Vol.91 15 November 2014 ISSN 0029-8018
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AN INTERNATIONAL J O U R N A L O F R E S E A R C H AND DEVELOPMENT
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O c e a n E n g i n e e r i n g 9 1 ( 2 0 1 4 ) 2 6 3 - 2 7 2
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Ocean Engineering
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g
Propulsive factors in waves: A comparative experimental study for
an Open and a ducted propeller
Anirban Bhattacharyya *, Sverre Steen
Department of Marine Tectinology, Nonvegian University of Science and Technology (NTNU), Trondheim, Norway
C r o s s M a r k A R T I C L E I N F O Article history: R e c e i v e d 19 D e c e m b e r 2 0 1 3 A c c e p t e d 1 0 S e p t e m b e r 2 0 1 4 A v a i l a b l e o n l i n e 2 O c l : o b e r 2 0 1 4 Keywords: P r o p u l s i o n f a c t o r s W a v e D u c t e d p r o p e l l e r A B S T R A C T T h e e s t i m a t i o n o f p r o p u l s i v e f a c t o r s i n w a v e s i s e s s e n t i a l t o u n d e r s t a n d t h e p r o p u l s i o n c h a r a c t e r i s t i c s f o r a s h i p i n a c t u a l s e a c o n d i t i o n s . W h i l e t h e p r o p u l s i o n f a c t o r s o f o p e n p r o p e l l e r s i n w a v e s h a v e b e e n e s t a b l i s h e d b y s i g n i f i c a n t e x p e r i m e n t a l w o r k , s i m i l a r s t u d y f o r t h e d u c t e d p r o p u l s i o n c a s e c o u l d n ' t b e f o u n d . T h e p r o p e l l e r - d u c t i n t e r a c t i o n h a s a c o n s i d e r a b l e i n f l u e n c e o n t h e p r o p u l s i v e f a c t o r s f o r a d u c t e d p r o p e l l e r , a n d h e n c e d e m a n d s a s e p a r a t e s t u d y f o r u n d e r s t a n d i n g t h e e f f e c t o f w a v e s . T h i s p a p e r p r e s e n t s t h e r e s u l t s o f m o d e l p r o p u l s i o n t e s t s w i t h a 1 2 0 m c a r g o v e s s e l a t t w o F r o u d e n u m b e r s u n d e r d i f f e r e n t p r o p e l l e r l o a d i n g s f o r a s e r i e s o f h e a d s e a c o n d i t i o n s , c a r r i e d o u t w i t h b o t h a d u c t e d a n d a n o p e n p r o p e l l e r . C o m p a r i s o n o f p r o p u l s i v e f a c t o r s i n w a v e s s h o w s t h a t , t h e r e a r e d i f f e r e n c e s i n b o t h m a g n i t u d e a n d i n s o m e c a s e s — a l s o t r e n d b e t w e e n a n o p e n a n d a d u c t e d p r o p e l l e r , m o s t n o t a b l y i n t h e e f f e c t i v e w a k e f r a c t i o n , w h e r e t h e d u c t e d p r o p e l l e r s h o w s a s t r o n g e r i n f l u e n c e o f t h e w a v e s . 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 s f o u n d t o b e i n d e p e n d e n t o f p r o p e l l e r l o a d i n g i n a b r o a d r a n g e o f l o a d i n g s . T h e p r o p e l l e r e f f i c i e n c y d o e s n ' t s e e m t o b e m u c h i n f l u e n c e d b y t h e w a v e s , e x c e p t f o r t h e e f f e c t o f c h a n g e o f p r o p u l s i o n p o i n t . © 2 0 1 4 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 . 1. I n t r o d u c t i o n D u c t e d p r o p e l l e r s have l a r g e l y b e e n a d o p t e d i n t o a range o f m a r i n e vessels a f t e r Luigi Stipa a n d l a t e r L u d w i g K o r t ( 1 9 3 4 ) d e m o n s t r a t e d t h a t a f o i l - s h a p e d s h r o u d o r a n a c c e l e r a t i n g d u c t c o u l d be used to increase t h e p r o p u l s i v e e f f i c i e n c y f o r h e a v i l y l o a d e d p r o p e l l e r s .
W h e n a s h i p is o p e r a t i n g i n waves, t h e r e is a n a u g m e n t i n t h e t o t a l resistance o f t h e ship causing a s h i f t i n t h e p r o p u l s i o n p o i n t . T h e change i n f l o w c o n d i t i o n s a n d pressure i n waves also i m p a c t s t h e p r o p u l s i o n f a c t o r s . M o o r a n d M u r d e y ( 1 9 7 0 ) p l o t t e d t h e self-p r o self-p u l s i o n f a c t o r s f o r c o n v e n t i o n a l self-p r o self-p e l l e r s o n d i f f e r e n t s h i self-p m o d e l s i n b o t h l o a d e d a n d ballast c o n d i t i o n s f o r a series o f r e g u l a r w a v e c o n d i t i o n s . I t is c o n c l u d e d t h a t - t h e d i v e r g e n c e o f t h e p r o p u l s i o n factors f r o m s t i l l w a t e r values is h i g h e s t at a w a v e -l e n g t h e q u a -l t o t h e s h i p -l e n g t h , a n d r e t u r n s t o s t i -l -l w a t e r v a -l u e at a r o u n d t h e w a v e l e n g t h e q u a l to t h r i c e t h e l e n g t h o f t h e s h i p . The v a r i a t i o n o f s e l f - p r o p u l s i o n f a c t o r s w i t h w a v e - l e n g t h f o r a c o n t a i n e r s h i p at f o u r d i f f e r e n t Froude n u m b e r s (Fn 0.15 t o 0.30) p r e s e n t e d b y N a k a m u r a a n d N a i t o ( 1 9 7 5 ) s h o w t h a t t h e w a k e • C o r r e s p o n d e n c e t o : D e p a r t m e n t o f M a r i n e T e c h n o l o g y , N T N U , 7 4 9 1 T r o n d h e i m , N o r w a y T e l . : + 4 7 4 0 3 8 3 9 2 7 . E-mail address: a n i r b a n . b h a t t a c h a t y y a ® n m u . n o ( A . B h a t t a c h a r y y a ) . h t t p : / / d x . d o i . O r g / 1 0 . 1 0 1 6 / j . o c e a n e n g . 2 0 1 4 . 0 9 . 0 2 0 0 0 2 9 - 8 0 1 8 / © 2 0 1 4 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 . v e l o c i t i e s increased u n d e r w a v e c o n d i t i o n s w h i c h is m o s t p r o -n o u -n c e d at w a v e l e -n g t h s c o r r e s p o -n d i -n g t o t h e -n a t u r a l p e r i o d s i -n p i t c h , as also p o i n t e d o u t b y Faltinsen et a l . ( 1 9 8 0 ) . The t h r u s t d e d u c t i o n f r a c t i o n decreased w i t h i n c r e a s i n g w a v e h e i g h t ; h o w -ever, t h e t r e n d has b e e n reverse i n c e r t a i n s m a l l ranges o f a m p l i t u d e a n d Froude n u m b e r s . W h i l e t h e e f f e c t o f waves o n t h e r e l a t i v e r o t a t i v e e f f i c i e n c y has b e e n n e g l i g i b l e , t h e r e are p r o -n o u -n c e d r e d u c t i o -n s i -n t h e o p e -n w a t e r e f f i c i e -n c y , especially at t h e c r i t i c a l w a v e l e n g t h range. Bhattachai-yya ( 1 9 7 8 ) m e n t i o n s t h e r e d u c t i o n s i n o p e n w a t e r e f f i c i e n c y values as t h e e f f e c t o f changes i n p r o p e l l e r l o a d i n g d u e t o t h e a d d e d resistance i n waves. I n a l m o s t a l l t h e cases, t h e q u a s i - p r o p u l s i v e e f f i c i e n c y values i n w a v e s are l o w e r t h a n t h e c a l m w a t e r v a l u e , m o r e so at h i g h e r w a v e a m p l i t u d e s a n d at w a v e l e n g t h s close t o t h e l e n g t h o f t h e s h i p . For t h e d u c t e d p r o p e l l e r s , i n - d e p t h studies h a v e b e e n c a r r i e d o u t r e g a r d i n g t h e p r o p u l s i o n f a c t o r s i n c a l m w a t e r . E n g l i s h a n d R o w e ( 1 9 7 3 ) h a d discussed t h e e f f e c t s o f d u c t e 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 o n p r o p u l s i v e e f f i c i e n c y f o r t a n k e r s , w h e r e , an i n c r e -m e n t i n h u l l e f f i c i e n c y d u e t o h i g h e r w a k e f r a c t i o n f o r t h e d u c t e d p r o p e l l e r h a d r e s u l t e d i n a h i g h e r p r o p u l s i v e e f f i c i e n c y c o m p a r e d t o a n o p e n p r o p e l l e r Based o n s y s t e m a t i c tests w i t h d i f f e r e n t d u c t p r o f i l e s c a r r i e d o u t at N e t h e r l a n d s S h i p M o d e l Basin, O o s t e r v e l d ( 1 9 7 3 ) c o n c l u d e d t h a t t h e increase i n p r o p u l s i v e e f f i c i e n c y b y u s i n g a n a c c e l e r a t i n g d u c t e d p r o p e l l e r is o b t a i n e d b y t h e increase
2 6 4 A. Bhattacharyya, S. Steen / Ocean Engineenng 91 (2014) 263-272 N o m e n c l a t u r e TD [ N ] d u c t t h r u s t Q [ N m ] p r o p e l l e r t o r q u e n [Hz] p r o p e l l e r r o t a t i o n a l speed KT [ d i m e n s i o n l e s s ] t h r u s t c o e f f i c i e n t {T/pn^D'^) f o r o p e n Lpp [ m ] l e n g t h b e t w e e n p e r p e n d i c u l a r s p r o p e l l e r B [ m ] b r e a d t h ( m o u l d e d ) KTP [ d i m e n s i o n l e s s ] p r o p e l l e r t h r u s t c o e f f i c i e n t TAP/FP [ m ] d r a f t s at a f t a n d f o r e p e r p e n d i c u l a r s {Tp/pn?D^) f o r d u c t e d p r o p e l l e r A [ m ^ ] v o l u m e d i s p l a c e m e n t KTD [ d i m e n s i o n l e s s ] d u c t t h r u s t c o e f f i c i e n t {To/pn^D'^) [ d i m e n s i o n l e s s ] t o t a l t h r u s t c o e f f i c i e n t {T/pn^D'^) f o r CB [ d i m e n s i o n l e s s ] b l o c k c o e f f i c i e n t Kr.Tot [ d i m e n s i o n l e s s ] d u c t t h r u s t c o e f f i c i e n t {To/pn^D'^) [ d i m e n s i o n l e s s ] t o t a l t h r u s t c o e f f i c i e n t {T/pn^D'^) f o r V [ m / s ] speed d u c t e d D [ m ] p r o p e l l e r d i a m e t e r p r o p e l l e r X [ m ] w a v e l e n g t h ( m o d e l scale) [ d i m e n s i o n l e s s ] t o r q u e c o e f f i c i e n t {Q_lpn^D^) Ca [ m ] w a v e a m p l i t u d e t [ d i m e n s i o n l e s s ] t h r u s t d e d u c t i o n f r a c t i o n
f [ d i m e n s i o n l e s s ] advance c o e f f i c i e n t based o n m o d e l w [ d i m e n s i o n l e s s ] m e a n Taylor w a k e f r a c t i o n speed (V/nD) [ d i m e n s i o n l e s s ] r e l a t i v e r o t a t i v e e f f i c i e n c y F [ N ] t o w f o r c e 'lo [ d i m e n s i o n l e s s ] o p e n w a t e r e f f i c i e n c y FD [ N ] f r i c t i o n c o r r e c t i o n f o r c e ( t o w f o r c e at p r o p u l s i o n >lH [ d i m e n s i o n l e s s ] h u l l e f f i c i e n c y p o i n t ) 'ID [ d i m e n s i o n l e s s ] q u a s i - p r o p u l s i v e e f f i c i e n c y T [ N ] t o t a l t h r u s t ( p r o p e l l e r t h r u s t f o r t h e o p e n p r o p e l l e r ) IE [ H z ] w a v e e n c o u n t e r f r e q u e n c y o f o p e n w a t e r e f f i c i e n c y o f t h e p r o p u l s i o n system. He h a d m e n -t i o n e d -t h e p r o b a b i l i -t y o f -t h e n o z z l e o p e r a -t i n g less e f f e c -t i v e l y b e h i n d a h u l l c o m p a r e d t o o p e n w a t e r d u e t o n o n - u n i f o r m i n f l o w , w h i c h m a y cause s o m e parts o f t h e n o z z l e t o have n o c i r c u l a t i o n . M i n s a a s e t a l . ( 1 9 7 3 ) t e s t e d d u c t e d p r o p e l l e r s o n large t a n k e r m o d e l s a n d o b t a i n e d h i g h e r t h r u s t d e d u c t i o n a n d l o w e r r e l a t i v e -r o t a t i v e e f f i c i e n c y c o m p a -r e d t o c o n v e n t i o n a l o p e n p -r o p e l l e -r s , t h e l a t t e r b e i n g a t t r i b u t e d t o t h e decrease o f d u c t t h r u s t i n t h e b e h i n d c o n d i t i o n . For s i m p l i c i t y , i t has b e e n a s s u m e d i n o u r w o r k , t h a t t h e t i m e averaged values o f t h e p r o p e l l e r o p e n w a t e r characteristics i n w a v e s are s i m i l a r t o t h e s t i l l w a t e r values ( M c C a r t h y et al., 1 9 6 1 ; N a k a m u r a a n d N a i t o , 1977). This has a l l o w e d t h e use o f c a l m w a t e r o p e n w a t e r characteristics f o r t h e e s t i m a t i o n s o f p r o p u l s i v e factors f o r a l l t e s t c o n d i t i o n s . T h e p u r p o s e o f t h i s w o r k is t o e s t i m a t e a n d d o c u m e n t t h e change i n p r o p u l s i v e f a c t o r s i n w a v e s f o r a d u c t e d p r o p e l l e r a n d v i e w i t as a c o m p a r i s o n w i t h a n o p e n p r o p e l l e r , w h i c h m a y h e l p t h e e s t i m a t i o n o f p r o p u l s i v e characteristics i n w a v e s f o r ships fitted w i t h d u c t e d p r o p e l l e r s t h a t o p e r a t e f r e -q u e n t l y i n h e a v y seas. 2. M o d e l tests T h e m o d e l tests w e r e c a r r i e d o u t w i t h a 1:22.629 scale m o d e l o f t h e s i n g l e - s c r e w 120 m cargo vessel i n t h e large t o w i n g t a n k ( l e n g t h : 2 6 0 m , b r e a d t h : 10 m , d e p t h : 5 m ) at T h e M a r i n e T e c h -n o l o g y Ce-ntre, T r o -n d h e i m , N o r w a y . T h e vessel, d e s i g -n e d b y Rolls Royce was t e s t e d a l t e r n a t e l y w i t h a c o n v e n t i o n a l o p e n p r o p e l l e r a n d a d u c t e d p r o p e l l e r o f s l i g h t l y s m a l l e r d i a m e t e r , u s i n g a r u d d e r i n b o t h cases. B o t h t h e p r o p e l l e r s w e r e stock p r o p e l l e r s d e s i g n e d T a b l e 1 P r i n c i p a l p a r t i c u l a r s o f t h e 1 2 0 m c a r g o v e s s e l . F u l l s c a l e M o d e l s c a l e l.(jp 1 1 7 . 6 m 5 , 1 9 7 m B 2 0 . 8 m 0 . 9 1 9 m TAP/FP 5 . 5 m / 5 . 5 m 0 . 2 4 3 m / 0 . 2 4 3 m A 8 8 3 2 . 7 0 . 7 6 2 m ^ C B 0 . 6 5 7 0 . 6 5 7 V 1 3 . 4 / 9 . 4 k n o t 1 . 4 4 9 / 1 . 0 1 6 m / s
b y MARINTEK. The p r i n c i p a l p a r t i c u l a r s are s h o w n i n Table 1, and t h e details o f t h e stock p r o p e l l e r s are m e n t i o n e d i n Table 2. The m o d e l was tested a t Froude n u m b e r s 0.203 ( p r i m a r y speed) and 0.142. The m o d e l o f t h e s t a n d a r d 19A t y p e a c c e l e r a t i n g d u c t h a d a l e n g t h o f 7 7 3 m m a n d an i n n e r d i a m e t e r o f 195.5 m m . Details of t h e r e g u l a r waves tested are p r o v i d e d i n Table 3.
2.J. Test set-up
A transverse b e a m was m o u n t e d a m i d s h i p s w i t h w i r e s i n a
c r o w - f o o t a r r a n g e m e n t (Fig. 1 ) t o c o n n e c t t h e m o d e l t o t h e t o w i n g carriage.
The m o d e l w a s t o w e d i n t h e a r r a n g e m e n t s h o w n i n Fig. 1, and t h e p r o p e l l e r speed set t o ' d i f f e r e n t levels t o o b t a i n d i f f e r e n t t o w f o r c e values. I n this sense, t h e t e s t m e t h o d c o r r e s p o n d s t o the so-c a l l e d B r i t i s h m e t h o d f o r p r o p u l s i o n tests. T h e t o w f o r so-c e , p r o p e l l e r t h r u s t , t o r q u e a n d r p m w e r e m e a s u r e d a l o n g w i t h i n c o m i n g wave e l e v a t i o n , s h i p m o d e l m o t i o n s a n d accelerations. T h e p r o p e l l e r t h r u s t a n d t o r q u e w e r e m e a s u r e d w i t h a Cussons p r o p e l l e r T a b l e 2 D e t a i l s o f s t o c k p r o p e l l e r s u s e d f o r m o d e l t e s t s . O p e n p r o p e l l e r D u c t e d p r o p e l l e r N u m b e r o f b l a d e s 4 4 P r o p e l l e r d i a m e t e r 1 8 5 . 6 m m 1 7 8 . 3 m m P i t c h r a t i o a t r / R = 0 . 7 0 . 9 7 5 1 . 2 9 7 B l a d e a r e a r a t i o 0 . 5 1 5 0 . 6 9 7 T a b l e 3 D e t a i l s o f r e g u l a r w a v e s t e s t e d ( h e a d sea c o n d i t i o n s : m o d e l s c a l e ) .
W a v e s XILpp Co/l-pp r ( s ) (aim)
1 0 . 5 5 0 . 0 0 7 7 1 . 3 5 3 0 . 0 4 2 0 . 8 0 . 0 1 5 4 1 . 6 3 2 0 . 0 8 3 1 0 . 0 0 7 7 1 . 8 2 4 0 . 0 4 4 1 0 . 0 1 5 4 1 . 8 2 4 0 . 0 8 5 1 0 . 0 2 3 1 1 . 8 2 4 0 . 1 2 6 1 . 2 0 . 0 1 5 4 1 . 9 9 9 0 . 0 8 7 1 . 5 0 . 0 1 5 4 2 . 2 3 4 0 . 0 8 8 1 . 9 0 . 0 1 5 4 2 . 5 1 5 0 . 0 8
A . Bhattacharyya, S. Steen / Ocean Engineermg 91 (2014) 263-272 2 6 5
F i g . 1. ( a ) D i a g r a m f o r m o d e l s e t - u p i n c r o w - f o o t a r r a n g e m e n t , ( b ) S t e r n a r r a n g e m e n t w i t h t h e d u c t e d p r o p e l l e r a n d r u d d e r .
d y n a m o m e t e r . W t i e n f i t t e d w i t h d u c t , d u c t t h r u s t a n d v e r t i c a l f o r c e w e r e m e a s u r e d u s i n g t w o o n e - c o m p o n e n t l o a d cells i n c o m b i n a t i o n . The t a n k has a m a x i m u m t o w i n g speed o f 10 m / s . The d o u b l e f l a p w a v e m a k e r generates a m a x i m u m w a v e h e i g h t o f 0.9 m ( r e g u l a r w a v e ) a n d a range o f w a v e p e r i o d s 0 . 8 - 5 s. Waves w e r e m e a s u r e d b o t h a t a l o c a t i o n 10 m f r o m t h e w a v e m a k e r a n d i n a l o c a t i o n 2 m t o t h e side o f FP o f t h e m o d e l . The w a v e p r o b e f o l l o w i n g t h e m o d e l was o f t h e acoustic t y p e , so as t o o b t a i n r e l i a b l e w a v e m e a s u r e m e n t s a t f o r w a r d speed. 2.2. Test procedure T h e o p e n w a t e r tests f o r b o t h t h e p r o p e l l e r s are c o n d u c t e d i n c a l m w a t e r c o n d i t i o n . For t h e m o d e l p r o p u l s i o n tests, t h e FD f o r each m o d e l speed is c a l c u l a t e d f r o m t h e d i f f e r e n c e i n t o t a l resistance c o e f f i c i e n t s b e t w e e n m o d e l and f u l l scales. For each t e s t r u n c o r r e s p o n d i n g t o a c e r t a i n m o d e l speed, f o u r values o f n w e r e chosen: a l o w v a l u e v e r y close t o zero t h r u s t , t w o i n t e r -m e d i a t e values c o v e r i n g t h e c a l c u l a t e d t o w force, a n d a h i g h e r n t o ensure t h a t t h e m o d e l is t o w e d at t h e f u l l scale p r o p u l s i o n p o i n t i n m o s t o f t h e d e s i g n w a v e c o n d i t i o n s . For b e t t e r c o m p a r i s o n , t h e n values w e r e k e p t c o n s t a n t f o r a p a r t i c u l a r p r o p e l l e r i n a l l t h e w a v e c o n d i t i o n s . To c o m p a r e p r o p u l s i v e c o e f f i c i e n t s as f u n c t i o n s o f w a v e s a n d m o t i o n s , separate f r o m t h e e f f e c t o f p r o p e l l e r l o a d i n g , t h e n values w e r e k e p t c o n s t a n t f o r a p a r t i c u l a r p r o p e l l e r i n a l l t h e w a v e c o n d i t i o n s . Corrections w e r e a p p l i e d t o t h e m e a s u r e d t h r u s t a n d t o r q u e d u e t o effects o f t h e h u b drag, s h a f t i d l e t o r q u e a n d o t h e r f i t t i n g s , a c c o r d i n g t o t h e s t a n d a r d p r o c e d u r e s n o r m a l l y a p p l i e d f o r c a l m w a t e r p r o p u l s i o n tests (ITTC, 2 0 0 2 a ) ,
2.3. Analysis of test results
Each rime w i n d o w was selected t o i n c l u d e a m i n i m u m o f t e n c o m p l e t e w a v e cycles i n such a w a y t h a t a l l t h e m e a s u r e d variables a t t a i n e d a steady state f o r a l l t h e f o u r values o f p r o p e l l e r r p s i n a single r u n . T h e steady states w e r e i d e n t i f i e d b y v i s u a l i n s p e c d o n o f t h e data, a n d l a t e r c o m p a r i n g t h e m e a n values u s i n g m u l t i p l e rime w i n d o w s w i t h i n t h e same range o f data. T h e values w e r e t h e n used f o r s u b s e q u e n t analysis f o r a l l t h e t e s t variables.
2.3.3. Pitch motions T h e response a m p l i t u d e o p e r a t o r (RAO) f o r p i t c h is o b t a i n e d as t h e . r a d o o f t h e s t a n d a r d d e v i a d o n (SD) o f t h e m e a s u r e d p i t c h response t o t h e s t a n d a r d d e v i a t i o n o f t h e c a l i b r a t e d w a v e a m p l i -t u d e . RAOresponse = SDresponse/SDwave (1) 2.3.2. Propulsive performance FD has b e e n c a l c u l a t e d as: FD = 0.5p,„Sn,V\^ + k) [Cf,n - (Cft -t- A C p ) ] (2)
S,„ is t h e m o d e l w e t t e d surface area, V is t h e m o d e l speed,
Cfm a n d Cfs are t h e f r i c t i o n a l resistance c o e f f i c i e n t s i n m o d e l a n d f u l l scale r e s p e c d v e l y c a l c u l a t e d u s i n g ITTC 57 c o r r e l a d o n l i n e . T h e f o r m f a c t o r k a n d t h e r o u g h n e s s a l l o w a n c e are c a l c u l a t e d u s i n g e m p i r i c a l f o r m u l a e suggested b y MARINTEK: k = 0.6(p+75q)^, where (p^iCB/LwOVeFAP+Tpp) x B (3) here, LWL is t h e l e n g t h at t h e w a t e r l i n e o f t h e vessel. ACp = [110.31 X ( H X V s ) ° 2 i - 4 0 3 . 3 3 ] x C% (4)
H is the h u l l r o u g h n e s s i n |am ( H = 1 5 0 |.im), a n d Vs is t h e s h i p
speed i n m / s .
For any p a r t i c u l a r w a v e c o n d i t i o n , t h e r e l a t i o n b e t w e e n t o t a l m o d e l resistance (RT), F, a n d T can be w r i t t e n as:
RT=TO-t)+F (5)
here, t is a s s u m e d c o n s t a n t f o r d i f f e r e n t p r o p e l l e r l o a d i n g s u n d e r t h e same w a v e c o n d i t i o n . This a s s u m p t i o n is v e r i f i e d b y p l o t t i n g F against T at d i f f e r e n t p r o p e l l e r r p s ( l o a d i n g ) f o r a l l w a v e c o n d i -tions, w h e r e g o o d l i n e a r c o r r e l a t i o n s have been o b t a i n e d f o r b o t h o p e n a n d d u c t e d p r o p e l l e r cases. The p l o t f o r d u c t e d p r o p e l l e r ( F n = 0 . 2 0 3 ) is s h o w n i n Fig. 2. For a l l t h e t e s t runs, Rj is c a l c u l a t e d by l i n e a r e x t r a p o l a t i o n o f F t o t h e p o i n t w h e r e t h e t o t a l t h r u s t is zero (FT^O): a n d t is o b t a i n e d as t h e slope o f t h e l i n e . Bare h u l l resistance tests w e r e n o t c o n d u c t e d f o r t h i s w o r k . T h i s p r o c e d u r e o f d e t e r m i n i n g RT a n d f solely f r o m l o a d v a i y i n g p r o p u l s i o n tests have b e e n d e s c r i b e d i n details by H o l t r o p ( 2 0 0 1 ) a n d l a t e r b y Bose ( 2 0 0 8 ) . I t has b e e n m e n t i o n e d b y H o l t r o p ( 2 0 0 1 ) t h a t t h e t o w i n g force a t zero t h r u s t is f o u n d to c o r r e s p o n d w e l l w i t h t h e m o d e l resistance {FT^OIRT is a r o u n d 1.02).
KT is c a l c u l a t e d f r o m t h e m e a s u r e d t h r u s t v a l u e s , a n d t h r u s t
i d e n t i t y (ITTC, 2011) is used, w h e r e t h i s v a l u e o f KT is used as a n i n p u t data t o f i n d t h e o p e n w a t e r advance c o e f f i c i e n t C/o). t o r q u e c o e f f i c i e n t (/CQO). a n d o p e n w a t e r e f f i c i e n c y {>7o) f r o m t h e o p e n w a t e r d i a g r a m . For c o n v e n i e n c e o f e s t i m a t i o n , t h e o p e n w a t e r p r o p e l l e r characteristics f o r c a l m w a t e r are used f o r a l l t h e t e s t cases, t h e j u s t i f i c a t i o n f o r w h i c h is m e n t i o n e d i n S e c t i o n 3.2. For each t e s t r u n , t h e p r o p u l s i v e c o e f f i c i e n t s are c a l c u l a t e d u s i n g t h e f o l l o w i n g f o r m u l a e . w = l - J o / r (6) s nR = K(^o/KQ. (7) ;7„ = ( l - f ) / ( l - w ) (8)
2 S 6 A . Bhattacharyya, S. Steen / Ocean Engineering 91 (2014) 263-272 100 80 60 40 -20 -40 ,, V L = 0.55 L l A / L = 0 . a X V L = l ( ? a = 0 . 0 8 ) A / L = 1.2 + A / L = 1.5 A / L = 1.9 C A L M W A T E R :;-K 10 20 30 1 ' S ü - - . , . e a 7 0 T h r u s t ( N ) F i g . 2 . P l o t o f t o w f o r c e as f u n c t i o n o f t o t a l t h r u s t f o r d u c t e d p r o p e l l e r : Fn = 0 . 2 0 3 . ' / d = Ih X ' / h X no (9)
The p r o p u l s i o n p o i n t n values are e s t i m a t e d b y i n t e r p o l a t i o n / e x t r a p o l a t i o n at t h e c a l c u l a t e d F d f o r t h e c o r r e s p o n d i n g speed. The c o r r e s p o n d i n g Kj a n d KQ are t h e n used t o find t h e p r o p u l s i o n f a c t o r s u s i n g the above equations.
The a d d e d resistance f o r a n y w a v e c o n d i t i o n is c a l c u l a t e d b y s u b t r a c t i n g t h e t o t a l resistance o b t a i n e d f o r c a l m w a t e r f r o m t h e t o t a l resistance f o r t h e c o r r e s p o n d i n g w a v e
Ridded = '^r.wave — l^T.calm (10)
The a d d e d resistance c o e f f i c i e n t ( d w ) i n waves is o b t a i n e d b y
CAW=R.AAed/iPgfi^^B^/hp) (11)
3. Results a n d d i s c u s s i o n s
I n t h i s section, t h e a d d e d resistance, t h r u s t a n d t o r q u e c o e f f i -cients, a n d p r o p u l s i o n f a c t o r s f o r b o t h t h e d u c t e d a n d t h e o p e n p r o p e l l e r are discussed i n details. For a l l t h e p l o t legends, DP means d u c t e d p r o p e l l e r , w h i l e OP m e a n s o p e n p r o p e l l e r . As t h e w a v e a m p l i t u d e w a s n o t s i m i l a r f o r t h e d i f f e r e n t w a v e l e n g t h s tested, t h e p l o t s are p r e s e n t e d f o r t h e range 0.8 < XjLpp < 1.9 f o r Lpp=0.0154. The l o w e r w a v e l e n g t h c o n d i t i o n ( ; i / L p p = 0 , 5 5 ) has b e e n s h o w n i n t h e tables.
3.3. A d d e d resistance
The p l o t s f o r a d d e d resistance c o e f f i c i e n t f o r b o t h t h e p r o p u l -sion cases are s h o w n i n Fig. 3. The m a x i m u m a d d e d resistance is n o t i c e d a r o u n d / l / i p p = l f o r b o t h t h e speeds tested, t h e values b e i n g h i g h e r at t h e g r e a t e r speed. The p i t c h i n g m o t i o n s are m o s t severe i n t h e range 1 < Xjlpp < 1.5. A d e t a i l e d a c c o u n t o f t h e seakeeping p r o p e r t i e s o f t h e ship m o d e l w i t h b o t h o p e n a n d d u c t e d p r o p e l l e r s are g i v e n b y B h a t t a c h a r y y a a n d Steen ( 2 0 1 4 ) .
3.2. Thrust and torque coefficients
SH -D P; F n = 0 . 2 0 3 4 -O P : F n = 0 . 2 0 3 3 -D P : F n = 0 . 1 4 2 - O P : F n = 0 . 1 4 2 0 . 5 1 1.5 F i g . 3 . P l o t o f a d d e d r e s i s t a n c e c o e f f i c i e n t as f u n c t i o n o f w a v e l e n g t h (Co/ipp= 0 . 0 1 5 4 ) . 0 . 4 0.3 0 . 2 . ( 0 . 1 0 , 0 - 0 . 1 • J * = 1 . 4 5 )i X » J * = 0 . 9 -A A J * = 0 . 6 6 K J * = 0 . 5 5
•
na a
P r o p u l s i o n p o i n t 0 . 5 1% F i g . 4 . T h r u s t c o e f f i c i e n t as f u n c d o n o f w a v e l e n g t h f o r o p e n p r o p e l l e r : F n = 0 . 2 0 3 , C „ / i p p = 0 . 0 1 5 4 . 0 . 0 6 0 . 0 4 0 . 0 2 A 0 . 0 0 • J * = 1 . 4 5 B J * = 0 . 9 A J * = 0 . 6 6 X J * = 0 . 5 5 - P r o p u l s i o n p o i n t A A 0 . 5 1 1.5 F i g . 5 . T o r q u e c o e f f i c i e n t as f u n c d o n o f w a v e l e n g t h f o r o p e n p r o p e l l e r : F n = 0 . 2 0 3 , f „ / L p p = 0 . 0 1 5 4 . Due t o t h e m o t i o n s o f t h e s h i p i n waves, t h e r e is a p r o b a b i l i t y o f loss o f t h e p r o p e l l e r t h r u s t a n d t o r q u e d u e t o v e n t i l a t i o n a n d emergence o f t h e p r o p e l l e r f r o m the w a t e r . As m e n t i o n e d b y Faltinsen et a l . ( 1 9 8 0 ) , t h e use o f c a l m w a t e r o p e n w a t e r char-acteristics f o r the e s t i m a t i o n o f p r o p u l s i o n f a c t o r s i n waves is q u e s t i o n a b l e , as t h e o p e n w a t e r Kj a n d KQ values w o u l d be r e d u c e d u n d e r w a v e c o n d i t i o n s i n case o f v e n t i l a t i o n a n d p r o p e l l e r emergence. Figs. 4 - 7 s h o w t h e e f f e c t o f w a v e o n Kj a n d KQ ( o b t a i n e d f r o m m e a s u r e d T a n d Q v a l u e s ) f o r b o t h o p e n a n d d u c t e d p r o p e l l e r s at d i f f e r e n t p r o p e l l e r l o a d i n g s . As a m e a s u r e o f t h e p r o p e l l e r l o a d i n g , t h e n o n - d i m e n s i o n a l c o e f f i c i e n t J* is used. For p u r p o s e o fc o m p a r i s o n , t h e c a l m w a t e r values are p l o t t e d o n t h e v e r t i c a l axis at
XlLpp = 0 t h r o u g h o u t Section 3. For a l l t h e cases, t h e e f f e c t o f waves
is p r o n o u n c e d f o r t h e h i g h e s t / * value, c o r r e s p o n d i n g t o the l o w e s t v a l u e o f n, t h e h i g h e s t r e d u c t i o n s o c c u r near / l / I p p = l a n d 1.2. H o w e v e r w i t h t h e increase o f n, t h e r e are lesser r e d u c t i o n s i n KT a n d KQ, a n d a t t h e l o w e s t ƒ* value, Kj a n d KQ are s i m i l a r t o t h o s e of c a l m w a t e r r u n s . The notable p o i n t h e r e is t h a t f o r m o s t o f the c r i t i c a l w a v e l e n g t h s t h e p r o p u l s i o n p o i n t s o c c u r at l o w f values (see Table 4 ) , w h e r e t h e loss i n KT a n d KQ c o m p a r e d t o c a l m w a t e r are m i n i m u m , w h i l e f o r n o n - c r i t i c a l c o n d i t i o n s ( l i k e XILpp=t9), t h e p r o p u l s i o n p o i n t o c c u r at h i g h e r / ' values, w h e r e t h e p r o p u l -s i o n p o i n t KT a n d KQ value-s are clo-se t o t h o -s e i n c a l m w a t e r
A. Bhattacharyya, S. Steen / Ocean Engineermg 91 (2014) 263-272 2 6 7 0 . 7 0 . 6 0 . 5 0 . 4 ^ 0 . 3 0 . 2 - 1 0 . 1 0 . 0 * J * = 1 . 4 1 O J * = 1 . 0 7 A J * = 1 . 0 2 X J * = 0 . 6 9 . P r o p u l s i o n p o i n t I • 0 . 5 1 1.5 F i g . 6 . T o t a l t h r u s t c o e f f i c i e n t as f u n c t i o n o f w a v e l e n g t h f o r d u c t e d p r o p e l l e r : F n = 0 . 2 0 3 , f „ / i p p = 0 . 0 1 5 4 . 0 . 2 0 . 1 0 . 0 - 0 . 1 f J * = 1 . 4 1 B J * = 1 . 0 7 A J * = 1 . 0 2 : : J * = 0 . 6 9 0 . 5 A 1 1.5 F i g . 8. D u c t t h r u s t c o e f f i c i e n t as f u n c t i o n o f w a v e l e n g t h f o r d u c t e d p r o p e l l e r : F i i = 0 . 2 0 3 , f o / L p p = 0 . 0 1 5 4 . 0 . 1 0 0 . 0 8 0 . 0 6 0 . 0 4 0 . 0 2 0 . 0 0 H J * = 1 . 0 7 A J * = 1 . 0 2 ) C J * = 0 . 6 9 ti ;< >ï * ft A
• * -*
A 0 . 5 1 VL„, 1.5 F i g . 7 . T o r q u e c o e f f i c i e n t as f u n c t i o n o f w a v e l e n g t h f o r d u c t e d p r o p e l l e r : F n = 0 . 2 0 3 , f a / L p p = 0 . 0 1 5 4 . T a b l e 4 P r o p u l s i o n p o i n t / ' v a l u e s f o r d i f f e r e n t t e s t c a s e s . F n = 0 . 2 0 3 F l l = 0 . 1 4 2 D P O P D P O P W a v e 0 . 0 4 0 . 5 5 0 . 9 3 9 0 . 7 9 7 0 . 8 8 3 0 . 7 6 5 0 . 0 8 0.8 0 . 7 1 2 0 . 5 8 6 0 . 5 9 6 0 . 4 6 3 0 . 0 8 1 0 . 5 9 7 0 . 5 0 3 0 . 4 9 9 0 . 3 7 7 0 . 0 8 1.2 0 . 6 1 8 0 . 5 2 0 0 . 5 8 9 0 . 4 7 3 0 . 0 8 1.5 0 . 7 4 5 0 . 6 2 7 0 . 7 2 6 0 . 6 0 0 0 . 0 8 1.9 0 . 8 8 4 0 . 7 5 0 0 . 9 1 1 0 . 7 5 4 C a l m w a t e r 1 . 0 5 6 0 . 8 8 0 1 . 0 3 5 0 . 8 7 1c o n d i t i o n s . Hence i t can be said t h a t , an a u t o - c o r r e c t i o n exists w h e n p r o p u l s i o n f a c t o r s at t h e p r o p u l s i o n p o i n t are e s t i m a t e d f o r a n y w a v e c o n d i t i o n . This j u s t i f i e s t h e use o f c a l m w a t e r o p e n w a t e r characteristics f o r e s t i m a t i o n s i n w a v e s . For t h e o p e n p r o p e l l e r , t h e r e is l i t t l e loss i n t h r u s t a n d t o r q u e values o v e r t h e e n t i r e J* range f o r d i f f e r e n t w a v e c o n d i t i o n s . H o w e v e r , t h e d u c t e d p r o p e l l e r gave larger r e d u c t i o n s o f t h r u s t , m o s t l y at t h e c r i t i c a l w a v e l e n g t h s , t h o u g h t h e t o r q u e values s h o w e d less e f f e c t . F u r t h e r i n v e s t i g a t i o n s w i t h t h e d u c t t h r u s t c o e f f i c i e n t (KTD) i n Fig. 8 s h o w s t h a t t h i s t h r u s t r e d u c t i o n is m a i n l y caused b y t h e decrease i n KTD w h i c h s h o w s c o n s i d e r a b l e i n f l u e n c e o f w a v e c o n d i t i o n s o n a b r o a d e r r a n g e o f f . F i g . 9 . O p e n w a t e r d i a g r a m ( d u c t e d p r o p e l l e r ) . F i g . 1 0 . O p e n w a t e r d i a g r a m ( o p e n p r o p e l l e r ) .
The p r o p u l s i o n p o i n t ƒ* values a t b o t h t h e speeds are p r e s e n t e d i n Table 4 f o r b o t h t h e p r o p e l l e r s .
3.3. Propulsive factors
The p l o t s f o r o p e n w a t e r characteristics f o r t h e d u c t e d a n d o p e n p r o p e l l e r s o b t a i n e d f r o m o p e n w a t e r tests (ITTC, 2 0 0 2 b ) c o n d u c t e d at MARINTEK are p r o v i d e d i n Figs. 9 a n d 10. I n t h e f o l l o w i n g sections, t h r u s t i d e n t i t y is used t o calculate t h e p r o p u l -s i o n f a c t o r -s at t w o value-s o f Fn f o r t h e o p e n a n d d u c t e d p r o p e l l e r -s (as m e n t i o n e d i n Section 2.3.2).
The p r o p u l s i o n p o i n t s f o r t w o cases: t h e d u c t e d p r o p e l l e r ( F n = 0 . 2 0 3 ) a n d t h e o p e n p r o p e l l e r (Fn = 0.142) a t / l / L p p = l r e q u i r e d
2 6 8 h. Bhattacharyya, S. Steen / Ocean Engineerins 91 (2014) 263-272
m u c h e x t r a p o l a t i o n ( e v i d e n t f r o m Figs. 6 a n d 7 f o r t h e f i r s t case), a n d hence o m i t t e d i n t h e plots. I n a d d i t i o n , the v a r i a t i o n i n p r o p u l s i v e factors w i t h p r o p e l l e r l o a d i n g i n t h e range 0.8 < A/
Lpp < 1.9 has been p l o t t e d f o r b o t h propellers at t h e h i g h e r speed.
The l o w e s t n c o r r e s p o n d i n g t o h i g h e s t / * value, gives v e r y l o w KQ a n d ?;o a n d hence n o t considered f o r t h e plots o f ?7R, Ï / Q and I]D.
3.3.1. Thrust deduction fraction
The assumption o f constant t at d i f f e r e n t propeller loadings f o r t h e same wave c o n d i t i o n is j u s t i f i e d f r o m Fig. 2. C o n t r a i y to the m a j o r i t y o f e x p e r i m e n t a l results i n p u b l i s h e d literature, the plots s h o w n i n Figs. 11 a n d 12 s h o w consistent increase i n t h r u s t d e d u c t i o n values f o r w a v e conditions c o m p a r e d t o c a l m w a t e r values f o r b o t h the o p e n and ducted p r o p e l l e r cases. The values obtained w i t h t h e ducted p r o p e l l e r w e r e c o m p a r a t i v e l y higher f o r all the w a v e conditions. The m a x i m u m t h r u s t d e d u c t i o n occurs near t h e A/Lpp= 1 range w h e r e the m a x i m u m p i t c h m o t i o n s w e r e noticed.
The p l o t o f t h r u s t d e d u c t i o n values w i t h w a v e a m p l i t u d e is s h o w n i n Fig. 12. I t s h o w s a clear i n c r e m e n t a t t h e h i g h e r w a v e a m p l i t u d e , m o r e so f o r t h e d u c t e d p r o p e l l e r . The o p e n p r o p e l l e r h o w e v e r , shows a s l i g h t decrease i n t a t t h e l o w e r w a v e a m p l i t u d e ( f j l p p = 0.0077).
Previous w o r k s suggest a d e p e n d e n c y o n Fn f o r the value o f t i n waves f o r a n o p e n propeller, a n d a n increase i n t is possible at some w a v e l e n g t h s . A n increase i n t f o r d u c t e d propellers c o m p a r e d t o c o n v e n t i o n a l o p e n propellers b e h i n d t h e same h u l l has b e e n r e p o r t e d b y Minsaas et al. (1973). I t has b e e n argued here that, t h e active r e g i o n o f t h e d u c t is closer to t h e h u l l t h a n t h e propeller, w h i c h makes t m o r e d e p e n d e n t o n t h e change i n d u c t t h r u s t c o m p a r e d to a change i n p r o p e l l e r t h r u s t . Plots o f d u c t t h r u s t close t o p r o p u l s i o n p o i n t / " value (Fig. 13) shows a consistent increase i n
0 . 3 0 . 2 0 . 1 0 . 0 • D P : F n = 0 . 2 0 3 a O P : F n = 0 . 2 0 3 A D P : F n = 0 . 1 4 2 X O P : F n = 0 . 1 4 2 • A 0 . 5 1 1.5 F i g . 1 1 . T h r u s t d e d u c t i o n as f u n c t i o n o f w a v e l e n g t h f o r f „ / L p p = 0 . 0 1 5 4 . 0.3 0 . 2 0 . 1 <f D P : F n = 0 . 2 0 3 ra O P : F n = 0 . 2 0 3 A D P : F n = 0 . 1 4 2 X O P : F n = 0 . 1 4 2 0 . 0 0 5 0 . 0 1 ?./Lpp 0 . 0 1 5 0 . 0 2 0 . 3 5 0 . 3 0 . 2 5
...
^ 0 . 1 5 0 . 1 0 . 0 5 0 s 11 s ... ™ 4 — • i < • • W a v e s J ' = l , 0 2 a W a v e s _ J * = 0 . 6 9 — - - C a l m w a t a r j * = 1 . 0 2 C a l m « ; a t e r _ J * = 0 . 6 9 • • W a v e s J ' = l , 0 2 a W a v e s _ J * = 0 . 6 9 — - - C a l m w a t a r j * = 1 . 0 2 C a l m « ; a t e r _ J * = 0 . 6 9 • W a v e s J ' = l , 0 2 a W a v e s _ J * = 0 . 6 9 — - - C a l m w a t a r j * = 1 . 0 2 C a l m « ; a t e r _ J * = 0 . 6 9 ) 0 . 5 ] I 1 5 2 A/Lpp F i g . 1 3 . D u c t t h r u s t as a p r o p o r t i o n o f t o t a l t h r u s t f o r d i f f e r e n t c o n d i t i o n s : F n = 0 . 2 0 3 , f „ / i p p = 0 . 0 1 5 4 . 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 < A 1 ! X 1 n J • DP: Fn=0.203 • O P ; F n - 0 . 2 0 3 A DP: Fn=0.142 X O P : Fn=0.142 • DP: Fn=0.203 • O P ; F n - 0 . 2 0 3 A DP: Fn=0.142 X O P : Fn=0.142 0 . 5 1 A / L p p 1.5 F i g . 14. 0 . 0 1 5 4 . P r o p u l s i o n p o i n t w a k e f r a c t i o n as f u n c t i o n o f w a v e l e n g t h f o r fo/ipp= 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 - • — X / L p p = O . S - i & - V L p p = 1 . 0 - A - X / L p p = 1 . 2 - K - A / L p p = 1 . 5 X / L p p = 1 . 9 — C a l m W a t e r - • — X / L p p = O . S - i & - V L p p = 1 . 0 - A - X / L p p = 1 . 2 - K - A / L p p = 1 . 5 X / L p p = 1 . 9 — C a l m W a t e r A O - • — X / L p p = O . S - i & - V L p p = 1 . 0 - A - X / L p p = 1 . 2 - K - A / L p p = 1 . 5 X / L p p = 1 . 9 — C a l m W a t e r 0 . 0 0 . 4 0 . 8 J * = V / n D 1.2 1.6 F i g . 1 2 . T h r u s t d e d u c t i o n as f u n c t i o n o f w a v e a m p l i t u d e f o r AjLpp^l. F i g . 1 5 . W a k e f r a c t i o n as f u n c t i o n o f ƒ w i t h d u c t e d p r o p e l l e r : f n = 0 . 2 0 3 , Co/lpp= 0 . 0 1 5 4 .d u c t t h r u s t c o m p o n e n t i n waves c o m p a r e d t o c a l m water. This can be a t t r i b u t e d t o t h e h i g h e r t i n waves f o r t h e d u c t e d p r o p e l l e r .
3.3.2. Wake fraction
The b e h a v i o u r o f w at t h e p r o p u l s i o n p o i n t calculated using t h r u s t i d e n t i t y ( p l o t t e d i n Fig. 14) shows s i g n i f i c a n t d i f f e r e n c e b e t w e e n t h e t w o p r o p e l l e r cases. For the o p e n propeller, w is l o w e r i n waves t h a n t h e c a l m w a t e r values, the greater r e d u c t i o n s b e i n g a r o u n d the c r i t i c a l w a v e l e n g t h s , s i m i l a r to t h e results o f e x p e r i -m e n t s presented b y N a k a -m u r a et al. a n d M o o r et al. H o w e v e r , the ducted p r o p e l l e r s h o w e d an increase i n w at t h e p r o p u l s i o n p o i n t i n waves c o m p a r e d to c a l m w a t e r values, the h i g h e r i n c r e m e n t s being
A. Bhattacharyya. S. Steen / Ocean Engineermg 9 1 (2014) 263-272 2 6 9 0.4 0.3 5 0.2 0 . 1 0.0 • • — A / L p p = 0 . 8 - K - V L p p = 1 . 0 - A - V L p p = 1 . 2 - ; c - A / L p p = 1 . 5 . - ! * > - - A / L p p = 1 . 9 — — C a l m W a t e r - K - V L p p = 1 . 0 - A - V L p p = 1 . 2 - ; c - A / L p p = 1 . 5 . - ! * > - - A / L p p = 1 . 9 — — C a l m W a t e r 0.0 0.5 1.0 J * = V / n D 1.5 2.0 F i g . 1 6 . W a k e f r a c t i o n as f u n c t i o n o f J' w i t h o p e n p r o p e l l e r : f n = 0 . 2 0 3 , f „ / l p p = 0 . 0 1 5 4 . 1,4 1.2 -H 0 . 8 0 . 6 --* DP; Fn=0,203 P O P ; Fn=0.203 i , DP; Fn=0.142 ï<OP: Fn=0.142 --* DP; Fn=0,203 P O P ; Fn=0.203 i , DP; Fn=0.142 ï<OP: Fn=0.142 n , • 1 1 L 1 • 3 0 . 5 1 1.5 F i g . 1 8 . P r o p u l s i o n p o i n t r e l a t i v e r o t a t i v e e f f i c i e n c y as f u n c t i o n o f w a v e l e n g t h f o r f „ / i . p p = 0 . 0 1 5 4 . 0.5 0.4 0 . 3 - W 0 . 2 - W 0 . 1 0.0 — * - A / L p p = 0 . 8 - i ï - A / L p p = 1 . 0 - ^ - A / L p p = l , 2 - ) f ~ A / L p p = 1 . 5 A / L p p = 1 . 9 • C a l m W a t e r 0 . 0 0 . 4 F i g . 1 7 . W a k e f r a c t i o n ( t o r q u e i d e n t i t y ) as f u n c t i o n o f J' w i t h d u c t e d p r o p e l l e r : F n = 0 . 2 0 3 , f „ / / . p p = 0 . 0 1 5 4 .
i n the range 0 . 8 < A / L p p < 1 . 2 . For the d u c t e d propeller, a d e p e n -dency o n Fn is observed, m o r e f o r t h e wave c o n d i t i o n s , w h e r e h i g h e r values o f w are o b t a i n e d a t the l o w e r speed.
Figs. 15 and 16 s h o w t h a t , at each i n d i v i d u a l l o a d i n g ( f value), w has a h i g h e r value i n c a l m w a t e r t h a n a l m o s t a l l wave c o n d i t i o n s . W i t h increase i n n, a substantial increase o f w is observed f o r t h e d u c t e d p r o p e l l e r i n a c e r t a i n range o f l o a d i n g (Fig. 15). This leads t o h i g h p r o p u l s i o n p o i n t w values f o r the wave c o n d i t i o n s due t o increase o f p r o p u l s i o n p o i n t l o a d i n g i n waves (see Table 4 ) . For t h e o p e n p r o p e l l e r Fig. 14 i m p l i e s a steady decrease i n w w i t h n.
To c o n f i r m t h e p a t t e r n o f w w i t h l o a d i n g f o r t h e d u c t e d p r o p e l l e r , calculations w e r e p e r f o r m e d u s i n g t o r q u e i d e n t i t y (ITTC, 2011), w h e r e s i m i l a r observations w e r e m a d e (Fig. 17).
The slight i n c r e m e n t o f d u c t t h r u s t coefficients i n waves compared t o calm w a t e r should not be t h e reason f o r such higher p r o p u l s i o n p o i n t w a k e values i n waves. An interesting explanation has been presented by English and Rowe (1973) that, a ducted propeller has t h e p r o p e r t y o f being able to induce a greater p r o p o r t i o n o f the b o u n d a r y layer f l u i d t h r o u g h the duct compared t o a conventional screw b e h i n d t h e same hull, w h i c h should lead to higher w a k e fraction. I f this a r g u m e n t holds true, this effect should be higher i n waves due to increase o f the p r o p u l s i o n p o i n t l o a d i n g w h i c h w o u l d result i n m u c h higher w a k e fractions as observed i n our case.
1.2 1.1 + H 1.0 0 . 9 0 . 8 — * — A / L p p = 0 . 8 - a - A / L p p = 1 . 0 - r é - A / L p p = 1 . 2 - i < - A / L p p = 1 . 5 k : ; - A / l p p = 1 . 9 - - C a l m W a t e r 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 J * = V / n D 1.0 1.2 F i g . 1 9 . R e l a t i v e r o t a t i v e e f f i c i e n c y as f u n c t i o n o f J* w i t h d u c t e d p r o p e l l e r : F n = 0 . 2 0 3 , f o / L p p = 0 . 0 1 5 4 . 1.2 1.1 1 . 0 - H 0 . 9 0 . 8 1 — * - A / L p p = 0 . 8 - g - A / L p p = 1 . 0 - * - A / L p p = 1 . 2 - M - \ / l p p = 1 . 5 - s ; - V L p p = l . 9 • ~ — C a l m W a t e r — * - A / L p p = 0 . 8 - g - A / L p p = 1 . 0 - * - A / L p p = 1 . 2 - M - \ / l p p = 1 . 5 - s ; - V L p p = l . 9 • ~ — C a l m W a t e r — * - A / L p p = 0 . 8 - g - A / L p p = 1 . 0 - * - A / L p p = 1 . 2 - M - \ / l p p = 1 . 5 - s ; - V L p p = l . 9 • ~ — C a l m W a t e r t t 0 . 0 0 . 2 0 . 4 0 . 6 J * = V / n D 0 . 3 1.0 F i g . 2 0 . R e l a t i v e r o t a t i v e e f f i c i e n c y as f u n c t i o n o f / w i t h o p e n p r o p e l l e r : F n = 0 . 2 0 3 , C„/Z.pp=0.0154. For t h e d u c t e d p r o p e l l e r , t h e r e is a s l i g h t increase o f TJR w i t h J* i n a c e r t a i n range o f p r o p e l l e r l o a d i n g as s h o w n i n Fig. 19. T h i s can be e x p l a i n e d i n t h e l i g h t o f d u c t t h r u s t v a r i a t i o n w i t h l o a d i n g (Section 3.4) H o w e v e r , f o r t h e o p e n p r o p e l l e r . Fig. 2 0 s h o w s n e g l i g i b l e v a r i a t i o n s o f riR w i t h J* f o r a l l t h e c o n d i t i o n s t e s t e d .
3.3.3. Relative rotative efficiency
The value o f p r o p u l s i o n p o i n t rjR is v e r y close t o 1, a n d t h e r e is l i t t i e change oitjR i n w a v e c o n d i t i o n s f r o m t h e c a l m w a t e r value, as w a s also observed b y M o o r et a l . a n d N a k a m u r a e t al. Fig. 18 also s h o w l i t t i e d i f f e r e n c e b e t w e e n t h e o p e n a n d d u c t e d p r o p e l l e r s i n t h i s regard f o r a l l t h e tested c o n d i t i o n s . 3.3.4. Hull efficiency T h e r e is l i t t l e d i f f e r e n c e i n r]n b e t w e e n t h e o p e n a n d d u c t e d p r o p e l l e r cases a t t h e p r o p u l s i o n p o i n t f o r t h e c a l m w a t e r c o n d i -t i o n a n d a-t A/Lpp&g-t; 1.2 (Fig. 2 1 ) . H o w e v e r , h i g h w a k e v a l u e s h a v e r e s u l t e d i n large rjn at 0.8 < A/Lpp < 1.2 f o r t h e d u c t e d p r o p e l l e r at t h e l o w e r speed. As t h e t h r u s t d e d u c t i o n is a s s u m e d t o be c o n s t a n t
2 7 0 A. Bhattacharyya, S. Steen / Ocean Engineering 91 (2014) 263-272 F i g . 2 1 . 0 . 0 1 5 4 . • DP: Fn=0.203 - B O P ; F n = 0 . Z 0 3 A D P ; F n = 0 . 1 4 2 • DP: Fn=0.203 - B O P ; F n = 0 . Z 0 3 A D P ; F n = 0 . 1 4 2 X O P ! Fn=0.142 1 n 1 , n [ 0.5 1 1.5 P r o p u l s i o n p o i n t h u l l e f f i c i e n c y as f u n c t i o n o f w a v e l e n g t h f o r f o / L p p = 1.4 1.3 1.2 1 . 1 1.0 0.9 0 . 8 —*—V.pp=0.8 ~ s - V i . p p = i - 0 - * - V L p p = 1 . 2 - H - A / L p p = 1 . 5 - - VLpp=1.9 - — C a l m W a t e r 0 . 0 0 . 4 F i g . 2 2 . H u l l e f f i c i e n c y as f u n c t i o n o f / w i t h d u c t e d p r o p e l l e r : F n = 0 . 2 0 3 , f o / L p , 0 . 0 1 5 4 . 1.1 1.0 0 . 9 0 . 8 — # — A / L p p = 0 . 8 - ~ . 3 - A / L p p = 1 . 0 - * - V l - P P = 1 . 2 - « - A / L p p = 1 . 5 :- A / L p p = 1 . 9 - C a l m W a t e r 0 . 0 0 . 5 1.0 J * = V / n D 1.5 2 . 0 F i g . 2 3 . H u l l e f f i c i e n c y as f u n c t i o n o f / w i t h o p e n p r o p e l l e r : Fn = 0 . 2 0 3 , f . / i p p = 0 . 0 1 5 4 . w i t h p r o p e l l e r l o a d i n g , t h e I]H v s f p l o t s i n Figs. 22 a n d 23 f o l l o w s i m i l a r t r e n d s as t h o s e o f w vs _ƒ*, w h e r e at a l l p r o p e l l e r l o a d i n g s , rjH is m i n i m u m a r o u n d XlLpp=\.2 f o r b o t h t h e o p e n a n d d u c t e d p r o p e l l e r s . T h e h i g h h u l l e f f i c i e n c y f o r d u c t e d p r o p e l l e r f o r Xj Lpp=1 i n Fig. 21 is caused b y a l o w v a l u e o f f f o r t h i s p o i n t , a n d t h e
f a c t t h a t t h e d u c t e d p r o p e l l e r gave large w a k e values f o r h i g h p r o p e l l e r l o a d i n g s .
3.3.5. Open water efficiency
A t t h e p r o p u l s i o n p o i n t s (Fig. 2 4 ) i]o s h o w a s l i a r p decrease i n t h e c r i t i c a l w a v e l e n g t h range (0.8 < A/Lpp < 1.2), a n d g r a d u a l l y c o n v e r g e t o t h e c a l m w a t e r values at h i g h e r w a v e l e n g t h s . W i t h i n t h e same range, a s l i g h t increase w i t h Fn is o b s e r v e d f o r b o t h t h e
1.0 0 . 8 0 . 5 0 . 4 0 . 2 0 . 0 1 + DP: Fn=0.203 • OP: Fn=0.203 A DP: Fn=0.142 X O P : Fn=0.142 1 + DP: Fn=0.203 • OP: Fn=0.203 A DP: Fn=0.142 X O P : Fn=0.142 I ' l r,
1
• 1
1 0.5 1 A / l p p 1.5 F i g . 2 4 . P r o p u l s i o n p o i n t o p e n w a t e r e f f i c i e n c y as f u n c d o n o f w a v e l e n g t h f o r C „ / / . p p = 0 . 0 1 5 4 . 1.0 0 . 8 0 . 6 0 . 4 0 . 2 0 . 01
X i, 1 X * DP: Fn=O.203 • OP: Fn:=a.203 A D P : F n = 0 . 1 4 2 X O P : Fn=0.142 * DP: Fn=O.203 • OP: Fn:=a.203 A D P : F n = 0 . 1 4 2 X O P : Fn=0.142 0.5 1 A / L p p 1.5 F i g . 2 5 . P r o p u l s i o n p o i n t q u a s i - p r o p u l s i v e e f f i c i e n c y as f u n c t i o n o f w a v e l e n g t h f o r f „ / i p p = 0 . 0 1 5 4 . t w o p r o p u l s i o n cases, b u t t h e d i f f e r e n c e b e t w e e n t h e m f o r t h e c a l m w a t e r as w e l l as d i f f e r e n t w a v e c o n d i t i o n s is n e g l i g i b l e .T h e o p e n w a t e r e f f i c i e t i c y values s h o w n i n Fig. 2 4 are f o u n d f r o m the c a l m w a t e r o p e n w a t e r d i a g r a m . Thus, t h e d e v i a t i o n f r o m t h e c a l m w a t e r value is p u r e l y d u e t o change o f p r o p u l s i o n p o i n t so t h e r e d u c t i o n i n i-jo is j u s t d u e t o t h e p r o p e l l e r e f f i c i e n c y d r o p p i n g w i t h increased l o a d i n g . I n l i g h t o f t h e s t a t e m e n t s i n f o r i n s t a n c e Faltinsen et a l . ( 1 9 8 0 ) , i t is o f i n t e r e s t t o check w h e t h e r t h e p r o p e l l e r e f f i c i e n c y is a f f e c t e d b y t h e w a v e s t e s t e d . W i t h our m e t h o d o f analysis, w h e r e w e use t h e c a l m w a t e r o p e n w a t e r d i a g r a m f o r analysis, t h e c h a n g e o f p r o p e l l e r e f f i c i e n c y due to w a v e s s h o u l d t h e n s h o w u p as a c h a n g e o f r e l a t i v e r o t a t i v e e f f i c i e n c y IJR w i t h waves. R e f e r r i n g t o Figs. 1 8 - 2 0 w e c a n say t h a t t h e e f f e c t o f w a v e s o n t h e p r o p e l l e r e f f i c i e n c y is n o t s i g n i f i c a n t i n t h e c u r r e n t tests. T h a t m i g h t be r e l a t e d t o t h e f a c t t h a t p r o p e l l e r v e n t i l a t i o n ( o r p r o p e l l e r e m e r g e n c e ) w a s n ' t obsei-ved i n t h e tests. 3.3.6. Quasi-propulsive efficiency T h e p r o p u l s i o n p o i n t / / D values (Fig. 2 5 ) s h o w s t r o n g i n f l u e n c e o f w a v e c o n d i t i o n s , t h e m i n i m u m values are a r o u n d 0.8 < XjLpp < 1.2, w h i c h is m a i n l y d u e t o l o w ijo values. T h e c a l m w a t e r values are s i m i l a r f o r t h e o p e n a n d d u c t e d p r o p e l l e r s a t b o t h speeds, w h i l e ?/D is h i g h e r f o r t h e d u c t e d p r o p e l l e r i n a l m o s t all w a v e c o n d i t i o n s , l a r g e l y due t o h i g h e r i]n c o m p a r e d t o t h e open p r o p e l l e r . T h e i n f l u e n c e o f Fn f o r b o t h p r o p e l l e r s c a n be t r a c e d to t h e c o r r e s p o n d i n g i-jo values.
T h e p l o t s o f t]D w i t h ƒ f o r Fn = 0.203 (Figs. 2 6 a n d 2 7 ) show i n c r e a s i n g t r e n d s i n t h e range p l o t t e d f o r b o t h t h e p r o p e l l e r s . This is largely d u e t o the t r e n d o f i]o i n t h i s range ( p r o p e l l e r o p e n w a t e r characteristics. Figs. 9 and 10). The slope is h i g h e r f o r t h e open
A. Bhattacharyya, S. Steen / Ocean Engineering 91 (2014) 263-272 2 7 1 1.0 0 . 6 0 . 2 0 . 0 --1 — * - A / L p p = 0 . 8 ~ B — A / L p p = 1 . 0 - * - V L p p = 1 . 2 - ^ i - A / L p p = 1 . 5 - A / L p p = 1 . 9 - - C a l m W a t e r --1 — * - A / L p p = 0 . 8 ~ B — A / L p p = 1 . 0 - * - V L p p = 1 . 2 - ^ i - A / L p p = 1 . 5 - A / L p p = 1 . 9 - - C a l m W a t e r --1 — * - A / L p p = 0 . 8 ~ B — A / L p p = 1 . 0 - * - V L p p = 1 . 2 - ^ i - A / L p p = 1 . 5 - A / L p p = 1 . 9 - - C a l m W a t e r --1 — * - A / L p p = 0 . 8 ~ B — A / L p p = 1 . 0 - * - V L p p = 1 . 2 - ^ i - A / L p p = 1 . 5 - A / L p p = 1 . 9 - - C a l m W a t e r 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 J * = V / n D 1.0 1 . 2 F i g . 2 6 . Q u a s i - p r o p u l s i v e e f f i c i e n c y as f u n c t i o n o f f w i t h d u c t e d p r o p e l l e r : f n = 0 . 2 0 3 , f „ / l „ p = 0 . 0 1 5 4 . 1.0 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 1 — * — V l . P P = 0 . 8 • - • - V L p p = 1 . 0 - A - A / L p p = 1 . 2 - ^ < - A / L p p = 1 . 5 - ; ! t - A / L p p = 1 . 9 - C a l m W a t e r 1 — * — V l . P P = 0 . 8 • - • - V L p p = 1 . 0 - A - A / L p p = 1 . 2 - ^ < - A / L p p = 1 . 5 - ; ! t - A / L p p = 1 . 9 - C a l m W a t e r 1 — * — V l . P P = 0 . 8 • - • - V L p p = 1 . 0 - A - A / L p p = 1 . 2 - ^ < - A / L p p = 1 . 5 - ; ! t - A / L p p = 1 . 9 - C a l m W a t e r 0 . 0 0 . 2 0 . 4 0 . 6 J * = V / n D 0 . 8 1.0 F i g . 2 7 . Q u a s i - p r o p u l s i v e e f f i c i e n c y as f u n c t i o n o f ƒ * w i t h o p e n p r o p e l l e r : f n = 0 . 2 0 3 , i-„/ipp=0-0154. T a b l e 6 P r o p u l s i v e f a c t o r s a t F n = 0 . 1 4 2 . 0 . 5 5 0 . 8 1.0 1.0 1.2 1.5 1.9 C a l m 0 . 0 0 7 7 0 . 0 1 5 4 0 . 0 0 7 7 0 . 0 1 5 4 0 . 0 1 5 4 0 . 0 1 5 4 0 . 0 1 5 4 w a t e r t D P 0 . 1 5 5 0 . 1 5 5 0 . 1 8 2 0 . 2 2 4 0 . 2 1 8 0 . 2 1 0 0 . 1 7 4 0 . 1 7 4 O P 0.116 0 . 1 2 3 0.118 0 . 1 4 0 0.153 0.153 0 . 1 3 6 0 . 1 3 0 I V D P 0 . 3 4 6 0 . 3 8 1 0 . 3 5 5 0 . 4 5 7 0 . 3 7 7 0 . 3 5 0 0 . 3 3 4 0 . 3 1 9 O P 0 . 2 6 1 0 . 2 5 4 0 . 2 3 7
-
0 . 2 8 9 0 . 2 3 7 0 . 2 5 4 0 . 2 8 6 'III D P 1 . 2 9 3 1 . 3 6 5 1 . 2 6 8 1 . 4 2 9 1.255 1.216 1 . 2 4 0 1.213 OP 1.197 1.175 1.157-
1.191 1.110 1.159 1.219 llR D P 0 . 9 9 5 0 . 9 7 1 0 . 9 6 5 0 . 9 5 9 0 . 9 9 6 0 . 9 8 7 1 . 0 2 9 0 . 9 9 5 llR O P 0 . 9 8 6 0 . 9 6 2 1 . 0 0 8-
0 . 9 6 5 1 . 0 2 6 0 . 9 9 0 1 . 0 2 5 'lo D P 0 . 5 5 6 0 . 4 2 0 0 . 5 0 5 0 . 3 3 4 0 . 4 1 8 0 . 4 9 4 0 . 5 7 0 0 . 6 0 3 OP 0 . 5 7 1 0 . 4 0 1 0 . 4 9 2-
0 . 3 9 3 0 . 4 9 4 0 . 5 6 9 0 . 5 9 8 'ID D P 0 . 7 1 5 0 . 5 5 6 0 . 6 1 9 0 . 4 5 8 0 . 5 2 3 0 . 5 9 3 0 . 7 2 7 0 . 7 2 7 'ID OP 0 . 6 7 4 0 . 4 5 4 0 . 5 7 3-
0 . 4 5 2 0 . 5 6 2 0 . 6 5 2 0 . 7 4 7 - B e h i n d Hull_Fn=0,Z03 - B e h i n d H u l L F n = 0 . 1 4 2 - O p e n W a t e r 0 . 2 5 0 . 2 0 . 1 5 0 . 1 0 . 0 5 0 - 0 . 0 5 - 0 . 1 F i g . 2 8 . P l o t o f d u c t t h r u s t a g a i n s t t o t a l t h r u s t f o r o p e n w a t e r a n d b e h i n d h u l l c o n d i t i o n s . T a b l e 5 P r o p u l s i v e f a c t o r s a t f n = 0 . 2 0 3 . 0 . 5 5 0 . 0 0 7 7 0 . 8 0 . 0 1 5 4 1.0 0 . 0 0 7 7 1.0 0 . 0 1 5 4 1.2 0 . 0 1 5 4 1.5 0 . 0 1 5 4 1.9 0 . 0 1 5 4 C a l m w a t e r t D P 0 . 1 7 3 0 . 1 7 8 0 . 1 8 1 0 . 2 1 1 0 . 2 0 8 0 . 2 0 3 0 . 1 8 0 0 . 1 7 4 OP 0.118 0 . 1 2 2 0.116 0 . 1 6 0 0 . 1 4 8 0 . 1 3 2 0 . 1 2 6 0 . 1 3 2 w D P 0 . 3 1 9 0 . 3 2 1 0 . 3 3 9 - 0 . 3 6 5 0 . 3 1 5 0 . 3 0 5 0 . 3 0 7 O P 0 . 2 7 6 0 . 2 6 4 0 . 2 6 3 0 . 2 4 6 0 . 2 5 0 0 . 2 5 2 0 . 2 6 7 0 . 2 8 8 'IH D P 1.213 1.210 1 . 2 3 9-
1 . 2 4 6 1.164 1.180 1.193 O P 1.217 1.193 1 . 2 0 0 1.115 1.135 1.160 1.192 1 . 2 2 0 'IR D P 1.017 0 . 9 9 1 0 . 9 8 7 _ 0 . 9 6 1 1 . 0 3 0 1 . 0 3 5 1 . 0 0 0 'IR O P 1.011 1 . 0 2 5 1 . 0 0 0 0 . 9 9 4 1 . 0 0 5 1 . 0 0 4 1.013 1 . 0 1 1 'lo D P 0 . 5 8 2 0 . 5 0 2 0 . 5 3 2-
0 . 4 3 8 0 . 5 1 7 0 . 5 7 1 0 . 6 0 4 'lo O P 0 . 5 7 4 0 . 4 7 5 0 . 5 3 6 0 . 4 2 4 0 . 4 3 4 0 . 5 0 9 0 . 5 6 1 0 . 5 9 1 'ID D P 0 . 7 1 8 0 . 6 0 1 0 . 6 5 0-
0 . 5 2 5 0 . 6 1 9 0 . 6 9 8 0 . 7 2 1 'ID O P 0 . 7 0 6 0 . 5 8 1 0 . 6 4 3 0 . 4 7 0 0 . 4 9 5 0 . 5 9 3 0 . 6 7 8 0 . 7 2 9p r o p e l l e r as the c o n t r i b u t i o n s f r o m rjH are h i g h e r at h i g h e r ƒ values c o m p a r e d t o the d u c t e d p r o p e l l e r w h e r e has a n o p p o s i t e t r e n d .
3.3.7. Propulsive factors comparison
I n Tables 5 a n d 6, t h e p r o p u l s i v e factors f o r b o t h t h e d u c t e d a n d t h e o p e n p r o p e l l e r are p r e s e n t e d a t t h e d i f f e r e n t w a v e c o n d i t i o n s as w e l l as i n c a l m w a t e r f o r b o t h t h e speeds. -•-J*=1.41 -'a-J*=l,07 -*-J*=1.02 —;::-J'=0.69
\
\
U H = i 1 > 0 . 5 1 1.5 2 X / L p p F i g . 2 9 . S p e c t r a l e n e r g y r a t i o s f o r d u c t t h r u s t a t 2 f e : F n = 0 . 2 0 3 .3.4. Investigations with duct thrust
For c a l m w a t e r c o n d i t i o n s , KJD at b o t h speeds are p l o t t e d against KT,Tot a l o n g w i t h t h e c o r r e s p o n d i n g o p e n w a t e r values i n Fig. 2 8 . A t l o w values o f t o t a l t h r u s t c o r r e s p o n d i n g t o l o w p r o p e l l e r l o a d i n g , t h e d u c t t h r u s t is h i g h e r i n b e h i n d c o n d i t i o n f o r b o t h t h e t e s t e d speeds. H o w e v e r , a t h i g h e r l o a d i n g s , t h e o p e n w a t e r d u c t t h r u s t s are s l i g h t l y h i g h e r . Thus, i f t h r u s t i d e n t i t y is used, t h e i n c r e m e n t i n d u c t t h r u s t w i t h p r o p e l l e r l o a d i n g is less w h e n p l a c e d b e h i n d t h e h u l l , c o m p a r e d t o o p e n w a t e r c o n d i t i o n . T h i s c a n e x p l a i n t h e decrease i n t]R f o r ^ t h e d u c t e d p r o p e l l e r at h i g h e r l o a d i n g s as o b s e r v e d f r o m Fig. 16. For w a v e c o n d i t i o n s , f r e q u e n c y analysis o f t h e d u c t t h r u s t reveals a s t r o n g i n f l u e n c e o f t h e second h a r m o n i c o f t h e w a v e
2 7 2 A. Bhattacharyya, S. Steen / Ocean Engineermg 91 (2014) 263-272 T a b l e 7 P r e c i s i o n l i m i t s ( 9 5 % v a l u e s ) f r o m m o d e l t e s t s . P a r a m e t e r P. (%) P ^ ( % ) n 0 . 0 3 5 0 . 0 1 8 T 1.255 0 . 6 2 7 FD 1 . 7 6 4 0 . 8 8 2 Q 2 . 2 4 0 1.120 t 6 . 6 5 1 3 . 3 2 6 w 1.943 0 . 9 7 1 'IR 1 . 9 5 8 0 . 9 7 9 'IH 1.537 0 . 7 6 9 '10 0 . 1 4 4 0 . 0 7 2 •ID 2 . 6 8 7 1 . 3 4 4 e n c o u n t e r f r e q u e n c y i n ttie range 1 < / l / I p p < 1.5, w l i e r e t h e m o t i o n s are s t r o n g . This is d u e t o the d y n a m i c l i f t i n g f o r c e g e n e r a t e d b y t h e duct, t w i c e i n a c o m p l e t e w a v e cycle, a c o m p o -n e -n t o f w h i c h is v i s u a l i z e d as t h r u s t . A p l o t o f t h e spectral e -n e r g y a t 2fE p r e s e n t e d as a r a t i o o f t h e spectral e n e r g y at f o r d i f f e r e n t w a v e l e n g t h s is s h o w n i n Fig. 2 9 . 4. U n c e r t a i n t y I n t h i s paper, a c o m p a r a t i v e s t u d y o f p r o p u l s i o n f a c t o r s i n w a v e s has b e e n p r e s e n t e d f o r a d u c t e d and a c o n v e n t i o n a l o p e n p r o p e l l e r . A s s u m i n g t h a t t h e e x p e r i m e n t a l bias f o r b o t h cases is t h e same, t h e p r e c i s i o n e r r o r is a g o o d r e p r e s e n t a t i o n o f t h e u n c e r t a i n t y o f t h e c o m p a r i s o n . The p r e c i s i o n l i m i t s f o r d i f f e r e n t m e a s u r e d a n d c a l c u l a t e d p a r a m e t e r s are e s t i m a t e d as percentages o f t h e m e a n values as s h o w n i n Table 7 b y t a k i n g f o u r r e p e t i t i o n s a t t h e m o d e l speed o f 1.449 m / s f o r w a v e - 4 (A/Lpp = l ,
CalLpp=0.0154). The p r e c i s i o n l i m i t f o r a single r u n (P^) a n d f o r
t h e m e a n o f 4 r u n s ( P ^ ) are g i v e n by:
P;, = tSx; P^ = t S , / ^ / ] V (12) here, Sx is t h e s t a n d a r d d e v i a t i o n , N is t h e n u m b e r o f samples ( i n
t h i s case, 4 ) , a n d t = 2 . 3 5 3 f o r 95% accuracy is o b t a i n e d f r o m S t u d e n t ' s t - d i s t r i b u t i o n , f o r DOF ( N - l ) = 3 .
The u n c e r t a i n t i e s are w e l l w i t h i n acceptable l i m i t s f o r a l l t h e cases. H o w e v e r , f o r t h e e s t i m a t i o n s o f values at t h e p r o p u l s i o n p o i n t s , a d d i t i o n a l u n c e r t a i n t i e s d u e t o i n t e r p o l a t i o n s / e x t r a p o l a -t i o n s are i n v o l v e d .
5. C o n c l u s i o n s
The p r o p u l s i v e factors i n a range o f r e g u l a r h e a d sea c o n d i t i o n s are i n v e s t i g a t e d f o r a d u c t e d p r o p e l l e r , a n d t h e results are c o m p a r e d t o a c o n v e n t i o n a l o p e n p r o p e l l e r . The w a y t h e s t u d y is p e r f o r m e d a l l o w s f o r s e p a r a t i n g t h e e f f e c t o f w a v e s f r o m t h e e f f e c t o f increased p r o p e l l e r l o a d i n g d u e t o t h e a d d e d resistance o f t h e w a v e s . C o m p a r a t i v e e s t i m a t i o n s reveal t h a t i n s p i t e o f t h e s i m i l a r c a l m w a t e r values, the p r o p u l s i v e factors i n w a v e s can v a r y l a r g e l y b e t w e e n a n o p e n a n d a q u i t e s i m i l a r d u c t e d p r o p e l l e r . For e x a m p l e , at A/Lpp=1.2, t h e v a l u e o f / / D f o r t h e d u c t e d p r o p e l l e r is h i g h e r b y 6% a n d 15% at F n = 0 . 2 0 3 a n d 0.142 r e s p e c t i v e l y c o m p a r e d t o t h e o p e n p r o p e l l e r i n s p i t e o f t h e l a t t e r h a v i n g a s l i g h t l y h i g h e r iju v a l u e i n c a l m w a t e r . One o f t h e m o s t i n t e r e s t i n g findings is t h e n a t u r e o f change o f e f f e c t i v e w a k e f r a c t i o n i n w a v e s .
w h i c h has s i g n i f i c a n t d i f f e r e n c e s b e t w e e n t h e t w o cases. For d i f f e r e n t w a v e c o n d i t i o n s , t h e d u c t e d p r o p e l l e r s h o w s s i g n i f l c a n t increase i n w at t h e p r o p u l s i o n p o i n t c o m p a r e d to c a l m w a t e r , the d i f f e r e n c e b e i n g m o r e w h e n t h e w a v e l e n g t h is close t o t h e ship l e n g t h . This i n f l u e n c e s t h e c o m p u t e d values o f h u l l e f f l c i e n c y % a n d q u a s i - p r o p u l s i v e e f f i c i e n c y tjo. Except f o r t h e e f f e c t o f c h a n g e o f p r o p u l s i o n p o i n t , t h e i n f l u e n c e o f w a v e s o n t h e p r o p e l l e r e f f i c i e n c y seems t o be i n s i g n i f i c a n t i n t h i s case, p r o b a b l y because t h e r e i s n ' t a n y n o t a b l e p r o p e l l e r v e n t i l a t i o n . This c o n c l u s i o n is d r a w n f r o m t h e f a c t t h a t IIR r e m a i n s close t o u n i t y f o r a l l w a v e c o n d i t i o n s . The t h r u s t d e d u c t i o n is f o u n d to be i n d e p e n d e n t o f p r o p e l l e r l o a d i n g . Especially f o r t h e d u c t e d p r o p e l l e r , t h e t h r u s t d e d u c t i o n increase w i t h i n c r e a s i n g ship m o t i o n s , so t h a t i t reaches a m a x -i m u m f o r w a v e s t h a t have a p p r o x -i m a t e l y t h e same l e n g t h as t h e s h i p . The d u c t t h r u s t i n b e h i n d c o n d i t i o n is f o u n d t o f a l l b e l o w t h e o p e n w a t e r v a l u e at h i g h l o a d i n g s . The d y n a m i c l i f t i n g e f f e c t o f the d u c t is p r o m i n e n t i n t h e spectral e n e r g y c o m p o n e n t o f t h e second h a r m o n i c o f 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 . This s t u d y can be o f i m p o r t a n c e i n d e t e r m i n i n g t h e p r o p u l s i v e p e r f o r m a n c e o f ships i n waves w h e r e d u c t e d p r o p e l l e r s are used.
A c l m o w l e d g e m e n t s
The w o r k p r e s e n t e d here is financed b y Rolls-Royce U n i v e r s i t y T e c h n o l o g y C e n t r e (UTC) at t h e D e p a r t m e n t o f M a r i n e Technology, N T N U . The financial s u p p o r t a n d t h e p e r m i s s i o n t o p u b l i s h data f o r t h e Rolls-Royce-designed h u l l are a p p r e c i a t e d . W e also appreciate the h e l p f r o m MARINTEK t e c h n i c i a n s f o r c o n d u c t i n g t h e m o d e l tests. R e f e r e n c e s B h a t t a c h a r y y a , R., 1 9 7 8 . D y n a m i c s o f M a r i n e V e h i c l e s . J o h n W i l e y a n d S o n s , I S B N : 0 - 4 7 1 - 0 7 2 0 6 - 0 , p p . 2 3 0 - 2 3 1 . . B h a t t a c h a r y y a , A . , S t e e n , S., 20l4. I n f l u e n c e o f d u c t e d p r o p e l l e r o n s e a k e e p i n g i n w a v e s . O c e a n E n g . h t t p : / / d x . d o i . O r g / 1 0 . 1 0 1 6 / j . o c e a n e n g . 2 0 1 4 . 0 9 . 0 1 6 , i n p r e s s . B o s e , N . , 2 0 0 8 . M a r i n e P o w e r i n g P r e d i c t i o n a n d P r o p u l s o r s . i s b n : 0 - 9 3 9 7 7 3 - 6 5 - l , S N A M E . E n g l i s h , J . W . , R o w e , S.J., 1 9 7 3 . S o m e a s p e c t s o f d u c t e d p r o p e l l e r p r o p u l s i o n . I n : R I N A S y m p o s i u m o n d u c t e d p r o p e l l e r s . H o l t r o p , J., 2 0 0 1 . E x t r a p o l a t i o n o f p r o p u l s i o n t e s t s f o r s h i p s w i t h a p p e n d a g e s a n d c o m p l e x p r o p u l s o r s . M a r . T e c h n o l . 3 8 ( 3 ) , 145-157. K o r t , L , 1 9 3 4 . D e r n e u e D i l s e n s c h r a u b e n - A n t r i e b , W e r f t - R e d e r e i - H a f e n , J a h r g a n g 1 5 , H e f t 4 . F a l t i n s e n , O . M . , M i n s a a s , K.J., L i a p i s , N . , S k j o r d a l , S.O., 1 9 8 0 . P r e d i c t i o n o f r e s i s t a n c e a n d p r o p u l s i o n o f a s h i p i n a s e a w a y . I n : P r o c e e d i n g s o f 1 3 t h S y m p o s i u m o n N a v a l H y d r o d y n a m i c s , T o k y o , J a p a n , p . 5 0 5 - 5 2 9 . m c , 2 0 0 2 a . P r o p u l s i o n , P e r f o r m a n c e U n c e r t a i n t y A n a l y s i s , E x a m p l e f o r P r o p u l s i o n T e s t , R e c o m m e n d e d P r o c e d u r e 7 . 5 - 0 2 - 0 3 - 0 1 . 2 R e v 0 0 . ITTC, 2 0 0 2 b . T e s t i n g a n d E x t r a p o l a t i o n M e t h o d s P r o p u l s i o n , P r o p u l s o r O p e n W a t e r T e s t . R e c o m m e n d e d P r o c e d u r e 7 . 5 - 0 2 - 0 3 - 0 2 . 1 R e v 0 1 . ITTC 2 0 1 1 . 1 9 7 8 ITPC P e r f o r m a n c e P r e d i c t i o n M e t h o d , R e c o m m e n d e d P r o c e d u r e 7 5 - 0 2 - 0 3 - 0 1 . 4 R e v 0 2 . M c C a r t h y J.H., N o r l e y W . H . , O b e r , G.L., M a y 1 9 6 1 . T h e P e r f o r m a n c e o f a S u b m e r g e d P r o p e l l e r i n R e g u l a r W a v e s . D T M B R e p . 1 4 4 0 . M i n s a a s , K.J., J a c o b s e n , G . M . , O k a m o t o , H . , 1 9 7 3 . T h e d e s i g n o f l a r g e d u c t e d p r o p e l l e r s f o r o p t i m u m e f f i c i e n c y a n d m a n o e u v r a b i l i t y . P a r t 1. I n : R I N A S y m p o s i u m o n d u c t e d p r o p e l l e r s . M o o r , D.1., M u r d e y , D . C , 1 9 7 0 . M o t i o n s a n d p r o p u l s i o n o f s i n g l e s c r e w m o d e l s i n h e a d seas. P a r t I I . R I N A . Q, T r a n s . 1 1 2 ( 2 ) . N a k a m u r a , S., N a i t o , S., 1 9 7 7 . P r o p u l s i v e P e r f o r m a n c e o f a C o n t a i n e r S h i p i n W a v e s . J . S N . A . 15 ( 1 ) , 2 4 - 4 8 . O o s t e r v e l d , M . W . C , 1 9 7 3 . D u c t e d p r o p e l l e r c h a r a c t e t i s t i c s . I n : R I N A S y m p o s i u m o n d u c t e d p r o p e l l e r s .
