c o r d i n g t o a c o m p a r i s o n o f t h e t h r u s t m e a s u r e d i n c a l m w a t e r i n t h e ocean b a s i n a n d t h e o r i g i n a l c a l m w a t e r t e s t r e s u l t s . C o n s i d e r i n g t h e changes t o t h e m o d e l , a n increase o f m o d e l resistance o f 3.89 N at V o = 1.769 m/s is used t o c o r r e c t t h e o r i g i n a l c a l m w a t e r resistance c u r v e . The increase o f resistance a m o u n t s t o 5.6% o f t h e o r i g i n a l c a l m w a t e r resistance a t t h i s speed. A n a d d i t i o n a l m o d e l resistance c o e f f i c i e n t is c o m p u t e d f o r t h i s speed, a n d resistances a t o t h e r speeds are c o r r e c t e d b y a s s u m i n g t h a t t h e a d d i t i o n a l m o d e l resistance c o e f f i c i e n t varies l i n e a r l y w i t h s p e e d :
ACr
3.89 (8)w h e r e \ / o = 1 . 7 6 9 m / s . The m o d i f i e d f u l l scale c a l m w a t e r resis-t a n c e c o e f f i c i e n resis-t resis-t h a resis-t w e a p p l y i n resis-t h e n u m e r i c a l s i m u l a resis-t i o n s is t h e n o b t a i n e d b y Eq. ( 9 ) : CT5_mod iRed = (Cm, - f A C , - , , , )- 0 , 6 7 X Cs (9) T h e t o w r o p e f o r c e c o e f f i c i e n t C^ is expressed as t h e d i f f e r e n c e b e t w e e n t h e m o d e l a n d f u l l scale t o t a l resistance c o e f f i c i e n t s CS= CT, „ - CJ S . Due t o t h e p r e v i o u s l y m e n t i o n e d t h r u s t loss o f t h e f a n u s e d t o a p p l y t h e t o w r o p e f o r c e , e f f e c t i v e t o w r o p e f o r c e is o n l y 0.67 t i m e s t h e f a n f o r c e m e a s u r e d d i r e c t l y . M o d i f i e d f u l l scale c a l m w a t e r resistance is c a l c u l a t e d asR7-s_modiried = Crs.modined X ( l / 2 ) p V ^ S . F i n a l l y , t h e p o l y n o m i a l e x p r e s s i o n f o r m o d i f i e d resistance i n t h e e n t i r e speed range f r o m 0 t o 18 k n o t s is e s t a b l i s h e d . Rjs = a^ xV^ + a2xV^ + a3xV+a4 (10)
w h e r e 0 , - 0 4 are regression c o e f f i c i e n t s . T h e r e s u l t i n g f u l l scale c a l m w a t e r resistance a p p l i e d i n n u m e r i c a l s t u d y is s h o w n as Fig. 5. For t h e s i m u l a t i o n s , w e have a d o p t e d 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 f r o m t h e o r i g i n a l c a l m w a t e r p r o p u l s i o n tests ( f = 0 . 1 5 1 ) . T h e t h r u s t d e d u c t i o n o b t a i n e d f r o m t h e c a l m w a t e r r u n s i n t h e o c e a n b a s i n , u s i n g t h e c o r r e c t e d resistance c u r v e o f Fig. 5, w a s s l i g h t l y h i g h e r . The d i f f e r e n c e is f o u n d t o be caused m a i n l y b y s t e e r i n g , a n d -since t h i s e f f e c t is i n h e r e n t l y i n c l u d e d i n t h e s i m u l a t i o n , 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 w i t h o u t t h i s e f f e c t is a p p l i e d .
3.2. Added resistance due to waves
i
W a v e forces i n regular waves are m a i n l y c o m p o s e d o f first o r d e r w a v e e x c i t a t i o n forces and second order w a v e d r i f t forces i n surge, s w a y a n d y a w directions. In the c u r r e n t calculations, t h e y are p r e calculated f o r d i f f e r e n t vessel speeds, headings a n d w a v e f r e q u e n cies b e f o r e t h e t i m e d o m a i n s i m u l a t i o n started. W h e n the s i m u l a -t i o n s-tar-ted, b o -t h firs-t a n d second order forces f r o m -the w a v e s w e r e calculated b y i n t e r p o l a t i o n i n the i n p u t dataset at each time step. Radiation forces are calculated b y means o f a c o n v o l u t i o n i n t e g r a l u s i n g r e t a r d a t i o n f u n c t i o n s t o account f o r t h e m e m o r y effects.
3.2. J. First order wave exciting force
T h e first o r d e r w a v e e x c i t a t i o n f o r c e is d e s c r i b e d b y t r a n s f e r f u n c t i o n a n d t h e w a v e e l e v a t i o n . FlJ>„, = H'"(co,0)C(co,0,x,y) (11) w h e r e H'^'(a),0) is first o r d e r t r a n s f e r f u n c t i o n . It d e p e n d s o n w a v e f r e q u e n c y co a n d p r o p a g a t i o n d i r e c t i o n G. ((oj,9,x,y) is w a v e e l e v a t i o n w h i c h is t h e f u n c t i o n o f w a v e f r e q u e n c y , p r o p a g a t i o n d i r e c t i o n a n d p o s i t i o n i n space.
3.2.2. Second order wave drift force on a ship in oblique waves
As s h o w n i n fig. 6, t h e s h i p is m o v i n g a l o n g a p r e d e t e r m i n e d d i r e c t i o n a l o n g its x - a x i s w i t h speed V. T h e w a v e s are c o m i n g f r o m a d i r e c t i o n at angle / i w i t h p r o p a g a t i n g s p e e d c-. A d d e d w a v e resistance R-x, transverse d r i f t f o r c e R-y a n d m e a n y a w m o m e n t i n surge, s w a y a n d y a w d i r e c t i o n s are c o m p o s e d o f t h e m e a n s e c o n d o r d e r w a v e force o n a s h i p i n o b l i q u e w a v e s . These f o r c e s are c a l c u l a t e d b y t h e m e t h o d p r o p o s e d b y L o u k a k i s a n d 500000 300000 4 200000 Vs (l<n)
Sclavounos ( 1 9 7 8 ) . T h i s m e t h o d is based o n the e x t e n s i o n o f t h e m e t h o d p r o p o s e d b y 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 ) w h i c h is o r i g i n a l l y o n l y f o r head w a v e s .
The Gerritsma and B e u k e l m a n m e t h o d is based o n the energy p r i n c i p l e u n d e r the a s s u m p t i o n t h a t radiated energy is equal t o t h e w o r k o f added resistance done and the energy contained i n t h e d a m p i n g waves. A n d this energy relationship can be expressed b y E q . ( 1 2 ) .
(-R,)(V-C)T= { b 3 3 - V ^ ^ [URzi'dc ( 1 2 )
w h e r e cu a n d T are e n c o u n t e r f r e q u e n c y and encounter p e r i o d o f the w a v e , VRZ is t h e v e r t i c a l relative v e l o c i t y o f each ship section, 033 a n d ^33 are t w o - d i m e n s i o n a l sectional added mass a n d d a m p i n g coefficients.
The l e f t p a r t o f Eq. ( 1 2 ) c o r r e s p o n d s t o t h e w o r k t r a n s f e r r e d t o t h e fluid p l u s t h e a m o u n t o f e n e r g y t r a n s f e r f r o m t h e i n c i d e n t w a v e s d u e t o t h e i r d i r e c t i o n o f p r o p a g a t i o n p l u s r a d i a t e d w a v e s a l l a r o u n d t h e s h i p .
This approach can be e x t e n d e d to calculate the added resistance a n d d r i f t force i n o b l i q u e regular waves. I t is k n o w n t h a t i f one considers a c o n t r o l surface fixed i n space and s u r r o u n d i n g the ship, t h e energy i n f l u x a n d e f f l u x t h r o u g h i t are equal w h e n the c o n t r i b u t i o n o f the d i f f r a c t e d waves and ship generated waves are t a k e n i n t o consideration. T h e v e l o c i t y o f the ship relative t o the fixed c o n t r o l surface is VR = V - c a n d can be resolved i n t o Vmand VRT w h i c h are parallel a n d n o r m a l to c respectively. The m o t i o n o f t h e s h i p w i t h VRT does n o t m a k e c o n t r i b u t i o n to the m e a n h o r i z o n t a l force RT. w h i c h consequently has the d i r e c t i o n o f the i n c i d e n t w a v e p r o p a g a t i o n . T h e n (-RT)VRH shall represent the net energy g i v e n b y t h e ship to t h e fluid {(-Rx)V) and the energy radiated all a r o u n d the s h i p (RTC). T h e n the l e f t p a r t o f Eq. ( 1 6 ) can be r e w r i t t e n as:
P = {-RT}VRH={-RT){-C-VC0Sli) ( 1 3 ) T h e final expressions w h i c h a l l o w us t o c a l c u l a t e t h e a d d e d r e s i s t a n c e a n d d r i f t f o r c e i n o b l i q u e w a v e s are: (-RT){-C-VC0S P ) = P35 +P26-FP4+2P24 ( 1 4 ) |i?y| = |R7-sin/;| (17) w h e r e P35, P26, P4. a n d P24 are e n e r g y r a d i a t e d by s h i p h e a v e - p i t c h m o t i o n , s w a y - y a w m o t i o n , r o l l m o t i o n and s w a y - r o l l m o t i o n d u r i n g one e n c o u n t e r p e r i o d . 3.3. Thrust force T h e m o d e l was e q u i p p e d w i t h m o d e l s o f Rolls-Royce A z i p u i l p u l l i n g a z i m u t h i n g t h r u s t e r s . I n t h e t i m e - d o m a i n s i m u l a t i o n s , a n e m p i r i c a l l y based p r o p u l s i o n m o d e l is a p p l i e d t o r e p r e s e n t t h e forces f r o m t h e t h r u s t e r s . T h e s i m u l a t i o n m o d e l is based o n a s y s t e m a t i c series o f o p e n w a t e r tests w i t h a m o d e l o f t h e A z i p u i l t h r u s t e r (Berg, 2 0 0 2 ) , w h e r e a d v a n c e n u m b e r , a z i m u t h angle a n d p r o p e l l e r p i t c h was v a r i e d s y s t e m a t i c a l l y . The m o d e l tests r e p o r t e d by Berg ( 2 0 0 2 ) w e r e p e r f o r m e d b y MARINTEK, w h i l e t h e i m p l e m e n t a t i o n o f the p r o p u l s i o n m o d e l w a s d o n e b y Rolls-Royce M a r i n e . The e m p i r i c a l p r o p u l s i o n m o d e l is i n t e r p o l a t i n g i n t h e m o d e l tests i n t a b u l a r f o r m t o p r o d u c e p r o p e l l e r o p e n w a t e r c u r v e f o r the i n s t a n t a n e o u s v a l u e o f p i t c h a n d a z i m u t h angle. T h e p r o p u l s i o n m o d e l also c o n t a i n s a c o r r e c t i o n t e r m t o a c c o u n t f o r t h e flow s t r a i g h t e n i n g e f f e c t o f t h e h u l l w h e n t h e s h i p is t u r n i n g , b u t t h i s c o r r e c t i o n is w i t h o u t i m p o r t a n c e f o r t h e cases s t u d i e d here.
A n electric e n g i n e m o d e l is selected here f o r t h e t w o t h r u s t e r s . A n a z i m u t h rate l i m i t e r w i t h m a x i m u m rate l i m i t a l l o w e d f o r s i g n a l 7.5 deg/s ( i n f u l l scale) is a p p l i e d t o get r i d o f t h e h i g h f r e q u e n c y o s c i l l a t i o n s . Gear r a t i o a n d s h a f t loss are i n c l u d e d by e d i t i n g e n g i n e - p r o p e l l e r gear r a t i o a n d m e c h a n i c a l e f f i c i e n c y . Also t h e t o t a l m o m e n t o f i n e r t i a o f the engine a n d p r o p e l l e r are c o n s i d e r e d i n o r d e r t o g i v e t h e c o r r e c t r a t e o f change o f e n g i n e r e v o l u t i o n s . A h e a d i n g c o n t r o l l e r is used t o keep t h e d e s i r e d h e a d i n g a n d keep t h e p o w e r c o n s t a n t . This is t h e s a m e w a y as d o n e i n the m o d e l tests.
3.4. Speed loss due to loss of thrust
k 7 k
l'^T| = - ( P 3 5 + P 2 6 + P 4 ) + f P24 ( 1 5 )
|R;(| = |R7-C0S/(| ( 1 6 ) / s
Fig. 6. Definition of iieading direction and mean second order force.
T h r u s t loss i n w a v e s d u r i n g m a n e u v e r i n g c a n m a i n l y be a t t r i b u t e d t o v e n t i l a t i o n a n d s t e e r i n g . V e n t i l a t i o n t y p i c a l l y occurs i n r o u g h w a v e c o n d i t i o n s , w h e n large r e l a t i v e m o t i o n s b e t w e e n sea s u r f a c e and s h i p o c c u r at t h e p r o p e l l e r p o s i t i o n s . V e n t i l a t i o n leads to a s i g n i f i c a n t d r o p i n t h r u s t , a c c o m p a n i e d b y a s l i g h t l y l o w e r d r o p i n t h e p r o p e l l e r t o r q u e . V e n t i l a t i o n is n o t d e t e c t e d i n t h e r e p o r t e d tests. C a l c u l a t i o n o f h y d r o d y n a m i c forces o n t h e a z i m u t h t h r u s t e r d u e t o s t e e r i n g is i l l u s t r a t e d Fig. 7. As s h o w n i n Fig. 7, t h e t h r u s t e r has a s t e e r i n g angle a r e l a t i v e t o t h e s h i p fixed c o o r d i n a t e s y s t e m Os-XsYs, w h e r e l i s t h e t o t a l t h r u s t f o r c e a l o n g t h e p r o p e l l e r s h a f t . T h e t h r u s t e r is r o t a t e d a c c o r d i n g t o t h e r u d d e r angle d e m a n d . T h e n t h e force i n the l o n g i t u d i n a l d i r e c t i o n a n d t h e t r a n s v e r s e d i r e c t i o n r e l a t i v e t o t h e s h i p c a n be expressed by Eqs. ( 1 8 ) a n d ( 1 9 ) : f longitudinal = r X cos a ( 1 8 ) f transverse = T X s i n a ( 1 9 ) It c a n be seen t h a t t h e r e w i l l be a r e d u c e d f o r c e i n t h e l o n g i t u d i n a l d i r e c t i o n o f t h e s h i p i f t h e a z i m u t h u n i t has an a z i m u t h angle o t h e r t h a n zero. T h e t h r u s t T w i l l also c h a n g e w h e n t h e a z i m u t h angle a change. As a r e s u l t o f t h e c h a n g i n g t h r u s t f o r c e i n l o n g i t u d i n a l d i r e c t i o n Fiongitudinai. t h e balance b e t w e e n t h r u s t a n d resistance w i l l c h a n g e so t h a t vessel speed d r o p s u n t i l a n e w e q u i l i b r i u m p o i n t is e s t a b l i s h e d at a l o w e r speed v a l u e .
1. Chuang, S. Steen / Ocean Engineering 64 (2013) 88-99 95
4. Discussion of the results
Speed r e d u c t i o n o f a vessel i n o b l i q u e w a v e s d u e t o a d d e d w a v e resistance a n d s t e e r i n g is s t u d i e d . Results o b t a i n e d f r o m m o d e l tests and n u m e r i c a l s i m u l a t i o n s are c o m p a r e d i n t h i s s e c t i o n . N u m e r i c a l s i m u l a t i o n s are c a r r i e d o u t i n f u l l scale, so t h e d a t a are scaled t o m o d e l scale i n o r d e r t o m a k e c o m p a r i s o n w i t h t h e e x p e r i m e n t a l data. I n t h e n u m e r i c a l s i m u l a t i o n s , t h r u s t d e d u c t i o n and w a k e f r a c t i o n are a l w a y s set e q u a l to the c a l m w a t e r values, since t h e c a l m w a t e r values are m u c h closer t o t h e r e a l i t y t h a n the c a l c u l a t e d values. C a l c u l a t e d t h r u s t d e d u c t i o n s are o n l y used w h e n a p p l y i n g t h e a t t a i n a b l e speed p r e d i c t i o n m e t h o d .
4.1. Testing the attainable speed prediction method
T h e a t t a i n a b l e speed p r e d i c t i o n m e t h o d as e x p l a i n e d i n S e c t i o n 1.3 has t o be t e s t e d . The t e s t i n g can be based o n cases w h e r e w e have b o t h c o n v e r g e d a n d n o n - c o n v e r g e d runs i n t h e s a m e c o n d i t i o n . Table 6 lists t h e e r r o r b e t w e e n t h e a t t a i n a b l e speed w h i c h w e r e m e a s u r e d d i r e c t i y f r o m t h e c o n v e r g e d test a n d t h e p r e d i c t e d speed f o u n d f r o m t h e n o n - c o n v e r g e d test u s i n g b o t h c a l c u l a t e d a n d c a l m w a t e r t h r u s t d e d u c t i o n f a c t o r i n t h e c o r r e c t i o n procedure.
I t c a n be seen f r o m Table 6 t h a t a t t a i n a b l e speed can be w e l l p r e d i c t e d , a n d t h a t t h e m e t h o d is s e n s i t i v e t o t h e t h r u s t d e d u c -tion f a c t o r a p p l i e d . U s i n g t h e c a l c u l a t e d t h r u s t d e d u c t i o n gives b e t t e r p r e d i c t i o n t h a n c a l m w a t e r t h r u s t d e d u c t i o n i n t h e f i r s t t h r e e cases. The reason is p r o b a b l y t h a t t h e c a l c u l a t e d t h r u s t d e d u c t i o n already i n c l u d e s t h e e r r o r o f t h e c a l c u l a t e d a d d e d w a v e resistance. U s i n g t h e c a l c u l a t e d t h r u s t d e d u c t i o n i n Eq. ( 2 ) t o p r e d i c t t h e converged speed w i l l p a r t l y cancel t h e e r r o r o f
X s
Fig. 7. Azimuth thruster with steering angle.
c a l c u l a t e d a d d e d resistance. The d e p e n d e n c e o f t h e choice o f t h r u s t d e d u c t i o n f r a c t i o n o n the c o n v e r g e d speed p r e d i c t i o n m e t h o d is f u r t h e r d e m o n s t r a t e d i n t h e f o l l o w i n g s e c t i o n .
4.2. Comparison of experimental and theoretical results
I n t h i s section, m o d e l tests and n u m e r i c a l results are compared. I n the n u m e r i c a l s i m u l a t i o n s , t h r u s t d e d u c t i o n is a l w a y s set equal t o the c a l m w a t e r t h r u s t d e d u c t i o n value 0 . 1 5 1 . Fig. 8 compares t h e attainable speed i n a heading of 30 deg and a speed o f 1.769 m/s. I t is f o u n d t h a t n u m e r i c a l results agree w e l l w i t h t h e e x p e r i m e n t a l data. I n t h e case w i t h ship l e n g t h d i v i d e d b y w a v e l e n g t h equal t o 1.29, t h e m o d e l speed is still decreasing d u r i n g t h e test. So t h e c o r r e c t i o n m e t h o d suggested i n Section 1.3 was a p p l i e d o n this p o i n t , b o t h w i t h t h r u s t d e d u c t i o n f r o m c a l m w a t e r ( t _ c a ( m = 0 . 1 5 1 ) and f o u n d u s i n g calculated Raw {t_calculated=0.304). I t can be seen t h a t t h e p r e d i c t e d speed is sensitive to the d i f f e r e n t t h r u s t d e d u c t i o n factors applied. Fig. 9 shows propeller r e v o l u t i o n s per second f o r t h e same runs as r e p o r t e d i n Fig. 8. C o m p u t e d and m e a s u r e d propeller speed generally agree w e l l w i t h each othei^; w h i c h is o n t h e basis t h a t f o r c a l m w a t e r c o n d i t i o n RPS_explRPS_numerical=1.0283. Fig. 10 compares t h e u n i t thruster force i n l o n g i t u d i n a l d i r e c t i o n i n waves. I t can be seen t h a t n u m e r i c a l t h r u s t has a g o o d corre-spondence w i t h e x p e r i m e n t a l t h r u s t i n all w a v e c o n d i t i o n s . I n still w a t e r c o n d i t i o n , t h e u n i t t h r u s t measured f r o m t h e m o d e l p r o p u l -sors a n d the e f f e c t i v e f a n force is equal to 1.0466 t i m e s o f the u n i t t h r u s t force f o u n d i n the n u m e r i c a l simulations. This s m a l l d e v i a t i o n i n c a l m w a t e r c o u l d be due to inaccuracy i n the n e t f a n force, o r i n the c a l m w a t e r resistance, w h e r e f o r instance t h e n e g l e c t i n g o f air resistance w i l l i n t r o d u c e a s m a l l error. Since p o w e r c o n t r o l was applied, w h e n t h e m o d e l speed reach its l o w e s t p o i n t a t L / A = l , t h e highest c o r r e s p o n d i n g t h r u s t force w e r e obtained. This t r e n d is w e l l c a p t u r e d b o t h i n m o d e l tests a n d i n n u m e r i c a l s i m u l a t i o n s .
Figs. 1 1 - 1 3 p r e s e n t s t h e c o m p a r i s o n s b e t w e e n m o d e l t e s t a n d n u m e r i c a l s i m u l a t i o n results i n t e r m s o f speed, RPS a n d l o n g -i t u d -i n a l t h r u s t f o r c e -i n 6 0 deg h e a d -i n g angle w -i t h a n -i n -i t -i a l speed o f 1.769 m/s. Attainable speed m e t h o d w e r e a p p l i e d i n the w a v e c o n d i t i o n w i t h L / ; . = l (t_ca/cu/ated=0.0461), 1.29 (^t_calculated= 0.1646) and 2.4 ( L c a ; a j / a f e d = 0 . 1 6 4 6 ) .
Figs. 1 4 - 1 6 c o m p a r e s t h e speed, p r o p e l l e r r e v o l u t i o n a n d u n i t t h r u s t i n t h e l o n g i t u d i n a l d i r e c t i o n b e t w e e n m o d e l t e s t a n d n u m e r i c a l s i m u l a t i o n results i n t h e case w i t h c a l m w a t e r speed 1.769 m / s a n d h e a d i n g angle 9 0 deg. T h e c o r r e c t i o n p r o c e d u r e d e s c r i b e d i n S e c t i o n 1.3 is used f o r t h e cases w h e n s h i p l e n g t h d i v i d e d b y w a v e l e n g t h is e q u a l t o 0.66 ( t _ c a / c u ( a t e d = 0 . 1 7 4 ) a n d 2.4 ( t _ c a / c u / a f e d = 0 . 2 6 8 6 ) . G e n e r a l l y g o o d a g r e e m e n t is a c h i e v e d b e t w e e n m o d e l t e s t a n d n u m e r i c a l s i m u l a t i o n s . It c a n be seen f r o m Figs. 11 a n d 14 t h a t t h e n u m e r i c a l l y c a l c u l a t e d a t t a i n a b l e speeds are a l w a y s s l i g h t l y l o w e r t h a n t h e e x p e r i m e n t a l values, The m o s t l i k e l y reason t o cause t h i s is t h a t t h e n u m e r i c a l t h r u s t v a l u e is a l w a y s l o w e r t h a n t h e e x p e r i m e n t a l values, w h i c h can be seen f r o m Figs. 13 a n d 16. A l s o a p p l y i n g m o r e a c c u r a t e m e t h o d t o p r e d i c t w a v e resistance m i g h t have t h e p o t e n t i a l t o i m p r o v e t h e r e s u l t s .
Table 6
Testing attainable speed prediction method.
H ( m ) T ( s ) Heading (deg) Vo (m/s) l/_measured (m/s) V_test (tjalculate) Errori (%) V_test it_calm) Error2 (%)
0.121 2.383 30 1.769 1.4702 1.4658 - 0 . 3 0 2 7 1.4326 - 2 . 5 5 5 7 0.121 3.071 30 1.769 1.7681 1.7898 1.2298 1.7722 0.2328 0.121 2.579 60 1.769 1.6927 1.6803 - 0 . 7 3 2 4 1.6374 - 3 . 2 6 9 3 0.121 2.088 90 1.769 1.7523 1.7571 0.2745 1.8123 3.4222
2.0-1 1.9 1.8 1.7
I
1.5 E > 1.4¬ 1.3 -1.2 1.1 1.0 Vo=1.769 m/s Dir=30 degreeV
• exp (measured) O exp (t_calculate) exp (t_calm) -a— numerical —I I I I I —I —I —1— I—•—I—.—I—,—I—1—I —1— I — . — , — . — , 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.0 1.9 1.8 1.7Ê 1,5
E > 1,4 1.3 1.2 1.1 1.0 Vo=1.769m/s Dir=60 degree • exp (measured) exp (t_calculate) A exp (t_calm) numerical — I — ' — I— ' —I— ' —I— ' —I — ' — 1 — ' — I — ' — I — ' — 1 — ' — I— ' —I — ' — I 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1,8 2.0 2,2 2.4 2,6Fig. 8. Attainable speed in waves with initial speed 1.769 m/s, heading 30 deg, pig. 11. Attainable speed in waves with initial speed 1.769 m/s, heading 60 deg.
11.0 10.5 10,0 - I 9.5 ^ 9,0 tr 8.5 8.0 -1 7.5 7.0 exp - Numerical Vo=1,769 m/s Dir=30 (degree) - I ' I ' I I I I I 1 1 1 1 i 1 1 1 r 1 1 1 r—I 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2.0 2.2 2.4 2,6 UI
Fig. 9. Propeller revolutions per second in waves with initial speed 1.769 m/s, heading 30 deg. 12.0 -1 11,5 11.0 10.5 10,0 -w Q. 9,5 -CC 9,0 8,5 8,0 7,5 7,0 -exp Numerical Vo=1.769 m/s Dir=60 (degree) 1—'—1—'—r - 1 —'— I — ' — r 0,2 0,4 0.6 0.8 1.0 1.2 1,4 1,6 1.8 2.0 2.2 2.4 2.6 UX
Fig. 12. Propeller revolutions per second in waves with initial speed 1.769 m/s, heading 60 deg. 54 52 50 48 46 44 42 40 -38 36 34 32 - I 30 28 26 exp • Numefical Vo=1,769 m/s Dir=30 (degree) 0,2 0,4 0,6 1.2 1,4 UX 1.6 1.8 2.0 2.2 2.4 2.6 48 46 44 42 40 38 36 34 Vo=1,769 m/s Dir=60 ( d e g r e e ) —\— 1.0 0,2 0.4 0.6 0.8 1.4 UX 1.6 l.i 2.0 2.2 2.4 2,6
Fig. TG. Unit thrust in longitudinal direction in waves with initial speed 1.769 m/s, Fig. 13. Unit thrust in longitudinal direction in waves with initial speed 1.769 m/s, heading 30 deg. heading 60 deg.
Z. Chuang, S. Steen / Ocean Engineering 64 (2013) 88-99 97 2.0-, 1.9-1.8¬ 1.7- 1.6- 1.5-E
>
1.4- 1.3- 1.21.1 - 1.0-Vo=1.769 m/s Dir=90 degree exp (measured) -exp (t_calculate) -exp (t_calm) -numerical —' 1 ' ) ' — I— ' —I— ' —i— • —I— • —I— I —I— I —1— 1— I — • — 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 L\;vFig. 14. Attainable speed in>Waves witli initial speed 1,769 m/s in Beam Sea.
50 -1 48 46 40
ra
,E 38 -a S 36-1 p 34 - 3 2 - 1 m2 30
sz 28 26 • exp - c - Numerical Vo=1.769 m/s Dir=90 (degree) — I — ' — I — ' — I — ' — I— I —I— ' —I — ' — I— I —I — ' — I— ' — \ — ' —I—f 0,2 0,4 0,6 0.8 1.0 1,2 1,4 1,6 1.8 2,0 2,2 2.4 UXFig. 16. Unit thrust in longitudinal direction in waves with initial speed 1.769 m/s in Beam Sea, 12.0 -1 11,5 11.0 10,5 10.0 -CO • Q- 9,5 -01 9,0 8,5 8,0 7,5 7,0 -exp Numerical Vo=1.769 m/s Dir=90 (degree) 1 — ' — I — ' — I — ' — I — ' — I — ' — I — ' — I — ' — I— ' —I— ' —I — • — r 0.2 0.4 0,6 0.8 1.0 1,2 1,4 1,6 1,8 2.0 2,2 2.4 UX
Fig. 15. Propeller revolutions per second in waves with initial speed 1.769 m/s in Beam Sea. 2.0 -| 1.9 1.8 1.7 1,6 1,5 1,4 1.3 -V m 1.2 1,1 1,0 0,9 0,8 0,7 -0,6 - T ' 1 ' 1 ' r 0 00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0.16 wave elevation (m)
Fig. 17. Attainable speed with varying wave elevations in Odeg, 30 deg heading conditions in waves with period 2,383 s.
4.3. Relation between attainable speed, wave condition and iieading angles Fig. 17 s l i o w s t l i e r e l a t i o n b e t w e e n a t t a i n a b l e speeds a n d w a v e e l e v a t i o n i n 0 a n d 3 0 deg w a v e h e a d i n g a t a p o w e r c o r r e s p o n d i n g t o c a l m w a t e r speed o f 1.769 m / s . W a v e e l e v a t i o n s ( a m p l i t u d e s ) w e r e 0.0302 m , 0.0604 m a n d 0 . 1 2 0 7 m . Speed loss data f o r 0 d e g h e a d i n g is o b t a i n e d f r o m C h u a n g a n d Steen ( 2 0 1 1 ) . I t has t o be p o i n t e d o u t t h a t t h e o r i g i n a l e x p e r i m e n t a l d a t a f o r 0 d e g h e a d i n g i n C h u a n g a n d Steen ( 2 0 1 1 ) w a s o b t a i n e d w i t h o u t i n c l u d i n g t o w r o p e f o r c e i n t h e m o d e l test. T h e d a t a w e r e c o r r e c t e d u s i n g t h e m e t h o d p r o p o s e d i n t h e p a p e r c o n s i d e r i n g t h e e f f e c t o f t o w r o p e f o r c e . Speed is d r o p p i n g w i t h t h e i n c r e a s e o f w a v e e l e v a t i o n , since a d d e d w a v e resistance is p r o p o r t i o n a l t o w a v e e l e v a t i o n s q u a r e d . A l s o , w h e n h e a d i n g increase t h e d r o p i n speed is r e d u c e d .
Figs. 18 a n d 19 p r e s e n t t h e m o d e l t e s t speed loss i n p e r c e n t a g e i n w a v e s w i t h c o r r e s p o n d i n g c a l m w a t e r speed Vo e q u a l t o 1.769 m / s a n d h e a d i n g angles 30, 6 0 , 90 a n d 1 5 0 deg. T h r u s t d e d u c t i o n f a c t o r based o n c a l c u l a t e d a d d e d resistance w a s u s e d i n Fig. 18 a n d c a l m w a t e r t h r u s t d e d u c t i o n f a c t o r w a s a p p l i e d i n Fig. 19 w h e n m a k i n g p r e d i c t i o n f o r t h e a t t a i n a b l e s p e e d at t h e e n c i r c l e d d a t a p o i n t s .
Fig. 2 0 r e p r e s e n t s t h e speed loss i n p e r c e n t a g e i n w a v e s w i t h c o r r e s p o n d i n g c a l m w a t e r speed V o = 1 . 1 9 1 m/s w i t h 3 0 d e g a n d 6 0 d e g headings. I t c a n be c o n c l u d e d t h a t w i t h i n h e a d a n d b o w sea c o n d i t i o n s i n c r e a s i n g t h e h e a d i n g a n g l e can g r e a t l y r e d u c e speed r e d u c t i o n . By c o m p a r i n g Figs. 1 8 - 2 0 , i t is s h o w n t h a t t h e s p e e d loss i n p e r c e n t a g e is i n c r e a s i n g w i t h t h e d e c r e a s i n g o f t h e c o r r e s p o n d i n g c a l m w a t e r speed i n t h e same sea c o n d i t i o n s .
4.4. Contribution to totcil speed loss from added wave resistance and from steering
Speed d r o p f o r a ship sailing i n o b l i q u e waves is d u e t o added w a v e resistance {R^J a n d due t o loss o f t h r u s t i n t h e l o n g i t u d i n a l d i r e c t i o n ( ( r - F| o „ g , , d i n a i ) x d - t ) , w h e r e
